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Imperial College LondonDepartment of Chemical Engineering

Ring Formation in a Statistical AssociatingFluid Theory Framework

Sara Antunes Febra

A thesis submitted for the degree ofDoctor of Philosophy and the Diploma ofMembership of Imperial College, July 2018

Declaration of originality

I, Sara Antunes Febra, declare that this thesis is a product of my own work and that any ideas

from the work of other people are fully acknowledged.

Copyright

The copyright of this thesis rests with the author and is made available under a Creative

Commons Attribution-Non Commercial-No Derivatives licence. Researchers are free to copy,

distribute or transmit the thesis on the condition that they attribute it, that they do not use it

for commercial purposes and that they do not alter, transform or build upon it. For any reuse

or distribution, researchers must make clear to others the licence terms of this work.

i

Abstract

Hydrogen bonds (HB) form, most commonly, between independent molecules (intermolecular

HB), leading to the formation of linear or branched chain-like networks, which can extend in

open form and can include ring-like networks (closed loops). In addition, hydrogen bonds may

involve atoms in different parts of the same molecule (intramolecular HB), on occasion leading

to bent X-H...X conformations in smaller molecules (e.g., Schiff bases) where strong steric

conditions apply, or from within large macromolecules (polymers [1,2,3,4,5] and proteins [6,7,8])

with little constraint from the covalent bonds otherwise binding the atoms. The formation of

HB leads to long-lived molecular aggregates and the macroscopic manifestation of these two

types of HB bond can be rather different and striking [9, 10,11,12,13].

The direct result of the formation of hydrogen bonds is the decrease in the availability of

the donor/acceptor sites of the chemical groups involved to associate. In the development of

equations of state (EOSs) based on Wertheim’s thermodynamic perturbation theory (TPT),

including the statistical associating fluid theory (SAFT), the formation of any ring clusters,

that are formed by inter- or intramolecular hydrogen bonding, is typically neglected. As a

consequence, the applicability of SAFT-like EOSs is limited to systems where the anomalies

arising from ring formation are insignificant. Previous attempts to extend the TPT formalism

to account for rings have not provided all the answers.

The issue is addressed here whereby the TPT treatment is extended to account for ring formation,

under the approximation that the appropriate many-body distribution function of the ring

aggregate can be expressed as a power of a (pair) radial distribution function in a homogeneous

fluid. The theory developed in this thesis constitutes an improvement to the existing theories

of association in that it can be used to account for the competition between free monomers,

linear-chain, branched-chain, inter- and intramolecular ring aggregates by hydrogen bonding.

The theory requires as input the size of rings formed and one extra parameter per ring type to

capture the probability associated with the two sites in a chain molecule/aggregate meeting

each other. The resulting generic framework is applicable to mixtures with an arbitrary number

of association sites and ring types.

The newly developed treatment is then compared to the standard framework for reference, to

examine the impact of ring formation on the phase equilibria of model systems and to model

iii

the solubility of ring-forming statins in simple alcohols. The formation of both inter- and

intramolecular rings is favoured by the increase of the association energy, low temperature and

low density. For fixed parameters, the formation of rings may result in either the enhancement

or the decrease of the solubility of statins, depending on the nature of the solvent.

iv

Acknowledgements

I would like to express my gratitude in the first place to Professor Amparo Galindo who helped

with my application and the Department of Chemical Engineering of Imperial College London

for funding my research. I thank all my three supervisors Professor Amparo Galindo, Professor

Claire Adjiman and Professor George Jackson who welcomed me into the Molecular Systems

Engineering (MSE) research group, for the inspiration, kindness, positivity. You are truly

extraordinary and it is an absolute privilege to work with you. I thank Vassilis, Esther and

Simon who eased my beginning in this journey. I thank Dr. Andrew Haslam for the science, the

tea and the walks. I thank all my friends from the office, I am so lucky to have you around me

every day to share the news, the frustrations, happiness and the occasional beer. You made the

office feel like home. In particular I thank my friends Smitha for the contagious good energy,

Daniel for the good common sense and the music, Nikos for the nonsense and the music too,

Suela, Hajar, Ed, Maz, Spiros, Karl, Srikanth, Tom, Harry for the best coffee, Coni, Eliana,

Georgia, Redz, Isaac, Lupe, Carmelo, Jason, Silvia, Fonzie, Lauren, Dave, Stefanos, and Fabian.

Ailo came along late but made a lasting difference. Thank you for everything. I have shared

some of the best moments with Daniela and Joana, you are amazing. Pedro, Joana and Jelena,

the London experience would not have been half as good without you, thank you for being in

my life. A big thank you to all my remote friendships whom I keep in my heart, Barbara, Saras,

Catarinas, Maria and Leonor. I finally thank my parents, sister, grandmother and family for

always believing in me. I feel so grateful.

v

Dedication

To my parents, who taught me how to dream.

vii

“You’re dealing with highly complex systems wherein everything is interacting with everythingelse.”

–Warren Buffet–

“Everything should be made as simple as possible, but not simpler.”

–Albert Einstein–

“’Cause if you like it then you should have put a ring on it.”

–Beyoncé Knowles–

viii

Contents

Abstract iii

Acknowledgements v

1 INTRODUCTION 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 WERTHEIM’S THERMODYNAMIC PERTURBATION THEORY 6

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 The thermodynamic perturbation theory of first order . . . . . . . . . . . . . . . 11

2.3 The statistical associating fluid theory (SAFT) . . . . . . . . . . . . . . . . . . . 29

2.3.1 The association contribution to the residual free energy . . . . . . . . . . 32

2.3.2 The chain contribution to the residual Helmholtz free energy . . . . . . . 35

2.4 Literature review of the extension of first-order thermodynamic perturbation

theory (TPT1) to account for ring formation . . . . . . . . . . . . . . . . . . . 40

2.4.1 Sear and Jackson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4.2 Ghonasgi, Perez, and Chapman . . . . . . . . . . . . . . . . . . . . . . . 50

ix

x CONTENTS

2.4.3 Comparison between the approaches of Sear and Ghonasgi . . . . . . . . 55

2.4.4 Further contributions and applications . . . . . . . . . . . . . . . . . . . 57

2.5 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3 UNIFIED THEORY TO ACCOUNT FOR RING FORMATION 62

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.2 Molecular model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2.1 Spherical molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.2.2 Fully flexible chain molecules . . . . . . . . . . . . . . . . . . . . . . . . 70

3.2.3 Inter- and intramolecular potentials . . . . . . . . . . . . . . . . . . . . . 70

3.3 The Helmholtz free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.3.1 Non-associating monomers . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.3.2 Free monomers and open chain aggregates . . . . . . . . . . . . . . . . . 75

3.3.3 Free monomers, open chains and intramolecular ring aggregates . . . . . 76

3.3.4 Free monomers, open chains, inter- and intramolecular ring aggregates . 77

3.4 Law of mass action equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.4.1 The fundamental graph sum ∆c(0) . . . . . . . . . . . . . . . . . . . . . . 81

3.4.2 The distribution of bonding states . . . . . . . . . . . . . . . . . . . . . . 88

3.5 Summary and formulation for mixtures . . . . . . . . . . . . . . . . . . . . . . . 94


4 INTERMOLECULAR RINGS IN PURE SYSTEMS 99

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.2 SAFT-VR square-well equation of state . . . . . . . . . . . . . . . . . . . . . . . 101

CONTENTS xi

4.3 Validation - the sticky limit of a ring cluster . . . . . . . . . . . . . . . . . . . . 102

4.3.1 Residual Helmholtz free energy from the formation of ring only aggregates103

4.3.2 The “sticky limit” in a fluid of rings formed by intermolecular hydrogen

bonding only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.4 The end-to-end-distribution function . . . . . . . . . . . . . . . . . . . . . . . . 108

4.5 Effect of intermolecular ring formation . . . . . . . . . . . . . . . . . . . . . . . 112

4.5.1 Impact of the ring size . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.5.2 Impact of the association energy . . . . . . . . . . . . . . . . . . . . . . . 114

4.5.3 Two-site model versus three-site model . . . . . . . . . . . . . . . . . . . 117


5 INTRAMOLECULAR HB IN MIXTURES AND APPLICATIONS IN SLE121

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.2 SAFT-VR Mie and SAFT-γ Mie EOSs . . . . . . . . . . . . . . . . . . . . . . . 125

5.3 Effect of the intramolecular ring formation on fluid properties . . . . . . . . . . 132

5.3.1 Model system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.3.2 Impact of intramolecular ring formation on a pure chain fluid . . . . . . 134

5.3.3 Impact of the chain length on the vapour–liquid equilibrium of a pure

ring-forming chain fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

5.3.4 Influence of intramolecular ring formation on the vapour–liquid and liquid–

liquid behaviour of a binary system . . . . . . . . . . . . . . . . . . . . . 144

5.4 Modelling solid–liquid equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . 147

5.4.1 The heat capacity in solubility calculations . . . . . . . . . . . . . . . . . 152

5.4.2 Effect of the W parameter on solid–liquid equilibria . . . . . . . . . . . . 155

5.5 Case study – ring formation in binary systems of statins and alcohols . . . . . . 158

5.5.1 Transferability of W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

5.5.2 Impact of the alcohol length on the ring fraction . . . . . . . . . . . . . . 165

5.5.3 Impact of W on ring fraction and solubility . . . . . . . . . . . . . . . . 166

5.5.4 Impact of W on solvent ranking . . . . . . . . . . . . . . . . . . . . . . . 169


6 CLOSING REMARKS 173

6.1 Summary of research achievements described in the thesis . . . . . . . . . . . . . 173

6.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

6.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

A Derivation of the property of independence between sites 178

B Molecular graphs 181

C Direct derivation of the law of mass action equations 186

C.1 Solving the type (I) equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

C.2 Solving the type (II) equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

C.3 Solving the type (III) equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

C.4 Final expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

C.4.1 The C coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

D Group like and unlike parameters for use in the SAFT-γ Mie EOS 194

Bibliography 197

xii

List of Tables

2.1 Value of the last term of Equation (2.104) according to the size of Γ . . . . . . . 34

4.1 Calculated correction factors and contact values for the corrected end-to-end

distribution function for chains of 2 to 10 links. . . . . . . . . . . . . . . . . . . 111

4.2 Parameters for the square-well spheres of the model fluids. . . . . . . . . . . . . 113

5.1 Like and unlike parameters in SAFT-VR Mie (modified version to account for

ring formation) molecular models. . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.2 Group like and unlike parameters in SAFT-γ Mie (modified version to account

for ring formation) molecular models. . . . . . . . . . . . . . . . . . . . . . . . . 128

5.3 Pure and unlike parameters for the model systems, where i, j are the component

indexes (1 for the chain and 2 for the sphere), σ is the segment diameter. The

interaction between a pair of segments is characterised by a Lennard-Jones

potential, for which the repulsive and attractive exponents are λrij = 12 and

λaij = 12, respectively, and εdisp is the depth of the potential well. The association

is characterised by a square-well potential of depth εHB and the bonding volume

KHB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4 Main approximations used to calculate ln asati (T, P,xsat). . . . . . . . . . . . . . 151

5.5 Compounds and respective %ARDx calculated at saturation temperature of

298.15 K from Equation (5.24). . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

xiii

5.6 Group make-up of simvastatin (SVS) and lovastatin (LVS) molecules according

to the SAFT-γ Mie approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

5.7 Literature data [14] characterising the solid state at atmospheric pressure of the

two statins used in the calculations. . . . . . . . . . . . . . . . . . . . . . . . . . 160

5.8 Average Absolute Relative Deviations (AARD%) between reported experimental

data [15] and SAFT-γ Mie predictions for the solubility of LVS in alcohols both

in the absence (“No rings”) and in the presence of rings (“With rings”) with a

fitted W ∗ per solvent. NP is the total number of experimental points. . . . . . . 166

C.1 System of the mass action equations for given number of sites |Γ |. . . . . . . . . 187

D.1 Group association energies εHBkl,ab and bonding volume parameters KHB

kl,ab for use to

model lovastatin, simvastatin and alcohols in the SAFT-γ Mie group-contribution

approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

D.2 Like group parameters for use to model lovastatin, simvastatin and alcohols with

the SAFT-γ Mie group-contribution approach: ν∗k is the number of segments, Skis the shape factor, λr

kk is the Mie repulsive exponent, λakk is the Mie attractive

exponent, σkk is the segment diameter, εdispkk is the dispersion energy of the Mie

potential characterising the interaction of two k groups (the k in the denominator

is the Boltzmann constant), and NST,k represents the number of association site

types on group k, with nk,H and nk,e1 denoting the number of association sites of

type H and e1, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

D.3 Group dispersion interaction energies εdispkl and Mie repulsive exponent λr

kl for

use to model lovastatin, simvastatin and alcohols with the SAFT-γ Mie group-

contribution approach. The unlike segment diameter σkl is obtained from the

arithmetic combining rule [16] and all unlike Mie attractive exponents λakl = 6.0000;

these are not shown in the table. CR indicates that the λrkl is obtained from a

combining rule [16]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

xiv

List of Figures

2.2.1 Illustrations of some of the constraints of TPT1. . . . . . . . . . . . . . . . . . 26

2.3.1 SAFT contributions to Helmholtz free energy. . . . . . . . . . . . . . . . . . . . 30

2.3.2 Each chain is formed by monomers that associated selectively. . . . . . . . . . . 36

2.4.1 Distance and relevant angles between two associative segments. . . . . . . . . . 42

2.4.2 intermolecular association into linear aggregates and rings of size τ = 4. . . . . . 42

2.4.3 Step 1: Formation of chain molecules of length m = 4 from hard sphere segments.

Step 2: inter- and intramolecular association. . . . . . . . . . . . . . . . . . . . . 46

2.4.4 Angles and distances to take into consideration of a chain molecule of m = 4

hard spheres with one association site on each terminal segment. . . . . . . . . . 51

3.2.1 Constraints of TPT1 [17]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.2.2 Examples of association aggregates that can be found in a pure fluid of spherical

molecules with the molecular structure represented in a). A green check mark or

red cross indicates whether the aggregate is captured by the theory presented in

this work. The spherical component has an arbitrary set of sites Γ = A,B, ... of

length |Γ |, and can form loop structures consisting of τ molecules. The aggregates

represented are a) monomer, b) open chain aggregate, c) AB intermolecular ring

with τ molecules, d) branched BC intermolecular ring with τ molecules, e) AB

intermolecular ring associated to a BC intermolecular ring, f) loop aggregate

that involve more than one pair of sites. . . . . . . . . . . . . . . . . . . . . . . 69

xv

xvi LIST OF FIGURES

3.2.3 Examples of association aggregates that can be found in a pure chain fluid with

the molecular structure represented in a). A green check mark or red cross

indicates whether the aggregate is captured by the theory presented in this work.

The chain molecule has an arbitrary set of sites Γ = A,B, ... of length |Γ |,

a number of segments m and can form loop structures formed by τ molecules.

The aggregates represented are a) monomer, b) CE intramolecular ring, c) open

chain aggregate, d) DE intermolecular ring with τ = 2, e) AE intramolecular

ring associated to an CE intramolecular ring, f) branched AE intramolecular

ring, g) and h) loop aggregates that involve more than one pair of sites. . . . . . 71

3.2.4 Scheme of position of molecule (r1) and site vectors dA and dB that are function

of the molecular orientation and conformation Ω1. . . . . . . . . . . . . . . . . . 72

4.1.1 Methanol intermolecular ring of size τ = 4. . . . . . . . . . . . . . . . . . . . . . 100

4.3.1 Chain molecule composed of m segments with a distance between segments of σ. 103

4.3.2 The molecules are composed ofm = 2 segments and have a set of sites Γ = A,B

that can only promote the formation of rings of size τ = 4. Increasing the

association energy to infinity and reducing association range to zero results in

irreversible bonding. Note that the number of segments composing the chain

molecule m and the ring size τ are arbitrary. . . . . . . . . . . . . . . . . . . . . 105

4.3.3 Description of the coexistence packing fractions as a function of reduced tem-

perature for a two-site spherical model that can only associate in intermolecular

rings of size τ = 6. The symbols represent the critical points and the continuous

curves the calculations with the theory: non-associating spheres in red, increasing

association energy from bottom to top and covalent hexamer cycle in orange at

the top. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.4.1 Possible conformation of a freely-jointed chain of 30 links (m = 31) or in other

words, a random walk of 30 steps. . . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.5.1 Types of association aggregates in a two-site sphere: a) linear chain aggregate; b)

ring by intermolecular hydrogen bonding (InterMHB) with τ = 4; and c) ring

by InterMHB with τ = 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

LIST OF FIGURES xvii

4.5.2 Types of association aggregates in a three-site sphere (AAB): a) linear/branched

chain aggregate; b) ring by InterMHB with τ = 4 with possible branching; and

c) ring by InterMHB with τ = 6 with possible branching. . . . . . . . . . . . . 113

4.5.3 Fractions of molecular bonding states of an associating fluid of two-site spherical

molecules in the liquid phase. The three scenarios are represented with the curves

in black for scenario 0, pink for scenario 1 and blue for scenario 2. The fractions

of molecules forming linear aggregates, fopen, for each scenario are represented

by dashed curves and the fractions of molecules forming intermolecular rings of

size 4, ξ4 (scenario 1) and of size 6, ξ6 (scenario 2) by continuous curves. These

curves correspond to SAFT-VR SW predictions. . . . . . . . . . . . . . . . . . . 114

4.5.4 Fractions of molecular bonding states of an associating fluid of two-site spherical

molecules in the vapour phase. The three scenarios are represented with the curves

in black for scenario 0, pink for scenario 1 and blue for scenario 2. The fractions

of molecules forming linear aggregates, fopen, for each scenario are represented

by dashed curves and the fractions of molecules forming intermolecular rings of

size 4, ξ4 (scenario 1) and of size 6, ξ6 (scenario 2) by continuous curves. These

curves correspond to SAFT-VR SW predictions. . . . . . . . . . . . . . . . . . . 115

4.5.5 Phase diagrams T ∗ versus η for an associating fluid of two-site spherical molecules

that can only form linear aggregates (scenario 0) at various association energies. 116

4.5.6 Phase diagrams T ∗ versus η for an associating fluid of two-site spherical molecules

that can form both linear aggregates and intermolecular rings of size 4 (scenario

1) at various association energies. . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.5.7 Fractions of molecular bonding states bonded in the saturated liquid and vapour

phases (red points correspond to the critical points) for an associating fluid of

two-site spherical molecules in a) scenario 0, and b) scenario 1. The colours

correspond to the different association energies. . . . . . . . . . . . . . . . . . . 117

4.5.8 Coexistence volumes for a a) two-site, and a b) three-site model, and vapourisation

enthalpies for a c) two-site, and a d) three-site model. The curves correspond

to an associating fluid of spherical molecules in scenario 0 (black), in scenario 1

(pink), in scenario 2 (blue) and in scenario 3 (green). . . . . . . . . . . . . . . . 118

xviii LIST OF FIGURES

5.1.1 2-methoxyethanol in intramolecular ring form. . . . . . . . . . . . . . . . . . . . 122

5.1.2 Representation of 3D models of a) 2-acetyl-1-naphtol and b) 1-acetyl-2-naphtol. 123

5.1.3 Compressibility factor Z as a function of the association energy εHB/(kT ) at

packing fraction η = 0.05. The symbols represent simulation data and the curves

represent theory: TPT1 (dashed curve and squares) and TPT1+intramolecular

rings (continuous curve and downward-pointing triangles). The diagram was

made using data extracted from the original publication of Ghonasgi et al. [18]. . 124

5.2.1 Molecular structure of thymol on the left and respective homonuclear model used

in SAFT-VR Mie calculations on the right. The model is composed of three

segments and three association sites of two types, one H and two e1. . . . . . . . 126

5.2.2 Molecular structure of ibuprofen on the left and respective group contribution

(GC) model with an underlying 3D structure used in SAFT-γ Mie calculations

on the right. The groups (highlighted with different colours and descriptions) are

fused and are made up of one segment each. Each group involved in the aromatic

cycle contains one association site e1 and the group COOH contains three site

types: one H, two e1 and two e2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.2.3 Chemical structure of 1,2,4-butanetriol. . . . . . . . . . . . . . . . . . . . . . . . 130

5.3.1 The molecules are modelled as Lennard-Jones chains (a)/spheres (b) that consist

of segments (red large sphere of diameter σ), decorated with sites A (green)

and/or B (blue) located off-centre in the segments at rd = 0.4σ from the segment

centre of mass. The volume of the association sites defines the bonding volume.

When two sites overlap, the pair interaction energy is taken to be equal to −εHBab .

Association is promoted by the site pair AB. The chain molecule (a) is involved

in intramolecular association and self-association into open aggregates as well as

in cross-association with the spherical component (b). . . . . . . . . . . . . . . . 133

LIST OF FIGURES xix

5.3.2 Reduced vapour pressures P ∗ = Pσ311/ε

disp11 as function of reduced temperature

T ∗ = kT/εdisp11 of the non-associating fluid (dashed black curve) and the associating

fluids with W ∗ = 0 (continuous black curve), with W ∗ = 10 (continuous green

curve), with W ∗ = 100 (continuous blue curve) and with W ∗ = 107 (continuous

pink curve). All curves were calculated using SAFT-VR Mie modified for ring

formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.3.3 Clausius-Clapeyron representation of reduced vapour pressures P ∗ = Pσ311/ε

disp11

as function of reduced temperature T ∗ = kT/εdisp11 of the non-associating fluid

(dashed black curve) and the associating fluids with W ∗ = 0 (continuous black

curve), with W ∗ = 10 (continuous green curve), with W ∗ = 100 (continuous blue

curve) and with W ∗ = 107 (continuous pink curve). All curves were calculated

using SAFT-VR Mie modified for ring formation. . . . . . . . . . . . . . . . . . 135

5.3.4 Reduced vapourisation enthalpies ∆hvap,∗ = ∆hvap/(NkT ) as function of reduced

temperature T ∗ = kT/εdisp11 of the non-associating fluid (dashed black curve) and

the associating fluids with W ∗ = 0 (continuous black curve), with W ∗ = 10

(continuous green curve), with W ∗ = 100 (continuous blue curve) and with W ∗ =

107 (continuous pink curve). All curves were calculated using SAFT-VR Mie

modified for ring formation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

5.3.5 Coexistence packing fractions η across reduced temperatures T ∗ = kT/εdisp11 for

the non-associating fluid (dashed black curve) and the associating fluids with

W ∗ = 0 (continuous black curve), with W ∗ = 10 (continuous green curve), with

W ∗ = 100 (continuous blue curve) and with W ∗ = 107 (continuous pink curve).

All curves were calculated using SAFT-VR Mie modified for ring formation. . . 136

xx LIST OF FIGURES

5.3.6 Relative fractions of molecules in each bonding state in the vapour phase a) across

reduced temperature T ∗ = kT/εdisp11 at fixed packing fraction η = 0.007 and b)

across packing fraction at fixed reduced temperature T ∗ = 2.0. The fraction of

molecules in open-chain aggregates is represented by a coloured area, the fraction

of intramolecular rings is represented by a square-patterned area and the fraction

of monomers by a white area. The three columns refer to the associating systems

in the absence of rings, i.e., with W ∗ = 0, in grey tones (left-hand side), with

W ∗ = 10 in green tones (middle) and with W ∗ = 100 in blue tones (right-hand

side). All curves were calculated using SAFT-VR Mie modified for ring formation. 138

5.3.7 Relative fractions of molecules in each bonding state in the liquid phase a) across

reduced temperature T ∗ = kT/εdisp11 at fixed packing fraction η = 0.300 and b)

across packing fraction at fixed reduced temperature T ∗ = 2.0. The fraction of

molecules in open-chain aggregates is represented by a coloured area, the fraction

of intramolecular rings is represented by a square-patterned area and the fraction

of monomers by a white area. The three columns refer to the associating systems

in the absence of rings, i.e., with W ∗ = 0, in grey tones (left-hand side), with

W ∗ = 10 in green tones (middle) and with W ∗ = 100 in blue tones (right-hand

side). All curves were calculated using SAFT-VR Mie modified for ring formation. 139

5.3.8 Representations of vapour–liquid phase behaviour for pure chain fluids consisting

of m = 5 segments (orange curves), m = 8 segments (green curves) and m = 10

segments (blue curves). For each chain length it is shown the non-associating

system (short dashes), and the associating with W ∗ = 0 (continuous), with

W ∗ = 10 (dash-dot-dash) and with W ∗ = 100 (long dashes). All curves were

calculated using SAFT-VR Mie modified for ring formation. . . . . . . . . . . . 141

5.3.9 Fractions of free molecules (monomers) along the saturation vapour and liquid

lines for pure chain fluids consisting of m = 5 segments (orange curves), m = 8

segments (green curves) andm = 10 segments (blue curves). For each chain length

it is shown the associating systems with W ∗ = 0 (continuous), with W ∗ = 10

(dash-dot-dash) and with W ∗ = 100 (dashes). All curves were calculated using

SAFT-VR Mie modified for ring formation. . . . . . . . . . . . . . . . . . . . . . 142

LIST OF FIGURES xxi

5.3.10Fractions of molecules in open-chain aggregates along the saturation vapour and

liquid lines for pure chain fluids consisting of m = 5 segments (orange curves),

m = 8 segments (green curves) and m = 10 segments (blue curves). For each

chain length it is shown the associating systems with W ∗ = 0 (continuous),

with W ∗ = 10 (dash-dot-dash) and with W ∗ = 100 (dashes). All curves were


5.3.11Fraction of molecules in intramolecular rings (ξ1) along the saturation vapour and

liquid lines for pure chain fluids consisting of m = 5 segments (orange curves),

m = 8 segments (green curves) and m = 10 segments (blue curves). For each

chain length it is shown the associating systems with W ∗ = 0 (continuous),

with W ∗ = 10 (dash-dot-dash) and with W ∗ = 100 (dashes). All curves were


5.3.12vapour–liquid equilibrium (VLE) and liquid–liquid equilibrium (LLE) diagrams

for the binary system and distribution of bonding states of the chain molecule,

for which the molecular models are described in Section 5.3.1. Four systems

are considered consisting of the non-associating (black dashed curves in a), b)

and c)) and the associating with W ∗ = 0 (black continuous curves in a), b)

and c)), W ∗ = 10 (colourful continuous curves in b)) and W ∗ = 100 (colourful

continuous curves in c)). The fractions of chain molecules in all four bonding

states along the saturation curves are represented in the central and right-hand

side diagrams. The central diagrams correspond to the saturated lines of the

VLE’s vapour phase (V) and the LLE’s right branch (L1) and the right-hand

side diagrams correspond to the saturated lines of the VLE’s liquid phase (L)

and the LLE’s left branch (L2). The bonding states of the chain molecule consist

of monomer (‘free’), intramolecular ring (‘ring’), open self-association (‘open, s’)

if the molecules associate with other chain and open cross-association (‘open, c’)

if the molecules associate with a spherical molecule. . . . . . . . . . . . . . . . 146

5.4.1 Thermodynamic cycle neglecting pressure effects to solve for the enthalpy and

entropy of fusion of solid i at a temperature T lower than its normal melting

point T fusi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xxii LIST OF FIGURES

5.4.2 Experimental heat capacity data of pure acetic acid by [19] (upward-pointing

triangles), [20] (downward-pointing triangles), [21] (circles) and [22] (squares). . 151

5.4.3 Representation of the estimated error from neglecting the heat capacity contribu-

tion to solubility calculations (a) as a function of the quotient between operating

and fusion temperatures and (b) as a function of f(T ) given by Equation (5.25).

The points represent the estimated error for the compounds in Table 5.5 for

T = 298.15 K labelled accordingly. . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5.4.4 Prediction of the solid-liquid equilibria in binary systems a function of temperature

at P = 0.100 MPa. The symbols represent the experimental data and the

continuous curves the prediction with the SAFT-γ Mie approach accounting for

(black) and neglecting (blue) the heat capacity term for: (a) stearic acid in

benzene (squares [23]); (b) ibuprofen in ethanol (squares [24], upward-pointing

triangles [25], downward-pointing triangles [26], circles [27], diamonds [28] and

pentagons [29]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5.4.5 Prediction of the solid-liquid equilibria in a binary system of water and methanol

as a function of temperature at P = 0.100 MPa. The symbols represent the

experimental data (circles [30], downward-pointing triangles [31] and squares

[32]) and the continuous curves the prediction with the SAFT-γ Mie approach

accounting for (black) and neglecting (blue) the heat capacity term. . . . . . . . 155

5.4.6 Scheme of molecular models of solute, solvent 1 and solvent 2. . . . . . . . . . . 156

5.4.7 Range of solid–liquid equilibria curves for W ∗ between 1 and the limit of complete

ring formation in various solvent systems and energies of association. . . . . . . 157

5.4.8 Solute fractions X0 (dashed curves) and ξ1 (continuous curves) in various solvent

systems with W ∗ = 1: S1 (orange), S2 (blue), 80 % S1 (green), and 40 % S1 (pink).157

5.5.1 Chemical structures of lovastatin (LVS) and simvastatin (SVS), highlighting the

groups (colourful spheres) and association sites (black e1 and white H spheres)

considered in the SAFT-γ Mie approach. Association may occur only between

association sites of different types (black-white association). . . . . . . . . . . . 160

LIST OF FIGURES xxiii

5.5.2 Solubility predictions in mole fraction (x) of LVS (a) and SVS (b) in various

alcohols at ambient pressure p = 0.1 MPa as a function of temperature. The

continuous curves represent the predictions of the SAFT-γ Mie approach for

the solubilities in ethanol (pink), 1-propanol (purple), 1-butanol (dark blue),

1-pentanol (light blue), 1-hexanol (green), 1-heptanol (orange) and 1-octanol

(red). The empty circles represent the experimental solubility of a) LVS by

Nti-Gyabaah et al. [15] and Sun et al. [33] and of b) SVS by Aceves-Hernández

et al. [34] in the various solvents. The filled circles represent the experimental

solubility of b) SVS by Nti-Gyabaah et al. [35] in the various solvents. . . . . . 163

5.5.3 Solubility predictions of LVS (a) and SVS (b) in ethanol at ambient pressure

p = 0.1 MPa as a function of temperature. The solubility xstatin is given in

molar fraction. The dashed and the continuous black curves represent the

predictions with the SAFT-γ Mie approach with and without the “switching off

sites” treatment, respectively. The coloured lines represent predictions with the

SAFT-γ Mie approach including the modified association term to account for

ring formation, with values for the W ∗ parameter of 1 (purple), 10 (pink), 102

(blue), 103 (green) and 104 (orange). The experimental data is represented by

circles [33] in a) and [34] in b) and by diamonds [15] in a) and [35] in b). . . . . 164

5.5.4 Fractions of molecules of LVS a) bonded inter- and intramolecularly in the

absence of ring formation (dashed curves) and in the presence of ring formation

withW ∗ = 100 (continuous curves), and b) in chain and ring form withW ∗ = 100,

in the various solvents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

5.5.5 Relationship between a) the fraction of LVS in ring form and the parameter W ∗,

and b) the solubility of LVS as a function of W ∗, in various alcohols at constant

temperature 312 K and ambient pressure 0.1 MPa. . . . . . . . . . . . . . . . . . 168

5.5.6 Solubility of LVS as a function of ring fraction in various alcohols at 312 K. . . . 170

5.5.7 Solubility predictions of LVS in various alcohols at ambient pressure p = 0.1 MPa

as a function of temperature. The continuous curves represent the predictions

of the SAFT-γ Mie approach modified to account for ring formation for the

solubilities in ethanol with W ∗ = 104 (pink), 1-propanol with W ∗ = 103 (purple),

1-butanol with W ∗ = 20 (dark blue), 1-pentanol with W ∗ = 103 (light blue),

1-hexanol with W ∗ = 103 (green) and 1-octanol with W ∗ = 104 (red). The

symbols represent the experimental solubility of LVS [15, 33]. . . . . . . . . . . . 170

B.0.1Each chain is formed by segments that associated selectively. . . . . . . . . . . . 181

B.0.2Each associating chain molecule is formed as described in Figure B.0.1. . . . . . 182

xxiv

List of Acronyms

API active pharmaceutical ingredient

CPA cubic plus association

DFT density functional theory

EOS equation of state

FG functional group

GC group contribution

HB hydrogen bond

InterMHB intermolecular hydrogen bonding

IntraMHB intramolecular hydrogen bonding

LLE liquid–liquid equilibrium

LVS lovastatin

MDF multi-density formalism

PT perturbation theory

SAFT statistical associating fluid theory

SI steric incompatibility

SLE solid–liquid equilibrium

SVS simvastatin

SW square-well

TPT thermodynamic perturbation theory

TPT1 first-order thermodynamic perturbation theory

VLE vapour–liquid equilibrium

VLLE vapour–liquid–liquid equilibrium

VR variable range

xxv

Chapter 1

INTRODUCTION

The accurate description of properties of fluid systems is crucial to an efficient design and

optimisation of products/processes relevant to industry. Experimental data can be difficult,

costly and time-consuming to obtain and, as such, a variety of mathematical methods have been

developed for the purpose of minimising this effort, which can differ greatly in approach and

form. The fact that the physical and chemical properties of compounds are function of their

molecular structures has inspired multiple mathematical approaches from first principles. The

advantage of developing a theory from first principles, detached from experimental error as much

as possible, is the added clarity that the use of approximations provides to the understanding of

the underlying physical behaviour. Accordingly, the conditions of validity are known, which

allow for the possibility of carrying out modifications and extensions in a physically supported

way. Moreover, a type of statistical-mechanical treatment and equation of state (EOS) based

on a well-defined Hamiltonian (the total energy of the system in terms of coordinates and

momenta), allows for the testing of the theory against molecular simulation [36], which is very

useful in particular when the model systems do not directly represent reality, or in case of

inaccuracy/scarcity/unattainability of experimental data.

The first EOS to successfully describe simultaneously gas and liquid phases was the van der

Waals EOS [37], marking the beginning of the cubic EOSs1. The empirical modifications to1Cubic equations of state are called as such because they can be rewritten as a cubic function of volume [38].

1

2 Chapter 1. INTRODUCTION

the van der Waals EOS that followed were designed to improve its accuracy and applicability.

Although cubic EOSs can be used to accurately describe various aspect of the fluid-phase

equilibria of mixtures, they are empirical correlations [38]. A lot of effort has been expended

on the development of EOSs with a more rigorous physical foundation, which allow one to

tackle systems of increasing complexity. Modelling plays a central role across petrochemical, fine

chemical, pharmaceutical and polymer industry, in the design of new products and processes that

involve complex systems with anisotropic interactions, such as the hydrogen bonding. In the early

days of quantum mechanics, Lennard-Jones and Pople [39, 40] described hydrogen bonding as a

type of molecular association in terms of a ‘crude’ but chemically intuitive representation based

on the hybridisation of atomic orbitals into molecular orbitals. Hydrogen bonding molecules

must typically contain an electronegative atom (X = O, N, F, S) bonded to a hydrogen atom XH;

the remaining orbitals on atom X corresponding to the so-called ‘lone pairs’ of electrons can then

interact with the slight positive charge at the centre of the XH bond leading to a localised, and

therefore directional, XH− X hydrogen-bonding interaction. And currently, the hydrogen bond

is defined as an “attractive interaction between a hydrogen atom from a molecule or a molecular

fragment X− H in which X is more electronegative than H, and an atom or a group of atoms

in the same or a different molecule, in which there is evidence of bond formation”, according

to IUPAC’s recommendation [41]. Short-ranged directional attractive interactions, such as

those associated with hydrogen bonding, lead to the formation of association aggregates. The

hydrogen bonding has a marked influence on thermodynamic properties and phase behaviour of

associating fluids (e.g., water and water mixtures [42,43]), which deviate significantly from those

of simple fluids. Wertheim’s thermodynamic perturbation theory (TPT) [17,44,45,46] is one of

the simplest and most successful approaches for treating strongly associating molecules from

first principles. Shortly after its original formulation, Wertheim’s TPT formalism was recast into

an EOS by Chapman et al. [47,48,49,50] and presented as the statistical associating fluid theory

(SAFT). SAFT constitutes a major advance in the development of a theoretical framework for

modelling complex associating fluids. Despite the success of SAFT, there is still the need for

improvements to enhance the accuracy or extend the capabilities of the methodology.

1.1. Motivation 3

1.1 Motivation

The applicability of the SAFT family of EOSs in their standard formulation is limited by the

simplifying approximations applied in the development of Wertheim’s first-order thermodynamic

perturbation theory (TPT1). One of these constraints implies the absence of ring-like association

aggregates considered by the theory, i.e. one assumes that only linear or branched chains are

formed by the associating molecules. Ring structures in association aggregates can be found

in the presence of at least a hydrogen donor group (OH, COOH, H2O, NH, SH, . . . ) and a

receptor group (O, N, S, aromatic C, . . . ) which can be part of the same or different molecules.

Examples of molecules that may exhibit the formation of ring-like aggregates include simple

compounds like diols, ether glycols, hydrogen fluoride, and complex compounds like the active

pharmaceutical ingredients (APIs) ibuprofen, phenyl acids, ibuprofen, statins, among others.

The formation of association aggregates has a strong effect on the fluid behaviour and hence, it

is crucial to accurately capture the distribution of the various molecular aggregates with an

arbitrary structure and still be compatible with the framework developed by Wertheim. Efforts

have been devoted to relaxing some of the constraints imposed by the approximations inherent

in TPT1 but EOSs of the SAFT family still incorporate the standard treatment of hydrogen

bonding developed by Wertheim and extended to mixtures by Joslin et al. [51], which neglects

the existence of inter- or intramolecular ring-like association aggregates. Indeed, a general

expression to treat the formation of ring aggregates that is compatible with a SAFT framework

is still missing and it is the core goal of the work presented in this thesis.

1.2 Objectives

The objectives of the research presented here are listed below:

• Examine and summarise the major contributions to the extension of TPT1 in systems

which form ring aggregates.

• Derive a formalism to include intermolecular ring aggregates in a theory which can be

readily implemented in any EOS that accounts for the association contribution to the

4 Chapter 1. INTRODUCTION

Helmholtz free energy as a perturbation, given a reference system with known properties.

• Derive a formalism to include intramolecular ring aggregates in a theory which can be

readily implemented implemented in any EOS that accounts for the association contribution

to the Helmholtz free energy as a perturbation, given a reference system with known

properties.

• Study the effect of ring formation in the thermodynamic properties and fluid-phase

behaviour of associating systems.

• Assess the influence of the difference in the heat capacities of the liquid and solid states of

a pure solute in solubility predictions.

• Apply the newly developed theory for ring aggregates to a case study of solubility of

statins in alcohol and comparison with the results obtained from the standard TPT1.

1.3 Thesis outline

This thesis is structured as follows:

• Chapter 2 – Wertheim’s TPT1 is reviewed from the graphical expansion perspective,

highlighting the approximations taken in its development. Next, the SAFT EOS is

described briefly and a closer look is taken at how TPT1 is translated into the association

and chain terms of the EOS. A comprehensive literature review of all contributions to the

extension of TPT1 to account for the formation of ring aggregates is given at the end of

the chapter, including a detailed description of the most relevant.

• Chapter 3 – TPT1 is extended to a formalism that can be used to account for any type

of aggregate. Simple mass equations are derived for the set of aggregates consisting of

linear-chain, branched-chain, intermolecular (any sizes) and intramolecular rings formed

in associating fluids.

• Chapter 4 – the re-normalisation constant for the end-to-end distribution function is

derived, and the newly developed theory is implemented in the SAFT-VR SW EOS, and

then used in calculations for simple model molecules that form intermolecular rings of

1.3. Thesis outline 5

specified sizes.

• Chapter 5 – the impact of the heat capacity contribution in the solubility prediction is

examined. The SAFT-γ Mie is then modified and the properties of simple model chain

molecules that form intramolecular rings are assessed. Furthermore we evaluate the ring

contribution to the solid–liquid equilibria in real binary systems of statins and alcohols.

• Chapter 6 – The key achievements presented in the thesis are summarized, highlighting

the main conclusions and recommendations/directions for future work.

Chapter 2

WERTHEIM’S THERMODYNAMIC

PERTURBATION THEORY

In this Chapter, the first-order thermodynamic perturbation theory (TPT1) of Wertheim is

reviewed, which is fundamental to the framework of this work. Next, the statistical associating

fluid theory (SAFT), an equation of state (EOS) that stemmed from TPT1, is briefly introduced.

Finally, the attempts at the extension of TPT1 to account for ring formation are mentioned,

and the most relevant are described in detail.

2.1 Introduction

The field of thermodynamics comprises the relationships between the macroscopic properties

of a system in equilibrium. Closely related is the field of statistical mechanics, which provides

the description of the observed values of the macroscopic properties from the way particles

making up the system interact with each other. The macroscopic properties result from the

time-averaged behaviour of its comprising particles. An alternative approach proposed by Gibbs

to obtain the properties is followed – the ensemble average. An ensemble consists of a large

number of subsystems, each being a replica of the macroscopic system of interest. At the

microscopic level, each of these replicas may have a different state (i.e. different positions and

6

2.1. Introduction 7

momenta of the particles) but they are in the same macroscopic state. According to the ergodic

hypothesis, the ensemble average of a quantity is equal to the time average of the quantity.

Consider an isolated, macroscopic system consisting of N identical spherical particles at temper-

ature T enclosed in a volume V . The canonical NV T ensemble, the most commonly used, is

characterised by each member of the ensemble having the same number of particles, volume and

temperature. In the canonical ensemble the probability (P ) of the system having a particular

configuration (N particles in positions r1, r2, ..., rN ) and a momenta distribution (p1,p2, ...,pN )

is given as

P (r1, r2, ..., rN ,p1,p2, ...,pN) = 1h3NN !

exp(−βH(r1, r2, ..., rN ,p1,p2, ...,pN))QN

, (2.1)

where h is the Planck’s constant, β = 1/kT , with the Boltzmann constant k,H is the Hamiltonian,

the total energy of the system given by the sum of the kinetic energy (K(N)) and the potential

energy (comprising Φ(N), the interparticle potential energy and V (N)ext , the potential energy arising

from the interactions of the particles with an external field), and QN is the canonical partition

function given as

QN = 1N !Λ3N

∫exp

[−β

(Φ(N)(r1, r2, ..., rN) + V (N)

ext (r1, r2, ..., rN))]

dr1 dr2... drN ,

= ZNN !Λ3N ,

=QidealN

ZNV N

,

(2.2)

where QidealN is the canonical partition function for an ideal gas, Λ is the de Broglie wavelength

that includes the translational, the rotational, and the quantal parts of the molecular partition

function and ZN is the configuration integral given as

ZN =∫

exp[−β

(Φ(N)(r1, r2, ..., rN) + V (N)

ext (r1, r2, ..., rN))]

dr1 dr2... drN , (2.3)

which equals V N for an ideal gas. The partition function provides with the link between the

8 Chapter 2. WERTHEIM’S THERMODYNAMIC PERTURBATION THEORY

macroscopic properties and the molecular behaviour through the relation

A = −kT lnQN , (2.4)

where A is the Helmholtz free energy. Once A is known, all other thermodynamic properties

can be obtained as its derivatives, such as the pressure (p),

P = −(∂A

∂V

)N,T

, (2.5)

the entropy (S),

S = −(∂A

∂T

)N,V

, (2.6)

the chemical potential (µ),

µ =(∂A

∂N

)V,T

, (2.7)

the heat capacity (cV ),

cV = −T(∂2A

∂T 2

)N,V

, (2.8)

etc. Similar methods can be used to obtain thermodynamic functions with other ensembles. The

grand canonical µV T ensemble is used to model open homogeneous systems. It is characterised by

each member of the ensemble having the same chemical potential µ, volume V and temperature

T . The grand partition function Ξ is related to the canonical partition function by

Ξ =∑N≥0

exp(βNµ)QN . (2.9)

For the ideal gas, the Helmholtz free energy is obtained from the substitution of QidealN = V N

N !Λ3N

in Equation (2.4):

Aideal = −kTN[ln(ρΛ3)− 1

], (2.10)

where ρ is the density given as N/V . The equation of state for the ideal gas is calculated from

2.1. Introduction 9

the relation Equation (2.5) and is given as

P ideal = NkT

V, (2.11)

The equation of state for interacting particles is less straightforward as the configuration integral

must be considered. The probability of finding any particle in position r1, another in r2... and

another in rn, is determined by the quantity

ρ(n)(r1, r2, . . . rn) = N !(N − n)!ZN

∫exp

[−β

(Φ(N)(r1, r2, ..., rN)

+V (N)ext (r1, r2, ..., rN)

)]drn+1 . . . drN ,

(2.12)

also known as the n-particle density. In a homogeneous fluid ρ(1)(r1) = ρ = N/V . Of particular

interest are the n-body distribution functions g(n)(r1, r2, . . . rn) which measure the deviation of

the fluid structure from that of a uniform distribution of particles in space. These are given by

g(n)(r1, r2, . . . rn) = ρ(n)(r1, r2, . . . rn)∏ni=1 ρ

(1)(ri), (2.13)

with the most important quantity (and best known) being the two-body distribution function

given as

g(2)(r1, r2) = ρ(2)(r1, r2)ρ(1)(r1)ρ(1)(r2) . (2.14)

If the system is both isotropic and homogeneous, then the two-body distribution reduces to the

radial/pair distribution function which is given by

g(r) = ρ(r)ρ, (2.15)

where r = |r2 − r1| measures the distance from a particle. Once the structure and Helmholtz

energy of a given reference fluid is known, an approach to obtain the equation of state of a

dense fluid is given by perturbation theories.

In perturbation theory, a property of a target system is calculated by summing a perturbation

contribution to a reference value. This reference value is extracted from a reference system which


we assume to have full knowledge of and shares a similar structure with the target system. Due

to the structure of a tightly packed fluid (liquid not too close to the critical point) being mostly

governed by repulsive interactions [52], a repulsive system is usually used as the reference. One

can often split the intermolecular pair potential (φ) of a fluid in a steep short-range repulsion

(φrep) and a smoothly varying longer range attraction (φatt) so that

φ = φrep + φattr, (2.16)

where φrep is taken as the reference potential and the attractive forces are modelled as a

perturbation given by φattr. Since the structure and thermodynamic properties of a hard-sphere

fluid are well known [36,52], unlike the repulsive part of a general potential, one step further may

involve the mapping of the repulsive fluid onto a hard-sphere fluid with an effective diameter

given as a function of temperature (dependent on the target potential considered) [53,54,55].

The free energy (A) of a target fluid with a potential of the form in Equation (2.16) can thus be

obtained by perturbing a hard-sphere fluid,

A− AHS = ∆A1 + ∆A2 + ..., (2.17)

where AHS is the free energy for a hard-sphere reference and ∆Ai is an attractive perturbation

according to a perturbation theory such that of Barker and Henderson [53] or that of Chandler–

Weeks–Andersen [56].

When associating fluids are considered, highly-directional and short-range interactions (such as

hydrogen bonds) are added as a perturbation to the intermolecular potential. The thermodynamic

perturbation theory (TPT) is the theory of interest to this work as it treats this additional

perturbation originating from hydrogen bonding. The TPT was developed by Michael S.

Wertheim in his landmark papers [17,44,45,46], tailored to handle complex fluids that exhibit

bonding interactions, namely hydrogen bonding fluids, fluids of chain molecules and polymers.

In Wertheim’s TPT the reference system is the non-associating version of the real system and

it contains all repulsive and van der Waals interactions between its particles according to a

certain defined reference potential.

2.2. The thermodynamic perturbation theory of first order 11

Wertheim was influenced by the ideas of Andersen [57], the author of one of the earliest theories

of association, who first considered that the repulsive cores of two associated molecules prevent

their simultaneous association to a third molecule. Andersen reformulated the cluster expansion

method of Mayer [58, 59] in terms of the singlet density (ρ) to describe the formation of dimers

and oligomers by intermolecular potentials of hydrogen bonding type both in dilute and dense

fluids. However the method was complicated to apply. Later, Høye and Olaussen [60] extended

Andersen’s approach and applied it to solvation in a binary mixture. They used an activity

expansion instead of a density expansion which proved to simplify the expansion further as it

enabled more term cancellations. The activity expansion had already been used by Chandler

and Pratt [61] in chemical equilibria, intramolecular structures and intermolecular correlation

structures but with physical clusters [62] (as opposed to clusters of the Mayer expansion). Høye

and Olaussen [60] suggested that the cluster expansion should converge more rapidly, if done

in terms of the overall and non-associated molecular densities instead of only overall densities,

introducing the concept of multiple density taken up and extended by Wertheim.

Wertheim used perturbation theory to derive an expansion of the Helmholtz free energy in

terms of multiple densities related to the formation of association clusters. The first order

approximation (TPT1) reduces the many-body problem to a two-body problem (in general,

TPTn involves (n + 1)–body distribution functions).The derivation of the free energy of an

associating fluid is presented next according to Wertheim’s TPT1 and it is in large part a

summary of the outstanding review by Zmpitas and Gross [63], but focusing on the restrictions

of TPT1.

2.2 The thermodynamic perturbation theory of first or-

der

Consider an associating system composed of N spheres interacting through a pair potential

φ(12). A molecule i may have many degrees of freedom, that includes three translational and a

number of orientational ones. These are included in the positional vector ri and orientational


angles that are included in the configuration vector Ωi. Here, (1) is a compact notation for

(r1,Ω1) and (12) = (r1,Ω1, r2,Ω2) accordingly. The theory neglects cooperative effects [64] (i.e.

when the formation of a hydrogen bond strengthens or weakens others), and therefore the total

intermolecular potential, Φ(N)(1...N), is given by the sum of the pairwise additive interactions

between pairs of molecules as

Φ(N)(1...N) =N∑n=1

N∑m>n

φ(mn). (2.18)

Each pair of molecules contributes with a core interaction, φref , which Wertheim considered to

be hard-sphere1 of diameter σ, as in

φref(12) = φHS(12) =

∞ , |r2 − r1| < σ

0 , otherwise, (2.19)

and an associating interaction promoted by off-centred association sites a indexing the sites of

molecule 1 and b indexing those of molecule 2, according to a square-well potential φab, similar

to Equation (2.16),

φ(12) = φref(12) +∑a

∑b

φSWab (12), (2.20)

with the difference that the reference fluid does not have to be fully repulsive and the perturbation

is of generic hydrogen bond (HB) type. The perturbation is typically considered to be a square

well of range rcab with well depth of εHB

ab given as

φSWab (12) =

−εHB

ab , |r2 + db(Ω2)− r1 − da(Ω1)| < rcab

0 , else, (2.21)

where da(Ω1) is the displacement vector of site a from the centre of molecule (1) and db(Ω2) is

the displacement vector of site b from the centre of molecule (2).

In the interest of general applications, the system is additionally subjected to an external field,

1However, the reference interaction is not limited to the hard-sphere. For instance, in the SAFT-γ Mieequation of state the Mie potential is used, as we will explore in Section 2.3.


Vext (such as the confining potential) allowing for the treatment of inhomogeneity. The intrinsic

Helmholtz free energy (A) is the part of the Helmholtz free energy (A) that accounts for the

interactions between particles only, excluding the effect of the external field [52]. The intrinsic

free energy is given by the functional of the density profile

A[ρ] =A[ρ]−∫ρ(1)Vext(1) d(1)

=∫ρ(1) (µ− Vext(1)) d(1) + Ω[ρ]

(2.22)

where µ is the chemical potential including the external forces from the field and the difference

(µ − Vext(1)) equals the intrinsic chemical potential. The grand potential Ω, whose natural

variables are volume V , temperature T and chemical potential µ, is given by the logarithm of

the grand canonical partition function, Ξ, as

Ω = −kT lnΞ. (2.23)

Wertheim’s result is an expression for the Helmholtz free energy in terms of multiple densities

related to the distribution of association clusters. The starting point is an activity expansion of

the grand partition function [52,65] and graphical methods are used in order to translate the

activity dependence into number density dependence. The grand partition function is given

as [52]

Ξ =∑N≥0

[exp(βNµ)N !Λ3N

∫exp

(−βΦ(N)(1...N)

)exp

(−β

N∑m=1Vext(m)

)N∏m=1

d(m)]. (2.24)

The local activity (z(1)) is given as

z(1) = Λ−3 exp [β (µ− Vext(1))] , (2.25)

which impliesN∏i=1

z(i) = Λ−3N exp (βNµ) exp(−β

N∑i=1Vext(i)

), (2.26)


that allows the partition function to be rewritten as the functional of the activity as

Ξ[z] =∑N≥0

∫ 1N !

(N∏i=1

z(i))

exp−β N∑

j=1

N∑i>j

φ(ij) N∏i=1

d(i) . (2.27)

Our goal however is to represent the partition function as a functional of density to substitute

in Ω [ρ] = −kT lnΞ [ρ]. To that effect, the relation between the singlet number density function

ρ(1) and activity, given by [52]

ρ(1) = 1Ξ[z]

δΞ[z]δ ln z(1) , (2.28)

is used to write the variation of the logarithm of the grand partition function, δ lnΞ[z] =∫ρ(1)δ ln z(1) d(1), which can be integrated to obtain [52]

lnΞ[ρ] = −∫ρ(1)

(ln ρ(1)z(1) − 1

)d(1) + c(0)[ρ]. (2.29)

The functional c(0)[ρ] is defined through its derivative as

δc(0)[ρ] =∫

ln ρ(1)z(1)δρ(1) d(1), (2.30)

and it is related to the single-particle direct correlation function, c(1), defined as

c(1) = δc(0)[ρ]δρ(1) = ln ρ(1)

z(1) . (2.31)

Finally, we are in the position of writing the intrinsic Helmholtz free energy as a functional of

density. By substituting Equations (2.23), (2.26) and (2.29) into Equation (2.22) we obtain

βA[ρ] =∫ρ(1)

[ln(Λ3ρ(1)

)− 1

]d(1)− c(0)[ρ], (2.32)

where the integral term in Equation (2.32) is the ideal contribution to the Helmholtz free

energy [52] and −kTc(0)[ρ] is the correction term for the real contribution. At this point, the

description of A[ρ] does not capture the contribution from hydrogen bonding in associating

fluids explicitly but it is useful to apply to the reference system, where no association takes


place as [44,63]

βAref [ρ] =∫ρ(1)

[ln(Λ3ρ(1)

)− 1

]d(1)− c(0)

ref [ρ]. (2.33)

The intrinsic residual Helmholtz free energy is given as the difference

β (A[ρ]−Aref [ρ]) = c(0)[ρ]− c(0)ref [ρ]

= ∆c(0)[ρ].(2.34)

Once association sites are considered, moving to a graphical language is convenient due to the ease

of the representation of the pairwise additive potential interactions in multiple body problems.

The diagrams of graph theory also facilitate the application of the TPT1 approximations and

steric constraints through graph cancellation [17, 63, 66]. Before looking at association, we

write the expressions for the reference fluid in Equations (2.27)–(2.31) in graphical language.

This allows us to get acquainted with the procedure of translating activity expansions into

density expansions through a well known graph manipulation method known as topological

reduction [52,63].

The expression for the partition function (Equation (2.27)) includes the Boltzmann factor

e(12) = exp(−βφ(12)), which can be rewritten as

e(12) = exp[−β(φref(12) +∑a

∑b

φab(12))], (2.35)

after we substitute the reference and association contributions to the pair potential explicitly

(Equation (2.20)). Using the product property of exponentials, Equation (2.35) becomes

e(12) = eref(12)∏a∈Γ

∏b∈Γ

eab(12), (2.36)

where eref is the Boltzmann factor for the reference potential and eab for the association potential

between sites a and b. The Mayer f -function, is defined as usual as f(12) = e(12)− 1 [52], i.e.

f(12) = fref(12) + eref(12)∏a∈Γ

∏b∈Γ

(fab(12) + 1)− 1 , (2.37)


where fref(12) = (eref(12) − 1) and fab(12) = (eab(12) − 1). The spherical molecules in the

reference fluid thus interact through eref-bonds only. The activity expansion of the grand

partition function, according to Equation (2.27), is thus given explicitly as

Ξ[z] = 1 +∫z(1) d(1) + 1

2!

∫z(1)z(2)e(12) d(1) d(2)

+ 13!

∫z(1)z(2)z(3)e(12)e(23)e(13) d(1) d(2) d(3) + . . . ,

(2.38)

which is equivalent to the graphical representation given as

Ξ[z] = 1 + + 12! + 1

3! + 14! + . . . , (2.39)

where the shaded nodes ( ) are field points, each representing a single molecule activity

z integrated over the respective positional and orientational coordinates. The zig-zag edges

( ) connecting nodes represent the eref-bonds (the only type of e-bonds in the reference

fluid). The value of each diagram (integrals in Equation (2.38)) is divided by the symmetry

number of the graph (pre-factors of the integrals in Equation (2.38)). The symmetry num-

ber is given by the number of ways to label the field points (shaded nodes) of a diagram

without changing both its value and the bonds between nodes. For example, the integral∫z(1)z(2)z(3)z(4)e(12)e(23)e(24) d(1) d(2) d(3) d(4) can be represented in graphical language

as . If we were to label all nodes of this diagram we would have the following six

equivalent diagrams:1

2 3

4

,1

2 4

3

,3

2 1

4

,3

2 4

1

,4

2 3

1

and4

2 1

3

. The symmetry

number (S) of is thus S = 6. Substituting the e-bonds for f -bonds in Equation (2.39)

(e(12) = f(12) + 1) and applying the exponentiation theorem [52,63], the logarithm of the grand

partition function is rewritten as

lnΞ[z] = + 12 + 1

2 + 16 + 1

2 + 16 + 1

8 +. . . , (2.40)

where the straight-line edges ( ) connecting nodes represent the fref-bonds (the only

f -bonds in the reference fluid).


In order to express c(0)ref as a functional of density, as required in the free energy expression

(Equation (2.33)), a topological reduction is carried out, where the activity z-nodes ( ) are

substituted for density ρ-nodes ( ). For that effect, the relationship between the particle

number density and activity (Equation (2.28)) is used. Applying the functional derivative

lemma [52,63] to Equation (2.40) results in

δ lnΞ[z]δz(1) = 1 + 1 +

1

+ 12 1

+ 12

1

+1

+1

+ . . . , (2.41)

where a labelled position is all that remains in the place of the z-node that vanished as a result

of the differentiation. The value of an absent node is 1 and its position is not an integration

variable. The graph 1 thus represents the integral∫z(2)fref(12) d(2).

After substituting Equation (2.41) in Equation (2.28) the expression for density as a function of

activity is finally obtained so that

ρ(1) = z(1)1 + 1 +

1

+ 12 1

+ 12

1

+1

+1

+ . . .

= 1 + 1 +1

+ 12 1

+ 12

1

+1

+1

+ . . . ,

(2.42)

where the white nodes are called root points and, unlike field points, their coordinates are held

constant during the integration. Accordingly, the ρ-node is defined as

=∫ρ(1) d(1)

= + + + 12 + 1

2 + + + . . .

(2.43)

Following Equation (2.31) the direct correlation function is given as ln ρ(1)z(1) which can be

written in diagrammatic language, after application of the exponentiation theorem [52,63] to

Equation (2.42), as

c(1) = 1 +1

+ 12

1

+1

+ 12

1

+ 12

1

+ 12

1

+ . . . (2.44)


We proceed with a topological reduction using the definition of a ρ-node in Equation (2.43).

Due to the fact that a single ρ-node includes an infinite sum of graphs with z-nodes, the single

particle correlation function can be written as

c(1) = 1 + 12

1

+ 12

1

+ 12

1

+ 12

1

+ 16

1

+ . . . , (2.45)

where each graph is irreducible and includes many graphs of the activity expansion. As an

example, the simplest graph in the sum is given as

1 = 1 +1

+1

+ 12

1

+ 12

1

+ . . . (2.46)

A sequence of adjacent nodes and bonds connecting them is called a path. A diagram is said to

be connected if there is a path between any pair of its nodes. A node is called an articulation

node if its removal results in the separation of a connected diagram in two or more parts, of

which at least one contains no white nodes. For instance, consider the graph1

, after the

removal of the bottom right node (which is an articulation node), we are left with1

.

Lastly, a diagram is irreducible if it does not contain articulation nodes. For instance, consider

the graph1

, after the removal of the bottom right node (which is not an articulation

node), we are left with1

, which is a connected graph. According to Equation (2.30),

Equation (2.45) must be recovered from the functional derivative of c(0)[ρ] with respect to ρ(1),

which enables the expression for c(0)[ρ] being given as

c(0)ref [ρ] = 1

2 + 16 + 1

8 + 14 + 1

24 + . . . , (2.47)

concluding the derivation of the expression for the intrinsic Helmholtz free energy for the

non-associating reference fluid (Equation (2.33)).

We now turn to a pure associating fluid and follow an analogous rationale to find the expression

for the intrinsic residual Helmholtz free energy due to hydrogen bonding (AWert[ρ]), the result

of TPT1 [17,48,63]. In an associating fluid, apart from fref-bonds and eref-bonds, there are also


fab-bonds ( ) connecting labelled points situated in the nodes that represent association

sites. Considering a two-site model fluid, the z-node is now represented by . The small

points are coloured differently (white and black) to indicate that the sites are distinguishable,

i.e. labelled. In this work we follow the convention of using capital letters of the latin alphabet

to label individual sites. In the target fluid, Equation (2.37) for a fluid with sites A (white) and

B (black) becomes

f(12) = fref(12) + eref(12)fAB(12) + eref(12)fBA(12), (2.48)

or in graph form

= + + , (2.49)

after we applied the following bonding restrictions: 1) a pair of particles can be linked by at most

one hydrogen bond (no double bonding); 2) each site can be involved in at most one hydrogen

bond at a time, assured by imposing a sufficiently small interaction range rcab (Equation (2.21));

3) association between same type sites, i.e. A− A or B −B, is not permitted. In the words of

Andersen [57], “when the cancellation theorem is applied, it has a devastating effect, eliminating

diagrams with a variety of topological features”, such as , and .

After substituting Equation (2.49) in Equation (2.40) the logarithm of the grand partition

function for the associating fluid is rewritten as

lnΞ[z] =

+ + + + 13 + + + 1

4 + . . .

+ 12 + 1

2 + 16 + 1

2 + 16 + 1

8 + . . .

+ + + + + + + + . . .

(2.50)

Wertheim follows a multi-density formalism (MDF), where molecules in different bonding states

are treated as different species. Within this formalism, the number density is given as a sum of


the number density of molecules in the different bonding states, so that

ρ(1) =∑α⊆Γ

ρα(1), (2.51)

where we sum all densities of particles bonded at the sites in the subset α of the set of all

sites Γ = A,B. The number density of molecules is thus given by the sum of the density of

monomers ρ0(1), the density of molecules with site A bonded, the density of molecules with site

B bonded and the density of molecules with the two sites bonded, as in

ρ(1) = ρ0(1) + ρA(1) + ρB(1) + ρAB(1). (2.52)

Following Equation (2.28), we differentiate the logarithm of the grand partition function with

respect to the activity to obtain ρ(1) and find that it can be given as the sum of the four

contributions in Equation (2.52):

ρ0(1) = 1 + 1 +1

+ 12 1

+ 12

1

+1

+1

+ . . . (2.53)

ρA(1) = 1 +1

+1

+1

+1

+1

+1

+ . . . (2.54)

ρB(1) = 1 +1

+1

+1

+1

+1

+1

+ . . . (2.55)

ρAB(1) =1

+1

+1

+1

+1

+1

+1

+ . . . (2.56)

In this formalism, instead of single-density functionals [ρ], we strive for multi-density functionals,

represented by [ρ0, ρA, ρB, ρAB], which is written in compact notation as [ρ] [63]. As

previously done to obtain Equation (2.29), the variation of the lnΞ[z] is considered first and

then an integration in activity is carried out to obtain the functional lnΞ[ρ]. The relations

δ ln z(1) = δ ln ρ0(1) − δ ln ρ0(1)z(1) and δ ln ρ0(1) = −ρ0(1)

ρ(1) δ(ρ(1)ρ0(1)

)+ δρ(1)

ρ(1) are used to write


the differential as

δ lnΞ[z] = δ∫ρ(1) d(1)−

∫ρ0(1)δ

(ρ(1)ρ0(1)

)d(1)−

∫ρ(1)δc0(1) d(1), (2.57)

where we define [17,63]

c0(1) = ln ρ0(1)z(1) . (2.58)

The second term of Equation (2.57) can be rewritten as

∫ρ0(1)δ

(ρ(1)ρ0(1)

)d(1) =

∫ρ0(1)δ

(ρA(1)ρ0(1)

)d(1)+

∫ρ0(1)δ

(ρB(1)ρ0(1)

)d(1)+

∫ρ0(1)δ

(ρAB(1)ρ0(1)

)d(1)

(2.59)

using Equation (2.52). For a particular site a ∈ Γ , the fraction ρaρ0

defines the function ca, which

can be derived graphically by decomposing the graph sum for ρa into the product of ca with ρ0

ρA(1)ρ0(1) = cA(1) = 1 +

1

+1

+1

+1

+1

+1

+. . . (2.60)

ρB(1)ρ0(1) = cB(1) = 1 +

1+

1+

1+

1+

1

+1

+. . . (2.61)

In general, for any number of sites in Γ , cα is defined as [17]

ργ(1)ρ0(1) =

∑α1,...αM∈P(γ)

M∏i=1

cαi(1), (2.62)

where γ ⊆ Γ and ∅ 6= α ⊆ γ. Here the sum is over all possible partitions of the subset γ and

the product is over the M elements in a partition. Equations (2.60) and (2.61) are particular

cases of Equation (2.62) and the last particular case for a two-site model is given as

ρAB(1)ρ0(1) = cA(1)cB(1) + cAB(1)

=1

+1

+1

+1

+1

+1

+1

+ . . .

(2.63)


At this point cα is expressed as an activity expansion with z-nodes and therefore a topological

reduction is necessary to express it in a multi-density expansion. Substituting the c-functions in

Equation (2.59) we obtain

∫ρ0(1)δ

(ρ(1)ρ0(1)

)d(1) =

∫ρ0(1)δcA(1) d(1) +

∫ρ0(1)δcB(1) d(1)

+∫ρ0(1)δ (cA(1)cB(1) + cAB(1)) d(1),

(2.64)

which is equivalent to

∫ρ0(1)δ

(ρ(1)ρ0(1)

)d(1) =

∫ρ0(1)

[δcA(1) + δcB(1)

+(ρA(1)ρ0(1) δcB(1) + ρB(1)

ρ0(1) δcA(1) + δcAB(1))]

d(1),(2.65)

and after factoring out the cα-functions, is rewritten as

∫ρ0(1)δ

(ρ(1)ρ0(1)

)d(1) =

∫(ρ0(1) + ρA(1))δcB(1) + (ρ0(1) + ρB(1))δcA(1) + ρ0(1)δcAB(1) d(1).

(2.66)

Wertheim [17] defines the collective densities σα where α is a subset of sites of Γ as

σα(1) =∑γ⊆α

ργ(1), (2.67)

where the sum is over all subsets γ of α, including the empty set ∅. The limiting cases are the

density of free molecules given as σ∅ = ρ0 and the total number density given as σΓ = ρ. Using

the definition of collective densities (Equation (2.67)) in Equation (2.66), we obtain

∫ρ0(1)δ

(ρ(1)ρ0(1)

)d(1) =

∑∅6=α⊆Γ

∫σΓ−α(1)δcα(1) d(1). (2.68)

After substituting Equations (2.58) and (2.68) in Equation (2.57), the logarithm of the grand

partition function is rewritten as

δ lnΞ[z] = δ∫ρ(1) d(1)−

∑α⊆Γ

∫σΓ−α(1)δcα(1) d(1). (2.69)


The variation on both sides of Equation (2.69) is reversed to obtain

lnΞ[ρ] = −∫ ρ(1)

(ln ρ0(1)

z(1) − 1)

+∑∅6=α⊆Γ

σΓ−α(1)cα(1) d(1) + c(0)[ρ], (2.70)

which is the density functional analogous to Equation (2.29) but for associating fluids in a

multi-density formalism. The cα-functions for ∅ 6= α ⊆ Γ are defined as

cα(1) = δc(0)[ρ]δσΓ−α(1) , (2.71)

where the subset Γ − α includes all sites in Γ apart from the sites in α. The collective densities

σα (α ⊆ Γ ) can be calculated self-consistently through the relation given in Equation (2.62).

Equations (2.23), (2.25) and (2.70) are inserted in the Helmholtz free energy (Equation (2.22))

to obtain

βA[ρ] =∫ ρ(1)

(lnΛ3ρ0(1)− 1

)+

∑∅6=α⊆Γ

σΓ−α(1)cα(1) d(1)− c(0)[ρ]. (2.72)

The last step is to find the forms of cA, cB, cAB and c(0) as functionals of multi-density to be

used in Equation (2.72). For this effect, a topological reduction is carried out where infinite

sums of z-nodes graphs are substituted with σα-nodes, with α ⊆ Γ . In this work, a σα-node

is represented by regardless of α, where the value of α will be clear from the context. A

σα-node has the sites in the set Γ − α bonded. Using the definition of collective density in

Equation (2.67), and the expressions for ρ0, ρA, ρB and ρAB (Equations (2.53)–(2.56)), it is

possible to define the σα-nodes ( ) as

=

∫ρ0(1) d(1), if all sites in the node are bonded

,∫σA(1) d(1), if the node’s site A is free (at least)

( ),

∫σB(1) d(1), if the node’s site B is free (at least)

( ),

∫ρ(1) d(1), if all sites in the node are free

( ).

(2.73)


After the topological reduction the c-functions Equations (2.60), (2.61) and (2.63) are rewritten

as

cA(1) = 1 +1

+1

+1

+1

+1

+1

+ . . . , (2.74)

cB(1) = 1 +1

+1

+1

+1

+1

+1

+ . . . , (2.75)

cAB(1) =1

+1

+1

+1

+1

+1

+11

+ . . . , (2.76)

where the nodes are of the different types included in Equation (2.73). A graph containing

fab-bond(s) represents a dimer if it includes one fab-bond, a trimer if it includes two adjacent

fab-bonds, ... or an s-mer if it includes (s− 1) adjacent fab-bonds. It is very important to note

that even the simplest dimer in the cA expression (Equation (2.75)):

1 = 1 +1

+1

+1

+1

+1

+1

+ . . . , (2.77)

includes all possible lengths of association chain aggregates and no closed loops. This is important

to realise because the z-graphs are the graphs that describe the kind of association clusters that

are accounted for. Indeed, a dimer σ-graph accounts for s-mers of molecules for any arbitrary

integer s.

The expression for the functional c(0)[ρ] in the target fluid is recovered from Equation (2.71)

and is given as

c(0)[ρ] = 12 + +1

6 + + +13 +1

8 +. . . (2.78)

The change in residual Helmholtz free energy due to hydrogen bonding as a function of the


distribution of multi-density (bonding states) is thus written as

β (A[ρ]−AR[ρ]) = βAWert[ρ],

=∫ (

ρ(1) ln ρ0(1)ρ(1) + ρ(1) +Q(1)

)d(1)−

(c(0)[ρ]− c(0)

R [ρ]),

(2.79)

withQ(1) =− ρ(1) +

∑∅6=α⊆Γ

σΓ−α(1)cα(1),

=−∑a∈Γ

σΓ−a(1) + ρ0(1)∑

γ1,...γM∈P(Γ ),M≥2

(−1)M(M − 2)!M∏i=1

σγi(1)ρ0(1) .

(2.80)

Here, Q(1) is a function of the number of sites and its last expression was obtained after

expressing the cα in terms of σα in a non-trivial way from Equations (2.62) and (2.67) [17,63].

The first sum is over all sites in Γ and the second sum is over all possible partitions of the set

Γ (P(Γ )), i.e. ways to divide up Γ into pairwise disjoint subsets. The elements of the partition

of Γ into M subsets are γ1, γ2, . . . , γM and the condition M ≥ 2 ensures that each partition

considered must contain two or more elements.

The graphs that are devoid of association bonds in c(0)[ρ] compose c(0)ref [ρ]. Hence, upon

subtraction of the reference term c(0)ref [ρ] from c(0)[ρ] obtained in Equation (2.78), the

diagrams with interactions of the dispersion type only (fref(12)) vanish and

c(0)[ρ]− c(0)ref [ρ] = ∆c(0)[ρ]

= + + + 13 + + 1

2 + . . .

(2.81)

Up until now, the expressions are exact given the initial approximations in Equation (2.49).

However, in order to use ∆c(0)[ρ] analytically, approximations must be made to truncate the

infinite sum into a finite number of contributions. The initial approximations in the TPT are

related to steric incompatibilities (SI) and were considered from the start of our derivation,

upon the application of Equation (2.37) in the definition of f(12), Equation (2.48).

When a bond between two sites located in different molecules is formed, the repulsive cores


of the two molecules prevent a third molecule from coming inside the bonding range without

overlap of cores, as that would imply a very high energy cost. This is a geometry constraint that

was introduced early on in the derivation of the theory which simplified the problem by graph

cancellation. This steric constraint had already been used by Andersen [57,67]. This effect is

illustrated in Figure 2.2.1a and is named SI1 by Wertheim [17]. Illustrated in Figure 2.2.1b

is a related constraint stating that the angle between two sites in a molecule is not small

enough to allow the sites to bond to the same site in a different molecule. This steric effect

is named SI2W [17], where W stands for weak, and together with the SI1 constitutes the

condition of single bonding at each site, excluding all graphs with more than one hydrogen

bond per site and more than two sites per hydrogen bond. Multiple bonding between molecules

is also not allowed in the original TPT framework which is a reasonable constraint when we

consider hard cores with two sites of very short association range and a large enough angle

between them. This constraint is the strong form of the steric incompatibility SI2, named

SI2S [17], and it is illustrated in Figure 2.2.1c. This constitutes the condition of single bonding

between each pair of molecules. These are the only approximations taken in the theory at

(a) (b) (c) (d)

Figure 2.2.1: Illustrations of some of the constraints of TPT1, [17]. (a) according to Wertheim’sSI1, two hard cores of associated molecules prevent a third hard core to come into the associationrange of the bonded sites. (b) according to Wertheim’s SI2W, an associative bond involvesexactly two sites, and therefore a site is unable connect to two at the same time. (c) accordingto Wertheim’s SI2S, double-bonding is not permitted, only single-bonding between two particlesis allowed. (d) Only contributions from tree-like clusters are included in ∆c(0) and closed loopsare neglected.

this point. Further considerations include the absence of steric hindrance between association

sites, which results in association being independent of bond angle and in the lack of bond

cooperativity. Furthermore in the single-chain approximation [46,63], Wertheim neglects the


interaction between the association clusters, meaning that the molecules interact with each

other at the molecular level only with interactions existent in the reference fluid (fref(12)). This

approximation implies the exclusion of diagrams including more than one (s≥ 2)-mer such as

which are entropically unfavourable. Although z-graphs with inter-s-mer interactions,

e.g., , are still included in the σα-nodes, Wertheim is unsure of the extent to which the

inter-s-mer interactions are accounted for [46], as the terms that vanished with the single chain

approximation may be very important. Equation (2.81) is now rewritten as

∆c(0)[ρ] = + + + 13 + + + + . . . (2.82)

It is not feasible to include all lengths of the s-mers in our ∆c(0) expression and therefore it

must be truncated. Moreover, each s-mer requires the knowledge of the s-body correlation

function g(s)ref (1...s). At the first order of approximation of TPT, i.e. TPT1, only two-body

interactions are considered and therefore all contributions to the irreducible graph sum(∆c(0)

)with more than a single association bond such as are neglected. Despite this, chain

aggregates of any length are still being accounted for in dimer graphs indirectly such as

according to Equation (2.77) and the fact that the diagrams in the pair correlation function

which are most important for calculating the extent of hydrogen bonding are those in which

the roots are directly connected by a hydrogen bond [57]. Indeed, TPT1 accounts for linear

(molecular model of two or more sites) and branched (molecular model of three or more sites)

chain clusters for which only two-body correlation functions are needed. Hence, the diagrams

containing interactions between associated clusters beyond that of the reference fluid, including

closed loops of association bonds, as depicted in Figure 2.2.1d, vanish and the fundamental

graph sum is rewritten as

∆c(0)[ρ] = + + + + + 12 + + . . . (2.83)


This is the final form of the fundamental graph sum, which can be written in integral form as

∆c(0)[ρ] =∫σA(2)σB(2)fAB(12)gref(12) d(2), (2.84)

where the pair distribution function, g(12) = ρ(2)(12)/ρ(1)ρ(2), is given as [63]

g(2)(12) = 1 2 +1

2+

1

2+2

1

2+ 1

21

2+2

1

2+ 1

21

2+. . . , (2.85)

whose value is typically taken at contact (gref(σ)) since the associative interactions are short

ranged. According to Equation (2.71), the cα for a two-site model within a standard TPT1

framework are thus written as

cA(1) =∫σA(2)fAB(12)gref(12) d(2), (2.86)

cB(1) =∫σB(2)fAB(12)gref(12) d(2), (2.87)

and

cAB(1) = 0. (2.88)

At last, the residual Helmholtz free energy in Equation (2.79) for two-site particles is rewritten

as

βAWert[ρ] =∫ [

ρ(1) ln ρ0(1)ρ(1) + ρ(1) +

(−σA(1)− σB(1) + σA(1)σB(1)

ρ0(1)

)]d(1)

− 12

∫(σA(1)cB(1) + σB(1)cA(1)) d(1),

(2.89)

where the σα are calculated self-consistently through Equations (2.62), (2.67) and (2.71), which

result in the “law of mass action” equations given as

σA(1)ρ0(1) = 1 + cA(1) = 1 +

∫σA(2)fAB(12)gref(12) d(2), (2.90)

σB(1)ρ0(1) = 1 + cB(1) = 1 +

∫σB(2)fAB(12)gref(12) d(2), (2.91)

2.3. The statistical associating fluid theory (SAFT) 29

andρ(1)ρ0(1) = 1 + cA(1) + cB(1) + cA(1)cB(1) + cAB(1) = σA(1)σB(1)

(ρ0(1))2 . (2.92)

From thermodynamics we know that at equilibrium we expect the minimum of AWert and

therefore an equivalent procedure to obtain the law of mass action equations would be minimising

AWert with respect to all σα, for α ⊂ Γ, α 6= Γ .

2.3 The statistical associating fluid theory (SAFT)

The statistical associating fluid theory (SAFT) [50,68] used to refer to a single EOS inspired

by Wertheim’s TPT1 that accounted for the ideal, the hard-sphere interactions, non-sphericity

and hydrogen bonding contributions in associating fluids [47,48,49,69]; today, however, it refers

to a family of EOSs of variations of the original formulation that were developed to tackle

fluids of increasing complexity. The development of the original SAFT [47, 48, 49, 69] was

motivated by the need for a rigorous algebraic framework to model properties of associating

fluids and indeed, since its conception in the late 1980s, the original SAFT formulation has

undergone extensions and has become more and more popular due to its success in application

to complex fluid systems. To name but a few applications, the use of SAFT has been proven

valuable in the characterization of the behaviour of small molecules such as xenon [70], carbon

dioxide [71] and water [72], of more complex systems of aqueous solutions of alcohols [73],

alkanes, perfluoralkanes [74, 75], organic and pharmaceutically relevant compounds [76, 77]

through to downstream industry and carbon capture and storage [78, 79], in fluid design in

organic Rankine cycle processes, [80], polymer solutions and blends [81], biofuels [82], room

temperature ionic liquids and amino acids and extensions have been made to describe dipolar,

quadrupolar fluids, [83], and electrolytes, [84].

SAFT consists of an algebraic equation to compute the Helmholtz free energy A of a fluid

system, given an intermolecular potential and a molecular model. The standard form of SAFT,

(as originally developed but no longer valid for all versions such as the case of the perturbed


chain version (PC-SAFT) [85]) consists of four contributions given as

A = Aideal + Amono + Achain + Aassoc, (2.93)

as depicted Figure 2.3.1. The model system consists of N =NC∑i

Ni molecules, where NC is the

Aideal Amonomer Achain Aassoc

a) b) c) d)

Figure 2.3.1: SAFT contributions to Helmholtz free energy.

number of components in the system, Ni = xiN is the number of molecules of species i, xi is

the fraction of species i. Each molecule of species i consists of mi segments. The ideal term

originates from the ideal monoatomic gas equation [86] and it was previously given for a pure

fluid in Equation (2.10). It is given for mixtures as

Aideal = NkT

NC∑i

xi ln ρiΛ3i

− 1. (2.94)

The monomer term [87],

Amono = NkT

NC∑i

ximi

amono

= NSkT(aHS + adisp

),

(2.95)

accounts for the repulsive and dispersive interactions between the segments that compose

the molecules (NS). This term results from the perturbation of a hard-sphere to account for

attractive dispersion between the segments, and is the sum of a hard-sphere term aHS [88] and an

attractive term adisp. The main difference between the various versions of the SAFT EOS is in

the expression for the monomer term, which depends on the intermolecular potential considered

and on the perturbation expansion adopted. For example, in the SAFT-HS approach [76,77,89],

the monomer free energy in a pure component fluid is given by the augmented van der Waals


expression as in

aHS + adisp = 4η − 3η2

(1− η)2 −αvdWρSkT

, (2.96)

where η =[NS(4π/3)(σ/2)3

]/V is the packing fraction (volume occupied by molecules divided

by total system volume), ρS is the number density of segments and αvdW is the attractive van

der Waals constant.

The association term is simply the generalised expression for TPT1 found in Section 2.2 for

mixtures and multiple sites. It is a perturbation to the monomer fluid, accounting for highly-

directional and short-ranged associative interactions in the fluid such as hydrogen bonds or

dipole-dipole interactions in highly polar but aprotic compounds such as acetone [90]. The

chain term is also a perturbation term with the monomer fluid as reference and accounts for the

non-sphericity of molecules. The chain term is the limit of complete bonding of TPT1 and in

the subsections of this section we derive the exact expressions for Achain and Aassoc.

Within a SAFT framework, the scheme of the molecular models used may fall into different

categories:

• United-atom or all-atom segments – in the united-atom approach, segments represent

groups of atoms, such as functional groups that characterize the molecules, opposed to

the all-atom model, where there is one segment per atom;

• Tangent or fused segments – in the tangent model, the molecule is formed by a number

of segments m tangentially bonded, and in the fused model, m takes a noninteger value,

since the segments are effectively fused to form the molecule;

• Homo- or heteronuclear segments – in the homonuclear framework, all the segments

of a molecule are of the same type, unlike in the heteronuclear model, where a molecule can

be composed by segments of different types, which are described by their own parameters

(size, energy, dispersive range, hydrogen bonding energy and association range).

Other effects may be accounted for with additional excess contributions to the sum in Figure 2.3.1.

Excellent reviews of different SAFT-based approaches and applications can be found in references


[50,68,91,91,92,93,94,95,96] together with comparisons of their relative performance in different

applications.

2.3.1 The association contribution to the residual free energy

In the last subsection, we focused on the two-site model (Γ = A,B) as the inclusion of more

sites would make the problem cumbersome and not aid in the understanding of the derivation of

TPT1. We will now consider the association contribution arising from the interaction between

multiple sites in a homogeneous fluid. This contribution is extremely relevant in associating

fluids and inspired the development of sophisticated EOSs for associating fluids in the late

1980s, also known as EOSs of the SAFT family, see Section 2.3. We consider a pure system as

illustration, because the extension to mixtures is straightforward [51].

At this early point, it is helpful to introduce a change of variables from number densities to

fractions of molecules with set of sites α free as it eases the analysis. The fraction of molecules

with the set of sites α free is thus given by Xα, conventionally referred to as Xα:

Xα =

σΓ−αρ

, α ⊂ Γ

X0 = ρ0

ρ, α = Γ

. (2.97)

In a fluid with Γ = A,B, the number density σA denotes the density of molecules with all

sites free or only site A bonded, equivalently, it corresponds to the density of molecules with site

B free (σΓ−B). Dividing this density by the overall number density ρ thus yields the fraction of

molecules with site B free, represented by XB.

Consider a pure associating fluid with the set of sites Γ = A,B, .... The molecules can either

be spherical or chain-like, as the molecular structure does not interfere in the derivation or

the results. The residual Helmholtz free energy in Equation (2.79) is rewritten in terms of the

variables Xα for the homogeneous system by removing the dependency of the densities on the


position and orientation (i) as in

AWert = NkT

(lnX0 + 1 + Q

ρ− ∆c(0)

N

), (2.98)

withQ

ρ= −

∑A∈Γ

XA +X0∑

γ1,...,γM∈P(Γ ),M≥2

(−1)M (M − 2)!M∏i=1

XΓ−γi

X0, (2.99)

and the generalisation of Equation (2.84) for any number of associating site pairs

∆c(0)

N= 1

2∑a∈Γ

∑b∈Γ

ρXa∆abXb. (2.100)

Here, ∆ab represents the association strength between site a and site b and is given by

∆ab =∫gref(12)fab(12) d(2). (2.101)

The property of independence between sites is an important feature of TPT1 and its derivation

is included in Appendix A. It states that the probability of an association site being bonded is

independent of the bonding state of all the other association sites. In other words, the fraction

of molecules with sites a and b simultaneously free is given by the product between the fraction

of molecules with a free and the fraction of molecules with site b free. In the general case, it is

written as

Xα =∏a∈α

Xa =∏

β1,...,βM∈P(α)

Xβi , (2.102)

for α ⊆ Γ, α 6= ∅. This definition includes the particular case

X0 =∏a∈Γ

Xa =∏

β1,...,βM∈P(Γ )

Xβi , (2.103)

which defines the fraction of monomers (all sites free) as the product between the fractions

of molecules with free sites in all subsets of any given P (Γ ) (partition of the set of sites Γ ).

The function Q can be simplified by using the relation X0 = XΓ−γiXγi which results from


Equation (2.103):

Q

ρ= −

∑a∈Γ

Xa +X0∑

γ1,...,γM∈P(Γ ),M≥2

(−1)M (M − 2)!M∏i=1

1Xγi

= −∑a∈Γ

Xa +∑

γ1,...,γM∈P(Γ ),M≥2

(−1)M (M − 2)!. (2.104)

Despite the rather complicated look, the sum over the partition elements in Equation (2.104)

has a simple result2 described by conjecture in Table 2.1 and Q = −∑a∈Γ

Xa + |Γ | − 1. After

Table 2.1: Value of the last term of Equation (2.104) according to the size of Γ .Γ A A,B A,B,C A,B,C,D . . . A,B,C, . . .∑

γ1,...,γM∈P(Γ ),M≥2

(−1)M (M − 2)! 0 1 2 3 . . . |Γ |−1

substituting the expressions for Q and ∆c(0), Equation (2.98) is rewritten as

AWert = NkT

∑a∈Γ

(lnXa +Xa + 1)− 12∑a∈Γ

∑b∈Γ

ρXa∆abXb

. (2.105)

We minimise the Helmholtz free energy in all densities to find the distribution of bonding states.

Minimising in σΓ−a = ρXa at fixed ρ, is equivalent to minimising in fractions of free sites Xa,

∂ (βAWert/N)∂Xa

∣∣∣∣∣Xb 6=a

= 0

= 1Xa

+ 1−∑b∈Γ

ρ∆abXb,

(2.106)

which provides the expression for the law of mass action equation system given by

Xa = 11 +∑

b∈Γ ρ∆abXb

. (2.107)

Lastly, a simple form of AWert only valid at chemical equilibrium is found by inserting the law

2The result of this conjecture can also be reached in an exact approach by considering the limit of noassociation, when the residual AWert = 0, see e.g., [97]


of mass action equation in the ∆c(0) term of Equation (2.105):

AWert = NkT∑a∈Γ

(lnXa + Xa + 1

2

). (2.108)

This is thus the expression for the contribution of hydrogen bonding Aassoc to the Helmholtz

free energy of the system A. The extension of the theory for mixtures is obtained when we sum

over all species and is given by Joslin et al. [51] as

Aassoc = AWert = NkTNC∑i

xi

NST,i∑a

si,a

(lnXi,a −

1−Xi,a

2

) , (2.109)

with

Xi,a =1 + ρ

NC∑j

NST,j∑b

xjXj,b∆ij,ab

−1

, (2.110)

where NC represents the number of components, NST,i represents the number of site types in

component i and si,a represents the number of sites of type a in component i.

The residual free energy due to association, Aassoc modelled by TPT1 takes into account

association clusters limited to linear or branched chains, and neglects the contribution from

the formation of any type of closed loop aggregates. Additionally, steric self-hindrance is not

accounted for in the formation of the association bonds, since the activity of each bonding site

is independent of each other, so double hydrogen bonding [98] and cooperation effects [64] are

also neglected in the theoretical framework. Despite its limitations, TPT1 has been widely

applied and resulted in extremely successful in equations of state [93], such as the cubic plus

association (CPA), [99], and the statistical associating fluid theory (SAFT) [49,69,100,101],

which is described in Section 2.3.

2.3.2 The chain contribution to the residual Helmholtz free energy

The main differences in nature between a hydrogen and a chemical bond are the strength and

the inter-segmental distance. We therefore expect that increasing the association energy and

reducing the inter-segmental distance should approximate a hydrogen bond to a covalent bond.


TPT1 can thus be used to estimate the contribution from the formation of a covalent chain

to the residual Helmholtz free energy Achain by taking the limit of complete association, an

observation made by Wertheim [102]. This limit of complete bonding is also referred to as the

“sticky limit”.

Consider a fluid system of spherical molecules interacting isotropically with each other through

a reference potential. The spheres have association sites A and B that promote the formation of

linear open chain aggregates only. For finite bonding strengths, a distribution of chain lengths

with average length m is expected in a system in equilibrium, where m is the number of monomer

units in a chain. Both this distribution and the average chain length vary with the bonding

strength. Wertheim [102] observed very good agreement between theory and simulation, but

discrepancies become larger as chain length increases, showing significant deviations at m ≥ 16.

Chapman et al. [47, 66] derived an expression based on TPT1’s limit of complete bonding to

account for the chain contribution to the total Helmholtz free energy. We here derive the chain

contribution for chains of a single length m in a pure system, since the extension to multiple

chain lengths and components is straightforward.

We consider a fluid consisting of an m-component mixture of spherical segments, with N

segments of each component, as in Figure 2.3.2.

1 1

3m

m...2...2 m...1

2

Figure 2.3.2: Each chain is formed by m monomers that associated selectively; sphere (1) can

bond to sphere (2) only, sphere (2) can bond to sphere (1) and sphere (3) only, ... and sphere

(m) can bond to sphere (m− 1) only. Upon increase of the association energy to infinity and

decrease of the association range to an infinitely small value, the spheres become irreversibly

bonded. In other words, they form a covalent chain.

These individual segments interact through a reference potential and have embedded two


attraction sites, A and B, represented as blue and orange spheres, respectively. The monomers

will bond between themselves to form N chains with component 1 in position (1), component 2

in position (2) and similarly up to component m in position (m). In order to form the chains we

restrict the bonding interactions permitted, and accordingly, sphere (1) can only bond to sphere

(2), sphere (2) to (1) and (3),... and sphere (m) to sphere (m− 1). In practice, we control the

bonding with the specification of εHBAB > 0 only for the pairs of sites A in sphere in position (i)

and B in sphere in position (i+ 1) of adjacent segments, for i = [1,m− 1]. It is thus implied

that sites A in component 1 and B in component m are always free, which is equivalent to

remove them as their existence does not affect the properties of the fluid.

The contribution of a fluid with this composition in terms of excess free energy, regardless of the

bonding configurations allowed is given by Equation (2.79), as seen in Section 2.2. Our starting

point is thus given by

βAWert[ρ] =m∑α=1

∫ ρ(α)(1) ln ρ(α)0 (1)ρ(α)(1) + ρ(α)(1) +Q(1)

d(1)−∆c(0)[ρ], (2.111)

with

Q(1) = −σ(α)Γ−A(1)− σ(α)

Γ−B(1) + σ(α)A (1)σ(α)

B (1)ρ

(α)0 (1)

, (2.112)

for two sites, where the superscript (α) represents the species. Considering the formation of

linear chains only, the fundamental graph sum becomes

∆c(0)[ρ] =m−1∑α=1

∫σ

(α)Γ−A(1)∆(α,α+1)

AB (12)σ(α+1)Γ−B (2) d(1) d(2), (2.113)

where ∆AB(ij) = gref(ij)fAB(ij). We obtain the law of mass action equations by minimising

the free energy with respect to the different bonding states γ = A, B, ∅ of all α = [1,m]

species, so that δAWert/δσ(α)γ = 0. This is equivalent to minimisation with respect to the

fractions of segments with free sites Xγ , with γ = B, A, A,B respectively, and results


in the following equations:

δβAWert

δσ(α)B

= 0⇒

X

(α)B (1)

X(α)0 (1)

− 1 =∫ρ(α+1)(2)X(α+1)

B (2)∆(α,α+1)AB (12) d(2) , α = [1,m− 1]

X(m)B (1) = X

(m)0 (1) , α = m

,

(2.114)

δβAWert

δσ(α)A

= 0⇒

X

(1)A (1) = X

(1)0 (1) , α = 1

X(α)A (1)

X(α)0 (1)

− 1 =∫ρ(α−1)(2)X(α−1)

A (2)∆(α−1,α)AB (12) d(2) , α = [2,m]

,

(2.115)

andδβAWert

δρ(α)0

= 0⇒ X(α)0 (1) = X

(α)A (1)X(α)

B (1), α = [1,m] . (2.116)

Since there is exactly one segment of each species per chain, all species share the same num-

ber density ρ, and therefore the superscript (α) in ρ can be dropped. Inserting relations

Equation (2.114) and Equation (2.115) in Equation (2.113) results in

∆c(0)[ρ] = 12

m−1∑α=1

∫ρ(1)

X(α)A (1)

X(α)B (1)

X(α)0 (1)

− 1+X

(α+1)B (1)

X(α+1)A (1)

X(α+1)0 (1)

− 1 d(1),

(2.117)

which after some algebra allows us to rewrite Equation (2.111) in a simple form given by

βAWert[ρ] =m∑α=1

∫ρ(1)

(lnX(α)

0 (1) + 1− 12X

(α)A (1)− 1

2X(α)B (1)

)d(1), (2.118)

for the inhomogeneous system and by

AWert[ρ] = NkTm∑α=1

(lnX(α)

0 + 1− 12X

(α)A −

12X

(α)B

), (2.119)

in the homogeneous case. These are the inhomogeneous and homogeneous results, respectively,

for the association contribution of a mixture fluid to the free energy as obtained in the previous

section, where N is the number of segments of each species.

In the “sticky limit”, the energy εHBAB is taken to infinity and the range of association rcAB goes

to zero as the spheres come into contact. This means that for α = [1,m− 1], the fractions of


free sites X(α)A and X(α+1)

B tend to zero, X(α)0 tends to zero even faster for α = [1,m] because

of Equation (2.116) and the integral ∆(α,α+1)AB tends to infinity. For the sake of mathematical

consistency, we consider ∆(α,α+1)AB to be a very large but finite number and the fractions of

molecules with free sites to be very small, negligible, but not zero. Considering the homogeneous

system, the law of mass action equations can now be rewritten as

δβAWert

δσ(α)B

= 0⇒

X

(α)B

X(α)0

= ρX(α+1)B ∆

(α,α+1)AB , α = [1,m− 1]

X(m)B = X

(m)0 , α = m

, (2.120)

δβAWert

δσ(α)A

= 0⇒

X

(1)A = X

(1)0 , α = 1

X(α)A

X(α)0

= ρX(α−1)A ∆

(α−1,α)AB , α = [2,m]

, (2.121)

andδβAWert

δρ(α)0

= 0⇒ X(α)0 = X

(α)A X

(α)B , α = [1,m] . (2.122)

The strength of association ∆(α,γ)AB is approximated to a product between three contributions, the

bonding volume K(α,γ)AB , F (α,γ)

AB =[exp

(βε

HB, (α,γ)AB

)− 1

]and the pair distribution function for

the reference fluid at contact g(α,γ)ref (σ) [87]. Since all the species are equivalent, the superscript

(α, γ) on ∆(α,γ)AB can be dropped. Using Equations (2.120) and (2.121) to express X(α)

0 results in

m∑α=1

lnX(α)0 = ln

m∏α=1

X(α)0

= 12

lnm−1∏α=1

X(α)B

ρX(α+1)B ∆AB

X(m)B + lnX(1)

A

m∏α=2

X(α)A

ρX(α−1)A ∆AB

= 1

2(lnX(1)

B X(m)A

)− (m− 1) ln(ρ∆AB).

(2.123)

In the resulting chain there is no association (see initial figure), and therefore, the fractions of

molecules with free sites are either negligible (formally, 0), for the covalent bond points, or 1, for

the two sites in the extremities, X(m)A and X(1)

B . Setting X(m)A = X

(1)B = 1 and X(α)

A = X(α+1)B = 0


for α ∈ 1, 2, . . . ,m− 1, we can rewrite Equation (2.119) as

AWert[ρ] = NkT [m− 1− (m− 1) ln(ρ∆AB)] . (2.124)

The number density of the chain molecules ρ is here given by N

Vand the segment number

density by Nm

V. Lastly, the residual part of the free energy contribution due to chain formation

Achain for chains of size m is found by subtracting the ideal part for which gref (σ) = 1. The only

term in ∆AB dependent on density is the distribution function, and thus we can write

Achain[ρ] = NkT [m− 1− (m− 1) ln(ρ∆AB)−m+ 1 + (m− 1) ln(ρKABFAB)] (2.125)

since. The final expression for the residual chain contribution to the Helmholtz free energy is

thus given as

Achain = −NkT (m− 1) ln gref (σ) . (2.126)

This expression can be reached by an alternative approach by Zhou and Stell [103] that involves

the relationship between the Helmholtz free energy and the two-body cavity function [104].

2.4 Literature review of the extension of TPT1 to ac-

count for ring formation

The approximations carried out in the development of the TPT1, explored at the end of

Section 2.2, imply certain limitations of the theory. Firstly, only linear and branched chain

aggregates are considered in the theory, while loop formation in the aggregates (ring-like

aggregates) are neglected. This restriction excludes intramolecular hydrogen bonding [18,105,

106,107] relevant in systems involving polymers or proteins for instance, and intermolecular ring

formation by association such as that observed experimentally in HF systems [108]. Secondly,

cooperative effects are not accounted for, which implies that all (m − 1) bonds in a chain

aggregate of m molecules have the same likelihood of forming, whereas in reality the formation of

2.4. Literature review of the extension of TPT1 to account for ring formation 41

a bond may strengthen/weaken the bonds in aggregates of three or more molecules [64,109,110].

Thirdly, the actual geometry of the molecule and association sites is not captured by the

molecular model [111], as molecules are considered to be fully flexible and the sites to be

independent of each other. Lastly, double bonding is not accounted for which prevents TPT1

from accounting for doubly bonded dimers [112,113], which are quite commonly observed in

systems with carboxylic acids [114,115].

Despite its success in the description of a range of complex associating fluids, there have been

many efforts to relax the constraints of TPT1. In this work we strive to lift the constraint that

prevents ring formation. In this section we start by presenting the two different approaches to

account for ring formation developed by Sear and Jackson [105] and Ghonasgi et al. [107]. In

the end, we mention only briefly other relevant works since these were based on the former two

approaches. Both Sear and Jackson [105] and Ghonasgi et al. [107] extended the theory to flexible

hard chains with two association sites in the terminal segments that associate intramolecularly.

While Ghonasgi et al. followed a mass balance approach between bonded and non-bonded

state, Sear and Jackson incorporated the ring graphs in the fundamental graph sum ∆c(0)

(Equation (2.83)).

2.4.1 Sear and Jackson (1994)

Sear and Jackson developed a theoretical framework to account for ring formation in a fluid

system. They started by extending the theory for bonded hard spheres of Amos and Jackson [116]

to flexible rings of hard spheres [117]. Next, focusing on a pure system of a two-site model, they

extended Wertheim’s TPT1 to account for the intermolecular hydrogen bonding in a fluid of

associating hard spheres into linear aggregates and intermolecular rings of one size [105]. In

a separate approach, they have also derived an expression for the change in Helmholtz free

energy due to the formation of a covalent chain that is capable of associating into linear chain

aggregates and into intramolecular rings [105]. In this section, we focus on the theories for the

competition between ring and linear chain association. Since the theories were derived for an

inhomogeneous system, the variables have a dependence on position.


Association into linear chains and intermolecular rings

The system considered consists of associating hard spheres with two sites labelled A and B for

distinction, as represented in Figure 2.4.1. θB1 is the angle between the vector from the centre

θB1θA2

r12

ξ

Figure 2.4.1: Distance and relevant angles between two associative segments.

of segment (1) to site B, θA2 is the angle between the vector from the centre of segment (2) to

site A and the vector connecting the two molecular centres. The angle between the vectors

connecting the centre of the sphere to each site is ξ, and consistently with a Wertheim formalism

devoid of structural constraints, is allowed to vary freely. The association scheme obeys the

steric constraints of TPT1 presented in Figure 2.2.1, with the constraint in Figure 2.2.1d

relaxed, as intermolecular rings are allowed to be formed while competing with the formation of

open linear chains, as depicted in Figure 2.4.2.

Linearchain

Intermolecularring

Figure 2.4.2: intermolecular association into linear aggregates and rings of size τ = 4.

The pair interaction potential between two molecules φ,

φ(12) = φHS(12) + φHBAB(12), (2.127)

as in Equation (2.20), is given by the sum of a reference contribution, which in this case is the

hard sphere potential φHS,

φHS(r12) =

∞, r12 < σ

0, r12 ≥ σ

, (2.128)


and an associative contribution, defined as usual by an asymmetric short-range square well

potential φHBAB between the sites A and B to mimic the hydrogen bond, given by

φHBAB(12) =

− εHB

AB, r12 < rc; θB1 < θc; θA2 < θc

0, otherwise, (2.129)

where r12 is the distance between segments in oriented positions (1) and (2), rc is the cut-off

distance, the maximum separation between two segments that still permits association, and θc

is the cut-off angle, beyond which association is no longer allowed.

The only association allowed is A−B, i.e. εHBAA = εHB

BB = 0. These sites can bond and contribute

for the formation of linear chains or rings of a specified size τ , in a reversible way. In a system

with two sites there are four possible bonding states to which a number density is associated.

The density of molecules with all sites free, i.e. monomers is given by ρ0, with only site A bonded

is given by ρA, with only site B bonded is given by ρB and with sites A and B simultaneously

bonded is given by ρAB. The composite densities defined previously in Equation (2.67) are thus

given by σA = ρ0 + ρA, σB = ρ0 + ρB and the total number density ρ at position (1) is given by

ρ(1) = ρ0(1) + ρA(1) + ρB(1) + ρAB(1). (2.130)

The distribution of bonding states is obtained from minimization of the free Helmholtz energy

derived in the Section 2.2 obtained in Equations (2.79) and (2.80):

AWert

kT[ρ] =

∫ (ρ(1) ln ρ0(1)

ρ(1) + ρ(1)− σA(1)− σB(1) + σA(1)σB(1)ρ0(1)

)d(1)−∆c(0). (2.131)

This expression is general for any order of the TPT and any type of aggregates considered,

since only the fundamental graph sum, ∆c(0), is aggregate specific. The linear chains are already

accounted for in the standard TPT1 formulation, and in order to extend TPT1 to account for

the formation of intermolecular rings, Sear and Jackson proposed the addition of a τ -mer ring


graph to the fundamental graph sum in Equation (2.82). The fundamental graph sum

∆c(0) = ∆c(0)linear + ∆c(0)

ring, (2.132)

is thus composed of two contributions. The ∆c(0)linear accounts for association linear aggregates

and is given by

∆c(0)linear =

∫σA(1)∆AB(12)σB(2) d(1) d(2), (2.133)

where the quantity ∆AB(ij) is a product of the f -Mayer function with the hard-sphere pair

distribution function given as

∆AB(ij) = gHS(ij)fAB(ij), (2.134)

and the ∆c(0)ring accounts for associated intermolecular rings of size τ and is given by

∆c(0)ring = 1

τ

∫gHS(123...τ)fAB(12)...fAB(τ − 1, τ)fAB(τ1)

τ∏i=1

ρ0(i) d(i). (2.135)

Ring graphs are irreducible and this expression is for size τ only, which means that each size

requires such contribution to be included in the term ∆c(0)ring. Since the molecules composing a

ring are indistinguishable from each other, the symmetry number of a ring of size τ is equal to

τ . The factor 1/τ is thus included in ∆c(0)ring to avoid double counting. The τ -body distribution

function (gHS(123...τ )) is typically approximated to a product of two-body distribution functions

and given as

∆c(0)ring = 1

τ

∫∆AB(12)∆AB(23)...∆AB(τ − 1, τ)∆AB(τ1)

τ∏i=1

ρ0(i) d(i), (2.136)

in the case of Sear and Jackson [105]. Upon minimization of the free energy with respect to

each bonding state, δ [AWert/ (kT )]δρ0

= 0, δ [AWert/ (kT )]δσA

= 0 and δ [AWert/ (kT )]δσB

= 0 the law of

mass action equations are obtained:

ρ(1)ρ0(1) −

σA(1)σB(1)(ρ0(1))2 =

∫ 1ρ0(1) d(1)

τ∏j=1

fAB(j, j + 1)gHS(j, j + 1)ρ0(j) d(j) (2.137)


σB(1)ρ0(1) − 1 =

∫fAB(1, 2)gHS(1, 2)σB(2) d(2) (2.138)

σA(1)ρ0(1) − 1 =

∫σA(2)fAB(1, 2)gHS(1, 2) d(2) (2.139)

The insertion of these relations back in the equation for the Helmholtz free energy and application

of some algebra leads to the expression valid at equilibrium distribution of bonding states,

AWert

kT=∫ [

ρ(1) ln(ρ0(1)ρ(1)

)+ ρ(1)

(1− 1

τ

)− 1

2 (σA(1) + σB(1)) + 1τ

σA(1)σB(1)ρ0(1)

]d(1).

(2.140)

The equivalent for the homogeneous system is given by

AWert

NkT= ln ρ0

ρ+ 1− 1

τ− 1ρσA + σ2

A

τρρ0(2.141)

where we used the result from symmetry σA = σB. The mass action equations in the homogeneous

regime are given byρ

ρ0− σAσB

ρ20

= (∆AB)τ Wτ−1ρτ−10 , (2.142)

andσAρ0− 1,= σA∆AB (2.143)

with ∆AB given by

∆AB = KABFABgHS(σ). (2.144)

Since the sites interact with each other through a square well potential, FAB = exp (εAB,inter/kT )−

1) is constant throughout KAB, the bonding volume. WhereWτ−1(r12) is the end-to-end function

for a freely jointed chain given by Treloar [118,119], and it represents the probability per unit

of volume that the two ends of a chain with τ − 1 links have a separation r12. The prod-

uct Wτ−1(σ)KABgHS(σ) is the fraction of molecules in the intramolecular bonding orientation

according to the approximation. The function Wτ−1(r12) is given by [119]

Wτ−1(r12) = (τ − 1)(τ − 2)8πr12σ2

k∑j=0

(−1)jj!(τ − 1− j)!

[τ − 1− (r12/σ)− 2j

2

]τ−3

, (2.145)


with k being the integer satisfying

τ − 1− (r12/σ)2 − 1 ≤ k <

τ − 1− (r12/σ)2 . (2.146)

Equation (2.141) is thus the residual contribution of intermolecular association into linear

association chains and rings of size τ to the Helmholtz free energy.

Association into linear chains and intramolecular rings

This theory accounts for the formation of chain molecules from hard spheres and for the

association of said chain molecules into linear association aggregates and intramolecular rings,

see Figure 2.4.3. Since we are accounting for the formation of a chain molecule of size m, the

Step 1:Chain

formationStep 2:Intermolecularassociation

Step 2:Intramolecularassociation

4

2

1

2

3

32

3

1

2

3

4

4321

4 3 21 3 2

144

3214321

1

4

1

4

Figure 2.4.3: Step 1: Formation of chain molecules of length m = 4 from hard sphere segments.Step 2: inter- and intramolecular association.

derivation includes the association between spherical segments taken to the limit of complete

bonding, see Section 2.3.2 for the sticky limit approach. The system considered thus consists of

hard spheres with two sites, just as before, see Figure 2.4.1 but with the difference that these

spheres are now of m different species. The number of molecules of each species is the same for

all species. Each species corresponds to a unique segment position in the chain molecule, e.g., a

chain of m = 4 spheres is formed from the covalent bonding between 4 species of spheres.


The pair interaction potential between two molecules of species β and γ, φ(β,γ),

φ(β,γ)(12) = φ(β,γ)HS (12) + φ

HB, (β,γ)AB (12), (2.147)

as before, is given by the sum of the hard sphere potential φ(β,γ)HS ,

φ(β,γ)HS (r12) =

∞, r12 < σ

0, r12 ≥ σ

, (2.148)


potential between the sites A and B, φHB, (β,γ)AB , to mimic the hydrogen bond

φHB, (β,γ)AB (12) =

− εHB, (β,γ)

AB , r12 < rc; θB1 < θc; θA2 < θc


Despite belonging to m different species, all spheres are identical, but site A of species (i) can

only bond to site B of species (i+ 1), where (β+ 1) = (1) for β = m and (β−1) = (m) for β = 1

by convention. For (i) ranging from (1) to (m− 1) this association bond will be taken to the

limit of complete bonding in order to form a chain with a free site A in the segment occupying

position (1) and a free site B in the segment positioned in sphere (m). The hydrogen bond

between species (1) and (m) can bridge two different molecules (intermolecular association) or

the end segments of the same chain molecule (intramolecular association) in a reversible way.

The multiple densities and association strengths are specific to the species involved, and therefore

it is now necessary to have the superscript (β) or (β, γ) indicating the species. The expression

for the free Helmholtz energy of Wertheim derived in Section 2.2 for mixtures

AWert

kT[ρ] =

m∑β=1

∫ ρ(β)(1) ln ρ(β)0 (1)ρ(β)(1) + ρ(β)(1)− σ(β)

A (1)− σ(β)B (1)

+σ(β)A (1)σ(β)

B (1)ρ

(β)0 (1)

d(1)−∆c(0)

(2.150)

is the starting point. The differences from Equation (2.131) are the sum over all m species and


the definition of the ∆c(0):

∆c(0) = ∆c(0)chain + ∆c(0)

ring + ∆c(0)inter, (2.151)

which includes the contributions from the chain formation ∆c(0)chain, the intramolecular ring

formation ∆c(0)ring and the intermolecular association between chains ∆c(0)

inter. The ring graph for

an intramolecular ring formed by a hydrogen bond between the terminal segments of a chain of

size m,

∆c(0)ring =

∫∆

(12)AB (12)∆(23)

AB (23)...∆(m−1,m)AB (m− 1,m)∆(m,1)

AB (m1)m∏i=1

ρi0(i) d(i), (2.152)

is very similar to the ring seen before for an intermolecular ring of size τ seen in Equation (2.136),

however we do not divide by the number of segments in the ring, because the symmetry number

is 1 in the case of an intramolecular ring. This is due to each node being distinguishable from

the others by their unique position/species in a covalent chain. The quantity ∆β,γAB(ij)

∆(β,γ)AB (ij) = g

(β,γ)HS (ij)f (β,γ)

AB (ij), (2.153)

corresponds to the interaction between site A of segment species β in position (i) and site B of

segment species γ in position (j). The chain graph is given by

∆c(0)chain =

m−1∑β=1

∫σ

(β)A (1)∆(β,β+1)

AB (12)σ(β+1)B (2) d(1) d(2) (2.154)

where we sum over all pairs of adjacent positions in the chain and the intermolecular association

that leads to linear aggregates of any size is given by

∆c(0)inter =

∫σ

(m)A (1)∆(m,1)

AB (12)σ(1)B (2) d(1) d(2). (2.155)

Minimizing the free energy expression in Equation (2.150) with respect to all densities results in


the mass action equations given below:

ρ(β)(1)ρ

(β)0 (1)

− σ(β)A (1)σ(β)

B (1)(ρ

(β)0 (1)

)2 =∫ 1ρ

(β)0 (β) d(β)

m∏i=1

∆(i,i+1)AB (i, i+ 1)ρ0(i) d(i), β = [1,m] , (2.156)

σ(β)A (1)ρ

(β)0 (1)

=

∫σ

(β+1)A (2)∆(β,β+1)

AB (12) d(2) , β = [1,m− 1]

1 +∫σ

(β+1)A (2)∆(β,β+1)

AB (12) d(2) , β = m

, (2.157)

and

σ(β)B (1)ρ

(β)0 (1)

=

∫σ

(β−1)B (2)∆(β−1,β)

AB (12) d(2) , β = [2,m]

1 +∫σ

(β−1)B (2)∆(β−1,β)

AB (12) d(2) , β = 1, (2.158)

where the approximation 1+∫∆

(β,β+1)AB (12) d(2) ≈

∫∆

(β,β+1)AB (12) d(2) was taken for all covalent

links, i.e. for β = [1,m− 1]. The species superscript in total density ρ(α)(1) can be omitted

because there is the same number of molecules per species. Hence, the number density of chain

molecules is equal to the number density of each segment:

ρ(1) = ρ(1)(1) = ρ(2)(1) = · · · = ρ(m)(1). (2.159)

Inserting Equations (2.156)–(2.158) in Equation (2.150), we obtain after some algebra the

expression for the Helmholtz free energy at chemical equilibrium for an inhomogeneous fluid:

AWert

kT=∫ ρ(1) ln

σ(1)A (1)

∆m−1 (ρ(1))m(1 + σ

(1)A (1)∆AB

)+mρ(1)− σ(1)

A (1)

−∆ABWmσ

(1)A (1)

1 + σ(1)A (1)∆AB

d(1), (2.160)

where we used the relationship σ(1)A = σ

(m)B from symmetry. The strength of association ∆AB is

given by Equation (2.144) (∆AB = KABFABgHS(σ)) and ∆ is the covalent limit of ∆AB. In a

homogeneous fluid, the densities are no longer function of the coordinates and Equation (2.160)


can be written as

AWert

NkT= ln

(σA

∆m−1ρm (1 + σA∆AB)

)+m− 1

ρ

(σA +∆ABWm−1

σA1 + σA∆AB

), (2.161)

with the respective mass action equation given by

ρ− σA(1 + σA∆AB) = ∆ABWm−1

(σA

1 + σA∆AB

). (2.162)

Equation (2.161) accounts for the contribution of chain formation, inter- and intramolecular

association to the Helmholtz free energy. Note that even though AWert = A− Aref , it is not a

residual since limV→∞

6= 0.

2.4.2 Ghonasgi, Perez, and Chapman (1994)

Ghonasgi and co-workers developed a theoretical framework to account for intramolecular associ-

ation in a pure system of a two-site model. They started by looking at intramolecular association

in the absence of intermolecular association, carried out Monte Carlo simulations, derived an

expression for the change in Helmholtz free energy due to the formation of an intramolecular

hydrogen bond and also looked at its limit at complete bonding to find the contribution from

cyclic molecules [107]. Later, they focused on the competition between intramolecular rings and

linear chain aggregates [18]. In this section, we focus on the intermolecular association in the

formation of linear association aggregates competing with intramolecular association. Since

the theory was derived for a homogeneous system, the variables do not have a dependence on

position.

The molecular model considered consisted of flexible hard chains with one association site in each

end of the chain, as shown in Figure 2.4.4. The associating sites were oriented on the terminal

segments perpendicularly to the bond that the terminal segments form with the neighbouring

segments. After labelling the sites A and B, only association A − B was allowed, inter- or

intramolecularly. All relevant distances and angles are represented in Figure 2.4.4. The distance

rij is that between segments i and j of the same (rintraij ) or different molecules (rinter

ij ), σ is the


B

Arintra1m

B

θintraB1

rinter2m

rinterm2r

rinterm1

θinterB1

θinterAm

θintraAm

A

Figure 2.4.4: Angles and distances to take into consideration of a chain molecule of m = 4 hardspheres with one association site on each terminal segment.

segment diameter, θinterB1 and θinter

Am are the angles between the vector connecting the centre of a

terminal segment to its respective site and the vector that connects the centres of the terminal

segments of different molecules. Analogously, θintraB1 and θintra

Am are the angles between the vector

connecting the centre of a terminal segment to its respective site and the vector that connects

the centres of the terminal segments of the molecule.

The pair interaction potential between two molecules φinter,

φinter = φHS + φHB, interAB , (2.163)

as in Equation (2.20), is given by the sum of a reference contribution, which in this case is the

hard sphere potential φHS,

φHS(rij) =

∞, rij < σ

0, rij ≥ σ

, (2.164)



potential φHB, interAB between the sites A and B to mimic the hydrogen bond, given by

φHB, interAB (rinter

1m , θinterB1 , θinter

Am ) =

− εHB, inter

AB , rinter1m < rc; θinter

B1 < θc; θinterAm < θc

0, otherwise. (2.165)

Similarly, the pair interaction potential between segments of the same molecule φintra

φintra = φHS + φHB, intraAB , (2.166)

is obtained from the sum of the repulsive contribution, given by Equation (2.164), and the

associative φHB, intraAB ,

φHB, intraAB (rintra

1m , θintraB1 , θintra

Am ) =

− εHB, intra

AB , rintra1m < rc; θintra

B1 < θc; θintraAm < θc


which is only non-zero for the interaction between the terminal segments.

Following the concept of multi-density presented in Section 2.2, the total number density of

molecules ρ is given by the sum of the number densities of molecules in all the different bonding

states:

ρ = ρ0 + ρA + ρB + ρinterAB + ρintra

AB . (2.168)

Using the definition of fraction of molecules with sites free in Equation (2.97), we have the fraction

of monomers, X0 = ρ0/ρ, the fraction of molecules not bonded intermolecularly, (ρ0 + ρintraAB )/ρ,

the fraction of molecules not bonded intramolecularly, X intra0 = (ρ0 + ρA + ρB + ρinter

AB )/ρ, the

fraction of molecules not bonded at site A, XA = (ρ0 + ρB)/ρ, and the fraction of molecules not

bonded intermolecularly at site A, (X interA = ρ0 + ρB + ρintra

AB )/ρ. Also, by symmetry, ρA = ρB

and XA = XB.

At the ideal gas level, the pressure is proportional to the number of “species”, i.e. monomers,

dimers, trimers, and all other aggregates accounted for in the theory. Each intermolecular bond

reduces the number of “species” by one unlike the intramolecular bonds which do not affect the


number of aggregates in the fluid. The average number of aggregates N is then given by

N = N[1− 1

2(1−X inter

A

)− 1

2(1−X inter

B

)]= NX inter

A (2.169)

The condition for chemical equilibrium imposes that the chemical potential of a monomer is the

same as that of any size of cluster or ring, i.e. µ0 = µbonded = kT ln(ρ0Λ3

). The Helmholtz free

energy of the model is thus given by

A = N0µ0 +Nintraµintra +Ndimerinter µ

dimerinter + . . .− PV = kTN ln ρ0Λ3 − kTN, (2.170)

The association contribution is thus given by

Aassoc

NkT= A− Aideal

NkT= ln

(ρ0

ρ

)−X inter

A + 1

= lnX0 −X interA + 1.

(2.171)

The equations to calculate the fractions of free sites are obtained through mass balances. The

fraction of molecules intramolecularly bonded is given by

1−X intra0 = X0∆

intraAB (2.172)

where,

∆intraAB = DintraF

intraAB , (2.173)

is the strength of the intramolecular association, F intraAB = exp

[−εHB, intra

AB /(kT )]− 1 and Dintra

is the fraction of molecules of the reference fluid in the bonding direction. The parameter Dintra

is a function of density only, and is given by

Dintra = 1(4/3)π(m− 1)3σ3

∫bondingvolume

gintraHS (1m) d(1m), (2.174)

and is obtained by simulations of the associating fluid with the association energies set equal

to zero. It is assumed that the intramolecular distribution function gintraHS is the same as in the


reference fluid.

The fraction of molecules intramolecularly bonded can also be obtained from the subtraction of

the molecular fraction not bonded at A, XA, from the fraction not intermolecularly bonded at

A, X interA as in

1−X intra0 = X inter

A −XA, (2.175)

or from the subtraction of the molecular fraction of monomers, X0, from the fraction not

intermolecularly bonded X inter0 ,

1−X intra0 = X inter

0 −X0. (2.176)

Lastly, in order to make use of relationships valid in TPT1, we decouple the intramolecular

from the intermolecular association by defining X inter′A as the fraction of molecules in the subset

not inter-molecularly bonded at A of the set of not intramolecularly bonded at A. The quantity

X inter′A is thus a conditional probability given by

X inter′A = X inter

A |X intra0 = X inter

A ∩X intra0

X intra0

= XA

X intra0

(2.177)

We can use the property of independence of TPT1 that states that the likelihood of a particular

site associating is independent of the bonding state of other association sites, see Appendix A.

This property is only valid in the absence of rings and allows us to express X0 as

X0 = X inter′A X inter′

A X intra0 . (2.178)

Another result from TPT1 is the definition of fraction of molecules with site A free given that

the molecule is not involve in a ring:

X inter′A = 1

1 + ρXA∆interAB

, (2.179)


where the strength of intermolecular association between sites A and B is given by [48]

∆interAB = 4πginter

HS (σ)KABFinterAB . (2.180)

The bonding volume KAB is a measure of the volume where sites A and B on the two different

molecules overlap and F interAB is given by

F interAB = exp (εAB,inter/kT )− 1. (2.181)

The six unknown variables X0, X inter0 , X intra

0 , XA, X interA and X inter′

A can thus be found from the

six Equations (2.172) and (2.175)–(2.179). Equation (2.171) thus accounts for the contribution

of inter- and intramolecular association to the Helmholtz free energy. This is not its residual

format as its the limit at low density is not zero.

2.4.3 Comparison between the approaches of Sear and Ghonasgi

In order to compare the two theories, we first need to isolate the contribution of the formation

of linear chains and intramolecular rings by association from the expression found by Sear

and Jackson in Equation (2.161) and secondly, express both theories in terms of XA. Sear’s

expression can be written in fractions of free sites as

AWert

NkT= ln

(XA

(∆ρ)m−1 (1 + ρXA∆AB)

)+m−XA −

XA

1 + ρXA∆AB

∆ABWm−1. (2.182)

The respective mass action equation in Equation (2.162) can also be rewritten as

1−XA(1 + ρXA∆AB) = ∆ABWm−1

(XA

1 + ρXA∆AB

)(2.183)

In order to isolate the association part of Equation (2.161), we turn off the association by setting

the association energy to 0, ∆AB = 0, and therefore having all sites free, XA = 1, obtaining

AWert

NkT= −(m− 1) ln (∆ρ) +m− 1, (2.184)


the precursor of the chain contribution. Subtracting Equation (2.184) from Equation (2.182),

leads to the association contribution which is given by

AassocSearNkT

= ln(

XA

1 + ρXA∆AB

)+ 1−XA −

XA

1 + ρXA∆AB

∆ABWm−1. (2.185)

We can express the association contribution found by Ghonasgi et al. in the format of Equa-

tion (2.185). Using Equations (2.177)–(2.179) the fraction of monomers X0 is expressed in terms

of XA as

X0 = XA

1 + ρXA∆interAB

, (2.186)

and using Equations (2.172), (2.175) and (2.186) the fraction of molecules with site A not

intermolecularly bonded X interA is expressed as

X interA = XA + XA

1 + ρXA∆interAB

∆intraAB , (2.187)

where the intermolecular association strengths are equivalent in both methods and are given as,

∆AB = ∆interAB = KABFABgHS(σ). (2.188)

Inserting Equations (2.186) and (2.187) in the free energy expression in Equation (2.171) allows

us to rewrite it as

AassocGhonasgi

NkT= ln

(XA

1 + ρXA∆interAB

)+ 1−XA −

XA

1 + ρXA∆interAB

∆intraAB + 1. (2.189)

The respective mass action equation is obtained by combining Equations (2.172), (2.177), (2.179)

and (2.186) and is written as

1−XA(1 + ρXA∆interAB ) = ∆intra

AB

(XA

1 + ρXA∆interAB

). (2.190)

Comparing the expressions to account for the competition between intermolecular association into

linear chains and intramolecular association of the two research groups, namely the free energy in

Equations (2.185) and (2.189), and the mass action Equations (2.162) and (2.190), it is clear that


they are equivalent, as previously observed by García-Cuéllar and co-workers [120] and Avlund et

al. [121]. The only difference resides in the definition of the strength of intramolecular association,

∆intra. In Ghonasgi’s approach, ∆intra = DintraFintraAB , which can be obtained from simulation

of the reference fluid. In Sear’s approach we use the intermolecular pair distribution function

corrected with the parameter Wm−1 taken from polymer chemistry [119], ∆intra = ∆ABWm−1,

hence preserving the association parameters. The values for the intramolecular association

strength calculated by the two methods showed good agreement for fully flexible chains of hard

spheres [106]. However,W can be regarded as an extra parameter to fit and it is possible to define

different energies of association and bonding volumes for the inter- and intramolecular types of

association. Extending these theories to multiple sites seemed not to be straightforward with

Ghonasgi’s method of mass balances, whereas Sear’s approach seemed to be more methodical

and therefore was the one followed by the author.

2.4.4 Further contributions and applications

Following the work of Ghonasgi et al. [18,107], García-Cuéllar and Chapman [120,122] extended

the theory to a aqueous-like solution of a model telechelic polymer with linear-chain and

intramolecular aggregates. The polymer is modelled as a flexible linear hard tetramer with one

association site on each terminal segment and the solvent is modelled a hard sphere with four

association sites. Good agreement with simulation data was obtained.

Galindo et al. [108] applied the theory for intermolecular ring formation of Sear and Jackson to

pure systems of HF and showed that the inclusion of the competition between rings and linear

chains in the association clusters is key to capture the maximum in the enthalpy of vapourisation

of HF. The accuracy of Wertheim’s theory extended to account for intermolecular ring formation

[105] is extensively tested in a series of studies by Tavares and co-workers [123,124,125,126].

Tavares et al. [123] applied the theory [105] to two patchy sites (AA) on spherical particles and

looked at chain length and ring size distribution across a range of temperatures and densities,

obtaining good agreement with simulation. The theory [123] was extended by Rovigatti et

al. [124] to three patchy sites (AAB) and by Tavares et al. [125] to 12 patchy sites (2A, 10B)


on spherical particles which enabled the possibility of branching in aggregates. The theory

considered A − A and A − B hydrogen bonding with ring formation by A − A association.

Remarkably, the vapour–liquid equilibria predictions in [124] were consistent with the Monte

Carlo simulations of AAB ring-forming models carried out by Almarza [9] which predicted

two critical points (lower and upper) in a single-component system found in a limited range of

ratios between the association energies corresponding to the A− A and A−B hydrogen bonds.

The existence of a reentrant vapour phase in these studies is explained by the formation of

non-/weakly interacting intermolecular rings. Similarly, the theory in [125] was also in good

agreement with Monte Carlo simulations unless the angle between ring forming sites was small

(leading to the formation of small rings), in which the agreement was found to be only qualitative,

possibly due to the formation of AB rings being non-negligible. The further extension to AB

rings enabled the formation of networks (adjacent loops) was done by Tavares et al. [126], using

the correspondence between TPT1’s free energy and the free energy of an ideal mixture of

self-assembling clusters. The agreement with simulation was much improved.

Avlund et al. [97, 121,127] extended the framework for intramolecular rings of Sear and Jackson

and Ghonasgi et al. [18,105] to any number of sites with the constraint that all rings formed

in species i had to involve a common association site. In other words, if the set of association

sites of component i Γi = A,B,C,D, rings promoted by intramolecular hydrogen bonding

(IntraMHB) between pairs A − B and C − D cannot be accounted for simultaneously by

the theory of Avlund et al. [121], but rings involving the pairs A− B, A− C and A−D can

be accounted for simultaneously because they all involve a common association site (A in this

case). The authors used the theory [121] to predict phase behaviour of binary mixtures of

glycol ethers in water and alkanes. The glycol ethers were modelled with two oxygen sites

and one hydrogen site. The authors did not observe a significant improvement in accuracy by

considering the new theory in comparison to TPT1. Since they used the sPC-SAFT EOS [128]

to provide the reference properties, all parameters were estimated individually for the cases of

open-chain aggregates both in the presence and absence of intramolecular rings. The end-to-end

distribution function [118,119], presented in Section 4.4, was used for the parameter W = Wn,

which is probably not a good approximation to real molecules, since it was developed for freely


jointed chains and is most accurate for very long chains. In the end, the differences seen between

TPT1 and TPT1+intramolecular rings in the prediction of phase equilibria was probably due

mostly to the different parameters than to the theories.

The restriction of a single bond per pair of molecules of TPT1 is relaxed by Sear and Jackson [112],

where doubly bonded dimers are considered following a similar line of thought as that for ring

formation, the only difference being the approximation taken for the extra graph added to the

fundamental graph sum. Janeček and Paricaud [114,115] extended this theory to mixtures and

applied it successfully in the prediction of the phase behaviour of pure and aqueous solutions

of carboxylic acids. Note however that in the supporting information of [115] the extension to

molecules with multiple sites (more than two) is unfortunately not correct. The authors carried

the extension by summing over association sites, a procedure that would be correct if the sites

were independent, which is not the case in general.

More recently, Marshall et al. [98,110] used the approach by Sear and Jackson for intermolecular

ring formation [105] in studies of two-site associating fluids which exhibit bond cooperativity

considering effects of steric hindrance and double bonding [112]. The influence of these effects

in the cluster distribution was in good agreement between the theory and simulation results.

The authors also extended the theory for intramolecular hydrogen bonding [105] to interfacial

systems [129], by considering its inhomogeneous form in conjunction with density functional

theory. The theory developed was successfully used in the study of surface tension and partition

coefficients of four-segment chain molecules exhibiting inter- and intramolecular hydrogen

bonding. Despite the excellent agreement with simulation, the theory showed to be very

demanding computationally for molecules with more than five segments. A way around this

was found shortly after by the authors [130] who considered Monte Carlo ensemble averages

to compute some of the integrals, which speeded up the numerics. This Monte Carlo density

functional theory was used in the study of the effect of chain length and confinement in the

competition between inter- and intramolecular association in two-site chain molecules. All

theories by Marshall et al. mentioned are limited to two-site molecules. Ballal and Chapman [131]

extended the inhomogeneous approach of Marshall et al. [129] to mixtures of associating chains

with multiple sites explicitly in a water-like solvent modelled with a four-site hard sphere. The


theory introduces a parameter ψ defined as the “number of association sites on the end segment

−1”, which is larger than 1 for a number of sites equal or superior to 3. The theory was in good

agreement with simulation data, however, we found a mistake in the expression for the mass

action equations for the general case of multiple association sites in the solute: equation (14) of

the paper [131] or (4.11) of the doctoral thesis [132], would imply that for a non-associating

system,

ρ(i)(ri) = ρ(i)0 (ri)Ψ (i), (2.191)

where ρ(i)(ri) and ρ(i)0 (ri) are the total and monomer densities (respectively) of segment of species

i. This equation does not seem to be correct as the density of molecules should be matching the

density of monomers (free molecules) at the limit of no association, but only does for Ψ (i) = 2.

This of course does not affect their results since Ψ (i) = 2 for the case considered [131].

Despite all great progress in the extension of Wertheim’s TPT1 formalism to capture the impact

of the formation of ring-like clusters from hydrogen bonding, an algebraic description of this

effect that is generally valid in associating systems of arbitrary number of components and

association sites is still missing. Furthermore, all previous studies to the knowledge of the authors

focused on rings formed by hydrogen bonds of a single nature, either inter- or intramolecular,

overlooking the competition between these two types of rings that possibly exist in systems of

short diols, dicarboxylic acids and other molecules that exhibit hydrogen bond forming groups

in appropriate positions and orientations.

2.5 Concluding remarks

Wertheim proposed a perturbative formalism for associating fluids based on a Mayer cluster

expansion in terms of activities. The activity expansion of the grand partition function allowed

to write explicitly the contribution from each association cluster and apply graph cancellations

early on in the derivation. A key feature of the theory is the concept of multiple densities that

arise from considering the molecules in different bonding states as being of different species.

Resulting from the theory is an elegant expression for the Helmholtz free energy as function of

2.5. Concluding remarks 61

densities of molecules with given association sites free. The multiple densities are calculated

self-consistently through the law of mass action equations that are naturally retrieved from the

minimization of the expression of the Helmholtz free energy. The simplest form of the theory is

the first order approximation (TPT1) which requires only the two-body distribution function

for the reference fluid. The theory has proven to be very powerful and inspired the development

of the equation of state named statistical associating fluid theory (SAFT). The equations of this

family consist of an algebraic expression for the Helmholtz free energy of fluid systems composed

of contributions: the ideal term, a reference term (typically hard-sphere) and perturbation

terms that include contributions from intermolecular potential, non-sphericity of the molecules,

hydrogen bonding interactions, etc. Despite the multitude of versions of SAFT currently in use

and development, most share the standard association term given by the TPT1 and hence the

applicability of the EOSs is constrained by the TPT1’s constraints, in particular the neglect

of ring clusters. The success of TPT1 gave rise to extensions and applications that stimulate

further extensions. The extension of the theory to ring formation has been done for limited

cases [18,105,120,121,131] but a unified extension treating models with unlimited number of

sites and inter- and intramolecular ring configurations is still missing, which is the main goal of

this work and is treated in the next section.

Chapter 3

UNIFIED THEORY TO ACCOUNT

FOR RING FORMATION

The proposed theory, object of the work of this thesis, is treated in this Chapter. A particularly

pedagogical approach is adopted, which starts with counting the association aggregates based

on the types of hydrogen bonds and the definition of the inter- and intramolecular potentials in

the fluid. Next, the residual contribution to the Helmholtz free energy arising from association

is derived. The fundamental graph sum typical to TPT1 is modified by summing in the ring

graphs and the law of mass action equations are derived. The Chapter is concluded after a

summary and the formulation for mixtures.

3.1 Introduction

The hydrogen bond (HB) is defined as an “attractive interaction between a hydrogen atom from

a molecule or a molecular fragment X− H in which X is more electronegative than H, and an

atom or a group of atoms in the same or a different molecule, in which there is evidence of bond

formation”, according to IUPAC’s recommendation [41]. Attractive short-ranged directional

interactions, such as the hydrogen bond, lead to the formation of association aggregates that

highly influences the behaviour of associating fluids (e.g., water and water mixtures [42,43]),

62

3.1. Introduction 63

being the reason why this deviates from that of simple fluids. The association aggregates can be

distinguished by the arrangement of the constituting molecules connected by hydrogen bonds,

e.g., linear chains of size n consist of n− 2 middle molecules with two hydrogen bonds and two

end-molecules with one hydrogen bond, and branched chains of size n consist of n ≥ 4 molecules,

with no loops and more than two end-molecules. These are the two most obvious association

aggregates, which exclude ring-like aggregates. However, when an HB donor and an HB acceptor

are in proximity on the same molecule, an equilibrium may exist between closed conformations

in which an intramolecular hydrogen bond is formed, creating a temporary (intramolecular)

ring system, and open conformations (linear/branched chains) in which the polar groups are

exposed to other molecules. Other types of competing ring systems are formed by the interaction

between donor and acceptor pairs of different molecules, also known as intermolecular rings.

Accounting for the competition between these different aggregates enables a better understanding

of the thermo-physical behaviour of fluid systems: the reentrant vapour-phase behaviour of

patchy colloids that form intermolecular rings [9]; the higher solubility in non-polar solvents of

ring forms in comparison to the open forms which are more water-soluble; the enhancement of

lipophilicity of molecules with intramolecular HB and consequent promotion of cell membrane

permeability [12]; the impact of small structural changes on the metabolism of molecules due to

stress, disruption, or hindrance of intramolecular hydrogen bonding [13]; the folding of proteins

(aggregation of misfolded proteins that escape the cellular quality-control mechanisms is a

common feature of a wide range of highly debilitating and increasingly prevalent diseases, such

as Alzheimer’s, Parkinson’s and type II diabetes [6]), as well as the binding and specificity of

ligands [7, 8]; the stabilization of polymer structures [1, 2, 3] and the effects on their cloud-point

pressures [4, 5]; and the impact of the competition between intramolecular bonding and linking

to the adsorbent [133] in adsorption efficiency of separation processes, as extra energy is required

to break the intramolecular bond. The formation of ring-like structures, originating both from

inter- or intramolecular hydrogen bonding, has thus found a plethora of interest within the

scientific community, in particular in drug design and delivery [134], medicinal chemistry, and

process and solvent design [135,136].

As more complex systems are considered, a continuous effort is put on the theoretical description

64 Chapter 3. UNIFIED THEORY TO ACCOUNT FOR RING FORMATION

of anisotropic interactions such as is the case of the hydrogen bond. A deeper understanding

will lead to a higher efficacy and accuracy of core modelling tools of product and process

design. Over a century ago, Dolezalek [137] proposed a chemical approach to deal with hydrogen

bonding. He considered that molecules converted into a new species through aggregation in

a chemical reaction fashion. He was able to explain many of the observed deviations from

Raoult’s law in solutions. The main drawback of this approach is that the association aggregates

need to be specified a priori. The refining of molecular-based association theories provides a

path towards the understanding of the distribution of association aggregates. Contrasting to

Dolezalek, in the late 1980s Wertheim [17,44,45,46] proposed the thermodynamic perturbation

theory (TPT) to account with the residual contribution from short range and highly directional

interactions in a fluid, with the inevitable formation of short-lived clusters of molecules. Some

constraints were imposed in the theory to mimic common steric effects, namely exactly two

association sites being involved in a bond and the prohibition of both double bonding and

closed-loop aggregates. The formation of hydrogen bonds leads to a distribution molecular

bonding states, which is directly obtained from the minimisation of the free energy in a system

of fixed number of molecules N , volume V and temperature T . Wertheim’s TPT1, due to

its simplicity and allowing for a parameter free investigation of the formation of polydisperse

polymers [46], has been combined with the combinatorial Flory-Stockmeyer approach [138] in

calculations of bonding probability, cluster distribution and percolation threshold [139]; the

agreement between TPT1 and simulation results for the self-assembly of patchy particles into

polymers chains is remarkable [140]. Furthermore, the need for a rigorous and reliable equation

of state (EOS) to model the thermo-physical behaviour of complex associating mixtures inspired

the recast of Wertheim’s TPT of first order (TPT1) into an EOS, the statistical associating

fluid theory (SAFT). The original approach was published in 1988/89 [47, 48, 49] and since,

there has been a continuous effort on its improvement, which gave rise to numerous variations

constituting today the family of SAFT EOS [50]. The assumptions made in TPT1 limit the

association aggregates being considered by the EOS, such as the neglecting of all terms involving

the formation of rings by inter- and intramolecular hydrogen bonding.

Addressing this issue, two research groups, Sear and Jackson [105, 106], and Ghonasgi et


al. [18, 107], relaxed the constraint that imposed the absence of ring formation. The groups

developed, independently, expressions to account for intramolecular association in terms of

free energy in a Wertheim-like approach. Both groups considered a pure fluid of fully flexible

hard chain molecules of m segments with two association sites A and B located in the terminal

segments. Inter- and intramolecular hydrogen bonding between A and B sites was considered,

whereas A−A and B−B site-site association was prohibited. The authors note that the correct

term for the intramolecular bonding contains the intramolecular site-site correlation function

for the ends of the chain. In [105], this is approximated to a product of the intermolecular

correlation function with a parameter W that accounts for the probability of the two sites in a

molecule to find themselves in associative proximity, and in [107] it is retrieved from simulation.

Sear and Jackson adopted a rather formal approach through the modification of the fundamental

graph sum of Wertheim by adding the ring integral and then proceeded with a minimisation

of the residual Helmholtz free energy by functional derivation. Ghonasgi et al. opted for the

alternative approach of mass balance. The applicability of the theories is limited to pure systems

of chain molecules with two association sites that exhibit competition between intramolecular

hydrogen bonding and open-chain association. These theories shown to be equivalent [97,122].

Sear and Jackson also looked at the formation of intermolecular rings from a specified number of

associating spherical molecules and its competition with the formation of open-chain aggregates

[105]. Galindo et al. [108] applied this theory to pure systems of HF and showed that the

inclusion of the competition between rings and linear chains in the association clusters is key to

capture the maximum in the enthalpy of vaporization of HF. Sear and Jackson also developed

a theory for doubly bonded dimers [112] following a similar line of thought as that for ring

formation, the only difference being the approximation taken for the extra graph added to the

fundamental graph sum. Janecek and Paricaud [114,115] extended this theory to mixtures and

applied it successfully in the prediction of the phase behaviour of pure and aqueous solutions

of carboxylic acids. However, in the respective supporting information [115] the extension to

multiple sites suggested is not correct.

Following the work of Ghonasgi et al. [18], Chapman and co-workers [120, 122] extended the

theory to mixtures and applied it to model a telechelic polymer solution, where the polymer was


modelled as a flexible linear hard tetramer with one associating site on each terminal segment

and the solvent was modelled as a hard sphere with four association sites. Good agreement with

simulation data was obtained, proving the value of the theory for polymer chemistry.

Avlund et al. [121] was the first extending the intramolecular hydrogen bonding theories [18,105]

to more than two sites. However, the theory is not general as the formation of multiple rings is

limited to a single ring forming site that can associate with other sites into the formation of open-

chain aggregates or intramolecular rings. The theory used Sear and Jackson’s approximation for

the intramolecular association strength [105]. Avlund et al. [127] proceeded with the application

of sPC-SAFT [128] equation of state modified with the new association contribution to predict

phase behaviour of binary mixtures of glycol ethers in water and alkanes. Surprisingly, the new

theory did not seem to bring significant improvement to the predictions for the systems analysed.

However, other factors may have impacted the predictions, such as the parameters chosen and

the use of the end-to-end function for freely jointed chains, which is arguably unsuitable for real

molecules.

More recently, Marshall et al. [129, 130] developed a Monte Carlo density functional theory [18]

to study the effect of chain length and confinement on inter- and intramolecularly associating

two-site chain molecules. Ballal and Chapman [131] extended the inhomogeneous approach of

Marshall to mixtures of associating chains with multiple sites explicitly in a water-like solvent

modelled with a four-site hard sphere. The theory was in good agreement with simulation data.

Despite all progress in the extension of Wertheim’s TPT1 formalism to capture the impact of the

formation of rings by association, an algebraic description of this effect that is generally valid

in associating systems of arbitrary number of components and associating sites is still missing.

Moreover, the studies described focused on the formation of rings formed by hydrogen bonds of

a single nature, either intermolecular or intramolecular, overlooking the competition between

these two types of rings that possibly exist in systems of short diols, dicarboxylic acids and

more complex components that exhibit hydrogen bond forming groups in appropriate positions

and orientations. In this work, Wertheim’s TPT1 [17,44, 45,46] is extended to a unified theory

to account for the competition between the formation of intramolecular rings, intermolecular

3.2. Molecular model 67

rings of many sizes and open-chain aggregates by hydrogen bonding. The expression obtained

refers to the residual contribution to the Helmholtz free energy from the formation of hydrogen

bonds and is of general applicability to real associating mixture systems with multiple sites.

The focus of this study is the theoretical framework which we illustrate with two examples of

model systems after modifying one of the latest versions of the SAFT family, the statistical

associating fluid theory of variable range employing a Mie potential (SAFT-VR-Mie) [55]. The

theory is however applicable to the multitude of EOS employing the Wertheim association term,

e.g., CPA [99], PC-SAFT [85], soft-SAFT [141,142], SAFT-γ Mie [16].

3.2 Molecular model

In this study we consider first two pure model fluids: a fluid of spherical molecules (Figure 3.2.2);

and a fluid of flexible chains of m tangent homonuclear spherical segments (Figure 3.2.3). An

arbitrary number of association sites, labelled with upper-case letters of the latin alphabet

A,B, . . ., are introduced to mediate aggregate formation (hydrogen bonds) and the set of all

sites in a molecule is given by Γ = A,B,C, . . .. The number of sites in a molecule is thus given

by the size of this set |Γ |. The bonding constraints followed are those of TPT1 (Figure 3.2.1).

When a bond between two sites located in different segments is formed, the repulsive cores of

the two segments prevent a third segment to come inside the bonding range without overlap of

cores (steric incompatibility 1), as that would imply a very high energy cost. The angle between

two sites in a molecule is not small enough to allow for simultaneous bonding to the same site

in a different molecule (weak version of steric incompatibility 2). Together with SI1 constitutes

the condition of single bonding at each site. Lastly, multiple bonding between molecules is not

allowed, (strong version of steric incompatibility 2), which is a reasonable constraint when we

consider hard cores with two sites of very short range and a large angle between them. SI2W

and SW2S constitute the condition of single bonding between each pair of molecules. The

positions of the sites are not required to be specified in the theory, however it is assumed that

their configuration is such to respect the steric incompatibilities described. Accordingly, if a

molecule has only one site, association results in dimers only. In the case of two sites, molecules


(a) Steric incompatibility 1(SI1) – two hard cores of as-sociated molecules preventa third hard core to comeinto the association rangeof the bonded sites.

(b) Weak version of stericincompatibility 2 (SI2W)– an associative bond in-volves exactly two sites,and therefore a site is un-able connect to two at thesame time.

(c) Strong version of stericincompatibility 2 (SI2S) –double-bonding is not per-mitted, only single-bondingbetween two particles is al-lowed.

Figure 3.2.1: Constraints of TPT1 [17].

are able to form linear chain aggregates (dimers, trimers, tetramers,...) as well as rings of any

size. Lastly, open chains (i.e. linear and branched chains) as well as ring aggregates may be

formed for the case of three or more sites (Figures 3.2.2 and 3.2.3). The associatinWith regards

to the formation of rings, they can be formed inter- or intramolecularly and a parameter τ is

used to indicate the number of molecules taking part in the formation of a ring, which is equal

to one in the case of an intramolecular ring.

In the molecular model considered no limit for the number of association sites that are located in

a given segment or molecule is imposed, and the precise positions of the sites in a given segment

are not specified. We highlight that the theoretical framework presented is also compatible with

other molecular models, such as fused heteronuclear chains [16,143], and mixtures. In mixture

fluids, the composition within the rings is limited to one component, meaning that we do not

allow for molecules of different components to be part of the same ring. We leave behind mixed

component rings such as those formed by acetylene and hydrogen fluoride [144]. Also, we allow

for a single pair of association sites to promote the formation of each ring. These limitations

are imposed for simplicity of the framework, however, they can easily be relaxed. For clarity,

ring formation in each of the components of a mixture is accounted for.


...AB

C ...

a) b)

c) d)

e) f)

A

...C

B

C

......

C

B

C

...AC

...

A

AB

BA

BB

A

...C

C

1

τ1

2 1

...

...

τ

1

2

BA

BA

B

C...

B

A...

...

C

...C

A

C

A

...

BA

...C

BC

...

AB

...

AB

C

1

2

...

τ

τ

1

2

...

A

...

CBB

C...

ABC

...

A

BC ...

A B

A

...C

B

A

...C B

BC ...

A C

B

A

...

...C

A

τ2

A

BC

...

Figure 3.2.2: Examples of association aggregates that can be found in a pure fluid of sphericalmolecules with the molecular structure represented in a). A green check mark or red crossindicates whether the aggregate is captured by the theory presented in this work. The sphericalcomponent has an arbitrary set of sites Γ = A,B, ... of length |Γ |, and can form loopstructures consisting of τ molecules. The aggregates represented are a) monomer, b) open chainaggregate, c) AB intermolecular ring with τ molecules, d) branched BC intermolecular ringwith τ molecules, e) AB intermolecular ring associated to a BC intermolecular ring, f) loopaggregate that involve more than one pair of sites.

3.2.1 Spherical molecules

In a fluid of spheres (Figure 3.2.2), association can only result in open aggregates or intermolecular

rings. The open aggregates can be linear chains or branched chains (Figure 3.2.2b). The

intermolecular rings are formed by association between τ ≥ 2 molecules of a given component

into a closed loop, see examples in subfigures c), d) and e) of Figure 3.2.2. A molecule may be

able to form rings of different sizes, as illustrated in Figure 3.2.2e where a ring is of size τ1 and

the other of size τ2. As only single bonding is permitted between two molecule, the number of


molecules involved in an intermolecular ring must be at least three (τ = 3). As in our treatment

we allow for a single pair of association sites per ring, aggregates of mixed site pairs such that

represented in Figure 3.2.2f are excluded. When a pair of sites a, b in a component promotes

the formation of rings of multiple sizes, the arbitrary number of ring sizes is given by NRS,ab.

3.2.2 Fully flexible chain molecules

In a chain fluid of m ≥ 2 segments (Figure 3.2.3), association can not only result in open

aggregates and intermolecular rings as before, but also in intramolecular rings. The same rules

for the formation of intermolecular rings described previously for spherical molecules apply to

chain molecules too. However, if the sites of a pair involved in a ring belong to different segments

of a chain molecule, an intermolecular ring can be formed between two molecules (τ = 2) as in

Figure 3.2.3d). An intramolecular ring involves a single chain molecule, i.e. τ = 1, with at least

two association sites (|Γ | ≥ 2) located in different segments, as illustrated in sunfigures b), e)

and f) of Figure 3.2.3.

3.2.3 Inter- and intramolecular potentials

In order to capture the effects of both inter- and intramolecular association, there are two distinct

segment-segment potentials to be considered: φinter, the pair interaction potential between two

segments in different molecules, and φintra, the intramolecular potential between two segments in

the same molecule. The intermolecular pair potential is given as the sum of a reference potential

and the perturbative association potential [44]:

φinter(r12,Ω1,Ω2) = φref(r12,Ω1,Ω2) +∑a∈Γ

∑b∈Γ

φHB,interab (r12,Ω1,Ω2) (3.1)

where r12 represents the vector between the centres of two molecules with centres of mass in

positions r1 and r2, and Ω1 and Ω2 are vectors containing the respective angles characterising

the molecular orientation and conformation, including all angles subtended by the association

sites. The reference potential φref is only dependent on the distance between each segment of


3

2...

m1

A

D

...

EC

B

1m

...3

2

...

E

DC

AB

a) b)

c) d)

e)

g) h)

f)

1

AB

CD

3

2

m

...

E

...

E

...DC

BA ...3

...

3

BA

C DE

...

D

E...

A B

E

...

BDC

BA

E

...

C

DA

...

EA B

C D

D A B

CE ...

...3

...3

m

A

D

B

C

E

...

E

D

...

C

B A

EC

DEA

B

C

BA

...D

...

11

1

1

1

1

1

1

1

1

1

2 2

2

2 22

2

2

2 2

2

3

3

3

3

3

3

3

...

...

......

...

...

...

m

m

m

m

m mm

m

m

m

C

Figure 3.2.3: Examples of association aggregates that can be found in a pure chain fluid withthe molecular structure represented in a). A green check mark or red cross indicates whether theaggregate is captured by the theory presented in this work. The chain molecule has an arbitraryset of sites Γ = A,B, ... of length |Γ |, a number of segments m and can form loop structuresformed by τ molecules. The aggregates represented are a) monomer, b) CE intramolecularring, c) open chain aggregate, d) DE intermolecular ring with τ = 2, e) AE intramolecular ringassociated to an CE intramolecular ring, f) branched AE intramolecular ring, g) and h) loopaggregates that involve more than one pair of sites.

molecule in position r1 and each segment of molecule in position r2 in case of a spherically

symmetric reference fluid. Otherwise, an angular dependency is also included in φref . The


AdB

dAr1

B

Figure 3.2.4: Scheme of position of molecule (r1) and site vectors dA and dB that are functionof the molecular orientation and conformation Ω1.

short-ranged directional interaction that is used to model association interactions between sites

a and b in molecules 1 and 2 (hydrogen bond type), is given by

φHB,interab (r12,Ω1,Ω2) =

−εHB

ab , |r12 + db(Ω2)− da(Ω1)| < rcab

0 , else, (3.2)

where εHBab is the well depth of the square-well interaction, da(Ω1) is the displacement vector of

site a from the centre of the molecule in position r1 (Figure 3.2.4), db(Ω2) is the displacement

vector of site b from the centre of the molecule in position r2, |r12 + db(Ω2) − da(Ω1)| =

|(r2 + db(Ω2))− (r1 + da(Ω1))| denotes the centre-centre distance between the association sites

a and b in molecules in coordinates (r1,Ω1) and (r2,Ω2) respectively, and rcab is the range of

the association interaction.

The intramolecular potential is defined as

φintra(Ω1) = φref(Ω1) + 12∑a∈Γ

∑b∈Γ

φHB,intraab (Ω1), (3.3)

where the relative position between segments and sites are given by the orientational/conformational

vector Ω1 and the factor 1/2 is included to prevent double counting, since the associating sites

3.3. The Helmholtz free energy 73

of the pair ab are located in the same molecule.

φHB, intraab (Ω1) =

−εHB

ab , |db(Ω1)− da(Ω1)| < rcab

0 , otherwise, (3.4)

where the potential is defined as a function of the distance and orientation between the segments

of molecule in coordinates r1 with the intramolecularly associative sites a and b given by the

vector |db(Ω1)− da(Ω1)|. The intramolecular potential is non-zero only for the pair of sites in

the molecule that are sterically (site-site distance > rcab) and energetically able of association

(εHBab > 0).

3.3 The Helmholtz free energy

The Helmholtz free energy of the associating pure fluid A = Nµ − PV , when considering

explicitly molecules in a non-associated state (monomers), involved in dimers, chain aggregates,

intramolecular rings, intermolecular rings of size τ , and any other cluster species that are present

in the fluid can be written as [18,145]

A = N0µ0 +Ndimerµdimer +Ntrimerµtrimer + ...+NRS∑R=1

NτRµR +Nintraµintra − PV, (3.5)

where

N = N0 +Ndimer +Ntrimer + ...+NRS∑R=1

NτR +Nintra (3.6)

is the number of molecules in the fluid, N0 is the number of monomers, Ndimer is the number

of molecules in chain dimers, Ntrimer is the number of molecules in chain trimers, NτR is the

number of molecules which are intermolecularly bonded into rings of size τR ≥ 2 and Nintra is the

number of molecules with an intramolecular hydrogen bond. Furthermore, µ0 is the chemical

potential of the monomers, µdimer is the chemical potential of the dimers, Ntrimer is the chemical

potential of the trimers, µτR is the chemical potential of intermolecular rings of size τR, µintra

is the chemical potential of intramolecular rings, P is the pressure and V is the volume. At


thermodynamic equilibrium, the chemical potential of a given molecule regardless of cluster

arrangements, including monomers, must be equal, which allows us to write

A = Nµ0 − PV. (3.7)

Following an observation by Andersen, [57,67], the form of the combinatorial terms in the cluster

expansion of associating fluids is independent of the density, and therefore the Helmholtz free

energy can be determined by considering the limit of low density. The results obtained at the

low density limit will therefore be valid for higher densities where the dispersion effects are also

considered. It is thus possible to derive an expression for the association term for an associating

fluid considering the Dalton’s law [18,108], i.e. assuming proportionality between the pressure

and the number of aggregate species

Nspecies = N0 + Ndimer

2 + Ntrimer

3 + ...+NRS∑R=1

NτR

τR+Nintra, (3.8)

so that

A = Nµ0 −NspecieskT. (3.9)

The ideal chemical potential of monomers is given by [52]

µ0 = kT ln (ρ0Υ (T )) , (3.10)

where ρ0 = N0/V is the number density of monomers and Υ (T ) is a function of temperature

that includes the translational, rotational, and quantal parts of the molecular partition function

(the de Broglie wavelength) as well as other contributions from molecular configurations [104].

The number of aggregate species in the system Nspecies and consequently the expression for

the free energy A depends on the type of association clusters considered. We are interested

in capturing the Helmholtz free energy of a model fluid in which association into linear and

branched chains, as well as inter- and intramolecular rings may occur. For pedagogical reasons,

we start by considering the simplest case of a non-associating fluid and build up to systems of

increasing complexity before reaching our target fluid.


3.3.1 Non-associating monomers

Since no association takes place, the only cluster species present is the monomer, i.e. Nspecies =

N0 = N and ρ0 = ρ. It therefore follows from Equation (3.9) that the free energy in a

non-associating ideal system A0 is given by

A0 = NkT [ln (ρΥ (T ))− 1] , (3.11)

as correspondent to the ideal gas [52].

3.3.2 Free monomers and open chain aggregates

Consider a fluid with ten molecules, if two molecules associate to form a dimer, nine species

remain, if further association occurs with another molecule to form a trimer, Nspecies = 8 and

so on. In the general case, the number of species in a fluid where the molecules associate only

in chains, branching possible but not rings, is reduced by one per association bond formed.

It is helpful now to define the fraction of molecules with sites in the set α ⊆ Γ free as Xα.

Note that the sites free considered in Xα include but are not limited to the subset α. In this

notation, XΓ corresponds to the fraction of molecules with all sites free, i.e. the fraction of

free monomer molecules, and it is conventionally referred to as X0 [146] (note that X0 6= X∅,

conventionally X∅ = 1). The fraction of molecules with (at least) site a free is thus given by

Xa, conventionally referred to as Xa, and the fraction of molecules with a bonded is given by

(1−Xa). This means that for a model the set of sites Γ , the number of species is

Nspecies = N − 12N

∑a∈Γ

(1−Xa) , (3.12)

and the corresponding form of the free energy

A = NkT

[ln (ρ0Υ (T ))− 1 + 1

2∑a∈Γ

(1−Xa)], (3.13)


where the factor 1/2 is included to not double count the number of bonds, as each bond involves

two sites.

3.3.3 Free monomers, open chains and intramolecular ring aggre-

gates

The formation of an intramolecular bond does not change the number of aggregate species in

the system. Defining the fraction of molecules with site a not bonded in an open chain as Xopena ,

the number of species and the free energy can be rewritten as [18]


∑a∈Γ

(1−Xopena ) , (3.14)

and

A = NkT

[ln (ρ0Υ (T ))− 1 + 1

2∑a∈Γ

(1−Xopena )

]. (3.15)

The sum of the fraction of molecules with site a bonded in open chains and the fraction of

molecules with site a bonded into intramolecular rings defining(1−X intra rings

a

)equals the total

fraction of molecules with site a bonded(1−Xa). Therefore, for each site a a relation

(1−Xopena ) +

(1−X intra rings

a

)= 1−Xa, (3.16)

can be written, which can be substituted in Equation (3.15) to obtain

A = NkT

[ln (ρ0Υ (T ))− 1 + 1

2∑a∈Γ

(1−Xa)−12∑a∈Γ

(1−X intra rings

a

)], (3.17)

where the last term of the expression corresponds to the fraction of intramolecular rings ξ1:

ξ1 = 12∑a∈Γ

(1−X intra rings

a

). (3.18)


3.3.4 Free monomers, open chains, inter- and intramolecular ring

aggregates

Since there are τ molecules in each ring, the formation of intermolecular rings of size τ reduces

the number of species by τ − 1, so that it is possible to rewrite


∑a∈Γ

(1−Xopena )−

NRS∑R=1

(τR − 1)×NτR , (3.19)

where NτR is the number of rings of size τR ≥ 1 given by

NτR = ξτ, RN

τR, (3.20)

with ξτ, R as the fraction of molecules in rings of size τR given by

ξτ, R = 12∑a∈Γ

(1−Xrings size τR

a

), (3.21)

where Xrings size τRa is the fraction of molecules with site a not in rings of size τR. Note that

ξτ, R = ξ1 and Xrings size τRa = X intra rings

a for τR = 1. Using Equation (3.20) allows to rewrite

Equation (3.19) as

Nspecies = N

1− 12∑a∈Γ

(1−Xopena )−

NRS∑R=1

(τR − 1)τR

× 12∑a∈Γ


a

) . (3.22)

Analogously to Equation (3.16), the relation

(1−Xopena ) +

NRS∑R=1


a

)= 1−Xa, (3.23)

is true and upon insertion in Equation (3.22), the number of species

Nspecies = N

1− 12∑a∈Γ

(1−Xa) +NRS∑R=1

1τR× ξτ, R

(3.24)


is used to write the free Helmholtz energy of our fluid at low density as

A = NkT

ln (ρ0Υ (T ))− 1 + 12∑a∈Γ

(1−Xa)−NRS∑R=1

1τRξτ, R

. (3.25)

Our goal is however to find the residual contribution that arises in the general case of inter- and

intramolecular association Aassoc. In the ideal limit, chain connectivity is preserved, allowing

intramolecular interactions, including intramolecular rings. The residual contribution is thus

obtained by subtracting the contribution from unbonded molecules given by Equation (3.11),

and any remaining of the ideal contributions from the total in Equation (3.25).

Aassoc = (A− A0)− limV→∞

(A− A0)

= Aassoc − limV→∞

Aassoc,

(3.26)

where the limit is taken at constant N and Aassoc = A− A0 is given as

Aassoc = NkT

lnX0 + 12∑a∈Γ

(1−Xa)−NRS∑R=1

1τR× ξτ, R

. (3.27)

Equations (3.26) and (3.27) provide a way to calculate the free energy at chemical equilibrium

but at this point the variables X0, Xα, α ⊆ Γ and the fraction of molecules in rings of size τR,

ξτ, R are still unknown. At this point we turn to the original approach of Wertheim.

3.4 Law of mass action equations

In order to derive the distribution of bonding states of the molecules in the fluid, we make use

of Wertheim’s TPT1 [63] and extend it to account for ring formation. Wertheim introduced the

concept of multi-density, which translates into treating molecules in different bonding states as

different pseudo-species. In his formalism, there is a density associated to molecules with site a

bonded, ρa, with sites a and b bonded, ρab, and so on and so forth. More generally the number

density of molecules with sites in the set α bonded in a multi-density formalism is ρα, where

α is a subset of the set of sites Γ , including the empty set ∅ that conventionally corresponds

3.4. Law of mass action equations 79

to the density of free monomers (ρα = ρ0, for α = ∅). Following Sear and Jackson [117], we

start by considering the inhomogeneous case, where the densities are dependent on the position

coordinates r and the orientation and conformation vector Ω for clarity. We make use of

(1), the short-hand notation for (r1,Ω1) which refers to all degrees of freedom of molecule 1.

Accordingly, we define (12) = (r1, r2,Ω1,Ω2) = (r1, r12,Ω1,Ω2).

The local number density ρ(1) of particles in coordinates (1)(∫

ρ(1)d(1) = N)can be written

as the sum of all bonding state densities as

ρ(1) =∑α⊆Γ

ρα(1), (3.28)

and we note that the density of molecules with sites free are the driving force densities for

association to take place. We thus define σα(1) as the density of molecules that are not bonded

plus the ones that are bonded exactly through one, some or all sites in the set α. Accordingly,

σΓ−α(1) is the density of molecules with (at least) the sites in the set α free, e.g., for Γ = A,B,

σΓ−A(1) = σB(1) and is given by the sum ρ0(1) + ρB(1). More generally,

σα(1) =∑δ⊆α

ρδ(1), (3.29)

with the special cases σ0(1) = ρ0(1) and σΓ (1) = ρ(1).The density of molecules with sites in set

α free is simply related to the fraction of molecules with set of sites α free as

σΓ−α(1) = ρ(1)Xα(1), (3.30)

for α ⊂ Γ and the fraction of monomers

σΓ−α(1) = ρ0(1) = ρ(1)X0(1), (3.31)

for α = Γ .

In the last section, an expression for the chemical equilibrium of the Helmholtz free energy was

derived. Now, we turn to the Helmholtz free energy due to association as given by Wertheim [46]


following his proposed TPT:

AWert [ρ] = kT∫ [

ρ(1) ln(ρ0(1)ρ(1)

)+ ρ(1) +Q(1)

]d(1)− kT∆c(0) [ρ] , (3.32)

where the notation ρ = ρ0, ρA, . . . highlights the dependency of the free energy on all

bonding state densities (multi-density formalism). The quantity AWert, named after Wertheim,

is an excess energy that arises from the hydrogen bonds, in other words, it is the difference in

free energy between the real (associating) system and the reference (non-associating) system,

AWert = A− Aref . All interactions but the hydrogen bonding are included in the reference fluid,

such as the chain connectivity in case of a chain molecule, the repulsive and the dispersion

attractive interactions. This is a general expression valid for all possible aggregates (i.e. inter-

and intramolecular rings as well as open chain aggregates). Equation (3.32) provides a different

way to express the association contribution to the free energy found in Equation (3.27). The

difference between the two equations is that Equation (3.32) is a general expression before

chemical equilibrium is imposed (and reduces to Equation (3.27) in chemical equilibrium),

which enables the calculation of the distribution of bonding states by minimisation of the free

energy [147] in order to all densities ρ.

The function Q results from a cumbersome derivation found originally in [17] and recently

re-derived in an excellent review of Wertheim’s thermodynamic perturbation theory by Zmpitas

and Gross [63], and is given as

Q(1) = −∑a∈Γ

σΓ−a(1) + ρ0(1)∑

γ1,...γM∈P(Γ ) ,M≥2

(−1)M(M − 2)!M∏i=1

σγi(1)ρ0(1) . (3.33)

Here, the first sum is over all sites in Γ and the second sum is over all possible partitions of

the set Γ (P(Γ )), i.e. ways to divide up Γ into pairwise disjoint subsets. The elements of the

partition of Γ into M subsets are γ1, γ2, . . . , γM and the condition M ≥ 2 ensures that each

partition considered must contain two or more elements.

In Wertheim’s formalism, each density ρα for α ⊆ Γ is defined in terms of a sum of graphs1.1Graph theory and lemmas for manipulations of graphs are given in the literature [52,63].


The functional ∆c(0) = c(0) − c(0)ref in Equation (3.32) is referred to by Wertheim [17] as the

fundamental graph sum, where c(0) is the correction quantity to the free energy due to cluster-

forming interactions, while c(0)ref is the contribution from interactions existing in the reference

fluid. It is therefore the difference ∆c(0) that concerns us.

3.4.1 The fundamental graph sum ∆c(0)

The fundamental graph sum ∆c(0) relates the possible bonded states of the molecular model

to Nspecies. In our current work it contains the sum of the contributions from the formation of

open chains, linear or branched (∆c(0)open), and from the formation of rings promoted by inter-

and/or intramolecular association (∆c(0)ring) [129], i.e.

∆c(0) [ρ] = ∆c(0)open [ρ] + ∆c(0)

ring [ρ] , (3.34)

where ∆c(0)ring [ρ] comprises the contributions arising from the formation of inter- and intramolec-

ular ring aggregates

∆c(0)ring [ρ] = ∆c(0)

inter ring [ρ] + ∆c(0)intra ring [ρ] . (3.35)

∆c(0) in the formation of open-chain aggregates

The fundamental graph sum includes all the irreducible σα-graphs responsible for the different

aggregates. The graphs corresponding to open-chain aggregates of a size of three (trimers) or s

particles (s-mers) are reducible to the graphs containing an association bond between a single

pair of particles (dimers), as seen in Section 2.2. A ring, however, is an irreducible graph and

therefore it is not accounted for by the same graphs corresponding to open chains. Accordingly,

in TPT1 all graphs are discarded except those which contain associations between a single pair


of segments. The graph sum for open-chain aggregates ∆c(0)open [ρ] (see )

∆c(0)open [ρ] =1

2∑a∈Γ

∑b∈Γ

∫σΓ−a(1)σΓ−b(r1 + r12,Ω2)gref(r12,Ω1,Ω2)

× fab(r12,Ω1,Ω2) d(r1) d(r12) d(Ω1) d(Ω2)

(3.36)

thus accounts for the formation of all open chain aggregates of any size.

As seen in Section 2.2, in the case of a spherical molecular model with a set of sites Γ = A,B,

Section 3.4.1 can be written graphically with σα-nodes as (see Equation (2.83))

∆c(0) [ρ] = + + + + + 12 + + . . .

However, our model molecule may be non-spherical and instead consisting of an arbitrary number

of segments to mimic its non-sphericity. This may complicate the graphical representation, but

the integrals studied in Section 2.2 are still valid. The graph sum corresponding to Section 3.4.1

thus differs from the one used by Wertheim [102] in the sense that it involves molecular graphs

(see Appendix B), simplifying the analysis, a concept found in the work of Marshall et al. [148]

who showed the relation between segment-based and molecular graphs.

It is possible to physically interpret Section 3.4.1: considering Γ = A,B, . . ., the formation of

an association bond between sites A and B in different molecules is a function of the number

density of molecules with sites A and B free, σΓ−A and σΓ−B respectively, and the likelihood

of these sites forming a bond given by the product between the radial distribution of the

reference fluid gref , and the Mayer f -function of the ab interaction, expressed as usual by

fAB(r12,Ω1,Ω2) = exp(−βφHB,inter

AB (r12,Ω1,Ω2))− 1. The molecules interact across all volume

and configurations, and therefore their coordinates are integrated over all space and orientations.

A sum over all sites a and b is carried out and a factor of half is included to avoid counting the

same pair of sites twice. Any pair of sites is allowed to associate provided that the respective

εHBab is larger than zero.


∆c(0) in the formation of open-chain and ring aggregates

In order to account for the formation of ring aggregates we modify the fundamental graph sum

by relaxing the TPT1 approximation of Wertheim that neglects all graphs containing more that

one association bond. Following Sear and Jackson [117], in addition to the diagrams present in

Section 3.4.1, we now include the irreducible ring graphs in the fundamental graph sum.

Since in TPT1’s formalism the angles between segments are not accounted for explicitly and a full

flexibility of the molecular bonds is assumed, the pairs of sites that promote ring formation are

an input to the theory, in particular through a parameter Wab that is related to the probability

of bonding between the pair of sites a, b. The intermolecular ring contribution from clusters of

the types represented in subfigures c), d) and e) of Figure 3.2.2 and Figure 3.2.3d, is written

consistently with that found in [106] after being generalised to an arbitrary number of sites as

∆c(0)inter ring [ρ] =1

2∑a∈Γ

∑b∈Γ

NRS,ab∑R=1

1τR

∫ τR∏i=1

[gref(ri,i+1,Ωi,Ωi+1)fab(ri,i+1,Ωi,Ωi+1)]

× σΓ−ab(1) d(r1)τR−1∏j=1

[σΓ−ab(r1 + r12 + ...+ rj,j+1,Ωj+1)

× d(rj,j+1)]τR∏k=1

d(Ωk),

(3.37)

where the number of molecules per intermolecular ring τR ≥ 2 for all R. In a ring aggregate we

consider that the molecules are in contact2 only with the preceding and succeeding molecules.

Accordingly, in this work, we use the approximation of Sear and Jackson [117] for the τR-body

distribution function, which is given in terms of pair distribution functions, one per pair of sites

in contact,

g(τR)(12 . . . τR) =τR∏i=1

g(ri,i+1,Ωi,Ωi+1), (3.38)

where the convention (i+ 1) = 1, for i = τR is used. In case of a spherical molecular model with

a set of sites Γ = A,B (the case explored in Section 2.2), the fundamental graph sum would

2The value for the pair distribution function is typically taken at contact value since the association interactionsare short ranged.


now be given graphically as

∆c(0)[ρ] = + + + + + 12 + . . .

+ 13 + + + . . .

+ 14 + 1

2 + 14 + . . . ,

(3.39)

where the first row includes linear open chain aggregates of any length and the second row

includes intermolecular ring clusters of any size. Although the z-graphs with closed-loop

hydrogen bonds, e.g., , are included in the σα-nodes of Equation (2.83), these graphs

are insufficient to account fully to the ring structures [46]. In order to account for the ring

clusters, one must add the irreducible rings graphs to the fundamental graph sum (second line

of Equation (3.39)) [105], which are the leading terms to account for ring aggregates. In the case

of a number of sites of three or more, there is the possibility of branched rings. Since branched

rings are reducible to ring graphs, they are accounted for implicitly in Equation (3.39).

The analysis of the expression for ∆c(0)ring [ρ] follows the same rationale as that for ∆c(0)

open [ρ]:

in order to form an intermolecular ring of size τ , τ molecules must come close enough to form τ

links between them. For the sake of simplicity in Equation (3.37), the position index (τ + 1)

refers to the position index (1). It is thus required that the sites a and b of each molecule involved

are simultaneously free, correctly oriented and that the interaction parameter εab is larger than

zero. As we can have rings of different sizes we sum over R, the ring size index. We divide by

τR to avoid double counting, since there are τR ways to number the positions of molecules in

a ring of size τR. Later we will see that the last link, gref(rτ,1,Ω1,Ωτ )fab(rτ,1,Ωτ ,Ω1), has a

different nature, as its formation turns an open chain cluster into a ring. ∆c(0)inter ring [ρ], as

it is formulated in Equation (3.37), ensures that a given intermolecular ring is composed of

molecules of the same species through hydrogen bonds promoted by a single pair of ring forming

sites. However, it is possible and straightforward to rewrite Equation (3.37) in a way to account

for rings of mixed composition or structures that involve more than one pair of sites like those


in Figure 3.2.2f, Figure 3.2.3g and Figure 3.2.3h.

In its turn, the intramolecular association contribution from clusters of the types represented in

3.2.2c and subfigures b), e) and f) of Figure 3.2.3, is given as

∆c(0)intra ring [ρ] = 1

2∑a∈Γ

∑b∈Γ−a

∫gintra

ref (Ω1)f intraab (Ω1)σΓ−ab(1) d(r1) d(Ω1) (3.40)

where gintraref (Ω1) is the intramolecular distribution function for the two segments that carry the ab-

ring-forming sites of a molecule in the reference fluid, and f intraab (Ω1) = exp

(−βφHB,intra

ab (Ω1))−1.

The intramolecular radial distribution function contains information about the likelihood of

the segments of molecule at coordinates (1) that contain sites a and b coming together at a

close enough distance to allow for the formation of a bond. The graph in Equation (3.40) looks

different from the one published by Sear and Jackson [105] as we are using molecular instead of

segment graphs [148], see Appendix B. This time, one only association link per ring is formed,

and therefore we require the number density of molecules in coordinates (1) with simultaneously

sites a and b free, σΓ−ab(1), and the likelihood of these sites forming a bond which is given by

the product of the radial distribution of the reference fluid, gref(Ω1) with the Mayer f -function.

∆c(0)intra ring [ρ], as it is formulated in Equation (3.40), ensures that at most one intramolecular

hydrogen bond is formed at a time. However, it is possible to include more than one hydrogen

bond forming in an intramolecular aggregate at the same time by modifying Equation (3.40).

Since the interest of this work resides at the homogeneous level, the multiple densities can be

treated as independent of position and orientation. Following the definition of the fraction

of molecules with the set of sites α free (Xα), according to Equation (3.30) the free energy

expression (Equation (3.32)) and function Q (Equation (3.33)), are rewritten as

AWert = NkT

[lnX0 + 1 + Q

ρ− ∆c(0)

N

](3.41)

andQ

ρ= −

∑a∈Γ

Xa +X0 ×∑

γ1,...,γM∈P(Γ ),M≥2

(−1)M (M − 2)!M∏i=1

XΓ−γiX0

. (3.42)


In a homogeneous system, the contributions to the fundamental graph sum ∆c(0)open, ∆c(0)

inter ring

and ∆c(0)intra ring are obtained as

∆c(0)open [ρ]N

= 12∑a∈Γ

∑b∈Γ

ρXaXb∆ab, (3.43)

from Section 3.4.1 with

∆ab =∫gref(r12,Ω1,Ω2)fab(r12,Ω1,Ω2) d(r12) d(Ω1) d(Ω2), (3.44)

and∆c(0)

inter ring [ρ]N

= 12∑a∈Γ

∑b∈Γ

NRS,ab∑R=1

1τR∆ringab,Rρ

τR−1 (Xab)τR , (3.45)

from Equation (3.37) with

∆ringab,R =

∫ τR∏i=1

[gref(ri,i+1,Ωi,Ωi+1)fab(ri,i+1,Ωi,Ωi+1)]τR−1∏j=1

d(rj,j+1)τR∏k=1

d(Ωk). (3.46)

where τR ≥ 2 for all ring type indices R. We can think about the formation of an intermolecular

ring in two steps; first, a chain cluster of τ molecules with τ − 1 association links between sites a

in position ri and b in ri+1 is formed. Once the chain is formed, there is a probability of the two

ends of the chain to find themselves at a close enough distance to form the last link to complete

the ring structure. We follow previous work [117] to capture the probability of the sites of the

molecules at the extremities coming into contact by introducing a parameter Wab,R [105], which

we shall call the non-normalised probability of ring formation. Hence, we approximate

∆ringab,R ≈ (∆ab)τRWab,R (3.47)

that corresponds to having τ − 1 links given by ∆ab each and one last by ∆abWab,R. The

parameter Wab,R has dimensions of inverse volume and, in this work, at a first approximation, it

is considered to be independent of density, temperature and composition. This idea stems from

the fact that the parameter was introduced to describe the chain-closing (ring-forming) last link

multiplied by (∆ab), a term that is already function of temperature, density and composition.


Lastly, for the case of an intramolecular bond,

∆c(0)intra ring

N[ρ] = 1

2∑a∈Γ

∑b∈Γ−a

Xab∆ringab , (3.48)

from Equation (3.40) with

∆ringab =

∫gintra

ref (Ω1)f(Ω1) d(Ω1). (3.49)

The association link in an intramolecular ring is of similar nature to the last link in an

intermolecular ring described previously and accordingly, we approximate

∆ringab ≈ ∆abWab. (3.50)

Comparing Equations (3.45) and (3.48) with the definitions given in Equations (3.47) and (3.50)

we realise that the intramolecular ring is a special case of the intermolecular ring for τ = 1, i.e.

one molecule per ring. As a result, the contribution from the formation of inter- or intramolecular

rings to the fundamental graph sum can be written as one single term with τR ≥ 1:

∆c(0)ring [ρ]N

=∆c(0)

inter ring [ρ]N

+∆c(0)

intra ring [ρ]N

≈ 12∑a∈Γ

∑b∈Γ

NRS,ab∑R=1

1τR

(∆ab)τRWab,RρτR−1 (Xab)τR

(3.51)

which we can relate to the fraction of molecules in rings of size τR as

ξτ, R = 12∑a∈Γ

∑b∈Γ

(∆ab)τRWab,RρτR−1 (Xab)τR (3.52)

as required in Equation (3.27). Substituting Equations (3.42), (3.43) and (3.51) in Equation (3.41)


we write the general expression for AWert as

AWert =NkT

lnX0 + 1−∑a∈Γ

Xa +X0∑

γ1,...,γM∈P(Γ ),M≥2

(−1)M (M − 2)!M∏i=1

XΓ−γi

X0

− 12∑a∈Γ

∑b∈Γ

ρXaXb∆ab −12∑a∈Γ

∑b∈Γ

NRS,ab∑R=1

1τR


.(3.53)

3.4.2 The distribution of bonding states

The chemical equilibrium conditions that establish the self-consistent values ofX0, Xa, . . . , Xab, . . .

are the ones that make AWert stationary with respect to these parameters [102]:

∂AWert

∂Xδ

∣∣∣∣∣Xγ

= 0, γ ∩ δ = ∅, (3.54)

noting that the number of variables Xδ for a component with a set of sites Γ is 2|Γ | − 1, a

number which grows exponentially with the tha number of sites |Γ |: X0 for one site, X0, XA

and XB for |Γ | = 2, X0, XA, XB, XC , XAB, XAC , XBC for |Γ | = 3, etc. Minimizing the free

energy with respect to each of these variables will generate an increasingly large and rather

complex system of equations. Fortunately, it is analytically solvable. In this section, a compact

formulation to determine the value of Xδ, resulting from the minimisation of the free Helmholtz

energy for systems with both chains and rings is presented. In the next section we discuss first

the case in the absence of rings, in which the well-known expressions used in the statistical

associating fluid theory (SAFT) [47,48,49] family of equations of state are recovered.

Open chain clusters

In the original TPT1 formalism of Wertheim [17,44,45,46], only association into open-chain clus-

ters is considered, so that ∆c(0) = ∆c(0)open (Equation (3.43)). Moreover, all site-site interactions

are independent of each other (as can be mathematically proven as a result of the minimisation

procedure) so that the likelihood of the bonding state of an association site is independent of


the bonding state of all other association sites [48]. TPT1’s property of independence between

sites allows one to write

Xα =∏a∈α

Xa =∏

β1,...,βM∈P(α)

Xβi , (3.55)

for any α ⊆ Γ , which includes the special case

X0 =∏a∈Γ

Xa =∏

β1,...,βM∈P(Γ )

Xβi . (3.56)

We highlight that this property is not applicable in systems where rings are formed, as the

formation of rings imply that at two sites are not independent of each other, i.e. if A and B

can bond into a ring, then XAB 6= XAXB. If no rings are allowed, we can use the independence

property to define X0 = XΓ−γXγ, and after elimination of X0 rewrite Equation (3.42) as

Q

ρ= −

∑a∈Γ

Xa +∑

γ1,...,γM∈P(Γ ),M≥2

(−1)M (M − 2)!. (3.57)

Despite the rather complicated look, the sum over the partition elements in Equation (3.57) is

equal to |Γ | − 1 [97]. After substituting in the expressions for Q (Equation (3.42)) and ∆c(0)

(Equation (3.43)), AWert (Equation (3.41)) can be rewritten as

AWert = NkT

∑a∈Γ

(lnXa −Xa + 1)− 12∑a∈Γ

∑b∈Γ

ρXa∆abXb

. (3.58)

By minimisation of the free energy with respect to the density σΓ−a = ρXa, which is equivalent

to minimisation in fractions of molecules with free site a, i.e.

∂AWert

∂Xa

∣∣∣∣∣Xb 6=a

= 0

= 1Xa

− 1−∑b∈Γ

ρ∆abXb

, (3.59)


we find the system of equations determining the site fractions:

Xa = 11 +∑

b∈Γ ρ∆abXb

, (3.60)

which collectively is known as the law of mass action. In the ideal limit, Xa = 1 for all a ∈ Γ ,

which results in limV→∞

AWert = 0. AWert is thus a residual contribution from association Aassoc

and a simple form is found by substituting Equation (3.60) (law of mass action equation) in

Equation (3.58):

Aassoc = NkT∑a∈Γ

(lnXa + 1−Xa

2

), (3.61)

which agrees with the expression for Aassoc presented previously in Equations (3.26)–(3.27) for

the residual free energy contribution due to association in open chains only.

Open chain and ring clusters

As we include the ring contribution in the fundamental graph sum, the sites are no longer

independent and the relations in Equations (3.55) and (3.56) are no longer valid. Upon

minimisation of the free energy in Equation (3.53) with respect to each Xδ for δ ⊆ Γ , the

fractions of molecules with the sites in the set δ free given is by Wertheim [17] as

Xδ = X0∑

ψ⊆Γ−δ

∑(γ1,γ2,...∈P(ψ)

)∏i

cγi , (3.62)

where the first summation is over all subsets ψ of the set Γ − δ, including the empty set ∅

for which we follow the convention that Xδ = X0 for δ = ∅. The second summation is over

partitions of ψ with elements γ. The derivation of this equation is not trivial, and we include

an alternative way to obtain it in Appendix C, different from that followed by Wertheim [17,63]

but based on the same principle of minimisation of the free energy. All the site fractions in

Equation (3.62) are re-expressed in terms of cγ’s, a new set of intensive variables, defined by

Wertheim as

cδ = ∂(∆c(0)/N)∂Xδ

, (3.63)


which in our model defines

ca = ρ∑b∈Γ

Xb∆ab (3.64)

from Equation (3.43) for |δ| = 1, and

cab =NRS,ab∑R=1

(∆ab)τRWab,RρτR−1 (Xab)τR−1 (3.65)

from Equation (3.51) for |δ| = 2 in the case of a, b being a ring-forming pair of sites (i.e.

Wab > 0). If a is not involved in ring formation, then Wab = 0 for all b ∈ Γ and all ring sizes and

cab = 0. For any δ ≥ 3, we have cδ = 0 as neither the graphs for open chains (Equation (3.43))

nor the graphas for rings (Equation (3.51)) depend on. Applying these constraints allows us

to rewrite Equation (3.62) in a simpler form where we cancelled all c−functions are cancelled

apart from ca and cab:

Xδ = X0∑(

γ1,γ2,...∈P(Γ−δ) with|γj |∈1,2

)∏j

Θ(γj) (3.66)

where

Θ(γ) =

1 + ca , if γ = a, a ∈ Γ

cab , if γ = a, b, (a, b) ∈ Γ(3.67)

The closed system of Equations (3.64)–(3.67) provide us with a route to calculate the fractions of

molecules with any subset (of Γ ) of sites free that may be of the interest for different applications,

such as the approach to reaction modelling [149]. However, for the calculation of the free energy

residual in Equation (3.27), as well as of any of its derivatives, only X0, Xa and Xab, for

a, b ∈ Γ are required. The law of mass action to calculate X0 is obtained when we set δ = ∅ in

Equation (3.66). Noting that X∅ = 1, it is written explicitly in terms of c−functions as

(X0)−1 =∑(

γ1,γ2,...∈P(Γ ) with|γj |∈1,2

)∏j

Θ(γj). (3.68)

Here, we derive expressions that are simpler than Equation (3.66) for Xa andXab by re-expressing

Equation (3.68) into contributions. In any particular partition of Γ an arbitrary site a appears


either in an element γ = a or an element γ = ab for any b different than a. The sum over

the partitions of the set of association sites Γ can thus be separated into two disjoint sums; the

sums where site a appears in terms of the form (1 + ca) and the sums where site a appears in

terms of the form cab:

(X0)−1 = (1 + ca)×∑(

γ1,γ2,...∈P(Γ−a) with|γj |∈1,2

)∏j

Θ(γj) +∑b∈Γ

cab ×∑(

γ1,γ2,...∈P(Γ−ab) with|γj |∈1,2

)∏j

Θ(γj). (3.69)

The two sums over partitions can be translated into quotients of molecular fractions over

monomer fractions using Equation (3.66) and Equation (3.69) is rewritten as

(X0)−1 = (1 + ca)Xa

X0+∑b∈Γ

cabXab

X0, (3.70)

which we multiply by X0 to obtain

Xa = 1−∑b∈Γ cabXab

1 + ca, (3.71)

for all a in the set Γ . Here,∑b∈Γ

cabXab is the fraction of molecules with site a involved in a

ring and Xaca is the conditional fraction of molecules that are not involved in open chain

formation given that they were not involved in ring formation in the first place. The sum

of these two quantities corresponds naturally to the fraction of molecules with site a bonded

given by (1−Xa) and thus Equation (3.71) corresponds to the kind of mass balances seen in

Ghonasgi’s approach [18, 121, 122]. A similar analysis is followed to find an equally friendly

expression for Xab from Equation (3.68). In any particular partition of Γ two arbitrary sites a

and b may appear in an element γ = ab, a may appear in an element γ = a, b may appear

in an element γ = b, or both a and b may appear in elements γ of size |γ| = 1. The sum

over the partitions of the set of association sites Γ can thus be separated into the sums where

sites ab appear in the term cab, the sums where site a appears in terms of the form (1 + ca)

and the sums where site b appears in terms of the form cb. However we are double counting

the partitions where a and b both appear in terms corresponding to |γ = 1|, and therefore we


discount them at the end:

(X0)−1 = cab ×∑(

γ1,γ2,...∈P(Γ−ab) with|γj |∈1,2

)∏j

Θ(γj) + (1 + ca)×∑(

γ1,γ2,...∈P(Γ−a) with|γj |∈1,2

)∏j

Θ(γj)

+ (1 + cb)×∑(

γ1,γ2,...∈P(Γ−b) with|γj |∈1,2

)∏j

Θ(γj)− (1 + ca)(1 + cb)×∑(

γ1,γ2,...∈P(Γ−ab) with|γj |∈1,2

)∏j

Θ(γj). (3.72)

Once again we use Equation (3.66) to rewrite Equation (3.72) as

(X0)−1 = cabXab

X0+ (1 + ca)

Xa

X0+ (1 + cb)

Xb

X0− (1 + ca)(1 + cb)

Xab

X0, (3.73)

which can be simplified to

Xab = 1− (1 + ca)Xa − (1 + cb)Xb

cab − (1 + ca)(1 + cb), (3.74)

for all pairs a, b in Γ . We have now reached our target expressions for the fraction of molecules

with site a free (Equation (3.71)), with sites a and b free (Equation (3.74)) and the fraction of

monomers (Equation (3.68)). Lastly, for validation sake, we substitute the law of mass equations

in Equation (3.53) to recover Equation (3.27). Substituting the relations in Equation (3.62)

back in Equation (3.53) leads to the free energy at the chemical equilibrium given by

AWert = NkT

lnX0 + 1ρ

∑δ⊆Γδ 6=∅

Xδcδ −∆c(0)

N

, (3.75)

which is a relation derived by Wertheim [17] with general validity for any definition of ∆c(0). In

our case, ∑δ⊆Γδ 6=∅

Xδcδ =∑a∈Γ

Xaca + 12∑a∈Γ

∑b∈Γ

Xabcab, (3.76)


since all cα, for α ≥ 3 are zero and

∆c(0)

N= 1

2∑a∈Γ

∑b∈Γ

ρXaXb∆ab + 12∑a∈Γ

∑b∈Γ

NRS,ab∑R=1

1τR

(∆ab)τRWab,RρτR−1 (Xab)τR . (3.77)

After inserting Equation (3.71) in Equation (3.76) and Equations (3.76) and (3.77) in Equa-

tion (3.75), we can write the free energy at chemical equilibrium in terms of molecular fractions

as inAWert =NkT

[lnX0 + 1

2∑a∈Γ

(1−Xa)

−12∑a∈Γ

∑b∈Γ

NRS,ab∑R=1

1τR


,(3.78)

which matches the expression for Aassoc we found previously in Section 3.3.4 (Equation (3.27)).

We could have used this same procedure involving Equation (3.62) in the original TPT1 (where

the formation of rings is absent and therefore cab = 0) to obtain the association residual energy

(Equation (3.61)) and respective law of mass action equation (Equation (3.60)). We did not do it

so that we could highlight the property of independence of sites (Equation (3.55)) that allowed

for the short demonstration. This property results from Equation (3.62) and, for completeness,

it is shown in Appendix A how it is only true in the association into open-chain aggregates,

failing as soon as the first ring forms.

3.5 Summary and formulation for mixtures

The residual term to account for the formation of branched chain clusters in a fluid of associating

molecules is obtained from the usual Wertheim TPT1 expression in Equation (3.61). Its

extension to mixtures of NC components is provided by Joslin et al. [51] as

Aassoc

NkT=

NC∑i=1

xi

lnXi,0 + 12∑a∈Γi

(1−Xi,a) , (3.79)

where the sum is over the NC of the mixture, xi is the component molar fraction and Xi,0 is the

fraction of monomers of component i and is given by the product of all fractions of individual

3.5. Summary and formulation for mixtures 95

sites free of component i as in Equation (3.56) which are given as

Xi,a = 11 + ρ

∑NCj=1 xj

∑b∈Γj Xj,b∆ij,ab

(3.80)

for all sites a in the site-set Γi. In the case of a fluid mixture of associating molecules taking

into account open-chain and ring aggregates, where rings are composed only of on species,

Equation (3.78) is rewritten as

Aassoc = Aassoc − limV→∞

Aassoc, (3.81)

with

Aassoc =NkTNC∑i=1

xi

lnXi,0 + 12∑a∈Γi

1−Xi,a

−∑b∈Γi

NRS,ab∑R=1

1τRρτR−1 (Xi,ab)τR (∆ii,ab)τRWi,ab,R

.(3.82)

Note that the sums are over individual sites a and b in the set Γi, rather than over sites types

as it is usually seen in literature [55]. The reason for this lies in the fact that the angle and site

location dependency of steric effects are not accounted for explicitly in this framework and it

may be the case that just some sites of a certain type are ring forming sites. The parameter W

can be approximated by the exact expression for freely jointed chains [105,127], extracted from

simulation [107,123] or estimated from experimental data. The fraction of molecules with the

sites free in the set δ is now generally given as

Xi,δ = Xi,0∑(

γ1,γ2,...∈P(Γi−δ) with|γj |∈1,2

)∏j

Θi(γj), (3.83)

with Θi(γ) defined as

Θi(γ) =

1 + ci,a , if γ = a, a ∈ Γi

ci,ab , if γ = a, b, (a, b) ∈ Γi

, (3.84)


with

ci,a = ρNC∑j=1

xj∑b∈Γj

Xj,b∆ij,ab, (3.85)

and

ci,ab =NRS,ab∑R=1

(∆ii,ab)τRWi,ab,oρτR−1 (Xi,ab)τR−1 , (3.86)

where ci,ab is only non-zero if there the sites a and b, both situated on molecule of species i,

constitute a ring forming pair. In particular, the expressions for Xi,0, Xi,a and Xi,ab required in

Equation (3.82) are rewritten as

(Xi,0)−1 =∑(

γ1,γ2,...∈P(Γi) with|γj |∈1,2

)∏j

Θi(γj), (3.87)

Xi,a = 1−∑b∈Γi ci,abXi,ab

1 + ci,a, (3.88)

and

Xi,ab = 1− (1 + ci,a)Xi,a − (1 + ci,b)Xi,b

ci,ab − (1 + ci,a)(1 + ci,b). (3.89)

Once the integrated association strength ∆ij,ab is defined, the residual association term (Equa-

tion (3.26)) can be analytically calculated. At the ideal gas level, the intermolecular potential

φinter = 0 and

limV→∞

Aassoc = NkTNC∑i=1

xi

lnX ideali,0 + 1

2∑a∈Γi

1−X ideali,a −

∑b∈Γi

X ideali,ab ∆

idealii,abWi,ab,intra

, (3.90)

where the superscript ‘ideal’ is shorthand for the the low density limit(

limV→∞

)of the variable

under it. The ideal limits for Equations (3.83)–(3.89) are given as

X ideali,δ = X ideal

i,0∑(

γ1,γ2,...∈P(Γi−δ) with|γj |∈1,2

)∏j

Θideali (γj), (3.91)


Θideali (γ) =

1 , if γ = a, a ∈ Γi

cideali,ab , if γ = a, b, (a, b) ∈ Γi

. (3.92)

cideali,a = 0, (3.93)

cideali,ab = ∆ideal

ii,abWi,ab,intra, (3.94)

where Wi,ab,intra is the W parameter correspondent to the intramolecular ring between sites a

and b of component i, as only the rings of intramolecular type can be present in the ideal gas,

(X ideali,0 )−1 =

∑(γ1,γ2,...∈P(Γi) with|γj |∈1,2

)∏j

Θideali (γj), (3.95)

X ideali,a = 1−

∑b∈Γi

cideali,ab X

ideali,ab , (3.96)

and

X ideali,ab =

1−X ideali,a −X ideal

i,b

cideali,ab − 1 . (3.97)


Wertheim’s TPT1 is extended to a general formalism that allows for the consideration of

additional types of association clusters, given an expression for the number of aggregates and the

definition of the fundamental graph sum. The main focus is the addition of ring-like aggregates

to the open-chain aggregates in TPT1: intramolecular rings, promoted by conformational

preferences or flexibility of a molecule, that bring two non-contiguous electronegative molecular

groups capable of hydrogen bonding to an association distance in the right orientation; and

intermolecular rings, promoted by an hydrogen bond between two molecules of an open-chain

aggregate. The extension requires two new parameters per ring type: the ring size τ and

a W parameter. The parameter W captures the probability of two segments of the same

molecule or chain aggregate carrying a given pair of ring-forming sites meeting each other to

enable the formation of a ring. In a first approximation, W is considered to be independent of


density, temperature and composition, i.e., it is specific for a pair of sites ab in a component i

(single-component rings) capable of associating to form a ring of a given size τR ≥ 1: Wi,ab,τR .

In a system with ring formation the amount of linear chain aggregates is being overestimated

without the extended theory. The formation of rings of both inter- and intramolecular nature

implies a decrease in the association between two different components and as such is expected to

impact the phase equilibria curves. The extended theory is expected to improve the representation

of associating systems that exhibit ring formation. In the next Chapters the extent of the impact

of the theoretical extension on the results of the calculations of thermo-physical properties is

assessed. In Chapter 4 the focus is placed on the formation of intermolecular rings on model

systems of spheres and in Chapter 5 the formation of intramolecular rings on model systems of

chain molecules and real mixtures is studied. The decision of having one chapter dedicated to

each type of rings was to facilitate the understanding of the isolated effects of the consideration

of each ring type.

Chapter 4

INTERMOLECULAR RINGS IN

PURE SYSTEMS

In this Chapter, the newly developed theory is illustrated with its application on model systems

at phase coexistence. The calculations are carried out with the EOS statistical associating fluid

theory (SAFT) for potentials of variable range for a square-well fluid, i.e. SAFT-VR SW EOS,

modified with the new association term, which is introduced at the beginning. Next, the theory

for two sites is validated using the “sticky limit” in a two-site fluid with only rings as association

aggregates. The end-to-end distribution function for freely jointed chains as well as a small

correction are described, to approximate the parameter W for fully flexible chains. Using this

approximation for the W parameter, the influence of association energy and number of sites in

the formation of intermolecular rings of two sizes and in coexistence properties is studied. Note

that no intramolecular rings are considered in this Chapter.

4.1 Introduction

A ring formed by intermolecular hydrogen bonding (InterMHB) happens when a hydrogen

bond is formed between two molecules of an open-chain aggregate. An example of a tetramer

cycle is given in Figure 4.1.1 for methanol. The intermolecular rings may be present in systems

99

100 Chapter 4. INTERMOLECULAR RINGS IN PURE SYSTEMS

Figure 4.1.1: Methanol intermolecular ring of size τ = 4.

where usual hydrogen bonding takes place, i.e. in compounds with the group X− H in which X is

more electronegative than H (examples of X are the atoms O, S, F and N). For alcohols, evidence

for significant formation of rings has been found using a range of methods: quantum mechanical

density functional theory (DFT) [150,151], ab initio calculations [152,153], Fourier-transform

infrared spectroscopy [153], molecular dynamics [154], Monte Carlo simulations and X-ray

scattering experiments [155]. Hydrogen fluoride has also been reported to exhibit relevant ring

formation, especially in the vapour phase [108, 156]. The distribution of association clusters has

a large impact on the physical properties of fluids, and therefore, the nature of hydrogen-bonds

can sometimes be inferred indirectly from their profound influence on the properties of fluids.

Sear and Jackson extended TPT1 to account for InterMHB rings [105], as described in

Section 2.4.1, which is a special case of the theory developed in Chapter 3, corresponding to

a single component with two sites and one allowed ring size. The theory was applied by the

authors to a spherical model fluid and by Galindo et al. [108] to pure hydrogen fluoride. Sear

and Jackson [105] observed that the formation of rings of size τ = 4 impacts the calculations

the most and that the equilibrium between rings and open chains is shifted towards the rings

for higher association energies. They have also found that the vapour pressure is increased by

allowing for ring formation and that only at near-critical temperatures the fraction of rings

in the liquid phase be higher than in the saturated vapour phase. Galindo et al. [108] used

SAFT-VR SW [87] in the calculation of pure properties of hydrogen fluoride and could reproduce

the maximum in the vapourisation enthalpy vs temperature diagram only when the competition

between intermolecular rings of size τ = 4 and open-chain aggregates was considered.

4.2. SAFT-VR square-well equation of state 101

In this Chapter, the newly developed theoretical approach is used in conjunction with the

SAFT-VR SW EOS and validated by applying the sticky limit to an intermolecular ring. The

EOS used in the calculations is described next, prior to the presentation and discussion of the

results.

4.2 SAFT-VR square-well equation of state

Gil-Villegas et al. [87, 157] developed the SAFT-VR approach to describe chain molecules of

hard core monomers with an arbitrary attractive potential of variable range, such as the square

well, Yukawa or Sutherland potentials. The variable range feature implied the introduction of a

new parameter for the intermolecular potential range, λ. In this model, two segments of species

i and j interact via a square-well potential of variable attractive range given as

φ(ij) =

∞ , rij < σij,

−εdispij , σij ≤ rij < λσij,

0 , else.

(4.1)

This EOS has the typical structure of four contributions, as explored in Section 2.3 (Equa-

tion (2.93)), and the main features are the treatment of the monomer term and the molecular

model. Following the Barker and Henderson perturbation theory [54] up to second order, in

SAFT-VR SW, the segment Helmholtz free energy (amono) is expressed as a series expansion in

the inverse powers of the temperature as in

amono = aHS + adisp,

= aHS + a1/kT + a2/(kT )2,

(4.2)

assuming a hard sphere system as the reference with attractive terms as perturbations. The first

term of the expansion (a1) corresponds to the monomer-monomer attractive part of the potential

energy averaged over the reference fluid structure through the distribution function gHS (hard-


sphere), and the second (a2), to the fluctuations of the attractive energy as a consequence of the

compression of the fluid due to the effects of the attraction well, using the local compressibility

approximation [54]. A first order high-temperature expansion is used to express the radial

distribution function (gmono(σ)) [54, 87], for components i and j:

gmonoij (σij) = gHS

ij (σij) + g1(σij)εdispij /kT (4.3)

In the extension of the theory to mixtures, the Barker and Henderson perturbation theory

is combined with the free energy of the hard sphere mixture reference by Boublík [158] and

Mansoori et al. [159].

The molecular model consists of a homonuclear chain of tangentially bonded spherical segments

corresponding to united atom models. The number of segments in the chain is represented by m

which does not have to be an integer, as it simply provides an indication of the non-sphericity of

the molecule. The molecules may contain square-well associating sites that mediate directional

and short-range interactions, which are captured in the association term (Aassoc). We use the

SAFT-VR SW in all calculations presented in this chapter with the newly developed association

term summarised in Section 3.5.

4.3 Validation - the sticky limit of a ring cluster

The framework presented can also be used to obtain a free energy expression for the formation

of covalent bonds in a chain, Achain. This was an observation made by Wertheim [102] and it is

useful in the validation of our theoretical framework. Consider a fluid ternary mixture, where

one component has two sites A and B, other component has only site A and the last component

has only site B. The quantity of the three components exist in stoichiometric proportions

consistent with Figure 4.3.1. Upon increase of the association energy to infinity and decrease

of the association range to 0, the limit of irreversible bonding, also known as “sticky limit”, is

4.3. Validation - the sticky limit of a ring cluster 103

1 m...2

σ

1 ...2 mrc → 0εHB →∞

Figure 4.3.1: Chain molecule composed of m segments with a distance between segments of σ.

realised. The resulting expression for a chain of m tangentially bonded segments is [47,160]

Achain = NkT [−(m− 1) ln(gref(σ))] , (4.4)

where gref(σ) is the pair distribution function of the reference non-bonded fluid at contact, and

it is noted that there are m− 1 links in an m–segments chain, and the formation of a covalent

bond results in a change of − ln(gref(σ)) in the residual free energy. To verify our theoretical

framework, we proceed to extract the contribution of the formation of a covalent bond in an

associating fluid with two sites where only rings of size τ are formed (no open-chain aggregates

or rings of different size are considered).

4.3.1 Residual Helmholtz free energy from the formation of ring

only aggregates

We start by revisiting the method to predict the form of the free Helmholtz energy at equilibrium

in a fluid of two-site molecules with ring formation, given by Equation (3.9). Only rings of size

τ are considered. The number of species is now given by

Nspecies = N − (τ − 1)×Nτ , (4.5)

where Nτ is the number of rings of size τ ≥ 1 given by

Nτ = ξτN

τ, (4.6)


with ξτ as the fraction of molecules in rings of size τ given by Equation (3.52):

ξτ = ∆τWρτ−1 (X0)τ ,

= (1−X0),(4.7)

as a molecule is either bonded in a ring or free. Inserting Equation (4.6) in Equation (4.5), the

number of species is thus rewritten as

Nspecies = N

[1− (τ − 1)

τ× ξτ

]. (4.8)

The association strength ∆, given in Equation (3.44), can be approximated as [48]

∆ = KFgref(σ), (4.9)

where K is the bonding volume and F =[exp(εHB/kT )− 1

]for the case of a square-well site-site

interaction. It thus follows from Equation (3.9) that the free energy is given by

A = NkT[ln (ρ0Υ (T ))− 1 + τ − 1

τ(1−X0)

]. (4.10)

The residual contribution due to association is obtained upon subtraction of the ideal contribution

that comprises the reference non-bonded system (Equation (3.11)) and the limit of low density

at constant N (i.e. V →∞) contribution:


Aassoc, (4.11)

where

Aassoc = NkT[lnX0 +

(1− 1

τ

)(1−X0)

], (4.12)

where X0 is given by Equation (4.7). Another way to obtain this expression would be considering

cA = cB = 0 in Equations (3.68) and (3.75).


4.3.2 The “sticky limit” in a fluid of rings formed by intermolecular

hydrogen bonding only

Consider a first system (1) consisting of the fluid described in Section 4.3.1. The fluid consists

of N(1) two-site molecules with formation of associated rings of size τ only. The molecules

are formed by m ≥ 1 segments and an exemplifying representation is found in Figure 4.3.2.

Considering a monomer non-associating reference fluid, the association residual in system (1) is

given by Equations (4.10) and (4.11) and the chain residual contribution in system (1) is given

by Equation (4.4) as

Achain(1) = N(1)kT [−(m− 1) ln(gref(σ))] . (4.13)

In the sticky limit of a system, the association sites become infinitely attractive points at

the surface of the segments, the contact points, and the residual properties of system (1)

must be equal to those of a second system (2) composed of N(2) = N(1)/τ rings with mτ

segments that are covalently bonded and non–associating, i.e. τAchain(1) + lim

stickyAassoc

(1) = Achain(2) .

The compressibility factor contribution from association for the first system Zassoc(1) is obtained

rc → 0εHB →∞

System 1 System 2

Figure 4.3.2: The molecules are composed of m = 2 segments and have a set of sites Γ = A,Bthat can only promote the formation of rings of size τ = 4. Increasing the association energy toinfinity and reducing association range to zero results in irreversible bonding. Note that thenumber of segments composing the chain molecule m and the ring size τ are arbitrary.

as the density derivative of the free energy for the formation of ring aggregates only (Aassoc(1) from

Equation (4.11)),

Zassoc(1) = ρ(1)

∂(Aassoc

(1) /N(1)kT)

∂ρ(1)

∣∣∣∣∣∣T

,

= ρ(1)∂X0

∂ρ(1)

( 1X0− 1 + 1

τ

),

(4.14)


where the density of system (1) is given by ρ(1) = N(1)/V . Using Equation (4.7) we can write

the terms ∂X0/∂ρ(1) and X0 as

∂X0

∂ρ(1)= −

(ρ(1)

)τ−1(X0)τ ∆τW

((τ − 1)

(ρ(1)

)−1+ τ

(∂∆/∂ρ(1)

)∆−1

)1 + τ

(ρ(1)

)τ−1(X0)τ−1 ∆τW

, (4.15)

and

X0 = 11 +

(ρ(1)

)τ−1Xτ−1

0 ∆τW, (4.16)

and substitute them back in Equation (4.14) to obtain

Zassoc(1) = −

(ρ(1)

)τ(X0)τ ∆τW

τ

((τ − 1)

(ρ(1)

)−1+ τ

∂∆

∂ρ(1)∆−1

). (4.17)

In the limit of complete bonding, the molecules of system (1) become part of (τ ×m)–segment

rings where the segments are covalently and tangentially bonded at r12 = σ. The chain

contribution to the compressibility factor in system (2) is written as

Zchain(2) = lim

εHB→∞rc→0

Zassoc(1) + τZchain

(1) , (4.18)

where

Zchain(1) = ρ(1)

∂(Achain

(1) /(N(1)kT ))

∂ρ(1)

∣∣∣∣∣∣T

,

= −(m− 1)ρ(1)∂ ln gref(σ)∂ρ(1)

.

(4.19)

From the observation of the law of mass action Equation (4.16) we can deduce that as the

bonding strength tends to infinity, the fraction of all sites free tends to 0 and ∆ tends to

infinity. Since the only term in ∆ dependent on density is gref , the chain contribution to the

compressibility factor is written as

Zchain(2) = 1− τ

(1 + ρ(2)

∂ ln gref(σ)∂ρ(2)

)+ τZchain

(1) , (4.20)


where the density of system (2) is given by ρ(2) = N(2)/V and the limit

limεHB→∞rc→0

Xτ0∆

τ = 1(ρ(1)

)τ−1W, (4.21)

was used. The residual contribution from the formation of the covalent links which is obtained

as:Zchain

(2) =Zchain(2) − lim

V→∞Zchain

(2) ,

=− τρ(2)∂ ln gref(σ)∂ρ(2)

− τ(m− 1)ρ(1)∂ ln gref(σ)∂ρ(1)

,

(4.22)

and since ρ(1) = ρ(2)/τ , it can be rewritten as

Zchain(2) = −τρ(2)

∂ ln gref(σ)∂ρ(2)

− τ(m− 1)ρ(2)∂ ln gref(σ)∂ρ(2)

. (4.23)

Lastly, the residual contribution of system (2) to the free energy is given by integration of Zchain(2)

in density as

Achain(2) = N(2)kT [−(τ ×m) ln gref(σ)] , (4.24)

which is consistent with the chain contribution in Equation (4.4) as there are (τ ×m) links in a

ring composed of τ molecules of m segments and the formation of a covalent bond results in a

change of − ln(gref(σ)) in the free energy. We note that the result obtained in Equation (4.24)

is the same as if we had considered the parameter W to be a function of density. Considering

other association clusters that a ring is also possible, but it would complicate the mathematics.

The general result for any cluster considered is Achain = −(“number of links”)× ln(gref(σ)).

We now investigate the “sticky limit” for a spherical fluid with two sites (A and B) that can

form intermolecular rings of size τ = 6 and no other aggregates. Not all versions of SAFT

allow for this test because of the way bond length is defined in a soft potential in the chain

and association terms, but SAFT-VR SW does. The SAFT-VR SW approach [87] was used in

the calculation of the ideal and reference contributions together with the association residual

obtained in the previous section (Equations (4.7), (4.11) and (4.12)). Calculations of coexistence

packing fractions were carried out at various increasing association energies εHB until a limit


behaviour (“sticky limit”) was reached. The coexistence packing fractions at the limit at

complete association were found to match those respective to the hexamer covalent cycle, as

given by Equation (4.24), where m = 1 and τ = 6. The curves obtained were plotted in a

diagram that is presented in Figure 4.3.3.

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

02351020εHB/εSW=

hexamer cycle

T∗

η

Figure 4.3.3: Description of the coexistence packing fractions as a function of reduced temperaturefor a two-site spherical model that can only associate in intermolecular rings of size τ = 6. Thesymbols represent the critical points and the continuous curves the calculations with the theory:non-associating spheres in red, increasing association energy from bottom to top and covalenthexamer cycle in orange at the top.

4.4 The end-to-end-distribution function

A chain molecule can assume different configurations and in order to estimate the distance

between two segments, we may turn to statistics and consider the average over all configurations.

The end-to-end vector r1m is the sum of all bond vectors in a chain of m segments and is given

as [119]

r1m =m−1∑i=1ri,i+1, (4.25)

where ri,i+1 is the bond vector between particles in position (i) and (i+ 1) of the chain. The

distribution of the vector between particles in positions (i) and (j) is given by Wij(r), and thus

4.4. The end-to-end-distribution function 109

Wij(r) dr is the probability of particle in position (j) being located within the volume dr that

is situated at the end of the vector r starting in position (i). In order to capture the probability

of two sites that are part of the same molecule or cluster, separated by n links, being within

bonding distance, Sear and Jackson [105,106] considered the freely jointed chain (Figure 4.4.1),

also known as random flight, for which Treloar [118,119] developed a closed form expression

for the end-to-end distribution function Wn(r) (which is an alternative notation for W1m(r)).

A freely jointed chain consists of a linear model of n = m − 1 bonds of one specified fixed

r1m

r12

Figure 4.4.1: Possible conformation of a freely-jointed chain of 30 links (m = 31) or in otherwords, a random walk of 30 steps.

length and free rotation at each junction between bonds. Its nature implies that the directions

of any pair of bonds is uncorrelated and thus allows for overlap of particles. The end-to-end

distribution function is given as [119]

Wn(r) = n(n− 1)8πrσ2

k∑j=0

(−1)jj!(n− j)!

[n− (r/σ)− 2j

2

]n−2

, (4.26)

where r = |r| and k is the integer constrained by the condition

n− (r/σ)2 − 1 ≤ k <

n− (r/σ)2 . (4.27)

The probability density is normalised over all distances between end segments:

∫ ∞0

Wn(r)4πr2 dr = 1. (4.28)

However, r can only be any value between contact (r = σ) and a maximum value corresponding

to a fully stretched chain (r = nσ). A correction is proposed by Sear and Jackson [106] to


prevent the overlaping of the end segments (r = [0, σ]). The rescaled probability density Qn(r)

is thus given by ∫ nσ

σQn(r)4πr2 dr = 1, (4.29)

with

Qn(r) = cWn(r), (4.30)

where c ∈ IR is the rescaling constant. The normalising factor c is not dependent on σ as can

be easily seen by doing the variable substitution u = r/σ in Equation (4.29), and thus we will

assume σ = 1. We proceed with the integration in order to determine c; the integral is in

the form of a sum with each term corresponding to a range of r for which the parameter k is

constant:

1c

=n(n− 1)2

∫ 5+(−1)n+1

2

1

kmax∑j=0

(−1)jrj!(n− j)!

[n− r − 2j

2

]n−2dr + · · ·

+∫ (n−2)

(n−4)

k=1∑j=0

(−1)jrj!(n− j)!

[n− r − 2j

2

]n−2dr +

∫ n

(n−2)

r

n!

[n− r

2

]n−2dr

, (4.31)

with kmax being the integer constrained by the condition

n

2 − 2 < kmax ≤n

2 − 1. (4.32)

The first term in the sum consists of the integral with limits 1 and either 2 or 3, depending

if n, the number of links, is an even or an odd number, respectively, and it corresponds to a

maximum value of the parameter k (kmax). The number of terms in the sum is given as kmax + 1.

The terms following the first correspond to decreasing values of k and the respective limits of

integration increase by 2 until the last term in the sum which corresponds to k = 0 and has

limits r = [n − 2, n]. The terms thus cover the whole range of possible values for r. After

4.4. The end-to-end-distribution function 111

solving the integrals, Equation (4.31) can be rewritten as

1c

=n(n− 1)2

kmax∑j=0

(−1)jj!(n− j)!

×[

22−n

n(n− 1) (−2j + n− r)n−1 (2j − n(r + 1) + r)] 5+(−1)n+1

2

1+ . . .

+k=1∑j=0

(−1)jj!(n− j)!

[22−n

n(n− 1) (−2j + n− r)n−1 (2j − n(r + 1) + r)](n−2)

(n−4)

+ 1n!

[22−n

n(n− 1) (n− r)n−1 (−n(r + 1) + r)]n

(n−2)

, (4.33)

where the last term can be simplified to

1n!

[22−n

n(n− 1) (n− r)n−1 (−n(r + 1) + r)]n

(n−2)= 1n!

2n(n− 1) [n(n− 1)− (n− 2)] , (4.34)

concluding the expression for the constant c in terms of the length of the chain n only. This

correction factor is only significant for shorter chains, as can be seen in Table 4.1. Given the

Table 4.1: Calculated correction factors and contact values for the corrected end-to-end distri-bution function for chains of 2 to 10 links.chainlength (n)

2 3 4 5 6 7 8 9 10

c 1.333 1.200 1.136 1.101 1.078 1.064 1.053 1.045 1.039σ3cWn×102 5.31 4.77 2.83 2.19 1.72 1.41 1.18 1.01 0.877

small range typically considered for the hydrogen bond (rcab), in order for a bond to occur, the

two segments decorated with the associating sites need to be at an inter-segmental distance

very close to σ. Neglecting the variation of Wn within the bonding volume, i.e. the volume of

overlap of the bonding sites, the end-to-end distribution function (Equation (4.26)) at contact

(r = σ) can thus be written as

Wn(σ) = n(n− 1)8πσ3

k∑j=0

(−1)jj!(n− j)!

[n− 1− 2j

2

]n−2, (4.35)


withn− 3

2 ≤ k <n− 1

2 . (4.36)

4.5 Effect of intermolecular ring formation

Consider a fluid of spheres with two (Γ = A,B) or three (Γ = A,A,B) association sites

that promote hydrogen bonding into linear (both two- and three-site models) or branched (only

the three-site model) chain aggregates and intermolecular rings of size τ with the parameter

W = Wτ−1 given by the end-to-end distribution function at contact. The new inputs to the

extended association energy are thus the ring sizes allowed. Only A−B association is permitted,

i.e., εHBAA = εHB

BB = 0. The aggregates existing in the fluid with two sites are pictured in Figure 4.5.1

and the aggregates in the three-site model fluid in Figure 4.5.2. The bonding sites are assumed

(a) Linear chain aggregate. (b) intermolecular ring of 4molecules.

(c) intermolecular ring of 6 molecules.

Figure 4.5.1: Types of association aggregates in a two-site sphere: a) linear chain aggregate; b)ring by InterMHB with τ = 4; and c) ring by InterMHB with τ = 6.

to be off-centre by a distance rd/σ, and have a square-well cut-off range of rcAB/σ. Additionally

the spheres interact through a square-well potential of depth εdisp and range λ. The model

system was inspired in the SAFT-VR SW model for hydrogen fluoride in literature [108], since it

is established that this compound exhibits intermolecular ring formation [161,162], particularly

in the vapour phase. The parameters used in the calculations are the ones presented in Table 4.2,

unless said otherwise. All calculations presented in this section use the SAFT-VR SW EOS in

combination with a gPROMS solver [163]. The reduced units used in the calculations are the

4.5. Effect of intermolecular ring formation 113

(a) Branched chain aggregate. (b) intermolecular ring of 4molecules.

(c) intermolecular ring of 6 molecules.

Figure 4.5.2: Types of association aggregates in a three-site sphere (AAB): a) linear/branchedchain aggregate; b) ring by InterMHB with τ = 4 with possible branching; and c) ring byInterMHB with τ = 6 with possible branching.

Table 4.2: Parameters for the square-well spheres of the model fluids.λ εHB

AB/εdisp rc

AB/σ rd/σ

1.8 20 1 0.25

reduced temperature T ∗ = kT/εdisp, the packing fraction η = (π/6)σ3/V , the reduced volume

V ∗ = V/σ3 and the reduced vapourisation enthalpy ∆hvap,∗ = ∆hvap/(NkT ), where ∆hvap is

the vapourisation enthalpy.

4.5.1 Impact of the ring size

The effect of the number of molecules forming the ring (τ) was evaluated by testing three

scenarios for the association aggregates of the two-site model fluid:

• scenario 0 – linear chains (Figure 4.5.1a);

• scenario 1 – linear chains (Figure 4.5.1a) and rings of 4 molecules (Figure 4.5.1b);

• scenario 2 – linear chains (Figure 4.5.1a) and rings of 6 molecules (Figure 4.5.1c).

In Figures 4.5.3 and 4.5.4 we plot the evolution of the fractions of bonding states in each

association scenario for a non-saturated liquid and a non-saturated vapour, respectively. The

pink and blue curves represent the bonding fractions in scenario 1 and scenario 2, respectively.

Despite the high association energy that reduced the fraction of free monomers in the liquid


0

0.2

0.4

0.6

0.8

1

2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

2 2.5 3 3.5 4 4.5 5

ξ τ;f

open

T ∗

(a) Fractions of bonding states of a two-sitemolecule at η = 0.384 in the liquid phase.

0.35 0.4 0.45 0.5 0.55 0.6 0.65

ξ τ;f

open

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

0.35 0.4 0.45 0.5 0.55 0.6 0.65η

(b) Fractions of bonding states of a two-sitemolecule at T ∗ = 3 in the liquid phase.

Figure 4.5.3: Fractions of molecular bonding states of an associating fluid of two-site sphericalmolecules in the liquid phase. The three scenarios are represented with the curves in blackfor scenario 0, pink for scenario 1 and blue for scenario 2. The fractions of molecules forminglinear aggregates, fopen, for each scenario are represented by dashed curves and the fractions ofmolecules forming intermolecular rings of size 4, ξ4 (scenario 1) and of size 6, ξ6 (scenario 2) bycontinuous curves. These curves correspond to SAFT-VR SW predictions.

phase to zero, it is possible to observe from Figures 4.5.3 and 4.5.4 that both chain aggregates

(dashed curves) and ring fraction (continuous curves) are favoured by low temperatures and high

densities. Even at very high association energy, the rings are present in a very small, almost

negligible, fraction in the liquid phase. In the vapour phase (Figure 4.5.4), for scenario 1, the

ring form is favoured over the chain aggregate at low temperatures and high densities, but the

opposite is observed at high temperatures and low densities. In this study, rings of size 4 are

much more abundant than rings of size 6. This is in agreement with past literature that suggests

that the extent of ring formation is larger for smaller ring size τ [105,108].

4.5.2 Impact of the association energy

Next, we present predictions of properties of the coexisting vapour and liquid of a pure fluid.

To this effect the equilibrium criteria must be verified and are given by:

T L = TV = T, (4.37a)

P (T, V L, nL) = P (T, V V, nV), (4.37b)


2 2.5 3 3.5 40

0.050.10.150.20.250.30.350.4

00.050.10.150.20.250.30.350.4

2 2.5 3 3.5 4

ξ τ;f

open

T ∗

(a) Fractions of bonding states of a two-site moleculeat η = 3.5× 10−6 in the vapour phase.

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.40

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

ξ τ;f

open

η × 103

(b) Fractions of bonding states of a two-site moleculeat T ∗ = 3 in the vapour phase.

Figure 4.5.4: Fractions of molecular bonding states of an associating fluid of two-site sphericalmolecules in the vapour phase. The three scenarios are represented with the curves in blackfor scenario 0, pink for scenario 1 and blue for scenario 2. The fractions of molecules forminglinear aggregates, fopen, for each scenario are represented by dashed curves and the fractions ofmolecules forming intermolecular rings of size 4, ξ4 (scenario 1) and of size 6, ξ6 (scenario 2) bycontinuous curves. These curves correspond to SAFT-VR SW predictions.

µ(T, V L, nL) = µ(T, V V, nV), (4.37c)

where the superscripts L and V refer to the liquid and vapour phases, respectively, and nα is the

number of moles in phase α. Here, µ, the chemical potential, and other relevant properties, are

obtained from derivatives of the Helmholtz free energy given by SAFT-VR SW theory modified

to account for ring formation, according to thermodynamic relations such as

µ(T, V, n) = ∂A

∂n

∣∣∣∣∣T,V

, P (T, V, n) = −∂A∂V

∣∣∣∣∣T,n

and S(T, V, n) = −∂A∂T

∣∣∣∣∣V,n

.

The effect of the formation of rings in the vapour–liquid envelope at increasing association

energy was evaluated by testing two scenarios for the association aggregates of the two-site

model fluid:

• scenario 0 – linear chains (Figure 4.5.1a);

• scenario 1 – linear chains (Figure 4.5.1a) and rings of 4 molecules (Figure 4.5.1b).

The results for scenario 0 are presented in Figure 4.5.5 and those for scenario 1 in Figure 4.5.6.

At high association energies (εHB/εdisp ≥ 10), when the formation of rings is significant, the


1

2

3

4

5

6

7

8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

02102050100200εHB/εSW=

T∗

η

Figure 4.5.5: Phase diagrams T ∗ versus η for an associating fluid of two-site spherical moleculesthat can only form linear aggregates (scenario 0) at various association energies.

1

2

3

4

5

6

7

8

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

02102050100200εHB/εSW=

T∗

η

Figure 4.5.6: Phase diagrams T ∗ versus η for an associating fluid of two-site spherical moleculesthat can form both linear aggregates and intermolecular rings of size 4 (scenario 1) at variousassociation energies.


branch of the saturated vapour is shifted to the right and the critical point is decreased. This

effect is more and more marked the higher the association energy is, which is a consequence of

the increase in ring fraction. The evolution of the fractions of molecules in ring form and in

chain aggregates along the saturation lines of Figure 4.5.6 can be observed in Figure 4.5.7.

00.10.20.30.40.50.60.70.80.91

0.05 0.1 0.15 0.2 0.25 0.3

22050

ξ 4;f

open

η

εHB/εSW=

(a) Fractions of molecules in linear aggregates (sce-nario 0) for an associating fluid of two-site spheres forthree association energies.

00.10.20.30.40.50.60.70.80.91

0.05 0.1 0.15 0.2 0.25 0.3ξ 4

;fop

enη

22050

εHB/εSW=

(b) Fractions of molecules in linear aggregates (dashed)and in rings of size 4 (continuous) for an associatingfluid of two-site spheres for three association energies.

Figure 4.5.7: Fractions of molecular bonding states bonded in the saturated liquid and vapourphases (red points correspond to the critical points) for an associating fluid of two-site sphericalmolecules in a) scenario 0, and b) scenario 1. The colours correspond to the different associationenergies.

In Figure 4.5.7a, the fractions of molecules bonded in the scenario 0 (absence of ring formation)

for three energies along the saturation curve are represented. Similarly, in Figure 4.5.7b, the

fractions of molecules in chain aggregates (dashed curve) are represented alongside the fraction

of molecules in ring form (continuous curve). At εHB/εdisp = 2 the fraction of rings is negligible,

but it increases with the association energy, especially at low packing fractions (vapour phase).

The higher the association energy, the quicker the fraction of molecules in rings increases with

decreasing density.

4.5.3 Two-site model versus three-site model

Lastly, we studied the effect of the number of association sites in the formation of rings in

saturation properties by testing four scenarios for the association aggregates of the two-site and

the three-site model fluids:


• scenario 0 – linear chains (Figures 4.5.1a and 4.5.2a);

• scenario 1 – linear chains (Figures 4.5.1a and 4.5.2a) and rings of 4 molecules (Figures 4.5.1b

and 4.5.2b);

• scenario 2 – linear chains (Figures 4.5.1a and 4.5.2a) and rings of 6 molecules (Figures 4.5.1c

and 4.5.2c);

• scenario 3 – linear chains (Figures 4.5.1a and 4.5.2a), rings of 4 molecules (Figures 4.5.1b

and 4.5.2b), and rings of 6 molecules (Figures 4.5.1c and 4.5.2c).

In Figure 4.5.8, the the coexixtence packing fractions and vapourisation enthalpies for the four

scenarios are represented for the associating fluid of two-site spherical molecules in a) and c)

and for the three-site in b) and d).

0 100 200 300 400 500 600 700 8002.5

3

3.5

4

4.5

5

2.5

3

3.5

4

4.5

5

0 100 200 300 400 500 600 700 800

T∗

V ∗

(a) Coexistence volumes in a fluid with two associationsites (Γ = A,B). for the four scenarios.

0 100 200 300 400 500 600 700 8003.5

4

4.5

5

5.5

6

3.5

4

4.5

5

5.5

6

0 100 200 300 400 500 600 700 800

T∗

V ∗

(b) Coexistence volumes in a fluid with three associa-tion sites (Γ = 2A,B). for the four scenarios.

2.5 3 3.5 4 4.5 5024681012

024681012

2.5 3 3.5 4 4.5 5

∆h

vap,∗

T ∗

(c) Enthalpies of vapourisation in a fluid with twoassociation sites (Γ = A,B) for the four scenarios.

2.5 3 3.5 4 4.5 5024681012

024681012

2.5 3 3.5 4 4.5 5

∆h

vap,∗

T ∗

(d) Enthalpies of vapourisation in a fluid with threeassociation sites (Γ = 2A,B) for the four scenarios.

Figure 4.5.8: Coexistence volumes for a a) two-site, and a b) three-site model, and vapourisationenthalpies for a c) two-site, and a d) three-site model. The curves correspond to an associatingfluid of spherical molecules in scenario 0 (black), in scenario 1 (pink), in scenario 2 (blue) andin scenario 3 (green).


As can be seen from Figure 4.5.8, very little difference can be observed between scenario 0 (black

curve) and scenario 2 (blue curve) and between scenario 1 (pink curve) and scenario 3 (green

curve) due to the insignificant fraction of rings of size 6. What is also evident is that the impact

of ring formation is reduced with the increase of association sites. The case of a water-type

model with four sites (Γ = A,A,B,B) was also tested but it is not shown here. In the water

case the curves were coinciding with each other, i.e., no significant difference between the new

theory and the standard TPT1 was observed at εHBAB/ε

disp.


The new theory described in Chapter 3 was implemented and the impact of the formation of

intermolecular rings with τ1 = 4 and τ2 = 6 in addition to linear and branched chains in the

phase behaviour of fluids of spherical molecules with two and three association sites was studied.

The study covered both pure single phase and vapour–liquid coexistence properties. After the

parameter W = Wτ−1 was approximated by the end-to-end distribution function at contact, the

only new inputs after the extension of the theory were the size of the rings forming.

The effects of the competition between ring and chain formation by hydrogen bonding were

in accordance with the past literature [105,108]. The formation of rings decreases the critical

temperature, increases the packing fraction of the saturated vapour phase and increases the

vapour pressure across all temperatures. The rings are most abundant in the vapour phase,

but unless the energy of association is very high (εHB/εdisp) ≥ 10, and the model only has two

sites, no significant impact in the fluid properties is expected. The effect of ring formation

in systems of three or more bonding sites showed to be negligible. This effect was expected

as Ghonasgi and Chapman had already showed with a study on associating fluids with four

bonding sites, [164], that Wertheim’s theory without accounting for rings matches with high

precision the simulation results for these same systems. The properties prediction for a four-site

model of water was not improved by using the new theory. Using a fitted W instead of the

end-to-end distribution function and including 3D clusters (e.g. prism [165]) in the fundamental


graph sum to try to capture the network nature of water could potentially yield better property

prediction. Moreover, it is expected that the theory’s representation of the number of HBs

and ring formation in water molecules can benefit from a theoretical extension accounting for

cooperativity effects [165,166].

It is expected to observe a larger effect of the ring formation in mixtures, since the degree

of self-association of a ring-forming compound is increased and consequently the degree of

cross-association with the other compounds in the mixture is decreased. Mixtures are thus

covered in the next Chapter for the intramolecular hydrogen bonding case.

Chapter 5

INTRAMOLECULAR HB IN

MIXTURES AND APPLICATIONS

IN SLE

The effect of temperature and density on the formation of intramolecular rings and subsequent

effect on phase equilibria is the central focus of this Chapter. Additionally, the impact of

the heat capacity term in solubility calculations is assessed, prior to the study of the effect

of intramolecular rings in solubility curves. The calculations are carried out using either the

SAFT-VR Mie EOS or the SAFT-γ Mie EOS (both described in this Chapter). The Chapter

closes after the analysis of a case study: the solubility of two statins (simvastatin and lovastatin)

in simple alcohols and the influence of the extent of the ring form of these compounds on their

solid–liquid behaviour. Note that no intermolecular rings are considered in this Chapter.

5.1 Introduction

In a bulk solution, intermolecular hydrogen bonding (InterMHB) can lead to self-association

aggregates, if it occurs between molecules of the same components, or cross-association aggregates,

if it occurs between molecules of different components. Differently, an intramolecular hydrogen

121

122 Chapter 5. INTRAMOLECULAR HB IN MIXTURES AND APPLICATIONS IN SLE

bond is that occurring within one single molecule and leads to the formation of a ring. A

ring formed by intramolecular hydrogen bonding (IntraMHB) is promoted by conformational

preferences or flexibility of a molecule that bring two non-contiguous electronegative atoms, one

of them connected to a hydrogen, to an association distance and orientation. An example of an

intramolecular ring is given in Figure 5.1.1 for 2-methoxyethanol.

Figure 5.1.1: 2-methoxyethanol in intramolecular ring form.

Quantitative experimental measurements of IntraMHB have been successfully accomplished

[167,168,169,170]. The experimental evidence for IntraMHB can also be observed indirectly

through its impact on bulk properties. Indeed, IntraMHB plays a critical role in chemical and

biological molecules, by stabilizing structures, facilitating the protein folding, increasing the

hydrophobicity or lipophilicity of molecules, and affecting the physical properties by competing

with intermolecular hydrogen bonding (InterMHB) [1, 2, 3, 6, 7, 8, 12, 13]. Even though

IntraMHB does not cause the molecules to repel each other, it can make them bind together

less strongly by hindering InterMHB, which can have a dramatic effect on the behaviour of

associating fluids. The effects of this competition were shown to affect the cloud-point pressures

of telechelic polymers by Gregg et al. [4, 5]. In polar solvents, intermolecular association shifts

the cloud points to higher pressures, opposing the effect of the formation of intramolecular rings,

which decreases the cloud-point pressures.

Another example of the dependence of bulk properties on IntraMHB was the focus of extensive

experimental studies carried out by Domańska and co-workers [10,11,171]. They studied the

impact of the interplay between self-association, cross-association and intramolecular association

on solubility. A solvent mixture is said to have a positive or a negative synergistic effect on the

solubility of a solute if the solubility of the solute in the mixed solvent is higher or lower than in

all pure solvents, respectively [172]. Compounds that exhibit stronger cross-association with

polar solvents in comparison to self- and intramolecular association do not tend to form pure


component clusters in abundance and an enhanced solubility effect (positive synergistic effect)

in binary solvent systems is not observed, such as 1-acetyl-2-naphthol in the solvent mixture

hexane and 1-butanol [10,11]. But the opposite is also observed, as a positive synergistic effect

appears to be promoted by the formation of a stable intramolecular hydrogen bond, such as

2-acetyl-1-naphthol in the solvent mixture cyclohexane and 1-butanol [10]. Both 2-acetyl-1-

naphthol (Figure 5.1.2a) and 1-acetyl-2-naphthol (Figure 5.1.2b) are capable of forming an

intramolecular hydrogen bond. However, the proximity of the peri hydrogen to the CH3 in the

1-acetyl substituent, forces the acetyl out of the plane of the naphthalene core [173], as depicted

in Figure 5.1.2b) which interferes with the formation of the intramolecular hydrogen bond. The

(a) 2-acetyl-1-naphtol. (b) 1-acetyl-2-naphtol.

Figure 5.1.2: Representation of 3D models of a) 2-acetyl-1-naphtol and b) 1-acetyl-2-naphtol.

intramolecular bond in component (a) is thus a lot more stable than the bond in (b) which

prevents it from being disrupted easily in an associating solvent. This explains why component

(a) exhibits similar solubility in both pure cyclohexane and pure 1-butanol and component (b)

does not. The solubility enhancement effect in a mixture of cyclohexane and 1-butanol, only

detected for component (a), decreases with the decrease in temperature, which is explained

by the increase of the degree of self-association of 1-butanol. The discussion of the solubility

results was supported by direct evidence for the existence of IntraMHB in these components

obtained by ultraviolet absorption spectroscopy [10].


We now turn towards the theoretical approaches. Sear and Jackson [105] and Ghonasgi and

co-workers [18, 120, 122] extended TPT1 to account for IntraMHB rings, as described in

Sections 2.4.1 and 2.4.2, for particular cases of the theory in Chapter 3 where IntraMHB

occurs in a two-site chain molecule. The theory was applied by the authors to a chain molecule of

m = 6 segments with one association site at each extremity capable of promoting the formation

of intramolecular rings and linear-chain aggregates. The authors [105] found that the formation

of intramolecular rings is favoured with the decrease of density and increase of association

energy (just as seen for intermolecular rings in Chapter 4). The authors [18, 120, 122] also

observed that intramolecular ring molecules have lower compressibility factor than monomer

chains (non-associated) but greater than linear-chain aggregates, as can be seen in Figure 5.1.3

where data taken from [18] is plotted.

Figure 5.1.3: Compressibility factor Z as a function of the association energy εHB/(kT ) atpacking fraction η = 0.05. The symbols represent simulation data and the curves representtheory: TPT1 (dashed curve and squares) and TPT1+intramolecular rings (continuous curveand downward-pointing triangles). The diagram was made using data extracted from the originalpublication of Ghonasgi et al. [18].

In this Chapter, we will use our theoretical approach firstly on model pure and mixed systems of

simple chain molecules capable of forming intramolecular rings, and lastly on real binary systems

of statins and alcohols. The main interest is to understand how the saturation curves shift with

the inclusion of intramolecular rings in the calculations, mostly solid–liquid equilibrium (SLE),

but also VLE and LLE curves. The EOSs used in the calculations are described next, prior to

the presentation and discussion of the results.

5.2. SAFT-VR Mie and SAFT-γ Mie EOSs 125

5.2 SAFT-VR Mie and SAFT-γ Mie EOSs

Due to the simplified representation of intermolecular forces, potentials with infinitely repulsive

cores (such as the SW) are limited in the range of properties that can be described accurately.

With the aim of improving the description of fluid phase behaviour and second-order derivatives,

Lafitte et al. [55,174] proposed an extension of SAFT-VR SW [87,157] to chain molecules formed

from spherical segments interacting through soft potentials: Mie potentials [175]. The Mie

potential is a generalisation of the Lennard-Jones potential with repulsion (λrkl) and attraction

(λakl) parameters between segment k in molecule in coordinates (1) and segment l in molecule in

coordinates (2) and is given as

φMiekl (12) = Cklε

dispkl

[(σklrkl

)λrkl

−(σklrkl

)λakl

], (5.1)

with

Ckl = λrkl

λrkl − λa

kl

(λrkl

λakl

) λakl

λrkl−λa

kl

, (5.2)

where rkl = |r2 + dl(Ω2)− r1 + dk(Ω1)| is the distance between segments k and l, dk(Ω1) is

the vector connecting the centre of mass of molecule in position r1 containing segment k and

centre of mass of segment k, analogously, dl(Ω2) is the vector connecting the centre of mass of

molecule in position r2 and centre of mass of segment l.

The approach is based on an application of Barker and Henderson’s [54] high-temperature

perturbation theory (PT) expansion of the Helmholtz free energy Amono, of a fluid composed of

Mie segments and the expansion is truncated at third order (previous SAFT-VR truncated at

second order), in which the third perturbation term is given by an empirical expression [55]. This

introduction of a third order of perturbation enabled a great improvement on the accuracy of

the fluid behaviour in the near-critical region [55]. A high-temperature perturbation expansion

(up to second order) is used to represent the radial distribution function for the monomer (Mie)

fluid gmono.

In SAFT-VR Mie, a molecule is modelled as a chain of spherical united-atom segments tangen-


tially bonded, just as the example for thymol given in Figure 5.2.1. SAFT-VR Mie models are

e1

e1

H

Figure 5.2.1: Molecular structure of thymol on the left and respective homonuclear model usedin SAFT-VR Mie calculations on the right. The model is composed of three segments and threeassociation sites of two types, one H and two e1.

united-atom and therefore, the number of segments in the chain does not represent the number

of carbon atoms. There is not a direct relationship between the number of atoms and the number

of segments, since the latter is a parameter, estimated from experimental data. The interactions

between molecules of the same compound (i) require pure-interaction (like) parameters and

those between molecules of different compounds (i and j) require cross-interactions (unlike)

parameters. All these are listed in Table 5.1. The segment diameter of component i is given by

σii, the number of Mie segments per chain by mi (may not be an integer), the Mie’s potential

depth by εdispii , the repulsive and attractive exponents by λr

ii and λaii, respectively, the number

of sites of type a by ni,a, the association potential depth by εHBii,ab, the range of the association

interaction by rcii,ab, the size of the ring formed by association between sites a and b of index R by

τi,ab,R and the probability density relative to the formation of a ring of index R is given byWi,ab,R.

The unlike parameters have analogous definitions for the interactions between segments/sites

that are in molecules of different components i and j. There are combining rules to obtain the

Table 5.1: Like and unlike parameters in SAFT-VR Mie (modified version to account for ringformation) molecular models.

pure-interaction parameters cross-interaction parameters

all compounds σii,mi, εdispii , λr

ii, λaii εdisp

ij , λrij, λ

aij

assoc. compounds ni,a, εHBii,ab, r

cii,ab εHB

ij,ab, rcij,ab

ring-forming assoc. compounds τi,ab,R,Wi,ab,R

unlike parameters that may involve an extra parameter kij used to calculate εdispij or γij used to


calculate λrij. The attractive exponent is usually taken as λa

ii = λaij = 6.

Lymperiadis et al. [176, 177] considered fused-segment models instead of tangential, which

implied the segments no longer had to be spherical. The EOS resulting from their work is of

GC formalism, denoted SAFT-γ. Later, Papaioannou et al. [16] developed SAFT-γ Mie with

the aim of using the underlying model of the successful SAFT-VR Mie [55] in the GC model

formalism of the SAFT-γ [176, 177]. A GC model treats a fluid system at the level of the

functional groups (FGs) that make up the molecules, which are characterized by a set of like

and unlike group parameters.

In SAFT-γ Mie EOS [16, 73, 178], the monomer term is computed using the same approach

followed in SAFT-VR Mie, while being based on a different molecular model. In SAFT-γ Mie,

a molecule is composed of different groups, each modelled as a chain of fused united-atom

segments, just as the example for ibuprofen given in Figure 5.2.2.

e1e1

e2e2

H

e1

e1

e1e1

e1

e1COOH CH

CH3

CH3

CH3

aCCH2

aCHaCH

aCCH

aCHaCH

Figure 5.2.2: Molecular structure of ibuprofen on the left and respective GC model with anunderlying 3D structure used in SAFT-γ Mie calculations on the right. The groups (highlightedwith different colours and descriptions) are fused and are made up of one segment each. Eachgroup involved in the aromatic cycle contains one association site e1 and the group COOHcontains three site types: one H, two e1 and two e2.

The interaction between segments of the group k is characterised by a set of like group parameters,

and the interaction between segments of group k and group l are characterised by a set of

unlike group parameters. The number of occurrences of group k in compound i is given by

νi,k. The ring-forming associating compounds are also characterised by the compound-specific

parameters τi,kk,ab,R and Wi,kk,ab,R or τi,kl,ab,R and Wi,kl,ab,R if the sites a and b are in different

groups k and l. All other parameters are group-specific and are listed in Table 5.2. The number


Table 5.2: Group like and unlike parameters in SAFT-γ Mie (modified version to account forring formation) molecular models.

pure-interaction parameters cross-interaction parameters

all groups νi,k, ν∗k , Sk, σkk, ε

dispkk , λr

kk, λakk εdisp

kl , λrkl, λ

akl

assoc. groups nk,a, εHBkk,ab, Kkk,ab εHB

kl,ab, Kkl,ab

ring-forming compounds τi,kl,ab,R,Wi,kl,ab,R

of occurrences of a given group k in a molecule of component i is given by νi,k, the number of

identical segments in group k is given by ν∗k and the extent to which the segments of a given

group k contribute to the overall molecular properties is characterised by the shape factor Sk.

The bonding volume characteristic of the association between sites a and b is given by Kkl,ab.

The remaining parameters in Table 5.2 have analogous definitions as those given before for the

SAFT-VR Mie parameters, but are given in terms of groups instead of compounds. The table

of currently available parameters is published in the literature [73,179,180].

The properties of the system are thus obtained by considering the separate contributions from the

groups. The great advantage of a GC model, provided that suitable parameters are available, is

its predictive capability, since from the description of a few groups, the thermodynamic properties

and phase equilibria of a large number of pure or mixture compounds can be predicted without

further fitting [50]. Moreover, it usually allows for a parametrisation solely based on pure

component experimental data, and thus less experimental data input is needed. There are

exceptions to this capability, such as water or acetone mixtures, since both water and acetone

are groups and naturally do not exist in any other molecule.

The contribution of each group k to the overall molecular properties is weighted through the

introduction of the parameter Sk (shape factor). In the monomer and chain terms, the number

of segments is replaced by the effective number of segments (weighted through the introduction

of the parameter Sk), and the association term after the extension to account for the formation

of rings is given as

Aassoc = NkTNC∑i=1

xi

NG∑k=1

νi,k

NST,k∑a=1

nk,a

(lnXi,k,a + 1−Xi,k,a

2

), (5.3)


with

Xi,k,a = 11 + ρ

∑NCj=1 xj

∑NGl=1 νj,l

∑NST,lb=1 nl,bXj,l,b∆ij,kl,ab

, (5.4)

where NC is the number of components, NG is the number of groups, νi,k is the number of

groups k in a molecule of species i, NST,k is the number of site types, nk,a is the number of sites

of type a in group k, Xi,k,a is the fraction of molecules of species i with (at least) one site of

type a of group k free and ∆ij,kl,ab is the association strength between a site of type a on group

k in molecule i and a site of type b on group l in molecule j.

The association term accounting for ring formation (Equations (3.83)–(3.97)) can be directly

incorporated in the SAFT-VR Mie and SAFT-γ Mie approaches. The expressions below are

general for both these approaches where for the case of SAFT-VR Mie, the dependency on the

group is omitted as NG = 1. The notation proposed is slightly different to that of the original

approaches:


Aassoc, (3.81)

with

Aassoc =NkTNC∑i=1

xi

lnXi,0 + 12

∑k∈NG,i

∑a∈Γk

1−Xi,k,a

−∑

l∈NG,i

∑b∈Γl

NRS,ab∑R=1

1τRρτR−1 (Xi,kl,ab)τR (∆ii,kl,ab)τRWi,kl,ab,R

,(3.82)

and Equations (3.83)–(3.97). In the original SAFT-VR Mie and SAFT-γ Mie approaches [16,55],

a is the index of a site type; in the modified version to account for ring formation, a is the

index of an individual site. The reason for the adoption of this notation is that sites of the

same type (e.g., a) in groups of the same type (e.g., k) are not necessarily equivalent, as one

of the sites a might have the proximity to a site b required to promote ab-ring formation and

the other not. Indeed, two sites of type a each in a group of type k might not be equivalent

when ring formation is considered. As an example, consider the 1,3,4-butanetriol represented in

Figure 5.2.3. There are six ring configurations of three different sizes in 1,2,4-butanetriol to be

considered; the formation of rings is promoted by IntraMHB between three OH groups with

two site types each (H and e1). To each of these rings is associated a parameter W that might

have a different value for each of the six ring configurations.


Figure 5.2.3: Chemical structure of 1,2,4-butanetriol.

Consider the interaction between site a in group k in component i and site b in group l in

component j. In the case of a reference Mie fluid of spherical segments and square-well association

sites the integrated association parameter ∆ij,kl,ab can be expressed in a factorised form as [55]

∆ij,kl,ab = Fij,kl,abKij,kl,abIij,kl, (5.5)

where the magnitude of the Mayer f -function of the square-well interaction between sites a, b is

denoted as

Fij,kl,ab = exp(βεHB

ij,kl,ab

)− 1. (5.6)

In this work, the association kernel Iij,kl used is a two-dimensional polynomial that results from

the numerical evaluation and mapping of the integral of the radial distribution function of the

reference Mie fluid valid for a range of densities and temperatures developed by Dufal et al. [181]

and is given as

Iij,kl(T ∗, ρ∗) =y+z∑y=0

≤10∑z=0

cyz [ρ∗]y [T ∗]z . (5.7)

The dimensionless polynomial is parametrised with 66 cyz parameters found in [182] and is a

function of the dimensionless temperature T ∗ = kT/εij and density ρ∗ = ρσ3ij, where εij is the

dispersion energy and σij is the segment diameter averaged for the multicomponent mixture

according to the combining rules detailed in [181]. As the polynomial was developed for fixed

values of the cut-off range (rc) of the square-well interaction between site pairs, and rd = 0.4σij ,

the distance of each association site from the centre of the corresponding segment, the ‘bonding

volume’ parameter Kij,kl,ab serves to correct for the geometry of the intermolecular potential of

the different molecular models.

The ci,kl,ab at the ideal limit, according to the integrated association strength in Equation (5.5),


is given by

cideali,kl,ab = Fii,kl,abKii,kl,abI

idealii,kl Wi,kl,ab,intra, (5.8)

because the only term in ∆ij,ab dependent on the density is the radial distribution function,

which is captured by the association kernel Iij,kl. It follows from Equation (5.7), that the ideal

limit of the kernel is defined as

I idealii,kl (T ∗) =

z≤10∑z=0

c0l [T ∗]z , (5.9)

since the only terms of the polynomial that do not vanish at low density are the ones corresponding

to exponent zero on the density dimension.

The quantity Wn is the probability density of the intersegmental distance between the segments

decorated with the ring forming sites being σ. The extent of ring formation in a model fluid

can thus be controlled by the value of the parameter W : the higher the W , the larger the

extent of the ring formation. The expression for Wn [118, 119] given in Equation (4.35) was

derived for long freely jointed chains of n = m− 1 equal links of fixed length, assuming that all

orientations of a link in space are equally likely. The use of this expression in the prediction

of phase behaviour of HF accounting for the formation of intermolecular rings proved to be

valuable as shown in the study by Galindo et al. [108]. However, due to the molecular internal

structure constraints, we do not expect Wn (Equation (4.35)) to be a good approximation in

the modelling of real fluids exhibiting intramolecular hydrogen bonding. Alternatively, W can

be estimated from simulation [107] or experimental data.

We predict the coexisting properties of fluid phase equilibrium, where the relations,

T α = T β = T, (5.10a)

P (T, V α,nα) = P (T, V β,nβ), (5.10b)

µi(T, V α,nα) = µi(T, V β,nβ), (5.10c)

are true. Here, µi, the chemical potential of component i in each phase α and β, and other


relevant properties are obtained from derivatives of the Helmholtz free energy given as in:

µi(T, V,n) = ∂A

∂ni

∣∣∣∣∣T,V,nj 6=i

, P (T, V,n) = −∂A∂V

∣∣∣∣∣T,n

, and S(T, V,n) = −∂A∂T

∣∣∣∣∣V,n

,

where ni is the number of moles of component i.

5.3 Effect of the intramolecular ring formation on fluid

properties

The impact of the chain length and parameter W in the extent of ring formation, together with

the influence of the ring formation in the phase behaviour of fluid systems are analysed. In this

study, we explore a range of W ∗ = W/Wn, where Wn is given by the value of the end-to-end

distribution function for a freely jointed chain of n links at contact.

The theory is first applied to a pure fluid of chain molecules (Figure 5.3.1a), and next to a binary

mixture where a second component is added to the fluid consisting a spherical molecule with

one association site (Figure 5.3.1b). The chain molecules associate in linear-chain aggregates

and intramolecular rings. The spherical molecule can only associate with one of the sites of the

chain molecule.

5.3.1 Model system

We consider a two-site m-segment chain model illustrated in Figure 5.3.1a, and a single-site

spherical model illustrated in Figure 5.3.1b. The molecular segments are modelled as Lennard-

Jones spheres with diameter σ and well-depth εdisp. A − B association takes place (A − A

and B − B are not permitted) and can result in the formation of linear-chain aggregates

and intramolecular rings (note that branched chains cannot occur in a two–site fluid). The

formation of intermolecular rings is not being taken into account in this study for simplicity.

The parameters for the pure systems of both components and respective cross-interaction are

5.3. Effect of the intramolecular ring formation on fluid properties 133

defined in Table 5.3, in reduced form. We use the modified SAFT-VR Mie approach in the

A

B

(1) (2)(3)

(...)

(m)

(a) Chain molecule.

A

(b) Sphere molecule.

Figure 5.3.1: The molecules are modelled as Lennard-Jones chains (a)/spheres (b) that consistof segments (red large sphere of diameter σ), decorated with sites A (green) and/or B (blue)located off-centre in the segments at rd = 0.4σ from the segment centre of mass. The volume ofthe association sites defines the bonding volume. When two sites overlap, the pair interactionenergy is taken to be equal to −εHB

ab . Association is promoted by the site pair AB. The chainmolecule (a) is involved in intramolecular association and self-association into open aggregatesas well as in cross-association with the spherical component (b).

Table 5.3: Pure and unlike parameters for the model systems, where i, j are the componentindexes (1 for the chain and 2 for the sphere), σ is the segment diameter. The interactionbetween a pair of segments is characterised by a Lennard-Jones potential, for which the repulsiveand attractive exponents are λr

ij = 12 and λaij = 12, respectively, and εdisp is the depth of the

potential well. The association is characterised by a square-well potential of depth εHB and thebonding volume KHB.

ij σij/σ11 εdispij /εdisp

11 εHBij /ε

disp11 KHB

ij /σ311

11 1.0 1.0 4.0 4.722 1.0 1.6 – –12 1.0 1.2 4.8 4.7

calculations for model molecules (Sections 5.3 and 5.4.2). The chosen A− B association energy

of the chain molecule (εHB/εdisp = 4) is low when compared with usual associating components

such as alcohols (e.g., εHB/εdisp ≈ 6.7 for 1-butanol [55]). Nevertheless, it is sufficiently large to

be used in the study of the impact of the inclusion of ring formation in the calculations. The

graphical representations use reduced units, namely the reduced temperature T ∗ = kT/εdisp11 ,

the reduced pressure P ∗ = Pσ311/ε

disp11 , the packing fraction η = (π/6)σ3

11NavNm/V and the

reduced vapourisation enthalpy ∆hvap,∗ = ∆hvap/(NkT ), where Nav is the Avogadro’s number

and ∆hvap is the vapourisation enthalpy of the chain molecule.


5.3.2 Impact of intramolecular ring formation on a pure chain fluid

We consider five pure five-segment chain fluids. A non-associating system, an associating system

in the absence of rings (W ∗ = 0), and associating systems with intramolecular ring formation

where the parameter W ∗ takes the values 10, 100 and 107. The parameter W ∗ represents

W/W4, where W4 is the density probability at contact for a chain of five segments defined

in Equation (4.35), and W is the parameter used in the calculations, taken as density and

temperature independent. Upon the increase of the value of W ∗, the equilibrium distribution

of aggregates is shifted towards the ring form, i.e., there is an increase of the fraction of chain

molecules in ring form (ring fraction). Any value of W ∗ larger than a certain limit where all the

chain molecules are in intramolecular ring conformation produces the same phase equilibrium

diagrams. In the fluids studied, W ∗ = 107 was observed to be past this limit and was used to

calculate the properties corresponding to the fluid consisting of all molecules in the ring form.

In Figures 5.3.2–5.3.5, different representations of vapour–liquid coexistence for the five fluids

considered are presented. The selected small association energy (εHB11 /ε

disp11 = 4) results in

a reduced difference between the curves for the non-associating fluid and the linear-chain

aggregates, without compromising the analysis.

In Figures 5.3.2 and 5.3.3, the vapour pressure is shown in the usual PT representation and

in the logarithmic Clausius-Clapeyron representation. In comparison with the non-associating

fluid, the associating fluid without ring formation exhibits a higher critical point and lower

vapour pressure across all temperature range. The relative difference between the two curves

increases with the decrease in temperature and consequent increase in associated molecules. As

the formation of rings is considered, the increase of the parameter W ∗ results in an increase of

the critical temperature and decrease of the vapour pressure at high temperatures. Opposing

this trend, an increase in the vapour pressure is observed at low temperatures (approx. below

T ∗=1.5), where the curves sit between those relative to the non-associating and the associating

W ∗ = 0. This inversion of trends in vapour pressures already observed in Figure 5.3.3 is most

evident in Figure 5.3.4, where the enthalpy is plotted against temperature.


0.00

0.01

0.02

0.03

0.04

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

P*

T*

− no sites

− no rings

− W*=10

− W*=100

− W*=107

0.00

0.01

0.02

0.03

0.04

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Figure 5.3.2: Reduced vapour pressures P ∗ = Pσ311/ε

disp11 as function of reduced temperature

T ∗ = kT/εdisp11 of the non-associating fluid (dashed black curve) and the associating fluids with

W ∗ = 0 (continuous black curve), with W ∗ = 10 (continuous green curve), with W ∗ = 100(continuous blue curve) and with W ∗ = 107 (continuous pink curve). All curves were calculatedusing SAFT-VR Mie modified for ring formation.

−17

−15

−13

−11

−9

−7

−5

−3

0.4 0.5 0.6 0.7 0.8 0.9 1.0

ln

P*

1/T*

− no sites

− no rings

− W*=10

− W*=100

− W*=107

−17

−15

−13

−11

−9

−7

−5

−3

0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 5.3.3: Clausius-Clapeyron representation of reduced vapour pressures P ∗ = Pσ311/ε

disp11 as

function of reduced temperature T ∗ = kT/εdisp11 of the non-associating fluid (dashed black curve)

and the associating fluids with W ∗ = 0 (continuous black curve), with W ∗ = 10 (continuousgreen curve), with W ∗ = 100 (continuous blue curve) and with W ∗ = 107 (continuous pinkcurve). All curves were calculated using SAFT-VR Mie modified for ring formation.


0

5

10

15

20

25

30

35

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

∆h

vap

,*

T*

− no sites

− no rings

− W*=10

− W*=100

− W*=107

0

5

10

15

20

25

30

35

1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

Figure 5.3.4: Reduced vapourisation enthalpies ∆hvap,∗ = ∆hvap/(NkT ) as function of reducedtemperature T ∗ = kT/εdisp

11 of the non-associating fluid (dashed black curve) and the associatingfluids with W ∗ = 0 (continuous black curve), with W ∗ = 10 (continuous green curve), withW ∗ = 100 (continuous blue curve) and with W ∗ = 107 (continuous pink curve). All curves werecalculated using SAFT-VR Mie modified for ring formation.

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0.0 0.1 0.2 0.3 0.4 0.5

T*

η

− no sites

− no rings

− W*=10

− W*=100

− W*=107

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

0.0 0.1 0.2 0.3 0.4 0.5

Figure 5.3.5: Coexistence packing fractions η across reduced temperatures T ∗ = kT/εdisp11 for the

non-associating fluid (dashed black curve) and the associating fluids with W ∗ = 0 (continuousblack curve), with W ∗ = 10 (continuous green curve), with W ∗ = 100 (continuous blue curve)and with W ∗ = 107 (continuous pink curve). All curves were calculated using SAFT-VR Miemodified for ring formation.


The coexistence packing fractions in Figure 5.3.5 are examined. The packing fraction of the

saturated liquid of the associating fluid is higher than that of the non-associating fluid across

all temperatures. As the formation of rings is considered, the increase of the parameter W ∗

results in an increase of the packing fraction of the saturated liquid across all temperatures. At

low temperatures the curves approximate that of W ∗ = 0 as a response to an increase in the

intermolecular association.

In order to understand better the curve trends we analyse the effect of temperature and

packing fraction on the distribution of aggregates; we focus on the single phase in order to

decouple the density from the temperature effects. Molecules can be in one of three bonding

states: free monomer, bonded in a linear-chain aggregate, or intramolecularly bonded (ring).

Therefore, the fractions of monomers X0, of chain forming molecules (1 − XopenA ) and of

molecules intramolecularly bonded (1−X intra ringsA ) sum up exactly to 1. Taking advantage of

the disjointed bonding states, the fractions of three possible bonding states are represented as

areas in Figures 5.3.6 and 5.3.7. The fractions of each bonding state of the fluid molecules are

plotted against temperature (constant packing fraction) and against packing fraction (constant

temperature) for the associating fluid in three columns corresponding to (from left to right) the

absence of rings (W ∗ = 0) and the associating fluid with W ∗ = 10 and W ∗ = 100.

The decrease of temperature favours association of both inter- and intramolecular type. In

Figure 5.3.7, it is observed that in the vapour phase, intermolecular association only takes place

in the system without rings (grey area) at sufficiently low temperatures, while most molecules

will be preferably in the free monomer state (white area). The molecules are far apart which

hinders the formation of intermolecular hydrogen bonds. Moreover, if the molecules have the

option of forming rings (square pattern), the association sites become less available to associate

intermolecularly, and IntraMHB is the most favourable. By observation of Figure 5.3.6b,

IntraMHB is a weak function of density if the density is low enough to prevent InterMHB.

As seen in Figure 5.3.7a, while the fraction of molecules intermolecularly associated tends to 1

in the absence of ring formation, this is not the case when it is competing with the formation

of rings. This is due to the decrease in energy inherent in the formation of a hydrogen bond.


(a) η = 0.007.

(b) T ∗ = 2.0.

Figure 5.3.6: Relative fractions of molecules in each bonding state in the vapour phase a)across reduced temperature T ∗ = kT/εdisp

11 at fixed packing fraction η = 0.007 and b) acrosspacking fraction at fixed reduced temperature T ∗ = 2.0. The fraction of molecules in open-chainaggregates is represented by a coloured area, the fraction of intramolecular rings is representedby a square-patterned area and the fraction of monomers by a white area. The three columnsrefer to the associating systems in the absence of rings, i.e., with W ∗ = 0, in grey tones (left-handside), with W ∗ = 10 in green tones (middle) and with W ∗ = 100 in blue tones (right-hand side).All curves were calculated using SAFT-VR Mie modified for ring formation.


(a) η = 0.300.

(b) T ∗ = 2.0.

Figure 5.3.7: Relative fractions of molecules in each bonding state in the liquid phase a) acrossreduced temperature T ∗ = kT/εdisp

11 at fixed packing fraction η = 0.300 and b) across packingfraction at fixed reduced temperature T ∗ = 2.0. The fraction of molecules in open-chainaggregates is represented by a coloured area, the fraction of intramolecular rings is representedby a square-patterned area and the fraction of monomers by a white area. The three columnsrefer to the associating systems in the absence of rings, i.e., with W ∗ = 0, in grey tones (left-handside), with W ∗ = 10 in green tones (middle) and with W ∗ = 100 in blue tones (right-hand side).All curves were calculated using SAFT-VR Mie modified for ring formation.


Regardless of the value of W ∗, at sufficiently low temperatures, the fraction of monomers is zero

and all molecules are in ring form, both in the vapour (see Figure 5.3.6a) and liquid phases.

This can be explained by the fact that at low temperatures, the entropic loss involved in the

formation of a ring is not important and therefore the ring form is energetically more favourable,

since both the monomer and the linear chain aggregates have two free sites regardless of its size.

Lastly, according to Figure 5.3.7, the extent of both inter- and intramolecular association

decreases with the decrease of packing fraction. However, while the intermolecular association

decreases very noticeably, the dependence of intramolecular association on the density is very

small.

5.3.3 Impact of the chain length on the vapour–liquid equilibrium

of a pure ring-forming chain fluid

In this section, the effect of the chain length on the phase equilibrium trends observed in

the previous section is examined. We consider three sizes of the chain molecule presented

in Figure 5.3.1a: m = 5, m = 8 and m = 10. For each size, four pure fluids are studied:

the non-associating, and the associating with W ∗ = 0, W ∗ = 10 and W ∗ = 100. Just as

in Section 5.3.2, the definition of W ∗ is based on the end-to-end distribution function of a

freely jointed chain of four links (W4); W ∗ = W/W4, where W is the parameter used in the

calculations, taken as density and temperature independent.

In Figure 5.3.8, different representations of vapour–liquid coexistence for the pure fluids consid-

ered are presented using the parameters listed in Table 5.3. The critical temperature increases

with the chain size and the formation of rings. The difference between the curves of the

non-associating fluid and the associating fluid without ring formation is less significant for

larger chains. Similarly, the difference in vapour pressures, packing fractions and vapourisation

enthalpies between associating systems with and without rings is also decreased with the increase

in chain length.

The fractions of molecules in each of the bonding states for all associating systems are represented


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

2.0

2.2

2.4

2.6

2.8

3.0

3.2

P* × 10

2

T*

m =

10

m =

8

m =

5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

2.0

2.2

2.4

2.6

2.8

3.0

3.2

(a)Reduced

vapo

urpressuresP∗

=Pσ

3 11/ε

disp

11.

−35

−30

−25

−20

−15

−10

−50 0

.30.4

0.5

0.6

0.7

0.8

0.9

1.0

ln P*

1/T

*

m =

10

m =

8

m =

5

−35

−30

−25

−20

−15

−10

−50 0

.30.4

0.5

0.6

0.7

0.8

0.9

1.0

(b)Clausius-Clape

yron

loga

rithm

icrepresentatio

n.

1.0

1.5

2.0

2.5

3.0

0.0

0.1

0.2

0.3

0.4

0.5

T*

η

m =

10

m =

8

m =

5

1.0

1.5

2.0

2.5

3.0

0.0

0.1

0.2

0.3

0.4

0.5

(c)Coexistence

packingfractio

ns.

0

10

20

30

40

50

60 1

.01.5

2.0

2.5

3.0

∆hvap,*

T*

m =

10

m =

8 m

= 5

0

10

20

30

40

50

60 1

.01.5

2.0

2.5

3.0

(d)R

educed

vapo

urisa

tionenthalpies

∆h

vap,∗

=∆h

vap/(NkT

).

Figu

re5.3.8:

Rep

resentations

ofvapo

ur–liquidph

asebe

haviou

rforpu

rechainflu

idsconsist

ingofm

=5segm

ents

(orang

ecu

rves),m

=8

segm

ents

(green

curves)an

dm

=10

segm

ents

(bluecu

rves).

Foreach

chainleng

thit

isshow

ntheno

n-associatingsystem

(sho

rtda

shes),

andtheassocia

tingwi

thW∗

=0(con

tinuo

us),wi

thW∗

=10

(dash-do

t-dash)

andwi

thW∗

=10

0(lo

ngda

shes).

Allcu

rves

were

calcu

lated

usingSA

FT-V

RMie

mod

ified

forrin

gform

ation.


in Figures 5.3.9–5.3.11.

liquid phase

vapour phase

Figure 5.3.9: Fractions of free molecules (monomers) along the saturation vapour and liquidlines for pure chain fluids consisting of m = 5 segments (orange curves), m = 8 segments (greencurves) and m = 10 segments (blue curves). For each chain length it is shown the associatingsystems with W ∗ = 0 (continuous), with W ∗ = 10 (dash-dot-dash) and with W ∗ = 100 (dashes).All curves were calculated using SAFT-VR Mie modified for ring formation.

In Figure 5.3.9, the evolution of the fraction of monomers can be examined. In the associating

systems with W ∗ = 0 the saturated vapour phase consists of monomers (X0 = 1) due to the

low density that hinders intermolecular association. As the parameter W ∗ is increased and the

formation of rings is promoted, the fraction of monomers in the vapour phase decreases. At high

temperatures, the extent of association is smaller than at low temperatures in both liquid and

vapour phases, but across the whole temperature range there are consistently more monomers

in the vapour than in the liquid phase. Considering the different chain lengths, the fraction

of monomers in the vapour phase is independent of these for a fixed W ∗; the differences are

significant in the liquid phase however, especially for low values of W ∗ and in the absence of ring

formation. The shortest chain length (m = 5) produces the smallest amount of monomers in

the liquid phase of a fluid without ring formation but this gap is closed, and eventually inverted

with the increase of W ∗.

In Figures 5.3.10 and 5.3.11, the fractions of molecules in open-chain aggregates and intramolecu-


liquid phase

vapour phase

Figure 5.3.10: Fractions of molecules in open-chain aggregates along the saturation vapourand liquid lines for pure chain fluids consisting of m = 5 segments (orange curves), m = 8segments (green curves) and m = 10 segments (blue curves). For each chain length it is shownthe associating systems with W ∗ = 0 (continuous), with W ∗ = 10 (dash-dot-dash) and withW ∗ = 100 (dashes). All curves were calculated using SAFT-VR Mie modified for ring formation.

Figure 5.3.11: Fraction of molecules in intramolecular rings (ξ1) along the saturation vapourand liquid lines for pure chain fluids consisting of m = 5 segments (orange curves), m = 8segments (green curves) and m = 10 segments (blue curves). For each chain length it is shownthe associating systems with W ∗ = 0 (continuous), with W ∗ = 10 (dash-dot-dash) and withW ∗ = 100 (dashes). All curves were calculated using SAFT-VR Mie modified for ring formation.


lar rings along the saturation curves, respectively, can be examined. It is visible from Figure 5.3.10

that the extent of intermolecular association is negligible in the vapour phase, regardless of

the formation of rings and the size of the chain molecule. Considering the liquid phase, more

chain aggregates are formed for the smaller chain lengths. When W ∗ > 0 is considered, the

fraction of intermolecular hydrogen bonds increases with the decrease of temperature only up

to a certain point, after which it starts decreasing. With the increase of W ∗, the formation

of chain aggregates in the liquid phase becomes negligible too. The reason for this decrease

can be explained by the increase of ring formation, observed in Figure 5.3.11. In Figure 5.3.11,

it is also observed that at W ∗ = 10, the fraction of rings in the vapour phase is lower than

that in the liquid phase at high temperatures but increases with the decrease of temperature,

eventually crossing the liquid curve. At low temperatures there are more rings in the vapour

than in the liquid phase because, in the vapour phase the ring formation is not competing

with the intermolecular hydrogen bonding and in the liquid phase it is. At high values of W ∗

(W ∗ = 100) the crossing of the vapour and liquid curves is no longer observed because the

fraction of open-chain aggregates is insignificant in both phases.

5.3.4 Influence of intramolecular ring formation on the vapour–liquid

and liquid–liquid behaviour of a binary system

Liquid–liquid equilibrium is of particular importance for unit operations involving immiscible

systems such as extraction processes (separation of compounds based on their solubility in two

immiscible liquids). In this section, the liquid–liquid and vapour–liquid diagrams for four binary

mixtures consisting of the model molecules represented in Figure 5.3.1 are presented; the four

binary systems are mixtures of the spherical component (Figure 5.3.1b) with a non-associating,

or associating with W ∗ = 0 (no rings), W ∗ = 10 or W ∗ = 100 five-segment chain component

(Figure 5.3.1a). The parameters selected (see Table 5.3) were such to promote partial miscibility

(liquid-liquid equilibrium). The phase equilibria results at P ∗ = 0.001, together with the

distribution of bonding states of the chain molecule, are compiled in Figure 5.3.12.

In Figure 5.3.12a we compare the phase behaviour of the non-associating with the associating


system without ring formation. These two fluids serve as reference for the associating fluid

with W ∗ = 10 in Figure 5.3.12b and the associating fluid with W ∗ = 100 in Figure 5.3.12c.

Each diagram consists of four parts labelled “V” for saturated vapour, “L” for saturated liquid,

“L1” for saturated liquid 1 and “L2” for saturated liquid 2. A vapour–liquid–liquid equilibrium

(VLLE) curve is present in the system where no association takes place (Figure 5.3.12a) as a

result of the intersection between the VLE and the LLE regions. Next to the phase diagrams,

the distributions of the bonding states of the chain molecule are plotted along the saturated

lines for the associating fluids with W ∗ = 0 (Figure 5.3.12a), with W ∗ = 10 (Figure 5.3.12b)

and with W ∗ = 100 (Figure 5.3.12c). The associating chain molecule has three possible bonding

states: non-bonded (monomer), bonded to a spherical molecule (cross-associated), bonded

intermolecularly to at least other chain molecule and no chain molecules (self-associated) and

intramolecularly bonded (ring) if W ∗ > 0. The fractions of molecules in each of the bonding

states must sum up to 1.

In Figure 5.3.12a, it is observed that the spheres are not as much miscible with the non-associating

chains (dashed curves) as they are with the associating chains with W ∗ = 0 (continuous black

curves). Additionally, both bubble and dew curves of the non-associating system are shifted

down in relation to the associating system. The fraction of chain molecules bonded increases

with the decrease in temperature in both the saturated vapour and liquid phases, while being

negligible in the vapour. In particular, the fraction of cross-associated molecules increases with

the enrichment of the solution in spherical molecules and it decreases with the depletion.

We now turn to the systems with formation of intramolecular rings in Figure 5.3.12b for W ∗ = 10

and in Figure 5.3.12c for W ∗ = 100. With the increase of W ∗ and consequent increase of the ring

formation, an increase of the upper critical solution temperature and widening of the miscibility

gap are observed, approximating the associating fluid’s LLE curve to that of the non-associating

fluid. Moreover, the increase of W ∗ produces a similar effect in the VLE’s saturated liquid

branch (purple curve) that also approaches that of the non-associating fluid. An upwards shift

of the boiling temperature of the chain fluid occurs and the dew curve (orange curve) is not

affected. The overall fraction of associated chains increases considerably along the VLE curves

(orange and purple) but remains fairly constant along the LLE curves (red and blue). However,


(a) Associating fluid in the absence of ring formation (W ∗ = 0).

(b) Associating fluid with W ∗ = 10.

(c) Associating fluid with W ∗ = 100.

Figure 5.3.12: VLE and LLE diagrams for the binary system and distribution of bondingstates of the chain molecule, for which the molecular models are described in Section 5.3.1. Foursystems are considered consisting of the non-associating (black dashed curves in a), b) and c))and the associating with W ∗ = 0 (black continuous curves in a), b) and c)), W ∗ = 10 (colourfulcontinuous curves in b)) and W ∗ = 100 (colourful continuous curves in c)). The fractions ofchain molecules in all four bonding states along the saturation curves are represented in thecentral and right-hand side diagrams. The central diagrams correspond to the saturated lines ofthe VLE’s vapour phase (V) and the LLE’s right branch (L1) and the right-hand side diagramscorrespond to the saturated lines of the VLE’s liquid phase (L) and the LLE’s left branch(L2). The bonding states of the chain molecule consist of monomer (‘free’), intramolecular ring(‘ring’), open self-association (‘open, s’) if the molecules associate with other chain and opencross-association (‘open, c’) if the molecules associate with a spherical molecule.

5.4. Modelling solid–liquid equilibrium 147

the extent of cross-association is reduced substantially which explains the upwards movement of

the LLE binodal with the increase of W ∗. Lastly, while in the absence of ring formation, the

fraction of molecules in self-associated open aggregates increased with the enrichment of the

chain component in the saturated liquid 2 phase (L2), a decrease is observed upon the increase

of W ∗.

5.4 Modelling solid–liquid equilibrium

Separation and purification (e.g., distillation, absorption, extraction and crystallisation) consti-

tute essential unit operations in cross-industry processes. Due to the large volumes of solvents

these and other stages of a given process involve, solvent selection is often determinant in the cost

of a particular process. In particular in the pharmaceutical industry, the typical manufacture

process of a drug consists of reaction, extraction, distillation, crystallisation, washing and drying

stages, all of which involve solvents. Kolár et al. [183] state that over 30 % of the efforts of

industrial modellers and experimentalists in the thermodynamics department are put into solvent

selection. Solvent selection is intimately linked to solubility theory as well as solvation promotion

(form change), i.e., solvents may provide with structure stability, bond formation and impact

physical properties of compounds. Solvents may have crucial relevance in the maximisation

of the product’s quality and yield as well as in the operating conditions of the manufacturing

process. In practice, the selection of solvents and anti-solvents for crystallisation mostly relies

on experience, analogy and experimental testing [183]. Testing solvents takes time and resources.

Hence, representatives are selected (solvent classes), leaving out many potentially better options.

Moreover, it is not always possible to generate experimental data due to material constraints,

which tend to occur in the early development phases or in case of impurities. While good

experimental measurements are essential for parameter estimation procedures, it is desirable to

keep experiments to a minimum. As a result, theoretical approaches used in the prediction of

solubility play a key role in today’s drug development process.

The solubility of a compound in a pure or mixed solvent system at a given T and P is defined


by its concentration or any other measure of its proportion in the saturated solution [184]. It is

an equilibrium property and as such, the chemical potential of solute i (µi) in the pure solid

state (S) at T and P equals that in the saturated solution (sat) at temperature T , pressure P

and concentration xsat corresponding to the saturation conditions as in

µSi (T, P, xS

i = 1) = µsati (T, P,xsat). (5.11)

The chemical potential of the solute in a non-ideal saturated solution can be written as

µsati (T, P,xsat) = µ,Li (T, P, xL

i = 1) + RT ln(asati (T, P,xsat)

), (5.12)

where µ,Li is the chemical potential of species i in its reference state (pure liquid), xsat is the

composition vector of the saturated solution and asati is the activity of i in solution given by the

product of the activity coefficient γsati with the molar fraction of i in the saturated solution xsat

i

as in

asati (T, P,xsat) = γsat

i (T, P,xsat)xsati . (5.13)

The reference state of pure i is a liquid at the same temperature T and pressure p of the real

system. It is in fact a supercooled liquid, since T is below T fusi , the fusion temperature of the

solute [185]. Analogously, the chemical potential of i in the solid µSi is defined as

µSi (T, P, xS

i = 1) = µ, Si (T, P, xSi = 1) + RT ln

(aSi (T, P, xS

i = 1)), (5.14)

which equals to µ, Si , since the activity of i in pure solid phase aSi = 1. Combining Equa-

tions (5.11), (5.12) and (5.14), we obtain

ln asati =

(µsati − µ

,Li

)RT

=

(µSi − µ

,Li

)RT

=

(µ, Si − µ

,Li

)RT

, (5.15)

where the thermodynamic state has been omitted for compactness.

The activity of the solute in the saturated solution is thus related to its partial molar Gibbs


energy of fusion ∆gfusi at the system conditions by

ln asati (T, P,xsat) = −∆gfus

i (T, P )RT

, (5.16)

and using the fundamental thermodynamics relation ∆G = ∆H − T∆S we obtain

ln asati (T, P,xsat) = −∆hfus

i (T, P )RT

+ ∆sfusi (T, P )R

. (5.17)

It is not possible experimentally to directly measure the fusion enthalpy ∆hfusi (T, P ) and entropy

∆sfusi (T, P ) at a temperature lower than the fusion temperature. We thus typically resort to the

use of the thermodynamic cycle [186] in Figure 5.4.1, that enables the expression of ∆hfusi (T, P )

and ∆sfusi (T, P ) as functions of known solute specific quantities. We assume incompressibility

of the solution and ambient pressure, thus neglecting pressure effects, a derivation of which can

be found elsewhere [187,188]. The thermodynamic cycle is expressed in terms of enthalpy and

∆sb−→c = ∆sfusi (T = T fus

i )Step 2: Melting process

∆hb−→c = ∆hfusi (T = T fus

i )

∆ha−→d = ∆hfusi (T < T fus

i )∆sa−→d = ∆sfus

i (T < T fusi )

∆ha−→b =∫ T fus

i

TcSp,i dT

∆sa−→b =∫ T fus

i

T

cSp,i

TdT

∆hc−→d = −∫ T fus

i

TcLp,i dT

∆sc−→d = −∫ T fus

i

T

cLp,i

TdT

a d

cb

T

(solid)T fusi

(solid)

(liquid)T fusi

(supercooledT

Step 1:

the solidHeating up

Step 3:

the liquidCooling down

liquid)

Figure 5.4.1: Thermodynamic cycle neglecting pressure effects to solve for the enthalpy andentropy of fusion of solid i at a temperature T lower than its normal melting point T fus

i .

entropy changes, according to which the Gibbs free energy for the path of interest a −→ d can

be calculated by the alternative path a −→ b −→ c −→ d:

• a −→ b, the solid is heated from the operating temperature T to its melting point T fusi ;

• b −→ c, the solid undergoes fusion at its melting point and turns into a (pure) liquid;

• c −→ d, the liquid is cooled down to the operating temperature and turns into a supercooled

(pure) liquid.

The changes in enthalpy and entropy of the pure compound i upon fusion at the saturation


temperature of the mixture are thus given respectively as

∆hfusi (T, P ) =

∫ T fusi

TcSp,i(T, P ) dT + ∆hfus

i

(T fusi , P

)−∫ T fus

i

TcLp,i(T, P ) dT, (5.18)

and

∆sfusi (T, P ) =

∫ T fusi

T

cSp,i(T, P )T

dT + ∆sfusi

(T fusi , P

)−∫ T fus

i

T

cLp,i(T, P )T

dT, (5.19)

where cSp,i and cL

p,i are the heat capacity of the pure solid i and pure liquid i. By definition, at

the melting point, ∆gfusi

(T fusi , P

)= 0, and thus

∆hfusi

(T fusi , P

)= T fus

i ∆sfusi

(T fusi , P

). (5.20)

Lastly, inserting Equations (5.18)–(5.20) in Equation (5.17), the master equation for solid-liquid

equilibrium is obtained:

ln asati (T, P,xsat) =

∆hfusi

(T fusi , P

)R

(1T fusi

− 1T

)

+ 1RT

∫ T fusi

T∆cp,i(T, P ) dT − 1

R

∫ T fusi

T

∆cp,i(T, P )T

dT,

(5.21)

where the difference between heat capacities of the liquid and solid state is given as

∆cp,i(T, P ) = cLp,i(T, P )− cS

p,i(T, P ). (5.22)

It is not straightforward to obtain experimentally the heat capacity for the pure supercooled

liquid (cLp,i) at temperatures below its melting point (T < T fus

i ). Moreover, very little is known

about the form of the liquid heat capacity curve below the melting point for a supercooled

liquid. This discredits the validity of an extrapolation of the liquid heat capacity of an organic

compound [189]. In Figure 5.4.2, experimental heat capacity data [19,20,21,22] for liquid and

solid acetic acid are represented, where the discontinuity happens at the fusion temperature.

There are significantly different extrapolation curves that can be drawn for the liquid heat

capacity at T < T fusi . For this reason, approximations for the difference between liquid and


020406080100120140160

0 50 100 150 200 250 300 350 400020406080100120140160

0 50 100 150 200 250 300 350 400T/K

c p,i/J

K−

1mol−

1

Figure 5.4.2: Experimental heat capacity data of pure acetic acid by [19] (upward-pointingtriangles), [20] (downward-pointing triangles), [21] (circles) and [22] (squares).

solid heat capacities are commonly used and the most popular are summarized in Table 5.4. In

Table 5.4: Main approximations used to calculate ln asati (T, P,xsat).

References lnasati (T, P,xsat) Approximation

[15, 25, 189,190,191,192,193,194,195,196]

∆hfusi

(T fusi , P

)R

(1T fusi

− 1T

)− ∆cp,i(T, P )

R

(ln T

fusi

T− T fus

i

T+ 1) (A) ∆cp,i(T, P ) ≈ ∆cp,i(T fus

i , P )

[23, 171,183,187,190,197,198,199,200,201,202]

∆hfusi

(T fusi , P

)R

(1T fusi

− 1T

)(B) (∆cp,i(T, P ) ≈ 0) or/and T fus

i

T≈ 1

[189,203,204,205]

∆sfusi

(T fusi , P

)R

ln(

T

T fusi

)(C) ∆cp,i(T, P ) ≈ ∆cp,i(T fus

i , P ) ≈ ∆sfusi (T fus

i , P )

approximation (A) it is assumed that the difference between the heat capacities is insensitive

to temperature, equal to its value at the temperature of fusion (T fusi ). In the absence of heat

capacity data, ∆cp,i(T, P ) can be estimated by group contribution methods [206]. Alternatively,

the whole term involving heat capacity can be neglected (second line of Equation (5.21)) or the

∆cp,i(T fusi , P ) can be approximated to ∆sfus

i (T fusi , P ), which correspond to approximations (B)

and (C), respectively.

If data for the difference in heat capacity exists in literature, approximation (A) is better

than approximation (C). It is surprisingly common to come across publications neglecting the


contribution from the difference in heat capacities (approximation (B)), while citing Prausnitz et

al. [190], where (B) is simply stated as a possible approximation to make. However, studies in

literature [202,206,207] have shown that in many cases this contribution makes a significant

impact on the calculated solubility. Before we look into the formation of rings in solid–liquid

equilibria, we present a brief evaluation of the impact of the heat capacity term in the solubility

calculations by comparing approximations (A) and (B), as a correct theory is key to the accurate

modelling of solubility. Indeed, the application of an improved theory to capture association

effects in solid–liquid equilibria is of little use if the solubility calculations are wrong.

5.4.1 The heat capacity in solubility calculations

Let xsat,YESi be the solubility accounting for the ∆cp,i(T fus

i , P ) term according to approximation

(A) in Table 5.4 and xsat,NOi the solubility calculated while neglecting the contribution of this

term according to approximation (B). The relative change in solubility expressed as the absolute

relative deviation in percentage terms (%ARDx) incurred while neglecting the heat capacity

term can be calculated as

%ARDx =∣∣∣∣∣x

sat,NOi − xsat,YES

i

xsat,YESi

∣∣∣∣∣× 100

=∣∣∣∣∣γ

sat,YESi

γsat,NOi

exp[

∆cp,i(T fusi , P )R

(ln T

fusi

T− T fus

i

T+ 1

)]− 1

∣∣∣∣∣× 100,(5.23)

where γsat,YESi and γsat,NO

i are the activity coefficients corresponding to the theory accounting

and neglecting the heat capacity term, respectively. It is possible to get a reasonable estimation

of this error by considering the case close to ideal solubility (γsati ≈ 1) or by approximating

γsat,YESi /γsat,NO

i ≈ 11:

%ARDx ≈∣∣∣∣∣exp

[∆cp,i(T fus

i , P )R

(ln T

fusi

T− T fus

i

T+ 1

)]− 1

∣∣∣∣∣ ,≈ |exp [−f(T )]− 1| ,

(5.24)

1This was tested in the examples and temperature ranges studied in this section, in which it showed to be agood approximation. These tests are not shown in this work.


with

f(T ) = −∆cp,i(T fusi , P )R

(ln T

fusi

T− T fus

i

T+ 1

). (5.25)

The expression for the absolute relative deviation obtained in Equation (5.24) is convenient as it

requires the input of only two solute-specific properties: the difference in heat capacities and the

fusion temperature. We applied Equation (5.24) to six selected compounds and estimated the

expected %ARDx upon neglecting the heat capacity term. The inputs and estimated %ARDx

for the six compounds are compiled in Table 5.5. In Figure 5.4.3a, the %ARDx are plotted as a

function of T/T fusi and in Figure 5.4.3b as a function of f(T ) defined in Equation (5.25).

Table 5.5: Compounds and respective %ARDx calculated at saturation temperature of 298.15 Kfrom Equation (5.24).

# Compound T fusi /K ∆cp,i(T fus

i , P )/(J/(K mol)) %ARDx

(1) diethylstilbestrol [206] 441.8 43.6 37.1(2) stearic acid [208] 342.5 29.3 3.5(3) ibuprofen [206] 347.2 47.3 6.8(4) paracetamol [202] 442.1 87.4 60.7(5) benzoic acid [202] 395.2 58.4 26.4(6) anthracene [209,210] 489.2 6.28 10.5

0

20

40

60

80

0.4 0.5 0.6 0.7 0.8 0.9 1

1005030155

(1)

(2)(3)

(4)

(5)

(6)

%ARDx

T/T fusi

(a)

0

20

40

60

80

100

0 0.2 0.4 0.6 0.8 1

(1)

(2)(3)

(4)

(5)

(6)

%ARDx

f(T )(b)

Figure 5.4.3: Representation of the estimated error from neglecting the heat capacity contribu-tion to solubility calculations (a) as a function of the quotient between operating and fusiontemperatures and (b) as a function of f(T ) given by Equation (5.25). The points represent theestimated error for the compounds in Table 5.5 for T = 298.15 K labelled accordingly.

For a fixed temperature, the larger the fusion temperature and difference between the liquid


and solid heat capacities, the larger is the weight of the heat capacity contribution. Neglecting

the heat capacity term is thus expected to impact the solubility calculations in any solvent at

298.15 K of diethylstilbestrol and paracetamol the most, with an %ARDx of 37.1 % and 60.7 %,

respectively. Although anthracene has the highest fusion temperature, it also exhibits the lowest

∆cp,i among the six compounds, which results in an estimated %ARDx of around 10 %. If

either the fusion temperature or the heat capacity difference is sufficiently small, a very small

difference between approximations (A) and (B) is expected, such is the case of stearic acid with

an estimatimated %ARDx of 3.5 %.

The solubility of stearic acid in benzene and ibuprofen in ethanol were calculated with

SAFT-γ Mie [16] and are represented in Figure 5.4.4 as a function of temperature. In both

260

280

300

320

340

0 0.1 0.2 0.3 0.4

T/K

xstearic acid(a) Stearic acid solubility in benzene.

220

240

260

280

300

320

340

0 0.1 0.2 0.3 0.4 0.5

T/K

xibuprofen(b) Ibuprofen solubility in ethanol.

Figure 5.4.4: Prediction of the solid-liquid equilibria in binary systems a function of temperatureat P = 0.100 MPa. The symbols represent the experimental data and the continuous curvesthe prediction with the SAFT-γ Mie approach accounting for (black) and neglecting (blue) theheat capacity term for: (a) stearic acid in benzene (squares [23]); (b) ibuprofen in ethanol(squares [24], upward-pointing triangles [25], downward-pointing triangles [26], circles [27],diamonds [28] and pentagons [29]).

Figures 5.4.4a and 5.4.4b, the curves for xsat,NOi and xsat,YES

i as function of T are repre-

sented in black and blue, respectively. The points are experimental data from the litera-

ture [23, 24, 25, 26, 27, 28, 29]. In agreement with our %ARDx calculations, the impact of the

heat capacity contribution in the solubility calculations of stearic acid and ibuprofen is similar

(of the order of 6 % at 298.15 K). This is more evident in the ibuprofen diagram since this

compound is more soluble in this particular system. A last diagram was included in this study


for the binary system of water and methanol, in which the effect of the heat capacity is very

evident (Figure 5.4.5). Water [211, 212] has ∆cp,i = 36.1 J/(K mol) and T fusi = 273.15 K and

190

220

250

280

0 0.2 0.4 0.6 0.8

T/K

xCH3OH

Figure 5.4.5: Prediction of the solid-liquid equilibria in a binary system of water and methanolas a function of temperature at P = 0.100 MPa. The symbols represent the experimental data(circles [30], downward-pointing triangles [31] and squares [32]) and the continuous curves theprediction with the SAFT-γ Mie approach accounting for (black) and neglecting (blue) the heatcapacity term.

methanol [213,214] has ∆cp,i = 17.1 J/(K mol) and T fusi = 178.45 K. The low melting point of

methanol indicates that no significant effect of the heat capacity is expected to be observed in

the methanol branch of the solid–liquid equilibrium curve. Since the existence of a peritectic

point is suggested in the literature [215], we focus on the solid–liquid branch between pure ice

and the miscible water–methanol liquid mixture, which is shown in Figure 5.4.5. An %ARDx of

34 % is observed at 190 K.

It is conclusive from this study that neglecting the heat capacity term in solubility calculations

may introduce a significant error. Next, the effect of the W parameter on the solubility curves

of associating model systems is studied.

5.4.2 Effect of the W parameter on solid–liquid equilibria

We look at the effect of ring formation on the solid–liquid curve by considering the solubility

of a chain molecule in two pure and mixed solvents. Schemes of the three model molecules

are depicted in Figure 5.4.6. Each model consists of five equivalent segments that interact


through a Lennard-Jones potential. The solute contains two association sites A and B that

promote inter- and intramolecular hydrogen bonds, and no intermolecular rings. Solvent 1 (S1)

is non-associating and solvent (S2) has one site C that promotes cross-association with site B

of the solute.

Solute Solvent 1 (S1) Solvent 2 (S2)

Figure 5.4.6: Scheme of molecular models of solute, solvent 1 and solvent 2.

We are interested in the direction of the shift of the solubility curve upon increase of the ring

fraction in solution, that we control by varying the value for W . We consider the system with

ring formation with W between 1 and an arbitrarily high number that ensures that the solute

percentage in ring form is 100 %. We also look at the effect of the association energies εHBAB and

εHBBC : in the first scenario εHB

AB = εHBBC = 10εdisp, in the second εHB

AB = εHBBC = 20εdisp and in the

third εHBAB = 8εdisp and εHB

BC = 10εdisp. For each energy scenario we consider four solvent systems:

pure solvent 1 (non-associating), pure solvent 2 (associating), the mixture 80 % S1/20 % S2, and

40 % S1/ 60 % S2. We compile the solubility diagrams obtained in three columns corresponding

to three energy scenarios in Figure 5.4.7.

In Figure 5.4.8, the fractions of solute in free monomer (dashed curves) and ring form (continuous

curves) corresponding to the saturation curves (composition not fixed) for W ∗ = 1 are presented.

The different colours indicate the solvent system.

When we consider the chain aggregates only, the solubility is higher in the system with the

most attractive interactions between the solute and the solvent system, which is the case of

pure solvent 2, represented in blue in Figure 5.4.7. As we add solvent 1 to the mixtures, the

solubility decreases, as less and less hydrogen bonds between solute and the solvent exist, and

therefore the solubility in pure solvent 1 is the lowest.

Let us consider the diagram a) in Figure 5.4.7. The outer limit of the orange curve corresponds

to the solubility in the pure solvent 1 with W ∗ = W/W4 = 1. As we increase the value of W ,


i)e)a)

j)f)b)

c) g) k)

l)h)d)

εHBAB = εHB

BC = 10εdisp εHBAB = εHB

BC = 20εdisp εHBAB = 8εdisp; εHB

BC = 10εdisp

Figure 5.4.7: Range of solid–liquid equilibria curves for W ∗ between 1 and the limit of completering formation in various solvent systems and energies of association.

11.21.41.61.8

0 0.2 0.4 0.6 0.8 1X0; ξ1

T∗

(a) εHBAB = εHB

BC = 10εdisp

11.21.41.61.8

0 0.2 0.4 0.6 0.8 1X0; ξ1

T∗

(b) εHBAB = εHB

BC = 20εdisp

11.21.41.61.8

0 0.2 0.4 0.6 0.8 1X0; ξ1

T∗

(c) εHBAB = 8εdisp; εHB

BC = 10εdisp

Figure 5.4.8: Solute fractions X0 (dashed curves) and ξ1 (continuous curves) in various solventsystems with W ∗ = 1: S1 (orange), S2 (blue), 80 % S1 (green), and 40 % S1 (pink).


the curve naturally shifts in the direction of higher solubility, since a ring does not have sites

free and is therefore more soluble in a non-associating solvent that a chain with two free sites.

The black curve is common to all diagrams and represents the limit at complete ring formation

(100 % solute in ring form) regardless of the ratio between solvents, because at this limit there

are no intermolecular hydrogen bonds. Upon doubling εHBAB, in diagram e), interestingly the

shaded area increases. This is explained by the fact that even though a higher association energy

favours the ring formation (compare continuous orange curves in Figure 5.4.8a and Figure 5.4.8b)

which are more soluble than free monomer or chains, the small fraction of solute molecules that

is not in ring form has free sites that are too energetic. This impacts the solubility more than

the formation of rings, and the result is a decrease in solubility.

Differently, when considering the green solvent mixture (80 %S1), in diagrams b) and f), the

effect observed is an increase of solubility, which is mostly due to more intermolecular bonds

between solute and solvent 2 forming. It is important to realise that after increasing the

percentage of S2 in the solvent mixture enough, the trend is inverted and the increase of ring

fraction reduces the solubility, as observed in the last two rows of Figure 5.4.7. This is explained

by the competition between ring formation and cross-association between solute and solvent 2.

The take away message from this study is that accounting for ring formation can impact

significantly the solubility in both non-associating and associating solvents. The shifting

direction of the solubility curve can be hard to predict as it depends on the balance between

inter- and intramolecular attractive forces. Increasing the value of W can also have opposite

effects in the solid–liquid equilibrium of a given system such as the example in diagram j).

5.5 Case study – ring formation in binary systems of

statins and alcohols

The work presented in this thesis did not involve re-estimation of parameters, which hindered the

analysis of real molecules. In the SAFT-γ Mie framework, being a GC method, the parameters

5.5. Case study – ring formation in binary systems of statins and alcohols 159

are often estimated based on data for pure compounds only. This means that compounds that

form intramolecular hydrogen bonds, such as diols and ether glycols, were included in some of

the estimations, and consequently some of the parameters may be capturing the ring formation

in an effective way; clearly, this is unfortunate, as some key parameters then are not compatible

with the new theory. Moving forward, it is advisable to re-estimate some of the parameters,

but for now we proceed with a study of the effect of the ring formation from a qualitative

perspective.

In this last section we use the modified SAFT-γ Mie EOS in the solubility prediction of two

statins, lovastatin (LVS) and simvastatin (SVS), in a range of simple alcohols with a carbon

chain length between 2 and 8. The statins are 3-hydroxy-3-methylglutaryl coenzyme A reductase

inhibitors and constitute a family of APIs used to lower thfe low-density lipoprotein (LDL)

cholesterol in the blood and reduce its production inside the liver [216,217]. Both LVS and SVS

are prodrugs, i.e. they are metabolised within the body after administration in order to become

active drugs. These drugs are activated by in vivo hydrolysis of the lactone ring [218]. This

lactone ring is common to both statins and it has been suggested by Hutacharoen [180] that it

promotes intramolecular hydrogen bonding; however, there is no experimental data available

supporting or contradicting this hypothesis. The chemical structures for LVS and SVS differ

essentially in one methyl group as seen in Figure 5.5.1 and the breakdown of the two compounds

into constituent groups according to the SAFT-γ Mie EOS is shown in Table 5.6.

Table 5.6: Group make-up of simvastatin (SVS) and lovastatin (LVS) molecules according tothe SAFT-γ Mie approach.Solute CH3 CH2 CH CH= C= cCH2 COO OH Clovastatin(LVS)

4 3 8 3 1 3 2 1 0

simvastatin(SVS)

5 3 7 3 1 3 2 1 1

The experimental data for the solid state of the statins used in the calculations, i.e., the fusion

enthalpy, fusion temperature and the heat capacity difference as seen in approximation (A)

shown in Table 5.4, were those used by Hutacharoen [180] and it is summarised in Table 5.7.

The solvents we consider in the prediction of the solubility of statins of this section are 1-alcohols


H

e1

e1

e1e1

e1

e1

(a) lovastatin (LVS)

H

e1

e1

e1e1

e1

e1

(b) simvastatin (SVS)

Figure 5.5.1: Chemical structures of lovastatin (LVS) and simvastatin (SVS), highlighting thegroups (colourful spheres) and association sites (black e1 and white H spheres) considered inthe SAFT-γ Mie approach. Association may occur only between association sites of differenttypes (black-white association).

Table 5.7: Literature data [14] characterising the solid state at atmospheric pressure of the twostatins used in the calculations.Solute T fus

i /K ∆hfusi

(T fusi

)/(J/mol) ∆cp,i(T fus

i )/(J/(K mol))

lovastatin(LVS)

445.6 43169 177

simvastatin(SVS)

412.6 32170 149


with a number of carbons between 2 and 8. In the SAFT-γ Mie framework, 1-alcohols are

composed of a CH3 and CH2 groups and a CH2OH or a OH group. Recent work has been based

on using CH2OH [73] but the interaction between the groups CH2OH and OH is currently under

development, which is the reason why here we have used the group OH only. The statins and

1-alcohols are modelled with two types of association sites, the OH group contains two e1 sites

and one H site and the acetate group (COO) contains two e1 sites. Association may occur only

between association sites of different types. In a mixture of LVS/SVS with an alcohol, hydrogen

bonding occurs between sites e1 of groups COO and site H of group OH. No fitting was carried

out. The group parameters used as well as the details of the estimation procedure followed can

be found in the most recent work by Hutacharoen et al. [73, 180], and elsewhere [16, 90, 179].

For clarity, the parameters are compiled in Appendix D.

We start by considering the solubility predictions of LVS and SVS in 1-alcohols with chain

length between 3 and 8 using SAFT-γ Mie in its standard formulation [16]. According to

Nti-Gyabaah et al. [15] the experimental solubility of LVS in the six different 1-alcohols is

ordered as ethanol < octanol < hexanol ≈ pentanol < propanol < butanol, peaking in butanol.

No other sources were found supporting or not this ranking, but the solubility of LVS in ethanol

measured by Sun et al. [33] was in agreement with that measured by Nti-Gyabaah et al. [15].

The solubility of SVS was similarly measured in the six alcohols by both Nti-Gyabaah et al. [35]

and Aceves-Hernández et al. [34], who disagreed both in the values and the solvent ranking of

the solubility of SVS. According to Nti-Gyabaah et al. [35], the solvent ranking corresponding

to the solubility of SVS does not match that found for LVS [35] and is ordered as ethanol <

butanol ≈ propanol < pentanol ≈ octanol < hexanol, peaking in hexanol. On the other hand,

the solubility measurements of Aceves-Hernández et al. [34] indicate that the solvent ranking

for the solubility of SVS matches that found previously for LVS [35], i.e., ethanol < octanol

< hexanol ≈ pentanol < propanol < butanol, peaking in butanol. The theoretical predictions

neglecting ring formation and the experimental data are presented in Figure 5.5.2.

The theory seems to capture both the range of solubility and the solubility trends, but it fails to

predict the solvent ranking, regardless of the experimental source. According to SAFT-γ Mie,

the solvent ranking is the same for the solubility of LVS and SVS where the solvents are


ordered by decreasing chain length, i.e., octanol < heptanol < hexanol < pentanol < butanol

< propanol < ethanol. The theoretical solubility ranking is explained by the fact that the

dispersion interactions between the statins and the alcohols become more relevant with the

increase of the alkyl chain of the alcohols. The experimental data is reduced and shows a high

level of disagreement which, but in case in case it is correct, one possible explanation for the

unusual solvent rankings could be that the extent of ring formation of a statin in solution

depends on the alcohol used as solvent.

At the end of this section, we attempt to use the ring modified version of SAFT-γ Mie to

reproduce the solvent ranking suggested by Nti-Gyabaah et al. [15] for the solubility of LVS in

the six 1-alcohols. We attempt this by manipulating the value of the parameter W . Before that,

we take a closer look at the effect of the parameter W on ring formation and solubility and the

transferability of W between the two statins, since these exhibit the same ring type.

5.5.1 Transferability of W

In the modelling of the statins, Hutacharoen [73] considered an effective treatment for the

intramolecular hydrogen bond. In this treatment the association sites that are involved in

intramolecular association or are sterically hindered due to a preferred conformation of the

molecule, are switched off. It is an effective method since it only treats one possible effect

of intra-molecular association which is the prevention of the sites involved in a ring from

associating intermolecularly. This method has shortcomings, as it does not account for the

energy of an intra-molecular bond and it involves a decision of binary nature which does not

reflect real systems: accounting for intermolecular association (no rings) or turning the sites

off (no association). Moreover, there is no systematic way to know if one should apply this

approach or not.

We now predict the solubility of the statins in ethanol, in Figures 5.5.3a and 5.5.3b, with the

modified version of SAFT-γ Mie to account for intramolecular hydrogen bonding (IntraMHB),

while varying W ∗ = W/W5 by factors of 10.


270

280

290

300

310

320

330

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

ethanolpropanolbutanolpentanolhexanolheptanoloctanol

T/K

xlovastatin

(a) Solubility of LVS in various alcohols.

270

280

290

300

310

320

330

0.00 0.02 0.04 0.06 0.08 0.10


T/K

xsimvastatin

(b) Solubility of SVS in various alcohols.

Figure 5.5.2: Solubility predictions in mole fraction (x) of LVS (a) and SVS (b) in variousalcohols at ambient pressure p = 0.1 MPa as a function of temperature. The continuous curvesrepresent the predictions of the SAFT-γ Mie approach for the solubilities in ethanol (pink),1-propanol (purple), 1-butanol (dark blue), 1-pentanol (light blue), 1-hexanol (green), 1-heptanol(orange) and 1-octanol (red). The empty circles represent the experimental solubility of a) LVSby Nti-Gyabaah et al. [15] and Sun et al. [33] and of b) SVS by Aceves-Hernández et al. [34]in the various solvents. The filled circles represent the experimental solubility of b) SVS byNti-Gyabaah et al. [35] in the various solvents.


270

280

290

300

310

320

330

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

W ∗ = 104 103 102 10 1

T/K

xlovastatin

(a) Solubility of LVS in ethanol.

270

280

290

300

310

320

330

0 0.02 0.04 0.06 0.08 0.1

W ∗ = 104 103 102 10 1

T/K

xsimvastatin

(b) Solubility of SVS in ethanol.

Figure 5.5.3: Solubility predictions of LVS (a) and SVS (b) in ethanol at ambient pressurep = 0.1 MPa as a function of temperature. The solubility xstatin is given in molar fraction.The dashed and the continuous black curves represent the predictions with the SAFT-γ Mieapproach with and without the “switching off sites” treatment, respectively. The coloured linesrepresent predictions with the SAFT-γ Mie approach including the modified association termto account for ring formation, with values for the W ∗ parameter of 1 (purple), 10 (pink), 102

(blue), 103 (green) and 104 (orange). The experimental data is represented by circles [33] in a)and [34] in b) and by diamonds [15] in a) and [35] in b).


The best W ∗ found to fit the experimental data for LVS [15, 33] was around 225 and for SVS,

based on the data of Aceves-Hernández et al. [34] (circles in Figure 5.5.3b), was around 100. The

best W ∗ values for both statins in ethanol are fairly similar and the curves with increasing W ∗

seem to be approaching the limit of complete ring formation (solutes in 100 % ring form) in a

similar way too. The results are encouraging but not conclusive with regards to the value of W ∗

being a transferable parameter between molecules with the same ring type in a specific solvent.

In order to draw a conclusion on the transferability of the parameter W ∗, further investigation

on ring formation in different systems for which the trustworthy experimental data is available

must be carried out.

5.5.2 Impact of the alcohol length on the ring fraction

In Figure 5.5.4a the bonded fraction of LVS is represented at solid–liquid saturation conditions.

The dashed curves are coloured by alcohol according to the key and correspond to calculations

of the standard SAFT-γ Mie accounting for open-chain aggregates only. After including the

contribution from intramolecular rings in the theory with W ∗ = 100 we observe a higher fraction

of molecules bonded.

We break the fraction of bonded molecules with W ∗ = 100 into chains and rings in Figure 5.5.4b.

If one ranks the solvents by the extent of InterMHB in LVS, the same ranking is obtained

when considering the absence or the presence of rings, which is octanol < heptanol < hexanol <

pentanol < butanol < propanol < ethanol. However, the shape of the curves corresponding

to the extent of intermolecular association as function of temperature is markedly different.

When accounting for ring formation, the increase in the number of intermolecular bonds is

not monotonous with the decrease in temperature for octanol and heptanol, which exhibit a

maximum at ≈ 390 K and ≈ 380 K, respectively. Depending on the solvent, the fraction of

molecules in ring form along the saturation curve has different trends too, as it seems to increase

slightly in longer alcohols and decrease substantially in shorter ones. The solvent ranking by

the extent of IntraMHB is the opposite to that found for the extent of InterMHB due to

the lower density of competing OH groups in the surroundings of a molecule of LVS in longer


alcohols than in shorter ones.

5.5.3 Impact of W on ring fraction and solubility

The ring fraction of LVS as a function of W ∗ is shown for mixtures with the various alcohols in

Figure 5.5.5a. The plot suggests that the same value of W may lead to different ring fractions

which are strongly dependent on the composition of a system. The highest discrepancy between

ring fractions seems to be obtained for lower values of W . Moreover, the increase of W ∗ (ring

fraction) has different effects on the solubility depending on the solvent we are considering, as

can be seen in Figure 5.5.5b. It is also clear that for a fixed W ∗, the solvent ranking is the same

apart from ethanol, which can drop two positions at a W ∗ > 500.

Considering the experimental data [15, 33] in Figure 5.5.2a, SAFT-γ Mie without the ring

contribution overestimates the solubility of LVS in ethanol and propanol and underestimates it

in all the other alcohols. According to Figure 5.5.5b, considering ring formation results in a

decrease of solubility in all alcohols but heptanol and octanol. Accounting for ring formation

(W > 0) can thus improve the predictions for the solubility of LVS in ethanol, propanol and

octanol but not in the other alcohols. The AARDs respective to the solubility predictions with

SAFT-γ Mie in its original and modified for rings versions are presented in Table 5.8. The degree

Table 5.8: Average Absolute Relative Deviations (AARD%) between reported experimentaldata [15] and SAFT-γ Mie predictions for the solubility of LVS in alcohols both in the absence(“No rings”) and in the presence of rings (“With rings”) with a fitted W ∗ per solvent. NP is thetotal number of experimental points.Solvent Ethanol Propanol Butanol Pentanol Hexanol Octanol All

NP 19 9 10 9 7 10 64

No rings 109 % 11.8 % 23.3 % 12.0 % 25.7 % 40.1 % 48.4 %

With rings 5.01 % 8.38 % 23.3 % 12.0 % 25.7 % 32.5 % 15.9 %W ∗ 225 10 0 0 0 109

of improvement in these three solvents is dependent on the W selected. Following Figure 5.5.5b,

the solubility of octanol increases slighly with the increase of W without ever reaching a value


260280300320340360380400420440

0.8 0.84 0.88 0.92 0.96 1


1−X0, lovastatin

T/K

(a) Extent of association in the absence (dashed curves), i.e. W ∗ = 0, and presence(continuous curves) of ring formation with W ∗ = 100.

260280300320340360380400420440

0 0.2 0.4 0.6 0.8 1

260280300320340360380400420440

0 0.2 0.4 0.6 0.8 1fchains; ξ1

T/K

chains rings

(b) Fraction of LVS in open-chain aggregates and intramolecular rings withW ∗ = 100.

Figure 5.5.4: Fractions of molecules of LVS a) bonded inter- and intramolecularly in theabsence of ring formation (dashed curves) and in the presence of ring formation with W ∗ = 100(continuous curves), and b) in chain and ring form with W ∗ = 100, in the various solvents.


0.00.10.20.30.40.50.60.70.80.91.0

0 200 400 600 800 1000


ξ1

W∗

(a) Ring fraction of LVS as a function of W ∗ in various alcohols at 312 K.

0.0040.0050.0060.0070.0080.0090.0100.0110.0120.0130.014

0 200 400 600 800 1000


xlo

vast

atin

W ∗

(b) Solubility of LVS as a function of W ∗ in various alcohols at 312 K.

Figure 5.5.5: Relationship between a) the fraction of LVS in ring form and the parameter W ∗,and b) the solubility of LVS as a function of W ∗, in various alcohols at constant temperature312 K and ambient pressure 0.1 MPa.


as high as the experimental one. For this reason we chose W ∗ = 109 for lovastatin in ethanol,

a value high enough to ensure 100 % of LVS in ring form. The W for ethanol and propanol

were fitted to the experimental data. For the remaining three alcohols, the contribution of the

ring form of LVS was neglected (W ∗ = 0). The average AARD for the solubility of LVS in

the different alcohols considering the 64 experimental points is improved from 48.4 % to 15.9 %

upon consideration of ring formation.

5.5.4 Impact of W on solvent ranking

As mentioned before, the alcohol ranking obtained experimentally [15,33,34] for LVS may not

be accurate due to the paucity of data and the order of magnitude of the solubility being so low.

Nevertheless, as an exercise, we close this chapter with an attempt to reproduce the solvent

ranking obtained experimentally.

The ranking obtained in the absence of ring formation (Figure 5.5.2a) will not be affected by

considering a fixed W ∗ (with the exception of ethanol), as is clear from Figure 5.5.5. We will

thus select a W ∗ per solvent, assuming that the value for W ∗ is not a function of the solvent

system. For this effect, a helpful relationship to look at is that describing the solubility of LVS

at 312 K as a function of the ring fraction, which is shown in Figure 5.5.6. Here, we can see

how the solubility in ethanol drops considerably with the formation of rings and the opposite is

observed for octanol. After studying Figure 5.5.6, we can infer that to order the solubilities

consistently with the experimental solvent ranking, one must assume that the extent of ring

formation (ξ1) is the highest in ethanol and octanol and the lowest in pentanol and hexanol,

which is somewhat unexpected. Assuming that W ∗ is a function of the solvent, there are

plausible explanations supporting that extent of LVS in ring form is at its highest in the shorter

and longer alcohols and at its lowest in butanol. If shorter alcohols (ethanol and propanol)

have higher propensity to self-associate than the longer-chain ones [219], these are expected

to compete less for association with the ring-forming sites in LVS, enabling a higher ξ1. In

addition, longer alcohols are more susceptible to steric hindrance and consequently are expected

too associate less with the ring-forming sites of LVS, enabling a higher ξ1. Setting W ∗ = 104 to


0.0040.0050.0060.0070.0080.0090.0100.0110.0120.0130.014

0 0.2 0.4 0.6 0.8 1


xlo

vast

atin

ξ1

Figure 5.5.6: Solubility of LVS as a function of ring fraction in various alcohols at 312 K.

270

280

290

300

310

320

330

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014


T/K

xlovastatin

Figure 5.5.7: Solubility predictions of LVS in various alcohols at ambient pressure p = 0.1 MPaas a function of temperature. The continuous curves represent the predictions of the SAFT-γ Mieapproach modified to account for ring formation for the solubilities in ethanol with W ∗ = 104

(pink), 1-propanol with W ∗ = 103 (purple), 1-butanol with W ∗ = 20 (dark blue), 1-pentanolwith W ∗ = 103 (light blue), 1-hexanol with W ∗ = 103 (green) and 1-octanol with W ∗ = 104

(red). The symbols represent the experimental solubility of LVS [15, 33].


ethanol, W ∗ = 103 to 1-propanol, W ∗ = 20 to 1-butanol, W ∗ = 103 to 1-pentanol, W ∗ = 103

1-hexanol and W ∗ = 104 to 1-octanol, the solubility predictions with the newly developed

theory obtained are represented in Figure 5.5.7. We succeeded in matching the ranking with

the exception of ethanol for which could be due to proximity effects. With this, we come to the

end of our qualitative analysis of ring impact in real systems.


The final objective of this work is to improve the property prediction of EOSs compatible

with the framework of Wertheim. Accordingly, the impact of the new theory accounting with

intramolecular in addition to intermolecular hydrogen bonds (no intermolecular rings included)

was studied in models systems in pure single phase and coexistence densities, as well as in a

binary system at VLE, LLE and SLE.

In order to ensure that the calculations of the solubility were correct, the impact of neglecting

the heat capacity term in the solid–liquid equilibrium working equation was assessed. The

method to estimate the impact of the heat capacity term in the calculations only required the

knowledge of two pure solute properties: the difference in heat capacity of the liquid and the

solid (∆cp,i(T fusi )) at the melting point and the melting point (T fus

i ). The heat capacity difference

term was shown to be negligible in the solubility modelling in calculations at temperatures

close to the solute’s melting point or in solutes with a low value for the heat capacity difference.

A possible indication that this term might be significant is if in the solubility calculations

neglecting the term, the saturation temperature is overestimated, since adding the heat capacity

contribution results in a downwards shift of the saturation curve.

The effect of the competition between IntraMHB and InterMHB was studied in two-site

model systems using SAFT-VR Mie and the results were in accordance with the past literature

[18, 105, 121, 131]. The formation of intramolecular rings increases the critical temperature, and

the vapour pressure at low temperature. At high temperature, there are less free monomers

and the rings are less volatile than the monomers which results in the decrease of the vapour


pressure. It was shown that accounting for rings can impact significantly all phase equilibria

predictions and the vapour phase seems to be the most affected as here the intermolecular

association is usually very limited.

The theory was also applied to solubility of lovastatin and simvastatin in several 1-alcohols

using SAFT-γ Mie. The availability of experimental data was not abundant and presented a

limitation, since the solvent ranking suggested was not supported by more than two studies [15,34].

No re-estimation of parameters accounting with the new framework was carried out, which

together with the paucity of experimental data restrained the evaluation of the predictions

from being conclusive. Nevertheless, the results were optimistic, suggesting that the value of

W is transferable across molecules with the same ring type. A fixed value of W can result in

different ring fractions depending on the temperature, density and mixture composition. For

this reason we hope that W is independent of all these variables, but it requires further study,

since the study with real molecules was not conclusive. Finally, accounting with rings can

change the ranking order of solvents (relevant for solvent selection procedures) and can change

the predictions in different directions depending on the system under consideration.

Chapter 6

CLOSING REMARKS

One limitation of the statistical associating fluid theory (SAFT) is that inherited from

Wertheim’s first-order thermodynamic perturbation theory (TPT1): neglecting the forma-

tion of associated ring-like aggregates, impairing an accurate description of the properties of

systems where the fraction of molecules involved in hydrogen-bonded rings is significant, e.g.,

in systems comprising diols, hydrogen fluoride, ether glycols, glycerol, phenyl acids, ibuprofen,

ketoprofen, di-carboxylic acids, lovastatin, simvastatin, mevastatin.

6.1 Summary of research achievements described in the

thesis

Going back to the foundations of TPT1, the fundamental graph sum has been modified

to account explicitly for the ring clusters, based on a method previously used by Sear and

Jackson [105].

Within the context of our approximations, the intramolecular ring and the intermolecular ring

are accounted for in a unified treatment, the only difference between the two types of rings being

that the former has size τ = 1 and the latter τ ≥ 2. The new residual association term, coupled

with the modified mass action equations, can be employed to describe the competition between

173

174 Chapter 6. CLOSING REMARKS

different association aggregates, namely monomers, linear-chain, branched-chain, intramolecular

rings and different sizes of intermolecular rings. We have, for the first time, extended the

association residual term (stemming from TPT1) for mixtures with an arbitrary number of

components and association sites, to account for the simultaneous presence of ring aggregates

of different types, including those formed by intermolecular hydrogen bonding (InterMHB)

between an arbitrary number of molecules and those formed by intramolecular hydrogen bonding

(IntraMHB) between any two sites within a given molecule. One extra parameter per ring type

is introdiced in the theory: Wi,ab,R, the density probability of the two site-carrying segments,

part of the same molecule or open-chain aggregate, to be at bonding distance.

The newly developed theory is incorporated in two existing SAFT equations of state: SAFT-VR SW

and SAFT-γ Mie, and tested for a number of representative model and real systems. In order

to assess the theory, we focus on the simplest cases that had already been considered in previous

work, including additional effects, such as the value of W and the increase on the number of

intermolecular-ring-forming sites. The analysis is completed with a study of the solubility of

lovastatin and simvastatin in a range of 1-alcohols, examining both solvent and ring formation

effects in an attempt to reproduce an unusual trend observed experimentally.

The novelty of the research presented in the thesis can be summarised by the key contributions

described below.

• The first-order thermodynamic perturbation theory (TPT1) and previous extensions to

account for ring formation by association are collected and summarised in chapter 1, where

the capabilities and limitations of the methodology are highlighted.

• A systematic procedure to derive the expression for the residual association contribution

to the Helmholtz free energy of fluid systems is presented. This procedure consists of

counting the number of aggregates. The procedure is based on those previously presented

for specific cases [18,108] but is generic and valid for all different aggregate kinds.

• The translation of Wertheim’s formalism into a framework that allows one to account for

any kind of aggregates as contributors to the free Helmholtz energy is undertaken.

6.2. Conclusions 175

• The newly developed framework is applied to account for ring aggregates both of inter- and

intramolecular nature in an unified theory. The unified theory is of equivalent simplicity to

the TPT1 and written in a format compatible or use in any EOS that accounts with the

association contribution to the Helmholtz free energy as a perturbation, given a reference

system with known properties.

• The effect associated with the formation of inter- and intramolecular rings in pure and

binary mixtures at both single-phase and coexistence conditions is studied;

• The impact of the difference in the liquid and solid capacities of the pure solute in solubility

predictions is assessed.

• The newly developed theory is applied to a case study of the solubility of statins in simple

alcohols and the predictions are compared with the results obtained from the standard

TPT1.

6.2 Conclusions

The framework developed here provides the expressions for the extension to rings in addition to a

generic formulation to make further future extensions to account for other association aggregates,

such as mixed-component rings, double bonding or any other aggregate corresponding to an

irreducible graph. The residual association Helmholtz free energy developed in the current work

can be applied to systems where rings are formed by association of inter- or intramolecular

nature and the treatment reduced to the standard TPT1 description in the absence of ring

formation.

The increase in the value of the parameter W promotes an increase in ring formation since

it is related to the probability of the two site-carrying segments of a chain finding each other.

As a first approximation, the parameter W is considered to be independent of temperature,

density and fluid composition. A deeper analysis is required to assess its dependence on the

mixture components. A good approximation for W in the formation of intermolecular rings

is the end-to-end distribution function as suggested by Sear and Jackson [105]. In the case of

176 Chapter 6. CLOSING REMARKS

intramolecular rings, we suggest that the parameter is estimated from target experimental data.

The formation of intramolecular rings is favoured by the increase of the association energy

between the sites for which Wab > 0, low temperature and low density. The formation of

intramolecular rings leads to an increase in the vapour–liquid critical temperature. In a pure

system, a system with ring-like aggregates is less volatile than one of monomers or chains.

However, at low temperatures, the vapour pressure is higher in an associating fluid with rings

because the one without contains long open-chain aggregates.

With regards to the formation of intermolecular rings, these too are favoured by the increase

of the association energy between the sites for which Wab > 0, low temperature and low

density (but below a certain density, intermolecular rings are no longer formed). The formation

of intermolecular rings decreases the critical point and increases the packing fraction of the

saturated gas phase at a given temperature. The highest proportion of rings is observed for

small ring sizes such as τ = 4. Accounting for the formation of intermolecular rings does not

appear to have much impact in the calculations for models with three or more sites.

6.3 Future work

Currently, a publication based on Chapter 3 entitled “Ring Formation in a Statistical Associating

Fluid Theory Framework” is in the process of being submitted, and applications of the theory

to real systems of chemical/pharmaceutical interest will be the subject of future publications.

In the application of the theory to real systems with a modified SAFT-γ Mie treatment there

was a main obstacle. The group-contribution parameters are usually estimated with the original

theory employing experimental data including compounds that form ring-like aggregates. Indeed,

the ring effects are most likely captured in the parameters in an effective manner. In this respect,

it will be particularly interesting to re-estimate the association parameters for the alcohols

(OH, CH2OH) and the acetate COO groups. Only when the parameter estimation procedure is

consistent with the underlying theory will it be possible to draw conclusive statements about

the transferability of the parameter W and evaluate the extent to which the predictions are

6.3. Future work 177

improved. Moreover, it would be highly desirable to compare our theory to simulation results

for freely jointed and rigid chains and understand if the parameter W is in fact function of

density, temperature or composition.

The framework presented here could be applied to extend the association term further to account

for doubly bonded dimers for multiple sites [112, 115]. This has been attempted before by

Janeček and Paricaud [220], but their approach is not appropriate for a molecule with more

than 2 sites.

Despite the room for further theoretical improvements, the EOSs of the SAFT family and in

particular the SAFT-γ Mie EOS, at its current state, are extremely powerful methods for property

prediction of fluids since they allow for fast calculations (being mostly algebraic approaches)

with high accuracy. The capabilities of SAFT are of key interest for both academic research

and industrial product/process design and optimisation and can save enormous amount of time

and human/material resources currently spent to run experiments. Thus, the most valuable

investment at the moment is probably in the parametrisation of more molecules/molecular

groups, for which high quality of experimental data is paramount.

Appendix A

Derivation of the property of

independence between sites

Given two disjoint sets of association sites γ1 and γ2, independence means that

Xγ1∪γ2 = Xγ1Xγ2 , (A.1)

where Xγ is the fraction of molecules with sites γ free. We can write the fractions of molecules

with the sites in the set δ free as [17]

Xδ

X0=

∑ψ⊆Γ−δ

∑(γ1,γ2,...∈P(ψ)

)∏i

cγi , (A.2)

where the first sum includes the improper set ∅ and we follow the convention that the product

equals 1 for ψ = ∅. The functions cδ are defined as [44]

cδ = 1N

∂∆c(0)

∂Xδ

. (A.3)

Considering the fundamental graph sum studied in this work

∆c(0)

N= 1

2∑a∈Γ

∑b∈Γ

ρXaXb∆ab + 12∑a∈Γ

∑b∈Γ

NRS∑o=1

1τo

(∆ab)τoWab,oρτo−1 (Xab)τo , (A.4)

178

179

we find that

ca = ρ∑b∈Γ

Xb∆ab (A.5)

for |δ| = 1, and

cab =∑b∈Γ

NRS∑o=1

(∆ab)τoWab,oρτo−1 (Xab)τo−1 (A.6)

for |δ| = 2 in the case of a being a ring forming site. If a is not involved in ring formation, then

Wab = 0 for all b ∈ Γ and all ring sizes and cab = 0. Finally, since ∆c(0) is not a function of Xδ

for any |δ| ≥ 3, we have that

cδ = 0, |δ| ≥ 3. (A.7)

In the absence of ring formation, we can rewrite Equation (3.62) as

Xδ

X0=

∏a∈Γ−δ

(1 + ca), (A.8)

and since 1 + ca = XΓ−a/X0,Xδ

X0=

∏a∈Γ−δ

XΓ−a

X0. (A.9)

By convention, X∅ = 1, which means that

1X0

=∏a∈Γ

XΓ−a

X0=

∏a∈Γ−δ

XΓ−a

X0

∏a∈δ

XΓ−a

X0= Xδ

X0

XΓ−δ

X0(A.10)

i.e., X0 = XδXΓ−δ. Using the same argument, we also have the slightly more general equality

Xδ =∏

β1,...,βM∈P(δ)

Xβi . (A.11)

Let us now consider the case where two sites, A and B, can form a ring. In that case, cAB 6= 0,

180 Appendix A. Derivation of the property of independence between sites

and Equation (A.8) is no longer valid. Going back to Equation (A.2), we find that

Xδ

X0=

∏a∈Γ−δ

(1 + ca) + cAB∏

a∈Γ−δa6=A,B

(1 + ca) , if A and B ∈ Γ − δ

∏a∈Γ−δ

(1 + ca) , otherwise

(A.12)

In particular, we have that

XA

X0=

∏a∈Γ−A

(1 + ca) (A.13)

XB

X0=

∏a∈Γ−B

(1 + ca) (A.14)

XAB

X0=

∏a∈Γ−A,B

(1 + ca) (A.15)

1X0

=∏a∈Γ

(1 + ca) + cAB∏

a∈Γ−A,B(1 + ca) (A.16)

Due to the presence of cAB in the expression for 1/X0, we have

( 1X0

)(XAB

X0

)6=(XA

X0

)(XB

X0

),

and therefore also XAB 6= XAXB. Independency of sites therefore is not generally true for

systems with ring-forming molecules. We note that other relationships between the Xδ variables

exist. For example, one can show that XΓ−ABXAB = X0.

Appendix B

Molecular graphs

Wertheim’s theory was developed around spherical molecules with association sites that promote

hydrogen bonding. The graphical aggregates used in the expansions are referent to such models.

However, being constrained to a segment basis implies that to treat non-spherical molecules, the

chain formation contribution must be accounted for too. And in order to isolate the association

contribution, the chain contribution is discounted posteriorly. In this appendix we describe the

procedure to go from segment-based graphs to molecular graphs. Considering molecular graphs

from the start, as done in Chapter 3 is a much simpler alternative to derive the association

contribution of a fluid, since the chain contribution does not have to be included and therefore,

the different segments do not have to be acknowledged explicitly.

1 1

3m

m...2...2 m...1

2

Figure B.0.1: Each chain is formed by m monomers that associated selectively; sphere (1) can

bond to sphere (2) only, sphere (2) can bond to sphere (1) and sphere (3) only, ... and sphere

(m) can bond to sphere (m − 1) only. Upon the increase of the association energy and the

decrease of the association range, the spheres become irreversibly bonded.

181

182 Appendix B. Molecular graphs

To form a non-associating chain (Figure B.0.1), we take te sticky limit of the bonds as seen

previously. The segments belong to different species that bond according their sequence in the

chain molecule, i.e. there are m segment species in a chain of m segments and site B of segment

(1) bonds only to site B of (2), A of (2) to B of (3) ... and B of (m− 1) to A of (m). For that

effect, the segments are considered explicitly:

AchainWert =NkT

[(lnX(1)

B + 1−X(1)B

)+(lnX(m)

A + 1−X(m)A

)

+m−1∑α=2

lnX(α)0 + 1−X(α)

A −X(α)B + X

(α)A X

(α)B

X(α)0

− ∆c∗chainN

, (B.1)

with∆c∗chainN

=m−1∑α=1

ρX(α)A X

(α+1)B ∆, (B.2)

where the superscript (α) indicates the segment species and ∗ indicates the quantity is segment-

based. Considering the sticky limit, as we have done in Section 2.3.2, the chain contribution is

obtained.

Now consider that the resulting molecule is associating and has an association site in each end

(Figure B.0.2). If the association can only result in the formation of linear chains, we introduce

a new contribution to the fundamental graph sum (∆c∗open).

1 1

3m

m...2...2 m...1

2

Figure B.0.2: Each associating chain molecule is formed as described in Figure B.0.1.

We can write the contribution from chain and hydrogen bond formation as

Achain+openWert = NkT

m∑α=1

lnX(α)0 + 1−X(α)

A −X(α)B + X

(α)A X

(α)B

X(α)0

− ∆c∗chainN

−∆c∗open

N

,(B.3)

183

with∆c∗open

N= ρX

(m)A X

(1)B ∆AB. (B.4)

Each molecule has a segment of each species and only one site A (first segment) and one

site B (last segment). For this reason, the fraction of segments (1) with site A free (X(1)A ) is

equivalent to the fraction of molecules with site A free (XA), and in the same way, X(m)B = XB.

Furthermore, in the absence of rings, X(α)0 = X

(α)A X

(α)B , and therefore,

AopenWert =Achain+open

Wert − AchainWert ,

=NkT[ln(X

(1)A X

(m)B

)−X(1)

A −X(m)B + 2−

∆c∗open

N

],

=NkT[lnX0 + 1−XA −XB + XAXB

X0− ∆copen

N

],

(B.5)

with∆copen

N= ρXAXB∆AB, (B.6)

where the last equality and ∆copen are written in terms of fractions of molecules and the

chain term is not included. We finally consider association into linear-chain aggregates and

intramolecular rings. The expression for all contributions is given in terms of fractions of

segments and segment-based graphs as,

Achain+open+ringWert =NkT

m∑α=1

lnX(α)0 + 1−X(α)

A −X(α)B + X

(α)A X

(α)B

X(α)0

−∆c∗chain

N−

∆c∗open

N−

∆c∗ring

N

],

(B.7)

with the segment-based ring graph given as

∆c∗ring

N= X

(1)0 X

(2)0 . . . X

(m)0 (ρ∆)m−1WAB∆AB. (B.8)

184 Appendix B. Molecular graphs

Using,

law of mass action equations

for the covalent links

ρX

(α+1)B ∆ = X

(α)B

X(α)0,

ρX(α)A ∆ = X

(α+1)A

X(α+1)0

,

α = 1, . . . ,m− 1 (B.9)

and

law of mass action equations

for the hydrogen bonds

c∗A = ρX(1)B ∆AB = X

(m)B

X(m)0− 1,

c∗B = ρX(m)A ∆AB = X

(1)A

X(1)0− 1,

1− X(α)A X

(α)B

X(α)0

= (ρ∆)m−1cABm∏i=1

X(i)0 , α = 1, . . . ,m,

(B.10)

allowed us to write the relationship,

(ρ∆)m−1X(1)0 X

(2)0 . . . X

(m)0 = 1

2

X(m)B

1 + c∗A+ X

(1)A

1 + c∗B

. (B.11)

Using these relations in Equation (B.7) leads to the association contribution from linear-chain

aggregates and intramolecular hydrogen bonds in terms of fractions of molecules, as in

Aopen+ringWert =Achain+open+ring

Wert − AchainWert ,

=NkT[lnX0 + 1−XA −XB + XAXB

X0− ∆copen

N− ∆cring

N

],

(B.12)

with∆cring

N= XABWAB∆AB. (B.13)

Here, as given by Equation (3.87),

X0 = 1cAB + (1 + cA)(1 + cB) , (B.14)

185

with cAB, cA and cB defined in Equation (3.85) and Equation (3.86) as

cA = ρXB∆AB, (B.15)

cB = ρXA∆AB, (B.16)

and

cAB = WAB∆AB. (B.17)

Appendix C

Direct derivation of the law of mass

action equations

We now give a direct derivation of the mass action equations, without appealing to Equation (3.62)

from Wertheim. This lets us see more clearly how the mass action equations come about by

minimisation of the free energy.

The expression AWert/(NKT ) = lnX0 + 1 + Q/ρ − ∆c(0)/N is a general expression true for

arbitrary values of the site fractions Xδ, and at equilibrium the site fractions are such as to

minimize this free energy. The resulting stationarity conditions that determine the site fractions

Xδ give rise to three types of mass action equations, according to the size of δ and Γ . These

equations are analysed and solved in the following sections, with an overview of the stationarity

conditions given in Table C.1. We begin by differentiating Q/ρ, defined in Equation (3.42). The

derivative ∂(Q/ρ)/∂Xδ with 2 ≤ |δ| ≤ |Γ | − 1 is given by

∂(Q/ρ)∂Xδ

∣∣∣∣∣Xα, α 6=δ

= ∂

∂Xδ

∑γ1,...,γM∈P(Γ ),M≥2

(−1)M (M − 2)! (X0)−M+1M∏i=1

XΓ−γi

,

= ∂

∂Xδ

∑γ1,...,γM−1∈P(δ),M≥2

(−1)M (M − 2)!Xδ (X0)−(M−1)M−1∏i=1

XΓ−γi

,(C.1)

186

187

Table C.1: System of the mass action equations for given number of sites |Γ |.

|Γ |∂ (AWert/(NkT ))

∂Xδ

= 0∂(Q/ρ)∂Xδ

|δ| type

1 X−10 + ∂(Q/ρ)

∂X0=∂(

∆c(0)open/N

)∂X0

−1 |Γ | (I)

2 X−10 + ∂(Q/ρ)

∂X0=∂(

∆c(0)ring/N

)∂X0

∑γ1,...,γM∈P(Γ )

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X0−X−1

0 |Γ | (II)

∂(Q/ρ)∂Xδ

=∂(

∆c(0)open/N

)∂Xδ

−1 +∑

γ1,...,γM∈P(δ)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X01 (I)

3 X−10 + ∂(Q/ρ)

∂X0= 0

∑γ1,...,γM∈P(Γ )

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X0−X−1

0 |Γ | (III)

∂(Q/ρ)∂Xδ

=∂(

∆c(0)ring/N

)∂Xδ

∑γ1,...,γM∈P(δ)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X02 (II)

∂(Q/ρ)∂Xδ

=∂(

∆c(0)open/N

)∂Xδ

−1 +∑

γ1,...,γM∈P(δ)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X01 (I)

≥ 4 X−10 + ∂(Q/ρ)

∂X0= 0

∑γ1,...,γM∈P(Γ )

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X0−X−1

0 |Γ | (III)

∂(Q/ρ)∂Xδ

= 0∑

γ1,...,γM∈P(δ)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X0∈ [3, |Γ |〉 (III)

∂(Q/ρ)∂Xδ

=∂(

∆c(0)ring/N

)∂Xδ

∑γ1,...,γM∈P(δ)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X02 (II)

∂(Q/ρ)∂Xδ

=∂(

∆c(0)open/N

)∂Xδ

−1 +∑

γ1,...,γM∈P(δ)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X01 (I)

188 Appendix C. Direct derivation of the law of mass action equations

where in the second equality we have only summed over the partitions for which Xδ appears in

the product, yielding,

∂(Q/ρ)∂Xδ

∣∣∣∣∣Xα, α 6=δ

=∑

γ1,...,γM∈P(δ)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X0, (C.2)

where we essentially redefined M . Moreover, for |δ| = 1,

∂(Q/ρ)∂Xa

= −1 +∑

γ1,...,γM∈P(a)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X0 (C.3)

and for |δ| = |Γ | we have

∂(Q/ρ)∂X0

= ∂

∂X0

∑γ1,...,γM∈P(Γ ),M≥2

(−1)M (M − 2)! (X0)−M+1M∏i=1

XΓ−γi

=

∑γ1,...,γM∈P(Γ ),M≥2

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X0

=∑

γ1,...,γM∈P(Γ )

(−1)M+1 (M − 1)!M∏i=1

XΓ−γi

X0− (X0)−1

(C.4)

We also need the derivatives of ∆c(0)/N = ∆c(0)open [ρ] + ∆c(0)

ring [ρ] with respect to Xδ, which

falls into three categories: |δ| = 1, |δ| = 2, and when |δ| ≥ 3. These cases will be referred to as

type (I), type (II) and type (III), respectively. By inspection we conclude that for the one-site

model we have one equation of type (I), for the two-site model we have two of type (I) and one

of type (II), and for the model with three or more sites we have |Γ | of type (I), |Γ |× (|Γ | − 1) /2

of type (II), and the remaining are of type (III). This is summarized in Table C.1. At this point,

it is helpful to define the quantities

Yδ = σδρ0, δ ⊆ Γ (C.5)

C.1. Solving the type (I) equations 189

which can be expressed in terms of the fractions of free sites as

Yδ =

X−10 , δ = Γ

1 , δ = ∅

XΓ−δ

X0, otherwise

(C.6)

We now discuss the solution of the equations of type (I), (II) and (III).

C.1 Solving the type (I) equations

These equations correspond to

∂ (βAWert/N)∂Xa

= 0, for all a ∈ Γ (C.7)

for which we need the derivative

∂(∆c(0)

open/N)

∂Xa

=∑b∈Γ

ρXb∆ab (C.8)

Equation (C.8) reminds us of the resulting law of mass action equations obtained for chain

clusters in equation (3.60). For this reason we define

(Xopena )−1 = 1 +

∑b∈Γ

ρXb∆ab (C.9)

where (Xopena )−1 corresponds to the fraction of molecules not involved in rings that has site a

free. The type (I) law of mass action equation

− 1 +∑

γ1,...,γM∈P(a)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γiX0

=∑b∈Γ

ρXb∆ab (C.10)


translates into the important result

XΓ−a

X0= 1 +

∑b∈Γ

ρXb∆ab

⇔ Ya = (Xopena )−1

(C.11)

The parameter Ya is thus related to the fraction of molecules with at most site a bonded, given

that a can be non-bonded or associating into open chain clusters.

C.2 Solving the type (II) equations

Corresponds to∂ (βAWert/N)

∂Xab

= 0, for all a, b ∈ Γ, a 6= b (C.12)

for which we need the derivative

∂(∆c(0)

ring/N)

∂Xab

=NRS,ab∑R=1

(∆ab)τoWab,oρτo−1 (Xab)τo−1 = Rab (C.13)

where the last equality is the definition of the quantity Rab. Substituting equation (C.13) in the

type (II) equation we obtain

−1 +∑

γ1,...,γM∈P(ab)

(−1)M+1 (M − 1)!M∏i=1

XΓ−γiX0

=NRS,ab∑R=1

(∆ab)τoWab,oρτo−1 (Xab)τo−1

⇔ XΓ−ab

X0= (Xopen

a )−1 (Xopenb )−1 +Rab

(C.14)

which can be written as

Yab = YaYb +Rab (C.15)

The term Rab is related to the fraction of molecules with a and b bonded into ring clusters. The

parameter Yab is thus related to the fraction of molecules with at most a and b bonded, given

that a and b can be non-bonded or associating into branched chain and ring clusters.

C.3. Solving the type (III) equations 191

C.3 Solving the type (III) equations

Corresponds to∂ (βAWert/N)

∂Xδ

= 0, for all δ ⊆ Γ, |δ| ≥ 3 (C.16)

Writing the partition that has only one element, δ, explicitly, leads to the result for type (III)

XΓ−δ

X0=

∑γ1,...,γM∈P(δ),M≥2

(−1)M (M − 1)!M∏i=1

XΓ−γiX0

(C.17)

or in terms of Yδ

Yδ =∑

γ1,...,γM∈P(δ),M≥2

(−1)M (M − 1)!M∏i=1

Yγi (C.18)

The type (III) equations thus show how we can recover Yδ from the variables Yα for which α ⊆ δ

and |α| < |δ|. Indeed, by a recursive procedure, we can obtain Yδ, |δ| ≥ 3 as a function of only

Yab’s and Ya’s. We illustrate this relation for the simplest examples of |δ| = 3 and |δ| = 4:

Yabc = YaYbc + YbYac + YcYab − 2YaYbYc

= Ya (YbYc +Rbc) + Yb (YaYc +Rac) + Yc (YaYb +Rab)− 2YaYbYc

= YaRbc + YbRac + YcRab + YaYbYc

(C.19)

Yabcd =YaYbcd + YbYacd + YcYabd + YdYabc + YabYcd + YacYbd + YadYbc

− 2YabYcYd − 2YacYbYd − 2YadYbYc − 2YcdYaYb − 2YbdYaYc − 2YbcYaYd

+ 6YaYbYcYd

=RabRcd +RacRbd +RadRbc + YaYbRcd + YaYcRbd + YaYdRbc + YbYcRad

+ YbYdRac + YcYdRab + YaYbYcYd

(C.20)

The parameter Yδ is thus related to the fraction of molecules whose bonded sites are the ones

present in the set δ at most.


C.4 Final expression

The sites can be non-bonded or bonded by association into open chain or ring clusters, and so

they can be partitioned into monomer and chain (Ya) as well as ring (Rab) contributions. We

can therefore write a general expression for Yδ as the sum over all partitions of the sites in δ

into chain and ring contributions:

Yδ =∑(

γ1,γ2,...∈P(δ) with|γj |∈1,2

)C (γ1, γ2, ...)∏j

Θ(γj), (C.21)

where

Θ(γ) =

Ya , if γ = a, a ∈ Γ

Rab , if γ = a, b, (a, b) ∈ Γ. (C.22)

The C’s in equation (C.21) are the coefficients of the terms of the sum to obtain Yδ; they arise

from the sum of the pre-factors (−1)M (M − 1)! in equation (C.18) and therefore are function of

the partition γ1, γ2, .... In the specific examples of |δ| = 3 and |δ| = 4, presented in equations

(C.19) and (C.20) respectively, these coefficients take the value of one for all terms. However, it

is not straightforward to prove that this is still the case for the cases with |δ| ≥ 5. In the next

section we show how all C’s are equal to one for all terms of Yδ, independently of the size |δ|,

just as seen for the specific cases of |δ| = 3 and |δ| = 4.

C.4.1 The C coefficients

Looking at the structure of the expression for the parameter Yδ, we can distinguish two types of

terms for an odd value of |δ|, “mixed" and “pure open":

Yabcd··· = YaRbc + . . .+ YcRab︸︷︷︸mixed

+YaYbYc · · ·︸︷︷︸pure open

C.4. Final expression 193

and three types for an even value of |δ|, “mixed", “pure open" and “pure ring":

Yabc··· = RabRcd + . . .+RadRbc︸︷︷︸pure ring

+YaYbRcd + . . .+ YcYdRab︸︷︷︸mixed

+YaYbYcYd · · ·︸︷︷︸pure open

The coefficients for the “pure open" terms have to be 1 since Equation (C.21) must reduce to

the TPT1’s relationship between Y ’s in the absence of rings. Next, we show that the coefficient

is 1 for pure ring terms. The only partitions in the expression that give rise to “pure ring” terms

of the form RabRcd · · · are those from partitions for which each of the pairs A,B and C,D etc.

appear in the same subset of the partition γ of Equation (C.18). It is as if we were treating

each specific pair A,B as an inseparable entity “AB”, the same as reducing the pair of sites to

a “single site”. Consider the case where A can only associate into a ring aggregate with B, C

with D and so on and so forth. The rings are now considered to be independent to each other

and analogously to the result obtained for TPT1, we would get the relation

Yδ = Y“AB”Y“CD”Y“EF” · · · (C.23)

for δ = A,B,C,D, · · · with an even |δ|. As we have seen previously for the independence

relation in Equation (3.55), the coefficients are always 1.

Consider now the effect of “turning off” a site a, i.e. letting the association strength between a

and all other sites go to zero. Then all terms of the form Rab, b ∈ Γ will go to zero, Ya will go

to 1, and Yabcd··· will go to Ybcd···. In general, by turning off sites, we can turn “mixed" terms

into “pure ring" terms, for which we have shown that the coefficients are 1.

Appendix D

Group like and unlike parameters for

use in the SAFT-γ Mie EOS

In this appendix the parameters used to model lovastatin, simvastatin and alcohols with a

SAFT-γ Mie approach found in [16,179,180] are compiled.

Table D.1: Group association energies εHBkl,ab and bond-

ing volume parameters KHBkl,ab for use to model lovas-

tatin, simvastatin and alcohols in the SAFT-γ Mie group-

contribution approach.

group k site a of k group l site b of l (εHBkl,ab/k)/K KHB

kl,ab/Å3

COO e1 OH H 1920.10 114.85

OH e1 OH H 2161.00 54.396

194

195

Table D.2: Like group parameters for use to model lo-

vastatin, simvastatin and alcohols with the SAFT-γ Mie

group-contribution approach: ν∗k is the number of seg-

ments, Sk is the shape factor, λrkk is the Mie repulsive

exponent, λakk is the Mie attractive exponent, σkk is the

segment diameter, εdispkk is the dispersion energy of the Mie

potential characterising the interaction of two k groups

(the k in the denominator is the Boltzmann constant),

and NST,k represents the number of association site types

on group k, with nk,H and nk,e1 denoting the number of

association sites of type H and e1, respectively.

group k ν∗k Sk λr

kk λakk σkk/Å (εdisp

kl /k)/K NST,k nk,H nk,e1

CH3 1 0.57255 15.050 6.0000 4.0773 256.77 - - -

CH2 1 0.22932 19.871 6.0000 4.8801 473.39 - - -

CH 1 0.07210 8.0000 6.0000 5.2950 95.621 - - -

CH= 1 0.20037 15.974 6.0000 4.7488 952.54 - - -

C= 1 0.15334 8.0000 6.0000 4.0335 1500.0 - - -

cCH2 1 0.24751 20.386 6.0000 4.7852 477.36 - - -

COO 1 0.65264 31.189 6.0000 3.9939 868.92 1 - 2

OH 1 0.96342 20.702 6.0000 2.7998 410.31 2 1 2

C 1 0.04072 8.0000 6.0000 5.2950 50.020 - - -

196 Appendix D. Group like and unlike parameters for use in the SAFT-γ Mie EOS

Table D.3: Group dispersion interaction energies εdispkl and

Mie repulsive exponent λrkl for use to model lovastatin,

simvastatin and alcohols with the SAFT-γ Mie group-

contribution approach. The unlike segment diameter σkl

is obtained from the arithmetic combining rule [16] and

all unlike Mie attractive exponents λakl = 6.0000; these

are not shown in the table. CR indicates that the λrkl is

obtained from a combining rule [16].

group k group l (εdispkl /k)/K λr

kl group k group l (εdispkl /k)/K λr

kl

CH3 CH3 256.77 15.050 CH OH 198.08 CR

CH3 CH2 350.77 CR CH C 2.0000 CR

CH3 CH 387.48 CR CH= CH= 952.54 15.974

CH3 CH= 252.41 CR CH= C= 1195.3 CR

CH3 C= 281.40 CR CH= cCH2 398.35 CR

CH3 cCH2 355.95 CR CH= COO 818.79 CR

CH3 COO 402.75 CR CH= OH 625.17 CR

CH3 OH 314.67 CR C= C= 1500.0 8.0000

CH3 C 339.91 CR C= cCH2 846.19 CR

CH2 CH2 473.39 19.871 C= COO 868.11 CR

CH2 CH 506.21 CR C= OH 784.51 CR

CH2 CH= 459.40 CR cCH2 cCH2 477.36 20.386

CH2 C= 286.58 CR cCH2 COO 498.60 CR

CH2 cCH2 469.67 CR cCH2 OH 376.57 CR

CH2 COO 498.86 CR cCH2 C 0.0000 CR

CH2 OH 396.27 CR COO COO 868.92 31.189

CH2 C 300.07 CR COO OH 490.95 CR

Continued on next page

197

Table D.3 – Continued

group k group l (εdispkl /k)/K λr

kl group k group l (εdispkl /k)/K λr

kl

CH CH 95.621 8.0000 COO C 0.0000 CR

CH CH= 502.99 CR OH OH 410.31 20.702

CH C= 378.72 CR OH C 0.0000 CR

CH cCH2 570.45 CR C C 50.020 8.0000

CH COO 353.65 CR

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