thermodynamic principles of self-assembly 계면화학 8 조 최민기, liu di ’ nan, 최신혜...
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Thermodynamic Principles of Self-assemblyThermodynamic Principles of Self-assembly
계면화학 8 조
최민기 , Liu Di’nan, 최신혜
Chapter 16
16.1 Introduction16.1 Introduction
* Understanding Self-assembly by using statistical thermodynamics
* Association colloids or complex fluids – ‘fluid-like’ micelles, bilayers
* Forces in micelles, bilayers - van der Waals, hydrophobic, hydrogen-bonding,
screened electrostatic interactions
Micelle Inverted micelles
Bilayer Bilayer vesicle
16.2 Fundamental thermodynamic equations of self-assembly
= N + kT log X1= 20 + ½ kT log ½X2 = 3
0 + ⅓ kT log⅓ X3 = . . .
or= N = 0
N + (kT/N) log (XN/N) = constant, N= 1, 2, 3, …,
monomers dimers trimers
N : mean chemical potential of a molecule in an aggregate of aggregation number N
0N : standatd part of the chemical potential (the mean interaction free energy per molecule)
XN : concentration (activity) of molecules in aggregates
Rate of association = k1X1N,
Rate of dissociation = kN(XN/N)
In equilibrium, both rates are same .
k1X1N = kN(XN/N)
∴k1/kN = (XN/N)X -N
Equilibrium constant is also given by,
K= k1/kN= exp[-N(0N- 0
1)/kT]
We can combine two equations to obtain
XN = N{X1 exp[(01- 0
N)/kT]}N
More generally,
XN= N{(XM/M)exp[M(01- 0
N)/kT]}N
Law of Mass Action (Alexander and Johnson, 1950)
16.3 Conditions Necessary for the Formation of Aggregates
XN= NX1N
* Aggregates formation is originated from, different cohesive energies
between the molecules in the ‘aggregate’ and the ‘dispersed states’ (N0 < 1
0 )
∴ X1<1 => XN<<X1
Most of molecules will be in the monomer state
{N
0 decreases progressively
N0 has a minimum at some finite N
* For 10 = 2
0 = 30 = . . . = N
0 ( No cohesive force )
* The exact function of N0 => mean size, polydispersity of aggregates
16.4 Variation of N0 with N for simple structures of different geometries
: RODS, DISCS, and SPHERES
* Geometrical shape of the aggregate determines the way N0 decreases with N
SpheresSpheresDiscsDiscs
RodsRods
NN0 = - (N-1) αkT : total interaction free energy
N0 = - ( 1- 1/N ) αkT = ∞
0 + αkT / N
bond energy α kT
1. One-dimensional aggregates (rods)
* Linear chains of identical molecules or monomer units in equilibrium
with monomers in solution.
‘Bulk’ energy of a molecule in an infinite aggregates
* the number of molecules per disc ∝ πR2
∝ N
* the number of unbonded molecules in the rim) ∝ 2πR
∝ N1/2
2. Two-dimensional aggregates (discs, sheets)
R
* NN0 = - (N–N1/2) αkT : total interaction free energy
N0 = - ( 1- 1/N1/2) αkT = ∞
0 + αkT / N1/2
3. Three-dimensional aggregates (spheres)
* the number of molecules per disc ∝ 4/3πR3
∝ N
* the number of unbonded molecules in the rim) ∝ 4πR2
∝ N2/3
R
* NN0 = - (N–N2/3) αkT : total interaction free energy
N0 = - ( 1- 1/N1/3) αkT = ∞
0 + αkT / N1/3
α : constant dependent on the strength of the intermolecular interactionsp : number dependent on the shape or dimensionality of the aggregates
<General Expression> N
0 =∞0 + αkT /NP
Relation Between Surface energy and intermolecular interactions
Consider the droplets of hydrocarbon in water (sphere-shape)
N = 4πR3/3v v : volume per molecule
The total free energy of sphere = N ∞0 + 4πR2γ
γ: Interfacial energy per unit area Hence, N
0 = ∞0 + 4πR2 γ/N = ∞
0 + 4 π γ (3v/4 π)2/3
= ∞0 + αkT / N 1/3
∴ α = 4πγ(3v/4π)2/3 / kT = 4πr2 γ / kT
Interfacial energy is proportional to intermolecular forces!
16.5 The critical micelle concentration (CMC)
(X1)crit = CMC ≈ exp[-(10 - N
0)/kT]
or
(X1)crit = CMC ≈ e-α
XN = N{X1 exp[(10- N
0) / kT]}N
= N{X1 exp[α(1 – 1/NP)]}N ≈ N[X1eα]N
X1 > X2 > X3 > . . . .for all α∴ X1 ≈ C
‘At what concentration will aggregates form?’
* For low monomer concentrations X1 , X1 exp[(10- N
0) / kT] or X1eα << 1
* Since XN cannot exceed unity, X1 exp[(1
0- N0) / kT < 1
Once X1 approaches exp[ -(10 - N
0)/kT ] or e-α, it cannot increase no further!
16.6 Infinite aggregates (Phase separation)
Nature of aggregates depend on shape
XN = N[X1 eα]Ne- α N1/2 for discs (p=1/2)
XN = N[X1 eα]Ne- α N2/3 for spheres (p=1/3)
Above the CMC (X1eα ≈1)
XN ≈Ne-α N
XN ≈Ne-α N
1/2
2/3
As N increases above certain limit (N>5), XN decreases exponentially.
Then, ‘where do the molecules go above the CMC?’
Infinite size aggregate (N →∞) at the CMC , ‘phase separation’
Such a transition to a separated phase occurs whenever p<1 .
(X1)crit ≈ e-α ≈e- 4πr2 γ / kT
Relation between intermolecular interaction and CMC (solubility)
By measuring CMC (solubility), we could obtain α value.
above which oil will separate out into a bulk oil phase
Ex) Strength of hydrophobic interaction
{Simple hydrocarbon : 3.8 kJ/mol per CH2 increment
Amphiphiles : 1.7~2.8 kJ/mol per CH2 increment
Consider the case where p=1,
XN = N[X1eα]N e-α
: Above CMC, X1eα ~1 → XN N for small N∝
XN ~ 0 for large N
Therefore, distribution is Highly polydisperse.
p<1 : as occurs for simple discs or spheres abrupt phase transition to one infini
tely sized aggregate occurs at the CMC and the concept of a size distribution d
oes not arise
p>1 : Impossible to occur
How polydispersity comes about?
{
16.7 Size Distribution of Self-assembled Structures
C = ∑ XN = ∑ N [X1eα]Ne-α
= e- α[X1eα + 2(X1e α)2 +3(X1e α)3 + …. ]
= X1/(1 - X1eα)2 By using approximation, ∑ NχN = χ/(1- χ)2
Thus,
N=1
∞
X1 = (1+2Ceα) – 1+4Ceα 2Ce2α
X1≈ (1 - 1/ Ceα )e-α ≤ e-α (CMC) for low C
This function peaks at ∂XN/ ∂N = 0, Nmax = M = Ceα
XN = N(1 - 1/ Ceα )Ne-α ≈ N e-N/ Ceα for large N
XN = N[X1eα]N e-α
* The expectation value of N, <N> =∑NXN/∑XN = ∑NXN/C
<N> = 1 + 4Ceα
≈ 1 below the CMC
≈ 2 Ceα = 2M above the CMC
* The density of aggregates above the CMC
XN/N = Const.e-N/M for N > M
16.8 More Complex Amphiphilic Structures
* The value of p is actually not constant for complex molecules (like amphiphiles)
* Complex amphiphilic molecules can assemble into more complex shapes such as vesicles, interconnected rods or three-dimensional ‘periodic structures
{Directionalites of binding forces
Flexible molecule structures N
0
N
Complex molecules
Simple molecules
* { Hydrocarbons: infinite aggregate formation (phase separation)
Amphiphiles: finite aggregate formation (micellization)
N=M
σ = kT / 2MΛ
CMC ≈ exp[- (10 - M
0) / kT
The variation of N0 about M
0 can usually be expressed in the parabolic form:
N0 - M
0 = Λ(ΔN)2 , where ΔN = (N-M)
XM
For the case when N0 has a minimum value at N=M, CMC is given by
M{ }N exp[ -M Λ(ΔN)2 kT]N/M
XN =
Gaussian Distribution of Aggregation number N
Su
rfac
tan
t co
nce
ntr
atio
n X
N
M N
MonomersX1=CMC
Micelles
At the CMC
Below the CMC
Distribution above the CMC
* Attractive / repulsive forces between aggregates → structural phase transitions
<Structural transitions>
1. Strong Repulsive forces (electrostatic, steric or hydration forces)
Phase transition : to get apart within a confined volume of solution,
16.9 Effects of interactions between aggregates : Mesophases and mutilayers
* Interaggregate interactions cannot be ignored at high concentration!
H1
V1
L
Liquid phase + crystal
hexagonal
cubic Ia3d
lamellar
2. Attractive forces
* Between uncharged amphiphilic surfaces
(nonionic, zwitter ionic, for charged headgroups in high salt concentration)
10 + kT log X1 = 0
M + (kT / M)log (XM / M)log (XM/M) = 0M + (kT/M)log(XM/M)
monomer micelles/vesicles liposomes/superagregates
XM/M = {(XM/M)exp[M(M0 - M
0)kT]}M/M
The concentration at which XM = XM is therefore,
(XM)crit = M exp[-M(0M - 0
M)/kT]
M : aggregation number in micelle or vesiclesM : aggregation number in liposomes