discrete perturbation theory for continuous soft-core potential fluids
TRANSCRIPT
Discrete perturbation theory for continuous soft-core potential fluidsL. A. Cervantes, G. Jaime-Muñoz, A. L. Benavides, J. Torres-Arenas, and F. Sastre Citation: The Journal of Chemical Physics 142, 114501 (2015); doi: 10.1063/1.4909550 View online: http://dx.doi.org/10.1063/1.4909550 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/142/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermodynamic of fluids from a general equation of state: The molecular discrete perturbation theory J. Chem. Phys. 140, 234504 (2014); 10.1063/1.4882897 Molecular dynamics study of one-component soft-core system: Thermodynamic properties in the supercooledliquid and glassy states J. Chem. Phys. 138, 144503 (2013); 10.1063/1.4799880 Perturbation theory for multipolar discrete fluids J. Chem. Phys. 135, 134511 (2011); 10.1063/1.3646733 Coarse-grained models for fluids and their mixtures: Comparison of Monte Carlo studies of their phasebehavior with perturbation theory and experiment J. Chem. Phys. 130, 044101 (2009); 10.1063/1.3050353 A global investigation of phase equilibria using the perturbed-chain statistical-associating-fluid-theoryapproach J. Chem. Phys. 123, 014908 (2005); 10.1063/1.1948374
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THE JOURNAL OF CHEMICAL PHYSICS 142, 114501 (2015)
Discrete perturbation theory for continuous soft-core potential fluidsL. A. Cervantes,1,a) G. Jaime-Muñoz,2,b) A. L. Benavides,2,c) J. Torres-Arenas,2,d)
and F. Sastre2,e)1Departamento de Infraestructura, Universidad de Guanajuato, Noria Alta, Guanajuato, CP 36000, México2División de Ciencias e Ingenierías, Universidad de Guanajuato, Loma del Bosque 103, Colonia Lomasdel Campestre, León, Guanajuato, CP 37150, México
(Received 30 September 2014; accepted 9 February 2015; published online 16 March 2015)
In this work, we present an equation of state for an interesting soft-core continuous potential [G.Franzese, J. Mol. Liq. 136, 267 (2007)] which has been successfully used to model the behaviorof single component fluids that show some water-type anomalies. This equation has been obtainedusing discrete perturbation theory. It is an analytical expression given in terms of density, temperature,and the set of parameters that characterize the intermolecular interaction. Theoretical results for thevapor-liquid phase diagram and for supercritical pressures are compared with previous and newsimulation data and a good agreement is found. This work also clarifies discrepancies betweenprevious Monte Carlo and molecular dynamics simulation results for this potential. C 2015 AIPPublishing LLC. [http://dx.doi.org/10.1063/1.4909550]
I. INTRODUCTION
Since the beginning of this century, the interest in under-standing the phenomenon of liquid and glass polymorphismin single component fluids has attracted the attention of thescientific community. For instance, it is well known thathelium exhibits two critical points: the common gas-liquid anda liquid-liquid one. Recently, experimental and/or theoreticalevidence has been published, showing that other importantsubstances could also show this anomalous behavior. Besides,the appearance of more than one type of glass has also beenobserved in real single component systems.1
For the case of some single components substances, thereare some good atomistic potential models that are able toreproduce some of the water-type anomalies and polymor-phism, but they can only be studied by simulation techniquesthat require high performance computers.2,3 Another way ofstudying these atypical systems is by using some effectiveisotropic pair potentials models that can be handled bydifferent statistical-mechanics techniques. It seems that a goodeffective potential, able to predict some of these anomaliesmust have two characteristic lengths, as, for example, thediscrete and soft-core continuous versions of the square-shoulder + square-well (SS + SW) and the Jagla ramppotentials.4–16
In the context of fluids, a good and efficient statistical-mechanics technique able to generate analytical equationsof state is the perturbation theory.17–19 An example of thisapproach is Discrete Perturbation Theory (DPT) which hasbeen used to study non-polar and polar systems.20,21 Examplesof DPT applications for systems that exhibit water-type
a)Electronic mail: [email protected])Electronic mail: [email protected])Electronic mail: [email protected])Electronic mail: [email protected])Electronic mail: [email protected]
anomalies are the discontinuous SS + SW potential10 andthe Jagla ramp potential.11 For the discontinuous SS + SWpotential, DPT has predicted multiple fluid-fluid transitionsbut not some other water-like anomalies, e.g., the maximumdensity anomaly. Besides, it has been shown that DPT canbe applied not only to hard-core discontinuous potentialsbut it also has a good performance for hard-core continuouspotentials.22,23
The main advantages of using DPT are (1) it providesan analytical expression for the Helmholtz free-energy as afunction of density, temperature, and the set of parametersthat characterize the intermolecular potential, which allowsto obtain all the thermodynamic properties in a straightfor-ward way, and (2) it is a very efficient tool to generatethermodynamic properties when compared, for instance, withsimulation techniques.
Using a molecular dynamics (MD) study, Franzese8,9,24,25
has found that a soft-core continuous version of this SS + SWpotential can predict both multiple critical points and themaximum density anomaly for some particular set of potentialparameters. Besides, recently, integral equations theory hasbeen used for this potential together with Monte Carlo (MC)simulations. Some discrepancies between MD and MC datahave been found for the critical points.16 To our knowledge,no analytical equation of state has been developed for thispotential.
The DPT application to soft-core continuous potentialshas only been done for the Lennard-Jones potential.20,26 TheFranzese potential is a suitable example to test the performanceof the theory for other soft-core potentials.
In this first study, DPT will be used to analyze thevapor-liquid phase diagram near the critical point and singlephase pressures at the super-critical region. We have selectedthis study region because DPT is an inverse temperatureperturbation expansion and as a consequence it convergesfaster at high temperatures. Even though we are limiting our
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114501-2 Cervantes et al. J. Chem. Phys. 142, 114501 (2015)
study to the critical and super-critical zone, super-criticalfluids have interesting properties that can be used in thedesign of new processes: they behave near the critical point assolvents, exhibiting a high compressibility and with transportproperties intermediate between gases and liquids. Besides,in an astronomical context, they could model the physicalproperties inside the bigger gaseous planets.27 The super-critical region has also been recently studied with moreaccurate experiments and simulations and its topology is moreintriguing than expected.28
The work is organized as follows. In Sec. II, two differentversions of the Franzese potential are presented. Section IIIprovides a brief description of DPT. For both potentials, inSec. IV, DPT predictions for critical data and super-criticalpressures are given together with a comparison with new andavailable simulation data. The article is closed in Sec. V withthe main conclusions of this work.
II. THE FRANZESE POTENTIAL
The Franzese potential8,9,24 is a continuous version ofa discontinuous hard-core SS + SW potential and can beexpressed reduced with respect to an appropriate energy scaleas
uF(x) = U(x)|UA| =
U∗R1 + exp(∆(x − R∗R))
−U∗A exp *,−(x − R∗A)2
2δ2A
+-+U∗A
(1x
)24
, (1)
where x = r/σ, with σ being the corresponding SS + SWhard-core diameter. The reduced energy parameters aredefined as U∗R = UR/|UA|, U∗A = UA/|UA| = 1. For this poten-tial, the reduced characteristic lengths are R∗R = RR/σ andR∗A = RA/σ, ∆ is a parameter that controls the steepness ofthe repulsive part of the potential, and δA is the approximateattractive width for the Gaussian function defining theattractive well. In the limit of high ∆ values, the SS + SWpotential that originates the Franzese potential is recovered.
As can be seen, this potential represents a family ofpotentials characterized by a set of parameters. In previousworks, the selected parameters were those of the originalpaper,8 and for different ∆ values. In this work, we willconsider the same cases: U∗R = 2.0, R∗R = 1.6, R∗A = 2.0, δ∗2A= 0.1 and ∆ = 15, 30, 100, 300, 500.
It is important to remark that Franzese, in order to treat thispotential with MD, added a constant and a linear term to theoriginal potential to have both the potential and its derivativeequal to zero at the cutoff (xc = 3.0). This new version of thepotential for the case of ∆ = 15 is
uFMD = uF(x) + C + Bx, (2)
with C = 0.208 876 and B = −0.067 379 4. The same con-stants were used for different ∆ values since all satisfy with agood precision the same conditions at the cutoff.29
In this work, we will consider both potentials, uF(x) anduFMD(x). In Figure 1, both potentials are shown for the case∆ = 15. As can be noticed, the potentials are not identical.
FIG. 1. The Franzese potential uF (solid line) and the modified Franzesepotential uFMD (dashed line).
III. DISCRETE PERTURBATION THEORY
For a system of N spherical particles of diameter σ con-tained in a volume V , which are interacting with an arbitrarydiscrete potential, DPT expresses the excess Helmholtz free-energy as a high-temperature expansion to l th-order20 as
AN kT
=AHS (η)N kT
+
lm=1
ni=1
βm
×�aSm(η,λi, ϵ i) − aS
m(η,λi−1, ϵ i)� , (3)
where AHS is the free energy for the hard-sphere referencefluid, β = 1/kT , k is the Boltzmann’s constant, T is thetemperature, n is the total number of steps conforming thediscrete potential, η is the hard-sphere packing fraction η= (π/6)ρ∗, where ρ∗ = (N/V )σ3 is the reduced density, andaSm are the m th-order perturbation terms for a SW (ϵ i < 0) or
square shoulder (ϵ i > 0) fluid.Equation (3) only requires the knowledge of the SW free-
energy perturbation terms since we assume that the SW andSS potentials are related through
aSSi (η,λ, ϵ) = (−1)iaSW
i (η,λ, ϵ). (4)
The excess Helmholtz free-energy can be rewritten in amore compact expression as an expansion of the inverse of thereduced temperature, T∗ = kT/|UA|, which is given by
aex ≡ Aex
N kT= aHS(η) + a1(η,λi, ϵ i)
T∗
+a2(η,λi, ϵ i)
T∗2+
a3(η,λi, ϵ i)T∗3
+ · · ·. (5)
From this expression, one can obtain any thermodynamicproperty. For instance, the compressibility factor, Z = 1+ η
(∂aex
∂η
), can also be expressed as a high-temperature
expansion,
Z = 1 + zHS(η) + 1T∗
z1(η,λi, ϵ i)
+1
T∗2z2(η,λi, ϵ i) + 1
T∗3z3(η,λi, ϵ i) + · · ·, (6)
where zi = η(∂ai∂η
).
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114501-3 Cervantes et al. J. Chem. Phys. 142, 114501 (2015)
A reduced pressure, P∗ = Pσ3
|UA| , can be obtained fromZ , since P∗ = Z ρ∗T∗. The reduced chemical potential canbe obtained by adding the Helmholtz free-energy and thecompressibility factor, µ∗ = µ/kT = a + Z . These expressionsare useful to obtain the coexistence curve. We solved numer-ically the system of non-linear equations obtained from gas-liquid equilibrium conditions at a fixed reduced temperatureT∗,
P∗(ρ∗gas) = P∗(ρ∗liquid),µ∗(ρ∗gas) = µ∗(ρ∗liquid).
(7)
Up to this point, DPT expressions are suitable for anyhard-core plus an arbitrary discrete potential. Since in thiswork we are interested in two soft-core potentials, DPTrequires a new approximation to account of the potentialsoftness. One possibility is to use the Barker and Hendersonrelation18 that provides an expression for a temperaturedependent diameter
d = σ
0(1 − e−βu(r ))dr. (8)
New reduced quantities can be defined with this neweffective diameter, ρ△ = ρd3 and P△ = Pd3
|UA| . All above DPTthermodynamic properties remain the same, except that theywill be evaluated using this new reduced density.
The implementation of DPT to the Franzese potentialsrequires a discrete version of the potentials in terms ofstep functions. (See Figure 2 for a picture of this kind ofdiscretization.) The number of partitions n was estimated asn = (xc − 1)/b, where b defines the width of the steps and weselected b = 0.14. In this work, the middle point on each stepwas used to evaluate the potential. Besides, DPT requires acutoff selection. Since the range of both potentials is fixed to3 so the same value was used in DPT.
In Eq. (5), the hard-sphere term has been evaluated usingthe Carnahan-Starling equation.30 The terms aSW
i are the mainingredient in DPT and can be obtained from different sources
FIG. 2. Discrete version of the Franzese potential. Notice that discretizationstarts at x = r/σ = 1.
(simulation data, integral equations, perturbation theory, etc.).DPT requires the knowledge of these expressions to any orderand for any λ value up to the selected cutoff, however, thisinformation is limited in the literature. There are only a fewavailable analytical SW expressions according to the rangeof the potential and at most to fourth-order.31–33 Third- andfourth-order terms in some of these expressions are obtainedfrom the correlation of simulation data with high uncertaintiesand in general are not so accurate for all the ranges anddensities required by DPT.
For the potentials under study, because the selected cutoffis 3, no long-range terms ai are required. Besides, since theselection of the step width was 0.14 and because the middlepoint of the first step is at 1.07, DPT only requires terms ai for1.07 ≤ λ ≤ 3.0. In this work, the expressions for a1,a2, anda3 from Espíndola-Heredia et al.32 have been used. We haveassumed the validity of these expressions for 1.07 ≤ λ ≤ 3.0.The explicit form of these ai terms is given in the Appendix.
IV. RESULTS
In order to test the performance of DPT applied topotentials uF and uFMD, supercritical pressures and vapor-liquid phase diagrams will be compared against available andnew complementary simulation data.
The new simulations were MC type only for the potentialuFMD for the case ∆ = 15 in the NVT ensemble in order toobtain supercritical pressures. The number of particles wasN = 1372 in a cubic box with periodic boundary conditions.In our simulations, the system reached equilibrium after25 000 cycles. Each cycle consisted of N attempts of particlemovements. The averaged results were obtained over 50 000cycles. In all runs, the trial move acceptance ratios were alwaysaround 40%.
To estimate the Barker and Henderson diameter, wesolved numerically Eq. (8). The effective diameter was esti-mated for both potentials for different ∆ values and for 0.8≤ T∗ ≤ 10. For this calculation, we have used the continuousversion of the potentials.
Results were very similar for both potentials and all ∆values considered so we used the same polynomial
d∗ =dσ= 1 − 0.002 853T∗ − 0.001 046T∗2 + 0.000 077T∗3.
(9)
Although thermodynamic properties were calculated interms of ρ△ and P△, in order to compare them with simulationdata, we returned to the original reduced densities ρ∗ andpressures P∗ with the transformations: ρ∗ = ρ△/d∗3 and P∗
= P△/d∗3.Before exploring the super-critical region, DPT predic-
tions for the critical points were analyzed. Since for obtainingcritical data from perturbation theories, it is important toinclude high-order perturbation terms35,36 we have used DPTto second-order (DPT2) and to third-order (DPT3). Vapor-liquid equilibrium equations for each potential in regionsclose to their critical temperatures have been solved. Barkerand Henderson diameter was included in DPT2 and DPT3,
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114501-4 Cervantes et al. J. Chem. Phys. 142, 114501 (2015)
TABLE I. Critical vapor-liquid data for the Franzese potential from DPT and MC simulations of Huš andUrbic.16
MC DPT2 DPT3
∆ T ∗ ρ∗ P∗ T ∗ ρ∗ P∗ T ∗ ρ∗ P∗
15 1.13 ± 0.02 0.085 ± 0.005 0.0255 ± 0.002 1.13 0.0711 0.0249 1.072 0.0718 0.023930 1.19 ± 0.02 0.084 ± 0.005 0.0290 ± 0.002 1.17 0.0704 0.0256 1.113 0.0723 0.0251100 1.25 ± 0.02 0.085 ± 0.005 0.0310 ± 0.002 1.24 0.0732 0.0280 1.185 0.0758 0.0278300 1.24 ± 0.02 0.086 ± 0.005 0.0291 ± 0.002 1.25 0.0737 0.0283 1.191 0.0761 0.0281500 1.24 ± 0.02 0.085 ± 0.005 0.0304 ± 0.002 1.25 0.0737 0.0283 1.191 0.0761 0.0281
however, in this region it had no effect (similar results werefound with and without it).
DPT2 and DPT3 critical data are given in Tables I and IItogether with reported simulation data9,16 for each potential.In Table I, MC and DPT results for the uF potential areshown. The MC simulation errors on critical temperatures,densities, and pressures are at most 0.02, 0.005, and 0.002,respectively.34 Table II includes MD and DPT results for theuFMD potential. Different ∆ values were considered. We haveseparated MC and MD simulation data intentionally, in orderto try to give a possible explanation about the discrepanciesfound by Huš and Urbic16 between MC and MD critical data,as will be discussed at the end of this section.
In Figure 3, the vapor-liquid phase diagram (T∗ vs ρ∗) ispresented for different ∆ values. The case for ∆ = 500 wasnot included in this figure since it is almost equal to the case∆ = 300, as can be seen in both tables. In this figure, DPT2and DPT3 predictions for both potentials together with MCand MD simulation data of Hus and Urbic16 and Vilaseca andFranzese,9 respectively, are shown.
As it can be observed, for each ∆ case, it is clear that MCand MD data for the critical temperature are different, the samehappens with DPT predictions. DPT2 and DPT3 predictionsfor uF potential are closer to the central values of MC data andDPT2 and DPT3 predictions for uFMD are closer to the centralvalues of MD data.
Comparing DPT2 and DPT3 results for each potential, onaverage for all the ∆ cases considered, DPT2 predicts betterthe critical temperature when compared with its correspondingsimulation data. This result was not expected since normallythe inclusion of higher order terms in perturbation theoriesimproves the predictions of critical data.
For both potentials, the critical temperature predicted byDPT increases as ∆ increases, but the change ratio is clearly
slower for larger ∆ values, this behavior is consistent with thesimulation results.
Considering the critical density, it can be seen that MCdata are within the error bars of MD data. For each potential,differences in the critical density are less pronounced betweenDPT2 and DPT3 than for the case of critical temperature. Inall cases, DPT3 predictions are slightly better than DPT2 oneswhen compared with simulation values.
In Figure 4, the vapor-liquid phase diagram (P∗ vs ρ∗)is presented for different ∆ values. Since MD data havebig uncertainties, it is not possible to obtain conclusiveinformation from these results. DPT2 and DPT3 predictionsfor uF are different from those of uFMD. Again DPT2 andDPT3 results for uF are closer to MC data. The effect ofincluding a third-order term in DPT is not relevant for thecritical pressures.
For uF, and ∆ ≥ 100, DPT2 and MC data predict criticaldata very close to the critical data of the SS + SW potentialthat originates this family of potentials with critical values:Tc = 1.24 ± 0.01, ρ∗ = 0.09 ± 0.02, and P∗ = 0.03 ± 0.01.37
This behavior was expected since as ∆ increases potentialsshould reach as a limiting case this SS + SW potential.However, DPT and MD predictions for uFMD do not recoverthe critical data of this limiting potential.
As can be observed from the vapor-liquid phase diagrams(Figures 4 and 5), the introduction of the third-order termin DPT has not improved the critical values when comparedwith simulation data. This effect of third-order terms in aperturbation expansion was not expected, but as mentionedin the introduction, it is difficult to find accurate analyticalexpressions for 3rd- or higher-order SW perturbation termsavailable in the literature. The a3 SW term used in thiswork comes from simulation data correlation with largeuncertainties difficult to fit in a single expression as a function
TABLE II. Critical vapor-liquid data for the FMD potential from DPT and MD simulations of Vilaseca andFranzese.9
MC DPT2 DPT3
∆ T ∗ ρ∗ P∗ T ∗ ρ∗ P∗ T ∗ ρ∗ P∗
15 0.95 ± 0.06 0.08 ± 0.03 0.019 ± 0.008 0.989 0.0604 0.0188 0.871 0.0653 0.018230 1.01 ± 0.07 0.08 ± 0.03 0.022 ± 0.008 1.029 0.0604 0.0196 0.910 0.0662 0.0193100 1.06 ± 0.04 0.08 ± 0.03 0.025 ± 0.005 1.083 0.0626 0.0213 0.975 0.0689 0.0214300 1.06 ± 0.05 0.09 ± 0.02 0.027 ± 0.009 1.087 0.0628 0.0214 0.980 0.0691 0.0216500 1.08 ± 0.06 0.09 ± 0.03 0.027 ± 0.008 1.087 0.0628 0.0214 0.980 0.0691 0.0216
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114501-5 Cervantes et al. J. Chem. Phys. 142, 114501 (2015)
FIG. 3. Phase diagram (T ∗ vs ρ∗) forthe potentials considered in this workfor different ∆ values. The two curves totop correspond to DPT2 (solid line) andDPT3 (dashed line) predictions for uF.The couple of curves to bottom are pre-dictions of DPT2 (solid line) and DPT3(dashed line) for uFMD. MD simulationdata from Vilaseca and Franzese9 (solidsquare) and MC simulation data of Hušand Urbic16 (solid diamond).
of λ and density (see Fig. 3 of Espíndola-Heredia et al.32).Besides, DPT approximation could also be responsible of thisnot expected effect.
In order to analyze the performance of DPT in the super-critical region, reduced pressures as a function of density forfour super-critical temperatures are presented in Figure 5 forboth potentials. DPT2 and DPT3 predictions overlap in allcases, so we just show DPT2 results. We also have includedMC data of Huš and Urbic15 and our new simulation data.
As can be seen, the agreement between DPT2 andsimulation is good as long as they are compared with their
corresponding potential simulation data. In general, thetheory predicts the same tendency of the simulation data,pressure rises as temperature, and density raises. As expectedfrom an inverse temperature perturbation expansion, theagreement between DPT2 and simulation data improves asthe temperature increases. In this figure, for each potential,the corresponding predictions without Barker and Hendersondiameter are shown as dotted lines. As can be seen, animprovement is obtained with the inclusion of the Barker andHenderson diameter for the higher densities and temperatures,but systematic overestimation of the theory with respect to
FIG. 4. Phase diagram (P∗ vs ρ∗) forthe potentials considered in this workfor different ∆ values. The two curves totop correspond to DPT2 (solid line) andDPT3 (dashed line) predictions for uF.The couple of curves to bottom are pre-dictions of DPT2 (solid line) and DPT3(dashed line) for uFMD. MD simulationdata from Vilaseca and Franzese9 (solidsquare) and MC simulation data of Hušand Urbic16 (solid diamond).
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114501-6 Cervantes et al. J. Chem. Phys. 142, 114501 (2015)
FIG. 5. Reduced pressures as a func-tion of reduced density for the poten-tials considered in this work for sev-eral supercritical reduced temperaturesT ∗ for the case ∆= 15. For each case,from top to bottom, DPT2 results foruF (solid line) and DPT2 results foruFMD (dashed line). Dotted lines repre-sent DPT2 predictions without Barkerand Henderson diameter for each po-tential. MC simulation data of Huš andUrbic15 are shown with solid circles andMC data of this work with solid trian-gles up.
the simulation data is observed. A better approximation thanBarker and Henderson diameter to account for the softness ofthe potentials is required.
Discrepancies between some simulation techniques fora given potential are sometimes due to the way the originalpotential is modified to avoid technical simulation problems.38
For instance, Trokhymchuk and Alejandre39 discussed thispoint for the case of a Lennard-Jones fluid and found that apossible reason, among others, for the discrepancies betweeninterfacial tension MC and MD predictions is that MC usesa truncated potential, while MD uses a truncated force, i.e., aderivative of the potential. The difference between potentialsuF and uFMD is the term C + Bx, and this is possible the originfor the discrepancies between the two versions of the Franzesepotential, which is reinforced by DPT predictions.
V. CONCLUSIONS
Discrete perturbation theory has been applied to a familyof soft-core continuous potentials. The vapor-liquid phasediagram near the critical point and super-critical pressuresfor two versions of the Franzese potential have been obtainedand compared with simulation results and a good agreementwas found. Discrepancies between simulation data previouslyobtained by Monte Carlo and molecular dynamics for thesepotentials have been clarified. We are working on improvingthis theory to study the low temperature region where a secondcritical point and some interesting anomalies appear.
ACKNOWLEDGMENTS
We thank Dr. T. Urbic for providing us his simulationdata. We also thank the financial support from CONACYT(México): Project No. 152684 and Universidad de Guanajuato(México) Grant No. 56-060. A.L.B. also thanks CONACYT
(México) Convocatoria 2014 de Estancias Sabáticas Na-cionales, Estancias Sabáticas al Extranjero y Estancias Cortaspara la Consolidación de Grupos de Investigación.
APPENDIX: SQUARE-WELL HELMHOLTZFREE-ENERGY PERTURBATION TERMS
The first three-order perturbation terms as obtained byEspíndola-Heredia et al.32 used in this work are presented.
The first coefficient from Eq. (3) can be expressed as
a1 =
3i=2
αi,1(λ)(
6ηπ
) (i−1)+
4i=1
γi(λ)(
6ηπ
) (i+2), (A1)
here, α2,1 is given by
α2, i = −2π3i!
(λ3 − 1). (A2)
The α3,1 is given by
α3,1 = −(π
6
)2
P1(λ), λ ≤ 2P1(2), λ > 2
, (A3)
where P1(λ) is the 6th order polynomial
P1(λ) = λ6 − 18λ4 + 32λ3 − 15. (A4)
As for the γn, their expressions are
γn = γn,1λ + γn,2λ2 + Rn(λ)/Qn(λ), (A5)
which are valid for n ≤ 4, Rn(λ) is
Rn = γn,3 +
8j=4
γn, j(λ3 − 1) j−2 (A6)
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114501-7 Cervantes et al. J. Chem. Phys. 142, 114501 (2015)
TABLE III. γn, j coefficients.32
j γ1, j γ2, j γ3, j γ4, j
1 −59.046 4 214.316 −225.479 65.050 42 26.098 −88.139 4 88.820 2 −25.0963 26.445 4 273.3 250.472 74.309 54 7.401 36 95.975 9 90.260 6 26.215 35 11.074 3 71.122 8 57.027 4 18.439 76 −5.491 52 −40.265 6 −33.237 6 −10.089 17 0.781823 5.94069 4.99527 1.502438 −0.031 975 1 −0.238 42 −0.195 714 −0.057 6949 0.827 621 −2.175 58 1.846 77 −1.871 5410 0.605 635 −1.292 55 0.998 13 −1.016 8211 −0.254 959 0.554 993 −0.440 314 0.445 24712 0.037 711 1 −0.085 754 3 0.070 879 3 −0.072 510 713 −0.002 108 96 0.004 925 11 −0.004 162 74 0.004 278 6214 0.000 045 232 8 −0.000 107 067 0.000 091 729 1 −0.000 094 972 3
and Qn(λ) is
Qn = γn,9 +
14j=10
γn, j(λ3 − 1) j−7. (A7)
The coefficients γn, j are listed in Table III.The second coefficient from Eq. (3) is given by
a2 = χ(η,λ) expξ2(λ)6η
π+ ϕ1(λ)
(6ηπ
)3
+ ϕ2(λ)(
6ηπ
)4, (A8)
with
χ = α2,2(λ)6ηπ
*,1 −
(6ηπ
)2
/1.5129+-, (A9)
where α2,2 is given by Eq. (A2). The functions ϕi are given by
ϕi =
7n=0
ϕi,nλn, (A10)
whose coefficients ϕi,n are listed in Table IV. The function ξ2is given by
ξ2 = α3,2(λ)/α2,2(λ), (A11)
where α3,2 is
TABLE IV. ϕn, j and θ3, j coefficients.32
j ϕ1, j ϕ2, j θ3, j
0 −1 320.19 1 049.76 . . .1 5 124.1 −4 023.29 −945.5972 −8 145.37 6 305.95 1 326.613 6 895.8 −5 265.42 −471.6884 −3 381.42 2 553.84 . . .5 968.739 −727.3 23.227 16 −151.255 113.631 −2.634 777 9.985 92 −7.562 66 . . .
α3,2 =
(π
6
)2
P2(λ) − P1(λ)/2, λ ≤ 2
−172+ P4(λ), λ > 2
. (A12)
P1 is the polynomial given by Eq. (A4), P2 and P4 are also6th order polynomials defined as
P2(λ) = −2λ6 + 36λ4 − 32λ3 − 18λ2 + 16, (A13)
P4(λ) = 32λ3 − 18λ2 − 48. (A14)
Finally, the third coefficient is given by
a3 = α2,36ηπ
expξ3
6ηπ+ K3(η,λ)
, (A15)
where K3 is given by
K3(η,λ) =(
6ηπ
)2 4n=1 θ3,nλ
n
1 + 6ηπ
7n=5 θ3,nλ
n−4. (A16)
The coefficients θ3,n are contained in Table IV. Thefunction ξ3 is given by
ξ3 = α3,3(λ)/α2,3(λ), (A17)
where α2,3 is given by Eq. (A2) and α3,3 is
α3,3 =
(π
6
)2
P2(λ) − P1(λ)/6 − P3(λ), λ ≤ 2
−176+ P4(λ) − P5(λ), λ > 2
. (A18)
P1, P2, and P4 are given by Eqs. (A4), (A13), and (A14),respectively. While P3 and P5 are defined by
P3(λ) = 6λ6 − 18λ4 + 18λ2 − 6, (A19)
P5(λ) = 5λ6 − 32λ3 + 18λ2 + 26. (A20)
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