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Excess chemical potential and entropy for pure fluids ARTICLE in THE JOURNAL OF CHEMICAL PHYSICS · NOVEMBER 2003 Impact Factor: 2.95 · DOI: 10.1063/1.1623184
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Excess chemical potential and entropy for pure fluids Jean-Marc Bomont Citation: The Journal of Chemical Physics 119, 11484 (2003); doi: 10.1063/1.1623184 View online: http://dx.doi.org/10.1063/1.1623184 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/119/21?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Excess entropy scaling of dynamic quantities for fluids of dumbbell-shaped particles J. Chem. Phys. 133, 104506 (2010); 10.1063/1.3477767 A consistent calculation of the chemical potential for dense simple fluids J. Chem. Phys. 124, 206101 (2006); 10.1063/1.2198807 Diffusivity, excess entropy, and the potential-energy landscape of monatomic liquids J. Chem. Phys. 124, 014507 (2006); 10.1063/1.2140282 Transient cavities and the excess chemical potentials of hard-spheroid solutes in dipolar hard-sphere solvents J. Chem. Phys. 123, 154501 (2005); 10.1063/1.2062027 Calculating excess chemical potentials using dynamic simulations in the fourth dimension J. Chem. Phys. 111, 3387 (1999); 10.1063/1.479622
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JOURNAL OF CHEMICAL PHYSICS
VOLUME 119, NUMBER 21
1 DECEMBER 2003
LETTERS TO THE EDITOR The Letters to the Editor section is divided into three categories entitled Notes, Comments, and Errata. Letters to the Editor are limited to one and three-fourths journal pages as described in the Announcement in the 1 July 2003 issue.
NOTES Excess chemical potential and entropy for pure fluids Jean-Marc Bomonta) Equipe de Chimie et Biochimie The´oriques, UMR CNRS-UHP 7565, Universite´ Henry Poincare´-Nancy I, F-54509 Vandoeuvre-Le`s-Nancy, France
共Received 25 July 2003; accepted 9 September 2003兲 关DOI: 10.1063/1.1623184兴 Recently, we published1 a study of soft-sphere fluid properties using the Ornstein–Zernike-type integral equations with a self-consistent closure based on a new bridge function B (2) (r). This closure includes only one thermodynamic consistency condition and a partitioning scheme for the pair potential founded on the merging process of the test-particle insertion method. The calculations involve the renormalized indirect correlation function ␥ * (r) as described in a prior work.2 The results obtained were highly accurate as compared to available bridge function simulation data for a wide range of densities. In this note, we take advantage of the accuracy of our bridge function to propose a simple method to calculate the excess chemical potential and the entropy in a thermodynamically consistent manner. We summarize the thermodynamic consistencies in the following. For soft spheres interacting through a pair potential v (r), the essential thermodynamic properties3 are the energy u, the pressure P, and the isothermal compressibility T . While the excess internal energy reads u⫽  U ex /N⫽
 2
冕
共1兲
v共 r 兲 g 共 r 兲 dr,
the pressure from the virial route ( v ) has to be equal to the pressure from the compressibility route 共c兲. This is the first thermodynamic consistency condition4 that reads

冏
Pv ⫽ T ⬅1⫺ T
冕
c 共 r 兲 dr⫽ 
冏
Pc , T
共2兲
where c(r) and g(r)⫽h(r)⫹1 are the direct and pair correlation functions, respectively,  ⫽1/k B T the reciprocal temperature, and the number density. Calculation of the excess chemical potential  ex and entropy s is less straightforward and requires a special analysis. The latter can be evaluated5 by using the thermodynamic equality s⫽u⫹  P v / ⫺1⫺  ex ,
共3兲
and  ex has to satisfy the second thermodynamic consistency condition6 that reads 0021-9606/2003/119(21)/11484/3/$20.00

冏
ex ⫽⫺ T
冕
共4兲
c 共 r 兲 dr.
This is the Gibbs–Duhem relation for pure fluids. The excess chemical potential can be expressed exclusively in terms of the correlation functions.7 Namely,
 ex⫽
冕冋
⫺h 共 r 兲 ⫹ ␥ 共 r 兲 ⫹B 共 2 兲 共 r 兲 ⫹
册
h共 r 兲 关 ␥共 r 兲 2
⫹B 共 2 兲 共 r 兲 ⫹B 共 1 兲 共 r 兲兴 dr,
共5兲
where ␥ (r) is the indirect correlation function and B (1) (r) is the so-called bridge functional of the first kind or the ‘‘one particle’’ bridge functional. On the one hand, as can be seen, the excess chemical potential depends explicitly on the bridge function B (2) (r) and it has been shown that its calculation is mainly affected by the contribution of B (2) (r) inside the core 共98% in the case of hard-sphere fluid8兲. That is the reason why an accurate bridge function like the one we use is necessary to the calculation of  ex . On the other hand, B (1) (r) is an infinite series of irreducible diagrams and, unfortunately, cannot be summed up exactly.7 Little is known about it, excepted that at low density B (1) (r)⫽ 31 B (2) (r) and at higher densities, the link between B (1) (r) and B (2) (r) remains still unknown. Then, an accuracy of the chemical potential and entropy calculations is determined by the Ornstein–Zernike solution and the approximation of B (1) (r). We examine the Lennard-Jones 共LJ兲 fluid using our closure B 共 2 兲 共 r 兲 ⫽ 关 1⫹2 ␥ * 共 r 兲 ⫹ f ␥ * 2 共 r 兲兴 1/2⫺1⫺ ␥ * 共 r 兲 ,
共6兲
which has been proven to be accurate.1 The mixing parameter f is to ensure the pressure consistency condition, while we propose the approximation B (1) (r)⫽ ␣ (T, )B (2) (r). Once the correlation functions, the excess internal energy, the pressure, and the isothermal compressibility are calculated with respect to the first thermodynamic consistency condition, the parameter ␣ (T, ) is iterated until  ex / satisfies the second thermodynamic consistency condition
11484
© 2003 American Institute of Physics
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J. Chem. Phys., Vol. 119, No. 21, 1 December 2003
Excess chemical potential and entropy for pure fluids
11485
TABLE I. Thermodynamic properties of the Lennard-Jones fluid at supercritical temperatures. Simulation data are from Ref. 9. The results in the last column have been taken in Ref. 10 共on the left兲 and in Ref. 7 共on the right兲. This work
Simulation
*
␣ coh(T, )
 ex
⫺s
0.1 0.3 0.5 0.7 0.9
0 0.1705 0.1275 0.1845 0.1740
0.125 0.551 1.356 2.851 5.450
0.161 0.531 0.963 1.491 2.050
0.1 0.3 0.5 0.7 0.9
0 0.0030 0.1140 0.1570 0.1715
⫺0.060 0.009 0.523 1.971 5.155
0.1 0.35 0.5 0.7 0.9
0 0 0 0.111 0.147
⫺0.576 ⫺1.568 ⫺1.818 ⫺0.928 2.928
Theories
 ex
⫺s
 ex
0.129 0.562 1.371 2.883 5.582
0.161 0.534 0.986 1.532 2.190
0.124 0.542 1.315 2.699 5.044
0.124 0.542 1.334 2.807 5.473
T * ⫽2.74 0.188 ⫺0.056 0.610 0.022 1.142 0.536 1.796 1.996 2.564 5.229
0.189 0.615 1.133 1.801 2.642
⫺0.062 0.001 0.479 1.801 4.651
⫺0.062 ⫺0.002 0.488 1.898 5.130
T * ⫽1.35 0.285 ⫺0.571 0.965 ⫺1.557 1.402 ⫺1.802 2.317 ⫺1.060 3.454 2.633
0.282 0.955 1.379 2.199 3.316
⫺0.578 ⫺1.456 ⫺1.858 ⫺1.103 2.362
⫺0.579 ⫺1.468 ⫺1.882 ⫺1.107 2.670
T * ⫽5
within 1%. At the end of the iteration cycle ␣ (T, ) ⫽ ␣ coh(T, ) and the excess chemical potential reads
coh  ex ⫽
冕冋
⫺h 共 r 兲 ⫹ ␥ 共 r 兲 ⫹B 共 2 兲 共 r 兲 ⫹
册
⫹ 共 1⫹ ␣ coh共 T, 兲兲 B 共 2 兲 共 r 兲兴 dr.
h共 r 兲 关 ␥共 r 兲 2 共7兲
Then the entropy s is calculated with the help of Eq. 共3兲. It is a matter of fact that the proposed method does not require knowledge of the primitive of B (2) (r) with respect to ␥ * (r) as is the case with the direct formula of Lee.6
The results for the Lennard-Jones fluid with the dimensionless parameters * ⫽ 3 共 being the soft-sphere diameter兲 and T * ⫽k B T/ are presented in Table I. For sake of comparison, we have included some simulation data9 and recent calculations of the integral equation theory.7,10 The results are in close agreement altogether from low density up to * ⫽0.9. It is obvious that our method improves by several percent the previous results that we obtained by using the same bridge function in conjunction with the direct formula of Lee. In Fig. 1, along the isotherm T * ⫽1.5, our results are in excellent agreement with those of MC11 and do compare very well with the equation of state of Nicolas et al.12 At * ⫽0.9, the calculated excess chemical potential takes the value 3.17, which is in very good agreement with Lee’s five parameters ZSEP integral equation6 result that is 3.05. Yet the results are rather sensitive to the approximation of B (1) (r). The state-dependent parameter ␣ coh(T, ) presents a nontrivial density dependence due to the changing local liquid structure. Nevertheless, it is noticeable that it takes zero value at low density which is not surprising, since for this range of densities, B (2) (r) is almost zero. Our approach presents several advantages: 共i兲 It allows one to satisfy the Gibbs–Duhem relation in a simple way, 共ii兲 it is more tractable than the direct formula proposed in Ref. 6, and 共iii兲 it can be used in conjunction with any consistent 共or not兲 integral equation. The method presented here and applied to our thermodynamically consistent closure has proven its ability to predict accurate thermodynamic properties of the LJ fluid. In addition, this approach could be used to predict the vapor–liquid coexistence boundaries and the self-diffusion coefficient13 for dense liquids.
FIG. 1. Excess chemical potential 共full line兲 as a function of density along the T * ⫽1.5 isotherm calculation with our integral equation and the proThe author would like to thank sincerely Lloyd L. Lee posed method. The triangles are those obtained by using the equation of and J. L. Bretonnet for stimulating discussions and Sylvia state of Nicolas et al. 共Ref. 12兲 and the symbols stand for MC results 共Ref. Gomez for her kind interest. 11兲. This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
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11486 a兲
J. Chem. Phys., Vol. 119, No. 21, 1 December 2003
Electronic mail:
[email protected] J. M. Bomont and J. L. Bretonnet, J. Chem. Phys. 119, 2188 共2003兲. 2 J. M. Bomont and J. L. Bretonnet, J. Chem. Phys. 114, 4141 共2001兲. 3 G. A. Martynov, Fundamental Theory of Liquids. Method of Distribution Functions 共Adam Hilger, Bristol, 1992兲. 4 E. Lomba, M. Alvarez, L. L. Lee, and Almarza, J. Chem. Phys. 104, 4180 共1996兲. 5 A. B. Schmidt, J. Chem. Phys. 99, 4225 共1993兲. 6 L. L. Lee, J. Chem. Phys. 107, 7360 共1997兲. 1
Jean-Marc Bomont G. Sarkisov, J. Chem. Phys. 114, 9496 共2001兲. L. L. Lee, J. Chem. Phys. 97, 8606 共1992兲. 9 J. K. Johnson, J. A. Zallweg, and K. E. Gubbins, Mol. Phys. 78, 591 共1993兲. 10 N. Choudhury and S. K. Ghosh, J. Chem. Phys. 116, 8517 共2002兲. 11 L. L. Lee, D. Ghonasgi, and E. Lomba, J. Chem. Phys. 104, 8058 共1996兲. 12 J. J. Nicolas, K. E. Gubbins, W. B. Streett, and D. J. Tidesley, Mol. Phys. 37, 1429 共1979兲. 13 M. Dzugutov, Nature 共London兲 381, 137 共1996兲. 7 8
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