Calculating thermodynamic properties from perturbation theory I. An analytic representation of square-well potential hard-sphere perturbation theory

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Ž. Fluid Phase Equilibria 154 1999 1 1 Calculating thermodynamic properties from perturbation theory I.

accepted September 1998 Abstract The Barker Henderson macroscopic compressibility approximation of the second-order perturbation term is improved by assuming that the numbers of molecules in every

1 Ž. Fluid Phase Equilibria Calculating thermodynamic properties from perturbation theory I. An analytic representation of square-well potential hard-sphere perturbation theory Bing-Jian Zhang ) Department of Chemistry, Zheiang UniÕersity, Hangzhou, Zheiang, 317, China Received 1 June 1998; accepted September 1998 Abstract The Barker Henderson macroscopic compressibility approximation of the second-order perturbation term is improved by assuming that the numbers of molecules in every two neighbour shells are correlated, based upon the original assumptions. The results are better than those for the original macroscopic compressibility and local compressibility approximation, especially at high densities. A simple analytic representation of square-well potential hard-sphere perturbation theory is derived based upon this improvement. The method is tested by calculating thermodynamic properties with the four-term truncated form, and the results are in good agreement with those of Monte Carlo and Molecular Dynamics simulation. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Statistical mechanics; Perturbation theory; Equation of state; Excess properties; Heat capacity 1. Introduction Perturbation theories have provided a decisive step in understanding and calculating the thermodynamic properties of classical fluids. Of a number of perturbation theories, the Barker Henderson second-order and higher-order expansion w1,x is often used pertaining to the fluids consisting of spheres or sphere-chains, such as the perturbation theory of Chang and Sandler w3,4x or the recent perturbed hard-sphere-chain equation of state by Hino and Prausnitz wx 5. In this paper, an improved second-order and higher-order term is proposed on the basis of the Barker Henderson macroscopic compressibility Ž mc. approximation, and a simple analytic representation of the square-well potential hard-sphere perturbation theory is derived based upon this ) Tel.: q ; fax: q ; zb@public.hz.z.cn r99r$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. Ž. PII: S

2 ( ) B.-J. ZhangrFluid Phase Equilibria improvement. The method will be expended to fluids with a complex potential and to non-spherical molecules in subsequent publications.. Theoretical For a one-component system formed by N spheres in a volume V at a temperature T, expressing the potential energy of the studied system as a combination of the potential energy of the reference system plus the perturbation energy, and if the perturbation energy is given by the sum of pair interactions, the first-order perturbation term of the Helmholtz energy of the system may be expressed on the basis of the distribution function g Ž r. of the reference system. H A1r NkT sp Nrb up r g r r dr 1 Ž. Ž. Ž. Ž. where r is the distance of the interacting particles, u Ž r. P is the perturbation pair interaction, bsž kt. y1, k is the Boltzmann constant and rsnrv is the particle density. The second-order perturbation terms, include three- and four-body distribution functions, and no satisfactory approximations are available. Barker and Henderson wx 1 divided the range of intermolecular distances into intervals Ž r, r.,... Ž r, r. 1 i iq1,..., etc., and by taking the limit when the interval width goes to zero, the continuous description is recovered. If Ni represents the numbers of molecules in a spherical shell i surrounding other central molecules and if u Ž r. P i can be regarded as having a constant value u in the spherical shell of interval Ž r, r., Ž is, 1,,... etc.. i i iq1, the perturbation expansion of the second-order can be written in the form Ý Ž. ÝŽ. Ž. ² : ² : ² :² : 3 b Ž AyA. sb N u y b r NN y N N u u qo b Ž. i i i i i i i, where A and A are the Helmholtz energy of the studied and the reference system, respectively. Angular brackets ²: are used to indicate the average over the configurations of the reference system. From Eqs. Ž 1. and Ž., one obtains H r i i ² N : sp Nr g Ž r. r dr Ž 3. r iy1 Barker and Henderson wx 1 assumed that these shells were large macroscopic volumes, and the numbers of molecules in different shells would be uncorrelated: ² N yn : y² N:² N : s Ž i/. Ž 4. i i The fluctuation of the number of molecules in a given shell would be given by ² N : y² N : s² N: ktž ErrE P. Ž 5. i i i where P is the pressure. Eqs. Ž 3. Ž 5. are substituted into Eq. Ž., and returning to the continuous description, the second-order perturbation term A rž NkT. is obtained A Er syprb Hu P Ž r. kt g Ž r. r dr. Ž 6. NkT E P r

3 ( ) B.-J. ZhangrFluid Phase Equilibria The approximation for the second-order perturbation term given in Eq. Ž 6. is referred to as macroscopic compressibility Ž mc.. Similar expressions can be obtained for the higher-order terms. Just as Barker and Henderson said, despite its intuitive appeal and convenience, the mc approximation has not been overly successful wx. A rž NkT. calculated from Eq. Ž 6. using the square-well potential is too small at intermediate and high densities. This is also true for the 6 1 potential. Further, the local compressibility Ž lc. approximation is only slightly better than Eq. Ž 6.. As Barker and Henderson realized, in mc approximation, there is a problem in that by taking the limit as the interval width Ž r, r., Ž is,1,..., etc.,. i iq1 tends to zero, the shells were not large macroscopic volumes. The number of molecules in neighbouring shells would be correlated! Assuming that the number of molecules in every two neighbour shells is correlated, and the correlation coefficient w of the number of molecules in every two neighbour shells is directly proportional to the particle density and the hard volumes of molecules in this two neighbour shells. For hard-core molecules, according to the definition of the correlation coefficient ² NN : y² N:² N : p ws sk r s skh Ž 7. ² N : y² N: ² N : y² N : i iq1 i iq1 3 6 i i iq1 iq1 ( ( where K is the proportion coefficient; s is the hard-core diameter; hsprs 3 r6. Fig. 1. The second-order perturbation term A rž NkT. of Helmholtz energy for the square-well potential system at ls1.5, b s1. The solid curve gives the results obtained from Eq. Ž 1.; the linear gives mc approximate using Eq. Ž 6. and PPP gives lc approximate wx ; the symbols v are the MD data wx 9 and are the MC data w1 x.

4 4 ( ) B.-J. ZhangrFluid Phase Equilibria Fig.. Radial distribution function g Ž r. of the hard-spheres. The broken curves give the results obtained from Eq. Ž 13. and the symbols ^, I, are the MC results w13x for hs.5p,.1p,.15p, respectively. To determine K, we further assume that the number of molecules in every two neighbour shells is approximately linear correlated with a probability one when the fluid has condensed ŽIn this conditions h s.493 wx. 7. So w s1. Substituting this value in Eq. Ž 7. con con leads to a value of K, i.e., K(1rh con. The mean square difference value of the number of molecules in every two neighbour shells is sufficiently small for fluids. Eq. Ž 7. can be approximated by ² NN : y² N:² N :(Kh ² N : y² N : Ž 8. i iq1 i iq1 i i The number of molecules in non-neighbouring different shells would be uncorrelated. By taking the limit as the interval width tends to zero, the number of shells n would be large, and ny1(n, n ( n. Substituting Eqs. Ž 5. and Ž 8. into the second term of equation Eq. Ž. gives iq1 i n ny1 ² : ² :² : ² : ² : ÝŽ NN y N N. u u s Ýž N y N / u q Ý Ž² NN : i i i i i i i iq1 i, is1 is1 y² N:² N : u u. i iq1 i iq1 n ² : ² : Ý i i i is1 n Er Ž. ( 1q Kh N y N u Ž.Ý i i s 1q Kh ² N: kt u Ž 9. E P is1

5 ( ) B.-J. ZhangrFluid Phase Equilibria Fig. 3. The first-, second- and third-order perturbation terms A, Ž is1,, 3. i, of Helmholtz energy for the square-well potential system at ls1.5, r s1. The broken curves give the results obtained from Eq. Ž 14. and the symbols ^, I, are the MC results wx 9 for first-, second- and third-order terms, respectively. Eqs. Ž 3. and Ž 9. are substituted into Eq. Ž., and returning to the continuous description, the second-order perturbation term A rž NkT. is obtained as follows A Er syprb Ž 1q Kh.Hu P Ž r. kt g Ž r. r dr. Ž 1. NkT E P r The improved mc approximation for the second-order perturbation term is given by Eq. Ž 1.. It is important to note that the only difference between Eqs. Ž 6. and Ž 1. is the factor Ž1 q Kh., i.e., when Ks orr Eq. Ž 1. is identical to Eq. Ž 6.. For the higher-order perturbation terms Ž i). another approximation given by Barker and Henderson wx is used iy1 i 1Ž 1. Ž. A sa A ra ri! 11 This approximation for Ai results in an improvement over the second-order perturbation theory. In this work, we consider the square-well perturbation potential for a hard-sphere reference system, u s Ž sfrfls. P. is the depth of the well. l is the reduced well width of the square-well. Substituting Eqs. Ž 1. and Ž 1. into Eq. Ž 11., leads to 3 AyA b 1 b 1 b l sy1ha q q qppp H g Ž R. R d R Ž 1. NkT a! a 3! a 1

6 6 ( ) B.-J. ZhangrFluid Phase Equilibria Fig. 4. The compressibility factors PV rnkt for the square-well potential system at ls1.5. The broken curves give Ž. w x the results obtained from Eq. 15 and the symbols ^, I, are the MC results 15 for b s.5, 1, 1.5, respectively. where A and A are the configuration Helmholtz energy of the studied system and the reference system, R s rrs, a sž1 q Kh. y1 Ž EPrEr.. A new form of the radial distribution function similar to that of Song et al. wx 6 is used 1 h Ý h s1 1yh g Ž R. s g Ž 13. where g1s1. This expansion is convergent for the same values of the fluid density as those of the virial series, and the coefficients of the series are related to those of the virial series. Nevertheless, because of the denominators in each of its terms, Eq. Ž 13. should converge much faster than number density series as shown by Barboy and Gelbart wx 8. Here h is independent of R for the hard-sphere l reference system. We can write C s H g R d R, and substitute Eq. Ž 13. into Ž Finally, the Helmholtz energy A, can be obtained relative to its value A in the standard state. AyA b h Ý sy1a exp y1 C Ž 14. NkT a 1yh s1

7 ( ) B.-J. ZhangrFluid Phase Equilibria By usual derivations, one obtains the other thermodynamic properties, such as the equation of state PV PV E AyA y sh NkT NkT Eh ž NkT / ½ Ea h E h sy1h Ý C qa Ý C Eh 1yh Eh ž 1yh / the internal energy U = s1 s1 b b Ea b h Ý 5 a a Eh a s1 1yh exp y 1 y exp C Ž 15. UyU E AyA b h Ý sb s1b exp C Ž 16. NkT Eb NkT a 1yh and the constant-volume heat capacity C V s1 CV E AyA Ž b. b h syb s1 exp Ž 17 Ý ž. NkT Eb NkT a a 1yh s1 / Eq. Ž 14. is a good approximation of the Helmholtz energy for a square-well potential and it has the advantage of being analytical. We can expect, moreover, that truncated forms of Eqs. Ž 14. Ž 17. might even describe fluids at liquid-like densities well. These expectations are explored and corroborated in Section Results and discussion In Fig. 1 we compare our values of A rž NkT. vs. h calculated from Eq. Ž 1. with that of Barker Henderson mc approximation calculated from Eq. Ž 6.. The molecular-dynamics Ž MD. results of Alder et al. wx 9 and the Monte Carlo Ž MC. values of Barker and Henderson w1x are also shown in Fig. 1. All results are for the case of ls1.5, r s1. For g Ž r. and Ž Er rep., the values derived from the solution of the Percus Yevick equation for hard-spheres w11x are used, the values of Ž ErrEP. being those given by the compressibility equation isotherm. Fig. 1 shows how the improvement of the mc approximation agree with the available computer simulation data for A rž NkT.. The results indicate that our improvement of the mc approximation are reasonable, especially at intermediate and high densities. To obtained simple analytical expressions, the four-term truncated series of Eq. Ž 13. is applied. According to Ree et al. w1 x, the coefficient of the second power for Eq. Ž 13. has been calculated from the virial density expansion of the radial distribution function for the hard-sphere reference system. Ž. Ž. 3 gs7y6ry.5r R- 18

8 8 ( ) B.-J. ZhangrFluid Phase Equilibria Similarly, by approximation, g3 and g4 are given by y1 y g3s1.y6.41 R q39.89r Ž R-. Ž 19. g4sy9.1q13.69rq4.587r Ž R-. Ž. Substituting Eqs. Ž 18. Ž. into Eq. Ž 13., values of g Ž R. for 1-R- are obtained, which are in good agreement with the simulation values w13x for hard-spheres at h s.5p,.1p,.15p, respectively, as shown in Fig.. The excess Helmholtz energy from Eq. Ž 14. is well at all densities. In Fig. 3, we show results from Eq. Ž 14. using the first-order term as well as calculations including the second- and third-order terms. The values of Ž Er rep. are derived from the Carnahan Starling equation w14x for hard-spheres. The contributions from A 1, A and A3 bring the model into good agreement with the computer simulations wx 9. The equation of state, excess internal energy, and constant-volume heat capacity of the square-well fluid calculated from analytic formula Eqs. Ž 15. Ž 17. are compared with computer simulation data w9,15x in Figs. 4 6, respectively. The agreement is good for densities below that of freezing Ž hf All these results are for the case of the square-well potential, ls1.5, and the Carnahan Starling equation of state w14x is used for the hard-sphere reference system. Calculations were also made for the square-well fluid with l s Similar agreement has been obtained between calculated results and simulation data. In this paper, we have applied the improved perturbation theory based on that of Barker and Henderson to the square-well potential fluids, and show that it agrees with the results of computer Fig. 5. The excess internal energy Ž UyU. rž NkT. for the square-well potential system at ls1.5. The broken curves give the results obtained from Eq. Ž 16. and the symbols ^, I, are the MC results w15x b s.5, 1, 1.5, respectively.

9 ( ) B.-J. ZhangrFluid Phase Equilibria Fig. 6. The constant-volume heat capacity for the square-well potential system at ls1.5, h s6p r18. The broken curves Ž. wx give the results obtained from Eq. 17 and the symbols ^ are the MC results 9. simulations. We have also applied the model to non-ideal fluids with complex potentials and the non-sphericals molecules. These calculations will be reported in subsequent publications. 4. List of symbols A Helmholtz free energy Ai i-order perturbation term for the Helmholtz free energy C Coefficient in Eq. Ž 14. i CV Constant-volume heat capacity g Ž r. Radial distribution function of the reference system g The coefficient of Eq. Ž 13. for g Ž r. K Proportion coefficient in Eq. Ž 7. k Boltzmann constant N Number of molecules Ni Number of molecules in spherical shell i n Number of shells P Pressure R Reduced distance of interacting particles defined as R s rrs r Distance of the interacting particles T Absolute temperature Ž K.

10 1 ( ) B.-J. ZhangrFluid Phase Equilibria U Internal energy u Ž r. P Perturbation pair interaction potential ui Pair interaction potential in the spherical shell i V Volume Greek symbols a Correcting factor defined as asž1q Kh. y1 Ž EPrEr. b Boltzmann factor defined as b sž kt. y1 Depth of the square-well h Reduced density defined as hsprs 3 r6 l Reduced well width of the square-well r Number density s Hard-sphere Ž core. diameter w Correlation coefficient Subscript Reference system Acknowledgements The authors wish to thank Professor Y.C. Hou for many discussions and suggestions. This work Ž. was supported by the Natural Science Foundation of China Zheiang Proect: 951. References wx 1 J.A. Barker, D. Henderson, J. Chem. Phys. 47 Ž wx J.A. Barker, D. Henderson, Rev. Mod. Phys. 48 Ž wx 3 J. Chang, S.I. Sandler, Mol. Phys. 81 Ž wx 4 J. Chang, S.I. Sandler, Mol. Phys. 81 Ž wx 5 T. Hino, J.M. Prausnitz, Fluid Phase Equilib. 138 Ž wx 6 Y.H. Song, S.M. Lambert, J.M. Prausnitz, Macromolecules 7 Ž wx 7 V.C. Aguilera-Navarro, J. Stati. Phys. 3 Ž wx 8 B. Barboy, W.M. Gelbard, J. Chem. Phys. 71 Ž wx 9 B.J. Alder, D.A. Young, M.A. Mark, J. Chem. Phys. 56 Ž w1x J.A. Barker, D. Henderson, Annu. Rev. Phys. Chem. 3 Ž w11x W.R. Smith, D. Henderson, Mol. Phys. 19 Ž w1x F.H. Ree, R.N. Keeler, S.L. McCarthy, J. Chem. Phys. 44 Ž w13x J.A. Barker, D. Henderson, Mol. Phys. 1 Ž w14x N.F. Carnahan, K.E. Starling, J. Chem. Phys. 51 Ž w15x D. Henderson, W.G. Madden, D.D. Fitts, J. Chem. Phys. 64 Ž

Three Semi-empirical Analytic Expressions for the Radial Distribution Function of Hard Spheres

Commun. Theor. Phys. (Beijing, China) 4 (2004) pp. 400 404 c International Academic Publishers Vol. 4, No. 3, March 5, 2004 Three Semi-empirical Analytic Expressions for the Radial Distribution Function