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

Transcription

1 Commun. Theor. Phys. (Beijing, China) 4 (2004) pp c International Academic Publishers Vol. 4, No. 3, March 5, 2004 Three Semi-empirical Analytic Expressions for the Radial Distribution Function of Hard Spheres SUN Jiu-Xun, CAI Ling-Cang, 2 WU Qiang, 2 and JING Fu-Qian 2 Department of Applied Physics, University of Electronic Science and Technology, Chengdu 60054, China 2 Laboratory for Shock Wave & Detonation Physics Research, Southwest Institute of Fluid Physics, P.O. Box 99-02, Mianyang 62900, China (Received February 3, 2003) Abstract Three simple analytic expressions satisfying the limitation condition at low densities for the radial distribution function of hard spheres are developed in terms of a polynomial expansion of nonlinear base functions and the Carnahan Starling equation of state. The simplicity and precision for these expressions are superior to the well-known Percus Yevick expression. The coefficients contained in these expressions have been determined by fitting the Monte Carlo data for the first coordination shell, and by fitting both the Monte Carlo data and the numerical results of Percus- Yevick expression for the second coordination shell. One of the expressions has been applied to develop an analytic equation of state for the square-well fluid, and the numerical results are in good agreement the computer simulation data. PACS numbers: Ce Key words: radial distribution function, hard spheres, equation of state, square-well fluids Introduction The perturbation theories such as the Barker Henderson theory, [] the Weeks Chandler Anderson theory, [2] the Ree theory, [3] and the Ross theory [4] are most frequently used in the research of thermodynamic properties for fluids both at the normal condition and at the condition high temperatures and densities. The perturbation theories require knowledge of the equation of state (EOS) and the radial distribution function (r.d.f.) of a reference hard-sphere fluid. For this fluid, the Carnahan Starling (CS) EOS [5] combines simplicity and accuracy. As for the r.d.f., there are analytical expressions available for the r.d.f. from the solution of the Percus Yevick (PY) integral equation. [6,7] However, except that the Laplace transformation of PY r.d.f. is simple enough, its expression in coordinate space is too complicated to be convenient for practical applications, and this results the perturbation schemes manifestly failing to provide a generally analytic and applicable EOS even for the simplest square-well or Sutherland fluids. So many theories developed subsequently prefer using the numerical table given by Troop and Bearman to using the analytic PY expression, [8] such as the mode expansion theory for the electrolyte solutions [9] and the renormalization theory for simple fluids. [0] Moreover, it is known that the PY solution is not sufficiently accurate, particularly for radial distances close to contact. Therefore, a considerable number of procedures have been developed to improve the r.d.f. obtained from integral equation theories. [ 5] In most cases, the resulting r.d.f. becomes more complicated than the PY expression or even nonanalytical, which makes its use impractical, particularly in the context of perturbation theories. In recent years, people tend to develop semi-empirical analytic expressions for the r.d.f. of hard spheres, for example, Zhang [6,7] and Largo and Solana [8] have developed their expressions. But they have not carefully selected the form of the expressions, and the developed expressions have several shortcomings. The most serious shortcoming is that the two expressions cannot satisfy the limitation condition, which is the r.d.f. should tend to one as the density of fluid tends to zero. The second shortcoming for Zhang s expression [6,7] is that the coefficients were not properly fitted, thus the error contrast to the Monte Carlo (MC) simulation data [9] is even larger than that of the PY expression. The second shortcoming for Largo and Solana s expression [8] is that it contains too many coefficients, for example, 28. Its complexity for practical applications is competitive to the numerical table of Troop and Bearman. [8] The third is that the two expressions have not used any information of CS EOS, so one cannot reproduce the CS EOS by using the two expressions. The last is that they can only give the expression of r.d.f. for the first coordination shell, but cannot give the expression outside the first coordination shell for the complexity. Now we develop three analytic expressions, which can overcome the above-mentioned four shortcomings and can combine the simplicity, accuracy, and analyticity in it, and such expressions are believed to be very useful for many practical applications. By using these expressions, most of The project supported by National Natural Science Foundation of China under Grant Nos and , and by the Science and Technology Foundation for the Youth of the University of Electronic Science and Technology of China under Grant No. YF020703

2 No. 3 Three Semi-empirical Analytic Expressions for the Radial Distribution Function of Hard Spheres 40 the present perturbation theories may become simple analytic ones, and one expression has been applied to develop an analytic equation of state for the square-well fluid. In Sec. 2, the analytic expression is proposed. In Sec. 3, the analytic EOS for the square-well fluid is developed. At last, the numerical results are given in Sec Development of Analytic Expressions The three expressions of the r.d.f. that we propose are + x k g(x) = ( η) m g(k) m (x), x < 3, () m=, x 3, where k = 0,, and 2 correspond to the three expressions, respectively, η = πρd 3 /6 is the packing fraction corresponding to the number density ρ for spheres of diameter d, and x is the radial coordination reduced to the hard sphere diameter d. The expressions in Eq. () have some theoretical foundations. First, they satisfy the limitation at low densities, i.e., g(x) tends one as η tends zero. Second, in terms of the expression of g() from the CS EOS, [5] the obtained g() can be reformulated as the following form, g() = 0.5η 2.5η = + ( η) 3 ( η) + 2η2 ( η) η3 ( η) 3. (2) It is obvious that the form of Eq. () is in accordance Eq. (2). It should be pointed out that the CS EOS is quite accurate at low and intermediate densities but at higher densities it starts to slightly deviate from the computer simulation data. In the last decade the Kolafa [9] and other equations have been shown having higher precision for hard sphere system at higher densities. [20] However, Mulero, et al. [2] have pointed that such complicated EOS only gives good description for some properties of the hard sphere system but for other properties gives worse results. The most important reason for the researches of the hard sphere system is that it has been taken as the reference system for most of thermodynamic perturbation theories, but Mulero, et al. [2] have also shown that the more complicated EOS for hard spheres does not always give better results for the perturbation system. The simple CS EOS may give better results than most of the more complicated EOS. Although the CS EOS slightly deviates from the computer simulation data at higher densities, Ree [3] and Ross [4] have shown that a perturbation theory being applicable for fluids at high densities or all fluid densities will most probably be a variational theory. For such a variational theory the packing fraction ay always take finite values, which makes the CS EOS the most appropriate. Thus we have selected the CS EOS as a foundation in this work. In previous works, [6 8] people tend to directly expand the relevant functions of coordination g m (k) (x) as polynomials of x. Such expansions are slowly convergent, for example, in order to reach an acceptable fitting precision, Largo and Solana had to retain 28 terms in their expansion, [8] and the expansion of Zhang, only retaining several terms, results in very poor fitting precision. [6,7] Instead of expanding g m (k) (x) as polynomials of x, we propose expanding g m (k) (x) as polynomials of nonlinear base functions, and g (0) g () 4 C (0) mn(s s 4 ) n, C () mn(s s 4 ) n, C (2) mn(s s 4 ) n (3) s = exp(x ), ( x < 2) ; (4) g (0) g () 4 D (0) mn(s s 7 ) n, D (2) mn(s s 7 ) n, D (2) mn(s s 7 ) n (5) s = exp(x 2) (2 x < 3). (6) Such expansions are rapidly convergent, and we find that retaining five terms for k = 0, retaining four terms for k = and 2 can give best fitting results, respectively. In order to determine six coefficients C (k) m0 and D(k) m0, we need the expressions for g() and g(2). The expression for g() is given in Eq. (2), and the expression for g(2) is determined by fitting the MC data [9] at x = 2 and is given as g(2) = 0.34η ( η) + η 2 ( η) η3 ( η) 3. (7) Comparing Eq. (3) Eq. (2), and Eq. (5) Eq. (7), C (k) m0 and D(k) m0 can be easily determined. Other coefficients n 0 are determined by fitting the MC data [9] for C mn (k) and by fitting both the MC [9] and the PY data [8] for D mn. (k) The fitting procedure, which contains two steps, is simple and straightforward. In the first step, we keep x invariable and fit three g m (x) at every x value. In the second step, we fit g m (x) by using Eqs. (3) and (5), respectively. The fitted coefficients are listed in Table. The totally average errors of the three expressions and the PY expression for 344 MC data points in the interval x < 2 are 0.77%,.04%,.09%, and.32%, respectively, and these errors of the three expressions for 488

3 402 SUN Jiu-Xun, CAI Ling-Cang, WU Qiang, and JING Fu-Qian Vol. 4 MC and PY data points in the interval 2 x < 3 are 0.62%, 0.96%, and 0.98%, respectively. The error comparison of our two expressions PY expression in the most important range ( x < 2) contrast to the MC data is shown in Fig.. The figure shows that our expressions give fairly well and improved results as compared the PY expression, especially near the contact. Fig. Error comparison for the r.d.f. of hard spheres calculated by using several analytic expressions, at ρd 3 = 0.8 (a) and 0.9 (b). Open circles are Percus Yevick expression, crisscrosses are analytic expression in this paper k = 0, and diamonds are analytic expression k = 2. The MC data [9] used to determine the coefficients are from Barker and Henderson, although the data are slightly old, they are cited by many latest works. [4,5] We think this has several reasons. The most important is that such data are scarce, for people do not like doing repeated work after Barker and Henderson. [9] The second is that the errors in a perturbation theory from r.d.f. of reference hard spheres mainly come from the neighbor of the contact, and once the CS EOS is used to determine the r.d.f. at contact, the data from BH are acceptable. The third are the good agreement of improved expression of Tang and Lu the MC data shows the MC data must have reasonable precision. In some theories, such as the WCA perturbation theory, [2] the direct correlation function c(x) is also needed. The function c(x) derived from PY integral equation is c(x) = λ + λ 2 x + λ 3 x 3, (8) where three coefficients λ i in PY integral equation theory have been given by Wertheim [6] and Thiele. [7] Since figure shows that our expression for r.d.f. near the contact is fairly accurate, we would like to use it to improve the direct correlation function. The method is to treat three coefficients λ i as adjustable parameters, and determine them by using the continuity of g(x) and c(x) as well as the first and second derivatives at contact. The obtained λ i is given in the following, g () = g () = m= m= λ 3 = 6 g (), λ 2 = g () 3λ 3, λ = g() λ 2 λ 3 (9) ( η) m [5C(k) m kc(k) m ], ( η) m [50C(k) m2 + (5 0k) C (k) m + k(k + )C(k) m0 ]. (0)

4 No. 3 Three Semi-empirical Analytic Expressions for the Radial Distribution Function of Hard Spheres Application to Square-Well Fluid The square-well potential is the simplest and useful model potential. With three adjustable parameters, it can model practical thermodynamic properties for a variety of simple fluids. Many authors have researched its properties using various methods. Barker and Henderson gave the MC simulation results a long time before. [22] White has selected it as working model in the renormalization theory for simple fluids very recently. [0] In order to check the applicability of our expressions, we apply one of our expressions to the square-well fluids depth ε and variable width λ (λ 2). For simplicity, we select the expression k = 2. In the second-order Barker Henderson perturbation theory, [] the free energy can be expressed in the form F NkT = F 0 NkT + F NkT T r + F 2 NkT T 2 r () T r = kt/ε, where ε is the energy parameter of the potential. The subscript 0 refers to the hard-sphere reference fluid, while F NkT = 2η F 2 NkT = 6ηQ where L(λ) = + L m (λ) = g(x)x 2 dx = 2ηL(λ), (2) g(x)x 2 dx = 6ηQL(λ), (3) g(x)x 2 dx = 3 (λ3 ) m= + C (2) m + C (2) m2 +C (2) m3 ( η) m L m(λ), (4) m (x)dx = C (2) (λ ) m0 ( q + 4 q 4 5 ) 4 ( 2 q q 3 8 q 8 25 ) 24 ( 3 q q q q ) (5) q = exp(λ ), and in the so-called macro-compressibility approximation, ( ) ρ ( η) 4 Q = kt = P ( + 2η) 2 (4 η)η 3. (6) 0 The compressibility factor can be obtained from Eq. (9) in the form P V NkT = η ( F ) = P 0V η NkT NkT + P V + P 2V, (7) NkT T r NkT where P V L(λ) = 2ηL(λ) 2η2, (8) NkT η P 2 V NkT = 6ηQL(λ) 6η2[ L(λ) Q η + Q L(λ) η and L(λ) η = m= T 2 r ], (9) m ( η) m+ L m(λ), (20) Q η = ( η)3 (8 + 20η 4η 2 ) [( + 2η) 2 (4 η)η 3 ] 2. (2) 4 Numerical Results and Conclusive Remarks From the expressions derived previously, we have calculated the first- and second-order perturbation free energy and the compressibility factor of square-well fluid λ =.5 as a function of the reduced density ρd 3 and the temperature factor βε = ε/kt, respectively. Results are compared in Figs. 2 and 3 simulation data. The situation is just similar to Ref. [22], that is, for the first-order perturbation free energy and the compressibility factor, the agreement is as good as the results from Barker Henderson calculation, which uses the PY expression. The large discrepancy for the second-order perturbation has been attributed to the perturbation theory itself by Barker and Henderson, rather than to the inaccuracy of the analytic expression of r.d.f. developed. Fig. 2 F (a) and F 2 (b) for the square-well fluid λ =.5. Filled circles are simulation data from Ref. [9], solid lines are Eq. (5) Eqs. (6) (9).

5 404 SUN Jiu-Xun, CAI Ling-Cang, WU Qiang, and JING Fu-Qian Vol. 4 Fig. 3 Compressibility factor Z = P V /NkT for the square-well fluid, λ =.5, at ρd 3 = 0.6 (a) and 0.85 (b), filled circles are simulation data from Ref. [9], solid lines are Eq. (5) Eqs. (6) (9). In summary, we have shown that it is possible to obtain simple analytic expressions for the r.d.f. of hard spheres high precision by carefully selecting the fitting functions. The precision of the analytic expressions is higher than the well-known PY expression. By using the expressions, most of present perturbation theories can become simple analytic ones, and this has been implemented for the simplest square-well fluids. The extension to continuous potentials, such as the Lennard-Jones potential is straightforward and is to be done. Alternatively, thermodynamic properties of fluids continuous potentials could be obtained from those of an equivalent square-well fluid its potential parameters suitably determined. References [] J.A. Barker and D. Henderson, J. Chem. Phys. 47 (967) 474. [2] J.D. Weeks, D. Chandler, and H.C. Andersen, J. Chem. Phys. 54 (97) [3] F.H. Ree, J. Chem. Phys. 64 (976) 460. [4] M. Ross, J. Chem. Phys. 7 (979) 567. [5] N.F. Carnahan and K.E. Starling, J. Chem. Phys. 5 (969) 635. [6] M.S. Wertheim, Phys. Rev. Lett. 0 (963) 32. [7] E. Thiele, J. Chem. Phys. 39 (963) 474. [8] G.J. Troop and R.J. Bearman, J. Chem. Phys. 42 (965) [9] H.C. Anderson and D. Chandler, J. Chem. Phys. 55 (97) 497. [0] J.A. White, J. Chem. Phys. 3 (2000) 580. [] W.R. Smith and D. Henderson, Mol. Phys. 9 (970) 4. [2] J. Chang and S.I. Sandler, Mol. Phys. 8 (994) 735. [3] S. Bravo Yuste and A. Santos, Phys. Rev. A43 (99) 548. [4] S. Bravo Yuste, M. López de Haro, and A. Santos, Phys. Rev. E53 (996) [5] Y. Tang and C.-Y. Lu, J. Chem. Phys. 00 (994) [6] B.J. Zhang, Chem. J. Chinese Univ. 6 (995) 440. [7] B.J. Zhang, Chem. Phys. Lett. 296 (998) 266. [8] J. Largo and J.R. Solana, Fluid Phase Equil. 67 (2000) 2. [9] J.A. Barker and D. Henderson, Mol. Phys. 2 (97) 87. [20] J. Kolafa, quoted by T. Boublik, Mol. Phys. 59 (986) 37. [2] A. Mulero, C. Galán, and F. Cuadros, J. Chem. Phys. (999) 486. [22] J.A. Barker and D. Henderson, Rev. Mod. Phys. 48 (976) 587.