Russian Physics Journal, Vol. 53, No. 2, 2010
CALCULATION OF DISTRIBUTION FUNCTIONS FOR TWOCOMPONENT RIGID-SPHERE MOLECULES OF A FLUID Yu. A. Bogdanova, S. A. Gubin, S. B. Viktorov, A. V. Lyubimov, and V. A. Shargatov
UDC 536.715
Analytical expressions are derived and a computational algorithm and program for calculating distribution functions of rigid spheres gij(r) necessary for calculations of the thermodynamic parameters of a binary fluid mixture are developed in the context of perturbation theory using the procedure based on the inversion of the Laplace transform for functions rgij(r) obtained from the Percus–Yevick equation. Keywords: distribution function of molecules, fluid, Monte Carlo (MC) method, molecular dynamics, Laplace transform, equations of state, statistical mechanics, two-component system, Helmholtz energy, supercritical conditions, perturbation theory.
INTRODUCTION To solve problems in geophysics, astrophysics for modeling of the state of matter in the atmospheres and internal planetary layers, and physics and chemistry of detonation and shock waves, the composition and thermodynamic parameters of multicomponent multiphase chemical systems at high pressures and temperatures must be calculated. The realistic character of the examined thermodynamic quantities depends primarily on the accuracy of the employed equations of state. Most of the chemical systems studied in applied problems consist entirely or substantially of gaseous components under supercritical conditions. Such systems are called fluids. The problem of obtaining the equations of fluid state that describe reliably the properties of the multicomponent fluid mixtures at high pressures and temperatures is important for thermodynamic calculations of such systems. The derivation of theoretically substantiated equations of state for dense fluid systems using modern methods of statistical mechanics and realistic molecular interaction potentials is of significant interest. These equations of state provide good agreement with the results of Monte Carlo (MC) simulation and molecular dynamics, and they can be used to predict the properties of fluid systems at high pressures. Many theoretical models of fluid systems have been developed by the present time, and work on their refinement is underway. The best modern perturbation theories [1‒3] and integral equations [4] allow the thermodynamic properties of dense one-component fluids with different intermolecular potentials to be calculated in excellent agreement with the results of Monte Carlo (MC) simulation and molecular dynamics. Definite successes have been achieved in the development of methods describing high-density fluid mixtures, but equations of state for systems with any arbitrary number of components with accuracy as high as that of one-component theories have not yet been derived. The derivation of reliable theoretical equations of state for a multicomponent fluid is of great scientific and practical importance. The present work is the first stage of the derivation of these equations of state, and its results can be used to calculate the state parameters of two-component fluids. The perturbation theory with the help of which it became possible to substantiate physically calculations of the thermodynamic fluid properties at high pressures suggests that the Helmholtz energy (and hence any other thermodynamic characteristic obtained by differentiation of the Helmholtz energy) of an ensemble of molecules at preset temperature T and volume V is the sum of two components. First of them is the Helmholtz energy of an ideal gas
Moscow Engineering Physics Institute, Moscow, Russia, e-mail:
[email protected]. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 10‒17, February, 2010. Original article submitted June 17, 2008. 114
1064-8887/10/5302-0114 ©2010 Springer Science+Business Media, Inc.
at the same T and V values, and the second component is an addition caused by the intermolecular interaction. Therefore, one of the primary goals of the perturbation theory is exact calculation of the additional Helmholtz energy. To calculate it in the context of the perturbation theory, the radial distribution function of molecules g0(r) must be known. Here 4πr2g0(r)dr characterizes the probability of finding any pair of molecules of the basic system at distances ranging from r to r + dr, which specifies the physical meaning of the function g0(r). Therefore, calculation of the molecule distribution functions for different fluid models is of significant interest. The present work is aimed at derivation of analytical expressions for calculation of the radial molecule distribution functions and development of a procedure for calculating the radial distribution functions for rigid spheres of a two-component system. One of the most successful theories used for calculation of the molecule distribution functions is the approximate Percus–Yevick theory which gives the Percus–Yevick equation for the radial molecule distribution function. Lebowitz [5] obtained the Laplace transforms for the radial rigid sphere distribution function of a binary mixture in the entire range of intermolecular distances, but the inverse Laplace transforms reported by Lebowitz were calculated only at points of rigid sphere contacts. Using these expressions at the points of contact, Lebowitz and Rolinson calculated some thermodynamic properties of a binary rigid sphere mixture. Smith and Henderson [6] estimated numerically the well-known solution [7, 8] for the radial rigid-sphere distribution function of the fluid g(r) in the Percus–Yevick approximation. The calculated radial distribution functions were presented in [9] only graphically and cover a limited range of intermolecular distances without analytical form of the distribution functions. The corresponding solution were also presented in [10]; however, the analytical expressions for the distribution functions covered only a small range of intermolecular distances and were incorrect (the formulas were wrong). Therefore, it is impossible to use expressions presented in [10] to calculate gij(r). Based on the foregoing, we can conclude that in the literature there are no exact analytical expressions suitable for calculations of the molecule distribution functions gij(r) for two-component solid-sphere fluids. In the present work, analytical expressions for the radial molecule distribution function of a two-component fluid are derived and a procedure applicable, in contrast to [9], to any arbitrary distances r between the centers of the rigid spheres with different diameters is developed. These expressions can be used to confirm the results of Smith and Henderson [6] obtained for a one-component fluid that are applicable for r values for which the procedure suggested in [6] does not work. ANALYTICAL EXPRESSION FOR THE RADIAL DISTRIBUTION FUNCTION OF RIGID-SPHERE MOLECULES IN A BINARY MIXTURE Proceeding from the condition that the integral ∫r(gij(r) - 1)dr remained finite, Lebowitz [5] derived analytical expressions for the Laplace transform of three functions gij(r) for a binary mixture:
Gij ( s) = 12(η1η2 )1/ 2
∞
∫ exp(− sr )rgij (r )dr ,
(1)
Rij
G11 ( s) = s [h − L2 ( s) exp ( sR22 )]/ D( s) , 3 3 G21 ( s) = (η1η2 )1/ 2 s 2 exp(sR12 ){[0.75(η2 R22 − η1R11 )( R22 − R11 ) − R12 (1 + 0.5ξ)]s − (1 + 2ξ)}/ D( s) ,
G22 ( s) = s [h − L1 ( s ) exp ( sR11 )]/ D( s),
(2)
where
h = 36(η1η2 )( R22 − R11 )2 , 115
D( s) = h − L1 ( s) exp( sR11 ) − L2 ( s ) exp( sR22 ) + S ( s ) exp( s ( R11 + R22 )) , 2 L1 ( s) = 12η2 [(1 + 0,5ξ) + 1.5η1R11 ( R22 − R11 )]R22 s 2 + (12η2 (1 + 2ξ) − hR11 ) s + h , 2 L2 ( s) = 12η1[(1 + 0.5ξ) + 1.5η2 R22 ( R22 − R11 )]R11s 2 + (12η1 (1 + 2ξ) − hR22 )s + h , 2 2 2 2 S ( s) = h + [12(η1 + η2 )(1 + 2ξ) − h( R11 + R22 )]s − 18(η1 R11 + η2 R22 ) s 2 2 −6(η1 R11 + η2 R22 )(1 − ξ) s 3 + (1 − ξ) 2 s 4 ,
ηi =
π ρi , 6
3 3 ξ = η1R11 + η2 R22 .
(3)
In Eq. (3), ρi = Ni/V is the molecular density of the ith component, Rii is the effective diameter of the rigid sphere for each component, and R12 = (R11 + R22)/2. Representing 1/D(s) in the form of a geometrical progression, we have
[ D( s )]−1 =
exp(− s ( R11 + R22 )) ⎡ I ( s) ⎤ ⎢1 − S ( s ) ⎥ S (s) ⎣ ⎦
−1
n
=
exp(− s( R11 + R22 )) ∞ ⎡ I ( s ) ⎤ ∑⎢ ⎥ , S ( s) n =0 ⎣ S ( s) ⎦
where I(s) = L2(s)exp(–sR11) + L1(s)exp(‒sR22) – hexp(–s(R11 + R22)). Taking the Laplace transform, the exact solution ∞
rgij (r ) = ∑ ∑ n =0 k
1 C +ι∞ φk ( s ) exp{s[r − Ψk ( R11 , R22 )]} ds ∫ 2πι C −ι∞ S ( s)
(4)
was obtained, where φk(s) is a polynomial and ψk(R11, R22) is a linear combination of R11 and R22 with real coefficients. If r < ψk, the integral is equal to zero. When r > ψk, the integral can be calculated with the help of residues of the four poles of the order n + 1. Each quantity φk(s) is one of the terms of the expansion [I(s)]‒n multiplied by the corresponding coefficient in the definition of Gij(s). Each term must be differentiated n times and calculated in each pole. This labor-consuming analytical work was carried out by Throop and Bearman [9]. As a result, the functions gij(r) were obtained for r < 3Rmin + Rii, and in [10] they were obtained for r ≤ 5Rii. No analytical expressions and numerical tables were presented in [9, 10]. Taking into account that the analytical expression for gij(r) is required to calculate the free energy of the binary fluid in the context of the perturbation theory, it must be derived for the functions gij(r). In the present work, the following final expression was derived for the radial rigid-sphere distribution functions of the binary mixture based on the inverse Laplace transform of functions (1): n 1 n i i j d n−k ∑ ∑ C nC i (−h)n −i ∑ C nk n − k ds p =1 n = 0 n !i = 0 j = 0 k =0 4
N
rgij (r ) = ∑ ∑
m
×{mn1h(r − β) m exp( s (r − β)) + mn2 ∑ C lm l =0
where for g11 we have
116
d
k ⎡ L j ( s ) Li − j ( s ) ⎤ d n−k ⎢ 1 n +12 ⎥ ∑ C km n − k [ s] ds ⎢⎣ G p ( s ) ⎥⎦ s = Z m = 0 p
m −l
ds
m −l
( L0 )( r − γ )l exp( s ( r − γ ))} |s = Z p ,
(5)
mn1 =
1 1 , mn2 = − , 12η1 12η1
γ = α + R11 , L0 = L2 , Rmin = R11 ; for g12 we have
mn1 = 0 , mn2 = 1 , γ = α + R12 ,
{
}
3 3 L0 = 0.75(η2 R22 − η1 R11 )( R2 − R1 ) − R12 (1 + 0.5ξ) s 2 − (1 + 2ξ) s ,
Rmin = R12 ; and for g22 we have mn1 =
1 1 , mn2 = − , 12η2 12η2
γ = α + R22 , L0 = L1 , Rmin = R22 ,
α = R11 (n − j ) + R22 (n − i + j )
.
The N value here depends on r: N = 0, q 1, N = [q], q > [q] > 1, where q = r/Rmin. The last term of sum (5) represents a real reminder, and it is difficult enough to calculate it. Throop and Bearman [9] estimated it analytically up to N = 4. However, using the Leibniz rule of calculating derivatives for L1j, L2i–j, and Gp–(n+1), recurrent formulas can be used to estimate these derivatives without calculation of the coefficients. Thus, for L1 and L2 being the quadratic forms, the expression
117
dn ds
[ L ( s )] j = n
m
∑ unrj ,
r =0
holds true, where ⎡n⎤ m=⎢ ⎥ ⎣2⎦ , unj+1, r = ( j − n + r )
L' j L'' un,r + (n − 2r + 2) ' unj,r −1 , L L j u0,0 = Lj .
On the other hand, Gp(s) is a cubic polynomial whose zeros Zp have been known. Therefore, the formula
dm ds
[G p ( s )]− n −1 = m
(−1) m m ! n +1
[G p ( s )]
m m −i
∑ ∑ C in +iC nj + jC nm+−mi −−ji − j ( s − a)−i ( s − b)− j ( s − c)i + j − m
i =0 j =0
holds true, where a, b, and c are three zeros of the function Gp(s). Throop and Bearman [9] demonstrated that there exist two real and two imaginary Zp values; therefore, all calculations were performed in the complex plane. The above-described procedure is applicable, in particular, for η1 = 0, η2 = 0, or R11 = R22. In these cases, the results obtained can be compared with the available exact values of the distribution function for a one-component system. However, the functions gijPY(r) given by Eq. (5) calculated from the Percus–Yevick equation deviate from the exact rigid-sphere distribution functions gijMC(r) calculated by the Monte Carlo method of computer simulation. As well as for the one-component fluid, the following discrepancies between the functions gijPY(r) and their exact values gijMC(r) were observed: 1) Values of the functions gijPY(r) at contact points were considerably underestimated in comparison with the functions gijMC(r). 2) Oscillations of the functions gijPY(r) after the first minimum were not in phase with oscillations of the exact MC functions. For the one-component system, Verlet and Weis [11] suggested an approximate expression for the radial rigidsphere distribution function based on the solution of the Percus–Yevick equation that involved the correction term Δg(r) to adjust it to the results of MC simulation. In this work, we suggest the following procedures for the two-component rigid-sphere system. 1) The effective rigid-sphere diameters were calculated for gijPY(r) from the formula
Rijw = Rij (1 − ξ /16)1/ 3 .
gij
(6)
This condition allowed us to make oscillations of the function gijPY(r) in phase with those of the exact function (r). It should be noted that in this case, the rigid sphere diameters Rijw remain additive. 2) The deviation of gijPY(r) in the vicinity of the contact points was avoided by adding the function Δgij(r):
HS
gij (r ,{R},{ρ}) = H (r − Rij ) gijPY (r ,{R w },{ρ} + Δgij (r ) , ⎧⎪1, x ≥ 0, H ( x) = ⎨ ⎪⎩0, x < 0,
118
(7)
Δgij (r ) =
Aij r
exp[−μij (r − Rij )]cos μij (r − R ij ) .
(8)
The parameters Aij were calculated from exact values of the functions gij(r) at contact points: Aij Rij
(
)
= gij ( Rij ,{R},{ρ}) − gijPY Rij ,{R w },{ρ} ,
ij = 11, 12, 22 .
(9)
To calculate the function gij(Rij) in Eq. (9), we used the approximate scheme of calculating exact values of gij(Rij) at contact points according to the Carnahan‒Starling rule [13, 14]: 1 PY 2 gij ( Rij , ξ) + gijSPT ( Rij , ξ) , 3 3
gijCS ( Rij , ξ) =
where gijPY(Rij) are radial molecule distribution functions at contact points obtained from the Lebowitz solution for rigid sphere mixtures, and gijSPT(Rij) are the corresponding functions obtained in the context of Scaled Particle Theory (SPT) [15]. To calculate the second parameter μij in Eq. (8), the Fourier transform of the total correlation function hij(r) = gij(r) ‒ 1 must be taken. After substitution of the correlation function into Eq. (7), we can write
hijHS (k , ξ) = hijPY (k , ξ w ) + δhij0 (k ) + δhij' (k ) , where hijPY (k , ξw ) is the corresponding Fourier transform for the functions and
δhij0 (k ) = ∫ Δij (r ) exp(ikr )dr , δhij' (k ) = −
R
4π ij w PY ∫ r sin(kr ) gij (r ,{R },{ρ})dr . k Rw ij
For a one-component fluid, Verlet and Weis established that the isothermal compressibility of the rigid sphere system was close to that obtained for the Percus–Yevick high-density approximation. As demonstrated in [11], this implies that δhij0 (0) + δhij' (0) = 0 . This conclusion remains also valid for rigid sphere mixtures [12]. Therefore, expression
δhij0 (0) + δhij' (0) = 0 yields the condition for finding μij. Hence we obtain μij Rij ≈
24 Aij / Rij PY ξ w gij ( Rijw ,{R w },{ρ})
.
(10)
COMPARISON WITH THE RESULTS OF MC SIMULATION
To estimate the accuracy of the suggested procedure of calculating the solid-sphere molecule distribution functions for the two-component fluid, the results of calculation by the developed program were compared with the data
119
Fig. 1. Radial rigid-sphere distribution functions for the two-component mixture with ρ = 1.0825, ξ = 0.49, R22/R11 = 0.9, and x1 = x2 = 0.5. of Monte Carlo simulation obtained by Lee and Levesque [12] for three cases each comprising a critical value of one of the parameters: • High density and nearly identical rigid sphere diameters ρ = 1.0825 (ξ = 0.49, R22/R11 = 0.9, and x1 = x2 = 0.5), •
High density and different rigid sphere diameters ρ = 1.8225 (ξ = 0.49, R22/R11 = 0.3, and x1 = x2 = 0.5),
•
Different molar fractions of the components ρ = 1.1266 (ξ = 0.47, R22/R11 = 0.9, x1 = 0.25, and x2 = 0.75).
Results of comparison are shown in Figs. 1‒3, where the following designations have been used: 1) Points show results of MC simulation, 2) Solid curves show results of calculations by the method suggested in this work with the use of the correction term Δg(r), 3) Dash-dotted curves show results calculated from the Percus–Yevick equation, 4) Dashed curves in Fig. 2 show results of Lee and Levesque calculations [12]. From Figs. 1‒3 it can be seen that the radial distribution functions with allowance for the correction term show better agreement with the results of Monte Carlo simulations than the results of calculations from the Percus–Yevick equation. 1) At contact points, the corrected values of the radial molecule distribution functions (the solid curves) show better agreement with the results of MC simulation (points) then highly underestimated values of the radial molecule distribution functions obtained from the Percus–Yevick equation (the dash-dotted curves).
120
Fig. 2. Radial rigid-sphere distribution functions of the two-component mixture with ρ = 1.8225, ξ = 0.49, R22/R11 = 0.3, and x1 = x2 = 0.5.
Fig. 3. Radial rigid-sphere distribution functions of the two-component mixture with ρ = 1.1266, ξ = 0.47, R22/R11 = 0.9, x1 = 0.25, and x2 = 0.75. 2) Oscillation phases of the corrected radial molecule distribution functions (the solid curves) coincide with those from the results of MC calculations (points), whereas oscillation phases of the radial molecule distribution 121
functions calculated from the Percus–Yevick equation (the dash-dotted curves) do not coincide with those from the results of MC calculations (points). We note also that the radial molecule distribution functions show better agreement with the MC data than the results of Lee and Levesque calculations [12]. Thus, the corrected procedure for calculating the radial molecule distribution functions for the two-component fluid can be used to calculate the thermodynamic properties of fluid binary mixtures at high pressures and temperatures, for example, to calculate the parameters and composition of condensed system detonation products.
CONCLUSIONS
1. Analytical expressions have been derived for the radial rigid-sphere distribution functions gij(r) of the binary mixture based on the inverse Laplace transforms of the functions rgij(r) calculated form the Percus–Yevick equation. 2. The procedure for calculating the rigid-sphere distribution functions gij(r) of the binary mixture with any arbitrary distances r between the centers of the rigid spheres having different diameters with the correction term Δg(r) was developed. 3. The computational algorithm and the program for calculating the distribution functions gij(r) of the binary rigid-sphere mixture were developed. 4. The correctness of the analytical expressions for the radial molecule distribution function, computational algorithm, and program are confirmed by complete agreement of the calculated rigid-sphere distribution functions gij(r) for one- and two-component mixtures with the results of MC simulation. REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
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M. Ross, J. Chem. Phys, 71, No. 4, 1567‒1571 (1979). H. S. Kang, C. S. Lee, T. Ree, and F. H. Ree, J. Chem. Phys., 82, No. 1, 414‒423 (1985). W. Byers-Brown and T. V. Horton, Mol. Phys., 63, No. 1, 125‒138 (1988). G. Zerah and J.-P. Hansen, J. Chem. Phys., 84, No. 4, 2336‒2343 (1986). J. L. Lebowitz, Phys. Rev., A133, No. 4, 895 (1964). W. R. Smith and D. Henderson, Mol. Phys., 19, 411 (1970). M. S. Wertheim, Phys. Rev. Lett., 10, 321 (1963). E. Thiele, J. Сhem. Phys., 39, 474 (1963). G. J. Throop and R. J. Bearman, J. Сhem. Phys., 42, 2838 (1965). P. J. Leonard, D. Henderson, and J. A. Barker, Mol. Phys., 21, 107 (1971). L. Verlet and J.-J. Weis, Phys. Rev., A5, No. 2, 939 (1972). L. L. Lee and D. Levesque, Mol. Phys., 26, No. 6, 1351 (1973). G. A. Mansoori, N. F. Carnahan, K. E. Starling, and T. W. Leland, Jr., J. Chem. Phys., 54, 1523 (1971). N. F. Carnahan and K. E. Starling, J. Chem. Phys., 51, 635 (1969). J. L. Lebowitz, E. Helfand, and E. Praestgaard, J. Chem. Phys., 43, 774 (1965).
