Generalized Integral Equations of Classical Fluids
standing for many systems, we shall be better able t o predict parameters, and so AG', for other mixtures. 5. Conclusion I have shown t h a t certain theoretical equations, based on a reasonable model for binary solutions, are in quantitative agreement with very accurate experimental excess enthalpy data. I have shown t h a t these equations, with others for the excess entropy, also lead to good agreement with experimental Gibbs energies of mixing. The parameters in the equations are all reasonably related to molecular and intermolecular properties. Activities, vapor pressures, solubilities, and other measurable thermodynamic properties of solutions are of course readily deducible by rigous equations from the Gibbs energies of mixing.
Acknowledgment. I gratefully acknowledge my indebtedness to the scientists whose fine experimental data I have used and t o those with whom I have discussed some of the theoretical aspects of my theory. I also acknowledge the fact that some of the theoretical concepts have previously been used by others. Some pertinent references to this have been given in earlier papers in this series. Finally, I express my gratitude to the Paint Research Institute for some financial assistance. References and Notes (1) Presented at h169th National Meeting of the American Chemical Society, Philadelphia, Pa., April 1975. See Am. Chem. Soc.. Div. Org. Coat. Plast. Pap., 35, 283 (1975). (2) M. L. Huggins, J. Phys. Chem., 74, 371 (1970).
1321 (3) M. L. Huggins, Polymer, 12, 389 (1971). (4) M. L. Huggins, J. Phys. Chem., 75, 1255 (1971). (5) M. L. Huggins, J. Paint Techno/., 44, 55 (1972). (6) M. L. Huggins in "International Review of Science, Physical Chemistry. Series Two, Volume 8, Macromolecular Science", C. E. H.'Bawn. Ed., Butterworths, London, 1975. (7) Fourth InternationalConference on Chemical Thermodynamics,Montpellier. France, Aug 27. 1975. (8)R. H. Stokes, K. N. Marsh, and R. P. Tomiins, J. Chem. Thermo@m., 1, 21 1 (1969). (9) M.B. Ewing, K. N. Marsh. R. H. Stokes, and C. W. Tuxford, J. Chem. Thermodyn., 2,-751 (1970). (10) J. M. Sturtevant and P. A. Lyons, J. Chem. Thermodyn., I , 201 (1969). (11) K. N. Marsh and R. P. Tomlins, Trans. Faraday SOC.,66, 783 (1970). (12) M. B. Ewing and K. N. Marsh, J. Chem. Thermodyn., 2, 351 (1970). (13) K. N. Marsh and R . H. Stokes, J. Chem. Thermodyn., 1, 223 (1969). (14) A. E. P. Watson, I. A. McLure, J. E. Bennett, and G. C. Benson, J. Phys. Chem.. 69, 2751 (1965). (15) D. E. G. Jones, I. A. Weeks, and G. C. Benson. Can. J. Chem., 49, 2481 (1971). (16) J. Polak, S. Murakami, V. T. Lam, and G. C. Benson, J. Chem. Eng. Data, 15, 323 (1970). (17) S.Murakami, V. T. Lam, and G. C. Benson, J. Chem. Thermodyn., 1,397 (1969). (18) M. L. Huggins. J. Chem. Phys.. 9, 440 (1941). (19) M. L. Huggins, Colloid Symposium Preprint (1941); J. Phys. Chem., 46, 151 (1942). (20) M. L. Huggins, Ann. N.Y. Acad. Sci., 41, 1 (1942). (21) M. L. Huggins, J. Am. Chem. SOC..64, 1712 (1942). (22) M. L. Huggins, "Physical Chemistry of High Polymers", Wiley, New York, N.Y., 1958. (23) P. J. Flory. J. Chem. Phys., 10, 51 (1942). (24) G. Scatchard, S. E. Wood, and J. M. Mochel, J. Am. Chem. SOC.,62, 712 (1940). (25) G. Scatchadd and L. B. Ticknor. J. Am. Chem. SOC., 74,3724 (1952). (26) G. Scatchard, S.E. Wood. and J. M. Mochel, J. ptys. Chem., 43, 119 (1939). (27) S. E. Wood and A. E. Austin, J. Am. Chem. SOC.,67, 480 (1945). (28) K. N. Marsh, Trans. faraday SOC.,64, 883 (1968). (29) G. Scatchard, S.E. Wood, and J. M. Mochel. J. Am. Chem. Soc., 61, 3206 (1939). , (30) T. Boublik, V. T. Lam, S.Murakami. and G. C. Benson, J. Phys. Chem., 73, 2556 (1969).
Generalized Integral Equations of Classical Fluids H. S. Chung Central Research Division Laboratory, Mobil Research and Development Corporation, Princeton, New Jersey 08540 (Received October 6. 1975) Publication costs assisted by Mobil Research and Development Corporation
Parametrized integral equations of classical fluids have been derived within the framework of functional differentiation. T h e first-order theory is examined in detail for rigid spherical and Gaussian molecules. In the region of lower densities, it appears that the lowest order theory is numerically quite similar, in the case of rigid spheres, to the schemes of Rowlinson and of Hurst which are based on different (diagrammatic) arguments. For the Gaussian molecules, the virial coefficients up to the fourth are given correctly by this method. Extension of this parametrization t o second and higher order theories is indicated.
I. Introduction Functional analysis offers a powerful technique for systematically improving the integral equations of classical fluids. Thus, by retaining the quadratic term in the functional Taylor expansion, Verlet' has indicated how the original PercusYevick and hypernetted chain (hereafter referred to as PY-1 and HNC-1) theories may be generalized t o what are subsequently known as the PY-2 and HNC-2 equations. One of
these second generation theories, the PY-2, has been extensively studied and the most important conclusions drawn from this work2 is that ". . . when the PY-1 equation is a decent first approximation, that is for high temperatures or around the critical point, the PY-2 equation significantly improves over those results, and is a useful equation. On the other hand, for dense fluids at low temperatures where the PY-1 and HNC-1 equations are poor, the PY-2 equation is also bad . . ." In view of these assertions, it seems that further progress in the theory The Journal of Physical Chemistry, Vol. 80, No. 12, 1976
1322
H, S, Chung
of fluids may be more readily attained, not in the retention of the numerically difficult, still higher order terms of the functional expansion, but in the formulation of better first approximations upon which these more accurate equations are based. There have been several investigations of this nature, the more significant being those of Rowlinson,3 Carley and Lado,“ Rushbrooke and Hutchinson,6 and Hurstn6The common feature among these studies is the introduction into the appropriate integral equation a parameter whose value may be determined within the framework of the theory. T h e justifications for the parametrization are often couched in “diagrammatic” terms, e.g., the heuristic summation of a fraction of graphs of a certain class. Perhaps the most cogent argument in favor of these procedures is the fact that thermodynamic functions and virial coefficients thus derived are often superior to those of the original unparametrized theory. Despite some improvements, the modified theories are still not sufficiently accurate a t all temperatures and densities so extensions of these techniques are evidently desirable. However, in terms of the diagrammatic formulations, it is not immediately apparent how this kind of parametrization can be further developed in a systematic fashion. Functional analysis provides such a method. It is the purpose of this study to examine the parametrization approach within the context of functional differentiation.
11. T h e P a r a m e t r i z e d I n t e g r a l Equations In the grand canonical ensemble, the n-particle generic distribution function p ( ” ) ( l , .-, n ) is related to that of the ( n + 1)particles by p(n+’)(l, n l)/p(,)(l, n ) = p(’)(n 11V“) ..a,
+
..e,
+
expl-PIUt + V ( n l t ) ] ) (2) t. Here, we have followed essentially the notations and formalism of Rice and Gray.7 Briefly, P = l l k T where k is the Boltzmann constant and T the absolute temperature. The quantity (1,-., n ) denotes (Ri, -, R,) where (R;) = (Xi,Yi, Zi),the positional coordinates of the ith particle: d ( N ) = dR1 -ad R N where N is the total number of molecules in the system. p(l)(n 1 1U,)is the singlet distribution function when the system is subjected to an external potential field arising from n fixed molecules. U ( n l t ) represents the interaction of n particles in the field o f t particles. Finally E and z are, respectively, the grand partition function and activity. Let us consider the function F ( 2 ( U ( 1 ) ) 5 & / J ( ” ( 4 ( 1 ( 1 ) ) e x p b u ( l , ~(where ) u is the pair potential) as a functional of G(3(U(1)) 2 p(’)(31U ( 1 ) )in which ‘‘0”is a parameter whose value will be determined by methods described later. By performing a functional Taylor expansion, one finds F(21 U(1)) = F(2) 21 7 J-Jd(t)
Z=
t>O
+
Through straightforward manipulation, it can be readily shown that
T h e function c(2,3) is the direct correlation function which is related to the pair distribution function g(2,3) by the Ornstein-Zernike equation g(1,3)
- 1 = ~ ( 1 , 3+) p J d(2) ~ ( 2 , 3 (g(1,2) ) - 1)
(5)
Neglecting quadratic and higher order terms in the Taylor expansion and simply combining eq 3-5, one arrives a t the following parametrized integral equation:
Q(1,2) = PS d(3) (g(2,3)
- l)[(ln
CY)
(g(1,3)
- 1) - Q(1,3)) (6a)
and
Q(i,j) =
g(l,i)expl4u(i,i)l
-1
(6b)
It is interesting to note that if the function Q(i,j)is expanded in a series and only the leading term is retained, i.e.
Q ( i , j )= (In a ) In I(g(i,j) exp[@u(i,j)ll
(7)
one finds the replacement of Q ( i j ) by that given in eq 7 leads to the HNC-1 equation which is of course independent of the parameter a.On the other hand, if a is set equal to e, eq 6 is reduced to the PY-1 equation. Thus, this parametrized integral equation encompasses both the HNC-1 and PY-1 formalisms and it is in this sense that it resembles the method of Rowlinson which has a diagrammatic basis. I t should be emphasized that more refined versions of the theory, which bear the same relation to the PY-2 and HNC-2 equations as the present one does to the PY-1 and HNC-1, may be constructed from this scheme by keeping additional terms of the functional Taylor expansion. Therefore, this method provides a means of systematically correcting for the current approximate integral equations of fluids. There are two ways of demonstrating the improvements resulting from this parametrization vis-a-vis the PY-1 and HNC-1 equations. T h e first is the complete solution of eq 6 for all temperatures and densities. This involves a means of uniquely specifying the parameter cy, e.g., through the minimization of the Helmholtz free energy. This will form the subject of a separate communication. The second is the study of the low density properties of the integral equation via the virial coefficients. In the following sections, we shall consider this approach for rigid spherical and Gaussian molecules.
111. The Virial Coefficients
A. Rigid Spherical Molecules. We seek the solution of eq 6 for the radial distribution function g( 1,2) in the form of a density expansion g(1,2) = (exp[-8u(l,2)])(1
+ k2= l gi(1,2)pk)
(8)
The coefficients appearing in the expansion of the compressibility factor (the symbols have their customary significance)
PPIp = 1 + Bp
+ Cp2 + Dp3 + E p 4 + . . .
(9)
may be expressed in terms of the functions gk(1,2). We find that g1(1,2) and consequently the coefficients up to the third (C) are given correctly and that g2(1,2) and g3(1,2) may be written as: The Journal of Physical Chemistry, V d . 80, No. 12, 1976
Generalized Integral Equations of Classical Fluids
1323
TABLE I: Comparison of Virial Coefficients of Rigid Spherical Molecules Derived from Various Integral Equations of Fluids
a
Method
K
DP*
Dc+
Dp* = Dc* Dp* = Dexact* Dc* = Dexact* PY-1 HNC-1 Exact"
-0.165 637 -0.189 184 -0.113 188
0.2824 0.2869 0.2721
0.2824 0.2803 0.2869
0.0844 0.0845 0.0843
0.1119 0.1105 0.1149
0.2500 0.4453 0.2869
0.2969 0.2092 0.2869
0.0859 0.1447 0.1103
0.1211 0.0493 0.1103
E,'
Ec+
References 8.
TABLE 11: Comparison of Virial Coefficients of Gaussian Molecules Derived from Various Integral Equations of Fluids" Method K DP* Dc* E,* E,+
-0.646 4464 -0.646 4446 -0.646 4474
D,* = Dc* Dp* = Dexact* Dc* = Dexact* PY-1 HNC-1 Exact 0
-0.1255 -0.1255 -0.1255
-0.1255 -0.1255 -0.1255
-0.0050 -0.0050 -0.0050
0.0439 0.0439 0.0439
-0.1540 -0.1098 -0.1255
-0.0732 -0.1540 -0.1255
0.0683 -0.0408 0.0133
0.0139 0.0554 0.0133
Please note that the values of E,* for the PY-1 and HNC-1 theories were incorrectly given in ref 11.
- (K/2) gi2(1,2) g3(1,2) = g3"-'(1,2) - K I(1,2) - K g1(1,2) gzPy-'(1,2) + (K/6)(2K + 1) g13(l,2); K = (In a) - 1 g2(1,2) = gzPy-'(1,2)
fi3(1
+ f23) gi2(1,2)
(11)
(12)
For rigid spherical molecules f(r;j) = -1 if rij < u = 0 if rij > u
(13)
T h e superscript PY-1 in gk(1,2) denotes quantities t o be evaluated according to the PY-1 theory. Next, we designate the virial coefficients obtained from the pressure equation
PPIp = 1 - (2p7$/3) by C, D,, E,, tion
Jm
dr r 3 g ( r ) (du(r)/dr)
(14)
. . . , and those from the compressibility equa-
@-'ap/aP = 1
+ 4prrJmdr
r 2 (g(r) - 1)
(15)
by C,, D,, E,, etc. The formulas for D , and D , deduced from the above equations for a system of rigid spherical molecules may finally be expressed as D,+ = Dp'py-' - (25K/128)
(16a)
and
D,* = DC*"-'
+ (2357K/26 880);
Dk* = Dk/B3 (16b)
Similarly, the equations for the fifth virial coefficients assume the following form:
E,* = Ep*'Y-I - K I * ( l ) - (5K/8)Dp*PY-1 + (125K ( K 1))/3072 (17) and
+
(18)
where
T h e function 1(1,2) is defined in terms of fi; (= exp -pu(i,j) - 1): I(1,2) = Jd(3)
+ K(56 269 - 20 954K)/1075 200
E,* =
(10)
Ek* =
I * ( l ) = I(1)/B3
and I(R) = (r3/R)[(87/4480) - (2357R/22 680) (251R2/5760)]; 1 I R I 2 (19)
+
I(R) = (r3/R)[(3159/4480) - (2673R/840) + (2187R2/640) - (9R3/8) - (15R4/64) (9R5/40) - (31R6/960) - (17R7/2520) (5Rs/2688) - (R"/51 84011; 2 I R I 3
+
+
I(R) = 0; R
>3
From these equations, it is evident that once the parameter
K (or a ) is specified, the coefficients can be immediately evaluated. There are three ways of determining K within the present framework: (a) by requiring the coefficients derived from eq 14 and 15 be identical, i.e., Dp = D,. This is a selfconsistency argument; (b) by setting D , = D,,,,t(known), and (c) by letting D , = Dexact(known).Table I summarizes the results of considering these three cases. From this table, one notes that the virial coefficients derived from this first-order parametrized integral equation are in significantly better accord with the known values than either the PY-1 or the HNC-1 theory. Also, the self-consistent ( D , = D , case) results are quite similar to those of the formally different theories of Rowlinson and Hurst, i.e., the values of the fourth virial coefficient are identical but those of the fifth vary somewhat. I t appears that, in the region of lower fluid densities, our procedure is numerically analogous to these diagrammatic schemes. B. Gaussian Molecules. Whereas the rigid spherical molecules we have just considered interact with a "hard" repulsive force which is infinite a t a distance corresponding to the diameter of the molecules, we now wish to examine the case of a "soft" repulsive force, one in which the Mayer f function is given by The Journal of Physical Chemistry. Vol. BO. No. 12, 1976
H. s. Chung
1324
E,* = Ep*PY-l + K[2(3)-5'2 Here, r is the intermolecular separation and a a size parameter whose value is to be selected so that the resulting second virial coefficient is the same as that of the rigid spheres of diameter n. This system has been analyzed in various context by Uhlenbeck and Ford,g Helfand and Kornegay,lo and Chung and Espenscheid." In the evaluation of the fourth and fifth virial coefficients, cluster integrals over multidimensional Gaussians are involved; these integrals may be computed exactly for this model. For example, a p point diagram with 12 f bonds has the general form (21) J(1,2) = (-l)kJ . . . J d ( 3 ) . . . d ( p ) exp[- ZMijri-rj] , . LJ
where M is a p X p cluster matrix whose elements are -1 if points i and j are connected by a f bond = 0 otherwise (22)
Mij =
and Mii is equal to the number o f f bonds emanating from point i. Helfand and Kornegay showed that J(,.)
= (-1) k ( a3R3/2)p-2(M11;22)-3/2
+
and
The Journal of Physical Chemistry, Vol. 80. No. 12, 1976
(254
E,* = E,*Py-' + K[(3)-5/2 - (53/3)(5)-5/2 + (48/5)(11)-3/2 - (32/5)(21)-3/2] + [(2/3)(5)-5/2 - (2/5)(3)-5/2]K2 (25b) The values of these virial coefficients are presented in Table 11. Again, one finds that the parametrized integral equation leads to large improvements over the PY-1 and HNC-1 equations. In fact, for this molecular model, the fourth virial coefficients are in complete agreement with the exact, known results. The fifth virial coefficients derived from the three methods of specifying K are numerically the same and they are comparable to those of the PY-1 and HNC-1 theories. Further comparisons a t higher densities must await the solution of the integral equations for various molecular models. A t present, suffice it to say that, with the method of functional differentiation, successively better parametrized theories can be developed. I t is anticipated, for example, that the second-order theory (inclusion of the quadratic term in the Taylor expansion) of the parametrized integral equation will provide more satisfactory results a t all temperatures and densities than the PY-2 and HNC-2 equations. References and Notes
where M" is the 1,l minor of M and is the 1,1;2,2 second minor. Using this result, we found that for the Gaussian model D,* = Dp*l'Y-l - (2-9/2)K (244
D,* = D,*PY-~ ( K / ~ ) (-I 2-319
- (8/3)(5)-5/2
+ 48(11)-5/2 - 160(21)-5/2] + (2/3)(5)-5/2K 2
(24b)
L. Verlet. Physica. 30, 95 (1964). L. Verlet and D. Levesque, Physica, 36, 254 (1967). J. S.Rowlinson, Mol. Phys., 9, 217 (1965). D. D. Carley and F. Lado. Phys. Rev., 137, A42 (1965). G. S.Rushbrooke and P. Hutchinson. Physica. 29, 675 (1963). C. Hurst, Proc. Phys. Soc., 88, 193 (1965). S.A. Rice and P. Gray, "The Statistical Mechanics of Simple Liquids", Interscience, New Ywk. N.Y., 1965. F. H. Ree, R. N. Keeler, and S.L. McCarthy. J. Chem. Phys., 44, 3407 (1966). G. E. Uhlenbeck and G. W. Ford, "Studies in Statistical Mechanics", Vol. I, J. deBoer and G. E. Uhlenbeck,Ed., North Holland Publishing Co., Amsterdam, 1962. E. Helfand and R. L. Kornegay, Physica, 30, 1481 (1964). H. S. Chung and W. F. Espenscheid, Mol. Phys., 14(4), 317 (1968).
