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An interpretation of the mean spherical approximation
Volume 5 3A, number 5
14 July 1975
PHYSICS LETTERS
AN INTERPRETATION OF THE MEAN SPHERICAL APPROXIMATION E. MARTINA * and F. Del RIO Universidad Autbnoma Metropolitanta, Apdo. 55-534, Mixico 13, D.F. Received 22 May 197.5 The mean spherical approximation is shown to be derivable from a perturbation expansion where the reference system follows the Percur-Yevick approximation. Then the MSA may be interpreted as a first-order expansion in which the perturbation is averaged with respect to the low-density reference distribution function. This represents a generalization of an equivalent interpretation previously made with respect to the free energy of a polar fluid.
The mean spherical approximation (MSA) [l] has been used in recent years to study a variety of systems including electrolytes and polar fluids [2]. It is the purpose of this letter to generalize an interpretation of the MSA that arises in the study of the Helmholtz free energ: of a polar liquid [3]. According to this view, the influence of the dipole-dipole interaction is taken into account as an average with respect to the zero density radial distribution function. We will show that this interpretation is also applicable to the distribution functions for a wide class of potentials. The MSA is defined for intermolecular potentials u(r) of the form u(r)==,
r
(14
u(r)=ul(r),
r>o;
(lb)
where u is the distance of closest approach between two particles. Then the MSA is defined by the relations
VI &)=O, c(r)=-&(r),
[4], since the former reduces to the latter in the case ul(r) = 0. We will consider a system with an intermolecular potential which can be written as the sum of two parts: a short range term uu(r) which vanishes beyond r = u, and a perturbation u1 (r). Hence u(r) = uo(r) + hul(r) ;
h being a coupling parameter of the type used in Zwanzig’s perturbation theory [5]. Now we may expand c(r) in a Taylor series in X around A = 0 to obtain c(r) = co(r) + c;(r)& + (l/2) ci(r)X2 + ... ,
r>a;
(24 (2b)
where g(r) is the radial distribution function and c(r) is the direct correlation function defined in terms of the total correlation function h(r) = g(r) - 1 by the well known Ornstein-Zernike (OZ) expression. Substituting the eqs. (2) in the OZ relation, one obtains an integral equation which has been solved for various cases [2]. The MSA integral equation is related to the hard-sphere solution of the Percus-Yevick equation
(4)
where the subscript zero indicates the h = 0 values and the primes denote differentiation with respect to X. We will calculate eq. (4) to first order in the PercusYevick approximation. The PY approximation may be written for c(r) showing the explicit dependence on the coupling parameter as c(r)=
r
(3)
]exp(-&r-&h)
-
lI(l+SA(r)),
(5)
where Sk(r) is the sum of the set of series diagrams defined in the cluster expansion of the correlation function h(r) [e.g. 61. Differentiating eq. (5) and substituting into eq. (4) one obtains c(r) = co(r) - Pul(r)
exp (-Puu)(l+Su) (6)
+ [exp(-/3uo)
- l]Sb .
In eq. (6), X has been put equal to 1 and all the terms of order higher than the first have been neglected. Since we are using the PY approximation for the unperturbed potential, eq. (6) may be written as
* Supported by CONACYT. 355
PHYSICS LETTERS
Volume 53A, number 5
c(r) =c&)
-pUl(r)g&)
+ [exp(-I%)
- 11$1 . (7)
If the reference system is chosen to be made of hard spheres of diameter u, then for r < u, go(r) = 0, and from eq. (7) we see that 60;
c(r) = co(r) -s;(r);
(8)
We can use this equation to find g(r) for r < u. Since g(r) = 1t c(r) t S(r), by the same expansion leading to eq. (4) we find, at X = 1. g(r)=ltc(r)tSo(r)tS~(r)+....
(9)
Substitution of c(r) from eq. (8) into eq. (9) leads to g(r) = go(r), r < a; and so g(r) = 0;
r
(10)
which is identical to the first condition of the MSA, eq. (2a). Moreover, the PY hard-sphere direct correlation ’ function co(r) is zero for r > a; from this fact and eq. (7) we may express c(r) as r>u.
c(r) = -pUl(r)g&);
(11)
Therefore, eqs. (10) and (11) are the result of two approximations: 1) To take the perturbation potential ul(r) to first order. And 2) To consider the hard-sphere reference system in the Percus-Yevick approximation. The fact that eq. (10) is identical to the first of the MSA conditions has already been pointed out. To obtain the second condition, eq. (2b), one simply takes the terms which are first order in ul(r) to be equal to those of a structureless fluid. The first-order term in u1 (r) appears in eq. (1 l), this means that the factor go(r) in this equation has to be put equal to 1, as for a very low density reference fluid. Then eq. (11) becomes c(r) = -@l(r),
356
r>u.
(12)
14 July 1975
Eq. (12) is iden t ical to the second condition of the MSA, eq. (2b). Conclusion. We have shown how the MSA in the form of eq. (2) can be obtained from the Percus-Yevick formalism to which it is related. The derivation consists of the following steps: Make a perturbation expansion to first order in the perturbative potential, take the reference system to be formed of hard spheres, treat the reference, zero&order terms within the Percus Yevick approximation, and consider all properties of the reference fluid that appear in the first order term in their low density limit. Therefore, the mean spherical approximation, for those cases that consider a hard core potential of the type of eq. (l), can always be regarded as a first order perturbation theory in which the perturbation f&es” the reference system in its structureless low density limit. The authors wish to thank Professor G.S. Rushbrooke for his valuable suggestions and criticism.
References
111J.L. Lebowitz and J.K. Percus, Phys. Rev. 144 (1966) 25 1. VI E. Waisman and J.L. Lebowitz, J. Chem. Phys. 52 (1970) 4307; M.S. Wertheim, J. Chem. Phys. 55 (1971) 4291; E. Waisman, Molec. Phys. 25 (1973) 45; S.A. Adelman and J.M. Deutch, J. Chem. Phys. 60 (1974) 3935. 131 G.S. Rushbrooke, G. Stell and J.S. Huye, Molec. Phys. 26 (1973) 1199. 141 J.K. Percus and G.J. Yevick, Phys. Rev. 110 (1958) 1. [51 R. Zwanzig, J. Chem. Phys. 22 (1954) 1420. [61 S.A. Rice and P. Gray, Statistical Mechanics of simple liquids (Interscience, New York, 1965); chapter 2.
14 July 1975
PHYSICS LETTERS
AN INTERPRETATION OF THE MEAN SPHERICAL APPROXIMATION E. MARTINA * and F. Del RIO Universidad Autbnoma Metropolitanta, Apdo. 55-534, Mixico 13, D.F. Received 22 May 197.5 The mean spherical approximation is shown to be derivable from a perturbation expansion where the reference system follows the Percur-Yevick approximation. Then the MSA may be interpreted as a first-order expansion in which the perturbation is averaged with respect to the low-density reference distribution function. This represents a generalization of an equivalent interpretation previously made with respect to the free energy of a polar fluid.
The mean spherical approximation (MSA) [l] has been used in recent years to study a variety of systems including electrolytes and polar fluids [2]. It is the purpose of this letter to generalize an interpretation of the MSA that arises in the study of the Helmholtz free energ: of a polar liquid [3]. According to this view, the influence of the dipole-dipole interaction is taken into account as an average with respect to the zero density radial distribution function. We will show that this interpretation is also applicable to the distribution functions for a wide class of potentials. The MSA is defined for intermolecular potentials u(r) of the form u(r)==,
r
(14
u(r)=ul(r),
r>o;
(lb)
where u is the distance of closest approach between two particles. Then the MSA is defined by the relations
VI &)=O, c(r)=-&(r),
[4], since the former reduces to the latter in the case ul(r) = 0. We will consider a system with an intermolecular potential which can be written as the sum of two parts: a short range term uu(r) which vanishes beyond r = u, and a perturbation u1 (r). Hence u(r) = uo(r) + hul(r) ;
h being a coupling parameter of the type used in Zwanzig’s perturbation theory [5]. Now we may expand c(r) in a Taylor series in X around A = 0 to obtain c(r) = co(r) + c;(r)& + (l/2) ci(r)X2 + ... ,
r>a;
(24 (2b)
where g(r) is the radial distribution function and c(r) is the direct correlation function defined in terms of the total correlation function h(r) = g(r) - 1 by the well known Ornstein-Zernike (OZ) expression. Substituting the eqs. (2) in the OZ relation, one obtains an integral equation which has been solved for various cases [2]. The MSA integral equation is related to the hard-sphere solution of the Percus-Yevick equation
(4)
where the subscript zero indicates the h = 0 values and the primes denote differentiation with respect to X. We will calculate eq. (4) to first order in the PercusYevick approximation. The PY approximation may be written for c(r) showing the explicit dependence on the coupling parameter as c(r)=
r
(3)
]exp(-&r-&h)
-
lI(l+SA(r)),
(5)
where Sk(r) is the sum of the set of series diagrams defined in the cluster expansion of the correlation function h(r) [e.g. 61. Differentiating eq. (5) and substituting into eq. (4) one obtains c(r) = co(r) - Pul(r)
exp (-Puu)(l+Su) (6)
+ [exp(-/3uo)
- l]Sb .
In eq. (6), X has been put equal to 1 and all the terms of order higher than the first have been neglected. Since we are using the PY approximation for the unperturbed potential, eq. (6) may be written as
* Supported by CONACYT. 355
PHYSICS LETTERS
Volume 53A, number 5
c(r) =c&)
-pUl(r)g&)
+ [exp(-I%)
- 11$1 . (7)
If the reference system is chosen to be made of hard spheres of diameter u, then for r < u, go(r) = 0, and from eq. (7) we see that 60;
c(r) = co(r) -s;(r);
(8)
We can use this equation to find g(r) for r < u. Since g(r) = 1t c(r) t S(r), by the same expansion leading to eq. (4) we find, at X = 1. g(r)=ltc(r)tSo(r)tS~(r)+....
(9)
Substitution of c(r) from eq. (8) into eq. (9) leads to g(r) = go(r), r < a; and so g(r) = 0;
r
(10)
which is identical to the first condition of the MSA, eq. (2a). Moreover, the PY hard-sphere direct correlation ’ function co(r) is zero for r > a; from this fact and eq. (7) we may express c(r) as r>u.
c(r) = -pUl(r)g&);
(11)
Therefore, eqs. (10) and (11) are the result of two approximations: 1) To take the perturbation potential ul(r) to first order. And 2) To consider the hard-sphere reference system in the Percus-Yevick approximation. The fact that eq. (10) is identical to the first of the MSA conditions has already been pointed out. To obtain the second condition, eq. (2b), one simply takes the terms which are first order in ul(r) to be equal to those of a structureless fluid. The first-order term in u1 (r) appears in eq. (1 l), this means that the factor go(r) in this equation has to be put equal to 1, as for a very low density reference fluid. Then eq. (11) becomes c(r) = -@l(r),
356
r>u.
(12)
14 July 1975
Eq. (12) is iden t ical to the second condition of the MSA, eq. (2b). Conclusion. We have shown how the MSA in the form of eq. (2) can be obtained from the Percus-Yevick formalism to which it is related. The derivation consists of the following steps: Make a perturbation expansion to first order in the perturbative potential, take the reference system to be formed of hard spheres, treat the reference, zero&order terms within the Percus Yevick approximation, and consider all properties of the reference fluid that appear in the first order term in their low density limit. Therefore, the mean spherical approximation, for those cases that consider a hard core potential of the type of eq. (l), can always be regarded as a first order perturbation theory in which the perturbation f&es” the reference system in its structureless low density limit. The authors wish to thank Professor G.S. Rushbrooke for his valuable suggestions and criticism.
References
111J.L. Lebowitz and J.K. Percus, Phys. Rev. 144 (1966) 25 1. VI E. Waisman and J.L. Lebowitz, J. Chem. Phys. 52 (1970) 4307; M.S. Wertheim, J. Chem. Phys. 55 (1971) 4291; E. Waisman, Molec. Phys. 25 (1973) 45; S.A. Adelman and J.M. Deutch, J. Chem. Phys. 60 (1974) 3935. 131 G.S. Rushbrooke, G. Stell and J.S. Huye, Molec. Phys. 26 (1973) 1199. 141 J.K. Percus and G.J. Yevick, Phys. Rev. 110 (1958) 1. [51 R. Zwanzig, J. Chem. Phys. 22 (1954) 1420. [61 S.A. Rice and P. Gray, Statistical Mechanics of simple liquids (Interscience, New York, 1965); chapter 2.