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Journal of Molecular Structure (Theochem) 493 (1999) 241–247 www.elsevier.nl/locate/theochem The solvation shell in ionic solutions: variational mean spherical scaling approximation L. Blum a,b,*, E.S. Velázquez b,c a Department of Physics, University of Puerto Rico, P.O. Box 23343, Ro Piedras, PR 00931, USA Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106-4030, USA c Department of Physics, University of Puerto Rico, Mayaguez Campus, P.O. Box 9016, Mayaguez, PR 00681-9016, USA b Received 3 December 1998; received in revised form 16 February 1999; accepted 2 July 1999 Abstract The non-specific solvent effects in neutral dipolar (or better yet, effective dipolar) solvents were discussed in the literature with success. However the situation is not so clear when the solvent is an electrolyte, where the reaction field approach is no longer valid. In the present communication we present a theory based on the Mean Spherical Scaling Approximation in which the system is encapsulated in a ionic cloud that is represented by an equivalent capacitor of shape determined by precise prescriptions derived from exact relations and sum rules. The construction of the equivalent capacitor depends on the geometry of the system, on the nature of the solvent and on the nature and concentration of the ions. The treatment proposed here satisfies exact relations like the Stillinger–Lovett sum rules, the perfect screening sum rules, and the high and low coupling conditions and constitutes an interpolation scheme between exact high density and low density limits in it simplest form. The calculation is illustrated with an example of an ellipsoidal cavity. q 1999 Elsevier Science B.V. All rights reserved. Keywords: Ionic solvation; Solvation shells; Variational mean spherical approximation; Scaling theory 1. Introduction The computational description of molecular systems and reactions in solution was given a fair amount of attention [1–9] and a number of very interesting approaches were developed. In most of them the system is encapsulated in a region, which may or may not include some of the solvent (water) molecules, and the exterior is treated as a continuum dielectric. Different kinds of boundary conditions were used ranging from metallic to pure dielectric. In our present contribution we discuss only systems in electrolytic solutions. We would like to show that * Corresponding author. Tel.: 1 1-809-764-0000. E-mail address: l_blum@upr1.upr.clu.edu (L. Blum) the Mean Spherical Approximation (MSA) or more properly the Mean Spherical Scaling Approximation (MSSA) [10] and its recent improvements provide a procedure that is coherent and simple. Just as the Debye–Huckel (DH) theory [11,12], the MSSA is a linearized Poisson–Boltzmann approximation for electrolytes, but unlike the DH, the MSSA takes into account the excluded volume of all the ions, those in the center and those in the ionic cloud. The MSSA also admits analytical solutions for the primitive model (continuum dielectric solvent) [13], the iondipole model [14,15], the ion-YUKAGUA model [16] and more recently for inhomogeneous confined systems. All of these solutions are expressed in terms of a screening parameter (or 0166-1280/99/$ - see front matter q 1999 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(99)00245-6 242 L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247 more generally a matrix) that is related to the DH screening parameter k . The MSSA satisfies a number of exact conditions: 1. Stillinger–Lovett sum rule [17,18] This rule is obeyed by all approximations that are symmetric and have ions with excluded volume. It is violated by the DH theory and all Poisson–Boltzmann theories that take the ions in the ionic atmosphere as points. 2. Perfect screening sum rule [19] Ionic solutions are mixtures of charged particles, the ions and the neutral solvent particles, most commonly water, which has an asymmetric charge distribution, a large electric dipole, and higher electric moments. Because of the special nature of these forces the charge distribution around a given ion and the thermodynamics do satisfy a series of conditions or sum rules. One remarkable property of mixtures of classical charged particles is that because of the very long range of the electrostatic forces, they must create a neutralizing atmosphere of counterions, which shields perfectly any charge or fixed charge distribution. Otherwise the partition function and therefore all the thermodynamic functions, will be divergent [19]. The size of the region where this charge shielding occurs depends not only on the electrostatics, but also on all the other interactions of the system. This also means that for any approximation the internal energy of the ionic system can be expressed as the sum of grounded metal capacitors surrounding each charge density. If this is a spherical ion then the capacitor is a spherical capacitor, and then the exact form of the energy is e2 X zpi ri zi ; DE 2 1 i 1=Gi 1 si zpi 1 is the effective charge, we will use where b 1=kT, the usual Boltzmann thermal factor, 1 is the dielectric constant, e is the elementary charge, and ions i have charge, diameter and density ezi , si , ri , respectively. For the continuum dielectric primitive model Gi for alli. Here g ù rs2 ez2 =1kT is the plasma coupling parameter. 3. High r -High g : Onsager limits [20] The Onsagerian limit is obtained by setting Gi ! ∞ in our previous expression. When the ionic concentration goes to infinity and at the same time the charge diverges, then the limiting energy for the spherical ion is bounded by DE 2 e2 X zp ri zi i : 1 i si 2 4. Low r -Low g : DH limits This exact limit requires that the screening parameter satisfy 2G ! k with k2 m 4pbe2 X r z2 : 1 j1 j j 3 It also means that the excess entropy must be of the form DS MSA 2k G3 : 3p 4 All the known analytical MSSA solutions for the dimers and the polymers satisfy this relation at infinite dilution and at any given concentration. We conjecture that there is an ‘universality’ principle for the excess entropy for any arbitrarily shaped system, and this is the basis of our approximation. 5. Low r -High g : Binding limits [21] The Binding Mean Spherical Approximation (BIMSA) obtained by solving the Wertheim form of the Ornstein– Zernike (OZ) equation satisfies exactly the correct DH limits for both the associated and dissociated dimers or n-mers of any composition. This means that if component 1 forms a n-mer the DH limiting law must satisfy k2 m 4pbe2 X rj z2j 1 r1 nz1 2 : 1 j2 5 This limiting law is not satisfied by any closure of the regular OZ equation, but only for closures of the Werteim–Ornstein–Zernike (WOZ) equation [22–26]. Treating the solvent effects in quantum mechanics is often a challenge because of the difficulties in defining precisely the limits of the solvent ‘cell’ (or capsule) in which the reaction takes place. The cavity size and shape is not defined in most cases, and the continuum region is also difficult to assert [27–30]. When the solvent is an ionic solution, however, a very L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247 clear set of prescriptions can be formulated. This is so because of the very special nature of the Coulomb forces in classical mechanics, which forces perfect screening of the unbalanced charges in the system [19]. In our treatment we use the MSA and the BIMSA in which the cavity size is defined by a variational expression, and the cavity shape is defined from geometrical excluded volume considerations. All of these are ionic strength dependent parameters, and satisfy the required asymptotic limiting expressions [13,31,32]. For systems with Coulomb and screened Coulomb interactions in a variety of mean spherical approximations it is known that the solution of the OZ equation is given in terms of a single screening parameter G . This includes the ‘primitive’ model of electrolytes, in which the solvent is a continuum dielectric, but also models in which the solvent is a dipolar hard sphere, and much more recently the YUKAGUA model of water that has the correct tetrahedral structure. The MSA can be deduced from a variational principle in which the energy is obtained from simple electrostatic considerations and the entropy is a universal function. For the primitive model it is DS 2k G3 ; 3p where G is MSA the screening parameter and in general it will be of the form The internal energy for a given object is written as DE G. Then G is determined by 2bDE G 1 G3 = 3p 0: 2G 2A 0: 2G The analytical solution of the MSSA can also be derived from a variational principle in which the screening parameter is obtained from the minimization of a trial function [13]. The general principle was formulated by Baxter [33] for the Percus-Yevick approximation, by Chandler [34] for the MSA, and by one of us for the soft MSA (SMSA) [35,36]. We extend the variational approach for the screening parameters [13,37] to arbitrarily shaped objects. 6 In Section 2 we review previous results of the MSSA obtained for dimers from the WOZ equation. In Section 3 we discuss a variational form, the Variational Mean Spherical Scaling Approximation (VMSSA) that will satisfy all exact sum rules described above. 2. The BIMSA for dimers In this section we review our previous results for dimers [38,39]. Consider the case of dimers formed in a solution containing m species of ions of diameters s i and charges zi. The dimers are formed by species 1 and 2, and the rest of the ions do not associate. Now we remark that the excess internal energy is a function of the set of the aNi xNi : DE 3 e2 X zk X rk aN xN 2 zk 1 k sk N0 k k or DE 2 DS S G; which is independent of the form of the cavity in this approximation, and G is now the scaling matrix. We propose in this article that for a general form of the cavity, the scaling matrix G is obtained from the variational principle 243 1 e2 X GT zk 1 hT sk rk zk 1 k 1 1 G T sk 3 e2 X zk X rk aN xN ; 1 k sk N0 k k 7 with hT 2 X p X r k sk aNk xNk : 2D k N0 8 Now we will consider the case of the charged chain constituted by the particles 1; 2; …; n with z1 ^z2 … zn and s1 s2 … sn . In this case, from Eq. (7) we deduce [21] " ! e2 z1 G 12a 1 rz 2 DE 2 1 1 1 1 1 Gs 2s 1 1 Gs2 # X 2 G ri zi : 9 1 1 1 Gs i 244 L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247 The closure relation obtained directly from the boundary condition of the WOZ is " ! G2 be 2 z1 z2 1 2 a r1 z1 1 p 1 1 1 Gs2 1 1 Gs3 # X 2 1 ; 10 ri zi 1 1 1 Gs2 i where the degree of dissociation is 0 , a , 1 and it is obtained from the dimerization equilibrium relation Ka r1 1 2 a2 : a 11 It can be verified that Eq. (10) does satisfy the variational relation (6). It can also be verified that conditions 3 and 4 are satisfied for all values of a . To our knowledge, no other theory does this. However condition 2 is not satisfied correctly in the limit of perfect association, a 0, because one always gets the result of independent particles. an incomplete elliptic integral of the first kind [42] C 2l ; F fum 15 where F fum l 1 2 Zf 0 q c20 2 a20 ; 16 17 a0 ; c0 18 c20 2 b20 : c20 2 a20 19 f cos21 m 1 2 m sin 2 u21=2 du; For our model capacitor C G 2l G ; F f Gum G 20 where l G, f G, and m G are obtained by letting 3. The variational mean spherical scaling approximation for ellipsoids a0 ! a G a0 1 s0 1 1 2 2G 21 We consider an ellipsoidal shell of semiaxes a0 , b0 , c0 centered at the origin of a Cartesian coordinate system. The distance of closest approach along the x; y; z axes of a sphere of radius s 0 are a0 1 s 0 =2, b0 1 s 0 =2, c0 1 s 0 =2, respectively. Based on the spherical capacitor model for hard spheres, we take the excess internal energy to be b0 ! b G b0 1 s0 1 1 2 2G 22 c0 ! c G c0 1 s0 1 1 2 2G 23 DE 2 e2 X 2 1 ; rz 21 i i i C G 12 where C G is the capacitance of the model capacitor. The capacitance of an ellipsoid of semiaxes a0 , b0 , c0 is [40,41] C Z∞ 0 2 R21 j dj ; where q Rj j 2 a20 j 2 b20 j 2 c20 : 13 in Eqs. (16)–(19). Note that the excess internal energy has the expected form in the limit G ! ∞, since lim C G C∞ ; G!∞ 24 where C∞ is the capacitance of a conductor shaped as the boundary, i.e. an ellipsoid of semiaxes a0 s 0 =2, b0 1 s 0 =2, c0 1 s 0 =2 and therefore lim DE 2 G!∞ e2 X 2 1 rz : 21 i i i C ∞ 25 The excess entropy of the system is DS 2k 14 This result can be conveniently rewritten in terms of G3 ; 3p 26 and thus we find the excess Helmholtz free energy DA DE 2 TDS: 27 L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247 Fig. 1. Concentration vs. screening parameter for oblate (O) and prolate (P) ellipsoids. The ratio of the largest semiaxis to the smallest is 5 in both cases. The closure relation obeyed by the VMSSA can be written in the form dDA 0; 28 Fig. 3. Excess Helmholtz free energy versus screening parameter for oblate (O) and prolate (P) ellipsoids. The ratio of the largest semiaxis to the smallest is 5 in both cases. and from Eq. (20), we obtain 2 1 2 C G C G l G 2G l G 2G ! 1 2 F f Gum G : 2 F f Gum G 2G from which 2 G2 DE 2 : pb 2G 245 31 29 Furthermore, Using Eq. (12) we have 2 e X 2 1 2 E C G; rz 2G D 21 i i i C 2 G 2G 2 30 1 2 1 ; l G 2 l G 2G 2Gm G 32 where m G 1 1 G a0 1 c0 ; 33 and 2 1 2f F fum 1 2G n G 2G F fum E fum 1 2m 2m 1 2 m ! sin 2f 2m ; 2 4 1 2 mn G 2G 2 34 Fig. 2. Excess internal energy versus screening parameter for oblate (O) and prolate (P) ellipsoids. The ratio of the largest semiaxis to the smallest is 5 in both cases. where the dependence of f G and m G on G has not been explicitly written for compactness, E fum is the incomplete elliptic integral of the second kind, and q n G 1 2 m G sin2 f G: 35 246 L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247 It is straightforward to find 2 c 2 a0 1=2 f G 1=2 0 ; 2G G 1 1 2Gc0 m1=2 G 2 b 2 a0 c0 2 b0 m G 0 : 2G c0 2 a0 m2 G 36 37 Since we can write k ; DE 2 8pbC G 38 the closure relation (6) takes the form 2 C G 28G2 C 2 G: 2G 39 Fig. 1 shows the dependence of the concentration on the screening parameter, while Figs. 2 and 3 show the excess internal energy and the excess Helmholtz free energy respectively. In each of these figures two curves are shown corresponding to a prolate ellipsoid with a0 b0 1; c0 5 and to an oblate ellipsoid with a0 1=5, b0 c0 1. 4. Discussion We have already shown that the excess internal energy (12) satisfies the Onsager limit (condition 3). We will now show that the same expression provides the correct form in the limit of low coupling, i.e. we should obtain 2 e X 2 DE 2 rz ; lim G!0 2G 1 i i i 2 40 which implies that we have to show lim 2 G!0 1 2 C G 2: C 2 G 2G 41 We first obtain the following series expansions for G!0 p c 0 2 a0 l G 1 O G1=2 ; 42 2G1=2 p f G 2 c0 2 a0 G1=2 1 O G3=2 c 2 b0 1 O G: m G 0 c 0 2 a0 F uum0 u 1 m0 u3 1 O u5 ; 6 45 to obtain C G 2 k2 Since f G ! 0 and m G ! m0 , where m0 is a fixed value, we need the following series, valid for u!0 1 1 O 1: 2G 46 To obtain the limit of the derivative of C G we use Eq. (31). The first term in parenthesis is, using Eqs. (32) and (33) 1 2 1 1 O 1: l G 2 l G 2G 2G 47 The limit of the second term in parenthesis is obtained from Eq. (34). First using Eqs. (35), (36) and (43) we obtain n G 1 1 O G so that the first term of Eq. (34) is p c 0 2 a0 1 2 f G 1 O 1: n G 2G G1=2 48 49 For the reason stated above when finding the limit of F f Gum G, we need E u um0 u 2 m0 u3 1 O u5 : 6 50 Furthermore, since all three terms inside the parenthesis in Eq. (34) are of order f , and consequently of order G 1/2, and since from Eq. (37) we see that the limit of the derivative of m Gk is of order 1, we conclude that the first term of Eq. (34), which is of order G 21/2 is the leading term and therefore p c 0 2 a0 2 F f Gum G 1 O 1: 51 2G G1=2 Substitution in Eq. (31) leads to 2 1 C G 2 2 1 O 1; 2G 2G 52 which combined with Eq. (46) gives 43 lim 2 G!0 44 1 C2 G 2 C G 2; 2G as we expected. 53 L. Blum, E.S. Velázquez / Journal of Molecular Structure (Theochem) 493 (1999) 241–247 Acknowledgements This research was supported in part by the National Science Foundation under Grant No. PHY-94-07194. L.B. thanks the NSF for support through grants PHY94-07194, CHE-95-13558 and EPSCoR OSR-9452893. We also thank J. Given, J. 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