Analytical solution of the associative mean spherical approximation for the ion-dipole model

A simple electrolyte in a polar solvent is modelled by a mixture of polar hard spheres and equal diameter charged hard spheres with the possibility of ionic dimerization. The analytical solution of the associative mean spherical approximation (AMSA) for this model is derived to its full extent. E...

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Datum:	1998
Hauptverfasser:	Holovko, M.F., Kapko, V.
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Sprache:	English
Veröffentlicht:	Інститут фізики конденсованих систем НАН України 1998
Schriftenreihe:	Condensed Matter Physics
Online Zugang:	http://dspace.nbuv.gov.ua/handle/123456789/118929
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Zitieren:	Analytical solution of the associative mean spherical approximation for the ion-dipole model / M.F. Holovko, V. Kapko // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 239-255. — Бібліогр.: 44 назв. — англ.

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spelling	irk-123456789-1189292017-06-02T03:03:25Z Analytical solution of the associative mean spherical approximation for the ion-dipole model Holovko, M.F. Kapko, V. A simple electrolyte in a polar solvent is modelled by a mixture of polar hard spheres and equal diameter charged hard spheres with the possibility of ionic dimerization. The analytical solution of the associative mean spherical approximation (AMSA) for this model is derived to its full extent. Explicit expressions for pair correlation functions and dielectric constant in terms of the AMSA are established. Some numerical calculations illustrate the role of ionic association. Простий електроліт моделюється в полярному розчиннику сумішшю полярних твердих сфер і однакового розміру заряджених твердих сфер із можливою іонною димеризацією. Подано аналітичний розв’язок асоціативного середньо-сферичного наближення для цієї моделі (АССН). Приводяться точні вирази для парних кореляційних функцій і діелектричної константи в термінах АССН. Роль іонної асоціативності ілюструється числовими результатами. 1998 Article Analytical solution of the associative mean spherical approximation for the ion-dipole model / M.F. Holovko, V. Kapko // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 239-255. — Бібліогр.: 44 назв. — англ. 1607-324X DOI:10.5488/CMP.1.2.239 PACS: 05.20.-y http://dspace.nbuv.gov.ua/handle/123456789/118929 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	A simple electrolyte in a polar solvent is modelled by a mixture of polar hard spheres and equal diameter charged hard spheres with the possibility of ionic dimerization. The analytical solution of the associative mean spherical approximation (AMSA) for this model is derived to its full extent. Explicit expressions for pair correlation functions and dielectric constant in terms of the AMSA are established. Some numerical calculations illustrate the role of ionic association.
format	Article
author	Holovko, M.F. Kapko, V.
spellingShingle	Holovko, M.F. Kapko, V. Analytical solution of the associative mean spherical approximation for the ion-dipole model Condensed Matter Physics
author_facet	Holovko, M.F. Kapko, V.
author_sort	Holovko, M.F.
title	Analytical solution of the associative mean spherical approximation for the ion-dipole model
title_short	Analytical solution of the associative mean spherical approximation for the ion-dipole model
title_full	Analytical solution of the associative mean spherical approximation for the ion-dipole model
title_fullStr	Analytical solution of the associative mean spherical approximation for the ion-dipole model
title_full_unstemmed	Analytical solution of the associative mean spherical approximation for the ion-dipole model
title_sort	analytical solution of the associative mean spherical approximation for the ion-dipole model
publisher	Інститут фізики конденсованих систем НАН України
publishDate	1998
url	http://dspace.nbuv.gov.ua/handle/123456789/118929
citation_txt	Analytical solution of the associative mean spherical approximation for the ion-dipole model / M.F. Holovko, V. Kapko // Condensed Matter Physics. — 1998. — Т. 1, № 2(14). — С. 239-255. — Бібліогр.: 44 назв. — англ.
series	Condensed Matter Physics
work_keys_str_mv	AT holovkomf analyticalsolutionoftheassociativemeansphericalapproximationfortheiondipolemodel AT kapkov analyticalsolutionoftheassociativemeansphericalapproximationfortheiondipolemodel
first_indexed	2025-07-08T14:55:24Z
last_indexed	2025-07-08T14:55:24Z
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fulltext	Condensed Matter Physics, 1998, Vol. 1, No 2(14), p. 239–255 Analytical solution of the associative mean spherical approximation for the ion-dipole model M.F.Holovko, V.Kapko Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii St., UA–290011 Lviv–11, Ukraine Received May 19, 1998 A simple electrolyte in a polar solvent is modelled by a mixture of polar hard spheres and equal diameter charged hard spheres with the possibility of ionic dimerization. The analytical solution of the associative mean spherical approximation (AMSA) for this model is derived to its full extent. Explicit expressions for pair correlation functions and dielectric constant in terms of the AMSA are established. Some numerical calculations illustrate the role of ionic association. Key words: association, ion-dipole model, mean spherical approximation PACS: 05.20.-y 1. Introduction The mean spherical approximation (MSA) is an analytical theory for electrolyte solutions in the ionic approach [1-3], as well as in the ion-molecular approach [4- 12]. The solution of MSA defines screening potentials which play an important role in the construction of the optimized cluster expansion for free energy and correlation functions [13-15]. The MSA corresponds to the well-known Debye- Huckel (DH) theory [16] in the low coupling Mayer limit (small ionic concentrations and low nonideality of electrostatic interactions) The MSA is also asymptotically correct in the high coupling Onsager limit (high densities and strong nonideality of electrostatic interactions) [17], where, unlike the DH theory, it satisfies the exact Onsager bounds [18] for the Helmholtz free energy and the internal energy of the system. However, the MSA (similar to the DH theory) is a linearized theory (in the strength of electrostatic interactions and in the inverse temperature) that fails at low temperature and does not capture the full second ionic virial coefficient or the full effect of ion pairing. Moreover, the MSA, as a usual integral equation theory, is derived from the Mayer density expansion and does not satisfy the high coupling c© M.F.Holovko, V.Kapko 239 M.F.Holovko, V.Kapko low-density limits. For strongly associating liquids an infinite number of terms in the density expansion need to be included in order to obtain the correct low-density limit, while only a few terms of the activity expansion are enough for this purpose [18-19]. A simple and natural way to correct the theory of electrolyte solutions was proposed by Bjerrum [20] who developed the theory of an ionic association in conjunction with the DH theory. In general, an ionic association occurs due to strong Coulomb interactions or as a result of the chemical mechanism caused by a true chemical bond formation. However, the first realization of an ionic association was done at the thermodynamical level with the application of the original Bjerrum idea, which is exactly correct only for point ions [21]. Recently, a multidensity integral equation theory which correctly accounts for the effects of association in ionic systems has been proposed [20-23]. It is based on the multidensity formalism for associating fluids of Wertheim [36-37] and combines a description in terms of the activity and density expansions. The description of ionic associations in the multidensity formalism with the extension MSA began from the two-density version which was referred to in [26] as an associative MSA (AMSA). The analytical solution of the AMSA was found for an ionic dimerization model [26] and then for an arbitrary mixture of dimerizing ions [27], for an ionic restricted primitive model [26], for a shielding sticky ionic model [32], for a highly asymmetric ionic model [29] and then it was generalized for polymerizing ions [29- 30] and also for ionic fluids with an arbitrary multiple bonding [31]. The explicit expression for thermodynamical properties of a mixture of dimerizing ions was obtained with the application of the exponential approximation for the contact values of a non-associated part of pair distribution functions g00ab which was used for the calculation of the degree of association [36-37]. Such an approximation yields the correct Bjerrum limit for very dilute solutions and the AMSA corresponds to the DH theory with the Bjerrum corection for an ionic association in the limit of point ions. Moreover, as we have recently shown [21], the results of the AMSA can be supplemented by the Ebeling-Grigo [38] choice of the degree of association which provide the capture of the full second ionic virial coefficient. As a result, the solution of the AMSA of associating ionic systems is asymptotically correct in the Onsager high coupling and high density limits, as well as in a high coupling total association and low density limits. The application of the AMSA to the electrolyte solution has been focused up to this time only on the development of an ionic approach. The application of the AMSA to an ion-molecular model was done for any case. The simplest such model is an ion-dipole mixture which has been solved analytically in the MSA for a restricted case of equal sizes of ions and dipoles [4-6, 10] and also for a more general case [7-9, 12-14]. From this article we begin to generalize the solution of the MSA for an ion- dipole model for the AMSA case. In general, in ion-molecular systems, ion-mo- lecular and molecular associations can exist, except for an ionic association. An ion-molecular association describes ionic solvation effects and a molecular associ- ation can be connected with H-bonding effects. Important aspects of molecular 240 AMSA for ion-dipole model systems with hydrogen bonds connected with the proton transfer were developed by Stasyuk and co-workers [39]. However, in this article for simplification we re- strict the consideration of the ion-dipole model only to an ionic association. A more general ion-diple model with molecular and ion-molecular association will be considered elsewhere [41]. The article is organized as follows. In the second section an ion-dipole model with an ionic association is introduced and a general scheme of the analytical solution of the AMSA for this model is presented. In the third section explicit expressions for pair correlation functions and the dielectric constant are obtained for the AMSA solution. Some explicit numerical results, as well as some conclusions are given in the fourth section. 2. The model and the general scheme of the AMSA solution The considered model consists of a three-component mixture of hard spheres with embedded point dipoles and equal size hard spheres with embedded point charges fulfilling the charge neutrality condition. The oppositely charged hard spheres may have associative interactions, which are unique for dimerizing parti- cles. The potential model employed in our study can be presented in the form: Uxy(X1, X2) = UHS xy (r12) + Uel xy(X1, X2) + (1− δxy)U dim xy (z12), (1) where UHS xy (r12) is the potential of the hard spheres of diameters σ, Uel xy(X1, X2) are the pairwise electrostatic interaction potentials, X1 denotes set of coordinates of particle 1 and Ω1 is a set of the Euler angles necessary to define the orientation of the molecule, r12 is the interparticle distance, Udim xy (z12) is the associative potential which appears due to a sticky point site placed on the surface of each hard ion. Here z12 is the distance between the sticky sites of two ions at a given centre-centre separation and orientation. Our solution is based upon the Baxter-Wertheim factorization technique and follows the scheme developed earlier. The AMSA can be expressed in terms of the Wertheim-Ornstein-Zernike (WOZ) equation for mixtures hαβ xy (X1, X2) = cαβxy (X1, X2) + ∑ z ∑ γδ ∫ dX3h αγ xz (X1, X3)ρ γδ z cδβzy(X3, X2) (2) together with AMSA closure relations hαβ xy (X1, X2) = −δα0δβ0, r12 < 1 cαβxy (X1, X2) = −δα0δβ0 ∗ βTU el xy(X1, X2) + (3) δα1δβ1(1− δxy)Bg00+−(r12)δ(r − 1+), r12 > 1, where we put σ = 1 and consider σ as a unit of length, βT = 1/(kbT ) is the Boltzmann thermal factor, B is the strength parameter of the association between sticky points placed on the surface, δ(r) is the Dirac delta function. hαβ xy (X1, X2) 241 M.F.Holovko, V.Kapko and cαβxy (X1, X2) are a pair correlation function and a direct correlation function, respectively, for species x and y in positions −→r 1 and −→r 2. For dipolar particles these functions will depend on the orientation. Thus, the notation hαβ xy (X1, X2) includes the relative distance r12 =\| −→r 1−−→r 2 \|, as well as possible orientations ŝ1 and ŝ2 of dipole moments, with ŝ being a unit vector along the dipolar moment, dX3 denotes an integration over the position r3 and orientations ŝ3 (with ∫ dŝ3 = 1 4π ∫ dΩ = 1). The indices α, β, γ, δ have the values 0, 1 for ions and 0 for dipoles and indicate the degree of association of the corresponding particle. The case α = 0 corresponds to an unbonded particle, α = 1 – to the particle with a bonded site. For simplicity we will omit index ”α” for dipoles. ρx is the density of the particle of type x, which is separated into two densities ρz = ρ0z + ρ1z, where ρ0z is the density of monomer particles, ρ1z - the density of dimers; ρ00z = ρz, ρ 01 z = ρ10z = ρ0z, ρ 11 z = 0. Owing to the pair potential (1) and to the condition of electroneutrality ρ+ = ρ−, ρ 0 + = ρ0−. For dipoles ρd = ρ0d. The monomer and total ionic densities are related by a self-consistent relation [26] ρi = ρ0i + 2π(ρ0i ) 2B g00+−(1+), (4) where ρi = ρ+ + ρ−,ρ 0 i = ρ0+ + ρ0−. All orientation-dependent functions are presented in the orientation-invariant form [10,14,41] fαβ xy (X12) = ∑ mnl fαβ xy (r12 \|mnl)Φmnl(Ω1,Ω2,Ωr12), (5) where Φmnl(Ω1,Ω2,Ωr12) = ∑ αβλ l! ( mnl αβλ ) / ( mnl 000 ) ∗ Dm 0α(Ω1)D n 0β(Ω2)D l 0λ(Ωr12), (6) fαβ xy (r12 \|mnl) = (2m+ 1)(2n+ 1)(2l + 1) l! ( mnl 000 ) ∑ mnl ( mnl αβλ ) ∗ ∫ dΩ1dΩ2dΩr12D ∗m 0α (Ω1)D ∗n 0β(Ω2)D ∗l 0λ(Ωr12)f αβ xy (X12). Here the standard notation for the Wigner 3 − j symbol is applied; Dm 0α(Ω) are generalized spherical functions. In our particular case this reduces to fαβ ii (X12) = fαβ ii (r \|000), fα id(X12) = fα id(r \|000) + fα id(r \|011) Φ011(Ω2,Ωr), (7) fdd(X12) = fdd(r \|000) + fdd(r \|110) Φ110(Ω1,Ω2) + fdd(r \|112) Φ112(Ω1,Ω2,Ωr), where “i” and “d” denote ions and dipoles, respectively. 242 AMSA for ion-dipole model We have Φ011(Ω2,Ωr) = r̂ ŝ2, Φ110(Ω1,Ω2) = ŝ1 ŝ2, (8) Φ112(Ω1,Ω2,Ωr) = 3 (ŝ1 r̂)(ŝ2 r̂)− (ŝ1 ŝ2), where r̂ is a unit vector along −→r and ŝ - along the dipolar moment. The set of WOZ equations can be divided into two formally independent sets of equations by introducing the following linear combinations: h (S)αβ ii (r) = 1 2 [ hαβ ++(r \|000) + hαβ +−(r \|000) ], hαβ(r \|000) = 1 2 [ hαβ ++(r \|000)− hαβ +−(r \|000) ], h (S)α id (r) = 1 2 [ hα +d(r \|011) + hα −d(r \|011) ], hα(r \|011) = 1 2 [ hα +d(r \|011)− hα −d(r \|011) ], (9) h (S) dd (r) = hdd(r \|000), h(r \|110) = hdd(r \|110), h(r \|112) = hdd(r \|112). Similar combinations we have for cµνxy(r \|mnl). We begin from the solution of a set of equations for the electrostatic part. 2.1. Solution of a set of equations for the electrostatic part The closure relations (3) for the electrostatic part of the correlation functions can be written in the following form: hαβ(r \|mnl) = 0ifr < 1, cαβ(r \|000) = − δα0δβ0 βT e2 r − 1 2 δα1δβ1B g00+−(1+) δ(r − 1+), cα(r \|011) = δα0 βT e pd r2 , (10) cα(r \|110) = 0ifr > 1, c(r \|112) = βT p2 d r3 , where e and pd are an ionic charge and a dipole moment, respectively. The WOZ has to be solved in the Fourier space. Therefore, the three-dimensio- nal Fourier transform of hαβ mnl(r) (and cαβmnl(r)) is a Hankel transform of the order l: hαβ mnl(k) = 4πil ∞ ∫ 0 r2drjl(kr)h αβ mnl(r), (11) where jl(kr) is a spherical Bessel function of the order l. As it was shown earlier for the MSA case [6-11], for the solution of the WOZ equation we can introduce 243 M.F.Holovko, V.Kapko the functions Jαβ(r \|mn λ ) = 2π [ρmρn] 1 2 [(2m+ 1)(2n+ 1)] 1 2 ∑ l l! ( mnl λ− λ0 ) / ( mnl 000 ) ∗ ∞ ∫ \|r\| r′ Pl ( r r′ ) hαβ(r′ \|mnl)dr′, (12) where 0 6 λ 6 min(m,n), Pl(x) is the Legendre polynomial of the order l. The similar functions Sαβ(r \|mn λ ) should be introduced as transformations of cαβ(r \|mnl). As a result, the WOZ equation divides into two different sets of equations in the Fourier space, which can be written as equations between J ′ 11(r) = J(r \|111 ) and S ′ 11(r) = S(r \|111 ) and a matrix equation with Ĵ(r) = Jαβ(r \|mn 0 ) and Ŝ(r) = Sαβ(r \|mn 0 ) In this way the WOZ equation reduces to Ĵ(k)− Ŝ(k) = Ĵ(k)× σ̂ × Ŝ(k) (13) together with J ′ 11(k)− S ′ 11(k) = −J ′ 11(k)S ′ 11(k), (14) where Ĵ(k), Ŝ(k), J ′(k) and S ′(k) are one-dimensional Fourier transforms of the corresponding functions Ĵ(r), Ŝ(r), J ′(r) and S ′(r), the matrix σ̂ =   1 xi 0 xi 0 0 0 0 1   (15) and xi = ρ0i / ρi. (16) From (10) and (12), the AMSA closure condition can be written as Ĵ(r) =   b000 b010 0 b100 b110 0 0 0 b′0  +   0 0 b01 0 0 b11 −b01 −b11 0   r +   0 0 0 0 0 0 0 0 b2   r2 ifr < 1. (17) J ′ 11 = b′′0 − 1 2 b2 r 2ifr < 1, (18) where from equation (12), bαβ0 = 2π ρi ∞ ∫ 1 rhαβ(r \|000) dr, bα1 = 2π √ ρiρd/3 ∞ ∫ 1 hα(r \|011) dr, 244 AMSA for ion-dipole model b2 = 2π ρd ∞ ∫ 1 h(r \|112)/r dr, (19) b′0 = 2π ρd 3 ∫ ∞ 1 r[ h(r \|110)− h(r \|112) ] dr, b′′0 = 2π ρd 3 ∫ ∞ 1 r[ h(r \|110) + 1 2 h(r \|112) ] dr. Ŝ(r) =   −d2 0 e−µr 2µ 0 d0 d2 2 e−µr 0 −zΘ(1− r) 0 −d0 d2 2 e−µr 0 0   ifr > 1 (20) S ′ 11(r) = 0 ifr > 1, (21) where d20 = 4π βT e2 ρi, d 2 2 = 4π 3 βT p2d ρd, z = π B ρi g 00 +−(1+), (22) µ is a factor introduced to avoid divergences. In what follows the limit µ → 0 is considered. Here it must be noted that the decoupled equation (14) together with the closure relations (18) and (21) are nothing but a transformed hard spheres Percus-Yevick equation in which the hard sphere density can be expressed as ρ = −b2/2π. (23) Equation (13) together with the closure relations (17) and (20) are suitably solved by means of the Baxter-Wertheim factorization method. This factorization is given by 1̂− σ̂ × Ŝ(k) = σ̂ × Q̂(k)× σ̂ × Q̂T (−k), (24) where σ̂ × Q̂(k) = 1̂− ∞ ∫ 0 σ̂ × Q̂(r) eikr dr. (25) 1̂ is a unity 3 × 3 matrix, and the symbol T denotes a transposed matrix. Q̂(r) will be a simple matrix function of the form: Q̂(r) = Q̂0 − Â e−µr, (26) where Â is a constant matrix and Q̂0(r) is a short-ranged matrix function Q̂0(r) = 0ifr > 1. (27) Explicit relations between Q̂(r) and the quantities Ĵ(r) and Ŝ(r) can be obtained in r space. Ŝ(r) = Q̂(r)− ∞ ∫ 0 dt Q̂(r + t) × σ̂ × Q̂T (t) (28) 245 M.F.Holovko, V.Kapko and Ĵ(r) = Q̂(r) + ∞ ∫ 0 dt Ĵ(r − t)× σ̂ × Q̂(t). (29) By analysing equation (28) and taking into account the closure (20), we obtain: Q̂0(1+)− Q̂0(1−) = Ẑ (30) where Â =   a1 a2 a3 0 0 0 0 0 0   , (31) and Ẑ =   0 0 0 0 z 0 0 0 0   . (32) The solution of the AMSA reduces to the calculation of the matrix factor function Q̂(r). Using some algebra we find its elements, which can be presented in the form: Q̂0(r) = Q̂(1)(r − 1) + Q̂(2)(r2 − 1)− Ẑ, (33) Q (1) 11 = 1 ∆ (D1 a1 − 1 2 (b01) 2), Q (1) 12 = 1 ∆ (D1 a2 − 1 2 b01(b 1 1 + xib 0 1z)), Q (1) 13 = 1 ∆ (D1 a3 + b01(1 + 1 3 b2)), Q (1) 21 = 1 ∆ (D2 a1 − 1 2 b01b 1 1), Q (1) 22 = 1 ∆ (D2 a2 − 1 2 b11(b 1 1 + xib 0 1z)), Q (1) 23 = 1 ∆ (D2 a3 + b11(1 + 1 3 b2)), Q (1) 31 = 1 ∆ (D3 a1 − b01(1− 2 3 b2)), Q (1) 32 = 1 ∆ (D3 a2 − (b11 + xib 0 1z)(1− 2 3 b2)), (34) Q (1) 33 = 1 ∆ (D3 a3 − b01 2 (b01 + 2xib 1 1)− (b2) 2 2 ), Q (2) 11 = Q (2) 12 = Q (2) 13 = 0, Q (2) 21 = Q (2) 22 = Q (2) 23 = 0, Q (2) 31 = 1 ∆ (D4 a1 − b01b2 2 ), Q (2) 32 = 1 ∆ (D4 a2 − b2 2 (b11 + xib 0 1z)), Q (2) 33 = 1 ∆ (D4 a3 + b2(1 + 1 3 b2)), , 246 AMSA for ion-dipole model where D1 = J0(1− 1 6 b2) 2 + xib 0 1 4 (J0b 1 1 − J1b 0 1)− b01 3 (b01 + xib 1 1)(1− b2 24 ), D2 = J1(1− 1 6 b2) 2 − 1 4 (b01 + xib 1 1)(J0b 1 1 − J1b 0 1)− b11 3 (b01 + xib 1 1)(1− b2 24 ), D3 = −1 2 [ J0(b 0 1 + xib 1 1) + J1xib 0 1 ] (1− 2 3 b2) + b01 6 (b01 + xib 1 1) ∗ (b01 + 2xib 1 1) + b2 4 (b01 + xib 1 1), (35) D4 = −1 4 [ J0(b 0 1 + xib 1 1) + J1xib 0 1 ] b2 − b01 8 (b01 + xib 1 1) ∗ (b01 + 2xib 1 1)− 1 2 (b01 + xib 1 1)(1 + b2 3 ), ∆ = (1− b2 6 )2 + b01 4 (b01 + 2xib 1 1), J0 = b000 + xi b 01 0 , J1 = b100 + xi b 11 0 . The contact value of the function h00(1+ \|000) follows from (29) and (12) h00(1+ \|000) = 1 2π∆ [[( (1− 1 6 b2) 2 + 1 4 xib 0 1b 1 1 ) J0− 1 4 xi(b 0 1) 2J1 − 1 3 b01(b 0 1 + xib 1 1)(1− 1 24 b2) ] ∗ a1 − 1 2 (b01) 2 ] . (36) 2.2. Solution of a set of equations for ĥ (S)(r) A set of equations for a non-electrostatic part of correlation functions ĥ(S)(r) can be written in the Fourier space. ĥ(S)(k) = ĉ(S)(k) + ĉ(S)(k)× ρ̂× ĥ(S)(k), (37) where ĥ(S)(k), ĉ(S)(k) and ρ̂ are matrices defined as follows: ĥ(S)(k) =    h (S) 00 ii (k) h (S) 01 ii (k) h (S) 0 id (k) h (S) 10 ii (k) h (S) 11 ii (k) h (S) 1 id (k) h (S) 0 di (k) h (S) 1 di (k) h (S) dd (k)    , (38) ρ̂ =   ρi 0 0 0 ρi 0 0 0 ρd  × σ̂, (39) 247 M.F.Holovko, V.Kapko ĥ (S)αβ xy (k), ĉ (S)αβ xy (k) denote the Fourier transforms of the correlation functions ĥ (S)αβ xy (r) and ĉ (S)αβ xy (r), respectively. A set equations (37) with closure conditions h(S)αβ xy (r) = − δα0δβ0ifr < 1, (40) c(S)αβxy (r) = 1 2 δxyδα1δβ1B g00+−(1+) δ(r − 1+)ifr > 1 has a form similar to the mixture of hard spheres and dimerizing hard spheres. This is a particular case of mixture of dimerizing hard spheres which was solved analytically in [42]. The only exception is that the closure (40) includes function 1 2 g00+−(r) instead of g (S) 00 xy (r). This difference does not, however, change the general scheme of the analytical solution in both cases. The solution can be presented in the form: Q̂(S)(r) = Q̂S(1)(r − 1) + Q̂S(2)(r2 − 1) + Ẑ, (41) Q S(1) 11 = Q S(1) 13 = Q S(1) 31 = Q S(1) 33 = − 3πη (1− η)2 , Q S(2) 11 = Q S(2) 13 = Q S(2) 31 = Q S(2) 33 = π(1 + 2η) (1− η)2 , (42) Q S(1) 12 = Q S(1) 32 = −Q S(2) 12 = −Q S(2) 32 = − πzρ0i 1− η , Q S(1) 21 = Q S(1) 22 = Q S(1) 23 = Q S(2) 21 = Q S(2) 22 = Q S(2) 23 = 0, where η = π 6 (ρi + ρd). The contact value for g (S) 00 ii (r): g (S) 00 ii (1+) = 1 + h (S) 00 ii (1+) = 1 + η/2 (1− η)2 . (43) Finally, using the results of the previous section we have the expression for g00+−(1+) g00+−(1+) = 1 + η/2 (1− η)2 − h00(1+ \|000), (44) where h00(1+ \|000) is given in (36). 2.3. A set of nonlinear equations After a substitution of equation (26) in the k-space into (24) the matrix equa- tion is obtained 1̂− σ̂ × Ŝ(k) = σ̂ × Q̂0(k)× σ̂ ∗ Q̂0T (−k) + σ̂ × Q̂0(k)× σ̂ × ÂT 1 ik − σ̂ × Â× σ̂ × Q̂0 T (−k) 1 ik + σ̂ × Â× σ̂ × ÂT 1 k2 . (45) From the asymptotic behaviour of Ŝ(r), contained in equations (20), we can obtain in [10] (for small k and µ = 0): 1̂− σ̂ ∗ Ŝ(k) =   d20 1 k2 −S12(k) − αS22(k) d0d2 1 ik αd20 1 k2 1− αS12(k) αd0d2 1 ik −d0d2 1 ik −S32(k) d22 + y21   , (46) 248 AMSA for ion-dipole model where y1 = (1− 1 6 b2)/(1 + 1 12 b2) 2. Comparing equations (45) and (46), when k equals zero, the following set of equations is obtained: a21 + 2αa1a2 + a23 = d20, (47) K31(a1 + αa2) +K32 ∗ αa1 + (K33 − 1) ∗ a3 = d0d2, (48) K2 31 + 2αK31 ∗K32 + (K33 − 1)2 = d22 + (1− 1 6 b2) 2 (1 + 1 12 b2)4 , (49) a1(K21 + αK22) + a2(αK21 − 1) + a3K23 = 0, (50) where Kmn = 1 ∫ 0 Q0 mn(r) dr = −1 2 QS(1) mn − 2 3 QS(2) mn . (51) Taking into account equations (33), (28) and (26) we can write lim r→0+ d dr Ŝ(r) = Q̂(1) + 1 2 Â× σ̂ × ÂT − Ẑ × σ̂ × ÂT + Q̂(1) × σ̂ × ÂT + Q̂(2) × σ̂ × ÂT − Q̂(1) × σ̂ × Ẑ − Q̂(2) × σ̂ × Ẑ + 1 2 Q̂(1) × σ̂ × Q̂(1)T + 1 2 Q̂(2) × σ̂ × Q̂(2)T + 1 3 Q̂(2) × σ̂ × Q̂(1)T + (52) 2 3 Q̂(1) × σ̂ × Q̂(2)T . Similarly to ionic systems with an association [26] we have lim r→0+ dSmn(r) dr = 0forS11, S12, S21, S22, S33. (53) We additionally have five equations. We can rewrite equation (14) as xi + 2x2 i ρiz = 1 (54) Equations (47-50), (53), (54) and (22) are a set of eleven equations for ten vari- ables a1, a2, a3, J0, J1, b 0 1, b 1 1, b2, xi, z. In order to solve this system of equations one can apply the numerical methods (a standard Newton-Raphson technique, for ex- ample). It is necessary to note, that among the set of equations (53) for S12 and S22 and equation (50) only two are independent. We cannot prove this analytically due to the complexity of the obtained system. But numerical calculations confirm the identity of these sets. 3. The pair distribution functions In this section we intend to develop explicit expressions to compute the corre- lation functions. As usual, we have jmn(r) = dJmn(r) dr . (55) 249 M.F.Holovko, V.Kapko From equation (29) for r > 1 jmn(r) = ∑ lk 1 ∫ 0 jml(r − t)σlkQ 0 kn(t) dt− ∑ lk r ∫ 1 jml(t)σlkAkn dt+ δm2ρ 0 i z Q 0 1n(r − 1)Θ(2− r)− cman, (56) where c1 = J0, c2 = J1 + ρ0i z, c3 = −(b01 + αb11). To obtain jmn(r) for r > 1, we may use the iterative scheme of Perram [43]. Knowing these functions it is possible to find the pair distribution function ac- cording to the scheme: H00(r) = − 1 2πr (j11(r) + α (j12(r) + j21(r) + αj22(r))) , H01(r) = − 1 2πr (j13(r) + αj23(r)) , H10(r) = −H01(r), (57) H11(r) = − 1 2πr j33(r), H ′ 11(r) = ρhpy(r)byρ = − b2 2π , H000(r) = H00(r), H011(r) = √ 3H01(r), (58) H110(r) = H11(r) + 2H ′ 11(r), H112(r) = H11(r)−H ′ 11(r), h000(r) = H000(r)/ρi, h110(r) = H110(r)/ρd, (59) h011(r) =  H011(r)− 1 r2 r ∫ 0 H011(R)R dR   / √ ρiρd, h112(r) =  H112(r)− 3 r3 r ∫ 0 H112(R)R2 dR   /ρd. The latter are connected with the functions (9) by the relations: h000(r) = h00(r \|000) + xi ( h01(r \|000) + h10(r \|000) + xih 11(r \|000) ) , h011(r) = h0(r \|011) + xih 1(r \|011), h110(r) = h(r \|110), (60) h112(r) = h(r \|112). 250 AMSA for ion-dipole model From (9) and (8) we can define the total pair distribution functions g++(r) = g−−(r) = 1 + h (S) ii (r)− h000(r), g+−(r) = 1 + h (S) ii (r)− h000(r), g+d(r, θ2) = 1 + h (S) id (r) + h011(r) cos(θ2), (61) gdd(r, θ1, θ2) = 1 + h (S) dd (r) + h110(r) cos(θ12), +h112(r)(3 cos(θ1) cos(θ2)− cos(θ12)). Using the Aldeman expression [44] the dielectric constant can be written in the form: ǫ = 1 + d22 (1 + 1 12 b2) 4 (1− 1 6 b2)2 . (62) 4. Discussion of the numerical results In this section we shall consider some AMSA numerical results of interest for electrolytes. The influence of ionic dimerization on the screening potentials h000(r), h011(r), h110(r) and h112(r) is represented in figure 1. In this figure and the following 1.5 2.0 2.5 3.0 3.5 -2 0 2 r h 000 2 3 -2 -1 0 1 h 011 r 2 3 -0.2 0.0 0.2 0.4 r h 110 2 3 -0.1 0.0 0.1 0.2 0.3 0.4 r h 112 Figure 1. The electrostatic part of the correlation functions of the ion-dipole system with an ionic dimerization: - - - - - B=0; ——— B=1000. 251 M.F.Holovko, V.Kapko ones we have used a set of dimensionless parameters to describe the model, namely: βi = βT e 2 σ , βd = βTp 2 d σ3 , ρ∗ = Ni +Nd V σ3, and ci = Ni Ni +Nd . (63) We put ρ∗ = 0.43975, βi = 2982.0, βd = 38.851, which corresponds to aqueous electrolyte solutions of singly charged ions with the ionic size σ = 2.76Å at normal temperature and pressure. The data are represented for ionic concentration ci = 10−2 and for two values of the association parameter Bas: 1) for non-associative case Bas = 0 (dotted line); 2) for Bas = 103 which, in accordance with (4), corresponds to the monomer fraction ρ0i /ρi = 0.04 (solid line). The ionic curves exhibit a well-known minimum at r=2 [6,11,14] which shows the stability of positive ion-solvent molecule - negative ion triplets. h011(r) describes the formation of an ionic solvation shell. The ionic dimeriza- tion leads to a more structural form of the functions h000(r),h011(r) and h112(r), but not of h110(r). A more structured form of the screening potential is in agree- ment with the numerical solution of nonlinear integral equations for an ion-dipole model without an association. It means that association essentially improves the results of the usual MSA. An essential role of an assocation for ionic distribution functions is demonstrated in figure 2. We should note that due to an association, g++(r) includes a nonregular term 1 4π ρi−ρ0i ρi δ(r−1) which is not presented in figure 2. 2 3 -1 0 1 2 3 r g++ 2 3 -2 -1 0 1 2 g+ - r Figure 2. The ionic pair distribution functions: - - - - - B=0; ——— B=1000. The dependence of the fraction of undimerized ions xi = ρ0i ρi as a function of ionic concentration is presented in figure 3. Similarly to the pure ionic model with the increasing ionic concentration, the process of ionic dimerization becomes more intensive. The above result is consistent with the Le Chatelier principle, since the process of olimerization can be treated as an exothermic reaction. Finally, the dependence of the dielectric constant on the ionic concentration is represented in figure 4. As in the MSA, the dielectric constant decreases, but 252 AMSA for ion-dipole model due the ionic dimerization this decrease is slowler compared with a non-associative case. 0.0 4.0x10 -5 8.0x10 -5 0.4 0.6 0.8 c x i 0.0 4.0x10 -3 8.0x10 -3 70 75 80 c ε Figure 3. The fraction of undimerized ions vs an ionic concentration. Figure 4. The dielectric constant as a function of the ionic concentration: - - - - - B=0; ——— B=1000. 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Phys., 1976, vol. 64, p. 724. 254 AMSA for ion-dipole model Аналітичний розв’язок асоціативного середньо-сферичного наближення для іонно-дипольної моделі М.Ф.Головко, В.І.Капко Інститут фізики конденсованих систем НАН України, 290011 м. Львів, вул. Свєнціцького, 1 Отримано 19 травня 1998 р. Простий електроліт моделюється в полярному розчиннику сумішшю полярних твердих сфер і однакового розміру заряджених твердих сфер із можливою іонною димеризацією. Подано аналітичний розв’я- зок асоціативного середньо-сферичного наближення для цієї моделі (АССН). Приводяться точні вирази для парних кореляційних функцій і діелектричної константи в термінах АССН. Роль іонної асоціативності ілюструється числовими результатами. Ключові слова: асоціативність, іонно-дипольна модель, середньо-сферичне наближення PACS: 05.20.-y 255 256

Analytical solution of the associative mean spherical approximation for the ion-dipole model

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