Multi-component mixture of dipolar hard spheres with surface adhesion ∗

Mean spherical approximation (MSA) [1] occupies a special place among the liquid state integral equation theories [2–4] due to the availability of the analytical solution of the corresponding Ornstein-Zernike (OZ) equation for a number of simple albeit nontrivial models (see, for example [2–14]). In the present study we propose the method, which can be used to describe the properties of the fluid of multicomponent dipolar hard spheres with surface adhesion. It is based on the analytical solution of the corresponding version of the MSA with the surface adhesion accounted for following Baxter [15]. In the relevant publications of Blum and coworkers multi-component sticky hard-sphere ion-dipole mixture with orientationally dependent stickiness for dipoles of the same size [16] and one-component hard-sphere system with anisotropic adhesion of arbitrary symmetry and electric multipoles [17] have been studied.

In the present study we propose the method, which can be used to describe the properties of the fluid of multicomponent dipolar hard spheres with surface adhesion.It is based on the analytical solution of the corresponding version of the MSA with the surface adhesion accounted for following Baxter [15].In the relevant publications of Blum and coworkers multi-component sticky hard-sphere ion-dipole mixture with orientationally dependent stickiness for dipoles of the same size [16] and one-component hard-sphere system with anisotropic adhesion of arbitrary symmetry and electric multipoles [17] have been studied.

The model
We consider M -component adhesive hard-sphere fluid with the number density of each species s ρ s = N s /V , hard-sphere diameter σ s and dipolar moment p s .
In addition to hard-sphere and dipolar interaction there is the so-called "sticky" interaction, which is characterized by the adhesion parameter Λ st (Ω 12 ), where Ω 12 = (Ω 1 , Ω 2 ), Ω 1 is a set of Euler angles that give the orientation of the molecule 1.
For the model at hand MSA consists of the following OZ equation and closure conditions where h st (X 12 ) = g st (X 12 ) − 1 is the pair correlation function, g st (X 12 ) is the pair distribution function, c st (X 12 ) is the direct correlation function, U st (X 12 ) is the electrostatic pair potential, r 12 is the distance between the particles, is the unit vector and δ(r) is the Dirac delta-function.Solution of the present MSA problem is based on the Wiener-Hopf factorization technique developed by Wertheim [18] and Baxter [19,20].In order to consistently account for the orientation dependencies, the technique developed by Blum and co-workers [6][7][8] is utilized.According to this method all orientation dependent functions are presented in orientational-invariant form where the linear symmetry of the dipoles is accounted for and Ω r 12 is a set of Euler angles, which defined the orientation of the vector r 12 .This vector connects the centers of masses of the particles 1 and 2.
Here a standard notation is used for the 3-j Wigner symbols and for generalized spherical functions.

General solution
All three sets of equations ( 14) are of the same form.Omitting the indices we have where the function J st (r) has a jump discontinuity at r = σ st : Using matrix notation we have where I is the unit matrix, the symbol " • " denotes matrix multiplication, square brackets denote matrices of the order M , ρ 1/2 st = δ st ρ 1/2 , δ st is the Kronecker delta.Using Wiener-Hopf factorization method we have where the upper index T denotes matrix transpose.Wertheim-Baxter factorization correlation functions are of the following form After the inverse Fourier transform for the equations ( 23) and (24) we have where the upper and lower integration limits are represented by the smallest and largest numbers in braces, respectively.Using the closure conditions ( 17) and taking into account the relation ( 19) from the equation (27) at r < σ st we have

Wertheim-Baxter factorization correlation function
Using the results obtained in the previous section and taking into account the rotationally-invariant indices m, n and χ Wertheim-Baxter factorization correlation functions can be written as follows: This set of equations have to be supplemented by the additional equation for the parameter b 112 st,2 .This equation can be obtained from the closure conditions for Introducing the quantities equation ( 31) can be written as follows where Thus, we obtain a closed set of equations ( 30) and (33) for the coefficients of the Wertheim-Baxter factorization correlation function and for the dipole-dipole interaction parameter b 112 st,2 .

Dielectric constant. Numerical calculations
The expression for the dielectric constant of the dipolar mixture was obtained in [7,21].For the model at hand we have where Table 1.Dielectric properties of the two-component dipolar hard-sphere fluid.
Hard spheres of species 1 has a diameter σ 1 = 1 and hard-spheres size of species 2 is σ 2 = 2 1/3 σ 1 .The reduced total density is ρ * = ρ 1 σ 3 1 + ρ 2 σ 3 2 = 0.8.The dipole moment of species 1 and 2 are such that when ρ 2 = 0 (pure fluid 1) then y = 2.5 and when ρ 1 = 0 (pure fluid 2) then y = 1.5, where y = 4π/9β s ρ s p 2 s .Here Λ mnl st is treated as a fitting parameter.To calculate the dielectric constant of the system, the solution of the set of equations (30) and (33) has to be obtained.This is a set of highly nonlinear equations, which can be solved only using the numerical methods, for example using the Newton's method.To use this method it is important to have a sufficiently accurate initial guess for the unknowns of the problem.In our study we start from the twocomponent version of the model with hard spheres of the same size and dipolar moment.For such a model, the dipole-dipole interaction parameter b 112 st,2 is the same for both species and can be obtained from the corresponding nonlinear equation.All the rest of the unknown coefficients can be easily obtained from b 112 st,2 .Gradually changing the ratio of the hard-sphere sizes and dipolar moments one can get solution of the set of equations (30) and (33) for the two-component system with arbitrary hard-sphere sizes and dipolar moments.
For the sake of illustration in table 1 we present the dependance of the dielectric constant of the two-component mixture on the concentration of the first species X 1 = ρ 1 /ξ 0 with the value of the adhesion constant Λ mnl st chosen so as to fit the corresponding MC values.Good agreement between MC and theoretical predictions in the whole range of concentrations with only one value for Λ mnl st shows that the model at hand can be used to correlate the experimental data with the dielectric properties of multi-component polar fluids.