Analytical static structure factor for a two-component system interacting via van der Waals potential

Abstract

In this work, we derive analytical solutions for static structure factor for homogeneous and isotropic solution composed of two components interacting with each other via simple van der Waals potential which is inversely proportional to the sixth power of the distance between the two components. We assume that the first component is the solvent and the second component is the dissolved material which has low concentration or density in comparison to the concentration of the solvent, i.e. dilute solution, which is common for macromolecular fluids. The presented solution is obtained using direct and inverse Fourier transforms in solving Ornstein–Zernike equations (OZE) for multicomponent systems. We calculated isothermal compressibilities as a function of concentration and comparison with other simulation and theoretical results are presented. We believe that the presented solution can be useful in studying biomolecular fluids and other soft matter fluids.

Appendices

Appendix

Generalisation

We can generalise our calculation to higher concentrations and in this case we use eqs (12)–(15) to derive the following formulas:

$$\begin{aligned}&S_{11} (q)=x_1 +x_1 \rho _1 \left( {\begin{array}{l} \left[ {[4\pi \beta D_{11} \sigma _{11}^3 M_{11} (q_{01} )][1-4\pi \beta D_{11} \rho _1 \sigma _{11}^3 M_{11} (q_{01} )]^{-1}} \right] \\ +\left[ {\begin{array}{l} [16\pi ^{2}\rho _2 [\beta D_{12} \sigma _{12}^3 M_{12} (q_{012} )]^{2}][1- \\ 8\pi \beta D_{11} \sigma _{11}^3 \rho _1 M_{11} (q_{01} ) \\ -4\pi \beta D_{22} \rho _2 \sigma _{22}^3 M_{22} (q_{02} ) \\ +32\pi ^{2}\beta ^{2}\rho _1 \rho _2 D_{11} \sigma _{11}^3 M_{11} (q_{01} )D_{22} \sigma _{22}^3 M_{22} (q_{02} ) \\ +16\pi ^{2}\rho _1^2 [\beta D_{11} \sigma _{11}^3 M_{11} (q_{01} )]^{2} \\ -16\pi ^{2}\rho _1 \rho _2 [\beta D_{12} \sigma _{12}^3 M_{12} (q_{012} )]^{2} \\ -64\pi ^{3}\beta ^{3}\rho _1^2 \rho _2 [D_{11} \sigma _{11}^3 M_{11} (q_{01} )]^{2}D_{22} \sigma _{22}^3 M_{22} (q_{02} ) \\ +64\pi ^{3}\beta ^{3}\rho _1^2 \rho _2 D_{11} \sigma _{11}^3 M_{11} (q_{01} )[D_{12} \sigma _{12}^3 M_{12} (q_{012} )]^{2}]^{-1} \\ \end{array}} \right] \\ \end{array}} \right) \\&S_{12} (q)=[4\pi x_1 \rho _2 \beta D_{12} \sigma _{22}^3 M_{22} (q_{012} )] \left[ {\begin{array}{l} 1-4\pi \beta \rho _1 D_{11} \sigma _{11}^3 M_{11} (q_{01} ) \\ -4\pi \beta \rho _2 D_{22} \sigma _{22}^3 M_{22} (q_{02} ) \\ +16\pi ^{2}\beta ^{2}\rho _1 \rho _2 D_{11} \sigma _{11}^3 M_{11} (q_{01} )D_{22} \sigma _{22}^3 M_{22} (q_{02} ) \\ -4\pi \rho _1 \rho _2 [\beta D_{12} \sigma _{22}^3 M_{22} (q_{012} )]^{2} \\ \end{array}} \right] ^{-1}\\&S_{22} (q)=x_2 +\rho _2 x_2 \left( {\begin{array}{l} \left[ {[4\pi \beta D_{22} \sigma _{22}^3 M_{22} (q_{02} )][1-4\pi \rho _2 \beta D_{22} \sigma _{22}^3 M_{22} (q_{02} )]^{-1}} \right] + \\ \left[ {\begin{array}{l} [16\pi ^{2}\rho _2 [\beta D_{12} \sigma _{22}^3 M_{22} (q_{012} )]^{2}][1- \\ 8\pi \beta \rho _2 D_{22} \sigma _{22}^3 M_{22} (q_{02} ) \\ -4\pi \beta \rho _1 D_{11} \sigma _{11}^3 M_{11} (q_{01} ) \\ +32\pi ^{2}\beta ^{2}\rho _1 \rho _2 D_{11} \sigma _{11}^3 M_{11} (q_{01} )D_{22} \sigma _{22}^3 M_{22} (q_{02} ) \\ +16\pi ^{2}\rho _2^2 [\beta D_{22} \sigma _{22}^3 M_{22} (q_{02} )]^{2} \\ -16\pi ^{2}\rho _1 \rho _2 [\beta D_{12} \sigma _{22}^3 M_{22} (q_{012} )]^{2} \\ -64\pi ^{3}\beta ^{3}\rho _2^2 \rho _1 [\beta D_{22} \sigma _{22}^3 M_{22} (q_{02} )]^{2}D_{11} \sigma _{11}^3 M_{11} (q_{01} ) \\ +64\pi ^{3}\beta ^{3}\rho _{21}^2 \rho _1 D_{22} \sigma _{22}^3 M_{22} (q_{02} )[D_{12} \sigma _{22}^3 M_{22} (q_{012} )]^{2}]^{-1} \\ \end{array}} \right] \\ \end{array}} \right) \end{aligned}$$

and $M_{\alpha \beta } (q_0 )$ are the same functions as in eqs (34)–(36).

Computational implementation

We have used Wolfram Mathematica online [18] resources to implement the computation using cloud computing. Users need to have registered software to be able to access these capabilities. To do so we did the following:

• 1. We defined the needed parameters like $\sigma _{11}$, $\sigma _{22}$, $D_{11}$ etc., calculate $\beta$ for specific temperature to be used in $S_{ij}$ and in M functions.

• 2. We typed equations [31–33] for low densities and the general case equations as they appear in the Appendix, on Mathematica software. Note that only online Mathematica package will allow editing such long equations.

• 3. Then we used online Wolfram Mathematica to calculate the results $S_{ij}$ for every q (will take from half an hour to an hour).