Long-range interactions in model colloidal dispersions with surface charge distributions

Abstract

The theoretical model, which is based on the density functional approach, has been developed for studying the electrostatic properties of colloidal dispersions (modified colloidal primitive model; MCPM) containing hard-spherical ions with a homogeneous surface charge distribution. The mean-spherical approximation for the multi-component charged ions has been used to account for the electrostatic ion correlation effects. The present study reflects the importance of the charge distribution of macroion in determining the ionic structure and mean forces acting on the macroions. Compared with the MCPM, the colloidal primitive model (CPM), on which the colloidal charge is assumed to be in the center of particle, shows very long-range electrostatic properties and mean force acting on the macroions because of the strong cross correlation between the hard-sphere contribution and the Coulomb interaction. The long-range attractions and repulsions in the charged colloidal dispersions originate from the entropic effects and are found at high packing fractions of the colloidal ions.

Appendix: Electronic residual contribution \(c^{(2)}_{{\mathrm{el}},ij}(r, \rho _{i})\)

We used an analytic expression to calculate the electronic residual contribution based on the solution in the mean-spherical approximation, which yields reasonable accuracy [32,33,34]. The main difference between the MSA solution for size-asymmetric electrolytes with center–center interactions and the present solution for uniformly charged hard-sphere ions is the Coulomb interaction between two ions [10, 11, 24,25,26]. In the MSA approach, the two-particle, direct, correlation function (DCF) [32,33,34] for uniformly charged hard-sphere ions, $c_{{\mathrm{chs}},ij}^{(2)}(r, \rho _{i})$, can be expressed as a sum of the hard-sphere and the electronic residual contributions:

$$\begin{aligned} c_{{\mathrm{chs}},ij}^{(2)}(r, \rho _{i})= & {} c_{{\mathrm{hs}},ij}^{(2)}(r, \rho _{i})+{{\beta e^{2}Z_{i}Z_{j}} \over {\epsilon }} A_{ij}(r) \theta (d_{ij}-r)\nonumber \\&+{{e^{2}Z_{i}Z_{j}} \over {r}}\theta (r-d_{ij}), \end{aligned}$$

(14)

where $d_{ij}=(d_{i}+d_{j})/2$ and $c_{{\mathrm{hs}},ij}^{(2)}(r, \rho _{i})$ is the two-particle DCF of the Percus–Yevick integral equation for the hard spheres [35, 36]. $\theta (x)$ is the Heaviside step function: $\theta (x)=1$ for $x>0$, and $\theta (x)=0$ for $x<0$. For the MCPM, the coefficients $A_{ij}(r)$ becomes

$$\begin{aligned} {\begin{array}{llll} A_{ij}(r, \rho _{i})&{}=&{} \alpha _{ij}-Z_{i}Z_{j}/r , &{}\quad 0 \le r \le |d_{i}-d_{j}|/2 \\ &{}=&{} \beta _{ij}/r-Z_{i}Z_{j}/r-\gamma _{ij}+r\delta _{ij}+r^{3}\xi _{ij} ,&{}\quad |d_{i}-d_{j}|/2 \le r \le d_{ij} \\ &{}=&{}0, &{}\quad r > d_{ij} \end{array}} \end{aligned}$$

(15)

with

$$\begin{aligned} \alpha _{ij}= & {} -2[-Z_{i}n_{j}+x_{i}s_{j}-a_{i}s_{j}^{2}/3], \end{aligned}$$

(16)

$$\begin{aligned} \beta _{ij}= & {} (d_{i}-d_{j})[(x_{i}+x_{j})(s_{i}-s_{j})/4\nonumber \\&-(a_{i} -a_{j})[(s_{i}+s_{j})^{2}-4n_{i}n_{j}]/16], \end{aligned}$$

(17)

$$\begin{aligned} \gamma _{ij}= & {} (x_{i}-x_{j})(n_{i}-n_{j})+(x_{i}^{2}+x_{j}^{2})\Gamma \nonumber \\&+(a_{i}+a_{j})n_{i}n_{j}-(a_{i}s_{j}^{2}+a_{i}s_{j}^{2})/3, \end{aligned}$$

(18)

$$\begin{aligned} \delta _{ij}= & {} x_{i}s_{i}/a_{i}+x_{j}s_{j}/a_{j}+n_{i}n_{j}-(s_{i}^{2}+s_{j}^{2})/2, \end{aligned}$$

(19)

$$\begin{aligned} \xi _{ij}= & {} [(s_{i}/a_{i})^{2}+(s_{j}/a_{j})^{2}]/6, \end{aligned}$$

(20)

where $s_{i}=n_{i}+\Gamma x_{i}$ and $x_{i}=Z_{i}+n_{i}d_{i}$ [34]. The $\Gamma$ and the $n_{i}$ functions are obtained numerically to satisfy the algebraic equations

$$\begin{aligned} \Gamma ^{2}= & {} {{\beta \pi e^{2}} \over {\epsilon }} \sum _{i=+,-,M} \rho _{i}(Z_{i}+n_{i}d_{i})^{2}\quad \mathrm{and}\nonumber \\&-x_{i}\Gamma =n_{i}+c\,d_{i}\sum _{i=+,-,M}\rho _{i}d_{i}x_{i} \end{aligned}$$

(21)

with $c=(\pi /2)[1-\pi /6\sum _{i=+,-,M}\rho _{i}d_{i}^{3}]^{-1}$.

The interaction between two charged hard-sphere ions for overlapping separation $u_{ij}(r)$ is given by [10, 11, 26]

$$\begin{aligned} u_{ij}(r)={{ e^{2}Z_{i}Z_{j}}\over {\epsilon }}B_{ij}(r)\theta (d_{ij}-r) + {{e^{2}Z_{i}Z_{j}} \over {r}}\theta (r-d_{ij}), \end{aligned}$$

(22)

where the function $B_{ij}(r)$, which represents the interaction potential between the ions [10, 11, 24,25,26], i.e., the macroion–macroion, macroion–ion, and ion–ion interactions, is simply given by

$$\begin{aligned} B_{\mathrm{MM}}(r)= & {} {{2d_{\mathrm{M}}-r} \over {d_{\mathrm{M}}^{2}}},\quad 0 \le r \le d_{\mathrm{M}} \nonumber \\ B_{-+}= & {} B_{++}=B_{--}={1 \over r},\quad 0 \le r \le d \nonumber \\ B_{-M} = B_{+M}= & {} {2 \over d_{\mathrm{M}}},\quad 0 \le r \le d_{\mathrm{M}}/2 \nonumber \\= & {} {1 \over r},\quad d_{\mathrm{M}}/2 \le r \le (d+d_{\mathrm{M}})/2. \end{aligned}$$

(23)

Then, the electronic residual contribution between the hard-sphere contribution and the Coulomb interaction $c_{{\mathrm{el}},ij}^{(2)}(r, \rho _{i})$ becomes

$$\begin{aligned} c^{(2)}_{{\mathrm{el}},ij}(r, \rho _{i})= & {} c_{{\mathrm{chs}},ij}^{(2)}(r, \rho _{i}) -c_{{\mathrm{hs}},ij}^{(2)}(r, \rho _{i})+\beta u_{ij}(r) \nonumber \\= & {} -{{\beta e^{2}Z_{i}Z_{j}} \over {\epsilon }} \bigg [ A_{ij}(r)+B_{ij}(r) \bigg ]\theta (d_{ij}-r). \end{aligned}$$

(24)

Notice here that for hard-sphere ions with a charge embedded at the center of the sphere (CPM), Eq. (24) exactly recovers the MSA result for size-asymmetric electrolytes, is derived by Blum and Hiroike [32,33,34].