Renormalized jellium model for colloidal mixtures

In an attempt to quantify the role of polydispersity in colloidal suspensions, we present an efficient implementation of the renormalized jellium model for a mixture of spherical charged colloids. The different species may have different size, charge, and density. Advantage is taken from the fact that the electric potential pertaining to a given species obeys a Poisson's equation that is species independent; only boundary conditions do change from one species to the next. All species are coupled through the renormalized background (jellium) density, that is determined self-consistently. The corresponding predictions are compared to the results of Monte Carlo simulations of binary mixtures, where Coulombic interactions are accounted for exactly, at the primitive model level (structureless solvent with fixed dielectric permittivity). An excellent agreement is found.

Figure 1

Behavior of the effective charge $f(X,Y)$ as a function of screening, as encoded in $X$. The quantity $Y$ denotes the bare charge of the macroion under study, so that the upper curve, showing $f(X,\infty )$ corresponds to the saturation value studied in Ref. [23]. Practically, $f(X,Y)a/{\ell }_B$ is the effective jellium charge of a sphere having radius $a$, bare charge $Ya/{\ell }_B$, at a packing fraction $X/f(X,Y)$ (monocomponent case).

Figure 2

Illustration of the method employed to find the solution of Eq. (11), for ${\tilde{Z}}_{\text{bare}}^1={\tilde{Z}}_{\text{bare}}^2=4,a_2=2a_1$, and $\tilde{\eta }={10}^{-2}$. The continuous curves show the effective charge $f(X,4)$ (lower curve, indexed “species 1”) and $2f(4X,2)$ (upper curve, indexed “species 2”). Depending on the mixture composition, the weighted average of both with weights $x_1$ and $x_2=1-x_1$ are shown with the dashed lines. These are the master curves, corresponding to the right-hand side of Eq. (11), to be considered for all possible $\tilde{\eta }$. The linear curves show $X/\tilde{\eta }$ for two values of the dressed packing fraction (${\tilde{\eta }}_a={10}^{-2}$ and ${\tilde{\eta }}_b=2\times {10}^{-2}$). For an equimolar mixture ($x_1=1/2$), the effective background charge is shown, by the circle (case $\tilde{\eta }={10}^{-2}$) and by the square (case $\tilde{\eta }=2\times {10}^{-2}$).

Figure 3

Like-size binary mixture of a weakly charged species with ${\tilde{Z}}_{\text{bare}}^1=1$ and a strongly charged species (limit ${\tilde{Z}}_{\text{bare}}^2\to \infty$). The packing fraction is $\eta =\tilde{\eta }={10}^{-5}$. When changing the mixture composition, the allowed range for ${\tilde{Z}}_{\text{eff}}^2$ is displayed by the vertical double arrow on the left-hand side).

Figure 4

Binary case. Log-linear plot. Here ${\tilde{Z}}_{\text{bare}}^1={\tilde{Z}}_{\text{bare}}^2$ are both saturated (divergent), $\tilde{\eta }={10}^{-3},a_2/a_1=1/3$. The weighted average Eq. (10) lies in between the two thick curves upon changing the composition $x_1$ from 0 (in which case it corresponds to the “species 2” bottom curve) to 1 (in which case it coincides to the “species 1” upper curve). As a consequence, the values $X$ can lie between the two vertical dashed lines, from which the allowed range for ${\tilde{Z}}_{\text{back}}$ can be read on the $y$ axis and falls in between the two horizontal dashed lines. As in Fig. 3, the allowed range is thus shown by the vertical arrow.

Figure 5

Same as described in the caption of Fig. 4 but for $a_2/a_1=3$.

Figure 6

Effect of salt on the screening function $f_{\text{salt}}$ appearing in Eq. (18). Here, the reduced charge in chosen equal to 10, and we show $f_{\text{salt}}(X,10,z)$ as a function of $X$ (the jellium background dimensionless charge), for different salinities $z$.

Figure 7

Saturation charge, ${\tilde{Z}}_{\text{sat}}^2$, as a function of the total density of colloids in the no salt case. The dependence on $x_1$ is not very strong. In these case ${\tilde{Z}}_{\text{bare}}^1=1$ (left) and ${\tilde{Z}}_{\text{bare}}^1=20$ (right).

Figure 8

Saturation charge in the no salt case as a function of ${\tilde{Z}}_{\text{bare}}^1$ for a value of the total fraction $\tilde{\eta }={10}^{-3}$ and with $a_1=a_2$ (so that $\eta ={10}^{-3}$ as well).

Figure 9

(Left) Saturation charge ${\tilde{Z}}_{\text{sat}}^2$ as a function of the radius ratio $a_2/a_1$, for a system with $\tilde{\eta }={10}^{-3},{\tilde{Z}}_{\text{bare}}^1=5$ and $x_1=0.5$. (Right) ${\tilde{Z}}_{\text{sat}}^2{a}_1/a_2$ for a system with $\tilde{\eta }={10}^{-3}$ and different values of ${\tilde{Z}}_{\text{bare}}^1$ and $x_1$.

Figure 10

Osmotic pressure for a system consisting of two kinds of colloidal particles with the same charge ${\tilde{Z}}_{\text{bare}}^i=6.4$, as a function of $a_2/a_1$.

Figure 11

(Left) Osmotic pressure for a system consisting of two kinds of colloidal particles with the same radius $a_1=a_2$, as a function of ${\tilde{Z}}_{\text{bare}}^2$, with ${\tilde{Z}}_{\text{bare}}^1=6.4$. (Right) Osmotic pressure, changing the size ratio, keeping a constant surface charge density for both colloids. Here, $a_1$ is fixed, ${\tilde{Z}}_{\text{bare}}^i=6.4$, and $a_2$ changes.