Electrostatic interactions between charge regulated spherical macroions

Abstract

We study the interaction between two charge regulating spherical macroions with dielectric interior and dissociable surface groups immersed in a monovalent electrolyte solution. The charge dissociation is modelled via the Frumkin-Fowler-Guggenheim isotherm, which allows for multiple adsorption equilibrium states. The interactions are derived from the solutions of the mean-field Poisson-Boltzmann type theory with charge regulation boundary conditions. For a range of conditions we find symmetry breaking transitions from symmetric to asymmetric charge distribution exhibiting annealed charge patchiness, which results in like-charge attraction even in a univalent electrolyte—thus fundamentally modifying the nature of electrostatic interactions in charge-stabilized colloidal suspensions.

Graphic abstract

Appendices

Appendix A: Alternative forms of the free energy

In this appendix we derive alternative forms of the free energy that do not contain the derivatives of the electrostatic potential.

Inserting the Euler-Lagrange equations back into the free energy Eq. (7) we obtain the equilibrium free energy as

$$\begin{aligned} {{{\mathcal {E}}}} [\psi (\textbf{r}), \phi (\textbf{r})]= & {} \frac{k_BT~\kappa _D^2}{4 \pi \ell _B} \!\int _{V}\!\!\!d^3\textbf{r}\Big ( \frac{1}{2}\beta e_0 \psi \sinh {\beta e_0 \psi } - \cosh {\beta e_0 \psi }\Big ) \nonumber \\{} & {} {+} \sum _{i=1,2}\oint _{S_i}\!\!\!d^2\textbf{r}\left( {-}\frac{1}{2} \psi \frac{\partial f_{CR}(\psi , \phi )}{\partial \psi } {+} f_{CR}(\psi , \phi )\right) .~~~ \nonumber \\ \end{aligned}$$

(A1)

This can be rewritten in a fully symmetric form that as far as we know has not yet been derived in the PB literature. In fact, introducing

$$\begin{aligned} f_{PB}(\psi ) = - \frac{k_BT~\kappa _D^2}{4 \pi \ell _B} \cosh {\beta e_0 \psi (\textbf{r})} \end{aligned}$$

(A2)

which is proportional to the (negative) osmotic pressure of the mobile ions, we obtain

$$\begin{aligned} {{{\mathcal {F}}}} [\psi (\textbf{r})]= & {} \int _{V} d^3\textbf{r}\Big (-\frac{1}{2} \psi \frac{\partial f_{PB}(\psi )}{\partial \psi } + f_{PB}(\psi ) \Big ) \nonumber \\{} & {} + \sum _{i=1,2}\oint _{S_i} d^2\textbf{r}~ \left( -\frac{1}{2} \psi \frac{\partial f_{CR}(\psi , \phi )}{\partial \psi } + f_{CR}(\psi , \phi )\right) .\nonumber \\ \end{aligned}$$

(A3)

This form of the free energy is particularly apt for numerical calculations since it contains only the electrostatic potential and the surface charge fraction, but does not contain any derivatives of these fields. We can rework this form of the free energy further by inserting the result of Eqs. (12) that yields

$$\begin{aligned}{} & {} f_{CR}(\psi , \phi ) = \nonumber \\{} & {} \quad = n_0 k_BT \Big ( - \frac{1}{2}\beta e \psi + \frac{1}{2}\chi \phi ^2 - \ln {\left( 1 + e^{- \beta e \psi + \chi \phi + \alpha }\right) }\Big ), ~\nonumber \\ \end{aligned}$$

(A4)

a form valid at each of the macroion surfaces 1, 2. From here it also follows that

Fig. 8

a Plot of the charge asymmetry $\vert \phi _{1}-\phi _2\vert$ for $A = \kappa _D \ell _B/(4\pi )\frac{e^{-\kappa _D D}}{\kappa _D D } = 3$ as a function of $\alpha$ and $\chi$ for two point macroions. The case $\vert \phi _{1}-\phi _2\vert = 0$, corresponds to a symmetric branch of the solution, while the charge symmetry broken state corresponds to $\vert \phi _{1}-\phi _2\vert \ne 0$. The line represents the critical dissociation “isotherm” $\alpha = -{\textstyle \frac{1}{2}}\chi$. b Plot of the force $-\partial _D{{{\mathcal {F}}}}(D)$ in the units of $\kappa _D^2 \ell _B/(4\pi )$ as a function of $u = \kappa _D D$. $\alpha = -2$ and $\chi = 10, 7, 6, 5, 4$ (top to bottom curves). The charge symmetry transitions between the symmetric and asymmetric branches of the solution are now translated into a discontinuous jump in the interaction force from repulsion to attraction. Note that this discontinuity moves to larger spacing as $\chi$ decreases

$$\begin{aligned}{} & {} -\frac{1}{2} \psi _i \frac{\partial f_{CR}(\psi , \phi )}{\partial \psi } = -\frac{1}{2} \psi \sigma \nonumber \\{} & {} \quad = -\frac{1}{2} e n_0~\psi \left( \phi - \frac{1}{2}\right) \nonumber \\ \end{aligned}$$

(A5)

valid at each of the macroion surfaces 1, 2 and therefore combining the two together we remain with

$$\begin{aligned}{} & {} -\frac{1}{2} \psi \frac{\partial f_{CR}(\psi , \phi )}{\partial \psi } + f_{CR}(\psi , \phi ) \nonumber \\{} & {} \quad = n_0\left( - \frac{1}{2}\beta e \psi + \frac{1}{2}\chi \phi ^2 - \ln {\left( 1 + e^{- \beta e \psi + \chi \phi + \alpha }\right) }\right. \nonumber \\{} & {} \quad \left. -\frac{1}{2} \beta e\psi \left( \phi - \frac{1}{2}\right) \right) . \end{aligned}$$

(A6)

again valid at each of the macroion surfaces 1, 2. Inserting this into Eq. (A1) we finally obtain the form of the free energy most suitable for numerical calculations

$$\begin{aligned} {{{\mathcal {E}}}} [\psi (\textbf{r}), \phi (\textbf{r})] =&\frac{k_BT~\kappa _D^2}{4 \pi \ell _B} \int _{V} d^3\textbf{r}\Big ( \frac{1}{2}~\beta e_0 \psi (\textbf{r}) ~\sinh {\beta e_0 \psi (\textbf{r}) } \nonumber \\&- \cosh {\beta e_0 \psi (\textbf{r}) }\Big ) \nonumber \\&+ n_0 k_BT \sum _{i=1,2}\oint _{S_i} d^2\textbf{r} \Big ( - \frac{1}{4}\beta e \psi + \frac{1}{2}\chi \phi ^2 \nonumber \\&- \ln {\left( 1 + e^{- \beta e \psi + \chi \phi + \alpha }\right) } -\frac{1}{2} \beta e\psi \phi \Big ). \nonumber \\ \end{aligned}$$

(A7)

This is a rather simple free energy expression that can be straightforwardly used in numerical calculations specifically in the context of charge regulation. It remains valid with or without the $\chi$ term, and therefore also in the case of the Langmuir dissociation isotherm for $\chi = 0$. Again, we note that since this free energy contains no derivatives, it is a practical formulation for numerical computation.

Note that in the absence of CR—for constant values of the surface charge densities $\sigma _{1,2}$—the above free energy reduces to the form equivalent to the one derived by Overbeek [59, 92]

$$\begin{aligned} {{{\mathcal {E}}}} [\psi (\textbf{r}), \phi (\textbf{r})]= & {} \frac{k_BT~\kappa _D^2}{4 \pi \ell _B} \int _{V} d^3\textbf{r}\Big ( \frac{1}{2}\beta e_0 \psi (\textbf{r}) ~\sinh {\beta e_0 \psi (\textbf{r}) } \nonumber \\{} & {} - \cosh {\beta e_0 \psi (\textbf{r}) }\Big ) + \frac{1}{2} \sum _{i=1,2}\oint _{S_i} d^2\textbf{r}~ \sigma _i \phi _i. \nonumber \\ \end{aligned}$$

(A8)

Appendix B: Point charge limit

We proceed by casting the electrostatic part of the free energy of the two macroions in the limit of point charges into a much simplified form that can be derived from the Casimir charging process

$$\begin{aligned}{} & {} \lim _{\left( \oint _{S_1} \!\!\!d^2\textbf{r} \longrightarrow 4\pi a^2\right) }\!\!\!\!\!\!\!\!\!\!{{{\mathcal {F}}}}_{ES}[e_1, e_2] = \int _0^{e_1} \!\!\!\!\!\psi (e_1) de_1 + \int _0^{e_2} \!\!\!\!\!\psi (e_2) de_2,\nonumber \\ \end{aligned}$$

(B1)

where the limit indicates the point-like macroion approximation, with $e = e_0 ~(\phi -{\textstyle \frac{1}{2}})$, and the electrostatic potential is given by the DH expression for two point charges separated by D, yielding finally the electrostatic free energy in the form

$$\begin{aligned} {{{\mathcal {F}}}}_{ES} [\psi (\textbf{r})]\simeq & {} \frac{e_0^2(\phi _1-{\textstyle \frac{1}{2}})^2}{4\pi \varepsilon _0\varepsilon _w a} + \frac{e_0^2(\phi _2-{\textstyle \frac{1}{2}})^2}{4\pi \varepsilon _0\varepsilon _w a} \nonumber \\ {}{} & {} \quad + \frac{e_0^2(\phi _1-{\textstyle \frac{1}{2}})(\phi _2-{\textstyle \frac{1}{2}})}{4\pi \varepsilon _0\varepsilon _w} \frac{e^{-\kappa _D D}}{D}. \end{aligned}$$

(B2)

Clearly, the terms linear in $\phi _{1,2}$ simply renormalize $\alpha$ and the terms quadratic in $\phi _{1,2}$ renormalize $\chi$ in the expression for the total free energy, Eq. (15),

$$\begin{aligned}{} & {} \alpha _{1,2} \longrightarrow {{\tilde{\alpha }}}_{1,2} = \alpha _{1,2} - \frac{e_0^2}{8\pi \varepsilon _0\varepsilon _w a} - \frac{e_0^2 }{4\pi \varepsilon _0\varepsilon _w} \frac{e^{-\kappa _D D}}{D} \nonumber \\{} & {} \chi _{1,2} \longrightarrow {{\tilde{\chi }}}_{1,2} = \chi _{1,2} + \frac{e_0^2}{8\pi \varepsilon _0\varepsilon _w a}\,, \end{aligned}$$

(B3)

A renormalized and rescaled free energy is then of the form

$$\begin{aligned}{} & {} {{{\mathcal {F}}}} [\phi _1, \phi _2] \simeq \frac{\kappa _D \ell _B}{4\pi } \frac{e^{-\kappa _D D}}{\kappa _D D }\left( \phi _1-\frac{1}{2}\right) \left( \phi _2-\frac{1}{2}\right) - \nonumber \\{} & {} \quad - {{\tilde{\alpha }}}_1 \phi _1 - \frac{1}{2} {{\tilde{\chi }}}_1 \phi _1^2 + \phi _1\ln \phi _1+(1-\phi _1)\ln (1-\phi _1)+ \nonumber \\{} & {} \quad - {{\tilde{\alpha }}}_2 \phi _2 - \frac{1}{2} {{\tilde{\chi }}}_2 \phi _2^2 + \phi _2\ln \phi _2+(1-\phi _2)\ln (1-\phi _2), \nonumber \\ \end{aligned}$$

(B4)

where $\ell _{B}$ is again the Bjerrum length. One should note the difference between the above free energy and the Langmuir isotherm model used in Adžić et al. [47,48,49]. The equilibrium state is obtained numerically—by minimizing ${{{\mathcal {F}}}} [\phi _1, \phi _2]$ with respect to $\phi _{1,2}$:

$$\begin{aligned} \frac{\partial {{{\mathcal {F}}}} [\phi _1, \phi _2]}{\partial \phi _{1,2}(D)} = 0. \end{aligned}$$

(B5)

The equilibrium free energy exhibits a separation dependence ${{{\mathcal {F}}}} [\phi _1(D), \phi _2(D)] \longrightarrow {{{\mathcal {F}}}}(D)$, and the interaction force is $f=- \partial _D {{{\mathcal {F}}}}(D)$.

The dependence of free energy on the (dimensionless) separation $\kappa _D D$ is shown in Fig. 8. Figure 8a displays the charge asymmetry proportional to $\vert \phi _{1}-\phi _2\vert$ as a function of $(\alpha ,\chi )$ at fixed D. In fact the case $\vert \phi _{1}-\phi _2\vert = 0$, corresponds to a symmetric branch of the solution, while the $\vert \phi _{1}-\phi _2\vert \ne 0$ corresponds to charge symmetry broken state. The line in Fig. 8a represents the critical dissociation “isotherm” $\alpha = -{\textstyle \frac{1}{2}}\chi$. Clearly, there is an island of asymmetry in the see of symmetric charge partitioning. The boundary of this island of asymmetry exhibits either a continuous or discontinuous transition from the symmetric to an asymmetric state, which is reflected in the behavior of ${{{\mathcal {F}}}}(D)$ in Fig. 8b that shows the interaction force dependence on $\kappa _D D$ for fixed $(\alpha ,\chi )$.