2.1. Electrokinetic model

The steady-state electrokinetic model is the one adopted by Hill (Reference Hill2015) for nanoparticle gel electrophoresis. It comprises the Poisson equation

(2.1)\begin{equation} {-}\epsilon \epsilon_0 \nabla^2 \psi = \rho_m + \rho_f, \end{equation}

a total of $N$ ion-conservation equations with Nernst–Plank fluxes

(2.2)\begin{equation} 0 ={-} \boldsymbol{\nabla}\boldsymbol{\cdot}\left(n_i \boldsymbol{u} - D_i \boldsymbol{\nabla} n_i - z_i e \frac{D_i}{k_B T} n_i \boldsymbol{\nabla}\psi \right) \quad (i = 1,\ldots, N) \end{equation}

and solvent momentum (inertia-free) and mass (incompressible) conservation equations

(2.3a,b)\begin{equation} 0 = \eta \nabla^2 \boldsymbol{u} - \boldsymbol{\nabla} p - \frac{\eta}{\ell^{2}} \boldsymbol{u} - \rho_m \boldsymbol{\nabla} \psi,\quad \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u} = 0, \end{equation}

where $\rho _f$ is the charge density fixed to the hydrogel skeleton, and

(2.4)\begin{equation} \rho_m = \sum_{i=1}^N z_i e n_i \end{equation}

is the net (mobile) charge density of ions in the fluid phase.

The functions to be calculated are the electrostatic potential $\psi$, the $N$ mobile-ion concentrations $n_i$, and the fluid velocity $\boldsymbol {u}$ and pressure $p$. The parameters in the foregoing equations are the thermal energy $k_B T$, dielectric permittivity $\epsilon \epsilon _0$ (assumed to be uniform due to the cavities and hydrogel being predominantly occupied by water), ion diffusion coefficients $D_i$, ion charges $z_i e$ (valence $z_i$ and fundamental charge $e$), fluid shear viscosity $\eta$ and hydrogel (Brinkman) permeability $\ell ^2$.

The permeability and fixed charged density are prescribed functions of radial distance $r$ from the cavity centre: $\ell ^{-2} = n_s(r) 6 {\rm \pi}a_s$ and $\rho _f = - \chi e n_s(r)$, where

(2.5)\begin{equation} \frac{n_s(r)}{n_s^\infty} = 1 - \frac{1}{2}\mbox{erfc}[(r - R) / \delta] \end{equation}

is the concentration of Stokes resistance centres (with radius $a_s$), and $\chi$ is the fraction of Stokes resistance centres (bearing a single negative charge). Equation (2.5) is a convenient phenomenological means to insert a spherical cavity, avoiding cumbersome boundary conditions to couple the interior and exterior domains, also enabling one to vary the sharpness of the transition. Although there are no experimental measures available, the cavities in real hydrogels likely have a more gradual transition than assumed (for simplicity) herein. All the results reported below were undertaken with $\delta = 1$ nm and cavity radii in the range $R = 1$–$1000$ nm. Accordingly, the Brinkman length undergoes a rapid variation from $\infty$ inside the cavity to a prescribed finite value in the bulk hydrogel

(2.6)\begin{equation} \ell = \frac{1}{\sqrt{n_s^\infty 6 {\rm \pi}a_s}}. \end{equation}

Note that, when $\ell$ and $\rho _f^\infty$ are considered independent variables,

(2.7)\begin{equation} \chi = \frac{-\rho^\infty_f / e}{n_s^\infty} ={-}(\rho^\infty_f / e) 6 {\rm \pi}a_s \ell^2 \end{equation}

may be (unphysically) greater than one. For example, varying $\ell$ from $0.1$ to $100$ nm with $a_s = 0.15 \times 10^{-10}$ m, $n_s^\infty$ varies from $600$ M to $0.6$ mM, whereas $-\rho ^\infty _f / e$ will be prescribed in the range $0.1$–$10$ mM. Note that these estimates of $n_s^\infty$ neglect the hydrodynamic interactions between the Stokes resistance centres, which increase the effective value of $a_s$ when their volume fraction $n^\infty _s 4 {\rm \pi}a_s^3 / 3$ is not sufficiently small. In very dense polymers, however, ion mobilities will be smaller than based on their limiting molar conductivities (as assumed herein). The dielectric constant of the hydrogel would also be smaller than the value for water (assumed herein).

The electrokinetic model, which is nonlinear, is solved by perturbing the variables from their equilibrium values, calculated with the application of either an applied pressure gradient $\boldsymbol {P} = \boldsymbol {\nabla }{p}$ as $r \rightarrow \infty$ or an electric field $\boldsymbol {E} = - \boldsymbol {\nabla } \psi$ as $r \rightarrow \infty$. More general solutions with both fields acting are obtained by linear superposition.

The equilibrium values, identified with superscripts $0$, are functions of radial position: $\boldsymbol {u}^0 = 0$ with $\psi ^0$, $n_i^0$ and $p^0$ solving the nonlinear Poisson–Boltzmann system

(2.8)$$\begin{gather} -\epsilon \epsilon_0 \nabla^2 \psi^0 = \sum_{i=1}^N z_i e n_i^0 + \rho_f, \end{gather}$$

(2.9)$$\begin{gather}0 ={-} \boldsymbol{\nabla}\boldsymbol{\cdot}\left(- D_i \boldsymbol{\nabla} n_i^0 - z_i e \frac{D_i}{k_B T} n_i^0 \boldsymbol{\nabla}\psi^0 \right) \quad (i = 1,\ldots, N), \end{gather}$$

(2.10)$$\begin{gather}0 ={-} \boldsymbol{\nabla} p^0 - \sum_{i=1}^N z_i e n_i^0 \boldsymbol{\nabla} \psi^0, \end{gather}$$

with boundary conditions

(2.11a,b)\begin{equation} \frac{\mbox{d} n_i^0}{\mbox{d} r} = 0 \quad\mbox{and}\quad \frac{\mbox{d} \psi^0}{\mbox{d} r} \rightarrow 0 \quad\mbox{at } r =0, \end{equation}

and

(2.12a,b)\begin{equation} n_i^0 \rightarrow n_i^\infty \quad \mbox{and}\quad \psi^0 \rightarrow 0 \quad\mbox{as } r \rightarrow \infty. \end{equation}

Note that bulk electroneutrality demands $\rho ^\infty _f + \rho _m = 0$ at $r = \infty$. Moreover, it is assumed that there is no external electrolyte bath, which would alter the bulk ion concentrations by inducing a non-zero Donnan potential in the hydrogel (with respect to the bath) (Doi Reference Doi2013). The equilibrium solution is accomplished using finite differences with an adaptive mesh (Hill, Saville & Russel Reference Hill, Saville and Russel2003) and equilibrium pressure $p^0 = - \sum _{i=1}^N z_i e n_i^0 \psi ^0$ (furnished by the equilibrium momentum equation).

Next, the equations are solved with boundary conditions for two linearly independent solutions. In this work, both solutions have

(2.13)\begin{equation} n_i \rightarrow n_i^\infty \quad \mbox{as } r \rightarrow \infty, \end{equation}

since it is assumed that there are no macroscopic ion-concentration gradients (with vanishing Donnan potential). As noted by O'Brien & Perrins (Reference O'Brien and Perrins1984), this may be justified when the forcing time scale is large compared with the micro-scale relaxation times (e.g. micro-diffusion time scale $\sim R^2 / D_i$) and short enough to avoid the build-up of substantial ion-concentration gradients. For example, on the macro-scale with characteristic length $L$, the diffusion time $\tau _d \sim L^2/ D_i$ with electro-migration time $\tau _e \sim L k_B T / (|\boldsymbol {E}| |z_i| e D_i)$. Thus, with $|\boldsymbol {E}| = |\Delta V| / L$, where $\Delta V$ is the voltage across the sample, we find $\tau _e \sim \tau _d / |\Delta V|^*$ where $|\Delta V|^* = |\Delta V| |z_i| e / (k_B T)$.

One linearly independent solution comes from the application of an electric field $\boldsymbol {E}$ in the absence of a mean pressure gradient $\boldsymbol {P} = 0$

(2.14)\begin{equation} \psi \rightarrow - \boldsymbol{r} \boldsymbol{\cdot} \boldsymbol{E} \quad \mbox{as } r \rightarrow \infty. \end{equation}

As readily verified from the momentum equation, this generates a uniform far-field flow $\boldsymbol {u} \rightarrow - \boldsymbol {E} \rho ^\infty _f \ell ^2 / \eta$ as $r \rightarrow \infty$. The other linearly independent solution comes from the application of a pressure gradient $\boldsymbol {P}$ in the absence of a mean electric field $\boldsymbol {E} = 0$

(2.15)\begin{equation} p \rightarrow \boldsymbol{r} \boldsymbol{\cdot} \boldsymbol{P} \quad \mbox{as } r \rightarrow \infty, \end{equation}

which generates a uniform far-field flow $\boldsymbol {u} \rightarrow - \boldsymbol {P} \ell ^2 / \eta$ as $r \rightarrow \infty$.

Solutions of the linearized model have the general form (deduced by linearity and symmetry considerations)

(2.16)$$\begin{gather} \psi(\boldsymbol{r}) = \psi^0(r) - \boldsymbol{r} \boldsymbol{\cdot} \boldsymbol{E} + \hat{\psi}(r) \boldsymbol{e}_r \boldsymbol{\cdot} \boldsymbol{X}, \end{gather}$$

(2.17)$$\begin{gather}n_i (\boldsymbol{r}) = n_i^0(r) + \hat{n}_i(r) \boldsymbol{e}_r \boldsymbol{\cdot} \boldsymbol{X}, \end{gather}$$

(2.18)$$\begin{gather}\boldsymbol{u} ={-} \frac{\ell^2}{\eta} \boldsymbol{P} - \rho^\infty_f \frac{\ell^2}{\eta} \boldsymbol{E} + \boldsymbol{\nabla} \times \boldsymbol{\nabla} \times h(r) \boldsymbol{X}, \end{gather}$$

where $\boldsymbol {X} = \boldsymbol {E}$ or $\boldsymbol {P}$, thus solving

(2.19)$$\begin{gather} -\epsilon \epsilon_0 \nabla^2 \psi' = \rho'_m + \rho'_f, \end{gather}$$

(2.20)$$\begin{gather}0 ={-} \boldsymbol{\nabla}\boldsymbol{\cdot}\left(n^0_i \boldsymbol{u} - D_i \boldsymbol{\nabla} n'_i - z_i e \frac{D_i}{k_B T} n'_i \boldsymbol{\nabla} \psi^0 - z_i e \frac{D_i}{k_B T} n^0_i \boldsymbol{\nabla} \psi' \right) \quad (i = 1,\ldots, N), \end{gather}$$

(2.21)$$\begin{gather}0 = \eta \nabla^2 \boldsymbol{u} - \boldsymbol{\nabla} p' - \frac{\eta}{\ell^{2}} \boldsymbol{u} - \rho'_m \boldsymbol{\nabla} \psi ^0 - \rho^0_m \boldsymbol{\nabla} \psi',\quad \boldsymbol{\nabla}\boldsymbol{\cdot}\boldsymbol{u} = 0, \end{gather}$$

where

(2.22)\begin{equation} \rho'_m = \sum_{i=1}^N z_i e n'_i \end{equation}

is the perturbed mobile-ion charge density. Note that primed variables denote the perturbation from equilibrium, i.e. $(\cdot )' = (\cdot ) - (\cdot )^0$, and that continuity of the perturbed variables at the origin requires

(2.23)\begin{equation} \hat{\psi} = \hat{n}_i = h_r(r=0) = 0 \quad \mbox{at } r = 0. \end{equation}

Then the velocity, constructed from the scalar function $h(r)$ above, may be written (radial and tangential components)

(2.24)\begin{equation} \boldsymbol{u} = \left(\boldsymbol{U}_0 - 2 \frac{h_r}{r} \boldsymbol{X} \right) \boldsymbol{\cdot} \boldsymbol{e}_r \boldsymbol{e}_r + \left[ \boldsymbol{U}_0 - \left(h_{rr} + \frac{h_r}{r}\right)\boldsymbol{X} \right] \boldsymbol{\cdot} \boldsymbol{e}_\theta \boldsymbol{e}_\theta, \end{equation}

where the uniform far-field velocity

(2.25)\begin{equation} \boldsymbol{U}_0 ={-} \frac{\ell^2}{\eta} \boldsymbol{P} - \rho^\infty_f \frac{\ell^2}{\eta} \boldsymbol{E}. \end{equation}

The equations and boundary conditions are solved using finite differences with an adaptive mesh (Hill et al. Reference Hill, Saville and Russel2003). The pressure is eliminated by taking the curl of the momentum equation and noting that the velocity written in terms of $h(r)$ satisfies the continuity equation. Pertinent aspects of this numerical solution are highlighted in § 2.5. First, details on how the solutions of this model are used to derive the averaged fluxes for cavity doped hydrogels are presented.

2.2. Averaged current density

The linearly independent solutions of the electrokinetic model detailed in § 2.1 provide asymptotic coefficients that are shown in this section to prescribe the averaged fluxes in a hydrogel that bears randomly distributed cavities with cavity volume fraction $\phi \ll 1$. The averaging undertaken here is inspired by the methodology of Saville (Reference Saville1979) for colloidal dispersions. As shown in Appendix A, this is equivalent to the ensemble averaging applied by Koch & Brady (Reference Koch and Brady1985) to hydrodynamic dispersion in fixed beds of spheres. Volume averaging over the entire composite is reduced to the evaluation of unconditionally convergent volume integrals that enclose a single cavity in an unbounded hydrogel. Then, following a methodology introduced by O'Brien (Reference O'Brien1981), these are transformed to surface integrals that can be expressed in terms of the dipolar disturbances of a single cavity in an unbounded hydrogel. In contrast to colloidal dispersions, the averaged current density for a cavity doped hydrogel requires explicit knowledge of the fluid-velocity disturbances.

The averaged electrical current density (steady state) is written as the volume average of the advective, electro-migrative and diffusive current fluxes

(2.26)\begin{align} \boldsymbol{I} &= \frac{1}{V} \int_{V} \sum_{i=1}^{N} z_i e \boldsymbol{j}_i \,\mbox{d}V \nonumber\\ &= \frac{1}{V} \int_{V} \sum_{i=1}^{N} \left[z_i e n_i^\infty \boldsymbol{u} - z_i e D_i \boldsymbol{\nabla}{n_i} - (z_i e)^2 \frac{D_i}{k_B T} n^\infty_i \boldsymbol{\nabla}{\psi}\right] \,\mbox{d}V \nonumber\\ &\quad + \frac{1}{V} \int_{V} \sum_{i=1}^{N} \left[z_i e \boldsymbol{j}_i - z_i e n_i^\infty \boldsymbol{u} + z_i e D_i \boldsymbol{\nabla}{n_i} + (z_i e)^2 \frac{D_i}{k_B T} n^\infty_i \boldsymbol{\nabla}{\psi}\right]\mbox{d}V, \end{align}

where the total flux of the $i$th mobile ion is

(2.27)\begin{equation} \boldsymbol{j}_i = n_i \boldsymbol{u} - D_i \boldsymbol{\nabla}{n_i} - z_i e \frac{D_i}{k_B T} n_i \boldsymbol{\nabla}{\psi}. \end{equation}

Note that the integrands are written in terms of bulk variables that yield unconditionally convergent integrals for random cavity configurations that are dilute enough for their Debye layers not to overlap (Saville Reference Saville1979). This requires a cavity volume fraction $\phi \ll [1 + 1 / (\kappa R)]^{-1/3}$. Note that the velocity disturbances in the hydrogel (Brinkman medium) decay as $\boldsymbol {u}' \sim r^{-3}$, reflecting the gradient of a dipolar decay of the accompanying pressure perturbation in the far field (Brinkman Reference Brinkman1947).

In a uniform hydrogel without cavities,

(2.28)\begin{equation} \boldsymbol{I} = \sum_{i=1}^{N} \left( z_i e n_i^\infty \boldsymbol{U} + (z_i e)^2 \frac{D_i}{k_B T} n^\infty_i \boldsymbol{E} \right), \end{equation}

where $\boldsymbol {U} = \boldsymbol {U}_0$ comprises the electro-osmotic and pressure driven terms in (2.25). With bulk electro-neutrality, the current density in a uniform (cavity free) hydrogel is

(2.29)\begin{equation} \boldsymbol{I} = \sigma_\infty \left(1 + \frac{{\rho^\infty_f}^2 \ell^2}{\sigma_\infty \eta}\right) \boldsymbol{E} + \frac{\rho^\infty_f \ell^2}{\eta} \boldsymbol{P}, \end{equation}

where

(2.30)\begin{equation} \sigma_\infty = \sum_{i=1}^{N} (z_i e)^2 \frac{D_i}{k_B T} n^\infty_i \end{equation}

is the electro-osmosis-free conductivity.

Note that the electro-migrative current density in a hydrogel for which the current is dominated by the counter-ions is $\sim \rho ^\infty _f D_i E e / (k_B T)$, and the advective/electro-osmotic current density is $\sim \rho ^\infty _f u_{eo} = {\rho ^\infty _f}^2 E \ell ^2 / \eta$. Thus, the ratio of the electro-osmotic to electro-migrative currents is

(2.31)\begin{equation} \sim \mbox{Pe}_g = \frac{u_{eo}}{D_i E e / (k_B T)} = \frac{\rho^\infty_f \ell^2 k_B T}{\eta D_i e} = 6 {\rm \pi}a_i \ell^2 \rho^\infty_f / e, \end{equation}

where $D_i = k_B T / (6 {\rm \pi}\eta a_i )$ with $a_i$ the Stokes radius of a counter-ion. This ratio may be large or small, depending on the fixed charge density and permeability, both of which will be varied systematically when the results are presented.

Now returning to the cavity filled hydrogel, (2.26) may be written

(2.32)\begin{align} \boldsymbol{I} &= \sum_{i=1}^{N} \left[ z_i e n_i^\infty \langle \boldsymbol{u} \rangle - (z_i e)^2 \frac{D_i}{k_B T} n^\infty_i \langle \boldsymbol{\nabla}{\psi} \rangle \right] \nonumber\\ &\quad + \frac{1}{V} \int_{V} \sum_{i=1}^{N} \left[z_i e (n_i^0 - n^\infty_i) \boldsymbol{u} - (z_i e)^2 \frac{D_i}{k_B T} (n_i^0 - n^\infty_i) \boldsymbol{\nabla}{\psi}' - (z_i e)^2 \frac{D_i}{k_B T} n_i' \boldsymbol{\nabla}{\psi}^0\right]\mbox{d}V, \end{align}

where $\langle \cdot \rangle$ denotes the volume average (or ensemble average). Note that the integrands in the second line vanish beyond the diffuse layer of each cavity because $n_i^0 - n^\infty _i$ and $\boldsymbol {\nabla }{\psi }^0$ are exponentially small there. It follows that the volume $V$ over which the integrals are evaluated in (2.32) can be replaced by an integral over the volume of a single cavity in an unbounded hydrogel $V'$. This amounts to replacing $V^{-1} \int _{V}$ with $n \int _{V'}$ (assuming all integrals/cavities are equal), where $n = 3 \phi / (4 {\rm \pi}R^3)$ is the cavity number density. This is equivalent to integrating the conditionally averaged fields $\langle \cdot \rangle _1$ in ensemble averaging when the dispersed phase is dilute (Hinch Reference Hinch1977; Koch & Brady Reference Koch and Brady1985).

Having now established that the averaging in the second line of (2.32) can be evaluated in terms of unconditionally convergent integrals, we may now draw on the following identities (with $\boldsymbol {\nabla }\boldsymbol {\cdot }{\boldsymbol {u}} = \boldsymbol {\nabla }\boldsymbol {\cdot }{\boldsymbol {j}_i} = 0$) to evaluate the volume integrals from (2.26):

(2.33)\begin{equation} \int_V \boldsymbol{\nabla}{(\boldsymbol{\cdot})} \mbox{d}V \rightarrow \int_A (\boldsymbol{\cdot}) \boldsymbol{e}_r \,\mbox{d}A, \end{equation}

and

(2.34a,b)\begin{equation} \int_V \boldsymbol{j} \,\mbox{d}V \rightarrow \int_A \boldsymbol{x} \boldsymbol{j} \boldsymbol{\cdot} \boldsymbol{e}_r \,\mbox{d}A \quad \mbox{and}\quad \int_V \boldsymbol{u} \,\mbox{d}V \rightarrow \int_A \boldsymbol{x} \boldsymbol{u} \boldsymbol{\cdot} \boldsymbol{e}_r \,\mbox{d}A. \end{equation}

In this manner, the surface integrals may be evaluated using only the asymptotic far-field decay of the spatial perturbations. This avoids having to numerically evaluate volume integrals from the single cavity in an unbounded hydrogel (approximating conditionally averaged fields $\langle \cdot \rangle _1$). The volume integral in (2.32) may be written

(2.35)\begin{align} & \frac{3 \phi}{4 {\rm \pi}R^3} \int_{A} \sum_{i=1}^{N} z_i e \boldsymbol{x} \left[(n^0_i - n^\infty_i) \boldsymbol{u} - D_i \boldsymbol{\nabla}{(n^0_i + n_i')} - z_i e \frac{D_i}{k_B T} (n_i^0 \boldsymbol{\nabla}{\psi'} + n'_i \boldsymbol{\nabla}{\psi^0}) \right] \boldsymbol{\cdot} \boldsymbol{e}_r \,\mbox{d}A \nonumber\\ &\quad + \frac{3 \phi}{4 {\rm \pi}R^3} \int_{A} \sum_{i=1}^{N} z_i e \left[ D_i (n^0_i + n_i') \boldsymbol{e}_r + z_i e \frac{D_i}{k_B T} n^\infty_i (\psi' + \psi^0) \boldsymbol{e}_r \right] \mbox{d}A, \end{align}

with $A$ the area of a concentric sphere with radius $r \rightarrow \infty$ where $\psi ^0$ and $n_i^0 - n_i^\infty$ are exponentially small. Noting the respective even–odd symmetries of $n_i^0$ and $n_i'$, and $\psi ^0$ and $\psi _i'$, the total current density becomes

(2.36)\begin{align} \boldsymbol{I} &= \sum_{i=1}^{N} \left[ z_i e n_i^\infty \langle \boldsymbol{u} \rangle + (z_i e)^2 \frac{D_i}{k_B T} n^\infty_i \boldsymbol{E} \right] \nonumber\\ &\quad +\frac{3 \phi}{R^3} \sum_{i=1}^{N} \lim_{r \rightarrow \infty} \left[z_i e D_i \left \langle (n_i' - \boldsymbol{x} \boldsymbol{\nabla}{n_i'} \boldsymbol{\cdot}) \boldsymbol{e}_r \right \rangle_\theta + (z_i e)^2 \frac{D_i}{k_B T} n^\infty_i \left \langle (\psi' - \boldsymbol{x} \boldsymbol{\nabla}{\psi'} \boldsymbol{\cdot}) \boldsymbol{e}_r \right \rangle_\theta\right] r^2, \end{align}

where

(2.37)\begin{equation} \lim_{r \rightarrow \infty} \left \langle (n_i' - \boldsymbol{x} \boldsymbol{\nabla}{n_i'} \boldsymbol{\cdot}) \boldsymbol{e}_r \right \rangle_\theta r^2 = \lim_{r \rightarrow \infty} \hat{n}^E_i r^2 \boldsymbol{E} - \frac{\ell^2}{\eta} \lim_{r \rightarrow \infty} \hat{n}^U_i r^2 \boldsymbol{P} \equiv J^E_i \boldsymbol{E} - \frac{\ell^2}{\eta} J^U_i \boldsymbol{P}, \end{equation}

and

(2.38)\begin{equation} \lim_{r \rightarrow \infty} \left \langle (\psi' - \boldsymbol{x} \boldsymbol{\nabla}{\psi'} \boldsymbol{\cdot}) \boldsymbol{e}_r \right \rangle_\theta r^2 = \lim_{r \rightarrow \infty} \hat{\psi}^E_i r^2 \boldsymbol{E} - \frac{\ell^2}{\eta} \lim_{r \rightarrow \infty} \hat{\psi}^U_i r^2 \boldsymbol{P} \equiv D^E \boldsymbol{E} - \frac{\ell^2}{\eta} D^U \boldsymbol{P}. \end{equation}

Note that $\langle \cdot \rangle _\theta = \int (\cdot )\,\mbox {d}A / (4 {\rm \pi}r^2)$ denotes a directional average over the surface of a concentric sphere with radius $r$, thus identifying dipole strengths/asymptotic coefficients for the concentration ($J_i^X$), and electrostatic-potential ($D^X$) disturbances.

In the next section, the averaged fluid velocity $\langle \boldsymbol {u}\rangle$ in (2.36) is shown to be

(2.39)\begin{equation} \langle \boldsymbol{u} \rangle ={-} \frac{\ell^2}{\eta} \boldsymbol{P} - \rho^\infty_f \frac{\ell^2}{\eta} \boldsymbol{E} + \phi \frac{\ell^2}{\eta}\frac{3C^U}{R^3} \boldsymbol{P} - \phi \frac{3C^E}{R^3} \boldsymbol{E}, \end{equation}

where

(2.40)\begin{equation} C^X = \lim_{r \rightarrow \infty} h^X_r r^2 \end{equation}

are the dipole strengths/asymptotic coefficients for the pressure disturbances. This enables (2.36) to be written

(2.41)\begin{align} \boldsymbol{I} &={-} \rho^\infty_f \left[ - \frac{\ell^2}{\eta} \boldsymbol{P} - \rho^\infty_f \frac{\ell^2}{\eta} \boldsymbol{E} + \phi \frac{\ell^2}{\eta}\frac{3C^U}{R^3} \boldsymbol{P} - \phi \frac{3C^E}{R^3} \boldsymbol{E} \right] \nonumber\\ &\quad + \sigma_\infty \left[ \boldsymbol{E} + \frac{3 \phi}{R^3} D^E_e \boldsymbol{E} - \frac{\ell^2}{\eta} \frac{3 \phi}{R^3} D^U_e \boldsymbol{P} \right], \end{align}

where

(2.42)\begin{equation} D^X_e = D^X + \frac{1}{\sigma_\infty} \sum_{i=1}^{N} z_i e D_i J^X_i \end{equation}

are termed effective electrostatic dipole strengths/asymptotic coefficients. Note that the superscripts $U$ of the asymptotic coefficients above denote that the coefficient is calculated when a uniform far-field velocity $\boldsymbol {U}_0 = - \boldsymbol {P} \eta / \ell ^2$ is prescribed with $\boldsymbol {E} = 0$.

2.3. Averaged fluid momentum

In the sections above, the velocity disturbance $\boldsymbol {u}' = \boldsymbol {u} - \boldsymbol {U}_0$ decays as $r^{-3}$, so it is tempting to calculate the average in the first line of (2.32) as

(2.43)\begin{equation} \langle \boldsymbol{u} \rangle = \frac{3 \phi}{4 {\rm \pi}R^3} \int_{V} (\boldsymbol{u} - \boldsymbol{U}_0) \,\mbox{d}V + \boldsymbol{U}_0 = \frac{3 \phi}{R^3} \lim_{r \rightarrow \infty} \langle \boldsymbol{x} (\boldsymbol{u} - \boldsymbol{U}_0) \boldsymbol{\cdot} \boldsymbol{e}_r \rangle_\theta r^2 + \boldsymbol{U}_0, \end{equation}

which, from the far-field decay of $\boldsymbol {u}'$ in (2.24), furnishes

(2.44)\begin{equation} \langle \boldsymbol{u} \rangle ={-} \frac{\ell^2}{\eta} \boldsymbol{P} - \rho^\infty_f \frac{\ell^2}{\eta} \boldsymbol{E} + \phi \frac{\ell^2}{\eta} \frac{2 C^U}{R^3} \boldsymbol{P} - \phi \frac{2 C^E}{R^3} \boldsymbol{E}. \end{equation}

However, this average does not respect Onsager reciprocity, thus motivating consideration of the averaged fluid momentum equation, which, as shown below, furnishes (2.39) (with prefactor $3$ multiplying $C^X$).

According to ensemble averaging (Hinch Reference Hinch1977; Koch & Brady Reference Koch and Brady1985), the fields $\boldsymbol {u}$, $\psi$ and $\rho _m$ above (for a single cavity in an unbounded hydrogel) become approximations of their conditionally averaged counterparts $\langle \cdot \rangle _1$, i.e. the average of $(\cdot )$ over an ensemble of dilute, random cavity configurations, conditioned on a single cavity centred at the origin. Ensemble averaging the fluid momentum equation gives

(2.45)\begin{equation} \langle \boldsymbol{\nabla} p \rangle + \langle \eta/\ell^2 \rangle \langle \boldsymbol{u} \rangle - \langle \rho_{f} \rangle \langle \boldsymbol{\nabla}{\psi}\rangle={-}\langle (\eta / \ell^2)' \boldsymbol{u}' \rangle - \langle \rho_m' \boldsymbol{\nabla} \psi' \rangle, \end{equation}

where $(\cdot )' = (\cdot ) - \langle \cdot \rangle$, not the perturbation from equilibrium $(\cdot ) - (\cdot )^0$ adopted in the sections above for the single cavity in an unbounded hydrogel. For dilute cavity doped hydrogels, the averages on the right-hand side of (2.45) may be approximated as

(2.46)\begin{equation} \langle (\eta / \ell^2)' \boldsymbol{u}' \rangle \approx n \frac{\eta}{\ell^2_\infty} \int_{r =0}^\infty \left\langle \frac{\ell^2_\infty}{\ell^2} - 1\right\rangle_1 \langle \boldsymbol{u}' \rangle_1 \,\mbox{d}\boldsymbol{r}, \end{equation}

and

(2.47)\begin{equation} \langle \rho_m' \boldsymbol{\nabla} \psi' \rangle \approx n \rho_f^\infty \int_{r=0}^\infty \left\langle \frac{\rho_m}{\rho_f^\infty} + 1 \right\rangle_1 \langle \boldsymbol{\nabla} \psi' \rangle_1 \,\mbox{d}\boldsymbol{r}, \end{equation}

where $\boldsymbol {r} = r \boldsymbol {e}_r$ is position relative to the cavity centre (based on a uniform probability of cavity centres). To avoid having to numerically evaluate these volume integrals, a means to express them in terms of the asymptotic coefficient measuring the far-field decay of $\langle p' \rangle _1$ is sought, as follows. Note that this will transform the averages $\langle \eta / \ell ^2 \rangle$ and $\langle \rho _{f}^\infty \rangle$ on the left-hand side of (2.45) to their bulk counterparts, $\eta / \ell _\infty ^2$ and $\rho _f^\infty$, respectively.

First, the volume integral in (2.46) for incompressible flow with radially varying permeability may be written

(2.48)\begin{equation} \langle (\eta / \ell^2)' \boldsymbol{u}' \rangle ={-} n \int_{r = 0}^\infty r \langle \boldsymbol{u}' \rangle_1 \boldsymbol{\cdot} \boldsymbol{e}_r \boldsymbol{e}_r \frac{\partial (\eta \ell^{{-}2})}{\partial r} \mbox{d}\boldsymbol{r}. \end{equation}

Next, by drawing on the Maxwell stress tensor (Russel et al. Reference Russel, Saville and Schowalter1989) for a uniform dielectric $\epsilon \epsilon _0 \boldsymbol {\nabla } \psi \boldsymbol {\nabla } \psi - \epsilon \epsilon _0 \boldsymbol {\nabla } \psi \boldsymbol {\cdot } \boldsymbol {\nabla } \psi \boldsymbol {I} / 2$ (electrical stress on the medium comprising the charged polymer and fluid) and bulk electroneutrality, it can be shown that

(2.49)\begin{equation} \langle \rho_m' \boldsymbol{\nabla} \psi' \rangle ={-}\langle \rho'_f \boldsymbol{\nabla} \psi' \rangle, \end{equation}

enabling the volume integral in (2.47) to be written

(2.50)\begin{equation} \langle \rho_m' \boldsymbol{\nabla} \psi' \rangle = n \int_{r = 0}^\infty \frac{\partial \rho_f}{\partial r} \boldsymbol{e}_r \langle \psi' \rangle_1\,\mbox{d}\boldsymbol{r}. \end{equation}

To derive a more general representation of (2.45) in terms of the asymptotic coefficients $C^X$, valid for cavities with arbitrary radial permeability and radial charge-density profiles, let us now consider the single cavity in an unbounded hydrogel. Here, we will evaluate the force $\boldsymbol {F}_1$ on the fluid within a large volume $V$ that encloses the single cavity addressed in the sections above (single-cavity problem). Accordingly, this analysis adopts the previous notation by which $(\cdot )' = (\cdot ) - (\cdot )^0$ denotes the perturbation from equilibrium (not the disturbance from an average). The force, which vanishes in the absence of fluid inertia, is

(2.51)\begin{equation} \boldsymbol{F}_1 ={-} \int_V \frac{\eta}{\ell^2} \boldsymbol{u} \,\mbox{d}V + \int_A (\boldsymbol{T}_h + \boldsymbol{T}_e) \boldsymbol{\cdot} \boldsymbol{n} \,\mbox{d}A + {\int_V \rho_f \boldsymbol{\nabla}{\psi}\,\mbox{d}V}, \end{equation}

where the integral of the electrical body force $-\rho _f \boldsymbol {\nabla }{\psi }$ on the right-hand side corrects the surface integral of the Maxwell/electrical stress tensor $\boldsymbol {T}_e$ (the total electrical force on the fluid and hydrogel). In the far field, $- (\eta / \ell _\infty ^2) \boldsymbol {U}_0 - \boldsymbol {P} - \rho _f^\infty \boldsymbol {E} = 0$, so $- (\eta / \ell _\infty ^2) \boldsymbol {u}' - \boldsymbol {\nabla } p' + \rho _f^\infty \boldsymbol {\nabla }{\psi '} \sim 0$ (neglecting $O(r^{-2})$ smaller viscous shear stresses), which may be integrated (in the radial direction), giving a far-field pressure disturbance

(2.52)\begin{align} p'(\boldsymbol{r}) ={-} \int_{r}^\infty \left(\rho_f^\infty \boldsymbol{\nabla} \psi' - \frac{\eta}{\ell_\infty^2} \boldsymbol{u}' \right) \boldsymbol{\cdot} \boldsymbol{e}_r\,\mbox{d}r = \sum_{X = P,E} \left(\rho_f^\infty D^X + \frac{\eta}{\ell_\infty^2} C^X \right) r^{{-}2} \boldsymbol{X} \boldsymbol{\cdot} \boldsymbol{e}_r. \end{align}

Now, since $\boldsymbol {F}_1 = 0$ (steady state, absence of fluid inertia), (2.51) and (2.52) furnish

(2.53)\begin{align} & - \boldsymbol{U}_0 \frac{\eta}{\ell_\infty^2} \int_{r=0}^\infty \left(\frac{\ell_\infty^2}{\ell^{2}} - 1\right) \mbox{d}V - \rho_f^\infty \boldsymbol{E} \int_{r=0}^\infty \left(\frac{\rho_f}{\rho_f^\infty} - 1\right) \mbox{d}V \nonumber\\ &\qquad + \sum_{X=P,E} \left[\int_{r=0}^\infty 2 h_r^X \boldsymbol{X} \boldsymbol{\cdot} \boldsymbol{e}_r \boldsymbol{e}_r \frac{\partial (\eta \ell^{{-}2})}{\partial r} \mbox{d}V - \int_{r=0}^\infty \hat{\psi}^X \boldsymbol{X} \boldsymbol{\cdot} \boldsymbol{e}_r \boldsymbol{e}_r \frac{\partial\rho_f}{\partial r} \mbox{d}V \right] \nonumber\\ &\quad = 4 {\rm \pi}\frac{\eta}{\ell_\infty^2} \sum_{X=P,E} C^X \boldsymbol{X}. \end{align}

The integrands of the volume integrals inside the square brackets of (2.53) are equivalent to their counterparts in (2.48) and (2.50) ($\langle \boldsymbol {u}' \rangle _1 \boldsymbol {\cdot } \boldsymbol {e}_r \equiv (\boldsymbol {u} - \boldsymbol {U}_0) \boldsymbol {\cdot } \boldsymbol {e}_r = - 2 \sum _{X=P,E} h_r^X r^{-1} \boldsymbol {X} \boldsymbol {\cdot } \boldsymbol {e}_r$ and $\langle \psi ' \rangle _1 \equiv \psi ^0 + \sum _{X=P,E} \psi ^X \boldsymbol {X} \boldsymbol {\cdot } \boldsymbol {e}_r$). Therefore, multiplying (2.53) by the cavity number density $n$, and noting

(2.54a,b)\begin{align} \left\langle \frac{\eta}{\ell_\infty} \right\rangle - n \frac{\eta}{\ell_\infty^2} \int_{r=0}^\infty \left(\frac{\ell_\infty^2}{\ell^{2}} - 1\right) \mbox{d}V = \frac{\eta}{\ell_\infty^2},\quad \langle \rho_f \rangle - n \rho_f^\infty \int_{r=0}^\infty \left(\frac{\rho_f}{\rho_f^\infty} - 1\right) \mbox{d}V =\rho_f^\infty \end{align}

transforms (2.45) to (2.39).

Equation (2.39) is the same as derived by Hill (Reference Hill2006) for uncharged hydrogels doped with charged nanoparticles. In that case, the asymptotic coefficients $C^X$ measure the force monopole ($\boldsymbol {F}^X = - 4 {\rm \pi}C^X \boldsymbol {X} \eta / \ell ^2$) for a particle that is immobilized by mechanical contact with the hydrogel. In (2.39), the term involving $C^X$ represents a body force equal to $\boldsymbol {F}^X$ times the cavity number density $n$. Of course, (2.39) may be applied to the purely hydrodynamic limit of the model, i.e. of pressure-driven flow past a spherical cavity in a Brinkman medium. To this end, Davis & Stone (Reference Davis and Stone1993) ascertained the drag force by integrating the hydrodynamic traction over the surface of porous spheres (of which a cavity is a special case). Their formula for the force on a cavity is readily verified to yield the same force as furnished by the far-field decay of the velocity disturbance (measured by $C^U$ herein) according to Hill (Reference Hill2006): $\boldsymbol {F}^U = - 4 {\rm \pi}C^U \boldsymbol {U} \eta / \ell ^2$. Explicit formulas for $C^{U}$ (albeit without electrical influences) are provided in § 3.2. Note that Davis & Stone used the force to self-consistently determine the effective permeability of packed beds of porous spheres (not a cavity filled Brinkman medium), thus implicitly averaging the fluid velocity by an averaged momentum/force balance.

2.4. Onsager relationships

The foregoing volume averaged current density may be derived from a general Onsager principle by which the macroscopic/averaged fluxes ($\boldsymbol {I}$, $\boldsymbol {U}$) are linearly related to the macroscopic/average gradients ($-\boldsymbol {E}$, $\boldsymbol {P}$) that drive these fluxes (Doi Reference Doi2013). Adopting a power-series expansion in the cavity volume fraction for all macroscopic terms, e.g. $\boldsymbol {U} = \boldsymbol {U}_0 + \phi \boldsymbol {U}_1 + \cdots$ and $\boldsymbol {I} = \boldsymbol {I}_0 + \phi \boldsymbol {I}_1 + \cdots$, the linear response may be written

(2.55) \begin{equation} \boldsymbol{U} ={-}(\ell^2 / \eta + \phi \alpha_{UP}) \boldsymbol{P} + (-\rho^\infty_f \ell^2 /\eta + \phi \alpha_{UE}) \boldsymbol{E} + O(\phi^2), \end{equation}

and

(2.56) \begin{equation} \boldsymbol{I} ={-}(-\rho^\infty_f \ell^2 / \eta + \phi \alpha_{IP}) \boldsymbol{P} + (\sigma_\infty + {\rho^\infty_f}^2 \ell^2 / \eta + \phi \alpha_{IE})\boldsymbol{E} + O(\phi^2), \end{equation}

where $\alpha _{ij}$ are (dimensional) Onsager matrix coefficients for the cavity contributions to the macroscopic fluxes. Specifically, $\phi \alpha _{UP}$ furnishes the mass flux proportional to the pressure gradient in the absence of an electric field, which is the cavity contribution to the effective hydrodynamic permeability. Similarly, $\phi \alpha _{IE}$ is the cavity contribution to the conventional electrical conductivity (when there is no pressure gradient), and $\phi \alpha _{UE} = \phi \alpha _{IP}$ is the cavity contribution to average electro-osmotic flow (in the absence of an average pressure gradient). The Onsager matrix must be symmetric to respect time-reversal symmetry of the fluctuations at equilibrium (Doi Reference Doi2013). For the present problem (e.g. with no ion-concentration or temperature gradients), this demands only that the two off-diagonal terms are equal, i.e.

(2.57)\begin{equation} \alpha_{UE} = \alpha_{IP}.\quad (\mbox{Onsager reciprocity}) \end{equation}

Note that $\phi \alpha _{IP}$ furnishes the cavity contribution to the electric current from flow driven by a pressure gradient (in the absence of an electric field).

Equating the foregoing Onsager fluxes to their averaged counterparts above furnishes

(2.58a,b)$$\begin{gather} \alpha_{UP} ={-} \frac{\ell^2}{\eta} \frac{3 C^U}{R^3},\quad \alpha_{UE} ={-} \frac{3 C^E}{R^3}, \end{gather}$$

(2.59a,b)$$\begin{gather}\alpha_{IP} = \frac{\ell^2 \sigma_\infty}{\eta}\frac{3 D_e^U}{R^3} + \rho^\infty_f \frac{\ell^2}{\eta} \frac{3 C^U}{R^3},\quad \alpha_{IE} = \sigma_\infty \frac{3D_e^E}{R^3} + \rho^\infty_f \frac{3 C^E}{R^3}, \end{gather}$$

where $\alpha _{UE} = \alpha _{IP}$. As noted above, the reciprocity relation provides a valuable consistency check on the calculations of $C^X$ and $D^X$ and $J_i^X$.

To relate the current density $\boldsymbol {I}$ to measurements, there are two pertinent experimental configurations. One is conducted with zero pressure gradient, for which the ratio

(2.60)\begin{equation} \frac{I}{E} = \sigma_{P=0} = (\sigma_\infty + {\rho^\infty_f}^2 \ell^2 / \eta) (1 + \Delta \sigma_{P =0} \phi + \cdots), \end{equation}

thus defining a (dimensionless) conductivity increment

(2.61)\begin{equation} \Delta \sigma_{P = 0} = \frac{\dfrac{\rho^\infty_f}{\sigma_\infty } \dfrac{3 C^E}{R^3} +\dfrac{3 D^E_e}{R^3}}{1 + \dfrac{{\rho^\infty_f}^2 \ell^2}{ \sigma_\infty \eta}}. \end{equation}

The other is conducted with impenetrable electrodes so that there is zero average flow, for which the ratio

(2.62)\begin{equation} \frac{I}{E} = \sigma_{U=0} = \sigma_\infty (1 + \Delta \sigma_{U = 0} \phi + \cdots), \end{equation}

thus defining a conductivity increment

(2.63)\begin{equation} \Delta \sigma_{U = 0} = \frac{3 D^E_e}{R^3} + \frac{\rho^\infty_f \ell^2}{\eta}\frac{3 D^U_e}{R^3} . \end{equation}

Note that the conductivity increment counterpart for colloidal dispersions is (Saville Reference Saville1979; O'Brien Reference O'Brien1981; Hill et al. Reference Hill, Saville and Russel2003)

(2.64)\begin{equation} \Delta \sigma = (\sigma / \sigma_\infty - 1) / \phi = \frac{3 D_e^E}{R^3} - M \frac{3 D_e^U}{R^3}, \end{equation}

where $M = V / E = - U / E$ is the particle electrophoretic mobility. Of course, for immobile particles ($M = 0$), the conductivity increment depends only on the effective electric-field-induced dipole strength, as is the case for ideal conducting solid composites. When we examine the results for cavity doped hydrogels, the conductivity increments will be compared with a greatly simplified analysis based on the conductivities of the cavity interior and exterior for electro-migration of mobile ions, as predicted by Donnan equilibrium between the hydrogel and cavity. This completely neglects electro-osmosis and the other diffuse-layer polarization effects captured by the full electrokinetic model.

2.5. Numerics

The numerical solution is challenging due to several disparate length scales, exemplified by the ratios $\kappa R$ and $R / \ell$, and the coupled diffusion, advection and electro-migration. Here, numerical stiffness is tackled using the method of Hill (Reference Hill2006) for the electrokinetics of spherical nanoparticles in hydrogels. This code, which employs finite-difference approximations and an adaptive mesh (Hill et al. Reference Hill, Saville and Russel2003), was modified to solve the equations on $0 \le r \le r_{max}$. Based on Hill et al. (Reference Hill, Saville and Russel2003), the equations are scaled using a characteristic velocity $u^* =(k_B T / e)^2 \epsilon \epsilon _o / (\eta a)$, where $a$ is the radius of a colloidal sphere. The counterpart for the present problem is $u_c = \kappa (k_B T / e) \rho ^\infty _f \ell ^2 / \eta = u^* \kappa a (\kappa \ell )^2 \rho ^\infty _f / (2 I e)$, so $a$ becomes an arbitrary/numerical parameter/length that only affects the numerical solution via its influence on the ion Péclet numbers, for example. Of course, owing to linearity, the value of $u_c$ should not affect the physical solution. This was verified to be so, and the dimensionless quantities gleaned from the flow/pressure-driven flows herein are appropriately scaled by a factor $\kappa a$ so that their resulting $O(1)$ magnitudes are independent of $a$ (thus depending on $R$).

Numerical accuracy was tested by examining Onsager reciprocity. Here, the coefficients $C^U$ and $C^E$ of the velocity disturbances were found to be susceptible to the choice of $u_c$ (or $a$), irrespective of other numerical factors, such as the spatial discretization and the maximum radial extent of the computations, $r_{max}$, which must be very large to capture the far-field asymptotic decays of all the perturbations. The ratio of the off-diagonal Onsager coefficients $\alpha _{UE} / \alpha _{IP}$ was initially found to deviate (by a ratio of the order of one) from one. This did not have a significant or even perceptible impact on the resulting conductivity increments (under conditions where electro-osmosis is weak), but the calculation of $C^U$ and $C^E$ (and therefore the Onsager coefficients) was tremendously improved by setting $u_c$ to a very large value. The resulting calculations were then readily verified to satisfy Onsager reciprocity. This also stabilized computations for large cavities ($R \gtrsim 100$ nm), lending further confidence to the numerical accuracy of the results reported herein.

Calculations with $R / \ell \lesssim 1$ and $\kappa R \lesssim 1$ exhibited notable systematic departures of $\alpha _{UE} / \alpha _{IP}$ from one. In such cases, however, the velocity disturbances, as measured by $C^U$ and $C^E$ were very weak, and so the conductivity increments were not susceptible to numerical inaccuracies in $C^U$ and $C^E$. Furthermore, because the cavities under these conditions present vanishing disturbances, the breakdown seems to reflect numerical stiffness.

Further checks of the numerical accuracy were undertaken by comparing the Onsager coefficient $\alpha _{UP}$ for weakly charged hydrogels with exact analytical formulas derived by solving the Stokes and Brinkman equations for a spherical cavity within a continuous Brinkman medium. These formulas are provided in the results section, and are compared with the numerically calculated Onsager counterpart $\alpha _{UP}$.

Note that the maximum radial extent for the computations reported herein was set to $\kappa r_{max} = 20 \kappa R + 40$ with $10\,000$ grid points for each of the $7$ functions (potential, $4$ ion concentrations, and two scalar functions that prescribe the velocity, $h_r$ and $g = h_{rrr}$). This enabled all the scalar fields to robustly establish their asymptotic power-law decay at $r = r_{max}$. Failing to do so generally produced poor mesh convergence and a notable violation of Onsager reciprocity (as revealed by $\alpha _{UE} / \alpha _{IP}$ deviating from one).

2.6. Donnan–Maxwell model

This section begins by establishing how the bulk electrolyte composition is prescribed to be electroneutral, also establishing independent analytical calculations of the Donnan potential $\psi _D$, and cavity and hydrogel conductivities $\sigma _c$ and $\sigma _\infty$. These are used to compute what will be termed a Donnan–Maxwell conductivity increment $\Delta \sigma _{DM}$, which neglects electro-osmosis and space-charge perturbations.

While Donnan equilibrium (Doi Reference Doi2013) is anticipated when the cavity radius $R$ is large compared with the Debye length $\kappa ^{-1}$, i.e. when $\kappa R \gg 1$, it will be seen that this is not sufficient for the accompanying Donnan–Maxwell conductivity to approximate the full electrokinetic model. Nevertheless, it still provides a valuable benchmark with which to interpret electro-osmosis and space-charge polarization effects emerging from the full model.

As a specific illustrative example, let us consider a four-component electrolyte, comprising H$^+$, Cl$^-$, Na$^+$ and OH$^-$ ions. Assuming, for simplicity, that the fixed charge density $\rho ^\infty _f$ is independent of the pH (i.e. a strong electrolyte), electro-neutrality in the bulk hydrogel demands

(2.65)\begin{equation} 10^{-\text{pH}} - [\mbox{Cl}^-] + [\mbox{Na}^+] - 10^{\text{pH} - 14} + \rho^\infty_f / (10^3 N_A e) = 0, \end{equation}

where $[\cdot ]$ explicitly denotes an ion concentration in $\mbox {M}=\mbox {mol}$ l$^{-1}$, and $\mbox {pH} = -\log _{10}{[\mbox {H}^+]}$ with $[\mbox {H}^+] [\mbox {OH}^-] = 10^{-14}$ M$^2$ ($N_A$ is Avogadro's number).

If the pH and $\rho ^\infty _f$ are adopted as independent variables, then $[\mbox {Na}^+]$ or $[\mbox {Cl}^-]$ may be prescribed according to the sign of

(2.66)\begin{equation} 10^{-\text{pH}} - 10^{\text{pH} - 14} + \rho^\infty_f / (10^3 N_A e), \end{equation}

thus ascertaining $[\mbox {Cl}^-]$ or $[\mbox {Na}^+]$, respectively, from the electro-neutrality condition. At $\text {pH} = 7$, for example, this identifies Na$^+$ (Cl$^-$) counter-ions for a negatively (positively) charged hydrogel.

Thus, with an electro-neutral bulk, the equilibrium electrostatic potential $\psi ^0$ decays to zero when $\kappa (r - R) \ll 1$. Inside the cavity, $\psi ^0$ increases toward (minus) the Donnan potential $\psi _D$ as $r \rightarrow 0$. In a sufficiently large cavity ($\kappa R \gg 1$), electro-neutrality requires

(2.67)\begin{equation} \sum_i^Nz_i e n_{i}^\infty \exp({z_i \psi_D e / (k_B T)}) = 0, \end{equation}

and in the bulk hydrogel,

(2.68)\begin{equation} \sum_i^Nz_i e n_{i}^\infty + \rho^\infty_f = 0. \end{equation}

Solving these furnishes the Donnan potential $\psi _D$, to which $-\psi ^0$ in a finite cavity approaches when $\kappa R \gg 1$.

For a hydrogel with a negatively charged skeleton ($\psi _D < 0$) bearing H$^+$ and Na$^+$ counter-ions and (added NaCl), the general Donnan condition above furnishes

(2.69)\begin{equation} \psi_D = \frac{k_B T}{2 e} \ln{\frac{n_{{\rm Cl}^-}^\infty + n_{{\rm OH}^-}^\infty}{n_{{\rm H}^+}^\infty + n_{{\rm Na}^+}^\infty}} = \frac{k_B T}{2 e} \ln{\frac{[\mbox{Cl}^-] + 10^{\text{pH}-14}}{10^{-\text{pH}} + [\mbox{Na}^+]}}, \end{equation}

with either (depending on the sign of $10^{-\text {pH}} - 10^{\text {pH} - 14} + {\rho ^\infty _f} / (10^3 N_A e)$)

(2.70)\begin{equation} [\mbox{Cl}^-] = [\mbox{Na}^+] + 10^{-\text{pH}} - 10^{\text{pH} - 14} + {\rho^\infty_f} / (10^3 N_A e), \end{equation}

or

(2.71)\begin{equation} [\mbox{Na}^+] = [\mbox{Cl}^-] -10^{-\text{pH}} + 10^{\text{pH} - 14} - {\rho^\infty_f} / (10^3 N_A e). \end{equation}

Note that the electrolyte conductivity in a large ($\kappa R \gg 1$) cavity is

(2.72)\begin{equation} \sigma_c = \sum_i^N(z_i e)^2 n_{i}^\infty \exp({z_i \psi_D e / (k_B T)}) \frac{D_i}{k_B T}. \end{equation}

Expressing this in terms of independent variables ${\rho ^\infty _f}$ and $[\mbox {Cl}^-]$ gives

(2.73)\begin{align} \sigma_c &= \sum_i^N (z_i e)^2 n_{i}^\infty \left(\frac{n_{{\rm Cl}^-}^\infty + n_{{\rm OH}^-}^\infty}{n_{{\rm H}^+}^\infty + n_{{\rm Na}^+}^\infty} \right)^{z_i / 2} \frac{D_i}{k_B T}, \nonumber\\ &= \frac{e^2}{k_B T}\left(n_{{\rm H}^+}^\infty D_{{\rm H}^+} + n_{{\rm Na}^+}^\infty D_{{\rm Na}^+}\right) \left( \frac{n_{{\rm Cl}^-}^\infty + n_{{\rm OH}^-}^\infty}{n_{{\rm H}^+}^\infty + n_{{\rm Na}^+}^\infty} \right)^{1 / 2} \nonumber\\ &\quad + \frac{e^2}{k_B T} \left(n_{{\rm Cl}^-}^\infty D_{{\rm Cl}^-}+ n_{{\rm OH}^-}^\infty D_{{\rm OH}^-}\right) \left(\frac{n_{{\rm H}^+}^\infty + n_{{\rm Na}^+}^\infty}{n_{{\rm Cl}^-}^\infty + n_{{\rm OH}^-}^\infty} \right)^{1 / 2}, \end{align}

or

(2.74)\begin{align} \frac{\sigma_c k_B T}{10^3 N_A e^2} &= \left([\mbox{Cl}^-] D_{{\rm Cl}^-} + 10^{\text{pH} - 14} D_{{\rm OH}^-}\right) \left( \frac{ [\mbox{Cl}^-] + 10^{\text{pH} - 14} - \frac{{\rho^\infty_f}}{10^3 N_A e}}{[\mbox{Cl}^-] + 10^{\text{pH} - 14}} \right)^{1 / 2} \nonumber\\ &\quad + \left(10^{-\text{pH}} D_{{\rm H}^+} + ([\mbox{Cl}^-] -10^{-\text{pH}} + 10^{\text{pH} - 14} -\frac{{\rho^\infty_f}}{10^3 N_A e}) D_{{\rm Na}^+}\right) \nonumber\\ &\quad \times \left( \frac{[\mbox{Cl}^-] + 10^{\text{pH} - 14}}{ [\mbox{Cl}^-] + 10^{\text{pH} - 14} - \frac{{\rho^\infty_f}}{10^3 N_A e}} \right)^{1 / 2}. \end{align}

For example, when $\mbox {pH} = 7$ with $- \rho ^\infty _f / e$ and $n_{{\rm Cl}^-}^\infty \gg 10^{-7}$ M, (2.73) reduces to

(2.75)\begin{equation} \sigma_c \approx \frac{n_{{\rm Cl}^-}^\infty e^2}{k_B T} (D_{{\rm Cl}^-} + D_{{\rm Na}^+}) \left(1 - \frac{{\rho^\infty_f}}{n_{{\rm Cl}^-}^\infty e} \right)^{1 / 2}, \end{equation}

so when there is an excess of added salt (NaCl) such that $- \rho ^\infty _f \ll n_{{\rm Cl}^-}^\infty e$, the conductivity is that of a NaCl electrolyte. On the other hand, when $- \rho ^\infty _f \gg n_{{\rm Cl}^-}^\infty e$ (counter-ion dominated hydrogel), $\sigma _c\sim [\mbox {Cl}^-]^{1/2}$, i.e.

(2.76)\begin{equation} \sigma_c \approx \frac{e^2}{k_B T} (D_{{\rm Cl}^-} + D_{{\rm Na}^+}) \left(- n_{{\rm Cl}^-}^\infty {\rho^\infty_f} / e \right)^{1 / 2}.\quad (- \rho^\infty_f \gg n_{{\rm Cl}^-}^\infty e) \end{equation}

The foregoing Donnan equilibrium provides a basis for interpreting the conductivity of cavity doped hydrogels, albeit limited to advection-free conduction in large, space-charge-free cavities. The conductivity of the hydrogel accompanying (2.75) is

(2.77)\begin{equation} \sigma_\infty \approx \frac{n_{{\rm Cl}^-}^\infty e^2}{k_B T} \left[D_{{\rm Cl}^-} + \left(1 - \frac{{\rho^\infty_f}}{n_{{\rm Cl}^-}^\infty e} \right) D_{{\rm Na}^+} \right].\end{equation}

Thus, the classical Maxwell analysis (Russel et al. Reference Russel, Saville and Schowalter1989) for a spherical inclusion with conductivity $\sigma _c$ embedded in an unbounded continuum with conductivity $\sigma _\infty$ furnishes a Donnan–Maxwell electrostatic dipole strength

(2.78)\begin{equation} \frac{D_{DM}}{R^3} = \frac{\sigma_c / \sigma_\infty - 1}{\sigma_c / \sigma_\infty + 2}, \end{equation}

and accompanying conductivity increment

(2.79)\begin{equation} \Delta \sigma_{DM} \equiv (\sigma / \sigma_\infty - 1) / \phi = \frac{3 D_{DM}}{R^3}, \end{equation}

where

(2.80)\begin{equation} \sigma_c / \sigma_\infty = \frac{(D_{{\rm Cl}^-}/D_{{\rm Na}^+} + 1) \left(1 - \dfrac{{\rho^\infty_f}}{n_{{\rm Cl}^-}^\infty e} \right)^{1 / 2}}{D_{{\rm Cl}^-} / D_{{\rm Na}^+} + 1 - \dfrac{{\rho^\infty_f}}{n_{{\rm Cl}^-}^\infty e}}. \end{equation}

Interestingly, for a symmetrical electrolyte, such as KCl, $\sigma _c / \sigma _\infty$ would span the range $1$–$0$ as $-{\rho ^\infty _f} / (n_{{\rm Cl}^-}^\infty e) > 0$ increases from $0$ to $\infty$, so $\Delta \sigma _{DM}$ spans the range $0$ to $-3/2$. For NaCl, for which $D_{{\rm Cl}^-}/D_{{\rm Na}^+} \approx 1.5$, however, $\sigma _c / \sigma _\infty$ may be $> 1$ when $-{\rho ^\infty _f} / (n_{{\rm Cl}^-}^\infty e) \ll 1$, thus furnishing a positive increment with sufficient added salt.

As will be shown below, the dipole strength furnished by the full electrokinetic model often deviates considerably from (2.78). This reflects additional ion conduction mechanisms, namely electro-osmotic advection and, perhaps, ion diffusion at finite values of $\kappa R$. Note that the foregoing Maxwell analysis is based on the absence of a space charge (the charge is confined to the interface at $r = R$). Thus, for example, it furnishes a perfectly uniform electric field inside the cavity, whereas the full electrokinetic model often reveals a non-zero space charge inside and outside the cavity.