2.1. Boundary conditions for a drop

In the boundary conditions adopted for a drop, the interface has no electrical capacitance and no monopole surface charge. It is also impervious or impenetrable to ions, and is therefore referred to as ideally polarisable. This gives the interfacial boundary conditions

(2.8)\begin{equation} [\phi]=0, \quad [\boldsymbol{D}\boldsymbol{\cdot}\boldsymbol{n}]=0, \quad \mbox{and} \quad \boldsymbol{j}^{({\pm})} \boldsymbol{\cdot}\boldsymbol{n}=0 \quad \mbox{on} \ S , \end{equation}

respectively. Here $S$ denotes the sharp interface between the two phases, with outward unit normal $\boldsymbol {n}$, and $[\,\cdot\, ]\equiv [\,\cdot\, ]^{\partial \varOmega _{ex}}_{\partial \varOmega _{in}}$ denotes the jump across $S$, with the convention that it is the limit as $S$ is approached from the exterior domain $\varOmega _{ex}$ minus the limit as $S$ is approached from the interior domain $\varOmega _{in}$.

Continuity of velocity of the solvent and the kinematic boundary condition imply that

(2.9)\begin{equation} [\boldsymbol{u}]=0 \quad \mbox{and} \quad \left(\boldsymbol{u}-\frac{\partial\boldsymbol{x}_{s}}{\partial t}\right)\boldsymbol{\cdot}\boldsymbol{n}=0 \quad \mbox{on} \ S, \end{equation}

where $\boldsymbol {x}_{s}$ is any point on $S$. The stress-balance boundary condition is that

(2.10)\begin{equation} [(\boldsymbol{\mathsf{T}}_{H} + \boldsymbol{\mathsf{T}}_{M})\boldsymbol{\cdot}\boldsymbol{n}] = \sigma(\kappa_{1}+\kappa_{2})\boldsymbol{n} \quad \mbox{on} \ S,\end{equation}

where

(2.11a,b)\begin{equation} \boldsymbol{\mathsf{T}}_{H}={-}p\boldsymbol{\mathsf{I}}+2\mu\boldsymbol{\mathsf{e}} \quad \mbox{and} \quad \boldsymbol{\mathsf{T}}_{M}= \varepsilon\big(\boldsymbol{\nabla} \phi \otimes \boldsymbol{\nabla} \phi-\tfrac{1}{2}|\boldsymbol{\nabla}\phi|^{2}\boldsymbol{\mathsf{I}}\big) . \end{equation}

Here $\boldsymbol{\mathsf{T}}_{H}$ is the stress tensor for a Newtonian fluid with viscosity $\mu$ and $(\boldsymbol {e})_{ij}=(\partial _{x_{j}}u_{i}+\partial _{x_{i}}u_{j})/2$ is the symmetric part of the velocity gradient. We use $\boldsymbol{\mathsf{T}}_{M}$ to denote the part of the Maxwell stress tensor due to the electric field, which has been written in terms of the potential $\phi$, $\sigma$ is a constant surface tension and $\kappa _{1}$ and $\kappa _{2}$ are the principal curvatures of $S$. With $\boldsymbol {n}$ pointing outward on $S$, the principal curvatures are taken to be positive when the curve of intersection of $S$ by a plane containing $\boldsymbol {n}$ and a principal direction is convex on the side to which $\boldsymbol {n}$ points, and is negative otherwise.

Under equilibrium conditions there is typically a difference in the electric potential between distinct phases that occurs at and near the interface between them. It is caused by a difference in the affinity for charge carriers of the phases and is referred to as an inner or Galvani potential (Hunter Reference Hunter2001). The boundary conditions

(2.12)\begin{equation} c^{(+)}|_{\partial\varOmega_{in}} = l^{(+)} c^{(+)}|_{\partial\varOmega_{ex}} \quad \mbox{and} \quad c^{(-)}|_{\partial\varOmega_{in}} = l^{(-)} c^{(-)}|_{\partial\varOmega_{ex}} \quad \mbox{on} \ S , \end{equation}

where $l^{(+)}$ and $l^{(-)}$ are constant partition coefficients, imply the presence of a non-zero Galvani potential when $l^{(+)}\neq l^{(-)}$. The notation $c^{(\pm )}|_{\partial \varOmega _{in,ex}}$ denotes evaluation of the ion concentration $c^{(\pm )}$ in the limit as $S$ is approached from the interior domain $\varOmega _{in}$ or exterior domain $\varOmega _{ex}$. Boundary conditions of this type have been considered by, for example, Zholkovskij, Masliyah & Czarnecki (Reference Zholkovskij, Masliyah and Czarnecki2002) and by Mori & Young (Reference Mori and Young2018).

2.2. Boundary conditions for a cell or vesicle

Mori, Liu & Eisenberg (Reference Mori, Liu and Eisenberg2011) introduce a set of phenomenological boundary conditions for the behaviour of a cell or vesicle, where the interior and exterior phases are separated by a membrane, and these are summarised here. Similar boundary conditions for a membrane have been considered by Lacoste et al. (Reference Lacoste, Menon, Bazant and Joanny2009). The membrane is treated as a sharp interface $S$ between the phases that nonetheless has electromechanical structure. Its initial equilibrium reference configuration, denoted by $S_{ref}$, has a Lagrangian or material point coordinate system $\boldsymbol {\theta }$, and at later times its position is described by $\boldsymbol {x}=\boldsymbol {X}(\boldsymbol {\theta }, t)$.

The same notation for interfacial quantities is used for both a drop and a vesicle. Here we have: $[\,\cdot\, ]$ for the jump in a quantity across the membrane, $\cdot |_{\partial \varOmega _{in,ex}}$ for evaluation as the side $\partial \varOmega _{in,ex}$ of the membrane is approached or simply on $S$ for a quantity that is continuous.

If the membrane is semi-permeable or porous to the aqueous solvent it allows a flow of solvent in the normal direction that occurs by osmosis and is assumed to be proportional to the jump in the solvent's partial pressure across the membrane. Hence,

(2.13)\begin{equation} \boldsymbol{u}-\left.\frac{\partial\boldsymbol{X}}{\partial t}\right|_{\boldsymbol{\theta}} ={-} {\rm \pi}_{m} \, [ p- k_{B}T (c^{(+)}+c^{(-)}) ] \, \boldsymbol{n} \quad \mbox{on} \ S , \end{equation}

where ${\rm \pi} _{m} \geqslant 0$ is an effective membrane porosity and the partial pressure of the ions is given by the ideal gas law.

The trans-membrane flux for each ion species is taken to be proportional to the jump in its electrochemical potential across the membrane surface. Mori et al. (Reference Mori, Liu and Eisenberg2011) give the boundary condition

(2.14)\begin{equation} c^{({\pm})} \left(\boldsymbol{v}^{({\pm})} - \left.\frac{\partial\boldsymbol{X}}{\partial t}\right|_{\boldsymbol{\theta}}\right) \boldsymbol{\cdot} \boldsymbol{n} ={-} g^{({\pm})} [ k_{B}T \ln c^{({\pm})} \pm e \phi ] \quad \mbox{on} \ S , \end{equation}

where $g^{(\pm )}\geqslant 0$ is a gating constant for each ion species, and the velocity on the left-hand side, $\boldsymbol {v}^{(\pm )}-\partial _{t}\boldsymbol {X}|_{\boldsymbol {\theta }}$, is the velocity of ions relative to the membrane material.

In the analysis that follows, a boundary condition is needed for the normal component of the molecular ion flux $\boldsymbol {j}^{(\pm )}\boldsymbol {\cdot }\boldsymbol {n}$ in the electrolyte solution immediately adjacent to the membrane, where the flux $\boldsymbol {j}^{(\pm )}$ is given by (2.3), i.e. $\boldsymbol {j}^{(\pm )}=c^{(\pm )}(\boldsymbol {v}^{(\pm )} - \boldsymbol {u})$. If the normal ion flux $\boldsymbol {j}^{(\pm )}\boldsymbol {\cdot }\boldsymbol {n}$ in the electrolyte and the normal ion flux across the membrane given by (2.14) are not equal, then charge is not conserved at the membrane boundaries $\partial \varOmega _{in,ex}$ but accumulates there. We note that this occurs if both (2.13) and (2.14) are applied to a membrane that is semi-permeable to solvent, i.e. when ${\rm \pi} _{m} > 0$. This is verified by subtracting (2.13) multiplied by the ion concentration $c^{(\pm )}$ from (2.14) to form the electrolyte ion flux $\boldsymbol {j}^{(\pm )}\boldsymbol {\cdot }\boldsymbol {n}$, and noting that the concentration of each ion species is not continuous but has a jump across the membrane.

This difficulty is easily resolved by modifying the trans-membrane flux boundary condition (2.14) to read

(2.15)\begin{equation} c^{({\pm})} (\boldsymbol{v}^{({\pm})} - \boldsymbol{u}) \boldsymbol{\cdot} \boldsymbol{n} ={-} g^{({\pm})} [ k_{B}T \ln c^{({\pm})} \pm e \phi ] \quad \mbox{on} \ S , \end{equation}

which conserves charge across the membrane and its boundaries and holds in general, that is, for all ${\rm \pi} _{m} \geqslant 0$. When the membrane is impermeable to solvent (i.e. ${\rm \pi} _{m}=0$) the boundary condition (2.13) reduces to no-slip between the electrolyte and membrane surface, and (2.14) and (2.15) are identical, but when the membrane is semi-permeable to solvent (i.e. ${\rm \pi} _{m} > 0$), the solvent and ion species cross the membrane at different rates via distinct pores or channels, and (2.15) must be applied instead.

The membrane has electrical capacitance because it can maintain a jump in the potential across its outer faces, between which the potential varies linearly with normal distance. The jump in the potential is referred to as the trans-membrane potential and is denoted by $[\phi ]$. The membrane has no net monopole charge, in the sense that at distinct points along the normal the surface charge densities on opposite faces have equal magnitude and opposite polarity, cancelling each other. The charge on each face is related to the trans-membrane potential by the membrane capacitance per unit area $C_{m}$. Analogous to the first two boundary conditions of (2.8) for a drop, but modified to express continuity of electric displacement and the capacitance of the membrane, we have the vesicle boundary conditions

(2.16)\begin{equation} \varepsilon \boldsymbol{\nabla} \phi\boldsymbol{\cdot}\boldsymbol{n}|_{\partial\varOmega_{in}} = \varepsilon \boldsymbol{\nabla} \phi\boldsymbol{\cdot}\boldsymbol{n}|_{\partial\varOmega_{ex}} = C_{m} [\phi] \quad \mbox{where} \ C_{m} =\frac{\varepsilon_{m}}{d} . \end{equation}

Here $\varepsilon _{m}$ is the membrane permittivity and $d$ is its thickness.

Two specific membrane capacitance models of Mori et al. (Reference Mori, Liu and Eisenberg2011) are

(2.17) \begin{equation} C_{m} = \left\{\begin{array}{@{}ll} C_{m}^{0} & (a) , \\ C_{m}^{0} \delta A (\boldsymbol{X}) & (b) . \end{array} \right. \end{equation}

These hold if, for example, the bulk material of the membrane is incompressible, and then (2.17a) applies when the membrane surface is locally inextensible and (2.17b) applies when it is locally extensible. The capacitance per unit area in the initial undeformed or reference state is $C_{m}^{0}$, which is constant for a homogeneous membrane. The ratio of the local surface area in the deformed state to its value in the initial reference state following a material point $\boldsymbol {x}=\boldsymbol {X}(\boldsymbol {\theta }, t)$ is denoted by $\delta A (\boldsymbol {X})$. Bulk incompressibility implies that the volume of a membrane material element ${d}(\boldsymbol {X}) \delta A (\boldsymbol {X})$ is conserved during deformation, so that for the locally inextensible surface of (2.17a) $\delta A (\boldsymbol {X})=1$ with ${d}(\boldsymbol {X})$ and $C_{m}^{0}$ conserved, whereas for (2.17b) ${d}(\boldsymbol {X})^{-1} \propto \delta A (\boldsymbol {X})$.

The stress-balance boundary condition is

(2.18)\begin{align} [(\boldsymbol{\mathsf{T}}_{H} + \boldsymbol{\mathsf{T}}_{M})\boldsymbol{\cdot}\boldsymbol{n}] = 2\sigma H \boldsymbol{n} - \boldsymbol{\nabla}_{s} \sigma - \kappa_{b} \left( (2H-C_{0})\left(H^{2}-K+\frac{C_{0}H}{2}\right) + {\nabla}_{s}^{2} H \right) \boldsymbol{n} \end{align}

on $S$. The vesicle membrane has a bending stiffness that is expressed in terms of the bending modulus $\kappa _{b}$, the mean curvature $H=(\kappa _{1}+\kappa _{2})/2$, the Gaussian curvature $K=\kappa _{1}\kappa _{2}$ and a spontaneous curvature $C_{0}$. The spontaneous curvature $C_{0}$ is twice the mean curvature of the membrane material in a traction-free equilibrium state, see, for example, Vlahovska, Podgorski & Misbah (Reference Vlahovska, Podgorski and Misbah2009) and Seifert (Reference Seifert1999), and it is usually set to zero. Here, $\boldsymbol {\nabla }_{s}$ denotes the surface gradient operator, ${\nabla }_{s}^{2}$ is the Laplace–Beltrami or surface Laplacian operator and the convention for the sign of the principal curvatures $\kappa _{1}$ and $\kappa _{2}$ is the same as that in (2.10).

The tension in the membrane surface is denoted by $\sigma$ in (2.18), but, in contrast to its appearance in (2.10) for a drop, the value of $\sigma$ is not a given constant. For a locally inextensible membrane $\sigma$ is determined by imposing inextensibility as a constraint, namely

(2.19)\begin{equation} \boldsymbol{\nabla}_{s}\boldsymbol{\cdot}\left( \frac{\partial}{\partial t} \boldsymbol{X}_{s}(\boldsymbol{\theta}, t)\right) =0 , \end{equation}

or, equivalently, $\delta A (\boldsymbol {X})=1$, for all $t>0$. Here $\partial _{t}\boldsymbol {X}_{s}$ denotes the tangential projection of the velocity $\partial _{t}\boldsymbol {X}$ of a material particle onto $S$. For a locally extensible membrane a stress–strain equation of state must be introduced, see, for example, Barthès-Biesel (Reference Barthès-Biesel2011). The membrane tension is initially constant, but during deformation it can vary from point to point on $S$, leading to the term $\boldsymbol {\nabla }_{s} \sigma$ in (2.18).

We note that Mori et al. (Reference Mori, Liu and Eisenberg2011) considered a membrane bending stress given by the variational derivative of a general energy functional. The specific choice of the functional due to Helfrich (Reference Helfrich1973) has been adopted in many studies of cell and vesicle deformation, and leads to the bending stress term on the right-hand side of (2.18). Barthès-Biesel (Reference Barthès-Biesel2011) gives a concise description of the different mechanical properties of vesicle, cell and capsule membranes.

2.3. The far-field and initial conditions

The electric field that induces flow and deformation is applied at time $t=0$. It can be spatially uniform and constant, or more general, so that

(2.20) \begin{equation} \boldsymbol{E}\rightarrow \left\{ \begin{array}{@{}ll} E_{\infty}\boldsymbol{e}_{z} & \mbox{for a uniform field} \\ -\boldsymbol{\nabla} \phi_{\infty} & \mbox{in general} \end{array} \right.\quad \mbox{as} \ |\boldsymbol{x}| \rightarrow \infty \ \mbox{for} \ t > 0, \end{equation}

where $\phi _{\infty }$ is a solution of Laplace's equation. In addition,

(2.21)\begin{equation} \boldsymbol{u}(\boldsymbol{x}, t) \rightarrow 0 \quad \mbox{and} \quad c^{({\pm})}(\boldsymbol{x}, t) \rightarrow \ \mbox{const. as}\ |\boldsymbol{x}|\rightarrow \infty \ \mbox{for} \ t > 0 . \end{equation}

A drop is assumed to have relaxed to a spherical shape for $t<0$, with no applied field and equilibrium initial conditions given by

(2.22a,b)\begin{equation} \boldsymbol{u}(\boldsymbol{x}, 0)=0 \quad \mbox{and} \quad c^{({\pm})}(\boldsymbol{x}, 0)= \mbox{piecewise constant for all}\ \boldsymbol{x} . \end{equation}

However, when there is a non-zero Galvani potential difference between the two phases the $t<0$ equilibrium relations for the potential and ion concentrations are revised to accommodate an electrical double layer; this is considered later, in § 5.3.

2.4. Non-dimensionalisation

Quantities used to put the equations in non-dimensional form are listed in table 1. The length scale $a_{*}$ is the initial drop radius, or a similar length scale for a vesicle. The velocity scale $U_{*c}=D_{*}/\lambda _{*}$ is the ion-diffusive scale, where $D_{*}$ is a representative ion diffusivity and $\lambda _{*}$ is the Debye layer screening thickness. The time scale $\tau _{*c}=a_{*}/U_{*c}$ is the scale on which charge in the Debye layers changes in response to changes in the applied electric field, i.e. the charge-up time scale. The scale for the pressure $p$ is the scale of the Stokes flow regime, and the scale for the potential is the thermal voltage $k_{B}T/e$, where $k_{B}$ is the Boltzmann constant and $T$ is the temperature.

Table 1.

Variables are made non-dimensional by the quantities shown.

All material and rate parameters in the two phases are made non-dimensional by a single representative value. For example, the ion concentrations are made non-dimensional by a representative ambient ion concentration $c_{*}$, and the salt concentration is made non-dimensional by $s_{*}$. Similarly, $D^{(\pm )}, k_{d}, k_{a}, \varepsilon$, $\sigma$ and $\mu$ are made non-dimensional by $D_{*}, k_{d*}, k_{a*}, \varepsilon _{*}$, $\sigma _{*}$ and $\mu _{*}$, respectively. To observe equilibrium between the ion and salt concentrations, the relation

(2.23)\begin{equation} k_{d*}s_{*}=k_{a*}c_{*}^{2} \end{equation}

holds. The Debye layer thickness $\lambda _{*}$ is given by

(2.24)\begin{equation} \lambda_{*}^{2}=\frac{\varepsilon_{*}k_{B}T}{2c_{*}e^{2}} . \end{equation}

Dimensionless groups that appear are

(2.25a)$$\begin{gather} \tau=\frac{k_{d*}a_{*}}{U_{*c}} \gg 1, \quad \alpha= \frac{c_{*}}{s_{*}} \gg 1, \quad \epsilon=\frac{\lambda_{*}}{a_{*}} \ll 1, \end{gather}$$

(2.25b)$$\begin{gather}\varDelta=\frac{\sigma_{*}}{\mu_{*}U_{*c}}, \quad \nu=\frac{\varepsilon_{*}(k_{B}T/e)^{2}}{\mu_{*}D_{*}} \quad \mbox{and} \quad \varPsi = \frac{e\varPhi_{*\infty}}{k_{B}T} . \end{gather}$$

The role of $\tau$ and $\alpha$ is considered in the following, in § 3. Except for this, the analysis is carried out in the limit of small Debye layer thickness, $\epsilon \rightarrow 0$, with $\varDelta$, $\nu$ and $\varPsi$ all of order $O(1)$. The quantity $\varDelta$ is the ratio of the capillary velocity $\sigma _{*}/\mu _{*}$ to the ion-diffusive velocity $U_{*c}$. The group $\nu \epsilon$ is the ratio of electrostatic stress to viscous stress, and is sometimes referred to either as the inverse of a Mason number or as an electrical Hartmann number. If the Helmholtz–Smoluchowski slip velocity $U_{*HS}$ is based on the thermal voltage, then $U_{*HS}=\varepsilon _{*}(k_{B}T/e)^{2}/\mu _{*}a_{*}$, and the group $\nu \epsilon = U_{*HS}/U_{*c}$ is also the ratio of the two velocity scales. The parameters $\epsilon$, $\varDelta$ and $\nu$ are all fixed for a given choice of electrolytes and initial inclusion size. In contrast, the parameter $\varPsi$ is the ratio of the change in potential difference $\varPhi _{*\infty } = E_{*\infty }a_{*}$ on the inclusion length scale to the thermal voltage, and is a control parameter.

In the scaling regime of this study, the Péclet number $Pe=U_{*c}a_{*}/D_{*}$ does not appear independently, because the velocity scale for $U_{*c}$ implies that $Pe=\epsilon ^{-1}$.

The same variable and parameter names are now used to express the governing equations and boundary conditions in non-dimensional form. The molecular ion flux $\boldsymbol {j}^{(\pm )}=c^{(\pm )}(\boldsymbol {v}^{(\pm )} - \boldsymbol {u})$ of (2.3), recast via (2.1b) and the relation $D^{(\pm )}=b^{(\pm )}k_{B}T$, is

(2.26)\begin{equation} \boldsymbol{j}^{({\pm})} ={-} \epsilon D^{({\pm})}( \boldsymbol{\nabla} c^{({\pm})} \pm c^{({\pm})}\boldsymbol{\nabla} \phi ) , \end{equation}

in non-dimensional form. Conservation of ions (2.1) and salt (2.2), together with the expression (2.4) for the ion kinetic rate term $\mathcal {R}$ and the incompressibility condition of (2.7a,b) give

(2.27)$$\begin{gather} (\partial_{t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla})c^{({\pm})}-\epsilon D^{({\pm})}\boldsymbol{\nabla}\boldsymbol{\cdot} ( \boldsymbol{\nabla} c^{({\pm})}\pm c^{({\pm})}\boldsymbol{\nabla} \phi) = \tau \alpha^{{-}1} (k_{d}s-k_{a}c^{(+)}c^{(-)}) , \end{gather}$$

(2.28)$$\begin{gather}(\partial_{t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla})s-\epsilon D_{s}\nabla^{2} s ={-} \tau (k_{d}s-k_{a}c^{(+)}c^{(-)}) . \end{gather}$$

The Poisson equation (2.6) is

(2.29)\begin{equation} \epsilon^{2}\nabla^{2}\phi={-}\frac{q}{\varepsilon} , \quad \mbox{where}\ q = \frac{c^{(+)}-c^{(-)}}{2} , \end{equation}

so that $q$ is a normalised charge density. The continuity and Stokes equation are

(2.30a,b)\begin{equation} \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}=0\quad \mbox{and} \quad -\boldsymbol{\nabla} p + \mu {\nabla}^{2}\boldsymbol{u} + \varepsilon \nu \epsilon \nabla^{2}\phi\boldsymbol{\nabla}\phi = 0 , \end{equation}

where the body force is written in terms of the potential via (2.5a–d).

2.4.1 Drop boundary conditions

In non-dimensional form, from (2.5a–d) and (2.26), the interfacial boundary conditions (2.8) on $S$ become

(2.31a,b)\begin{equation} [\phi]=0, \quad [\varepsilon \boldsymbol{\nabla} \phi\boldsymbol{\cdot}\boldsymbol{n}] = 0 , \end{equation}

and

(2.32)\begin{equation} \boldsymbol{j}^{({\pm})} \boldsymbol{\cdot}\boldsymbol{n}=0 , \quad \mbox{or equivalently} \ ( \boldsymbol{\nabla} c^{({\pm})}\pm c^{({\pm})}\boldsymbol{\nabla} \phi)\boldsymbol{\cdot} \boldsymbol{n}=0 . \end{equation}

Continuity of velocity and the kinematic condition are formally unchanged, namely

(2.33)\begin{equation} [\boldsymbol{u}]=0 \quad \mbox{and} \quad \left(\boldsymbol{u} - \frac{\partial\boldsymbol{x}_{s}}{\partial t}\right)\boldsymbol{\cdot}\boldsymbol{n}=0 \quad \mbox{on} \ S . \end{equation}

The stress-balance boundary condition becomes

(2.34)\begin{equation} [(\boldsymbol{\mathsf{T}}_{H} + \nu\epsilon\boldsymbol{\mathsf{T}}_{M})\boldsymbol{\cdot}\boldsymbol{n}] = \varDelta(\kappa_{1}+\kappa_{2})\boldsymbol{n} \quad \mbox{on} \ S,\end{equation}

where

(2.35a,b)\begin{equation} \boldsymbol{\mathsf{T}}_{H}={-}p\boldsymbol{\mathsf{I}}+2 \mu \boldsymbol{\mathsf{e}} \quad \mbox{and} \quad \boldsymbol{\mathsf{T}}_{M}= \varepsilon \big(\boldsymbol{\nabla} \phi \otimes \boldsymbol{\nabla} \phi-\tfrac{1}{2}|\boldsymbol{\nabla}\phi|^{2}\boldsymbol{\mathsf{I}}\big) . \end{equation}

The boundary conditions (2.12) for a Galvani potential (when $l^{(+)}\neq l^{(-)}$) are formally unchanged, namely

(2.36)\begin{equation} c^{(+)}|_{\partial\varOmega_{in}} = l^{(+)} c^{(+)}|_{\partial\varOmega_{ex}} \quad \mbox{and} \quad c^{(-)}|_{\partial\varOmega_{in}} = l^{(-)} c^{(-)}|_{\partial\varOmega_{ex}} \quad \mbox{on} \ S . \end{equation}

2.4.2 Vesicle boundary conditions

In non-dimensional form, the trans-membrane ion flux $\boldsymbol {j}^{(\pm )} \boldsymbol {\cdot }\boldsymbol {n}$ is given by

(2.37)\begin{equation} c^{({\pm})} (\boldsymbol{v}^{({\pm})}-\boldsymbol{u}) \boldsymbol{\cdot} \boldsymbol{n} ={-} \epsilon g^{({\pm})} [ \ln c^{({\pm})} \pm \phi ] \quad \mbox{on} \ S , \end{equation}

where the gating constant $g^{(\pm )}$ of (2.15) has been made non-dimensional and scaled by $g_{*}={c_{*}D_{*}}/{k_{B}Ta_{*}}$. The flow of solvent across the membrane is given by

(2.38)\begin{equation} \boldsymbol{u} - \left.\frac{\partial\boldsymbol{X}}{\partial t}\right|_{\boldsymbol{\theta}} ={-} \epsilon {\rm \pi}_{m} \left[ \frac{2\epsilon}{\nu} p - (c^{(+)}+c^{(-)}) \right] \boldsymbol{n} \quad \mbox{on} \ S , \end{equation}

where the effective porosity ${\rm \pi} _{m}$ of (2.13) has been made non-dimensional and scaled by ${\rm \pi} _{*}={D_{*}}/{k_{B}Tc_{*}a_{*}}$. These relations contrast with the boundary conditions (2.32) and (2.33) for an ideally polarisable drop with a sharp impervious interface; the cell has a small $O(\epsilon )$ flux of ions and solvent across the membrane when it is semi-permeable to these species.

The relations (2.16) that express continuity of electric displacement across the membrane become

(2.39)\begin{equation} \varepsilon \epsilon \boldsymbol{\nabla} \phi\boldsymbol{\cdot}\boldsymbol{n}|_{\partial\varOmega_{in}} = \varepsilon \epsilon \boldsymbol{\nabla} \phi\boldsymbol{\cdot}\boldsymbol{n}|_{\partial \varOmega_{ex}} = C_{m} [\phi] . \end{equation}

As a lipid bilayer or similar biomembrane has a thickness of the order of $5$–$8$ nm, which is the same order of magnitude as the thickness of the neighbouring Debye layers, the membrane thickness has been made non-dimensional by $\lambda _{*}$ and the membrane capacitance per unit area, $C_{m}$ or $C_{m}^{0}$ of (2.17), has been made non-dimensional by $\varepsilon _{*}/\lambda _{*}$. This accounts for the $\epsilon$-scaling in (2.39).

The stress-balance boundary condition is now

(2.40)\begin{equation} [(\boldsymbol{\mathsf{T}}_{H} + \nu \epsilon \boldsymbol{\mathsf{T}}_{M})\boldsymbol{\cdot}\boldsymbol{n}] = 2 \sigma H \boldsymbol{n} - \boldsymbol{\nabla}_{s} \sigma - \kappa_{b} (2H(H^{2}-K) + {\nabla}_{s}^{2} H ) \boldsymbol{n} , \end{equation}

where the tension in the membrane and the bending modulus have been made non-dimensional by $\mu _{*}U_{*c}$ and $\mu _{*}U_{*c}a_{*}^{2}$ respectively, and the spontaneous curvature $C_{0}$ has been set to zero. The membrane inextensibility constraint is formally unchanged, namely

(2.41)\begin{equation} \boldsymbol{\nabla}_{s}\cdot\left( \frac{\partial}{\partial t}\boldsymbol{X}_{s}(\boldsymbol{\theta}, t) \right) =0 . \end{equation}

2.4.3 Initial and far-field conditions

The initial conditions are

(2.42)\begin{equation} \phi(\boldsymbol{x}, 0^{+}) ={-} \varPsi z, \quad \boldsymbol{u}(\boldsymbol{x}, 0) = 0 \quad \mbox{and} \quad c^{({\pm})}(\boldsymbol{x}, 0) = c_{in,ex} \quad \mbox{on} \ \varOmega_{in,ex} , \end{equation}

except that the first and third relations are revised for a drop when it has a non-zero Galvani potential, as considered in § 5.3. Here $c_{in}$ and $c_{ex}$ are the initial dimensionless ion concentrations on $\varOmega _{in}$ and $\varOmega _{ex}$, respectively. A drop is initially spherical with radius 1, whereas a vesicle has a known initial equilibrium configuration $S_{ref}$. Far from the inclusion, for a uniform applied field

(2.43)\begin{equation} \phi(\boldsymbol{x}, t) \sim{-} \varPsi z, \quad \boldsymbol{u}(\boldsymbol{x}, t) \rightarrow 0, \quad \mbox{and} \quad c^{({\pm})}(\boldsymbol{x}, t)\rightarrow c_{ex} \ \mbox{as} \ |\boldsymbol{x}|\rightarrow \infty \ \mbox{for} \ t > 0 . \end{equation}

For a more general applied field $\phi$ approaches a specified solution $\phi _{\infty }$ of Laplace's equation that has magnitude $\varPsi$.

5.1. The intrinsic coordinate system

To resolve the dynamics of the Debye layers, we introduce a local surface-fitted or intrinsic orthogonal curvilinear coordinate system. This has tangential coordinates $\xi _{1}$ and $\xi _{2}$ that are aligned with the principal directions on $S$ and normal coordinate $n$, with $n=0$ on $S$ and $n>0$ in $\varOmega _{ex}$. The origins of the Eulerian and intrinsic coordinate systems are $O$ and $O^{\prime }$, respectively, and the position vector $\boldsymbol {x}$ of a point $P$ in space relative to $O$ is written in the two coordinate systems as

(5.1)\begin{equation} \boldsymbol{x}=\boldsymbol{X}(\xi_{1}, \xi_{2}, t) + n \boldsymbol{n}(\xi_{1}, \xi_{2}, t). \end{equation}

Here $\boldsymbol {x}=\boldsymbol {X}(\xi _{1}, \xi _{2}, t)$ is the parametric equation of $S$ and $\boldsymbol {n}$ is its outward unit normal. As $\xi _{1}$ and $\xi _{2}$ are principal directions on $S$ they define (locally) an orthogonal coordinate system on it with associated unit tangent vectors $\boldsymbol {e}_{i}=({1}/{a_{i}})({\partial \boldsymbol {X}}/{\partial \xi _{i}})$, where $a_{i}=|{\partial \boldsymbol {X}}/{\partial \xi _{i}}|$ for $i=1, 2$. With the convention that $\boldsymbol {n}=\boldsymbol {e}_{1}\times \boldsymbol {e}_{2}$, Rodrigues’ formula implies that the principal curvatures $\kappa _{i}$ satisfy ${\partial \boldsymbol {n}}/{\partial \xi _{i}}=\kappa _{i}({\partial \boldsymbol {X}}/{\partial \xi _{i}})$, and the change in $\boldsymbol {x}$ corresponding to increments in the intrinsic coordinates with time fixed is

(5.2)\begin{equation} {\rm d}\boldsymbol{x}=l_{1}\,{\rm d}\xi_{1}\, \boldsymbol{e}_{1}+l_{2}\,{\rm d}\xi_{2}\, \boldsymbol{e}_{2}+{\rm d}n \, \boldsymbol{n}, \quad \mbox{where} \ l_{i}=a_{i}(1+n\kappa_{i}) \quad (i=1, 2). \end{equation}

Expressions for vector differential operators and the rate of strain tensor in a general orthogonal curvilinear coordinate system can be found in, for example, Batchelor (Reference Batchelor2000, Appendix 2).

Dependent variables within the Debye layers are denoted by uppercase letters, with terms in an expansion in integer powers of $\epsilon$ denoted by subscripts, so that for the ion concentrations $c^{(\pm )}=C^{(\pm )}=C_{0}^{(\pm )}+\epsilon C_{1}^{(\pm )}+\cdots$ and for the potential $\phi =\varPhi = \varPhi _{0}+\epsilon \varPhi _{1}+\cdots$, for example. The analysis of the Debye layers is based on a local normal coordinate $N$ defined by

(5.3)\begin{equation} n=\epsilon N , \end{equation}

where $N=O(1)$ as $\epsilon \rightarrow 0$.

5.2. The Gouy–Chapman solution

The leading-order ion concentrations and potential are given by the Gouy–Chapman solution. Reviews of this are given by, for example, Russel et al. (Reference Russel, Saville and Schowalter1989) and Hunter (Reference Hunter2001), and it is summarised here because it is the foundation for the analysis that follows. In the present context, all dependent variables depend parametrically on the tangential coordinates $\xi _{1}$ and $\xi _{2}$, and on time $t$.

In terms of local variables, at leading order, the ion conservation equations (3.2) and the interface condition (2.32) for a drop, or (2.37) with (2.26) for a vesicle, imply that

(5.4a)$$\begin{gather} \partial_{N} (\partial_{N}C_{0}^{({\pm})}\pm C_{0}^{({\pm})}\partial_{N}\varPhi_{0})=0 \quad N \neq 0, \end{gather}$$

(5.4b)$$\begin{gather}\mbox{with} \quad \partial_{N}C_{0}^{({\pm})}\pm C_{0}^{({\pm})}\partial_{N}\varPhi_{0} =0 \quad \mbox{on} \ S . \end{gather}$$

An integration immediately gives

(5.5)\begin{equation} \partial_{N}C_{0}^{({\pm})}\pm C_{0}^{({\pm})}\partial_{N}\varPhi_{0} =0 \quad \mbox{for all} \ N . \end{equation}

This has a simple interpretation in terms of the molecular ion flux (2.26), which has local expansion $\boldsymbol {j}^{(\pm )}=\boldsymbol {J}_{0}^{(\pm )}+\epsilon \boldsymbol {J}_{1}^{(\pm )} + O(\epsilon ^{2})$ in the Debye layers, and where the leading-order flux $\boldsymbol {J}_{0}^{(\pm )}$ is in the normal direction. Equation (5.5) states that $\boldsymbol {J}_{0}^{(\pm )}$ is zero throughout the layers, with diffusion and electromigration of ions in equilibrium at this level of approximation.

The requirement of matching with the outer regions is that

(5.6a,b) \begin{equation} C_{0}^{({\pm})} \rightarrow \left\{\begin{array}{@{}ll} c_{0}|_{S_{+}} & \mbox{as} \ N\rightarrow \infty \\ c_{0}|_{S_{-}} & \mbox{as} \ N\rightarrow -\infty \end{array} \right.\quad \mbox{and} \quad \varPhi_{0}\rightarrow \left\{\begin{array}{@{}ll} \phi_{0}|_{S_{+}} & \mbox{as} \ N\rightarrow \infty \\ \phi_{0}|_{S_{-}} & \mbox{as} \ N\rightarrow -\infty \end{array} \right. , \end{equation}

where an $S_{+}$ or $S_{-}$ subscript denotes, respectively, the limit of a quantity in the outer regions as $S$ is approached from the exterior of the drop (as $n\rightarrow 0^{+}$) or from the interior (as $n\rightarrow 0^{-}$). We have just shown at (4.5a,b) that $c_{0}|_{S_{+}}=c_{ex}$ and $c_{0}|_{S_{-}}=c_{in}$ are constant over regions where the flow enters the Debye layers, at least at early times, but are to be determined and satisfy (4.4b) elsewhere.

Integration of (5.5) with the matching conditions (5.6a,b) gives the Boltzmann distribution between the ion concentrations and the potential, namely

(5.7a)\begin{equation} C_{0}^{(+)}=c_{0}|_{S_{{\pm}}}e^{-\eta_{0}} , \quad C_{0}^{(-)}=c_{0}|_{S_{{\pm}}}e^{\eta_{0}} , \\ \end{equation}

where

(5.7b) \begin{equation} \eta_{0}= \left\{\begin{array}{@{}ll} \varPhi_{0}-\phi_{0}|_{S_{+}} & \mbox{for} \ N>0, \\ \varPhi_{0}-\phi_{0}|_{S_{-}} & \mbox{for} \ N<0. \end{array} \right. \end{equation}

Here $\eta _{0}$ is the first term in the expansion for $\eta =\varPhi -\phi |_{S_{\pm }}$, which is the excess potential in the Debye layers.

From (2.29), the charge density $q$ in the layers is therefore given at leading order by $Q_{0}=-c_{0}|_{S_{\pm }}\sinh \eta _{0}$, and the potential $\varPhi _{0}$ satisfies the Poisson–Boltzmann equation

(5.8)\begin{equation} \varepsilon\partial_{N}^{2}\varPhi_{0}= c_{0}|_{S_{{\pm}}} \sinh (\varPhi_{0}-\phi_{0}|_{S_{{\pm}}}). \end{equation}

This can be integrated once on multiplying by $\partial _{N}\varPhi _{0}$ and using the matching conditions (5.6a,b) to find

(5.9)\begin{equation} \varepsilon (\partial_{N}\varPhi_{0})^{2}=2 c_{0}|_{S_{{\pm}}} (\cosh (\varPhi_{0}-\phi_{0}|_{S_{{\pm}}})-1) . \end{equation}

For a drop, the first of the interfacial boundary conditions (2.31a,b), that $[\phi ]=0$, states that the potential is continuous across $S$ with a surface value denoted by $\lim _{N\rightarrow 0^{\pm }}\varPhi _{0}=\varPhi _{0}|_{S}$. A second integration then gives the Gouy–Chapman solution in the form

(5.10)\begin{equation} \tanh \tfrac{1}{4}(\varPhi_{0}-\phi_{0}|_{S_{{\pm}}}) = \tanh \tfrac{1}{4} (\varPhi_{0}|_{S}-\phi_{0}|_{S_{{\pm}}}) \exp\left({{\mp}\sqrt{( c_{0}|_{S_{{\pm}}}/\varepsilon )} N}\right), \end{equation}

where the upper choice of signs hold for $N>0$ and the lower choice for $N<0$. For a vesicle, its membrane capacitance implies that there can be a non-zero jump in the potential across the vesicle membrane, with $\lim _{N\rightarrow 0^{-}}\varPhi _{0} \equiv \varPhi _{0}|_{\partial \varOmega _{in}} \neq \lim _{N\rightarrow 0^{+}}\varPhi _{0} \equiv \varPhi _{0}|_{\partial \varOmega _{ex}}$. This gives the Gouy–Chapman solution

(5.11a) $$\begin{gather} \tanh \tfrac{1}{4}(\varPhi_{0}-\phi_{0}|_{S_{+}}) = \tanh \tfrac{1}{4} (\varPhi_{0}|_{\partial\varOmega_{ex}}-\phi_{0}|_{S_{+}}) \exp\left({- \sqrt{( c_{0}|_{S_{+}}/\varepsilon )} N}\right)\quad \mbox{for} \ N>0 , \end{gather}$$

(5.11b) $$\begin{gather}\tanh \tfrac{1}{4}(\varPhi_{0}-\phi_{0}|_{S_{-}}) = \tanh \tfrac{1}{4} (\varPhi_{0}|_{\partial\varOmega_{in}}-\phi_{0}|_{S_{-}}) \exp\left({ \sqrt{( c_{0}|_{S_{-}}/\varepsilon )} N}\right) \quad \mbox{for} \ N <0 . \end{gather}$$

With the same convention for the choice of signs as in (5.10), the branch of the square root of (5.9) is found by considering the behaviour as $N\rightarrow \pm \infty$, which gives

(5.12)\begin{equation} \partial_{N}\varPhi_{0}={\mp} 2 \sqrt{\frac{c_{0}|_{S_{{\pm}}}}{\varepsilon}} \sinh \frac{(\varPhi_{0}-\phi_{0}|_{S_{{\pm}}})}{2} . \end{equation}

At this point, we introduce the $\zeta$-potential for each layer. This is the excess potential $\eta$, which is defined after (5.7), evaluated as $S$ is approached from the exterior or interior phase. For a drop, with its continuous surface potential $\varPhi _{0}|_{S}$ at $S$, the layer $\zeta$-potentials are, therefore,

(5.13a,b)\begin{equation} \zeta_{0+} = \varPhi_{0}|_{S}-\phi_{0}|_{S_{+}} \quad \mbox{and} \quad \zeta_{0-} = \varPhi_{0}|_{S}-\phi_{0}|_{S_{-}} . \end{equation}

The second of the drop interfacial boundary conditions (2.31a,b), which is the statement of Gauss's law across the interface, then implies the relation

(5.14)\begin{equation} \sqrt{\varepsilon_{ex} c_{0}|_{S_{+}}}\sinh \frac{\zeta_{0+}}{2} ={-} \sqrt{\varepsilon_{in} c_{0}|_{S_{-}}} \sinh \frac{\zeta_{0-}}{2} . \end{equation}

Recall that the electric permittivity, as a material parameter, is assumed to be piecewise constant and may take different values, $\varepsilon _{in}$ and $\varepsilon _{ex}$, in the two distinct phases. For a vesicle, with its jump in potential across the membrane surface, the Debye layer $\zeta$-potentials are defined by

(5.15a,b)\begin{equation} \zeta_{0+} = \varPhi_{0}|_{\partial\varOmega_{ex}}-\phi_{0}|_{S_{+}} \quad \mbox{and} \quad \zeta_{0-} = \varPhi_{0}|_{\partial\varOmega_{in}}-\phi_{0}|_{S_{-}} . \end{equation}

The relation analogous to (5.14) is then given by the interfacial condition (2.39), namely

(5.16)\begin{equation} \sqrt{\varepsilon_{ex} c_{0}|_{S_{+}}}\sinh \frac{\zeta_{0+}}{2} ={-} \sqrt{\varepsilon_{in} c_{0}|_{S_{-}}} \sinh \frac{\zeta_{0-}}{2} ={-} \frac{C_{m}}{2} (\varPhi_{0}|_{\partial\varOmega_{ex}}-\varPhi_{0}|_{\partial\varOmega_{in}}) . \end{equation}

A relation that will simplify an expression in the hydrodynamics of the Debye layers is given by taking the tangential derivative $\partial _{\xi _{i}}$ of (5.9) and using (5.8), namely

(5.17)\begin{equation} \partial_{N}^{2}\varPhi_{0}\partial_{\xi_{i}}\varPhi_{0} - \partial_{N}\varPhi_{0}\partial^{2}_{\xi_{i}N}\varPhi_{0} = \partial_{N}^{2}\varPhi_{0}\partial_{\xi_{i}}\phi_{0}|_{S_{{\pm}}} - \frac{(\partial_{N}\varPhi_{0})^{2}}{2} \partial_{\xi_{i}} \ln c_{0}|_{S_{{\pm}}} . \end{equation}

The charge per unit area contained within each Debye layer can be found by integrating the Poisson equation (2.29) with respect to $N$. Integration across each layer implies that at leading order the normalised charge density per unit area (i.e. with the $1/2$ normalisation factor in the volume charge density $q$ included) is $\varepsilon _{ex}\partial _{N}\varPhi _{0}|_{\partial \varOmega _{ex}}$ for $N>0$ (i.e. in $\varOmega _{ex}$) and is $- \varepsilon _{in}\partial _{N}\varPhi _{0}|_{\partial \varOmega _{in}}$ for $N<0$ (i.e. in $\varOmega _{in}$). It follows from relations (2.31a,b) for a drop and (2.39) for a vesicle that, as anticipated, at each point on $S$ the layer charges are equal and opposite for all time $t>0$ under dynamic conditions, that is, after the applied electric field is imposed, as well as under the $t<0$ equilibrium conditions of a Galvani potential.

From (5.12), the amount of the leading-order normalised charge density in the exterior layer, for example, is given by minus two times: the term on the left-hand side of (5.14) for a drop and the term on the far left-hand side of (5.16) for a vesicle. The charge densities are of opposite sign to the layer $\zeta$-potentials.

5.3. Reduced expressions for the equilibrium Galvani and $\zeta$-potentials for a drop

The equilibrium interface conditions (2.36) for a drop, together with the Boltzmann distribution (5.7) and the definition of the $\zeta$-potentials (5.13a,b), give the relations

(5.18)\begin{equation} c_{0}|_{S_{-}} = l^{(+)} c_{0}|_{S_{+}} \exp({[\phi_{0}]^{S_{+}}_{S_{-}}}) = l^{(-)} c_{0}|_{S_{+}} \exp({-[\phi_{0}]^{S_{+}}_{S_{-}}}) \end{equation}

between the ambient ion concentrations immediately outside the Debye layers $c_{0}|_{S_{\pm }}$, the jump in potential across the outer edges of the Debye layer pair $[\phi _{0}]^{S_{+}}_{S_{-}}$ and the ion partition coefficients $l^{(\pm )}$. A re-arrangement implies that

(5.19)\begin{equation} [\phi_{0}]^{S_{+}}_{S_{-}} = \frac{1}{2} \ln \left(\frac{l^{(-)}}{l^{(+)}}\right) , \end{equation}

where the equilibrium potential jump $[\phi _{0}]^{S_{+}}_{S_{-}}$ is the Galvani potential, and the ion concentrations are related by

(5.20)\begin{equation} c_{0}|_{S_{-}}=c_{0}|_{S_{+}} \sqrt{l^{(+)}l^{(-)}} . \end{equation}

Expressions for the separate layer $\zeta$-potentials can be found when the relations (5.18) are rewritten in terms of $\zeta _{0+}$ and $\zeta _{0-}$ and the pair $c_{0}|_{S_{-}}, \zeta _{0-}$ is eliminated in favour of the pair $c_{0}|_{S_{+}}, \zeta _{0+}$ (or vice versa) in (5.14). This gives

(5.21a,b)\begin{equation} \zeta_{0+} = \ln \left( \frac{1+ \sqrt{\dfrac{\varepsilon_{in} l^{(+)}}{\varepsilon_{ex}}}}{1+ \sqrt{\dfrac{\varepsilon_{in} l^{(-)}}{\varepsilon_{ex}}}} \right) \quad \mbox{and} \quad \zeta_{0-} = \ln \left( \frac{1+ \sqrt{\dfrac{\varepsilon_{ex}}{\varepsilon_{in}l^{(+)}}}}{1+ \sqrt{\dfrac{\varepsilon_{ex}}{\varepsilon_{in}l^{(-)}}}} \right) . \end{equation}

It follows that when $l^{(-)}>l^{(+)}$ the exterior layer has $\zeta$-potential $\zeta _{0+}<0$ and carries net positive charge, whereas the interior layer has $\zeta _{0-}>0$ and net negative charge. The converse holds when $l^{(-)}< l^{(+)}$.

5.4. Debye layer hydrodynamics

The hydrodynamics in the Debye layers is described by considering the Eulerian fluid velocity $\boldsymbol {u}$ in the intrinsic frame of § 5.1. If $P(\xi _{1}, \xi _{2}, n)$ is fixed in space and $Q(\xi _{1}, \xi _{2}, 0)$ is the projection of $P$ onto $S$ in the normal direction, then the fluid velocity $\boldsymbol {u}$ at $P$ is written in terms of its tangential and normal projections in the intrinsic frame as

(5.22)\begin{equation} \boldsymbol{u}= \boldsymbol{u}_{t} + u_{p}\boldsymbol{n} , \quad \mbox{where} \ \boldsymbol{u}_{t}=u_{t1}\boldsymbol{e}_{1}+u_{t2}\boldsymbol{e}_{2} . \end{equation}

In the Debye layers it is useful to compare the fluid velocity $\boldsymbol {u}$ at $P$ to the fluid velocity on the interface at $Q$, which in terms of its tangential and normal projections is $\boldsymbol {u}_{s}+u_{n}\boldsymbol {n}$. The difference between the fluid velocity at $P$ and $Q$ is denoted similarly by $\boldsymbol {v}_{t} + v_{p} \boldsymbol {n}$, so that

(5.23a)$$\begin{gather} u_{p}=u_{n}+v_{p} , \end{gather}$$

(5.23b)$$\begin{gather}u_{ti}=u_{si}+v_{ti} , \quad i=1,2. \end{gather}$$

The interfacial velocity components $u_{n}$ and $u_{si}$ are necessarily independent of the normal coordinate $n$, and the dependence of the remaining components on $n$ expresses the shear within the Debye layers. We find in the following that the relative velocity components $v_{ti}$ and $v_{p}$ are all of order $O(\epsilon )$ there.

The scale of the normal component $v_{p}$ follows from the incompressibility condition $\boldsymbol {\nabla } \boldsymbol {\cdot } \boldsymbol {u}=0$, so that temporarily we set $v_{p}=\epsilon W_{0}+ O(\epsilon ^{2})$. In the intrinsic frame, the exact expression of the incompressibility condition is

(5.24)\begin{equation} \partial_{\xi_{1}}(l_{2}u_{t1})+\partial_{\xi_{2}}(l_{1}u_{t2})+\partial_{n}(l_{1}l_{2}u_{p})=0 . \end{equation}

In the layers the dependent variables and the scale factors $l_{i}$ of (5.2) depend on the local normal coordinate $N=n/\epsilon$. So that, with the convention that terms in a local expansion of a dependent variable are written in uppercase, i.e. $u_{ti}=U_{ti0}+\epsilon U_{ti1}+O(\epsilon ^{2})$ etc., the leading-order expression of incompressibility is

(5.25)\begin{equation} \frac{1}{a_{1}a_{2}} \left( \partial_{\xi_{1}}(a_{2}U_{t10})+\partial_{\xi_{2}}(a_{1}U_{t20}) \right) + (\kappa_{1}+\kappa_{2})u_{n0} + \partial_{N}W_{0}=0 . \end{equation}

We now turn to local expansion of the forced Stokes equation from (2.30a,b). When written in terms of $N$, the electrostatic body force has a normal component that is $O(\epsilon ^{-2})$, and this can only be balanced by a normal pressure gradient of the same order. The local expansion for the pressure is therefore

(5.26)\begin{equation} p=\frac{P_{{-}1}}{\epsilon}+P_{0} + \cdots . \end{equation}

With the differential operators expressed in the local intrinsic coordinate frame, an estimate for the normal and tangential components of each term in the Stokes equation can be found, and this is listed in table 2.

Table 2.

Order of magnitude estimate for the terms in the Stokes equation. The estimate for the normal component of the viscous force is revised after considering the tangential component, as explained in the text.

The leading-order tangential component of the Stokes equation, at order $O(\epsilon ^{-2})$, gives

(5.27)\begin{equation} \partial_{N}^{2}U_{ti0}=0 , \quad i=1,2 . \end{equation}

As a homogeneous solution for the tangential velocity $U_{i0}$ that is linear in $N$ cannot match with the $O(1)$ velocity field away from the interface, the leading-order fluid velocity in the layer has a plug flow profile and is equal to the tangential velocity at the interface, namely

(5.28)\begin{equation} U_{ti0}=u_{si0}(\xi_{1}, \xi_{2}, t) , \quad i = 1,2 . \end{equation}

In other words, there is no mechanism to support a boundary layer profile in the tangential velocity.

As a consequence of (5.28) the normal component of the viscous force in the Stokes equation is smaller than first estimated, and is $O(1)$ as listed as revised in table 2. Further, all terms of (5.25) are now seen to be independent of the local normal coordinate $N$, so that integration to find $v_{p}=\epsilon W_{0}+O(\epsilon ^{2})$ is straightforward.

To find $v_{p}$, note first that the $\xi$-derivative terms of (5.25) are the surface divergence of the tangential fluid velocity on the interface $\boldsymbol {u}_{s}$, at leading order, i.e. $\boldsymbol {\nabla }_{s}\boldsymbol {\cdot } \boldsymbol {u}_{s0}$. For a drop with an impervious interface the relative velocity $v_{p}$ is zero on the interface $S$, where $N=0$, so that integration of (5.25) with respect to $N$ gives

(5.29a)\begin{equation} v_{p} (\xi_{1}, \xi_{2}, N, t) = \epsilon \partial_{n}v_{p}|_{S} N + O(\epsilon^{2}) , \end{equation}

where

(5.29b)\begin{equation} \partial_{n}v_{p}|_{S} ={-} (\boldsymbol{\nabla}_{s} \boldsymbol{\cdot} \boldsymbol{u}_{s0}+(\kappa_{1}+\kappa_{2})u_{n0}) . \end{equation}

For a semi-permeable vesicle membrane the relative velocity $v_{p}$ on the interface is small, of order $O(\epsilon )$, and given by (2.38). Integration of (5.25) with respect to $N$ then gives

(5.30a)\begin{equation} v_{p} (\xi_{1}, \xi_{2}, N, t) = \epsilon (v_{p}|_{S} + \partial_{n}v_{p}|_{S} N ) + O(\epsilon^{2}) , \end{equation}

where

(5.30b)\begin{equation} v_{p}|_{S} ={-} {\rm \pi}_{m} \left[\frac{2}{\nu} P_{{-}1} - (C_{0}^{(+)} + C_{0}^{(-)}) \right] . \end{equation}

The notation in (5.30a) and (5.30b) differs slightly from that elsewhere in this study in the expression of an $\epsilon$-scaling: (5.30a) gives the leading-order value of $v_{p}$ on the interface as $v_{p}(N=0) = \epsilon v_{p}|_{S}$, where $v_{p}|_{S}$ of (5.30b) is order $O(1)$. This choice will ease notation later. The derivative term $\partial _{n}v_{p}|_{S}$ is also order $O(1)$.

The expression (5.29b) for $\partial _{n}v_{p}|_{S}$ holds for flow quantities on and immediately adjacent to $S$ for both the drop and vesicle, and has a simple interpretation in terms of flow field kinematics for incompressible flow. The group of terms $\boldsymbol {\nabla }_{s} \boldsymbol {\cdot } \boldsymbol {u}_{s0}+(\kappa _{1}+\kappa _{2})u_{n0}$ on the right-hand side of (5.29b) is the surface divergence of all components of the interfacial fluid velocity, $\boldsymbol {u}_{s0}+ u_{n0}\boldsymbol {n}$, including the normal velocity of $S$; see, for example, Weatherburn (Reference Weatherburn1927) and note the different convention for the sign of the principal curvatures. Given the local conservation of volume of a fluid element and the definition of $v_{p}$ at (5.23a), $\partial _{n}v_{p}|_{S}$ is the rate of extension or contraction of an infinitesimal fluid line element normal to the interface.

The normal component of the Stokes equation, to leading order, implies that

(5.31)\begin{equation} {-}\partial_{N}P_{{-}1}+ \varepsilon\nu\partial_{N}^{2}\varPhi_{0}\partial_{N}\varPhi_{0}=0 , \end{equation}

and the solution that tends to zero as $N\rightarrow \pm \infty$, which confines the large $O(\epsilon ^{-1})$ pressure term to the Debye layers, is

(5.32)\begin{equation} P_{{-}1}=\frac{\varepsilon\nu}{2}(\partial_{N}\varPhi_{0})^{2} . \end{equation}

At the next order, the normal component of the Stokes equation is

(5.33)\begin{equation} -\partial_{N}P_{0} + \varepsilon \nu (\partial_{N}^{2}\varPhi_{0}\partial_{N}\varPhi_{1} + \partial_{N}^{2}\varPhi_{1}\partial_{N}\varPhi_{0} +(\kappa_{1}+\kappa_{2}) (\partial_{N}\varPhi_{0})^{2} ) =0 , \end{equation}

which has solution

(5.34)\begin{equation} P_{0} = \varepsilon \nu \left( \partial_{N} \varPhi_{0} \partial_{N}\varPhi_{1} +(\kappa_{1}+\kappa_{2})\int_{{\pm} \infty}^{N}(\partial_{N^{\prime}}\varPhi_{0})^{2}\,{\rm d}N^{\prime}\right) + p_{0}|_{S_{{\pm}}}. \end{equation}

Here, the terms in parenthesis decay exponentially as $N \rightarrow \pm \infty$ because of the exponential decay of the excess potential implied by the Gouy–Chapman solution (5.10) or (5.11), and $p_{0}|_{S_{\pm }}=\lim _{n\rightarrow 0^{\pm }}p_{0}$ is the pressure in the outer regions immediately outside the Debye layers.

We resolve the velocity profile in the Debye layers further by considering the tangential components of the Stokes equation at order $O(\epsilon ^{-1})$, and this result is needed to evaluate the tangential viscous stress on both sides of the interface. As the components $(u_{t1}, u_{t2}, u_{p})$ of the Eulerian velocity (5.22) are known to be independent of $N$ at order $O(1)$ the expression for the tangential viscous force at order $O(\epsilon ^{-1})$ simplifies. The component of the Stokes equation in the $\xi _{1}$-direction is therefore

(5.35)\begin{equation} -\frac{1}{a_{1}}\partial_{\xi_{1}}P_{{-}1} + \mu \partial_{N}^{2}U_{t11} + \varepsilon\nu \partial_{N}^{2}\varPhi_{0} \frac{1}{a_{1}}\partial_{\xi_{1}}\varPhi_{0}=0 , \end{equation}

with an analogous expression for the component in the $\xi _{2}$-direction. If $P_{-1}$ is eliminated in favour of $\varPhi _{0}$ using (5.32) and the relation (5.17) is used, we find that

(5.36)\begin{equation} \partial_{N}^{2}U_{t11}={-}\frac{\varepsilon\nu}{\mu a_{1}} \left( \partial_{N}^{2}\varPhi_{0}\partial_{\xi_{1}}\phi_{0}|_{S_{{\pm}}} - \frac{(\partial_{N}\varPhi_{0})^{2}}{2} \, \partial_{\xi_{1}} \ln c_{0}|_{S_{{\pm}}} \right) . \end{equation}

This has the solution, in terms of the excess potential $\eta _{0}=\varPhi _{0}-\phi _{0}|_{S_{\pm }}$ in each layer,

(5.37)\begin{align} U_{t11} &={-} \frac{\varepsilon\nu}{\mu a_{1}} \left( \eta_{0} \, \partial_{\xi_{1}}\phi_{0}|_{S_{{\pm}}} - 4 \ln \left( \cosh \frac{\eta_{0}}{4}\right) \partial_{\xi_{1}} \ln c_{0}|_{S_{{\pm}}} \right) \nonumber\\ &\quad + N \partial_{n}u_{10}|_{S_{{\pm}}} + u_{t11}|_{S_{{\pm}}} . \end{align}

As $N\rightarrow \pm \infty$ this satisfies the matching condition that $U_{t11} \sim N \partial _{n}u_{10}|_{S_{\pm }} + u_{t11}|_{S_{\pm }}$, and the terms in parentheses tend to zero exponentially. The tangential fluid velocity on the interface at order $O(\epsilon )$, $u_{s11}$, will remain undetermined but an expression for the $O(\epsilon )$ jump in tangential velocity across the combined Debye layer pair can be found by taking the limit of (5.37) as $N\rightarrow 0^{\pm }$. In constructing (5.37) the antiderivative $\varepsilon \int (\partial _{N}\varPhi _{0})^{2} /2 \, {\rm d}N$ as found after (A10) has been used, from which a second integration follows.

Consideration of the stress-balance boundary condition, which is given by (2.34) for a drop or (2.40) for a vesicle, together with the expressions (2.35a,b) for the stress tensors, confirms that there is no net traction on either side of the interface at order $O(\epsilon ^{-1})$. The hydrodynamic and electrostatic tractions separately each have an order $O(\epsilon ^{-1})$ component in the normal direction, but these cancel as a consequence of (5.32). This point is taken up in Appendix B.

5.5. Reduced expressions for the trans-membrane ion flux and solvent osmotic flow speed for a vesicle

The trans-membrane ion flux, $\boldsymbol {j}^{(\pm )}\boldsymbol {\cdot }\boldsymbol {n}$ on $S$, is given in terms of the jump in the electrochemical potential across the membrane by (2.37), which we write here as

(5.38)\begin{equation} \boldsymbol{j}^{({\pm})} \boldsymbol{\cdot} \boldsymbol{n} ={-} \epsilon g^{({\pm})} [ \ln c^{({\pm})} \pm \phi ] \quad \mbox{on} \ S . \end{equation}

When the flux $\boldsymbol {j}^{(\pm )}$ within the Debye layers is given the local expansion $\boldsymbol {j}^{(\pm )}=\boldsymbol {J}_{0}^{(\pm )}+\epsilon \boldsymbol {J}_{1}^{(\pm )}+O(\epsilon ^{2})$ we have the fundamental result that $\boldsymbol {J}_{0}$ is zero, as noted below (5.5). Then, when the electrochemical potential on the right-hand side of (5.38) is written in terms of the local variables $C_{0}^{(\pm )}$ and $\varPhi _{0}$, the Boltzmann distribution (5.7a,b) implies that

(5.39)\begin{equation} \boldsymbol{J}_{1}^{({\pm})}\boldsymbol{\cdot}\boldsymbol{n} ={-}g^{({\pm})} \left( \ln \left(\frac{c_{0}|_{S_{+}}}{c_{0}|_{S_{-}}}\right) \pm (\phi_{0}|_{S_{+}}-\phi_{0}|_{S_{{-}}}) \right) \quad \mbox{on} \ S . \end{equation}

This is written in a more compact form as

(5.40)\begin{equation} j|_{S}^{({\pm})} ={-} g^{({\pm})} [ \ln c_{0} \pm \phi_{0} ]^{S_{+}}_{S_{-}} \quad \mbox{on} \ S , \end{equation}

where $j|_{S}^{(\pm )}$ denotes $\boldsymbol {J}_{1}^{(\pm )}\boldsymbol {\cdot }\boldsymbol {n}$ evaluated on $S$, i.e. when $N=0$ and $[\,\cdot \,]^{S_{+}}_{S_{-}}$ denotes the jump in a quantity across the outer edges of the Debye layer pair.

To interpret this we note that, in the Debye layers, the flux $\boldsymbol {J}_{0}$ being zero and the Boltzmann distribution together imply that the electrochemical potential is independent of $N$ to leading order, so that its jump across the membrane in (5.38) is equal to the jump across the outer edges of the electrical double layer in (5.40).

Similarly, for a semi-permeable membrane the scaled osmotic flow speed $v_{p}|_{S}$ of (5.30a,b) can be written in terms of quantities immediately outside or at the outer edges of the Debye layers. Inside the layers, the absolute pressure is given by (5.32) with (5.9) whereas the dissolved salt concentration is given by (5.7a,b). It turns out that their difference, which is the partial pressure of the solvent, is independent of $N$, and the expression for the jump across the membrane surface simplifies, to give

(5.41)\begin{equation} v_{p}|_{S} = 2{\rm \pi}_{m} [ c_{0} ]^{S_{+}}_{S_{-}} , \end{equation}

which is proportional to the difference in the total ion concentrations at the outer edges of the layers and is independent of the layer $\zeta$-potentials.

6.1. Debye layer ion transport for a drop

For a drop with an interface that is impermeable to ions the boundary condition (2.32) implies that the normal flux term $\boldsymbol {J}_{1}^{(\pm )}\boldsymbol {\cdot }\boldsymbol {n}$ vanishes at $N=0$, that is,

(6.5)\begin{equation} \partial_{N}C_{1}^{({\pm})} \pm (C_{0}^{({\pm})}\partial_{N}\varPhi_{1} + C_{1}^{({\pm})} \partial_{N}\varPhi_{0})=0 \quad \mbox{on} \ S . \end{equation}

The interface is also impermeable to flow of the immiscible liquids that it separates, so that the relative normal fluid velocity $v_{p}|_{S}=0$, and an integration of (6.4) gives

(6.6)\begin{align} &\partial_{N}C_{1}^{({\pm})} \pm (C_{0}^{({\pm})}\partial_{N}\varPhi_{1} + C_{1}^{({\pm})} \partial_{N}\varPhi_{0}) \nonumber\\ &\quad =\frac{1}{D^{({\pm})}} \int_{0}^{N} \left(\partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} + \partial_{n}v_{p}|_{S}N^{\prime} \partial_{N^{\prime}} \right) C_{0}^{({\pm})} \,{\rm d}N^{\prime} . \end{align}

This expresses a balance between the electrodiffusive flux of ions in the normal direction and advection of ions within the layers in integrated form.

To form the limit of (6.6) as $N\rightarrow \pm \infty$ we match with the outer regions, which requires that: $\partial _{N}C_{1}^{(\pm )}\rightarrow \partial _{n}c_{0}|_{S_{\pm }}$ and $\partial _{N}\varPhi _{1}^{(\pm )}\rightarrow \partial _{n}\phi _{0}|_{S_{\pm }}$; both $C_{0}^{(\pm )}\rightarrow c_{0}|_{S_{\pm }}$ and $\varPhi _{0}\rightarrow \phi _{0}|_{S_{\pm }}$, where the approach to the limit is exponential in $N$ from (5.7) and (5.10). The limit of the left-hand side is therefore $\partial _{n}c_{0}|_{S_{\pm }}\pm c_{0}|_{S_{\pm }}\partial _{n}\phi _{0}|_{S_{\pm }}$, in which the upper (lower) choice of the sign that sits between the two groups of terms refers to the positive (negative) ions, respectively, so that it is now written as $\partial _{n}c_{0}|_{S_{\pm }} (\pm ) c_{0}|_{S_{\pm }}\partial _{n}\phi _{0}|_{S_{\pm }}$.

To see that the integral on the right-hand side of (6.6) converges as $N\rightarrow \pm \infty$ and to evaluate it in terms of the layer $\zeta$-potentials, etc., we recall from (3.2) and § 4 that in the outer regions, leading-order ion conservation requires that $(\partial _{t}+\boldsymbol {u}_{0}\boldsymbol {\cdot }\boldsymbol {\nabla })c_{0}=0$ for all $n$, including the limit $n\rightarrow 0^{\pm }$ in which $c_{0}=c_{0}|_{S_{\pm }}$. In the local intrinsic coordinate system, from (6.3a,b) this becomes

(6.7)\begin{equation} \left( \partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} \right) c_{0}|_{S_{{\pm}}}=0 , \end{equation}

because $c_{0}|_{S_{\pm }}$ is independent of $N$. When this is subtracted from the integrand of (6.6), in the limit we have

(6.8)\begin{align} &\partial_{n}c_{0}|_{S_{{\pm}}} ({\pm}) c_{0}|_{S_{{\pm}}}\partial_{n}\phi_{0}|_{S_{{\pm}}} \nonumber\\ &\quad = \frac{1}{D^{({\pm})}} \left( (\partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s}) \int_{0}^{{\pm}\infty} (C_{0}^{({\pm})}-c_{0}|_{S_{{\pm}}}) \, {\rm d}N + \partial_{n}v_{p}|_{S} \int_{0}^{{\pm}\infty} N\partial_{N} C_{0}^{({\pm})} \,{\rm d}N \right) . \end{align}

The last integral on the right-hand side of (6.8) can be found by integration by parts, where because of the exponential approach of $C_{0}^{(\pm )}$ to $c_{0}|_{S_{\pm }}$ as $N\rightarrow \pm \infty$ the antiderivative of $\partial _{N}C_{0}^{(\pm )}$ is chosen to be $C_{0}^{(\pm )}-c_{0}|_{S_{\pm }}$, so that $\int _{0}^{\pm \infty }N\partial _{N} C_{0}^{(\pm )} \,{\rm d}N = - \int _{0}^{\pm \infty } (C_{0}^{(\pm )}-c_{0}|_{S_{\pm }}) \,{\rm d}N$. In physical terms both integrals in (6.8) are now, up to a sign, the excess ion concentration integrated across a layer. As the excess potential $\eta _{0}$ is a monotone function of $N$ and of the same sign as the $\zeta$-potential, the variable of integration can be changed from $N$ to $\eta _{0}$, from which (5.7) and (5.12) give

(6.9)\begin{equation} \int_{0}^{{\pm}\infty} (C_{0}^{({\pm})}-c_{0}|_{S_{{\pm}}}) \,{\rm d}N ={\pm} 2 \sqrt{\varepsilon c_{0}|_{S_{{\pm}}}} \left( \exp({({\mp})\zeta_{0\pm}/2}) -1 \right) . \end{equation}

When (6.8) is written out for each ion species separately, their sum and difference is formed and (6.9) is used, we find a pair of boundary conditions to be applied at the outer edge of each Debye layer. To do so, recall that in (6.8) and (6.9) when a choice of sign appears in parenthesis, as in $(\pm )$ and $(\mp )$, the upper (lower) choice of sign refers to the positive (negative) ion species, respectively. When a choice of sign appears without parenthesis, the upper (lower) choice of sign refers to the Debye layer in the exterior (interior) phase, where $N>0$ ($N<0$). This gives

(6.10)\begin{equation} \partial_{n}c_{0}|_{S_{{\pm}}} ={\pm} \left( \partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} - \partial_{n} v_{p}|_{S} \right) \sqrt{\varepsilon c_{0}|_{S_{{\pm}}}} \left( \frac{\exp\left({\dfrac{-\zeta_{0\pm}}{2}}\right)\,{-}\,1}{D^{(+)}} + \frac{\exp\left({\dfrac{\zeta_{0\pm}}{2}}\right)\,{-}\,1}{D^{(-)}}\right)\!, \end{equation}

and

(6.11)\begin{align} &c_{0}|_{S_{{\pm}}}\partial_{n}\phi_{0}|_{S_{{\pm}}}\nonumber\\ &\quad ={\pm} \left(\partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} - \partial_{n} v_{p}|_{S}\right) \sqrt{\varepsilon c_{0}|_{S_{{\pm}}}} \left( \dfrac{\exp\left({\dfrac{-\zeta_{0\pm}}{2}}\right)-1}{D^{(+)}} - \frac{\exp\left({\dfrac{\zeta_{0\pm}}{2}}\right)-1}{D^{(-)}}\right) . \end{align}

To conclude this section we note that for a drop the permittivity $\varepsilon$ and ion diffusivities $D^{(\pm )}$ may differ between the interior and exterior phases. When this occurs, the material properties are ascribed an $ex$ subscript (or $in$ subscript) when relations (6.10) and (6.11) are applied as boundary conditions on $S_{+}$ (or $S_{-}$) for the exterior (or interior) phase.

6.2. Debye layer ion transport for a vesicle

When compared with a drop, some steps of the analysis for ion transport in the Debye layers need to be modified for a vesicle, because its membrane can be semi-permeable to ion gating and solvent osmosis channels.

First, the expression (6.4) for ion conservation in the Debye layers can be written in terms of the normal ion flux $\boldsymbol {J}_{1}^{(\pm )}\boldsymbol {\cdot }\boldsymbol {n}$ as

(6.12)\begin{equation} \partial_{N} ( - \boldsymbol{J}_{1}^{({\pm})}\boldsymbol{\cdot}\boldsymbol{n} ) = \left( \partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} + (v_{p}|_{S} + \partial_{n}v_{p}|_{S}N) \partial_{N} \right) C_{0}^{({\pm})} . \end{equation}

This includes the trans-membrane osmotic flow velocity $v_{p}|_{S}$ in the ion advection terms on the right-hand side.

Next, the trans-membrane ion flux $\boldsymbol {J}_{1}^{(\pm )}\cdot \boldsymbol {n}$ evaluated on $S$, i.e. when $N=0$, is denoted by $j|_{S}^{(\pm )}$ and given in reduced form by (5.40). An integration of (6.12) with respect to $N$, with the ion flux restored to terms of the ion concentrations and potential, gives

(6.13)\begin{align} &\partial_{N}C_{1}^{({\pm})} \pm (C_{0}^{({\pm})}\partial_{N}\varPhi_{1} + C_{1}^{({\pm})} \partial_{N}\varPhi_{0}) \nonumber\\ &\quad =\frac{-j|_{S}^{({\pm})}}{D^{({\pm})}} + \frac{1}{D^{({\pm})}} \int_{0}^{N} \left(\partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} + (v_{p}|_{S} + \partial_{n}v_{p}|_{S}N^{\prime}) \partial_{N^{\prime}} \right) C_{0}^{({\pm})} \,{\rm d}N^{\prime} . \end{align}

The procedure of matching with the outer regions and subtracting (6.7) from the integrand remains unchanged, with the result that the analog for a vesicle of (6.8) is

(6.14)\begin{align} \partial_{n}c_{0}|_{S_{{\pm}}} ({\pm}) c_{0}|_{S_{{\pm}}}\partial_{n}\phi_{0}|_{S_{{\pm}}} &=\frac{-j|_{S}^{({\pm})}}{D^{({\pm})}} + \frac{1}{D^{({\pm})}} \left( (\partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s}) \int_{0}^{{\pm}\infty} (C_{0}^{({\pm})}-c_{0}|_{S_{{\pm}}}) \, {\rm d}N \right.\nonumber\\ &\qquad\left.+ \int_{0}^{{\pm}\infty} ( v_{p}|_{S} + \partial_{n}v_{p}|_{S} N ) \partial_{N} C_{0}^{({\pm})} \,{\rm d}N \right) . \end{align}

As the osmotic flow velocity $v_{p}|_{S}$ is independent of $N$, the additional integral term is simply the change in the ion concentrations across either Debye layer, which is given by (5.7a,b) and (5.15a,b) as $c_{0}|_{S_{\pm }}(1-\exp ({(\mp )\zeta _{0\pm }}))$.

When (6.14) is written for each ion species, and then their sum and difference is formed, we find that for a vesicle the boundary conditions (6.10) and (6.11) are modified to become

(6.15)\begin{align} \partial_{n}c_{0}|_{S_{{\pm}}} &={-} \frac{1}{2} \left( \frac{j|_{S}^{(+)}}{D^{(+)}} + \frac{j|_{S}^{(-)}}{D^{(-)}} \right) + \frac{v_{p}|_{S} c_{0}|_{S_{{\pm}}}}{2} \left( \frac{1-{\rm e}^{-\zeta_{0\pm}}}{D^{(+)}} + \frac{1-{\rm e}^{\zeta_{0\pm}}}{D^{(-)}} \right) \nonumber\\ &\quad \pm \left( \partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} - \partial_{n} v_{p}|_{S} \right) \sqrt{\varepsilon c_{0}|_{S_{{\pm}}}} \left( \frac{\exp\left({\dfrac{-\zeta_{0\pm}}{2}}\right)-1}{D^{(+)}} + \frac{\exp\left({\dfrac{\zeta_{0\pm}}{2}}\right)-1}{D^{(-)}}\right) , \end{align}

and

(6.16)\begin{align} c_{0}|_{S_{{\pm}}}\partial_{n}\phi_{0}|_{S_{{\pm}}} &={-} \frac{1}{2} \left( \frac{j|_{S}^{(+)}}{D^{(+)}} - \frac{j|_{S}^{(-)}}{D^{(-)}} \right) + \frac{v_{p}|_{S} c_{0}|_{S_{{\pm}}}}{2} \left( \frac{1-{\rm e}^{-\zeta_{0\pm}}}{D^{(+)}} - \frac{1-{\rm e}^{\zeta_{0\pm}}}{D^{(-)}} \right) \nonumber\\ &\quad \pm \left(\partial_{t} +\boldsymbol{v}_{s0}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} - \partial_{n} v_{p}|_{S} \right) \sqrt{\varepsilon c_{0}|_{S_{{\pm}}}} \left( \frac{\exp\left({\dfrac{-\zeta_{0\pm}}{2}}\right)-1}{D^{(+)}} - \frac{\exp\left({\dfrac{\zeta_{0\pm}}{2}}\right)-1}{D^{(-)}}\right) , \end{align}

where the upper (lower) choice of sign refers to the outer edge of the Debye layer in the exterior (interior) phase, where $N>0$ ($N<0$).

Expressions for the additional trans-membrane ion flux contributions to (6.15) and (6.16), written in reduced form via (5.40), are found on noting that

(6.17)\begin{equation} \frac{j|_{S}^{(+)}}{D^{(+)}} \pm \frac{j|_{S}^{(-)}}{D^{(-)}} ={-} \left( \frac{g^{(+)}}{D^{(+)}} \pm \frac{g^{(-)}}{D^{(-)}} \right) [ \ln c_{0} ]^{S_{+}}_{S_{-}} - \left( \frac{g^{(+)}}{D^{(+)}} \mp \frac{g^{(-)}}{D^{(-)}} \right) [ \phi_{0} ]^{S_{+}}_{S_{-}} . \end{equation}

Similarly, for the osmotic flow contribution we recall from (5.41) that $v_{p}|_{S} = 2{\rm \pi} _{m} [ c_{0} ]^{S_{+}}_{S_{-}}$.

For a vesicle, the same solvent occupies both the exterior ($\varOmega _{ex}$) and interior ($\varOmega _{in}$) phases and we consider single cation and anion species, so that material properties such as the permittivity $\varepsilon$ and ion diffusivities $D^{(\pm )}$ are constant throughout.

6.3. Use of the ion transport relations as boundary conditions

As noted in § 4 the parts of $\varOmega _{in}$ and $\varOmega _{ex}$ in the outer regions, outside the Debye layers, are charge-neutral to high order, where the field equations are

(6.18a)$$\begin{gather} \nabla^{2}\phi_{0} = 0 , \quad (\partial_{t}+\boldsymbol{u}_{0}\boldsymbol{\cdot}\boldsymbol{\nabla})c_{0}=0 , \end{gather}$$

(6.18b)$$\begin{gather}-\boldsymbol{\nabla} p_{0} +\mu{\nabla}^{2}\boldsymbol{u}_{0} = 0 \quad \mbox{and} \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u}_{0} = 0 . \end{gather}$$

Except for the appearance of the fluid velocity $\boldsymbol {u}_{0}$ in the transport equation for $c_{0}$, the field equations are uncoupled.

The relation (6.11) for a drop or (6.16) for a vesicle is a Neumann boundary condition for the potential $\phi _{0}$ of the Laplace equation that is applied at the outer edge of each Debye layer, that is, $S_{+}$ in $\varOmega _{ex}$ and $S_{-}$ in $\varOmega _{in}$. It depends on the local fluid velocity and on the local ion concentration $c_{0}$ and $\zeta$-potential $\zeta _{0}$ at each point of the respective Debye layer edges. For a vesicle, the boundary condition also depends on the jump in both the ion concentration and potential across the double layer pair, per (6.17) and the expression for $v_{p}|_{S}$ below it.

The relation (6.10) for a drop or (6.15) for a vesicle is a boundary condition of mixed type for the ion concentration $c_{0}$ of the transport equation at (6.18a). It depends on the local fluid velocity and the layer $\zeta$-potential. For a vesicle it also depends on the jump in the ion concentration and on the jump in the potential across the double layer. We note that, especially for a drop, the boundary condition cannot be applied exactly on the interface $S$, because the characteristics of the transport equation are particle paths satisfying ${{\rm d}\kern0.06em \boldsymbol {x}}/{{\rm d}t}=\boldsymbol {u}_{0}$ that are tangential to $S$ in the Lagrangian frame and, therefore, do not enter the outer regions. Instead, for both a drop and a vesicle the boundary condition is applied only on the outflow regions of the Debye layers, where the quantity $\partial _{n}v_{p}|_{S}$ introduced in § 5.4 is positive, at some fixed $\epsilon >0$ and $N\neq 0$. Further details of this construction will be given elsewhere.

7.1. The net force on the interface and the integral equation for a drop

The traction on the interface $S$ due to the hydrodynamic field appears in the first integral on the right-hand side of (7.8) and can be expressed in terms of the capillary stress and electrostatic potential from the stress-balance boundary condition for a drop (2.34) and the Maxwell stress tensor (2.35a,b). This gives the exact relation

(7.9)\begin{equation} [\boldsymbol{\mathsf{T}}_{H}\cdot \boldsymbol{n}] = \varDelta (\kappa_{1}+\kappa_{2})\boldsymbol{n} - \nu\epsilon \big[ \varepsilon \big(\tfrac{1}{2}(\partial_{n}\phi^{2}-| \boldsymbol{\nabla}_{s}\phi |^{2})\boldsymbol{n} + \partial_{n}\phi\boldsymbol{\nabla}_{s}\phi\big)\big] , \end{equation}

where the permittivity $\varepsilon$ in the jump operator is, in general, discontinuous across $S$. As the drop interface is sharp, with no surface charge and no electrical capacitance, the electrostatic potential $\phi$ and its tangential gradient $\boldsymbol {\nabla }_{s}\phi$ are continuous across $S$, and, per the second of (2.31a,b), $[\varepsilon \partial _{n}\phi ]=0$. On the right-hand side of (7.9), the last term is therefore zero. Recalling the $\epsilon$-scaling and local expansion of $\phi$ in the Debye layers, we find that the preceding term, containing $|\boldsymbol {\nabla }_{s}\phi |^{2}$, is of order $O(\epsilon )$, so that the hydrodynamic traction on $S$ is

(7.10)\begin{equation} [\boldsymbol{\mathsf{T}}_{H}\cdot \boldsymbol{n}] = \varDelta (\kappa_{1}+\kappa_{2})\boldsymbol{n} - \frac{\nu}{2\epsilon} [ \varepsilon (\partial_{N}\varPhi_{0})^{2}] \boldsymbol{n} -\nu [ \varepsilon \partial_{N}\varPhi_{0}\partial_{N}\varPhi_{1}] \boldsymbol{n} + O(\epsilon) . \end{equation}

It turns out that the last two terms on the right-hand side of (7.10) are cancelled by contributions to the total force on the electric charge distribution within the layers, which is given by the last integral on the right-hand side of (7.8) containing the Coulomb force density $\boldsymbol {f}$. This integral is recast as a surface integral by performing the integration in the normal direction over the $\epsilon$-support of the Debye layers, for which we give details in Appendix A.

The result of this analysis is that the integral equation (7.8) can be written as

(7.11) \begin{align} u_{j}(\boldsymbol{x}_{0}) &= \frac{-1}{4{\rm \pi}(\mu_{ex}+\mu_{in})}\int_{S}{\mathsf{G}}_{{\mathsf{ij}}}(\boldsymbol{x}, \boldsymbol{x}_{0})[\hat{{\mathsf{T}}}_{{\mathsf{ik}}}(\boldsymbol{x})]n_{k}(\boldsymbol{x}) \,{\rm d}S_{\boldsymbol{x}} \nonumber\\ &\quad +\frac{(\mu_{ex}-\mu_{in})}{4{\rm \pi}(\mu_{ex}+\mu_{in})} \int^{PV}_{S} u_{i}(\boldsymbol{x}) {\mathsf{T}}_{{\mathsf{ijk}}}(\boldsymbol{x}, \boldsymbol{x}_{0}) n_{k}(\boldsymbol{x}) \,{\rm d}S_{\boldsymbol{x}} \nonumber\\ &\quad- \frac{1}{4{\rm \pi}(\mu_{ex}+\mu_{in})}\int_{S} n_{k}(\boldsymbol{x})\frac{\partial}{\partial x_{k}} {\mathsf{G}}_{{\mathsf{ij}}}(\boldsymbol{x}, \boldsymbol{x}_{0}) \nu \varTheta(\boldsymbol{x}) n_{i}(\boldsymbol{x}) \,{\rm d}S_{\boldsymbol{x}} . \end{align}

Here, $[\hat {\boldsymbol{\mathsf{T}}}\boldsymbol {\cdot }\boldsymbol {n}]$ is a modified surface traction, given by

(7.12)\begin{equation} [\hat{\boldsymbol{\mathsf{T}}}\boldsymbol{\cdot}\boldsymbol{n}] = (\kappa_{1}+\kappa_{2})\left( \varDelta - \nu\varTheta \right)\boldsymbol{n} + \nu \boldsymbol{\nabla}_{s}\varTheta , \end{equation}

and where $\varTheta$ is given in terms of the layer $\zeta$-potentials and other parameters by

(7.13)\begin{equation} \varTheta= 4 \left( \surd({\varepsilon_{ex}c_{0}|_{S_{+}}}) \sinh^{2}\frac{\zeta_{0+}}{4} + \surd({\varepsilon_{in}c_{0}|_{S_{-}}}) \sinh^{2}\frac{\zeta_{0-}}{4} \right) . \end{equation}

The quantity $\varTheta$ is defined in Appendix A at (A10) by $\varTheta =\int _{-\infty }^{\infty }\varepsilon ({\partial _{N}\varPhi _{0}^{2}}/{2})\,{\rm d}N$, which brings to mind the electrostatic energy density per unit area in the Debye layers. To verify this conjecture, at leading order, the dimensions can be restored to (7.11) to (7.13), when it is found that they still hold with the only modifications that: $\nu$ is set to 1 in (7.11) and (7.12), $\varDelta$ is replaced by the surface tension $\sigma$ in (7.12), and the $\zeta$-potentials in (7.13) are replaced by $\zeta _{0\pm }/\phi _{*}$ so as to remain dimensionless. Then the scale by which $\varTheta$ is made dimensionless is found to be $\varepsilon _{*}\phi _{*}^{2}/\lambda _{*}$, which is consistent with the $O(1/\epsilon )$ magnitude of the potential gradient and the $\epsilon$-width of the Debye layers.

7.2. Modification of the net force on the interface and the integral equation for a vesicle

Relative to a drop with constant uniform surface tension, the stress in a vesicle membrane includes the effects of bending stiffness and a surface tension $\sigma$ that varies so as to maintain local conservation of area via the inextensibility constraint $\boldsymbol {\nabla }_{s}\boldsymbol {\cdot }(\partial _{t}\boldsymbol {X}_{s}(\boldsymbol {\theta }, t))=0$ of (2.41). This modifies the stress-balance boundary condition (7.9) by replacing the capillary stress $\varDelta (\kappa _{1}+\kappa _{2})\boldsymbol {n}$ on the right-hand side of the relation with the membrane stress

(7.14)\begin{equation} \boldsymbol{\tau}_{m} = 2 \sigma H \boldsymbol{n} - \boldsymbol{\nabla}_{s} \sigma - \kappa_{b} (2H(H^{2}-K) + \nabla_{s}^{2}H) \boldsymbol{n} , \end{equation}

per (2.34) and (2.40).

The second modification to the analysis for a vesicle membrane concerns the jump in the tangential Maxwell stress, which is proportional to $[\varepsilon \partial _{n}\phi \boldsymbol {\nabla }_{s}\phi ]$ in (7.9). For a membrane that has no net monopole charge the electric displacement is continuous, so that $[\varepsilon \partial _{n}\phi ]=0$ per the boundary condition (2.39). However, the trans-membrane potential $[\phi ]$ is non-zero and varies around the membrane surface $S$, so that $[\boldsymbol {\nabla }_{s}\phi ]\neq 0$, and the tangential Maxwell stress jump is therefore also non-zero. However, the electrostatic force in the Debye layers has a component that, up to the order of calculation, exactly cancels this contribution to the interfacial traction. This point is made in the analysis of Appendix A, below (A14) at item (i).

It follows that the integral equation (7.11) also holds for a vesicle, but with the surface traction (7.12) replaced by

(7.15) \begin{equation} [\hat{\boldsymbol{\mathsf{T}}}\boldsymbol{\cdot}\boldsymbol{n}] = \{ (\kappa_{1}+\kappa_{2}) (\sigma - \nu\varTheta) - \kappa_{b} (2H(H^{2}-K) + {\nabla}_{s}^{2} H)\} \boldsymbol{n} - \boldsymbol{\nabla}_{s} (\sigma - \nu\varTheta) . \end{equation}

The Stokes dipole integral remains unchanged. However, as noted below (6.17) for example, because the same solvent and ion species occupy both phases, the permittivity is either the same constant throughout or nearly so, so that in the expression (7.13) for the energy density $\varTheta$, $\varepsilon _{in}=\varepsilon _{ex}$.

9.1. Initial conditions

The initial conditions of (2.42), for which the applied field is uniform, are

(9.1)\begin{equation} \phi(\boldsymbol{x}, 0^{+}) ={-} \varPsi z, \quad \boldsymbol{u}(\boldsymbol{x}, 0) = 0 \quad \mbox{and} \quad c^{({\pm})}(\boldsymbol{x}, 0) = c_{in,ex} \ \mbox{on} \ \varOmega_{in,ex} . \end{equation}

An exception to this is a drop that has a non-zero Galvani potential. Then, for $t<0$ the potential is piecewise constant with jump across $S$ given by (5.19), namely

(9.2)\begin{equation} [\phi]^{S_{+}}_{S_{-}} = \frac{1}{2} \ln \left(\frac{l^{(-)}}{l^{(+)}}\right) . \end{equation}

The potential at $t=0^{+}$ is given by superimposing the applied potential. The ambient ion concentrations on $\varOmega _{in}$ and $\varOmega _{ex}$ for $t<0$ are related by (5.20), that is

(9.3)\begin{equation} c_{in} = c_{ex} \sqrt{l^{(+)}l^{(-)}} , \end{equation}

where the ion partition coefficients $l^{(\pm )}$ are known. The $\zeta$-potentials for $t<0$ that correspond to this are given by (5.21a,b). A drop is initially spherical with radius 1, whereas a vesicle has a known initial equilibrium configuration $S_{ref}$.

9.2. Far-field conditions

Far from a drop or vesicle, for a uniform applied field

(9.4)\begin{equation} \phi(\boldsymbol{x}, t) \sim{-} \varPsi z, \quad \boldsymbol{u}(\boldsymbol{x}, t) \rightarrow 0, \quad \mbox{and} \quad c^{({\pm})}(\boldsymbol{x}, t)\rightarrow c_{ex} \ \mbox{as} \ |\boldsymbol{x}|\rightarrow \infty \ \mbox{for} \ t > 0 . \end{equation}

For a more general applied field $\phi$ approaches a specified solution $\phi _{\infty }$ of Laplace's equation that has magnitude $\varPsi$. This may be time-dependent on the scale of the charge-up time.

9.3. Kinematic condition

Time update of the interface position is determined by the kinematic condition. For a drop this is given by

(9.5)\begin{equation} \left(\boldsymbol{u} - \frac{\partial\boldsymbol{x}_{s}}{\partial t}\right)\boldsymbol{\cdot}\boldsymbol{n}=0 \quad \mbox{on}\ S , \end{equation}

per (2.33), at any point $\boldsymbol {x}_{s}$ on $S$. The continuity of velocity across $S$, $[\boldsymbol {u}]=0$, is built-in to both the integral equation for $\boldsymbol {u}$ and the reduced stress-balance boundary condition.

For a vesicle with a semi-permeable membrane $S$ having equation $\boldsymbol {x}=\boldsymbol {X}(\boldsymbol {\theta }, t)$, we have the boundary condition

(9.6)\begin{equation} \boldsymbol{u} - \left.\frac{\partial\boldsymbol{X}}{\partial t}\right|_{\boldsymbol{\theta}} = \epsilon v_{p}|_{S} \, \boldsymbol{n} \quad \mbox{on } S, \ \mbox{ where} \ v_{p}|_{S} = 2 {\rm \pi}_{m} [c]^{S_{+}}_{S_{-}} , \end{equation}

per (2.38) and (5.41) via (5.30b). The normal component of this is a slightly modified version of the relation (9.5) for a drop; when the permeability ${\rm \pi} _{m}>0$ there is a small $O(\epsilon )$ osmotic flux and change in vesicle volume on the charge-up time scale. The tangential projection of (9.6) implies continuity of tangential fluid velocity across $S$, i.e. $[\boldsymbol {u}_{s}]=0$, which is already built-in to other components of the model where needed, but, in addition, it ensures no slip between the membrane and adjacent fluid. That is, $\boldsymbol {u}_{s}= \partial _{t}\boldsymbol {X}_{s}|_{\boldsymbol {\theta }}$, so that local conservation of membrane area or incompressibility implies the additional constraint that

(9.7)\begin{equation} \boldsymbol{\nabla}_{s}\boldsymbol{\cdot}\left( \frac{\partial}{\partial t}\boldsymbol{X}_{s}(\boldsymbol{\theta}, t) \right) =0 \end{equation}

of (2.41). It also ensures that the tangential fluid velocity $\boldsymbol {u}_{s}$ is such that $\boldsymbol {\nabla }_{s}\boldsymbol {\cdot }\boldsymbol {u}_{s}=0$.

9.4. Inflow–outflow condition

The rate of extension of an infinitesimal fluid element based on and normal to $S$ is denoted by $\partial _{n}v_{p}|_{S}$, and given in terms of surface data by (5.29b), which is

(9.8)\begin{equation} \partial_{n}v_{p}|_{S} ={-} (\boldsymbol{\nabla}_{s} \boldsymbol{\cdot} \boldsymbol{u}_{s}+(\kappa_{1}+\kappa_{2})u_{n}) . \end{equation}

For a vesicle with an incompressible membrane, from the comment below (9.7), this simplifies to

(9.9)\begin{equation} \partial_{n}v_{p}|_{S} ={-} (\kappa_{1}+\kappa_{2})u_{n} . \end{equation}

9.5. $\zeta$-potentials and potential jumps

The definition of the $\zeta$-potentials for a drop, for which there is no jump in the potential across its sharp interface, is given at (5.13a,b), from which we have

(9.10)\begin{equation} [\phi]^{S_{+}}_{S_{-}} = \zeta_{-}-\zeta_{+} . \end{equation}

For a vesicle, which can sustain a jump in the potential across its membrane, the definition of the $\zeta$-potentials is given at (5.15a,b), from which

(9.11)\begin{equation} [\phi]^{S_{+}}_{S_{-}} -[\phi]_{m} = \zeta_{-}-\zeta_{+} . \end{equation}

In this summary, the jump in a quantity across the faces of the membrane, which earlier was denoted by $[\,\cdot\, ] = [\,\cdot\, ]^{\partial \varOmega _{ex}}_{\partial \varOmega _{in}}$, is now denoted by $[\,\cdot\, ]_{m}$ instead.

9.6. Gauss's relation between the $\zeta$-potentials

The drop interface holds no monopole charge, from which Gauss's law implies the relation (5.14) between the $\zeta$-potentials and the ion concentrations at the outer edges of the Debye layers, namely

(9.12)\begin{equation} \sqrt{\varepsilon_{ex} c|_{S_{+}}}\sinh \frac{\zeta_{+}}{2} ={-} \sqrt{\varepsilon_{in} c|_{S_{-}}} \sinh \frac{\zeta_{-}}{2} . \end{equation}

This also implies that the charge per unit area in the back-to-back Debye layers is equal and opposite at all points. For a vesicle, Gauss's law relates the charge held in the layers to the membrane capacitance $C_{m}$ and the trans-membrane potential $[\phi ]_{m}$, per (5.16), namely

(9.13)\begin{equation} \sqrt{\varepsilon_{ex} c|_{S_{+}}}\sinh \frac{\zeta_{+}}{2} ={-} \sqrt{\varepsilon_{in} c|_{S_{-}}} \sinh \frac{\zeta_{-}}{2} ={-} \frac{C_{m}}{2} [\phi]_{m} . \end{equation}

At this point we see that, relative to a drop, the model for a vesicle contains an additional variable $[\phi ]_{m}$ and an additional constraint in (9.13).

9.7. Transport relations for the potential

For a drop, the relation (6.11) for ion transport in the Debye layers gives the boundary condition on $S_{+}$ that

(9.14)\begin{align} c|_{S_{+}}\partial_{n}\phi|_{S_{+}} = \left(\partial_{t} +\boldsymbol{v}_{s}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} - \partial_{n} v_{p}|_{S} \right) \sqrt{\varepsilon_{ex} c|_{S_{+}}} \left( \frac{\exp\left({\dfrac{-\zeta_{+}}{2}}\right)-1}{D^{(+)}_{ex}} - \frac{\exp\left({\dfrac{\zeta_{+}}{2}}\right)-1}{D^{(-)}_{ex}}\right) . \end{align}

Similarly, on $S_{-}$ we have

(9.15)\begin{align} c|_{S_{-}}\partial_{n}\phi|_{S_{-}} ={-} \left(\partial_{t} +\boldsymbol{v}_{s}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} - \partial_{n} v_{p}|_{S} \right) \sqrt{\varepsilon_{in} c|_{S_{-}}} \left( \frac{\exp\left({\dfrac{-\zeta_{-}}{2}}\right)-1}{D^{(+)}_{in}} - \frac{\exp\left({\dfrac{\zeta_{-}}{2}}\right)-1}{D^{(-)}_{in}}\right) . \end{align}

These give the normal derivative $\partial _{n}\phi$ in terms of flow field quantities on $S$ and the $\zeta$-potential and ion concentration at the outer edge of each respective Debye layer, $S_{+}$ or $S_{-}$. This can be used either as data for the integral equations (8.1) and (8.2) for $\phi$ or for the construction of a solution to Laplace's equation

(9.16)\begin{equation} \nabla^{2}\phi = 0 \end{equation}

by some other means.

For a vesicle, analogous data for $\partial _{n}\phi$ in the exterior and interior phase are given by (6.16), which includes a contribution from the trans-membrane ion flux (6.17) and osmotic solvent flux (5.41). These trans-membrane effects introduce an additional coupling via the jumps in the potential $[\phi ]^{S_{+}}_{S_{-}}$ and ion concentration $[c]^{S_{+}}_{S_{-}}$ across the outer edges of the double layer pair.

9.8. Transport relations for the ion concentrations

For a drop, the relation (6.10) for Debye layer ion transport gives the boundary condition on $S_{+}$ that

(9.17)\begin{align} \partial_{n}c|_{S_{+}} = \left( \partial_{t} +\boldsymbol{v}_{s}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} - \partial_{n} v_{p}|_{S} \right) \sqrt{\varepsilon_{ex} c|_{S_{+}}} \left( \frac{\exp\left({\dfrac{-\zeta_{+}}{2}}\right)-1}{D^{(+)}_{ex}} + \frac{\exp\left({\dfrac{\zeta_{+}}{2}}\right)-1}{D^{(-)}_{ex}}\right) , \end{align}

and on $S_{-}$

(9.18)\begin{align} \partial_{n}c|_{S_{-}} ={-} \left( \partial_{t} +\boldsymbol{v}_{s}\boldsymbol{\cdot}\boldsymbol{\nabla}_{s} - \partial_{n} v_{p}|_{S} \right) \sqrt{\varepsilon_{in} c|_{S_{-}}} \left( \frac{\exp\left({\dfrac{-\zeta_{-}}{2}}\right)-1}{D^{(+)}_{in}} + \frac{\exp\left({\dfrac{\zeta_{-}}{2}}\right)-1}{D^{(-)}_{in}}\right) . \end{align}

These are boundary conditions for the transport equation

(9.19)\begin{equation} (\partial_{t}+\boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{\nabla})c=0 \end{equation}

in both the exterior and interior phase outer regions that, as noted in § 6.3, must be applied away from or off $S$ at some chosen small value of the normal coordinate $n \lessgtr 0$. Further, they can only be applied on regions of outflow from the Debye layers to the bulk, which are such that the quantity $\partial _{n}v_{p}|_{S}$ of (9.8) is positive, i.e. where $\partial _{n}v_{p}|_{S}>0$, because otherwise the problem for the ion transport equation outside the Debye layers is over-determined.

For a vesicle, analogous boundary conditions are given by (6.15), which include the influence of trans-membrane ion and osmotic flux terms given by (6.17) and (5.41). In the outflow condition $\partial _{n}v_{p}|_{S}>0$ that determines where the boundary conditions can be applied, membrane incompressibility implies that $\partial _{n}v_{p}|_{S}$ is given by the relation (9.9).

The boundary conditions for both a drop and a vesicle show coupling from the flow field on $S$ and the layer $\zeta$-potentials to the ion concentration in the outer regions away from the Debye layers. For a vesicle, additional coupling is induced when the trans-membrane fluxes are included.

9.9. Determination of the flow field

The fluid velocity $\boldsymbol {u}$ on the interface can be found from the integral equation (7.11) together with its ancillary components. For a drop these are the net or modified traction (7.12) and the electrostatic energy density (7.13). For a vesicle the modified traction is given by (7.15), which includes the effects of membrane bending stiffness and incompressibility, and in the energy density the electrical permittivity is constant throughout the interior and exterior phases.

Instead of solution via an integral equation, the unforced equations of Stokes flow

(9.20a,b)\begin{equation} -\boldsymbol{\nabla} p + \mu {\nabla}^{2}\boldsymbol{u} = 0 \quad \mbox{and} \quad \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{u} = 0 , \end{equation}

where $\mu =\mu _{in}$ on $\varOmega _{in}$ and $\mu =\mu _{ex}$ on $\varOmega _{ex}$, can be solved by some other means, for example by Lamb's general solution, with the reduced stress-balance boundary conditions for a drop or vesicle that are given in Appendix B. In this formulation, because the model is closed at leading order, the tangential fluid velocity is continuous across the outer edges of the double layer pair but there is a jump in pressure and there can be a mismatch in the viscosity $\mu _{in}\neq \mu _{ex}$.

Either formulation shows coupling to the flow field on the interface from the electrostatic and ion concentration fields.

10.1. Sample numerical simulations for the flow about a drop

Table 3 lists representative values of material properties for a typical aqueous electrolyte solution, together with values for the charge-up time scale $\tau _{*c}$ and velocity scale $U_{*c}$, and dimensionless groups. As already noted, these scales are much faster than the scale of the Helmholtz–Smoluchowski slip velocity, which is given in the table by $U_{*}$ based on an applied field strength of $100\ {\rm V} \ {\rm cm}^{-1}$.

Table 3.

Representative scales and dimensionless groups.

The surface tension $\sigma _{*}$ listed in the table is for a water–nitrobenzene interface, and this is a common choice for experiments on ITIES, see, for example, Samec (Reference Samec1988). This fluid pair was taken for the sample numerical solutions, which imply dimensionless drop parameters $\varepsilon _{in} = 0.45$ for permittivity and $\mu _{in}=1.8$ for viscosity with suspending phase values $\varepsilon _{ex}=\mu _{ex}=1$. The ion diffusivities $D_{in,ex}^{(\pm )}$ and ambient concentrations $c_{in}$ and $c_{ex}$ were all set to unity for the computational runs, and the integral equation formulation of (7.11)–(7.13) for the fluid velocity and (8.1)–(8.3) for the potential were used.

Sample data from the simulations are shown in figures 3 and 4. The drop and surroundings are initially at rest with no Galvani potential, and a uniform applied DC field is applied impulsively at $t=0^{+}$ with potential $\phi _{\infty }=-\varPsi z$, which maintains a geometry that is axisymmetric about the $z$-axis. The applied field charges up the double layer pair and sets up the familiar quadrupolar flow pattern about the drop from the initial zero equilibrium state.

Figure 3.

Numerical simulation of the electrokinetic flow induced in and around a drop by a uniform applied field with potential $\phi _{\infty }=-\varPsi z$ and dimensionless strength $\varPsi = 5$. (a) The geometry is axisymmetric about the $z$-axis, with the flow magnitude and direction indicated by arrows, at time $t=2.0$. (b) The $\zeta$-potential in the exterior Debye layer $\zeta _{+}$ versus normalised arc length $\bar {s}$ along the interface in a meridional plane. This is shown from drop pole to pole with $\bar {s} \in [0, 1]$ at a sequence of times from $t=0.5$ to $t=2.0$. (c) The tangential fluid velocity $u_{s}$ on the drop interface versus normalised arc length $\bar {s}$ at the same sequence of times.

Figure 4.

The data here are analogous to those shown in figure 3 but at increased applied field strength $\varPsi = 10$. (a) The axisymmetric, quadrupolar flow field at time $t=2.0$. (b) The exterior $\zeta$-potential $\zeta _{+}$ and (c) the tangential fluid velocity $u_{s}$ on the interface, both shown versus the normalised pole-to-pole arc length $\bar {s}$, at times from $t=0.5$ to $t=2.0$ as indicated.

Figures 3 and 4 show parallel data at different applied field strengths, with $\varPsi =5$ in figure 3 and $\varPsi =10$ in figure 4. Panel $(a)$ shows the flow magnitude and direction with the interface profile at the final, near steady state time $t=2.0$. The deformation is small due to the large value of the inverse capillary number $\varDelta$. With centre-to-pole radius $L$ and centre-to-equator radius $B$, $(L, B) = (1.0011, 0.9994)$ when $\varPsi =5$ and $(L, B) = (1.0021, 0.9986)$ when $\varPsi =10$, so that the shape is slightly prolate. The flow direction is from pole to equator, as is familiar in the context of electrokinetic flow.

Panel $(b)$ of the figures shows data for the $\zeta$-potential in the exterior layer $\zeta _{+}$ versus arc length measured from pole to pole in a meridional plane, normalised so that $\bar {s}\in [0, 1]$. In the absence of a Galvani potential, the layer $\zeta$-potentials and charge are zero at $t=0^{+}$ and increase with time. Figure 3(b) shows that the potential profile is close to a cosine at all times when $\varPsi =5$, but at $\varPsi =10$ figure 4(b) shows the profile departs from this and saturates, especially at later times, with an abrupt change in sign or polarity near the drop equator at $z=0$ and $\bar {s}=0.5$. This effect is also seen in the data of panel $(c)$ of both figures for the tangential fluid velocity on the interface $u_{s}$ versus $\bar {s}$. Here the profile is close to a sine only at $t=0.5$ and $\varPsi =5$, and departs from it at later times or with increase in $\varPsi$.

10.2. Comparison with electrohydrodynamic models derived from the Poisson–Nernst–Planck equations

In common with this electrokinetic study, two studies that are based on the Poisson–Nernst–Planck and Stokes equations, but which diverge from it in that they derive the electrohydrodynamic Taylor–Melcher leaky dielectric model in specific limits, have been given by Schnitzer & Yariv (Reference Schnitzer and Yariv2015) and by Mori & Young (Reference Mori and Young2018). Here we give a short synopsis and comparison.

The study by Schnitzer & Yariv (Reference Schnitzer and Yariv2015) adopts the limit $1/\epsilon \gg \varPsi \gg 1$, proposed by Baygents & Saville (Reference Baygents and Saville1989), in which Debye layers are thin and the strength of the imposed field is asymptotically large. A steady-state and fixed spherical drop geometry are assumed. Importantly, sorption of ions at the sharp interface is included. This limit leads to the presence of Debye layers on either side of an interfacial surface charge distribution. In a first approximation, the potential difference across the Debye layers is relatively weak and the charge they contain completely screens the interfacial charge; the combined layer–interface structure is spherically symmetric with zero net or apparent charge. The symmetry is broken in a second approximation, and the combined layer–interface structure carries non-zero net charge that is related to the normal component of the electric field at its outer edges. The final flow scale that appears is the slip velocity $U_{*}$ of table 3, and convection of surface charge does not appear in their macro-scale model.

This contrasts with the limit taken in our study, where $\epsilon \ll 1$ but $\varPsi =O(1)$. Here there is a large potential difference across each side of the electrical double layer, so that each side contains substantial separated charge, and there is no symmetry of the charge distribution tangential to the interface at leading order. For a drop, we omit ion sorption at the interface; nonetheless an imposed electric field induces a flow that is set up on the charge-up time scale $\tau _{*c}$.

In common with our study for a drop, the study by Mori & Young (Reference Mori and Young2018) also omits interfacial ion sorption and surface charge, but relaxes our assumption of an ideally polarisable interface. Instead, the no-ion flux boundary condition $\boldsymbol {j}^{(\pm )} \boldsymbol {\cdot }\boldsymbol {n}=0$ on $S$ of (2.32) is replaced by continuity of the molecular ion and salt fluxes, so that, e.g., $[\boldsymbol {j}^{(\pm )} \boldsymbol {\cdot }\boldsymbol {n}]=0$. Their study also includes the presence of a non-zero Galvani potential.

Mori & Young (Reference Mori and Young2018) then pass directly to the limit in which the rate of dissociation of salt into its component ions is slow, namely the limit of a weak electrolyte $\alpha \ll 1$, where the ratio of bulk concentration of ions to combined salt is small. In this limit, the concentration of combined salt is piecewise constant, and conservation of ions is replaced by a single nonlinear drift-diffusion equation for the conservation of ionic charge $q=c^{(+)}-c^{(-)}$. This charge diffusion model is then further reduced in the thin Debye layer limit, $\epsilon \ll 1$. The velocity scale is the slip velocity based on the thermal voltage, $U_{*HS}=\varepsilon _{*}(k_{B}T/e)^{2}/\mu _{*}a_{*}$ as introduced below (2.25b), with corresponding time scale $a_{*}/U_{*HS}$. Different scaling regimes for the applied potential, Péclet number and capillary number in terms of $\epsilon$ are found to lead to the macro-scale descriptions of the Taylor–Melcher model, with or without surface charge convection. As in Schnitzer & Yariv (Reference Schnitzer and Yariv2015), the potential difference across the outer sides of the Debye layers and the separated charge they contain is relatively small.