2.1 Bulk inner and outer regions

The bulk oil and aqueous phase beyond the diffuse layer are electrically neutral, and so their dynamics are governed by the unsteady Stokes equations ($Re=Va/\unicode[STIX]{x1D708}\ll 1$)

$$\begin{eqnarray}-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70C}\boldsymbol{u}=-\unicode[STIX]{x1D735}p+\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{u}\quad \text{with }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\end{eqnarray}$$

where the density $\unicode[STIX]{x1D70C}$ and shear viscosity $\unicode[STIX]{x1D702}$ (kinematic viscosity $\unicode[STIX]{x1D708}=\unicode[STIX]{x1D702}/\unicode[STIX]{x1D70C}$) for the oil and aqueous phases are hereafter distinguished using subscripts $i$ and $o$ (denoting inside and outside the drop) when appropriate. With the velocity $V\sim ME\sim 10^{-5}~\text{m}~\text{s}^{-1}$ (taking $M\sim 10^{-8}~\text{m}^{2}~\text{s}^{-1}~\text{V}^{-1}$ with $E\sim 10^{3}~\text{V}~\text{m}^{-1}$) and $\unicode[STIX]{x1D708}\sim 10^{-6}~\text{m}^{2}~\text{s}^{-1}$, a small Reynolds number is readily achieved for submicrometre-sized drops. Moreover, drops may be considered spherical, since the Laplace pressure dominates the inertial stress: balancing an $O(\unicode[STIX]{x1D714}\unicode[STIX]{x1D70C}Va)$ inertial stress with an $O(\unicode[STIX]{x1D6FE}/a)$ Laplace pressure (interfacial tension $\unicode[STIX]{x1D6FE}\sim 0.01~\text{N}~\text{m}^{-1}$) suggests a spherical interface when $a\lesssim \sqrt{\unicode[STIX]{x1D6FE}_{0}/(\unicode[STIX]{x1D714}\unicode[STIX]{x1D70C}_{o}ME)}\sim 1~\text{cm}$ with $\unicode[STIX]{x1D714}/(2\unicode[STIX]{x03C0})\sim 1~\text{kHz}$.

Symmetry, linearity and incompressibility demand solutions of the form

$$\begin{eqnarray}\boldsymbol{u}=\unicode[STIX]{x1D735}\times [f(r)\boldsymbol{X}\times \boldsymbol{e}_{r}]=(f_{r}+fr^{-1})\boldsymbol{X}+(-f_{r}+fr^{-1})\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}\boldsymbol{e}_{r},\end{eqnarray}$$

with

$$\begin{eqnarray}p=\hat{p}(r)\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r},\end{eqnarray}$$

where $\boldsymbol{X}=\boldsymbol{U}$ or $\boldsymbol{E}$, and $\boldsymbol{e}_{r}$ is a radial unit vector with $\boldsymbol{e}_{\unicode[STIX]{x1D703}}$ its tangential counterpart.

For mathematical convenience, we consider solutions for a stationary sphere at the origin with either a far-field translation of the fluid $\boldsymbol{U}$ with electric field $\boldsymbol{E}=0$ or a finite electric field $\boldsymbol{E}$ with stationary far-field fluid $\boldsymbol{U}=0$. These far-field boundary conditions define what are termed the $U$- and $E$-problems, and their linear superposition will be used to construct a solution that satisfies boundary conditions that are compatible with the particle equation of motion.

Substituting the fluid velocity (which is divergence-free) and pressure above into the Stokes equations requires

(2.2)$$\begin{eqnarray}\hat{p}=\text{i}\unicode[STIX]{x1D6FA}r^{2}(f_{r}r^{-1}+fr^{-2})+rf_{rrr}+3f_{rr}-2f_{r}r^{-1}+2fr^{-2},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D714}\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D702}$ (square of the reciprocal viscous penetration depth). Note that the continuity equation furnishes $\unicode[STIX]{x1D6FB}^{2}p=0$, so $\hat{p}=Ar^{-2}+Br$, where $A$ and $B$ are constants that can be ascertained from (2.2).

The general solution, leading to (2.3) and (2.4) for $f$ and (2.5) and (2.6) for $\hat{p}$ below, is found as follows. Taking the curl of the momentum equation gives

$$\begin{eqnarray}-\text{i}\unicode[STIX]{x1D6FA}\mathscr{L}(f)=\mathscr{L}^{2}(f),\end{eqnarray}$$

where

$$\begin{eqnarray}\mathscr{L}=\frac{\unicode[STIX]{x2202}^{2}}{\unicode[STIX]{x2202}r^{2}}+\frac{2}{r}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}-\frac{2}{r^{2}}.\end{eqnarray}$$

Defining

$$\begin{eqnarray}g=\mathscr{L}(f)\end{eqnarray}$$

gives

$$\begin{eqnarray}-\text{i}\unicode[STIX]{x1D6FA}g=g_{rr}+2g_{r}/2-2g/r^{2},\end{eqnarray}$$

for which

$$\begin{eqnarray}g=c_{2}\text{y}_{1}(\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}}r)-c_{1}\text{j}_{1}(\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}}r)\end{eqnarray}$$

and

$$\begin{eqnarray}f_{rr}+2f_{r}/r-2f/r^{2}=g.\end{eqnarray}$$

The solution of this equation (in which $\text{y}_{1}$ and $\text{j}_{1}$ are spherical Bessel functions) has four integration constants for each domain ($c_{1}$–$c_{4}$, $c_{1}^{\prime }$–$c_{4}^{\prime }$).

However, in the inside domain, removing singularities at the origin, and requiring zero radial velocity at $r=a$, furnish

(2.3)$$\begin{eqnarray}f=c_{4}^{\prime }r+c_{1}^{\prime }\frac{\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}r\cos (\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}r)-\text{sin}(\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}r)}{\unicode[STIX]{x1D6FA}_{i}^{2}r^{2}}\quad \text{for }r<a,\end{eqnarray}$$

where

$$\begin{eqnarray}c_{4}^{\prime }=c_{1}^{\prime }a\frac{\sin (\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}a)-\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}a\cos (\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}a)}{\unicode[STIX]{x1D6FA}_{i}^{2}a^{4}}.\end{eqnarray}$$

Similarly, in the outside domain, removing singularities in the far field and requiring zero radial velocity as $r\rightarrow a$, furnish

(2.4)$$\begin{eqnarray}f=\frac{U}{2X}r+c_{3}r^{-2}+c_{1}\frac{(\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{o}}r+\text{i})\text{e}^{(\text{i}-1)\sqrt{\unicode[STIX]{x1D6FA}_{o}/2}r}}{\unicode[STIX]{x1D6FA}_{o}^{2}r^{2}}\quad \text{for }r>a,\end{eqnarray}$$

where

$$\begin{eqnarray}c_{3}=-\frac{U}{2X}a^{3}-c_{1}a\frac{(\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{o}}a+\text{i})\text{e}^{(\text{i}-1)\sqrt{\unicode[STIX]{x1D6FA}_{o}/2}a}}{\unicode[STIX]{x1D6FA}_{o}^{2}a}.\end{eqnarray}$$

The inner and outer flows are therefore determined to within one unknown scalar coefficient each: $c_{1}^{\prime }$ and $c_{1}$. We will prescribe these using boundary conditions arising from analysis of the interfacial region, as undertaken in § 2.2 below. Similarly to the fluid properties, subscripts $i$ and $o$ attached to $\unicode[STIX]{x1D6FA}$ denote inside and outside the drop.

With the entire fluid velocity (outside the interfacial region) now prescribed by just two scalar constants of integration, equation (2.2) furnishes

(2.5)$$\begin{eqnarray}\hat{p}=-{\textstyle \frac{2}{3}}\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}c_{1}^{\prime }r\quad \text{for }r<a\end{eqnarray}$$

and

(2.6)$$\begin{eqnarray}\hat{p}=\text{i}\unicode[STIX]{x1D6FA}_{o}\left(\frac{U}{X}r-c_{3}r^{-2}\right)\quad \text{for }r>a.\end{eqnarray}$$

These confirm that $p$ satisfies Laplace’s equation. Note that $\unicode[STIX]{x1D6FA}$ and the integration constants must be prescribed separately for the inside and outside domains, which are coupled by boundary conditions at $r=a$ that account for electro-osmotic and Marangoni flow in the interfacial region. For future reference, note that the tangential velocity $\boldsymbol{u}_{\unicode[STIX]{x1D703}}$ and traction $\boldsymbol{t}_{\unicode[STIX]{x1D703}}$ (on the surface with inward unit normal $-\boldsymbol{e}_{r}$) inside the interface (i.e. at the interface, on the drop side, $r=a_{-}$) are

$$\begin{eqnarray}\boldsymbol{u}_{\unicode[STIX]{x1D703}}(r=a_{-})=f_{r}\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}}=c_{1}^{\prime }aV_{i}(\unicode[STIX]{x1D6FA}_{i}a^{2})\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}},\end{eqnarray}$$

where

(2.7)$$\begin{eqnarray}V_{i}(\unicode[STIX]{x1D6FA}_{i}a^{2})=\frac{(3-\text{i}\unicode[STIX]{x1D6FA}_{i}a^{2})\sin (\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}a)-3\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}a\cos (\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}a)}{\unicode[STIX]{x1D6FA}_{i}^{2}a^{4}}\end{eqnarray}$$

and

$$\begin{eqnarray}\boldsymbol{t}_{\unicode[STIX]{x1D703}}(r=a_{-})=-\unicode[STIX]{x1D702}_{i}f_{rr}(a)\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}}=-\unicode[STIX]{x1D702}_{i}c_{1}^{\prime }T_{i}(\unicode[STIX]{x1D6FA}_{i}a^{2})\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}},\end{eqnarray}$$

where

(2.8)$$\begin{eqnarray}\displaystyle & & \displaystyle \hspace{-9.60004pt}T_{i}(\unicode[STIX]{x1D6FA}_{i}a^{2})\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{(6\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}a-(-1)^{3/4}\unicode[STIX]{x1D6FA}_{i}^{3/2}a^{3})\cos (\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}a)+(3\text{i}\unicode[STIX]{x1D6FA}_{i}a^{2}-6)\sin (\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}a)}{\unicode[STIX]{x1D6FA}_{i}^{2}a^{4}}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Outside the interface (i.e. at the interface, on the aqueous side, $r=a_{+}$), the tangential velocity and traction (surface with outward unit normal $\boldsymbol{e}_{r}$) are

$$\begin{eqnarray}\boldsymbol{u}_{\unicode[STIX]{x1D703}}(r=a_{+})=f_{r}\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}}\quad \text{and}\quad \boldsymbol{t}_{\unicode[STIX]{x1D703}}(r=a_{+})=\unicode[STIX]{x1D702}_{o}f_{rr}\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}},\end{eqnarray}$$

where

$$\begin{eqnarray}f_{r}(r=a_{+})=\frac{3U}{2X}-c_{1}aV_{o}(\unicode[STIX]{x1D6FA}_{o}a^{2})\end{eqnarray}$$

and

$$\begin{eqnarray}f_{rr}(r=a_{+})=-\frac{3U}{Xa}+c_{1}T_{o}(\unicode[STIX]{x1D6FA}_{o}a^{2}),\end{eqnarray}$$

which define

$$\begin{eqnarray}V_{o}(\unicode[STIX]{x1D6FA}_{o}a^{2})=\frac{\text{e}^{(\text{i}-1)\sqrt{\unicode[STIX]{x1D6FA}_{o}/2}a}}{\unicode[STIX]{x1D6FA}_{o}a^{2}}\end{eqnarray}$$

and

$$\begin{eqnarray}T_{o}(\unicode[STIX]{x1D6FA}_{o}a^{2})=[3-(\text{i}-1)\sqrt{\unicode[STIX]{x1D6FA}_{o}/2}a]V_{o}(\unicode[STIX]{x1D6FA}_{o}a^{2}).\end{eqnarray}$$

The coefficient $c_{3}$ for the outside flow measures the (dipole) strength of the decaying contribution to the pressure, which also measures the slowest-decaying contribution to the velocity disturbance, e.g. as identified in (2.4). Note that integrating the total hydrodynamic traction over the surface of a sphere furnishes a force

$$\begin{eqnarray}\boldsymbol{F}(r=a_{+})=-{\textstyle \frac{4}{3}}\unicode[STIX]{x03C0}a^{2}\unicode[STIX]{x1D702}_{o}\boldsymbol{X}[\text{i}\unicode[STIX]{x1D6FA}_{o}(rf_{r}+f)+rf_{rrr}+f_{rr}-6f_{r}r^{-1}+6fr^{-2}],\end{eqnarray}$$

where the terms premultiplied by $\text{i}\unicode[STIX]{x1D6FA}_{o}$ come from the pressure/normal traction. This formula does not include the electrical stress, which is non-zero on the surface of the drop. This physics enters via the interfacial region and its coupling to the outer flows, as addressed in § 2.2 below. A much more convenient evaluation of the force is available that requires only knowledge of $c_{3}$ (measuring the far-field decay of the outer velocity disturbance), furnishing (Mangelsdorf & White Reference Mangelsdorf and White1992)

$$\begin{eqnarray}\boldsymbol{F}=6\unicode[STIX]{x03C0}\unicode[STIX]{x1D702}_{o}a\boldsymbol{X}{\textstyle \frac{2}{3}}\text{i}\unicode[STIX]{x1D6FA}_{o}a^{2}c_{3}a^{-3}.\end{eqnarray}$$

Accompanying the foregoing hydrodynamic flows is an electrostatic potential $\unicode[STIX]{x1D713}^{\prime }$, which can be shown (at sufficiently high frequency) to satisfy Laplace’s equation everywhere, i.e. in the bulk and interfacial regions:

(2.9)$$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}^{\prime }=0.\end{eqnarray}$$

Thus, linearity, symmetry and the far field require

$$\begin{eqnarray}\unicode[STIX]{x1D713}^{\prime }=r[-(E/X)+\hat{d}_{\unicode[STIX]{x1D713}}r^{-3}]\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}\quad \text{for }r>a,\end{eqnarray}$$

where we will term $\hat{d}_{\unicode[STIX]{x1D713}}$ the (electrostatic) dipole strength. This measures electrical polarization of the interfacial region, which has contributions from the charge-density perturbations in the diffuse layer and at the interface due to lateral transport of charged surfactant. To be finite inside the drop and continuous across the interface (continuous tangential electric field (Baygents & Saville Reference Baygents and Saville1991a)), the potential inside the drop must be

$$\begin{eqnarray}\unicode[STIX]{x1D713}^{\prime }=r[-(E/X)+\hat{d}_{\unicode[STIX]{x1D713}}a^{-3}]\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}\quad \text{for }r<a.\end{eqnarray}$$

Perturbations to the equilibrium interfacial charge and momentum balances are addressed below: together, these determine the presently unknown scalars, $\hat{d}_{\unicode[STIX]{x1D713}}$, $c_{1}$ and $c_{1}^{\prime }$ (and therefore $c_{3}$).

2.2 Inner/interfacial region

This inner region provides matching conditions for the foregoing solutions in the bulk regions. Here, in addition to mass and momentum conservation, we must consider the conservation of ions and surfactant molecules in the aqueous phase, and of surfactant molecules adsorbed on the interface. These differential relationships are coupled, and we will show that their solution leads to a set of linear algebraic relationships which, when combined with those in § 2.1, solve the full dynamic electrokinetic model.

We begin with a species conservation equation for any molecular species (subscript $i$) that is subject to advection, diffusion and electromigration:

(2.10)$$\begin{eqnarray}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }\,\boldsymbol{j}_{i}=-\text{i}\unicode[STIX]{x1D714}c_{i}^{\prime }\quad \text{with }\boldsymbol{j}_{i}=-D_{i}\unicode[STIX]{x1D735}c_{i}^{\prime }-\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{0}z_{i}ec_{i}^{\prime }\frac{D_{i}}{k_{B}T}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\prime }z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+\boldsymbol{u}c_{i}^{0}.\end{eqnarray}$$

Note that we have linearized the flux with respect to perturbations defined by $c_{i}=c_{i}^{0}+c_{i}^{\prime }$ and $\unicode[STIX]{x1D713}=\unicode[STIX]{x1D713}^{0}+\unicode[STIX]{x1D713}^{\prime }$, where $c_{i}$ denotes the concentration of a species with $c_{i}^{0}$ the equilibrium concentration, and $c_{i}^{\prime }$ the perturbation induced by the forcing $\boldsymbol{X}$.

Similarly, $\unicode[STIX]{x1D713}^{0}$ and $\unicode[STIX]{x1D713}^{\prime }$ denote the equilibrium electrostatic potential and perturbation. As usual, $D_{i}$, $z_{i}$, $e$ and $k_{B}T$ are the diffusion coefficient, valence, fundamental charge and thermal energy. Note that the equilibrium quantities are functions of radial position, for which tangential gradients vanish in the formulae below. As is well known, the equilibrium state satisfies the nonlinear Poisson–Boltzmann equation (Russel et al. Reference Russel, Saville and Showalter1989):

$$\begin{eqnarray}\unicode[STIX]{x1D716}\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D713}^{0}(r)=-\mathop{\sum }_{i}z_{i}ec_{i}^{0}(r),\end{eqnarray}$$

where ($\unicode[STIX]{x1D716}\unicode[STIX]{x1D716}_{0}$ is the dielectric permittivity)

$$\begin{eqnarray}c_{i}^{0}(r)=c_{i}^{\infty }\text{e}^{-\unicode[STIX]{x1D713}^{0}(r)z_{i}e/k_{B}T}.\end{eqnarray}$$

The radial and tangential components of $\boldsymbol{j}_{i}$ are

$$\begin{eqnarray}\boldsymbol{j}_{i,r}=\left[-D_{i}\unicode[STIX]{x1D735}c_{i}^{\prime }-\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{0}z_{i}ec_{i}^{\prime }\frac{D_{i}}{k_{B}T}-\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\prime }z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+\boldsymbol{u}c_{i}^{0}\right]\boldsymbol{\cdot }\boldsymbol{e}_{r}\boldsymbol{e}_{r}\end{eqnarray}$$

and

$$\begin{eqnarray}\boldsymbol{j}_{i,\unicode[STIX]{x1D703}}=\left[-D_{i}\unicode[STIX]{x1D735}c_{i}^{\prime }-\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\prime }z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+\boldsymbol{u}c_{i}^{0}\right]\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}}.\end{eqnarray}$$

These can be used to develop conservation equations for an interfacial control volume (thickness ${\sim}\unicode[STIX]{x1D705}^{-1}\ll a$, as identified by the region bounded by dashed lines in figure 1):

$$\begin{eqnarray}-\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }\left\langle -D_{i}\unicode[STIX]{x1D735}_{s}c_{i}^{\prime }-\unicode[STIX]{x1D735}_{s}\unicode[STIX]{x1D713}^{\prime }z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+\boldsymbol{u}_{\unicode[STIX]{x1D703}}c_{i}^{0}\right\rangle -\boldsymbol{j}_{r}\boldsymbol{\cdot }\boldsymbol{e}_{r}=\langle -\text{i}\unicode[STIX]{x1D714}c_{i}^{\prime }\rangle ,\end{eqnarray}$$

where $\unicode[STIX]{x1D735}_{s}$ is the surface gradient operator, $\boldsymbol{u}_{\unicode[STIX]{x1D703}}=\boldsymbol{u}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}}$, and

$$\begin{eqnarray}\langle \cdot \rangle =\int _{r^{\prime }=a}^{r}\cdot \,\text{d}r^{\prime }.\end{eqnarray}$$

Linearity and symmetry require the general forms

$$\begin{eqnarray}c_{i}^{\prime }=d_{i,c}(r)\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}\quad \text{and}\quad \unicode[STIX]{x1D713}^{\prime }=d_{\unicode[STIX]{x1D713}}(r)\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r},\end{eqnarray}$$

with

$$\begin{eqnarray}\boldsymbol{u}_{r}=2fr^{-1}\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}\boldsymbol{e}_{r}\quad \text{and}\quad \boldsymbol{u}_{\unicode[STIX]{x1D703}}=(f_{r}+fr^{-1})\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}}.\end{eqnarray}$$

Note that, within the diffuse layer, the function $f$ here is not the one calculated in the section above (outer domain), since the momentum equation for the diffuse layer includes an electrical body force that drives electro-osmotic flow.

With the foregoing forms of the perturbation functions, the conservation equations (one for each species) for the diffuse layer become

(2.11)$$\begin{eqnarray}\left\langle 2r^{-2}(\text{i}\unicode[STIX]{x1D714}r^{2}/2-D_{i})d_{i,c}-2r^{-1}d_{\unicode[STIX]{x1D713}}z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+2r^{-1}(f_{r}+fr^{-1})c_{i}^{0}\right\rangle \boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}=\boldsymbol{j}_{i,r}\boldsymbol{\cdot }\boldsymbol{e}_{r},\end{eqnarray}$$

where

$$\begin{eqnarray}\boldsymbol{j}_{i,r}\boldsymbol{\cdot }\boldsymbol{e}_{r}=\left[-D_{i}d_{i,c,r}-\unicode[STIX]{x1D713}_{r}^{0}z_{i}ed_{i,c}\frac{D_{i}}{k_{B}T}-d_{\unicode[STIX]{x1D713},r}z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+2fr^{-1}c_{i}^{0}\right]\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}.\end{eqnarray}$$

Note that we have prescribed zero flux on the oil side of the oil–water interface, assuming that the integrals are finite when $r\gtrsim a+\unicode[STIX]{x1D705}^{-1}=a_{+}$.

Applying (2.11) to the surfactant adsorbed on the interface (taking the thickness of the control volume to zero with a Dirac-delta concentration $(c^{0}+c^{\prime })\unicode[STIX]{x1D6FF}(r-a)$), where (as required by linearity and symmetry)

$$\begin{eqnarray}c^{\prime }=d_{c}\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}\end{eqnarray}$$

is the surface-concentration perturbation, gives

$$\begin{eqnarray}(\text{i}\unicode[STIX]{x1D714}a^{2}/2-D)d_{c}-d_{\unicode[STIX]{x1D713}}zec^{0}\frac{D}{k_{B}T}+a(f_{r}+fa^{-1})c^{0}=0\quad \text{at }r=a,\end{eqnarray}$$

where the absence of a subscript $i$ attached to $c^{\prime }$, $c^{0}$, $d_{c}$, $D$ and $z$ distinguishes the adsorbed surfactant from the non-adsorbed species; it follows that $c^{0}$ is now a surface concentration (molecules per unit area). Note that this balance captures tangential transport (electromigration, diffusion and advection) of the adsorbed surfactant, neglecting exchange with the oil and aqueous phases. Here, the approximation is motivated by the high frequency of the oscillating electric field. Thus, with diffusion-limited exchange kinetics, we anticipate an $O(D/a^{2})$ limit on the frequency below which the model may break down.

The constant $d_{c}$ measures the concentration polarization of the interface, which is forced by advection and electromigration, and relaxed by diffusion. The surface divergence of the interface velocity is

(2.12)$$\begin{eqnarray}\unicode[STIX]{x1D735}_{s}\boldsymbol{\cdot }\boldsymbol{u}_{\unicode[STIX]{x1D703}}=-2a^{-1}(f_{r}+fa^{-1})\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}=-2a^{-1}c_{1}^{\prime }V_{i}(\unicode[STIX]{x1D6FA}_{i}a^{2})X\cos \unicode[STIX]{x1D703}.\end{eqnarray}$$

A charge conservation relationship for the interfacial control volume is obtained by multiplying the species conservation equations above by the respective charge $z_{i}e$ and summing over all species:

(2.13)$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{i}\left\langle -2r^{-2}D_{i}d_{c,i}-2r^{-2}d_{\unicode[STIX]{x1D713}}z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+2r^{-1}(f_{r}+fr^{-1})c_{i}^{0}\right\rangle z_{i}e\nonumber\\ \displaystyle & & \displaystyle \quad -\,\mathop{\sum }_{i}\left[-D_{i}d_{c,i,r}-\unicode[STIX]{x1D713}_{r}^{0}z_{i}ed_{c,i}\frac{D_{i}}{k_{B}T}-d_{\unicode[STIX]{x1D713},r}z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+2fr^{-1}c_{i}^{0}\right]z_{i}e=-\text{i}\unicode[STIX]{x1D714}d_{\unicode[STIX]{x1D70E}},\qquad\end{eqnarray}$$

where

$$\begin{eqnarray}d_{\unicode[STIX]{x1D70E}}=\mathop{\sum }_{i}z_{i}e\langle d_{c,i}\rangle\end{eqnarray}$$

measures the net charge density (per unit area). Recall that we have taken the radial flux of all species at $r=a_{-}$ to be zero, so the second sum is to be evaluated at $r\sim a+\unicode[STIX]{x1D705}^{-1}=a_{+}$.

Similarly to the radially averaged species conservation equation above, the (perturbed) Poisson equation

$$\begin{eqnarray}-\unicode[STIX]{x1D735}\boldsymbol{\cdot }(\unicode[STIX]{x1D716}\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\prime })=\mathop{\sum }_{i}z_{i}ec_{i}^{\prime }\end{eqnarray}$$

may be written as

$$\begin{eqnarray}-\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\prime }\boldsymbol{\cdot }\boldsymbol{e}_{r}|_{a_{+}}+\unicode[STIX]{x1D716}_{i}\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\prime }\boldsymbol{\cdot }\boldsymbol{e}_{r}|_{a_{-}}-\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}\unicode[STIX]{x1D6FB}_{s}^{2}\langle \unicode[STIX]{x1D713}^{\prime }\rangle =\mathop{\sum }_{i}z_{i}e\langle c_{i}^{\prime }\rangle ,\end{eqnarray}$$

which ($\unicode[STIX]{x1D716}_{i}\unicode[STIX]{x1D716}_{0}$ and $\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}$ are the dielectric permittivities of the inside (oil) and outside (aqueous) phases), using the foregoing forms of the perturbation functions, becomes

(2.14)$$\begin{eqnarray}-\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}d_{\unicode[STIX]{x1D713},r}|_{a_{+}}+\unicode[STIX]{x1D716}_{i}\unicode[STIX]{x1D716}_{0}d_{\unicode[STIX]{x1D713},r}|_{a_{-}}+\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}2a^{-2}\langle d_{\unicode[STIX]{x1D713}}\rangle =\mathop{\sum }_{i}z_{i}e\langle d_{c,i}\rangle =d_{\unicode[STIX]{x1D70E}}.\end{eqnarray}$$

We can now eliminate $d_{\unicode[STIX]{x1D70E}}$ from the Poisson and charge-conservation equations (2.13) and (2.14), furnishing

(2.15)$$\begin{eqnarray}\displaystyle & & \displaystyle \text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}d_{\unicode[STIX]{x1D713},r}|_{a_{+}}-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{i}\unicode[STIX]{x1D716}_{0}d_{\unicode[STIX]{x1D713},r}|_{a_{-}}+\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}2a^{-2}\langle d_{\unicode[STIX]{x1D713}}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{i}\left\langle -2r^{-2}D_{i}d_{c,i}-2r^{-2}d_{\unicode[STIX]{x1D713}}z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+2r^{-1}(f_{r}+fr^{-1})c_{i}^{0}\right\rangle z_{i}e\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\mathop{\sum }_{i}\left[-D_{i}d_{c,i,r}-\unicode[STIX]{x1D713}_{r}^{0}z_{i}ed_{c,i}\frac{D_{i}}{k_{B}T}-d_{\unicode[STIX]{x1D713},r}z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+2fr^{-1}c_{i}^{0}\right]z_{i}e\nonumber\\ \displaystyle & & \displaystyle \qquad -\,2a^{-2}Dzed_{c}-2a^{-2}d_{\unicode[STIX]{x1D713}}(ze)^{2}c^{0}\frac{D}{k_{B}T}+2a^{-1}(f_{r}+fa^{-1})c^{0}ze.\end{eqnarray}$$

Note that sums are now over all mobile species in the diffuse layer, and the last line (tangential transport of the surfactant at the interface) is the separate contribution (to the interfacial charge balance from radial and tangential fluxes) of the adsorbed species at the interface with $f$ (i.e. the advective velocity) evaluated at $r=a_{-}$ for the oil phase (where $f=0$ due to the radial velocity vanishing at the interface).

Next, based on a thin diffuse layer, we note that the third term on the left-hand side of (2.15) will be $O(\unicode[STIX]{x1D705}^{-1}/a)$ smaller than the other terms, and may therefore be neglected. Moreover, O’Brien (Reference O’Brien1988) has analysed the other radial and tangential fluxes in the second and third lines of equation (2.15), for thin diffuse layers, showing that they can be expressed in terms of the ‘surface’ (hereafter termed ‘diffuse layer’) and bulk conductivities, $K_{s}$ and $K_{\infty }$, with the electrostatic potential inside the diffuse layer well approximated by the electrostatic potential in the outer domain, which satisfies Laplace’s equation. Note that O’Brien’s $K_{s}$ accounts for charge transport within the (thin) diffuse layer and its exchange with the outer bulk domain. Thus, drawing on his analysis of current in the diffuse layer, we set

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{i}\left\langle -2r^{-2}D_{i}d_{c,i}-2r^{-2}d_{\unicode[STIX]{x1D713}}z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+2r^{-1}(f_{r}+fr^{-1})c_{i}^{0}\right\rangle z_{i}e\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\mathop{\sum }_{i}\left[-D_{i}d_{c,i,r}-\unicode[STIX]{x1D713}_{r}^{0}z_{i}ed_{c,i}\frac{D_{i}}{k_{B}T}-d_{\unicode[STIX]{x1D713},r}z_{i}ec_{i}^{0}\frac{D_{i}}{k_{B}T}+2fr^{-1}c_{i}^{0}\right]z_{i}e\nonumber\\ \displaystyle & & \displaystyle \quad =2K_{s}a^{-1}(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})-K_{\infty }(E/X+2\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,2a^{-1}(f_{r}+fa^{-1})\mathop{\sum }_{i}\langle c_{i}^{0}\rangle z_{i}e-2fa^{-1}\mathop{\sum }_{i}c_{i}^{0}z_{i}e,\nonumber\end{eqnarray}$$

where we have used (§ 2.1) $d_{\unicode[STIX]{x1D713}}(r=a_{+})=-a(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})$, $d_{\unicode[STIX]{x1D713},r}(r=a_{+})=-(E/X+2\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})$ and $d_{\unicode[STIX]{x1D713},r}(r=a_{-})=-(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})$. Note that the last two terms capture the advective contribution to the current arising from the interfacial mobility. The first of these is the tangential velocity of the interface multiplied by the net charge of the diffuse layer, and the second is the radial velocity of the interface multiplied by the net (vanishing) charge outside the diffuse layer. From the boundary conditions in § 2.1, the only non-zero term is

$$\begin{eqnarray}2a^{-1}f_{r}\mathop{\sum }_{i}\langle c_{i}^{0}\rangle z_{i}e=-2a^{-1}f_{r}c^{0}ze,\end{eqnarray}$$

which, by overall electroneutrality of the interfacial control volume (interface and diffuse layer), is equal and opposite to the advective current of the adsorbed species at the interface. Equation (2.15) therefore reduces to

$$\begin{eqnarray}\displaystyle & & \displaystyle -\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}(E/X+2\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})+\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{i}\unicode[STIX]{x1D716}_{0}(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})\nonumber\\ \displaystyle & & \displaystyle \quad =2K_{s}a^{-1}(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})-K_{\infty }(E/X+2\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,2a^{-2}Dzed_{c}+2a^{-1}(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})(ze)^{2}c^{0}\frac{D}{k_{B}T}.\nonumber\end{eqnarray}$$

For a rigid interface with immobile surface charge ($d_{c}=D=0$), this furnishes O’Brien’s electrostatic dipole strength:

(2.16)$$\begin{eqnarray}\hat{d}_{\unicode[STIX]{x1D713}}a^{-3}=\frac{E}{X}\frac{\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{i}\unicode[STIX]{x1D716}_{0}+2K_{s}a^{-1}-K_{\infty }}{-2\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}-\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{i}\unicode[STIX]{x1D716}_{0}+2K_{s}a^{-1}+2K_{\infty }},\end{eqnarray}$$

which clearly vanishes for the problem in which the drop translates in the absence of an applied electric field. This is a consequence of the thin-double-layer approximations.

Table 1.

Representative set of model parameters ($T=25\,^{\circ }\text{C}$) and characteristic frequencies, e.g. for the spectra plotted in figures 5 and 7. The variables that can be varied experimentally are the oil volume fraction $\unicode[STIX]{x1D719}$, drop size $a$, total surfactant concentration $c_{\infty ,0}$ and added salt concentration $c_{-}^{\infty }$. All other quantities are derived from these and known material properties, as detailed in the main text. Here $I_{s}$ is the ionic strength for the added salt (NaCl) with $I$ the total ionic strength (SDS and NaCl).

We must now address momentum conservation for the interfacial control volume. Here we note that the viscous diffusion time for the diffuse layer $\unicode[STIX]{x1D705}^{-2}/\unicode[STIX]{x1D708}\lesssim O(10^{-17}/10^{-6})=O(10^{-11})~\text{s}$ (this is the shortest relaxation time, table 1), justifying quasi-steady hydrodynamics within the diffuse layer at frequencies $f\lesssim 100~\text{MHz}$. Accordingly, the fluid velocity outside the diffuse layer slips relative to the interface by the well-known Smoluchowski-slip velocity (Russel et al. Reference Russel, Saville and Showalter1989; Hunter Reference Hunter2001):

$$\begin{eqnarray}\boldsymbol{u}_{S}=M_{S}\unicode[STIX]{x1D735}\unicode[STIX]{x1D713}^{\prime }\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}},\end{eqnarray}$$

where the interfacial-slip mobility $M_{S}=\unicode[STIX]{x1D701}\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}/\unicode[STIX]{x1D702}_{o}$, with $\unicode[STIX]{x1D701}=\unicode[STIX]{x1D713}^{0}(r=a)$ the equilibrium electrostatic potential at the interface ($\unicode[STIX]{x1D702}_{o}$ is the aqueous-phase shear viscosity).

As highlighted by Hunter (Reference Hunter2001), the Smoluchowski-slip velocity emerges from a neglect (in the momentum balance) of ion-concentration perturbations in the diffuse layer. He shows that this may be questionable for highly charged interfaces, but the question still remains as to the magnitude of such errors. To assess these, we compare in figure 2 O’Brien’s dynamic mobility formula ((3.2) below, requiring $\unicode[STIX]{x1D714}a^{2}/D\gtrsim 1$, $\unicode[STIX]{x1D701}e/k_{B}T\gtrsim 1$, $\unicode[STIX]{x1D705}a\gg 1$) with the standard electrokinetic model (Mangelsdorf & White Reference Mangelsdorf and White1992), as computed by the MPEK software package (Hill, Saville & Russel Reference Hill, Saville and Russel2003). Note that the surface potentials ($|\unicode[STIX]{x1D701}|\gg k_{B}T/e$) and ionic strengths are representative of the values adopted below for SDS-stabilized oil-in-water emulsions. As cautioned by Hunter (Reference Hunter2001), for these extremely highly charged interfaces, O’Brien’s formula – and therefore the theory for fluid spheres advanced here – departs somewhat from the numerically exact solutions in the megahertz range. Also consistent with Hunter’s analysis is the error diminishing with increasing $\unicode[STIX]{x1D705}a$.

Figure 2.

Dynamic electrophoretic mobility spectra (magnitude and phase) for large, highly charged rigid spheres. Calculations are according to the standard electrokinetic model (solid lines) and O’Brien’s dynamic mobility formula ((3.2) with $D=0$; dashed lines): $\unicode[STIX]{x1D701}=-224~\text{mV}$, $I=4.63~\text{mM}$ ($\unicode[STIX]{x1D705}a=90$, blue); $\unicode[STIX]{x1D701}=-215~\text{mV}$, $I=8.50~\text{mM}$ ($\unicode[STIX]{x1D705}a=120$, red); and $\unicode[STIX]{x1D701}=-200~\text{mV}$, $I=23.0~\text{mM}$ ($\unicode[STIX]{x1D705}a=200$, yellow). Other parameters: $a=400~\text{nm}$, $\unicode[STIX]{x1D70C}_{i}=768~\text{kg}~\text{m}^{-3}$, $\unicode[STIX]{x1D70C}_{o}=998~\text{kg}~\text{m}^{-3}$, $\unicode[STIX]{x1D716}_{i}=2$, $\unicode[STIX]{x1D716}_{o}=78$, aqueous NaCl electrolyte at $T=298~\text{K}$. Vertical lines identify the relaxation frequencies $D\unicode[STIX]{x1D705}^{2}$ (solid) and $D/a^{2}$ (dashed) with $D=D_{\text{Na}^{+}}=1.33\times 10^{-9}~\text{m}^{2}~\text{s}^{-1}$. Errors in the megahertz range can be attributed to ion-concentration perturbations in the diffuse layers, and thus a breakdown of the Smoluchowski-slip velocity formula (Hunter Reference Hunter2001).

Note that the net electrical force acting on the interface and diffuse layer is zero, so the tangential tractions on the interfacial control volume sum to zero:

$$\begin{eqnarray}\boldsymbol{t}_{\unicode[STIX]{x1D703}}(r=a_{-})+\boldsymbol{t}_{\unicode[STIX]{x1D703}}(r=a_{+})-\unicode[STIX]{x1D6FE}^{0}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D735}_{s}c^{\prime }=0,\end{eqnarray}$$

with tangential slip (across the diffuse layer)

$$\begin{eqnarray}\boldsymbol{u}_{\unicode[STIX]{x1D703}}(r=a_{-})+\boldsymbol{u}_{S}=\boldsymbol{u}_{\unicode[STIX]{x1D703}}(r=a_{+}).\end{eqnarray}$$

The hydrodynamic tractions, e.g. $\boldsymbol{t}_{\unicode[STIX]{x1D703}}(r=a_{+})=\{-p\boldsymbol{I}+\unicode[STIX]{x1D702}_{o}[\unicode[STIX]{x1D735}\boldsymbol{u}+(\unicode[STIX]{x1D735}\boldsymbol{u})^{\text{T}}]\}\boldsymbol{\cdot }\boldsymbol{e}_{r}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}}$, are evaluated according to the fluid velocity in the bulk outer regions, and the Marangoni traction $-\unicode[STIX]{x1D6FE}^{0}\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D735}_{s}c^{\prime }=-\unicode[STIX]{x1D6FE}^{0}\unicode[STIX]{x1D6FD}d_{c}a^{-1}\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{\unicode[STIX]{x1D703}}\boldsymbol{e}_{\unicode[STIX]{x1D703}}$ is from the perturbation in surface tension $\unicode[STIX]{x1D6FE}^{\prime }$ that accompanies the perturbation in the interfacial concentration $c^{\prime }$. Thus, with $\unicode[STIX]{x1D6FE}^{0}$ the equilibrium surface tension, which will be prescribed according to an equilibrium isotherm, we define

$$\begin{eqnarray}\unicode[STIX]{x1D6FE}^{0}\unicode[STIX]{x1D6FD}=-\left.\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}}{\unicode[STIX]{x2202}c}\right|_{c^{0}}>0\end{eqnarray}$$

via the linearization $\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D6FE}^{0}(1-\unicode[STIX]{x1D6FD}c^{\prime })$. Note that the radial traction need not be continuous: with the assumption of a spherical interface, the pressure jump is assumed to be balanced by the Laplace pressure (equilibrium value $2\unicode[STIX]{x1D6FE}/a$). Pozrikidis (Reference Pozrikidis1998) has shown that the pressure jump across the interface of an uncharged oscillating spherical drop is independent of position, in which case the motion does not induce a shape change. (Under steady translation, Taylor & Acrivos (Reference Taylor and Acrivos1964) showed that such drops transform to oblate spheroids and then spherical caps with increasing Weber number.) More generally, we have

(2.17)$$\begin{eqnarray}\unicode[STIX]{x0394}p=[\hat{p}(a_{-})-\hat{p}(a_{+})]\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}=\left[-\frac{2}{3}\sqrt[4]{-1}\sqrt{\unicode[STIX]{x1D6FA}_{i}}c_{1}^{\prime }a-\text{i}\unicode[STIX]{x1D6FA}_{o}a\left(\frac{U}{X}-c_{3}a^{-3}\right)\right]\boldsymbol{X}\boldsymbol{\cdot }\boldsymbol{e}_{r}.\end{eqnarray}$$

Finally, combining the foregoing momentum boundary conditions with the other independent conservation relationships, we must solve the following linear system for$\hat{d}_{\unicode[STIX]{x1D713}}$, $d_{c}$, $c_{1}^{\prime }$ and $c_{1}$ (or $c_{3}$):

(2.18)$$\begin{eqnarray}(\text{i}\unicode[STIX]{x1D714}a^{2}/2-D)d_{c}+a(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})zec^{0}\frac{D}{k_{B}T}+af_{r}(r=a_{-})c^{0}=0,\end{eqnarray}$$

(2.19)$$\begin{eqnarray}\displaystyle & & \displaystyle -\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}(E/X+2\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})+\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D716}_{i}\unicode[STIX]{x1D716}_{0}(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})\nonumber\\ \displaystyle & & \displaystyle \quad =2K_{s}a^{-1}(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})-K_{\infty }(E/X+2\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,2a^{-2}Dzed_{c}+2a^{-1}(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})(ze)^{2}c^{0}\frac{D}{k_{B}T},\end{eqnarray}$$

(2.20)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D702}_{o}f_{rr}(r=a_{+})=\unicode[STIX]{x1D702}_{i}f_{rr}(r=a_{-})+\unicode[STIX]{x1D6FE}^{0}\unicode[STIX]{x1D6FD}d_{c}a^{-1}, & \displaystyle\end{eqnarray}$$

(2.21)$$\begin{eqnarray}\displaystyle & \displaystyle f_{r}(r=a_{-})-M_{S}(E/X-\hat{d}_{\unicode[STIX]{x1D713}}a^{-3})=f_{r}(r=a_{+}), & \displaystyle\end{eqnarray}$$

where $a_{+}\sim a+\unicode[STIX]{x1D705}^{-1}$ and $a_{-}\sim a-\unicode[STIX]{x1D6FF}$ (see figure 1). Equations (2.18)–(2.21) may be termed, respectively, interfacial adsorbed surfactant conservation, interfacial charge conservation, interfacial tangential momentum conservation, and diffuse-layer tangential slip relationships. Recall that $f$ and its radial derivatives are readily obtained from (2.3) and (2.4). Note that the surface conductivity $K_{s}$ emerging from O’Brien’s matching of current between the diffuse layer and bulk electrolyte depends on the electrolyte composition and interfacial charge density and, therefore, surface potential $\unicode[STIX]{x1D701}$ (O’Brien Reference O’Brien1986, Reference O’Brien1988). In this paper, $\unicode[STIX]{x1D701}$, $\unicode[STIX]{x1D6FD}$, $c^{0}$ and $\unicode[STIX]{x1D6FE}^{0}$ are furnished by a surface adsorption isotherm (detailed below) that links these variables to the composition of the bulk surfactant-containing electrolyte.

2.3 Model solution and mobility

The equations above become too cumbersome to solve in closed form, and so we evaluate the solution numerically as a linear system

$$\begin{eqnarray}\boldsymbol{f}(\boldsymbol{x})=\mathop{\sum }_{j=1}^{4}x_{j}\boldsymbol{a}_{j}-\boldsymbol{b}=\mathbf{0}\end{eqnarray}$$

for which $x_{j}$ are the unknowns and

$$\begin{eqnarray}\boldsymbol{a}_{j}=\boldsymbol{f}(\boldsymbol{e}_{j})+\boldsymbol{b}\quad \text{with }\boldsymbol{b}=-\boldsymbol{f}(0),\end{eqnarray}$$

where $\boldsymbol{e}_{j}$ is a vector for which the $j$th entry is $1$ and the others are $0$. It follows that

$$\begin{eqnarray}\boldsymbol{x}=[\boldsymbol{a}_{1},\boldsymbol{a}_{2},\boldsymbol{a}_{3},\boldsymbol{a}_{4}]^{-1}\boldsymbol{\cdot }\boldsymbol{b}.\end{eqnarray}$$

To compute the electrophoretic mobility, we solve the system with $U=1$ and $E=0$, and $E=1$ and $U=0$, furnishing, for example, $c_{3}^{U}$ and $c_{3}^{E}$, respectively. Then, since the particle equation of motion is (Mangelsdorf & White Reference Mangelsdorf and White1992)

$$\begin{eqnarray}-\text{i}\unicode[STIX]{x1D714}\frac{4\unicode[STIX]{x03C0}a^{3}}{3}(\unicode[STIX]{x1D70C}_{i}-\unicode[STIX]{x1D70C}_{o})\boldsymbol{V}=4\unicode[STIX]{x03C0}\text{i}\unicode[STIX]{x1D714}\unicode[STIX]{x1D70C}_{o}(c_{3}^{U}\boldsymbol{V}+c_{3}^{E}\boldsymbol{E}),\end{eqnarray}$$

where the particle velocity $\boldsymbol{V}=-\boldsymbol{U}$, the dynamic mobility (complex-valued $M=V/E$) is

(2.22)$$\begin{eqnarray}M=\frac{-3c_{3}^{E}a^{-3}}{-3c_{3}^{U}a^{-3}+\unicode[STIX]{x1D70C}_{i}/\unicode[STIX]{x1D70C}_{o}-1}.\end{eqnarray}$$

This linear superposition can be used to construct, for example, the flow and dipole strengths under electrophoresis:

$$\begin{eqnarray}\boldsymbol{u}=\boldsymbol{u}^{E}-M\boldsymbol{u}^{U}\quad \text{and}\quad \hat{d}_{\unicode[STIX]{x1D713}}=\hat{d}_{\unicode[STIX]{x1D713}}^{E}-M\hat{d}_{\unicode[STIX]{x1D713}}^{U},\end{eqnarray}$$

and so on.

2.4 Surface tension and adsorption isotherm

The surface tension and its dependence on the electrolyte composition are extremely important contributions to the foregoing electrokinetic model. In addition to prescribing the Marangoni stress, the isotherm prescribes the surface-charge density and $\unicode[STIX]{x1D701}$-potential.

Here we consider the equilibrium existing between a flat interface between oil and water containing bulk concentrations of an anionic surfactant $c_{\infty }$ (valence $z=\pm 1$) and added ‘$1$-$1$’ salt with bulk concentration $c_{-}^{\infty }$ (anion concentration). It is assumed that only the surfactant anion adsorbs to the interface, at an equilibrium concentration $c^{0}=\hat{\unicode[STIX]{x1D6E4}}$. The system is exemplified by an aqueous electrolyte of sodium dodecylsulfate (SDS, anion DS$^{-}$ and cation Na$^{+}$) and sodium chloride (Na$^{+}$, Cl$^{-}$), at surfactant concentrations below the c.m.c. We must also assume here that an ionic surfactant is strongly dissociated, existing in solution as unimers (e.g. DS$^{-}$). With adsorption to the interface is the formation of a diffuse layer of ions. The concentrations in excess of or below the bulk values contribute to the surface excesses of each species, and it is these that all contribute to how the surface tension/free energy of the interface varies with the bulk composition.

For this system, it can be shown that the equilibrium surface tension may be evaluated from the Gibbs thermodynamic relationship (Eriksson & Ljunggren Reference Eriksson and Ljunggren1989) as

(2.23)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}(c_{\infty },c_{-}^{\infty }) & = & \displaystyle \unicode[STIX]{x1D6FE}(0,0)-k_{B}T\int _{0}^{c_{\infty }}\frac{\hat{\unicode[STIX]{x1D6E4}}}{c_{\infty }}\,\text{d}c_{\infty }\nonumber\\ \displaystyle & & \displaystyle -\,4k_{B}T\int _{0}^{c_{\infty }}\{\cosh [\unicode[STIX]{x1D701}e/(2k_{B}T)]-1\}\sqrt{\frac{k_{B}T\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}}{2(c_{\infty }+c_{-}^{\infty })e^{2}}}\,\text{d}c_{\infty },\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}(0,0)$ is the surface tension of the oil–water interface with no surfactant or added salt. We use (2.23) to numerically evaluate the partial derivative $\unicode[STIX]{x1D6FD}$ for the Marangoni stress in the electrokinetic model: first, $c^{0}=\hat{\unicode[STIX]{x1D6E4}}(c_{\infty },c_{-}^{\infty })$ and the equilibrium surface tension $\unicode[STIX]{x1D6FE}^{0}=\unicode[STIX]{x1D6FE}(c_{\infty },c_{-}^{\infty })$ are computed for two bulk surfactant concentrations that are very close to $c_{\infty }$, and then $\unicode[STIX]{x2202}\unicode[STIX]{x1D6FE}^{0}/\unicode[STIX]{x2202}c^{0}$ is approximated using a centred finite-difference formula. This prescription implicitly assumes that the dynamic (non-equilibrium) perturbations in the interfacial surface tension arise solely from (lateral transport-induced) perturbations in the adsorbed surfactant concentration.

The first integral in (2.23) captures the free energy of the adsorbed layer (electrostatic, entropic and enthalpic contributions), and the second is the contribution from the diffuse layer (electrostatic and translational entropic free energies) with adsorption isotherm (Prosser & Franses Reference Prosser and Franses2001)

(2.24)$$\begin{eqnarray}\hat{\unicode[STIX]{x1D6E4}}=\frac{c_{\infty }\unicode[STIX]{x1D6E4}}{\text{e}^{ze\unicode[STIX]{x1D701}/k_{B}T}n+c_{\infty }},\end{eqnarray}$$

where the surface-charge density (Gouy–Chapman formula (Russel et al. Reference Russel, Saville and Showalter1989))

$$\begin{eqnarray}\hat{\unicode[STIX]{x1D6E4}}ze=2\sqrt{2k_{B}T\unicode[STIX]{x1D716}_{o}\unicode[STIX]{x1D716}_{0}(c_{-}^{\infty }+c_{\infty })}\sinh [|z|e\unicode[STIX]{x1D701}/(2k_{B}T)]\end{eqnarray}$$

and (model parameters)

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}=1/(\unicode[STIX]{x03C0}a_{s}^{2})\quad \text{and}\quad n=\text{e}^{\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}/(\unicode[STIX]{x03C0}a_{s}^{2}\unicode[STIX]{x1D6FF}).\end{eqnarray}$$

The integrals are evaluated numerically using iteration at each integrand evaluation to ascertain how the integrand changes with respect to the integration variable $c_{\infty }$. In deriving the isotherm, $a_{s}$ is interpreted as the ‘excluded-volume’ radius of an adsorbed surfactant molecule, and $\unicode[STIX]{x1D6FF}$ is a length that characterizes the thickness of the interface, ultimately prescribing the translational free energy. Moreover, $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}$ is the change in enthalpic free energy (scaled with $k_{B}T$) when transferring a DS$^{-}$ ion from the bulk solvent/water to the interface.

The calculations undertaken in this paper have, as prescribed parameters (fitted to experimental data to be published elsewhere), $\unicode[STIX]{x1D6FE}(0,0)=0.047~\text{N}~\text{m}^{-1}$, $\unicode[STIX]{x0394}\unicode[STIX]{x1D716}\approx -19$ (value for similarly sized hydrocarbons transferred between water and oil) and $a_{s}=2.42$  Å (solution radius of an SO$_{4}^{2-}$ ion to represent the DS$^{-}$ polar head group in water). Then, setting $n=\text{e}^{\unicode[STIX]{x0394}\unicode[STIX]{x1D716}}/(\unicode[STIX]{x03C0}a_{s}^{2}\unicode[STIX]{x1D6FF})\approx 9.29\times 10^{-4}~\text{mmol}~\text{l}^{-1}$ as a single adjustable parameter, we find $\unicode[STIX]{x1D6FF}\approx 0.225a_{s}\approx 0.54$  Å, which, as expected, is small, suggesting a compact interface with weak fluctuations in the protrusion of DS$^{-}$ head groups from the oil–water interface.

This physically motivated approach to prescribing the model parameters provides an excellent fit to data for salt concentrations ${\lesssim}20~\text{mmol}~\text{l}^{-1}$ and SDS concentrations below the c.m.c. (varying with salt concentration) at which point the surface tension becomes constant with further addition of surfactant, departing from the theory, which predicts a continuous decline (unphysical negative surface tension). Surface tension above the c.m.c. is ${\sim}0.2\unicode[STIX]{x1D6FE}(0,0)$ (Prosser & Franses Reference Prosser and Franses2001), which can be used as a guide with which to bound the isotherm to concentrations below the c.m.c.

Note that numerically integrating (2.24) yields the surface tension as furnished by formulae of Borwankar & Wasan (Reference Borwankar and Wasan1988) (which require an iterative solution) to ascertain, for example, the $\unicode[STIX]{x1D701}$-potential. A numerical integration enables the model to be modified, e.g. to model electrostatic excluded-volume effects. Nevertheless, for the results presented in this paper, the model is applied in its most basic form to compute the equilibrium surface-charge density $zec^{0}$, equilibrium surface tension $\unicode[STIX]{x1D6FE}^{0}$, surface-tension gradient (with respect to surface concentration) $\unicode[STIX]{x1D6FD}$ and equilibrium $\unicode[STIX]{x1D701}$-potential, as required for the electrokinetic model. These are plotted in figure 3 versus the bulk SDS concentration for three representative concentrations of added salt. Note that calculations at SDS concentrations above those for which $\unicode[STIX]{x1D6FE}^{0}\lesssim 0.2\unicode[STIX]{x1D6FE}(0,0)$ are unphysical extrapolations of the model, since it does not account for micelle formation. We also note that the volume fraction of Na$^{+}$ ions at the interface can reach values of the order of close packing at high SDS concentrations and higher added salt concentrations. It may therefore be prudent to examine the role of ion-steric effects on the adsorption isotherm, e.g. as undertaken by Khair & Squires (Reference Khair and Squires2009) for interfaces with prescribed surface-charge density using the Bikerman mean-field model.

Figure 3.

Equilibrium characteristics of the SDS–hexadecane–water interface versus the bulk SDS concentration: added salt concentrations $c_{-}^{\infty }=1$  mM (solid), $5$  mM (dashed) and $20$  mM (dash-dotted). Note that the model is extrapolated beyond its range of validity at each ionic strength, i.e. beyond the c.m.c. to values of $c_{\infty }$ for which $\unicode[STIX]{x1D6FE}^{0}/\unicode[STIX]{x1D6FE}(0,0)\lesssim 0.2$.

Before proceeding to the results, note that sonication of oil and water with a strongly adsorbing surfactant produces an extremely large surface area, so the concentration of surfactant in the aqueous phase can be significantly lower than initially prepared. Here, we invoke the isotherm equation (2.24) with a material balance on the oil and surfactant phases to link the overall surfactant concentration $c_{\infty ,0}$ to the actual/equilibrium bulk concentration $c_{\infty }$, assuming equilibrium and a monodisperse emulsion with drop radius $a$. In the absence of a direct measurement, $a$ becomes the single unknown when comparing theoretical predictions of the dynamic mobility with experiments. In practice, dynamic light scattering or acoustic attenuation may be used to independently ascertain $a$ and estimate its distribution.

Mass balances on the oil and surfactant (comparing the as-prepared and equilibrium states) furnish

$$\begin{eqnarray}3\unicode[STIX]{x1D719}\hat{\unicode[STIX]{x1D6E4}}/a+c_{\infty }(1-\unicode[STIX]{x1D719})=c_{\infty ,0},\end{eqnarray}$$

where we recall that $\unicode[STIX]{x1D719}$ is the oil volume fraction. Owing to the nonlinear manner in which the equilibrium surface concentration $c^{0}=\hat{\unicode[STIX]{x1D6E4}}$ depends on $c_{\infty }$ (using (2.24)), this is solved numerically for $c_{\infty }$ using a standard iterative algorithm that also furnishes $\unicode[STIX]{x1D6FE}^{0}=\unicode[STIX]{x1D6FE}^{0}(c_{\infty },c_{-}^{\infty })$, $\unicode[STIX]{x1D6FD}$, $c^{0}=\hat{\unicode[STIX]{x1D6E4}}(c_{\infty },c_{-}^{\infty })$ and $\unicode[STIX]{x1D701}$. As exemplified by pioneering electrokinetic interpretations of the ESA for dispersions of solid particulates (O’Brien Reference O’Brien1990), such a measurement furnishes mobility magnitude and phase spectra, so, at least in principle, such a measurement may furnish the particle size (O’Brien et al. Reference O’Brien, Cannon and Rowlands1995).