# Electrophoretic mobility of a water-in-oil droplet separately affected
by the net charge and surface charge density

Yuki Uematsu<EMAIL_ADDRESS>Department of Physics, Kyushu
University, Motooka 744, Fukuoka 819-0395, Japan  PRESTO, Japan Science and
Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama 332-0012, Japan  Hiroyuki
Ohshima  Faculty of Pharmaceutical Sciences, Tokyo University of Science, 2641
Yamazaki Noda, Chiba 278-8510, Japan

###### Abstract

Water-in-oil emulsions and droplets exhibit completely different physico-
chemical properties compared to oil-in-water emulsions and droplets. Thus,
directly applying a standard theoretical model to water-in-oil systems cannot
describe these anomalous properties. Here, the electrophoretic mobility of a
water-in-oil droplet is analytically investigated using Debye-Hückel
linearization and neglecting Marangoni effect. The resulting electrophoretic
mobility is shown to be separately dependent on the net charge of the droplet
and the surface charge density at the droplet interface. Furthermore, when the
net charge is negligible, electrophoretic mobility is proportional to the
surface charge density with a negative coefficient. This indicates that the
internal electric double layer inversely contributes the electrophoresis. This
theory is applied to experimental data of water-in-oil emulsions and droplets
in the literature, and qualitative and quantitative verification of the theory
is discussed.

##  I Introduction

Emulsion stability is essential for chemical processing and basic surface
science [1, 2, 3, 4, 5, 6], and it can be partially explained by the
Derjaguin-Landau-Veerwey-Overbeek (DLVO) theory [7, 8]. However, emulsions can
coarsen via coalescence [9] and Ostwald ripening [10, 11], which are unique to
liquid droplets [6], in addition to sedimentation, creaming, and flocculation
[12, 13]. Ionic and non-ionic surfactants are typically used to obtain stable
emulsions, and the stability of emulsions are partially determined by the
chemical nature and partition of the surfactants, as well as the interactions
between the droplets [6]. Although the composition of the two immiscible
liquids is one of the factors that control the continuous phase (water or
oil), the hydrophile-lipophile-balance (HLB) of the stabilizing surfactant
plays a crucial role in determining the continuous phase and phase coexistence
[2, 3, 5, 6]. Surfactant-free emulsions have been considered unstable after
emulsification due to creaming (sedimentation), flocculation, coalescence, and
Ostwald ripening [6]. However, recent experimental studies have shown that
surfactant-free emulsions are stable for some time [14, 15, 16, 17, 18, 19,
20, 21, 22], and the stability of such surfactant-free oil-in-water emulsions
is believed to be determined by the electrophoretic mobility (i.e., zeta
potential) of the droplet interface [20, 21, 22].

To date, most of the studies mentioned in the literature have focused on oil-
in-water emulsions, and studies on water-in-oil emulsions are less common due
to their low stability. However, compared to oil-in-water emulsions, water-in-
oil emulsions and droplets exhibit unique physicochemical properties [23, 24,
25, 26, 27, 28, 29, 30]. In water-in-oil emulsions, the continuous phase has a
low dielectric constant, and the application of the DLVO theory to the
stability is not straightforward [24, 25]. The sign and the magnitude of the
electrophoretic mobility of surfactant-stabilized water-in-oil emulsions has
been measured [23, 24, 26, 27, 28]. However, the relationship between emulsion
stability and the charge of the stabilizing surfactants remains unclear [24].
For example, water-in-benzene emulsions show positive electrophoretic mobility
even when anionic stabilizers are used [24]. Furthermore, surfactant-free
water-in-oil emulsions were recently obtained using ultrasound irradiation
[29, 30], and they are also stable for a while [29]. Nevertheless, to the best
of our knowledge, the electrophoretic mobility of surfactant-free water-in-oil
emulsions has not been reported.

As in water-in-oil emulsions, the electrophoretic mobility of water-in-oil
droplets has been directly measured by pipetting a water droplet into the oil
and applying a strong DC electric field [31, 32, 33, 34, 35, 36, 37, 38, 39].
Immediately after pipetting, the water droplet somehow has a positive
electrophoretic mobility [35, 36]. In these studies, the net charge of the
droplet $Q$ is obtained from electrophoretic mobility $\mu$ by equating
$\mu=Q/\gamma$, where $\gamma$ is the hydrodynamic friction coefficient
depending on the model (e.g., Hadamard-Rybczynski friction [40, 41], Stokes
friction [42], and the wall-effect-included friction coefficient [43, 39]).
From a theoretical viewpoint, the application of $\mu=Q/\gamma$ is valid for
the point-charge limit as known in the Hückel formula for the electrophoretic
mobility of a charged solid sphere [44]. Further, the convection induced by
the electric double layer inside the water droplet should significantly
contribute the electrophoretic mobility of the water-in-oil droplet, and thus,
the effective charge converted from the electrophoretic mobility, defined by
$Q_{\mathrm{eff}}=\gamma\mu$, should be different from the net charge $Q$.
Many theories for calculating the electrophoretic mobility of liquid droplets
have been constructed thus far [45, 46, 47, 48, 49, 50, 51, 52, 48, 53, 54,
55, 56]. Linear approximation in both electrostatics and applied electric
field was first used for the droplet electrophoretic mobility whereas the
surface charge gradient (Marangoni effect) was neglected [45]. With fully
nonlinear treatment in electrostatics, the electrophoretic mobility of a
droplet has been numerically obtained for more general cases including surface
charge gradient [46]. However, experimental comparison has been limited [57,
58] except for mercury droplets [50] and surfactant-stabilized oil-in-water
emulsions [54, 55, 56], and more specific theory and experimental comparison
for water-in-oil droplets have not been done yet. Therefore, a simple formula
for the electrophoretic mobility of a water-in-oil droplet is necessary to
discuss experimental data in the literature.

In this study, we reformulate the theory on the electrophoresis of droplets
using Debye-Hückel linearization with neglecting the surface charge gradient
[45] and calculate the electrophoretic mobility of a water-in-oil droplet as a
function of net charge and surface charge density. Because our results are
based on linear electrostatics, the effective charge deduced from our formula
is not quantitatively correct compared to the model including nonlinear
electrostatics [46,54-56]. Therefore, adopting our formula to interpret
experimental data needs great care. Nevertheless, our results are still
significant because our theory reveals the uniqueness of the electrophoretic
mobility of water-in-oil droplets compared to oil-in-water droplets or solid
particles. For the conventional theory of electrophoresis of solid particles,
the most important parameter is the surface charge density (or zeta potential)
[59, 60]. However, the water-in-oil droplet has an additional parameter: the
net charge of the droplet. When the oil phase is insulating, the net charge of
the droplet can be considered a parameter independent of the surface charge
density. Using the derived electrophoretic mobility formula, we discuss the
experimental results for water-in-oil emulsions and a water droplet in
insulating oil [24, 31, 32, 36, 35, 37, 38, 39]. According to our theory, the
positive electrophoretic mobility of a water-in-benzene emulsion stabilized by
an anionic surfactant is naturally explained by the negative linear
coefficient of the electrophoretic mobility with respect to the surface charge
density. Furthermore, our theory suggests that the effective charge extracted
from the electrophoretic mobility approaches a half of the total net charge of
the droplet in the limit of large radius, whereas the experiment indicates
that they are identical. However, to date, experimental studies have been
inconclusive due to the difficulties in independent and accurate measurements
of the total net charge [61] and the hydrodynamic friction coefficient [39].
We hope that this paper helps explain the unique electrokinetics of water-in-
oil emulsions and droplets.

##  II Model

The model is similar to the work of Booth [45]. The analytic calculation of
the electrophoretic mobility for a water-in-oil droplet is performed with the
following assumptions. (1) A liquid droplet with radius $R$ is fixed at the
center of the spherical coordinate and surrounded by another liquid. The
interface is fixed when external flow and electric fields are applied. (2) The
electrostatic potential is the sum of the solution of the linearized
equilibrium Poisson-Boltzmann equation and the potential of an insulating
droplet under an external electric field. This is equivalent to neglecting the
relaxation effect in the case of spherical solid particles [60]. In the case
of liquid droplets, the external electric field or flow induces the surface
charge gradient, and thus, Marangoni stress contributes to the force balance.
Solving fully nonlinear electrostatics and including Marangoni effect were
performed in earlier works [46, 54, 55, 56], but in this paper nonlinear
electrostatics and Marangoni effect are neglected for simplicity. (3) The
inertia terms in the electrohydrodynamic equations are neglected. This is
equivalent to concerning the electrophoretic velocity linear with the applied
electric field. (4) The internal and external liquids are incompressible, and
the dielectric constants and viscosities are represented by
$\varepsilon_{\mathrm{in}}$, $\varepsilon_{\mathrm{out}}$,
$\eta_{\mathrm{in}}$, and $\eta_{\mathrm{out}}$. Additionally, the bulk salt
concentration of the external liquid is zero, and the ion partition across the
interface is neglected. (5) We do not explicitly consider ionic and non-ionic
surfactants (or equivalently impurities) at the interface. These surface-
active molecules significantly alter the surface tension and surface charge
density, and thus, they affect the hydrodynamic and electrostatic boundary
conditions. Instead, the surface charge density is infinitesimal (Debye-Hückel
linearization) and it remains uniform and constant under external flow and
electric fields (neglecting Marangoni effect). Additionally, slip boundary
condition (continuity of the tangential velocity and stress at the interface)
is used. The validity of these assumptions is discussed further in appendix.

Under these assumptions, the steady-state low-Reynolds-number
electrohydrodynamic equations are as follows:

$\displaystyle\varepsilon_{0}\nabla^{2}\psi$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-\rho(\boldsymbol{r})/\varepsilon_{\mathrm{in}}&\textrm{for
}r<R\\\ -\rho(\boldsymbol{r})/\varepsilon_{\mathrm{out}}&\textrm{for }r>R\\\
\end{array}\right.,$ (3)
$\displaystyle\nabla\cdot\overleftrightarrow{\boldsymbol{\sigma}}$
$\displaystyle=$ $\displaystyle 0,$ (4)
$\displaystyle\nabla\cdot\boldsymbol{u}$ $\displaystyle=$ $\displaystyle 0,$
(5) $\displaystyle\nabla\cdot\boldsymbol{j}_{+}$ $\displaystyle=$
$\displaystyle 0,$ (6) $\displaystyle\nabla\cdot\boldsymbol{j}_{-}$
$\displaystyle=$ $\displaystyle 0,$ (7)

where $\psi$ is the local electrostatic potential, $\rho(\boldsymbol{r})$ is
the ionic charge density, $\overleftrightarrow{\boldsymbol{\sigma}}$ is the
stress tensor, $\boldsymbol{u}$ is the velocity field, and
$\boldsymbol{j}_{\pm}$ is the ionic current for cations and anions,
respectively. The stress tensor comprises the hydrodynamic and Maxwell stress
tensor as
$\overleftrightarrow{\boldsymbol{\sigma}}=\overleftrightarrow{\boldsymbol{\sigma}_{\mathrm{H}}}+\overleftrightarrow{\boldsymbol{\sigma}_{\mathrm{M}}}$,
which are defined by

$\overleftrightarrow{\boldsymbol{\sigma}_{\mathrm{H}}}=\left\\{\begin{array}[]{ll}\displaystyle-p\boldsymbol{I}+\eta_{\mathrm{in}}\left(\nabla\otimes\boldsymbol{u}+(\nabla\otimes\boldsymbol{u})^{t}\right)&\textrm{for
}r<R\\\ \\\
\displaystyle-p\boldsymbol{I}+\eta_{\mathrm{out}}\left(\nabla\otimes\boldsymbol{u}+(\nabla\otimes\boldsymbol{u})^{t}\right)&\textrm{for
}r>R\end{array}\right.,$ (8)
$\overleftrightarrow{\boldsymbol{\sigma}_{\mathrm{M}}}=\left\\{\begin{array}[]{ll}\displaystyle-\frac{\varepsilon_{\mathrm{in}}\varepsilon_{0}}{2}(\nabla\psi)^{2}\boldsymbol{I}+\varepsilon_{\mathrm{in}}\varepsilon_{0}\left(\nabla\psi\otimes\nabla\psi\right)&\textrm{for
}r<R\\\ \\\
\displaystyle-\frac{\varepsilon_{\mathrm{out}}\varepsilon_{0}}{2}(\nabla\psi)^{2}\boldsymbol{I}+\varepsilon_{\mathrm{out}}\varepsilon_{0}\left(\nabla\psi\otimes\nabla\psi\right)&\textrm{for
}r>R\end{array}\right.,$ (9)

where $\boldsymbol{I}$ is the unit tensor. The charge density is defined by

$\rho(\boldsymbol{r})=\left\\{\begin{array}[]{ll}e\left(c_{\mathrm{+,in}}-c_{\mathrm{-,in}}\right)&\textrm{for
}r<R\\\ e\left(c_{\mathrm{+,out}}-c_{\mathrm{-,out}}\right)&\textrm{for
}r>R\end{array}\right.,$ (10)

where $e$ is the elementary charge and $c_{\mathrm{\pm,in}}$ and
$c_{\mathrm{\pm,out}}$ are the local cation and anion concentrations inside or
outside the droplet. Although ionic currents consist of convection and
conduction, a linear approximation of the surface charge density and the
surface potential does not require explicit equations for ionic currents [59,
45, 60].

The boundary conditions for electrophoresis are

$\displaystyle\boldsymbol{u}\cdot\boldsymbol{e}_{r}|_{r=R}$ $\displaystyle=$
$\displaystyle 0,$ (11)
$\displaystyle\boldsymbol{u}\cdot\boldsymbol{e}_{\theta}|_{r=R+0}-\boldsymbol{u}\cdot\boldsymbol{e}_{\theta}|_{r=R-0}$
$\displaystyle=$ $\displaystyle 0,$ (12)
$\displaystyle\sigma_{r\theta}|_{r=R+0}-\sigma_{r\theta}|_{r=R-0}$
$\displaystyle=$ $\displaystyle 0,$ (13)
$\displaystyle\boldsymbol{j}_{\pm}\cdot\boldsymbol{e}_{r}|_{r=R}$
$\displaystyle=$ $\displaystyle 0$ (14) $\displaystyle\nabla p|_{r\to\infty}$
$\displaystyle=$ $\displaystyle 0,$ (15) $\displaystyle
c_{i,\mathrm{out}}|_{r\to\infty}$ $\displaystyle=$ $\displaystyle
c_{\mathrm{out}}^{\infty},$ (16) $\displaystyle\boldsymbol{u}|_{r\to\infty}$
$\displaystyle=$ $\displaystyle U\boldsymbol{e}_{z},$ (17)
$\displaystyle\psi|_{r\to\infty}$ $\displaystyle=$ $\displaystyle-
Er\cos\theta,$ (18)

where $U$ is the external uniform velocity, $E$ is the externally applied
electric field, and $c_{\mathrm{out}}^{\infty}$ is the salt concentration
outside the droplet, which is substituted by zero later. Eq. 11 is the fixed
interface condition, eq. 12 is the continuity of the tangential velocity at
the interface, eq. 13 is the continuity of the tangential stress at the
interface, and eq. 14 is the condition that the ions cannot penetrate the
interface. The boundary condition for $\sigma_{rr}$ is unnecessary because of
its fixed interface [42]. When the surface charge gradient is considered, an
additional boundary condition is necessary for the balance between ion
diffusion and migration in the tangential direction to the interface [46, 54,
55, 56], which is further explained in Appendix.

When the electric field ($E=0$) is switched off and the uniform flow $U$ is
applied, the hydrodynamic equations and boundary conditions provide the
velocity field and the frictional force of the liquid droplet in an external
uniform flow, where the force is given by

$F_{z}=\int_{r=R}(\overleftrightarrow{\boldsymbol{\sigma}}\cdot
d\boldsymbol{S})_{z}=\int_{r\to\infty}(\overleftrightarrow{\boldsymbol{\sigma}_{\mathrm{H}}}\cdot
d\boldsymbol{S})_{z}=\gamma U,$ (19)

where $\int_{r>R}\nabla\cdot\overleftrightarrow{\boldsymbol{\sigma}}d^{3}r=0$
and
$\int_{r\to\infty}\overleftrightarrow{\boldsymbol{\sigma}}_{\mathrm{M}}\cdot
d\boldsymbol{S}=0$ are used and

$\gamma=2\pi\eta_{\mathrm{out}}R\frac{3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}}}{\eta_{\mathrm{in}}+\eta_{\mathrm{out}}},$
(20)

is the Hadamard-Rybczynski friction coefficient [40, 41]. When the external
uniform flow is switched off ($U=0$) and the electric field $E$ is applied,
the equations provide the force induced by the electric field. The effective
charge $Q_{\mathrm{eff}}$ of the liquid droplet is then defined by the force
divided by the electric field,

$Q_{\mathrm{eff}}=\frac{F_{z}}{E}=\frac{1}{E}\int_{r\to\infty}(\overleftrightarrow{\boldsymbol{\sigma}}\cdot
d\boldsymbol{S})_{z}.$ (21)

If the surrounding liquid has no ions,
$\int_{r\to\infty}(\overleftrightarrow{\boldsymbol{\sigma}_{\mathrm{M}}}\cdot
d\boldsymbol{S})_{z}=QE$ does not disappear. However, when the surrounding
liquid at least has counterions,
$\int_{r\to\infty}(\overleftrightarrow{\boldsymbol{\sigma}_{\mathrm{M}}}\cdot
d\boldsymbol{S})_{z}$ disappears. This difference is discussed later.
Consequently, because the electrophoresis is a linear problem, the
electrophoretic mobility $\mu=-U/E$ is obtained by the force balance condition
$F_{z}=0$ where $E$ and $U$ are the applied external fields. Therefore,

$\mu=\frac{Q_{\mathrm{eff}}}{\gamma}.$ (22)

In the following sections, we calculate the equilibrium state and linear
response within the approximation of a low surface charge and a low net charge
(Debye-Hückel linearization).

###  II.1 Equilibrium state

At equilibrium ($U=E=0$), all quantities are spherically symmetric, and the
ions are distributed according to the Boltzmann distribution:

$\displaystyle c_{\pm,\mathrm{in}}(r)$ $\displaystyle=$ $\displaystyle
c^{R}_{\pm,\mathrm{in}}\mathrm{e}^{\mp(\Psi^{\mathrm{eq}}(r)-\Psi_{R})},$ (23)
$\displaystyle c_{\pm,\mathrm{out}}(r)$ $\displaystyle=$ $\displaystyle
c^{\infty}_{\mathrm{out}}\mathrm{e}^{\mp\Psi^{\mathrm{eq}}(r)},$ (24)

where $c_{\mathrm{\pm,\mathrm{in}}}^{R}$ is the local concentration of cations
or anions at the internal interface, $c^{\infty}_{\mathrm{out}}$ is the bulk
ion concentration outside the droplet,
$\Psi^{\mathrm{eq}}(r)=e\psi^{\mathrm{eq}}(r)/k_{\mathrm{B}}T$ is the
dimensionless equilibrium electrostatic potential, and
$\Psi_{R}=e\psi_{R}/k_{\mathrm{B}}T$ is the dimensionless potential at $r=R$.
Note that $c_{\mathrm{+,\mathrm{in}}}^{R}$ and
$c_{\mathrm{-,\mathrm{in}}}^{R}$ are generally different, and the partition
equilibrium of the ion species across the interface is not considered. The
equilibrium potential profile $\Psi^{\mathrm{eq}}(r)$ is calculated by the
Poisson-Boltzmann equation

$\begin{split}&\frac{1}{r^{2}}\frac{d}{dr}\left[r^{2}\frac{d}{dr}\Psi^{\mathrm{eq}}(r)\right]=\\\
\\\
&\left\\{\begin{array}[]{ll}-\kappa_{+,\mathrm{in}}^{2}\mathrm{e}^{-(\Psi^{\mathrm{eq}}-\Psi_{R})}+\kappa_{-,\mathrm{in}}^{2}\mathrm{e}^{\Psi^{\mathrm{eq}}-\Psi_{R}}&\textrm{for
}r<R\\\ \kappa^{2}_{\mathrm{out}}\sinh\Psi^{\mathrm{eq}}&\textrm{for
}r>R\end{array}\right.,\end{split}$ (25)

where

$\displaystyle\kappa_{i,\mathrm{in}}^{2}$ $\displaystyle=$
$\displaystyle\frac{e^{2}c_{i,\mathrm{in}}^{R}}{\varepsilon_{\mathrm{in}}\varepsilon_{0}k_{\mathrm{B}}T},$
(26) $\displaystyle\kappa^{2}_{\mathrm{out}}$ $\displaystyle=$
$\displaystyle\frac{2e^{2}c^{\infty}_{\mathrm{out}}}{\varepsilon_{\mathrm{out}}\varepsilon_{0}k_{\mathrm{B}}T}.$
(27)

$\kappa_{\mathrm{out}}^{-1}$ is the thickness of the electric double layer
outside the droplet. The boundary condition is
$d\Psi^{\mathrm{eq}}/dr|_{r=0}=0$, $\Psi^{\mathrm{eq}}|_{r\to\infty}=0$,
$\Psi(R)=\Psi_{R}$.

The surface charge density is defined by

$\sigma_{R}=-\varepsilon_{0}\left[\left.\varepsilon_{\mathrm{out}}\frac{d\psi^{\mathrm{eq}}}{dr}\right|_{R+0}-\left.\varepsilon_{\mathrm{in}}\frac{d\psi^{\mathrm{eq}}}{dr}\right|_{R-0}\right],$
(28)

and the net charge of the droplet is defined by

$\begin{split}Q&=4\pi
R^{2}\left(-\varepsilon_{\mathrm{in}}\varepsilon_{0}\left.\frac{d\psi^{\mathrm{eq}}}{dr}\right|_{r=R-0}+\sigma_{R}\right)\\\
&=-4\pi
R^{2}\varepsilon_{\mathrm{out}}\varepsilon_{0}\left.\frac{d\psi^{\mathrm{eq}}}{dr}\right|_{r=R+0}\end{split}$
(29)

where the surface charge density is assumed to be ion adsorption from the
water phase rather than the oil phase. The Poisson-Boltzmann equation 25 is
linearized and solved as follows:

$\begin{split}&\Psi^{\mathrm{eq}}(r)=\\\
&\left\\{\begin{array}[]{ll}\displaystyle\Psi_{R}+\frac{\kappa_{\mathrm{+,in}}^{2}-\kappa_{\mathrm{-,in}}^{2}}{\kappa_{\mathrm{in}}^{2}}\left(1-\frac{R\sinh\kappa_{\mathrm{in}}r}{r\sinh\kappa_{\mathrm{in}}R}\right)&\textrm{for
}r<R\\\ \\\
\displaystyle\Psi_{R}\frac{R}{r}\mathrm{e}^{-\kappa_{\mathrm{out}}(r-R)}&\textrm{for
}r>R\end{array}\right.,\end{split}$ (30)

where
$\kappa_{\mathrm{in}}^{2}=\kappa_{+,\mathrm{in}}^{2}+\kappa_{-,\mathrm{in}}^{2}$.
Within the limit of $c_{\mathrm{out}}^{\infty}\to 0$ (water-in-oil droplet),
eq. 30 for $r>R$ approaches $\Psi_{R}(R/r)$, which is the exact solution of
the Poisson equation, and thus the condition of the small net charge
($|\Psi_{R}|\ll 1$) becomes unnecessary.

Figure 1:  Typical profiles of $\psi^{\mathrm{eq}}(r)$ with
$c_{\mathrm{out}}^{\infty}=0$ and $Q\geq 0$ and their schematic illustrations
of the ion distribution. The vertical broken lines are the location of the
droplet interface and the horizontal lines are $\psi^{\mathrm{eq}}=0$. (a)
$\sigma_{R}=Q/4\pi R^{2}$, (b) $\sigma_{R}<Q/4\pi R^{2}$, (c)
$\sigma_{R}>Q/4\pi R^{2}$. (d) $Q=0$ and $\sigma_{R}<0$. The ions on the
droplet interface in the illustrations denote the surface charge, whereas the
ions between the interface and inner dashed circles denote the electric double
layer. Inside the dashed circles, the ionic charge density is almost zero
because the potential profile $\psi^{\mathrm{eq}}(r)$ is flat.

Typical electrostatic profiles and their schematic illustrations are plotted
in Fig. 1. Because the surface charge density, $\sigma_{R}$, and the net
charge, $Q$, are the independent parameters, qualitatively different profiles
of the electrostatic potential exist for a water-in-oil droplet. When the
charge of the droplet is distributed only at the interface (Fig. 1a), the
potential is flat inside the droplet. In (b), the rest of the droplet charge,
$Q-4\pi R^{2}\sigma_{R}$, is positive, and the electric double layer forms
inside the droplet, whereas in (c) the rest of the droplet charge is negative,
and thus, the charge of the double layer inside is negative. In (d), the net
charge of the droplet is zero, and thus, the potential outside is flat. In the
illustrations, the ions on the droplet interface denote the surface charge,
whereas the ions between the interface and inner dashed circles denote the
electric double layer. Inside the dashed circles, the ionic charge density is
almost zero because the potential profile $\psi^{\mathrm{eq}}(r)$ at this
range is flat for all cases of (a) to (d). Experimental classification of
water-in-oil droplets or emulsions into these four cases shown in Fig. 1 is
still difficult because no reliable method to control and measure the surface
charge density and net charge.

###  II.2 Linear response

When an external uniform flow and/or an external electric field is applied,
the deviations of all quantities from the equilibrium state become
axisymmetric, and thus the velocity field can be described using the function
$h(r)$ as [60]

$\boldsymbol{u}(r,\theta)=-\frac{2h(r)\cos\theta}{r}\boldsymbol{e}_{r}+\frac{\sin\theta}{r}\frac{d}{dr}[rh(r)]\boldsymbol{e}_{\theta}.$
(31)

Note that the function $-rh(r)\sin^{2}\theta$ is the Stokes stream function.
The velocity field under an external uniform flow without an electric field
was previously analytically solved [40, 41]. Therefore, in this work, the
velocity field under an external electric field without an external uniform
flow ($U=0$) is studied. As we are focusing on low surface charge densities
and low net charges, it is unnecessary to explicitly consider the deviations
of the ionic concentration fields from the equilibrium distribution [59];
thus, eqs. 6, 7, and 14 are not solved. Instead, these approximations impose
the use of the electric potential profile given by

$\delta\psi(r,\theta)=\left\\{\begin{array}[]{ll}\displaystyle-(1+\lambda)Er\cos\theta&\textrm{for
}r<R\\\ \\\
\displaystyle-\left(1+\lambda\frac{R^{3}}{r^{3}}\right)Er\cos\theta&\textrm{for
}r>R\end{array}\right..$ (32)

The coefficient $\lambda$ was classically understood as the difference in
conductivity
$(\sigma_{\mathrm{out}}-\sigma_{\mathrm{in}})/(2\sigma_{\mathrm{out}}+\sigma_{\mathrm{in}})$
where $\sigma_{\mathrm{in}}$ and $\sigma_{\mathrm{out}}$ are the
conductivities of the inner and outer liquids [59, 45]. However, if the ions
cannot penetrate the other phase (eq. 14), this coefficient is $\lambda=1/2$
and is independent of the conductivities $\sigma_{\mathrm{in}}$ and
$\sigma_{\mathrm{out}}$, and the dielectric constants
$\varepsilon_{\mathrm{in}}$ and $\varepsilon_{\mathrm{out}}$ [51]. For mercury
droplets, the coefficients are different inside ($\lambda=-1$) and outside
($\lambda=1/2$) due to the perfect polarization of the droplet [51, 52]. Using
eqs. 31 and 32, the hydrodynamic equations, eqs. 4 and 5, are reduced to the
equation for $h(r)$ as follows [60]:

$L^{2}h(r)=-f(r)E,$ (33)

where

$L=\frac{d}{dr}\left(\frac{1}{r^{2}}\frac{d}{dr}r^{2}\right)=\frac{d^{2}}{dr^{2}}+\frac{2}{r}\frac{d}{dr}-\frac{2}{r^{2}},$
(34)

and

$\begin{split}f(r)&=\left\\{\begin{array}[]{ll}\displaystyle\frac{\varepsilon_{\mathrm{in}}\varepsilon_{0}\kappa_{\mathrm{in}}^{2}}{\eta}(1+\lambda)\frac{d\psi^{\mathrm{eq}}}{dr}&\textrm{for
}r<R\\\ \\\
\displaystyle\frac{\varepsilon_{\mathrm{out}}\varepsilon_{0}\kappa_{\mathrm{out}}^{2}}{\eta}\left(1+\lambda\frac{R^{3}}{r^{3}}\right)\frac{d\psi^{\mathrm{eq}}}{dr}&\textrm{for
}r>R\end{array}\right..\end{split}$ (35)

The solution of eq. 33 is

$\begin{split}&h(r)=\\\ &\left\\{\begin{array}[]{l}\displaystyle
A_{1}\frac{r}{R}+A_{2}\frac{R^{2}}{r^{2}}+A_{3}\frac{r^{3}}{R^{3}}+A_{4}-\int^{r}_{0}G(r,r^{\prime})f(r^{\prime})Edr^{\prime}\\\
\textrm{for }r<R\\\ \\\ \displaystyle
B_{1}\frac{r}{R}+B_{2}\frac{R^{2}}{r^{2}}+B_{3}\frac{r^{3}}{R^{3}}+B_{4}-\int^{r}_{\infty}G(r,r^{\prime})f(r^{\prime})Edr^{\prime}\\\
\textrm{for }r>R\\\ \end{array}\right.,\end{split}$ (36)

where $A_{n}$ and $B_{n}$ are constants and

$G(r,r^{\prime})=-\frac{r^{\prime 5}}{30r^{2}}+\frac{r^{\prime
3}}{6}-\frac{rr^{\prime 2}}{6}+\frac{r^{3}}{30}.$ (37)

The boundary conditions, eqs. 11–13 and 17–18 for $(U,E)=(0,E)$, can be
converted to $B_{1}=B_{3}=0$, $A_{2}=A_{4}=0$, $B_{2}+B_{4}-D_{0}=0$,
$A_{1}+A_{3}-C_{0}=0$, $A_{1}+3A_{3}-C_{1}=-2B_{2}-D_{1}$, and

$6\eta_{\mathrm{in}}A_{3}-\eta_{\mathrm{in}}C_{2}=6\eta_{\mathrm{out}}B_{2}-\eta_{\mathrm{out}}D_{2}-\sigma_{R}(1+\lambda)R^{2},$
(38)

where $C_{n}$ and $D_{n}$ are defined as

$\displaystyle\displaystyle C_{n}$ $\displaystyle=$ $\displaystyle
R^{n}\int^{R}_{0}\left(\frac{d}{dR}\right)^{n}G(R,r)f(r)dr,$ (39)
$\displaystyle\displaystyle D_{n}$ $\displaystyle=$ $\displaystyle
R^{n}\int^{R}_{\infty}\left(\frac{d}{dR}\right)^{n}G(R,r)f(r)dr.$ (40)

These equations yield $B_{4}$ as

$\begin{split}B_{4}&=\frac{\eta_{\mathrm{in}}(3C_{0}-3C_{1}+C_{2}+6D_{0}+3D_{1})}{6(\eta_{\mathrm{in}}+\eta_{\mathrm{out}})}\\\
&+\frac{\eta_{\mathrm{out}}(6D_{0}-D_{2})-\sigma_{R}(1+\lambda)R^{2}}{6(\eta_{\mathrm{in}}+\eta_{\mathrm{out}})}.\end{split}$
(41)

Because the hydrodynamic force is obtained as $8\pi\eta_{\mathrm{out}}B_{4}$
[60], the effective charge $Q_{\mathrm{eff}}$ is

$Q_{\mathrm{eff}}=Q+8\pi\eta_{\mathrm{out}}B_{4},$ (42)

where $Q$ is due to long-range Maxwell stress when the surrounding liquid
contains no ions (even counterions). In this case, $\kappa_{\mathrm{out}}=0$,
and thus, $f(r)=0$ for $r>R$. Conversely, if the surrounding liquid contains
counterions (this corresponds to $\kappa_{\mathrm{out}}\to 0$ not
$\kappa_{\mathrm{out}}=0$)), $Q=0$ is obtained because of the Gauss’s law in
electrostatics. Later, we demonstrate that these two limiting situations
provide the same electrophoretic mobility.

The resulting electrophoretic mobility $\mu=Q_{\mathrm{eff}}/\gamma$ is broken
down into five parts:

$\mu=\mu_{1}+\mu_{2}+\mu_{3}+\mu_{4}+\mu_{5},$ (43)

where $\mu_{1}$ to $\mu_{4}$ correspond to $3C_{0}-3C_{1}+C_{2}$,
$6D_{0}+3D_{1}$, $6D_{0}-D_{2}$, and $-\sigma_{R}(1+\lambda)R^{2}$ terms in
eq. 41, and $\mu_{5}$ corresponds to $Q/\gamma$. Each term is explicitly
obtained as

$\displaystyle\mu_{1}$ $\displaystyle=$
$\displaystyle-\frac{\varepsilon_{\mathrm{out}}\varepsilon_{0}(1-H(\kappa_{\mathrm{in}}R))}{3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}}}\left[(1+\kappa_{\mathrm{out}}R)\psi_{R}-\frac{R\sigma_{R}}{\varepsilon_{\mathrm{out}}\varepsilon_{R}}\right],$
$\displaystyle\mu_{2}$ $\displaystyle=$
$\displaystyle\frac{2\eta_{\mathrm{in}}\varepsilon_{\mathrm{out}}\varepsilon_{0}\psi^{\mathrm{eq}}_{R}}{\eta_{\mathrm{out}}(3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}})}\left[1+\frac{F_{1}(\kappa_{\mathrm{out}}R)}{2}\right],$
(45) $\displaystyle\mu_{3}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{\mathrm{out}}\varepsilon_{0}\psi^{\mathrm{eq}}_{R}}{3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}}}\left[(1+\kappa_{\mathrm{out}}R)+\frac{4}{3}-\frac{2F_{2}(\kappa_{\mathrm{out}}R)}{3}\right],$
(46) $\displaystyle\mu_{4}$ $\displaystyle=$
$\displaystyle-\frac{R\sigma_{R}}{3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}}},$
(47) $\displaystyle\mu_{5}$ $\displaystyle=$
$\displaystyle\frac{2\varepsilon_{\mathrm{out}}\varepsilon_{0}\psi_{R}}{3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}}}\left[1+\frac{\eta_{\mathrm{in}}}{\eta_{\mathrm{out}}}\right].$
(48)

We define the functions according to the work of Booth [45] as

$\displaystyle F_{1}(x)$ $\displaystyle=$ $\displaystyle
1+6E_{5}(x)\mathrm{e}^{x}-15E_{7}(x)\mathrm{e}^{x},$ (49) $\displaystyle
F_{2}(x)$ $\displaystyle=$ $\displaystyle 2-6E_{5}(x)\mathrm{e}^{x},$ (50)
$\displaystyle F_{3}(x)$ $\displaystyle=$
$\displaystyle(\mathrm{e}^{x}(x-1)+\mathrm{e}^{-x}(x+1))^{-1},$ (51)
$\displaystyle F_{4}(x)$ $\displaystyle=$
$\displaystyle\mathrm{e}^{x}(x^{2}-3x+3)-\mathrm{e}^{-x}(x^{2}+3x+3),$ (52)

where $E_{n}(x)=\int^{\infty}_{1}\mathrm{e}^{-xt}t^{-n}dt$. Additionally, a
function is newly introduced by

$H(x)=5F_{3}(x)F_{4}(x)/x^{2}.$ (53)

The shapes of the functions $F_{1}(x)$, $F_{2}(x)$, and $H(x)$ are plotted in
Fig. 2.

Each electrophoretic mobility term is described below: $\mu_{1}$ is the
contribution from the double layer inside the droplet because this originates
from $C_{n}$. $\mu_{2}$ is the contribution from the double layer outside the
droplet, and in the solid particle limit
($\eta_{\mathrm{in}}/\eta_{\mathrm{out}}\to\infty$), only this term survives
and approaches Henry’s formula [59]. $\mu_{3}$ is another contribution from
the double layer outside the droplet, but it only exists for the liquid
droplet. For mercury droplets, this contribution is dominant [50, 51, 52].
$\mu_{4}$ is the contribution from the continuity of the Maxwell tensor, which
vanishes for mercury droplets [50, 51, 52]. In Booth’s work [45], the effect
of surface charge density on the electrophoretic mobility was mentioned, but
no explicit mobility equation containing $\sigma_{R}$ was derived. In
addition, at that time, the coefficient $\lambda$ was considered
$(\sigma_{\mathrm{out}}-\sigma_{\mathrm{in}})/(2\sigma_{\mathrm{out}}+\sigma_{\mathrm{in}})$
[59, 45], but $\lambda=1/2$ is used here.

The last term $\mu_{5}$ is the contribution from the net charge of the
droplet, which is unnecessary if the surrounding liquid has counterions. From
eqs. 45, 46, and 48, we obtain

$(\mu_{2}+\mu_{3})|_{\kappa_{\mathrm{out}}R\to 0}=\mu_{5},$ (54)

indicating that both cases, $\kappa_{\mathrm{out}}\to 0$ and
$\kappa_{\mathrm{out}}=0$, have the same electrophoretic mobility. Therefore,
it is unnecessary to consider the two situations, which are hereafter referred
to as water-in-oil droplet, separately. The surface potential $\psi_{R}$ of a
water-in-oil droplet can be calculated by using the net charge $Q$ as

$\psi_{R}=\frac{Q}{4\pi\varepsilon_{\mathrm{out}}\varepsilon_{0}R}.$ (55)

Therefore, we can rewrite the electrophoretic mobility of the water-in-oil
droplet, $\mu=\mu_{1}+\mu_{4}+\mu_{5}$, using $\sigma_{R}$ and $Q$ as

$\mu=\frac{Q/2\pi
R}{3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}}}\left[\frac{1}{2}+\frac{\eta_{\mathrm{in}}}{\eta_{\mathrm{out}}}+\frac{H(\kappa_{\mathrm{in}}R)}{2}\right]-\frac{\sigma_{R}RH(\kappa_{\mathrm{in}}R)}{3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}}}.$
(56)

If we multiply $\mu$ by $\gamma$, we obtain the effective charge

$Q_{\mathrm{eff}}=Q\frac{2\eta_{\mathrm{in}}+\eta_{\mathrm{out}}(1+2H(\kappa_{\mathrm{in}}R))}{2(\eta_{\mathrm{in}}+\eta_{\mathrm{out}})}-2\pi
R^{2}\sigma_{R}\frac{\eta_{\mathrm{out}}H(\kappa_{\mathrm{in}}R)}{\eta_{\mathrm{in}}+\eta_{\mathrm{out}}}.$
(57)

Eqs. 56 and 57 clearly demonstrate that the electrophoretic mobility of a
water-in-oil droplet is separately affected by the net charge $Q$ and the
surface charge density $\sigma_{R}$. In the next section, we discuss the
results using eqs. 56 and 57.

Before proceeding to the following section, we discuss the validity of
assuming that the oil is ion-free [62, 63, 64, 65, 66]. The ion partition
between the water and oil phases depends on the chemical properties of the
ions and the oil [64, 65, 66]. For example, the Gibbs transfer energy of
alkali metals and halide ions from water to 1,2-dichloroethane (dielectric
constant $\sim 10$) is approximately 30–50 kJ/mol (12–20$k_{\mathrm{B}}T$)
[64]. This indicates that the ion concentration in the oil phase is
significantly lower than that in the aqueous phase where the Boltzmann factor
is at most $10^{-5}$. In addition to these ions, it is assumed that some of
the ionic surfactants are not dissolved in the low-dielectric oil phase. For
example, the Gibbs transfer energy of dodecylsulfate ions from water to
nitrobenzene (dielectric constant $\sim 35$) is $+5\,$kJ/mol
($2k_{\mathrm{B}}T$) [65], and the HLB of sodium dodecylsulfate is estimated
by $\sim 40$ [5]. Due to the different Gibbs transfer energies of the cations
and anions in the water droplet, a Galvani potential difference is created
[66], which can contribute to electrophoretic mobility [67]. However, in this
study, we do not consider this contribution for simplicity. Ion penetration at
the interface is unlikely because the conductivity of the oil phase is
negligible.

Figure 2:  Shape of functions $F_{1}(x)$, $F_{2}(x)$, and $H(x)$.

##  III Results and discussion

###  III.1 Limiting equations for mobility

First, we take the limit of $\kappa_{\mathrm{in}}R\to 0$ (small droplet) and
$\kappa_{\mathrm{in}}R\to\infty$ (large droplet) of eq. 57 using the
relations, $H(x\to 0)=1-x^{2}/35$ and $H(x\to\infty)=5/x$. Then, we obtain

$\displaystyle Q_{\mathrm{eff}}|_{\kappa_{\mathrm{in}}R\to 0}$
$\displaystyle=$ $\displaystyle
Q\frac{2\eta_{\mathrm{in}}+3\eta_{\mathrm{out}}}{2(\eta_{\mathrm{in}}+\eta_{\mathrm{out}})}-2\pi
R^{2}\sigma_{R}\frac{\eta_{\mathrm{out}}}{\eta_{\mathrm{in}}+\eta_{\mathrm{out}}},$
(58) $\displaystyle Q_{\mathrm{eff}}|_{\kappa_{\mathrm{in}}R\to\infty}$
$\displaystyle=$ $\displaystyle
Q\frac{2\eta_{\mathrm{in}}+\eta_{\mathrm{out}}}{2(\eta_{\mathrm{in}}+\eta_{\mathrm{out}})}-\frac{10\pi
R\sigma_{R}}{\kappa_{\mathrm{in}}}\frac{\eta_{\mathrm{out}}}{\eta_{\mathrm{in}}+\eta_{\mathrm{out}}}.$

When $\sigma_{R}=Q/4\pi R^{2}$, eq. 58 approaches to $Q_{\mathrm{eff}}=Q$ like
Hückel formula whereas the second term in eq. LABEL:eq:55 vanishes. Therefore,
eq. LABEL:eq:55 suggests that the effective charge of a large water droplet in
low-molecular-weight insulating oil
($\eta_{\mathrm{out}}\approx\eta_{\mathrm{in}}$) is $Q_{\mathrm{eff}}=(3/4)Q$.

The viscosity of the oil can be changed by choosing the molecular weight of
the oil. For example, at room temperature, benzene and toluene are less
viscous than water $\sim(2/3)\eta_{\mathrm{water}}$ where
$\eta_{\mathrm{water}}$ is the water viscosity, whereas the viscosity of
silicone oil ranges from $\eta_{\mathrm{water}}/2$ to
$10^{6}\eta_{\mathrm{water}}$. In addition, our theory can be applied to water
droplets (even oil droplets) in the gas phase [68]. At room temperature, air
has a viscosity of about $\eta_{\mathrm{water}}/50$. Thus, the two limits of
$\eta_{\mathrm{out}}/\eta_{\mathrm{in}}\to 0$ (droplet in gas [68]) and
$\eta_{\mathrm{out}}/\eta_{\mathrm{in}}\to\infty$ (viscous oil) are worth
considering. Thus, we obtain

$\displaystyle Q_{\mathrm{eff}}|_{\eta_{\mathrm{out}}/\eta_{\mathrm{in}}\to
0}$ $\displaystyle=$ $\displaystyle Q,$ (60) $\displaystyle
Q_{\mathrm{eff}}|_{\eta_{\mathrm{out}}/\eta_{\mathrm{in}}\to\infty}$
$\displaystyle=$ $\displaystyle Q\frac{1+2H(\kappa_{\mathrm{in}}R)}{2}-2\pi
R^{2}\sigma_{R}H(\kappa_{\mathrm{in}}R).$

Another important limit is large or small droplets surrounded by viscous oil,
and in this limit

$\displaystyle
Q_{\mathrm{eff}}|_{\eta_{\mathrm{out}}/\eta_{\mathrm{in}}\to\infty,\kappa_{\mathrm{in}}R\to
0}$ $\displaystyle=$ $\displaystyle\frac{3Q}{2}-2\pi R^{2}\sigma_{R},$ (62)
$\displaystyle
Q_{\mathrm{eff}}|_{\eta_{\mathrm{out}}/\eta_{\mathrm{in}}\to\infty,\kappa_{\mathrm{in}}R\to\infty}$
$\displaystyle=$ $\displaystyle\frac{Q}{2}-\frac{10\pi
R\sigma_{R}}{\kappa_{\mathrm{in}}}.$ (63)

When $\sigma_{R}=Q/4\pi R^{2}$, eq. 62 approaches to $Q_{\mathrm{eff}}=Q$ like
Hückel formula whereas the second term in eq. 63 vanishes. Therefore, eq. 63
suggests that the effective charge of a large water droplet in high-viscous
insulating oil is a half of the net charge, $Q_{\mathrm{eff}}=Q/2$.

Subsequently, we consider the case of $Q=0$ as shown in Fig. 1d. In principle,
the net charge can leak into the external liquid by the relaxation time
$\tau=\varepsilon_{\mathrm{out}}\varepsilon_{0}/\sigma_{\mathrm{out}}$.
Substituting typical values for insulating oils,
$\varepsilon_{\mathrm{out}}=2$ and $\sigma_{\mathrm{out}}=10^{-13}\,$S/m, we
obtain $\tau\sim 100\,$s [34]. Therefore, within minutes of preparing the
emulsions and droplets, the net charge will be negligible. The electrophoretic
mobility in this case is

$\mu=-\frac{\sigma_{R}RH(\kappa_{\mathrm{in}}R)}{3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}}},$
(64)

indicating that the electrophoretic mobility has the inverse sign of the
surface charge density. With the limit $\kappa_{\mathrm{in}}R\to\infty$, eq.
64 reduces to

$\mu|_{\kappa_{\mathrm{in}}R\to\infty}=-\frac{5\sigma_{R}}{\kappa_{\mathrm{in}}(3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}})}$
(65)

Because in this limit
$\sigma_{R}=-\varepsilon_{\mathrm{in}}\varepsilon_{0}\kappa_{\mathrm{in}}\psi_{0}$
where $\psi_{0}$ is the potential at the center, eq. 65 reduces

$\mu=\frac{5\varepsilon_{\mathrm{in}}\varepsilon_{0}\psi_{0}}{3\eta_{\mathrm{in}}+2\eta_{\mathrm{out}}},$
(66)

indicating that the double layer inside the droplet adversely affects
electrophoresis, which has not been pointed out thus far to the best of our
knowledge.

###  III.2 Discussion of the experimental data in the literature

First, we test our theory using the experimental data obtained by Albers and
Overbeek [24]. They prepared a water-in-benzene emulsion using a metallic soap
as a stabilizer, and the radius was about $1\,\mu$m to $4\,\mu$m. Metallic
soaps generally dissolve better in organic liquids than in water. However, as
metallic soaps do not dissociate in the oil phase more than in the aqueous
phase, the dissociated multivalent cations and fatty acids may be better
distributed in the water phase than the oil phase. Therefore, we assume that
our theory can be applied to this experiment.

For simplicity, we assume
$\eta_{\mathrm{in}}=\eta_{\mathrm{out}}=1\,$mPa$\cdot$s, $R=3\,\mu$m,
$\varepsilon_{\mathrm{in}}=80$, $\varepsilon_{\mathrm{out}}=2$. Water-in-
benzene emulsions stabilized by various metallic oleates exhibit positive
electrophoretic mobilities about $+10^{-10}\,$m2/V$\cdot$s ($\zeta=+10\,$mV
based on the Hückel formula
$\mu=2\varepsilon_{\mathrm{out}}\varepsilon_{0}\zeta/3\eta_{\mathrm{out}}$)
and the effective charge is $Q_{\mathrm{eff}}=5\times 10^{-18}\,$C $=30e$.
Because $R=3\,\mu$m and the maximum Debye length of the
$\mathrm{CO_{2}}$-saturated water is $300\,$nm, we set
$H(\kappa_{\mathrm{in}}R)=1/2$ according to Fig. 2. Then, the surface charge
density is obtained as $\sigma_{R}=-0.3\,\mu$C/m2 from eq. 64, which is
quantitatively realistic. Here, we assume $Q=0$ because the net charge of the
emulsion droplet is seemed to be relaxed by the leak and partition of charges.
Therefore, the positive zeta potential observed for a water-in-benzene droplet
stabilized by metallic soaps can be explained by the adsorption of soap anions
at the interface. Because our theory neglects the Marangoni effect, this
explanation is not conclusive, and other explanations are also possible, for
example, strong partition of multivalent cations in the water droplet, or
incomplete dissociation of metallic soaps with multivalent cations [24]. We
propose an experiment to validate our theory using a water-in-oil droplet
stabilized by small amount of monovalent anionic and cationic surfactants that
dissolve or dissociate only in the aqueous phase (HLB is high).

Second, we examine recent experiments on the electrophoretic mobility of
surfactant-free water-in-oil droplets prepared by pipette injection [31, 32,
35, 36, 39]. Before the droplet comes into contact with the electrode, its
electrophoretic mobility is usually positive [35, 36]. The origin of this
positive electrophoretic mobility is believed to be the net charge that
compensates for the negative surface charge on the pipette tip surface [36,
69]. In the work by Schoeler et. al [35], $Q_{\mathrm{eff}}\sim 1\,$pC for a
water droplet in silicone and paraffin oil
($\eta_{\mathrm{out}}/\eta_{\mathrm{in}}\sim 100$) with radius $R=0.6\,$mm,
whereas Choi et. al observed the effective charge $Q_{\mathrm{eff}}\sim
100\,$pC for a water droplet in silicone oil
($\eta_{\mathrm{out}}/\eta_{\mathrm{in}}\sim 50$) with radius $R\approx
1\,$mm. After the droplet comes into contact with the electrode, the water
droplet becomes charged with the sign of the electrode. When a fully
conductive sphere comes into contact with the electrode, the charge acquired
by the sphere is $Q_{\mathrm{M}}=(\pi^{2}/6)(4\pi
R^{2})\varepsilon_{\mathrm{out}}\varepsilon_{0}E$ [70]. Thus far, Choi et. al
measured the net charge of pipette-injected water droplets by the Faraday cup
method, and they obtained $Q_{\mathrm{eff}}=Q$ for deionized water droplets
using the Hadamard-Rybczynski friction coefficient [36]. Furthermore, Im
investigated the wall effect on hydrodynamic friction and obtained
$Q_{\mathrm{eff}}=Q_{\mathrm{M}}$ using the wall-effect-included Stokes
friction [39]. Our theory, eq. 63, predicts $Q_{\mathrm{eff}}=Q/2$ when the
surface charge of a droplet is negligible, and thus the experimental data
disagree with our theory. However, for a water droplet, the charge extracted
from the electric current signal is usually smaller than $Q_{\mathrm{M}}$ [32,
61]. The hydrodynamic friction usually deviates from the Hadamard-Rybczynski
friction coefficient because of the wall effect [35, 39] and unexpected
impurities at the interface [33, 39]. Impurities at the oil-water and air-
water interfaces have been proposed by other studies to explain various
experimental phenomena [71, 72, 19]. If our theory is applied to the Stokes
friction (no slip boundary), the results are completely replaced by Henry’s
classical result [59]. Eqs. 12 and 13 are replaced by
$\boldsymbol{u}\cdot\boldsymbol{e}_{\theta}|_{r=R+0}=0$ and
$\boldsymbol{u}\cdot\boldsymbol{e}_{\theta}|_{r=R-0}=0$; therefore, the inner
flow of the droplet is independent of the outer flow. If the surrounding
liquid contains no ions, even if the radius of the droplet is as large as
$R\approx 1\,$mm, $Q_{\mathrm{eff}}=Q$ recovers. In the context of the
electrophoresis of water-in-oil droplet, $Q_{\mathrm{eff}}=Q$ was an
assumption, and it was not examined itself [32, 33, 36, 35]. Therefore, to
experimentally validate $Q_{\mathrm{eff}}=Q/2$, the experimental setup should
be designed to prevent contamination and to provide independent measurements
of the hydrodynamic friction and the electrostatic net charge of a millimeter-
scale droplet.

##  IV Conclusions

We analytically calculated the electrophoretic mobility of a water-in-oil
droplet with Debye-Hückel approximation. The resulting electrophoretic
mobility depends separately on the net charge of the droplet and the surface
charge density at the droplet interface. When the net charge is negligible,
the electrophoretic mobility is proportional to the surface charge density
with a negative coefficient. This indicates that the internal electric double
layer adversely contributes to electrophoresis.

Our theory is used to discuss the experimental data of water-in-oil emulsions
and droplets in the literature. The positive electrophoretic mobility of a
water-in-benzene emulsion stabilized by metallic soaps can be explained by the
negative surface charge of the absorbed soap ions. Recent electrophoretic
mobility measurements of water-in-oil droplets suggest that the effective
charge deduced from electrophoresis is identical to the net charge of the
droplet. Conversely, our theory indicates that the effective charge should be
half the net charge when the surface charge density is negligible. However, in
experiments, the Hadamard-Rybczynski friction is quite difficult to achieve,
and the wall effect and unexpected impurities at the interface often modify
the hydrodynamic friction [35, 39]. The accurate electric measurement of the
net charge of the droplet is another issue when comparing the effective
electrophoretic charge with the net charge [61]. Thus, it is still unclear
what determines the electrophoretic mobility of water-in-oil droplets in
experiments.

Finally, we briefly mention the outlook on the electrophoresis of water-in-oil
droplets. In this paper, ion adsorption was not included in the model.
However, surface charge density is generally caused by ion adsorption [73].
Ion partition to the oil phase would also be important because it creates a
Galvani potential difference across the interface, even when the oil phase
concentration is extremely low [66, 67]. To include such effects in the model,
the Poisson-Boltzmann and hydrodynamic equations must be treated completely
nonlinearly. Marangoni flow, the stagnant cap of the surfactants at the
interface, and droplet deformation are other important effects on
electrophoresis [33, 54, 74].

## acknowledgement

This work was supported by JST, PRESTO Grant Number JPMJPR21O2, Japan. YU
thanks JSPS KAKENHI Grant Numbers 18KK0151 and 20K14430, Sasakawa Scientific
Research Grant from The Japan Science Society (2021-3001), Grant from Kurita
Water and Environment Foundation (21E006), and Qdai-jump Research Program from
Kyushu University (R3-01302) for their support.

##  V Appendix

In this appendix, we briefly discuss how the Marangoni effect modifies the
steady-state low-Reynolds-number electrohydrodynamic equations [46]. The ionic
currents inside the droplet and at the interface are given by

$\displaystyle\boldsymbol{j}_{\pm}$ $\displaystyle=$ $\displaystyle
c_{\pm,\mathrm{in}}\boldsymbol{u}-\omega_{\pm}c_{\pm,\mathrm{in}}\nabla\mu_{\pm},$
(67) $\displaystyle\boldsymbol{j}_{\pm}^{\mathrm{s}}$ $\displaystyle=$
$\displaystyle\Gamma_{\pm}\boldsymbol{u}^{\mathrm{s}}-\omega_{\pm}^{\mathrm{s}}\Gamma_{\pm}\nabla^{\mathrm{s}}\mu^{\mathrm{s}}_{\pm}$
(68)

where $\omega_{\pm}$ and $\omega_{\pm}^{\mathrm{s}}$ are the ionic mobilities
inside the droplet and at the interface, and $\Gamma_{\pm}$ is the surface
concentration of each ions. The chemical potentials $\mu_{\pm}$ and
$\mu_{\pm}^{\mathrm{s}}$ are defined by

$\displaystyle\mu_{\pm}$ $\displaystyle=$ $\displaystyle k_{\mathrm{B}}T\ln
c_{\pm,\mathrm{in}}\pm e\psi,$ (69) $\displaystyle\mu^{\mathrm{s}}_{\pm}$
$\displaystyle=$ $\displaystyle k_{\mathrm{B}}T\ln(\Gamma_{\pm}/z^{*})\pm
e\psi+\Delta G_{\pm},$ (70)

where $\Delta G_{\pm}$ is the adsorption energy, and $z^{*}$ is the thickness
of the adsorption layer. The equilibrium adsorption isotherm of ions can be
obtained by the relation $\mu_{\pm}|_{r=R-0}=\mu_{\pm}^{\mathrm{s}}$, and this
yields

$\Gamma_{\pm}=K_{\pm}c_{\pm}|_{r=R-0},$ (71)

where $K_{\pm}=z^{*}\mathrm{e}^{-\Delta G_{\pm}/k_{\mathrm{B}}T}$ is the
adsorption coefficient. The adsorption-desorption dynamics can be included by
the reaction equations [50], while in this paper we assume it is rapid
according to the previous work [46]. Then, the surface charge density is given
by

$\begin{split}\sigma_{R}&=e(\Gamma_{+}-\Gamma_{-})\\\
&=e(K_{+}c_{+}|_{r=R-0}-K_{-}c_{-}|_{r=R-0}),\end{split}$ (72)

and thus, under the assumption of rapid adsorption-desorption dynamics, the
surface charge gradient is directly coupled to the distortion of the electric
double layer. The conservation law for the ions at the interface is,
therefore,

$\nabla^{\mathrm{s}}\cdot\boldsymbol{j}^{\mathrm{s}}_{\pm}+(\boldsymbol{j}_{\pm}|_{r=R+0}-\boldsymbol{j}_{\pm}|_{r=R-0})\cdot\boldsymbol{e}_{r}=0$
(73)

and for water-in-oil droplet (no ion outside the droplet) with rapid
adsorption-desorption equilibrium we have

$\displaystyle\boldsymbol{j}_{\pm}\cdot\boldsymbol{e}_{r}|_{r=R+0}$
$\displaystyle=$ $\displaystyle 0,$ (74)
$\displaystyle\boldsymbol{j}_{\pm}\cdot\boldsymbol{e}_{r}|_{r=R-0}$
$\displaystyle=$
$\displaystyle\nabla^{\mathrm{s}}\cdot\boldsymbol{j}^{\mathrm{s}}_{\pm},$ (75)

instead of eq. 14. The boundary condition of the tangential stress tensor is

$\sigma_{r\theta}|_{r=R+0}-\sigma_{r\theta}|_{r=R-0}=k_{\mathrm{B}}T\sum_{i=\pm}(\nabla^{\mathrm{s}}\Gamma_{i})\cdot\boldsymbol{e}_{\theta}$
(76)

instead of eq. 13. To verify the assumption of neglecting the Marangoni effect
(right hand side of eqs. 75 and 76), it is necessary to solve full equations
with nonlinrar electrostatics and to evaluate each term in specific parameter
sets [46]. Because the Marangoni effect is usually proportional to the
Marangni number $\Gamma_{\pm}R/\eta_{\mathrm{in}}\omega^{\mathrm{s}}$, it can
be neglected when the surface concentration is small as
$\Gamma_{\pm}\ll\eta_{\mathrm{in}}\omega^{\mathrm{s}}/R$. For oil-in-water
droplets with anionic surfactants, full calclations and verifications were
performed for MHz frequency [54, 55]. However, for the water-in-oil droplets,
a clear analytical or numerical study to explain how the Marangoni effect
modifies the electrophoretic mobility is still lack, and thus, it remains a
future problem.

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