# Universal Translational and Rotational Mobility Expressions of Phoretic and
Self-phoretic Particles with Arbitrary Interaction Potentials

Arkava Ganguly∗1  Souradeep Roychowdhury∗1  Ankur Gupta1
<EMAIL_ADDRESS>1Department of Chemical and Biological Engineering,
University of Colorado Boulder, Boulder, CO 80303, USA

###### Abstract

The mobility of externally-driven phoretic propulsion of particles is
evaluated by simultaneously solving the solute conservation equation,
interaction potential equation, and the modified Stokes equation. While
accurate, this approach is cumbersome, especially when the interaction
potential decays slowly compared to the particle size. In contrast to external
phoresis, the motion of self-phoretic particles is typically estimated by
relating the translation and rotation velocities with the local slip velocity.
While this approach is convenient and thus widely used, it is only valid when
the interaction decay length is significantly smaller than the particle size.
Here, by employing the Lorentz reciprocal theorem, we combine the benefits of
two approaches and derive unified mobility expressions with arbitrary
interaction potentials such that the expressions predict the translation and
rotation velocities for both externally driven and self-propelling particles.
We show that these expressions can conveniently recover the well-known
mobility relationships of external electrophoresis and diffusiophoresis for
arbitrary double-layer thickness. Additionally, we show that for a spherical
microswimmer, our derived expressions relax to the slip velocity calculations
in the limit of thin interaction lengthscales. We also employ the derived
mobility expressions to calculate the velocities of an autophoretic Janus
particle. We find that there is significant dampening in the translation
velocity even when the interaction length is an order of magnitude larger than
the particle size. Finally, we study the motion of a catalytically self-
propelled particle, while it also propels due to external concentration
gradients, and demonstrate how the two propulsion modes compete with each
other.

**footnotetext: These authors contributed equally to this work

##  1 Introduction

Phoretic phenomena, i.e., the movement of particles in response to an external
field (Anderson, 1989; Velegol et al., 2016; Khair, 2022), is pivotal for a
range of applications such as separation of biomacromolecules (Heller, 2001;
Lee et al., 2012), purification of nucleic acids from whole blood (Persat et
al., 2009), measurement of zeta potential (Doane et al., 2012; Shin et al.,
2017a), banding of colloidal particles (Abécassis et al., 2008; Banerjee &
Squires, 2019; Raj et al., 2023b), membrane-less water filtration (Shin et
al., 2017b) and understanding biological pattern formation (Alessio & Gupta,
2023), among others. In contrast, self-phoretic particles, also known as
microswimmers, respond to a self-generated field gradient (Paxton et al.,
2004; Howse et al., 2007; Ebbens & Howse, 2010; Popescu et al., 2016; Ganguly
et al., 2023). Microswimmers are extensively studied for applications in
targeted drug delivery (Xuan et al., 2014; Luo et al., 2018), environmental
remediation (Gao et al., 2013; Wang et al., 2019), remote sensing of toxic
chemicals (Esteban-Fernández de Ávila et al., 2016), the autonomous motion of
microbots (Zarei & Zarei, 2018; Hu et al., 2022), and collective behavior of
active colloids (Palacci et al., 2013; Takatori & Brady, 2016; Illien et al.,
2017). The mobility expressions of phoretic and self-phoretic processes are
identical, with the key distinction being that the origin of field gradients
in the two processes is different. In phoretic processes, this field gradient
is externally imposed on colloidal particles, while in self-phoretic particles
they are locally generated by the particles themselves typically through
surface reactions or other mechanisms.

Studies on external electrophoretic motion have focused on the dependence of
electrophoretic mobility on the effect of particle shape (Yoon & Kim, 1989;
Solomentsev & Anderson, 1994), surface heterogeneity (Fair & Anderson, 1992;
Velegol et al., 1996), finite double-layer thickness (Henry, 1931; OB́rien &
White, 1978), and more recently, strong deformation of double-layers (Khair,
2018, 2022) and charge reversal (Kubíčková et al., 2012; Gupta et al., 2020a).
Similarly, researchers have predicted the dependence of diffusiophoretic
mobility (Anderson, 1989; Brady, 2011) on finite double-layer thickness
(Prieve et al., 1984; Keh & Wei, 2000), surface chemistry (Gupta et al.,
2020b), and multiple electrolytes (Gupta et al., 2019; Alessio et al., 2021).
Studies on self-phoretic systems (Ramaswamy, 2010; Moran & Posner, 2017) focus
on the impact of particle shape (Shklyaev et al., 2014; Nourhani & Lammert,
2016; Poehnl et al., 2020; Daddi-Moussa-Ider et al., 2021; Ganguly & Gupta,
2023; Raj et al., 2023a; Lee et al., 2023), active patch shape (Lisicki et
al., 2018; Lee et al., 2021), surface interaction (Sharifi-Mood et al., 2013)
and finite Péclet number (Michelin & Lauga, 2014).

Broadly speaking, there are two approaches for predicting the mobilities
described above. The first approach solves the coupled solute conservation
equations and the modified Stokes equation (Henry, 1931; OB́rien & White,
1978; Prieve et al., 1984; Prieve & Roman, 1987; Anderson, 1989; Keh & Wei,
2000; Sharifi-Mood et al., 2013; Khair, 2018, 2022; Gupta et al., 2019) and
employs a force-free and torque-free condition to arrive at the translation
velocity, $\mathbf{U}$, and rotational velocity, $\mathbf{\Omega}$, of the
particle. Thus the above approach requires resolving the interaction potential
simultaneously with the hydrodynamic equations. While exact and powerful, the
methodology described above is cumbersome for analytical results when the
particle-fluid interaction potential decays at much larger length scales than
the particle size. Further, the solution strategy needs to be revised whenever
the interaction potential changes, making it less convenient to be integrated
into other analyses.

The second approach employs the reciprocal theorem in the thin interaction
length limit (Stone & Samuel, 1996; Brady, 2011; Michelin & Lauga, 2014;
Lisicki et al., 2018; Masoud & Stone, 2019; Poehnl et al., 2020; Ganguly &
Gupta, 2023; Raj et al., 2023a). In this limit, it is assumed that there
exists a slip velocity, $\mathbf{u}_{s}$, at the interface of the inner region
where the interaction potential is non-zero, and the outer region where the
interaction potential is zero. This allows us to treat the outer problem as a
classical Stokes flow problem with a slip boundary condition. Consequently,
$\mathbf{U}$ and $\mathbf{\Omega}$ can be represented as surface integrals of
appropriate functions of $\mathbf{u}_{s}$. This approach based on the
reciprocal theorem was first utilized by Stone & Samuel (1996) to study the
impact of distortions in spherical microswimmers. This methodology is
particularly powerful because, unlike the first method, computing $\mathbf{U}$
and $\mathbf{\Omega}$ is relatively straightforward and agnostic to the
mechanistic origin of $\mathbf{u}_{s}$. However, this approach is valid only
when the interaction potential length is significantly smaller than the
particle size, restricting its applicability. Additionally, it requires the
knowledge of $\mathbf{u}_{s}$ a priori, and most studies have to rely on a
lumped mobility parameter to estimate the value of $\mathbf{u}_{s}$ (Michelin
& Lauga, 2014; Lisicki et al., 2018; Poehnl & Uspal, 2021; Ganguly & Gupta,
2023; Raj et al., 2023a).

In this work, we seek to unify the benefits realized through the two
approaches. To do so, we take inspiration from prior literature on the
inertial correction to Stokes flow (Brenner & Cox, 1963; Hinch, 1991; Leal,
2007), where the first-order corrections include a body force term from the
leading order and reciprocal theorem is employed to find the resulting motion.
In section 2, we take a similar approach to obtain a general mobility
expression for an arbitrary particle shape, subjected to an osmophoretic
(combination of osmotic and phoretic force) body force $\mathbf{b}$.
Subsequently, we take our expression to the thin interaction length scale
limit and retrieve the mobility expressions in Stone & Samuel (1996), see
section 3. In section 3, we also retrieve the expression for both the
electrophoretic mobility of translation of a charged spherical particle in an
external electric field, obtained by Henry (1931). Additionally, we derive the
diffusiophoretic mobility of a charged spherical particle in an externally
imposed solute gradient at finite interaction lengths, as first obtained by
Keh & Wei (2000). Finally in section 4, we apply our result to study the
autophoretic motion of spherical microparticles with catalytic caps. We study
how translation velocity depends on the cap size, the surface interaction
potential, and the interaction length relative to particle size. Since our
methodology works for both externally driven and self-propelling particles, we
also study particle propulsion by both modes simultaneously; see section 5.
These model problems demonstrate the wide applicability of the expressions
derived in this manuscript. Finally, in section 6 we summarize the key
findings from our work and outline future ideas.

##  2 Derivation of the unified mobility expression

In this section, we derive the translation velocity ($\mathbf{U}$) and
rotational velocity ($\mathbf{\Omega}$) of an arbitrary particle with surface
$S_{p}$ immersed in a fluid of volume $V$ due to an arbitrary osmophoretic
body force $\mathbf{b}$, see Fig 1a. The particle surface is defined via the
vector $\mathbf{x}_{S}$ relative to the center of mass of the particle. We
define $\mathbf{e}_{r}$ as the outward unit normal to the particle surface,
$r$ is the distance from the center of mass of the particle and $\mathbf{r}$
is the position vector defined from the center of mass of the particle. The
fluid velocity around the particle can be resolved through the modified Stokes
equation, defined as,

$\mathbf{\nabla}\cdot\bm{\sigma}+\mathbf{b}=\bm{0},$ (1)

where $\bm{\sigma}$ is the hydrodynamic stress tensor. The velocity field
$\mathbf{u}$ is assumed to decay to zero in the far-field,
$\mathbf{u}|_{r\to\infty}\to 0$. At the particle surface, the fluid obeys a
no-slip, rigid body boundary condition,
$\mathbf{u}|_{S_{p}}=\mathbf{U}+\mathbf{\Omega}\times\mathbf{x}_{S}$. To
obtain $\mathbf{U}$ and $\mathbf{\Omega}$ using Lorentz reciprocal theorem
(Masoud & Stone, 2019), we define an auxiliary Stokes flow
($\hat{\mathbf{U}},\hat{\mathbf{\Omega}}$) while preserving particle geometry
with the same no-slip rigid surface,
$\hat{\mathbf{u}}|_{S_{p}}=\hat{\mathbf{U}}+\hat{\mathbf{\Omega}}\times\mathbf{x}_{S}$,
and far-field decay, $\hat{\mathbf{u}}|_{r\to\infty}\to 0$, boundary
conditions.

Using the Lorentz reciprocal theorem, we can relate the phoretic problem
($\mathbf{U}$,$\mathbf{\Omega}$,$\mathbf{b}$) and the auxiliary problem
($\hat{\mathbf{U}}$,$\hat{\mathbf{\Omega}}$,$\hat{\mathbf{b}}$) to be

$\displaystyle\int_{S_{p}}\mathbf{e}_{r}\cdot\bm{\sigma}\cdot\hat{\mathbf{u}}\;{\rm
dS}-\displaystyle\int_{S_{p}}\mathbf{e}_{r}\cdot\hat{\bm{\sigma}}\cdot\mathbf{u}\;{\rm
dS}=\displaystyle\int_{V}\hat{\mathbf{u}}\cdot\mathbf{b}\;{\rm
dV}-\displaystyle\int_{V}\mathbf{u}\cdot\hat{\mathbf{b}}\;{\rm dV}.$ (2)

Substituting in the expressions of the fluid velocities, $\mathbf{u}|_{S_{p}}$
and $\hat{\mathbf{u}}|_{S_{p}}$, at the particle surface we can simplify Eq.
(2). Additionally, we assume that there is no body force in the auxiliary
problem, ${\mathbf{\hat{b}}}=\bm{0}$. Thus we can rewrite Eq. (2) to be,

$\displaystyle\int_{S_{p}}\mathbf{e}_{r}\cdot\bm{\sigma}\cdot\left(\hat{\mathbf{U}}+\hat{\mathbf{\Omega}}\times\mathbf{x}_{S}\right){\rm
dS}-\displaystyle\int_{S_{p}}\mathbf{e}_{r}\cdot\hat{\bm{\sigma}}\cdot\left(\mathbf{U}+\mathbf{\Omega}\times\mathbf{x}_{S}\right){\rm
d}S=\displaystyle\int_{V}\hat{\mathbf{u}}\cdot\mathbf{b}\;{\rm dV}.$ (3)

Since the inertia of the particle is negligible, for both the phoresis and the
auxiliary problem, the particle is force and torque-free. For the phoretic
propulsion, the hydrodynamic force and torque is balanced by the osmophoretic
force and torque, or

$\underbrace{\displaystyle\int_{S_{p}}\mathbf{e}_{r}\cdot\bm{\sigma}\;{\rm
dS}}_{\text{hydrodynamic}}\ \ \ \
\underbrace{-\displaystyle\int_{V}\mathbf{b}\;{\rm
dV}}_{\text{osmophoretic}}=\mathbf{0},$ (4)
$\underbrace{\displaystyle\int_{S_{p}}\mathbf{x}_{s}\times\mathbf{e}_{r}\cdot\bm{\sigma}\;{\rm
dS}}_{\text{hydrodynamic}}\ \ \ \
\underbrace{-\displaystyle\int_{V}\mathbf{r}\times\mathbf{b}\;{\rm
dV}}_{\text{osmophoretic}}=\mathbf{0},$ (5)

where the negative sign in front of the osmophoretic term comes because the
osmophoretic force on the particle is equal in magnitude to the osmophoretic
force on the fluid but opposite in sign (Brady, 2011). For the auxiliary
system, we balance the hydrodynamic and external forces and torques, or

$\underbrace{\displaystyle\int_{S_{p}}\hat{\bm{\sigma}}\cdot\mathbf{e}_{r}\;{\rm
dS}}_{\text{hydrodynamic}}\ \ \ \ +\underbrace{\hat{\mathbf{F}}_{\rm
ext}}_{\text{external}}=\bm{0},$ (6)
$\underbrace{\displaystyle\int_{S_{p}}\mathbf{x}_{s}\times\hat{\bm{\sigma}}\cdot\mathbf{e}_{r}\;{\rm
dS}}_{\text{hydrodynamic}}\ \ \ \ +\underbrace{\hat{\mathbf{L}}_{\rm
ext}}_{\text{external}}=\bm{0},$ (7)

where $\hat{\mathbf{F}}_{\rm ext}$ and $\hat{\mathbf{L}}_{\rm ext}$ are the
external force and torque required to move the particle in the auxiliary
problem, which is to be determined. We note that Eqs. (4)-(5) are different
from Eqs. (6)-(7) since the phoretic propulsion is induced by $\mathbf{b}$
whereas in the auxiliary problem motion is caused by $\hat{\mathbf{F}}_{\rm
ext}$ and $\hat{\mathbf{L}}_{\rm ext}$.

To calculate $\mathbf{U}$ and $\mathbf{\Omega}$ for a given $\mathbf{b}$ from
Eq. (3) - (7), we need to express, $\hat{\mathbf{F}}_{\rm ext}$ ,
$\hat{\mathbf{L}}_{\rm ext}$ and $\mathbf{\hat{u}}$ in the auxiliary problem
as functions of $\mathbf{\hat{U}}$ and $\mathbf{\hat{\Omega}}$. To do so, we
use a resistance formulation to write,

$\begin{bmatrix}\hat{\mathbf{F}}_{\rm ext}\\\ \hat{\mathbf{L}}_{\rm
ext}\end{bmatrix}=\begin{bmatrix}\mathbf{R}_{FU}&\mathbf{R}_{F\Omega}\\\
\mathbf{R}_{LU}&\mathbf{R}_{L\Omega}\end{bmatrix}\cdot\begin{bmatrix}\hat{\mathbf{U}}\\\
\hat{\mathbf{\Omega}}\end{bmatrix},$ (8)

where the resistance matrices $\mathbf{R}_{FU}$, $\mathbf{R}_{F\Omega}$,
$\mathbf{R}_{LU}$, and $\mathbf{R}_{L\Omega}$ relate the driving force
($\hat{\mathbf{F}}_{\rm ext}$) and torque ($\hat{\mathbf{L}}_{\rm ext}$) to
the translational ($\mathbf{\hat{U}}$) and rotational velocity
($\mathbf{\hat{\Omega}}$). Further, we describe $\hat{\mathbf{u}}$ as

$\hat{\mathbf{u}}=\mathbf{D}_{T}\cdot\hat{\mathbf{U}}+\mathbf{D}_{R}\cdot\hat{\mathbf{\Omega}}\times\mathbf{r},$
(9)

where $\mathbf{D}_{T}$ is the translation disturbance tensor and
$\mathbf{D}_{R}$ is the rotation disturbance tensor. Eqs. (8) - (9) combined
with Eqs. (6) - (7) provide necessary information to simplify Eq. (3) as a
function of $\mathbf{\hat{U}}$ and $\mathbf{\hat{\Omega}}$.

Next, we choose convenient values of $\mathbf{\hat{U}}$ and
$\mathbf{\hat{\Omega}}$ to simplify Eq. (3). Specifically, we use six
auxiliary flow problems, pure translation ($\hat{\mathbf{\Omega}}=\mathbf{0}$
and $\hat{\mathbf{U}}$ = $U_{0}\mathbf{e}_{1}$, $U_{0}\mathbf{e}_{2}$,
$U_{0}\mathbf{e}_{3}$) and pure rotation ($\hat{\mathbf{U}}=\mathbf{0}$ and
$\hat{\mathbf{\Omega}}=U_{0}/a\mathbf{e}_{1}$, $U_{0}/a\mathbf{e}_{2}$,
$U_{0}/a\mathbf{e}_{3}$) with $U_{0}$ being the characteristic velocity scale
and $a$ being the characteristic particle length, to obtain

$\begin{bmatrix}\mathbf{R}_{FU}&\mathbf{R}_{F\Omega}\\\
\mathbf{R}_{LU}&\mathbf{R}_{L\Omega}\end{bmatrix}\cdot\begin{bmatrix}{\mathbf{U}}\\\
{\mathbf{\Omega}}\end{bmatrix}=\begin{bmatrix}\displaystyle\int_{V}\left(\mathbf{D}_{T}-\mathbf{I}\right)\cdot\mathbf{b}\;{\rm
dV}\\\
\displaystyle\int_{V}\left(\mathbf{D}_{R}-\mathbf{I}\right)\cdot\mathbf{r}\times\mathbf{b}\;{\rm
dV}\end{bmatrix},$ (10)

By inverting the resistance tensor, we obtain a mobility formulation that
resolves $\mathbf{U}$ and $\mathbf{\Omega}$ in terms of volume integrals of
$\mathbf{b}$,

$\begin{bmatrix}\mathbf{U}\\\
\mathbf{\Omega}\end{bmatrix}=\begin{bmatrix}\mathbf{M}_{UF}&\mathbf{M}_{UL}\\\
\mathbf{M}_{\Omega F}&\mathbf{M}_{\Omega
L}\end{bmatrix}\cdot\begin{bmatrix}\displaystyle\int_{V}\left(\mathbf{D}_{T}-\mathbf{I}\right)\cdot\mathbf{b}\;{\rm
dV}\\\
\displaystyle\int_{V}\left(\mathbf{D}_{R}-\mathbf{I}\right)\cdot\mathbf{r}\times\mathbf{b}\;{\rm
dV}\end{bmatrix}$ (11)

where the matrices $\mathbf{M}_{FU}$, $\mathbf{M}_{F\Omega}$,
$\mathbf{M}_{LU}$, and $\mathbf{M}_{L\Omega}$ are the corresponding mobility
tensors.

Physically, Eq. (11) is insightful as it helps parse apart the difference
between the phoretic problem and the auxiliary problem. The rightmost term is
the effective force and torque on the particle due to phoretic interactions
and has two contributions: (1) the term associated with the identity tensor
($\mathbf{I}$) is the osmophoretic force and torque acting on the particle,
and (2) the term associated with the disturbance tensors ($\mathbf{D}_{T}$,
$\mathbf{D}_{R}$) is the hydrodynamic correction to the distribution of body
forces around the particle. This correction arises because the phoretic
interactions near the particle surface lead to an additional compensating
fluid motion (Brady, 2011) causing a long-range hydrodynamic disturbance. This
effect is not captured in the definition of the hydrodynamic mobility tensor
and thus manifests separately. If the terms associated with disturbance
tensors were not present, Eq. (11) is essentially identical to Eq. (8) with
osmophoretic force on the particle as the external force.

We note that Eq. (11) is valid only when $\mathbf{b}$ is independent of
$\mathbf{U}$ and $\mathbf{\Omega}$ and is thus suitable for systems with small
Péclet number ($\mathrm{Pe}\ll 1$), where $\mathrm{Pe}={U_{0}L}/{D}$, and $D$
is the diffusivity of the solute. The distinction from prior work, such as
Stone & Samuel (1996); Michelin & Lauga (2014); Lisicki et al. (2018); Poehnl
et al. (2020); Poehnl & Uspal (2021); Ganguly & Gupta (2023), that utilize
Lorentz reciprocal theorem is that they invoke the thin interaction length
limit and apply the analysis in the outer region where $\mathbf{b}=\bm{0}$;
see Fig. 1b. Consequently, they do not arrive at Eq. (11) but rather represent
$\mathbf{U}$ and $\mathbf{\Omega}$ in terms of a slip velocity at the particle
surface $\mathbf{u}_{s}$.

We acknowledge that similar results have been presented in Khair (2018) and
Brady (2021). However, in Khair (2018), $\mathbf{b}$ only focused on the
electrophoretic contributions. In contrast, Brady (2021) argued that
$\mathbf{b}$ could include both osmotic and phoretic contributions. Building
on the work by Brady (2021), we systematically illustrate how both phoretic
and osmotic contributions to the body force term are required to reconcile a
broad range of results in the literature and arrive at universal mobility
relationships. We thus refer to $\mathbf{b}$ as an osmophoretic body force.

For a spherical particle (see Duprat & Stone (2016) for derivation), the
relevant hydrodynamic parameters are
$\mathbf{D}_{T}=\frac{3a}{4r}\left(\mathbf{I}+\mathbf{e}_{r}\mathbf{e}_{r}\right)+\frac{a^{3}}{4r^{3}}\left(\mathbf{I}-3\mathbf{e}_{r}\mathbf{e}_{r}\right)$,
$\mathbf{D}_{R}=\frac{a^{3}}{r^{3}}\mathbf{I}$,
$\mathbf{M}_{UF}=\frac{1}{6\pi\mu a}\mathbf{I}$, $\mathbf{M}_{UL}=0$,
$\mathbf{M}_{\Omega F}=0$ and $\mathbf{M}_{\Omega L}=\frac{1}{8\pi\mu
a^{3}}\mathbf{I}$, where $a$ is the radius of the sphere, $\mu$ is the fluid
viscosity, $r$ is the radial distance from the center of the sphere, and
$\mathbf{e}_{r}$ is the radial vector pointing away from the center.
Substituting, these definitions of the hydrodynamic disturbance and mobility
in Eq. (11) we obtain,

$\mathbf{U}=\frac{1}{6\pi\mu
a}\displaystyle\int_{V}\left[\left(\frac{3a}{2r}-\frac{a^{3}}{2r^{3}}-1\right)\mathbf{b}_{\perp}+\left(\frac{3a}{4r}+\frac{a^{3}}{4r^{3}}-1\right)\mathbf{b}_{\parallel}\right]\;{\rm
dV},$ (12) $\mathbf{\Omega}=\frac{1}{8\pi\mu
a^{3}}\displaystyle\int_{V}r\left(\frac{a^{3}}{r^{3}}-1\right)\mathbf{e}_{r}\times\mathbf{b}_{\parallel}\;{\rm
dV},$ (13)

where the body force is decomposed into
$\mathbf{b}=\mathbf{b}_{\perp}+\mathbf{b}_{\parallel}$. The perpendicular
subscript denotes the component normal to the sphere and the parallel
subscript denotes the component parallel to the surface. Eq. (12) \- (13) is a
key result of this article. We extensively validate this result in the next
section and show it relaxes to the various well-known expressions present in
the literature, for both microswimmers and externally-driven particles.
Further, we employ this expression to study a microswimmer in the arbitrary
interaction layer limit and a microswimmer driven by an external gradient in
addition to its self-propelling mode of swimming.

##  3 Validation

###  3.1 Simplification at the limit of the thin interaction length scale

In this subsection, we aim to simplify Eq. (12) - (13) in the limit of the
thin interaction length scale for a spherical microswimmer and recover the
equations discussed in Stone & Samuel (1996).

We proceed to simplify Eqs. (12)-(13) at the thin interaction length limit,
${\lambda}/{a}\ll 1$, where $\lambda$ is the interaction lengthscale. To this
end, we define a stretched radial coordinate $\rho={(r-a)}/{\lambda}$. Next,
we expand and re-write Eqs. (12)-(13) in orders of ${\lambda}/{a}$.
Subsequently, the leading order contribution to the translation and rotation
velocities are obtained to be,

$\mathbf{U}=-\frac{\lambda}{4\pi\mu
a^{2}}\displaystyle\int_{V}\rho\mathbf{b}_{\parallel}\;{\rm dV},$ (14)
$\mathbf{\Omega}=-\frac{3\lambda}{8\pi\mu
a^{3}}\displaystyle\int_{V}\rho\mathbf{e}_{r}\times\mathbf{b}_{\parallel}\;{\rm
dV}.$ (15)

Since the volume of interest at the thin interaction limit is a spherical
shell of thickness $\lambda$ surrounding the particle, we can rewrite the
differential volume element to be, ${\rm dV}=\lambda{\rm d\rho dS}$ and the
volume integrals as,

$\mathbf{U}=-\frac{\lambda^{2}}{4\pi\mu
a^{2}}\displaystyle\int_{S_{p}}\left[\displaystyle\int_{0}^{\infty}\rho\mathbf{b}_{\parallel}\;{\rm
d\rho}\right]\;{\rm dS},$ (16) $\Omega=-\frac{3\lambda^{2}}{8\pi\mu
a^{3}}\displaystyle\int_{S_{P}}\mathbf{e}_{r}\times\left[\displaystyle\int_{0}^{\infty}\rho\mathbf{b}_{\parallel}\;{\rm
d}\rho\right]\;{\rm d}S.$ (17)

Figure 1: Two approaches to finding the velocity of a particle by resolving
the fluid velocity at the particle surface. (a) Obtain the fluid velocity near
the particle surface by resolving the modified Stokes equation with an
arbitrary body force ($\mathbf{b}$). The body force, $\mathbf{b}$, depends on
charge ($\rho$), salt ($s$), and interaction potential ($\phi$). (b) When the
interaction length is small $\lambda/a\ll 1$, the velocity near the fluid
surface, at the outer edge of the interaction layer,
$\mathbf{u}_{\mathrm{s}}$, is taken to be the velocity at the particle
surface. The slip velocity, $\mathbf{u}_{\mathrm{s}}$, depends on the lumped
mobility ($\mathcal{M}$), which depends on the interaction between the surface
and solute, and the solute concentration at the vicinity of the surface ($c$).

In the thin interaction limit, the shear force is balanced by the parallel
body force, or

$\frac{\mu}{\lambda^{2}}\frac{\partial^{2}\mathbf{u}_{\parallel}}{\partial\rho^{2}}+\mathbf{b}_{\parallel}=0.$
(18)

We note that in Eq. (18), since $\mathbf{b}$ is the osmophoretic force,
$\mathbf{b}_{\parallel}$ also includes the excess osmotic term. Multiplying
Eq. (18) by $\rho$ and employing integration by parts, we arrive at

$\displaystyle\int_{0}^{\infty}\rho\mathbf{b}_{\parallel}\;{\rm
d\rho}=\frac{\mu}{\lambda^{2}}\mathbf{u}_{\parallel}|_{0}^{\infty}=\frac{\mu}{\lambda^{2}}\mathbf{u}_{s},$
(19)

where $\mathbf{u}_{s}=\mathbf{u}_{\parallel,\infty}-\mathbf{u}_{\parallel,0}$,
is the phoretic slip velocity. Substituting Eq. (19) into Eqs. (16)-(17), we
get the widely used result derived in Stone & Samuel (1996),

$\mathbf{U}=-\frac{1}{4\pi a^{2}}\displaystyle\int_{S_{P}}\mathbf{u}_{s}\;{\rm
dS},$ (20) $\mathbf{\Omega}=-\frac{3}{8\pi
a^{3}}\displaystyle\int_{S_{P}}\mathbf{e}_{r}\times\mathbf{u}_{s}\;{\rm dS},$
(21)

where the integral is over the surface of the sphere.

###  3.2 Electrophoretic mobility at arbitrary interaction lengthscales

To further demonstrate the applicability of our framework, we utilize Eqs.
(12)-(13) to determine the electrophoretic mobility of a sphere in the Debye-
Hückel limit for arbitrary Debye length (Henry, 1931; Teubner, 1982; Kim &
Karrila, 2013). We assume a homogeneous sphere of radius, $a$, immersed in a
binary monovalent electrolytic solution such that the electrical permittivity
of the solution is denoted as $\varepsilon$. Our objective is to analyze the
motion of the particle with a given surface zeta potential driven by an
external electric field. First, we assume that the surface zeta potential,
$\zeta$, falls in the Debye-Hückel limit, ${e\zeta}/{k_{B}T}\ll 1$, where $e$
is the charge of an electron, $k_{B}$ is the Boltzmann constant, and $T$ is
the absolute temperature. We also assume an electric field disturbance of
$\mathbf{E}_{\infty}$ far away from the particle such that
$\mathbf{E}_{\infty}=\epsilon E_{0}\mathbf{e}_{z}$, where $\epsilon$ is a
small parameter and physically indicates that the length scale of far-field
potential decay is much larger than the particle size. Note that $\varepsilon$
is the electrical permittivity and should not be confused with $\epsilon$,
which is a small parameter in our analysis. Finally, the total osmophoretic
body force ($\mathbf{b}$) driving the particle arises through a combination of
the electrostatic interaction and net osmotic pressure in the fluid, or

Figure 2: Methodology to validate proposed mobility expressions for a charged
particle with a zeta potential ($\zeta$) in the Debye-Hückel limit for (a)
electrophoresis with an external field $\mathbf{E}_{\infty}=\epsilon
E_{0}\mathbf{e}_{z}$ and (b) diffusiophoresis with externally imposed solute
gradient $\bm{\nabla}s_{\infty}=2\epsilon c_{0}\mathbf{e}_{z}$. The
expressions of dimensionless osmophoretic force $\tilde{\mathbf{b}}$ are
provided. Substituting the appropriate $\mathbf{b}$ in Eq. (12) enables us to
recover mobility relationships that otherwise require cumbersome calculations.

$\mathbf{b}=-e\left(c_{+}-c_{-}\right)\bm{\nabla}\phi-
k_{B}T\bm{\nabla}\left(c_{+}+c_{-}\right),$ (22)

where $c_{+}$ and $c_{-}$ are the concentrations of the positive and negative
electrolytic species respectively and $\phi$ is the electric potential. As a
convenient choice, we can represent the solute concentrations in terms of net
charge, $\rho=e\left(c_{+}-c_{-}\right)$, and salt, $s=c_{+}+c_{-}$, and
rewrite the body force to be $\mathbf{b}=-\rho\bm{\nabla}\phi-
k_{B}T\bm{\nabla}s$.

To appropriately derive the particle motion, we are required to obtain the
solutions to $\rho$, $s$, and $\phi$ for a given $\mathbf{E}_{\infty}$ and
$\zeta$. As mentioned earlier, we assume that the species are monovalent,
$z_{\pm}=\pm 1$ and have diffusivities $D_{\pm}$, the species balance is given
by the steady Nernst-Planck equations,

$\mathbf{u}\cdot\bm{\nabla}c_{+}=D_{+}\left[\nabla^{2}c_{+}+\frac{e}{k_{B}T}\bm{\nabla}\cdot\left(c_{+}\bm{\nabla}\phi\right)\right]$
(23)
$\mathbf{u}\cdot\bm{\nabla}c_{-}=D_{-}\left[\nabla^{2}c_{-}-\frac{e}{k_{B}T}\bm{\nabla}\cdot\left(c_{-}\bm{\nabla}\phi\right)\right].$
(24)

For $\mathrm{Pe}=aU/D\ll 1$ ($U$ is the velocity scale for the particle,
$D=\frac{2D_{+}D_{-}}{D_{+}+D_{-}}$ is the ambipolar diffusivity), we ignore
the convective effects. Consequently, Eqs. (23) and (24) can be rewritten in
terms of $\rho$ and $s$ as

$\nabla^{2}s+\frac{1}{k_{B}T}\bm{\nabla}\cdot\left(\rho\bm{\nabla}\phi\right)=0,$
(25)
$\nabla^{2}\rho+\frac{e^{2}}{k_{B}T}\bm{\nabla}\cdot\left(s\bm{\nabla}\phi\right)=0.$
(26)

Finally, the system of equations is closed by using Poisson’s equation to
resolve the electric potential,

$-\varepsilon\nabla^{2}\phi=\rho.$ (27)

In the far-field, at $r\to\infty$, the potential gradient is the externally
imposed electric field and the fluid is electroneutral, or

$\left.-\bm{\nabla}\phi\right|_{r\rightarrow\infty}=\epsilon
E_{0}\mathbf{e}_{z}.$ (28) $\left.\rho\right|_{r\rightarrow\infty}=0.$ (29)

Moreover, in the far-field $c_{+}=c_{-}=c_{0}$, where $c_{0}$ is a
characteristic solute concentration, we can write $s$ to follow

$\left.s\right|_{r\rightarrow\infty}=2c_{0}$ (30)

At the particle surface, at $r=a$, the electrostatic potential is equal to the
zeta potential at the surface, or

$\left.\phi\right|_{r=a}=\zeta.$ (31)

Additionally, there is no salt or charge flux normal to the particle surface,

$\mathbf{e}_{r}\cdot\left[\bm{\nabla}s+\frac{\rho}{k_{B}T}\bm{\nabla}\phi\right]_{r=a}=0,$
(32)
$\mathbf{e}_{r}\cdot\left[\bm{\nabla}\rho+\frac{se^{2}}{k_{B}T}\bm{\nabla}\phi\right]_{r=a}=0.$
(33)

Eqs. (25)-(33) are non-dimensionalized using the following appropriate scales,

$\tilde{\bm{\nabla}}=a\bm{\nabla},\;\;\;\tilde{\nabla}^{2}=a^{2}\nabla^{2},\;\;\;\tilde{\phi}=\frac{e\phi}{k_{B}T},\;\;\;\tilde{\rho}=\frac{\rho}{ec_{0}},\;\;\;\tilde{s}=\frac{s}{c_{0}},\;\;\;\tilde{r}=\frac{r}{a}.$
(34)

Thus the non-dimensional Poisson-Nernst-Planck equations are given as,

$\displaystyle\tilde{\nabla}^{2}\tilde{s}+{\tilde{\bm{\nabla}}}\cdot\left(\tilde{\rho}\tilde{\bm{\nabla}}\tilde{\phi}\right)=0,$
(35)
$\displaystyle\tilde{\nabla}^{2}\tilde{\rho}+\tilde{\bm{\nabla}}\cdot\left(\tilde{s}\tilde{\bm{\nabla}}\tilde{\phi}\right)=0,$
(36)
$\displaystyle\tilde{\nabla}^{2}\tilde{\phi}=-\frac{\kappa^{2}}{2}\tilde{\rho},$
(37)

where $\kappa=\left(\frac{2a^{2}e^{2}c_{0}}{\varepsilon k_{B}T}\right)^{1/2}$
is the dimensionless inverse Debye length. In the far field, thus

$\displaystyle-\left.\tilde{\bm{\nabla}}\tilde{\phi}\right|_{\tilde{r}\rightarrow\infty}=\epsilon\tilde{E}_{0}\mathbf{e}_{z},$
(38) $\displaystyle\left.\tilde{\rho}\right|_{\tilde{r}\rightarrow\infty}=0,$
(39) $\displaystyle\left.\tilde{s}\right|_{\tilde{r}\rightarrow\infty}=2,$
(40)

where $\tilde{E}_{0}=aeE_{0}/\left(k_{B}T\right)$ is the non-dimensional
electric-field. Similarly, at the particle surface, the non-dimensional
boundary conditions read

$\displaystyle\mathbf{e}_{r}\cdot\left[\tilde{\bm{\nabla}}\tilde{s}+\tilde{\rho}\tilde{\bm{\nabla}}\tilde{\phi}\right]_{\tilde{r}=1}=0,$
(41)
$\displaystyle\mathbf{e}_{r}\cdot\left[\tilde{\bm{\nabla}}\tilde{\rho}+\tilde{s}\tilde{\bm{\nabla}}\tilde{\phi}\right]_{\tilde{r}=1}=0,$
(42)
$\displaystyle\left.\tilde{\phi}\right|_{\tilde{r}=1}=\frac{e\zeta}{k_{B}T}=\tilde{\zeta}.$
(43)

As we discuss later, it is more appropriate to write Eq. (43) as a constant
charge boundary condition, which renders the gradient of the potential to be
constant instead. However, for the weak-field, these surface boundary
conditions are equivalent and hence we retain the constant potential boundary
condition for simplicity.

For the remainder of the calculation until Eq. (80), we will drop the tilde
superscript in Eqs. (35)-(43) for convenience and restore the dimensions once
the non-dimensional calculations are complete. We expand $\phi$, $\rho$, and
$s$ in the small parameters $\zeta$ and $\epsilon$ as

$\phi=\phi_{00}+\zeta\phi_{01}+\epsilon\left(\phi_{10}+\zeta\phi_{11}\right),$
(44)
$\rho=\rho_{00}+\zeta\rho_{01}+\epsilon\left(\rho_{10}+\zeta\rho_{11}\right),$
(45) $s=s_{00}+\zeta s_{01}+\epsilon\left(s_{10}+\zeta s_{11}\right).$ (46)

The asymptotic expansions in Eqs.(44)-(46) are substituted into Eqs.
(35)-(43), and the corresponding equations are solved at each asymptotic
order.

$\bm{O(1)}$: An uncharged particle without any electric field. The governing
equations and boundary conditions are obtained to be

$\displaystyle\nabla^{2}s_{00}+\bm{\nabla}\cdot\left(\rho_{00}\bm{\nabla}\phi_{00}\right)=0,$
(47)
$\displaystyle\nabla^{2}\rho_{00}+\bm{\nabla}\cdot\left(s_{00}\bm{\nabla}\phi_{00}\right)=0,$
(48) $\displaystyle\nabla^{2}\phi_{00}=-\frac{\kappa^{2}}{2}\rho_{00}.$ (49)
$\displaystyle\mathbf{e}_{r}\cdot\left[\bm{\nabla}s_{00}+\rho_{00}\bm{\nabla}\phi_{00}\right]=0,\;\;\;\mathrm{at}\;\;r=1,$
(50)
$\displaystyle\mathbf{e}_{r}\cdot\left[\bm{\nabla}\rho_{00}+s_{00}\bm{\nabla}\phi_{00}\right]=0,\;\;\;\mathrm{at}\;\;r=1,$
(51) $\displaystyle\phi_{00}=0,\;\;\;\mathrm{at}\;\;r=1.$ (52) $\displaystyle
s_{00}=2,\;\;\;\mathrm{at}\;\;r\to\infty,$ (53)
$\displaystyle\rho_{00}=0,\;\;\;\mathrm{at}\;\;r\to\infty,$ (54)
$\displaystyle\bm{\nabla}\phi_{00}=0,\;\;\;\mathrm{at}\;\;r\to\infty,$ (55)

The system of Eqs. (47)-(53) have a trivial solution, i.e., $\phi_{00}=0$,
$\rho_{00}=0$, $s_{00}=2$. Physically, the solution simply implies that the
ion concentration is uniform because the particle is uncharged and there is no
electric field.

$\bm{O(\epsilon)}$: Perturbation due to the electric field for an uncharged
particle. The governing equations for the salt and charge dynamics, and
electrostatic potential after substituting the expressions of $\rho_{00}$,
$s_{00}$, and $\phi_{00}$ are given as,

$\nabla^{2}s_{10}=0,$ (56) $\nabla^{2}\rho_{10}+2\nabla^{2}\phi_{10}=0,$ (57)
$\nabla^{2}\phi_{10}=-\frac{\kappa^{2}}{2}\rho_{10}.$ (58)

The corresponding reduced boundary conditions when $r\to\infty$ are

$s_{10}=0,$ (59) $\rho_{10}=0,$ (60)
$-\bm{\nabla}\phi_{10}=E_{0}\mathbf{e}_{z},$ (61)

and when $r=1$, they read

$\mathbf{e}_{r}\cdot\bm{\nabla}s_{10}=0,$ (62)
$\mathbf{e}_{r}\cdot\left[\bm{\nabla}\rho_{10}+2\bm{\nabla}\phi_{10}\right]=0.$
(63) $\mathbf{e}_{r}\cdot\bm{\nabla}\phi_{10}=0,$ (64)

where the derivative of the potential is set to be zero to ensure that there
is no excess charge on the surface; see discussion below Eq. (43).
Mathematically, at $O(\epsilon)$, the charge on the particle surface is zero.
Through Gauss’s law, the no surface charge boundary condition necessitates
that $\mathbf{e}_{r}\cdot\bm{\nabla}\phi_{10}=0$. Hence, Eq. (63) implies that
$\mathbf{e}_{r}\cdot\bm{\nabla}\rho_{10}=0$ at the particle surface. Along
with Eqs. (57) and (60), we obtain $\rho_{10}=0$. Similarly, Eqs. (56), (59)
and (62) reveal $s_{10}=0$.

Since $\rho_{10}=0$, $\phi_{10}$ is governed by the Laplace equation, and the
solution with appropriate boundary conditions reads (Griffiths, 2005),

$\phi_{10}\left(r,\theta\right)=-E_{0}r\left(1+\frac{1}{2r^{3}}\right)\cos\theta.$
(65)

Physically, this order implies that perturbed potential and corresponding
electric field lines get modified due to the geometry of the particle but
there is no charge and salt accumulation.

$\bm{O(\zeta)}$: Perturbation of a charged particle without an external
electric field. The equations governing the charge, salt, and potential at
$O(\zeta)$ are analogous to Eqs. (56)-(58) at $O(\epsilon)$ but have different
boundary conditions. The governing equations are

$\displaystyle\nabla^{2}s_{01}=0,$ (66)
$\displaystyle\nabla^{2}\rho_{01}+2\nabla^{2}\phi_{01}=0,$ (67)
$\displaystyle\nabla^{2}\phi_{01}=-\frac{\kappa^{2}}{2}\rho_{01}.$ (68)

As $r\to\infty$ the net charge, salt, and electric potential gradient all
decay to zero, or

$\displaystyle s_{01}=0,$ (69) $\displaystyle\rho_{01}=0,$ (70)
$\displaystyle\bm{\nabla}\phi_{01}=0.$ (71)

At the particle surface, $r=1$, we obtain

$\displaystyle\mathbf{e}_{r}\cdot\bm{\nabla}s_{01}=0,$ (72)
$\displaystyle\mathbf{e}_{r}\cdot\left[\bm{\nabla}\rho_{01}+2\bm{\nabla}\phi_{01}\right]=0,$
(73) $\displaystyle\phi_{01}=1.$ (74)

Eqs. (66), (69), and (72) yield $s_{01}=0$. The remainder of the equations
reveal

$\phi_{01}(r)=\frac{1}{r}e^{-\kappa\left(r-1\right)}$ (75)
$\rho_{01}(r)=-\left(\frac{2}{r}\right)e^{-\kappa\left(r-1\right)}$ (76)

Physically, the results indicate the distribution of potential and charge
arising due to a charged particle.

$\bm{O(\epsilon\zeta)}$: Perturbation of both imposed electric field and
surface zeta potential. To fully resolve the body force to the order of
$O(\epsilon\zeta)$, we have to obtain $s_{11}$, this can be observed by the
expansion of Eq. (22) and collecting the respective orders. The governing
equations and boundary conditions for salt, $s_{11}$, are given by,

$\nabla^{2}s_{11}+\bm{\nabla}\cdot\left(\rho_{01}\bm{\nabla}\phi_{10}\right)=0,$
(77) $s_{11}\to 0,\;\;\;\mathrm{at}\;r\to\infty,$ (78)
$\mathbf{e}_{r}\cdot\left[\bm{\nabla}s_{11}+\rho_{01}\bm{\nabla}\phi_{10}\right]=0\;\;\;\mathrm{at}\;r=1.$
(79)

Substituting in the expression of $\rho_{01}$ from Eq. (76) and $\phi_{10}$
from Eq. (65) we can solve for $s_{11}(r,\theta)$. The salt dynamics are in
the form of $s_{11}=E_{0}f(r)\cos\theta$ where $f(r)$ is

$\displaystyle f(r)=$
$\displaystyle\frac{1}{r^{2}}\left(\frac{\kappa}{6}-\frac{5}{3}-\frac{10}{\kappa}-\frac{2}{\kappa^{2}}\right)+\frac{\kappa^{2}}{3}e^{\kappa}\textrm{Ei}\left(-\kappa
r\right)+\frac{2}{\kappa^{2}r^{2}}e^{-\kappa\left(r-1\right)}-$ (80)
$\displaystyle\frac{4}{3\kappa
r^{2}}e^{-\kappa(r-1)}+\frac{\kappa}{3}e^{-\kappa(r-1)}-\frac{1}{3r}e^{-\kappa(r-1)}+\frac{2}{\kappa
r}e^{-\kappa(r-1)},$

where $\textrm{Ei}()$ is the elliptic integral.
Reintroducing dimensions. At this stage, we restore the dimensions and
reintroduce the tilde for dimensionless variables. The total osmophoretic body
force $\mathbf{b}$ is made dimensionless by writing
$\mathbf{b}=\frac{k_{B}Tc_{0}}{a}\tilde{\mathbf{b}}=\frac{\varepsilon\left(k_{B}T\right)^{2}\kappa^{2}}{2e^{2}a^{3}}\tilde{\mathbf{b}}$.
The first relevant order of $\tilde{\mathbf{b}}$ for electrophoresis is
$\epsilon\tilde{\zeta}$ because it is the order at which a charged particle is
being driven by an electric field. To this end, we write

$\displaystyle\begin{split}\tilde{\mathbf{b}}=-\epsilon\tilde{\zeta}\left[\tilde{\rho}_{00}\tilde{\bm{\nabla}}\tilde{\phi}_{11}+\tilde{\rho}_{11}\tilde{\bm{\nabla}}\tilde{\phi}_{00}+\tilde{\rho}_{01}\tilde{\bm{\nabla}}\tilde{\phi}_{10}+\tilde{\rho}_{10}\tilde{\bm{\nabla}}\tilde{\phi}_{01}+\tilde{\bm{\nabla}}\tilde{s}_{11}\right].\end{split}$
(81)

Based on the solutions at different orders, it is straightforward to see that
$\mathbf{\tilde{b}}$ reduces to

$\mathbf{\tilde{b}}=-\epsilon\tilde{\zeta}\left[\tilde{\rho}_{01}\tilde{\bm{\nabla}}\tilde{\phi}_{10}+\tilde{\bm{\nabla}}\tilde{s}_{11}\right].$
(82)

We note that the body force term of $\tilde{s}_{11}$ integrates out to zero in
the calculation of $\mathbf{U}$ and $\mathbf{\Omega}$ for electrophoresis and
is generally not included in prior analyses. However, we retain this term for
consistency as it does become crucial for diffusiophoretic phenomena, as we
detail in section 3.3.

After substituting the values of $\tilde{\rho}_{01}$, $\tilde{\phi}_{10}$ and
$\tilde{s}_{11}$, the resultant equation in dimensional form is

$\displaystyle\mathbf{b}=-\frac{\epsilon\varepsilon
E_{0}\zeta\kappa^{2}}{2a^{2}}$
$\displaystyle\left\\{\left[\frac{2}{\tilde{r}}e^{-\kappa(\tilde{r}-1)}\left(1-\frac{1}{\tilde{r}^{3}}\right)\cos\theta+\frac{df(\tilde{r})}{d\tilde{r}}\cos\theta\right]\mathbf{e}_{r}\right.$
(83)
$\displaystyle\left.-\left[\frac{2}{\tilde{r}}e^{-\kappa(\tilde{r}-1)}\left(1+\frac{1}{2\tilde{r}^{3}}\right)\sin\theta+\frac{f(\tilde{r})}{\tilde{r}}\sin\theta\right]\mathbf{e}_{\theta}\right\\}.$

Eq. (83) is substituted into Eq. (12) to obtain the translational velocity to
be

$\mathbf{U}=\mathcal{M}\mathbf{E}_{\infty},$ (84)

where the mobility $\mathcal{M}$ after restoring dimensions is

$\mathcal{M}=\frac{\varepsilon\zeta}{6\mu}\left[(1+\kappa)+(12-\kappa^{2})\displaystyle\int_{1}^{\infty}\frac{e^{\kappa(1-t)}}{t^{5}}{\rm
d}t\right],$ (85)

which is the seminal result of Henry (1931) for arbitrary double layer
thickness, and has also been reported by Teubner (1982) and Kim & Karrila
(2013). For a homogeneous sphere, our analysis reveals
$\mathbf{\Omega}=\mathbf{0}$, as expected. We emphasize that it is
straightforward to extend the calculations to heterogeneous spheres (Velegol
et al., 1996; Teubner, 1982) and obtain results for electrorotation in
arbitrary double-layer thicknesses, which otherwise requires considerable
effort.

###  3.3 Electrolytic diffusiophoretic mobility at arbitrary interaction
lengthscales

Next, we focus on the process of electrolytic diffusiophoresis in the Debye-
Hückel limit and for arbitrary double-layer thickness. We assume that the
external concentration gradient of a binary monovalent electrolyte is given as
$\bm{\nabla}s_{\infty}=2\epsilon\bm{\nabla}c_{0}$, where $\epsilon$ is a small
parameter, much like electrophoresis. Here, $\mathbf{b}$ is required to be
expanded to an additional higher order of $\epsilon\tilde{\zeta}^{2}$. The
term on the order $O\left(\epsilon\tilde{\zeta}\right)$ is identical to
electrophoresis and represents the electrophoretic component of the
diffusiophoretic mobility. The second term on the order O
$\left(\epsilon\tilde{\zeta}^{2}\right)$ denotes the chemiphoretic component.
We employ the expression of $\mathbf{b}$, derived in this sub-section, to Eqs.
(12)-(13). This allows us to retrieve the expression of the translation
velocity of a charged spherical particle in an unbounded solution of a
symmetrically charged electrolyte for an arbitrary double-layer thickness,
which otherwise requires considerable efforts; see Keh & Wei (2000).

We acknowledge the electrokinetic equations used to describe such
diffusiophoretic systems are analogous to our treatment of the motion of
electrophoretically propelled particles in section 3.2. However, the key
mechanistic difference is the presence of an external gradient of solute
$\bm{\nabla}s_{\infty}$ instead of an imposed electric field
$\mathbf{E}_{\infty}$; this results in a change of boundary conditions and
subsequently the solutions at different asymptotic orders. To preserve the
pedagogical nature of our manuscript, we will re-derive the electrophoretic
contribution and subsequently solve for the chemiphoretic contribution to the
osmophoretic body force term and attempt to emphasize key physical and
mathematical differences between the derivations laid out in sections 3.2 and
3.3.

Consider a colloidal particle with a surface zeta potential, $\zeta$, in an
external solute gradient of a symmetric binary electrolyte. We assume that the
electrolytes are monovalent such that $z_{\pm}=\pm 1$. The ions are assumed to
have different diffusivities $D_{+}\neq D_{-}$. The governing equations of the
concentration of the ionic species, $c_{\pm}$, are identical to Eqs. (23)-(24)
and the interaction potential is governed by Poisson’s equation, as given in
Eq. (27). However, the far-field boundary conditions are different.
Specifically, as $r\to\infty$, we assume that the concentration of the ionic
species is linear with position $z$, or

$c_{\pm}=c_{0}\left(1+\frac{\epsilon z}{a}\right),$ (86)

Further, it is assumed that the electric current in the far field is zero,
which yields (Prieve et al., 1984; Velegol et al., 2016; Gupta et al., 2019)

$-\bm{\nabla}\phi_{\infty}=\epsilon\beta\frac{k_{B}T}{ae}=\epsilon
E_{0}\mathbf{e}_{z},$ (87)

where $\beta=\frac{D_{+}-D_{-}}{D_{+}+D_{-}}$ and
$E_{0}=\beta\frac{k_{B}T}{ae}$. The electric field is thus induced due to
unequal diffusivities and a non-zero salt gradient.

Similar to electrophoresis, modification of the governing equations in terms
of charge, $\rho=e\left(c_{+}-c_{-}\right)$, salt, $s=c_{+}+c_{-}$, and
potential result in Eqs. (25)-(27). In the far field, the boundary conditions
for $\phi$ and $\rho$ are identical to Eqs. (28) and (29) with the
aforementioned definition of $E_{0}$. However, the boundary condition of $s$
is modified to

$s|_{r\to\infty}=2c_{0}\left(1+\frac{\epsilon z}{a}\right).$ (88)

Note that $\epsilon/a=\bm{\nabla}\log s_{\infty}$. The boundary conditions are
identical at the particle surface, i.e., Eqs. (31)-(33). The objective is to
solve $\rho$, $s$, and $\phi$ with the modified boundary conditions above and
subsequently evaluate the total osmophoretic body force, following the
procedure used to obtain Eq. (22). We non-dimensionalize the equations using
the same scales as Eq. (34) and also define
$\mathbf{b}=\frac{k_{B}Tc_{0}}{a}\tilde{\mathbf{b}}=\frac{\varepsilon\left(k_{B}T\right)^{2}\kappa^{2}}{2e^{2}a^{3}}\tilde{\mathbf{b}}$,
where the definition of $\kappa$ is also identical.

For simplicity, we drop the tilde from our analysis until Eq. (105) and
reintroduce them afterward. Thus the non-dimensional osmophoretic body force
is given as $\mathbf{b}=-\rho\bm{\nabla}\phi-\bm{\nabla}s$. We expand $\rho$,
$s$ and $\phi$ until $O(\epsilon\zeta^{2})$, and solve the equations at each
order.

$\bm{O(1)}$: An uncharged particle without any external salt gradient. The
results at this order are identical to electrophoresis and thus yield
$s_{00}=2$, $\rho_{00}=0$, and $\phi_{00}=0$, indicating a uniform
concentration of ion with no charge and potential.

$\bm{O(\epsilon)}$: Perturbation of the external salt concentration to an
uncharged particle. This order is distinct compared to electrophoresis since
the far-field boundary condition for salt is different, while the remainder of
the equations and boundary conditions are identical. We note that the boundary
condition for the electric field is similar to electrophoresis since we have
defined $E_{0}$. The solution simply reduces to zero and uniform charge
density $\rho_{10}=0$, while both salt and potential follow the Laplace
equation. The results read

$\displaystyle
s_{10}\left(r,\theta\right)=2r\left(1+\frac{1}{2r^{3}}\right)\cos\theta.$ (89)
$\displaystyle\phi_{10}=-E_{0}r\left(1+\frac{1}{2r^{3}}\right)\cos\theta.$
(90)

Physically, at this order, a gradient in the salt concentration far away
perturbs the salt field and induces a potential field if diffusivity asymmetry
is present ($E_{0}\neq 0$ only when $\beta\neq 0$). However, since the surface
is uncharged, $\rho_{10}=0$.

$\bm{O(\zeta)}$: Perturbation in the surface charge of the particle without an
external field. Since there is no external field at this order, the solution
is identical to electrophoresis with $s_{01}=0$ and

$\phi_{01}(r)=\frac{1}{r}e^{-\kappa(r-1)},$ (91)
$\rho_{01}(r)=-\frac{2}{r}e^{-\kappa(r-1)}.$ (92)

This order represents the potential and charge profiles due to the surface
charge of the particle. However, there is no salt accumulation at this order
since the reduction in the co-ion concentration is balanced by the increase in
the counter-ion concentration.

$\bm{O(\epsilon\zeta)}$: Perturbation in both the imposed salt concentration
and surface charge. The governing equations for charge ($\rho_{11}$), salt
($s_{11}$), and potential ($\phi_{11}$) are

$\displaystyle\nabla^{2}s_{11}+\bm{\nabla}\cdot\left[\rho_{01}\bm{\nabla}\phi_{10}\right]=0,$
(93)
$\displaystyle\nabla^{2}\rho_{11}+\bm{\nabla}\cdot\left[s_{10}\bm{\nabla}\phi_{01}+2\bm{\nabla}\phi_{11}\right]=0,$
(94) $\displaystyle\nabla^{2}\phi_{11}=-\frac{\kappa^{2}}{2}\rho_{11}.$ (95)

The boundary conditions at the particle surface, $r=1$, are

$\displaystyle\mathbf{e}_{r}\cdot\left[\bm{\nabla}s_{11}+\rho_{01}\bm{\nabla}\phi_{10}\right]=0,$
(96)
$\displaystyle\mathbf{e}_{r}\cdot\left[\bm{\nabla}\rho_{11}+s_{10}\bm{\nabla}\phi_{01}+2\bm{\nabla}\phi_{11}\right]=0,$
(97) $\displaystyle\mathbf{e}_{r}\cdot\bm{\nabla}\phi_{11}=0.$ (98)

Again, there is no external field as $r\to\infty$. Substituting in the
solutions obtained in $O(\epsilon)$, and $O(\zeta)$ we begin to solve
$s_{11}$, $\rho_{11}$, and $\phi_{11}$. Further, we can separate the $r$ and
$\theta$ contributions by redefining
$s_{11}\left(r,\theta\right)=f_{s_{11}}(r)\cos\theta$,
$\rho_{11}\left(r,\theta\right)=f_{\rho_{11}}(r)\cos\theta$, and
$\phi_{11}\left(r,\theta\right)=f_{\phi_{11}}(r)\cos\theta$ and solve the
equations numerically; see Appendix A. It is possible to find analytical
solutions to Eqs. (A1)-(A6), similar in form to Eq. (80). In the scope of our
current paper, we choose to resolve the dynamics at $O(\epsilon\zeta)$ and
$O(\epsilon\zeta^{2})$ numerically. For an analytical derivation of such
higher-order effects the reader is directed to Keh & Wei (2000).

$\bm{O(\epsilon\zeta^{2})}$: First-order perturbation in salt field and
second-order perturbation in surface charge. We only seek to solve $s_{12}$ at
this order since it is the only quantity required to resolve the body force up
to $O(\epsilon\zeta^{2})$; see Eq. (104). The equation governing the dynamics
of $s_{12}$ is,

$\nabla^{2}s_{12}+\bm{\nabla}\cdot\left[\rho_{01}\bm{\nabla}\phi_{11}+\rho_{11}\bm{\nabla}\phi_{01}\right]=0,$
(99)

with

$\mathbf{e}_{r}\cdot\left[\bm{\nabla}s_{12}+\rho_{01}\bm{\nabla}\phi_{11}+\rho_{11}\bm{\nabla}\phi_{01}\right]_{r=1}=0,$
(100)

and $s_{12}(r\rightarrow\infty,\theta)=0$. We write $s_{12}$ as
$s_{12}\left(r,\theta\right)=f_{s_{12}}(r)\cos\theta$ and solve the equations
numerically; see Appendix A. As discussed previously, we solve Eqs. (A7),
(A8), and the far field constraint numerically.

Restoring dimensions. We now restore dimensions and reintroduce tilde to
describe dimensionless variables. Therefore, we write the body force
$\mathbf{b}=\frac{\varepsilon\left(k_{B}T\right)^{2}\kappa^{2}}{2e^{2}a^{3}}\tilde{\mathbf{b}}$
such that

$\displaystyle\tilde{\mathbf{b}}=\epsilon\tilde{\zeta}\mathbf{\tilde{b}}_{11}+\epsilon\tilde{\zeta}^{2}\mathbf{\tilde{b}}_{12}$
(101)
$\displaystyle\tilde{\mathbf{b}}_{11}=-\left(\tilde{\rho}_{01}\bm{\tilde{\nabla}}\tilde{\phi}_{10}+\bm{\tilde{\nabla}}s_{11}\right)$
(102)
$\displaystyle\tilde{\mathbf{b}}_{12}=-\left(\tilde{\rho}_{01}\bm{\tilde{\nabla}}\tilde{\phi}_{11}+\tilde{\rho}_{11}\bm{\tilde{\nabla}}\tilde{\phi}_{01}+\bm{\tilde{\nabla}}\tilde{s}_{12}\right)$
(103)

At this point, some comments are in order. We note that $\tilde{\mathbf{b}}$
could also include a term at the $O(\epsilon)$ since $s_{10}\neq 0$. However,
$\tilde{\mathbf{b}}$ only includes excess osmotic pressure and not the osmotic
pressure itself. This is because the osmotic pressure contribution due to
$\bm{\nabla}s_{\infty}$ would lead to particle motion even in the absence of
phoretic interactions. Only the terms at subsequent orders are included to
ignore this effect. Further, we highlight that both $\tilde{\phi}_{10}$ and
$\tilde{s}_{11}$ are proportional to $\tilde{E}_{0}$. Therefore, the
$O\left(\epsilon\tilde{\zeta}\right)$ term only depends on $E_{0}$ and is
referred to as the electrophoretic contribution. Since $\tilde{\rho}_{01}$,
$\tilde{\phi}_{10}$ and $\tilde{s}_{11}$ are identical to electrophoretic
solution, the $O(\epsilon\tilde{\zeta})$ is equal to the one described earlier
in Eq. (85).

Figure 3: Comparison of proposed mobility expressions of diffusiophoretic
mobility in Eq. (106) with the mobility reported in Keh and Wei Keh & Wei
(2000). Quantitative agreement of both (a) $\Theta_{1}\left(\kappa\right)$ and
(b) $\Theta_{2}\left(\kappa\right)$ is observed.

However, in contrast, for the $O(\epsilon\tilde{\zeta}^{2})$ contribution,
$\tilde{\rho}_{01}$, $\tilde{\phi}_{11}$, $\tilde{\rho}_{11}$,
$\tilde{\phi}_{01}$ and $\tilde{s}_{12}$ are independent of $E_{0}$.
Furthermore, $\tilde{\phi}_{11}$, $\tilde{\rho}_{11}$ and $\tilde{s}_{12}$ are
all proportional to $\tilde{s}_{10}$, which is consistent with the prior
literature (Anderson, 1989; Keh & Wei, 2000; Gupta et al., 2019).

Since $\tilde{\mathbf{b}}_{11}$ is identical to the electrophoretic motion, we
focus our attention on $\tilde{\mathbf{b}}_{12}$, which reads

$\tilde{\mathbf{b}}_{12}=-\left(\rho_{01}\frac{df_{\phi_{11}}}{dr}+f_{\rho_{11}}\frac{d\phi_{01}}{dr}+\frac{df_{s_{12}}}{dr}\right)\cos\theta\mathbf{e}_{r}-\left(\frac{\rho_{01}f_{\phi_{11}}}{r}+\frac{f_{s_{12}}}{r}\right)\sin\theta\mathbf{e}_{\theta}$
(104)

Upon substituting the value of $\mathbf{b}$ in Eq. (12), the translation
velocity of the particle could be simplified to read

$\displaystyle\mathbf{U}=\mathcal{M}\bm{\nabla}\log s_{\infty},$ (105)

where

$\mathcal{M}=\frac{\varepsilon}{\mu}\left[\frac{k_{B}T}{e}\beta\zeta\Theta_{1}\left(\kappa\right)+\frac{\zeta^{2}}{8}\Theta_{2}\left(\kappa\right)\right],$
(106)

where $\Theta_{1}$ and $\Theta_{2}$ are evaluated numerically; see Appendix A.
Figure 3 demonstrates good quantitative agreement between the values obtained
in Keh & Wei (2000) and our results.

This section highlighted the generality of Eqs. (12) and (13). We showed they
are able to recover the mobilities for microswimmers in thin interaction layer
limit, electrophoresis for arbitrary double-layer thickness, and electrolytic
diffusiophoresis for arbitrary double-layer thickness.

##  4 Autophoretic motion of microswimmers

In this section, we use the formula obtained in Eqs. (12)-(13) to study the
translation of Janus-like particles with a spherical cap, see Fig.4a. The key
novelty of our analysis is that Eqs. (12)-(13) do not impose the restriction
on interaction lengthscale. As we show later, if th interaction length is
comparable to particle size, the particle velocity is significantly impacted.

We define catalytic surface activity through a non-dimensional outward surface
flux of strength, $J$ (scaled by a characteristic flux $Dc_{0}/a$, where $D$
is the diffusivity of the solute, $c_{0}$ is a reference concentration of the
solute and $a$ is the particle radius). The non-dimensional interaction length
is characterized by $\kappa^{-1}$ (scaled by $a$). Lastly, the size of the
catalytic cap is controlled by the polar angle $\theta_{0}$ such that
$\theta_{0}=0$ indicates no catalytic cap on the particle and
$\theta_{0}=\frac{\pi}{2}$ represents a hemispherical cap.

We assume a Helmholtz-like equation governs the interaction potential ($\phi$,
scaled by $k_{B}T$) with a constant surface potential ($\phi_{0}$) and a far-
field decay. While Helmholtz potential is not representative of different
surface interactions possible, it provides a convenient choice to explore the
impact of $\kappa$ and thus has been chosen for this analysis. We note that
our analysis can be easily extended to other interaction potentials provided
that the integrals in Eqs. (11) are convergent.

To resolve particle translation for a given surface activity and interaction,
we write

$\nabla^{2}\phi=\kappa^{2}\phi,$ (107) $\phi=\phi_{0},\;\;\;r=1,$ (108)
$\phi\to 0,\;\;\;r\to\infty.$ (109)

Solute transport is governed by diffusion and phoretic interactions with the
following boundary conditions

Figure 4: (a) Self-phoretic Janus particle where propulsion is controlled by
the size of the spherical cap $\theta_{0}$, the reactive flux $J$ and the
interaction length scale $\kappa^{-1}$. (b) $U$ vs $\theta_{0}$ for different
$\kappa$ values demonstrates a maximum velocity for $\theta_{0}=\frac{\pi}{2}$
irrespective of $\kappa$. (c) $\frac{U\kappa^{2}}{U_{0}}$ versus $\kappa$
asymptotically approaches the thin limit calculations as $\kappa\to 100$.
However, considerable dampening is observed even for $\kappa=O(10)$. The
values are reported for $\theta_{0}=\frac{\pi}{2}$. (d) The dampening of
$\frac{U\kappa^{2}}{U_{0}}$ with $\kappa$ is observed for all $\theta_{0}$.
The value $U_{0}$ is the asymptotic limit of $U\kappa^{2}$ from the thin
interaction layer calculations.

$\bm{\nabla}\cdot\left(\bm{\nabla}c+c\bm{\nabla}\phi\right)=0,$ (110)
$-\mathbf{n}\cdot\left(\bm{\nabla}c+c\bm{\nabla}\phi\right)=J,\;\;\;r=1,$
(111) $c\to 0,\;\;\;r\to\infty.$ (112)

Our model problem is illustrated in Fig. 4(a). We solve the coupled equations
(107)-(112) numerically. A scaled model geometry was constructed with the
particle radius given to be $a=1$ and an outer radius of $r|_{\infty}=20$,
representing the far field. The interaction potential ($\phi$) and solute
concentration ($c$) were defined by Eq. (107)-(112). To obtain the translation
velocity through Eq. (12), we define the body force
$\mathbf{b}=-c\mathbf{\nabla}\phi-\mathbf{\nabla}c$. Note that the arbitrary
body force has both a phoretic and an osmotic contribution.

We first analyze the effects of the spherical cap size ($\theta_{0}$) and
interaction length scale ($\kappa^{-1}$) for a fixed surface flux ($J=1$) and
surface potential ($\phi_{0}=-1$); see Fig. 4. First, we note that the
particle moves in the direction of the catalytic cap, since when $\phi_{0}=-1$
the particle is attracted towards regions of higher solute concentration. From
Fig. 4(b), we observe that the propulsion velocity is maximum for
$\theta_{0}=\frac{\pi}{2}$. This is in agreement with prior observations in
the literature, Golestanian et al. (2007); Michelin & Lauga (2014); Popescu et
al. (2018) and others. Additionally, the dependence of translation velocity on
the cap size is symmetric about $\theta_{0}=\frac{\pi}{2}$. As described in
Michelin & Lauga (2014), when $\theta_{0}=\frac{\pi}{2}$, for small Péclet
numbers, the sharpest concentration gradients are located near the equator,
consequently leading to a larger slip velocity over an extended inert surface.
In contrast, for smaller or larger catalytic caps, the aforementioned solute
front is closer to the pole and thus involves a smaller share of the particle
surface in generating slip velocities and therefore a smaller swimming speed.

Thin interaction layer: We also study the dependence of propulsion velocity on
the interaction length scale ($\kappa^{-1}$); see Fig. 4(c). To compare our
calculations with the thin interaction layer limit, we perform analogous
calculations following the approach of Anderson (1989) and Derjaguin et al.
(1947). We consider diffusive transport of solute $\nabla^{2}c=0$ through the
fluid volume. The surface flux condition is given by
$-\mathbf{n}\cdot\mathbf{\nabla}c=J$ with a far-field decay condition. The
phoretic slip at the particle surface is defined as,

$\mathbf{u}_{\mathrm{slip}}=-\mathbf{\nabla}_{s}c\displaystyle\int_{1}^{\infty}(r-1)\left[\exp\left(-\phi\right)-1\right]\
dr,$ (113)

where the non-dimensional $\mathbf{u}_{\mathrm{slip}}$ is scaled with
$kTc_{0}a/\mu$. After that, we refer to Stone & Samuel (1996), also derived
using our approach in Eq. (20), to obtain the translation velocity
$\mathbf{U}^{\rm thin}$ to be,

$\mathbf{U}_{\mathrm{thin}}=-\frac{1}{4\pi}\displaystyle\int_{S}\mathbf{u}_{\mathrm{slip}}{\rm
d}S.$ (114)

It is well-known in diffusiophoretic literature (Golestanian, 2019) that for
$\kappa\gg 1$, $\mathbf{U}_{\mathrm{thin}}\propto
1/\kappa_{\mathrm{thin}}^{2}$. Consequently, we introduce a scaled velocity
expression,
$\mathbf{U}_{0}=\mathbf{U}_{\mathrm{thin}}\kappa^{2}_{\mathrm{thin}}$ which
becomes constant as $\kappa\to\infty$. As shown in Fig. 4(c), we observe that
the velocity ratio $\mathbf{U}\kappa^{2}/\mathbf{U}_{0}$ significantly
decreases when the interaction limit becomes comparable to particle size,
i.e., $\kappa\rightarrow 1$. We find that even for $\kappa=O(10)$, the
velocity is reduced by almost a factor of 2. We only reach the thin
interaction limit for $\kappa=O(10^{2})$. This observation is consistent with
passive diffusiophoretic literature where even moderately thin double layers
can significantly reduce the diffusiophoretic velocity; see Fig. 3(b). Our
analysis highlights that even for autophoretic swimmers, this effect can be
observed, and using a thin interaction limit could overestimate the velocity
for moderately thin interaction thickness. To ensure that our trends of
reductions in velocity are consistent for other conditions, in Fig. 4(d), we
observe that reduction in velocity for smaller $\kappa$ values is consistent
for all $\theta_{0}$ values.

##  5 Autophoretic swimmers with external solute gradients

To further demonstrate the applicability of Eq. (12) in scenarios where it is
difficult to use a slip velocity approach, we modify our model problem to
include an external solute flux in the fluid bulk. The model geometry is
preserved as shown in Fig. 5a. The concentration in the far field is
$\bm{\nabla}c_{\infty}=J_{\mathrm{ext}}\mathbf{e}_{z}$, where $\mathbf{e}_{z}$
is the z-coordinate basis vector in the universal Cartesian frame of reference
and $J_{\mathrm{ext}}$ is a free parameter used to control the direction and
strength of this external field. We note that all additional parameters have
been appropriately non-dimensionalized as per the discussion in section 4. We
analyze the variations of the local surface flux ($J$), external solute flux
($J_{\mathrm{ext}}$), and interaction length scale $\kappa$.

Figure 5: (a) A self-phoretic particle also driven by external
$\bm{\nabla}c_{\infty}$. (b) $U_{z}$ vs. $\bm{\nabla}c_{\infty}$ shows that
both the particle speed and direction depend on the competing effects of
external and self-propulsion modes. We define
$(\bm{\nabla}c_{\infty})_{\mathrm{crit}}$ as the value when the particle
motion was arrested despite the presence of concentration gradients due to
surface activities. (c) $(\bm{\nabla}c_{\infty})_{\mathrm{crit}}$ vs $\kappa$.
We observe an increase in the magnitude of the external flux needed to arrest
motion as we approach thin interaction length limits.

For a given surface flux $J=1$, it is observed that the translation velocity
is linear with the imposed solute flux, $\bm{\nabla}c_{\infty}$. The particle
moves with a velocity $\mathbf{U}=U_{z}\mathbf{e}_{z}$. The direction of the
propulsion is governed by the relative magnitudes of different contributions
of the body force terms. In the absence of an external flux, the body force
contributions due to the phoretic activity cause the particle to move in the
positive z-direction. In our analysis, we have ignored the excess osmotic
contribution ($\bm{\nabla}c$) from the external solute gradient and only
considered the phoretic contribution. This is done to ignore the motion of the
particle only due to $\bm{\nabla}c_{\infty}$ when $\phi=0$. For
$J_{\mathrm{ext}}>0$, one can observe that force terms arising from the
gradients in active and passive concentration fields are in the same
direction, hence $U_{z}>0$ as in Fig. 5(b). Alternatively when
$J_{\mathrm{ext}}<0$, the phoretic force terms arising out of the passive
solute concentration field compete with forcing arising due to activity. This
leads to a direction reversal below a threshold concentration gradient,
$(\bm{\nabla}c_{\infty})_{\mathrm{crit}}$, where $U_{z}<0$.
$(\bm{\nabla}c_{\infty})_{\mathrm{crit}}$ is the external solute gradient
necessary to arrest particle motion due to activity. The linearity of the
results in Fig. 5b is due to the computations being performed in the weak
field limit. To inspect the effects of interaction potential length, we
obtained the $(\bm{\nabla}c_{\infty})_{\mathrm{crit}}$ values at $J=1$ while
varying over $\kappa$. A qualitative agreement is observed between the results
in Fig. 5(c) and the $\kappa$-dependence of velocity ($U$). Further, we see
that a stronger external flux is necessary to arrest motion as we approach the
thin interaction length limit. This is a consequence of increased phoretic
mobilities at the thin interaction length limit for a given surface activity;
see Fig. (3).

##  6 Conclusion

The main result of our work is Eq. (11) which allows us to find the particle
velocities, and consequently its trajectory, for a given particle shape,
interaction length, and surface activity. Further, we obtain Eqs. (12)-(13)
which are valid only for spherical particles. Using these equations we have
retrieved the mobility expressions for both electrophoretic and electrolytic
diffusiophoretic motion of colloidal spherical particles at arbitrary
interaction lengths at the low potential limit. Additionally, we have
asymptotically taken equations (12)-(13) to the thin interaction length limit
to represent the net translation and rotation of the particle in terms of a
phoretic slip velocity as in Stone & Samuel (1996). Finally, from our model
problem on the self-propulsion of spherical microswimmers, we observe a peak
in translational velocity as we approach a hemispherical catalytic coverage
with a velocity reduction for lower or higher coverages. At moderate
interaction lengths, $\kappa=O(10)$, a dampening in the translation velocity
is observed. However, for higher $\kappa$ values our calculations can retrieve
the thin interaction length limit results arising from equivalent slip
velocity calculations. Further, we can add an external passive solute
concentration gradient to our problem to understand the competing effects of
surface-generated and externally imposed solute concentration gradients. We
find configurations where the external flux can arrest motion induced by
surface activity ($\bm{\nabla}c_{\mathrm{crit}}$) and regions where the
propulsion induced by the passive external solute gradient aids with or
competes against the propulsion induced due to surface activity. A non-linear
decay of $(\bm{\nabla}c)_{\mathrm{crit}}$ with $\kappa$ is observed from our
analysis which is qualitatively similar to the $\kappa$-dependence of the
translation velocity, $\mathbf{U}$.

Beyond the result described in this paper, our work can be utilized to predict
$\mathbf{U}$ and $\mathbf{\Omega}$ for an arbitrarily shaped particle. For
such a calculation, one would need the appropriate expressions of the mobility
($\mathbf{M}$) and the disturbance tensors ($\mathbf{D}_{T}$,
$\mathbf{D}_{R}$), which might be possible to obtain analytically or
numerically. While this manuscript focuses on the mobility of a single
particle, a similar analysis could be extended to multiple particles and
particles under confinement.

The derived expression will be particularly useful for multiphysics
propulsion. For instance, it might be possible to induce propulsion of
particles using multiple modes such as a combination of electric fields and
other fields, such as electrodiffusiophoresis (Wang et al., 2022; Jarvey et
al., 2023). Another possibility is the inclusion of chemical kinetics at the
particle surface (Davis & Yariv, 2022) which will modify the solute problem
and thus consequently change $\mathbf{b}$.

## Acknowledgement

The authors acknowledge H.A. Stone and A.S. Khair for their input in the
preparation of this manuscript. Ankur G. thanks the National Science
Foundation (CBET - 2238412) CAREER award for financial support. Arkava G.
acknowledges the Teets Family Endowment in Nano-Technology Graduate Fellowship
and the Mukhopadhyay Graduate Research Fellowship. Souradeep R. acknowledges
the Mukhopadhyay Graduate Research Fellowship. The authors acknowledge the
donors of the American Chemical Society Petroleum Research Fund.

## Declaration of Interests

The authors report no conflict of interest.

##  Appendix A Numerical resolution of the radial components at
$O(\epsilon\zeta)$ and $O(\epsilon\zeta^{2})$

$\bm{\mathrm{O}(\epsilon\zeta)}$:The radial components of Eqs. (94)-(98) are,

$\displaystyle\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{df_{\rho_{11}}}{dr}\right)-\left(\frac{2}{r^{2}}+\kappa^{2}\right)f_{\rho_{11}}+\frac{2}{r^{2}}\frac{d}{dr}\left(r^{2}f_{s_{10}}(r)\frac{d\phi_{01}(r)}{dr}\right)=0,$
(A1)
$\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{df_{s_{11}}}{dr}\right)-\frac{2f_{s_{11}}}{r^{2}}+\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\rho_{01}(r)\frac{df_{\phi_{10}}(r)}{dr}\right)-\frac{2\rho_{01}(r)f_{\phi_{10}}(r)}{r^{2}}=0,$
(A2)
$\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{df_{\phi_{11}}}{dr}\right)-\frac{2f_{\phi_{11}}}{r^{2}}=-\frac{\kappa^{2}}{2}f_{\rho_{11}},$
(A3)

where $f_{\phi_{10}}(r)$ and $f_{s_{10}}(r)$ are the radial components of the
$O(\epsilon)$ solutions. The appropriate boundary conditions at the particle
surface are,

$\frac{df_{\rho_{11}}}{dr}+f_{s_{10}}(r)\frac{d\phi_{01}(r)}{dr}=0,$ (A4)
$\frac{df_{s_{11}}}{dr}+\rho_{01}(r)\frac{df_{\phi_{10}}(r)}{dr}=0,$ (A5)
$\frac{\mathrm{d}f_{\phi_{11}}}{\mathrm{d}r}=0$ (A6)

In the far field, the radial flux of $f_{\phi_{11}}$, $f_{\rho_{11}}$, and
$f_{s_{11}}$ all go to zero.

$\bm{\mathrm{O}(\epsilon\zeta^{2})}$: The radial dependence of
$s_{12}(r,\theta)$ is captured by,

$\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{df_{s_{12}}}{dr}\right)-\frac{2f_{s_{12}}}{r^{2}}+\frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\left\\{\rho_{01}(r)\frac{df_{\phi_{11}}(r)}{dr}+f_{\rho_{11}}(r)\frac{df_{\phi_{01}}(r)}{dr}\right\\}\right)-\frac{2f_{\phi_{11}}\rho_{01}}{r^{2}}=0,$
(A7)

with the boundary condition at the particle surface being

$\frac{df_{s_{12}}}{dr}+\rho_{01}\frac{df_{\phi_{11}}}{dr}+f_{\rho_{11}}\frac{d\phi_{01}}{dr}=0,$
(A8)

and $f_{s_{12}}=0$ in the far-field.

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