2. Odd Poisson–Nernst–Planck–Stokes equations

We consider an unbounded incompressible odd fluid, which is steady and quiescent. The fluid contains $N$ species of ions with number densities $n_i(\boldsymbol{r})$ , for $i=1,\ldots ,N$ , and total charge density $\rho (\boldsymbol{r})=\sum _{i=1}^Nz_i e\, n_i(\boldsymbol{r})$ . Here, $z_i$ is the valency of ion species $i$ , and $e$ is the elementary charge unit. In this charged chiral active fluid with domain $\mathcal{V}_{\!{f}}$ , we consider a rigid charged particle occupying a volume $\mathcal{V}_{\!{p}}$ , which is impenetrable to the free ions in the solution. Neglecting magnetic effects, the local electric field can be expressed as $\boldsymbol{E}(\boldsymbol{r})=-\boldsymbol{\nabla }\psi (\boldsymbol{r})$ , and the electrostatic potential $\psi (\boldsymbol{r})$ satisfies

(2.1) \begin{equation} \varepsilon {\nabla} ^2\psi (\boldsymbol{r})=-\rho (\boldsymbol{r}), \quad \boldsymbol{r}\in \mathcal{V}_{\!{f}}, \quad {\nabla} ^2\psi (\boldsymbol{r})=0, \quad \boldsymbol{r}\in \mathcal{V}_{\!{p}}, \end{equation}

where $\varepsilon$ is the homogeneous dielectric constant of the solvent. In general, the dielectric properties of an odd fluid are described by the dielectric tensor $\boldsymbol{\varepsilon }(\hat{\boldsymbol{\ell} })$ with tensorial structure $\boldsymbol{\varepsilon }(\hat{\boldsymbol{\ell} })=\varepsilon _\parallel \hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} }+\varepsilon _\perp ({\unicode{x1D644}}-\hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} })+\varepsilon _{{o}}(\boldsymbol{\epsilon }\boldsymbol{\cdot }\hat{\boldsymbol{\ell} })$ . Here, we assume that the self-spinning particles, yielding $\boldsymbol{\ell }\neq \boldsymbol{0}$ , are not polarisable, and their interaction with the solvent is isotropic, therefore to a good approximation, $\varepsilon =\varepsilon _\parallel =\varepsilon _\perp$ and $\varepsilon _{{o}}=0$ .

A schematic of the particle and the fluid is given in figure 1. The whole system is subjected to an external electric field $\boldsymbol{E}_{\textit{ext}}$ , which results in motion of the suspended particle. To compute this motion, we need to find the relation between $\rho (\boldsymbol{r})$ and the fluid velocity field $\boldsymbol{v}(\boldsymbol{r})$ . Therefore, we consider the total stress tensor $\boldsymbol{\sigma }(\boldsymbol{r};\hat{\boldsymbol{\ell} })=\boldsymbol{\sigma }_{\!{H}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })+\boldsymbol{\sigma }_{\!{E}}(\boldsymbol{r})$ , which consists of a hydrodynamic part

(2.2) \begin{gather} \boldsymbol{\sigma }_{\!{H}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })=-p(\boldsymbol{r}){\unicode{x1D644}}+2\eta _{{s}}{\unicode{x1D65A}}(\boldsymbol{r})+2\eta _{{o}}[{\unicode{x1D65A}}(\boldsymbol{r})\boldsymbol{\cdot }\boldsymbol{\epsilon }\boldsymbol{\cdot }\boldsymbol{\hat {\ell }}+\boldsymbol{\hat {\ell }}\boldsymbol{\cdot }\boldsymbol{\epsilon }\boldsymbol{\cdot }{\unicode{x1D65A}}(\boldsymbol{r})], \quad \boldsymbol{r}\in \mathcal{V}_{\!{f}} , \end{gather}

and the electrostatic Maxwell stress tensor

(2.3) \begin{gather} \boldsymbol{\sigma }_{\!{E}}(\boldsymbol{r})=\varepsilon \left [\boldsymbol{E}(\boldsymbol{r})\,\boldsymbol{E}(\boldsymbol{r})-\frac {1}{2}{E}(\boldsymbol{r})^2\,{\unicode{x1D644}}\right ]\!, \quad \boldsymbol{r}\in \mathcal{V}_{\!{f}} . \end{gather}

Here, $p(\boldsymbol{r})$ is the absolute pressure, and $e_{\alpha \beta }(\boldsymbol{r})=[\partial _\alpha v_\beta (\boldsymbol{r})+\partial _\beta v_\alpha (\boldsymbol{r})]/2$ is the strain-rate tensor. The fluid is characterised by a dynamic shear viscosity $\eta _{{s}}$ and an intrinsic spin angular momentum $\boldsymbol{\ell }=4\eta _{{o}}\boldsymbol{\hat {\ell }}$ , which in a spatially uniform non-equilibrium steady state has a magnitude proportional to the odd-viscosity coefficient $\eta _{{o}}$ (Banerjee et al. Reference Banerjee, Souslov, Abanov and Vitelli2017; Markovich & Lubensky Reference Markovich and Lubensky2021). In the creeping-flow regime, the balance of linear momentum is given by $\partial _\beta \sigma _{\alpha \beta }(\boldsymbol{r};\hat{\boldsymbol{\ell} })=0$ , with Greek indices running over Cartesian coordinates $\{x,y,z\}$ , and an implied summation when the indices are repeated. Together with the incompressibility condition and (2.1), we find for $\boldsymbol{r}\in \mathcal{V}_{\!{f}}$ that

(2.4) \begin{gather} \eta _{{s}}{\nabla} ^2\boldsymbol{v}(\boldsymbol{r})-\boldsymbol{\nabla }\!\tilde {p}(\boldsymbol{r})+\eta _{{o}}(\hat{\boldsymbol{\ell} }\boldsymbol{\cdot }\boldsymbol{\nabla })[\boldsymbol{\nabla }\times \boldsymbol{v}(\boldsymbol{r})]=-\rho (\boldsymbol{r})\boldsymbol{E}(\boldsymbol{r}), \quad \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{v}(\boldsymbol{r})=0, \end{gather}

with effective pressure $\tilde {p}(\boldsymbol{r})=p(\boldsymbol{r})+2\eta _{{o}}\hat{\boldsymbol{\ell} }\boldsymbol{\cdot }[\boldsymbol{\nabla }\times \boldsymbol{v}(\boldsymbol{r})]$ . Next, the steady flux of free ions $\boldsymbol{J}_{\!i}(\boldsymbol{r})$ is conserved, i.e. $\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{J}_{\!i}(\boldsymbol{r})=0$ , where

(2.5) \begin{gather} \boldsymbol{J}_{\!i}(\boldsymbol{r})=-{\unicode{x1D63F}}_i(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\left [\boldsymbol{\nabla }n_i(\boldsymbol{r})+\frac {z_ie}{k_{{B}}T}\,n_i(\boldsymbol{r})\,\boldsymbol{\nabla }\psi (\boldsymbol{r})\right ]+n_i(\boldsymbol{r})\,\boldsymbol{v}(\boldsymbol{r}), \quad \boldsymbol{r}\in \mathcal{V}_{\!{f}}, \quad i=1,\ldots ,N, \end{gather}

with $T$ the temperature, and $k_{{B}}$ the Boltzmann constant. The terms in (2.5) describe ionic diffusion, electromigration and advection, respectively. Furthermore, ${\unicode{x1D63F}}_i(\hat{\boldsymbol{\ell} })$ are the ionic diffusion tensors with a tensorial structure similar to the dielectric tensor. However, in this work, its precise form is irrelevant, as we will see later, in § 3. Equations (2.1), (2.4) and (2.5) constitute the simplest generalisation of the Poisson–Nernst–Planck–Stokes equations (Hunter Reference Hunter2001) to odd fluids. To the best of our knowledge, we are the first to introduce such a notion of odd electrokinetics.

Our goal is to analyse the steady motion of an ion-impenetrable particle with a no-slip surface $\mathcal{S}_{\!{p}}$ , described by

(2.6) \begin{equation} \boldsymbol{v}(\boldsymbol{r})=\boldsymbol{U}+\boldsymbol{\varOmega }\times \boldsymbol{r}, \quad \boldsymbol{\hat n}(\boldsymbol{r})\boldsymbol{\cdot }\boldsymbol{J}_{\!i}(\boldsymbol{r})=0, \quad i=1,\ldots ,N, \quad \boldsymbol{r}\in \mathcal{S}_{\!{p}}, \end{equation}

where $\boldsymbol{U}$ is its translational velocity, and $\boldsymbol{\varOmega }$ is its rotational velocity. Henceforth, $\boldsymbol{\hat n}(\boldsymbol{r})$ is a normal vector pointing from fluid to particle, with a caret denoting normalised vectors. We note that assessing the location of the slip plane can be quite complex in a charged odd fluid due to the presence of solvent, ionic and spinner degrees of freedom. Here, we view the location and existence of the slip plane as a given (experimental) input. Electrostatic boundary conditions depend on the charge functionality of the particle (insulating, charge-regulating or conducting) and will be specified later. Far-field conditions are

(2.7) \begin{align} \lim _{r\rightarrow \infty }\boldsymbol{v}(\boldsymbol{r})=\boldsymbol{0},\quad \lim _{r\rightarrow \infty }\psi (\boldsymbol{r})/r=-\boldsymbol{E}_{\textit{ext}}\boldsymbol{\cdot }\boldsymbol{\hat r}, \quad \lim _{r\rightarrow \infty }n_i(\boldsymbol{r})=n_i^\infty , \quad i=1,\ldots ,N, \end{align}

where the $n_i^\infty$ are constant bulk ion densities satisfying local charge neutrality $\sum _i z_in_i^\infty =0$ . The forces $\boldsymbol{F}_{{H,E}}$ and torques $\boldsymbol{T}_{{H,E}}$ that the particle exerts on the charged chiral active fluid are

(2.8) \begin{equation} \boldsymbol{F}_{\!j}=\int _{\mathcal{S}_{\!{p}}}\, \mathrm{d}S\, \boldsymbol{\sigma }_{\!j}(\boldsymbol{r})\boldsymbol{\cdot }\boldsymbol{\hat n}(\boldsymbol{r}), \quad \boldsymbol{T}_{\!j}=\int _{\mathcal{S}_{\!{p}}}\, \mathrm{d}S\, \boldsymbol{r}\times [\boldsymbol{\sigma }_{\!j}(\boldsymbol{r})\boldsymbol{\cdot }\boldsymbol{\hat n}(\boldsymbol{r})], \quad j={H,E}. \end{equation}

Since the particle is subjected to steady motion, we have $\boldsymbol{F}_{\!{H}}+\boldsymbol{F}_{\!{E}}=\boldsymbol{0}$ and $\boldsymbol{T}_{\!{H}}+\boldsymbol{T}_{\!{E}}=\boldsymbol{0}$ . We are interested in finding the relation between the translational velocity $\boldsymbol{U}$ and rotational velocity $\boldsymbol{\varOmega }$ with $\boldsymbol{E}_{\textit{ext}}$ , given by

(2.9) \begin{equation} \boldsymbol{U}=\boldsymbol{\mu }^{{tE}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{E}_{\textit{ext}}, \quad \boldsymbol{\varOmega }=\boldsymbol{\mu }^{{rE}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{E}_{\textit{ext}}. \end{equation}

These relations define the electrophoretic mobility tensor $\boldsymbol{\mu }^{{tE}}(\hat{\boldsymbol{\ell} })$ and the electrorotation tensor $\boldsymbol{\mu }^{{rE}}(\hat{\boldsymbol{\ell} })$ , which typically depend on the direction of intrinsic spin-momentum of the fluid $\hat{\boldsymbol{\ell} }$ . Here, we want to compute these tensors for small external electric fields – known in the literature as linear electrophoresis – where both tensors do not depend on $\boldsymbol{E}_{\textit{ext}}$ .

3. General result for electrophoretic mobility

For small electric fields, the equations are effectively linear, and we can use the Lorentz reciprocal theorem to find expressions for $\boldsymbol{\mu }^{{tE}}(\hat{\boldsymbol{\ell} })$ and $\boldsymbol{\mu }^{{rE}}(\hat{\boldsymbol{\ell} })$ by generalising the considerations of Teubner (Reference Teubner1982) to the odd case. We consider an incompressible flow of interest $\boldsymbol{v}(\boldsymbol{r})$ , and an incompressible auxiliary flow $\boldsymbol{v}^{(0)}(\boldsymbol{r})$ satisfying the same constitutive relation (2.2), but with different body forces $\boldsymbol{f}(\boldsymbol{r})$ and $\boldsymbol{f}^{(0)}(\boldsymbol{r})$ , respectively, acting on the fluid:

(3.1) \begin{equation} \partial _\beta \sigma _{{H},\alpha \beta }(\boldsymbol{r};\hat{\boldsymbol{\ell} })+f_\alpha (\boldsymbol{r})=0, \quad \partial _\beta \sigma _{{H},\alpha \beta }^{(0)}(\boldsymbol{r};\hat{\boldsymbol{\ell} })+f_\alpha ^{(0)}(\boldsymbol{r})=0. \end{equation}

Both flows do not necessarily satisfy the same boundary conditions and are related by the ‘odd’ Lorentz reciprocal theorem (Hosaka et al. Reference Hosaka, Golestanian and Vilfan2023), resulting from microscopic time reversibility, which leads to

(3.2) \begin{gather} \oint _{\mathcal{S}_{\!{p}}}\mathrm{d}S\, \boldsymbol{v}^{(0)}(\boldsymbol{r};-\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\sigma }_{\!{H}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\hat n}(\boldsymbol{r})+\int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, \boldsymbol{v}^{(0)}(\boldsymbol{r};-\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{f}(\boldsymbol{r})\nonumber \\ =\oint _{\mathcal{S}_{\!{p}}}\mathrm{d}S\, \boldsymbol{v}(\boldsymbol{r};\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\sigma }_{\!{H}}^{(0)}(\boldsymbol{r};-\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\hat n}(\boldsymbol{r})+\int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, \boldsymbol{v}(\boldsymbol{r};\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{f}^{(0)}(\boldsymbol{r}). \end{gather}

Due to the nature of odd viscosity, the auxiliary flow is to be evaluated at the time-reversed spin-momentum density $-\hat{\boldsymbol{\ell} }$ , as indicated by the second argument in the relevant quantities. The flow of interest with $\boldsymbol{f}(\boldsymbol{r})=\rho (\boldsymbol{r})\,\boldsymbol{E}(\boldsymbol{r})$ satisfies boundary conditions (2.6) and (2.7). For the auxiliary flow, we consider an uncharged particle suspended in an odd fluid with no free ions in the solution, $n_i(\boldsymbol{r})=0$ for $i=1,\ldots ,N$ , thus $\boldsymbol{f}^{(0)}(\boldsymbol{r})=\boldsymbol{0}$ . In the derivation of Teubner (Reference Teubner1982), the same stick boundary conditions for the auxiliary flow are taken on $\mathcal{S}_{\!{p}}$ as the flow of interest. However, for an odd fluid it is essential to set $\boldsymbol{v}^{(0)}(\boldsymbol{r})|_{\boldsymbol{r}\in \mathcal{S}_{\!{p}}}=\boldsymbol{U}^{(0)}+\boldsymbol{\varOmega }^{(0)}\times \boldsymbol{r}$ , with $\boldsymbol{U}\neq \boldsymbol{U}^{(0)}$ and $\boldsymbol{\varOmega }\neq \boldsymbol{\varOmega }^{(0)}$ , because the friction and mobility tensors contain antisymmetric contributions. Thus (3.2) becomes

(3.3) \begin{align} \boldsymbol{U}^{(0)}\boldsymbol{\cdot }\boldsymbol{F}_{\!{H}}-\boldsymbol{U}\boldsymbol{\cdot }\boldsymbol{F}_{\!{H}}^{(0)}+\boldsymbol{\varOmega }^{(0)}\boldsymbol{\cdot }\boldsymbol{T}_{\!{H}}-\boldsymbol{\varOmega }\boldsymbol{\cdot }\boldsymbol{T}_{\!{H}}^{(0)}=-\int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, \rho (\boldsymbol{r})\,\boldsymbol{v}^{(0)}(\boldsymbol{r};-\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{E}(\boldsymbol{r}). \end{align}

We add to this equation $\boldsymbol{U}^{(0)}\boldsymbol{\cdot }\boldsymbol{F}_{\!{E}}+\boldsymbol{\varOmega }^{(0)}\boldsymbol{\cdot }\boldsymbol{T}_{\!{E}}$ , and use that the fluid is force- and torque-free. Furthermore, by introducing the grand friction and mobility matrices

(3.4) \begin{equation} \begin{pmatrix} \boldsymbol{F}_{\!{H}}^{(0)}\\[5pt] \boldsymbol{T}_{\!{H}}^{(0)} \end{pmatrix}= \begin{pmatrix} \boldsymbol{\zeta }^{{tt}}(\hat{\boldsymbol{\ell} }) & \boldsymbol{\zeta }^{{tr}}(\hat{\boldsymbol{\ell} })\\[5pt] \boldsymbol{\zeta }^{{rt}}(\hat{\boldsymbol{\ell} }) & \boldsymbol{\zeta }^{{rr}}(\hat{\boldsymbol{\ell} }) \end{pmatrix} \begin{pmatrix} \boldsymbol{U}^{(0)} \\[5pt] \boldsymbol{\varOmega }^{(0)} \end{pmatrix}, \quad \begin{pmatrix} \boldsymbol{\mu }^{{tt}}(\hat{\boldsymbol{\ell} }) & \boldsymbol{\mu }^{{tr}}(\hat{\boldsymbol{\ell} })\\[5pt] \boldsymbol{\mu }^{{rt}}(\hat{\boldsymbol{\ell} }) & \boldsymbol{\mu }^{{rr}}(\hat{\boldsymbol{\ell} }) \end{pmatrix} =\begin{pmatrix} \boldsymbol{\zeta }^{{tt}}(\hat{\boldsymbol{\ell} }) & \boldsymbol{\zeta }^{{tr}}(\hat{\boldsymbol{\ell} })\\[5pt] \boldsymbol{\zeta }^{{rt}}(\hat{\boldsymbol{\ell} }) & \boldsymbol{\zeta }^{{rr}}(\hat{\boldsymbol{\ell} }) \end{pmatrix}^{-1} \! , \end{equation}

we find

(3.5) \begin{gather} \boldsymbol{\zeta }^{{tt}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{U}+\boldsymbol{\zeta }^{{tr}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\varOmega }=\int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, \rho (\boldsymbol{r})\,\boldsymbol{E}(\boldsymbol{r})\boldsymbol{\cdot }[{\unicode{x1D651}}(\boldsymbol{r};-\hat{\boldsymbol{\ell} })-{\unicode{x1D644}}\kern1.5pt], \end{gather}

(3.6) \begin{gather} \boldsymbol{\zeta }^{{rt}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{U}+\boldsymbol{\zeta }^{{rr}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\varOmega }=\int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, \rho (\boldsymbol{r})\,\boldsymbol{E}(\boldsymbol{r})\boldsymbol{\cdot }[{\unicode{x1D652}}(\boldsymbol{r};-\hat{\boldsymbol{\ell} })-\boldsymbol{r}\boldsymbol{\cdot }\boldsymbol{\epsilon }], \end{gather}

where we used the linearity of the equations describing the uncharged auxiliary fluid, following the convention of Happel & Brenner (Reference Happel and Brenner1983), to define

(3.7) \begin{equation} \boldsymbol{v}^{(0)}(\boldsymbol{r};\hat{\boldsymbol{\ell} })={\unicode{x1D651}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{U}^{(0)}+{\unicode{x1D652}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\varOmega }^{(0)}. \end{equation}

Furthermore, we used the symmetry property $\zeta _{\alpha \beta }^{ab}(\hat{\boldsymbol{\ell} })=\zeta _{\beta \alpha }^{ba}(-\hat{\boldsymbol{\ell} })$ for $a,b\in \{{t},{r}\}$ (Everts & Cichocki Reference Everts and Cichocki2024a ). Next, we expand the electrostatic potential $\psi (\boldsymbol{r})$ , to linear order in $\boldsymbol{E}_{\textit{ext}}$ , as

(3.8) \begin{equation} \psi (\boldsymbol{r})=\psi ^{{eq}}( \boldsymbol{r})+\boldsymbol{\varphi }\boldsymbol{\cdot }\boldsymbol{E}_{\textit{ext}}+\cdots , \end{equation}

where $\psi ^{{eq}}(\boldsymbol{r})$ is the equilibrium electrostatic potential (determined by $\boldsymbol{J}_{\!i}^{{eq}}(\boldsymbol{r})=\boldsymbol{0}$ , $\boldsymbol{v}^{{eq}}(\boldsymbol{r})=\boldsymbol{0}$ ). For the equilibrium situation, we consider particles with fixed zeta potential – the electrostatic potential at the slipping plane – i.e. $\psi ^{{eq}}(\boldsymbol{r})|_{\boldsymbol{r}\in \mathcal{S}_{\!{p}}}=\varPsi _0$ . Using (3.8), we find the following expansion for the electric body force:

(3.9) \begin{align} \rho (\boldsymbol{r})\,\boldsymbol{E}(\boldsymbol{r})=\varepsilon {\nabla} ^2\psi ^{{eq}}(\boldsymbol{r})\,\boldsymbol{\nabla }\psi ^{{eq}}(\boldsymbol{r})+\varepsilon [{\nabla} ^2\psi ^{{eq}}(\boldsymbol{r})\,\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol{r})+\boldsymbol{\nabla }\psi ^{{eq}}(\boldsymbol{r})\,{\nabla} ^2\boldsymbol{\varphi }(\boldsymbol{r})]\boldsymbol{\cdot }\boldsymbol{E}_{\textit{ext}}+\cdots . \end{align}

When inserted into (3.5) and (3.6), the first term on the right-hand side vanishes. At low zeta potentials $|e\varPsi _0/(k_{{B}}T)|\ll 1$ (the Henry approximation) and/or thin electric double layers (Smoluchowski limit), the other terms can be simplified. In this case,

(3.10) \begin{equation} {\nabla} ^2\boldsymbol{\varphi }(\boldsymbol{r})=\boldsymbol{0}, \quad \boldsymbol{\hat n}(\boldsymbol{r})\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol{r})=\boldsymbol{0}, \quad \boldsymbol{\varphi }(\boldsymbol{r})\rightarrow -\boldsymbol{r}+O(1/r^2), \quad r\rightarrow \infty . \end{equation}

Equations (3.5) and (3.6) simplify to

(3.11) \begin{gather} \boldsymbol{\zeta }^{{tt}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{U}+\boldsymbol{\zeta }^{{tr}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\varOmega }=\left \{\varepsilon \int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, {\nabla} ^2\psi ^{{eq}}(\boldsymbol{r})\, [{\unicode{x1D651}}^{\kern1pt \top} (\boldsymbol{r};-\hat{\boldsymbol{\ell} })-{\unicode{x1D644}} \kern1.5pt]\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol{r})\right \}\boldsymbol{\cdot }\boldsymbol{E}_{\textit{ext}}, \end{gather}

(3.12) \begin{gather} \boldsymbol{\zeta }^{{rt}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{U}+\boldsymbol{\zeta }^{{rr}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\varOmega }=\left \{\varepsilon \int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, {\nabla} ^2\psi ^{{eq}}(\boldsymbol{r})\,[{\unicode{x1D652}}^{\kern1pt \top} (\boldsymbol{r};-\hat{\boldsymbol{\ell} })+\boldsymbol{r}\boldsymbol{\cdot }\boldsymbol{\epsilon }] \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol{r})\right \}\boldsymbol{\cdot }\boldsymbol{E}_{\textit{ext}}, \end{gather}

with ${\top}$ denoting the tensor transpose. Equation (3.11) can be inverted to obtain the electrophoretic mobility and electrorotation tensor as defined in (2.9). Furthermore, note that (3.11) does not depend on ${\unicode{x1D63F}}_i(\hat{\boldsymbol{\ell} })$ .

4. Electrophoretic mobility for a sphere with uniform zeta potential

An uncharged sphere of radius $a$ in an odd fluid described by (2.2) exhibits no translational–rotational coupling (Everts & Cichocki Reference Everts and Cichocki2024a ). In this case, combining (2.9) with (3.11) gives, for the electrophoretic mobilities,

(4.1) \begin{gather} \boldsymbol{\mu }^{{tE}}(\hat{\boldsymbol{\ell} })=\boldsymbol{\mu }^{{tt}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\left \{\varepsilon \int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, {\nabla} ^2\psi ^{{eq}}(\boldsymbol{r})\,[{\unicode{x1D651}}^{\kern1.5pt\top} (\boldsymbol{r};-\hat{\boldsymbol{\ell} })-{\unicode{x1D644}} \kern1.5pt]\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol{r})\right \}\!, \end{gather}

(4.2) \begin{gather} \boldsymbol{\mu }^{{rE}}(\hat{\boldsymbol{\ell} })=\boldsymbol{\mu }^{{rr}}(\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\left \{\varepsilon \int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, {\nabla} ^2\psi ^{{eq}}(\boldsymbol{r})\,[{\unicode{x1D652}}^{\kern1.5pt\top} (\boldsymbol{r};-\hat{\boldsymbol{\ell} })+\boldsymbol{r}\boldsymbol{\cdot }\boldsymbol{\epsilon }] \boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol{r})\right \}\!. \end{gather}

Note that all quantities given in (4.1) and (4.2) are analytically known. The electrostatic quantities are

(4.3) \begin{equation} \psi ^{{eq}}(\boldsymbol{r})=\varPsi _0\frac {a}{r}\text{e}^{\kappa (a-r)}, \quad \boldsymbol{\varphi }(\boldsymbol{r})=-\! \left (1+\frac {a^3}{2r^3}\right )\boldsymbol{r}, \end{equation}

where $\kappa ^{-1}$ is the Debye screening length, with $\kappa ^2=e^2/(\varepsilon k_{{B}}T)\sum _{i=1}^N z_i^2 n_i^\infty$ . For electrorotation, ${\unicode{x1D652}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })=(a^3/r^3)(\boldsymbol{r}\boldsymbol{\cdot }\boldsymbol{\epsilon })$ (Hosaka et al. Reference Hosaka, Chatzittofi, Golestanian and Vilfan2024), thus it follows that ${\mu }^{{rE}}_{\alpha \beta }(\hat{\boldsymbol{\ell} })=0$ for all $\alpha ,\beta \in \{x,y,z\}$ . For electrophoresis, the tensor ${\unicode{x1D651}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })$ is analytically known in closed form (Meissner-Oszer et al. Reference Meissner-Oszer, Cichocki and Everts2026), and can be expressed as

(4.4) \begin{equation} {\unicode{x1D651}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })=\mathcal{L}_0{\unicode{x1D642}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\zeta }^{{tt}}(\hat{\boldsymbol{\ell} }), \quad \mathcal{L}_0 = \sum _{{k}=0}^\infty \frac {a^{2{k}}}{(2{k}+1)!}\big({\nabla} ^2\big)^{{k}}, \end{equation}

with ${\unicode{x1D642}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })$ the odd Oseen tensor (Everts & Cichocki Reference Everts and Cichocki2024a ). Although it is known analytically, we do not need its explicit form. The only relevant property is that it can be expressed as ${\unicode{x1D642}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })=(a/r)\,{\unicode{x1D65C}}(\boldsymbol{\hat r};\hat{\boldsymbol{\ell} })$ ; see Appendix A. The final quantity that we need is

(4.5) \begin{equation} \boldsymbol{\mu }^{{tt}}(\hat{\boldsymbol{\ell} })= \frac {1}{6\pi \eta _{{s}}a} \left [m_\parallel (\gamma )\,\hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} }+m_\perp (\gamma )\,({\unicode{x1D644}}-\hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} })+m_{{o}}(\gamma )\,(\boldsymbol{\epsilon }\boldsymbol{\cdot }\hat{\boldsymbol{\ell} })\right ] \!. \end{equation}

The closed-form analytical expressions for $m_\parallel (\gamma )$ , $m_\perp (\gamma )$ and $m_{{o}}(\gamma )$ as functions of arbitrary values of $\gamma =\eta _{{o}}/\eta _{{s}}$ can be found in Everts & Cichocki (Reference Everts and Cichocki2024a ). For $\gamma \rightarrow 0$ , we have $m_\parallel (\gamma )\rightarrow 1$ , $m_\perp (\gamma )\rightarrow 1$ and $m_{{o}}(\gamma )\rightarrow 0$ , retrieving the Newtonian fluid result for the translational mobility.

4.1. Hückel limit

First, we consider the limit $\kappa a\ll 1$ . This is the case of an infinite electric double layer, and has been first analysed for simple fluids by Hückel (Reference Hückel1924). We are considering stationary motion, so there is a force balance between the electric force and the hydrodynamic drag force, $\boldsymbol{\zeta }^{{tt}}(\hat {\boldsymbol{\ell }})\boldsymbol{\cdot }\boldsymbol{U}=\textit{Ze}\boldsymbol{E}_{\textit{ext}}$ , or $\boldsymbol{U}=\textit{Ze}\,\boldsymbol{\mu }^{{tt}}(\hat {\boldsymbol{\ell }})\boldsymbol{\cdot }\boldsymbol{E}_{\textit{ext}},$ with $\textit{Ze}$ the total charge of the particle. In equilibrium, with constant-potential boundary conditions on $\mathcal{S}_{\!{p}}$ , we have $\textit{Ze}=4\pi \varepsilon a\varPsi _0$ for $\kappa a\ll 1$ . We conclude that the electrophoretic mobility is $\boldsymbol{\mu }^{{tE}}(\hat {\boldsymbol{\ell }})=4\pi \varepsilon a\varPsi _0\,\boldsymbol{\mu }^{{tt}}(\hat {\boldsymbol{\ell }})$ , with components

(4.6) \begin{align} \boldsymbol{\mu }^{{tE}}(\hat {\boldsymbol{\ell }})=\frac {2\varepsilon }{3\eta _{{s}}}\varPsi _0\left [m_\parallel (\gamma )\,\hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} }+m_\perp (\gamma )\,({\unicode{x1D644}}-\hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} })+m_{{o}}(\gamma )\,(\boldsymbol{\epsilon }\boldsymbol{\cdot }\hat{\boldsymbol{\ell} })\right ]\!, \quad \kappa a \ll 1. \end{align}

Equivalently, one can neglect ${\unicode{x1D651}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })$ in (4.1) (Teubner Reference Teubner1982) and by global charge neutrality one would find the same result. Note that for $\gamma \rightarrow 0$ , we retrieve the Newtonian result for the Hückel limit. Moreover, the electrophoretic mobility tensor contains distinct contributions in the parallel, perpendicular and axial (i.e. odd) components, in contrast to the case of a homogeneously charged sphere suspended in a Newtonian fluid where $\boldsymbol{\mu }^{{tE}}(\hat {\boldsymbol{\ell }})$ is isotropic.

4.2. Smoluchowski limit

For thin electric double layers, $\kappa a \gg 1$ , first analysed for Newtonian fluids by Smoluchowski (Reference Smoluchowski1903), we transform (4.1) by two partial integrations and applying the boundary condition ${\unicode{x1D651}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })|_{\boldsymbol{r}\in \mathcal{S}_{\!{p}}}={\unicode{x1D644}}$ . The volume term can be neglected in this large-screening limit, and we find

(4.7) \begin{equation} \boldsymbol{\mu }^{{tE}}({\hat{\boldsymbol{\ell} }})=-\varepsilon \varPsi _0\int _{\mathcal{S}_{\!{p}}}{\rm d}S\, \boldsymbol{\mu }^{{tt}}(\hat {\boldsymbol{\ell }})\boldsymbol{\cdot }\left \{[\boldsymbol{\hat n}(\boldsymbol{r})\boldsymbol{\cdot }\boldsymbol{\nabla }]{\unicode{x1D651}}^{\kern1.5pt \top} (\boldsymbol{r};-{\hat{\boldsymbol{\ell} }})\right \}\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol{r}). \end{equation}

For a sphere $\boldsymbol{\hat n}=-\boldsymbol{\hat r}$ and after using (4.4), we find that (4.7) simplifies to

(4.8) \begin{equation} \boldsymbol{\mu }^{{tE}}({\hat{\boldsymbol{\ell} }})=-6\pi \varepsilon a^2\varPsi _0\left [\frac {\partial }{\partial r}\overline {\mathcal{L}_0{\unicode{x1D642}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }({\unicode{x1D644}}-\boldsymbol{\hat r}\boldsymbol{\hat r})}\right ]_{r=a}. \end{equation}

Here, we denote a surface average over the unit sphere $S^2$ as $\overline {(\ldots )}=(4\pi )^{-1}\int _{S^2}{\rm d}^2\boldsymbol{\hat r}\, (\ldots )$ . Now, for fixed $\alpha$ , we may view $\mathcal{L}_0G_{\alpha \beta }(\boldsymbol{r};\hat{\boldsymbol{\ell} })$ as an incompressible vector field that is constant on $\mathcal{S}_{\!{p}}$ . Therefore, its normal component has a vanishing gradient on $\mathcal{S}_{\!{p}}$ , which means that $({\partial }/{\partial r})\overline {\mathcal{L}_0{\unicode{x1D642}}(\boldsymbol{r};\hat{\boldsymbol{\ell} })\boldsymbol{\cdot }\boldsymbol{\hat r}\boldsymbol{\hat r}}$ vanishes on $\mathcal{S}_{\!{p}}$ . The remaining part equals $-1/a$ upon using (A2); see Appendix A. We find $\boldsymbol{\mu }^{{tE}}({\hat{\boldsymbol{\ell} }})=6\pi \varepsilon a\varPsi _0\,\boldsymbol{\mu }^{{tt}}(\hat{\boldsymbol{\ell} })$ with components

(4.9) \begin{equation} \boldsymbol{\mu }^{{tE}}({\hat{\boldsymbol{\ell} }}) =\frac {\varepsilon }{\eta _{{s}}}\varPsi _0\left [m_\parallel (\gamma )\,\hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} }+m_\perp (\gamma )\,({\unicode{x1D644}}-\hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} })+m_{{o}}(\gamma )\,(\boldsymbol{\epsilon }\boldsymbol{\cdot }\hat{\boldsymbol{\ell} })\right ] \!, \quad \kappa a\gg 1. \end{equation}

Note that this result is not necessarily restricted to the case $|e\varPsi _0/(k_{{B}}T)|\ll 1$ , since it also applies to particles with a local radius of curvature much larger than $\kappa ^{-1}$ . Furthermore, we retrieve the Newtonian result for the Smoluchowski limit when $\gamma \rightarrow 0$ . Interestingly, the fluid anisotropy still persists in $\boldsymbol{\mu }^{{tE}}({\hat{\boldsymbol{\ell} }})$ even for short Debye screening lengths. This is in sharp contrast to charged anisotropic particles in isotropic Newtonian fluids, where the electrophoretic mobility is isotropic and independent of particle shape under the same screening conditions.

4.3. Henry approximation

Next, we analyse electrophoresis for intermediate Debye screening length and small zeta potentials. Combining (4.1) and (4.4) gives

(4.10) \begin{equation} \boldsymbol{\mu }^{{tE}}(\hat {\boldsymbol{\ell }})=\varepsilon \int _{\mathcal{V}_{\!{f}}}\mathrm{d}V\, {\nabla} ^2\psi ^{{eq}}(r)\,[\mathcal{L}_0{\unicode{x1D642}}(\boldsymbol{r};\hat {\boldsymbol{\ell }})-\boldsymbol{\mu }^{{tt}}(\hat {\boldsymbol{\ell }})]\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol {r}). \end{equation}

We write $\int _{\mathcal{V}_{\!{f}}}\mathrm{d}V=\int _a^\infty \mathrm{d}r\, r^2\int _{S^2}{d}^2\boldsymbol{\hat r}$ , and we obtain

(4.11) \begin{equation} \boldsymbol{\mu }^{{tE}}(\hat {\boldsymbol{\ell }})=4\pi \varepsilon \kappa ^2a\varPsi _0\int _a^\infty {\rm d}r\, r\text{e}^{\kappa (a-r)}\left \{\,\overline {[\mathcal{L}_0{\unicode{x1D642}}(\boldsymbol {r};\hat {\boldsymbol{\ell }})]\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol { r})}+\boldsymbol{\mu }^{{tt}}(\hat {\boldsymbol{\ell }})\right \}\!, \end{equation}

where we used that $\overline {\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol{r})}=-{\unicode{x1D644}}$ . After applying the results from (A2) and (A10) (see Appendix A), we find

(4.12) \begin{equation} \overline {\mathcal{L}_0{\unicode{x1D642}}(\boldsymbol{r};\hat {\boldsymbol{\ell }})\boldsymbol{\cdot }\boldsymbol{\nabla }\boldsymbol{\varphi }(\boldsymbol { r})}=\left (-\frac {a}{r}+\frac {a^4}{4r^4}-\frac {a^6}{4r^6}\right )\boldsymbol{\mu }^{{tt}}(\hat {\boldsymbol{\ell }}). \end{equation}

We conclude that $\boldsymbol{\mu }^{{tE}}(\hat {\boldsymbol{\ell }})=6\pi \varepsilon a\varPsi _0f(\kappa a)\,\boldsymbol{\mu }^{{tt}}(\hat {\boldsymbol{\ell }})$ , with components

(4.13) \begin{gather} \boldsymbol{\mu }^{{tE}}(\hat {\boldsymbol{\ell }})=\frac {\varepsilon }{\eta _{{s}}}f(\kappa a)\,\varPsi _0\left [m_\parallel (\gamma )\,\hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} }+m_\perp (\gamma )\,({\unicode{x1D644}}-\hat{\boldsymbol{\ell} }\hat{\boldsymbol{\ell} })+m_{{o}}(\gamma )\,(\boldsymbol{\epsilon }\boldsymbol{\cdot }\hat{\boldsymbol{\ell} })\right ] \!, \end{gather}

and the Henry function $f$ (Henry Reference Henry1931) defined by

(4.14) \begin{equation} f(x)=\frac {2}{3}x^2\text{e}^x\int _1^\infty {\rm d}t\, t\text{e}^{-xt}\left (1-\frac {1}{t}+\frac {1}{4t^4}-\frac {1}{4t^6}\right )=1-\text{e}^x[5E_7(x)-2E_5(x)], \end{equation}

for $x\gt 0$ . Here, $E_{{k}}(x)=x^{{k}-1}\int _x^\infty {\rm d}t\, {\text{e}^{-t}}/{t^{{k}}}$ are exponential integrals. The function $f$ has the properties $\lim _{x\rightarrow 0}f(x)=2/3$ and $\lim _{x\rightarrow \infty }f(x)=1$ . Therefore, (4.13) reproduces both the Hückel limit (4.6), for $\kappa a\ll 1$ , and the Smoluchowski limit (4.9), for $\kappa a\gg 1$ . The result for a Newtonian fluid is retrieved for $\gamma \rightarrow 0$ as well. Again, the electrophoretic mobility tensor contains distinct parallel, perpendicular and axial components, while the modification by the Henry function is identical to that in isotropic Newtonian fluids. We note that for $\gamma \ll 1$ , $m_{\perp ,\parallel }(\gamma )=O(\gamma ^2)$ , while $m_{{o}}(\gamma )=\gamma /2+O(\gamma ^3)$ (Everts & Cichocki Reference Everts and Cichocki2024a ). In this regime, the linear growth of the axial component of $\boldsymbol{\mu }^{{tE}}(\hat{\boldsymbol{\ell} })$ with $\gamma$ , which is proportional to the steady-state intrinsic rotation-rate vector of the fluid particles (Markovich & Lubensky Reference Markovich and Lubensky2021), is an important experimental signature of odd electrophoresis.

We have also obtained (4.13) from a numerical computation using (4.1), (4.3) and (4.5), with the exact closed-form solution for ${\unicode{x1D651}}(\boldsymbol{r};\hat {\boldsymbol{\ell }})$ listed in Meissner-Oszer et al. (Reference Meissner-Oszer, Cichocki and Everts2026). Considering the complicated analytical form of this quantity shows the power of the singularity representation (4.4): it showcases how numerics can be completely circumvented due to the uncharged translating sphere having constant tractions on its surface. By the same properties, we find the remarkable result that $\boldsymbol{\mu }^{{tE}}(\hat{\boldsymbol{\ell} })$ is proportional to $\boldsymbol{\mu }^{{tt}}(\hat{\boldsymbol{\ell} })$ for a charged sphere within the Henry approximation and/or Smoluchowski limit.