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Propulsion of a Spheroidal Self-Diffusiophoretic Particle in Complex Fluids

Qiang Zhao1, Guangpu Zhu2, Lailai Zhu2, On Shun Pak3 and Yi Man1

1 Peking University
2 National University of Singapore
3 Santa Clara University

20th U.S. National Congress on Theoretical and Applied Mechanics

June 22, 2026, Pasadena

Active Janus Particles in Viscoelastic Fluids

Viscoelastic biofluids — the natural habitat of many swimmers

Self-diffusiophoretic propulsion — an asymmetric coating sets up a self-generated solute gradient that drives motion

How does fluid elasticity change the propulsion speed itself?

Viscoelasticity Reshapes Janus-Particle Dynamics

Stronger rotational diffusion with faster motion

Gomez-Solano et al., PRL, 2016

Propulsion reduced in polymer solutions

Saad & Natale, Soft Matter, 2019

Active motion suppressed in polymer solutions

Raman et al., ACS Phys. Chem. Au, 2023

Experiments reveal rich but noise-dominated dynamics — the intrinsic propulsion speed is hard to isolate. To probe it cleanly, we turn to theory.

Viscoelastic Propulsion of a Spherical Janus Particle

Datt et al., J. Fluid Mech., 2017

Weakly viscoelastic limit → second-order fluid model

The first-order viscoelastic correction is antisymmetric in surface coverage.

For a half-coated sphere, the leading-order speed correction vanishes.

But what if the Janus particle is not spherical?

From Spheres to Spheroids

Sphere — correction vanishes at half-coating

?

Spheroid — geometric anisotropy

We vary two control parameters: eccentricity \(e\) (shape) and coverage \(\zeta_0\) (activity).

Model: a spheroidal self-diffusiophoretic particle

  • Solve the concentration and flow fields in prolate spheroidal coordinates \((\tau,\zeta)\)
  • Shape parameter: eccentricity \(e=\sqrt{a^2-b^2}/a\)
  • Activity parameter: catalytic coverage \(\zeta_0\)

Concentration gradients drive surface slip

Surface flux \(\to\) Concentration field

\[ \frac{\partial C}{\partial t} + \boldsymbol{u}\cdot\nabla C = D\nabla^2 C, \quad \left. D\boldsymbol{n}\cdot\nabla C \right|_{\tau=\tau_0,\zeta<\zeta_0} = -A \]

\(C\): solute concentration; \(\boldsymbol{u}\): fluid velocity; \(D\): solute diffusivity; \(A\): surface activity / fixed solute flux on the active cap; \(\boldsymbol{n}\): unit normal.

Tangential gradient \(\to\) Phoretic slip

\[ \boldsymbol{u}_s = M(\boldsymbol{I}-\boldsymbol{n}\boldsymbol{n})\cdot \nabla C \]

\(M\): phoretic mobility; \((\boldsymbol{I}-\boldsymbol{n}\boldsymbol{n})\): tangential projection operator.

Slip velocity \(\to\) Force-free propulsion

\[ \boldsymbol{u}(\tau=\tau_0) = \boldsymbol{u}_s(\zeta)+\boldsymbol{U}, \]

\[ \int_S \boldsymbol{n}\cdot\boldsymbol{\sigma}\,\mathrm{d}S = \boldsymbol{0} \]

biological propulsion analogy

Fluid rheology: second-order fluid model

Flow equations

\[ \nabla\cdot\boldsymbol{\sigma}=\boldsymbol{0}, \qquad \nabla\cdot\boldsymbol{u}=0, \qquad \boldsymbol{\sigma}=-p\boldsymbol{I}+\boldsymbol{T} \]

\(\boldsymbol{\sigma}\): total stress; \(p\): pressure; \(\boldsymbol{T}\): deviatoric stress.

Second-order fluid model

\[ \boldsymbol{T} = \mu\dot{\boldsymbol{\gamma}} -\frac{\Psi_1}{2}\boldsymbol{G} +\Psi_2\dot{\boldsymbol{\gamma}}\cdot\dot{\boldsymbol{\gamma}}, \]

\[ \begin{aligned} \boldsymbol{G} =& \boldsymbol{u}\cdot\nabla\dot{\boldsymbol{\gamma}} -(\nabla\boldsymbol{u})^{\mathrm{T}}\cdot\dot{\boldsymbol{\gamma}} -\dot{\boldsymbol{\gamma}}\cdot\nabla\boldsymbol{u} \end{aligned} \]

\(\mu\): viscosity; \(\dot{\boldsymbol{\gamma}}\): shear-rate tensor; \(\Psi_1,\Psi_2\): first and second normal-stress coefficients; \(\boldsymbol{G}\): steady upper-convected derivative.

Non-dimensionalization

length scale: semi-major axis of the particle \(a\),

velocity scale: \(MA/D\)

Peclet number (advection/diffusion): \(\mathrm{Pe}=\dfrac{MAa}{D^2}\)

Deborah number (fluid elasticity): \(\mathrm{De}=\dfrac{MA\Psi_1}{2\mu D a}\)

Diffusion-dominated solute transport

Diffusion-dominated solute field

\[ \nabla^2 c=0, \qquad \left.\boldsymbol{n}\cdot\nabla c\right|_{\tau=\tau_0} = \begin{cases} -1, & \zeta\le \zeta_0 \quad \text{active},\\ 0, & \zeta>\zeta_0 \quad \text{inert}, \end{cases} \qquad c|_{\tau\to\infty}=0 \]

\(c\): dimensionless relative solute concentration; \(\tau=\tau_0\): particle surface.

Series solution and induced slip

\[ c(\tau,\zeta) = \sum_{n=0}^{\infty} \rho_n Q_n(\tau)P_n(\zeta) \]

\[ \implies u_s(\zeta) = \tau_0\sum_{n=1}^{\infty} B_n\frac{P_n^1(\zeta)}{\sqrt{\tau_0^2-\zeta^2}} \]

\(P_n,Q_n\): Legendre functions; \(\rho_n\): coefficients set by the flux boundary condition; \(B_n=-\rho_n Q_n(\tau_0)\).

Note: finite-\(\mathrm{Pe}\) coupling

At finite \(\mathrm{Pe}\), advection couples the solute field back to the flow. Then the rheology can also affect the concentration and the slip generated inside the interaction layer.

Choudhary et al., J. Fluid Mech., 2020

Newtonian baseline

Weakly viscoelastic: expand in \(\mathrm{De}\ll 1\)

\[ \{\boldsymbol{u}, p, \boldsymbol{U}\} = \{\boldsymbol{u}_0, p_0, \boldsymbol{U}_0\} + \mathrm{De}\{\boldsymbol{u}_1, p_1, \boldsymbol{U}_1\} + \mathcal{O}(\mathrm{De}^2) \]

The zeroth order is the Newtonian problem, valid for any eccentricity, with a known speed:

\[ U_0 = -\frac{\tau_0}{2} \int_{-1}^{1} u_s \frac{\sqrt{1-\zeta^2}}{\sqrt{\tau_0^2-\zeta^2}} \,\mathrm{d}\zeta \]

Poehnl et al., J. Phys.: Condens. Matter, 2020; Popescu et al., Eur. Phys. J. E, 2010

Linearity makes the Newtonian speed symmetric about half-coating: \(U_0(\zeta_0)=U_0(-\zeta_0)\).

First-order viscoelastic correction

Viscoelasticity enters as a forcing term, set by the Newtonian flow:

\[ \boldsymbol{\sigma}_1 = -p_1\boldsymbol{I} + \dot{\boldsymbol{\gamma}}_1 + \boldsymbol{A}, \qquad \boldsymbol{A} = -(\boldsymbol{G}_0 + b\,\dot{\boldsymbol{\gamma}}_0\cdot\dot{\boldsymbol{\gamma}}_0) \]

An auxiliary towing problem + the reciprocal theorem give \(U_1\) directly, without solving the first-order flow:

\[ U_1 = -\frac{(\tau_0^2+1)\operatorname{coth}^{-1}\tau_0-\tau_0}{4\tau_0^2} \int_{\tau_0}^{\infty}\int_{-1}^{1} (\boldsymbol{A}:\nabla\hat{\boldsymbol{u}})(\tau^2-\zeta^2)\,\mathrm{d}\zeta\,\mathrm{d}\tau \]

\(\nabla\hat{\boldsymbol{u}}\): velocity gradient of the auxiliary (towing) flow.

The spherical cancellation is a symmetry

In a second-order fluid, a mirror reflection must be paired with De → −De:

\[U(\zeta_0, \mathrm{De}) = U(-\zeta_0, -\mathrm{De})\]

Half-coating is its own mirror ⟹ U even in De: the O(De) correction vanishes, leading correction is O(De²).

Eccentricity revives the correction

Half-coated particle: \(\zeta_0=0\)

  • Sphere: \(U_1=0\), protected by symmetry
  • Spheroid: eccentricity breaks the cancellation
  • Slender particles: pronounced speed enhancement

Breaking the coverage symmetry

Antisymmetry lost as \(e\) grows

Phase diagram: enhancement vs reduction

Eccentricity breaks the coverage antisymmetry, and the enhancement regime broadens with elongation.

Why the cancellation breaks

  • The first-order speed is set by a local kernel: \(U_1 \propto \int_V\) 𝒦 \(\mathrm{d}V\), where 𝒦 \(= \boldsymbol{A}:\nabla\hat{\boldsymbol{u}}\) (blue enhances, red reduces)

Sphere — 𝒦 antisymmetric, cancels exactly ⟹ \(U_1=0\)

Spheroid — curvature localizes 𝒦, antisymmetry broken ⟹ \(U_1\neq0\)

Summary

  • Spheres — the half-coated correction vanishes, protected by reflection symmetry
  • Spheroids — eccentricity breaks it; slender particles are strongly enhanced
  • General principle — symmetry breaks only when both shape anisotropy and fluid nonlinearity are present

Qiang Zhao

First author · PhD student, Peking University