Man,Yi – Propulsion of a Spheroidal Self-Diffusiophoretic Particle in Complex Fluids

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Propulsion of a Spheroidal Self-Diffusiophoretic Particle in Complex Fluids

Qiang Zhao, Guangpu Zhu, Lailai Zhu, On Shun Pak and Yi Man

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

June 22, 2026, Pasadena

Non-Newtonian Biofluids

Bansil et al., Front. Immunol., 2013

Polymeric microstructure of human mucus

Viscous and elastic responses coexist across timescales

Polymer-containing biofluids, such as mucus, exhibit heterogeneous microstructures and viscoelastic rheology

Rheology Alters Biological Swimming

Patteson et al., Sci. Rep., 2015

Elastic stresses suppress inefficient body wobbling, allowing E. coli to swim faster

Tung et al., Sci. Rep., 2017

Fluid viscoelasticity strengthens sperm–sperm interactions and promotes collective swimming

Self-Diffusiophoretic Janus Particles

Saad and Natale, Soft Matter, 2019

A catalytic Janus particle with asymmetric surface coating

Moran and Posner, Annu. Rev. Fluid Mech., 2017

Asymmetric surface reactions create a self-generated solute gradient that drives diffusiophoretic motion

Viscoelasticity Reshapes Janus-Particle Diffusion

Gomez-Solano et al., Phys. Rev. Lett., 2016

Rotational diffusion increases dramatically with increasing viscoelasticity and eventually saturates

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

Translational diffusion falls far below the Stokes–Einstein prediction as polymer concentration increases

These pronounced changes have motivated theoretical studies of active-particle dynamics in viscoelastic fluids.

Viscoelastic Propulsion of a Spherical Janus Particle

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

The first-order viscoelastic correction is antisymmetric with respect to the surface coverage
For the commonly studied half-coated particle, the first-order speed correction vanishes

An apparent theory–experiment tension

Weakly viscoelastic theory predicts no speed change for a half-coated sphere, whereas experiments report reduced propulsion in viscoelastic fluids.

Saad and Natale, Soft Matter, 2019

What Changes for a Non-Spherical Janus Particle?

Spherical Janus particle

First-order correction is antisymmetric with respect to surface coverage

Spheroidal Janus particle

How geometric anisotropy modifies viscoelastic propulsion remains unclear

Does the spherical cancellation persist for a non-spherical Janus particle?
How do eccentricity and surface coverage control viscoelastic propulsion?

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; \(D\): diffusivity; \(A\): surface activity / fixed solute flux on the active cap.

Tangential gradient \(\to\) Phoretic slip

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

\(M\): phoretic mobility; the projection extracts the tangential concentration gradient.

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

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

\(\boldsymbol{u}\): fluid velocity; \(\boldsymbol{\sigma}\): 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; \(\dot{\boldsymbol{\gamma}}\): Newtonian 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): \(\mathrm{De}=\dfrac{MA\Psi_1}{2\mu D a}\)

\(M\): phoretic mobility; \(A\): surface activity.

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; \(\tau=\tau_0\) denotes the particle surface.

Series solution and induced slip

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

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

\(P_n\) and \(Q_n\): Legendre functions; \(\rho_n\) is determined from 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

Weakly non-Newtonian expansion

Weak elasticity permits a perturbation expansion

For \(\mathrm{De}\ll1\), the propulsion problem is expanded about its Newtonian limit:

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

Subscript \(0\) denotes the Newtonian base state, while subscript \(1\) denotes the leading viscoelastic correction.

The Newtonian base state is already known

The concentration field determines the surface slip, and previous solutions provide the corresponding Newtonian flow \(\boldsymbol{u}_0\) and propulsion speed \(U_0\) for a spheroidal particle.

Concentration field

\(\longrightarrow\)

Surface slip

\(\longrightarrow\)

Known Newtonian flow \(\boldsymbol{u}_0\)

Viscoelasticity enters through an additional first-order stress

The first-order nonlinear contribution is determined by the Newtonian flow:

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

\(b=-2\Psi_2/\Psi_1\): viscoelastic parameter; \(\boldsymbol{A}\): known forcing term that drives the first-order correction.

Reciprocal theorem gives the speed correction

A virtual towing problem provides the resistance needed to determine \(U_1\)

We consider a rigid spheroid of the same geometry translating in a Newtonian fluid:

\[ \hat{\boldsymbol{u}}(\tau=\tau_0) = \hat{U}\boldsymbol{e}_z, \qquad \hat{\boldsymbol{F}} = \int_S \boldsymbol{n}\cdot\hat{\boldsymbol{\sigma}}\,\mathrm{d}S \]

\((\hat{\boldsymbol{u}},\hat{\boldsymbol{\sigma}})\) denote the known auxiliary velocity and stress fields.

The reciprocal theorem avoids solving the first-order flow

The reciprocal theorem eliminates \(\boldsymbol{u}_1\) by linking the \(\mathcal{O}(\mathrm{De})\) problem to the auxiliary flow:

\[ \hat{\boldsymbol{F}}\cdot\boldsymbol{U}_1 = \int_V \boldsymbol{A} : \nabla\hat{\boldsymbol{u}}\,\mathrm{d}V \]

The kernel characterizes the local power-density contribution associated with the viscoelastic stress.

The first-order propulsion correction follows directly

Using the known force on the auxiliary spheroid, we obtain:

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

Geometric isotropy leads to antisymmetry

The spherical-limit slip determines \(U_1\)

\[ u_s(\zeta)=\sum_{n=1}^{\infty} B_n P_n^1(\zeta), \quad U_1 = (b - 1) \sum_{n=1}^{\infty} K_n B_n B_{n+1} \]

\(B_n\) denotes the phoretic modes, while \(K_n\) is determined analytically.

Spherical isotropy relates complementary coverages

Reflecting the coating distribution about the equator gives

\[ u_s(-\zeta;-\zeta_0)=u_s(\zeta;\zeta_0), \quad U(\zeta_0,De)=U(-\zeta_0,-De) \]

and therefore

\[ U_{2n+1}(-\zeta_0)=-U_{2n+1}(\zeta_0), \quad U_{2n}(-\zeta_0)=U_{2n}(\zeta_0) \]

For a sphere, the first-order viscoelastic correction is antisymmetric in surface coverage and therefore vanishes at half coverage.

Eccentricity-induced enhancement for half-coated particles

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

Sphere: \(U_1=0\) by symmetry
Spheroid: eccentricity breaks the cancellation
Slender shapes show pronounced enhancement

Eccentricity-induced symmetry breaking

Shape anisotropy breaks the antisymmetry of \(U_1\)

Larger eccentricity broadens the enhancement regime

Broken kernel cancellation enhances propulsion

Spherical geometry yields antisymmetric local contributions
Prolate geometry localizes gradients and breaks the cancellation

Summary

Spherical symmetry

For spheres, the first-order correction is antisymmetric and vanishes at half coverage.

Shape-induced enhancement

Slender spheroids can exhibit pronounced speed enhancement.

Tunable propulsion

Shape and surface coverage jointly determine whether viscoelasticity enhances or suppresses propulsion.

Particle shape provides a direct control parameter for active propulsion in viscoelastic fluids.

Acknowledgments

Qiang Zhao

Ph.D. Student

College of Engineering, Peking University