[Paper Review] Self-Diffusiophoresis in the Advection Dominated Regime

Gareth P. Alexander, Andrea J. Liu|arXiv (Cornell University)|Jul 19, 2011

NMR spectroscopy and applications3 citations

TL;DR

This paper investigates self-diffusiophoresis in the advection-dominated regime (infinite Péclet number), showing that particles self-propel via asymmetric solute gradients generated by non-diffusing filaments. Unlike the low-Péclet diffusive limit, steady-state motion requires only two of four low-Pe scenarios, with speed scaling as √α (activity) and sin³(½θₚ) (patch size), indicating spontaneous symmetry breaking in uniformly active particles.

ABSTRACT

In both biological and artificial systems, concentration gradients can serve as a convenient mechanism for manipulating particles and generating motility. Particles that interact with a solute will move along its gradient; if they themselves generate the gradient, this mechanism provides a means of self-propulsion. We consider a version of this type of motility appropriate to certain biological systems where polymeric filaments provide the concentration gradient. As the filament diffusion is small, this corresponds to a regime of large Péclet number where the motion is dominated by the effects of fluid advection. The nature of such concentration-gradient-driven motion in the advective regime differs in certain fundamental respects from the same process at low Péclet number. In particular, we show that out of four broad scenarios of steady state motion at low Péclet number, only two remain viable in the strongly advecting limit.

Motivation & Objective

• To understand self-propulsion mechanisms in biological systems where solute diffusion is negligible, such as actin-based motility in Listeria and ParA-driven chromosome segregation in Caulobacter.
• To determine the conditions for steady-state self-diffusiophoresis when fluid advection dominates over diffusion (high Péclet number).
• To identify how the scaling of propulsion speed with surface activity and patch size differs from the classical diffusive (Pe = 0) limit.
• To clarify the role of solute-surface interactions (attractive vs. repulsive) in enabling sustained motion under advective transport.

Proposed method

• Uses a minimal axisymmetric model of a spherical colloid with a localized active patch producing solute at rate α.
• Applies boundary layer analysis to model fluid flow and slip velocity at the particle surface, assuming solute interactions are short-ranged (δ ≪ a).
• Solves the Stokes equations with a body force term proportional to the solute concentration gradient and interaction potential V(r).
• Derives the slip velocity from the tangential flow in the boundary layer, accounting for advective transport via the Péclet number limit (Pe → ∞).
• Analyzes the solute conservation equation in the limit of vanishing diffusion, focusing on advective fluxes and steady-state concentration profiles.
• Compares results to the low-Pe limit, highlighting differences in symmetry, scaling, and viability of motion scenarios.

Experimental results

Research questions

• RQ1 Which of the four low-Péclet motility scenarios remain viable in the advection-dominated regime (Pe → ∞)?
• RQ2 How does the propulsion speed scale with surface activity α in the high-Péclet limit compared to the diffusive limit?
• RQ3 What determines the directionality and stability of motion when the active patch covers the entire particle surface (θₚ → π)?
• RQ4 How do attractive versus repulsive solute-surface interactions influence steady-state self-diffusiophoresis under advective transport?
• RQ5 What is the role of boundary layer flow in distorting solute profiles and altering the stall force in high-Pe systems?

Key findings

• Only two of the four low-Péclet motility scenarios remain viable in the high-Péclet limit: repulsive producers and attractive consumers of non-diffusing solutes.
• The propulsion speed scales as √α in the advective regime, contrasting with the linear √α scaling in the diffusive limit, due to the quadratic dependence of advective flux on slip velocity.
• Speed scales as sin³(½θₚ) with active patch size θₚ, meaning uniformly active particles (θₚ → π) can spontaneously break symmetry and move, unlike in the diffusive limit.
• The logarithmic divergence in the concentration profile is regularized by a mollifier ln(a/δ), removing unphysical singularities and confirming robustness of the high-Pe scaling.
• Spontaneous symmetry breaking is predicted in uniformly active particles at Pe → ∞, consistent with observations of directional motility in small actin-driven colloids.
• The model suggests that self-diffusiophoresis remains a viable mechanism for motility in systems like Listeria and Vibrio, where filamentous proteins have negligible diffusion.