Enhancement and Suppression of Active Particle Movement Due to Membrane Deformations

Microswimmers and active colloids often move in confined systems, including those involving interfaces. Such interfaces, especially at the microscale, may deform in response to the stresses of the flow created by the active particle. We develop a theoretical framework to analyze the effect of a nearby membrane due to the motion of an active particle whose flow fields are generated by force-free singularities. We demonstrate our result on a particle represented by a combination of a force dipole and a source dipole, while the membrane resists deformation due to tension and bending rigidity. We find that the deformation either enhances or suppresses the motion of the active particle, depending on its orientation and the relative strengths between the fundamental singularities that describe its flow. Furthermore, the deformation can generate motion in new directions.

💡 Research Summary

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The paper presents a comprehensive theoretical framework for understanding how a nearby deformable membrane influences the motion of an active particle in a low‑Reynolds‑number fluid. The authors model the particle as a spherical, force‑free swimmer that generates flow fields represented by a combination of a force dipole (stresslet) and a source (mass) dipole. The membrane is described by the linearized Helfrich model, incorporating both surface tension (T) and bending rigidity (κ_B), and is assumed to deform only in the out‑of‑plane direction.

Using the Lorentz reciprocal theorem, the authors derive a general integral expression (Eq. 11) for the first‑order correction to the particle’s swimming velocity, V₁, caused by membrane deformation. This expression links the known flow and stress fields of the particle near a rigid wall (v₀, σ₀), the deformation field u, and the stress field of a model problem (a Stokeslet near a rigid wall, denoted ˆσ). Because the governing Stokes equations are linear, the result can be superposed for any number of singularities, making the formula applicable to a broad class of active swimmers.

The paper then applies the formalism to two concrete cases. First, a self‑propelled swimmer (V_act ≠ 0) whose far‑field flow is a superposition of a stresslet of strength |D| and a source dipole of strength q. The authors introduce a dimensionless parameter Q = q/(|D| h) that measures the relative importance of the source dipole, and an orientation angle α that defines the dipole axis with respect to the membrane normal. The interaction with a flat rigid wall already modifies the swimming velocity (Eq. 15), producing both parallel (x) and perpendicular (z) components that depend on α and Q.

Next, the deformation‑induced correction V₁ is evaluated by solving the membrane equation (κ_B∇⁴ − T∇²) u_z = σ_zz⁰ and inserting the resulting u into the integral (Eq. 11). The authors nondimensionalize lengths by a reference height h₀, define a small deformation parameter Λ = |D| κ_B/(η h³), and assume Λ ≪ 1. Numerical integration shows that the sign and magnitude of V₁ are highly sensitive to (i) the dipole orientation α, (ii) the strength ratio Q, and (iii) the tension‑to‑bending ratio T h²/κ_B. When bending rigidity dominates (small T h²/κ_B), the membrane deforms more strongly, leading to a nonlinear increase of V₁; when tension dominates, deformation is suppressed and V₁ remains small.

A key finding is that membrane deformation can either enhance (speed up) or suppress (slow down) the swimmer, depending on the combination of α and Q. Moreover, for certain parameter sets the correction introduces a non‑zero component in a direction that was absent in the undeformed case, effectively generating motion along a new diagonal direction. This “directional switching” is a purely hydrodynamic‑elastic effect and does not require any external torque.

The second case concerns “shakers” – active particles that exert stresses on the fluid but do not have a self‑propulsion velocity (V_act = 0). For shakers the first‑order correction V₁ scales quadratically with the source dipole strength (∝ q²), reflecting the fact that the induced flow is itself generated by the membrane deformation. The analysis predicts a lift force that can push the particle away from or toward the membrane depending on whether the stresslet or source dipole dominates.

Overall, the study demonstrates that even modest membrane deformations can have a profound impact on microswimmer dynamics, producing both quantitative changes in speed and qualitative changes in trajectory. The results are relevant to biological contexts such as microorganisms swimming near cell membranes, to engineered systems involving soft interfaces (e.g., oil‑water emulsions, polymer films), and to the design of microrobots that exploit soft boundaries for navigation or cargo transport. The authors conclude by suggesting extensions to nonlinear deformations, non‑spherical swimmers, and many‑body interactions as promising directions for future research.