Two-phase hydrodynamic model of active colloid motion

The paper presents a two-phase hydrodynamic model for the numerical simulation of collective motion in a thin layer of active colloids containing spherical microswimmers. The model accounts for three fundamental mechanisms governing the dynamics of the active colloid: the random motion of the microswimmers, their mutual collisions, and their interaction with the surrounding fluid phase. The accurate resolution of the characteristic time scales associated with each mechanism is crucial for reproducing the different dynamic modes. The model reproduces two primary modes of motion: Brownian and collective, as well as the transition between them. It is demonstrated that hydrodynamic interactions begin to play a significant role when the microswimmer velocity exceeds a critical threshold. At this point, the kinetic energy transferred to the fluid phase is sufficient to generate a noticeable feedback effect on the swimmers’ motion. Conversely, a further increase in microswimmers’ velocity enhances the role of collisions, causing the system to revert from a collective mode back to a Brownian-like state. A similar transition occurs at higher volume fractions of microswimmers within the colloid.

💡 Research Summary

The paper introduces a two‑phase hydrodynamic framework for simulating the collective dynamics of active colloids composed of spherical microswimmers confined to a thin layer. Unlike traditional single‑phase approaches, the authors treat the solid particles and the surrounding liquid as separate phases: particles are followed in a Lagrangian manner while the fluid is solved on an Eulerian grid using the incompressible Navier‑Stokes equations. Each particle experiences a constant‑magnitude active force whose direction undergoes stochastic rotation (modeling rotational diffusion), elastic collisions with neighboring particles, and Stokes drag proportional to the velocity difference between particle and fluid. The drag time scale (Stokes time τ_St = m_p/(3πμd_p)) couples the particle and fluid dynamics.

The fluid momentum equation is rewritten in terms of a stagnation energy potential H = |u|²/2 – p/ρ, yielding ∂u/∂t + F + ∇H = 0, where F contains pressure gradient, viscous diffusion, and the body force contributed by all particles (f_b = (1/ρV) Σ m_p du_p/dt). The resulting Poisson equation for H (∇²H = –∇·F) is solved with a multigrid method, and a predictor‑corrector scheme advances velocity and pressure.

A key contribution is the explicit treatment of disparate time scales. The authors define τ = Δt_f/Δt_p, the ratio of the fluid time step to the particle sub‑step. When τ = 1, particle reorientation and momentum transfer to the fluid occur on the same time scale; τ > 1 allows many particle direction updates within one fluid step, thereby resolving collisions and hydrodynamic feedback more accurately. Numerical experiments on a circular domain with prescribed circular particle trajectories demonstrate that only for τ ≳ 10 does the particle motion remain close to the target path and generate a well‑defined vortex in the fluid. For τ = 1 the particles spiral outward, and the fluid remains essentially stagnant.

The main simulation domain is a 2‑D square (3.2 mm × 3.2 mm) populated with 40 µm diameter magnetite particles suspended in water (ρ = 1 g cm⁻³, μ = 9 × 10⁻⁴ Pa·s). Particle speed u_p is varied from 0.25 to 5 mm s⁻¹, and volume fractions n_p from 0.2 to 0.6 (≈1600–4900 particles). By scanning τ, u_p, and n_p, the authors construct a phase diagram that identifies three distinct dynamical regimes:

1. Brownian‑like regime – particles move randomly against an almost quiescent fluid; occurs at low τ (insufficient time‑scale separation) and low u_p.
2. Transient regime – an initial collective vortex forms but later collapses back to Brownian‑like motion as collisions dominate.
3. Collective regime – sustained vortex and directed flow arise when hydrodynamic coupling outweighs collisions; typically for τ ≥ 10 and u_p above a critical threshold (~0.5 mm s⁻¹).

Increasing u_p further (≈1.3 mm s⁻¹) amplifies collision frequency, causing the system to revert to a Brownian‑like state despite the higher propulsion. Raising the particle volume fraction widens the transitional band and shifts the collective window to lower u_p values.

These findings mirror experimental observations in bacterial suspensions and magnetically driven microswimmers, where increasing propulsion speed first triggers flocking or vortex formation, then leads to disorder at very high activity due to steric hindrance. The two‑phase model thus captures the non‑monotonic dependence of collective behavior on activity and concentration without resorting to ad‑hoc source terms.

The authors conclude that accurate resolution of both particle and fluid time scales is essential for reproducing collective phenomena in active matter. The framework is readily extensible to more complex geometries (e.g., channels, capillaries), external fields, or three‑dimensional configurations, offering a versatile tool for future studies of active suspensions and emulsions.