Disorder-induced persistent random motion and trapping of microswimmers

Microorganisms ofter move in confined, disordered environments, where hydrodynamic couplings can modify their transport behavior. Using extensive finite-element simulations, we investigate the dynamics of microswimmers – modeled as squirmers – in two-dimensional disordered porous media by resolving the full hydrodynamic interactions. We reveal that the deterministic coupling between activity, hydrodynamics, and disorder is sufficient to generate effective diffusive transport. Strong pushers and pullers become localised in the porous medium either by trapping at corners or dynamic trapping, depending on swimmer type and obstacle packing fraction. Squirmers can escape from dynamic traps, leading to a prominent ``hopping-and–trapping’’ dynamics. Strikingly, we find a pusher-puller asymmetry in the trapping probability that can be reversed by short-range swimmer-obstacle interactions, highlighting the sensitivity of transport to near-field effects.

💡 Research Summary

This study investigates the transport of self‑propelled microswimmers in a two‑dimensional disordered porous medium by fully resolving the hydrodynamic interactions using finite‑element simulations. The swimmers are modeled as spherical squirmers of radius a whose surface slip velocity is prescribed by u_S = B₁(1 + β p·n)(nn − I)·p, where B₁ sets the free‑space speed U = B₁/2 and β is the dimensionless squirming parameter (β < 0 for pushers, β > 0 for pullers, β = 0 for neutral swimmers). The porous matrix consists of N non‑overlapping disks of radius R = 4a placed at random, with packing fractions ϕ = 0.15 (dilute) and ϕ = 0.45 (dense). A short‑range repulsive potential enforces a minimal swimmer‑obstacle distance h_cut = δ a, with δ = 1/20 or 1/4, effectively increasing the obstacle radius to R_eff = R + a + h_cut. For ϕ = 0.45 the effective packing fraction reaches ≈0.69, i.e., close to the percolation threshold.

The Stokes equations (μ∇²u = ∇p, ∇·u = 0) are solved with no‑slip conditions on the obstacles and force‑ and torque‑free conditions on the swimmer. Periodic boundary conditions emulate an infinite medium. For each (ϕ, δ, β) combination, 128 trajectories are generated across four independent realizations of the porous matrix, yielding statistically robust data.

Two distinct dynamical regimes emerge:

1. Exploring (diffusive) regime – Swimmers initially move ballistically (MSD ≈ U²t²). After a characteristic time τ = (R + a)/U, encounters with obstacles reorient the swimmers, leading to a crossover from ballistic to either normal diffusion (MSD ∝ t) or anomalous scaling. In dense media with δ = 1/20, only neutral squirmers remain diffusive at long times; pushers show slight sub‑diffusion due to tangential alignment with obstacles, while pullers retain higher persistence. In dilute media, pullers (β > 0) display pronounced super‑diffusion (MSD ∝ t^α, α > 1) because they experience fewer reorientations, whereas pushers are more frequently redirected.

2. Localized (trapping) regime – Swimmers become confined either statically (fixed at a corner) or dynamically (quasi‑periodic orbits between two or more obstacles). Dynamic traps are not closed loops; each cycle is slightly displaced, allowing eventual escape (“hopping”). The confinement radius ρ (gyration radius) distinguishes the two: ρ < a indicates static trapping, ρ ≥ a indicates dynamic trapping. The trapping probability P(β) rises with |β|, but increasing the repulsive cutoff (δ = 1/4) suppresses pusher trapping and can even reverse the pusher‑puller asymmetry, highlighting the sensitivity to near‑field interactions. Survival probability S(t) (the probability of not being trapped up to time t) decays roughly exponentially; the median trapping time ˜T varies strongly with β, ϕ, and δ. Strong pushers (β = −8) are trapped almost immediately in dense media, whereas neutral and weak pullers can explore for long periods before localization.

The key insight is that deterministic hydrodynamic coupling with a disordered obstacle field alone can generate effective diffusion, sub‑diffusion, super‑diffusion, and various trapping mechanisms without any stochastic tumbling. The pusher‑puller asymmetry, the obstacle density, and the range of short‑range repulsion together dictate whether a swimmer will percolate through the medium or become localized. These findings elucidate how microorganisms navigate complex natural habitats and provide design principles for synthetic microrobots that must either avoid entrapment or exploit it for targeted delivery.