Two-phase hydrodynamic model of active colloid motion

TL;DR

The paper tackles the challenge of modeling collective motion in thin layers of active colloids by proposing a two-phase hydrodynamic framework that separately tracks discrete microswimmers and the surrounding fluid. It resolves three coupled mechanisms— active propulsion, swimmer collisions, and fluid feedback— allowing the system to exhibit Brownian-like, collective, and transient motion regimes. The study demonstrates that hydrodynamic interactions become significant above a critical swimmer velocity and that higher collision rates can suppress collective order, with phase boundaries shifting with swimmer concentration. The approach reproduces qualitative features seen in experiments on self-organizing vortices in confined active suspensions and underscores the importance of resolving multiple time scales for predicting dynamic states. This framework provides a versatile tool for simulating active colloids and informs strategies to harness collective motion in microfluidic and biomedical applications.

Abstract

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.

Figures (8)

• Figure 1: Schematic of the test problem. The green dots indicate the particles' initial positions (particles diameter $d_p=40$$\mu$m), and the arrows represent the particles' velocity vectors.
• Figure 2: Results of the test problem simulation. Top row: particle positions at time instances from 0 to 1 s, $\Delta t=10$ ms. Bottom row: fluid streamlines at t = 1 s. Columns correspond to (a) $\tau=1$, (b) $\tau=10$, and (c) $\tau=100$.
• Figure 3: Diagram of dynamic modes in terms of microswimmers velocity ($u_p$) and parameter $\tau$ for $n_p=$ 0.2 (solid lines) and $n_p=$ 0.6 (dashed lines). 1 -- Brownian-like motion, 2 -- transient mode, 3 -- collective motion.
• Figure 4: Analysis of the flow patterns developed in Mode 1 ($u_p=1$ mm/s, $n_p=0.2$, $\tau=5$). (a) Time histories of the mean square displacement of particles ($MSD$, red line), the average particle velocity ($u_p$, green line), and the exponent $\alpha$ characterizing the scaling $MSD \propto t^{\alpha}$ (blue line). (b) Streamlines of the fluid phase flow. (c) Instantaneous spatial distribution of particles. (d) Velocity vectors of the particles. (e) Stream traces illustrating the collective motion of the particles. Frames (b)--(e) correspond to time $t = 2$ s.
• Figure 5: Energy spectra for the particulate phase (a) and the fluid phase (b). The red curves correspond to Mode 1 ($u_p=1$ mm/s, $n_p=0.2$, $\tau=5$), and the green curves to Mode 3 ($u_p=1$ mm/s, $n_p=0.2$, $\tau=50$). The dashed lines indicate the reference slopes $E(k) \propto k^{-5/3}$ and $E(k) \propto k^{-3}$. Arrows mark the dissipation ranges. Here, $k$ is the wavenumber and $\delta$ is the mean path of the particles.