The interplay between chemo-phoretic interactions and crowding in active colloids

Abstract

Many motile microorganisms communicate with each other and their environments via chemical signaling which leads to long-range interactions mediated by self-generated chemical gradients. However, consequences of the interplay between crowding and chemotactic interactions on their collective behavior remain poorly understood. In this work, we use Brownian dynamics simulations to investigate the effect of packing fraction on the formation of non-equilibrium structures in a monolayer of diffusiophoretic self-propelled colloids as a model for chemically active particles. Focusing on the case when a chemical field induces attractive positional and repulsive orientational interactions, we explore dynamical steady-states of active colloids of varying packing fractions and degrees of motility. In addition to collapsed, active gas, and dynamical clustering steady-states reported earlier for low packing fractions, a new phase-separated state emerges. The phase separation results from a competition between long-range diffusiophoretic interactions and motility and is observed at moderate activities and a wide range of packing fractions. Our analysis suggests that the fraction of particles in the largest cluster is a suitable order parameter for capturing the transition from an active gas and dynamical clustering states to a phase-separated state.

Fig. 1. Schematics of chemical field mediated interactions between self-catalytic Janus colloids for different cases of translational ζ_tr and rotational ζ_rot chemotactic mobility parameters in the situation when the colloids act as chemical sinks. (a) When ζ_tr > 0, the colloids move towards each other, (b) when ζ_tr < 0, the colloids move away from each other, (c) when ζ_rot > 0 the colloids rotate towards each other, and (d) when ζ_rot < 0 they rotate away from each other.

Fig. 2. (a) State diagram of self-phoretic colloids in the Péclet-packing fraction (Φ–Pe) representation for translational and rotational mobility coefficients ζ_tr = 15.4 and ζ_rot = −0.38, respectively. We distinguish four distinct dynamical states: active gas (red region), dynamical clusters (green region), CMIPS (yellow region) and collapsed state (blue region). Panels (b–e) depict representative snapshots for collapsed, chemotactic phase separated, dynamical clustering and active gas states, respectively, obtained at fixed Φ = 0.1 but different propulsion speeds Pe = 5, 16, 21 and 30 as given on the top of each snapshot.

Fig. 3. Probability distribution function of (a) cluster-size P(n) and (b) local packing fraction P(Φ) for active colloids of overall packing fraction Φ = 0.1 at Pe = 16 corresponding to a phase-separated state with mean cluster size N^avg_c ≈ 4000, Pe = 21 forming dynamical clusters with N^avg_c ≈ 4 and Pe = 30 in a gas state with N^avg_c ≈ 2.8. The dashed lines in panel (a) correspond to the fits of P(n) with the function a₀n^−βe^(−n/n₀) with a₀ = 0.83, β = 2.9 and n₀ = 10 for the active gas with Pe = 30, a₀ = 0.5, β = 2.01 and n₀ = 10 for the dynamic clusters with Pe = 21 and a₀ = 0.89, β = 2.6 and n₀ = 10 for the active gas part of the phase-separated state with Pe = 16.0.

Fig. 4. Probability distribution of local density P(ϕ) at Pe = 30 for (a) self-phoretic colloids at Φ = 0.1 (active gas), Φ = 0.2 (dynamical clustering) Φ = 0.3 and 0.5 (phase-separated states), respectively. (b) Active Brownian particles at Φ = 0.1 and 0.3 (active gas) and Φ = 0.5 (motility-induced phase-separated state).

Fig. 5. Snapshots of phase-separated states found at packing fraction Φ = 0.5 and Pe = 25 for (a) chemotactic and (b) active Brownian particles, where represents the average cluster size in the dilute phase, and q^†₆ gives the average hexatic order parameter of the dense giant cluster. The hexatic domains are defined as the regions where |Ψ^(k)₆| > 0.5 and color coded based on the phase α of local hexatic order parameter Ψ^(k)₆. Particles colored in light pink are either particles which belong to clusters of 5% or less of the biggest cluster or their hexatic order parameter is |Ψ^(k)₆|<0.5.

Fig. 6. Chemotactic phase-separated states found at packing fraction Φ = 0.2 at different activities (a) Pe = 12, (b) Pe = 12 and (c) Pe = 25. Here, represents the average cluster size outside the biggest cluster and q^†₆ gives the mean hexatic order parameter inside the biggest cluster. The colors represent the phase of hexatic domains α and light pink colored particles show either particles which belong to clusters of 5% or less of the biggest cluster or particles for which the hexatic order parameter is |Ψ^(k)₆| < 0.5.

Fig. 7. (a) Mean cluster size averaged in time N^avg_c and (b) ratio of the largest cluster size averaged in time N^max_c relative to the total number of particles N = 10⁴ of self-phoretic colloids (ζ_tr = 15.4 and ζ_rot = −0.38) as a function of the packing fraction Φ for various Péclet numbers 10 ≤ Pe ≤ 30 as given in the legend. The dashed lines in panel (a) correspond to N^avg_c = 3 and 10, respectively. The dashed lines in panel (b) show N^max_c/N for ABP particles with periodic boundary conditions with Pe = 21, 25 and 30 with identical color codes as the chemotactic particles.

Fig. 8. Time-averaged 6-fold-bond orientational order parameter q₆ as a function of the packing fraction Φ for Pe = 5, 10, 12, 14, 16, 18, 21, 25 and 30 for chemotactic particles with ζ_tr = 15.4 and ζ_rot = −0.38 (continuous lines) and for ABP particles ζ_tr = 0 and ζ_rot = 0 at Pe = 21, 25 and 30 (dashed lines). The inset shows q₆ as a function of Pe at Φ = 0.2, 0.5 and 0.7.

Fig. 9. Plot of the exponent α of the relation ΔN ∼ N̄^α as a function of the packing fraction Φ for Pe = 10, 12, 14, 16, 18, 21, 25 and 30 with ζ_tr = 15.4 and ζ_rot = −0.38 (continuous lines) and for ABPs with periodic boundary conditions at Pe = 21, 25 and 30 (dashed lines).