Inferring non-equilibrium interactions from tracer response near confined active Janus particles

Abstract

Chemically active Janus particles sustain non-equilibrium spatial variations in the chemical composition of the suspending solution; these induce hydrodynamic flow and (self-)motility of the particles. Direct mapping of these fields has so far proven to be too challenging. Therefore, indirect methods are needed, e.g., deconvolving the response of "tracer" particles to the activity-induced fields. Here, we study experimentally the response of silica particles, sedimented at a wall, to active Pt/silica Janus particles. The latter are either immobilized at the wall, with the symmetry axis perpendicular or parallel to the wall, or motile. The experiments reveal complex effective interactions that are dependent on the configuration and on the state of motion of the active particle. Within the framework of a coarse-grained model, the behavior of tracers near an immobilized Janus particle can be captured qualitatively once activity-induced osmotic flows on the wall are considered.

Fig. 1. Active-passive interaction for an immobilized Pt-silica Janus particle in an axis-perpendicular configuration.

(A) Tracked trajectories of silica tracers (radius R_tr = 1 μm) near a Janus particle without (left) and with (right) H₂O₂. (B) Time-lapse images showing the attraction and retention of a tracer to an active Janus particle. (C) In-plane steady-state distribution of silica tracers (yellow regions) of radius R_tr = 1 μm around an active (6% H₂O₂) Janus particle (the gray disk at the center). (D) Snapshot showing multiple-rings accumulation of silica tracers around an active Janus particle; for reasons of visual clarity, a color overlay was added over each ring. (E) Tracked trajectories and (F) mean square displacement (MSD) of silica tracers starting from various locations around the center of the active particle. The insets in (F) show representative trajectories for each of the three classes of MSDs. (G) Snapshot showing the emergent multiple-rings structure for silica tracers of radius R_tr = 0.5 μm around an active Janus particle. (H) In-plane steady-state distribution of silica tracers (yellow regions) of radius R_tr = 0.5 μm around an active Janus particle (the gray disk at the center). The bright spot at the right edge is a spurious feature due to a tracer particle stuck at the wall. Except for (F), the scale bars correspond to 5 μm.

Fig. 2. Active-passive interaction for an immobilized Janus particle in an axis-parallel configuration.

(A and C) Snapshots showing accumulation of silica tracers around an active (4% H₂O₂) Janus particle and formation of circular ring segments for tracers of radius R_tr = 1 (A) and 0.5 μm (C). (B and D) In-plane steady-state distribution of silica tracers around an active Janus particle in an axis-parallel configuration for tracers of radius R_tr = 1 (B) and 0.5 μm (D). The origin of the in-plane system of coordinates is set at the center of the active particle. The strongly attractive circular ring segment at the inert side of the Janus particle (the bright yellow at the left) is accompanied by the formation of a spatially extended zone completely depleted of tracers at the active, Pt-coated, side (deep blue region at the right).

Fig. 3. Active-passive interactions for an active particle in an axis-parallel configuration in motile and jammed states.

(A) Time-lapse images showing the interaction between a motile Janus particle (green) and a tracer (red). (B) Tracked trajectories (comoving frame of reference) of tracers interacting with a motile active particle. The color code depicts time progression from initial (deep blue) to final (deep red) state. (C) Two-dimensional histogram of contact points between the active particle and incoming tracers. (D and E) Depletion zone at the Pt cap of an active Janus particle jammed within a dense monolayer of tracers (R_tr = 0.5 μm), at 3 (D) and 6% (E) H₂O₂ concentration. (F) Steady-state probability distribution (pdf) of in-plane location of tracers around an active (3% H₂O₂) Janus particle in an axis-parallel configuration, in a jammed state: attraction (yellow spots) at the silica side and repulsion, i.e., depletion zone (the deep blue region at the right), at the Pt side. (G and H) Illustration of the automated tracking of shape and area of the depletion zone; the snapshots are separated by 6 s. (I) Histogram and (J) time dependence of the area of the depletion zone for H₂O₂ concentrations of 3 (cyan) and 6% (red).

Fig. 4. Chemical field, hydrodynamic flow, and the emergent tracer dynamics around an immobile active Janus particle in a cap-down configuration near a wall with phoretic mobility coefficient b_w = b_i.

Left column (A to C): Dimensionless solute density (color-coded) and hydrodynamic flow streamlines corresponding to the immobile silica/Pt Janus particle located at h/R = 1.1. Right column (D to F): x-y in-plane flow response of large (R_tr = 1.0 μm, h_tr = 0.41) silica tracers (b_t = b_i) to the activity of the Janus particles (gray disks, as viewed from above). The color-coded background shows the corresponding Pe_t values. The corresponding values β_i are indicted at the left of each row. Red circles indicate the in-plane exclusion zone (of radii ∼1.2R), which is due to the tracer–Janus particle hard-core interaction.

Fig. 5. Chemical field, hydrodynamic flow, and the emergent tracer dynamics around an immobile active Janus particle in a cap-up configuration near a wall with phoretic mobility coefficient b_w = b_i.

(A to C) Dimensionless solute density (color-coded) and hydrodynamic flow streamlines corresponding to the immobile silica/Pt Janus particle located at h/R = 1.1. (D to I) x − y in-plane flow response of (D to F) small (R_tr = 0.5 μm, h_tr = 0.21) and (G to I) large (R_tr = 1.0 μm, h_tr = 0.41) silica tracers (b_t = b_i) to the activity of the Janus particles (gray disks, as viewed from above). The color-coded background shows the corresponding Pe_t values. The corresponding values β_i are indicted at the left of each row. Red circles (D to I) indicate the in-plane exclusion zone, which is due to the tracer–Janus particle hard-core interaction. Green dashed circles in (B), (C), (E), and (F) mark the location at which the motion of the tracers changes direction: toward (away from) the Janus particle if the tracer is inside (outside) this domain. Bottom row (J and K): Signed Péclet number versus radial distance for the small (J) and large (K) tracers, for the three values β_i = 0.1, 0.5, and 0.9. The dotted lines indicate the values sg(v_t) Pe_t = ± 1. The magenta markers and numbers indicate the ends of close-packed tracer layers (rings).

Fig. 6. Active-passive interaction for an immobilized Janus particle in an axis-parallel configuration for different surface types.

(A) Snapshots showing accumulation of silica tracers around an active Janus particle and formation of circular ring segments for tracers of radius R_tr = 1 μm for different surface types (glass, Ti, and mica). The tracers indicated in red are tightly bound to the active particle, tracers in yellow are weakly bound to the active particle, and the tracers in green are unbound in the solution. (B) Probability density of the silica tracer particles (R_tr = 1 μm) accumulated near the active Janus particle on the three different surface types (glass, Ti, and mica). The magenta markers indicate the ends of close packed tracer layers (rings). Color code: Deep color corresponds to high probability, white to the “far” constant value (inverse of the average tracer density). (C) Signed Péclet number for a tracer with z_tr/R = 0.41 in the vicinity of a cap-up Janus particle with β_t = β_i = 0.2. Negative values of sg(v_t) Pe_t indicate radial attraction. Roughly speaking, the effective attraction dominates the Brownian motion when sg(v_t) Pe_t < −1, and this condition is used to define the extent of the region of effective attraction. The various curves correspond to different values of the wall phoretic mobility ratio β_w, such as would be obtained with different substrate materials. The dotted lines indicate the values sg(v_t) Pe_t = ±1. The magenta markers and numbers indicate the ends of close-packed tracer layers (rings).

Fig. 7. Chemical field, hydrodynamic flow, and the emergent tracer dynamics around an immobile active Janus particle in a cap-parallel configuration near a wall with phoretic mobility coefficient b_w = b_i for tracer size R_tr = 0.5 μm.

Left column: x − y in-plane (z_t/R = h_tr = 0.21) and dimensionless (in units of c₀ := κR/D) solute density (color-coded) and hydrodynamic flow streamlines corresponding to the immobile silica/Pt Janus particle located at h/R = 1.1. Right column: x-y in-plane (z_t/R = h_tr = 0.21) tracer-flow response of silica particles to the activity of the Janus particle and the corresponding Pe_t values (color-coded background). The results shown correspond to different ratios β_i of the phoretic mobilities of the inert (silica) and active (Pt) sides; the phoretic mobility of the tracer is equal to that of the inert part of the Janus particle. The red circles indicate the in-plane (h_tr = 0.21) exclusion zone (of radii ∼ 0.8R) for the center of the tracer due to the hard-core interaction of the tracer (radius 0.2R) with the Janus particle of radius R. The half-black–half-gray disk shows the cut by the h_tr plane through the Janus particle.

Fig. 8. Chemical field, hydrodynamic flow, and the emergent tracer dynamics around an immobile active Janus particle in a cap-parallel configuration near a wall with phoretic mobility coefficient b_w = b_i for tracer size R_tr = 1.0 μm.

Left column: x − y in-plane (z_t/R = h_tr = 0.41) and dimensionless (in units of C₀) solute density (color-coded) and hydrodynamic flow streamlines corresponding to the immobile silica/Pt Janus particle located at h/R = 1.1. Right column: x-y in-plane (z_t/R = h_tr = 0.41) tracer-flow response of silica particles to the activity of the Janus particle and the corresponding Pe_t values (color-coded background). The results shown correspond to different ratios β_i of the phoretic mobilities of the inert (silica) and active (Pt) sides; the phoretic mobility of the tracer is equal to that of the inert part of the Janus particle. The red circles indicate the in-plane (h_tr = 0.41) exclusion zone (of radii ∼ 1.2R) for the center of the tracer due to the hard-core interaction of the tracer (radius 0.4R) with the Janus particle of radius R. The half-black–half-gray disk shows the cut by the h_tr plane through the Janus particle.