Modeling chemo-hydrodynamic interactions of phoretic particles: A unified framework

Phoretic particles exploit local self-generated physico-chemical gradients to achieve self-propulsion at the micron scale. The collective dynamics of a large number of such particles is currently the focus of intense research efforts, both from a physical perspective to understand the precise mechanisms of the interactions and their respective roles, as well as from an experimental point of view to explain the observations of complex dynamics as well as formation of coherent large-scale structures. However, an exact modeling of such multiparticle problems is difficult, and most efforts so far rely on the superposition of far-field approximations for each particle's signature, which are only valid asymptotically in the dilute suspension limit. A systematic and unified analytical framework based on the classical Method of Reflections (MoR) is developed here for both the Laplace and Stokes' problems to obtain the higher-order interactions and the resulting velocities of multiple phoretic particles, up to any order of accuracy in the radius-to-distance ratio $\varepsilon$ of the particles. Beyond simple pairwise chemical or hydrodynamic interactions, this model allows us to account for the generic chemo-hydrodynamic couplings as well as $N$-particle interactions ($N\ge 3$). The ${\varepsilon }^5$-accurate interaction velocities are then explicitly obtained, and the resulting implementation of this MoR model is discussed and validated quantitatively against exact solutions of a few canonical problems.

Figure 1

Notations used for geometric description of the arrangement of any two Janus particles $j$ and $k$. The Janus particles comprise active (white) and inert (black) parts.

Figure 2

Illustration of three-particle chemical interactions arising from a single reflection of the concentration field. The chemical source from particle 1 (left) induces a chemical dipole (right) at the surface of particle 2. In turn, this corrected field and its gradient (arrows) induce a drift of particles 1 and 3.

Figure 3

Illustration of three-particle hydrodynamic interactions resulting from a single hydrodynamic reflection: The stresslet induced by the self-propulsion of particle 1 (a) induces a reflected stresslet at particle 2 (b). In turn, this modifies the hydrodynamic environment of particle 1 and 3 and induces their hydrodynamic drift (red arrow). The velocity magnitude (color) and direction (white arrow) are shown. Note that a rotation is also induced but scales as $O\left({\varepsilon }^6\right)$ and is neglected here.

Figure 4

Illustration of one of the dominant chemo-hydrodynamic interactions resulting from a single reflection of the concentration field: (a) the chemical source from particle 1 induces a quadrupolar correction of the concentration field near particle 2 (b). This source quadrupole induces a hydrodynamic stresslet (c) which is responsible for the drift of particles 1 and 3 (red arrows). In (a) and (b), the concentration fields are shown, while (c) shows the velocity magnitude (color) and direction (white arrows).

Figure 5

Interactions of two aligned Janus particles: (left) flow velocity magnitude obtained using BEM and (right) concentration field obtained analytically (Appendix pp2). Both Janus particles have positive mobility ($M=1$) and equal unit radius, with three fourths of their surface releasing solute at a fixed rate ($A=1$, white region) while the rest of their surface is inert ($A=0$, black region). The particles have a contact distance $d_c=d-2=0.5$, and swim toward their inert cap when isolated (i.e., along $+{\mathbf{e}}_z$).

Figure 6

Translation of two aligned Janus particles (see Fig. 5). (a) Swimming velocities of particle 1 (blue) and particle 2 (red) as a function of their contact distance $d_c=d-2$, as obtained analytically (solid), using MoR (dashed) or far-field models (dotted). The reference self-propulsion velocity of an isolated particle, $U^{\text{self}}=0.1875$, is also shown (dot-dashed). (b) Error magnitude $\left|\mathrm{\Delta }U\right|$ in the velocity prediction of far-field and MoR models with respect to the analytical solution.

Figure 7

Translation of two aligned Janus particles (see Fig. 5). (a) Evolution in time of the particles' velocity for an initial separation distance $d_c=0.5$. The predictions for MoR and far-field modes are computed at the relative positions described analytically. (b) Trajectories of the particles. Positions of particles at $t=0$ and $t=75$ are shown.

Figure 8

Coplanar trajectories of two Janus particles obtained using BEM (solid), MoR (dashed), and far-field (dotted). The particles' center-to-center distance is initially $d=4a$. Particles have uniform mobility $M=1$ and hemispherical activity with $A=1$ on their active half (black) and $A=0$ on their inert cap (white). Note that the $y$ axis is directed into the plane of the paper.

Figure 9

(a) Horizontal, (b) vertical, and (c) rotation velocity of Janus particle 1 for two coplanar particles (see Fig. 8), as obtained from BEM (solid), MoR(dashed), and the far-field model. At each time, the comparison between the prediction of the different models is performed for the same geometric configuration of the particles (i.e., that obtained from BEM simulations). The translation velocity is scaled by the self-propulsion velocity of an isolated Janus particle. Note that there is no angular velocity in the far-field model for all separations. The corresponding velocities of particle 2 are obtained using the planar symmetry of the problem with respect to $z=0$.

Figure 10

Comparison of the effects of the different interactions on the trajectory of two coplanar particles using the ${\varepsilon }^5$-accurate MoR model. The trajectories obtained with all interactions (solid), purely chemical interactions only (dotted), purely hydrodynamic interaction only (dash-dotted), and chemical and hydrodynamic interactions (i.e., without chemohydrodynamic interactions, dashed) are shown. Particles' location and their orientation at various instances of time are shown graphically.

Figure 11

Comparison of the trajectories of five Janus particles predicted by BEM (solid), MoR (dashed), and far-field models (dotted). The initial positions (at $t=0$) of the numbered particles and their predicted positions at $t=117$ are shown as well. Note that the $y$ axis is directed into the plane of the paper.

Figure 12

Evolution of the particles' velocities during the planar interactions of five Janus particles (Fig. 11): (left) Horizontal, (center) vertical, and (right) rotation velocities of particles 2 (top) and 5 (bottom) predicted by BEM (solid), MoR(dashed), and far-field models (dotted). The same particles' positions (obtained from the BEM simulations) are considered for all three models.

Figure 13

Collective dynamics of 25 Janus particles with uniform positive mobility and hemispherical activity. A set of 10 particles (in red) are arranged on a circle of radius 10 units, aligned offset from the radial direction by an angle $0.05\pi$, and 15 particles (in blue) are arranged on a circle of radius 15 units with the same angular offset from the radial direction. Particles' position computed using the MoR framework of Sec. 5 are shown for various instances of time, $t$.