Exact Phoretic Interaction of Two Chemically Active Particles

We study the nonequilibrium interaction of two isotropic chemically active particles taking into account the exact near-field chemical interactions as well as hydrodynamic interactions. We identify regions in the parameter space wherein the dynamical system describing the two particles can have a fixed point—a phenomenon that cannot be captured under the far-field approximation. We find that, due to near-field effects, the particles may reach a stable equilibrium at a nonzero gap size or make a complex that can dissociate in the presence of sufficiently strong noise. We explicitly show that the near-field effects originate from a self-generated neighbor-reflected chemical gradient, similar to interactions of a self-propelling phoretic particle and a flat substrate.

Figure 1

Examples for the behavior of the relative velocity versus the gap size in different regimes. The solid lines correspond to the exact solution and the dashed lines represent the far-field approximation. The parameter sets are as follows: (a) ${\tilde{\alpha }}_1=-3$, ${\tilde{\alpha }}_2={\tilde{\mu }}_1={\tilde{\mu }}_2=1$; (b) ${\tilde{\alpha }}_1={\tilde{\alpha }}_2={\tilde{\mu }}_1={\tilde{\mu }}_2=1$; (c) ${\tilde{\alpha }}_1={\tilde{\mu }}_1=-0.4$, ${\tilde{\alpha }}_2={\tilde{\mu }}_2=1$; (d) ${\tilde{\alpha }}_1=-2$, ${\tilde{\mu }}_1=3$, ${\tilde{\alpha }}_2={\tilde{\mu }}_2=1$. Each arrow shows the direction of the velocity for the corresponding particle, and double arrows indicate the larger speed. The velocities of individual particles are shown in the insets; the dotted lines show the value zero. See the videos in the Supplemental Material [58].

Figure 2

The regime diagram for the interactions of two chemically active particles in the activity-mobility parameter space, when ${\alpha }_2{\mu }_2>0$. If ${\alpha }_2{\mu }_2<0$, the map needs to be inverted, namely, regime I changes to regime II, regime III changes to regime IV, and vice versa. The dashed lines show the regime boundaries when the hydrodynamic interactions are neglected.

Figure 3

(a) The variation of fixed-point (FP) location (${\mathrm{\Delta }}_{\mathrm{FP}}$) as a function of ${\tilde{\mu }}_1$ for ${\tilde{\mu }}_2=1$, and ${\tilde{\alpha }}_2=1$. The lines in red are unstable fixed points with ${\tilde{\alpha }}_1=3$ ($○$), ${\tilde{\alpha }}_1=2.5$ ($\ominus$), ${\tilde{\alpha }}_1=2$ ($\oplus$), and ${\tilde{\alpha }}_1=1.5$ ($\otimes$). Those in black are stable fixed points with ${\tilde{\alpha }}_1=0.5$ ($●$), ${\tilde{\alpha }}_1=0$ ($\oslash$), ${\tilde{\alpha }}_1=-0.5$ ($\odot$), and ${\tilde{\alpha }}_1=-0.75$ ($⊚$). (b) The mean first-passage time for particles in regime IV, in units of ${\tau }_0=R^2/D_c^{\infty }$, corresponding to $D_c^{\infty }/R{V}_0=0.05$, where $D_c^{\infty }=D_c(\mathrm{\Delta }\to \infty )$. Here, ${\tilde{\alpha }}_2=1$, and ${\tilde{\mu }}_2=1$, with ${\tilde{\alpha }}_1=3.5$ ($○$), ${\tilde{\alpha }}_1=4$ ($\ominus$), and ${\tilde{\alpha }}_1=4.5$ ($\oplus$). (c) The ratio of the collision time obtained from the exact solution, $t^{\text{exact}}$, and the corresponding value from the far-field solution $t^{\mathrm{ff}}$ for ${\tilde{\alpha }}_2=1$ and ${\tilde{\mu }}_2=1$, with ${\tilde{\alpha }}_1=-2.1$ ($○$), ${\tilde{\alpha }}_1=-3.1$ ($\ominus$), and ${\tilde{\alpha }}_1=-4.1$ ($\oplus$).