Self-phoretic oscillatory motion in a one-dimensional channel

Abstract

We study a simple model for a particle that is active due to self-phoresis and that has been proposed to model symmetric camphor grains. The particle generates a concentration field through the continuous emission of a chemical substance, and its motion is driven by gradients of this field as it diffuses within a confined channel whose ends perfectly reflect the chemical. The reflection of the chemical field leads to an effective confinement of the particle, which itself is reflected before encountering the channel ends. The system displays a transition from a passive state, where the particle rests at the channel midpoint, to an active state characterized by highly regular, non-chaotic oscillations. We analytically construct the phase diagram and derive the oscillation frequency and amplitude in the vicinity of the transition. A perturbative analysis perfectly describes the dynamics of the particle even for oscillations as large as half the channel size. Furthermore, we develop an analysis which explains the mechanism of particle reflection close to the channel edges in the regime of large activity.

Figures (7)

• Figure 1: (a) Stability boundary between oscillatory (above) and static (below) phases in the $\{ \mu^*,\lambda^* \}$ plane. The blue line shows the analytical prediction; green points denote the numerically determined phase boundary, and red crosses indicate simulations that remain static at long times. (b) Critical frequency at the transition, $\omega_c^*$, versus absorption $\mu^*$. The blue line is the analytical prediction and green points are numerical results.
• Figure 2: Numerical solutions of $j[X,V, \mu, D, L,\lambda]=0$, with $j := V- \frac{\lambda}{D} \theta(X,V, \mu, D, L)$. For $\mu, D, L =1$ and $\lambda=40$ the equation admits at most three real roots depending on $X$. A physically invalid root at $v_1=-20$ exists for all $X$. For $0<X<X_c$ there are two positive roots at $v_2,v_3$ with $v_2 < v_3$. Beyond a critical value of $X_c$ no positive roots exist, and for $X<0$ the root $v_2$ becomes negative. Curves shown correspond to $X=-0.5$ (orange), $X= 0.915< X_c$ (blue), $X=X_c=0.9326$ (green) and $X=0.945 >X_c$ (red).
• Figure 3: Representative numerical results as trajectories on the $(X_\tau^*,V_\tau^*)$ plane. The orange circular point indicates the initial position at $(0.1,0)$. (a) For $\mu^*=1, \lambda^*= 5$: Oscillatory motion that settles onto a stable limit cycle after transient growth. Numerics shown by solid purple line and theory by red dashed line. (b) For $\mu^*=1, \lambda^*= 2$: Overdamped oscillations that decay to the rest state at $X_\tau^*=0$.
• Figure 4: A snapshot of the chemical concentration field released by the particle, $c^*(x^*,\tau)$ for $\tau=25.5, \mu^*=1,\lambda^*=5$.
• Figure 5: For $\mu^*=1, \lambda^* =30$: (a) Numerical simulation phase portrait in the active regime far from the phase transition $(\lambda^*\gg \lambda_c^*)$. Black arrows indicate the particle's trajectory direction for increasing time. (b) Comparison of numerics with the theoretical prediction from the large $\lambda^*$ analysis. Green circular points represent valid numerical solutions to equation \ref{['j']}, orange points represent invalid solutions with $|X^*_\tau| >X_c$, and red points indicate the critical solution with $X_\tau^*=X_c$.