Intrinsic speed characteristics of a self-propelled camphor disk under repulsive perturbations

TL;DR

This study investigates a camphor-based self-propelled rotor perturbed by a fixed camphor disk using a one-dimensional reaction–diffusion–drag framework with a distance-dependent repulsive potential. It shows that rotor speed depends asymmetrically on the distance to the perturbation, a behavior captured by simulations across several potential shapes and analytically derived in the weak-perturbation limit by a co-moving-frame perturbation theory. The analytical results provide a closed-form velocity perturbation and demonstrate robustness of the asymmetry for small perturbations, aligning with experimental observations and challenging Hamiltonian energy-conservation models for this dissipative active-matter system. The findings highlight the importance of camphor transport and surface-tension gradients in binary interactions and offer a tractable approach for understanding nonreciprocal dynamics in simple active systems.

Abstract

Camphor is a well-studied material capable of generating self-propelled motion at a water surface, and the resulting dynamics can exhibit a wide range of behaviors. Here, we analyze a one-dimensional model describing a mobile camphor disk perturbed by a second localized camphor source. The interaction between the rotor and the perturbing disk is represented by a distance-dependent potential. The study is motivated by experiments in which a camphor rotor interacts with a fixed camphor disk placed on the water surface. Numerical simulations of the model reproduce the essential features of the experimentally observed position-dependent rotor velocity for all considered forms of the potential. For weak perturbations, we derive analytical solutions valid for arbitrary potential profiles. Both the simulations and the analytical results demonstrate a pronounced asymmetry in the rotor velocity depending on whether the rotor approaches or recedes from the perturbation.

Figures (7)

• Figure 1: Experiments illustrating a mobile arm (rotor) propelled by camphor and perturbed by a fixed camphor disk placed at a distance $R_f \cong 15~\mathrm{mm}$ from the axis. (a) Geometry of the setup. (b,c) Two overlapped snapshots of rotor positions around the perturbation. The blue arc or circle shows the trajectory of the black marker, and arrows indicate the direction of motion. The angle $\theta=0$ corresponds to the location of the fixed camphor disk. (b) Arm reflection for $R \cong 17~\mathrm{mm}$. The time difference between rotors below and above the perturbation is $1.14~\mathrm{s}$. The mobile camphor disk was reflected at the angles $\theta_1 \sim 0.30~\mathrm{rad}$ and $\theta_2 \sim -0.33~\mathrm{rad}$. (c) Continuous rotation for $R \cong 22~\mathrm{mm}$. The time difference between the lower and upper rotor positions is $0.4~\mathrm{s}$. The length of the thick trajectory segment indicates $x_c$ for the upper rotor position.
• Figure 2: Rotor velocity $v_c(x_c)$ as a function of the propelling disk position $x_c$. The green and blue points mark results for a rotor moving towards and away from the perturbation, respectively. (a) $15900$ points representing velocities measured during $318~\mathrm{s}$ (almost $130$ periods) long observation of rotor strongly perturbed by the fixed disk ($R \cong 17~\mathrm{mm}$). (b) $10500$ points representing velocities measured during $210~\mathrm{s}$ (over $100$ rotations) long observation of the rotor weakly perturbed by the fixed disk ($R \cong 22~\mathrm{mm}$). (c) Scaled difference between the local velocity $v_c(x_c)$ and its stationary value $(v_c(x_c)-V)/V$ calculated form the data shown in (b). Here $V= 70~\mathrm{mm/s}$. (d) as (c) but for perturbation of rotor rotating clockwise ($R = 21.4~\mathrm{mm}$). The data were collected over a duration of $90~\mathrm{s}$, during which the rotation period was approximately $1.5~\mathrm{s}$ and $V= 98~\mathrm{mm/s}$.
• Figure 3: Numerical simulation results of the steady state for a non-perturbed case, i.e. $U(x) \equiv 0$. (a) Stationary speed $v_s$ of the camphor disk as a function of $\eta$. (b-d) The steady-state surface camphor concentration profiles $u(x)$ for $\eta = 0.09$(b), $0.07$(c), and $0.05$(d). In all cases, the camphor disk was located at $x_c = 0$. For $\eta = 0.09$ the disk remained stationary at this position. For $\eta = 0.05$ and $0.07$ it was moving in the positive $x$ direction with constant speeds $V = v_s(\eta = 0.07) = 0.777$ and $v_s(\eta = 0.05) = 1.94$.
• Figure 4: Numerical simulation results on the effect of the potential amplitude on the camphor disk motion. Here, we adopted the Gaussian potential $\mathcal{U}_G(x)$ (Eq. \ref{['UG']}). (a,b) Time series of the camphor disk position $x_c$ for $U_0 = 0.005$(a) and $0.01$(b). Unidirectional motion and reciprocal motion were observed in (a) and (b), respectively. (c,d) Camphor disk velocity $v_c = d x_c/dt$ as a function of its location $x_c$. (c) and (d) correspond to (a) and (b), respectively. (e,f) Profiles of the concentration field $u(x)$ in the case of unidirectional and reciprocal motions corresponding to (c) and (d), respectively. The time difference between subsequent profiles is 3 time unit.
• Figure 5: Numerical simulation results with different forms of the potential and their amplitude. The camphor disk velocity $v_c = d x_c/dt$ is shown as a function of its location $x_c$. (a,b) Exponential potential $\mathcal{U}_e(x)$ (Eq. \ref{['UE']}). (c,d) Piecewise linear potential $\mathcal{U}_1(x)$ (Eq. \ref{['U1']}). (e,f) Piecewise quadratic potential $\mathcal{U}_2(x)$ (Eq. \ref{['U2']}). The unidirectional motions for $U_0 = 0.005$(a,c,e) and the reciprocal motions for $U_0 =0.01$(b,d,f) are shown.