Gradient dynamics model for chemically driven running drops

TL;DR

The paper develops a thermodynamically consistent gradient-dynamics framework for chemically driven running drops with reversible substrate adsorption. Starting from a closed N-field gradient-dynamics description of a drop height and multi-species concentrations, the authors chemostat species in the drop and ambient fluid to obtain a reduced open-system model in which sustained propulsion emerges from a nonequilibrium wettability contrast driven by external chemical potentials μ_a and μ_c. Self-propulsion arises when μ_a ≠ μ_c and is organized by drift-pitchfork bifurcations, linking propulsion to the competition between adsorption and desorption dynamics of the substrate species. Numerical simulations on a 1D domain illustrate rest versus run states and map how the propulsion strength depends on reaction rates, adsorbate diffusion, and chemostat values, highlighting the thermodynamic consistency and control afforded by the chemostatting approach.

Abstract

We present a thermodynamically consistent model for chemically driven running drops on a solid substrate with reversible substrate adsorption of a wettability-changing chemical species. We consider drops confined to a vertical gap, thereby allowing us to first obtain a gradient dynamics description of the closed system, corresponding to a set of coupled dynamical equations for the drop profile and the chemical concentration profiles of species on the substrate and in both fluids (drop, ambient medium). Chemostatting the species in the drop and the ambient medium, we then derive a reduced model for the dynamics of the drop and the adsorbate on the substrate. When the externally imposed chemical potentials are distinct, the system is driven away from thermodynamic equilibrium, allowing for sustained drop self-propulsion across the substrate due to a wettability contrast maintained by chemical reactions. We numerically study the resulting running drops and show how they emerge from drift-pitchfork bifurcations.

Figures (6)

• Figure 1: Sketch of a chemically driven running drop confined to a vertical gap of height $d$. The local film height is denoted by $h(\vec{x}, t)$. The concentration of particles in the drop, on the substrate and in the surrounding medium are given by $a(\vec{x}, t), b(\vec{x}, t)$ and $c(\vec{x}, t)$ (respectively yellow, red and green). Particles may adsorb from the drop onto the substrate or dissolve from the substrate into the surrounding medium with the reaction rates $r_1$ and $r_2$, respectively. Particles that adsorb onto the substrate render it less wettable, thereby causing the drop to move along the wettability gradient with velocity $v$
• Figure 2: Comparison of (a, c) relaxational dynamics converging to a resting drop and (b, d) persistent out-of-equilibrium dynamics converging to a self-propelled drop. Panels (a) and (b) [(c) and (d)] show space-time representations of the drop profile [adsorbate profile] for an initially symmetric drop with a small adsorbate concentration gradient (i.e., a wettability gradient) underneath the drop. In the passive relaxational case, the initial concentration gradient is equilibrated and the overall adsorbate concentration gradually increases. In the contact line regions, interfaces are formed in the concentration profile as the system relaxes to thermodynamic equilibrium, where the drop is at rest. In the persistently out-of-equilibrium case, the initial concentration gradient beneath the drop is amplified while adsorbate is continuously removed from the surrounding substrate. Ultimately, drops self-propel across the substrate with constant velocity. The external chemical potentials are (a, c) $\mu_a=\mu_c=\ln 6$ and (b, d) $\mu_a=\ln 6, \mu_c=\ln 0.5$. The remaining parameters are $W=1, D_b=0.001, r_1=0.001, r_2=0.001, h_0 = 2.4, \Delta h = 0.6,\sigma_b= 1, \lambda = 0.5$. The periodic computational domain is [0, 2000]
• Figure 3: Snapshots of the adsorbate profile at different times in the persistently out-of-equilibrium case in Fig. \ref{['fig:passive_active']}. Panel (a) corresponds to the initial condition. As time progresses, the initially small concentration gradient beneath the drop is amplified [panels (b) and (c)] and develops into a profile that increases across the drop and then decays behind it [panel (d)]. In (d), the profile corresponds to stable uniform drop motion
• Figure 4: Emergence of drop self-propulsion in dependence of the external chemical potentials $\mu_a$ and $\mu_c$ that characterize the chemostats. (a) Parameter scan of time simulations in the $(e^{\mu_a}, e^{\mu_c})$-plane. Colors indicate the drop velocity in the converged state. The passive case $\mu_a=\mu_b$ is represented as a dotted line. Self-propelled drops emerge at a line of supercritical drift-pitchfork bifurcations (DP, white solid line). Panels (b) and (c) respectively show the velocity $v$ in dependence of $e^{\mu_a}$ and $e^{\mu_c}$ near the bifurcation, i.e., for horizontal and vertical cuts in (a) (visualized as white arrows). The drift-pitchfork bifurcations are marked with black filled circles. The remaining parameters and the initial condition are as in Fig. \ref{['fig:passive_active']}
• Figure 5: Concentration profiles of stationary [(a), (g), (f), (l)] and self-propelled drops [(b)-(e), (h)-(k)] in dependence of the reaction rates $r_1$ and $r_2$. Arrows indicate moving drops. In the left [right] column, the reaction rate $r_1$ [$r_2$] increases from top to bottom. As $r_1$ is increased at fixed $r_2=1.5\times 10^{-3}$, the overall concentration beneath the drop is increased until a saturation value is reached. This corresponds to a plateau in $b$ across the drop, see (e). When $r_2$ is increased at fixed $r_1=1.5\times 10^{-3}$, the adsorbate concentration on the substrate away from the drop is reduced, reaching a lower plateau value, see (j) and (k). When values of either $r_1$ or $r_2$ are too small [(a), (g)] or too large [(f), (l)], drops are stationary. The remaining parameters are as in the self-propelled case in Fig. \ref{['fig:passive_active']}, except for $D_b=0.1$ and $\Delta h=0.35$ in (a)-(f)