Gradient dynamics model for chemically driven running drops

1. Problem Statement

The paper addresses the modeling of chemically driven running drops on solid substrates. Specifically, it focuses on scenarios where a drop contains a chemical species that can reversibly adsorb onto the substrate, altering its wettability.

• The Challenge: Previous models for such systems (e.g., Refs. [14, 15]) often relied on "one-sided" coupling assumptions where the chemical reaction kinetics and diffusion were not thermodynamically consistent with the fluid dynamics. These models assumed ideal mass-action kinetics unaffected by the local film thickness or pressure, leading to a lack of thermodynamic consistency.
• The Gap: While a thermodynamically consistent model for irreversible substrate coverage was proposed in Ref. [17], it was shown that such systems are inherently relaxational. In the irreversible case, drops eventually stop moving because they run into their own "trail" of altered wettability or because the system relaxes to equilibrium. There was no thermodynamically consistent framework to explain sustained, indefinite self-propulsion in the reversible case where the substrate can recover its original state.

2. Methodology

The authors employ a gradient dynamics framework, a formalism that ensures thermodynamic consistency by deriving evolution equations from a free energy functional and a dissipation potential.

• System Geometry: The model considers a sessile drop confined within a vertical gap of height $d$ between two parallel plates. This confinement is crucial for capturing the dynamics of the chemical species in the surrounding ambient fluid in a closed system.
• Chemical Species: Three species are modeled:
  • A: Species dissolved in the drop.
  • B: Species adsorbed on the substrate.
  • C: Species dissolved in the surrounding ambient medium.
  • Reactions: Two reversible monomolecular reactions occur: $A \rightleftharpoons B$ (adsorption under the drop) and $B \rightleftharpoons C$ (desorption into the ambient medium away from the drop).
• Free Energy Functional ($F$): The authors construct a free energy functional that includes:
  • Mesoscopic wetting energy (dependent on film height $h$ and adsorbate density $b$).
  • Osmotic contributions from species A, B, and C.
  • Interfacial tension and gradient penalties (stiffness) for the film height and concentration profiles.
• Gradient Dynamics Equations: The time evolution of the fields (film height $h$ and concentrations) is derived using the general gradient dynamics form:
  $$\partial_t \psi = \nabla \cdot \left( Q \nabla \frac{\delta F}{\delta \psi} \right) - M \frac{\delta F}{\delta \psi} + R$$
  where $Q$ and $M$ are mobility matrices, and $R$ represents reaction currents obeying local detailed balance.
• Chemostatting (Non-equilibrium Driving): To drive the system out of equilibrium, the authors treat the drop (Species A) and the ambient medium (Species C) as chemostats with fixed, distinct chemical potentials ($\mu_a \neq \mu_c$). This creates a continuous flux of particles: adsorption occurs under the drop (driven by $\mu_a$), and desorption occurs away from the drop (driven by $\mu_c$).
• Reduction: By assuming the drop and ambient medium act as infinite reservoirs, the authors derive a reduced model describing only the drop profile ($h$) and the adsorbate density on the substrate ($b$).

3. Key Contributions

1. Thermodynamic Consistency: The paper presents the first thermodynamically consistent gradient dynamics model for chemically driven running drops with reversible substrate coverage. It rigorously couples the hydrodynamics of the thin film with the reaction-diffusion dynamics of the chemical species.
2. Mechanism for Sustained Propulsion: The authors demonstrate that by imposing distinct chemical potentials ($\mu_a > \mu_c$), the system is driven far from equilibrium. This allows for sustained, indefinite self-propulsion, a phenomenon impossible in the purely relaxational irreversible case or in models with a single chemostat.
3. Bifurcation Analysis: The study identifies the emergence of self-propelled states via supercritical drift-pitchfork bifurcations. The transition from a resting drop to a moving drop occurs when the external chemical potentials cross a specific threshold line in the parameter space.
4. Critical Wettability Contrast: Unlike models with externally imposed wettability gradients (where motion occurs for arbitrarily small gradients), this self-organized system requires a finite critical wettability difference across the drop to initiate motion. This arises because the wettability profile is dynamic and coupled to the drop's motion.

4. Key Results

• Numerical Simulations: Using finite-element methods, the authors simulated the reduced model.
  • Relaxational Case ($\mu_a = \mu_c$): The system relaxes to a resting drop with a symmetric adsorbate profile.
  • Non-equilibrium Case ($\mu_a > \mu_c$): The system evolves into a state where the drop moves at a constant velocity. The adsorbate density ($b$) accumulates under the front of the drop and is depleted behind it, creating a self-sustained wettability gradient that drives the motion.
• Bifurcation Diagrams:
  • Varying $\mu_a$ and $\mu_c$ reveals a line of supercritical drift-pitchfork bifurcations. The drop velocity $v$ scales as $v \propto |\mu - \mu_{crit}|^{1/2}$ near the onset.
  • The onset of motion depends on a critical concentration difference $\Delta b_0$ between the region under the drop and the adsorption layer.
• Reaction Rate Dependence:
  • Self-propulsion only occurs within an intermediate range of reaction rates ($r_1, r_2$).
  • If rates are too slow, the gradient cannot be established. If too fast, the system saturates or becomes symmetric, halting motion.
  • The velocity exhibits a maximum at intermediate reaction rates.
• Diffusivity Effects: The nature of the bifurcation (supercritical vs. subcritical) depends on the adsorbate diffusivity ($D_b$). Lower diffusivity can lead to subcritical bifurcations and multistability (coexistence of resting and moving states).

5. Significance

• Theoretical Advancement: This work bridges the gap between non-equilibrium thermodynamics and thin-film hydrodynamics. It provides a rigorous framework for studying active matter systems where chemical reactions drive mechanical motion, correcting the thermodynamic inconsistencies of previous phenomenological models.
• Experimental Relevance: The model explains how chemically driven drops can achieve sustained motion without external fields, provided there is a source (drop) and a sink (ambient medium) with different chemical potentials. This is relevant for understanding biological condensates, catalytic droplets, and self-propelled micro-swimmers.
• Predictive Power: The identification of critical wettability contrasts and the dependence on reaction rates offers specific, testable predictions for experimentalists designing systems for controlled drop transport.
• Future Directions: The authors suggest extending the model to include Marangoni effects, liquid substrates, and more complex reaction schemes (e.g., cooperative binding), opening avenues for studying complex active hydrodynamic patterns.

In summary, the paper establishes that reversible substrate coverage combined with distinct chemical potentials in the drop and ambient medium is the key to achieving thermodynamically consistent, sustained self-propulsion of liquid drops, governed by a specific bifurcation structure.