Hydrodynamic theory of wetting by active particles

TL;DR

This work develops a deterministic hydrodynamic theory for active wetting by self-propelled particles, modeled as an active lattice gas interacting with a smooth permeable barrier under periodic boundary conditions. By taking a hydrodynamic limit that suppresses fluctuations, the authors obtain exact mean-field equations and demonstrate that active wetting exhibits equilibrium-like critical wetting features, including a logarithmically diverging wetting film width and a square-root scaling of the asymmetry, while also revealing intrinsically nonequilibrium effects such as a spontaneous ratchet current across the barrier. The surface phase diagram reveals four steady states (fully-wet, partially-wet, unpinned, homogeneous) and a critical wetting transition at a finite barrier height $\epsilon^*$ that strengthens with activity, $\text{Pe}$. The nonequilibrium phenomena—notably the ratchet current, bulk-density deviations from binodal values, and a novel dynamical pathway between wetting states—underscore how activity modifies surface phenomena beyond equilibrium analogies. The framework connects active wetting to passive wetting in a precise, mean-field sense, and points to extensions to higher dimensions, the inclusion of noise, and experimental realizations with optically controlled colloids as promising directions.

Abstract

The accumulation of self-propelled particles on repulsive barriers is a widely observed feature in active matter. Despite being implicated in a broad range of biological processes, from biofilm formation to cytoskeletal movement, wetting of surfaces by active particles remains poorly understood. In this work, we study this active wetting by considering a model comprising an active lattice gas, interacting with a permeable barrier under periodic boundary conditions, for which an exact hydrodynamic description is possible. We consider a hydrodynamic scaling limit that eliminates dynamical noise while retaining microscopic interpretability, enabling a precise characterisation of steady-states and their transitions. We demonstrate that the accumulation of active particles has remarkable similarities to equilibrium wetting, and that active wetting transitions display all the salient characteristics of the equilibrium critical wetting transition -- despite fundamental differences in underlying microscopic dynamics. However, our framework also enables the investigation of subtle but important nonequilibrium effects in active wetting, including a spontaneous ratchet effect which leads to a global steady-state current, departure of bulk densities from their binodal values, and a novel dynamical transition pathway. Our results provide an intrinsically nonequilibrium framework in which to study active wetting, precisely demonstrating the connection to passive wetting while clarifying the nonequilibrium consequences of activity.

Figures (7)

• Figure 1: (a) Diagram of equilibrium wetting on a solid surface. A liquid droplet rests on the solid in coexistence with the vapour phase, forming a contact angle $\theta_w$. The interfacial tensions $\gamma_{sl}$, $\gamma_{sv}$, and $\gamma_{lv}$ correspond to solid–liquid, solid–vapour, and liquid–vapour interfaces, respectively. (b) Schematic of wetting states in a slit geometry with a fixed mass of fluid. (b.1) Fully wet: macroscopic liquid films coat both walls. (b.2) Partially wet: a macroscopic liquid film forms on one side. (b.2) Dry: no liquid adsorption at the walls.
• Figure 2: Phase diagram of the active lattice gas. The binodal (coexistence curve) is obtained by solving \ref{['eq:ABP_evolution']}-\ref{['eq:J_act']} numerically, and characterising the phase-separated steady-states. The spinodal is obtained from Mason2023, it is compared with numerical stability results for \ref{['eq:ABP_evolution']}-\ref{['eq:J_act']} starting from weakly perturbed homogeneous states (coloured points, no data shown for $\bar{\rho} < 0.4$). The critical point where the spinodal and binodal curves intersect is ($\bar{\rho}_\text{crit}, \text{Pe}_\text{crit}) = (0.659, 4.33)$. As is usual for active phase separation Cates2025_APS, these binodals (dashed curves) do not depend on the value $\bar{\rho}$ so long as it lies between them. See Appendix \ref{['sec:binodal_spinodal_convergence']} for further details.
• Figure 3: Schematic illustration of the system geometry. The system occupies a two-dimensional periodic domain, with a smooth permeable barrier centred at $x=0$. The barrier is represented by an external potential $V(x)$ of finite width $w_b$ and height $\epsilon$, which locally repels particles while allowing finite probability of crossing. Active particles move with self-propulsion $v_0$ along the $\vb{e}_\theta$-direction and interact only via excluded volume. The orientation-dependent probability density $f(\vb{x}, \theta, t)$ is translationally invariant in the transverse direction(s), so the wetting behaviour is fully characterised by the one-dimensional profile shown here.
• Figure 4: Wetting of a barrier by the active lattice gas. (a) Surface phase diagram in the $(\epsilon,\text{Pe})$ plane. Each point represents a numerical solution (simulation) of the hydrodynamic equation, whose final state is classified with the routine described in Section \ref{['subsec:numerical_implementation']}, which also explains the estimation of the phase boundaries. The blue line is the onset of MIPS; the green line is the wetting transition; the dark purple line likewise represents unpinning of the liquid at $\epsilon = 0$. The structure of this surface phase diagram does not depend on $\bar{\rho}$, as long as it remains inside the spinodal. See Section \ref{['subsec:numerical_implementation']} for details of numerics. (b-e) Snapshots of the steady-state densities $f(x, \theta)$ and $\rho(x)$ for the starred points shown in the phase diagram, which illustrate the four phases. The upper plots show the density profile $\rho(x)$, and the potential $V(x)$; the lower plots show $f(x, \theta)$ (colour) and ${\cal J}(x, \theta)$ (arrows -- scaled proportionally to $\mathcal{J}^{1/4}$). The magnified section of (c) shows a linear density gradient throughout the liquid bulk, in the partially-wet state; it shows $x\in [0,6]$.
• Figure 5: The wetting transition between fully- and partially-wet states is critical. All plots are at $\text{Pe} = 10$. (a) The width of the liquid film on the "dry" wall, estimated by measuring the distance between the inflection points of $\rho$ at the liquid-boundary and liquid-vapour interfaces. This increases as the transition is approached. Results are shown for a system of size $\ell_s=20$, so maximum width of (symmetric) liquid layers is approximately $6.3$ (using the lever rule). For an infinite system we expect $\zeta$ to diverge at $\epsilon = \epsilon^*$. (b) Asymmetry $\mathcal{A}$ as a function of $\epsilon$, scanning both up (blue) and down (orange), to check for hysteresis. The total scan time is $T_\text{tot} = 2 \times 10^5$; we use a system size $\ell_s = 5$, smaller than in (a), such that the scan is quasi-static and the behaviour of $\mathcal{A}$ is not too sharp (see Appendix \ref{['app:wetting']}). The dashed grey line is a fit to \ref{['eq:Aeq-sqrt']}. We estimate $\epsilon^* \approx 6.01$. The hysteresis loop is observed to shrink as the scan time is increased. (c) Linear stability of the fully-wet state with the same parameters. We plot the exponential growth rate of the asymmetry $\dot{\mathcal{A}}/\mathcal{A}$ as a function of barrier height $\epsilon$. Positive values correspond to partial wetting (the fully-wet state is unstable), and we estimate $\epsilon^* \simeq 6.01$ from a linear fit. The growth rate is measured at $t=50$, when asymmetry growth is approximately linear and $\mathcal{A}$ remains small.