Probing wetting properties with self-propelled droplets

TL;DR

This paper addresses wetting metastability on pillared substrates by embedding active matter concepts into a 3-state Cellular Potts Model. By introducing an activity term that yields self-propelled, memory-guided droplet motion, the authors show that increasing activity allows the droplet to overcome free-energy barriers and explore multiple metastable states predicted by passive landscapes. A threshold activity $\mu_t$ marks the transition to a unique dry state, with $\mu_t$ decreasing as roughness increases, providing a quantitative metastability metric. Wetting observables such as $\langle \theta_C \rangle$ and $\langle f \rangle$ quantify state accessibility, while mean squared displacement (MSD) offers complementary dynamical insight though is not a standalone metastability probe. Overall, the work proposes a scalable framework for probing metastability and guiding substrate design, with potential experimental validation using active droplets.

Abstract

Wetting phenomena are relevant in several technological applications, particularly those involving hydrophobic or hydrophilic surfaces. Many substrates support multiple wetting states depending on surface conditions or droplet history, a behavior known as metastability. This feature is crucial both for its theoretical complexity and for its relevance in practical applications that rely on controlling metastable states. While several experimental and computational techniques have been developed to study metastability, they tend to be complex or computationally expensive. In this work, we introduce an alternative approach based on concepts from active matter physics. We investigate the wetting behavior of a droplet placed on a pillared surface using a 3-state Cellular Potts model with a polarity term that mimics a self-propelled droplet. Applying this model to a pillared substrate with known metastable wetting states, we demonstrate that increasing activity enables the droplet to traverse free energy barriers, explore consecutive metastable states, and eventually suppress metastability entirely. Our results show that activity reduces the disparity between dry and wet states and provides a reliable framework for identifying and quantifying metastability through contact angle measurements.

Figures (6)

• Figure 1: (a) Definition of the geometric parameters of the pillared surface: roughness ratio $r=1+4hw/(a+w)^2$, where $a$ is the inter-pillar distance, $w$ is the pillar width and $h$ is the pillar height. (b) Schematic representation of the free energy $\mathcal{F}$ as a function of the water fraction penetrating the substrate $f$, based on results from two distinct pillared surfaces in Ref. AMI_Marion2021. The light-green curve, which exhibits a single minimum, corresponds to a surface with a high roughness ratio $r$, whereas the dark-green curve, displaying multiple local minima, represents substrates with lower $r$. (c) Cross-sectional representations of the water droplet at each minimum of $\mathcal{F}$, arranged from the driest to the wettest state (I–IV) with its respective contact angle (shown in red)).
• Figure 2: (a) Schematic cross-section of a 3D droplet over a pillared substrate in the Cellular Potts Model (CPM). Each pixel corresponds to a different state ($s_i=0, 1, 2$). (b) Representation of the activity term, Eq. \ref{['act']}. The displacement of the droplet’s center of mass ($\vec{P}$), in orange, is measured at intervals of $\Delta t_P$ (in Monte Carlo Steps, MCS). The vector $\vec{c}$(red) represents an effective displacement of a water pixel at the interface (see text for more details).
• Figure 3: Two initial wetting configurations used in the Monte Carlo simulations: one in the dry state, refereed as D$^0$, with $\theta_c = 180$°(a); and one in the wet state, W$^0$where $\theta_c = 90$°(b).
• Figure 4: Wetting properties for three roughness values $r$. Top row: most frequent contact angle $\langle \theta_c \rangle$ as a function of activity $\mu$. Error bars represent the standard deviation around the mean value $\langle \theta_C \rangle$. Middle row: cross-sections of the droplet for different values of $\mu$. Bottom row: Fraction of water penetrating the substrate, $\langle f \rangle$, as a function of $\mu$, with error bars representing the standard deviation around $\langle f \rangle$. All plots are shown for two initial conditions: D$^0$ and W$^0$. The differences in $\langle \theta_C \rangle$ and $\langle f \rangle$ between these initial conditions indicate that substrates with $r=2.18$ and $r=1.78$ exhibit metastability, whereas $r=3$ corresponds to a surface with a single energy minimum. Horizontal dotted lines indicate the values of $\theta_C$ associated with the free energy minima reported in Ref. AMI_Marion2021.
• Figure 5: Threshold value $\mu_t$ is shown as a function of the substrate’s geometric parameters. Horizontal (bottom) represents the roughness ratio $r$, while the top axis indicates corespondent interpillar distance $a$. When $\mu_t = 0$, the substrate exhibits a single energy minimum. For $\mu_t > 0$, multiple minima emerge, signaling a transition from a non-metastable substrate to one that supports metastable states. Circles indicate actual data points, while the orange cross represents a prediction based on a linear fit given by $\mu_t=-13.1~r+34.7$ and shown as a dashed line.