Migration and spreading of a droplet driven by a chemical step

TL;DR

This work analyzes droplet motion on a horizontal substrate featuring a sharp chemical step, using lubrication theory and Navier slip to resolve contact-line singularities. It uncovers a two-stage dynamics common to both 2D and 3D droplets: first a migration stage across the border driven by wettability contrast, then an asymmetric spreading stage on the hydrophilic side with a pinned border that regularizes curvature through slip. In 2D, matched asymptotics yield explicit relations for the migration velocity $\delta$ and length $L$, with an equilibrium length $L_{eq}=2/\sqrt{K}$; for 3D, simulations show similar qualitative behavior but significant lateral-flow effects, such as $W\sim t^{1/2}$ growth in width and non-monotonic $L(t)$. The findings illuminate how chemical steps and border pinning govern transient and final droplet morphologies, with implications for designing wettability-patterned surfaces and microfluidic control of sessile droplets.

Abstract

The chemical step is an elementary pattern in chemically heterogeneous substrates, featuring two regions of different wettability separated by a sharp border. Within the framework of lubrication theory, we investigate droplet motion and the contact-line dynamics driven by a chemical step, with the contact-line singularity addressed by the Navier slip condition. For both two-dimensional (2D) and three-dimensional (3D) droplets, two successive stages are identified: the migration stage, when the droplet traverses both regions, and the asymmetric spreading stage, when the droplet spreads on the hydrophilic region while being constrained by the border. For 2D droplets, we present a matched asymptotic analysis which agrees with numerical solutions. In the migration stage, a 2D droplet can exhibit translational motion with a constant speed. In the asymmetric spreading stage, the contact line at the droplet rear is pinned at the border. We show that a boundary layer still exists near the pinned contact line, across which the slope is approximately constant, whereas the curvature would diverge in the absence of slip. For 3D droplets, our numerical simulations show that the evolution is qualitatively analogous to the 2D case, while being significantly affected by the lateral flow. At early times, the contact line on the hydrophilic region advances linearly and spreads transversely according to a power law $t^{1/2}$. The droplet length and width exhibit non-monotonic variations due to the lateral flow. Eventually, the droplet detaches from the border and reaches equilibrium at the hydrophilic substrate.

Figures (9)

• Figure 1: Schematic of a droplet driven by a chemical step. The chemical step is jointed by two homogeneous substrate. The left-right equilibrium contact angles $\theta_1$ and $\theta_2$ satisfy $\theta_1>\theta_2$, representing a hydrophilic substrate on the right-hand side of a hydrophobic one.
• Figure 2: Evolution of a 2D droplet on a chemical step with $K=0.5$. (a) Evolution of droplet profile. (b) The droplet length $L$ as a function of time. At $t\approx 155$, the receding contact line reaches the wettability border at $x=0$, signifying the end of the migration stage and the onset of the asymmetric spreading stage. After a rapid elongation, $L$ maintains constant during the migration stage. Then $L$ increases towards the equilibrium length $L_{eq}$ on the hydrophilic substrate during the asymmetric spreading stage.
• Figure 3: Comparison of the theoretical and numerical results during the steady migration stage. Variation of (a) the droplet length $L$ and (b) the migration velocity $\delta$ as a function of $K$. The finite-$K$ theory corresponds to \ref{['finKL']} and \ref{['finKDel']}; the small-$K$ theory corresponds to \ref{['finKDel']} and \ref{['smaKmatch']}. (c) Migration velocity $\delta$ as a function to $L$ with the theoretical curve given by \ref{['finKDel']}.
• Figure 4: Temporal evolution of $L$ converging towards equilibrium during the asymmetric spreading stage. The theory is validated by the numerical results for $K\gtrapprox0.2$ and $K=0$.
• Figure 5: Microscopic properties of the inner region near the pinned contact line for $K=0.5$. (a) Evolution of the contact angle $\partial_xh(0,t)$ with the theoretical curve given by \ref{['thetaTheo']}. (b) Snapshot of the surface curvature $\partial^2_xh(x,t)$ at $t=50$. The outer solution represents the second-order derivative of \ref{['eq:hpin']} together with \ref{['sprh0']} and \ref{['sprh1']}; the inner solution corresponds to \ref{['eq:curvature']}.