Simulating droplet adhesion on superhydrophobic surfaces

TL;DR

This work presents a Surface Evolver–based numerical framework to simulate droplet probe microscopy on pillared superhydrophobic surfaces, capturing adhesion, compression, and detachment through four stages: snap-in, advance, recede, and detach. The model reproduces the characteristic sawtooth variation in the adhesion force, including a maximum before detachment, and demonstrates good agreement with non-evaporating experimental data while revealing evaporation-related shifts due to weight loss. By incorporating Cassie-Baxter wetting and energy dissipation during contact line jumps, the approach provides quantitative insight into CL dynamics, morphologies, and force responses, enabling surface characterization and design of hydrophobic textures. The framework is extensible to different pillar geometries, area fractions, and evaporation scenarios, offering a versatile tool for predicting droplet–surface interactions in force-probe and AFM–scale experiments.

Abstract

A numerical model is proposed to simulate the adhesion, compression, and subsequent detachment of a micro-liter droplet from a superhydrophobic surface composed of chemically homogeneous pillars arranged in a periodic fashion, replicating a typical force probe microscopy experiment. We observe that as the droplet is pulled away from the surface, the net vertical force varies in a typical sawtooth manner with peculiar peaks and troughs, characteristic of the surface. The force first reaches a maximum before the droplet detaches from the surface with a comparatively lower force. The force variation predicted by the numerical model is in good agreement with the experimental results of Kumar et al. [1]. We also studied the effect of evaporation on the variation in the adhesion force by simulating an evaporating droplet on a superhydrophobic surface. For an evaporating droplet, the numerically predicted maximum and detachment force magnitudes are in good agreement with those obtained experimentally when we take into account the change in the droplet weight as it evaporates. The proposed method will be useful for the quantitative analysis and design of a variety of superhydrophobic surfaces and will pave the way for more accurate surface characterization based on droplet adhesion force measurements.

Figures (8)

• Figure 1: (a) Pictorial representation of a typical droplet probe microscopy simulation. The method is depicted in 4 key steps (approach stage is not shown here): (1) Snap-in - the droplet contacts the surface, (2) Advance - the droplet is pushed against the surface by lowering the disk, (3) Recede - the droplet is pulled away from the surface by moving the disk in the opposite direction, and (4) Detach - the droplet detaches from the surface. (b) Schematic of the simulation domain. Due to symmetry, only one-eighth of the full droplet is simulated to reduce the computational time. The pillars in contact with the droplet are shown in yellow, the pillars from which the CL has depinned are shown in magenta and the pillars untouched by the liquid are shown in gray/white colors. Symmetry boundary conditions is applied on the left and right walls of the domain. Young's angle boundary condition is applied at the contact line. In equilibrium, the droplet maintains the condition of Young's angle boundary condition at the CL and the liquid-air interface being a surface of constant mean curvature ($\kappa_1+\kappa_2$ being constant).
• Figure 2: Front and bottom view of the equilibrium droplet morphologies during the advance and recede stages obtained via numerical simulation on a superhydrophobic surface with $\theta_{\rm{A}}=111.2$°, $\theta_{\rm{R}}=98.7$° and $\phi=0.05$ and a droplet of volume $V=1.6$$\mu$L. In the bottom view of the droplet the red, magenta and white colors are used to indicate the pillar tops in contact with the liquid, pillar tops where the CL has depinned and the pillar tops which were not in contact with the liquid at any point of time during the entire process, respectively. The high curvature of the liquid-air interface lying between the pillars can be seen in the bottom view of the equilibrium morphologies during the recede stage.
• Figure 3: Depiction of the dynamics of the interface during the advance and recede stages respectively for a 1.6 $\mu$L droplet on a superhydrophobic surface with $\theta_{\rm{A}}=111.2$°, $\theta_{\rm{R}}=98.7$° and $\phi=0.05$. (a) Equilibrium interfacial morphologies showing how the interface advances. The CL stays pinned on a set of pillars while the liquid-air interface curves forward until it contacts the next set of pillars, thereby advancing the CL. The equilibrium interfacial morphologies on the $x-z$ plane is shown in (b). The pinned states are shown by dashed-dotted lines while the equilibrium states just before and after the advancement of the CL are shown by the solid black and solid red lines respectively. (c) Shows the apparent CL measured at a distance of 1 $\mu$m from the pillar tops. The CL starts as a circle ($t_0$) and eventually acquires a shape resembling that of an octagon as it advances (eg. $t_1 \to t_3$. ). During the recede stage the CL moves in its characteristic stick-slip motion, executing jumps between the first and the second critical states shown by solid back and red lines, respectively, in (d) which shows the equilibrium interfacial morphologies on the $x-z$ plane. The corresponding CL morphologies (at $\Delta z=1$$\mu$m) are shown in (e). (f) Equilibrium interfacial morphologies corresponding to the critical states ii, iii and iv in (d). The yellow colored pillars indicated pinned pillars, magenta represents the depinned pillars, and the white color represents the pillars which were never contacted by the liquid.
• Figure 4: Variation in the net vertical force $F$ with time $t$ (in simulation units), between a droplet and a superhydrophobic surface during a typical droplet probe microscopy simulation with maximum and detachment forces shown by empty and filled circles respectively. The force variation for the recede stage is shown for both the moving disk (in red) and evaporation (in green) modes. The simulation time for the evaporation mode is scaled down to fit in the same plot. Inset shows the sharp drop in the force during snap-in and two CL advancement moments. The maximum and detachment forces obtained by each approach are comparable. The force curves were generated for a 1.6 $\mu$L droplet on a surface with $\theta_{\rm{A}}=111.2$°, $\theta_{\rm{R}}=98.7$° and $\phi=0.05$.
• Figure 5: Variation in the net vertical force ($F$, in micronewtons) with time ($t$, in seconds) measured experimentally and calculated via energy minimization simulations. For both the moving disk and evaporation modes, simulation time is scaled linearly to match the initial slope of the experimental force curve during the recede stage. $F$ vs $t$ for a droplet of volume $V=1.6$$\mu$L and a superhydrophobic surface with $\theta_{\rm{A}}=111.2$°, $\theta_{\rm{R}}=98.7$° and pillar area fractions $\phi=0.05$ and $\phi=0.30$ are shown in (b) and (c) respectively. The force variation with an additional term for the loss in droplet weight due to evaporation (i.e. $F+\Delta W$) is also plotted in green in (b) and (c). (d) $F$ vs $t$ variation from liimatainen2017mappingliimatainen2017mapping (in black) and simulation results for the evaporation mode considering change in droplet weight (in green) and without considering change in droplet weight (in red) due to evaporation. The droplet has an initial volume of $V=1.5$$\mu$L and the surface has $\theta_{\rm{A}}=111.0$°, $\theta_{\rm{R}}=85.1$° and $\phi=0.016$.