Water droplet friction and rolling dynamics on superhydrophobic surfaces

The study addresses how to accurately quantify the friction (slipperiness) of water droplets on superhydrophobic surfaces as these surfaces approach the extreme non-wetting limit. Conventional contact angle goniometry infers friction from advancing and receding contact angles but becomes unreliable on highly slippery and superhydrophobic surfaces due to small hysteresis and optical distortions near the contact region. The authors propose direct force measurements using micropipette force sensors to overcome these limitations, enabling precise quantification of very low friction forces and allowing investigation of internal droplet dynamics (rolling versus sliding) on superhydrophobic substrates.

Prior approaches include tilt-stage and oscillating-droplet tribometry, and cantilever-based methods using glass capillaries or polymer tubes to measure droplet friction on hydrophobic, SOCAL, and lubricated surfaces. However, previous cantilevers are wide relative to droplet size, perturbing droplet shape and internal flow, hindering detailed fluid dynamics studies. Particle image velocimetry studies on tilted planes reported water droplets transitioning from roll-slip on hydrophobic to pure sliding on superhydrophobic surfaces at high velocities. Contact angle goniometry suffers from large errors for high contact angles and small hysteresis, limiting its utility for next-generation super-slippery surfaces.

Micropipette force sensors (MFS) were fabricated by pulling glass capillaries (0.75/1 mm i.d./o.d.) using a puller and cutting to 1.9–2.5 cm cantilever length with a microforge. Pipettes with inner/outer tip radii ~15/20 µm were water-filled and calibrated by correlating their horizontal deflection to known droplet weights dispensed from the pipette tip, yielding spring constants k = 2.5–40 nN µm⁻¹ with ~1–3% relative error. Friction experiments: A water droplet was dispensed from the vertically mounted MFS onto the superhydrophobic sample on a motorized xyz stage. After recording a zero-force baseline (~5 s), the substrate was translated at constant speed (typically 0.1 mm s⁻¹; acceleration 4 mm s⁻²) while imaging at 50 fps. The kinetic friction force F was taken as the difference between the average baseline and the steady kinetic plateau from the pipette deflection (F = k Δx). Contact region diameter D and advancing/receding angles during the kinetic regime were analyzed with custom Matlab code. Static friction bumps were observed on some samples but excluded from kinetic analyses. Superhydrophobic model substrates: Five etched silicon surfaces (A–E) were produced via maskless cryogenic deep reactive ion etching with varying SF6/O2 flows and forward powers to tune topography from sparse spikes (A) to dense grass (E), followed by PECVD fluoropolymer coating. Surface characterization included SEM (top and tilted views), AFM (10 µm scans), and confocal microscopy to assess contact and estimate solid fraction (φs ≈ 0.06 ± 0.03 for spikes A; 0.47 ± 0.05 for grass E). Contact angle goniometry provided advancing/receding angles for comparison. PIV experiments: Water droplets seeded with 5 µm polystyrene tracers were used to visualize internal flow while simultaneously measuring friction with a stiffer pipette (k ≈ 40 nN µm⁻¹). Substrate speeds v = 0.1–1.9 mm s⁻¹ and droplet radii R = 0.8–1.6 mm were tested; PIVLab was used to extract average angular velocity ω. The droplet outline edges were excluded due to opacity. Data analysis included comparison to the theoretical lateral adhesion force from contact angles (FLA = 2γD(cos θr − cos θa) × 12, i.e., Eq. (1) as given), linear fits of F vs D, relative error propagation, and scaling analysis for rolling regime dissipation (Pη ~ η φs l ω² with l = D/2, balanced by Pc ~ F v).

• MFS achieved direct friction force resolution as low as ~4 nN, about 2–25× better than prior cantilever-based methods and up to three orders of magnitude more precise than friction inferred from contact angle goniometry on very slippery surfaces. • On the most slippery etched silicon spikes surface (sample A), a millimetric water droplet (weight ~10 µN) experienced a kinetic friction force of 7 ± 4 nN. The corresponding dimensionless friction was Fμ/(γ D) = (4 ± 3) × 10⁻⁴, lower than previously reported for micropillars, SOCAL, silicone nanofilaments, lubricated surfaces, and underwater-SOCAL benchmarks. • Friction scaled linearly with contact region diameter D across all samples. Linear fit slopes Fμ/D were: E (grass) 2.7 ± 0.4 nN µm⁻¹, D 1.6 ± 0.3 nN µm⁻¹, C 0.6 ± 0.2 nN µm⁻¹, B 0.11 ± 0.04 nN µm⁻¹, A (spikes) 0.03 ± 0.02 nN µm⁻¹. • The experimentally measured friction forces agreed quantitatively with the theoretical lateral adhesion force computed from measured advancing/receding angles using Eq. (1), but the relative error of CAG-derived forces (δF_CAG) exceeded that of MFS (δF_MFS) by factors ranging from ~10 up to ~1000 as surfaces became more slippery. • PIV revealed that slowly moving water droplets on superhydrophobic surfaces roll without slip, with angular velocity ω = v/R. A transition to roll-slip occurred beyond a critical ωc, after which ω saturated while slip speed increased. Critical values were ωc = 0.79 ± 0.15 s⁻¹ (spikes A) and 1.3 ± 0.3 s⁻¹ (grass E). • In the pure rolling regime (v/R < ωc), energy dissipation is dominated by viscous losses near the liquid-solid interface and balances contact-line dissipation, leading to the scaling ω ~ (F v / (η φs l))^(1/2). Experimental data collapsed onto a line of slope 1/2 in a log-log plot of ω versus F v/(η φs l), consistent with the scaling. • The ratio of translational to rotational kinetic energy indicated a transition to roll-slip at Et/Er ≈ 4 on both surfaces, with rotational energy exceeding translational for v/R < 0.25 ± 0.10 s⁻¹.

Direct force measurement with micropipette force sensors overcomes the accuracy limitations of contact angle goniometry for characterizing superhydrophobic coatings approaching the extreme non-wetting limit. The precise, minimally invasive MFS enables quantification of ultralow friction forces and reveals that water droplets can exhibit pure rolling on superhydrophobic surfaces at low v/R, contradicting earlier high-speed inclined-plane observations of dominant sliding. The linear F vs D scaling and agreement with Eq. (1) validate the friction mechanism, while the superior precision of MFS demonstrates that contact angle-based inference is inadequate for distinguishing highly slippery surfaces. The observed rolling-to-roll-slip transition, its dependence on surface solid fraction, and the ω scaling with F, viscosity, and contact geometry provide mechanistic insight into dissipation pathways and the conditions under which slip emerges. These findings establish the dimensionless friction F/(γ D) as a robust benchmark for slipperiness and offer a path to rational design and comparison of next-generation liquid-repellent surfaces.

Micropipette force sensors enable direct, highly sensitive quantification of droplet friction on superhydrophobic surfaces, outperforming contact angle goniometry by up to three orders of magnitude in precision on super-slippery samples. Etched silicon spikes achieved a record-low dimensionless friction F/(γ D) ≈ (4 ± 3) × 10⁻⁴, corresponding to 7 ± 4 nN for a millimetric water droplet, surpassing state-of-the-art liquid-like and lubricated coatings. Combining MFS with PIV uncovered a previously unexplored regime in which water droplets roll without slip, transitioning to roll-slip above a critical angular velocity, with a scaling law linking rotation to frictional dissipation and interfacial properties. Future work could extend these measurements to broader material classes (e.g., diverse textures, chemistries, and lubricant-infused systems), explore wider speed and size ranges, quantify static friction with higher resolution, and leverage the dimensionless friction benchmark to guide the engineering of ultra-low-friction and durable liquid-repellent surfaces.

The MFS approach approaches a lower detection limit for very small droplets and on the most slippery samples, increasing relative errors; some static friction events were likely below the resolution and thus not fully captured. PIV measurements excluded non-transparent droplet edge regions, potentially missing near-contact-line flow details. Solid fraction estimation on irregular etched structures carries uncertainty due to image thresholding and resolution limits. Experiments focused on relatively low velocities compared to inclined-plane studies, and findings may not directly extrapolate to high-speed, inertia-dominated regimes.