Femtosecond-laser writing of silicon produces laser-induced periodic surface structures (LIPSS) whose near-subwavelength ripple topography acts as a controllable roughness parameter for wettability. Varlamova, Reif, Stolz, Borcia, Borcia and Bestehorn (Eur. Phys. J. B 92, 91, 2019) showed that a stochastic phase-field ('noisy surface') model of wetting — roughness represented as random height fluctuations with a single isotropic correlation length — reproduces the measured wetting of silicon LIPSS, but only in the hydrophilic regime. The authors themselves flagged the open issues this leaves: the model does not represent the anisotropic ripple orientation of real LIPSS, it is unvalidated for hydrophobic and superhydrophobic (Cassie-Baxter-type) states, the role of surface chemistry was not probed, and the goal of model-guided droplet-transport surfaces built from graded LIPSS roughness was suggested but not quantitatively demonstrated. Later phase-field work by the model's own authors turned to deterministic ratchet substrates under vibration, and other numerical studies treat idealized reliefs or pillared substrates rather than stochastic anisotropic LIPSS topography; experimental gradient-driven droplet transport on laser-structured surfaces has not been coupled to the validated model. The problem — extending the model to anisotropic LIPSS roughness, validating it in hydrophobic regimes with chemistry controlled, and using it to design graded LIPSS surfaces for controlled droplet transport — remains open.

Background

When a solid surface is rough, its apparent wettability deviates from that of a chemically identical flat surface. Two classical idealizations describe the limiting behaviors: in the Wenzel regime the liquid fully follows the rough topography, so roughness amplifies the intrinsic wetting preference of the material; in the Cassie-Baxter regime the liquid rests on a composite of solid protrusions and trapped air, which strongly promotes water repellency. The intrinsic wettability of the flat, chemically homogeneous surface is characterized by the Young contact angle θY\theta_Y set by the interfacial energies of the three phase pairs. Laser-induced periodic surface structures (LIPSS, also called ripples) are grating-like topographies with sub-micrometer periods that self-organize when a solid is irradiated by many femtosecond-laser pulses; on silicon (100), irradiation at 790 nm wavelength, 100 fs pulse duration, and fluences of 0.4–1.2 J/cm^2 produces ripples with periods of about 500–700 nm and amplitudes of a few tens of nanometers, with a well-defined orientation relative to the laser polarization (perpendicular for the common low-spatial-frequency type, parallel in some regimes). Because this roughness is written into the surface in a single, scalable laser step, LIPSS are attractive as a control parameter for wettability. To connect topography to wetting quantitatively, the cited study used a phase-field (diffuse-interface) model, in which the liquid–vapor interface is represented by an order-parameter field with a thin transition layer and droplet shapes and dynamics follow from a free-energy functional and its gradient flow. In this framework the rough solid was represented not by an explicit grating but by a stochastic 'noisy surface': random height fluctuations characterized by an amplitude and a single isotropic correlation length (the length over which height fluctuations decorrelate). This model line was established by Borcia, Borcia and Bestehorn (Phys. Rev. E 78, 066307, 2008) for drops on arbitrarily wetting substrates and validated against laser-structured steel and silicon by Borcia, Borcia, Bestehorn, Varlamova, Hoefner and Reif (Langmuir 35, 928–934, 2019), who found that increasing the noise amplitude amplifies the intrinsic behavior — hydrophilic surfaces become more wettable — in the Wenzel sense. The source study (Varlamova et al., 2019) combined this model with dedicated experiments on silicon LIPSS of increasing irradiation dose and reported excellent agreement between simulation and measurement. That validated-but-limited state defines the present problem. Real LIPSS are anisotropic — the ripples have a preferred orientation relative to the laser polarization, so the roughness is direction-dependent — whereas the validated model uses isotropic noise with one correlation length; how to extend it to anisotropic roughness, and what directional wetting it predicts (for example, different apparent contact angles parallel versus perpendicular to the ripples), is unresolved. The existing validation covers only hydrophilic surfaces (θY<90°\theta_Y < 90°); hydrophobic and superhydrophobic regimes, where composite Cassie-Baxter states, trapped air, contact-angle hysteresis and pinning dominate, are untested for this model class on LIPSS — and on silicon these regimes are entangled with time-dependent surface chemistry (native oxide growth, ambient hydrocarbon adsorption). Finally, the design goal that motivated the work — controlled droplet transport on wettability or roughness gradients, demonstrated empirically on laser-treated silicon as early as Sun, Zhao, Han and Gu (Thin Solid Films 516, 4059–4063, 2008) — has not been combined with the validated model into a quantitative, model-guided design of graded LIPSS surfaces.

Problem Statement

The problem is to turn the phase-field description of wetting that is validated for isotropically rough, hydrophilic LIPSS on silicon into a predictive, experimentally validated tool for anisotropic LIPSS topography across wetting regimes, sufficient to design droplet-transport surfaces. The open question has three linked deliverables serving this single objective: (1) extend the model's roughness representation from isotropic noise with a single correlation length to anisotropic roughness encoding the LIPSS ripple orientation relative to the laser polarization, and determine the resulting directional wetting behavior — for example, the difference of apparent contact angles and drop shapes parallel versus perpendicular to the ripples; (2) validate the model in the hydrophobic and superhydrophobic regimes, including composite Cassie-Baxter-type states, whereas the existing validation is limited to θY<90°\theta_Y < 90° — which requires controlling or characterizing surface chemistry (native oxide, possible hydrocarbon contamination) so that topographic effects are separable from chemical ones; and (3) use the extended, validated model together with graded roughness or wettability gradients to quantitatively design LIPSS-structured silicon surfaces that achieve controlled droplet transport (prescribed direction, and predictable motion thresholds or velocities). The separate question of the LIPSS formation mechanism (predicting pattern evolution a priori from multi-pulse irradiation) is a distinct problem and is not part of this boundary.

References

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