DropFENO: A Data-Free, Physics-Based Operator-Learning Framework for Droplet Equilibria on Mixed-Wettability Patterned Surfaces
Ganesh Sahadeo Meshram, Partha Pratim Chakrabarti, Suman Chakraborty
We study whether the equilibrium of a drop on a chemically patterned wall, and the capillary forces that the pattern exerts on it, can be learned from the free energy alone, without solution data. We introduce DropFENO (Droplet Free-Energy Neural Operator), in which Fourier (FNO), tensor-factorised Fourier (TFNO) and wavelet (WNO) neural operators are trained in two and three dimensions on the diffuse-interface free energy of the drop, with its volume and centroid constraints imposed as penalties. The predictions are tested with the virtual-work (Hellmann--Feynman) theorem: the force exerted by the pattern on a held drop is a wall integral of the equilibrium field which, for a drop straddling a wettability step, reduces identically to the contact-line law $F=σΔ\cosθ\,L$. DropFENO reproduces this law and the energy landscape of a drop crossing a step, and reproduces the pinning of the receding contact line less accurately. The residual error lies in the soft modes of the capillary energy (small displacements of the interface and the contact line). This explains why the energy is predicted accurately while contact-line positions are not, and why on striped walls the operators return Cassie-like states instead of the edge-pinned states whose anisotropic apparent angles the reference minimiser shares with lattice Boltzmann simulations. On held-out patterned walls the median energy of the predictions lies $\NdEmedRandFnoTwo$ of the free-drop energy above the best known minimum (FNO, two dimensions), at about one hundredth of the cost of a per-instance minimisation, and a refinement by a classical solver started from the prediction returns the reference state in almost all cases.
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