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Description
We present an inverse-design method that determines the chemical heterogeneity profile of a surface required to transport a droplet along a prescribed path. Droplet dynamics are simulated by employing a reduced-order model that reduces the lubrication equation to a system of ODEs for the Fourier harmonics of the contact line, and integrating this stiff system with an implicit adaptive Runge–Kutta scheme (Kvaerno 5/4) via diffrax. Designing the surface profile so that the simulated droplet reproduces a prescribed path is an inverse problem whose cost is dominated by the dimensionality of the design space. To make this problem tractable, we use adjoint gradients obtained using JAX's automatic-differentiation framework, which makes the computational cost largely independent of the number of design parameters. The chosen cost function penalizes the Fréchet distance between the simulated centroid trajectory and the prescribed path, as well as deformation of the contact line from a circle, the latter guarding against numerical instabilities. We validate the inverse-design method on reference paths of increasing complexity, from a straight line to a sinusoid, recovering centroid trajectories that accurately reproduce each target, with wall-clock times ranging from ~13 minutes to under 2 hours 40 minutes. We further demonstrate passive droplet sorting: droplets below or above a critical volume move in opposite directions under a single designed profile, all without the application of external fields or a droplet detection mechanism. The same differentiable-solver pattern applies broadly to PDE-constrained inverse design in computational fluid dynamics.