Energy landscape and disorder-induced slow relaxation of droplets on heterogeneous wetting surfaces

We develop an energy landscape framework for droplet equilibrium and relaxation dynamics on heterogeneous wetting surfaces. For a two-dimensional cylindrical droplet straddling a sharp wettability boundary, energy minimization of the Gibbs free energy functional ΔG(L1,L2) yields the spreading condition L2/|L1|=S(θ1)/S(θ2), where S(θ)=sin θ tan θ, showing that contact line partitioning depends exclusively on interfacial energy ratios and is independent of the Bond number. Two disorder models are examined. For a random contact angle field with amplitude σ and correlation length ξ, Monte Carlo simulations reveal that increasing disorder broadens the energy barrier distribution and drives a transition from exponential (β=1) to stretched exponential relaxation (β≈0.67–0.71), with the stretching exponent related to the mean barrier height by β≈Teff/E¯. For a contact angle hysteresis model with window Δθhyst, a metastable band forms whose width grows linearly with Δθhyst; the relaxation timescale shifts asymmetrically between advancing and receding scenarios while the stretching exponent remains invariant (β≈0.51). These results establish quantitative connections between disordered wetting and the statistical physics of glassy systems, and provide design criteria for functional surfaces requiring either droplet mobility or strong pinning.