Motion of a contact line on a heterogeneous surface

Using the lubrication approximation, we calculate the dynamics of a contact line on a periodic heterogeneous plate which is advanced at constant force F or constant velocity u. For a constant force, motion starts when F exceeds a threshold Fc which is simply related to the static advancing (receding) contact angle. If the heterogeneity is smooth, the force increases with velocity as F−Fc∝u2. Alternating patches of constant wettability produce a linear relation. For constant velocity experiments, we identify weak and strong pinning regimes. In the weak pinning regime, the threshold force is zero and the force–velocity relation approaches that of a uniform surface. In the strong pinning regime, the threshold F0 is finite and approaches Fc as the strength of the heterogeneity increases. For smooth heterogeneity, F−F0∝u2/3, while alternating patches produce a linear response. The relevance of these results to experimental surfaces with random heterogeneity is discussed.