Motion of a drop on a horizontal solid surface with a wettability gradient

The motion of drops of tetraethylene glycol in a wettability gradient present on a silicon surface is investigated experimentally and theoretically. The gradient was formed by exposing clean silicon surfaces to a source of dodecyltrichlorosilane vapor. The static contact angles were measured as a function of position and used to characterize the local wettability gradient. The Reynolds, capillary, and Bond numbers in the experiments were relatively small. The measured migration velocities of drops over a range of sizes demonstrated the complex nature of the variation of the velocity with position on the gradient surface in response to the changes in the driving force and the resistance to the motion. The results are organized and interpreted using a simple quasi-steady hydrodynamic model in which inertial effects and deformation due to gravity as well as motion are neglected so that the shape is approximated by a spherical cap. Two approaches are used to estimate the hydrodynamic resistance experienced by the drop. In the "wedge approximation" the drop is modeled as a collection of wedges; the drag on each wedge is calculated from a solution for Stokes flow. In the second approach, lubrication theory is employed while retaining the exact shape of the drop. A slip boundary condition is used in a region close to the contact line to relax the usual stress singularity. The results from the wedge approximation and lubrication theory are indistinguishable at contact angles ≤ 30°.; The theoretical model based on the wedge approximation describes the qualitative features of the shape of the curve of velocity versus position along the gradient surface. A detailed investigation of the remaining discrepancy does not support the hypothesis of a missing resistance due to either contact line dissipation or an underestimation of the hydrodynamic drag. Instead, it is concluded that a reduction in the driving force due to contact angle hysteresis is the most likely reason. The quantitative differences are accommodated by approximately accounting for the influence of hysteresis.