The hydrodynamic force experienced by a spherical-cap drop moving on a solid surface is obtained from two approximate analytical solutions and used to predict the quasi-steady speed of the drop in a wettability gradient. One solution is based on approximation of the shape of the drop as a collection of wedges, and the other is based on lubrication theory. Also, asymptotic results from both approximations for small contact angles, as well as an asymptotic result from lubrication theory that is good when the length scale of the drop is large compared with the slip length, are given. The results for the hydrodynamic force also can be used to predict the quasi-steady speed of a drop sliding down an incline.
The motion of drops of decane on horizontal poly(dimethylsiloxane) (PDMS)-coated glass surfaces resulting from a temperature gradient on the surface is studied experimentally, and a theoretical description of the thermocapillary motion of spherical-cap drops on a horizontal solid surface obtained using the lubrication approximation also is presented. The drop size and the applied temperature gradient are varied in the experiments, and the measured velocities of the drops are compared with predictions from the model. The scalings of the velocity with drop size and with the applied temperature gradient are predicted correctly by the theoretical model, even though the actual velocities are smaller than those predicted. The influence of contact angle hysteresis, which leads to a critical drop size below which drops do not move, is found to be minimal. Unlike in previous studies (Chen, J. Z.; Troian, S. M.; Darhuber, A. A.; Wagner, S. J. Appl. Phys. 2005, 97, 014906; Brzoska, J. B.; Brochard-Wyart, F.; Rondelez, F. Langmuir 1993, 9, 2220), this small critical drop size appears to be independent of the applied temperature gradient. Results also are presented on the deformation of the contact lines of the moving drops in the form of an aspect ratio, and correlated with the temperature difference across the footprints of the drops and the capillary number.
Results from experiments performed on the motion of drops of tetraethylene glycol in a wettability gradient present on a silicon surface are reported and compared with predictions from a recently developed theoretical model. The gradient in wettability was formed by exposing strips cut from a silicon wafer to dodecyltrichlorosilane vapors. Video images of the drops captured during the experiments were subsequently analyzed for drop size and velocity as functions of position along the gradient. In separate experiments on the same strips, the static contact angle formed by small drops was measured and used to obtain the local wettability gradient to which a drop is subjected. The velocity of the drops was found to be a strong function of position along the gradient. A quasi-steady theoretical model that balances the local hydrodynamic resistance with the local driving force generally describes the observations; possible reasons for the remaining discrepancies are discussed. It is shown that a model in which the driving force is reduced to accommodate the hysteresis effect inferred from the data is able to remove most of the discrepancy between the observed and predicted velocities.
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