Langmuir 1992,8, 3038-3039
3038
Dynamics of Partial Wetting Eli Ruckenstein Department of Chemical Engineering, State University of New York at Buffalo, Buffalo,New York 14260-4200 Received June 29, 1992
It is shown that the slip velocity at the leading edge of a drop calculated on the basis of the gradient of the chemicalpotential along the solid-liquid interface is given by an expressionsimilar to that obtained
by de Gennes on the basis of a strict hydrodynamic model. In a recent paper,l Brochard-Wyart and de Gennes divide in two classes the models that have been proposed to describe the motion of a contact line when the wetting angle 6 deviates from ita equilibrium value 6,: (i) the Eyring approach, employed by Blake and Haynes,2which involves the microscopic jump of a single molecule at the tip, and (ii) de Gennes’hydrodynamic approach3that emphasizes the viscous losses inside an assumed liquid wedge of angle 6. In 1977,Ruckenstein and Dunn4 derived an equation for the slip velocity during wetting of solids, and the goal of the present paper is to compare their approach to those in refs 2 and 3, and particularly to show that their equation can be reduced for small values of 6 to that derived by de Gennes. In essence, in ref 4 it was shown that near the contact line a net translational velocity parallelto the solid is generated by a force caused by the gradient of the chemical potential in the liquid along the solid-liquid interface. In order to understand the origin of this gradient of the chemicalpotential, let us focus on a molecule located in the liquid on the solid-liquid interface and consider first that the range of interaction forces between this molecule and other molecules of the liquid is smaller than the distanceto any of the points on the liquid-gasinterface. In this case, the molecule does not feel the presence of the liquid-gas interface and ita chemical potential can be equated to that in a semi-infinite liquid. If, however, a molecule on the solid-liquid interface is selectedfor which the range of interaction forces is larger, the molecule at the chosen point is additionally subject to a field that is generated because the number of molecules with which it interacts at ita left differs from those with which it interacts at ita right. This field, equal to the difference between the real interaction potential and that for a semi-infinite liquid, decays to zero for large enough distances from the leading edge, but becomes increasingly important when the leading edge is approached because liquid molecules are replaced by gas molecules within the range of the interaction forces. Consequently, near the leading edge along the solid surface there is a nonuniform region in which the interaction potential of a molecule located on the solid surface with all the other molecules varies with the distance to the leading edge. The chemical potential per unit volume can be written in the form P = Po
+ 40
where p~ is the chemical potential in a point of the solid surface corresponding to a semifinite layer of liquid, (1) Brochard-Wyart,F.; de Gennes, P. G. Adu. Colloid Interface Sci. 1992, 39, 1. (2) Blake, T. D.; Haynes, J. M. J . Colloid Interface Sci. 1969,30,421. ( 3 ) de Gennes, P. G.Kolloid Polym. Sci. 1986, 264, 463. (4) Ruckenstein,E.; Dum, C. S. J . Colloid Interface Sci. 1977,59,135.
defined per unit volume, and +O is the additionalinteraction potential, also defined per unit volume. The gradient of the chemicalpotential along the interfacegeneratesa force F per molecule
where nL is the concentration in the liquid, in molecules per unit volume, and x the distance along the liquid-solid interface measured from the leading edge. For the slip velocity along the solid-liquid interface, one can use Einstein’s equation v, = qF
where q = D/kT is the mobility of the molecule, D is the surfacediffusion coefficient, k is the Boltzmann constant, and T is the temperature (K).Consequently
A more complex equation can also be derived: namely v, = 6C sinh (F612kT)
(3) where 6 is the distance between two potential wells located on the solid surface over which a molecule jumps and C = voe-E/kT,vo being the vibration frequency of a molecule in the potential well and E an activation energy. For F6I 2kT
