Langmuir 1991, 7, 3216-3218
3216
Collective Modes of a Contact Line F. Brochardt and P. G. de Gennes'J SRI Universitd Paris 6, 11 rue P . et M . Curie, 75230 Paris Cedex 05,France, and Laboratoire de Physique de la Matibre Condensde, Collbge de France 1 1 , place M , Berthelot, 75231 Paris Cedex 05, France Received March 14, 1991. In Final Form: May 24, 1991 We consider a contact line between a solid, a nonvolatile liquid, and a gas that is deformed sinusoidally . discuss these times with a wavelength 2r/q. This relaxes back to a straight line within a time T ~ We for systems with a small contact angle Be K ) , while Xmin is a microscopic cutoff, dependent on the long-range forces present near the solid surface: for small 8,and attractive van der Waals forces, one findd2
xmin= aOe+
(7)
where a is a molecular size. For practical purposes, the logarithmic factor 1 in eq 6 can be treated as a constant (1-12). The most important factor in eq 5 is the factor fF1, indicating that (at fixed V) the dissipation in a thin wedge becomes very large. All our discussion will be focused on small amplitudes and small velocities; then, in eq 5, we may replace 0 by its equilibrium value 0 -* Be. Let us now consider a mode (q)of amplitude uq,velocity L,, and write that the elastic energy is burnt according to eq 5
This gives an exponential relaxation:
L, = - (1/7 ( )1u9
(9)
with a relaxation rate
(a) In the limit q
> K , this is of the unusual form l/Tq
= cq
(11)
where c is a characteristic velocity c = (1/30(7ee3/7)
I
q-1
1-
Figure 3. Screening region for the vorticity in the collective modes of a wedge. Only a "skin"(hatched),of depth X below the surface, is involved. Out of the skin region, the fluid is static.
(b) When q < K , ure recover a more traditional regime for a line, combiningtension 3 and viscous friction, namely,
117, = 9 2 .
111. Mixed Modes with Inertia and Viscosity (1) Absence of Inviscid Modes. The advancing wedge of Figure l a is always associated with some shear flow. This can be understood simply when 0
