Line Tension Effect upon Static Wetting

 Abstract:  Adding simply, in the classical capillary model, a constant line density of energy along the contact line leads to ill-posed equilibrium problems. Then, when line tension is present, the equilibrium configuration minimizes a different energy : the "relaxed" energy, which explicitly depends on the presence of surface phases (i.e. infinitesimal films) on the boundary of the container. This formulation enable us to describe the modifications of the Young's law and then of equilibrium configurations which are due to line tension.

Keywords : Line tension, Static wetting, Relaxation.
 

In the simplest model for capillarity, one consider two phases A and B lying in a rigid container O. One, at least, of the phases is incompressible and a constant surface energy sAB is concentrated on the interface SAB which divides the two phases. The wetting properties of the wall W of the container O are taken into account by considering constant surface energies sAW and sBW   concentrated on the contact surfaces SAW, SBW of the phases A and B on the wall. The contact line LC, defined as the intersection of the interface and the wall of the container , plays an important role for describing equilibrium conditions. The associated contact angle q is defined as the angle made by the interface and the wall, more precisely made by normal vector of the interface SAB external with respect to A and the normal vector of the wall W external with respect to O(cf. figure 1).
What are the equilibrium conditions in that case? And before all, is this minimization problem a well-posed problem? These are the questions we will discuss in the sequel. We will show that the problem is, in general, ill-posed. The associated well-posed  problem (the minimization of the "relaxed" energy) cannot be formulated without considering surface phases on the wall. This notion of surface phases is connected with the notion of wetting (or dewetting) films. Rigorous proofs will not be given here : interested readers can refer to [5].
 

1 Back to the no line-tension case.

Let us first consider the classical case when no line tension is present. The equilibrium conditions are well known [6] : (i) the interface has a constant mean curvature, (ii) the contact angle q
 is constant along the contact line and is given by the Young's law. Clearly, when |sAw- sBW|> sAB, the Young's law cannot be satisfied. Different attitudes are possible in this situation :

 i) one can first consider that the wetting inequality |sAw- sBW|<=sAB  holds in every physical case;
ii) or one can assume that there is no contact between the phase A and the wall if this inequality is not satisfied.

Both attitudes cannot be entirely correct : many cases have been described in which the wetting inequality is not satisfied and, when the volume of the phase A is sufficiently large, the contact between A and the wall cannot be avoided.
Assume, for instance, that sAw- sBW> sAB . In that case the minimization of the energy E is a ill-posed problem. Indeed, let us consider a minimizing sequence (one can imagine either a slow motion of the phases toward the equilibrium state or a numerical descent method for searching the minimum of the functional). In some geometric case as the one represented in figure 2, the limit of the minimizing sequence may not be a minimizer.
From a microscopic (infinitesimal) point of view SAW is empty and attitude (ii) is correct. From a macroscopic point of view, the equilibrium configuration is the limit of the minimizing sequence and SAW is not empty : A does not minimizes the original energy E but a "relaxed" energy E given by

Then the energy E one has to consider from a macroscopic point of view satisfies the wetting inequality and attitude (i) is correct.
The difference between the original energy E and the relaxed one E takes into account the existence of a microscopic (infinitesimal) film of phase B between phase A and the wall. Of course, extra physical arguments may bound the thinness of this film and modify its energy (which is simply here the sum of the energies sAB+ sBW. Such arguments are not necessary at this point and do not change fundamentally our conclusions.
The remarkable fact we much emphasize is that the relaxed energy E has the same form as the original one E. Owing to this "miracle" one can ignore in this model the presence of films along the wall by considering only, from the very beginning, energies which satisfy the wetting condition. The relaxation of the model when line tension is present is not so simple...