Relaxation of a moving contact line and the Landau-Levich effect

– The dynamics of the deformations of a moving contact line is formulated. It is shown that an advancing contact line relaxes more quickly as compared to the equilibrium case, while for a receding contact line there is a corresponding slowing down. For a receding contact line on a heterogeneous solid surface, it is found that a roughening transition takes place which formally corresponds to the onset of leaving a Landau-Levich film. We propose a phase diagram for the system in which the phase boundaries corresponding to the roughening transition and the depinning transition meet at a junction point, and suggest that for sufficiently strong disorder a receding contact line will leave a Landau-Levich film immediately after depinning. When a drop of liquid spreads on a solid suface, the contact line, which is the common borderline between the solid, the liquid, and the corresponding equilibrium vapor, undergoes a rather complex dynamical behavior. This dynamics is determined by a subtle competition between the mutual interfacial energetics of the three phases, dissipation and hydrodynamic flows in the liquid, and the geometrical or chemical irregularities of the solid surface [1]. A most notable feature of contact lines, which is responsible for their novel dynamics, is their anomalous elasticity as noticed by Joanny and de Gennes [2]. For length scales below the capillary length, which is usually of the order of 1 mm, a contact line deformation of wave vector k, denoted as h(k) in Fourier space, will distort the surface of the liquid over a distance |k|−1. Assuming that the surface deforms instantaneously in response to the contact line, the elastic energy cost for the deformation can be calculated from the surface tension energy stored in the distorted area, and is thus proportional to |k|, namely Ecl = γθ 2 2 ∫ dk 2π |k||h(k)| 2, in which γ is the surface tension and θ is the contact angle [2]. The anomalous elasticity leads to interesting equilibrium dynamics, corresponding to when the contact line is perturbed from its static position, as studied by de Gennes [3]. Balancing dEcl dt and the dissipation, which for small contact angles is dominated by the hydrodynamic dissipation in the liquid nearby the contact line, he finds that each deformation mode relaxes (∗) Permanent address: Institute for Advanced Studies in Basic Sciences Zanjan 45195-159, Iran.