Axisymmetric evolution of thin films in spin coating

Spin coating is the process of generating a uniform coating film on a substrate by centrifugal forces during rotation. In the framework of lubrication theory, we investigate the axisymmetric film evolution and contact-line dynamics in spin coating on a partially wetting substrate. The contact-line singularity is regularized by imposing a Navier slip model. The interface morphology and the contact-line movement are obtained by numerical solution and asymptotic analysis of the lubrication equation. The results show that the evolution of the liquid film can be classified into two modes, depending on the rotational speed. At lower speeds, the film eventually reaches an equilibrium state, and we provide a theoretical description of how the equilibrium state can be approached through matched asymptotic expansions. At higher speeds, the film exhibits two or three distinct regions: a uniform thinning film in the central region, an annular ridge near the contact line, and a possible Landau-Levich-Derjaguin-type (LLD-type) film in between that has not been reported previously. In particular, the LLD-type film occurs only at speeds slightly higher than the critical value for the existence of the equilibrium state, and leads to the decoupling of the uniform film and the ridge. It is found that the evolution of the ridge can be well described by a two-dimensional quasi-steady analysis. As a result, the ridge volume approaches a constant and cannot be neglected to predict the variation of the contact-line radius. The long-time behaviours of the film thickness and the contact radius agree with derived asymptotic solutions.