Modelling effects of surface tension on surface topology in spin coatings for integrated optics and micromechanics

Surface profiles of multilayer coatings of spin-on glass over topography often display unexpected features, which have been attributed to the effects of surface tension. Several nonlinear numerical models of this have been developed and are compared here. The most comprehensive model, which is used as a yardstick, is based on a time-dependent two-dimensional (2D) model of two-phase fluid flow during spinning, which includes the effects of convection, surface tension, concentration-dependent diffusion and viscosity and evaporation of a volatile liquid phase. The mathematical model is a system of a time-dependent, 2D, convective-diffusion equation for the volatile phase and the Navier-Stokes equations for the non-volatile phase in a non-rectangular space domain with a moving boundary. A robust, efficient numerical method based on the transformation of the physical domain with a moving boundary into a rectangular computational domain that is invariant in time was developed. The problem in the transformed coordinates is solved by the implicit finite-difference method; an unconditionally monotone approximation is used for the convective-diffusion terms. Two further simplified models requiring enormously reduced CPU time are also discussed and compared. The capabilities of the models are demonstrated with some practical examples of spin-coat planarization.