Spin coating of non-Newtonian fluids

Spin coating is a process which is widely used to apply uniform thin solid films to extended flat substrates. In many cases, the coating material is polymeric and is applied in the form of a solution from which the solvent evaporates. Centrifugal force drives the fluid to flow radially off the spinning substrate, and this flow may be non-Newtonian. A theoretical description is obtained for the unsteady non-Newtonian flow, and the diffusion of solvent through the film, which is driven by evaporation at the free surface. This provides an analytical estimate of the final film thickness: , in which hf is the final film thickness, ν0 is the kinematic viscosity, D0 is the coefficient of solvent diffusion and is the critical shear rate for onset of shear thinning of the viscosity, which is assumed to follow a power-law for intermediate shear rates with index n = 1/α. The rotation speed is Ω and a is the radius of the substrate. The prediction is in qualitative agreement with experimental data for polymide solutions that indicate hf σ Ω−b with b ≈ 0.8. The theoretical description begins with a dimensional analysis of the governing equations. The Reynolds number and film thickness are small, so that the lubrication approximation applies. The Peclet number is large so that solvent depletion is initially confined to a thin region near the free surface of the film and the fluid is homogeneous. The Deborah number is very small so that the flow is quasi-viscometric, and shear thinning is characterized by a Weissenberg number, which may be small or large. A general constitutive law for this flow requires only a single function relating shear stress and shear rate. We consider the general case and obtain an analytic solution for the film thickness profile and flow for arbitrary initial conditions; this solution quickly asumptotes to a self-similar form. We consider the diffusion of solvent in the evaporation-driven boundary layer at the free surface and obtain a similarity solution for the boundary layer concentration profile. The boundary layer thickness satisfies a hyperbolic evolution equation which is solved numerically; this solution is also asymptotically self-similar. We consider rheological models which show Newtonian behavior at low shear rates with a transition to power-law shear thinning at moderate shear rates—the truncated power law, Ellis and Carreau-Yasuda models. For α < 3, the Ellis model leads to unphysical results; the Carreau-Yasuda model can be used for any α provided the Newtonian/power law transition is sufficiently rapid. For the models used, the flow and diffusion in the film show Newtonian behavior near the axis of rotation and power-law behavior far from the axis. In particular, the film is flat near the axis of rotation and decreases in thickness as η(1−α)/(1 + α) for large values of the scaled radial coordinate η. Flow off the edge of the substrate is drastically reduced once the solute depleted region penetrates through the film. The time for this to occur is estimated and the residual solute determines the average final film thickness; it is speculated that the final film may be relatively uniform. The results are presented in terms of a ‘final’ Weissenberg number, which characterizes the importance of shear thinning in the final stages of flow. When Wif is small, previously published Newtonian results are recovered; when Wif is large, the power-law result presented above is obtained. If spinning is stopped before the process is complete, the resulting film is more non-uniform, and the thickness depends more strongly on spin speed, hf σ Ω−2α/(α + 1).