Spin Coating of a Non-volatile Newtonian Fluid
The simplest analysis of the key elements of spin coating concerns the spin coating of a nonvolatile Newtonian fluid. Only the primary results will be provided here; an excellent treatment is given in detail by Middleman¹. A simplified schematic of the analysis is illustrated in Figure 2 at right.

After numerous assumptions, the equation of motion reduces simply to:

{1}

This equation nicely illustrates the balance of viscous and centrifugal forces which is at the center of this analysis.

Further development yields the film thickness as a function of material and process parameters:

{2}

where H is the thickness of the film, H₀ is its initial thickness, and t is the spinning time. Note that at long times, the film thickness is independent of its initial thickness. This is particularly helpful in making the process more robust.

Refinement of the Analysis
Both the volatility of the solvent and non-Newtonian fluid flow are important in actual spin coating operations. Indeed, these effects are coupled because the solvent diffusivity and solution viscosity are functions of the concentration of polymer. Flack et al.² have provided a more detailed analysis which includes these effects.

The equation set is sufficiently compled so that a numical solution is required. As might be expected, the effects of convection are dominant early in the process while evaporation of the solvent becomes more important at later times. Once again, the initial amount of solution placed on the substrate has little effect on the final thickness of the film at long times.

1.   S. Middleman, What is the Title?, Publisher (Year).
2.   W. Flack, D. Soong, A. Bell, and D. Hess, "A Mathematical Model for Spin Coating Polymer Resists," J. Appl. Phys., 56, 1199 (1984).