7-28. Spin Coating Various manufacturing processes require that a thin, uniform coating of a liquid be created on a solid substrate. This may be done by applying a certain amount of the liquid on top of the substrate and then spinning it rapidly. As shown in Emslie et al. (1958), such liquid films level quickly, after which they gradually thin. This allows the fluid dynamics during most of the coating step to be mod- eled as in Fig. P7-28. Spinning the substrate at an angular velocity o causes the liquid to flow out- ward, thinning the film as its thickness h(t) remains uniform. (a) Assuming that ve = wor, it has been proposed that the r-momentum equation be reduced to av az Air Liquid h(t) Substrate Figure P7-28. Spin coating. Rotation of a horizontal substrate causes radial flow of a liquid film, which thins over time. 314 UNIDIRECTIONAL AND NEARLY UNIDIRECTIONAL FLOW where the centrifugal acceleration term takes the place of a radial pressure gradient. Use this to determine v,(r, z, t) and the radial flow rate per unit of circumference, h(t) 9(r,t) = [v,dz. 0 (b) Use a macroscopic balance to derive another relationship between h(t) and q(r,t). Show that dh 2w2 dt and solve for h(t), assuming that h(0) ho (c) Use order-of-magnitude estimates to identify the criteria that must be met to: (i) Regard the flow as pseudosteady; (ii) Neglect the remaining inertial terms in the r-momentum equation; (iii) Neglect the absent viscous term in the r-momentum equation; (iv) Neglect a P/ar; and (v) Assume that ve is independent of z. Items (ii) and (v) involve a Reynolds number, how should Re be defined? (d) For photoresist coatings (polymer solutions), representative propertics and operating conditions arev=0.001-0.01 m/s, ho= 1 mm, hp (final thickness) = 1-10 m, c = 1,000-5,000 rpm, R (substrate radius) = 10 cm, and tp (process time) = 10-60 s. How well are the criteria in part (c) met in this type of application?