Film flow on a rotating disk
TL;DR: In this article, the authors analyzed how a film of uniform thickness thins when the disk is set in steady rotation and showed that the leading order term in the long-time-scale solution for the transient film thickness is a lower bound for film thickness for all time.
Abstract: Unsteady liquid film flow on a rotating disk is analyzed by asymptotic methods for low and high Reynolds numbers. The analysis elucidates how a film of uniform thickness thins when the disk is set in steady rotation. In the low Reynolds number analysis two time scales for the thinning film are identified. The long‐time‐scale analysis ignores the initial acceleration of the fluid layer and hence is singular at the onset of rotation. The singularity is removed by matching the long‐time‐scale expansion for the transient film thickness with a short‐time‐scale expansion that accounts for fluid acceleration during spinup. The leading order term in the long‐time‐scale solution for the transient film thickness is shown to be a lower bound for film thickness for all time. A short‐time analysis that accounts for boundary layer growth at the disk surface is also presented for arbitrary Reynolds number. The analysis becomes invalid either when the boundary layer has a thickness comparable to that of the thinning film, or when nonlinear effects become important.
Citations
More filters
••
TL;DR: Dip coating is a simple old way of depositing onto a substrate, especially small slabs and cylinders, a uniform thin film of liquid for solidification into a coating.
Abstract: Dip coating is a simple old way of depositing onto a substrate, especially small slabs and cylinders, a uniform thin film of liquid for solidification into a coating. The basic flow is steady, and in it film thickness is set by the competition among viscous force, capillary (surface tension) force and gravity. Thickness and uniformity can be sensitive to flow conditions in the liquid bath and gas overhead. The faster the substrate is withdrawn, the thicker the film deposited. This can be countered by using volatile solutes and combining rapid enough drying with the basic liquid flow. Then the physics grows more complicated, theoretical prediction of process performance more difficult, and control of the process more demanding. Outside product RD actually it is often modified.
458 citations
••
12 Oct 2005
TL;DR: In this paper, the fundamental physical and chemical processes governing the conventional spincoating process and methodologies for the preparation of spin-coated polymer thin films are discussed and a range of advanced applications and recent developments within this field are reviewed with focus on engineering chemical and topological structure during the coating process.
Abstract: Spin-coating is widely employed for the highly reproducible fabrication of thin film coatings over large areas with high structural uniformity. Research in recent years has extended the scope of spin-coating by chemically engineering the interface of support and solution to obtain specific structural order in the resulting supported thin films. This review will discuss both the fundamental physical and chemical processes governing the conventional spin-coating process and describe methodologies for the preparation of spin-coated polymer thin films. Furthermore, a range of advanced applications and recent developments within this field will be reviewed with focus on engineering chemical and topological structure during the coating process.
338 citations
••
TL;DR: In this paper, the process of spin coating is described, with particular attention to applications in microelectronics, and the physical mechanisms involved in the process are discussed and those mechanisms that affect the final state are identified, viz., centrifugal and viscous forces, solute diffusion, and solvent evaporation.
Abstract: The process of spin coating is described, with particular attention to applications in microelectronics. The physical mechanisms involved in the process are discussed and those mechanisms that affect the final state are identified, viz., centrifugal and viscous forces, solute diffusion, and solvent evaporation: A model is proposed that incorporates only the latter mechanisms, with viscosity and diffusivity depending on solute concentration. The evaporation of solvent during spinning causes the solution viscosity to increase and the flow is reduced. The thickness of the final solid film is related to the thickness of a diffusion boundary layer near the free surface. The model predicts the final dry film thickness in terms of the primary process variables, spin speed, and initial polymer concentration. A similarity boundary‐layer analysis leads to a simple approximate result for the final film thickness that is consistent with limited experimental data, hf ∼KC0(ν0D0)1/4/Ω1/2, where K is a number of order un...
317 citations
••
TL;DR: In this article, the creeping flow of a liquid film along an inclined periodic wall of arbitrary geometry is considered, and the boundary-integral method for Stokes flow is extended to two-dimensional flows involving free surfaces.
Abstract: The creeping flow of a liquid film along an inclined periodic wall of arbitrary geometry is considered. The problem is formulated using the boundary-integral method for Stokes flow. This method is extended to two-dimensional flows involving free surfaces, and is implemented in an iterative numerical procedure. Detailed calculations for flow along a sinusoidal wall are perfomed. The free-surface profile is studied as a function of flow rate, inclination angle, wave amplitude, and surface tension, and is compared with previous asymptotic solutions. The results include streamline patterns, velocity profiles and wall-shear-stress distributions, and establish criteria for flow reversal. For specified wall geometry, the asymptotic behaviour for very small flow rates is shown to be a strong function of surface tension. It is demonstrated that these results are valid in a qualitative sense for general wall geometries. The analogy between gravity-driven flow and the flow of a liquid layer on a rotating disk (spin coating) is also discussed.
136 citations
••
TL;DR: In this paper, the influence of polymer molecular weight, molecular weight distribution, and polymer-solvent interactions on the thickness and topography of spin-coated polymer films was examined.
Abstract: The influence of polymer molecular weight, molecular weight distribution, and polymer-solvent interactions on the thickness and topography of spin-coated polymer films was examined. For films prepared from dilute solutions, highly volatile solvents or fair or “poor” solvents for the polymer adversely affect film surfaces causing nonuniformities (waves) to appear. However, if the concentration of these solutions is increased to approximately the concentration at which entanglements are formed, nearly uniform films are produced even if the solvent employed is highly volatile, such as dichloromethane. When toluene is employed as the solvent, which has a relatively low volatility and therefore forms nearly flat film surfaces, films prepared from dilute solution were found to have thicknesses, h, proportional to η Ω−0.49 for polystyrene and η Ω−0.49 for poly(methylmethacrylate) where ηo is the zero-shear rate solution viscosity and Ω is the rotational speed at which the films were prepared. These results suggest that the exponents associated with ηo and Ω may be nearly independent of the type of polymer used as long as flat films are produced. Finally, the molecular weight parameter most important in controlling final film thickness for films made from dilute solutions is Mv, the viscosity-average molecular weight.
119 citations
References
More filters
•
01 Jan 1981
TL;DR: In this paper, the authors introduce the notion of forced Oscillations of the Duffing Equation and the Mathieu Equation for weakly nonlinear systems with quadratic and cubic nonlinearities.
Abstract: Algebraic Equations. Integrals. The Duffing Equation. The Linear Damped Oscillator. Self-Excited Oscillators. Systems with Quadratic and Cubic Nonlinearities. General Weakly Nonlinear Systems. Forced Oscillations of the Duffing Equation. Multifrequency Excitations. The Mathieu Equation. Boundary-Layer Problems. Linear Equations with Variable Coefficients. Differential Equations with a Large Parameter. Solvability Conditions. Appendices. Bibliography. Index.
3,020 citations
••
2,029 citations
••
TL;DR: In this paper, it was shown that initially irregular fluid distributions tend toward uniformity under centrifugation, and means of computing times required to produce uniform layers of given thickness at given angular velocity and fluid viscosity are demonstrated.
Abstract: Equations describing the flow of a Newtonian liquid on a rotating disk have been solved so that characteristic curves and surface contours at successive times for any assumed initial fluid distribution may be constructed. It is shown that centrifugation of a fluid layer that is initially uniform does not disturb the uniformity as the height of the layer is reduced. It is also shown that initially irregular fluid distributions tend toward uniformity under centrifugation, and means of computing times required to produce uniform layers of given thickness at given angular velocity and fluid viscosity are demonstrated. Contour surfaces for a number of exemplary initial distributions (Gaussian, slowly falling, Gaussian plus uniform, sinusoidal) have been constructed. Edge effects on rotating planes with rising rims, and fluid flow on rotating nonplanar surfaces, are considered.
696 citations
••
TL;DR: In this article, a model for the description of thin films prepared from solution by spinning is presented, and the thickness of the film and the time of drying can be calculated as functions of various processing parameters.
Abstract: A model is presented for the description of thin films prepared from solution by spinning. Using only the centrifugal force, linear shear forces, and uniform evaporation of the solvent, the thickness of the film and the time of drying can be calculated as functions of the various processing parameters. The model is compared with experimental results obtained on positive photoresists and excellent agreement is obtained. When adequate care are is taken, the liquid forms a level surface during spinning, and the film thickness becomes uniform and independent of the size of the substrate. The film thickness h shows the following dependence on spin speed f, initial viscosity ν0, and evaporation rate e:h∝f−2/3νo1/3e1/3, and e is proportional to f1/2.
628 citations