Thin Liquid Film Flow on a Rotating Annular Disk: Asymptotic Solution
01 Jan 2002-pp 511-522
TL;DR: In this article, the axisymmetric flow of a Newtonian fluid associated with the spreading of a thin liquid film on a rotating annular disk is considered and the effects of surface tension and gravity terms are included.
Abstract: We consider the axisymmetric flow of a Newtonian fluid associated with the spreading of a thin liquid film on a rotating annular disk. The effects of surface tension and gravity terms are included. The asymptotic solution for the free surface of the thin film is found using an expansion for the film thickness in powers of a small parameter characterizing the thickness of the film and applying the method of matched asymptotic expansion. This solution can be used to calculate the thickness of the film, the velocity field, and the pressure at any point on the disk with good accuracy. Numerical results are presented for a specific initial distribution of the film thickness. Many features of the spin-coating thinning process are captured by our asymptotic solution. We also produce results which are in excellent agreement with the experimental findings of Daughton and Givens (6) and Hwang and Ma (9).
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TL;DR: In this article, the Von Karman alterations have been used to alter the basic equations into the set of nonlinear differential equations and the elegant homotopy analysis method (HAM) was used to obtain the solution of the system.
18 citations
TL;DR: In this article, the axisymmetric flow of a viscous heat conducting uniform liquid film on a rotating disk is considered and the governing equations are solved numerically, and the model described highlights the effects of thermocapillary force and the effect of different spin-up protocols on the rate of thinning of the film.
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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
TL;DR: In this article, it was shown that a thin film with small dynamic contact angle and driven by an external body force is unstable to the formation of fingers in the direction perpendicular to the main flow.
Abstract: We show that a thin film with small dynamic contact angle and driven by an external body force is unstable to the formation of fingers in the direction perpendicular to the main flow. The instability is largest in the capillary region near the contact line, where the force due to surface tension is comparable to the viscous and gravitational forces. The fastest growing wavelength is calculated in the limit of small-amplitude disturbances. These instabilities may be related to finger patterns observed in gravitational flows and spinning drops.
322 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