THIN FILM FLOW SIMULATION ON A ROTATING DISC

Transcription

1 European Congress on Computational Metods in Applied Sciences and Engineering (ECCOMAS 2012) J. Eberardsteiner et.al. (eds.) Vienna, Austria, September 10-14, 2012 THIN FILM FLOW SIMULATION ON A ROTATING DISC Petr Vita 1, Bernard F. W. Gscaider 2, Doris Prieling 3 and Helfried Steiner 3 1 Department Mineral Resources and Petroleum Engineering University of Leoben, Austria petr.vita@unileoben.ac.at 2 ICE Strömungsforscung GmbH Leoben, Austria bernard.gscaider@ice-sf.at 3 Institute of Fluid Mecanics and Heat Transfer Graz University of Tecnology, Austria {prieling, steiner}@fluidmec.tu-graz.ac.at Keywords: Computational Fluid Dynamics, tin film approximation, impinging jet, Finite Area Metod, OpenFOAM TM Abstract. Te film flow wit an impinging jet on a rotating disk is studied numerically in order to model a wafer etcing process, an important application in te semiconductor industries. Series of numerical studies based on te Volume-of-Fluid (VoF) metod were performed and evaluated against bot experimental data and analytical solutions. Te conclusion from tis was tat a transient two-pase 3D free-surface VoF-simulation wit a movable inlet is impractical for an industrial use due to very long computational times. Te tin film approximation (TFA) based on an integral metod, wic reduces te 3D nature of te problem into a 2D approximation by integrating te Navier-Stokes equations over te film tickness, was selected as a possible remedy. Te implementation of te TFA-model was carried out in te open-source software toolbox OpenFOAM TM using te Finite Area Metod (FAM). Te validation of te approac was done wit te ANSYS Fluent software and its VoF-implementation.

2 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner 1 INTRODUCTION Te cleaning, etcing and application of stripping cemicals on te semiconductor substrates (wafers) are crucial processes in te te semiconductor industries. One of te widespread tecnologies involved is a single-wafer wet-processing based on a spin processing tool as described by Junk[4]: Te wafer is placed on a cuck wit te side to be processed facing up inside of te spin processor camber. A nitrogen cusion protects te bottom side from any contamination. Wile te cuck and te wafer rotate, cemical mixtures are applied on te wafer by means of a dispenser in order to form a liquid layer on te wafer surface. Dependent on te process step, te dispenser is eiter in a fixed position (centre or off-centre) or is moving across te wafer surface. Te continuously supplied cemicals are spun off te wafer and collected in drain levels surrounding te cuck for recycling. Figure 1: Single-wafer spin processor Te ongoing trend in semiconductor industry is connected to smaller and smaller dimensions (32nm and beyond) of structures on te wafer. Tis development poses new callenges in removal of nano-particulate contamination witout damaging ig aspect ratio structures. Te better understanding of te involved penomena is an essential step in increasing quality and efficiency of te wafer processing. To meet tese callenges, te use of Computational Fluid Dynamics (CFD) is becoming inevitable. Te move from te single-wafer wet-processing inside te spin processor camber to a computational model involves an abstraction toward te rotating disk wit te movable impinging jet. Te formed tin film flow is ten studied numerically. Te work on te tin film flow simulation started wit a systematic study of several variations of Volume-of-Fluid metod. Bot commercial, ANSYS Fluent[1], and open source, OpenFOAM[6, 10], implementations were considered. Comparisons of 2D axisymmetric simulations were carried out against te experimental results by Tomas[8] and Ozar[7] on test cases wit simplified inflow conditions. Tese experiments allowed to select appropriate VoF-scemes for ongoing work. 2D axisymmetric simulations wit a more realistic vertical inflow were compared wit analytical solution based on tin film approximation by Kim[5] and available experimental data. Tese comparisons sowed a very good agreement of time-averaged numerical results wit analytical solutions. Furtermore it was sown tat te velocity profiles obtained from numerical solutions are in a very good agreement wit a polynomial velocity profiles from te analytical solutions, provided te flow was sufficiently far away from impinging jet. 2

3 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner Unfortunately 2D axisymmetric simulations are very limited in teir practical use as te offcentre application of te liquid films is more a rule tan an exception. Te off-centre impingement introduces a strong asymmetry into te flow and 3D simulations are tus unavoidable. 3D VoF simulations carried out sowed a very good agreement wit experiments conducted by Carwat[3] and Burns[2] as well a very good visual agreement wit a film structures observed on te experimental testbed. However some disadvantages exist. Te 3D VoF simulations are very dependent on te quality of computational meses since bad cell distributions can easily induce various artifacts in te solution. Te oter important aspect are mes sizes wit cell counts in te order of millions. Te sizes of computational meses directly correspond wit long simulation times wic render a 3D VoF approac impractical for industrial use. Te experience gained wit 3D VoF simulations igligted an urgent need for an innovative approac wic would drastically reduce simulation times wile providing a reasonably good solution of te tin film flow. 2 THIN FILM MODEL 2.1 Film flow Te film flow of interest is an incompressible flow of Newtonian fluid described by te Navier-Stokes and continuity equations: ( ) u ρ + (uu) = p + ρg + T + f (1) t u = 0 (2) were ρ is te fluid density, u is te flow velocity, p is te pressure, g is te gravity vector, T is te sear stress tensor and f represents various body forces. Te film flow problem and its governing equations are tree-dimensional in teir nature. However it would be very advantageous if one could reduce its dimensional complexity, for example by an integral metod, wile supplying te information lost in te process by means of supporting models, if needed. Suc a reduction of dimensions would directly translate into sorter simulation times. 2.2 Modelling assumptions Te fluid film on te rotating disk as some specifics tat can be exploited. Figure 2: Tin film model Te fluid film is being primarily driven by centrifugal forces. Tus te film flow velocity component in te normal direction can be neglected: u = (u, v, w), w 0 (3) 3

4 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner Te film tickness is very small, in te order of tent of millimetres wit extremes in te order of millimetres for te cases under consideration. Terefore we can assume a constant pressure across te film tickness. Te flow is assumed to be laminar wit polynomial velocity profiles. Air/liquid sear stress interactions at te film surface are neglected. Te gravity acts against te disk normal direction. Using above assumptions it is possible to transform te 3D film flow problem into te 2D tin film approximation. Te dependent variables in te approximation would be te film tickness and te film mean velocity u defined as: u = 1 u dz (4) 2.3 Momentum conservation equation Te transformation sall start wit te momentum conservation equation. But before starting wit te integration itself, te vertical component of te momentum equation is examined. Using te assumption of te negligible velocity component in te normal direction, te z- momentum equation collapses into: p = ρ g (5) z Considering te surface tension σ at te film surface (z = ), te pressure distribution over te film tickness can be directly expressed as: p = ρ g ( z) + σκ (6) were te first term is te ydrostatic pressure distribution and κ is te surface curvature approximated by: κ ( ) (7) Integration of te momentum equation (1) across te film tickness applying te modelling assumptions, te definition of te film mean velocity u and te pressure distribution p provides following intermediate formulation: ρ t (u) + ρ (uu) dz = (ρ g + σκ) τ disk + F (8) were τ disk represents te sear stress at te disk (z = 0): τ disk = µ u z (9) z=0 wit µ being te fluid dynamic viscosity. Te intermediate formulation contains two problematic terms to deal wit te advective term and te sear stress at te disk. Bot terms are dependent on te information lost in te integration. 4

5 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner 2.4 Velocity profile function Te weakness of TFA is limited information in te dimension reduced by te integration wat concerns all flow properties sowing strong variations tere, for example te flow velocity. Wile te film mean velocity u is resolved, te velocity distribution across te film tickness is not. In order to overcome te problem a polynomial velocity profile function u is introduced, a supporting model tat reconstructs te film velocity distribution as close as possible: u(x, y, z) = u(x, y, ξ) + ε u (10) u(x, y, ξ) = a 0 + a 1 ξ + a 2 ξ 2 + a 3 ξ 3, ξ 0, 1, z = ξ (11) were x and y represent coordinate directions aligned wit te disk, ξ is a normalised vertical coordinate and ε u denotes a modelling error. Te polynomial coefficients a i are determined by a set of boundary conditions: 1 0 u(ξ) dξ = ū u(ξ) ξ=0 = u disk u(ξ) ξ 2 u(ξ) ξ 2 = 0 ξ=1 = 0 (12) ξ=0 wic describe a connection to te film mean velocity u, von Neumann boundary condition at te free surface, a no slip boundary condition at te disk and an influence of te pressure gradient. Te boundary conditions lead to te following solution for te velocity profile function: ( 12 u(x, y, ξ) = u disk + (u u disk ) 5 ξ 4 ) 5 ξ3 (13) Figure 3: Examples of velocity profile function u(ξ) (u disk > u: blue velocity profiles, u disk < u: red velocity profiles) 2.5 Vertical velocity fluctuations Te relationsip between te film mean velocity u and te velocity distribution across te film tickness is depicted in te figure 4. 5

6 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner Figure 4: Vertical velocity fluctuations A concept of a vertical velocity fluctuation ũ is now introduced: u = u + ũ (14) A closer look on te mean film velocity u: u = 1 u dz = 1 (u + ũ) dz = 1 u dz + 1 ũ dz (15) reveals togeter wit a realisation tat te mean value is invariant to averaging: 1 u dz = u (16) an important identity of te vertical velocity fluctuation ũ: 1 ũ dz = 0 (17) Finally te vertical velocity fluctuation ũ can be expressed wit te elp of te velocity profile model (10): ũ = u u = lim (u(ξ) u) (18) εu Advective term Te advective term in te intermediate formulation of te momentum equation (8) is difficult to deal wit due te fact tat an average of te product of two functions is not te product of teir averages. Te complications are idden witin te vertical velocity fluctuations. Te advective term is expanded using te definition of te vertical velocity fluctuation (14) and simplified wit elp of te identities (16, 17): ( ) ρ (uu) dz = ρ uu + uũ + ũu + ũũ dz ( ) = ρ (uu) + ρ ũũ dz (19) } {{ } Differential Advection Using te velocity profile model (10) te differential advection part of advective term is approximated: ( ) [ 1 ] ρ ũũ dz = lim ρ (u(ξ) u) (u(ξ) u) dξ εu 0 0 [ ] (20) 213 = ρ 875 (u u disk) (u u disk ) 6

7 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner 2.7 Sear stress term Te sear stress term (9) works directly wit te velocity distribution across te film tickness. We apply te velocity profile model (10) in order to resolve it: τ disk = µ u z = lim µ u(ξ) εu 0 z=0 (ξ) ξ=0 = µ (21) 12 5 (u u disk) 2.8 Continuity equation In order to transform te continuity equation te beaviour of te film elevation at te free surface (z = ) as to be described: t + u z= x + v z= y w z= = 0 (22) Te continuity equation (2) is integrated applying te Leibniz integral rule. Te boundary terms are simplified wit te elp of te film elevation (22): 0 u dz = u dz + x 0 y = + (u) = 0 t 2.9 Impinging jet 0 ( ) v dz + w z= u z= x + v z= y Adding te impinging jet into te tin film model poses a real problem as te vertical momentum is being transfered into te tangential one in te impingement area. Tis violates assumptions of te film model and te solution in te impingement area and its surrounding is invalid and as to be provided by oter means. Two possible implementations of te impinging jet were considered a boundary condition moving in te domain and a fixation of te solution wit te source terms. Te source terms implementation was selected due to computational costs connected wit te movable boundary condition. Tere are some problems related to te impinging jet implementation troug te source terms tat ave to be addressed: Faces in te impingement area not resolving te exact inlet sape. Face boundaries are not aligned wit an inlet boundary. Total mass-flow correction is needed. Velocities varies along te inlet boundary. Furter te impinging jet induces a formation of a circular ydraulic jump[11]. Te ydraulic jump penomena is connected wit a flow reversing roller, see figure 5, wic locally breaks assumption of negligible normal velocity component and complicate velocity profile modelling. Te treatment of te ydraulic jump is going to be addressed in ongoing work. (23) 7

8 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner Figure 5: Hydraulic jump 2.10 Tin Film Approximation Te final form of te governing equations in te 2D tin film approximation is: t (u) + (uu + C) = 1 ρ p 1 ρ τ disk + S m (24) t + (u) = S m (25) were te momentum source S m and te mass source S m represent te moving impinging jet and te differential advection term C, te pressure p and te sear stress τ disk are given by: [ ] 213 C = 875 (u u disk) (u u disk ) (26) 3 RESULTS p = ρ g + σ ( ) (27) τ disk = µ 12 5 (u u disk) (28) Te implementation of te discussed TFA-model was carried out in te open-source software toolbox OpenFOAM TM using te Finite Area Metod code developed by Tuković[9]. Te FAM is a specialisation of te Finite Volume Metod (FVM) for flows on surfaces. Its implementation solves equations on a boundary patc of te volume mes. Figure 6: Polyedral mes Te validation of te approac was done wit te ANSYS Fluent software and its VoF implementation. Te selected validation cases covered bot centre and off-centre impingement. 8

9 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner Te fluid used was a wafer etcing agent Spinetc-D (ν = m 2 s 1 ) wit a constant inlet volumetric flow (Q = 1.5lpm). Te disk rotated in a counter-clockwise direction wit a constant rotation speed (Ω = 500rpm). No moving inlet was used due to excessive 3D VoF simulation times. Te 3D VoF simulation was using a locally refined computational mes wit 5 million cells. Te 2D TFA simulation used a flow neutral polyedral mes, see figure 6, wit just cells. (a) 3D VoF, xy-plane (b) 2D TFA, xy-plane Figure 7: Contours of te film tickness (a centre impingement) (a) xz-plane (b) yz-plane Figure 8: Instantaneous film tickness in cuts troug te impinging jet (a centre impingement) Te central impingement case, see figures 7 and 8, sows a very good agreement of 2D TFA results wit te 3D VoF simulation. A surprisingly good agreement is acieved even inside te critical region in te vicinity of te impinging jet. Altoug, 2D TFA solution is very smoot and sows a little of waviness opposite to te 3D VoF solution. Bot results are sowing grid artifacts in a form of rose petals. Te off-centre impingement case, see figures 9 and 10, sows a very good agreement of solutions outside of te impingement area were all model assumptions are met. Te ydraulic jump is owever under-predicted in te 2D TFA solution as te current velocity profile model 9

10 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner (a) 3D VoF, xy-plane (b) 2D TFA, xy-plane Figure 9: Contours of te film tickness (an off-centre impingement wit x = 30mm) (a) xz-plane (b) yz-plane Figure 10: Instantaneous film tickness in cuts troug te impinging jet (an off-centre impingement wit x = 30mm) does not account for te flow separation and te reverse roller. 2D TFA solution sows a little of waviness again. Te comparison of simulation times, see table 1, illustrates effectiveness of te 2D tin film approximation. Because te FAM implementation in te OpenFOAM is not parallelised yet, only single core computations are possible at te moment. Te projected 2D TFA computation performance on te 4 CPU cores assuming a linear speed-up would acieve an effective speedup by more tan a factor of tousand against 3D VoF simulation. Process time 3D VoF 2D TFA 1s 4 CPU cores 30days 1 CPU core 2ours Table 1: Simulation times 10

11 Petr Vita, Bernard F. W. Gscaider, Doris Prieling and Helfried Steiner 4 CONCLUSION AND OUTLOOK Te 2D tin film approximation, a viable model for film flows on te rotating disk, was presented. Te validation of te model was done wit 3D VoF metod. Te comparisons sow a very good agreement in areas were all model assumptions are met. However, te solution as to be prescribed in te impingement area by oter means. Te solution witin te strong influence of te impingement is generally invalid and its usability as to be judged on a case by case basis. Te solver based on te TFA model significantly reduced simulation times allowing its possible industrial application. Furter, te TFA model can be easily extended wit additional pysical models. Te ydraulic jump wit connected velocity profile modelling and te formation of surface waves are currently under investigation. REFERENCES [1] ANSYS, Inc. ANSYS Fluent Teory Guide Release 13.0, November [2] J. R. Burns, C. Ramsaw, and R. J. Jacuck. Measurement of liquid film tickness and te determination of spin-up radius on a rotating disc using an electrical resistance tecnique. Cemical Engineering Science, 58: , [3] A. F. Carwat, R. E. Kelly, and C. Gazley. Te flow and stability of tin liquid films on a rotating disk. Journal of Fluid Mecanics, 53:2: , [4] Markus Junk, Frank Holsteyns, Felix Staudegger, Cristiane Lecner, Hendrik Kulmann, Doris Prieling, Helfried Steiner, Bernard Gscaider, and Petr Vita. Simulations of liquid film flows wit free surface on rotating silicon wafers (rowaflowsim). In Te 10t International Conference on Modeling and Applied Simulation, pages Universita di Genova, September ISBN [5] Tae-Sung Kim and Moon-Un Kim. Te flow and ydrodynamic stability of a liquid film on a rotating disc. Fluid Dynamics Researc, 41(3): (28pp), June [6] OpenCFD Limited. OpenFOAM User Guide Version 1.6, July [7] B. Ozar, B. M. Cetegen, and A. Fagri. Experiments on te flow of a tin liquid film over a orizontal stationary and rotating disk surface. Experiments in Fluids, 34: , [8] S. Tomas, A. Fagri, and W. Hankey. Experimental analysis and flow visualization of a tin liquid film on a stationary and rotating disk. ASME Journal of Fluids Engineering, 113:73 80, [9] Željko Tuković and Hrvoje Jasak. Simulation of tin liquid film flow using openfoam finite area metod. Presented at 4t OpenFOAM Worksop, Montreal, Canada, June [10] H. G. Weller, G. Tabor, H. Jasak, and C. Fureby. A tensorial approac to computational continuum mecanics using object-oriented tecniques. Computers in Pysics, 12 (6): , November [11] Kenuke Yokoi and Feng Xiao. Mecanism of structure formation in circular ydraulic jumps: numerical studies of strongly deformed free-surface sallow flows. Pysica D: Nonlinear Penomena, 161(3-4): ,