FIDAP Numerical Modeling Scott Taylor

List of Topics Fixed Gap – Rigid Pad Fixed Gap – Deformable Pad Modified Step Free Surface Integration

1. Fixed Gap – Rigid Pad Model Length = 10 mm Rigid Pad no deformation Step dimensions 10 μm high 1 mm long Gap thickness = 20 μm

Boundary Conditions Velocity (x,y) Slurry Properties Pad = (0.278, 0) m/s or --- 70 RPM Wafer = (0) m/s Inlet/Outlet = (--, 0) m/s Slurry Properties Density = 1164 kg/m^3 Viscosity = 2 cp

Fixed Gap Width: 10 μm step Wafer Pad Fixed Gap Width: 10 μm step

Results Results for streamline, UX, UY are as expected. A change in magnitude of velocity only results in magnitude change of solution. Pressure contours need to be investigated.

Pressure Contour Large pressure variation at step face High (Low) pressure ‘pocket’ offset from corner Couette flow (no step) run as validation. No abnormal results Step sensitivity study

Step Sensitivity Step height increased to 30 μm. All other conditions the same

Step Sensitivity Step height decreased to 3 μm. All other conditions the same

Step Sensitivity Unexpected pressure contour most likely the result of sharp geometric discontinuity and not a genuine solution. Possible way to reduce is to introduce sloping sides, rather than sharp corner.

2. Fixed Gap – Deformable Pad Pad now modeled as a continuum instead of a line boundary. Pad Properties – Homogeneous & Isotropic Density = 630 kg/m^3 Young’s Modulus = 20 - 40E6 Mpa Poisson’s ratio = 0.3

Model WAFER SLURRY INLET OUTLET PAD Model is NOT to scale

Boundary Conditions Old method – Minimal BC UX wafer = 0.84 m/s UY inlet/outlet = 0 m/s DX/DY bottom of pad = 0 m Lack of BC’s allow FIDAP to get smoother results. Create ‘edge effects’ that are undesirable.

Boundary Conditions – New Method Pad given velocity Model ‘attachment’ of pad boundary to continuum help attain convergence. BC additions: UX pad = 0.278 m/s DY/DX pad bottom = 0 m: DY pad sides (left & right) = 0 m UX/UY wafer = 0 m

Discontinuity more apparent, but edge effects are eliminated, which will help with free surface integration.

General Results Deformation in X, Y directions small Order of nanometers Depends on E, υ, velocity Pressure Contours similar to rigid pad Deflections don’t appear to affect pressure distribution

3. Modified Step Slope given to step to reduce any errors due to discontinuity. Old New Angle reduced to 45 degrees from 90. NOTE: Currently, any model with the modified step has more nodes than the older model, but resolution near the step is decreased.

Pressure contour now located around step.

Deflection in Ydirection is very similar to 90 deg. step. Other results are as expected.

4. Free Surface FIDAP capable of coupling pad deformation with a movable wafer Force balance Moment balance Attempts to use ‘standard’ free surface rigid body motion unsuccessful. Solution diverges Model database related

Free Surface - Subroutine Using USRBCN user subroutine, surface position can be modified explicitly. Subroutine currently being written to work with wafer ‘step’. Subroutine successful for a flat wafer.

USRBCN Problems Not robust Substantial computational time Error prone Model locked Nodes Geometry Parameter changes difficult Substantial computational time Error prone Potential to inadvertently modify solution arrays

To Do Finish writing subroutine for models. Determine grid dependence. Gather results for variety of conditions. Complete thesis/manual

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