Derivation of Reynolds Equation in Cylindrical Coordinates Applicable to Pin-On-Disk and CMP

Proceedings of the STLE/ASME International Joint Tribology Conference IJTC2007 October 20-22, 2008, Miami, Florida, USA

DRAFT

IJTC2008-71245 DERIVATION OF REYNOLDS EQUATION IN CYLINDRICAL COORDINATES APPLICABLE TO PIN-ON-DISK AND CMP K.E. Beschorner Dept. of Bioengineering University of Pittsburgh Pittsburgh, PA 15261 [email protected]

C. F. Higgs III, Ph.D. Dept. of Mechanical Engineering Carnegie Mellon University Pittsburgh, PA 15213 [email protected]

ABSTRACT Traditional tribology references typically provide the cylindrical (or polar) Reynolds equation, which may not be applicable when entrainment velocities vary with radius and/or angle. However, entrainment velocities are known to vary with angle for some cases of pin-on-disk contact and chemical mechanical polishing (CMP). A form of Reynolds equation is derived in this manuscript from the Navier-Stokes equations without entrainment velocity assumptions. Two case studies, related to pin-on-disk and CMP, are presented and results from the derived form of Reynolds equation are compared with results from the traditionally used form. Pressure distributions obtained from the two forms of Reynolds equation varied greatly in magnitude and in pressure shape. Therefore, a new form of the cylindrical Reynolds equation derived in this manuscript is used when entrainment velocities are known to vary with radius or angle. INTRODUCTION Reynolds equation in cylindrical (or polar) form is used for numerous tribology applications. Popular tribology references list cylindrical Reynolds equation, which may not be valid when entrainment velocities vary relative to angle or radius [1, 2]. Pin-on-disk contact and chemical mechanical polishing represent two cases where entrainment velocities may vary with angle. This study derives a cylindrical form of the Reynolds equation from the Navier-Stokes equations without making assumptions regarding the entrainment velocities. The results from the developed Reynolds equation without entrainment velocity assumptions are compared with the results obtained with the traditional Reynolds equation for simulations of both a pin-on-disk and chemical mechanical polishing problem.

M.R. Lovell, Ph.D. Dept. of Industrial Engineering University of Pittsburgh Pittsburgh, PA 15261 [email protected]

METHODOLOGY The Reynolds equation is derived from the continuity and momentum Navier-Stokes equations based on constant density and viscosity assumptions [2]: Continuity Equation:

1 ∂ (rυr ) + 1 ∂υθ + ∂υz = 0 r ∂r r ∂θ ∂z

(1)

r-momentum Equation:

 ∂ν ∂ν ν ∂ν ∂ν ν2  ρ r + ν r r + θ r + ν z r + θ  ∂r r ∂θ ∂z r   ∂t ∂p ν 2 ∂ν   + − η ∇ 2 ν r − 2r − 2 θ  = 0 ∂r r r ∂θ  

(2)

θ-momentum Equation:

∂ν ν ∂ν ∂ν νν   ∂ν ρ θ + ν r θ + θ θ + ν z θ + r θ  ∂r r ∂θ ∂z r   ∂t 1 ∂p ν 2 ∂ν   + − η ∇ 2 ν θ − 2θ + 2 r  = 0 r ∂θ r r ∂θ  

(3)

∂2 1 ∂ 1 ∂ ∂ + + 2 + 2 2 r ∂r r ∂θ ∂r ∂θ 2

(4)

∇2 =

The Navier-Stokes equations can be simplified by applying the following assumptions [3]: 1. Viscous forces are much larger inertial and body forces. 2. Velocity gradients across the z-dimension are larger than in the r and θ directions. 3. Fluid is Newtonian. 4. Velocity flow is negligible in the z-direction and thus pressure gradient across the z-direction is negligible

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5. Steady-state conditions are assumed. 6. No slip of the fluid occurs on the surfaces. 7. Distances in the r-direction are large enough compared with the film thickness that velocity gradients across the fluid are much larger than ν2/r and ν/r2. The continuity equation (1) is multiplied by r*dz and then integrated across the film: h

∂

h

∂

∫ ∂r (rυ )dz + ∫ ∂θ (υ )dz = 0

(5)

θ

r

0

0

By implementing the above stated assumptions and the Leibniz rule of integration: h h  ∂ ∂h ∂  ( ) ( ) [ f x , y , z dz f x , y , h = − +  ∫ f (x, y , z )dz  (6) ∫0 ∂x ∂x ∂x  0  Equation (5) becomes: − (rυ r )z = h

∂h ∂ + ∂ r ∂r

∂h ∫ (rυr )dz − (υθ )z = h + h

0

∂θ

∂ (υ )dz = 0 ∂θ ∫0 θ h

(7)

By applying the stated assumptions, Eqs. (2) and (3) become:

∂p ∂  ∂ν r  = η  ∂r ∂z  ∂z 

h−z 2η h−z ν θ = −z 2η r

ν θ( a ) + ν θ ( b ) ∂ (h ) + ∂θ 2

(12)

CASE STUDY I: PIN-ON-DISK A pin-on-disk test operating in full film lubrication is evaluated using the derived Reynolds equation (11) and with the traditional form of Reynolds equation (12). To determine the velocity profile, the pin is assumed to be sufficiently far from the center of the disk such that the velocity profile is:

ν r ( a ) = − U sin(θ )

(13a)

ν θ(a ) = − U cos(θ )

(13b) where U represents the mean sliding speed of the disk relative to the pin. The film thickness of the fluid was defined by the pin shape and a minimum film thickness as:

h = B * r 2 + h0

(14)

The simulation parameters are defined in Table 1. (8a)

1 ∂p ∂  ∂ν θ  = η  (8b) r ∂θ ∂z  ∂z  Applying the appropriate boundary conditions, Eqs. (8a) and (8b) can be solved to determine the velocity profiles in the r and θ direction as:

ν r = −z

∂  h 3 ∂p  1 ∂  h 3 ∂p  ν r ( a ) + ν r ( b ) ∂  = (rh ) r + ∂r  12η ∂r  r ∂θ  12η ∂θ  ∂r 2

h−z ∂p + ν r(a) + ν r(b) h ∂r h−z ∂p + ν θ(a ) + ν θ(b ) h ∂θ

z h z h

(9a) (9b)

Radius (mm) 3.3

Table 1: Simulation parameters for pin-on-disk Curvature, Viscosity, Sliding speed, B (mm-1) h0 (mm) η (cP) U (mm*s-1) 0.011 0.02 10.7 50

The cylindrical Reynolds equation without entrainment velocity assumptions yields a much different pressure distribution than the traditional cylindrical Reynolds equation (Fig. 1). The peak pressure for the traditional Reynolds equation is about 3 times (2.2 kPa) as large as the proposed form (0.65kPa) derived in this work. In addition, the shape of the pressure distribution between the two solutions is slightly different.

The velocity profile can be integrated across the film thickness for substitution back into Eq. (7):

h 3 ∂p ν r ( a ) + ν r ( b ) h ∫0 ν r dz = − 12η ∂r + 2 h

h

∫ ν θ dz = − 0

h 3 ∂p ν θ ( a ) + ν θ ( b ) h + 12ηr ∂θ 2

(10a)

(10b)

By substituting Eq. (10) back into Eq. (7), the generalized form of Reynolds equation in cylindrical coordinates can be found: ∂  h 3 ∂p  1 ∂  h 3 ∂p  ∂h   = − ν r ( b )r r + − ∂r  12η ∂r  r ∂θ  12η ∂θ  ∂r (11) ∂h ∂  ν r ( a ) + ν r ( b )  ∂  ν θ (a ) + ν θ( b )    − ν θ( b ) + rh  + h  ∂θ ∂ r  2 2  ∂θ   The derived form of this Reynolds equation (11) has a very different right side of the equation than the form of Reynolds equation (11) of often adopted from tribology resources [1, 2]:

Figure 1: Solution to pin-on-disk problem using Eq. (11), left, and using Eq. (12), right. CASE STUDY II: CHEMICAL-MECHANICAL POLISHING Chemical-mechanical polishing (CMP) often consists of a rotating pad and rotating wafer. The cylindrical form of Reynolds equation is typically used because the velocity and film thickness profiles are simpler when expressed in this form. Therefore, a sample CMP problem was solved using the proposed form of Reynolds equation (11) and the traditional form of Reynolds equation (12). The entrainment velocities can

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be represented for the pad surface (a) and the wafer surface (b) as [4]:

ν r ( a ) = d sin (θ )ω p

(15a)

ν θ ( a ) = (r + d cos(θ )) * ω p

(15b)

ν r (b ) = 0

(15c)

ν θ( b ) = rω w (15d) In Eq. (15), d represents the center to center distance of the wafer and the pad; ωp is the rotational velocity of the pad and ωw is the rotational velocity of the wafer. The film thickness is given as: h(r , θ ) = h M − r sin α * cos θ − r sin β * sin θ (16) In Eq. (16), hM is the mean film thickness, α and β are the tilt angles of the wafer. Both forms of Reynolds equation were solved using the simulation parameters in Table 2. Diameter (mm) 200

Table 2: Simulation parameters for CMP d ωp ωw hM (mm) (RPM) (RPM) (µm) α (°) 150 100 50 100 0.015

β (°) 0.015

The solution of the cylindrical Reynolds equation for the CMP example without entrainment velocity assumptions, Eq. (11), yields much different pressures than the traditional cylindrical Reynolds equation (12) (Fig. 2). Specifically, the peak pressure found with Eq. (12) is 180 kPa, while the peak pressure for the traditional Reynolds equation is 420kPa. In addition, a subambient pressure region was generated when using the traditional form of Reynolds equation (12) that did not result from the proposed version of Reynolds equation (11).

Figure 2: Solution to CMP example using the derived Reynolds equation (11), left, and the traditional Reynolds equation (12), right.

CONCLUSIONS A form of the cylindrical Reynolds equation is presented here without entrainment velocity assumptions. This form of Reynolds equation is most useful for applications where νr is a function of r and/or νθ is a function of θ. In addition, simulations for both pin-on-disk and CMP case studies showed large pressure deviations between the proposed form of cylindrical Reynolds equation and the traditional Reynolds equation. Therefore, using the traditional form of Reynolds equation when entrainment velocities vary with the angle or radius may lead to incorrect and misleading results. NOMENCLATURE B: Curvature of the pin U: Sliding velocity of disk relative to pin d: Center-to-center distance of the pad to the wafer h: Film thickness h0: Minimum film thickness hM: Mean film thickness r: Cylindrical coordinate, r z: Cylindrical coordinate, z α, β: Angles of tilt for the wafer θ: Cylindrical coordinate, θ η: Viscosity ρ: Fluid density ν: Velocity ωp: Rotational speed of the pad ωw: Rotational speed of the wafer a: Refers to surface of (z=0) b: Refers to surface of (z=h) con: Refers to region of shoe-floor contact fl: Refers to region where lubricant is present t: Refers to total (combined fluid and contact regions) REFERENCES 1. Bhushan, B., Principles and Applications of Tribology. 1999: Wiley-Interscience. 2. Hamrock, B.J., B.O. Jacobson, and S.R. Schmid, Fundamentals of Fluid Film Lubrication. 2004: CRC Press. 3. Meyer, D., Reynolds Equation for Spherical Bearings. Journal of Tribology, 2002. 125: p. 203. 4. Park, S.-S., C.-H. Cho, and Y. Ahn, Hydrodynamic analysis of chemical mechanical polishing process. Tribology International, 2000. 33(10): p. 723-730.

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