Asymmetric fluid-structure dynamics in nanoscale imprint lithography

and the result is⎡ 12µ⎧τ = L⎢3 ⎨⎣ h ⎩23θDsec θ tanθ48h⎛ 2 ⎞6 6( ) ⎜θDsecθ h tanθ− + − ⎟5 5x x( x − x )+αβ⎜⎝30D5h4 4 C⎫⎤1 3 3 C22 2( x − x ) + ( x − x ) + ( x − x ) ⎥⎦⎛⎞⎜ h D C1tanθ+ ⎟[2.31]β αα β⎜⎟⎬β α⎝8 32µh⎠36µ⎭ 2⎟⎠βα2.4 THREE-DIMENSIONAL PROBLEM2.4.1 3D Pressure Distribution for Parallel, Rectangular PlatesIn the previous two-dimensional case, the plates were considered to beinfinite in the y direction. In the three-dimensional case, the plate dimensions arefinite. For a flat, rectangular plate moving parallel towards a flat surface withfluid completely filling the gap, i.e. there is no capillary effect; the pressuredistribution in the squeeze film is relatively complex owing to the introduction ofcorner effects and the absence of rotational symmetry. For a rectangular plate oflength L and width B where the shape ratio B/L gives a characteristic shapefactor, f ( B ), Hays assumed an infinite, double Fourier series solution for theLpressure distribution of the following formwhere[ M , N = 1, 3, 5, ..., ∞],p =∑∑∞ ∞M Nπxθ =LAMNsin Mθ sin Nφ[2.32]πyandφ = [Moore 1965]. This satisfies theBzero boundary conditions such that the pressure is the same atmospheric pressure35