Asymmetric fluid-structure dynamics in nanoscale imprint lithography

⎛⎜θDsec⎜⎝6C22= Pθ hDtanθ⎞− ⎟h ⎟⎠xβ12µ⎧−3 ⎨h ⎩3 3 h 2 2( x − x ) + ( x − x )αβD223θDsec θ tanθ−x8h⎛⎞⎜ hDC1tanθ+ ⎟2 C1xβ− xβ⎜⎟⎝2 8µh⎠12µ⎫⎬⎭α4ββ⎤⎥⎦⎛⎜θDsec+⎜⎝62⎫⎬⎭θ hDtanθ⎞− ⎟xh ⎟⎠( x , x )maxα βEquation 2.27 is valid for small values oftanθ. Note that equationh2.27 is a polynomial equation in x. It can be shown that in the limit, as θapproaches zero, the pressure distribution approaches that of a parallel plate(equation 2.23).In Figure 2.6, the pressure distributions for both parallel and nonparallelplates are presented. The pressure distribution for the case of nonparallel plates isskewed and generates a torque to correct the deviation of θ from zero. Figure 2.6shows that for small angles, the pressure distribution is nearly symmetric and thatthe location of the maximum pressure moves away from x = 0.The damping force resulting from the squeeze film pressure is obtained bythe integration of the pressure over the projected area of the wetted portion of thetemplate. This damping force is given byThe result of this integration is⎡f = L⎢⎣12µ⎧3 ⎨h ⎩23θDsec θ tanθ40hf=L xβ∫∫0 xα3βp(x)dxdy . [2.28]⎛ 2 ⎞5 5( ) ⎜θDsecθ h tanθ− + − ⎟4 4x x( x − x )+αβ⎜⎝24D4h⎟⎠+βα33