Asymmetric fluid-structure dynamics in nanoscale imprint lithography

pressure distribution is tedious and only the result follows. Integrating the twodimensionalReynolds equation once gives2dp 1 ⎧ ⎛ x⎫2⎞= ⎨12µ ⎜hxD+ θDsec θ⎟ + C3 1⎬. [2.25]dx h ⎩ ⎝ 2x⎠ ⎭Integrating again gives2( )⎪⎧ 2= 6µ hDh + 2xtanθ6µθDsec θ 3h+ 4hxtanθp(x)−+2 23 ⎨h tan tan θ ⎪⎩ 22xθh x[2.26]⎛ hx⎞⎫C1+ ln⎜ ⎬ − + C22h⎟⎝ α ⎠⎭2hxtanθFor the unsubmerged plate-surface system, meniscus effects at the outerperiphery of the squeeze film, and capillary pressure opposes the squeeze filmpressure [Moore 1965]. Assuming the liquid perfectly wets the surfaces of theplate and substrate, i.e., zero contact angles, the pressure at the boundary is given− 2γby p( x)= . Applying the boundary conditions p( x x )p( x x )h x− 2γ=β= , the integration constants, C 1 and C 2 , are obtained.hβh x= h + x tanθ, h x= h + x tanθ,αα− 6µhDC1=2tan θhC2− k=ββ26µθD sec θk = ,3tan θ2( h + 2xtanθ) ⎧( 3h+ 4hxtanθ)β2xβ⎪+ k⎨⎪⎩2h2xβ2( 3h+ 4hxtanθ) 6µhD( h + 2xtanθ)2h+α22xαh x αtanα2βθ,⎛ h+ ln⎜⎝ hxβxα⎞⎪⎫⎟⎬,⎠⎪⎭− 2γ=α= andhα31