Asymmetric fluid-structure dynamics in nanoscale imprint lithography

where A is the Hamaker constant (~ 10 -20 J), h avg is the average distance betweenthe plates (m), and P appplied is the applied pressure (Pa).Assuming the applied pressures on the order of 1 psi, h avg is about 4 nm whenvan der Waals forces are on the same order as the pressures of interest [Freeland2000]. Also, Moore documents that it has been agreed that molecular influencecan extend outward from a surface no more than 0.5 µin (13 nm). A base layerthickness of approximately 100 nm, well above these limits of continuum fluidmechanics, justifies a few of the following assumptions.In deriving the Reynolds equation, the assumptions that are to be made mustbe considered:1. Body forces are neglected, i.e. van der Waals forces.2. The pressure is constant through the thickness of the film. Since thethickness of the films considered here are about ten microns or smaller,while the length scales in the plane of the template are measured in∂pcentimeters, it is reasonable to assume that = 0 .∂z3. There is no slip at the boundaries. There has been much work on this andit is universally accepted [Cameron 1976].∂u4. The fluid is Newtonian, i.e. τ = µ . Shear stress is proportional to the∂zrate of shear strain. This assumption is valid when the lubrication is in thebulk regime (minimum film thickness above 10 nm).5. The flow is laminar. The Reynolds number based on gap height is lessthan one for the range of gap heights and velocities. Reh