Asymmetric fluid-structure dynamics in nanoscale imprint lithography

(ODE). With any numerical simulation both accuracy and stability of the methodin computing the solutions to the dynamics equations are importantconsiderations. The fourth order Runge-Kutta scheme has an increased region ofstability as compared with other explicit schemes such as second order Runge-Kutta and explicit Euler. The stability diagram for the fourth order Runge-Kuttascheme includes eigenvalues on the imaginary axis with a small portion involvingboth positive real eigenvalues when the imaginary component is nonzero, i.e.λ = λ + iλ where when λ ≠ 0 , λ > 0 [Collis 2000].criirThe problem is conditionally stable. Since the initial gap is small, thetime step must be chosen carefully to maintain numerical stability. Numericalinstability exists in the analytical solution to the fluid damping force and dampingtorque. Firstly, the film thickness cannot be less than zero. This would result inthe breakdown of the Reynolds equation. The pressure becomes infinite at zerofilm thickness. Furthermore, hard contact between the template and wafer cannotbe modeled by the Reynolds equation. Secondly, the damping force and torqueare dependent on the state variables, f ( z, z, D θ , θ D ) and τ ( z , z, D θ , θD). Thus thesolution could become unstable since the eigenvalues of the system change duringthe simulation. This phenomenon is observed during the simulations. Forexample, an initial time step of 1 × 10 -7 seconds or 0.1 microseconds will result ina stable solution for the first part of the simulation when the base layer thicknessis large. Then this time step must be decreased for the solution to remain stable.When the base layer thickness is below 500 nm, a time step of 1 nanosecond isrequired to maintain numerical stability. With a very small time step such as 1nanosecond, a 1 second simulation could easily extend over 24 hours in a C++implementation on an 850 MHz computer with 128 MB of RAM and 256KB oflevel 2 cache memory (L2 cache operations are faster than RAM). An adaptive67