Numerical simulation of solvent diffusion and reaction in deforming polymers: Applications to microlithography.

摘要

In this dissertation, an efficient, accurate and a robust simulation tool for coupled solvent diffusion and reaction in deforming polymers is developed. This development is motivated by the need to simulate difficult yet physically important regimes of solvent diffusion in polymers in two and three dimensions. In particular, Case II diffusion, a specific regime which is characterized by sharp solvent fronts, glass to rubber transition and polymer swelling is considered. As an application, silylation, an important process step in microlithography involving strong coupling between diffusion, reaction and deformation is modeled as a Case II diffusion phenomenon. Mathematically, this results in a highly nonlinear, strongly coupled problem. Therefore, development of a computational methodology for strongly, highly nonlinear problems in general and application to simulating solvent diffusion and reaction in polymers forms the scope of this dissertation.; As a first step, an existing 3-D Case II model and the accompanying balance laws are formulated in a dimensionless form. The Case II model in turn is extended to the silylation case. Constitutive relations in terms of a nonlinear flux law needed to produce sharp solvent and reaction fronts, a concentration dependent relaxation time of the polymer to model glass-rubber phase transitions and coupling terms between diffusion, reaction and deformation are provided. The resulting governing equations lead to a highly nonlinear, strongly coupled initial boundary value problem (IBVP). In the next step, an efficient and accurate numerical scheme for the IBVP is developed.; The IBVP is spatially discretized and abstracted into a system of Index 1 Differential algebraic-equations (DAEs). As an efficient alternative to monolithic time integration, fractional step (also known as operator splitting) methods are developed. Analytical results on orders of accuracy for fractional step methods are derived. For the first time, order reduction of two-pass splitting algorithms is proved. As a remedy to order reduction, a novel deferred correction method is developed and is shown to perform better than a highly optimized, state-of-the-art fifth order Runge Kutta scheme, RADAU5. Due to strong coupling, splitting schemes, in spite of their efficiency, can become inaccurate. This important shortcoming for strongly coupled problems is overcome by designing a novel error based adaptive hybrid scheme for time stepping Index 1 DAEs. By design, the hybrid scheme chooses from the best of monolithic and split choices at each time step in addition to the best stepsize. The result is an efficient, accurate and a robust time integrator which is applied to the diffusion-reaction-deformation problem. (Abstract shortened by UMI.)