Modeling nanoporosity development in thin film polymers for low-k (dielectric) applications
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The bubble growth dynamics subsequent to nucleation resulting from supersaturation of a polymer-dissolved gas system have been modeled. The model components include mass and momentum balance partial differential equations along with appropriate initial and boundary conditions. In addition the model equations also include thermodynamic equilibrium criteria along with equations of state, evolution of transport properties with concentration of dissolved gas and temperature, mechanical constitutive equations for polymer-dissolved gas/saturated gas phase. The model equations are based on the shell model of Arefmanesh et al. which envisages a concentric shell of polymer-dissolved gas surrounding every bubble nucleated. Bubble grows by diffusion of dissolved gas from this shell. The above model was modified to allow for diffusion of dissolved gas out of the shell in addition to diffusion into the bubble. The model equations are then solved using a finite difference approach. The results are presented for various growth conditions obtained by varying key variables in a parametric study. Detailed model equations include the Fox equation to model the reduction in the glass transition temperature, Tg, of the polymer upon plasticization. This change in the Tg affects material properties like viscosity (77), characteristic relaxation time(/l) and diffusivity (D) due to the softening effect. These material properties are temperature and dissolved gas concentration dependent and are modeled on the principles of time-temperature and time-concentration superposition. The shift factors have a Williams-Landell-Ferry (WLF) type dependence. A single element Maxwell fluid is used defme the polymer constitutive equation and diffusion is assumed to be Fickian, but with a nonconstant diffusivity. Equilibrium at the gas bubble and polymer shell interface is assumed to be defmed by Henry's law and the gas in the bubble is assumed to be ideal. A parametric analysis by varying film thickness ((^Q ) > temperature (T), diffusivity at the Tg DQ) and Henry's law constant ( ) is done, to investigate the effects of these variables on bubble growth dynamics.