5.1.3 Viscous Model

Another approach uses an incompressible viscous fluid in the oxide and nitride layer. In this case stress is calculated from the flow velocity [Zie92], i.e., the strain rate rather than the strain. In contrast to the previous model it must be solved self consistently with the oxidation growth due to the close coupling. The residual stress at the end of the oxidation is that one after the last step. Nevertheless, the evolution of the oxide shape is not influenced by the stress history.

To account for incompressible viscous flow the Navier Stokes equations have to be solved with the explicit solution of a pressure term which often leads to numerically bad conditioned systems:

$${\frac{\partial p}{\partial x}} \mu \left(\vphantom{\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}}\right. {\frac{\partial^2 u}{\partial x^2}} {\frac{\partial^2 u}{\partial y^2}} {\frac{\partial^2 u}{\partial z^2}} \left.\vphantom{\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} + \frac{\partial^2 u}{\partial z^2}}\right)$$ (5.11)

$${\frac{\partial p}{\partial y}} \mu \left(\vphantom{\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}}\right. {\frac{\partial^2 v}{\partial x^2}} {\frac{\partial^2 v}{\partial y^2}} {\frac{\partial^2 v}{\partial z^2}} \left.\vphantom{\frac{\partial^2 v}{\partial x^2} + \frac{\partial^2 v}{\partial y^2} + \frac{\partial^2 v}{\partial z^2}}\right)$$ (5.12)

$${\frac{\partial p}{\partial z}} \mu \left(\vphantom{\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}}\right. {\frac{\partial^2 w}{\partial x^2}} {\frac{\partial^2 w}{\partial y^2}} {\frac{\partial^2 w}{\partial z^2}} \left.\vphantom{\frac{\partial^2 w}{\partial x^2} + \frac{\partial^2 w}{\partial y^2} + \frac{\partial^2 w}{\partial z^2}}\right)$$ (5.13)

$${\frac{\partial u}{\partial x}} {\frac{\partial v}{\partial y}} {\frac{\partial w}{\partial z}}$$ (5.14)

where u, v, w denotes the velocity terms of the viscous material, $\mu$ the viscosity and p the hydrostatic pressure.