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TimeDomain CVD, Inc. |
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The two flows we've studied are actually the two extreme cases of flow in a "duct" or chamber, as shown below:
| In the "plug flow" regime, the real flow configuration has nearly constant velocity in most of the chamber, with a relatively thin "boundary layer" in which the gas or fluid is strongly affected by viscous drag from the walls. In plug flow the boundary layer is so small that we can ignore it to first order -- that is, the flow is "inviscid". In the Poiseuille regime, the two boundary layers have grown so much that they no longer exist as distinct regions, but have merged to give a smooth continuous variation in velocity. |
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In the intermediate region, we can treat the flow as taking place in two distinct regions: a boundary layer with low velocity, whose thickness we need to determine, and a "free stream" region outside the boundary where velocity (and other parameters such as concentration) are relatively fixed.
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The boundary layer formation is the result of diffusion of momentum from the wall into the stream. To first order, we can treat this problem just as we did the plug-flow reactor, by allowing diffusion to proceed perpendicular to the streamlines, and taking the gas flow velocity into account through a streamtime transformation t => x/U. We use the kinematic viscosity as the diffusivity of momentum.
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Note that the result can be expressed in terms of the Reynolds number -- as we noted, Re is a measure of the relative size of the momentum diffusion length and the system. Note also that the boundary layer grows as the square root of the distance downstream. |
In some cases the CVD problem can be treated using the boundary layer approximation as well: the concentration of precursors and products is taken to be constant (at least with respect to y) in the free stream, and transport to the wafer surface is by diffusion through the boundary layer. The result is generally similar to simple depletion but with a new "effective" surface reaction rate that takes into account diffusion through the boundary. |
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