Drive Diffusion Equations

Gaussian Approximation

The exact drive diffusion solution is unnecessarily complicated. A simplified form can be obtained by changing the initial condition for Fick's Laws.

The predeposition diffusion has a very high surface concentration, but is very shallow. This distribution can be approximated by a Dirac delta function - Δ(x).

Redefine the intial condition to:
N₂(x,0) = Qδ(x)

Solution for Fick's Laws using the delta function:

Substituting

This becomes:

Surface concentration (N₀₂):

S&M Correction

The S&M correction is an empirical correction factor that improves the Gaussian approximation when D₂t₂ is not much greater than D₁t₁.

Define

where:
Δ₁ = D₁t₁
Δ₂ = D₂t₂
Δ₃ = D₃t₃

Substitute Δ* for Δ₂

And the surface concentration is