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Kinetics and Diffusion
Ohring: Chapter 1, Sections 1.6, 8.1 - 8.2
= how fast it will happen
we will concentrate on mass transport= atoms diffusing through a solid
Fick's 1st Law = "stuff moves from where you have lots to where you have little"
Now let us consider the flux of atoms into and out of a particular volume in the solid.The fluxes (J) may be different at different positions (x) in the material.
rate of increase of matter in the region = rate of flow in - rate of flow out= (J1 - J2 ) Æx = - ¶J / ¶x
or this is also = rate of change of concentration in the volume
so . . .
Solve this equation as a boundary value problem (see example in Ohring p. 34-36.
for simplicity consider a cubic lattice
There is always a potential energy barrier to diffusion (activation energy).
What do we expect mathematically for the flux to the right (from position1 to 2):
Similarly we can find the flux to the left:
(note: if we had used the gas constant, R, instead of Boltzmann constant, k, then the energy would be the diffusion energy/mole)
actually D is typically NOT CONSTANT, D is a function of:
high diffusivity paths:
These are all more open structures with higher jump frequencies and lower energy barriers.
Usually the cross sectional areas of these are small compared to the rest of the film.
In general (but not necessarily at high temperature): DS > Dgb > D
for determining activation energies
start with Fick's First Law:
Diffusion can be changed by stress fields, electric fields, other energy gradients (interfaces)
[note: Ohring changes notation here: uses diffusion free energy per mole GD instead of diffusion energy per atom ED so all of the k's become R's.]
Examine what happens when we apply a field: