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## Kinetics and Diffusion

Ohring: Chapter 1, Sections 1.6, 8.1 - 8.2

Kinetics

= how fast it will happenwe will concentrate on mass transport

= atoms diffusing through a solid

Diffusion in one dimension - Fick's 1st and 2nd Laws

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.Mathematically:

rate of increase of matter in the region = rate of flow in - rate of flow out

= (J_{1}- J_{2}) Æx = - ¶J / ¶xor 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.

Atomic view

for simplicity consider a cubic latticeThere 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)

COMMENTS:

- diffusion increases with temperature

Diffusion Coefficients

- self diffusion (element A in A): D
_{A} - vacancy diffusion: D
_{V} - chemical diffusion (element A in B): D
_{AB} - grain boundary diffusion: D
_{gb} - surface diffusion: D
_{s}actually D is typically NOT CONSTANT, D is a function of:

- position
- jump frequency depends on local atomic arrangement and defects
- temperature gradients

- direction in lattice
- time
- defects, concentrations, temperature may vary with time

**high diffusivity paths:** - position
- grain boundaries
- three dimensional dislocation networks
- surfaces
These are all more open structures with higher jump frequencies and lower energy barriers.

- D
_{o}and E_{D}are different for these paths

Usually the cross sectional areas of these are small compared to the rest of the film.

In general (but not necessarily at high temperature): D

_{S}> D_{gb}> D - D

Arrhenius Plot

for determining activation energiesstart with Fick's First Law:

Other effects on diffusion

Diffusion can be changed by stress fields, electric fields, other energy gradients (interfaces)[note: Ohring changes notation here: uses diffusion free energy per mole G

_{D}instead of diffusion energy per atom E_{D}so all of the k's become R's.]

Examine what happens when we apply a field:

How fast do atoms diffuse?