In a random walk, the average distance moved from the starting point is proportional to the square-root of time. \(\overline x = \lambda \sqrt {\nu {\kern 1pt} t} \)

Diffusion occurs via two mechanisms, either through the movement of vacancies or interstitial atoms moving between different interstitial sites.

Fick’s 1st and 2nd Laws can be used to calculate the flux of atoms through a crystal structure under different conditions

\[J \equiv - D\left\{ {\frac{{\partial C}}{{\partial x}}} \right\}\;\;{\sf{Fick's}}\;{\sf{1st}}\;{\sf{Law}}\]

\[\frac{{\partial C}}{{\partial t}} = D\left\{ {\frac{{{\partial ^2}C}}{{\partial {x^2}}}} \right\}\;\;{\sf{Fick's}}\;{\sf{2nd}}\;{\sf{Law}}\]

where D is diffusivity

The rate of diffusion varies exponentially with temperature, following the equation;

\[D = {D_0}\exp \left( {\frac{{ - H*}}{{{k_B}T}}} \right)\]

Diffusivity also increases with the vacancy concentration in substitutional diffusion, which means that it is generally higher along grain boundaries and dislocations due to the disruption to the lattice.

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