Jacobi form

2013-08-11T17:30:36Z

165.124.167.212: /* Definition */

The '''effective diffusion coefficient''' (also referred to as the apparent diffusion coefficient) of a diffusant in [[atomic diffusion]] of solid [[polycrystalline]] materials like [[metal alloy]]s is often represented as a [[weighted average]] of the [[grain boundary diffusion coefficient]] and the [[lattice diffusion coefficient]].<ref name=Heitjans>P. Heitjans, J. Karger, Ed, “Diffusion in condensed matter: Methods, Materials, Models,” 2nd edition, Birkhauser, 2005, pp. 1-965.</ref> Diffusion along both the grain boundary and in the lattice may be modeled with an [[Arrhenius equation]]. The ratio of the grain boundary diffusion activation energy over the lattice diffusion activation energy is usually 0.4 - 0.6, so as temperature is lowered, the grain boundary diffusion component increases.<ref name=Heitjans /> Increasing temperature often allows for increased grain size, and the lattice diffusion component increases with increasing temperature, so often at 0.8Tmelt (of an alloy), the grain boundary component can be neglected.

==Modeling==
The effective diffusion coefficient can be modeled using Hart's equation when only grain boundary and lattice diffusion are dominant:

:<math> {D^{\mathrm{eff}}} = f</math> Dgb <math> + (1-f) </math>Dl.

where
:<math> {D^{\mathrm{eff}}} = </math> effective diffusion coefficient.
:Dgb = grain boundary diffusion coefficient.
:Dl = lattice diffusion coefficient.
:<math> f = \tfrac{q}{d} </math>.δ
:<math> q = </math> value based on grain shape, 1 for parallel grains, 3 for square grains.
:<math> d = </math> average grain size.
:δ <math> = </math> grain boundary width,often assumed to be 0.5 nm.

Grain boundary diffusion is significant in [[face centered cubic]] metals below about 0.8 Tmelt (Absolute). Line dislocations and other [[Crystallographic defect|crystalline defects]] can become significant below ~0.4 Tmelt in FCC metals.

== References ==
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== See also ==
* [[Kirkendall effect]]
* [[Phase transformations in solids]]
* [[Mass diffusivity]]

[[Category:Diffusion]]

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Jacobi form