Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 1 Part 3 ppt

References 35

For crystals with triclinic, monoclinic, and orthorhombic symmetry

all three principal diffusivities are different:

D1 = D2 = D3 . (2.17)

Among these crystal systems only for crystals with orthorhombic symmetry

the principal axes of diffusion do coincide with the axes of crystallographic

symmetry.

For uniaxial materials, such as trigonal, tetragonal, and hexagonal

crystals and decagonal or octagonal quasicrystals, with their unique axis

parallel to the x3-axis we have

D1 = D2 = D3 . (2.18)

For uniaxial materials Eq. (2.16) reduces to

D(Θ) = D1 sin2 Θ + D3 cos2 Θ , (2.19)

where Θ denotes the angle between diffusion direction and the crystal axis.

For cubic crystals and icosahedral quasicrystals

D1 = D2 = D3 ≡ D

and the diffusivity tensor reduces to a scalar quantity (see above).

The majority of experiments for the measurement of diffusion coefficients

in single crystals are designed in such a way that the flow is one-dimensional.

Diffusion is one-dimensional if a concentration gradient exists only in the

x-direction and both, C and ∂C/∂x, are everywhere independent of y and z.

Then the diffusivity depends on the crystallographic direction of the flow. If

the direction of diffusion is chosen parallel to one of the principal axis (x1,

or x2, or x3) the diffusivity coincides with one of the principal diffusivities

D1, or D2, or D3. For an arbitrary direction, the measured D is given by

Eq. (2.16).

For uniaxial materials the diffusivity D(Θ) is measured when the crys￾tal or quasicrystal is cut in such a way that an angle Θ occurs between the

normal of the front face and the crystal axis. For a full characterisation of

the diffusivity tensor in crystals with orthorhombic or lower symmetry mea￾surements in three independent directions are necessary. For uniaxial crystals

two measurements in independent directions suffice. For cubic crystals one

measurement in an arbitrary direction is sufficient.

References

1. A. Fick, Annalen der Phyik und Chemie 94, 59 (1855); Philos. Mag. 10, 30

(1855)

36 2 Continuum Theory of Diffusion

2. J.B.J. Fourier, The Analytical Theory of Heat, translated by A. Freeman, Uni￾versity Press, Cambridge, 1978

3. J. Crank, The Mathematics of Diffusion, 2nd edition, Oxford University Press,

1975

4. I.N. Bronstein, K.A. Semendjajew, Taschenbuch der Mathematik, 9. Auflage,

Verlag Harri Deutsch, Z¨urich & Frankfurt, 1969

5. J.F. Nye, Physical Properties of Crystals: their Representation by Tensors and

Matrices, Clarendon Press, Oxford, 1957

6. S.R. de Groot, P. Mazur, Thermodynamics of Irreversible Processes, North￾Holland Publ. Comp., 1952

7. J. Philibert, Atom Movement – Diffusion and Mass Transport in Solids, Les

Editions de Physique, Les Ulis, Cedex A, France, 1991

8. M.E. Glicksman, Diffusion in Solids – Field Theory, Solid-State Principles and

Applications, John Wiley & Sons, Inc., 2000