1. © meg/aol ‘02 Module 2: Diffusion In Generalized Media DIFFUSION IN SOLIDS Professor Martin Eden Glicksman Professor Afina Lupulescu Rensselaer Polytechnic Institute Troy, NY, 12180 USA

2. © meg/aol ‘02 Outline Diffusivity tensor Principal directions –Antisymmetric contribution –Symmetric contribution Diffusion in generalized media Cauchy relations Influence of imposed symmetry: Neumann’s Principle –Rotational symmetry operations –Isotropic materials –Cubic crystals –Orthotropic materials –Orthorhombic crystals –Monoclinic crystals –Triclinic crystals decreasing symmetry

3. © meg/aol ‘02 Diffusivity Tensor The diffusivity is defined operationally as the ratio of the flux magnitude to the magnitude of the concentration gradient. Equivalently, the diffusivity is the constant of proportionality between flux and gradient. Tensor form for Fick’s 1st law. The vector flux may be expanded as e3e3 e2e2 e1e1 Triad of unit vectors

4. © meg/aol ‘02 Diffusivity Tensor Elements comprising the matrix diffusivity. Fick’s 1st law in component form Cartesian coordinates

5. © meg/aol ‘02 Binary Diffusivity Tensor A = antisymmetric component S = symmetric component Any square matrix may be decomposed into the sum of a symmetric part and an antisymmetric part.

6. © meg/aol ‘02 Binary Diffusivity Tensor A = antisymmetric matrix: S = symmetric matrix:

7. © meg/aol ‘02 S = symmetric matrix: D ij = D ji Binary Diffusivity Tensor

8. © meg/aol ‘02 A = antisymmetric matrix: D ij = -D ji Binary Diffusivity Tensor

9. © meg/aol ‘02 Diffusion in Generalized Media Mass conservation symmetric responseantisymmetric response

10. © meg/aol ‘02 where the gradient operator is expressible as a column matrix Diffusion in Generalized Media In general:

11. © meg/aol ‘02 Diffusion in Generalized Media Antisymmetric response Order of differentiation is inconsequential!

12. © meg/aol ‘02 Cauchy relations The diffusivity is a symmetric tensor containing at most 6 elements: So the antisymmetric part contributes nothing!

13. © meg/aol ‘02 Neumann’s Symmetry Principle Tensor transformation rule (i, j, k, l=1, 2, 3) Neumann’s principle states that after any symmetry operation on the coordinate system Direction cosines transpose

14. © meg/aol ‘02 3-D Matrix Rotation: Implemented by Mathematica ® For an arbitrary rotation, , in 3-D about axis x 1  D ij Other arbitrary rotations about axes x 2 and x 3 must then be applied. TT

15. © meg/aol ‘02 Symmetry Operations for Diffusivity Tensors Four-fold rotation by  /2 about x 1 -axis

16. © meg/aol ‘02 Symmetry Operations for Diffusivity Tensors Two-fold rotation by  about x 1 -axis

17. © meg/aol ‘02 Symmetry Operations for Diffusivity Tensors Three-fold rotation by 2  /3 about x 1 -axis Important in hexagonal and rhombohedral systems.

18. © meg/aol ‘02 Symmetry Operations for Diffusivity Tensors No rotation Orthogonal coordinates Identity matrix: x1x1 x2x2 x3x3

19. © meg/aol ‘02 Isotropic Materials D is a scalar. —“Isotropy” is the lack of directionality— Flux vector, J, remains antiparallel to the applied concentration gradient,  C, and is invariant with respect to the gradient’s orientation within the material.

20. © meg/aol ‘02 Cubic Crystals Typical structure of many engineering materials. Includes FCC and BCC metals and alloys, and many cubic ceramic and mineralogical systems.

21. © meg/aol ‘02 Neumann’s principle applied to cubic symmetry Diffusivity tensor for cubic symmetry, where D 11 = D 22 Element-by-element comparison shows

22. © meg/aol ‘02 Diffusivity tensor for orthotropic materials (Tetragonal, Hexagonal, Rhombohedral) These crystals require two independent diffusivity elements.

23. © meg/aol ‘02 Diffusivity Tensor: orthorhombic, monoclinic, triclinic crystals Triclinic Symmetry arguments fail to reduce the number of independent elements in the diffusivity tensor of triclinic crystals. 6 elements are needed to describe diffusion responses in such low symmetry materials.This symmetry, although rare in engineering systems, exists in nature. Orthorhombic Monoclinic

24. © meg/aol ‘02 Exercise 1. The general diffusion response for a two dimensional lattice is Determine the forms of the diffusivity tensor for the following lattices (a) Square lattice (b) Rectangular lattice

25. © meg/aol ‘02 Matrix Transformations in 2-D x1x1 x2x2 x1x1 x 2  Matrix transformation rule: direction cosines

26. © meg/aol ‘02 Exercise (a) In 2-dimensions: The transformation matrix for an axis rotation of +π/2 is The diffusivity tensor in the rotated coordinate system Square lattice

27. © meg/aol ‘02 Exercise Rectangular lattice (b) In 2-dimensions: The transformation matrix for a mirror reflection is The diffusivity tensor in the transformed coordinate system 2 independent elements remain

28. © meg/aol ‘02 Exercise 2. Use the general transformation properties of the diffusivity tensor and show that in the cases of hexagonal, tetragonal, and rhombohedral crystals the mass flux and diffusivity are independent (orthotropic) of the orientation of the concentration gradient, providing that the gradient lies in the x 1 –x 2 plane. The transformation matrix for an arbitrary rotation, , about the x 3 –axis is given by

29. © meg/aol ‘02 Exercise Chemical gradient,  C, lying in the x 1 - x 2 plane, applied at angle  to the x 1 axis. The x 1, x 2, x 3, axes are rotated to make x 1 parallel to  C.

30. © meg/aol ‘02 Exercise The element D 11 in the rotated coordinate system is Orthotropic materials by definition have D 11 = D 22,

31. © meg/aol ‘02 Exercise: Implemented by Mathematica ® 3) A 2-D trapezoidal lattice with vectors a and b has the matrix diffusivity, [D ij ], given by Find the flux response, J.

32. © meg/aol ‘02 Exercise: Implemented by Mathematica ®

33. © meg/aol ‘02 Exercise J CC ~26.5°

34. © meg/aol ‘02 Key Points The diffusion coefficient, D, is in general a tensor quantity expressible in matrix form, [D ij ]. Physical and mathematical arguments shown that in 3-D, the diffusion matrix has at most 6 independent elements. In 2-D, at most 4 independent elements occur. Neumann’s principle may be applied to reduce the maximum number of diffusivity elements on the basis of crystallographic symmetry operations. Many engineering materials fortuitously often exhibit isotropic diffusion behavior. Crystal structure and texture have profound influences on the diffusion response of a material.