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Module 2: Diffusion In Generalized Media

DIFFUSION IN SOLIDS

Professor Martin Eden GlicksmanProfessor Afina Lupulescu

Rensselaer Polytechnic InstituteTroy, NY, 12180

USA

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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

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Diffusivity Tensor

J i =− Dij{ } ∂C∂xj

⎛ ⎝ ⎜

⎞ ⎠ ⎟ i, j =1,2,3( )

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.

J r( )=J 1e1 +J 2e2 +J 3e3

The vector flux may be expanded as e3

e2

e1

Triad of unit vectors

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Diffusivity Tensor

Dij[ ]=D11 D12 D13

D21 D22 D23

D31 D32 D33

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Elements comprisingthe matrix diffusivity.

J 1 =−D11∂C∂x1

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −D12∂C∂x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −D13∂C∂x3

⎛ ⎝ ⎜ ⎞

⎠ ⎟

J 2 =−D21∂C∂x1

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −D22∂C∂x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −D23∂C∂x3

⎛ ⎝ ⎜ ⎞

⎠ ⎟

J 3 =−D31∂C∂x1

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −D32∂C∂x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −D33∂C∂x3

⎛ ⎝ ⎜

⎞ ⎠ ⎟

⎫

⎬

⎪ ⎪ ⎪ ⎪

⎭

⎪ ⎪ ⎪ ⎪

Fick’s 1st law in component form Cartesian coordinates

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Binary Diffusivity Tensor

A = antisymmetric component

S = symmetric component Dij[ ]S ≡12 Dij +Dji( )= Dji[ ]S

Dij[ ]A ≡12 Dij −Dji( )=- Dji[ ]A

Dij[ ]S+Dij[ ]ADij[ ]=

Any square matrix may be decomposed into the sum of a symmetric part and an antisymmetric part.

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Binary Diffusivity Tensor

A = antisymmetric matrix:

S = symmetric matrix:

Dij[ ]S =D11

12 D12 +D21( ) 1

2 D13 +D31( )12 D21+D12( ) D22

12 D23 +D32( )

12 D31+D13( ) 1

2 D32 +D23( ) D33

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Dij[ ]A =0 1

2 D12 −D21( ) 12 D13−D31( )

12 D21−D12( ) 0 1

2 D23−D32( )12 D31−D13( ) 1

2 D32−D23( ) 0

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Dij[ ]S+Dij[ ]ADij[ ]=

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S = symmetric matrix:

Dij[ ]S =

D1112 D12 +D21( ) 1

2 D13 +D31( )

12 D21+D12( ) D22

12 D23 +D32( )

12 D31+D13( ) 1

2 D32 +D23( ) D33

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Dij= Dji

Binary Diffusivity Tensor

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A = antisymmetric matrix:

Dij[ ]A =

0 12 D12 −D21( ) 1

2 D13−D31( )

12 D21−D12( ) 0 1

2 D23−D32( )

12 D31−D13( ) 1

2 D32−D23( ) 0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Dij= -Dji

Binary Diffusivity Tensor

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Diffusion in Generalized Media

∂C∂t

=−∇⋅J

∂C∂t

=∇ ⋅ Dij[ ]S ∂C∂xj

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ej + Dij[ ]A ∂C

∂xj

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ej

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Mass conservation

∂C∂t

=−∇⋅− Dij[ ] ⋅ ∂C∂xj

⎛ ⎝ ⎜

⎞ ⎠ ⎟ ej

symmetric response antisymmetric response

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where the gradient operator is expressible as a column matrix

Diffusion in Generalized Media

∂C∂t

=∇ ⋅ Dij[ ]S ⋅∇ C+∇ ⋅ Dij[ ]A ⋅∇C

∇ ⋅ Dij[ ]S =

∂∂x1

∂∂x2

∂∂x3

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

⋅ S⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥

∇ ⋅ Dij[ ]A =

∂∂x1

∂∂x2

∂∂x3

⎛

⎝

⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟

⋅ A⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥

In general:

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Diffusion in Generalized Media

Antisymmetric response

∂C∂t

⎛ ⎝

⎞ ⎠

A

= ∂2 C∂x1

2⎛ ⎝ ⎜ ⎞

⎠ ⎟ 0( )+ ∂2C∂x1∂x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 12 D12 −D21( )[ ]+ ∂2 C

∂x1∂x3

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 12 D13−D31( )[ ]

+ ∂2C∂x2∂x1

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 12 D21−D12( )[ ]+ ∂2 C

∂x22

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 0( )+ ∂2 C∂x2∂x3

⎛ ⎝ ⎜ ⎞

⎠ ⎟ 12 D23−D32( )[ ]

+ ∂2 C∂x3∂x1

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 1

2 D31−D13( )[ ]+ ∂2 C∂x3∂x2

⎛ ⎝ ⎜

⎞ ⎠ ⎟ 1

2 D32 −D23( )[ ]+ ∂2 C∂x3

2⎛ ⎝ ⎜

⎞ ⎠ ⎟ 0( )

∂2C∂xi∂xj

⎛ ⎝ ⎜

⎞ ⎠ ⎟ =

∂2C∂xj∂xi

⎛ ⎝ ⎜

⎞ ⎠ ⎟

Order of differentiation is inconsequential!

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Cauchy relations

∂C∂t

⎛ ⎝

⎞ ⎠

A

=0

Dij[ ]=D11 D12 D13

D12 D22 D23

D13 D23 D33

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

The diffusivity is a symmetric tensor containing at most 6 elements:

So the antisymmetric part contributes nothing!

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Neumann’s Symmetry Principle

Dij'[ ]= ∂xi

'

∂xk

⎛ ⎝ ⎜ ⎞

⎠ ⎟ ⋅∂xj

'

∂xl

⎛ ⎝ ⎜

⎞ ⎠ ⎟ Dkl[ ] Tensor transformation

rule (i, j, k, l=1, 2, 3)

Dij'[ ]=αikα jl Dkl[ ]= α[ ] Dkl[ ] α[ ]T

Dij'[ ]= Dij[ ]

Neumann’s principle states that after any symmetry operationon the coordinate system

Direction cosines

transpose

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3-D Matrix Rotation: Implemented by Mathematica®

• For an arbitrary rotation, , in 3-D about axis x1

Dij

• Other arbitrary rotations about axes x2 and x3 must then be applied.

T

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Symmetry Operations for Diffusivity Tensors

Four-fold rotation by /2 about x1-axis

α[ ]=1 0 00 0 10 −1 0

⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥ x

3

x

(a)

2

x

1x1'

x2'

x3'

/2

x2

x3

x1

(b)x1

'

x2'

x3'

π/2

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Symmetry Operations for Diffusivity Tensors

Two-fold rotation by about x1-axis

α[ ]=1 0 00 −1 00 0 −1

⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥

x

3

x

(a)

2

x

1x1'

x2'

x3'

1

(c)

x2

x3

x

x3'

x1'

x2'

π

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Symmetry Operations for Diffusivity Tensors

Three-fold rotation by 2/3 about x1-axis

α[ ]=1 0 00 −1

23

2

0 − 32 −1

2

⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥

Important in hexagonal and rhombohedral systems.

x

3

x

(a)

2

x

1x1'

x2'

x3'

2/3

(d)

x3

x2

x1

x3'

x1'

x2'

2π/3

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Symmetry Operations for Diffusivity Tensors

No rotationOrthogonal coordinates

Identity matrix:

α[ ]=1 0 00 1 00 0 1

⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥

x

3

x

(a)

2

x

1x1'

x2'

x3'

x

3

x

(a)

2

x

1x1'

x2'

x3'

x1

x2

x3

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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.

Dij[ ]=D 0 00 D 00 0 D

⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥ =D .

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Cubic Crystals

x 1< 1 0 0 >

< 0 1 0 >

< 0 0 1 >x 3

x 2

Typical structure of many engineering materials. Includes FCC and BCC metals and alloys, and many cubic ceramic and mineralogical

systems.

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Neumann’s principle applied to cubic symmetry

Diffusivity tensor for cubic symmetry, where D11 = D22

Dij'[ ]=

D11 D13 −D12

D31 D33 D32

−D21 −D23 D22

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Dij'[ ]=

D11 0 00 D22 00 0 D22

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

D22 =D33

D12 =D13

D13 =−D21 =0D32 =−D23 =0.

Element-by-elementcomparison shows

/2

x2

x3

x1

(b)x1

'

x2'

x3'

π/2

=D (scalar)

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Diffusivity tensor for orthotropic materials (Tetragonal, Hexagonal, Rhombohedral)

Dij =0 i ≠j( )

Dij[ ]=D11 0 00 D11 00 0 D33

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

These crystals require two independent diffusivity elements.

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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 elementsare needed to describe diffusion responses in such low symmetry materials.This symmetry, although rare in engineering systems, exists in nature.

Orthorhombic Dij[ ]=D11 0 00 D22 00 0 D33

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

Monoclinic Dij[ ]=D11 0 D13

0 D22 0D13 0 D33

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥

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Exercise

J 1 =−D11∂C∂x1

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −D12∂C∂x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟

J 2 =−D12∂C∂x1

⎛ ⎝ ⎜ ⎞

⎠ ⎟ −D22∂C∂x2

⎛ ⎝ ⎜ ⎞

⎠ ⎟ .

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

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Matrix Transformations in 2-D

α[ ]≡ α11 α12

α21 α22

⎡ ⎣ ⎢

⎤ ⎦ ⎥ =

cosθ cos90−θ( )cos90+θ( ) cosθ

⎡ ⎣ ⎢

⎤ ⎦ ⎥ x1

x2

x 1

x 2

α[ ]T ≡ α11 α21

α12 α22

⎡ ⎣ ⎢

⎤ ⎦ ⎥ =

cosθ cos90+θ( )cos90−θ( ) cosθ

⎡ ⎣ ⎢

⎤ ⎦ ⎥

α[ ]⋅Dij ⋅ α[ ]T =α11 α11D11+α12D21( )+α12 α11D12 +α12D22( ) α21 α11D11+α12D21( )+α22 α11D12 +α12D22( )α11 α21D11+α22D21( )+α12 α21D12 +α22D22( ) α21 α21D11+α22D21( )+α22 α21D12 +α22D22( )

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Matrix transformation rule:

direction cosines

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Exercise

Dij[ ]= D11 D12

D21 D22

⎡ ⎣ ⎢

⎤ ⎦ ⎥

α[ ]= 0 1−1 0

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Dij'[ ]= D22 −D21

−D12 D11

⎡ ⎣ ⎢

⎤ ⎦ ⎥

x2

x1

(a) In 2-dimensions:

The transformation matrix for an axis rotation of +π/2 is

The diffusivity tensor in therotated coordinate system

Dij[ ]= D11 00 D11

⎡ ⎣ ⎢

⎤ ⎦ ⎥ =D11

Square lattice

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Exercise

αij[ ]= 1 00 −1

⎡ ⎣ ⎢

⎤ ⎦ ⎥

Dij[ ]= D11 00 D22

⎡ ⎣ ⎢

⎤ ⎦ ⎥

x2

x1

Rectangular latticeDij[ ]= D11 D12

D21 D22

⎡ ⎣ ⎢

⎤ ⎦ ⎥

(b) In 2-dimensions:

The transformation matrix for a mirror reflection is

The diffusivity tensor in thetransformed coordinate system

Dij'[ ]= D11 −D12

−D21 D22

⎡ ⎣ ⎢

⎤ ⎦ ⎥

2 independent elements remain

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Exercise

α[ ]=cosθ sinθ 0−sinθ cosθ 0

0 0 1

⎡

⎣ ⎢ ⎢ ⎢

⎤

⎦ ⎥ ⎥ ⎥

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 x1–x2 plane.

The transformation matrix for an arbitrary rotation, , about the x3–axis is given by

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Exercise

x1

x2

x3

x2'

∇C

x1'

Chemical gradient, C, lying in the x1- x2 plane, applied at angle to the x1

axis. The x1, x2

, x3 , axes

are rotated to make x1

parallel to C.

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Exercise

D11' =α11α11D11+α12α12D22,

D11' =cos2θD11+sin2θD22.

The element D11 in the rotated coordinate system is

Orthotropic materials by definition have D11= D22 ,

′ D 11= cos2θ+sin2θ( )D11

′ D 11=const.

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Exercise: Implemented by Mathematica®

3) A 2-D trapezoidal lattice with vectors a and b has the matrix diffusivity, [Dij], given by

Dij = 1 0.50.5 2.5

⎡ ⎣ ⎢

⎤ ⎦ ⎥ ×10−9 cm2/s

• Find the flux response, J. Given the trapezoidal lattice with lattice vectors a and b and a diffusivity tensor,{Dij}.

881* 10^-9 Cm̂ 2êS, 0.5 * 10 -̂ 9 Cm̂ 2 êS <,80.5 * 10^-9 Cm̂ 2êS , 2.5 * 10 -̂ 9 Cm̂ 2 êS <<êêN

: : 1.¥10-9 Cm2

S,

5.¥10-10 Cm2

S>, :

5.¥10-10 Cm2

S,

2.5¥10-9 Cm2

S>>

MatrixForm @%D

i

k

1.¥10-9 Cm2

S5.¥10-10 Cm2

S

5.¥10-10 Cm2

S2.5¥10-9 Cm2

S

y

{

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Exercise: Implemented by Mathematica®

A chemical gradient is applied in the form of the vector

gradC =882* 10^5 GramêCm̂ 4<, 80<<

The flux response is the vector J,

J =Dij . gradC

Dij.gradC

i

k

1.`*^-9 Cm2

S5.`*^-10 Cm2

S5.`*^-10 Cm2

S2.5`*^-9 Cm2

S

y

{

.882 * 10 ^ 5 Gram ê Cm ^ 4<, 80<<

::0.0002Gram

Cm2 S>, :

0.0001GramCm2 S

>>

MatrixForm @%D

i

k

0.0002GramCm2 S

0.0001GramCm2 S

y

{

© meg/aol ‘02

ExerciseThe angle of the flux is

j =ArcTan @H1* 10^-4L êH2* 10^- 4LD

ArcTanB12F

ArcTan A12E êê N

0.463648

0.4636476090008061` *180 êPi

26.5651

26.56505117707799`

26.56505117707799` Degrees

The flux magnitude is

MagJ = $ ik0.0002` Gram

Cm 2 Sy{^2 + i

k0.0001` Gram

Cm2 Sy{^2

0.000223607$Gram2

Cm4 S2

ScientificForm @%D

2.3607¥10-4

"2.3607 " ¥10"-4" G êHCm ^2 *SL* 10

J

C~26.5°

© meg/aol ‘02

Key Points• The diffusion coefficient, D, is in general a tensor quantity expressible in

matrix form, [Dij].• 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.