Tensors for Physics

Appendix: Exercises... 417

Furthermore, determine the anisotropy ratio R=(D‖−D⊥)/(D‖+ 2 D⊥),
the average diffusion coefficientD ̄ =^13 D‖+^23 D⊥and the geometric meanD ̃=

D
1 / 3
‖ D

2 / 3

⊥ .Discuss the cases Q>^1 and Q<^1 for prolate and oblate particles.

The fluxjtransforms like the position vector, viz. jμ = A−μν^1 /^2 jνA. The spatial
gradient in real space is linked with the nabla operator in affine space according

to∇ν=A^1 νκ/^2 ∇Aκ.
From the diffusion law in the affine space,jλA=−DAλκ∇νAρ, whereDAλκis the
diffusion tensor in that space, follows

jμ=A−μλ^1 /^2 jλA=−A−μλ^1 /^2 DAλκ∇Aνρ=−A−μλ^1 /^2 DλκAA−κν^1 /^2 ∇νρ.

Comparison with the diffusion law in real space andDλκA=D 0 δλκimplies

Dμν=A

− 1 / 2

μλ D

A

λκA

− 1 / 2

κν =A

− 1

μνD^0.

For uniaxial particles with their symmetry axis parallel to the director, one has

A−μν^1 =Q−^2 /^3

[

δμν+(Q^2 − 1 )nμnν

]

,

cf. (5.58). HereQ=a/bis the axes ratio of an ellipsoid with the semi-axesaand
b=c. Thus the above mentioned resultD‖=Q^4 /^3 D 0 ,D⊥=Q−^2 /^3 D 0 is obtained.
This implies

R=(D‖−D⊥)/(D‖+ 2 D⊥)=(Q^2 − 1 )/(Q^2 + 2 ),

and

D ̄=^1

3

D‖+

2

3

D⊥=

1

3

Q−^2 /^3 (Q^2 + 2 )D 0 ,

for the ansotropy ratioRand for the average diffusion coefficientD ̄. The geometric

mean diffusion coefficientD ̃=D‖^1 /^3 D⊥^2 /^3 =D 0 is independent of the axes ratioQ.
As expected intuitively, prolate particles withQ>1 diffuse faster in the direction
parallel ton, sinceD‖>D⊥, in this case. For prolate particles, pertaining toQ<1,
one hasD‖<D⊥. There the diffusion alongnis slower compared with the diffusion
in a direction perpendicular ton. The anisotropy ratioRis positive for prolate and
negative for oblate particles.