## interpretation of the tetragonal anisotropy functions

Exchange about the physics background, diffuse interface theory, etc..
deepumaj1
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### interpretation of the tetragonal anisotropy functions

Hi. Micress has the following tetragonal anisotropy functions in built:
# This anisotropic interaction is not yet implemented.
# Anisotropy of interfacial stiffness? (tetragonal)
# (1 - delta * (4*(nx^4 +nz^4) -3)
# * (1-nz^2 +nz^2*faktor)
# Coefficient delta (<1.), factor z/x (>0)? (2 REALS)
0.05 2.
# Anisotropy of interfacial mobility? (tetragonal)
# (1 + delta * (4*(nx^4 +nz^4) -3)
# * (1-nz^2 +nz^2*faktor)
# Coefficient delta (<1.), factor z/x (>0)? (2 REALS)
0.05 0.02
What are the meaning of the coefficients, delta and the factor? Does these coefficients(delta, factor) have any physical significance? Can anyone show the original reference papers where these functions are referenced? Thanks.

Deepu

deepumaj1
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### Re: interpretation of the tetragonal anisotropy functions

Deepu

janin
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Joined: Thu Oct 23, 2008 3:06 pm

### Re: interpretation of the tetragonal anisotropy functions

Hi Deepu,
the tetragonal anisotropy is a 4-fold cosine function 1 + delta*cos(4*theta) which can be elongated in z-direction by an elangation factor.
This means, you will get a cosine function with 1+ delta as maximum and 1-delta as minimun, if the elongation factor is 1. In the general case, the elongation factor stretches the whole function in z direction such that the z-maximum becomes factor * (1+delta).
A graphical illustration of the function has been given in a former thread:
http://board.micress.de/viewtopic.php?f ... tropy#p477

Note that this anisotropy corresponds to the stiffness, not to the interfacial energy itself.
The maximal value of the kinetic coefficient is commonly related to the minimum value of the stiffness.

This is a rather pragmatic appraoch and has never been published. If you prefer a more physical description of your anisotropy, you may use the 'harmonic expansion' feature in Micress.

More information can be found in the Micress user guide, volume 2, section 3.8.3.

Best regards,
Janin

deepumaj1
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Joined: Fri Feb 13, 2015 12:00 pm
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### Re: interpretation of the tetragonal anisotropy functions

Hi Janin,
Thanks for the explanation. What does the factor nx, ny and nz stands for. Does it represent the co-ordinates in the cartesian plane?

Deepu

janin
Posts: 31
Joined: Thu Oct 23, 2008 3:06 pm

### Re: interpretation of the tetragonal anisotropy functions

nx,ny,nz are the cartesian coordinates of the local interfacial normal vector transformed into the coordinate system of the anisotropic grain.

They can be transformed into spherical coordinates by:
nx = cos(phi) *sin(theta)
ny = sin(phi)*sin(theta)
nz = cos(theta)

In 2d this reduces to:
nx = sin(theta)
ny = 0
nz = cos(theta)

and inserted into the 2D anisotropy function this yields
a = 1 - delta * (4*(nx^4 +nz^4) -3)
= 1 - delta * (4*(cos(theta)^4 +sin(theta)^4) -3)
= 1 - delta * cos( 4 theta)
using cos(theta)^4 +sin(theta)^4 = 1/4 (cos(4 theta) + 3)

Regards,
Janin

Lilly
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### Re: interpretation of the tetragonal anisotropy functions

Hi Janin

I have a problem that why is the maximal value of the kinetic coefficient commonly related to the minimum value of the stiffness?

best regards,
Lilly

janin
Posts: 31
Joined: Thu Oct 23, 2008 3:06 pm

### Re: interpretation of the tetragonal anisotropy functions

I won't say that this statement can be generalized, but both the kinetic coefficient and the interfacial energy sigma depend on the specific crystal lattice and their anisotropy can commonly be described by very similar functions, e.g. for cubic systems in 2D we assume
a_kin = 1 + delta_kin * cos (4 theta) and a_sigma = 1 + delta_sigma * cos (4 theta).
So, here we assume that the interfacial energy sigma has maximal values where the kinetic coefficient has maximal values .

The stiffness sigma* can in 2D be evaluated by stiff = sigma + sigma'' (sigma'' here denotes the second derivate with respect to theta, in 3D this is more complex).
For the cubic anistropy function given above this yields:
stiff = 1 - 15 * delta_sigma * cos (4 theta) = 1 - delta_stiff * cos (4 theta) .
You can see that the stiffness has minimal values where the interfacial energy sigma (and also the kinetic coeffient) has maximal values.

Dendrite arms generally form in directions of minimal stiffness and maximal kinetic coefficient.

Regards,
Janin