anisotropy

[zhubq]

If we consider pure grain growth, according to the phase field equations, the effect of anisotropic mobility and anisotropic interface energy is the same. Is that strange? anisotropic energy could get non-120 triple angle. Is it physical that only anisotropic mobility could get non-120 triple angle?

[Bernd]

Dear zhubq,

What you say is correct. Presently we are using an antisymmetric approximation in the phase-field equation, which neglects higher order effects in triple junctions. Therefore, one will always get 120°, whichever interfacial energies are used!

In our next release which you will probably get in February, the higher order terms in triple (or higher) junctions will be optionally available. With the new model activated, anisotropy of the mobility and of the interface energy wíll no longer be equivalent and the angles will be correctly described!

Bernd

[zhubq]

So the new release is based on some tricks or the basic phase-field model has been modified? If the model is changed, what's the new form? Thank you.

[janin]

The improved phase-field equations are derived from the following multiphase free energy functional:

free_energy_functional1.jpg (9.43 KiB) Viewed 6786 times

with

free_energy_functional2.jpg (12.99 KiB) Viewed 6785 times

The comprehensive multiphase relaxation approach takes into account that the phase-fields are subject to the unity constraint which yields the following set of phase-field equations:

phase_field_equation1.jpg (11.2 KiB) Viewed 6786 times

where dG is the thermodynamic driving force and K are pairwise curvature contributions:
Compared to the former model we now consider additional junction forces J:

phase_field_equation3.jpg (7.68 KiB) Viewed 6786 times

These junction forces only act when interfaces with different interfacial energies adjoin and result in angles different
from 120°.

I still like add that this is the isotropic version. I could give the full anisotropic formulation, but I'm not sure that this is really what you asked for. In most cases of grain growth, different interfacial energies are caused by misorientation rather than by anisotropy (i.e. dependence on the growth direction).

Regards,
Janin Eiken

[zhubq]

do you have a paper to discuss this new model in detail? If so ,could you send a copy to me (zhubq@interchange.ubc.ca)

[janin]

The basic model for the isotropic case has been published in
J. Eiken , B.Böttger, I.Steinbach
Multiphase-Field approach for multicomponent alloys with extrapolation scheme for numerical application
Phys. Rev. E 73 066122 (2006)
In my new formulation I still have changed the mobility and the prefactor to the thermodynamic driving force. The comprehensive anisotropic derivation was part of my thesis but has not yet been published yet in a paper.
Regards,
Janin

[zhubq]

So now it is not perfect to do anisotropic simulation in current version of MICRESS , although there is input option when we write the input file?

[janin]

I think you mix up two different things: on the one hand you talk about anisotropy and on the other hand you talk about junction equilibria. Actually both effects are seldomly coupled in applied simulations. Do you really refer to anisotropy (dependence on growth direction) or rather to misorientation (difference between the adjacent grain orientations). Maybe you could specify what kind of simulations you are really interested in?

Janin

[zhubq]

Oh, yeah, i mixed them up. I mean misorientation dependence. For a triple junction, there are 3 interface, if the interface energy of each binary boundary is different, so what kind of profile could we get from MICRESS, still 120-120-120?

[janin]

In case of misorientation you are absolutetely right. The present implementation still cannot treat the effect of different interface energies on the junction angles. This problems will be overcome by the improved multiphase model in the new release. To illustrate the effects of misorientation, I attach some simple examples, which have all been started from the same initial situation:

Initial situation

multigrain1.jpg (23.11 KiB) Viewed 6745 times

You can see that, in the present release, the angles are always adjusted to values around 120°. However, as there are no explicit junction forces, they may slightly deviate, which even in the case of equal interfacial energies results in differences compared to the new release. As you assumed, it makes no difference in the present model to use either half the value for the mobility or half the value for the interfacial energy. ( It only matters, if there are explicit driving forces.)
The first implemented model already represented a good start to face misorientation in a very simple way, but of course it still needed improvement. As you can see, the new release will be capable of reproducing the angles correctly according to Young's law.

multigrain2.jpg (70.04 KiB) Viewed 6744 times

Regards,

Janin