Morphology of dendrites

[WTMuser]

Dear Bernd,

we performed 2D calculations of dendritic growth. The results indicate different morphologies in dependence on chemical composition (see Figure). The used setup (e.g. anisotropy coefficient, interfacial energy and mobility) was constant during investigation. Are the differences in morphology caused by solute drag model?

Thanks for your support.
Martin

[Bernd]

Hi Martin,

I an a bit confused with the term "solute drag model". For my understanding (and how it is used in MICRESS see here), a solute drag model is a sort of meso-model which takes into account the effect of impurities or solute during e.g. grain growth, without explicitly modeling the redistribution and diffusion of these solutes.
The effect which you see in your simulations is caused by the direct interplay of solute redistribution and diffusion. Obviously, in case a) there is a stronger segregation which results in a "more dendritic" morphology. You can call this effect "solute drag", but then MICRESS is the solute drag model...

So, apart from my problems with the wording, the answer is probably yes!

Bernd

[WTMuser]

Hi Bernd,

Starting from your explanation I would not call this effect "solute drag". If a more dendritic morphology is caused only by segregation, I would expect constitutional undercooling in front of a dendrite arm (dendritic growth due to destabilization). Regarding result files from Micress I could not observe a temperature drop in front of the dendrites.

(Up to now I assumed that the shape of dendrites is mainly caused by anisotropy coefficient, interfacial energy and interfacial mobility.)

Thanks for your support.
Martin

[Bernd]

Hi Martin,

but if you assume a constant cooling rate (as you do in MICRESS, I suppose), there cannot be any temperature drop in front of the dendrites - instead, the higher constitutional undercooling results in a slower growth of the interface in the concave regions where diffusion of solute away from the interface is disfavoured, and thus in a more dendritic shape!

Thus, apart from the factors you mentioned, composition plays an important role for morphology!

Bernd

[WTMuser]

Hi Bernd,

I agree with you. An increased constitutional undercooling results in a higher growth of convex regions / slower growth of concave regions and we will see a more dendritic shape. Is there any result file, which contains information about the constitutional undercooling at each step?

If there is no information about constitutional undercooling I would like to ask another question. How does Micress know that the diffusion of solute away from the interface of concave regions is disfavoured? Of course, diffusion is powered by the gradient. But maybe there is another specific parameter (corresponding to the shape), which rules the diffusion in Micress.

Martin

[Bernd]

Hi Martin,

MICRESS does not have a direct output for constitutional undercooling, because a reference state is needed for its calculation, like the liquidus temperature at the initial composition.
But if the kinetic undercooling is small - and this is what we assume in case of diffusion-limited growth - evaluation of the constitutional undercooling is quite simple: For a planar front, the difference between the liquidus temperature and the temperature at the front is exactly the constitutional undercooling. For a curved front, the difference between the constitutional undercooling at the concave and convex regions is only given by curvature undercooling. Curvature undercooling is directly reflected in the .driv output (only in case of diffusion limited growth!)! You just have to divide the driving force value by the transformation entropy.

MICRESS does not explicitly know that diffusion is favoured at convex regions and disfavoured at concave regions - this is just an implicit result of the 2D/3D diffusion equation!

Bernd

[WTMuser]

Dear Bernd,

I have a further question concerning the morphology of dendrites. Why does each dendrite displays 4 arms? How does Micress provide this "typical" morphology in detail? Only by diffusion/segregation?

Thanks for your answer.
Martin

[Bernd]

Hi Martin,

The four-fold symmetry of the dendrite is due to the anisotropy functions and is specified in the phase data for fcc with the keyword "cubic". More on anisotropy you can find e.g. here!

Bernd

[WTMuser]

Dear Bernd,

I have some more questions concerning the morphology of dendrites. Of course, the dendritic growth in MICRESS is ruled by anisotropic coefficients. The interfacial energy is ruled by an anisotropic coefficient, which is posted in the user manual (1-cos(4*theta)). The function of this coefficient is quite smooth and displays no instabilities. But obviously the calculated dendrite displays closely spaced tertiary arms (see Figure). That is why I am a little bit confused, because from my point of view the anisotropic coefficient should inhibit closely spaced tertiary dendrites.

The second question deals with the symmetry. I would expect a symmetrical morphology, because of symmetrical anisotropic functions (for mobility and interfacial energy). It is clear to see that the tertiary arms on the left side (see Figure) are less pronounced. So what is the reason for this finding?

Due to the user manual the calculation of the driving force (out_driv_force) is performed by the following equation: dG = dS*dT. The corresponding result file displays differences for concave and convex regions. Independent from concave or convex shaped morphologies the chemical composition at the phase boundary is just constant for a given time step. So I would expect no differences in dG.

Kind regards.
Martin

[Bernd]

Dear Martin,

Let me try to to answer your questions:

The first question is about the closely spaced tertiary dendrite arms. As you can see, both of them are growing in an 90° angle to the secondary arm direction (110). Due to the fourfold nature of the anisotropy function, this direction is also a preferred one, where the anisotropy coefficient has a maximum (or minimum, depending on whether it is kinetic or static). Thus it is easy to understand that tertiary arms are expected to grow.
What I do not understand is why closely spaced tertiary arms should be inhibited - their spatial proximity has nothing to do with their orientation! Essentially, the formation of tertiary arms is similar to the break-up of a planar front, and you would need to enter deeply into perturbation theory to understand with which characteristic length (corresponding to the distance between tertiary arms) break-up is taking place.

In principle you are right with respect to the symmetry of the problem. But if there is noise - and there is noise! - this noise will be greatly amplified if instabilities like those which lead to the formation of tertiary arms are involved. The noise comes primarily from coupling to Thermo-Calc, from the intended use of noise in the dG options, and also from some other less clearly defined numerical sources.
Thus, if you are simulating dendrites, you never should expect them to be very symmetric! By the way, there are persons claiming that without any type of noise, tertiary arm would even not form at all...

Third question: The chemical composition at the phase boundary is not constant! It depends on curvature according to the Gibbs-Thompson effect!

Best wishes

Bernd