Morphology of dendrites

[WTMuser]

Dear Bernd,

You mentioned that the chemical composition depends on the curvature (Gibbs-Thomson-effect). But I am still confused about the details. (What came first, the curvature or the chemistry?)

I would expect the following approach. The Gibbs-Thomson-effect is given by the following equation: dT=2*sigma*curvature/dS. Starting from this relation one can calculate the curvature undercooling for given parameters (curvature, entropy drop, interfacial energy). Calculations of the chemical composition c are performed by the use of a corresponding phase diagram and curvature undercooling dT. But how does MICRESS determine the necessary information for curvature and the entropy drop?

Kind regards.
Martin

[Bernd]

Dear Martin,

In MICRESS, it works the following way: Curvature leads to a driving force dG. This driving force moves the front, leading to a redistribution of solute and a composition change. If there was no diffusion, the movement would stop at the point where the driving force which is due to curvature is exactly compensated by the driving force due to the composition change. It would stay like that forever.
With diffusion, the situation is not stable and we observe Ostwald ripening.

This understanding is important for efficiently using MICRESS: The observation of positive and negative values of the driving force dG in concave and convex regions of the interface demonstrates that interface kinetics are fast compared to diffusion (i.e. we are close to the diffusion limit).
On the other hand, if the interface mobility is small, diffusion will "wipe out" the composition differences, and the kinetic undercooling will be displayed in the driving force output. That means, in cases which are close to the diffusion limit like in solidification (at least of the primary phase), the occurance of changes of dG from positive to negative are good a qualitative indication that the interface mobility is high enough.

Bernd

[WTMuser]

Hi Bernd,

I checked the chemical composition at the phase boundary by the use of *.c{k}Pha{i}. The attached Figure displays the concentration of hafnium in the liquid phase / interface region between liquid and solid. Unfortunately I could not find any differences between concave and convex shaped regions. All the other elements (Al, Ta, Ti, ...) confirm this finding.

Kind regards.
Martin

[Bernd]

Hi Martin,

I can imagine 2 reasons for your finding:

1.) Mobiity is too low and you are not close to the diffusion limit. This you would see in the .driv output where you should have alternating values of dG for the concave and convex regions. If dG is negative everywhere, there is still a considerable amount of kinetic undercooling, and the composition differences could be much smaller due to the dominance of diffusion over interface movement. If you use a temperature-dependent mobility, it can easily happen that the mobility drops a bit too fast towards the end of solidification...
Not being close enough to the diffusion limit does not necessarily mean that the simulation is wrong, but, depending on the amount of kinetic undercooling, formation of side branches could be suppressed.

2.) In the figure which you provided, the color scale for the liquid composition is quite coarse. If you "zoom in" to the liquid region with DP_MICRESS and make an automatic rescale, you will see the differences and how big they are. These differences, multiplied by the liquidus slope and summed up over all elements, should give exactly the difference in driving force, if you compare e.g. concave and convex regions. If this is not the case, I would be extremely astonished, because the driving force is calculated like that!
Please remember that there is (probably) some averaging of the driving force along the interface normal!

Bernd

[WTMuser]

Dear Bernd,

I have one further question concerning the interfacial energy (between solid and liquid phase). The proposed value for CMSX-4 is in the range of 50 mJ/m². Are there any references or is this an empirical value?

Kind regards
Martin

[Bernd]

Dear Martin,

who proposed this value? If you are referring to simulations from myself, I have to admit that I probably did not really care about the value

I personally do not have any source of information about the interface energy fcc/liquid for CMSX4. From a practical point of view, the exact value does not help much as long as the exact diffusion coefficients in the liquid phase are not known. So, what you need is a suitable combination of guesses for both variables.
I think a good way of calibrating these variables is by comparison of the secondary arm spacing with experiments. In case of alloy 718 and a 2D simulation in longitudinal section, I obtained a correct secondary arm spacing compared to experiments (Whitesell et. al, Met. Mat Trans 31B., 2000, p. 546) using all liquid diffusion coefficients as 1x10-5cm²/s and an interface energy of 2x10-5 J/cm² (i.e. 200 mJ/m²). But I don't know whether this is similar in CMSX4...

Bernd

[WTMuser]

Dear Bernd,

this value was stated in a predefined driv.-file (test case) regarding directional solidification of CMSX-4. In the last months I performed some calculations of interfacial energy (solid/liquid) by the use of NNBB-model. For CMSX-4 the calculated values are in the range of 100 mJ/m². I would like to compare these results with data from literature (unfortunately there is no data for multicomponent alloys - solid/liquid interfacial energies are only given for pure elements or binary systems).

Martin

[Bernd]

Yes, that is a pity, but it is due to the fact that measurement is extremely difficult. Please don't trust the value which was given in this reference .dri file!
Maybe, only careful calibration of simulation with good experiments can help to fill the gap...