Solidification of a dendrite unit cell

[SeanBöhm]

Hello everyone,

i have a question about the solidification of a dendrite in Ni-based alloys.
I would like to simulate a unit cell to get the distribution of elements in the solidified state.
As a setup for the simulation, I have set the domain length and width to half the dendrite spacing and placed a start seed in a corner, so I only have 1/4 of a dendrite to simulate. The Boundary conditions was set to be symmetrical at the left and lower boundary and to be isolating at the top and right boundary. I now have two problems with the simulation:
1. When a point is reached where the solid phase reaches the boundaries, regions appear where the rest of the liquid is trapped in small areas, resulting in extreme values of the elemental contribution.
2. The nucleation of the gamma prime phase works quite well, but after nucleation the phase grows very slowly, and I still have liquid phase trapped in the remaining areas where the concentration of the elements reaches extreme values.
Is there a trick to avoiding such phenomena?
In the attached images, the left shows the content of Re, the middle shows the content of Al and the right shows the phase.

If anyone has already had a similar problem and has found a way to solve it, I would be grateful for help

Thank you in advance,
Sean

[Bernd]

Dear Sean,

In principle, it is correct that rest liquid forms in shape of droplets along the symmetric boundary conditions. There, the dendrite virtually meets the neighboring dendrite, and the liquid film decomposes into droplets. I also cannot see extreme compositions, maybe they exist for other elements.
The reason why the liquid does not disappear right away after nucleation of gamma prime phase could be that other elements are accumulating, so that nucleation of other phases is required (depending on the type and composition of your Ni-based alloy). In case of CMSX-4, for example, solidification can indeed happen without further phases (if there is solid diffusion) but this also needs time and further cooling.
What I can see from your concentration distributions is that the morphology of gamma prime is unstable. This could have various reasons. I would start checking the following:

1.) Are there any error messages in the screen output (Thermo-Calc errors, MICRESS errors, messages about demixing)? They could perhaps give a hint where the problem comes from.
2.) Are other phases to be expected during solidification? A Thermo-Calc Scheil simulation is very helpful to see what can be expected. Having e.g. an element like C or B included and not allowing formation of carbides/borides can lead to serious problems due to extreme compositions.
3.) Check for negative compositions in all phases using the phase composition outputs. The most efficient method is to set the color scale to -0.001..+0.001 so that negative values appear as blue. Negative compositions can cause serious problems an can be easily overlooked.
4.) If diffusion is unstable, which easily may happen if multicomponent diffusion is simulated close to a spinoidal, this also shows up in the phase compositions (as chess pattern). It is favorable to write phase compositions in at%, then it is easier to identify the unstable components.
5.) Check phase compositions in the linearisation data (.TabLin). If there is unsteadyness for certain elements in a phase, this could point to a miscibility gap or indicate that gamma has switched to gamma prime (or the other way round).
6.) Sometimes phase transformation kinetics are negatively affected by certain numerical or physical parameters, like improper limits of time-stepping or missing diffusion data. Therefore it can be helpful to check the interface mobility (.mueS) and driving force (.driv) outputs.

If you don`t get further with the problem, please send me your complete input files, and I will have a look...

Bernd

[SeanBöhm]

Hi Bernd,

thank you for your tipps.
In fact, I had negative values in the areas of gamma prime solidification (in the areas where the broad interfaces occur).

I was able to solve the problem by adjusting the interface mobility.
However, this brings me to a further question. I have now been able to customise the interface mobility, but I have tried this with trial and error, and I also think it is still not perfect. Are there any tricks, rules or methods for determining this from the start on?

Best regards,

Sean

[Bernd]

Dear Sean,

In solidification (and also in many solid state reactions) the interface kinetics can be assumed to be diffusion controlled. This means that the physical interface mobility has no effect on the front velocity. However, in a phase-field simulation with artificially increased interface thickness there are artifacts which make it mobility dependent, and the front can even run faster than diffusion limited by partial overrunning of the pile-up of segregating elements.

The problem can (and should) be solved by using the "mob_corr" option in MICRESS, which automatically applies a correction on a given physical interface mobility to remove these artifacts. You enable the use of the mob_corr option by adding "redistribution_control" to the "phase_interaction" keyword. Then you enter a physical mobility value (i.e. something big enough like 10 cm4(Js)-1), and - at the end of the phase interaction block - "normal mob_corr" or "atc mob_corr" for every element. Doing so, MICRESS automatically will determine a numerical interface mobility which fits to diffusion limited transformation kinetics.

Bernd

[SeanBöhm]

Hello Bernd,

I have activated mob_corr, but I have the problem that when I set a large value, I have problems with nucleation (that's why I tried to lower the physical value and adjust it).

Although gamma prime nuclei form, they disappear again immediately afterwards.
This also results in error messages (but only in the last steps, the nucleation starts much earlier)

I have attached the scr output for you, maybe you can finde a hint here.

Kind regards,
Sean

[Bernd]

Hi Sean,

This does not make sense to me. Maybe, the problem comes from something else. I would e.g. like to see how your cooling curve looks like. With this information I could try to figure out more. Perhaps cooling is too steep for the low grid resolution, or the simulation starts at a too low temperature (1653K would already be much too low!). In the same way, the temperature range for nucleation could be chosen such that gamma-prime already nucleates with too high undercooling.
Finally, it could be that with the actual limits ("min./s") of the time-stepping you cut off the phase transformation kinetics.

Bernd

[Bernd]

Hi Sean,

just an important addition: You have selected an interface stiffness (!) of 10. J/cm2 for the 1/2-interface!

Bernd

[SeanBöhm]

Hello Bernd,

thanks for your answer, the interface stiffness was a stupid mistake, I swapped the lines in the driving file when entering it.
I have changed it, but unfortunately I still haven't got any further with my error.
I have made a few adjustments:
- finer grid resolution
- smaller time step
- used only the diagonal terms in the diffusion matrix
- smaller time interval for the linearisation of the database
- Experiments with the start temperature and the nucleation temperatures
- Averaging of the driving force

Up to a certain point, the simulation now works very well, nucleation also works well, at the beginning the gamma prime nuclei grow normally (see 265s).
However, at a certain point these annoying structures of the interface still appear. The gamma prime phase also forms strange structures at random ( 400s right upper corner).
I still don´t get any error massages.

Dou you have any further guess what could be the Problem?
Since the simulation already takes quite a long time, I would hesitate to try an even smaller time step and a finer grid.

Kind regards
Sean

[SeanBöhm]

Additionally here is the Driving File and the Temperature Profile.

Have a nice Weekend

Sean

[Bernd]

Hi Sean,

The problem you found is not so common, and it took me some time to analyze it. I am still not sure about the exact reasons. What happens is that the interface contracts to a sharp one, followed by a breakout of needle-like structures along the grid directions. Nevertheless, after optimizing some parameters about interface stability and also solving some probably unrelated issues, it seems to work properly:

1.) Put the driving force averaging back to its default value (avg 0.5) for all interfaces. Strong averaging (avg 1.) sometimes may lead to strange microstructures if the averaging direction logs into the grid directions. Furthermore, mobility correction has been calibrated to the default value, so that small deviations from the expected diffusion controlled kinetics could occur if the default is changed.

2.) Use full extent of interface stabilization (2.E-4 Jcm-2 for all interfaces). The value for interface stabilization (which is optionally given in the same line with the interface energy) should be at least 10 times bigger than the value of the interface energy to fully take advantage of interface stabilization. Our current experiences suggest that there are no negative consequences to be expected even if this value is chosen too large.

3.) The interface energy itself appears lower than realistic. I typically estimate solid-liquid interface energies in Ni-alloys as 1-2 E-5 J/cm2. For the solid-solid interface energy, a smaller value of 5.E-6 J/cm2 is realistic, which then leads to wetting effects, further reducing the tendency of forming liquid inclusions between γ and γ'. Thus, choosing a slightly higher value for the solid liquid interfaces strongly improves interface stability (I know that you have taken the values from the A006-example, which should be updated at some point...).

4.) Use a more adapted updating scope for thermodynamic data ("database fragment"). Averaging thermodynamic data to the full domain ("database global") can sometimes lead to inconsistencies, e.g. if the liquid phase separates into individual ponds which may have strongly different composition. I recently have seen such effects in an alloy 718 simulation, leading to a strong driving force gradient over the interface (visible in .driv output). This can lead to interface spreading or wrong coalescence behavior in some places, while in other places the interface tends to get sharper. Using "database fragment" forces averaging of thermodynamic data to interconnected regions only, so that ponds with different composition are treated individually.
This comes at the cost of increased time consumption for TQ-operations (see .TabP tabular output), which in your case is not significant. In cases where the number of individual precipitates or liquid ponds is very high, it may be better to restrict the relinearisation scope by a fixed distance instead (e.g. "database global 20."). The number and location of the thermodynamic scopes (averaging regions) can be visualized with help of the .refR output.

5.) Do not cut the interface mobility by choosing a too high value for the minimal phase-field time step in the numerical parameter section ("1.E-5 0.1). For all interface cells where one of the time step criteria is violated by forcing a larger time step value (check in .TabT tabular output), the interface mobility is automatically reduced such that stability remains guaranteed. If this happens not only in few cells but over large parts of the interface (check in .mueS output), there is a risk of altered interface kinetics and sometimes also of destabilization of the interface. Use time-step limitation during simulation setup only to avoid "hanging" of the simulation in case of numerical issues in single interface cells (min. time step is not reached most of the time) or to optimize performance of the final production run (min. time step is reached, but kinetics is still unaltered or acceptable).

6.) It has been proven that exactness and interface stability improves when having an interface thickness >2√2 cells (because diagonal neighbors come into play). Therefore I currently recommend a value of 2.85 cells instead of 2.5

7.) Unrelated, but important: The "diagonal_dilute" definition of diffusion coefficients in case of the γ'-phase requires definition of a suitable reference element ("dilute_matrix"). This model calculates effective diagonal terms of the diffusion matrix (which are always positive!) by reducing the composition of the flux element to a small value. By default, the matrix component (Ni in your case) is increased correspondingly, leading to problems in γ' when applied to elements like Ti, Ta, or Al: The too low content of γ'-formers resulting from the correction leads to numerical problems or switching of the single-phase equilibrium to disordered γ (resulting in wrong diffusion coefficients). Instead, for each γ' -forming element, another γ'-former has to be chosen as reference (preferentially the one with the highest content), like it is demonstrated in the A006-example.

As I said, I am not sure about the contribution of each of the above changes to solving your problem (although it may be interesting to figure that out systematically). But in the sum it looks good for me.

Bernd