## Requirement for Numerical Calibration: calculating tip undercooling

dendritic solidification, eutectics, peritectics,....
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### Requirement for Numerical Calibration: calculating tip undercooling

Dear all:

I want to ask two things in this post: 1st theories to calculate the tip undercooling and 2nd solutions to the theories to calculate the tip undercooling.

1)
In order to calibrate the numerical parameter (kinetic coefficient), I need to approximate before the tip temperature (undercooling). For this, one could use the theory developed by Kurz Giovanola and Trivedi KGT1. Another possibility is to use the model developed by Hunt2. The key difference between the two - afaik - is that they assume a different growth condition to determine the amount of undercooling. KGT: marginal stable. Hunt: maximum growth rate. I have trouble identifying, which of two theories would be suitable for a laser welding / slm solidification sceneario. Or do I compare pears with apples? 2)
As far as I understand KGT theory correctly, I need to numerically solve for a given temperature gradient the unique pair of tip radius R and growth rate V. With this information I get the Peclet Number and can solve the super saturation assumed to be equal to Ivantsov's solution. Lastly, I can calculate the solutal undercooling with that last puzzle piece. Values for Ivantsov's solution can be found in Fundamentals of Solidification by Kurz and Fisher. I wondered that I am probably not the first one trying to implement such a code: do any of you know a python/matlab/octave code for solving this, given the physical properties of the alloy? I am looking forward for your answers. Thanks.

1 https://doi.org/10.1016/0001-6160(86)90056-8
2 https://doi.org/10.1016/0022-0248(74)90127-4
---- Constantin

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### Re: Requirement for Numerical Calibration: calculating tip undercooling

Dear Constantin,

1) Indeed, those two theories could give you an indication about tip undercooling in a directional solidification regime. Both are approximations, because they treat a single dendrite tip and use some criteria to solve the equations. Even more criteria exist, i.e. "Optimum stability conjecture for the role of interface kinetics in selection of the dendrite operating state" by Robert F. Sekerka Journal of Crystal Growth 154 (1995) 377-385. KGT may be usefull for dendritic growth, because it includes some of the features relevant for rapid solidification. You may consider to use a different value of the stability constant sigma*, which is sigma*=1/4/Pi/Pi in the KGT model (and is "hidden" as 1/2/Pi in the paper), because this value should depend on at least surface tension anisotropy, according to microscopic solvability theory.

2) Yes, you need to numerically solve eq.(4) of the KGT-paper for the tip radius R, where Gc (and probably xi_c) would also depend on R. The resulting value for the tip radius will provide you the liquid concentration ahead of the tip und the solutal undercooling.

Best regards,

Laszlo