Hello everyone,
I am trying to make a prediction of the PDAS using a phase field.
My approach here is:
1. calculate the tip undercooling using the KGT model and a pseudo binary phase diagram like its done here.
2. Select the simulation domain so that the width corresponds to half the dendrite spacing.
The dendrite spacing is selected here so that it is within an expected stable range.
3.Calibrating the interface mobility so that the tip undercooling which was determined occurs in the simulation.
4.Vary the PADS to find out in which range the tip temperature is constant
The problem I have here is that the tip temperature does not have a constant value with different PADS, but always increases with the distance (see picture)
What could be the reason for this, or do I have a mistake in my approach?
With kind regards
Sean
Predicting of PDAS
Predicting of PDAS
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Re: Predicting of PDAS
Hi Sean,
In principle, the unit cell approach with KGT-calibrated interface mobility is a good basis for investigation of the PDAS. However, your conclusion, that the invariance of the tip temperature with changing PDAS (i.e. domain size) would indicate the stable PDAS value, is wrong. The tip undercooling here is an indicator for interaction with the neighbor dendrites which obviously vanishes for infinite PDAS - that is exactly what your curves show! But a "stable" PDAS value is one where every single dendrite cannot be overgrown by its neighbors, irrespective whether those have a larger or smaller individual "PDAS".
One way to use your unit-cell approach would be to separately determine the minimum stable PDAS (by having 2 or 4 direct neighbors in the unit-cell and reduce its size until neighbors vanish) or the maximum stable PDAS (having only one dendrite and observing observation of a new one by branching. The problem here is (apart from the high simulation times) that one would probably overestimate the stable PDAS range due to assuming too high symmetry in comparison to reality in a larger dendrite grid.
But there are other methods available, which are also based on simulations like yours. Interestingly, I myself just published a paper exactly on this topic:
https://doi.org/10.1016/j.commatsci.2024.112854
I am interested in further discussion with you about the methods proposed in my paper.
Bernd
In principle, the unit cell approach with KGT-calibrated interface mobility is a good basis for investigation of the PDAS. However, your conclusion, that the invariance of the tip temperature with changing PDAS (i.e. domain size) would indicate the stable PDAS value, is wrong. The tip undercooling here is an indicator for interaction with the neighbor dendrites which obviously vanishes for infinite PDAS - that is exactly what your curves show! But a "stable" PDAS value is one where every single dendrite cannot be overgrown by its neighbors, irrespective whether those have a larger or smaller individual "PDAS".
One way to use your unit-cell approach would be to separately determine the minimum stable PDAS (by having 2 or 4 direct neighbors in the unit-cell and reduce its size until neighbors vanish) or the maximum stable PDAS (having only one dendrite and observing observation of a new one by branching. The problem here is (apart from the high simulation times) that one would probably overestimate the stable PDAS range due to assuming too high symmetry in comparison to reality in a larger dendrite grid.
But there are other methods available, which are also based on simulations like yours. Interestingly, I myself just published a paper exactly on this topic:
https://doi.org/10.1016/j.commatsci.2024.112854
I am interested in further discussion with you about the methods proposed in my paper.
Bernd
Re: Predicting of PDAS
Hello Bernd,
thank you for your input, good to know that the first step of my approach was not so wrong.
I have read your paper, and I think the model with the distance criterion is quite good. If I apply it to my example, I also get meaningful values for the PDAS. But i have also an question about it.
We assume that the dendrite with the smallest distance from the tip to the upper corner will prevail, as this has the shortest path for a diffusion flow of the solutes.Shouldn't the minimum of the curve as shown in figure 8 in the paper be the state for the stable dendrite spacing, and not for the minimum PDAS? Since a larger PDAS also provides a further diffusion path for the solutes? In fact, in a real solidification there cannot be only one fixed distance for all dendrites, but shouldn't it rather be the case that the stable area is located around the minimum of the curve in figure 8?
As an Example, when we look at case 1 the Minimum is at at 279µm. When we look at a Dendrite with a PDAS with 260µm and it vanishes, the new PDAS is 520µm. The curve is not extendet up to this value, but im sure, that the distance for this PDAS is higher than for 260µm. So does an dendrite with 260µm really vanish, when the dendrite with 520µm has an higher distance for diffusion flow?
I am interested to hear your opinion on this.
Until then have nice holidays,
Sean
thank you for your input, good to know that the first step of my approach was not so wrong.
I have read your paper, and I think the model with the distance criterion is quite good. If I apply it to my example, I also get meaningful values for the PDAS. But i have also an question about it.
We assume that the dendrite with the smallest distance from the tip to the upper corner will prevail, as this has the shortest path for a diffusion flow of the solutes.Shouldn't the minimum of the curve as shown in figure 8 in the paper be the state for the stable dendrite spacing, and not for the minimum PDAS? Since a larger PDAS also provides a further diffusion path for the solutes? In fact, in a real solidification there cannot be only one fixed distance for all dendrites, but shouldn't it rather be the case that the stable area is located around the minimum of the curve in figure 8?
As an Example, when we look at case 1 the Minimum is at at 279µm. When we look at a Dendrite with a PDAS with 260µm and it vanishes, the new PDAS is 520µm. The curve is not extendet up to this value, but im sure, that the distance for this PDAS is higher than for 260µm. So does an dendrite with 260µm really vanish, when the dendrite with 520µm has an higher distance for diffusion flow?
I am interested to hear your opinion on this.
Until then have nice holidays,
Sean
Re: Predicting of PDAS
Dear Sean,
I think your approach of comparing the "stability" of the dendrite array before and after full overgrowth of the smaller dendrite is wrong.
One should rather focus on the question whether a smaller dendrite can growth side-to-side with a bigger one, even if there are small fluctuations of the front position of the smaller or bigger dendrite. If we consider that the competence between the two dendrites is a solutal one (like discussed more explicitly for the modified Hunt criterion), the situation during the fluctuation can be compared to a slight shift of the plane which separates the domain of the two dendrites: The equilibrium position of this plane would be located such that there is no flow between the domains, or - in case of the distance criterion - the distances d are equal.
Then, sticking to your example with the two dendrites with 520 and 260 µm, respectively, we must (besides other fluctuations) consider the case that the smaller dendrite would accidentally lag a bit behind. Then, the distance d is increased because we are on the left side of the stability curve in Fig. 8, and the PDAS would be slightly decreased. That means that this type of fluctuation would be automatically enhanced (positive feedback), which eventually must lead to the complete vanishing of the smaller dendrite.
This does not mean that smaller dendrite must always vanish if it is below the stability limit given by the minimum in FIg. 8. If the fluctuation goes in direction of a higher growth position, then the smaller dendrite must grow until reaching or exceeding the stability limit. This as well means that a PDAS of 260 µm was unstable, and therefore cannot exist in a stationary growth situation.
Another simpler argument is that any criterion which addresses overgrowth of dendrites (and thus the condition for increasing of the average PDAS) must be a criterion for the lower stability limit. The process of decreasing the PDAS, which is branching of dendrites by conversion of side branches to main trunks, is not yet considered. For this reason, such a criterion cannot describe the average PDAS including both processes, but only a lower limit.
Bernd
I think your approach of comparing the "stability" of the dendrite array before and after full overgrowth of the smaller dendrite is wrong.
One should rather focus on the question whether a smaller dendrite can growth side-to-side with a bigger one, even if there are small fluctuations of the front position of the smaller or bigger dendrite. If we consider that the competence between the two dendrites is a solutal one (like discussed more explicitly for the modified Hunt criterion), the situation during the fluctuation can be compared to a slight shift of the plane which separates the domain of the two dendrites: The equilibrium position of this plane would be located such that there is no flow between the domains, or - in case of the distance criterion - the distances d are equal.
Then, sticking to your example with the two dendrites with 520 and 260 µm, respectively, we must (besides other fluctuations) consider the case that the smaller dendrite would accidentally lag a bit behind. Then, the distance d is increased because we are on the left side of the stability curve in Fig. 8, and the PDAS would be slightly decreased. That means that this type of fluctuation would be automatically enhanced (positive feedback), which eventually must lead to the complete vanishing of the smaller dendrite.
This does not mean that smaller dendrite must always vanish if it is below the stability limit given by the minimum in FIg. 8. If the fluctuation goes in direction of a higher growth position, then the smaller dendrite must grow until reaching or exceeding the stability limit. This as well means that a PDAS of 260 µm was unstable, and therefore cannot exist in a stationary growth situation.
Another simpler argument is that any criterion which addresses overgrowth of dendrites (and thus the condition for increasing of the average PDAS) must be a criterion for the lower stability limit. The process of decreasing the PDAS, which is branching of dendrites by conversion of side branches to main trunks, is not yet considered. For this reason, such a criterion cannot describe the average PDAS including both processes, but only a lower limit.
Bernd