solidification of MPEA during LPBF

dendritic solidification, eutectics, peritectics,....
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Atur
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solidification of MPEA during LPBF

Post by Atur » Wed Jul 21, 2021 10:35 am

Dear All,

I am trying to simulate the solidification of a eutectic multi-principle element alloy (FCC/B2) under L-PBF conditions. My aim is to see what is happening at different cooling rates (phase fractions, shift in eutectic coupled zone etc.). After reaching to a certain cooling rate / growth rate , as expected, I observe non-equilibrium single phase solidification of B2 phase instead of eutectic solidification (experimentally validated). Micress tells me that at eutectic composotion, I have higher undercooling and driving force for B2 phase. My EDS results shows me that I have similar composition after solidification compared to the nominal composition of the Liquid. Therefore, I would expect solute trapping since the growth speeds might be larger/close to diffusion speed.

My question is; would it be possible to simulate a behaviour such as solute trapping? For example, can I set a constant setting where after reaching a certain growth speed I will observe some artefacts leading to "trapping"?

I attached one of the input file I was working on below. To note that, I dont have any mobility database so I entered diffusion coefficients constant and I am using a custom TDB for these simulations. I closed the atc and mob_corr options intentionally and observed different results at same cooling rates. I would be happy to hear your comments to further improve my simulations.

Regards,
Last edited by Atur on Wed Sep 22, 2021 10:10 am, edited 1 time in total.

Bernd
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Re: solidification of MPEA during LPBF

Post by Bernd » Wed Jul 21, 2021 6:27 pm

Dear Atur,

During typical L-PBF processes, the solidification speed is well below 1 m/s. For that reason, we do not expect solute trapping (in the sense of non-equilibrium partitioning at the interface) under these conditions. The MICRESS software is based on local quasi-equilibrium at the interface, which means that partitioning is assumed to be fast with respect to the phase transformation. Thus, MICRESS intrinsically is not capable of predicting the transition between trapping and no-trapping.

However, long before the typical velocities for solute trapping are reached, there is already a transition to a massive transformation. This occurs when the solutal undercooling of the dendrite tips reach the solidus temperature and is accompanied by a planar solidification front. Thus, no microstructures can be observed anymore. This morphological transition is still compatible with local equilbrium (comparable perhaps with NPLE-condition in steel, where slow elements are overrun without being able to produce a morphological instability despite being full partitioned), and thus can be simulated with MICRESS. Like in real solute trapping, you would not observe any partitioning anymore (post mortem).

If you want to simulate such processes with MICRESS, it is absolutely necessary to use "mob_corr" to ensure diffusion limited growth and to prevent artificial solute trapping. Alternatively, you would have to calibrate the interface mobility, which I don't see how you could do that under theses conditions. Both methods, "mob_corr" or calibration, lead to an extra artificial contribution to the front undercooling, with is the bigger the coarser the grid resolution is. If this contribution gets large compared to curvature undercooling, morphology is not correctly displayed anymore. Therefore, resolution needs to be high enough.

Looking at your input file, I noted that you use a much too coarse grid resolution. Based on my experiences with similar cases I would propose a value of Δx=0.01 µm. Furthermore, you absolutely should use "mob_corr", otherwise the interface mobility value of 10 cm4/(Js) is taken as numerical one, which is much too high and necessarily causes numerical problems and faster than diffusion-limited kinetics.
Finally, I have the feeling you should use "diagonal" extrapolation in order to avoid multibinary "demixing" effects, given that your alloy is far from dilute.

Bernd

Atur
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Re: solidification of MPEA during LPBF

Post by Atur » Thu Aug 05, 2021 6:57 pm

Dear Bernd,

Thank you for your detailed answer, I tried to correct my input and feel like already having some improvements . If you don't mind, I will have some basic questions to improve my understanding;

After checking the metastable extensions of the phase diagram, I can see that after a certain Al concentration B2_BCC solidus line is over the FCC solidus line, which actually explains single phase solidification when exceeding a certain cooling rate. As far as I understand, only a slight undercooling below the solidus line is sufficient to drive partitionless (massive) solidification at high Tgradient and growth rate (depending on composition and material). However I am still confused whether the liquid and solid should have almost the same free energy per unit volume in this case (DeltaG(l-->s) =0) to obtain this type of spontaneous single-phase solidification. Would that be necessary?

Moreover absolute morphological stability should be established in order to have these planar fronts. May I ask if it is directly related to the growth rate in this case? Is it the thing that when interface velocity exceeds a certain speed, the perturbations will diminish or become negligible in the end? Is this planar front somehow related to tension between solid and liquid?

I also attached 4 of my simulation results below. I selected two Al concentrations namely 15 at% and 16 at%. The reason I choose these compositions is one of them is slightly less (15 at% Al) than the crossing concentation point of metastable solidus lines and other one is slightly higher (16 at%).To note that I used same parameters where only the concentration and cooling rates are varied. I try to start with the initial microstructure with both fcc and b2_bcc grains to let the one with larger driving force to grow. Yet I experience melting of my initial grains when I change my cooling rate. In order to stabilize them I needed to increase their size up to 3 µm. Is there anything that I am missing regarding input parameters?

In lower Al content (15%) at larger cooling rates (attached as "15_high") I observed combined planar + cellular growth where the planar part is partitionless (same comp. with liquid) and other part allows segregations. Is this something relatd to my initial microstructure? When I decrease the cooling rate and hence growth speed (attached as "15_low"), I end up having dendritic growth. Eventhough I see the Al and Ni enrichement in interdendritic regions, I cant see any b2_bcc solidification. Is this related to fast moving interfaces?

In case of higher Al content (16%) at larger cooling rates (16_high), I can see single phase b2_bcc solidification together with a planar front and identical nominal composition of liqud, which makes sense re-thinking metastable solidus lines. When I decrease the growth speed (16_low), I can see fcc - b2_bcc where b2 phase is in interdendritic regions. Yet I dont understand why it looks like "banded" and why I dont get b2 solidifcation in case of lower Al content simulation considering everything is same but only composition.

I would be happy to hear your comments to further improve my understanding and simulations!

thank you and regards,
Attachments
15_high.JPG
15_high.JPG (129.14 KiB) Viewed 1712 times
15_low.JPG
15_low.JPG (183.82 KiB) Viewed 1712 times
16_low.JPG
16_low.JPG (134.63 KiB) Viewed 1712 times
16_high.JPG
16_high.JPG (122.23 KiB) Viewed 1712 times
Last edited by Atur on Wed Sep 22, 2021 10:09 am, edited 1 time in total.

Bernd
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Re: solidification of MPEA during LPBF

Post by Bernd » Sun Aug 08, 2021 7:08 pm

Dear Atur,

I think your view is correct: As soon as the front temperature falls below solidus, the transformation must switch to a massive one, because no diffusion is needed anymore, and the liquid composition can be "frozen" in (so that a solid with the same composition is formed). This switch to massive automatically leads to a planar front, because the solidification process now resembles that of a pure substance (without segregating alloy components), and the driving force ΔG then is identical to the difference in free energy of the two phases at this composition (and temperature).
That means that growth would now essentially be controlled by latent heat and fast heat diffusion, leading to much less morphological instability and much coarser structures. In a simulation without temperature coupling, growth would be controlled by the physical interface mobility only, and there would be no morphological instability at all.

In principle, phase-field models should automatically capture this behavior correctly, but only as long as the length of the diffusion fields at the front is much bigger than the interface thickness, and as long as the solidification velocity is not too high so that local equilibrium can be established at the interface (v <~1m/s). Especially the first condition, however, is difficult to realize, because very fine grid resolution would be needed, and simulation time would be excessive.

In MICRESS we therefore use "mob_corr" in order to correct for the numerical effects of a too big interface thickness on growth kinetics under non-massive transformation conditions. In order to remove the artificial "trapping" effect (the large diffuse interface actually tunnels through the solute pile-up), "mob_corr" reduces the interface mobility to ensure a correct front velocity. This works quite well if resolution is not too coarse, and the reduction of the interface mobility is modest.

When working at very low resolution, however, this mobility correction leads to a large artificial kinetic contribution (i.e. extra undercooling) which is not real. As a rule of thumb, we can say if this undercooling exceeds the order of curvature undercooling, morphology will be not be correct anymore. Furthermore, with rapid solidification processes, this extra undercooling can bring the front temperature below the solidus temperature, triggering a "false" transition to a massive transformation. The amount of the extra undercooling can be estimated based on the .driv output: When looking at flat parts of the interfaces, the driving force value indicates the amout of this extra undercooling (because under assumption of diffusion-controlled growth ΔG should be close to zero). If you look e.g. at your "15_high" case, the driving force is very similar at dendrite tips and at flat interface parts, indicating that the artificial undercooling is very dominant. Under these conditions we should not expect correct morphologies, and resolution must be increased (I assume that the massive mode at the left hand side would vanish then...).

Another numerical problem which probably would vanish with higher grid resolution is the loss of anisotropy (or better the dominance of grid anisotropy) which probably makes dendrites grow in z-direction in case of "16_low". In this context, it would perhaps be helpful to increase the interface thickness from 2.5 to 2.85 cells. Although this in principle decreases the "effective grid resolution", as I learned from my janin, a better numerical evaluation of interface curvature could outweigh this effect (as when using fd_correction, a further row of neighbor cells is taken into account if the interface thickness in cells is above 2 x 21/2).

Finally, the problem of vanishing initial grains could be due to the initial equilibrium, which is calculated at a higher temperature of 1700K. This means that the initial compositions are not in equilibrium at the start temperature of 1610K.

Bernd

Atur
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Re: solidification of MPEA during LPBF

Post by Atur » Fri Aug 13, 2021 12:05 pm

Dear Bernd,

Thank you for your reply and explanation.

I still have some confusion in certain points. Maybe I can formulate them in bullet points:

1) I re-scaled the interface thickness to 2.85 and grid spacing to 0.0025. Yet my .driv output says the driving force is still around -30 in dendrite tips, but there are no increase and decrease over simulation time. Does that mean that I should still increase my interface mobility or shall I tune some other parameters? Besides, what should be the acceptable range in .driv output? I can see that in some of my compositions (e.g. 16_high in my previous post) that the value in the interface is around -10. I wonder if it is the case that the extra undercooling is still significant?

2) I can see in my log file after linearisation of parameters for Liquid/solid interfaces I get an undercooling value for stable growth. Should I set my minimum undercooling accordingly for each phase in nucleation ?

3) I  am also confused about which bottom temperature I should select or whether I should use a moving frame option to see real massive solidification after a certain composition. I can see that after a certain composition below 1560 K the system crosses the B2_BCC metastable-solidus line. So I need a certain undercooling for this transformation to occur. For inducing a massive or let's say single phase metastable solidification, should I set my bottom temperature in a way that after the frame starts to move the value will somehow fall below to 1560K?

Kind regards,
Attachments
Capture.JPG
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Last edited by Atur on Wed Sep 22, 2021 10:08 am, edited 1 time in total.

Bernd
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Re: solidification of MPEA during LPBF

Post by Bernd » Mon Aug 16, 2021 10:47 am

Hi Atur,

I will try to answer following your bullet points:

1.) At the dendrite tip there is curvature present, so you always will have a high driving-force value (caused by the concentration change due to the Gibbs-Thomson effect). My point was to look at the driving-force value at planar parts of the interface, and compare them to the curved parts. As you can see in your .driv output, now with higher resolution the driving-force is even switching sign to positive values at kinks with negative curvature, which is a good sign that curvature effects are not erased by a large kinetic undercooling (like it was before with lower resolution).

2.) No. The minimum undercooling given in the .log-file after initialization of the thermodynamic interaction is a numerical value which depends on grid resolution and the interface energy. To the contrary, the critical undercooling which you specify in the data for further nucleation is (at least in principle) a physical value.
However, if you specify a critical undercooling which is lower than the numerical minimum undercooling, the seeds may not be able to grow. As workaround, you then should use the "analytical_curvature" model for nucleation (explained e.g. here).

3.) There is no general rule how to define the initial conditions. However, in such type of simulation, you typically start with a planar front which initially is at liquidus temperature, and during growth the front undercooling will automatically approach a stationary value. You would use moving_frame for following the dendrite tip in order to avoid that the front touches the top of the domain before stationary conditions have been reached.
In some cases, especially with low temperature gradients, reaching steady-state may take long time. Then it may be better to start already at a temperature which is close to the expected steady-state.

Bernd

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