Seed-density and seed-undercooling models

dendritic solidification, eutectics, peritectics,....
mauvec
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Seed-density and seed-undercooling models

Post by mauvec » Mon Mar 15, 2021 3:44 pm

Dear Bernd,

Thank you for your detailed response. It has been of great support.

With the intention to consolidate expertise on MICRESS, I departed from T002_AlCu_Equiaxed and A006_CMSX4 as examples to understand how to implement the seed-density and seed-undercooling models to mimic eventually dendrite evolution of a Ni-based superalloy under rapid cooling.

I utilized the T002_AlCu_Equiaxed as template to begin simulations. As a start, I only modified the corresponding sections where needed to include all the constituents of my system and of course being careful to follow the structure of the input data in MICRESS, e.g., component, diffusion, initial concentrations sections and so on. At the moment, it is worth to mention that I am only focusing on the liquid-to-solid transition without contemplating formation of secondary phases such as gamma-prime and/or borides/carbides, which I presume might be extremely difficult due to how Thermo-Calc treat composition sets for distinct phases in heavily alloyed systems. Anyways, when running the simulation with only adapting nominal composition of my system, the outcome is quite similar to the original case, i.e. formation of equiaxed grains. However, when I decided to modify the simulation domain to 15 micrometers x 15 micrometers with 0.1 grid spacing, no outcome is generated. Thus, I presume that the seed-density distribution needs to be re-adapted. So, I am wondering if there is way to do the latter according to the information that MICRESS generates in the different Tabs, i.e. TabC, TabD, TabF and TabK to construct a more appropiate seed-density distribution?


Regarding the A006_CMSX4 example, I also utilized this as template to grasp a better understanding on how to implement the seed-undercooling model to perform the liquid-to-solid transition as previously mentioned. By trying again to only adapt the nominal composition of my system, the outcome of this simulation is shown in the series of snapshots attached to this message. I am wondering if previously assistance on a similar outcome has been requested from MICREES by their multiple users. So far, I started to approach this issue by manipulating parameters of the DeltaG options, however, still cannot certify if this is the correct way. No errors are indicated when the simulation is progressing. So, may I ask you for assistance or any idea of what could be happening? and how again to use the multiple taps to spot errors and generate solutions?

Thank you for taking the time to read this.

Best regards,

Mauro
Attachments
snapshot3.png
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snapshot2.png (73.79 KiB) Viewed 567 times
snapshot1.png
snapshot1.png (27.14 KiB) Viewed 567 times
Last edited by mauvec on Wed Mar 24, 2021 5:29 pm, edited 1 time in total.

Bernd
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Re: Seed-densidy and seed-undercooling models

Post by Bernd » Mon Mar 15, 2021 7:05 pm

Dear Mauro,

Welcome to the MICRESS Forum.

Thank you for giving us here a detailed description of your initial approach to simulate rapid solidification of Ni-base superalloys. In principal it is good practice to start from the standard application examples T002_AlCu_Equiaxed and A006_CMSX4 example, which represent an equiaxed solidification type on one hand, and a directional type (considering isothermal cross-sections) on the other hand, and - in a first step - adapt the alloy composition to the system of your interest. Then, the second step can be to adapt to the process conditions of rapid cooling, which is somewhere in between, and which will need some fundamental considerations about the temperature coupling/boundary conditions which are suitable for the process.

Obviously, you already managed to take the first steps which are necessary for changing the alloy:
  • deciding on the elements which should be included.
  • creating a new GES5 file which contains all the selected elements and at least the LIQUID and FCC_L12 phase
  • add or change the elements in the MICRESS input file everywhere where needed (definition of elements, initial composition, ...)
  • decide on the source of diffusion coefficients for the elements in the solid and liquid phase
In principle, as long as you consider only the higher temperature range where only a solid (fcc) and a liquid phase are expected, there are not so many problems to be expected, but still there are some points which need consideration:
  • Ni-based alloys are "high-alloyed" systems, i.e. you should expect strong interactions between the elements (no dilute system). Thus, it is recommendable to use "diagonal extrapolation" from the beginning (see also e.g. here).
  • With the changing alloy chemistry also the liquidus temperature is changing. So, it is necessary to adjust the initial temperature accordingly.
  • Diffusion coefficients in the melt are generally not well-known. Therefore, it is justified to use a constant value of e.g 10-5 cm2/s for all elements, even if estimated values could also be obtained from the diffusion database. In my opinion this is not a problem, because typically there is anyway a calibration of liquid/fcc-interface energy needed to obtain correct values for the secondary arm spacing (which at the same time include 1st order calibration of the liquid diffusivity). If you nevertheless prefer taking diffusion coefficients from the database, you should either include the full diffusion matrix (because of element interactions), or effective diagonal terms using "diagonal_dilute".
  • In case segregation (or more exactly the "growth restriction factor") of the alloy is strongly different from that of the example you are starting from, the solidification velocity (and thus the diffusion length) will be different, and the length and time scale of the simulation needs to be adapted. Generally, the diffusion length should not be smaller than the interface thickness to prevent strong artefacts like wrong interface kinetics, solute trapping, and numerical instability. Using "mob_corr" helps for automatically adjusting the interface mobility such that diffusion limited kinetics is achieved.
To answer your concrete question about the seed_density model: There is an output the number and size distribution of the seed particles how it is initialized at the beginning of the simulation, and which strongly depends on the domain size. You find it in the .log output in the section which belongs to the nucleation input:

Seed type 1 will be described by the nucleation model seed_density
Integer for randomization :77777
10 classes are read for the critical seed radius
Defined seed density classes:
(Specify radius [micrometers], Seed density [cm**-3])
Class 1: 0.45000 0.50000E+03
Class 2: 0.30000 0.10000E+04
Class 3: 0.25000 0.25000E+04
Class 4: 0.18000 0.50000E+04
Class 5: 0.15000 0.10000E+05
Class 6: 0.12000 0.25000E+05
Class 7: 0.10000 0.45000E+05
Class 8: 0.08000 0.70000E+05
Class 9: 0.07000 0.12500E+06
Class 10: 0.06000 0.25000E+06
Resulting seed density distribution:
Class 1: 0 seed(s), 3.7500E-01 < radii < 5.2500E-01 [micrometers]
Class 2: 0 seed(s), 2.7500E-01 < radii < 3.7500E-01 [micrometers]
Class 3: 0 seed(s), 2.1500E-01 < radii < 2.7500E-01 [micrometers]
Class 4: 0 seed(s), 1.6500E-01 < radii < 2.1500E-01 [micrometers]
Class 5: 0 seed(s), 1.3500E-01 < radii < 1.6500E-01 [micrometers]
Class 6: 1 seed(s), 1.1000E-01 < radii < 1.3500E-01 [micrometers]
Class 7: 1 seed(s), 9.0000E-02 < radii < 1.1000E-01 [micrometers]
Class 8: 1 seed(s), 7.5000E-02 < radii < 9.0000E-02 [micrometers]
Class 9: 2 seed(s), 6.5000E-02 < radii < 7.5000E-02 [micrometers]
Class 10: 3 seed(s), 5.5000E-02 < radii < 6.5000E-02 [micrometers]
For this seed type 8 seed positions have been generated



The simulation results for your modification of A006_CMSX4 example look like crashed. If this is not due to a missing adjustment of the initial temperature, it may be any other error you could have made during adaption of the input file (e.g. wrong assignment of the primary phase, etc.). It would be probably easier for me to find it, if you could attach your input file together with the .ges5 file to your next post.

Best wishes
Bernd

mauvec
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Re: Seed-density and seed-undercooling models

Post by mauvec » Wed Mar 17, 2021 10:38 pm

Dear Bernd,

Thank you for these detailed information.

After going carefully through your message, manual and some discussions in the forum regarding the kinetic coefficient, it is suggested that this parameter only needs to be defined high enough so it does not slow the interface motion when considering a diffusion-controlled phase transformation, which can be assumed to be the case for the A006_CMSX4 example (isothermal sections).

It is also mentioned that the ‘mobility correction’ (mob_corr) option for diffusion-controlled phase transformations does not apply where spatial resolution is not sufficient, and artificial low mobility values must be applied to avoid numerical instabilities. The previous could be the case for my simulation since after modifying exclusively the kinetic coefficients from 1.0 to a) 1E-02 and b) 1E-03, the interface experienced changes as it can be observed in the attached snapshots.

For these simulations, I also implemented a cooling rate of –50 K/s. Whether as a ‘physical’ and/or ‘stabilisation’ parameter, I am trying to understand how the kinetic coefficient should be manipulated within this framework since it seems to have multiple repercussions not only for numerical reasons but also for secondary arm formation. As the simulation continues evolving it is evident that more options need to get refined in order to get a better descriptive result. Please find attached (in a private message) together the input and GES5 files that I implemented for these simulations.

Regarding segregation, when looking at the concentration outcome of the different constituents, the partitioning behaviour looks qualitatively consistent with EDX experimental evidence, which is encouraging. Precisely, part of the motivation of implementing MICRESS is to contrast quantitatively the local micro-segregation from simulations against experimental evidence. Probably it is a bit too early, but I tried to implement the advanced options ‘criterion higher’ and ‘criterion lower’ to remark the concentration differences between grain interiors and the melt for some constituents. As already expected, several errors were indicated. So, I am wondering if this is strictly incorrect to perform the aforementioned before moving towards secondary-phase formation at these regions?

Finally, as you previously explained, the seed size distribution depends strongly on the domain size. So may I ask you if there is a procedure to adjust the seed-distribution according to the latter?

Best regards,

Mauro
Attachments
snapshot-B.png
B) mu = 1E-03
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snapshot-A.png
A) mu = 1E-02
snapshot-A.png (583.64 KiB) Viewed 553 times
Last edited by mauvec on Wed Mar 24, 2021 5:29 pm, edited 1 time in total.

Bernd
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Re: Seed-densidy and seed-undercooling models

Post by Bernd » Thu Mar 18, 2021 1:14 am

Dear Mauro,

I think you indicated yourself where the problem is why you need to reduce the interface mobility in order to get numerically stable simulations: The example A006 has been designed to show directional dendritic solidification in an isothermal cross section through the mushy zone. In such a setup, the selection of the primary dendrite spacing cannot be predicted but must be imposed by definition of the initial grain positions. This was done for typical investment casting or Bridgman furnace conditions with a low cooling rate.
What you did is not only change the alloy composition from CMSX-4 to another Ni-base alloy but at the same time you also increased the cooling rate by nearly a factor of 100. That means that the expected microstructure will be a lot finer, and also the assumption of the primary spacing is far from something realistic. Generally, if the provided grid resolution is not sufficient to resolve the expected microstructure, the simulation will either crash (when using the recommended high mobility together with mob_corr), or the microstructure must be artificially coarsened by assuming a lower mobility than that needed for diffusion limited growth. This seems to be exactly what you observe.
Thus, whenever you change the typical diffusion length scale by either changing the cooling rate, the diffusion coefficients, or the interface energy in a wider range, you absolutely must adjust the length scale of the simulation by changing the grid resolution (and the primary spacing in this specific case). I expected you would keep the cooling rate constant in the first step, therefore I still did not talk about that in my previous post...

The second issue you raised is the advanced new options ‘criterion higher’ and ‘criterion lower’. They represent just one of several ways to restrict the composition range by raising specific errors in the calculation of quasi-equilibrium. This is only necessary in cases, where the phase description in the database shows miscibility gaps which divide the useful composition ranges into so-called "composition sets". This is something very common for the FCC_L12 or FCC_A1 phase description, because often we find separated phase occurences of fcc close to the pure elements (al, ni, cu etc.) which are separated by intermetallic phase regions. Furthermore, MC-cabides and gamma-prime phase in Ni-based alloys are also described by the same phase in the database. In MICRESS simulations we cannot accept what Thermo-Calc often does, namely switching between these composition sets (which we consider as different phases). This requires defining rules to recognize whether you are still in the correct composition region or not. You will find more about these specific problem of composition sets here or here.

Finally, you asked how to adjust the seed density distribution to the simulation domain. This question is at the same time very simple but also very difficult: The simple part is that you can just change the values of the seed density for each class in the nucleation data part of the input file. There is no automatic way how to achieve a certain number of potential seed places, but you can easily see the effect directly in the .log file output without having to wait for the simulation to finish.
The difficult part is how to know which seed density distribution you really have in the melt of an alloy, and to predict the effect of changes of the density of potential nuclei to the real nuclei which will form during simulation...

Bernd

mauvec
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Re: Seed-density and seed-undercooling models

Post by mauvec » Wed Mar 24, 2021 5:28 pm

Dear Bern,

Thank you for this information. In this regard, I am looking now for formation of MC-type carbides (FCC_12#3) at the interface between FCC_L12/LIQUID. To perform the aforementioned, I considered the following,

For the GES file:
  • Before saving the GES5 file (attached together with the 'TCM' file), I utilised the AMEND_PHASE_DESCRIPTION and MAJOR_CONSTITUENT commands to set new major constituents on each sub-lattice of the composition set of FCC_L12#3.
For the input-file:
  • Five stoichiometric components, including Carbon, were defined for FCC_L12#3 phase.
  • Keyword 'limits' was implemented to redefine minimum and maximum composition range of Carbon (at.%) in FCC_L12#3.
  • 'Diagonal' scheme was utilised as extrapolation method.
  • Because of using stoichiometric conditions, 'solubility_on' option was set on.
  • Criterion higher was implemented to restrict composition in LIQUID, FCC_L12 and FCC_L12#3.
  • Set of 'diff_comp_sets' to allow adding the redefined composition set FCC_L12#3.
It is worth to mention that I implemented the 'seed-density' model for nucleation of FCC_L12. I also implemented a constant cooling rate of -50 K/s and no thermal gradient. Although I am not still using parameter values that mimic SLM conditions, I did not consider FCC_L12#2 (gamma-prime) for simulations due its very small size found in experimental evidence and also well reported in the literature (~ 5 nm). In the former and the latter, MC-type carbides render a ~50 nm particle size. Therefore, I considered the 'stabilisation' model for nucleation of FCC_L12#3, since it neglects curvature as long as the particle still remains small than the grid resolution.

When performing simulations considering either full diffusion matrix or only diagonal terms ('input' files attached), several errors occur when FCC_L12#3 starts growing ('log' files attached) and simulation hardly progresses as it can be seen in the TabF file,

Image

From the lists of errors, 1010001 and 1010005 are the most persistent. May I ask you about the meaning of these, and in general how to interpret the messages 'trying hard phases 1 0', 'level: 4' and 'zp = #####' ?

Image

From all of these, I am wondering if I implemented incorrectly the stoichiometric conditions and therefore, modifications should be adapted accordingly? Would it be worth to simplify the system by not including certain constituents? and finally, shall I try to use finer grid resolution in order to capture the formation of the FCC_L12#3 and therefore, increment updating times for re-linearisation intervals in database, phase diagram, diffusion and nucleation sections?

Many thanks,

Mauro

Bernd
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Re: Seed-density and seed-undercooling models

Post by Bernd » Thu Mar 25, 2021 5:22 pm

Dear Mauro,

This is a quite long list of questions, reflecting the fact that you address a quite complex alloy system. Note that you obviously tried to include images which are not shown in the forum. Please try to edit your post so that the images are visible, or send them again in a separate post.

So, let me try to address your questions one by one. It is worth mentioning that I got additional information including the input file per PM.

1.) General setup:
In principal, the setup which you explain sounds reasonable, including many details on specific numerical parameters which are used to cope with the complexity of the system. However, I don't understand your thermal setup. Generally, depending on the temperature gradient, solidification processes can be subdivided into directional and equiaxed, while reality often lies in between. However, the distinction is essential because of the role of latent heat.
The parameters "cooling rate" and "temperature gradient" typically are strongly linked by the heat conduction equation. Only in special extreme cases like in a Bridgman furnace, the two variables can be decoupled, and latent heat can be neglected. Then you typically have a strong temperature gradient, a low cooling rate, and a directional dendritic or cellular solidification.
On the other hand, in equiaxed solidification, temperature gradients are small, and latent heat is dominant. In MICRESS, instead of defining a cooling rate, you rather define a heat extraction rate, or perform a more complex type of temperature coupling by use of a 1d-temperature field.
What you try to do is to use no temperature gradient together with fast cooling, using the seed-density model (which is a specific model for nucleation of equiaxed grains or heterogeneous nucleation). This is a combination which does not make sense, neither for directional nor for equiaxed solidification. Note that the combination of high cooling rate and no temperature gradient in our application example A006_CMSX4_dri is a special case because it is a 2D-Simulation perpendicular to the temperature gradient. This makes sense only if we assume the directional dendrites coming from the bottom at predefined positions corresponding to their primary distance. The seed density model for primary nucleation does not make sense in this context (only for nucleation in front of directional dendrites if the temperature gradient is in the simulation plane!).

2.) Stoichiometric condition for MC
You mentioned that you defined only part of the elements as stoichiometric in MC (FCC_L12#3). I recommend using stoichiometric condition for all elements, because the matrix element Ni typically has a very low solubility in MC. Thus, redistribution of elements in MC against Ni is numerically dangerous for all elements.

3.) "solubility_on/solubility_off"
This is a switch keyword which applies a specific stoichiometric condition with solubility to those elements which are defined as stoichiometric after this keyword:

2 1-5 7 11
solubility_on
2 6 8 9 10

The shown input will define components 1,2,3,4,5,7,11 as "normal" stoichiometric", while 6,8,9,10 are defined as "stoichiometric with solubility" in phase 2.
Stoichiometric elements will be considered to have a fix composition during redistribution always. If they have "solubility", their composition furthermore can be used as condition in thermodynamic calculations and their slopes can be obtained. Without solubility (default) this will not be done for stoichiometric elements. The usage is discussed e.g. here.

4. diff_comp_sets <ph_1 .. ph_i>
This option allows adding further composition sets of the same phase (e.g. FCC_L12) to the single-phase equilibria which are calculated for obtaining the diffusion data. It can be helpful to allow getting diffusion data even if other FCC_L12 phases are stable. Please note that this option is only needed for those phases which occur with different composition sets in your simulation and for which you access diffusion data from database (only phase 1 in your case!).

5.) diagonal / diagonal_dilute
The question whether to use only diagonal or also off-diagonal diffusion data in MICRESS depends on the extent of interactions between the elements: In strictly dilute alloys the off-diagonal terms are usually small and can be neglected, while in high-alloyed systems, using diagonal terms only can be wrong or even impossible (they may be negative in some extreme cases like for the γ/γ' spinodal). Thus, in case of Ni-based alloys, you must include off-diagonal terms in one or the other way, if you do not want to make significant errors. Using the full diffusion matrix (keywords "multi / multi_plus") yields the highest accuracy, but are often numerically delicate especially in Ni-based alloys.
If diffusion in some phases is not so essential for the simulation outcome (e.g. all solid phases in solidification simulation, small particles like MC, γ'), a useful alternative is to apply effective diagonal diffusion by using the dilute limit for each element separately (keyword "diagonal_dilute"). Under these conditions, the off-diagonal terms automatically go to zero, and diagonals are always positive. Please also note that for FCC_L12#3 (MC) no diffusion data can be obtained from the database!
For the liquid phase, for computational efficiency and in view of the uncertainty of the database values for this phase, it is legitimate to use a constant value of 1.E-5 cm2/s instead. Effects e.g. on secondary arm spacing or morphology in most cases anyway need to be calibrated via the interfacial energy.

6.) criterion_higher / criterion_lower
This is a relative type of composition limit which forces an error when violated by any kind of TQ-operation. Like the other limit keywords (limits, sum_limits, ordered/disordered), it should be used carefully, and characteristic error numbers are given on screen output.
In your case, due to the improper use of this option, you get corresponding error messages already shortly after nucleation of phase 1!

7.) Disregarding elements
It is always reasonable to check whether certain elements can be neglected in a MICRESS simulation, because in helps tremendously reducing calculation times and the risk of numerical issues. Often, disregarding an element is linked to disregarding a corresponding phase, like e.g. for B and the corresponding borides. Conversely, the choice of an element also requires including corresponding relevant phases, if their formation temperature is reached.
The important preconditions thus are:
- low composition
- low thermodynamic impact (e.g. in Thermo-Calc Scheil simulation)
- not relevant for important phases.
In your case, B and Zr could be easily removed if there is no specific interest in these elements.


Bernd

mauvec
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Re: Seed-density and seed-undercooling models

Post by mauvec » Mon Mar 29, 2021 10:41 pm

Dear Bernd,

Thank you for your comments and clarification regarding the thermal setup. I completely misunderstood on how to implement several concepts for 2D simulations of cross-isothermal sections without temperature gradient. The intention of using the 'seed-density' model was to create first a random distribution of grains within a certain domain and then control the distribution of these with the 'shield distance' and 'shield time' options. All the previous mentioned also to mimic the grain size and distribution according to experimental observations (micrograph attached in a PM). In this regard, would it make sense to create a microstructure-file (as explained in section 4.2 of the Manual) in order to mimic the aforementioned? So coordinates for deterministic grain positioning can be specified at the beginning of the simulation.

From your last message with comments and corrections, I also realised how many mistakes and terrible implementation of keywords and/or options I made during the design of the input file. So I really thank you for your support and patience on this. I am now more aware about pros and cons of retrieving data from databases. On continuing with MC-type carbide formation at the interface, the following snapshot corresponds to the isothermal cross-section but now departing again from the 'initial microstructure' setup implemented in your application example A006_CMSX4.
image1.png
image1.png (70.46 KiB) Viewed 484 times
The number of errors diminished considerably, and only few were indicated (please find attached in a PM the .scr file). In spite of the latter, the volume fraction of FCC_L12#3 in the simulation remains relatively imperceptible (See 'TabF' file), and therefore undetectable in the simulation domain. Experimentally, these carbides render a particle size of ~50 nm approximately. So I am wondering if a finer grid resolution should be implemented which suits better to microstructural features shown in the micrograph separately attached.
image2.png
image2.png (77.5 KiB) Viewed 484 times
In one of my previous messages, I also mentioned that experimentally, gamma-prime (FCC_L12#2) phase render a considerably small particle size at grain interiors and boundary regions of 5 and 50 nm, respectively. However, regardless the small size of this phase, I am wondering if this should be included for quasi-equilibrium calculations?

I also noticed that concentration of Ti and W render negative values in the 'TabLin' file. Might this be happening because of a bad implementation of stoichiometric conditions? To perform this simulation I considered the following from your previous message,
  • Made distinction between elements defined as stoichiometric with solubility in MC (FCC_#3).
  • Implemented 'limit' of C for FCC_L12 between 0-20 at%.
  • Utilised constant values of 1E-05 cm2/s for the LIQUID phase. Considered full diffusion matrix (multi_plus) for FCC_L12, and no diffusion for FCC_L12#3.
  • Implemented exclusively 'criterion_higher' for C between MC(FCC_#3) and FCC_L12.
  • Disregarded B and Zr from simulations.
I hope I did not forget and/or missed anything this time.

Thank you for taking the time to read this.

Best regards,

Mauro

Bernd
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Re: Seed-density and seed-undercooling models

Post by Bernd » Tue Mar 30, 2021 3:06 am

Dear Mauro,

This already looks much better, congratulation!
As you said, there are still some (few) errors appearing in the .scr-file. Error 20 refers to nucleation of phase 2 from the melt, so I guess there might be some regions where the liquid phase got negative for one of the carbide forming elements (you can check this easily using the .c*pha0 output and manually rescaling to +-0.01 % - then you will easily find areas of negative concentration as blue zones, probably around carbides). If you find such regions, they are probably caused by over-extrapolation of linearized phase-diagram data, and thus theoretically can be avoided by sufficiently frequent updating of the 1/2-linearisation data. In practice, we apply "penalty" terms (if we find extended negative regions), which are less problematic in terms of performance. Probably, this corresponds to the negative values of TI and W which you mentioned.
With "microstructure file" in Section 4.2 of the MICRESS manual we mean "real" microstructures which are created from and resemble experimental microstructures, and which are stored pixel-wise in ASCII-files. What you probably propose instead is to extract only the central coordinates of each dendrite from a cross-sectional image of a directionally solidified microstructure, which makes perfect sense, right?

Best wishes
Bernd

mauvec
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Re: Seed-density and seed-undercooling models

Post by mauvec » Tue Apr 06, 2021 1:18 pm

Dear Bernd,

Yes, I meant extracting the central coordinates. While simulations running in the background, may I ask you if it is possible to produce temperature profiles in MICRESS by solving the transient heat transfer equation?

Best regards,

Mauro

Bernd
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Joined: Mon Jun 23, 2008 9:29 pm

Re: Seed-density and seed-undercooling models

Post by Bernd » Tue Apr 06, 2021 6:33 pm

Hi Mauro,

Yes, this is possible if you use the 1d-temperature field which I mentioned above. This automatically includes the calculation of latent heat, but requires knowledge or iterative determination of the thermophysical properties H(T), Cp(T) and λ(T) for the regions outside the microstructure domain (see here).

Bernd

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