Looking for an example to simulate three phases in a ternary alloy using phase field method

[zuzup...@gmail.com]

Hi,

I know there is an example using KKS (kks_example_noflux.i), which solves two phases in a binary system. What I want to do here is kind of similar, but to extend it to three phases in a ternary system. The ternary system is Ce-Fe-Sb, the gibbs free energy of the system is available.  To be more specific, I am trying to simulate the three phases in the ternary alloy shown in the picture below.

Is there an example available in the repository? I don't think kks_example_ternary.i (or here) is the one I can use, because kks focuses on a binary system.

Thank you!

Wesley

[zuzup...@gmail.com]

Correction.

Perhaps kks_example_ternary.i is the one I could use. There are three components, but there are only two phases (liquid and solid). Is this valid to include the third phase and make it happen?

 

Thank you!

Wesley

[Larry K. Aagesen Jr]

Hi Wesley,

We have a 3-phase KKS model with an example in the MOOSE repository here:

moose/modules/phase_field/test/tests/KKS_system/kks_multiphase.i

There are 3 order parameters, 1 for each phase, rather than the single order parameter in the original KKS model. The derivation of the formulation is in this paper:

https://doi.org/10.1016/j.commatsci.2017.02.017

However it is for a binary and therefore only has one solute species; since you are interested in a ternary system, you would need to add an additional set of variables to track the additional solute species. You can have a look at the differences between kks_example_noflux.i and kks_example_ternary.i to understand what needs to be added. The following from the MOOSE documentation site will also be helpful:

https://www.mooseframework.org/modules/phase_field/MultiPhase/KKSMultiComponentExample.html

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From: moose...@googlegroups.com <moose...@googlegroups.com> on behalf of zuzup...@gmail.com <zuzup...@gmail.com>
Sent: Monday, September 21, 2020 2:40 PM
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Subject: [EXTERNAL] Re: Looking for an example to simulate three phases in a ternary alloy using phase field method

 

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[zuzup...@gmail.com]

Hi Dr. Larry Aagesen,

Thanks for the suggestions. I am going through all the materials, meanwhile I have some quick questions for now:

(1) The Fig.6 in that ref shows a tri-junction, how would it be able to create a tri-junction (i.e., how to designate eta1 in the top-right, eta2 in the left and eta3 in the bottom-right) ? The  kks_multiphase.i example uses SmoothCircleIC to setup a circle condition.

(2) In that ref, it reads "As seen in Fig. 6, at equilibrium a stable tri-junction is formed." What's the criteria to say it is "at equilibrium" when performing the simulation ?

(3) In kks_multiphase.i, per my understanding, eta1 is in the middle circle, eta2 is the rest, eta3 is set to be 0 throughout the domain. How to understand that eta3 = 0?  (If eta3 = 0, how can we tell this phase exist?) 

(4) How to understand the concept of phase concentration (c_i). In that ref, it says "effectively the fractions of the total concentration held in a given phase". 

This is important because the free energy input is a function of phase concentration, i.e., f_i(c_i). In kks_multiphase.i, for example,  the free energy of phase 1 is set to '20*(c1-0.2)^2', so it means that the equilibrium concentration is 0.2 in this phase, correct? If this is correct, then we will have the term "(phase concentration - equilibrium concentration)^2" for the free energy, and the equilibrium concentration can be obtained by the common tangent construction. That being said, if I have a phase A2B3 in the system, and the concentration of specie A is c, so I will include something like "(c1-c1_eq)^2", where c1_eq is the equilibrium concentration of A based on the common tangent construction. Does this correct?

Thanks.

[Larry K. Aagesen Jr]

Hi Wesley, please see in line responses below:

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From: moose...@googlegroups.com <moose...@googlegroups.com> on behalf of zuzup...@gmail.com <zuzup...@gmail.com>

Sent: Tuesday, September 22, 2020 4:33 PM
To: moose-users <moose...@googlegroups.com>
Subject: Re: [EXTERNAL] Re: Looking for an example to simulate three phases in a ternary alloy using phase field method

 

Hi Dr. Larry Aagesen,

Thanks for the suggestions. I am going through all the materials, meanwhile I have some quick questions for now:

(1) The Fig.6 in that ref shows a tri-junction, how would it be able to create a tri-junction (i.e., how to designate eta1 in the top-right, eta2 in the left and eta3 in the bottom-right) ? The  kks_multiphase.i example uses SmoothCircleIC to setup a circle condition.

There is no IC to create the exact shape of a tri-junction; but you can use the BoundingBoxIC to set the order parameters in rectangles that are close to the final shape in the ICs, then allow the simulation to run to achieve the equilibrium trijunction shape.

(2) In that ref, it reads "As seen in Fig. 6, at equilibrium a stable tri-junction is formed." What's the criteria to say it is "at equilibrium" when performing the simulation ?

If you allow the system to run long enough, no further evolution in the microstructure will be observed. That is what is meant by "at equilibrium" (to be more precise this may be only metastable equilibrium).

(3) In kks_multiphase.i, per my understanding, eta1 is in the middle circle, eta2 is the rest, eta3 is set to be 0 throughout the domain. How to understand that eta3 = 0?  (If eta3 = 0, how can we tell this phase exist?) 

If eta3 = 0 everywhere in the initial conditions, that phase does not exist in the microstructure as defined in the ICs. However having eta3 defined in the model allows you to put in that phase if you need to.

(4) How to understand the concept of phase concentration (c_i). In that ref, it says "effectively the fractions of the total concentration held in a given phase". 

This is important because the free energy input is a function of phase concentration, i.e., f_i(c_i). In kks_multiphase.i, for example,  the free energy of phase 1 is set to '20*(c1-0.2)^2', so it means that the equilibrium concentration is 0.2 in this phase, correct? If this is correct, then we will have the term "(phase concentration - equilibrium concentration)^2" for the free energy, and the equilibrium concentration can be obtained by the common tangent construction. That being said, if I have a phase A2B3 in the system, and the concentration of specie A is c, so I will include something like "(c1-c1_eq)^2", where c1_eq is the equilibrium concentration of A based on the common tangent construction. Does this correct?

Phase concentration in the KKS formulation is a bit tricky to get an intuitive understanding of. The key thing to remember is the phase concentrations are not the physical solute concentrations at each position. The physical concentration of solute at each position is determined by interpolating the phase concentrations with the interpolation function. That physical concentration is what is transported by the diffusion equation in the KKS model. The phase concentrations are only used to determine the free energy/chemical potential of that solute in the given phase. It may be helpful to read the original KKS paper (Phys Rev E, 60, 7186, 1999) to understand this concept better if you have not already done so.

To view this discussion on the web visit https://groups.google.com/d/msgid/moose-users/b02603b3-e6c4-4fd6-a779-6ece8b69df9dn%40googlegroups.com.

[zuzup...@gmail.com]

Hi Dr. Larry Aagesen,

With your explanation, I am able to reproduce something similar. There is an interesting observation here: the shape of the phases 1 and 2 are initially set to be symmetric (left), but they are not necessarily to be symmetric after the evolution (right). Is this explainable ? Is this similar to what we discussed in this thread?

The second question is:

In that ref., in the paragraph right above eq(26), it reads that the three-phase concentrations reduces to that of the two-phase model, i.e.,  eq (25) reduces to eq (16). Is this the correct understanding? If so, it should use KKSPhaseConcentration to present eq (16) in the model. However, the kks_multiphase.i uses KKSMultiPhaseConcentration to enforce c = h_1*c_1 + h_2*c_2 + h_3*c_3, thus presents eq (25). How to understand this?

Thanks.

Wesley

[zuzup...@gmail.com]

Correction (marked red) and additional comments (blue) are as below.

On Thursday, September 24, 2020 at 9:28:34 PM UTC-4 zuzup...@gmail.com wrote:

Hi Dr. Larry Aagesen,

With your explanation, I am able to reproduce something similar. There is an interesting observation here: the shape of the phases 2 and 3 (not phases 1 and 2) are initially set to be symmetric (left), but they are not necessarily to be symmetric after the evolution (right). Is this explainable ? Is this similar to what we discussed in this thread?

The second question is:

In that ref., in the paragraph right above eq(26), it reads that the three-phase concentrations reduces to that of the two-phase model, i.e.,  eq (25) reduces to eq (16). Is this the correct understanding? If so, it should use KKSPhaseConcentration to present eq (16) in the model. However, the kks_multiphase.i uses KKSMultiPhaseConcentration to enforce c = h_1*c_1 + h_2*c_2 + h_3*c_3, thus presents eq (25). How to understand this?

The third question is:

There are three phases, and the minima of the free energy of each phases are at Zero. To implement the real free energy, the free energy of each phase needs to be revised into something similar below. It contains two parts: the first part is the gibbs energy expression as a function of temperature normally calculated by Calphad, the second part is the parabola. So now we have the parabola, and need to include the first part. At a particular temperature, this part is just a negative value. However, if this part is added, the results are not as beautiful as that Tri-junction. I believe it is because the minima of the free energy of each phases are NOT at Zero. I realized the model is not used properly but at this point I do not know how to proceed...

 

Thanks.

Wesley