Questions about phase field using kks model

[Wesley]

Hi phase filed experts from MOOSE team:

I am exploring phase field method with kks model, and have tried to extend the example into my case.

The example I found in MOOSE folder is kks_example.i. With this example, I am able to extend to my case, briefly described as below.

(1) There are two phases in the system, phase 1 is inside phase 2, both phases are binary intermetallic compound;

(2) For each phase, the free energy as a function of temperature is included;

(3) The phase evolution at one temperature (1000K) is shown as below, we see that phase 1 shrinks, and more phase 2 is present.

Just one question: what determines the shrink of the inside phase?

Some thoughts are listed below:

(1) Did the shrink of the phase 1 mean something? Why does it shrink? Does this explanation below correct:

  

--> The free energy of phase 2 is lower than that of phase 1 (this can be easily calculated), so more phase 2 is preferable, so phase 1 shrink. 

I thought this maybe one of the reasons. However it seems not very correct because I have another binary system in which the free energy of phase 1 is lower that of phase 2. I was expecting that phase 1 will be expand, but the results show that phase 1 still shrink. At this point, I don't understand what will determine the shrink of the phase 1.

(2) With this question, I am looking at the kks_example.i. Similar results are observed as shown below, in my opinion, the phase inside is shrink too. I cannot judge it from the point view of free energy, because the free energy of the phase is a function of concentration (i.e. free energy of outside phase = (0.1-cm)^2, free energy of inside phase = (0.9-cd)^2).

[Daniel Schwen]

Two things to consider: 

1. the interfacial energy: the reduction in interfacial area can reduce the total free energy of the system

2. The initial condition: the color scale in your plots is changing. The IC at t=0 is 60%/40% and the equilibrium concentrations are 90%/10%

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[Larry K. Aagesen Jr]

Hi Wesley,

A couple of things may be contributing to phase 1 shrinking. The first is what you are setting the concentration in the matrix and precipitate phases to be in the initial conditions. You mentioned that you are simulating a binary system and each phase is an intermetallic. In this case the equilibrium composition is dictated by the common tangent construction based on the free energies of the two phases. (If you're not familiar with this construction you can read about it in a book as the Phase Transformations book by Porter and Easterling, or I'm sure there are good on line resources if you just search "binary alloy common tangent" or something like that). So, if you don't set the initial composition in each phase to the equilibrium values determined by the common tangent construction, solute will move from one phase to another until each phase is at the equilibrium composition, and the volume fraction of each phase will change as that happens. So this is one potential cause of phase 1 shrinking.

Another factor that may be causing phase 1 to shrink is the Gibbs-Thomson effect. The presence of a curved interface will cause the equilibrium composition of solute on both sides of the interface to change from the composition determined from the common tangent by an amount proportional to the interfacial energy and curvature of the interface. This shift can be an additional cause of the shrinkage. So I would suggest first making sure you know what the equilbrium composition of the two phases should be in the absence of the Gibbs-Thomson effect. Then you can run a simulation of two-phase coexistence with a flat interface to make sure you have the initial compositions set to their equilibrium values - if this is the case then the flat interface between the two phases should not move. Then following that you can go back to the circular precipitate particle; and any shrinking you see would be due to the Gibbs-Thomson effect. If you want to prevent this from happening you can increase the initial composition to account for the shift due to Gibbs-Thomson; with a little digging you can find the analytical expression for the size of the shift and increase your initial composition in each phase by that amount.

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From: moose...@googlegroups.com <moose...@googlegroups.com> on behalf of Wesley <zuzup...@gmail.com>
Sent: Monday, May 11, 2020 11:52 AM
To: moose-users <moose...@googlegroups.com>
Subject: [EXTERNAL] Questions about phase field using kks model

 

[Wesley]

Hi Dr. Daniel and Dr. Larry,

 

Thanks for your comments.

 

I realize that the phase evolution depends on more than one factors, such as bulk energy, interfacial energy, elastic energy. etc.. At this point, I understand Dr. Daniel's comment 1.

 

For the ICs, Dr. Larry's comment makes sense to me. Per my understanding, if the initial composition in each phase is not the equilibrium value, it will change until it reaches the equilibrium. I think this is what has happened in the kks_example.i. I put more time steps (set to 50) and finally c will be around 0.431 throughout the domain, quickly shown as below. We can tell the matrix and inside phase, but the values are very close. 

Then, how to understand Dr. Daniel's comment 2:" ...the equilibrium concentrations are 90%/10%"

The two values are represent in the codes:

  # Free energy of the matrix
  [./fm]
    type = DerivativeParsedMaterial
    f_name = fm
    args = 'cm'
    function = '(0.1-cm)^2'
    outputs = oversampling
  [../]
  # Free energy of the delta phase
  [./fd]
    type = DerivativeParsedMaterial
    f_name = fd
    args = 'cd'
    function = '(0.9-cd)^2'
    outputs = oversampling
  [../]

i.e. free energy of matrix = (0.1-cm)^2, free energy of delta phase = (0.9-cd)^2). I understand it in this way: if they are binary compounds, then the compound A1B9 is in the matrix and the compound A9B1 in delta phase (inside phase), and c is the concentration of A.

So what's the "equilibrium concentration" here? Is it (1) or (2)?

(1) 0.431 for all the domain. This means A become homogenized everywhere.

(2) 0.1 for cm, 0.9 for cd. This refers to the compound composition (as it's natural property).

For my case, c does not homogenized like the example, it can still be clearly differentiated in different phases, and the values are not that close.

May I ask one more questions:

 

(1) In the kks_example.i., it uses KKSCHBulk kernel. In the kks model definitions (https://mooseframework.org/modules/phase_field/MultiPhase/KKSDerivations.html), it says

 " KKSCHBulk is the non-split version, which is not fully implemented**. "

What does that "not fully implemented" mean? I see ** there, and am expecting to see more explanation/annotation but I cannot find.

For the Gibbs-Thomson effect, thanks for pointing out but I will need more time to digest and reply.

Thanks.

[Larry K. Aagesen Jr]

The equilibrium compositions are given by the common tangent between the free energy curves of the two phases. So, for this test since they are both parabolas with minima at 0, the common tangent is just a horizontal line, the equilibrium composition of the matrix is at c_m = 0.1 and the equilibrium composition of the delta phase is c_d = 0.9. The initial compositions are not set to those values in this test, and the order parameter values are also not set to the equilibrium values in the initial conditions. Sorry that this may be a little confusing but the reason this is done is so that more evolution will occur in the short duration of a test (2 time steps) - if we initialize the system with equilibrium conditions not much happens so the test is not very sensitive to potential errors in new code that we are trying to detect. So for purposes of learning the fundamentals of phase-field modeling I would suggest you start with the examples in the directory moose/modules/phase_field/examples/ rather than in the tests.

---

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Sent: Monday, May 11, 2020 4:23 PM
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Subject: [EXTERNAL] Re: Questions about phase field using kks model

 

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[Wesley]

Hi Larry,

Thanks for your reply. Your explanations are clear. Yes I am moving forward to to the examples in the file you suggested. I understand the test files were created for testing purpose, and they may be not the good examples to learn, I apologize for taking your time to explain.

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[Wesley]

Hi Dr. Larry,

You mentioned the following:

Then you can run a simulation of two-phase coexistence with a flat interface to make sure you have the initial compositions set to their equilibrium values - if this is the case then the flat interface between the two phases should not move. 

So I have tried this example (kks_example_noflux.i). At the interface, x=0, and eta = 0.5. The interface does not move. Everything looks perfect so I have moved forward and have developed my app following that example. Accordingly, I have changed the free energy of the two phases. However, I notice that in my case the interface moves. I plot some of the time step of eta below, as you can see, the interface moves to the right, meaning the shrinkage of the right phase. In addition, I notice that eta does not equal to 0.5 when x=0. I don't know why it is different from the example, would you be able to help me find out the reason?

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[Larry K. Aagesen Jr]

Hi Wesley,

The change in interface position is most likely due to the change in phase free energies that you made. When you change the free energies the equilibrium composition in each phase changes. I would plot the composition at the end and make sure the composition in each phase matches what you get from the common tangent construction. You can also look at the chemical potential variable, it should be constant throughout when the system reaches equilibrium and its value should be equal to the slope of the common tangent line.

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Sent: Friday, May 15, 2020 10:18 AM
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Subject: Re: [EXTERNAL] Questions about phase field using kks model

 

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[Wesley]

Hi Dr. Larry,

Thanks for the hints!

(1) In my case, after 90 time step, the composition in each phase are 0.4116 for Ce4Sb3 and 0.3261 for Ce2Sb, as shown in the left picture below. Those values match the values calculated from the common tangent construction as shown in in the right picture.

(2) Also the chemical potential (w) matches the tangent slope.

I am trying to understand  "the change in interface position is most likely due to the change in phase free energies that you made".

At a fixed temperature, the change of the phase free energy comes from the change of the concentration because of the concentration term (something like  k/2*(c_Si - 0.4)^2 in your report (eq.2)). At the beginning, the concentration term is 0 in bulk phase (because  I set the concentration to its nominal value. For example, c_Sb=3/7 in Ce4Sb3, c_Sb=1/3 in Ce2Sb. It is set up in the initial condition). However, at the interface region the concentration varied (so the concentration term is not 0). This will make the free energy change, and the whole system is being affected. As a result, the in interface position moves. And finally, the system will reach equilibrium, the the interface does not move. 

Does this correctly interpret your words?

I have the common tangent construction of the two phases

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[Larry K. Aagesen Jr]

OK good, it looks like the system is behaving as expected based on the phase diagram. In my earlier comment I meant that the interface position is changing from its initial value because you are no longer using the free energies with minima at c=0.1 and c=0.9 and minima at 0 as you were before. I can answer a bit more specifically now based on the phase free energies provided. If you had set c = 0.4116 on the left side and c = 0.3261 on the right side in your initial conditions you should not see any interface motion (or very minimal based on exactly how it is divided between the two phases in the interface in the initial conditions), because the two phases are in equilibrium at their equilibrium compositions. However in your initial conditions you have set c~0.43 on the left side and c~0.33 on the right side. The system wants to come to equilibrium with  c = 0.4116 on the left side and c = 0.3261 on the right side. It does that by transferring solute from the left to the right side. But since your initial condition is farther from the equilibrium composition on the left side than it is on the right side, and solute must be conserved, to get to having both phases at equilbrium compositions the Ce4Sb3 phase must grow in size. You can also review the tie line construction from an intro materials science textbook to understand phase fractions to understand this better.

---

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Sent: Wednesday, May 20, 2020 2:32 PM

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[Wesley]

Thanks Larry, the explanation makes sense to me.

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[Wesley]

Hi Dr. Larry,

I think the takeaway from the previous discussion is, we have two phases in the system, the initial composition is not the equilibrium composition, if the difference between the initial condition and the equilibrium composition is bigger, this phase should grow.

I have an exception for this as below. This is a system for U4Sb3 and USb, the Sb concentration in IC and equilibrium value are different. The initial value is the input of the code. The equilibrium value is from common tangent construction ( the bottom left figure), the concentration in the simulation matches that. The chemical potential matches the slop. If the statement above is correct, U4Sb3 should grow because it has a bigger difference as compared to USb phase (see the top table, the value in Diff column). However, the USb phase is growing (see the bottom right figure). Is there any misunderstand here?

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[Larry K. Aagesen Jr]

Hi Wesley,

You are setting the initial composition very close to equilibrium in this case. So in this case the initial interface motion may be dependent more on how you have interpolated the composition field across the interface in the initial conditions. 

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[Wesley]

Hi Dr. Larry,

I have been struggling a lot for this question. You were trying to help me understand it deeper, and gave the explanation. Unfortunately, I can not prove such statement in my cases, the statement is:

There are two phases in the system, the initial composition is not the equilibrium composition, if the difference between the initial condition and the equilibrium composition is bigger, this phase should grow.

I have tired the following modifications and comparisons:

(1) You said the initial composition and equilibrium composition was too closed. So I tried to increase the difference by setting c_Sb=0.44 in left side, the difference with equilibrium is 0.0122 (was 0.0008). In both cases, the motions are the same (moving to the left, meaning that USb growth).  

(2) In addition, I tried to use different IC functions. In the previous one the IC function was like that in this example (kks_example_noflux.i). at the interface it is smooth. Later I tried another IC, it is simply put eta = 1 at left side and eta = 0 at right side, at the interface there is a jump. The trends are still the same. 

(3) I have done 4 cases in total, two of them do not follow that statement, the results are listed below.

To me, the acceptable explanation is: the phase with more negative free energy will grow.

If you have more comments, please let me know. 

Thanks.

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[Larry K. Aagesen Jr]

Wesley, the differences you are using between equilibrium and your initial condition are still small. As I mentioned before, how you are interpolating the composition field between the phases may be outweighing the differences between initial and equilibrium compositions in determining the net composition. You mentioned using different IC functions on the order parameter eta, but it is the initial composition field that I was referring to. The IC on the order parameter does not control how much solute is in the system. To get a better understanding and build your intuition, I would stick with one alloy system and vary parameters rather than considering 4 different alloys. You could try increasing the difference between initial and equilibrium composition in one phase only systematically, and you should see that phase grow more (assuming you set initial composition above equilibrium composition). Also try computing the net system composition by integrating the composition variable using a postprocessor, and compare the phase fraction obtained from simulation to the value predicted by the lever rule.

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[Wesley]

Hi Dr. Larry,

Thanks for your response.

Let’s stick on one system and do the tests systematically. The IC of the eta and concentration are setting to something below, basically it means that there is a jump at the interface.

[Functions]

  [ic_func_eta]

    type = ParsedFunction

    value = '1*if((x<0&x>=-50),1,0)'

  []

  [ic_func_c]

    type = ParsedFunction

    value = '0.4278*if((x<0&x>=-50),1,0)+0.4998*if((x>=0&x<=50),1,0)'

  []

[]

 

I have performed three tests. In test 1, the concentration is pretty much the equilibrium concentration. In other two tests, the concentration of the left phase increase while the right phase does not change. In final state, the composition in each phase and the chemical potential match the common tangent construction. As we discussed, the left phase should grow because of the difference between IC and equilibrium, but it’s not, it’s shrinking, quickly shown in the pictures below. At this point, I do not know if there are something wrong with code.

As for the phase fraction, I used ElementIntegralVariablePostprocessor for variable eta and can quickly get the fraction of the left phase, as summarized in the table below.

 

Case	Sb concentration		Left phase fraction
	U4Sb3 (Left)	USb (Right)	Initial	Final
Test 1 (Equilibrium)	0.4278	0.4998	0.5	0.4961
Test 2	0.44	0.4998	0.5	0.4120
Test 3	0.46	0.4998	0.5	0.2741

I am looking forward to your comments.

Thanks!

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[Larry K. Aagesen Jr]

Hi Wesley,

The system is behaving exactly as I would expect. You have set the composition in the U4Sb3 phase above its equilibrium value in initial conditions, and USb at its equilibrium composition. So, in order for both phases to have their equilibrium composition, you have to transfer solute from the U4Sb3 phase to the USb phase. As you do that the USb phase wants to stay at its equilibrium composition so it will grow. You cannot generalize that "whatever phase doesn't have the equilibrium composition in initial conditions is going to grow," it depends on whether the initial composition is above or below the equilibrium composition. I would again suggest you review the lever rule to understand how to predict phase fractions based on net system composition before delving too much further into simulations.

---

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[Wesley]

Hi Dr. Larry,

Thanks for the comments. I think I have understood the lever rule. I use it to get the phase fraction for some systems, and have the data compared to what I got from MOOSE, as listed in the table below. There are some difference (<1.51%). Do you think the difference is acceptable?

Thanks!

Binary System	Initial composition		Equilibirum composition		Fraction of phase 1
	Phase 1	Phase 2	Phase 1	Phase 2	Lever rule	MOOSE	Diff
U4Sb3 (phase 1) + USb (phase 2)	0.44	0.4998	0.4278	0.4998	0.4153	0.4120	-0.79%
U4Sb3 (phase 1) + USb (phase 2)	0.46	0.4998	0.4278	0.4998	0.2764	0.2741	-0.83%
U4Sb3 (phase 1) + USb (phase 2)	0.4286	0.5	0.4278	0.4998	0.4931	0.4892	-0.78%
Ce4Sb3 (phase 1) + CeSb (phase 2)	0.4286	0.5	0.4168	0.4966	0.4048	0.402	-0.68%
Ce2Sb (phase 1) + Ce4Sb3 (phase 2)	0.3333	0.4289	0.3261	0.4116	0.3567	0.3621	1.51%

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[Larry K. Aagesen Jr]

Yes, that seems like pretty good agreement, given that there is some uncertainty in exactly where one phase begins and the other ends in the diffuse interface approach.

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[Wesley]

Hi Dr. Larry,

Thanks for your clarification, your help is highly appreciated!

Regards,

Wesley

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