N-order parameter KKS model

[Johan Hektor]

Hi,

I've just started implementing a KKS type phase field model for simulating growth of intermetallic compounds in the Cu-Sn system (see attached paper) in Moose. I want to use one order parameter for each grain of each phase. I've found the file kks_multiphase.i which is a KKS model with 3 order parameters. What would be a reasonable approach for extending that file to N-order parmeters? From what I understand the kernels (e.g. KKSMultiACBulkC) are written in general way, but the switching function is not. If I write an appropriate switching function, can I add more order parameters to my simulation just by copy+paste in the input file?

Cheers

Johan Hektor

[Aagesen, Larry K]

The attached paper gives a description of a multi-phase KKS model. We have implemented the switching functions they use in Eq. 10 for a different project. The kernel is in the phase-field module and is called ACSwitching and it gives the derivative of that switching function with respect to a particular order parameter. I haven't looked at this paper in detail in a little while, but I think there are some changes in the Cahn-Hilliard equation implementation that would need to happen too.

Please let us know how this progresses for you, a multi-phase KKS model is something we would be interested in having in the framework so we would be willing to work with you to implement it and and incorporate your changes into the framework.

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[Johan Hektor]

Thank you for your reply! Could you elaborate a bit on what needs to change in the implementation of the Cahn-Hilliard equation?

I'll update you on any progress/problems.

On Monday, May 1, 2017 at 4:32:14 PM UTC+2, Aagesen, Larry K wrote:

The attached paper gives a description of a multi-phase KKS model. We have implemented the switching functions they use in Eq. 10 for a different project. The kernel is in the phase-field module and is called ACSwitching and it gives the derivative of that switching function with respect to a particular order parameter. I haven't looked at this paper in detail in a little while, but I think there are some changes in the Cahn-Hilliard equation implementation that would need to happen too.

Please let us know how this progresses for you, a multi-phase KKS model is something we would be interested in having in the framework so we would be willing to work with you to implement it and and incorporate your changes into the framework.

On Fri, Apr 28, 2017 at 9:50 AM, Johan Hektor <johan....@gmail.com> wrote:

Hi,

I've just started implementing a KKS type phase field model for simulating growth of intermetallic compounds in the Cu-Sn system (see attached paper) in Moose. I want to use one order parameter for each grain of each phase. I've found the file kks_multiphase.i which is a KKS model with 3 order parameters. What would be a reasonable approach for extending that file to N-order parmeters? From what I understand the kernels (e.g. KKSMultiACBulkC) are written in general way, but the switching function is not. If I write an appropriate switching function, can I add more order parameters to my simulation just by copy+paste in the input file?

Cheers

Johan Hektor

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[Aagesen, Larry K]

Actually, let me start out by saying that if you are dealing with only 2 (Cu-Sn) components you may be able to use the existing Cahn-Hilliard formulation. The paper I attached considers an arbitrary number of phases, grains,  and chemical components. However you might not need to modify the Cahn-Hilliard part if you only have two chemical components, in which case a single composition variable is all you need.

If you do need extra chemical components, let me start by pointing you in the direction of our Wiki so you can see how the single order parameter KKS works. Equation 32 from the original KKS paper is implemented in the framework. You can see how this is done here:

http://mooseframework.org/wiki/PhysicsModules/PhaseField/DevelopingModels/KKS/

Also have a look at one of the examples in the framework so you get an idea of what kernels are used and look at the source files for the relevant kernels. To extend this to multiple phases, you will need to similarly represent Eq. 14 and 15 in the Moelans paper. You should be able to use the existing CoupledTimeDerivative and SplitCHWRes kernels for Eq. 14 (I believe you will have to sum them so you will need C-1 equations and C-1 SplitCHWRes kernels in each equation). Eq. 15 is in a different form than we currently use in the KKSSplitCHCRes kernel. I think if you assume a constant chemical mobility you can put Eq. 15 in a similar form to what we currently use. This is probably a good starting point, once you get this working, if you need you can modify things for non-constant chemical mobility, but I would recommend starting with the simpler case first.

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[Johan Hektor]

For now, I'm only dealing with the binary Cu-Sn system so one composition variable is enough. 

I've been trying to derive the residual Cahn-Hilliard equation based on the Moelans paper. It seems that for constant mobility, M, the residual becomes the same as in the wiki: http://mooseframework.org/wiki/PhysicsModules/PhaseField/DevelopingModels/ http://mooseframework.org/wiki/PhysicsModules/PhaseField/DevelopingModels/

Except for one thing: In the Moleans paper there is no energy term depending on the gradient of the composition. In the end this makes the C-H equation a second order PDE instead of a fourth order. Does this make sense or have I missed something? I guess I can still solve the equation using the KKSSplit kernels if I put the \kappa_i=0 but maybe there is a better way?

[Aagesen, Larry K]

Yes you are correct about the Cahn-Hilliard equation being only 2nd order in the Moelans paper (no composition gradient term). The same is true in the original KKS paper, in fact, and that's part of the magic in how the chemical energy contribution to the interfacial energy is removed. I suppose maybe it's better to refer to it as a generalized diffusion equation rather than the Cahn-Hilliard equation for that reason...

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[Johan Hektor]

I've finally found some time to work on this. I've put together an input file with a KKS model for a binary system containing three phases (three order parameters and one composition). In principle it seems to work (one phase is growing at the expense of the others and the total energy is decreasing with time).
There are some things that need to be written in a nicer and more general way and I'm not sure if the non-constant mobilities are taken into consideration in the Jacobian but overall it seems promising.

[Chris Sewell]

Hey guys,

Did anyone make any more progress on this topic?

I'm just looking into a three-phase, 2 concentration KKS model, so any pointers to additional work done in the last year would be helpful/

Thanks in advance,

Chris Sewell

[Larry K. Aagesen Jr]

When you say you need 2 concentrations- are you looking at a two-species or three-species system? I'm assuming you mean three total species with two concentrations needed to represent two solute species but just wanted to confirm first before answering. What are the chemical species and phases?

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From: moose...@googlegroups.com <moose...@googlegroups.com> on behalf of Chris Sewell <sewellj...@gmail.com>
Sent: Thursday, December 6, 2018 3:19:03 AM
To: moose-users
Subject: Re: N-order parameter KKS model

 

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[Sewell, Christopher]

Yes it’s a 3 species film system: iron | iron-oxide | iron-sulfide | environment

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[Larry K. Aagesen Jr]

There's a couple of different routes you could go. You could use the existing 3-phase KKS model and add an additional concenration variable. The three-phase KKS model is described in 

Rapid multiphase-field model development using a modular free energy based approach with automatic differentiation in MOOSE/MARMOT

By: Schwen, D.; Aagesen, L. K.; Peterson, W.; et al.

COMPUTATIONAL MATERIALS SCIENCE   Volume: 132   Pages: 36-45   Published: MAY 2017

You can find more information about adding additional composition variables at:

https://mooseframework.inl.gov/modules/phase_field/MultiPhase/KKSMultiComponentExample.html

Alternatively, if the free energy versus composition dependence in your model can be described using either a parabolic or dilute solution model, you can use the multi-component, multi-phase grand potential model described in

Grand-potential-based phase-field model for multiple phases, grains, and chemical components

By: Aagesen, Larry K.; Gao, Yipeng; Schwen, Daniel; et al.

PHYSICAL REVIEW E   Volume: 98   Issue: 2     Article Number: 023309   Published: AUG 24 2018

There are pointers to specific examples of the model (input files in the MOOSE repository) in the paper.

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[Larry K. Aagesen Jr]

[Google]

Thanks, that’s a really helpful starting point. I’ll get reading :)

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[Chris Sewell]

Thanks again Larry,

A quick question regarding the Grand Potential model (I'm sure they'll be more to come).

Is it ok to mix parabolic and dilute solution free energy phases, e.g. a solid (eta1) interfaced with a liquid (eta2)?

[Larry K. Aagesen Jr]

I have tried this and it did seem to work OK. However to make sure, I would start with a binary system at a temperature between the solidus and liquidus, plot both free energies and use the common tangent construction to get the equilibrium compositions. Then simulate a two-phase system and make sure that when the system reaches equilibrium, the equilibrium compositions in the model match the compositions predicted by the common tangent construction.

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From: moose...@googlegroups.com <moose...@googlegroups.com> on behalf of Chris Sewell <sewellj...@gmail.com>

Sent: Monday, December 10, 2018 2:06:33 PM

To: moose-users
Subject: Re: N-order parameter KKS model

Thanks again Larry,

A quick question regarding the Grand Potential model (I'm sure they'll be more to come).

Is it ok to mix parabolic and dilute solution free energy phases, e.g. a solid (eta1) interfaced with a liquid (eta2)?

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