## Multiphase field equation problem

Exchange about the physics background, diffuse interface theory, etc..
shaojielv
Posts: 32
Joined: Thu Apr 01, 2021 2:16 pm
anti_bot: 333

### Multiphase field equation problem

Dear friend,
I met some problems in the learning of MICRESS software, and I hope I can get some help here.

First, I would like to know how the dynamic equation in MICRESS was derived. The equation can be found in the attachment. When I was deriving the equation, I noticed that it should be derived by Lagrange multiplier, but I did not find the relevant derivation process, so I hope to get help.

Second, the double/multi-obstacle functional used in MICRESS is not a double-well function, but the value range of the order parameter in the double-obstacle function is 0 to 1, so I think there is no derivative for the double-obstacle function at 0 and 1.Therefore, I have a lot of doubts here. If you can help me, I will be very grateful.

Finally, I wish you a happy life and smooth work
Lv Shaojie
Attachments
kinetic equation.docx

janin
Posts: 34
Joined: Thu Oct 23, 2008 3:06 pm

### Re: Multiphase field equation problem

Hi,

To question1: Relaxation approach
The MPF order parameters are linked by the unit sum constraint, i.e. each phase can only change its state in interaction with another phase. The given relaxation approach describes the evolution of the multiphase system towards a minimum of the free energy under comprehensive consideration of pairwise interactions with individually defined mobilities. A combined handling of the dependent derivatives based on a single Lagrange multiplier (used in other MPF models) is only valid under assumption of identical or locally averaged interface mobilities. In this simplified case, the individual dependent contributions can be extracted from the brackets and combined. This is however not suitable for realistic quantitative application. I attach some PPT-slides for more detailed information.

A comprehensive derivation of the MPF kinetic equations can be found in my thesis:
J.Eiken: A Phase-Field Model for Technical Alloy Solidification
https://www.researchgate.net/publicatio ... dification

To question 2: Multi-obstacle potential
The advantage of the multi-obstacle potential is that the interfaces have a finite and clearly defined thickness. Thus, the MPF equations have to be solved within restricted interface areas, only, and artificial ghost contributions of dormant faces are avoided. However, it is very difficult to mathematically formulate the model equations in the singular points and in fact, we never published this correctly, so far. The multi-obstacle potential combines an inner "multi-parabolic" potential ∑(Φ_α * Φ_β) with outer "obstacles". The inner multi-parabolic potential is defined with respect to those phases which are “locally interacting”, only. A phase is defined as “locally interacting” if the amount of the gradient of the respective order parameter differs from zero. Thus, a point with order parameter 0 or 1 may be either inside or outside the interface region. Inside, its finite derivatives are clearly defined by the multi-parabolic potential ∑(Φ_α * Φ_β), outside all associated interfacial mobilities and energies are defined as zero and therefore the theoretical ‘obstacle’ part of the free energy functional is irrelevant and never defined in practice.

The discretization of the (double-obstacle) PF-equation based on a dedicated Finite-difference scheme is outlined in:
https://www.researchgate.net/publicatio ... tion_error

With best regards,
Janin
Attachments
Eiken_mpf_relaxation-approach.pdf

shaojielv
Posts: 32
Joined: Thu Apr 01, 2021 2:16 pm
anti_bot: 333

### Re: Multiphase field equation problem

Dear Janin,

First of all, thank you very much for your reply. I don't quite understand the external "obstacle" you mentioned in the second question. Could you please explain it in detail?In addition, I can't open the link in question two, so I can't see the article you sent. I hope to get your help.

Thank you very much for taking time out of your busy schedule to answer my questions. Wish you a happy life.

Lv Shaojie