Hello researchers, I have a question about phase field models that I was hoping someone could help me or point me in the right direction.
I understand that phase field models generally start from a energy density function, which is then formulated into an "evolution" equation that describes how the system evolves in time. My question is, if I only have the phase field evolution model instead (a differential equation, describing the evolution of boundaries in time), how do I calculate the total free energy of the system at any given point?
Is this something you can help me with, or point me to some articles that can help?
Thanks very much,
Ali
Hi fellow researcher,
To be blunt, I don't have much experience with such models. As far as my knowledge goes, what is done is to calculate free energy expressions that present certain characteristics depending on the problem using various methods. e.g, non-linear conservation laws methods (see R. Knobel, An introduction to the mathematical theory of waves, Student Mathematical Library, IAS/PARK CITY MATH SUBSERIES, vol. 3, AMS, 1999.).
See this review
https://d1wqtxts1xzle7.cloudfront.net/32238846/cheng02-with-cover-page-v2.pdf?Expires=1661521422&Signature=OXq24fDRK3beoL5pUZYNKineEccVYm5-RvAeg4IPNYvmgYU8y3Fq35mS3iyoafHZkbNMV2VfRwYI9fpcgOYEpNGJ0ICrxWSIzJ~ypWKDOk~8s5bgYhur~LzrCdBzx44F1ttNW1625tSMpio1GC7~5Xfgu9THXntEdFwFvOMZArrnLYa8dtWhs6KRxvqisGAPVNFNCiDndxDmsKK8euqMu8kneBsOtIk0hj7xPKp8WXyS-466sLSBACEfp~iYAo4a7iqna0hfbPkLxvPKt9zzTix8VIu4SMo~Yl3X00BfqBnGPZpzmERhcR5wXRctktVsdHYv3aUe3Kz0UNbOl8H0Vw__&Key-Pair-Id=APKAJLOHF5GGSLRBV4ZA
I hope this have been a useful comment, best regards
Hello Ali,
I just came across your question. Here is a response, assuming that you still require an answer.
In general, you cannot determine the corresponding energy function if the only thing you're given is the evolution equation. This is because some models are non-variational (i.e., the evolution equation isn't a simple functional derivative of energy functional) either fully or partially. You may have other additions to the evolution equation that is not part of the energy at all but are added in an ad-hoc manner to achieve the "correct" properties. Also, as with regular calculus, there may be constants not observed in the evolution equation.
You may look into the Eq. 23 of the paper authored by Abinandanan and Haider (Philos Mag A 81 (2001) 2457) for an extended Cahn-Hilliard model, if you are still looking for an answer. From your query it seems the model you are dealing with is an Allen-Cahn type phase field model. In such a case, if you do not have any higher order terms, then by simply ignoring the third term in Eq. 23, you should be able calculate the total free energy of the system.
- Badges
- Science topic
- Similar topics
- Biological Science
- Bioecology
- Systems Ecology
- Landscape Ecology
- Connectivity