Posts Tagged ‘phase field modelling’

Some reading material on structural phase transformations!

June 16, 2008

Phase transformations and their modelling is an area of research that is of great interest to me. So, I found the latest journal club piece at iMechanica by Kaushik Dayal on the kinetics of structural phase transformations very interesting.

Dayal introduces four papers in the piece. The first one by Ericksen dates from 1975. The next two are more recent and are by Truskinovsky and Vainchtein (2005) and Hildebrand and Abayaratne (2007). The last one is by Kaushik Bhattacharya and is nearly a decade old (1999). While the first one is about the limitations of continuum models to study problems of this type, the second and third are of atomistic models, and the last is related to mesoscopic models.

Although I have heard of some of the authors (and, heard one of the authors long back in Bangalore on some of these problems — Kaushik Bhattacharya), all these papers and the ideas presented in them are new to me. I am planning to read at least a couple of them and blog about them in these pages later.

However, in the meanwhile, I also wanted to draw attention to another school that uses phase field models to study structural phase transformations (martensitic transformations) at the mesoscopic scale;  some of the important papers in this area (that I am aware of) are due to Khachaturyan and his co-workers and Roytburd and his co-workers (and, a Google Scholar search might also throw up plenty of other references). It might also be interesting to see how these continuum models compare (or differ) from the types of models that Dayal discusses in his piece.

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Impressions from TMS Annual Meeting 2008

March 12, 2008

TMS Annual Meeting is a rather large and varied affair; the program and abstract booklet, for example, runs into nearly 350 pages. So, not surprisingly, not only the quality and quantity of talks across different symposia are varied, but even within any given symposia, the talks are more varied in style, depth, philosophy and methodologies.

In any case, I have confined myself exclusively (barring a few exceptions) to the symposia on Computational Thermodynamics and Kinetics. Based on my attendance and understanding, here are my impressions (especially with regard to those areas of computational thermodynamics that interests me)

The attitude towards phase field modelling seems to be ambivalent; people like it because one does not have to track interfaces explicitly, and, in most of the problems of interest to the community — it is the interfaces and surfaces that are of interest and need to be tracked. However, phase field models are not satisfactory in cases where the defects that are of interest are at the atomic level, and/or, when such defects interact with extended defects such as grain boundaries and antiphase boundaries — microstructural evolution in irradiated systems, where a large number of point defects interact with extended defects and creep in Ni-base superalloys where dislocations interact with extended defects are a couple of examples that comes to my mind immediately.

I saw that the attempts to remedy the above-mentioned shortcoming of phase field models are generally of two types; one is the mixed method, in which, one decides to use both discrete and continuum descriptions either in parallel or in series. The other approach is to try and reformulate the phase field model based on the inputs from a lower level model — the most successful approach along these lines comes from the phase field crystal community; the general understanding in this community seems to be to consider phase field model as some approximation obtained from phase field crystal (which, in itself is considered as an approximation to the classical density function theory of freezing).

The two areas where a huge number of efforts have gone in and are still going on are in (a) extending phase field models (like, for example, the attempts to get phase field models for electrochemical problems, piezeoelectric problems, and hydrodynamic problems) and (b) extending phase field crystal models to study plasticity, coupled electric and magnetic material problems and so on. The works in these areas are also at some level look very glamorous to me since getting a formulation and getting it working to solve some important, model problem is the surest way to attain some immortality, and even a citation classic.

I have also found that most of the phase field models that are presented are not on model systems, and in those rare cases where such a result was presented the author(s) were apologetic; on the other hand, there have been several attempts to either use values from some database, or to derive the input parameters using a lower level model such as molecular dynamics or phase field crystal.

The numerical methods used for solving the phase field methods are also becoming more complex and involved — parallel computations, moving meshes and adaptive meshes seem to be the norm — in fact, I was told by Jim Warren that one has to bite that bullet at some point — the earlier the better.

The symposium was not without its own fireworks, either. When Martin Glicksman tried to argue that stochastic noise is not needed for dendritic side branching but just the anisotropy in interfacial energy is sufficient, Alain Karma told him “In your next presentation, I dare you to present a system with 2% anisotropy evolved for a long time, and show us a plot of velocity versus time”.

Finally, for those of you who are so inclined, some of the open problems that got mentioned explicitly are (and, frankly, I think some of them are very involved, if not impossible, and might also need access and facility with fairly advanced mathematical tools — mostly, linear and non-linear analyses):

  • An approximate model for quantum density function theory along the lines of PFC which is an approximate model for classical density function theory — such a model can incorporate information about chemical bonds I understand — I can also see why such a model would incorporate information about magnetic, optic and other such properties;
  • Derivation of phase field models using entropy based formulations that include hydrodynamics; and,
  • Imposing some kind of constraints on the density fields in the usual PFC.

The case of the curious reference

October 11, 2007

Today, I came across a reference to what is called Landau-Lifshitz-Gilbert equation; a little bit of googling got me the reference to Gilbert’s paper:

T.L. Gilbert, A Lagrangian formulation of the gyromagnetic equation of the magnetic field, Phys. Rev. 100 (1955) 1243.

However, when I tried Phys. Rev. site, I got the error message:

Phys. Rev. 100 1243
No data available for this citation

A valid journal, volume, and page or article id are required. Please try again.

So, I thought there probably is something wrong with the page numbers; I thought a Google Scholar search might help me get the correct volume and/or page number. That didn’t help me either:

[CITATION] A Lagrangian formulation of gyromagnetic equation of the magnetization field
TL Gilbert – Phys. Rev, 1955
Cited by 272Related ArticlesWeb Search

Now, that is strange; there is only citation (and, a whopping 272 at that), but, no link to the article itself. Surely, there is no way so many of them got it wrong.

I went back to PROLA and browsed through the volume 100 of Phys. Rev. to find that pp. 1236-1272 are missing; however, the stuff that is published on p.1235 of the issue indicates that the missing pages should contain some abstracts, and probably Gilbert’s paper is one among them.

This suspicion was confirmed when I found this page:

T.L. Gilbert. Phys. Rev., 100:1243, 1955. [Abstract only; full report, Armor Research Foundation Project No. A059, Supplementary Report, May 1, 1956] (unpublished).

Wow! Finally, a couple of researchers who not only took the pains to locate the article (abstract, in this case) but also to note it for those who might be trying to hunt it down.

Of course, I do not understand why Phys. Rev does not host pdf pages of these abstracts in their archive. In any case, a visit to the library helped me get the abstract, which reads as follows:

D6. A Lagrangian formulation of the gyromagnetic equation of the magnetization field. T. L. Gilbert, Armour Research Foundation of Illinois Institute of Technology.–The gyromagnetic equation, d{\mathbf {M}}/dt = \gamma {\mathbf {M}} \times {\mathcal{G}}, for the motion of the magnetization field {\mathbf {M(r)}}, in a ferromagnetic material can be derived from a variation principle, as first shown by Doering.1 Here {\mathcal{G}} is the effective internal field, including the magnetic field and contributions from exchange, anisotropy, and magnetoelastic effects. Using the variational principle, the equations of motion can be recast into a Lagrangian form. This makes possible a consistent derivation of the equations of motion of the magnetization field and other fields to which it may be coupled (e.g., the displacement field of the lattice and the electromagnetic field). It also permits the introduction of viscous damping effects in a consistent manner using the Rayleigh dissipation function. It is shown that viscous damping of the magnetization fields leads to an equation of motion which reduces to the Landau-Lifshitz equation only when the damping is small. It is also shown that this Lagrangian formalism permits the introduction f damping due to disaccomodation in a consistent and very general way.

1 W. Doering, Z. Naturforsch. 3a, 374 (1948)

So, there are a couple of morals to this story: sometimes, if it is good enough, a paragraph like above can get you hundreds of citations; and, if you find some pages of Phys Rev are missing (before 1955; a note in 1955, Vol. 100, issue 4 notes that they will not be published thenceforth), it probably is an abstract of some meeting, and you can only get it in the hard copy format.

Well, the successful resolution of the mystery calls for a cup of coffee, don’t you think! See you around.

PS: For those of you who are interested in using LLG equation to numerically solve domain evolution in giant magnetostrictive materials (of course, using phase field methods — you knew it was coming, didn’t you?), here is a paper:

Title: Phase-field microelasticity theory and micromagnetic simulations of domain structures in giant magnetostrictive materials

Authors: J.X. Zhang and L.Q. Chen

Abstract
:

A computational model is proposed to predict the stability of magnetic domain structures and their temporal evolution in giant magnetostrictive materials by combining a micromagnetic model with the phase-field microelasticity theory of Khachaturyan. The model includes all the important energetic contributions, including the magnetocrystalline anisotropy energy, exchange energy, magnetostatic energy, external field energy, and elastic energy. While the elastic energy of an arbitrary magnetic domain structure is obtained analytically in Fourier space, the Landau–Liftshitz–Gilbert equation is solved using the efficient Gauss–Seidel projection method. Both Fe81.3Ga18.7 and Terfenol-D are considered as examples. The effects of elastic energy and magnetostatic energy on domain structures are studied. The magnetostriction and associated domain structure evolution under an applied field are modeled under different pre-stress conditions. It is shown that a compressive pre-stress can efficiently increase the overall magnetostrictive effect. The results are compared with existing experiment measurements and observations.

Have fun!