A tutorial introduction
Introduction
Following the examples set by Rapaport in The Art of Molecular Dynamics Simulation and Press et al., in their Numerical Recipes in C: The Art of Scientific Computing, this article on phase field modelling is accompanied by source codes for carrying out numerical calculations using Cahn-Hilliard (CH) and Cahn-Allen (CA) equations for microstructural evolution. This, I hope, would help the reader build her own set of programs for doing phase field calculations using FFTW (Fastest Fourier Transform in the West). The source codes are distributed under GPL and please go through the license agreement before using these code segements in your programs. It would be nice if you drop me a line when you use these code segements in your programs. Your comments/suggestions are also most welcome.
Microstructural evolution and the equations of Cahn
Let us consider the description of a microstructure - composition or order parameter as a function of spatial coordinates. The change in composition or order parameter as a function of time describes the microstructural evolution. The composition of a given alloy system has to remain constant while there is no such constraint on the order parameter that describes, say, whether the crystal is ordered or not. The variables like composition are called "conserved" while those like the order parameters are called "non-conserved".
Cahn and his coworkers have introduced the equations for the study of the microstructural evolution in systems which are described by conserved as well as non-conserved variables. The Cahn-Hilliard equation (CH) is used for the study of microstructures with conserved variables while the Cahn-Allen equation (CA) is used for the study of microstructures with non-conserved variables.
Spinodal decomposition
Spinodal decomposition is the classic example of the application of Cahn-Hilliard equation to the study of microstructural evolution. In this context, the microstructure is described by the composition field, and, CH is but a modified diffusion equation.
Spinodal decomposition (in a binary system) is a mechanism by which a homogeneous alloy spontaneously decomposes into two isostructural (or at best very slightly different) phases. Such a "spontaneous" decomposition happens because at the composition range where it happens, any composition fluctuation, however small it be, would result in the lowering of the free energy of the system. As the two phases are formed, interfaces are created between the two phases, and these interfaces cost energy. Thus, the system would choose a wavelength for the decomposed structure which is a compromise between the energy gain due to decomposition and the energy loss due to the interface formation.
Spinodal line (which is the locus of the composition points in the phase diagram which satisfy the condition that the second derivative of the free energy with respect to composition is zero), identifies the temperatures and compositions at which spinodal decomposition can take place. In this context, "spinodal line" can be understood to be the boundary between the unstable and metastable parts of a two-phase region. There is a nice mechanical anlogue to metastability (and a historical note as to the origin of the word spinodal) in the 1967 Institute of Metals lecture - "On spinodal decomposition" by JW Cahn.
A C code based on FFTW which does CH spinodal decomposition calculations is available for download. It is called muSE - mu as in "micro" and SE for structural evolver. A short write-up describing the formulation, and numerical implementation is also available for download in PS and PDF format. This is the first version for the manual and any comments, suggestions, and/or clarifications you may have are most welcome.
Order-disorder transition
Order-disorder transition is the classic example of the application of Cahn-Allen equation to the study of microstructural evolution. In this context, the microstructure is described by an order parameter field, and, CA describes the evolution of this field with time.
Over and above the crystalline order, alloys could be "ordered" - By this we mean that specific type of atoms occupy specific sites of the crystallaine lattice. For example, take NiAl. It crystallises into B2 crystal structure - which is ordered bcc. The Ni atoms occupy the cube corners while Al atoms occupy the cube centre (or vice versa since these two sites are equivalent).
At high enough temperatures, due to the considerations of configuratioal entropy, it is almost always preferable for the system to be disordered (even though in the NiAl case, for example, till about its melting temperature the alloy remains ordered). This transition in the the site occupancy that switches from specific to random is known as the "order-disorder" transition. The order-disorder transition is described in terms of an "order parameter" which takes a value of unity in the ordered state and the value of zero in the disordered state.
A C code based on FFTW which does CA order-disorder calculations is available for download. It is called muSE - mu as in "micro" and SE for structural evolver. I will upload a manual describing the formulation, non-dimensionalisation and other implementation details soon. Till then, the short tutorial that accompanies the download should be sufficient to get the code running.
Some pointers to the relevant literature
Phase field models are used to solve a wide variety of problems. There are several nice reviews that are published in the past few years. The following are some papers of interest:
(1) K. Thornton, J. Agren, and P.W. Voorhees
Modelling the evolution of phase boundaries in solids at the meso- and nano-scales
Acta materialia , VOl. 51, pp. 5675-5710, 2003.
This article appeared in the golden jubilee issue of Acta titled "Selected Topics in Materials Science and Engineering: Past, Present and Future", edited by S. Suresh. The reference list is exhaustive with 307 references. This review discusses the differences between the sharp and diffuse interface models and hints at some future topics of research in microstructural evolution.
(2) Long-Qing Chen
Phase-field models for microstructural evolution
Annual Review of Materials Research, Vol. 32, pp. 113-140, 2002.
Chen has written several review articles on phase field models - with specific reference to solid-solid phase transformations - and is one of the most active players in phase field research. This review (with 191 references) is worth taking a look at. When you are at it, you might also want to read the following review in the same volume:
(3) W.J. Boettinger et al.,
Phase-field simulation of solidification
Annual Review of Materials Research, Vol. 32, pp. 163-194, 2002.
I have not read much of phase field solidification literature myself. But, my colleague Subhradeep Chatterjee (Deep, as we call him) recommends this review as well as the reviews in the April 2004 issue of Journal of Materials (JOM), if you are interested in solidification related microstructures and their modelling using phase field models. (Deep himself is working on phase field modelling of solidification, and some day, his codes and thesis might also be uploaded/linked to in these pages).