Journal Club for 15 June 2008: Kinetics of Structural Phase Transformations

A structural phase transformation is the transformation of a crystal structure due to the displacement of atoms, with no diffusion and no changes in the relative positions. These transformations are often induced by changes in temperature; however stress, electromagnetic fields and other loads can also induce transformations. The schematic below shows a cubic to tetragonal transformation. The crystal is cubic at high temperature and tetragonal at low temperature. Due to the symmetry of the cube stretching along any of its axes, there are 3 possible variants or twins at low temperature.

The phenomenon of twinning is closely related to structural phase transformations. A twin plane is an interface with different twins on either side of the interface. Twinning can compete with plasticity in some settings. The schematic below shows twinning of 2 tetragonal twins

By considering the different twins as a distinct phases, the concepts and methods developed for phase transformations can be applied with minor modifications to twin plane motion. Hence, in the continuum mechanics literature, the term phase transformation often includes twinning.

A common example of a phase transformation is in iron: it has a body-centered cubic (BCC) structure at room temperature, called ferrite, that transforms to face-centered cubic (FCC) austenite structure above 1200K. The iron-carbon phase diagram is full of interesting examples. Structural transformations and twin motion also drive the interesting behavior of shape-memory alloys (coupling deformation and temperature) and ferroelectrics (coupling deformation and electric fields - the recent summary by Chad Landis provides more information).

Send a blank email to temporary.for.imechanica@gmail.com if you need a PDF of any of these papers.

1. Equilibrium of bars, by J. L. Ericksen. Journal of Elasticity, 5(3-4), 1975.

This paper by Ericksen first applied continuum elasticity to the modeling of phase transformations. In the process, it also exposed the inadequacy of continuum theory in describing such phenomena.

Prior to this work, the Landau-Ginzburg framework had been applied to phase transformations of various types (such as superconductivity). However, these did not deal with structural phase transformations that carefully considered deformations. Further, this framework typically had been applied to uniformly transforming specimens, not considering the phase transformation process as being at different stages in different regions, ie, a field problem. In this paper, Ericksen introduces the idea of modeling structural phase transformations using continuum mechanics, and, in the process, showed that classical continuum mechanics is inadequate to describe such phenomena.

The outline of this paper is as follows. In a one-dimensional bar, a strain-energy density with 2 minima is studied. Each minimum corresponds to a different crystal structure. For certain values of the stress, both phases can coexist in the bar with interfaces (phase boundaries) separating them. However, the positions and numbers of these interfaces are not predictable as an outcome of the balance laws of continuum mechanics. This is a surprising result; typical materials modeled in classical elasticity have convex strain energy and have unique solutions for the simple BCs that Ericksen considers.

In equilibrium, this non-uniqueness can be resolved by going beyond the balance laws and further requiring that the system be at an absolute energy minimum (similar to the Maxwell construction in Van der Walls fluids). However, when elastodynamics is used to model the kinetics of phase transformations, the situation is much more complex.

Work by Abeyaratne and Knowles and others showed that further material information is required, beyond that contained in the strain energy. That is, given a continuum elastodynamics problem with non-convex strain energy, the solutions may contain interfaces, and the velocity and number of these interfaces is not unique; one possible way of specifying this material information to precisely achieve uniqueness is to add the kinetic relation for the phase boundaries (how fast interfaces move) and the nucleation criterion (when new phase boundaries nucleate). This viewpoint is the subject of the recent book by Abeyaratne and Knowles, and the key issues are discussed in Sections 4 to 6 here.

2. Kinetics of martensitic phase transitions: Lattice model, by L. Truskinovsky and A. Vainchtein. SIAM Journal on Applied Mathematics, 66(2): 533-553, 2005.

A major focus in recent years is to extract the kinetic relation for the phase boundary from atomic models, ie, to do for the kinetic relation what the Cauchy-Born rule does for strain energy density. Truskinovsky and Vainchtein study a 1D chain of atoms. The bond potentials are highly simplified, and this permits the use of analytical techniques. By using a traveling-wave form as an ansatz for a phase-boundary, the structure of a phase boundary as well as the kinetic relation (relating the velocity to the driving force) are found. In principle, such a procedure can provide input to the continuum model.
 

3. An atomistic investigation of the kinetics of detwinning, by F. E. Hildebrand and R. Abeyaratne. J. Mech. Phys. Solids (2007).

(preprint attached below) 

The closed-form solutions of Truskinovsky and Vainchtein provide much insight. But their closed form comes at a cost; a key missing component is higher dimensionality. Hildebrand and Abeyaratne use MD techniques to examine the motion of a twin boundary in a 2D collection of atoms. The potentials are of Lennard-Jones type. While these are not as accurate as current potentials, they are simple and yet sufficiently capture a lot of interesting mechanics of the atomic motion. One observation is the motion of ledges or steps across the width of the phase boundary, rather than the phase boundary propagating as a uniform interface.

4. Phase boundary propagation in heterogeneous bodies, by K. Bhattacharya. Proc. Royal Soc. London A, 455:757-766, 1999.

The kinetic relations that are calculated from MD provide a mesoscopic view of phase boundary motion, ie, it allows us to go from atoms to a continuum but still at a small scale. At this small scale, phase boundaries can interact with other defects in the material, for example inclusions. Bhattacharya studies a continuum phase boundary with a simple kinetic relation interacting with a large number of defects, and derives the effective kinetics that would be observed at much larger scales. One interesting feature that comes out of the analysis naturally is the stick-slip nature of the effective phase boundary motion.