## Lamellar-rod transition, stress effect on nucleation and growth and thermodynamics of domain formation

### March 9, 2011

[1] Dynamic effects in the lamellar–rod eutectic transition

S Liu et al

Critical experiments in the Al–Cu system are carried out to establish the conditions for the stability of rod and lamellar eutectics. It is shown that the instability of a lamella initiates locally through the formation of a sinusoidal perturbation, and the fastest growing wavelength of perturbation, which corresponds to the rod spacing, is related to the local lamella spacing. The instabilities in adjacent lamellae are observed to be out of phase to give rise to a hexagonal arrangement of rods at the transition. The specific relationship found between the unstable lamella spacing and the resulting rod spacing at the transition is then taken into account to develop a general model of the rod–lamellar transition which also includes the relative undercooling and the presence of a spacing distribution. A microstructure map is presented which defines the regimes of rod, lamellar and mixed structures, which is shown to be in good agreement with the experimental results.

W Guo et al

Abstract

The effect of a superimposed stress on the coarsening of interacting Ni4Ti3 particles is studied using the multi-phase field method. It is found that the thickness/diameter ratio of a Ni4Ti3 particle in a (1 1 1)B2 plane increases with an increasing [1 1 1]B2 stress component. The particle shape can change from a disk to a sphere with increasing applied stress. It is also found that diffusional and mechanical interactions between two Ni4Ti3 particles can promote the nucleation of new particles. This provides an explanation for the autocatalytic nature of nucleation reported previously. Compressive stresses along [1 1 1]B2 increase the volume fraction and growth velocity of the Ni4Ti3 particles of the (1 1 1)B2 plane. Misoriented particles disappear during particle growth. The simulation results are discussed in the light of previous experimental results.

Research highlights

► Nucleation and growth of Ni4Ti3 precipitates in NiTi shape memory alloys is studied by multi-phase field simulations. ► A model of for thermodynamically consistent treatment of stoichiometric phases is proposed and applied in the present study. ► External compressive stress is predicted to change the morphology of the precipitates and to favor variants whose axis is parallel to the direction of stress. ► Autocatalytic nucleation of a chain of precipitates is explained by the trade of between solutal and strain related deviation for thermodynamic equilibrium.

[3] Thermodynamics of formation of tetragonal and rhombohedral heterophase polydomains in epitaxial ferroelectric thin films

Y Ouyang et al

Abstract

In this work, the thermodynamics of formation of tetragonal and rhombohedral heterophase polydomains in ferroelectric films is explained by the theory of elastic domains. The energetics of the heterophase polydomain microstructure are analyzed. The three major energy terms determining the crystalline orientation of the interdomain interface, i.e. interdomain elastic energy, interdomain electrostatic energy and domain interface energy, are investigated and compared. The crystalline orientation of the elastically best fitting plane between the two phases is analytically solved under an isotropic approximation of elasticity. It is found that a {1 1 2} type of domain interface minimizes interdomain elastic energies. Using available material parameters, it is found that the {1 1 2} domain interface prevails in Pb(Zr, Ti)O3, Pb(Mg1/3 Nb2/3)O3–PbTiO3 and BiFeO3 heterophase polydomains under zero applied electric field, as elastic energy is the dominant factor of interdomain interactions in all three systems. On the other hand, an increasing interdomain electrostatic energy under a poling field may induce a different domain interface, which is beneficial to extrinsic electromechanical responses.

Research highlights

► Tetragonal and rhombohedral phases coexist in ferroelectric films as elastic domains. ► Elastic energy is the dominant factor in determining the as-grown microstructure. ► A left angle bracket1 1 2right-pointing angle bracket domain interface prevails in the as-grown heterophase polydomain film. ► An increasing electrostatic energy under field may induce a different microstructure. ► Evolution of the microstructure under field enhances electromechanical responses.

## Triple lines and nucleation

### February 5, 2010

**Title**: Kinetics of phase transformations in the peridynamic formulation of continuum mechanics

**Auhors**: Kaushik Dayal and Kaushik Bhattacharya

**Source**: Journal of Mechanics and Physics of Solids, 54, 9, September 2006, pp. 1811-1842.

**Abstract**:

We study the kinetics of phase transformations in solids using the peridynamic formulation of continuum mechanics. The peridynamic theory is a nonlocal formulation that does not involve spatial derivatives, and is a powerful tool to study defects such as cracks and interfaces.

We apply the peridynamic formulation to the motion of phase boundaries in one dimension. We show that unlike the classical continuum theory, the peridynamic formulation does not require any extraneous constitutive laws such as the kinetic relation (the relation between the velocity of the interface and the thermodynamic driving force acting across it) or the nucleation criterion (the criterion that determines whether a new phase arises from a single phase). Instead this information is obtained from inside the theory simply by specifying the inter-particle interaction. We derive a nucleation criterion by examining nucleation as a dynamic instability. We find the induced kinetic relation by analyzing the solutions of impact and release problems, and also directly by viewing phase boundaries as traveling waves.

We also study the interaction of a phase boundary with an elastic non-transforming inclusion in two dimensions. We find that phase boundaries remain essentially planar with little bowing. Further, we find a new mechanism whereby acoustic waves ahead of the phase boundary nucleate new phase boundaries at the edges of the inclusion while the original phase boundary slows down or stops. Transformation proceeds as the freshly nucleated phase boundaries propagate leaving behind some untransformed martensite around the inclusion.

**Notes**: Via iMechanica.

What is peridynamics? Here is an answer (that I got by googling):

Peridynamics is a theory of continuum mechanics that is formulated in terms of integral equations rather than partial differential equations. It assumes that particles in a continuum interact across a finite distance as in molecular dynamics. The integral equations remain valid regardless of any fractures or other discontinuities that may emerge due to loading. In contrast, the differential equations of classical theory break down when a discontinuity appears. Peridynamics predicts the deformation and failure of structures under dynamic loading, especially failure due to fracture. Cracks emerge spontaneously as a result of the equations of motion and material model and grow in whatever direction is energetically favorable. The implementation does not require a separate law that tells cracks when and where to grow.

I know that in the pre-spinodal decomposition days, there was some talk about how spinodal region is actually a region of instability, since the barrier to nucleation goes to zero as one approaches the spinodal points resulting in infinite nuclei. However, it is not clear to me if something similar is meant by “nucleation as dynamic instability” in this paper. On the other hand, the phase boundary as a travelling wave sounds similar to what one finds in phase field models. A paper that I have to read more carefully and understand!