## Dislocation-crack interaction and recrystallised grains

### March 31, 2011

Stress relaxation at a crack tip relies on the material’s ability to generate dislocations. Despite the extensive literature devoted to crack–dislocation interaction, no one has yet explained how dislocations appear and multiply in order to build a fully plastic zone. Here we will show how a simple event, such as the intersection of a unique incoming dislocation with a crack front, induces the generation of new dislocations: this effect is called “stimulated emission”. Submitted to the applied crack stress field, these dislocations can repeat the stimulation process step by step all along the crack front, through a cross-slip mechanism. Such a rapidly increasing rate of dislocations nucleation leads to a sudden growth of the plastic zone (avalanche).

S Wang et al

A model is proposed for post-recrystallization grain size. The model is based on the coarsening of subgrain networks as present after deformation and recovery. It is shown that the orientation spread in the subgrain network is the key variable in predicting the density of abnormal subgrains and, hence, the recrystallized grain size. The model explains the strong dependence of the post-recrystallization grain size on prior strain and the lack of a dependence on the annealing temperature.

## Discontinuous grain growth, Dislocation dynamics

### March 5, 2011

S Tsuchiya et al

Abstract

The formation mechanism of as-cast coarse columnar γ grain (CCG) structure in hyperperitectic carbon steels is investigated by means of a rapid unidirectional solidification method. This method achieves cooling conditions similar to those in the vicinity of a practically continuously cast slab surface. The microstructural observation of the quenched samples indicates that the CCG structure develops from the mold side along the direction of the temperature gradient. In the solidifying samples, fine columnar γ grains (FCG) always exist ahead of the CCG region. Instead of continuous growth into large grains, FCG always shrink and vanish as a result of the growth of CCG initially formed near the mold side. Therefore, the grain size at a fixed point in the ingot discontinuously changes from the FCG to the CCG. The validity of this process is supported by numerical analyses. This finding is in marked contrast to the assumption made in conventional grain growth analysis on the CCG structure.
Highlights

► We examine the formation process of as-cast coarse columnar γ grains (CCG) in steels. ► Fine columnar γ grains (FCG) exist ahead of the CCG region during the solidification. ► The FCG do not continuously grow into the CCG and they always shrink. ► We find that the CCG develop by the mechanism of the discontinuous grain growth.

L Capolungo et al

Adaptative meshing finite-element-based discrete dislocation dynamics simulations are employed to predict dislocation junction formation in magnesium as well as their resulting strength. Apart from coplanar and collinear interactions, all possible interactions between basal, prismatic and pyramidal slip are considered. Among others it is found that while non-coplanar prismatic junctions are more likely than basal–prismatic junctions, the latter are more stable. However, pyramidal–prismatic junctions appear more stable than pyramidal–basal junctions. Finally, non-coplanar pyramidal junctions are more likely than any other junction formation, and these junctions also appear to be amongst the strongest.

## Cemented carbide: grain growth

### January 8, 2011

Carbide grain growth in cemented carbides

K Mannesson et al

Abnormal grain growth is often observed in cemented carbides during sintering, but cannot be understood in terms of the classical LSW theory. In this work the grain growth behavior during sintering at 1430 °C is studied both experimentally and by means of computer simulations. A model based on several processes—2-D nucleation of growth ledges, mass transfer across the interface and long-range diffusion coupled in series—is formulated and the equations are solved numerically. Both computer simulations and experimental studies reveal that the grain growth behavior is strongly influenced by the initial size distribution.

## Stress induced martensitic transformation and shadowing effects on growth of thin films

### June 26, 2010

In situ TEM observation of stress-induced martensitic transformations and twinning processes in CuAlNi single crystals

N Zarubova et al

Stress-induced martensitic transformations and twinning processes were studied in thin foils of CuAlNi single crystals strained in situ in a transmission electron microscope. The nucleation and growth of the martensite plates were monitored for three transformation processes known from bulk experiments: (i) the transformation of austenite into 2H martensite at low-stress levels; (ii) the twinning/detwinning processes in 2H martensite; and (iii) the transformation between austenite and 18R martensite at higher stress levels. The morphology of the austenite/martensite habit planes was examined, and the existence of planar interfaces between a single variant of 2H martensite and austenite on the microscopic level was proven.

The role of self-shadowing on growth and scaling laws of faceted polycrystalline thin films

C Ophus et al

We investigate, via both experiment and simulation, the effects of self-shadowing on the growth of faceted polycrystalline thin films. Faceted aluminum thin films were sputtered and the anomalous scaling behaviour of their surfaces was characterized. To understand the causes of this anomalous behavior, growth of faceted thin films was simulated by coupling a level set construction to a ballistic deposition model. The angular distribution function of deposition flux was varied to control the degree of self-shadowing. We show how differing degrees of self-shadowing strongly modify film surface morphologies and compare these results with experimental findings.

## Enhanced multiferroic properties and shear coupled grain growth

### June 16, 2010

F Yan et al

BiFeO3 (BFO) and La-doped BFO (BLFO) thin films are grown on Pt/TiO2/SiO2/Si substrate using pulsed laser deposition. The domain structure of BFO and BLFO are investigated via piezoresponse force microscopy. Highly enhanced ferroelectric properties with great remanent polarization (Pr) of 102 μC/cm2 and decreased leakage current density are obtained via La-doping. Magnetic property is also increased by La doping ascribed to spatial homogenization of spin arrangement. The mechanisms for the enhancement of ferroelectric and ferromagnetic characteristics are discussed.

T Gorkaya et al

A novel set-up developed to continuously observe and measure stress-driven grain boundary migration is presented. A commercial tensile/compression scanning electron microscope hot stage was utilized for in situ observations of mechanically loaded samples at elevated temperatures up to 850 °C by recording orientation contrast images of bicrystal surfaces. Two sample holders were designed and fabricated for applying a shear stress to the boundary in bicrystals of various geometries. The results of the first measurements in Al bicrystals are presented.

## Grain growth and neutron Larmor diffraction

### March 17, 2010

V.Yu. Novikov

The ability of second phase particles to migrate along with grain boundaries is shown to be determined not only by the particle mobility but also by the migration rate of the grain boundary where they locate. This leads to a duality in the mobile particle behaviour: they behave as either movable or immovable depending on the boundary migration rate. In the first case, they reduce the boundary mobility; in the second one they decrease the driving force for boundary migration. It is demonstrated by numerical modeling that mobile particles with low mobility can suppress grain growth even in nanocrystalline material, the limiting grains size being several times smaller than in the case of randomly distributed immobile particles. It is also shown that the Zener solution to the problem of the grain growth retardation by disperse particles is a specific case of the proposed approach.

J. Repper et al

Neutron Larmor diffraction (LD) is a high-resolution diffraction technique based on the Larmor precession of polarized neutrons. In contrast to conventional diffraction, LD does not depend on the accurate measurement of Bragg angles, and thus the resolution is independent of the beam collimation and monochromaticity. At present, a relative resolution for the determination of the crystal lattice spacing d of Δd/dnot, vert, similar10-6 is achieved, i.e. at least one order of magnitude superior to conventional neutron or X-ray techniques. This work is a first step to explore the application of LD to high-resolution problems in the analysis of residual stresses, where both the accurate measurement of absolute d values and the possibility of measuring type II and III stresses may provide additional information beyond those accessible by conventional diffraction techniques. Data obtained from Inconel 718 samples are presented.

## Some interesting papers in recent issues of Acta

### November 6, 2009

A Novick-Cohen et al

Using numerical computations and asymptotic analysis, we study the effects of grain grooves on grain boundary migration in nanofilms, focusing for simplicity on axisymmetric bicrystals containing an embedded cylindrical grain located at the origin. We find there is a critical initial grain radius, R*, such that if RR*, groove growth during grain shrinkage leads to film break-up. The central cross-section of the grain boundary profile is seen to be parabolic, and an ordinary differential equation which depends on the tilt angle and the groove depth is seen to govern the location of the groove root. Near the annihilation–pinch-off transition, temporary stagnation occurs; thereafter, the shrinking grain accelerates rapidly, then disappears.

Q Y Qiu et al

The phase stability of ultra-thin (0 0 1) oriented ferroelectric PbZr1–xTixO3 (PZT) epitaxial thin films as a function of the film composition, film thickness, and the misfit strain is analyzed using a non-linear Landau–Ginzburg–Devonshire thermodynamic model taking into account the electrical and mechanical boundary conditions. The theoretical formalism incorporates the role of the depolarization field as well as the possibility of the relaxation of in-plane strains via the formation of microstructural features such as misfit dislocations at the growth temperature and ferroelastic polydomain patterns below the paraelectric–ferroelectric phase transformation temperature. Film thickness–misfit strain phase diagrams are developed for PZT films with four different compositions (x = 1, 0.9, 0.8 and 0.7) as a function of the film thickness. The results show that the so-called rotational r-phase appears in a very narrow range of misfit strain and thickness of the film. Furthermore, the in-plane and out-of-plane dielectric permittivities ε11 and ε33, as well as the out-of-plane piezoelectric coefficients d33 for the PZT thin films, are computed as a function of misfit strain, taking into account substrate-induced clamping. The model reveals that previously predicted ultrahigh piezoelectric coefficients due to misfit-strain-induced phase transitions are practically achievable only in an extremely narrow range of film thickness, composition and misfit strain parameter space. We also show that the dielectric and piezoelectric properties of epitaxial ferroelectric films can be tailored through strain engineering and microstructural optimization.

E A Lazar et al

We describe a method for evolving two-dimensional polycrystalline microstructures via mean curvature flow that satisfies the von Neumann–Mullins relation with an absolute error O(Δt2). This is a significant improvement over a different method currently used that has an absolute error O(Δt). We describe the implementation of this method and show that while both approaches lead to indistinguishable evolution when the spatial discretization is very fine, the differences can be substantial when the discretization is left unrefined. We demonstrate that this new front-tracking approach can be pushed to the limit in which the only mesh nodes are those coincident with triple junctions. This reduces the method to a vertex model that is consistent with the exact kinetic law for grain growth. We briefly discuss an extension of the method to higher spatial dimensions.

R Besson

The aim of this work is to give the independent-point-defect thermodynamics of ordered compounds a sufficiently general flavour, adapted to and working for multicomponent alloys. Generalizing previous approaches, we first show that an appropriate description for a crystal with point defects allows treatment of the practically important pressure and defect volume parameters in the grand canonical framework, the equivalence of which is explicited with the closer to experiments isothermal–isobaric conditions. Since industrial applications often involve multialloyed compounds, we then derive an operational tool for atomic-scale investigations of long-range order alloys with complex crystallographies and multiple additions.

J Gruber et al

A critical event model for the evolution of number- and area-weighted misorientation distribution functions (MDFs) during grain growth is proposed. Predictions from the model are compared to number- and area-weighted MDFs measured in Monte Carlo simulations with anisotropic interfacial properties and several initial orientation distributions, as well as a dense polycrystalline magnesia sample. The steady-state equation of our model appears to be a good fit to all data. The relation between the grain boundary energy and the normalized average boundary area is discussed in the context of triple junction dynamics.

A L Genau and P W Voorhees

Spatial correlations of interfacial curvature are compared for symmetric and asymmetric two-phase mixtures produced following spinodal decomposition as given by a numerical solution to the Cahn–Hilliard equation in three dimensions. By calculating radial distribution functions of the density of interfacial area as a function of the mean interfacial curvature of these bicontinuous microstructures, it is found that long-range diffusive interactions, in combination with the morphology of the system, yield a variety of correlations and anticorrelations over a range of length scales. The asymmetric mixtures show some similarities to the symmetric mixtures, as well as other unique features.

## Grain rotation, growth, nucleation, magnetic thin films, stress, …

### August 4, 2009

Some recent papers from scripta:

F Yang and W Yang

A kinetic model is developed to quantify the rate of grain rotations driven by either grain boundary energy or stress. The critical roles of triple junctions and grain shape are emphasized. The size effects for the rotation rate are analyzed. As the grain size decreases, the model predicts shifts in the dominating driving forces and dissipation mechanisms.

S S West et al

A 30% deformed aluminum sample was mapped non-destructively using Three-Dimensional X-ray Diffraction (3DXRD) before and after annealing to nucleation of recrystallization. Nuclei appeared in the bulk of the sample. Their positions and volumes were determined, and the crystallographic orientations were compared with the orientations of the deformed grains. It was found that nuclei with new orientations can form and their orientations have been related to the dislocation structure in the deformed grains.

J Ciulik and E M Taleff

Dynamic abnormal grain growth (DAGG) is a newly discovered phenomenon which can be used to produce large single crystals from polycrystalline material in the solid state at temperatures above approximately half the melting temperature. The unique aspect of DAGG, compared to previously understood abnormal grain growth phenomena, is the requirement of plastic straining for initiation and propagation of abnormal grain growth. Our findings demonstrate that DAGG can be used to produce large single crystals of molybdenum in the solid state.

J Torrens-Serra et al

The kinetic data obtained from the analysis of experimental measurements of nanocrystallization in Fe65Nb10B25 metallic glass are used to successfully estimate the molten alloy viscosity, Fe23B6 crystallization driving force and solid-liquid interface energy in the framework of the classical theory of nucleation and growth. We use a Vogel-Fulcher-Tamman law for the viscosity and linear temperature dependence for the crystallization driving force and interfacial energy. A negative temperature coefficient for the crystal-melt interfacial energy is obtained. Both the thermal stability and the glass forming ability of this alloy are discussed.

X N Xu et al

The miscibility gap (MG) of the Au–Pt binary system in the temperature range 600–1050 °C has been experimentally determined by the diffusion couple technique. The results show that the determined MG deviates from the currently accepted one, which shifts to the Au-rich side of the Au–Pt system. Based on the present experimental data, the Au–Pt system has been thermodynamically reassessed, with the result that the critical point of the miscibility gap is not, vert, similar1200 °C at 56 at.% Pt, in contrast to the currently accepted 1260 °C at 61 at.% Pt. The chemical and coherent spinodals of the Au–Pt system have been thus calculated.

C Garcio-Mateo et al

It is possible by means of high-resolution dilatometry, together with a model based on isotropic dilatation and atomic volumes, to estimate the dislocation density introduced in the microstructure as a consequence of the isothermal decomposition of austenite into bainitic ferrite. The relatively high dislocation density associated with this microstructure is attributed to the fact that the shape deformation accompanying this displacive transformation is accommodated by plastic relaxation.

This paper reports an investigation of the structure, magnetic phase transition and magneto-optical properties of FeRh0.95Pt0.05 thin film. A first-order magnetic phase transition occurs at a temperature around 180 °C, accompanied by a lattice expansion in the c-axis. The effect of substitution on the phase transition in ordered FeRh-based alloy systems is discussed. The nucleation and growth mechanism of the phase transition is quite similar to that of the crystallization of solids. In addition, the Kerr rotation spectrum was also studied.

## Abnormal grain growth

### July 17, 2009

J Dennis et al

Significant abnormal grain growth has been observed in an Al–3.5 wt.% Cu alloy at temperatures where the volume fraction of small CuAl2 particles was less than about 0.01. The initial fine-grained material had a weak crystallographic texture and there was no indication that any special boundaries were involved in the abnormal growth. Island grains isolated within the abnormal grains also showed no indication of special orientation relationships with their surrounding grains. Measurements indicated that the island grains initially had a size advantage over other matrix grains. The fraction of pinning phase was much lower at abnormal grain boundaries than at boundaries in the fine-grained matrix into which they were growing. A variety of simulations were made, including attempts to model that difference in pinning phase distribution, but none of these were successful in predicting abnormal grain growth.

## Continuum models of grain boundaries

### October 19, 2007

There are two important classes of models for the study of grain growth, namely, multiple order parameter models in which each allowed orientation is assigned an order parameter, and vector valued phase field models in which only a few order parameters are introduced. Here are the links to some papers that discuss vector valued phase field models:

1. Vector-valued phase field model for crystallization and grain boundary formation. R Kobayashi, J A Warren, and W C Carter, Physica D, 119, 415-423 (1998)

We propose a new model for calculation of the crystalliztation and impingement of many particles with differing orientations. Based on earlier phase field models, a vector order parameter is introduced, and thus orientation of crystal/disordered interfaces can be determined relative to a crystalline frame. This model improves upon previous attempts to describe this phenomenon, as it requires far fewer equations of motion, and is energetically invariant under rotations. In this report a one-dimensional simulation of the model will be presented along with preliminary investigations of two-dimensional simulations.

2. A phase field model of the impingement of solidifying particles. J A Warren, W C Carter and R Kobayashi, Physica A, 261, 159-166 (1998)

We propose a model of the impingement of solidifying crystalline particles, the ensuing grain boundary formation, and grain coarsening. This model improves upon previous theoretical descriptions of this phenomenon, in that it has the proper behavior under rotations and is easy to implement numerically. Also, insight into the model is straightforward since the parameters are physically motivated, and anisotropy in both the liquid–solid and grain boundary energies can be introduced in a natural manner. A one dimensional analytic solution is presented.

3. Modeling grain boundaries using a phase-field technique. J A Warren, R Kobayashi, and W C Carter, Journal of Crystal Growth, 211, 18-20 (2000)

We propose a two-dimensional phase-field model of grain boundary dynamics. One-dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are computed. By comparison with microscopic models of dislocation walls, insights into the physical accuracy of this model can be obtained. Indeed, for a particular choice of functional dependencies in the model, the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley (Phys. Rev. 78 (1950) 275).

4. A continuum model of grain boundaries. R Kobayashi, J A Warren, and W C Carter, Physica D, 140, 141-150 (2000)

A two-dimensional frame-invariant phase field model of grain boundaries is developed. One-dimensional analytical solutions for a stable grain boundary in a bicrystal are obtained, and equilibrium energies are computed. With an appropriate choice of functional dependencies, the grain boundary energy takes the same analytic form as the microscopic (dislocation) model of Read and Shockley [W.T. Read, W. Shockley, Phys. Rev. 78 (1950) 275]. In addition, dynamic (one-dimensional) solutions are presented, showing rotation of a small grain between two pinned grains and the shrinkage and rotation of a circular grains embedded in a larger crystal.

5. Phase field model of premelting of grain boundaries. A E Lobkovsky, and J A Warren, Physica D, 164, 202-212 (2002).

We present a phase field model of solidification which includes the effects of the crystalline orientation in the solid phase. This model describes grain boundaries as well as solid–liquid boundaries within a unified framework. With an appropriate choice of coupling of the phase field variable to the gradient of the crystalline orientation variable in the free energy, we find that high-angle boundaries undergo a premelting transition. As the melting temperature is approached from below, low-angle grain boundaries remain narrow. The width of the liquid layer at high-angle grain boundaries diverges logarithmically. In addition, for some choices of model coupling, there may be a discontinuous jump in the width of the fluid layer as function of temperature.

6. Nucleation and bulk crystallization in binary phase field theory. L Granasy, T Boerzsoenyi, and Pusztai, Physical Review Letters, 88, 20, 206105-1–206105-4 (2002)

We present a phase field theory for binary crystal nucleation. In the one-component limit, quantitative agreement is achieved with computer simulations (Lennard-Jones system) and experiments (ice-water system) using model parameters evaluated from the free energy and thickness of the interface. The critical undercoolings predicted for Cu-Ni alloys accord with the measurements, and indicate homogeneous nucleation. The Kolmogorov exponents deduced for dendritic solidification and for “soft impingement” of particles via diffusion fields are consistent with experiment.

7. Extending phase field models of solidification to polycrystalline materials. J A Warren, R Kobayashi, A E Lobkovsky, and W C Carter, Acta Materialia, 6035-6058 (2003)

We present a two-dimensional phase field model of grain boundary statics and dynamics. We begin with a brief description and physical motivation of the crystalline phase field model. The description is followed by characterization and analysis of several microstructural implications: the grain boundary energy as a function of misorientation, the liquid–grain–grain triple junction behavior, the wetting condition for a grain boundary and stabilized widths of intercalating phases at these boundaries, and evolution of a polycrystalline microstructure by solidification and impingement, followed by both grain boundary migration and grain rotation. Simulations that demonstrate these implications are presented, with a description of the numerical methods that were used to obtain them.

8. Equations with singular diffusivity. R Kobayashi and Y Giga, Journal of Statistical Physics, 95, 5/6, 1187-1220 (1999)

Recently models of faceted crystal growth and of grain boundaries were proposed based on the gradient system with nondifferentiable energy. In this article, we study their most basic forms given by the equations $u_t=(u_x/|u_x|)_x$ and $u_ t=(1/a)(a u_x/|u_x|)_x$ , where both of the related energies include a $|u_x|$ term of power one which is nondifferentiable at $u_x=0$. The first equation is spatially homogeneous, while the second one is spatially inhomogeneous when $a$ depends on $x$. These equations naturally express nonlocal interactions through their singular diffusivities (infinitely large diffusion constant), which make the profiles of the solutions completely flat. The mathematical basis for justifying and analyzing these equations is explained, and theoretical and numerical approaches show how the solutions of the equations evolve.

9. Sharp interface limit of a phase field model of crystal grains. A E Lobkovsky and J A Warren, Physical Review R, 63, 051605-1 — 051605-10 (2001)

We analyze a two-dimensional phase field model designed to describe the dynamics of crystalline grains. The phenomenological free energy is a functional of two order parameters. The first one reflects the orientational order, while the second reflects the predominantally local orientation of the crystal. We consider the gradient flow of this free energy. Solutions can be interpreted as ensembles of grains (in which the orientation is constant in space) separated by grain boundaries. We study the dynamics of the boundaries as well as the rotation of the grains. In the limit of an infinitely sharp interface, the normal velocity of the boundary is proportional to both its curvature and its energy. We obtain explicit formulas for the interfacial energy and mobility, and study their behavior in the limit of a small misorientation. We calculate the rate of rotation of a grain in the sharp interface limit, and find that it depends sensitively on the choice of the model.

10. Phase field modeling of polycrystalline freezing. T Pusztai, G Bortel, and L Granasy, Materials Science and Engineering A, 413/414, 412-417 (2005)

The formation of two and three-dimensional polycrystalline structures are addressed within the framework of the phase field theory. While in two dimensions a single orientation angle suffices to describe crystallographic orientation in the laboratory frame, in three dimensions, we use the four symmetric Euler parameters to define crystallographic orientation. Illustrative simulations are performed for various polycrystalline structures including simultaneous growth of randomly oriented dendritic particles, the formation of spherulites and crystal sheaves.

11. Phase field theory of polycrystalline solidification in three dimensions. T Pusztai, G Bortel, and L Granasy, Europhysics Letters, 71 (1), 131-137 (2005)

A phase field theory of polycrystalline solidification is presented that describes the nucleation and growth of anisotropic particles with different crystallographic orientation in three dimensions. As opposed to the two-dimensional case, where a single orientation field suffices, in three dimensions, a minimum number of three fields are needed. The free energy of grain boundaries is assumed to be proportional to the angular difference between the adjacent crystals expressed here in terms of the differences of the four symmetric Euler parameters. The equations of motion for these fields are obtained from variational principles. Illustrative calculations are performed for polycrystalline solidification with dendritic, needle and spherulitic growth morphologies.

To give a short introduction to these papers:

Almost all the papers are related to solidification and the problem of impingement of different nuclei, which results in the grain structure when the solidification is complete. Thus, the problem of grain growth is incidental in all these papers; however, by modifying the bulk free energy density (by making sure that there is only one minimum which corresponds to the solid state), and dropping the thermal evolution equations (isothermal simulations), one can obtain equations that pertain to pure grain growth.

The idea behind papers 1-7, and 10-11 is that one can specify the crystalline orientations completely by giving an order parameter (say, $\phi$) which denoted the bulk of the grain (unity in the grain interior and less than unity at the grain boundaries), and an orientation parameter(s) field (say, $\theta$, in the 2D case — Ref. 1-7, or, say, $q_i$, in the 3D case, where $q_i$ represents an unit quaternion — Ref. 10-11). Ref.10-11 also show that the representation in terms of quaternion order parameters can be reduced to that of a single order parameter $\theta$ in 2D. These order parameters are evolved according to the Allen-Cahn equations meant for non-conserved order parameters.

While representing the orientation in terms of the quaternions or orientational order parameter $\theta$, the bulk free energy of the system can only depend on the crystallanity parameter $\phi$, and the gradients in $\phi$ and $\theta$ or $q_i$, since the different orientations are all energetically favourable, and none is preferred over another.

In the following, for simplicity’s sake, let us consider a 2D model; the extension of the discussion to 3D is straightforward.

To obtain stable grain boundaries of finite width in thes models with orientational order parametes, we also need to introduce $|\nabla \theta|$ in the free energy, in addition to the usual $|\nabla \theta|^{2}$ terms.

The introduction of a term of the type $|\nabla \theta|$ leads to an evolution equation which contains a term of the type $\nabla \theta/|\nabla \theta|$; this leads to a singular diffusivity in the bulk of the grains since in the bulk $|\nabla \theta|$ is zero. While such a singular diffusivity allows for grain rotations in a natural manner in these models, it leads to both numerical and analytical difficulties.

The mathematical basis of dealing with singular diffusivities are dealt with in Ref. 8, while, the asymptotic analysis on these systems is performed in Ref. 9. And, Ref. 7, which is a review contains the details of the nuanced numerical implementations.

Finally, a couple of points that are problematic about these models (as far as my understanding of them goes):

(1) Ref. 1-7 and 9, deal with the problem as if the coordinate frame of reference used in the calculations is circular polar, which leads to extra terms of the type $\phi^{2}$ in the evolution equations. I believe they are extraneous, and should be dropped.

(2) The details of the addition or subtraction of $2 \pi$ terms in the Ref. 7 are again an artifact, I believe. In a true 3D case with quaternions (or a reduction thereof to 2D), such terms should not appear int he evolution equations.

(3) Though these models are capable of incorporating rotations, they might also lead to unphysical rotation events.

Before I end this post: soon, I will do a post on the other type of grain growth models with multiple order parameters, and how they compare with these vector order parameter models. I will also publish C codes of numerical implementation of these models. See you around!