## Movement of Sigma boundaries under stress

### September 6, 2011

Migration of Σ7 tilt grain boundary in Al under an applied external stress

Molodov et al

Stress driven migration of symmetrical tilt grain boundaries with misorientations close to 38.2°, 73.4° and 135.6° rotations related to the same special Σ7{132} CSL boundary was measured by in-situ observations in a scanning electron microscope. Contrary to expectations and theoretical predictions, the investigated boundaries moved under an applied stress, but their motion did not produce shear. The three crystallographically equivalent Σ7 boundaries were found to behave different with respect to migration rate and its temperature dependence.

## Grain growth and neutron Larmor diffraction

### March 17, 2010

[1] On grain growth in the presence of mobile particles

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.

[2] Neutron Larmor diffraction measurements for materials science

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 recent papers from scripta:

[1] Kinetics and size effect of grain rotations in nanocrystals with rounded triple junctions

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.

[2] Direct non-destructive observation of bulk nucleation in 30% deformed aluminum

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.

[3] Dynamic abnormal grain growth: A new method to produce single crystals

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.

[4] Evaluation of the liquid-solid interfacial energy from crystallization kinetic data

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.

[5] Experimental study of the miscibility gap and calculation of the spinodal curves of the Au–Pt system

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.

[6] Estimation of dislocation density in bainitic microstructures using high-resolution dilatometry

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.

[7] Magnetic phase transition and magneto-optical properties in epitaxial FeRh0.95Pt0.05 (0 0 1) single-crystal thin film

W Lu et al

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

Abnormal grain growth in Al–3.5Cu

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.

May be this can be a case study for the course?

M J Santofimia et al

The quenching and partitioning (Q&P) process is a new heat treatment for the creation of advanced high-strength steels. This treatment consists of an initial partial or full austenitization, followed by a quench to form a controlled amount of martensite and an annealing step to partition carbon atoms from the martensite to the austenite. In this work, the microstructural evolution during annealing of martensite–austenite grain assemblies has been analyzed by means of a modeling approach that considers the influence of martensite–austenite interface migration on the kinetics of carbon partitioning. Carbide precipitation is precluded in the model, and three different assumptions about interface mobility are considered, ranging from a completely immobile interface to the relatively high mobility of an incoherent ferrite–austenite interface. Simulations indicate that different interface mobilities lead to profound differences in the evolution of microstructure that is predicted during annealing.

## 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:

- 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.

- 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.

- 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).

- 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.

- 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.

- 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.

- 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.

- 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 and , where both of the related energies include a term of power one which is nondifferentiable at . The first equation is spatially homogeneous, while the second one is spatially inhomogeneous when depends on . 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.

- 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.

- 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.

- 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, ) 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, , in the 2D case — Ref. 1-7, or, say, , in the 3D case, where 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 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 , the bulk free energy of the system can only depend on the crystallanity parameter , and the gradients in and or , 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 in the free energy, in addition to the usual terms.

The introduction of a term of the type leads to an evolution equation which contains a term of the type ; this leads to a singular diffusivity in the bulk of the grains since in the bulk 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 in the evolution equations. I believe they are extraneous, and should be dropped.

(2) The details of the addition or subtraction of 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!

## Some notes on GB migration — I

### March 27, 2007

The following are the notes of Chapter 3 of *Grain boundary migration in metals: Thermodynamics, kinetics, applications* by Guenter Gottstein and Lasar S Shvindlerman, CRC Press, New York 1999.

**Grain boundary displacement vs. non-zero diffusive flux across a boundary **

Grain boundaries are structures that separate regions of the same phase and crystal structure which differ in their orientation. Thus, a displacement of grain boundary corresponds to the growth of one at the expense of another grain. This displacement also distinguishes the grain boundary migration from the shrinkage/growth of grains due to a diffusive motion.

Consider a case in which a non-zero flux across the grain boundary leads to a growth of a grain at the cost of the other. In such a case, with respect to an external frame of reference, the grain boundary remains stationary, while the opposite faces of the grains move. On the other hand, during the grain boundary displacement, when atoms from one grain make a jump to another, they change their orientation to that of the grain to which they are jumping to. Thus, there is a non-zero net exchange of lattice sites across the boundary, which results in the grain boundary displacement.

**Do grain boundaries have specific properties?**

Grain boundaries are usually defined to be layers of defined thickness separating phases of different orientations and the grain boundary phase is assumed to have an associated energy, entropy or mobility. However, in reality grain boundaries are only regions of discontinuities in crystal orientation; thus, for example, depending on the constraints of the adjacent crystalline surfaces, the grain boundary mobility can be different for motions in opposite directions.

**Is there a theory of grain boundary migration? **

No; all theories are attempts to describe grain boundary motion in terms of the rate of atoms crossing the grain boundary with net energy gain.

Consider a very narrow–single layer wide– grain boundary; thus, each atomic jump displaces the boundary by the diameter of an atom . The grain boundary velocity in such a system is given by where are the jump frequencies in the respective directions.

If there is no difference in the Gibbs free energy of the two crystallites, , and hence there is no boundary migration. This leads us to the question as to what leads to such differences in Gibbs free energy; in other words, what is the driving force for GB migration? That question has to wait for the next post.