Four Questions About Triple Lines

W Craig Carter et al

The identification of triple lines as one of a hierarchy of defects is presented. There are several distinct cases of triple junctions, and these are sorted into classes. Viewpoints about open questions and directions for future research are offered, including 1) the effect of induced order on the structure and energy of a triple line; 2) considerations of interfacial (complexion) transitions; 3) suggestions of methods for the direct measurement of triple-line energy; 4) observations of the possibility of triple-line anisotropy.

Incorporating Diffuse-interface Nuclei in Phase-field Simulations

T W Heo et al

We propose a computational framework for incorporating diffuse-interface critical nuclei in phase-field simulations. Using a structural transformation as an example, we first generate a table of diffuse-interface critical nuclei. We then incorporate them in phase-field simulations through the explicit nucleation algorithm. The temporal growth kinetics of the introduced nuclei is obtained by numerically solving the Allen-Cahn equation. The results are analyzed by comparing to the phase transformation kinetics using the classical nucleation and normal growth theory and the Kolmogorov-Johnson-Mehl-Avrami equation.

[1] The effects of grain grooves on grain boundary migration in nanofilms

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.

[2] Misfit strain–film thickness phase diagrams and related electromechanical properties of epitaxial ultra-thin lead zirconate titanate films

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.

[3] A more accurate two-dimensional grain growth algorithm

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.

[4] Point defects in multicomponent ordered alloys: Methodological issues and working equations

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.

[5] Misorientation texture development during grain growth. Part II: Theory

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.

[6] Spatial correlations in symmetric and asymmetric bicontinuous structures

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.

Title: Solving phase field equations using a meshless method

Authors: J X Zhou and M E Li

Source: Communications in numerical methods in engineering, Vol. 22, Issue 11, pp. 1109-1115.


The phase field equation is solved by using a meshless reproducing kernel particle method (RKPM) for the very first time. The 1D phase field equation is solved using different grid sizes and various time steps at a given grid size. The method can give accurate solutions across the interface, and allows a larger time step than explicit finite-difference method. The 2D phase field equation is computed by the present method and a classic shrinking of a circle is simulated. This shows the powerfulness and the potential of the method to treat more complicated problems.

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!

Title: Spectral implementation of an adaptive moving mesh method for phase-field equations

Authors: W M Feng, P Yu, S Y Hu, Z K Liu, Q Du and L-Q Chen

Source: Journal of Computational Physics, Vo. 220, Issue 1, 20 December 2006, pp. 498-510.


Phase-field simulations have been extensively applied to modeling microstructure evolution during various materials processes. However, large-scale simulations of three-dimensional (3D) microstructures are still computationally expensive. Among recent efforts to develop advanced numerical algorithms, the semi-implicit Fourier spectral method is found to be particularly efficient for systems involving long-range interactions as it is able to utilize the fast Fourier transforms (FFT) on uniform grids. In this paper, we report our recent progress in making grid points spatially adaptive in the physical domain via a moving mesh strategy, while maintaining a uniform grid in the computational domain for the spectral implementation. This approach not only provides more accurate treatment at the interfaces requiring higher resolution, but also retains the numerical efficiency of the semi-implicit Fourier spectral method. Numerical examples using the new adaptive moving mesh semi-implicit Fourier spectral method are presented for both two and three space dimensional microstructure simulations, and they are compared with those obtained by other methods. By maintaining a similar accuracy, the proposed method is shown to be far more efficient than the existing methods for microstructures with small ratios of interfacial widths to the domain size.

Phase field models are used, among other things, for a study of grain growth [1,2]. In one type of phase field models used to study grain growth, each grain orientation is represented by an order parameter \eta_{i}, where the index i runs from 1 to N, where N is the total number of orientations present in the system [3,4]. Though this method is simple to implement and has been used extensively, it is computationally intensive. So, recently there had been at least two attempts to come up with an efficient numerical implementation:

  1. Efficient numerical algorithm for multiphase field simulations. Srikanth Vedantam and B S V Patnaik, Phys Rev E 73, 016703, 2006.

    Phase-field models have emerged as a successful class of models in a wide variety of applications in computational materials science. Multiphase field theories, as a subclass of phase-field theories, have been especially useful for studying nucleation and growth in polycrystalline materials. In theory, an infinite number of phase-field variables are required to represent grain orientations in a rotationally invariant free energy. However, limitations on available computational time and memory have restricted the number of phase-field variables used in the simulations. We present an approach by which the time and memory requirements are drastically reduced relative to standard algorithms. The proposed algorithm allows us the use of an unlimited number of phase-field variables to perform simulations without the associated burden on computational time or memory. We present the algorithm in the context of coalescence free grain growth.

  2. Sparse data structure and algorithm for the phase field method. J Gruber, N Ma, Y Wang, A D Rollett, and G S Rohrer, Modelling and Simulation in Materials Science and Engineering, 14, 1189, 2006.

    The concepts of sparse data structures and related algorithms for phase field simulations are discussed. Simulations of polycrystalline grain growth with a conventional phase field method and with sparse data structures are compared. It is shown that memory usage and simulation time scale with the number of nodes but are independent of the number of order parameters when a sparse data structure is used.

The source code for a C++ implementation of the method described in the paper of Gruber et al is available for download here (for non-profit scientific research purposes).

The idea behind these implementations is rather simple. Consider an arbitrary mesh point in a simulation cell. The mesh point either lies in the bulk of a given grain, or it lies in the interface. If it lies in the bulk, all the order parameters except the one corresponding to the grain orientation are zero, and there is nothing much to be done about the calculation at that point. On the other hand, if it lies in the interface, the total number of order parameters which have non-zero values are still a very small number as compared to the total number of orientations present in the entire system. Hence, if there is a database such that for any point (and its neighbours) we have the information of non-zero order parameters, then the calculation can be made more efficient.

Take a look and have fun!


[1] Phase field methods for microstructure evolution. Long-Qing Chen, Annu. Rev. Mater. Res., 32, 113, 2002.

[2] Modeling grain boundaries using a phase-field technique. J A Warren, R Kobayashi, and W Craig Carter, J Cryst Growth, 211, 1, 18, 2000.

[3] A novel computer simulation technique for modeling grain growth. Long-Qing Chen, Scripta Metallurgica et Materialia, 32, 1, 115, 1995.

[4] A phase field concept for multiphase systems. I Steinbach, F Pezzolla, B Nestler, M Seeszelberg, R Prieler, G J Schmitz, and J L L Rezende, Physica D, 94, 3, 135, 1996.

Here is a follow-up paper on the unconditionally stable time step for diffuse interface methods.

Paper: Maximally fast coarsening algorithms

Authors: Mowei Cheng and Andrew D. Rutenberg


We present maximally fast numerical algorithms for conserved coarsening systems thatare stable and accurate with a growing natural time step \Delta t = A t_{s}^{2/3}. We compare the scaling structure obtained from our maximally fast conserved systems directly against the standard fixed time-step Euler algorithm, and find that the error scales as \sqrt{A}—so arbitrary accuracy can be achieved. For non-conserved systems, only effectively finite time steps are accessible for similar unconditionally stable algorithms.

Paper: Fast and accurate coarsening simulation with an unconditionally stable time step

Authors: Benjamin P. Vollmayr-Lee and Andrew D. Rutenberg


We present Cahn-Hilliard and Allen-Cahn numerical integration algorithms that are unconditionally stable and so provide significantly faster accuracy-controlled simulation. Our stability analysis is based on Eyre’s theorem and unconditional von Neumann stability analysis, both of which we present. Numerical tests confirm the accuracy of the von Neumann approach, which is straightforward and should be widely applicable in phase-field modeling. For the Cahn-Hilliard case, we show that accuracy can be controlled with an unbounded time step \Delta t that grows with time t as \Delta t \sim t^{\alpha}. We develop a classification scheme for the step exponent \alpha and demonstrate that a class of simple linear algorithms gives \alpha = 1/3. For this class the speedup relative to a fixed time step grows with N, the linear size of the system, as N/\ln N. With conservative choices for the parameters controlling accuracy and finite-size effects we find that an 8192^{2} lattice can be integrated 300 times faster than with the Euler method.

The Eyre’s theorem referred to in the abstract is described in this report (ps file).