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!

It is well known that fluctuations play a crucial role in fluid mixing in turbulent flows, and that continuum models have difficulty in capturing the fluid mixing in such flows. In the latest issue of PNAS, Kadau et al report on their atomistic simulation results of fluid mixing in turbulent systems. In addition, they also use magnetic levitation Rayleigh-Taylor instability experimental results to show that their atomistic results are in better qualitative and quantitative agreement with experiments. Here is the abstract of their paper:

A ubiquitous example of fluid mixing is the Rayleigh–Taylor instability, in which a heavy fluid initially sits atop a light fluid in a gravitational field. The subsequent development of the unstable interface between the two fluids is marked by several stages. At first, each interface mode grows exponentially with time before transitioning to a nonlinear regime characterized by more complex hydrodynamic mixing. Unfortunately, traditional continuum modeling of this process has generally been in poor agreement with experiment. Here, we indicate that the natural, random fluctuations of the flow field present in any fluid, which are neglected in continuum models, can lead to qualitatively and quantitatively better agreement with experiment. We performed billion-particle atomistic simulations and magnetic levitation experiments with unprecedented control of initial interface conditions. A comparison between our simulations and experiments reveals good agreement in terms of the growth rate of the mixing front as well as the new observation of droplet breakup at later times. These results improve our understanding of many fluid processes, including interface phenomena that occur, for example, in supernovae, the detachment of droplets from a faucet, and ink jet printing. Such instabilities are also relevant to the possible energy source of inertial confinement fusion, in which a millimeter-sized capsule is imploded to initiate nuclear fusion reactions between deuterium and tritium. Our results suggest that the applicability of continuum models would be greatly enhanced by explicitly including the effects of random fluctuations.

Take a look!

Here are a few papers that discuss the renormalization group asymptotic analysis of PDEs in general, and phase field methods in particular:

  1. Renormalization group and singular perturbations: Multiple scales, boundary layers, and reductive perturbation theory. L-Y Chen, Nigel Goldenfeld, and Y Oono. Phys Rev E, 54, 1, 376, July 1996.

    Perturbative renormalization group theory is developed as a unified tool for global asymptotic analysis. With numerous examples, we illustrate its application to ordinary differential equation problems involving multiple scales, boundary layers with technically difficult asymptotic matching, and WKB analysis. In contrast to conventional methods, the renormalization group approach requires neither ad hoc assumptions about the structure of perturbation series nor the use of asymptotic matching. Our renormalization group approach provides approximate solutions which are practically superior to those obtained conventionally, although the latter can be reproduced, if desired, by appropriate expansion of the renormalization group approximant. We show that the renormalization group equation may be interpreted as an amplitude equation, and from this point of view develop reductive perturbation theory for partial differential equations describing spatially extended systems near bifurcation points, deriving both amplitude equations and the center manifold.

  2. Renormalization group approach to multiscale simulation of polycrystalline materials using the phase field crystal model. Nigel Goldenfeld, Badrinarayan P Athreya, and Jonathan A Dantzig. Phys Rev E, 72, 020601(R), 2005.

    We propose a computationally efficient approach to multiscale simulation of polycrystalline materials, based on the phase field crystal model. The order parameter describing the density profile at the nanoscale is reconstructed from its slowly varying amplitude and phase, which satisfy rotationally covariant equations derivable from the renormalization group. We validate the approach using the example of two-dimensional grain nucleation and growth.

  3. Renormalization-group theory for the phase-field crystal equation. Badrinarayan P Athreya, Nigel Goldenfeld, and Jonathan A Dantzig. Phys Rev E, 74, 011601, 2006.

    We derive a set of rotationally covariant amplitude equations for use in multiscale simulation of the two-dimensional phase-field crystal model by a variety of renormalization-group (RG) methods. We show that the presence of a conservation law introduces an ambiguity in operator ordering in the RG procedure, which we show how to resolve. We compare our analysis with standard multiple-scale techniques, where identical results can be obtained with greater labor, by going to sixth order in perturbation theory, and by assuming the correct scaling of space and time.

  4. Reductive use of renormalization group. K Nozaki, Y Oono, and Y Shiwa. Phys Rev E, 62, R4501, 2000.

    It has been recognized that singular perturbation and reductive perturbation can be unified from the renormalization group (RG) theoretical point of view. However, the recognition has been only formal in the sense that it has not given us any new insight nor provided any new technical advantage over the usual RG approach. With our approach, the proto RG method proposed here, we can clearly show that system reduction is the key to singular perturbation methods. The approach also makes the calculation of singular perturbation results more transparent than the conventional RG approach. Consequently, for example, a consistent and easy RG derivation of the rotational covariant Newell-Whitehead-Segel equation is possible.

Of the three, the first one, is the earliest and is also a good read. Further, it also puts several asymptotic analysis techniques–methods of multiple scales, boundary layers, asymptotic mathcing, WKB–in perspective, and discusses them in the renormalization group (RG) framework. And, the last one gives an algorithm with five steps to obtain RG equations (Hat tip: KuoAn Wu for the pointer).

Some other links of interest: Prof. Goldenfeld’s homepage; Prof. Jonathan A Dantzig’s homepage; Wiki page on renormalization.