Phase field model

The phase-field model and generalizations are now widely used for simulations of dendritic growth and solidification [71-76] and even hydro-dynamic flow with moving interfaces [78,79]. One can even use the phase-field model to treat the growth of faceting crystals [77]. More details will be given later. 

FIG. 8 Compact seaweed originating from the simulation of an isotropic phase by a phase-field model [120]. A doublon structure is just about to emerge from the chaotic background. 

Braun, S. Cornell, R. Sekerka. Phase field models for anisotropic interfaces. Phys Rev 48 10X6, 1993. 

A. Boesch, H. Miiller-Krumbhaar, O. Shochet. Phase field models for moving boundary problems Controlling metastability and anisotropy. Z Physik B 97 161, 1995. 

A. Karma, W.-J. Rappel. Phase field method for computationally efficient modeling of solidification with arbitrary interface kinetics. Phys Rev E 55 R3017, 1996 A. Karma, W.-J. Rappel. Quantitative phase field modeling of dendritic growth in two and three dimensions. Phys Rev E 57 4111, 1998. 

A. Karma. Phase field model of eutectic growth. Phys Rev E 49 2245, 1994. J. Burton, R. Prim, W. Slichter. J Chem Phys 27 1987, 1953. 

M. K. Venkitachalam, L.-Q. Chen, A. G. Khachaturyan, G. L. Messing. A multiple-component order parameter phase field model for anisotropic grain growth. Mater Sci Eng A 238 94, 1997. 

J.A. Warren, R. Kobayashi, A.E. Lobovsky, and W.C. Carter. Extending phase field models of solidification to polycrystalline materials. Acta Mater., 51(20) 6035-6058, 2003. 

O. Penrose and P.C. Fife. On the relation between the standard phase-field model and a thermodynamically consistent phase-field model. Physica D, 69(1-2) 107-113, 1993. 

S.Y. Hu and L.Q. Chen. A phase-field model for evolving microstructures with strong elastic inhomogeneity. Acta Mater., 49(11) 1879-1890, 2001. 

Jeong, J.-H., N. Goldenfeld, and J. Dantzig. 2001. Phase field model for three dimensional dendritic growth with fluid flow. Phys. Rev. E 64 041602. 

Vetsigian, K., and N. Goldenfeld. 2003. Computationally efficient phase-field models with interface kinetics. Phys. Rev. E 68 60601. 

Yet another promising line of research lies in creating mesoscopic representations whose fundamental scale is somewhere within the 1010 range of distance scales and for which one defines closed (or fully consistent) equations of motion. At the macroscopic limit, hydrodynamic models are a very successful and standard example. More recent approaches include the Cahn-Hillard coarse-grained models and phase-field models. In some cases, one aims to ascertain the degree to which the systems exhibit self-similarity at various length scales hence the lack of a specific parameterization—which would be necessary using reduceddimensional models—is not of much importance. 

Several hybrid simulations on crystal growth can be found in recent literature. Examples include dendritic solidification by coupling finite-different discretization of a phase field model to a MC simulation (Plapp and Karma, 2000), coupling a finite difference for the melt with a cellular automata for the solidification (Grujicic et al., 2001), a DSMC model for the fluid phase with a Metropolis-based MC for the surface to address cluster deposition onto substrates (Hongo et al., 2002 Mizuseki et al., 2002), a step model for the surface processes coupled with a CFD simulation of flow (Kwon and Derby, 2001) (two continuum but different feature scale models), an adaptive FEM CVD model coupled with a feature scale model (Merchant et al., 2000), and one-way coupled growth models in plasma systems (Hoekstra et al., 1997). Some specific applications are discussed in more detail below. 

Fig. 21.4 Temperature dependencies of normalized Raman intensities of TO2 (solid triangles) and TO4 (open triangles) phonons for (a) SLs [(BaTi03)2(SrTi03)4] x 40 and [(BaTi03)5(SrTi03)4] x 25 (b) SLs [(BaTi03)g(SrTi03)4] x 40 and [(BaTi03)8(SrTi03)4] x 10. The dash-dotted lines are fits to linear temperature dependence, (c) and (d) - three-dimensional phase-field model calculations of polarization as a function of temperature in the same superlattice...

Figure 21.5 shows the Tc in superlattices determined by the Raman data, variable-temperature XRD, and the phase-field model, as a function of the BaTiOs and SrTiOs layer thicknesses. The XRD measurements provide data on the temperature dependence of the lattice parameters, and the observed noticeable change in the slope of the out-of-plane c-axis lattice parameter due to the... 

Fig. 21.5 Tc dependence on layer thicknesses n and m in superlattices (BaTi03) / (SrTi03) j. Blue triangles and red circles are for m = 4 and m = 13, respectively. Open squares show the values obtained from variable temperature X-ray diffraction measurements. Solid lines are from the three-dimensional phase-field model calculations, dashed lines -simulations assuming a single domain in the BaTi03 layers. The dash-dotted line shows in bulk BaTi03 (After Li et al. [150])...

Fig. 21.12 Ferroelectric phase transition temperature, Tc in strained BaTiOs films on SrTiOs substrates as a function of the film thickness, as determined from Raman data for all films studied (symbols). The dashed-dotted line is a result of the phase-field model calculation with open circuit boundary conditions. Dotted line shows In bulk BaTlOs (After Tenne et al. [48])...

Li YL, Hu SY, Liu ZK, Chen LQ (2001) Phase-field model of domain structures in ferroelectric thin films. Appl Phys Lett 78 3878... 

Fig. 10.46. Phase field model for two-dimensional grain growth (adapted from Chen and Wang (1996)). Figure shows grain distribution at various times in the evolution of the grain structure with (a) t = 1000, (b) t = 3000, (c) t = 5000 and (d) t = 8000. Time is measured in units of how many integration steps have taken place.

Fig. 10.47. Temporal evolution of mean grain size in phase field model of grain growth (adapted from Chen and Yang (1994)). Plots are of logarithm of average grain area as a function of time, with the two curves corresponding to four (crosses) and thirty-six (circles) different order parameter fields.

Fig. 12.20. Results from phase field model of solidification (adapted from Provatas et at. (1998)).

Phase Field Models of Two-Phase Microstructures Three Dimensions. It has... 

Fig. 12.22. Representation configurations from the phase field model of phase separation in a three-dimensional binary alloy (adapted from Sagui et al. (1998)). The progression from (a) through (g) represents different choices of the parameters characterizing the elastic response of the two media, with (a) corresponding to a treatment in which elastic effects are ignored. In those cases where there is more than one picture associated with a given letter, these represent snapshots at different times in the simulation.

Phase Field Modeling for a Single-Component System Sectorization and Ripple Formation in sPP... 

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