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Phase field theory of crystal nucleation and polycrystalline growth: A review

Published online by Cambridge University Press:  01 February 2006

L. Gránásy*
Affiliation:
Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
T. Pusztai
Affiliation:
Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
T. Börzsönyi
Affiliation:
Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
G. Tóth
Affiliation:
Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
G. Tegze
Affiliation:
Research Institute for Solid State Physics and Optics, H-1525 Budapest, Hungary
J.A. Warren
Affiliation:
National Institute of Standards and Technology, Gaithersburg, Maryland 20899
J.F. Douglas
Affiliation:
National Institute of Standards and Technology, Gaithersburg, Maryland 20899
*
a)Address all correspondence to this author. e-mail: grana@szfki.hu This paper was selected as the Outstanding Meeting Paper for the 2004 MRS Fall Meeting Symposium JJ Proceedings, Vol. 859E.
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Abstract

We briefly review our recent modeling of crystal nucleation and polycrystalline growth using a phase field theory. First, we consider the applicability of phase field theory for describing crystal nucleation in a model hard sphere fluid. It is shown that the phase field theory accurately predicts the nucleation barrier height for this liquid when the model parameters are fixed by independent molecular dynamics calculations. We then address various aspects of polycrystalline solidification and associated crystal pattern formation at relatively long timescales. This late stage growth regime, which is not accessible by molecular dynamics, involves nucleation at the growth front to create new crystal grains in addition to the effects of primary nucleation. Finally, we consider the limit of extreme polycrystalline growth, where the disordering effect due to prolific grain formation leads to isotropic growth patterns at long times, i.e., spherulite formation. Our model of spherulite growth exhibits branching at fixed grain misorientations, induced by the inclusion of a metastable minimum in the orientational free energy. It is demonstrated that a broad variety of spherulitic patterns can be recovered by changing only a few model parameters.

Type
Outstanding Meeting Paper
Copyright
Copyright © Materials Research Society 2006

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References

REFERENCES

1.Jin, L-W., Claborn, K.A., Kurimoto, M., Geday, M.A., Maezawa, I., Sohraby, F., Estrada, M., Kaminsky, W. and Kahr, B.: Imaging linear birefringence and dichroism in cerebral amyloid pathologies. Proc. Natl. Acad. Sci. U S A 100, 15297 (2003).Google Scholar
2.Proc. Royal Soc. Discussion Meeting on Nucleation Control, edited by Greenwood, G.W., Greer, A.L., Herlach, D.M., and Kelton, K.F., Philos. Trans. 361 (2003).Google Scholar
3.Swope, W.C. and Andersen, H.C.: 106—particle molecular-dynamics study of homogeneous nucleation of crystals in a supercooled atomic liquid. Phys. Rev. B 41, 7042 (1990).CrossRefGoogle Scholar
4.Wolde, P.R. ten and Frenkel, D.: Homogeneous nucleation and the Ostwald step rule. Phys. Chem. Chem. Phys. 1, 2191 (1999).Google Scholar
5.Wolde, P.R. ten, Ruiz-Montero, M.J. and Frenkel, D.: Numerical eviedence for bcc ordering at the surface of critical fcc nucleus. Phys. Rev. Lett. 75, 2714 (1995).CrossRefGoogle Scholar
6.Auer, S. and Frenkel, D.: Prediction of absolute crystal-nucleation rate in hard-sphere colloids. Nature 409, 1020 (2001).Google Scholar
7.Davidchack, R.L. and Laird, B.B.: Direct calculation of the hard-sphere crystal/melt interfacial free energy. J. Chem. Phys. 108, 9452 (1998).CrossRefGoogle Scholar
8.Oxtoby, D.W.: Density-functional methods in the statistical mechanics of materials. Annu. Rev. Mater. Res. 32, 39 (2002).Google Scholar
9.Shen, Y.C. and Oxtoby, D.W.: Bcc symmetry in the crystal-melt interface of Lennard-Jones fluids examined through density-functional theory. Phys. Rev. Lett. 77, 3585 (1996).CrossRefGoogle ScholarPubMed
10.Gránásy, L. and Oxtoby, D.W.: Cahn–Hilliard theory with triple parabolic free energy. II. Nucleation and growth in the presence of a metastable crystalline phase. J. Chem. Phys. 112, 2410 (2000).CrossRefGoogle Scholar
11.Lee, K. and Losert, W.: Private communication (2004).Google Scholar
12.Nobel, B.D. and James, P.F.: Private communication (2003).Google Scholar
13.Ferreiro, V., Douglas, J.F., Warren, J.A. and Karim, A.: Growth pulsation in symmetric dendritic crystallization in thin polymer blend films. Phys. Rev. E 65, 051606 (2002).Google Scholar
14.Ryshchenkow, G. and Faivre, G.: Bulk crystallization of liquid selenium—Primary nucleation, growth-kinetics and modes of crystallization. J. Cryst. Growth 87, 221 (1988).Google Scholar
15.Ojeda, M. and Martin, D.C.: High-resolution microscopy of PMDA-ODA poly(imide) single crystals. Macromol. 26, 6557 (1993).CrossRefGoogle Scholar
16.Padden, F.J. and Keith, H.D.: Crystalline morphology of synthetic polypeptides. J. Appl. Phys. 36, 2987 (1965).Google Scholar
17.Boettinger, W.J., Warren, J.A., Beckermann, C. and Karma, A.: Phase-field simulation of solidification. Ann. Rev. Mater. Res. 32, 163 (2002).CrossRefGoogle Scholar
18.Hoyt, J.J., Asta, M. and Karma, A.: Atomistic and continuum modeling of dendritic solidification. Mater. Sci. Eng. Rep. R41, 121 (2003).Google Scholar
19.Keith, H.D., Padden, F.J. Jr.: A phenomenological theory of spherulitic crystallization. J. Appl. Phys. 34, 2409 (1963).CrossRefGoogle Scholar
20.Goldenfeld, N.: Theory of spherulitic solidification. J. Cryst. Growth 84, 601 (1987).Google Scholar
21.Nagarajan, K. and Myerson, A.S.: Molecular dynamics of nucleation and crystallization of polymers. Cryst. Growth Design 1, 131 (2005).Google Scholar
22.Yamamoto, T.: Molecular dynamics modeling of polymer crystallization from the melt. Polymer 45, 1357 (2004).CrossRefGoogle Scholar
23.Gránásy, L., Börzsönyi, T. and Pusztai, T.: Nucleation and bulk crystallization in binary phase field theory. Phys. Rev. Lett. 88, 206105 (2002).Google Scholar
24.Gránásy, L., Pusztai, T., Tóth, G., Jurek, Z., Conti, M. and Kvamme, B.: Phase field theory of crystal nucleation in hard-sphere liquid. J. Chem. Phys. 119, 10376 (2003).Google Scholar
25.Gránásy, L., Pusztai, T., Warren, J.A., Börzsönyi, T., Douglas, J.F. and Ferreiro, V.: Growth of ‘dizzy dendrites’ in a random field of foreign particles. Nat. Mater. 2, 92 (2003).Google Scholar
26.Gránásy, L., Pusztai, T., Börzsönyi, T., Warren, J.A. and Douglas, J.F.: A general mechanism of polycrystalline growth. Nat. Mater. 3, 645 (2004).Google Scholar
27.Gránásy, L., Pusztai, T. and Warren, J.A.: Modelling polycrystalline solidification using phase field theory. J. Phys.: Condens. Matter 16, R1205 (2004).Google Scholar
28.Gránásy, L., Pusztai, T., Tegze, G., Warren, J.A. and Douglas, J.F.: On the growth and form of spherulites. Phys. Rev. E 72, 011605 (2005).Google Scholar
29.Gránásy, L., Pusztai, T., Tegze, G., Warren, J.A., and Douglas, J.F.: Polycrystalline patterns in far-from-equilibrium freezing: A phase field study. Philos. Mag. A (in press).Google Scholar
30.Kobayashi, R., Warren, J.A. and Carter, W.C.: Vector-valued phase field model for crystallization and grain boundary formation. Physica D 119, 415 (1998).CrossRefGoogle Scholar
31.Pusztai, T., Bortel, G. and Gránásy, L.: Phase field theory of polycrystalline solidification in three dimensions. Europhys. Lett. 71, 131 (2005).Google Scholar
32.Kobayashi, R. and Warren, J.A.: Modeling the formation and dynamics of polycrystals in 3D. Physica A 356, 127 (2005).Google Scholar
33.Kobayashi, R. and Giga, Y.: Equations with singular diffusivity. J. Stat. Phys. 95, 1187 (1999).Google Scholar
34.Warren, J.A., Kobayashi, R., Lobkovsky, A.E. and Carter, W.C.: Extending phase field models of solidification to polycrystalline materials. Acta Mater. 51, 6035 (2003).Google Scholar
35.Roy, A., Rickman, J.M., Gunton, J.D. and Elder, K.R.: Simulation study of nucleation in a phase-field model with nonlocal interactions. Phys. Rev. E 56, 2610 (1998).Google Scholar
36.Elder, K.R., Drolet, F., Kosterlitz, J.M. and Grant, M.: Stochastic eutectic growth. Phys. Rev. Lett. 72, 677 (1994).CrossRefGoogle ScholarPubMed
37.Castro, M.: Phase-field approach to heterogeneous nucleation. Phys. Rev. B 67, 035412 (2003).Google Scholar
38.Cacciuto, A., Auer, S. and Frenkel, D.: Solid-liquid interfacial free energy of small colloidal hard-sphere crystals. J. Chem. Phys. 119, 7467 (2003).Google Scholar
39.Mu, Y., Houk, A. and Song, X.: Anisotropic interfacial free energies of the hard-sphere crystal-melt interfaces. J. Phys. Chem. B 109, 6500 (2005).CrossRefGoogle ScholarPubMed
40.Tóth, G. Investigation of crystal nucleation in the hard sphere system. Diploma Thesis Technical University of Budapest, Hungary (2004).Google Scholar
41.Girshick, S.L. and Chiu, C.P.: Kinetic nucleation theory—A new expression for the rate of homogeneous nucleation from an ideal supersaturated vapor. J. Chem. Phys. 93, 1273 (1990).Google Scholar
42.Gránásy, L.: Diffuse interface theory for homogeneous vapor condensation. J. Chem. Phys. 104, 5188 (1996).Google Scholar
43.Magill, J.H.: Review spherulites: A personal perspective. J. Mater. Sci. 36, 3143 (2001).Google Scholar
44.Khoury, F.: The spherulitic crystallization of isotatic polypropylene from solution: On the evolution of monoclinic spherulites from dendritic chain-folded precursors. J. Res. Natl. Bur. Stand. 70A, 29 (1966).Google Scholar
45.Keller, A. and Waring, J.R.: The spherulitic structure of crystalline polymers. Part III. Geometrical factors in spherulitic growth and the fine-structure. J. Polymer Sci. 17, 447 (1955).Google Scholar
46.Magill, J.H.: Crystallization of poly-(tetramethyl-p-silphenylene)-solixane polymers. J. Appl. Phys. 35, 3249 (1964).Google Scholar
47.Pusztai, T., Bortel, G. and Gránásy, L.: Phase field theory modeling of polycrystalline freezing. Mater. Sci. Eng. A 413–414, 412 (2005).Google Scholar