American Institute of Mathematical Sciences

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July  2011, 16(1): 173-188. doi: 10.3934/dcdsb.2011.16.173

Dynamic phase transition for binary systems in cylindrical geometry

I-Liang Chern 1, and  Chun-Hsiung Hsia 2,

1. 

Department of Mathematics and Taidar Institute of Mathematical Science, National Taiwan University, Taipei, 10617, Taiwan
2. 

Department of Mathematics and Taidar Institute of Mthematical Science, National Taiwan University, Taipei, 10617, Taiwan

Received  February 2010 Revised  August 2010 Published  April 2011

In this article, we present a dynamic phase transition and stability analysis for the Cahn-Hilliard equations in cylindrical geometry. Two types of phase transitions (the continuous type and the jump type) are determined explicitly in terms of relevant physical and geometric parameters.
Keywords: binary system., bifurcation, Phase transition, stability.
Mathematics Subject Classification: 37G3.
Citation: I-Liang Chern, Chun-Hsiung Hsia. Dynamic phase transition for binary systems in cylindrical geometry. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 173-188. doi: 10.3934/dcdsb.2011.16.173
References:
[1]

Stephen J. Blundell and Katherine M. Blundell, "Concepts in Thermal Physics,", Oxford University Press, (2008). Google Scholar

[2]

C. M. Elliott and H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix,, Physica D, 109 (1997), 242. doi: 10.1016/S0167-2789(97)00066-3. Google Scholar

[3]

J. E. Hilliard, Spinodal decomposition,, in Phase Transformations, (1970), 497. Google Scholar

[4]

T. Ma and S. Wang, "Bifurcation Theory and Applications,", vol. \textbf{53} of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53 (). Google Scholar

[5]

T. Ma and S. Wang, Dynamic phase transition theory in PVT systems,, Indiana Univ. Math. J., 57 (2008), 2861. doi: 10.1512/iumj.2008.57.3630. Google Scholar

[6]

T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741. doi: 10.3934/dcdsb.2009.11.741. Google Scholar

[7]

T. Ma and S. Wang, Phase separation of binary systems,, Phys. Rev. A., 388 (2009), 4811. Google Scholar

[8]

J. Moser, A rapidly convergent iteration method and nonlinear partial differential equation. I,, Ann. Sc. Norm. Super. Pisa, 20 (1966), 265. Google Scholar

[9]

L. E. Reichl, "A Modern Course in Statistical Physics," (Second ed.), A Wiley-Interscinece Publication. New York: John Wiley & Sons Inc. 1998., (1998). Google Scholar

show all references

References:
[1]

Stephen J. Blundell and Katherine M. Blundell, "Concepts in Thermal Physics,", Oxford University Press, (2008). Google Scholar

[2]

C. M. Elliott and H. Garcke, Diffusional phase transitions in multicomponent systems with a concentration dependent mobility matrix,, Physica D, 109 (1997), 242. doi: 10.1016/S0167-2789(97)00066-3. Google Scholar

[3]

J. E. Hilliard, Spinodal decomposition,, in Phase Transformations, (1970), 497. Google Scholar

[4]

T. Ma and S. Wang, "Bifurcation Theory and Applications,", vol. \textbf{53} of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, 53 (). Google Scholar

[5]

T. Ma and S. Wang, Dynamic phase transition theory in PVT systems,, Indiana Univ. Math. J., 57 (2008), 2861. doi: 10.1512/iumj.2008.57.3630. Google Scholar

[6]

T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamics for binary systems,, Discrete Contin. Dyn. Syst. Ser. B, 11 (2009), 741. doi: 10.3934/dcdsb.2009.11.741. Google Scholar

[7]

T. Ma and S. Wang, Phase separation of binary systems,, Phys. Rev. A., 388 (2009), 4811. Google Scholar

[8]

J. Moser, A rapidly convergent iteration method and nonlinear partial differential equation. I,, Ann. Sc. Norm. Super. Pisa, 20 (1966), 265. Google Scholar

[9]

L. E. Reichl, "A Modern Course in Statistical Physics," (Second ed.), A Wiley-Interscinece Publication. New York: John Wiley & Sons Inc. 1998., (1998). Google Scholar

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