# American Institute of Mathematical Sciences

March  2014, 19(2): 607-627. doi: 10.3934/dcdsb.2014.19.607

## Dynamic transitions for Landau-Brazovskii model

 1 School of Mathematical Sciences, Peking University, Beijing, 100871, China, China 2 LMAM, CAPT and School of Mathematical Sciences, Peking University, Beijing, 100871, China

Received  June 2013 Revised  September 2013 Published  February 2014

We study the disordered and ordered phase transitions modeled by the Landau-Brazovskii (LB) equation using the dynamic transition theory. It is shown that the linear instability of the disordered phase always leads to phase transitions to ordered phases, and type of transitions is dictated by the sign of a nondimensional parameter, measuring the strengths of the quadratic and cubic nonlinearity of the model. By the center manifold reduced theory, we analysis the lower dimension system. The analysis shows that the second-order disordered-ordered phase transition occurs in two ways: one is due to the combined affect of the surface energy and the rectangular geometry of the spatial domain, and another other is due to mixed transition. For the mixed transition, there exist two regions of fluctuations where the first and second-order transitions occur respectively. We also present the general results of the phase transition for the rectangle domain as if the first eigenvalue is simple, which is the most likely to happen. It is shown that for case of second order transition, the spatial period of the ordered phases is essentially independent of the size of the domain, and for the first order transition case, the disordered phase undergoes a drastic change, leading to more richer ordered patterns. In addition, we show the stability of some classical ordered phases near the homogeneous phase, like HPC, BCC and Doulbe Gyroid phase.
Citation: Hao Zhang, Kai Jiang, Pingwen Zhang. Dynamic transitions for Landau-Brazovskii model. Discrete and Continuous Dynamical Systems - B, 2014, 19 (2) : 607-627. doi: 10.3934/dcdsb.2014.19.607
##### References:
 [1] S. A. Brazovskii, Phase transition of an isotropic system to a nonuniform state, In 30 Years of the Landau Institute - Selected Papers, volume 11 of World Scientific Series in 20th Century Physics, pages 109-113. World Scientific Publishing, 1995. doi: 10.1142/9789814317344_0016. [2] M. Brunella and M. Miari, Topological equivalence of a plane vector field with its principal part defined through newton polyhedra, Journal of differential equations, 85 (1990), 338-366. doi: 10.1016/0022-0396(90)90120-E. [3] A. D. Bruno, Local Methods in Nonlinear Differential Equations, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61314-2. [4] G. H. Fredrickson and E. Helfand, Fluctuation effects in the theory of microphase separation in block copolymers, Journal Chemical Physics, 87 (1987), 697-705. doi: 10.1063/1.453566. [5] H. Jenkins, Chemical Thermodynamics at a Glance, Wiley Online Library, 2008. doi: 10.1002/9780470697733. [6] E. I. Kats, V. V. Lebedev and A. R. Muratov, Weak crystallization theory, Physics Reports, 228 (1993), 1-91. doi: 10.1016/0370-1573(93)90119-X. [7] L. Leibler, Theory of microphase separation in block copolymers, Macromolecules, 13 (1980), 1602-1617. doi: 10.1021/ma60078a047. [8] H. Liu, T. Sengul, S. Wang and P. Zhang, Dynamic Transitions and Pattern Formations for Cahn-Hilliard Model with Long-Range Repulsive Interactions, to be published, 2013. [9] T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamic for binary systems, Discrete and Continuous Dynamical Systems, Series B, 11 (2009), 741-784. doi: 10.3934/dcdsb.2009.11.741. [10] T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, 2014. doi: 10.1007/978-1-4614-8963-4. [11] A. Shi, Nature of anisotropic fluctuation modes in ordered systems, Journal of Physics: Condensed Matter, 11 (1999), 10183-10197. doi: 10.1088/0953-8984/11/50/311. [12] A. Shi, J. Noolandi and R. C. Desai, Theory of anisotropic fluctuations in ordered block copolymer phases, Macromolecules, 29 (1996), 6487-6504. doi: 10.1021/ma960411t. [13] X. Zhang and P. Zhang, An efficient numerical method of Landau-Brazovskii model, Journal of Computational Physics, 227 (2008), 5859-5870. doi: 10.1016/j.jcp.2008.02.021.

show all references

##### References:
 [1] S. A. Brazovskii, Phase transition of an isotropic system to a nonuniform state, In 30 Years of the Landau Institute - Selected Papers, volume 11 of World Scientific Series in 20th Century Physics, pages 109-113. World Scientific Publishing, 1995. doi: 10.1142/9789814317344_0016. [2] M. Brunella and M. Miari, Topological equivalence of a plane vector field with its principal part defined through newton polyhedra, Journal of differential equations, 85 (1990), 338-366. doi: 10.1016/0022-0396(90)90120-E. [3] A. D. Bruno, Local Methods in Nonlinear Differential Equations, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-642-61314-2. [4] G. H. Fredrickson and E. Helfand, Fluctuation effects in the theory of microphase separation in block copolymers, Journal Chemical Physics, 87 (1987), 697-705. doi: 10.1063/1.453566. [5] H. Jenkins, Chemical Thermodynamics at a Glance, Wiley Online Library, 2008. doi: 10.1002/9780470697733. [6] E. I. Kats, V. V. Lebedev and A. R. Muratov, Weak crystallization theory, Physics Reports, 228 (1993), 1-91. doi: 10.1016/0370-1573(93)90119-X. [7] L. Leibler, Theory of microphase separation in block copolymers, Macromolecules, 13 (1980), 1602-1617. doi: 10.1021/ma60078a047. [8] H. Liu, T. Sengul, S. Wang and P. Zhang, Dynamic Transitions and Pattern Formations for Cahn-Hilliard Model with Long-Range Repulsive Interactions, to be published, 2013. [9] T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamic for binary systems, Discrete and Continuous Dynamical Systems, Series B, 11 (2009), 741-784. doi: 10.3934/dcdsb.2009.11.741. [10] T. Ma and S. Wang, Phase Transition Dynamics, Springer-Verlag, 2014. doi: 10.1007/978-1-4614-8963-4. [11] A. Shi, Nature of anisotropic fluctuation modes in ordered systems, Journal of Physics: Condensed Matter, 11 (1999), 10183-10197. doi: 10.1088/0953-8984/11/50/311. [12] A. Shi, J. Noolandi and R. C. Desai, Theory of anisotropic fluctuations in ordered block copolymer phases, Macromolecules, 29 (1996), 6487-6504. doi: 10.1021/ma960411t. [13] X. Zhang and P. Zhang, An efficient numerical method of Landau-Brazovskii model, Journal of Computational Physics, 227 (2008), 5859-5870. doi: 10.1016/j.jcp.2008.02.021.
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