# 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 & 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, (1995), 109. doi: 10.1142/9789814317344_0016. Google Scholar [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. doi: 10.1016/0022-0396(90)90120-E. Google Scholar [3] A. D. Bruno, Local Methods in Nonlinear Differential Equations,, Springer-Verlag, (1989). doi: 10.1007/978-3-642-61314-2. Google Scholar [4] G. H. Fredrickson and E. Helfand, Fluctuation effects in the theory of microphase separation in block copolymers,, Journal Chemical Physics, 87 (1987), 697. doi: 10.1063/1.453566. Google Scholar [5] H. Jenkins, Chemical Thermodynamics at a Glance,, Wiley Online Library, (2008). doi: 10.1002/9780470697733. Google Scholar [6] E. I. Kats, V. V. Lebedev and A. R. Muratov, Weak crystallization theory,, Physics Reports, 228 (1993), 1. doi: 10.1016/0370-1573(93)90119-X. Google Scholar [7] L. Leibler, Theory of microphase separation in block copolymers,, Macromolecules, 13 (1980), 1602. doi: 10.1021/ma60078a047. Google Scholar [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). Google Scholar [9] T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamic for binary systems,, Discrete and Continuous Dynamical Systems, 11 (2009), 741. doi: 10.3934/dcdsb.2009.11.741. Google Scholar [10] T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, (2014). doi: 10.1007/978-1-4614-8963-4. Google Scholar [11] A. Shi, Nature of anisotropic fluctuation modes in ordered systems,, Journal of Physics: Condensed Matter, 11 (1999), 10183. doi: 10.1088/0953-8984/11/50/311. Google Scholar [12] A. Shi, J. Noolandi and R. C. Desai, Theory of anisotropic fluctuations in ordered block copolymer phases,, Macromolecules, 29 (1996), 6487. doi: 10.1021/ma960411t. Google Scholar [13] X. Zhang and P. Zhang, An efficient numerical method of Landau-Brazovskii model,, Journal of Computational Physics, 227 (2008), 5859. doi: 10.1016/j.jcp.2008.02.021. Google Scholar

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, (1995), 109. doi: 10.1142/9789814317344_0016. Google Scholar [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. doi: 10.1016/0022-0396(90)90120-E. Google Scholar [3] A. D. Bruno, Local Methods in Nonlinear Differential Equations,, Springer-Verlag, (1989). doi: 10.1007/978-3-642-61314-2. Google Scholar [4] G. H. Fredrickson and E. Helfand, Fluctuation effects in the theory of microphase separation in block copolymers,, Journal Chemical Physics, 87 (1987), 697. doi: 10.1063/1.453566. Google Scholar [5] H. Jenkins, Chemical Thermodynamics at a Glance,, Wiley Online Library, (2008). doi: 10.1002/9780470697733. Google Scholar [6] E. I. Kats, V. V. Lebedev and A. R. Muratov, Weak crystallization theory,, Physics Reports, 228 (1993), 1. doi: 10.1016/0370-1573(93)90119-X. Google Scholar [7] L. Leibler, Theory of microphase separation in block copolymers,, Macromolecules, 13 (1980), 1602. doi: 10.1021/ma60078a047. Google Scholar [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). Google Scholar [9] T. Ma and S. Wang, Cahn-Hilliard equations and phase transition dynamic for binary systems,, Discrete and Continuous Dynamical Systems, 11 (2009), 741. doi: 10.3934/dcdsb.2009.11.741. Google Scholar [10] T. Ma and S. Wang, Phase Transition Dynamics,, Springer-Verlag, (2014). doi: 10.1007/978-1-4614-8963-4. Google Scholar [11] A. Shi, Nature of anisotropic fluctuation modes in ordered systems,, Journal of Physics: Condensed Matter, 11 (1999), 10183. doi: 10.1088/0953-8984/11/50/311. Google Scholar [12] A. Shi, J. Noolandi and R. C. Desai, Theory of anisotropic fluctuations in ordered block copolymer phases,, Macromolecules, 29 (1996), 6487. doi: 10.1021/ma960411t. Google Scholar [13] X. Zhang and P. Zhang, An efficient numerical method of Landau-Brazovskii model,, Journal of Computational Physics, 227 (2008), 5859. doi: 10.1016/j.jcp.2008.02.021. Google Scholar
 [1] Matteo Novaga, Enrico Valdinoci. The geometry of mesoscopic phase transition interfaces. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 777-798. doi: 10.3934/dcds.2007.19.777 [2] Hayato Chiba, Georgi S. Medvedev. The mean field analysis of the kuramoto model on graphs Ⅱ. asymptotic stability of the incoherent state, center manifold reduction, and bifurcations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3897-3921. doi: 10.3934/dcds.2019157 [3] 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 [4] Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799 [5] Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems - A, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249 [6] Jun Yang. Coexistence phenomenon of concentration and transition of an inhomogeneous phase transition model on surfaces. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 965-994. doi: 10.3934/dcds.2011.30.965 [7] Pavel Krejčí, Jürgen Sprekels. Long time behaviour of a singular phase transition model. Discrete & Continuous Dynamical Systems - A, 2006, 15 (4) : 1119-1135. doi: 10.3934/dcds.2006.15.1119 [8] Mauro Garavello. Boundary value problem for a phase transition model. Networks & Heterogeneous Media, 2016, 11 (1) : 89-105. doi: 10.3934/nhm.2016.11.89 [9] Mauro Garavello, Francesca Marcellini. The Riemann Problem at a Junction for a Phase Transition Traffic Model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5191-5209. doi: 10.3934/dcds.2017225 [10] Pierluigi Colli, Antonio Segatti. Uniform attractors for a phase transition model coupling momentum balance and phase dynamics. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 909-932. doi: 10.3934/dcds.2008.22.909 [11] Wei-Ming Ni, Xuefeng Wang. On the first positive Neumann eigenvalue. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 1-19. doi: 10.3934/dcds.2007.17.1 [12] Robert Brooks and Eran Makover. The first eigenvalue of a Riemann surface. Electronic Research Announcements, 1999, 5: 76-81. [13] Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure & Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839 [14] Katarzyna Grabowska, Paweƚ Urbański. Geometry of Routh reduction. Journal of Geometric Mechanics, 2019, 11 (1) : 23-44. doi: 10.3934/jgm.2019002 [15] Francesca Marcellini. Existence of solutions to a boundary value problem for a phase transition traffic model. Networks & Heterogeneous Media, 2017, 12 (2) : 259-275. doi: 10.3934/nhm.2017011 [16] Kousuke Kuto, Tohru Tsujikawa. Stationary patterns for an adsorbate-induced phase transition model I: Existence. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1105-1117. doi: 10.3934/dcdsb.2010.14.1105 [17] Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873 [18] Stefano Bianchini, Alberto Bressan. A center manifold technique for tracing viscous waves. Communications on Pure & Applied Analysis, 2002, 1 (2) : 161-190. doi: 10.3934/cpaa.2002.1.161 [19] Alain Miranville. Some mathematical models in phase transition. Discrete & Continuous Dynamical Systems - S, 2014, 7 (2) : 271-306. doi: 10.3934/dcdss.2014.7.271 [20] Ulisse Stefanelli. Analysis of a variable time-step discretization for a phase transition model with micro-movements. Communications on Pure & Applied Analysis, 2006, 5 (3) : 659-673. doi: 10.3934/cpaa.2006.5.659

2018 Impact Factor: 1.008

## Metrics

• HTML views (0)
• Cited by (3)

• on AIMS