American Institute of Mathematical Sciences

Advanced
August  2009, 24(3): 979-1003. doi: 10.3934/dcds.2009.24.979

Shell structure as solution to a free boundary problem from block copolymer morphology

Xiaofeng Ren 1,

1. 

Department of Mathematics, The George Washington University, 2115 G Street, Washington, DC 20052

Received  November 2007 Revised  March 2008 Published  April 2009

A shell like structure is sought as a solution of a free boundary problem derived from the Ohta-Kawasaki theory of diblock copolymers. The boundary of the shell satisfies an equation that involves its mean curvature and the location of the entire shell. A variant of Lyapunov-Schmidt reduction process is performed that rigorously reduces the free boundary problem to a finite dimensional problem. The finite dimensional problem is solved numerically. The problem has two parameters: $a$ and $\gamma$. When $a$ is small, there are a lower bound and a sequence such that if $\gamma$ is greater than the lower bound and stays away from the sequence, there is a shell like solution.
Keywords: diblock copolymer, shell pattern..
Mathematics Subject Classification: 35R35, 82B24, 82D6.
Citation: Xiaofeng Ren. Shell structure as solution to a free boundary problem from block copolymer morphology. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 979-1003. doi: 10.3934/dcds.2009.24.979
[1]

Michael Helmers, Barbara Niethammer, Xiaofeng Ren. Evolution in off-critical diblock copolymer melts. Networks & Heterogeneous Media, 2008, 3 (3) : 615-632. doi: 10.3934/nhm.2008.3.615
[2]

Jean-Philippe Lessard, Evelyn Sander, Thomas Wanner. Rigorous continuation of bifurcation points in the diblock copolymer equation. Journal of Computational Dynamics, 2017, 4 (1&2) : 71-118. doi: 10.3934/jcd.2017003
[3]

Thomas Wanner. Computer-assisted equilibrium validation for the diblock copolymer model. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 1075-1107. doi: 10.3934/dcds.2017045
[4]

Ian Johnson, Evelyn Sander, Thomas Wanner. Branch interactions and long-term dynamics for the diblock copolymer model in one dimension. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3671-3705. doi: 10.3934/dcds.2013.33.3671
[5]

Peng-Fei Yao. On shallow shell equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 697-722. doi: 10.3934/dcdss.2009.2.697
[6]

G.W. Hunt, Gabriel J. Lord, Mark A. Peletier. Cylindrical shell buckling: a characterization of localization and periodicity. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 505-518. doi: 10.3934/dcdsb.2003.3.505
[7]

Tania Biswas, Sheetal Dharmatti. Control problems and invariant subspaces for sabra shell model of turbulence. Evolution Equations & Control Theory, 2018, 7 (3) : 417-445. doi: 10.3934/eect.2018021
[8]

Inger Daniels, Catherine Lebiedzik. Existence and uniqueness of a structural acoustic model involving a nonlinear shell. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 243-252. doi: 10.3934/dcdss.2008.1.243
[9]

Xiaofeng Ren, Chong Wang. A stationary core-shell assembly in a ternary inhibitory system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (2) : 983-1012. doi: 10.3934/dcds.2017041
[10]

Shumin Li, Masahiro Yamamoto, Bernadette Miara. A Carleman estimate for the linear shallow shell equation and an inverse source problem. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 367-380. doi: 10.3934/dcds.2009.23.367
[11]

Sylvia Anicic. Existence theorem for a first-order Koiter nonlinear shell model. Discrete & Continuous Dynamical Systems - S, 2019, 12 (6) : 1535-1545. doi: 10.3934/dcdss.2019106
[12]

Chufen Wu, Dongmei Xiao, Xiao-Qiang Zhao. Asymptotic pattern of a migratory and nonmonotone population model. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1171-1195. doi: 10.3934/dcdsb.2014.19.1171
[13]

Elena Izquierdo-Kulich, Margarita Amigó de Quesada, Carlos Manuel Pérez-Amor, Magda Lopes Texeira, José Manuel Nieto-Villar. The dynamics of tumor growth and cells pattern morphology. Mathematical Biosciences & Engineering, 2009, 6 (3) : 547-559. doi: 10.3934/mbe.2009.6.547
[14]

Yu Zheng, Li Peng, Teturo Kamae. Characterization of noncorrelated pattern sequences and correlation dimensions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (10) : 5085-5103. doi: 10.3934/dcds.2018223
[15]

Julien Cividini. Pattern formation in 2D traffic flows. Discrete & Continuous Dynamical Systems - S, 2014, 7 (3) : 395-409. doi: 10.3934/dcdss.2014.7.395
[16]

Yuan Lou, Wei-Ming Ni, Shoji Yotsutani. Pattern formation in a cross-diffusion system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1589-1607. doi: 10.3934/dcds.2015.35.1589
[17]

Peter Rashkov. Remarks on pattern formation in a model for hair follicle spacing. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1555-1572. doi: 10.3934/dcdsb.2015.20.1555
[18]

Daniel Wetzel. Pattern analysis in a benthic bacteria-nutrient system. Mathematical Biosciences & Engineering, 2016, 13 (2) : 303-332. doi: 10.3934/mbe.2015004
[19]

Rui Peng, Fengqi Yi. On spatiotemporal pattern formation in a diffusive bimolecular model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 217-230. doi: 10.3934/dcdsb.2011.15.217
[20]

Tian Ma, Shouhong Wang. Dynamic transition and pattern formation for chemotactic systems. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2809-2835. doi: 10.3934/dcdsb.2014.19.2809

2018 Impact Factor: 1.143

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences