Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-05T04:13:56.976Z Has data issue: false hasContentIssue false
  • English
  • Français

Computing Optimal Interfacial Structure of Modulated Phases

Part of: Statistical mechanics, structure of matter Equilibrium statistical mechanics

Published online by Cambridge University Press:  05 December 2016

Jie Xu ,
Chu Wang ,
An-Chang Shi  and
Pingwen Zhang
Jie Xu*
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
Chu Wang*
Affiliation:
Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA
An-Chang Shi*
Affiliation:
Department of Physics and Astronomy, McMaster University, Hamilton, Ontario L8S4M1, Canada
Pingwen Zhang*
Affiliation:
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
*
*Corresponding author. Email addresses:[email protected] (J. Xu), [email protected] (C. Wang), [email protected] (A.-C. Shi), [email protected] (P. Zhang)
*Corresponding author. Email addresses:[email protected] (J. Xu), [email protected] (C. Wang), [email protected] (A.-C. Shi), [email protected] (P. Zhang)
*Corresponding author. Email addresses:[email protected] (J. Xu), [email protected] (C. Wang), [email protected] (A.-C. Shi), [email protected] (P. Zhang)
*Corresponding author. Email addresses:[email protected] (J. Xu), [email protected] (C. Wang), [email protected] (A.-C. Shi), [email protected] (P. Zhang)
Get access

Abstract

We propose a general framework of computing interfacial structures between two modulated phases. Specifically we propose to use a computational box consisting of two half spaces, each occupied by a modulated phase with given position and orientation. The boundary conditions and basis functions are chosen to be commensurate with the bulk phases. We observe that the ordered nature of modulated structures stabilizes the interface, which enables us to obtain optimal interfacial structures by searching local minima of the free energy landscape. The framework is applied to the Landau-Brazovskii model to investigate interfaces between modulated phases with different relative positions and orientations. Several types of novel complex interfacial structures emerge from the calculations.

Keywords

Interfacemodulated phasemetastable statecompatibilityLandau-Brazovskii model

MSC classification

Secondary: 82B24: Interface problems; diffusion-limited aggregation 82-08: Computational methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 1 , January 2017 , pp. 1 - 15
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Li, W. and Müller, M.. Defects in the Self-Assembly of Block Copolymers and Their Relevance for Directed Self-Assembly. Annu. Rev. Chem. Biomol. Eng., 6:187–216, 2015.Google Scholar
[2] Cahn, J. W. and Hilliard, J. E.. Free Energy of a Nonuniform System. I. Interfacial Free Energy. J. Chem. Phys., 28(2):258–267, 1958.Google Scholar
[3] Cahn, J. W. and Hilliard, J. E.. Free Energy of a Nonuniform System. III. Nucleation in a Two-Component Incompressible Fluid. J. Chem. Phys., 31(3):688–699, 1959.Google Scholar
[4] McMullen, W. E. and Oxtoby, D. W.. The equilibrium interfaces of simple molecules. J. Chem. Phys., 88(12):7757–7765, 1988.CrossRefGoogle Scholar
[5] Talanquer, V. and Oxtoby, D. W.. Nucleation on a solid substrate: A density-functional approach. J. Chem. Phys., 104(4):1483–1492, 1996.Google Scholar
[6] Talanquer, V. and Oxtoby, D. W.. Nucleation in a slit pore. J. Chem. Phys., 114(6):2793–2801, 2001.Google Scholar
[7] Chen, Z. Y. and Noolandi, J.. Numerical solution of the Onsager problem for an isotropicnematic interface. Phys. Rev. A, 45(4):2389–2392, 1991.CrossRefGoogle Scholar
[8] McMullen, W. E. and Moore, B. G.. Theoretical Studies of the Isotropic-Nematic Interface. Mol. Cryst. Liq. Cryst., 198(1):107–117, 1991.Google Scholar
[9] Rogers, T. M. and Desai, R. C.. Numerical study of late-stage coarsening for off-critical quenches in the Cahn-Hilliard equation of phase separation. Phys. Rev. B, 39(16):11956–11964, 1989.Google Scholar
[10] Liu, C. and Shen, J.. A phase field model for the mixture of two incompressible fluids and its approximation by a Fourier-spectral method. Physica D, 179(3-4):211–228, 2003.Google Scholar
[11] Netz, R. R., Andelman, D., and Schick, M.. Interfaces of Modulated Phases. Phys. Rev. Lett., 79(6):1058–1061, 1997.CrossRefGoogle Scholar
[12] Masten, M. W.. Kink grain boundaries in a block copolymer lamellar phase. J. Chem. Phys., 107(19):8110–8119, 1997.Google Scholar
[13] Tsori, Y., Andelman, D., and Schick, M.. Defects in lamellar diblock copolymers: Chevronand ω-shaped tilt boundaries. Phys. Rev. E, 61(3):2848–2858, 2000.CrossRefGoogle Scholar
[14] Duque, D., Katsov, K., and Schick, M.. Theory of T junctions and symmetric tilt grain boundaries in pure and mixed polymer systems. J. Chem. Phys., 117(22):10315–10320, 2002.CrossRefGoogle Scholar
[15] Jaatinen, A., Achim, C. V., Elder, K. R., and Ala-Nissila, T.. Thermodynamics of bcc metals in phase-field-crystal models. Phys. Rev. E, 80:031602, 2009.CrossRefGoogle ScholarPubMed
[16] Belushkin, M. and Gompper, G.. Twist grain boundaries in cubic surfactant phases. J. Chem. Phys., 130:134712, 2009.Google Scholar
[17] Elder, K. R. and Grant, M.. Modeling elastic and plastic deformations in nonequilibrium processing using phase field crystals. Phys. Rev. E, 70:051605, 2004.Google Scholar
[18] Kyrylyuk, A. V. and Fraaije, J. G. E. M.. Three-Dimensional Structure and Motion of Twist Grain Boundaries in Block Copolymer Melts. Macromolecules, 38:8546–8553, 2005.Google Scholar
[19] Pezzutti, A. D., Vega, D. A., and Villar, M. A.. Dynamics of dislocations in a two-dimensional block copolymer system with hexagonal symmetry. Phil. Trans. R. Soc. A, 369:335–350, 2011.CrossRefGoogle Scholar
[20] Yamada, K. and Ohta, T.. Interface between Lamellar and Gyroid Structure in Diblock Copolymer Melts. Journal of the Physical Society of Japan, 76(8):084801, 2007.CrossRefGoogle Scholar
[21] Wang, C., Jiang, K., Zhang, P., and Shi, A. C.. Origin of epitaxies between ordered phases of block copolymers. Soft Matter, 7:10552–10555, 2011.CrossRefGoogle Scholar
[22] Fredrickson, G.H.. The equilibrium theory of inhomogeneous polymers. Clarendon Press, Oxford, 2006.Google Scholar
[23] Merkle, K. L.. High-Resolution Electron Microscopy of Grain Boundaries. Interface Sceince, 2:311–345, 1995.Google Scholar
[24] E, W., Ren, W., and Eijnden, E. V.. String method for the study of rare events. Phys. Rev. B, 66:052301, 2002.CrossRefGoogle Scholar
[25] E, W., Ren, W., and Eijnden, E. V.. Simplified and improved string method for computing the minimum energy paths in barrier-crossing events. J. Chem. Phys., 126:164103, 2007.Google Scholar
[26] Brazovskii, S. A.. Phase transition of an isotropic system to a nonuniform state. Sov. Phys.- JETP, 41(1):85–89, 1975.Google Scholar
[27] Fredrickson, G. H. and Helfand, E.. Fluctuation effects in the theory of microphase separation in block copolymers. J. Chem. Phys., 87(1):697–705, 1987.CrossRefGoogle Scholar
[28] Kats, E. I., Lebedev, V. V., and Muratov, A. R.. Weak crystallization theory. Physics reports, 228(1):1–91, 1993.CrossRefGoogle Scholar
[29] Zhang, P. and Zhang, X.. An efficient numerical method of Landau-Brazovskii model. J. Comput. Phys., 227:5859–5870, 2008.Google Scholar
[30] Schulz, M. F., Bates, F. S., Almdal, K., and Mortensen, K.. Epitaxial Relationship for Hexagonal-to-Cubic Phase Transition in a Block Copolymer Mixture. Phys. Rev. Lett., 73(1):86–89, 1994.Google Scholar
[31] Hajduk, D. A., Gruner, S. M., Rangarajan, P., Register, R. A., Fetters, L. J., Honeker, C., Albalak, R. J., and Thomas, E. L.. Observation of a Reversible Thermotropic Order-Order Transition in a Diblock Copolymer. Macromolecules, 27:490–501, 1994.Google Scholar
[32] Bang, J. and Lodge, T. P.. Mechanisms and Epitaxial Relationships between Close-Packed and BCC Lattices in Block Copolymer Solutions. J. Phys. Chem. B, 107(44):12071–12081, 2003.Google Scholar
[33] Park, H. W., Jung, J., Chang, T., Matsunaga, K., and Jinnai, H.. New Epitaxial Phase Transition between DG and HEX in PS-b-PI. J. Am. Chem. Soc., 131:46–47, 2009.Google Scholar
[34] Cheng, X., Lin, L., E, W., Zhang, P., and Shi, A. C.. Nucleation of Ordered Phases in Block Copolymers. Phys. Rev. Lett., 104:148301, 2010.Google Scholar
[35] Barzilai, J. and Borwein, J. M.. Two-point step size gradient methods. IMA J. Numer. Anal., 8:141–148, 1988.Google Scholar
[36] Zhou, B., Gao, L., and Dai, Y. H.. Gradient Methods with Adaptive Step-Sizes. Computational Optimization and Applications, 35:69–86, 2006.CrossRefGoogle Scholar