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Computing Optimal Interfacial Structure of Modulated Phases

  • Jie Xu (a1), Chu Wang (a2), An-Chang Shi (a3) and Pingwen Zhang (a1)
Abstract
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.

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Corresponding author
*Corresponding author. Email addresses: rxj_2004@126.com (J. Xu), chuw@math.princeton.edu (C. Wang), shi@mcmaster.ca (A.-C. Shi), pzhang@pku.edu.cn (P. Zhang)
References
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[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., 6:187216, 2015.
[2] Cahn J. W. and Hilliard J. E.. Free Energy of a Nonuniform System. I. Interfacial Free Energy. J. Chem. Phys., 28(2):258267, 1958.
[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):688699, 1959.
[4] McMullen W. E. and Oxtoby D. W.. The equilibrium interfaces of simple molecules. J. Chem. Phys., 88(12):77577765, 1988.
[5] Talanquer V. and Oxtoby D. W.. Nucleation on a solid substrate: A density-functional approach. J. Chem. Phys., 104(4):14831492, 1996.
[6] Talanquer V. and Oxtoby D. W.. Nucleation in a slit pore. J. Chem. Phys., 114(6):27932801, 2001.
[7] Chen Z. Y. and Noolandi J.. Numerical solution of the Onsager problem for an isotropicnematic interface. Phys. Rev. A, 45(4):23892392, 1991.
[8] McMullen W. E. and Moore B. G.. Theoretical Studies of the Isotropic-Nematic Interface. Mol. Cryst. Liq. Cryst., 198(1):107117, 1991.
[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):1195611964, 1989.
[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):211228, 2003.
[11] Netz R. R., Andelman D., and Schick M.. Interfaces of Modulated Phases. Phys. Rev. Lett., 79(6):10581061, 1997.
[12] Masten M. W.. Kink grain boundaries in a block copolymer lamellar phase. J. Chem. Phys., 107(19):81108119, 1997.
[13] Tsori Y., Andelman D., and Schick M.. Defects in lamellar diblock copolymers: Chevronand ω-shaped tilt boundaries. Phys. Rev. E, 61(3):28482858, 2000.
[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):1031510320, 2002.
[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.
[16] Belushkin M. and Gompper G.. Twist grain boundaries in cubic surfactant phases. J. Chem. Phys., 130:134712, 2009.
[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.
[18] Kyrylyuk A. V. and Fraaije J. G. E. M.. Three-Dimensional Structure and Motion of Twist Grain Boundaries in Block Copolymer Melts. Macromolecules, 38:85468553, 2005.
[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, 369:335350, 2011.
[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.
[21] Wang C., Jiang K., Zhang P., and Shi A. C.. Origin of epitaxies between ordered phases of block copolymers. Soft Matter, 7:1055210555, 2011.
[22] Fredrickson G.H.. The equilibrium theory of inhomogeneous polymers. Clarendon Press, Oxford, 2006.
[23] Merkle K. L.. High-Resolution Electron Microscopy of Grain Boundaries. Interface Sceince, 2:311345, 1995.
[24] E W., Ren W., and Eijnden E. V.. String method for the study of rare events. Phys. Rev. B, 66:052301, 2002.
[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.
[26] Brazovskii S. A.. Phase transition of an isotropic system to a nonuniform state. Sov. Phys.- JETP, 41(1):8589, 1975.
[27] Fredrickson G. H. and Helfand E.. Fluctuation effects in the theory of microphase separation in block copolymers. J. Chem. Phys., 87(1):697705, 1987.
[28] Kats E. I., Lebedev V. V., and Muratov A. R.. Weak crystallization theory. Physics reports, 228(1):191, 1993.
[29] Zhang P. and Zhang X.. An efficient numerical method of Landau-Brazovskii model. J. Comput. Phys., 227:58595870, 2008.
[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):8689, 1994.
[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, 27:490501, 1994.
[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):1207112081, 2003.
[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., 131:4647, 2009.
[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.
[35] Barzilai J. and Borwein J. M.. Two-point step size gradient methods. IMA J. Numer. Anal., 8:141148, 1988.
[36] Zhou B., Gao L., and Dai Y. H.. Gradient Methods with Adaptive Step-Sizes. Computational Optimization and Applications, 35:6986, 2006.
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Communications in Computational Physics
  • ISSN: 1815-2406
  • EISSN: 1991-7120
  • URL: /core/journals/communications-in-computational-physics
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