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Computing Optimal Interfacial Structure of Modulated Phases

Published online by Cambridge University Press:  05 December 2016

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:rxj_2004@126.com (J. Xu), chuw@math.princeton.edu (C. Wang), shi@mcmaster.ca (A.-C. Shi), pzhang@pku.edu.cn (P. Zhang)
*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)
*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)
*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)
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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.

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

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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., 6:187216, 2015.CrossRefGoogle ScholarPubMed
[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.CrossRefGoogle 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):688699, 1959.CrossRefGoogle Scholar
[4] McMullen, W. E. and Oxtoby, D. W.. The equilibrium interfaces of simple molecules. J. Chem. Phys., 88(12):77577765, 1988.CrossRefGoogle Scholar
[5] Talanquer, V. and Oxtoby, D. W.. Nucleation on a solid substrate: A density-functional approach. J. Chem. Phys., 104(4):14831492, 1996.CrossRefGoogle Scholar
[6] Talanquer, V. and Oxtoby, D. W.. Nucleation in a slit pore. J. Chem. Phys., 114(6):27932801, 2001.CrossRefGoogle Scholar
[7] Chen, Z. Y. and Noolandi, J.. Numerical solution of the Onsager problem for an isotropicnematic interface. Phys. Rev. A, 45(4):23892392, 1991.CrossRefGoogle ScholarPubMed
[8] McMullen, W. E. and Moore, B. G.. Theoretical Studies of the Isotropic-Nematic Interface. Mol. Cryst. Liq. Cryst., 198(1):107117, 1991.CrossRefGoogle 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):1195611964, 1989.CrossRefGoogle ScholarPubMed
[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.CrossRefGoogle Scholar
[11] Netz, R. R., Andelman, D., and Schick, M.. Interfaces of Modulated Phases. Phys. Rev. Lett., 79(6):10581061, 1997.CrossRefGoogle Scholar
[12] Masten, M. W.. Kink grain boundaries in a block copolymer lamellar phase. J. Chem. Phys., 107(19):81108119, 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):28482858, 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):1031510320, 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.CrossRefGoogle ScholarPubMed
[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.CrossRefGoogle ScholarPubMed
[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.CrossRefGoogle 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, 369:335350, 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, 7:1055210555, 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, 2:311345, 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.CrossRefGoogle Scholar
[26] Brazovskii, S. A.. Phase transition of an isotropic system to a nonuniform state. Sov. Phys.- JETP, 41(1):8589, 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):697705, 1987.CrossRefGoogle Scholar
[28] Kats, E. I., Lebedev, V. V., and Muratov, A. R.. Weak crystallization theory. Physics reports, 228(1):191, 1993.CrossRefGoogle Scholar
[29] Zhang, P. and Zhang, X.. An efficient numerical method of Landau-Brazovskii model. J. Comput. Phys., 227:58595870, 2008.CrossRefGoogle 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):8689, 1994.CrossRefGoogle 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, 27:490501, 1994.CrossRefGoogle 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):1207112081, 2003.CrossRefGoogle 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., 131:4647, 2009.CrossRefGoogle ScholarPubMed
[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.CrossRefGoogle ScholarPubMed
[35] Barzilai, J. and Borwein, J. M.. Two-point step size gradient methods. IMA J. Numer. Anal., 8:141148, 1988.CrossRefGoogle Scholar
[36] Zhou, B., Gao, L., and Dai, Y. H.. Gradient Methods with Adaptive Step-Sizes. Computational Optimization and Applications, 35:6986, 2006.CrossRefGoogle Scholar
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