Strain and defects in oblique stripe growth

Kelly Chen; Zachary Deiman; Ryan Goh; Sally Jankovic; Arnd Scheel

doi:10.1137/21M1397210

Strain and defects in oblique stripe growth

Kelly Chen
, Zachary Deiman
, Ryan Goh
, Sally Jankovic
, Arnd Scheel

School of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

We study stripe formation in two-dimensional systems under directional quenching in a phase-diffusion approximation including nonadiabatic boundary effects. We find stripe formation through simple traveling waves for all angles relative to the quenching line using an analytic continuation procedure. We also present comprehensive analytical asymptotic formulas in limiting cases of small and large angles as well as small and large quenching rates. Of particular interest is a regime of small angle and slow quenching rate which is well described by the glide motion of a boundary dislocation along the quenching line. A delocalization bifurcation of this dislocation leads to a sharp decrease of strain created in the growth process at small angles. We complement our results with numerical continuation reliant on a boundary-integral formulation. We also compare results in the phase-diffusion approximation numerically to quenched stripe formation in an anisotropic Swift - Hohenberg equation.

Original language	English (US)
Pages (from-to)	1236-1260
Number of pages	25
Journal	Multiscale Modeling and Simulation
Volume	19
Issue number	3
DOIs	https://doi.org/10.1137/21M1397210
State	Published - 2021

Bibliographical note

Funding Information:
\ast Received by the editors February 5, 2021; accepted for publication (in revised form) May 6, 2021; published electronically August 5, 2021. https://doi.org/10.1137/21M1397210 Funding: The first, second, and fourth authors were partially supported through grant NSF DMS-1907391. The third author was partially supported through NSF-DMS 2006887. \dagger Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA ([email protected]). \ddagger School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (deima008@ umn.edu, [email protected], [email protected]). \S Department of Mathematics and Statistics, Boston University, Boston, MA 02215 USA (rgoh@ bu.edu).

Publisher Copyright:
© 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.

Keywords

Defect
Directional quenching
Pattern formation
Phase-diffusion equation
Swift - Hohenberg equation

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10.1137/21M1397210

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title = "Strain and defects in oblique stripe growth",

abstract = "We study stripe formation in two-dimensional systems under directional quenching in a phase-diffusion approximation including nonadiabatic boundary effects. We find stripe formation through simple traveling waves for all angles relative to the quenching line using an analytic continuation procedure. We also present comprehensive analytical asymptotic formulas in limiting cases of small and large angles as well as small and large quenching rates. Of particular interest is a regime of small angle and slow quenching rate which is well described by the glide motion of a boundary dislocation along the quenching line. A delocalization bifurcation of this dislocation leads to a sharp decrease of strain created in the growth process at small angles. We complement our results with numerical continuation reliant on a boundary-integral formulation. We also compare results in the phase-diffusion approximation numerically to quenched stripe formation in an anisotropic Swift - Hohenberg equation.",

keywords = "Defect, Directional quenching, Pattern formation, Phase-diffusion equation, Swift - Hohenberg equation",

author = "Kelly Chen and Zachary Deiman and Ryan Goh and Sally Jankovic and Arnd Scheel",

note = "Funding Information: \textbackslash{}ast Received by the editors February 5, 2021; accepted for publication (in revised form) May 6, 2021; published electronically August 5, 2021. https://doi.org/10.1137/21M1397210 Funding: The first, second, and fourth authors were partially supported through grant NSF DMS-1907391. The third author was partially supported through NSF-DMS 2006887. \textbackslash{}dagger Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA ([email protected]). \textbackslash{}ddagger School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (deima008@ umn.edu, [email protected], [email protected]). \textbackslash{}S Department of Mathematics and Statistics, Boston University, Boston, MA 02215 USA (rgoh@ bu.edu). Publisher Copyright: {\textcopyright} 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.",

year = "2021",

doi = "10.1137/21M1397210",

language = "English (US)",

volume = "19",

pages = "1236--1260",

journal = "Multiscale Modeling and Simulation",

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TY - JOUR

T1 - Strain and defects in oblique stripe growth

AU - Chen, Kelly

AU - Deiman, Zachary

AU - Goh, Ryan

AU - Jankovic, Sally

AU - Scheel, Arnd

N1 - Funding Information: \ast Received by the editors February 5, 2021; accepted for publication (in revised form) May 6, 2021; published electronically August 5, 2021. https://doi.org/10.1137/21M1397210 Funding: The first, second, and fourth authors were partially supported through grant NSF DMS-1907391. The third author was partially supported through NSF-DMS 2006887. \dagger Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139 USA ([email protected]). \ddagger School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (deima008@ umn.edu, [email protected], [email protected]). \S Department of Mathematics and Statistics, Boston University, Boston, MA 02215 USA (rgoh@ bu.edu). Publisher Copyright: © 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.

PY - 2021

Y1 - 2021

N2 - We study stripe formation in two-dimensional systems under directional quenching in a phase-diffusion approximation including nonadiabatic boundary effects. We find stripe formation through simple traveling waves for all angles relative to the quenching line using an analytic continuation procedure. We also present comprehensive analytical asymptotic formulas in limiting cases of small and large angles as well as small and large quenching rates. Of particular interest is a regime of small angle and slow quenching rate which is well described by the glide motion of a boundary dislocation along the quenching line. A delocalization bifurcation of this dislocation leads to a sharp decrease of strain created in the growth process at small angles. We complement our results with numerical continuation reliant on a boundary-integral formulation. We also compare results in the phase-diffusion approximation numerically to quenched stripe formation in an anisotropic Swift - Hohenberg equation.

AB - We study stripe formation in two-dimensional systems under directional quenching in a phase-diffusion approximation including nonadiabatic boundary effects. We find stripe formation through simple traveling waves for all angles relative to the quenching line using an analytic continuation procedure. We also present comprehensive analytical asymptotic formulas in limiting cases of small and large angles as well as small and large quenching rates. Of particular interest is a regime of small angle and slow quenching rate which is well described by the glide motion of a boundary dislocation along the quenching line. A delocalization bifurcation of this dislocation leads to a sharp decrease of strain created in the growth process at small angles. We complement our results with numerical continuation reliant on a boundary-integral formulation. We also compare results in the phase-diffusion approximation numerically to quenched stripe formation in an anisotropic Swift - Hohenberg equation.

KW - Defect

KW - Directional quenching

KW - Pattern formation

KW - Phase-diffusion equation

KW - Swift - Hohenberg equation

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U2 - 10.1137/21M1397210

DO - 10.1137/21M1397210

M3 - Article

AN - SCOPUS:85114027761

SN - 1540-3459

VL - 19

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EP - 1260

JO - Multiscale Modeling and Simulation

JF - Multiscale Modeling and Simulation

IS - 3

ER -

Strain and defects in oblique stripe growth

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