{"title":"Coupled surface diffusion and mean curvature motion: An axisymmetric system with two grains and a hole","authors":"Katrine Golubkov, A. Novick-Cohen, Yotam Vaknin","doi":"10.1090/qam/1691","DOIUrl":null,"url":null,"abstract":"Thin polycrystalline solid state films, which are used in many technological applications, can exhibit various phenomena, such as wetting, dewetting, and hole formation. We focus on a model system containing two contacting grains which surround a hole. For simplicity, the system is assumed to be axisymmetric, to be supported by a planar substrate and to be bounded within an inert semi-infinite cylinder. We assume that the exterior surfaces of the grains evolve by surface diffusion and the grain boundary between the adjacent grains evolve by motion by mean curvature. Boundary conditions are imposed following W.W. Mullins, 1958. Parametric formulas are derived for the steady states, which contain two nodoids describing the exterior surfaces, which are coupled to a catenoid which describes the grain boundary. At steady state, the physical parameters of the system may be prescribed via two angles, \n\n \n β\n \\beta\n \n\n, the angle between the exterior surface and the grain boundary, and \n\n \n \n θ\n c\n \n \\theta _c\n \n\n, the contact angle between the exterior surface and the substrate; additionally, there are two dimensionless geometric parameters which must satisfy certain constraints. We prove that if \n\n \n \n β\n ∈\n (\n π\n \n /\n \n 2\n ,\n π\n )\n \n \\beta \\in (\\pi /2, \\pi )\n \n\n and \n\n \n \n \n θ\n c\n \n =\n π\n \n \\theta _c=\\pi\n \n\n, then there exists a continuum of steady states. Numerical calculations indicate that steady state profiles can exhibit physical features, such as hillock formation; a fuller numerical study of the steady states and their properties recently appeared in Zigelman and Novick-Cohen [J. Appl. Phys. 134 (2023), 135302], which relies on the formulas and results derived here.","PeriodicalId":20964,"journal":{"name":"Quarterly of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Quarterly of Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/qam/1691","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
Thin polycrystalline solid state films, which are used in many technological applications, can exhibit various phenomena, such as wetting, dewetting, and hole formation. We focus on a model system containing two contacting grains which surround a hole. For simplicity, the system is assumed to be axisymmetric, to be supported by a planar substrate and to be bounded within an inert semi-infinite cylinder. We assume that the exterior surfaces of the grains evolve by surface diffusion and the grain boundary between the adjacent grains evolve by motion by mean curvature. Boundary conditions are imposed following W.W. Mullins, 1958. Parametric formulas are derived for the steady states, which contain two nodoids describing the exterior surfaces, which are coupled to a catenoid which describes the grain boundary. At steady state, the physical parameters of the system may be prescribed via two angles,
β
\beta
, the angle between the exterior surface and the grain boundary, and
θ
c
\theta _c
, the contact angle between the exterior surface and the substrate; additionally, there are two dimensionless geometric parameters which must satisfy certain constraints. We prove that if
β
∈
(
π
/
2
,
π
)
\beta \in (\pi /2, \pi )
and
θ
c
=
π
\theta _c=\pi
, then there exists a continuum of steady states. Numerical calculations indicate that steady state profiles can exhibit physical features, such as hillock formation; a fuller numerical study of the steady states and their properties recently appeared in Zigelman and Novick-Cohen [J. Appl. Phys. 134 (2023), 135302], which relies on the formulas and results derived here.
期刊介绍:
The Quarterly of Applied Mathematics contains original papers in applied mathematics which have a close connection with applications. An author index appears in the last issue of each volume.
This journal, published quarterly by Brown University with articles electronically published individually before appearing in an issue, is distributed by the American Mathematical Society (AMS). In order to take advantage of some features offered for this journal, users will occasionally be linked to pages on the AMS website.