A new diffuse-interface model for step flow in epitaxial growth

Rätz, Andreas

doi:10.1093/imamat/hxu012

Cited by 8 publications

(13 citation statements)

References 23 publications

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“…In this contribution it is used for a new diffuseinterface approximation of a Stefan-like free boundary problem for osmotic swelling, which has not been numerically investigated yet. Besides [23], the current paper provides a further example of the flexibility of the diffuse domain method with respect to applicability to more classical free boundary problems.…”

Section: Discussionmentioning

confidence: 99%

“…In such a diffuse-interface approximation the interface is implicitly given by a phase-field function, which is obtained, roughly speaking, by smearing out the indicator function of Ω + on a short length O(ε) for some small parameter ε > 0. The diffuse domain method [9,10,11] offers a powerful tool for the approximation of free boundary problems [23,25] and coupled bulk-surface PDE-systems [27,24]. It will be used here, in order to numerically simulate the one-and two-phase models previously introduced.…”

Section: 2mentioning

confidence: 99%

“…Again, we choose initial conditions related to the ones of the sharp-interface model as well as boundary conditions for ϕ and u ± on ∂Ω. We remark that this approach is similar to the proposed in [23], where a related phase-field model has been used in order to approximate a Stefan-like free boundary ("Burton-Cabrera-Frank") step flow model in epitaxial growth.…”

Section: Diffuse-interface Approximationsmentioning

confidence: 99%

“…We closely follow the asymptotic expansions in [11,23] and apply the setup in [23]. The outer solution is an approximation to the solution in the solvent, the inner solution zooms in at the interface.…”

Section: Matched Asymptotic Expansionsmentioning

confidence: 99%

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Diffuse-Interface Approximations of Osmosis Free Boundary Problems

Raźtz¹

2016

SIAM J. Appl. Math.

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Abstract. Free boundary problems based on mass conservation and surface tension with application in osmotic swelling are the topic of this contribution. We introduce new phase-field approximations of such models, in order to numerically investigate properties of the solutions. Formal justification of the proposed approximations is provided by matched asymptotic expansions supported by numerical tests reproducing the convergence for shrinking interface thickness.

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Section: Discussionmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Diffuse-interface Approximationsmentioning

confidence: 99%

Section: Matched Asymptotic Expansionsmentioning

confidence: 99%

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Diffuse-Interface Approximations of Osmosis Free Boundary Problems

Raźtz¹

2016

SIAM J. Appl. Math.

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“…This approach considered only isotropic edge energies and kinetic coefficients and did not consider van der Waals interactions. Further, the DDM used in [19] did not use a second-order accurate formulation and thus was only first order accurate in (and h assuming ∝ h).…”

Section: Introductionmentioning

confidence: 99%

Modeling the vertical growth of van der Waals stacked 2D materials using the diffuse domain method

Guo

Price

Shenoy

et al. 2020

Modelling Simul. Mater. Sci. Eng.

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Vertically-stacked monolayers of graphene and other atomically-thin 2D materials have attracted considerable research interest because of their potential in fabricating materials with specifically-designed properties. Chemical vapor deposition has proved to be an efficient and scalable fabrication method. However, a lack of mechanistic understanding has hampered efforts to control the fabrication process beyond empirical trial-and-error approaches. In this paper, we develop a general multiscale Burton-Cabrera-Frank (BCF) type model of the vertical growth of 2D materials to predict the necessary growth conditions for vertical versus in-plane (monolayer) growth of arbitrarily-shaped layers. This extends previous work where we developed such a model assuming the layers were fully-faceted (Ye et al., ACS Nano, 11, 12780-12788, 2017). To solve the model numerically, we reformulate the system using the phase-field/diffuse domain method that enables the equations to be solved in a fixed regular domain. We use a second-order accurate, adaptive finitedifference/nonlinear multigrid algorithm to discretize and solve the discrete system. We investigate the effect of parameters, including the van der Waals interaction energies between the layers, the kinetic attachment rates, the edge-energies and the deposition flux, on layer growth and morphologies. While the conditions that favor vertical growth generally follow an analytic thermodynamic criterion we derived for circular layers, the layer boundaries may develop significant curvature during growth, consistent with experimental observations. Our approach provides a mechanistic framework for controlling and optimizing the growth multilayered 2D materials.

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Analysis of the Diffuse Domain Method for Second Order Elliptic Boundary Value Problems

2015

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The diffuse domain method for partial differential equations on complicated geometries recently received strong attention in particular from practitioners, but many fundamental issues in the analysis are still widely open. In this paper we study the diffuse domain method for approximating second order elliptic boundary value problems posed on bounded domains, and show convergence and rates of the approximations generated by the diffuse domain method to the solution of the original second order problem when complemented by Robin, Dirichlet or Neumann conditions.The main idea of the diffuse domain method is to relax these boundary conditions by introducing a family of phase-field functions such that the variational integrals of the original problem are replaced by a weighted average of integrals of perturbed domains. From an functional analytic point of view, the phase-field functions naturally lead to weighted Sobolev spaces for which we present trace and embedding results as well as various type of Poincaré inequalities with constants independent of the domain perturbations. Our convergence analysis is carried out in such spaces as well, but allows to draw conclusions also about unweighted norms applied to restrictions on the original domain. Our convergence results are supported by numerical examples.

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A new diffuse-interface model for step flow in epitaxial growth

Cited by 8 publications

References 23 publications

Diffuse-Interface Approximations of Osmosis Free Boundary Problems

Diffuse-Interface Approximations of Osmosis Free Boundary Problems

Modeling the vertical growth of van der Waals stacked 2D materials using the diffuse domain method

Analysis of the Diffuse Domain Method for Second Order Elliptic Boundary Value Problems

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