On the convergence to critical scaling profiles in submonolayer deposition models

Fernando P. da Costa; João T. Pinto; Rafael Sasportes

doi:10.3934/krm.2018053

Article Contents

2018, Volume 11, Issue 6: 1359-1376. Doi: 10.3934/krm.2018053

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On the convergence to critical scaling profiles in submonolayer deposition models

1.
Departamento de Ciȇncias e Tecnologia, Universidade Aberta, Lisboa, Portugal
2.
Centro de Análise Matemática, Geometria e Sistemas Dinâmicos, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal
3.
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Lisboa, Portugal

* Corresponding author: F.P. da Costa
* Corresponding author: F.P. da Costa

Received: June 30, 2017

Published: June 2018

Partially funded by FCT/Portugal through project RD0447/CAMGSD/2015.

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Abstract

In this work we study the rate of convergence to similarity profiles in a mean field model for the deposition of a submonolayer of atoms in a crystal facet, when there is a critical minimal size $ n≥ 2$ for the stability of the formed clusters. The work complements recently published related results by the same authors in which the rate of convergence was studied outside of a critical direction $ x = τ$ in the cluster size $ x$ vs. time $ \tau $ plane. In this paper we consider a different similarity variable, $ ξ : = (x-\tau )/\sqrt \tau $, corresponding to an inner expansion of that critical direction, and prove the convergence of solutions to a similarity profile $ Φ_{2,n}(ξ)$ when $ x, \tau \to +∞$ with $ ξ$ fixed, as well as the rate at which the limit is approached.

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Mathematics Subject Classification: Primary: 34C11, 34C20, 34C45, 34D05; Secondary: 82C21.

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Figure 1. Lines with $\xi = \text{constant}$ (full), and with $\eta = \text{constant}$ (dashed) in the $(j,\tau)$ plane. The values used for these parameters are the following, in counterclockwise direction: $\xi = 5.0, 2.0, 1.0, 0.5, 0.0, -0.3, -0.5, -0.7, -0.9,$ and $\eta = 1/4,$ $1/3,$ $1/2,$ $1,$ $2,$ $3,$ $4$.

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Figure 2. Graph of the similarity profile $\Phi_{2,n}(\xi)$ for different values of $n$.

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