Article Contents
Article Contents

# On the convergence to critical scaling profiles in submonolayer deposition models

• * Corresponding author: F.P. da Costa
Partially funded by FCT/Portugal through project RD0447/CAMGSD/2015.
• 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.

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

 Citation:

• 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 $\eta = 1/4,$ $1/3,$ $1/2,$ $1,$ $2,$ $3,$ $4$.

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

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