Distributional fixed-point equations for island nucleation in one dimension: the inverse problem

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Abstract

The self-consistency of the distributional fixed-point equation (DFPE) approach to understanding the statistical properties of island nucleation and growth during submonolayer deposition is explored. We perform kinetic Monte Carlo simulations, in which point islands nucleate on a one-dimensional lattice during submonolyer deposition with critical island size $i$, and examine the evolution of the inter-island gaps as they are fragmented by new island nucleation. The DFPE couples the fragmentation probability distribution within the gaps to the consequent gap size distribution (GSD), and we find a good fit between the DFPE solutions and the observed GSDs for $i = 0, 1, 2, 3$. Furthermore, we develop numerical methods to address the inverse problem, namely the problem of obtaining the gap fragmentation probability from the observed GSD, and again find good self-consistency in the approach. This has consequences for its application to experimental situations where only the GSD is observed, and where the growth rules embodied in the fragmentation process must be deduced.
LanguageEnglish
Article number052801
Number of pages8
JournalPhysical Review E
Volume98
Issue number5
DOIs
Publication statusPublished - 12 Nov 2018

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Fixed-point Equation
Fragmentation
Inverse problems
Nucleation
One Dimension
Self-consistency
Inverse Problem
nucleation
Kinetic Monte Carlo
fragmentation
Statistical property
Probability distributions
Numerical methods
Probability Distribution
Monte Carlo Simulation
Numerical Methods
Kinetics
kinetics

Keywords

  • distributional fixed-point equation
  • DEPE
  • island nucleation
  • submonolayer deposition
  • Monte Carlo simulations

Cite this

@article{cc29f8a6875c4746aba38d5b2b2e7624,
title = "Distributional fixed-point equations for island nucleation in one dimension: the inverse problem",
abstract = "The self-consistency of the distributional fixed-point equation (DFPE) approach to understanding the statistical properties of island nucleation and growth during submonolayer deposition is explored. We perform kinetic Monte Carlo simulations, in which point islands nucleate on a one-dimensional lattice during submonolyer deposition with critical island size $i$, and examine the evolution of the inter-island gaps as they are fragmented by new island nucleation. The DFPE couples the fragmentation probability distribution within the gaps to the consequent gap size distribution (GSD), and we find a good fit between the DFPE solutions and the observed GSDs for $i = 0, 1, 2, 3$. Furthermore, we develop numerical methods to address the inverse problem, namely the problem of obtaining the gap fragmentation probability from the observed GSD, and again find good self-consistency in the approach. This has consequences for its application to experimental situations where only the GSD is observed, and where the growth rules embodied in the fragmentation process must be deduced.",
keywords = "distributional fixed-point equation, DEPE, island nucleation, submonolayer deposition, Monte Carlo simulations",
author = "Hrvojka Krcelic and Michael Grinfeld and Mulheran, {Paul A.}",
year = "2018",
month = "11",
day = "12",
doi = "10.1103/PhysRevE.98.052801",
language = "English",
volume = "98",
journal = "Physical Review E",
issn = "1539-3755",
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T1 - Distributional fixed-point equations for island nucleation in one dimension

T2 - Physical Review E

AU - Krcelic, Hrvojka

AU - Grinfeld, Michael

AU - Mulheran, Paul A.

PY - 2018/11/12

Y1 - 2018/11/12

N2 - The self-consistency of the distributional fixed-point equation (DFPE) approach to understanding the statistical properties of island nucleation and growth during submonolayer deposition is explored. We perform kinetic Monte Carlo simulations, in which point islands nucleate on a one-dimensional lattice during submonolyer deposition with critical island size $i$, and examine the evolution of the inter-island gaps as they are fragmented by new island nucleation. The DFPE couples the fragmentation probability distribution within the gaps to the consequent gap size distribution (GSD), and we find a good fit between the DFPE solutions and the observed GSDs for $i = 0, 1, 2, 3$. Furthermore, we develop numerical methods to address the inverse problem, namely the problem of obtaining the gap fragmentation probability from the observed GSD, and again find good self-consistency in the approach. This has consequences for its application to experimental situations where only the GSD is observed, and where the growth rules embodied in the fragmentation process must be deduced.

AB - The self-consistency of the distributional fixed-point equation (DFPE) approach to understanding the statistical properties of island nucleation and growth during submonolayer deposition is explored. We perform kinetic Monte Carlo simulations, in which point islands nucleate on a one-dimensional lattice during submonolyer deposition with critical island size $i$, and examine the evolution of the inter-island gaps as they are fragmented by new island nucleation. The DFPE couples the fragmentation probability distribution within the gaps to the consequent gap size distribution (GSD), and we find a good fit between the DFPE solutions and the observed GSDs for $i = 0, 1, 2, 3$. Furthermore, we develop numerical methods to address the inverse problem, namely the problem of obtaining the gap fragmentation probability from the observed GSD, and again find good self-consistency in the approach. This has consequences for its application to experimental situations where only the GSD is observed, and where the growth rules embodied in the fragmentation process must be deduced.

KW - distributional fixed-point equation

KW - DEPE

KW - island nucleation

KW - submonolayer deposition

KW - Monte Carlo simulations

UR - https://arxiv.org/abs/1806.03776

UR - https://journals.aps.org/pre/

U2 - 10.1103/PhysRevE.98.052801

DO - 10.1103/PhysRevE.98.052801

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JO - Physical Review E

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