Capture-zone Distribution Research Articles

Capture-zone areas in submonolayer nucleation: Effects of dimensionality and short-range interactions

Simulation results for the asymptotic scaled capture-zone distribution (CZD) for the case of irreversible nucleation and growth of point islands are presented for substrate dimension d=1 , 2, 3, and 4 and compared with a recent conjecture based on the Wigner distribution. Poor agreement is found between the predicted Wigner distributions and the asymptotic CZD in the limit of infinite DF (corresponding to the ratio of monomer hopping rate D to deposition rate F ). Our results also indicate that for d=2 and 3 the asymptotic CZD for point islands is independent of model details and dimension. However, for d=1 and d=4 the resulting distribution is significantly more sharply peaked. We also find that in contrast to the island-size distribution, for which mean-field-like behavior is observed in d=3 and above, the asymptotic CZD is significantly broadened by fluctuations even in d=4 .

Capture-Zone Scaling in Island Nucleation: Universal Fluctuation Behavior

In island nucleation and growth, the distribution of capture zones (in essence proximity cells) can be described by a simple expression generalizing the Wigner surmise (power-law rise, Gaussian decay) from random matrix theory that accounts for spacing distributions in a host of fluctuation phenomena. Its single adjustable parameter, the power-law exponent, can be simply related to the critical nucleus of growth models and the substrate dimensionality. We compare with extensive published kinetic Monte Carlo data and limited experimental data. A phenomenological theory elucidates the result.

Upper critical dimension for irreversible cluster nucleation and growth in the point-island regime

We compare the results of kinetic Monte Carlo (KMC) simulations of a point-island model of irreversible nucleation and growth in four dimensions (4D) with the corresponding mean-field (MF) rate-equation predictions for the monomer density, island density, island-size distribution (ISD), capture-number distribution (CND), and capture-zone distribution (CZD), in order to determine the critical dimension d(c) for mean-field behavior. The asymptotic behavior is studied as a function of the fraction of occupied sites (coverage) and the ratio DF of the monomer hopping rate D to the (per site) monomer creation rate F. Excellent agreement is found between our KMC simulation results and the MF rate equation results for the average island and monomer densities. For large D/F, the scaled CND and CZD do not depend on island size, in good agreement with the MF prediction, while the scaled ISD also agrees well with the MF prediction except for a slight difference at the peak values. Coupled with previous results obtained in d = 3 , these results indicate that for growth in the point-island regime, the upper critical dimension for irreversible cluster nucleation and growth is equal to 4.

Island-size distribution and capture numbers in three-dimensional nucleation: Comparison with mean-field behavior

The scaling of the monomer and island densities, island-size distribution (ISD), capture-number distribution (CND), and capture-zone distribution is studied as a function of the fraction of occupied sites (coverage) and ratio $D∕F$ of the monomer hopping rate $D$ to the (per site) monomer creation rate $F$ in a three-dimensional (3D) point-island model of irreversible nucleation and island growth. Our model is a 3D analog of submonolayer growth and may also be viewed as a simplified model of the early stages of vacancy cluster nucleation and growth under irradiation. Good agreement is found between mean-field (MF) rate-equation results for the average island and monomer densities and our simulation results. In addition, due to the decreased influence of correlations and fluctuations in 3D, the scaled CND depends only weakly on the island-size. As a result, the scaled ISD is significantly sharper than obtained in 2D and diverges with increasing $D∕F$. However, for large $D∕F$ both the scaled ISD and the scaled CND differ from the MF prediction. In particular, the scaled ISD diverges more slowly than the MF prediction while the asymptotic divergence occurs at a value of the scaled island size which is larger than the MF prediction. These results are further supported by an analysis of the asymptotic scaled capture-number distribution.

Stochastic determination of three‐dimensional capture zones for a fully penetrating well

A stochastic methodology is used to investigate transport of a conservative solute by steady flow to a fully penetrating pumping well in a three‐dimensional heterogeneous formation. The spatially variable hydraulic conductivity is modeled as a stationary random function of anisotropic two‐point covariance. Although mean transport is radial, heterogeneity produces three‐dimensional velocity fluctuations which are shown to affect solute recovery and the capture zone distribution. Expressions accurate to the first order in log conductivity variance σY2 are derived for travel time variance, capture zone extent, and solute recovery under the following simplifying assumptions: (1) the vertical integral scale of conductivity is much less than the horizontal integral scale, (2) the depth of the formation is much greater than the vertical integral scale of conductivity, (3) the well bore diameter is much smaller than the horizontal scale of conductivity, (4) longitudinal and lateral local dispersion are negligible while the effect of transverse dispersion is examined. Travel time variance στ2 for transport in convergent flow in a three‐dimensional aquifer is shown to be greater than στ2 in either constant mean flow in a three‐dimensional formation or radially convergent flow in a heterogeneous two‐dimensional medium. Transverse dispersion between layers in a sedimentary formation is found to increase mixing between flow paths leading to a marginal reduction in travel time variance. Because of negative correlation between mass flux and travel time, solute retrieval at the well occurs earlier from a constant flux solute source than from an instantaneous plume. The relationship between initial plume length and relative plume spreading is illustrated for simple plume geometries.

Bayesian methodology for stochastic capture zone delineation incorporating transmissivity measurements and hydraulic head observations

The paper presents a methodology to estimate the uncertainty in the prediction of capture zones using transmissivity measurements and hydraulic head observations. We use a stochastic approach to parameterise the transmissivity field. By treating the parameters of the stochastic model as random variables we account for the fact that they are unknown and that a simple deterministic process cannot model their genesis. The method requires the definition of a prior probability density function for the mean value and for the covariance parameters of the log transmissivity field. In a first phase, log transmissivity measurements are incorporated into Bayes' theorem updating the prior densities to yield the posterior densities for the model parameters. Conditional realisations of the log transmissivity field are generated using parameter sets obtained by Monte Carlo sampling from the posterior parameter distributions. The second phase consists of updating the posterior probabilities of the conditional log transmissivity fields using the hydraulic head observations through a second application of Bayes' theorem. Then, for each realisation of the log transmissivity field, the capture zone is determined using particle tracking, which is weighted by the posterior probabilities of the respective log transmissivity field. The set of weighted capture zones results in a capture zone probability distribution. We show an application to a hypothetical flow system consisting of a single abstraction well in a confined aquifer under a regional background gradient and compare the worth of log transmissivity measurements and head observations by incorporating the different data types into the procedure sequentially. Results show that head observations are more effective in reducing the spread of the predictive capture zone distribution, whereas the transmissivity measurements are more valuable in predicting the actual location of the true capture zone.

Kinetics of submonolayer epitaxial growth

Molecular beam epitaxy is an important method for growing thin-films and nanostructures. One of the scientific challenges is to understand the fundamental processes that control the evolution of thin film structure and morphology. The results of kinetic Monte Carlo simulations carried out to study the dependence of the submonolayer scaled island-size distribution on the critical island-size are presented and compared with experiments. A recently developed method which involves a self-consistent coupling of evolution equations for the capture-zone distributions with rate-equations for the island-size distribution is also described. Our method explicitly takes into account the existence of a denuded (“capture”) zone around every island and the correlations between the size of an island and the corresponding average capture zone, and has been used to develop a quantitative rate-equation approach to irreversible submonolayer growth on a two-dimensional substrate. The resulting predictions for the capture-zones, capture numbers, and island-size distributions are in excellent agreement with experimental results and kinetic Monte Carlo simulations.

Self-consistent rate equation theory of cluster size distribution in aggregation phenomena

Cluster nucleation and growth by aggregation is the central feature of many physical processes, from polymerization and gelation in polymer science, flocculation and coagulation in aerosol and colloidal chemistry, percolation and coarsening in phase transitions and critical phenomena, agglutination and cell adhesion in biology, to island nucleation and thin-film growth in materials science. Detailed information about the kinetics of aggregation is provided by the time dependent cluster size-distribution, a quantity which can be measured experimentally. While the standard Smoluchowski rate-equation approach has been in general successful in predicting average quantities like the total cluster density, it fails to account for spatial fluctuations and correlations and thus predicts size distributions that are in significant disagreement with both experiments and kinetic Monte Carlo simulations. In this work we outline a new method which takes into account such correlations. We show that by coupling a set of evolution equations for the capture-zone distributions with a set of rate-equations for the island densities one may obtain accurate predictions for the time- and size-dependent rates of monomer capture. In particular, by using this method we obtain excellent results for the capture numbers and island-size distributions in irreversible growth on both one- and two-dimensional substrates.

Rate-equation approach to island size distributions and capture numbers in submonolayer irreversible growth

We present a quantitative rate-equation approach to irreversible submonolayer growth on a two-dimensional substrate. Our method explicitly takes into account the existence of a denuded (``capture'') zone around every island and the correlations between the size of an island and the corresponding average capture zone. The evolution of the capture-zone distributions is described by a set of Voronoi-area evolution equations, which are coupled to the usual rate equations for the island densities through local rates of monomer capture. The combined set of equations leads to a fully self-consistent calculation of the size- and coverage-dependent capture numbers. The resulting predictions for the average capture-zone and capture-number distributions are in excellent agreement with experimental results and Monte Carlo simulations. As a result, the corresponding scaled island size distributions and their dependence on coverage and deposition rate are also accurately predicted in the precoalescence regime.

Rate-equation approach to island capture zones and size distributions in epitaxial growth.

Understanding and predicting the effects of correlations between island size and the rate of monomer capture has been shown to be the central problem in predicting the island-size distribution in submonolayer growth. Here we summarize a method which involves a self-consistent coupling of evolution equations for the capture-zone distributions with rate equations for the island-size distribution. The method has been successfully applied to irreversible submonolayer growth in both one and two dimensions to predict the size-dependent capture numbers and island-size distributions.