Cluster Perimeter Research Articles

Macrostructural compartmentation in rat renal inner medulla

Three-dimensional architecture of vasculature and nephron segments in rat renal inner medulla (IM) was assessed with digital reconstructions from physical sections. Descending vasa recta (DVR), ascending vasa recta (AVR), descending thin limbs (DTLs), ascending thin limbs (ATLs), and collecting ducts (CDs) were identified with antibodies against segment-specific proteins associated with solute and water transport (UT-B, PV-1, AQP1, ClC-K1, and AQP2) by indirect immunofluorescence. Countercurrent exchange by IM vasculature sustains interstitial solute gradients. However, exchange occurs only at the perimeter of CD clusters since DVR are absent within these CD clusters. A symmetrical set of typically four fenestrated, branched or unbranched AVR abuts each CD at nearly all levels of the IM. AVR adhere to each CD continuously for as much as one mm in length or more. 54% of the CD surface area abuts AVR. Electron microscopy shows close contact between AVR and CDs at adherence areas. Microvilli arising from AVR endothelia connect to CDs at these areas. ATLs are distributed nearly uniformly amongst the CDs within each CD cluster. ATLs and AVR are sufficiently close together to form interstitial compartments delimited by AVR, ATLs and CDs. This architectural arrangement and apparent isolation of these compartments from outlying DTLs and DVR raise questions regarding their function. For one, it raises the possibility that lateral solute diffusion from ATLs and CDs and diffusion into AVR could be preferentially restricted to these areas. An additional factor increasing and defining compartmentation, and thereby restricting diffusive exchange, would be interstitial cell architecture. (NIH DK16294)

Simulation of damage accumulation kinetics with a probabilistic cellular automaton

The kinetics of damage accumulation and the evolution of damage clusters in loaded materials are simulated with a probabilistic cellular automaton controlled by three probabilities, namely, by the probabilities of free-cell occupation, cluster perimeter growth, and coalescence of clusters separated by a critical distance. The automaton algorithm is realized with Microsoft Visual Basic 6.0 as a single-document Windows application connected with Microsoft Excel, which is used as an automaton client to save and process output data. The operation of the automaton is illustrated by the example of the kinetics of damage accumulation and the evolution of a damage cluster structure compared for two simulation scenarios.

Thiophene hydrodesulfurization activity of MoS2/Al2O3catalysts prepared via exfoliation. The effect of cobalt

In contrast to the bulk MoS2, the structure of the active phase of the catalyst prepared via exfoliation is shown to exhibit a number of distortions, which form a superstructure of the type of the charge density waves with a quasi-period of 30 A. Due to such distortions, the ions of cobalt firmly chemisorb on the basal plane of MoS2 occupying, along the perimeter of the MoS2 clusters, the regions with a lower energy of the Mo-S interaction. A synergetic increase in the catalyst activity due to the introduction of cobalt was observed up to the atomic ratios of Co/(Co+Mo) = 0.33, which is possible for the nanoparticles of MoS2 with a diameter 200 A, only if cobalt adsorbs on the basal plane.

Surface-state mediated three-adsorbate interaction: electronic nature and nanoscale consequences

The interaction energy of a three-adsorbate cluster on a surface consists of the sum of pair interactions plus a trio contribution. The trio interaction is produced by interference of electrons which propagate around the cluster perimeter, d 123. We investigate such interactions for a noble-metal (1 1 1) surface, where the isotropic Shockley surface-state band produces cluster-interaction energies significant for the low-temperature adsorbate dynamics. We provide experimentally testable interaction-energy predictions, specified by the s-wave phase shift δ F that characterizes the standing-wave patterns seen in scanning-tunneling microscopy (STM) images. Compared with the adsorbate-pair interactions, the trio contribution exhibits a slightly weaker amplitude and a slightly faster asymptotic decay. We discuss how the trio interaction can affect the cluster geometry by introducing an additional angular dependence in the adsorbate interaction.

Cluster size distributions: signatures of self-organization in spatial ecologies.

Three different lattice-based models for antagonistic ecological interactions, both nonlinear and stochastic, exhibit similar power-law scalings in the geometry of clusters. Specifically, cluster size distributions and perimeter-area curves follow power-law scalings. In the coexistence regime, these patterns are robust: their exponents, and therefore the associated Korcak exponent characterizing patchiness, depend only weakly on the parameters of the systems. These distributions, in particular the values of their exponents, are close to those reported in the literature for systems associated with self-organized criticality (SOC) such as forest-fire models; however, the typical assumptions of SOC need not apply. Our results demonstrate that power-law scalings in cluster size distributions are not restricted to systems for antagonistic interactions in which a clear separation of time-scales holds. The patterns are characteristic of processes of growth and inhibition in space, such as those in predator-prey and disturbance-recovery dynamics. Inversions of these patterns, that is, scalings with a positive slope as described for plankton distributions, would therefore require spatial forcing by environmental variability.

Anomalous dynamic scaling in locally conserved coarsening of fractal clusters

We report two-dimensional phase-field simulations of locally conserved coarsening dynamics of random fractal clusters with fractal dimension D=1.7 and 1.5. The correlation function, cluster perimeter, and solute mass are measured as functions of time. Analyzing the correlation function dynamics, we identify two different time-dependent length scales that exhibit power laws in time. The exponents of these power laws do not show any dependence on D; one of them is apparently the "classical" exponent 1/3. The solute mass versus time exhibits dynamic scaling with a D-dependent exponent, in agreement with a simple scaling theory.

Geometrical Properties of Gel and Fluid Clusters in DMPC/DSPC Bilayers: Monte Carlo Simulation Approach Using a Two-State Model

In this paper the geometrical properties of gel and fluid clusters of equimolar dimyristoylphosphatidylcholine/distearoylphosphatidylcholine (DMPC/DSPC) lipid bilayers are calculated by using an Ising-type model (Sugar, I. P., T. E. Thompson, and R. L. Biltonen. 1999. Biophys. J. 76:2099–2110). The model is able to predict the following properties in agreement with the respective experimental data: the excess heat capacity curves, fluorescence recovery after photobleaching (FRAP) threshold temperatures at different mixing ratios, the most frequent center-to-center distance between DSPC clusters, and the fractal dimension of gel clusters. In agreement with the neutron diffraction and fluorescence microscopy data, the simulations show that below the percolation threshold temperature of gel clusters many nanometer-size gel clusters co-exist with one large gel cluster of size comparable with the membrane surface area. With increasing temperature the calculated effective fractal dimension and capacity dimension of gel and fluid clusters decrease and increase, respectively, within the (0, 2) interval. In the region of the gel-to-fluid transition the following geometrical properties are independent from the temperature and the state of the cluster: 1) the cluster perimeter linearly increases with the number of cluster arms at a rate of 8.2 nm/arm; 2) the average number of inner islands in a cluster increases with increasing cluster size, S, according to a power function of 0 .00427 × S 1 .3 ; 3) the following exponential function describes the average size of an inner island versus the size of the host cluster, S: 1 + 1 .09(1 − e −0 .0072 × S). By means of the equations describing the average geometry of the clusters the process of the association of clusters is investigated.

Mode transition between growth and decomposition of oxides on Si(001): Kinetically determined critical coverage for oxidation

Effects of preoxidation on the reaction kinetics of oxygen molecules at Si(001) surface have been investigated by real-time ultraviolet photoelectron spectroscopy. A mode transition from decomposition to growth of surface oxides was found to exist at a certain initial oxide coverage, which is kinetically, not energetically, determined. By considering a change of balance between Si adatom and oxygen-monomer fluxes at the perimeter of oxide clusters, this mode transition is quantitatively described as a bifurcation of an autocatalytic-reaction rate equation.

Geometrical multifractality of the perimeter of DLA clusters

The geometrical multifractality of diffusion-limited aggregation (DLA) clusters is investigated by evaluating the D q spectrum for q⩾0 using the standard box-counting technique. Using the cluster points themselves as input to the algorithm, deviations were found from the expected multifractal scaling. However on examining the geometric scaling properties of the cluster perimeter, such deviations were found to be significantly reduced, thus allowing a reliable D q spectrum to be calculated.

Path-Crossing Exponents and the External Perimeter in 2D Percolation

2D Percolation path exponents $x^{\cal P}_{\ell}$ describe probabilities for traversals of annuli by $\ell$ non-overlapping paths, each on either occupied or vacant clusters, with at least one of each type. We relate the probabilities rigorously to amplitudes of $O(N=1)$ models whose exponents, believed to be exact, yield $x^{\cal P}_{\ell}=({\ell}^2-1)/12$. This extends to half-integers the Saleur--Duplantier exponents for $k=\ell/2$ clusters, yields the exact fractal dimension of the external cluster perimeter, $D_{EP}=2-x^{\cal P}_3=4/3$, and also explains the absence of narrow gate fjords, as originally found by Grossman and Aharony.

Diffusion of small clusters on metal (100) surfaces: Exact master-equation analysis for lattice-gas models

Exact results are presented for the surface diffusion of small two-dimensional clusters, the constituent atoms of which are commensurate with a square lattice of adsorption sites. Cluster motion is due to the hopping of atoms along the cluster perimeter with various rates. We apply the formalism of Titulaer and Deutch [J. Chem. Phys. {bold 77}, 472 (1982)], which describes evolution in reciprocal space via a linear master equation with dimension equal to the number of cluster configurations. We focus on the regime of rapid hopping of atoms along straight close-packed edges, where certain subsets of configurations cycle rapidly between each other. Each such subset is treated as a single quasiconfiguration, thereby reducing the dimension of the evolution equation, simplifying the analysis, and elucidating limiting behavior. We also discuss the influence of concerted atom motions on the diffusion of tetramers and larger clusters. {copyright} {ital 1999} {ital The American Physical Society}

Kinetics of the total cluster perimeter in thin film nucleation on solid surfaces

The time evolution of the total perimeter of clusters growing on a surface has been described on the basis of the JMAK (Johnson-Mehl-Avrami-Kolmogoroff) statistical theory. A general formula, which can be easily extended to any space dimension, is obtained. When particular nucleation functions and the cluster growth law are considered the kinetics of the perimeter can be explicitly calculated and moreover, it can be expressed as a function of the covered surface. Experimental data on the efficiency of a Cu/CuO x model catalyst, towards imide formation, have been satisfactorily described by the model. Moreover, the growth of Ag on a GaAs(110) surface studied via photoelectron spectroscopy has been qualitatively explained by the proposed model. In the model case of cylindrical clusters the knowledge of the evolution of the total perimeter allows the entire area of the film to be evaluated.

Mechanism for diffusion of nanostructures and mesoscopic objects on surfaces

We have studied the diffusion of mesoscopic objects created on surfaces by applying time rescaling Monte Carlo algorithms. It is shown that under the effect of very small forces (0.01–0.03 eV/lattice constant), these nanostructures reach large velocities and long displacements are observable. The main mechanism involved in this displacement is the fast movement of the particles at the cluster perimeter, i.e. at an adequate temperature, for instance Au at room temperature, Au atoms move very fast along the cluster periphery and the application of a small driving force gives rise to the displacement of the object as a whole. Atomic metallic ion emission (AMIE) experiments are presented and explained in terms of our MC results.

Cluster hull algorithms for large systems with small memory requirement

Improved hull walking algorithms for two-dimensional percolation are proposed. In these algorithms a walker explores the external perimeter of percolation clusters. With our modifications very large systems (sizeL) can be studied with finite and small memory requirement and in computation time τ∼L7/4. Applications in determining spanning probabilities, continuum percolation, and percolation with nonuniform occupation probability are pointed out.

Introduction to Percolation Theory

Preface to the Second Edition Preface to the First Edition Introduction: Forest Fires, Fractal Oil Fields, and Diffusion What is percolation? Forest fires Oil fields and fractals Diffusion in disordered media Coming attractions Further reading Cluster Numbers The truth about percolation Exact solution in one dimension Small clusters and animals in d dimensions Exact solution for the Bethe lattice Towards a scaling solution for cluster numbers Scaling assumptions for cluster numbers Numerical tests Cluster numbers away from Pc Further reading Cluster Structure Is the cluster perimeter a real perimeter? Cluster radius and fractal dimension Another view on scaling The infinite cluster at the threshold Further reading Finite-size Scaling and the Renormalization Group Finite-size scaling Small cell renormalization Scaling revisited Large cell and Monte Carlo renormalization Connection to geometry Further reading Conductivity and Related Properties Conductivity of random resistor networks Internal structure of the infinite cluster Multitude of fractal dimensions on the incipient infinite cluster Multifractals Fractal models Renormalization group for internal cluster structure Continuum percolation, Swiss-cheese models and broad distributions Elastic networks Further reading Walks, Dynamics and Quantum Effects Ants in the labyrinth Probability distributions Fractons and superlocalization Hulls and external accessible perimeters Diffusion fronts Invasion percolation Further reading Application to Thermal Phase Transitions Statistical physics and the Ising model Dilute magnets at low temperatures History of droplet descriptions for fluids Droplet definition for the Ising model in zero field The trouble with Kertesz Applications Dilute magnets at finite temperatures Spin glasses Further reading Summary Numerical Techniques

Nucleation and Growth of Adsorbed Layer Self‐Consistent Approach Based on Kolmogoroff‐Avrami Model

AbstractThe kinetic model for the growth of adsorbed monolayer is developed in order to describe the growth process beginning from the initial stage of nucleation up to the formation of a continuous film. The nucleation of clusters on the uncovered substrate region is studied within the frame of kinetic theory of condensation. Direct coalescence of clusters is described by means of Kolmogoroff‐Avrami model for two‐dimensional crystallization. The basic self‐consistent equation for the dynamics of adatom supersaturation is derived. Numerical solutions for supersaturation, mean substrate coverage, boundary perimeter, and total number of clusters are obtained. Depending on the numerical value of the governing parameter, three known experimental growth modes are determined. In the regime of incomplete condensation the analytical solutions for the basic kinetic and morphological characteristics of the growing layer are obtained, which are in agreement with numerical results.

Scaling of Overhang Distribution of Invasion Percolation Fronts

We analyse simulations of invasion percolation with a gradient in two dimensions. The fronts of the invaded structures are examined using the overhang size distribution introduced by Hansen et al. [1]. We observe that overhang sizes h are distributed according to a power law: nh ~ h-a where a is measured to be a = 2.3 ± 0.1. The fractal dimension of the external perimeter of invasion percolation clusters is known to be De ≃ 1.35. We argue that in general a = Da + 1, where Da is the fractal dimension of the perimeter sampled by particles coming from outside the cluster and moving in straight lines paralles to the gradient and the overhangs.

Crossover from standard to reduced hull for random percolation.

The hull (external perimeter) of percolation clusters in two dimensions is a fractal object with fractal dimension DH=74 or DH\ensuremath{'}=43 depending on how it is constructed. Here it is shown that to each hull construction there corresponds a characteristic length R* such that the fractal dimension of the hull is DH on length scales smaller than R* and DH\ensuremath{'} on length scales larger than R*. If r0 is the elementary length used for constructing the percolation cluster and r (>r0) is the elementary length used for constructing the hull, then R* diverges as (r-r0)-a as r\ensuremath{\rightarrow}r0, with a\ensuremath{\cong}1.37.

Probability density of the 2D percolation cluster perimeter

The authors study the form of the probability density P(r, t) of the end-to-end distance r of a segment of length t of the perimeter of the 2D percolation cluster at criticality. They find that P(r, t) behaves as P(r, t) varies as t-2dH/ exp(-(r/t1dH/)delta ), where delta =1/(1-1/dH). The exponent dH=7/4 is the fractal dimension of the hull (the external perimeter of a percolation cluster). This result fits very well with recent numerical data.

On growth walks with self-avoiding constraints

The authors study a general class of growth walks in two dimensions with a self-avoiding constraint, which can be considered as 'dressed' self-avoiding walks. A walker travels on a square lattice and chooses at each step one of the three forward directions with probability p1, p2 and p3 if the directions are allowed. The self-avoiding constraint is obtained by 'dressing' the walk with 'black' and 'white' particles after each step such that the walk generates the external perimeter of arbitrary clusters consisting of black or white particles. They discuss the critical values of p1, p2 and p3 which allow for infinite walks and study the end-to-end distance r(t) and the corresponding distribution function N(r, t). Furthermore, they introduce a bias field in growth process and discuss how the time evolution of the walk is modified by the bias.