Growth kinetics in multicomponent fluids

The hydrodynamic effects on the late-stage kinetics in spinodal decomposition of multicomponent fluids are examined using a lattice Boltzmann scheme with stochastic fluctuations in the fluid and at the interface. In two dimensions, the three- and four-component immiscible fluid mixture (with a 1024² lattice) behaves like an off-critical binary fluid with an estimated domain growth oft ^0.4±0.03 rather thant ^1/3 as previously estimated, showing the significant influence of hydrodynamics. In three dimensions (with a 256³ lattice), we estimate the growth ast ^0.96±0.05 for both critical and off-critical quenches, in agreement with phenomenological theory.     

Interface structure in colored DLA model

We propose a model for the simultaneous diffusion-limited growth of two clusters A and B, where the growth of one cluster screens the growth of the other one. We consider the possibility that the A and B clusters can penetrate into each other in course of their growth in different spatial dimensions and express the conjecture that the A-B boundary is flat in all dimensions. Using an electrostatic analogy, we compute some spatial characteristics of the clusters. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 7, 504–509 (10 October 1996) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Growth of tellurium clusters in presence of metal impurities

The growth of small tellurium clusters in helium and the influence of a metal impurity (dysprosium atoms) on the cluster size distribution are investigated in a double laser vaporization source. A model describing the role of the carrier gas as collision partner is presented, emphasizing the crucial influence of the gas pressure on cluster formation. Changes in cluster reactivity due to dysprosium addition are discussed in terms of ionic structures Dy ^{3 +}(Te _N)^{3 -} containing a radical electron. Received 28 November 2000     

The Birth of the Infinite Cluster:¶Finite-Size Scaling in Percolation

We address the question of finite-size scaling in percolation by studying bond percolation in a finite box of side length n, both in two and in higher dimensions. In dimension d= 2, we obtain a complete characterization of finite-size scaling. In dimensions d>2, we establish the same results under a set of hypotheses related to so-called scaling and hyperscaling postulates which are widely believed to hold up to d= 6. As a function of the size of the box, we determine the scaling window in which the system behaves critically. We characterize criticality in terms of the scaling of the sizes of the largest clusters in the box: incipient infinite clusters which give rise to the infinite cluster. Within the scaling window, we show that the size of the largest cluster behaves like n ^d π _n , where π _n is the probability at criticality that the origin is connected to the boundary of a box of radius n. We also show that, inside the window, there are typically many clusters of scale n ^d π _n , and hence that “the” incipient infinite cluster is not unique. Below the window, we show that the size of the largest cluster scales like ξ ^d π_ξ log(n/ξ), where ξ is the correlation length, and again, there are many clusters of this scale. Above the window, we show that the size of the largest cluster scales like n ^d P _∞, where P _∞ is the infinite cluster density, and that there is only one cluster of this scale. Our results are finite-dimensional analogues of results on the dominant component of the Erdős–Rényi mean-field random graph model. Received: 6 December 2000 / Accepted: 25 May 2001     

Transport in a disordered medium: Analysis and Monte Carlo simulation

We consider a classical stochastic model describing particle transport on a lattice with randomly distributed nearest-neighbor transition rates. Applying an effective medium theory to the model, we determine average properties related to the particle's dynamics ind-dimensions. In particular, we calculate the mean-square displacement, and the fourth moment of the displacement in one-, two- and three dimensions. The results compare favorably with Monte Carlo simulations of the model. We also present preliminary results for the velocity autocorrelation function.An aspect of the bond percolation problem, which is a special case of the stochastic model is investigated; the average inverse cluster size, <N _c ^–1>, is calculated. In one dimension the expression for this quantity is exact and in higher dimensions our results are very accurate not too close to the percolation concentration.     

Dimensionally hybrid Green's functions for impurity scattering in the presence of interfaces

A previous calculation of Green's function (E-H)^-1 for an interface Hamiltonian which interpolates between two and three dimensions is generalized to include scattering from perturbations both on and off the interface.     

Critical acceleration of lattice gauge simulations

We present a stochastic cluster algorithm that drastically reduces critical slowing down forZ ₂ lattice gauge theory in three dimensions. The dynamical exponentz is reduced fromz>2 (standard Metropolis algorithm) tozO.73. The Monte Carlo pseudodynamics acts on the gauge-invariant flux tubes that are known to be the relevant large-scale low-energy excitations. A comparison of our results with known results for the 3D Ising model and ⁴ model supports the conjecture of universality classes for stochastic cluster algorithms.     

Influence of radiation-induced clusters on transport properties of silicon

Summary A calculation method for the scattering cross-section σ of charged carriers on radiation-induced cluster defects has been developed using a spherical cluster model with rectangular potential barrier shape, of radius and height of 15 nm and 0.6 eV, respectively. Values of the cluster cross-section around 2·10⁻¹¹ cm² have been obtained for charged carrier energies from 10⁻⁴ eV to over 600 eV. Applying the relaxation-time approximation of the Boltzmann equation, the influence of clusters on silicon transport properties has been observed to be close to the acoustic-phonon one. The dependence of the Hall factor on radiation-induced clusters has been determined numerically for temperatures ranging from 5 K to 400 K. The results indicate that the presence of clusters of such dimensions would not change significantly the Hall coefficientR _H.     

Scaling exponents for kinetic roughening in higher dimensions

We discuss the results of extensive numerical simulations in order to estimate the scaling exponents associated with kinetic roughening in higher dimensions, up tod=7 + l. To this end, we study the restricted solid-on-solid growth model, for which we employ a novel fitting ansatz for the spatially averaged height correlation function¯G(t)t ² to estimate the scaling exponent. Using this method, we present a quantitative determination of ind=3 + 1 and 4+1 dimensions. To check the consistency of these results, we also compute the interface width and determine andx from it independently. Our results are in disagreement with all existing theories and conjectures, but in four dimensions they are in good agreement with recent simulations of Forrest and Tang for a different growth model. Above five dimensions, we use the time dependence of the width to obtain lower bound estimates for. Within the accuracy of our data, we find no indication of an upper critical dimension up tod = 7 + 1.     

A new cluster model interpretation of the anomalous lifetime of <Superscript>14</Superscript>C

A new cluster model solution to the long-standing nuclear structure problem of describing the anomalously long lifetime of ¹⁴C is presented. Related beta-decay data for ¹⁴O to states in ¹⁴N, gamma-decay data between low-lying positive parity states in ¹⁴N and the elastic and inelastic magnetic dipole electron scattering from ¹⁴N data are all shown to be very accurately described by the model. The shapes of the beta spectra for the A = 14 system are also well reproduced by the model. The model invokes four-nucleon tetrahedral symmetric spatial correlations arising from three- and four-nucleon interactions, which yields a high degree of SU(4) singlet structure for the clusters and a tetrahedral intrinsic shape for the doubly magic ¹⁶O ground state. The large quadrupole moment of the ¹⁴N ground state is obtained here for the first time and arises because of the almost 100% d-wave deuteron-like-hole cluster structure inherent in the model.     

Fluctuations in the Composite Regime of a Disordered Growth Model

 We continue to study a model of disordered interface growth in two dimensions. The interface is given by a height function on the sites of the one-dimensional integer lattice and grows in discrete time: (1) the height above the site x adopts the height above the site to its left if the latter height is larger, (2) otherwise, the height above x increases by 1 with probability p _x . We assume that p _x are chosen independently at random with a common distribution F, and that the initial state is such that the origin is far above the other sites. Provided that the tails of the distribution F at its right edge are sufficiently thin, there exists a nontrivial composite regime in which the fluctuations of this interface are governed by extremal statistics of p _x . In the quenched case, the said fluctuations are asymptotically normal, while in the annealed case they satisfy the appropriate extremal limit law. Received: 6 November 2001 / Accepted: 8 April 2002 Published online: 6 August 2002     

Cluster growth poised on the edge of break-up: Size distributions with small exponent power-laws

In many naturally occurring growth processes, cluster size distributions of power-law form n(s)∝s^−τ with small exponents 0<τ<1 are observed. We suggest here that such distributions emerge naturally from cluster growth, where size dependent aggregation is counterbalanced by size dependent break-up. The model used in the study is a simple reaction kinetic model including only monomer-cluster processes. It is shown that under such conditions power-law size distributions with small exponents are obtained. Therefore, the results suggest that the ubiquity of small exponent power-law distributions is related to the growth process, where aggregation driven cluster growth is poised on the edge of cluster break-up.     

Phase Separation in Random Cluster Models I: Uniform Upper Bounds on Local Deviation

This is the first in a series of three papers that addresses the behaviour of the droplet that results, in the percolating phase, from conditioning the planar Fortuin-Kasteleyn random cluster model on the presence of an open dual circuit Γ₀ encircling the origin and enclosing an area of at least (or exactly) n ². (By the Fortuin-Kasteleyn representation, the model is a close relative of the droplet formed by conditioning the Potts model on an excess of spins of a given type.) We consider local deviation of the droplet boundary, measured in a radial sense by the maximum local roughness, MLR(Γ₀), this being the maximum distance from a point in the circuit Γ₀ to the boundary ∂ of the circuit’s convex hull; and in a longitudinal sense by what we term maximum facet length, MLF(Γ₀), namely, the length of the longest line segment of which the polygon ∂ is formed. The principal conclusion of the series of papers is the following uniform control on local deviation: that there are constants 0 < c < C < ∞ such that the conditional probability that the normalized quantity n ^−1/3(log n )^−2/3MLR lies in the interval [c, C] tends to 1 in the high n-limit; and that the same statement holds for n ^−2/3 (log n )^−1/3 MLF. In this way, we confirm the anticipated n ^1/3 scaling of maximum local roughness, and provide a sharp logarithmic power-law correction. This local deviation behaviour occurs by means of locally Gaussian effects constrained globally by curvature, and we believe that it arises in many radially defined stochastic interface models, including growth models belonging to the Kardar-Parisi-Zhang universality class.     

Real-time icosahedral to fcc structure transition during CoPt nanoparticles formation

Nucleation and growth of supported CoPt nanoparticles were studied in situ and in real time by combined grazing incidence small-angle x-ray scattering (GISAXS) and grazing incidence x-ray diffraction (GIXD). GISAXS provides morphological features of nanoparticles as a function of size, shape and correlation distance between particles, while GIXD allows the determination of the atomic structure. We focus on the formation of ultrasmall CoPt nanoparticles, in the 1–4 nm size range at 500^○C. The structural analysis method based on the Debye equation is coupled with cluster model calculations performed by Monte Carlo simulations using a semi-empirical tight-binding potential to interpret diffraction spectra and structural transitions. Our results show that the cluster structure evolution during the growth is size-dependent and composition-dependent, yielding an icosahedral to fcc structure transition.     

On the local structure of the phase separation line in the two-dimensional Ising system

We investigate the structure of the phase separation line between the pure phases in the two-dimensional Ising model, the liquid and vapor phase in lattice gas language, at low temperatures. The fluctuations in the location of this line are known to diverge in the thermodynamic limit, something which is also believed to happen to the continuum liquid-vapor interface in three dimensions (in the absence of the gravitational field). We show that despite this global divergence it is possible to define precisely the local structure of the phase separation line. This has a finite, exponentially small, width at low temperatures which is related by a central limit theorem⁽¹⁾ to the width of the global fluctuations on the appropriate (divergent) length scale. The latter has been computed explicitly⁽²⁾ for all temperatures below the critical temperatureT _c, where it diverges as (T _c –T)^–1/2. We also prove a Gibbs formula for the surface tension at low temperature, which relates it to the local structure of the phase separation line.Supported in part by NSF grant No. M^rPHY 78-15920 and MCS78-01885.On leave from: Departement de Physique Théorique, Université de Louvain, Belgium.     

Nucleation and stability of defect clusters: New energetic model in the case of substituted wustites

Abstract

For wüstite Fe_1?z O (z < 0.08) an energetic model accounts for the stability of cubic defect clusters (m/n) which are partly ordered in the crystal. The Gibbs energy G_T (N) associated with clusters, including their distorted envelope, is expressed as a sum of a volume term in N ³ and of surface terms in N ⁴; N is the number of bonds characteristic of the cluster size. In the case of a (10/4) type cluster, this energy is negative and minimum for N_m ranging between 4 and 5, when the volume and surface energies range between specific values. Using simple assumptions, a volume energy ?0.80 eV per vacancy is found in accordance with the value of stabilization energy calculated by theorists for the (10/4) cluster. The substitution of Fe²⁺ by Ca²⁺ should lead to a decrease of cluster size; this has been recently suggested by neutron diffraction studies.     

Cubic potential models for cluster radioactivity

Cluster radioactivity is a process by which nuclei equal and heavier than the α-particle is emitted spontaneously. The clusters usually emitted in this process are the α-particle, carbon, oxygen, neon, magnesium, silicon etc. When the mass of the cluster becomes comparable with the mass of the daughter, symmetric fission takes place. Thus the cluster radioactivity is an intermediate process between the well known α-decay and the spontaneous fission. In earlier years such cluster radioactivity was found mostly in actinide nuclei like radium, uranium etc. Very recently it has been predicted that such decays are possible in a new region around ¹¹¹Ba. There has been an exciting experimental detection of the emission of ¹²C from ¹¹¹Ba leading to ¹⁰²Sn, which is attracting a lot of attention recently. To study the phenomenon of cluster radioactivity there are various theoretical models in vogue. The existing models generally fall under two categories: the unified fission model (UFM) and the preformed cluster model (PCM). The physics of the UFM and the PCM are completely different. The UFM considers cluster radioactivity simply as a barrier penetration phenomenon in between the fission and the α-decay without worrying about the cluster being or not being preformed in the parent nucleus. In the PCM clusters are assumed to be preborn in a parent nucleus before they could penetrate the potential barrier with a given Q-value. The basic assumption of the UFM is that heavy clusters as well as the α-particle have equal probability of being preformed. In PCM, clusters of different sizes have different probabilities of their being preformed in the parent nucleus. We have developed three fission models during the last decade using the cubic potential for the pre-scission region. The use of these models in the study of cluster radioactivity in both the actinide and barium regions will be discussed in this talk in comparison with the other existing theories.     

Internal Diffusion-Limited Aggregation: Parallel Algorithms and Complexity

The computational complexity of internal diffusion-limited aggregation (DLA) is examined from both a theoretical and a practical point of view. We show that for two or more dimensions, the problem of predicting the cluster from a given set of paths is complete for the complexity class CC, the subset of P characterized by circuits composed of comparator gates. CC-completeness is believed to imply that, in the worst case, growing a cluster of size n requires polynomial time in n even on a parallel computer. A parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths, and then corrects defects in the guessed cluster through a nonlocal annihilation process. The parallel running time of the relaxation algorithm for two-dimensional internal DLA is studied by simulating it on a serial computer. The numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n. Thus the computational resources needed to grow large clusters are significantly less on average than the worst-case analysis would suggest. For a parallel machine with k processors, we show that random clusters in d dimensions can be generated in ((n/k+logk)n ^2/d ) steps. This is a significant speedup over explicit sequential simulation, which takes (n ^1+2/d ) time on average. Finally, we show that in one dimension internal DLA can be predicted in (logn) parallel time, and so is in the complexity class NC.