Nonscaling and source-induced scaling behaviour in aggregation model of movable monomers and immovable clusters

Supercluster states and phase transitions in aggregation-fragmentation processes

We study the evolution of aggregates triggered by collisions with monomers that either lead to the attachment of monomers or the break-up of aggregates into constituting monomers. Depending on parameters quantifying addition and break-up rates, the system falls into a jammed or a steady state. Supercluster states (SCSs) are very peculiar nonextensive jammed states that also arise in some models. Fluctuations underlie the formation of the SCSs. Conventional tools, such as the van Kampen expansion, apply to small fluctuations. We go beyond the van Kampen expansion and determine a set of critical exponents quantifying SCSs. We observe continuous and discontinuous phase transitions between the states. Our theoretical predictions are in good agreement with numerical results.

Finite-size effects in addition and chipping processes

We investigate analytically and numerically a system of clusters evolving via collisions with clusters of minimal mass (monomers). Each collision either leads to the addition of the monomer to the cluster or the chipping of a monomer from the cluster, and emerging behaviors depend on which of the two processes is more probable. If addition prevails, monomers disappear in a time that scales as lnN with the total mass N≫1, and the system reaches a jammed state. When chipping prevails, the system remains in a quasistationary state for a time that scales exponentially with N, but eventually, a giant fluctuation leads to the disappearance of monomers. In the marginal case, monomers disappear in a time that scales linearly with N, and the final supercluster state is a peculiar jammed state; i.e., it is not extensive.

Nonextensive Supercluster States in Aggregation with Fragmentation

Systems evolving through aggregation and fragmentation may possess an intriguing supercluster state (SCS). Clusters constituting this state are mostly very large, so the SCS resembles a gelling state, but the formation of the SCS is controlled by fluctuations and in this aspect, it is similar to a critical state. The SCS is nonextensive, that is, the number of clusters varies sublinearly with the system size. In the parameter space, the SCS separates equilibrium and jamming (extensive) states. The conventional methods, such as, e.g., the van Kampen expansion, fail to describe the SCS. To characterize the SCS we propose a scaling approach with a set of critical exponents. Our theoretical findings are in good agreement with numerical results.

Hopf bifurcation in addition-shattering kinetics

In aggregation-fragmentation processes, a steady state is usually reached. This indicates the existence of an attractive fixed point in the underlying infinite system of coupled ordinary differential equations. The next simplest possibility is an asymptotically periodic motion. Never-ending oscillations have not been rigorously established so far, although oscillations have been recently numerically detected in a few systems. For a class of addition-shattering processes, we provide convincing numerical evidence for never-ending oscillations in a certain region U of the parameter space. The processes which we investigate admit a fixed point that becomes unstable when parameters belong to U and never-ending oscillations effectively emerge through a Hopf bifurcation.

Steady oscillations in aggregation-fragmentation processes

We report surprising steady oscillations in aggregation-fragmentation processes. Oscillating solutions are observed for the class of aggregation kernels K_{i,j}=i^{ν}j^{μ}+j^{ν}i^{μ} homogeneous in masses i and j of merging clusters and fragmentation kernels, F_{ij}=λK_{ij}, with parameter λ quantifying the intensity of the disruptive impacts. We assume a complete decomposition (shattering) of colliding partners into monomers. We show that an assumption of a steady-state distribution of cluster sizes, compatible with governing equations, yields a power law with an exponential cutoff. This prediction agrees with simulation results when θ≡ν-μ<1. For θ=ν-μ>1, however, the densities exhibit an oscillatory behavior. While these oscillations decay for not very small λ, they become steady if θ is close to 2 and λ is very small. Simulation results lead to a conjecture that for θ<1 the system has a stable fixed point, corresponding to the steady-state density distribution, while for any θ>1 there exists a critical value λ_{c}, such that for λ<λ_{c}, the system has an attracting limit cycle. This is rather striking for a closed system of Smoluchowski-like equations, lacking any sinks and sources of mass.

Stationary mass distribution and nonlocality in models of coalescence and shattering

We study the asymptotic properties of the steady state mass distribution for a class of collision kernels in an aggregation-shattering model in the limit of small shattering probabilities. It is shown that the exponents characterizing the large and small mass asymptotic behavior of the mass distribution depend on whether the collision kernel is local (the aggregation mass flux is essentially generated by collisions between particles of similar masses) or nonlocal (collision between particles of widely different masses give the main contribution to the mass flux). We show that the nonlocal regime is further divided into two subregimes corresponding to weak and strong nonlocality. We also observe that at the boundaries between the local and nonlocal regimes, the mass distribution acquires logarithmic corrections to scaling and calculate these corrections. Exact solutions for special kernels and numerical simulations are used to validate some nonrigorous steps used in the analysis. Our results show that for local kernels, the scaling solutions carry a constant flux of mass due to aggregation, whereas for the nonlocal case there is a correction to the constant flux exponent. Our results suggest that for general scale-invariant kernels, the universality classes of mass distributions are labeled by two parameters: the homogeneity degree of the kernel and one further number measuring the degree of the nonlocality of the kernel.

Phase transitions in systems with aggregation and shattering

We consider a system of clusters made of elementary building blocks, monomers, and evolving via collisions between diffusing monomers and immobile composite clusters. In our model, the cluster-monomer collision can lead to the attachment of the monomer to the cluster (addition process) or to the total breakup of the cluster (shattering process). A phase transition, separating qualitatively different behaviors, occurs when the probability of shattering events exceeds a certain threshold. The novel feature of the phase transition is the dramatic dependence on the initial conditions.

First passage times in homogeneous nucleation: Dependence on the total number of particles

Motivated by nucleation and molecular aggregation in physical, chemical, and biological settings, we present an extension to a thorough analysis of the stochastic self-assembly of a fixed number of identical particles in a finite volume. We study the statistics of times required for maximal clusters to be completed, starting from a pure-monomeric particle configuration. For finite volumes, we extend previous analytical approaches to the case of arbitrary size-dependent aggregation and fragmentation kinetic rates. For larger volumes, we develop a scaling framework to study the first assembly time behavior as a function of the total quantity of particles. We find that the mean time to first completion of a maximum-sized cluster may have a surprisingly weak dependence on the total number of particles. We highlight how higher statistics (variance, distribution) of the first passage time may nevertheless help to infer key parameters, such as the size of the maximum cluster. Finally, we present a framework to quantify formation of macroscopic sized clusters, which are (asymptotically) very unlikely and occur as a large deviation phenomenon from the mean-field limit. We argue that this framework is suitable to describe phase transition phenomena, as inherent infrequent stochastic processes, in contrast to classical nucleation theory.

Kinetics of self-assembly via facilitated diffusion: Formation of the transcription complex

We present an analytically solvable model for self-assembly of a molecular complex on a filament. The process is driven by a seed molecule that undergoes facilitated diffusion, which is a search strategy that combines diffusion in three dimensions and one dimension. Our study is motivated by single-molecule-level observations revealing the dynamics of transcription factors that bind to the deoxyribonucleic acid at early stages of transcription. We calculate the probability that a complex made up of a given number of molecules is completely formed, as well as the distribution of completion times, upon the binding of a seed molecule at a target site on the filament (without explicitly modeling the three-dimensional diffusion that precedes binding). We compare two different mechanisms of assembly where molecules bind in sequential and random order. Our results indicate that while the probability of completion is greater for random binding, the completion time scales exponentially with the size of the complex; in contrast, it scales as a power law or slower for sequential binding, asymptotically. Furthermore, we provide model predictions for the dissociation and residence times of the seed molecule, which are observables accessible in single-molecule tracking experiments.

Cluster-size dynamics: A phenomenological model for the interaction between coagulation and fragmentation processes

We present a novel phenomenological modeling approach to describe the growth of clusters as the result of the interaction between cluster coagulation and fragmentation. The cluster-size growth (CSG) model tracks the evolution of cluster-sizes rather than the concentrations of clusters with different sizes as in the Smoluchowski and Becker-Döring coagulation-fragmentation equations. Our modeling perspective allows for a description of cluster growth in realistic systems by using a significantly smaller number of differential equations to describe their dynamics. We used dynamical system tools (phase-plane analysis) and numerical simulations to investigate the CSG model dynamics and to understand how the model parameters describing the coagulation and fragmentation processes determine balances between these two processes that create non-zero stationary cluster size distributions.

Driven Brownian coagulation of polymers

We present an analysis of the mean-field kinetics of Brownian coagulation of droplets and polymers driven by input of monomers which aims to characterize the long time behavior of the cluster size distribution as a function of the inverse fractal dimension, a, of the aggregates. We find that two types of long time behavior are possible. For 0≤a<1/2 the size distribution reaches a stationary state with a power law distribution of cluster sizes having exponent 3/2. The amplitude of this stationary state is determined exactly as a function of a. For 1/2<a≤1, the cluster size distribution never reaches a stationary state. Instead a bimodal distribution is formed in which a narrow population of small clusters near the monomer scale is separated by a gap (where the cluster size distribution is effectively zero) from a population of large clusters which continue to grow for all time by absorbing small clusters. The marginal case, a=1/2, is difficult to analyze definitively, but we argue that the cluster size distribution becomes stationary and there is a logarithmic correction to the algebraic tail.

Instantaneous gelation in Smoluchowski's coagulation equation revisited

We study the solutions of the Smoluchowski coagulation equation with a regularization term which removes clusters from the system when their mass exceeds a specified cutoff size, M. We focus primarily on collision kernels which would exhibit an instantaneous gelation transition in the absence of any regularization. Numerical simulations demonstrate that for such kernels with monodisperse initial data, the regularized gelation time decreases as M increases, consistent with the expectation that the gelation time is zero in the unregularized system. This decrease appears to be a logarithmically slow function of M, indicating that instantaneously gelling kernels may still be justifiable as physical models despite the fact that they are highly singular in the absence of a cutoff. We also study the case when a source of monomers is introduced in the regularized system. In this case a stationary state is reached. We present a complete analytic description of this regularized stationary state for the model kernel, K(m(1),m(2)) = max{m(1),m(2)}(ν), which gels instantaneously when M → ∞ if ν>1. The stationary cluster size distribution decays as a stretched exponential for small cluster sizes and crosses over to a power law decay with exponent ν for large cluster sizes. The total particle density in the stationary state slowly vanishes as [(ν-1)log M](-1/2) when M → ∞. The approach to the stationary state is nontrivial: Oscillations about the stationary state emerge from the interplay between the monomer injection and the cutoff, M, which decay very slowly when M is large. A quantitative analysis of these oscillations is provided for the addition model which describes the situation in which clusters can only grow by absorbing monomers.

In situ liquid-cell electron microscopy of colloid aggregation and growth dynamics

We report on real-time observations of the aggregation of gold nanoparticles using a custom-made liquid cell that allows for in situ electron microscopy. Process kinetics and fractal dimension of the aggregates are consistent with three-dimensional cluster-cluster diffusion-limited aggregation, even for large aggregates, for which confinement effects are expected. This apparent paradox was resolved through in situ observations of the interactions between individual particles as well as clusters at various stages of the aggregation process that yielded the large aggregates. The liquid cell described herein facilitates real-time observations of various processes in liquid media with the high resolution of the electron microscope.

Progress of nanocrystalline growth kinetics based on oriented attachment

The crystal growth mechanism, kinetics, and microstructure development play a fundamental role in tailoring the materials with controllable sizes and morphologies. The classical crystal growth kinetics-Ostwald ripening (OR) theory is usually used to explain the diffusion-controlled crystal growth process, in which larger particles grow at the expense of smaller particles. In nanoscale systems, another significant mechanism named "oriented attachment (OA)" was found, where nanoparticles with common crystallographic orientations directly combine together to form larger ones. Comparing with the classical atom/molecular-mediated crystallization pathway, the OA mechanism shows its specific characteristics and roles in the process of nanocrystal growth. In recent years, the OA mechanism has been widely reported in preparing low-dimension nanostructural materials and reveals remarkable effects on directing and mediating the self-assembly of nanocrystals. Currently, the interests are more focused on the investigation of its role rather than the comprehensive insight of the mechanism and kinetics. The inner complicacy of crystal growth and the occurrence of coexisting mechanisms lead to the difficulty and lack of understanding this growth process by the OA mechanism.In this context, we review the progress of the OA mechanism and its impact on materials science, and especially highlight the OA-based growth kinetics aiming to achieve a further understanding of this crystal growth route. To explore the OA-limited growth, the influence of the OR mechanism needs to be eliminated. The introduction of strong surface adsorption was reported as the effective solution to hinder OR from occurring and facilitate the exclusive OA growth stage. A detailed survey of the nanocrystal growth kinetics under the effect of surface adsorption was presented and summarized. Moreover, the development of OA kinetic models was systematically generalized, in which the "molecular-like" kinetic models were built to take the OA nanocrystal growth behavior as the collision and reaction between molecules. The development of OA growth kinetics can provide a sufficient understanding of crystal growth, and the awareness of underlying factors in the growth will offer promising guidance on how to control the size distribution and shape development of nanostructural materials.

Mechanisms of diffusional nucleation of nanocrystals and their self-assembly into uniform colloids

We survey our research on modeling the mechanisms of control of uniformity in the growth of nanosize and colloid size particles. The former are produced as nanocrystals by burst nucleation from solution and the latter are formed by self-assembly (aggregation) of the nanocrystals. In the colloid particle synthesis, the two dynamical processes are coupled, both governed by diffusional transport of the respective building blocks (monomers). The interrelation of the two processes allows for the synthesis of narrow size distribution colloid dispersions, which are of importance in many applications. We first review a mathematical model of diffusive cluster growth by the capture of monomer "singlets." We then analyze burst nucleation of nanoparticles in solution. Finally, we couple it to the secondary process of aggregation of nanoparticles to form colloids and discuss various aspects of the modeling of particle size distribution, as well as other features of the processes considered.

Oriented Attachment Kinetics for Ligand Capped Nanocrystals: Coarsening of Thiol-PbS Nanoparticles

In this work, the growth kinetics of thiol-capped PbS nanoparticles was studied. Two-stage growth process was observed, which was controlled first by oriented attachment (OA) mechanism and then by the hybrid Ostwald ripening (OR) and OA mechanism. Different from the NaOH-ZnS system, where OA will occur between any two multilevel nanoparticles, an OA kinetic model only considering the attachment related to original particles was fitted well with the experimental results. Analysis reveals that this model may be a universal one to describe the OA crystal growth process of nanocrystals capped with easily destroyed ligands, such as thiol-ZnS in the previous report. The OA crystal growth characteristics determined by the surface agent were discussed and compared. We propose that with stronger surface capping, the OR growth of nanocrystals is hindered, which facilitates the size controlling via OA kinetics during nanosynthesis.

A Multistep Oriented Attachment Kinetics: Coarsening of ZnS Nanoparticle in Concentrated NaOH

Crystal growth of ZnS nanoparticles during hydrothermal coarsened in 4 M NaOH occurs via a two-stage process. In the first stage, the primary particles grow into a size over hundred times of the original volume. The initial growth rate can be fitted by an asymptotic curve. High-resolution transmission electron microscope (HRTEM) data indicate that in this stage, crystal growth mainly occurs via a multistep crystallographically specific oriented attachment (OA). The higher the coarsening temperature, the earlier the first stage ends. In the second stage, an abrupt transition from asymptotic to square parabola growth kinetics occurs. The crystal growth data can be fitted by a standard Ostwald ripening (OR) model consistent with growth controlled by dissolution/precipitation of ions in solution. HRTEM data indicate that a minor amount of OA-based growth also occurs in the early period of the second stage. A new multistep OA kinetics model analogous to the reaction between molecules was proposed to illustrate the asymptotic growth in the first stage of coarsening. The effect of concentrated NaOH was discussed and proved to be the key that hindered the OR process, attributing to the almost exclusive pure OA-based growth of ZnS particles in the first stage.

Exchange-driven growth

We study a class of growth processes in which clusters evolve via exchange of particles. We show that depending on the rate of exchange there are three possibilities: (I) Growth-clusters grow indefinitely, (II) gelation-all mass is transformed into an infinite gel in a finite time, and (III) instant gelation. In regimes I and II, the cluster size distribution attains a self-similar form. The large size tail of the scaling distribution is Phi(x) approximately exp(-x(2-nu)), where nu is a homogeneity degree of the rate of exchange. At the borderline case nu=2, the distribution exhibits a generic algebraic tail, Phi(x) approximately x(-5). In regime III, the gel nucleates immediately and consumes the entire system. For finite systems, the gelation time vanishes logarithmically, T approximately [lnN](-(nu-2)), in the large system size limit N--> infinity. The theory is applied to coarsening in the infinite range Ising-Kawasaki model and in electrostatically driven granular layers.

Organization of growing random networks

The organizational development of growing random networks is investigated. These growing networks are built by adding nodes successively, and linking each to an earlier node of degree k with an attachment probability A(k). When A(k) grows more slowly than linearly with k, the number of nodes with k links, N(k)(t), decays faster than a power law in k, while for A(k) growing faster than linearly in k, a single node emerges which connects to nearly all other nodes. When A(k) is asymptotically linear, N(k)(t) approximately tk(-nu), with nu dependent on details of the attachment probability, but in the range 2<nu<infinity. The combined age and degree distribution of nodes shows that old nodes typically have a large degree. There is also a significant correlation in the degrees of neighboring nodes, so that nodes of similar degree are more likely to be connected. The size distributions of the in and out components of the network with respect to a given node--namely, its "descendants" and "ancestors"-are also determined. The in component exhibits a robust s(-2) power-law tail, where s is the component size. The out component has a typical size of order ln t, and it provides basic insights into the genealogy of the network.

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.

Mechanism of Formation of Monodispersed Colloids by Aggregation of Nanosize Precursors

It has been experimentally established in numerous cases that precipitation of monodispersed colloids from homogeneous solutions is a complex process. Specifically, it was found that in many systems nuclei, produced rapidly in a supersaturated solution, grow to nanosize primary particles (singlets), which then coagulate to form much larger final colloids in a process dominated by irreversible capture of these singlets. This paper describes a kinetic model that explains the formation of dispersions of narrow size distribution in such systems. Numerical simulations of the kinetic equations, with experimental model parameter values, are reported. The model was tested for a system involving formation of uniform spherical gold particles by reduction of auric chloride in aqueous solutions. The calculated average size, the width of the particle size distribution, and the time scale of the process agreed reasonably well with the experimental values. Copyright 1999 Academic Press.