Flocculation and percolation in reversible cluster-cluster aggregation

Off-lattice dynamic Monte-Carlo simulations were done of reversible cluster-cluster aggregation for spheres that form rigid bonds at contact. The equilibrium properties were found to be determined by the life time of encounters between two particles (t_e). t_e is a function not only of the probability to form or break a bond, but also of the elementary step size of the Brownian motion of the particles. In the flocculation regime the fractal dimension of the clusters is d_f=2.0 and the size distribution has a power law decay with exponent τ=1.5. At larger values of t_e transient gels are formed. Close to the percolation threshold the clusters have a fractal dimension d_f=2.7 and the power law exponent of the size distribution is τ=2.1. The transition between flocculation and percolation occurs at a characteristic weight average aggregation number that decreases with increasing volume fraction.     

Off lattice simulations of cluster-cluster aggregation in dimensions 2–6

Cluster-cluster aggregation has been simulated in dimensions two to six using both linear and brownian cluster trajectories. Relatively efficient off lattice algorithms have allowed large clusters to be generated and values for the fractal dimensionalities of the aggregates have been obtained without finite concentration effects. The values for the fractal dimensionality are in good aggreement with lattice model simulations for euclidean dimensionalities 2–4. The effective dimensionality (D_β) obtained from the dependence of the radius of gyration on cluster size increases with increasing cluster size for all of our models (particularly for d ≥ 4). For clusters in the accessible size range (up to 10³-10⁴) D_β is slightly larger for cluster-cluster aggregation via linear trajectories than for brownian trajectories. For cluster-cluster aggregation via brownian trajectories, the limiting (large cluster size) fractal dimensionality is estimated to be 1.46 ± 0.04 for d=2,1.82 ± 0.10 for d = 3, 2.10 ± 0.15 for d = 4, 2.35 ± 0.15 for d = 4, 2.65 ± 0.25 for d = 6. For cluster- cluster aggregation via linear trajectories, the limiting fractal dimensionality is estimated to be 1.55 ± 0.04 for d = 2, 1.91 ± 0.10 for d = 3≥ 2.5 ± 0.06 for d = 5 and ≥2.64 ± 0.05 for d = 6.     

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold p _c ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 1.733. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as p → p _c with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution P _s(p) show power law behaviour with | p - p _c| in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of P _s(p). The results obtained are in good agreement with other model calculations. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

Diffusion on two-dimensional percolation clusters with multifractal jump probabilities

By means of Monte Carlo simulations we studied the properties of diffusion limited recombination reactions (DLRR's) and random walks on two dimensional incipient percolation clusters with multifractal jump probabilities. We claim that, for these kind of geometric and energetic heterogeneous substrata, the long time behavior of the particle density in a DLRR is determined by a random walk exponent. It is also suggested that the exploration of a random walk is compact. It is considered a general case of intersection ind euclidean dimension of a random fractal of dimension D_F and a multifractal distribution of probabilities of dimensionsD _q (q real), where the two dimensional incipient percolation clusters with multifractal jump probabilities are particular examples. We argue that the object formed by this intersection is a multifractal of dimensionsD' _q =D _q +D _F -d, for a finite interval ofq.     

A study of radiative properties of fractal soot aggregates using the superposition T-matrix method

We employ the numerically exact superposition T-matrix method to perform extensive computations of scattering and absorption properties of soot aggregates with varying state of compactness and size. The fractal dimension, D_f, is used to quantify the geometrical mass dispersion of the clusters. The optical properties of soot aggregates for a given fractal dimension are complex functions of the refractive index of the material m, the number of monomers N_S, and the monomer radius a. It is shown that for smaller values of a, the absorption cross section tends to be relatively constant when D_f<2 but increases rapidly when D_f>2. However, a systematic reduction in light absorption with D_f is observed for clusters with sufficiently large N_S, m, and a. The scattering cross section and single-scattering albedo increase monotonically as fractals evolve from chain-like to more densely packed morphologies, which is a strong manifestation of the increasing importance of scattering interaction among spherules. Overall, the results for soot fractals differ profoundly from those calculated for the respective volume-equivalent soot spheres as well as for the respective external mixtures of soot monomers under the assumption that there are no electromagnetic interactions between the monomers. The climate-research implications of our results are discussed.     

Evolution of the electronic properties of transition metal nanoclusters on graphite surface

The electronic properties of nanoclusters of transition (Ni, Co, Cr) and noble (Au, Cu) metals deposited on the surface of highly oriented pyrolytic graphite (HOPG) are studied using the method of X-ray photoelectron spectroscopy. The laws of variation of a change ΔE _b in the binding energies of core-level electrons in the initial (ΔE _i) and final (ΔE _f) states of atoms in nanoclusters, the intrinsic widths γ of photoelectron lines, and their singularity indices α as functions of the metal cluster size d are determined. A qualitative difference in behavior of the ΔE _i(d) and α(d) values in metals of the two groups (Ni, Cr versus Co, Cu) is found. The values of the final-state energy (ΔE _f < 0) and the line width (Δγ > 0) in the clusters of all metals studied vary in a similar manner. It is shown that a significant contribution to E _i is due to a transfer of the valence-shell electrons at the cluster-substrate interface, which is caused by the contact potential difference. The value of an uncompensated charge per nanocluster is determined as a function of the cluster size and the number of atoms in the cluster. The behavior of ΔE _f(d) is controlled by the Coulomb energy of a charged cluster and by a decrease in the efficiency of electron screening, which is different in the metals studied. The broadening of photoelectron lines is determined by a spread of the cluster sizes and by lower electron screening in the final Fermi system. An asymmetry of the core-level electron spectra of nanoclusters can be explained using notions about the electron-hole pair excitation near the Fermi level. The effect of the structure of the density of electron states in the d band of transition metals on the asymmetry of photoelectron lines is considered and it is concluded that this structure near the Fermi level qualitatively changes with a decrease in the nanocluster size. The obtained results indicate that the behavior of the electron subsystem of clusters of the d-metals in a size range of 2–10 nm under consideration is close to the behavior of a normal Fermi system.     

A finite fractal cluster theory of 1/f noise in percolation systems near metal-insulator transition

The problem of 1/f noise in thin metal films and metal-insulator composites in the scaling fractal regime near percolation threshold is considered. The correspondence between a percolation transition and a second order phase transition is extended from the point of view of electronic polarization and electrical fluctuations. The charge fluctuations on finite fractal clusters are argued to be analogous to spontaneous order parameter fluctuations in phase transitions, being correlated upto percolation correlation length. The charge relaxation times are shown to be related to the cluster sizes having distribution function of the formg()^–b , whereb is connected to Euclidean and fractal dimensionalities and critical exponents. This produces the 1/f noise spectrum. Below percolation threshold, the nodes-links-blobs picture is invoked such that the blobs represent metallic conductances of the finite clusters and the links are tunnelling conductances between them through narrowest barrier regions. Above threshold, the finite cluster network is visualized as connected to the infinite cluster through narrowest tunnelling regions. The correlated spontaneous charge fluctuation on finite fractal clusters is held responsible for conductance fluctuation on either side of the metal-insulator transition via tunnelling processes. Finally, the scaling behaviour of noise magnitude near percolation threshold is explained.     

On the melting of gold nanoclusters formed by pulsed laser deposition on different substrates

The experimental results are presented for the backscattering of 500-eV electrons on Au nanoclusters formed on the surface of highly oriented pyrolytic graphite HOPG(0001) and amorphous SiO₂. It has been found that the measured intensity of the elastically backscattered electrons nonmonotonically depends on the size of nanoclusters. It has been shown that the observed features can be explained by an increase in the rms deviation of the atoms of the Au nanocluster with a decrease in its size. The difference in the dependence of the rms deviation of atoms on the size of the nanoclusters formed on the surfaces of HOPG(0001) and amorphous SiO₂ is qualitatively explained by an increase in the roughness of the nanocluster surface accompanying their formation under the strongly nonequilibrium conditions of pulsed laser deposition.     

Anomalous scaling behaviour of cobalt cluster size distributions on graphite,epitaxial graphene and carbon-rich (6√3 × 6√3)R30°

The influence of the underlying interface on adsorption of cobalt (Co) is investigated by comparing the nucleation and growth of Co at room temperature on three carbon (C) surfaces, i.e. highly oriented pyrolytic graphite (HOPG), epitaxial graphene/SiC(0001) (hereafter abbreviated as EG) and precursor of EG i.e. C-rich (6√3 × 6√3)R30°/SiC(0001) (hereafter abbreviated as 6√3). On all three surfaces, Co adopts Volmer–Weber growth mode via formation of three-dimensional dome-shaped nanoclusters. Co clusters formed on 6√3 surface are smaller but denser than Co/HOPG or Co/EG. Scaling analysis reveals a critical nucleus size, i* = 1 (atom) and the smallest stable cluster (i* + 1) would be a dimer. Co/HOPG and Co/EG have the same order of magnitude for their cluster densities and sizes. Scaling analyses however show that the i* for Co/EG (i* = 3) is larger than Co/HOPG (i* = 0) and in this respect the smallest stable cluster would be tetramer and monomer respectively. This difference is attributed to the influence of an interface situated between graphene and SiC bulk. It appears that EG is more inert than HOPG towards the adsorption of Co and may act as a better substrate to host Co clusters.     

Effect of nanostructuring of the Ge1 − x Mn x single-crystal alloy on the percolation and cluster ferromagnetism

Nanostructuring of the Ge_0.99Mn_0.01 single-crystal alloy with the formation of nanowires 60 nm in diameter has been revealed to bring about an increase in the temperature of the percolation transition to the ferromagnetic state to T _C1 ≈ 45 K and a broadening of the distribution of Curie temperatures in Ge _n Mn _m clusters T _C2 ≈ 125?270 K as compared to Ge_0.98Mn_0.02 thin single-crystal films for which T _C1 ≈ 6 K and T _C2 ≈ 270 K.     

Spatial scaling in fracture propagation in dilute systems

The geometry of fracture patterns in a dilute elastic network is explored using molecular dynamics simulation. The network in two dimensions is subjected to a uniform strain which drives the fracture to develop by the growth and coalescence of the vacancy clusters in the network. For strong dilution, it has been shown earlier that there exists a characteristic time t_c at which a dynamical transition occurs with a power law divergence (with the exponent z) of the average cluster size. Close to t_c, the growth of the clusters is scale-invariant in time and satisfies a dynamical scaling law. This paper shows that the cluster growth near t_c also exhibits spatial scaling in addition to the temporal scaling. As fracture develops with time, the connectivity length xi of the clusters increases and diverges at t_c as xi ∼ (t_c − t)^−ν, with ν = 0.83 ± 0.06. As a result of the scale-invariant growth, the vacancy clusters attain a fractal structure at t_c with an effective dimensionality d_f ∼ 1.85 ± 0.05. These values are independent (within the limit of statistical error) of the concentration (provided it is sufficiently high) with which the network is diluted to begin with. Moreover, the values are very different from the corresponding values in qualitatively similar phenomena suggesting a different universality class of the problem. The values of ν and d_f supports the scaling relation z = νd_f with the value of z obtained before.     

Morphology Development in Multi‐Component Polymer Blends: I Composition Effect on Phase Morphology in PP/PET Polymer Blends

The relationship of the phase morphology of polypropylene/polyethylene‐terephthalate (PP/PET) blends and their corresponding compatibilized blends with composition was investigated using digital image analysis. A diameter, d _g , was defined and calculated to discuss the phase morphology of this polymer blend system. A figure‐estimation method was introduced to determine the width of the distribution of d _g . Based on the method, it is proven that the distribution of d _g obeys a log‐normal distribution and consequently, the distribution width, σ was calculated. Further, a fractal dimension, D _f , was introduced to describe the distribution of main sizes of the particles of the dispersed phase. The results showed that, while d _g increased with the concentration of the dispersed phase, σ and D _f show different dependence relations on composition;σ increases monotonously but D _f shows a maximum at a PET content of 30%, indicating that, even though the whole size distribution is much broader, the distribution of the main body of size becomes more uniform when the content of PET is less than 30%.     

Different W cluster deposition regimes in pulsed laser ablation observed by in situ scanning tunneling microscopy

We report on how different cluster deposition regimes can be obtained and observed by in situ scanning tunneling microscopy by exploiting deposition parameters in a pulsed laser deposition process. Tungsten clusters were produced by nanosecond pulsed laser ablation in Ar atmosphere at different pressures and deposited on Au(1 1 1) and HOPG surfaces. Deposition regimes including cluster deposition-diffusion-aggregation, cluster melting and coalescence and cluster implantation were observed, depending on background gas pressure and target-to-substrate distance which influence the kinetic energy of the ablated species. These parameters can thus be easily employed for surface modification by cluster bombardment, deposition of supported clusters and growth of films with different morphologies. The variation in cluster mobility on different substrates and its influence on aggregation and growth mechanisms has also been investigated.     

Exploration of fractal nature of WO3 nanowire aggregates

The morphology of WO₃ aggregates formed by irregular nanoparticles (D∼40 nm) and nanowires of different aspect ratios (2, 4, 6, and 10 μm nominal lengths) dispersed in commonly used polar solvents without dispersant agents is investigated using a small-angle light scattering technique and by means of fractal theory. Nanoparticles form compact spherical aggregates (D_f∼2.6), whereas 2 μm nanowires with low aspect ratio (L/D∼10) follow a slow cluster-cluster aggregation mechanism with no discernable change in fractal dimension (D_f=2.1) monitored in an extended period of 6 months, despite a notable growth in size (R_g=2.3-3.1 μm). For higher aspect ratio nanowires, scattered intensity profiles, which migrate towards the Porod regime, qualitatively obey the Lorenz-Mie theory predictions. The 10 μm nanowires with very high aspect ratio (L/D∼250) are observed to form stable dispersions in a time span of 6 days. Analytical methods based on spherical primary particle formulations predict D_f=1.9, 1.7, and 1.4 for 4, 6, and 10 μm nanowires, respectively.     

Effects of Monomer Size Distribution on the fractal dimensionality of diffusion-limited aggregates

Computer simulations of diffusion-limited aggregation (DLA) for monomers to investigate the effects of size and of lognormal distribution on the fractal dimensionality of the aggregates were conducted on a two-dimensional lattice. The results show the DLA clusters posses multifractal characteristics. For clusters consisting of monodisperse monomers, the bifurcation point on the graph of the pair correlation function (PCF) for each cluster is located right at the monomers size under investigation The textural dimension (D_f1) has a stable value of about 1.65, whereas the structural dimension (D_f2) decreased with increase in monomer size. For the cases with monomers in log-normal distributions, the textural dimension is around 1.67; however, the structural dimension decreases with increasing polydispersity of monomer size.     

An Infinite Number of Effectively Infinite Clusters in Critical Percolation

An infinite number of effectively infinite clusters are predicted at the percolation threshold, if effectively infinite means that a cluster's mass increases with a positive power of the lattice size L. All these cluster masses increase as L _D with the fractal dimension D = d – /v, while the mass of the rth largest cluster for fixed L decreases as 1/r , with = D/d in d dimensions. These predictions are confirmed by computer simulations for the square lattice, where D = 91/48 and = 91/96.     

Influence of the size distribution of granules and of their attractive interaction on the percolation threshold in granulated alloys

Numerical simulation was applied to study the influence of the size distribution of granules and the interaction between them on the percolation threshold in granulated metal-insulator alloys. An alloy model was considered in which metal granules have two characteristic sizes, l and L (with L>l), and the size distribution of granules of greater size L having an average value of approximately L ₀ is described by a normal distribution with a standard deviation d, by a step function with a halfwidth d, or by a delta function. A model with attraction between granules and mechanisms of trapping of an additional granule by an already developed cluster with a characteristic trapping range R was also considered. The percolation threshold significantly grows with the ratio L ₀/l and with R for both two-and three-dimensional cases and tends to flattening at large L/l or R. The calculated results make it possible to explain the high percolation threshold observed for the majority of granulated alloys.     

Height-height correlations for surface growth on percolation networks

The height-height correlations of the surface growth for equilibrium and nonequilibrium restricted solid-on-solid (RSOS) model were investigated on randomly diluted lattices, i.e., on infinite percolation networks. It was found that the correlation function calculated over the chemical distances reflected the dynamics better than that calculated over the geometrical distances. For the equilibrium growth on a critical percolation network, the correlation function for the evolution time t?1 yielded a power-law behavior with the power ζ^′, associated with the roughness exponent ζ via the relation ζ=ζ^′d_f/d_l, with d_f and d_l being, respectively, the fractal dimension and the chemical dimension of the substrate. For the nonequilibrium growth, on the other hand, the correlation functions did not yield power-law behaviors for the concentration of diluted sites x less than or equal to the critical concentration x_c.     

Molecular trajectory algorithm for random walks on percolation systems at criticality in two and three dimensions

Random walk simulations based on a molecular trajectory algorithm are performed on critical percolation clusters. The analysis of corrections to scaling is carried out. It has been found that the fractal dimension of the random walk on the incipient infinite cluster is d_w=2.873±0.008 in two dimensions and 3.78 ± 0.02 in three dimensions. If instead the diffusion is averaged over all clusters at the threshold not subject to the infinite restriction, the corresponding critical exponent k is found to be k=0.3307±0.0014 for two-dimensional space and 0.199 ± 0.002 for three-dimensional space. Moreover, in our simulations the asymptotic behaviors of local critical exponents are reached much earlier than in other numerical methods.     

Particle aggregation versus cluster aggregation in high dimensions

We distinguish two different types of irreversible aggregation-accretion of individual particles and successive aggregation of clusters of comparable size. In aggregation of particles which follow trajectories of fractal dimensionD ₁, we show that physical limits on the aggregation rate impose a lower bound on the fractal dimensionD ₀ of the aggregate. Ind-dimensional space,D ₀{d–D}₁ + 1. Thus aggregation of ballistic particles, withD ₁ = 1, is not fractal. By contrast, cluster aggregates appear to attain a finite, limitingD ₀ in high dimensions. We present a soluble model with this property, and argue that it should agree with Sutherland's binary aggregation model in high dimensions. For this model,D ₀ depends continuously on a parameter; the exponent is not universal.