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.     

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     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Structural properties and scaling of the radius of gyration of two-dimensional star-branched polymers grown by diffusion

We built up star-branched polymers, whose morphology is fully determined by diffusion, with p=1,3,6 and 12 branches with a total of 30,000 monomer units. We investigated their structural properties by calculating the monomer-monomer correlation functions. A detailed finite size scaling analysis of the radius of gyration was also performed to determine the exponent and the corrections to scaling. From these results we calculated the fractal dimension of the branched aggregates and obtained: d_f=1.222(7), for the linear chain, d_f=1.2305(8), d_f=1.247(8) and d_f=1.261(8) for the three, six and twelve branches polymer, respectively.     

Changes in pore size distribution inside sludge under various ultrasonic conditions

The pore size distribution is quite significant for determining the transport capacity of heat and moisture in sludge during the drying process. It is crucial to investigate the transformation of the pore size in sludge under sonication. In this paper, the microstructures of pores inside sludge before and after ultrasonic treatment with various ultrasonic conditions were observed using a microscope. Fractal geometry and image analysis were combined to quantitatively identify the evolution of pore size in sludge undergoing various acoustic energy densities and treatment times. The surface fractal dimension (d_f) was applied to characterize the pore size distribution of sludge. The results confirmed that sonication has a positive influence on the characteristics of pore structure inside the sludge and that the average pore size increases with increasing ultrasonic energy level, as determined by both acoustic energy density and treatment time. The d_f appropriately characterizes and quantifies the evolution of the pore size distribution of sludge under various ultrasonic conditions. This work is quite valuable for further investigating and evaluating moisture removal in the sludge drying process assisted by ultrasonic treatment.     

Glass formation in amorphous SiO<Subscript>2</Subscript> as a percolation phase transition in a system of network defects

Thermodynamic parameters of defects (presumably, defective SiO molecules) in the network of amorphous SiO₂ are obtained by analyzing the viscosity of the melt with the use of the Doremus model. The best agreement between the experimental data on viscosity and the calculations is achieved when the enthalpy and entropy of the defect formation in the amorphous SiO₂ network are H _d =220 kJ/mol and S _d =16.13R, respectively. The analysis of the network defect concentration shows that, above the glass-transition temperature (T _g ), the defects form dynamic percolation clusters. This result agrees well with the results of molecular dynamics modeling, which means that the glass transition in amorphous SiO₂ can be considered as a percolation phase transition. Below T _g , the geometry of the distribution of network defects is Euclidean and has a dimension d=3. Above the glass-transition temperature, the geometry of the network defect distribution is non-Euclidean and has a fractal dimension of d _f =2.5. The temperature T _g can be calculated from the condition that percolation arises in the defect system. This approach leads to a simple analytic formula for the glass-transition temperature: T _g =H _d /((S _d +1.735R). The calculated value of the glass-transition temperature (1482 K) agrees well with that obtained from the recent measurements of T _g for amorphous SiO₂ (1475 K).     

Structure and electrical conductivity of ultrathin Ni-Cu films

The structure, phase composition, morphology, and electrical conductivity of Ni-Cu alloy ultrathin films having a thickness of d = 1?10 nm and a Cu concentration of 10–95 at % have been studied. All films are shown to be fcc Ni-Cu alloys; they have an island structure with an island size of 1.5–2 nm in the as-deposited films and of about 20 nm in the films annealed to 700 K. The electrical conductivity of the films depends on their thickness and morphology. For films with d ≈ 1 nm, the electrical conductivity is thermally activated with an activation energy E _a ≈ 0.086?0.095 eV. Films with d > 3 nm exhibit the metallic temperature dependence of electrical conductivity with a positive temperature coefficient of resistivity.     

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