Fractal Dimension Of Clusters Research Articles

Langevin agglomeration of nanoparticles interacting via a central potential

Nanoparticle agglomeration in a quiescent fluid is simulated by solving the Langevin equations of motion of a set of interacting monomers in the continuum regime. Monomers interact via a radial rapidly decaying intermonomer potential. The morphology of generated clusters is analyzed through their fractal dimension df and the cluster coordination number. The time evolution of the cluster fractal dimension is linked to the dynamics of two populations: small (k≤ 15) and large (k>15) clusters. At early times monomer-cluster agglomeration is the dominant agglomeration mechanism (d(f)=2.25) , whereas at late times cluster-cluster agglomeration dominates (d(f)=1.56). Clusters are found to be compact (mean coordination number of ∼5), tubular, and elongated. The local compact structure of the aggregates is attributed to the isotropy of the interaction potential, which allows rearrangement of bonded monomers, whereas the large-scale tubular structure is attributed to its relatively short attractive range. The cluster translational diffusion coefficient is determined to be inversely proportional to the cluster mass and the (per-unit-mass) friction coefficient of an isolated monomer, a consequence of the neglect of monomer shielding in a cluster. Clusters generated by unshielded Langevin equations are referred to as ideal clusters because the surface area accessible to the underlying fluid is found to be the sum of the accessible surface areas of the isolated monomers. Similarly, ideal clusters do not have, on average, a preferential orientation. The decrease in the numbers of clusters with time and a few collision kernel elements are evaluated and compared to analytical expressions.

Slow salt-induced aggregation of citrate-covered silver particles in aqueous solutions of cellulose derivatives

In this work, the salt-induced aggregation of bare and polymer-covered silver particles has been studied with the aid of light scattering and UV-visible spectroscopy. Light scattering on a suspension of bare silver particles at a low salt concentration shows that the cluster fractal dimension d f changes from 1.6 to 2 in the course of the aggregation process, whereas no restructuring of the clusters is observed at a higher salinity where d f ≈ 1.6. The growth of the clusters over time can be described by a power law R h ∝ t α , where R h is the apparent hydrodynamic radius. The UV-visible experiments revealed that increasing the size of the bare silver particles lead to a significant broadening and red-shift of the absorbance band, whereas for salt-induced growth of Ag clusters, a blue-shift and broadening was observed. Addition of salt to a suspension of silver particles and hydroxyethylcellulose divulged a slower broadening of the surface plasmon peak than without polymer.

Some issues in polymer nanocomposites: Theoretical and modeling opportunities for polymer physics

Some issues in polymer nanocomposites: Theoretical and modeling opportunities for polymer physics

Cluster size distribution in percolation theory and fractal Cantor dust

Results of numerical simulation of cluster size distribution in the site percolation problem are presented. These results disagree with the theoretical data obtained on the basis of the standard drop model of finite cluster structure, in particular, they give a different value of exponent zeta (lnn{s} approximately -s{zeta}). Therefore, a more precise fractal model for describing the structure of clusters in a percolation system is proposed. The consideration is based on the solution of a kinetic equation for the number of finite clusters. In the framework of the proposed approach (fractal model together with kinetic equation), a correct value of exponent zeta is obtained and an explanation is given to the dependence of this exponent on the fraction of occupied sites p, which was revealed by numerical simulations. Additionally, a relation is established between the characteristics of cluster size distribution and fractal dimension of the Cantor dust constructed on the percolation cluster.

Master Curves for Aggregation and Gelation: Effects of Cluster Structure and Polydispersity

A parametric study of the effects of cluster structure and polydispersity on the kinetics of aggregation and gelation is presented. The aggregation kinetics is described in terms of master curves characterizing the evolution of suitable dimensionless averages (measurable by light scattering) of the underlying cluster mass distribution (CMD), as a function of a suitable dimensionless time. Such master curves are shown to be dependent only on two dimensionless parameters: the cluster fractal dimension (df) and the ratio between Brownian and shear aggregation rates (κ*). Shear aggregation of fractal clusters leads to higher-order moments of the CMD diverging to infinity in a finite time, which is usually called runaway. It is determined that the parameter space is split into two distinct regions: either where aggregates gel because of space filling before runaway occurs or where gelation occurs as a consequence of runaway. It is also determined that cluster polydispersity and the fractal dimension signific...

Stratification of colloidal aggregation coupled with sedimentation

One of the consequences of sedimentation in colloidal aggregation is the stratification of the system in the sense that, after a sufficiently long elapsed time, the large clusters lie preferentially at the bottom zones of the confinement prism, and the structural and dynamical quantities describing the aggregates depend on the depth at which they are measured. A few years ago a computer simulation using particles for colloidal aggregation coupled with sedimentation was proposed by the author [A. E. González, Phys. Rev. Lett. 86, 1243 (2001)]. In that simulation, due to computational limitations, the mentioned quantities were averaged over all clusters in the prism, independently of the depth at which they were located, in order to have good statistics for the evaluation of the cluster fractal dimension and the cluster size distribution function. In this work we present a computer simulation using particles of colloidal aggregation coupled with sedimentation, for which the clusters in the simulation box represent those clusters inside a layer at a fixed depth and of arbitrary thickness in the prism. It would then be possible to compare the results with an eventual validation experiment, in which an aggregating sample is sipped out with a pipette at a fixed depth in the prism and subjected to further studies, or with a light scattering study in which the laser beam is focused at a fixed depth in the system. We confirm the acceleration of the aggregation rate, followed by a slowing down, compared with an aggregating system driven purely by diffusion (DLCA). In the present system, the large clusters when drifting downwards sweep smaller ones, which in turn occlude the holes and cavities of these large clusters, increasing in this way their compacticity. We also confirm that (i) in some cases of sedimentation strengths and layer depths, the mean width (perpendicular to the gravitational field direction) and the mean height of the large settling clusters scale with the size as a power law, with the same scaling power, in some range of cluster sizes. This leads to self-similar clusters with an appreciably higher fractal dimension (d{f}) than the d{f} of DLCA clusters, a case that we called the "sweeping scaling regime" in earlier works. However, the present system is much richer than DLCA in that (ii) there are some other cases for which the parallel and perpendicular scaling powers differ, leading to anisotropic self-affine clusters. (iii) There are further cases for which only the mean width or the mean height scale as a power law, leading again to anisotropic clusters. Finally, (iv) there are still some cases for which neither the mean width nor mean height scale as a power law with the size. In the last (ii), (iii), and (iv) cases the large settling clusters are anisotropic and non-self-similar, and a fractal dimension cannot be defined for them, as found recently by some other authors for case (iii); however, their "compacticity" should be greater than that for DLCA clusters, in a yet undefined way.

Computer simulation of growing fractal nanodendrites by using of the multi-directed cellular automatic device

Computer simulation of self-assembly growth of fractal nanodendrites is suggested by using a new model of multi-directed cellular automatic device. The novelty is that in a framework of the multiparticle diffusion limited aggregation (DLA) model, atoms stick to the growing cluster according to specific kinematics multi-directional contact bonds. The aggregation is limited by initial concentration of particles confined inside definite nano-sized enclosures. Self-organizing process results in the formation of fractal dendrite nanostructures. From results of computer experiments, some equations for cluster fractal dimension D, depending on the initial concentration C of diffusing particles in 2d and 3d space enclosures, are deduced. Correlations between cluster fractal dimension D and number of aggregated particles N are found also.

Kinetics of Aggregation and Gelation in Colloidal Dispersions

Control of aggregation and gelation of colloidal particles are among important issues in processing of disperse systems. We present an experimentally validated approach to kinetic modelling based on master curves for aggregation and gelation. The master curves depend on basic physicochemical properties of primary particles, which are reflected in terms of the cluster fractal dimension and kernel kinetic exponent. We also analyse interplay between aggregation and diffusion at small length scales in the process of formation of gel domains in imperfectly mixed colloidal dispersions.

Orientation and rupture of fractal colloidal gels during start-up of steady shear flow

The transient structural evolution of polystyrene colloidal gels with fractal structure is quantified during start-up of steady shear flow by time-resolved small-angle light scattering and rheometry. Three distinct regimes are identified in the velocity-gradient plane: structural orientation, network breakup, and cluster densification. Structural anisotropy in the first regime is a universal function of applied strain. Flow cessation in this regime shows a lack of structural relaxation for Pe⪡1, where Pe is the Peclet number. In the second regime, the anisotropy attains a maximum value before monotonically decreasing. The volume fraction dependence of the critical strain for maximum anisotropy follows the scaling: 1+0.6γc,r∼ϕ(1−x)(3−D). Here x and D are the backbone and cluster fractal dimensions, respectively. This scaling agrees with the simple model of a gel network that ruptures after the cluster backbone is extended affinely to its full length. Rheological measurements demonstrate that the maximum an...

Multifractal concentrations of inertial particles in smooth random flows

Collisionless suspensions of inertial particles (finite-size impurities) are studied in two- and three-dimensional spatially smooth flows. Tools borrowed from the study of random dynamical systems are used to identify and to characterize in full generality the mechanisms leading to the formation of strong inhomogeneities in the particle concentration.Phenomenological arguments are used to show that in two dimensions, the positions of heavy particles form dynamical fractal clusters when their Stokes number (non-dimensional viscous friction time) is below some critical value. Numerical simulations provide strong evidence for the presence of this threshold in both two and three dimensions and for particles not only heavier but also lighter than the carrier fluid. In two dimensions, light particles are found to cluster at discrete (time-dependent) positions and velocities in some range of the dynamical parameters (the Stokes number and the mass density ratio between fluid and particles). This regime is absent in the three-dimensional case for which evidence is that the (Hausdorff) fractal dimension of clusters in phase space (position–velocity) always remains above two.After relaxation of transients, the phase-space density of particles becomes a singular random measure with non-trivial multiscaling properties, whose exponents cannot be predicted by dimensional analysis. Theoretical results about the projection of fractal sets are used to relate the distribution in phase space to the distribution of the particle positions. Multifractality in phase space implies also multiscaling of the spatial distribution of the mass of particles. Two-dimensional simulations, using simple random flows and heavy particles, allow the accurate determination of the scaling exponents: anomalous deviations from self-similar scaling are already observed for Stokes numbers as small as .

Analysis and Control of a Turbulent Coagulator

Recent developments of particle sizing sensors and control algorithms together with a better understanding of the physics of coagulation processes foster the application of advanced control systems to aggregating particulate systems in order to ensure the product quality despite disturbances and model errors. A model containing the main features of coagulation dynamics in shear flows, including aggregation, breakage, and time evolution of the cluster fractal dimension, was introduced and tuned to reproduce experimental data from the literature. Using this model, a set of particulate products, in terms of the mean size and fractal dimension, that can be obtained with different shear rate policies was identified. Finally, a control algorithm, the batch model predictive control, was applied to the system. The algorithm exploits the fact that batch processes are run repetitively and the availability of online measurements. The controller, tested with simulations, was able to guarantee the product quality in terms of the mean radius of gyration by adjusting the shear rate according to the current measurements and the results obtained in previous batches.

Colloidal aggregation with sedimentation: concentration effects.

The results of computer models for colloidal aggregation, that consider both Brownian motion and gravitational drift experienced by the colloidal particles and clusters, are extended to include concentrations spanning three orders of magnitude. In previous publications and for a high colloidal concentration, it was obtained that the aggregation crosses over from diffusion-limited colloidal aggregation (DLCA) to another regime with a higher cluster fractal dimension and a speeding up followed by a slowing down of the aggregation rate. In the present work we show, as the concentration is decreased, that we can still cross over to a similar regime during the course of the aggregation, as long as the height of the sample is increased accordingly. Among the differences between the mentioned new regimes for a high and a low colloidal concentration, the cluster fractal dimension is higher for the high concentration case and lowers its value as the concentration is decreased, presumably reaching for low enough concentrations a fixed value above the DLCA value. It is also obtained the fractal dimension of the sediments, arising from the settling clusters that reach the bottom and continue a 2D-like diffusive motion and aggregation, on the floor of the container. For these clusters we now see two and sometimes three regimes, depending on concentration and sedimentation strength, with their corresponding fractal dimensions. The first two coming from the crossover already mentioned, that took place in the bulk of the sample before the cluster deposition, while the third arises from the two-dimensional aggregation on the floor of the container. For these bottom clusters we also obtain their dynamical behavior and aggregation rate.

Simulations of colloidal aggregation with short- and medium-range interactions

Extensive numerical simulations are used to simulate the aggregation of colloidal particles confined in a two-dimensional space. The colloidal particles experience short- or medium-range repulsions coming from a potential barrier that inhibits the aggregation of the particles into the primary minimum of the potential. When the potential barrier is short ranged, we find the usual reaction-limited colloidal aggregation (RLCA) or diffusion-limited colloidal aggregation cluster fractal dimensions for a high or low barrier, respectively. However, for medium-ranged, shallow barriers, for which the aggregation takes a long time as in the RLCA case, a very low fractal dimension is obtained, reaching values as low as 1.2 for the longer-ranged potentials considered. Nevertheless, as the aggregation proceeds, the cluster fractal dimension crosses over to a value close to the RLCA one, indicating that the small clusters of low fractal dimensionality act as the aggregating units of an RLCA system, given that they take a long time to react. A correspondence is made between our results and the experimental results by Hurd and Schaefer (Phys. Rev. Lett. 54 (1985) 1043).

Aggregation of protein-coated colloidal particles: Interaction energy, cluster morphology, and aggregation kinetics

The salt-induced aggregation of polystyrene particles, partially coated with monomeric bovine serum albumin molecules, has been studied. For this purpose, the cluster fractal dimension df and the van Dongen–Ernst homogeneity parameter λ have been measured by means of static and dynamic light scattering techniques. The theoretical analysis indicates that aggregation takes places at an interaction energy minimum predicted by a modified Derjaguin–Landau–Verwey–Overbeeck (DLVO) theory when additional osmotic and elastic terms are considered. It could be shown that the depth of the energy minimum is directly related to df and λ. Moreover, the particle–particle distance within the cluster could also be assessed and compared with the interparticle distance at which the DLVO-predicted minimum appears.

Techniques for characterization of fractal clusters using X-ray microscopy

X-ray microscopy is well suited to study cluster formation of colloidal particles experimentally. Presenting different techniques for characterizing the structure of colloidal clusters we show how the fractal dimension of the clusters can be calculated from the X-ray micrographs, which may be considered as projections of these clusters. Dealing with simple projections we show that the fractal dimension is not preserved in the projection, but about Δd=0.15 smaller than the fractal dimension itself. This method can be applied to clusters with d≤1.9. Generally colloidal particles have a non zero transmission for X-rays leading different greyscales of the pixels in X-ray micrographs. These greyscales can be used to calculate the fractal dimension in the interval 1.8≤d≤2.2 exceeding the principle limit d=2 given for simple projections Combining both methods the fractal dimension of colloidal clusters can be calculated in a broad range.

Visualization and measurement of multiphase flow in porous media using light transmission and synchrotron x-rays.

Non-aqueous phase liquids enter the vadose zone as a result of spills or leaking underground storage facilities, thus contaminating groundwater resources. Measuring the contaminant concentrations is important in assessing the risk to human health and the environment and to develop effective remediation. This research presents the development and application of the light transmission method (LTM) for three-phase flow systems, aimed at investigating unstable fingered flow in a soil-air-oil-water system. The LTM uses the hue and intensity of light transmitted through a slab chamber to measure fluid content, since total liquid content is a function of both hue and light intensity. Evaluation of the LTM is obtained by comparing experiments with LTM and synchrotron X-rays. The LTM captures the spatial resolution of the fluid contents and can provide new insights into rapidly changing, two-phase and three-phase flow systems. Application of the LTM as a visualization technique for environmental and physical phenomena is noted. Visualization by LTM of groundwater remediation by surfactants as well as visualization of model cluster growth and fractal dimensions was also explored.

Concentration effects on two- and three-dimensional colloidal aggregation

By means of extensive numerical simulations of diffusion-limited colloidal aggregation in two and three dimensions, we have found the concentration dependence of the structural and dynamical quantities. Both on- and off-lattice simulations were used in 2D to check the independence of our results on the simulational algorithms and on the space structure. The range in concentration studied spanned two-and-a-half orders of magnitude, in both dimensionalities. In two dimensions, it was found that the cluster fractal dimension difference from the zero-concentration value shows a linear increase with the concentration, while this increase is of a square root type for the three-dimensional case. For the exponent z, defining the increase of the weight-average cluster size as a function of time, the difference from the zero-concentration value in three dimensions is again of a square root type increase with concentration, while in two dimensions this increase goes as the 0.6 power of the concentration. We give arguments for the drastic change in the power laws for the case of the fractal dimension, when going from two to three dimensions, and for the small change for the case of the kinetic exponent z. We also present the master curves for the scaling of the cluster size distribution and their dependence on concentration, in both dimensionalities.

Colloidal aggregation in the presence of a gravitational field

We present the results of a computational model for colloidal aggregationthat considers the Brownian motion, sedimentation and deposition experienced by the colloidal particles and clusters.Among our results, we find that for intermediate strengths ofdownward gravitational drift, the aggregation crosses overfrom diffusion-limited colloidal aggregation to another type ofaggregation with a higher cluster fractal dimension, Df. Wealso get a critical gelation concentration that is higher by severalorders of magnitude than for the non-drifting case. Wealso found a speeding up followed by a slowing down of theaggregation rate and an algebraically decaying cluster sizedistribution. As the drift strength becomes much higher, the newfractal dimension is reduced due to the anisotropy of theclusters, becoming more elongated along the vertical direction.We finally present the scaling shown by the cluster sizedistribution for all the drift strengths studied.

Two-Dimensional Colloidal Aggregation: Concentration Effects

Extensive numerical simulations of diffusion-limited (DLCA) and reaction-limited (RLCA) colloidal aggregation in two dimensions were performed to elucidate the concentration dependence of the cluster fractal dimension and of the different average cluster sizes. Both on-lattice and off-lattice simulations were used to check the independence of our results on the simulational algorithms and on the space structure. The range in concentration studied spanned 2.5 orders of magnitude. In the DLCA case and in the flocculation regime, it was found that the fractal dimension shows a linear-type increase with the concentration φ, following the law: df=dfo+aφc. For the on-lattice simulations the fractal dimension in the zero concentration limit, dfo, was 1.451±0.002, while for the off-lattice simulations the same quantity took the value 1.445±0.003. The prefactor a and exponent c were for the on-lattice simulations equal to 0.633±0.021 and 1.046±0.032, while for the off-lattice simulations they were 1.005±0.059 and 0.999±0.045, respectively. For the exponents z and z′, defining the increase of the weight-average (Sw(t)) and number-average (Sn(t)) cluster sizes as a function of time, we obtained in the DLCA case the laws: z=zo+bφd and z′=z′o+b′φd′. For the on-lattice simulations, zo, b, and d were equal to 0.593±0.008, 0.696±0.068, and 0.485±0.048, respectively, while for the off-lattice simulations they were 0.595±0.005, 0.807±0.093, and 0.599±0.051. In the case of the exponent z′, the quantities z′o, b′, and d′ were, for the on-lattice simulations, equal to 0.615±0.004, 0.814±0.081, and 0.620±0.043, respectively, while for the off-lattice algorithm they took the values 0.598±0.002, 0.855±0.035, and 0.610±0.018. In RLCA we have found again that the fractal dimension, in the flocculation regime, shows a similar linear-type increase with the concentration df=dfo+aφc, with dfo=1.560±0.004, a=0.342±0.039, and c=1.000±0.112. In this RLCA case it was not possible to find a straight line in the log–log plots of Sw(t) and Sn(t) in the aggregation regime considered, and no exponents z and z′ were defined. We argue however that for sufficiently long periods of time the cluster averages should tend to those for DLCA and, therefore, their exponents should coincide with z and z′ of the DLCA case. Finally, we present the bell-shaped master curves for the scaling of the cluster size distribution function and their evolution when the concentration increases, for both the DLCA and RLCA cases.

Aggregation kinetics and structure of cryoimmunoglobulins clusters

Cryoimmunoglobulins are pathological antibodies characterized by a temperature-dependent reversible insolubility. Rheumatoid factors (RF) are immunoglobulins possessing anti-immunoglobulin activity and usually consist of an IgM antibody that recognizes IgG as antigen. These proteins are present in sera of patients affected by a large variety of different pathologies, such as HCV infection, neoplastic and autoimmune diseases. Aggregation and precipitation of cryoimmunoglobulins, leading to vasculiti, are physical phenomena behind such pathologies. A deep knowledge of the physico-chemical mechanisms regulating such phenomena plays a fundamental role in biological and clinical applications. In this work, a preliminary investigation of the aggregation kinetics and of the final macromolecular structure of the aggregates is presented. Through static light scattering techniques, the gyration radius R g and the fractal dimension D m of the growing clusters have been determined. However, while the initial aggregation mechanism could be described using the universal reaction-limited cluster–cluster aggregation (RLCCA) theory, at longest times from the beginning of the process, the RLCCA theory fails and a restructuring of clusters is observed together with an increase of the cluster fractal dimension D m up to a value D m∼3. The time t n, at which the restructuring takes place, and the final cluster size can be modulated by varying the quenching temperature.