Diffusion-limited Colloid Aggregation Research Articles

SERS study on the aggregation mechanisms resulting from the orientation of dipyridinic derivatives on gold nanoparticles

In this work, was studied the adsorption and orientation of three dipyridinic derivatives 9,10-bis-((E)-2-(pyridin-4-yl)vinyl)anthracene (DPAC), 1,4-bis-((E)-2-(pyridin-4-yl)vinyl)naphthalene (DPNA-T) and 2,6-bis-((E)-2-(pyridin-4-yl)vinyl)naphthalene (DPNA-L) on gold nanoparticles, using Surface Enhanced Raman Scattering (SERS). Systematic modification in the shapes of the bifunctional systems (Cross-shape, T-shape and Linear-shape) shows changes significant in the preferential orientation of these analytes on the nanostructured gold surface. Additional data from UV–vis measurements and TEM images are in agreement with the Reaction Limited Colloid Aggregation (RLCA) mechanisms for DPAC and DPNA-T and Diffusion Limited Colloid Aggregation (DLCA) mechanisms for DPNA-L, showing that for the same analyte concentration, the aggregation mechanism depends on the molecular shape. These results allow us to rationalize the fundamental aspects involved in the development of devices based on plasmonic resonance with potential applications in the field of molecular electronics.

Inhibited radiation transmittance and enhanced thermal stability of silica aerogels under very-high temperature

Silica aerogels are applied extensively in thermal insulation due to their extra-low density and thermal conductivity. However, this can be heavily undermined under high temperature, especially beyond 700–800 °C, due to not only a high transparency for radiation heat transfer, but also an instability caused by nanoparticle collapse. The present study demonstrated a solution to this problem by two steps. Firstly, the size effect of nanoparticles is explored by experimentally preparing unconventional samples composed of large-diameter particles (larger than 20 nm). The prepared sample microstructures and radiative characteristics are investigated by SEM apparatus and FTIR measurements, respectively. Additionally, based on the fractal structures reconstructed using the Diffusion-Limited Colloid Aggregation (DLCA) method, the effects of particle size on thermal stability of silica aerogels are theoretically investigated and the light shielding performance is numerically analyzed by the Generalized Multiparticle Mie-solution (GMM) method. The simulation and experimental results indicate that the enhanced size of nanoparticles can improve the radiative inhibition and thermal stability property under high temperature significantly. However, the ideal above should be refined as it brings an increasing contribution of conduction heat transfer. Thus, a novel structure of core-mantle nanoparticle is proposed to retain the excellent resistance in radiation but avoid an increased thermal conduction. The present work may pave a new direction for developing aerogels under extreme temperature conditions, compared with the widely-used opacifier addition.

Self-assembly of ellipsoidal particles at fluid-fluid interfaces with an empirical pair potential

Colloidal particles adsorbed at fluid-fluid interfaces interact via mechanisms that can be specific to the presence of interfaces, for instance, lateral capillary interactions induced by nonspherical particles. Capillary interactions are highly relevant for self-assembly and the formation of surface microstructures, however, these are very challenging to model due to the multibody nature of capillary interactions. This work pursues a direct comparison between our computational modelling approach and experimental results on surface microstructures formed by ellipsoidal particles. We begin by investigating the accuracy of using pairwise interactions to describe the multibody capillary interaction by contrasting exact two- and three-particle interaction energies and we find that the pairwise approximation appears reasonable for the experimentally relevant configurations studied. We then develop an empirical pair potential and use it in Monte-Carlo type simulations to efficiently model the structure formation process for relevant particle properties such as aspect ratio, contact angle and surface coverage, and succeed in reproducing our experimental observations where we spread sterically-stabilised ellipsoidal particles onto an oil-air interface at high surface coverage. At lower surface coverages, we find that the self-assembly process falls into the diffusion-limited colloid aggregation universality class.

Coagulation behavior of humic acid in aqueous solutions containing Cs+, Sr2+ and Eu3+: DLS, EEM and MD simulations.

The coagulation behaviors of humic acid (HA) with Cs+ (10-500 mM), Sr2+ (0.8-10.0 mM) and Eu3+ (0.01-1.0 mM) at different pH values (2.8, 7.1 and 10.0) were acquired through a dynamic light scattering (DLS) technique combined with spectroscopic analysis and molecular dynamic (MD) simulations. The coagulation rate and the average hydrodynamic diameter (<Dh>) increased significantly as the concentration of nuclides increased. <Dh> could be scaled to time t as <Dh>∝ ta at higher Sr2+ concentrations, which shows that HA coagulation is consistent with the diffusion-limited colloid aggregation (DLCA) model. Trivalent Eu3+ induced HA coagulation at a much lower concentration than bivalent Sr2+ and monovalent Cs+. The coagulation value ratio of Sr2+ and Eu3+ to Cs+ is almost proportional to Z-6, indicating that the HA coagulation process is generally consistent with the Schulze-Hardy rule. Spectroscopic analysis indicated that the complexation between nuclides and carboxylic/phenolic groups of HA molecules played important roles in the coagulation of HA. MD modelling suggested that Sr2+ and Eu3+ ions increased the coagulation process through the formation of intra- or inter-molecular bridges between negatively charged HA molecules, whereas for Cs+, no inter-molecular bridges were formed. This work offers new insight into the interactions between HA and radionuclides and provides a prediction for the roles of HA in the transportation and elimination of radionuclides in severely polluted environments.

Amino acid induced fractal aggregation of gold nanoparticles: Why and how

Gold colloids are the object of many studies as they are reported to have potential biological sensing, imaging and drug delivery applications. In the presence of certain amino acids the aggregation of the gold nanoparticles into linear structures is observed, as highlighted by the appearance of a second plasmon band in the UV–Vis spectra of the colloid. The mechanism behind this phenomenon is still under debate. In order to help elucidate this issue, the interaction between gold colloids and different amino acids, modified amino acids and molecules mimicking their side-chain was monitored by UV–Vis absorption, DLS and TEM. The results show that phenomenon can be rationalized in terms of the Diffusion Limited Colloid Aggregation (DLCA) model which gives rise to the fractal aggregation colloids. The global charge of the compound, which influences the ionic strength of the solution, and the ease with which the compound can interact with the GNPs and affect their surface potential, are, the two parameters which control the DLCA regime. Calculations based on the Derjaguin, Landau, Verwey and Overbeek (DLVO) theory confirm all the experimental observations.

A ‘Universal’ Bottom-up Sol-gel Based Synthesis of Pzt (1-x)/x Submicrometric Structures: 1. Dynamic Light Scattering Characterization of Precursor Stability and Aggregation Kinetics

This work proposes an extension of a sol-gel based synthesis route for obtaining highly hydrophobic, uniformly nanosized and stable precursor solutions of lead zirconate titanate [Pb(Zr1-xTix)O3 or PZT (1-x)/x] for any given (1-x)/x Zr:Ti ratio. Several representative PZT compositions were chosen and their precursors were studied by taking into account the time dependence of the mean particles size andthe predominant aggregation kinetics. Size distributions were recorded by the Dynamic Light Scattering technique and the aggregation kinetics was explored by considering a Diffusion-Limited Colloidal Aggregation model. Transmission electron microscopy was also used to characterize some morphological features of our resultant precursor nanoparticles. Our final results supports the universalization of this approach to any PZT composition and its extension to other material systems. This work is intended to be the first on a series dealing with the sol-gel based synthesis of any PZT material at any scale using a bottom-up approach.

Linking self-assembly, rheology, and gel transition in attractive colloids

We propose a microscopic framework based on nonequilibrium statistical mechanics to connect the microscopic level of particle self-assembly with the macroscopic rheology of colloidal gelation. The method is based on the master kinetic equations for the time evolution of the colloidal cluster size distribution, from which the relaxation time spectrum during the gelation process can be extracted. The relaxation spectrum is a simple stretched-exponential for irreversible diffusion-limited colloidal aggregation gelation, with a stretching exponent df/3, where df is the mass fractal dimension. As opposed to glassy systems, the stretched-exponential relaxation does not result from quenched disorder in the relaxation times, but from the self-assembly kinetics in combination with the fractal character of the process. As the master kinetic equations for colloidal aggregation do not admit bond-percolation solutions, the arrest mechanism is driven by the interconnection among fractal clusters when excluded volume becomes active, i.e., at sufficiently high packing of clusters. The interconnections between rigid clusters decrease the soft modes of the system and drive a rigidity-percolation transition at the cluster level. Using the Boltzmann superposition principle, the creep and the full rheological response can be extracted for both irreversible and thermoreversible colloidal aggregation. In the case of thermoreversible gelation, the attraction energy is finite and plays the role of the control parameter driving a nonequilibrium phase transition into a nonequilibrium steady-state (the gel). A power-law spectrum coexisting with a stretched-exponential cut-off is predicted leading to power-law rheology at sufficiently high frequency. Our theory is in good agreement with experimental data of different systems published by other authors, for which no theory was available.

Coagulation Kinetics of Humic Aggregates in Mono- and Di-Valent Electrolyte Solutions

Coagulation behaviors of humic acids (HAs) aggregates in electrolyte solutions at different pHs, valences and concentrations of electrolyte cations were investigated using dynamic light scattering technique in combination of other analytical tools. For monovalent electrolyte sodium chloride (NaCl) solution, at its low concentrations the average hydrodynamic radius (<Rh>) of aggregates kept nearly constant. However, at high NaCl concentrations, <Rh> could be scaled to the time t as <Rh> ∝ t(a), suggesting a diffusion-limited colloid aggregation (DLCA). The coagulation value of NaCl in a buffer at pH 7.1 was calculated to be in a range of 61.3-84.4 mM. Divalent cation Mg(2+) was far more effective in enhancing the HA coagulation, as evidenced by a lower coagulation value (between 1.0 and 1.7 mM) and a more rapid coagulation rate. Such an enhancement could be explained by the combined effects of electrostatic repulsion, complexation and bridging. The highest coagulation rate (d<Rh>/dt) and coagulation value at different pHs followed the order of: acidic > neutral > alkaline, and alkaline > neutral > acidic, respectively. Such a difference was associated with the extent of hydrogen bond and electrostatic repulsion at different protonation/deprotonation states of carboxyl and phenolic -OH groups. Transmission electron microscopic imaging reveals that HA was predominantly globular aggregates with a rough periphery at pH 5.26, and was changed to smooth spherical particles at pH 10.00. These results are useful for better understanding the coagulation behaviors of HAs in both natural and engineered aqueous systems.

Alcohol induced structural and dynamic changes in β-lactoglobulin in aqueous solution: A neutron scattering study

Structural and dynamic properties of β-lactoglobulin (β-LG) were revealed as a function of alcohol concentration in ethanol- and trifluoroethanol(TFE)-water mixtures with circular dichroism (CD), small-angle neutron scattering (SANS) and quasi-elastic neutron scattering (QENS). The CD spectra showed that an increase in TFE concentration promotes the formation of the β-sheet structure of β-LG. The SANS-intensities were fitted using form factors for two attached spheres for the native and native-like states of the protein. At higher alcohol concentrations, where aggregation takes place, a form factor modelling diffusion limited colloidal aggregation (DLCA) was employed. The QENS-data were analyzed in terms of internal motions for all alcohol concentrations. While low concentrations of TFE (10% (v/v)) lead to an increase of the mean square amplitudes of vibrations <u2> and a retention of a native-like structure — but not to an increase of the characteristic radius of proton diffusion processes a. Addition of 20% (v/v) of TFE induces aggregation, going along with a further increase of <u2>. Further increase of TFE concentration to 30% (v/v) changes the nanoscale structure of the oligomeric nucleate, but induces no further significant changes in <u2>. The present study underlines the necessity of methods sensitive to the dynamics of a system to obtain a complete picture of a molecular process.

Measuring Agglomerate Size Distribution and Dependence of Localized Surface Plasmon Resonance Absorbance on Gold Nanoparticle Agglomerate Size Using Analytical Ultracentrifugation

Agglomeration of nanoparticles during measurements in relevant biological and environmental media is a frequent problem in nanomaterial property characterization. The primary problem is typically that any changes to the size distribution can dramatically affect the potential nanotoxicity or other size-determined properties, such as the absorbance signal in a biosensor measurement. Herein we demonstrate analytical ultracentrifugation (AUC) as a powerful method for measuring two critical characteristics of nanoparticle (NP) agglomerates in situ in biological media: the NP agglomerate size distribution, and the localized surface plasmon resonance (LSPR) absorbance spectrum of precise sizes of gold NP agglomerates. To characterize the size distribution, we present a theoretical framework for calculating the hydrodynamic diameter distribution of NP agglomerates from their sedimentation coefficient distribution. We measure sedimentation rates for monomers, dimers, and trimers, as well as for larger agglomerates with up to 600 NPs. The AUC size distributions were found generally to be broader than the size distributions estimated from dynamic light scattering and diffusion-limited colloidal aggregation theory, an alternative bulk measurement method that relies on several assumptions. In addition, the measured sedimentation coefficients can be used in nanotoxicity studies to predict how quickly the agglomerates sediment out of solution under normal gravitational forces, such as in the environment. We also calculate the absorbance spectra for monomer, dimer, trimer, and larger gold NP agglomerates up to 600 NPs, to enable a better understanding of LSPR biosensors. Finally, we validate a new method that uses these spectra to deconvolute the net absorbance spectrum of an unknown bulk sample and approximate the proportions of monomers, dimers, and trimers in a polydisperse sample of small agglomerates, so that every sample does not need to be measured by AUC. These results demonstrate the potential utility of AUC to characterize NP agglomeration and sedimentation for nanotoxicity and biosensor studies, as well as to characterize NP agglomerate size and absorbance to improve LSPR and surface-enhanced Raman spectroscopy based biosensors.

Stable nanoparticle aggregates/agglomerates of different sizes and the effect of their size on hemolytic cytotoxicity

To study the toxicity of nanoparticles under relevant conditions, it is critical to disperse nanoparticles reproducibly in different agglomeration states in aqueous solutions compatible with cell-based assays. Here, we disperse gold, silver, cerium oxide, and positively-charged polystyrene nanoparticles in cell culture media, using the timing between mixing steps to control agglomerate size in otherwise identical media. These protein-stabilized dispersions are generally stable for at least two days, with mean agglomerate sizes of ∼23 nm silver nanoparticles ranging from 43–1400 nm and average relative standard deviations of less than 10%. Mixing rate, timing between mixing steps and nanoparticle concentration are shown to be critical for achieving reproducible dispersions. We characterize the size distributions of agglomerated nanoparticles by further developing dynamic light scattering theory and diffusion limited colloidal aggregation theory. These theories frequently affect the estimated size by a factor of two or more. Finally, we demonstrate the importance of controlling agglomeration by showing that large agglomerates of silver nanoparticles cause significantly less hemolytic toxicity than small agglomerates.

Fractal Growth of PAMAM Dendrimer Aggregates and Its Impact on the Intrinsic Emission Properties

The dynamics of aggregation between electrostatically bound carboxylate and amine terminated poly(amidoamine) {PAMAM} dendrimers have been investigated in aqueous medium. Unlike the covalently connected tecto (dendrimer) analogues, aggregates of electrostatically bound PAMAM dendrimers can self-assemble in a dendritic pattern, generating fascinating fractal structures. The kinetic studies of the aggregation in water by dynamic light scattering (DLS) experiments suggest that diffusion limited colloidal aggregation (DLCA) is the prevailing mechanism for the fractal growth between 3.5 generation carboxylate terminated PAMAM dendrimer and fourth generation amine terminated PAMAM dendrimer. The fractal aggregation observed in aqueous medium was further corroborated by scanning and transmission electron microscopic studies. Furthermore, unprecedented enhancement in the intrinsic emission intensity from PAMAM dendrimer was observed associated with the dendritic aggregation of the dendrimer units. Most importantly, the present study suggests that electrostatic self-assembly can be utilized as an effective synthetic strategy to generate highly stable, nano- to microscale higher order complex structures from PAMAM dendrimers, which significantly reduces the nonradiative pathways of the excitons, resulting in many fold enhancements in the intrinsic emission intensity from the system.

Fractal aggregates in tennis ball systems

We present a new practical exercise to explain the mechanisms of aggregation of somecolloids which are otherwise not easy to understand. We have used tennis balls tosimulate, in a visual way, the aggregation of colloids under reaction-limited colloidaggregation (RLCA) and diffusion-limited colloid aggregation (DLCA) regimes.We have used the images of the cluster of balls, following Forrest and Witten’spioneering studies on the aggregation of smoke particles, to estimate their fractaldimension.

Formation of Stable Mesoglobules by a Thermosensitive Dendronized Polymer

Wepresentastudyontheformationofstablemesoglobulesbyathermosensitivedendronized polymer in aqueous solution. The polymer consists of a polymethacrylate backbone (Mn=0. 34� 10 6 , Mw= 1.1� 10 6 g/mol) with ethoxy-terminated oligoethylene oxide (OEO) dendrons of second generation appended onto each repeat unit. The collapse of the dendronized chains as well as their aggregation with increasing temperature is followed by monitoring the hydrodynamic radius Rh as a function of temperature with dynamic light scattering. The aggregation of the polymer stops at a certain size that depends on the heating rate leading to the formation of stable mesoglobules. Cryogenic transmission electron microscopy demon- stratesthatthesemesoglobulesadoptasphericalshapeandarerathermonodisperse.Itisfoundthatfordilute concentrations the size of the mesoglobules at 50 C only weakly depends on concentration. Moreover, no hysteresis of the formation of mesoglobules is found; that is, Rh is solely a function of temperature for both heating and cooling runs. The kinetics of the early stage of mesoglobule formation can be monitored by measurements of Rh as the function of time. It can be modeled in terms of diffusion-limited colloid aggregation (DLCA). The transition from the DLCA to the stable mesoglobules is found to be well-defined. Differentpossibleoriginsofthestabilitythatpreventsfurtheraggregationarediscussed.Aweakelectrostatic stabilizationcausedbychargesthatarecomplexedbytheOEOdendronsistheprimaryreasonforthemarked transition from the DLCA to the stable mesoglobules. Most probably, an additional slowing down of the coalescence of the mesoglobules is effected by vitrification.

Broadband light scattering measurements of the time evolution of the fractal dimension of smectite clay aggregates

The characterization of the initial stages of complex aggregate flocculation is a central problem of colloidal science and environmental studies. A fast, white light scattering technique for monitoring aggregation kinetics is used to provide a unique probe of the changes that transpire during flocculation. The time evolution of an accreting smectite clay aggregate is monitored on the basis of changes in the fractal dimension (Df) initiated by the addition of an electrolyte (MgCl2). The evaluation of the Df change reveals for the first time three distinct stages in aggregate formation that characterize smectite flocculation: a rapid, initial jump; followed by a slow decrease; and a final gradual increase, eventually reaching a plateau. The three stages in the time evolution of Df correlate with (1) collapse of loosely packed face-to-edge structures and the formation of compact face-to-face composite structures; (2) the merging of the face-to-face composites; and (3) aggregation of the composites. Stage 3 eventually reaches a quasi-equilibrium condition. The quasi-equilibrium value of Df varies from 1.86 in 1 mM MgCl2—corresponding to diffusion-limited colloidal aggregation—to 2.20 in 57 mM MgCl2—corresponding to reaction-limited colloidal aggregation. These unexpected results originate from the formation of intermediate composite particles during aggregation and the increase in the compactness of the intermediate composite particles with electrolyte concentration. The new white light optical technique demonstrated here is non-invasive and has significant implications in the investigation of the growth dynamics of colloidal systems.

Planar Voronoi cells: the violation of Aboav's law explained

In planar cellular systems $m\_n$ denotes the average sidedness of a cell neighboring an $n$-sided cell. Aboav's empirical law states that $nm\_n$ is linear in $n$. A downward curvature is nevertheless observed in the numerical $nm\_n$ data of the Random Voronoi Froth. The exact large-N expansion of $m\_n$ obtained in the present work, {\it viz.} $m\_n=4+3(\pi/n)^{{1/2}}+...$, explains this curvature. Its inverse square root dependence on $n$ sets a new theoretical paradigm. Similar curved behavior may be expected, and must indeed be looked for, in experimental data of sufficiently high resolution. We argue that it occurs, in particular, in diffusion-limited colloidal aggregation on the basis of recent simulation data due to Fern\'andez-Toledano {\it et al.} [{\it Phys. Rev. E} {\bf 71}, 041401 (2005)] and earlier experimental results by Earnshaw and Robinson [{\it Phys. Rev. Lett.} {\bf 72}, 3682 (1994)].

Controlled Clustering and Enhanced Stability of Polymer-Coated Magnetic Nanoparticles

The clustering and stability of magnetic nanoparticles coated with random copolymers of acrylic acid, styrenesulfonic acid, and vinylsulfonic acid has been studied. Clusters larger than 50 nm are formed when the coatings are made using too low or too high molecular weight polymers or using insufficient amounts of polymer. Low-molecular-weight polymers result in thin coatings that do not sufficiently screen van der Waals attractive forces, while high-molecular-weight polymers bridge between particles, and insufficient polymer results in bare patches on the magnetite surface. The stability of the resulting clusters is poor, but when an insufficient polymer is used as primary coating, and a secondary polymer is added to coat remaining bare magnetite, the clusters are stable in high salt concentrations (>5 M NaCl), while retaining the necessary cluster size for efficient magnetic recovery. The magnetite cores were characterized by TEM and vibrating sample magnetometry, while the clusters were characterized by dynamic light scattering. The clustering and stability are interpreted in terms of the particle-particle interaction forces, and the optimal polymer size can be predicted well on the basis of these forces and the solution structure and hydrophobicity of the polymer. The size of aggregates formed by limited polymer can be predicted with a diffusion-limited colloidal aggregation model modified with a sticking probability based on fractional coating of the magnetite cores.

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).

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.