Diffusion-limited Cluster-cluster Aggregation Research Articles

Self-Assembly of Ligated Gold Nanoparticles: Phenomenological Modeling and Computer Simulations

We study the assembly of ligated gold nanoparticles by both phenomenological modeling and computer simulations for various ligand chain lengths. First, we develop an effective nanoparticle-nanoparticle pair potential by treating the ligands as flexible polymer chains. Besides van der Waals interactions, we incorporate both the free energy of mixing and elastic contributions from compression of the ligands in our effective pair potentials. The separation of the nanoparticles at the potential minimum compares well with experimental results of gold nanoparticle superlattice constants for various ligand lengths. Next, we use the calculated pair potentials as input to Brownian dynamics simulations for studying the formation of nanoparticle assembly in three dimensions. For dodecanethiol ligated nanoparticles in toluene, our model gives a relatively shallower well depth and the clusters formed after a temperature quench are compact in morphology. Simulation results for the kinetics of cluster growth in this case are compared with phase separations in binary mixtures. For decanethiol ligated nanoparticles, the model well depth is found to be deeper, and simulations show hybrid, fractal-like morphology for the clusters. Cluster morphology in this case shows a compact structure at short length scales and a fractal structure at large length scales. Growth kinetics for this deeper potential depth is compared with the diffusion-limited cluster-cluster aggregation (DLCA) model.

Mechanism of suppression of protein aggregation by α-crystallin.

This review summarizes experimental data illuminating the mechanism of suppression of heat-induced protein aggregation by α-crystallin, one of the small heat shock proteins. The dynamic light scattering data show that the initial stage of thermal aggregation of proteins is the formation of the initial aggregates involving hundreds of molecules of the denatured protein. Further sticking of the starting aggregates proceeds in a regime of diffusion-limited cluster-cluster aggregation. The protective effect of α-crystallin is due to transition of the aggregation process to the regime of reaction-limited cluster-cluster aggregation, wherein the sticking probability for the colliding particles becomes lower than unity.

Study of kinetics of thermal aggregation of mitochondrial aspartate aminotransferase by dynamic light scattering: protective effect of α-crystallin

Thermal aggregation of aspartate aminotransferase from pig heart mitochondria (mAAT) has been studied at various temperatures and various protein concentrations by dynamic light scattering. The character of the dependence of protein aggregate size on time indicates that aggregation of mAAT proceeds in the regime of diffusion-limited cluster-cluster aggregation. Suppression of mAAT aggregation by alpha-crystallin is due to transition of the aggregation process into the regime of reaction-limited cluster-cluster aggregation. Realization of this regime of aggregation means that the sticking probability for the colliding particles is less than unity.

Aggregation of magnetic holes in a rotating magnetic field

We experimentally investigated field-induced aggregation of nonmagnetic particles confined in a magnetic fluid layer when rotating magnetic fields were applied. After application of a magnetic field rotating in the plane of the fluid layer, the single particles start to form two-dimensional clusters, like dimers, trimers, and more complex structures. These clusters aggregated again and again to form bigger clusters. During this nonequilibrium process, a broad range of cluster sizes was formed, and the scaling exponents z and z;{'} of the number of clusters N(t) approximately t;{-z;{'}} and average cluster size S(t) approximately t;{z} were calculated. The process could be characterized as diffusion-limited cluster-cluster aggregation. We found that all sizes of clusters that occurred during an experiment fall on a single curve, as the dynamic scaling theory predicts. However, the characteristic scaling exponents z;{'},z and crossover exponents Delta were not universal. A particle tracking method was used to find the dependence of the diffusion coefficients D_{s} on cluster size s . The cluster motions show features of Brownian motion. The average diffusion coefficients D_{s} depend on the cluster size s as a power law D_{s} proportional, variants;{gamma} where values of gamma as different as gamma=-0.62+/-0.19 and gamma=-2.08+/-0.51 were found in two of the experiments.

DLCA AND LANGEVIN DYNAMICS APPROACHES OF SOL GEL TRANSITION: A COMPARISON VIA THE FRACTAL DIMENSION DURING AGGREGATION

We compared two models of aggregation which represent the sol gel transition of base-catalyzed colloidal gels. The two models are a Langevin dynamics approach of this transition and the DLCA model (diffusion limited cluster-cluster aggregation). For large numbers of particles, we compared the fractal dimension of the gelling network via the structure factor of the bulk. We observed that the Langevin dynamics approach did give more realistic fractal dimension than the DLCA model. This is due to the fact that Langevin dynamics evolves as a function of a physical time.

Constant flux relation for diffusion-limited cluster-cluster aggregation

In a nonequilibrium system, a constant flux relation (CFR) expresses the fact that a constant flux of a conserved quantity exactly determines the scaling of the particular correlation function linked to the flux of that conserved quantity. This is true regardless of whether mean-field theory is applicable or not. We focus on cluster-cluster aggregation and discuss the consequences of mass conservation for the steady state of aggregation models with a monomer source in the diffusion-limited regime. We derive the CFR for the flux-carrying correlation function for binary aggregation with a general scale-invariant kernel and show that this exponent is unique. It is independent of both the dimension and of the details of the spatial transport mechanism, a property which is very atypical in the diffusion-limited regime. We then discuss in detail the "locality criterion" which must be satisfied in order for the CFR scaling to be realizable. Locality may be checked explicitly for the mean-field Smoluchowski equation. We show that if it is satisfied at the mean-field level, it remains true over some finite range as one perturbatively decreases the dimension of the system below the critical dimension, d_{c}=2 , entering the fluctuation-dominated regime. We turn to numerical simulations to verify locality for a range of systems in one dimension which are, presumably, beyond the perturbative regime. Finally, we illustrate how the CFR scaling may break down as a result of a violation of locality or as a result of finite size effects and discuss the extent to which the results apply to higher order aggregation processes.

Noncovalent columnar structures based on β-cyclodextrin

A new method for the synthesis of associates of cyclodextrins (CDs) of the columnar type consisting of the precipitation of CDs from aqueous solutions into acetone at lowered temperatures is developed. It is shown that columnar structures exist in both a crystalline state and in aqueous solutions. Hydrodynamic radii and molecular masses of noncovalent columnar structures (NCSs) in aqueous solutions are determined by the dynamic and static light scattering methods. The degree of association of noncovalent columnar polymers is ∼40. It is revealed the NCS associates based on β-CD are stable and their hydrodynamic radius Rh is equal to 100 ± 10 nm. The kinetics of interactions of initial β-CD and NCS with poly(propylene oxide) (PPO) is studied. The pattern of kinetic curves of Rh growth upon interaction between NCS and PPO indicates that the aggregation of the particles of polymer inclusion complex proceeds in the regime of reaction-limited cluster-cluster aggregation. Kinetic curves describing the interaction processes between β-cyclodextrin and PPO are characterized by the presence of induction period t0. At t > t0, Rh ∞ t0.56 which is typical for the diffusion-limited cluster-cluster aggregation. Schemes of the formation of polymer inclusion complexes between initial β-CD or NCS and poly(propylene oxide) are proposed. Comparison of kinetic data on the complexation of β-CD in solution in the form of associates of two types with PPO demonstrates that columnar forms of associates are reactive species acting as macroreceptors.

Solid-state dewetting-mediated aggregation of nanoparticles

The deposition of Au onto thin Xe films at a low temperature leads to cluster formation. The subsequent Xe sublimation results in cluster aggregation and delivery to the substrate in a process known as buffer-layer-assisted growth. Previously, this process was described in terms of a diffusion-limited cluster-cluster aggregation process during layer-by-layer desorption of the buffer. Instead, significant diffusion, restructuring, and dewetting of the Xe occur prior to desorption, and this leads to cluster aggregation. Cluster motion and aggregation are driven by capillary forces as the dewetting film retreats and sublimes. Kinetic Monte Carlo simulations reproduce the experimentally observed particle shapes and size distributions, and they provide additional insight into the interaction of the particles with the dewetting front. The presence of nanoscale particles on the film inhibits dewetting and significantly alters the shape of the front.

Filler size effects on the conductivity of polymer nanocomposites: Semiconductive phthalocyanine nanoparticles in epoxy matrices

Three Cobalt(III) phthalocyanine (Phthalcon) powders with different particle sizes and chemical compositions, but almost equal XRD spectra and powder conductivity were synthesized and used as conductive fillers in crosslinked epoxy matrices. Two of these Phthalcons are new compounds. The relation between the conductivity of the composites and the type and amount of filler used was determined. The influence of particle size and chemical composition on this relation appeared to be minimal. These composites had a percolation threshold of 0.9 vol % and a maximum volume conductivity of 10−7 S/cm. Detailed analysis showed that the particle networks have very similar fractal structures and that they are likely to be formed by diffusion limited cluster-cluster aggregation during processing. Evidence is presented that these particle networks are formed at an early stage of crosslinking and that the charge transfer between particles in the networks is neither limited by the Phthalcon particle size, nor by the presence of polymer matrix between the particles. The maximum volume conductivity of these composites is likely to be limited by the amount of filler used, the crystal structure defects on the particle surface, and the fractality and the imperfection of the particle networks. The impact of these findings on the conductivity of other polymer nanocomposites is discussed. © 2008 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 46: 1079–1093, 2008

Scaling in the aggregation dynamics of a magnetorheological fluid

We present experimental results on the aggregation dynamics of a magnetorheological fluid, namely, an aqueous suspension of micrometer-sized superparamagnetic particles, under the action of a constant uniaxial magnetic field using video microscopy and image analysis. We find a scaling behavior in several variables describing the aggregation kinetics. The data agree well with the Family-Vicsek scaling ansatz for diffusion-limited cluster-cluster aggregation. The kinetic exponents z and z' are obtained from the temporal evolution of the mean cluster size S(t) and the number of clusters N(t), respectively. The crossover exponent Delta is calculated in two ways: first, from the initial slope of the scaling function; second, from the evolution of the nonaggregated particles, n1(t). We report on results of Brownian two-dimensional dynamics simulations and compare the results with the experiments. Finally, we discuss the differences obtained between the kinetic exponents in terms of the variation in the crossover exponent and relate this behavior to the physical interpretation of the crossover exponent.

Kinetics of thermal aggregation of glycogen phosphorylase b from rabbit skeletal muscle: Mechanism of protective action of α‐crystallin

The kinetics of thermal aggregation of glycogen phosphorylase b (Phb) from rabbit skeletal muscle have been studied by dynamic light scattering (0.08M Hepes, pH 6.8, containing 0.1M NaCl; 48 degrees C). The hydrodynamic radius of the start aggregates determined from the initial linear parts of the dependences of the hydrodynamic radius (R(h)) on time was found to be 16.7 +/- 1.0 nm. At rather high values of time, the R(h) value for the protein aggregates becomes proportional to t(1/1.8) = t(0.56) suggesting that the aggregation process proceeds in the regime of diffusion-limited cluster-cluster aggregation. In the presence of alpha-crystallin, a protein possessing the chaperone-like activity, the process of protein aggregation switches to the regime of reaction-limited cluster-cluster aggregation as indicated by the exponential dependence of the R(h) value on time. It was shown that the addition of alpha-crystallin raises the rate of thermal inactivation of Phb. These data in combination with the results of the study of interaction of Phb with alpha-crystallin by analytical ultracentrifugation suggest that alpha-crystallin interacts with the intermediates of unfolding of the Phb molecule.

Effect of α-crystallin on thermal aggregation of glycogen phosphorylase b from rabbit skeletal muscle

Thermal aggregation of rabbit skeletal muscle glycogen phosphorylase b (Phb) has been investigated using dynamic light scattering under conditions of a constant rate of temperature increase (1 K/min). The linear behavior of the dependence of the hydrodynamic radius on temperature for Phb aggregation is consistent with the idea that thermal aggregation of proteins proceeds in the kinetic regime wherein the rate of aggregation is limited by diffusion of the interacting particles (the regime of "diffusion-limited cluster-cluster aggregation"). In the presence of alpha-crystallin, a protein exhibiting chaperone-like activity, the dependence of the hydrodynamic radius on temperature follows the exponential law; this suggests that the aggregation process proceeds in the kinetic regime where the sticking probability for colliding particles becomes lower than unity (the regime of "reaction-limited cluster-cluster aggregation"). Based on analysis of the ratio between the light scattering intensity and the hydrodynamic radius of Phb aggregates, it has been concluded that the addition of alpha-crystallin results in formation of smaller size starting aggregates. The data on differential scanning calorimetry indicate that alpha-crystallin interacts with the intermediates of the unfolding process of the Phb molecule. The proposed scheme of thermal denaturation and aggregation of Phb includes the stage of reversible dissociation of dimers of Phb into monomers, the stage of the formation of the starting aggregates from the denatured monomers of Phb, and the stage of the sticking of the starting aggregates and higher order aggregates. Dissociation of Phb dimer into monomers at elevated temperatures has been confirmed by analytical ultracentrifugation.

Application of the DLCA model to “natural” gels: The allophanic soils

Volcanic soil comprises weathering products such as allophane, originating from a leaching process of volcanic ashes and glasses. These soils are interesting in terms of mitigation of the greenhouse effect (C sequestration), because they are known for accumulating more C than non-volcanic soils. Allophanes are natural amorphous silicates and have physical features very close to those of synthetic gels. Knowledge of the allophanic soil structure is required to understand the sequestration mechanism. In this paper, nitrogen adsorption-desorption experiments, measured on allophanic soil samples, show that the hydraulic diameter (Dh) is shifted towards smaller size while the pore volume (Vp) and specific surface area (S) increase, when the allophane content of the soil increases. We introduce a numerical model to simulate the structure of this “natural gel”. The algorithm is based on Diffusion-Limited Cluster-Cluster Aggregation in which larger particles hinder the DLCA. As a function of the relative content of allophane (gel) and larger particles, the textural properties (Vp, S, Dh) of the different simulated structure are calculated using a simple triangulation method.

Computer simulation of diffusion-limited cluster-cluster aggregation with an Epstein drag force

The motion of particles, dispersed in a medium, between collisions with each other can, in limiting situations, be either ballistic (straight line) or diffusive (random walker). The diffusive regime can be divided into two distinct subregimes. The "continuum regime" exhibits Stokes-Einstein-type diffusion (no-slip surface boundary condition) with a frictional coefficient proportional to the particle size (linear dimension). The "Epstein regime," as we shall refer to it, is characterized by a frictional coefficient proportional to the particle cross-sectional area, hence an Epstein-type diffusion (slip surface). The purpose of the current study is to illuminate the dynamics of dilute-limit aggregation in the Epstein regime. We present results from low volume fraction Monte Carlo simulations of cluster-cluster aggregation in the Epstein regime with the particle motion based on each particle's cross-sectional area. Our findings indicate that aggregates grown under Epstein conditions have a fractal dimension of approximately 1.8, similar to that of diffusion-limited cluster-cluster aggregates (DLCA) in the continuum regime. The kinetic exponent z in the Epstein regime is found to be z approximately 0.8, lower than its value for both the continuum regime DLCA (z = 1) and for the ballistic cluster aggregation regime (z approximately 2). Cluster size distribution data for Epstein systems are found to scale at large cluster sizes with exponents consistent with the kinetic data. A scaling argument for predicting the kinetic exponent and kernel homogeneity based on the mass or size dependence of the particle velocity and collision cross section is presented and is seen to give accurate results for dilute and intermediate values of particle volume fractions not only for the current study, but also for work done by other researchers with various choices for the aggregation kernel.

Three-dimensional Ising model confined in low-porosity aerogels: A Monte Carlo study

The influence of correlated impurities on the critical behavior of the three-dimensional (3D) Ising model is studied using Monte Carlo simulations. Spins are confined into the pores of simulated aerogels (diffusion-limited cluster-cluster aggregation) in order to study the effect of quenched disorder on the critical behavior of this magnetic system. Finite-size scaling is used to estimate critical couplings and exponents. Long-range correlated disorder does not affect the critical behavior. Asymptotic exponents differ from those of the pure 3D Ising model, but it is impossible, with our precision, to distinguish them from the randomly diluted Ising model.

Structures of DNA-Linked Nanoparticle Aggregates

The room-temperature structure of DNA-linked gold nanoparticle aggregates is investigated using a combination of experiment and theory. The experiments involve extinction spectroscopy measurements and dynamic light scattering measurements of aggregates made using 60 and 80 nm gold particles and 30 base-pair DNA. The theoretical studies use calculated spectra for models of the aggregate structures to determine which structure matches the observations. These models include diffusion-limited cluster-cluster aggregation (DLCA), reaction-limited cluster-cluster aggregation (RLCA), and compact (nonfractal) cluster aggregation. The diameter of the nanoparticles used in the experiments is larger than has been considered previously, and this provides greater sensitivity of spectra to aggregate structure. We show that the best match between experiment and theory occurs for the RLCA fractal structures. This indicates that DNA hybridization takes place under irreversible conditions in the room-temperature aggregation. Some possible structural variations which might influence the result are considered, including the edge-to-edge distance between nanoparticles, variation in the diameter of the nanoparticles, underlying lattice structures of on-lattice compact clusters, and positional disorders in the lattice structures. We find that these variations do not change the conclusion that the room-temperature structure of the aggregates is fractal. We also examine the variation in extinction at 260 nm as temperature is increased, showing that the decrease in extinction at temperatures below the melting temperature is related to a morphological change from fractal toward compact structures.

Stability of binary colloids: kinetic and structural aspects of heteroaggregation processes.

This review reports on recent advances in our knowledge about the stability of binary colloids. We focus not only on experimental results but also discuss theoretical and simulation studies regarding kinetic and structural aspects of heteroaggregation processes arising in such systems. In the first part of this work, heteroaggregation of oppositely charged particles is reviewed. When the interactions are short ranged, binary diffusion-limited cluster-cluster aggregation takes place. In this case, the short time behavior of the system follows the Hogg, Healy and Fuerstenau (HHF) theory. At long times, however, stable aggregates may form and remain in the system. Furthermore, cluster discrimination is observed, clusters that differ only by one constituent particle were found to behave quite differently. When the range of the interactions is increased, the latter effects become more pronounced. The fractal dimension of heteroaggregates is, in general, smaller than the values reported for fast and slow homoaggregation processes. In some cases, even values close to unity were obtained. This means that heteroaggregates have an open branched structure that may approach a chain-like morphology. In the second part of this work, we briefly discuss similar effects arising in heteroaggregation phenomena due to differences in particle size and chemical composition. The third part of this review tackles recent developments in the field of equilibrium phase diagrams of binary colloids. In the last section, the relatively small number of papers about heteroaggregation processes in two-dimensional systems is also discussed.

Formation and structure of stable aggregates in binary diffusion-limited cluster-cluster aggregation processes

Binary diffusion-limited cluster-cluster aggregation processes are studied as a function of the relative concentration of the two species. Both, short and long time behaviors are investigated by means of three-dimensional off-lattice Brownian Dynamics simulations. At short aggregation times, the validity of the Hogg-Healy-Fuerstenau approximation is shown. At long times, a single large cluster containing all initial particles is found to be formed when the relative concentration of the minority particles lies above a critical value. Below that value, stable aggregates remain in the system. These stable aggregates are composed by a few minority particles that are highly covered by majority ones. Our off-lattice simulations reveal a value of approximately 0.15 for the critical relative concentration. A qualitative explanation scheme for the formation and growth of the stable aggregates is developed. The simulations also explain the phenomenon of monomer discrimination that was observed recently in single cluster light scattering experiments.

Pore Structure Simulation of Gels with a Binary Monomer Size Distribution

The pore size distribution in silica gels can be tailored by the addition of silica soot particles during the gel formation. We introduce a numerical model in order to simulate the structure of this “composite gel”. The algorithm is based on Diffusion-Limited Cluster-Cluster Aggregation model with an initial binary distribution of monomer sizes. The textural properties of the simulated gels are calculated using a simple triangulation method. Nitrogen adsorption-desorption experiments show that with the powder addition the mean pore size is shifted towards larger pore size and the specific surface area decreases. Numerical results of the mean pore size, specific surface area, and particles are in good agreement with experimental data. Because of these textural properties this new type of gels and aerogels has larger permeability and interesting properties as host matrix. The composite gels and the numerical model could also be helpful to simulate the natural allophanic gel found in volcanic soils.

Persistence in systems with conserved order parameter

We consider the low-temperature coarsening dynamics of a one-dimensional Ising ferromagnet with conserved Kawasaki-like dynamics in the domain representation. Domains diffuse with size-dependent diffusion constant, $D(l) \propto l^\gamma$ with $\gamma = -1$. We generalize this model to arbitrary $\gamma$, and derive an expression for the domain density, $N(t) \sim t^{-\phi}$ with $\phi=1/(2-\gamma)$, using a scaling argument. We also investigate numerically the persistence exponent $\theta$ characterizing the power-law decay of the number, $N_p(t)$, of persistent (unflipped) spins at time $t$, and find $N_{p}(t)\sim t^{-\theta}$ where $\theta$ depends on $\gamma$. We show how the results for $\phi$ and $\theta$ are related to similar calculations in diffusion-limited cluster-cluster aggregation (DLCA) where clusters with size-dependent diffusion constant diffuse through an immobile `empty' phase and aggregate irreversibly on impact. Simulations show that, while $\phi$ is the same in both models, $\theta$ is different except for $\gamma=0$. We also investigate models that interpolate between symmetric domain diffusion and DLCA.