Reaction-limited Cluster Aggregation Research Articles

Modeling Colloidal Particle Aggregation Using Cluster Aggregation with Multiple Particle Interactions.

In this study, we investigate the aggregation dynamics of colloidal silica by generating simulated structures and comparing them to experimental data gathered through scanning transmission electron microscopy (STEM). More specifically, diffusion-limited cluster aggregation and reaction-limited cluster aggregation models with different functions for the probability of particles sticking upon contact were used. Aside from using a constant sticking probability, the sticking probability was allowed to depend on the masses of the colliding clusters and on the number of particles close to the collision between clusters. The different models of the sticking probability were evaluated based on the goodness-of-fit of spatial summary statistics. Furthermore, the models were compared to the experimental data by calculating the structures' fractal dimension and mass transport properties from simulations of flow and diffusion. The sticking probability, depending on the interaction with multiple particles close to the collision site, led to structures most similar to the STEM data.

Modelling network formation in folded protein hydrogels by cluster aggregation kinetics.

Globular folded protein-based hydrogels are becoming increasingly attractive due to their specific biological functionality, as well as their responsiveness to stimuli. By modelling folded proteins as colloids, there are rich opportunities to explore network formation mechanisms in protein hydrogels that negate the need for computationally expensive simulations which capture the full complexity of proteins. Here we present a kinetic lattice-based model which simulates the formation of irreversibly chemically crosslinked, folded protein-based hydrogels. We identify the critical point of gel percolation, explore the range of network regimes covering diffusion-limited to reaction-limited cluster aggregation (DLCA and RLCA, respectively) network formation mechanisms and predict the final network structure, fractal dimensions and final gel porosity. We reveal a crossover between DLCA and RLCA mechanisms as a function of protein volume fraction and show how the final network structure is governed by the structure at the percolation point, regardless of the broad variation of non-percolating cluster masses observed across all systems. An analysis of the pore size distribution in the final network structures reveals that, approaching RLCA, gels have larger maximal pores than the DLCA counterparts for both volume fractions studied. This general kinetic model and the analysis tools generate predictions of network structure and concurrent porosity over a broad range of experimentally controllable parameters that are consistent with current expectations and understanding of experimental results.

Through-Bond-Driven Through-Space Interactions in a Fullerene C60 Noncovalent Dyad: An Unusual Strong Binding between Spherical and Planar π Electron Clouds and Culmination of Dyadic Fractals.

Hydrogen-bond-induced π-depletion as a criterion for π-stacking, a configurationally unique noncovalent strategy enabled an unconventional strong binding between the spherical N-fulleropyrrolidine (NFP) and the planar distributions of π electron clouds of three substituted pybates to form noncovalent fulleropyrrolidino-4-(pyrenyl) butanoate dyads of large computed interaction energies, varying between 37.49 and 44.93 kcal/mol. The geometrical distortion/bending of the alkyl tail of pybate in the noncovalent dyad was experimentally corroborated via UV-vis absorption spectroscopy, which translated into spectral broadening along with pronounced shifts in the n-π* transitions of the oxy-substituted pyrene in different solvents, ensuring through-bond interactions. Facile electron transfer through H-bond influenced the dynamic dispersive forces to be active, revealing the supremacy of through-bond over through-space interactions. The analyses of intermolecular forces using an absolutely localized molecular orbital-based energy decomposition analysis (ALMO-EDA) scheme revealed intricate insights into the intermolecular interactions and characteristic charge transfer; the dominance of forward electron transfer (pybate to NFP) over the reverse in offering stabilization was noted. Charge transfer was investigated further from natural bond orbital (NBO) and absolutely localized molecular orbital-based charge-transfer analysis (ALMO-CTA) methods, establishing the supremacy of donor-to-acceptor electron transfer over the reverse (acceptor-to-donor) one. The characteristic self-assembly of the noncovalent dyad in suitable solvents led to the formation of fractal networks via reaction-limited cluster aggregation with a fractal dimension of 2.37. Adoption of constrained molecular dynamics simulations indicated probable wrapping of pybates around NFP, leading to fractal-like assembly.

Influence of ionomer adsorption on agglomerate structures in high-solid catalyst inks

Because the agglomerate structures of carbon-black-supported platinum nanoparticles (Pt/C) in catalyst inks affect their productivity and performance, identifying the structural parameters that control the dispersion state and stability of Pt/C agglomerates is important. This study aimed to identify agglomerate structures of Pt/C in high-solidity catalyst inks by examining the rheological behavior, fractal dimension, and electrical conductivity of the inks. In inks containing 48–75% water, the amount of adsorbed ionomers decreased with decreasing water content, resulting in increases in the shear viscosity (η), storage modulus (G′), and electrical conductivity (Σ). The fractal dimension of the agglomerates (df) was 1.7–2.1, suggesting that Pt/C agglomerates form a rigid three-dimensional network via a diffusion-limited cluster aggregation process. Conversely, in inks containing 89% water, the amount of adsorbed ionomers increased, leading to the lowest values for η, G′, and Σ. The df value was 2.6, suggesting that Pt/C agglomerates formed a weak-link network via a reaction-limited cluster aggregation process, which was inhibited by the steric and electrostatic repulsion effects of the adsorbed ionomers. The results of this study demonstrate that agglomeration can be controlled through ionomer adsorption by Pt/C, leading to changes in the rheological behavior and stability of catalyst inks.

Effect of metal ion solubility on the oxidative assembly of metal sulfide quantum dots.

The versatility of the oxidative assembly method for the creation of 2D and 3D quantum dot (QD) architectures represents both an opportunity and a challenge as a method enabling controlled placement of chemically distinct QDs in multicomponent systems. The opportunity lies in the ability to independently tune the kinetics of the different components so that they are similar (leading to well-mixed systems) or different (enabling gradient or phase-segregated composites) using a wide range of variables; the challenge lies in understanding those variables and how their interplay affects the overall kinetics. Here, we show that the identity of the cation in the sulfide matrix (M = Cd2+ vs Zn2+) plays a large role in the kinetics of assembly of mass spectrometry QDs, attributed to differences in solubility. Time resolved dynamic light scattering is used to monitor the hydrodynamic radius, R¯h. ZnS shows an exponential growth associated with reaction-limited cluster aggregation (RLCA), whereas CdS demonstrates a significant induction period (10-75 min) followed by a growth step that cannot be distinguished between RLCA and diffusion limited cluster aggregation. These data correlate with relative solubilities of the nanoparticles, as probed by free-cation concentration. Data also confirm prior studies showing that cubic-closest-packed (ccp) lattices are kinetically slow relative to hexagonally closest-packed (hcp); using the slope of the ln R¯h vs time plot for the rate constant, the values of 0.510 s-1 and 3.92 s-1 are obtained for ccp ZnS and hcp ZnS, respectively. Thus, both the structure and the solubility are effective levers for adjusting the relative reactivity of QDs toward oxidative assembly.

Self-Assembly of Fibrinogen in Aqueous, Thrombin-Free Solutions of Variable Ionic Strengths.

Fibrinogen not only forms fibrin networks if assisted by thrombin but also exhibits self-assembly in dilute aqueous solutions in the absence of thrombin. It could be shown that self-assembly can be triggered in a controlled way by diluting the ionic strength set to a value of 0.14 M NaCl in the starting solutions. The present work unravels the mechanism of this self-assembly process by means of a combination of time-resolved multiangle static and dynamic light scattering and atomic force microscopy. Analysis was carried out as a function of the ionic strength adjusted by the drop in ionic strength and at variable salt compositions at a given final ionic strength. Composition was varied by changing the ratio of NaCl and phosphate buffer. The self-assembly induced by the drop of the ionic strength depends on the final value. The lower the final ionic strength gets, the faster is the self-assembly process. The variation of the salt composition at a given ionic strength has only a marginal effect, which depends on the ionic strength. The self-assembly obeys a step-growth process, where any intermediate cluster can coalesce with any other cluster. Interpretation of the data with a kinetic model based on the approach of von Smoluchowski follows a diffusion-limited cluster aggregation at ionic strength values lower than 30 mM. At an ionic strength of 30 mM, the model has to take into account a size dependence of the rate constant, and at 60 mM a transition is observed to a reaction-limited cluster aggregation.

Colloidal particle aggregation in three dimensions.

Colloidal systems are of importance not only for everyday products, but also for the development of new advanced materials. In many applications, it is crucial to understand and control colloidal interaction. In this paper, we study colloidal particle aggregation of silica nanoparticles, where the data are given in a three-dimensional micrograph obtained by high-angle annular dark field scanning transmission electron microscopy tomography. We investigate whether dynamic models for particle aggregation, namely the diffusion limited cluster aggregation and the reaction limited cluster aggregation models, can be used to construct structures present in the scanning transmission electron microscopy data. We compare the experimentally obtained silica aggregate to the simulated postaggregated structures obtained by the dynamic models. In addition, we fit static Gibbs point process models, which are commonly used models for point patterns with interactions, to the silica data. We were able to simulate structures similar to the silica structures by using Gibbs point process models. By fitting Gibbs models to the simulated cluster aggregation patterns, we saw that a smaller probability of aggregation would be needed to construct structures similar to the observed silica particlestructure.

New Insight into Cluster Aggregation Mechanism during Polymerization-Induced Self-Assembly by Molecular Dynamics Simulation.

Investigations of polymerization-induced self-assembly in emulsion were conducted using molecular dynamics simulations. Using umbrella sampling and the weighted histogram analysis method algorithm, we calculated the interaction free energy between different self-assembled copolymer aggregates. In the presence of poly(ethylene glycol) (PEG) side chains at 80 °C, an attractive interaction between the copolymer micelles is observed. This attractive well is followed, in some case, by a repulsive barrier depending on the position of the PEG side chains. The strength of this repulsive barrier controls the aggregation kinetics: a strong repulsive barrier leads to slower aggregation rate and thus larger and denser clusters (i.e., reaction-limited cluster aggregation). These clusters then coalesce into large vesicles due to the presence of interstitial water molecules in the cluster. Inversely, a weak repulsive barrier causes rapid aggregation, which gives loose and ramified clusters (i.e., diffusion-limited cluster aggregation) that coalesce after swelling with a hydrophobic monomer, leading to tubular nanostructures and small vesicles. This new mechanism approach can explain the change of morphology from spheres to fibers and vesicles depending on the polymer architecture in the case of polymerization-induced self-assembly (PISA) in emulsion.

On the permeability of colloidal gels

We reexamine and refine analytical theories for permeability in colloidal networks, with particular focus on constants and identification of approximations. The new theories are compared against numerical simulations of Stokes flow through the networks and reveal nearly quantitative power-law predictions for both pore size and permeability at low volume fractions, with systematic deviations observed only at high volume fractions. Comparison with two previously published experimental data sets yields mixed results: in one case, very good agreement is found, while in the other, only the scaling is correctly predicted. In fractal gel networks, the permeability is commonly modeled as a power-law function of volume fraction, with the fractal dimension of the network determining the power-law exponent. To quantitatively probe the influence of gel structure on permeability, we investigate this relation in structures generated by diffusion-limited cluster aggregation (DLCA) and reaction-limited cluster aggregation (RLCA) and, for contrast, non-overlapping uniform random dispersions of particles. Geometric analyses are used to determine network pore size distributions, fractal dimensions, and percolation characteristics. High-fidelity simulations of the slow viscous flow of Newtonian fluids are used to obtain first-principles-based velocity and fields and hence network permeabilities. Interestingly, the effective pore size that determines permeability is found to be somewhat larger than that measured by a method based on the insertion of spherical probes. Empirical inclusion of a fractal dimension dependence on volume fraction is found to yield quantitative results for permeabilities over the entire volume fraction range studied, in both DLCA and RLCA materials.

Diffusion- and reaction-limited cluster aggregation revisited.

We simulated irreversible aggregation of non-interacting particles and of particles interacting via repulsive and attractive potentials explicitly implementing the rotational diffusion of aggregating clusters. Our study confirms that the attraction between particles influences neither the aggregation mechanism nor the structure of the aggregates, which are identical to those of non-interacting particles. In contrast, repulsive particles form more compact aggregates and their fractal dimension and aggregation times increase with the decrease of the temperature. A comparison of the fractal dimensions obtained for non-rotating clusters of non-interacting particles and for rotating clusters of repulsive particles provides an explanation for the conformity of the respective values obtained earlier in the well established model of diffusion-limited cluster aggregation neglecting rotational diffusion and in experiments on colloidal particles.

Silica-Resorcinol-Formaldehyde Aerogels Nanostructure Modelling

The work is devoted to the study and modelling of the internal nanostructure of silica-resorcinol-formaldehyde aerogels. These aerogels are highly porous (up to 98 %) functional materials with a high specific surface area, low thermal conductivity, and good sound insulation. This material is of interest for use in thermal insulation, gas sorption, and also as a soundproof material. In addition, silica-resorcinol-formaldehyde aerogels are a material for further pyrolysis to obtain silica-carbon aerogels. Several aerogel samples experimental data were analysed. The experimental data includes amount of solvent, silica to resorcinol-formaldehyde ratio, pore size distribution, density, and specific surface area. The size and volume of different pores and specific surface area vary depending on the gelation conditions of the samples. This fact indicates the gelation conditions have a significant effect on the final aerogel structure. However, the input parameters of the model are the pore size distribution and sample density. Analysis and comparison of two methods for the generation of porous structures of silica-resorcinol-formaldehyde aerogels including Diffusion-Limited Cluster Aggregation (DLCA) and Reaction-Limited Cluster Aggregation (RLCA) are carried out and the advantages of these two methods are shown in this paper.

Growth and Aggregation Regulate Clusters Structural Properties and Gel Time.

When particles undergo aggregation, very often they form structures that can be described with fractal geometry concepts. The spatial organization of the particles embedded in these aggregates, quantified by means of their fractal dimension, plays a key role in the clusters' diffusion and aggregation process. Fractal dimensions are typically known for some specific, ideal aggregation scenarios, such as for diluted diffusion limited cluster aggregation (DLCA) or reaction limited cluster aggregation (RLCA). The situation becomes significantly more complicated as soon as the initial particle concentration increases and fractal-dimension changing phenomena, such as particle growth, occur. In this frame, the aim of the present work is twofold: (i) to investigate the clusters' spatial organization in a scenario where growth and aggregation occur simultaneously and (ii) to assess the corresponding aggregation kinetics. To this end, an ad hoc Monte Carlo model has been developed. Both DLCA and RLCA regimes have been explored at several initial primary particle concentrations and for different growth rates. The results were discussed in terms of the characteristic times of growth (τG) and aggregation (τA), as well as the rate at which the structural properties change, vR. It was then possible to propose empirical correlations in the form dm = dm (vR) and tgel = tgel (τA, τG) to relate the evolution of the mass mobility exponent dm and the gel times to the simultaneously occurring processes of growth and aggregation.

Salt-induced cluster formation of gold nanoparticles followed by stopped-flow SAXS, DLS and extinction spectroscopy.

In this work we investigate the salt-induced cluster formation of monodisperse, spherical gold nanoparticles possessing negative surface charge. Time dependent dynamic light scattering was used to determine the critical coagulation concentration and to study the kinetics of cluster growth. Cryo-TEM as well as small angle X-ray scattering allowed to obtain insights into the cluster morphology at various salt concentrations. The time dependent cluster growth was monitored by stopped-flow small angle X-ray scattering experiments and UV-Visible extinction measurements simultaneously. Our results reveal the transition from reaction limited cluster aggregation (RLCA) at low salt concentrations to diffusion limited cluster aggregation (DLCA) at high salt concentrations. We were able to correlate changes of the plasmon resonance position of the gold nanoparticles to the cluster size. By comparison to theoretical calculations using Mie theory, we found indication that stronger resonance shifts during cluster growth can be related to smaller inter-particle spacings.

Modeling simultaneous deposition and aggregation of colloids

Solid formation and particle deposition in chemical and pharmaceutical processes often leads to heat exchanger or reactor fouling, compromising operability. The aim of this work is to propose a mathematical framework, based on population balance equations, to describe the simultaneous colloid aggregation and deposition. The problem is dealt with by accounting for two distinct cluster mass distributions, the first one describing the free clusters, while the second one the deposited aggregates. Three main events are considered: (i) aggregation of two free clusters, (ii) deposition of free clusters on the reactor wall (i.e. a free cluster turns into a deposited one), and (iii) aggregation of a free cluster with a deposited one. To describe the aggregation of free clusters, three typical aggregation kernels (i.e. diffusion-limited and reaction-limited cluster aggregation as well as simple shear aggregation) have been used. Starting from those kernels, aggregation laws to describe the interaction between a free and a deposited cluster were proposed and a simple cluster-mass dependent deposition law was employed. The simulation results are then discussed in the frame of one dimensionless parameter, function of the relevant characteristic time of the process (i.e. aggregation and deposition).

In situ time-resolved small-angle X-ray scattering observation of the fractal aggregation process in tin alkoxide polymeric solution

SnO2 transparent gels have been synthesized from alkoxide precursor in toluene-2-propanol solvents. The chemical reactivity of transition metal alkoxides must be controlled in order to obtain sols and gels. In tin alkoxide based systems, this control can be achieved through complexation by a chelating agent such as acetylacetone. The gelation of the sols has been studied by in situ small-angle X-ray scattering (SAXS) measurements at the European Synchrotron Radiation Facility in Grenoble on the BM02 beamline. After the addition of water, primary particles are created and stick progressively together to form fractal aggregates. The primary particles are continually created during the aggregation process, which causes an evolution of the fractal dimension of the aggregate during gelation. This evolution is similar whatever the chemical composition is, meaning that the aggregation is ruled by one process which has been identified as reaction-limited cluster aggregation. Nevertheless, the final size of the aggregates is dependent on the chemical composition of the sols.

Evidence of viscoplastic behavior of exfoliated graphite nanofluids.

The rheological behavior of ethylene glycol-based nanofluids containing exfoliated graphite nanoplatelets has been carried out using a cone-plate Physica MCR rheometer. Initial experiments based on flow curves were carried out, the flow curves were based on the controlled shear stress model, these tests show that the studied nanofluids present non-Newtonian shear thinning behavior with yield stress. Furthermore, linear viscoelastic experiments were conducted in order to determine the viscoelastic behavior: using strain sweep and frequency sweep tests the storage and loss modulus were determined. The fractal dimension (Df) was estimated from the suspension static yield-stress and volume fraction (ϕ) dependence, and was determined to be Df = 2.36, a value consistent with a process of aggregation of RLCA type (reaction limited cluster aggregation). This value is unusual if compared with other nanofluids, and can be regarded as a result of the bidimensionality of the suspended nanoplatelets. Finally, creep-recovery tests and mechanical models confirm the viscoplastic nature of our nanofluids, a feature never shown so far for this type of systems, increasing the solid-like character in the range of concentrations studied if compared with other nanofluids reported in the literature. This is a result of the combination of a remarkable internal structure and strong interactions, which evidence an unexpected behaviour sharing many solid-like features.

From static micrographs to particle aggregation dynamics in three dimensions.

Studies on colloidal aggregation have brought forth theories on stability of colloidal gels and models for aggregation dynamics. Still, a complete link between developed frameworks and obtained laboratory observations has to be found. In this work, aggregates of silica nanoparticles (20 nm) are studied using diffusion limited cluster aggregation (DLCA) and reaction limited cluster aggregation (RLCA) models. These processes are driven by the probability of particles to aggregate upon collision. This probability of aggregation is one in the DLCA and close to zero in the RLCA process. We show how to study the probability of aggregation from static micrographs on the example of a silica nanoparticle gel at 9 wt%. The analysis includes common summary functions from spatial statistics, namely the empty space function and Ripley's K-function, as well as two newly developed summary functions for cluster analysis based on graph theory. One of the new cluster analysis functions is related to the clustering coefficient in communication networks and the other to the size of a cluster. All four topological summary statistics are used to quantitatively compare in plots and in a least-square approach experimental data to cluster aggregation simulations with decreasing probabilities of aggregation. We study scanning transmission electron micrographs and utilize the intensity-mass thickness relation present in such images to create comparable micrographs from three-dimensional simulations. Finally, a characterization of colloidal silica aggregates and simulated structures is obtained, which allows for an evaluation of the cluster aggregation process for different aggregation scenarios. As a result, we find that the RLCA process fits the experimental data better than the DLCA process.

Surfactant-assisted synthesis of mono-dispersed cubic BaTiO3 nanoparticles

In this study, poly(vinylpyrrolidone)-assisted synthesis of mono-dispersed BaTiO3 nanoparticles have been reported. The various processing parameters, namely, refluxing temperature, KOH concentration, and poly(vinylpyrrolidone) concentration, have been varied, and the effects on the growth of BaTiO3 particles have been analyzed systematically. X-ray diffraction studies indicated that poly(vinylpyrrolidone) did not affect the crystal structure, but rather influenced the crystal lattice structure. In addition, the use of surfactant poly(vinylpyrrolidone) hindered the agglomeration of the nanoparticles, and facilitated the formation of mono-dispersed core–shell organic/inorganic hybrid nanocomposite. Furthermore, the mineralizer KOH promoted the dissolution of reactants and promoted the crystallization of BaTiO3 particles. Accordingly, the dissolution-precipitation scheme was believed to be the mechanism underlying the formation of BaTiO3 particles. This was further substantiated by the experimental observations, which indicated that the nucleation and crystallization of the particles was affected by the KOH concentration in the reaction system. Finally, the formation of mono-dispersed core–shell nanocomposites proceeded via reaction limited cluster aggregation. We believe that the method proposed in this study could be extended for the synthesis of mono-dispersed nanoparticles for industrial applications.

Population Balance Modeling of Aggregation and Coalescence in Colloidal Systems

A complex interplay between aggregation and coalescence occurs in many colloidal polymeric systems and determines the morphology of the final clusters of primary particles. To describe this process, a 2D population balance equation (PBE) based on cluster mass and fractal dimension is solved, employing a discretization method based on Gaussian basis functions. To prove the general reliability of the model and to show its potential, parametric simulations are performed employing both diffusion‐limited‐cluster aggregation (DLCA) and reaction‐limited‐cluster‐aggregation (RLCA) kernels and different coalescence rates. It turns out that in both DLCA and RLCA regimes, a faster coalescence leads to smaller sized and more compact clusters, whereas a slow coalescence promotes the formation of highly reactive fractals, resulting in larger aggregates.

Transition from Fractal to Spherical Aggregates of Globular Proteins: Brownian-Like Activation and/or Hydrodynamic Stress?

We report on the structure of whey protein aggregates formed by a short heating coupled to shear at high temperatures (80-120) and neutral pH in scale-up processing conditions, using gel filtration chromatography, light scattering, small angle neutron scattering, and cryogenic transmission electron microscopy. The results are interpreted in terms of coexistence of residual non-aggregated proteins and aggregates. The characteristics of aggregates such as the size, the aggregation number and the shape evidence two different morphologies. Whereas aggregates formed at 80 °C show a selfsimilar structure down to a length scale of the monomer with a fractal dimension typical for reaction limited cluster aggregation (D~2.2), aggregates formed at higher temperature show a spherical morphology, with the structure from small angle neutron scattering data best modelled with the form factor of a polydisperse sphere. We compare the structure of these aggregates to that of aggregates formed in quiescent conditions at lab scale. The structure transition is interpreted in terms of a non-trivial interplay between three perturbation factors: interparticle interaction, temperature and shear.