Constant bond breakup probability model for reversible aggregation processes

A Population Balance Model for Shear-Induced Polymer-Bridging Flocculation of Total Tailings

Shear-induced polymer-bridging flocculation is widely used in the solid–liquid separation process in cemented paste backfill, beneficial to water recycling and tailings management in metal mines. A flocculation kinetics model based on Population Balance Model (PBM) is proposed to model the polymer-bridging flocculation process of total tailings. The PBM leads to a system of ordinary differential equations describing the evolution of the size distribution, and incorporates an aggregation kernel and a breakage kernel. In the aggregation kernel, a collision frequency model describes the particle collision under the combined effects of Brownian motions, shear flow, and differential sedimentation. A semi-empirical collision efficiency model with three fitting parameters is applied. In the breakage kernel, a new breakage rate coefficient model with another three fitting parameters is introduced. Values of the six fitting parameters are determined by minimizing the difference between experimental data obtained from FBRM and modeling result through particle swarm global optimization. All of the six fitting parameters vary with flocculation conditions. The six fitting parameters are regressed with the flocculation factors with six regression models obtained. The validation modeling demonstrates that the proposed PBM quantifies well the dynamic evolution of the floc size during flocculation under the given experimental setup. The investigation will provide significant new insights into the flocculation kinetics of total tailings and lay a foundation for studying the performance of the feedwell of a gravity thickener.

Quantification of energy input required for chitin nanocrystal aggregate size reduction through ultrasound

Nanoparticles have been claimed to contribute efficiently to e.g. the mechanical strength of composite materials when present as individual particles. However, these particles tend to aggregate. In this paper we prepare nanocrystals from chitin, a product with high potential added value for application in bio-based materials, and investigate the effect of ultrasound on de-aggregation. Chitin nanocrystals with a length ~ 200 nm and a diameter ~ 15 nm, were obtained via acid hydrolysis of crude chitin powder. Freeze drying resulted in severe aggregation and after redispersion sizes up to ~ 200 µm were found. Ultrasound treatment was applied and break up behaviour was investigated using static light scattering, dynamic light scattering, and laser diffraction. Our results suggest that the cumulative energy input was the dominant factor for chitin nanocrystal aggregate breakup. When a critical energy barrier of ~ 100 kJ/g chitin nanocrystals was exceeded, the chitin nanocrystal aggregates broke down to nanometre range. The break up was mostly a result of fragmentation: the aggregation energy of chitin nanocrystal aggregates was quantified to be ~ 370 kJ/g chitin nanocrystals and we hypothesize that mainly van der Waals interactions and hydrogen bonds are responsible for aggregation.

Thermoplastic bio-nanocomposites: From measurement of fundamental properties to practical application

Although the discovery of plastic has revolutionized materials used in many industries and by consumers, their non-biodegradable nature has led to one of the greatest problems of our times: plastic waste in the environment. Bioplastics which are biobased and biodegradable, have been suggested as alternatives for their fossil based counterparts, but their properties often do not meet the requirements that standard plastics would, and are in clear need of improvement. One way to do so is by the addition of nanoparticles which, when homogeneously dispersed, have been reported to result in great improvements. However, in practice, homogenous distribution of nanoparticles is not that trivial due to their tendency to aggregate, also after addition to the polymer matrix. Although theoretical frameworks to prevent this process are available, we feel that the options explored in practice are often rather trial and error in nature. For that reason, we review the theories available, aiming to facilitate the design of the nanocomposites for a sustainable future. We first discuss thermodynamic frameworks which revolve around nanoparticle aggregation. To minimize nanoparticle aggregation, the nanoparticle and polymer can be selected in such a way that they have similar polar and dispersive surface energies. The second part is dedicated to nanocomposite processing, where kinetic effects act on the nanocomposite material therewith influencing its final morphology, although it is good to point out that other factors such as reaggregation also affect the final nanocomposite morphology. The third section is dedicated to how nanoparticles affect the polymer matrix to which they are added. We describe how interactions at an atomic scale, result in the formation of an interphasial region which ultimately leads to changed bulk material properties. From these three sections, we conclude that three parameters are often overlooked when designing nanocomposites, namely the surface energies of the nanoparticles and polymers, the aggregation bond energy or strength, and the interphase region. Therefore, in the fourth section we provide an overview of techniques to identify these three parameters. We finish with a summery and outlook for the design of bio nanocomposites, where we bring all insights from the previous four sections together.

Nonequilibrium continuous phase transition in colloidal gelation with short-range attraction

The dynamical arrest of attractive colloidal particles into out-of-equilibrium structures, known as gelation, is central to biophysics, materials science, nanotechnology, and food and cosmetic applications, but a complete understanding is lacking. In particular, for intermediate particle density and attraction, the structure formation process remains unclear. Here, we show that the gelation of short-range attractive particles is governed by a nonequilibrium percolation process. We combine experiments on critical Casimir colloidal suspensions, numerical simulations, and analytical modeling with a master kinetic equation to show that cluster sizes and correlation lengths diverge with exponents  ~1.6 and 0.8, respectively, consistent with percolation theory, while detailed balance in the particle attachment and detachment processes is broken. Cluster masses exhibit power-law distributions with exponents  -3/2 and  -5/2 before and after percolation, as predicted by solutions to the master kinetic equation. These results revealing a nonequilibrium continuous phase transition unify the structural arrest and yielding into related frameworks.

Tuning mitochondrial structure and function to criticality by fluctuation-driven mechanotransduction

Cells in vascular walls are exposed to blood pressure variability (BPV)-induced cycle-by-cycle fluctuations in mechanical forces which vary considerably with pathology. For example, BPV is elevated in hypertension but reduced under anesthesia. We hypothesized that the extent of mechanical fluctuations applied to vascular smooth muscle cells (VSMCs) regulates mitochondrial network structure near the percolation transition, which also influences ATP and reactive oxygen species (ROS) production. We stretched VSMCs in culture with cycle-by-cycle variability in area strain ranging from no variability (0%), as in standard laboratory conditions, through abnormally small (6%) and physiological (25%) to pathologically high (50%) variability mimicking hypertension, superimposed on 0.1 mean area strain. To explore how oxidative stress and ATP-dependent metabolism affect mitochondria, experiments were repeated in the presence of hydrogen peroxide and AMP-PNP, an ATP analog and competitive inhibitor of ATPases. Physiological 25% variability maintained activated mitochondrial cluster structure at percolation with a power law distribution and exponent matching the theoretical value in 2 dimensions. The 25% variability also maximized ATP and minimized cellular and mitochondrial ROS production via selective control of fission and fusion proteins (mitofusins, OPA1 and DRP1) as well as through stretch-sensitive regulation of the ATP synthase and VDAC1, the channel that releases ATP into the cytosol. Furthermore, pathologically low or high variability moved mitochondria away from percolation which reduced the effectiveness of the electron transport chain by lowering ATP and increasing ROS productions. We conclude that normal BPV is required for maintaining optimal mitochondrial structure and function in VSMCs.

Self-assembling outside equilibrium: emergence of structures mediated by dissipation

Self-assembly under non-equilibrium conditions may give rise to the formation of structures not available at equilibrium.

Aggregation in colloidal suspensions: effect of colloidal forces and hydrodynamic interactions

The forces acting in colloidal suspensions and affecting their stability and aggregation kinetics are considered. The approximations used for these forces in numerical simulations and the importance of the balanced account for both colloidal forces and hydrodynamic interactions are discussed. As an example the results of direct numerical simulations of kinetics of aggregation either with account for hydrodynamic interaction between particles or without it are compared by varying the parameters of the interaction potential between particles and fraction of solid. Simulations are based on the Langevin equations with pairwise interaction between particles and take into account Brownian, hydrodynamic and colloidal forces. It is confirmed that the neglecting of hydrodynamic interaction results in an accelerated growth of aggregates. The results of numerical simulations of aggregation kinetics are compared with well known analytical solutions.

Endocytic pathway rapidly delivers internalized molecules to lysosomes: an analysis of vesicle trafficking, clustering and mass transfer

Lysosomes play a critical role in intracellular drug delivery. For enzyme-based therapies, they represent a potential target site whereas for nucleic acid or many protein drugs, they represent the potential degradation site. Either way, understanding the mechanisms and processes involved in routing of materials to lysosomes after cellular entry is of high interest to the field of drug delivery. Most therapeutic cargoes other than small hydrophobic molecules enter the cells through endocytosis. Endocytosed cargoes are routed to lysosomes via microtubule-based transport and are ultimately shared by various lysosomes via tethering and clustering of endocytic vesicles followed by exchange of their contents. Using a combined experimental and numerical approach, here we studied the rates of mass transfer into and among the endocytic vesicles in a model cell line, 3T3 fibroblasts. In order to understand the relationship of mass transfer with microtubular transport and vesicle clustering, we varied both properties through various pharmacological agents. At the same time, microtubular transport and vesicle clustering were modeled through diffusion-advection equations and the Smoluchowski equations, respectively. Our analysis revealed that the rate of mass transfer is optimally related to microtubular transport and clustering properties of vesicles. Further, the rate of mass transfer is highest in the innate state of the cell. Any perturbation to either microtubular transport or vesicle aggregation led to reduced mass transfer to lysosome. These results suggest that in the absence of an external intervention the endocytic pathway appears to maximize molecular delivery to lysosomes. Strategies are discussed to reduce mass transfer to lysosomes so as to extend the residence time of molecules in endosomes or late endosomes, thus potentially increasing the likelihood of their escape before disposition in the lysosomes.

Monte Carlo algorithm for simulating reversible aggregation of multisite particles

We present an efficient and exact Monte Carlo algorithm to simulate reversible aggregation of particles with dedicated binding sites. This method introduces a data structure of dynamic bond trees to record clusters and sequences of bond formations. The algorithm achieves a constant time cost for processing cluster association and a cost between O(logM) and O(M) for processing bond dissociation in clusters with M bonds. The algorithm is statistically exact and can reproduce results obtained by the standard method. We applied the method to simulate a trivalent ligand and a bivalent receptor clustering system and obtained an average scaling of O(M(0.45)) for processing bond dissociation in acyclic aggregation, compared to a linear scaling with the cluster size in standard methods. The algorithm also demands substantially less memory than the conventional method.

Formation of stable clusters in colloidal suspensions

The experimental evidence and theoretical explanations of stable cluster formation in colloidal suspensions are reviewed. The clusters form in the intermediate range between a stable suspension built up by singlets and the irreversible coagulation or gelation of the suspension. The stable clusters develop as a result of a balance between competing short range attraction and long range repulsion between colloidal particles or due to reversible flocculation in the shallow secondary potential well. Heteroaggregation in binary colloids can also result in formation of stable clusters.

Self-diffusion of reversibly aggregating spheres

Reversible diffusion limited cluster aggregation of hard spheres with rigid bonds was simulated and the self-diffusion coefficient was determined for equilibrated systems. The effect of increasing attraction strength was determined for systems at different volume fractions and different interaction ranges. It was found that the slowing down of the diffusion coefficient due to crowding is decoupled from that due to cluster formation. The diffusion coefficient could be calculated from the cluster size distribution and became zero only at infinite attraction strength when permanent gels are formed. It is concluded that so-called attractive glasses are not formed at finite interaction strength.

Aggregation kinetics of latex microspheres in alcohol–water media

We report zeta potential and aggregation kinetics data on colloidal latex particles immersed in water-alcohol media. Zeta potential values show absolute maxima for volume fractions of alcohol of 0.10 and 0.05 for ethanol and 1-propanol, respectively. For methanol, no maximum of the absolute value of the zeta potential was found. Aggregation kinetics was studied by means of a single-cluster optical sizing equipment and for alcohol volume fractions ranging from 0 to 0.1. The aggregation processes are induced by adding different potassium bromide concentrations to the samples. We expected to find a slowdown of the overall aggregation kinetics for ethanol and 1-propanol, and no significant effect for methanol, as compared with pure water data. That is, we expected the zeta potential to govern the overall aggregation rate. However, we obtained a general enhancement of the aggregation kinetics for methanol and 1-propanol and a general slowdown of the aggregation rate for ethanol. In addition, aggregation data under ethanol show a slower kinetics for large electrolyte concentration than that obtained for intermediate electrolyte concentration. We think that these anomalous behaviors are linked to layering, changes in hydrophobicity of particle surfaces due to alcohol adsorption, complex ion-water-alcohol-surface structuring, and competition between alcohol-surface adsorption and alcohol-alcohol clustering.

Linking Phase Behavior and Reversible Colloidal Aggregation at Low Concentrations: Simulations and Stochastic Mean Field Theory

We have studied the link between the kinetics of clustering and the phase behavior of dilute colloids with short range attractions of moderate strength. This was done by means of computer simulations and a theoretical kinetic model originally developed to deal with reversible colloidal aggregation. Three different regions of the phase diagram were accessed. For weak attractions, a gas phase of small clusters in equilibrium forms in the system. For intermediate attractions, the system undergoes liquid-gas separation, which is signatured by the formation of a few large droplike aggregates, a gas phase of small clusters, and an overall kinetics where a few seeds succeed in explosively growing at long times, after a lag time. Finally, for very strong attractions, fractal unbreakable clusters form and grow following DLCA-like (diffusion limited cluster aggregation) kinetics; liquid-gas separation is prevented by the strength of the bonds, which do not allow restructuration. Good qualitative and quantitative agreement is found between the dynamic simulations and the kinetic model in all the three regions.

Two-dimensional colloidal aggregation mediated by the range of repulsive interactions

We study the effect of the interaction's range on the structural and kinetic properties of a computer-simulated two-dimensional aggregating colloidal system. For this purpose, we considered that the particles of the system interact through a repulsive Yukawa potential which depends on two parameters: the value of the interaction potential between particles in contact V0 and the range of the interaction kappa(d) . We observed that the increase of the interaction range or V0 provokes the arrangement of the small aggregates in linear structures. The repulsive interactions have also a strong influence on the kinetic behavior of the coagulation process. Indeed, they induce the formation of three different time-separated aggregation regimes. In the first regime (at early states) the aggregation is dominated by the range of the repulsive forces, and the cluster-cluster repulsion increases with the cluster size. The second regime (at intermediate times) is reached when the average cluster size is larger than the interaction range. Here, the cluster-cluster repulsions do not grow anymore with the cluster size, so the probability of overcoming the repulsive barrier is the same for all clusters. This corresponds with the so-called reaction-limited-cluster-aggregation regime, where more than one collision between the clusters is needed to form a bond. The third aggregation regime is found at long aggregation times. In this region the coagulation is mainly determined by the diffusion time and the kinetics becomes diffusion controlled. A physical interpretation for the transition between chain structures and the typical fractals aggregates from the point of view of the range of the interactions is discussed. Moreover, a method has been developed in order to obtain the effect of the interactions with a non-negligible range over the aggregation rates directly from the simulations. The relation between these different regions with the parameters of the interaction potential V0 and kappa(d) is analyzed.

Phase separation and percolation of reversibly aggregating spheres with a square-well attraction potential

Reversible aggregation of spheres is simulated using a novel method in which clusters of bound spheres diffuse collectively with a diffusion coefficient proportional to their radius. It is shown that the equilibrium state is the same as with other simulation techniques, but with the present method more realistic kinetics are obtained. The behavior as a function of volume fraction and interaction strength was tested for two different attraction ranges. The binodal and the percolation threshold were determined. The cluster structure and size distribution close to the percolation threshold were found to be consistent with the percolation model. Close to the binodal phase separation occurred through the growth of spherical dense domains, while for deep quenches a system spanning network is formed that coarsens with a rate that decreases with increasing attraction. We found no indication for arrest of the coarsening.

Dynamic imaging with MRI contrast agents: quantitative considerations

Time-resolved MRI has had enormous impact in cognitive science and may become a significant tool in basic biological research with the application of new molecular imaging agents. In this paper, we examine the temporal characteristics of MRI contrast agents that could be used in dynamic studies. We consider "smart" T1 contrast agents, T2 agents based on reversible aggregation of superparamagnetic nanoparticles and sensors that produce changes in saturation transfer effects (chemical exchange saturation transfer, CEST). We discuss response properties of several agents with reference to available experimental data, and we develop a new theoretical model that predicts the response rates and relaxivity changes of aggregation-based sensors. We also perform calculations to define the extent to which constraints on temporal resolution are imposed by the imaging methods themselves. Our analysis confirms that some small T1 agents may be compatible with MRI temporal resolution on the order of 100 ms. Nanoparticle aggregation T2 sensors are applicable at much lower concentrations, but are likely to respond on a single second or slower timescale. CEST agents work at high concentrations and temporal resolutions of 1-10 s, limited by a requirement for long presaturation periods in the MRI pulse sequence.

Flocculation and percolation in reversible cluster-cluster aggregation

Off-lattice dynamic Monte-Carlo simulations were done of reversible cluster-cluster aggregation for spheres that form rigid bonds at contact. The equilibrium properties were found to be determined by the life time of encounters between two particles (te). te is a function not only of the probability to form or break a bond, but also of the elementary step size of the Brownian motion of the particles. In the flocculation regime the fractal dimension of the clusters is df=2.0 and the size distribution has a power law decay with exponent tau=1.5. At larger values of te transient gels are formed. Close to the percolation threshold the clusters have a fractal dimension df=2.7 and the power law exponent of the size distribution is tau=2.1. The transition between flocculation and percolation occurs at a characteristic weight average aggregation number that decreases with increasing volume fraction.

Cluster-cluster aggregation controlled by the number of intercluster connections: kinetics of aggregation and cluster mass frequency

The aggregation of colloids in the presence of hydrodynamic forces was investigated, employing a numerical model that took into account the masses of the individual clusters and the number of intercluster connections established when two clusters stuck together. The number of possible connections was determined by analyzing all the possible nonoverlapping configurations of stuck clusters. This operation was done for a couple of clusters of various masses, taking into account the assembly of clusters of even and uneven masses. The formulation of the constraints established a certain hierarchy in the sticking on a basis compatible with the irregular fracture model of Horwatt and co-workers. As a result, the permanent sticking of large clusters required the formation of a large number of connections, whereas that of small clusters might be realized even with a small number of connections. Thus, the aggregation started with the features of the standard reaction-limited process and this cluster growth became progressively inhibited as a result of the prevailing effects of the connection constraints. The cluster-mass frequency showed the emergence at least of a second population whose bell-shaped mass distribution was superimposed on the monotonically decreasing distribution resulting from the reaction-limited aggregation process. The results of the numerical study were confronted with those previously obtained in the aggregation of hydrated polystyrene latex particles dispersed in 1 M sodium chloride solution. The two striking features--the aggregate growth kinetics and the mass distribution function--were common to the computer-generated clusters and the latex aggregates.

Irreversible versus reversible aggregation: Mean field theory and experiments

Colloidal aggregation processes arising at different electrolyte concentrations were studied by means of experiments and confronted with theoretical predictions of different kinetic aggregation models. For this purpose, aqueous dispersions of relatively large polystyrene microspheres were chosen as experimental systems. Aggregation was induced by adding KBr electrolyte to the initially stable particle dispersions. During the aggregation processes, the cluster-size distribution was monitored by means of single cluster light scattering. Analyzing the time evolution of the monomer concentration, we found that the processes arising even at moderate electrolyte concentrations cannot be described by pure time-independent irreversible aggregation models. Hence, alternative models such as time-dependent irreversible aggregation and several reversible aggregation models were also tested. The model that considers a time-dependent sticking probability was found to fit the data quite satisfactorily. Nevertheless, the fitted was so slow that it seems not very likely to find such a behavior in real systems. The aggregation-fragmentation models reported in the literature were unable to reproduce the experimental observations. Hence, a more realistic reversible aggregation model was developed. This model accounts also for reenforced or double bonds between the constituent particles. The corresponding fit improved significantly and reached the same quality as the time-dependent model. Moreover, the obtained fitting parameters were in qualitative agreement with the DLVO predictions and so, reversible aggregation seems to be a more reasonable explanation for the experimental data than time-dependent irreversible aggregation. However, no definite statement on the possible secondary bond fragmentation mechanism may be made since both the applied shear stress in the measuring cell and thermal fluctuations can cause weaker bonds to break.