Simple analytic solution of the rate equations for the early-stage aggregation kinetics of colloidal particles

The potential energy of the interaction between two approaching colloidal particles obtained by the DLVO theory can exhibit a maximum, a primary minimum, and a secondary minimum on the potential curve of the interparticle interaction energy. Behrens and Borkovec (J Colloid Interface Sci 225: 460, 2000) considered a set of coupled nonlinear differential rate equations for the early-stage aggregation kinetics of colloidal particles by taking into account the influence of the secondary minimum and derived an approximate solution to the rate equations as well as their exact numerical solutions. In the present article, an improved simple analytic solution is derived for these rate equations. The obtained solution, which involves two distinct (fast and slow) exponentially decay constants, is found to be in excellent agreement with numerical solutions to the rate equations with negligible errors.     

Bimodal colloidal mixtures: from fast to slow aggregation regions

A Brownian dynamics simulation has been used to investigate the aggregation kinetics of bimodal colloidal mixtures with similar surface chemistries but different sizes, driven by the DLVO interaction potential. The time evolution of structural formation is examined by the mean number of neighbors under fast and slow aggregation regions. It was found that the electrolyte ionic strength affects the kinetic pattern of colloidal aggregation. Under the high electrolyte ionic strength conditions (fast aggregation), the selective aggregation of the least stable single component can take place in the early stage, while the other component is enriched in this least stable component in the later stage. With the ionic strength decreasing (towards the slow aggregation), the hybrid aggregation (selective aggregation and heteroaggregation) gradually dominates the aggregation kinetics. Also in the early stage, this evolves to the heteroaggregation of different components under lower ionic strength conditions. The volume fraction has no obvious influence on this kinetic pattern in the early stage.     

The effect of the salt concentration and counterion valence on the aggregation of latex particles at the air/water interface

An experimental study on colloidal aggregation in two dimensions is presented. This study shows that a high amount of electrolyte concentration is necessary to screen the particle interactions and to induce the aggregation process. Our results indicate that the stability of the colloidal particles, with a diameter of 735 nm, increases when they are trapped at the air-water interface. The reason for this stability is the existence of long-range repulsive interactions between the external parts of the particles that are propagated at the air phase. The subphase electrolyte concentration that separates the slow aggregation rate region from the fast aggregation rate region, the critical coagulation concentration (C.C.C.), has been determined for counterions with a different valence. Two regimes can be distinguished: at low salt concentration the aggregation process becomes slower and the aggregation is reaction limited. At high ionic strength the repulsive interactions between the immersed part of the particles are very weak and the aggregation rate tends to grow. However, because of the aerial repulsive interactions, pure diffusion-limited cluster aggregation is never found.     

Breakdown of colloid filtration theory: role of the secondary energy minimum and surface charge heterogeneities

The mechanisms and causes of deviation from the classical colloid filtration theory (CFT) in the presence of repulsive Derjaguin-Landau-Verwey-Overbeek (DLVO) interactions were investigated. The deposition behavior of uniform polystyrene latex colloids in columns packed with spherical soda-lime glass beads was systematically examined over a broad range of physicochemical conditions, whereby both the fluid-phase effluent particle concentration and the profile of retained particles were measured. Experiments conducted with three different-sized particles in a simple (1:1) electrolyte solution reveal the controlling influence of secondary minimum deposition on the deviation from CFT. In a second series of experiments, sodium dodecyl sulfate (SDS) was added to the background electrolyte solution with the intent of masking near-neutrally charged regions of particle and collector surfaces. These results indicate that the addition of a small amount of anionic surfactant is sufficient to reduce the influence of certain surface charge inhomogeneities on the deviation from CFT. To verify the validity of CFT in the absence of surface charge heterogeneities, a third set of experiments was conducted using solutions of high pH to mask the influence of metal oxide impurities on glass bead surfaces. The results demonstrate that both secondary minimum deposition and surface charge heterogeneities contribute significantly to the deviation from CFT generally observed in colloid deposition studies. It is further shown that agreement with CFT is obtained even in the presence of an energy barrier (i.e., repulsive colloidal interactions), suggesting that it is not the general existence of repulsive conditions which causes deviation but rather the combined occurrence of "fast" and "slow" particle deposition.     

Heteroaggregation in binary mixtures of oppositely charged colloidal particles

Heteroaggregation (or heterocoagulation) rate constants have been measured in mixtures of well-characterized colloidal particles of opposite charge with multiangle static and dynamic light scattering. This technique permits routine measurements of absolute heteroaggregation rate constants, also in the presence of homoaggregation. Particularly with multiangle dynamic light scattering, one is able to estimate absolute heteroaggregation rate constants accurately in the fast aggregation regime for the first time. Heteroaggregation rate constants have also been measured over a wide range of parameters, for example, ionic strength and different surface charge densities. Amidine latex particles, sulfate latex particles, and silica particles have been used for these experiments, and they were well characterized with respect to their charging and homoaggregation behavior. It was shown that heteroaggregation rate constants of oppositely charged particles increase slowly with decreasing ionic strength, and provided the surface charge is sufficiently large, the rate constant is largely independent of the surface charge. These trends can be well described with DLVO theory without adjustable parameters.     

流体力学半径对估算胶体微球聚集速率常数的影响

胶体粒子聚集速率常数实验值远低于理论值一直是被普遍关注的问题.聚集速率常数的理论推导是基于粒子的几何半径来考虑的,但决定粒子扩散速率及聚集速率的应该是粒子的流体力学半径(大于几何半径),因而它是使聚集速率常数实验值低于理论值的因素之一.影响流体力学半径的因素很多,其中,带电粒子在溶液中因表面存在双电层,会明显增大流体力学半径,造成聚集速率减慢.而双电层的厚度又随溶液中离子强度的不同而改变.本工作在聚集速率的公式中引入了修正因子,即几何半径与其流体力学半径之比,以修正由于用几何半径代替流体力学半径带来的误差.其中几何半径和流体力学半径可以分别用扫描电镜(SEM)和动态光散射(DLS)来测定.以两种粒径的聚苯乙烯带电微球为例,考察了在不同离子强度下,该误差的大小.结果发现,对于半径为30 nm的微球,用流体力学半径计算的慢聚集速率常数比理论值偏低约8%.该误差随离子强度增加而减少.对于快聚集情况,流体力学半径对聚集速率基本没有影响.     

Salt-regulated attraction and repulsion of spherical polyelectrolyte brushes towards polyelectrolyte multilayers

Adsorption of colloidal particles presents an interesting alternative to the modification of surfaces using covalent coupling or physisorption of molecules. However, to tailor the properties of these materials full control over the effective particle-substrate interactions is required. We present a systematic investigation of the adsorption of spherical polyelectrolyte brushes (SPB) onto polyelectrolyte multilayers (PEM). A brush layer grafted from colloidal particles allows the incorporation of various functional moieties as well as the precise adjustment of their adsorption behaviour. In the presence of oppositely charged surfaces the amount of adsorbed SPB monotonically increases with the ionic strength, whereas equally charged substrates efficiently prevent colloidal attachment below a threshold salt concentration. We found that the transition from the osmotic to the salted brush regime at approximately 100 mM coincided with a complete loss of substrate selectivity. In this regime of high ionic strength, attractive secondary interactions become dominant over electrosteric repulsion. Due to the soft polyelectrolyte corona a surface coverage exceeding the theoretical jamming limit could be realized. Both the adsorption kinetics and the resulting thin film morphologies are discussed. Our study opens avenues for the production of two-dimensional arrays and three-dimensional multilayered structures of SPB particles.     

Self-assembly of doublets from flattened polymer colloids

Bottom-up fabrication methods are used to assemble strong yet flexible colloidal doublets. Part of a spherical particle is flattened, increasing the effective interaction area with another particle having a flat region. In the presence of a moderate ionic strength, the flat region on one particle will preferentially "bond" to a flat region on another particle in a deep (≥10 kT) secondary energy minimum. No external field is applied during the assembly process. Under the right conditions, the flat-flat bonding strength is ≥10× that of a sphere-sphere interaction. Not only can flat-flat bonds be quite strong, but they are expected to remain freely rotatable and flexible, with negligible energy barriers for rotation because particles reside in a deep secondary energy minimum with a ~20-30 nm layer of fluid between the ~1 μm radius particles. We present a controlled technique to flatten the particles at room temperature, the modeling of the interparticle forces for flattened spheres, and the experimental data for the self-assembly of flat-flat doublets.     

Theoretical studies of the early stage coagulation kinetics for a charged colloidal dispersion

We study the early stage coagulation kinetics for a charged colloidal dispersion which is here modeled by an effective two-body colloid-colloid potential. The colloidal system was physically prepared by choosing sets of colloidal parameters varying in particular the Hamaker constant and the particle's size. The kinetics of coagulation process was driven by the addition of an indifferent electrolyte and assumed to proceed in two quasi-steady steps. In the first step, colloidal particles are destabilized by the presence of a second potential minimum to diffuse from a bulk-stabilized liquid phase to a flocculated phase. In the second step, we assume that different entities are found in the second potential minimum. The entities comprise secondary dimers, secondary dimers undergoing redispersion, and monomers still in singlet states. If, under favorable condition, this kind of interaction-driven diffusive motion continues, a fraction of the secondary dimers will be induced to undergo primary dimers formation in the first deep minimum. Whether or not the latter process occurs is determined either energetically by the potential barrier falling below a prescribed value, say of 15k(B)T, or/and the second potential minimum becoming negligibly small (with a magnitude coagulation transition and would throw a fresh light on the use of both the energy and the kinetic criteria for understanding the colloidal stability such as those observed in the liquid-liquid coexistence.     

Structural evolution of colloidal crystals with increasing ionic strength

We have directly observed the structural evolution of colloidal crystals as a function of increasing ionic strength using confocal scanning laser microscopy. Silica colloids were sedimented onto a glass substrate in deionized water to create large, single domain crystals. The solution ionic strength was then increased by one of three methods of controlled electrolyte addition: (1) direct injection of electrolyte solutions, (2) single step diffusion of electrolyte solutions through a dialysis membrane, and (3) multiple step diffusion of electrolyte solutions of increasing ionic strength through a dialysis membrane. During direct injection of electrolyte solutions, initially large, single domain colloidal crystals were shear melted and then evolved into polycrystalline structures at low ionic strengths and gels at higher ionic strengths. Diffusion of electrolyte solutions though dialysis membranes in a single step produced gradient-driven transport that also melted initial single domain crystals to yield polycrystalline and gel structures similar to the injection approach. Interestingly, the multistep diffusion of several electrolyte solutions through dialysis membranes facilitated retention of large, single domain crystals even as particles came into adhesive contact. This was achieved by reducing the contraction rate of the crystalline lattice to allow sufficient time for diffusion-limited configurational rearrangements to occur within the evolving structure. These mechanically robust, single domain colloidal crystals may find important applications as templates for photonic materials and sensors.     

Charging and aggregation of latex particles by oppositely charged dendrimers

Poly(amidoamine) (PAMAM) dendrimers were shown to adsorb strongly on negatively charged latex particles, and their effect on the particle charge and aggregation behavior was investigated by light scattering and electrophoretic mobility measurements. Time-resolved simultaneous static and dynamic light scattering was used to measure absolute aggregation rate constants. With increasing dendrimer dose, the overall charge could be tuned from negative to positive values through the isoelectric point (IEP). The aggregation is fast near the IEP and slows down further away. With decreasing ionic strength, the region of fast aggregation narrows and the dependence of the aggregation rate on the dendrimer dose is more pronounced. Surface charge heterogeneities become important for higher dendrimer generations. They widen the fast aggregation region, reduce the dependence of the aggregation rate on the dendrimer dose, and lead to an acceleration of the rate in the fast aggregation regime near the IEP. The ratio of the dendrimer charge and the particle charge exceeds the stoichiometric ratio of unity substantially and further increases with increasing generation. The tentative interpretation of such superstoichiometric charge neutralization involves coadsorption of anions and the finite thickness of the adsorbed dendrimer layer.     

Local Structure Evolution in Particle Network Formation Studied by Brownian Dynamics Simulation

The effect of solid content and colloidal interactions on the structure of forming networks of colloidal particles is studied by Brownian dynamics simulation. The different situations are compared in terms of the pair distribution function and the distribution of nearest neighbors around each particle. The results indicate that, in fast coagulation, the higher solid contents lead to a freezing-in of the liquid structure. Nevertheless, this effect can be reduced substantially by the introduction of a shallow secondary minimum and an energy barrier in the interaction potential. However, the structures resulting from such slow coagulation show a substantial degree of porosity, larger than those produced at the same solid content but by fast coagulation. It is also shown how the porosity (defined on a few particle diameters) is reflected in the distribution of nearest neighbors around the center particle, i.e., the very local conformation in the particle network. Fractal analysis shows that, at the relatively high volume fractions considered in this study, no intermediate fractal regime exists. Copyright 2000 Academic Press.     

Surface functionalization of silica nanoparticles supports colloidal stability in physiological media and facilitates internalization in cells

The influence of the surface functionalization of silica particles on their colloidal stability in physiological media is studied and correlated with their uptake in cells. The surface of 55 ± 2 nm diameter silica particles is functionalized by amino acids or amino- or poly(ethylene glycol) (PEG)-terminated alkoxysilanes to adjust the zeta potential from highly negative to positive values in ethanol. A transfer of the particles into water, physiological buffers, and cell culture media reduces the absolute value of the zeta potential and changes the colloidal stability. Particles stabilized by L-arginine, L-lysine, and amino silanes with short alkyl chains are only moderately stable in water and partially in PBS or TRIS buffer, but aggregate in cell culture media. Nonfunctionalized, N-(6-aminohexyl)-3-aminopropyltrimethoxy silane (AHAPS), and PEG-functionalized particles are stable in all media under study. The high colloidal stability of positively charged AHAPS-functionalized particles scales with the ionic strength of the media, indicating a mainly electrostatical stabilization. PEG-functionalized particles show, independently from the ionic strength, no or only minor aggregation due to additional steric stabilization. AHAPS stabilized particles are readily taken up by HeLa cells, likely as the positive zeta potential enhances the association with the negatively charged cell membrane. Positively charged particles stabilized by short alkyl chain aminosilanes adsorb on the cell membrane, but are weakly taken up, since aggregation inhibits their transport. Nonfunctionalized particles are barely taken up and PEG-stabilized particles are not taken up at all into HeLa cells, despite their high colloidal stability. The results indicate that a high colloidal stability of nanoparticles combined with an initial charge-driven adsorption on the cell membrane is essential for efficient cellular uptake.     

The effect of environmental conditions on the steric stabilization of casein micelles

In an attempt to characterize the steric stabilizing sheath around the casein micelles of bovine milk, photon correlation spectroscopy techniques have been used to measure the micellar radius on exposure to ethanolic buffers of varying pH, ionic strength and calcium concentration. It is shown that on exposure to alcohol, the stabilizing protein sheath undergoes dimensional collapse and that immediately prior to aggregation, a minimum or core radius is reached, characteristic of the diluting buffer conditions. Defining barrier thickness as the difference between the micellar radius in alcohol-free buffer and this minimum radius, the same linear relationship is observed between barrier thickness and the critical ethanol concentration required to reach the core radius and induce subsequent aggregation, whether those variations in barrier thickness were achieved by altering the pH, ionic strength or calcium level of the buffer. Considering the initial rate of response to added ethanol as a measure of barrier strength, it is observed that thicker barriers are weaker whereas thinner barriers are more resistant to collapse and hence intrinsically stronger. This paradox is qualitatively resolved by considering the stabilizing sheath to possess some of the characteristics of a weak or soft gel, whose rigidity or extent of cross-linking is influenced by the variations in buffer conditions.     

Ion-specific colloidal aggregation: population balance equations and potential of mean force

Recently reported colloidal aggregation data obtained for different monovalent salts (NaCl, NaNO(3), and NaSCN) and at high electrolyte concentrations are matched with the stochastic solutions of the master equation to obtain bond average lifetimes and bond formation probabilities. This was done for a cationic and an anionic system of similar particle size and absolute charge. Following the series Cl(-), NO(3)(-), SCN(-), the parameters obtained from the fitting procedure to the kinetic data suggest: (i) The existence of a potential of mean force (PMF) barrier and an increasing trend for it for both lattices. (ii) An increasing trend for the PMF at contact, for the cationic system, and a practically constant value for the anionic system. (iii) A decreasing trend for the depth of the secondary minimum. This complex behavior is in general supported by Monte Carlo simulations, which are implemented to obtain the PMF of a pair of colloidal particles immersed in the corresponding electrolyte solution. All these findings contrast the Derjaguin, Landau, Verwey, and Overbeek theory predictions.     

Role of the secondary minimum on the flocculation rate of nondeformable droplets

The kinetic stability of suspensions is usually associated with a decrease in the flux of flocculating particles due to the action of a repulsive potential. However, previous calculations on bitumen drops suggest the possible occurrence of relatively fast aggregation rates in systems with large electrostatic barriers for primary minimum flocculation. This indicates a strong effect of the secondary minimum in the process of aggregation. Here, emulsion stability simulations (ESS) are used to study the aggregation behavior of 11 systems showing different depths of the secondary minimum and three particle sizes. Micron size drops (as those of Bitumen emulsions) usually exhibit deep secondary minima, which rarely occur between nanometer size particles. At high surfactant concentrations, these drops do not coalesce but can still show fast aggregation rates caused by irreversible secondary-minimum flocculation. On the other hand, the extent of coalescence in nanometer-size systems markedly depends on the height of the repulsive barrier. Furthermore, the secondary minimum of these smaller particles is usually shallow, causing reversible aggregation or no aggregation at all. In this article, the consequences of the referred behaviors on the magnitude of the stability ratio are discussed.     

A new view on chemistry of solids in solution--cryo energy-filtered transmission electron microscopy (cryo-EFTEM) imaging of aggregating palladium colloids in vitreous ice

It is shown that by using cryo-transmission electron microscopy (cryo-TEM) it is possible to image the aggregation behaviour of nanoparticles while they are still in solution. This technique has allowed the study of the arrangement of colloidal palladium particles in solution by preparing the specimen by the plunge-freezing technique. This method of rapidly cooling the specimen avoids rearrangement of the particles during specimen preparation. The palladium particles were identified by energy-filtered cryo-TEM. The aggregation of particles in solution was studied as a function of pH and ionic strength. The results can be used as recommendations for colloidal solutions intended for deposition of single particles.     

The sphere-rod transition of micelles of dodecyldimethylammonium bromide in aqueous NaBr solutions,and the effects of counterion binding on the micelle size,shape and structure

Micelle size and shape of dodecyldimethylammonium bromide have been determined by measurement of light scattering from its aqueous NaBr solutions. In water and in the presence of NaBr up to 0.07 M, the Debye plots give straight lines with positive slopes, and spherical micelles having molecular weight less than 30 000 are formed. At higher NaBr concentrations, the Debye plots decrease with increasing micelle concentration, indicating the aggregation of the primary spherical micelles into larger secondary micelles. The molecular weight and the radius of gyration of the secondary micelles increase with increasing NaBr concentration, and the relation between molecular weight and radius of gyration suggests that they are rodlike and flexible. Linear logarithmic relations between micelle molecular weight and ionic strength hold for spherical and rodlike micelles, respectively, and the threshold concentration of NaBr for the sphere-rod transition is located at 0.07 M. The spherical micelle of dodecyldimethylammonium ions has a size more than 20 surfactant ions larger in NaBr solutions than in NaCl solutions, and their rodlike micelle has a shorter length in NaBr solutions than in NaCl solutions, when compared at an identical aggregation number, indicating 2 more surfactant ions in its cross-section.