Von Smoluchowski Research Articles

Kinetics and energetics of specific intermolecular interactions.

Taking into account the energy vs. distance functions of the aspecific (macroscopic) repulsion that usually prevails between antigen (Ag) and antibody (Ab) molecules in polar media, as well as the specific (microscopic) attraction between epitope and paratope of Ag and Ab, it proved possible to determine the kinetic constants (von Smoluchowski, 1917; Hammes, 1978) of Ag-Ab interactions, from the surface properties of Ag, Ab and the aqueous medium. The kinetic constants thus found correlate well with experimentally determined kinetic constants in comparable systems, and confirm the importance of the influence of the concentration of one of the reagents (e.g. the Ab) on the kinetic association constant (Van Regenmortel et al., 1994), which is largely due to steric hindrance. Applying the same energy vs. distance approach to the influence of temperature (T) on Ag-Ab reactions, it ensues that the familiar occurrence of an apparent 'enthalpy-entropy compensation' in aqueous media is in fact the relatively gratuitous outcome of a complex set of effects caused by an increase in T, on the total free energy, the hydration energy and, as a result, on the inter-epitope-paratope distance. A close correlation exists between the outcome of these surface-thermodynamic analyses and experimental results.

A new method for calculating kinetic constants within the Rayleigh - Gans - Debye approximation from turbidity measurements

Using the Rayleigh - Gans - Debye (RGD) approximation and a von Smoluchowski kinetics, a simple theory has been developed which provides a complete fitting of the absorbance versus time curve in a homocoagulation process. Contributions from pairs of particles with zero, one, and two particles between them have been taken into account for aggregates of any size. A description of the curve in the first 20 seconds is obtained for all cases, and in general it is possible to reproduce the curve correctly for the first minute. Finally, we present a technique based on this theory for adjusting the kinetic constant in a homocoagulation process.

Scaling invariance of the von Smoluchowski rate law

The kinetics of flocculation for colloidal suspensions that occur in natural environments have often been studied in terms of the von Smoluchowski rate law. Recent theoretical studies have analyzed scale-invariant flocculation processes leading to the formation of fractal structures which, in turn, have been observed in experiments on synthetic colloids. In this paper, these processes are described in terms of the scaling invariance of the von Smoluchowski equation itself and are shown to be consistent with asymptotic limits of two of its well known exact solutions, corresponding to transport- or reaction-controlled flocculation. The resulting self-similar forms of these solutions and the associated fractal properties of the floccules are discussed in the context of proposed universality classes. The results are applied to the characterization of heterogeneous colloids found in natural aqueous systems.

Dynamic modeling of limited particle coagulation in emulsion polymerization

In emulsion polymerization, a limited degree of particle coagulation may occur. Particle coagulation is caused by a loss of colloidal stabilization when the surface coverage of emulsifier on the particles drops below a critical value. It has been demonstrated experimentally in a previous article that the time scale for particle coagulation is small compared to the time scale for particle growth by polymerization and absorption of monomer. This indicates that the coagulation process can probably be described by von Smoluchowski kinetics. Based on this result, a comprehensive dynamic model for the simulation of limited particle coagulation in emulsion polymerization has been developed. It has been shown that there is a reasonable agreement between simulations with the dynamic model and experimental data (e.g., conversion time history, particle number, and particle size distribution). © 1996 John Wiley & Sons, Inc.

Kinetic Model for the Simultaneous Processes of Flocculation and Coalescence in Emulsion Systems

We consider an emulsion system in which the two interrelated processes of flocculation and coalescence take place simultaneously. The flocculation is described in terms of von Smoluchowski's theory, whereas the coalescence is related to the rupture of the liquid films between the drops. Since the latter process takes place in aggregates, we adopt for it a kinetic scheme which formally resembles that of parallel chemical reactions. A set of differential equations governing the overall kinetics is formulated and solved for several particular cases. The rate constants of flocculation and coalescence stand as model parameters. Solutions for the number of single drops vs time are obtained in the asymptotic regimes of fast and slow coalescence. In the case of linearly built aggregates we derive an explicit expression for the total number of drops in the system. For arbitrary aggregates the complete set of kinetic differential equations is solved numerically. The results are compared with those obtained by other authors through particular averaged models of flocculation and coalescence. The theory can be employed to investigate the role of different factors for the emulsion stability: the solubility of the surfactant, the adsorption and the Gibbs elasticity of the layer, the surface viscosity etc.

Distinguishing different paths for rearrangements on surfaces

There exist a variety of theoretical models that can be employed in the description of the rearrangements on surfaces. Prominent examples are the coarsening model of Lifshitz and Slyozow and the diffusion aggregation process due to von Smoluchowski. We perform computer simulations on the merging of pores in a two-dimensional Ising model with the goal of determining the conditions under which the system follows one of these paths during reconstruction. The parameters that are varied during the investigation are the size of the pores, their separation and the temperature of the system. It is found that with increasing temperature the transition line moves towards larger distance and radii and broadens for high temperatures.

Electrophoresis of proteins in uncoated capillaries with amines and amino sugars as electrolyte additives

The effect of triethylamine and triethanolamine in the running electrolyte on the electroosmotic flow and the migration of four standard basic proteins in bare fused-silica capillaries was examined. At pH 2.5 the direction of the electroosmotic flow was anodal with both additives and at constant ionic strength its magnitude increased with increasing additive concentration. The observations are in qualitative agreement with a theoretical model which is based on the Gouy—Chapman—Stern theory and incorporates a Langmuir-type relationship and also the Von Smoluchowski expression for the electroosmotic mobility, with the approximation to identify the zeta potential with the potential at the Stern plane. From the independence of the electrophoretic mobilities of the additive concentration and type the absence of interactions between the four basic proteins and the two alkylamines is inferred. The utility of glucosamine and galactosamine as additives for the capillary electrophoresis of basic proteins is also demonstrated and their effectiveness is compared with that of the alkylamines.

Electroosmosis of polymer solutions in fused silica capillaries.

The classical von Smoluchowski equation predicts that the electroosmotic mobility generated by the wall zeta potential could be suppressed if the viscosity of the solution adjacent to the wall were extremely high. When performing runs in capillaries filled with polymer solutions (2% methyl cellulose solutions with viscosities of 25 cP), however, one consistently finds that the quenching of electroosmotic mobility is substantially less than predicted by the von Smoluchowski relationship. The electroosmotic flow is progressively suppressed with subsequent electrophoretic runs, suggesting a "dynamic coating" of the polymers onto the capillary wall. This progressive reduction of electroosmotic mobility tends to a plateau value which is still substantially higher than the value derived on the basis of the von Smoluchowski relationship. The following explanation is proposed: due to the very high shear rate in the electric double layer, the polymer molecules change their orientation and/or conformation, which lowers the fluid viscosity in this region. A scaling equation for electroosmotic mobility taking into account the non-Newtonian properties of polymer solutions is derived. It predicts electric field dependence of the electroosmotic mobility as the shear rate in the double layer is proportional to the electric field. Experimental measurements confirm the dependence of the electroosmotic mobility on the electric field.

Polystyrene Latex Coagulation: A Study Using Centrifugation in Density Gradients

Coagulation of polystyrene latices was followed by using a combination of centrifugation in density gradients and light-scattering densitometry. Second-order rate constants (k 11) were obtained under various conditions. At 25° and 45°C under 0.5 mol dm -3 NaCl, they agreed with the predictions of von Smoluchowski's theory, corrected for viscous interactions. However, some discrepancies with theory were found under other conditions. Monomer and oligomer concentrations decayed more slowly than predicted, with time. Detectable, steady concentrations of these species were found at t → ∞. The formation and decay of large ( n > 15) aggregates was also followed.

Diffusion control in blood coagulation. Activation of factor X by factors IXa/VIIIa assembled on human monocyte membranes.

This study examines mechanisms that regulate the activation of blood coagulation proteases on intact cell membranes. The activation of factor X by factors IXa and VIIIa assembled on viable monocytes is presented as a biologically relevant model for membrane-dependent proteolysis of coagulation zymogens. The hypothesis that this reaction is limited by diffusion was tested by comparing predicted with observed concentration dependence, temperature dependence, and effective rate coefficient. Rates of factor X catalysis were measured using a chromogenic substrate specific for the product, factor Xa. The value of KR and of K1/2, i.e. concentrations giving half-maximal rates in reciprocal functional titrations with substrate and enzyme, respectively, were directly correlated with the concentration of the titrated component. Arrhenius plots constructed over temperatures encompassing 10-35 degrees C were biphasic with downward concavity. Apparent activation energies were 6.01 +/- 0.93 and 35.84 +/- 8.9 kcal/mol for the interval above and below the inflection point, respectively. The effective rate coefficient calculated from apparent kinetic parameters was 3.58 +/- 0.1 x 10(12) M-1 s-1. This rate is similar to the maximal rate of collision between factor X molecules and the monocyte, i.e. 2.9 x 10(12) M-1 s-1 estimated from the steady-state von Smoluchowski equation for uniformly reacting spherical particles. The observed agreement between predicted and experimental results indicates that under biologically relevant conditions, the rate of factor X activation by the intrinsic protease is controlled by diffusion of factor X toward the catalytic site.

Diffusive motion in a fractal medium in the presence of a trap.

There have been many studies of transport in an unbounded fractal medium. Here we discuss a number of quantities related to the concentration of reactants diffusing in fractal media in the presence of a trapping point. This investigation is suggested by an extension of the Smoluchowski model to calculate reaction rates in such media [M. von Smoluchowski, Phys. Z. 16, 321 (1915); 17, 557 (1917); 17, 585 (1917)]. Results, some analytic and some based on a scaling argument, are given for the flux into the trap, which is the analog of the reaction rate, in addition to the concentration profile in the neighborhood of the trap and the time dependence of the distance between the trap and the nearest untrapped particle. The results of our theory are found to be in good agreement with Monte Carlo and exact enumeration calculations for the concentration profile on a Sierpinski gasket and on an infinite percolation cluster at criticality. Some scaling and numerical results are reported for the situation in which the traps move in the presence of fixed particles.

Kinetics of crystallization and flocculation of calcium fluoride

The kinetics of growth of calcium fluoride crystals from supersaturated solutions has been investigated using the constant composition method. During the reactions, a fluoride specific ion electrode was used to control the addition of titrants containing calcium and fluoride ions in order to maintain constant supersaturation. The quadratic dependence of crystal growth rate upon the relative supersaturation suggests a spiral growth mechanism. Particle size measurement, both during the growth kinetics experiments and at equilibrium in saturated solutions of calcium fluoride, showed that considerable flocculation takes place. In the former case agglomeration rather than aggregation occurred, following the DLVO theory as developed by von Smoluchowski. The calculated Hamaker constant was 6.0·10 −21 J.

Rapid coagulation of polystyrene and poly (vinyl acetate) particles: Influence of the initial particle number

The time evolution of singlets, doublets and triplets was measured by single-particle light scattering for a monodisperse hard sphere latex (polystyrene) and a monodisperse soft sphere latex [poly (vinyl acetate)] under conditions of rapid coagulation. The experimental curves were fitted to calculated curves by introducing reversibility into the von Smoluchowski theory by variation of the coagulation and deaggregation constant. It was shown that the coagulation constant was independent of the initial particle number.

Kinetics of magnetic flocculation. II. Flocculation of coarse particles

For pt.I see ibid., vol.23, p.1447 (1990). In theoretical studies of flocculation kinetics one usually assumes that the flocculation process reaches a steady state very rapidly. This assumption is invalid when a long-range attraction between the particles is present. In this paper the authors show that such a situation occurs for magnetic flocculation, provided that the particles are not too small. A full mathematical description of such a process is very difficult. As an alternative, a modified version of von Smoluchowski's model for rapid flocculation is presented, which describes the qualitative features of their experimental observations of aqueous Mn2O3 colloids with a particle diameter of about 0.8 mu m quite well. In particular this model suggests that the deviations from the 'steady-state' predictions increase very rapidly with increasing particle size, making these results almost always incorrect for many practical applications of magnetic flocculation.

Single particle optical sizing (SPOS): II. Hydrodynamic forces and application to aggregating dispersions

The shear and extension forces occurring during the hydrodynamic focusing in our SPOS instrument which is described in the preceding paper ( E. G. M. Pelssers, M. A. Cohen Stuart, and G. J. Fleer, J. Colloid Interface Sci. 137, 350 (1990) ), are analyzed and compared with estimates for the binding forces between particles in salt coagulation and polymer flocculation. It is found that the hydrodynamic forces are several orders of magnitude smaller than these binding forces, so that the aggregate size distribution is not changed during the measurement. Also the additional orthokinetic aggregation in the apparatus is shown to be negligible. The aggregate size distribution of a coagulating dispersion could be monitored very accurately up to heptaplets. The intensity scattered under 5° by aggregates of latex particles of 350 nm radius is found to scale as V 1.70, where V is the (coalesced) volume of the aggregates. The coagulation of latex by salt follows the overall bimolecular kinetics as predicted by M. Von Smoluchowski ( Phys. Z. 17, 557, 585 (1916) and Z. Phys. Chem. 92, 129 (1917)) .

Kinetics of bridging flocculation. Role of relaxations in the polymer layer

From experimental data on the kinetics of flocculation of polystyrene latex by means of poly(ethylene oxide) it is deduced that the well known classical aggregation theory of von Smoluchowski does not apply to bridging flocculation. A new kinetic model is therefore developed. It has as a key element the flattening of the polymer molecules with time, by which they lose the ability to form bridges. We take this into account by introducing a characteristic time of reconformation. The only other parameter is a minimum coverage of ‘active’ polymer chains per particle, necessary for bridge formation. Using reasonable values for these two parameters, we find that the model describes the experimental data very well.

Light-scattering measurements of slow aggregation in colloids: Deviations from asymptotic time scaling.

We report measurements on the reversible and irreversible kinetics of aggregation in a colloid of polystyrene particles, performed by quasielastic light scattering. The experiment was made with small to moderate concentrations of the simple electrolyte inducing the aggregation in order to observe both the reversible flocculation and the slow, irreversible diffusion-limited coagulation. In the latter case we observe deviations from asymptotic scaling in time. The experimental results are well described by a numerical solution of the von Smoluchowski equation for aggregation limited by Brownian diffusion.

Coagulation in cell suspensions: Extensions of the von Smoluchowski model

Recently, a long-range interactive force between erythrocytes has been proposed (Rowlands et al., 1981, 1982a,b) based on an apparent increase in the rate of aggregation of erythrocytes in an aqueous suspension over that predicted by one model of Brownian aggregation. Here, we examine the assumptions underlying this model and propose modifications compatible with the biological constraints on the model. The refined model is represented as a series of coupled differential equations representing the change in particle number density as a function of time. Numerical solution of these equations is consistent with the absence of an intercellular force.

Flocculation kinetics of the didisperse system-computer simulation of flocculation by the random coalescence model.

The process of Brownian flocculation of suspended particles has been simulated based on the random coalescence model for polydisperse systems by using a digital computer. The conformity between von Smoluchowski's and Muller's theories was examined. It was found that the simplifying assumption by which the concept of collision probability is introduced into the theories was adequate. Since no such assumption was necessary in the present model, the model can be applied to more complicated polydisperse systems. Moreover, this model can be used to obtain not only the total number of particles formed but also the particle size distribution at any period in the flocculation process. The results indicate that the process is governed by a layering mechanism such that the flocculation is promoted by a preferential cohesion of small particles to large aggregates.

Reactivity of the hydrated electron with (ω-carboxylatoalkyl)ferricenium zwitterions. A radiation chemical study

Rate constants have been determined for the reaction of the hydrated electron with the substituted ferricenum zwitterions obtained by one-electron oxidation of 3-ferrocenylpropanoate, 4-ferrocenylbutanoate and 5-ferrocenylpentanoate anions. These values are all close to 3 × 1010 dm3 mol–1 s–1 and since, on the basis of the equation of von Smoluchowski, this implies that reaction occurs when the centres of the reactants are at a separation of at least 7 Å, it would appear that these reactions are diffusion-controlled. Thus the sandwiching of the iron atom between two cyclopentadienyl groups does not appreciably alter the high reactivity towards the hydrated electron of the ferric part, whose complexes with inorganic ligands are known to react at the diffusion-controlled limit.