Coagulation Theory Research Articles

Irreversible bimolecular reactions with inertia: from the trapping to the target setting at finite densities

We investigate numerically pseudo-first-order irreversible bimolecular reactions of the type A + B → B between hard spheres undergoing event-driven Brownian dynamics. We study the encounter rate and the survival probability of A particles as functions of the packing fraction ϕ in the trapping (a single particle diffusing among static non-overlapping traps) and target (many traps diffusing in the presence of a single static target particle) settings, as well as in the case of diffusing traps and particles (full mobility). We show that, since inertial effects are accounted for in our simulation protocol, the standard Smoluchowski theory of coagulation of non-interacting colloids is recovered only at times greater than a characteristic time Δt, marking the transition from the under-damped to the over-damped regime. We show that the survival probability S(t) decays exponentially during this first stage, with a rate 1/τ0 ∝ ϕ. Furthermore, we work out a simple analytical expression that is able to capture to an excellent extent the numerical results for t < Δt at low and intermediate densities. Moreover, we demonstrate that the time constant of the asymptotic exponential decay of S(t) for diffusing traps and particles is , where kS = 4π(DA + DB)Rρ is the Smoluchowski rate. Detailed analyses of the effective decay exponent β = d [log(−logS(t))]/d (logt) and of the steady-state encounter rate reveal that the full mobility and trapping problem are characterized by very similar kinetics, rather different from the target problem. Our results do not allow one to ascertain whether the prediction S(t) ∝ exp(−at3/2) (a = const.) as t → ∞ for the trapping problem in 3D is indeed recovered. In fact, at high density, S(t) is dominated by short encounter times, which makes it exceedingly hard to record the events corresponding to the exploration of large, trap-free regions. As a consequence, at high densities the steady-state rate simply tends to 1/τ0. Finally, we work out an analytical formula for the rate that shows a remarkable agreement with the numerics up ϕ = 0.4.

Thermodynamic and kinetic effect of organic solvent on the nucleation of nesquehonite

Nesquehonite crystallization experiments were conducted in H2O and water/dimethylformamide (DMF) mixture to examine solvent effect on the nucleation kinetics. Results show a reduction of nucleation induction time in the presence of DMF and a positive correlation between nucleation rate and DMF concentration. Analysis in the context of the classical nucleation theory reveals an unexpected increase in the surface energy of nesquehonite upon the solvent change, implying that kinetic factors, rather than surface energetics, is the driving force behind the resultant enhancement in nucleation rate. Further analyses suggest that kinetic acceleration under higher surface energy conditions is possible if nucleation proceeds through a cluster aggregation mechanism instead of critical nucleus formation. Fitting the Smoluchowski’s coagulation theory to the experimental data seems to provide support to the operation of the non-classical nucleation mechanism as it shows that the cluster association rate is independent of the sizes of the precursors but positively correlated to the DMF concentration, consistent with the scenario of growing population of sub-critical nuclei and ensuing higher probability of cluster collision/aggregation leading to an accelerated nucleation.

Coag-Flocculation Kinetics of &amp;lt;i&amp;gt;Mucuna sloanei&amp;lt;/i&amp;gt; Seed for Phosphorus Removal from Waste Water

Mucuna sloanei (MS) an environmentally friendly biomass was used as a coagulant for treatment of phosphorus containing waste water. The study evaluates the coag-flocculation efficiency of MS and its functional kinetic parameter response to varying pH and dosage of the waste water effluent. Coag-flocculation reaction order α, coag-flocculation rate constant K, and coagulation period τ1/2 were determined. The maximum coag-flocculation performance (97.4%) is recorded at rate constant, K of 1.24 × 10-4 l/mg·min, dosage of 400 mg/l, pH of 8 and coagulation period τ1/2 of 0.100 min while the minimum (61%) is recorded at K of 3 × 10-5 l/gm·min, dosage 100 mg/l, pH of 2 and τ1/2 of 8.900 mins. The results confirm that MS coagulant is an effective coagulant obeying the theory of fast coagulation in the conditions of the experiments.

Photoinduced Disaggregation of TiO2 Nanoparticles Enables Transdermal Penetration

Under many aqueous conditions, metal oxide nanoparticles attract other nanoparticles and grow into fractal aggregates as the result of a balance between electrostatic and Van Der Waals interactions. Although particle coagulation has been studied for over a century, the effect of light on the state of aggregation is not well understood. Since nanoparticle mobility and toxicity have been shown to be a function of aggregate size, and generally increase as size decreases, photo-induced disaggregation may have significant effects. We show that ambient light and other light sources can partially disaggregate nanoparticles from the aggregates and increase the dermal transport of nanoparticles, such that small nanoparticle clusters can readily diffuse into and through the dermal profile, likely via the interstitial spaces. The discovery of photoinduced disaggregation presents a new phenomenon that has not been previously reported or considered in coagulation theory or transdermal toxicological paradigms. Our results show that after just a few minutes of light, the hydrodynamic diameter of TiO2 aggregates is reduced from ∼280 nm to ∼230 nm. We exposed pigskin to the nanoparticle suspension and found 200 mg kg−1 of TiO2 for skin that was exposed to nanoparticles in the presence of natural sunlight and only 75 mg kg−1 for skin exposed to dark conditions, indicating the influence of light on NP penetration. These results suggest that photoinduced disaggregation may have important health implications.

PHYSICO-CHEMICAL THEORY OF THE COAGULATION WATER TREATMENT. THE INFLUENCE OF VARIOUS FACTORS ON THE PROCESS (REPORT 2)

From the standpoint of a new physical-chemical water treatment coagulation theory, proposed by the author, the theoretical basis of the hydrolysis of coagulants and coagulation process of pollutants of natural waters to produce potable water condition.The influence of temperature, pH, ionic composition, mixing, additives and opacifiers flocculants on the process of coagulation. Analyzed the different ways of feeding coagulants to purify water. Particular attention is paid to the contact coagulation, which allows to intensify the process of drinking water, reducing reagent consumption and improve the quality of treated water.

Research of the microstructure and precipitation strengthening in a high-temperature Fe–Ni superalloy

The paper describes the relationships between the kinetics of precipitation and growth of the intermetallic phase γ′ - Ni3(Al,Ti) and the strengthening magnitude obtained in a high-temperature Fe–Ni superalloy of the A-286 type. In order to accomplish the goal of the study, the author used the LSW coagulation theory and Brown and Ham's conventional analysis of strengthening by ordered particles. The samples were subjected to a solution heat treatment at 980°C/2h/water and then aged at 715, 750 and 780°C, with holding times 0.5-500 h. The heat-treated samples were subjected to structural analyses (TEM, X-ray diffraction) and analyses of mechanical properties (hardness test, static tensile test). Direct measurements on the electron micrographs allowed to calculate the structural parameters of the γ′ phase, i.e. mean diameter, volume fraction and mean distance between particles. In accordance with the LSW theory, linear dependencies of changes in mean diameter as a function of ageing time (t1/3) were elaborated and the activation energy (E) of the γ′ phase coagulation process was determined. The author carried out analyses of strengthening and flow stress (Δτ0) increases as a function of the particle size of the γ′ phase and determined the value of the antiphase boundary energy (yapb) for the analyzed Fe–Ni alloy. It was found that the value of APB energy of phase γ′ depended significantly on the alloy ageing temperature. Such a dependence of the quantity yapb on the ageing temperature for the investigated alloy was explained by an increase in the degree of internal arrangement of phase γ′ and by increasing sizes of ordered domains as a result of coalescence.

Modeling of particle generation in laser ablation plasma

In this study, we used the general dynamic equation (GDE) for aerosols, combined with particle charging by positive ions and electrons, to simulate particle generation in laser ablation plasmas. The population balance equations based on Brownian coagulation and diffusion charging theory were represented by using a discrete-sectional model. The temporal evolution of the particle size distribution was simulated at three different initial number concentrations of neutral monomers, positive ions and electrons, as well as at two different gas pressures.The simulated average particle size increased with time and with increasing initial concentration of neutral monomer and gas pressure. The particle growth was affected by the particle charging when both number ratios of positive ion to neutral monomer and of electron to neutral monomer are higher than 0.1. Simulated particle size distributions were compared with SEM images of particles collected at different sampling points and gas pressures in actual plasmas operating at conditions similar to those simulated. Changes of particle size with increasing distance between the target and the sampling point and with increasing the pressure were well reproduced by simulations. However, particle diameters calculated were much smaller than SEM images because physical values, such as, initial concentrations of monomer, positive ions and electrons and the recombination probability of electron and positive ions, used in calculations were invalid.

Effect of adsorbed poly(ethylene glycol) on the gelation evolution of Laponite suspensions: Aging time-polymer concentration superposition

The effects of the concentration of polyethylene glycol (PEG) and NaCl on the gelation kinetics of aqueous Laponite suspensions were investigated. The PEG concentration cp was increased from 0.063wt.% to 1.0wt.%, and the NaCl concentration cs was increased from 3.0mM to 5.0mM. The gelation process was monitored with the small amplitude oscillatory shear (SAOS) rheology. At each NaCl concentration, the time curves of storage and loss moduli G′ and G″ at different PEG concentrations were shifted horizontally to construct a master curve. Master curves for different NaCl concentrations were shifted further into a main master curve. The results were understood in terms of an increase in steric repulsion with increasing cp. A model based on the theory of interaction-limited coagulation was proposed to describe the dependence of the shift factor on the PEG concentration, taking into account of the interactions between particles. The quantitative description of the observed results has been achieved with this model.

Computer simulation of the polydispersive sol coagulation process

AbstractThis work reports on a computer‐simulated investigation of the coagulation rate for a system comprising spherical sol and coagulant particles. The discussed experiment positively verified the functioning of the simulated coagulation system, where the aggregation process proceeds in line with the particle‐cluster model as a rapid and perikinetic coagulation process that satisfies the Smoluchowski equation. The rate of the simulated coagulation process satisfies kinetic equations for both first‐order and second‐order reactions. Selected concepts and models of the coagulation theory have been also verified. © 2012 Canadian Society for Chemical Engineering

Next approximation of the random walk theory for Brownian coagulation

The new approach to Brownian coagulation theory, developed in the authors’ previous papers and based on the simple random walk theory (with a fixed elementary walk distance of migrating particles), is further improved by more realistic consideration of random walks with stochastically distributed lengths. The interpolation expressions derived in the simple random walk approach avoiding semi-empirical assumptions of the traditional models, are justified (within the calculation accuracy) by new calculation results obtained in the next approximation of the random walk theory. The developed approach being applied to the problem of heavy vapour molecules condensation in a large trap, results in the improved description of the condensation rate in the transition regime.

PLANETESIMALS IN DEBRIS DISKS OF SUN-LIKE STARS

Observations of dusty debris disks can be used to test theories of planetesimal coagulation. Planetesimals of sizes up to a couple thousand kms are embedded in these disks and their mutual collisions generate the small dust grains that are observed. The dust luminosities, when combined with information on the dust spatial extent and the system age, can be used to infer initial masses in the planetesimal belts. Carrying out such a procedure for a sample of debris disks around Sun-like stars, we reach the following two conclusions. First, if we assume that colliding planetesimals satisfy a primordial size spectrum of the form dn/ds ~ s^{-q}, observed disks strongly favor a value of q between 3.5 and 4, while both current theoretical expectations and statistics of Kuiper belt objects favor a somewhat larger value. Second, number densities of planetesimals are two to three orders of magnitude higher in detected disks than in the Kuiper belt, for comparably-sized objects. This is a surprise for the coagulation models. It would require a similar increase in the solid surface density of the primordial disk over that of the Minimum Mass Solar Nebula, which is unreasonable. Both of our conclusions are driven by the need to explain the presence of bright debris disks at a few Gyrs of age.

Field-theory methods in coagulation theory

Coagulating systems are systems of chaotically moving particles that collide and coalesce, producing daughter particles of mass equal to the sum of the masses involved in the respective collision event. The present article puts forth basic ideas underlying the application of methods of quantum-field theory to the theory of coagulating systems. Instead of the generally accepted treatment based on the use of a standard kinetic equation that describes the time evolution of concentrations of particles consisting of a preset number of identical objects (monomers in the following), one introduces the probability W(Q, t) to find the system in some state Q at an instant t for a specific rate of transitions between various states. Each state Q is characterized by a set of occupation numbers Q = {n 1, n 2, ..., n g , ...}, where n g is the total number of particles containing precisely g monomers. Thereupon, one introduces the generating functional Ψ for the probability W(Q, t). The time evolution of Ψ is described by an equation that is similar to the Schrodinger equation for a one-dimensional Bose field. This equation is solved exactly for transition rates proportional to the product of the masses of colliding particles. It is shown that, within a finite time interval, which is independent of the total mass of the entire system, a giant particle of mass about the mass of the entire system may appear in this system. The particle in question is unobservable in the thermodynamic limit, and this explains the well-known paradox of mass-concentration nonconservation in classical kinetic theory. The theory described in the present article is successfully applied in studying the time evolution of random graphs.

Stability and coagulation of natural diamond sols in LiCl solutions

The aggregate stability of natural diamond sols in LiCl aqueous solutions in the acidic region of pH values has been investigated using flow ultramicroscopy. The range of variations in parameters of structural forces for natural diamond in these solutions has been determined. The possibility of applying the Smoluchowski theory and the Muller theory of reversible coagulation to the description of the kinetics of coagulation and the equilibrium aggregate state of polydisperse diamond sols has been discussed.

Thermophoretically Dominated Aerosol Coagulation

A theory of aerosol coagulation due to size-dependent thermophoresis is presented. This previously overlooked effect is important when local temperature gradients are large, the sol population is composed of particles of much greater thermal conductivity than the carrier gas, with mean diameters much greater than the prevailing gas mean free path, and an adequate "spread" in sizes (as in metallurgical mists or fumes). We illustrate this via a population-balance analysis of the evolution of an initially log-normal distribution when this mechanism dominates ordinary Brownian diffusion.

Study on the Integrated Vortex-Grid Clarification Process and its Application

Based on Micro-vortex coagulation and Hazen theory, the process of integrated vortex grid clarification bears four characteristics. Firstly, there are a lot of micro-whirling reactors whose surfaces hole rate is designed by the raw water quality in the first and second flocculation zone. Then, an inclined-tube in the precipitate zone is installed. Thirdly, a sludge concentration zone in the tank is also equipped. Lastly, by applying the low return or non-return technologies and changing some inner structure which will reduce the total energy cost. We have not only achieved a more efficient use of the cubage of the reactors, but also reduced the total energy consumption. The paper argues that in comparison to other clarification technology, the Micro-vortex clarification process deserves wide application for its various advantages, such as a higher coagulation efficiency, a shorter reactivity time, a better quality of finished water, a stronger adaptive capability, a more convenient construction and the like.

Development of the new approach to the Brownian coagulation theory: Transition regime

The new approach for calculation of the collision frequency function in the two limiting regimes (continuous and free molecular) proposed in the previous paper [1] is extended to the general case including the intermediate transition regime. The collision frequency function in the transition regime is numerically calculated without using semi-empirical assumptions (from the literature). The best fit to the numerically calculated points can be attained with the new interpolation formulas that have a simpler form and higher accuracy in comparison with the semi-empirical models.

On the theory of Brownian coagulation

The traditional diffusion approach for calculation of the collision frequency function for coagulation of Brownian particles is critically analyzed and shown to be valid only in the particular case of coalescence of small particles with large ones and inapplicable to calculation of the coalescence rate for particles of comparable sizes. It is shown that coalescence of Brownian particles generally occurs in the kinetic regime (realized under condition of homogeneous spatial distribution of particles), however, the expression for the collision frequency function in the continuum mode of the kinetic regime formally coincides with the standard expression derived in the diffusion regime for the particular case of large and small particles. This explains the validity of the traditional form of the coagulation rate equation in a wide range of parameters, corresponding to the continuum mode. Transition from the continuum to the free molecular mode can be described by the interpolation expression derived within the new analytical approach with fitting parameters that can be specified numerically, avoiding semi-empirical approach of existing models.

Coag-Flocculation Kinetics and Functional Parameters Response of Mucuna Seed Coagulant to pH Variation in Organic Rich Coal Effluent Medium

The coag-flocculation performance of Mucuna Seed Coagulant as affected by pH variation in coal washery effluent has been investigated at room temperature using various dosages of unblended Mucuna Seed Coagulant. In addition, coag-flocculation parameters such as Coagflocculation reaction order α, αth order coag-flocculation constant K, Collision factor for Brownian Transport βBr, Smoluchowski constant KR,, Collision Efficiency ep, and Coagulation Period τ1/2 were determined. Turbidity measurement was employed using the nephelometric (turbidimetric) standard method while Mucuna Seed Coagulant preparation was adopted from the works of Adebowale and Adebowale (2007). The maximum Mucuna Seed Coagulant parameter performance is recorded at α of 2, K of 8.3334 x 10-3 m3/kg.s, dosages of (0.15 kg/m3; 0.2 kg/m3 ; 0.25 kg/m3); pH of 6 and τ1/2 of 1.7339 sec while the minimum parametric performance is recorded at α of 1; K of 6.3001 x 10-4 s-1; dosage of 0.2kg/m3; pH of 8 and τ1/2 of 1100.2161seconds. The minimum value of coag-flocculation efficiency E (%) recorded is greater than 88.00 %. Conclusively; Mucuna Seed Coagulant is an effective coagulant obeying the theory of fast coagulation at the conditions of the experiment.

The effects of organic additives on induction time and characteristics of precipitated calcium carbonate

The effects of an organic additive, polyoxyethylene sorbitan trioleate (Tween 85), on the induction time for the precipitation of calcium carbonate are experimentally and theoretically investigated. Calcium carbonate was precipitated from aqueous solutions of K 2CO 3 and Ca(NO) 3 at moderate supersaturations ranging between 5 and 16 with and without the organic additive. Experimentally it has been noticed that the induction period for CaCO 3 precipitation increases at low supersaturation and is also influenced by temperature. An increase of the induction time was noticed when Tween 85 was added in the system. The “cluster coagulation model” proposed by Qian and Botsaris (1997), which combines nucleation models and coagulation theory, was used to explain the effects of operating parameters on the induction time in terms of interfacial energy and cluster sizes.

Kinetics of Complex Formation between DNA and Cationically Charged Cylindrical Brush Polymers Observed by Stopped Flow Light Scattering

The complex formation of pUC19 DNA and a cylindrical brush polymer with quaternized poly(vinyl pyridinium) side chains is investigated by stopped flow light scattering (LS). In highly dilute solutions up to two kinetic processes are observable, depending on the mole fraction of anionic charges, x(anion) . For x(anion) < 0.2 and x(anion) > 0.4-0.5 only one kinetic process was identified leading to the largest complexes at x(anion) = 0.4 well below charge stoichiometry. For 0.2 < x(anion) < 0.4 two kinetic growth processes were identified: The initially formed complexes on the time scale of a few hundred milliseconds keep growing by an unusual fractal growth process until after several minutes they become stable. This last fractal process does not follow the classical Smoluchowski coagulation theory as monitored by LS up to 90 s after mixing.