Expanding spherical shell kinetic model for diffusion-controlled solid-state reactions

A reaction kinetic model has been derived for a solid-state diffusion-controlled reaction in a system of spherical particles of A isolated in a matrix of B. This model is analogous to the Ginstling-Brounsthein model except that the diffusion of A is the rate-controlling step. The model, therefore, resulting in an expanding spherical shell of product AB, is

where t = time, x = fraction reaction, K = reaction rate constant.     

Diffusion-controlled reactions: hydrodynamic interaction between charged, uniformly reactive spherical reactants

In this work, different models of hydrodynamic interaction (HI) are examined in the diffusion-controlled reaction between uniformly reactive charged spherical particles. In addition to Oseen "stick" and "slip" models of HI, one is considered that accounts for the disturbance of fluid flow by the ions around one reactive partner as they interact with a neighboring reactive species. This interaction is closely related to the "electrophoretic effect" in electrokinetics and can be described by a fairly simple electrophoretic, or E-tensor. These models are applied to the electron-transfer quenching reaction of Ru(bpy)3(2+) and methyl viologen (MV2+) over a wide range of NaCl concentrations (Chiorboli, C. et al., J. Phys. Chem. 1988, 92, 156). The back reaction is also considered. From a comparison of the salt dependence of the model and experimental rates, it is concluded that the "E-tensor" model works best and ignoring HI altogether works worst. The Oseen "stick" and "slip" models fall between these.     

Mathematical model for the growth of phases in binary multiphase systems upon isothermic annealing

A phenomenological mathematical model of the formation and growth of phases in a binary multiphase system with allowance for factors influencing the process of diffusion in a binary system is presented. It is shown that phases can grow for a certain time at different ratios between diffusion parameters according to a parabolic law that depends on the duration of isothermic annealing. They then slow their growth after successor phases appear at their interface with one component and can completely disappear from a diffusion layer or begin to grow again, but only at a rate slower than during their initial formation. The dependence of the thickness of each phase layer in a multiphase diffusion zone on the duration of isothermic annealing and the ratio between the diffusion parameters in neighboring phases is obtained. It is established that a certain ratio between the phase growth and rates of dissolution with allowance for the coefficients of diffusion in each phase and the periods of incubation can result in the complete disappearance of one phase as early as the onset of the growth of phase nuclei and be interpreted as a process of reaction diffusion.     

A general theoretical model for electron transfer reactions in complex systems

In this paper we present a general theoretical-computational model for treating electron transfer reactions in complex atomic-molecular systems. The underlying idea of the approach, based on unbiased first-principles calculations at the atomistic level, utilizes the definition and the construction of the Diabatic Perturbed states of the involved reactive partners (i.e. the quantum centres in our perturbation approach) as provided by the interaction with their environment, including their mutual interaction. In this way we reconstruct the true Adiabatic states of the reactive partners characterizing the electron transfer process as the fluctuation of the electronic density due to the fluctuating perturbation. Results obtained by using a combination of Molecular Dynamics simulation and the Perturbed Matrix Method on a prototypical intramolecular electron transfer (from 2-(9,9'-dimethyl)fluorene to the 2-naphthalene group separated by a steroidal 5-α-androstane skeleton) well illustrate the accuracy of the method in reproducing both the thermodynamics and the kinetics of the process.     

Reversible adsorption of spherical particles from binary mixtures: long-time behaviour

Formation of monolayers of spherical particles in processes with reversible adsorption from mixtures of large and small particles was simulated in computer experiments. Computer program was based on an algorithm that took into account random sequential adsorption, desorption and lateral diffusion of adsorbed particles (RSA–DLD model). Computer experiments were performed for systems with rate constants of particle adsorption at least 10³ times higher than rate constants of desorption. In processes with very fast adsorption and slow desorption, formation of monolayer can be divided into two stages. During the first stage, the total surface coverage (the coverage with particles of both types) increases very fast and becomes very close to that at equilibrium. During the second stage, the total coverage changes very slowly and the system approaches equilibrium mainly by the replacement of large particles with the small ones. A simple kinetic model for evolution of the monolayer composition during the second stage has been proposed. Kinetic equations related to this model allow the determination of large particles’ desorption rate constants on the basis of changes in the surface concentrations of adsorbed large and small microspheres. The validity of the model has been tested comparing large particles’ desorption rate constants values that had been used for simulations with values of the corresponding rate constants determined using analytical equations, with a view to analysing the simulation results. To cite this article: S. Slomkowski et al., C. R. Chimie 6 (2003).     

Nucleation in binary polymer blends: effects of foreign mesoscopic spherical particles

We study nucleation in binary polymer blends in the presence of mesoscopic spherical particles using self-consistent field theory, considering both heterogeneous and homogeneous nucleation mechanisms. Heterogeneous nucleation is found to be highly sensitive to surface selectivity and particle size, with rather subtle dependence on the particle size. Particles that preferentially adsorb the nucleating species generally favor heterogeneous nucleation. For sufficiently strong adsorption, barrierless nucleation is possible. By comparing the free energy barrier for homogeneous and heterogeneous nucleation, we construct a kinetic phase diagram.     

A versatile method for the production of monodisperse spherical particles and hollow particles: templating from binary core-shell structures

Using the porosity of a binary exotemplate, with mesoporous core-shell structure (SiO2@ZrO2), opens a new pathway to produce hybrid core-shell spheres, composite hollow spheres, and porous hollow spheres -- all monodisperse in size.     

Understanding supported reactions in spherical compartments: a general algorithm to model and determine rate constants, diffusion coefficients, and spatial product distributions

A general algorithm allowing the numerical modeling of the time and space dependence of product formation in spherical reaction volumes is described. The algorithm is described by the complete set of mass balance equations. On the basis of these equations, the effects of the diffusion coefficient, reaction rate, bead size, reagent excess, and packing density of the resin beads on the overall reaction rates are determined for second-order reactions. Experimental data of reaction progress are employed to calculate reaction rates and diffusion coefficients in polymer-supported reactions. In addition, the conditions for shell-like product formation are determined, and various strategies for the radial patterning of resin beads are compared. The effect of diffusion on polymer-supported enzyme-catalyzed reactions of the Michaelis-Menten type is treated, as well. Finally, the effects of typical nonideal solid-phase phenomena, namely, the inhomogeneity of rate constants and the concentration dependence of diffusion coefficients, on overall rates are discussed.     

Radiation-induced solid-state polymerization in binary systems. II. Relationship between polymerization rate and physical structure of binary systems

The radiation-induced solid-state polymerization of binary systems consisting of acrylic monomer (acrylamide, acrylic acid) and organic compounds was investigated. In the previous paper on binary systems the authors reported that the rate of polymerization increased in the solid state (eutectic mixture systems). The mechanism of rate increase has been investigated by examination of phase diagrams, viscosities, and surface tension of the binary systems. Viscosity and surface tension are the measure of the molecular interaction of the two-component systems. In addition, the effect of linear crystal growth rate and half maximum width of the x-ray diffraction diagram of the crystallization process were determined. The larger the molecular interaction between the two components, the slower the linear crystal growth rate of monomer. The size of the monomer crystal decreases and the dislocation density of the monomer crystals increases in systems with large molecular interaction. Consequently it can be concluded that the physical structure of a binary solid system is the most important parameter determining the rate increase of solid-state polymerization. Dislocation on the grain boundary is more important than defects inside of the crystal lattice. It was found that the acceleration of polymerization rate is large in binary systems with larger molecular interaction. In some systems such as organic acid—amide systems with strong hydrogen bonds, glassy phases may be formed in which monomer may readily polymerize at very low temperatures.     

Diffusion-controlled instabilities and stationary structures in a model for a ferroin-catalyzed Belousov-Zhabotinskii reaction

A model for the bromate-ferroin-bromomalonic acid reaction has been used to examine the conditions under which the homogeneous state loses stability as a result of Turing bifurcation. Stationary periodic structures with small or large amplitudes can exist. There is also a structure modification effect.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 24, No. 4, pp. 440–449, July–August, 1988.I am indebted to A. M. Zhabotinskii and Yu. A. Kuznetsov for useful discussions, and also to I. I. Goryanin for providing access to his software suite for defining a curve in n-dimensional space.     

Dependence of growth of the phases of multiphase binary systems on the diffusion parameters

A mathematical model of the diffusion interaction of a binary system with several phases on the equilibrium phase diagram is presented. The theoretical and calculated dependences of the layer thickness of each phase in the multiphase diffusion zone on the isothermal annealing time and the ratio of the diffusion parameters in the neighboring phases with an unlimited supply of both components were constructed. The phase formation and growth in the diffusion zone during “reactive” diffusion corresponds to the equilibrium state diagram for two components, and the order of their appearance in the diffusion zone depends only on the ratio of the diffusion parameters in the phases themselves and on the duration of the incubation periods. The dependence of phase appearance on the incubation periods, annealing time, and difference in the movement rates of the components across the interface boundaries was obtained. An example of the application of the model for processing the experimental data on phase growth in a two-component three-phase system was given.     

A model for the stability of films stabilized by randomly packed spherical particles

Particle stabilized thin films occur in a range of industrial applications where their properties affect the efficiency of the process concerned. However, due to their dynamic and unstable nature they are difficult to observe experimentally. As such, a tractable way of gaining insight into the fundamental aspects of this complicated system is to use computer simulations of particles at interfaces. This paper presents modeling results of the effect of nonuniform packing of spherical particles on the stability of thin liquid films. Surface Evolver was used to model cells containing up to 20 particles, randomly packed in a thin liquid film. The capillary pressure required to rupture the film for a specific combination of particle arrangement, packing density, and contact angle was identified. The data from the periodic, randomly packed models has been used to find a relationship between particle packing density, contact angle, and critical capillary pressure which is refined to a simple equation that depends on the film loading and contact angle of the particles it contains. The critical capillary pressure for film rupture obeys the same trends observed for particles in regular 2D and 3D packing arrangements. The absolute values of P*(crit), however, are consistently lower than those for regular packing. This is due to the irregular arrangement of the particles, which allows for larger areas of free film to exist, lowering the critical capillary pressure required to rupture the film.     

The intensification of rapid reactions in multiphase systems using slug flow in capillaries

A multiphase microreactor based upon the use of slug flow through a narrow channel has been developed. The internal circulation, which is stimulated within the slugs by their passage along the channel, is responsible for a large enhancement in the interfacial mass transfer and the reaction rate. Mass transfer performance data has been obtained for a glass chip-based reactor in a 380 microm wide channel by monitoring the extraction of acetic acid from kerosene slugs as they moved along the reactor channel. Finally, the data was compared with that provided from other inter-phase contacting techniques.     

γ- and ultraviolet-radiation-induced solid-state polymerization of maleimide in a few binary systems

The solid-state polymerization of maleimide by γ- and ultraviolet irradiation was carried out in binary systems with succinimide, maleic anhydride, and acenaphthylene. Polymaleimide obtained from the solid-state polymerization of maleimide by γ-rays was amorphous, while that obtained from the solid-state polymerization by ultraviolet rays was highly crystalline. In the maleimide–succinimide system the rate of polymerization reached a maximum nearly at the eutectic composition when the polymerization was carried out by γ-irradiation. With ultraviolet irradiation the rate of polymerization became higher with increasing content of succinimide in the feed. In the maleimide–maleic anhydride system a copolymer of both constituents was formed by γ-irradiation, but almost no homopolymer was produced. On the other hand, two kinds of polymers, a crystalline copolymer and an amorphous one, were produced by ultraviolet irradiation. The results were compared with those obtained from the copolymerization in solution. In the maleimide-acenaphthylene system the main products with ultraviolet irradiation was the dimer of acenaphthylene.     

A general model for electron-transfer reactions via electronic intermediate states

A theory for electrochemical electron-transfer reactions involving surface states as intermediates is developed. The electronic interactions are treated by perturbation theory. The electron transfer takes place by resonance exchange at a fixed solvent configuration on the reaction hypersurface, which is defined by the intersection of the three potential energy surfaces. An explicit expression for the reaction rate is derived. The relation of this work to other theories and to experiments is discussed.     

Distribution of colloidal particles in a spherical cavity

The spatial distribution of colloidal particles in a confined space is frequently a key issue to many phenomena of practical significance. This problem is investigated by considering the distribution of colloidal particles in a spherical cavity under the conditions of relatively large cavities, low cavity and colloidal particles potentials, and low monovalent electrolyte and colloidal concentrations. The analytical expression for the particle-cavity pair interaction energy is derived under various surface conditions. The results obtained are used to evaluate the direct correlation functions in the hypernetted chain approximation employed for the resolution of an Ornstein-Zernike equation. For a fixed particle number concentration at the center of a cavity, we make the following conclusions: (i) the spatial distribution of particles increases in an oscillatory manner with the distance away from the cavity surface, (ii) increasing the particle-cavity pair interaction energy has the effect of reducing the free space of particles inside a cavity, and (iii) the greater the pair interaction energy between two particles, the higher the average concentration of particles.     

Wall effects on terminal falling velocity of spherical particles moving in a Carreau model fluid

Experimental verification of our previous numerical simulation of wall effects on the terminal falling velocity of spherical particles moving slowly along the axis of a cylindrical vessel filled with a Carreau model fluid is presented. Dependences of the wall correction factor F _W on the sphere to tube ratio d/D and on the dimensionless Carreau model parameters m, Λ, and η _r were obtained using a finite element method. Calculated data of the wall correction factor were compared with the results of our new falling sphere experiments. The experiments were carried out in six types of cylindrical Perspex columns (16 mm, 21 mm, 26 mm, 34 mm, 40 mm, and 90 mm in diameter) filled with aqueous solutions of polymers exhibiting different degrees of shear thinning and elasticity. Seventeen types of spherical particles (1–8 mm in diameter) made of glass, ceramics, steel, lead, and tungsten carbide were used for the drop tests. Measurements of the liquid flow curves, primary normal stress differences, oscillatory, creep and recovery, stress relaxation, and stress growth tests were carried out on the rheometer Haake MARS (Thermo Scientific). A good agreement between numerically and experimentally obtained F _W data was found.     

Discrimination of the kinetic model of overlapping solid-state reactions from non-isothermal data

A method has been developed that allows the deconvolution of up to 15 overlapping solidstate reactions without previous assumptions. Both the kinetic parameters and the reaction mechanisms fitted by the unit reactions can be determined from a series of non-isothermal experiments carried out at different heating rates.

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Zusammenfassung Es wurde eine methode zur Dekonvolution von bis zu 15 einander überlappenden Festphasenreaktionen entwickelt, die keinerlei vorherige Annahmen notwendig macht. Mittels einer Reihe nichtisothermer Experimente mit unterschiedlichen Aufheizgeschwindigkeiten können sowohl die kinetischen Parameter als auch der Reaktionsmechanismus der Teilreaktionen bestimmt werden.

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