Kinetics Of Diffusion-limited Reactions Research Articles

Diffusion-Limited Reaction Kinetics of a Reactant with Square Reactive Patches on a Plane

We present a simple reaction model to study the influence of the size, number, and spatial arrangement of reactive patches on a reactant placed on a plane. Specifically, we consider a reactant whose surface has an N × N square grid structure, with each square cell (or patch) being chemically reactive or inert for partner reactant molecules approaching the cell via diffusion. We calculate the rate constant for various cases with different reactive N × N square patterns using the finite element method. For N = 2, 3, we determine the reaction kinetics of all possible reactive patterns in the absence and presence of periodic boundary conditions, and from the analysis, we find that the dependences of the kinetics on the size, number, and spatial arrangement are similar to those observed in reactive patches on a sphere. Furthermore, using square reactant models, we present a method to significantly increase the rate constant by sequentially breaking the patches into smaller patches and arranging them symmetrically. Interestingly, we find that a reactant with a symmetric patch distribution has a power–law relation between the rate constant and the number of reactive patches and show that this works well when the total reactive area is much less than the total surface area of the reactant. Since our N × N discrete models enable us to examine all possible reactive cases completely, they provide a solid understanding of the surface reaction kinetics, which would be helpful for understanding the fundamental aspects of the competitions between reactive patches arising in real applications.

Horizontal permeable reactive barriers with zero-valent iron for preventing upward diffusion of chlorinated solvent vapors in the unsaturated zone

Chlorinated solvents are extensively used in many activities and hence in the past decades impacted a large number of sites. The presence of these contaminants in groundwater is challenging particularly for the management of the vapor intrusion pathway. In this work we examine the potential feasibility of using horizontal permeable reactive barriers (HPRBs) placed in the unsaturated zone to treat chlorinated solvent vapors emitted from groundwater. Zero-valent iron (ZVI) powders, partially saturated with water and characterized by different specific surface areas (SSA), were tested, alone or mixed with sand, in lab-scale batch reactors using TCE as model compound. Depending on the type of iron powder used, a reduction of TCE concentration in the vapor phase from approximately 35% up to 99% was observed after 3 weeks of treatment. The best performance in terms of TCE reduction was obtained using the ZVI characterized by the intermediated values of the specific surface area (SSA). This finding, which is in contrast with the results generally observed in in aqueous solutions, was tentatively attributed to a non-selective higher reactivity of the fine-grained iron samples with water and dissolved oxygen (with a consequent iron passivation) or to the occurrence of a diffusion-limited reaction kinetics. Based on the first-order kinetic degradation rate constants estimated from the experimental data, a horizontal barrier of 1 m containing ZVI or a mixture of ZVI and sand can potentially lead to an attenuation of TCE vapors over 99%.

Electrokinetic mixing in electrode-embedded multiwell plates to improve the diffusion limited kinetics of biosensing platforms

Rapid and accurate biosensing with low concentrations of the analytes is usually challenged by the diffusion limited reaction kinetics. Thus, as a remedy, long incubation times or excess amounts of the reagents are employed to ensure the reactions to go to completion. Therefore, mixing becomes both a serious problem and necessity to overcome that diffusion limitation and homogenize the samples, especially for the biochemical reactions that take place in multiwell plates. Because the current mixing platforms such as shakers/vortexers, sonicators, magnetic stirrers and acoustic mixers have disadvantages including, but not limited to, being invasive/harfmul to the samples, causing the samples to splash out or stick to the walls of the wells and allowing foreign compartments to enter the solutions in the wells. Here we propose a noninvasive and safer (considering the risk of sample loss) technology that provides electrokinetic-mixing (EKM) of the reagents placed in electrode-embedded multiwell plates where the incubation times, or in other words, the time required for the desired molecules to meet in stationary solutions, can be reduced substantially. In order to demonstrate the power of this innovation, in this specific case, a simple Förster resonance energy transfer (FRET) based quenching bioplatform was adopted, where a molecular beacon DNA (MB) modified with sulfhydryl (–SH) and fluorescein (FITC) dye at opposite terminals was incubated with 10 nm sized gold nanoparticles (AuNPs) in the wells of an electrode-embedded multiwell plate, in which a printed circuit board (PCB) was attached at the bottom to control the liquid flows by EKM. When the MB binds to AuNPs through thiolate chemistry in the solution, FITC dye comes in close proximity to the AuNP surface and the emission is quenched via FRET principle. Thus, this quenching percentage over time was our comparison parameter for the mixing and no mixing cases to demonstrate the impact of mixing on the quenching kinetics. This reaction was conducted with different concentrations of AuNPs to observe the impact of mixing on MB quenching kinetics when the concentrations of the AuNPs were increased. Total quenching efficiency could go up to 90% in the presence of the AuNPs and it took about 60 min to reach stability. When the EKM was involved, fluorescence quenching time for the MBs could be reduced by up to 4.1 times. Thus, it was demonstrated that this technology may improve the kinetics of the diffusion limited biological reactions take place in multiwell plates substantially so that it may be adopted in various different sensing platforms for rapid measurements.

Effects of the Size, the Number, and the Spatial Arrangement of Reactive Patches on a Sphere on Diffusion-limited Reaction Kinetics: A Comprehensive Study.

We investigate how the size, the number, and the spatial arrangement of identical nonoverlapping reactive patches on a sphere influence the overall reaction kinetics of bimolecular diffusion-limited (or diffusion-controlled) reactions that occur between the patches and the reactants diffusing around the sphere. First, in the arrangement of two patches, it is known that the overall rate constant increases as the two patches become more separated from each other but decreases when they become closer to each other. In this work, we further study the dependence of the patch arrangement on the kinetics with three and four patches using the finite element method (FEM). In addition to the patch arrangement, the kinetics is also dependent on the number and size of the patches. Therefore, we study such dependences by calculating the overall rate constants using the FEM for various cases, especially for large-sized patches, and this study is complementary to the kinetic studies that were performed by Brownian dynamics (BD) simulation methods for small-sized patches. The numerical FEM and BD simulation results are compared with the results from various kinetic theories to evaluate the accuracies of the theories. Remarkably, this comparison indicates that our theory, which was recently developed based on the curvature-dependent kinetic theory, shows good agreement with the FEM and BD numerical results. From this validation, we use our theory to further study the variation of the overall rate constant when the patches are arbitrarily arranged on a sphere. Our theory also confirms that to maximize the overall rate constant, we need to break large-sized patches into smaller-sized patches and arrange them to be maximally separated to reduce their competition.

Quasi-Isothermal External Short Circuit Tests Applied to Lithium-Ion Cells: Part II. Modeling and Simulation

Measurement data gained from quasi-isothermal external short circuit tests on single-layered pouch-type Li-ion cells presented in the first part of this combined work was used to validate a well-known homogenized physical-chemical model for different electrode loadings, cell temperatures, initial cell voltages, and external short circuit resistances. Accounting for diffusion-limited reaction kinetics, effective solid phase diffusion coefficients, and one representative active material particle size within each electrode, the model is capable of describing the experimentally observed characteristic change in magnitudes of current and heat generation rate throughout the short circuit. Underlying mechanisms for the observed characteristics are studied by evaluating the predicted concentration distribution across the electrodes and separator and by calculating the cell polarization due to ohmic losses, diffusion processes, and reaction kinetics. The importance of mass transport in the solid and liquid phase limiting reaction kinetics is discussed and evaluated in the context of a sensitivity analysis. Concentration dependent transport properties, electrode tortuosity, particle size, and electrode energy density are affecting different stages of a short circuit. Simulation results suggest a strong impact of electrode design on the short circuit dynamics allowing for an optimization regarding a cell’s energy and power characteristics whilst guaranteeing a high short circuit tolerance.

Theory of curvature-dependent kinetics of diffusion-limited reactions and its application to ligand binding to a sphere with multiple receptors.

We present a simple theory that explains how surface curvature affects the reaction kinetics of diffusion-limited reactions on spherically curved surfaces. In this theory, we derive a quadratic equation under the conditions that the rate constant satisfies the Hill and Smoluchowski rate constants at the lowest and highest curvatures, respectively, and that at a certain intermediate curvature, there should be a maximum value of the rate constant, which was recently found in our previous work. We find that the result obtained from our theory is in good agreement with the corresponding one obtained from numerical calculation. In addition, we show that our theory can be directly applied to the Šolc-Stockmayer model of axially symmetric reactants, which can be considered as a spherical reactant with a single reaction site. Furthermore, we discuss using our theory to improve the formula for the rate constant in the Berg-Purcell ligand-binding model of a cell membrane covered by multiple receptors. Our simple theory yields insight into the effect of curvature on diffusion-influenced reactions and provides a useful formula for easily and quantitatively evaluating the curvature effect.

On the theory of point defect recombination in crystals

A new approach to the diffusion-limited reaction kinetics for particles migrating by random walks on discrete lattice sites and reacting when two particles occupy the same site is extended to a more general case of a large reaction radius and applied to the problem of the recombination rate of point defects in cubic lattices. Numerical calculations correctly reproduce the analytic expressions in the limit cases considered in previous work and in the general case represent a stepwise dependence of the reaction rate on the recombination radius.

Kinetics of diffusion-limited reactions with biased diffusion in percolating to compact substrates

We study through Monte Carlo simulations, the kinetics of the two-species diffusion-limited reaction model with same species excluded volume interaction in substrates embedded on a square lattice ranging in occupancy from a fractal percolating structure to the compact limit. We study the time evolution of the concentration of single-particle species for various values of substrate occupancies, 0.5927460, 0.61, 0.63, 0.65, 0.7, 0.8, and 1, where the first value corresponds to the percolating probability of the square lattice. We show that in the diffusion-limited reaction regime, the kinetics strongly depends on the presence of a bias along a particular square lattice direction, representing the net effect of a driving field. We were able to explain the slow dynamics at high values of the driving field in terms of traps appearing in diluted substrates, particularly at the percolation threshold.

Electrokinetically Controlled DNA Hybridization Microfluidic Chip Enabling Rapid Target Analysis

Biosensors and more specifically biochips exploit the interactions between a target analyte and an immobilized biological recognition element to produce a measurable signal. Systems based on surface nucleic acid hybridization, such as microarrays, are particularly attractive due to the high degree of selectivity in the binding interactions. One of the drawbacks of this reaction is the relatively long time required for complete hybridization to occur, which is often the result of diffusion-limited reaction kinetics. In this work, an electrokinetically controlled DNA hybridization microfluidic chip will be introduced. The electrokinetic delivery technique provides the ability to dispense controlled samples of nanoliter volumes directly to the hybridization array (thereby increasing the reaction rate) and rapidly remove nonspecific adsorption, enabling the hybridization, washing, and scanning procedures to be conducted simultaneously. The result is that all processes from sample dispensing to hybridization detection can be completed in as little as 5 min. The chip also demonstrates an efficient hybridization scheme in which the probe saturation level is reached very rapidly as the targets are transported over the immobilized probe site enabling quantitative analysis of the sample concentration. Detection levels as low as 50 pM have been recorded using an epifluorescence microscope.

Monte?Carlo model of the diffusion-limited reaction kinetics in microheterogeneous media. Comparison with the experimental plots of the forward and backward electron phototransfer kinetics on the pore diameter in silica gel

The kinetics of the diffusion-limited decay reaction A + B → B was simulated by the Monte—Carlo method on a two-dimensional square lattice with defects presented by randomly distributed sites. The cases were considered where [B] ≫ [A] at the random initial distribution (quenching reaction) and [B] = [A] with the initial distribution of the A and B particles on neighboring sites (geminate recombination). The kinetic curves were approximated by the simplest analytical equation [A]/[A]0 = (1 − ϕ)exp[−(kt)1−h] + ϕ (where k and ϕ are constants). The plots of the heterogeneity parameter (h) and time-averaged first-order rate constant vs. concentration of defects (p) or B particles (in the case of quenching) were obtained and compared with similar correlations obtained earlier by the experimental study of the kinetics of forward (quenching reaction) and backward (geminate recombination) electron phototransfer on the surface of different porous silica gels. The experimental plots of h vs. silica gel porosity are in satisfactory agreement with the plots of h vs. p in the model space, if the fraction of volume inaccessible for reactants, calculated from the free silica gel volume, is chosen as the p parameter for silica gel.

Membrane-based nanoscale proteolytic reactor enabling protein digestion, peptide separation, and protein identification using mass spectrometry.

A miniaturized trypsin membrane reactor housed inside a commonly used capillary fitting is developed and demonstrated for enabling rapid and sensitive protein identification by on-line proteolytic digestion and analysis of protein digests using nano-ESI-MS and MALDI-MS. The design and assembly of the capillary fitting-based trypsin membrane reactor are straightforward and highly robust, without the need for expensive fabrication technology and procedures. The resultant protein digests can also be further concentrated and resolved using capillary reversed-phase liquid chromatography or transient capillary isotachophoresis/zone electrophoresis prior to the mass spectrometric analysis in an integrated platform. By comparing these results with the results obtained from our previous studies using plastic microfluidics (Gao et al., Anal. Chem. 2001, 73, 2648-2655), significant reduction in dead volume and sample consumption can be achieved using this newly developed tryptic digestion station. This nanoscale reaction system enables rapid proteolytic digestion in seconds instead of hours for a protein concentration of less than 10(-8) M, consumes very little sample (< or = 5 fmol), and offers capillary interfaces with various separation and mass spectrometry techniques. The ultrafast enzymatic turnover for attaining complete peptide coverage in protein identification is contributed by the highly porous structure of the membrane media, providing excessive trypsin loading while eliminating the constraints of diffusion-limited reaction kinetics.

Transverse NMR relaxation as a probe of mesoscopic structure.

Transverse NMR relaxation in a macroscopic sample is shown to be extremely sensitive to the structure of mesoscopic magnetic susceptibility variations. Such a sensitivity is proposed as a novel kind of contrast in the NMR measurements. For suspensions of arbitrary-shaped paramagnetic objects, the transverse relaxation is found in the case of a small dephasing effect of an individual object. Strong relaxation rate dependence on the objects' shape agrees with experiments on whole blood. Demonstrated structure sensitivity is a generic effect that arises in NMR relaxation in porous media, biological systems, as well as in kinetics of diffusion limited reactions.

Stochastic gating influence on the kinetics of diffusion-limited reactions

We study how the kinetics of diffusion-influenced reactions is modified when the reactivity of species fluctuates in time (stochastically gated) with emphasis on the many-particle aspect of the problem. Because of the fact that the dynamics of ligand binding to proteins originally motivated the problem, it is considered in that context. Recently, Zhou and Szabo [J. Phys. Chem. 100, 2597 (1996)] have demonstrated many-particle effects in the problem and found that the kinetics of reaction between a gated protein with a large number of ligands significantly differs from that between a protein and gated ligands. With our approach, the difference between the kinetics of ligand-gated and protein-gated reactions appears formally the same as the difference between the target and trapping problems despite the origin of the corresponding effects and their manifestations are distinctly different. A simple approximate method to treat the many-particle effects is proposed. The theory is applied to a particular two-state gating model. Explicit analytical expressions for the protein survival probability are obtained. We show that (1) for ligand-gated reactions, gating is effectively accounted for by the appropriate reduction of the species reactivity and (2) for protein-gated reactions, the survival probability changes its time behavior from exponential (fast gating) to nonexponential (slow gating). The role of intensity and asymmetry of the gate motion is discussed.

Diffusion-limited reaction kinetics in nanofabricated porous model catalysts

Electron beam lithography makes it possible to fabricate ideal, representative models of supported, porous catalysts (with pore and particle dimensions and separations of ~10 nm and up). This allows systematic exploration of various scientifically and technically important phenomena associated with porous, supported catalysts. In the present work, we have derived general equations describing the reaction kinetics on such model catalysts in combination with mass-transport corrections. Depending on the geometrical parameters and boundary conditions the reaction rate may be limited by bulk or Knudsen diffusion of reactants inside pores or by diffusion in the regions near or far from the support surface. All these regimes are treated in detail. The results will help both to guide and interpret experiments with nanofabricated catalysts.

Influence of external steady source structure on particle distributions and kinetics of diffusion-limited reactions. II. A + B → 0 simulations

Monte Carlo simulations were performed to study the effect of a steady external source structure (i.e., particle correlation and vertical reactions) on diffusion-limited A + B → 0 reactions at steady state. Several methods were developed to describe the spatial organization of the system. They are the distributions of aggregates and inter-particle distance (“gap”), and a parameter based on the number of boundaries between A-rich and B-rich domains. The correlation of the particles in the steady external source reduces the local fluctuation in the particle landing process. The vertical reaction restrains the organization of particles, and does not allow it to reach total segregation. The degree of segregation affects the steady state kinetic behavior. The simulation results are consistent with existing theoretical predictions for the reaction order, the correlation length, and the segregation size.

Influence of external steady source structure on particle distributions and kinetics of diffusion-limited reactions. I. A + A .fwdarw. 0 simulations

The effect of a steady external source and its correlation structure on the particle organization patterns and the kinetics of A+A→0 reactions at steady state was studied by Monte Carlo simulations. The simulation results are compared with existing theoretical predictions for the reaction order and the self-ordering of particles. The spatial organization of the system is described by the interparticle distance («gap») distribution. The pair correlation length of particles in the external source alters the particle distribution at the steady state and affects the macroscopic rate law

Low dimensional reaction kinetics and self-organization

Diffusion-limited reaction kinetics becomes anomalous not only for fractals, with their anomalous diffusion, but also for low-dimensional (one and two) and disperse media, where the random walk is compact. We focus on annihilation, recombination and trapping reactions under non-equilibrium steady state (steady source) or batch (big bang) conditions. The typical reactions are: A + A → Products, A + B → Products and A + C → Products. We are interested in the global rate laws, and their relation to particle-particle distributions (e.g., pair-correlation and nearest-neighbor distribution functions) and in local rate laws (if definable). Anomalous reaction kinetics (more than classical kinetics) is particularly sensitive to initial conditions, source term structure, conservation laws (e.g., equal densities for A and B), excluded volume effects, and medium size, dimensionality and anisotropy. Analytical formalisms, scaling arguments, computer (and supercomputer) simulations and experiments (on chemical and physical reactions) all play an important role in the newly emerging picture.

Mutual influence of traps on the death of a Brownian particle

The role of many-body effects in the kinetics of diffusion-limited reactions has been studied in the simplest situation, i.e. in the case of death of a Brownian particle in two stationary traps. For this purpose, the survival probability of the particle was calculated using two methods: the traditional one, where the many-body effects are neglected, and a more precise one which allows one to take these effects into account. Comparison of the results obtained showed that neglecting many-body effects can lead to both over- and under-estimated values of the survival probability of the particle during a time interval t. The role of these effects in a more realistic situation when there are many traps is discussed.

TDPAC studies of the kinetics of diffusion-limited reactions in dilute AgPt alloy

The TDPAC studies of In-Pt pairs formation in dilute AgPt alloy in nonequilibrium state are performed. The concentration of the In-Pt complexes is measured for different annealing temperatures and times. The results are compared with calculations based on the model of diffusion controlled bimolecular reactions. The conclusions about diffusion enhancement due to radiation damage and the interaction of impurities with radiation defects are drawn.

Reaction at traps involving two diffusing species: Application to atomic–molecular hydrogen conversion at defects

Kinetics of diffusion-limited reactions are derived for the case in which there are two mobile species A and M which can convert to one another at a trap or catalysis site. At the same time the standard reaction of A getting trapped and detrapped is allowed to compete with the A⇄M reaction. Nonlinearities associated with trap saturation and the model where M is a diatomic molecule of A are included. Rate equations are derived for the bulk concentrations of A and M. Excellent agreement is obtained with simulation data when account is taken of the correlation between trap occupation and concentration immediately outside the trap. The equations are specifically designed for trapping, recombination, and dissociation of hydrogen atoms and molecules at defects; but they should have a broad range of applicability so that problems with several species and reactions can be handled with the same degree of accuracy as the Smoluchowski theory for trapping and detrapping of a single species. Results are accurate and general enough to include the nonlinearities introduced by trap saturation and trap-concentration-dependent rate constants.