Time-dependent Rate Coefficient Research Articles

Target and trapping problems: From the ballistic to the diffusive regime

The target and trapping problems refer to the reaction A+B→B. In the target problem a single A particle is fixed in space and surrounded by B’s allowed to move freely, while in the trapping problem the B’s are fixed and the A is able to move. Exact solutions are found for both problems in the ballistic regime, in all dimensions. We show that the solution of the target problem provides a mean-field approximation to the solution of the trapping problem, not only in the diffusive regime, but also for arbitrary noise. This approximate solution works well in the diffusive regime, but not when motion is ballistic, since it breaks down at very early times. We show that the time-dependent rate coefficients in both the target and trapping problems remain finite at t=0 for arbitrarily strong noise intensities. This behavior is in contrast to the diffusion theory prediction that the coefficient diverges at t=0. A recently developed model that discretizes the velocity, allowing only three values, ±v and 0, is used to study the reaction kinetics of both the trapping and target problems in one dimension over the entire range of noise intensities. The solutions are used to study the effects of noise intensity on the mean survival time. We show that in the target problem this time decreases monotonically with increasing noise, while in the trapping problem this time exhibits a turnover behavior. We argue that a similar turnover occurs in the one-dimensional trapping problem when particle motion is governed by a Langevin equation.

Quantitative determination of a sodium dodecyl sulfate micellar radius through positron annihilation lifetime spectroscopy

Rigorous equations are established to describe the lifetime decay spectrum expected for positronium (Ps) atoms reacting with a solute by diffusion, taking into account the time dependence of the encounter rate coefficient. To assess the validity of the equations, these are applied to the case of Ps trapping into sodium dodecyl sulfate micellar aggregates in water studied by positron annihilation lifetime spectroscopy (LS). The LS spectra are satisfactorily analysed either on this new basis or in a conventional way, in terms of simple decaying exponentials. Several hypotheses are examined to derive the radius of the micelles from the data, by considering furthermore that the Ps atoms exist in the solution either in a quasi-free or in a bubble state. In complete accordance with the theoretical expectations, the combination of a time dependent encounter rate coefficient together with the bubble state assumption leads to a very good agreement between the radius thus determined (1.72 nm) and the micellar hydrophobic radius known from other techniques (about 1.84 nm).

Quantum dynamics of unimolecular dissociation reaction HFCO→HF+CO

Quantum dynamics calculations of the title reaction were performed within planar 4D and 5D models to investigate how the dissociation rate is affected by the change of total energy or vibrational character of the initial states. In the 4D calculations, the possibility for observing the mode specificity in the in-plane modes was examined by generating the initial states that are locally excited with respect to specific vibrational modes and comparing the dissociation rates for these states. In order to generate the appropriate initial states, we employed the spectral filter method combined with the zero-order vibrational Hamiltonian in the bond coordinates. Comparison was also made between the dissociation rate and the intramolecular vibrational energy redistribution (IVR) rate. In the 5D calculations, we have focused on the CH overtone excited states and examined how the dissociation dynamics is changed by the total energy that is above or below the dissociation threshold. We observed quite large fluctuations in the time-dependent rate coefficient for the fifth CH overtone state, which was found to occur due to the resonant oscillations of the CH stretching and bending energies as well as the closeness of the total energy to the dissociation threshold.

Dissolution kinetics of Ca-maleate crystals: evaluation for biotransformation reactor design

In order to develop a bioreactor for solid to solid conversions the biocatalytic conversion of solid Ca-maleate to solid Ca-D-malate is studied. The dissolution of Ca-maleate is the first step in this process and is described here. A kinetic model, based on the interfacial-barrier theory and the diffusion-layer theory, was developed which describes the increase in Ca-maleate concentration due to dissolution with the help of the time-dependent parameters. According to the model two processes contribute to the dissolution of Ca-maleate·H2O crystals: (1) the dissolution (and dissociation) reaction of Ca-maleate at the solid–liquid interface, characterized by a time-independent reaction rate coefficient, and (2) the transport of Ca2+ and maleate2− across a boundary liquid film, characterized by a time-dependent mass-transfer rate coefficient. In addition, the surface of a crystal and the driving force are time-dependent variables. Since Ca-maleate·H2O crystals are not uniform, a crystal-size distribution was also used in the model. The effects of stirring speed, temperature, pH, and initial Ca2+ concentration on the dissolution rate of Ca-maleate·H2O crystals were determined experimentally in order to evaluate the model. The model fitted the data well (R2>0·97). In order to determine whether the overall dissolution process was reaction or transport controlled, a method based on overall reaction and transport rates (per unit of driving force) was developed. This showed that the dissolution of Ca-maleate was reaction controlled. Temperature influenced the reaction rate coefficient the most; it ranged from 5·7×10−6 m s−1 at 10°C to 67×10−6 m s−1 at 60°C. The reaction rate coefficient was also influenced by the pH and the initial Ca2+ concentration, but, as expected, hardly by the stirring speed. Simplifying the model by omitting the time-dependent mass-transfer rate coefficient and by assuming uniform crystals, resulted in only slightly worse fits of the data with R2 being at most 0·004 smaller. © 1988 Society of Chemical Industry

Quenching of excited states of pyrene derivatives by amphiphilic nitroxide radicals in cationic micellar solutions. Dynamics and location of the guest molecules in the aggregates

Steady-state and nanosecond time-resolved studies have been performed on fluorescence quenching of excited pyrene sulfonic acid and pyrene butyric acid (PBA and PSA, respectively, in the form of sodium salts) by n-doxylstearic methyl esters ( n-DSE, n=5, 10, 12) in aqueous solution of cationic micelles of hexadecyltrimethylammonium chloride (HTAC, 0.1 M). The results are compared with those reported earlier for pyrene (P) as a luminophore (Szajdzinska-Pietek and Wolszczak, Chem. Phys. Lett., 270 (1997) 527). It is shown that PBA/ n-DSE and PSA/ n-DSE luminophore/quencher pairs, unlike P/ n-DSE pairs, are not suitable for determination of the micellar aggregation number, neither by the ‘quasi-static’ method of Turro—Yekta nor by the dynamic method of Infelt—Tachiya. It is suggested that the fluorescence decay curves in the examined systems can be adequately described in terms of dispersive kinetics using the time-dependent rate coefficient for intramicellar quenching process. The average location of the luminophore and the quencher molecules in the aggregates is discussed, considering previous findings of electron spin echo modulation studies of n-doxylstearic acid spin probes in micellar systems.

Effects of a quantum-mechanically driven two-state gating mode on the diffusion-influenced bimolecular reactions

The reduced distribution function formalism of diffusion-influenced bimolecular reactions is generalized to incorporate a quantum-mechanical gating mode in the description. An analytical expression for the reaction rate coefficient is obtained in the Laplace transform domain for a general initial condition. For a simple reaction model, the time-dependent reaction rate coefficient and the product yield are calculated numerically for two representative initial conditions. Dependence of the rate coefficient and the product branching ratio on various reaction parameters is discussed.

Deterministic and Stochastic Asymptotic Behavior of Nonequilibrium Chemical Systems with Time-Dependent Rate Coefficients

We analyze the long time limit of the nonequilibrium solutions of a system of multivariable nonlinear kinetic equations with time-dependent rate coefficients, as achieved, for example, by temporal variation of the temperature. If the characteristic time scale attached to the change of rate coefficients is smaller than the relaxation time to equilibrium, then the system is constrained to evolve away, possibly far, from equilibrium. However, after a sufficiently large time the system forgets its past: in the long run all solutions of the kinetic equations tend toward a special (normal) solution which depends on the previous values of the rate coefficients, but it is independent of the initial state of the system. The normal solution may be very different from the equilibrium solution. The occurrence of this type of time-dependent normal regime for the deterministic kinetic equations is intimately connected to a similar behavior of the stochastic evolution equation of the system. In the long run all solutio...

A stochastic approach to nonequilibrium chain reactions in disordered systems: Breakdown of eikonal approximation

A stochastic description of chain reactions occurring in disordered systems is suggested by considering a statistical distribution of time-dependent rate coefficients. The possibilities of constructing a thermodynamic formalism for non-equilibrium chain reactions are investigated by testing the validity of the eikonal approximation in the thermodynamic limit. If the fluctuations of the rate coefficient are restricted to a finite range, then for large systems the probability of concentration fluctuations obeys the eikonal scaling condition, which makes possible the development of a nonequilibrium thermodynamic formalism. For an infinite range of variation of the rate coefficient, however, the eikonal scaling does not hold anymore: the probability of concentration fluctuations has a long tail of the negative power-law type and the system displays statistical fractal features. The passage from the stochastic eikonal behavior to the fractal scaling is characterized by a change in the deterministic kinetic equations of the process: in the eikonal regime the effective reaction order with respect to the active intermediate is 1, whereas for fractal scaling it is equal to 2. Due to this change in the effective reaction order for fractal scaling, the reaction is much faster than in the eikonal regime and the explosion threshold may be reached after a finite time interval.

First- and Second-Order Sensitivity Analysis of a Photochemically Reactive System (a Green's Function Approach)

Local sensitivity analysis of air quality models permits an estimation of the effects of parameter variations, can help assess uncertainties, and distinguishes important from unimportant model features. While local sensitivity analysis in atmospheric chemistry is usually conducted to the first-order, this work includes second-order sensitivity analysis. A Green's function method is used to compute the sensitivities of a photochemically reactive system. Time-dependent photolytic reaction rate coefficients are speci fied using two-parameter functions of time. The Green's function is used directly to study the temporal dependence of ozone concentrations on the NOx concentrations. ROG-limited (ROG/NOx = 6) and NOx-limited (ROG/NOx = 24) conditions are both evaluated. The importance of NOx, formaldehyde, and aromatic chemistry is emphasized by the high ozone sensitivity to the initial concentrations of these species and to the reaction rate parameters of NO2 photolysis, NO recombination with ozone (NO + O3 → NO2 + O2), formaldehyde photolysis (HCHO + hν → 2HO2 + CO), and NO2 + OH → HNO3. The parameters that ozone concentration is most sensitive to were treated with second-order sensitivity analysis. A comparison of ozone concentration changes predicted by second-order sensitivity analysis with concentration changes computed directly shows that second-order sensitivity predictions are valid for parameter variations as large as ±50% of the parameter's nominal value, whereas first-order predictions are valid for parameter variations of at most ±25%. Even though all of the NOx in these simulations is introduced via initial concentrations, the Green's function reveals that the temporal pattern of NOx emission is likely to influence ozone concentrations.

Single-particle survival in parallel gated trapping

Any chemical reaction A+B\ensuremath{\rightarrow}C whose progress is modulated by another reaction of the form A\ensuremath{\rightleftarrows}A is said to be gated. The gating reaction A\ensuremath{\rightleftarrows}A represents a reversible fluctuation from a reactive state A to an unreactive state A not reacting with B. Reversibly blocked chemical reactions, conformational fluctuations in proteins, and reactions occurring within biomembranes or involving biological molecules have all been studied recently in contexts related to gating. This article calculates certain trapping rates and mean survival times in the presence of a single gated trap. Unlike previous methods, the formalism in this paper is based directly on trapping rates and not on Green's functions. The trapping rate formalism leads quite naturally to explicit solutions for some recently developed ('parallel') gating models, solutions that might be quite difficult to derive within a Green"s-function formalism. These solutions give time-dependent rate coefficients for parallel gated chemical reactions.

Theory and simulation of the time-dependent rate coefficients of diffusion-influenced reactions

A general formalism is developed for calculating the time-dependent rate coefficient k(t) of an irreversible diffusion-influenced reaction. This formalism allows one to treat most factors that affect k(t), including rotational Brownian motion and conformational gating of reactant molecules and orientation constraint for product formation. At long times k(t) is shown to have the asymptotic expansion k(infinity)[1 + k(infinity) (pie Dt)-1/2 /4 pie D + ...], where D is the relative translational diffusion constant. An approximate analytical method for calculating k(t) is presented. This is based on the approximation that the probability density of the reactant pair in the reactive region keeps the equilibrium distribution but with a decreasing amplitude. The rate coefficient then is determined by the Green function in the absence of chemical reaction. Within the framework of this approximation, two general relations are obtained. The first relation allows the rate coefficient for an arbitrary amplitude of the reactivity to be found if the rate coefficient for one amplitude of the reactivity is known. The second relation allows the rate coefficient in the presence of conformational gating to be found from that in the absence of conformational gating. The ratio k(t)/k(0) is shown to be the survival probability of the reactant pair at time t starting from an initial distribution that is localized in the reactive region. This relation forms the basis of the calculation of k(t) through Brownian dynamics simulations. Two simulation procedures involving the propagation of nonreactive trajectories initiated only from the reactive region are described and illustrated on a model system. Both analytical and simulation results demonstrate the accuracy of the equilibrium-distribution approximation method.

Dynamics of colloidal suspensions

Recent work in the theory of interacting Brownian particles is reviewed. The dynamics of a suspension of Brownian particles is shown to be dominated by slow diffusive processes with a continuous spectrum of relaxation times. Typical time-correlation functions have a long-time tail corresponding to a superposition of long wavelength diffusive modes. The long-time behavior can be extrapolated over a wide range of time by an approximate representation of the Laplace transform of the time-correlation function. As an illustrative example the time-dependent rate coefficient of a diffusion-controlled reaction is discussed.

One‐Dimensional Virus Transport in Porous Media With Time‐Dependent Inactivation Rate Coefficients

A model for virus transport in one‐dimensional, homogeneous, saturated porous media is developed, accounting for virus sorption and inactivation of liquid phase and adsorbed viruses with different time dependent rate coefficients. The virus inactivation process is represented by a pseudo first‐order rate expression. The pseudo first‐order approximation is shown to simulate available experimental data from three virus inactivation batch studies better than the frequently employed constant rate inactivation model. Model simulations indicated that the pseudo first‐order approximation, compared to the constant inactivation, leads to extended survival of viruses and, consequently, more distant migration. Results from a parameter sensitivity analysis demonstrated that estimation of pseudo first‐order inactivation rate coefficients from field observations requires data collection near the source of virus contamination during initial stages of virus transport.

Time-Dependent Rate Coefficients from Brownian Dynamics Simulations

The kinetics of a diffusion-influenced reaction is described by a time-dependent rate coefficient k(t). In this paper, we implement an algorithm for calculating k(t), introduced by one of us, in the UHBD software package. In the implementation, the electrostatic force exerted on a diffusing substrate by an enzyme is obtained from a finite-difference solution of the Poisson equation. The rate coefficient is obtained by starting the substrate in the active site and monitoring its survival probability using Brownian dynamics simulations. The technique is applied to the binding of superoxide to Cu/Zn superoxide dismutase. The long-time limit of k(t) is found to be in agreement with both the experimental value and that calculated using an algorithm designed specifically for finding k(∞).

Molecular and phenomenological rate coefficients of fast reactions in solutions

The paper is concerned with the problem of formulating chemical rate equations for reversible reactions in solution in terms of concentration-independent, phenomenological rate coefficients. These time-dependent rate coefficients approach, after an initial transient, the rate constants that can be obtained in a relaxation experiment. We start with the coupled evolution equations for the macroscopic concentrations, and for the two-particle distribution functions describing association–dissociation (A+B=C), bimolecular isomerization (A+B=B+C), and double decomposition (A+B=C+D). The effects of interparticle forces and long-ranged reactivity are included. We derive general identities linking the reactants and products radial distribution functions. For association–dissociation this leads to relations among the molecular rate coefficients which are valid for both contact and long ranged reactivities. For the other two reaction types, we were able to derive analogous relations only for contact reactivities. We demonstrate how the phenomenological rate coefficients can be defined via the solutions of the corresponding diffusional boundary-value problems. This approach is quite general, and valid for both contact and long-ranged reactivities and if interaction forces are included.

Transient effects in diffusion-controlled absorption by a nonuniform sink of arbitrary constitution

The time-dependent rate coefficient for absorption of diffusing particles by a nonuniform sink of arbitrary constitution is studied on the basis of the Smoluchowski equation. It is shown that the amplitude of the long-time tail can be expressed in terms of the asymptotic rate coefficient. The time-dependent rate coefficient is expressed as the sum of the asymptotic steady-state coefficient and an integral of purely decaying exponentials with a positive weight function. It is shown that the rate coefficient can be represented accurately over a wide range of time by an approximation to its Laplace transform, characterized by a small number of poles in the complex square root of frequency plane.

A fractal analysis of external diffusion limited first-order kinetics for the binding of antigen by immobilized antibody

A fractal analysis of external diffusion limited first-order kinetics for the binding of antigen in solution by immobilized antibody on a fibreoptic biosensor indicates that as the fractal parameter (measure of “disorder” on the surface) increases the rate of binding and the amount of antigen bound to the antibody on the surface decreases. The fractal analysis and exponential type binding rate coefficients are used to analyze the influence of time-dependent binding rate coefficients on external diffusion limited kinetics. A decrease in the binding rate coefficients with time decreases the Damkohler number (decrease in the mass transfer limitations) leading to an increase in the rate of binding and the amount of antigen bound to the antibody on the surface, as expected. An increase in the (exponential) binding rate coefficient with time leads to unusual shapes of the binding curves. The time-dependent binding rate coefficients provide a more realistic picture of the binding of antigen in solution to the antibody covalently attached to the surface, and should assist in the control and manipulation of these interactions at the surface. A value of the fractal dimension of the surface of 2·96 to 2·97 obtained for our system characterizes the anomalies in the reaction-diffusion system and the heterogeneity of the surface.

Brownian dynamics study of the influences of electrostatic interaction and diffusion on protein-protein association kinetics

A unified model is presented for protein-protein association processes that are under the influences of electrostatic interaction and diffusion (e.g., protein oligomerization, enzyme catalysis, electron and energy transfer). The proteins are modeled as spheres that bear point charges and undergo translational and rotational Brownian motion. Before association can occur the two spheres have to be aligned properly to form a reaction complex via diffusion. The reaction complex can either go on to form the product or it can dissociate into the separate reactants through diffusion. The electrostatic interaction, like diffusion, influences every step except the one that brings the reaction complex into the product. The interaction potential is obtained by extending the Kirkwood-Tanford protein model (Tanford, C., and J. G. Kirkwood. 1957. J. Am. Chem. Soc. 79:5333-5339) to two charge-embedded spheres and solving the consequent equations under a particular basis set. The time-dependent association rate coefficient is then obtained through Brownian dynamics simulations according an algorithm developed earlier (Zhou, H.-X. 1990. J. Phys. Chem. 94:8794-8800). This method is applied to a model system of the cytochrome c and cytochrome c peroxidase association process and the results confirm the experimental dependence of the association rate constant on the solution ionic strength. An important conclusion drawn from this study is that when the product is formed by very specific alignment of the reactants, as is often the case, the effect of the interaction potential is simply to scale the association rate constant by a Boltzmann factor. This explains why mutations in the interface of the reaction complex have strong influences on the association rate constant whereas those away from the interface have minimal effects. It comes about because the former mutations change the interaction potential of the reaction complex significantly and the latter ones do not.

Extended mechanism of excimer formation

A modified excimer formation mechanism using an encounter complex, in which the main reaction takes place, is presented and the rate coefficients of the elementary reaction steps are discussed. The kinetic description by ordinary rate equations fails in the case of excimer formation with a time-dependent rate coefficient caused by non-stationary diffusion. Convolution kinetics remain valid in that case and provide decay curves and quantum yields.

Diffusive dynamics on potential energy surfaces: Nonequilibrium CO binding to heme proteins

Theory and practice of reaction dynamics on two-dimensional potential energy surfaces is investigated. Nonequilibrium multidimensional barrier crossing, occurring when the initial density is located near the ridgeline separating reactants and products, is treated by solving the time-dependent Smoluchowski equation as a function of diffusion anisotropy. For a locally separable potential, and slow relaxation in the perpendicular mode, the problem reduces to a one-dimensional Smoluchowski equation with a sink term. It may be further approximated as a first-order reaction with a time-dependent rate coefficient. These approximations are compared with exact two-dimensional propagations on a potential surface representing CO binding to α heme. The intermediate-time power-law decay of the survival probability is analyzed with the aid of the above approximations. The power also shows some kind of critical behavior near the isotropic diffusion limit, where these approximations are no longer valid. For fast relaxation, a nonmonotonic survival probability is observed. The long time decay of the survival probability is governed by the equilibrium rate coefficient. We calculate its anisotropy dependence and compare it with two asymptotic expansions for the lowest eigenvalue of the Smoluchowski operator−for the one-dimensional sink-Smoluchowski operator and the fully two-dimensional operator. The difference between the fast relaxation limit of these one- and two-dimensional equations may provide a quantitative explanation for previous problems in extrapolating to high temperatures using the Agmon–Hopfield model. The implications of these results to heme protein dynamics are discussed and new experiments are proposed.