Theory Of Diffusion-controlled Reactions Research Articles

A numerical study of the generalized Steklov problem in planar domains

Abstract We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry.

Diffusion in a porous medium with dead ends: An analysis by methods of the theory of diffusion-controlled reactions

The problem of the diffusion of a particle in a porous medium with dead ends entering which is related to overcoming high entropy barriers is considered. Dead ends effectively decelerate migration, because, while residing in them, a particle does not participate in diffusion transfer. A new approach to the problem was suggested. The approach was based on the possibility of splitting the diffusion process into separate well-defined stages thanks to the presence of high entropy barriers. An analysis of these stages was performed using methods developed in the theory of diffusion-controlled reactions. The new approach was used to calculate the effective diffusion coefficient (characterizing the limiting particle migration deceleration caused by the presence of dead ends in a porous medium) and estimate the time of its establishment.

Excess electrons scavenging in n-dodecane solution: The role of tunneling of electron from its localized state to acceptor

The quenching of delayed radiofluorescence of n-dodecane solutions of N, N, N′, N′-tetramethyl-1,4-phenylenediamine by both electron acceptors and electric field was studied. The values of excess electron mobility and rate constants of electron scavenging by different acceptors were determined over the temperature range of 265–343 K. The data were analyzed with the theory of diffusion-controlled reactions with distant transfer of electron to an acceptor. Evidence was obtained that no enthalpy and entropy of activation were needed for the electron scavenging.

Photochemical bimolecular reaction between biphenyl and carbon tetrachloride: observed ultrafast kinetics and diffusion-controlled reaction model

Ultrafast bimolecular reaction kinetics between photoexcited biphenyl and carbon tetrachloride, observed by femtosecond time-resolved fluorescence spectroscopy, is successfully interpreted by theories of diffusion-controlled reactions. The obtained decay kinetics is well explained by Smoluchowski's theory of diffusion-controlled reactions when the parameter R, distance between the reactants, is 0.39 nm. A modified kinetic theory by Collins and Kimball also explains the results satisfactorily, when R is 0.40 nm and the bimolecular reaction rate constant k act is 3.4×10 11 dm 3 mol −1 s −1 . It is suggested that molecular motions in solution for a time period of a few picoseconds is described by diffusion.

Diffusion-controlled reactions: A revisit of Noyes’ theory

Noyes’ theory of diffusion-controlled reactions is revisited in great details. First, it is shown that this theory provides an interesting alternative approach to perform molecular dynamics simulations for diffusion-controlled reactions. With this approach, reaction rate can be determined from simulations of nonreactive equilibrium systems. No annihilation procedure is needed to simulate the reaction event. Provided that encounters with different reactants are strictly uncorrelated, the reaction rate can be determined more directly and accurately than by the methods that compute the survival probability. We describe in detail the method for accurately determining the key quantity in Noyes’ theory, i.e., the first recollision probability, from molecular dynamics simulations. It will also be shown that arguments similar to those in Noyes’ theory allow us to establish an exact relation (under the same assumptions of absence of correlations) between the distribution function of a reacting system at the encounter distance and that of a nonreactive equilibrium system. This relation can be used to fix the boundary condition at the reaction distance in the approaches based on a diffusion equation. New insights have been gained into the usefulness of the recollision probability. The recollision probability also provides a very useful tool for characterizing quantitatively some dynamic features of the cage effect for reactions in dense liquids. Finally, the method presented here may also be used to calculate reaction rates for diffusion-controlled reactions in systems where the dynamics cannot be described by a diffusion equation.

Ultrafast Bimolecular Reaction Kinetics between S1 trans-Stilbene and Carbon Tetrachloride Studied by Sub-Picosecond Time-Resolved Visible Absorption Spectroscopy

Abstract Ultrafast bimolecular reaction kinetics between S1 trans-stilbene and carbon tetrachloride is monitored by sub-picosecond time-resolved visible absorption spectroscopy. The observed kinetics is well explained either by Smoluchowski’s theory or by Collins-Kimball’s theory of diffusion-controlled reactions. These results suggest that the molecular motion in solution for a few picosecond time period is well described by diffusion.

Theory of Diffusion-Controlled Reactions in Polymers under Flow

We extend the theoretical framework of de Gennes for treating diffusion-controlled reactions in complex fluid systems to situations in which a steady, linear flow is externally imposed. Consideration is restricted to systems that are dilute in reactive sites and for which the reaction occurs instantaneously and irreversibly when a pair of sites are brought in close contact. Illustrative calculations are presented for the long-time intermolecular trapping rate constant in polymer melts where a small fraction of chains have a single terminally attached reactive group. Both shear and extensional flows are considered, as well as Rouse and reptation models of chain dynamics. Extensions of the analysis may have applications in modeling free radical termination processes and reactive blending.

Effective diffusion coefficient and diffusion-controlled reactions in insulating solids with defects

Abstract The expressions for effective diffusion coefficient are obtained in the mean field approximation for two-phase system for spatial dimensions of 1, 2 and 3. The existence of potential barrier for diffusion on the phase boundary was taken into account via the boundary conditions. Obtained formulae could be applied in the theory of diffusion-controlled reactions and for interpreting the experimental data on defect diffusion in two-phase media.

Reactions at the gas-phase-porous-solid interface: Mechanisms and effects of surface geometry around room temperature

The luminescence quenching of Ru(bpy) 2+ 3 by molecular oxygen on a porous silica and controlled porous glass was studied at the 253–353 K temperature range. The reaction was observed to take place by a mixture of an Eley-Rideal (ER) and a Langmuir-Hinshelwood mechanism. The relative contribution of the two mechanisms was calculated and the ER component was found to dominate at higher temperatures. The lack of effect of the average pore diameter (apd) on the rate of the quenching reaction supports our contention that the Knudsen diffusion controlled reaction theory developed elsewhere is incorrect. In addition we show that, for porous solids, diffusion controlled reaction theory is not applicable in principle below the Knudsen regime.

Some properties of a fractal-time continuous-time random walk in the presence of traps

The CTRW has often been applied to problems related to transport in a statistically homogeneous disordered medium, which means that there are no traps or reflecting boundaries to be found in the medium. Two physical applications, one to the migration of photons in a turbid medium and the second to the theory of diffusion-controlled reactions in a random medium, suggest that it might be useful to study properties of the CTRW, particularly as they refer to survival probability in the presence of a trap or a trapping surface. We calculate a number of these properties when the pausing-time density is asymptotically proportional to a stable law, i.e.,ψ(t)∼T α+1 as (t/T)→∞, where 0<α<1. A forthcoming paper will establish the correspondence between properties of the CTRW and proprties of random walkers on a fractal with trapping boundaries.

Stereospecificity of quenching of excited aromatic molecules by electron acceptors

On the example of fluorescence quenching of perylene by organic electron acceptors we have shown that the radius of the quenching sphere depends on the electron affinity of the oxidizing agent. The deactivation of excited molecules of m-dinitrobenzene was achieved by tunnel electron transfer at a distance, which was significantly greater than the sum of the radii of the reagent molecules. The fluorescence quenching with phthalic anhydride, a weaker electron acceptor, occurs on contact between molecules. The kinetics of this process are described by the Smoluchowski theory of diffusion-controlled reactions. The rate of fluorescence quenching with carbon tetrachloride and nitromethane in polar solvents also increases with the increase of the coefficients of mutual diffusion, but the formal reaction radii of these processes are smaller than the sum of the radii of the interacting particles, which is due to the stereospecifity of the reaction. It was shown that the degree of stereospecificity increases with the decrease of the reaction radius.

Boundary layer effects on the rate of diffusion-controlled reactions

The description using the Smoluchowski equation, on which the traditional theory of diffusion-controlled reactions is based, presupposes local equilibrium. Hence it must break down in the vicinity of an absorbing wall, where a kinetic boundary layer is formed. We explore in this paper the effects of this boundary layer on the reaction rate. We start from a description using the Klein-Kramers equation and use two approximate treatments of the boundary layer that gave satisfactory results for the analogous one-dimensional problem. The simplest one leads to a simple analytic correction factor in the Smoluchowski-Debye reaction rate that becomes appreciable for particle radii not very large compared to a typical length, the velocity persistence length. The correction factor always decreases the reaction rate. For constant of piecewise constant potentials between the reacting particles we also work out a somewhat more elaborate approximation, based on Grad's thirteen-moment equations. It yields effects of the same order of magnitude as the simple theory; the differences between the two approximate treatments should give an indication of the errors still contained in each of them.

Quantum field methods in the theory of diffusion-controlled reactions

The diagrammatic perturbation technique for the kinetic theory of classical reacting systems with diffusion is developed. It is further applied to investigation of recombination-type reactions in media of one, two, and three dimensions. The effective rates of this reaction are calculated, covering the whole range from the slow to the fast (diffusion-controlled) regimes.

Boundary conditions and reversibility in diffusion controlled reactions. II. A three state model for steady state evaporation of a spherical drop

A simple three state model of the reaction complex was introduced into reversible diffusion controlled reaction theory and the flux balance equations were derived for all times. The long time asymptote of this relation was used to describe the evaporation from and condensation on a spherical drop immersed in a dilute gas infinitely dilute in vapor. The molecular velocity distribution is required to satisfy the linearized steady-state Boltzmann equation with the molecular interaction taken to be Maxwell’s potential. The solution to this and the boundary condition imposed by the flux-balance equations of the three state model was approximated by a four mode ansatz. The coefficients of these modes were determined by four moment equations which could be shown to satisfy a variation principle due to Cercignani. Two of the modes corresponded to two of the components of the usual Chapman–Enskog ‘‘normal’’ solution to the problem and were found to be dominant at long distances, falling off as r−1 and r−2. The other two modes were discontinuous in velocity, and, outside a boundary layer several mean free paths thick, were found to decay as r−4 and r−3. The boundary condition at the drop surface was found to be of the usual radiation boundary condition type used in diffusion controlled reaction theory, modified by a factor dependent on mean free path and reaction probability.

The kinetic boundary layer for the Klein-Kramers equation; A new numerical approach

We explore a numerical technique for determining the structure of the kinetic boundary layer of the Klein-Kramers equation for noninteracting Brownian particles in a fluid near a wall that absorbs the Brownian particles. The equation is of interest in the theory of diffusion-controlled reactions and of the coagulation of colloidal suspensions. By numerical simulation of the Langevin equation equivalent to the Klein-Kramers equation we amass statistics of the velocities at the first return to the wall and of the return times for particles injected into the fluid at the wall with given velocities. The data can be used to construct the solutions of the standard problems at an absorbing wall, the Milne and the albedo problem. We confirm and extend earlier results by Burschka and Titulaer, obtained by a variational method vexed by the slow convergence of the underlying eigenfunction expansion. We briefly discuss some further boundary layer problems that can be attacked by exploiting the results reported here.

Exact algorithm ford-dimensional walks on finite and infinite lattices with traps. III. Role of lattice valency in influencing the efficiency of diffusion-controlled reactions

In this paper we continue our development of an exact lattice theory of diffusion-controlled reactions. We show how general theorems drawn from the theory of finite Markov processes may be brought to bear on the approach elaborated in our earlier work [Phys. Rev. Lett. 47, 1500 (1981); Phys. Rev. B 26, 4166 (1982)] and, in addition, we exploit well-known diagram and generating-function methods (in particular, those based on the adjacency-walk matrix) to gain further insight into the statistics underlying the processes considered in this paper. We consider reactions in which the diffusing molecule encounters a single reaction center and reacts there, irreversibly, upon first encounter. We also consider the situation where the diffusing particle may, at any of the sites surrounding the reaction center, form an activated complex with a coreactant situated there and, with finite probability, be removed irreversibly from the system. In each case we focus on the problem of reaction efficiency and determine the average number of steps required before a diffusing particle undergoes, eventually, an irreversible reaction. We report extensive new (and exact) results for hexagonal lattices and consider explicitly the role of spatial extent and dimensionality as well as the influence of passive versus active boundary conditions. By comparing the results obtained for hexagonal lattices for $d=2 \mathrm{and} 3$ with those reported earlier for square and/or cubic lattices, quantitative conclusions can be drawn on the role of lattice valency in influencing the efficiency of reactiondiffusion processes. A principal, general conclusion of this study concerns the efficiency of reaction when there exists the possibility of reactant deactivation at the $N\ensuremath{-}1$ sites surrounding the reaction center. We find that a 5% probability of reaction at these adjacent sites effectively erases distinctions between lattices subject to different boundary conditions or characterized by different valencies, i.e., the process becomes kinetically controlled.

Diffusion controlled reactions of gases with surfaces and state dependent reactivity. II. Diffusion slip and Knudsen corrections

An approximate procedure is outlined for incorporating nonequilbrium velocity distributions and free molecular flow effects into the boundary conditions customarily used in diffusion controlled reaction theory. This is applied to the theory of surfaces reacting with gases with state dependent reactivity [S. Hess and L. Monchick, J. Chem. Phys. 77, 307 (1982)], and is found to lead to formulas that are at least qualitatively correct over long ranges of Knudsen number, i.e., that go over into the correct vacuum rate as the mean free path becomes very large.

Diffusion-controlled reactions among stationary sinks

We summarize results of some of our calculations in diffusion-controlled reaction theory. We derive the transport equation describing a diffusing species which can react with a set of randomly distributed spherical sinks. Both the form of the transport equation and the dependence on sink volume fraction of the reaction rate and the effective diffusion coefficient are discussed.

Theory of diffusion-controlled reactions: Formulation of the bulk reaction rate in terms of the pair probability

We have formulated the bulk reaction rate rigorously, namely, in terms of the pair probability. Although Shlesinger criticized the usual method of calculating the bulk reaction rate, our result shows that the usual method is correct, at least with respect to the point raised by him. By use of the new formalism we have also analyzed the Wilemski-Fixman model for diffusion-controlled reactions. The present result is in contrast with their approximate one, indicating that the approximation used by them is inadequate.

Theory of diffusion-controlled reactions on spherical surfaces and its application to reactions on micellar surfaces

Diffusion-controlled reactions of a pair of particles on a spherical surface are studied. An analytical expression for the pair survival probability is derived in the case of a pair of freely diffusing particles. The decay of the pair survival probability is found to be well approximated by a single exponential. A simple approximate method for calculating the decay time which characterizes the exponential is presented. The decay time of a pair of particles interacting with each other by an arbitrary force is derived by use of this method. The results are applied to analyze available experimental data on reactions occurring on micellar surfaces, including the dismutation reaction of Br2− on the surface of cationic micelles and that of Ag2+ on the surface of anionic micelles, and the diffusion coefficients of these species on the micellar surfaces are estimated.