Nearest-neighbor distance distributions and self-ordering in diffusion-controlled reactions. II. A+B simulations

Steady state of a two-species annihilation process with separated reactants

We describe the steady state of the annihilation process of a one-dimensional system of two initially separated reactants A and B. The parameters that define the dynamical behavior of the system are the diffusion constant, the reaction rate, and the deposition rate. Depending on the ratio between those parameters, the system exhibits a crossover between a diffusion-limited (DL) regime and a reaction-limited (RL) regime. We found that a key quantity to describe the reaction process in the system is the probability p(x_{A},x_{B}) to find the rightmost A (RMA) particle and the leftmost B (LMB) particle at the positions x_{A} and x_{B}, respectively. The statistical behavior of the system in both regimes is described using the density of particles, the gap length distribution x_{B}-x_{A}, the marginal probabilities p_{A}(x_{A}) and p_{B}(x_{B}), and the reaction kernel. For both regimes, this kernel can be approximated by using p(x_{A},x_{B}). We found an excellent agreement between the numerical and analytical results for all calculated quantities despite the reaction process being quite different in both regimes. In the DL regime, the reaction kernel can be approximated by the probability to find the RMA and LMB particles in adjacent sites. In the RL regime, the kernel depends on the marginal probabilities p_{A}(x_{A}) and p_{B}(x_{B}).

Logarithmic speed-up of relaxation in A-B annihilation with exclusion

We show that the decay of the density of active particles in the reaction A+B→0 in one dimension, with exclusion interaction, results in logarithmic corrections to the expected power law decay, when the starting initial condition (i.c.) is periodic. It is well known that the late-time density of surviving particles goes as t^{-1/4} with random initial conditions, and as t^{-1/2} with alternating initial conditions (ABABAB⋯). We show that the decay for periodic i.c.'s made of longer blocks (A^{n}B^{n}A^{n}B^{n}⋯) do not show a pure power-law decay when n is even. By means of first-passage Monte Carlo simulations, and a mapping to a q-state coarsening model which can be solved in the independent interval approximation (IIA), we show that the late-time decay of the density of surviving particles goes as t^{-1/2}[ln(t)]^{-1} for n even, but as t^{-1/2} when n is odd. We relate this kinetic symmetry breaking in the Glauber Ising model. We also see a very slow crossover from a t^{-1/2}[ln(t)]^{-1} regime to eventual t^{-1/2} behavior for i.c.'s made of mixtures of odd- and even-length blocks.

Extinction and survival in two-species annihilation

We study diffusion-controlled two-species annihilation with a finite number of particles. In this stochastic process, particles move diffusively, and when two particles of opposite type come into contact, the two annihilate. We focus on the behavior in three spatial dimensions and for initial conditions where particles are confined to a compact domain. Generally, one species outnumbers the other, and we find that the difference between the number of majority and minority species, which is a conserved quantity, controls the behavior. When the number difference exceeds a critical value, the minority becomes extinct and a finite number of majority particles survive, while below this critical difference, a finite number of particles of both species survive. The critical difference Δ_{c} grows algebraically with the total initial number of particles N, and when N≫1, the critical difference scales as Δ_{c}∼N^{1/3}. Furthermore, when the initial concentrations of the two species are equal, the average number of surviving majority and minority particles, M_{+} and M_{-}, exhibit two distinct scaling behaviors, M_{+}∼N^{1/2} and M_{-}∼N^{1/6}. In contrast, when the initial populations are equal, these two quantities are comparable M_{+}∼M_{-}∼N^{1/3}.

Fast chemical reaction and multiple-scale concentration fields in singular vortices

Two species involved in a simple, fast reaction tend to become segregated in patches composed of a single of these reactants. These patches are separated by a boundary where the stoichiometric condition is satisfied and the reaction occurs, fed by diffusion. Stirred by advection, this boundary and the concentration fields within the patches may tend to present multiple-scale characteristics. Based on this segregated state, this paper aims at evaluating the temporal evolutions of the length of the boundary and diffusive flux of reactants across it, when concentrations presenting initial self-similar fluctuations are advected by a singular vortex. First the two sources of singularity, i.e., the self-similar initial conditions and the singular vortex, are considered separately. On the one hand, self-similar initial conditions are imposed to a diffusion-reaction system, for one- and two-dimensional cases. On the other hand, an imposed singular vortex advects initially on/off concentration fields, in combination with diffusion and reaction. This problem is addressed analytically, by characterizing the boundary by a box-counting dimension and the concentration fields by a Hölder exponent, and numerically, by direct numerical simulations of the advection-diffusion-reaction equations. Second, the way the two sources hang together shows that, depending on the self-similar properties of the initial concentration fields, the vortex promotes the chemical activity close to its inner smoothed-out core or close to the outer region where the boundary starts to spiral. For all the considered situations, the length of the boundary and the global reaction speed are found to evolve algebraically with time after a short transient and a good agreement is found between the analytical and numerical scaling laws.

Reaction front in an A+B-->C reaction-subdiffusion process

We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.

Monte carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation

Conventional equations for enzyme kinetics are based on mass-action laws, that may fail in low-dimensional and disordered media such as biological membranes. We present Monte Carlo simulations of an isolated Michaelis-Menten enzyme reaction on two-dimensional lattices with varying obstacle densities, as models of biological membranes. The model predicts that, as a result of anomalous diffusion on these low-dimensional media, the kinetics are of the fractal type. Consequently, the conventional equations for enzyme kinetics fail to describe the reaction. In particular, we show that the quasi-stationary-state assumption can hardly be retained in these conditions. Moreover, the fractal characteristics of the kinetics are increasingly pronounced as obstacle density and initial substrate concentration increase. The simulations indicate that these two influences are mainly additive. Finally, the simulations show pronounced S-P segregation over the lattice at obstacle densities compatible with in vivo conditions. This phenomenon could be a source of spatial self organization in biological membranes.

Reaction front structure in the diffusion-limited A+B model with initially randomized reactants

Subtle features of the reaction front formation in the A+B-->0 reaction are reported for the initially random and equal A+B reactant distribution. Three nonclassical parameters (initial linewidth, minimum, and maximum), for each interparticle gap and nearest neighbor distance distributions, are derived, as a function of time, using Monte Carlo simulations. These empirical front measures and their temporal scaling exponents are compared with the previously studied ones for the reactant interparticle distributions.

Exciton microscopy and reaction kinetics in restricted spaces

We describe the development of a new biologically non-invasive ultraresolution light microscopy, based on combining the energy transfer "spectral ruler" method with the micro-movement technology employed in scanning tunneling microscopy (STM). We use near-field scanning optical microscopy, with micropipettes containing crystals of energy packaging donor molecules in the tips that can have apertures below 5 nm. The excitation of these tips extends near field microscopy well beyond the 50 nm limit. The theoretical resolution limit for this spectrally sensitive light microscopy is well below 1 nm. Exciton microscopy is ideally suited for kinetic studies that are spatially resolved on the molecular scale, i.e., at a single molecule site. Moreover, the successful operation of the scanning exciton tip depends on an understanding of reaction kinetics in restricted spaces. In contrast to the many recent reviews on scanning tip microscopies, there is no adequate review of the recent revolutionary developments in the area of reaction kinetics in confined geometries. We thus attempt such a review in this paper. Reactions in restricted spaces rarely get stirred vigorously by convection and are thus often controlled by diffusion. Furthermore, the compactness of the Brownian motion leads to both anomalous diffusion and anomalous reaction kinetics. Elementary binary reactions of the type A + A----Products, A + B----Products and A + C----C + Products are discussed theoretically for both batch and steady-state conditions. The anomalous reaction orders and time exponents (for the rate coefficients) are discussed for various situations. Global and local rate laws are related to particle distribution functions. Only Poissonian distributions guarantee the classical rate laws. Reactant self-organization leads to interesting new phenomena. These are demonstrated by theory, simulations, and experiments. The correlation length of reactant production affects the self-ordering length-scale. These effects are demonstrated experimentally, including the stability of reactant segregation observed in chemical reactions in one-dimensional spaces, e.g., capillaries and microcapillaries. The gap between the reactant A (cation) and B (anion) actually increases in time, and extends over millimeters. Excellent agreement is found among theory, simulation, and experiment for the various scaling exponents.