Interparticle distribution functions and rate equations for diffusion-limited reactions

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.