Diffusion-collision model kinetics

Bashford D Weaver D. L. and Karplus M. Diffusion-collision model for the folding kinetics of the lambda-repressor operatorbinding domain. J. Biomol. Str. Dyn. (1984) 1 1243-1255. 

In the diffusion-collision model of protein folding, the kinetics of microdomains with inherent structure-forming... 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

In this case no detailed collision kinetics are involved. The collision parameters rnm and the parameters in the Maxwellian post-collision velocity distribution f m are derived from experimentally determined gas viscosity or diffusivity, and the collisional invariants, respectively. Usually this term is negligible in present experiments, but exceptions exist [16]. In particular for ITER, and the high collisionality there, these terms are expected to become more relevant. However, due to the BGK-approximations made, their implementation into the models does not require further discussion here. 

The above results illustrate the utility of multiparticle Brownian dynamics for the analysis of diffusion controlled polymerizations. The results presented here are, however, qualitative because of the assumption of a two-dimensional system, neglect of polymer-polymer interactions and the infinitely fast kinetics in which every collision results in reaction. While the first two assumptions may be easily relaxed, incorporation of slower reaction kinetics by which only a small fraction of the collisions result in reaction may be computationally difficult. A more computationally efficient scheme may be to use Brownian dynamics to extract the rate constants as a function of polymer difflisivities, and to incorporate these in population balance models to predict the molecular weight distribution [48-50]. We discuss such a Brownian dynamics method in the next section. 

Sintering kinetics of model-suppoted catalysts are generally coielated by a GPLE of the form -d(D/Djj)/dt = kjCD/D -Dg /Dg) where m = 2. This result has important mechanistic implications since a number of fundamental processes such as emission of atoms frcnn crystallites, diffusion of adatoms on a support, collision of crystallites, w recmibination metal atoms may involve second-order processes. 

Particle sintering involves atoms that escape and diffuse away from crystallites, eventually being captured by other particles (267), or crystallite diffusion on the surface and subsequent collision and coalescence with larger particles (262). The latter model is unlikely for particles over 50 A (267). Since practical supported catalysts have a broad particle size distribution, a combination of atom and small crystallite diffusion is probable. That more than one process is responsible for sintering is also apparent from the power law kinetics of the aging process (267-264) ... 

This balance equation can also be derived from kinetic theory [101], In the Maxwellian average Boltzman equation for the species s type of molecules, the collision operator does not vanish because the momentum mgCs is not an invariant quantity. Rigorous determination of the collision operator in this balance equation is hardly possible, thus an appropriate model closure for the diffusive force is required. Maxwell [65] proposed a model for the diffusive force based on the principles of kinetic theory of dilute gases. The dilute gas kinetic theory result of Maxwell [65] is generally assumed to be an acceptable form for dense gases and liquids as well, although for these mixtures the binary diffusion coefficient is a concentration dependent, experimentally determined empirical parameter. 

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