Diffusion Classical approach

There was therefore a clear need to assess the assumptions inherent in the classical kinetic approach for determining surface-catalysed reaction mechanisms where no account is taken of the individual behaviour of adsorbed reactants, substrate atoms, intermediates and their respective surface mobilities, all of which can contribute to the rate at which reactants reach active sites. The more usual classical approach is to assume thermodynamic equilibrium and that surface diffusion of reactants is fast and not rate determining. 

The stochastic motion of particles in condensed matter is the fundamental concept that underlies diffusion. We will therefore discuss its basic ideas in some depth. The classical approach to Brownian motion aims at calculating the number of ways in which a particle arrives at a distinct point m steps from the origin while performing a sequence of z° random steps in total. Consider a linear motion in which the probability of forward and backward hopping is equal (= 1/2). The probability for any sequence is thus (1/2). Point m can be reached by z° + m)/2 forward plus (z° m)/2 backward steps. The number of distinct sequences to arrive at m is therefore... 

In this section we will present briefly some relevant aspects of the classical approach to model diffusion in polymers along with representative features and results of the most often cited of these models. 

The BCF volume diffusion model has already been discussed in Sect. 5.1.2. The classical approach is to consider the diffusion of lattice ions from the... 

Studies of adsorption and diffusion processes and of framework dynamics necessitate an explicit consideration of the periodicity of the system. For such studies, classical models involving force fields are generally used in which the interactions between the atoms in the system are represented by potential functions. The third class of models are hybrid models, which as mentioned previously try to combine the advantages of the quantum and classical approaches explicit treatment of the electronic structure for certain parts of the system like the catalytic site and its direct environment, combined with a classical treatment of the rest of the system. Each of these models is now described in turn. 

As seen in the earlier discussion of Onsager s theory (section 6.2), in the classical approach diffusion is a response to a concentration gradient. If species i is diffusing then the flux of i is given by... 

In this context, it is worthwhile to recall the quantum jump approach developed in the quantum optics community. In this approach, an emission of a photon corresponds to a quantum jump from the excited to the ground state. For a molecule with two levels, this means that right after each emission event, = 0 (i.e., the system is in the ground state). Within the classical approach this type of wave function collapse never occurs. Instead, the emission event is described with the probability of emission per unit time being Fp (t), where Pee(0 is described by the stochastic Bloch equation. At least in principle, the quantum jump approach, also known as the Monte Carlo wave function approach [98-103], can be adapted to calculate the photon statistics of a SM in the presence of spectral diffusion. 

We need to understand what controls the rate of a phase transformation. We can monitor both chemical and structural changes to address the sometimes subtle question— which change (chemistry or structure) occurs first The answer depends on why the phase change itself occurs. The experimental techniques we use are those given in Chapter 10, so we just give some specific illustrations here. The classical approach used to study the kinetics of solid-state reactions between two ceramic oxides is to react a bulk diffusion couple in much the same way as, for example, when studying the Kirkendall effect in metals. 

One of such tendencies is polymers synthesis in the presence of all kinds of fillers, which serve simultaneously as reaction catalyst [26, 54]. The second tendency is the chemical reactions study within the framework of physical approaches [55-59], from which the fractal analysis obtained the largest application [36]. Within the framework of the last approach in synthesis process consideration such fundamental conceptions as the reaction prodrrcts stracture, characterized by their fractal (Hausdorff) dimension [60] and the reactionary medium connectivity, characterized by spectral (fracton) dimension J [61], were introduced. In its titrrt, diffusion processes for fractal reactions (strange or anomalous) differ principally from those occurring in Euclidean spaces and described by diffusion classical laws [62]. Therefore the authors [63] give transesterification model reaction kinetics description in 14 metal oxides presence within the framework of strange (anomalous) diffusion conception. 

The diffusion coefficient Dj of solute 1 in solvent 2 at infinitely dilute solution is a fundamental property. This is different from the self-diffusion coefficient Dq in pure liquid. Both Di and Dq are important properties. The classical approach to Di can be done based on Stokes and Einstein relation to give the following equation... 

The classical approach consists of eliminating one of the two variables, say the flow density, for writing only one equation, by assuming a constant scalar diffusivity through space ... 

An integration constant has been voluntarily added for allowing this relation to be inverted, as explained earlier in the classical approach, in order to retrieve Equation G7.5. This integration constant is the unperturbed concentration far from the place where the diffusional process occurs, meaning that this model applies to infinite (or semi-infinite) transient diffusion. 

In the absence of significant molecular motion the spectra can be calculated simply as a powder-like super-imposition of the individual molecular static lines of Lorentzian shape from all over the sample. These lines are then positioned into the spectrum according to Eq. (5) as in Ref. [6j. To include also dynamic effects, such as fluctuations of molecular long axes (defining the scalar order parameter S and the director n) and translational molecular diffusion, it is convenient to use a semi-classical approach with the time-dependent deuteron spin Hamiltonian [25] where the H NMR line shape I cj) is calculated as the Fourier transform of... 

To deal with ET in organic semiconductors, one has to incorporate the coherent motion of electron in the multi-states. The single two-state rate model developed for the donor-acceptor system may not be used straightforwardly. Here, we display a time-dependent wavepacket diffusion (TDWPD) approach for the charge carrier dynamics. In the approach, the nuclear vibrational motions are dealt with the semi-classical fluctuations on the electronic energies of molecules. In this way, we can apply the approach to the nanoscale organic crystals. 

The common and classical approach to considering pore diffusion limitations is the utilization of an effectiveness factor as a single parameter, which was developed by Damkoehler, Thiele and Zddovkh in the 1930s (Damkoehler, 1936,1937a, 1937b, 1939 Thiele, 1939 Zeldowitsch, 1939). However, an exact calculation of the effectiveness factor is only possible for simple power law kinetics, isothermal particles, or simple reaction networks, for example, for two parallel or serial reactions, as described in many textbooks (e.g., Froment and Bischoff, 1990 or Levenspiel, 1996,... 

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