Microscopic diffusion coefficient

The derivative of U with respect to position is the regular force acting on the particle. However, this so-called stochastic approach results in similar Nernst-Planck type equations albeit with microscopic diffusion coefficient differing from the traditional macroscopic diffusion coefficient. The microscopic diffusion coefficient, D, is related to f, through D = k T)/ fim).)... 

The diffusion coefficients of the migrating particles discussed above (proportional to the mobihties of ionic or electronic defects, see Eq. (6.15)) are not directly accessible experimentally to measurement and are sometimes termed microscopic diffusion coefficients. It is a complication, in principle, that, on account of the requirement of electroneutrality, we cannot simply move the charge carrier, under consideration, through the solid. However, there are various possibilities for measuring phenomeno-... 

The combination of microscopic and macroscopic information is made possible by what can be called parameter imaging . In the general sense, it consists of the encoding of properties such as spectral line shifts, relaxation times, diffusion coefficients, etc., in the image by suitable combination of corresponding modules into one pulse sequence. Parameter images are to be distinguished from mere... 

Alkyl radicals react in solution very rapidly. The rate of their disappearance is limited by the frequency of their encounters. This situation is known as microscopic diffusion control or encounter control, when the measured rate is almost exactly equal to the rate of diffusion [230]. The rate of diffusion-controlled reaction of free radical disappearance is the following (the stoichiometric coefficient of reaction is two [233]) ... 

To obtain a more complete description, we need to find an analytic expression for the pre-exponential factor Dq of the diffusion coefficient by considering the microscopic mechanism of diffusion. The most straightforward approach, which neglects correlated motion between the ions, is given by the random-walk theory. In this model, an individual ion of charge q reacts to a uniform electric field along the x-axis supplied, in this case, by reversible nonblocking electrodes such that dCj(x)/dx = 0. Since two... 

Combining the above descriptions leads to a picture that describes the experimentally observed concentration dependence of the polymer diffusion coefficient. At low concentrations the decrease of the translational diffusion coefficient is due to hydrodynamic interactions that increase the friction coefficient and thereby slow down the motion of the polymer chain. At high concentrations the system becomes an entangled network. The cooperative diffusion of the chains becomes a cooperative process, and the diffusion of the chains increases with increasing polymer concentration. This description requires two different expressions in the two concentration regimes. A microscopic, hydrodynamic theory should be capable of explaining the observed behavior at all concentrations. 

The diffusion coefficient is related to a microscopic variable through a Green-Kubo relation... 

We have shown that the microscopic expression for the polymer diffusion coefficient. Equation 2, is the starting point for a discussion of diffusion in a wide range of polymer systems. For the example worked out, polymer diffusion at theta conditions, the resulting expresssion describes the experimental data without adjustable parameters. It should be possible to derive expressions for diffusion... 

Thus, microscopically, the diffusion coefficient may be interpreted as one-sixth of the jumping distance squared times the overall jumping frequency. Since / is of the order 3 x 10 ° m (interatomic distance in a lattice), the jumping frequency can be roughly estimated from D. For D m /s such as Mg diffusion in... 

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