Single diffusion coefficient

Strictly the diffusion coefficient D measured for any type of binary system A/B is in fact the resultant effect of two partial diffusivities and D, representing respectively the diffusivity of A into B and of B intO/4. For most practical purposes, however, a single diffusion coefficient is sufficient to define a given diffusion system. 

When applied to a volume-fixed frame of reference (i.e., laboratory coordinates) with ordinary concentration units (e.g., g/cm3), these equations are applicable only to nonswelling systems. The diffusion coefficient obtained for the swelling system is the polymer-solvent mutual diffusion coefficient in a volume-fixed reference frame, Dv. Also, the single diffusion coefficient extracted from this analysis will be some average of concentration-dependent values if the diffusion coefficient is not constant. 

In DTI, diffusion is no longer described by a single diffusion coefficient, but by an array of nine coefficients that fully characterize how diffusion in space varies according to direction (Basser 1995). With this approach, diffusion anisotropy can be exploited to provide details on tissue microstructure and fiber tracts (le Bihan 2003). To obtain sufficient information on the direction of diffusion, the full diffusion tensor needs to be sampled [for a review on theoretical foundations of DTI see Basser and Jones (2002)]. 

Lawrence Stamper Darken (1909-1978) subsequently showed (Darken, 1948) how, in such a marker experiment, values for the intrinsic diffusion coefficients (e.g., Dqu and >zn) could be obtained from a measurement of the marker velocity and a single diffusion coefficient, called the interdiffusion coefficient (e.g., D = A ciiD/n + NznDca, where N are the molar fractions of species z), representative of the interdiffusion of the two species into one another. This quantity, sometimes called the mutual or chemical diffusion coefficient, is a more useful quantity than the more fundamental intrinsic diffusion coefficients from the standpoint of obtaining analytical solutions to real engineering diffusion problems. Interdiffusion, for example, is of obvious importance to the study of the chemical reaction kinetics. Indeed, studies have shown that interdiffusion is the rate-controlling step in the reaction between two solids. 

In order to elucidate the PS diffusion behaviour in PMMA gel, the PFGStE aH NMR measurements have been made with varying values of the diffusion time A. For samples A1 and A2, the diffusion coefficients for the fast and slow diffusion components (as indicated by Dfast and Dsiow, respectively) and the corresponding fractions (/fast and/siow, respectively) are determined by using Equation (2), and Dfast and Dsiow are plotted in Figure 14A and B. The single diffusion coefficients of... 

Whereas mutual diffusion characterizes a system with a single diffusion coefficient, self-diffusion gives different diffusion coefficients for all the particles in the system. Self-diffusion thereby provides a more detailed description of the single chemical species. This is the molecular point of view [7], which makes the selfdiffusion more significant than that of the mutual diffusion. In contrast, in practice, mutual diffusion, which involves the transport of matter in many physical and chemical processes, is far more important than self-diffusion. Moreover mutual diffusion is cooperative by nature, and its theoretical description is complicated by nonequilibrium statistical mechanics. Not surprisingly, the theoretical basis of mutual diffusion is more complex than that of self-diffusion [8]. In addition, by definition, the measurements of mutual diffusion require mixtures of liquids, while self-diffusion measurements are determinable in pure liquids. 

The hop-diffusion pattern caimot be found in liposomes or membrane blebs. In these membranes, the membrane molecules show simple Brownian diffusion with a single diffusion coefficient (55). 

Because the assumption of simple Brownian diffusion breaks down, the diffusion in biomembranes cannot be described by a single diffusion coefficient. For instance, FRAP experiments in the plasma membrane showed that the observed translational diffusion rates depend on the size of the initial photobleached spot, which is also inconsistent with a simple Singer-Nicolson model. 

The logarithm of the ratio of the dynamic structure factor and the static structure factor is linear in time for diffusive motion described by a single diffusion coefficient ... 

The diffusion process is described now by a single diffusion coefficient D ... 

Additional approaches concern the determination of single diffusion coefficients at various polymerization stages and while using polymeric support for the covalent fixing of the metallocene components. 

Kirkendal Effect When we previously discussed the transient interdiffiision of two semi-infinite bodies (material A and material B, respectively), we explicitly specified that the diffusion of A in B and that of B in A were identical and could therefore be described by a single diffusion coefficient. In many solids, however, this is not true. For example, the diffusivity of zinc in copper is much larger than the diffiisivity of copper in zinc. If a block of brass (a copper-zinc alloy) and a block of pure copper are bonded together at high temperatures, the zinc atoms will diffuse out of the brass and into the copper at a much faster rate than the copper atoms diffuse into the brass block. The net result is that the effective interface between the brass and copper blocks moves toward the brass, as illustrated schematically in Figure 4.17. This phenomenon is known as the Kirkendal effect and it occurs in many solid-state systems. 

As seen above, in almost every system whose relaxations can be characterized by a single diffusion coefficient Dp, the concentration dependence of Dp is described by a stretched exponential in c, see Eq. 9.4. If Dp instead followed a power law... 

The diffusion of sodium chloride can be accurately described by a single diffusion coefficient. Somehow this does not seem surprising, because we always refer to sodium chloride as if it were a single solute and ignore the knowledge that it ionizes. We get away with this selective ignorance because the sodium and chloride ions diffuse at the same rate. If they did not do so, we could easily separate anions from cations. 

Tracer diffusion is readily measured by preparing a bilayer film sample in which the upper layer is very much thinner than the lower layer, say 20 and 1000 nm, respectively. As diffusion rapidly results in the diflusant becoming highly dilute, the measurement essentially characterizes the tracer diffusion. The diffusion is often simplified to a single diffusion coefficient, D, which is obtained by fitting the concentration profile (x) with eqn (4] ... 

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