Macroscopic transport diffusivities

The first theoretical attempts in the field of time-resolved X-ray diffraction were entirely empirical. More precise theoretical work appeared only in the late 1990s and is due to Wilson et al. [13-16]. However, this theoretical work still remained preliminary. A really satisfactory approach must be statistical. In fact, macroscopic transport coefficients like diffusion constant or chemical rate constant break down at ultrashort time scales. Even the notion of a molecule becomes ambiguous at which interatomic distance can the atoms A and B of a molecule A-B be considered to be free Another element of consideration is that the electric field of the laser pump is strong, and that its interaction with matter is nonlinear. What is needed is thus a statistical theory reminiscent of those from time-resolved optical spectroscopy. A theory of this sort was elaborated by Bratos and co-workers and was published over the last few years [17-19]. 

The dynamics of highly diluted star polymers on the scale of segmental diffusion was first calculated by Zimm and Kilb [143] who presented the spectrum of eigenmodes as it is known for linear homopolymers in dilute solutions [see Eq. (77)]. This spectrum was used to calculate macroscopic transport properties, e.g. the intrinsic viscosity [145], However, explicit theoretical calculations of the dynamic structure factor [S(Q, t)] are still missing at present. Instead of this the method of first cumulant was applied to analyze the dynamic properties of such diluted star systems on microscopic scales. 

The Fisher relation (38) has a structure similar to a fluctuation dissipation relation in statistical mechanics It relates a macroscopic transport coefficient, the hydrodynamic speed, to the diffusion tensor and to the statistical properties of... 

The motion of ions through solids results in both charge as well as mass transport. Whereas charge transport manifests itself as ionic conductivity in the presence of an applied electric field, macroscopic mass transport (diffusion) occurs in a concentration gradient. Both ionic conductivity and diffusion arise from the presence of point defects in solids (Section 5.2). For a solid showing exclusive ionic conduction, conductivity is written as... 

Dr. Sanfeld refers to phenomena that concern membranes and surface behaviors rather than bulk solutions whether such behaviors play a role in the motion of fluids in cells remains an open question. On the other hand, the coupling of chemical and hydrodynamic effects is not a necessary condition to the existence of rotations and macroscopic transport of matter. We have seen in the lecture of Professor Prigogine that such effects can result more simply from the coupling of chemical reactions with transport phenomena like diffusion. 

On the large side, the concept of diffusion also can be applied to macroscopic transport. This process is called turbulent diffusion. Turbulent diffusion is not based on thermal molecular motions, but on the mostly irregular (random) pattern of currents in water and air. 

Before continuing with the discussion on the dynamics of SE s in crystals and their kinetic consequences, let us introduce the elementary modes of SE motion. In a periodic lattice, a vacant neighboring site is a necessary condition for transport since it allows the site exchange of individual atomic particles to take place. Rotational motion of molecular groups can also be regarded as an individual motion, but it has no macroscopic transport component. It may, however, promote (translational) diffusion of other SE s [M. Jansen (1991)]. 

Most important macroscopic transport properties (i.e., permeabilities, solubilities, constants of diffusion) of polymer-based membranes have their foundation in microscopic features (e.g., free-volume distribution, segmental dynamics, distribution of polar groups, etc.) which are not sufficiently accessible to experimental characterization. Here, the simulation of reasonably equilibrated and validated atomistic models provides great opportunities to gain a deeper insight into these microscopic features that in turn will help to develop more knowledge-based approaches in membrane development. 

The relative insensitivity of this type of diffusion criterion to particle shape and to assumption of exact kinetics, has been discussed in connection with the macroscopic reactant diffusion problem on catalyst granules (7). The condition (15) is a general order-of-magnitude criterion defining the physical conditions of intimacy between the component systems for no mass-transport inhibition. It defines a requirement for realizing the formal kinetics of polystep reactions. 

As early as 1815 it was observed qualitatively that whenever a gas mixture contains two or more molecular species, whose relative concentrations vary from point to point, an apparently natural process results which tends to diminish any inequalities in composition. This macroscopic transport of mass, independent of any convection effects within the system, is defined as molecular diffusion. 

To obtain the mean free path Xp, we recall that in Section 9.1, using kinetic theory, we connected the mean free path of a gas to measured macroscopic transport properties of the gas such as its binary diffusivity. A similar procedure can be used to obtain a particle mean free path A,p from the Brownian diffusion coefficient and an appropriate kinetic theory expression for the diffusion flux. Following an argument identical to that in Section 9.1, diffusion of aerosol particles can be viewed as a mean free path phenomenon so that... 

Molecular dynamics (MD) simulations have been used to simulate non-equilibrium binary diffusion in zeolites. Highly anisotropic diffusion in boggsite provides evidence in support of molecular traffic control. For mixtures in faujasite, Fickian, or transport, diffusivities have been obtained from equilibrium MD through appropriate correlation functions and used in macroscopic models to predict fluxes through zeolite membranes under co- and counterdiffusion conditions. For some systems, MD cannot access the relevant time scales for diffusion, and more appropriate simulation techniques are being developed. 

A simple classification of the main macroscopic techniques is shown in Table 1, and this provides a useful framework for our review. Macroscopic measurements generally yield transport diffusivities, although variants of the techniques, using isotopically tagged tracers, can be devised to measure self-diffusivities. The large majority of the macroscopic techniques involve transient measurements. Steady-state or quasi-steady-state methods, notably membrane permeation and catalyst effectiveness measurements, have been demonstrated, but their application has been limited to a few systems. 

Since most microscopic techniques measure self-diffusion, whereas the macroscopic techniques generally measure transport diffusion, direct comparisons between the measured diffusivities are not meaningful, except in the... 

Molecular movement under non-equilibrium conditions (i.e. under the influence of differences in the overall concentration) is associated with a macroscopic particle transfer and is generally referred to as transport diffusion. Transport diffusion may be measured under both steady-state conditions (e.g., by studying the permeation rates through zeolite membranes [187-... 

Despite extensive work in the last decade, large discrepancies still persist between the various experimental techniques which measure diffusion in zeohtes. One of the difficulties is that one has to compare self-diffusivities, obtained by PFG NMR or QENS methods, with transport diffusivities derived from macroscopic experiments. The transport diffusivity is defined as the proportionahty factor between the flux and a concentration gradient (Fick s first law)... 

Equation (5-9) is identical with the relation obtained from collision theory, Eq. (2-33) for k = 1 (unlike reactants), and for a reaction whose steric factor p is unity. It is also worth noting that the synthesis of the macroscopic rate constant /f(T) in Eq. (5-6) from the microscopic quantities in Eqs. (5-2) to (5-5) strongly parallels portions of the rigorous derivation of the macroscopic transport coefficients, e.g., those of viscosity, thermal conductivity, or diffusion [4, 5]. 

The main objective of this chapter is to establish the relation between the macroscopic equations like (3.1) and (3.5), the mesoscopic equations (3.2) and (3.3), etc., and the underlying microscopic movement of particles. We will show how to derive mesoscopic reaction-transport equations like (3.2) and (3.3) from microscopic random walk models. In particular, we will discuss the scaling procedures that lead to macroscopic reaction-transport equations. As an example, let us mention that the macroscopic reaction-diffusion equation (3.1) occurs as a result of the convergence of the random microscopic movement of particles to Brownian motion, while the macroscopic fractional equation (3.5) is closely related to the convergence of random walks with heavy-tailed jump PDFs to a-stable random processes or Levy flights. 

For the purpose of macroscopic transport of water, self-diffusion coefficients of water are converted to Fickian diffusion or the chemical diffusion... 

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