Transport and Self-Diffusion

At a specified concentration level the transport diffusivity (D) relating the flux to the concentration gradient, in accordance with Eq. (5.1), is in general different from the self-diffusivity S ) which defines the rale of tracer exchange of marked molecules under equilibrium conditions in a system with no 

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Another attempt to correlate transport and self-diffusivities has been based on a generalization of the Stefan-Maxwell formulation of irreversible thermodynamics [111-113]. By introducing various sets of parameters describing the facility of exchange between two molecules of the same and of different species, the resulting equations are more complex than eqs 27 and 28 They may be shown, however, to include these relations as special cases... 

NMR PFG measurements determine the tracer or self-diffusivity (D ) under equilibrium conditions with no concentration gradient. n any sorption rate measurement it is the transport diffusivity under the influence of a concentration gradient which is measured. In general these two quantities are not the same but the relationship between them can be established from irreversible thermodynamics. (17,18) In the low concentration limit the thermodynamic correction factor vanishes and the transport and self diffusivities should approach the same limit. Since ZLC measurements are made at low concentrations within the Henry s Law region the diffusivity values should be directly comparable with the NMR self-dif fusivities. ... 

The experimental methods for measurement of transport and self-diffusion in zeolite crystals (and in other microporous materials) are reviewed. Large discrepancies between distent techniques are commonly observed and appear to be related to the scale of the measurements, suggesting that structural defects may be more important than is generally believed. 

Some 30 years ago, transport properties of molten salts were reviewed by Janz and Reeves, who described classical experimental techniques for measuring density, electrical conductance, viscosity, transport number, and self-diffusion coefficient. 

In general, the coefficients of transport diffusion (D) and self-diffusion (D ) under the conditions of singlecomponent adsorption are assumed to be correlated by the equation... 

In order to establish the prerequisites for interpreting surface layer formation on metals it appears necessary to deal somewhat extensively with transport processes in crystalline substances wdth lattice defects and to explore the relationship between transport of defects and self-diffusion. 

The fact that this empirical law applies so widely clearly is trying to tell us something. It is that the basic factors determining viscous flow and self-diffusion are the same for all liquids that do not have to break bonds before they undergo transport. 

A summary of the major experimental techniques that have been applied to study diffusion in microporous solids is given in Table 1. These can be classified according to the nature of the measurement (transient or steady state), the scale of the measurement (micro, meso or macro) and the type of process (transport or self-diffusion). Additional details with complete references are given in recent reviews [10-12]. 

In liquid salts the isotope factor is in principle measurable by electrotransport (electromigration). Since many such measurements are available (13), it is especially tempting to study thermotransport in molten ionic media. One must bear in mind, however, the possibility of the following complications (a) the mechanisms of electrotransport, thermotransport, and self-diffusion may be non-identical see Reference 16) b) the isotope factors, as determined by electrotransport, are dependent, via transport numbers, on the reference system and (c) severe experimental difficulties may be encountered in liquid salt thermotransport, mainly corrosion and convection effects. 

From the NMR tracer desorption and self-diffusion data (second and third lines of Table I), one obtains the relation Timm > TmlL. In the example given, intercrystalline molecular exchange is limited, therefore, by transport resistances at the surface of the individual crystals. Combined NMR and high-resolution electron microscopy studies 54) suggest that such surface barriers are caused by a layer of reduced permeability rather than by a mere deposit of impenetrable material on the crystal surface, although that must not be the case in general. 

As stated in the earlier section on Generalized Equations of Motion, we would ultimately like to find a set of equations of motion in the form of Eq. [91] to compute transport coefficients such as the shear viscosity and self diffusion... 

Physical Mechanisms. The simplest interpretation of these results is that the transport coefficients, other than the thermal conductivity, of the water are decreased by the hydration interaction. The changes in these transport properties are correlated the microemulsion with compositional phase volume 0.4 (i.e. 60% water) exhibits a mean dielectric relaxation frequency one-half that of the pure liquid water, and ionic conductivity and water selfdiffusion coefficient one half that of the bulk liquid. In bulk solutions, the dielectric relaxation frequency, ionic conductivity, and self-diffusion coefficient are all inversely proportional to the viscosity there is no such relation for the thermal conductivity. The transport properties of the microemulsions thus vary as expected from simple changes in "viscosity" of the aqueous phase. (This is quite different from the bulk viscosity of the microemulsion.)... 

Equilibrium MD simulations of self-diffusion coefficients, shear viscosity, and electrical conductivity for C mim][Cl] at different temperatures were carried out [82] The Green-Kubo relations were employed to evaluate the transport coefficients. Compared to experiment, the model underestimated the conductivity and self-diffusion, whereas the viscosity was over-predicted. These discrepancies were explained on the basis of the rigidity and lack of polarizability of the model [82], Despite this, the experimental trends with temperature were remarkably well reproduced. The simulations reproduced remarkably well the slope of the Walden plots obtained from experimental data and confirmed that temperature does not alter appreciably the extent of ion pairing [82],... 

The transport coefficients like viscosity, thermal conductivity and self-diffusivity for a pure mono-atomic gas and the diffusivity for binary mixtures obtained from the rigorous Chapman-Enskog kinetic theory with the Lennard-Jones interaction model yield (e.g., [39], sect 8.2 [5], sects 1-4, 9-3 and 17-3) ... 

Even if a proton jump along a bond is very rapid, the conductivity is explained by the Grotthuss mechanism (it is not the same proton that jumps). Please notice that this jump is not a rotation it takes place along the bond from one minimum of the potential to another, creating, for short time, an ion H30+ and a correspondent OH-. But, I insist, their concentration is very small. It is out of question to see such species, for example, in scattering experiments. Their impact on the transport properties (self-diffusion, molecular rotations) is totally negligible. 

The transport diffusivity and self-diffusivity are related by the Darken relation,... 

Computer simulations such as molecular dynamics (MD) and Brownian dynamics (BD) permit the study of transport properties. Self-diffusion coefficients can easily be obtained by differentiation of mean-square displacements or by integration of the velocity self-correlation functions of the ion. In contrast, the evaluation of conductivity by means of cross-correlation functions is cumbersome and computer-time-consuming and can only scarcely be executed. 

Fig. 2.8. Reciprocal of the apparent value of as determined from viscosity (—) and self-diffusion coefficients (- -) using hard-sphere transport theory (Table 2.2). Transport coefficient data from H. H. Landolt and R. Bomstein, Zahlenwerte and Funktionen, 6th ed. (Berlin Springer, 1969), Vol. Ill, Part 5a, pp. 3, 516.

Experimental diffusion coefficients, as obtained from time-lag measiu ements, report a transport diffusion coefficient which carmot be obtained from equilibrium MD simulation. Comparisons made in the simulation literatme are typically between time-lag diffusion coefficients (even calculated for glassy polymers without correction for dual-mode contributions and self-diffusion coefficients. As discussed above, mutual diffusion coefficients can be obtained directly from equilibrium MD simulation but simulation of transport diffusion coefficients require the use of NEMD methods, that are less commonly available and more computationally expensive [117]. 

Whether, however, one can conclude that computer simulation is a practical tool for calculating transport coefficients in general is open to question. It is more difficult to obtain the viscosity and thermal conductivity to within a reasonable error than it is to deter-min and self-diffusion, and the latter is not a trivial calculation. ... 

Hellmann, R., Bich, E., Vogel, E., Dickinson, A. S., and Vesovic, V, Calculation of the Transport and Relaxation Properties of Methane. I. Shear Viscosity, Viscomagnetic Effects, and Self-Diffusion, / Chem. Phys. 129,064302, 2008. 

Ohkubo et study electronic properties structure and transport coefficients (conductivity and self-diffusion) of a molten acLi20-(1 — x)B203 system using first-principles MD simulations performed with their own finite element density functional theoiy code, FEMTECK and PFG NMR measurements. For diffusion the first-principles simulation results were in better agreement with experiment than that obtained from classical simulations. 

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