Diffusivity tensor

In an isotropic medium, D is a scalar, which may be constant or dependent on time, space coordinates, and/or concentration. In anisotropic media (such as crystals other than cubic symmetry, i.e., most minerals), however, diffusivity also depends on the diffusion direction. The diffusivity in an anisotropic medium is a second-rank symmetric tensor D that can be represented by a 3 x 3 matrix (Equation 3-25a). The tensor is called the diffusivity tensor. Diffusivity along any given direction can be calculated from the diffusivity tensor (Equation 3-25b). Each element in the tensor may be constant, or dependent on time, space coordinates and/or concentration. 

The tensor has three degrees of freedom and is represented by a 3x3 symmetric matrix (Dxx, Dxy, Dxz, Dyx, Dyy, Dyz, Dzx, Dzy, Dzz) (see Fig. 7.5a). To obtain the full diffusion tensor, diffusion measurements in all of these directions are required (minus those that are mathematically the same, i.e. Dxy=Dyx) therefore, the diffusion gradients need to be applied in at least six different non-collin-ear directions (at high b-values) to sample the full tensor data plus an additional image that needs to be sampled at a low b-value. 

Brenner Edwards (1993) have examined mixed systems of materials of the same thermal diffusivity but differing thermal conductivity, emphasizing that the resultant effective diffusivity of the mixed system on the macroscale is not a constant scalar diffusivity but rather an anisotropic tensor diffusivity. This point is illustrated by the simple example of a laminated material, composed of alternating layers of equal diffusivity air a and liquid metal m of thicknesses and / , which extends to infinity in all three spatial directions. Applying the formulas for parallel and series conductivities, the effective diffusivity in the direction parallel to the strata is just a = a., = a . In the limit where > k, the relation for the effective diffusivity in the direction perpendicular to the strata reduces to (Brenner c Edwards 1993)... 

Dw = hydrodynamic dispersion mechanisms tensor (diffusion + mechanical dispersion)... 

Basser P J, Mattiello J and LeBihan D 1994 MR diffusion tensor speotrosoopy and imaging Biophys. J. 66 259-67... 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

Die-drawing 143 Die-drawn fibers 144 Differential polarizability tensor 91 Differential scanning calorimetry 63, 165 Diffusion 189... 

As expected from continuum theory, the friction and diffusion coefficients are replaced In Inhomogeneous fluid by tensors whose symmetry reflects that of the Inhomogeneous media. 

In order to overcome these problems, one approach, originally developed by Whitaker [420], Slattery [359], and Anderson and Jackson [17], involves the method of volume averaging. Using volume averaging theory, Whitaker and coworkers [193,264,268,337,436] found the effective diffusion tensor for a two-phase system to be given by... 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

Trinh, S Locke, BR Arce, P, Effective Diffusivity Tensors of Point-Like Molecules in Anisohopic Media by Monte Carlo Simulation, Transport in Porous Media in review, 2001. 

This simple relaxation theory becomes invalid, however, if motional anisotropy, or internal motions, or both, are involved. Then, the rotational correlation-time in Eq. 30 is an effective correlation-time, containing contributions from reorientation about the principal axes of the rotational-diffusion tensor. In order to separate these contributions, a physical model to describe the manner by which a molecule tumbles is required. Complete expressions for intramolecular, dipolar relaxation-rates for the three classes of spherical, axially symmetric, and asymmetric top molecules have been evaluated by Werbelow and Grant, in order to incorporate into the relaxation theory the appropriate rotational-diffusion model developed by Woess-ner. Methyl internal motion has been treated in a few instances, by using the equations of Woessner and coworkers to describe internal rotation superimposed on the overall, molecular tumbling. Nevertheless, if motional anisotropy is present, it is wiser not to attempt a quantitative determination of interproton distances from measured, proton relaxation-rates, although semiquantitative conclusions are probably justified by neglecting motional anisotropy, as will be seen in the following Section. 

D. LeBihan, J. F. Mangin, C. Poupon, C.A. Clark, S. Pappata, N. Molko, H. Chabriat 2001, (Diffusion tensor imaging concepts and applications),/. Magn. Reson. Imag. 13, 534—546. 

Some micro- and mesoporous materials exhibit anisotropic pore structures, which may yield different values for the diffusivities in the three orthogonal spatial directions. In such systems, the self-diffusion should be described by a diffusion tensor rather than by a single scalar self-diffusion coefficient. By measuring over a... 

Fig. 3.1.4 Anisotropic self-diffusion of water in and filled symbols, respectively). The horizon-MCM-41 as studied by PFG NMR. (a) Depen- tal lines indicate the limiting values for the axial dence of the parallel (filled rectangles) and (full lines) and radial (dotted lines) compo-perpendicular (circles) components of the axi- nents of the mean square displacements for symmetrical self-diffusion tensor on the inverse restricted diffusion in cylindrical rods of length temperature at an observation time of 10 ms. / and diameter d. The oblique lines, which are The dotted lines can be used as a visual guide, plotted for short observation times only, repre-The full line represents the self-diffusion sent the calculated time dependences of the...

The spatial and temporal evolution of the concentration field is dependent on the velocity field vector v(r,t), the diffusion tensor D(r,t) and any reactions occurring in the system R(r,t). Non-dimensionalization of Eqn. (5.1.4) generates the Pedet... 

The equations of motion (75) can also be solved for polymers in good solvents. Averaging the Oseen tensor over the equilibrium segment distribution then gives = l/ n — m Y t 1 = p3v/rz and Dz kBT/r sNY are obtained for the relaxation times and the diffusion constant. The same relations as (80) and (82) follow as a function of the end-to-end distance with slightly altered numerical factors. In the same way, a solution of equations of motion (75), without any orientational averaging of the hydrodynamic field, merely leads to slightly modified numerical factors [35], In conclusion, Table 4 summarizes the essential assertions for the Zimm and Rouse model and compares them. 

Key Words Carbon-13 spin relaxation, T, Measurements, Nuclear Overhauser effect, Rotation-diffusion tensor, HOESY experiments... 

If the considered molecule cannot be assimilated to a sphere, one has to take into account a rotational diffusion tensor, the principal axes of which coincide, to a first approximation, with the principal axes of the molecular inertial tensor. In that case, three different rotational diffusion coefficients are needed.14 They will be denoted as Dx, Dy, Dz and describe the reorientation about the principal axes of the rotational diffusion tensor. They lead to unwieldy expressions even for auto-correlation spectral densities, which can be somewhat simplified if the considered interaction can be approximated by a tensor of axial symmetry, allowing us to define two polar angles 6 and

symmetry axis of the considered interaction) in the (X, Y, Z) molecular frame (see Figure 5). As the tensor associated with dipolar interactions is necessarily of axial symmetry (the relaxation vector being... 

Figure 5 Definition of the polar angles in the principal axis system of the diffusion-rotation tensor.

Fortunately, in the case of a rotational diffusion tensor with axial symmetry (such molecules are denoted "symmetric top"), some simplification occurs. Let us introduce new notations D// = Dz and D = Dx = Dy. Furthermore, we shall define effective correlation times ... 

It can be noticed that at least two independent relaxation parameters in the symmetric top case, and three in the case of fully anisotropic diffusion rotation are necessary for deriving the rotation-diffusion coefficients, provided that the relevant structural parameters are known and that the orientation of the rotational diffusion tensor has been deduced from symmetry considerations or from the inertial tensor. 

Figure 15 The model molecule used to demonstrate the possibilities of HOESY experiments in terms of carbon-proton distances and reorientational anisotropy. To a first approximation, the molecule is devoid of internal motions and its symmetry determines the principal axis of the rotation-diffusion tensor. Note that H, H,., H,-, H,/ are non-equivalent. The arrows indicate remote correlations.

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