The diffusion tensor

According to Kubo (10), the general quantum-mechanical expression for the diffusion tensor is 

Let us consider the Kubo expression in somewhat more detail for the case of excitons in a molecular crystal. We will restrict ourselves to the presence of one exciton. The position of its center of gravity then reads 

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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... 

Derivation of the Diffusion Tensor when Protein Structure is Known 295... 

The anisotropy of the overall tumbling will result in the dependence of spin-relaxation properties of a given 15N nucleus on the orientation of the NH-bond in the molecule. This orientational dependence is caused by differences in the apparent tumbling rates sensed by various internuclear vectors in an anisotropically tumbling molecule. Assume we have a molecule with the principal components of the overall rotational diffusion tensor Dx, Dy, and l)z (x, y, and z denote the principal axes of the diffusion tensor), and let Dx< Dy< Dz. 

Equation (10) allows the determination of the principal values of the diffusion tensor if the orientation of its principal axes frame is known. The latter information is included there in implicit form, via Eq. (11). The problem is that neither the principal axes nor principal values are known a priori and are to be determined simultaneously, as outlined below. 

When the structural information is available, then the components of the diffusion tensor can be derived from the minimization of the target function,... 

What Can We Do when Protein Structure is not Known Preliminary Characterization of the Diffusion Tensor... 

It should be mentioned that rotational anisotropy of the molecule will result in an increase in the R2 values for NH vectors having particular orientation with respect to the diffusion tensor frame [46]. This increase could be misinterpreted as conformational exchange contributions, and, vice versa, small values of Rex, usually of the order or 1 s 1 or less, could be mistaken for the manifestation of the rotational anisotropy. Therefore, identification of residues subjected to conformational exchange is critical for accurate analysis of relaxation data. Additional approaches are necessary to distinguish between the two effects. As suggested earlier [27] (see also Ref. [26]), a comparison between R2 and the cross-correlation rate r]xy could serve this purpose, as tjxy contains practically the same combination of spec-... 

Equation (51) is highly nonlinear. One of the key approximations made to enable analytical calculations is the preaveraging approximation, whereby the diffusion tensor D[R(i) - R s )] is replaced by its average over chain configurations. 

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... 

For binary diffusion in an isotropic medium, one diffusion coefficient describes the diffusion. For binary diffusion in an anisotropic medium, the diffusion coefficient is replaced by a diffusion tensor, denoted as D. The diffusion tensor is a second-rank symmetric tensor representable by a 3 x 3 matrix ... 

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 diffusion equation in an anisotropic medium is complicated. Based on the definition of the diffusivity tensor, the diffusive flux along a given direction (except along a principal axis) depends not only on the concentration gradient along this direction, but also along other directions. The flux equation is written as F = —D VC (similar to Fick s law F= -DVC but the scalar D is replaced by the tensor D), i.e.. 

The size of a surface available for field ion microscope study of surface diffusion is very small, usually much less than 100 A in diameter. The random walk diffusion is therefore restricted by the plane boundary. For a general discussion, however, we will start from the unrestricted random walk. First, we must be aware of the difference between the chemical diffusion coefficient and the tracer diffusion coefficient. The chemical diffusion coefficient, or more precisely the diffusion tensor, is defined by a generalized Fick s law as... 

Here C is the concentration vector and D(C) is the diffusivity tensor defined by (3.1.15a). Thus, locally electro-neutral electro-diffusion without electric current is exactly equivalent to nonlinear multicomponent diffusion with a diffusivity tensor s being a rational function of concentrations of the charged species. 

Zibrowius et al. (77) used an NMR spin-echo attenuation technique to estimate the ratio of the diffusion tensor element related to motion along the y-axis (the straight channel) and the average of the other two elements. For a methane loading of 8 molecules per unit cell, the ratio was estimated to be less than 5 at room temperature ... 

The diffusion equation for the motion of all the m quenchers and the fluorophor considers the rate of change of the density n to be due to migration of the fluorophor and each quencher out of their respective volume elements or to quenching of the excited fluorophor. When the flux of a quencher or fluorophor is in the same direction as the concentration gradient and hydrodynamic effects are unimportant, the diffusion tensor is diagonal and eqn. (211) becomes... 

The diffusion tensor Djk is real and symmetric, and for small enough p it is positive definite. Therefore it possesses a real symmetric square root. From Eq. (22), we see that this square root is in fact the tensor [Pg.332]

The error concerned an explicit formula for the translational diffusion coefficient. Kirkwood calculated the diffusion tensor as the projection onto chain space of the inverse of the complete friction tensor he should have projected the friction tensor first, and then taken the inverse. This was pointed out by Y. Ikeda, Kobayashi Rigaku Kenkyushu Hokoku, 6, 44 (1956) and also by J. J. Erpenbeck and J. G. Kirkwood, J. Chem. Phys., 38, 1023 (1963). An example of the effects of the error was given by R. Zwanzig, J. Chem. Phys., 45, 1858 (1966). In the present article this question does not come up because we use the complete configuration space. 

The arrangement of Si04 tetrahedra in three dimensions, and thus the degree of Si-O-Si interlinkage, determines the diffusivity tensor of the components to a large extent. If, in the network of tetrahedra, Si4+ is replaced by a cation of lower valence (e.g., Als+), a corresponding amount of network modifying cations is... 

D is called the diffusivity tensor and acts as an object that connects one vector to another (e.g., the flux vector with the gradient vector). This connection can be written in matrix form as in Eq. 4.57. The diffusivity tensor D is symmetric (i.e., Dtj = Dji) for any underlying material symmetry. 

The diffusivity tensor has special forms for particular choices of coordinate axes if the diffusing body itself has special symmetry (e.g., if it is crystalline). Neumann s principle states ... 

A consequence of Neumann s symmetry principle is that direct tensor Onsager coefficients (such as in the diffusivity tensor) must be symmetric. This is equivalent to the addition of a center of symmetry (an inversion center) to a material s point group. Thus, the direct tensor properties of crystalline materials must have one of the point symmetries of the 11 Laue groups. Neumann s principle can impose additional relationships between the diffusivity tensor coefficients Dij in Eq. 4.57. For a hexagonal crystal, the diffusivity tensor in the principal coordinate system has the form... 

According to Eq. 4.63, the diffusivity tensor in any other rotated system (indicated by a prime) will have the form... 

Consider two-dimensional anisotropic diffusion in an infinite thin film where the initial condition consists of a point source of atoms located at x = 2 = 0. The diffusivity tensor D, in arbitrary units, in the (2 1, 2) coordinate system is... 

Therefore, the diffusivity tensor in the principal axes system is... 

Solution. Adopting a principal axis system in which k is parallel to the c-axis and i and j lie in the basal plane, the diffusivity tensor will have the form given in Eq. 4.66 and the flux will be given by... 

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