Molecular diffusion tensor

As the existence of MChA can be deduced by very general symmetry arguments and the effect does not depend on the presence of a particular polarization, one may wonder if something like MChA can also exist outside optical phenomena, e.g. in electrical conduction or molecular diffusion. Time-reversal symmetry arguments cannot be applied directly to the case of diffusive transport, as diffusion inherently breaks this symmetry. Instead, one has to use the Onsager relation. (For a discussion see, e.g., Refs. 34 and 35.) For any generalized transport coefficient Gy (e.g., the electrical conductivity or molecular diffusion tensor) close to thermodynamic equilibrium, Onsager has shown that one can write... 

The dynamic terms in eqn (1) depend upon the assumptions used to describe the motion. For the intermolecular motion a diffusive process is assumed (rotation through a sequence of small angular steps). In that case intermolecular reorientation can be characterized by two rotational correlation times, and Tru. The correlation time for reorientation of the symmetry axis of a molecular diffusion tensor is Tr, while Tru refers to rotation about the axis. For the intramolecular motion a random jump process is assumed. Thus, isomerization occurs through jumps between different conformations with an average lifetime Tj. 

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. 

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... 

The connection between anisotropic molecular motion and nuclear relaxation was derived by Woessner as early as 1962 [161]. Accordingly, the dipole-dipole relaxation time of a carbon nucleus is a function of the diagonal components R, R2, and R3 of the rotational diffusion tensor and the cosines X, p, and v of the angles assumed by the C —H bonds relative to the principal axes of this tensor ... 

If the position of the principal axes of the rotational diffusion tensor were known with respect to the molecular coordinates, then the motion of the molecule could be calculated from the measured relaxation times. With simple molecules, however, it is possible to interpret the Tt values qualitatively in terms of an anisotropic motion. 

Abstract We use Nuclear Magnetic Resonance relaxometry (i.e. the frequency variation of the NMR relaxation rates) of quadrupolar nucleus ( Na) and H Pulsed Gradient Spin Echo NMR to determine the mobility of the counterions and the water molecules within aqueous dispersions of clays. The local ordering of isotropic dilute clay dispersions is investigated by NMR relaxometry. In contrast, the NMR spectra of the quadrupolar nucleus and the anisotropy of the water self-diffusion tensor clearly exhibit the occurrence of nematic ordering in dense aqueous dispersions. Multi-scale numerical models exploiting molecular orbital quantum calculations, Grand Canonical Monte Carlo simulations, Molecular and Brownian Dynamics are used to interpret the measured water mobility and the ionic quadrupolar relaxation measurements. 

For asymmetric-top molecules, all three principal values of the rotational diffusion tensor are required to describe the molecular dynamics hence at least three different T, values of geometrically nonequivalent carbons are required to solve the three independent simultaneous equations derived by Woessner45 ... 

A more rigorous treatment of the variable-temperature, 3C Tl data has been used to describe the overall molecular motion of the glucose derivative, namely 1,6-an-hydro-/3-D-glucopyranose (31) in solution.147 This rigid molecule contains a number of nonequivalent 13C- H vectors and is amenable to a rigorous quantitative treatment by means of Eq. 29. Table VI summarizes the diagonal elements of the rotational diffusion tensor, that is, the rotational diffusion constants Dx,Dy, and Dz obtained upon diagonalization of the rotational diffusion tensor with respect to... 

Latour LL, Warach S (2002) Cerebral spinal fluid contamination of the measurement of the apparent diffusion coefficient of water in acute stroke. Magn Reson Med 48 478-486 Latour LL, Svoboda K, Mitra PP, Sotak CH (1994) Time-depen-dent diffusion of water in a biological model system. Proc Natl Acad Sci USA 91 1229-1233 Le Bihan D (1995) Molecular diffusion, tissue microdynamics and microstructure. NMR Biomed 8 375-386 Le Bihan D (2003) Looking into the functional architecture of the brain with diffusion MRI. Nat Rev Neurosci 4 469-480 Le Bihan D, van Zijl P (2002) From the diffusion coefficient to the diffusion tensor. NMR Biomed 15 431-434 Le Bihan D, Mangin JF, Poupon C, Clark CA, Pappata S, Molko... 

In all of the above descriptions and equations, it was assumed that molecular tumbling is isotropic in the solution. This is an idealized case that can be approximated by real proteins only in fortunate instances. In general, proteins exhibit totally anisotropic tumbling7 but in special cases, the simplification of only an axially symmetric motion is used, for which two of the molecular axes are assumed to be identical and thus the corresponding rotational diffusion tensor can be simplified. 

As indicated, the flux may be expressed either in units of molecules/m2 s or in units of kg/m2 s. Here, p and n are the density and number density of air, respectively, and K is called the eddy diffusion coefficient. This quantity must be treated as a tensor because atmospheric diffusion is highly anisotropic due to gravitational constraints on the vertical motion and large-scale variations in the turbulence field. Eddy diffusivity is a property of the flowing medium and not specific to the tracer. Contrary to molecular diffusion, the gradient is applied to the mixing ratio and not to number density, and the eddy diffusion coefficient is independent of the type of trace substance considered. In fact, aerosol particles and trace gases are expected to disperse with similar velocities. 

The results simplify considerably if the body-fixed axis system is a principal axis system for the polarizability tensor as well as for the rotational diffusion tensor. In this case Ag = At = A5 = 0 in Eq. (7.5.27). Then the spectrum consists of only two Lorentzians. Many asymmetric diffusors do have enough symmetry to rigorously satisfy this condition-for instance, planar molecules with at least one two fold rotation axis in the molecular plane. Others may have these axes so close together that Ag = A4 = A5 = 0 and the spectrum effectively consists of only two Lorentzians. In any particular application, it must be kept in mind that the spectrum might very well be the five-Lorentzian form given by the Fourier transform of Eq. (7.5.25). 

In Chapter 7 an application of C-13 NMR to study rotational motion of molecules is briefly described. When combined with depolarized light scattering, the magnetic resonance results yield values for the components of the rotational diffusion tensors of some symmetric top molecules. In some circumstances, NMR and ESR methods allow measurement of the relaxation times of the molecular angular momentum (e.g., see McClung and Kivelson, 1968). 

Here and (p denote the orientation of the principal axes of the diffusion tensor with respect to the applied field gradients. For quantitative analysis of Eq. (38) it is essential that in the case of ZSM-5 the three principal elements of the diffusion tensor are not independent of each other. Under the assumption that molecular propagation from one channel intersection to an adjacent one is independent of the diffusing molecule s history (in other words, independent of the original... 

A large variety of zeolites (e.g., ZSM-12, -22, -23, -48, AIPO4-5, -8, -11, and VPI-5) contain systems of parallel channels with diameters on the order of the molecular dimensions. Molecular propagation in this type of adsorbent represents a special case of diffusion anisotropy, since the main elements of the diffusion tensor referring to the plane perpendicular to the direction of the channel system are equal to zero. In the first PFG NMR diffusion measurements of methane in ZSM-12 and AIPO4-5, only a lower limit on the order of I0 m"s could be determined for diffusivity in the channel direction (160). This value is two orders of magnitude below the diffusivity of methane in the straight channels of zeolite ZSM-5 (see Fig. 14). Since the channel diameters of ZSM-12... 

The evolution of the orientational distribution results from the orientational hole burning, orientational redistribution, and rotational diffusion. If we assume that the trans and cis molecules are anisometrically shaped and that they may be characterized by uniaxial molecular polarizability tensors (cigar-shaped molecules), then the evolution of the angular distribution has the general form... 

In order to determine the diffusion tensor that best fits a given trajectory, we use a procedure that was developed for the analysis of NMR relaxation data [45-47], and which is well-suited for adaptation to the analysis of MD simulations. This uses the average (or "effective") correlation time for a particular vector, n (e.g. a backbone N-H bond vector, or a randomly chosen direction in the molecular frame), which can be defined as... 

In contrast to the X and A type zeolites, the framework of ZSM-5 is of non-cubic structure. Hence, as a consequence of the interrelation between zeolite structure and molecular mobility, molecular diffusion in different crystallographic directions has to proceed at different rates so that, strictly speaking, molecular diffusion must be described by a diffusion tensor rather than by the diffusion coefficient, i.e. a scalar quantity. 

The magnetization was only taken as an example. Many other properties (dielectric susceptibility, electric and thermal conductivity, molecular diffusion, etc.) are also described by second rank tensors of the same (quadrupolar) type Microscopically, such properties can be described by single-particle distribution functions, when intermolecular interaction is neglected. There are also properties described by tensors of rank 3 with 3 = 27 components (e.g., molecular hyperpolarizability Yijk) and even of rank 4 (e.g., elasticity in nematics, ATiju) with 3 = 81 components. Microscopically, such elastic properties must be described by many-particle distribution functions. 

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