Diffusion ellipsoid

As it is difficult to display tensor data, the concept of diffusion ellipsoids has been proposed (le Bihan 2003). An ellipsoid is a three-dimensional... 

Fig. 7.4 Relationship between anisotropic diffusion, diffusion ellipsoids, and diffusion tensor. In an isotropic environment (a), diffusion is equal in aU directions and can be characterized by diagonal elements (D, D, and D ) all of which have the same value D. In anisotropic diffusion (b and c), the diffusion tensor is geometrically equivalent to an ellipsoid, with the three eigenvectors of the tensor matrix set as the minor and major axis of the ellipsoid...

Other techniques such as X-ray diffusion or small angle neutron diffusion are also used in attempts to describe the size and form of asphaltenes in crude oil. It is generally believed that asphaltenes have the approximate form of very flat ellipsoids whose thicknesses are on the order of one nanometer and diameters of several dozen nanometers. 

This is obvious for the simplest case of nondeformable anisotropic particles. Even if such particles do not change the form, i.e. they are rigid, a new in principle effect in comparison to spherical particles, is their turn upon the flow of dispersion. For suspensions of anisodiametrical particles we can introduce a new characteristic time parameter Dr-1, equal to an inverse value of the coefficient of rotational diffusion and, correspondingly, a dimensionless parameter C = yDr 1. The value of Dr is expressed via the ratio of semiaxes of ellipsoid to the viscosity of a dispersion medium. 

Fig. 46 Schematic diagram of elemental process during transition from Hex cylinder to bcc sphere (i) undulation of interface (a, b), (ii) break-up of cylinders into ellipsoids (b, c), (iii) relaxation of domains from ellipsoids into spheres (c, d), and (iv) relaxation in junction distribution to attain uniform distribution (d, e). Pole where concentration of junction points is low may work as memory of grain conservation upon reverse transition from bcc sphere to Hex cylinder. Small arrows in part (b) indicate diffusion of chemical junctions along interface in process (ii). From [136], Copyright 2000 American Chemical Society...

In the diffuse reflectance mode, samples can be measured as loose powders, with the advantages that not only is the tedious preparation of wafers unnecessary but also diffusion limitations associated with tightly pressed samples are avoided. Diffuse reflectance is also the indicated technique for strongly scattering or absorbing particles. The often-used acronyms DRIFT or DRIFTS stand for diffuse reflectance infrared Fourier transform spectroscopy. The diffusely scattered radiation is collected by an ellipsoidal mirror and focussed on the detector. The infrared absorption spectrum is described the Kubelka-Munk function ... 

In the particular case of prolate and oblate ellipsoids, the number of exponentials is reduced to three because two of the three axes are equivalent. The rotation diffusion coefficients around the axis of symmetry and the equatorial axis are denoted Di and D2, respectively. The emission anisotropy can then be written as... 

Perrin R, Mouvement Brownien d un ellipsoide. II. Rotation libre et depolarisation des fluorescences. Translation et diffusion de molecule ellipsoidales, J. Phys. Radium, 7, 1, 1936. 

Figure 2.9 is a plot of possible combinations of hydration and asymmetry for protein particles in water. Similar curves could be drawn for other materials as well. For the human hemoglobin molecule discussed in Table 2.1, the combination of sedimentation and diffusion measurements gives an /// value that lies within the domain defined by the 1.15 and 1.20 contours of Figure 2.9. The current picture of the structure of human hemoglobin, deduced from x-ray diffraction studies, suggests that the molecule may be regarded as an ellipsoid with height, width, and depth equal to 6.4, 5.5, and 5.0 nm, respectively. Applying these dimensions to the dispersed unit leads us to describe the particle as being hydrated to the extent of about 0.4-0.5 g water (g protein)...

Ellipsoidal or rod-shaped molecules have two different rotary diffusion constants while, if the dimensions of the molecules are different along all three axes, three constants must be specified.36... 

Cylinders, Ellipsoids, and Elliptical Paraboloids. The diffusion-limited growth of particles whose planar intercepts are conic sections can also be analyzed by the scaling method. For example, the scaling function appropriate for a cylinder is T] = r/y/t.4 The solution for the growth of a cylinder is obtained in Exercise 20.5. 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

Eq. (3.21) discussed in Section 3.3.2 is only valid if the motion of the molecules under study has no preferential orientation, i.e. is not anisotropic. Strictly speaking, this applies only for approximately spherical bodies such as adamantane. Even an ellipsoidal molecule like trans-decalin performs anisotropic motion in solution it will preferentially undergo rotation and translation such that it displaces as few as possible of the other molecules present. This anisotropic rotation during translation is described by the three diagonal components Rlt R2, and R3 of the rotational diffusion tensor. If the principal axes of this tensor coincide with those of the moment of inertia - as can frequently be assumed in practice - then Rl, R2, and R3 indicate the speed at which the molecule rotates about its three principal axes. 

The rotational diffusion constant in water at 25° and neutral pH as measured by electric birefringence (258) is 230 X 105 sec-1 or 0.73 X HT8 sec as a relaxation time. For a hydrodynamic ellipsoid of dimensions 66 X 22 A and a molecular weight of 14,000, the calculated relaxation tilde is 0.72 X 10-8 sec. However, the apparent asymmetry of the molecule from the X-ray structure corresponds to an axial ratio of no more than 2 1 rather than 3 1. 

To the extent that the X-ray structure can be represented by an ellipsoid of revolution, it appears to be an oblate ellipsoid with an axial ratio no greater than 1 2. The approximate dimensions are 25 X 45 X 45 A. From molecular weight and free diffusion data the frictional ratio /// is 1.26. This corresponds to an unhydrated prolate ellipsoid with an axial ratio of 5.2 or an oblate ellipsoid with a ratio 0.18. If the hydration is assumed to be 0.34 g H20/g protein, the axial ratio values would be 3.0 and 0.33, respectively. The maximum hydration for a sphere would be 0.7 or 0.55 g/g for a prolate ellipsoid of axial ratio 2. 

Figure 12. Optical arrangement by Fuller and Griffiths (25) for diffuse reflectance spectroscopy P, paraboloidal mirror E, ellipsoidal mirror S, sample D, detector...

In addition to translational Brownian motion, suspended molecules or particles undergo random rotational motion about their axes, so that, in the absence of aligning forces, they are in a state of random orientation. Rotary diffusion coefficients can be defined (ellipsoids of revolution have two such coefficients representing rotation about each principal axis) which depend on the size and shape of the molecules or particles in question28. 

After, the essential features of a mechanical model of adsorption and diffusion to characterize, e.g., the transport of a contaminant with rainwater through the soil will be outlined in particular, the model consists of a fluid carrier of an adsorbate, the adsorbate in the liquid state and an elastic skeleton with ellipsoidal microstructure it means that each pore has different microdeformation along principal axes, namely a pure strain, but rotates locally with the matrix of the material (see [5, 6]). 

As an example we consider the flow of a fluid/adsorbate mixture through the big pores of a skeleton, thought like an elastic solid with an ellipsoidal microstructure, and propose suitable constitutive equations to study the coupling of adsorption and diffusion under isothermal conditions in particular, we insert the concentration of adsorbate and its gradient in the usual variables, other than microstructural ones. Finally, the expression of the dissipation shows clearly its dependence on the adsorption and the diffusion, other than on the micro-structural interactions. The model was already applied by G. and Palumbo [7] to describe the transport of pollutants with rainwater in soil. 

These models consider the mechanisms of formation of oscillations a mechanism involving the phase transition of planes Pt(100) (hex) (lxl) and a mechanism with the formation of surface oxides Pd(l 10). The models demonstrate the oscillations of the rate of C02 formation and the concentrations of adsorbed reactants. These oscillations are accompanied by various wave processes on the lattice that models single crystalline surfaces. The effects of the size of the model lattice and the intensity of COads diffusion on the synchronization and the form of oscillations and surface waves are studied. It was shown that it is possible to obtain a wide spectrum of chemical waves (cellular and turbulent structures and spiral and ellipsoid waves) using the lattice models developed [283], Also, the influence of the internal parameters on the shapes of surface concentration waves obtained in simulations under the limited surface diffusion intensity conditions has been studied [284], The hysteresis in oscillatory behavior has been found under step-by-step variation of oxygen partial pressure. Two different oscillatory regimes could exist at one and the same parameters of the reaction. The parameters of oscillations (amplitude, period, and the... 

When the overall motion is not isotropic, the diagonal elements of the rotational diffusion tensor are no longer equivalent and rotation about the three principal axes of the diffusion tensor may be described by different diffusion coefficients or correlation times. For anisotropic motion, the correlation time in Eqs. 16 and 25 is an effective correlation time, r ff, containing contributions from the various modes of reorientation. Partitioning of the various components of rff can be achieved through appropriate dynamic models. The simplest case of anisotropic motion is that for a symmetric-top molecule. The r ff of a rigid ellipsoid is expressed in terms of two parameters, Dn and DL these two parameters respectively describe the rotational diffusion about the C3 symmetry axis (major axis) and the two perpendicular axes (minor axes), which are assumed to be equivalent25-44 (Fig. 4) ... 

Woessner47 has also treated the case of a methyl group attached to an axially symmetric ellipsoid, whereas Levy et al.66 derived equations for the methyl internal rotation superposed on a fully anisotropic motion. The effect of anisotropic reorientation can dramatically alter the relationship between rigidly held methine, methylene, and methyl C—H vectors. Deviation from the ratio T,(CH)/ T,(CH3) = 3 can be considerable, depending on the relative orientation of C—H vectors with respect to the principal diffusion axes. 

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