Diffusion, anisotropic

Extensive studies of anisotropic diffusion have also been reported [193,266,267]. 

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. 

In the most general case of a completely anisotropic diffusion tensor, six parameters have to be determined for the rotational diffusion tensor three principal values and three Euler angles. This determination requires an optimization search in a six-dimensional space, which could be a significantly more CPU-demanding procedure than that for an axially symmetric tensor. Possible efficient approaches to this problem suggested recently include a simulated annealing procedure [54] and a two-step procedure [55]. 

In the absence of accurate structural information, the analysis based on anisotropic diffusion as discussed above cannot be applied. The use of the isotropic overall model is still possible (see below) because it does not require any structural knowledge. However, the isotropic model has to be validated, i.e. the degree of the overall rotational anisotropy has to be determined prior to such an analysis. 

Despite the insights which the dynamics have provided into surface diffusion, additional studies will be required to hilly characterize surface dynamics. In particular, the role of surface reconstructions on diffusion has not bwn fully explored, and additional studies of the relationship between anisotropic diffusion and step stability are currently needed. 

The above-mentioned results can be easily extended to anisotropic diffusion characterized by a diffusion tensor D . In this case the hydrodynamic speed is given by... 

If the diffusion medium is isotropic in terms of diffusion, meaning that diffusion coefficient does not depend on direction in the medium, it is called diffusion in an isotropic medium. Otherwise, it is referred to as diffusion in an anisotropic medium. Isotropic diffusion medium includes gas, liquid (such as aqueous solution and silicate melts), glass, and crystalline phases with isometric symmetry (such as spinel and garnet). Anisotropic diffusion medium includes crystalline phases with lower than isometric symmetry. That is, most minerals are diffu-sionally anisotropic. An isotropic medium in terms of diffusion may not be an isotropic medium in terms of other properties. For example, cubic crystals are not isotropic in terms of elastic properties. The diffusion equations that have been presented so far (Equations 3-7 to 3-10) are all for isotropic diffusion medium. 

Example 4.1. Suppose olivine and garnet are in contact and olivine is on the left-hand side (x<0). Ignore the anisotropic diffusion effect in olivine. Suppose Fe-Mg interdiffusion between the two minerals may be treated as one dimensional. Assume olivine is a binary solid solution between fayalite and forsterite, and garnet is a binary solid solution between almandine and pyrope. Hence, Cpe + CMg= 1 for both phases, where C is mole fraction. Let initial Fe/(Fe- -Mg) = 0.12 in olivine and 0.2 in garnet. Let Xq = (Fe/Mg)gt/ (Fe/Mg)oi = 3, >Fe-Mg,oi = 10 ° mm+s, and Dpe-Mg,gt =... 

Reich M., Ewing R.C., Ehlers T.A., and Becker U. (2007) Low-temperature anisotropic diffusion of helium in zircon implication for zircon (U-Th)/He thermochronometry. Geochim. Cosmochim. Acta 71, 3119-3130. 

As just mentioned, there are a large number of unsolved problems in membrane biophysics, including the questions of local anisotropic diffusion, hysteresis, protein-lipid phase separations, the role of fluctuations in membrane fusion, and the mathematical problems of diffusion in two dimensions Stokes paradox). 

Expand Fick s law and Gauss theorem (Eq. 18-12) to three dimensions and derive Fick s second law for the general situation that the diffusivities Dx, Dy, and Dz are not equal (anisotropic diffusion) and vary in space. Show that the result can then be reduced to Eq. (1) of Box 18.3 provided that D is isotropic (Dx = Dy=Dz) and spatially constant. 

The vector notation is dropped when there is only one spatial dimension. Unless indicated otherwise, the diffusion is assumed to be isotropic in this chapter. Anisotropic diffusion is treated in Section 4.5. 

The anisotropic form of Fick s law would seem to complicate the diffusion equation greatly. However, in many cases, a simple method for treating anisotropic diffusion allows the diffusion equation to keep its simple form corresponding to isotropic diffusion. Because Dtj is symmetric, it is always possible to find a linear coordinate transformation that will make the Dij diagonal with real components (the eigenvalues of D). Let elements of such a transformed system be identified by a hat. Then... 

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... 

The dynamics of block copolymers melts are as intriguing as their thermodynamics leading to complex linear viscoelastic behaviour and anisotropic diffusion processes. The non-linear viscoelastic behaviour is even richer, and the study of the effect of external fields (shear, electric. ..) on the alignment and orientation of ordered structures in block copolymer melts is still in its infancy. Furthermore, these fields can influence the thermodynamics of block copolymer melts, as recent work has shown that phase transition lines shift depending on the applied shear. The theoretical understanding of dynamic processes in block copolymer melts is much less advanced than that for thermodynamics, and promises to be a particularly active area of research in the coming years. 

Fig. 2.50 An entangled diblock copolymer chain in a lamellar morphology with a sinusoidal composition profile (Lodge and Dalvi 1995). Entanglement constraints are indicated by x. Coupling between the thermodynamic forces produced by the composition gradients and replation dynamics leads to an anisotropic diffusion coefficient.

As indicated above, much interest exists in dynamic behavior of thin aligned layers of nematic liquid crystals. It is not surprising to find, therefore, that measurement of the anisotropy of transport properties has been the objective of many studies of thermotropic systems. The literature on anisotropic thermal conductivity in nematic liquid crystals has been reviewed recently by Rajan and Picot (12). Among the studies of anisotropic diffusion are those of Yun and Fredrickson (13), Bline... 

Monteil J and Beghdadi A 1999 A new interpretation and improvement of the nonlinear anisotropic diffusion for image enhancement. IEEE Transactions on Pattern Analysis and Machine Intelligence 21(9), 940-946. 

If there is a systematic (i.e., highly ordered) tissue substructure such as in white matter, diffusion is usually more restricted in one than in another direction, i.e., the molecular mobility of water is not the same in all directions. In white matter, diffusion is less restricted parallel to than perpendicular to fiber tracts. If diffusion is different along various directions, then it is termed anisotropic diffusion. In stroke imaging the avoidance of the confounding effects of anisotropy is a common goal. However,... 

BFig. 7.4. Trace ADC maps. Averaging of ADC maps obtained with diffusion gradients applied in the three orthogonal directions in space (x,y,z) yields trace ADC maps with relative suppression of anisotropic diffusion. (Courtesy of Prof. M. E. Moseley, Stanford University)... 

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