Diffusion equation anisotropic

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. 

Self-diffusion and tracer diffusion are described by Equation 3-10 in one dimension, and Equation 3-8 in three dimensions. For interdiffusion, because D may vary along a diffusion profile, the applicable diffusion equation is Equation 3-9 in one dimension, or Equation 3-7 in three dimensions. The descriptions of multispecies diffusion, multicomponent diffusion, and diffusion in anisotropic systems are briefly outlined below and are discussed in more detail later. 

In the general case of three-dimensional multicomponent diffusion in an anisotropic medium (such as Ca-Fe-Mg diffusion in pyroxene), the mathematical description of diffusion is really complicated it requires a diffusion matrix in which every element is a second-rank tensor, and every element in the tensor may depend on composition. Such a diffusion equation has not been solved. Because rigorous and complete treatment of diffusion is often too complicated, and because instrumental analytical errors are often too large to distinguish exact solutions from approximate solutions, one would get nowhere by considering all these real complexities. Hence, simplification based on the question at hand is necessary to make the treatment of diffusion manageable and useful. 

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.. 

This equation is the same as the diffusion equation in isotropic media with D = 1. Hence, theoretically, solutions in isotropic media can be applied to diffusion problems in anisotropic media after these transformations. However, it must be realized that the transformed coordinates may correspond to strange and unin-... 

For three-dimensional diffusion in an anisotropic medium, theoretically it is possible to transform the diffusion equation to a form similar to that in an isotropic system. However, in practice, the transformed equation is rarely used, and diffusion is often simplified to be along the fastest diffusion direction. 

To treat this more quantitatively, it is necessary to know the behavior of Fe-Mg interdiffusion in olivine. Olivine crystals are usually equidimensional. The diffusion is anisotropic with Dc -- 4Da -- 5Db (Buening and Buseck, 1973), where the subscripts refer to crystallographic directions. Although the difference in diffusivity is not very large, for simplicity, diffusion in olivine is often treated to occur only along the c-axis. The Fe-Mg interdiffusivity in olivine along the c-axis at 1253-1573 K may be expressed as (Equation 3-147)... 

Recently, Steiger and Keizer [259b] have discussed the theory of reactions between anisotropically reactive species in considerable detail. They illustrated their analysis by using a diffusion equation approach to solve for the rate coefficient for reaction between species which displayed dipolar reactivity. The rate coefficient was reduced by approximately 15% from the Smoluchowski value [eqn. (19)]. Berdnikov and Doktorov [259c] have also analysed the rate of reactions between a spherical reactant having a reactive site, which is a spherical shell of semi-angle 60, and a spherical symmetric reactant. Again, these reactants were not allowed to rotate. Approximate analytic expressions were obtained for the rate coefficient, which was a factor feit less than the Collins and Kimball expression, where... 

Equation 4.2 can take various forms, depending upon the behavior of D. The simplest case is when D is constant. However, as discussed below, D may be a function of concentration, particularly in highly concentrated solutions where the interactions between solute atoms are significant. Also, D may be a function of time for example, when the temperature of the diffusing body changes with time. D may also depend upon the direction of the diffusion in anisotropic materials. 

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... 

Macroscopic diffusion model is based on underlying microscopic dynamics and should reflect the microscopic properties of the diffusion process. A single diffusion equation with a constant diffusion coefficient may not represent inhomogeneous and anisotropic diffusion in macro and micro scales. The diffusion equation from the continuity equation yields... 

The pairwise Brownian dynamics method has several advantages over numerical methods based on Smoluchowski s [9] approach (e.g., finite element method), and we discuss these here. The primary advantage of the method is the ease of mathematical formulation even for cases involving complex reaction site geometries, hydrodynamic interactions, charge effects, anisotropic diffusion and flow fields. Furthermore the method obviates the need to solve complex diffusion equations to obtain the concentration field from which the rate constant is calculated in the Smoluchowski method. In contrast, the rate constant is obtained directly in the pairwise Brownian dynamics method. The effective rate constants for different reaction conditions may be obtained from a single simulation this is not possible using the finite element method. 

We discuss the diffusion process in three dimensions in the context of an anisotropic convection-diffusion equation for the density of particles. Our goal is to obtain the probabilistic solution to the initial value problem... 

The anisotropic convection-diffusion equation (3.311) can be written in the form of a backward equation, dpjdt = Lp, if... 

It is again supposed that translational diffusion motions of molecules can be described by a diffusion equation. The theory of spin relaxation by translational diffusion can, in principle, be formulated [7.16]. The review by Kruger [7.57] provides an exhaustive description and interpretation of the behavior of mass diffusion in different thermotropic mesophases. The mass diffusion is anisotropic in mesophases and, in general, will be given by a second-rank tensor D, the symmetry of which is related to the symmetry of the mesophase under consideration. For a uniaxial system with the z axis along the director, the translational diffusion tensor is... 

The solutions to the anisotropic diffusion equation can be written as a series expansion, each term of which can be associated with a particular relaxation time. For a harmonic perturbation of the rotational distribution function, as occurs in a dielectric relaxation experiment with an ac electric field, it was found that a single relaxation time was sufficient to describe the relaxation of p, and this could be expressed in terms of the relaxation time Xq) for in the absence of a nematic potential by ... 

The full solution of the rotational diffusion equation including a general single particle potential of D , symmetry has been investigated [36], and it is found that the dipole correlation function, which can be related to the permittivity as a function of frequency, is a sum over many exponential terms each characterised by a different relaxation time. Extending Eq. (20) for an anisotropic fluid gives ... 

In fact, one is usually dealing with real systems, which are not purely one (or two- or three-dimensional) but present some anisotropy of their transport properties, so that three diffusion coefficients should be defined along the three axes. The diffusion equation, Eq. (9), has to be solved in the general case of anisotropic diffusion. 

This orientational degeneracy is of course raised in anisotropic systems such as in the case of the patterns formed during catalysis at single crystal surfaces [25] or in metals under irradiation [16-18]. The diffusion coefficients of some species are then highest along some crystallographic directions. In a 2D uniaxial medium the reaction-diffusion equations are now ... 

If one allows for anisotropic frictional forces by retaining the friction tensor fin equation (51), and allowing for anisotropic Brownian motion by allowing the Maxwellian velocity distribution to be skewed (so that = — (kT/ F)[(5/5ry) f F]), then the diffusion equation and stress tensor expressions become... 

The Curtiss-Bird theory for concentrated systems is structured quite differently, taking as the starting point the general phase-space formalism as shown in Figure 8. The polymer molecule is modeled throughout as a Kramers chain, and the restricted motion of the molecule is described by means of an anisotropic Stokes law expression for the links and an anisotropic Brownian motion. Despite the great differences between the Doi-Edwards and Curtiss-Bird theories, some of the key results are very similar. For example, both theories lead to a diffusion equation for a segment of the polymer chain as follows ... 

The difference in phase retardation did not exceed the experimental error of 0.001 rad. A monotonic increase of the photoinduced phase retardation 5 from zero level at r = 0 (isotropic state) up to the saturation value So (anisotropic state) was observed. The experimental values are in good agreement with the results of numerical calculations, using diffusion equation (2.1), shown in Figure 2.8 by solid lines. The saturation photoinduced phase retardation level was found to be proportional to the power of UV illumination W, which is in agreement with the results of numerical calculations (see Figure 2.9). 

---