Diffuse planes

A driving force for diffusion includes any influence that increases the jump frequency. Examples of driving forces include chemical potential, thermal, and stress gradients. If a chemical potential gradient is present, the flux of species i at the diffusion plane is given by = —D (dc-Jdx), where D is the intrinsic diffusion coefficient of species i. Assuming Henry s law for the tracer element in binary alloys, Darken30 has shown that... 

We now consider a simple extension of the presentations in Secs. 8-10 and 8-11 to analyze a medium where reflection, transmission, and absorption modes are all important. As in Sec. 8-10, we shall analyze a system consisting of two parallel diffuse planes with a medium in between which may absorb, transmit, and reflect radiation. For generality we assume that the surface of the transmitting medium may have both a specular and a diffuse component of reflection. The system is shown in Fig. 8-58. 

Fig. 8-61 Radiation netwo- tor infinite parallel planes separated by a transmit ,ng specular-diffuse plane...

There are three possibilities corresponding to the dimension of the distribution. The first is a ID concentration distribution (d = 1), in which the diffusing species spreads evenly in the z directions from an initial line pulse at z = 0 on the xz plane. In this case, the variable r in (6-37) is the Cartesian variable z. The second case is a circularly symmetric distribution for c (d = 2), which evolves by diffusion on a plane from an initial compact planar pulse. In this case, r in (6 37) is the radial component of a polar (or cylindrical) coordinate system that lies in the diffusion plane. The third case is a spherically symmetric distribution corresponding to d = 3, which evolves at long times from a compact 3D pulse that diffuses outward into the frill 3D space. In this case, r is the radial variable of a spherical coordinate system. To obtain the long-time form of the distribution we must solve (6-37), but subject to the integral constraint that the total amount of the diffusing species is constant, independent of time ... 

The surface of a macromolecule forms a two-dimensional diffusion plane. As pointed out by Adam and Delbruck (1968), the probability of encounter between mobile ligand and a target on surface can be higher than encounter in three-dimensional space. 

Tr fast regime, the kinetics is influenced by both physical and chemical properties and the reaction is localized to a zone on either side of the diffusion plane. 

T,j 2> Tr instantaneous regime, the process is limited by the molecular-scale mixing and the reaction takes place in the diffusion plane. 

Let us first consider the set of equidistant diffuse streaks (c) perpendicular to the director [la,b, 30]. These streaks are very similar to those observed on the X-ray scattering patterns of the nematic phases of main-chain LCPs discussed in Sect. 3.2 but they must be interpreted differently because the SmA phase shows (quasi) long-range positional order. These diffuse streaks also correspond to the intersection with the Ewald sphere of a set of equidistant diffuse planes. But here this set represents the Fourier transform of uncorrelated rows of side-chains displaced along the director from their equilibrium position inside the layers (Fig. 13). 

Fig. 14. X-ray scattered intensity in a direction parallel to the director. Each peak represents a diffuse plane. Their intensities are modulated dashed line) hy a factor sin (27i s.u) related to the displacement vector u...

Figure 2.6 Schematic of diffusion planes in the electrolyte solution. (For color version of this figure, the reader is referred to the online version of this book.)...

Before we go further, it is necessary to introduce another important concept, the Fick s second law. This law deals with the change of reactant concentration with time during the diffusion. It can be deduced using Figure 2.6. At the diffusion Plane 1, the oxidant diffusion rate can be expressed as ... 

However, because there is only a linear periodicity (ID order), in the reciprocal space the maxima are sharp only along a line (Figure 1.4). In the two other directions, they are extended into diffuse planes. 

Pick s second law of diffusion is for a non-steady state or transient conditions in which dCj/t 0. Using Crank s model [23] for the rectangular element shown in Figure 4.1 yields the fundamental differential equations for the rate of concentration. Consider the central plane as the reference point in the rectangular volume element and assume that the diffusing plane at position 2 moves along the x-direction at a distance x-da from position 1 and x-Hdi to position 3. Thus, the rate of diffusion that enters the volume element at position 1 and leaves at position 3 is... 

Derive Pick s second law if the volume element in Pigure 4.2 has a unit cross-sectional area and the diffusing plane is located between x and a -I- dar, where J is the entering molar flux at x and Jx+[Pg.152]

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