Self-diffusion profiles

Figure 9.10 (a) Normalised velocity profiles for different concentration solutions of polyfethylene oxide) in water obtained using dynamic NMR microscopy. The concentrations increase in equal steps from 0.5% (w/v) ( ) to 4.5% (w/v) ( ). (b) The polymer self-diffusion profile for the highest concentration solution in units of 10 m s" Note that this was obtained in a separate experiment so that the capillary wall does not fall at precisely the same pixel as in (a), (c) Water solvent velocity and (d) diffusion maps for the 4.5% (w/v) poly(ethylene oxide) solution. (From Y. Xia and P.T. Callaghan [18] and reproduced by permission of the American Chemical Society.)... 

Theoretical studies of diffusion aim to predict the distribution profile of an exposed substrate given the known process parameters of concentration, temperature, crystal orientation, dopant properties, etc. On an atomic level, diffusion of a dopant in a siUcon crystal is caused by the movement of the introduced element that is allowed by the available vacancies or defects in the crystal. Both host atoms and impurity atoms can enter vacancies. Movement of a host atom from one lattice site to a vacancy is called self-diffusion. The same movement by a dopant is called impurity diffusion. If an atom does not form a covalent bond with siUcon, the atom can occupy in interstitial site and then subsequently displace a lattice-site atom. This latter movement is beheved to be the dominant mechanism for diffusion of the common dopant atoms, P, B, As, and Sb (26). 

In this study the water temperature is changed to other values using the Pb(WW) and J(WW) parameters shown in Table 3.2. A profile of self-diffusion as a function of temperature can be derived from these results. 

Figure 5.3.5 displays dynamic NMR microscopy of xenon gas phase Poiseuille flow with an average velocity of 25 mm s-1 and self-diffusion coefficient of 4.5 mm2 s-1 at 130 kPa xenon gas pressure with numerical simulation (A) and experimental flow profiles (B-D) of xenon gas. 

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. 

Based on these observations, the diffusivity extracted from isotopic fraction profiles is usually regarded to be similar to intrinsic diffusivity or self-diffusivity even in the presence of major element concentration gradients. That is, the multicomponent effect does not affect the length of isotopic fraction profiles (but it affects the isotopic fractions and the interface position). On the other hand, the diffusion of a trace or minor element is dominated by multicomponent effect in the presence of major element concentration gradients. 

During self-diffusion in a pure material, whether a gas, liquid, or solid, the components diffuse in a chemically homogeneous medium. The diffusion can be measured using radioactive tracer isotopes or marker atoms that have chemistry identical to that of their stable isotope. The tracer concentration is measured and the tracer diffusivity (self-diffusivity) is inferred from the evolution of the concentration profile. 

Equation 5.18 offers a convenient technique for measuring self-diffusion coefficients. A thin layer of radioactive isotope deposited on the surface of a flat specimen serves as an instantaneous planar source. After the specimen is diffusion annealed, the isotope concentration profile is determined. With these data, Eq. 5.18 can be written... 

Powerful methods for the determination of diffusion coefficients relate to the use of tracers, typically radioactive isotopes. A diffusion profile and/or time dependence of the isotope concentration near a gas/solid, liq-uid/solid, or solid/solid interface, can be analyzed using an appropriate solution of - Fick s laws for given boundary conditions [i-iii]. These methods require, however, complex analytic equipment. Also, the calculation of self-diffusion coefficients from the tracer diffusion coefficients makes it necessary to postulate the so-called correlation factors, accounting for nonrandom migration of isotope particles. The correlation factors are known for a limited number of lattices, whilst their calculation requires exact knowledge on the microscopic diffusion mechanisms. 

Figure 5.3 depicts the Arrhenius plots of the apparent self-diffusion coefficient of the cation (Dcation) and anion (Oanion) for EMIBF4 and EMITFSI (Figure 5.3a) and for BPBF4 and BPTFSI (Figure 5.3b). The Arrhenius plots of the summation (Dcation + f anion) of the cationic and anionic diffusion coefficients are also shown in Figure 5.4. The fact that the temperature dependency of each set of the self-diffusion coefficients shows convex curved profiles implies that the ionic liquids of interest to us deviate from ideal Arrhenius behavior. Each result of the self-diffusion coefficient has therefore been fitted with VFT equation [6].

The temperature dependencies of the viscosity (Figure 5.6) and the summation of the self-diffusion coefficient (Dcation + Oanion) (Figure 5.4) interestingly show the contrasted profiles with the indication of inverse relationship between viscosity and self-diffiision coefficient. This can be explained in terms of the Stokes-Einstein equation, which correlates the self-diffusion coefficient (Dcation Danion) with viscosity (q) by the following relationship ... 

Further to this, another experiment was carried out (Run 2). In this experiment, the feed volume was changed to 3 L (from 6 F in Run 1), sweep volume to 1 L (from 1.5 L), and the membrane area to 370 cm (from 415 cm ). It was observed that the sweep side concentration profile for these self-diffusion coefficient values matched reasonably well with the experimental findings [51]. The results are shown in Figure 34.22c. The self-diffusion coefficient of H" " ions has been investigated earlier... 

We think first of two polymorphs with no chemical variability. If the materials showed no self-diffusion, the profile of radial compressive stress would have a step. But creeping materials do show self-diffusion (because if wafers can behave as in Figure 1.2b they can also behave as in Figure 1.2a). Hence, a real profile is a curve, with no step. The effect of the interface diminishes exponentially away from the interface, and the length scale over which it diminishes to 1/e of its maximum value is a multiple of the material s characteristic length Lq. 

Figure 1.3 Profiles of compressive stress in the neighborhood of an interface (a) in an imaginary non-diffusing material (b) in a material that show self-diffusion.

For plane strain, if the two polymorphs deformed without diffusion, would need to equal -I- tr x)/2 but if self-diffusion occurs, this stress condition no longer leads to plane strain. Where cylindrical constriction requires a simple exponential profile of compressive stress, plane-strain deformation at an interface requires a more complicated curve. 

Suppose the profile of is as shown in Figure 16.5 that is, suppose it continues to be of the same basic type as in Chapter 13. By itself, such a profile would drive self-diffusion of the ensemble (A, B)X across the interface without change of composition, just as in Chapter 13, with length scales B fixed by (2N K y. The material that travels is specifically wafers normal to z the interface moves with respect to remote points because of nonuniform strain rates e, as shown in Figure 13.2d. 

It has been emphasized, particularly in Chapter 11, that if a material can creep it can self-diffuse although for some purposes Figure 13.1b is an adequate approximation, there must be a scale on which profiles more like Figure 13.1c are seen. Thus we shall wish in due course to allow for self-diffusion and for the mechanical and chemical effects that are entailed. But even without self-diffusion, already a complication appears because of the fact that inclusion and host are likely to have different densities and thus different molar volumes. 

For analysis, let us assume that the concentration profile migrates inward as shown in Figure 16.1b with an error-function profile whose width increases in proportion to (time), and let us assume for a start that the stress state has no effect on the migration of potassium. Let us assume that the compressive stress at any point tends to rise because of this potassium concentration effect but tends to fall because of creep or relaxation of the glassy host. And let us assume that at small distances (less than 1 pm) from the surface, stress is relieved also by self-diffusion of the glass, so that right... 

If we now consider two velocity profiles, one at time /, and the other at time /2, with % > i> we might expect them to reduce to a single universal (self-similar) profile if we were to scale the distance from the wal I v with the length scale = (/ ) of the diffusion process, that is,... 

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