Summary of Diffusivities

Four different types of diffusivities are summarized in Table 3.1. These include the self-diffusivity in a pure material, D the self-diffusivity of solute i in a binary system, Df, the intrinsic diffusivity of component i in a chemically inhomogeneous system, Dand the interdiffusivity, D, in a chemically inhomogeneous system. These diffusivities are applicable only in certain reference frames which are also listed in Table 3.1. In the remainder of this book, the type of diffusivity under discussion will be identified by these symbols when this information is relevant. When a diffusivity is identified in this manner, it may be assumed that the diffusion under consideration is being described in the proper corresponding frame. 

Di is the composition-dependent intrinsic diffusivity of component i in a chemically inhomogeneous system. In a binary system, it relates the flux of component i to its corresponding concentration gradient via Fick s law in a local C-frame (which is fixed with respect to the local bulk material of the diffusing system) and is moving with a velocity v with respect to the corresponding V-frame. The Di are related to D as indicated. 

D is the composition-dependent interdiffusivity in a chemically inhomogeneous system. In a binary system, it relates the flux of either component 1 or 2 to its corresponding concentration gradient via Fick s law in a V-frame. 

A gradient in electrostatic potential can produce a driving force for the mass diffusion of a species, as discussed in Section 2.2.2. Two examples of this are the potential-gradient-induced diffusional transport of charged ions in ionic conductors such as those used in solid-electrolyte batteries and the electron-current-induced diffusion of interstitial atoms in metals. 

---

Table 4.9 gives a summary of diffusion data for divalent cations in aluminous garnets, with the relative activation volume. Although estimation of activation volume is still largely uncertain, its evaluation is essential when dealing with the wide baric regimes encountered in petrologic studies. As shown in the third column of table 4.9, the presence of AF implies substantial modifications of Qj on the kbar scale of pressure. 

Tanner s reviews on the migration of Rn in the ground (Tanner, 1964a, 1978, 1980) contain a detailed description of the processes responsible for Rn emanation and migration and an excellent summary of diffusion studies and coefficients therefore this topic will not be discussed here in detail. The emanation processes of recoil and diffusion release Rn into pore spaces of minerals and rocks, permitting diffusion and mechanical transport to move Rn away from its source. Typical diffusion constants of Rn in various media are listed in Table 1 l-I. From the many diffusion studies carried out it is evident that diffusion alone cannot move Rn beyond about 8 m within the lifetime of a Rn atom. A simple approximation of the distance-time relationship is given by = pDt, where S = distance, D = diffusion coefficient and t = time. Thus, assuming a typical diffusion constant of 0.05 cm /sec., after six half-lives or 23 days, only 1.5% of a Rn... 

Table I. Summary of Diffusion Coefficient and Wavelength Results From Several Studies...

Table 4. Summary of diffusion data from select glass and...

Figure 3.14 Summary of diffusion parameters for NiO, after Atkinson etal The Ni data were obtained at unit activity of oxygen and the O data at an oxygen partial pressure of 0.2.

TABLE 1 A Summary of Diffusivity Values from Studies Involving Single-Phase Adsorbents with Significant Porosity ... 

Numerical values for solid diffusivities D,j in adsorbents are sparse and disperse. Moreover, they may be strongly dependent on the adsorbed phase concentration of solute. Hence, locally conducted experiments and interpretation must be used to a great extent. Summaries of available data for surface diffusivities in activated carbon and other adsorbent materials and for micropore diffusivities in zeolites are given in Ruthven, Yang, Suzuki, and Karger and Ruthven (gen. refs.). 

Many authors contributed to the field of diffusion and chemical reaction. Crank (1975) dealt with the mathematics of diffusion, as did Frank-Kamenetskii (1961), and Aris (1975). The book of Sherwood and Satterfield (1963) and later Satterfield (1970) discussed the theme in detail. Most of the published papers deal with a single reaction case, but this has limited practical significance. In the 1960s, when the subject was in vogue, hundreds of papers were presented on this subject. A fraction of the presented papers dealt with the selectivity problem as influenced by diffitsion. This field was reviewed by Carberry (1976). Mears (1971) developed criteria for important practical cases. Most books on reaction engineering give a good summary of the literature and the important aspects of the interaction of diffusion and reaction. 

Draxler, R. R., "A Summary of Recent Atmospheric Diffusion Experiments." National Oceanic and Atmospheric Administration Technical Memorandum ERL ARL-78. Silver Spring, MD, 1979. 

It is seen that if the diffusivity is to be correlated with the molecular weight, then a knowledge of the density of the solute is also necessary. The result of the correlation of the reciprocal of the diffusivity of the 69 different compounds to the product of the cube root of the molecular volume and the square root of the molecular weight is shown in Figure 1. A summary of the errors involved is shown in Figures 2 and 3... 

We will give a brief summary of the general principles of diffusion and preassociation in electrophilic substitutions before discussing the values in Table 3-1 (for more comprehensive reviews see North, 1964 and Ridd, 1978). A precursor A (HN02 in... 

Many CVD reactions are being investigated for the deposition of carbides and nitrides, particularly for titanium nitride for semiconductor applications, such as diffusion barrier. The following is a summary of the metallo-organic precursors and deposition condition presently used in development or production of these materials. 

Table 2 Fate and effects of metals in a stream receiving a point-source of metals (upper part of the table) or diffuse input via urban runoff (lower part of the table). Summary of the expected influence of four different hydrological situations base-flow in a rainy period a flood after a rainy period low-flow after a long period of low rainfall (water scarcity) and a flood produced after this drought. Metal concentration (M) metal retention efficiency (measured on the basis of the nutrient spiraling concept) exposure (dose and duration) bioaccumulation (in fluvial biofilms) and metal sensitivity (of biofihns)...

In summary, we found that the students received lower scores on items that were at different pressures than on items with the same amount of pressure. Also, we found that although the students learned the concept of diffusion in their seventh grade biology class, they did not generate the conception of diffusion in a submicro-scopic maimer. Instead, they tended to conceptualize the diffusion of the particles in a more intuitive way (the heavier object sinking to the bottom of the container) than in a scientific model that was designed to delineate the random nature of the particle motion. 

Summary of experimental data Film boiling correlations have been quite successfully developed with ordinary liquids. Since the thermal properties of metal vapors are not markedly different from those of ordinary liquids, it can be expected that the accepted correlations are applicable to liquid metals with a possible change of proportionality constants. In addition, film boiling data for liquid metals generally show considerably higher heat transfer coefficients than is predicted by the available theoretical correlations for hc. Radiant heat contribution obviously contributes to some of the difference (Fig. 2.40). There is a third mode of heat transfer that does not exist with ordinary liquids, namely, heat transport by the combined process of chemical dimerization and mass diffusion (Eq. 2-162). 

Table 1 Summary of Experimental Methods for Evaluating Diffusion Coefficients and Investigating Mass Transport Processes of Pharmaceutical Interest... 

The summary of Pe values for the steroids as a function of stirring rates is found in Table 11 and their correlations with log PC (n-octanol-water) in Figure 20. The transport kinetics of the relatively hydrophilic hydrocortisone and dexa-methasone are controlled by passive diffusion across the cell monolayer. On the other hand, the Pe values of testosterone and progesterone are highly dependent on stirring rate. The results for testosterone are used to obtain the relationships between the effective permeability coefficients of the ABL on the donor and receiver sides and the stirring rate, using the linear expression (see Eq. (69)]... 

This entire reaction is reversed when the blood reaches the lungs. Because carbon dioxide is eliminated by ventilation, the reaction is pulled to the left. Bicarbonate ions diffuse back into the red blood cells. The hemoglobin releases the hydrogen ions and is now available to load up with oxygen. The bicarbonate ions combine with the hydrogen ions to form carbonic acid, which then dissociates into carbon dioxide and water. The carbon dioxide diffuses down its concentration gradient from the blood into the alveoli and is exhaled. A summary of the three mechanisms by which carbon dioxide is transported in the blood is illustrated in Figure 17.8. 

Summary of Experimental Results for the Diffusion Coefficient of 218Po... 

In Eq. (17), Dip(Ci) is the concentration-dependent diffusion coefficient its form of concentration dependence is affected by the structural characteristics of the polymer carrier. A selective summary of the various forms of the diffusion coefficient is provided in Table I. 

In this paper, we have given a brief summary of our recent work on heat conduction in one dimensional systems. We have shown that strong chaos is sufficient but not strictly necessary for the validity of the Fourier heat law. Indeed linear mixing can be sufficient to induce a diffusive process which ensures normal heat conductivity. 

This chapter starts with a short introduction on the skin barrier s properties and the methods employed for analyzing experimental data. This is followed by an overview of several selected approaches to predict steady-state diffusion through the skin. Then a few approaches that approximate the structural complexity of the skin by predicting drug diffusion in biphasic or even multiphasic two-dimensional models will be presented. Finally, the chapter concludes with a short summary of the many variables possibly influencing drug permeation and penetration. 

Table 1 presents a summary of the solutions obtained in this section. Of primary interest at this point is comparison of the forms of the Lagrangian and Eulerian expressions, in particular the relationships between the eddy diffusivities and the plume dispersion variances. For the slender-plume cases, for example, the Lagrangian and Eulerian expressions are identical if... 

Summary of the n Values Found in the Diffusion-Controlled Nuclei Growth Model for Different Growth Geometries and Nucleation Rates. 

The plan of this chapter is the following. Section II gives a summary of the phenomenology of irreversible processes and set up the stage for the results of nonequilibrium statistical mechanics to follow. In Section III, it is explained that time asymmetry is compatible with microreversibility. In Section IV, the concept of Pollicott-Ruelle resonance is presented and shown to break the time-reversal symmetry in the statistical description of the time evolution of nonequilibrium relaxation toward the state of thermodynamic equilibrium. This concept is applied in Section V to the construction of the hydrodynamic modes of diffusion at the microscopic level of description in the phase space of Newton s equations. This framework allows us to derive ab initio entropy production as shown in Section VI. In Section VII, the concept of Pollicott-Ruelle resonance is also used to obtain the different transport coefficients, as well as the rates of various kinetic processes in the framework of the escape-rate theory. The time asymmetry in the dynamical randomness of nonequilibrium systems and the fluctuation theorem for the currents are presented in Section VIII. Conclusions and perspectives in biology are discussed in Section IX. 

---