Phenomenological Treatment of Diffusion

Palladium is an interesting metal because it permits the rapid transport of hydrogen through its lattice stmcture via an atomic diffusion mechanism. Thus, a thin Pd membrane can be used as a selective filter for the separation or purification of hydrogen gas. This technology has potentially important implications for a number of industrial chemical conversion applications. 

Pick s first law of diffusion deals with the diffusional transport of matter under steady-state conditions. Steady state means that the concentration profile of the diffusing species does not vary as a function of time  

Pick s first law indicates that the flux of a diffusing species i is proportional to its concentration gradient. Pick s first law expresses the fundamental concept that matter 

Pick s first law (as well as Pick s second law) can be widely applied to model many diffusion processes in gases, liquids, or solids. The equations do not change between these phases— what changes is the diffusivity. Diffusivities tend to be high in gases, lower in liquids, and very low in solids, thereby capturing the differences in relative speed of diffiisional transport between these three phases of matter. In solids, diffusivities tend to be on the order of 10 -10 cm /s, which means that even diffusion over relatively small distances (e.g., micrometers) can take hours, days, or even longer. A detailed derivation of Pick s first law based on an atomistic picture of diffusion is provided in Section 4.5.3. 

A simple example showing how Pick s first law can be applied to model the steady-state diffusion of hydrogen in Pd is provided below. 

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Bitter [6] has provided an interesting new phenomenological treatment of diffusion, which can be especially useful in calculating mixture diffusivities in polymers. 

Figure 5-11 illustrates the results of an oxide interdiffusion experiment. Clearly, the transport coefficients are not single valued functions of composition. From the data, one concludes that for a given composition, the chemical diffusion coefficients depend both on time and location in the sample [G. Kutsche, H. Schmalzried (1990)]. Let us analyze this interdiffusion process in the ternary solid solution Co. O-Nq. O, which contains all the elements necessary for a phenomenological treatment of chemical transport in crystals. The large oxygen ions are almost immobile and so interdiffusion occurs only in the cation sublattice of the fee crystal. When we consider the following set ( ) of structure elements... 

It is intended to restrict the present discussion to the transport processes of diffusion and conduction and their interconnection. (The laws of hydrodynamic flow will not be described, mainly because they are not particular to the flow of electrolytes they are characteristic of the flow of all gases and liquids, i.e., of fluids.) The initial treatment of diffusion and conduction will be in phenomenological terms then the molecular events underlying these transport processes will be explored. 

How many ions travel a distance a , how many, x, etc. In other words, how are the ions spatially distributed after a time t, and how does the spatial distribution vary with time This spatial distribution of ions will be analyzed, but only after a phenomenological treatment of o steady-state diffusion is presented. 

This is the Einstein relation. It is probably the most important relation in the theory of the movements and drift of ions, atoms, molecules, and other submicroscopic particles. It has been daived ho e in an atomistic way. It will be recalled that in the phenomenological treatment of the diffusion coefficient (Section 4.2.3), it was shown that... 

In the phenomenological treatment of the directed drift that the field brings, we take the attitude that there is a stream of cations going toward the negative electrode and anions going toward the positive one. We now neglect the random diffusive movements they do not contribute to the vectorial flow that produces an electrical current. 

A more rigorous way to generalize Pick s law is to use phenomenological equations based on linear irreversible thermodynamics. In this treatment of an N-component system, the diffusive flux of component i is (De Groot and Mazur,... 

Approximate treatment of the many-particle effects in reversible bimolecular reactions has been undertaken in several papers (see for a review [78]) we would like also to note here pioneering studies of Ovchinnikov s group [79-82] and Kang and Redner s paper [83]. The former approach was discussed above in Section 2.1.2.3 where the kinetics of the approach to equilibrium for the simple reaction A B + B (dissociation and association of molecules A) was shown to approach the equilibrium as t 3/2. Note also that in the paper [84] a new elegant quantum-field formalism has been developed for the first time and applied to the diffusion-controlled reactions in the fluctuation regime its results agree completely with the phenomenological estimate (2.1.61). 

The treatment so far has been phenomenological and therefore the dependence of the diffusion coefficient on factors such as temperature and type of ion can be theoretically understood only by an atomistic analysis. The quantity D can be understood in a fundamental way only by probing into the ionic movements, the results of which show up in the macroscopic world as the phenomenon of diffusion. What are these ionic movements, and how do they produce diffusion The answering of these two questions will constitute the next topic. 

Some of the phenomenological coefficients relating forces and fluxes are already familiar from less general treatments of the subject. For example, the phenomenological coefficient relating a concentration gradient and a mass transfer flux is the diffusion coefficient. Other phenomenological coefficients are related to the ionic mobility, the coefficient of thermal conductivity, and the solvent viscosity. These are discussed in more detail later in this chapter. 

The two standard approaches in any treatment of kinetics [28] are to explain the system in terms of me thermodynamic driving forces (namely, VjJ.) or in terms of the fnndamental rate eqnations. The rate equations can be fnrther subdivided into an atomistic, or microscopic, approach that accounts for individual molecules as they go through the various processes (adsorption, desorption, diffusion, capture, and release) or a phenomenological, or macroscopic, explanation that looks for correlations and the so-called scaling laws over large distances (much larger than the lattice spacing). 

In the classical diffusion theory the adsorption term of bentonite is commonly treated by the distribution factor K. Instead of this macroscopic phenomenological treatment we propose a microscale HA procedure which exactly represents the edge adsorption characteristics at the edges of clay minerals. 

The unified treatment of gas phase and dense phase reactions is, in principle, possible in all cases in which the proper chemical reaction is the rate-determining process. In the situation in which the diffusion of reactants plays an essential role, a phenomenological description of the complicated phenomena in condensed media is certainly more appropriate from the practical point of view. In this respect, the recent development of the "theory of encounters" seems to be very promising /202/. 

In this section the transport of ions in an electrical field and their diffusion in a concentration or activity gradient will be treated. The expressions derived are valid for the fluxes of each type of ion or electron separately. From these expressions equations for an interconnected transport of different types of particles can be derived. In the following a phenomenological treatment and an outlook on the statistical treatment will be given. 

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