Phenomenological diffusion coefficients

The diffusion coefficients of the migrating particles discussed above (proportional to the mobihties of ionic or electronic defects, see Eq. (6.15)) are not directly accessible experimentally to measurement and are sometimes termed microscopic diffusion coefficients. It is a complication, in principle, that, on account of the requirement of electroneutrality, we cannot simply move the charge carrier, under consideration, through the solid. However, there are various possibilities for measuring phenomeno- 

Let us consider the three principal solutions (Pig. 6.17) to this electroneutrality dilemma . The three respective diffusion coefficients to be derived have to be carefully differentiated. 

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Section IA summarizes the molecular model of diffusion of Pace and Datyner (1 2) which proposes that the diffusion of gases in a polymeric matrix is determined by the cooperative main-chain motions of the polymer. In Section IB we report carbon-13 nmr relaxation measurement which show that the diffusion of gases in poly(vinyl chloride) (PVC) - tricresyl phosphate (TCP) systems is controlled by the cooperative motions of the polymer chains. The correlation of the phenomenological diffusion coefficients with the cooperative main-chain motions of the polymer provides an experimental verification for the molecular diffusion model. 

Section IIA summarizes the physical assumptions and the resulting mathematical descriptions of the "concentration-dependent (5) and "dual-mode" ( 13) sorption and transport models which describe the behavior of "non-ideal" penetrant-polymer systems, systems which exhibit nonlinear, pressure-dependent sorption and transport. In Section IIB we elucidate the mechanism of the "non-ideal" diffusion in glassy polymers by correlating the phenomenological diffusion coefficient of CO2 in PVC with the cooperative main-chain motions of the polymer in the presence of the penetrant. We report carbon-13 relaxation measurements which demonstrate that CO2 alters the cooperative main-chain motions of PVC. These changes correlate with changes in the diffusion coefficient of CO2 in the polymer, thus providing experimental evidence that the diffusion coefficient is concentration dependent. 

The experimentally determined leads to a phenomenological diffusion coefficient that is in the order of 10 cm s , assuming an interface thickness of 10 A. 

The expression for kdiff is essentially the same as that derived by Kakiuchi for an ion-transfer reaction given in Eq. (82). It thus immediately follows that the apparent good agreement between experimentally determined ion-transfer rate constants and Kakiuchi s model is also consistent with the Marcus model. Furthermore, the small phenomenological diffusion coefficient needed to fit the data to Eqs. (80 to 84) may simply be explained by a small... 

Based on Saxena s efforts (Saxena, 2001) and the adjustment on the timescales of phase separation to make numerical simulation data coincide with hght scattCT-ing results, the following phenomenological diffusion coefficients for the ternary PMAA/MAA/water (7 wt%/3 wt%/90 wt%) system at 80°C were obtained ... 

Frequency spectra in Fig. 1.4.39 can be seen to be qualitatively the same as those of qt3 and qt4, and almost the same as that of qt6 (Table 1.4.2). This is expected since values of Py are all the same. However, composition profiles in Fig. 1.4.40 are more qualitatively the same as those of qtl, qt2, qt5, and qt7 due to the combination of phenomenological diffusion coefficients obtained from experimental data. They indicate higher concentrations of MAA in the polymer-rich phase domains, which should result in more polymerization reaction and enhanced coarsening from those domains in the corresponding reactive system. 

Microbalances with alphanumerical display and electronic data recording systems allow one to observe the approach to equilibrium for gas adsorption processes in porous sorbent samples. Typical relaxation times can be one or several seconds, minutes, hours, and -sometimes - even days, cp. helium adsorption data Sect. 2 of Chap. 1. Hence gravimetric measurements do allow one to check whether an adsorption system actually has reached its equilibrium state, i. e. these measurements deliver in principle also information concerning the kinetics of the adsorption process, represented for example by (phenomenological) diffusion coefficients, cp. Sect. 2.3 and Sect. 4.4 and [3.27, 3.48). 

Useful forms of Tick s law in dilute solutions are shown in Table 2.1-2. Each equation closely parallels that suggested by Fick, that is, Eq. 2.1-1. Each involves the same phenomenological diffusion coefficient. Each will be combined with mass balances to analyze the problems central to the rest of this chapter. 

A parabolic rate law will also be obtained if part or even all, of the diffusion through the product layer is by grain boundary diffusion rather than diffusion through the volume of each grain. The volume diffusion coefficient is quite simply defined as the phenomenological coefficient in Fick s laws. The grain boundary diffusion must be described by a product, DbS, where S is the grain... 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

The phenomenological treatment allows us to identify the diffusion coefficients with the gradients in chemical potentials such that 3)... 

DSi are the principal or main diffusion coefficients and Dy the interaction or cross-diffusion coefficients. These diffusion coefficients can be expressed by phenomenological coefficients and chemical potentials as follows ... 

Chemical diffusion has been treated phenomenologically in this section. Later, we shall discuss how chemical diffusion coefficients are related to the atomic mobilities of crystal components. However, by introducing the crystal lattice, we already abandon the strict thermodynamic basis of a formal treatment. This can be seen as follows. In the interdiffusion zone of a binary (A, B) crystal having a single sublattice, chemical diffusion proceeds via vacancies, V. The local site conservation condition requires that /a+/b+7v = 0- From the definition of the fluxes in the lattice (L), we have... 

In order to investigate the relation between the phenomenological transport coefficients Ljj and Z) we formulate for the isotope (tracer) diffusion of A in A... 

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... 

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