Fick diffusion

The Stem-Volmer equations discussed so far apply to solutions of the luminophore and the quencher, where both species are homogeneously distributed and Fick diffusion laws in a 3-D space apply. Nevertheless, this is a quite unusual situation in fluorescent dye-based chemical sensors where a number of factors provoke strong departure from the linearity given by equation 2. A detailed discussion of such situations is beyond the scope of this chapter however, the optosensor researcher must take into account the following effects (where applicable) ... 

The second approach employs a detailed reaction model as well as the diffusion of EG in solid PET [98, 121-123], Commonly, a Fick diffusion concept is used, equivalent to the description of diffusion in the melt-phase polycondensation. Constant diffusion coefficients lying in the order of Deg, pet (220 °C) = 2-4 x 10 10 m2/s are used, as well as temperature-dependent diffusion coefficients, with an activation energy for the diffusion of approximately 124kJ/mol. 

The crystal surface may be regarded as covered with a layer of saturated solution of a definite thickness through which the products have to diffuse. If the actual solvation of the solid proceeds rapidly in comparison to the process of diffusion the rate of solution wfil he essentially that of diffusion, and can accordingly be expressed by the Fick diffusion equation, the rate of solution per unit area of interface being given by... 

Fick (F2) proposed one of the more useful relationships for describing molecular material transport. The Fick diffusion coefficient is identified... 

In the case of transport in the gas phase it is often convenient to use the Maxwell diffusion coefficient (M4), which is related to the Fick diffusion coefficient in the following way (04) for a phase that may be treated as an ideal solution (L4) ... 

The conservation principle applied to unsteady, nonuniform, laminar, two-dimensional flow results in the following expression if the Fick diffusion coefficient is considered to be isotropic ... 

Schmidt numbers vary widely depending on the value of the Fick diffusion coefficient, which in the case of gases is strongly dependent on the concentration of the diffusing component. It should be noted that in a mixture the Schmidt number is not a property solely of either the component or... 

If the above treatment is formulated in terms of Fick diffusion coefficients Dls D2 53) (cf. the introductory section for the difference between a thermodynamic and a Fick diffusion coefficient), the expressions equivalent to Eqs. (13) and (14) are respectively... 

The conditions basic to the Fickian sorption were that (1) D is a function of ct only and (2) a constant surface concentration is maintained during sorption. So long as we wishes to retain the Fick diffusion equation as the basis of the discussion, any attempt for the theoretical interpretation of non-Fickian characteristics must abandon either or both of these conditions. In this section we give a brief account of a theory which involves an alternation of condition (1). It is due originally to Crank and Park (1951). 

The behavior of the Fick diffusion coefficient in nonideal systems may be complicated, while the Maxwell-Stefan diffusion coefficients behave quite well, and are always positive for binary systems. In nonideal binary systems, the Fick diffusivity varies with concentration. As seen in Figure 6.1, water-acetone and water-ethanol systems exhibit a minimum diffusivity at intermediate concentrations. Table 6.1 displays the dependency of binary diffusivity coefficients on concentration for selected alkenes in chloroform at 30°C and 1 atm. As the nonideality increases, mixture may split into two liquid phases at certain composition and temperature. 

Figure 6.1. Concentration dependence of the Fick diffusivity for the binary mixtures (a) water (A)—acetone (B) at 45°C, (b) water (A)—ethanol (B) at 40°C in terms of water mole fraction (Tyn and Calus, 1975).

The description of diffusion may be complex in mixtures with more than two components. Diffusion coefficients in multicomponent mixtures are usually unknown, although sufficient experimental and theoretical information on binary systems is available. The Maxwell-Stefan diffusivities can be estimated for dilute monatomic gases from D k Dkl when the Fick diffusivities are available. The Maxwell diflfusivity is independent of the concentration for ideal gases, and almost independent of the concentration for ideal liquid mixtures. The Maxwell-Stefan diffusivities can be calculated from... 

In the HSDM model, pore diffusion is neglected and it is assumed that surface diffusion is the dominant mechanism of intraparticle mass transfer. The imsteady-state diffusion process within the particle is taken into accoimt by a diffusion process model assuming a constant matrix of Fick diffusivity. The variation of q,- with the distance along the column and the time is governed by the diffusion equation [49-51]... 

Fick diffusion takes place when the pore diameter is much greater than the mean free path of the molecule. In this case, the collisions between the molecules dominate and the collision between the molecule and the walls of the pore is negligible in Figure 2.15a. The driving force of the Fick diffusion is the concentration gradient of the gaseous species and the diffusivity discussed above. 

Figure 2.15. Gas diffusion types within a pore (a) fick diffusion, (b) transition diffusion and (c) Knudsen diffusion...

The Knudsen diffusivity is proportional to the radius of the pore and the temperature but inversely proportional to the mole mass of the gas. This implies that small gaseous species exhibit high diffusivity. Unlike the Fick diffusivity, the Knudsen diffusivity is independent of pressure. 

According to Equation (5.7) the increase in processing temperature and preform thickness leads to larger values of Valid poor density uniformity. As stated previously, CVI processes preferably operate in a chemical-reaction-controlled regime where the ratio of k/D is small. For Fick diffusion, discussed in Section2.3.1, the diffusivity D is inversely proportional to the pressure and thus operates at lower pressures. Furthermore, coarser pore structures correspond to more uniform deposition. Figure 5.5 shows the microstructures of C/SiC composites prepared at different 9 numbers. 

A few typical values of the Fick diffusion coefficients are listed in Table 3.1. Although it may not be discerned from this small sample of values, the diffusion coefficient in an ideal gas mixture is independent of the mixture composition, inversely proportional to pressure, and varies with the absolute temperature to around the 1.5 power. More extensive listings are provided by Reid et al. (1987) and by Cussler (1984). The most comprehensive collection of 50... 

The matrix [D] of Fick diffusion coefficients is a square matrix of dimension n — 1 X n — 1... 

The Fick diffusion coefficients may be termed practical in the sense that the binary coefficient P and the corresponding multicomponent diffusion coefficients can be obtained from composition profiles measured in a diffusion apparatus. The measurement of binary and multicomponent diffusion coefficients, a subject with an extensive literature, is beyond the scope of this book. The interested reader is referred to Dunlop et al. (1972), Cussler (1976) and Tyrrell and Harris (1984) for descriptions of techniques and summaries of experimental results. Most experimental data are reported for [P ]. This matrix must be... 

TABLE 3.2 Fick Diffusion Coefficients in the System Acetone(l)-Benzene(2)-Methanol(3) at 25°C"... 

Example 3.2.1 Fick Diffusion Coefficients for the System Acetone-Benzene-Methanol... 

The Fick diffusion coefficients for the system acetone(l)-benzene(2)-methanol(3) in the volume average reference velocity frame are given in Table 3.2. Calculate the elements of [D] in the molar average reference velocity frame. 

It follows that the Fick diffusion coefficient must tend towards zero as the spinodal curve is approached. This has been experimentally confirmed for a few systems, the data of Haase and Siry (1968) for the systems water-triethylamine and n-hexane-nitrobenzene are shown in Figs. 3.2 and 3.3 (see, also, Claesson and Sundeldf, 1957 Myerson and Senol, 1984). Vitagliano et al. (1980) and Clark and Rowley (1986) determined spinodal compositions by extrapolating diffusivity data to zero. 

Figure 3.2. Fick diffusion coefficient D as a function of temperature for the system water-triethyl-amine. Measured data for Fick diffusivity D at constant composition = critical composition X2 = 0.0874. Critical temperature = 18.3°C. Data from Haase and Siry (1968).

We see that, for a binary system, the Fick diffusivity D and the Maxwell-Stefan diffusivity 67... 

Examples illustrating the use of Eq. 4.1.1 and 4.1.2 are given by Reid et al. (1977, 1987) and by Danner and Daubert (1983). The same authors describe methods for estimating Fick diffusion coefficients for gases at high pressure. 

Figure 4.1. a Concentration dependence of the Fick diffusivity D and the Maxwell-Stefan D for the system ethanol(l)-water(2). Data from Tyn and Calus (1975b). (6) Composition dependence of Fick D and Maxwell-Stefan D for the system acetone(l)-benzene(2). Data from Anderson et al. (1958) and Cullinan and Toor (1965). (c) Composition dependence of Fick D and Maxwell-Stefan D for diffusion in triethylamine(l)-water(2). Data from Dudley and Tyrell (1973). (J) Fick diffusion coefficient for the system methanol-n-hexane at 40°C measured by Clark and Rowley (1986). 

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