Composition Dependence of Diffusion

Figure 3-37 Compositional dependence of diffusivities. (a) Fe-Mg interdiffusivity along the c-axis in olivine as a hinction of fayalite content at P — O.l MPa and log /b2 =-6.9 0.1. Diffusion data are extracted using Boltzmann analysis. Some of the nonsmoothness is likely due to uncertainty in extracting interdiffusivity using the Boltzmann method. Data are from Chakraborty (1997). (b) Ar and CO2 diffusivity in melt as a function of H2O content. Data are from Watson (1991b) and Behrens and Zhang (2001).

Koiwa, 1992 Bakker et al., 1992). Only recently a model, which is based on a combination of two mechanisms, has been proposed for describing the composition dependence of diffusion in B2 phases (Kao and Chang, 1993). It should be noted that the understanding of the basic diffusion processes for other intermetallics is still less than for NiAl and other B2 phases (Wever, 1992). Diffusion studies of multi-component systems are rare, and with respect to NiAl base alloys only data for the ternary phase (Ni,Fe)Al are available from a systematic study of the system Ni-Al-Fe (Cheng and Dayananda, 1979 Dayananda, 1992). Recently, the effect of Cr on diffusion in NiAl has been studied (Hopfe et al., 1993). 

Cullinan presented an extension of Cussler s cluster diffusion the-oiy. His method accurately accounts for composition and temperature dependence of diffusivity. It is novel in that it contains no adjustable constants, and it relates transport properties and solution thermodynamics. This equation has been tested for six very different mixtures by Rollins and Knaebel, and it was found to agree remarkably well with data for most conditions, considering the absence of adjustable parameters. In the dilute region (of either A or B), there are systematic errors probably caused by the breakdown of certain implicit assumptions (that nevertheless appear to be generally vahd at higher concentrations). 

Figure 10. Composition dependence of the chemical diffusion coefficient in the Li44Sn phase at ambient temperature [43).

Figure 1 is a schematic diagram illustrating a typical composition dependence of the mutual diffusion coefficient for a polymer-penetrant system [8], Here the penetrant is apparently a good solvent for the polymer since the entire composition range is realized. Note that four regions can be distinguished. In the... 

The dependence of diffusivity in silicate melts on composition is related to how melt structure (including degree of polymerization and ionic porosity) depends on composition. One the one hand, as Si02 concentration increases, the melt becomes more polymerized and the viscosity increases. Hence, diffusivity of most structural components, such as Si02 and AI2O3, decreases from basalt to rhyolite. On the other hand, as Si02 content increases, the ionic porosity increases. The increasing He diffusivity from basalt to rhyolite to silica, opposite to the viscosity... 

Figure 5-25 (a) Diffusion profile across a diffusion couple for a given cooling history. This profile is an error function even if temperature is variable as long as D is not composition dependent, (b) Diffusion profile across a miscibility gap for a given cooling history. Because the interface concentration changes with time, each half of the profile is not necessarily an error function. 

Fig. 34. Dependence of diffusion coefficient Dci)r (corrected for solvent viscosity) on solvent composition for PBLG in mixtures of DCA and EDC at 25° C (103). The dashed line...

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. 

The extension of ideal phase analysis of the Maxwell-Stefan equations to nonideal liquid mixtures requires the sufficiently accurate estimation of composition-dependent mutual diffusion coefficients and the matrix of thermodynamic factors. However, experimental data on mutual diffusion coefficients are rare, and prediction methods are satisfactory only for certain types of liquid mixtures. The thermodynamic factor may be calculated from activity coefficient models such as NRTL or UNIQUAC, which have adjustable parameters estimated from experimental phase equilibrium data. The group contribution method of UNIFAC may also be helpful, as it has a readily available parameter table consisting of mam7 species. If, however, reliable data are not available, then the averaged values of the generalized Maxwell-Stefan diffusion coefficients and the matrix of thermodynamic factors are calculated at some mean composition between x0i and xzi. Hence, the matrix of zero flux mass transfer coefficients [k ] is estimated by... 

Concentrated, Binary Mixtures of Nonelectrolytes Several correlations that predict the composition dependence of DAB are summarized in Table 5-15. Most are based on known values of DAB and Dba- In fact, a rule of thumb states that, for many binary systems, DAB and Dba bound the Dab vs. xa curve. Cullman s [8] equation predicts diffusivities even in lieu of values at infinite dilution, but requires accurate density, viscosity, and activity coefficient data. 

The spontaneous mixing of the two polymers will transpire at a rate which reflects the degree of miscibility of the system. As X approaches the critical value for phase separation, "thermodynamic slowing down" of the interdiffusion will occur [12]. The rate of increase of the scattering contrast reflects the proximity of the system to criticality, as well as the strong composition dependence of the glass transition temperature of the blend. Extraction of a value for either the self diffusion constants [13,14] or the interaction parameter is not feasible from the presently available data. 

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). 

A number of studies suggested that the composition dependence of the Fick diffusivity can be reasonably well represented by an equation with the form... 

It is further noted that the use of interfacial mass flux weighted transfer terms is generally not convenient treating multicomponent reactive systems, because the phase change processes are normally not modeled explicitly but deduced from the species composition dependent joint diffusive and convective interfacial transfer models. Moreover, the rigorous reaction kinetics and thermodynamic models of mixtures are always formulated on a molar basis. 

In practice, (f) can be calculated by inserting experimental copolymerization rates into Eq. (7.64). The values of (j> thus obtained are frequently greater than unity, and these deviations are ascribed to polar effects that favor cross-termination over homotermination. However, this is not always unambiguous, since the apparent cross-termination factor may vary with monomer feed composition in a given system [25,26]. It is clear also that termination reactions are at least partially diffusion controlled [27,28]. A dependence of segmental diffusivity on the structure of macroradicals is to be expected and dependence of diffusion controlled termination on copolymer composition seems reasonable. It is therefore plausible that the value of the overall termination rate constant ku in copolymerizations should be functions of fractions F and Fi) of the comonomers incorporated in the copolymer. An empirical expression for ku has thus been proposed [27] ... 

Foam fractionation Fractional extraction Fractionation, seeDistillation Free-volume theory of diffusion Freezing-point determination Fugacity of nitrogen standard state Fugacity coefficient composition dependence of acetic acid vapor... 

Composition Dependence of Dab- The diffusion coefficient of a binary solution is a function of composition, ff the true driving Force for ordinary diffusion is the isothermal, isoharic gradient of chemical potential, then the binary diffusivity can be written in the form... 

Vignes30 correlated the composition dependence of binary diffusion coefficients in terms of their infinite-dilution values and this thermodyxamic correction factor. 

Some representative curves from his paper are also shown in Fig. 2.3-4. The correlation appears to be quite successful unless the mixtures are stroagjy associating (e.g,. CH3C1-acetone). Lefflerand Cellinan31 have provided a derivation of Eq- (2.3-20) based on Absolute Rale Theoiy while Gainer12 has used Absolute Rate Theoiy to provide a different correlation for the composition dependence of binery diffusion coefficients. 

FIGURE 2.3-4 Composition dependence of binary diffusion coefficients in liquids, (in units of cm2/ ) (a) acetone (A)-CC 4, (b) methanol (A)-HjO (c) ethanol (A)-H20. (d) acetone (A)-CH3CI. Reprinted with permission from A. Vigtics. lnd, Eng. Chem. Fundam,, 5, 198 (1966). Copyright 1966 Amarican Chemical Secisty,... 

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