Diffusion composition dependence

In general, tests have tended to concentrate attention on the ability of a flux model to interpolate through the intermediate pressure range between Knudsen diffusion control and bulk diffusion control. What is also important, but seldom known at present, is whether a model predicts a composition dependence consistent with experiment for the matrix elements in equation (10.2). In multicomponent mixtures an enormous amount of experimental work would be needed to investigate this thoroughly, but it should be possible to supplement a systematic investigation of a flux model applied to binary systems with some limited experiments on particular multicomponent mixtures, as in the work of Hesse and Koder, and Remick and Geankoplia. Interpretation of such tests would be simplest and most direct if they were to be carried out with only small differences in composition between the two sides of the porous medium. Diffusion would then occur in a system of essentially uniform composition, so that flux measurements would provide values for the matrix elements in (10.2) at well-defined compositions. 

The generalized Stefan-Maxwell equations using binary diffusion coefficients are not easily applicable to hquids since the coefficients are so dependent on conditions. That is, in hquids, each Dy can be strongly composition dependent in binary mixtures and, moreover, the binaiy is strongly affected in a multicomponent mixture. Thus, the convenience of writing multicomponent flux equations in terms of binary coefficients is lost. Conversely, they apply to gas mixtures because each is practically independent of composition by itself and in a multicomponent mixture (see Taylor and Krishna for details). 

The structure and composition of diffusion coatings depends of necessity on the metal or alloy from which the article is made. Thus, for example, it is not possible to speak of chromised coatings generally the material into which chromium is diffused must be specified. Some data on methods of application and properties of commercially chromised irons and steels are given in Table 12.4. 

Measurements were also made of the potential-composition behavior, as well as the chemical diffusion coefficient, and its composition dependence, in each of the intermediate phases in the Li-Sn system at 415 °C [39]. 

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

Whereas D is a physical property of the system and a function only of its composition, pressure and temperature, Ed, which is known as the eddy diffusivity, is dependent on the flow pattern and varies with position. The estimation of Ep presents some difficulty, and this problem is considered in Chapter 12. 

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... 

Other than in polymer matrix composites, the chemical reaction between elements of constituents takes place in different ways. Reaction occurs to form a new compound(s) at the interface region in MMCs, particularly those manufactured by a molten metal infiltration process. Reaction involves transfer of atoms from one or both of the constituents to the reaction site near the interface and these transfer processes are diffusion controlled. Depending on the composite constituents, the atoms of the fiber surface diffuse through the reaction site, (for example, in the boron fiber-titanium matrix system, this causes a significant volume contraction due to void formation in the center of the fiber or at the fiber-compound interface (Blackburn et al., 1966)), or the matrix atoms diffuse through the reaction product. Continued reaction to form a new compound at the interface region is generally harmful to the mechanical properties of composites. 

The rate of copolymerization, unlike the copolymer composition, depends on the initiation and termination steps as well as on the propagation steps. In the usual case both monomers combine efficiently with the initiator radicals and the initiation rate is independent of the feed composition. Two different models, based on whether termination is diffusion-controlled, have been used to derive expressions for the rate of copolymerization. The chemical-controlled termination model assumed that termination proceeds with chemical control, that is, termination is not diffusion-controlled [Walling, 1949]. But this model is of only historical interest since it is well established that termination in radical polymerization is generally diffusion-controlled [Atherton and North, 1962 Barb, 1953 Braun and Czerwinski, 1987 North, 1963 O Driscoll et al., 1967 Prochazka and Kratochvil, 1983] (Sec. 3-10b). 

Other general cases in binary systems are referred to as interdiffusion or binary diffusion. For example, Fe-Mg diffusion between two olivine crystals of different Xpo (mole fraction of forsterite Mg2Si04) is called Fe-Mg interdiffusion. Inter-diffusivity often varies across the profile because there are major concentration changes, and diffusivity usually depends on composition. 

In summary, the diffusion behavior of both H2O and CO2 demonstrates the importance of understanding the role of speciation in diffusion, and the very different consequences due to that role. Diffusion of a single-species component (such as Ar) usually does not depend on its own concentration (when the concentration is low), but depends on the melt composition. For a multispecies component, speciation affects the diffusion behavior. For H2O, speciation makes the diffusion behavior very complicated even at low H2O concentrations, total H2O diffusivity still depends on H2O content (because the species concentrations are not proportional), in addition to the dependence on melt composition. If species concentrations are proportional to each other and hence to the total concentration of the component, then the diffusivity is independent of the concentration of the component, as in the case of CO2 diffusion. Many multispecies components probably satisfy this condition that the concentrations of... 

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).

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. 

Various forms of diffusion coefficients are used to establish the proportionality between the gradients and the mass flux. Details on determination of the diffusion coefficients and thermal diffusion coefficients is found in Chapter 12. Here, however, it is appropriate to summarize a few salient aspects. In the case of ordinary diffusion (proportional to concentration gradients), the ordinary multicomponent diffusion coefficients Dkj must be determined from the binary diffusion coefficients T>,kj. The binary diffusion coefficients for each species pair, which may be determined from kinetic theory or by measurement, are essentially independent of the species composition field. Calculation of the ordinary multicomponent diffusion coefficients requires the computation of the inverse or a matrix that depends on the binary diffusion coefficients and the species mole fractions (Chapter 12). Thus, while the binary diffusion coefficients are independent of the species field, it is important to note that ordinary multicomponent diffusion coefficients depend on the concentration field. Computing a flow field therefore requires that the Dkj be evaluated locally and temporally as the solution evolves. 

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... 

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 adsorbed species, which are considered to be adatoms, can diffuse to favorable low-energy sites and react, or they can be emitted into the gas phase. At sufficiently low temperatures, adatoms may have insufficient energy to diffuse and react or to be emitted into the gas phase. These adatoms will be codeposited with the compound film as crystal defects or as a second solid phase. As a result of these competing processes in the surface reaction zone, the growth rate and film composition depend on the flux and energy of the incident species and on the substrate temperature. 

In Section I we introduce the gas-polymer-matrix model for gas sorption and transport in polymers (10, LI), which is based on the experimental evidence that even permanent gases interact with the polymeric chains, resulting in changes in the solubility and diffusion coefficients. Just as the dynamic properties of the matrix depend on gas-polymer-matrix composition, the matrix model predicts that the solubility and diffusion coefficients depend on gas concentration in the polymer. We present a mathematical description of the sorption and transport of gases in polymers (10, 11) that is based on the thermodynamic analysis of solubility (12), on the statistical mechanical model of diffusion (13), and on the theory of corresponding states (14). In Section II we use the matrix model to analyze the sorption, permeability and time-lag data for carbon dioxide in polycarbonate, and compare this analysis with the dual-mode model analysis (15). In Section III we comment on the physical implication of the gas-polymer-matrix model. 

The particle surface itself can also affect molecular velocities in two ways. First, molecular velocities of the rebounding molecules will depend on the type of rebound (whether specular or diffuse). Since the fraction of molecules rebounding either specularly or diffusively will depend on both particle and gas composition, these two factors should also be of importance in determining the thermal force. 

Even the binary system diffusivities in liquid mixtures are composition dependent. Therefore, in multicomponent liquid mixtures with n components, predictions of the diffusion coefficients relating flows to concentration gradients are empirical. The diffusion coefficient of dilute species i in a multicomponent liquid mixture, Dim, may be estimated by Perkins and Geankoplis equation... 

We may describe multicomponent diffusion by (1) the Maxwell-Stefan equation where flows and forces are mixed, (2) the Chapman-Cowling and Hirschfelder-Curtiss-Bird approaches where the diffusion of all the components are treated in a similar way, and (3) a reference to a particular component, for example, the solvent or mass average (baiycentric) definition. Frames of reference in multicomponent system must be clearly defined. Binary diffusion coefficients are often composition dependent in liquids, while they are assumed independent of composition for gases. 

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... 

Table 7.11 shows the thermal diffusion ratios obtained from Onsager s reciprocal rules for toluene (1), chlorobenzene (2), and bromobenzene (3) at 1 atm and 35°C. The heats of transport for the ternary mixtures are shown in Tables 7.12 and 7.13. For the ternary mixture of toluene (l)-chlorobenzene (2)-bromobenzene (3), the heats of transport are tabulated at 298 and 308 K. The temperature- and composition-dependent heats of transport values are fitted by the following equations by Platt et al. (1982) with a deviation below 5% ... 

In this equation ep is the porosity of the catalyst pellet and yp the tortuosity of the catalyst pores as discussed in Chapter 3 (the rest of the symbols are as defined before). From this formula it follows that the effective diffusion coefficient depends on both the gas composition and the pressure. Since we know the pressure as a function of the concentration, Equation 7.74 provides the effective diffusion coefficient as a function of the concentration. If we define... 

For this case Equations 7.132 and 7.133 cannot be used, because the effective diffusion coefficients depend on the gas composition and, because of that, so do the numbers etc. 

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. 

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