Binary diffusion processes

Fig. 1. Generation of a gap for the penetrant with subsequent collapse of the volume that previously housed the penetrant, emphasizing the mutual nature of the binary diffusion process.

If a liquid system containing at least two components is not in thermodynamic equilibrium due to concentration inhomogenities, transport of matter occurs. This process is called mutual diffusion. Other synonyms are chemical diffusion, interdiffusion, transport diffusion, and, in the case of systems with two components, binary diffusion. 

The variation of efficiencies is due to interaction phenomena caused by the simultaneous diffusional transport of several components. From a fundamental point of view one should therefore take these interaction phenomena explicitly into account in the description of the elementary processes (i.e. mass and heat transfer with chemical reaction). In literature this approach has been used within the non-equilibrium stage model (Sivasubramanian and Boston, 1990). Sawistowski (1983) and Sawistowski and Pilavakis (1979) have developed a model describing reactive distillation in a packed column. Their model incorporates a simple representation of the prevailing mass and heat transfer processes supplemented with a rate equation for chemical reaction, allowing chemical enhancement of mass transfer. They assumed elementary reaction kinetics, equal binary diffusion coefficients and equal molar latent heat of evaporation for each component. 

First a derivative is given of the equations of change for a pure fluid. Then the equations of change for a multicomponent fluid mixture are given (without proof), and a discussion is given of the range of applicability of these equations. Next the basic equations for a multicomponent mixture are specialized for binary mixtures, which are then discussed in considerably more detail. Finally diffusion processes in multicomponent systems, turbulent systems, multiphase systems, and systems with convection are discussed briefly. 

After this formal discussion of chemical diffusion, let us now turn to some more practical aspects. In order to compare the formal theory with experiment, we have to carefully define the reference frame for the diffusion process, which is not trivial in the case of binary or multicomponent diffusion. To become acquainted with the philosophy of this problem, we deal briefly with defining a suitable reference frame in a binary system. Since only one (independent) transport coefficient is needed to describe chemical diffusion in a binary system, then according to Eqn. (4.57) we have in a one-dimensional system... 

First we treat diffusion processes within the homogeneous phase. The presence of a temperature gradient in binary fluid mixtures and polymer blends requires an extension of Fick s diffusion laws, since the mass is not only driven by a concentration but also by a temperature gradient [76] ... 

In the great majority of cases, a line of the markers located in the zinc phase displaces a few micrometres aside from a line located in the other phases, indicative of the crack formation at the interface with zinc. To understand the further course of the reaction-diffusion process after the rupture of any reaction couple, it is necessary first to analyse the growth kinetics of the same compound layer in different reaction couples of a multiphase binary system. This will be done in the next chapter. 

The consideration of the reaction-diffusion process in binary heterogeneous systems, carried out in this book, is actually based upon the two simple and obvious assumptions ... 

It appears relevant to note that many workers tend to overestimate the significance of thermodynamic predictions concerning the direction of the reaction-diffusion process. In fact, however, those only bear a likelihood character. Even if the free energy of formation of one compound from its constituents is -200 kJ mol-1, while that of the other is -20 kJ mol1, this does not necessarily mean, as often (tacitly or directly) assumed, that the former will occur first and the more so that its growth rate must be ten times greater than that of the latter. As exemplified with the growth rate of a compound layer in various diffusion couples of the same multiphase binary system, the opposite may well take place. 

In this case, it is well known that the process occurs in steady state. To understand this process, one must consider it as a special case of binary diffusion, where the diffusivity of the Pd atoms is zero. Consequently, the frame of reference is the fixed coordinates of the solid Pd thin film. The interdiffusion or chemical diffusion coefficient is the diffusivity of the mobile species [20], that is, hydrogen. Then, the hydrogen flux in the Pd thin film is given by... 

The process is also known as chemical diffusion, interdiffusion, transport diffusion, and in the case of systems with two components, binary diffusion. 

The product in equation (11) may be related to the binary diffusion coefficients by considering the limiting case of a constant-pressure process in a two-component system with no body forces, for which equation (11) reduces to... 

Finally, to facilitate the mathematical developments, it is convenient to assume for the gas phase that the second-order Soret and Dufour diffusion processes are unimportant the heat capacities at constant pressure Cp, and the thermal conductivity coeflBcients fc, are constants the binary diffusion coeflBcients D are equal for all pairs of species and the thermal and mass diflFusion rates are equal such that the Lewis number is unity. 

For binary systems we must in addition take account of the condition of stability with respect to diffusion processes (15.105). It is in fact this condition which here determines the stability of the system. In 8 we shall show why it is that the condition of mechanical stability plays no part in determining the equilibrium in a binary system. 

In this chapter we discuss a novel fast diffusion process that was experimentally discovered by Yasuda, Mori, and co-workers (YM) [7] in nano-sized binary metal clusters, because it is supposed to be a typical manifestation of an anomalous diffusion process peculiar to microclusters. The authors have reported some numerical results of MD simulation on rapid alloying (RA) [8]. Special attention is paid to the diffusion of atoms in cluster [9]. The aim of the present work is to realize RA by elucidating what kind of diffusive motion is relevant to RA. 

Rapid alloying (RA) is a fast diffusion process that was experimentally discovered by Yasuda, Mori, and co-workers (YM) in binary microclusters. By using an evaporator, they deposited individual solute atoms (Cu) on the surface of host nano-sized clusters that are supported by amorphous carbon him at room temperature. YM observed the alloying behavior with a transmission electron microscope in in situ condition as schematically described in Fig. 1. In Ref. 7 it is demonstrated that Au clusters promptly changed into highly concentrated, homogeneously mixed (Au-Cu) alloy clusters. RA is similarly observed in various nano-sized binary clusters, such as (Au-Ni), (In-Sb), (Au-Zn), and (Au-Al) [7]. They examined the presence and absence of RA for clusters of different sizes. Consequently, YM summarize the unusual features of RA as follows ... 

We compare this diffusion process with the same rapid model protein diffusing on a binary PIP2-depleted membrane containing only monovalent (PS) lipids (Figure 5a). Clearly, acidic (PS) lipids segregate around the macroion to a much lesser extent compared to the ternary mixture, resulting in low energetic barriers to adsorbate motion. Hence, the diffusion of the macroion here is less restricted compared to that seen for the ternary mixture. 

For process engineering calculations it is almost inevitable that experimental values of D or f), even if available in the literature, will not cover the entire range of temperature, pressure, and concentration that is of interest in any particular application. It is, therefore, important that we be able to predict these coefficients from fundamental physical and chemical data, such as molecular weights, critical properties, and so on. Estimation of gaseous diffusion coefficients at low pressures is the subject of Section 4.1.1, the correlation and prediction of binary diffusion coefficients in liquid mixtures is covered in Sections 4.1.3-4.1.5. We do not intend to provide a comprehensive review of prediction methods since such are available elsewhere (Reid et al., 1987 Ertl et al., 1974 Danner and Daubert, 1983) rather, it is our purpose to present a selection of methods that may be useful in engineering calculations. 

In order to analyze multicomponent diffusion processes we must be able to solve the continuity equations (Eq. 1.3.9) together with constitutive equations for the diffusion process and the appropriate boundary conditions. A great many problems involving diffusion in binary mixtures have been solved. These solutions may be found in standard textbooks, as well as in specialized books, such as those by Crank (1975) and Carslaw and Jaeger (1959). 

The solution of multicomponent diffusion problems is a little more complicated than the solution of binary diffusion problems because the differential equations governing the process are coupled. In the early 1960s a versatile and powerful method of solving multicomponent diffusion problems was developed independently by Toor (1964a) and by Stewart and Prober (1964). The method they proposed is described and illustrated in this chapter. 

Some assumptions regarding the constancy of certain parameters are usually in order to facilitate the solution of the diffusion equations. For the binary diffusion problems discussed in Chapters 5 (as well as later in Chapters 8-10), we assume the binary Fick diffusion coefficient can be taken to be a constant. In the applications of the linearized theory presented in the same chapters, we assume the matrix of multicomponent Fick diffusion coefficients to be constant. If, on the other hand, we use Eq. 6.2.1 to model the diffusion process then we must usually assume constancy of the effective diffusion coefficient if... 

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