Transport mutual diffusion

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. 

From the applications point of view, mutual diffusion is far more important than self-diffusion, because the transport of matter plays a major role in many physical and chemical processes, such as crystallization, distillation or extraction. Knowledge of mutual diffusion coefficients is hence valuable for modeling and scaling-up of these processes. 

Various diffusion coefficients have appeared in the polymer literature. The diffusion coefficient D that appears in Eq. (3) is termed the mutual diffusion coefficient in the mixture. By its very nature, it is a measure of the ability of the system to dissipate a concentration gradient rather than a measure of the intrinsic mobility of the diffusing molecules. In fact, it has been demonstrated that there is a bulk flow of the more slowly diffusing component during the diffusion process [4], The mutual diffusion coefficient thus includes the effect of this bulk flow. An intrinsic diffusion coefficient, Df, also has been defined in terms of the rate of transport across a section where no bulk flow occurs. It can be shown that these quantities are related to the mutual diffusion coefficient by... 

For practical purposes, the mutual diffusion coefficient is the quantity commonly reported to characterize diffusional transport in pharmaceutical systems. It is thus the purpose of investigators to determine this quantity experimentally. To this end, both sorption and permeation methods are commonly used. 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

We will now describe the application of the two principal methods for considering mass transport, namely mass-transfer models and diffusion models, to PET polycondensation. Mass-transfer models group the mass-transfer resistances into one mass-transfer coefficient ktj, with a linear concentration term being added to the material balance of the volatile species. Diffusion models employ Fick s concept for molecular diffusion, i.e. J = — D,v ()c,/rdx, with J being the molar flux and D, j being the mutual diffusion coefficient. In this case, the second derivative of the concentration to x, DiFETd2Ci/dx2, is added to the material balance of the volatile species. 

There are a number of quantitative features of Eq. (14) which are important in relation to rapid diffusional transport in binary systems. The mutual diffusion coefficient is primarily dependent on four parameters, namely the frictional coefficient 21 the virial coefficients, molecular weight of component 2 and its concentration. Therefore, for polymers for which water is a good solvent (strongly positive values of the virial coefficients), the magnitude of (D22)v and its concentration dependence will be a compromise between the increasing magnitude of with concentration and the increasing value of the virial expansion with concentration. 

In this experiment the mutual diffusion coefficients for the Ar—CO2 and He—CO2 systems are to be measured using a modified Loschmidt apparatus. These transport coefficients are then compared with theoretical values calculated with hard-sphere collision diameters. 

Whereas mutual diffusion characterizes a system with a single diffusion coefficient, self-diffusion gives different diffusion coefficients for all the particles in the system. Self-diffusion thereby provides a more detailed description of the single chemical species. This is the molecular point of view [7], which makes the selfdiffusion more significant than that of the mutual diffusion. In contrast, in practice, mutual diffusion, which involves the transport of matter in many physical and chemical processes, is far more important than self-diffusion. Moreover mutual diffusion is cooperative by nature, and its theoretical description is complicated by nonequilibrium statistical mechanics. Not surprisingly, the theoretical basis of mutual diffusion is more complex than that of self-diffusion [8]. In addition, by definition, the measurements of mutual diffusion require mixtures of liquids, while self-diffusion measurements are determinable in pure liquids. 

Once an appropriate frame of reference is chosen, a two components (A, B) system may be described in terms of the mutual diffusion coefficient (diffusivity of A in B and vice versa). Unfortunately, however, unless A and B molecules are identical in mass and size, mobility of A molecules is different with respect to that of B molecules. Accordingly, the hydrostatic pressure generated by this fact will be compensated by a bulk flow (convective contribution to species transport) of A and B together, i.e., of the whole solution. Consequently, the mutual diffusion coefficient is the combined result of the bulk flow and the molecules random motion. For this reason, an intrinsic diffusion coefficient (Da and Db), accounting only for molecules random motion has been defined. Finally, by using radioactively labeled molecules it is possible to observe the rate of diffusion of one component (let s say A) in a two component system, of uniform chemical composition, comprised of labeled and not labeled A molecules. In this manner, the self-diffusion coefficient (Da) can be defined [54]. Interestingly, it can be demonstrated that both Da and Da are concentration dependent. Indeed, the force/acting on A molecule at point X is [1]... 

D Me-S surface alloy and/or 3D Me-S bulk alloy formation and dissolution (eq. (3.83)) is considered as either a heterogeneous chemical reaction (site exchange) or a mass transport process (solid state mutual diffusion of Me and S). In site exchange models, the usual rate equations for the kinetics of heterogeneous reactions of first order (with respect to the species Me in Meads and Me t-S>>) are applied. In solid state diffusion models, Pick s second law and defined boundary conditions must be solved using Laplace transformation. 

Equation (5.15) concerns self-diffusion, and mass transport involves mutual diffusion if the solute diffuses in one direction, then the solvent does so in the opposite one. A kind of average diffusion coefficient must be taken, and only for low solute concentrations is it about equal to the self-diffusion coefficient of the solute, taking the viscosity of the solvent. This is primarily because solute concentration will generally affect the viscosity of the solution in most cases it is higher than that of the solvent. 

Mutual diffusion may go along with a change in volume, since many solute-solvent mixtures have a different volume (mostly a smaller volume) than the sum of that of both components. This implies that the frame of reference moves for instance, the original interface between two layers of different concentration moves, and this means that some transport by flow occurs also. 

While "kinetics" means only time-dependent, the terminology "adsorption dynamics" includes the coupling of transport by diffusion and hydrodynamic fields. It comprises surface concentration changes, movement in the adsorption layer and correlation between the distribution of surface concentration and velocities along the surface. The adjacent liquid bulk is involved in the diffusion and hydrodynamic flows which exhibit mutual interrelation. The term "dynamic adsorption layer" refers only to the non-equilibrium state of the adsorption layer. 

In real situations, where the concentration of solute is finite, the mutual diffusion coefficient is often the relevant measure of transport rate. Mutual diffusion coefficients provide a quantitative measure of the rate of molecular diffusion when gradients are present i.e., when solute and solvent molecules are both diffusing in an attempt to eliminate differences in chemical potential. The mutual diffusion coefficient is defined by Fick s law (recall Equation 3-19) ... 

The diffusivity (D) defined in this way is not necessarily independent of concentration. It should be noted that for diffusion in a binary fluid phase the flux (/) is defined relative to the plane of no net volumetric flow and the coefficient D is called the mutual diffusivity. The same expression can be used to characterize migration within a porous (or microporous) sohd, but in that case the flux is defined relative to the fixed frame of reference provided by the pore walls. The diffusivity is then more correctly termed the transport diffusivity. Note that the existence of a gradient of concentration (or chemical potential) is implicit in this definition. 

In the case of systems containing ionic liquids, components and chemical species have to be differentiated. The system methanol/[BMIM][Pp6], for example, consists of two components (methanol and [BMIMJfPPe]) but three chemical species (methanol, [BMIMJ+ and [PPe] ), assuming that [BMIM][Pp6] is completely dissociated. If [BMIM][PFis] is not completely dissociated, one has a fourth species, the undissociated [BMIM][PFis]. From this it follows that the diffusive transport can be described with three and four flux equations, respectively. But the fluxes of [BMIMJ+ and [PFg]" are not independent because of electroneutrality in each volume of the system. Furthermore, the flux of [BMIMJIPFe] is not independent of the flux of the ions because of the dissociation equilibrium. Thus, the number of independent fluxes is reduced to one, and the system can be described with only one mutual diffusion coefficient. In addition, one has four self-diffusion coefficients Ds(methanol), Ds([BMIM]+), Ds([PF6] ), and Ds([BMIM](PF6]), so that five diffusion coefficients are necessary to describe the system completely. 

Transport equations of electrolyte and single ion conductance, self- and mutual-diffusion, and transference numbers can be obtained either from Onsager s orntinuity equation or from Onsager s fundamental equations of irreversible processes. Many publications deal with this matter, especially with electrolyte conductance. For monographs, review articles, surveys of results and recent contributions in this field see Refs. Recent extamons of in-... 

In Eq. (1), Ja is the diffusive flux of species A (mass or moles per unit area per unit time), a vector quantity Dab is the binary or mutual diffusion tensor describing the diffusion of A in a mixture of A and B and Vc a is the spatial concentration gradient of A. If diffusion is isotropic, then it may be characterized by a scalar value Z ah- Pick s law is a phenomenological description of diffusion on a macroscopic scale. It is useful in the design and analysis of processes like chromatography that involve nonequilibrium mass transport, as mathematical models of chromatography concern themselves with how fast a solute penetrates into the stationary phase, i.e., the flux. 

Popov, V.V., Determination of Mutual Diffusion-coefficients of Metallic Elements in Austenite and Ferrite of Fe-Cr-C, Fe-Mo-C and Fe-W-C Systems (in Russian), Fiz. Met. Metalloved., 79(4), 94-103 (1995) (Experimental, Transport Phenomena, 23)... 

Tables 7.2 and 7.3 display the heats of transports and thermal diffusion ratio (Kj) of chloroform in binary mixtures with selected alkanes and of toluene (1), chlorobenzene (2), and bromobenzene at 30 °C and 1 atm. Concentration-dependent thermal conductivity, mutual diffusion coefficients, and heats of transport of alkanes in chloroform and in carbon tetrachloride are given by Rowley et al. (1988). The polynomial fits to these coefficients for the alkanes in chloroform and in carbon tetrachloride are used to estimate the degree of coupling and the thermal diffusion ratio Kn from Eqns (7.46) and (7.47), and shown in Figures 7.1 and 7.2 (Demirel and Sandler, 2002). The thermal conductivity and the thermodynamic factors for the hexane-carbon tetrachloride mixture have been predicted by the local composition model of NRTL.

The calculation of diffusion coefficients from equations based on some models describing the movement of matter in electrolyte solutions, in the end, a process contributing to the knowledge of their stmcture, provided we have accurate experimental data to test these equations. Thus, to understand the behavior of transport process of these aqueous systems, experimental mutual diffusion coefficients have been compared with those estimated using several equations, resulting from different models. 

---