Interdiffusion coefficients

Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient. 5-46... 

Mutual Diffusivity, Mass Diffusivity, Interdiffusion Coefficient Diffusivity is denoted by D g and is defined by Tick s first law as the ratio of the flux to the concentration gradient, as in Eq. (5-181). It is analogous to the thermal diffusivity in Fourier s law and to the kinematic viscosity in Newton s law. These analogies are flawed because both heat and momentum are conveniently defined with respec t to fixed coordinates, irrespective of the direction of transfer or its magnitude, while mass diffusivity most commonly requires information about bulk motion of the medium in which diffusion occurs. For hquids, it is common to refer to the hmit of infinite dilution of A in B using the symbol, D°g. 

The interdiffusion coefficient has been measured for this mixture by Boardman and Wild,5 by Waldmann,88 and by Schafer and his colleagues.69, 70 Figure 16 shows that the coefficient calculated... 

The resolution of infra-red densitometry (IR-D) is on the other hand more in the region of some micrometers even with the use of IR-microscopes. The interface is also viewed from the side (Fig. 4d) and the density profile is obtained mostly between deuterated and protonated polymers. The strength of specific IR-bands is monitored during a scan across the interface to yield a concentration profile of species. While in the initial experiments on polyethylene diffusion the resolution was of the order of 60 pm [69] it has been improved e.g. in polystyrene diffusion experiments [70] to 10 pm by the application of a Fourier transform-IR-microscope. This technique is nicely suited to measure profiles on a micrometer scale as well as interdiffusion coefficients of polymers but it is far from reaching molecular resolution. 

The Stokes-Einstein equation predicts that DfxITa is independent of the solvent however, for real solutions, it has long been known that the product of limiting interdiffusion coefficient for solutes and the solvent viscosity decreases with increasing solute molar volume [401]. Based upon a large number of experimental results, Wilke and Chang [437] proposed a semiempirical equation,... 

By selecting the reference properly, the diffusion coefficients DA and DB can be made equal to each other. This value is termed the mutual diffusion (or interdiffusion) coefficient Dab- The reference frame is one across which no change in volume occurs (fixed volume) ... 

Fig. 67. Experimental variations of the interdiffusion coefficient Qi(Q)/Q2 as a function of Q at the total polymer concentration c = 0.34 g/cm3 for the two investigated sytems ( ) diblock copolymer PSD-PSH 561/ben-zene + d-benzene ( ) mixture of homopolymers PSH155/PSD425/d-benzene. The solid lines are visual aids. (Reprinted with permission from [171]. Copyright 1989 American Chemical Society, Washington)...

The above sections have focused upon homogeneous systems with a constant composition in which tracer diffusion coefficients give a close approximation to selfdiffusion coefficients. However, a diffusion coefficient can be defined for any transport of material across a solid, whether or not such limitations hold. For example, the diffusion processes taking place when a metal A is in contact with a metal B is usually characterized by the interdiffusion coefficient, which provides a measure of the total mixing that has taken place. The mixing that occurs when two chemical compounds, say oxide AO is in contact with oxide BO, is characterized by the chemical diffusion coefficient (see the Further Reading section for more information). 

The interdiffusion coefficient, D , between Cs and Na for example, can be defined by the Nernst-Plank Equation,... 

Tominaga, T. and Matsumoto, S. Limiting interdiffusion coefficients of some hydroxylic compounds in water from 265 to 433... 

If there is simultaneous diffusion of more than one component in the crystal, the flux of A in direction X depends on the individual diffusivities of all diffusing components (Darken, 1948), and the individual diffusivity coefficient in equations 4.87 and 4.88 is replaced by interdiffusion coefficient D i.e., for the simultaneous diffusion of two ions A and B,... 

Mn Mgi )2Si04 olivine mixture obtained by Morioka (1981). The concentration profiles are symmetric, indicating that the diffusivities are independent of the concentration of the diffusing ion. On the contrary, possible asymmetry in the diffusion profiles indicate that concentration depends significantly on the diffusing cations. In this case, the interdiffusion coefficient can be obtained by the Boltzmann-Matano equation ... 

Unfortunately, the interpretation of Giletti et al. (1978) does not solve the problem of differential diffusivities of 0 and To do this, their experimental results should be interpreted in terms of interdiffusion of 0 and O. Application of Pick s first law to interdiffusion of the two species would in fact lead to the definition of an interdiffusion coefficient/), so that... 

D thus depends on a, which, according to Eqn. (5.131), is determined by the assumptions which govern the equilibration of oxygen (d//G = 0) or the vacancies (d v = 0). In other words, the interdiffusion coefficient D depends directly on the mode and extent of the point defect equilibration. 

The first two terms in the bracket correspond to the customary chemical free energy density and the gradient energy respectively. The third term takes into account the ballistic flux. D is the Darken interdiffusion coefficient (Eqn. (4.78)), but adapted to the radiation induced increased defect concentration of the alloy. 

D (chemical) interdiffusion coefficient e0 charge of electron (= 1.6 1(T19 C) e electron in crystal... 

Earlier theories proposed (Boyd et al., 1947 Kressman and Kitchener, 1949) that Fick s first law with a constant interdiffusion coefficient could be utilized as the flux equation ... 

Helfferich, F. (1963). Ion exchange kinetics. IV. Demonstration of the dependence of the interdiffusion coefficient on ionic composition, J. Phys. Chem. 67, 1157-1158. 

This coefficient describes a diffusion process under the influence of a gradient in the chemical composition. When two diffusing species mix together, their rate of mixing depends on the diffusion rates of both species. Consequently, the interdiffusion coefficient is defined to measure this rate of mixing in relation to a laboratory frame of reference [13]. In this sense, the relation defining the interdiffusion coefficient, deduced by Darkens [29], is... 

As we have commented previously for metals, the diffusion in concentration gradients is described with the chemical diffusion coefficient, or interdiffusion coefficient. In this case, it is possible to consider that the interdiffusion and the intrinsic diffusion coefficients are equivalent, since we have only the movement of one species, that is, oxygen, by the vacancy mechanism. Subsequently, applying the Einstein relation... 

Lawrence Stamper Darken (1909-1978) subsequently showed (Darken, 1948) how, in such a marker experiment, values for the intrinsic diffusion coefficients (e.g., Dqu and >zn) could be obtained from a measurement of the marker velocity and a single diffusion coefficient, called the interdiffusion coefficient (e.g., D = A ciiD/n + NznDca, where N are the molar fractions of species z), representative of the interdiffusion of the two species into one another. This quantity, sometimes called the mutual or chemical diffusion coefficient, is a more useful quantity than the more fundamental intrinsic diffusion coefficients from the standpoint of obtaining analytical solutions to real engineering diffusion problems. Interdiffusion, for example, is of obvious importance to the study of the chemical reaction kinetics. Indeed, studies have shown that interdiffusion is the rate-controlling step in the reaction between two solids. 

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