Chemical interdiffusion coefficient

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

In the form of eq. (5-30), Pick s second law applies only to one-dimensional problems in an isotropic medium. The index i on the diffusion coefficient has been removed in order to make it clear that this is no longer the component diffusion coefficient D,-, but rather, it is the chemical interdiffusion coefficient. Normally, the chemical interdiffusion coefficient will be a function of the individual component diffusion coefficients Di because of the coupling of the fluxes in the lattice system. When local thermodynamic equilibrium prevails, the coefficients Di are, in turn, unique functions of the composition. From the thermodynamics of irreversible processes it can be shown [6] that in binary systems there is only one independent transport coefficient, and in general, in n-component systems there can only be (n - 1) /2 independent transport coefficients. 

It has thus been shown by means of two examples how the phenomenological laws of diffusion are formulated and what procedures must be followed in special cases in order to express the chemical interdiffusion coefficient in terms of the component diffusion coefficients or, as in eq. (5-39), in terms of the diffusion coefficients of point defects. 

The subsequently discussed ternary nitrides are mostly high-temperature materials. Preparation reactions of multicomponent Th-N-X mixtures must allow for very slow chemical interdiffusion coefficients in solids. Many preparations can be made by powder metallurgical techniques involving multiple comminution, mixing, and elevated-temperature annealing, e.g., formation of (Th,U) N solid solutions from the binary compounds. 

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). 

Chemical diffusion has been treated phenomenologically in this section. Later, we shall discuss how chemical diffusion coefficients are related to the atomic mobilities of crystal components. However, by introducing the crystal lattice, we already abandon the strict thermodynamic basis of a formal treatment. This can be seen as follows. In the interdiffusion zone of a binary (A, B) crystal having a single sublattice, chemical diffusion proceeds via vacancies, V. The local site conservation condition requires that /a+/b+7v = 0- From the definition of the fluxes in the lattice (L), we have... 

The vacancy flux and the corresponding lattice shift vanish if bA = bB. In agreement with the irreversible thermodynamics of binary systems i.e., if local equilibrium prevails), there is only one single independent kinetic coefficient, D, necessary for a unique description of the chemical interdiffusion process. Information about individual mobilities and diffusivities can be obtained only from additional knowledge about vL, which must include concepts of the crystal lattice and point defects. 

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... 

In discussing AO-BO interdiffusion, we saw that the two independent fluxes of this ternary system can lead to different chemical diffusion coefficients D. They depend upon the constraints which define the physical situation (e.g., VjuG = 0 or Vy/v = 0). The analysis of this relatively simple and fundamental situation is already rather complex. The complexity increases further if diffusion occurs between heterovalent components of compound crystals. This diffusion process is important in practice (e.g., heterovalent doping) and its treatment in the literature is not always adequate. We therefore add a brief outline of the relevant ideas for a proper evaluation of D. 

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. 

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... 

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... 

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... 

With the help of Equation 5.107, as was previously done with Equation 5.86, we obtain a transport or chemical diffusion coefficient that is a result of Fick s laws. We now interpret the meaning of this coefficient if we consider diffusion in a microporous solid, as a special case of binary diffusion, where A is the mobile species and the diffusivity of the microporous framework atoms is zero, then, the frame of reference are the fixed coordinates of the porous solid consequently, we have a particular case of interdiffusion where the diffusion coefficient is simply the diffusivity of the mobile species [12,20],... 

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. 

Most of the known IE kinetic problems have been solved by the use of a single mass-balanced diffusion equation [1-3,7-11,14-24,34-43]. They are, on this basis, identified as one component systems and the diffusion rate for the invading B ion is controlled by the concentration gradient of this ion alone. In these cases the effective interdiffusion coefficient depends on the ion concentrations and the equilibrium constants of the chemical reaction between both ions in the ion exchanger [2-3,7-12,16-22,23,23,30,32,34,42,32-34]. 

Diffusion of Water in poly[PFSA] Membranes To describe diffusion of water through the membrane in the presence of a water activity gradient, an appropriate interdiffusion coefficient must be determined. Experimental methods used to study diffusion of water in these polymers, such as radiotracer and pulsed gradient spin-echo NMR techniques, probe intrad-iffusion coefficients, often referred to as tracer or self-diffusion coefficients, determined in the absence of a chemical potential gradient. Intra- and interdiffusion coefficients are related for the case of diffusion of a small molecule in a polymeric matrix as follows [28] ... 

The other major category of D values are termed interdiffusion or chemical diffusion coefficients. Experimentally, Ci does change significantly and D is a function of Ci. Diffusion of one or more chemical species is dependent on the opposing diffusion of another species in order to maintain a constant matrix volume and/or electrical neutrality. The diffusion in olivine of Mg in one direction and the complementary diffusion of Fe in the opposite direction represent one example. Rarely does this type of experiment employ the use of isotopically labeled species. However, in some cases isotopically-enriched H2O (T, D and/or O) has been used where the composition of the solid (melt) became significantly modified by incorporation of water into the structure. 

In binary substitutional alloys, it is possible for one atomic species to diffuse faster than the other. This unequal diffusion (the Kirkendall effect [83-85]) is consistent with the vacancy mechanism of diffusion. Each component in a binary alloy (e.g., A and B) has its own diffusion coefficient, and the Kirkendall effect is due to the fact that D for A is different from D for B. An overall chemical diffusion coefficient D (interdiffusion coefficient) can be defined as... 

The measured interdiffusion coefficients differ fundamentally from the self-diffusion coefficients D because the presence of the chemical concentration gradient under which the former is measured imposes a bias on the otherwise random motion. Thermodynamic considerations give an expression for the partial chemical diffusion coefficient as... 

In this equation, which is known as the Darken equation, X indicates the mole fraction of Co or of Ni. The equation assumes local equilibrium everywhere and that D is a chemical or interdiffusion coefficient in a chemical potential gradient. The matrix is Co Ni O. D is plotted as a function of concentration for both Ni and Co diffusing in the mixed oxide at 1300°C and 1445°C in Figure 25.15a. 

There are various diflEiision coefficients (i) self-diflfusion coefficient, (ii) tracer diffusion coefficient, (iii) lattice diffusion coefficient, (iv) grain boundary diffusion coefficient, (v) surface diffusion coefficient, (vi) defect diffusion coefficients, and (vii) chemical, effective or interdiffusion coefficient. 

Figure 3.41. Concentration dependence of the diffusion coefficient Z), normalized by Dj = Dj at c -> 0 for polyvinylpyrrolidone in aqueous solutions from Figure 3.40 s data. Fast (/) and slow (2) modes. Curve 3 is the interdiffusion coefficient from the first cumulant ACi at q -> 0 (Burchard and Elsele, 1984) [Reprinted with permission from W.Burchard, M.Eisele. Pure and Appl. Chem. 56 (1984) 1379-1390. Copyright 1984 by the American Chemical Society]...

The relationship for the interdiffusion coefficient (Rqiiation 2.4-47) can be written as a function of the first derivative of the chemical potential of the ith component with respect to the concentration of any component by mean.s of Fx uations 1.1.2-51, 52. 

---