Vacancy diffusion constant

The vacancy diffusion constant (Dv) of LSCF was calculated using Eq. (36.3) and compared with the results of Xu and Thomson. Again, a good match between the approach over powder and the semipermeability data by Xu and Thomson [32] was observed. 

The diffusion constant for Cu can also be discussed using this model. Because diffusion by singly charged cation vacancies, and by neutral cation vacancies, are both important, we may write... 

It should be noted that relation (2.51) is valid within the sudden approximation. However, the relaxation of heavy particle impurities typically involves motion that is slow compared with vibrations of the host lattice (i.e., the tunneling takes place in the adiabatic limit). The net effect of the adiabatic approximation is to renormalize the effective moment of inertia of the particle. This approach was used, for example, to describe vacancy diffusion in light metals. The evolution of the rate constant from Arrhenius behavior to the low-temperature plateau was described within the framework of one-dimensional tunneling of a... 

According to the accepted model it can be supposed that diffusion of elementary vacancies with diffusion coefficient Dv occurs. Then rv(r,r) would be a solution of the diffusion equation and in the case of cylindrical symmetry rv(r,f) depends only on the axial co-ordinate r and on t. The film periphery is in equilibrium with the bulk phase and close to it Tv(r,f) does not depend on time. It is also supposed that at the moment of film formation (t = 0) the concentration of vacancies is constant in the whole film. This yields... 

Figure 5 Multiscale approach to understand rate of CO2 diffusion into and CH4 diffusion out of a structure I hydrate, (left) Molecular simulation for individual hopping rates, (middle) Mesoscale kinetic Monte Carlo simulation of hopping on the hydrate lattice to determine dependence of diffusion constants on vacancy, CO2 and CH4 concentrations, (right) Macroscopic coupled non-linear diffusion equations to describe rate of CO2 infusion and methane displacement. Graph from Stockie. ...

Here gp is the Gibbs free energy to form a vacancy, k is the Boltzmann constant, and T is the temperature. Diffusion in a crystal lattice occurs by motion of atoms via jumps between these defects. For example, vacancy diffusion - the most common mechanism in close-packed lattices such as face-centered cubic fee) metals, occurs by the atom jumping into a neighboring vacancy. The diffusion coefficient, D, therefore will depend upon the probability that an atom is adjacent to a vacancy, and the probability that it has sufficient energy to make the jump over the energy barrier into the vacancy. The first of these probabilities is directly proportional to c,. and the... 

Self-Diffusion. The self-diffusion coefficient D of liquid PH3 and PD3 in sealed tubes was determined by spin echo measurements of the nuclei H, h, and ip from 139 K to ambient temperature. Numerical values for both phosphanes are given by D(cm2/s) = 5.18x10" exp(-413/T) at temperatures up to 200 K. Above this temperature, D rises faster with temperature to reach 1.5x10 cm /s at 293 K. Attempts to correlate D and the viscosity t failed except at the lowest temperatures [11]. The self-diffusion constant of plastic crystalline PH3 was estimated for a vacancy diffusion mechanism on the basis of the spin-lattice relaxation time. The increase from D = 2x10" to 1x10" cm /s between 103 and 138 K is typical for a plastic crystal. A diffusion activation energy of 19 kJ/mol was estimated [12]. 

In general, functions 0 and F are not known. To determine these functions one may use as the input values such experimental data as lattice parameters, elastic constants, energy of the vacancy formation, sublimation energy, and the stacking fault energy. Thermal expansion factors, heat of solution, phonon frequencies, surface energy, and energy of the vacancy diffusion can also be included in quantities used for determination of the functions 0 and F. 

The limitation of this approach is that when the activation energy for atom transport is in excess of kT there is little probability of an atom or vacancy migrating during a simulation run, and the diffusion constant will appear to be zero. Thus, constraints need to be introduced to force the atom to move. One approach for identifying the diffusion... 

When there is a fluctuation in the concentration, the polymer under study tries to move to fill the vacancy in the low-concentration region, but it is impossible for the whole chain to move simultaneously due to the topological constraints. Instead, a blob plays a role of the moving unit. It can diffuse into the neighborhood to restore the concentration back to the average value. This movement can be seen as a diffusion of a rigid sphere of radius f in the solvent, so that the diffusion constant Dc is estimated to be... 

Electrical conduction and the diffusion of copper vacancies in non-stoichiometric cuprous selenide was studied at 500 to 850C. It was found that the diffusion constant of the Cu vacancies had extremely high absolute values. Its temperature dependence could be described by ... 

Hereinafter, we consider the vacancy diffusivity as constant everywhere. Then, the change in migration energy is negligible and we can treat the dependence of the vacancy concentration profile only on the energy profile Ey (r) via... 

Here, Do is the diffusion constant of vacancy diffusion and Qex is the activation energy for vacancy migration i. e., the exchange of a vacancy with a neighbouring atom. 

From studies of the effect of Cd-dopants on the ionic conductivity it could be concluded that cationic Frenkel defects predominate in AgBr. Thus the diffusion was therefore expected to involve both vacancy diffusion and transport of interstitial ions. The experimentally measured ratios of Dt/Dr varied from 0.46 at 150 °C to 0.67 at 350°C. For vacancy diffusion a constant ratio of 0.78 (=f) would have been expected, and the diffusion mechanism could thus be ruled out. For interstitial diffusion f=l, and this mechanism could also be excluded. 

The last equation is known as Pick s law. D(T) is the diffusion constant it has dimensions [length] [time] and depends on the temperature T. If motion of vacancies and interstitials were the only possible means for diffusion in a solid, then the diffusion constant would be... 

---