Vacancies volume change

The oxygen vacancies then diffuse to the gas interface where they are annihilated by reaction with adsorbed oxygen. The important point, however, is that metal is consumed and oxide formed in the same reaction zone. The oxide drift has thus only to accommodate the net volume difference between the metal and its equivalent amount of oxide. In theory this net volume change could represent an increase or a decrease in the volume of the system, but in practice all metal oxides in which anionic diffusion predominates have a lower metal density than that of the original metal. There is thus a net expansion and the oxide drift is away from the metal. 

Cell and hole models were used to formulate equations of state for polymer liquids or to discuss isothermal expansion and compressibility of the systems [Hory et al., 1964 Simha, 1977 Dee and Walsh, 1988]. In the models, chain segments are placed on lattice sites. All sites are completely occupied in cell models, and volume changes of the system are related solely to changes in cell volume. Hole models as used by Simha and Somcynsky allow for both lattice vacancies and changes in cell volume. 

To study the volume change (compressibility) of the melts of semiflexible polymers, we consider a special case where the solvent component is the vacancy. Because the volume occupied by the vacancy (the free volume) is Noa = (N —nNi)a, the mixing free energy is given by Amix F (iV - n Mi, ATi) in (2.143). The pressure is therefore derived by the differentiation... 

The vacancy volume can be defined in the following way. If an initially perfect crystal is altered by inserting a number density p of vacancies per unit volume throughout its interior without changing the number of atoms in the crystal, then the crystal undergoes a stress-free isotropic expansion if its boundaries are unconstrained. If the change in volume per unit volume is expressed as pfivac, then fivac is the volume of a vacancy its value is commonly assumed to be between one-half and one times the atomic volume of the material. If the distribution is not dilute, then the value of vacancy volume could depend on the density itself. 

The volume change associated with the annihilation of vacancies located at grain boundaries oriented normally to the plane of a thin film was discussed in Section 1.8.6. Although a volume change occurs at any site in a thin film where a vacancy is annihilated, this process need not always result in the generation of an internal stress. Provide a simple mechanistic justification for each of the following statements. 

In the code LIFE-1, porosity closure is assumed to be controlled by stress-directed diffusion of vacancies from the pores, with the grain boundaries of the fuel acting as sinks (42, 56). The volume change rate for this process is... 

There are two ways in which the volume occupied by a sample can influence the Gibbs free energy of the system. One of these involves the average distance of separation between the molecules and therefore influences G through the energetics of molecular interactions. The second volume effect on G arises from the contribution of free-volume considerations. In Chap. 2 we described the molecular texture of the liquid state in terms of a model which allowed for vacancies or holes. The number and size of the holes influence G through entropy considerations. Each of these volume effects varies differently with changing temperature and each behaves differently on opposite sides of Tg. We shall call free volume that volume which makes the second type of contribution to G. 

Here, ct is the shear stress in the transition liquid layer, y oi is a molecular volume, h is Planck s constant, AG is the free energy change of the movement of a molecule into a vacancy, and A/Zyac is the enthalpy of formation of a vacancy. The rate of deformation of a hquid is the strain rate, y [see Eq. (2)], so the right-hand side of Eq. (34) can be used to estimate the viscosity of the transition layer. 

Several points are to be noted. Firstly, pores and changes of sample dimension have been observed at and near interdiffusion zones [R. Busch, V. Ruth (1991)]. Pore formation is witness to a certain point defect supersaturation and indicates that sinks and sources for point defects are not sufficiently effective to maintain local defect equilibrium. Secondly, it is not necessary to assume a vacancy mechanism for atomic motion in order to invoke a Kirkendall effect. Finally, external observers would still see a marker movement (markers connected by lattice planes) in spite of bA = bB (no Kirkendall effect) if Vm depends on composition. The consequences of a variable molar volume for the determination of diffusion coefficients in binary systems have been thoroughly discussed (F. Sauer, V. Freise (1962) C. Wagner (1969) H. Schmalzried (1981)]. 

Solution. First, calculate the energy change contributed by the excess vacancies which are eliminated to relieve the strain due to the dilatation eh. If V is the cluster volume, AV = 3enV. The number of vacancies required is then N = 3e iV7n and the free-energy change due to the removal of these vacancies is therefore... 

Next, calculate the free-energy change due to the destruction of the additional vacancies which are removed to the point where the rate of buildup of elastic strain due to their annihilation is just equal to the rate at which energy is given up by the vacancy annihilation. If N vacancies are destroyed in this fashion, the volume of matrix removed... 

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