Volume-vacancy model

An account of the mechanism for creep in solids placed under a compressive hydrostatic suess which involves atom-vacancy diffusion only is considered in Nabano and Hemirg s (1950) volume diffusion model. The counter-movement of atoms and vacancies tends to relieve the effects of applied pressure, causing extension normal to the applied sU ess, and sluinkage in the direction of the applied sU ess, as might be anticipated from Le Chatelier s principle. The opposite movement occurs in the case of a tensile sU ess. The analysis yields the relationship... 

For the calculation of the net adsorption enthalpies of transactinides on metal surfaces the partial molar enthalpies of solution and the enthalpy of displacement are required. These values can be obtained using the semi-empirical Miedema model [66-70] and the Volume-Vacancy or Surface-Vacancy model [32,70,71]. Data for these calculations are given in [34,72,73]. 

Cohen and Turnbull s critical free-volume fluctuations picture of selfdiffusion in dense liquids is similar to the vacancy model of self-diffusion in crystals. However, in crystals individual vacancies exist and retain their identity over long periods of time, whereas in liquids the corresponding voids are ephemeral. The free volume is distributed statistically so that at any given instance there is a certain concentration of molecule-sized voids in the liquid. However, each such void is short-lived, being created and dying in continual free-volume fluctuations. The Frenkel hole theory of liquids ignores this ephemeral, statistical character of the free volume. 

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. 

Figure 1.9. Comparison between the observed values of the heat capacity at constant volume and those calculated using the cellular and vacancies model by Eyring etal. [EYR 61]...

At subcritical potentials a single oxidation process with participation of only electronegative component takes place on the surface of the alloy. The surface layer is saturated with nonequilibrium defects (mainly vacancies), maintains morphological stability and represents a diffusion zone in which the atomic fraction of the noble component gradually increases as we approach the interface with the solution [6-9], According to the volume-diffusion model [10, 11], the formation of such zone is limited by the time-dependent interdiffusion of alloy components for the vacancy mechanism. 

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. 

Thus, contributions include accounting for adsorbent heterogeneity [Valenzuela et al., AIChE J., 34, 397 (1988)] and excluded pore-volume effects [Myers, in Rodrigues et al., gen. refs.]. Several activity coefficient models have been developed to account for nonideal adsorbate-adsorbate interactions including a spreading pressure-dependent activity coefficient model [e.g., Talu and Zwiebel, AIChE h 32> 1263 (1986)] and a vacancy solution theory [Suwanayuen and Danner, AIChE J., 26, 68, 76 (1980)]. 

Here (3 is a function of vp, the number of sites active toward recombination in the recombination sphere. (In [110] the concentration and recombination volumes were expressed in units of the volume v0 of the unit cell, and n0 coincides with the fraction of sites or interstiatial sites occupied respectively by vacancies or interstitial atoms.) We note that, in the model being discussed, the cell itself in which a vacancy occurs is considered inactive with respect to recombination of an interstitial on it. 

There have been a number of computer simulations of block copolymers by Binder and co-workers (Fried and Binder 1991a,ft), and this work was reviewed in Binder (1994). Although computer simulations are limited due to the restriction on short chain lengths that can be studied, finite size effects and equilibration problems at low temperatures, the advantages are that the models are perfectly well characterized and ideal (monodisperse, etc.) and microscopic details of the system can be computed (Binder 1994). In the simulations by Binder and co-workers, diblocks were modelled as self- and mutually-avoiding chains on a simple cubic lattice, with chain lengths N = 14 to 60 for/ = 1.A purely repulsive pairwise interaction between A and B segments on adjacent sites was assumed. A finite volume fraction of vacancies was included to speed the thermal equilibration process (Binder 1994). 

One more isotherm equation that could be helpful for the determination of the micropore volume is the osmotic isotherm of adsorption. Within the framework of the osmotic theory of adsorption, the adsorption process in a microporous adsorbent is regarded as the osmotic equilibrium between two solutions (vacancy plus molecules) of different concentrations. One of these solutions is generated in the micropores, and the other in the gas phase, and the function of the solvent is carried out by the vacancies that is, by vacuum [26], Subsequently, if we suppose that adsorption in a micropore system could be described as an osmotic process, where vacuum, that is, the vacancies are the solvent, and the adsorbed molecules the solute, it is possible then, by applying the methods of thermodynamics to the above described model, to obtain the so-called osmotic isotherm adsorption equation [55] ... 

Structural effects the free volume model. Demonstrative examples of the role of free spaces on Ps formation in solids are provided by solids in which no Ps is formed when pure, and where the Ps yield increases as some dopant impurity is added. This is the case for p-terphenyl, in which the Ps yield increases as either chrysene or anthracene are added. Both dopant molecules, when introduced in the p-terphenyl matrix, promote the formation of extrinsic defects having roughly the size of a naphthalene molecule [42], Similarly, doping the ionic KIO4 matrix by lOj ions induces the formation of oxygen vacancies which promote the formation of Ps [43],... 

Positronium in condensed matter can exist only in the regions of a low electron density, in various kinds of free volume in defects of vacancy type, voids sometimes natural free spaces in a perfect crystal structure are sufficient to accommodate a Ps atom. The pick-off probability depends on overlapping the positronium wavefunction with wavefunctions of the surrounding electrons, thus the size of free volume in which o-Ps is trapped strongly influences its lifetime. The relation between the free volume size and o-Ps lifetime is widely used for determination of the sub-nanovoid distribution in polymers [3]. It is assumed that the Ps atom is trapped in a spherical void of a radius R the void represents a rectangular potential well. The depth of the well is related to the Ps work function, however, in the commonly used model [4] a simplified approach is applied the potential barrier is assumed infinite, but its radius is increased by AR. The value of AR is chosen to reproduce the overlap of the Ps wavefunction with the electron cloud outside R. Thus,... 

While so far we have considered the volume fraction computer simulations [84-99,101-103,107], it also is useful to consider the melt as compressible and then the pressure controls the vacancy concentration. This interpretation is the starting point of the lattice fluid model [108-110] of polymer mixtures and related models [111, 112], but will not be pursued further here. 

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