Frenkel value

The intercept on the adsorption axis, and also the value of c, diminishes as the amount of retained nonane increases (Table 4.7). The very high value of c (>10 ) for the starting material could in principle be explained by adsorption either in micropores or on active sites such as exposed Ti cations produced by dehydration but, as shown in earlier work, the latter kind of adsorption would result in isotherms of quite different shape, and can be ruled out. The negative intercept obtained with the 25°C-outgassed sample (Fig. 4.14 curve (D)) is a mathematical consequence of the reduced adsorption at low relative pressure which in expressed in the low c-value (c = 13). It is most probably accounted for by the presence of adsorbed nonane on the external surface which was not removed at 25°C but only at I50°C. (The Frenkel-Halsey-Hill exponent (p. 90) for the multilayer region of the 25°C-outgassed sample was only 1 -9 as compared with 2-61 for the standard rutile, and 2-38 for the 150°C-outgassed sample). 

FIG. 2 Growth rates as a function of the driving force A//. Comparison of theory and computer simulation for different values of the diffusion length and at temperatures above and below the roughening temperature. The spinodal value corresponds to the metastability limit A//, of the mean-field theory [49]. The Wilson-Frenkel rate WF is the upper limit of the growth rate. 

In essentially all of the prior formulations of TDDFT a complex Lagrangian is used, which would amount to using the full expectation value in Eq. (2.9), not just the real part as in our presentation. The form we use is natural for conservative systems and, if not invoked explicitly at the outset, emerges in some fashion when considering such systems. A discussion of the different forms of Frenkel s variational principle, although not in the context of DFT, can be found in (39). 

Beginning with values of ]r) and ]v) at time 0, one calculates the new positions and then the new velocities. This method is second-order in At, too. For additional details, see Allen, M. R, and D. J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford (1989) Frenkel, D., and B. Smit, Understanding Molecular Simulation, Academic Press (2002) Haile, J. M., Molecular Dynamics Simulation, Wiley (1992) Leach, A. R., Molecular Modelling Principles and Applications, Prentice-Hall (2001) Schlick, T., Molecular Modeling and Simulations, Springer, New York (2002). 

Fig. 1 Melting temperatures of polymers k%Tm/Ec) with variable Ev/Ec values. The line is calculated from Eq. 10 and the circles are the simulation results obtained from the onset of crystallization on the cooling curves of disorder parameters, in a short-chain (r = 32) system (occupation density is 0.9375 in a 32-sized cubic box) with a template substrate (Hu and Frenkel, unpublished results)...

Fig. 23 Parallel tempering of the free-energy curves in the overlapping windows as a function of the number of molten units for a single 1024-mer at a temperature of 2.967 p/A b. The y-axis is not for the absolute value of the free energy but for the relative distribution of the free energy (Hu and Frenkel, unpublished results)...

Note that due to the sinusoidal function of the shear stress in Frenkel s model V Yb= p/(4dc)). The corresponding maximum value of the shear energy, W y is given by... 

The maximum shear strain in engineering units is fb=2/J, which corresponds to the value p(4dc) l in the Frenkel model discussed in Sect. 2.5.1. The relative displacement, xy of adjacent chains shown in Fig. 57 is limited by the maximum value 2/3 for larger displacements the attracting force decreases rapidly and failure is initiated. It is further assumed that A0(tb) [Pg.85]

Some experimental values for the formation enthalpy of Frenkel defects are given in Table 2.2. As with Schottky defects, it is not easy to determine these values experimentally and there is a large scatter in the values found in the literature. (Calculated values of the defect formation energies for AgCl and AgBr, which differ a little from those in Table 2.2, can be found in Fig. 2.5.)... 

Frenkel defects on the anion sublattice show only anion migration and hence have fa close to 1. The alkali halides NaF, NaCl, NaBr, and KC1 in which Schottky defects prevail and in which the cations and anions are of similar sizes have both cation and anion contributions to ionic conductivity and show intermediate values of both anion and cation transport number. 

The approximations to use depend upon the pressure regime and the values of the equilibrium constants. This oxide is an insulator under normal conditions, and so, in the middle region of the diagram, Frenkel equilibrium is dominant, that is, Kf > Ke and the electroneutrality equation is approximated by... 

Frenkel and Schottky defect equilibria are temperature sensitive and at higher temperatures defect concentrations rise, so that values of Ks and Kv, increase with temperature. The same is true of the intrinsic electrons and holes present, and Kc also increases with temperature. This implies that the defect concentrations in the central part of a Brouwer diagram will move upward at higher temperatures with respect to that at lower temperatures, and the whole diagram will be shifted vertically. 

Intrinsic Frenkel disorder, in which some of the oxygens are displaced into normally unoccupied sites, is responsible for the oxide ion conduction in, for example, Zr2Gd207, Fig. 2.11. The interstitial oxygen concentration is rather low, however, and is responsible for the low value of the preexponential factor and for the rather low (by -Bi203 standards ) conductivity. 

From classic thermodynamics alone, it is impossible to predict numeric values for heat capacities these quantities are determined experimentally from calorimetric measurements. With the aid of statistical thermodynamics, however, it is possible to calculate heat capacities from spectroscopic data instead of from direct calorimetric measurements. Even with spectroscopic information, however, it is convenient to replace the complex statistical thermodynamic equations that describe the dependence of heat capacity on temperature with empirical equations of simple form [15]. Many expressions have been used for the molar heat capacity, and they have been discussed in detail by Frenkel et al. [4]. We will use the expression... 

Table 5.1 lists some enthalpy-of-formation values for Schottky and Frenkel defects in various crystals. 

Whether Schottky or Frenkel defects are found in a crystal depends in the main on the value of A//, the defect with the lower A// value predominating. In some crystals it is possible for both types of defect to be present. 

The quantity Uo characterizes the degree of aggregation only on the average. Therefore it is important in each particular case to analyze also the spatial distribution of the defects. Thus, the low-temperature accumulation of the Frenkel defects in the two- and three-dimensional cases was simulated in [36, 114] and the obtained values of Uo for d = 2 considerably exceed the same in [113]. In contrast to the latter, the authors of [36, 114] used a circle as the recombination region. In its turn, the values of u0 obtained in [115] are considerably larger than those found in [114]. We note that it was assumed in [115] that, when an interstitial atom occurs at a site where the recombination spheres of several vacancies overlap, it recombines with the closest vacancy. This demonstrates very well how any details, insignificant at first glance, can affect considerably the accumulation kinetics. 

Figure 7.15 shows the joint correlation function of similar defects in the region of concentration saturation as calculated from the results of the simulation. We see that the fraction of the close Frenkel defects (of the type of dimer F2-centres) exceeds by approximately threefold the value expected in the Poisson distribution, which agrees well with the analytical theory presented in Section 7.1 for the annihilation mechanism (see also [31, 111]) and with actual experiments for alkali halide crystals [13]. 

The value of the pre-exponential factor is very important in Frenkel s theory of pre-transition states and heterophase fluctuations [10]. However, simple qualitative arguments appear to show that the effects considered by... 

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