Activation energy jumping

The end of secondary carbonization is the formation of pure carbon corresponding to a jump in activation energy. This jump is manifested by a sndden wiping-out of residual intralayer defects, such as vacancies or interstitial atoms [122,123]. Polygonization occurs simultaneously with this activation energy jump. 

In tire transition-metal monocarbides, such as TiCi j , the metal-rich compound has a large fraction of vacairt octahedral interstitial sites and the diffusion jump for carbon atoms is tlrerefore similar to tlrat for the dilute solution of carbon in the metal. The diffusion coefficient of carbon in the monocarbide shows a relatively constairt activation energy but a decreasing value of the pre-exponential... 

Following the general trend of looldng for a molecular description of the properties of matter, self-diffusion in liquids has become a key quantity for interpretation and modeling of transport in liquids [5]. Self-diffusion coefficients can be combined with other data, such as viscosities, electrical conductivities, densities, etc., in order to evaluate and improve solvodynamic models such as the Stokes-Einstein type [6-9]. From temperature-dependent measurements, activation energies can be calculated by the Arrhenius or the Vogel-Tamman-Fulcher equation (VTF), in order to evaluate models that treat the diffusion process similarly to diffusion in the solid state with jump or hole models [1, 2, 7]. 

The practical importance of vacancies is that they are mobile and, at elevated temperatures, can move relatively easily through the crystal lattice. As illustrated in Fig. 20.21b, this is accompanied by movement of an atom in the opposite direction indeed, the existence of vacancies was originally postulated to explain solid-state diffusion in metals. In order to jump into a vacancy an adjacent atom must overcome an energy barrier. The energy required for this is supplied by thermal vibrations. Thus the diffusion rate in metals increases exponentially with temperature, not only because the vacancy concentration increases with temperature, but also because there is more thermal energy available to overcome the activation energy required for each jump in the diffusion process. 

In molecular doped polymers the variance of the disorder potential that follows from a plot of In p versus T 2 is typically 0.1 eV, comprising contributions from the interaction of a charge carrier with induced as well as with permanent dipoles [64-66]. In molecules that suffer a major structural relaxation after removal or addition of an electron, the polaron contribution to the activation energy has to be taken into account in addition to the (temperature-dependent) disorder effect. In the weak-field limit it gives rise to an extra Boltzmann factor in the expression for p(T). More generally, Marcus-type rates may have to be invoked for the elementary jump process [67]. 

Value of Activation Volume (V ) Per Jumping Segment and Activation Energy (AH) for Different Interfacial Agent Contents... 

Figure 2. Plot of the desorption rate, molecules/sec, (solid circles) and the Integrated number of molecules desorbed (solid line) for an adsorbate with a desorption activation energy of 20Kcal/mole and a preexponentlal of 10 sec-. The temperature jump shown In Figure 1 was used for this calculation.

For the jump of an ion into a hole, a certain energy barrier mnst be overcome with the activation energy A. The rate of this process (or valne of condnctivity) is subject to temperature dependence, according to the well-known Arrhenins equation (see Section 14.1) ... 

Notice that the energy of the ion is the same at the beginning and the end of the jump the energy required to make the jump, E, is known as the activation energy for the jump. This means that the temperature dependence of the mobility of the ions can be expressed by an Arrhenius equation ... 

SRPAC spectra of Fig. 9.33a with a model that allows random, single-activated jumps of the EFG on a cone (Fig. 9.33b) was possible over the entire temperature range. This random jump cone model follows an Arrhenius law with activation energy a = (20.1 0.8) kJ moP and frequency factor A = (5.5 1.6) 10 s and it yields a cone opening angle of about 47° above 380 K [81]. 

The values derived in this way for the diffusion coefficients exhibit surprising agreement with the experimental values, even for small ions, better than a coincidence in the order of magnitude. More detailed theoretical analysis indicates that the formation of holes required for particle jump is analogous to the formation of holes necessary for viscous flow of a liquid. Consequently, the activation energy for diffusion is similar to that for viscous flow. 

The high conductivity of (3-alumina is attributed to the correlated diffusion of pairs of ions in the conduction plane. The sodium excess is accommodated by the displacement of pairs of ions onto mO sites, and these can be considered to be associated defects consisting of pairs of Na+ ions on mO sites plus a V N l on a BR site (Fig. 6.12a and 6.12b). A series of atom jumps will then allow the defect to reorient and diffuse through the crystal (Fig. 6.12c and 6.12d). Calculations suggest that this diffusion mechanism has a low activation energy, which would lead to high Na+ ion conductivity. A similar, but not identical, mechanism can be described for (3"-alumina. 

Here is the distance between the two sites and r(E) is the jump time corresponding to an activation energy E ... 

There remains an interpretation of ta to be found, ta exhibits an activation energy of about 0.43 0.1 eV, about three times as high as the C-C torsional barrier of 0.13 eV. The discrepancy must reflect the influence of the interactions with the environment and therefore ta appears to correspond to relaxation times most likely involving several correlated jumps. The experimental activation energy is in the range of that for the NMR correlation time associated with correlated conformational jumps in bulk PIB [136] (0.46 eV) and one could tentatively relate ta to the mechanism underlying this process (see later). 

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