AB model

Table 5.3. Evaluation scheme of two-way ANOVA (ab model single measurements in each point the index point marks that levels over which is actually added up or averaged, respectively)...

In the case that interactions prove to be insignificant, it should be gone over to the ab model the estimations of which for the various variance components is more reliable than that of the 2ab model. A similar scheme can be used for three-way ANOVA when the factor c is varied at two levels. In the general, three-way analysis bases on block-designed experiments as shown in Fig. 5.1. 

Tab. 3.1 Main data defining the optimized structure of the FeP(AB) and FeP(lm)(AB) models investigated (AB = CO, NO, 02). Distances are given in A, angles in degrees, and energies in kcal mob1. Porphyrin nitrogens are denoted Np and Nc refers to one of the nitrogen atoms of the axial imidazole. The experimental values correspond to X-ray structures of heme models [18].

AB Model II Detculed Compaurisons with Classic Detonation Theory 577... 

Most of the simulations reviewed were performed in two dimensions, although some simulations of three-dimensional detonations are discussed in Sec. 3.1.2. Equilibrated molecular crystals for AB Model I in both two and three dimensions are shown in Fig. 2. This model has slightly longer bond lengths than either Model II or III, but otherwise the starting crystal structures are similar for all three models. To model an infinite crystal, almost all detonation simulations were carried out with periodic boundary conditions enforced perpendicular to the direction of the shock propagation. 

In this section we first present a series of results (Sec. 3.1) which show that molecular dynamics simulations can be used to directly link atomic scale chemistry to the continuum theory of detonations. We then show (Secs. 3.2 and 3.3) that complex initiation behavior can arise even within the simple AB Model I system. Taken together, the results reviewed in this section demonstrate that simulations using REBO potentials provide a powerful probe of the interplay between the continuum properties of shockwaves and the atomic scale chemistry induced in the initiation and propagation of condensed phasn detonations. 

AB Model II Detailed Comparisons with Classic Detonation Theory... 

Fig.6. The B2(110) surface average (solid lines) and sub-lattice (dotted lines) concentrations of the segregant in AB model alloy as a function of reduced temperature calculated in the FCEM approximation for different segregation/order factors r (indicated near the plots). The difference in sub-lattice concentrations corresponds to the surface LRO parameter that vanishes at the surface transition temperature Tg that coincides with the bulk transition temperature T independently of r.

In Ref. 51, this task was attained. For one orientation of the crystal of 8 for which the signals of interest and those of the impurities (originating mainly from incompletely deuterated methyl groups) do not overlap a clear discrimination between the DQR and AB models could be made. Examples of the DQR fits are shown in Fig. 10. For two temperatures, a comparison of the AB and DQR fits for selected echo spectra is shown in Figs. 11 and 12. 

Fig. 11. A compai ison of selected experimental solid echo spectra of 8 at 28 K (middle column) with the best fit theoretical spectra obtained using the DQR (left column) and AB model (right column). In the AB fits, the amplitudes of the peak in the centre are systematically overestimated. At this temperature, the value of k,/k = c — 1.95 was obtained in the DQR fit. (Reproduced from Ref. 51. Copyright, 2001, with permission from Elsevier).

At ambient temperatures, the stochastic motions of methyl groups are generally fast on the NMR time scale. The only exceptions repotted so far involve the methyl group in 9-methyltriptycene derivatives (see Fig. 13) in which the torsional energy barriers concerned can be very high they can exceed 40 k.l/mol. In the latter compounds, the stochastic dynamics of the methyl rotor can be frozen out on the time-scale of NMR experiments in liquids. The liquid-phase NMR investigations on a number of 9-methyltriptycene derivatives, performed in the period up to the early 1990s, were reviewed by Oki. The experimental spectra of the methyl protons were interpreted in terms of the AB theory there were no mentions to the possible inadequacies of the AB model in the description of the observed line shapes. 

According to Ref. 51, the status of the AB approach in the low-temperature studies can be described as follows. Beyond the obvious instance where (i) c = kt/k approaches 1, use of the AB model is fully legitimate in the case where (ii) A —> oo (it is the latter instance that is addressed in the already mentioned line shape theory by Heuer" ). 

Fig. 17. The Arrhenius plots for the rate constants A (diamonds), A/3 (stars), and SAciass (fuU triangles) for 8, reported in Ref. 51. The arrow marks the singularity on the Adass (T) curve the occurrence of which is an indirect indication of the inadequacy of the AB model to the system investigated. On the other hand, the kifT) and kfT) curves are smooth. A non-Arrhenius behaviour of kifT) is remarkable.

In the latter paper, a throughout NMR study on a single crystal of acetylsalicylic acid selectively C-labelled in the methyl group is also reported. Despite the fact that in the temperature range investigated condition (ii) is perfectly fulfilled for the above material (at that range, the values of A approach 300 MHz), the observed spectra cannot be interpreted in terms of the AB model unless one admits that below 25 K the values of A ciass increase with further decrease of temperature. It must be stressed that use of the DQR instead of the AB model would offer no help to rationalize that strange behaviour in the limit of Id/Al 1, where d = max[ld( H- H)l, H)l], the calculated spectra are insensitive to... 

As far as experimental investigations of the ammonium ion dynamics are concerned, the present authors are aware of only two reports where both the coherent and stochastic effects are taken into account in the interpretation of the observed spectra. Both the reports involve H spectra of a single crystal of variously deuterated ammonium persulfate. For this material it was found that at temperatures below 40 K, the relevant dynamics are dominated by re-orientations about one of the possible re-orientation axes of the ND4 ions. Since it is a three-fold axis, the picture seen in the spectra is essentially similar to that for a deuterated methyl group. The experimental H spectra concerned could be fairly reproduced in terms of the AB model (see Fig. 19). 

Molecular dynamics studies of diatomic model detonations were first carried out by Karo and Hardy in 1977 [14]. They were soon followed by other groups [15, 16]. These early studies employed predissociative potentials, in which the reactant dimer molecules are metastable and can dissociate exothermically. More realistic models, combining an endothermic dissociation of reactants with an exothermic formation of product molecules, were introduced by White and colleagues at the Naval Research Laboratory and U.S. Naval Academy, first using a LEPS (London-Eyring-Polanyi-Sato) three-body potential for nitric oxide [17], and later a Tersoff-type bond-order potential [18] for a generic AB model, loosely based on NO [19, 20]. 

Tliis technique is commonly used for connecting PVC and ABS pipes, for acrylic in the manufacture of display items and gluing polystyrene and ABS model kits. 

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