Schwarz model

In AOT microemulsions, where the aqueous core of the droplets also contains counterions, a considerable part of the dielectric response to the applied fields originates from the redistribution of the counterions. As mentioned in Sec. II, the counterions near th charged surface can be distributed between the Stem layer and the Gouy-Chapman diffuse double layer (28-31). The distribution of counterions is essentially determined by their concentration and the geometry of the water core. Thus, for very large droplets the diffuse double layer peters out and the polarization can be described by the Schwarz model (32). However, as already mentioned, this approach is more relevant to the dielectric behavior of emulsions than to that of micro emulsions. 

Pacchioni has recently carried out calculations on the low-lying states of Sn2 and Pb2. This author gives the impression that he is the first to carry out a comparative ab initio Cl calculation on these systems. We would like to clarify this further. First, his calculation starts with the Hafner-Schwarz model potentials in comparison to our relativistic ab initio potentials derived from numerical Dirac-Fock solutions of the atoms. Pacchioni s calculations ignore spin-orbit interaction. Our calculations include spin-orbit interaction in a relativistic Cl scheme in comparison to the non-relativistic Cl of Pacchioni. Thus, he obtains a Z), approximately twice the experimental value which he corrects by a semi-empirical scheme to arrive at a value close to our calculated value with a relativistic Cl. Our calculations have clearly demonstrated the need to carry out an intermediate-coupling Cl calculation for Pbj as a result of large spin-orbit contamination. Calculations without spin-orbit, such as Pacchioni s, have little relationship to the real Pb2 molecule. 

Ludden TM, Beal SL, Sheiner LB. Comparison of the Akaike Information Criterion, the Schwarz criterion and the F test as guides to model selection. /PAar-macokinet Biopharm 1994 22 431-45. 

In 1990, Schroder and Schwarz reported that gas-phase FeO" " directly converts methane to methanol under thermal conditions [21]. The reaction is efficient, occuring at 20% of the collision rate, and is quite selective, producing methanol 40% of the time (FeOH+ + CH3 is the other major product). More recent experiments have shown that NiO and PtO also convert methane to methanol with good efficiency and selectivity [134]. Reactions of gas-phase transition metal oxides with methane thus provide a simple model system for the direct conversion of methane to methanol. These systems capture the essential chemistry, but do not have complicating contributions from solvent molecules, ligands, or multiple metal sites that are present in condensed-phase systems. 

The second assumption has been effectively invalidated by the discovery of the hydrated electron. However, the effects of LET and solute concentration on molecular yields indicate that some kind of radical diffusion model is indeed required. Kuppermann (1967) and Schwarz (1969) have demonstrated that the hydrated electron can be included in such a model. Schwarz (1964) remarked that Magee s estimate of the distance traveled by the electron at thermalization (on the order of a few nanometers) was correct, but his conjecture about its fate was wrong. On the other hand, Platzman was correct about its fate—namely, solvation—but wrong about the distance traveled (tens of nanometers). 

Discovery of the hydrated electron and pulse-radiolytic measurement of specific rates (giving generally different values for different reactions) necessitated consideration of multiradical diffusion models, for which the pioneering efforts were made by Kuppermann (1967) and by Schwarz (1969). In Kuppermann s model, there are seven reactive species. The four primary radicals are eh, H, H30+, and OH. Two secondary species, OH- and H202, are products of primary reactions while these themselves undergo various secondary reactions. The seventh species, the O atom was included for material balance as suggested by Allen (1964). However, since its initial yield is taken to be only 4% of the ionization yield, its involvement is not evident in the calculation. 

Schwarz s model is a multiradical extension of the Ganguly-Magee model with some additional improvements, to be described later. Schwarz assumes that initially—that is, 10 11 s after the act of energy deposition in water—there appear five species, namely eh, H, OH, H30+, and H2. Their initial yields, indicated by superscript zero, are related by charge conservation and material balance. Thus, there are three independent initial yields, taken to be those of eh, H, and Hr The initial yield of H2 is identified with the unscavengable molecular hydrogen yield. No mechanism of its production is speculated, except that it is not formed by radical recombination. For the gaussian distribution of the radicals, two initial... 

TABLE 7.1 Reaction Scheme in Schwarz s Diffusion Model... 

While fitting five adjustable parameters to four sets of experimental data may not seem surprising, the strength of the diffusion model lies in predicting a much wider body of experimental results. Of these, the most important are the variations of molecular yields with LET and solute concentration. Since these calculated variations agree quite well with experiment, no further comment is necessary except to note that calculations often require normalization, so that only relative yields can be compared with experiment. One main reason is that the absolute yields often differ from laboratory to another for the same experiment. Thus, Schwarz s theoretical predictions have reasonable normalization constants, which, however, are not considered as new parameters. In the next subsection, we will consider some experimental features that could possibly be in disagreement with the diffusion model. 

Four observation were thought to be in disagreement with the diffusion model (1) the lack of a proportional relationship between the electron scavenging product and the decrease of H2 yield (2) the lack of significant acid effect on the molecular yield of H2 (3) the relative independence from pH of the isotope separation factor for H2 yield and (4) the fact that with certain solutes the scavenging curves for H2 are about the same for neutral and acid solutions. Schwarz s reconciliation follows. 

Mahlman and Sworski (1967) found that curves for scavenging of H2 by NOj- have nearly the same shape for neutral and 0.1 M acid solutions, over the concentration range 1 mM to 0.4 M of the solute, despite the fact that nitrate reacts much faster with eh than with H. Schwarz points out, however, that when two solutes (H+ and N03-) compete for the same intermediate (eh), the extrapolation to zero scavenger concentration is not a valid procedure. The calculation of the diffusion model agrees with experiment over the entire N03- concentration range. Further, the model predicts a lower H2 yield at very low scavenger concentrations. 

It has been suggested that a sensitive test of the diffusion model would be found in the evolution of the eh yield (Schwarz, 1969). Early measurements by Hunt and Thomas (1967) and by Thomas and Bensasson (1967) revealed -6% decay within the first 10 ns and 15% decay in 50 ns. The diffusion theory of Schwarz predicts a very substantial decay ( 30%) in the first nanoseconds for instantaneous energy deposition. Schwarz (1969) tried to mitigate the situation by first integrating over pulse duration (-4.2 ns) and then over the detector response time (-1.2 ns). This improved the agreement between theory and experiment somewhat, but a hypothesis of no decay in this time scale would also agree with experiment. Thus, it was decided that a crucial test of the diffusion theory would... 

The first subnanosecond experiments on the eh yield were performed at Toronto (Hunt et al., 1973 Wolff et al., 1973). These were followed by the subnanosecond work of Jonah et al. (1976) and the subpicosecond works of Migus et al. (1987) and of Lu et al. (1989). Summarizing, we may note the following (1) the initial (-100 ps) yield of the hydrated electron is 4.6 0.2, which, together with the yield of 0.8 for dry neutralization, gives the total ionization yield in liquid water as 5.4 (2) there is -17% decay of the eh yield at 3 ns, of which about half occurs at 700 ps and (3) there is a relatively fast decay of the yield between 1 and 10 ns. Of these, items (1) and (3) are consistent with the Schwarz form of the diffusion model, but item (2) is not. In the time scale of 0.1-10 ns, the experimental yield is consistently greater than the calculated value. The subpicosecond experiments corroborated this finding and determined the evolution of the absorption spectrum of the trapped electron as well. 

Following Schwarz (1969), we may draw the following conclusions regarding the diffusion model of water radiolysis ... 

Lane, G. L., Schwarz, M. P., and Evans, G. M., Modelling of the Interaction Between Gas and Liquid in Stirred Vessels . Proceedings of the 10th European Conference on Mixing, Delft, Netherlands, 197-204 (2000). 

The values of the ESP at the nuclear positions, as obtained from the electron and Hartree-Fock structure amplitudes for the mentioned crystals (using a K-model and corrected on self-potential) are given in table 2. An analysis shows that the experimental values of the ESP are near to the ab initio calculated values. However, both set of values in crystals differ from their analogs for the free atoms [5]. It was shown earlier (Schwarz M.E. Chem. Phys. Lett. 1970, 6, 631) that this difference in the electrostatic potentials in the nuclear positions correlates well with the binding energy of Is-electrons. So an ED-data in principle contains an information on the bonding in crystals, which is usually obtaining by photoelectron spectroscopy. 

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