An atmospheric pressure chemical ionization study of the positive and negative ion chemistry of the hydrofluorocarbons 1,1-difluoroethane (HFC-152a) and 1,1,1,2-tetrafluoroethane (HFC-134a) and of perfluoro-n-hexane (FC-72) in air plasma at atmospheric pressure

A report is given on the ionization/dissociation behavior of the title compounds within air plasmas produced by electrical corona discharges at atmospheric pressure: both positive and negative ions were investigated at different temperatures using atmospheric pressure chemical ionization mass spectrometry (APCI-MS). CHF(2)CH(3) (HFC-152a) undergoes efficient ionic oxidation to C(2)H(5)O(+), in which the oxygen comes from water present in the plasma. In contrast, CF(3)CH(2)F (HFC-134a) does not produce any characteristic positive ion under APCI conditions, its presence within the plasma being revealed only as a neutral ligand in ion-molecule complexes with ions of the background (H(3)O(+) and NO(+)). Analogously, the perfluorocarbon FC-72 (n-C(6)F(14)) does not produce significant positive ions at 30 degrees C: at high temperature, however, it undergoes dissociative ionization to form many product ions including C(3)F(6)(+), C(2)F(4)(+), C(n)F(2n+1)(+) and a few families of oxygen containing cations (C(n)F(2n+1)OH(2)(+), C(n)F(2n)OH(+), C(n)F(2n-1)O(+), C(n)F(2n-1)O(2)H(2)(+), C(n)F(2n-2)O(2)H(+)) which are suggested to derive from C(n)F(2n+1)(+) in a cascade of steps initiated by condensation with water followed by steps of HF elimination and H(2)O addition. Negative ions formed from the fluoroethanes CHF(2)CH(3) and CF(3)CH(2)F (M) include complexes with ions of the background, O(2)(-)(M), O(3)(-)(M) and some higher complexes involving also water, and complexes of the fluoride ion, F(-)(H(2)O), F(-)(M) and higher complexes with both M and H(2)O also together. The interesting product O(2)(-)(HF) is also formed from 1,1-difluoroethane. In contrast to the HFCs, perfluoro-n-hexane gives stable molecular anions, M(-), which at low source temperature or in humidified air are also detected as hydrates, M(-)(H(2)O). In addition, in humidified air F(-)(H(2)O)(n) complexes are also formed. The reactions leading to all major positive and negative product ions are discussed also with reference to available thermochemical data and relevant literature reports. The effects on both positive and negative APCI spectra due to ion activation via increasing V(cone) are also reported and discussed: several interesting endothermic processes are observed under these conditions. The results provide important information on the role of ionic reactions in non-thermal plasma processes.     

Metrics for probability distributions and the trend to equilibrium for solutions of the Boltzmann equation

This paper deals with the trend to equilibrium of solutions to the spacehomogeneous Boltzmann equation for Maxwellian molecules with angular cutoff as well as with infinite-range forces. The solutions are considered as densities of probability distributions. The Tanaka functional is a metric for the space of probability distributions, which has previously been used in connection with the Boltzmann equation. Our main result is that, if the initial distribution possesses moments of order 2+, then the convergence to equilibrium in his metric is exponential in time. In the proof, we study the relation between several metrics for spaces of probability distributions, and relate this to the Boltzmann equation, by proving that the Fourier-transformed solutions are at least as regular as the Fourier transform of the initial data. This is also used to prove that even if the initial data only possess a second moment, then _v>R f(v, t) v² dv0 asR, and this convergence is uniform in time.     

Asymptotic approximation of a highly oscillatory integral with application to the canonical catastrophe integrals

We consider the highly oscillatory integral $F(w):={\int }_{-\infty }^{\infty }{e}^{iw(t^{K+2}+e^{i\theta }{t}^p)}g(t)dt$ for large positive values of w, , K and p positive integers with $1\le p\le K$, and $g(t)$ an entire function. The standard saddle point method is complicated and we use here a simplified version of this method introduced by López et al. We derive an asymptotic approximation of this integral when $w\to +\infty$ for general values of K and p in terms of elementary functions, and determine the Stokes lines. For $p\ne 1$, the asymptotic behavior of this integral may be classified in four different regions according to the even/odd character of the couple of parameters K and p; the special case $p=1$ requires a separate analysis. As an important application, we consider the family of canonical catastrophe integrals ${\mathrm{\Psi }}_K(x_1,x_2,\text{…},x_K)$ for large values of one of its variables, say $x_p$, and bounded values of the remaining ones. This family of integrals may be written in the form $F(w)$ for appropriate values of the parameters w, θ and the function $g(t)$. Then, we derive an asymptotic approximation of the family of canonical catastrophe integrals for large $|x_p|$. The approximations are accompanied by several numerical experiments. The asymptotic formulas presented here fill up a gap in the NIST Handbook of Mathematical Functions by Olver et al.