Mathematical model

SF6 plasma composition

The calculation of the composition by a kinetics model has several aims: i) It enables an indirect check to be made of the calculation of the direct and the reverse rates by comparing the composition obtained with the results in the literature obtained by another method. This stage therefore validates our subroutine for the calculation of micro-reversibility which is then used throughout arc decay. ii) The composition obtained is tabulated versus temperature and enables the stationary state density field to be set as an initial condition for the 2-D model. iii) The equilibrium composition is used as a reference to study deviations from equilibrium during arc decay. iv) From the composition and the values of the reaction rates obtained in the stationary state, we can estimate the reaction rates and the relaxation times of the various chemical reactions. These data are useful for interpretation as they indicate the most probable reactions over a given temperature range.

Here, we present the assumptions made to calculate the composition with the kinetics model.The medium is homogeneous and in thermal equilibrium; the energy distribution functions of all species are Maxwellian; the reaction rates are solely determined by the mean stationary temperature; there are no external forces; the pressure is constant. The results presented here are for P = 10E5 Pa.

For temperatures between 2100 K and 12000 K, we considered 19 species: (e-, S, S-, S+, S2, S2+, F, F-, F+, F2, F2+, SF, SF-, SF+, SF4, SF5, SF6, SF2, SF3). In order to avoid excessive calculation times, the minor species, such as (SF5+, SF4+, SF3+, SF2+, F2+, S2+...) were ignored. Similarly, the negative ions (F2-, S2-, SF6-, SF5 and SF4-), which were present in very small amounts over the temperature range considered (2100 K < T < 12000 K) were not taken into account. A preliminary study of the reactions showed that these species are only weakly involved in electron capture processes. Sixty-six chemical reactions were taken into account and have been described in [1]. Most of the direct reaction rates proceed from reference [2], whereas the reverse rates were computed by micro-reversibility requiring the calculation of the partition functions.

The conservation equation for species 'i' is given by (1).The terms Cai and Dai describe the chemical reaction rates and were calculated previously [1, 3]. In equilibrium conditions, the creation term is equal to the loss term and the SF6 plasma composition is calculated by equation (1) which comes down to(2).The model is composed of 19 reaction rate equations. In fact these equations (2), written for the stationary state, are not linearly independent. Other relations exist to link the particle densities: the perfect gas law, electrical neutrality and stoichiometric equilibrium between S and F in the plasma. We thus obtained a table of plasma densities for temperatures between 12000 K and 2100 K with a step of 50 K.

Calculation gives the equilibrium composition which may be used to analyse the results of the kinetic model and to estimate the relaxation times (also called reaction time constants). As an example, figure 1 shows the variations of the equilibrium particle densities versus the temperature at atmospheric pressure (105 Pa). Between 12000 K and 4500 K, the electrons and the S+ ions constitute the majority of the charged species. Below 4500 K and down to 2500 K, S2+and F- are the major ions. It can be noted that the diatomic species SF et S2 are more abundant than F2 at temperatures between 3500 K and 6000 K. In order to validate our kinetic model, we compared the composition obtained with that calculated using a thermodynamic model [4]. We found a good agreement between the two series of results and with the results in the literature [2].

Figure 1: Variations of densities in SF6 plasma equilibrium.