Introduction: The classical kinetic theory models were the obvious choice for early DSMC work, but were found to have serious shortcomings that have been overcome by phenomenological models that have been introduced in the context of the DSMC method.

   For elastic collisions, the classical hard sphere and Maxwell molecules lead to fixed and unrealistic temperature variations of the coefficient of viscosity and should not be used for problems that involve large variations in temperature.  The power- law models are physically realistic and accurately predict the transport properties.  On the other hand, they are computationally difficult and, because an arbitrary cut-off must be applied to the effective molecular radius, they do not lead to unambiguous values for the collision rate and mean free path.  The developers of kinetic theory had attributed the success of the power-law models to the realistic modeling of the scattering distribution in collisions and made little or no mention of the variation with temperature of the effective molecular size that was implicit in their equations.  However, as real gas flows were more intensively studied, it was found that the measurable effects of molecular model correlate almost exactly with the changes in the effective molecular size and that the scattering law has a relatively small effect.  This led to the introduction of the variable hard sphere, or VHS, model which combines the unrealistic but simple isotropic scattering of a hard sphere with a realistic variation of diameter with the relative speed of the collision pair.  This model leads to accurate results for viscosity and heat conduction coefficients, but the Schmidt number that depends on the diffusion coefficient does not match the measured values in real gases.  The variable soft sphere, or VSS, model includes an empirical distortion of the isotropic scattering that can be adjusted to also match the measured Schmidt number.  The VSS extension is required only for those applications to gas mixtures in which diffusion is important factor.

   Classical kinetic theory did not lead to any useful models for inelastic collisions.  For example, the rough sphere model retains the deficiencies of the hard sphere model with regard to the transport properties, it is unable to deal with the quantum effects that cause most gases to have fewer than three rotational degrees of freedom and has a fixed and unrealistically fast rotational relaxation time.  This problem was solved in the context of the DSMC method by the introduction of the Larsen-Borgnakke model for rotation which can be regarded as the archetypal phenomenological procedure.  It is an add-on to the VHS model and, for a fraction of the collisions that is chosen to match the measured rotational relaxation rate, the post-collision rotational energies are selected from the equilibrium distribution that corresponds to the collision energy.  The original L-B model was extended to the vibrational modes, but it assumed that the internal energies were continuously distributed and was barely satisfactory.  This situation was transformed by the introduction  of the quantum version of the Larsen-Borgnakke model for vibration.

New molecular models for DSMC

The extension of the quantum vibration model  to include dissociation was first presented at the RGD26 meeting as a physically realistic extension of the model.  The results were so encouraging that an attempt was made to develop phenomenological models for the other reactions.  That for the endothermic exchange and chain reactions is analogous to the dissociation condition, but has only recently evolved to a fully satisfactory state.  The new chemisty model was termed the Quantum-Kinetic or Q-K model in my RGD27 paper, but the procedures for exothermic reactions remained unsatisfactory.   Intensive work over the past year has led to a fully satisfactory theory.

The implementation of the Q-K model is incomparably faster than that of the existing TCE model.

Unlike the TCE model, The Q-K model predicts rather than depends on rate equations.

Unlike the TCE model, the Q-K model does not assume equilibrium distribution functions.

The latest developments with regard to Q-K model have been described in a recent paper The Q-K model for gas-phase chemical reaction rates that has recently been published in the Physics of Fluids.

NOTE: There is a typo in the text following Eq. (5) in that the symmetry factor is 1 for unlike and 2 for like molecules.  Also, in the first paragraph in the second column of the following page, imax should be i*.

The supplementary material for the paper included an interactive graphical program for the evaluation of the parameters that ensure that the Q-K rates are consistent with the equilibrium constant.   The program can also be used to compare the Q-K rate equations with those in the existing reaction databases.