A moment method with closures based on Gaussian quadrature formulas is proposed to solve the Boltzmann kinetic equation with a hard-sphere collision kernel for mono-dispersed particles. Different orders of accuracy in terms of the moments of the velocity distribution function are considered, accounting for moments up to seventh order. Quadrature-based closures for four different models for inelastic collision-the Bhatnagar-Gross-Krook, ES-BGK, the Maxwell model for hard-sphere collisions, and the full Boltzmann hard-sphere collision integral-are derived and compared. The approach is validated studying a dilute non-isothermal granular flow of inelastic particles between two stationary Maxwellian walls. Results obtained from the kinetic models are compared with the predictions of molecular dynamics (MD) simulations of a nearly equivalent system with finite-size particles. The influence of the number of quadrature nodes used to approximate the velocity distribution function on the accuracy of the predictions is assessed. Results for constitutive quantities such as the stress tensor and the heat flux are provided, and show the capability of the quadrature-based approach to predict them in agreement with the MD simulations under dilute conditions.

The kinetics of the crystallization of metals from the melt under superfast cooling velocities realized in the treatment of the materials by ultrashort laser pulses is investigated. The size distribution function for crystalline phase nuclei is analytically found within the framework of the classical kinetic equation in case of superfast temperature changing. The average number of particles in the crystalline nuclei and relative volume of the crystalline phase are determined as functions of thermodynamic and laser treatment regime parameters. Good agreement is observed with experimental results for ultrashort laser pulses induced micro- and nanostructures production.