Kinetic Theory of Soft Matter. The Penetrable-Sphere Model

The penetrable-sphere model has been introduced in the literature to describe the peculiar thermodynamic behavior of some colloidal systems. In this model the interaction potential is $\phi(r)=\epsilon>0$ if the two spheres are overlapped ($r\sigma$). In this paper the shear viscosity, thermal conductivity, and self-diffusion coefficients of a dilute gas of penetrable spheres are evaluated. It is found that the effective collision frequency $\nu(T^*)$ grows as $\sqrt{T^*}$ up to $T^*\equiv k_BT/\epsilon\simeq 0.25$, reaches a maximum at $T^*\simeq 0.415$ and then decays as ${T^*}^{-3/2}\log T^*$ for large temperatures. The results are applied to the hydrodynamic profiles in the steady Fourier and Couette flows.Comment: 6 pages, 4 figures; to appear in Rarefied Gas Dynamics: 24th International Symposium (AIP Conference Proceedings

A simple model kinetic equation for inelastic Maxwell particles

The model of inelastic Maxwell particles (IMP) allows one to derive some exact results which show the strong influence of inelasticity on the nonequilibrium properties of a granular gas. The aim of this work is to propose a simple model kinetic equation that preserves the most relevant properties of the Boltzmann equation (BE) for IMP and reduces to the BGK kinetic model in the elastic limit. In the proposed kinetic model the collision operator is replaced by a relaxation-time term toward a reference Maxwellian distribution plus a term representing the action of a friction force. It contains three parameters (the relaxation rate, the effective temperature of the reference Maxwellian, and the friction coefficient) which are determined by imposing consistency with basic exact properties of the BE for IMP. As a consequence, the kinetic model reproduces the true shear viscosity and predicts accurate expressions for the transport coefficients associated with the heat flux. The model can be exactly solved for the homogeneous cooling state, the solution exhibiting an algebraic high-energy tail with an exponent in fair agreement with the correct one.Comment: 6 pages, 2 figures; presented in the 25th International Symposium on Rarefied Gas Dynamics (Saint-Petersburg, Russia, July 21-28, 2006

Poiseuille flow in a heated granular gas

We consider a dilute gas of inelastic hard spheres enclosed in a slab under the action of gravity along the longitudinal direction. In addition, the gas is subject to a white-noise stochastic force that mimics the effect of external vibrations customarily used in experiments to compensate for the collisional cooling. The system is described by means of a kinetic model of the inelastic Boltzmann equation and its steady-state solution is derived through second order in gravity. This solution differs from the Navier-Stokes description in that the hydrostatic pressure is not uniform, normal stress differences are present, a component of the heat flux normal to the thermal gradient exists, and the temperature profile includes a positive quadratic term. As in the elastic case, this new term is responsible for a bimodal shape of the temperature profile. The results show that, except for high inelasticities, the effect of inelasticity on the profiles is to slightly decrease the quantitative deviations from the Navier-Stokes results.Comment: 18 pages, 5 figures; minor changes; to be published in JS

Low-temperature and high-temperature approximations for penetrable-sphere fluids. Comparison with Monte Carlo simulations and integral equation theories

The two-body interaction in dilute solutions of polymer chains in good solvents can be modeled by means of effective bounded potentials, the simplest of which being that of penetrable spheres (PSs). In this paper we construct two simple analytical theories for the structural properties of PS fluids: a low-temperature (LT) approximation, that can be seen as an extension to PSs of the well-known solution of the Percus-Yevick (PY) equation for hard spheres, and a high-temperature (HT) approximation based on the exact asymptotic behavior in the limit of infinite temperature. Monte Carlo simulations for a wide range of temperatures and densities are performed to assess the validity of both theories. It is found that, despite their simplicity, the HT and LT approximations exhibit a fair agreement with the simulation data within their respective domains of applicability, so that they complement each other. A comparison with numerical solutions of the PY and the hypernetted-chain approximations is also carried out, the latter showing a very good performance, except inside the core at low temperatures.Comment: 14 pages, 8 figures; v2: some figures redone; small change