Thermophoretic force and velocity of nanoparticles in the free molecule regime

We extend our previous gas-kinetic theory analysis of drag force in a uniform temperature field [Li and Wang, Phys. Rev. E. 68, 061206 (2003); 68, 061207 (2003)] to particle transport in fluids with nonuniform temperature. Formulations for drag and thermophoretic forces are proposed for nanoparticle transport in low-density gases. We specifically consider the influence of nonrigid body collision due to van der Waals or other forces between the particle and gas molecules and find that these forces play a notable role for particles a few nanometers in size. It is shown that the present formulations can be easily reduced to the classical result of Waldmann [Z. Naturforsch. A 14a, 589 (1959)] by assuming rigid body collision. From the force formulations we also obtain the equation governing the thermophoretic velocity. This velocity is found to be highly sensitive to the potential energy of interactions between gas molecules and particle, and as such Waldmann's thermophoretic velocity is not expected to be accurate for nanosized particles.