Modelling and numerical methods for nanoparticles in a gas phase
Mathematics
Technical Thermodynamics
Final Report Abstract
The project aimed at modeling and simulation of the transport of nanoparticles in a surrounding gas. Due to the small size of the particles, the gas has to be modelled by kinetic equations, specifically the Boltzmann equation. In the first part of the project, the thermophoresis and thermophoretic orientation of complex nanoparticles has been studied. The problem was reduced to its essential building blocks by considering a Janus particle consisting of two hemispheres with different momentum accommodation coefficients in a temperature gradient. The force and the torque on the particle were computed in the free-molecular flow regime, assuming either a Chapman-Enskog distribution with a temperature gradient or a particle arranged between parallel plates of different temperatures. The drift velocity of the particle was obtained by balancing the thermophoretic force and the drag force on the particle. For not too large temperature gradients, the distribution function for the particle orientation can be obtained by assigning a mean temperature to the particle and deriving a potential energy function depending on the Euler angles. The distribution function for the particle orientation is then given as the Boltzmann distribution corresponding to this potential energy function. A first step in supplementing the analytical results by numerical simulations was the formulation of a Langevin equation for the rotational degrees of freedom. The motion is driven by the torque resulting from the collisions of gas molecules with the sphere, supplemented by a stochastic term representing thermal fluctuations and a damping term. For the case that the Knudsen number is no longer large, i.e. in the transition flow regime, it will no longer be possible to evaluate the torque and the rotational resistance analytically. In that case numerical simulations of the Boltzmann equation have been performed to determine these quantities. A comparison between the analytically computed distribution function for the particle orientation and the results obtained with the Boltzmann solver shows a reasonably good agreement for a Knudsen number of 10, while deviations become visible at Kn = 1. A main result is that a Janus particle with different momentum accommodation coefficients on its two hemispheres assumes a preferred orientation in a temperature gradient, that is, the side with the higher momentum accommodation coefficient points towards the colder region. The translation of the particle, i.e. its thermophoretic motion, counteracts this effect by the torque due to the temperature gradient. These phenomena could be of practical relevance in gas phase processes where nanoparticles are exposed to large temperature gradients, for example when nanoparticles are deposited on a cold surface. In the second part of the project, the translational motion of a spherical particle immersed in a rarefied gas with varying distance to a planar wall was investigated. In the general case, the rarefied gas was simulated by solving the Boltzmann equation. For the case of free molecular flow, results based on Fredholm integral equations were obtained. For the latter, analytical expressions for the momentum transfer due to the bombardment of molecules on the surface of the sphere were derived as a function of the velocity of the sphere and its distance from the wall. The momentum transfer includes molecules originating from the planar surface and from the ambient gas far away from the surface. The integral equations were solved numerically based on a discretization of the particle surface and the plane. The numerical results for arbitrary Knudsen numbers were obtained using DSMC simulations of the Boltzmann equation. Three cases were considered. The first one is the motion of a sphere in an unbounded domain. The other cases are the motion of a sphere moving parallel and normal to a planar wall, corresponding to the problem under study. In the free molecular flow regime, a very good agreement between the results obtained by solving the Fredholm integral equations and the Boltzmann equation is obtained. It was observed that while in the case of continuum flow the drag force increases steeply when the distance to the wall decreases, the increase of the drag force is much more moderate in the free molecular flow regime. It was also observed that the solution of the Fredholm integral equations still agrees with the DSMC results obtained for Kn = 10 quite well. Even at a Knudsen number of 1 the drag force only increases slowly when the distance to the wall decreases. Overall, the conclusion is that for a particle immersed in a rarefied gas close to a surface, the drag coefficient increases much more slowly than in the continuum regime when the distance to the wall decreases.
Publications
- Thermophoresis of Janus particles at large Knudsen numbers, Phys. Rev. Fluids, 3, 094202, 2018
T. Baier, S. Tiwari, S. Shrestha, A. Klar, S. Hardt
(See online at https://doi.org/10.1103/PhysRevFluids.3.094202) - A meshfree method for the BGK model for rarefied gas dynamics, Int. J. Adv. Eng. Sci. Appl. Math., 11(3), 187-197, 2019
S. Tiwari, A. Klar, G. Russo
(See online at https://doi.org/10.1007/s12572-019-00254-5) - Drag force on spherical particle moving near a plane wall in highly rarefied gas, J. Fluid Mech., 883, A47, 2020
P. Goswami, T. Baier, S. Tiwari, C. Lv, S. Hardt and A. Klar
(See online at https://doi.org/10.1017/jfm.2019.921) - Interaction of rigid body motion and rarefied gas dynamics based on the BGK model, Mathematics in Engineering, 2,2, 203-229, 2020
S. Tiwari, A. Klar, G. Russo
(See online at https://doi.org/10.3934/mine.2020010)
