利用者:Komeet92/sandbox

2.2. Particle thermophoresis in gases Thermally driven transport in aerosols has been known since the seminal observation by Tyndall [18] who, by visually observing the light scattered by hazes, noticed that the suspended dust particles tend to avoid hot surfaces. Although a deep connection to the Soret effect obviously exists, the theoretical analysis of particle thermophoresis in gases proceeded along a totally independent route (with the noticeable exception of the work by Chapman [19]). The basic mechanisms underlying thermophoresis in gases are very different, depending on the ratio Kn = λ/a between the mean free path λ and the particle size a (the Knudsen number). When (very low density gases), the problem is actually very similar to thermal diffusion in a gas mixture where one of the two components (the particles) has an exceedingly large size compared to the other. We are more interested in the opposite limit, (gas at moderate pressure), which is properly called the 'quasi-hydrodynamic regime'. The basic ideas allowing us to understand thermophoresis in this regime stem from the last paper by Maxwell [20], where stresses in rarefied gases arising from inequalities of temperature are thoroughly investigated5. The key result of this paper is that, in a homogeneous gas, no longitudinal (pressure) or transverse stresses are associated with temperature gradients. The situation is however very different when a bounding solid surface is present. Let us indeed assume that the gas is bounded by a planar surface S, with a temperature gradient parallel to S, and consider those molecules that lie within a mean free path λ from S (and therefore suffer no molecular collision before hitting the wall). Since the impacting molecules are not specularly reflected (outgoing molecules have a partially random momentum distribution, due to thermalization with S), each molecule transfers a momentum Δp to the wall. A careful evaluation of the total momentum exchange requires however taking into account corrections to the equilibrium distribution f0(v) of the molecular speed, which must be included to account for dissipative processes. From the Boltzmann equation, one finds at first order [22]

m is the molecular mass, C is a normalization constant, and the thermal gradient is taken along z. If this is made, the total rate of momentum transfer turns out to be larger for those molecules coming from the hot side: therefore a net longitudinal momentum transfer takes place, pulling Stowards the cold side. Using Maxwell's result, Epstein [23] was able to calculate the steady-state thermophoretic velocity acquired by particle with thermal conductivity κp, embedded in a gas having thermal conductivity κg, viscosity η, and number density ρ:

Since Epstein's seminal contribution, extensive theoretical and experimental work on thermophoresis in gases has been performed (as general reviews, see, for instance [24–26]), which even spurred the development of widely used thermophoretic soot sampling methods [27]. Furthermore, the intimate connection between thermophoresis and thermal diffusion in gas mixtures has been thoroughly investigated [28]. For our purposes, however, it is sufficient to stress some basic features of the mechanism driving thermophoresis in the quasi-hydrodynamic regime. (i) The gas exerts on the surface a purely tangential stress. Therefore, within a surface layer with a thickness of the order of λ, the pressure tensor is anisotropic. (ii) Because onlytangential stresses are involved, even if it is a surface effect, the total force on the particle FT scales only with a, and not a2 (so that vT = FT/f, where f is the friction coefficient, is size independent6). (iii) In hydrodynamic terms, vT can eventually be seen as a slip velocity of the particle, with a slip length . (iv) Particle bulk properties enter the problem only through the thermal conductivity κp, that, together with κg, determines the local temperature field around the particle via the heat equation.