Phoretic forces on convex particles from kinetic theory and nonequilibrium thermodynamics

Abstract

In this article we derive the phoretic forces acting on a tracer particle, which is assumed to be small compared to the mean free path of the surrounding nonequilibrium gas, but large compared to the size of the surrounding gas molecules. First, we review and extend the calculations of Waldmann ͓Z. Naturforsch. A 14A, 589 ͑1959͔͒ using half-sphere integrations and an accommodation coefficient characterizing the collision process. The presented methodology is applied to a gas subject to temperature, pressure, and velocity gradients. Corresponding thermophoretic, barophoretic, and rheophoretic forces are derived, and explicit expressions for spherical particles are compared to known results. Second, nonequilibrium thermodynamics is used to join the diffusion equation for the tracer particle with the continuum equations of nonisothermal hydrodynamics of the solvent. So doing, the distinct origin of the thermophoretic and barophoretic forces is demonstrated. While the latter enters similarly to an interaction potential, the former is given by flux-flux correlations in terms of a Green-Kubo relation, as shown in detail.

Unifying kinetic approach to phoretic forces and torques onto moving and rotating convex particles

“…The range of applicability of this assumption has been extensively discussed. 27 Further taking into account the uncrossability of the tracer particle it has been shown earlier 1,26 that the negative force onto a surface element, under the above mentioned assumptions, becomes 55…”

“…In Secs. II and III we provide background and extend the results obtained via kinetic theory 1 to include angular motion. The approach takes into account an accommodation coefficient ␣ characterizing the collision process ͑elastic versus diffusive, 0 ഛ ␣ ഛ 1, introduced in Ref.…”

“…1 Thermophoretic forces caused by a temperature gradient induce dust structures, are of central interest in the scavenging of aerosol particles in clouds, when droplets and/or ice crystals grow or evaporate, during the fall of hydrometeors, 2,3 influence the velocity measurements in particle image velocimetry measurements, 4 and have been discussed in the context of nonequilibrium thermodynamics, 1,5,6 and in terms of a kinetic description, [7][8][9][10] leading to a stochastic differential equation for a tracer particle in a solvent. 1,11 In contrast to thermophoretic forces, barophoretic effects caused by pressure gradients, 12,13 and rheophoretic forces caused by velocity field gradients have been considered less frequently and not yet elaborated in an extensive, complete, or unifying fashion for nonspherical tracers.…”

“…1 The same reference embedded the dynamics of the tracer particle into a wider description that encompasses the continuum equations for the nonisothermal hydrodynamics of the solvent. Beyond-equilibrium thermodynamics 22 had been used to examine the phoretic effects from a nonkinetic perspective.…”

“…17,18 In this article we will present explicit expressions for the diverse phoretic forces on dissolved ellipsoidal, cylindrical, and cuboidal tracer particles derived from forces on surface elements. 1 These results will be obtained via kinetic theory in a conceptually transparent manner for particles of arbitrary convex shape, and for arbitrary solvent velocity distribution functions by halfsphere integrations over solvent velocity distribution functions before and after a collision with the tracer particle. The simultaneous incorporation of thermo-, baro-, and rheophoresis is complete in the sense that it accounts for inhomogeneities in all of the independent field variables of a nonisothermal hydrodynamic description of the solvent, but our approach equally allows to study the "nonhydrodynamic" regime.…”