A laser-Doppler velocimeter (LDV) study of velocity profiles in the laminar boundary layer adjacent to a heated flat plate revealed that the seed particles used for the LDV measurements were driven away from the plate surface by thermophoretic forces, causing a particle-free region within the boundary layer of approximately one half the boundary-layer thickness. Measurements of the thickness of this region were compared with particle trajectories calculated according to several theories for the thermophoretic force. It was found that the theory of Brock, with an improved value for the thermal slip coefficient, gave the best agreement with experiment for low Knudsen numbers, λ/R = O(10−1), where λ is the mean free path and R the particle radius.Data obtained by other experimenters over a wider range of Knudsen numbers are compared, and a fitting formula for the thermophoretic force useful over the entire range 0 [les ] λ/R [les ] ∞ is proposed which agrees within 20% or less with the majority of the available data.

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Thermophoresis of particles in a heated boundary layer

Experimental evidence of thermophoresis in liquids is reported for 1.011 and 0.790 μm latex spheres in water and n-hexane. The thermophoretic velocity, vth, was related to the particle and liquid properties by an expression similar to Epstein's gas equation: V th =-0.26 kμ 2k+k p ρT ▽T .

Thermophoresis in liquids

A revision of Newton's law of viscosity appearing in the role of the deviatoric stress tensor in the Navier–Stokes equation is proposed for the case of compressible fluids, both gaseous and liquid. Explicitly, it is hypothesized that the velocity v appearing in the velocity gradient term ∇ v in Newton's rheological law be changed from the fluid's mass-based velocity v m , the latter being the velocity appearing in the continuity equation, to the fluid's volume velocity v v , the latter being a stand-in for the fluid's volume current (volume flux density n v ). A similar v m → v v alteration is proposed for the velocity v appearing in the no-slip tangential velocity boundary condition at solid surfaces. These proposed revisions are based upon both experiment and theory, including re-interpretation of the following three items: (i) experimental “near-continuum” thermophoretic and other low Reynolds number phoretic data for the movement of suspended particles in fluids under the influence of mass density gradients ∇ ρ , caused either by temperature gradients in single-component fluids undergoing heat transfer or by species concentration gradients in inhomogeneous two-component mixtures undergoing mass transfer; (ii) the hierarchical re-ordering of the Burnett terms appearing in the Chapman–Enskog gas-kinetic theory perturbation expansion of the viscous stress tensor from one of being based upon small Knudsen numbers to one of being based upon small Mach numbers; (iii) Maxwell's (1879) ubiquitous v m -based “thermal creep” or “thermal stress” slip boundary condition used in nonisothermal gas-kinetic theory models, recast in the form of a v v -based no-slip condition. The v v vs. v m dichotomy in the case of compressible fluids is shown to lead to a fundamental distinction between the fluid's tracer velocity as recorded by monitoring the spatio-temporal trajectory of a small non-Brownian particle deliberately introduced into the fluid, and the fluid's “optical” or “colorimetric” velocity as monitored, for example, by the introduction of a dye into the fluid or by some photochromic- or fluorescence-based scheme in circumstances where the individual fluid molecules are themselves responsive to being probed by light. Explicitly, it is argued that the fluid's tracer velocity, representing a strictly continuum nonmolecular notion, is v v , whereas its colorimetric velocity, which measures the mean velocity of the molecules of which the fluid is composed, is v m .

Navier–Stokes revisited

Laser-based tractor beams move hollow glass particles over tens of centimetres. The laser beam has become an indispensable tool for the controllable manipulation and transport of microscopic objects in biology, physical chemistry and condensed matter physics. In particular, ‘tractor’ laser beams can draw matter towards a laser source and perform, for instance, all-optical remote sampling. Recent advances in lightwave technology have already led to small-scale experimental demonstrations of tractor beams1,2,3,4. However, the realization of long-range tractor beams has not gone beyond the realm of theoretical investigations5,6,7,8,9. Here, we demonstrate the stable transfer of gold-coated hollow glass spheres against the power flow of a single inhomogeneously polarized laser beam over tens of centimetres. Additionally, by varying the polarization state of the beam we can stop the spheres or reverse the direction of their motion at will.

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A long-range polarization-controlled optical tractor beam

The chapter deals with hot Brownian particles and swimmers as two examples for Brownian motion very far from equilibrium. Thanks to the strong scale separation between the Brownian particles and the solvent atoms, substantial theoretical progress could be made along the lines first laid out by Einstein, yielding exact analytical predictions for hot Brownian dynamics by coarse graining. Wherever these predictions were tested, they were found in excellent agreement with experimental observations and simulation data. The chapter is as well dealing with the steering of hot swimmers by Maxwell-demon type methods summarily known as photon nudging. They enable experimentalists to tailor their interactions and to create micron-sized swarms of active particles serving as a microscopic laboratory for studying large-scale biological phenomena.

Hot Brownian Motion

Theory of the photophoretic motion of the large-size volatile aerosol particle

Theory of thermophoresis of large aerosol particles

respectively, where R is the radius of the aerosol particle and X~ is the mean free path of the molecules of the i th component of the gas mixture. The theory of diffusiophoresis (and thermophoresis) of small aerosol particles was developed by Derjaguin and Bakanov (1, 2) taking into consideration the fact tha t the velocity distribution of the molecules impinging upon the particle is not disturbed by the presence of an aerosol particle. By the use of the expressions obtained by Chapman and Enskog (3) for the velocity distribution of the molecules in the volume of the gas, the following formula ~ was obtained in reference 1:

Diffusiophoresis of large aerosol particles

A theory is derived for the photophoretic motion of a small aerosol particle, whose radius is much smaller than the wavelength of the light. The radiation pressure is taken into account; the effects of the incomplete accommodation of the energy and momentum of the gas molecules in collisions with the surface of the particle are also taken into account.

Yu. I. Yalamov

Papers

Theory of the photophoretic motion of the large-size volatile aerosol particle

Theory of thermophoresis of large aerosol particles

Diffusiophoresis of large aerosol particles

Motion of small aerosol particle in a light field

Authors' reply to Brock's discussion remarks on the theory of thermophoresis of large aerosol particles