## Abstract

This Letter presents a study on photophoresis of a microparticle in gaseous media with focus on the effect of thermal stress slip, which is deemed as one of factors causing deviations of previous theoretical predictions from measurements. The present modified theory agrees well with the measurements and, combining with 1 order-of-magnitude analysis, demonstrates the significance of the thermal stress slip in photophoresis of a particle, especially, of small radius. With the physical mechanisms addressed, the parametric analysis reveals that this interfacial thermal effect becomes more pronounced with reducing thermal conductivity of the particle and increasing Knudsen number as well.

© 2010 Optical Society of America

A particle in a gaseous medium subjected to an intensive light beam absorbs and scatters light and then turns the absorbed electromagnetic energy into thermal energy within the particle. Asymmetric energy distribution and in turn uneven heating at the gas-particle interface lead to a force driving the particle into photophoretic motion. To deal with this subject in rarefied gas, the Knudsen number ($Kn=\text{molecular}$ mean free path/particle radius) is one of the key parameters. Theoretical analyses on particle photophoresis in the flow regimes with $Kn$ up to $O\left({10}^{-1}\right)$, e.g. [1, 2, 3], have been performed with simple slip condition. Although this class of works makes remarkable progress in developing the theoretical framework of particle photophoresis, systematic deviations from the measurements can be observed as that presented in [2].

For rarefaction phenomena, Sone [4] demonstrated that the variation in the normal gradient of the gas temperature along the gas–solid interface, i.e., $\partial \left(\partial T/\partial n\right)/\partial s$ (*n* and *s* stand for normal and tangential directions, respectively), may result in a tangential velocity slip, which is now named thermal stress slip flow. Based on the Maxwell’s work [5], Lockerby *et al.* [6] developed a second-order slip boundary condition with thermal stress effect included, and used it to solve the natural convection of rarefied gas between two noncoaxial cylinders. However, the thermal stress slip has not ever been considered in studies of particle photophoresis. It is deemed as a probable reason for the deviations between previous predictions and measured data and is worthwhile of further investigation.

Figure 1 is a schematic diagram of an isotropic homogenous particle suspended in a gas medium with a monochromatic, parallel, and linearly polarized light beam along the *z* direction. Since the relative motion of the particle to fluid is very slow and the particle-fluid temperature difference is small, the heat transfer between the gas and the particle is assumed dominated by heat conduction. The gas temperature distribution $\left({T}_{g}\right)$ is governed by the Laplace equation, ${\nabla}^{2}{T}_{g}=0$. Whereas the temperature distribution inside the radiation-absorbing particle $\left({T}_{p}\right)$ is described by the Poisson equation, ${\nabla}^{2}{T}_{p}=-Q\left(r,\theta \right)/{k}_{p}$, where ${k}_{p}$ is the thermal conductivity of the particle, and $Q\left(r,\theta \right)$ is the radiant-absorption heat generation function. For a plane monochromatic incident light, $Q\left(r,\theta \right)=\left(4\pi {n}_{p}{\kappa}_{p}I/\lambda \right)\left({\left|\mathbf{E}\left(r,\theta \right)\right|}^{2}/{\left|{\mathbf{E}}_{0}\right|}^{2}\right)$ [2], where ${n}_{p}$ and ${\kappa}_{p}$ consist of the complex refractive index $\left({m}_{p}={n}_{p}+{\kappa}_{p}i\right)$ of the particle, *λ* and *I*, respectively, denote wavelength and intensity of the light, and ${\mathbf{E}}_{0}$ the incident electric field. The normalized internal electric energy, $B\left(\varsigma ,\theta \right)={\left|\mathbf{E}\left(r,\theta \right)\right|}^{2}/{\left|{\mathbf{E}}_{0}\right|}^{2}$ with $\varsigma \equiv r/R$, can be evaluated by the method proposed previously, e.g. [7].

In the far field, $r\to \infty $, the fluid temperature approaches the bulk gas temperature ${T}_{0}$, i.e., ${T}_{g}\to {T}_{0}$. On the particle surface, $r=R$, heat flux continuity ${k}_{g}\partial {T}_{g}/\partial r={k}_{p}\partial {T}_{p}/\partial r$ and temperature jump ${T}_{g}-{T}_{p}={c}_{tj}l\partial {T}_{g}/\partial r$ are imposed, where ${k}_{g}$ is the thermal conductivity of the gas, ${c}_{tj}$ the temperature jump coefficient, and *l* the molecular mean-free path. The fluid temperature distribution in spherical coordinates satisfying the above boundary conditions is ${T}_{g}={T}_{0}+\left[I{J}_{1}/{k}_{p}\left(1+2{k}_{g}/{k}_{p}+2{c}_{tj}Kn\right)\right]\cdot \left({R}^{3}/{r}^{2}\right)\text{cos \hspace{0.17em}}\theta $, where ${J}_{1}$ is the so-called asymmetry factor and ${J}_{1}=\left(6\pi R{n}_{p}{\kappa}_{p}/\lambda \right){\int}_{0}^{1}{\int}_{0}^{\pi}B\left(\varsigma ,\theta \right){\varsigma}^{3}\text{\hspace{0.17em} cos \hspace{0.17em}}\theta \text{\hspace{0.17em} sin \hspace{0.17em}}\theta d\theta d\varsigma $ [8]. The sign and magnitude of ${J}_{1}$ determine the direction and strength of the photophoretic motion.

For low velocity of the particle relative to the surrounding gas, the flow around a sphere is described by the Stokes equation, $\nabla p=\mu {\nabla}^{2}\mathbf{v}$ with *μ* denoting viscosity, **v** velocity vector, and *p* pressure. The far field velocity condition is $\mathbf{v}={V}_{0}\text{\hspace{0.17em} cos \hspace{0.17em}}\theta {\mathbf{e}}_{\mathbf{r}}-{V}_{0}\text{\hspace{0.17em} sin \hspace{0.17em}}\theta {\mathbf{e}}_{\mathbf{\theta}}$, where ${V}_{0}$ is the gas velocity relative to the particle. On the particle surface, the radial velocity is ${v}_{r}=0$. For the tangential velocity, we adopt the modified slip boundary condition of Lockerby *et al.* [6] with second-order terms neglected but the thermal stress term retained. In the present nomenclature, it is depicted as

*σ*denoting the tangential momentum accommodation coefficient, ${c}_{ts}$ the thermal stress slip coefficient, and ${c}_{tc}$ the thermal creep coefficient.

With the characteristic quantities of temperature difference $\Delta {T}_{c}$, length *R*, and velocity ${V}_{0}$, order-of-magnitudes of two ratios are analyzed. One, ${\mathfrak{R}}_{1}$, is the ratio of the thermal stress term, ${c}_{m}l\left[\left({c}_{ts}\mu /\rho {T}_{0}R\right)\cdot {\partial}^{2}{T}_{g}/\partial r\partial \theta \right]$, to the momentum exchange term, ${c}_{m}l\left[r\partial \left({v}_{\theta}/r\right)/\partial r\right]$, and then ${\mathfrak{R}}_{1}\sim {c}_{ts}\left(\mu /\rho R{V}_{0}\right)\cdot \left(\Delta {T}_{c}/{T}_{0}\right)$. The other, ${\mathfrak{R}}_{2}$, is the ratio of the thermal stress to the thermal creep ${c}_{tc}\left(\mu /\rho {T}_{0}R\right)\cdot \partial {T}_{g}/\partial \theta $, and one has ${\mathfrak{R}}_{2}\sim \left({c}_{m}\cdot {c}_{ts}/{c}_{tc}\right)\cdot \left(l/R\right)$. Both ratios have magnitudes inversely proportional to the particle radius *R*. It discloses that the thermal stress slip effect becomes increasingly important with reducing particle size.

Using the governing equations and boundary conditions above, the force exerted on the particle can be determined by

The photophoretic velocity ${V}_{ph}=-{V}_{0}$ can be determined with zero force exerted on the particle at steady state. Consider the viscosity given by $\mu =\rho \overline{v}l/2$ [10], where $\overline{v}=\sqrt{8{k}_{B}T/\pi M}$ is the mean molecular speed, ${k}_{B}$ the Boltzmann’s constant, and *M* the gas molecular mass. Then from Eq. (2), one has

*α*. For a particle of one-order lower absorption, ${m}_{p}=1.57+0.0475i$, the level of the energy absorbed by the particle is lower and the thermal stress effect relatively weaker. The difference between the theories with (present) and without (Mackowski) thermal stress effect also increases with reducing particle size. At continuum limit with thermal stress effect vanished, two theories coincide. The present theory not only catches the key trend that indicated by the order-of-magnitude analysis but is also appropriate in quantitative accuracy.

The dimensionless photophoretic velocity ${V}_{ph}^{\ast}$ in Eq. (4) is a function of two major parameters. One is $Kn\equiv l/R$ characterizing the rarefaction of the gas or the influence of the particle size. The other is the relative thermal conductivity of the particle, ${k}^{\ast}\equiv {k}_{p}/{k}_{g}$. In Fig. 3 , the values of ${V}_{ph}^{\ast}$ with and without considering thermal stress slip in the parameter ranges of ${10}^{-3}\le Kn\le {10}^{-1}$ and ${10}^{-2}\le {k}^{\ast}\le {10}^{2}$ are presented. It is disclosed that the photophoretic mobility increases with $Kn$ due to drag reduction with rarefaction. The thermal stress slip effect on ${V}_{ph}^{\ast}$ becomes more noticeable at high $Kn$ or, alternatively, small particle size. The large curvature of the micro-particle surface causes uneven temperature and heat flux distributions and then enhances the thermal stress slip effect. For example, at $Kn=0.1$, without considering thermal stress slip effect may underestimate ${V}_{ph}^{\ast}$ about 16%. As $Kn$ reduces to the continuum limit, results in Fig. 3 reveal that ${V}_{ph}^{\ast}$ and the thermal stress effect both vanish. A particle of higher thermal conductivity (or ${k}^{\ast}$) has better ability in spreading heat over the particle. The uneven thermal characteristics over the particle surface can be alleviated and, in turn, the influence of thermal stress weakened as ${k}^{\ast}$ increases.

In summary, with consideration of thermal stress effect on slip flow, we have developed a modification on the previous theory of particle photophoresis. Comparison with the previous experimental data shows the validity of the present results. Our theoretical expressions combining with an order-of-magnitude analysis demonstrate the significance of the thermal stress slip in micro-particle photophoresis and disclose the reason for deviation of the previous theory at small particle size. Parametric analysis reveals that the influence of thermal stress becomes more pronounced with a reducing particle thermal conductivity and/or increasing Knudsen number. The underlying physical mechanisms are also appropriately addressed. These results are helpful to further understanding of fundamentals of particle photophoresis and to related practical applications as well.

This study was supported by National Science Council of the Republic of China (Taiwan) (NSCT) through the grants NSC-98-2221-E-035-068-MY3 and NSC96-2221-E-264-002. The co-author (W. K. Li) would like to gratefully acknowledge Dr. D. A. Lockerby for the useful discussion.

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