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=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(101), 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., (T/n)/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 (Tg) is governed by the Laplace equation, 2Tg=0. Whereas the temperature distribution inside the radiation-absorbing particle (Tp) is described by the Poisson equation, 2Tp=Q(r,θ)/kp, where kp is the thermal conductivity of the particle, and Q(r,θ) is the radiant-absorption heat generation function. For a plane monochromatic incident light, Q(r,θ)=(4πnpκpI/λ)(|E(r,θ)|2/|E0|2) [2], where np and κp consist of the complex refractive index (mp=np+κpi) of the particle, λ and I, respectively, denote wavelength and intensity of the light, and E0 the incident electric field. The normalized internal electric energy, B(ς,θ)=|E(r,θ)|2/|E0|2 with ςr/R, can be evaluated by the method proposed previously, e.g. [7].

In the far field, r, the fluid temperature approaches the bulk gas temperature T0, i.e., TgT0. On the particle surface, r=R, heat flux continuity kgTg/r=kpTp/r and temperature jump TgTp=ctjlTg/r are imposed, where kg is the thermal conductivity of the gas, ctj the temperature jump coefficient, and l the molecular mean-free path. The fluid temperature distribution in spherical coordinates satisfying the above boundary conditions is Tg=T0+[IJ1/kp(1+2kg/kp+2ctjKn)](R3/r2)cos  θ, where J1 is the so-called asymmetry factor and J1=(6πRnpκp/λ)010πB(ς,θ)ς3  cos  θ  sin  θdθdς [8]. The sign and magnitude of J1 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, p=μ2v with μ denoting viscosity, v velocity vector, and p pressure. The far field velocity condition is v=V0  cos  θerV0  sin  θeθ, where V0 is the gas velocity relative to the particle. On the particle surface, the radial velocity is vr=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

vθ=cml[rr(vθr)+1rvrθctsμρT0R2Tgrθ]+ctcμρT0RTgθ,
where cm is defined as cm(2σ)/σ with σ denoting the tangential momentum accommodation coefficient, cts the thermal stress slip coefficient, and ctc the thermal creep coefficient.

With the characteristic quantities of temperature difference ΔTc, length R, and velocity V0, order-of-magnitudes of two ratios are analyzed. One, R1, is the ratio of the thermal stress term, cml[(ctsμ/ρT0R)2Tg/rθ], to the momentum exchange term, cml[r(vθ/r)/r], and then R1cts(μ/ρRV0)(ΔTc/T0). The other, R2, is the ratio of the thermal stress to the thermal creep ctc(μ/ρT0R)Tg/θ, and one has R2(cmcts/ctc)(l/R). 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

F=6πRμV0(1+2cmKn1+3cmKn)4πμ2RIJ1ρT0kgctc+2ctscmKn(1+3cmKn)(2+k+2ctjkKn),
where kkp/kg is the particle-to-gas thermal conductivity ratio indicating the relative importance of the particle thermal conductivity. The first term on the right hand side is the Stokes drag force with slip correction in rarefied gas, and the second term is the photophoretic force (Fph). Coefficients cm=1.14, ctj=2.18, and ctc=1.17 determined previously from kinetic theory [9] are adopted in the analysis. We take the coefficient cts=1 [6] in the analysis but, for comparison, cts=0 is used for the situation with the thermal stress slip ignored.

The photophoretic velocity Vph=V0 can be determined with zero force exerted on the particle at steady state. Consider the viscosity given by μ=ρv¯l/2 [10], where v¯=8kBT/πM is the mean molecular speed, kB the Boltzmann’s constant, and M the gas molecular mass. Then from Eq. (2), one has

Vph=2RIJ13kg2kBπMT0Kn(ctc+2ctscmKn)(1+2cmKn)(2+k+2ctjkKn).
With the definition of VphVph/[(2RIJ1/3kg)2kB/πMT0], the photophoretic velocity Vph can be normalized in the following dimensionless form:
Vph=Kn(ctc+2ctscmKn)(1+2cmKn)(2+k+2ctjkKn).
In Fig. 2 , the present predictions of photophoretic force Fph/Mg from Eq. (2) are compared with the experimental data [11] and Mackowski’s theory [2]. For a particle of complex refraction index mp=1.57+0.38i, the Mackowski’s results deviate from the measurements obviously and the deviation becomes more pronounced at small particle size (α2πR/λ). Whereas the present predictions with thermal stress considered agree the measured data quite well even at a small α. For a particle of one-order lower absorption, mp=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 Vph in Eq. (4) is a function of two major parameters. One is Knl/R characterizing the rarefaction of the gas or the influence of the particle size. The other is the relative thermal conductivity of the particle, kkp/kg. In Fig. 3 , the values of Vph with and without considering thermal stress slip in the parameter ranges of 103Kn101 and 102k102 are presented. It is disclosed that the photophoretic mobility increases with Kn due to drag reduction with rarefaction. The thermal stress slip effect on Vph 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 Vph about 16%. As Kn reduces to the continuum limit, results in Fig. 3 reveal that Vph and the thermal stress effect both vanish. A particle of higher thermal conductivity (or k) 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 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.

 

Fig. 1 Physical model of a microspherical particle photophoresis in a gaseous medium.

Download Full Size | PPT Slide | PDF

 

Fig. 2 Comparison of the theoretical predictions of photophoretic force with and without thermal stress slip and previous experimental data at various particle size parameters.

Download Full Size | PPT Slide | PDF

 

Fig. 3 Normalized photophoretic velocity with and without thermal stress slip at various conditions of relative thermal conductivity of the particle k and Knudsen number Kn.

Download Full Size | PPT Slide | PDF

1. L. D. Reed, J. Aerosol Sci. 8, 123 (1977). [CrossRef]  

2. D. W. Mackowski, Int. J. Heat Mass Transfer 32, 843 (1989). [CrossRef]  

3. H. J. Keh and F. C. Hsu, J. Colloid Interface Sci. 289, 94 (2005). [CrossRef]   [PubMed]  

4. Y. Sone, Phys. Fluids 15, 1418 (1972). [CrossRef]  

5. J. C. Maxwell, Philos. Trans. R. Soc. Lond. 170, 231 (1879). [CrossRef]  

6. D. A. Lockerby, J. M. Reese, D. R. Emerson, and R. W. Barber, Phys. Rev. E 70, 017303 (2004). [CrossRef]  

7. P. W. Dusel, M. Kerker, and D. D. Cooke, J. Opt. Soc. Am. 69, 55 (1979). [CrossRef]  

8. Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, J. Colloid Interface Sci. 57, 564 (1976). [CrossRef]  

9. L. Talbot, R. K. Cheng, R. W. Schefer, and D. R. Willis, J. Fluid Mech. 101, 737 (1980). [CrossRef]  

10. E. H. Kennard, Kinetic Theory of Gases (McGraw-Hill, 1938).

11. S. Arnold and M. Lewittes, J. Appl. Phys. 53, 5314 (1982). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. L. D. Reed, J. Aerosol Sci. 8, 123 (1977).
    [CrossRef]
  2. D. W. Mackowski, Int. J. Heat Mass Transfer 32, 843 (1989).
    [CrossRef]
  3. H. J. Keh and F. C. Hsu, J. Colloid Interface Sci. 289, 94 (2005).
    [CrossRef] [PubMed]
  4. Y. Sone, Phys. Fluids 15, 1418 (1972).
    [CrossRef]
  5. J. C. Maxwell, Philos. Trans. R. Soc. Lond. 170, 231 (1879).
    [CrossRef]
  6. D. A. Lockerby, J. M. Reese, D. R. Emerson, and R. W. Barber, Phys. Rev. E 70, 017303 (2004).
    [CrossRef]
  7. P. W. Dusel, M. Kerker, and D. D. Cooke, J. Opt. Soc. Am. 69, 55 (1979).
    [CrossRef]
  8. Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, J. Colloid Interface Sci. 57, 564 (1976).
    [CrossRef]
  9. L. Talbot, R. K. Cheng, R. W. Schefer, and D. R. Willis, J. Fluid Mech. 101, 737 (1980).
    [CrossRef]
  10. E. H. Kennard, Kinetic Theory of Gases (McGraw-Hill, 1938).
  11. S. Arnold and M. Lewittes, J. Appl. Phys. 53, 5314 (1982).
    [CrossRef]

2005

H. J. Keh and F. C. Hsu, J. Colloid Interface Sci. 289, 94 (2005).
[CrossRef] [PubMed]

2004

D. A. Lockerby, J. M. Reese, D. R. Emerson, and R. W. Barber, Phys. Rev. E 70, 017303 (2004).
[CrossRef]

1989

D. W. Mackowski, Int. J. Heat Mass Transfer 32, 843 (1989).
[CrossRef]

1982

S. Arnold and M. Lewittes, J. Appl. Phys. 53, 5314 (1982).
[CrossRef]

1980

L. Talbot, R. K. Cheng, R. W. Schefer, and D. R. Willis, J. Fluid Mech. 101, 737 (1980).
[CrossRef]

1979

1977

L. D. Reed, J. Aerosol Sci. 8, 123 (1977).
[CrossRef]

1976

Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, J. Colloid Interface Sci. 57, 564 (1976).
[CrossRef]

1972

Y. Sone, Phys. Fluids 15, 1418 (1972).
[CrossRef]

1879

J. C. Maxwell, Philos. Trans. R. Soc. Lond. 170, 231 (1879).
[CrossRef]

Arnold, S.

S. Arnold and M. Lewittes, J. Appl. Phys. 53, 5314 (1982).
[CrossRef]

Barber, R. W.

D. A. Lockerby, J. M. Reese, D. R. Emerson, and R. W. Barber, Phys. Rev. E 70, 017303 (2004).
[CrossRef]

Cheng, R. K.

L. Talbot, R. K. Cheng, R. W. Schefer, and D. R. Willis, J. Fluid Mech. 101, 737 (1980).
[CrossRef]

Cooke, D. D.

Dusel, P. W.

Emerson, D. R.

D. A. Lockerby, J. M. Reese, D. R. Emerson, and R. W. Barber, Phys. Rev. E 70, 017303 (2004).
[CrossRef]

Hsu, F. C.

H. J. Keh and F. C. Hsu, J. Colloid Interface Sci. 289, 94 (2005).
[CrossRef] [PubMed]

Keh, H. J.

H. J. Keh and F. C. Hsu, J. Colloid Interface Sci. 289, 94 (2005).
[CrossRef] [PubMed]

Kennard, E. H.

E. H. Kennard, Kinetic Theory of Gases (McGraw-Hill, 1938).

Kerker, M.

Kutukov, V. B.

Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, J. Colloid Interface Sci. 57, 564 (1976).
[CrossRef]

Lewittes, M.

S. Arnold and M. Lewittes, J. Appl. Phys. 53, 5314 (1982).
[CrossRef]

Lockerby, D. A.

D. A. Lockerby, J. M. Reese, D. R. Emerson, and R. W. Barber, Phys. Rev. E 70, 017303 (2004).
[CrossRef]

Mackowski, D. W.

D. W. Mackowski, Int. J. Heat Mass Transfer 32, 843 (1989).
[CrossRef]

Maxwell, J. C.

J. C. Maxwell, Philos. Trans. R. Soc. Lond. 170, 231 (1879).
[CrossRef]

Reed, L. D.

L. D. Reed, J. Aerosol Sci. 8, 123 (1977).
[CrossRef]

Reese, J. M.

D. A. Lockerby, J. M. Reese, D. R. Emerson, and R. W. Barber, Phys. Rev. E 70, 017303 (2004).
[CrossRef]

Schefer, R. W.

L. Talbot, R. K. Cheng, R. W. Schefer, and D. R. Willis, J. Fluid Mech. 101, 737 (1980).
[CrossRef]

Shchukin, E. R.

Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, J. Colloid Interface Sci. 57, 564 (1976).
[CrossRef]

Sone, Y.

Y. Sone, Phys. Fluids 15, 1418 (1972).
[CrossRef]

Talbot, L.

L. Talbot, R. K. Cheng, R. W. Schefer, and D. R. Willis, J. Fluid Mech. 101, 737 (1980).
[CrossRef]

Willis, D. R.

L. Talbot, R. K. Cheng, R. W. Schefer, and D. R. Willis, J. Fluid Mech. 101, 737 (1980).
[CrossRef]

Yalamov, Y. I.

Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, J. Colloid Interface Sci. 57, 564 (1976).
[CrossRef]

Int. J. Heat Mass Transfer

D. W. Mackowski, Int. J. Heat Mass Transfer 32, 843 (1989).
[CrossRef]

J. Aerosol Sci.

L. D. Reed, J. Aerosol Sci. 8, 123 (1977).
[CrossRef]

J. Appl. Phys.

S. Arnold and M. Lewittes, J. Appl. Phys. 53, 5314 (1982).
[CrossRef]

J. Colloid Interface Sci.

Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, J. Colloid Interface Sci. 57, 564 (1976).
[CrossRef]

H. J. Keh and F. C. Hsu, J. Colloid Interface Sci. 289, 94 (2005).
[CrossRef] [PubMed]

J. Fluid Mech.

L. Talbot, R. K. Cheng, R. W. Schefer, and D. R. Willis, J. Fluid Mech. 101, 737 (1980).
[CrossRef]

J. Opt. Soc. Am.

Philos. Trans. R. Soc. Lond.

J. C. Maxwell, Philos. Trans. R. Soc. Lond. 170, 231 (1879).
[CrossRef]

Phys. Fluids

Y. Sone, Phys. Fluids 15, 1418 (1972).
[CrossRef]

Phys. Rev. E

D. A. Lockerby, J. M. Reese, D. R. Emerson, and R. W. Barber, Phys. Rev. E 70, 017303 (2004).
[CrossRef]

Other

E. H. Kennard, Kinetic Theory of Gases (McGraw-Hill, 1938).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Physical model of a microspherical particle photophoresis in a gaseous medium.

Fig. 2
Fig. 2

Comparison of the theoretical predictions of photophoretic force with and without thermal stress slip and previous experimental data at various particle size parameters.

Fig. 3
Fig. 3

Normalized photophoretic velocity with and without thermal stress slip at various conditions of relative thermal conductivity of the particle k and Knudsen number K n .

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

v θ = c m l [ r r ( v θ r ) + 1 r v r θ c t s μ ρ T 0 R 2 T g r θ ] + c t c μ ρ T 0 R T g θ ,
F = 6 π R μ V 0 ( 1 + 2 c m K n 1 + 3 c m K n ) 4 π μ 2 R I J 1 ρ T 0 k g c t c + 2 c t s c m K n ( 1 + 3 c m K n ) ( 2 + k + 2 c t j k K n ) ,
V p h = 2 R I J 1 3 k g 2 k B π M T 0 K n ( c t c + 2 c t s c m K n ) ( 1 + 2 c m K n ) ( 2 + k + 2 c t j k K n ) .
V p h = K n ( c t c + 2 c t s c m K n ) ( 1 + 2 c m K n ) ( 2 + k + 2 c t j k K n ) .

Metrics