Abstract

In the present study, combining the conventional photothermal analysis and the concept of interaction of solvent molecules in interfacial layer used for thermophoresis in liquid, a theory for photophoresis of a hydrophobic particle suspended in liquids is developed. To characterize hydrophobicity of the micro-particle, slip length in Navier’s formula is used as an index. Analytical expressions are derived and a parametric analysis for photophoretic velocity is performed with emphasis on the influences of particle characteristics such as size, optical properties, hydrophobicity, and thermal conductivity. Heat source function and the corresponding asymmetry factor at various conditions are evaluated to interpret the mechanisms of negative and positive photophoresis and the conditions for transition between them. The present theory discloses that the particle surface hydrophobicity or fluid slippage at particle-liquid interface may lead to a remarkable enhancement in the particle photophoretic velocity in liquids. Higher particle thermal conductivity and larger size of liquid molecules both result in weaker photophoretic motion.

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  8. Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, “Theory of the Photophoretic Motion of Large-size Volatile Aerosol Particle,” J. Colloid Interface Sci. 57(3), 564–571 (1976).
    [CrossRef]
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  12. C. Y. Soong, W. K. Li, C. H. Liu, and P. Y. Tzeng, “Effect of Thermal Stress Slip on Micro-particle Photophoresis in Gaseous Media,” accepted for publication in Opt. Lett. (2010)
    [CrossRef] [PubMed]
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    [CrossRef]
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  23. H. Monjushiro, M. Tanaka, and H. Watarai, “Periodic Expansion-contraction Motion of Photoabsorbing Organic Droplets during Laser Photophoretic Migration in Water,” Chem. Lett. 32(3), 254–255 (2003).
    [CrossRef]
  24. M. Tamagawa, H. Monjushiro, and H. Watarai, “Microgravity Laser-photophoresis of High Density Microparticles in Water,” Colloid Surf. A 220(1-3), 279–284 (2003).
    [CrossRef]
  25. M. Tanaka, H. Monjushiro, and H. Watarai, “Laser photophoretic migration with periodic expansion-contraction motion of photo-absorbing microemulsion droplets in water,” Langmuir 20(25), 10791–10797 (2004).
    [CrossRef] [PubMed]
  26. H. Watarai, H. Monjushiro, S. Tsukahara, M. Suwa, and Y. Iiguni, “Migration analysis of micro-particles in liquids using microscopically designed external fields,” Anal. Sci. 20(3), 423–434 (2004).
    [CrossRef] [PubMed]
  27. N. V. Malai, “Effect of Motion of the Medium on the Photophoresis of Hot Hydrosol Particles,” Fluid Dyn. 41(6), 984–991 (2006).
    [CrossRef]
  28. M. Han, “Thermophoresis in Liquids: a Molecular Dynamics Simulation Study,” J. Colloid Interface Sci. 284(1), 339–348 (2005).
    [CrossRef] [PubMed]
  29. J. L. Anderson, “Colloid Transport by Interfacial Forces,” Annu. Rev. Fluid Mech. 21(1), 61–99 (1989).
    [CrossRef]
  30. A. Regazzetti, M. Hoyos, and M. Martin, “Experimental Evidence of Thermophoresis of Non-Brownian Particles in Pure Liquids and Estimation of Their Thermophoretic Mobility,” J. Phys. Chem. B 108(39), 15285–15292 (2004).
    [CrossRef]
  31. S. N. Semenov, “Mechanism of Particle Thermophoresis in Pure Solvents,” Philos. Mag. 83(17-18), 2199–2208 (2003).
    [CrossRef]
  32. S. N. Semenov and M. Schimpf, “Thermophoresis of dissolved molecules and polymers: Consideration of the temperature-induced macroscopic pressure gradient,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(1), 011201 (2004).
    [CrossRef] [PubMed]
  33. J. Baudry, E. Charlaix, A. Tonck, and D. Mazuyer, “Experimental Evidence for a Large Slip Effect at a Nonwetting Fluid-solid Interface,” Langmuir 17(17), 5232–5236 (2001).
    [CrossRef]
  34. J. T. Cheng and N. Giordano, “Fluid flow through nanometer-scale channels,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(3), 031206 (2002).
    [CrossRef] [PubMed]
  35. Y. Zhu and S. Granick, “Rate-dependent slip of Newtonian liquid at smooth surfaces,” Phys. Rev. Lett. 87(9), 096105 (2001).
    [CrossRef] [PubMed]
  36. P. Huang, J. S. Guasto, and K. S. Breuer, “Direct Measurement of Slip Velocities Using Three-dimensional Total Internal Reflection Velocimetry,” J. Fluid Mech. 566, 447–46 (2006).
    [CrossRef]
  37. D. C. Tretheway and C. D. Meinhart, “Apparent Fluid Slip at Hydrophobic Microchannel Walls,” Phys. Fluids 14(3), L9 (2002).
    [CrossRef]
  38. D. C. Tretheway and C. D. Meinhart, “A Generating Mechanism for Apparent Fluid Slip in Hydrophobic Microchannels,” Phys. Fluids 16(5), 1509 (2004).
    [CrossRef]
  39. Y. Ren and D. Stein, “Slip-enhanced Electrokinetic Energy Conversion in Nanofluidic Channels,” Nanotechnology 19(19), 195707 (2008).
    [CrossRef] [PubMed]
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  44. E. P. C. Mes, W. Th. Kok, and R. Tijssen, “Prediction of Polymer Thermal Diffusion Coefficients from Polymer-Solvent Interaction Parameters: Comparison with Thermal Field Flow Fractionation and Thermal Diffusion Forced Rayleigh Scattering Experiments,” Int. J. Polym. Mater. 8, 133–153 (2003).
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    [CrossRef]
  46. A. Parola and R. Piazza, “Particle thermophoresis in liquids,” Eur Phys J E Soft Matter 15(3), 255–263 (2004).
    [CrossRef] [PubMed]
  47. A. Würger, “Temperature dependence of the soret motion in colloids,” Langmuir 25(12), 6696–6701 (2009).
    [CrossRef] [PubMed]
  48. S. N. Semenov and M. E. Schimpf, “Molecular thermodiffusion (thermophoresis) in liquid mixtures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 041202 (2005).
    [CrossRef] [PubMed]
  49. M. E. Schimpf and S. N. Semenov, “Mechanism of Polymer Thermophoresis in Nonaqueous Solvents,” J. Phys. Chem. B 104(42), 9935–9942 (2000).
    [CrossRef]
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  51. S. Fayolle, T. Bickel, and A. Würger, “Thermophoresis of charged colloidal particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(4), 041404 (2008).
    [CrossRef] [PubMed]
  52. A. Würger, “Transport in charged colloids driven by thermoelectricity,” Phys. Rev. Lett. 101(10), 108302 (2008).
    [CrossRef] [PubMed]
  53. J. Morthomas and A. Würger, “Thermophoresis at a Charged Surface: the Role of Hydrodynamic Slip,” J. Phys. Condens. Matter 21(3), 035103 (2009).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  56. A. B. Pluchino and S. Arnold, “Comprehensive model of the photophoretic force on a spherical microparticle,” Opt. Lett. 10(6), 261–263 (1985).
    [CrossRef] [PubMed]

2010

C. Y. Soong, W. K. Li, C. H. Liu, and P. Y. Tzeng, “Effect of Thermal Stress Slip on Micro-particle Photophoresis in Gaseous Media,” accepted for publication in Opt. Lett. (2010)
[CrossRef] [PubMed]

2009

V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009).
[CrossRef] [PubMed]

A. S. Desyatnikov, V. G. Shvedov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Photophoretic manipulation of absorbing aerosol particles with vortex beams: theory versus experiment,” Opt. Express 17(10), 8201–8211 (2009).
[CrossRef] [PubMed]

A. Würger, “Temperature dependence of the soret motion in colloids,” Langmuir 25(12), 6696–6701 (2009).
[CrossRef] [PubMed]

J. Morthomas and A. Würger, “Thermophoresis at a Charged Surface: the Role of Hydrodynamic Slip,” J. Phys. Condens. Matter 21(3), 035103 (2009).
[CrossRef] [PubMed]

2008

S. Fayolle, T. Bickel, and A. Würger, “Thermophoresis of charged colloidal particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(4), 041404 (2008).
[CrossRef] [PubMed]

A. Würger, “Transport in charged colloids driven by thermoelectricity,” Phys. Rev. Lett. 101(10), 108302 (2008).
[CrossRef] [PubMed]

Y. Ren and D. Stein, “Slip-enhanced Electrokinetic Energy Conversion in Nanofluidic Channels,” Nanotechnology 19(19), 195707 (2008).
[CrossRef] [PubMed]

C. Haisch, C. Kykal, and R. Niessner, “Photophoretic velocimetry for the characterization of aerosols,” Anal. Chem. 80(5), 1546–1551 (2008).
[CrossRef] [PubMed]

2007

A. Würger, “Thermophoresis in colloidal suspensions driven by Marangoni forces,” Phys. Rev. Lett. 98(13), 138301 (2007).
[CrossRef] [PubMed]

2006

P. Huang, J. S. Guasto, and K. S. Breuer, “Direct Measurement of Slip Velocities Using Three-dimensional Total Internal Reflection Velocimetry,” J. Fluid Mech. 566, 447–46 (2006).
[CrossRef]

N. V. Malai, “Effect of Motion of the Medium on the Photophoresis of Hot Hydrosol Particles,” Fluid Dyn. 41(6), 984–991 (2006).
[CrossRef]

2005

M. Han, “Thermophoresis in Liquids: a Molecular Dynamics Simulation Study,” J. Colloid Interface Sci. 284(1), 339–348 (2005).
[CrossRef] [PubMed]

S. N. Semenov and M. E. Schimpf, “Molecular thermodiffusion (thermophoresis) in liquid mixtures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 041202 (2005).
[CrossRef] [PubMed]

2004

A. Parola and R. Piazza, “Particle thermophoresis in liquids,” Eur Phys J E Soft Matter 15(3), 255–263 (2004).
[CrossRef] [PubMed]

D. C. Tretheway and C. D. Meinhart, “A Generating Mechanism for Apparent Fluid Slip in Hydrophobic Microchannels,” Phys. Fluids 16(5), 1509 (2004).
[CrossRef]

A. Regazzetti, M. Hoyos, and M. Martin, “Experimental Evidence of Thermophoresis of Non-Brownian Particles in Pure Liquids and Estimation of Their Thermophoretic Mobility,” J. Phys. Chem. B 108(39), 15285–15292 (2004).
[CrossRef]

S. N. Semenov and M. Schimpf, “Thermophoresis of dissolved molecules and polymers: Consideration of the temperature-induced macroscopic pressure gradient,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(1), 011201 (2004).
[CrossRef] [PubMed]

M. Tanaka, H. Monjushiro, and H. Watarai, “Laser photophoretic migration with periodic expansion-contraction motion of photo-absorbing microemulsion droplets in water,” Langmuir 20(25), 10791–10797 (2004).
[CrossRef] [PubMed]

H. Watarai, H. Monjushiro, S. Tsukahara, M. Suwa, and Y. Iiguni, “Migration analysis of micro-particles in liquids using microscopically designed external fields,” Anal. Sci. 20(3), 423–434 (2004).
[CrossRef] [PubMed]

2003

H. Monjushiro, M. Tanaka, and H. Watarai, “Periodic Expansion-contraction Motion of Photoabsorbing Organic Droplets during Laser Photophoretic Migration in Water,” Chem. Lett. 32(3), 254–255 (2003).
[CrossRef]

M. Tamagawa, H. Monjushiro, and H. Watarai, “Microgravity Laser-photophoresis of High Density Microparticles in Water,” Colloid Surf. A 220(1-3), 279–284 (2003).
[CrossRef]

S. N. Semenov, “Mechanism of Particle Thermophoresis in Pure Solvents,” Philos. Mag. 83(17-18), 2199–2208 (2003).
[CrossRef]

E. P. C. Mes, W. Th. Kok, and R. Tijssen, “Prediction of Polymer Thermal Diffusion Coefficients from Polymer-Solvent Interaction Parameters: Comparison with Thermal Field Flow Fractionation and Thermal Diffusion Forced Rayleigh Scattering Experiments,” Int. J. Polym. Mater. 8, 133–153 (2003).

2002

H. Monjushiro, K. Takeuchi, and H. Watarai, “Anomalous Laser Photophoretic Behavior of Photo-Absorbing Organic Droplets in Water,” Chem. Lett. 31(8), 788–789 (2002).
[CrossRef]

D. C. Tretheway and C. D. Meinhart, “Apparent Fluid Slip at Hydrophobic Microchannel Walls,” Phys. Fluids 14(3), L9 (2002).
[CrossRef]

J. T. Cheng and N. Giordano, “Fluid flow through nanometer-scale channels,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(3), 031206 (2002).
[CrossRef] [PubMed]

2001

Y. Zhu and S. Granick, “Rate-dependent slip of Newtonian liquid at smooth surfaces,” Phys. Rev. Lett. 87(9), 096105 (2001).
[CrossRef] [PubMed]

J. Baudry, E. Charlaix, A. Tonck, and D. Mazuyer, “Experimental Evidence for a Large Slip Effect at a Nonwetting Fluid-solid Interface,” Langmuir 17(17), 5232–5236 (2001).
[CrossRef]

Q. Fu and W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40(9), 1354–1361 (2001).
[CrossRef]

2000

M. E. Schimpf and S. N. Semenov, “Mechanism of Polymer Thermophoresis in Nonaqueous Solvents,” J. Phys. Chem. B 104(42), 9935–9942 (2000).
[CrossRef]

H. Monjushiro, A. Hirai, and H. Watarai, “Size Dependence of Laser-Photophoretic Efficiency of Polystyrene Microparticles in Water,” Langmuir 16(22), 8539–8542 (2000).
[CrossRef]

1999

K. I. Morozov, “Thermal Diffusion in Disperse Systems,” J. Exp. Theor. Phys. 88(5), 944–946 (1999).
[CrossRef]

1996

A. Hirai, H. Monjushiro, and H. Watarai, “Laser Photophoresis of a Single Droplet in Oil in Water Emulsions,” Langmuir 12(23), 5570–5575 (1996).
[CrossRef]

1993

V. Chernyak and S. Beresnev, “Photophoresis of Aerosol Particles,” J. Aerosol Sci. 24(7), 857–866 (1993).
[CrossRef]

1989

D. W. MacKowski, “Photophoresis of Aerosol Particles in the Free Molecular and Slip-flow Regimes,” Int. J. Heat Mass Transfer 32(5), 843–854 (1989).
[CrossRef]

J. L. Anderson, “Colloid Transport by Interfacial Forces,” Annu. Rev. Fluid Mech. 21(1), 61–99 (1989).
[CrossRef]

1985

A. B. Pluchino and S. Arnold, “Comprehensive model of the photophoretic force on a spherical microparticle,” Opt. Lett. 10(6), 261–263 (1985).
[CrossRef] [PubMed]

1984

A. A. Gukasyan and Y. I. Yalamov, “Photophoretic Motion of Moderately Large Aerosol Particle,” J. Russ. Laser Res. 5(2), 308 (1984).
[CrossRef]

1983

A. B. Pluchino, “Photophoretic force on particles for low Knudsen number,” Appl. Opt. 22(1), 103–106 (1983).
[CrossRef] [PubMed]

1982

S. Arnold and M. Lewittes, “Size Dependance of the Photophoretic Force,” J. Appl. Phys. 53(7), 5314–5319 (1982).
[CrossRef]

M. Kerker and D. D. Cooke, “Photophoretic Force on Aerosol Particles in the Free Molecule Regime,” J. Opt. Soc. Am. 72(9), 1267–1272 (1982).
[CrossRef]

1980

S. Arnold and Y. Amani, “Broadband photophoretic spectroscopy,” Opt. Lett. 5(6), 242–244 (1980).
[CrossRef] [PubMed]

S. Arnold, Y. Amani, and A. Orenstein, “Photophoretic Spectrometer,” Rev. Sci. Instrum. 51(9), 1202–1204 (1980).
[CrossRef]

1979

P. W. Dusel, M. Kerker, and D. D. Cooke, “Distribution of Absorption Centers within Irradiated Spheres,” J. Opt. Soc. Am. 69(1), 55–59 (1979).
[CrossRef]

1977

L. D. Reed, “Low Knudsen Number Photophoresis,” J. Aerosol Sci. 8(2), 123–131 (1977).
[CrossRef]

A. Akhtaruzzaman and S. P. Lin, “Photophoresis of Absorbing Particles,” J. Colloid Interface Sci. 61(1), 170–182 (1977).
[CrossRef]

1976

Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, “Theory of the Photophoretic Motion of Large-size Volatile Aerosol Particle,” J. Colloid Interface Sci. 57(3), 564–571 (1976).
[CrossRef]

1975

S. P. Lin, “On Photophoresis,” J. Colloid Interface Sci. 51(1), 66–71 (1975).
[CrossRef]

N. T. Tong, “Experiments on Photophoresis and Thermophoresis,” J. Colloid Interface Sci. 51(1), 143–151 (1975).
[CrossRef]

1973

N. T. Tong, “Photophoretic Force in the Free Molecule and Transition Regimes,” J. Colloid Interface Sci. 43(1), 78–84 (1973).
[CrossRef]

1967

G. M. Hidy and J. R. Brock, “Photophoresis and the Descent of Particles into the Lower Stratosphere,” J. Geophys. Res. 72(2), 455–460 (1967).
[CrossRef]

1964

C. Orr, Jr. and E. Y. H. Keng, “Photophoretic Effects in the Stratosphere,” J. Atmos. Sci. 21(5), 475–478 (1964).
[CrossRef]

Akhtaruzzaman, A.

A. Akhtaruzzaman and S. P. Lin, “Photophoresis of Absorbing Particles,” J. Colloid Interface Sci. 61(1), 170–182 (1977).
[CrossRef]

Amani, Y.

S. Arnold and Y. Amani, “Broadband photophoretic spectroscopy,” Opt. Lett. 5(6), 242–244 (1980).
[CrossRef] [PubMed]

S. Arnold, Y. Amani, and A. Orenstein, “Photophoretic Spectrometer,” Rev. Sci. Instrum. 51(9), 1202–1204 (1980).
[CrossRef]

Anderson, J. L.

J. L. Anderson, “Colloid Transport by Interfacial Forces,” Annu. Rev. Fluid Mech. 21(1), 61–99 (1989).
[CrossRef]

Arnold, S.

A. B. Pluchino and S. Arnold, “Comprehensive model of the photophoretic force on a spherical microparticle,” Opt. Lett. 10(6), 261–263 (1985).
[CrossRef] [PubMed]

S. Arnold and M. Lewittes, “Size Dependance of the Photophoretic Force,” J. Appl. Phys. 53(7), 5314–5319 (1982).
[CrossRef]

S. Arnold, Y. Amani, and A. Orenstein, “Photophoretic Spectrometer,” Rev. Sci. Instrum. 51(9), 1202–1204 (1980).
[CrossRef]

S. Arnold and Y. Amani, “Broadband photophoretic spectroscopy,” Opt. Lett. 5(6), 242–244 (1980).
[CrossRef] [PubMed]

Baudry, J.

J. Baudry, E. Charlaix, A. Tonck, and D. Mazuyer, “Experimental Evidence for a Large Slip Effect at a Nonwetting Fluid-solid Interface,” Langmuir 17(17), 5232–5236 (2001).
[CrossRef]

Beresnev, S.

V. Chernyak and S. Beresnev, “Photophoresis of Aerosol Particles,” J. Aerosol Sci. 24(7), 857–866 (1993).
[CrossRef]

Bickel, T.

S. Fayolle, T. Bickel, and A. Würger, “Thermophoresis of charged colloidal particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(4), 041404 (2008).
[CrossRef] [PubMed]

Breuer, K. S.

P. Huang, J. S. Guasto, and K. S. Breuer, “Direct Measurement of Slip Velocities Using Three-dimensional Total Internal Reflection Velocimetry,” J. Fluid Mech. 566, 447–46 (2006).
[CrossRef]

Brock, J. R.

G. M. Hidy and J. R. Brock, “Photophoresis and the Descent of Particles into the Lower Stratosphere,” J. Geophys. Res. 72(2), 455–460 (1967).
[CrossRef]

Charlaix, E.

J. Baudry, E. Charlaix, A. Tonck, and D. Mazuyer, “Experimental Evidence for a Large Slip Effect at a Nonwetting Fluid-solid Interface,” Langmuir 17(17), 5232–5236 (2001).
[CrossRef]

Cheng, J. T.

J. T. Cheng and N. Giordano, “Fluid flow through nanometer-scale channels,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 65(3), 031206 (2002).
[CrossRef] [PubMed]

Chernyak, V.

V. Chernyak and S. Beresnev, “Photophoresis of Aerosol Particles,” J. Aerosol Sci. 24(7), 857–866 (1993).
[CrossRef]

Cooke, D. D.

M. Kerker and D. D. Cooke, “Photophoretic Force on Aerosol Particles in the Free Molecule Regime,” J. Opt. Soc. Am. 72(9), 1267–1272 (1982).
[CrossRef]

P. W. Dusel, M. Kerker, and D. D. Cooke, “Distribution of Absorption Centers within Irradiated Spheres,” J. Opt. Soc. Am. 69(1), 55–59 (1979).
[CrossRef]

Desyatnikov, A. S.

V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009).
[CrossRef] [PubMed]

A. S. Desyatnikov, V. G. Shvedov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Photophoretic manipulation of absorbing aerosol particles with vortex beams: theory versus experiment,” Opt. Express 17(10), 8201–8211 (2009).
[CrossRef] [PubMed]

Dusel, P. W.

P. W. Dusel, M. Kerker, and D. D. Cooke, “Distribution of Absorption Centers within Irradiated Spheres,” J. Opt. Soc. Am. 69(1), 55–59 (1979).
[CrossRef]

Fayolle, S.

S. Fayolle, T. Bickel, and A. Würger, “Thermophoresis of charged colloidal particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 77(4), 041404 (2008).
[CrossRef] [PubMed]

Fu, Q.

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P. Huang, J. S. Guasto, and K. S. Breuer, “Direct Measurement of Slip Velocities Using Three-dimensional Total Internal Reflection Velocimetry,” J. Fluid Mech. 566, 447–46 (2006).
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C. Orr, Jr. and E. Y. H. Keng, “Photophoretic Effects in the Stratosphere,” J. Atmos. Sci. 21(5), 475–478 (1964).
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C. Haisch, C. Kykal, and R. Niessner, “Photophoretic velocimetry for the characterization of aerosols,” Anal. Chem. 80(5), 1546–1551 (2008).
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M. Han, “Thermophoresis in Liquids: a Molecular Dynamics Simulation Study,” J. Colloid Interface Sci. 284(1), 339–348 (2005).
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G. M. Hidy and J. R. Brock, “Photophoresis and the Descent of Particles into the Lower Stratosphere,” J. Geophys. Res. 72(2), 455–460 (1967).
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H. Monjushiro, A. Hirai, and H. Watarai, “Size Dependence of Laser-Photophoretic Efficiency of Polystyrene Microparticles in Water,” Langmuir 16(22), 8539–8542 (2000).
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A. Hirai, H. Monjushiro, and H. Watarai, “Laser Photophoresis of a Single Droplet in Oil in Water Emulsions,” Langmuir 12(23), 5570–5575 (1996).
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A. Regazzetti, M. Hoyos, and M. Martin, “Experimental Evidence of Thermophoresis of Non-Brownian Particles in Pure Liquids and Estimation of Their Thermophoretic Mobility,” J. Phys. Chem. B 108(39), 15285–15292 (2004).
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P. Huang, J. S. Guasto, and K. S. Breuer, “Direct Measurement of Slip Velocities Using Three-dimensional Total Internal Reflection Velocimetry,” J. Fluid Mech. 566, 447–46 (2006).
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H. Watarai, H. Monjushiro, S. Tsukahara, M. Suwa, and Y. Iiguni, “Migration analysis of micro-particles in liquids using microscopically designed external fields,” Anal. Sci. 20(3), 423–434 (2004).
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V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009).
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E. P. C. Mes, W. Th. Kok, and R. Tijssen, “Prediction of Polymer Thermal Diffusion Coefficients from Polymer-Solvent Interaction Parameters: Comparison with Thermal Field Flow Fractionation and Thermal Diffusion Forced Rayleigh Scattering Experiments,” Int. J. Polym. Mater. 8, 133–153 (2003).

Krolikowski, W.

V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009).
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A. S. Desyatnikov, V. G. Shvedov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Photophoretic manipulation of absorbing aerosol particles with vortex beams: theory versus experiment,” Opt. Express 17(10), 8201–8211 (2009).
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Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, “Theory of the Photophoretic Motion of Large-size Volatile Aerosol Particle,” J. Colloid Interface Sci. 57(3), 564–571 (1976).
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C. Haisch, C. Kykal, and R. Niessner, “Photophoretic velocimetry for the characterization of aerosols,” Anal. Chem. 80(5), 1546–1551 (2008).
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S. Arnold and M. Lewittes, “Size Dependance of the Photophoretic Force,” J. Appl. Phys. 53(7), 5314–5319 (1982).
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C. Y. Soong, W. K. Li, C. H. Liu, and P. Y. Tzeng, “Effect of Thermal Stress Slip on Micro-particle Photophoresis in Gaseous Media,” accepted for publication in Opt. Lett. (2010)
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A. Akhtaruzzaman and S. P. Lin, “Photophoresis of Absorbing Particles,” J. Colloid Interface Sci. 61(1), 170–182 (1977).
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S. P. Lin, “On Photophoresis,” J. Colloid Interface Sci. 51(1), 66–71 (1975).
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Liu, C. H.

C. Y. Soong, W. K. Li, C. H. Liu, and P. Y. Tzeng, “Effect of Thermal Stress Slip on Micro-particle Photophoresis in Gaseous Media,” accepted for publication in Opt. Lett. (2010)
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D. W. MacKowski, “Photophoresis of Aerosol Particles in the Free Molecular and Slip-flow Regimes,” Int. J. Heat Mass Transfer 32(5), 843–854 (1989).
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N. V. Malai, “Effect of Motion of the Medium on the Photophoresis of Hot Hydrosol Particles,” Fluid Dyn. 41(6), 984–991 (2006).
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Martin, M.

A. Regazzetti, M. Hoyos, and M. Martin, “Experimental Evidence of Thermophoresis of Non-Brownian Particles in Pure Liquids and Estimation of Their Thermophoretic Mobility,” J. Phys. Chem. B 108(39), 15285–15292 (2004).
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J. Baudry, E. Charlaix, A. Tonck, and D. Mazuyer, “Experimental Evidence for a Large Slip Effect at a Nonwetting Fluid-solid Interface,” Langmuir 17(17), 5232–5236 (2001).
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D. C. Tretheway and C. D. Meinhart, “A Generating Mechanism for Apparent Fluid Slip in Hydrophobic Microchannels,” Phys. Fluids 16(5), 1509 (2004).
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D. C. Tretheway and C. D. Meinhart, “Apparent Fluid Slip at Hydrophobic Microchannel Walls,” Phys. Fluids 14(3), L9 (2002).
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E. P. C. Mes, W. Th. Kok, and R. Tijssen, “Prediction of Polymer Thermal Diffusion Coefficients from Polymer-Solvent Interaction Parameters: Comparison with Thermal Field Flow Fractionation and Thermal Diffusion Forced Rayleigh Scattering Experiments,” Int. J. Polym. Mater. 8, 133–153 (2003).

Monjushiro, H.

M. Tanaka, H. Monjushiro, and H. Watarai, “Laser photophoretic migration with periodic expansion-contraction motion of photo-absorbing microemulsion droplets in water,” Langmuir 20(25), 10791–10797 (2004).
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H. Watarai, H. Monjushiro, S. Tsukahara, M. Suwa, and Y. Iiguni, “Migration analysis of micro-particles in liquids using microscopically designed external fields,” Anal. Sci. 20(3), 423–434 (2004).
[CrossRef] [PubMed]

H. Monjushiro, M. Tanaka, and H. Watarai, “Periodic Expansion-contraction Motion of Photoabsorbing Organic Droplets during Laser Photophoretic Migration in Water,” Chem. Lett. 32(3), 254–255 (2003).
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M. Tamagawa, H. Monjushiro, and H. Watarai, “Microgravity Laser-photophoresis of High Density Microparticles in Water,” Colloid Surf. A 220(1-3), 279–284 (2003).
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H. Monjushiro, K. Takeuchi, and H. Watarai, “Anomalous Laser Photophoretic Behavior of Photo-Absorbing Organic Droplets in Water,” Chem. Lett. 31(8), 788–789 (2002).
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H. Monjushiro, A. Hirai, and H. Watarai, “Size Dependence of Laser-Photophoretic Efficiency of Polystyrene Microparticles in Water,” Langmuir 16(22), 8539–8542 (2000).
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A. Hirai, H. Monjushiro, and H. Watarai, “Laser Photophoresis of a Single Droplet in Oil in Water Emulsions,” Langmuir 12(23), 5570–5575 (1996).
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K. I. Morozov, “Thermal Diffusion in Disperse Systems,” J. Exp. Theor. Phys. 88(5), 944–946 (1999).
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J. Morthomas and A. Würger, “Thermophoresis at a Charged Surface: the Role of Hydrodynamic Slip,” J. Phys. Condens. Matter 21(3), 035103 (2009).
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C. Haisch, C. Kykal, and R. Niessner, “Photophoretic velocimetry for the characterization of aerosols,” Anal. Chem. 80(5), 1546–1551 (2008).
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C. Orr, Jr. and E. Y. H. Keng, “Photophoretic Effects in the Stratosphere,” J. Atmos. Sci. 21(5), 475–478 (1964).
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A. Parola and R. Piazza, “Particle thermophoresis in liquids,” Eur Phys J E Soft Matter 15(3), 255–263 (2004).
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A. Parola and R. Piazza, “Particle thermophoresis in liquids,” Eur Phys J E Soft Matter 15(3), 255–263 (2004).
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A. B. Pluchino, “Photophoretic force on particles for low Knudsen number,” Appl. Opt. 22(1), 103–106 (1983).
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L. D. Reed, “Low Knudsen Number Photophoresis,” J. Aerosol Sci. 8(2), 123–131 (1977).
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A. Regazzetti, M. Hoyos, and M. Martin, “Experimental Evidence of Thermophoresis of Non-Brownian Particles in Pure Liquids and Estimation of Their Thermophoretic Mobility,” J. Phys. Chem. B 108(39), 15285–15292 (2004).
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Ren, Y.

Y. Ren and D. Stein, “Slip-enhanced Electrokinetic Energy Conversion in Nanofluidic Channels,” Nanotechnology 19(19), 195707 (2008).
[CrossRef] [PubMed]

Rode, A. V.

A. S. Desyatnikov, V. G. Shvedov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Photophoretic manipulation of absorbing aerosol particles with vortex beams: theory versus experiment,” Opt. Express 17(10), 8201–8211 (2009).
[CrossRef] [PubMed]

V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009).
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S. N. Semenov and M. Schimpf, “Thermophoresis of dissolved molecules and polymers: Consideration of the temperature-induced macroscopic pressure gradient,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(1), 011201 (2004).
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S. N. Semenov and M. E. Schimpf, “Molecular thermodiffusion (thermophoresis) in liquid mixtures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 041202 (2005).
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S. N. Semenov and M. E. Schimpf, “Molecular thermodiffusion (thermophoresis) in liquid mixtures,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 72(4), 041202 (2005).
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S. N. Semenov and M. Schimpf, “Thermophoresis of dissolved molecules and polymers: Consideration of the temperature-induced macroscopic pressure gradient,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 69(1), 011201 (2004).
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S. N. Semenov, “Mechanism of Particle Thermophoresis in Pure Solvents,” Philos. Mag. 83(17-18), 2199–2208 (2003).
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M. E. Schimpf and S. N. Semenov, “Mechanism of Polymer Thermophoresis in Nonaqueous Solvents,” J. Phys. Chem. B 104(42), 9935–9942 (2000).
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Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, “Theory of the Photophoretic Motion of Large-size Volatile Aerosol Particle,” J. Colloid Interface Sci. 57(3), 564–571 (1976).
[CrossRef]

Shvedov, V. G.

V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009).
[CrossRef] [PubMed]

A. S. Desyatnikov, V. G. Shvedov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Photophoretic manipulation of absorbing aerosol particles with vortex beams: theory versus experiment,” Opt. Express 17(10), 8201–8211 (2009).
[CrossRef] [PubMed]

Soong, C. Y.

C. Y. Soong, W. K. Li, C. H. Liu, and P. Y. Tzeng, “Effect of Thermal Stress Slip on Micro-particle Photophoresis in Gaseous Media,” accepted for publication in Opt. Lett. (2010)
[CrossRef] [PubMed]

Stein, D.

Y. Ren and D. Stein, “Slip-enhanced Electrokinetic Energy Conversion in Nanofluidic Channels,” Nanotechnology 19(19), 195707 (2008).
[CrossRef] [PubMed]

Sun, W.

Q. Fu and W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40(9), 1354–1361 (2001).
[CrossRef]

Suwa, M.

H. Watarai, H. Monjushiro, S. Tsukahara, M. Suwa, and Y. Iiguni, “Migration analysis of micro-particles in liquids using microscopically designed external fields,” Anal. Sci. 20(3), 423–434 (2004).
[CrossRef] [PubMed]

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H. Monjushiro, K. Takeuchi, and H. Watarai, “Anomalous Laser Photophoretic Behavior of Photo-Absorbing Organic Droplets in Water,” Chem. Lett. 31(8), 788–789 (2002).
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Tamagawa, M.

M. Tamagawa, H. Monjushiro, and H. Watarai, “Microgravity Laser-photophoresis of High Density Microparticles in Water,” Colloid Surf. A 220(1-3), 279–284 (2003).
[CrossRef]

Tanaka, M.

M. Tanaka, H. Monjushiro, and H. Watarai, “Laser photophoretic migration with periodic expansion-contraction motion of photo-absorbing microemulsion droplets in water,” Langmuir 20(25), 10791–10797 (2004).
[CrossRef] [PubMed]

H. Monjushiro, M. Tanaka, and H. Watarai, “Periodic Expansion-contraction Motion of Photoabsorbing Organic Droplets during Laser Photophoretic Migration in Water,” Chem. Lett. 32(3), 254–255 (2003).
[CrossRef]

Tijssen, R.

E. P. C. Mes, W. Th. Kok, and R. Tijssen, “Prediction of Polymer Thermal Diffusion Coefficients from Polymer-Solvent Interaction Parameters: Comparison with Thermal Field Flow Fractionation and Thermal Diffusion Forced Rayleigh Scattering Experiments,” Int. J. Polym. Mater. 8, 133–153 (2003).

Tonck, A.

J. Baudry, E. Charlaix, A. Tonck, and D. Mazuyer, “Experimental Evidence for a Large Slip Effect at a Nonwetting Fluid-solid Interface,” Langmuir 17(17), 5232–5236 (2001).
[CrossRef]

Tong, N. T.

N. T. Tong, “Experiments on Photophoresis and Thermophoresis,” J. Colloid Interface Sci. 51(1), 143–151 (1975).
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N. T. Tong, “Photophoretic Force in the Free Molecule and Transition Regimes,” J. Colloid Interface Sci. 43(1), 78–84 (1973).
[CrossRef]

Tretheway, D. C.

D. C. Tretheway and C. D. Meinhart, “A Generating Mechanism for Apparent Fluid Slip in Hydrophobic Microchannels,” Phys. Fluids 16(5), 1509 (2004).
[CrossRef]

D. C. Tretheway and C. D. Meinhart, “Apparent Fluid Slip at Hydrophobic Microchannel Walls,” Phys. Fluids 14(3), L9 (2002).
[CrossRef]

Tsukahara, S.

H. Watarai, H. Monjushiro, S. Tsukahara, M. Suwa, and Y. Iiguni, “Migration analysis of micro-particles in liquids using microscopically designed external fields,” Anal. Sci. 20(3), 423–434 (2004).
[CrossRef] [PubMed]

Tzeng, P. Y.

C. Y. Soong, W. K. Li, C. H. Liu, and P. Y. Tzeng, “Effect of Thermal Stress Slip on Micro-particle Photophoresis in Gaseous Media,” accepted for publication in Opt. Lett. (2010)
[CrossRef] [PubMed]

Watarai, H.

M. Tanaka, H. Monjushiro, and H. Watarai, “Laser photophoretic migration with periodic expansion-contraction motion of photo-absorbing microemulsion droplets in water,” Langmuir 20(25), 10791–10797 (2004).
[CrossRef] [PubMed]

H. Watarai, H. Monjushiro, S. Tsukahara, M. Suwa, and Y. Iiguni, “Migration analysis of micro-particles in liquids using microscopically designed external fields,” Anal. Sci. 20(3), 423–434 (2004).
[CrossRef] [PubMed]

H. Monjushiro, M. Tanaka, and H. Watarai, “Periodic Expansion-contraction Motion of Photoabsorbing Organic Droplets during Laser Photophoretic Migration in Water,” Chem. Lett. 32(3), 254–255 (2003).
[CrossRef]

M. Tamagawa, H. Monjushiro, and H. Watarai, “Microgravity Laser-photophoresis of High Density Microparticles in Water,” Colloid Surf. A 220(1-3), 279–284 (2003).
[CrossRef]

H. Monjushiro, K. Takeuchi, and H. Watarai, “Anomalous Laser Photophoretic Behavior of Photo-Absorbing Organic Droplets in Water,” Chem. Lett. 31(8), 788–789 (2002).
[CrossRef]

H. Monjushiro, A. Hirai, and H. Watarai, “Size Dependence of Laser-Photophoretic Efficiency of Polystyrene Microparticles in Water,” Langmuir 16(22), 8539–8542 (2000).
[CrossRef]

A. Hirai, H. Monjushiro, and H. Watarai, “Laser Photophoresis of a Single Droplet in Oil in Water Emulsions,” Langmuir 12(23), 5570–5575 (1996).
[CrossRef]

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A. Würger, “Temperature dependence of the soret motion in colloids,” Langmuir 25(12), 6696–6701 (2009).
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J. Morthomas and A. Würger, “Thermophoresis at a Charged Surface: the Role of Hydrodynamic Slip,” J. Phys. Condens. Matter 21(3), 035103 (2009).
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A. A. Gukasyan and Y. I. Yalamov, “Photophoretic Motion of Moderately Large Aerosol Particle,” J. Russ. Laser Res. 5(2), 308 (1984).
[CrossRef]

Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, “Theory of the Photophoretic Motion of Large-size Volatile Aerosol Particle,” J. Colloid Interface Sci. 57(3), 564–571 (1976).
[CrossRef]

Zhu, Y.

Y. Zhu and S. Granick, “Rate-dependent slip of Newtonian liquid at smooth surfaces,” Phys. Rev. Lett. 87(9), 096105 (2001).
[CrossRef] [PubMed]

Anal. Chem.

C. Haisch, C. Kykal, and R. Niessner, “Photophoretic velocimetry for the characterization of aerosols,” Anal. Chem. 80(5), 1546–1551 (2008).
[CrossRef] [PubMed]

Anal. Sci.

H. Watarai, H. Monjushiro, S. Tsukahara, M. Suwa, and Y. Iiguni, “Migration analysis of micro-particles in liquids using microscopically designed external fields,” Anal. Sci. 20(3), 423–434 (2004).
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Annu. Rev. Fluid Mech.

J. L. Anderson, “Colloid Transport by Interfacial Forces,” Annu. Rev. Fluid Mech. 21(1), 61–99 (1989).
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Q. Fu and W. Sun, “Mie theory for light scattering by a spherical particle in an absorbing medium,” Appl. Opt. 40(9), 1354–1361 (2001).
[CrossRef]

A. B. Pluchino, “Photophoretic force on particles for low Knudsen number,” Appl. Opt. 22(1), 103–106 (1983).
[CrossRef] [PubMed]

Chem. Lett.

H. Monjushiro, K. Takeuchi, and H. Watarai, “Anomalous Laser Photophoretic Behavior of Photo-Absorbing Organic Droplets in Water,” Chem. Lett. 31(8), 788–789 (2002).
[CrossRef]

H. Monjushiro, M. Tanaka, and H. Watarai, “Periodic Expansion-contraction Motion of Photoabsorbing Organic Droplets during Laser Photophoretic Migration in Water,” Chem. Lett. 32(3), 254–255 (2003).
[CrossRef]

Colloid Surf. A

M. Tamagawa, H. Monjushiro, and H. Watarai, “Microgravity Laser-photophoresis of High Density Microparticles in Water,” Colloid Surf. A 220(1-3), 279–284 (2003).
[CrossRef]

Eur Phys J E Soft Matter

A. Parola and R. Piazza, “Particle thermophoresis in liquids,” Eur Phys J E Soft Matter 15(3), 255–263 (2004).
[CrossRef] [PubMed]

Fluid Dyn.

N. V. Malai, “Effect of Motion of the Medium on the Photophoresis of Hot Hydrosol Particles,” Fluid Dyn. 41(6), 984–991 (2006).
[CrossRef]

Int. J. Heat Mass Transfer

D. W. MacKowski, “Photophoresis of Aerosol Particles in the Free Molecular and Slip-flow Regimes,” Int. J. Heat Mass Transfer 32(5), 843–854 (1989).
[CrossRef]

Int. J. Polym. Mater.

E. P. C. Mes, W. Th. Kok, and R. Tijssen, “Prediction of Polymer Thermal Diffusion Coefficients from Polymer-Solvent Interaction Parameters: Comparison with Thermal Field Flow Fractionation and Thermal Diffusion Forced Rayleigh Scattering Experiments,” Int. J. Polym. Mater. 8, 133–153 (2003).

J. Aerosol Sci.

L. D. Reed, “Low Knudsen Number Photophoresis,” J. Aerosol Sci. 8(2), 123–131 (1977).
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J. Appl. Phys.

S. Arnold and M. Lewittes, “Size Dependance of the Photophoretic Force,” J. Appl. Phys. 53(7), 5314–5319 (1982).
[CrossRef]

J. Atmos. Sci.

C. Orr, Jr. and E. Y. H. Keng, “Photophoretic Effects in the Stratosphere,” J. Atmos. Sci. 21(5), 475–478 (1964).
[CrossRef]

J. Colloid Interface Sci.

Y. I. Yalamov, V. B. Kutukov, and E. R. Shchukin, “Theory of the Photophoretic Motion of Large-size Volatile Aerosol Particle,” J. Colloid Interface Sci. 57(3), 564–571 (1976).
[CrossRef]

N. T. Tong, “Photophoretic Force in the Free Molecule and Transition Regimes,” J. Colloid Interface Sci. 43(1), 78–84 (1973).
[CrossRef]

S. P. Lin, “On Photophoresis,” J. Colloid Interface Sci. 51(1), 66–71 (1975).
[CrossRef]

A. Akhtaruzzaman and S. P. Lin, “Photophoresis of Absorbing Particles,” J. Colloid Interface Sci. 61(1), 170–182 (1977).
[CrossRef]

N. T. Tong, “Experiments on Photophoresis and Thermophoresis,” J. Colloid Interface Sci. 51(1), 143–151 (1975).
[CrossRef]

M. Han, “Thermophoresis in Liquids: a Molecular Dynamics Simulation Study,” J. Colloid Interface Sci. 284(1), 339–348 (2005).
[CrossRef] [PubMed]

J. Exp. Theor. Phys.

K. I. Morozov, “Thermal Diffusion in Disperse Systems,” J. Exp. Theor. Phys. 88(5), 944–946 (1999).
[CrossRef]

J. Fluid Mech.

P. Huang, J. S. Guasto, and K. S. Breuer, “Direct Measurement of Slip Velocities Using Three-dimensional Total Internal Reflection Velocimetry,” J. Fluid Mech. 566, 447–46 (2006).
[CrossRef]

J. Geophys. Res.

G. M. Hidy and J. R. Brock, “Photophoresis and the Descent of Particles into the Lower Stratosphere,” J. Geophys. Res. 72(2), 455–460 (1967).
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J. Opt. Soc. Am.

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Langmuir

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Figures (10)

Fig. 1
Fig. 1

Physical model of a micro spherical particle photophoresis

Fig. 2
Fig. 2

Comparison of present calculations of asymmetry factor J 1 with previous results of a particle of m p = 1.57 + 0.048i in air assumed non-absorbing and having refractivity of n a = 1.

Fig. 3
Fig. 3

Variations of asymmetry factor J 1 with size parameter α for particle photophoresis in water with (a) m p = 1.57 + κ pi; and (b) m p = 2.0 + κ pi, where κ p = 0.001, 0.05, and 0.5

Fig. 4
Fig. 4

Normalized heat source function for the pattern of negative photophoresis dominant with particles of m p = 1.57 + 0.001i. The values of the particle size parameter are (a) α = 2; (b) α = 5; (c) α = 10; and (d) α = 20

Fig. 5
Fig. 5

Normalized heat source function for the positive photophoresis prevailing pattern with particles of m p = 1.57 + 0.5i. The values of the particle size parameter are (a) α = 2; (b) α = 5; (c) α = 10; and (d) α = 20

Fig. 6
Fig. 6

Normalized heat source function for the normal reversion of photophoresis pattern with particles of m p = 1.57 + 0.05i. The values of the particle size parameter are (a) α = 0.5; (b) α = 5; (c) α = 8cr ; and (d) α = 30

Fig. 7
Fig. 7

Comparison of present calculations of photophoretic velocity with previous results of polystyrene particles with light intensity I = 173 and 240 W/mm2 and optical characteristics of m p = 1.59 + κ p i, where κ p = 0.1 to 0.4

Fig. 8
Fig. 8

Dimensionless photophoretic velocity versus k* for various values of slip length, L S * = 0-1 × 10 3 or L S = 0-10nm, with particle radius R = 1μm and water molecular radius r 0 = 1.54 Å.

Fig. 9
Fig. 9

Dimensionless photophoretic velocity versus the slip length for various values of k*, k* = 0.1, 1, 5, and 10

Fig. 10
Fig. 10

Effects of molecule size of solvent on photophoretic velocity with particle radius R = 1μm at the slip lengths L S * = 1 × 10 4 , 1 × 10 3 , and 1 × 10 2

Equations (26)

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V S = L S τ 0 μ ,
2 T f = 0.
2 T p = Q ( r , θ ) k p ,
Q ( r , θ ) = Re [ 1 2 ( E × H ) ] = 1 2 σ E E * .
Q ( r , θ ) = 4 π n p κ p I m f 2 λ | E ( r , θ ) | 2 | E 0 | 2 = 4 π n p κ p I m f 2 λ B ( r , θ ) ,
r = R :    T f = T p   ;   k f T f r = k p T p r ;
r :    T f T 0 ,
T f = T 0 + [ I J 1 k f ( k p / k f + 2 ) ] R 3 r 2 cos θ ,
J 1 = 6 π R n p κ p m f 2 λ 0 1 0 π B ( ς , θ ) ς 3 cos θ sin θ d θ d ς .
E r = E 0 cos ϕ m 2 α 2 n = 1 i n + 1 ( 2 n + 1 ) d n P n ( 1 ) ( cos θ ) ψ n ( m α ) ;
E θ = E 0 cos ϕ m α n = 1 i n 2 n + 1 n ( n + 1 ) [ c n ψ n ( m α ) P n ( 1 ) ( cos θ ) sin θ i d n ψ n ( m α ) P n ( 1 ) ( cos θ ) sin θ ] ;
E ϕ = E 0 sin ϕ m α n = 1 i n 2 n + 1 n ( n + 1 ) [ c n ψ n ( m α ) P l ( 1 ) ( cos θ ) sin θ i d n ψ n ( m α ) P l ( 1 ) ( cos θ ) sin θ ] ,
c n = m p ξ n ( m f α ) ψ n ( m f α ) m p ξ n ( m f α ) ψ n ( m f α ) m p ξ n ( m f α ) ψ n ( m p α ) m f ξ n ( m f α ) ψ n ( m p α ) ;
d n = m p ξ n ( m f α ) ψ n ( m f α ) m p ξ n ( m f α ) ψ n ( m f α ) m f ξ n ( m f α ) ψ n ( m p α ) m p ξ n ( m f α ) ψ n ( m p α ) .
p = μ 2 v .
Φ = A 6 [ r 0 d + 1 2 + d / r 0 + ln ( d / r 0 2 + d / r 0 ) ] ,
p = β T Φ v 0 T f ,
V B = Δ V + ( L S / μ ) Δ τ 1 + 2 L S / R .
Δ V = ln 3 12 β T A r 0 2 μ v 0 [ I J 1 k f ( k p / k f + 2 ) ] sin θ ;
Δ τ = β T A r 0 v 0 ( ln 3 1 ) 3 [ I J 1 k f ( k p / k f + 2 ) ] sin θ .
V B = β T A r 0 2 12 μ v 0 [ I J 1 k f ( k p / k f + 2 ) ] ln 3 + 4 ( ln 3 1 ) L S / r 0 1 + 2 L S / R sin θ .
V p h = β T A r 0 2 18 μ v 0 k f I J 1 ln 3 + 4 ( ln 3 1 ) L S / r 0 ( k p / k f + 2 ) ( 1 + 2 L S / R ) .
V p h * = r 0 * [ ln 3 r 0 * + 4 ( ln 3 1 ) L S * ] ( 2 + k * ) ( 1 + 2 L S * ) ,
L S * = 0  :  V p h , 0 * = ln 3 r 0 * 2 2 + k * ;
L S *  :  V p h , * = 2 ( ln 3 1 ) r 0 * 2 + k * .
V p h * V p h , 0 * = 1 + 4 ( 1 1 / ln 3 ) L S * / r 0 * 1 + 2 L S * .

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