Abstract

This paper presents a detailed investigation of the motion of individual micro-particles in a moderately-viscous liquid in direct response to a local, laser-induced temperature gradient. By measuring particle trajectories in 3D, and comparing them to a simulated temperature profile, it is confirmed that the thermally-induced particle motion is the direct result of thermophoresis. The elevated viscosity of the liquid provides for substantial differences in the behavior predicted by various models of thermophoresis, which in turn allows measured data to be most appropriately matched to a model proposed by Brenner. This model is then used to predict the effective force resulting from thermophoresis in an optical trap. Based on these results, we predict when thermophoresis will strongly inhibit the ability of radiation pressure to trap nano-scale particles. The model also predicts that the thermophoretic force scales linearly with the viscosity of the liquid, such that choice of liquid plays a key role in the relative strength of the thermophoretic and radiation forces.

© 2011 OSA

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    [PubMed]
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2009

H. Brenner, “A nonmolecular derivation of Maxwell’s thermal creep boundary condition in gases and liquids via application of the LeChatelier-Braun principle to Maxwell’s thermal stress,” Phys. Fluids 21(5), 053602 (2009).

O. Jovanovic, “Photophoresis: light-induced motion of particles suspended in gas,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 889–901 (2009).

R. Di Leonardo, F. Ianni, and G. Ruocco, “Colloidal attraction induced by a temperature gradient,” Langmuir 25(8), 4247–4250 (2009).
[PubMed]

T. P. Otanicar, P. E. Phelan, and J. S. Golden, “Optical properties of liquids for direct absorption solar thermal energy systems,” Sol. Energy 83(7), 969–977 (2009).

2008

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[PubMed]

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008).

R. Piazza and A. Parola, “Thermophoresis in colloidal suspensions,” J. Phys. Condens. Matter 20(15), 153102 (2008).

2007

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).

2006

S. Duhr and D. Braun, “Thermophoretic depletion follows Boltzmann distribution,” Phys. Rev. Lett. 96(16), 168301 (2006).
[PubMed]

Y. Roichman, A. Waldron, E. Gardel, and D. G. Grier, “Optical traps with geometric aberrations,” Appl. Opt. 45(15), 3425–3429 (2006).
[PubMed]

J. K. Platten, “The Soret effect: a review of recent experimental results,” J. Appl. Mech. 73(1), 5–15 (2006).

S. Duhr and D. Braun, “Why molecules move along a temperature gradient,” Proc. Natl. Acad. Sci. U.S.A. 103(52), 19678–19682 (2006).
[PubMed]

2005

H. Brenner, “Navier-Stokes revisited,” Physica A 349(1-2), 60–132 (2005).

H. Brenner and J. R. Bielenberg, “A continuum approach to phoretic motions: thermophoresis,” Physica A 355(2-4), 251–273 (2005).

H. Brenner, “Kinematics of volume transport,” Physica A 349(1-2), 11–59 (2005).

2004

T. Sun and A. S. Teja, “Density, viscosity and thermal conductivity of aqueous solutions of propylene glycol, dipropylene glycol, and tripropylene glycol between 290 K and 460 K,” J. Chem. Eng. Data 49(5), 1311–1317 (2004).

S. Duhr, S. Arduini, and D. Braun, “Thermophoresis of DNA determined by microfluidic fluorescence,” Eur Phys J E Soft Matter 15(3), 277–286 (2004).
[PubMed]

S. 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).
[PubMed]

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

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).

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).

2003

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 1005–1017 (2003).

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 1019–1029 (2003).

E. Fällman and O. Axner, “Influence of a glass-water interface on the on-axis trapping of micrometer-sized spherical objects by optical tweezers,” Appl. Opt. 42(19), 3915–3926 (2003).
[PubMed]

2000

M. E. Schimpf and S. N. Semenov, “Mechanism of polymer thermophoresis in nonaqueous solvents,” J. Phys. Chem. B 104(42), 9935–9942 (2000).

1992

B. C. Hoke and E. F. Patton, “Surface tensions of propylene glycol + water,” J. Chem. Eng. Data 37(3), 331–333 (1992).

1981

E. Ruckenstein, “Can phoretic motions be treated as interfacial tension gradient driven phenomena,” J. Colloid Interface Sci. 83(1), 77–81 (1981).

1973

G. S. McNab and A. Meisen, “Thermophoresis in liquids,” J. Colloid Interface Sci. 44(2), 339–346 (1973).

1964

K. K. Kundu and M. N. Das, “Autoprotolysis constants of ethylene glycol and propylene glycol and dissociation constants of some acids and bases in the solvents at 30° C,” J. Chem. Eng. Data 9(1), 82–86 (1964).

Arduini, S.

S. Duhr, S. Arduini, and D. Braun, “Thermophoresis of DNA determined by microfluidic fluorescence,” Eur Phys J E Soft Matter 15(3), 277–286 (2004).
[PubMed]

Axner, O.

Bielenberg, J. R.

H. Brenner and J. R. Bielenberg, “A continuum approach to phoretic motions: thermophoresis,” Physica A 355(2-4), 251–273 (2005).

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).

Branczyk, A. M.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).

Braun, D.

S. Duhr and D. Braun, “Why molecules move along a temperature gradient,” Proc. Natl. Acad. Sci. U.S.A. 103(52), 19678–19682 (2006).
[PubMed]

S. Duhr and D. Braun, “Thermophoretic depletion follows Boltzmann distribution,” Phys. Rev. Lett. 96(16), 168301 (2006).
[PubMed]

S. Duhr, S. Arduini, and D. Braun, “Thermophoresis of DNA determined by microfluidic fluorescence,” Eur Phys J E Soft Matter 15(3), 277–286 (2004).
[PubMed]

Brenner, H.

H. Brenner, “A nonmolecular derivation of Maxwell’s thermal creep boundary condition in gases and liquids via application of the LeChatelier-Braun principle to Maxwell’s thermal stress,” Phys. Fluids 21(5), 053602 (2009).

H. Brenner and J. R. Bielenberg, “A continuum approach to phoretic motions: thermophoresis,” Physica A 355(2-4), 251–273 (2005).

H. Brenner, “Navier-Stokes revisited,” Physica A 349(1-2), 60–132 (2005).

H. Brenner, “Kinematics of volume transport,” Physica A 349(1-2), 11–59 (2005).

Brock, R. S.

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).

Bustamante, C.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[PubMed]

Chemla, Y. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[PubMed]

Das, M. N.

K. K. Kundu and M. N. Das, “Autoprotolysis constants of ethylene glycol and propylene glycol and dissociation constants of some acids and bases in the solvents at 30° C,” J. Chem. Eng. Data 9(1), 82–86 (1964).

Dholakia, K.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008).

Di Leonardo, R.

R. Di Leonardo, F. Ianni, and G. Ruocco, “Colloidal attraction induced by a temperature gradient,” Langmuir 25(8), 4247–4250 (2009).
[PubMed]

Dienerowitz, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008).

Duhr, S.

S. Duhr and D. Braun, “Thermophoretic depletion follows Boltzmann distribution,” Phys. Rev. Lett. 96(16), 168301 (2006).
[PubMed]

S. Duhr and D. Braun, “Why molecules move along a temperature gradient,” Proc. Natl. Acad. Sci. U.S.A. 103(52), 19678–19682 (2006).
[PubMed]

S. Duhr, S. Arduini, and D. Braun, “Thermophoresis of DNA determined by microfluidic fluorescence,” Eur Phys J E Soft Matter 15(3), 277–286 (2004).
[PubMed]

Fällman, E.

Gardel, E.

Golden, J. S.

T. P. Otanicar, P. E. Phelan, and J. S. Golden, “Optical properties of liquids for direct absorption solar thermal energy systems,” Sol. Energy 83(7), 969–977 (2009).

Grier, D. G.

Heckenberg, N. R.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 1005–1017 (2003).

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 1019–1029 (2003).

Hoke, B. C.

B. C. Hoke and E. F. Patton, “Surface tensions of propylene glycol + water,” J. Chem. Eng. Data 37(3), 331–333 (1992).

Hoyos, 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).

Hu, X.-H.

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).

Ianni, F.

R. Di Leonardo, F. Ianni, and G. Ruocco, “Colloidal attraction induced by a temperature gradient,” Langmuir 25(8), 4247–4250 (2009).
[PubMed]

Jacobs, K. M.

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).

Jovanovic, O.

O. Jovanovic, “Photophoresis: light-induced motion of particles suspended in gas,” J. Quant. Spectrosc. Radiat. Transf. 110(11), 889–901 (2009).

Knöner, G.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).

Kundu, K. K.

K. K. Kundu and M. N. Das, “Autoprotolysis constants of ethylene glycol and propylene glycol and dissociation constants of some acids and bases in the solvents at 30° C,” J. Chem. Eng. Data 9(1), 82–86 (1964).

Loke, V. L. Y.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).

Lu, J. Q.

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).

Ma, X.

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X.-H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).

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).

Mazilu, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics 2(1), 021875 (2008).

McNab, G. S.

G. S. McNab and A. Meisen, “Thermophoresis in liquids,” J. Colloid Interface Sci. 44(2), 339–346 (1973).

Meisen, A.

G. S. McNab and A. Meisen, “Thermophoresis in liquids,” J. Colloid Interface Sci. 44(2), 339–346 (1973).

Moffitt, J. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[PubMed]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75(9), 2787–2809 (2004).

Nieminen, T. A.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 1005–1017 (2003).

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 1019–1029 (2003).

Otanicar, T. P.

T. P. Otanicar, P. E. Phelan, and J. S. Golden, “Optical properties of liquids for direct absorption solar thermal energy systems,” Sol. Energy 83(7), 969–977 (2009).

Parola, A.

R. Piazza and A. Parola, “Thermophoresis in colloidal suspensions,” J. Phys. Condens. Matter 20(15), 153102 (2008).

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

Patton, E. F.

B. C. Hoke and E. F. Patton, “Surface tensions of propylene glycol + water,” J. Chem. Eng. Data 37(3), 331–333 (1992).

Phelan, P. E.

T. P. Otanicar, P. E. Phelan, and J. S. Golden, “Optical properties of liquids for direct absorption solar thermal energy systems,” Sol. Energy 83(7), 969–977 (2009).

Piazza, R.

R. Piazza and A. Parola, “Thermophoresis in colloidal suspensions,” J. Phys. Condens. Matter 20(15), 153102 (2008).

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

Platten, J. K.

J. K. Platten, “The Soret effect: a review of recent experimental results,” J. Appl. Mech. 73(1), 5–15 (2006).

Regazzetti, A.

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).

Roichman, Y.

Rubinsztein-Dunlop, H.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 1005–1017 (2003).

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79-80, 1019–1029 (2003).

Ruckenstein, E.

E. Ruckenstein, “Can phoretic motions be treated as interfacial tension gradient driven phenomena,” J. Colloid Interface Sci. 83(1), 77–81 (1981).

Ruocco, G.

R. Di Leonardo, F. Ianni, and G. Ruocco, “Colloidal attraction induced by a temperature gradient,” Langmuir 25(8), 4247–4250 (2009).
[PubMed]

Schimpf, M.

S. 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).
[PubMed]

Schimpf, M. E.

M. E. Schimpf and S. N. Semenov, “Mechanism of polymer thermophoresis in nonaqueous solvents,” J. Phys. Chem. B 104(42), 9935–9942 (2000).

Semenov, S.

S. 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).
[PubMed]

Semenov, S. N.

M. E. Schimpf and S. N. Semenov, “Mechanism of polymer thermophoresis in nonaqueous solvents,” J. Phys. Chem. B 104(42), 9935–9942 (2000).

Smith, S. B.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77(1), 205–228 (2008).
[PubMed]

Stilgoe, A. B.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A, Pure Appl. Opt. 9(8), S196–S203 (2007).

Sun, T.

T. Sun and A. S. Teja, “Density, viscosity and thermal conductivity of aqueous solutions of propylene glycol, dipropylene glycol, and tripropylene glycol between 290 K and 460 K,” J. Chem. Eng. Data 49(5), 1311–1317 (2004).

Teja, A. S.

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Supplementary Material (1)

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Figures (8)

Fig. 1
Fig. 1

Schematic diagrams of (a) the optical system used to manipulate and track particles in a liquid, and (b) the liquid-filled, square cross-section, glass capillary sample cell. The cell was held suspended in air by a rigid clamp, located 2.0 cm above the laser focus in the positive y-direction. Gravity was directed along the negative y-direction.

Fig. 2
Fig. 2

Time-lapsed video (Media 1) of the motion of two silica microspheres, (A) and (B), suspended in propylene glycol, in response to a focused laser beam. Each second in the video corresponds to thirty seconds in the experiment. The 1/e2 intensity limits of the laser beam, calculated using Eqs. (1) and (2), have been overlaid in gray. The thicknesses of the cell walls and liquid region between were 25 µm and 50 µm, respectively. Microsphere (A) falls into the beam, becomes trapped in (x,y), and is subsequently pushed along the beam 24 µm before being released from the 2D trap. Microsphere (B) falls past the beam without becoming trapped, but experiences a similar push when closest to the beam axis. Particle size is exaggerated due to imperfect focus.

Fig. 3
Fig. 3

Positions of (a) microsphere (A) and (b) microsphere (B), as a function of time. The origin of the coordinate system coincided with both the beam axis and the beam waist. Microsphere (A) became trapped in 2D, at (x,y,z) = (0,0,z), and was then pushed in the positive z-direction. Microsphere (B) was not trapped, but was also pushed in the positive z-direction when within 8 µm of the beam axis.

Fig. 4
Fig. 4

Measured z-velocity versus axial position, compared to (a) the z-velocity predicted by radiation pressure, and (b) the local intensity of a Gaussian beam. In (a), the curve labeled “ideal” is based on rigorous calculations for an ideal beam, while the curve labeled “aberrated” is a qualitative estimate of the impact of weak spherical aberration.

Fig. 5
Fig. 5

Simulated (a) temperature T and (b) magnitude of the temperature gradient in the z-direction |∂T/∂z|, in the y = 0 plane. The black square marks the liquid-glass boundary, and had sides 50 µm long. The peak temperature was 0.60 K above ambient. The white regions of (b) represent regions where |∂T/∂z| exceeds 13,000 K/m.

Fig. 6
Fig. 6

(a) Comparison of the measured z-velocity of microsphere (A) along the beam axis with predicted velocities due to thermophoresis, radiation pressure, and the combined effect of thermophoresis and radiation pressure, and (b) the predicted thermophoretic z-velocity of microsphere (B) versus y position. The radiation pressure curve in (a) matches the estimated plot for an aberrated beam in Fig. 4(a).

Fig. 7
Fig. 7

Simulated z-velocity of the liquid in sample cell, resulting from the transfer of optical momentum to the liquid due to optical absorption. Velocities shown are for the y = 0 plane. The black square marks the liquid-glass boundary, which had sides 50 µm long.

Fig. 8
Fig. 8

Predicted values of the peak radiation pressure force and the effective thermophoretic force, along the beam axis, for an individual silica nano-sphere in pure (a) propylene glycol and (b) deionized water. For the range of particle radii shown, the thermophoretic force dominates in propylene glycol. For deionized water the thermophoretic force is substantially reduced, but still dominates for smaller particle radii.

Tables (2)

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Table 1 Properties Used for Thermal Simulations

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Table 2 Material Properties Used for Model Comparison

Equations (24)

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I ( x , y , z ) 2 P π ω 2 ( z ) exp [ 2 ( x 2 + y 2 ) ω 2 ( z ) ]
ω ( z ) = ω 0 1 + ( z λ 0 π n l ω 0 2 ) 2 ,
ω 0 = λ 0 π n l tan θ 0 l λ 0 π tan θ 0 a ,
v R P = F R P 6 π η R ,
Q = 2 α I
( k T ) + Q = 0 ,
T 1 = T 2
k 1 n ^ T 1 = k 2 n ^ T 2
k c n ^ T = h ( T T ) ,
v T = D T T 0 ,
f a b s ( 2 α I n l / c ) z ^ .
D T M = 0.13 η ρ T [ 1 + ( k p / 2 k l ) ]
D T S = ( ln 3 ) A p A l 64 η r l [ 1 + ( k p / 2 k l ) ]
D T R = 2 ξ 3 η γ T
D T D = R σ e f f 2 λ D H 6 η ε R ε 0 T ( 1 T ε R ε R T )
D T B = k l β ρ c p [ 1 + ( k p / 2 k l ) ]
| T | < k B T 6 π η R 2 D T ,
| v T | < k B T 6 π η R 2 ,
F T e q = 6 π η R v T .
F T e q = 6 π η R D T T 0 .
F R P = 2 π R 3 c ( n p 2 n l 2 n p 2 + 2 n l 2 ) I + 128 π 5 n l R 6 3 λ 0 4 c ( n p 2 n l 2 n p 2 + 2 n l 2 ) 2 I ,
F a b s z = f a b s d x d y = 2 α P n l z ^ c .
F f l o w z = p z ( π a c h a n n e l 2 ) z ^ = 4 π η v max ,
v max ~ α P n l z ^ 2 π η c .

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