## Laser-induced thermophoresis of individual particles in a viscous liquid |

Optics Express, Vol. 19, Issue 11, pp. 10571-10586 (2011)

http://dx.doi.org/10.1364/OE.19.010571

Acrobat PDF (1366 KB)

### 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

## 1. Introduction

*within a particle*, and the resulting temperature gradient (known as the indirect photophoretic force), has been well explored for particles suspended in a gas [12]. Indeed, this force forms the basis for the “radiometric barrier” which must be overcome to trap particles at low gas pressure [13]. In liquids, laser-induced thermo-diffusion of large collections of nanoparticles has also been studied, but primarily in the context of thermally-induced variations in steady-state particle concentration (i.e. the Soret effect) rather than the motion of individual particles. Empirical and theoretical models describing thermo-diffusion, however, tend to differ from the observed motion of individual particles in a liquid [3,14]. Studies of laser-induced thermophoresis of individual particles in liquids have also been performed [10

10. 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]

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

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

18. 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]

10. 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]

## 2. Experiment

*(x,y)*, and the other

*(y,z)*. For the

*(y,z)*view, a lower, 0.28 numerical aperture lens was used to provide greater depth of field. 3D video was acquired by triggering the exposure of both cameras simultaneously, at 30 frames per second with a common signal, and recording the images on a computer. Illumination of the cell was performed using a pair of white light emitting diodes (LEDs), which were chosen to prevent sample heating due to infrared absorption of the illumination source. The divergence of the trap beam was measured to be 36° in air, by profiling the beam intensity with a z-scanned mirror. The optical power of the trap beam was also measured to be 38 mW in air. After accounting for Fresnel reflections in the sample cell the power was estimated to be 35 mW in the liquid.

*Δn = n*between the particle and liquid, where

_{p}-n_{l},*n*and

_{p}*n*are the refractive indices of the particle and liquid, respectively. This was accomplished by mixing dry silica microspheres (

_{l}*n*= 1.45 [20]) with 99.5% pure propylene glycol (1,2 propanediol) (

_{p}*n*= 1.43 [20]), forming a low-concentration, low index contrast (

_{l}*Δn*= 0.02) suspension. The suspension was then placed in the sample cell via capillary action. In the small-particle limit, radiation pressure can be separated into gradient and scattering force components which scale roughly as

*Δn*and

*(Δn)*, respectively [17]. Thus, compared to polystyrene spheres in water (

^{2}*Δn*= 0.25) [21], the radiation pressure in the experiment should have been substantially reduced. Furthermore, since the optical absorption of propylene glycol is similar to that of water [22], as is its thermal diffusivity, the temperature distribution within the cell should have been comparable to that of a water-filled cell. An additional advantage of propylene glycol was that it provided a relatively high viscosity, 0.055 kg/m·s, which served to dampen Brownian motion of the particles in the experiment, and to suppress convective flow.

^{2}intensity limits of the optical beam, as calculated by approximating a Gaussian beam profile of the formwhere

*P*is the optical power,

*λ*is the vacuum wavelength,

_{0}*ω*is size of the beam waist, and

_{0}*θ*and

_{0l}*θ*are the beam divergences in liquid and air, respectively [23]. Using the measured beam divergence in air, the waist size was estimated to be 0.4 µm. The location of the beam waist in the video, which corresponded to the origin of the coordinate system, was calibrated by moving the cell in

_{0a}*z*with a precision micrometer to focus on different microspheres with camera 1, and noting their locations in the camera 2 view. Since the beam waist coincided with the image focus (in air), this calibration was taken as a reasonable estimate of the beam waist position. The location of the beam axis (

*x,y*) = (0, 0) was readily determined by imaging the reflected beam profile with camera 1. With this calibration, the central axis of the capillary tube coincided with the coordinates (

*x,z*) = (−1.6,14) µm.

*y*direction, which was due to a combination of gravitational force (resisted by viscous drag) and a creeping laminar flow resulting from evaporation from the open ends of the capillary. Upon nearing the laser beam, microsphere

**A**was pulled rapidly into the center of the beam, becoming trapped in

*x*and

*y*, and subsequently pushed along the beam a distance of 24 µm before being released from the trap and resuming its downward motion. In the same timeframe, microsphere

**B**passed by the optical beam, with no noticeable change in direction except when nearest the optical beam.

**A**and

**B**in 3D as a function of time, which were determined by particle tracking analysis of the image sequences from both cameras. As shown, microsphere

**A**was initially pulled into the beam and pushed rapidly towards the waist, and then upon passing the waist exhibited slow but relatively constant velocity along the direction of the beam. Past the beam waist, it moved with an average velocity 0.12 µm/s over a 19 µm distance, while at the same time remaining trapped in

*x*and

*y*. Microsphere

**B**, on the other hand, which was offset from the beam axis by −5.5 µm in the x-direction, and therefore never trapped, exhibited a similar 0.11 µm/s velocity when within 8 µm of the beam axis. This was despite the fact that microspheres

**A**and

**B**never came in contact with each other, as their radii were 1.6 µm and 1.7 µm, respectively (as measured using camera 1), but their closest center-to-center spacing was 7.5 µm. The difference in y-velocity between the two microspheres, despite their similar size, can be attributed to the creeping fluid flow in the negative y-direction, which would be expected to exhibit a parabolic velocity profile. This flow was estimated to have a peak y-velocity of 0.2 µm/s, by tracking the y-velocities of various microspheres of different size and location within the cell, and correcting for terminal velocity due to gravity.

## 3. Analysis

**A**is plotted versus position along the beam axis in Fig. 4(a) . This curve was calculated by first fitting the position versus time data to a smoothed spline and then differentiating, which helped to reduce the effects of noise (though noise due to mechanical vibrations and particle tracking error was accentuated by differentiation, the overall trend is clear in both Figs. 3 and 4). As shown, microsphere

**A**experienced a rapid acceleration towards the beam waist (at

*z*= 0), which then all but vanished upon reaching the waist. Such behavior is qualitatively consistent with a 3D optical trap, which pushes a particle towards an equilibrium point located just beyond the beam waist via radiation pressure [17]. The motion observed beyond and away from the waist, in the region

*z>0*, was, however, inconsistent with a stable 3D trap.

### 3.1 Radiation pressure

**A**, using an approach based on Nieminen [24–26]. Since details of these calculations are not of immediate importance, the methodology has been summarized in Appendix A. Here it suffices to note that the calculations were performed assuming a 0.4 µm beam waist, 1.6 µm particle radius, 35 mW optical power, and liquid and particle refractive indices of 1.43 and 1.45, respectively. The calculated optical force

**on the microsphere was subsequently converted to a velocity**

*F*_{RP}**using Stokes’ drag relationwhere**

*v*_{RP}*η*is the dynamic viscosity of the liquid, 0.055 kg/m·s [27], and

*R*the radius of the sphere, 1.6 µm.

**A**as a function of position along the beam axis is plotted in Fig. 4(a). As shown, in the region

*z ≤ 0*the measured velocity was in reasonably good agreement with that predicted due to radiation pressure, particularly in terms of its magnitude. On the opposite side of the waist, however, agreement was extremely poor. One could ask whether this discrepancy was due to an incorrect assumption of a perfect, aberration-free optical beam in the radiation pressure calculations. It is known that even slight spherical aberration along the optical path, which tends to blur the focused spot, will significantly weaken the axial restoring force on the positive

*z*side of the waist [28

28. 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]

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

*z≤ 0*region remains largely unchanged, while the reverse force for

*z>0*is significantly reduced. Based on data presented in such works, we have included a

*qualitative estimate*of how the velocity of the microspheres might have been impacted by slight spherical beam aberration. This serves to illustrate that the axial restoring force for

*z>0*due to radiation pressure may have been significantly weaker than that predicted for the ideal beam. This is an important point, because the discrepancy between radiation pressure calculations and experiment in the region

*z>0*, both in magnitude and direction, force us to consider additional contributions to the motion of the particle, which must ultimately be reconciled with the restoring force due to radiation pressure. Spherical aberration was also to be expected in the experiment, since the lens was aberration-corrected for a 170 µm thick coverslip, but the experimental focus was only 39 µm into the sample cell. In the following subsection it will be shown that this qualitative estimate leads to a consistent description of the observed motion, in which radiation pressure dominates for

*z*<0 and thermophoresis dominates for

*z*>8 µm, despite reasonable uncertainty due to aberration in the region between.

**A**to the normalized intensity along the beam axis, defined as

*I*. As shown, the rapid

_{norm}≡ I(0,0,z)/ I(0,0,0)*1/z*roll-off in intensity for large z was very dissimilar to the relatively constant velocity observed from

^{2}*z*= 0 to 18 µm. Based on this simple observation, a direct relationship between the local intensity and the observed motion in the range

*z > 0*could be ruled out. This was important, because it allowed a wide range of effects to be confidently ruled out, including indirect photophoresis, momentum transfer due to optical absorption within the particle, and radiation pressure, all of which would have produced forces proportional to either the local intensity or its gradient. Based on this result, it was necessary to consider an indirect relationship between the local intensity and the observed motion in the region

*z > 0.*

### 3.2 Thermophoresis

*y*= ± 1.6 mm in the simulation, were modeled as thermal insulators. For reasons to be given shortly, the temperature profile in the cell was modeled without taking the presence of any microspheres into account.

*xz*plane (

*y*= 0). The magnitude of the z-component of the temperature gradient, |

*∂T/∂z|,*is also plotted in Fig. 5(b). Note that the peak simulated temperature in the cell was only 0.60 K above the ambient temperature. This is significant because it indicates that temperature variations in the cell only weakly-perturbed the material properties in the experiment.

*∇T*, where

_{0}*T*is the temperature in the

_{0}*absence*of the particle. Under such a condition, each model yields an expression of the formfor the thermophoretic velocity of a spherical particle in a liquid. The thermal diffusivity

*D*takes different forms for the different models, and will be discussed in greater detail in the following section. The present emphasis is on comparing the form of Eq. (10) to measured data.

_{T}*∇T*instead vary little over the extent of the particle to appropriately apply Eq. (10). This condition is reasonably well-satisfied for a 3.2 µm diameter particle near the center of the cell, and thus the latter portion of the trajectory of microsphere

_{0}**A**. Closer to the beam waist, however, variations in the temperature gradient over the extent of this microsphere were expected to introduce error when applying Eq. (10). For microsphere

**B**the rapid variation in the

*x*and

*y*components of the temperature gradient (nonexistent for microsphere

**A**) would also have introduced error when using Eq. (10). Thus, in the following it is understood that the application of Eq. (10) to the temperature profile of Fig. 5(a) provides approximate results, except for the latter portion of the trajectory of microsphere

**A**where error should be negligible. Furthermore, the presence of each microsphere in the liquid would have altered the temperature profile, not only due to heat conduction, which is accounted for in

*D*, but also by altering the local heat density

_{T}*Q*, due to the negligible optical absorption of the particle. This effect was most pronounced when the sphere is at the beam focus, but elsewhere relatively weak.

**A**at different points along the beam axis was calculated using Eq. (10) and the simulated temperature gradient, and is plotted in Fig. 6(a) . For this calculation a thermal diffusivity of

*D*= 22 µm

_{T}^{2}/s·K was assumed in order to provide a best fit to the measured data, which will later be shown to be consistent with theory. Note that velocity predictions have been omitted very close to the beam waist, where

*∇T*varied substantially over the extent of a 3.2 µm diameter microsphere, and where the (relatively low) absorption of the microsphere would have most impacted the thermal profile. Figure 6(a) also plots the measured velocity of microsphere

_{0}**A**, and the estimated velocity due to radiation pressure from Fig. 4(a) for an aberrated beam. The most important result of Fig. 6(a) is that thermophoresis accounts for the observed nonzero velocity of the particle far from the beam waist, where radiation pressure falls out of consideration. Also significant is that the combined effect of radiation pressure and thermophoresis, plotted as the sum of the two velocities in the figure, adequately describes the relatively constant velocity observed in the experiment, over a range where the local intensity varied by over an order of magnitude. Although the radiation pressure in Fig. 6(a) was merely an estimate (for

*z*>0), the resulting agreement between theory and experiment provides justification for its use.

*z*> 3 µm was dominated by thermophoresis, but it does not constitute conclusive proof. The case can be strengthened, however, by considering the trajectory of microsphere

**B.**This sphere fell continuously along a path that remained constant in

*x*(

*x*= −5.5µm), but varied slightly in

*z*(

*z*= 13 to 17 µm), due to a positive z-velocity that was only discernable over the limited range

*y*= −6 to 6 µm. For the purpose of comparison, Fig. 6(b) plots the thermophoretic z-velocity predicted for a similar path, along the line (

*x, z*) = (−5.5, 15) µm. This figure shows that thermophoresis should only have been significant over a very similar range of

*y*. The slight discrepancy between the predicted 8 µm FWHM interaction region and the measured 12 µm range was easily attributable to the variation in

*∇T*over the 3.4 µm diameter of the sphere. Such agreement between the measured particle velocity and the local temperature gradient, along multiple axes, provides strong support for the conclusion that the observed motion in the region z > 3 µm was due to thermophoresis.

### 3.3 Transfer of optical momentum to the liquid

32. 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**n*, where

_{l}/c**is the Poynting vector and**

*S**c*the vacuum speed of light, it can easily be shown that absorption of optical momentum by the liquid produces to the local force density

## 4. Discussion

*D*used in the preceding calculation with that of various theoretical models is in order. For this purpose, we consider models proposed by McNab and Meisen [4], Schimpf and Semenov [3,6], Ruckenstein [8,9

_{T}9. A. Parola and R. Piazza, “Particle thermophoresis in liquids,” Eur Phys J E Soft Matter **15**(3), 255–263 (2004). [PubMed]

10. 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]

*D*,

_{TM}, D_{TS}*D*,

_{TR}*D*and

_{TD}*D*, respectively, are given by

_{TB}*ρ, β, c*,

_{p}*ε*,

_{R}*r*and

_{l}*λ*represent the liquid’s density, volumetric thermal expansion coefficient, isobaric specific heat, relative permittivity, molecular radius, and Debye length, respectively, and

_{DH}*k*is the particle’s thermal conductivity. The Hamaker constants for the particle and liquid are denoted as

_{p}*A*and

_{p}*A*. Table 2 lists the relevant quantities for propylene glycol and silica, as well as the computed thermal diffusivities. As the interfacial tension of the propylene glycol-silica interface

_{l}*γ,*and the interaction layer thickness

*ξ*were not known [1],

*D*was excluded from Table 2, but is discussed below. As the effective surface charge per unit area of the microsphere

_{TR}*σ*was also unknown,

_{eff}*D*is also discussed independently below.

_{TD}### 4.1 The McNab-Meisen model

*D*exceeds the experimentally-fit value of22 µm

_{TM}^{2}/s·K by over two orders of magnitude, so this model could not be regarded as consistent with the measured data.

### 4.2 The Schimpf-Semenov dipole-dipole model

*D*was over two orders of magnitude lower than the experimentally-fit value of 22 µm

_{TS}^{2}/s·K. Thus, the Schimpf-Semenov model was inconsistent with the thermophoretic motion observed in the experiment.

### 4.3 The Ruckenstein model

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

^{−5}N/m·K for propylene glycol. In order to match

*D*to the experimentally-fit value 22 µm

_{TR}^{2}/s·K, the interaction layer thickness ξ then needed to be 26 nm. By way of comparison, if the same procedure were performed for water and silica, using the measured thermal diffusivity 22 µm

^{2}/s·K for silica in pure, deionized water [14], and the interfacial tension ∂γ/∂T ≈1.6x10

^{−4}N/m·K [20], the interaction layer thickness would instead have been 0.18 nm. Given the implausibility that the interaction layer thickness for interfacial tension should vary so substantially between the two similar liquids, the Ruckenstein model was not regarded as a viable explanation for the observed thermophoretic motion.

### 4.4 The Duhr-Braun ionic-shielding model

### 4.5 The Brenner model

*D*was within a factor of 1.8 of the experimentally-fit value 22µm

_{TB}^{2}/s·K. This small discrepancy could be attributed to the non-uniformity of

*∇T*in the experiment, or inaccuracy of the absorption coefficient

_{0}*2α*and/or heat transfer coefficient

_{l}*h*used in the thermal model. It also may be due to the fact that Eq. (16) did not account for “generation of volume” in the experiment due to optical absorption (for a discussion of this effect see [38]). Nonetheless, the relatively good agreement demonstrates a sound theoretical basis for the thermophoretic calculations presented in Section 3. Additionally, these results provide support for the Brenner model of thermophoresis, which has not been rigorously tested due to the scarcity of experimental data for thermophoretic motion of individual particles in a liquid [3,5].

## 5. Extension to small particles in an optical trap

*effective*thermophoretic force acting on a spherical particle in a temperature gradient, defined in analogy to Eq. (4), as

*R.*Such scaling is also supported by a body of work on thermo-diffusion, which indicates that

*D*is for the most part independent of

_{T}*R*in the quasi-hydrodynamic limit [1] (with the notable exception of strongly-electrolytic solutions [10

**103**(52), 19678–19682 (2006). [PubMed]

*R*. Thus, with decreasing particle size, thermophoresis can eventually dominate.

^{3}^{3}. At the same position the temperature gradient was 0.041 K/µm. Using these values as an example, along with material properties already provided, the magnitudes of both the effective thermophoretic force and the maximum force due to radiation pressure were calculated for propylene glycol and silica, from Eqs. (20) and (21), and

*D*= 22 µm

_{T}^{2}/s·K, and plotted versus particle radius in Fig. 8(a) . This figure also plots the force due to radiation pressure for a polystyrene particle with refractive index 1.58. Also included are the effective thermophoretic forces for power absorption coefficient of 19 m

^{−1}, which closely matches that of both propylene glycol and water at 1064 nm, and the power absorption coefficient 4900 m

^{−1}, matching that of water at 1480 nm. Note that the latter two curves provide a comparison of the

*relative*impact of thermophoresis and radiation pressure. However, in practice, temperature constraints may limit the optical power that can be used, in which case both forces would decrease linearly with

*P*.

*R*≤ 100 nm). For a Gaussian beam, the effective thermophoretic force is directed away from the focus, such that thermophoresis serves to inhibit trapping in 3D. Thus, the ability (or inability) of a Gaussian beam to trap small particles depends not only on random thermal motion, governed by

*k*, but also on the effective thermophoretic force.

_{B}T^{2}/s·K [14], refractive index, 1.33 [20], and dynamic viscosity, 0.00089 kg/m·s [20]. For simplicity, the intensity and temperature gradients used in these calculations were identical to those used for propylene glycol (0.105 W/µm

^{3}and 0.041 K/µm, respectively), although both would have been slightly different in water. As shown in Fig. 8, the effective thermophoretic force is predicted to be much weaker in deionized water than in propylene glycol. This is a consequence of the fact that the thermal diffusivities of the two liquids, both 22 µm

^{2}/s·K, are remarkably similar. According to Eq. (20), for an identical particle radius and temperature gradient, the effective thermophoretic force should thus be roughly sixty times larger in propylene glycol than in deionized water (due to the relatively low viscosity of water). That is, the thermophoretic force scales with viscosity, since the thermal diffusivity, apparently, does not. Such a concept may seem counterintuitive, but would appear to be consistent with Brenner’s theories on the nature of viscous stress in a non-isothermal fluid, and thermophoresis in general [3,11,38].

## 6. Conclusion

## Appendix A

*et al*in [24]. Briefly, the incident beam is written as a sum of vector spherical wavefunctions (VSWFs) where the coefficients of the VSWFs that correspond to the incident Gaussian beam are calculated using a point-matching procedure in the far field [25]. First, the far field limit for a paraxial y-polarized Gaussian beam as a function of the beam waist is used to estimate the field at a defined set of points. The VSWFs are then calculated at each of those points and the resulting overdetermined system of equations is solved in a least-squares sense for the coefficients of the VSWF basis.

**P**, are written as a function of the incident beam coefficients,

**A**, and the T-matrix as

**P=TA**. For a spherical particle, the T-matrix is diagonal with elements given by the usual Mie coefficients. Given the incident and scattered coefficients, the force and torque on the particle can be computed [24].

## Appendix B

## References and links

1. | R. Piazza and A. Parola, “Thermophoresis in colloidal suspensions,” J. Phys. Condens. Matter |

2. | J. K. Platten, “The Soret effect: a review of recent experimental results,” J. Appl. Mech. |

3. | H. Brenner, “Navier-Stokes revisited,” Physica A |

4. | G. S. McNab and A. Meisen, “Thermophoresis in liquids,” J. Colloid Interface Sci. |

5. | H. Brenner and J. R. Bielenberg, “A continuum approach to phoretic motions: thermophoresis,” Physica A |

6. | M. E. Schimpf and S. N. Semenov, “Mechanism of polymer thermophoresis in nonaqueous solvents,” J. Phys. Chem. B |

7. | 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. |

8. | E. Ruckenstein, “Can phoretic motions be treated as interfacial tension gradient driven phenomena,” J. Colloid Interface Sci. |

9. | A. Parola and R. Piazza, “Particle thermophoresis in liquids,” Eur Phys J E Soft Matter |

10. | S. Duhr and D. Braun, “Why molecules move along a temperature gradient,” Proc. Natl. Acad. Sci. U.S.A. |

11. | 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 |

12. | O. Jovanovic, “Photophoresis: light-induced motion of particles suspended in gas,” J. Quant. Spectrosc. Radiat. Transf. |

13. | A. Ashkin, |

14. | 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 |

15. | S. Duhr and D. Braun, “Thermophoretic depletion follows Boltzmann distribution,” Phys. Rev. Lett. |

16. | R. Di Leonardo, F. Ianni, and G. Ruocco, “Colloidal attraction induced by a temperature gradient,” Langmuir |

17. | K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. |

18. | J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. |

19. | M. Dienerowitz, M. Mazilu, and K. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophotonics |

20. | W. M. Haynes, ed., |

21. | 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. |

22. | T. P. Otanicar, P. E. Phelan, and J. S. Golden, “Optical properties of liquids for direct absorption solar thermal energy systems,” Sol. Energy |

23. | A. Yariv, |

24. | 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. |

25. | T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focused laser beams,” J. Quant. Spectrosc. Radiat. Transf. |

26. | 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. |

27. | 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 |

28. | 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. |

29. | Y. Roichman, A. Waldron, E. Gardel, and D. G. Grier, “Optical traps with geometric aberrations,” Appl. Opt. |

30. | J. P. Holman, |

31. | F. W. Schmidt, R. E. Henderson, and C. H. Wolgemuth, |

32. | S. Duhr, S. Arduini, and D. Braun, “Thermophoresis of DNA determined by microfluidic fluorescence,” Eur Phys J E Soft Matter |

33. | T. Cosgrove, |

34. | J. Berg, |

35. | B. C. Hoke Jr and E. F. Patton, “Surface tensions of propylene glycol + water,” J. Chem. Eng. Data |

36. | 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 |

37. | H. J. Butt, K. Graf, and M. Kappl, |

38. | H. Brenner, “Kinematics of volume transport,” Physica A |

39. | H. Bruus, |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(170.4520) Medical optics and biotechnology : Optical confinement and manipulation

(350.5340) Other areas of optics : Photothermal effects

(160.4236) Materials : Nanomaterials

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: March 14, 2011

Revised Manuscript: April 18, 2011

Manuscript Accepted: May 1, 2011

Published: May 13, 2011

**Virtual Issues**

Vol. 6, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Ross T. Schermer, Colin C. Olson, J. Patrick Coleman, and Frank Bucholtz, "Laser-induced thermophoresis of individual particles in a viscous liquid," Opt. Express **19**, 10571-10586 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10571

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### References

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