## Hydrodynamic measurement of Brownian particles at a liquid-solid interface by low-coherence dynamic light scattering

Optics Express, Vol. 18, Issue 7, pp. 7390-7396 (2010)

http://dx.doi.org/10.1364/OE.18.007390

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### Abstract

The hydrodynamics of Brownian particles close to a wall is investigated using low-coherence dynamic light scattering. The diffusion coefficient of the particles in a suspension is measured as a function of distance from the wall. A sudden reduction in the diffusion coefficient near the interface is clearly observed using this method. The theoretically predicted wall-drag effect is experimentally confirmed when the influence of the spatial resolution due to the finite coherence length of the light source is accounted for. The space-dependent dynamics of Brownian particles under the wall-drag effect is obtained for the first time using our spatially resolved dynamic light scattering technique.

© 2010 OSA

## 1. Introduction

1. H. Brenner, “The slow motion of a sphere through a viscous fluid towards a plane surface,” Chem. Eng. Sci. **16**(3-4), 242 – 251 (1961). [CrossRef]

2. A. J. Goldman, R. G. Cox, and H. Brenner, “Slow viscous motion of a sphere parallel to a plane wall – I Motion through a quiescent fluid,” Chem. Eng. Sci. **22**(4), 637 – 651 (1967). [CrossRef]

3. M. I. M. Feitosa and O. N. Mesquita, “Wall-drag effect on diffusion of colloidal particles near surfaces: A photon correlation study,” Phys. Rev. A **44**(10), 6677 – 6685 (1991). [CrossRef] [PubMed]

4. L. Lobry and N. Ostrowsky, “Diffusion of Brownian particles trapped between two walls: theory and dynamic-light scattering measurements,” Phys. Rev. B **53**(18), 12050 – 12056 (1996). [CrossRef]

5. P. G. Cummins and E. J. Staples, “Particle size measurements on turbid latex systems using heterodyne intensity autocorrelation spectroscopy,” J. Phys. E Sci. Instrum. **14**(10), 1171 – 1177 (1981). [CrossRef]

6. K. H. Lan, N. Ostrowsky, and D. Sornette, “Brownian dynamics close to a wall studied by photon correlation spectroscopy from an evanescent wave,” Phys. Rev. Lett. **57**(1), 17 – 20 (1986). [CrossRef] [PubMed]

8. M. Hosoda, K. Sakai, and K. Takagi, “Measurement of anisotropic Brownian motion near an interface by evanescent light-scattering spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(5), 6275 – 6280 (1998). [CrossRef]

9. D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. **23**(5), 319 – 321 (1998). [CrossRef]

11. K. Ishii, R. Yoshida, and T. Iwai, “Single-scattering spectroscopy for extremely dense colloidal suspensions by use of a low-coherence interferometer,” Opt. Lett. **30**(5), 555 – 557 (2005). [CrossRef] [PubMed]

10. G. Popescu and A. Dogariu, “Dynamic light scattering in localized coherence volumes,” Opt. Lett. **26**(8), 551 – 553 (2001). [CrossRef]

11. K. Ishii, R. Yoshida, and T. Iwai, “Single-scattering spectroscopy for extremely dense colloidal suspensions by use of a low-coherence interferometer,” Opt. Lett. **30**(5), 555 – 557 (2005). [CrossRef] [PubMed]

12. H. Xia, K. Ishii, and T. Iwai, “Hydrodynamic radius sizing of nanoparticles in dense polydisperse media by low-coherence dynamic light scattering,” Jpn. J. Appl. Phys. **44**(8), 6261 – 6264 (2005). [CrossRef]

## 2. Experimental setup

13. K. Ishii, T. Iwai, and S. Nakamura, “Numerical analysis of a path-length-resolved spectrum of time-varying scattered light field,” J. Opt. Soc. Am. A **25**(3), 718 – 724 (2008). [CrossRef]

*L*= 0) is defined such that the path length of the reference beam coincides with that of the light reflected from the glass−suspension interface. The optical path length

*L*can be varied by moving the reference mirror. Thus, the autocorrelation function of the scattered light amplitude is measured by the low-coherence DLS technique while the depth of the probe point can be varied by adjusting the position of the reference mirror.

*τ*is the delay time,

*D*is the diffusion coefficient, and

*q*is the magnitude of the scattering vector, and the relaxation time is defined as

*D*can be estimated by nonlinear fitting the normalized autocorrelation function obtained by experiments using Eq. (1). In the experimental configuration depicted in Fig. 1, only the component of the diffusion coefficient perpendicular to the wall is estimated. By varying the probe position in the scattering volume, the diffusion coefficient can be estimated as a function of distance from the wall. 10 vol% suspensions of monodisperse polystyrene particles with mean radii of 0.24, 0.40, 0.55, and 1.49 μm manufactured by Sigma-Aldrich Corporation were used as colloidal suspensions. We use the dense suspension to demonstrate LCDLS can be applied to highly scattering media without dilution, which is one of unique feature.

## 3. Results and discussion

*L*= 4, 8, and 15 μm. The solid lines indicate the results of fitting the experimental data with the single exponential decay function given by Eq. (1). The vertical axis in Fig. 2 has a logarithmic scale. In this plot, all the experimentally obtained autocorrelation functions decrease approximately linearly, in accordance with the single exponential function. According to literature [13

13. K. Ishii, T. Iwai, and S. Nakamura, “Numerical analysis of a path-length-resolved spectrum of time-varying scattered light field,” J. Opt. Soc. Am. A **25**(3), 718 – 724 (2008). [CrossRef]

14. K. K. Bizheva, A. M. Siegel, and D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: The transition to diffusing wave spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **58**(6), 7664 – 7667 (1998). [CrossRef]

*L*comes from the constraint of Brownian motion of the particles due to the wall-drag effect close to the interface. The auto correlation function for

*L*= 4 deviates from the single-exponential decrease in the region of long delay time. This comes from the decrease of signal-to-noise ratio because a part of the detection area is out of the sample and detected signal is weak compared with other cases.

*k*

_{B}is the Boltzmann constant,

*T*is the absolute temperature,

*η*

_{0}is the viscosity of water, and

*R*is the particle radius. In concentrated media, hydrodynamic interactions between particles also play a significant role in particle diffusion. Thus, the hydrodynamic interaction between particles should be taken into account so that the diffusion coefficient depends on the volume fraction of the medium, which is given by [15

15. G. K. Batchelor, “Brownian diffusion of particles with hydrodynamic interaction,” J. Fluid Mech. **74**(01), 1 (1976). [CrossRef]

*ϕ*is the volume fraction of the medium. The decrease of the diffusion coefficient in dense media has been experimentally observed by means of LCDLS [16

16. H. Xia, K. Ishii, T. Iwaii, H. Li, and B. Yang, “Dynamics of interacting Brownian particles in concentrated colloidal suspensions,” Appl. Opt. **47**(9), 1257 – 1262 (2008). [CrossRef] [PubMed]

1. H. Brenner, “The slow motion of a sphere through a viscous fluid towards a plane surface,” Chem. Eng. Sci. **16**(3-4), 242 – 251 (1961). [CrossRef]

*R*is the particle radius, and

*s*is the distance between the particle and the interface. Figure 3 shows the ratio of

*D*

_{h}for spherical particles as a function of the distance

*s*calculated numerically using Eq. (4) for radii 0.235 and 1.485 μm. The ordinate represents the wall-drag effective diffusion coefficients normalized by

*D*

_{h}, while the abscissa shows the distance between the particle and the interface. Brownian motion is constrained close to the interface, especially for motion perpendicular to the interface, and the diffusion coefficient decreases drastically in that region. The wall-drag effect increases with increasing particle size.

*D*is influenced by the spatial resolution of the low-coherence DLS. The effective diffusion coefficient, which is defined as the diffusion coefficient

*D*

_{w⊥}under the influence of the wall-drag effect, is convoluted with the light source coherence function

*γ*(

*l*) as follows:

*D*

_{h}, which is the diffusion coefficient of the particles in a 10 vol% latex suspension. Here, the distance

*s*between the particle and the interface is equal to the depth of the probe position, which is the optical path length divided by the refractive index of the suspension,

*n*= 1.33). The solid lines are calculated from Eq. (7). Figure 5 shows that the diffusion coefficient decreases suddenly as the probe position becomes closer to the wall. The measured diffusion coefficient is in reasonable agreement with the theoretical prediction for the wall-drag effect when the spatial resolution of the system is taken into account.

17. Y. Wang, Y. Zhao, J. S. Nelson, Z. Chen, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber,” Opt. Lett. **28**(3), 182 – 184 (2003). [CrossRef] [PubMed]

## 4. Conclusions

## References and links

1. | H. Brenner, “The slow motion of a sphere through a viscous fluid towards a plane surface,” Chem. Eng. Sci. |

2. | A. J. Goldman, R. G. Cox, and H. Brenner, “Slow viscous motion of a sphere parallel to a plane wall – I Motion through a quiescent fluid,” Chem. Eng. Sci. |

3. | M. I. M. Feitosa and O. N. Mesquita, “Wall-drag effect on diffusion of colloidal particles near surfaces: A photon correlation study,” Phys. Rev. A |

4. | L. Lobry and N. Ostrowsky, “Diffusion of Brownian particles trapped between two walls: theory and dynamic-light scattering measurements,” Phys. Rev. B |

5. | P. G. Cummins and E. J. Staples, “Particle size measurements on turbid latex systems using heterodyne intensity autocorrelation spectroscopy,” J. Phys. E Sci. Instrum. |

6. | K. H. Lan, N. Ostrowsky, and D. Sornette, “Brownian dynamics close to a wall studied by photon correlation spectroscopy from an evanescent wave,” Phys. Rev. Lett. |

7. | N. Garnier and N. Ostrowsky, “Brownian dynamics in a confined geometry. Experiments and numerical simulations,” J. Phys. II France |

8. | M. Hosoda, K. Sakai, and K. Takagi, “Measurement of anisotropic Brownian motion near an interface by evanescent light-scattering spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

9. | D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. |

10. | G. Popescu and A. Dogariu, “Dynamic light scattering in localized coherence volumes,” Opt. Lett. |

11. | K. Ishii, R. Yoshida, and T. Iwai, “Single-scattering spectroscopy for extremely dense colloidal suspensions by use of a low-coherence interferometer,” Opt. Lett. |

12. | H. Xia, K. Ishii, and T. Iwai, “Hydrodynamic radius sizing of nanoparticles in dense polydisperse media by low-coherence dynamic light scattering,” Jpn. J. Appl. Phys. |

13. | K. Ishii, T. Iwai, and S. Nakamura, “Numerical analysis of a path-length-resolved spectrum of time-varying scattered light field,” J. Opt. Soc. Am. A |

14. | K. K. Bizheva, A. M. Siegel, and D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: The transition to diffusing wave spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

15. | G. K. Batchelor, “Brownian diffusion of particles with hydrodynamic interaction,” J. Fluid Mech. |

16. | H. Xia, K. Ishii, T. Iwaii, H. Li, and B. Yang, “Dynamics of interacting Brownian particles in concentrated colloidal suspensions,” Appl. Opt. |

17. | Y. Wang, Y. Zhao, J. S. Nelson, Z. Chen, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber,” Opt. Lett. |

**OCIS Codes**

(290.5820) Scattering : Scattering measurements

(290.5850) Scattering : Scattering, particles

(290.7050) Scattering : Turbid media

(300.6310) Spectroscopy : Spectroscopy, heterodyne

**ToC Category:**

Scattering

**History**

Original Manuscript: January 4, 2010

Revised Manuscript: March 5, 2010

Manuscript Accepted: March 15, 2010

Published: March 25, 2010

**Citation**

Katsuhiro Ishii, Toshiaki Iwai, and Hui Xia, "Hydrodynamic measurement of Brownian particles at a liquid-solid interface by low-coherence dynamic light scattering," Opt. Express **18**, 7390-7396 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7390

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### References

- H. Brenner, “The slow motion of a sphere through a viscous fluid towards a plane surface,” Chem. Eng. Sci. 16(3-4), 242 – 251 (1961). [CrossRef]
- A. J. Goldman, R. G. Cox, and H. Brenner, “Slow viscous motion of a sphere parallel to a plane wall – I Motion through a quiescent fluid,” Chem. Eng. Sci. 22(4), 637 – 651 (1967). [CrossRef]
- M. I. M. Feitosa and O. N. Mesquita, “Wall-drag effect on diffusion of colloidal particles near surfaces: A photon correlation study,” Phys. Rev. A 44(10), 6677 – 6685 (1991). [CrossRef] [PubMed]
- L. Lobry and N. Ostrowsky, “Diffusion of Brownian particles trapped between two walls: theory and dynamic-light scattering measurements,” Phys. Rev. B 53(18), 12050 – 12056 (1996). [CrossRef]
- P. G. Cummins and E. J. Staples, “Particle size measurements on turbid latex systems using heterodyne intensity autocorrelation spectroscopy,” J. Phys. E Sci. Instrum. 14(10), 1171 – 1177 (1981). [CrossRef]
- K. H. Lan, N. Ostrowsky, and D. Sornette, “Brownian dynamics close to a wall studied by photon correlation spectroscopy from an evanescent wave,” Phys. Rev. Lett. 57(1), 17 – 20 (1986). [CrossRef] [PubMed]
- N. Garnier and N. Ostrowsky, “Brownian dynamics in a confined geometry. Experiments and numerical simulations,” J. Phys. II France 1(10), 1221 – 1232 (1991). [CrossRef]
- M. Hosoda, K. Sakai, and K. Takagi, “Measurement of anisotropic Brownian motion near an interface by evanescent light-scattering spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(5), 6275 – 6280 (1998). [CrossRef]
- D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion within highly scattering media,” Opt. Lett. 23(5), 319 – 321 (1998). [CrossRef]
- G. Popescu and A. Dogariu, “Dynamic light scattering in localized coherence volumes,” Opt. Lett. 26(8), 551 – 553 (2001). [CrossRef]
- K. Ishii, R. Yoshida, and T. Iwai, “Single-scattering spectroscopy for extremely dense colloidal suspensions by use of a low-coherence interferometer,” Opt. Lett. 30(5), 555 – 557 (2005). [CrossRef] [PubMed]
- H. Xia, K. Ishii, and T. Iwai, “Hydrodynamic radius sizing of nanoparticles in dense polydisperse media by low-coherence dynamic light scattering,” Jpn. J. Appl. Phys. 44(8), 6261 – 6264 (2005). [CrossRef]
- K. Ishii, T. Iwai, and S. Nakamura, “Numerical analysis of a path-length-resolved spectrum of time-varying scattered light field,” J. Opt. Soc. Am. A 25(3), 718 – 724 (2008). [CrossRef]
- K. K. Bizheva, A. M. Siegel, and D. A. Boas, “Path-length-resolved dynamic light scattering in highly scattering random media: The transition to diffusing wave spectroscopy,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 58(6), 7664 – 7667 (1998). [CrossRef]
- G. K. Batchelor, “Brownian diffusion of particles with hydrodynamic interaction,” J. Fluid Mech. 74(01), 1 (1976). [CrossRef]
- H. Xia, K. Ishii, T. Iwaii, H. Li, and B. Yang, “Dynamics of interacting Brownian particles in concentrated colloidal suspensions,” Appl. Opt. 47(9), 1257 – 1262 (2008). [CrossRef] [PubMed]
- Y. Wang, Y. Zhao, J. S. Nelson, Z. Chen, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography by broadband continuum generation from a photonic crystal fiber,” Opt. Lett. 28(3), 182 – 184 (2003). [CrossRef] [PubMed]

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