## Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy |

Optics Express, Vol. 19, Issue 9, pp. 8051-8065 (2011)

http://dx.doi.org/10.1364/OE.19.008051

Acrobat PDF (4144 KB)

### Abstract

We discuss a new method for simultaneously probing translational, rotational, and vibrational dynamics in dilute colloidal suspensions using digital holographic microscopy (DHM). We record digital holograms of clusters of 1-*μ*m-diameter colloidal spheres interacting through short-range attractions, and we fit the holograms to an exact model of the scattering from multiple spheres. The model, based on the T-matrix formulation, accounts for multiple scattering and near-field coupling. We also explicitly account for the non-asymptotic radial decay of the scattered fields, allowing us to accurately fit holograms recorded with the focal plane located as little as 15 *μ*m from the particle. Applying the fitting technique to a time-series of holograms of Brownian dimers allows simultaneous measurement of six dynamical modes — three translational, two rotational, and one vibrational — on timescales ranging from 10^{−3} to 1 s. We measure the translational and rotational diffusion constants to a precision of 0.6%, and we use the vibrational data to measure the interaction potential between the spheres to a precision of ∼50 nm in separation distance. Finally, we show that the fitting technique can be used to measure dynamics of clusters containing three or more spheres.

© 2011 OSA

## 1. Introduction

2. M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud’homme, M. Beck, and D. A. Weitz, “Investigating the microenvironments of inhomogeneous soft materials with multiple particle tracking,” Phys. Rev. E **64**, 061506 (2001). [CrossRef]

3. M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled f-actin solutions,” Phys. Rev. Lett. **91**, 158302 (2003). [CrossRef] [PubMed]

5. E. Andablo-Reyes, P. Díaz-Leyva, and J. L. Arauz-Lara, “Microrheology from rotational diffusion of colloidal particles,” Phys. Rev. Lett. **94**, 106001 (2005). [CrossRef] [PubMed]

8. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

10. S. M. Anthony, M. Kim, and S. Granick, “Translation-rotation decoupling of colloidal clusters of various symmetries,” J. Chem. Phys. **129**, 244701 (2008). [CrossRef]

11. D. Mukhija and M. J. Solomon, “Translational and rotational dynamics of colloidal rods by direct visualization with confocal microscopy,” J. Colloid Interface Sci. **314**, 98–106 (2007). [CrossRef] [PubMed]

*μ*m

^{3}imaging volume — limit the technique to highly viscous solvents. Dynamic light scattering can non-perturbatively probe translational dynamics over timescales ranging from tens of nanoseconds to tens of minutes, and depolarized dynamic light scattering can measure rotational dynamics of optically anisotropic particles, including colloidal clusters [12, 13

13. M. Hoffmann, C. S. Wagner, L. Harnau, and A. Wittemann, “3D brownian diffusion of submicron-sized particle clusters,” ACS Nano **3**, 3326–3334 (2009). [CrossRef] [PubMed]

13. M. Hoffmann, C. S. Wagner, L. Harnau, and A. Wittemann, “3D brownian diffusion of submicron-sized particle clusters,” ACS Nano **3**, 3326–3334 (2009). [CrossRef] [PubMed]

14. C. M. Sorensen, R. C. Mockler, and W. J. O’Sullivan, “Multiple scattering from a system of brownian particles,” Phys. Rev. A **17**, 2030–2035 (1978). [CrossRef]

15. T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. **41**, 1829–1839 (2002). [CrossRef]

16. F. C. Cheong and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express **18**, 6555–6562 (2010). [CrossRef] [PubMed]

17. Y. Pu and H. Meng, “Intrinsic aberrations due to mie scattering in particle holography,” J. Opt. Soc. Am. A **20**, 1920–1932 (2003). [CrossRef]

18. F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express **18**, 13563–13573 (2010). [CrossRef] [PubMed]

## 2. Experimental methods

*μ*m-diameter, surfactant-free, sulfate-stabilized polystyrene (PS) spheres (Invitrogen) and 80-nm-diameter poly-

*N*-isopropylacrylamide (PNIPAM) hydrogel particles, synthesized according to [20

20. M. Andersson and S. L. Maunu, “Structural studies of poly(N-isopropylacrylamide) microgels: Effect of SDS surfactant concentration in the microgel synthesis,” J. Polym. Sci., Part B: Polym. Phys . **44**, 3305–3314 (2006). [CrossRef]

^{−5}, and the approximate weight fraction of PNIPAM is 0.05. At room temperature, the PNIPAM particles are highly swollen with water, making them index-matched and effectively invisible. We use equal proportions of H

_{2}O and D

_{2}O to density-match the polystyrene spheres, and we add 15 mM NaCl to screen electrostatic interactions and 0.1% w/w Pluronic P123 triblock copolymer surfactant to stabilize the particles. Because the PNIPAM is index-matched, we treat everything in the system other than the PS particles as an optically homogeneous solvent with refractive index

*n*= 1.3349, as measured with an Abbé refractometer.

21. G. Meng, N. Arkus, M. P. Brenner, and V. N. Manoharan, “The free-energy landscape of clusters of attractive hard spheres,” Science **327**, 560–563 (2010). [CrossRef] [PubMed]

*μ*m-diameter surfactant-free, sulfate-stabilized polystyrene spheres (Invitrogen). We suspend the spheres, at a volume fraction of about 8 × 10

^{−4}, in a 0.1 M NaCl solution, which reduces the stability conferred by the charged sulfate groups. After loading the colloidal suspension into a glass sample cell, we use the optical trap to bring three particles close enough together for them to irreversibly bind into a triangular cluster. The binding is likely due to the van der Waals force. Because of the strength of the van der Waals interaction, we treat the trimers as rigid bodies with no vibrational modes.

*μ*m strips of polyaryletheretherketone (DuPont Teijin) as spacers between the glass surfaces. For the dimer system, we prevent depletion interactions between the PS particles and the glass surfaces by coating both the slides and cover slips used with PNIPAM. This is done by first silanizing the glass surfaces by immersion in a 1% w/w solution of 3-methylacryloxypropyl-trimethoxysilane (98%, Sigma) in anhydrous ethanol for 24 hours at room temperature. Next, the surfaces are rinsed with ethanol, dried with compressed nitrogen, and heated in an oven at 110° C for one hour. Finally, the slides and coverslips are immersed in an aqueous suspension of 100-nm-diameter PNIPAM particles for at least 24 hours at room temperature. After this procedure, the PNIPAM particles do not desorb from the surfaces.

## 3. Theoretical background

### 3.1. Translational and rotational Brownian motion

**D**and the rotational diffusion tensor

_{t}**D**[12, 22]. For an axisymmetric body like a dimer, consisting of two strongly interacting colloidal spheres, the full formalism simplifies considerably.

_{r}**D**can be diagonalized and has only two unique nonzero elements:

_{t}*D*

_{‖}, describing diffusion parallel to the axis of rotational symmetry, and

*D*

_{⊥}, describing diffusion perpendicular to the symmetry axis. There is also a single measurable rotational diffusion constant,

*D*

*. While in general the rotational and translational motions are coupled, such coupling is negligible if 3(*

_{r}*D*

_{‖}–

*D*

_{⊥})/(

*D*

_{‖}+ 2

*D*

_{⊥}) ≪ 1 [9

9. Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science **314**, 626–630 (2006). [CrossRef] [PubMed]

*D*

_{‖},

*D*

_{⊥}, and

*D*

*requires solving the Stokes equations for the hydrodynamic drag coefficients. For clusters of spheres, there are no known analytical solutions. But there are analytical solutions for prolate spheroids of semimajor axis*

_{r}*a*and semiminor axis

*b*[23

23. F. Perrin, “Mouvement brownien d’un ellipsoide-I. Dispersion diélectrique pour des molécules ellipsoidales,” J. Phys. Radium **7**, 497–511 (1934). [CrossRef]

*η*at temperature

*T*, and where

*r*≡

*a/b*is the aspect ratio, and

*S*is a dimensionless geometrical factor given by Similarly,

*D*is given by

_{r}### 3.2. Vibrational Brownian motion

*P*(

*r*) of center-to-center distances

*r*allows us to determine the interaction potential

*U*(

*r*) through the Boltzmann distribution [6

6. J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. **82**, 4352–4355 (1999). [CrossRef]

## 4. Analysis techniques

### 4.1. Qualitative overview of holograms from multiple particles

*μ*m-diameter spheres of relative refractive index

*m*= 1.2, corresponding to polystyrene in water. The relative refractive index is defined as

*m*≡

*n*

_{p}*/n*

*, where*

_{med}*n*

*is the refractive index of the particle and*

_{p}*n*

*is the refractive index of the surrounding medium. The single-sphere hologram in Fig. 3(a) consists of a circular fringe pattern, the center of which encodes the position of the sphere in a plane perpendicular to the optical axis. As described by Lee*

_{med}*et al.*[19], the circular fringes in the single-sphere hologram depend sensitively on the particle’s position along the optical axis as well as its size and refractive index. Thus this hologram contains information about the particle’s position in all 3 dimensions as well as its scattering properties.

**E**

*as the sum of the scattered fields from each of the two spheres: Then we can determine the intensity*

_{scat}*I*at any point on the hologram, following Lee

*et al.*[19]: where the hologram is normalized by the incident intensity |

**E**

*|*

_{inc}^{2}, measured from a field of view containing no particles. Here,

*α*≈ 1 is a scaling factor proportional to 1/|

**E**

*| that allows us to account for fluctuations in the intensity of the incident light, and*

_{inc}**ê**is the polarization vector of the incident light. When

**E**

_{scat,}_{1}and

**E**

_{scat,}_{2}add destructively, such that

**E**

*= 0, both the second and third terms of Eq. (10) will vanish and*

_{scat}*I*≈ 1, as is indeed the case in the fringes.

### 4.2. Quantitative modeling of holograms from multiple particles

26. M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by non-spherical particles: a review,” J. Quant. Spectrosc. Radiative Transfer **55**, 535–575 (1996). [CrossRef]

27. J. Baumgartl and C. Bechinger, “On the limits of digital video microscopy,” Europhys. Lett. **71**, 487493 (2005). [CrossRef]

### 4.3. Implementation and hologram fitting

28. D. W. Mackowski and M. I. Mishchenko, “Calculation of the t matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A **13**, 2266–2278 (1996). [CrossRef]

*l*beyond which the series expansions for

**S**can be truncated with negligible loss of precision and calculates the scattering coefficients

*a*

_{klp,}_{⊥}and

*a*

_{klp,}_{‖}. SCSMFO1B also contains a subroutine that performs the sums in Eqs. (12)–(15) using the asymptotic spherical wave approximation for the

*R*

*. We modified this subroutine to use the exact radial dependence on*

_{lp}29. C. B. Markwardt, “Non-linear least squares fitting in IDL with MPFIT,” http://arxiv.org/abs/0902.2850 (2009).

*α*(see Eq. (10)). Note that while three Euler angles are needed to describe the orientation of a general rigid body, two suffice to describe a dimer due to rotational symmetry about its long axis. Because our spheres do not interpenetrate, and also because SCSMFO1B cannot handle overlapping spheres, our fitting code constrains the center-to-center separation to be larger than the sum of the radii of both spheres.

## 5. Results and discussion

### 5.1. Fitting dimer holograms

*G*: where the summations run over the pixels of the normalized recorded hologram

*H*

*and the best fit model*

_{norm}*H*

*, and*

_{mod}*n*

*is the number of fit parameters. For the hologram shown in Fig. 5(b),*

_{params}*G*= 3.596 × 10

^{−4}.

### 5.2. Dimer translational motion

9. Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science **314**, 626–630 (2006). [CrossRef] [PubMed]

23. F. Perrin, “Mouvement brownien d’un ellipsoide-I. Dispersion diélectrique pour des molécules ellipsoidales,” J. Phys. Radium **7**, 497–511 (1934). [CrossRef]

^{−2}s, suggesting a noise floor at about 10

^{−15}m

^{2}. The noise floor gives an estimate of the tracking precision, showing that we are measuring center-of-mass displacements to a precision of 30 nm or better in all three dimensions.

*D*

_{‖}and

*D*

_{⊥}. Fits to the particle-frame mean-square displacements over nearly three decades of lag time yield

*D*

_{‖}= (1.79 ± 0.02) × 10

^{−13}m

^{2}s

^{−1}and

*D*

_{⊥}= (1.72±0.01) × 10

^{−13}m

^{2}s

^{−1}. Thus, we find

*D*

_{‖}/

*D*

_{⊥}= 1.04±0.02. In the prolate spheroid model, this ratio depends only on the spheroid’s aspect ratio and is independent of temperature or solvent viscosity. For a spheroid of aspect ratio 2, the model gives

*D*

_{‖}/

*D*

_{⊥}= 1.145, in reasonable agreement with our measurement, given that our dimers are not actually spheroids. Also, we find 3(

*D*

_{‖}–

*D*

_{⊥})/(

*D*

_{‖}+ 2

*D*

_{⊥}) ≈ 0.04, which supports our previous assertion that coupling between translation and rotation is negligible. Again, these results agree with the expected physics for this system, supporting the validity of the technique.

### 5.3. Dimer rotational motion

*D*

*by analyzing 〈Δ*

_{r}**u**

^{2}(

*τ*)〉, as discussed in Section 3.1. Figure 8 shows the data and a best fit to Eq. (3), yielding

*D*

*= 0.208 ± 0.002 s*

_{r}^{−1}.

*ε*

^{2}〉, or twice the mean-squared error in the measurement of the axis unit vector

**u**[8

8. J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. **179**, 298–310 (1996). [CrossRef]

*δθ*is about 0.06 radians.

*r*= 2, we can determine the effective spheroid semimajor axis

*a*by calculating

*D*

_{‖}/

*D*

*, which depends on*

_{rot}*a*but not

*η*. We can then determine

*η*from

*D*

_{‖}(Eq. (4)). We find that the effective semimajor axis is

*a*= 1.01

*μ*m and the viscosity is

*η*= 1.97 cP. Measurements of the solvent viscosity with a Cannon-Manning capillary viscometer gave

*η*= 3.01 ± 0.01 cP. Values for both

*a*and

*η*are close to the measured values. The agreement is reasonable given that the prolate spheroid model is only approximate for dimers, and the viscosity likely depends on the shear rate.

### 5.4. Dimer vibrational motion

*U*(

*r*), shown in Fig. 9. The measured potential is qualitatively consistent with what we expect for this system. At short range, we expect a van der Waals attraction and an electrostatic repulsion. The sum of these two competing interactions should lead to a potential well, which is what we observe.

30. S. Asakura and F. Oosawa, “Interaction between particles suspended in solutions of macromolecules,” J. Polym. Sci. **33**, 183–192 (1958). [CrossRef]

31. A. Vrij, “Polymers at interfaces and the interactions in colloidal dispersions,” Pure Appl. Chem. **48**, 471–483 (1976). [CrossRef]

*U*are meaningful, as we are unable to observe the particles at large separations where they are noninteracting. Also, the results depend strongly on the fitted radii: the particle radii are encoded in low spatial frequency variations in the magnitude of the hologram fringes, which can lead to a large uncertainty, on the order of 10-100 nm. This is why there are a few frames in which the measured separation distance is smaller than 0.95

*μ*m. Using the T-matrix technique to accurately determine pair potentials will require optimizing the fitting technique and improving the fidelity of the low spatial frequencies in the holograms.

*k*, consistent with our observations that the dimers do not break apart for at least several minutes.

_{B}T### 5.5. Fitting trimer holograms

*G*= 6.556 × 10

^{−4}, where

*G*is the goodness of fit statistic defined in Eq. (18). This shows that the technique should scale to even larger clusters of spheres.

## 6. Conclusions

*μ*m from the particles, rather than being limited to holograms recorded in the far field.

32. M. Doi and S. F. Edwards, “Dynamics of rod-like macromolecules in concentrated solution. Part 1,” J. Chem. Soc. Faraday Trans. 2 **74**, 560–570 (1978). [CrossRef]

33. W. J. Wiscombe, “Improved mie scattering algorithms,” Appl. Opt. **19**, 1505–1509 (1980). [CrossRef] [PubMed]

*μ*m) spheres or of clusters comprising many spheres. Another limitation to extending our technique to more than a few spheres is the large number of parameters that enter the scattering model. Any fitting algorithm must therefore be able to contend with many local minima. But the growing availability of computing clusters and cloud computing should alleviate many of these concerns.

*λ*) that the holograms show interparticle interference fringes. Moreover, the approach we illustrate can be extended to other nonspherical colloids. For example, other T-matrix codes can calculate the exact scattering from bodies of revolution such as spheroids or finite cylinders [34

34. M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. **39**, 1026–1031 (2000). [CrossRef]

## Acknowledgments

## References and links

1. | J. Perrin, “Mouvement brownien et réalité moléculaire,” Ann. Chim. Phys. |

2. | M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud’homme, M. Beck, and D. A. Weitz, “Investigating the microenvironments of inhomogeneous soft materials with multiple particle tracking,” Phys. Rev. E |

3. | M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled f-actin solutions,” Phys. Rev. Lett. |

4. | A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. |

5. | E. Andablo-Reyes, P. Díaz-Leyva, and J. L. Arauz-Lara, “Microrheology from rotational diffusion of colloidal particles,” Phys. Rev. Lett. |

6. | J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. |

7. | M. G. Nikolaides, A. R. Bausch, M. F. Hsu, A. D. Dinsmore, M. P. Brenner, C. Gay, and D. A. Weitz, “Electric-field-induced capillary attraction between like-charged particles at liquid interfaces,” Nature (London) |

8. | J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. |

9. | Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science |

10. | S. M. Anthony, M. Kim, and S. Granick, “Translation-rotation decoupling of colloidal clusters of various symmetries,” J. Chem. Phys. |

11. | D. Mukhija and M. J. Solomon, “Translational and rotational dynamics of colloidal rods by direct visualization with confocal microscopy,” J. Colloid Interface Sci. |

12. | B. J. Berne and R. Pecora, |

13. | M. Hoffmann, C. S. Wagner, L. Harnau, and A. Wittemann, “3D brownian diffusion of submicron-sized particle clusters,” ACS Nano |

14. | C. M. Sorensen, R. C. Mockler, and W. J. O’Sullivan, “Multiple scattering from a system of brownian particles,” Phys. Rev. A |

15. | T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. |

16. | F. C. Cheong and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express |

17. | Y. Pu and H. Meng, “Intrinsic aberrations due to mie scattering in particle holography,” J. Opt. Soc. Am. A |

18. | F. C. Cheong, B. J. Krishnatreya, and D. G. Grier, “Strategies for three-dimensional particle tracking with holographic video microscopy,” Opt. Express |

19. | S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express |

20. | M. Andersson and S. L. Maunu, “Structural studies of poly(N-isopropylacrylamide) microgels: Effect of SDS surfactant concentration in the microgel synthesis,” J. Polym. Sci., Part B: Polym. Phys . |

21. | G. Meng, N. Arkus, M. P. Brenner, and V. N. Manoharan, “The free-energy landscape of clusters of attractive hard spheres,” Science |

22. | J. K. Dhont, |

23. | F. Perrin, “Mouvement brownien d’un ellipsoide-I. Dispersion diélectrique pour des molécules ellipsoidales,” J. Phys. Radium |

24. | M. Doi and S. F. Edwards, |

25. | C. F. Bohren and D. R. Huffman, |

26. | M. I. Mishchenko, L. D. Travis, and D. W. Mackowski, “T-matrix computations of light scattering by non-spherical particles: a review,” J. Quant. Spectrosc. Radiative Transfer |

27. | J. Baumgartl and C. Bechinger, “On the limits of digital video microscopy,” Europhys. Lett. |

28. | D. W. Mackowski and M. I. Mishchenko, “Calculation of the t matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A |

29. | C. B. Markwardt, “Non-linear least squares fitting in IDL with MPFIT,” http://arxiv.org/abs/0902.2850 (2009). |

30. | S. Asakura and F. Oosawa, “Interaction between particles suspended in solutions of macromolecules,” J. Polym. Sci. |

31. | A. Vrij, “Polymers at interfaces and the interactions in colloidal dispersions,” Pure Appl. Chem. |

32. | M. Doi and S. F. Edwards, “Dynamics of rod-like macromolecules in concentrated solution. Part 1,” J. Chem. Soc. Faraday Trans. 2 |

33. | W. J. Wiscombe, “Improved mie scattering algorithms,” Appl. Opt. |

34. | M. I. Mishchenko, “Calculation of the amplitude matrix for a nonspherical particle in a fixed orientation,” Appl. Opt. |

**OCIS Codes**

(120.5820) Instrumentation, measurement, and metrology : Scattering measurements

(180.6900) Microscopy : Three-dimensional microscopy

(090.1995) Holography : Digital holography

**ToC Category:**

Microscopy

**History**

Original Manuscript: January 24, 2011

Revised Manuscript: April 1, 2011

Manuscript Accepted: April 3, 2011

Published: April 12, 2011

**Virtual Issues**

Vol. 6, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Jerome Fung, K. Eric Martin, Rebecca W. Perry, David M. Kaz, Ryan McGorty, and Vinothan N. Manoharan, "Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy," Opt. Express **19**, 8051-8065 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-9-8051

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### References

- J. Perrin, “Mouvement brownien et réalité moléculaire,” Ann. Chim. Phys. 18, 1–114 (1909).
- M. T. Valentine, P. D. Kaplan, D. Thota, J. C. Crocker, T. Gisler, R. K. Prud’homme, M. Beck, and D. A. Weitz, “Investigating the microenvironments of inhomogeneous soft materials with multiple particle tracking,” Phys. Rev. E 64, 061506 (2001). [CrossRef]
- M. L. Gardel, M. T. Valentine, J. C. Crocker, A. R. Bausch, and D. A. Weitz, “Microrheology of entangled f-actin solutions,” Phys. Rev. Lett. 91, 158302 (2003). [CrossRef] [PubMed]
- A. I. Bishop, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical microrheology using rotating laser-trapped particles,” Phys. Rev. Lett. 92, 198104 (2004).
- E. Andablo-Reyes, P. Díaz-Leyva, and J. L. Arauz-Lara, “Microrheology from rotational diffusion of colloidal particles,” Phys. Rev. Lett. 94, 106001 (2005). [CrossRef] [PubMed]
- J. C. Crocker, J. A. Matteo, A. D. Dinsmore, and A. G. Yodh, “Entropic attraction and repulsion in binary colloids probed with a line optical tweezer,” Phys. Rev. Lett. 82, 4352–4355 (1999). [CrossRef]
- M. G. Nikolaides, A. R. Bausch, M. F. Hsu, A. D. Dinsmore, M. P. Brenner, C. Gay, and D. A. Weitz, “Electric-field-induced capillary attraction between like-charged particles at liquid interfaces,” Nature (London) 420, 299–301 (2002). [CrossRef]
- J. C. Crocker and D. G. Grier, “Methods of digital video microscopy for colloidal studies,” J. Colloid Interface Sci. 179, 298–310 (1996). [CrossRef]
- Y. Han, A. M. Alsayed, M. Nobili, J. Zhang, T. C. Lubensky, and A. G. Yodh, “Brownian motion of an ellipsoid,” Science 314, 626–630 (2006). [CrossRef] [PubMed]
- S. M. Anthony, M. Kim, and S. Granick, “Translation-rotation decoupling of colloidal clusters of various symmetries,” J. Chem. Phys. 129, 244701 (2008). [CrossRef]
- D. Mukhija and M. J. Solomon, “Translational and rotational dynamics of colloidal rods by direct visualization with confocal microscopy,” J. Colloid Interface Sci. 314, 98–106 (2007). [CrossRef] [PubMed]
- B. J. Berne and R. Pecora, Dynamic Light Scattering (Plenum Press, 1985).
- M. Hoffmann, C. S. Wagner, L. Harnau, and A. Wittemann, “3D brownian diffusion of submicron-sized particle clusters,” ACS Nano 3, 3326–3334 (2009). [CrossRef] [PubMed]
- C. M. Sorensen, R. C. Mockler, and W. J. O’Sullivan, “Multiple scattering from a system of brownian particles,” Phys. Rev. A 17, 2030–2035 (1978). [CrossRef]
- T. M. Kreis, “Frequency analysis of digital holography with reconstruction by convolution,” Opt. Eng. 41, 1829–1839 (2002). [CrossRef]
- F. C. Cheong and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express 18, 6555–6562 (2010). [CrossRef] [PubMed]
- Y. Pu and H. Meng, “Intrinsic aberrations due to mie scattering in particle holography,” J. Opt. Soc. Am. A 20, 1920–1932 (2003). [CrossRef]
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