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-μm-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 μm 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 Optical Society of America
Measuring the translational and rotational dynamics of colloidal particles is central to many experiments in soft condensed matter physics. Perrin’s measurements of translational diffusion in 1909  confirmed Einstein’s theory of Brownian motion and helped establish the fluctuation-dissipation relation. More recently, measurements of the translational Brownian motion of probe particles embedded in viscoelastic media have led to a better understanding of the structure and mechanisms of stress relaxation in materials such as actin networks [2, 3]. Current microrheological techniques also make use of rotating, optically anisotropic spheres [4, 5].
Measuring vibrational dynamics can give equally rich insight into colloidal forces. For example, the study of the correlated Brownian motion of two colloidal particles interacting through a depletion attraction revealed an oscillatory component of the pair potential at high depletant concentrations . Vibrations of strongly interacting colloidal particles on the surface of an emulsion droplet have also been used to characterize a still-mysterious attractive interaction between like-charged particles at a liquid-liquid interface .
Although there are several optical tools to study colloidal dynamics, none can easily and simultaneously probe translation, rotation, and vibration in three dimensions. Video microscopy is a mature technique for measuring dynamics of dense colloidal suspensions, but it is limited to two-dimensional (2D) motion in the microscope’s focal plane [8–10]. When optical tweezers are used to confine the particles to the focal plane, interpreting pair potential measurements from video microscopy requires carefully measuring and subtracting the tweezer’s optical potential. Confocal microscopy can effectively image three-dimensional (3D) translational and rotational Brownian motion, as demonstrated by Mukhija and Solomon for polystyrene spheroids . But large acquisition times — tens of seconds for a ∼ 4000 μm3 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]. But the depolarized scattering signal from most colloids is much weaker than the polarized scattering signal, and it is easily overwhelmed by depolarization from multiple scattering [13, 14]. Furthermore, these light scattering techniques yield ensemble-averaged measurements that are convolved with the polydispersity of the scatterers.
Digital holographic microscopy (DHM) could overcome many of these limitations. In in-line DHM, light scattered from the object being imaged interferes with transmitted light, forming an easily-captured 2D hologram that encodes 3D information. One way to recover the 3D information from a hologram of weakly scattering objects is Rayleigh-Kirchoff reconstruction [15, 16]. But the imaging precision of reconstruction techniques, particularly along the optical axis, is limited by distortions in the reconstructed volume due to strong scattering [17, 18]. The technique also does not explicitly account for the optical properties of the scatterers. Thus it is difficult to use reconstruction methods to accurately measure the orientation of nonspherical scatterers whose aspect ratios are close to unity, or to measure ∼ 10 nm changes in the separation distance between two nearby particles. An alternative to reconstruction involves directly fitting a hologram using a scattering model. Fitting holograms formed by a single spherical particle to the Lorenz-Mie scattering solution allows one to measure the particle position to nanometer-scale precision . But so far, the fitting technique has only been used to study isolated spherical particles.
Here we show that with an appropriate scattering model, in-line DHM can simultaneously probe the translational, rotational, and vibrational motion of multiple micron-sized, strongly-interacting colloidal particles. We use an exact model for light scattering from multiple particles that includes all multiple scattering and near-field coupling effects. We demonstrate that this model accurately fits digital holograms of Brownian sphere dimers, allowing us to measure colloidal dynamics over more than 3 decades of timescales, ranging from milliseconds to seconds, with nanometer-scale precision. By fitting holograms of non-axisymmetric structures such as colloidal trimers, we show that the technique can quantify dynamics of colloidal objects that are more complex than those previously studied with DHM.
2. Experimental methods
Our experimental apparatus is an in-line digital holographic microscope built on the body of a Nikon TE-2000 inverted microscope (Fig. 1). In in-line DHM, a collimated laser beam illuminates a sample, and a microscope objective collects both the light scattered from the sample as well as the transmitted light, which serves as a reference beam. The objective’s focal plane lies further along the axis than the sample. In our apparatus, light from a 658 nm laser diode (Opnext HL6535MG with Thorlabs TED200C temperature controller and LDC201C current controller) is coupled to the microscope through a single-mode fiber, then collected by a 10x, 0.25 NA Newport objective, and collimated by a 0.59 NA long working distance condenser (Nikon). We image with a 60x, 1.20 NA Plan Apo water immersion objective (Nikon); the water immersion minimizes spherical aberration due to the index mismatch between the glass coverslip and our sample. We record images at 500 frames per second with a Photon Focus MVD-1024E-160 12-bit monochrome camera, and capture them with an EPIX PIXCI E4 frame grabber controlled by a PC.
Our holographic microscope also contains an optical tweezer. Light from a fiber-coupled 830 nm laser diode (Sanyo DL-8142-201, with Thorlabs TCM1000T temperature controller and LD1255 current controller) is collected by another 10x, 0.25 NA Newport objective. The light then passes through the back aperture of the 60x Nikon objective used for imaging and is focused in the sample. We use the trap only to isolate particles and form clusters in our samples; it remains off during the measurements.
We study two colloidal systems in our experiments. The first consists of dimers of colloidal spheres interacting through a short-range depletion force, and the second consists of trimers of colloidal spheres interacting through a van der Waals force. The dimer system contains 0.99-μm-diameter, surfactant-free, sulfate-stabilized polystyrene (PS) spheres (Invitrogen) and 80-nm-diameter poly-N-isopropylacrylamide (PNIPAM) hydrogel particles, synthesized according to . The volume fraction of PS in the sample is 2 × 10−5, 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 H2O and D2O 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.
The PNIPAM particles induce a depletion attraction between the PS spheres. We choose a depletion interaction because it is well-studied and has been used to assemble physically interesting clusters . The mechanism of the interaction is qualitatively illustrated in Fig. 2.
The trimer system contains 1.3-μm-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.
We image all of our samples in glass cells made from a standard microscope slide and a #1 cover slip. We use 76-μm 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
The translational and rotational diffusion of an arbitrarily-shaped, isolated colloidal particle can be described by two 3 × 3 tensors: the translational diffusion tensor Dt and the rotational diffusion tensor Dr [12, 22]. For an axisymmetric body like a dimer, consisting of two strongly interacting colloidal spheres, the full formalism simplifies considerably. Dt can be diagonalized and has only two unique nonzero elements: 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 r. While in general the rotational and translational motions are coupled, such coupling is negligible if 3(D ‖ – D ⊥)/(D ‖ + 2D ⊥) ≪ 1 [9, 12].
D ‖, D ⊥, and D r are related to measurable quantities, the mean-square displacements. A vector displacement Δx can be resolved into components parallel and perpendicular to the dimer’s orientation vector u. The mean-square displacements and are given by :24]:
Predicting the diffusion constants D ‖, D ⊥, and D r 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 a and semiminor axis b . We use these solutions to approximate the diffusion constants for our dimers. For a prolate spheroid suspended in a solvent of viscosity η at temperature T,
3.2. Vibrational Brownian motion
Pairs of particles that interact strongly will also exhibit vibrational motion about a potential minimum. The vibrations are in general overdamped. If the particles are in equilibrium with the fluid, measuring the probability distribution P(r) of center-to-center distances r allows us to determine the interaction potential U(r) through the Boltzmann distribution :
4. Analysis techniques
We first discuss some general features of holograms of multiple spherical particles and show qualitatively how they encode information about the size, refractive index, and position of each particle. We then explain how the positions can be precisely determined by fitting to an exact electromagnetic scattering model. Fitting a series of hologram frames allows us to independently measure translation, rotation, and vibration.
4.1. Qualitative overview of holograms from multiple particles
Figure 3 shows five simulated holograms for 1-μm-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 med, where n p is the refractive index of the particle and n med 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 et al. , 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.
Holograms of two or more particles contain additional information. The holograms in Fig. 3(b)–3(d) contain linear as well as circular fringes. The linear fringes, which encode information about the relative orientation and separation distance between the two particles, arise from interference between light scattered from each particle. To understand the origin of these fringes, it is helpful to consider the limit of weak scattering. In this limit, we can approximate the scattered electric field of a dimer E scat as the sum of the scattered fields from each of the two spheres:19]: Eq. (10) will vanish and I ≈ 1, as is indeed the case in the fringes.
The linear fringe pattern is analogous to that seen in double-slit interference. Thus, as the particles separate, the linear fringes move closer together (Fig. 3(b) and 3(c)). Moreover, the fringes reflect the position and orientation of the dimer: in Fig. 3(d) the fringes are wider apart than in Fig. 3(b) because the dimer is projected onto a plane perpendicular to the incident light direction. We note that for the range of particle sizes and separations in Fig. 3, the sphere separations are still on the order of the wavelength of light, so we do not see two separate sets of circular fringes. Separate circular fringes occur only when the particles are more than a few wavelengths apart.
The trimer hologram in Fig. 3(e) shows the same trends; we see a roughly circular fringe pattern, but the pattern of fringes arising from the interference between the light scattered from the three particles is significantly more complicated.
4.2. Quantitative modeling of holograms from multiple particles
Although the linear interference fringes give an approximate measure of the separation between particles, we can more precisely determine the particle positions by fitting to a scattering model that accounts not only for interference, but also for multiple scattering and near-field effects. The near-field effects, a consequence of the coupling of evanescent fields, are relevant when the surfaces of the particles are within a wavelength of one another. These effects cannot be modeled by superposing the Lorenz-Mie solutions  for each sphere.
To capture all the scattering effects, we use a numerical T-matrix superposition method whose results converge to the exact solutions of Maxwell’s equations for multiple spheres . Figure 4 illustrates why this is necessary. It shows that there are significant differences between holograms calculated from Lorenz-Mie superposition and the exact T-matrix scattering solution. The differences are likely due to near-field coupling, which becomes increasingly relevant at small interparticle separations. The near-field effects are neglected in the analysis of video microscopy data, limiting the accuracy of measurements of small particle separations . Reconstruction techniques for analyzing digital holograms do not consider such coupling either.
4.3. Implementation and hologram fitting
We use the Fortran 77 T-matrix code SCSMFO1B.FOR by Mackowski and Mishchenko  to calculate the exact field scattered from clusters of two or three spheres. This program calculates the scattered fields from a cluster of non-overlapping spheres in a fixed orientation; the spheres do not need to touch, and they may have different sizes.25]. The four elements of S are series expansions: 28], and the Rlp are defined as 25] and  in that we do not asymptotically approximate the spherical Hankel functions in the R lp as spherical waves of the form (–i)l +1 exp(in med k o r) /n med k o r. This is necessary because the hologram plane, which lies in the focal plane of our microscope, can be as close as ∼15 μm to the centers of mass of our clusters. For micron-sized particles, we must expand the series for S up to order l > 10. Thus l 2 /nmed k 0 r is of order unity, and the approximation of asymptotic radial dependence is invalid .
SCSMFO1B determines the maximum order 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 in the R lp. We modified this subroutine to use the exact radial dependence on .
We have wrapped the relevant portions of SCSMFO1B with additional code in Fortran 90 and Python/SciPy to enable rapid calculation of holograms while retaining ease of use. We use the Levenberg-Marquardt non-linear least squares algorithm to fit our holograms, as implemented in the Python translation of MPFIT, originally developed by C. Markwardt . Because the time between frames is small compared to the characteristic diffusion time, we can use the fitted parameter values from a given hologram frame as the initial values for the next frame. We do the fitting on up to 20 2.3-GHz, 64-bit processors simultaneously, using the Odyssey computational cluster at Harvard University.
In our analysis of dimers, each fit determines the following 10 parameters: relative refractive index (assuming that both spheres have the same refractive index), radii of each sphere, three center of mass coordinates, two Euler angles describing the dimer orientation, the center-to-center separation, and a normalization constant 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.
In our analysis of trimers, we fit for 9 parameters: the particle relative refractive index, sphere radius, three center of mass coordinates, three Euler angles, and a normalization constant. For simplicity we assume all spheres to be the same size, although the analysis code can be generalized to fit for the sizes of all three spheres individually.
5. Results and discussion
5.1. Fitting dimer holograms
We show in Fig. 5 that we can fit recorded holograms of dimers to model holograms computed with the T-matrix method. We find good quantitative agreement between the experiment and calculation. In particular, we quantify the goodness of fit with an appropriately normalized sum of squared residuals, G:Fig. 5(b), G = 3.596 × 10−4.
5.2. Dimer translational motion
To measure the dynamics, we analyze a 24.3 second trajectory of a colloidal dimer, recorded at 500 frames per second. From the same data set we obtain measurements of translational, rotational, and vibrational motion. We first examine the translational component.
As shown in Fig. 6, the distributions of dimer center-of-mass displacements in the laboratory frame appear Gaussian. This suggests that coupling between translational and rotational motion is negligible, as such coupling would manifest itself in non-Gaussian distributions at short times . The Gaussian form of the displacement distribution agrees with what we expect for particles with a small aspect ratio [12, 23].
We also see that the mean-square displacements start to deviate from linearity at times less than 10−2 s, suggesting a noise floor at about 10−15 m2. 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.
Measuring translations in the particle frame, as discussed in Section 3.1 and illustrated in Fig. 7, allows us to determine 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 m2s−1 and D ⊥ = (1.72±0.01) × 10−13 m2s−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 ‖ + 2D ⊥) ≈ 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
Once the centers of the particles are determined for each frame in the data set, we can use the results to measure the rotational dynamics as well. We measure D r by analyzing 〈Δu 2(τ)〉, as discussed in Section 3.1. Figure 8 shows the data and a best fit to Eq. (3), yielding D r = 0.208 ± 0.002 s−1.
The intercept in the fit of the data shown in Fig. 8 allows us to estimate the angular resolution of our technique. The intercept corresponds to 2〈ε 2〉, or twice the mean-squared error in the measurement of the axis unit vector u . Using the small-angle approximation , we find that the angular resolution δθ is about 0.06 radians.
Our measurements of diffusion constants, interpreted in the context of the prolate spheroid model, lead to reasonable model parameters and confirm the validity of the T-matrix fitting approach. Assuming an aspect ratio r = 2, we can determine the effective spheroid semimajor axis a by calculating D ‖/D rot, which depends on a but not η. We can then determine η from D ‖ (Eq. (4)). We find that the effective semimajor axis is a = 1.01 μm 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
The same data set also yields the vibrational motion between the particles. To show this, we invert the distribution of particle center-to-center separations to determine 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.
We do not make a quantitative comparison between our measured potential and a model potential, such as the Asakura-Oosawa potential [30, 31], because our system does not conform well to existing depletion models: our depletants are not hard spheres, nor is their concentration low. Moreover, we never observed the dimer break apart. Thus only differences in 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 μm. 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.
Nonetheless, the results are promising and qualitatively consistent with our expectations: the range of the measured potential is on the order of the depletant size, and the depth of the well is several kBT, consistent with our observations that the dimers do not break apart for at least several minutes.
5.5. Fitting trimer holograms
We have also fit holograms of trimers, as shown in Fig. 10. Here, we find good agreement between the experimental hologram and the fit model, even when all three spheres are constrained to be the same size. In particular, 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.
We have demonstrated that digital holograms of multiple spheres can be accurately fit through numerical scattering calculations. Obtaining accurate fits requires a T-matrix approach, which captures the near-field effects and multiple scattering from colloidal particles separated by a wavelength or less. By accounting for the non-asymptotic radial decay of the scattered fields, we are able to fit holograms taken with the focal plane located less than 15 μm from the particles, rather than being limited to holograms recorded in the far field.
Analysis of a time-series of holograms allows us to measure colloidal dynamics. For colloidal dimers, the technique allows us to simultaneously probe six dynamical modes — three translational, two rotational, and one vibrational — over a wide range of time scales. Compared to confocal microscopy, DHM can access a larger range of dynamical time scales, which may be particularly useful for complex materials that exhibit caging or other restricted diffusion effects . Also, unlike ensemble-averaged light scattering techniques, DHM probes local material properties.
Because no optical trap is needed, DHM may eventually prove to be simpler and more accurate than conventional microscopy for measuring interaction potentials. Also, smaller separation distances are accessible in DHM because the fitting technique can account for near-field effects. But further development of the fitting algorithms is needed to improve the precision of the fits.
The principle drawback of our method is the time it takes to analyze holograms. Currently we can analyze about 10 dimer holograms in 1 cpu-hour. The computation time scales roughly with the scattering object size , which makes it difficult to analyze holograms of very large (∼ 100 μm) 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.
Our technique may prove useful for microrheology, measurements of colloidal forces, or fundamental studies of Brownian motion. It applies to sphere clusters made out of materials other than polystyrene, so long as the clusters scatter enough light to be observed with DHM, and as long as the spheres are sufficiently large (diameter ∼ 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 . Overall, the combination of holographic microscopy and T-matrix fitting is a promising route for fast and accurate measurements of colloidal dynamics.
We thank Guangnan Meng and Thomas G. Dimiduk for useful discussions. This work was funded by the National Science Foundation through CAREER grant no. CBET-0747625 and through the Harvard MRSEC, grant no. DMR-0820484. Rebecca W. Perry is supported by a Graduate Research Fellowship from the National Science Foundation. Computations were performed on the Odyssey cluster, supported by the FAS Sciences Division Research Computing Group, and in part on the computational resources of the Center for Nanoscale Systems (CNS), a member of the National Nanotechnology Infrastructure Network (NNIN), which is supported by the National Science Foundation under NSF award no. ECS-0335765. CNS is part of the Faculty of Arts and Sciences at Harvard University.
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