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

## Contents

## General Information

**Authors:** Jerome Fung, K. Eric Martin, Rebecca W. Perry, David M. Kaz, Ryan McGorty, Vinothan N. Manoharan

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

Fig. 1 (From [1]) Fig. 2 (From [1]) Fig. 3 (From [1])

# Summary

## DHM

This paper discusses the application of digital holographic microscopy (DHM) to study the dynamics of single and multiple colloidal spheres. Translational dynamics of colloidal particles were useful in examining Brownian motion both historically and more recently as probes for viscoelastic media, while vibrational dynamics have elucidated interesting effects due to depletion attraction or at liquid-liquid interfaces. DHM offers improvements over other optical tools such as video microscopy, which is only two-dimensional, confocal microscopy, which is comparatively slower due to long acquisition times, or light-scattering methods, which often have limited resolution for clusters of multiple particles. Experiments using DHM demonstrate improvements in these areas by measuring ~50 nm separations at minimum time steps ~1 ms at distances down to 15 <math>\mu</math>m from the focal plane for systems of two to three colloidal spheres.

## 'Theory

The basic principle behind DHM is that laser light (in a regime well-approximated by plane waves) is shone on the object under study. The scattered light from the object interferes with the unscattered light from the beam, forming a hologram. This interference pattern is then magnified, captured by a microscope and recorded digitally by a camera. This hologram is then fitted to extract parameters of interest from the scattering object. Rather than using the Rayleigh-Kirchoff reconstruction, which can be thought of as back-projecting light through the hologram to reconstruct the location of the object, the authors chose the Lorenz-Mie scattering solution to generate predicted holograms, then use them to fit for spatial and optical properties of the scatterer (Fig. 1). This method gives results that are more precise than Rayleigh-Kirchoff reconstruction. For clusters of particles, due to multiple scattering and near-field effects, superposition of Lorenz-Mie solutions fail and a numerical T-matrix superposition method is used instead (Fig 2).

### Brownian motion

The diffusion of a colloidal particle is given by two <math>3\times3</math> tensors: <math>\vec{D_t}</math>, the translational diffusion tensor and <math>\vec{D_r}</math> the rotational diffusion tensor. <math>\vec{D}_t</math> can be diagonalized into two elements <math>D_{||}</math> and <math>D_\perp</math> for diffusion parallel and perpendicular to the axis of symmetry while <> the rotational diffusion is captured by a single constant <math>D_r</math>. In order to calculate these quantities, the authors' implementation of DHM allowed them to measure a particle's three-dimensional location and orientation (Euler angles), radii, index of refraction, and separation with its nearest neighbors. These diffusion constants may be calculated theoretically for a given object in a particular fluid and are related to vectorial mean square displacements, which may be determined experimentally. Vibrational Brownian motion arise from pair interactions of the colloidal spheres. Experimental measurements allow the determination of the interaction potential from distances modeled with the Boltzmann distribution.

## Experiment

In this experiment, the authors used 0.90 <math>\mu</math>m diameter sulfate-stabilized polystyrene spheres (PS) and 0.80 <math>\mu</math>m diameter poly-N-isopropylacrylamide (PNIPAM) hydrogel particles. The dilute solutions containing PS (2 x 10^{-5} volume fraction) and PNIPAM (0.05 weight fraction) were index matched in a solvent prepared so that n = 1.335. The PNIPAM particles create depletion attraction between the PS spheres of energy ~kT. For trimers, slightly larger PS particles (1.3-<math>\mu</math>m) at a volume fraction of <math>8\times10^{-4}</math> in a 0.1 NaCl solution area assembled using an optical trap. The imaging cells are made from standard microscope slides with spacers between cover slips with are appropriately prepared depending on the material under study.

## Results and Discussion

The authors found good quantitive agreement between the modeled and measured holograms for dimers. Using the hologram to measure translational and rotational motion, the authors found reasonable agreement between the modeled ratio of <math>D_{||}/D_{\perp}=1.145 \text{s}^{-1}</math> using an approximation of the dimers as prolate spheroids and the computed value of <math>D_{||}/D_{\perp}=1.04\pm0.02 \text{s}^{-1}</math>. For the rotational motion, the authors found that <math>D_r=0.208\pm0.002 \text{s}^{-1}</math> and the angular resolution of the method to be <math>\Delta\theta=0.06 rad</math>. These results that the T-matrix approach for fitting holograms is effective under the prolate spheroid approximation for a system of dimers. Inverting the particle separations give the measured pair potential for the dimers which exhibit a potential well, as expected, which the authors think is likely due to a van der Waals attraction and electrostatic repulsion for the system. Holograms of trimers were also successfully fitted and suggest that this technique may be extended to larger clusters of particles (Fig. 3). In summary, the authors successfully implemented DHM using a T-matrix approach whose time series data allowed measurements of the dynamics of colloidal dimers and trimers. The approach used in the study may be implemented for light-scattering particles of sufficient size and in principle, non-spherical shape. Additional work is needed to improve the fitting algorithms for the holograms both in terms of accuracy and computational efficiency. As the authors described, DHM seems to be a promising tool for measurements for microrheology, colloidal forces, and studies of Brownian motion.

# Keywords

Diffusion, Digital Holographic Microscopy, Brownian Motion, Interaction Potential

## References

[1] J. Fung, K. Martin, R. Perry, D. Kaz, R. McGorty, and V. Manoharan, "Measuring translational, rotational, and vibrational dynamics in colloids with digital holographic microscopy," Opt. Express 19, 8051-8065 (2011).