## Abstract

We use rotational photonic tweezers to access local viscoelastic properties of complex fluids over a wide frequency range. This is done by monitoring both passive rotational Brownian motion and also actively driven transient rotation between two angular trapping states of a birefringent microsphere. These enable measurement of high- and low-frequency properties, respectively. Complex fluids arise frequently in microscopic biological systems, typically with length scales at the cellular level. Thus, high spatial resolution as provided by rotational photonic tweezers is important. We measure the properties of tear film on a contact lens and demonstrate variations in these properties between two subjects over time. We also show excellent agreement between our theoretical model and experimental results. We believe that this is the first time that active microrheology using rotating tweezers has been used for biologically relevant questions. Our method demonstrates potential for future applications to determine the spatial-temporal properties of biologically relevant and complex fluids that are only available in very small volumes.

© 2017 Optical Society of America

## Corrections

Shu Zhang, Lachlan J. Gibson, Alexander B. Stilgoe, Itia A. Favre-Bulle, Timo A. Nieminen, and Halina Rubinsztein-Dunlop, "Ultrasensitive rotating photonic probes for complex biological systems: erratum," Optica**4**, 1372-1372 (2017)

https://www.osapublishing.org/optica/abstract.cfm?uri=optica-4-11-1372

## 1. INTRODUCTION

Currently there is ever-growing interest in studying fluid samples that are rare and often available only in very small volumes, for instance, biomedical fluids. These are typically micro- and nano-structured materials—as complex fluids that exhibit both viscous and elastic behaviors [1,2]. Furthermore, the microrheology of biological materials (e.g., cytoplasm and tear film) often enable studies of the relevant material properties linked to the development and diagnosis of diseased states [3]. Novel microrheological techniques therefore represent extremely useful tools for our understanding of biophysics and diagnosis of a variety of diseases.

Here, we first validate our microrheological technique based on rotational optical tweezers by testing it on known fluids. Next, we demonstrate that our technique is sensitive and accurate enough to provide useful measurements on biological fluids by measuring the viscoelastic properties of eye tear film coated on a contact lens. This evenly distributed multilayered fluid structure keeps a proper degree of hydration of the cornea, contributing to overall ocular health and comfort. A collection of problems associated with tear film instability is often referred to as dry eye syndrome [4,5]. However, the structure and function of the tear film are far from being understood. Notably, the dynamics of the tear film on corneal, conjunctival, and contact lens surfaces are different [6–9]. Hence, the properties of tears coated on contact lenses are important factors for successful contact lens use. As the thickness of tear films is only a few micrometers and reduces over time [10,11], there has not been any similar study with conventional methods conducted up to now on this eye fluid on a contact lens.

Optical tweezers have been successfully used for the investigation of rheological properties of complex fluids at microscopic length scales. In these methods, one measures the interaction between optically trapped particles and surrounding media [12–14]. The small scale and noninvasive nature of optical trapping systems make them well suited for the microrheological study of complex biological systems. Previously, optical tweezers have been employed for the active and passive microrheological measurements. Passive methods typically monitor the thermal fluctuation of a particle suspended in a fluid [1,15], while active techniques require an external force to act on a probe particle [16–19]. However, measurements of viscoelastic properties of complex fluids by passive microrheology techniques yield high-frequency results with high accuracy but often lack precise low-frequency information [20]. Some of the broadest frequency range microrheology data with optical tweezers has been created by combining translational Brownian motion with active microrheology by measuring the transient displacement of a bead moving between two optical traps [21], but did not strictly provide local measurements. Nearby surfaces influence the motion of the probe particle in the fluid, typically increasing the viscous drag over that which would be experienced in a free fluid without nearby surfaces. These effects are called “wall effects,” and can be large. For translational motion, the wall effects increase the viscous drag by over 5% for surfaces as far away as 10 times the radius of the probe particle [22]. Therefore, these methods become difficult to use when examining very small volumes of samples.

Furthermore, the ability to orient objects using optical tweezers offers new opportunities for studying fluid properties. Rotational microrheology techniques have greater spatial resolution compared to conventional methods, since the center of the probe can be tightly constrained and the rotational motion is less sensitive to boundary wall effects [23]. The viscous drag increases by less than 5%, even for a surface as close as one particle radius [22,24], for rotation about an axis parallel to the surface; the wall effect is even smaller when rotation is about an axis perpendicular to the surface [24], such that it is difficult to experimentally observe any effect until the probe particle touches the surface [16]. The rotational diffusion of microparticles has been used to extract the high-frequency complex dynamic modulus for viscoelastic fluids [25,26]. The next step in the application of optical tweezers for microrheology is to combine rotation and active–passive techniques to develop a new method with the highest spatial and temporal resolution.

## 2. APPARATUS AND NONLINEAR OPTICAL TORQUE THEORETICAL MODEL

Here we generalize the method using both passive and active rotational optical tweezers (Fig. 1), which enables measurements of viscoelasticity with high resolution and within broadband frequency range. The linearly polarized 1064 nm trapping laser beam (power $\approx 20\text{\hspace{0.17em}}\mathrm{mW}$ at the trap) is divided into two equal intensity beams by a polarizing beam splitter (PBS). In each path, an acousto-optic modulator (AOM, driven at 27 MHz) is used to control both the power and on/off state of the beam. A half-wave plate ($\lambda /2$) is used to rotate the linear polarization of the trap1 beam by $\pi /3$. Thus, a $\lambda /2$ wave plate and two AOMs provide a means to control the power as well as the polarization of the recombined beam. Then the two overlapping beams, with a fixed angle of $\pi /3$ between their planes of polarization, are focused into the sample through a water immersion objective ($60\times 1.2\text{\hspace{0.17em}}\mathrm{NA}$). The scattered light is collected by a condenser and then directed into two photodiode detectors (${D}_{3}$ and ${D}_{4}$). A circularly polarizing beam splitter is used so that detectors ${D}_{3}$ and ${D}_{4}$ measure the power of the left and right circularly polarized components of the transmitted trapping beam. Since the incident trapping beam is linearly polarized, this measurement of the spin angular momentum of the transmitted beam provides a measurement of the optical torque exerted on the probe particle [27]. A weak circularly polarized 633 nm He–Ne beam (power $\approx 20\text{\hspace{0.17em}}\mathrm{\mu W}$ at focal plane) is used to observe the angular displacement of the particle. As the beam passes through the probe particle, it becomes elliptically polarized, with the long axis of the polarization ellipse parallel to the optic axis of the probe particle. The beam is combined with the trapping beam by a dichroic mirror, separated from the trapping beam after collection by the condenser by a second dichroic mirror, and measured by two photodiode detectors (${D}_{1}$ and ${D}_{2}$). A linearly polarizing beam splitter is used so that detectors ${D}_{1}$ and ${D}_{2}$ measure the power of the two plane polarized components, in order to determine the direction of the polarization ellipse, thus providing a measurement of the orientation of the probe particle. To minimize optical forces caused by the tracking beam, the He–Ne beam is adjusted to underfill the objective lens, resulting in a relatively large focal spot ($\approx 1\text{\hspace{0.17em}}\mathrm{\mu m}$).

Measurements of the passive rotational diffusion of a birefringent microsphere trapped within either trap1 or trap2 provide information of the high-frequency viscoelastic properties of the fluid. In terms of the low-frequency fluid response, we measure active transient angular displacements of this probe, flipping between two overlapped traps (trap1 and trap2) with a fixed angle ${\varphi}_{0}=\pi /3$ between their directions of linear polarization. The two traps are alternately switched on and off (Fig. 1 inset) every few seconds (every 4 s for our experiments here). A spherical highly birefringent vaterite particle used as a probe causes the change in polarization of the beam passing through it [16]. Experimentally, we use measurements of the polarization of the transmitted light to determine the optical torque (trapping beam, measured by ${D}_{3}$ and ${D}_{4}$) and angular displacement of the probe particle (He–Ne beam, measured by ${D}_{1}$ and ${D}_{2}$).

In general, the frequency-dependent behavior of a viscoelastic fluid is described by the complex shear modulus, ${G}^{*}(\omega )$ [28], which consists of a real part, ${G}^{\prime}(\omega )$, the elastic storage modulus, and an imaginary part, ${G}^{\prime \prime}(\omega )$, the viscous loss modulus. Recent works have shown that measuring the rotational diffusion of a vaterite particle allows us to calculate the complex shear modulus at high frequency, which involves finding the normalized angular position autocorrelation function (NAPAF) $\mathrm{\varphi}$ defined by Eq. (1), which, containing a measurement of $\varphi (t)$, is a function of the time interval $t$ [26],

The complex shear modulus can be obtained by transforming the NAPAF with a unilateral Fourier transform defined by

A tilde represents a transformed variable, and $\chi $ is the rotational optical trap stiffness. The stiffness $\chi $ can be found using the equipartition theorem, with $\chi ={k}_{B}T/\u27e8\varphi {(t)}^{2}\u27e9$, where ${k}_{B}$ is Boltzmann’s constant and $T$ the absolute temperature [26]. Here, we assume that the angle is given in radians; if another unit of angle is used, the units can be converted to radians. Note that the stiffness $\chi $ is independent of the choice of units of angle, as can be seen by considering the equation of motion $I\ddot{\varphi}=\chi \varphi $, where any change to the units of angle affects both sides equally, and since the moment of inertia $I$ is independent of the choice of units of angle, so must be the stiffness $\chi $. Alternatively, $\chi $ can be found from the torque and angle measurements made during the active measurements; $\chi $ is the slope of the torque–angle curve at the angular equilibrium position [Fig. 2(c)]. Both of these methods of determining $\chi $ are independent of the rheological properties of the medium.

In a similar way, to resolve low-frequency viscoelasticity, we use a generalized Langevin equation to model the dynamics of the vaterite particle as it rotates between two traps over the angle of ${\varphi}_{0}$ between their directions of polarization. In principle, it is straightforward to convert a theoretical treatment of such active microrheology, e.g., [21], to the rotational case, by replacing linear quantities such as linear position, mass, and force with their rotational counterparts such as angular position, moment of inertia, and torque. For a spherical probe particle of radius $a$, the shape dependence of linear drag on velocity of $6\pi a$ (as appears in the well-known expression for Stokes drag) becomes the rotational equivalent $8\pi {a}^{3}$.

However, there is an additional complication. In such an active process, the effects of angular Brownian motion are mitigated by analyzing the average behavior of many flips between the traps. The effects of Brownian motion are also reduced by flipping over a larger angle. Given the wave-plate properties of the vaterite probe particles [29], the optical torque depends on the angle sinusoidally with a period of $\pi $, and can only be modeled as a linear spring for small angular displacements from the equilibrium angular orientation. The maximal signal-to-noise ratio is achieved by angularly offsetting the two traps by more than $\pi /4$ (we used ${\varphi}_{0}=\pi /3$), which is beyond the linear spring regime, and new theoretical analysis is required [30].

In Stokes flow, the moment of inertia is negligible, and the total torque can be assumed to be zero. We can further assume that Brownian motion can be eliminated by averaging over many flips. Since we only use a finite number of flips, the effect of Brownian motion is only reduced rather than eliminated completely, and an analysis including Brownian motion is presented in Ref. [30]; for simplicity, we only consider the limit where the effect of Brownian motion is eliminated. Accounting for a nonlinear torque exerted by the optical trap, dynamics of a probe with radius $a$ becomes

Assuming that the time it takes for the particle to flip between traps is much larger than the decay time of $\zeta (t)$, Eq. (4) can be linearized by making a transformation to a normalized variable,

Accordingly, in practice we set ${\varphi}_{0}$ to be $\pi /3$. We note that to reduce the error by 50%, the new theory only requires 4 times the number of flips, as opposed to the old model, which requires 8 times. As a result, our new active microrheology theory is less time-consuming and, hence, of great benefit to biological studies where the rheological properties can change over time.

## 3. RESULTS AND DISCUSSION

To verify our microrheological technique, we performed experiments in water and in aqueous dilutions of Celluvisc (Allergan), an artificial tear film. We found the viscosity of water to be $0.9\text{\hspace{0.17em}}\mathrm{mPa}\xb7\mathrm{s}$ for both passive and active measurements [Fig. 3(a)], in good agreement with the known viscosity of water near room temperature. Experiments were also performed in dilutions of 25% Celluvisc by weight and results [Fig. 3(b)] and were compared with conventional rheometry. We achieve good quantitative agreement between our method and conventional macrorheological data. We also experimentally confirm our active microrheology theory for nonlinear driving torque from both measurement of angular displacement and optical torque (Fig. 2). The optical torque in Fig. 2 is given as the torque efficiency ${Q}_{\tau}$, which is the torque in units of $\hslash $ per photon, and relates the torque $\tau $ and total beam power $P$ by $\tau ={Q}_{\tau}P/\omega $, where $\omega $ is the optical angular frequency of the trapping beam.

Thus, we have presented a working method to determine the wide-band viscoelastic properties of complex fluids using rotational optical tweezers.

We then examined the properties of the eye fluid film coated on contact lenses worn by two subjects [Figs. 4(a)–4(f)]. Experiments were performed at different times of the day (morning, midday, and afternoon) with two subjects of the same gender and age. Each individual set of data was obtained from contact lenses being worn for a few seconds before commencement of the experiment at room temperature. Figures 4(a)–4(c) illustrate the mean value of the complex shear modulus for each subject at three different times of day, and the results presented confirm the complex fluid structure of the tear film and show the distinct rheological properties of this structure for each subject. The data in Figs. 4(d)–4(f) are the average of the data of measurements from the two subjects to show the change in the viscoelasticity of the tear film over the day. Figure 4(f) shows that values of viscosity in the afternoon were smaller than values measured in the morning across a wide frequency range, and that the elasticity measured in the morning was lower than the values obtained in the afternoon in the low-frequency range, as shown in Fig. 4(d).

## 4. CONCLUSION

The potential of rotational optical tweezers as an accurate rheological technique to access the properties of fluids in highly confined environments has been demonstrated. Further experimental investigations are necessary to explore the properties of the tear film; other patient-related and environmental factors are known to affect the properties of the tear film [31,32]. The experimental methods and theoretical model for analysis presented here can simply be extended to other studies of biological fluids. We have demonstrated that our combined ultrasensitive, broad-frequency-response rotating photonic probe gives reliable, consistent, and very accurate spatially resolved measurements of properties of complex fluids on the microscale. This method can be used in the future studies of biologically relevant fluids such as cytoplasm.

## 5. METHODS

#### A. Materials and Sample Preparation

We synthesize vaterite crystals for our experiments, which are composed of the calcium carbonate mineral vaterite [33]. These microspherical particles with diameter of $\sim 3\text{\hspace{0.17em}}\mathrm{\mu m}$ are highly birefringent (with $\mathrm{\Delta}n=0.1$, which is the difference between the ordinary and extraordinary refractive indexes). Measurements of Celluvisc (Allergan), a 10 g/L aqueous solution of carboxymethylcelluose sodium solution, were carried out in dilutions of Celluvisc (25% by weight). To study the local viscoelastic properties of the tear film coated on contact lenses, we used daily disposable contact lenses (DAILIES AquaComfort Plus). The lens was placed on the eye, and after a few blinks, it was removed immediately from the eye and placed on a coverslip with a small adhesive spacer. We then cut the contact lens to fit the chamber, which also enabled us to assume that the small piece of the lens is flat, minimizing possible optical aberrations. Dried vaterite particles were then transferred to the contact lens, and the chamber was enclosed with a coverslip.

#### B. Angle and Torque Detection Scheme

The birefringent vaterite particle allows the exchange of spin angular momentum with the trapping beam, aligning the particle to a fixed orientation determined by the linear polarization vector of the trapping beam, or rotating with a constant torque in a circularly polarized beam [16].

A circularly polarized He–Ne beam is used to determine the angular position. The power of the beam is very low (20 μW), and this beam is underfilling the objective so that the torque exerted on the particle by this tracking beam is negligible compared to the trapping beam. When it passes through the vaterite particle, the polarization of the He–Ne beam changes from circular to elliptical. The long axis of the polarization ellipse of the elliptically polarized beam rotates with the optic axis of the particle [29]. By passing the transmitted He–Ne light through a polarizing beam splitter, we detect the power of each linearly polarized component at each detector (${D}_{1}$ and ${D}_{2}$ in Fig. 1). The normalized recorded voltage difference varies sinusoidally with twice the rotation rate of the particle, $\propto \mathrm{sin}\text{\hspace{0.17em}}2\varphi $. After each measurement, we introduce a quarter-wave plate into the infrared trapping beam path to make the probe particle spin at a constant rate, which allows us to find the exact relationship between the signals from ${D}_{1}$ and ${D}_{2}$ and the angular position $\varphi $. Since the power detected by ${D}_{1}$ and ${D}_{2}$ varies sinusoidally with the angle with period $\pi $, the angle of orientation can be found. Using this calibration, for both passive and active measurements, the rotational Brownian motion and the transient angular displacement of the particle are found. For the transient angular displacement, the directions of the planes of polarization of the two traps should be chosen so that the range of angles traversed during the flips lies on the close-to-straight section of the sinusoidal curve between a minimum and a maximum.

This measurement of angular position only yields the angle of the optic axis of the vaterite particle about the beam axis. Since the vaterite microspheres are positive uniaxial, they will align with their optic axes normal to the beam axis (i.e., with their optic axes horizontal) [16]. Since the vaterite microspheres are not exactly spherical, there will also be a torque acting to align their long axis with the beam axis. However, since the vaterite particles are highly birefringent and close to spherical, this alignment torque due to shape will be much smaller than the alignment torque due to their birefringence. If the calibration for using ${D}_{1}$ and ${D}_{2}$ to find the angle yields a sinusoidal curve, the orientation is sufficiently horizontal and stable for the particle to be used.

Since angular momentum is transferred between the trapping beam and the birefringent particle, the particle experiences a torque $\tau $ given by $\tau =\mathrm{\Delta}{\sigma}_{CP}P/\omega $, where $\mathrm{\Delta}{\sigma}_{CP}=({P}_{L}-{P}_{R})/P$ is the change in the degree of circular polarization of the beam, with ${P}_{L}$ and ${P}_{R}$ being the power of the left and the right circularly polarized components of the beam, $P$ the beam power, and $\omega $ the angular frequency of the light [29]. In practice, we measure the change in the degree of circular polarization of the trapping beam to determine the optical torque. When the linearly polarized trapping beam passes through the vaterite particle, the polarization of the beam changes. We use the combination of a quarter-wave plate and a polarizing beam splitter as a circularly polarizing beamsplitter to separate the transmitted light into two circularly polarized components, and measure their power with detectors ${D}_{3}$ and ${D}_{4}$ (see Fig. 1). Since the incident beam is linearly polarized, the two circularly polarized components are of equal power before passing through the vaterite particle. The difference between the signals collected by the photodiode detectors gives the change in the degree in circular polarization of the beam. When the particle is held in a linearly optical trap, in its equilibrium orientation when the optical torque is zero, we calibrate the two detectors (${D}_{3}$ and ${D}_{4}$) to make the difference in power equal to zero. Furthermore, the optical torque depends on the angle sinusoidally with a period of $\pi $. The active rotation is performed by rapidly switching the polarization of the trapping beam controlled by the AOMs. As the angle between the optic axis of the particle and the polarization is more than $\pi /4$ (${\varphi}_{0}=\pi /3$ in our experiment), the difference in power (${D}_{3}$ and ${D}_{4}$) first increases and reaches the maximum difference in power at the position when the angle becomes $\pi /4$, and then decreases down to zero when the particle reaches its new equilibrium orientation.

Noting that the spin angular momentum of a tightly focused circularly polarized beam is reduced by the focusing [34], one might wonder about the effect on this optical torque measurement. Since the angular momentum carried by the trapping beam is in the form of spin angular momentum (i.e., the circular or elliptical polarization of the beam), and this spin angular momentum is what is measured, any variation of the torque exerted on the vaterite sphere due to focusing of the beam, or any irregularities or defects in the structure of the vaterite from that which would be expected for a perfect vaterite sphere in a paraxial beam, is immaterial. The optical measurement provides us with the actual torque exerted on the particle.

## Funding

Australian Research Council (ARC) (DP140100753); China Scholarship Council (CSC) (201407510002); Australian Government Research Training Program Scholarship.

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