## Abstract

Optical tweezers have become a powerful tool to explore the viscoelasticity of complex fluids at micrometric scale. In the experiments, the Brownian trajectories of optically confined microparticles are properly analysed to provide the viscous and elastic moduli *G*′ and *G*″. Nevertheless, the elastic response of the medium is inherently superimposed on the trap stiffness itself. Usually, this drawback is removed by subtracting the elastic trap contribution from the measured medium response. However, it is clear that when trap and medium elasticity become comparable this procedure is no longer reliable. Still, there exists a wide class of complex fluids that exhibit a low elasticity (diluted biopolymers, Boger fluids, etc) for which alternative experimental approaches would be desirable. Herein we propose a new method based on blinking optical tweezers. It makes use of two independent laser beams: the first is used to trap a single bead while the second one, of very weak power, acts as probe to monitor its position with a quadrant photodiode. The trap laser intensity is modulated on-off: when the laser is off the bead follows a free diffusion trajectory that, hence, leads to an estimation of *G*′ and *G*″ free of the influence of the trap. We have successfully applied this technique to highly-diluted hyaluronic acid solutions (c < 0.1 mg/ml) reaching to measure very weak *G*′ modulus (~ 0.01 Pa) in a wide range of frequencies.

©2010 Optical Society of America

## 1. Introduction

Rheology is an interdisciplinary science that deals with deformation and flow of bulk materials. Generally speaking, when subjected to shear stress, these materials display both viscous and elastic responses. Relation between stress and strain is usually represented in terms of the frequency-dependent complex shear modulus *G*(*f*) which consists of a real part, *G*′(*f*), the elastic storage modulus, and an imaginary part, *G*″(*f*), the viscous loss modulus. For many years rheological investigations have been performed by conventional bulk rheometry [1, 2], which generally provides the macroscopic response of materials over a restricted frequency range (normally up to few Hz). Nevertheless, there exist many cases in which this approach is unsuitable. For instance, complex materials are usually inhomogeneous at mesoscopic scale: they contain extended molecular structures ranging from a few nanometers to several microns. In such cases conventional rheometry provides only average values of viscoelasticity. In other cases, such as for rare biomaterials, the small amounts of available material (microliters) represent another significant constraint. Moreover, during the recent years there has been a growing interest to study the viscoelasticity of living cells [3, 4, 5] which definitively cannot be performed with usual rheometers.

In order to overcome all these limits, several alternative techniques have been developed recently. These techniques, named *microrheology* [6, 7], are based on micrometric probes which non-invasively explore the local viscoelastic response in a frequency range much wider than rheometry requiring only few-microliters samples. In addition, a proper choice of the probe size makes it possible to explore the polymer network at different length scales [8]. Nowadays, numerous microrheological techniques exist such as, mentioning the most important ones, Video-Particle Tracking Microrheology, Atomic Force Microscopy, Diffusive Wave Spectroscopy, Fluorescence Recovery After Photobleaching, Magnetic Tweezers, and Optical Tweezers (OT) [6, 7]. In particular, OT, first developed by Ashkin et al. [9], has proved particularly interesting because it allows to confine micron-sized particles without any direct mechanical contact and to follow their interaction with the surrounding media by monitoring their Brownian motion. Optical Tweezers have been used in many experiments to measure the local viscosity of Newtonian fluids [10, 11, 12, 13] but also the viscoelasticity of complex fluids [14, 15, 16, 17].

Microrheology measurements based on OT have several advantages like the higher frequency range and the possibility to measure local heterogeneities, just to cite few of them, and they agree with the results obtained with classical rheometry on bulk materials. Nevertheless a critical point is when the elasticity of the sample is comparable with the stiffness of the optical trap. In fact the measured elasticity, as discussed below, is the sum of the sample elasticity and that coming from the optical potential. When the two are comparable it is not possible to measure the elastic modulus. This is an important limitation because there exists a wide class of complex fluids which exhibit low elasticity (diluted biopolymers, Boger fluids, low molecular weight polymer solutions, etc) and for which the possibility to measure their elastic modulus is of great relevance.

We propose a method to overcome this limitation, still using OT and its advantages. Unlike standard optical tweezers-based microrheology our method is based on two independent laser beams. The first one traps the particle, releases it for a certain time, and catches it again (Blinking Optical Tweezers, BOT) while the second one, of very weak power, monitors the particle position. BOT have already been used for several applications [18, 19, 20, 21] in combination with imaging devices for detection of the particle position. In particular it has been used to measure the diffusion of particles in low viscous fluids like water [18]. In fact in such fluids it is impossible to collect a large number of particle positions necessary to evaluate the diffusion coefficient, since the particle moves out of the focus plane very quickly. The use of BOT is necessary to catch and release the particle; while the particle is free, data can be acquired. However the use of imaging device strongly limits the acquisition bandwidth and as a consequence the extension of the frequency range in which the viscoelastic moduli can be measured. In the present work, we have extended the BOT technique to remove the elastic contribution of the optical trap itself and to estimate a very low elasticity using the high bandwidth of a quadrant photodiode.

#### 1.1. Theory

An optically trapped microsphere of radius *a* embedded in a viscous fluid is subjected to two kinds of forces: the elastic optical force which restores the equilibrium position of the particle in the trap and stochastic thermal forces caused by molecular collisions (Brownian motion). The scattering force which pushes the particle in the direction of the laser beam is relevant only for very weak trap [22] and, therefore could be neglected. The overall interaction can be described by the Langevin equation:

where *m* is the mass of the particle, κ is the trap stiffness, *F _{s}* is the stochastic thermal force due to the collisions of the particle with the surrounding molecules with <

*F*(

_{s}*t*) >= 0, and

*γ*is the Stokes hydrodynamic factor which is related to the viscosity

*η*by

*γ*= 6

*πη*

*a*. In this case the particle in the harmonic potential of the optical trap behaves like an overdamped oscillator and the inertial term of Eq. (1) can be neglected. The equation can be easily resolved in the frequency domain. The Power Spectral Density (PSD) of the stochastic position

*x*(

*t*) reads:

where *k _{B}* is the Boltzmann constant,

*T*the absolute temperature of the sample and

*f*is the corner frequency which is related to the trap stiffness and the viscosity by :

_{c}*f*=

_{c}*κ*/12

*π*

^{2}

*ηa*.

At low frequencies (long times), the PSD reaches a plateau due to the confinement of the particle while, at high frequencies (short times), decreases with a power law of *f*
^{-2}. Instead, for viscoelastic fluids the PSD tail follows a power law *f*
^{-β} where *β* depends on the polymer concentration and it is generally less than 2 [23]. Indeed, the analytical expression of the PSD is unknown.

The viscoelastic moduli can be obtained from the measurements of the particle thermal fluctuations, following method described below. The Fourier transform of the particle position 𝓕(*x*) is linearly related to the Fourier transform of the stochastic force, 𝓕(*F*), by the relation [24, 25]:

Here α(*f*) = α′(*f*) + α″(*f*) is the complex compliance which is related to the complex shear modulus *G*(*f*) through the generalized Stokes-Einstein relation (GSER):

The link between the PSD, *S*(*f*) and the imaginary part of the compliance α″(*f*) is provided by the fluctuation-dissipation theorem which states that:

The real part α′ can be computed using the Kramers-Kronig relations provided that α″ (*f*) is known on a wide frequency range. The relation is:

$$=4{\int}_{0}^{+\infty}\mathrm{cos}\left(2\pi ft\right)dt{\int}_{0}^{+\infty}\mathrm{\alpha \u2033}\left(\mathrm{f\prime}\right)\mathrm{sin}\left(2\pi \mathrm{f\prime}t\right)df\prime $$

Since α″ (*f*) is not known analytically the integral can only be computed numerically. Once α has been obtained, *G*′ and *G*″ can be obtained straightforwardly using Eq. (4).

When this procedure is applied to a particle trapped in an optical tweezers immersed in a Newtonian fluid, real and imaginary part of the shear modulus reduce to:

That is, the elastic modulus is frequency-independent and is related only to the trap stiffness, while the viscous modulus depends linearly with frequency and the slope is proportional to the sample viscosity.

When we estimate the elastic modulus in a complex viscoelastic medium, it results as the sum of the trap stiffness and the sample elasticity. Hence, to estimate the effective elastic contribution of the sample, *G*′_{sample}, the trap elasticity *G*′_{trap} should be subtracted from the measured storage modulus *G*′_{meas}, i.e. *G*′_{sample} = *G*′_{meas} - *G*′_{trap}. Clearly, *G*′_{sample} becomes unreliable when the two terms *G*′_{meas} and *G*′_{trap} are comparable within the experimental errors.

## 2. Experimental setup and calibration

A scheme of the experimental setup is shown in Fig. 1. The used lasers were both Nd:YAG lasers: the first one (Innoligth Mephisto 500) was used for trapping while the second one (LigthWave A125) to probe the bead position. The polarization of the two lasers was P and S, respectively. The power of the trapping laser was about 2 mW at the sample which gives a quite weak trap stiffness of the order of 10 pN/μm corresponding to a storage modulus of *G*′_{trap} ~ 1 Pa. As discussed below, this power allowed to satisfactory confine the bead and, at the same time, to reduce as well as possible the elastic contribution of the trap. On the contrary, the probe laser was still weaker (~ 100 μW) and we verified that its trapping effect was negligible. The two laser beams were made collinear using a polarizing cube beam splitter. A second polarizing cube beam splitter was placed after the sample chamber and the condenser lens so that the probe beam could be separated from the trapping beam and sent onto a In-GaAs quadrant photodiode (QP, Hamamatsu G6849) used as position sensor with a bandwidth of 250 kHz. The QP was properly calibrated (see below) and we checked that its response was linear for displacements from equilibrium position up to ± 300 nm. In our experiment, we trapped polystyrene microspheres (diameter= 1.00 ±0.01 μm) in hyaluronic acid (HA) of low molecular weight MW=155 kDa diluted in distilled water at low concentrations (typically < 0.2 mg/ml). The amplitude of the trapping laser was square-wave modulated with a shutter (risetime 1 ms). When the trapping laser is on, the particle is pushed around by the Brownian motion but remains confined in the trapping volume.

In order to estimate the calibration factor to convert the measured voltage signal from the QP into the length units, we first trapped polystyrene beads in pure water. The calibration factor was then determined by fitting PSD curve of the trapped beads with a Lorentzian [17] and subsequently used for the measurements in HA solutions.

The experiments in HA were carried out in two different operation modes: Continuous Wave OT (*CWOT*) in which the bead position is recorded while keeping the trapping laser continuously on, and Blinking OT (*BOT*), in which the Brownian trajectory is collected during a time interval *T _{OFF}* in which the trapping laser is off (see Fig. 2). During the interval

*T*the particle is animated only by thermal fluctuation. In our experiments the trapping laser was modulated at 4 Hz by using a mechanical shutter. Due to some electronics delay between the driving square wave signal and the shutter, the on/off times of the transmitted laser beam were different:

_{OFF}*T*= 0.14 s and

_{OFF}*T*= 0.1 s (see Fig. 2). Therefore, we used as trigger for the QP acquisition, the signal coming from a photodiode which monitored the transmitted trapping power after the sample chamber. The maximum bead displacement during the interval

_{ON}*T*was short enough to consider linear the response of the QP and, at the same time, to catch the bead when the trapping laser was on again. So we could repeat many cycles acquiring a large number of

_{OFF}*free*trajectories with the same bead; that allowed to improve the signal-to-noise ratio of the viscoelastic moduli measurements. The QP signals were recorded over a period of 4 s with a sampling time dt=25 μs (

*f*= 20 kHz). We computed the PSDs, only when the laser was off, that is 15 off-segments for each recorded trace. Then we averaged the PSDs over a set of N=20 traces (i.e. a total of 300 stochastic off-segments). Of course, in

_{max}*CWOT*mode we were able to measure the response at frequencies down to 1 Hz. We chose 20 s as total time and divided the records in 20 segments to increase the S/N ratio.

All the experiments were repeated with different beads and at different times. No significant differences were observed within the experimental errors.

## 3. Results and discussion

Figure 2 shows a typical stochastic signal trace recorded in a HA solution (c= 0.02 mg/ml) when the trapping laser is modulated on-off. In CWOT mode the stochastic bead displacement remains confined within ±30 nm while in *BOT* mode it reaches up to ±200 nm (i.e. well within the linear response, 300 nm, of the QP). We compare the two operating modes by calculating the mean square displacement (MSD) and the power spectral density. The PSDs for two HA concentrations (c=0.02 mg/ml and 2 mg/ml) are compared in Fig. 3. At high frequencies (short times) the PSDs for both modes agree and follow the same power law *f*
^{-β} with *β* ≈ 1.9 for both concentrations. The parameter *β* decreases asymptotically to 1.5 at the highest HA concentration (see ref [23]). Since *β* is slightly smaller than 2, the sample has prevalently a viscous behavior and its elasticity is quite low. If we analyse the low frequencies part of Fig. 3, the PSDs relative to both the operating regimes show a different behaviour: in the BOT mode they are straight lines while in the *CWOT* they reach a plateau due to the trap confinement. Clearly, the higher the HA concentration the larger the difference of the PSD in the two regimes. It is also interesting to compare the MSD: < Δ*x*
^{2}(*τ*) >=< [*x*(*t* + τ) - *x*(*t*)]^{2} > (see Fig. 4). At long time scales the MSDs obtained in *CWOT* mode reach a plateau due to the trap confinement while, at short time, the MSDs deviates from *BOT* mode behavior at very short lag times (*t* ≤ 4 ms). The viscoelastic moduli *G*′ and *G*″ were finally computed following the procedure described above and reported in refs. [24] and [23]. Both *G*′ and *G*″ were calculated for the two regimes discussed here. Figures 5 and 6 present the concentration dependence of viscous and elastic moduli, respectively.

We compare first *G*″(*f*) as calculated from the CWOT and BOT. Since *G*″(*f*) indicates the viscosity of the sample it should be not affected by the trap; indeed no differences are observed between the two modes. As expected, the viscous modulus, for all the concentrations investigated in this work, increases linearly with the frequency over all the investigated range. Thus the fluid viscosity does not depend on the frequency. However, at concentrations higher than 2 mg/ml and at frequencies higher than 1 kHz the viscosity starts to decrease as discussed in ref. [17]. This effect is known as *shear thinning* and is a common property of polymeric solutions.

More interesting is instead the discussion concerning the elastic modulus *G*′(*f*). The elastic modulus curves shown in Fig. 6 have an upper- and a lower-frequency limit. The lower limit, *f _{min}*, is essentially due to the total acquisition time (

*T*=20 s for

*CWOT*mode and

*T*= 0.14 s for

_{OFF}*BOT*mode) while the upper limit, fmax, arises from the finite sampling time

*dt*(

*f*= 0.5/

_{max}*dt*from the Nyquist theorem); it is also affected by the Kramers-Kronig integrals which tend to distort

*G*′ about one decade down from

*f*(see artifacts in Fig. 6 and discussions in ref. [24] and [23]).

_{max}As mentioned above, the measured elastic modulus *G*′_{meas} is given by two contributions: *G*′_{trap} and *G*′_{sample} which correspond to trap and sample elasticity. The trap stiffness can be calculated from the corner frequency of the PSD. Since the trap behaves as an ideal spring *G*′_{trap} is independent on the frequency *f* while normally *G*′_{sample} exhibits a dependence on the shear rate and usually drops to zero at low frequencies.

The trap contribution, at the lowest investigated concentration, completely dominates the elastic modulus so that, within the experimental noise, it becomes unreliable to extract *G*′_{sample} from the measured modulus *G*′_{meas}. At intermediate HA concentrations (0.1 mg/ml < *c* < 0.2 mg/ml) *G*′_{trap} dominates only at low frequencies, thus the correction *G*′_{meas} - *G*′_{trap} gives the correct estimation only for the high frequency part of the curve. For higher HA concentrations (*c* >0.2 mg/ml), *G*′_{sample} dominates over all frequency range and the subtraction of *G*′_{trap} provides correctly the elastic modulus.

*G*′_{meas} reaches the same value at low frequencies for all the concentrations explored in this work. Therefore we can assume this value represents the trap elasticity. We checked that, using the trap stiffness obtained from the PSD and Eq. (8), the same result was obtained. In particular we have estimated *G*′_{trap} as the average of the experimental points *G*′_{meas} in the interval between 0.5 and 2 Hz. As it can be noted in Fig. 6, *G*′_{sample} estimated following this procedure (dashed lines before and after the subtraction) results unreliable for the first two concentrations. On the contrary, in the *BOT* mode *G*′_{meas} (solid curve) provides directly the storage shear modulus of the sample *G*′_{sample}. At higher concentrations (*c* ~ 0.2 mg/ml), where the elastic modulus of the HA solution increases, the discrepancy between *G*′_{sample} estimated with the two modes vanishes. In particular, for quite large concentrations (*c* = 2 mg/ml) standard OT microrheology (*CWOT* mode) becomes a valid technique in good agreement with *BOT* mode in all the accessible frequency range of our experiment.

Hyaluronic acid is a polyelectrolyte polymer, characterized by two values of concentrations which mark the transition from dilute, semi-dilute and entangled regimes. In particular, we denote as *c*
^{*} and *c _{e}* the overlap and the critical concentrations, respectively. The overlap concentration is the border between the dilute and semidilute ranges and is determined as the point where the concentration inside a single coil equals the bulk concentration. In contrast, the critical concentration

*c*represents the entanglement concentration in the semidilute region, that is the limit between the unentangled and entangled regime in which the polymer chains start to overlap each other and to form a transient network. For the HA chain used in our experiment (MW=155 kDa) results:

_{e}*c*= 0.059 mg/ml and

^{*}*c*= 2.5 mg/ml [23]. Therefore the range of concentrations used in our experiment (0.02 - 2 mg/ml) is below the critical concentration

_{e}*c*at which the polymer chain entanglement starts and elastic behavior becomes very relevant. A part dilute polymers, as investigated in this work, there are interesting polymers where a direct and non-biased determination of the elasticity should be desirable. For instance, Boger fluids are polymer solutions that are characterized by a relatively weak elastic and a constant-viscosity. They are important because they enable elastic effects to be clearly separated from viscous ones. The separation is currently possible with experiments which are conducted with two fluids: a Boger fluid and a Newtonian fluid of the same viscosity. The difference in outcomes at the same flow rate, therefore, results from elasticity alone. We think that our method could be suitable to isolate weak elasticity in a direct and reliable way.

_{e}It is worth to mention that very recently a detailed study of Brownian motion in presence of a modulated optical trap has been proposed by Deng *et al*. [26]. In this work the authors determined how the position variance of a trapped particle is affected by a temporal modulation of the trap stiffness. Their analysis is limited to small modulation amplitudes, but gives information also for a fully modulated optical trap like the BOT here proposed. The main result is that at modulation frequencies below the corner frequency the position variance increases leading to systematic errors especially in force measurements. However, it is worth to be noted, that in our case the measurements is performed when the trap is switched off for a brief period, that is we get completely rid of the particle positions affected by the presence of the optical trap. In other words the effects discussed in ref.[26] should be taken into account when the whole position records is analyzed.

## 4. Conclusion

We have demonstrated a new technique which allows to measure the position of a bead trapped in an optical tweezers with high detection bandwidth. It makes use of two lasers, one of very weak power to monitor the particle displacements using a quadrant photodiode and the other to catch and release the particle. So far it is possible to follow the particle motion in absence of the restoring force of the optical tweezers. This technique is very useful for the determination of viscoelastic properties of complex fluids which have a weak elastic component like Boger fluids, low molecular weight polymer solutions and dilute solutions. The method is used to measure weak elasticity of the order of (10^{-2} Pa) which is not easily accessible with traditional rheometers and conventional OT technique.

The main restriction of our technique is the limited acquisition time, which is related to the free diffusion of particles within the detection range of the probe laser. This affects the minimum frequency in the elastic modulus curve (10 Hz in our experiment). But it should be noted, that in most cases the elasticity of very soft materials is relevant at high frequencies where our technique has no limitations. Moreover, to overcome this limitation it is possible to use materials with higher viscosity or to use larger particles. In both cases the *T _{OFF}* time can be increased, leading to a wider frequency range.

## Acknowledgments

We are very grateful to Prof. G. Marrucci, Prof. S. Guido and Dr. A. Jonas for helpful discussions and critical reading of the manuscript. Giulia Rusciano acknowledges CNISM for her research fellowship.

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