# Single-shot wideband active mircorheology using multiple-sinusoid modulated
Optical Tweezers

Avijit Kundu1 Raunak Dey1 Shuvojit Paul1,2 Ayan Banerjee1 1Department of
Physical Sciences, Indian Institute of Science Education and Research Kolkata,
Mohanpur Campus, Mohanpur, West Bengal, India  2Department of Physics,
University of Konstanz, Germany<EMAIL_ADDRESS>
###### Abstract

We employ multiple sinusoid modulated optical tweezers to measure the
frequency dependent rheological parameters of a linear viscoelastic fluid over
five decades of frequency in a single shot, hitherto not achieved using active
microrheology alone. Thus, we spatially modulate a trapped probe particle
embedded in a fluid medium with a combination of a square wave - which is by
definition a superposition of odd sinusoidal harmonics - and a linear
superposition of multiple sinusoids at a wideband frequency range, with
complete control over the amplitude, frequency and relative phase of the
modulating signals. For the latter, we selectively excite the particle by
larger amplitudes at high frequencies where the particle response is low,
thereby enabling wideband active microrheology with large signal-to-noise.
This mitigates the principal issue associated with conventional active
microrheology - which is low bandwidth - and also renders our method better in
terms of signal to noise, and faster compared to passive microrheology. We
determine the complex viscoelastic parameters of the fluid by extracting the
phase response (relative to input excitation) of the probe from the
experimentally recorded time series data of the probe displacement, and
employing well-known theoretical correlations thereafter. We test the efficacy
of our method by studying a linear viscoelastic media (polyacrylamide-water
solution) at different concentrations, and find good agreement of the measured
fluid parameters with known literature values.

††preprint: APS/123-QED

##  I Introduction

Complex fluids are ubiquitous in nature, and are essential to life itself due
to their various diverse manifestations in biological entities, including the
environments of cells themselves. In recent years, microrheology [1, 2, 3] has
proven to be an indispensable tool to study mechanical properties of complex
fluids at microscopic lengths scales [5, 6]. The technique generally employs
microscopic probe particles in fluids, and the motion of these probe particles
is measured to determine the fluid properties at different spatio-temporal
scales. There are several experimental methods which are used for
microrheological measurements including atomic force microscopy [7, 8],
optical tweezers [1, 9, 10], and free Brownian diffusers [11]. The use of
optical tweezers has specifically proven to be efficacious since they allow
controlled manipulation of the probe particles inside a given fluid, which may
be extremely useful in studying the heterogeneity in a biological sample [12],
tracking the local changes in a soft material over time [13], as well as
probing the interior of a cell [14].

Microrheology of complex fluids essentially consists of the determination of
the complex shear modulus
$G^{*}(\omega)[=G^{\prime}(\omega)+iG^{\prime\prime}(\omega)]$, where the real
part $G^{\prime}(\omega)$ of the shear modulus describes the elastic
properties (storage), while the imaginary part $G^{\prime\prime}(\omega)$
describes the fluid damping properties (loss). Conventionally, microrheology
is performed by tracking the Brownian motion of probe particles due to thermal
(passive) motion using photo-detectors, diffuse wave spectroscopy, light
scattering or video microscopy [15, 16, 17, 18, 19], and using the statistics
of the time series of the particles’ displacement to determine the complex
shear modulus $G^{*}{(\omega)}$ and viscosity $\eta^{*}{(\omega)}$ in the
frequency domain, or stress $\sigma(\tau)$ and strain rate
$\dot{\varepsilon}(\tau)$ for the time domain. Now, though the Brownian signal
has the components of all the frequencies embedded in the time series, the
signal power falls as $\frac{1}{f^{2}}$ ($f$ being the frequency), thus
rendering passive mircorheology challenging at high frequency regimes. As a
result, very often, long data-sets recorded over very long times are required
to obtain acceptable signal-to-noise ratio at high frequencies. On the other
hand, the response of a trapped probe particle can be enhanced by providing an
external perturbation to it. Modulating the position of the minimum of the
trap-potential sinusoidally (equivalent to an oscillating shear) [20] converts
the resultant motion of the particle into a superposition of Brownian and
active motion. This approach leads to a much higher signal at the modulating
frequency compared to the passive method, and is therefore known as active
microrheology. [21] The biggest issue with this technique is the long times
required to obtain a data over a large bandwidth, since in that case the
frequency of the perturbation needs to be varied continuously, with
$G^{*}(\omega)$ measured for each frequency [22, 23]. Any systematic changes
that occur during that time in the trap (induced by laser drifts or
fluctuations) or fluid properties, would introduce errors in the measured
values. To address such issues, several techniques have been recently
developed. These include flipping the probe particle between two time shared
traps [24], using ’chirp waves’ by continuously varying the modulation
frequency [25], square waves [10] which contain odd sinusoidal harmonics, or
even frequency modulation [26] over a certain bandwidth. However, none of
these single-shot active microrheology techniques are able to measure
$G^{*}(\omega)$ for more than two to three decades of frequency [27] (for
examples, 0.01-10 rad/sec or 1-1000 rad/sec), so that low bandwidth remains a
lacuna in single-shot active microrheology in comparison to the passive
technique, where frequency responses have been collected for five decades in
frequency, albeit with long measurement times and by employing a combination
of techniques such as diffuse wave spectroscopy and light scattering from free
probe particles [17, 18, 19], or even active and passive microrheology
performed in tandem using optical tweezers [24]. Indeed, the broad frequency
spectrum of thermally induced Brownian motion that is exploited in passive
microrheology to garner viscoelasticity measurements over wide frequency
ranges prompts the question whether this can also be achieved in active
microrheology in some manner. This is exactly what we address in this work.

In this paper, we develop a new technique of active microrheology where we
modulate a trapped colloidal probe particle using a “Multiple Sinusoids
Superposition Method”. Thus, we use a combination of square wave and a linear
superposition of sinusoidal signals of known amplitudes, frequencies, and
relative phases to modulate a single optically trapped probe particle to
measure the complex viscoelastic parameters of a linear viscoelastic fluid in
a single shot. We apply the square wave at low frequencies up to 500 rad/sec,
and the superposition of sines at higher frequencies beyond 500 rad/sec.
Earlier, we reported active microrheology using square wave excitation alone
in Ref. [10]. Here, we develop the sinusoidal excitation technique, where -
corresponding to our input excitation - the response of the particle is also a
superposition of all frequencies, and we extract specifically the phase
response at each excitation frequency by using a Discrete Fast Fourier
Transform (DFFT) algorithm on the recorded time series of the particle’s
displacement [28]. Since the particle response falls off at high frequencies
(compared to the corner frequency of the trap), we increase the modulation
amplitude correspondingly, so that a substantial response is still available
even at high frequencies. This innovation facilitates large signal-to-noise
measurements even at high modulation frequencies [29], thus enabling wide-band
active microrheology. However, we take care to ensure that the modulation
amplitudes are low enough to maintain the linear response of the viscoelastic
fluid, so that the excitation at a particular frequency elicits a response
only at the same frequency, which we then carry out for all frequencies of
modulation. We extract the complex viscoelastic parameters using a technique
we reported first in our previous work [10], and demonstrate our technique
first on water for basic calibration, and then on a water-polyacrylamide (PAM)
solution at different concentrations of PAM over a frequency range of $\sim
1.8-13900$ rad/sec, and observe that our measurements match literature values
at similar domains of polymer concentration and frequency [30, 31]. This
technique thus mitigates a crucial issue of active microrheology - that of
limited bandwidth - for linear viscoelastic fluids, and can also be evaluated
for performance inside biological cells for measurement of rheological
parameters. Also, to the best of our knowledge, high frequency rheology has
not yet been performed in the case of biological entities. Thus, there is
limited knowledge about the existence of interesting information at those
regimes - which maybe a lacuna given that several cellular processes and
neural phenomena happen in the few milliseconds regime.

##  II Microrheological analysis

We model our system around the well-known Langevin Equation for a spherical
colloidal particle trapped in a harmonic potential created by a Gaussian beam
in our case. We consider the fluid to be homogeneous around the locality of
the particle throughout the experiment and assume that there is no active flow
in our system. We modulate the minimum of the harmonic potential and detect
the position of the particle in a single direction, so that the single
variable differential equation in time for an independent degree of freedom
can be written as:

$m\frac{d^{2}x}{dt^{2}}=-\int_{-\infty}^{t}\gamma(t-t^{\prime})\dot{x^{\prime}}dt^{\prime}-k[x(t)-x_{0}(t)]+\xi(t)$
(1)

Here $m$ is the mass of the spherical polystyrene particle, $k$ is the
stiffness of the harmonic potential, $x_{0}(t)$ is the instantaneous position
of the potential minimum, $x(t)$ is the position of the centre of the
particle, and $\gamma(t)$ is the time dependent friction coefficient of the
viscoelastic fluid. The whole integral term on the right hand side of the
equation calculates the drag force that acts on the particle, and has
contribution from all past times. $\xi(t)$ represents the gaussian distributed
white noise having an auto-correlation given by
$\langle\xi(t)\xi(t^{\prime})\rangle=2k_{B}T\gamma(t-t^{\prime})$. Since the
inertia of the particle is damped out by the fluid (the inertial time constant
[32, 33] of a few microseconds is much lower than the time resolution of our
experiment), we ignore the inertial term of Eq. 14 and write it in Fourier
domain as,

$[-i\omega\gamma(\omega)+k]x(\omega)=kx_{0}(\omega)$ (2)

Note that the useable range of frequency is up to around 100 krad/sec after
which the inertia of the probe becomes significant [34] (see Supplementary
Information). The fluid being viscoelastic in nature has a complex drag
coefficient, so that
$\gamma(\omega)=\gamma^{\prime}(\omega)+i\gamma^{\prime\prime}(\omega)$, and
the phase response of the particle can be calculated as

$\tan(\phi)=-\frac{\omega\gamma^{\prime}(\omega)}{k+\omega\gamma^{\prime\prime}(\omega)}$
(3)

The negative sign in Eq.3 demonstrates that the phase response of the probe
particle lags the excitation signal. Now, for two different values of $k$, say
$k_{1}$ and $k_{2}$, we can extract two different values of the phase $\phi$,
namely $\phi_{1}$ and $\phi_{2}$, respectively. For the potential minimum
being modulated by $\omega_{0}$, we can then solve $\gamma^{\prime}(\omega)$
and $\gamma^{\prime\prime}(\omega)$

$\gamma^{\prime}(\omega_{0})=\frac{k_{1}-k_{2}}{\omega_{0}\left(\frac{1}{\tan(\phi_{1})}-\frac{1}{\tan(\phi_{2})}\right)}$
(4)

$\gamma^{\prime\prime}(\omega_{0})=\frac{\gamma^{\prime}(\omega_{0})}{\tan(\phi_{1})}-\frac{k_{1}}{\omega_{0}}$
(5)

Since, $G^{*}(\omega)$, the complex shear moduli of the linear viscoelastic
fluid is related to the drag coefficient of the fluid as
$G^{*}(\omega)=-i\omega\gamma(\omega)/6\pi a$, where $a$ is the radius of the
spherical probe, we can calculate the storage moduli (real part of G), loss
moduli (imaginary part of G) and viscosity of the fluid as

$G^{\prime}(\omega_{0})=\omega_{0}\gamma^{\prime\prime}(\omega_{0})/6\pi a$
(6)

$G^{\prime\prime}(\omega_{0})=\omega_{0}\gamma^{\prime}(\omega_{0})/6\pi a$
(7)

$\eta(\omega_{0})=\frac{\sqrt{G^{\prime}(\omega_{0})^{2}+G^{\prime\prime}(\omega_{0})^{2}}}{\omega_{0}}$
(8)

Now, we modulate the potential minimum by a linear superposition of sine waves
of known amplitudes ($A_{i}$), angular frequencies ($\omega_{i}$), and
relative phases ($\psi_{i}$). So,

$x_{0}(t)=\sum_{i}A_{i}sin(\omega_{i}t+\psi_{i})$ (9)

Using a Fourier decomposition on the final time series, each frequency
component can be isolated and the respective phase and amplitude components
extracted. The phase response of the particle (mentioned earlier as $\phi$)
would be the difference between extracted phase $\phi^{ext}_{i}$ and input
phase $\psi_{i}$, that is $\phi_{i}=\phi^{ext}_{i}-\psi_{i}$. For water, we
can show that $\phi_{i}=\tan^{-1}\left(\frac{f}{f_{c}}\right)$ where $f_{c}$
is the corner frequency of the operating trap. $f_{c}$ is given by

$f_{c}=\frac{k}{12\pi^{2}\eta_{0}a}$

where $\eta_{0}$ is the viscosity of water. By fitting the phase response of
the particle in water with varying frequency, we can find out the corner
frequency and stiffness of our trap in water.

From Eq. 2 we also calculate the amplitude response of the particle as

$\mid
x(\omega)\mid=\frac{kx_{0}(\omega)}{\sqrt{(k+\omega\gamma^{\prime\prime})^{2}+\omega^{2}\gamma^{\prime
2}}}$ (10)

For a simple, memory-less Newtonian fluid such as water,
$\gamma^{\prime\prime}=0$, and Eq. 10 simplifies to

$\mid x(\omega)\mid=\frac{kx_{0}(\omega)}{\sqrt{k^{2}+\omega^{2}\gamma^{\prime
2}}}$ (11)

Thus, the amplitude response of the particle is at a particular frequency, say
$\omega_{0}$, is proportional to $\omega_{0}^{-1}$

##  III Material and Methods

###  III.1 Multisine method

As mentioned earlier, we modulate the mininum of the harmonic trap with a
linear superposition of multiple sine waves of different frequencies,
amplitudes, and relative phase. We now describe our strategies about the
choice of these parameters:

####  III.1.1 Choice of amplitudes

We see from Eq. 10 that the amplitude response of the probe particle is
inversely proportional to the modulating frequency $\omega_{0}$. This poses a
challenge in obtaining high signal-to-noise at high frequencies- therefore, we
use input modulation amplitudes proportional to the input frequency. Our aim
is thus to obtain a constant response over the entire frequency range of
excitation in order to facilitate accurate phase and amplitude extraction.
However, using only a monotonically linearly increasing input amplitude
implies that the lower frequency components will be subjected to very low
excitation, which will detrimental for phase extraction at these frequencies,
especially for fluids with low concentrations. However, we have earlier shown
[10], that a square wave is essentially an infinite sum of the odd harmonics
of sine waves mathematically written as

$x_{0}(t)=\frac{4A}{\pi}\sum_{i=1}^{\infty}\frac{sin(2\pi(2i-1)ft)}{2i-1},$
(12)

where, $A$ is the amplitude of the square wave. As we see in Fig. 1, a square
wave has large input amplitude at low frequencies with the amplitude
decreasing linearly with increase in frequency. Thus, in our method of wide-
band microrheology, we use a square wave excitation for estimating the complex
viscoelastic parameters at the lower end of the frequency range, and the
increasing amplitude multisine excitation for the higher end of the frequency.
Considering that we actually use the sinusoidal composition of the square
waves for our method, we prefer to name this method of active microrheology as
the ’Multiple Sinusoids Superposition Method’ or MSSM. Our protocol finally
consists of a single programmed sequence of pulses used to modulate the probe
particle, which is made up of the following:

  1. 1.

Initially, we excite the particle for 100 seconds with a multi-sine signal in
the form of a square wave, which forms a truncated series of odd sines with
their amplitudes reducing with frequency. The frequencies we use are from
1-500 rad/sec roughly.

  2. 2.

For the next 100 seconds we provide a second multi-sine signal where
$A=constant\times\omega$, with $A$ and $\omega$ representing amplitude and
frequency, respectively. The frequencies we use in this approach are from
500-14000 rad/sec.

Figure 1: The normalised input amplitudes for two different kinds of multi-
sines are plotted on a logarithmic scale against the input frequencies. We see
that the monotonically increasing MSSM is better after $\sim 250$ rad/sec than
the multisine that follows square waves

####  III.1.2 Choice of individual frequencies

We can, in-principle, superpose any number of frequencies to obtain
rheological information of the fluid at great detail in frequency space. But
the superposed signal needs to be normalised to a certain peak-to-peak
amplitude such that when it is applied to the particle, the oscillation of the
particle remains constrained in the harmonic region of the trap, where the
linear response approximation is valid [35]. To choose the values of the
frequencies, we choose a series of prime numbers in order to ensure that the
response at one frequency is not mixed up in the next harmonic (though we are
working in the linear region of the VE fluid, where this should not occur in
theory). Also we observed that using prime-numbers minimizes the chances of
constructive interference in our input waveform, thus create stronger signal
at each individual frequency after normalisation. The frequency values
$f_{p}^{i}$ are therefore chosen as $f_{p}^{i}<f_{max}$, where $p$ is a prime
number and $i$ is a positive integer. In our case $f_{max}$ is 2197 Hz (13.8
Krad/sec). This frequency is close to an order of magnitude higher than the
corner frequency of our traps at two different powers and it is important to
note that the effects of the trap diminish rapidly at even higher frequencies.
Hence, we treat this frequency as a cut-off (note that the noise intensity
goes as $power\sim\frac{1}{f^{2}+f_{c}^{2}}$, and thus diminishes by two
orders of magnitude of the mean intensity if the operating frequency is
increased by one order). We also observed that the input phase has no
significant effect on the accuracy of the phase extraction. We have therefore
attempted experiments using linearly increasing phase difference, zero phase
difference, as well as random phase difference between individual components,
but have not observed any modifications in the nature of the results we
obtained.

Finally, we conclude that by combining two different genres of _multisines_
for equal time intervals, we are able to create a custom waveform of duration
over 200 seconds and perform single-shot wide-band microrheology.

##  IV Experimental design

We develop our optical tweezers setup (Fig. 2) around an inverted microscope
(Olympus IX71) with an oil immersion objective lens (Olympus 100x, 1.3
numerical aperture) and a semiconductor laser (Lasever, 500 mW max power) of
wavelength 1064 nm which is tightly focused on the sample. We modulate the
beam by using the first order diffracted beam off an acousto-optic modulator
(AOM), (Brimrose) placed at a plane conjugate to the focal plane of the
objective lens. The modulation amplitude is small enough such the power in the
first order diffracted beam is modified very minimally (around 1%) as the beam
is scanned. We employ a second stationary and co-propagating laser beam of
wavelength 780 nm and very low power (less than 5% of the trapping laser
power, such that the detection laser does not influence the motion of trapped
particle in any way) to track the probe particle’s position (detection laser),
which we determine from the back-scattered light that is incident on a
balanced detection system [36]. The balanced-detection-system together with a
data acquisition card record the probe displacement data into a computer. We
prepare a sample chamber of dimension around $20$ mm$\times$ $10$ mm$\times$
0.16 mm by attaching a cover slip to a glass slide by a double-sided tape
which contains our model viscoelastic fluid. The model fluid samples are
prepared by mixing polyacrylamide (PAM, flexible polyelectrolytes,
$M_{w}=(5-6)\times 10^{6}$ gm/mol, Sigma-Aldrich) into water along with mono-
dispersed spherical polystyrene probe particles of radii $1.5$ $\mu$m in very
low volume fraction ($\approx 0.1\%$) which we use as probes. We also prepare
the sample just before performing our experiments so that there are no ageing
issues associated with PAM. We use PAM solution at 5 different concentrations
in water: 0.01% w/w, 0.05% w/w, 0.1% w/w, 0.5% w/w and 1% w/w.

Figure 2: The schematic of our setup is presented here. The annotations are as
follows: HWP - half wave plate, AOM - acousto optical modulator, L - convex
lens of various focal lengths, PBS - Polarizing Beam Splitter, D - Dichroic
mirror, M - one sided mirror, PD - photodiode, DAQ - data acquisition card
(manufactured by National Instruments.)

For each viscoelastic sample, first, we trap a single probe particle around 30
$\mu$m away from the nearest wall to get rid of any surface effects [37] and
record its Brownian motion with sampling frequency 5 kHz over 20 s, to measure
the stiffness of the trap. Then, we modulate the trap center by a square
signal of very small peak-to-peak amplitude (130 nm) - so that the particle
always remains in the linear region of the trap. The fundamental frequency of
the square waves are set to 0.3 Hz (or 1.8 rad/sec) for all the cases. To
perform the experiments using the multisine excitation, we follow the same
protocol, only this time we use 100 seconds of observation for the multi-sine
signal keeping the sampling frequency fixed at 5 kHz. To achieve a high degree
of precision we maintained the experimental setup unaltered during the
experiment. The rheological parameters can depend on the trap stiffness, probe
size, and age of the sample– for which we have used freshly prepared solutions
of PAM everytime we performed an experiment.

##  V Results and Discussions

Equilibrium statistics allows us to measure the stiffness of the optical trap
using Maxwell’s Equipartition Theorem. This is free from the rheological
influence of the sample used [5, 38]. According to this theorem, the trap
stiffness is given by, $k=k_{B}T/\langle(x-\langle x\rangle)^{2}\rangle$.
Since the histogram of the positions follows a gaussian distribution, we find
out $\langle(x-\langle x\rangle)^{2}\rangle$ by fitting the data. That gives
us an accurate measure of $k$, for all the viscoelastic samples we have used
in our experiment. Before using the equipartition method to find out the trap
stiffnesses at two different powers, we need to find out the sensitivity of
our position detection system. Note that position calibration is independent
of the sample rheology. We have used the viscous drag method in water to find
out the corner frequency at two different powers and calibrations of our
setup. We use this method to calibrate our optical trap from the voltage (as
obtained in our detector) to particle displacement. We use the fact that the
phase lag of the particle response in water can be accurately fitted with
$\tan^{-1}\left(\frac{f}{f_{c}}\right)$, which gives us the value of $f_{c}$
\- the corner frequency. For two different powers of the trapping laser we
calculate the corner frequency as $116.6\pm 7.4$ Hz and $231.1\pm 20.0$ Hz,
for low and high trapping powers, respectively. From this, we find out the
trap stiffness from $k=12\pi^{2}\eta_{0}af_{c}$, which leads to trap
stiffnesses of $17.61\pm 1.12\ \mu N/m$ and $34.87\pm 3.02\ \mu N/m$,
respectively in water.

However, it should be noted that our method is built on a phase extraction
algorithm - we do not use the amplitude component due to the associated
signal-to-noise issues during extraction, especially for low concentrations of
the viscoelastic (PAM) component [10]. Nonetheless, we perform the spatial
calibration to determine the trap stiffness and to ensure that our trap
modulation leads to particle displacements that lie within the linear response
of the trap. We also check that the corner frequency changes linearly with
power so that we know that the trapping potentials we use are indeed harmonic.

It is important to note that for extraction of the phase response from the
data, trap calibration is not required. Indeed, our procedure for phase
extraction is entirely based on the _Discrete Fast Fourier Transform method_.
We take the Fourier transform of the entire time series data of length $N$ and
in accordance with the Nyquist criterion, dispose off half the data - so that
we are left with a complex time series of $N/2$ points. The ratio of the
imaginary to real components of each number in the Fourier transformed time
series gives the tangent of the phase value. Our task is essentially to find
the phase corresponding to the excitation frequency we are interested in. In a
similar vein, the amplitude response can be calculated as the modulus of each
number. The time for the analysis to run is highly dependent on the time taken
to compute to Fourier Transform, because rest of the operations such as
calculating rheological parameters from extracted phases have linear time
complexity. As a result, the time complexity of our entire method follows that
of a _Fast Fourier Transform_ as $\sim O(NlogN)$. Also, the high frequency
limit of our experiment is set by the corner frequency of our trap - which
presently - at the highest laser power we can couple into the trap, is around
232 Hz. Increasing both trap stiffness and sampling frequency will thus
facilitate the increase of the high frequency limit of this technique to
whatever level is desirable.

All the individual components of the multi-sine wave affect the probe particle
independently. We initially verified that we could extract individual
components using the DFT algorithm for a multisine wave, and also matched our
results with experiments using single sine waves of different frequencies.
Most importantly, we ensured that there was no crosstalk - i.e., no leaking of
the response at one frequency into another, which implied that the
displacement due to a particular force at a particular frequency was linearly
related to the response of the system. Thus, for the displacement $x_{o}(t)$
by the input perturbation $x_{i}(t)$, we have $x_{o}(t)=\int
dt^{\prime}\chi(t;t^{\prime})x_{i}(t^{\prime})$. Our system - comprising of
the trapped particle in a viscoelastic fluid gives - a linear response $\chi$
under small perturbations of the particle. This response function is invariant
under time translation, $\chi(t;t^{\prime})=\chi(t-t^{\prime})$, so that
performing a Fourier transform of $x_{o}(t)$, we obtain

$x_{o}(\omega)=\int dt^{\prime}\int dte^{i\omega
t}\chi_{oi}(t-t^{\prime})x_{i}(t^{\prime})=\chi_{oi}(\omega)x_{i}(\omega)$
(13)

This essentially is the general form of a linear response which we also see in
Eq. 10 for our special case, and this implies that the perturbation is ‘local’
in the frequency domain. In other words, the output response of the particle
is linearly independent at different frequencies for the small displacements
(of the order of tens of nanometers) we drive in our technique. In order to
verify that quantitatively, we determine the velocity of the particle
$x_{i}(t)$ using Euler’s differential method. We use this extracted velocity
to compute the Weissenberg number, and observe that in all our experiments, we
have $Wi<<1$. This indicates that we are indeed eliciting linear response from
the particle as a result of the effects on it from the viscoelastic fluid [39,
40, 3] (see Online Supplementary Information). We would like to emphasise that
this estimation is crucial - especially for soft materials having low yield
strain - in which case a trade-off between the linearity and magnitude of
response required to perform a reliable measurement may lead to a change in
the frequency range over which the viscoelastic properties can be measured -
especially at the higher end.

In order to experimentally validate our algorithms for phase and amplitude
extraction, we perform experiments in water, and observe that while
$G^{\prime}(\omega)$ is almost zero for the entire range of modulation
frequencies, $G^{\prime\prime}(\omega)$ follows the trend expected for water
theoretically. This is shown in Fig. 4b. We now move on to the PAM-water
mixtures, and determine the amplitude and phase responses of the trapped
probe, which we then compare to determine the parameter that would be the most
efficacious to use to measure the complex viscoelasticity of our samples. Now,
the amplitude response of the particle is defined as
$\frac{x_{o}(\omega)}{kx_{i}(\omega)}$, where, $x_{i}(\omega)$ and
$x_{o}(\omega)$ are the Fourier components of input signal provided to the
probe particle, and output signal from the particle as measured using photo-
diodes (both transformed in distance units, respectively). This amplitude
response can be theoretically calculated as shown in Eq. 10, but we do not
employ it to extract the rheological parameters of the fluid as mentioned
earlier. However we note that, for higher concentration samples the amplitude
response is also higher (See Online Supplementary information), as a result of
which, we are able to extract the whole broadband range of rheological
parameters for the 0.1% w/w, 0.5% w/w and 1% w/w PAM in water samples with our
MSSM technique only (without the aid of square wave modulation). For lower
concentration samples, signal-to-noise issues prevent us from obtaining
reliable values for $G^{\prime}(\omega)$ and $G^{\prime\prime}(\omega)$.

Figure 3: Phase response of the trapped probe particle embedded in the 0.01%
w/w sample to the Multisine perturbations plotted over the entire frequency
range in a semi-logarithmic graph. The two different curves are obtained for
different sets of measurements using two different powers of the trapping
laser.

(a) Storage moduli (b) Loss moduli

Figure 4: The storage moduli (a) and loss moduli (b) for 5 different
concentrations of PAM in water, (1% w/w, 0.5% w/w, 0.1% w/w, 0.05% w/w and
0.01% w/w) extracted using our technique and plotted against angular
frequency. We also have plotted the loss moduli of water using our method
along with theoretical values. As expected we see a gradual increase in both
of them with increasing frequency. Trendlines to confirm that $G^{\prime}$ and
$G"$ scale as $\omega^{2}$ and $\omega$, respectively, at low frequencies
where $G">G^{\prime}$, are also plotted in grey in each figure. Figure 5:
Viscosity for 5 different concentrations of PAM in water, (1% w/w, 0.5% w/w,
0.1% w/w, 0.05% w/w and 0.01% w/w) extracted using our technique and plotted
against angular frequency; standard error, obtained after averaging over 5
data sets, is shown as the error bar. We observe a gradual decrease in the
effective viscosity of the sample with increasing frequency of oscillations,
and after a cut-off frequency it reaches an asymptotic value. Figure 6: (a)
Signal-to-noise ratio over the entire frequency range, where (b) is the power
spectrum density (psd) from square wave modulation (SM), which shows the
amplitude response to be significantly high up to 500 rad/sec and (c) is the
psd from MSSM depicting the amplitude response to be significantly high after
500 rad/sec. All the graphs are plotted for the dataset 0.1% w/w PAM in water,
just taken as an example. Figure 7: The storage and loss moduli ($G^{\prime}$
and $G"$) are plotted in green squares and red dots respectively for 1%w/w PAM
in water using the protocol described in this paper. To compare we show the
values at the same concentration for a smaller particle size as reported in
Fig.5 of Tassieri et al. (Ref. [5]). We observe that the measured values
closely concur with the reported values, which further corroborates our
technique.

We now analyse the phase response of the probe more carefully. In Fig. 3, we
observe that the phase values for higher trap stiffness is less than the phase
values for the lower stiffness. This is understandable, since the phase
response is inversely proportional to corner frequency
$\left(\tan^{-1}\left(\frac{f}{f_{c}}\right)\right)$. We extract the phase, as
shown for one sample in Fig. 3, for 2 different powers for each concentration.
Using the measured phases and using Eqs. 4-8, we calculate the storage and
loss moduli for each sample. These are plotted in Fig. 4. It is clear that
with the increase in frequency of the probe particle - which means an increase
in its velocity through the fluid - both the storage and loss moduli of PAM
increase. This is because – as the probe oscillates at higher frequencies – it
deforms the polymeric mesh progressively, and also significantly enough for
the latter to affect the motion of the particle, and thus alter its response
to the drive. For low drive frequencies, the mesh is able to relax - so that
the particle does not see its effect. Similarly it is easy to understand that
with the increase in concentration of PAM in water, the overall density of the
mesh would increase, so that it would be able to impart more elastic energy to
the probe, as we observe in Fig. 4(a). In the same manner, the loss also
increases since dissipation is higher now, and we also observe in Fig. 4(b),
that all the measured values of $G^{\prime\prime}$ lie above the theoretical
value of $G^{\prime\prime}$ in water as expected (for water, a purely viscous
fluid, $G^{\prime\prime}(\omega)=\eta\omega$ and $G^{\prime}(\omega)\sim 0$,
since there is no polymeric constituent for storage). The effective viscosity
is therefore higher than that of water, and only approaches that value
asymptotically at high frequencies, as is clear from Fig. 5. For each of the
concentrations we have performed multiple measurements, and averaged over 5
data sets to ensure that our results are reliable. In Fig. 4(a) and (b) only
the mean is plotted in the graph, though, to avoid unnecessary congestion.
However, in Fig. 5, both the mean and the standard error in the measured
values of the viscosity have been plotted. In addition, as expected from the
theory, the shear moduli follows the Kramers-Kroenig relations, which is
demonstrated by the fact that $G^{\prime}$ and $G"$ scale as $\omega^{2}$ and
$\omega$, respectively, at low frequencies where $G">G^{\prime}$. We
explicitly show this in Fig. 4.[17].

An interesting observation in Fig. 4(a) is that $G^{\prime}$ saturates beyond
a particular frequency. Such saturation as a function of frequency has been
observed for entangled polymers in several other cases [1, 5, 24, 42, 43, 44].
We believe that this marks a transition from the polymer from the terminal
region to the rubbery plateau region as is described succinctly in Ref. [42].
Alternatively, it has also been understood as the saturation of $G^{\prime}$
beyond the frequency corresponding to the polymer time constant [43, 44]. The
time constant increases for increasing concentrations - as a result of which
the saturation frequency also lowers correspondingly. We also observed that
the saturation effect for the highest concentration (1% W/W) was not visible
at the modulation amplitude we used for other concentrations, i.e. around 130
nm. Reducing the amplitude to 70 nm, however, led to saturation of
$G^{\prime}$ as is visible in red solid line with star-shaped data points in
Fig. 4(a). This is because the condition of $W_{i}<<1$ required for the linear
viscoelastic regime is not enforced strongly enough for the modulation
amplitude of 130 nm (see more details and a plot of $G^{\prime}$ for two
modulation amplitudes in the Supplementary Information).

It is also interesting to quantitatively compare the signal-to-noise ratio for
a square wave and the multi-sine excitations, as we demonstrate in Fig. 6(a) -
(c). To generate the data shown in this figure, we actually excite the probe
by a square wave excitation, and a multi-sine wave till 13982 rad/sec. The
signal-to-noise is defined as the amplitude of detected response above the
thermal noise level, as obtained from the power spectral density. We define it
in decibel units as:

$S/N=10\log_{10}\left(\frac{\text{Power of signal}}{\text{Power of
noise}}\right)$

. Thus, we consider the noise power spectrum of the Brownian motion and
determine the amplitude of the response peak we obtain at a particular
excitation frequency, and its strength with respect to the noise floor defined
by the pure thermal motion. We determine the signal-to-noise values for both
square and multi-sine wave excitation and compare them in Fig. 6(a). Fig. 6(b)
and Fig. 6(c) shows the peaks in the power spectrum of square and multisine
respectively for the case 0.1% w/w PAM in water as an example. We observe that
at angular frequencies less than $\sim 500$ rad/sec, the square wave gives a
higher signal-to-noise ratio for the particle response - however beyond that,
the multi-sine wave gives continuously increasing signal-to-noise ratio. This
motivates us to use a square wave excitation till 500 rad/sec, and a multi-
sine excitation beyond that frequency. Another issue we consider is the
continuity in phase retrieval between the square wave excitation and multi-
sine excitation. Even here we observe that there is indeed a smooth continuity
between the extracted phase by square wave and that by the multi-sine
approach. This is borne out from the fact that the extracted complex
rheological parameters change smoothly, as is clear from Fig. 4, where the
data for frequencies less than 500 rad/sec is from a square wave excitation,
while that for higher frequencies is via the multi-sine excitation. Finally in
Fig 7, we compare the viscoleastic parameters that we have extracted with that
measured in the literature in Fig.5 of Tassieri et al. (Ref. [5]). We plot the
mean values of G’ and G” from multiple measurements and provide the standard
error obtained after averaging over 5 data sets. It is important to note that
the size of the probe particle used in Ref. [5] is $5\mu m$, which is greater
in size than the $3\mu m$ particles we use. Although the trend matches with
the reported literature value, the systematic deviation is probably due to the
difference in particle size, which may suggest that PAM has multiple different
length scales and is a multi-scale mesh [45].

The first crossover between the two shear moduli gives the time constant of
the fluid, from the inverse of the crossover frequency (which is $\approx 5$
ms in our case for 1% w/w PAM in water). Presence of multiple crossovers of
the shear moduli may suggest that PAM has multiple polymeric time constants
emerging from different length scales (multi-scale mesh). In Fig. 7, we
observe that the values of $G^{\prime}(\omega)$ and $G^{\prime\prime}(\omega)$
seem to approach each other - but it is difficult to infer conclusively about
the possibility of a crossover happening at even higher frequencies, since the
inertial effects of the medium start showing up [1] at that regime, which may
complicate the physics considerably. For a discussion on the inertial effects
in microrheology see the Online Supplementary Information. Clearly, more
experiments with different probe sizes are called for to clarify this matter.

##  VI Conclusion

In conclusion, we mitigate a major issue in active microrheology - that of
extracting rheological parameters of a fluid over a large bandwidth with high
signal-to-noise in a single shot. Thus, we propose a new technique, which we
name as the ’Multiple Sinusoids Superposition Method’ – where we use a
superposition of multiple sine waves to excite a colloidal probe particle
embedded in a PAM:water mixture at different concentrations to extract the
complex rheological parameters. For low frequencies, we use a square wave -
which is by definition a combination of odd harmonics of sines - while for
higher frequencies - we apply a superposition of sine waves with the
modulation amplitudes increasing with increasing excitation frequency in a
proportional manner. This ensures that the signal-to-noise extracted at high
frequencies does not diminish in spite of the low displacement of the probe
particle at those frequencies. We use only the phase component of the probe
response, which is advantageous since it does not require calibration of the
displacement sensitivity of our detection system, and is not susceptible to
spurious electronic noise which affects amplitude measurements considerably.
We extract the phase from a discrete fast Fourier transform that we perform on
the measured time series of the probe displacement, and obtain the complex
rheological parameters by repeating the measurement at two different trap
stiffnesses. Our extracted rheological parameters match reasonably well with
values available in literature, which acts as a good consistency check. The
method is fast, accurate, and is easily extendible to even higher frequencies
by employing optical traps of higher stiffnesses and larger sampling rate for
signal measurement. We are currently in the process of extending this method
to microrheology experiments inside biological cells[46], where it can
possibly extract information at frequency values hitherto inaccessible by
active microrheology.

## SUPPLEMENTARY MATERIAL

See Supplement file for supporting content.

###### Acknowledgements.

This study was supported by IISER Kolkata, teaching and research institute
supported by the Ministry of Human Resource Development, Govt. of India. A.K
and R.D acknowledge the Department of Science and Technology, Govt. of India
for the INSPIRE Fellowship and INSPIRE Scholarship respectively.

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Supplementary Information

Single shot wideband active mircorheology using multiple-sinusoid modulated
Optical Tweezers

Avijit Kundu1, Raunak Dey1, Shuvojit Paul1,2, Ayan Banerjee1

## A. Amplitude response

Although we completely use the phase response from the trapped probe to
compute the rheological parameters of the VE fluid, it principle this can also
be done from the amplitude response – which has its own drawbacks as discussed
in the paper. But it is intriguing to note that the amplitude response of the
particle is inversely proportional to the trapping power – which is further
corroborated by Eq. 10 of the main manuscript. The very increasing nature of
the shear modulus with concentration is linked to the fact that the amplitude
response is increases with increasing concentration of the samples. We
demonstrate this in Fig. 1, for two different concentrations and two different
laser powers.

Figure 1: The normalized amplitude responses are obtained from the time series
of particle displacement and plotted against the angular frequency of the
active probe particle. Here the normalized amplitude means the ratio of the
output signal to the input excitation for each frequency. We observe better
amplitude response in samples with higher concentrations. We have normalised
the curve, putting the maxima at 1, for clarity.

## B. Weissenberg Number

The Weissenberg number ($Wi$) which is defined as
$Wi=\frac{\dot{x}_{o}\lambda}{2a}$, where $\lambda$ is the relaxation time
constant of the fluid and $a$ is the radius of the trapped colloidal probe.
The method of obtaining the time constant is described in a previous work [3].
The Weissenberg number is very commonly used in the literature to denote the
ratio of the elastic to the viscous forces.

## C. Storage modulus with modulation amplitude

The response of the viscoelastic system lies in the linear regime when $Wi<<1$
[4]. The Weissenberg number depends on the velocity of the trapped particle
inside the solution. So we have modulated the particle with different
amplitudes 70 nm and 130 nm respectively, and extracted the storage modulus of
1% w/w PAM solution. At the higher modulation amplitude the Weissenberg number
is close to 0.1, where as for the lower amplitude it is close to 0.05. So
performing the experiment at the higher amplitude modulation may introduce
non-linearity in the response of the viscoelastic fluid at higher
concentrations. We have plotted in Fig. 4 both the storage moduli for
modulating the particle with different amplitudes, and it shows that with
higher amplitude-modulation the storage modulus does not saturate at high
frequency as is the case for lower amplitude, since we probably approach the
non-linear viscoelastic regime for the former. For this, we have used 70 nm
modulation for the highest concentration. In future we will expand our study
on the non-linear effects of viscoelastic fluids at higher concentrations.

Figure 2: Storage moduli of 1% w/w PAM solution extracted from modulating the
particle by different amplitudes.

## D. Inertia effect in viscoelastic fluid

We have neglected the inertia effect here as our measurement frequency range
obeys the following relation according to Ref. 33 in the main manuscript.

$\omega<<\sqrt{\pi^{2}G(\omega)/4a^{2}\rho_{f}}$ (14)

where $G(\omega)$ is the frequency dependent shear modulus, $a$ is the
particle radius, and $\rho_{f}$ is the density of the particle. In our case
the frequency is 100 krad/s and our measured frequency (14 krad/s is less than
that. However at very high frequency there still may be some influence of the
residual inertia of the fluid-probe system. Such effects can modify the
saturation behaviour of the shear-moduli frequencies over 10 kHz frequency.
Such effects are explicitly calculated in Ref. [5].

## E. Viscoelastic moduli with error bar

Figure 3: Storage moduli PAM solution in water for five different
concentrations of 0.01%, 0.05%, 0.1%, 0.5%, 1% w/w. Here storage moduli vary
with $\omega^{2}$. Figure 4: Loss moduli PAM solution in water for five
different concentrations of 0.01%, 0.05%, 0.1%, 0.5%, 1% w/w. Here the loss
moduli vary with $\omega$.

## References

  * [1] Department of Physical Sciences, IISER-Kolkata.
  * [2] University of Konstanz, Germany
  * [3] S. Paul, A. Kundu, and A. Banerjee, ”Active microrheology to determine viscoelastic parameters of Stokes-Oldroyd B fluids using optical tweezers,” Journal of Physics Communications 3, 035002 (2019).
  * [4] C. D. Chapman, R. M. Robertson-Anderson, ”Nonlinear microrheology reveals entanglement-driven molecular-level viscoelasticity of concentrated DNA,” Phys. Rev. Lett. 113, 098303 (2014).
  * [5] Grimm, M., Jeney, S., & Franosch, T. (2011). Brownian motion in a Maxwell fluid. Soft Matter, 7(5), 2076-2084.