Single-shot wideband active microrheology using multiple-sinusoid modulated optical tweezers

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 modulating signals, 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 variety of linear viscoelastic media—polyacrylamide-water solution, worm-like micelles, and polyethylene oxide—at different concentrations and find good agreement of the measured complex fluid parameters with the known literature values.

Figure 1

The normalized input amplitudes for two different kinds of multisines are plotted on a logarithmic scale against the input frequencies. We see that the monotonically increasing MSSM is better after $\sim 250$ rad/s than the multisine that follows square waves.

Figure 2

Schematic of our setup. 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).

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.

Figure 4

The (a) storage moduli and (b) loss moduli for five 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 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 $G^{\prime \prime }$ scale as ${\omega }^2$ and $\omega$, respectively, at low frequencies where $G^{\prime \prime }>G^{\prime }$ are also plotted in grey in each figure.

Figure 5

Viscosity for five 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 five 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 cutoff 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/s and (c) is the psd from MSSM depicting the amplitude response to be significantly high after 500 rad/s. 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 $G^{\prime \prime }$) 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.

Figure 8

Measurement of the complex viscoelastic moduli for aqueous CTAB/NaSal (worm-like micelles) solution. The storage and loss moduli ($G^{\prime }$ and $G^{\prime \prime }$) are plotted in black solid circles ($G^{\prime }$) and squares ($G^{\prime \prime }$), respectively. At low frequencies, the data match expected trends (dashed lines) for both moduli, i.e., $G^{\prime }\propto {\omega }^2$, and $G^{\prime \prime }\propto \omega$. We compared our results with work reported earlier for the same sample (Fig. 3 of Morishima et al.), where the existing data are displayed in green open circles ($G^{\prime }$) and red squares ($G^{\prime \prime }$) (Ref. [47]).

Figure 9

Measurement of the complex viscoelastic moduli for polyethylene oxide in water solution. The storage and loss moduli ($G^{\prime }$ and $G^{\prime \prime }$) are plotted in green dots and red squares, respectively, for a concentration of 0.05% w/w. At low frequencies, the data match expected trends (dashed lines) for both moduli, i.e., $G^{\prime }\propto {\omega }^2$, and $G^{\prime \prime }\propto \omega$.

Figure 10

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 lower concentrations. We normalize the curve, putting the maxima at 1, for clarity.

Figure 11

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

Figure 12

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 13

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 $\omega$.