Time-Resolved Mechanical Spectroscopy of Soft Materials via Optimally Windowed Chirps

The ability to measure the bulk dynamic behavior of soft materials with combined time and frequency resolution is instrumental for improving our fundamental understanding of connections between the microstructural dynamics and the macroscopic mechanical response. Current state-of-the-art techniques are often limited by a compromise between resolution in the time and frequency domains, mainly due to the use of elementary input signals that have not been designed for fast time-evolving systems such as materials undergoing gelation, curing, or self-healing. In this work, we develop an optimized and robust excitation signal for time-resolved mechanical spectroscopy through the introduction of joint frequency- and amplitude-modulated exponential chirps. Inspired by the biosonar signals of bats and dolphins, we optimize the signal profile to maximize the signal-to-noise ratio while minimizing spectral leakage with a carefully designed modulation of the envelope of the chirp, obtained using a cosine-tapered window function. A combined experimental and numerical investigation reveals that there exists an optimal range of window profiles (around 10% of the total signal length) that minimizes the error with respect to standard single-frequency sweep techniques. The minimum error is set by the noise floor of the instrument, suggesting that the accuracy of an optimally windowed-chirp (OWCh) sequence is directly comparable to that achievable with a standard frequency sweep, while the acquisition time can be reduced by up to 2 orders of magnitude, for comparable spectral content. Finally, we demonstrate the ability of this optimized signal to provide time- and frequency-resolved rheometric data by studying the fast gelation process of an acid-induced protein gel using repeated OWCh pulse sequences. The use of optimally windowed chirps enables a robust time-resolved rheological characterization of a wide range of soft materials undergoing rapid mutation and has the potential to become an invaluable rheometric tool for researchers across different disciplines.

Popular Summary

Many soft materials—such as curing epoxy, gelling yogurt, or self-healing artificial tissue—often undergo microscopic structural changes during their synthesis or assembly. To understand how these microscopic changes affect macroscopic behavior, researchers need a way to measure the mechanical properties of soft materials as they rapidly change. However, current experimental procedures often cannot provide the requisite time and frequency resolution. Inspired by sonar signals that bats and dolphins use for echolocation, we propose a novel technique based on oscillatory strain signals, with carefully designed frequency and amplitude modulation.

Our approach relies on commercially available instruments to provide unprecedented resolution in the characterization of material properties. Using a combination of numerical and experimental investigations on polymer solutions, wormlike micelles, and a rapidly gelling protein gel, we show that our specially modulated strain signals, which we call optimally windowed chirps, can provide a material mechanical spectrum with the same accuracy as standard methods while reducing the measurement time by as much as a factor of 100.

The use of optimally windowed chirps enables robust time-resolved dynamical mechanical characterization of a wide range of soft materials undergoing rapid changes in time, and our new protocol provides a new and potentially invaluable tool for materials researchers across a wide range of different disciplines.

Figure 1

(a) Example of a single-tone sine input: sinusoidal strain with constant amplitude ${\gamma }_0$ and frequency $\omega$. (b) Example of an exponential sine sweep or chirp: sinusoidal strain with constant amplitude and modulated frequency [see Eq. (2)]. (c) Comparison of measured viscoelastic moduli for a reference PIB solution (see Appendix pp1 for details) obtained by a classical discrete frequency sweep ($G^{\prime }$, solid line; $G^{\prime \prime }$, dashed line) with 30 points per decade (total experimental acquisition time greater than 30 min) and with the application of only one chirp signal ($G^{\prime }$, red solid line and filled circle; $G^{\prime \prime }$, red dashed line and circle) as described by Eq. (2) with $T=14\mathrm{s}$, ${\omega }_1=0.3\mathrm{rad}/\mathrm{s}$, ${\omega }_2=30\mathrm{rad}/\mathrm{s}$, and $TB=66$ at the same input strain amplitude ${\gamma }_0=6\%$.

Figure 2

Data presented are from experiments on the reference PIB solution. All chirp signals have the following parameters: ${\omega }_1=0.3\mathrm{rad}/\mathrm{s}$, ${\omega }_2=30\mathrm{rad}/\mathrm{s}$, ${\gamma }_0=0.06$, and $T=14\mathrm{s}$ ($TB\simeq 66$) but different degrees of tapering, i.e., different values of $r$. (a) Sine sweep in strain $\gamma (t)$ (as proposed by Ghiringhelli et al. [46]) with constant signal amplitude $r=0$ (grey line). The enclosing dashed line emphasizes the square nature of the signal envelope. Superposed is a windowed chirp with $r=50\%$ (blue line) together with its tapered envelope (dark blue dashed line). Both signals begin at time $t_w=1\mathrm{s}$ as described in the text. (b) Single-sided amplitude spectrum of the strain signal for the same values of the tapering parameter $r$ as in (a): $r=0$ (grey solid line and square) and $r=50\%$ (blue dashed line and circle). (c) Measured stress response as a function of the time corresponding to the input signals of (a): $r=0$ (grey line) and $r=50\%$ (blue line). (d) Single-sided amplitude spectrum of the stress signals shown in (c). Data from the measured stress ($r=0$, grey solid line and square; $r=50\%$, blue dashed line and circle) are compared to the error-free spectrum ($r=0$, black solid line; $r=50\%$, black dashed line) obtained in the frequency domain by multiplying the theoretical transfer function from the fractional Maxwell model, i.e., the complex modulus $G^{\star }(\omega )$ of the material, and the DFT of the experimental strain input. (e) Comparison of the viscoelastic moduli $G^{\star }(\omega )=G^{\prime }(\omega )+i{G}^{\prime \prime }(\omega )$ obtained by a classical frequency sweep ($G^{\prime }$, black solid line; $G^{\prime \prime }$, black dashed line) with the results obtained from a windowed chirp with different tapering parameters: $r=0$ ($G^{\prime }$, grey solid line and filled square; $G^{\prime \prime }$, grey dashed line and open square), $r=10\%$ ($G^{\prime }$, red solid line and filled triangle; $G^{\prime \prime }$, red dashed line and open triangle), and $r=50\%$ ($G^{\prime }$, blue solid line and filled circle; $G^{\prime \prime }$, blue dashed line and open circle). The inset shows $\tan \delta (\omega )=G^{\prime \prime }(\omega )/G^{\prime }(\omega )$ (the same color code and symbols as in the main graph).

Figure 3

Summary and comparison of experiments and numerical simulations on the reference PIB solution. The plots show the rms error in the determined storage modulus (a) and loss modulus (b), defined in Eqs. (6) and (7), as a function of the tapering parameter $r$. Black circles (${ϵ}_{G^{\prime }}$, filled circle; ${ϵ}_{G^{\prime \prime }}$, open circle) show the experimental results averaged over six different realizations. Error bars correspond to one standard deviation. Blue squares (${ϵ}_{G^{\prime }}$, filled; ${ϵ}_{G^{\prime \prime }}$, open) are the results from numerical simulations without noise, while green diamonds (${ϵ}_{G^{\prime }}$, filled; ${ϵ}_{G^{\prime \prime }}$, open) show the trend from simulations when Gaussian noise is added to the stress signal in order to mimic experimental conditions. The dashed line is an exponential fitting to the data for $r<1\%$, while the dotted line shows the quadratic divergence of the error for $r>15\%$.

Figure 4

Time- and frequency-resolved gelation of a casein gel obtained using an OWCh sequence with $r=10\%$, $T=14\mathrm{s}$, ${\omega }_1=0.3\mathrm{rad}/\mathrm{s}$, ${\omega }_2=30\mathrm{rad}/\mathrm{s}$, $TB=66$, $t_w=1\mathrm{s}$, and ${\gamma }_0=1\%$. (a) Evolution of the viscoelastic moduli ($G^{\prime }$, line; $G^{\prime \prime }$, dashed line) as a function of the gelation time for one selected test frequency ($\omega =10\mathrm{rad}/\mathrm{s}$). (b) Phase angle $\delta$ as a function of the time for different frequencies: $\omega =0.3\mathrm{rad}/\mathrm{s}$ (grey line), $\omega =10\mathrm{rad}/\mathrm{s}$ (brown line), and $\omega =20\mathrm{rad}/\mathrm{s}$ (black line). At the gel point (indicated by the pink right pointing triangle), all three frequencies pass through a single value of $\delta$, because the phase angle is independent of frequency [65, 66, 74]. (c) Viscoelastic moduli ($G^{\prime }$, closed symbols; $G^{\prime \prime }$, open symbols) and (d) phase angle as a function of frequency at different times during gelation: $t=1.21\mathrm{h}$ (blue filled circle), $t=1.31\mathrm{h}$ (violet filled square), $t=t_g=1.46\mathrm{h}$ (pink right pointing triangle), and $t=5.25\mathrm{h}$ (red filled diamond). Symbols and colors correspond to the points indicated in (a) and (b). At the gel point, the phase angle is invariant with the frequency (pink right pointing triangle), while before and after there is a more pronounced dependence of $\delta$ on $\omega$. The solid lines in (c) and (d) correspond to the model prediction obtained by assuming a mechanical model composed of a single fractional element as described in Sec. 4 (the same color code applies). (e) and (f) show, respectively, the evolution in the gel strength and average relaxation exponent as a function of time. The respective values at the gel point are indicated with the same symbol (pink right pointing triangle).

Figure 5

Fitting of the elastic and loss modulus measured with a classical frequency sweep ($G^{\prime }$, filled circle; $G^{\prime \prime }$, circle). This is the same data as in Fig. 1, determined with 30 points per decade but plotted with only ten points per decade for clarity. The superposed magenta lines ($G^{\prime }$, line; $G^{\prime \prime }$, dashed line) are best fits of the analytical expressions for the moduli based on the fractional Maxwell model presented in Sec. 3a, which give $\alpha =1$, $\mathbb{V}=\eta =18\mathrm{Pa}\mathrm{s}$, $\beta =0.36$, and $\mathbb{G}=50\mathrm{Pa}{\mathrm{s}}^{0.36}$. The inset shows the mechanical analog for the FMM, where one of the elements becomes effectively a classical viscous dashpot since $\alpha =1$.

Figure 6

The window function $w(t)$ is plotted for Tukey windows with (a) $n=1$ and (b) $n=4$. The different blue, green, magenta, and red colors correspond to $r=50\%$ (blue line), $r=100\%$ (green line), $r=150\%$ (magenta line), and $r=200\%$ (red line), respectively. (c) The computed errors for the complex modulus, from numerical simulations, are plotted versus the tapering parameter $r$ for different Tukey windows: $n=1$ (green dot filled circle), $n=2$ (blue dot filled circle), $n=3$ (magenta dot filled circle), and $n=4$ (red dot filled circle). The corresponding solid lines show the error predictions from Eq. (b4) with $c_1=0.3$ and $c_2=0.126$. The gray line represents the case of $n=0$.

Figure 7

(a) Viscoelastic moduli of the CPyCl micellar solution. Reference values ($G^{\prime }$, line; $G^{\prime \prime }$, dashed line) are measured with a sequence of sine steps at different frequencies. A comparison between results measured using optimally windowed chirps ($r=10\%$) with two different time-bandwidth products obtained by changing only the length of the signal $T$: $TB=8.6$ ($G^{\prime }$, blue filled triangle; $G^{\prime \prime }$, blue triangle) and $TB=107$ ($G^{\prime }$, green filled square; $G^{\prime \prime }$, green square). The frequency range is kept constant with ${\omega }_1=3\mathrm{rad}/\mathrm{s}$ and ${\omega }_2=30\mathrm{rad}/\mathrm{s}$ to guarantee a consistent comparison between chirps of different durations. Results obtained using a separate optimally windowed chirp ($r=10\%$) with the frequency range adjusted to the length of the signal ($T=14\mathrm{s}$, ${\omega }_1=0.3\mathrm{rad}/\mathrm{s}$, ${\omega }_2=30\mathrm{rad}/\mathrm{s}$, and $TB=66$) are also reported with circles ($G^{\prime }$, red filled circle; $G^{\prime \prime }$, red circle). The nominal strain for all signals (sine steps and chirps) is ${\gamma }_0=6\%$. (b) Trends of the error for the magnitude of the complex modulus as a function of varying time-bandwidth constant $TB$ for an optimally windowed chirp with $r=10\%$. Results for both the PIB solution (dot filled circle) and the micellar solution (dash filled star) are shown.