Nanoscale Broadband Viscoelastic Spectroscopy of Soft Materials Using Iterative Control

Abstract

A novel approach to nanoscale broadband viscoelastic spectroscopy is presented. The proposed approach utilizes the recently developed modeling-free inversion-based iterative control (MIIC) technique to achieve accurate measurement of the material response to the applied excitation force over a broad frequency band. Scanning probe microscope (SPM) and nanoindenter have become enabling tools to quantitatively measure the mechanical properties of a wide variety of materials at nanoscale. Current nanomechanical measurement, however, is limited by the slow measurement speed: the nanomechanical measurement is slow and narrow-banded and thus not capable of measuring rate-dependent phenomena of materials. As a result, large measurement (temporal) errors are generated when material is undergoing dynamic evolution during the measurement. The low-speed operation of SPM is due to the inability of current approaches to (1) rapidly excite the broadband nanomechanical behavior of materials, and (2) compensate for the convolution of the hardware adverse effects with the material response during high-speed measurements. These adverse effects include the hysteresis of the piezo actuator (used to position the probe relative to the sample); the vibrational dynamics of the piezo actuator and the cantilever along with the related mechanical mounting; and the dynamics uncertainties caused by the probe variation and the operation condition. In the proposed approach, an input force signal with frequency characteristics of band-limited white-noise is utilized to rapidly excite the nanomechanical response of materials over a broad frequency range. The MIIC technique is used to compensate for the hardware adverse effects, thereby allowing the precise application of such an excitation force and measurement of the material response (to the applied force). The proposed approach is illustrated by implementing it to measure the frequency-dependent plane-strain modulus of poly(dimethylsiloxane) (PDMS) over a broad frequency range extending over 3 orders of magnitude (~1 Hz to 4.5 kHz).

Appendix

We present the convergence analysis result of the MIIC algorithm in the presence of random noise/disturbance as follows.

Theorem 1

Kim and Zou [1]

Let G(jω) be a stable single-input-single-output (SISO), linear time invariant (LTI) system, and at each frequency ω , consider the system output z(t) to be affected by the disturbance and/or the measurement noise z _n (t) as (see Fig. 1 in [25])

$$ z(j\omega) = z_{l}(j\omega) + z_{n}(j\omega), $$

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where z _l (jω) denotes the linear part of the system response to the input u(jω), i.e. z _l (jω) = G(jω) u(jω), and z _n (jω) denotes the output component caused by the disturbances and/or the measurement noise. Then,

• 1. the ratio of the iterative input u _k (jω) to the desired input u _d (jω) is bounded in magnitude and phase, respectively, as

  $$ 1-\epsilon(\omega) \leq \lim\limits_{k \to \infty} \left| \frac{u_k(j\omega)}{u_d(j\omega)} \right| \leq \frac{1-\epsilon(\omega)}{1-2\epsilon(\omega)}, $$

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  $$ \lim\limits_{k \to \infty} \left| \angle \left( \frac{u_k(j\omega)}{u_d(j\omega)} \right) \right| \leq \sin^{-1}\left( \frac{\epsilon(\omega)}{1-\epsilon(\omega)}\right), $$

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  provided that the noise to signal ratio (NSR) as defined below, is upper-bounded by a less-than-half constant, ϵ(ω),

  $$ \left|\frac{z_{k,n}(j\omega)}{z_d(j\omega)}\right| \leq \epsilon(\omega) < 1/2, \forall k, $$

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  where the desired input u _d (jω) enables the linear part of the system output to exactly track the desired output, i.e., z _d (jω) = G(jω) u _d (jω), and z _k,n (jω) denotes the part of the output caused by disturbances and/or measurement noise in the kth iteration. Moreover, the relative tracking error is bounded as

  $$ \lim\limits_{k \to \infty} \left|\frac{z_{k}(j\omega)-z_{d}(j\omega)}{z_d(j\omega)} \right| \le \frac{2 \epsilon(\omega) (1 - \epsilon(\omega))}{1-2\epsilon(\omega)}; $$

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• 2. The use of the MIIC algorithm will improve the output tracking at frequency ω , i.e.,

  $$ \lim\limits_{k \to \infty} \left|\frac{z_{k}(j\omega)-z_{d}(j\omega)}{z_d(j\omega)} \right| < 1, $$

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  provided that the upper bound of the NSR is less than $1- \frac{\sqrt{2}}{2}\approx 0.3$ , i.e.,

  $$ \left|\frac{z_{k,n}(j\omega)}{z_d(j\omega)} \right| \leq \epsilon(\omega) <1- \frac{\sqrt{2}}{2}, \forall k. \\ $$

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