Determination of the complex-valued elastic moduli of polymers by electrical impedance spectroscopy for ultrasound applications

TL;DR

The paper presents ultrasound electrical impedance spectroscopy (UEIS) as a two-step inverse-method to determine frequency-independent complex-valued elastic moduli of polymers by fitting the entire impedance spectrum $Z(f)$ from $500\ \mathrm{Hz}$ to $5\ \mathrm{MHz}$ for a polymer ring mounted on a disk-shaped PZT. A axisymmetric finite-element model and a gradient-free optimization framework are used to extract both real and imaginary parts of the moduli, along with transducer and adhesive layer parameters, from measured impedance data. Validation is provided via ultrasonic-through-transmission (UTT) measurements on PMMA, showing good agreement for real parts (and derived speeds $c_{lo}$, $c_{tr}$) and attenuation, with larger uncertainties on the imaginary parts. The approach is low-cost, broadly applicable to polymers and related materials, and capable of capturing attenuation through off-resonance information, offering a practical tool for ultrasound device design and material characterization across temperatures and geometries.

Abstract

A method is presented for the determination of complex-valued compression and shear elastic moduli of polymers for ultrasound applications. The resulting values, which are scarcely reported in the literature, are found with uncertainties typically around 1 % (real part) and 6 % (imaginary part). The method involves a setup consisting of a cm-radius, mm-thick polymer ring glued concentrically to a disk-shaped piezoelectric transducer. The ultrasound electrical impedance spectrum of the transducer is computed numerically and fitted to measured values as an inverse problem in a wide frequency range, typically from 500 Hz to 5 MHz, both on and off resonance. The method was validated experimentally by ultrasonic through-transmission around 1.9 MHz. Experimentally, the method is arguably simple and low cost, and it is not limited to specific geometries and crystal symmetries. Moreover, by involving off-resonance frequencies, it allows for determining the imaginary parts of the elastic moduli, equivalent to attenuation coefficients. Finally, the method has no obvious frequency limitations before severe attenuation sets in above 100 MHz.

Figures (5)

• Figure 1: (a) A 3D sketch of the system consisting of a polymer ring (light gray) glued (green) to a transducer disk (dark gray) whit a quarter cut away for visibility. (b) The inset shows the 2D axisymmetric domain in the $r$-$z$ plane used for numerical simulations. The structured mesh is the one used at 5 MHz.
• Figure 2: The relative cost function sensitivity $\mathcal{S}(p_i)$ for the 17 piezoelectric material parameters $p_i$ obtained as an average from four Pz27 disks in the frequency interval 500 Hz$-$5 MHz is shown in the left side of the figure. $\mathcal{S}(p_i)$ for the four PMMA parameters, calculated from the average of four Pz27--PMMA--systems, are shown on the right. Corresponding real and imaginary parts are visualized in the same color, and the regions of high ($\mathcal{S}(p_i)>1$), medium ($0.1<\mathcal{S}(p_i)<1$) and low ($\mathcal{S}(p_i)<0.1$) sensitivity are highlighted by gray shadows. The sensitivity of the parameter $\varepsilon"_{11}$ is close to zero, as indicated by a black arrow.
• Figure 3: A flow chart of the steps in the fitting procedure to obtain complex-valued elastic moduli for a polymer sample. First, the electrical impedance spectrum $Z_\mathrm{exp}(f)$ of a Pz27 transducer disk is measured in the range $500~\textrm{Hz} < f_i < 5~\textrm{MHz}$. Then, the Pz27 parameters $p_i^\mathrm{pz27}$ are fitted in the same frequency range with increments of $\Delta f = 10$ kHz based on their sensitivities in descending order, with initial values from Ref. Kiyono2016. Next step is the measurement of $Z_\mathrm{exp}(f)$ for the PMMA ring glued to the Pz27 disk in the range $500~\textrm{Hz} < f_i < 5~\textrm{MHz}$. Then, coarse fitting of the $p_i^\mathrm{pmma}$ is performed in the range $500~\textrm{Hz} < f_i < 1~\textrm{MHz}$ with $\Delta f = 2$ kHz stepping, taking the initial values to be the average values of Refs. Hartmann1972Christman1972Sutherland1972Sutherland1978Carlson2003Simon2019Tran2016. Lastly, a final fitting of $p_i^\mathrm{pmma}$ is done in the the range $3.5~\textrm{MHz} < f_i < 5~\textrm{MHz}$ combined with previous range using steps of $\Delta f = 10$ kHz. A MATLAB-COMSOL sample script for the PMMA fitting procedure is presented in Sec. S2 of the Supplemental Material Note1.
• Figure 4: (a) Semilog plot of the measured (black) and simulated UEIS $|Z(f)|$ of an unloaded Pz27-0.5-10 disk. In the simulations are used the UEIS-fitted (orange) and initial literature (blue) Pz27 parameters listed in Tables \ref{['tab:Pz27_fit_real']} and \ref{['tab:Pz27_fit_imag']}. The gray region indicates the frequency range used in the fitting. (b) The logarithmic difference $\Delta_\mathrm{sim}^\mathrm{exp} = \log_{10}(|Z_\mathrm{exp}|/|Z_\mathrm{sim}|)$ between measured and simulated impedance spectrum. (c)-(f) Zoom-in on different regions showing the measured and simulated spectrum on a linear and re-normalized scale. Each region is indicated by a frame in both (a) and (b).
• Figure 5: (a) Semilog plot of the measured (black) and simulated UEIS $|Z(f)|$ of a PMMA-1.4-25 ring glued to a Pz27-0.5-10 disk by a 21-$\textrm{\textmu{}m}$-thick layer of NOA 86H glue. The UEIS-fitted simulation (orange) is computed using the UEIS parameter values listed in Tables \ref{['tab:Pz27_fit_real']}, \ref{['tab:Pz27_fit_imag']}, \ref{['tab:glue_fit']}, and \ref{['tab:PMMA_fit']}. The initial-value simulation is shown in blue. The gray regions indicate the frequency ranges used in the fitting. (b) The logarithmic difference $\Delta_\mathrm{sim}^\mathrm{exp} = \log_{10}(|Z_\mathrm{exp}|/|Z_\mathrm{sim}|)$ between measured and simulated impedance spectrum. (c)-(f) Zoom-in on different regions showing the measured and simulated spectrum on a linear and re-normalized scale. Each region is indicated by a frame in both (a) and (b).