††thanks: Equal share first-author††thanks: Equal share first-author

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

William N. Bodé<EMAIL_ADDRESS>Department of Physics, Technical University of
Denmark, DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark  Fabian
Lickert<EMAIL_ADDRESS>Department of Physics, Technical University of
Denmark, DTU Physics Building 309, DK-2800 Kongens Lyngby, Denmark  Per
Augustsson<EMAIL_ADDRESS>Department of Biomedical Engineering,
Lund University, Ole Römers väg 3, 22363 Lund, Sweden  Henrik Bruus
<EMAIL_ADDRESS>Department of Physics, Technical University of Denmark, DTU
Physics Building 309, DK-2800 Kongens Lyngby, Denmark

(30 September 2022)

###### 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.
The method is low cost, not limited to specific geometries and crystal
symmetries, and, given the developed software, easy to execute. The method has
no obvious frequency limitations before severe attenuation sets in above 100
MHz.

##  I Introduction

Numerical simulations play an important role when optimizing and predicting
piezoelectric device performance in applications including ultrasonic cleaning
[1], energy harvesting [2], inkjet printing [3], and acoustofluidics [4, 5].
To perform precise, accurate, and predictive simulations, well-characterized
material parameters such as the complex-valued elastic moduli are required.
Whereas material databases exist [6], and manufactures may provide some of the
required parameters, it is often not sufficient when attempting to perform
reliable simulations and predictions. Polymers are in this regard a
particularly challenging class of materials, since the elastic moduli of a
given polymer may depend on unspecified parameters such as the distribution of
polymer chain lengths and fabrication processes.

There exist a range of techniques to characterize an unknown material or
substance mechanically. Dynamic techniques such as resonant ultrasound
spectroscopy [7], transmission techniques [8, 9], impulse excitation [10],
laser vibrometry and triangulation [11, 12], as well as static techniques,
such as four-point bending, are widely used in various industries [13]. Those
methods however often rely on a few mechanical eigenmodes or resonance
frequencies of the material under study, a broad frequency spectrum due to a
narrow pulse in the time domain, or even static or low-frequency measurements.
Applications requiring actuation frequencies in the MHz-range however require
material properties that were measured in similar frequency intervals for an
accurate description of the system.

In this work we aim to extend the field of ultrasound spectroscopy [12, 14] by
utilizing an electrical impedance spectrum spanning a frequency range of
several MHz to obtain a full set of complex-valued elastic moduli of polymers.
With this technique, labeled ultrasound electrical impedance spectroscopy
(UEIS), a piezoelectric disk, driving vibrations in an attached polymer ring,
is used to characterize the complex-valued elastic compressional and shear
moduli of the polymer ring. Similar techniques have been used in the past to
fit piezoelectric material parameters by an inverse problem and numerical
optimization procedures on a free oscillating piezoelectric transducer [8, 15,
14, 16, 12, 17, 18]. Here, the same principles are used to fit elastic
material parameters. From the UEIS spectrum of a mass-loaded transducer, an
inverse problem is constructed to deduce the elastic moduli of the mass load.
The method is similar to those of Refs. [19, 12, 20], but by including an
automated whole-spectrum fit and complex parameter values, it extends the
previous method as suggested in the conclusion of Ref. [20]. Instead of a
thin-film transducer and manual fitting of few selected resonance peaks in the
impedance spectrum, the UEIS method makes use of several hundred impedance
values measured on a mechanically-loaded bulk transducer in the frequency
range from 500 Hz to 5 MHz, to extract both real and imaginary parts of the
complex-valued elastic moduli, and not just the real parts obtained in Ref.
[20]. The UEIS technique enables low-cost and in-situ measurements of elastic
moduli over a wide frequency range from low kHz to several MHz. It is easy to
execute, requiring only a disk-shaped piezoelectric transducer, a ring of the
unknown polymer sample, an impedance analyzer, and the developed fitting
software.

The paper is organized as follows. In Section II a brief overview of the
relevant theory is given, before in Section III the experimental and numerical
methodology of the UEIS technique is described in detail for polymer, glue,
and transducer. In Section IV we provide validation data based on ultrasonic-
through-transmission (UTT) measurements, before we in Section V present the
main results of the UEIS method in terms of the complex-valued
electromechanical parameters of the unloaded piezoelectric transducer and the
complex-valued elastic moduli of the UV-cured glue and the polymer ring. We
conclude in Section VI.

##  II Theoretical Background

We follow Ref. [21] and describe isotropic polymers using the standard linear
theory of elastic solids in the Voigt notation, in terms of the displacement
vector $\bm{u}$ of a given material point away from its equilibrium position,
and the $1\times 6$ strain $\bm{s}$ and stress $\bm{\sigma}$ column vectors
with the $6\times 1$ transposed row vectors
$\bm{s}^{\textsf{T}}=(\partial_{x}u_{x},\partial_{y}u_{y},\partial_{z}u_{z},\partial_{y}u_{z}+\partial_{z}u_{y},\partial_{x}u_{z}+\partial_{z}u_{x},\partial_{x}u_{y}+\partial_{y}u_{x})$
and
$\bm{\sigma}^{\textsf{T}}=(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{yz},\sigma_{xz},\sigma_{xy})$,
respectively. Representing the elastic moduli $C_{ik}$ by the tensor $\bm{C}$,
the constitutive equation for an elastic solid in the $\infty mm$-symmetry
class is [22],

$\displaystyle\bm{\sigma}$ $\displaystyle=\bm{C}\cdot\bm{s},$ (1a)
$\displaystyle\bm{C}$
$\displaystyle=\left(\begin{array}[]{cccccc}C_{11}&C_{12}&C_{13}&0&0&0\\\
C_{12}&C_{11}&C_{13}&0&0&0\\\ C_{13}&C_{13}&C_{33}&0&0&0\\\
0&0&0&C_{44}&0&0\\\ 0&0&0&0&C_{44}&0\\\ 0&0&0&0&0&C_{66}\\\
\end{array}\right).$ (1h)

For an isotropic polymer $C_{33}=C_{11}$, $C_{66}=C_{44}$, and
$C_{13}=C_{12}=C_{11}-2C_{44}$, so here, $\bm{C}$ is given only by the two
complex-valued elastic moduli $C_{11}$ and $C_{44}$, each with a real and
imaginary part, $C_{ik}=C^{\prime}_{ik}+\mathrm{i}C^{\prime\prime}_{ik}$,
relating to the propagation and attenuation of sound waves, respectively. Many
amorphous polymers, such as the injection-molded PMMA in this work, are
isotropic, but if not, such as semi-crystalline polymers [23], a $\bm{C}$
tensor with the appropriate lower symmetry must be used. Since only positive
power dissipation is allowed, the elastic moduli are restricted by the
constraint that the matrix $\operatorname{Im}(-\bm{C})$ must be positive
definite [24].

We also model the piezoelectric lead-zirconate-titanate (PZT) transducer in
the $\infty mm$-symmetry class [22], again following the notation of Ref.
[21]. Here, $\bm{u}$, $\bm{S}$, $\bm{\sigma}$, and $\bm{C}$ are supplemented
by the electric potential $\varphi$, the electric field
$\bm{E}=-\bm{\nabla}\varphi$, the dielectric tensor $\bm{\varepsilon}$, the
electric displacement field $\bm{D}=\bm{\varepsilon}\cdot\bm{E}$, and the
piezoelectric $3\times 6$ coupling tensor $\bm{e}$. The constitutive equation
becomes,

$\displaystyle\left(\begin{array}[]{c}\bm{\sigma}\\\
\bm{D}\end{array}\right)=\left(\begin{array}[]{cc}\bm{C}&-\bm{e}^{\textsf{T}}\\\
\bm{e}&\bm{\varepsilon}\end{array}\right)\,\left(\begin{array}[]{c}\bm{S}\\\
\bm{E}\end{array}\right)=\bm{M}\cdot\left(\begin{array}[]{c}\bm{S}\\\
\bm{E}\end{array}\right)$ (2i)
$\displaystyle\bm{e}=\left(\begin{array}[]{cccccc}0&0&0&0&e_{15}&0\\\
0&0&0&e_{15}&0&0\\\
e_{31}&e_{31}&e_{33}&0&0&0\end{array}\right)\\!,\;\bm{\varepsilon}=\left(\begin{array}[]{ccc}\varepsilon_{11}&0&0\\\
0&\varepsilon_{11}&0\\\ 0&0&\varepsilon_{33}\end{array}\right)\\!.$ (2p)

In the $\infty mm$-symmetry class, $C_{66}=\frac{1}{2}(C_{11}-C_{12})$, so the
coupling tensor $\bm{M}$ is given by the five complex-valued elastic moduli
$C_{11}$, $C_{12}$, $C_{13}$, $C_{33}$, and $C_{44}$, with
$C_{ik}=C^{\prime}_{ik}+\mathrm{i}C^{\prime\prime}_{ik}$, the two complex-
valued dielectric constants $\varepsilon_{11}$ and $\varepsilon_{33}$ with
$\varepsilon_{ik}=\varepsilon^{\prime}_{ik}+\mathrm{i}\varepsilon^{\prime\prime}_{ik}$,
and the three real-valued piezo-coupling constants $e_{31}$, $e_{33}$, and
$e_{15}$ with $e_{ik}=e^{\prime}_{ik}+0\mathrm{i}$. Since only positive power
dissipation is allowed, the coupling constants are restricted by the following
constraint on the matrix $\bm{K}$ [24],

$\bm{K}=\operatorname{Im}\left(\begin{array}[]{cc}-\bm{C}&\bm{e}^{\textsf{T}}\\\
\bm{e}&\bm{\varepsilon}\end{array}\right)\;\text{ must be positive definite}.$
(3)

We limit our analysis of the linear system to the time-harmonic response for a
given angular frequency $\omega=2\pi f$, where $f$ is the excitation frequency
of the system. Thus, any physical field $F_{\mathrm{phys}}(\bm{r},t)$ is given
by a complex-valued amplitude $F(\bm{r})$ as
$F_{\mathrm{phys}}(\bm{r},t)=\operatorname{Re}\big{[}F(\bm{r})\>\mathrm{e}^{-{\mathrm{i}\omega
t}}\big{]}$, and we need only to compute $F(\bm{r})$. In our model of a
polymer sample mounted on a PZT transducer having a bottom and top electrode,
the system is excited by the excitation voltage
$\varphi_{\mathrm{top}}^{\mathrm{phys}}=\operatorname{Re}\big{[}\varphi_{\mathrm{top}}\>\mathrm{e}^{-{\mathrm{i}\omega
t}}\big{]}$ as follows,

$\varphi_{\mathrm{bot}}=0~{}\textrm{V}\quad\text{ and
}\quad\varphi_{\mathrm{top}}=1~{}\textrm{V}.$ (4)

By introducing the density $\rho$ as an additional material parameter, the
governing equations for the time-harmonic displacement field $\bm{u}$ in the
polymer and in the PZT and for the quasi-electrostatic potential $\varphi$ in
the non-magnetic PZT without free charges, become

$\displaystyle\bm{\nabla}\cdot\bm{\sigma}=-\rho\omega^{2}\bm{u}\quad\text{ and
}\quad\bm{\nabla}\cdot\bm{D}=0.$ (5)

We neglect the effect of gravity in this formulation, as it only leads to a
minor deformation of the geometry. The stress- and charge-free boundary
conditions are imposed on free surfaces

$\bm{\sigma}\cdot\bm{n}=\bm{0}\quad\text{ and }\quad\bm{n}\cdot\bm{D}=0.$ (6)

The current density $\bm{J}$ in the PZT transducer is given by the
polarization $\bm{P}$ as

$\displaystyle\bm{J}=\partial_{t}\bm{P}=-\mathrm{i}\omega(\bm{D}-\varepsilon_{0}\bm{E}).$
(7)

Consequently, the electrical impedance $Z$ central to the UEIS method can be
computed via the flux integral of $\bm{J}$ through the surface
$\partial\Omega_{\mathrm{top}}$ with surface normal $\bm{n}$ as,

$Z=\frac{\varphi_{\mathrm{top}}-\varphi_{\mathrm{bot}}}{I},\quad\text{ with
}\quad I=\int_{\partial\Omega_{\mathrm{top}}}\bm{n}\cdot\bm{J}\>\mathrm{d}A.$
(8)

##  III Methodology

The ultimate goal is to develop and test a method for determination of the
complex-valued elastic moduli of polymers. However, to achieve an accuracy
level of about 1-5%, we need also to determine the mechanical and
electromechanical parameters of the piezoelectric transducer as well as the
elastic moduli of the glue used to mount the polymer sample on the transducer.

Table 1:  Measured thickness (TH) and outer diameter (OD) of the Pz27 disks (named Pz27-TH-OD). Measured TH, OD, and inner diameter (ID) of the rings made by the glue NOA 86H (named NOA86H-TH-OD) and of the polymer PMMA (named PMMA-TH-OD). Letters A, B, C are used as labels for samples with the same nominal dimensions. The precision is given as the standard deviation from six measurements.  Sample | TH | OD | ID
---|---|---|---
| (mm) | (mm) | (mm)
Pz27-0.5-6.35-A | $0.510(1)$ | $6.594(5)$ | –
Pz27-0.5-6.35-B | $0.502(1)$ | $6.587(2)$ | –
Pz27-0.5-10-A | $0.500(1)$ | $10.037(7)$ | –
Pz27-0.5-10-B | $0.492(2)$ | $10.039(3)$ | –
Pz27-0.5-10-C | $0.505(2)$ | $10.039(5)$ | –
NOA86H-1.4-20 | $1.5(2)$ | $19.22(6)$ | $1.90(2)$
PMMA-1.4-20-A | $1.428(2)$ | $19.981(5)$ | $1.98(1)$
PMMA-1.4-20-B | $1.440(4)$ | $20.017(8)$ | $1.94(1)$
PMMA-1.4-25-A | $1.427(5)$ | $24.98(1)$ | $1.92(1)$
PMMA-1.4-25-B | $1.437(5)$ | $24.94(1)$ | $1.94(2)$

###  III.1 Experimental procedure

Measurements of the electrical impedance spectrum $Z_{\mathrm{exp}}(f)$ were
performed using a Vector Network Analyzer Bode 100 (OMICRON electronics GmbH,
Klaus, Austria) in a linear interval with 10.000 steps between 500 Hz and 5
MHz, each frequency sweep taking less than 4 minutes. In a first step, the
electrical impedance of an unloaded Pz27 disk (Meggitt A/S, Kvistgaard,
Denmark) was characterized. The top and bottom electrode of the piezoelectric
disk were contacted through spring-loaded pins in a custom-made sample holder,
minimizing the contact force and area to a point in the center of the disk. In
the following step, using a thin layer of the UV-curable glue NOA 86H (Norland
Products, Jamesburg (NJ), USA), a polymer ring of known dimensions was glued
on top of a Pz27 disk. This ring was made from either the NOA 86H glue itself
or from polymethyl methacrylate (PMMA Diakon TD525, Lucite International,
Rotterdam, Netherlands). We aimed at making the glue layer as thin
($15$-$25~{}\textrm{\textmu{}m}$) and uniform as possible to reduce its
influence on the combined system, as studied by Bodé et al. [25]. This was
achieved by gently squeezing the system for a few seconds after assembly, just
before curing the glue by UV illumination. The glue was cured at a UV-
intensity of $15~{}\textrm{mW}/\textrm{cm}^{2}$ at 365 nm for 167 s to achieve
strong bonding between the transducer disk and the polymer ring. The glue NOA
86H was selected after performing experiments with several different
adhesives, as it enables good adhesion between the PZT and the polymer and
allows for good acoustic coupling due to acrylic-like properties in the cured
state with an attenuation comparable to that found for the polymer ring. After
curing, the electrical impedance of the polymer-loaded transducer was
measured. The small hole of the polymer ring allows contacting the transducer
disk using the above-mentioned spring-loaded pins. The average of three
impedance measurements, taking less than 12 minutes to obtain, was used both
for the unloaded and loaded case.

The diameter and thickness of the polymer ring and the Pz27 disk were measured
before assembling the system using an electronic micrometer with an accuracy
of $\pm 4~{}\textrm{\textmu{}m}$. The glue-layer thickness was obtained as the
measured total thickness of the assembled system minus the sum of the
individual thicknesses of the Pz27 disk and the polymer ring. The impedance
measurements were performed at $24~{}^{\circ}\\!\textrm{C}$ using a
combination of two different nominal transducer dimensions (diameter 6.35 mm
and 10 mm, thickness 0.5 mm) and two different nominal polymer ring dimensions
(diameter 20 mm and 25 mm, thickness 1.4 mm), yielding four transducer-polymer
systems with the dimensions listed in Tables 1 and 2.

###  III.2 Numerical model

The weak formulation of the finite element method (FEM) is used to implement
the governing equations in the software COMSOL Multiphysics [26] to simulate
the electrical impedance spectrum $Z_{\mathrm{sim}}(f)$ unloaded or loaded PZT
transducer. In particular, we use the weak form PDE interface as described in
our previous work [21, 5, 25]. The simulations are computed on a workstation
with a 12-core, 3.5-GHz central processing unit and 128 GB random access
memory. Third-order Lagrange polynomials are used as test functions for both
$\bm{u}$ and $\varphi$. The model consists of three domains: a piezoelectric
disk, a glue layer, and a polymer ring. Given the cylindrical geometry of the
assembled stack and the axisymmetric structure of the coupling tensors
$\bm{C}$ and $\bm{M}$ in Eqs. (1h) and (2i), the system can be reduced to an
axisymmetric model as shown in Ref. [27] and illustrated in Fig. 1. This
axisymmetrization reduces the computational time substantially. A suitable
mesh element size is found by the mesh convergence study presented in Sec. S1
of the Supplemental Material 111See Supplemental Material at [URL] for details
on the mesh convergence analysis, a study on the sensitivity versus elastic
moduli and frequency range, sample MATLAB and COMSOL scripts for the UEIS
fitting procedure, the corresponding data for the impedance spectra, and the
validation of the UEIS method by UTT and laser-Doppler vibrometry, which
includes Refs. [16, 29]., where in addition in Sec. S2, a COMSOL sample script
is presented.

Table 2:  The measured glue-layer thickness of the five studied PMMA-ring-on-Pz27-disk configurations. Pz27 disk |  | Polymer ring |  | Glue layer
---|---|---|---|---
Pz27-0.5-6.35-A |  | PMMA-1.4-20-A |  | 15 µm
Pz27-0.5-6.35-B |  | PMMA-1.4-25-A |  | 24 µm
Pz27-0.5-10-A |  | PMMA-1.4-25-B |  | 21 µm
Pz27-0.5-10-B |  | PMMA-1.4-20-B |  | 12 µm
Pz27-0.5-10-C |  | NOA86H-1.4-20 |  | 15 µm

Using the “LiveLink for MATLAB”-interface provided by COMSOL, the MATLAB
optimization procedures fminsearchbnd and patternsearch are used to fit the
material parameters such that $Z_{\mathrm{sim}}(f)$ is as close to
$Z_{\mathrm{exp}}(f)$ as possible. The fminsearchbnd algorithm [30] allows a
bounded search in parameter space. The patternsearch algorithm (part of the
Global Optimization Toolbox) makes twice as many function evaluations, but it
covers a larger region in parameter space and is better to locate the global
minimum for poor initial values. Both algorithms use a gradient-free direct
search and are therefore well suited for non-smooth numerical optimization
procedures. The algorithms require three inputs: (i) initial values, (ii)
upper and lower bounds, and (iii) a cost function to minimize. Based on the
measured and simulated electrical impedance values $Z_{\mathrm{exp}}(f_{i})$
and $Z_{\mathrm{sim}}(f_{i})$ obtained at $\sim 500$ frequencies $f_{i}$, we
define the cost function $\mathcal{C}$ as

$\mathcal{C}=\sqrt{{\sum_{i}\big{[}\log_{10}\\!\big{(}|Z_{\mathrm{exp}}(f_{i})|\big{)}-\log_{10}\\!\big{(}|Z_{\mathrm{sim}}(f_{i})|\big{)}\big{]}^{2}}}.$
(9)

Here, we use the logarithm, because $Z(f_{i})$, having many peaks, varies by
orders of magnitude as a function of $f_{i}$.

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.

###  III.3 Sensitivity analysis

To enhance the performance of our fitting procedure, we group the parameters
into sets of similar sensitivity based on the following sensitivity analysis
of the cost function $\mathcal{C}$ on each of the sixteen material parameters
$(p_{1},p_{2},\ldots p_{16})^{\mathrm{pz27}}=(C_{11}^{\prime}$,
$C_{12}^{\prime}$, $C_{13}^{\prime}$, $C_{33}^{\prime}$, $C_{44}^{\prime}$,
$\varepsilon_{11}^{\prime}$, $\varepsilon_{33}^{\prime}$, $e_{31}^{\prime}$,
$e_{33}^{\prime}$, $e_{15}^{\prime}$, $C^{\prime\prime}_{11}$,
$C^{\prime\prime}_{12}$, $C^{\prime\prime}_{13}$, $C^{\prime\prime}_{33}$,
$C^{\prime\prime}_{44}$, $\varepsilon^{\prime\prime}_{33})$ for the Pz27 disk
and on the four polymer parameters $(p_{1},\ldots
p_{4})^{\mathrm{polym}}=(C_{11}^{\prime}$, $C_{44}^{\prime}$,
$C^{\prime\prime}_{11}$, $C^{\prime\prime}_{44})$. The sensitivity analysis is
performed in the frequency range from 500 Hz to 5 MHz, with the initial value
$p_{i}^{0}$ taken from literature for a given parameter $p_{i}$, and therefore
the individual sensitivity values represent averages over the entire frequency
range. A more detailed study of the frequency dependency of the sensitivity is
shown in Sec. S2 of the Supplemental Material [28]. We use a discrete
approximation of the relative sensitivity
$\mathcal{S}(p_{i})=p_{i}\partial_{p_{i}}\mathcal{C}(p_{i})$ of
$\mathcal{C}(p_{i})$ based on a $\pm 10$% variation of $p_{i}$ around
$p_{i}^{0}$, while keeping the remaining parameters fixed at $p_{j}^{0}$,

$\mathcal{S}(p_{i})=\frac{\mathcal{C}(1.1p_{i})-\mathcal{C}(0.9p_{i})}{1.1-0.9},\;\;p_{j}=p_{j}^{0}\text{
for }\;j\neq i.$ (10)

The obtained sensitivities $\mathcal{S}$ for the $p_{i}^{\mathrm{pz27}}$ and
$p_{i}^{\mathrm{pmma}}$ parameters for Pz27 and PMMA, respectively, are shown
in Fig. 2. The Pz27 parameters are classified in three groups of high
$\mathcal{S}(p_{i})>1$, medium $0.1<\mathcal{S}(p_{i})<1$, and low
$\mathcal{S}(p_{i})<0.1$ sensitivity, respectively, and as described in the
following section, a robust fitting is obtained by fitting the parameters
group by group sequentially in descending order from high to low sensitivity.
Since all four PMMA parameters have a medium-to-high sensitivity we fit them
simultaneously in a single, undivided group.

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
$\varepsilon^{\prime\prime}_{11}$ is close to zero, as indicated by a black
arrow.

###  III.4 The UEIS fitting procedure

The first step in the UEIS fitting procedure is to measure and simulate the
electrical impedance $Z_{\mathrm{exp}}(f)$ and $Z_{\mathrm{sim}}(f)$,
respectively, of an unloaded Pz27 transducer disk and then following Refs.
[16, 17, 18] to fit the sixteen Pz27 parameters $p_{i}^{\mathrm{pz27}}$ in the
form of an inverse problem by minimizing the cost function $\mathcal{C}$. In
the second step, a characterized Pz27 disk is loaded by gluing on a given
polymer ring using the UV-curable glue NOA 86H. To characterize the glue, the
first studied polymer ring is made by the glue itself, and $Z(f)$ is used to
similarly fit the four glue parameters $p_{i}^{\mathrm{glue}}$. Subsequently,
using the characterized glue, a PMMA ring is glued to a characterized Pz27
disk, and $Z(f)$ is used to similarly fit the four PMMA parameters
$p_{i}^{\mathrm{pmma}}$. See the flow chart in Fig. 3.

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. [18]. 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. [31, 32, 33, 34,
35, 36, 37]. 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
[28].

For the unloaded Pz27 disk, the initial values of $p_{i}^{\mathrm{pz27}}$ are
taken from Ref. [18], and the $k$th iteration in the fit is divided into four
sub-steps: (1) Fit the six parameters $C_{11}^{\prime}$, $C_{12}^{\prime}$,
$C_{13}^{\prime}$, $C_{33}^{\prime}$, $\varepsilon_{33}^{\prime}$, and
$e_{33}^{\prime}$ of highest sensitivity $\mathcal{S}>1$ using the
fminsearchbnd algorithm in the range
$500~{}\textrm{Hz}<f_{i}<5~{}\textrm{MHz}$ in increments of 10 kHz with the
bounds set to $\pm 30~{}\%$, while keeping the remaining eleven parameters
fixed. (2) Check whether the cost function $\mathcal{C}_{k}$ of iteration $k$
deviates less than 1 % relative to $\mathcal{C}_{k-1}$ (the fit is converged
and $p_{i}^{\mathrm{pz27}}$ have been determined) or not (the fitting
continues). (3) Similarly, fit the five parameters $C_{44}^{\prime}$,
$e_{31}^{\prime}$, $e_{15}^{\prime}$, $C^{\prime\prime}_{11}$, and
$C^{\prime\prime}_{33}$ of medium sensitivity $0.1<\mathcal{S}<1$. (4)
Likewise, fit the last six parameters $\varepsilon_{11}^{\prime}$,
$C^{\prime\prime}_{12}$, $C^{\prime\prime}_{13}$, $C^{\prime\prime}_{44}$,
$\varepsilon^{\prime\prime}_{11}$, and $\varepsilon^{\prime\prime}_{33}$ of
low sensitivity $\mathcal{S}<1$ and move on to iteration $k+1$. If during the
fit a value of $p_{i}^{\mathrm{pz27}}$ is within 5 % of the pre-defined bound,
the latter is changed by 50 %. Furthermore, for each evaluation of the cost
function $\mathcal{C}$, it is checked if $\bm{K}$ in Eq. (3) is positive
definite, and if not we set $\mathcal{C}=\infty$.

For the glue ring, the initial values of the four parameters
$p_{i}^{\mathrm{glue}}$ are
$C_{11}^{\mathrm{glue}}=(4.7-0.47\mathrm{i})~{}\textrm{GPa}$ and
$C_{44}^{\mathrm{glue}}=(0.9-0.09\mathrm{i})~{}\textrm{GPa}$ inferred from
Young’s modulus of Ref. [38], the assumed value 0.38 of Poisson’s ratio, and
$C_{ik}^{\prime\prime}=0.1C_{ik}^{\prime}$. Moreover, the density of the glue
ring is measured. The fitting is divided into two sub-steps to increase
robustness and speed: (1) A coarse fit of the four parameters
$C_{11}^{\prime}$, $C_{44}^{\prime}$, $C_{11}^{\prime\prime}$, and
$C_{44}^{\prime\prime}$ using the patternsearch algorithm in the limited range
$500~{}\textrm{Hz}<f_{i}<1~{}\textrm{MHz}$ in increments of 2 kHz with the
bounds set to be $\pm 40~{}\%$ covering the typically observed range for
polymers [35, 9]. (2) A final fit of $C_{11}^{\prime}$, $C_{44}^{\prime}$,
$C_{11}^{\prime\prime}$, and $C_{44}^{\prime\prime}$ using the fminsearchbnd
algorithm in the combined ranges of $500~{}\textrm{Hz}<f_{i}<1~{}\textrm{MHz}$
and $3.5~{}\textrm{MHz}<f_{i}<5~{}\textrm{MHz}$ in increments of 2 kHz and 10
kHz, respectively, with the bounds set to $\pm 40~{}\%$, and with the coarse-
fit values used as initial values. If during the fit a value of
$p_{i}^{\mathrm{glue}}$ is within 5 % of the pre-defined bound, the bound is
changed by 5 %, see the Supplemental Material [28]. Furthermore, for each
evaluation of the cost function $\mathcal{C}$, it is checked if
$\operatorname{Im}(-\bm{C})$ is positive definite, and if not we set
$\mathcal{C}=\infty$.

For the PMMA ring, the initial values of $p_{i}^{\mathrm{pmma}}$ are taken to
be the average of the values reported in Refs. [31, 32, 33, 34, 35, 36, 37].
This average is used due to the lack of parameter values provided by the
manufacturer of our selected PMMA polymer. Since this PMMA consists of a
toughened acrylic compound, we expect that it deviates from standard PMMA
grades. Therefore we chose to use the average literature values only as
initial values in our fitting routine, and we refrain from comparing them with
the resulting UEIS values. Otherwise, the fitting procedure for the PMMA ring
is the same as the one for the glue ring.

Note that for the selected materials in the studied frequency range from 500
Hz to 5 MHz, and measured with relative accuracies from 1% to 5%, the
experimental results in Sections IV and V show that it is adequate to assume
frequency-independent parameters $p_{i}^{\mathrm{pz27}}$,
$p_{i}^{\mathrm{glue}}$ and $p_{i}^{\mathrm{pmma}}$. See further discussion in
Section V.3.

##  IV Ultrasonic-through-transmission (UTT) validation data

For the polymer PMMA, we have carried out ultrasonic-through-transmission
(UTT) measurements [8, 37] to acquire data for experimental validation of the
UEIS method. In UTT, a pulse, with center frequency $f_{c}$ and width $\Delta
f$ in the frequency domain and width $\Delta t$ in the time domain, is
transmitted through a polymer slab of thickness $d$ with its surface normal
$\bm{n}$ tilted an angle $\theta_{i}$ relative to the incident pulse and
emerged in water having the sound speed $c_{0}$. We have used
$f_{c}=1.90~{}\textrm{MHz}$, $\Delta f\approx 1~{}\textrm{MHz}$, and $\Delta
t\approx 2~{}\textrm{\textmu{}s}$. The UTT-method relies on the fact that at
normal incidence only longitudinal waves are transmitted, whereas above a
critical tilt angle $\theta_{\mathrm{crit}}=\sin^{-1}(c_{0}/c_{\mathrm{lo}})$
only transverse waves are transmitted in samples with $c_{\mathrm{lo}}>c_{0}$.
The longitudinal and transverse speed of sound, $c_{\mathrm{lo}}$ and
$c_{\mathrm{tr}}$, and the corresponding attenuation coefficients,
$\alpha_{\mathrm{lo}}$ and $\alpha_{\mathrm{tr}}$, of the slab can be
determined based on the difference $\Delta\tau$ of arrival times, with and
without the slab placed in the water,

$\displaystyle c_{\mathrm{lo}}$
$\displaystyle=\Big{[}1+\frac{c_{0}}{d}\>\Delta\tau\Big{]}^{-1}\>c_{0},$ (11a)
$\displaystyle c_{\mathrm{tr}}$
$\displaystyle=\Big{[}\sin^{2}\theta_{i}+\big{(}\frac{c_{0}}{d}\>\Delta\tau+\cos\theta_{i}\big{)}^{2}\Big{]}^{-\frac{1}{2}}\>c_{0},$
(11b) $\displaystyle\alpha_{\mathrm{lo}}$
$\displaystyle=\alpha_{\mathrm{0}}+\frac{1}{d}\ln{\\!\bigg{[}\frac{T_{\mathrm{lo}}A_{\mathrm{0}}}{A_{\mathrm{lo}}}\bigg{]}},$
(11c) $\displaystyle\alpha_{\mathrm{tr}}$
$\displaystyle=\alpha_{\mathrm{0}}\cos{(\theta_{s}-\theta_{i})}+\frac{1}{d}\ln{\\!\bigg{[}\frac{T_{\mathrm{tr}}A_{\mathrm{0}}}{A_{\mathrm{lo}}}\bigg{]}}\>\cos{\theta_{s}}.$
(11d)

Here, $T_{\mathrm{lo}}$ and $T_{\mathrm{tr}}$ are the longitudinal and
transverse transmission coefficients, $\theta_{s}$ is the refractive angle of
the shear wave, $A_{\mathrm{0}}$ is the amplitude of the direct signal, and
$A_{\mathrm{lo}}$ and $A_{\mathrm{tr}}$ are the longitudinal and transverse
amplitudes of the transmitted signal after passing through the sample. Using
the parameter values of water listed in Ref. [39], the attenuation coefficient
$\alpha_{0}$ of water is,

$\displaystyle\alpha_{0}(T,f)$
$\displaystyle=2\pi^{2}\bigg{[}\frac{4}{3}\eta_{0}(T)+\eta^{\mathrm{b}}(T)\bigg{]}\frac{f^{2}}{\rho_{0}c_{0}^{3}}$
$\displaystyle\approx\bigg{[}1-0.0249\frac{T-25~{}^{\circ}\\!\textrm{C}}{1~{}^{\circ}\\!\textrm{C}}\bigg{]}\>\frac{0.0217f^{2}}{1~{}\textrm{MHz}^{2}}\frac{\textrm{Np}}{\textrm{m}}.$
(12)

As we do not control the room temperature in our UEIS measurements, but simply
monitor it with a $1~{}^{\circ}\\!\textrm{C}$ uncertainty, we have used the
UTT experiments to determine the temperature dependence of the elastic moduli
of our PMMA sample. To this end, the UTT tank was filled with warm water at
temperature $T=31~{}^{\circ}\\!\textrm{C}$. Then over a period of 6 hours, as
the water steadily cooled to $T=23~{}^{\circ}\\!\textrm{C}$, the elastic
moduli were measured at regular intervals, corresponding to steps in
temperature of about $-0.5~{}^{\circ}\\!\textrm{C}$. As shown in Sec. S4 of
the Supplementary Material [28], the resulting longitudinal and transverse
speed of sound ($c_{\mathrm{lo}}$ and $c_{\mathrm{tr}}$) and attenuation
coefficients ($\alpha_{\mathrm{lo}}$ and $\alpha_{\mathrm{tr}}$) of PMMA at
the frequency $f_{c}=1.90~{}\textrm{MHz}$ are found to depend linearly on
temperature $T$ (in ${}^{\circ}\\!\textrm{C}$) as,

$\displaystyle c_{\mathrm{lo}}^{\mathrm{1.90\>MHz}}(T)$
$\displaystyle=\bigg{[}-11(2)\>\frac{T}{1~{}^{\circ}\\!\textrm{C}}+2743(46)\bigg{]}~{}\frac{\textrm{m}}{\textrm{s}},$
(13a) $\displaystyle c_{\mathrm{tr}}^{\mathrm{1.90\>MHz}}(T)$
$\displaystyle=\bigg{[}-4.5(4)\>\frac{T}{1~{}^{\circ}\\!\textrm{C}}+1267(9)\bigg{]}~{}\frac{\textrm{m}}{\textrm{s}},$
(13b) $\displaystyle\alpha_{\mathrm{lo}}^{\mathrm{1.90\>MHz}}(T)$
$\displaystyle=\bigg{[}1.7(4)\>\frac{T}{1~{}^{\circ}\\!\textrm{C}}+25(10)\bigg{]}~{}\frac{\textrm{Np}}{\textrm{m}},$
(13c) $\displaystyle\alpha_{\mathrm{tr}}^{\mathrm{1.90\>MHz}}(T)$
$\displaystyle=\bigg{[}7(2)\>\frac{T}{1~{}^{\circ}\\!\textrm{C}}+146(43)\bigg{]}~{}\frac{\textrm{Np}}{\textrm{m}}.$
(13d)

Here, the digits in the parentheses indicate $1\sigma$ uncertainties computed
from on the sum-of-square differences between measured data and regression-
line fits.

##  V Results of the UEIS method

###  V.1 UEIS-fitted material parameters for Pz27

First, we determine the sixteen material parameters $p_{i}^{\mathrm{pz27}}$
for the four unloaded Pz27 disks with nominal outer diameters 6.35 mm and 10.0
mm and the measured dimensions listed in Table 1. Using the UEIS method
described in Section III involving the measured and fitted impedance spectra
$Z_{\mathrm{exp}}(f)$ and $Z_{\mathrm{sim}}(f)$, we obtain the resulting
parameters listed in Tables 3 and 4. In Table 3 we compare the real part of
the obtained UEIS parameters to those provided in the literature (lit, [18])
and by the manufacturer (manf, [40]). The relative difference
$\Delta_{\mathrm{UEIS}}^{\mathrm{lit}}$ between UEIS and literature values is
in the range $\sim$2-8 %, whereas $\Delta_{\mathrm{UEIS}}^{\mathrm{manf}}$ is
higher, typically in the range $\sim$5-40 %. The deviations are overall
significant compared to the relative standard deviation
$\hat{\sigma}_{\mathrm{UEIS}}\lesssim 2$ % of the mean of the UEIS values.

Similarly, in Table 4 we compare the imaginary parts of the obtained UEIS
parameters to those provided in the literature (lit, [18]). Note that
$\varepsilon^{\prime\prime}_{11}$ is set to zero due to its low value and
sensitivity, and that $e^{\prime\prime}_{mj}=0$ by assumption. In general, the
imaginary parts are more difficult to measure than the real parts, which is
reflected in the high values of $\hat{\sigma}_{\mathrm{UEIS}}$ ($\lesssim$10
%), $\Delta_{\mathrm{UEIS}}^{\mathrm{lit}}$ ($\sim$5-50 %), but still with
significant deviation between UEIS values and the values provided in the
literature and by the manufacturer. The errors on the imaginary parts are
about one order of magnitude larger than the errors on the real parts. This is
in line with the previously found lower sensitivities of the former compared
to the higher sensitivities of the latter shown in Fig. 2. Relative deviations
of the initial values from the fitted values range from as little as 1.4 % for
$C^{\prime\prime}_{13}$ and up to 50 % for $C^{\prime\prime}_{12}$ and above
200 % for $\varepsilon_{33}^{\prime\prime}$. Despite those deviations from the
initial values, we find good convergence on the cost function and an excellent
agreement between the measured and fitted impedance spectrum for the Pz27
disk. The uniqueness of the sixteen material parameters
$p_{i}^{\mathrm{pz27}}$ is not guaranteed, but the simulated impedance
spectrum fits the measured one, and thus they provide an adequate estimate for
the subsequent determination of the polymer parameters
$p_{i}^{\mathrm{pmma}}$. In Fig. 4, an example is shown of the measured UEIS
spectrum and the resulting simulated UEIS spectrum for a Pz27 disk of diameter
10 mm and thickness 0.5 mm.

Table 3:  The real part (single prime) of the material parameters $p_{i}^{\mathrm{pz27}}$ of the four Pz27 disks of Table 1 obtained by the UEIS method with initial values from the literature [18]. The digits in brackets denote one standard deviation, $\hat{\sigma}_{\mathrm{UEIS}}$ is the relative standard deviation of the UEIS mean, and $\Delta_{\mathrm{UEIS}}^{\mathrm{lit}}$ and $\Delta_{\mathrm{UEIS}}^{\mathrm{manf}}$ is the relative deviation from UEIS of the values given by Ref. [18] (lit) and by the manufacturer (manf) [40], respectively. The density of Pz27 was measured to be $\rho_{\mathrm{pz27}}=7.7(1)$ g/cm3. Pz27 disk | $C^{\prime}_{11}$ | $C^{\prime}_{12}$ | $C^{\prime}_{13}$ | $C^{\prime}_{33}$ | $C^{\prime}_{44}$ | $\varepsilon^{\prime}_{11}$ | $\varepsilon^{\prime}_{33}$ | $e^{\prime}_{31}$ | $e^{\prime}_{33}$ | $e^{\prime}_{15}$
---|---|---|---|---|---|---|---|---|---|---
| (GPa) | (GPa) | (GPa) | (GPa) | (GPa) | ($\varepsilon_{0}$) | ($\varepsilon_{0}$) | (C/m2) | (C/m2) | (C/m2)
Pz27-0.5-6.35 (A)  | $127$ | $80.1$ | $82.3$ | $121$ | $20.3$ | $963$ | $780$ | $-5.3$ | $16.4$ | $11.8$
Pz27-0.5-6.35 (B) | $126$ | $78.8$ | $81.4$ | $122$ | $20.7$ | $957$ | $753$ | $-5.2$ | $15.9$ | $11.0$
Pz27-0.5-10 (A) | $123$ | $75.6$ | $77.3$ | $117$ | $20.8$ | $993$ | $839$ | $-5.0$ | $16.4$ | $10.8$
Pz27-0.5-10 (B) | $121$ | $74.7$ | $76.5$ | $117$ | $21.3$ | $1015$ | $842$ | $-5.1$ | $16.4$ | $10.4$
Mean of UEIS | $124(3)$ | $77(3)$ | $80(3)$ | $119(3)$ | $20.8(4)$ | $982(27)$ | $804(44)$ | $-5.2(1)$ | $16.3(3)$ | $11.0(6)$
Literature [18] | $117.64$ | $73.66$ | $73.46$ | $110.17$ | $20.41$ | $959.10$ | $816.61$ | $-5.19$ | $16.06$ | $11.59$
Manufacturer [40] | $147(4)$ | $105(3)$ | $94(2)$ | $113(3)$ | $23.0(6)$ | $1130(113)$ | $914(91)$ | $-3.1(2)$ | $16.0(8)$ | $11.6(6)$
$\hat{\sigma}_{\mathrm{UEIS}}(\%)$ | $1.1$ | $1.7$ | $1.8$ | $1.1$ | $1.0$ | $1.4$ | $2.8$ | $-1.5$ | $0.8$ |  $2.6$
$\Delta_{\mathrm{UEIS}}^{\mathrm{lit}}(\%)$ |  $-5.2$ | $-4.7$ | $-7.5$ | $-7.6$ | $-1.8$ | $-2.3$ | $1.6$ | $0.6$ | $-1.3$ | $5.5$
$\Delta_{\mathrm{UEIS}}^{\mathrm{manf}}(\%)$ |  $18$ | $36$ | $18$ | $-5.2$ | $11$ | $15$ | $14$ | $-40$ | $-1.7$ | $5.5$
Table 4:  The imaginary part (double prime) of the material parameters $p_{i}^{\mathrm{pz27}}$ of the four Pz27 disks of Table 1 obtained by the UEIS method with initial values from the literature [18] and the parameters $\varepsilon^{\prime\prime}_{11}$, $e^{\prime\prime}_{31}$, $e^{\prime\prime}_{33}$, and $e^{\prime\prime}_{15}$ set to zero by assumption. The digits in brackets denote one standard deviation, $\hat{\sigma}_{\mathrm{UEIS}}$ is the relative standard deviation of the UEIS mean, and $\Delta_{\mathrm{UEIS}}^{\mathrm{lit}}$ is the relative deviation from UEIS of the values of Ref. [18] (lit). Pz27 disk | $C^{\prime\prime}_{11}$ | $C^{\prime\prime}_{12}$ | $C^{\prime\prime}_{13}$ | $C^{\prime\prime}_{33}$ | $C^{\prime\prime}_{44}$ | $\varepsilon^{\prime\prime}_{11}$ | $\varepsilon^{\prime\prime}_{33}$ | $e^{\prime\prime}_{31}$ | $e^{\prime\prime}_{33}$ | $e^{\prime\prime}_{15}$
---|---|---|---|---|---|---|---|---|---|---
| (MPa) | (MPa) | (MPa) | (MPa) | (MPa) | ($\varepsilon_{0}$) | ($\varepsilon_{0}$) | (C/m2) | (C/m2) | (C/m2)
Pz27-0.5-6.35 (A)  | $-709$ | $370$ | $121$ | $-280$ | $-582$ | $0$ | $3.9$ | $0$ | $0$ | $0$
Pz27-0.5-6.35 (B) | $-628$ | $442$ | $122$ | $-377$ | $-596$ | $0$ | $2.5$ | $0$ | $0$ | $0$
Pz27-0.5-10 (A) | $-558$ | $448$ | $112$ | $-518$ | $-488$ | $0$ | $0.0$ | $0$ | $0$ | $0$
Pz27-0.5-10 (B) | $-510$ | $510$ | $90.7$ | $-538$ | $-486$ | $0$ | $6.3$ | $0$ | $0$ | $0$
Mean of UEIS | $-601(87)$ | $442(57)$ | $112(15)$ | $-428(122)$ | $-538(59)$ | $0$ | $3(3)$ | $0$ | $0$ | $0$
Literature [18] | $-460$ | $220$ | $110$ | $-400$ | $-400$ | $-$ | $10.72$ | $-$ | $-$ | $-$
$\hat{\sigma}_{\mathrm{UEIS}}(\%)$ | $-7.2$ | $6.4$ | $6.6$ | $-14$ | $-5.5$ | $-$ | $41$ | $-$ | $-$ |  $-$
$\Delta_{\mathrm{UEIS}}^{\mathrm{lit}}(\%)$ |  $-23$ | $-50$ | $-1.4$ | $-6.6$ | $-26$ | $-$ | $235$ | $-$ | $-$ | $-$
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 3 and 4. 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-µm-thick
layer of NOA 86H glue. The UEIS-fitted simulation (orange) is computed using
the UEIS parameter values listed in Tables 3, 4, 5, and 6. 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).

###  V.2 UEIS-fitted material parameters for glue

The parameters $p_{i}^{\mathrm{glue}}$ of the used UV-cured NOA 86H glue were
determined by the UEIS method as described in Section III.4 using a UV-cured
glue ring glued to a Pz27 disk with the dimensions listed in Tables 1 and 2.
The resulting values for $C_{11}$ and $C_{44}$ are presented in Table 5
together with the corresponding values for the sound speeds $c_{\mathrm{lo}}$
and $c_{\mathrm{tr}}$, the attenuation coefficients $\alpha_{\mathrm{lo}}$ and
$\alpha_{\mathrm{tr}}$, as well as Young’s modulus $E$ and Poisson’s ratio
$\nu$. The expressions for these additional parameters, valid for any
isotropic elastic material, are obtained by assuming frequency-independent
moduli $C_{11}$ and $C_{44}$ in the limit of weak attenuation,
$\big{|}C_{11}^{\prime\prime}\big{|}\ll C_{11}^{\prime}$ and
$\big{|}C_{44}^{\prime\prime}\big{|}\ll C_{44}^{\prime}$, and by introducing
the complex-valued wavenumbers
$k_{\mathrm{lo}}=\omega/c_{\mathrm{lo}}+\mathrm{i}\alpha_{\mathrm{lo}}$ and
$k_{\mathrm{tr}}=\omega/c_{\mathrm{tr}}+\mathrm{i}\alpha_{\mathrm{tr}}$,

$\displaystyle c_{\mathrm{lo}}$
$\displaystyle=\sqrt{\frac{C_{11}^{\prime}}{\rho}},\qquad$
$\displaystyle\alpha_{\mathrm{lo}}(f)$
$\displaystyle=\pi\sqrt{\frac{\rho}{\big{(}C_{11}^{\prime}\big{)}^{3}}}\big{|}C_{11}^{\prime\prime}\big{|}\>f,$
(14a) $\displaystyle c_{\mathrm{tr}}$
$\displaystyle=\sqrt{\frac{C_{44}^{\prime}}{\rho}},$
$\displaystyle\alpha_{\mathrm{tr}}(f)$
$\displaystyle=\pi\sqrt{\frac{\rho}{\big{(}C_{44}^{\prime}\big{)}^{3}}}\big{|}C_{44}^{\prime\prime}\big{|}\>f,$
(14b) $\displaystyle\nu$
$\displaystyle=\frac{\frac{1}{2}C_{11}^{\prime}-C_{44}^{\prime}}{C_{11}^{\prime}-C_{44}^{\prime}}$
$\displaystyle E$
$\displaystyle=\frac{3C_{11}^{\prime}-4C_{44}^{\prime}}{C_{11}^{\prime}-C_{44}^{\prime}}\>C_{44}^{\prime}.$
(14c)

Table 5:  The material parameters at $24~{}^{\circ}\\!\textrm{C}$ of the UV-cured NOA 86H glue determined by the UEIS method. The density of the glue was measured to be $\rho=1.3(2)\times 10^{3}~{}\textrm{kg}\>\textrm{m${}^{-3}$}$, and the attenuation coefficients are for the center frequency $f_{c}=1.90$ MHz of the UTT method described in Section IV. Parameter | Parameter | Parameter
---|---|---
$C_{11}^{\prime}=4.65~{}\textrm{GPa}$ | $c_{\mathrm{lo}}=1891~{}\textrm{m}/\textrm{s}$ |  $E=3.20~{}\textrm{GPa}$
$C_{44}^{\prime}=1.21~{}\textrm{GPa}$ | $c_{\mathrm{tr}}=\;\;965~{}\textrm{m}/\textrm{s}$ | $\nu=0.32$
$C_{11}^{\prime\prime}=-0.51~{}\textrm{GPa}$ | $\alpha_{\mathrm{lo}}(f_{c})=346~{}\textrm{Np}/\textrm{m}$ |
$C_{44}^{\prime\prime}=-0.12~{}\textrm{GPa}$ | $\alpha_{\mathrm{tr}}(f_{c})=613~{}\textrm{Np}/\textrm{m}$ |
Table 6:  The UEIS-fitted and UTT-measured elastic moduli for PMMA at $24~{}^{\circ}\\!\textrm{C}$ with a measured density $\rho=1162(4)$ kg/m3, and $\alpha_{\mathrm{lo}}$ and $\alpha_{\mathrm{tr}}$ evaluated at $1.90$ MHz. For UEIS, $\hat{\sigma}_{\mathrm{UEIS}}$ is the relative standard deviation of the UEIS mean. For UTT, the errors are based on $1\sigma$-prediction intervals around linear regression fits. $\Delta_{\mathrm{UEIS}}^{\mathrm{UTT}}$ is the deviation of the UTT values relative to the UEIS values. Param. | Unit | UEIS |  $\hat{\sigma}_{\mathrm{UEIS}}$ (%) | UTT |  $\Delta_{\mathrm{UEIS}}^{\mathrm{UTT}}$ (%)
---|---|---|---|---|---
$C_{11}^{\prime}$ | GPa | $7.18(4)$ | $0.6$ | $7.1(1)$ | $-1.1\rule{0.0pt}{10.76385pt}$
$C_{11}^{\prime\prime}$ | GPa | $-0.183(5)$ | $2.9$ | $-0.19(1)$ | $3.8$
$C_{44}^{\prime}$ | GPa | $1.553(8)$ | $0.5$ | $1.56(1)$ | $0.5$
$C_{44}^{\prime\prime}$ | GPa | $-0.111(7)$ | $6.3$ | $-0.098(6)$ | $-11.7$
$c_{\mathrm{lo}}$ | m/s | $2486(8)$ | $0.3$ | $2469(19)$ | $-0.7$
$c_{\mathrm{tr}}$ | m/s | $1156(4)$ | $0.3$ | $1160(4)$ | $0.3$
$\alpha_{\mathrm{lo}}$ | Np/m | $61(2)$ | $3.2$ | $66(4)$ | $8.2$
$\alpha_{\mathrm{tr}}$ | Np/m | $370(24)$ | $6.4$ | $322(18)$ | $-13.0$
$E$ | GPa | $4.23(2)$ | $0.5$ | $4.25(3)$ | $0.5$
$\nu$ | – | $0.362(1)$ | $0.4$ | $0.358(3)$ | $-1.1$

###  V.3 UEIS-fitted material parameters for PMMA

With the characterization of the Pz27 transducer disk and the glue completed,
we move on to the determination of the complex-valued elastic moduli $C_{11}$
and $C_{44}$ for PMMA, which in principle could have been any other elastic
polymer. We studied four PMMA polymer rings with the dimensions listed in
Table 1, all around 1.4 mm thick and with diameters of 20 or 25 mm, and glued
to Pz27 disks with the dimensions listed in Table 2.

The resulting UEIS-fitted parameters $C_{11}^{\prime}$, $C_{44}^{\prime}$,
$C_{11}^{\prime\prime}$, and $C_{44}^{\prime\prime}$ at 24
${}^{\circ}\\!\textrm{C}$ for the PMMA are listed in Table 6 together with the
corresponding values obtained by the UTT technique. The relative standard
deviation $\hat{\sigma}_{\mathrm{UEIS}}$ on the real parts is low ($\sim$0.5
%), and an order of magnitude higher on the imaginary parts (3-6 %). We find
good agreement between the UEIS and the UTT values, in all cases with relative
deviations
$\Delta_{\mathrm{UEIS}}^{\mathrm{UTT}}<3\hat{\sigma}_{\mathrm{UEIS}}$. In
terms of the derived sound speeds, $c_{\mathrm{lo}}$ and $c_{\mathrm{tr}}$,
and the derived Young’s modulus $E$ and Poisson’s ratio $\nu$, the relative
deviation of UTT values from UEIS values is around 0.5 %. For the longitudinal
and transverse attenuation $\alpha_{\mathrm{lo}}$ and $\alpha_{\mathrm{tr}}$
coefficients, the relative deviations of UTT values relative to UEIS values
are higher, around 7-15 %.

Again, likely due to the lower sensitivity of the $C_{11}^{\prime\prime}$ and
$C_{44}^{\prime\prime}$ coefficients, it proves more difficult to obtain the
imaginary parts of the elastic moduli than the real parts. Deviations of the
UTT values from the UEIS values, may in part be explained by the fact that the
UTT technique uses a frequency pulse with a width of 1 MHz around the center
frequency 1.90 MHz, whereas UEIS is based on an entire frequency spectrum from
500 Hz to 5 MHz using a single frequency at a time. However, whereas different
models exist, which assume a frequency-dependence of the elastic moduli of
PMMA [12], similar to the frequency dependencies measured in PDMS [9], we do
find it sufficient in the UEIS method to neglect the frequency-dependence of
the complex-valued elastic moduli of PMMA.

In Fig. 5 is shown an example of the measured and the simulated UEIS spectra
for a PMMA ring glued to a Pz27 disk. We find a good agreement between the
measured and the fitted simulated UEIS spectrum, and it can also be seen, how
even smaller features of the experimental impedance curve are captured in the
simulated frequency spectrum. Relative deviations
$\Delta_{\mathrm{sim}}^{\mathrm{exp}}=\log_{10}\frac{|Z_{\mathrm{exp}}|}{|Z_{\mathrm{sim}}|}$
up to $20~{}\%$ are found in regions near resonance peaks. However, zoom-ins
there show how a frequency-shift of a few percent can lead to high deviations
$\Delta_{\mathrm{sim}}^{\mathrm{exp}}$, while still maintaining good agreement
between measurement and simulation. For example, shifting a Lorentzian peak
with a Q-value of $Q=250$ by 1% of its resonance frequency, results in a
relative deviation $\sim 10\%$.

We furthermore studied the impact of small deviations in the thickness and
elastic moduli of the glue on the obtained coefficients for the polymer ring.
A change of the Young’s modulus $E_{\mathrm{glue}}$ by $\pm 1$ % leads to
changes in the real-valued coefficients $C_{11}^{\prime}$ and
$C_{44}^{\prime}$ by less than 0.05 %, while the $C_{11}^{\prime\prime}$ and
$C_{44}^{\prime\prime}$ coefficients change by 0.4 % and 0.6 % respectively.
In a separate numerical study, when changing the thickness of the glue layer
from 12 $\upmu\textrm{m}$ to 8 $\upmu\textrm{m}$, a relative deviation of 33.3
%, we observe a decrease in the real-valued coefficients $C_{11}^{\prime}$ and
$C_{44}^{\prime}$ by 0.8 % and 0.2 % respectively. The relative changes for
the imaginary-valued coefficients $C_{11}^{\prime\prime}$ and
$C_{44}^{\prime\prime}$ are slightly higher by +2.4 % and $-1.3$ %,
respectively, but still much lower when compared to the relative change in
thickness and well in line with the identified uncertainties of the parameters
listed in Table 6.

As a further validation of the UEIS method, we use the UEIS-fitted values
$p_{i}^{\mathrm{pmma}}$ to simulate selected resonance modes in the PMMA ring.
Subsequently, as shown in Sec. S5 in the Supplemental Material [28], we have
successfully compared these predicted modes with direct measurements of the
corresponding modes obtained by using a single-point laser-Doppler vibrometer
system VibroFlex Connect (Polytec, Waldbronn, Germany).

##  VI Conclusion

We have developed a method based on measured and simulated ultrasound
electrical impedance spectroscopy (UEIS) able to determine the frequency-
independent complex-valued elastic moduli of polymers. The method is a two-
step procedure: Firstly, the material parameters of the used, unloaded
piezoelectric transducer disk are fitted by an inverse problem, matching the
measured and simulated electrical impedance spectrum. Secondly, a polymer ring
is glued onto the transducer, and the same technique is used to fit the
complex-valued elastic moduli of the polymer. To evaluate its reproducibility,
the method was applied on four different system geometries involving the
polymer PMMA, achieving a relative error below $0.5\,\%$ for Young’s modulus
and Poisson’s ratio, and below $7\,\%$ for the attenuation coefficients. The
method was validated experimentally within the 3$\sigma$-level using
ultrasonic through-transmission on PMMA samples.

It is noteworthy that the model assumption of frequency-independent elastic
moduli $C_{ik}$ leads to simulated UEIS spectra $Z_{\mathrm{sim}}(f)$ that
predicts the measured UEIS spectra $Z_{\mathrm{exp}}(f)$ so well in the entire
frequency range from 500 Hz to 5 MHz as shown in Figs. 4 and 5 for Pz27 and
PMMA, as well as for the UV-curable glue NOA 86H (not shown). This frequency
independence leads to the linear frequency dependence of the attenuation
coefficients $\alpha_{\mathrm{lo}}$ and $\alpha_{\mathrm{tr}}$ exhibited in
Eq. (14), a linearity which can be contrasted with the $f^{2}$-dependence of
$\alpha_{0}$ in Newtonian fluids, Eq. (IV), and the non-integer powers
observed in typically softer materials, such as the $f^{1.456}$-dependence of
$\alpha_{\mathrm{lo}}$ and the $f^{0.924}$-dependence of
$\alpha_{\mathrm{tr}}$ observed in the rubber PDMS [9]. It is straightforward
to include such frequency-dependency of the elastic moduli in the UEIS model,
should materials with that property be studied. One simply modify the
respective moduli and coupling coefficients in the constitutive equations (1)
and (2) at the cost of extending the list of parameters $p_{i}$ with the
necessary parameters needed to describe the frequency dependency. For the
relatively stiff polymer PMMA, the elastic modulus tensor $\bm{C}$ can be
taken as frequency independent, whereas modeling softer, rubber-like
materials, the frequency dependency of $\bm{C}$ must be taken into account
[41].

The UEIS technique extends the existing field of resonance ultrasound
spectroscopy by making use of the electrical impedance spectrum over a wide
frequency range of several MHz involving both on-resonance and off-resonance
frequencies, it has no obvious frequency limitations before severe attenuation
sets in above 100 MHz, and it contains information of all relevant parameters
of the piezoelectric transducer disk, the glue layer, the polymer ring, and
the geometry of the assembled stack. Experimentally, the technique is low-
cost, easy-to-use, simple, and well-suited for materials used in ultrasound
applications. The recording of a given impedance spectrum takes less than 4
minutes. Afterwards, within about 1 minute, the impedance spectrum can be
loaded into our MATLAB script, and the automated UEIS fitting procedure is
executed. After a run time of about 10 hours, the resulting UEIS-fitted
impedance spectrum and the parameter values $p_{i}^{\mathrm{pz27}}$,
$p_{i}^{\mathrm{glue}}$, or $p_{i}^{\mathrm{pmma}}$ are delivered by the
software.

The UEIS technique is not limited to the chosen examples of Pz27, glue, and
PMMA, but it can in principle be used on other classes of elastic materials
including rubbers, glasses, and metals. We believe that the presented UEIS
technique will become a valuable and easy-to-use tool in the ultrasound
application fields mentioned in the introduction, by providing well-determined
parameter values for the materials used, namely the relevant complex-valued
elastic moduli at the relevant ultrasound frequencies.

##  VII Acknowledgments

We would like to thank Ola Jakobsson (Lund University) for setting up and
introducing us to the laser-Doppler vibrometer system, Axel Tojo (Lund
University) for his help with the UTT setup, Komeil Saeedabadi and Erik Hansen
(DTU) for providing and cutting the polymer rings, and Erling Ringgaard
(Meggitt) for useful discussions and for providing the Pz27 samples. This work
is part of the Eureka Eurostars-2 E!113461 AcouPlast project funded by
Innovation Fund Denmark, grant no. 9046-00127B, and Vinnova, Sweden’s
Innovation Agency, grant no. 2019-04500. Finally, the work was supported by
Independent Research Fund Denmark, Technology and Production Sciences, grant
no. 8022-00285B and the Swedish Foundation for Strategic Research, grant no.
FFL18-0122.

## References

  * Bretz  _et al._ [2005] N. Bretz, J. Strobel, M. Kaltenbacher, and R. Lerch, Numerical simulation of ultrasonic waves in cavitating fluids with special consideration of ultrasonic cleaning, in  _IEEE Int. Ultrason. Symp._ (2005) pp. 703–706.
  * Todaro  _et al._ [2017] M. T. Todaro, F. Guido, V. Mastronardi, D. Desmaele, G. Epifani, L. Algieri, and M. De Vittorio, Piezoelectric MEMS vibrational energy harvesters: Advances and outlook, Microelectron. Eng. 183, 23 (2017).
  * Singh  _et al._ [2010] M. Singh, H. M. Haverinen, P. Dhagat, and G. E. Jabbour, Inkjet printing-process and its applications, Adv. Mater. 22, 673 (2010).
  * Bodé  _et al._ [2020] W. N. Bodé, L. Jiang, T. Laurell, and H. Bruus, Microparticle acoustophoresis in aluminum-based acoustofluidic devices with PDMS covers, Micromachines 11, 292 (2020).
  * Lickert  _et al._ [2021] F. Lickert, M. Ohlin, H. Bruus, and P. Ohlsson, Acoustophoresis in polymer-based microfluidic devices: Modeling and experimental validation, J. Acoust. Soc. Am. 149, 4281 (2021).
  * [6] MatWeb, LLC, http://www.matweb.com/ (2022).
  * Migliori  _et al._ [1993] A. Migliori, J. Sarrao, W. M. Visscher, T. Bell, M. Lei, Z. Fisk, and R. Leisure, Resonant ultrasound spectroscopic techniques for measurement of the elastic moduli of solids, Physica B 183, 1 (1993).
  * Wang  _et al._ [1999] H. Wang, W. Jiang, and W. Cao, Characterization of lead zirconate titanate piezoceramic using high frequency ultrasonic spectroscopy, J. Appl. Phys. 85, 8083 (1999).
  * Xu  _et al._ [2020] G. Xu, Z. Ni, X. Chen, J. Tu, X. Guo, H. Bruus, and D. Zhang, Acoustic Characterization of Polydimethylsiloxane for Microscale Acoustofluidics, Phys. Rev. Applied 13, 054069 (2020).
  * Roebben  _et al._ [1997] G. Roebben, B. Bollen, A. Brebels, J. Van Humbeeck, and O. Van der Biest, Impulse excitation apparatus to measure resonant frequencies, elastic moduli, and internal friction at room and high temperature, Rev. Sci. Instrum. 68, 4511 (1997).
  * Willis  _et al._ [2001] R. Willis, L. Wu, and Y. Berthelot, Determination of the complex young and shear dynamic moduli of viscoelastic materials, J. Acoust. Soc. Am. 109, 611 (2001).
  * Ilg  _et al._ [2012] J. Ilg, S. J. Rupitsch, A. Sutor, and R. Lerch, Determination of Dynamic Material Properties of Silicone Rubber Using One-Point Measurements and Finite Element Simulations, IEEE T. Instrum. Meas. 61, 3031 (2012).
  * Radovic  _et al._ [2004] M. Radovic, E. Lara-Curzio, and L. Riester, Comparison of different experimental techniques for determination of elastic properties of solids, Mat Sci Eng A-Struct 368, 56 (2004).
  * Rupitsch and Lerch [2009] S. J. Rupitsch and R. Lerch, Inverse Method to estimate material parameters for piezoceramic disc actuators, Appl. Phys. A 97, 735 (2009).
  * Plesek  _et al._ [2004] J. Plesek, R. Kolman, and M. Landa, Using finite element method for the determination of elastic moduli by resonant ultrasound spectroscopy, J. Acoust. Soc. Am. 116, 282 (2004).
  * Pérez  _et al._ [2010] N. Pérez, M. A. B. Andrade, F. Buiochi, and J. C. Adamowski, Identification of elastic, dielectric, and piezoelectric constants in piezoceramic disks, IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 57, 2772 (2010).
  * Pérez  _et al._ [2014] N. Pérez, R. Carbonari, M. Andrade, F. Buiochi, and J. Adamowski, A FEM-based method to determine the complex material properties of piezoelectric disks, Ultrasonics 54, 1631 (2014).
  * Kiyono  _et al._ [2016] C. Y. Kiyono, N. Perez, and E. C. N. Silva, Determination of full piezoelectric complex parameters using gradient-based optimization algorithm, Smart Mater. Struct. 25, 025019 (2016).
  * Maynard [1996] J. Maynard, Resonant Ultrasound Spectroscopy, Physics Today 49, 26 (1996).
  * Steckel  _et al._ [2021] A. G. Steckel, H. Bruus, P. Muralt, and R. Matloub, Fabrication, characterization, and simulation of glass devices with $\mathrm{Al}\mathrm{N}$ thin-film transducers for excitation of ultrasound resonances, Phys. Rev. Applied 16, 014014, 1 (2021).
  * Skov  _et al._ [2019] N. R. Skov, J. S. Bach, B. G. Winckelmann, and H. Bruus, 3D modeling of acoustofluidics in a liquid-filled cavity including streaming, viscous boundary layers, surrounding solids, and a piezoelectric transducer, AIMS Mathematics 4, 99 (2019).
  * Ikeda [1996] T. Ikeda,  _Fundamentals of Piezoelectricity_ (Oxford University Press, London, UK, 1996).
  * Michler and Lebek [2016] G. H. Michler and W. Lebek,  _Electron microscopy of polymers_ (Wiley, Hoboken (NJ), 2016) pp. 277–293.
  * Holland [1967] R. Holland, Representation of Dielectric, Elastic, and Piezoelectric Losses by Complex Coefficients, IEEE T. Son. Ultrason 14, 18 (1967).
  * Bodé and Bruus [2021] W. N. Bodé and H. Bruus, Numerical study of the coupling layer between transducer and chip in acoustofluidic devices, J. Acoust. Soc. Am. 149, 3096 (2021).
  * Com [2019] COMSOL Multiphysics 5.5 (2019), http://www.comsol.com.
  * Steckel and Bruus [2021] A. G. Steckel and H. Bruus, Numerical study of acoustic cell trapping above elastic membrane disks driven in higher-harmonic modes by thin-film transducers with patterned electrodes, Phys. Rev. E submitted, 14 pages (2021), https://arxiv.org/abs/2112.12567.
  * Note [1] See Supplemental Material at [URL] for details on the mesh convergence analysis, a study on the sensitivity versus elastic moduli and frequency range, sample MATLAB and COMSOL scripts for the UEIS fitting procedure, the corresponding data for the impedance spectra, and the validation of the UEIS method by UTT and laser-Doppler vibrometry, which includes Refs. [16, 29].
  * Callister Jr [2007] W. D. Callister Jr,  _Materials Science and Engineering: An Introduction_ , seventh ed. (John Wiley & Sons, York, PA, 2007) p. 975.
  * John D’Errico [2012] John D’Errico, Matlab file exchange (2012), https://www.mathworks.com/matlabcentral/fileexchange/8277-fminsearchbnd-fminsearchcon, access 30 Sep 2022.
  * Hartmann and Jarzynski [1972] B. Hartmann and J. Jarzynski, Polymer sound speeds and elastic constants, Naval Ordnance Laboratory Report NOLTR 72-269, 1 (1972).
  * Christman [1972] D. Christman, Dynamic properties of poly (methylmethacrylate) (PMMA) (plexiglas), General Motors Technical Center, Warren (MI), USA Report No. DNA 2810F, MSL-71-24, 1, 1 (1972).
  * Sutherland and Lingle [1972] H. Sutherland and R. Lingle, Acoustic characterization of polymethyl methacrylate and 3 epoxy formulations, J. Appl. Phys. 43, 4022 (1972).
  * Sutherland [1978] H. Sutherland, Acoustical determination of shear relaxation functions for polymethyl methacrylate and Epon 828-Z, J. Appl. Phys. 49, 3941 (1978).
  * Carlson  _et al._ [2003] J. E. Carlson, J. van Deventer, and A. S. C. Carlander, Frequency and temperature dependence of acoustic properties of polymers used in pulse-echo systems, IEEE Ultrasonics Symposium , 885 (2003).
  * Simon  _et al._ [2019] A. Simon, G. Lefebvre, T. Valier-Brasier, and R. Wunenburger, Viscoelastic shear modulus measurement of thin materials by interferometry at ultrasonic frequencies, J. Acoust. Soc. Am. 146, 3131 (2019).
  * Tran  _et al._ [2016] H. T. Tran, T. Manh, T. F. Johansen, and L. Hoff, Temperature effects on ultrasonic phase velocity and attenuation in Eccosorb and PMMA, in  _2016 IEEE International Ultrasonics Symposium (IUS)_ (2016) pp. 1–4.
  * [38] _Norland Optical Adhesive 86H_ , Norland Products Inc., Jamesburg, NJ 08831, USA, https://www.norlandprod.com/adhesives/NOA86H.html, accessed 30 Sep 2022\.
  * Muller and Bruus [2014] P. B. Muller and H. Bruus, Numerical study of thermoviscous effects in ultrasound-induced acoustic streaming in microchannels, Phys. Rev. E 90, 043016 (2014).
  * [40] _Ferroperm matrix data_ , Meggitt A/S, Porthusvej 4, DK-3490 Kvistgaard, Denmark, https://www.meggittferroperm.com/materials/, accessed 30 Sep 2022\.
  * Pritz [1998] T. Pritz, Frequency dependences of complex moduli and complex poisson’s ratio of real solid materials, J. Sound Vib. 214, 83 (1998).