# Quantitative characterisation of the layered structure within lithium-ion
batteries using ultrasonic resonance

Ming Huanga a The Non-Destructive Evaluation Group, Department of Mechanical
Engineering, Imperial College London, London SW7 2AZ, UK b The Electrochemical
Science and Engineering Group, Department of Mechanical Engineering, Imperial
College London, London SW7 2AZ, UK c Breathe Battery Technologies Limited,
London SE1 7SJ, UK  Niall Kirkaldyb a The Non-Destructive Evaluation Group,
Department of Mechanical Engineering, Imperial College London, London SW7 2AZ,
UK b The Electrochemical Science and Engineering Group, Department of
Mechanical Engineering, Imperial College London, London SW7 2AZ, UK c Breathe
Battery Technologies Limited, London SE1 7SJ, UK  Yan Zhaoc a The Non-
Destructive Evaluation Group, Department of Mechanical Engineering, Imperial
College London, London SW7 2AZ, UK b The Electrochemical Science and
Engineering Group, Department of Mechanical Engineering, Imperial College
London, London SW7 2AZ, UK c Breathe Battery Technologies Limited, London SE1
7SJ, UK  Yatish Patelb a The Non-Destructive Evaluation Group, Department of
Mechanical Engineering, Imperial College London, London SW7 2AZ, UK b The
Electrochemical Science and Engineering Group, Department of Mechanical
Engineering, Imperial College London, London SW7 2AZ, UK c Breathe Battery
Technologies Limited, London SE1 7SJ, UK  Frederic Ceglaa a The Non-
Destructive Evaluation Group, Department of Mechanical Engineering, Imperial
College London, London SW7 2AZ, UK b The Electrochemical Science and
Engineering Group, Department of Mechanical Engineering, Imperial College
London, London SW7 2AZ, UK c Breathe Battery Technologies Limited, London SE1
7SJ, UK  Bo Lana111Corresponding author. E-mail addresses:
<EMAIL_ADDRESS>(M. Huang<EMAIL_ADDRESS>(N. Kirkaldy),
<EMAIL_ADDRESS>(Y. Zhao<EMAIL_ADDRESS>(Y. Patel),
<EMAIL_ADDRESS>(F. Cegla<EMAIL_ADDRESS>(B. Lan). Telephone
number: +44 20 7594 7227 (B. Lan). a The Non-Destructive Evaluation Group,
Department of Mechanical Engineering, Imperial College London, London SW7 2AZ,
UK b The Electrochemical Science and Engineering Group, Department of
Mechanical Engineering, Imperial College London, London SW7 2AZ, UK c Breathe
Battery Technologies Limited, London SE1 7SJ, UK

## Abstract

Lithium-ion batteries (LIBs) are becoming an important energy storage solution
to achieve carbon neutrality, but it remains challenging to characterise their
internal states for the assurance of performance, durability and safety. This
work reports a simple but powerful non-destructive characterisation technique,
based on the formation of ultrasonic resonance from the repetitive layers
within LIBs. A physical model is developed from the ground up, to interpret
the results from standard experimental ultrasonic measurement setups. As
output, the method delivers a range of critical pieces of information about
the inner structure of LIBs, such as the number of layers, the average
thicknesses of electrodes, the image of internal layers, and the states of
charge variations across individual layers. This enables the quantitative
tracking of internal cell properties, potentially providing new means of
quality control during production processes, and tracking the states of health
and charge during operation.

Keywords: Lithium-ion battery; ultrasonic resonance; characterisation; layer
structure; state of charge

##  1 Introduction

Lithium-ion batteries (LIBs) are already ubiquitous in electric vehicles,
consumer electronics, and energy storage devices 1, and their usages are
expected to be boosted even further by the upcoming governmental bans on
fossil-fuel vehicle sales in many countries 2, 3. Manufacturers are thus
incentivised to ramp up production and push performance of the batteries, for
larger capacity, faster charging speeds, and lower costs 4. However, LIB
safety can still be a concern, as exposed by the high-profile Samsung
cellphone failures 5, 6 and EVs catching fire 7, 8.

The push for performance demands reliable characterisation and monitoring of
states of charge (SOC) and health (SOH) of the batteries, while the assurance
of safety requires detection and elimination of manufacturing faults (e.g.
misaligned electrodes and poor cell constructions) during production 9.
Commonly deployed methods to infer battery states are based on electrical
signals 10, such as open-circuit voltage, internal resistance 11 and capacity
change 12, but these indirect estimations are purely model-based and can be
prone to inaccuracies. Lab-based X-ray has been employed to ensure safety in
production 13, while X-ray synchrotron 14, 15, 16 and computational tomography
17, 18, 19 have become powerful tools in research and development, but their
radiation hazards, high costs and practical limitations have also limited
their wider deployment.

Ultrasound has been recognised as an attractive candidate for rapid, cheap and
accessible examination of LIBs. The active materials within LIB electrodes
undergo changes in physical properties (e.g. wave speed and density) during
normal operation. The structure of the electrodes can also change as cells age
due to the effects of various degradation mechanisms. These changes can be
detected by ultrasound non-destructively. Indeed, this has spurred
considerable research employing various acoustic techniques. For instance,
Villevieille et al. 20 used acoustic emission to investigate in-operando the
structural and morphological changes of the electrodes. Ladpli et al. 21
demonstrated the feasibility of monitoring the SOC and SOH using acoustic
guided waves. Recently, the conventional time-of-flight (TOF) measurements
using the ultrasonic through-transmission or pulse-echo configurations have
been adopted in a range of investigations. The pioneer work by Hsieh et al. 22
showed clear correlations using experimental TOF measurements of 2.25MHz
ultrasound across all SOCs. Gold et al. 23 investigated the same issue with a
much lower frequency (200kHz), and proposed the second Biot mode in porous
materials as the theoretical model to explain the slow wave propagation speed.
In contrast, Davies et al. 24 used the classic Hashin-Shtrikman homogenisation
method for composites 25 to model the wave propagation through electrodes, and
applied machine learning to successfully extract SOC and SOH from ultrasonic
data. Similar experimental and theoretical frameworks were then successfully
applied to detect lithium metal plating 26 and to measure the effective
stiffness of the battery 27, among other things. Meanwhile, Robinson et al. 28
highlighted the need and advantages for such TOF measurements of battery
states and stiffnesses to be carried out in a spatially resolved fashion.

These studies have comprehensively proved the correlation between ultrasound
and the battery internal states. However, the fact that they are mostly based
on the through-transmission measurements of ultrasound means that they are
only able to deliver average estimations across the thickness without spatial
resolution. Given that a battery is a composite system of tens of layers with
contrasting properties, such averages of properties can be rather crude and
have limited accuracy. In addition, their common physical assumption that
electrodes are porous materials have actually found limited success against
experimental results (more discussions in section 3.1).

Recently there have been efforts to achieve spatial resolution in the
thickness direction. For example, Robinson et al. 9 demonstrated that a
certain reflection peak between the main front- and back-wall echoes, or the
lack of it, can be associated to a missing layer of an artificially
constructed battery, and used to image the profile of the said layer.
Similarly, by applying acoustic microscopy on a battery and gating the time-
domain signal to a certain range, Bauermann et al. 29 were able to obtain
high-quality images of the inner structures such as a fine mesh. Even though
these works mainly focused on the detection of features/defects instead of
characterisation of properties, they did point to the extra information
carried by the time-domain waveforms along the thickness.

We present a new methodology for layer-resolved characterisation of LIBs using
ultrasonic resonance, and its novelties are threefold. Firstly, it employs
simple, conventional ultrasound equipment to robustly acquire the resonant
time traces, which, compared to the transmitted signal mostly used so far in
literature, inherently carry more information. Secondly, a pivotal theoretical
model is established, from ground up, to comprehensively illustrate the
fundamental wave physics on how the resonance is formed from the reflections
from the repetitive internal layers. This key contribution opens the door for
new quantitative characterisation capabilities of the battery layers using
ultrasound, which are demonstrated in a range of applications as the last
aspect of the novelties.

The paper is organised to clearly explain the methodology and highlight the
contributions: section 2 firstly demonstrates robust, general experimental
observations of resonance. Then section 3 introduces the theoretical model to
explain the wave physics within each layer and the formation of ultrasonic
resonance from inter-layer reflections. The exciting possibilities of
quantitative characterisations of internal cell structures and states are then
illustrated by a variety of case studies in section 4. These include
estimating the number of layers, determining anode and cathode thicknesses,
constructing the image of individual layers in the thickness direction, and
tracking layer-level SOC changes during charging cycles. Section 5 concludes
this paper.

##  2 Experimental observation of ultrasonic resonance

The main experimental sample, a Kokam 7.5 Ah pouch cell (SLPB75106100), is a
typical LIB cell as illustrated in Fig. 1a. It has a periodic repetition of
internal layers, with each repetitive unit consisting of one Cu and one Al
current collector, two anodes and two cathodes, and one separator. In this
paper, the anode and cathode electrodes refer to the layer of active material
coated on the metal current collector. The thickness of each internal layer
has been destructively measured in the literature 30, 31, as summarised in
Table 1.

We performed ultrasonic tests on the cell in a pulse-echo configuration using
the contact and immersion setups in Fig. 1b and e. The contact setup had a 5
MHz (V109-RM, Olympus) and a 7.5 MHz (V121-RM, Olympus) probe in direct
contact with the cell, with water-based gel used as couplant to facilitate
wave transmission. The tests delivered the time-domain signals in Fig. 1c,
with their respective frequency-domain amplitude spectra (calculated via
Fourier transform) in Fig. 1d. The immersion setup used a 5 MHz immersion
probe (Harisonic I3-0504-S; default immersion probe throughout this work) to
examine the cell partly immersed in water. It received two reflected wave
packets shown in Fig. 1f, which correspond to the first and second reflections
from the cell, with their respective amplitude spectra in Fig. 1g. Note that
to process the time traces in Fig. 1f for amplitude spectra, the first several
cycles (light blue parts) need to be cut out, since they are generally
dominated by the front-wall (i.e. the outer surface of the cell facing the
probe, whereas the back-wall is the opposite surface of the cell) instead of
internal reflections.

Comparing the independent experimental results in Fig. 1d and g, it becomes
clear that despite their different experimental setups, probes, and centre
frequencies, all measurements gave very similar spectra with a pronounced
resonance of practically the same frequency. These confirm that the resonance
was indeed from the internal structures within cell.

Figure 1: Observation of ultrasonic resonance from contact and immersion tests
of an LIB pouch cell. a illustrates the pouch cell and its two repetitive
internal units. The fundamental element for forming ultrasonic resonance is
also highlighted. b illustrates the contact setup with a probe controlled by a
pulser/receiver, to generate waves into the cell and receive the reflections.
Water-based gel is used as couplant to facilitate wave transmission. c shows
the time-domain signals of 5 MHz and 7.5 MHz contact probes from the Kokam 7.5
Ah cell (example resonance peaks, with amplitudes alternately slightly higher
or lower than an exponential decay, are respectively from Cu and Al layers,
with explanations later), while d displays the respective amplitude spectra. e
shows the immersion setup with the cell partly immersed in water. f presents
the signal from the same cell acquired by a 5 MHz immersion probe placed 28.5
mm away from the cell, with the two wave packets representing the first and
second reflections. Light blue cycles are dominated by the packaging layer,
while dark blue and red ones indicate the main resonance. g shows the
amplitude spectra of the two packets after removing the cycles (light blue
parts in f) that are dominated by front-wall reflections.

We subsequently investigated the generality of the resonance, by carrying out
the same experiments on other LIB pouch cells, including a 210 mAh cell
(PL-651628-2C, AA Portable Power Corp.) that had been previously evaluated 24,
32, 33 and a Kokam 5 Ah cell (SLPB11543140H5) that had been destructively
analysed 34, 31. The acquired time- and frequency-domain results are shown in
Fig. A.1, which displayed strong resonance similar to those in Fig. 1. These
confirm that the ultrasonic resonance is not a property of one cell only, but
instead can be obtained in general cases.

##  3 Wave physics of the ultrasonic resonance

The experimentally observed resonance originates from reflections from the
repetitive layers within the battery cell. To explain the fundamental
mechanisms of this formation, three key components are outlined in this
section: the wave propagation in the separator and electrode layers, the
reverberations and total wave responses from a metal current collector layer,
and the formation of resonance by the interference of reflections. These are
covered consecutively in the three subsections below.

###  3.1 Wave propagation in separator and electrode layers

In the experiments, ultrasonic waves propagate within an individual layer at a
characteristic wave speed $c$, which is determined by the elastic properties
and density $\rho$ of the layer material. Across the cell, the wave goes
through layers with different properties, including metal sheets (including Cu
and Al current collectors and packaging layers), separators and electrodes;
their free-propagation speeds are provided in Table 1 (though the thin metal
sheets cause wave reverberations - details in the next subsection). The wave
speed in separator is calculated via the classic Biot model for fluid-
saturated porous media 35, 36, which accounts for the fact that separators are
usually porous polymers filled with liquid electrolytes 37, acting as an
ionically-conductive physical barrier between two electrodes. The calculation
details are provided in Appendix B.

Table 1: Layer properties of the Kokam 7.5 Ah cell. Units: speed $c$ (m/s),
density $\rho\,\mathrm{(kg/m^{3})}$, impedance $Z\,\mathrm{(Pa\cdot
s/m^{3})}$, thickness $d\,($\text{\,}\mathrm{\SIUnitSymbolMicro m}$)$, number
of layers $n$ (1).

Layer | $c$ | $\rho$ | $Z$ | $d$ |  $n$ 30
---|---|---|---|---|---
Cu | 4762 38 | 8940 38 | 4.26$\times 10^{7}$ | 14.7 30 | 24
Al | 6346∗ | 2700 39 | 1.71$\times 10^{7}$ | 15.1 30 | 25
Separator | 1209 | 1063 | 1.29$\times 10^{6}$ | 19.0 30 | 50
Anode (SOC=0)‡ | 1341 | 1909 | 2.56$\times 10^{6}$ | 64.2† (73.7 30) | 48
Cathode (SOC=0)‡ | 1093 | 4172 | 4.56$\times 10^{6}$ | 47.5† (54.5 30) | 48
Anode (SOC=1)‡ | 1443 | 1994 | 2.88$\times 10^{6}$ | 67.4†† (73.7 30) | 48
Cathode (SOC=1)‡ | 1136 | 3848 | 4.37$\times 10^{6}$ | 47.5† (54.5 30) | 48
Casing (Al) | 6346 | 2700 | 1.71$\times 10^{7}$ | 110§ | 2
Whole cell | 1360 (predicted from layers. Experimentally measured: 1404) |

  * ∗

Calculated with the self-consistent theory 40 using elastic constants from Qi
et al. 41

  * †

If calculated directly from the destructively obtained electrode layer
thicknesses, the total thickness of the cell (8.05 mm) would exceed the actual
value of 7.26 mm. Thus, we have proportionally scaled those electrode values
(metal layers and separator unchanged) for the total thicknesses to match.

  * ‡

The scaling of thicknesses also means the electrode porosity values need to be
adjusted, since the total volumes of the solid phases should remain
consistent. The values used here are 0.23 for anode and 0.19 for cathode,
which are derived from experimental results 30 with details below.

  * ††

This thickness has also considered a 5% expansion at SOC=1 compared to SOC=0
(the expansion of cathode layer is smaller than 1% and is thus neglected here)
42.

  * §

Measured in this work.

One potential contribution of this paper is the theoretical modelling of
electrodes. They are often treated as porous solids in literature, but the
wave speeds predicted by the Biot 35, 36 or composite homogenisation 25 models
(> 3000 m/s 23, 24) are in poor agreement with typical experimental results
(<1800 m/s 27). To explain this, Gold et al. 23 proposed that the experiments
actually obtained the shear mode wave of the Biot model, but it does not
explain why the first compressional mode, which should be much easier to
detect 43, were not observed in our experiments. These suggest that the porous
solid assumption may not be fundamentally accurate. Here we demonstrate that a
slurry model can deliver much closer estimations of the electrode material
properties than the homogenisation models. Physically, this is based on the
observations that the experimental compressional wave speeds are closer to
that of the liquid electrolyte than to those of the solid active materials
(unlike porous solid model predictions), and that the solid particles in
electrodes are spatially separate and are only held together by the liquid-
saturated soft polymer binder lattice 44, whose binding force is likely the
Van Der Waals force 45. From mechanics point of view, the latter observation
is different from typical rigid porous solids, e.g. ceramic or cured cement,
whose solid phases are atomically bound together and are thus much stronger;
rather, electrodes saturated in the liquid electrolyte may have more
similarities with sedimented sand in water, which is a typical concentrated
slurry.

This assumption enables the estimations of the wave speeds and densities for
the electrodes of the Kokam cell via a well-established theoretical model 46
as detailed in Appendix C. Using this model and the properties of the
electrolyte and the active materials in Table C.1, we obtained the results as
listed in Table 1. Note that these estimations are sensitive to the electrode
porosity (i.e. non-solid phase volume fraction), which could in turn be well-
informed from simple wave speed measurements (e.g. during manufacturing). The
predicted wave speeds in individual electrode layers are now shown to be
dominated by the liquid phase, and the overall wave speed through our sample
cell (1360 m/s) is very close to the experimental result (1404 m/s); in
addition, the predicted overall speeds of a combined anode and cathode layer
differ $\sim 10\%$ between the fully discharged (SOC=0) and charged (SOC=1)
states, which again agrees well with experiments 24. These substantiates the
suitability of the slurry model for the estimation of electrode properties.

###  3.2 Reverberations and total wave responses in a metal layer

We now proceed to examine the wave interactions with the metal current
collector layers. Generally, when incident upon an interface, part of the wave
is reflected back in the incident medium (denoted 0) while the rest is
transmitted through to the other medium (denoted 1). The amplitudes of these
two wave parts are described by the well-known reflection $R_{01}$ and
transmission $T_{01}$ coefficients 47:

$R_{01}=\frac{Z_{0}-Z_{1}}{Z_{0}+Z_{1}},\;T_{01}=\frac{2Z_{0}}{Z_{0}+Z_{1}},$
(1)

where the acoustic impedance $Z=\rho c$ is a function of the wave speed $c$
and density $\rho$ of the material. Notice that the reflection and
transmission depend not only on the impedance contrast across the boundary,
but also on which material the wave is incident from.

For a thin current collector, due to the close proximity of its boundaries,
the reflection and transmission at both boundaries would interfere and cause
reverberations in between. As exemplified in Fig. 2a for a metal layer bonded
with electrode on both sides, each reflection (or transmission) is separated
from its subsequent one by a phase shift corresponding to the round-trip
propagation through the layer thickness. The total macro responses of
reflection $R$ and transmission $T$ from the metal layer are summations of all
individual reflections and transmissions, given by48:

$\displaystyle R$
$\displaystyle=R_{01}+T_{01}R_{10}T_{10}e^{-2ikd}+T_{01}R_{10}^{3}T_{10}e^{-4ikd}+\ldots=R_{01}\frac{1-e^{-2ikd}}{1-R_{01}^{2}e^{-2ikd}},$
(2) $\displaystyle T$
$\displaystyle=T_{01}T_{10}e^{-ikd}+T_{01}R_{10}^{2}T_{10}e^{-3ikd}+\ldots=\frac{(1-R_{01}^{2})e^{-ikd}}{1-R_{01}^{2}e^{-2ikd}},$

with $k$ and $d$ denoting the wave number and thickness of the metal layer.

Figure 2: Ultrasonic reverberations and the resultant total signals of the
metal current collectors. a shows the individual reflected (R) and transmitted
(T) waves caused by multiple reverberations within the metal layer. Their
amplitudes are respectively described by $R_{01}$ and $T_{01}$ (subscript 0
before 1) for a wave travelling from 0 to 1; each subsequent reflection is
shifted by a phase related to the wave number $k$ and layer thickness $d$. b
illustrates the total reflection and transmission (summations of an infinite
number of individual reflected and transmitted waves) from the Cu and Al
layers of the Kokam cell.

Eq. 2 shows that a strong total reflection needs both strong individual
reflections (requiring large contrast of acoustic properties across the
boundary) and small enough phase shifts to form approximately constructive
interference (requiring thin thicknesses). Both are the case for the metal
layers inside LIBs in the considered MHz frequency range, but the
reverberations in electrodes and separators are negligible, due to the similar
acoustic impedances between the neighbouring anode-separator-cathode layers,
as can be seen from Table 1. Therefore, they can be effectively treated as a
combined thicker layer, within which a sound wave travels freely.

As a result, we only consider the reverberations in metal layers and neglect
those in electrodes and separators. For the Cu and Al layers of our cell, the
total reflected and transmitted waves calculated from Eq. 2 are exemplified in
Fig. 2b. They were calculated in the frequency domain using the properties in
Table 1, by applying the appropriate amplitude and phase modulations on each
individual frequency component of the incident wave, and transforming the
results back to the time domain. It is important to point out that they both
have phase shifts and amplitude changes relative to the incident wave, which
are caused by the reverberations. Regarding the phase shifts, it can be
mathematically proven from Eq. 2 that the reflected and transmitted waves have
a consistent $\pi/2$ phase difference:

$\phi_{T}=\phi_{R}+\pi/2,$ (3)

which is true irrespective of the material pairs, frequency, or layer
thickness. This relationship is an important step to arriving at the resonance
condition in the next subsection. For the amplitude changes, a prominent
observation is that the reflection from the Cu layer is much stronger in
amplitude than that from the Al layer in this case. This amplitude difference
is also evident from the experimental results in Fig. 1c, which shows slightly
alternating amplitudes of the peaks and could be used to distinguish the metal
layers. However, we emphasise that it is entirely possible for another battery
configuration to have stronger reflections from the Al layers, because it
depends on various factors including the acoustic impedance mismatch between
the metals and electrodes, the wave frequency and layer thickness.

###  3.3 Ultrasonic resonance formation

With the wave physics in individual layers and interfaces understood, we can
now derive the conditions needed for the ultrasonic resonance to form, which
can be simply based on the constructive interference of major reflections from
metal layers. For example, the first few major reflections and their wave
paths for the Kokam cell are illustrated in Fig. 3a. As discussed, the
reflected and transmitted signals from a metal layer have implicitly accounted
for the infinite reverberations within it, while only a single-trip
transmission is considered for each anode-separator-cathode combined layer.

Figure 3: Formation of resonance in an LIB. a illustrates the major
reflections that contribute to the formation of resonance. b displays the
actual major reflections when subjecting to the incident wave shown in Fig.
2b. c superposes the major reflections to form the total reflection as shown
in black. d illustrates the setup for full-waveform modelling with the
multilayered battery consisting of $N$ layers, bounded by semi-infinite media
0 and $N+1$; the model is formulated by satisfying the continuity of stress,
$\sigma_{n}=\sigma_{n+1}$, and velocity, $v_{n}=v_{n+1}$, across each
individual interface $n$. e compares the full-waveform result with that
predicted in c. f compares the respective reflection coefficients obtained by
the two analytical models in the frequency domain.

Due to their different wave paths, the major reflections incur different phase
shifts compared to the incident wave. For a resonance to form at a given
frequency, these reflections need to have an exact $2n\pi$ phase difference
between them, with the integer number $n$ denoting the order of resonance. For
instance, this requires the phase shifts $\phi_{u_{1}}$ and $\phi_{u_{2}}$ of
the first two reflections to satisfy:

$\phi_{u_{1}}=\phi_{u_{2}}+2n\pi$ (4)

The two phase shifts are given by $\phi_{u_{1}}=\phi_{\mathrm{rc}}$ and
$\phi_{u_{2}}=2\phi_{\mathrm{tc}}+2\phi_{\mathrm{e}}+\phi_{\mathrm{ra}}$,
where $\phi_{\mathrm{e}}$ is the phase shift in the combined anode-separator-
cathode layer, and $\phi_{\mathrm{rc}}$, as an example, is the reflection
(first subscript, $r$ for reflection and $t$ for transmission) from the Cu
(second subscript, $c$ for Cu and $a$ for Al) layer. Substituting these
expressions and, importantly, Eq. 3, into Eq. 4 results in the defining
relation for ultrasonic resonance:

$\phi_{\mathrm{ra}}+2\phi_{\mathrm{e}}+\phi_{\mathrm{rc}}=-(2n+1)\pi.$ (5)

Eq. 5 is formulated based on $u_{1}$ and $u_{2}$, but it can be proven that
the same equation would be similarly obtained from the phase relationship
between $u_{2}$ and $u_{3}$. These mean that $u_{1}$ and $u_{3}$ would
automatically satisfy the $2n\pi$ phase shift and form resonance, and so will
all subsequent reflections that have gone through a single round trip (i.e.
transmitted through multiple layers and only reflected once), e.g. between
$u_{2}$ and $u_{4}$. It thus transpires that the simple equation, Eq. 5, in
fact governs the general condition of resonance for any reflection pair,
regardless of whether the equations are constructed with Cu as the first layer
(as above) or Al.

Moreover, the resonance described in Eq. 5 originates from a Cu layer and its
neighbouring Al layer (e.g. $u_{1}$ and $u_{2}$), which is the main focus of
this paper. However, resonance can also form between a Cu layer and the next
Cu layer, e.g. $u_{1}$ and $u_{3}$ and so on. These considerations can
generalise Eq. 5 further into:

$\phi_{\mathrm{ra}}+2\phi_{\mathrm{e}}+\phi_{\mathrm{rc}}=-n\pi.$ (6)

When $n$ is odd, the resonance is from Cu-Al layers; whereas when $n$ is even,
the resonance is from two Cu or two Al layers. For example, the first and
third peaks of the black curve in Fig. 3f likely come from $n=0$ and $2$,
respectively.

So far we have considered all signals that make a straight single round-trip
to one metal layer, and now we prove that the waves which have been reflected
multiple times by the metal layers cannot form resonance. Let us consider an
exemplary signal $u_{3}^{*}$ that has a similar travelling distance with the
major round-trip reflection $u_{3}$ (interacted with metal layers 3 times),
but bounces back and forth between the Al and Cu layers. For these two signals
to form resonance, they need to satisfy

$\displaystyle
2\phi_{\mathrm{tc}}+4\phi_{\mathrm{e}}+2\phi_{\mathrm{ra}}+\phi_{\mathrm{rc}}+2n\pi$
$\displaystyle=2\phi_{\mathrm{tc}}+4\phi_{\mathrm{e}}+2\phi_{\mathrm{ta}}+\phi_{\mathrm{rc}}$
(7) $\displaystyle\text{i.e. }\phi_{\mathrm{ra}}+n\pi$
$\displaystyle=\phi_{\mathrm{ta}}$

Similarly, the requirement for the fourth reflections is:

$\phi_{\mathrm{rc}}+n\pi=\phi_{\mathrm{tc}}$ (8)

However, Eqs. 7 and 8 contradicts with the inherent relationship for any metal
layer shown in Eq. 3, which means they can never be satisfied, and that no
resonance can be formed between $u_{3}$ and $u_{3}^{*}$. Instead, the weaker
signal $u_{3}^{*}$ always has a phase difference of $\pi$ from the main
reflection $u_{3}$, i.e. it is always destructively interfering the main,
round-trip reflections and reducing the latter’s amplitudes. This also means
that in analysing the experimental resonance results from purely the phase
relationship point of view (amplitude of $u_{3}^{*}$ is < 2$\%$ of $u_{3}$),
as shown in Eq. 5, the multiply reflected signals like $u_{3}^{*}$ have
negligible contributions, and only the main round-trip reflections need to be
considered.

Using the layer properties in Table 1, the first-order main resonance ($n=1$)
of the Kokam cell at SOC=0 was estimated from Eq. 5 to be 4.15 MHz, agreeing
well with experimental results. The major time-domain reflections from the
first few metal layers, when subject to a 4.15 MHz centre-frequency incident
signal, are illustrated individually in Fig. 3b. They are plotted together in
coloured lines of Fig 3c, and through constructive interference, their
summation (the black line) evidently forms the main resonance as received in
experiments.

This simple physical model was rigorously evaluated against a full-waveform
model, which takes into account all possible wave events in a cell by
considering the continuity of velocity (compatibility, $v_{n}=v_{n+1}$) and
stress (equilibrium, $\sigma_{n}=\sigma_{n+1}$) across each individual
interface $n$ as outlined in Fig. 3d, with details in Appendix D. Here the
reflections between separators and electrodes are not considered, since they
were not observed experimentally. The time- and frequency-domain results of
the two methods are compared in Fig. 3e and f. The results demonstrate
reasonably good agreements between the two methods, especially for the
frequency domain. These prove that, despite its simplicity, the method in Eq.
5 can deliver very accurate predictions for the main resonant frequency. An
advantage for the full-waveform model, however, is its handling of amplitude,
which the simple model neglects. This has relevance for resonance amplitudes
and wave attenuation, and is potentially useful in the study of electrode
material degradation.

##  4 Application case studies

We have so far completed the outline of the physical model for analysing
ultrasonic resonance, which opens up various characterisation opportunities.
Firstly, the resonant frequency corresponds to the overall ultrasound
behaviour in the cell, so it enables quantitative evaluation of average cell
properties and states. Secondly, the time trace carries spatially-resolved
information about individual layers in the depth direction, i.e., signal peak
TOFs are associated with individual metal layers, and the reflections from Cu
and Al layers are distinguishable from their amplitudes. This allows for
characterisations of battery structures and states to a layer-resolved level.
The two aspects of the resonance is visible in e.g. Fig. 1d and g, where the
average cell property determines the central resonant frequency, and the
spatial variations contribute to the width of the frequency spectra. The
exciting characterisation possibilities are explored in the following section
as application case studies; note that they only require the prior knowledge
of the Cu, Al and separator layer thicknesses, which are assumed to be raw
production parameters and easily measurable.

###  4.1 Identifying number of electrode layers

We can estimate the total number of electrode layers (i.e. resonant elements)
inside a pouch cell from the ultrasonic resonance. When the cell is thin and
the resonance is formed throughout its thickness, the number of layers is
simply equal to the number of resonant peaks. When the cell is thick and the
resonance is only formed on the first 20 layers or so, this number can be
estimated by obtaining two phase shift values (physically equivalent to TOF)
of the propagated wave from the ultrasonic measurements: the phase shift of a
resonance element, and that of the whole cell. The number of layers is simply
estimated by dividing the latter with the former.

The first is the phase shift $\phi_{\mathrm{elem}}$ of a wave travelling a
round trip through the space of a resonant element, which consists of one Cu
and one Al current collector and a combined anode-separator-cathode layer. The
known thicknesses of the metal layers enable the phase shift
$\phi_{\mathrm{e}}$ of the combined layer to be obtained from experimental
resonance via Eq. 5. However, one important nuance here is that the phase
shifts of the metal layers $\phi_{\mathrm{tc}}^{*}$ and
$\phi_{\mathrm{ta}}^{*}$ (emphasised with asterisks) that contribute to
$\phi_{\mathrm{elem}}$ should be calculated from direct transmission through
the layers - unlike the phase shifts in Eq. 5, which implicitly include
multiple internal reverberations. This is because the back-wall echo (for
determining $\phi_{\mathrm{total}}$ below) is the first-arrival signal, which
requires the wave to go straight through each layer to satisfy the shortest
travel time. Considering the actual wave paths in individual layers of the the
Kokam cell, the phase change in a resonant element is given as
$\phi_{\mathrm{elem}}=\phi_{\mathrm{tc}}^{*}+2\phi_{\mathrm{e}}+\phi_{\mathrm{ta}}^{*}=-5.70$
rad.

The second is the total phase shift $\phi_{\mathrm{total}}$ of the wave
travelling a round trip through the whole cell. This can be obtained from the
reflections from the front- and back-wall of the cell. Such signals are
plotted in Fig. 4a for the Kokam cell, which were acquired using the pulse-
echo setup, with the 5 MHz probe placed 70.5 mm away from the cell. Despite
noticeably different amplitude spectra in Fig. 4b, the phase spectra of these
two signals in Fig. 4c show good linearity and estimate the phase difference
at the main resonance frequency to be $\phi=-269.64$ rad. This then needs to
exclude the phase delays caused by the two surface packaging layers of the
cell, and the phase reversal at the front-wall (due to wave incident from low-
impedance water to high-impedance cell), which eventually gives the total
round-trip phase change as $\phi_{\mathrm{total}}=-270.94$ rad.

Figure 4: Front- and back-wall reflections from the Kokam 7.5 Ah cell. a shows
the time-domain signals acquired with an immersion setup using a 5 MHz probe
placed 70.5 mm apart from the cell. Here the back-wall echo was acquired and
amplified from a battery-air interface (battery-water reflection was not
strong enough) created by water-sealing an air-filled container at the back-
wall. b and c present the respective amplitude and unwrapped phase spectra of
the front- and back-wall echoes.

The number $N$ of resonant elements (thus the number of electrode layer pairs)
of the cell, can be now calculated simply by
$N=\phi_{\mathrm{total}}/\phi_{\mathrm{elem}}=47.51$. This number is very
close to the actual number of 48 as counted by destructive tests 30, 31.
Moreover, excellent agreement of $N$ was obtained from further contact tests
on the same cell in Appendix E, proving the robustness and reproducibility of
the estimation.

###  4.2 Estimating average electrode thicknesses

Here the ultrasonic resonance is used to estimate another average property of
the examined cell, i.e., the average thicknesses of its anode and cathode
layers, $d_{\mathrm{an}}$ and $d_{\mathrm{ca}}$. This is achieved by
formulating and solving two independent equations, respectively from the phase
(or TOF) and thickness aspects, for $d_{\mathrm{an}}$ and $d_{\mathrm{ca}}$.

The first equation is based on the phase shift aspect. From the main resonance
formation condition in Eq. 5, the phase shift $\phi_{\mathrm{e}}$ in the
combined anode-separator-cathode layer, at the resonant frequency of
$f_{\mathrm{r}}\approx 4.17$ MHz, has been determined from time traces in the
previous subsection. By breaking $\phi_{\mathrm{e}}$ down to the individual
layers, and assuming that the wave speeds in the electrodes are known from the
slurry model, we arrive at the first equation:

$\phi_{\mathrm{e}}=-2\pi
f_{\mathrm{r}}(d_{\mathrm{an}}/c_{\mathrm{an}}+d_{\mathrm{ca}}/c_{\mathrm{ca}}+d_{\mathrm{s}}/c_{\mathrm{s}})=-2.78\,\mathrm{rad},$
(9)

where only $d_{\mathrm{an}}$ and $d_{\mathrm{ca}}$ are unknown, and all the
other parameters are given in Table 1.

The second equation is simply constructed from the thickness relationships.
The whole cell was measured to be 7.26 mm thick, while the thicknesses of all
packaging, separator, Cu and Al layers are summed up to be 1.90 mm based on
the measured values in Table 1. Subtracting the latter from the former, we
obtain the total thickness of the 48 layers of anode and cathode, written as:

$48(d_{\mathrm{an}}+d_{\mathrm{ca}})=7.26-1.90=5.36\,\mathrm{mm}.$ (10)

Solving these two equations gives the average thicknesses of
$d_{\mathrm{an}}=$69.18\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ and
$d_{\mathrm{ca}}=$42.49\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for a single
anode and cathode layer respectively. As a way of verification, we calculated
the nominal positive/negative electrode capacity ratio of the cell based on
the thicknesses estimated here and those destructively measured 30, under an
identical assumption that the anode and cathode layers have the same
proportion of active materials (with theoretical capacities of
$q_{\mathrm{an}}=372\,\mathrm{mAh/g}$ or $710\,\mathrm{mAh/cm^{3}}$ and
$q_{\mathrm{ca}}=274\,\mathrm{mAh/g}$ or $1143\,\mathrm{mAh/cm^{3}}$ 49). The
results are 1.01 calculated here versus 0.84 measured destructively, with the
former falling well into the typical design range of 1.0-1.2 50. Therefore, we
suspect that our estimations could be more accurate than destructive
measurements (though note that the estimation is sensitive to the accuracy of
wave speeds in electrodes), especially considering that the electrodes, which
are compressible materials, could expand considerably and unevenly when the
battery is dismantled.

###  4.3 Constructing image of internal layers

Now we utilise the layer-resolved characteristic of the resonant signal to
construct an image of the cell’s internal layers. For this purpose, we
performed a line scan (B-scan) of the Kokam 7.5 Ah cell using the immersion
setup, which allows the front surface of the cell to be captured
(alternatively, contact tests with a delay line can achieve the same purpose).
We used the 5 MHz immersion probe, which was placed 28.5 mm away from the cell
and scanned over a 40 mm line in the middle of the cell, with a step size of
0.125 mm. Local spot inspection (A-scan) signals were recorded at each
scanning step, delivering a sequence of 321 individual signals along the line.
All the results are plotted as a B-scan map in Fig. 5a, where each A-scan
signal (e.g. 5b) was plotted as a vertical line, with its varying amplitude
indicated by the changing colours. Thus, Fig. 5a illustrates the arrival time
variations across different scanned locations due to the spatially-varied
layer profile of the cell.

Figure 5: Constructing the image of battery internal layers using experimental
signals. a shows a B-scan of ultrasonic signals over a range of 40 mm on the
Kokam 7.5 Ah cell; the signals were acquired with the immersion setup shown in
Fig. 1e using a 5 MHz probe; the probe has a distance of 28.5 mm to the cell
and each signal is the second wave packet of the originally acquired signal. b
demonstrates the signal at point X as annotated in a. The colours in a
represent the amplitudes in b. Figure c plots the respective instantaneous
phase at point X; the black point corresponds to the value representing the
front surface of the cell; the green points have a phase difference of
$-\pi/2$ to the black point and represent the metal layers within the cell. d
displays a stack of instantaneous phases for the scanned positions in a; the
cell front surface is identified by a black line and the internal metal layers
by green lines. e illustrates the reconstructed layer profile of the cell.

Fig. 5b demonstrates well-formed resonance, with gradually decaying amplitude.
To use the time trace for layer reconstruction, the front surface and internal
metal layers of the cell are located from each testing position. This is
achieved through the analytical signal method, as outlined by Smith et al. 51:
if performed the Hilbert transform, the experimental time-domain signal is
transformed into a complex analytic signal, whose instantaneous amplitude,
phase and frequency of the analytical signal all have direct correspondence to
the locations of the reflection interfaces. The instantaneous phase curve of
the A-scan signal in Fig. 5b, given in Fig. 5c, is used below as an example,
but in general, the instantaneous amplitudes and frequencies can also be used
separately or collectively 51.

Following Smith et al51, the front surface of the cell should be located at
the peak of instantaneous amplitude, shown as the top solid point in Fig. 5c,
while the internal reflecting metal layers correspond to points with a phase
difference of $-\pi/2$ from the front surface, indicated by the subsequent
points in the figure. By doing so, the surface profile of the cell over the
scanning line is obtained as the top black line in Fig. 5d, and the internal
metal layer profiles as the subsequent green lines. Furthermore, the different
reflection amplitudes shown in Fig. 5b (also highlighted in Fig. 1c) allow the
Al and Cu layers to be differentiated.

With the metal layers located, the next step is to fill the anode, cathode and
separator layers in between. The separator thickness is known (Table 1), while
the electrode layers are based on the average thicknesses estimated from the
central resonant frequency in the preceding subsection. Note that due to
localised curvature (e.g. waves in non-normal directions travel longer
distances) and uneven thicknesses of the layers, the peak-to-peak gaps in the
time-domain signal may deviate slightly from the resonance estimation. In that
case, it is assumed that the ratio between the anode and cathode remains the
same as estimated in the previous subsection, and both thicknesses are scaled
linearly to fit the distance. The end result of this image construction is
plotted in Fig. 5e, where the individual internal layers of the cell are
successfully identified and tracked. For clarity, it only showcases several
shallow metal layers; however, the signals can reliably deliver structural
information of the first 10-15 peaks from one side (further peaks are limited
by lower amplitudes, but inspections can be performed on both sides).

###  4.4 SOC monitoring with layer resolution

The electrochemical reactions during charging and discharging of a battery
modify the key physical properties of the electrode layers, including elastic
constants, density and thickness, which affect, and thus could be detected by,
the ultrasonic resonance, potentially with single-layer resolution.

To investigate this, we performed 5 full charge-discharge cycles on the Kokam
7.5 Ah cell using a Biologic BCS-815 battery cycler. The cell was charged and
discharged with a constant current at a rate of 1C, with upper and lower
voltage limits of 4.2 and 2.7 V. A 10 minute rest period was applied after
each charge or discharge process, with the cell housed in a Binder thermal
chamber at 25 °C throughout. The cell SOC was calculated via coulomb counting
(sample interval of 2 ms), normalising the charge passed by the maximum
capacity obtained during the experiment. Over the whole period, the battery
was monitored using the 5 MHz contact probe (V109-RM, Olympus), which was
pressed against the cell surface by a spring mechanism to maintain consistent
contact. Ultrasonic signals were recorded every 10 seconds, thus the 10-hour
test (2 hours per full cycle) led to 3600 recordings, all plotted together as
a map changing with time in Fig. 6a. The upper half of the map highlights the
resonance, exemplified by the dark blue lines tracking the first few peaks;
while the lower half tracks the back-wall echoes, demonstrated by the orange
line. We emphasise that the latter delivers similar information to the
through-transmission configurations adopted in prior studies 22, 24, 27.

Figure 6: SOC monitoring using ultrasonic signals. a shows a map of ultrasonic
signals monitored every 10 seconds using a 5 MHz contact probe on the Kokam
7.5 Ah cell during the charging and discharging experiment. The upper half is
for reflections by the internal layers while the lower half for the back-wall
reflection. The dark blue (P1 to P8) and orange lines highlight the peaks of
the signals. The time points at X and Y correspond to SOC=0 and SOC=1,
respectively. b displays the respective signals at X and Y. c compares the
cyclic resonant frequency (top) and back-wall signal peak (middle) with the
SOC and voltage curves (bottom). The top figure of d shows the time
differences between neighbouring front-wall peaks versus cycling time, while
the bottom figure displays the respective averages (in blue line) and cyclic
changes (in orange) for all the individual peak pairs.

In Fig. 6a, the greatest contrast of the ultrasonic signals happen between
SOC=0 and 1, e.g. at the moments X and Y in the figure, and their time traces
are compared in Fig. 6b. The signal at SOC=1 has a noticeably shorter
propagation time, as if linearly compressed like a spring, compared to SOC=0.
This is due to increases in overall wave speeds through the electrode layers
with increasing SOC (Table 1).

As a result, the central resonant frequency of the time traces, which
implicitly accounts for all the analysed peaks through Fourier transform, has
similar sensitivity as the back-wall echo. This is illustrated in Fig. 6c,
where both the resonant frequency and the TOF of the back-wall reflection
display good correlation with coulomb counting and, in particular, voltage
estimations of SOC. Furthermore, the resonant frequencies at SOC=0 and SOC=1
predicted by our physical model, marked by the dashed lines in Fig 6c, match
almost perfectly with experimental results, further validating our analytical
model.

More excitingly, the resonance method allows monitoring of layer-by-layer
property changes during cycling. Since the resonance peaks track the positions
of metal layers, the gap between two neighbouring peaks gives information
about a single resonant element, i.e. one anode and one cathode layer. The
cyclic behaviours of the first seven elements are plotted in Fig. 6d, and
immediately noticeable are two prominent features. Firstly, the curves are
slightly shifted vertically from each other (plotted below in blue line as
’Average difference’). This indicates different travelling times through the
elements even without charging, likely caused by small variations of layer
thicknesses. Secondly, the cyclic changes (plotted in orange below) varies
considerably across layers. For example, the TOF changes for P8-P7 are almost
three times as much as P2-P1, and they appear to grow more pronounced deeper
into the battery. These observations strongly indicate that the state changes
during cycling are happening heterogeneously across layers. Indeed, similar
results were delivered by electro-thermal models 52, 53, which also revealed
that the variations were largely due to differences in resistance for each
layer, and the positive feedback with current and temperature. Even though the
fact that such heterogeneity exists in the battery bulk is experimentally
observable, e.g. by the different SOCs estimated by coulomb counting (which
gives the bulk SOC) and voltage (influenced by specific parts of the cell
which are at different SOCs than the bulk), there is currently no way to
estimate distributed SOC levels from normal electrochemical data. Therefore,
we believe that the layer-resolved SOC monitoring capability of the resonance
method (which will require dedicated studies to fully develop) offers a much-
need and powerful addition to the existing battery management tools.

##  5 Discussions and conclusion

In this paper, we have established an advanced methodology for characterising
the layer properties and states of LIBs from ultrasonic resonance. Robust
experimental acquisitions of resonant signals were achieved, and a
comprehensive theoretical model was established by analysing the wave physics
in individual layers of an LIB (a notable contribution is treating electrodes
as dense slurries) and the interference of their reflections to form the
resonance. We demonstrated high levels of accuracy of the developed approach
in comparison to experimental results (e.g. predicted resonant frequency error
$\sim$1% at different SOCs), and showcased its efficacy in quantitatively
characterising the number of electrode layers in the cell, the average
thickness of the anode and cathode layers, the images of internal structure
and SOC resolved up to layer level during electrochemical cycling.

The proofs of principle established in this paper open the door for a range of
exciting possibilities for further, more in-depth research, as well as real-
world applications. Specifically, the central resonant frequency enables
accurate and reliable inversion of battery SOC, and the variation of TOF
between resonant peaks in the depth direction may help understand the in-
operando temperature gradients or heterogeneity in electrochemical reactions.
The sensitivity of acoustic behaviours to material changes may facilitate the
monitoring of battery SOH by quantitative full-waveform studies. For example,
lithium plating could cause the resonance to shift to a higher frequency,
while degradation-associated cracks, dislocations and porosity changes of the
particles 9 may result in higher amplitude attenuation of the resonant peaks.
Volume expansion in electrode layers could also be determined from the
resonance, which may be of particular interest for next-generation, silicon-
based anode materials. Many of these investigations could be further
transported to the characterisation of cylindrical batteries which occupy a
large share of the market.

In addition, opportunities could emerge in developing new experimental
techniques for resonant signal acquisition. The bulky and intrusive probes
used in this paper could be replaced with thin and flexible micro-
electromechanical ones (already used on e.g. mobile phones), for permanently-
installed monitoring, or as an addition to the battery management system. Non-
contact, air-coupled probes could be used for in-production (e.g. for
electrode porosity) or in-service monitoring, to avoid using water or gel as
couplant. These developments can be explored in parallel with the research
topics outlined above, and can easily incorporate the latest theoretical
progress. Moreover, a growing set of experimental data will benefit both the
experimental and theoretical aspects of the work, since statistical analyses
and machine learning tools can be employed to enhance the accuracy of material
properties and aid physics-based interpretations, thus constantly improving
the reliability of the methodology.

## Acknowledgements

B.L. gratefully acknowledges the generous support from the Imperial College
Research Fellowship Scheme, M.H. the Non-Destructive Evaluation Group at
Imperial College, and N.K. Innovate UK through the WIZer project (Grant No.
104427). We would like to thank Prof Robert Smith of Bristol University for
helpful discussions on the resonance method, and Prof Peter Cawley of Imperial
College for reviewing the manuscript.

## Appendices

##  Appendix A Observation of ultrasonic resonance from two more LIBs

We conducted ultrasonic contact tests on two more LIB pouch cells, including a
210 mAh cell (PL-651628-2C, AA Portable Power Corp.) and a Kokam 5 Ah cell
(SLPB11543140H5). The acquired time- and frequency-domain results are shown in
Fig. A.1a and b for the former cell, and those for the latter are provided in
c and d. These results exhibit strong resonance similar to those in Fig. 1 of
the main text.

Figure A.1: Observation of ultrasonic resonance from two more LIBs. a shows
the time-domain signals of 2.25 MHz and 5 MHz contact probes from a 210 mAh
battery cell (PL-651628-2C, AA Portable Power Corp.), while b displays the
respective amplitude spectra. Similarly, c illustrates the time-domain signals
of a 5 MHz contact probe from a 5 Ah cell (SLPB11543140H5, Kokam) and d
presents the respective amplitude spectrum.

##  Appendix B Modelling separators using the Biot model

The separators in a LIB are usually porous solids filled with liquid
electrolytes 37, acting as an ionically-conductive physical barrier between
two electrodes. The Biot model 35, 36 is thus needed to describe the
propagation of ultrasonic waves within such a fluid-saturated porous
separator. This model predicts the existence of three waves, one shear and two
longitudinal. The shear wave is not involved since we consider longitudinal
waves only while the slow longitudinal wave barely contributes to the detected
ultrasonic signal due to its diffuse nature and excessive attenuation, so our
concern is with the fast longitudinal wave only. This fast wave can be treated
as independent of frequency in the frequency range utilised in this work,
which is lower than the transition frequency
$f_{\mathrm{t}}=\pi\mu/(4\rho_{\mathrm{L}}d^{2})=10\,\mathrm{MHz}$ 35. This
transition frequency estimation was based on the material parameters of the
Kokam 7.5 Ah battery, for which we used a large pore diameter 37
$d=0.5\,$\text{\,}\mathrm{\SIUnitSymbolMicro m}$$ for the porous polypropylene
solid for a cautious estimation, while used the viscosity 54
$\mu=4.2\,\mathrm{mPa\cdot s}$ and density 23
$\rho_{\mathrm{L}}=1270\,\mathrm{kg/m^{3}}$ for the liquid $\mathrm{LiPF_{6}}$
electrolyte 30. In this low-frequency range, the wave speed is given by 35
$c=\sqrt{H/\rho}$. The effective density of the medium is related to the
porosity $\nu$ and the densities $\rho_{\mathrm{S}}$ and $\rho_{\mathrm{L}}$
of the solid and liquid by
$\rho=(1-\nu)\rho_{\mathrm{S}}+\nu\rho_{\mathrm{L}}$. The subscript S here
refers to the homogeneous solid without pores, while the subscript P below
denotes the porous solid. $H=A+2N+R+2Q$ is the effective longitudinal modulus
of the medium, with $A$ and $N$ being the Lamé constants, $R$ the pressure
required for forcing a certain volume of the liquid into the medium whilst the
total volume remains constant, and $Q$ the coupling of volume change between
the solid and liquid. These four parameters are given by 55, 56:

$\small N=G_{\mathrm{P}},\;A=K_{\mathrm{P}}-2N/3+K_{\mathrm{L}}(1-\nu-
K_{\mathrm{P}}/K_{\mathrm{S}})^{2}/\nu_{\mathrm{eff}},\;Q=\nu
K_{\mathrm{L}}(1-\nu-
K_{\mathrm{P}}/K_{\mathrm{S}})/\nu_{\mathrm{eff}},\;R=\nu^{2}K_{\mathrm{L}}/\nu_{\mathrm{eff}},$
(B.1)

where $K_{\mathrm{S}}$ and $K_{\mathrm{L}}$ are the bulk moduli of the solid
and the liquid respectively.
$\nu_{\mathrm{eff}}=\nu+K_{\mathrm{L}}/K_{\mathrm{S}}(1-\nu-
K_{\mathrm{P}}/K_{\mathrm{S}})$ is an effective porosity. $K_{\mathrm{P}}$ and
$G_{\mathrm{P}}$ are the bulk and shear moduli of the porous solid, which are
related to the porosity and the properties of the homogeneous solid; we
determine these two parameters using the Mori-Tanaka mean field theory 57:

$\small
K_{\mathrm{P}}=4G_{\mathrm{S}}K_{\mathrm{S}}(1-\nu)/(4G_{\mathrm{S}}+3\nu
K_{\mathrm{S}}),\;G_{\mathrm{P}}=G_{\mathrm{S}}(8G_{\mathrm{S}}+9K_{\mathrm{S}})(1-\nu)/(8G_{\mathrm{S}}+9K_{\mathrm{S}}+6(2G_{\mathrm{S}}+K_{\mathrm{S}})\nu).$
(B.2)

By using the material properties in Table B.1, we calculated the wave speed
and density of the separator for the Kokam 7.5 Ah battery cell and the results
are given in Table 1.

Table B.1: Parameters for calculating the material properties of liquid-filled porous separator using the Biot model 35. The solid phase S is polypropylene and the liquid phase L is $\mathrm{LiPF_{6}}$ 30. $\nu$ |  $K_{\mathrm{S}}$ (GPa) |  $G_{\mathrm{S}}$ (GPa) | $\rho_{\mathrm{S}}\,\mathrm{(kg/m^{3})}$ |  $K_{\mathrm{L}}$ (GPa) | $\rho_{\mathrm{L}}\,\mathrm{(kg/m^{3})}$
---|---|---|---|---|---
0.508 30 | 2.2 58 | 0.3 58 | 850 58 | 1 23 | 1270 23

##  Appendix C Estimating electrode properties with the slurry model

We have proposed in section 3.1 that the battery electrodes should be modelled
as dense slurries. The acoustic model of a slurry is well-established, and the
wave speed can be calculated from 46:

$c=\sqrt{\frac{K}{\rho}},$ (C.1)

with the bulk modulus $K$ and the density $\rho$ of the slurry given by:

$\frac{1}{K}=\frac{\nu}{K_{\mathrm{S}}}+\frac{1-\nu}{K_{\mathrm{L}}},\;\rho=\nu\rho_{\mathrm{S}}+(1-\nu)\rho_{\mathrm{L}},$
(C.2)

where the subscripts of S and L respectively refer to the solid and liquid
phases of the slurry, and $\nu$ the volume fraction of the solid phase, which
is an important factor and needs to be pre-determined for the calculations.
For the Kokam 7.5 Ah cell, we used the volume fractions of 0.773 and 0.811 for
anode and cathode, which were obtained from the porosity data as determined by
mercury porosimetry 30. By substituting the material properties listed in
Table C.1 into Eqn. C.2, we obtain the bulk moduli and densities of the
electrodes as shown in the same table. Following this, the respective wave
speeds are calculated by Eqn. C.1 and given in Table 1.

Table C.1: Electrode properties of Kokam 7.5 Ah pouch cell: original solid and
liquid phases properties and predicted effective properties using a slurry
model 46. Units: bulk modulus $K$ (GPa), density $\rho\,\mathrm{(kg/m^{3})}$,
solid phase volume fraction $\nu$ (dimensionless).

Layer | Solid 24 |  $K_{\mathrm{S}}$ 24 |  $\rho_{\mathrm{S}}$ 24 | $\nu$ | Liquid 30 |  $K_{\mathrm{L}}$ 23 |  $\rho_{\mathrm{L}}$ 23 | $K$ | $\rho$
---|---|---|---|---|---|---|---|---|---
Anode (SOC=0) | $\mathrm{C}_{6}$ | 28.8 | 2260 |  $0.773^{\ddagger}$ (0.671 30) | $\mathrm{LiPF_{6}}$ | 1 | 1270 | 3.43 | 1909
Cathode (SOC=0) | $\mathrm{Li_{0.95}CoO_{2}}^{\dagger}$ | 88.9 | 4860 |  $0.811^{\ddagger}$ (0.704 30) | 4.98 | 4172
Anode (SOC=1) | $\mathrm{Li_{0.85}C_{6}}$ | 67.8 | 2210 |  $0.773^{\ddagger}$ (0.671 30) | 4.15 | 1994
Cathode (SOC=1) | $\mathrm{Li_{0.5}CoO_{2}}^{\dagger}$ | 82.4 | 4460 |  $0.811^{\ddagger}$ (0.704 30) | 4.96 | 3848

  * †

The cell has an as-built cathode of $\mathrm{Li(Ni_{0.4}Co_{0.6})O_{2}}$ 30,
31 containing nickel and cobalt; here we treat nickel and cobalt as a single
composition of cobalt because they have similar properties.

  * ‡

The volume fractions were obtained by scaling the destructively measured
values in parentheses in order for the total thickness of the cell to match
the actual value; the scaling was performed under the condition of volume
conservation for the solid phase.

##  Appendix D Full-waveform modelling using the transfer matrix method

The general idea of the full-waveform modelling is shown in Fig. 3e. Instead
of the individual time-domain reflections (which contain a broad frequency
bandwidth) from different layers and matching up their phases, the analysis
here is carried out in the frequency domain for individual frequencies and
then transformed back to the time domain. When the multilayered medium is
subject to a monochromatic longitudinal wave
$u_{0}^{+}(x)=a_{0}^{+}e^{-ik_{0}x}$, two wave components would arise in an
arbitrary layer $n$, propagating in the forward and backward directions. The
total displacement at $x$ in the layer is given by the summation of the two
components as
$u_{n}(x)=a_{n}^{+}e^{-ik_{n}(x-x_{n-1})}+a_{n}^{-}e^{ik_{n}(x-x_{n-1})}$.
$x_{n-1}$ and $x_{n}$ are the coordinates of the two interfaces. $a$ carries
both the amplitude and phase information of the wave. $k_{n}=\omega/c_{n}$ is
the wave number; $\omega$ is the angular frequency of the incident wave and
$c_{n}$ is the longitudinal wave speed of the layer. For simplicity, the time
harmonic $e^{-i\omega t}$ is neglected in the displacement expressions.

To calculate the actual wave $u_{n}(x)$, we need to determine the wave
amplitudes $a_{n}^{+}$ and $a_{n}^{-}$. The way to achieve this is to use the
fundamental fact that stress and velocity must be continuous across any
boundary to satisfy the equilibrium and compatibility conditions. In any layer
$n$, the stress and velocity are related to the displacement by:

$\displaystyle\sigma_{n}(x)$ $\displaystyle=M_{n}\frac{du_{n}(x)}{dx}=-i\omega
Z_{n}[a_{n}^{+}e^{-ik_{n}(x-x_{n-1})}-a_{n}^{-}e^{ik_{n}(x-x_{n-1})}],$ (D.1)
$\displaystyle v_{n}(x)$
$\displaystyle=\frac{du_{n}(x)}{dt}=-i\omega[a_{n}^{+}e^{-ik_{n}(x-x_{n-1})}+a_{n}^{-}e^{ik_{n}(x-x_{n-1})}].$

where $M_{n}$ is the longitudinal modulus of the layer and
$Z_{n}=M_{n}/c_{n}=\rho_{n}c_{n}$ is the acoustic impedance. For fluid, stress
corresponds to pressure and longitudinal modulus to bulk modulus. Therefore,
the continuity of stress and velocity on each boundary delivers two equations
for the wave amplitudes. For instance, the equations are as follows for
boundaries 0 (bordering layers $n=0$ and $n=1$) and 1 (bordering $n=1$ and
$n=2$):

$\displaystyle\sigma_{0}(x_{0})=\sigma_{1}(x_{0}):$ $\displaystyle\quad
Z_{0}a_{0}^{+}-Z_{0}a_{0}^{-}-Z_{1}a_{1}^{+}+Z_{1}a_{1}^{-}=0$ (D.2)
$\displaystyle v_{0}(x_{0})=v_{1}(x_{0}):$ $\displaystyle\quad
a_{0}^{+}+a_{0}^{-}-a_{1}^{+}-a_{1}^{-}=0$
$\displaystyle\sigma_{1}(x_{1})=\sigma_{2}(x_{1}):$ $\displaystyle\quad
Z_{1}e^{-ik_{1}d_{1}}a_{1}^{+}-Z_{1}e^{ik_{1}d_{1}}a_{1}^{-}-Z_{2}a_{2}^{+}+Z_{2}a_{2}^{-}=0$
$\displaystyle v_{1}(x_{1})=v_{2}(x_{1}):$ $\displaystyle\quad
e^{-ik_{1}d_{1}}a_{1}^{+}+e^{ik_{1}d_{1}}a_{1}^{-}-a_{2}^{+}-a_{2}^{-}=0$

Altogether, a system of $2(N+1)$ equations can be constructed by using the
continuity conditions for all $N+1$ boundaries. Practically, the incident
amplitude $a_{0}^{+}$ is given beforehand, and $a_{N+1}^{-}$ is zero because
no wave comes into the layers from the back face. As a result, the number of
equations is the same as the number of unknowns, which are $a_{0}^{-}$,
$a_{n}^{+}$ and $a_{n}^{-}$ ($n=1,2,\ldots,N$), and $a_{N+1}^{+}$; therefore,
solving the equation system gives the solutions for all unknown amplitudes.

Since we use the pulse-echo setup that sends and receives signals with the
same probe, we are only interested in the solution for $a_{0}^{-}$ and, in
particular, its ratio to the incident amplitude as characterised by the total
reflection coefficient $R=a_{0}^{-}/a_{0}^{+}$. For this reason, we have
formulated a very computationally-efficient transfer matrix scheme that solves
only for $R$ recursively. To achieve this, we write the stress and velocity at
the two interfaces $x_{n-1}$ and $x_{n}$ of the layer in matrix form (based on
Eq. D.1):

$\begin{bmatrix}\sigma(x_{n-1})\\\
v(x_{n-1})\end{bmatrix}_{n}=-i\omega\begin{bmatrix}-Z_{n}&Z_{n}\\\
1&1\end{bmatrix}\begin{bmatrix}a_{n}^{-}\\\ a_{n}^{+}\end{bmatrix},$ (D.3)

and

$\begin{bmatrix}\sigma(x_{n})\\\
v(x_{n})\end{bmatrix}_{n}=-i\omega\begin{bmatrix}-Z_{n}e^{ik_{n}d_{n}}&Z_{n}e^{-ik_{n}d_{n}}\\\
e^{ik_{n}d_{n}}&e^{-ik_{n}d_{n}}\end{bmatrix}\begin{bmatrix}a_{n}^{-}\\\
a_{n}^{+}\end{bmatrix},$ (D.4)

where $d_{n}$ is the layer thickness. Solving Eq. D.4 for
$[a_{n}^{-},a_{n}^{+}]^{\mathrm{T}}$ and substituting the solution into Eq.
D.3 would lead to the transfer matrix equation:

$\begin{bmatrix}\sigma(x_{n-1})\\\
v(x_{n-1})\end{bmatrix}_{n}=\begin{bmatrix}\cos(k_{n}d_{n})&iZ_{n}\sin(k_{n}d_{n})\\\
i\sin(k_{n}d_{n})/Z_{n}&\cos(k_{n}d_{n})\end{bmatrix}\begin{bmatrix}\sigma(x_{n})\\\
v(x_{n})\end{bmatrix}_{n},$ (D.5)

which relates (transfers) the stress and velocity of interface $x_{n}$ to
those of $x_{n-1}$.

In the bounding media $N+1$, owing to the absence of backward propagating wave
($a_{N+1}^{-}=0$), it follows from Eq. D.3 that
$\sigma_{N+1}(x_{N})/v_{N+1}(x_{N})=Z_{N+1}$ at the interface $N$. Considering
the continuity of stress and velocity at the interface $N$, we have
$\sigma_{N}(x_{N})/v_{N}(x_{N})=\sigma_{N+1}(x_{N})/v_{N+1}(x_{N})=Z_{N+1}$.
When the stress and velocity transfer to the interface $N-1$, an effective
impedance for the combination of layers $N$ and $N+1$ would be obtained from
Eq. D.5, given by:

$Z_{N}^{\mathrm{eff}}=\frac{\sigma_{N}(x_{N-1})}{v_{N}(x_{N-1})}=Z_{N}\frac{Z_{N+1}\cos(k_{N}d_{N})+iZ_{N}\sin(k_{N}d_{N})}{iZ_{N+1}\sin(k_{N}d_{N})+Z_{N}\cos(k_{N}d_{N})}.$
(D.6)

This procedure can be performed further towards shallower layers, and the
effective impedance of layer $n$ combined with all its deeper layers would be:

$Z_{n}^{\mathrm{eff}}=Z_{n}\frac{Z_{n+1}^{\mathrm{eff}}\cos(k_{n}d_{n})+iZ_{n}\sin(k_{n}d_{n})}{iZ_{n+1}^{\mathrm{eff}}\sin(k_{n}d_{n})+Z_{n}\cos(k_{n}d_{n})}.$
(D.7)

By using this recursive relation, we would eventually obtain the effective
impedance $Z_{1}^{\mathrm{eff}}=\sigma_{1}(x_{0})/v_{1}(x_{0})$ at the
interface 0. This leads to
$\sigma_{0}(x_{0})/v_{0}(x_{0})=Z_{1}^{\mathrm{eff}}$ as a result of the
continuity of stress and velocity. Substituting this relation into Eq. D.4 for
the bounding layer 0, we would obtain the total reflection coefficient as:

$R=\frac{a_{0}^{-}}{a_{0}^{+}}=\frac{Z_{0}-Z_{1}^{\mathrm{eff}}}{Z_{0}+Z_{1}^{\mathrm{eff}}},$
(D.8)

where $Z_{0}$ is the impedance of the bounding layer 0.

Therefore, upon applying an incident wave $u_{0}^{+}(x)=a_{0}^{+}e^{-ik_{0}x}$
on the multilayered medium, the reflected wave can be obtained as
$u_{0}^{-}(x)=a_{0}^{-}e^{ik_{0}x}$ with a complex amplitude
$a_{0}^{-}=Ra_{0}^{+}$. The reflection coefficient $R$ is given by Eq. D.8 and
the effective impedance therein is calculated using the recursive Eq. D.7. We
emphasise that the resulting solution takes into account all possible
reflections and reverberations since the physical continuity of stress and
velocity is considered at every single interface.

##  Appendix E Identifying the number of electrode layers using contact setup

Figure E.1: Front- and back-wall reflections from the Kokam 7.5 Ah cell using
contact setup. a illustrates the contact setup with a 20-mm thick perspex disc
placed between the ultrasonic probe and the cell to capture the front-wall
echo of the cell. b shows the time-domain signals acquired with a 5MHz contact
probe (V109-RM, Olympus), with the back-wall echo significantly amplified. c
and d present the respective amplitude and unwrapped phase spectra of the
front- and back-wall echoes.

In section 4.1, estimating the number of electrode layers was based on the
ultrasonic resonance signals acquired with the immersion setup. Here we
perform the same estimation for the Kokam 7.5 Ah cell using the contact setup
shown in Fig. E.1a. In the setup, a 20-mm thick perspex disc was used between
the probe and the cell in order for the front-wall echo of the cell to be
delayed and fully captured by the probe. The front- and back-wall reflections
acquired with a 5 MHz probe (V109-RM, Olympus) are provided in Fig. E.1b, with
the respective amplitude and phase spectra in c and d. At the main resonant
frequency, the phase difference between the two echos is $\phi=-272.63$ rad.
After excluding the phase delays caused by the two surface packaging layers of
the cell and the phase reversal at the front-wall, the phase change of the
wave travelling a round trip through the whole cell is
$\phi_{\mathrm{total}}=-273.88$ rad. The phase change in a resonant element of
the cell has been calculated in the main text, which is
$\phi_{\mathrm{elem}}=-5.70$ rad. So, the number $N$ of resonant elements
(thus the number of electrode layer pairs) of the cell is given by
$N=\phi_{\mathrm{total}}/\phi_{\mathrm{elem}}=48.05$. This number is
practically the same as that (47.51) obtained in the main text and is very
close to the actual number of 48 30, 31.

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