Characterization of mechanical properties of thin-film Li-ion battery electrodes from laser excitation and measurements of zero group velocity resonances

Lithium-ion (Li-ion) batteries are employed in many applications due to their high energy density, power, cycling performance, and long lifespan.¹ Despite the maturity of the technology, manufacturing cost and performance are still challenges faced by the battery industry as a whole. Improved quality control and nondestructive evaluation methods are needed to overcome these challenges.² 

As shown in Fig. 1, a single-sided battery electrode film is a bilayer structure consisting of a current collector layer and an electrode coating layer. The current collector layer is commonly made of thin aluminum for cathodes or copper for anodes. On top of the current collector, the coating layer is composed of relatively large active material particles, carbon additive and polymeric binder, and interparticle pore spaces. The structure of this mixture of materials can vary significantly due to manufacturing differences, causing heterogeneity on the micrometer as well as the millimeter scale.^3,4 Understanding, measuring, and controlling heterogeneity in battery electrodes are important to improve the performance of batteries, particularly with regard to life-cycle performance and safety.^5–7 

Mass loading, film thickness, and porosity are traditional metrics employed by manufacturers. These properties can be estimated from local mechanical properties such as Young’s modulus, density, and thickness of the coating layer. Local mechanical properties can be used as good indicators of battery film heterogeneity⁸ and for quality control. Because materials and processes change periodically during manufacturing, these properties can also be indicators of changing fabrication conditions. They are important properties to measure before the battery films are assembled into a full cell.

Acoustic methods have extensively been used to measure the mechanical properties of materials in plate structures.^8–17 In the past work, we used the acoustic resonances of a circularly clamped area of a battery electrode film, excited by a pulsed laser, to estimate Young’s modulus.⁸ Due to uneven clamping boundary conditions, this method had systematic variation that made it less-than-ideal to monitor changes in electrode properties.

Guided wave propagation has often been employed to investigate the mechanical properties or defects in one-layer or multilayer structures through Lamb waves.^9–11 For some Lamb wave modes, the group velocity is nearly zero at finite wavelengths.¹⁸ These modes are known as zero group velocity (ZGV) modes and exhibit a spatially local resonance at the ZGV frequencies. The ZGV frequencies are determined by the geometric arrangement of the materials in the structure and the mechanical properties of the materials. Due to the spatially local resonances, ZGV modes are ideal for estimating material properties that vary spatially on scales longer than the ZGV wavelength. ZGV modes can be remarkably well generated by a pulsed laser source and detected using a laser interferometer.¹² These modes have been measured in homogeneous plates to evaluate locally the thickness or Poisson’s ratio.^12,15 The deposited energy of the laser pulse is absorbed and transformed into mechanical waves through the thermoelastic effect.¹⁹ The amplitude of measured ZGV resonances depends on the total deposited energy, the shape of the pulse laser beam,²⁰ and the thermoelasticity of the interrogated material. ZGV modes have also been measured in bilayer plates¹³ and bilayer thin membranes.²¹ Because of the increased complexity of the guided waves, it is more difficult to exploit ZGV resonances in bilayer structures to evaluate the material properties.

In this work, we characterize the mechanical properties of Li-ion battery electrode films using ZGV resonances, which are excited by an ultrafast infrared laser and detected by a homodyne interferometer. The sensitivity of the ZGV mode response to changes of the mechanical properties of a battery coating is determined. These results indicate that this method can be used to distinguish among electrode films with different mechanical properties. Additionally, the mechanical effect of the calendering process is investigated. Young’s modulus of the battery coating increases significantly after calendering. While this work has been primarily motivated by Li-ion batteries, this work can be generalized to measure the ZGV Lamb modes of a thin “compliant” layer on a thin, rigid layer in a bilayer structure.

A single-sided battery film is a solid bilayer structure consisting of a metal current collector layer and a battery coating layer as shown in Fig. 2. The current collector materials are generally uniform and consist of a single metal layer, for example, a 20-$μm$-thick aluminum sheet for cathodes or a 10-$μm$-thick copper sheet for anodes. In a high-quality battery film, the current collector and battery coating are assumed to be rigidly held together.

The area of interest, interrogated by the mechanical waves excited by the laser pulse, is about 1 mm$2$⁠. Although the coating in the area of interest has a varied and complex microstructure, we assume that the area of interest is, mechanically, approximately homogeneous and isotropic. This assumption allows us to evaluate different regions because it is known that the battery coating is heterogeneous, and the material properties change from point to point.³ The layers above and below the film are considered as vacuum because the difference in acoustic impedance between air and the materials is very high. In Fig. 2, $ρi,Ei,νi,andhi$ represent the density, Young’s modulus, Poisson’s ratio, and thickness, respectively, of each layer denoted by $i$⁠. Layer 1 is the current collector, and layer 2 is the battery coating.

When the laser pulse illuminates the surface of the small target region of the current collector, the majority of the energy will be scattered and reflected, but deposited energy will be absorbed and produce mechanical waves because the thermalelastic effect causes stress difference along the surface. Such an impulse excitation induces a broadband set of Lamb wave modes.^12,22 These waves propagate through the materials and are reflected at the material interface and the free surfaces as illustrated in Fig. 2. L and S represent longitudinal and shear waves, respectively, and positive and negative signs denote the out of plane propagation direction. The velocities of longitudinal wave and shear wave are shown, respectively,²³ as

From these equations, it is clearly seen that the acoustic bulk wave velocities increase when $E$ increases and the velocities decrease when $ρ$ increases.

Lamb waves are dispersive, which means their phase velocity and group velocity depend on the angular frequency (⁠$ω=2πf$⁠, where $f$ is the frequency) as well as the elastic constants, densities, Poisson’s ratios, and thicknesses of the constituent materials. When the parameters $E$⁠, $ρ$⁠, $ν$⁠, and $h$ are known for each layer, a set of dispersion curves (wavenumber, $k$⁠, vs frequency, $f$⁠) can be calculated for the Lamb waves in the bilayer structure. Unlike Lamb waves in a single layer,^24,25 Lamb waves in a multilayered structure do not have analytical closed forms, which makes Lamb wave analysis for bilayers more challenging. Therefore, numerically solving for the dispersion curves is a more proper approach.

Numerical solutions for the dispersion curves have been very well-studied. One way is to use a widely acknowledged method: the semianalytical finite-element (SAFE)^26,27 method. When the SAFE method is applied to our system, it assumes that the guided waves propagate harmonically along the $x$ direction as shown in Fig. 2, meaning the displacement, stress, and strain are the same along the $y$ direction (orthogonal to $x$ and $z$ directions). This reduces the simulation complexity from 3-dimensions to 2-dimensions. The cross section is then discretized to approximate the solutions of the wave equations numerically^27,28 using the finite element method. Simulations show that the ZGV mode normally lies in the 3rd or the 4th mode of the dispersion curves.

The software GUIGUW 2.2 developed by Bocchini et al.²⁸ employing the SAFE method was used to simulate the dispersion curves for different battery films with varying layer thickness and mechanical properties. While real battery electrodes are generally composed of many particles with complex microstructural arrangements, in these simulations it is assumed that layers have locally homogeneous, isotropic mechanical properties.

For our simulations, the geometry in Fig. 2 is used and no material damping is taken into account. The simulation parameters are shown in Table I. The current collector (aluminum) listed in that table is used for all the simulations in this section. The density and Young’s modulus values of coating 1 and coating 2 are the measured and estimated values of battery electrode films from Dallon et al.⁸ Poisson’s ratio of coatings 1 and 2 is assumed to be the common value 0.3.²⁹ $Vl$ and $Vs$ are the velocities of longitudinal wave and shear wave calculated from Eqs. (1) and (2), respectively. The thicknesses of the coatings are chosen to be the same for comparison purposes. From the simulation parameters, two cases for different battery films are simulated. Case C1 is the battery film consisting of coating 1. Case C2 is the battery film consisting of coating 2.

Two reference cases of a single layer structure are also simulated for cases C1 and C2. Reference R1 is a single layer of coating 1, and reference R2 is a single layer of coating 2. Both reference cases are set to have the same thickness (⁠$h=46μm$⁠) as the total thicknesses of C1 and C2. Reference cases R1 and R2 are simulated to show how the dispersion curves change when part of the coating changes to the metal current collector.

In general, the density (⁠$ρ$⁠) and the thickness (⁠$h$⁠) of the coating layer and the current collector layer have the same order-of-magnitude. Poisson’s ratios of these two layers are also close. In contrast, the $E$ values for these two layers have more than one order-of-magnitude difference. Additionally, the $E$ values of the coatings themselves also have one order-of-magnitude difference. Cases C1 and C2 can be considered representative of different battery film conditions.

The simulated dispersion curves for cases C1 and C2 are shown in Figs. 3(a) and 3(b), respectively. Only the four lowest Lamb wave modes are shown in Fig. 3. These four modes are denoted as M1, M2, M3, and M4. The dispersion curves for references R1 and R2 are also shown in Figs. 3(a) and 3(b), respectively, for comparison (black dotted lines). The four lowest modes for both reference cases are commonly denoted as A0, S0, A1, and S1. A and S represent asymmetric modes and symmetric modes, respectively. M1–M4 modes exhibit a similar pattern to the pattern of the four lowest modes for their corresponding reference cases. It can be observed that both C1 and C2 have higher temporal frequency $f$ at a given wavenumber $k$ for each mode, respectively, compared to their reference cases. In general, it is suggested that each mode shifts to higher frequency at a given wavenumber (⁠$k$⁠), when part of the material (R1 and R2) changes into another material (C1 and C2) with higher acoustic velocities (⁠$Vl$ and $Vs$⁠). The M1–M4 modes cannot be represented using asymmetric and symmetric modes because there is no longer symmetry in Lamb waves propagation due to the wave interaction between the two different layers in C1 and C2.

In Fig. 3, the arrows denote ZGV points where the group velocity (⁠$vg=dω/dk$⁠)³³ of the waves becomes zero. The ZGV modes exist only where $vg$ changes from negative to positive on the dispersion curve.¹² Negative and positive signs of $vg$ represent the propagation direction of the group velocity of Lamb waves. Because the group velocity is zero, the energy confined in those modes will not propagate away from the source point which is called the ZGV point. Because the modes have small curvature at the ZGV point, they may be less pronounced in the response spectra. The corresponding nonzero wavenumbers (⁠$k$⁠) indicate that these ZGV modes have finite wavelengths given by $λ=2π/k$⁠. At the ZGV point, local resonances at ZGV frequencies occur after proper excitation. Detected ZGV resonances will show up as sharp peaks in the spectra of measured vibrations.¹² Thus, ZGV Lamb modes can be good indicators of the local mechanical properties of locally excited regions. Due to their nontraveling property, the detection point should be close to or coincident with the excitation point, as illustrated in Fig. 2.

As shown in Fig. 3(a), case C1 has only one ZGV mode, hereafter denoted as M4-ZGV. As shown in Fig. 3(b), case C2 has two ZGV modes denoted as M3-ZGV and M4-ZGV. The corresponding ZGV modes in references R1 and R2 are denoted as the S1-ZGV mode. Other higher ZGV modes can also exist in dispersion curves for both cases. Even though several ZGV modes can exist, some ZGV modes are easier to excite than others using laser impulse excitation, which is essentially broadband excitation of all frequencies. In general, the lowest ZGV modes are more likely to be excited and detected.³⁴ Both M4-ZGV mode and S1-ZGV mode are the 4th mode for each case, and both modes exhibit a similar pattern as shown in Figs. 3(a) and 3(b). From the simulations of Tofeldt and Ryden,³⁴ the M4-ZGV mode of a plate with continuous material variation across the thickness has a similar behavior and detectability as the S1-ZGV mode of a single isotropic layer structure. We extend those results to our structures by considering the film structure as an extreme case of their study. In case C1, the lowest ZGV mode appears to be the M4-ZGV mode. However, in case C2, two ZGV modes M3-ZGV and M4-ZGV exist and are closely related in frequency.

The sensitivity of the ZGV mode to variations in material parameters is explored. It is significant that the M3-ZGV and the M4-ZGV modes have a similar sensitivity in all cases. The overall approach is to keep all the mechanical properties constant except for one. The sensitivity of ZGV mode to Young’s modulus, thickness, and density is discussed.

The sensitivity of the ZGV modes to changes in Young’s modulus (⁠$E1$ or $E2$⁠) was calculated. In Fig. 4, the M3-ZGV and the M4-ZGV mode responses for both cases C1 and C2 to changes of Young’s modulus value of the battery coating layer (⁠$E2$⁠) or the current collector layer (⁠$E1$⁠) are shown. The geometry shown in Fig. 2 is used. In Fig. 4(a), $E2$ is simulated between 0.1 GPa and 15 GPa with an interval spacing of 0.5 GPa. In Fig. 4(b), $E1$ is simulated between 1 GPa and 150 GPa with an interval spacing of 5 GPa. It is shown that the M3-ZGV and M4-ZGV modes increase monotonically for both cases when $E1$ or $E2$ increases. The M3-ZGV and M4-ZGV modes converge when $E2$ is very low; they slightly diverge when $E2$ increases. M3-ZGV disappears when $E2$ is higher than 3 GPa and $E1$ equals to 70 GPa for case C1 as shown in Fig. 4(a). The M3-ZGV mode does not exist when $E1$ is in the lower range close to $E2$⁠, but the M3-ZGV mode appears when $E1$ is higher than 120 GPa for case C1 and 15 GPa for case C2. From Fig. 4, it is calculated that the M3-ZGV mode exists when the ratio of $E2/E1$ is greater than 4.2$%$ for case C1 and 2.3$%$ for case C2.

Figure 4 also shows that the M3-ZGV and M4-ZGV modes are mostly determined by $E2$ rather than $E1$ when $E1$ is sufficiently greater than $E2$⁠. When $E2/E1$ is approximately less than $12%$ for case C1 and $6%$ for case C2, changes in $E1$ will not result a significant shift in M3-ZGV and M4-ZGV modes as shown in Fig. 4(b). Since increasing Young’s modulus of coating increases the acoustic bulk wave velocities (⁠$Vl$ and $Vs$⁠) according to Eqs. (1) and (2), waves propagate through the structure in the $z$ direction faster, resulting in ZGV mode frequency increases.

For battery films, $E1$ is known, so this provides the opportunity to determine $E2$⁠, Young’s modulus of the battery coating layer from the detected ZGV mode.

The sensitivities of the ZGV modes to changes in the coating thickness, $h2$⁠, are shown in Fig. 5. The parameter $h2$ is simulated between 0 $μm$ and 50 $μm$ with an interval spacing of 5 $μm$⁠. M3-ZGV only appears for case C2. M3-ZGV and M4-ZGV modes decrease monotonically when the thickness of the coating layer increases. When acoustic bulk wave velocities are constant, increasing thickness increases the distance for acoustic bulk waves to travel through the whole structure in the $z$ direction, resulting in lower frequencies of the ZGV modes.

The sensitivities of the ZGV modes to changes in the coating density, $ρ2$⁠, are shown in Fig. 6. The density range is simulated between 0.1 g/cm$3$ and 10 g/cm$3$ with an interval spacing of 0.5 g/cm$3$⁠. M3-ZGV only appears for case C2. Both the M3-ZGV mode and M4-ZGV mode decrease monotonically when the density of the coating (⁠$ρ2$⁠) increases because the increasing coating density reduces the acoustic bulk wave velocities according to Eqs. (1) and (2). The decrease in the acoustic bulk wave velocities lengthens the time for waves to propagate through the structure in the $z$ direction, resulting in decreasing frequencies of the ZGV modes.

The sensitivities of ZGV modes to changes in the coating Poisson’s ratio, $ν2$⁠, are shown in Fig. 7. The Poisson’s ratio range is simulated between 0.2 and 0.4 with an interval spacing of 0.02. Both the M3-ZGV mode and M4-ZGV mode increase monotonically when Poisson’s ratio of the coating (⁠$ν2$⁠) increases. The acoustic bulk wave velocities described by Eqs. (1) and (2) indicate that as $ν2$ increases from 0.2 to 0.4, the longitudinal wave velocity $Vl$ increases and the shear wave velocity decreases. For these ZGV modes, the modes increase as Poisson’s ratio increases.

In summary, ZGV modes are dependent on the geometry of the materials as well as the mechanical properties of the materials. In particular, the observed models show that ZGV modes (M3-ZGV and M4-ZGV) are sensitive to the changes in mechanical properties of the coating material. The sensitivity of the ZGV modes offers an ideal method to determine the unknown properties of the coating.

According to Sec. II, the frequencies of the M3-ZGV mode and the M4-ZGV mode are very close to each other (less than 1 MHz difference). Our ZGV measurement has a resolution at 0.5 MHz. Damping additionally widens the spectral peaks. The difference between the M3-ZGV mode and M4-ZGV mode is within the uncertainty of the systematic error from our experimental setup. Since the M4-ZGV mode exists in all the simulations in Sec. II and is not dependent on Young’s modulus difference between the two layers, we assume the M4-ZGV mode as the measured ZGV resonance.

A diagram of the experimental arrangement is shown in Fig. 8. The pump laser source is an ultrafast diode-pumped-solid-state (DPSS) infrared (IR) laser (Standa STA-001) with an optical wavelength centered at 1050 nm. This pulsed laser provides a pulse energy around 600 $μJ$ within a 500 ps duration with a pulse repetition rate up to 50 Hz. Due to the high power of the pump laser pulse, two identical dichroic mirrors (DM1,2: Omega Optics EB00479B) with a 20$%$ transmitting rate for IR light at 1050 nm are used as a diffuser to sufficiently attenuate the energy focused on the current collector in order to avoid laser ablation of the metal films or the battery coatings. The detection laser is a polarized helium-neon laser (Uniphase 3067) with a rated power of 10 mW and an optical wavelength centered at 632.8 nm.

In this homodyne interferometer, two quarter wave plates (Newport 10RP0424) and a polarized beam splitter (Newport, 10BC16PC4) are used as a circulator to avoid retro-reflection of the helium-neon laser which can cause unpredictable power changes. The detection laser gets reflected while the pump laser goes through a dichroic mirror (DM3: Thorlabs DMLP900T), which has a 98$%$ reflection coefficient at wavelength 632.8 nm and a 96$%$ transmission coefficient at wavelength 1050 nm. Both the detection laser and pump laser are focused on the testing sample by a microscope objective (Newport M-10$×$⁠) with 10$×$ magnification. The beam spot sizes of the detection laser and pump laser are approximately 3 $μm$ and 5 $μm$⁠, respectively. These spot sizes were selected to produce good experimental results, but could be optimized for improved ZGV mode generation and detection.^20,35 The interferometer signal beam and reference beam are combined and pass through a polarizer (Newport 5511) at 45$°$ to show the interference pattern. Another microscope objective (Newport M-60$×$⁠) with 60$×$ magnification is used to expand the beam in order to observe the interference pattern during alignment procedures.

The interferometric signal from the sample is recorded by a fast amplified photo-detector (Newport 818BB21A) with a bandwidth from 30 kHz to 1.2 GHz and sampled by an oscilloscope (Lecroy WaveRunner 204Xi) using DC 50 $Ω$ coupling. The oscilloscope is triggered with a photodiode (Vishay 7511001ND) detecting the pump laser pulse and then records 2 $μs$ of signals, consisting of 0.2 $μs$ in pretriggering and 1.8 $μs$ in post-triggering. The oscilloscope sampling rate is set at 5 Gs/s. A preprocessing digital low-pass filter with a bandwidth of 151 MHz is applied to lower the noise level. The recorded signals are coherently averaged 2000 times in the time domain to increase the signal to-noise-ratio (SNR). It is noted that time gating could be used to reduce the influence of the IR pulse, but in these experiments all the data points were retained so that the full response was captured. Since the pulsed laser repetition rate is 50 Hz, this whole process takes about 1 min to measure one physical location on the sample film. The averaged data are transformed into the frequency domain in MATLAB, using a fast fourier transform (FFT) with a Hann window, to compute the power spectral density. Resonances are recognized by finding peaks in the power spectral density. The value of the peak frequency is determined as the mean of the Gaussian curve fitted to the peaks in the power spectral density.

Commercial-quality battery electrodes were obtained for testing. Cathode 1 consisted of an aluminum current collector with a coating of 92 wt. % Toda HE5050, 4 wt. % Timcal C45, and 4 wt. % Solvey 5130 PVDF binder. Cathode 1 was obtained in both uncalendered and calendered variations. Cathode 2 consisted of an aluminum current collector with a coating of 90 wt. % Toda NCM 523, 5 wt. % Timcal C45, and 5 wt. % Solvay 5130 PVDF binder. The anode is this study consisted of a copper current collector with a coating of 91.83 wt. % Superior Graphite SLC1520P, 2 wt. % Timcal C45 carbon, 6 wt. % Kureha 9300 PVDF binder, and 0.17 wt. % Oxalic Acid.

All measurements were performed at room temperature. Samples were clamped by two aluminum plates (⁠$6×6×3cm3$⁠) with a 1.5  cm diameter circular cutout in the center of both plates. The tested electrodes were cut into small pieces (⁠$2×2cm2$⁠) from the original electrode films to fit between the two aluminum plates. The two aluminum plates were clamped together using screws at the four corners and were slightly tightened together, just enough to hold the film rigidly. The pump laser was first approximately aligned by using a IR sensor card (Newport F-IRC1) to check if the IR laser beam and detection laser beam were overlapping behind the microscope objective (M-10$×$⁠). Then, the IR laser was aligned more carefully by minimizing time-of-flight of the first acoustic arrival (the highest displacement) after the IR pulse. By doing so, the excitation and detection points were approximately at the same point where the ZGV Lamb modes would exist. These procedures were repeated for each measurement at different spots on each sample. Visual inspection was used to confirm the ZGV modes were generated without any ablation of the electrode films.

The tested samples of different battery electrode films and current collectors are shown in Table III. The corresponding mechanical properties of the coating of battery films are shown in Table II. The notation (uncal) indicates that the samples were not calendered as received (though some films were subsequently calendered in our laboratory, as noted below). The film thickness was measured using a micrometer. The density of the battery coating was estimated by dividing the measured mass by the estimated volume. Poisson’s ratio was assumed to be the common value 0.3.²⁹ Each sample was tested at 5 different random spots to obtain the mean and standard deviation of the ZGV frequency response of each film.

Calendering is a process to compress electrode films prior to cell assembly. The calendering apparatus is shown in Fig. 9. By adjusting the gap between the two cylinders, the battery films were calendered to desired thicknesses. To avoid delamination of the battery coating layer, during the calendering process the films were calendered several times by steady reduction of the gap between the cylinders until the desired thickness was achieved. Thickness was measured with a micrometer (Mitutoyo IP65) with 0.001 mm resolution. Cathode 1 (uncal) was cut into square samples. To obtain a sequence of measurements during the calendering process, one sample went through a series of 10 cycles of (1) calendering, (2) thickness measurements, and (3) ZGV measurements. Other samples were calendered to a different thickness in one step, after which the thickness was measured followed by ZGV measurement.

The averaged ZGV measurements (from 2000 pulses) and the associated power spectral densities of a reading of noise, Cathode 1 and Anode 1 are shown in Figs. 10–12, respectively. There is a high voltage short duration jump denoted as an IR pulse in the time domain. This IR pulse generates a systematic peak in the power spectral density at approximately 21 MHz. The ZGV frequencies are at 22.5 MHz for Cathode 1 in Fig. 11 and 6.5 MHz for Anode 1 in Fig. 12. Except for the systematic IR peak in the power spectral density, only a single peak was observed in each battery film measurement.

The ZGV frequencies of different commercial-grade battery films and an aluminum current collector are shown in Table III. The results of three different aluminum films validate the accuracy of our method. A copper current collector was not measured because the ZGV frequency was higher than the bandwidth of our experimental setup. The resonance frequencies are shown as $f$ and s.d.$f$ representing the estimated mean and the standard deviation, respectively. The estimated wavelength of each ZGV mode is calculated by fitting the measured ZGV frequencies to simulated dispersion curves. Young’s modulus values of the battery coatings are calculated by fitting the experimental results in Table III to the theoretical models (discussed below).

The calendering test results are shown in Fig. 13. Changes of the resonance frequencies are seen during the calendering process. Error bars are estimated using a 95$%$ confidence interval. It is clear that the ZGV frequencies change significantly during the calendering process. The resonance frequency first drops by 2.23 MHz and then increases as the samples are calendered into thinner thickness. It is noted that the ZGV frequency of Cathode 1, which was calendered by the manufacturer, is significantly higher than ZGV frequency of Cathode 1 (uncal) that was calendered in our laboratory to the same apparent thickness.

The results show that the ZGV mode responses well to the mechanical property change during the calendering process, meaning the mechanical effect of calendering can be quantified.

With known Young’s modulus, the thicknesses of three different aluminum films were estimated as shown in Table III. The S1-ZGV mode of the current collectors of different cathodes has a 95$%$ confidence interval at $146.20±0.79$ MHz. The simulated result for a 20-$μm$-aluminum sheet is 142 MHz using the mechanical parameters in Table I. The thicknesses of the current collectors can be extracted from measured S1-ZGV frequencies.¹² 

The S1-ZGV modes of the current collectors of Cathode 1 and Cathode 2 (uncal) are different from the predicted value in Table III. The estimated thickness of current collector of Cathode 1 is $19.55±0.02μm$⁠, and the estimated thickness of current collector of Cathode 2 (uncal) is $19.31±0.09μm$⁠. The thickness measured with the micrometer is $20μm$ is in agreement with the value indicated by the manufacturer. The thicknesses of the current collectors estimated from the ZGV measurement agree very well (within 2.9$%$⁠) with the measurement from the micrometer and the manufacturer. The experimental results show that the ZGV measurement is accurate for measuring the thickness of current collectors.

Young’s modulus of the battery coating layer can be estimated from the measured ZGV frequencies, if the density and thickness are known independently. However, before estimating Young’s modulus of the battery coating, it is important to determine which ZGV mode to use during estimation.

A wide range of Young’s modulus value was used to generate the numerical relationships between Young’s modulus and the ZGV mode frequency. The measured ZGV frequencies are fitted to the simulated ZGV mode in order to estimate Young’s modulus of the coating. An example of estimation process is shown in Fig. 14. The estimated Young’s modulus is estimated to be 5.97 GPa when the ZGV frequency is 22.6 MHz.

The estimated Young’s modulus results for the coating of the measured battery film samples are listed in Table II. For each sample, the measured ZGV frequencies of the 5 interrogated positions were used individually to estimate the corresponding Young’s modulus at each spot. The mean and standard deviation (s.d.$E$⁠) of the estimated Young’s modulus was then calculated for each film sample. The results indicate that the frequency of ZGV Lamb modes is highly sensitive (resolution at hundreds of megapascal) to changes in Young’s modulus of battery film coatings.

After Young’s modulus is estimated, the corresponding wavenumber $k$ can be determined by examining the dispersion curve associated with that ZGV mode. From these wavenumbers, the corresponding wavelengths for each battery film sample are shown. The results of the calculated wavelengths for each battery film are shown in Table III. Anode 1 has the shortest wavelength around 200 $μm$⁠. Cathode 2 (uncal) has the longest wavelength around 700 $μm$⁠. The wavelength results demonstrate that the ZGV measurement is localized to an approximately $1×1mm2$ area of the film. This area of interrogation is on the same order as other measurements of heterogeneity.⁴ The measured area can be assumed to be part of an infinite plane until its position near the edge of the clamp boundary is on the same order as the ZGV mode wavelength.³⁵ This indicates that the measurement can be performed without the concern of the boundary conditions of the clamp shown in Fig. 8.

To validate that the estimated Young’s modulus of the battery film coating using the ZGV frequency was on the correct order-of-magnitude, we performed a compression test using a universal testing system (Instron 3345) to measure Young’s moduli of Cathode 1 and Cathode 1 (uncal). These two films were cut into many pieces, each approximately $8×8mm2$ and then stacked up together as shown in Fig. 15. 18 pieces of Cathode 1 and 20 pieces of Cathode 1 (uncal) were used in these tests. The force (⁠$N$⁠) to displacement (⁠$ΔL$⁠) relationships of Cathode 1 and Cathode 1 (uncal) are shown in Fig. 15. Young’s modulus is calculated using $E=(FL)/(AΔL)$⁠, where $F$ is the force, $L$ is the thickness of the battery coating, and $A$ is the area. Since the Young’s modulus difference between the battery coating and aluminum is very high, we assume that only the battery coating is compressed during the test. The estimated Young’s modulus from the compression experiment was $1.65±0.41$ GPa for Cathode 1 and $0.38±0.23$ GPa for Cathode 1 (uncal). The estimated Young’s moduli from these compression experiments have the same order-of-magnitude with estimated Young’s moduli from the ZGV measurement. Since the ZGV measurement is measured in a significantly shorter period of time than the compression test, the material reacts “harder” due to inertia.

The Young’s modulus values of different battery films apparently have been measured from the megapascal into the gigapascal range.^8,36,37 Forouzan et al.³⁶ measured the elasticity of coating (delaminated from the current collector) to be about 11.3 MPa. Nadimpalli et al.³⁷ calculated the elasticity of a cathode coating to be about 40 GPa using the rule of mixtures. Dallon et al.⁸ estimated the elasticity of the coating of three different cathodes from 0.344 GPa to 5.04 GPa using acoustic measurements. The Young’s modulus values estimated from the ZGV measurement are comparable to the measured results of Dallon et al. The measured elasticity difference of Cathode 1 is within 20$%$ of the previously estimated value by the alternative method.

Calendering is an important manufacturing step before battery films are packaged into cells. After the battery coating is dry, the coating layer surface is rough. Surface bumps and valleys have elevation differences of a few micrometers. In Fig. 16, the estimated thickness variation of the coating surface is approximately $4μm$⁠. Furthermore, the porosity and density of uncalendered coatings are not optimal for battery performance.^38,39 The calendering process can flatten the rough surface, increase the coating density, and decrease the porosity of the coating layer as shown in Figs. 17 and 18.

When the coating layer is calendered, the coating density increases. The density is estimated using

where $MA$ is a constant representing the loading of the coating, mass per unit area.

During calendering, it is expected that Poisson’s ratio would slightly change as the density increases. However, as shown in Fig. 7, when Poisson’s ratio of the battery coating changes slightly, the impact on the ZGV modes is negligible compared to the impact caused by the changes in density, thickness, and Young’s modulus. The ZGV mode changes less than 1 MHz if Poisson’s ratio of coating 1 changes less than 0.02 for case C2. In order to reduce the analysis complexity, Poisson’s ratio of the battery coating was assumed to be the common value of 0.3 throughout this analysis.²⁹ More sophisticated analyses using multiple modes would be necessary to resolve changes in Poisson’s ratio.

It is also expected that when the film is calendered, Young’s modulus should increase as the coating becomes harder and more compacted. However, Fig. 13 shows that this is not always the trend observed in the data. This indicates that the coating layer does not always behave mechanically like a simple, homogeneous solid.

Figure 13 shows that when Cathode 1 (uncal) coating was first calendered into a slightly thinner thickness (from $48μm$ to $45μm$⁠), the ZGV frequency decreased. A plausible explanation for this phenomenon originates in the rough surface and pores randomly distributed across the coating layer. Some large pores estimated from Fig. 16 are more than $19μm$ in length and $1μm$ in height. Lamb waves will not propagate efficiently through the pores because of the impedance mismatch between the pore air and coating material. This leads to a mode in which the effective thickness of the coating in which the Lamb waves propagate is smaller than the measured thickness.

To quantify how the effective thickness of the coating changes when Cathode 1 (uncal) is slightly calendered (from $48μm$ to $45μm$⁠), an illustrative model is shown in Fig. 19. The uncalendered battery film is a trilayer structure consisting of a current collector, an effective coating layer, and a mass loading layer. The mass loading layer is assumed to not propagate mechanical waves, and its mass is considered to be simply added to the effective coating layer mass. By calendering the coating layer by a few micrometers, the rough surface flattens, and the pores are compressed. These changes result in increased effective coating thickness (⁠$heff$⁠) and reduced mass loading layer thickness. Since the sample has the lowest ZGV frequency when it is calendered to $45μm$ as shown in Fig. 13, we assume that the mass loading layer disappears at that point. Since Lamb waves do not propagate in the mass loading layer, it is reasonable to simulate dispersion curves using a bilayer structure with higher density of the effective coating layer due to the mass loading effect, while considering Young’s modulus of the effective coating layer constant during this process. The density of the effective coating layer is estimated using Eq. (3), with $h=heff$⁠. Young’s modulus of the effective coating layer is estimated at 0.39 GPa when the sample is calendered to $45μm$⁠. Based on this estimation, the simulations indicate that the effective thickness is $18.3μm$ for Cathode 1 (uncal) at $48μm$⁠, and $35.9μm$ when the sample is calendered to $47μm$⁠.

Two simulated models with constant Young’s modulus assumed for the coating are shown in Fig. 13. The lowest ZGV frequency of the sample with thickness at $45μm$ is used to estimate the Young’s modulus value of the coating (⁠$E=0.39GPa$⁠). The density $ρ$ of the coating is calculated according to $h$ using Eq. (3). The corresponding M3-ZGV and M4-ZGV frequencies vs changes in thickness are then plotted. Similarly, the M4-ZGV frequency vs thickness is also plotted with an estimated $E=3.4$ GPa for the coating.

By comparing the simulated models with the experimental results in Fig. 13, it is evident that Young’s modulus of the coating changes significantly during the calendering process. The ZGV measurement can be used to effectively monitor these changes. Figure 20 shows the estimated Young’s modulus of the coating as determined from the M4-ZGV mode calculated with a constant Poisson’s ratio and the measured geometric parameters during the calendering process. Error bars are estimated using a 95$%$ confidence interval. Data are only presented for coating thicknesses less than 45 $μm$⁠.

In Fig. 21, the sensitivity of the ZGV frequency to the changes in Young’s modulus of the coating is calculated during the calendering process. The sensitivity (⁠$S$⁠) is calculated using $S=E/fZGV$⁠, where $E$ is the estimated Young’s modulus from Fig. 20 and $fZGV$ is the experimental ZGV frequency from Fig. 13. It can be observed that $fZGV$ becomes more sensitive to small changes in $E$ when the coating is calendered thinner, which makes the ZGV measurements more sensitive to changes in calendered films rather than uncalendered films.

In Fig. 13, it is evident that even though Cathode 1 (uncal) is calendered to the same thickness (⁠$26μm$⁠) as the industrially calendered Cathode 1, the ZGV frequencies of these two battery films are significantly different. In the industry process, the battery film is calendered soon after the coating layer has dried,³⁹ with heat applied during calendering, which appears to be capable of calendering the coating layer tighter, causing higher elasticity. In our work, the film was dried for a long time before the calendering process occurred. By comparing the SEM of Cathode 1 shown in Fig. 22 and the SEM of Cathode 1 (uncal) which was calendered to the same thickness, as shown in Fig. 18, the coating of the commercially calendered film appears to be more condensed than the film calendered in the laboratory, with generally smaller pores. The high compactness of the commercially calendered film would appear to have a higher Young’s modulus and, therefore, higher ZGV frequency.

The interferometric ZGV measurement demonstrated in this work can quickly and accurately measure the mechanical properties of battery films. The sensitivity of ZGV modes to changes of mechanical properties of the coating layer was simulated. Different commercial-grade battery films were characterized. With the combination of theoretical model and the interferometric measurements, Young’s modulus of the battery coating layer was estimated. The interferometric measurement can also be used to monitor changes during the calendering process. This noncontact and nondestructive evaluation technique is unique in obtaining local material properties, which can potentially be used to quantify the heterogeneity of battery films to improve the safety and performance of batteries.

This work was supported through the BMR program of the U.S. Department of Energy (DOE Project No. DE-AC02-05CH11231). We gratefully acknowledge Professors Stephen Shultz and Daniel Smalley for their assistance with the interferometer measurements. We would like to acknowledge the ANL CAMP facility and Bryant Polzin for providing electrodes for testing.