# Preliminary analysis of the electrical charge distribution within liquids using PEA

Matéo Dautremont, Paul Leblanc, Laurent Berquez, Thierry Paillat

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Matéo Dautremont, Paul Leblanc, Laurent Berquez, Thierry Paillat. Preliminary analysis of the electrical charge distribution within liquids using PEA. Journal of Electrostatics, 2025, 138, pp.104161. 10.1016/j.elstat.2025.104161. hal-05241401

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# Preliminary analysis of the electrical charge distribution within liquids using PEA

Matéo Dautremont<sup>a,*</sup>, Paul Leblanc<sup>a</sup>, Laurent Berquez<sup>b</sup>, Thierry Paillat<sup>a</sup>

<sup>a</sup> Institut PPRIME - Université de Poitiers – CNRS – ISAE ENSMA, 11, Boulevard Marie et Pierre Curie, 86073 Poitiers Cedex 9, France

<sup>b</sup> Laboratoire LAPLACE - Université Toulouse 3 - Paul Sabatier – CNRS, 118, route de Narbonne - bât 3R3 - 31062 Toulouse cedex 9, France

## Abstract

Measuring the distribution of the electrical charges in various materials is of great importance for numerous industrial applications. The development of measurement method has been widely studied in the case of dielectric solids. However, for liquids, the development of measurement techniques is not as much established. Liquid materials, as for solid, are used in wide range of industrial applications, such as transformers or electrochemical energy storage. In order to determine the charge distribution within liquid materials, measurement technique used for solids have been adapted to be applied to liquids. The pulsed electro-acoustic (PEA) method has recently been adapted to liquids. It has been applied to various liquids under electrical constraints. Its application to liquids without electrical constraint, with the aim of measuring the natural distribution of electric charges (electrical double layer), is not widely done. This work proposes to apply the PEA method to various liquids without subjecting it to an electrical constraint, in order to assess its capability to measure the electrical double layer. The influence of different parameters on the response is presented and the trends obtained are compared with the trends expected from a theoretical model.

**Keywords** – Pulsed Electro-Acoustic method (PEA), Liquid, Interface, Space charge, Electrical Double Layer (EDL)

## 1. Introduction

The measurement of trapped electric charges within dielectric solids has been a significant subject of study for many years. These electric charges cause reliability issue in many electrical systems, due to the local electric field they induce. To better understand the effect of these electric charges, different experimental methods have been developed. Currently, the most commonly used techniques are stimulus-based methods which rely on the application of a physical stimulus – of various kinds – to a sample in order to disturb the electric charges it contains and induce a physical response. The stimulus used can be thermal, mechanical or electrical and since the kind of stimulus change, the kind of physical response change too. It can be either mechanical or electrical. Based on these responses and an appropriate mathematical processing, the quantity and position of the charges can be determined. These various measurement methods and their application to dielectric solids are well established.

However, methods for measuring the charge distribution within liquid material are still marginal. With this in mind, the PPRIME Institute has been working on the development of new techniques of measurement. These new methods are based on adapting space charge measurement techniques, originally developed for solid dielectrics, to liquid materials.

These techniques have been adapted to measure the distribution of electric charges within the electric double layer (EDL) present at all solid/liquid interfaces. This charge distribution results from physicochemical reactions at the interface that remain unidentified, leading to the formation of two regions of electric charge: one located in the solid and the other, with opposite charge, in the liquid, in order to preserve overall electroneutrality. The characteristics of the EDL depend solely on the chemical composition of the two materials forming the interface.

The thermal step method (TSM) was the first explored techniques. It relies on the propagation of a thermal step through a liquid between 2 electrodes. This stimulus generates a current related to the charge within the sample. It has been applied to different dielectric liquids, such as mineral oil, silicone oil or cyclohexane and resulted in a unique and reproducible recorded current for each liquid. For the geometry of the cell used, only the signal from the excitation electrode is detectable. A drawback is the possible appearance of thermo-convective effect disturbing the distribution of electric charges within the liquid, which was studied by numerical simulation. Tests on more conductive liquids, such as water, failed to detect a signal, which is indicating a limitation of applicability of this method [1–3].

An electro-acoustic technique has been adapted to be applied to liquid sample is the Pressure Wave Propagation method (PWP). It is based on the propagation of a pulse acoustic wave, produced by a piezoelectric actuator, through a liquid sample sandwich between 2 electrodes in open circuit. The response of the charges to the acoustic wave is an electrical current or a voltage which contain information on the charges. With this method results have been obtained for a wide conductivity range of 10<sup>-2</sup> to 10<sup>-9</sup> S.m<sup>-1</sup>. For liquids having conductivity below 10<sup>-11</sup> S.m<sup>-1</sup>, no results were obtained, showing limitation in applicability of the technique at this time. The experimental results have been compared with numerical studies, showing a good agreement [4].

The PWP has also been applied to liquids under direct current (DC) constraint. The results made it possible to monitor over time the mechanisms of charge creation and movement, both at the interfaces and within the bulk of the liquid. The technique was applied to various liquids, including electrolytes (aqueous glycerol solution) and even dielectric liquids (mineral oil) [5].

The PPRIME Institute, in collaboration with the LAPLACE Laboratory, has recently been working on

<footnote>* Corresponding author: Matéo Dautremont
Email address: mateo.dautremont@univ-poitiers.fr</footnote>
adapting the pulsed electro-acoustic (PEA) method. This technique is based on applying a pulsed electric field across a sample. The charges then generate an acoustic wave, which allows for the determination of the volumetric charge density. This method has been widely applied to solid dielectrics and porous materials [6,7] (such as oil-impregnated papers). Its application to liquids under electrical stress has also been explored [8], however, in the context of measuring the EDL, its use remains relatively recent [9].

This document presents the new advancement concerning the measurement of the EDL by the mean of the PEA method adapted to liquid. The first part of the document gives theoretical explanations about the experimental method and the EDL. The second part relate the experimental setup and liquid samples used. And finally, a description and discussion of the results obtain is given.

## 2. Theoretical approach

### 2.1. Pulsed electro-acoustic principle: PEA

The PEA method is based on applying a voltage pulse in the kilovolt range, with a half-height duration of a few nanoseconds, as illustrated in Fig. 1. This system is assumed unidimensional along the $\vec{x}$ axis.

It consists in placing a sample (2) between 2 electrodes (1, 3).

<figure><img src="image_2.png" /><figcaption>Fig. 1. Schematic diagram of the PEA method.</figcaption></figure>

The electrode (1) is connected to a voltage pulse generator $V_{Pulse}$, and the electrode (3) is grounded. Applying a voltage pulse to the electrode (1) generates a pulsed electric field across the sample, creating an electrostatic force on the charges and a slight displacement around their equilibrium positions and thus generating a pressure wave. The wave propagates through the different part of the PEA cell, crossing several interfaces where a part of the wave is transmitted and the other is reflected. After passing through the grounded electrode, the wave reaches a piezoelectric material (4), that converts the mechanical stress of the acoustic wave into a voltage signal.

The grounded electrode has an important thickness to act as a waveguide introducing a controlled time delay. It allows the signal generated by the acoustic wave to be separated from the electromagnetic noise

produced by the excitation pulse. After passing through the piezoelectric sensor, the acoustic wave continues to propagate inside an acoustic absorber material (5), whose purpose is to avoid reflections at the back face of the piezoelectric material.

The electrical signal generated by the piezoelectric sensor is amplified and recorded by an oscilloscope. Through an appropriate deconvolution signal processing, the distribution of the space charge density within the sample can be determined.

The electrostatic force density that can be expressed by 5 terms [10] through the equation (1):

$$f = \underbrace{\rho \cdot E}_{\text{Coulombic force}} - \underbrace{\frac{1}{2} \cdot \frac{\partial \varepsilon}{\partial x} \cdot E^2}_{\text{Permittivity variation effect}} - \underbrace{\frac{1}{2} \cdot \frac{\partial (a \cdot E^2)}{\partial x}}_{\text{Electrostrictive effect}} + \underbrace{\pi \cdot \frac{\partial E}{\partial x}}_{\text{Permanent dipole density}} - \underbrace{\frac{\partial (e \cdot E)}{\partial x}}_{\text{Piezoelectric effect}} \quad (1)$$

where f is the volumetric force density, ρ the volumetric charge density, E is the electric field, ε the absolute permittivity, a the electrostriction coefficient, π the permanent dipole density and e is the piezoelectric coefficient.

The electrostatic density is the gradient of the electromagnetic stress tensor M [11]. Its expression is determined from equation (1), by replacing ρ with equation (2). Assuming that the piezoelectric effect is negligible, f can be expressed as follows in equation (3).

$$\rho = \frac{\partial}{\partial x}(\varepsilon \cdot E + \pi) \quad (2)$$

$$f = \frac{\partial M}{\partial x} = \frac{\partial \left( (\varepsilon \cdot E + \pi) \cdot E - \frac{1}{2} \cdot (\varepsilon + a) \cdot E^2 \right)}{\partial x} \quad (3)$$

When a voltage pulse ($\delta E$) is applied, the electric field in the sample becomes $E + \delta E$. This results in a variation in the electrostatic force density $f + \delta f$, as well as a variation in the electromagnetic stress tensor $M + \delta M$. The variation of the electromagnetic stress tensor $\delta M$ is then given by the equation (4):

$$\delta M = ((\varepsilon - a) \cdot E + \pi) \cdot \delta E + \frac{1}{2} \cdot (\varepsilon - a) \cdot \delta E^2 \quad (4)$$

The variation in electromagnetic stress consists of two terms: one proportional to the variation of the electric field, and the other proportional to the square of the electric field variation. The linear term provides information about the position and quantity of charges in the sample, through the electric field $E$, which represents the static electric field induced by the charges. The quadratic term is not correlated with the distribution of electric charges in the sample.

The PEA method can be used on solid materials under a continuous electric stress, allowing the observation of the electric charge behavior during polarization process (Fig. 1). A direct current generator $V_{DC}$ is connected in parallel with the pulse generator, with a decoupling capacitor C to protect the pulse generator. The DC electric field adds to the electric field induced by the charges in the sample. In the case where the continuous electric field is sufficiently larger than the pulsed electric field, the quadratic term of equation (4) is neglected [10].
## 2.2. Electrical Double Layer: EDL

When a solid comes into contact with a liquid – both initially electrically neutral – physicochemical reactions are induced that gradually polarize the solid/liquid interface until an electrostatic equilibrium is reached. This phenomenon occurs at all solid/liquid interfaces, regardless of the nature of the material involved. The distribution of electric charges at the equilibrium state is specific to each solid/liquid pair.

The charge distribution of the electrical double layer is composed of 2 parts: one located in the solid and the other, of opposite sign, in the liquid to preserve the global electroneutrality. This interfacial polarization has been the subject of numerous theoretical models. The commonly used theoretical model is the Stern [12] model (Fig. 2 & Fig. 3).

<figure><img src="image_3.png" /><figcaption>Fig. 2. Schematic diagram of the EDL.</figcaption></figure>

In the liquid part, the distribution can be divided into 2 distinct subregions. The first, located near the surface and with a thickness of the order of the atomic radius, is called the compact layer. In this region the potential decreases linearly from $\Psi_0$ at the interface to $\Psi_{0H}$ at the outer Helmholtz plane (Fig. 3). The 2nd region in the liquid part, further in the bulk, is named the diffuse layer. The volumetric charge density $\rho$ is defined by equation (5):

$$ \rho = \sum_{i} n_{i}^{0} \cdot e_{0} \cdot z_{i} \cdot \exp \left( - \left( \frac{e_{0} \cdot z_{i} \cdot \Psi}{k \cdot T} \right) \right) \quad (5) $$

where $n_i^0$ is the local concentration of the charged species i far from the interface, $e_0$ the elementary charge, $z_i$ the valence of species i, k the Boltzmann constant, and T is the temperature.

The distribution of the electric potential $\Psi$ in the diffuse layer is expressed by equation (6), where $\rho$ is defined by equation (5):

$$ \nabla^2 \Psi = - \frac{\rho}{\varepsilon_0 \cdot \varepsilon_r} \quad (6) $$

where $\varepsilon_0$ is the vacuum permittivity and $\varepsilon_r$ the relative permittivity.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Voltage distribution and volumetric charge density of the EDL.</figcaption></figure>

In the diffuse layer, the distribution of potential and volumetric charge density decreases according to a Boltzmann law, starting from the outer Helmholtz plane, where the volumetric charge density is $\rho_{wd}$ and the potential is $\Psi_{0H}$. The thickness of the diffuse layer is much greater than that of the compact layer. It is commonly approximated by the Debye length $\delta_0$, which is defined by the equation (7):

$$ \delta_0 = \left( \frac{\varepsilon_0 \cdot \varepsilon_r \cdot k \cdot T}{\sum_i (e_0^2 \cdot z_i^2 \cdot n_i^0)} \right)^{\frac{1}{2}} \quad (7) $$

66% of the charges in the diffuse layer are distributed within a distance of $\delta_0$ from the outer Helmholtz plane, and 95% within a distance of $3\delta_0$. The Debye length can also be expressed solely as a function of the properties of the liquid [4], including its permittivity, electrical conductivity ($\sigma_r$), and the mean ionic diffusion coefficient ($D_0$) (equation (8)).

$$ \delta_0 = \left( \frac{\varepsilon_0 \cdot \varepsilon_r \cdot D_0}{\sigma_r} \right)^{\frac{1}{2}} \quad (8) $$

According to equation (8), the diffuse layer increases in thickness as the conductivity of the liquid decreases. It is only a few nanometers for conductive liquids ($10^{-3}$ S.m$^{-1}$), and can reach several tens of micrometers for dielectric liquids ($10^{-12}$ S.m$^{-1}$).

In the solid part, located to the interface, the volume charge density is assumed to be constant over a thickness on the order of the atomic radius, and its magnitude ensures the preservation of overall electroneutrality across the solid/liquid interface (Fig. 2).

## 3. Experimental Setup

The experimental setup relies on a PEA cell made of 2 parts (Fig. 4): an excitation part and a measurement part. The excitation part includes an aluminum electrode (6), electrically insulated from the rest of the system by P.E.E.K. It is integrated into an aluminum body, which acts as both Faraday cage and a support for a micrometer (Newport® BM17.25) allowing to set the spacing between the waveguide electrode and the
excitation electrode and hence the sample thickness, with a precision of ±10 µm.

The measurement part includes an aluminum waveguide electrode (1) with a thickness of 15 mm, introducing a time delay of 2.3 µs. The sensor (2) used is made of polyvinylidene fluoride (PVDF) and has a thickness of 9 µm. In the cell, the absorber material (3) is also made of PVDF. The electrical signal produced by the sensor is amplified by 56 dB (Mini-Circuit® ZFL500N) before being recorded by an oscilloscope (Tektronix® MSO54B – 350 MHz – 6.25 GS/s). To extract the useful signal from the system's white noise, the measurement signal is obtained by averaging 2000 successive acquisitions at a frequency of 50 Hz.

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Schematic diagram of the PEA cell (1: waveguide electrode, 2: piezoelectric sensor, 3: acoustic absorber, 4: liquid sample, 5: excitation electrode, 6: electrical insulation).</figcaption></figure>

The voltage pulses $V_{\text{Pulse}}$ applied to the excitation electrode are generated by a laboratory-designed power supply. It is capable to produce a pulse with an amplitude ranging from 100 to 1200 V in 100 V increments (Fig. 5). The produced pulses have a temporal width of approximately 4 ns at half-height.

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Measurement of the electrical pulse at the output of the pulse generator.</figcaption></figure>

A 10 meters long coaxial cable is used to connect the generator and the excitation electrode. The propagation of the pulse in the cable induces a voltage drop. The magnitude of the pulses has been measured at the excitation electrode. These values are plotted as a

function of the control voltage set on the pulse generator (Fig. 6). The voltage measured at the electrode will be used to calculate the pulsed electric field applied to the liquid.

A 50 Ω terminator is inserted between the pulse generator and the excitation electrode to improve impedance matching with the measurement cell.

The pulse excitation frequency is set to 50 Hz using a function generator (Tektronix® AFG31022 – 25 MHz – 250 MS/s), which is also connected to the oscilloscope in order to synchronize the experimental setup. In the experimental setup used, no continuous electric field is applied to the liquid.

<figure><img src="image_7.png" /><figcaption>**Fig. 6.** Amplitude of the voltage pulse measured at the excitation electrode as a function of the control voltage of the pulse generator.</figcaption></figure>

The liquids used are ethanol (C₂H₆O) with a purity of over 99%, a mineral oil and a silicone oil. The electrical properties ($\sigma_r$ and $\varepsilon_r$) of these liquids have been measured by impedance spectroscopy (AMETEK® Modulab XM CHAS 08). The analysis of the real part of the permittivity obtained from spectroscopy measurements allows for the determination of the mean ionic diffusion coefficient ($D_0$), following the method proposed by Sidambarompoulé [13]. The results are summarized in the Table 1. Moreover, the also includes the sound speed ($v$) of the different liquids, provided by the manufacturer and compared to the literature.

All experiments were conducted at atmospheric pressure and ambient temperature (293 K).

<table><caption>**Table 1.** Electrical properties of the liquids.</caption><thead><tr><th></th><th>ε<sub>r</sub></th><th>σ<sub>r</sub></th><th>D<sub>0</sub></th><th>δ<sub>0</sub></th><th>v</th></tr><tr><th></th><th></th><th>[S·m<sup>-1</sup>]</th><th>[m<sup>2</sup>·s<sup>-1</sup>]</th><th>[µm]</th><th>[m·s<sup>-1</sup>]</th></tr></thead><tbody><tr><td>Ethanol</td><td>23.8</td><td>1.27·10<sup>-5</sup></td><td>1.05·10<sup>-9</sup></td><td>0.13</td><td>1.54·10<sup>3</sup></td></tr><tr><td>Mineral Oil</td><td>2.53</td><td>5.43·10<sup>-13</sup></td><td>1.24·10<sup>-10</sup></td><td>72.2</td><td>1.44·10<sup>3</sup></td></tr><tr><td>Silicone Oil</td><td>3.26</td><td>1.82·10<sup>-13</sup></td><td>5.75·10<sup>-10</sup></td><td>300</td><td>0.96·10<sup>3</sup></td></tr></tbody></table>

# 4. Results & Discussion

## 4.1. Impact of the nature of the liquids

Typical output signals obtained with the experimental setup described above are presented for different liquids and an electrode gap of 250 µm in Fig. 7 and Fig. 8. The measurement signal is delayed by the wave propagation time through the waveguide electrode, which is 2.3 µs. A
transient starts slightly earlier due to a delay in the
triggering of the measurement.

The transient consists of a negative peak, immediately followed by a positive peak. The amplitudes of these peaks are different. For both liquids, the negative peak (in absolute value) is larger than the positive peak. The temporal width of the negative peak is also shorter than that of the positive peak.

The synchronization of the signal with its passage
near interface 1 (waveguide electrode / liquid), as well as
its change in polarity, coincides with the presence of the
previously mentioned EDL charges. The positive peak
can be partly induced by an overshoot generated by the
acquisition chain.

<figure><img src="image_8.png" /><figcaption>**Fig. 7.** V<sub>PEA</sub> signal as a function of time for a mineral oil with an inter-electrode gap of 250 µm.</figcaption></figure>

The appearance of the signal originating from the 1st interface is theoretically determined based on the delay imposed by the waveguide. Using the inter-electrode gap and the speed of sound in the liquid (ν) (Table 1), the temporal position of the signal originating from the 2nd interface can be estimated.

<figure><img src="image_9.png" /><figcaption>**Fig. 8.** V<sub>PEA</sub> signal as a function of time for silicone oil with an inter-electrode gap of 250 µm.</figcaption></figure>

The signal generated at the 2^nd^ interface must travel through both the sample and the waveguide. Since liquids are poor acoustic propagators, the generated signal is significantly attenuated in amplitude. Furthermore, the signal must pass through the interface between the liquid and the waveguide electrode, where

the acoustic impedance mismatch leads to a weak wave
transmission. As a result, the signal from the second
interface cannot be distinguished from the system white
noise level.

The analysis of the measurements with mineral and silicone oils (Fig. 7 and Fig. 8) reveals clear differences, despite identical experimental conditions (250 µm inter-electrode gap and a command voltage of 1200 V). The overall signal amplitude for the tested mineral oil is lower than that obtained for silicone oil. The width of the positive peak – likely related to the thickness of the diffuse layer – is larger in the case of silicone oil, which is consistent with the estimated value of δ₀ for these 2 liquids (given in Table 1). The PEA system thus effectively provides a response specific to each liquid.

Electrostatic Charge Tendency (ECT) measurements [14], following the Westinghouse protocol, were carried out to confirm the polarity of the EDL at the aluminum/mineral oil and aluminum/silicone oil interfaces. In the liquid, positive charges were observed for mineral oil and negative charges for silicone oil. The polarity of the PEA signal is consistent with the ECT result for mineral oil, but not for silicone oil. These results will be discussed in the following sections.

4.2. Study of the influence of voltage excitation
magnitude

<figure><img src="image_10.png" /><figcaption>**Fig. 9.** V<sub>PEA</sub> signal for ethanol versus time for different excitation electric field values and a fixed inter-electrode gap of 200 µm, for ethanol.</figcaption></figure>

**Fig. 9** presents the V<sub>PEA</sub> signals obtained with ethanol for different voltage levels with a fixed inter-electrode gap of 200 µm. Changing the voltage amplitude at a fixed spacing corresponds to a variation in the pulsed electric field as indicated in the legend. The pulsed electric field is assumed to be uniform within the liquid.

The signal is globally similar to those obtained with
mineral and silicone oils (Fig. 7 and Fig. 8). The negative
peak amplitude is larger in absolute value than the
positive one for all tested stimuli. The positive peak has
a greater temporal width than the negative one. This
temporal width remains unchanged across different
stimulus amplitudes. The overall response amplitude
increases with the amplitude of the applied electric field
stimulus.

The temporal position of the second interface is
estimated at 2.43 µs, with some uncertainty due to
imprecise settings of the sample thickness and the
speed of sound. A low-magnitude negative peak appears just before the 2nd interface, with a polarity that does not match that of the liquid charges, as observed at the 1st interface. This aspect will be discussed in the following sections. This transient might also be related to the first interface. Electrical impedance mismatch between the pulse generator and the measurement cell leads to reflections of the stimulus pulse within the coaxial cable, even with the 50 Ω terminator. Due to its low magnitude, the signal induced at the second interface cannot be used for analysis.

The overall response amplitude increases with the magnitude of the applied electric field stimulus. **Fig. 10** and **Fig. 11** respectively show the recorded amplitudes of the minimum (plotted in absolute value) and the maximum of the transient signal originating from the 1st interface, as a function of the amplitude of the electric field stimulus.

<figure><img src="image_11.png" /></figure>

**Fig. 10.** Amplitude of the minimum as a function of the electric field pulse for inter-electrode gap of 200, 250 and 300 µm.

The minimum (in absolute value) of the PEA transient appear to increase with the electric field, following a second-order polynomial trend (Fig. 10). This behavior is particularly noticeable for an inter-electrode gap of 200 µm, but is less pronounced for other distances (250 and 300 µm), where the number of measurements was limited by the capabilities of the pulse generator. This evolution could support expression (4), which describes the relationship between the stimulus electric field and the acoustic response.

The first peak of the transient originating from the 1st interface is negative for all the measured liquids. For the silicone oil, this negative sign does not match the sign observed in the ECT measurement. The polynomial evolution, where the quadratic term is not depending on the electrical charge distribution, could be the origin of the inconsistency with the ECT measurement and hence confirming the domination of the quadratic term over the linear term.

For the study of the maximum (Fig. 11) of the PEA transient originating from the 1st interface leads to the same observations and conclusions obtained with the **Fig. 10**. Where the ECT measurement is not consistent with the positive polarity of the peak in the case of the silicone oil, showing an independency of the response with respect to the local polarity the charge distribution and hence confirming the domination of the quadratic

term over the linear term of the equation (4). The positive peak of the transient may also be partially attributed to an overshoot of the 1st peak, induced by the piezoelectric sensor and the acquisition chain.

<figure><img src="image_12.png" /></figure>

**Fig. 11.** Amplitude of maximum as a function of the electric field pulse for inter-electrode gap of 200 and 250 µm.

On the other hand, the response growth appears to be faster for 200 µm than for 250 µm. These unexplained differences are not due to attenuation or dispersion effects of the acoustic wave, since the first interface is not affected by changes in the inter-electrode gap, as the EDLs at both interfaces are fully separated. They are considered separated when the inter-electrode gap exceeds $10\delta_0$, which is the case for the inter-electrode distances tested with ethanol.

### 4.3. Study of the influence of the inter-electrode gap

The **Fig. 12** shows the time-domain response obtain for different ethanol sample thickness with a constant excitation voltage of 493 V.

<figure><img src="image_13.png" /></figure>

**Fig. 12.** $V_{PEA}$ signals as a function of time for different inter-electrode gap and an excitation voltage of 493 V.

The signals obtained are similar to the ones shown in **Fig. 7**, **Fig. 8** and **Fig. 9**. In the case of the **Fig. 12**, the amplitude of the electric field stimulus is modified by the change of inter-electrode gap. An increase of the spacing conducts to a decrease of the stimulus amplitude. As it can be seen, the global shape of the signal is the same for the different inter-electrode gap, indicating that the change has no effect on the charge
distribution. The amplitude of the response globally decreases with the increase in inter-electrode gap.

Changing the electrode spacing also alters the theoretical temporal position of the signal originating from the 2nd interface (liquid / excitation electrode). The theoretical temporal position can be seen in the **Fig. 12** for the different inter-electrode gap, represented by the dashed lines of color corresponding to the legend. However, due to a very low signal-to-noise ratio, the signal originating from the 2nd interface is not discernible. A way to observe the signal originating from the 2nd interface might be to increase strongly the electric field of stimulus. There are two possibilities to do it, the first is to increase the voltage amplitude of the pulse generator, and the second is to reduce the inter-electrode gap. The first method is limited by the pulse generator used, with a maximum of 1200 V at the output. In the second method, we are limited by the diffuse layer thickness. If the inter-electrode gap is too much reduced, the EDL at both interfaces might interfere.

The response amplitude for the maximum and minimum of the 1st interface is presented as a function of the inter-electrode gap for different voltage levels in **Fig. 13 & Fig. 14**. The trends shown in the two figures are globally similar.

<figure><img src="image_14.png" /><figcaption>**Fig. 13.** Amplitude of the minimum as a function of the sample thickness for different excitation voltage levels.</figcaption></figure>

The $V_{PEA}$ measurement decreases with increasing inter-electrode gap, or the decrease in electric field stimulus. The curve is decreasing rapidly for low inter-electrode gap, and at it becoming greater, the decrease speed become slower until the amplitude response seems to reach the level of noise of the measurement system. The response seems to evolve like an inverse function with the inter-electrode gap as variable but in a faster manner. The equation (4) is appropriate to describe this evolution.

For the different voltage levels tested, the curves have the same trend. It can also be noted that for both figures, the curves seem to superimpose.

The values of the inter-electrode gap have been chosen in function of the voltage levels in order to have corresponding (equal) electric fields of stimulus, as a consequence the measured response is the same for a considered electric field of stimulus. This is leading to the superimposition of the curves.

<figure><img src="image_15.png" /><figcaption>**Fig. 14.** Amplitude of the maximum as a function of the sample thickness for different excitation voltage levels.</figcaption></figure>

### 4.4. Study of the inter-electrode gap and voltage while remaining at constant electric fields

The impact on the amplitudes of the maximum and minimum of the signal generated at the first interface for various constant stimulus electric fields is presented in **Fig. 15 & Fig. 16**.

<figure><img src="image_16.png" /><figcaption>**Fig. 15.** Amplitude of the minimum as a function of the sample thickness for different electric field pulse levels.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 16.** Amplitude of the maximum as a function of the sample thickness for different electric field pulse levels.</figcaption></figure>
The trends observed in both figures are quite similar. A general decrease in the response amplitude is observed with the decrease in the stimulus electric field. A decrease in the amplitude of the response is also observed with the rise in sample thickness. This unexplained signal reduction cannot be attributed to attenuation or acoustic dispersion effects of the pressure wave, since those maximum and minimum are related to the 1st interface. According to equation (4), for a constant stimulus electric field $\delta E$, the resulting response should also remain constant. The results presented in Fig. 15 & Fig. 16 therefore do not appear to support equation (4).

## 5. Conclusion

The use of the PEA method for the measuring charge distribution in liquids has been assessed, highlighting its technical feasibility. This method was applied to the Electrical Double Layer (EDL) and yielded results that varied depending on experimental parameters and sample properties, indicating that the signals obtained are indeed specific each liquid. The detection of an electrical signal originating from the second interface was not successful. Improving the measurement system and selecting an optimal inter-electrode gap could potentially enable the detection of a physical response from this second interface. In some cases, the evolution of the measured response aligns with theoretical predictions (equation (4)), although certain observed trends remain unexplained.

In this study, the assumption was made that the piezoelectric sensor does not distort the acoustic wave during its conversion into an electrical signal; however, such distortion could account for the inconsistencies observed. Overall, it appears that measuring the electric double layer using the PEA method is not entirely straightforward due to the system's nonlinear response. Applying a continuous electric field across the sample, with an amplitude significantly greater than that of the impulse, could bring the system closer to a theoretically linear response. However, it is possible that such a high polarization field may overwhelm the field induced by the electric double layer, thereby masking its presence.

## Acknowledgement

This work is supported by the French Government program "Investissement d'Avenir" (LABEX INTERACTIF reference ANR-11-LABX-0017-01, EUR IntREE reference ANR-18-EURE-0010), European project PLATABAT (2022-23222510) and "Région Nouvelle Aquitaine".

## Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work the author(s) used ChatGPT in order to improve readability and language. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the published article.

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