# Use of a Distribution of Relaxation Times approach for impedance analysis of a channel electrode in microfluidics

Yann Bissessur, Hiu Mun Man, Jean Gamby

► To cite this version:

Yann Bissessur, Hiu Mun Man, Jean Gamby. Use of a Distribution of Relaxation Times approach for impedance analysis of a channel electrode in microfluidics. 2025. hal-05340585

HAL Id: hal-05340585

https://hal.science/hal-05340585v1

Preprint submitted on 31 Oct 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# Use of a Distribution of Relaxation Times approach for impedance analysis of a channel electrode in microfluidics

Yann Bissessur¹, Hiu Mun Man¹, and Jean Gamby¹,*

¹ Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologie, 91120, Palaiseau, France

**ABSTRACT.** A DNA-functionalized electrochemical microelectrode is investigated through electrochemical impedance spectroscopy and modeled using an equivalent electrical circuit. Traditional parameter extraction methods rely on multi-parameter regression, which can lead to ambiguous or non-physical solutions. In this study, we propose an alternative approach based on the Distribution of Relaxation Times (DRT) to analyze Electrochemical Impedance Spectroscopy (EIS) data. This method allows us to identify characteristic electrochemical processes without assuming a predefined circuit topology or overfitting the data. By linking features of the relaxation time distribution to physical properties of the system, such as charge transfer resistance, double-layer capacitance, and surface coverage, we demonstrate improved robustness and interpretability in parameter extraction. This work establishes a foundation for label-free, miniaturized microfluidic biosensing applications with enhanced theoretical rigor and reproducibility.

## I. INTRODUCTION.

Coupling electrochemistry with microfluidics offers significant advantages for molecular detection due to their high sensitivity and selectivity, portability, low cost and feasibility to miniaturize [1,2]. These platforms are particularly attractive for biosensing applications, where the ability to detect specific DNA strands at low concentrations is critical [3]. However, the interpretation of signals generated by such microchips remains challenging due to the delicate interplay of transports phenomena, interfacial kinetics and surface chemistry in highly confined microfluidic environments [4,5]. Robust experimental control as well as a well-defined theoretical framework are required to accurately analyze and untangle the complex events occurring near the electrode surface.

In this study, we adopted the Distribution of Relaxation Times (DRT) approach to analyze the electrochemical impedimetric response of a DNA-immobilized electrode under flow. Here we have focused on the analysis of Electrochemical Impedance Spectra (EIS) to study the interfacial processes through the frequency-dependent impedance response of the electrode. Commonly visualized in Nyquist plots, relevant information about the electrode can be extracted from EIS data via model-based interpretation. This involves fitting the experimental data to an equivalent circuit. Fig.1 shows a modified Randles equivalent circuit for modelling a partially blocked electrode, where each component represents a distinct physical process contributing to the impedance of the electrode [6,7]. For instance, this circuit accounts for ohmic resistance, double-layer capacitance, charge transfer kinetics, and mass transport effects. One key element is the $Q_{dl}$, which is

a Constant Phase Element (CPE), expressed as $Z_{Qdl} = \frac{1}{(Q_{dl} \cdot (j\omega)^{\alpha})}$, where the impedance represents the non-linear pseudo-capacitive behavior of the Gouy-Chapman-Stern double layer and $\alpha$ the phase in [0;1] is a transition parameter between capacitive and resistive behavior of the electronic exchange in the double layer [8,9]. The charge transfer resistance $R_{ct}$ is inversely proportional to the accessible surface area and is expressed as: $R_{ct} \propto (1-\theta)$, where $\theta$ is the surface coverage ratio [8]. It is the principal variant of a detection circuit. $R_e'$ represents the local ohmic resistance while $C_b$ represents the blocking capacitance for a modified electrode. $R_e$ represents the global ohmic resistance. The convection-diffusion impedance $Z_D$ is expressed as: $Z_D = \frac{\frac{\theta h (\sqrt{j\omega\tau_d})}{\sqrt{j\omega\tau_d}}$. This expression assumes a Nernstian depletion layer at the electrode surface. However, it must be noted that not all aspects of interfacial mass transport, particularly under turbulent or transitional flow conditions, can be captured using linear equivalent circuits [10].

As the dominant framework for interpreting EIS data in both classical and miniaturized electrochemical systems, numerous studies have employed combinations of resistive, capacitive, and diffusion-related elements in equivalent circuit modelling to describe interfacial phenomena, particularly in systems involving self-assembled monolayers, redox probes, and biomolecular recognition [11–14]. While such models can yield insight into charge transfer resistance, double-layer structure, and mass transport effects, they often require a prior knowledge about the electrochemical system for multi-parameter nonlinear fitting, which may produce non-unique solutions or parameters lacking clear physical meaning. Different
equivalent circuits could correspond to visually similar EIS plots. These challenges are exacerbated in microfluidic systems, where confinement, flow effects, and surface heterogeneity introduce additional complexity. These challenges motivate the development of alternative analytical frameworks that are both physically grounded and less reliant on overparameterized fitting procedures.

<figure><img src="image_2.png" /><figcaption>FIG. 1 The modified Randles equivalent circuit modelling the electrochemical system of a partially blocked electrode.</figcaption></figure>

Existing literature in the fields of batteries and fuel cells have demonstrated the ability of the DRT approach to resolve and extract meaningful kinetic parameters with greater clarity and less reliance on prior circuit assumptions [15–24]. Paul and coworkers have computed the DRTs functions numerically for lithium ion batteries within the Kramers-Krönig regime using Tikhonov regularization method [24]. The identifications of cell polarization processes in EIS spectra were achieved using DRT in high-temperature polymer electrolyte membrane fuel cells doped with phosphoric acid [16]. Song and Bazant presented a Distribution of Diffusion Times (DDT) inversion framework based on DRT analysis for the interpretation of nanostructured electrodes for energy conversion including a carbon nanotube supercapacitor, a silicon nanowire lithium-ion battery and a porous-carbon vanadium flow battery [25]. Piccioni et al. have employed a DRT approach based on lasso regression to analyze photoelectrochemical and photovoltaic systems [26]. A comprehensive review on the use of DRT for impedance analysis for batteries and fuel cells applications has been recently reported by Wang [27].

More recently, deep-neural-network-enhanced DRT was used to deconvolve real EIS data in various electrochemical systems including lithium-ion batteries and solid oxide fuel cells and proton exchange membrane fuel cells, suggesting the great potential to predict system performance using machine learning [28]. To the best of our knowledge, this approach has not been applied in channel electrode under flow.

In this study, we analyze the influence of flow rate on channel electrode using a theoretical approach based on equivalent circuit modelling in a microfluidic environment. To overcome the limitations of conventional multi-parameter regression techniques to extract circuit parameters. This framework enables a more transparent, physics-driven analysis of impedance data, facilitating the identification of meaningful electrochemical signatures related to hybridization and surface coverage. Finally, we tested our approach for partially blocked electrode in presence of self-assembled nucleic acids.

## II. METHODS

### A. Chemicals and reagents

Sodium chloride, potassium hydroxide, potassium hexacyanoferrate(III), and potassium hexacyanoferrate(II) trihydrate were purchased from Sigma Aldrich, France.

### B. Microfluidic chip and microelectrode fabrication

The microfluidic electrochemical platform with a two-electrode setup was fabricated in the clean room according to previous publications in our group [9,29–31]. Briefly, the electrode patterns were realized by SU-8 photolithography on a glass wafer, by spin-coating primer (hexamethyldisilazane) and negative photoresist AZ nLOF 2020 for 30 s at a speed of 4000 rpm s⁻¹ and an acceleration of 2000 rpm s⁻¹. The resin-coated wafer was baked at 105 °C for 1 min 30 s and exposed to UV light through hard contact mode. The post-exposure baking was carried out at 105 °C for 1 min 30 s, followed by light agitation in the developer MF-26. Metal deposition was carried out in an electron beam evaporator (Plassy MEB 550S) achieving 10/100 nm Ti/Pt on the glass wafer. The microfluidic channels of the device were realized by PDMS soft lithography on silicon wafer. The fluidic inlets and outlets were created using biopsy hole puncher (punch diameter = 0.5 mm). The glass chips and PDMS channels underwent O₂ plasma treatment for 30 s before assembly, followed by baking at 60 °C for at least 4 h.

The device architecture consists of a two-electrode system within a microfluidic channel. The counter electrode, characterized by a large surface area, functions as a near-zero-current reference, while the second smaller microband electrode serves as the working electrode (WE). The electrolyte solution contains a high concentration of a reversible redox couple, namely ferri-/ferrocyanide, which undergoes electron transfer at the WE, generating a measurable current. Due to the channel's small dimensions, the
flow is well approximated by a laminar Poiseuille profile. In such conditions, the concentration depletion zone formed near the electrode surface directly influences the local mass transport of redox species. Controlling this concentration profile is essential for ensuring measurement reproducibility (Fig. 2).

The electrodes are functionalized with a self-assembled monolayer (SAM) comprising a thiol molecule and a hybridizing agent. Specifically, mercaptohexanol (MCH) is employed for its strong affinity to noble metals, promoting a structured interface that enhances the specificity of the hybridizing agent. The latter typically consists of a single-stranded DNA (ssDNA) sequence, whose molecular recognition capabilities are defined by its base composition. Upon encountering a complementary target strand, hybridization occurs via the Watson-Crick base pairing (A-T and C-G), resulting in the formation of a double-helix DNA structure. As this hybridization process proceeds, previously unoccupied surface sites become occupied, altering the electrochemical processes at the interface. This evolution is characterized by the surface coverage parameter, $\theta$, which quantifies the ratio of the electrode surface occupied by hybridized DNA strands.

<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /><figcaption>FIG 2. (a) Different processes occurring near the electrode involved in charge transfer modelling. The dimensionality of the microchannel induces a Poiseuille flow along the x-axis, with $W_\alpha$ being the wall gradient, $Q$ is the total rate across a given section, $\delta_x$ is the thickness of the diffusion layer and $C_{Nernst}$ a Nernstian profile of concentration depletion near the electrode. (b) The Working Electrode (WE) being significantly smaller than the counter electrode (CE), it is used as a pseudo reference in a two-electrode setup.</figcaption></figure>

### C. Instrumentation and measurements

The microfluidic device was pre-treated by cycling repeatedly in 0.5 M KOH -0.2 V and -1.2 V at a scan rate of 0.05 V s⁻¹ for 10 cycles. All subsequent EIS

measurements were performed in a 20 mM equimolar solution ferri-/ferrocyanide containing 0.5 M NaCl at several flow rates (0.01, 0.05, 0.08, 0.1, 0.2, 0.5, 0.8 and 1 µL s⁻¹). EIS measurements were performed at VDC = 0 V and VAC = 10 mV within the frequency range of 1 MHz to 0.1 Hz. A potentiostat (Bio-Logic SP-300, controlled by the software EC-lab) was used for all electrochemical analyses. The flow rates were controlled by a syringe pump (Nemesys, Cetoni GmbH).

### D. EIS data fitting

The EIS data involve DNA probes in section 3.1 were extracted from our previous work [31]. The traditional EIS fitting approach was carried out in the software Measurement Model [32] to estimate several important parameters including the capacitance, ohmic resistance and the apparent charge transfer resistance.

## III. RESULTS & DISCUSSION

### A. Traditional EIS fitting approach

A common approach to analyzing experimental EIS data, and to physically interpreting the underlying physical processes, is to formulate a model hypothesis using an electrochemical equivalent circuit, and to fit the model parameters accordingly. In practice, one of the first challenges is to minimize the number of free parameters to be fitted. A typical EIS spectrum contains around thirty data points, of which only about twenty may be considered physically relevant to the proposed model. At the same time, a standard equivalent circuit model can include up to eight real parameters (see Fig. 1). Within this high-dimensional solution space, many regressions must be constrained within physically acceptable bounds.

The Measurement Model [32] provides a regression framework for equivalent circuits based on numerous prior studies. Based on the data in our previous work [31], specifically, the most accurate dataset of a DNA-immobilized electrode in Table S3<sub>electrode_1</sub>, we compared the regressions obtained from the study [31] with those produced by a python-based calculations (Table I). It can be noted that the parameter of interest $R_{ct,ap}$ (which is supposed to proportionally increase with the increase in surface coverage ratios) is one of the most stable fitted parameters with respect to the change of method. Nevertheless, the shape of the fitted curves as shown in Fig. 3, even for the data without significant anomalies, indicates minimizing the square deviation does not sufficiently resolve the underlying derivative features. As a result, the system's physical complexity is not fully captured by the fit as seen in Fig. 3. For example, while the experimental data
<table><caption>TABLE I. A table of log deviation of the different parameters fitting with between different methods.</caption><thead><tr><th></th><th>R<sub>e</sub></th><th>R'<sub>e</sub></th><th>C<sub>b</sub></th><th>Q<sub>dl</sub></th><th>α</th><th>R<sub>ct</sub></th><th>τ<sub>D</sub></th><th>R<sub>D</sub></th></tr></thead><tbody><tr><td>Mean</td><td>0.074</td><td>2.8</td><td>0.10</td><td>0.58</td><td>0.11</td><td>0.077</td><td>0.33</td><td>0.16</td></tr><tr><td>Standard deviation</td><td>0.037</td><td>1.3</td><td>0.13</td><td>0.20</td><td>0.031</td><td>0.036</td><td>0.47</td><td>0.14</td></tr></tbody></table>

clearly exhibit three distinct semi circles, the regression reproduces only two, despite the model having sufficient parameters to represent all three. One possible solution would be to set certain parameters: for an electrode at a given flow, $R_e, τ_D, R_D$ should be constant according to the theory. In reality, there is no good constraint on these parameters since a feedback loop on the regressed mean values amplifies the boundary errors (example: Supplementary Information in reference [31], values where $R_e' = 1$ and $α = 0.6$). The physical parameters setting these constants being unknown and/or likely to vary from one electrode to another, we are looking for an alternative method to deduce these constants from EIS.

<figure><img src="image_5.png" /><figcaption>FIG. 3 Comparison of two fitted EIS on the experimental points provided by [31].</figcaption></figure>

## B. DRT with DRTtools

DRT (Distribution of Relaxation Times) is an operation for deconvolving characteristic times over an impedance spectrum. It is defined as follows: according to the Kramers-Krönig (K-K) relations [33], for any impedance $Z(ω): R → C$ there exists a distribution $γ(τ): R^+ → R$ with normalised integral and two real numbers a and b such that:

$$Z(\omega) = b + a \int_{0}^{+\infty} \frac{\gamma(\tau)}{1+i\omega\tau} d\tau \qquad \text{Eq. (1)}$$

An elementary calculation allows us to check that the DRT $γ$ associated with an $R_0//C_0 + R_1$ is $γ = δ(τ-τ_0)$, and $a = R_1, b = R_0$ where $δ$ is the Dirac distribution and $τ_0 = R_0 C_0$. This shows the role of the DRT, revealing the characteristic times of the system. Instead, any linear system admits an impedance written as a rational fraction of $ω$, and its decomposition into

simple elements (SED) (assuming a non-degenerate case) reveals the characteristic times of the system in the spectrum of the relaxation time distribution. DRT, which is insensitive to the informational redundancy of $Z$ (K-K), is thus used in training models to increase the fidelity of parametric regressions of EIS [34].

DRTtools is a free access software developed by Ciucci et al. [35] performing gaussian peaks regression-DRT. **Fig. 4** shows the regression performed using this tool. The first graph compares the plots of experimental EIS versus reconstructed EIS according to the fitted DRT and **Eq. (1)**. Except for a high frequency phase drop, the fit is satisfactorily accurate. The calculated DRT reveals that for these experiments where only the flow rate varies, two peaks move with similar overall dynamics.

The **Fig. 5** is read as follows: along each curve, the log-resistance is indicated by the corresponding scale, but between two different curves, is an offset depending on the flow rate, with the corresponding second scale. The x-axis is the parameter $s$ which is log-characteristic time. These two peaks are named T2 and T3, and their time maxima can be fitted on an $s$ versus log-rate graph, as a linear function of slope 0.73 and uncorrelated $y$-intercepts (see **Fig. S1** in Supplemental Material). Given that the experiments consist in controlling only the flow rate within the microchannel, these two times can reasonably be attributed to a variation of $τ_D$. The phenomenon of the peak T2 splitting being seen at low rates could be hypothesised as a fluidic transition between two regimes for the shape of the depletion [30], but it might also be evidence for the removal of an accidental degeneration of the relaxation times.
<figure><img src="image_6.png" /><figcaption>FIG. 4 Comparison of DRT recalculated EIS (in red) with experimental data (in black), with the flow rates decreasing from a) to h), having the respective values of 1, 0.8, 0.5, 0.2, 0.1, 0.08, 0.05, 0.01 µL s⁻¹.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>FIG. 5 The log-DRT height (in Ω) is plotted against the log-characteristic-time to highlight the peaks in characteristic times. The offset between each curve is proportional to the flow rate (read the associated scale). In this one-parameter varying experiment, two characteristic times follow a coherent pattern they are T2 and T3. At low flow rates, a transition is observed corresponding to a transition to a state with non negligible diffusion when the diffusion layer's thickness increases.</figcaption></figure>

### C. DRT with DFT

In order to build up an extended perspective on the validity of modelling the complex processes taking place on a functionalized electrode, we wish to make the link between response time dynamics and parametric dynamics. To do this, we need an exact calculation of the DRT, for ideal EIS spectra. In fact, not all EIS plots generated in accordance with the equivalent circuit (Fig. 2) are well reconstructed by DRTtools as the regression of the first loop was lost at high frequencies (Fig. 4). To separate the exact times, the DRTtools DRT uses a Tichonov regularization that makes the problem invertible, but the quantification of stable regularization parameter $\lambda$ limits the finesse of the time resolution. To calculate the exact DRT, we proceed as follows: the change of variable, c.f. Eq. (1):

$$ \begin{cases} s = -\ln(\omega\tau_0) \\ z = \ln\left(\frac{\tau}{\tau_0}\right) \end{cases} $$

to where $\tau_0$ makes the problem non-dimensional. Defining the two functions $Z''(\omega\tau_0) = \Im(Z)(\omega)$ and $\Gamma(\omega\tau_0) = a\gamma(\omega)\tau$, then comes the result $Z''(s) = -\int_{R} \frac{\Gamma(z)}{2cosh(s-z)} dz$. But this is a convolution product and by deconvolving this product

in Fourier space (with the convention $TF(f)(\eta) = \int_{R} e^{-2i\pi\eta z} f(z) dz$), we obtain the following equality:

$$ \Gamma(s) = \frac{2}{\pi} TF^{-1}(ch(\pi^2\eta)TF(Z'')(\eta)) \qquad \text{Eq. (2)} $$

And for some values of $W$ as shown in Fig. S2 and Fig. S3

$$ \Gamma(s) \approx \frac{2}{\pi} TF^{-1}\left(\frac{\cosh(\pi^2\eta)TF(Z'')(\eta)}{1+W \cosh^2(\pi^2\eta)}\right) \qquad \text{Eq. (3)} $$

Where Eq. 3, comes from Wiener's method, and is more stable to compute numerically. If comparing the Wiener method and a Tichonov regularization, $\lambda$ is a root of $W$ [36]. The tuple of linear log-log regressions parameters on a coherent set of characteristic times of a system (vide infra) is called MAD (Maxima Asymptotic Dynamics) (Eq. (3)).

One time has few variations with respect to other parameters, it is the $\tau_D$ peaks system, generating a MAD slope $\frac{d \ln(\tau_{max})}{d \ln(\tau_D)} = 1$, independent from $\alpha$ (see Fig. S4).

However, the resolution of conduction-diffusion in 1D in a Poiseuille flow of rate $Q$, gives $\frac{d \ln(\tau_D)}{d \ln(Q)} = -\frac{2}{3}$. This could explain the coefficient of -0.73 calculated in the previous part. The difference between the experimental and the theoretical MAD slopes could be attributed a bias within DRTtools, or a limit of the calculation of the model of calculation of diffusion limit layer thickness. In Fig. 5, the MADs slopes are shown to follow several theorems (TH):

**TH.1** The MAD of a linear ideal model has only integer slopes (demonstrated by calculus on Eq. 3, it basically comes from the SED of a rational fraction obtained after a change of variables).

**TH.2** The mono-parametric (if only one parameter varies) MAD of a CPE, has its slopes integer multiples of its inverse phase $\frac{1}{\alpha}$. (demonstrated from calculus with complicated integrals).

**TH.3** The MAD of Nernstian diffusion impedance has slope 1 (asymptotic development + TH.1)

No relevant theorem has yet been found about the intercepts of MAD's which are expected to have their own dynamics as well as peaks heights (which
dynamics should be trivial if linear (see the calculus of
parameter b from DRT of R₀//C₀ + R₁ circuit).

In Fig. S5a, the ideal EIS spectrum has been produced using the parameters of Table S3<sub>electtrode_1</sub> in the Supplementary Information of our previous work in reference [31]. The likeness of the two spectra (Fig. S5a and b) confirms the stability of the previously selected parameter *W*. In Fig. S4 where $\tau_D$ varies shows that the non-linearities in the analytical expression of $\tau_D$ creates an accidental multiplicity, whereas the slope equality shows that multiples peaks describe the same phenomenon. Further calculations may deconvolute an expression for $\tau_D$ from these observations. Additionally in Fig. S6, the case where $C_b$ varies, the slope is completely linear and thus has an asymptotic MAD slope of 1.

In Fig. 6 where R<sub>ct</sub> varies shows how τ<sub>D</sub> could be
deconvoluted along with a regular experiment,
providing a new way to link the variations of R<sub>ct</sub> and
the variations of α. These three theorems provide an
initial analysis of the MAD. It therefore becomes
possible to check the linearity of a process at the
electrode (TH.1), but also to check the ability to
vary the parameters in isolation from each other and
their dependance in experimental conditions. Finally,
the parameter α in a regular EIS data set can be found
using TH.2, under the condition that the <i>a priori</i>
evolution of R<sub>ct</sub> (<i>via</i> θ) can be predicted. This
possibility of accessing Randles parameters via direct
calculations echoes Brug's study where a
comprehensive DRT analysis lead to a method
accessing the phase of the CPE through traditional
admittance studies [37]. However, the new numerical
methods make it possible and interesting to directly
study physical properties in the time-resistance plane
of DRT's. Further studies will enable us to move away
from regressed data to calculated data and greatly
stabilize the determination of circuit parameters.

<figure><img src="image_8.png" /><figcaption>FIG. 6 The case of variation in R<sub>ct</sub> illustrates TH.2 where the asymptotic slope is 1.32 approximately the 1/α. All parameters unless specified, have default value of probe 3<sub>1</sub> from the Supplementary Information of reference [31]: Re = 2563 Ω, Re' = 2823 Ω, τ<sub>D</sub> = 0.04 s, R<sub>D</sub> = 23902 Ω, Q<sub>dl</sub> = 4.3 × 10<sup>-8</sup> F s<sup>-α</sup>, α = 0.74, R<sub>ct</sub> = 49368 Ω, C<sub>b</sub> = 5.18 × 10<sup>-9</sup> F, W = 1 × 10<sup>-8</sup>.</figcaption></figure>

IV. CONCLUSIONS

In this work, we have demonstrated that the
conventional multi-parametric regression of Randles
equivalent circuit is inherently unstable and often
obscures the physical complexity of electrochemical
microfluidic systems under flow. By applying the
Distribution of Relaxation Times (DRT) approach, we
showed that the EIS data can be analyzed with higher
accuracy and interpretability, which enables us to
resolve the previously overlapping processes and
identify relevant parameters such as the charge
transfer resistance and the surface coverage ratio.
These results presented DRT as a powerful and robust
alternative to traditional regression-based methods,
providing more stable and physically meaning
parameters for microfluidics-based electrochemical
systems and advancing the development of machine
learning-based biosensing prediction models. Future
work will focus on systematic DRT deconvolution to
further stabilize multiparametric fits on Randle's
equivalent circuits and extend this framework to more
complexed microfluidic electrochemical
environments.

ACKNOWLEDGMENTS

The authors would like to thank the DIMELEC project
ANR-19-CE09-0016 for funding and RENATECH
clean room facilities at C2N, Palaiseau, France.
[1] D. G. Rackus, M. H. Shamsi, and A. R. Wheeler, Electrochemistry, biosensors and microfluidics: a convergence of fields, Chem. Soc. Rev. **44**, 5320 (2015).

[2] K. Ren, J. Zhou, and H. Wu, Materials for Microfluidic Chip Fabrication, Acc. Chem. Res. **46**, 2396 (2013).

[3] A. Sassolas, B. D. Leca-Bouvier, and L. J. Blum, DNA Biosensors and Microarrays, Chem. Rev. **108**, 109 (2008).

[4] A. Manz, D. J. Harrison, E. M. J. Verpoorte, James. C. Fettinger, A. Paulus, H. Lüdi, and H. M. Widmer, Planar chips technology for miniaturization and integration of separation techniques into monitoring systems: Capillary electrophoresis on a chip, Journal of Chromatography A **593**, 253 (1992).

[5] D. J. Beebe, G. A. Mensing, and G. M. Walker, Physics and applications of microfluidics in biology, Annu Rev Biomed Eng **4**, 261 (2002).

[6] A. M. Bond and R. D. Braun, Modern Polarographic Methods in Analytical Chemistry, J. Electrochem. Soc. **127**, 528C (1980).

[7] A. J. Bard, L. R. Faulkner, and H. S. White, *Electrochemical Methods: Fundamentals and Applications* (John Wiley & Sons, 2022).

[8] M. E. Orazem and B. Tribollet, *Equivalent Circuit Analogs*, in *Electrochemical Impedance Spectroscopy* (John Wiley & Sons, Ltd, 2008), pp. 153–162.

[9] M.-C. Horny, M. Freisa, C. Poujouly, P. Gonzalez-Losada, J. Le Gall, T.-H.-N. Dinh, D. Bouville, I. Le Potier, A. Pallandre, and J. Gamby, Electrochemical Nucleic Acid Biosensor on a Microfluidic Chip to Understand the Coupling between Electrochemistry and Microfluidics, J. Chem. Educ. **101**, 605 (2024).

[10] A. Ramos, A. González, A. Castellanos, N. G. Green, and H. Morgan, Pumping of liquids with ac voltages applied to asymmetric pairs of microelectrodes, Phys. Rev. E **67**, 056302 (2003).

[11] C. S. S. Sangeeth, A. Wan, and C. A. Nijhuis, Equivalent Circuits of a Self-Assembled Monolayer-Based Tunnel Junction Determined by Impedance Spectroscopy, J. Am. Chem. Soc. **136**, 11134 (2014).

[12] A. Piper, D. K. Corrigan, and A. R. Mount, An electrochemical comparison of thiolated self-assembled monolayer (SAM) formation and stability in solution on macro- and nanoelectrodes, Electrochemical Science Advances **2**, e2100077 (2022).

[13] H. Einati, A. Mottel, A. Inberg, and Y. Shacham-Diamand, Electrochemical studies of self-assembled monolayers using impedance spectroscopy, Electrochimica Acta **54**, 6063 (2009).

[14] S. Oliveira, B. V. Jones, P. Estrela, P. R. F. Rocha, and N. M. Reis, Real-Time Monitoring of Chemisorption of Antibodies onto Self-Assembled Monolayers Deposited on Gold Electrodes Using Electrochemical Impedance Spectroscopy, Langmuir **41**, 15974 (2025).

[15] M. A. Danzer, Generalized Distribution of Relaxation Times Analysis for the Characterization of Impedance Spectra, Batteries **5**, 3 (2019).

[16] A. Weiß, S. Schindler, S. Galbiati, M. A. Danzer, and R. Zeis, Distribution of Relaxation Times Analysis of High-Temperature PEM Fuel Cell Impedance Spectra, Electrochimica Acta **230**, 391 (2017).

[17] A. Kežionis and E. Kazakevičius, Some features of the analysis of broadband impedance data using distribution of relaxation times, Electrochimica Acta **349**, 136379 (2020).

[18] A. L. Gavrilyuk, D. A. Osinkin, and D. I. Bronin, The use of Tikhonov regularization method for calculating the distribution function of relaxation times in impedance spectroscopy, Russ J Electrochem **53**, 575 (2017).

[19] F. Ciucci and C. Chen, Analysis of Electrochemical Impedance Spectroscopy Data Using the Distribution of Relaxation Times: A Bayesian and Hierarchical Bayesian Approach, Electrochimica Acta **167**, 439 (2015).

[20] A. Kulikovsky, PEM fuel cell distribution of relaxation times: a method for the calculation and behavior of an oxygen transport peak, Phys. Chem. Chem. Phys. **22**, 19131 (2020).

[21] E. Ivers-Tiffée and A. Weber, Evaluation of electrochemical impedance spectra by the distribution of relaxation times, Journal of the Ceramic Society of Japan **125**, 193 (2017).

[22] J. Illig, M. Ender, T. Chrobak, J. P. Schmidt, D. Klotz, and E. Ivers-Tiffée, Separation of Charge Transfer and Contact Resistance in LiFePO4-Cathodes by Impedance Modeling, J. Electrochem. Soc. **159**, A952 (2012).

[23] J. Zhang, A. C. Welinder, A. G. Hansen, H. E. M. Christensen, and J. Ulstrup, Catalytic Monolayer Voltammetry and In Situ Scanning Tunneling Microscopy of Copper Nitrite Reductase on Cysteamine-Modified Au(111) Electrodes, J. Phys. Chem. B **107**, 12480 (2003).
[24] T. Paul, P. W. Chi, P. M. Wu, and M. K. Wu, Computation of distribution of relaxation times by Tikhonov regularization for Li ion batteries: usage of L-curve method, Sci Rep **11**, 12624 (2021).

[25] J. Song and M. Z. Bazant, Electrochemical Impedance Imaging via the Distribution of Diffusion Times, Phys. Rev. Lett. **120**, 116001 (2018).

[26] A. Piccioni, P. Vecchi, L. Vecchi, S. Grandi, S. Caramori, R. Mazzaro, and L. Pasquini, Distribution of Relaxation Times Based on Lasso Regression: A Tool for High-Resolution Analysis of IMPS Data in Photoelectrochemical Systems, J. Phys. Chem. C **127**, 7957 (2023).

[27] Z. Wang, Distribution of relaxation times for impedance analysis in batteries and fuel cells, Nat. Rev. Clean Technol. **1**, 453 (2025).

[28] E. Quattrocchi, B. Py, A. Maradesa, Q. Meyer, C. Zhao, and F. Ciucci, Deconvolution of electrochemical impedance spectroscopy data using the deep-neural-network-enhanced distribution of relaxation times, Electrochimica Acta **439**, 141499 (2023).

[29] M. Freisa, C. Poujouly, I. Le Potier, T. H. N. Dinh, L. Couraud, D. Bouville, C. Sella, L. Thouin, and J. Gamby, Highlighting the impact of blocking monolayers on DNA electrochemical sensors. Theoretical and experimental investigations under flow conditions, Electrochimica Acta **505**, 145006 (2024).

[30] C. Poujouly, P. Gonzalez-Losada, R. Boukraa, M. Freisa, J. Le Gall, D. Bouville, C. Deslouis, and J. Gamby, Diffusion–convection impedance for a micro-band electrode under microfluidic conditions, Electrochemistry Communications **137**, 107262 (2022).

[31] H. M. Man, C. Omar, M. Freisa, D. Bouville, T. Baptiste, A.-M. Haghiri-Gosnet, H. Jacquier, I. Le Potier, and J. Gamby, Microfluidics-Based Electrochemical Detection of Antimicrobial-Resistant DNA Sequence in Lysed Escherichia coli Medium, ACS Electrochem. **1**, 886 (2025).

[32] W. Watson and M. E. Orazem, EIS: Measurement Model Program, (2020).

[33] H. Schichlein, M. Feuerstein, A. Müller, A. Weber, A. Krügel, and E. Ivers-Tiffée, System Identification: A New Modelling Approach for SOFC Single Cells, Proc. Vol. **1999–19**, 1069 (1999).

[34] Z. Lyu, A. Sciazko, N. Shikazono, and M. Han, Enhanced temporal prediction of electrochemical impedance spectroscopy using

long short-term memory neural networks, Electrochimica Acta **508**, 145227 (2024).

[35] T. H. Wan, M. Saccoccio, C. Chen, and F. Ciucci, Influence of the Discretization Methods on the Distribution of Relaxation Times Deconvolution: Implementing Radial Basis Functions with DRTtools, Electrochimica Acta **184**, 483 (2015).

[36] N. Florsch, C. Camerlynck, and A. Revil, Direct estimation of the distribution of relaxation times from induced-polarization spectra using a Fourier transform analysis, Near Surface Geophysics **10**, 517 (2012).

[37] G. J. Brug, A. L. G. van den Eeden, M. Sluyters-Rehbach, and J. H. Sluyters, The analysis of electrode impedances complicated by the presence of a constant phase element, Journal of Electroanalytical Chemistry and Interfacial Electrochemistry **176**, 275 (1984).