# Current Transients in Blocking Electrochemistry Experiments Modeled by Analog Simulation of Diffusion

Arthur Langlard, Christine Thobie-Gautier, Mohammed Boujtita, Estelle Lebègue

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Arthur Langlard, Christine Thobie-Gautier, Mohammed Boujtita, Estelle Lebègue. Current Transients in Blocking Electrochemistry Experiments Modeled by Analog Simulation of Diffusion. Journal of Physical Chemistry A, 2025, 129 (26), pp.5881-5888. 10.1021/acs.jpca.5c02279. http://dx.doi.org/10.1021/acs.jpca.5c02279

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# Current Transients in Blocking Electrochemistry Experiments Modeled by Analog Simulation of Diffusion

Arthur Langlard,* Christine Thobie-Gautier, Mohammed Boujtita, and Estelle Lebègue*

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Cite This: J. Phys. Chem. A 2025, 129, 5881–5888

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Supporting Information

**ABSTRACT:** In blocking particle-impact electrochemistry, the single collision events often show current transients that do not immediately lead to the expected steady-state current but rather appear distorted or bouncy, with a short delay before reaching again a stable current. This artifact-looking increase of current up to a steady state, observed in both soft and hard insulating particle impacts, seems to be a physicochemical effect rather than a mechanical one. The reorganization of the diffusion layer around the disruptive insulating entity on the electrode surface is expected to induce a change in the shape of the measured

current and could be the cause of this effect. To investigate this, we present here an original analog simulation of the transient current flowing through the working ultramicroelectrode during a blocking impact. By modeling single electrochemical impacts using only diffusion, we obtained a similarly shaped signal thanks to an analog simulator based on an electric network composed of resistors and capacitors. We suggest that the relaxation of the diffusion layer after a blocking impact is the main cause of such distortion in the current transients.

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## INTRODUCTION

Since its formulation in 1822 by Joseph Fourier,¹ the heat equation has received considerable attention.²,³ This is partly because it describes isotropic diffusion processes in numerous fields, including electrochemistry where it is expressed as Fick's second law:⁴,⁵

$$ \frac{\partial c}{\partial t} = D \nabla^2 c \qquad (1) $$

expressing the spatial and temporal evolution of the concentration c of a chemical species in a solution with a constant diffusion coefficient, when no convection or migration occurs.

In blocking impact electrochemistry, insulating particles in solution come into contact with a working ultramicroelectrode (UME) and mask a patch of its electroactive surface.⁶,⁷ When the UME is polarized at a suitable potential and no particle blocks the electrode surface yet, a redox probe dissolved in solution is continuously oxidized or reduced at the electrode. There, a lasting current is measured, gradually reaching a steady value because of the growth of the diffusion layer. An “impact” of an insulating object will show up as a sharp decrease in current (Figure 1). Using UMEs, this phenomenon has been used as an analytical technique to detect miscellaneous microscopic or nanoscopic entities.⁸⁻¹¹ Moreover, the frequency of the collisions and the current drop can often be linked respectively to the number of insulating objects in solution and the size of the insulating object, leading to the development of quantitative sensors.¹²,⁸,¹³,⁵

When our group performed such experiments with liposomes, bacteria, and polystyrene spheres on disk-UMEs,¹⁴ we

<figure><img src="image_8.png" /><figcaption>**Figure 1.** Two types of insulating particles, 1,2-dimyristoyl-*sn*-glycero-3-phosphocholine (DMPC) liposomes and polystyrene (PS) spheres, are used to illustrate the hindrance of the oxidation of the Fe(CN)₆ 4⁻ redox probe at the impact location on the 10 µm diameter Pt disk-UME, which is polarized at +0.7 V vs Ag/AgCl. Insets highlight sporadic signals where the current drop overshoots the steady value. The thin trace is a rolling average over 5 points.</figcaption></figure>

noticed that collision events occasionally did not yield sharp current steps, but the end of the transient appeared distorted

Received: April 4, 2025

Revised: June 1, 2025

Accepted: June 13, 2025

Published: June 24, 2025

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or bouncy (see insets of Figure 1) and a delay (often less than 1 s) was needed to consider the impact done.

Despite its surprising aspect, records or depictions of this phenomenon seem to have eluded the literature. Probably because of its artifact look and intermittent occurrences, often happening on top of a slow declining current, which can be caused by the settling of the steady-state diffusion profile, or by a surface change of the electrode, such as hexacyanoferrate species adsorption¹⁵ or the formation of a platinum oxide film.¹⁶ A back-of-the-envelope calculation rules out the elastic nature of the object as a cause of this distorted step signal: objects larger than 100 nm, with Young's moduli higher¹⁷ than 1 MPa and masses lower than 10⁻¹² g show an oscillation period shorter than 1 µs when modeled by a damped harmonic oscillator (see Section 1 and Figure S1 of the Supporting Information), which is several orders of magnitude faster than the evolution time of the observed transient (0.1 to 1 s). Moreover, polystyrene beads, which are more rigid than phospholipid liposomes or bacteria, also showed this effect. Therefore, it seems that this discreet continuous increase of current reaching a steady state is a physicochemical effect rather than a mechanical one.

A blocking object disrupts the concentration gradient of the redox probe near the surface of the working electrode during the impact. The reorganization of this diffusion layer around the insulating entity is expected to modify the shape of the measured current and could be the cause of this effect. To investigate this, we wish to simulate the transient current flowing through the working electrode resulting from a blocking impact. While various digital methods have been used to simulate the electrochemical behavior of finite disk electrodes and UMEs,¹⁸⁻²² numerical simulations of transient events require challenging time discretization and stability. The commercially available COMSOL Multiphysics software is usually used for this type of simulations.²³⁻²⁹ Nonetheless, we consider its cost and closed sources unreasonable in regard to the promotion of accessible and open science. We decided to use an analog simulation instead of a digital one to circumvent these issues.

## EXPERIMENTAL METHODS

**Blocking Electrochemistry.** The following procedure is similar to the one we previously reported.¹⁴ All chemicals were reagent grade and used as purchased without further purification. Water used in each experiment was deionized water. Chloroform (>99%) was purchased from Fisher Scientific. Potassium ferrocyanide trihydrate (98.5%) was purchased from Acros Organics. Phosphate buffer (PB) solution at 1.0 M and pH 7.4 (25 °C) was purchased from Sigma-Aldrich and stored at 3 °C. Iron(III) nitrate nonhydrate was purchased from Acros Organics. The 1,2-dimyristoyl-*sn*-glycero-3-phosphocholine (DMPC) lipids were purchased as a powder from Sigma-Aldrich (Avanti Polar Lipids) and stored at -18 °C. Polystyrene spheres were obtained from Polyscience Europe GmbH.

As we previously reported,¹⁴ liposome suspensions were prepared by dissolving 5 mg DMPC lipid (powder) in chloroform (1 mL) for the complete dissolution of lipids. The homogeneous mixture was placed under a nitrogen (N₂) flow for 2 h for the complete evaporation of chloroform. The dry lipid film was hydrated by the addition of an aqueous solution (1 mL of 0.1 M K₄Fe(CN)₆ as a redox probe in 1.0 M PB), and then the mixture was sonicated for 1 min and heated

on a hot plate at 50 °C for 2 h under stirring. The DMPC liposomes were extruded using 400 nm diameter polycarbonate membranes from Avanti Polar Lipids. The extrusion of liposomes was carried out with the extruder set from Avanti Polar Lipids, which includes a mini-extruder, two syringes of 1 mL, polycarbonate membranes of 0.4 µm, and filter supports. The suspension of particles was passed through the extruder 15 times, which was kept warm at 50 °C, to obtain a suspension of DMPC liposomes. Then, size-exclusion chromatography (PD-10 Desalting Columns, Cytiva) was performed to remove the redox probe that was not encapsulated inside the liposomes. By using a 1.0 M PB aqueous solution at pH 7.4, the first fraction containing the liposomes was kept. The end is detected by the precipitation of a Prussian blue compound upon contact with an iron(III) nitrate aqueous solution, indicating the presence of ferrocyanide ions in the eluate. This final step for redox DMPC liposomes (encapsulating ferrocyanide ions) typically yields a nanomolar suspension of DMPC liposomes, which is stored at 3 °C for 3 days.

The electrochemical experiments were performed at room temperature (21 ± 2 °C) using an SP-300 potentiostat (BioLogic) with an ultralow current module and with a three-electrode cell placed in a Faraday cage (BioLogic FC-45) and using the EC-Lab software. The 10 µm diameter Pt disk UME purchased from CH Instruments (CHI107) was used as the working electrode. Pt wire was used as a counter electrode and pseudoreference electrode. For checking the stability of the Pt pseudoreference, all the potentials were controlled and reported with an Ag/AgCl (3 M NaCl) reference electrode, as previously described.¹⁴ The electrochemical aqueous solution was composed of 0.1 M K₄Fe(CN)₆ dissolved in 1.0 M PB, with a pH of 7.37 ± 0.05. For all chronoamperometry *i*-t curves recorded, the sample interval (in sampling time) was 10 ms. A shorter sample interval may help capture more subtle features of the current transient, if the signal-to-noise ratio of the instrumentation allows it. Before each electrochemical experiment, the Pt UME was mechanically polished using wetted fine-grit silicon carbide paper from Struers (4000-grit SiC) and washed in water.

**Electronic Setup.** A summary of the electronic circuit used for the simulation of 2D diffusion is given in Figure S2 and Figure S3, and its main electronic parts are listed in Table S1. In normal operation, this circuit drew a maximum current of 2 A from a power supply of 12 V. This total power of 24 W was dissipated over a surface with an area larger than 0.64 m², so we did not expect a temperature rise that can significantly change the values of the components.

For general use of the circuit, the output of the current-to-voltage converter was connected with a BNC cable to a Philips PM3232 (10 MHz bandwidth) oscilloscope. Pictures were taken with a Panasonic DMC-GX8 camera with a 0.5 s exposure time. For the following situations, a different oscilloscope was used.

**Time-Response Measurements.** The capture of waveforms was performed with a Tektronix 2205 (20 MHz bandwidth) oscilloscope with two probes GW Instek GTP-070B-4 (70 MHz bandwidth). Both probes were compensated and set to an attenuation ratio of 10:1. The voltage pulses were generated with an HP 8082A pulse generator.

**Data Acquisition and Video Animation.** The acquisition of waveforms was done with a GW Instek GDS-1054B (50 MHz bandwidth) oscilloscope with three probes GW Instek GTP-070B-4 (70 MHz bandwidth). All probes were
<figure><img src="image_11.png" /><figcaption>Figure 3. (A) Time response of the analog switch 74HC4066D chopping a 0.8 V voltage source (circuit in Figure S5). Trace A is the switching signal, trace B is the output of the switch when connected to ground with 470 Ω. The measured level is slightly below 0.8 V because of the resistive divider formed by the 50 Ω output impedance of the voltage source and the 470 Ω pull-down resistor. (B) Time response of the current-to-voltage converter (circuit in Figure S6). A pulse generator (trace A) is connected to the current-to-voltage converter via a 1 kΩ resistor, generating a pulsed current of 5 mA. Trace B is the output of the converter.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 4. Current measured when the RC network is configured to simulate planar (1D) diffusion. Similarly to the Cottrell equation, a linear relationship between the inverse of the squared current i and time is expected. Plotting versus time t instead of t⁻¹/² highlights the maximum duration of the experiment before the approximation of an infinite solution starts to fail. The simulation stops around 16 ms, when the voltage at the base of the grid switches back to E to reset the circuit to its initial state. Because the display of the oscilloscope was set to record the positive current, the negative spike of current is out of the oscilloscope's scale, explaining the lack of symmetry between the positive and negative surges of current. The signal was recorded four times with four different vertical sensitivities to avoid distortion caused by an overload of the input stage of the oscilloscope. The data plotted here is a composite data set of these four acquisitions, detailed in Table S2.</figcaption></figure>

their malleability, low cost, and compactness.³⁵ Unfortunately, digital simulations seldom reach the ease of interaction offered by analog simulations.

We wish to solve eq 1 in 2D thanks to the resistor-capacitor (RC) network³⁶,³²,³⁷ depicted in Figure 2A, which models the right half of the semi-infinite diffusion plane (see space discretization in Figure 2B). When all the switches are closed, an electric charge Q injected at any node of this network will diffuse through the resistors, with currents proportional to the voltage across each resistor R. The voltage U at each node is linked to the charge q stored in the nodal capacitor C by the relation q = UC. The physical quantity U, linearly related to the accumulation of a conserved quantity (q) causes the latter to

be transferred along the gradient of U, therefore the conserved quantity is described by an equation analog to Fick's second law:³⁸,³⁹

$$ \frac{\partial U}{\partial t} = \frac{l^2}{RC} \left( \frac{\partial^2 U}{\partial x^2} + \frac{\partial^2 U}{\partial y^2} \right) \qquad (2) $$

where l is the physical distance between the points of the system modeled by two adjacent nodes. A more rigorous proof is given in Section 3 and Figure S4 of the Supporting Information, and by Steere in a similar electric analog of heat flow.³⁷

As such, one can view this network as chained low-pass filters in two dimensions: the application of a voltage at one node will seem to propagate with delay to neighboring nodes, similarly to the concentration that seems to spread in solution as matter diffuses. Opening a switch prevents current from flowing from one node to its adjacent neighbor, and is used to simulate a diffusion path obstructed by a blocking object. The connected switches at the bottom of the network simulate the electrode, and the collected current can be measured with a current-to-voltage converter, represented here by an ammeter. Therefore, the resulting circuit can be used to compute the changes in the diffusion process when a still blocking object appears at the electrode.

We use this network to simulate 2D diffusion, with a mesh size of 64 × 64 nodes, R = 1.5 kΩ, and C = 4.7 nF. Figure S3 is a picture of the resulting electronic setup. The values of R and C were chosen so that the event is fast enough to be seamlessly seen on an oscilloscope when repeated. The short duration of one simulation cycle, approximately 32 ms, is convenient for continuously exploring the configurations space of different positions, sizes, and shapes of the blocking object, along with different sizes of the working electrode. For simulating a 2D diffusion-limited electrochemical reaction, the boundary conditions are listed in Table 1.

The response time of the switches is checked by connecting a voltage source to one side of a switch, and then triggering the circuit driving the switch while monitoring the current flowing through the switch (Figure S5). The resulting signal shows that the switches are fast enough to generate submicrosecond
<figure><img src="image_13.png" /><figcaption>**Figure 5.** Effect of the size of the simulated blocking object and its position on the simulated UME on the shape of the current transient coming out of the RC network, while the size of the electrode is kept at 32 pixels (px). The position of the impact is given in Table S3. Figure S8 shows the corresponding digitized signals. The blocking impact is generated approximately 0.5 ms after the triggering of the oscilloscope, by opening switches in the RC network as indicated in the top diagram (resistors and capacitors were omitted for clarity).</figcaption></figure>

blocking events (Figure 3A). The response time of the circuit measuring the current is also assessed by connecting a signal generator to its input through a resistor and then stepping the voltage to generate a fast change of current (Figure S6). The output of the circuit settles within less than 1 µs (Figure 3B), indicating that it is suitable for measuring submicrosecond current transients.

To prove that this RC network truly models the diffusion equation, we tried to simulate 1D diffusion in a semi-infinite plane to compare the obtained current with the known analytical solution. That is the Cottrell equation when considering the diffusion of chemical species. After configuring the network to model 1D diffusion (all switches closed, except the horizontal switches of the last column *m* = 63, Figure S7), the current *i* flowing through the base of the network is recorded while the voltage at the base of the grid is switched from *U* = *E* to *U* = 0. Plotting *i*<sup>-2</sup> versus time yields a linear relationship as expected by the Cottrell equation (Figure 4). Linearity fails at times higher than 6 ms because, as the size of the simulated diffusion layer increases, the finite network models less and less accurately a 1D semi-infinite solution.

The simulation of a blocking shape in the 64 × 64 network gives a current transient similar to the ones obtained in actual electrochemical blocking experiments. As expected, changing the simulated size of the working electrode changes the global amount of current, while varying the size and position of the blocking object changes the height of the current step (Figure 5).

As illustrated in Figure 5, the heights of the current steps are greater when the impacts are located near the edge of the simulated electrode. This is consistent with the literature, as the finite planar geometry of the electrode causes the current density to not be uniform across the electrode, but to increase from the center to the edge.⁵,⁴⁰ Therefore, identically sized objects generate different changes of current depending on their impact location.¹²,²⁵,⁴¹,²⁸

With *R* = 1.5 kΩ and *C* = 4.7 nF, the fixed ratio $I_D^2/I_A^2$ is about 7.1 µs. By changing both the size of the simulated electrode surface and of the blocking object (see Figure S9 and Table S4), it is possible to access different values of *D* as $I_D^2/I_A^2$ remains constant. This, however, causes a global change in the amount of current, as the number of paths available for the current to
<figure><img src="image_14.png" /><figcaption>**Figure 6.** Effect of different simulated diffusion coefficients *D* of the redox species on the shape of the current transient coming out of the RC network, when the impact of a flat insulating object is simulated. Changing the diffusion coefficient is done by scaling accordingly the size of the simulated electrode (black stripes) and the simulated object (red line). The blue text indicates the duration of half the current bend. Note that the vertical scale of each curve was adapted to display all current transients with the same apparent height. The blocking impact is generated approximately 1 ms after the triggering of the oscilloscope.</figcaption></figure>

flow is changed. When the simulated diffusion coefficient is increased across 1 order of magnitude, the relaxation process is faster with half-recovery times ranging from 0.74 to 0.41 ms (Figure 6).

Because the nodal voltages represent the concentrations of the species, mapping the voltages of the network gives a direct representation of the concentration distribution in solution (Figure 7A), which seems to agree with the numerical simulations of blocking spheres in the literature.¹⁹ The calculation of the current in each branch of the network via Ohm's law shows the flow of the redox species around the blocking object (Figure 7B), revealing the relaxation process as a reorganization of the diffusion layer and a slight increase of diffusion fluxes around the blocking object (Figure 7C, Figure S10 and video animation file). This diffusion flux map also shows the expected edge effect for a finite planar electrode.⁴⁰

The fixed ratio $l_D^2 = 7.1 \ \mu\text{s}$ of the network allows us to make convenient and real-time measurements on an oscilloscope, but it is not suitable for an immediate model of usual experiments ($D = 6.4 \times 10^{-10} \ \text{m}^2 \ \text{s}^{-1}$ for $\text{Fe(CN)}_6^{4-}$ and $l = 1.5 \times 10^{-7} \ \text{m}$ for a $10 \ \mu\text{m}$ diameter UME). Nevertheless, one can simulate the correct size of the working electrode, temporarily recording a current corresponding to a different diffusion coefficient. Extrapolating the transient current with the correct diffusion coefficient can be done qualitatively using Figure 6, which results in a similar effect.

<figure><img src="image_15.png" /><figcaption>**Figure 7.** (A) Simulated concentration profile around the blocking object obtained by measuring the voltage at each node of the RC network. (B) Simulated diffusion paths around the blocking object obtained by calculating the current in each branch of the RC network. (C) Simulation of the evolution of the diffusion layer over time after the impact of a blocking object onto an electrode. The four frames depict four distributions of the voltages at different moments during the simulation. Fluxes are drawn as gray arrows at the location of the impact to provide a visual reference for the edge of the blocking object. The electrode surface is represented by a gray line. The blocking impact is generated approximately 4 ms after the triggering of the oscilloscope.</figcaption></figure>
## ■ CONCLUSIONS

By modeling electrochemical nanoimpact processes using only diffusion, with an analog simulator composed of an RC network, we obtained a signal with a shape similar to the one recorded electrochemically. The use of analog means allowed us to avoid the sticky step of time discretization in numerical methods for transient events. In our circuit, the signal has a distorted end because a change of current pathways is forced by the opening switches, modeling an insulating object in solution. Therefore, we suggest that, in an analog fashion, the distortion of the step is not an instrumental artifact or caused by the elasticity of the particles, but can be explained by the relaxation of solute mass fluxes in the diffusion layer, which needs reorganization to accommodate for the insulating object. The described relaxation may not be the only reason for such phenomenon, as capacitive current induced by the local change of the structure of the electric double layer, or local convection caused by the movement of the particle, may also generate transient currents. Both of these effects were not modeled by the presented electrical circuit.

All the presented results stem from the simulation of a diffusion-limited electrochemical reaction implying a constant concentration of the redox species at the surface of the working electrode. In the case of a hypothetical kinetically limited reaction, there is no diffusion layer so this effect is not expected. We speculate that changing the potential of the working electrode²⁸ or adding a substrate to cause an EC' mechanism,⁴¹ both in order to promote a kinetically limited process, would limit the discussed bend of the current step.

This study is not limited to Fick's second law in single-impact blocking electrochemistry, but because it deals with the heat equation, its scope is quite broad. Such an effect can be expected to occur in diffusive processes in general, such as heat diffusion but also electron/hole diffusion in semiconductors,⁴² neutron diffusion,⁴³ or gas diffusion.⁴⁴

## ■ ASSOCIATED CONTENT

### ■ Supporting Information

The Supporting Information is available free of charge at
https://pubs.acs.org/doi/10.1021/acs.jpca.5c02279.

Elastic behavior modeled by a dampened harmonic
oscillator, experimental section, two-dimensional diffu-
sion equation in an RC network, time response of the
circuits, simulation of 1D diffusion, digitized current
transients, and variation of the diffusion coefficient;
animation of the evolution of the voltages in the RC
network simulating the diffusion of a redox probe during
a blocking impact at the electrode (PDF)

## ■ AUTHOR INFORMATION

### Corresponding Authors

Arthur Langlard – Nantes Université, CNRS, CEISAM, UMR 6230, Nantes F-44000, France; Email: arthur.langlard@univ-nantes.fr

Estelle Lebègue – Nantes Université, CNRS, CEISAM, UMR 6230, Nantes F-44000, France; orcid.org/0000-0002-8610-0891; Email: estelle.lebegue@univ-nantes.fr

### Authors

Christine Thobie-Gautier – Nantes Université, CNRS, CEISAM, UMR 6230, Nantes F-44000, France

Mohammed Boujtita – Nantes Université, CNRS, CEISAM, UMR 6230, Nantes F-44000, France; orcid.org/0000-0001-5604-3294

Complete contact information is available at:
https://pubs.acs.org/10.1021/acs.jpca.5c02279

### Notes

The authors declare no competing financial interest.

## ■ ACKNOWLEDGMENTS

This work was supported by the French National Research Agency (ANR-21-CE42-0007-01, ELIPOX).

## ■ ABBREVIATIONS

UME, ultramicroelectrode; DMPC, 1,2-dimyristoyl-*sn*-glycero-3-phosphocholine; PB, phosphate buffer; PS, polystyrene; RC, resistor-capacitor

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