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Spatially resolved impedance spectroscopy of a Nafion polyelectrolyte membrane is performed employing a conductive and Pt-coated tip of an atomic force microscope as a point-like contact and electrode. The experiment is conducted by perturbing the system by a rectangular voltage step and measuring the incurred current, followed by Fourier transformation and plotting the impedance against the frequency in a conventional Bode diagram. To test the potential and limitations of this novel method, we present a feasibility study using an identical hydrogen atmosphere at a well-defined relative humidity on both sides of the membrane. It is demonstrated that good quality impedance spectra are obtained in a frequency range of 0.2–1000 Hz. The extracted polarization curves exhibit a maximum current which cannot be explained by typical diffusion effects. Simulation based on equivalent circuits requires a Nernst element for restricted diffusion in the membrane which suggests that this effect is based on the potential dependence of the electrolyte resistance in the high overpotential region.

Proton exchange membrane fuel cells (PEMFCs) represent a promising energy conversion technology for mobile and portable applications. The development of deeper insight into the functionality of PEMFC components calls for new experimental methods. One of the most important components is the proton exchange membrane which warrants the performance of the PEMFC by its inherently high proton conductivity, and its longevity via its chemical and thermal stability. Different structural models of Nafion, mainly based on small-angle x-ray and small-angle neutron scattering experiments have been developed. A model derived by Schmidt-Rohr fits best the observed experimental data and excludes several other models [

For further investigations of the proton conducting domains, the so-called electrochemical atomic force microscopy (EC-AFM) was developed [

A variant of the AFM-based technique was reported by O’Hayre

Reaching a high spatial resolution with an AFM is difficult since the sample is strongly influenced by environmental parameters like temperature and RH. The resulting sample drift affects the resolution, and therefore a short measurement time is favorable. In the present, work, the setup of EC-AFM is extended to permit chronoamperometric experiments with high spatial and temporal resolution. The current response to a rectangular voltage step is measured as a function of time and converted to frequency space by Fourier transformation, allowing the impedance spectrum to be calculated and simulated using common equivalent circuits. For simplicity, the novel approach is tested by measurements of a hydrogen pump across a Nafion membrane as a model system.

The experimental setup consists of an atomic force microscope (Dimension 3100, Veeco) equipped with a conductive AFM module, enabling the measurement of currents between the tip and the sample in the pA to nA range. The construction of the measurement cell is sketched in ^{−1} with the same RH was provided for each chamber. For the measurement of transients the existing experimental setup is extended by a signal access module (SBOB3, Veeco) which provides access to all electrical signals of the AFM-system. The chronoamperometric experiments were performed using a data acquisition card from National Instruments (PCIe-6361) which was controlled by the LabVIEW software (Version 10.2). The main features of the in-house written data acquisition software are an adjustable sampling frequency, a variable duration of the measurement and a freely selectable voltage step.

Experimental set-up: A membrane separates two environmental chambers with gas atmospheres which can be controlled independently but were identical for the present work. The upper side is sealed off by a perfluorosilicone cap in which the cantilever holder is embedded. The cantilever deflection is measured through a small glass window. An external PC runs the chronoamperometric experiments.

All samples were prepared from Nafion^{®} which was activated following a standard procedure [_{2}O_{2} at 85 °C for 1 h and then rinsed with doubly distilled water for 15 min. Following this, it was placed in 0.5 M sulfuric acid at 85 °C for 1 h and subsequently rinsed again with doubly distilled water for 15 min. After drying in air, the membrane was coated on one side using a wet spraying preparation technique. The ink was prepared from platinum black (Hispec 1000, Johnson Matthey), doubly distilled water and Nafion Dispersion (10 wt. %, Aldrich) which was all mixed together and stirred for at least 12 h. The procedure leads to a catalyst loading of around 1 mg Pt per cm^{2} with 30 wt. % Nafion within the electrode. The coated membrane is stored in doubly distilled water for at least 24 h prior to use.

The transient current and the corresponding voltage signals are transformed to the frequency domain by Fourier transformation. The temporal resolution of the signal traces plays an important role. The highest defined frequency (with amplitude and phase) of a discrete signal is given by the sampling theorem and is equal to or smaller than half of the sampling rate. The lower frequency limit is given by the reciprocal duration of the measurement [

To calculate the impedance of the system, the voltage and the current signals, given by arrays of 10,000 points per second, are Fourier transformed (Equation (1)) using a numerical procedure described elsewhere [

The function H(t) represents the time dependent voltage or current signal which is transformed to the complex function H*(ν) in the frequency domain. Subsequently, the impedance (Z*) is calculated by complex division according to Equation (2), where

The AFM tip was placed on a non-conductive position of the Nafion 212 membrane in a hydrogen atmosphere at 78% RH. A series of voltage steps was applied and held for 5 s before it was set back to zero. The current and voltage signals were sampled with a rate of 100,000 s^{−1}. The waiting time before the next experiment was several seconds. The first voltage step was from 0 V to 0.1 V, and the upper value was incremented by 100 mV for each subsequent measurement.

Current traces measured in a series by increasing the potential jump from 0 V to the corresponding value given in the diagram. The measurements were performed at the same position in a hydrogen atmosphere with 76% RH. The dashed lines represent the fitted exponential decay with a time constant of around 0.24 ms.

All current traces show a fast increase followed by an exponential decay to zero current. This is the typical behavior of an ideal polarizable electrode which can be represented by an equivalent circuit consisting of a capacitor connected in series with an Ohmic resistance. When a step voltage is applied to this circuit, the current response I(t) is entirely capacitive and described by an exponential decay with a time constant τ = R_{S}∙C_{D} [

E represents the plateau voltage, C_{D} the capacity, and R_{S} reflects the Ohmic resistance of the solid electrolyte. The dashed lines in _{S}) value. With the extracted value, τ = 0.24 ms, it takes at least 1 ms until meaningful information can be observed from this system in a chronoamperometric experiment. In other words, the cell constant of the system restricts the high frequency limit to 1 kHz, which is chosen as the upper limit for the calculation of the impedance spectra.

To comply with this fact in accordance with the sampling theorem, the sampling rate for the subsequent measurements is chosen to be 10,000 points per second.

The total exchanged charge is obtained by calculating the areas under the curves in ^{−4} nF. Assuming a typical geometric capacitance of 30 µF cm^{−2} for platinum, the area is calculated to 0.37 µm^{2}. This corresponds to a radius of 346 nm if a circular contact area is assumed. As the radius of the tip is known to be about 10 nm, this contact area seems rather large. It is therefore likely that the observed capacitance is increased by other effects as for example pseudocapacitance due to hydrogen adsorption [

The small size of this electrode has a pronounced influence on the characteristics of the electrochemical system. The measured currents are in the pA to nA range, which in principle corresponds to a very low IR drop or permits using highly resistive electrolytes giving a moderate IR drop. In any case the use of a two-electrode system is meaningful. Additionally, regarding the highly resistive electrolyte, the high temporal resolution of 1 ms is a result of the small size of the electrode, which has a small capacity C_{D} and lowers the time constant according to τ = R_{S}∙C_{D}.

The reproducibility of the current-time traces were investigated by performing many measurement series for several experimental parameters. The aim was to verify a systematic behavior and find the optimum experimental parameters. Apart from the applied force between the tip and the sample and the RH of the gas atmosphere, the position on the sample and the recovery time between the measurements, which is a measure for the relaxation time of the membrane, were also varied. All measurements were carried out in a hydrogen atmosphere following a voltage step from 0.4 V to 0.5 V. The duration of a single measurement is 10.5 s whereby the voltage step was applied 0.5 s after the start. Five measurements were recorded for each set of parameters. The RH values were chosen to be 47%, 61% and 80%. For each RH value the measurements were taken at two different tip positions on the membrane so that a total of 6 different positions were investigated. On each position the force was varied from 10 nN to 30 nN and 50 nN, and also the time delay within the series of five measurements was varied from 10 s to 60 s and 180 s. Overall, 270 measurements were recorded.

To permit a suitable comparison, the relative error based on the standard deviation of the data in this measurement series is determined in the following way. The arithmetic mean,

The relative error

The relative errors (filled symbols) and the arithmetic mean of the relative errors (open symbols) are plotted in

Relative errors, calculated from several measurement series, are plotted as a function of the applied force between the tip, the relative humidity of the gas atmosphere, the tip position on the sample and the recovery time between the different measurements. The open symbols represent the arithmetic mean of the five measurements.

High contact forces can lead to damage of the membrane and also to high wear of the platinum catalyst layer on the tip. Hence, for the later experiments, the force was chosen to be 30 nN.

The increase of the RH leads to much higher relative errors. At 47% RH all values of ε_{x}

The current signals from position V are a factor two or three higher than those from position VI, indicating that the local proton conductivity at position V is significantly higher. This difference reflects the inhomogeneity of the membrane surface and therefore the suitability of this method to obtain spatially resolved measurements. To obtain similar relative errors for position V and position VI, the deviations between the measurements have to be a factor two or three higher as well, indicating that the deviations scale with (proton) current. Due to the electro-osmotic drag a certain amount of water molecules is dragged along with the protons. If the proton current is higher, the water distribution within the membrane is much more strongly affected. It is therefore suggested that the growing perturbation of the water distribution within the membrane with increasing proton flux is the dominating origin of the relative error at higher RH. Experimentally, the influence of the relaxation of the membrane due to the perturbed water distribution can be investigated by varying the recovery time between subsequent measurements. Regarding the measurements with 30 nN and 50 nN, an overall higher relative error is observed with a recovery time of 180 s. During further experiments it turned out that a recovery time of 30 s or 60 s leads to the best results.

A series of measurements was conducted at a fixed tip position on a Nafion 117 membrane in a hydrogen atmosphere with an RH of 71%. For the first measurement, the voltage step was applied from 0 V to 0.1 V, whereas for each subsequent measurement the plateau voltage is incremented by 100 mV. According to the reproducibility of the measurements discussed above, the delay time between the measurements was chosen to be 30 s and the tip was pressed onto the sample with a force of 30 nN. The corresponding current traces are shown in

Starting at 0 V, different voltage steps were applied to a Nafion 117 sample in a hydrogen atmosphere at RH = 71% and an applied force of 30 nN between the sample and the tip. The resulting current traces are shown. The break between the measurements was 30 s.

Shortly after the voltage step, a strong increase of the current is observed for each measurement, which is mainly based on the capacitive current of the cell. For the first 7 voltage steps up to the terminal voltage of 0.7 V, each subsequent current trace increases further. This behavior may be better seen if the polarization curves of this measurement series are extracted. Therefore different profiles at different delay times parallel to the voltage axis from

Current-voltage polarization curves derived at three different times after the beginning of the measurement as indicated in

From 0.1 V to at least 0.4 V all three curves follow an exponential increase. According to the theory of chronoamperometric experiments [

This is confirmed by the inset of

If the overpotential is very high in an electrochemical system with an aqueous electrolyte, the maximum current at long times reaches the diffusion limitation. This theoretical limiting current is independent of the applied voltage and is drawn schematically as a dotted line in

The polarization curves selected in

(

As can be seen from

The measurement with a voltage step to 0.3 V shows the typical behavior for a parallel RC-circuit. This behavior is representative for the voltage range from 0.1 V to 0.4 V. In the Nyquist plane the RC circuit manifests itself as a semi-circle with a radius that decreases with increasing potential for a Faradaic reaction. The simulated and experimental spectra of this voltage range are shown in the lower graph of

A distinctly different characteristic is observed for the voltage range between 0.8 V and 1.1 V. The impedance starts to increase in the low frequency region, as can be seen exemplarily in _{el}, shown in _{ct} representing the charge transfer resistance for the electrochemical reaction here mostly at the cathode, and a Nernst diffusion element

Equivalent circuits used for the simulations of the impedance spectra obtained from the data of

The Nernst diffusion is controlled by the Warburg coefficient W and the Nernst constant _{N}_{N}_{el}. The inductive behavior which is observed in the high frequency region for some spectra is simulated with an inductor L in series with R_{el}. A complex non-linear least square regression algorithm is used for the fitting procedure.

The results for the charge transfer resistance from the above measurement series (RH = 71%) are plotted in _{ct} from the potential is expected in the kinetically controlled region [

Simulated charge transfer resistance as a function of the voltage at different relative humidities, showing a pronounced decline with increasing voltage. The charge transfer resistance is smaller at higher relative humidity. In the section to the left of the dotted line the RC-circuit and to the right the modified Randles-circuit is used.

The impair of the performance of the oxygen reduction reaction was reported for a temperature dependent study of a proton exchange membrane fuel cell [

The simulated capacity shows little variation in the low overpotential region and reaches values of a few pF independent of the RH. For an overpotential >0.5 V up to 208 pF are observed for 89% RH, which is indicative of pseudocapacitance.

In _{el} decreases with increasing RH due to higher proton conductivity within the membrane. Another observed effect is the declining tendency of R_{el} with increasing voltage step. Typically the increase of the electrolyte resistance with increasing current density is reported for Nafion in a fuel cell and attributed to a dehydration of the membrane at the anode side [

Simulated electrolyte resistance plotted for different relative humidities as a function of the voltage step. The dotted line marks the change from the RC-circuit (left side) to the modified Randles circuit (right side) used for the simulation. With increasing RH the electrolyte resistance decreases. a) indicates R_{el} values which were fixed during the simulation of the impedance spectra.

Although the samples were conditioned using a common procedure, internal reorganization of the local membrane structure due to the proton flux is possible. This may lead to an improved water distribution within the membrane and in the end to a lower electrolyte resistance. For 89% RH in the high overpotential region Rel is almost constant near 75 MΩ. This level is also reached at high overpotential for 47% RH (1.1 V) and 71% RH (0.8 V), respectively, and it seems to be the lower limit for these experimental conditions.

The fitted values of the Warburg coefficient are plotted on a logarithmic scale against the voltage in

Simulated Warburg coefficients W plotted against the voltage for different relative humidities. With increasing RH the diffusion limitation decreases. An exponential increase with overpotential is observed as indicated by the fits (dashed lines).

The Warburg coefficient decreases with increasing relative humidity, indicating that the diffusion is less restrictive with higher RH. According to the model of Schmidt-Rohr [

Such an exponential relation is also observed for the Warburg parameters in

Another interesting fact is that the Nernst coefficient _{N}^{−1} and 3.2 s^{−1} for all investigated RH. A diffusion coefficient can be calculated from the definition of _{N}_{N}^{−1} D is calculated to be 0.98℘10^{−5} cm^{2} s^{−1}. This value is very close to the reported diffusion coefficients for water in fully humidified Nafion [

Although only absolute values are presented in this work, it should be noted that area-specific values can be obtained by normalizing with the contact area of the tip and the sample. Using the method presented by O’Hayre

An extension of the EC-AFM technique is developed and shown to be suitable to measure impedance spectra with good quality and high spatial resolution. In view of the acquired frequency range, this method is much faster than frequency-domain impedance techniques. A short measurement time is necessary if the system is strongly sensitive to external influences such as temperature and relative humidity and thus susceptible to drift. The characterization of the system reveals optimized parameters for the measurement technique. The limits of a meaningful Fourier transformation of the current and voltage traces are determined to range from 0.2 Hz to 1 kHz.

The simulations of the impedance spectra reveal detailed insight into the system. The local impedance is restricted by the charge transfer resistance in the low overpotential region. A decreasing current in the high overpotential region is observed in the polarization curves which were extracted from the current transients measured at different RH. There is evidence that this originates from a potential dependent electrolyte resistance, which is adequately described by a Nernst impedance element for restricted diffusion.

We gratefully acknowledge the Brennstoffzellen- und Batterie-Allianz Baden-Württemberg for financial support. WGB acknowledges funding from the Initiative and Networking fund of the Helmholtz Association.