Use of a small overpotential approximation to analyze Geobacter sulfurreducens biofilm impedance

Highlights

• Analyzed Geobacter sulfurreducens biofilm impedance in terms of conductance and capacitance.

• Normalization of the biofilm impedance is useful for characterizing changes during growth.

• Conductance had a linear response and CPE had a saturating response.

• Concluded that CPEs contain mass transfer effects in addition to redox capacitance.

Abstract

The electrochemical impedance of Geobacter sulfurreducens biofilms reflects the extracellular electron transfer mechanisms determining the rate of current output. Binned into two characteristic parameters, conductance and capacitance, biofilm impedance has received significant attention. The goal of this study was to evaluate a small overpotential approximation for extracellular electron transfer in G. sulfurreducens biofilms. Our motivation was to determine whether conductance over biofilm growth behaved linearly with respect to limiting current. Biofilm impedance was tracked during growth using electrochemical impedance spectroscopy (EIS) and electrochemical quartz crystal microbalance (eQCM). We showed that normalization of the biofilm impedance is useful for characterizing the changes during growth. When the conductance and capacitance were compared to the biofilm current, we found that: 1) conductance had a linear response and 2) constant phase elements (CPE) had a saturating response that coincided with the limiting current. We provided a framework using a simple iV relationship that predicted the conductance-current slope to be 9.57 V^-1. CPEs showed more variability across biofilm replicates than conductance values. Although G. sulfurreducens biofilms were used here, other electrochemically active biofilms exhibiting catalytic waves could be studied using the same methods.

Introduction

Geobacter sulfurreducens biofilms growing on electrodes are one of the primary model systems for studying electrochemically active biofilms. These biofilms consist of cells that aggregate onto an electrode surface, multiply, and allow the electrochemical conversion of acetate to electrons [16]. The electrochemistry of G. sulfurreducens biofilms has sparked many new ideas regarding the limits of extracellular electron transfer for use in biotechnology [14]. Adaptation of traditional electrochemical techniques to characterize extracellular electron transfer has provided many insights regarding the possibility of multiple types of conduction existing under biological conditions [7], [22]. In fact, the complexity of G. sulfurreducens electron transfer with mechanistic stratification [24] creates a challenging system to quantify through electrochemical means. Despite this, the availability of literature focused on improving or expanding the electrochemical techniques to study extracellular electron transfer is comparatively fewer [2], [6], [8].

One of the popular techniques adapted for investigating the electron transfer mechanisms of these biofilms is electrochemical impedance spectroscopy (EIS) [2], [5], [6], [9], [10], [12], [14], [15], [17]. The electrochemical impedance of G. sulfurreducens biofilms reflects the processes of extracellular electron transfer involved in the overall electron transfer rate measured as current output. Binned into two characteristic parameters, conductance and capacitance, electrochemical impedance is a powerful tool to simultaneously measure the resistance of the biofilms to electron transfer and their capacity to store electrons. Ambiguity in EIS can arise from the interpretation of the biofilm impedance. The transformation from biofilm impedance to resistance and capacitance requires the semiempirical use of a generic circuit network of resistors (R), capacitors (C), and constant phase elements (CPE) [13] such as those shown in Fig. 1. The impedance of G. sulfurreducens biofilm consists of at least two RC pairs [23]. More pairs could be used to improve the fit, but at the cost of additional ambiguity in the interpretation. Furthermore, the use of CPEs as a proxy for capacitance imposes several limitations on linking capacitance to the mechanism under study. First, the units of the CPE coefficient are not equivalent to the units of a capacitor. Second, depending on the value of fitting parameter α, the CPE impedance can become indistinguishable from an infinite Warburg element, resistor, or inductor. This reflects the purely mathematical origin of the CPE and its convenience in fitting “real” impedance data in electrochemical systems [4]. To the best of our knowledge, the link between biofilm parameters and CPE behavior has not been unequivocally proven. Thus, the conversion between CPEs and capacitors is often neglected, relegated to Supplementary Information, or assumed valid. In some cases, opting to use capacitors and sacrifice the quality of the fit yields a simpler yet equally effective interpretation [15].

The choice of the DC set potential can help strengthen the interpretation of the biofilm impedance if we select for discrete current regions along the potential axis. For example, the small overpotential approximation of the Butler-Volmer equation provides a linear basis for EIS [3]:

$$\text{i}\left(\eta \right)=\frac{nF{i}_o}{RT}\eta$$

where $\text{i}\left(\eta \right)$ is current (A), n is the stoichiometric coefficient, $R$ is the universal gas constant (J·mol⁻¹·K⁻¹), $T$ is temperature (K), $F$ is Faraday's constant (C·mol⁻¹), $i_o$ is the exchange current (A), and $\eta$ is overpotential (V). Equation (1) is only valid when η is less than 50 mV or less than 2RT/F. The use of Equation (1) requires a good estimation of the overpotential to obtain the conductance, $G$, or more commonly the charge transfer resistance, $R_{ct}$:

$$G\equiv \frac{di\left(\eta \right)}{d\eta }=\frac{nF{i}_o}{RT}=\frac{1}{R_{ct}}$$

For G. sulfurreducens biofilm, the above must be corrected to account for metabolic reactions (i.e. coupled chemical reactions). Here, we utilize the well-documented observation that G. sulfurreducens biofilms exhibit a single voltammetric wave [8]. We note the possibility that a superposition of waves exists and the catalytic wave is distributed because of metabolic variability [19] and mechanistic stratification [24] within the biofilm. However, we trade the metabolic complexity for mathematical simplicity in order to arrive at a useful result. Therefore, we consider the simplest case, where, at slow enough scan rates, the biofilm iV response approximates a catalytic electron transfer system (EC′) [3] that follows:

$$E\left(i\right)=E_{1/2}+\frac{RT}{nF}\ln \left(\frac{i_L-i}{i}\right)$$

This can be rearranged to represent current as:

$$i\left(E\right)=\frac{i_L}{1+\exp \left[\frac{nF}{RT}\left(E-E_{1/2}\right)\right]}$$

In our case, $i_L$ is the limiting current observed at a constant polarization at the top of the wave; it represents the maximum electron transfer rate for the biofilm at that time. Similar to Equation (1), Equation (4) can be linearly approximated around E_1/2 to obtain Equation (5) and its derivative, Equation (6):

$$i\left(\eta \right)=\frac{i_L}{2}-\frac{nF{i}_L}{4RT}\eta$$

$$G\equiv \frac{di\left(\eta \right)}{d\eta }=\frac{nF{i}_L}{4RT}$$

Similar to Equation (1), the linear approximation for Equation (5) holds only for small overpotentials, of less than 30 mV (10% error at 30 mV). The result applies universally to systems exhibiting EC′ behavior and is simple yet not widely employed in studying extracellular electron transfer. One of the major benefits of this approach is that the conductance is readily estimated from the limiting current, which is the most characterized parameter in the field of extracellular electron transfer. Another benefit is that it is amenable to EIS verification of the biofilm over time at an easily identifiable potential, E_1/2, using 5-mV_RMS amplitude sine waves. So long as Equation (5) holds, a plot of conductance vs. limiting current should yield a straight line with a slope of $\frac{nF}{4RT}$; evaluating for n = 1 and T = 303.15 K gives 9.57 V^-1.

The goal of this study was to evaluate a small overpotential approximation for extracellular electron transfer in G. sulfurreducens biofilms. Our motivation was to determine whether conductance during biofilm growth behaves linearly with respect to limiting current. The small overpotential approximation was assessed experimentally by running EIS centered on E_1/2 where E_1/2 was determined using square wave voltammetry. Thus, biofilm impedance was tracked during growth using EIS centered on E_1/2, with a wave amplitude of 5 mV_RMS. In order to find a suitable current region, we applied a normalization procedure to the biofilm impedance. We then estimated conductance and capacitance by fitting biofilm impedance to the impedance network shown in Fig. 1. We also correlated biofilm impedance with biofilm growth using an electrochemical quartz crystal microbalance (eQCM). The eQCM response, measured as a frequency shift, was a proxy for biofilm growth and allowed real-time monitoring of the transition from initial attachment to exponential growth. As a control, the eQCM analysis was supported by polymer deposition studies using polyaniline.