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Electrochemical Impedance Spectroscopy
Electrochemical Impedance Spectroscopy (EIS) is a powerful analytical technique used to characterize the electrochemical behaviour of systems by applying a small alternating current (AC) signal over a range of frequencies and measuring the resulting voltage response. The impedance is calculated by analysing the phase shift and magnitude of the current and voltage. EIS provides valuable information about various electrochemical processes, such as charge transfer resistance, double layer capacitance, and diffusion processes, by fitting the obtained data to equivalent circuit models. It is particularly useful for studying the electrochemical interfaces of electrodes, enabling the evaluation of their performance, stability, and the effects of surface modifications or roughness.•Complex impedance plots
In electrical impedance spectroscopy, historically based on the early use of impedance bridge circuits to determine impedances of systems under study, impedances are generally expressed in terms of the series resistance RS (Ω) and reactance XS (Ω) of the system as a function of the applied AC signal frequency. The reactance term can be either inductive or capacitive, depending on whether the current flowing through the impedance leads or lags the voltage across it.
A complex impedance plot, often termed a Nyquist Diagram in the literature, is a plot of individual XS versus RS values as a function of the applied signal frequency, thus forming a characteristic impedance locus or curve which can aid in understanding and modelling the electrical properties of the system under study.
As biological and electrochemical systems usually exhibit only capacitive reactance components, rather than inductive ones, it is common practice to plot -XS versus RS as a function of the applied signal frequency to facilitate study of the curves.
Below we will briefly overview some of the key electrochemical processes that occur at or near electrode interfaces and the equivalent circuit components/parameters used to model their electrical behaviours.•Double layer capacitance
When an electrode is placed into an electrolyte, a potential difference is established between the metal and the electrolyte. As a result, counter ions are attracted to the electrode surface. This is termed electrostatic or coulombic adsorption in electrochemistry [4].
In electrical engineering it is well known that a charged layer, say on the electrode metal, separated from another layer of opposite charge, say within the electrolyte, constitutes a capacitor capable of storing charge. This is found to be the case at the electrode-electrolyte interface and is termed the double layer capacitance, Cdl, in electrochemistry.•Charge transfer resistance
Current often manages to leak across the double layer due to any electrochemical reactions taking place, thus transferring charge across the interface. These reactions experience a charge transfer resistance, RCT, which can be thought of as being in parallel with the double layer capacitance and whose non-linear expression can be derived from the Butler-Volmer equation [4].
At its most basic, a biomedical electrode interface impedance can theoretically be represented by an equivalent circuit model comprising a double-layer capacitance in parallel with a charge transfer resistance, RCT. Both components will be in series with an electrolyte resistance Re, as well as the resistance of the leads and contacts.•Electrolyte resistance
The resistance of an electrolyte between electrodes depends on the electrolyte's ionic concentration, the type of ions involved, and the geometry of the area through which the current flows. In the present case, this resistance is expected to be relatively small compared to the impedance of the electrode interfaces, as physiological saline will be used to represent human sweat and tissues.•Overall simple equivalent circuit model and corresponding complex impedance characteristic
The equivalent circuit model of this most simple system, often termed the “three-component model”, is shown in Figure 1a. The complex impedance locus for such a simple system is shown in Figure 1b.
Based on the fundamental textbook electrochemistry outlined above, we expect to observe a semi-circular impedance locus with a low-frequency resistive intercept, which we will denote R0, and a high-frequency resistive intercept, which we will denote R∞ (Figure 2a).
At very high frequencies the impedance of the double layer capacitance, Cdl, tends towards zero, thus shorting out the contribution of the parallel charge transfer resistance, RCT. Thus, the only remaining contribution is the purely resistive electrolyte resistance, Re, leading to R=∞Re. At low frequencies, the impedance of the double layer capacitance tends towards infinity, leaving the contributions of Re and RCT, in series. Thus R=0Re+RCT.
We also expect a high-frequency intercept angle ϕ of 90°, which is indicative of ideal double layer capacitive behaviour (Figure 1a and Figure 2a).•Depressed circular arcs and “CPEs”
Obviously, real-world electrode interfaces are more complex than the simple, “lumped” three-component model presented above. In many biomedical (and other) electrode studies, the complex impedance plot does not form a perfect semi-circular arc, as is shown in Figure 1b. Instead, the plot is often approximately semi-circular, but with the centre of the semicircle lying below the real axis (Figure 2b), and hence the high frequency intercept angle, ϕ, is less than the expected 90°.
Cole (1940) [5] proposed the following, now widely used, empirical equation describing such “depressed” impedance loci (in his case, for similar arcs observed in tissues):Z=R∞+R0−R∞1+(jωτ)α
Where τ is a time constant equal to 1/(2πfc), and fc is considered the characteristic frequency of the system, defined as the point at which the imaginary component reaches its maximum, i.e. at the peak of the impedance arc;
α is a fractional power dependency related to the depression angle, such that ϕ=απ/2.
A simple interpretation of the above-mentioned behaviour is through a circuit model, as shown in Figure 3, where a resistance R∞ (or Re in our case) is in series with the parallel combination of a resistance, ΔR (or RCT in our case), which is equal to R0-R∞, and a rather unusual empirical, frequency-dependent “pseudo-capacitive” constant phase element (CPE) – sometimes referred to as a constant phase angle impedance, ZCPA, by some researchers [3], whereZCPA=K(jω)α
K is derived from the Cole equation such thatK=R0−R∞τα=RCTτα[Ωs−α]
Those who prefer to emphasise the capacitive nature of this empirical circuit element, particularly electrochemists, often use the following alternative expressions, which are mathematically equivalent to those above (in accordance with the expressions and parameters from ZView, the fitting software we will employ to model our results).ZCPE(ω)=1T(jω)P where T = 1/K and P is ZView's fractional power dependency, which is equivalent to Cole's alpha, P ≡α.
T is a measure of the magnitude of the pseudo-capacitance and has units of sP/Ω. (Note that capacitance, Farad, can be expressed as s/Ω, and hence the units of T can be expressed as F sP-1).
NOTE: In the software ZView, which is employed in this research, there is a related expression for this constant phase element, termed the “QPE”. In this case, the magnitude of its pseudo-capacitance in this case is denoted Q, and its frequency dependence denoted n (as opposed to α or P above, where n ≡ P ≡α).ZQPE(ω)=1(jωQ)n
In this version, Q has units of sΩ-1/n or F1/ns(n-1/n).
The changes between n, P and α can be somewhat confusing for readers. However, we have opted to adhere to the nomenclature used by ZView, given its widespread use in the literature. Indeed, there is considerable confusion in the literature regarding ZView's CPE and QPE circuit elements, as well as their respective units. In some instances, the units for T and, in particular, Q, are not even explicitly stated. Hence, our detailed review.
It is also noteworthy, incidentally, that an error appears in the “help section” of Zview4. In the description of the QPE element, the term ‘CPE’ is mistakenly used multiple times to describe its parameters. Nevertheless, this error does not appear in the PDF version of the manual.•Constant phase elements and diffusion impedances
CPEs and QPEs are empirical expressions which do not inherently provide a physical explanation of the phenomena being observed and modelled. Initially, we had not intended to present the Warburg diffusion impedance, which exhibits an impedance characteristic somewhat similar to these elements, as it has a constant phase angle of 45° (i.e. α=P=n=0.5), and possesses physical significance. This impedance arises at very low applied frequencies, where ions encounter increasing difficulty- or “impedance”- when diffusing from the bulk of the electrolyte to the electrode surface. In contrast, in biosignal monitoring, we are generally concerned with relatively high frequencies. This is particularly relevant in our study, where surface roughness effects (see later) dominates, and which we aim to exploit in the development of our novel miniaturised sensor systems.
However, we were compelled to adapt modified versions of Warburg diffusion impedance in ZView to represent and compare our unexpected experimental results in this preliminary study of porous, electrospun electrode structures. Additionally, one may argue that ions diffusing within the pores of the electrode surface experience an impedance somewhat analogous to those diffusing to an electrode surface from the bulk of the electrolyte. Thus, modified versions of the Warburg diffusion impedance may be considered appropriate, at least at this early stage of the research (Figure 4).
The diffusion of charged species towards the interface can introduce an impedance to current flow at lower frequencies, depending on the ion concentration and the distance they must travel to reach the interface. At high frequencies (corresponding to short time scales), the diffusion impedance remains low, as the proximity of the diffusing species facilitates shorter transport distances to the interface. Conversely, at lower frequencies (associated with longer time scales), the species must travel greater distances to the interface. This results in a gradual increase in the Warburg diffusion impedance, which eventually surpasses and regulates the charge transfer resistance.
The Warburg diffusion impedance, ZW, is found to have the following dependence on frequency [4]:ZW=σ/(jω)0.5 where ω is angular frequency (s−1) and σ is the mass transfer coefficient (Ω.s−0.5). Note its similarity to K in Equation (2), which is inversely proportional to the bulk concentration of the ions involved and their diffusion coefficients.
The diffusion impedance, ZW, has the impedance characteristic shown in Figure 5.
Note that the Warburg diffusion impedance behaves much like a constant phase element whose phase angle is limited to ϕ=45° (i.e. α=P=0.5, and ϕ=απ/2). Compare with Equation (4a) for ZCPE.
The above equation for the Warburg impedance (and the impedance locus shown in Figure 5) is valid only if the diffusion layer over which the ions must diffuse continues expanding towards infinity. However, this condition is not often met and at sufficiently low frequencies, the effect of the limited growth of the diffusion layer becomes significant (as the impedance cannot increase indefinitely, figure 6a). The expansion of the diffusion layer is limited either naturally (convection), or by design, for example by stirring the solution. This gives rise to finite diffusion behaviour, which modifies the response of the standard Warburg impedance. In this case, the impedance characteristic can be physically modelled by a finite-length RC transmission line, which is short-circuited at one end. This is why this element is commonly referred to as a “short” Warburg impedance WS. Using the parameters of the ZView software package, WS is expressed as:ZWs=Ws−Rtanh[(jωWs−T)0.5](jωWs−T)0.5 where:-WS-R is the purely resistive (diffusion resistance) value that ZWs approaches at low frequencies (see Figure 6a) (units: Ω);
-WS-R = σLD/D, where σ is the mass transfer coefficient (Ω.s−0.5), LD is the diffusion length (cm) and D is the average the diffusion coefficient of the diffusing species (cm2/s);
-WS-T is the diffusion time constant = LD2/D (units: s).
This characteristic is typically observed for diffusion through a layer of finite length. Its impedance tends to a limiting, purely resistive value at low frequencies, denoted Ws-R in ZView. At higher frequencies, the response is similar to that of a standard “infinite” Warburg impedance.
An alternative finite Warburg diffusion element is the “open” Warburg element, WO, which is associated with diffusion where one of the boundaries is “blocking” (i.e. purely capacitive). It is also analogous to wave transmission within a finite-length RC transmission line which is open-circuited at one end. Once again, this is why this element is commonly referred as an “open Warburg” impedance, WO [8].
The expression for the open Warburg element is given byZWo=Wo−Rcoth[(jωWo−T)0.5](jωWo−T)0.5 where:-WO-R is the diffusion resistance (units: Ω);
-WO-R = σLD/D, where σ is the mass transfer coefficient (Ω.s−0.5), LD is the diffusion length (cm) and D is the average value of the diffusion coefficients of the diffusing species (cm2/s);
-WO-T is the diffusion time constant = LD2/D (units: s).
Once again, at high frequencies, the response resembles that of a standard “infinite” Warburg impedance, displaying a characteristic line at 45° in the complex impedance plot (see Figure 6b). However, at very low frequencies, the real impedance Z′ tends towards a limit equal to WO-R/3 while Z″ continues to increase, similar to the behaviour of a capacitor.
As we will see later, de Levie proposed such a finite-length, open-circuited RC transmission line model (and hence, a corresponding mathematical or equivalent circuit model) to describe circular cylindrical pores of uniform diameter and of semi-infinite length filled with electrolyte in a porous electrode surface [9]. This behaviour has also been linked to ion diffusion within, for example, storage electrodes of lithium-ion batteries [8]. Hence, these “diffusional” impedance elements/models are highly relevant to our study of porous and rough-surfaced electrode-interface impedances.
The high-frequency phase angle, ϕ, of the above Warburg impedances is normally fixed at 45° due to the square root terms (i.e. power dependencies of 0.5) in the above diffusion equations. However, in real electrode systems, the theoretical high-frequency 45° angle is often not observed, and ϕ is frequently larger than expected. Fortunately, ZView provides an option to adjust the value of P (or n) as required.
ZView's more generalised expression for the finite “short-circuit” Warburg impedance therefore becomes:ZWs=Ws−Rtanh[(jωWs−T)Ws−P](jωWs−T)Ws−P
ZView's more generalised expression of the finite “open-circuit” Warburg impedance is given by:ZWo=Wo−Rcoth[(jωWo−T)Wo−P](jωWo−T)Wo−P
In ZView, with these generalised, finite Warburg expressions, P and hence ϕ (where ϕ = Pπ/2) are free to take on any value to best fit real experimental data.•Electrode surface roughness effects and some early models
Non-ideal capacitive behaviour, or the presence of depressed semi-circular impedance arcs, has been observed in a wide range of materials and interfaces across multiple disciplines, so much so that some researchers have proposed a “universal” power law [10]. As a result, we have reviewed findings and models from various fields in an effort to understand or at least adequately model our intriguing results.
The observation that, for very smooth surfaces (such as dropping mercury electrodes), the interface impedance closely resembles that of an ideal double layer capacitance (ϕ=90°) has led to the conclusion that non-ideal capacitive behaviour (ϕ<90°) in solid electrodes (including biomedical, in our case) is likely related to surface roughness effects [9]. However, it can also result from electrode surface heterogeneity or specific adsorption effects [4].
Many attempts have been made to model the effects of electrode surface roughness on the form and magnitude of the measured interface impedance. One of the earliest and most well-known model is that of De Levie [9], mentioned above, who suggested that the impedance, ZO, of cylindrical pores on an electrode surface could be represented using transmission lines, given by the expression:ZO=ZIRecoth(pl)
Where ZI is the interface impedance per unit length of the pore, Re is the electrolyte resistance per unit length of the pore, p−1 is the penetration depth, where p−1=(ZI/Re)0.5 and l is the pore depth.
At high frequencies, the penetration depth, p−1, will be small and the pore behaves as a semi-infinite RC transmission line, which can be modelled as the finite “open” Warburg element (see Equation (7) and Figure 6b). If one assumes that the interface impedance, in the absence of surface effects, is dominated by the double layer capacitance, i.e. Z=Il/jωCdl, Equation (10) tends to:ZOHF=(1−j)(Re2ωCdl)0.5
As before, the interface impedance in this case will exhibit a high-frequency phase angle of 45o. It is however important to note that the high-frequency impedance magnitude is proportional to the square root of that of the purely capacitive “smooth surfaced” impedance, ZI. This implies that the increase in surface area does not simply lead to a reduction in interfacial impedance. Instead, the presence of the square root term in the expression can lead to a marked reduction in the interface impedance. This is the reason why such surface topographies are of great interest in our effort to produce electrodes with small geometric areas while maintaining very low interfacial impedances.
Furthermore, the observed reduction in phase angle, approaching 45° in extreme cases, serves as a key indicator of this surface roughness “transmission line” effect - rather than being solely attributed to an increase in effective surface area.
Once again, at low frequencies, the penetration depth, p−1, becomes large, and the pore no longer behaves as a semi-infinite transmission line. Consequently, surface roughness effects will no longer dominate at these frequencies, and the impedance will gradually revert towards that of a smooth, ideally capacitive surface. As a result, the phase angle tends towards 90o at low frequencies, assuming that the effects of charge transfer resistance and other factors – such as specific adsorption, which is beyond the scope of this study – are ignored. See curve (2) in Figure 7, redrawn from Keiser et al. 1976 [11].
De Levie's relatively simple pore model successfully explains qualitatively many aspects of the interface impedances of solid biomedical electrodes. However, as previously pointed out, due to the many simplifications made, his model can only interpret constant phase angle behaviour where α (or P or n) = 0.5. This is generally not the case for most real electrode surfaces, which are not sufficiently rough to be considered porous.
Since then, more complex transmission line models have been explored. Poly-dimensional ladder networks [12], some incorporating hierarchical branching, have been proposed. Various distributions of pore depths, shapes, location, and widths have also been tested in an effort to improve agreement with experimental data (See review [13]).
Although these more complex models can more accurately reproduce measured impedances, many of them rely on arbitrary assumptions, and the increased number of variables involved often makes it difficult to draw meaningful conclusions. However, we can conclude from the work of Keiser et al. (1976) [11], as summarised in Figure 7, that the more occlusive the surface pores, or the more three-dimensional a surface is, the more pronounced the surface roughness effects. This is evident from the accentuated reduction in the magnitude of the high-frequency interfacial impedance and, in extreme cases, a high-frequency characteristic resembling that of a finite “short” Warburg element (see Figure 5). More recently, Cooper et al. [14], building upon and confirming the work of Keiser et al. [11], explained how pores with a broadening cross-section for the diffusing species - at a given depth - lead to a decrease in phase and a bending over of the impedance locus at a corresponding midfrequency range, below that of the standard Warburg characteristic. For a broadening of the pore cross-section closer to the pore mouth, the deviation from Warburg's 45° slope occurs at smaller penetration depths and therefore at higher frequencies, and vice-versa.
A related approach by Hitz and Lasia [15] involved modelling two slightly overlapping spherical pores, positioned one below the other, forming a continuous “figure-of-eight” cavity within the electrode surface. Hitz and Lasia observed that the effects of each spherical void on the resultant impedance locus manifest over different frequency ranges. At high frequencies, the signal penetrates the uppermost spherical void, reducing impedance in this range. At lower frequencies, the signal reaches the innermost spherical void, leading to a further impedance reduction at a lower frequency range.
Intuitively, based on these concepts (see the excellent review by [16]), one can infer that creating electrode structures with layers of interconnected voids should reduce interface impedances across a wide frequency spectrum. Given that small geometric electrode areas typically result in undesirably large interface impedances, we aim to counteract this by designing porous, sponge-like, “3D” electrode surfaces to achieve lower interface impedances.
Consequently, in this preliminary research, we investigated the potential use of electrospun electrode surfaces in order to create such advantageous sponge-like, “3D” structures for eventual integration into, for example, wearable monitoring systems.
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