On the physical interpretation of constant phase

Solid State Ionics 180 (2009) 922–927

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Solid State Ionics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / s s i

On the physical interpretation of constant phase elements M.R. Shoar Abouzari a, F. Berkemeier a,⁎, G. Schmitz a, D. Wilmer b,1 a b

Institut für Materialphysik, Westfälische Wilhelms-Universität Münster, Wilhelm-Klemm-Str. 10, D-48149 Münster, and Sonderforschungsbereich 458, Germany Institut für Physikalische Chemie, Westfälische Wilhelms-Universität Münster, Corrensstr. 30, D-48149 Münster, Germany

a r t i c l e

i n f o

Article history: Received 31 May 2008 Received in revised form 1 April 2009 Accepted 1 April 2009 Keywords: Impedance spectroscopy CPE CMR Equivalent circuit

a b s t r a c t Complex impedance spectra of ion-conducting lithium borate network glasses are used to study the deformed shape of impedance semicircles which are usually described by a parallel circuit of an ohmic resistor and a phenomenological Constant Phase Element (CPE). Based on the Concept of Mismatch and Relaxation (CMR) of Funke et al. which provides a theoretical treatment of the ion dynamics in disordered materials, a quantitative description of the experimental impedance spectra is presented. It takes into account the contribution of the static glass network by introducing an additional capacitor, and provides both, a physical interpretation of the CPE and an easy-to-handle mathematical formula to calculate the ’real’ capacity of a CPE. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Electrochemical impedance spectroscopy (EIS) is a well-known technique to determine the electrical properties of ionic materials [1]. In a typical EIS experiment, the complex electrical impedance Ẑ of a sample is measured as a function of frequency over a wide frequency range — typically several orders of magnitude [2]. From an experimental point of view, performing impedance analysis at frequencies below some MHz is a straightforward procedure, but the physical interpretation of the experimental data is quite often rather complex. A frequently used method to analyze impedance data is modelling the sample by an appropriate equivalent circuit and comparing the theoretical electrical response of this circuit with the experimental data. Graphically, this comparison is often performed in a so-called Nyquist diagram, i.e. a plot of the negative imaginary part of the impedance — Z″ versus its real part Z′ [1]. In a rather simple approach, the electrical properties of a solid-state ion conductor can be described by an equivalent circuit consisting of a parallel connection of an ohmic resistor R and a capacitor C. The Nyquist diagram of such a circuit gives a precise semicircle, centered on the Z′ axis at R2 ; 0 . However, in most cases the experimental data reveal a significant deviation from this simple R-C behaviour, becoming apparent in a deformation and broadening of the impedance semicircle [1,3–8].

⁎ Corresponding author. E-mail address: [email protected] (F. Berkemeier). 1 Now at Novocontrol Technologies, Obererbacherstr. 9, D-56414 Hundsangen, Germany. 0167-2738/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ssi.2009.04.002

A common phenomenological approach describes these broadened semicircles by a parallel connection of an ohmic resistor R and a so-called ’Constant Phase Element’ (CPE). Replacing the capacitor by the CPE but preserving the resistor in the circuit has the practical advantage that the d.c. conductivity can still be determined by directly fitting the parameter R. Since in most cases experimental results agree reasonably well with the R-CPE equivalent circuit, it is a widely-used empirical model, although its physical meaning is poorly understood [9–13]. Even worse, the frequency dependence of conductivity that is presumed by the CPE Ansatz is at variance with general statements of linear response theory [14]. Thus, the CPE element can be only justified by practicability but lacks any sounded physical basis at all. There are publications which attribute the deformation of the impedance semicircles to the roughness of the electrode-electrolyte interfaces or to inhomogeneities in the local distribution of defects in the vicinity of grain boundaries [9,10,13,15–17]. Simulating these structural properties by fractal models results indeed in broadened impedance semicircles. But nevertheless, these explanations are more or less restricted to crystalline materials, and the transfer to amorphous ion conductors is not completely satisfying. In this paper we follow a different route to explain the deviation of the EIS data from an ideal R-C behaviour: We take into account the frequency-dependence of the specific conductivity of ion conducting materials which is a direct consequence of the forward-backward correlation of the elementary hopping process of the ions. Based on the quantitative model provided by Funke and coworkers [18,19] in their ‘Concept of Mismatch and Relaxation’ (CMR), we suggest a modified equivalent circuit to describe the experimental impedance spectra. In this circuit, the CPE is replaced by a well sounded expression which no longer conflicts with linear response theory. A

M.R. Shoar Abouzari et al. / Solid State Ionics 180 (2009) 922–927

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behaviour of the real part of specific conductivity when approaching the dc limit: σ ðωÞ n − 1 ~ω : σ0

Fig. 1. EIS data of a 0.20Li2O · 0.80B2O3 glass at 220 °C. The solid line shows a fit of the data by an R-CPE equivalent circuit, the dashed line gives an ideal semicircle for comparison.

subsequent comparison with the R-CPE equivalent circuit leads to a better physical interpretation of the parameters of the latter. 2. Data analysis using an R-CPE equivalent circuit An equivalent circuit frequently used to model experimental impedance data of solid-state ion conductors consists of a parallel connection of an ohmic resistor R and a CPE. Its total impedance is given by 1 1 + R ˆZ CPE

Zˆ =

ð6Þ

To illustrate the difference between an ideal R-C equivalent circuit and an R-CPE circuit, Fig. 1 shows the Nyquist diagram of a typical impedance measurement obtained from a 0.20Li2O · 0.80B2O3 glass sample [25]. Compared to an ideal semicircle (dashed line) it is clearly observed that the experimental data represent a depressed one. A fitting of an R-CPE circuit to the EIS data according to Eqs. (3) and (4) is shown by the solid line, with the parameters R = 8.3 · 106 Ω, Q = 2.02 · 10− 11 snΩ− 1 and n = 0.897. 3. Experimental study of the CPE exponent — influence of ionic conduction and surface roughness To demonstrate the influence of the mobile ions on the CPE exponent n, we compare impedance spectra of two different types of network glasses: a high purity quartz glass sample with an inappreciable electronic and ionic conductivity (connected in parallel to an ohmic resistor of 100 kΩ, to anyhow facilitate electrical measurements) and a lithium borate glass of the composition 0.25Li2O · 0.75B2O3, i.e. a Li+conductor providing an appreciable ionic and negligible electronic

!−1 ;

ð1Þ

whereby the impedance of the CPE is defined via [1,20,21] 1 n = Q ðiωÞ : ˆZ CPE

ð2Þ

Here, i is the imaginary unit, Q is the pre-factor of the CPE, and n its exponent.2 Substitution of Eq. (2) into Eq. (1) and splitting Ẑ into its real and its imaginary part yields

Z V=

R 1 + RQωn cos nπ 2 n 2 1 + 2RQωn cos nπ 2 + ðRQω Þ

ZW = −

2 n R Qω sin nπ 2 : n 2 1 + 2RQωn cos nπ 2 + ðRQω Þ

ð3Þ

ð4Þ

The exponent n of the CPE may vary between 0 and 1 and is usually found around 0.8 [4,25]. For n = 1, Eq. (2) agrees to the impedance of an ideal capacitor, whereby Q is identified with the capacity C. For n b 1, the ‘real’ capacity C has to be calculated, e.g. according to a suggestion of Hsu [21]

C=R

1−n n

1

Q n:

ð5Þ

Although Eq. (5) is widely used, it does not give any physical interpretation of the pre-factor Q or the CPE itself, and it is only valid if n or the product R · Q is close to unity [3]. It should be further noted that the Ansatz of Eqs. (1) and (2) states a power law

2 It should be noted, that a similar but not equivalent expression to Eq. (1) has been introduced in the phenomenological power law of Jonscher [22,23] and in the Coupling Model of Ngai [24].

Fig. 2. Conductivity data obtained at a pure quartz glass and a lithium borate glass (0.25Li2O · 0.75B2O3). (a) CPE exponent n as a function of temperature. (b) Real part of the conductivity spectrum of the borate glass at 200 °C.

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conductivity [25,26]. Two cylindrical glass samples (diameter ≈10 mm, thickness ≈1 mm) of different surface quality are prepared for each type of glass to study the influence of surface roughness on the impedance spectra, and in particular on the CPE exponent n. While the surface of one sample is polished precisely, using 6 μm diamond paste, the surface of the other is roughly ground. Electrodes of an Al–8 at.%–Li alloy are deposited on the top and the bottom faces of the glass cylinders, by ionbeam sputtering [25]. Afterwards, impedance data are evaluated in the temperature range between 100 and 300 °C and values for R, Q, and n are obtained by fitting Eqs. (3) and (4). The results of these experiments are summarized in Fig. 2 (a). It is observed that in the case of the quartz glass samples the CPE exponent n is close to unity and independent of the surface roughness. (As a direct consequence, the corresponding Nyquist diagrams represent almost ideal semicircles.) Therefore, it is concluded that the rigid glass network behaves almost like an ideal capacitor, constant within the experimental frequency range (5 Hz to 2 MHz). Contrary to the quartz glass, the lithium borate glass samples reveal nvalues significantly below unity, but again no dependence on the surface roughness of the samples is found. Furthermore, all samples show the same weak dependence on temperature of only about 2% over the whole temperature range. Additional investigations of sputter-deposited thin film lithium borate glasses, presented in a former work [25,26], confirm this independence of the CPE exponent on the surface roughness: Electron micrographs of the interfaces between sputter-deposited glass films and metallic electrodes reveal atomically smooth boundaries, but EIS measurements result in n ≈ 0.7. The remarkable difference between the n-values of the quartz glasses and the lithium borate glasses, shown in Fig. 2 (a), suggests that the significant deformation of the impedance semicircles is caused by the presence of ionic charge carriers, which leads to a strong frequency-dependence of the electrical conductivity as shown in Fig. 2 (b), but it is not caused by different interface roughness. We may summarize as follows: While in the case of the quartz glass a frequency independent conductivity spectrum and a precise impedance semicircle is observed, the conductivity spectrum of the lithium borate glass reveals a strong dependence on frequency and simultaneously, a depressed impedance semicircle. Therefore, we state that the correlated forward-backward jumps of the ions are the main reason of the deformation of the impedance semicircles in the Nyquist diagram. 4. Impedance response and ion dynamics The Concept of Mismatch and Relaxation (CMR) of Funke et al. [18,19] describes the ionic conductivity of disordered materials quantitatively, and explains the frequency-dependent regime (dispersive regime) as a consequence of correlated forward-backward jumps of mobile ions.3 In the framework of the CMR model, it is assumed that the effective potential on each mobile ion consists of two different parts: a static potential, provided by the immobile glass network, and a time dependent potential, provided by the mobile ions. The jump of an ion to its neighbouring site causes a mismatch to the arrangement of the mobile ions nearby. To reduce this mismatch, either the neighbours rearrange or the ion jumps back to its original position. This leads to a forward-backward correlation of successive jumps and consequently, to a dispersive regime in the conductivity spectra of ionic materials. 3 Recently, the CMR model has been enhanced by Funke et al., resulting in the MIGRATION concept [27]. In the framework of this concept, a detailed description of both, the real and the imaginary part of the conductivity is given by three coupled rate equations. Because the CMR model and the MIGRATION concept mainly differ in the low frequency regime of the conductivity spectrum [28], they both approximately agree to each other within the frequency range of our experiments. Therefore, we restrict ourselves to the mathematically more simple CMR model.

The CMR describes the ion dynamics mathematically by two coupled rate equations −

dgðt Þ K = A g ðt Þ W ðt Þ dt

ð7Þ

−

dW ðt Þ dgðt Þ = − B W ðt Þ : dt dt

ð8Þ

Here, W (t) is a time-dependent correlation factor, representing the probability for an ion to be still in its new position occupied directly after the jump. It is supposed that a hop of a mobile ion happens at t = 0, and hence W (0) = 1. Furthermore, W (∞) is the fraction of successful elementary hops. The mismatch function g(t), with g(0) = 1, describes a normalised distance between the actual position of an ion and the position where its neighbours expect it to be. The parameter A is an internal frequency, proportional to the highfrequency limit of the specific conductivity σ(∞), and B determines the ratio σ(0)/σ(∞) = : exp(−B). Finally, the parameter K influences the shape of the conductivity spectra in the vicinity of the onset of the dispersion regime and is typically close to 2. In the framework of the CMR model, the relative specific conductivity as a function of the correlation factor is given by [19,28] σˆ ðωÞ =1+ σ ð∞Þ

Z

∞ 0

dW ðt Þ expð−iωt Þdt; dt

ð9Þ

with the angular frequency ω. A typical numerical solution of Eqs. (7) and (8) is shown in Fig. 3, with the parameter set A = 8 · 107 s− 1, B = 6, and K = 1.85. Splitting Eq. (9) into its real and imaginary parts results in Z σ ion V ðωÞ = σ ð∞Þ 1 + Z σWion ðωÞ = σ ð∞Þ

∞ 0

∞ 0

dW ðt Þ cosðωt Þdt dt

dW ðt Þ sinð−ωt Þdt: dt

ð10Þ

ð11Þ

The latter equations are based on a clear physical model and are formulated in agreement with the general requirements of linear response theory. Expanding the real part σ ′ (Eq. (10)) to low frequencies, it can be shown that σ ′ approaches the d.c. limit σ0 with quadratic power in ω [14]. Thus, in order to be consistent with the theory of linear response, the exponent n of the CPE used in Eq. (6) would have to be n = 2, whereas n b= 1 is usually necessary to fit experimental data over a reasonable range of frequencies. Quite

Fig. 3. Numerical solutions of Eqs. (7) and (8), obtained with the parameters A = 8 · 107 s− 1, B = 6, and K = 1.85.


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According to Eqs. (11) and (13), the ionic part of the dielectric constant ion is calculated via ion ðωÞ =

σ ð∞Þ ω 0

Z

∞ 0

dW ðt Þ sinð−ωt Þdt: dt

ð14Þ

At ω = ωp, which is the frequency in the maximum of the Nyquist diagram, ion is obtained numerically to 4.3. Consequently, the total relative dielectric constant at this frequency becomes r ωp = ion ωp + ∞ = 12:3: 5. Physical interpretation of the CPE behaviour

Fig. 4. Experimental conductivity spectra of a 0.20Li2O · 0.80B2O3 glass, obtained at 200 °C. The solid lines show the theoretical curves according to Eqs. (10) and (13), obtained with the fit parameters A = 8 · 107 s− 1, B = 6, K = 1.85, σ(∞) = 3.35 · 10− 6 Ω− 1 cm− 1, and ∞ = 8.0.

obviously, the CPE is unable to describe ion transport consistently with general physical considerations of ionic transport. While the long-ranged charge transport inside the glass samples is exclusively provided by the migration of ionic charge carriers, the dielectric constant characterizing the polarization of the sample comprises two parts: The first part, ion, arises from the contribution of the ionic motion and is considered via the CMR model, while the second part, ∞, is associated with vibrations of mobile ions, of the glass network, or of the electrons. Accordingly, we define the total capacity of the sample as S C = 0 ðion + ∞ Þ ; d

In the following we will use the relations summarized in the previous section to define an equivalent circuit that describes experimental impedance spectra and can replace the R-CPE circuit while preserving the advantage of a simple evaluation of the dc conductivity. As already mentioned, the dielectric constant of an ionconducting network glass can be split into two parts, one part due to the ionic motion, and a second part due to vibrational modes. Therefore, the desired circuit needs a parallel combination of two capacitors — Cion and C∞. While Cion strongly depends on frequency, C∞ is supposed to be constant over the measured frequency range. Similarly, we will split the real part of conductivity into two parts, a frequency-independent dc conductivity and a frequency-dependent one. Although long range charge transport is exclusively provided by mobile ions and therefore completely described by the CMR model, we will nevertheless separate the dc part, in order to preserve dc conductivity as a primary fitting parameter. Furthermore, separating

ð12Þ

in which S and d denote surface area and thickness of the glass samples, respectively. Consequently, the total imaginary part of the conductivity must read σ WðωÞ = σ Wion ðωÞ + σ W∞ = ω 0 ðion + ∞ Þ:

ð13Þ

Since the ionic motion inside the glass sample is considered by the CMR model, and the contribution of ∞ is taken into account via a simple capacitor, we will refer to this approach as the ‘CMR-C’ model. An experimental conductivity spectrum of a lithium borate glass sample of the composition of 0.20Li2O · 0.80B2O3 at a temperature of 200 °C is shown in Fig. 4. The solid lines represent a fit due to the CMR-C model, according to Eqs. (10) and (13), respectively, with the fit parameters A = 8 · 107 s− 1, B = 6, K = 1.85, σ(∞) = 3.35 · 10− 6 Ω− 1cm− 1, and ∞ = 8.0. As depicted in the Nyquist diagram in Fig. 5 (a), the complex impedance function defined by Eqs. (11) and (13) describe the experimental data, accurately. For comparison, the description by an R-CPE equivalent circuit is also shown. Although it seems that both models describe the experimental data comparatively well, the R-CPE model deviates from the experimental conductivity spectra in the high frequency range. This deviation is more clearly observed in a double logarithmic scale, as shown in Fig. 5 (b) and corresponds to the well-known deviation in the high-frequency regime between experimental conductivity data and theoretical data calculated using the power law approximation by Jonscher [22,23], or the Coupling Model introduced by Ngai [24]. In a recent publication, this deviation has also been pointed out by Funke and Banhatti [29].

Fig. 5. Comparison of the CMR-C model and the empirical R-CPE model to describe experimental impedance data obtained at a 0.20Li2O · 0.80B2O3 glass at 200 °C. The parameters of the R-CPE model are determined to R=2.04 · 107 Ω, Q=1.84 · 10− 11 snΩ− 1, and n=0.898. (a) Linear representation. (b) Double logarithmic representation to clarify the deviation of the R-CPE model from the experimental data in the high frequency range.

926


A comparison with the admittance of the CMR-C model (Eq. (15)) yields nπ 1 1 1 n + Q ω cos + = R 2 Rdc RðωÞ

ð21Þ

and nπ n Q ω sin = ω ðCion + C∞ Þ: 2

ð22Þ

Identifying R = Rdc, the CPE can be considered to be equivalent to the three basic elements, R(ω), Cion(ω), and C∞, as indicated in Fig. 6. Eq. (22) yields an analytical relation to interpret the pre-factor of the CPE in terms of the total capacity of the sample Fig. 6. Equivalent circuit to describe the Constant Phase Element in terms of the CMR-C model.

dc resistivity allows a direct comparison of the CPE with the remaining parts of the defined electric network, as presented below. Therefore, the resistance of the sample is considered by the two resistors, Rdc and R(ω), and the total admittance of a sample is written as 1 1 + iωðCion + C∞ Þ: Yˆ ðωÞ = + Rdc RðωÞ

ð15Þ

This circuit, representing the ion conductivity of the sample, is illustrated in Fig. 6. With exception of C∞, all elements of Eq. (15) can be expressed in terms of the correlation factor discussed in the previous section: Rdc is obtained using Eq. (10), when setting ω = 0 Rdc

1 d = σ Vð0Þ S

ð16Þ

1 d : = σ ð∞Þ W ð∞Þ S

The frequency-dependent part of the resistance, R(ω), is calculated via the conductivity difference σ ′(ω) − σ ′(0) RðωÞ =

σ ð∞Þ

1

R ∞

dW ðt Þ 0 dt

d : ðcosðωt Þ − 1Þdt S

ð17Þ

The ionic part of the capacity Cion(ω) is given by Eq. (11), using σ′′ = (ωC) · d/S Cion ðωÞ =

S σ ð∞Þ d ω

Z

∞ 0

dW ðt Þ sinð−ωt Þdt; dt

ð18Þ

and last but not least, C∞ is obtained by S C∞ = 0 ∞ : d

ð19Þ

Eqs. (15–19) define an equivalent circuit to describe the conductivity of an ion-conducting network glass in terms of the CMR-C model. Based on this set of mathematical expressions, we now will suggest a possible physical interpretation of the parameters of the Constant Phase Element. According to Eqs.(1) and (2), the admittance of the R-CPE circuit is given by 1 n Yˆ CPE ðωÞ = + Q ðiωÞ : R

ð20Þ

C ðωÞ = Cion ðωÞ + C∞ Q ðωÞ =

C ðωÞ ω

n−1

nπ : sin 2

ð23Þ

For the R-CPE circuit, the angular frequency at the maximum of the Nyquist diagram may be determined by setting the first derivative of Eq. (4) equal to zero ωp = ðQ RÞ

−1=n

:

ð24Þ

We substitute ω in Eq. (23) by Eq. (24) and obtain a relation between C and Q valid for frequencies in the range of ωp. C=R

1−n n

nπ 1 j Q n sin : 2 ω = ωp

ð25Þ

Eq. (25) represents a physically justified form of Eq. (5), introducing an additional correction factor sin(nπ/2). In the case of our experimental data, obtained at a 0.20Li2O · 0.80B2O3 glass at 120 °C, we find n = 0.7 which leads to a correction of ≈11%. 6. Conclusion In this work, we have used the Concept of Mismatch and Relaxation, introduced by Funke et al., to explain the deformation of impedance semicircles in the Nyquist diagram, in the case of ionconducting network glasses. Based on this model, we defined an equivalent circuit, named CMR-C model, which describes the experimental data accurately, and does not revert to the widely-used, but from a physical point of view unjustified Constant Phase Element. More precisely, our investigations have shown that this CMR-C model describes the broadened semicircles in a high grade of accuracy over the whole frequency range, while the common R-CPE model deviates from the experimental data in the high frequency regime. An analytical expression has been derived which relates the prefactor of the CPE to the ‘real’ capacity of the sample. The comparison of both theoretical approaches shows that the commonly used R-CPE model is nevertheless an appropriate work horse to describe experimental data at reasonably low frequencies and to determine the dc conductivity and the capacity of an ion conductor, if the proper physical interpretation is used. In the case of ion conducting network glasses, the deformation of the impedance semicircles is a consequence of the correlated forwardbackward jumps of the ions. Since the Concept of Mismatch and Relaxation describes the ionic motion in both, glassy materials and disordered crystalline ones, we expect our findings to be also valid in the case of crystalline ion conductors.


Acknowledgments We are grateful to the members of the Sonderforschungsbereich 458 for helpful discussions, and especially K. Funke and R. Banhatti (Institut für Physikalische Chemie, Westfälische Wilhelms-Universität Münster) for their help in understanding the theoretical background of the CMR model. This work was financially supported by the Deutsche Forschungsgemeinschaft within the Sonderforschungsbereich 458. References [1] J.R. Macdonald, Impedance Spectroscopy, Wiley, New York, 1987, p. 1. [2] C. Cramer, S. Brunklaus, Y. Gao, K. Funke, J. Phys.: Condens. Matter 15 (2003) S2309. [3] M.Sh. Abouzari, Ion-Conductivity of Thin-Film Li-Borate Glasses, Ph.D. Thesis, Westfälische Wilhelms-Universität Münster (2007). [4] F.H. Berkemeier, Ionleitende Borat- und Silikatglasschichten, Ph.D. Thesis, Westfälische Wilhelms-Universität Münster (2007). [5] E. Quartarone, P. Mustarelli, A. Magistris, M.V. Russo, I. Fratoddi, A. Furlani, Solid State Ionics 136-137 (2000) 667. [6] L. Tortet, J.R. Gavarri, J. Musso, G. Nihoul, Phys. Rev. B 58 (1998) 5390. [7] A.R. James, C. Prakash, G. Prasad, J. Phys. D: App. Phys. 39 (2006) 1635. [8] S. Gupta, N. Prasad, V. Wadhawan, Ferroelectrics 326 (2005) 43.

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