1 Introduction

Electrical properties and their measurements are significant in several areas of science. In particular, the experimental technique of impedance spectroscopy (IS) plays an important role in the investigation of several materials. With the availability of commercially made impedance bridges covering wide frequency ranges, impedance (conductivity) studies became popular among electrochemists and materials scientists [1]. For instance, one can examine in semiconducting materials how electrically active foreign atoms influence the electrical conductivity. For example, in ion-conducting materials with negligible electronic conduction, one can inspect the conductivity results in connection with the hopping motion of ions. For such materials and many others, the measurement of the electrical conductivity is an indispensable quantity in order to investigate charge carrier diffusion.

Commonly, the diffusive behavior is classified in terms of the mean square displacement (MSD), \(\langle r^2(t) \rangle\). The MSD is frequently related to random walks of particles and, in the Brownian (normal or usual) case, Einstein’s relation, \(\langle r^2(t) \rangle \propto t\), is valid; otherwise, an anomalous diffusion is present. For instance, in the field of solid-state ionics, \(\langle r^2(t) \rangle\) refers to the MSD of charge carriers [2,3,4,5] and may not be proportional to t. In fact, in ion-conducting materials, if the charge carriers perform simple random walks, the subsequent motion of them are uncorrelated and \(\langle r^2(t)\rangle\) increases linearly with t; if there is a correlation, \(\langle r^2(t)\rangle\) can be proportional to t only for very small and very large times [6]. When there is only one type of mobile ion, a way to obtain \(\langle r^2(t) \rangle\) is through frequency-dependent electrical conductivity, \(\sigma (\omega )\), via Kubo’s linear response theory [7,8,9]. Such approach has been successfully employed in the discussion of charge carrier diffusion in a variety of systems [5, 10,11,12,13,14,15,16,17,18,19]. Also, note that the MSD can be viewed as a key function to understand the dynamics of ions as it connects experimental spectra with model considerations [20, 21].

Typically, \(\sigma (\omega )\) is obtained directly from impedance spectroscopy (IS) [22,23,24] measurements. This procedure, which provides the electrical response of a system through the application of a small external voltage \({\mathcal{V}}\), is largely used to investigate a great diversity of materials, for instance, solids [25,26,27], liquids [28,29,30,31,32], fuel cells [33,34,35] and biomaterials [36,37,38]. In general, to investigate the dynamics of ion transport in materials, where more than one type of charge carrier might be relevant, \(\sigma (\omega )\) is a pivotal object to achieve a better understanding of charge carrier behavior [6], including the most diverse experimental conditions that can be imposed on a system. It is also important to mention that in many substances several types of ions may contribute to \(\sigma (\omega )\) [39]. For example, a well-documented case is the so-called mixed alkali effect, in which the conductivity dramatically decreases in alkali oxide glasses when some of the alkali ions are replaced by ions of another species [40, 41]. In spite of that, when a relatively simple measurement like IS is applied to different types of materials, one does not have any additional information about each type of ion and their precise relevance or contribution to \(\sigma (\omega )\) as well as the specific influence of the experimental conditions in several cases. Therefore, the goal here is to work around this limitation by exploring the possibility of considering a mean square displacement approximation that assumes the presence of several types of charge carriers. As a first application of this approximate MSD, we considered a liquid crystal. After obtaining IS data for the liquid crystal, we used the relationship between the MSD and \(\sigma (\omega )\) to identify anomalous and normal diffusion regimes. Lastly, our approximate MSD was employed in connection with well-known equivalent circuits models used for the analysis of impedance spectroscopy data, where we also identified anomalous and normal diffusive regimes.

In the following section, we present the theoretical aspects concerning the approximate mean square displacement. In Sect. 3, we apply the approximate MSD to a liquid-crystalline sample, and we also employ the approximate MSD in connection with equivalent circuits models. In the last section, we present the conclusions.

2 Methods

Supported by Kubo’s linear response theory [7], Roling and co-workers showed how to obtain \(\langle r^2(t) \rangle\) of the ionic diffusion directly from \(\sigma (\omega )\) if only one type of charge carrier in a bulk sample is present [8, 42, 43]. The resulting expression is

$$\begin{aligned} \langle r^2 (t) \rangle&= \frac{12 k_{\mathrm{B}} T V}{q^2 N \pi } \int _0^t {\mathrm{d}}t^{\prime } \int _0^\infty \frac{Re [ \sigma (\omega ) H_{R}(\omega )]}{\omega } \sin (\omega t^{\prime }) {\mathrm{d}} \omega , \end{aligned}$$
(1)

where \(k_{\mathrm{B}}\) is the Boltzmann constant, T is the absolute temperature, V is the volume of the sample, q is the ionic charge, N is the number of charges and \(H_{R}(\omega )\) is the generalized Haven ratio. In several situations, \(H_{R}(\omega )\) can be approximated by a constant, \(H_{R}\) [44]. Essentially, \(H_{R}\) indicates the degree of ionic dissociation [6, 28, 45, 46] and is defined as \(D/D_{\sigma }\), where D is the self-diffusion coefficient and \(D_{\sigma }\) is the diffusion coefficient. Generally, D is acquired via nuclear resonance measurements, while \(D_{\sigma }\) is obtained from conductivity measurements [47]. Another quantity that can be obtained is proportional to the mean square displacement of the center of charge of mobile ions, \(\langle R^2(t) \rangle\) [5, 7, 10, 11], given by

$$\begin{aligned} \langle R^2(t) \rangle = \frac{12 k_{\mathrm{B}} T V}{q^2 N \pi } \int _0^t {\mathrm{d}}t^{\prime } \int _0^\infty \frac{Re [ \sigma (\omega ) ]}{\omega } \sin (\omega t^{\prime }) {\mathrm{d}} \omega , \end{aligned}$$
(2)

where \({\mathbf{R}}\) is defined as

$$\begin{aligned} {\mathbf{R}} = \sqrt{N}\left( \frac{1}{N} \sum _{i=1}^N {\mathbf{r}}_i \right) , \end{aligned}$$
(3)

with \({\mathbf{r}}_i\) being the position of the ith charge and N being the number of charge carriers. If \(H_{R}(\omega )\) is constant, a comparison between Eqs. (1) and (2) enables us to write

$$\begin{aligned} \langle r^2(t) \rangle = H_{R} \, \langle R^2(t) \rangle . \end{aligned}$$
(4)

In general, since \(H_{R}\) is not known for several materials, it is assumed as constant [48], and \(\langle R^2(t) \rangle\) is used instead of \(\langle r^2(t) \rangle\), given their proportionality. This framework has been successfully employed in discussions of systems involving only one type of charge carriers [5, 10,11,12,13,14,15,16,17,18,19]. Here, we are going to consider a generalization of this scheme for when there may be multiple types of charge carriers.

In this generalization, we employ the center of charge displacement,

$$\begin{aligned} \varvec{{\mathcal{R}}}=\sqrt{N}\left( \frac{\sum _{i=1}^N q_i {\mathbf{r}}_i}{\sum _{i=1}^N q_i} \right) . \end{aligned}$$
(5)

Note that if only one type of charge carrier is present, \(q_i = q\), Eq. (3) is recovered. From Eq. (5), and adopting the procedure of Roling and co-workers [8], we can obtain the generalized mean square displacement of the mobile ions center of charge as a function of time in a similar way as in Eq. (2). This leads to

$$\begin{aligned} \langle {\mathcal{R}}^2(t)\rangle = \frac{12 N V k_{\mathrm{B}} T}{\left( \sum _{i=1}^N q_i \right) ^2 \pi }\int _0^t \int _0^\infty \frac{Re\{\sigma (\omega )\}}{\omega } \sin (\omega t^{\prime }){\mathrm{d}}\omega {\mathrm{d}}t^{\prime }, \end{aligned}$$
(6)

which is the approximate mean square displacement to be employed here.

If there are several types of ions and one of them is the most relevant in a diffusive process, we may think, as a first approximation, that \(\langle R^2(t) \rangle\) obtained from Eq. (2) could be seen as the MSD. However, when there are several relevant types of charge carriers, we propose to use \(\langle {\mathcal{R}}^2(t) \rangle\) given in Eq. (6) to investigate the MSD instead of employing Eq. (2). Considering the sample holder as a whole in an IS measurement, \(\sigma (\omega )\) captures, in addition to bulk effects, wall effects on electrodes among possible others. Similarly, because of Eq. (6), \(\langle {\mathcal{R}}^2(t) \rangle\) encompasses an analogous amount of aspects. Interestingly, similar effective approaches were considered before [6, 49], although not as directly applicable as \(\langle {\mathcal{R}}^2(t)\rangle\) proposed here. In fact, by putting the experimentally obtained \(\sigma (\omega )\) in Eq. (6), we can promptly investigate the behavior of \(\langle {\mathcal{R}}^2(t) \rangle\). Also, note that \(\langle {\mathcal{R}}^2(t) \rangle\) essentially has the same contents as \(Re\{\sigma (\omega )\}\) since there is basically an inverse Fourier transform in Eq. (6). These facts and the microscopic interpretation of \(\langle {\mathcal{R}}^2(t) \rangle\) point toward the use of it as a possible framework to investigate systems by using IS data.

Observe that if there is only one type of charge carrier in Eq. (6), \(q_i = q \ \forall \ i\), Eq. (2) is recovered. Similarly, if \(\sigma (\omega ) = \sigma\) (constant), it is verified that

$$\begin{aligned} \langle {\mathcal{R}}^2(t)\rangle&= \left( \frac{6 V k_{\mathrm{B}} T Re\{\sigma \}}{\left( \sum _{i=1}^N q_i \right) ^2 N} \right) t, \end{aligned}$$
(7)

i.e., the usual Brownian regime for \(\langle {\mathcal{R}}^2(t) \rangle\) is obtained.

The way \(\sigma\) depends on \(\omega\) is of considerable interest for different materials [22, 50]. One of the first works on this issue was the one published by Jonscher [51], who showed that different kinds of conductors exhibit a threshold frequency (approximate) from which \(\sigma (\omega )\) is not constant anymore. Nowadays, the presence of interactions and, consequently, correlations in the ions diffusion have been used to explain such type of frequency dependence on conductivity [22].

In view of this, the ionic conductivity has been commonly expressed as [2, 3, 22, 52]

$$\begin{aligned} \sigma (\omega ) = \sigma _0 \left[ 1 + \left( \frac{\omega }{\omega _0}\right) ^{p}\right] , \ \ \,{\text{with}}\,\ \ 0< p < 1, \end{aligned}$$
(8)

in which \(\sigma _0\) is the portion frequency—independent of \(\sigma (\omega )\) and \(\omega _0\) is a dispersion frequency. From Eq. (8), when \(\omega \lesssim \omega _0\), \(Re\{\sigma (\omega )\} \approx \sigma _0 =\) constant; therefore, the diffusion is usual (Eq. 7). On the other hand, if \(\omega \,\gtrsim\, \omega _0\), the diffusion is anomalous. In order to further understand this aspect, we consider the integrals in Eq. (6) with \(Re\{\sigma (\omega )\} \propto \omega ^{p}\). In this way, we have

$$\begin{aligned} \langle {\mathcal{R}}^2(t)\rangle \propto \int _0^t \int _0^\infty \frac{\omega ^{p}}{\omega } {\text{sin}}(\omega t^{\prime }) {\mathrm{d}}\omega {\mathrm{d}} t^{\prime } \propto t^{1-p}. \end{aligned}$$
(9)

This last proportionality can also be visualized via dimensional analysis. Indeed, as \(\sin (\omega t)\) is dimensionless, we have \(\omega \sim t^{-1}\) and consequently the integrals (9) in terms of t. As high (low) frequencies are related to short (long) times, by using the full Eq. (8), we obtain normal diffusion at long times and anomalous diffusion at short times as indicated by Eq. (9).

Particularly in anomalous diffusion, one usually has the power law

$$\begin{aligned} \langle r^2 (t) \rangle = A \, t^{\alpha }, \end{aligned}$$
(10)

with A constant. If \(\alpha > 1\), the process is superdiffusive; if \(\alpha < 1\), the process is subdiffusive; and \(\alpha =1\) refers to normal diffusion [53,54,55]. As pointed out before, in many cases \(\langle r^2(t) \rangle = H_{R} \langle R^2(t) \rangle\) and Eq. (2) has been successfully used to investigate diffusion in solid materials when there is only one type of charge carrier. In particular, it was found that the dynamics of mobile ions is subdiffusive in short timescales, approaching normal diffusion in long timescales [5, 8, 11,12,13,14,15,16,17,18, 42, 48, 56,57,58].

After the theoretical extension conducted along this section, we are considering the possibility of using Eq. (6) and, consequently, Eq. (2) as special case, as an effective mean square displacement to investigate situations related to electrical conductivity. In the next section, we discuss the application of the approximate mean square displacement \(\langle {\mathcal{R}}^2(t) \rangle\) to a liquid-crystalline sample and we also employ the approximate MSD in connection with equivalent circuits models.

3 Results and discussion

3.1 Application of \(\langle {\mathcal{R}}^2(t) \rangle\) to a liquid crystal

When analyzing the liquid crystal sample studied here, it was observed that a bit modified Jonscher’s expression seems to be more suitable for \(\sigma (\omega )\) data, namely

$$\begin{aligned} \sigma (\omega )=\sigma _0 \left[ 1+\left( \frac{\omega }{\omega _0}\right) ^{q}\right] ^{p/q}, \ \,{\text{with}}\,\ 0< p < 1 \ {\text{and}} \ q=2. \end{aligned}$$
(11)

This equation has the same asymptotic behavior observed in Jonscher’s formula (Eq. 8) and, when \(p=0\), one has normal diffusion, since \(\sigma (\omega )=\sigma _0\). In comparison with Eq. (8), one advantage of Eq. (11) is it fits better to data that present a more pronounced increase in \(\sigma (\omega )\) after the dispersion frequency.

Regarding the experimental procedures, a Solartron SI 1296 A impedance–gain phase analyzer was used to carry out impedance spectroscopy measurements. The frequency range employed was \(10^{3}\,{\mathrm{Hz}}\) to \(5\,{\mathrm{MHz}}\), with the external voltage amplitude fixed at \({\mathcal{V}}=10\,{\mathrm{mV}}\). The sample was placed between two stainless steel circular electrodes with an area of \(3.14\,{\mathrm{cm}}^2\), separated by a distance \(d=1.0\,{\mathrm{mm}}\), with the total volume of \(\nu = 0.314\,{\mathrm{ml}}\). In advance of starting the measurements, the surface of the electrodes was decontaminated in two steps: first, by washing with detergent and deionized water and second, by an ultrasonic bath in acetone (\({\mathrm{C}}_3{\mathrm{H}}_6{\mathrm{O}}\), molar mass = \(58.08\,{\mathrm{g\,mol}}^{-1}\)) for \(10\,{\mathrm{min}}\).

The first step is to perform the fit of \(\sigma (\omega )\) obtained via IS by using Eq. (11). Figure 1 shows \(\sigma (\omega )\) as a function of \(\omega\) for the investigated sample, a thermotropic liquid crystal E7, with temperature fixed at \(52\,^{\circ }{\mathrm{C}}\). This figure also shows the fit with Eq. (11).

Fig. 1
figure 1

Conductivity as a function of frequency for the thermotropic liquid crystal E7 sample

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The values of the parameters from the best fit (adjusted \(R^2=0.99\)) are \(\sigma _0 = 0.0948\,{{\Omega }}^{-1}\,{\mathrm{cm}}^{-1}\) [0.0947; 0.0949]\(_{95\%}\), \(\omega _0 = 1.0241 \times 10^{6}\,{\mathrm{Hz}}\) [\(0.9855 \times 10^{6}\); \(1.0628 \times 10^{6}\)]\(_{95\%}\) and \(p = 0.1747\) [0.1698; 0.1796]\(_{95\%}\), where the values between brackets are the 95% confidence interval. By using \(\langle {\mathcal{R}}^2(t) \rangle\) as an effective (approximate) mean square displacement, its calculation and comparison with Eq. (10) show a subdiffusion for \(\omega \,\gtrsim\, \omega _0\) (or short times) with the proportionality constant \(A = 0.0060\) [0.0059; 0.0061]\(_{95\%}\) and exponent \(\alpha = 0.827\) [0.826; 0.829]\(_{95\%}\), as shown in Fig. 2 (green dotted line). This behavior is probably because the charge carriers in the E7 sample find large molecular clusters [59] as they move through the material. For large times (\(\omega \lesssim \omega _0\)), the \(\langle {\mathcal{R}}^2(t)\rangle\) behavior approaches the usual regime, in which the fit parameters are \(A = 0.047\) [0.045; 0.050]\(_{95\%}\) and \(\alpha = 0.978\) [0.974; 0.983]\(_{95\%}\) (Fig. 2, red dashed line).

Fig. 2
figure 2

Diffusive behavior of the charge carriers in the thermotropic liquid crystal E7 sample

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This type of diffusive behavior has also been found when Eq. (2) is applied to solid samples. For example, for silver phosphate glasses, the silver ion migration presents \(\langle R^2(t) \rangle \propto t^{1-p}\) at short times, where the exponent p varies between 0.65 and 0.68 for different temperatures and compositions, while at large times \(\langle R^2(t) \rangle\) approaches the normal diffusive regime [12] (\(\propto t\)). Also, for silver ion dynamics in molybdophosphate glasses, \(\langle R^2(t) \rangle \propto t^{0.40}\) at short times, approaching the normal diffusive regime at large times [13]. For ions in lithium phosphate, bismuth borate and borotellurite glasses, a similar behavior was observed, as \(\langle R^2(t)\rangle \propto t^{1-p}\), with \(p = 0.64-0.70\) depending on the composition of the glasses [15,16,17,18].

Thereof, our results reveal that an anomalous diffusion of the charge carriers, at least when one considers \(\langle {\mathcal{R}}^2(t) \rangle\), may be present in the liquid crystal investigated here. Specifically, the exponent found for the anomalous case is smaller than one, which represents subdiffusive processes. This is an indication that the employment of \(\langle {\mathcal{R}}^2(t) \rangle\) can be useful in the study of liquid materials.

Although not explored here, another application of \(\langle {\mathcal{R}}^2(t)\rangle\) could be in the study of phase transitions. Conductivity as a function of frequency is constant for a frequency range, such as the initial plateau in Fig. 1. In general, the value of \(\sigma\) corresponding to this plateau is called \(\sigma _{\mathrm{dc}}\) (dc conductivity) [22]. In Eq. (7), we saw that, in a general way, \(\langle {\mathcal{R}}^2(t)\rangle = {\mathcal{D}}t\) when \(\sigma (\omega )=\sigma _{\mathrm{dc}}=\) constant. Along with the temperature dependence T in \({\mathcal{D}}\), \(\sigma _{\mathrm{dc}}\) also has a temperature dependence given by the Arrhenius relation [15]. In a first-order phase transition, for instance, \(\sigma _{\mathrm{dc}}\) may present a jump in its value as the temperature varies, which will directly correspond to a jump in the diffusion constant value \({\mathcal{D}}\). In this way, phase transitions could also be investigated by employing the scenario of the approximate mean square displacement \(\langle {\mathcal{R}}^2(t)\rangle\).

In the next subsection, we theoretically discuss the usage of the effective mean square displacement in equivalent circuits models used in impedance spectroscopy analysis.

3.2 Approximate mean square displacement in connection with equivalent circuits

Another approach widely used in connection with IS measurements is to fit the data by using equivalent circuits. These circuits are commonly composed by resistors and capacitors [22]. The simplest RC (resistor–capacitor) circuit is shown in Fig. 3.

Fig. 3
figure 3

Resistor–capacitor (RC) circuit

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This circuit provides a fairly simplified approximation of an electrolytic cell. Typically, the resistance represents the sample, and the capacitance corresponds to the electrodes. The connection with \(\langle {\mathcal{R}}^2(t)\rangle\) is made by obtaining the equivalent impedance \(Z_{\mathrm{eq}}\). For the RC circuit in Fig. 3, the association between the resistor with resistance R and the capacitor with capacitance C leads to

$$\begin{aligned} \frac{1}{Z_{\mathrm{eq}}} = \frac{1}{Z_{R}} + \frac{1}{Z_{C}} = \frac{1}{{R}} +i \omega {C}, \end{aligned}$$
(12)

in which was used that the resistive and capacitive impedance are \(Z_{R} = {R}\) and \(Z_{C} = 1/(i \omega {C})\), respectively. In turn, the conductivity is given by

$$\begin{aligned} \sigma (\omega ) \propto \frac{1}{Z_{\mathrm{eq}}}; \end{aligned}$$
(13)

therefore, \(Re\{\sigma (\omega )\} \propto Re\{1/Z_{\mathrm{eq}}\} = 1/R =\) constant. Thus, by means of Eq. (6), a system that is well represented by the RC circuit presents only the normal diffusion (see Eq. (7)).

Generally, the modeling of experimental data requires more elaborate equivalent circuits. An example is a Randles circuit, which is one of the simplest and most common models used for many aqueous, conductive and ionic solutions [60]. It includes only a solution resistance \(R_{{\mathrm{SOL}}}\), a parallel combination of a double-layer capacitor \(C_{{\mathrm{DL}}}\) and a charge transfer or polarization resistance \(R_{{\mathrm{CT}}}\) (Fig. 4).

Fig. 4
figure 4

Randles circuit

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The details of current distribution in the Randles circuit are largely similar to those in the RC circuit, with the necessary correction for the presence of the additional series resistor \(R_{{\mathrm{SOL}}}\). The simplified Randles model is often the starting point for other more complex models, mainly for charge-transfer kinetic analysis in highly conductive solution systems not impeded by migration and diffusion mass-transport effects [60]. The circuit analysis results in the expression for total impedance of the circuit as

$$\begin{aligned} Z_{\mathrm{eq}} = R_{{\mathrm{SOL}}} + \frac{R_{{\mathrm{CT}}}}{1+(\omega R_{{\mathrm{CT}}} C_{{\mathrm{DL}}})^2} - i \frac{\omega R_{{\mathrm{CT}}}^2 C_{{\mathrm{DL}}}}{1+(\omega R_{{\mathrm{CT}}} C_{{\mathrm{DL}}})^2}. \end{aligned}$$
(14)

From this, the real part of conductivity can be evaluated as

$$\begin{aligned} Re\{\sigma (\omega )\} \propto Re\left\{ \frac{1}{Z_{\mathrm{eq}}}\right\} = \frac{R_{{\mathrm{SOL}}} + B}{[R_{{\mathrm{SOL}}} + B]^2 + D^2}, \end{aligned}$$
(15)

where \(B = R_{{\mathrm{CT}}}/[1+(\omega R_{{\mathrm{CT}}} C_{{\mathrm{DL}}})^2]\) and \(D = \omega R_{{\mathrm{CT}}} C_{{\mathrm{DL}}}B\). From this result, the first analysis that can be conducted is to consider the limit case of \(R_{{\mathrm{SOL}}} \rightarrow 0\), which corresponds to the RC circuit case. At this limit, one obtains \(Re\{\sigma (\omega )\} \propto 1/R_{{\mathrm{CT}}} =\) constant; therefore, the diffusion is normal as expected. In a second analysis, the full Eq. (15) is considered and the integral in Eq. (6) can be numerically performed in order to obtain \(\langle {\mathcal{R}}^2(t) \rangle\). The result is illustrated in Fig. 5 with \(R_{{\mathrm{SOL}}}= 20\,{{\Omega }}\), \(R_{{\mathrm{CT}}}=350\,{{\Omega }}\) and \(C_{{\mathrm{DL}}}=30\,{\upmu }{\mathrm{F}}\).

Fig. 5
figure 5

Diffusive behavior of the charge carriers for a system described by a Randles circuit with typical values: \(R_{{\mathrm{SOL}}}= 20\,{{\Omega }}\), \(R_{{\mathrm{CT}}}=350\,{{\Omega }}\) and \(C_{{\mathrm{DL}}}=30\,{\upmu }{\mathrm{F}}\). The inset shows \(\langle {\mathcal{R}}^2(t) \rangle\) calculated assuming \(R_{{\mathrm{SOL}}}= 350\,{{\Omega }}\), \(R_{{\mathrm{CT}}}=20\,{{\Omega }}\) and \(C_{{\mathrm{DL}}}=30\,{\upmu }{\mathrm{F}}\)

Full size image

The system presents two separated normal diffusive regimes, which are mainly caused by the values of the resistances. For example, the inset in Fig. 5 shows \(\langle {\mathcal{R}}^2(t) \rangle\) calculated with \(R_{{\mathrm{SOL}}}= 350\,{{\Omega }}\), \(R_{\mathrm{CT}}=20\,{{\Omega }}\) and \(C_{{\mathrm{DL}}}=30\,{\upmu }{\mathrm{F}}\). For this set of values, it is difficult to visualize another diffusive regime than the normal one. In general, equivalent circuits with more resistive and capacitive elements present a richer scenario for \(\langle {\mathcal{R}}^2(t) \rangle\).

Another common equivalent circuit that has been used is the one with \(R_{{\mathrm{SOL}}}\) replaced by a constant phase element (CPE), as shown in Fig. 6. In this type of circuit, the RC part represents the bulk of the sample, and the CPE represents the effects on the electrodes surfaces [22, 61].

Fig. 6
figure 6

RC circuit connected in series with a constant phase element (CPE)

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The impedance of the CPE element is [60]

$$\begin{aligned} Z_{{\mathrm{CPE}}} = {\tilde{T}}(i\omega )^{-\nu }, \ \ \,{\text{with}}\,\ \ 0< \nu < 1, \end{aligned}$$
(16)

where \({\tilde{T}}\) and \(\nu\) are real phenomenological parameters that depend on the electrodes type and the solution, for the case of liquid samples. It can be noted that, when \(\nu \rightarrow 0\) (\(\nu \rightarrow 1\)), \(Z_{{\mathrm{CPE}}}\) is reduced to a resistance (capacitance). In general, the CPE element interpolates resistive and capacitive behaviors.

Instead of analyzing \(\sigma (\omega )\) for this full circuit, for simplicity’s sake, suppose that it comes down to just the CPE element. In this simplified scenario,

$$\begin{aligned} Re\{\sigma (\omega )\} \propto Re\left\{ \frac{1}{Z_{{\mathrm{CPE}}}}\right\} = \frac{\cos \left( \frac{\nu \pi }{2}\right) }{{\tilde{T}}}\,\omega ^\nu . \end{aligned}$$
(17)

Therefore, as done in Eq. (9), the CPE element corresponds to a subdiffusion of the charge carriers with a time exponent \(1-\nu\). This result indicates that systems which are best described by equivalent circuits with CPE elements present, to some degree, a subdiffusive aspect.

In general, following the line of reasoning shown above, we can connect \(\sigma (\omega )\) to \(\langle {\mathcal{R}}^2(t) \rangle\) for any equivalent circuit. However, a difficulty in using these circuits is the lack of a more detailed interpretation of the relation between the circuit elements and the physical system elements under consideration [22]. Note that, despite this difficulty, the use of \(Re\{\sigma (\omega )\} \propto Re\{1/Z_{\mathrm{eq}}\}\) enables us to write the MSD for the system via Eq. (6).

4 Conclusions

To put the subject matter in its proper perspective, and to make clear the nature of our study about the dynamics of charge carriers, we have highlighted some aspects. From Eq. (6), the impedance spectroscopy data can be directly translated into an approximate (effective) mean square displacement for charge carriers. Also, the electrical conductivity, \(\sigma (\omega )\), is basically connected with \(\langle {\mathcal{R}}^2(t) \rangle\) via a Fourier transform. Furthermore, after using a general equation as in Eq. (10), the \(\alpha\) exponent may quantitatively inform, at least approximately, the diffusive behavior of the full system. Thus, as an alternative of using only information from \(\sigma (\omega )\) to investigate materials, we also indicate the employment of \(\langle {\mathcal{R}}^2(t) \rangle\), mainly motivated by two attributes: \(\langle {\mathcal{R}}^2(t) \rangle\) is interpreted as a microscopic feature of the random walk of the charge carriers and, when the use of an Eq. (10) like is suitable, the properties of \(\sigma (\omega )\) are mapped into only two parameters, A and \(\alpha\). It must also be pointed out that the adequacy of \(\langle {\mathcal{R}}^2(t) \rangle\) to investigate the dynamics of charge carriers is supported by several experimental results on solids and glasses [5, 8, 10,11,12,13,14,15,16,17,18,19, 42, 48, 56,57,58], where it was possible to assume one type of charge carrier, and by our present investigation on a liquid crystal, considering the possibility of many types of charge carriers and several experimental effects on the sample holder. Concerning equivalent circuits, and the sometimes complex interpretation of their elements, we verified that the \(\langle {\mathcal{R}}^2(t) \rangle\) approach gives a very direct information regarding the diffusion, i.e., if it is normal or anomalous. From a theoretical point of view, these achievements give rise to the possibility of employing simple effective models assuming that anomalous diffusion is present or not. These facts enable us to claim that the effective anomalous diffusion approach explored here may be a simple, direct and concise route to investigate charge carrier dynamics in general, together with electrical properties of matter, from impedance spectroscopy data.