Comments on the electric modulus function
Introduction
The mechanism and dynamics of ion transport have remained a subject of great interest to researchers in many disciplines. The phenomena are difficult to characterize. The appearance of increasingly incisive experimental data and computer simulations on structure and dynamics has clarified many issues and advanced theoretical understanding of the problem, but much remains to be understood because of the complex structure of many crystals, glasses, and melts, within which the ions reside and diffuse.
The basic experimental technique for characterizing the dynamics of ion transport is electrical relaxation or admittance/impedance spectroscopy, in which an ac bridge or similar device is used to measure the equivalent conductance, G, and capacitance, C, of a material as a function of angular frequency, ω. All the experimental information regarding electrical relaxation at a given temperature is contained in G(ω) and C(ω) is the angular frequency. These quantities can be transformed into the complex permittivity ε∗(iω), complex conductivity σ∗(iω) = ie0ωε∗(iω), complex resistivity ρ∗(iω) = 1/σ∗(iω), and complex electric modulus M∗(iω) = 1/ε∗(iω) by the geometric cell constant. The electric modulus has been used by many researchers to analyze and interpret electrical relaxation data in a wide variety of materials, but almost since its inception it has drawn an unusual amount of criticism. We address these concerns here.
To place the issues in context, the properties and some representative successes of the electric modulus formalism are first described. The electric modulus M∗(iω) was first suggested by McCrum, Read and Williams [1] in their classic book on relaxation in polymers. These authors defined it as the reciprocal of the complex relative permittivity, in analogy with the mechanical shear and tensile moduli being the complex reciprocals of the shear and tensile compliances. The electric modulus completes the closed quartet of electrical relaxation functions referred to above, that is conveniently exhibited as [2]:The relaxation time for M∗(iω) and ρ∗(iω), τD, is the familiar Maxwell RC time constant, where the subscript D denotes the fact that it is a measure of the rate of decay of polarization at constant displacement field (i.e. the electric field):so that [3]:where 〈τD〉 denotes an average that is defined below. This relaxation time τD differs from the retardation time τE for polarization relaxation at constant electric field, appropriate for ε∗(iω) [3]:The ratio ε∞/ε0 is a function of the first and second moments of the distribution of relaxation times g(τD) [3]:that is analogous to the Lyddane–Sachs–Teller relation for under-damped optical phonons [4]:where ωT and ωL are the cutoff frequencies for transverse and longitudinal phonons, respectively (recall that ω0τ = 1 for critical damping). In addition to being distinct from τE, the time constant τD also differs from that in the Fermi gas expression for electronic conductivity in metals, which is directly proportional to σ0 [4]:where n is the number density of charge carriers of effective mass m and charge e. The distinction arises from τe being the time between electron scattering events (collisions with ions or other electrons, and umklapp), whereas τD can be interpreted as the residence time between jumps to and from adjacent sites. The latter concept is embodied in the microscopic description of conductivity relaxation by Scher and Lax (SL) [5], that generalizes the continuous time random walk (CTRW) model of Montroll and Weiss [6]:where is the mean square distance of ion hops and Φ(t) is the probability that an ion remains fixed on a site in the time interval [0, t]. Thus Φ(t) is the response function for relaxation of polarization at constant D, i.e. of the electric field. Eq. (8) has the same form as the phenomenological expression for σ′(ω) from the modulus representation:which strongly suggests that the electric modulus captures a fundamental aspect of conductivity. The different pre-factors in Eqs. (8), (9) merit a comment. The theoretical expression (8) is obtained from the mean square displacement, the ion-hopping rate, the frequency dependent diffusion constant, and the Nyquist theorem that generalizes the Nernst–Einstein relation between conductivity and diffusion constant to nonzero frequency. On the other hand, Eq. (9) is derived from a simple Maxwell relation. Both Φ(t) and ϕ(t) start from an initial value of unity and decrease monotonically towards zero with time. However, if Φ and ϕ are characterized by two different time constants, τSL and τD respectively, and the time dependencies of Φ(t/τT) and ϕ(t/τEMF) are written out explicitly, it can be shown [7] that Eqs. (8), (9) are identical whenThus the conductivity relaxation time τD is proportional to, not equal to, the ionic hopping correlation time τSL. Apart for the pre-exponential T−1 term, however, the temperature dependencies of τSL and τD are the same. They have essentially the same activation energy as long as this is much larger than kBT [8]. The dependency of τSL and τD on other variables such as the isotopic mass of the mobile ion is also the same [9]. Thus discussions of the activation energy [9], [10] and isotope mass dependency of τD [9], [10], [11] remain applicable to τSL. The shape of the dispersion (i.e. the form of the frequency dependence) of Im[M∗(iω)/M∞] given by analysis of electric modulus data is also clearly the same as that of the hopping ions embodied in Eq. (8), so that the frequency dependence of Im[M∗(iω)/M∞] is uniquely determined by ion dynamics. It is interesting to note that if τD is equated to a minimum vibrational lifetime of ca. 10−13 s and ε∞ = 5 is assumed for specificity, a maximum ionic conductivity of σ0,max ≈ e0ε∞/10−13 ≈ 400 S m−1 is obtained. This estimate agrees well with the fact that no totally ionic conductor has been observed to have a conductivity in excess of about of 103 S m−1.
In the past, Tunaley [12] made a literal interpretation of the CTRW procedure of SL as an ongoing renewal process which led to a frequency invariant D and σ. This difficulty is removed by a proper treatment of the CTRW problem as demonstrated by Lax and Scher [13] and others [14], [15], [16], [17]. A formal equivalence between averaged particle transport in disordered systems and the generalized master equation or the CTRW theory has been established by Klafter and Silbey [17].
The occurrence of the two time constants τD and τE for electrical relaxation is as it should be. Both time constants are needed for a complete description of polarization relaxation:where the functional form of f is of no importance here.
Two microscopic time constants are also predicted for a Debye dielectric with a single macroscopic time constant. The microscopic dipole correlation function predicted by the Fatuzzo and Mason treatment [18] of local field effects [19] isFor the more general case of the microscopic dipole correlation function [19],whereThus the microscopic dipole correlation function is also characterized by both permittivity and modulus time constants. Richert and Wagner [20] have also discussed the occurrence of two retardation times, and Friedman [21] noted that τE and τD can be derived from the Frohlich relation [22]Fulton [23] also observed that the longitudinal part of the polarization, PL, relaxes with time constant τD and that the transverse component, PT, relaxes with time constant τE. The electric modulus therefore draws appropriate attention to the basic issue of two time scales for electrical relaxation.
From an experimental point of view, the electric modulus is especially useful for analyzing electrical relaxation processes whose measurement is compromised by high capacitance effects, such as those due to inter-granular impedances [2] and electrode polarization [3]. This advantage rests on the fact that any capacitance Cs in series with any arbitrary impedance Z∗(iω) adds a constant to M′(ω) but does not affect M″(ω):(For electrochemical studies, of course, such high capacitance electrode effects are the subject of study and their suppression is the opposite of what is desired.) If an undesirable capacitance occurs in series with an element of an equivalent circuit, rather than in series with the circuit itself, M″(ω) can also sometimes suppress its effects if the capacitance is large compared with other capacitances in the circuit. An instructive example of this is found in the determination of the permittivity of supercooled water using the Maxwell–Wagner interfacial polarization effect [24]. For a volume fraction Φ of spheres of complex relative permittivity immersed in a continuous phase of relative permittivity , the measured permittivity is given by Wagner’s formula [25], [26] (Table 1)For water droplets in hexane or heptane, is real and (since the dielectric loss of water occurs at microwave frequencies, far above those at which Maxwell–Wagner relaxation occurs). For Φ ≪ 1 information about is obtained from the limiting high frequency limit of :which was used by Hasted and Shahidi [27] in their determination of the permittivity of supercooled water down to −35 °C, using volume fractions of water of 1% and 2%. Hodge and Angell [24] (who performed their measurements without prior knowledge of the earlier Hasted and Shahidi study) used a much higher volume fraction of water (30%), necessitated by their much lower instrumental sensitivity. Their dielectric loss spectra were very close to Debye-like in shape but were otherwise inconsistent with the simple Maxwell–Wagner formula: the maxima were about four times larger than predicted, and was positive rather than negative as required for . The failure of the Maxwell–Wagner formula was attributed to a high capacitance phenomenon associated in some way with the thin layer of emulsifying agent around the water droplets. The loss modulus spectra were also found to be nearly Debye-like in shape and their maximum values also increased with increasing temperature, but appropriately so in this case since is inversely proportional to the permittivity. Values of ε1 for water derived from agreed with the Hasted and Shahidi results to within the ±2% uncertainties claimed for each method (when systematic difference are eliminated the agreement is within 0.5%). This striking result can be accounted for in terms of a simplified equivalent circuit comprising a parallel RC element (R1C1), that simulates a water sphere with relative permittivity ε1 and conductivity σ1, in series with a capacitance Cs that simulates the surface layer of interfacial material, and a capacitance C2 in parallel with the series combination, that simulates the parallel dielectric path around the droplet through the suspending medium. The complex capacitance for this circuit isso thatThe maximum loss is and reflects the behavior of Cs and R1 ∝ 1/σ0 rather than of the desired C1 ∝ ε1. Eq. (20) also accounts for an otherwise unexplained (but unreported) shift in relaxation frequency when the suspending medium was changed [28], since this would be expected to affect the surfactant layer and Cs.
In the same limit, and for C1 ≫ C2 (a good approximation for water in heptane/hexane), electric modulus for the circuit iswhenceThe maximum value of M″(ω) is now (1/2C1), and contains the desired information about C1 independent of the value of Cs if Cs ≫ C ≫ C2.
Including the electric modulus in the mathematical toolbox for analyzing conductivity relaxation also yields valuable constraints on suitable functions for describing experimentally observed behavior. For example, comparison of observed electric modulus spectra with those derived from empirical functions such as those of Cole–Cole (CC) [29] and Davidson–Cole (DC) [30] yields the notable result that the ρ∗(iω) spectrum calculated from the DC form for M∗(iω) is remarkably well described by the CC function [25], so that the excellent fits of the CC function to the complex impedance data reported for alkali silicate glasses by Ravaine and Souquet [31] and others are consistent with the excellent DC fits to M∗(iω) data for the same alkali silicates demonstrated by Hodge and Angell [32]. A comparison between the two is shown in Fig. 1 for a representative pair of CC and DC shape parameters. However, the CC fits to experimental ρ∗(iω) data must be approximate, because if they were exact the modulus spectra would exhibit pathological high frequency properties that are never observed [33]. It can be shown [Appendix A] that if ρ∗(iω) adheres to the CC function then the high frequency limits of M′(ω) and M″(ω) areandEvidently some other function exists that is very similar to the CC function but which does not produce the high frequency divergence in M∗(iω). There are other difficulties with the CC function as well. For example, in the complex ρ(iω) plane neither the high nor low frequency limits of ρ″(ω) approaches the real axis at right angles, as required [30]. Also, if M∗(iω) adheres to the CC function then ε∗(iω) exhibits pathological properties at low frequencies [29] that are not observed [Appendix A]: ε0 → ∞ and σ0 → 0. These analyses involving the electric modulus function indicate that the CC function is unsuitable for conductivity relaxation and should be avoided.
Moynihan [34] has identified some further advantages of the electric modulus formalism. He showed that the complex plane plot of M″ vs. M′ is particularly effective for indicating important features of electrical relaxation behavior, and that it is ‘more efficacious in revealing high frequency, secondary relaxations… than are log σ′ vs. log f plots’. He also noted that empirical fitting functions for M∗ (KWW or other) can readily be compared with those for other properties (shear stress and enthalpy relaxation, for example).
Elliot [35], Roling [36], [37], Sidebottom [38], [39] and others [40] have all criticized the electric modulus function. We address their comments by first responding to the convenient list of issues raised by Elliot, since many of these are similar to those raised by Roling and Sidebottom.
(1) ‘…the modulus is… not a directly measured quantity; instead, it is a complicated function of the measured quantities ε′, ε″ and σ0. Thus, significant experimental errors in, say, one of these quantities can be propagated into both the real and imaginary parts of the complex modulus. … In my opinion, it is far better to analyze the data in the form of conductivity, a directly measured quantity, where no artificial frequency-dependent behavior is introduced, as can be the case in the modulus formalism.’
No artificial frequency dependencies are introduced by the modulus, because all members of the quartet of functions given in Eq. (1) are obtained from the same experimental data. The different functions simply emphasize different aspects of raw experimental data, in the same way that Fourier transforms do for example.
This statement also assumes that an admittance bridge is used to measure equivalent parallel values of Gp and Cp. The complex conductivity, σ∗(iω) and complex permittivity, ε∗(iω) are then obtained from the expressions σ∗(iω) = k[G(ω) + i/ωC(ω)] and ε∗(iω) = (k/e0)[C(ω) − iωG(ω)] = (1/C0)[C(ω) − iωG(ω)], where k is the geometric cell constant and e0 is the vacuum permittivity. However, readily available impedance bridges measure the equivalent series values Rs and Cs in terms of which ρ∗(iω) = (1/k)(Rs + 1/iωCs) and M∗(iω) = (ie0ω/k)(Rs + 1/iωCs) = C0/Cs + iRsC0ω, where C0 is the geometric cell capacitance. Richert and Wagner [20] have also described an experimental technique for directly determining M(t) in the time domain by measuring the decay of the electric field at constant dielectric displacement. In any event, the ‘complicated’ functions of whatever form is of little consequence now because of the very low cost of computing.
(2) ‘[at high frequencies] M″(ω) decreases to very small values and thus it is very difficult to discern [its] exact frequency dependence… with any degree of accuracy’.
This is no truer of the modulus than it is of the relative permittivity and specific conductivity. Assuming (with good accuracy) that the conductivity exhibits a power law frequency dependence (σ ∼ Aωα; α < 1), the high frequency limits of ε″ and M″ areso thatThus the high frequency behavior of M″(ω) is just as well defined as that of ε″(ω). In log–log plots of M″(ln ω) and σ′(ln ω) derived from the same data, the only difference is the trivial one of a positive slope for σ′(ln ω) and a negative slope for M″(ln ω).
(3) ‘…invariably, the curve for M″(ω) [calculated from the KWW function] differs appreciably from the measured data in the high frequency tail of the peak…’.
This is a statement about the fitting accuracy of the KWW function, and its relevance to the validity of M∗(iω) is not clear. The high frequency departure of best fit electric modulus functions calculated using the KWW function, compared with experimental data, is found in all ionic glasses [39] and also in melts [40]. This generally observed behavior led Moynihan to call it endemic to the glassy state [41]. These high frequency discrepancies do not significantly affect computed estimates of σ0 or ε0 from the fitted KWW distribution function, however. The conductivity σ0 is a function of the first moment of the distribution of relaxation times, 〈τD〉, and ε0 is in addition a function of the second moment [Eq. (5)], both of which are very weak functions of the short time components that determine the high frequency behavior. For the KWW function,andwhere Γ is the gamma function. The value of σ0 calculated from Eq. (27) agrees with the measured dc conductivity to within a few percent for a wide variety of materials [3], [42], [43]. It has also been shown for an ionic glass that the KWW function accounts for over 80% of the conductivity relaxation strength [41].
The high frequency KWW discrepancies for M″(ω) have the salutary effect of drawing attention to the possibility of additional high frequency processes. It was proposed as early as 1984 [11] that such departures are caused by the presence of an additional contribution to the loss modulus that does not contribute to the d.c. conductivity. Other papers have addressed the related high frequency departure of the nuclear spin-lattice relaxation rate from the KWW fit at low temperatures [44], [45], [46], [47]. This literature preceded the 1994 work of Elliott and coworkers (cited as Ref. [6] in [48]), that (evidently independently) reached the same conclusion.
(4) ‘The addition of a dispersive contribution to the conductivity merely changes the shape of the curve of M″(ω)’.
This is indisputable, but is considered as an advantage by many researchers because these changes point to phenomena that are not readily apparent in σ′(ω) because of the large range in conductivity and the necessity to use a detail-obscuring logarithmic scale. This sensitivity must surely be considered as advantageous for adducing the fine details of ionic conductivity. We have already noted that the high frequency power law dependence M″(ω) ∼ ωα−1 contains the same information as the power law behavior of the conductivity, σ′(ω) ∼ ωα.
(5) ‘The trend to more Debye-like behavior of M″(ω) with decreasing ion content is solely due to the dominance of σ0 and has nothing to do with an intrinsic reduction in ion–ion interactions’.
This remark refers to the large body of evidence that show an increase of β towards unity with decreasing ion content that has been interpreted in terms of a decreasing strength of ion–ion interactions. If only ion concentration is important it is not clear why the separation between ωσ and the frequency of maximum residual ε″(ω) should increase with temperature for all known ionic conductors. This implies a (slightly) greater activation energy for ωε than for ωσ, and since ωε is always higher a significant difference in pre-exponential factors is implied that is difficult to reconcile with their strongly coupled behavior.
Monte Carlo computer simulations of electrical relaxation in a disordered Coulomb lattice gas of ions, when analyzed in the modulus representation, reproduce just such an increase in β with decreasing strength of ion–ion interactions [49]. These Monte Carlo results suggest that ion–ion interactions are in fact rather important in determining the electrical relaxation of ionically conducting glasses. Similar conclusions were drawn from Monte Carlo simulation of the spin-lattice relaxation time of another disordered Coulomb lattice gas [49], where again it was found that a decrease of Coulomb interaction strength is accompanied by an increase in βSLR of the observed KWW correlation function . Comparison of conductivity relaxation and spin lattice relaxation data obtained from these Monte Carlo simulations, as well as from experiment [50], [51], [52], also shows that βSLR < βσ and τSLR ≫ τσ, where the suffix σ in βσ and τσ is added to indicate that they are respectively the KWW exponent and relaxation time of the KWW function for conductivity relaxation. The inequality βSLR < βσ is in agreement with experiment [53], [54] and theoretical considerations based on ion–ion interactions [55].
Another viewpoint that eschews ionic interactions as a cause for a broadening of the relaxation time distribution is that of Johari and Pathmanathan [56]. In their approach, separate contributions to M″(ω) of a Debye peak originating from σ0 and a (Davidson–Cole) peak originating from dipolar losses associated with ion pairs are assumed, and shown to account very well for reported M″(ω) data. The fits to the high frequency tail of M″(ω) were generally better than those given by the KWW distribution of conductivity times, as expected on purely statistical grounds because of two additional adjustable parameters. The values of ωσ and ωε were derived by least squares fitting but no interpretation was offered for why they are always so closely coupled. An analysis of M″(ω) in terms of a Debye peak originating from σ0 and a Davidson–Cole dielectric loss peak is given below [item (7)]. Ingram [57] has also opined ‘In glasses where there are few charge carriers, the modulus peak (which scales with σ0) is shifted to lower frequencies where no dispersion is experienced. The narrowing of the modulus is [therefore] identified as an artifact of the data analysis.’ This view ignores, however, the experimental fact that the dispersion in the relative permittivity (ε0/ε∞) is very well reproduced by the ratio of moments of the conductivity relaxation time distribution as determined from the modulus spectrum [Eq. (5)]. This is not a trivial observation, because the second moment is very sensitive to the long time components of this distribution. Thus, if a single relaxation time (Debye) modulus is observed then it is predicted, and found experimentally, that there is no dispersion in the permittivity, and therefore no dispersion to be separated from [58].
If the dominance of over σ0/e0ω over ε″(ω) − σ0/e0ω is attributed simply to a low concentration of ions, a corresponding reduction in σ0 and decrease in conductivity relaxation frequency ωσ = σ0/e0ε∞ is expected if the ionic mobilities are not appreciably affected by dilution. Why then does the residual dielectric loss always peak at a frequency above ωσ when only its magnitude is expected to change with ionic concentration? The modulus analysis draws desirable attention to the universal proximity of ωσ to ωε.
Roling [37] also noted that the temperature-modified dispersion in the relative permittivity, T[ε(ν) − ε∞], steadily increases with increasing ion content x. Since the frequency dependence of the conductivity changes with x, this result is in accord with the Kronig–Kramers relation and not in need of a separate explanation or interpretation. The electric modulus analysis provides a convenient and quantitatively accurate account of this effect for a distribution of conductivity relaxation times that broadens with increasing x, i.e. for ‘ …the shape of the conductivity master curves, i.e. the ionic relaxation mechanism, depend[ing] on [the composition parameter] x’ [37]. For a KWW function for M∗(iω) with non-exponentiality parameter β, for example, the high frequency power law exponent for conductivity is (1 − β) and the dispersion ε0/ε∞ is given byThus ε0/ε∞ is a sensitive test of both the M∗(iω) formalism and the KWW function [52]. The dispersion in the relative permittivity can even be estimated from the high frequency σ′ω and M″(ω) power law exponents alone. For frequencies about 2 decades above the σ′ crossover region, maximum and minimum values of the high frequency conductivity power law exponent extracted from Fig. 1 of Ref. [37] are roughly 0.71 and 0.62, corresponding to β = 0.29 and 0.38 respectively. Eq. (29) then predicts ε0/ε∞ to be 17 and 6.4, so that (ε0 − ε∞) = 16ε∞ and 5.4ε∞. These values cannot be directly compared with the reported experimental data, since the latter were in the form of (ε0 − ε∞)T spectra and no information on individual ln σ′(ln ν) spectra as a function of temperature was given. However, an indirect comparison can be made by assuming ε∞ to be a negligibly weak function of temperature compared with other properties, and inferring appropriate temperatures from the Arrhenius plots shown in Fig. 2 of Ref. [37]: about 360 and 650 K for the high and low sodium compositions, respectively. Thus the ratio of highest to lowest (ε0 − ε∞)T is predicted to be about (16/5.4)(360/650) ≈ 1.6, compared with about (4700/3400) ≈ 1.4 estimated from the data in Fig. 4 of Ref. [37].
(6) ‘Under certain circumstances two peaks may appear in M″(ω) even though a single relaxation process is operative’.
This statement refers to a paper by Funke et al. [59]. The first assertion is incorrect, and the second is unsupported speculation. Two peaks were not observed in the cited publication – rather, a single peak was followed by a monotonic increase in M″(ω) at higher frequencies, so that any high frequency maximum must have occurred beyond the experimental frequency window. The conductivity data strongly suggest that a resonance process occurs at frequencies above about 10 GHz, since the last five data points exhibit an increase in conductivity with frequency for which d(ln σ′)/dln(ω) > 1 that can only occur for resonance absorption. This increase signals that something unusual is indeed happening at frequencies above 10 GHz. Several other examples of such high frequency behavior in glassy fast ionic conductors have since been confirmed by Funke and coworkers [60], [61], [62], and others [63]. Conductivity data measured up to far-infrared frequencies for the molten salt CKN have clearly established the presence of a contribution to σ′(ω) which also has a resonance power-law dependence ωα(α > 1) [60]. High frequency data for CKN have also been discussed theoretically [61]. There is little doubt that interesting effects are happening at high frequencies in CKN and glassy fast ionic conductors, although at this time their physical origin is still a matter of debate [62], [64], [65]. Ironically, the high frequency resonance contribution to σ′(ω) was difficult even for the researchers who made the measurement to find from the complex conductivity representation of the data – if they had not eschewed an electric modulus representation of their data they could perhaps have been able to report their discovery as early as 1988.
(7) ‘Changing ε∞, while keeping the dispersive contribution of the conductivity constant, greatly changes the shape of the peak in M″(ω)’.
This claim referred specifically to the work of Dyre [66], but Roling [36], [37], Sidebottom et al. [38], [39] and Almond and West [40] have made similar remarks. We first address this issue in terms of the Dyre function for the complex conductivity, and then make additional remarks of broader generality. The Dyre function iswhere θ = arctan(ωτ) and the principle value of the complex logarithm (n = 0) has been adopted to avoid unphysical multi-valuedness. The Dyre function produces a low frequency limit of σ0 for the conductivity, a power law dependence ω+1 for the high frequency limit, a limiting high frequency relative permittivity of ε∞, and a limiting low frequency relative permittivity ε0 ofwhere limx→0[ln(1 + x)] = x and limx→0[arctan(ωτ)] = ωτ has been invoked. Thusis indeed a function of ε∞, and thus the distribution is also. However, if Eq. (3) is inserted into Eq. (32) then and τ is identified with 〈τD〉 than ε0/ε∞ = e0ε∞/2e0ε∞ + 1 = 3/2 and the distribution is indeed constant, corresponding to the specific functional form of σ(ω) = σ0[iωτ/ln(1 + iωτ)]. We also note that the two decade range in ε∞ used by Dyre to support his argument is unphysically large, since it corresponds to a one decade change in the real component of the refractive index. If a physically sensible range in ε∞ (say a factor of 4) is inserted into the Dyre equation, the changes in M″(ω) are almost imperceptible.
The conflict arises from τ in the Dyre equation not corresponding to τD, i.e. from a decoupling of τ from σ0. However, there is overwhelming experimental evidence that the crossover frequency ωX at which the increase in σ′(ω) above σ0 first occurs correlates very strongly with σ0 itself – ωX and σ0 always have the same activation energy, for example. The decoupling arises in the Dyre model from the separation of σ0 from the relaxation process. This is readily apparent from an elementary analysis of the simplest possible circuit for such a model: a series R1C1 combination with τ1 = R1C1, a resistance R0 in parallel with this to give a nonzero σ0, and a parallel capacitance C2 to produce a nonzero ε∞. The M″ spectrum for this circuit is the sum of a Debye peak from the parallel combination R0C2 (centered at ω0 = 1/R0C2 = σ0/e0ε∞) and the dielectric loss peak centered at ω1 = 1/τ1. Changing ε∞ alone therefore changes ω0 but not ω1, resulting in a different separation of peaks and a change in the shape of M″. Thus the issue is not so much with M∗ but rather with the equivalent circuit that is most appropriate for real materials. It is our view that the series RC representation adds needless complications because (i) separate ad hoc elements must be added to accommodate nonzero values of σ0 and ε∞; (ii) a forced approximate equality of R0 and R1 is required to reproduce the experimental fact that the observed conductivity relaxation frequency ωX is proportional to σ0 at all temperatures (this is of course the original motivation for introducing conductivity relaxation at constant displacement as a distinct process from dielectric relaxation at constant electric field); (iii) the value of ε∞ is assumed to be independent of relaxation dynamics, whereas electronic polarizability is expected to affect ion motion (because of dielectric screening and local field effects, for example). This couples C2 and C1 in a manner that is more consistent with experimental fact.
Roling [36], [37] has also asserted that ‘[M″(ω)] is influenced in a complex manner by [ε(ω → ∞)]’. This is indisputable if ε∞ alone is varied and all other variables are kept constant. This is as it should be, because such a variation will also alter ε0/ε∞ and this ratio is a direct reflection of the breadth of conductivity relaxation times [Eq. (5)]. On the other hand, if ε is varied at fixed ε0/ε∞, then the shape of M∗(iω) is unaffected and only its amplitude changes. Recall thatand that the limiting low frequency conductivity (σ0) contribution to ε″ is σ0/e0ω. Noting that σ0 = e0ε∞/〈τσ〉 so that σ0/e0ω = ε∞/ω〈τσ〉, and denoting the dispersion (ε0 − ε∞) by Δε for convenience, the components of ε∗ for an arbitrary distribution of residual permittivity retardation times (such as the DC function used by Johari and Pathmanathan [56]) areandwhere e′ and e″ are the normalized components of the arbitrary permittivity function. ThusFactoring out Δε from the numerator and Δε2 from the denominator, and dividing numerator and denominator by ε∞, yieldsEq. (37) indicates that the shape of M″ depends on three factors:
- (a)
;
- (b)
;
- (c)
the functional forms of e′ and e″.
Item (c) is not relevant to the present discussion because ε∞ and Δε have been factored out of e′ and e″. Items (a) and (b) are simple functions of the metric for the distribution of conductivity relaxation times. It is immediately apparent that if ε∞ is varied whilst keeping ε0/ε∞ constant, no change in the shape of M″ (or M′) occurs for any combination of functional form for e∗ and value of ε∞/Δε〈τσ〉. This invariance is illustrated in Fig. 2(A) using DC expressions for e′ and e″, in the form of plots of normalized plots of M″(ω) for three values of ε∞ [3], [6], [9] with ε0/ε∞ = 2. The Matlab® program is reproduced in Appendix B. The differences between successive plots are of the order of 10−16, and are displayed in Fig. 2(B).
(8) ‘… the most serious shortcoming of the modulus approach, in my view, is that essentially it can transform a quantity which is featureless and frequency-independent (i.e., the dc conductivity) into one which is frequency-dependent and exhibits a spectral feature (i.e., M″(ω) or M″(ω), the latter exhibiting a peak)…’.
Another view of this undeniable fact is that M″(ω) is actually advantageous because it displays differences that are essentially invisible in σ′(ω). As noted in the introduction, this feature has an exact correspondence with the renowned Maxwell model of viscoelastic relaxation. The viscoelastic functions of the Maxwell element are easily derived [67] and are summarized as follows:andThe complex fluidity, Φ∗ = 1/η∗, for a Maxwell element isand is the exact analog of the complex electrical conductivity of the ideal circuit of a resistance and capacitance in parallel. The analogous electrical and mechanical quantities are: {ε∗ ↔ J∗}, {M∗ ↔ G∗}, {σ∗ ↔ Φ∗}, {σ0 ↔ Φ0 = 1/η0}, and {M∞ ↔ G∞ = 1/J∞}. Thus the Maxwell model also ‘transforms’ a quantity that is featureless and frequency-independent, the fluidity, into one which is frequency-dependent and exhibits a spectral feature, G∗(iω) = G′(ω) + iG″(ω). As far as we know, no one has ever found fault with the Maxwell model on this account.
We now turn to further issues raised by others.
(9) Roling [37] has correctly stated that the loss modulus spectra narrow with decreasing ion concentration, but then went on to assert that the σ′(ln ω) spectra did not change. These two statements are mutually inconsistent, however, since both spectra are obtained from the same data:
(10) The scaling provided by the frequency variable advocated by Roling [37] is simply the elementary relation ω〈τ〉 = 1 for loss maxima. The maximum in M″(ln ω) occurs near the ‘crossover’ region of the conductivity, between the limiting low frequency constant σ0 and the high frequency power law behavior [38]. Specifically, the angular frequency ωmax at which the maximum in M″(ln ω) occurs is given byso thatInserting ω〈τσ〉 = 1 and the representative value T = 500 K into this expression yieldsThis is in the middle of the crossover region in Roling’s normalized Fig. 2. Thus M″(ln ω) draws attention to precisely one of the three effects features that Roling correctly identifies as requiring attention: ‘In the crossover regime from dc to dispersive conductivity, the shape of the conductivity master curves, i.e. the ionic relaxation mechanism, depends on [the composition parameter] x’.
(11) Fig. 7 in Ref. [36] reveals a misunderstanding of the concurrence of the frequency regions of maximum M″(ln ω) and crossover behavior in σ′(ln ω). This schematic figure shows maxima in M″(ln ω) for high and low number densities of mobile ions occurring at the same frequency (presumably by using an unidentified scaling factor for frequency), but shows the conductivity crossover behavior for the low mobile ion concentration material occurring at a higher frequency, near the high frequency tail of M″(ln ω) for the high ion content material. Since the breadths of typical spectra for high ion content materials are typically about 2 decades, Roling’s Fig. 7 indicates that the conductivity crossover region occurs a decade or so above the frequency of maximum M″(ln ω). This has never been reported for any material.
(12) It seems likely that the shift of the conductivity ‘master curves’ to higher values of (v/σ0T) = (ω〈τσ〉/2πe0ε∞T) as x decreases, as reported by Roling, is largely (entirely?) due the decrease in 〈τσ〉with decreasing x and increasing β, corresponding to ωmax increasing for ω〈τσ〉 = 1 – the average relaxation time for the KWW response function ϕ(t) = exp[−(t/τ0)β] increases rapidly with decreasing β:Unfortunately, it is not possible to apply a quantitative test of this suggestion because even if the criterion for defining the shift had not been omitted, the logarithmic scales for σ/σ0 and (v/σ0T) obscure too much needed detail.
Section snippets
Summary
There is nothing fundamentally incorrect about the electric modulus formalism. It draws appropriate attention to important details about conductivity relaxation behavior, it provides quantitative accounts of experimentally observed trends, and it provides sensible accounts of more of these trends than other formalisms.
Acknowledgment
We wish to thank J. Dyre, J.P. Johari, and D.L. Sidebottom for stimulating discussions.
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