Alternatives to Kronig-Kramers Transformation and Testing, and Estimation of Distributions

Two alternatives to Kronig-Kramers analysis of small-signal ac immittance data are discussed and illustrated using both synthetic and experimental data. The first, a derivative method of approximating imaginary-part response from real-part data, is found to be too approximate in regions where the imaginary-part varies appreciably with frequency. The second, a distribution of relaxation-times fitting method, is shown to be valuable for testing whether a data set satisfies the Kronig-Kramers relations and so is associated with a system whose properties are time-invariant. It also is valuable for estimating real- or imaginary-part response from the other part, usually with small error. Unlike Kronig-Kramers analysis, the second method usually requires no extrapolation outside the range of the measured data. Finally, this discrete-function method also allows one to estimate the distribution of relaxation times or activation energies associated with a given set of frequency-response data. This application is described and illustrated for both synthetic and experimental data and is shown to yield good but somewhat approximate results for the estimation of continuous distributions. It is particularly valuable for identifying response regions arising from a continuous distribution and distinguishing them from those associated with discrete time-constant response

Universality, the Barton, Namikawa, and Nakajima relation, and scaling for dispersive ionic materials

Many frequency-response analyses of experimental data for homogeneous glasses and single-crystals involving mobile ions of a single type indicate that estimates of the stretched-exponential beta1 shape parameter of the Kohlrausch K1 fitting model are close to 1/3 and are virtually independent of both temperature and ionic concentration. This model, which usually yields better fits than others, is indirectly associated with temporal-domain stretched-exponential response having the same beta1 parameter value. Here it is shown that for the above conditions several different analyses yield the important and unique value of exactly 1/3 for the beta1 of the K1 model. It is therefore appropriate to fix the beta1 parameter of this model at the constant value of 1/3, then defined as the U model. It fits data sets exhibiting conductive-system dispersion that vary with both temperature and concentration just as well as those with beta1 free to vary, and it leads to a correspondingly universal value of the Barton-Nakajima-Namikawa (BNN) parameter p of 1.65. Composite-model complex-nonlinear-least-squares fitting, including the dispersive U-model,the effects of the bulk dipolar-electronic dielectric constant, and of electrode polarization when significant, also leads to estimates of two hopping parameters that yield optimum scaling of experimental data that involve temperature and concentration variation.Comment: 30 pages, 6 figures. Expanded and rewrote discussion and important definitions, and added further support to conclusion

Comparison of methods for estimating continuous distributions of relaxation times

The nonparametric estimation of the distribution of relaxation times approach is not as frequently used in the analysis of dispersed response of dielectric or conductive materials as are other immittance data analysis methods based on parametric curve fitting techniques. Nevertheless, such distributions can yield important information about the physical processes present in measured material. In this letter, we apply two quite different numerical inversion methods to estimate the distribution of relaxation times for glassy \lila\ dielectric frequency-response data at 225 \kelvin. Both methods yield unique distributions that agree very closely with the actual exact one accurately calculated from the corrected bulk-dispersion Kohlrausch model established independently by means of parametric data fit using the corrected modulus formalism method. The obtained distributions are also greatly superior to those estimated using approximate functions equations given in the literature.Comment: 4 pages and 4 figure

On two incompatible models for dispersion in ionic conductors

The two models considered are the widely used 1973 original modulus formalism (OMF) of Moynihan and associates, and the later corrected modulus formalism (CMF). Both approaches involve a dispersive frequency-response model derived from Kohlrausch stretched-exponential temporal response, the KWW1 model, also termed the K1. A brief summary of the derivation of this model is followed by consideration of the small but crucial differences between OMF and CMF analysis approaches and the reasons why the OMF and an inferred physical basis for its behavior, variable correlation between mobile ions, are inappropriate. After discussions of some prior criticisms of the OMF approach, results of accurate least-squares fitting of experimental frequency-response data to OMF and CMF models for a variety of ionic materials illustrate these differences and demonstrate a crucial inconsistency of the OMF, one that critically falsifies it

Effective-medium model for nearly constant loss in ionic conductors

A complex quantitative model for nearly constant loss (NCL) is proposed based on an effective-medium approach. Unlike previous NCL response models, it satisfies the Kronig–Kramers transform relations. Here the effective-medium dielectric-level model depends directly on the concentration of mobile charge present and its complex dielectric response is identified as arising from electrical interactions between vibrating and/or hopping ions and the bulk matrix material. The parallel combination of the effective-medium response with dispersive hopping described by the Kohlrausch K1 model, a version of the corrected-modulus-formalism approach, leads to behavior that can represent dominant NCL at low temperatures well and, at higher temperatures, dispersive response followed by NCL. Complex nonlinear-least-squares fitting of experimental data sets that exhibit both types of response leads to excellent fits. Further, the effective-medium NCL model, which involves physically realizable response, can represent a wide range of NCL behavior analytically. Such behavior ranges from either approximate or exact power-law frequency dependence for both parts of the complex dielectric constant or to such response for its real part and very close to constant loss over a wide range of frequency for the associated imaginary part, as sometimes observed

Exact and approximate nonlinear least‐squares inversion of dielectric relaxation spectra

Three weighted, complex nonlinear least‐squares methods for the deconvolution of dielectric or conducting system frequency‐response data are described and applied to synthetic data and to dielectric data of n‐pentanol alcohol, water, and glycerol. The first method represents a distribution of relaxation times or transition rates by an inherently discrete function. Its inversion accuracy and resolution power are shown to be limited only by the accuracy of the data when the data themselves arise from a discrete distribution involving an arbitrary number of spectral lines. It is shown that those inversion methods employed here which allow the relaxation times to be free variables are much superior to those where these quantities are fixed. Furthermore, free‐τ methods allow unambiguous discrimination between discrete and continuous distributions, even for data with substantial errors. Contrary to previous conclusions, discrete distributions were determined for both n‐pentanol alcohol and water. A complex, continuous distribution estimate was obtained for glycerol. Algorithms for all approaches are incorporated in a readily available computer program. Serious problems with some previous dielectric inversion methods are identified. Finally, several possibilities are mentioned that may allow greater inversion resolution to be obtained for complex nonlinear least‐squares estimation of continuous distributions from noisy data

The Ngai coupling model of relaxation: Generalizations, alternatives, and their use in the analysis of non-Arrhenius conductivity in glassy, fast-ionic materials

The ionic conductivity of glassy, fast-ion-conducting materials can show non-Arrhenius behavior and approach saturation at sufficiently high temperatures [J. Kincs and S. W. Martin, Phys. Rev. Lett. 76, 20 (1996)]. The Ngai coupling model was soon applied to explain some of these observations [K. L. Ngai and A. K. Rizos, Phys. Rev. Lett. 76, 1296 (1996)], but detailed examination and generalization of the coupling model suggested the consideration of a related, yet different, approach, the cutoff model. Although both the coupling and cutoff models involve a shortest nonzero response time, τ c , and lead to single-relaxation-time Debye response at limiting short times and high frequencies, they involve different physical interpretations of their low- and high-frequency response functions. These differences are discussed; the predictions of both models in the frequency and time domains are compared; and the utility of both models is evaluated for explaining the non-Arrhenius conductivity behavior associated with the dispersed frequency response of z AgI +(1−z)[0.525 Ag 2 S+0.475B 2 S 3 :SiS 2 ] glass for z=0 and 0.4. The cutoff approach, using simulation rather than direct data fitting, yielded semiquantitative agreement with the data, but similar analysis using the coupling model led to poor results. The coupling model leads to an appreciable slope discontinuity at the τ c transition point between its two separate response parts, while the cutoff model shows no such discontinuity because it involves only a single response equation with a smooth transition at τ c to limiting single-relaxation-time response. The greater simplicity, utility, and generality of the cutoff model suggest that it should be the favored choice for analyzing high-conductivity data exhibiting non-Arrhenius behavior

Scaling and modeling in the analysis of dispersive relaxation of ionic materials

Problems with scaling of conductive-system experimental M dat ″ (ω) and σ dat ′ (ω) data are considered and resolved by dispersive-relaxation-model fitting and comparison. Scaling is attempted for both synthetic and experimental M ″ (ω) data sets. A crucial element in all experimental frequency-response data is the influence of the high-frequency-limiting dipolar-and-vibronic dielectric constant ε D∞ , often designated ε ∞ , and not related to ionic transport. It is shown that ε D∞ precludes scaling of M dat ″ (ω) for ionic materials when the mobile-charge concentration varies. When the effects of ε D∞ are properly removed from the data, however, such scaling is viable. Only the σ ′ (ω) and ε ″ (ω) parts of immittance response are uninfluenced by ε D∞ . Thus, scaling is possible for experimental σ ′ (ω) data sets under concentration variation if the shape parameter of a well-fitting model remains constant and if any parts of the response not associated with bulk ionic transport are eliminated. Comparison between the predictions of the original-modulus-formalism (OMF) response model of 1972–1973 and a corrected version of it that takes proper account of ε D∞ , the corrected modulus formalism (CMF), demonstrates that the role played by ε D∞ (or ε ∞ ) in the OMF is incorrect. Detailed fitting of data for three different ionic glasses using a Kohlrausch–Williams–Watts response model, the KWW 1 , for OMF and CMF analysis clearly demonstrates that the OMF leads to inconsistent shape-parameter (β 1 ) estimates and the CMF does not. The CMF KWW 1 model is shown to subsume, correct, and generalize the recent disparate scaling/fitting approaches of Sidebottom, León, Roling, and Ngai

Comparison and evaluation of several models for fitting the frequency response of dispersive systems

Using both simulated and experimental data, detailed comparisons are made between the different physical interpretations and responses of several important models commonly employed for fitting and analyzing conductive-system data sets, such as those for ionic glasses. Those considered are one following directly from stretched-exponential temporal response, designated the Kohlrausch K0; several ones indirectly associated with such stretched-exponential response: the original modulus formalism (OMF) model and corrected modulus formalism (CMF) ones; and the ZC model, one whose real-part conductivity expression has been termed “universal dynamic response.” In addition, several models involving dielectric dispersion, rather than resistive dispersion, are found to be less appropriate for the present data than are the CMF ones. Of the four main conductive-system models the CMF approach fits data for a wide variety of materials much better than do the others. The OMF is shown to be both experimentally and theoretically defective and leads to poor and inconsistent fitting results. The simple ZC model involves nonphysical low-frequency-limiting real-part conductivity response and is usually less appropriate even than the K0. High- and low-frequency expressions and fit results for the various dielectric elements are presented, along with discussion of characteristic, peak, and mean relaxation times for the various models, failing to confirm some proposed relations between these quantities suggested earlier