Reanalysis of the electrode polarization in electrolytic cells limited by blocking electrodes.

We evaluate the effect of ions on the electric response of an insulting liquid by means of the total electric polarization induced in a cell by an external field. The limiting surfaces are assumed blocking and identical and the ions pointike nonpolarizable charged particles. The analysis is limited to the case where the selective ionic adsorption is absent, in such a manner that in the absence of external electric field the sample is locally and globally neutral. We obtain formulas for the effective dielectric constant renormalized by the presence of the ions in the absence and presence of adsorption from the surfaces. Our results coincide with those obtained by means of the electric impedance of the cell. From the coincidence of the results relevant to the effective dielectric constant we infer that the ions in an insulating liquid do not have a conductive or dielectric nature. They are just electric charges dissolved in an insulating liquid.

Three to six ambiguities in immittance spectroscopy data fitting.

Several important ambiguities in immittance spectroscopy (IS) model data-fitting results are identified and illustrated by means of complex-nonlinear-least-squares (CNLS) fits of experimental and synthetic frequency response data. A well-known intrinsic ambiguity, following from Maxwell's electromagnetic equations, arises from the indistinguishability in external measurements of conduction and displacement currents. Usual fit models for either dielectric or conductive-system situations, such as the Davidson-Cole one, only involve a strength parameter, a dielectric constant, a characteristic relaxation time, and a fractional exponent and lead to no additional ambiguities. But the situation is different for more powerful and useful general models, such as ordinary or anomalous diffusion Poisson-Nernst-Planck ones: PNP and PNPA, used here, whose historical background, current status, and applicability are described and discussed herein. They apply to two different kinds of experimental IS situations and involve several additional, potentially free fit parameters, such as the mobilities of positive and negative charge carriers, and generation-recombination parameters that determine the partial or complete dissociation of a neutral entity of concentration N(0) into positive and negative charge carriers of equal concentration, c(0). Then, several additional ambiguities appear that may require information about the material system involved for their adequate resolution.

Effects of various boundary conditions on the response of Poisson-Nernst-Planck impedance spectroscopy analysis models and comparison with a continuous-time random-walk model.

Various electrode reaction rate boundary conditions suitable for mean-field Poisson-Nernst-Planck (PNP) mobile charge frequency response continuum models are defined and incorporated in the resulting Chang-Jaffe (CJ) CJPNP model, the ohmic OHPNP one, and a simplified GPNP one in order to generalize from full to partial blocking of mobile charges at the two plane parallel electrodes. Model responses using exact synthetic PNP data involving only mobile negative charges are discussed and compared for a wide range of CJ dimensionless reaction rate values. The CJPNP and OHPNP ones are shown to be fully equivalent, except possibly for the analysis of nanomaterial structures. The dielectric strengths associated with the CJPNP diffuse double layers at the electrodes were found to decrease toward 0 as the reaction rate increased, consistent with fewer blocked charges and more reacting ones. Parameter estimates from GPNP fits of CJPNP data were shown to lead to accurate calculated values of the CJ reaction rate and of some other CJPNP parameters. Best fits of CaCu(3)Ti(4)O(12) (CCTO) single-crystal data, an electronic conductor, at 80 and 140 K, required the anomalous diffusion model, CJPNPA, and led to medium-size rate estimates of about 0.12 and 0.03, respectively, as well as good estimates of the values of other important CJPNPA parameters such as the independently verified concentration of neutral dissociable centers. These continuum-fit results were found to be only somewhat comparable to those obtained from a composite continuous-time random-walk hopping/trapping semiuniversal UN model.

Transport process of ions in insulating media in the hyperbolic diffusion regime.

We extend the microscopic Poisson-Nernst-Planck theory of the effects of mobile charge carriers, completely blocked at the electrodes, on the properties of dielectric materials by incorporating a finite speed of response propagation. The usual microscopic theory is based on the assumption that the diffusion current is given by Fick's law, relating the current density with the gradient of concentration at the same time. On the contrary we assume that the flux of diffusing particles is delayed with respect to the concentration gradient, as suggested by extended thermodynamics formulations. We show that, in the hyperbolic diffusion regime approximation, new trends for the real and imaginary parts of the small-signal electrical impedance of a dielectric containing ions versus the frequency of the applied voltage are expected when the delay time is comparable with the Debye relaxation time. In particular, at sufficiently high frequencies the real part of the conductivity, normally independent of frequency, decreases toward zero because of the finite, rather than the infinite propagation speed present in the hyperbolic diffusion regime model.

Utility of continuum diffusion models for analyzing mobile-ion immittance data: electrode polarization, bulk, and generation-recombination effects.

Consequences of the well-known Poisson-Nernst-Planck (PNP) continuum equations of charge motion in liquids or solids for ordinary or anomalous diffusion are investigated for an electrochemical cell with completely blocking electrodes. Previous work is summarized and much of it is shown to be independent of earlier published results and incomplete, with little comparison made between ordinary and anomalous diffusion. Such comparison is provided here and also includes variation of the mobility ratio of the mobilities of positive and negative charges from equality to charge of only one sign mobile. New generation-recombination effects are demonstrated for a range of mobility ratios, with particular attention given to those present for the case of charge of only one sign mobile. No previous analyses of experimental data with PNP models using complex-least-squares fitting have been published. Here such a model is found to fit frequency response data well for a hydrogel and to lead to estimates of physically meaningful parameters such as the diffusion constant and ionic concentration. PNP analysis of a synthetic data set derived from experimental results for liquid electrolytes refutes claims made in the original publication dealing with it, but verifies and extends an interesting analysis equation proposed there. PNP fitting of data for solids, including ones showing colossal low-frequency-limiting dielectric constants, suggests that they may often be well described as arising from simple diffuse-charge double-layer effects, and that continuum microscopic models such as the PNP, in series with a conducting Debye response model, may be sufficient for fitting well an appreciable amount of data involving ion hopping and trapping behavior.

Comparison of some random-barrier, continuous-time random-walk, and other models for the analysis of wide-range frequency response of ion-conducting materials.

Several theoretical models are fitted to an exact wide-frequency-range data set representing a new random-free-energy, effective-medium expression for mobile-ion response at low temperatures. A continuous-time, random-walk K1 model, indirectly involving a stretched-exponential temporal correlation function, led to the best fit, much superior to those of two dielectric-dispersion models as well. Two types of approaches are compared for analyzing 0.4Ca(NO(3))(2).0.6KNO(3) (CKN) experimental data covering a very wide frequency range: a simplified conventional approach, usually involving only some or all of the real part of conductivity, denoted type 1, and an approach involving nonlinear least-squares fitting of full complex data over its entire range, denoted type 2. The type-2 analysis uses a composite fitting model involving the K1 and involves 10 free parameters needed to well represent electrode polarization, conductive-system dispersion, nearly constant loss, and limiting far-infrared vibrational effects. It confirmed that the latter were purely dielectric and led to sigma'' and epsilon'' boson peaks; included a mobile-charge explanation of the nearly constant loss region; and yielded reasonable values of the K1-model fractional exponent, beta(1), and plausible values of a completely blocking double-layer capacitance. The good type-2 fit parameter estimates were used to generate extrapolated model response over a 20-decade range at complex conductivity and complex relative permittivity levels, as well as their accurate slopes over that range. The maximum slope of the log-log sigma' curve failed to approximate well the value of 2 usually inferred from data of the present type but instead led to a novel double peak with peak slope values of about 1.6 and 1.7 before decreasing to zero at the limiting far-infrared plateau region of sigma' response. Nearly constant loss was found to be well described by the series combination of the bulk high-frequency-limiting dielectric constant of the material and a translational ionic-motion constant-phase-element expression, one whose inclusion was also needed for representing low-frequency electrode polarization effects. Further, this combination should dominate the full response at sufficiently low temperatures.

Analysis of dielectric and conductive dispersion above Tg in glass-forming molecular liquids.

Dynamics of the nonassociated supercooled liquids N-methyl-epsilon-caprolactam (NMEC) and glycerol in the frequency domain are investigated using full complex-nonlinear-least-squares fitting of immittance spectroscopy data for appreciable temperature ranges above the glass transition. Such fitting, not previously used for these materials, helps to identify physical processes responsible for the data and elements of their common behavior. Several different fitting models were applied to find a physically plausible best-fitting one to distinguish quantitatively between the dielectric effects of dipoles and the conductive effects of mobile ions. The utility of many composite fitting models was investigated, and although a pure conductive-system dispersive (CSD) fitting model led to good but physically unrealistic fits of all data sets, the dielectric-system dispersive (DSD) Davidson-Cole model best fitted the alpha-dispersion part of the responses. Nevertheless, the series combination of such a DSD model and a separate CSD model (one not associated with electrode effects) was found to yield much better fitting of the data for both materials. Although the CSD model plays somewhat the role of the conventional parallel DSD Johari-Goldstein beta-response, it is here in series and arises from mobile impurity-ion effects rather than from dipolar ones. Previous analyses of data of the present and other molecular materials have often involved two DSD models in parallel, but fitting with such a composite model led here to less physically plausible parameter values and ones with appreciably more uncertainties. Surprisingly, the series DSD and CSD composite-model fits led to comparable estimated values of the NMEC and glycerol dielectric strength parameters, as well as to the nearly equal small thermal activation energies of these parameters.

Dynamics of mobile ions: fitting of CKN frequency response data without an excess wing.

Ion dynamics effects and the resulting dispersed frequency response of conducting materials have often been explained in the past by a combination of the Moynihan original modulus formalism (OMF) and the Ngai coupling model (NCM). These incorrect approaches and their inappropriate conclusions are replaced by alternate, Kohlrausch-related physically reasonable conductive-system fitting and interpretation models that are then used for the analysis of both limited-range and wide-range data for the supercooled liquid 0.4Ca(NO3)2*0.6KNO3 (CKN). Detailed analysis of the limited-range 342 K data at the electric modulus immittance level shows that OMF fitting leads to an excess wing and that more appropriate models fit the data well without such a wing. Further, although such models allow estimation of the bulk dipolar dielectric constant of the material, as well as one associated only with mobile charges, they lead to implausibly small estimates of the important Kohlrausch K1 model shape parameter, beta1, and lead to an inadequate determination of its characteristic relaxation time. Therefore, wide-range CKN data sets extending to nearly 1012 Hz for the temperatures 342, 350, 356, and 361 K were very well-fitted with a more detailed composite model but one still involving K1 response. All model parameters were well-determined with no excess wings; beta1 estimates were all much closer to the universal value of 1/3; and the estimated model parameters led to a Boson peak beyond 1012 Hz, to very large thermal activation energies, and to evidence that the mobile charge concentration reached a saturation value at about 356 K. Such results do not support assumptions about variable ion-ion correlation, a mainstay of the OMF and NCM approaches. Finally, it is shown that although excess wings can sometimes be eliminated by using just an appropriate bulk fitting model and series blocking-electrode capacitor, as shown for the present narrow-range data, adequate fitting of the present wide-range data sets over their full spans of as much as 13 decades required the addition of an additional series dispersive-response model to the composite model. This addition seems likely to be required to take adequate account of the presence of more than one species of mobile charge in CKN.

Discrimination Between Equations of State.

Eight isothermal equations of state are analyzed to yield quantitative measures of the degrees to which equation pairs can he discriminated for real data, data of limited span and precision. Calculated curves allow one to assess the span and precision necessary in P-V data to allow unambiguous discrimination of various pairs. Some discussion is presented of bias and systematic error which may arise in least squares fitting. Using exact synthetic data, we also illustrate for seven equation pairs the very large relative systematic errors in parameter and standard deviation estimates which arise from such fitting of data of limited span with an incorrect but "close" equation model. General conclusions following from these results are discussed. Although the present work is principally concerned with discrimination between equations of state, its results are pertinent to the more general problem of choosing a "best" analytical model (linear or nonlinear) to represent experimental results.