Category: dielectric relaxation

Effective transport coefficients of anisotropic disordered materials

Post author By R Hilfer
Post date January 4, 2022

R. Hilfer, J. Hauskrecht

European Physical Journal B 95, 117 (2022)
https://doi.org/10.1140/epjb/s10051-022-00338-5

submitted on
Tuesday, January 4, 2022

A novel effective medium theory for homogenized transport coefficients of anisotropic mixtures of possibly anisotropic materials is developed. Existing theories for isotropic systems cannot be easily extended, because that would require geometric characterizations of anisotropic connectivity. In this work anisotropic connectivity is characterized by introducing a tensor that is constructed from a histogram of local percolating directions. The construction is inspired by local porosity theory. A large number of known and unknown generalized effective medium approximations for anisotropic media are obtained as limiting special cases from the new theory. Among these limiting cases the limit of strong cylindrical anisotropy is of particular interest. The parameter space of the generalized theory is explored, and the advanced results are applied to experiment.

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dielectric relaxation Disordered Systems Glasses Transport Processes

Excess wing physics and nearly constant loss in glasses

Post author By R Hilfer
Post date May 31, 2019

R. Hilfer

Journal of Statistical Mechanics: Theory and Experiment 2019, 104007 (2019)
https://doi.org/10.1088/1742-5468/ab38bc

submitted on
Friday, May 31, 2019

Excess wings and nearly constant loss are almost universal nonequilibrium phenomena in glass formers. Both lack an accepted theoretical foundation. A model-free and unified theoretical description for these phenomena is presented that encompasses also fast β-processes, emergent Debye peaks, and the relaxation strength of the boson peak. The theory is model-free in the same way as the classical Debye relaxation equation for orientational polarisation. It is based on generalizing time flow from translation semigroups to composite time translation-convolution semigroups. Composite translation-convolution fits have less parameters than traditional fits. They need only one dynamic scaling exponent, while four are needed in Havriliak-Negami fits. For glycerol the single dynamic exponent in the translation-convolution fit is found to be temperature-independent.

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Tags 2019, Excess wing

dielectric relaxation Glasses

Excess wings in broadband dielectric spectroscopy

Post author By R Hilfer
Post date July 15, 2014

S. Candelaresi, R. Hilfer

AIP Conference Proceedings 1637, 1283 (2014)
https://doi.org/10.1063/1.4907293

submitted on
Tuesday, July 15, 2014

Analysis of excess wings in broadband dielectric spectroscopy data of glass forming materials provides evidence for anomalous time evolutions and fractional semigroups. Solutions of fractional evolution equations in frequency space are used to fit dielectric spectroscopy data of glass forming materials with a range between 4 and 10 decades in frequency. It is shown that with only three parameters (two relaxation times plus one exponent) excellent fits can be obtained for 5-methyl-2-hexanol and for methyl-m-toluate over up to 7 decades. The traditional Havriliak-Negami fit with three parameters (two exponents and one relaxation time) fits only 4-5 decades. Using a second exponent, as in Havriliak-Negami fits, the α-peak and the excess wing can be modeled perfectly with our theory for up to 10 decades for all materials at all temperatures considered here. Traditionally this can only be accomplished by combining two Havriliak-Negami functions with 7 parameters. The temperature dependent relaxation times are fitted with the Vogel-Tammann-Fulcher relation which provides the corresponding Vogel-Fulcher temperatures. The relaxation times turn out to obey almost perfectly the Vogel-Tammann-Fulcher law. Computable expressions of time dependent relaxation functions are also reported.

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dielectric relaxation Fractional Calculus Fractional Time Theory of Time

Applications and Implications of Fractional Dynamics for Dielectric Relaxation

Post author By R Hilfer
Post date September 23, 2011

R. Hilfer

in: Recent Advances in Broadband Dielectric Spectroscopy
edited by: Y. Kalmykov
Springer, Berlin, 123 (2012)
10.1007/978-94-007-5012-8
978-94-007-5011-1

submitted on
Friday, September 23, 2011

This article summarizes briefly the presentation given by the author at the NATO Advanced Research Workshop on “Broadband Dielectric Spectroscopy and its Advanced Technological Applications”, held in Perpignan, France, in September 2011. The purpose of the invited presentation at the workshop was to review and summarize the basic theory of fractional dynamics (Hilfer, Phys Rev E 48:2466, 1993; Hilfer and Anton, Phys Rev E Rapid Commun 51:R848, 1995; Hilfer, Fractals 3(1):211, 1995; Hilfer, Chaos Solitons Fractals 5:1475, 1995; Hilfer, Fractals 3:549, 1995; Hilfer, Physica A 221:89, 1995; Hilfer, On fractional diffusion and its relation with continuous time random walks. In: Pekalski et al. (eds) Anomalous diffusion: from basis to applications. Springer, Berlin, p 77, 1999; Hilfer, Fractional evolution equations and irreversibility. In: Helbing et al. (eds) Traffic and granular flow’99. Springer, Berlin, p 215, 2000; Hilfer, Fractional time evolution. In: Hilfer (ed) Applications of fractional calculus in physics. World Scientific, Singapore, p 87, 2000; Hilfer, Remarks on fractional time. In: Castell and Ischebeck (eds) Time, quantum and information. Springer, Berlin, p 235, 2003; Hilfer, Physica A 329:35, 2003; Hilfer, Threefold introduction to fractional derivatives. In: Klages et al. (eds) Anomalous transport: foundations and applications. Wiley-VCH, Weinheim, pp 17– 74, 2008; Hilfer, Foundations of fractional dynamics: a short account. In: Klafter et al. (eds) Fractional dynamics: recent advances. World Scientific, Singapore,207, 2011) and demonstrate its relevance and application to broadband dielectric spectroscopy (Hilfer, J Phys Condens Matter 14:2297, 2002; Hilfer, Chem Phys 284:399, 2002; Hilfer, Fractals 11:251, 2003; Hilfer et al., Fractional Calc Appl Anal 12:299, 2009). It was argued, that broadband dielectric spectroscopy might be useful to test effective field theories based on fractional dynamics.

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dielectric relaxation diffusion electrical conductivity Heterogeneous Materials Porous Media

Quantitative comparison of meanfield mixing laws for conductivity and dielectric constant of porous media

Post author By R Hilfer
Post date June 4, 2002

R. Hilfer, J. Widjajakusuma, B. Biswal

Physica A 318, 319 (2003)
https://doi.org/10.1016/S0378-4371(02)01197-4

submitted on
Tuesday, June 4, 2002

Exact numerical solution of the electrostatic disordered potential problem is carried out for four fully discretised threedimensional experimental reconstructions of sedimentary rocks. The measured effective macroscopic dielectric constants and electrical conductivities are compared with parameterfree predictions from several mean field type theories. All these theories give agreeable results for low contrast between the media. Predictions from Local porosity theory, however, match for the entire range of contrast.

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dielectric relaxation Fractional Calculus Fractional Time Glasses

Experimental Evidence for Fractional Time Evolution in Glass Forming Materials

Post author By R Hilfer
Post date December 7, 2001

R. Hilfer

Chem.Phys. 284, 399 (2002)
https://doi.org/10.1016/S0301-0104(02)00670-5

submitted on
Friday, December 7, 2001

The infinitesimal generator of time evolution in the standard equation for exponential (Debye) relaxation is replaced with the infinitesimal generator of composite fractional translations. Composite fractional translations are defined as a combination of translation and the fractional time evolution introduced in [Physica A, 221 (1995) 89]. The fractional differential equation for composite fractional relaxation is solved. The resulting dynamical susceptibility is used to fit broad band dielectric spectroscopy data of glycerol. The composite fractional susceptibility function can exhibit an asymmetric relaxation peak and an excess wing at high frequencies in the imaginary part. Nevertheless it contains only a single stretching exponent. Qualitative and quantitative agreement with dielectric data for glycerol is found that extends into the excess wing. The fits require fewer parameters than traditional fit functions and can extend over up to 13 decades in frequency.

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dielectric relaxation Fractional Calculus Fractional Time Glasses

Fitting the excess wing in the dielectric α-relaxation of propylene carbonate

Post author By R Hilfer
Post date November 28, 2001

R. Hilfer

Journal of Physics: Condensed Matter 14, 2297 (2002)
https://doi.org/10.1088/0953-8984/14/9/318

submitted on
Wednesday, November 28, 2001

A novel fitting function for the complex frequency-dependent dielectric susceptibility is introduced and compared against other fitting functions for experimental broadband dielectric loss spectra of propylene carbonate taken from Schneider et al (Schneider U, Lunkenheimer P, Brand R and Loidl A 1999 Phys. Rev. E 59 6924). The fitting function contains a single stretching exponent similar to the familiar Cole–Davidson or Kohlrausch stretched exponential fits. It is compared to these traditional fits as well as to the Havriliak–Negami susceptibility and a susceptibility for a two-step Debye relaxation. The results for the novel fit are found to give superior agreement.

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dielectric relaxation Glasses Special Functions

H-function representations for stretched exponential relaxation and non-Debye susceptibilities in glassy systems

Post author By R Hilfer
Post date June 28, 2001

R. Hilfer

Physical Review E 65, 061510 (2002)
https://doi.org/10.1103/PhysRevE.65.061510

submitted on
Thursday, June 28, 2001

Analytical expressions in the time and frequency domains are derived for non-Debye relaxation processes. The complex frequency-dependent susceptibility function for the stretched exponential relaxation function is given for general values of the stretching exponent in terms of H-functions. The relaxation functions corresponding to the complex frequency-dependent Cole-Cole, Cole-Davidson, and Havriliak-Negami susceptibilities are given in the time domain in terms of H-functions. It is found that a commonly used correspondence between the stretching exponent of Kohlrausch functions and the stretching parameters of Havriliak-Negami susceptibilities are not generally valid.

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dielectric relaxation Glasses Nonequilibrium Special Functions

Analytical representations for relaxation functions of glasses

Post author By R Hilfer
Post date April 13, 2001

R. Hilfer

Journal of Non-Crystalline Solids 305, 122 (2002)
https://doi.org/10.1016/S0022-3093(02)01088-8

submitted on
Friday, April 13, 2001

Analytical representations in the time and frequency domains are derived for the most frequently used phenomenological fit functions for non-Debye relaxation processes. In the time domain the relaxation functions corresponding to the complex frequency dependent Cole–Cole, Cole–Davidson and Havriliak–Negami susceptibilities are also rep- resented in terms of H-functions. In the frequency domain the complex frequency dependent susceptibility function corresponding to the time dependent stretched exponential relaxation function is given in terms of H-functions. The new representations are useful for fitting to experiment.

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dielectric relaxation diffusion electrical conductivity fluid flow Porous Media

Effective Physical Properties of Sandstones

Post author By R Hilfer
Post date October 6, 1999

J. Widjajakusuma, R. Hilfer

in: IUTAM Symposium on Theoretical and Numerical Methods in Continuum Mechanics of Porous Materials
edited by: W. Ehlers
Solid Mechanics and Its Applications, vol. 87,Kluwer, Dordrecht, 113 (2001)
10.1007/0-306-46953-7
ISBN: 978-0-7923-6766-6

submitted on
Wednesday, October 6, 1999

In this paper we continue the investigation of the effective transport parameters of a digitized sample of Fontainebleau sandstone and three reconstruction models discussed previously in Biswal et. al., Physica A 273, 452 (1999). The effective transport parameters are computed directly by solving the disordered Laplace equation via a finite-volume method. We find that the transport properties of two stochastic models differ significantly from the real sandstone. Moreover, the effective transport parameters are predicted by employing local porosity theory and various traditional mixing-laws (such as effective medium approximation or Maxwell-Garnet theory). The prediction of local porosity theory is in good agreement with the exact result.

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Tags effective medium, sand stone

dielectric relaxation electrical conductivity Heterogeneous Materials

Macroscopic Dielectric Constant for Microstructures of Sedimentary Rocks

Post author By R Hilfer
Post date May 21, 1999

R. Hilfer, J. Widjajakusuma, B. Biswal

Granular Matter 2, 137 (1999)
https://doi.org/10.1007/s100359900035

submitted on
Friday, May 21, 1999

An approximate method to calculate dielectric response and relaxation functions for water saturated sedimentary rocks is tested for realistic three-dimensional pore space images. The test is performed by comparing the prediction from the approximate method against the exact solution. The approximate method is based on image analysis and local porosity theory. An empirical rule for the specification of the length scale in local porosity theory is advanced. The results from the exact solution are compared to those obtained using local porosity theory and various other approximate mixing laws. The calculation based on local porosity theory is found to yield improved quantitative agreement with the exact result.

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Tags effective medium, sand stone, sediments

dielectric relaxation electrical conductivity fluid flow Porous Media

Exact and Approximate Calculations for Conductivity of Sandstones

Post author By R Hilfer
Post date January 5, 1999

J. Widjajakusuma, C. Manwart, B. Biswal, R. Hilfer

Physica A 270, 325 (1999)
https://doi.org/10.1016/S0378-4371(99)00141-7

submitted on
Tuesday, January 5, 1999

We analyze a three-dimensional pore space reconstruction of Fontainebleau sandstone and calculate from it the effective conductivity using local porosity theory. We compare this result with an exact calculation of the effective conductivity that solves directly the disordered Laplace equation. The prediction of local porosity theory is in good quantitative agreement with the exact result.

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Tags effective medium, sand stone, sediments

dielectric relaxation diffusion electrical conductivity fluid flow Heterogeneous Materials Porous Media

Quantitative Prediction of Effective Material Properties of Heterogeneous Media

Post author By R Hilfer
Post date October 8, 1998

J. Widjajakusuma, B. Biswal, R. Hilfer

Computational Materials Science 16, 70 (1999)
https://doi.org/10.1016/S0927-0256(99)00047-6

submitted on
Thursday, October 8, 1998

Effective electrical conductivity and electrical permittivity of water-saturated natural sandstones are evaluated on the basis of local porosity theory (LPT). In contrast to earlier methods, which characterize the underlying microstructure only through the volume fraction, LPT incorporates geometric information about the stochastic microstructure in terms of local porosity distribution and local percolation probabilities. We compare the prediction of LPT and of traditional effective medium theory with the exact results. The exact results for the conductivity and permittivity are obtained by solving the microscopic mixed boundary value problem for the Maxwell equations in the quasistatic approximation. Contrary to the predictions from effective medium theory, the predictions of LPT are in better quantitative agreement with the exact results.

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Tags effective medium, local porosity theory

dielectric relaxation Disordered Systems Porous Media

Measurement of Local Porosities and Dielectric Dispersion for a Water Saturated Porous Medium

Post author By R Hilfer
Post date October 4, 1993

E. Haslund, B.D. Hansen, R. Hilfer, B. Nøst

Journal of Applied Physics 76, 5473 (1994)
https://doi.org/10.1063/1.357205

submitted on
Monday, October 4, 1993

The frequency‐dependent conductivity and dielectric constant of a salt‐water‐saturated porous glass specimen have been measured. The measurements cover the full frequency range of the Maxwell–Wagner dispersion. The experimental results have been compared with the recently introduced local porosity theory and with previous theories. For the purpose of comparing with the local porosity theory experimental measurements of local porosity distributions from digitized pore space images are presented. These experimental porosity distributions are then used for a first experimental test of local porosity theory. The comparison with previous theoretical expressions for the frequency‐dependent effective dielectric function shows that local porosity theory constitutes a significant improvement in the quantitative agreement.

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Tags local porosity theory

dielectric relaxation Porous Media

Geometric and Dielectric Characterization of Porous Media

Post author By R Hilfer
Post date October 12, 1990

R. Hilfer

Physical Review B 44, 60 (1991)
https://doi.org/10.1103/PhysRevB.44.60

submitted on
Friday, October 12, 1990

This paper introduces local porosity distributions and local percolation probabilities as well-defined and experimentally observable geometric characteristics of general porous media. Based on these concepts the dielectric response is analyzed using the effective-medium approximation and percolation scaling theory. The theoretical origin of static and dynamic scaling laws for the dielectric response including Archie’s law in the low-porosity limit are elucidated. The zero-frequency real dielectric constant is found to diverge as as a power law in the high-porosity limit with an exponent analogous to the cementation exponent. Model calculations are presented for the interplay between geometric characteristics and the frequency-dependent dielectric response. Three purely geometric mechanisms are identified, each of which can give rise to a large dielectric enhancement.

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