Havriliak Negami Research Articles

Analytical representations for relaxation functions of glasses

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 represented 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.

Dielectric relaxation and porosity determination of porous silicon

Results of the dielectric spectroscopy study of porous silicon (PS) samples in the frequency range, 20 Hz–1 MHz, and in the temperature range, 173–493 K, are presented. Three relaxation processes that dominate at low, moderate and high temperatures respectively were observed. The low temperature dispersion follows the Cole–Cole type with two activation energies of 0.28–0.4 and 0.2–0.3 eV for samples with thickness 20 and 30 μm, respectively. At moderate temperatures the time domain dielectric response function demonstrates stretch exponential behaviour associated with the percolation of charge carries near threshold. At temperatures above 400 K, we found a strong contribution of dc-conductivity, with approximately the same activation energy of 0.47±0.01 eV for both samples. An additional Havriliak–Negami relaxation process with typical relaxation times of the order of 10 −3 s is observed in this temperature interval. The dielectric response is found to be very sensitive to the geometrical micro- and meso-structural features of the PS. The dynamical aspects of the processes are discussed.

Broad-Band Dielectric Study on Poly(4-vinylphenol)/Poly(ethyl methacrylate) Blends

The dielectric relaxation behavior of miscible poly(4-vinylphenol)/poly(ethyl methacrylate) (PVPh/PEMA) blends has been studied using broad-band dielectric spectroscopy in the frequency domain from 10-2 to 107 Hz and at temperatures between 30 and 180 °C. The α and β processes are resolved from the dielectric spectra by modeling with Havriliak−Negami (HN) functions. PVPh enhances the splitting between the α and β processes and provides a better understanding of the splitting and merging mechanism. Arrhenius plots support the proposal that the α and β processes have the same molecular origin above the merging temperature. The local β process in the blends originates from side-chain rotation of PEMA and is only slightly influenced by PVPh. On the other hand, the α process includes contributions from both components, and its relaxation time increases rapidly with PVPh concentration. However, a fragility plot reveals that the presence of PVPh does not change the cooperativity beyond shifting Tg for the blends, which is inconsistent with the energy landscape model. The relaxation time for the merged αβ process also increases significantly as a result of the participation of PVPh. Because of the strong hydrogen bonding between PEMA and PVPh, the blends exhibit only slight dielectric broadening or concentration fluctuations.

Method for analysis of dynamic mechanical thermal analysis data using the Havriliak–Negami model

A new method is presented for analysis of single frequency dynamic mechanical thermal analysis (DMTA) data in the complex plane. The applicability of the method is restricted to cases where the polymer is thermorheologically simple and whose dynamic mechanical properties are well described by the Havriliak–Negami (HN) model. The method involved two steps: (1) the HN model was used to describe the shape of the complex plane representations of the DMTA data; and (2) the HN relaxation time (τ) was solved for at each temperature over which experimental measurements were made. This procedure resulted in the determination of four temperature independent HN parameters (α, β, E0, and E∞) and one temperature dependent parameter, τ(T). These model parameters were then used to calculate the dynamic mechanical properties over a range of temperatures and frequencies. The calculated moduli and loss factors were generally in good agreement with the experimental values for two elastomeric materials, neoprene and plasticized polyvinylchloride, that were subjected to the analysis procedure, over an 80° temperature range and three decades of frequency. It was also demonstrated that complex plane analysis of frequency multiplexed DMTA data could be used to calculate shift factors for time–temperature superposition. The corresponding master curves for storage modulus and loss factor created by horizontal shifts along the log(frequency) axis were smooth, providing additional support to the validity of the analysis procedure.

Relaxation of dynamically correlated clusters

In the frame of a new probabilistic approach to relaxation, the scenario of relaxation leading to the Havriliak–Negami and Kohlrausch–Williams–Watts responses of complex systems is presented. In this approach the macroscopic laws are related to the micro/mesoscopic stochastic characteristics of the relaxing systems. This provides a rigorous formulation of the energy-criterion argument, introduced by Jonscher to explain the commonly observed high-frequency fractional power law. The presented considerations reinforce the physical significance of the empirically found forms of relaxation, and open a new line of analysis of relaxation phenomena.

Dielectric relaxation of the alternating terpolymers of ethylene, propylene, and carbon monoxide

The alternating terpolymers of ethylene, propylene, and carbon monoxide were characterized by wide-angle x-ray scattering (WAXS), differential scanning calorimeter (DSC), thermally stimulated depolarization currents, and dielectric relaxation spectroscopy (DRS). At room temperature, the terpolymers were x-ray amorphous; however, the existence of weak endothermal effects over fairly broad temperature intervals above the glass transition suggested that they should be classified as “microcrystalline” (in a sense that the ultimate level of crystallinity lies beyond the resolution limits of ordinary WAXS instruments). The DRS data in the temperature intervals of α- and sub-glass relaxations were fitted quantitatively to the Havriliak–Negami equation. The characteristic parameters for the α-relaxation were consistent with classification of terpolymers as relatively fragile glass formers; however, their high fragility was attributed to steric constraints to the motion of chain dipoles by the residual network of microcrystals. Strong asymmetry of the dielectric sub-glass relaxation was regarded as additional experimental evidence for the interference of the spatial network of microcrystallites with the motion (in this case, noncooperative) of chain segments.

Dipolar relaxations in an epoxy–amine system

Abstract Dipolar relaxations in a reactive epoxy–amine system based on diglycidyl ether of bisphenol A with 4,4′-diaminodiphenylsulfone were studied with the Havriliak–Negami function. The system was cured isothermally at 140 °C using simultaneous kinetic and microdielectric studies. The relaxation time was calculated from the frequency fmax of the peak of the loss factor e′′ versus frequency f. A linear relation exists between the logarithm of the relaxation time τ and the glass transition temperature Tg. Then, logτ follows the Di Benedetto equation revisited by Pascault and Williams allowing the prediction of the relaxation time τ during cure. The unrelaxed permittivity at high frequency eu, the relaxed or static permittivity es and the skewness parameter β were found independent on the conversion or the curing time. The distribution parameter α decreases as curing time t increases. The difficulty of orientation of dipoles in the electric field due to vitrification is responsible of these behaviours.

Dielectric relaxation in liquid crystalline dimers

Dielectric measurements are reported for liquid crystalline dimers consisting of two mesogenic groups joined through ether links by a flexible alkyl chain. Results are presented for the odd-symmetric dimer, α,ω-bis[(4-cyanobiphenyl)-4′-yloxy]undecane (BCB.O11), and the odd-asymmetric dimer, α-[(4-cyanobiphenyl)-4′-yloxy]-ω-(4-decylanilinebenzylidene-4′-oxy) nonane (CB.O9O.10). The real and imaginary parts of the electric permittivities of aligned samples were measured as functions of frequency over the range 102–109 Hz at temperatures throughout the nematic phase. Measurements were fitted as a function of frequency to the Havriliak–Negami function, and yielded relaxation times and dielectric strengths for the relaxations at each of the temperatures studied. The static dielectric anisotropy for both materials was positive. For BCB.O11, the parallel and perpendicular components of the permittivity exhibited one low frequency and one high frequency relaxation, while the parallel component of the permittivity for CB.O9O.10 showed three relaxations, which were well-separated in frequency. A generic model of liquid crystalline dimers based on mixtures of linear and bent conformers with the tetrahedral angle has been proposed [A. Ferrarini et al., Chem. Phys. Lett. 214, 409 (1993)]. This model is used to provide a qualitative interpretation of the dielectric measurements reported. To obtain a satisfactory explanation of the results, it is necessary to include additional conformers having either a hairpin shape or a bent shape with an angle of 180° minus the tetrahedral angle. The temperature dependence measured for the strengths of the dielectric relaxations is explained in terms of changes of the order parameter and changes in the relative probabilities of different shaped conformers.

Incoherent quasi-elastic neutron scattering (IQENS) by ethylene glycol in confined space

With the aim of studying the confinement influence on the molecular mobility of H-bonded systems, high-resolution quasi-elastic neutron scattering data were obtained for ethylene glycol [EG: H(O–CH 2–CH 2)OH] in the bulk state at T=40°C and confined into 25 A ̊ pores of a sol–gel porous glass, at T=−35°C and 40°C. The experimental data are discussed in terms of the α-relaxation dynamics predicted by mode-coupling theory for supercooled liquids. We verified that an Havriliak–Negami profile, that is verified when a distribution of relaxation times is present, represents a good description for the measured QENS translational spectral line shape.

On the empirical functions describing the α-relaxation of glass-forming systems

In the present work a comparison among the different functions commonly used to describe the α-relaxation in glass-forming systems has been established. The main results obtained are: (i) By using a direct Fourier transformation we have corroborated previous results, which were obtained by different means, connecting the Havriliak–Negami (HN) and Kohlrausch–Williams–Watts (KWW) descriptions. (ii) An equivalence between the latter description and the one obtained from the Rajagopal distribution has been established in a very wide range. Thus, an expression for the distribution function of relaxation times, valid for most of the stretched exponential behaviours experimentally found, has been derived. (iii) Comparing these three functions with the dielectric relaxation data from different systems, we proposed a criteria for the choice of the most appropriate function depending on the particular type of the glass former system.

Dielectric behaviour of polycarbonate/poly(styrene-co-acrylonitrile) and polycarbonate/poly(methyl methacrylate) two-phase blends

Frequency and temperature dependencies of the complex permittivity ϵ* were investigated for two-phase blends polycarbonate/poly(styrene-co-acrylonitrile) (PC/SAN) and PC/poly(methyl methacrylate). The frequency dependencies of the ϵs* at all temperatures were described by the Havriliak-Negami (HN) equation and the conductivity contribution. The effect of mutual miscibility on the HN shape parameters and temperature dependencies of the peak frequences fm (at which loss component ϵ″ attains its maximum value) is discussed. The Böttcher equation is used for describing the effect of the blend composition on the reduced dielectric strength ΔϵT, the dipole moment μ and ϵ* values measured at the highest temperatures (in the conductivity region). The role of Maxwell-Wagner-Sillars polarization due to interfaces between ordered and disordered domains in PC is mentioned.

Energy criterion in interacting cluster systems

The high-frequency fractional power law of relaxation, seen in a wide range of materials, yields a constant ratio of the macroscopic energy lost per radian to the energy stored in the system, in the corresponding frequency range. For almost two decades, the above energy criterion has been supposed to imply the existence of similar microscopic properties which determine the observed power-law exponent. Here, a rigorous formulation of the energy-criterion argument is proposed in the frame of a new probabilistic approach to derive the Havriliak-Negami (HN) and Kohlraush-Williams-Watts (KWW) responses. In this approach the commonly observed macroscopic laws are related to the microscopic scenario of relaxation, and the energy-criterion interpretation is applied to the physical basis of the relation. The presented considerations reinforce the physical significance of the empirically found forms of relaxation, and open a new line of analysis of relaxation phenomena.

Investigation on dielectric properties of the polyetherketone nanocomposite with lead titanate ultrafines

The dielectric properties of polymer composites with inorganic nanoparticles were investigated. In the demonstration of the dielectric constant expression of the nanocomposite polymer, the dielectric contributions of the displacement polarization, the orientation polarization, and the space polarization in the nanocomposite polymer were all considered. In the demonstration, two dielectric relaxation models were used, that of Debye for inorganic nanoparticles and the Havriliak–Negami function model for polymers. Then the expression of the complex dielectric constant of the nanocomposite polymer was obtained by using Onsager's local field theory. Furthermore, the nanocomposite polymer thin films that consist of PbTiO3 nanoparticles and polyetherketone were prepared. The real parts of the dielectric constants were measured and calculated, respectively. It was found that our calculated results are in good agreement with our measured results. PACS Nos.: 77.20, 77.55, 81.20T

Study of the nature of glass transitions in the plastic crystalline phases of cyclo-octanol, cycloheptanol, cyanoadamantane and cis-1,2-dimethylcyclohexane

Cycloheptanol, cyclo-octanol, cyanoadamantane and cis-1,2-dimethylcyclohexane are known to form plastic crystals which can be supercooled to show a glass transition at a temperature Tg. The molecular dynamics in different plastic phases is studied in their supercooled states using dielectric spectroscopy (frequency range: 106 Hz–10−3 Hz) and differential scanning calorimetry (DSC) over a wide temperature range. The kinetic freezing of the various dielectric processes have been critically examined in relation to the Tg found in the DSC experiments. The plastic phase I of cyclo-octanol shows two Tg’s: one at 148.5 K and the other at 164 K, the former of which is not found in well annealed phase I. The dielectric α-modes correspond to the latter. Cycloheptanol exhibits many Tg’s for the different plastic phases. Unlike the cyclic alcohols, the dielectric spectra of cyanoadamantane and cis-1,2-dimethylcyclohexane is clearly found to follow the Havriliak–Negami equation, both of which interestingly show very little molecular mobility, often referred to as the secondary relaxation in their glassy states. In addition, the temperature dependence of the relaxation rates and the dielectric strengths are critically examined for various supercooled phases to gain an insight into the nature of the molecular mobility in those phases.

Nonexponential dielectric relaxation dynamics in supercooled liquid and glassy states of isoamyl bromide and 2-methylpentane mixtures

The dielectric spectra of mixtures of the polar solute isoamyl bromide in 2-methylpentane have been investigated in the frequency range 1 mHz to 1 MHz and in the temperature range approaching the glass transition temperature. The results obtained from the spectra are compared with those obtained recently [J. Chem. Phys. 111, 10979 (1999)] on pure isoamyl bromide. It is found that on increasing dilution with the nonpolar solvent, the width of the curves of the dielectric spectra increase significantly, and this is reflected in the increase in the nonexponential nature of the relaxation dynamics. This is found to be a consequence of the decrease in the cooperativity of the relaxation dynamics and or an increase in the heterogeneity of the solution. The data are found to fit the Havriliak–Negami equation extremely well. The data at low and high frequencies also fits the “universal law,” since the latter is a low and high frequencies limiting case of the Havriliak–Negami equation. The scaling parameters of this law are calculated for the 25 mol % solution of isoamyl bromide in 2-methylpentane, and these are shown to experimentally relate to the H–N parameters. The stretched exponential parameter, γ, is estimated as a function of the temperature and is shown to follow the equation γ≈a(T−T0). Vogel–Fulcher–Tammann equation fits the data of the relaxation peak frequency as a function of the inverse of absolute temperature for the various mixtures quite well, this being possibly a consequence of the temperature dependence of the stretched exponential parameter. The predictions from the mode coupling theory and those by Bertrand and Souletie are verified with the exception that the exponent is found to be much greater than predicted by these theories. The relative predominance of the Johari–Goldstein process in isoamyl bromide increases initially with dilution with 2-methylpentane and then disappears as the number density of the independent relaxors increases with further dilution.

Dielectric relaxation studies of miscible polycarbonate/polyester blends

Blends of bisphenol-A polycarbonate and a thermoplastic polyester (Kodar) are miscible as shown by single DSC and DMTA glass transition temperatures at all compositions. Blends of polycarbonate and Kodar also display a strong, single dielectric α-relaxation. Conclusions regarding the molecular nature of the blends, the state of mixing and molecular cooperativity can be made by both quantitative and semi-empirical analysis of dielectric relaxation data. There is an increased broadening of the loss spectra on blending as determined by both Havriliak–Negami (HN) and Kohlrausch–Williams–Watts (KWW) fits to the dielectric data, and by analysis of the DMTA loss peaks. An increase of the relaxation time on blending is attributed to the temperature of measurement being below the glass transition of the rigid PC-rich component of the blends and that this is likely to have an effect because of micro-heterogeneity in the blends. Local regions of composition rich in Kodar or polycarbonate exist in these blends and this allows the molecular motion of the PC molecules to dominate the relaxation behaviour of the blends. Concentration fluctuation is thus the dominant feature controlling the dielectric response in these miscible, amorphous blends.

Low frequency dielectric investigations into the relaxation behavior of frozen polyvinylpyrrolidone–water systems

The low frequency dielectric response of aqueous solutions containing 0, 1, 5, and 10% w/v polyvinylpyrrolidone (PVP) was studied to characterize the low temperature relaxation behavior of these systems. Complementary modulated temperature differential scanning calorimetry (MTDSC) studies allowed measurement of the glass transition temperature for these materials, corresponding to the behavior of the nonfrozen phase. Dielectric investigations in the frequency range of 106 to 10−2 Hz were performed on the systems in the liquid state, with a Maxwell–Wagner response noted for both the PVP solutions and water. The solid-phase responses were studied over a range of temperatures down to − 70°C, with a relaxation peak observed for the PVP systems in the kilohertz region. The spectra were modeled using the Havriliak–Negami equation and the corresponding relaxation times were calculated, with a satisfactory fit to the Arrhenius equation noted. The calculated activation energies were similar to literature values for the dielectric relaxation of water. It is suggested that the dielectric response is primarily a reflection of the relaxation behavior of the water molecules in the nonfrozen fraction, thereby indicating that the dielectric technique may yield insights into specific components of frozen aqueous systems. © 2001 Wiley-Liss, Inc. and the American Pharmaceutical Association J Pharm Sci 90:157–164, 2001

Dielectric relaxation studies on thermotropic side-chain liquid crystalline polymers

The dielectric activity of a thermotropic side-chain liquid crystalline polymer is analysed. The sub-glass relaxations are deconvoluted by two different empirically based equations (Fuoss–Kirkwood and Havriliak–Negami). The conductivity contributions are studied following a methodology proposed by Macdonald and Coelho. The effects of the orientation of the mesogenic group under electrical fields are also analysed. We propose a general method to estimate the curve of loss permittivity with a determined value of Sd based on the knowledge of two experimental curves, each corresponding to one well-known orientation. © 2001 Society of Chemical Industry

Structure, phase transition and molecular motions in (C5H5NH)BiCl4

The crystal structure at 293 K of the new pyridinium compound, (C5H5NH)BiCl4, has been determined by X-ray diffraction as monoclinic, space group Cc, Z = 4. The crystal is built up of one-dimensional (BiCl4−)n chains and pyridinium C5H5NH+ cations. A structural phase transition of first-order type is detected by differential scanning calorimetry (DSC) and dilatometric measurements at 114/110 K (on heating–cooling, respectively). Proton NMR second moment and spin–lattice relaxation time studies confirmed the order–disorder mechanism of the phase transition at 110 K. It was connected with the reorientational motion of the pyridinium cation. Dielectric investigations reveal absorption and dispersion in the audio-frequency region in both the high and low temperature phases. The experimental results were analysed in the high temperature phase in terms of the Havriliak–Negami formula. In the low temperature phase the Cole–Cole relation for a single relaxator was applied. Above the phase transition point the macroscopic relaxation time exhibits non-Arrhenius behaviour, whereas below Tc it is perfectly described by a pure Arrhenius relation.

Viscosity and shear response at the dynamic glass transition of glycerol

Viscosity measurements in a broad temperature range display a variation over 11 orders of magnitude and show a crossover at about T+=285 K, comparable with dielectric results from a Stickel evaluation. The index R, i.e., the product of dielectric loss peak maximum frequencies times the viscosity, divided by temperature, decreases by a factor of 3.5 while viscosity varies by seven orders of magnitude. Dynamic shear measurements on glycerol are reported for the temperature of T=192.5 K. The shape of the shear dispersion in the α relaxation zone was fitted with Havriliak–Negami and stretched exponential functions. The shape parameters show considerable differences between different relaxation methods. No indications of a separate (β) secondary relaxation were found below the glass temperature Tg.