Havriliak-Negami Relaxation Function Research Articles

Elucidation of the Mechanism of Phase Transition in a Zinc Formate Framework Templated by a Diammonium Cation—Structural, Phonon and Dielectric Studies

In this paper we present the synthesis method and a detailed description of the crystal structure, as well as thermal, dielectric and phonon properties, of the [CH3NH2CH2CH2NH2CH3][Zn2(HCOO)6] (dmenH2-Zn) metal organic framework. The negative charge of the anionic framework ([Zn2(HCOO)6]2-) is balanced by N,N′-dimethylethylenediamine (dmenH22+) ions located in the voids of the framework. Thermal analysis revealed that dmenH2-Zn underwent a reversible structural phase transition at around room temperature (Tc~300 K). The single-crystal X-ray diffraction showed that dmenH22+ templates were dynamically disordered at 295 K, since N-H…O bonds were too weak to surmount their thermally activated motions. Reduction in the temperature resulted in ordering of the dmenH22+ cations as a consequence of freezing of their reorientational movements. This behavior caused a symmetry change from P-31c (trigonal) to C 2/c (monoclinic). The mechanism of the observed phase transition of dmenH2-Zn compound was also investigated by temperature-dependent IR measurements. These spectroscopic studies showed that the ordering of the dmenH22+ ions also resulted in the distortion of the anionic framework. Dielectric investigations revealed the occurrence of the dipolar relaxation process clearly defined in the monoclinic phase. The asymmetric shape of the studied process, which indicated a non-Debye-like relaxation, was analyzed using the Havriliak–Negami relaxation function, leading to an Ea value of approximately 0.36 eV.

Trap-controlled fractal diffusion model of the Havriliak-Negami dielectric relaxation

A new model of non-Debye dielectric relaxation, which leads to the popular in broadband dielectric spectroscopy Havriliak-Negami expression for two-power-law dielectric response, is proposed. The new model is based on the multiple trapping model and the fractal diffusion model. Explicit analytical expressions for experimentally measured values such as dielectric strength, characteristic relaxation time, and power-law exponents, which include the microscopic parameters of the medium, are obtain. The comparison of the expression for the relaxation time with the experiment is demonstrated. A clear physical interpretation of the power-law exponents of the Havriliak-Negami relaxation function is found, one makes it possible to extend the area application of this function and to use it for the fitting of dielectric data, which exhibit atypical relaxation behavior.

Experimental and theoretical elucidation of dielectric behavior of MgO-r-GO nanocomposites

Dielectric behavior of MgO and r-GO containing (MgO-r-GO) nanocomposite is elucidated in the frequency range 1kHz-1MHz at fixed temperatures in the range 300–473K. Three MgO-r-GO nanocomposite samples are synthesized by molecular-level mixing technique by varying the r-GO content and are named as MgO-0.05G, MgO-0.1G and MgO-0.15G in the increasing order of r-GO content. At 1kHz and 300K, dielectric permittivity (ε′) values of MgO-0.05, MgO-0.1G and MgO-0.15G are measured as 10, 12 and 18, respectively, which enhanced to 358, 563 and 612, respectively at 473K. All the composites exhibited non-Debye relaxation which could be explained using Havriliak-Negami (HN) relaxation function. The enhanced ε′ values are found to vary as a function of both orientation and interfacial (Maxwell-Wagner) polarizations (due to the presence of oxygenated polar functional groups and defects). From the excellent fitting of the dielectric dispersion curves with HN relaxation function, it is inferred that at lower frequencies (and at 473K), the interfacial polarization is much stronger in comparison with the same at room temperature, which is attributed to the impeded motions of migrating charges (surface or free charge carriers) and their trapping within the large number of interfaces in the composites.

Low temperature dielectric relaxation study of aqueous solutions of diethylsulfoxide

In the present work, dielectric spectra of mixtures of diethylsulfoxide (DESO) and water are presented, covering a concentration range of 0.2 - 0.3 molar fraction of DESO. The measurements were performed at frequencies between 1 Hz and 10 MHz and for temperatures between 150 and 300 K. It is shown that DESO/water mixtures have strong glass-forming abilities. The permittivity spectra in these mixtures reveal a single relaxation process. It can be described by the Havriliak-Negami relaxation function and its relaxation times follow the Vogel-Fulcher-Tammann law, thus showing the typical signatures of glassy dynamics. The concentration dependence of the relaxation parameters, like fragility, broadening, and glass temperature, are discussed in detail.

Dielectric Relaxation Phenomena of Polylactic Acid with -Crystalline Chitin

-crystalline chitin was added to polylactic acid (PLA), and this PLA was then heat-treated at for one minute. The crystallinity of the heat-treated PLA increased to more than 40%, and its crystallization speed also increased significantly. The temperature dependency of these materials’ relative permittivity () and relative dielectric loss factor () was also examined. The dielectric absorption peak value in curve of the PLA to which chitin was added and was smaller than that of PLA without chitin. Additionally, the Havriliak-Negami relaxation function was used to produce approximation curves for the frequency dependency of and of chitin with PLA added at . As a result, the relaxation strength of the chitin with PLA added was smaller than that of the PLA without chitin, and the relaxation time of the chitin with PLA added was approximately 2.5 times larger than that of the PLA without chitin.

Biomonitoring of Environmental Pollution Using Dielectric Properties of Tree Leaves

In the present work, dielectric measurements were performed in plane-tree leaves collected from a polluted urban site and a natural unpolluted one, in order to investigate the sensitivity of dielectric relaxation spectroscopy to the detection of heavy metals pollution. Although heavy metal concentrations at the urban site are not found considerable higher than those at the natural site, the two samples exhibit different features in the recorded dielectric spectra. Evaluation of experimental data suggests that the dielectric modulus (M*(omega)) representation is the most suitable for accenting the different dielectric relaxation processes of each sample. The imaginary part of dielectric modulus M''(omega) was fitted using a three-term Havriliak-Negami relaxation function, with fitting parameters, which depend on the concentrations of heavy metals. The lower frequency relaxation process is attributed to the ionic conductivity of the samples, while the two others are due to different charge transport mechanisms of alpha-response. The investigation of plane-tree leaves in terms of their dielectric properties can be considered as a promising biomonitoring for environmental pollution.

High Dielectric Permittivity in Silica–Polyaniline Nanocomposites

Complex impedance and dielectric permittivity of wide band gap silica-conducting polyaniline nanocomposite have been investigated as a function of frequency and temperature at different compositions. Grain and grain boundary contributions are observed in impedance spectra. The results are interpreted in terms of two series connected equivalent circuit. The large dielectric permittivity approximately 7500 is found. Large value of permittivity is well described by Maxwell-Wagner polarization. Broad and asymmetric dielectric spectra are analyzed by Havriliak-Negami relaxation function.

Dielectric relaxation spectroscopy of reactively blended amorphous poly(ethylene terephthalate)–poly(ethylene naphthalate) films

Reactively blended poly(ethylene terephthalate)–poly(ethylene naphthalate) films of different compositions and degrees of ester exchange reaction have been investigated by dielectric relaxation spectroscopy. The α-relaxation peaks associated with the glass transition have been fitted by the semi-empirical Havriliak–Negami relaxation function to frequency scans. The parameter relating to relaxation broadness, β, and the relaxation strength, Δ ε, were quantified and were found to be a parameter sensitive to processing conditions. Both values were affected on a molecular level by concentration fluctuation and the molecular chain architecture (such as polymer chain blockiness). Molecular coupling of the blends was determined from the slope of log frequency maxima vs. reduced temperature T g/ T, as commonly done in coupling theory analysis. It was found that the initial materials, PET and PEN show very similar coupling behaviour. Molecular coupling of the blends was little affected by the blend composition or the degree of transesterification. Activation energies of molecular motions have also been determined and show a positive deviation from the rule-of-mixtures averages of the homopolymers which indicates greater chain hindrance to motion of both blocky or the random copolymers formed by the transesterification process.

Scaling behavior of the α relaxation in fragile glass-forming liquids under conditions of high compression

The effect of pressure variation on dynamics of alpha relaxation process in poly[(phenyl glycidyl ether)-co-formaldehyde] has been investigated both under isothermal (T=293 K) and isobaric (P=0.1, 60, 120, 180, and 240 MPa) conditions using broad band dielectric spectroscopy (10(-2) to 10(6) Hz). The alpha relaxation is analyzed by means of the Havriliak-Negami relaxation function which has two shape parameters (alpha and gamma) to characterize non-Debye behavior. As a result we found that the shape parameters of the dielectric function collected for different pressures fall on master curves constructed by plotting alpha and alphagamma against the logarithm of relaxation time. The scaling of shape parameters for different pressure stems from pressure independence of fragility. This provides strong experimental evidence supporting correlation of non-Debye behavior with non-Arrhenius relaxation under high pressure. From an analysis of the shape parameters of relaxation function, in terms of the Schonhals and Schlosser model, we drew conclusions that the molecular mobility of PPG is controlled in the same way by temperature and pressure. The relaxation times exhibit a clear non-Arrhenius behavior under isothermal and isobaric condition.

Dielectric relaxation spectroscopy of a styrene/ n-butyl methacrylate copolymer series

A series of styrene– n-butyl methacrylate copolymers is investigated by dielectric relaxation spectroscopy over a wide frequency range (10 −1 to 10 5 Hz). The relaxation observed is primarily due to the polar carbonyl side group of the n-butyl methacrylate units and behaves in the manner of an α-relaxation with the frequency–temperature locus showing non-Arrhenius behaviour. The relaxation peaks observed are analysed to determine the degree of intermolecular coupling as a function of composition and the broadness of the relaxation spectrum. Coupling is determined by measuring the slope of the log of the frequency maxima as a function of a reduced temperature, T g/ T, as is usual in coupling theory analysis. Intermolecular coupling is found to vary linearly with composition, the greater coupling occurring with increased styrene content. This is explained in terms of the interaction of the styrene units with adjacent chains. The broadness of the relaxation is quantified by the β parameter from the semi-empirical Havriliak–Negami relaxation function fitted to frequency scans. Increased styrene leads to narrower relaxations, the opposite effect to that expected from greater coupling. An explanation is proposed in terms of the range of secondary-bonded, molecular environments that polar units such as n-butyl methacrylate encourage. Increased content of the nonpolar styrene units would diminish this possibility.

Dynamic-Mechanical Study of the Dynamics of Polymer Blends near the Glass Transition

Poly(methyl acrylate) (PMA) and its blends with poly(4-hydroxystyrene) and a poly(4-hydroxy-co-4-methoxystyrene) have been studied by dynamic-mechanical spectroscopy at temperatures close to the calorimetric T g and frequencies between 0.3 and 30 Hz. The results for the imaginary part of the elastic modulus lead to broad master curves, which width increases with the content of the concentration of the second polymer. Even though the results can be described reasonably well using a Kohlrausch-Williams-Watts relaxation function, rather low values of the stretching parameter β are necessary in order to obtain good agreement with the experimental data. The Havriliak-Negami relaxation function leads to slightly worse agreement with experiment. The fragility parameter of the blends decreases with the concentration of PMA. The qualitative trend of the results may be interpreted in terms of the existence of concentration fluctuations in the blends

Non-Lorentzian Rayleigh spectra of bulk homopolymers far above the glass transition.

Non-Lorentzian depolarized Rayleigh-scattering spectra have been clearly observed by Fabry-Perot interferometry in bulk homopolymers, polyisoprene and 1,4-polybutadiene, far above the glass-transition temperature. We have developed two alternative phenomenological methods for analyzing this kind of spectra. First of all we have used a scattering model function based on the Havriliak-Negami relaxation function which, in fact, is compatible with a Kohlrausch-Williams-Watts relaxation function in the time domain. In the case of polyisoprene, the results obtained by this method are in good agreement with the obtained ones through other different relaxation techniques when a similar method of analysis is applied. Besides, we have extended to the frequency domain the contin algorithm which calculates an apparent distribution function of relaxation times from nonexponential time domain relaxation functions. The results obtained as well as the applicability of both methods are discussed.

Dynamics of the α-relaxation in glass-forming polymers. Study by neutron scattering and relaxation techniques

The dynamics of the α-relaxation in three different polymeric systems, poly(vinyl methyl ether) (PVME), poly(vinyl chloride) (PVC) and poly(bisphenol A, 2-hydroxypropylether) (PH) has been studied by means of relaxation techniques and quasielastic neutron scattering (backscattering spectrometers IN10 and IN13 at the ILL-Grenoble). By using these techniques we have covered a wide time scale ranging from mesoscopic to macroscopic times (10 -10 -10 1 s). For analyzing the experimental data we have developed a phenomenological procedure in the frequency domain based on the Havriliak-Negami relaxation function, which in fact implies a Kohlrausch-Williams-Watts relaxation function in the time domain. The results obtained indicate that the dynamics of the α-relaxation in a wide time scale shows a clear non-Debye behaviour. The shape of the relaxation functions is found to be similar for the different techniques used and independent of temperature and momentum transfer ( Q). Moreover, the characteristic relaxation times deduced from the fitting of the experimental data can also be described using only one Vogel-Fulcher functional form. Besides we found that the Q-dependence of the relaxation times obtained by QENS is given by a power law, τ( Q) ∞ Q - n ( n>2), n being dependent on the system, and that the Q-behaviour and the non-Debye behaviour are directly correlated. In the case of PVC, time of flight (TOF) neutron scattering experiments confirm these results in a shorter time scale (2×10 -11 -2× 10 -12 s). Moreover, TOF results also suggest the possibility of interpreting the “fast process” usually detected in glass-forming systems as a Debye-like short regime of the α-relaxation.

Segmental dynamics in polymer melts by relaxation techniques and quasielastic neutron scattering

The dynamics of the segmental α-relaxation in three different polymeric systems, poly(vinyl methy ether) (PVME), poly(vinyl chloride) (PVC) and poly(bisphenol A, 2-hydroxypropylether) (PH) has been studied by means of relaxation techniques and quasielastic neutron scattering (backscattering spectrometers IN10 and IN13 at the ILL-Grenoble). By using these techniques we have covered a wide timescale ranging from mesoscopic to macroscopic times (10−10–101s). For analyzing the experimental data we have developed a phenomenological procedure in the frequency domain based on the Havriliak–Negami relaxation function which in fact implies a Kohlrausch–Williams–Watts relaxation function in the time domain. The results obtained indicate that the dynamics of the α-relaxation in a wide timescale shows a clear non-Debye behaviour. The shape of the relaxation function is found to be similar for the different techniques used and independent of temperature and momentum transfer (Q). Moreover the characteristic relaxation times deduced from the fitting of the experimental data can also be described using only one Vogel-Fulcher functional form. Besides we found that the Q-dependence of the relaxation times obtained by QENS is given by a power law, τ(Q) ∝ Q−n (n > 2) n being dependent on the system, and that the Q-behaviour and the non-Debye behaviour are directly correlated. We discuss this correlation taking into account several data of the dynamics of the α-relaxation previously reported in the literature. We also outline a possible scenario for explaining this empirical correlation.

Interconnection between frequency-domain Havriliak-Negami and time-domain Kohlrausch-Williams-Watts relaxation functions.

The Kohlrausch-Williams-Watts (KWW) and the Havriliak-Negami (HN) relaxation functions have been widely used to describe the relaxation behavior of glass-forming liquids and complex systems over the last several years. The HN relaxation function is a frequency-domain function while the KWW function applies for the time domain. In a previous paper we discussed the interconnections between these two functions by presenting a method where we found that the best HN description in the frequency domain corresponds to a given KWW function in the time domain

Dynamics of the α relaxation of a glass-forming polymeric system: Dielectric, mechanical, nuclear-magnetic-resonance, and neutron-scattering studies

The dynamics of the \ensuremath{\alpha} relaxation in a glass-forming polymeric system, poly(vinyl methyl ether) (PVME) has been studied by means of dielectric and mechanical spectroscopies and nuclear magnetic resonance, as well as by means of quasielastic neutron scattering. By using these techniques we have covered a wide time scale ranging from mesoscopic to macroscopic times (${10}^{\mathrm{\ensuremath{-}}10--}$${10}^{1}$ sec). The dielectric and mechanical data have been interpreted in terms of a Havriliak-Negami relaxation function, ${\mathrm{\ensuremath{\Phi}}}_{\mathrm{HN}}$. Nuclear-magnetic-resonance data were interpreted by means of a spectral density function J(\ensuremath{\omega}) based on ${\mathrm{\ensuremath{\Phi}}}_{\mathrm{HN}}$. Neutron-scattering data were described in terms of a scattering law S(Q,\ensuremath{\omega}) which was also built starting from ${\mathrm{\ensuremath{\Phi}}}_{\mathrm{HN}}$. The results obtained from different experimental techniques indicate that the dynamics of the \ensuremath{\alpha} relaxation in PVME can be well described by means of (i) a common temperature-independent spectral shape and (ii) a common temperature behavior of the relaxation times. We deduce for the spectral shape very similar parameters by fitting the Havriliak-Negami relaxation function to the different experimental data. These shape parameters are found to be not very sensitive to changes of temperature. The resulting characteristic relaxation times follow a Vogel-Fulcher-like temperature behavior in the temperature range ${\mathit{T}}_{\mathit{g}}$-5 KT${\mathit{T}}_{\mathit{g}}$+150 K. Therefore, this implies a self-consistent description of the dynamics of the \ensuremath{\alpha} relaxation obtained by very different probes in PVME.

Relationship between the time-domain Kohlrausch-Williams-Watts and frequency-domain Havriliak-Negami relaxation functions.

The Kohlrausch-Williams-Watts (KWW) and the Havriliak-Negami (HN) relaxation functions have been widely used to describe the relaxation behavior of glass-forming liquids and complex systems. While the HN relaxation function is a frequency function, the natural domain of the KWW relaxation function is time (although it has also been used with frequency-domain spectroscopies). A relationship among the parameters of the two models is suggested by the fact that both models yield an accurate description of real data. Nevertheless, this relationship cannot be an analytical one, since it is known that the HN and the KWW relaxation functions are not exactly Fourier transforms of each other. In order to find out the nature of this relationship, a method which makes use of a distribution of relaxation times is proposed here. Numerical simulations following the KWW model have been assumed to describe the relaxation behavior in time; likewise, the HN description was assumed to be valid for the frequency domain. From this work, a connection among the parameters of both models is obtained, which is expected to be valid for those experimental data that can be described by either the KWW or the HN model. This is the case for most, if not all, measurements on the dynamics in complex systems and glass-forming liquids that frequently appear in the literature. The proposed procedure has been tested by using dielectric-spectroscopy measurements, both in frequency and time domains to study the \ensuremath{\alpha} relaxation in a glass-forming polymeric system, poly(hydroxy ether of bisphenol-A).

Dynamics of the α-process of polymer systems on a microscopical timescale. Neutron and nuclear magnetic resonance study

The dynamics of the α-process in poly(vinyl methyl ether) (PVME) on a microscopical timescale have been studied by means of nuclear magnetic resonance and incoherent quasi-elastic neutron scattering techniques. The experimental data have been interpreted in terms of a spectral density function, J( ω), and a scattering law, S( Q, ω), which were built startingl from the Havriliak-Negami relaxation function. The results obtained have been compared with the dielectric and mechanical relaxation behaviour of the α-process. It appears that the dynamics of the α-process of PVME on a wide time range (1 to 10 −10 s) can be described by using the same time-temperature behaviour and the same Havriliak-Negami ‘shape-parameters’.