Cole-Davidson Distribution Research Articles

Molecular motion of imidazolium cations and phase transitions in (ImH)2KCo(CN)6 with double perovskite structure

Temperature dependence of 1H nuclear magnetic resonance spin–lattice relaxation time T1 of (ImH)2KCo(CN)6, ImH+: imidazolium cation, is reported. The asymmetric T1 minimum observed in intermediate-temperature phase between 198 and 112 K is analyzed assuming the Cole-Davidson distribution of correlation time. Temperature dependences of 14N nuclear quadrupole resonance frequencies and spin–lattice relaxation times are also reported. The phase transitions at 198 and 112 K are shown to be of second and first order, respectively. The 14N relaxation is shown to be governed by the electric field gradient fluctuation due to the in-plane reorientational motion of the imidazolium cations.

Study on enthalpy relaxation of glassy polystyrene using Kohlrausch, Davidson-Cole and Havriliak-Negami distribution functions

The heat capacity data of PS after cooling at different rates were analyzed using heterogeneous kinetic model by combining exponential decay equation with various relaxation time distribution functions instead of the conventional method of combining the stretched relaxation function with Boltzmann superposition. It is found that the shape of the calculated heat capacity peak is highly correlated to the shape of the relaxation time distribution function. The fitting quality of the Davidson-Cole distribution function is worse compared with that of Kohlrausch distribution function. When the mirror functions of Kohlrausch and Davidson-Cole are used, the fitting quality improved greatly. The result indicates that relaxation kinetic of polymer may be described better by combining appropriate relaxation time spectrum with relaxation time model. This approach affords the possibility of taking the temperature and structure dependence of Kohlrausch exponent into consideration in the kinetic study of polymer in non-isothermal relaxations.

Dynamics of Molecular Crystals by Means of (1) H NMR Relaxometry: Dynamical Heterogeneity versus Homogenous Motion.

(1) H NMR relaxometry was used to reveal information on the dynamical properties of the molecular crystal (PyH)5 Bi2 Br11 (PyH=C5 H6 N, pyridinium cation), chosen as an example of a solid that exhibits a complex structure and rotational-like dynamics. Experimental studies were performed over a very broad frequency range, from 4 kHz to 40 MHz (referring to the (1) H resonance frequency) versus temperature. The extensive set of data was thoroughly analyzed in terms of two motional models differing with respect to the assumed mechanism (heterogeneous versus homogenous) of the motion of the PyH cations. A Cole-Davidson distribution of the correlation times describing the assumed motional heterogeneity was tested against a concept of two correlation times characterizing the rotation-like dynamics of the PyH cation around the perpendicular axes differing by about one order of magnitude. The parameters describing the dynamics of the cation, obtained by means of both models, were compared and discussed.

Relaxation and phase-transition characteristics of relaxor ferroelectric potassium lithium niobate

The electric modulus relaxations were considered to be coupled phenomena between the polarization fluctuations due to local symmetry breaking and ionic hopping through nearest neighbor sites. The Nb-rich potassium lithium niobate (K5.595Li3.125Nb11.28O30) crystals exhibited a ferroelectric diffused phase transition around the dielectric maximum temperature Tmax = 350. The electric modulus relaxations were characterized by using the Cole-Davidson distribution of the electric modulus relaxation times at frequencies ranging from 100 Hz to 1 MHz. Although the lattice constants along the a-axes and the c-axes and the tetragonal unit cell volume showed linear expansions with increasing temperature T, the axial ratio c/a decreased with increasing T at temperatures below the Burn’s temperature TB. The dielectric relaxation accompanied the high ionic conduction in the temperature range above T. The ac conductivity σ'(ω) was analyzed by using the formulae, where ωO is the crossover frequency. It was considered that a double-Arrhenius behavior of the dc conductivity σdc at temperatures around the Burn’s temperature TB was suspected of thermally activated motions in the random distribution of oxygen vacancies and lithium ions. The ac universality could result in a disordered configuration of the mobile ions, i.e., the dc conductivity at temperatures above TB. The slow relaxations of the nano-sized polar regions may contribute to the dc conductivity σdc at temperatures below TB. The characteristic relaxation time τCD showed a change in activation energy from 1.08 eV to 0.80 eV at temperatures around TB. The scaling factor increased in proportion to (TB − T) at TTB, but was almost constant at T >TB, where e'∞ is the high frequency dielectric constant and T is the temperature in Kelvin.

Sensitivity of 2H NMR spectroscopy to motional models: Proteins and highly viscous liquids as examples

In order to study to what extent mechanisms of molecular motion can be unambiguously revealed by (2)H NMR spectroscopy, (2)H spectra for proteins (chicken villin protein headpiece HP36, selectively methyl-deuterated at leucine-69, C(δ) D(3)) and binary systems of high viscosity (benzene-d(6) in tricresyl phosphate) have been carefully analyzed as illustrative examples (the spectra are taken from the literature). In the first case, a model of restricted diffusion mediated by jumps between rotameric orientations has been tested against jump- and free diffusion models which describe rotational motion combined with jump dynamics. It has been found that the set of (2)H spectra of methyl-deuterated at leucine-69 chicken villin protein headpiece HP36 can be consistently explained by different motional models as well as by a gaussian distribution of correlation times assuming isotropic rotation (simple brownian diffusion model). The last finding shows that when the possible distribution of correlation times is not very broad one might not be able to distinguish between heterogeneous and homogenous (but more complex) dynamics by analyzing (2)H lineshapes. For benzene-d(6) in tricresyl phosphate, the dynamics is heterogeneous and it has been demonstrated that a gaussian distribution of correlation times reproduces well the experimental lineshapes, while for a Cole-Davidson distribution the agreement is somewhat worse. For inquires into the sensitivity of quadrupolar NMR spectral analysis (by "quadrupolar NMR spectroscopy we understand NMR spectroscopy of nuclei possessing quadrupole moment), the recently presented theoretical approach [Kruk et al., J. Chem. Phys. 135, 224511 (2011)] has been used as it allows simulating quadrupolar spectra for arbitrary motional conditions by employing the stochastic Liouville equation.

Analysis of microsecond relaxation dynamics of proteins and viscous media by recording relaxation shifts of phosphorescence spectra

We studied the low-temperature dynamics of the Stokes shift of instantaneous phosphorescence spectra of eosin and eosin maleimide covalently bound to hemoglobin in a 66% v/v aqueous solution of glycerol under conditions of pulsed excitation. The kinetics of the Stokes shifts are described using the Cole-Davidson distribution function. From the experimental data, we obtain the parameters τ0 and β, which describe the Cole-Davidson distribution. Values of τ0 and β agree well with data obtained by other methods, and these parameters can be used to describe electron transfer reactions in polar solutions and proteins.

Dielectric properties of epoxy/short carbon fiber composites

The dielectric properties of epoxy/short carbon fiber composites at different concentrations 0, 5, 10 and 15% by weight, different thicknesses 2 and 4 mm, and frequency in the range from 20 Hz to 1 MHz were characterized. Scanning electron microscopy and differential scanning calorimetry were utilized. The alternating current (ac) electrical properties (complex impedance, dielectric constant, dielectric loss, real part of electric modulus, imaginary part of electric modulus, electrical conductivity, and relaxation time) were determined. It was found that the applied frequency, filler concentrations, and composite thickness affected the ac electrical properties of the epoxy/carbon fiber composites. The dielectric behaviors of the interfacial polarization between epoxy matrix and carbon fibers could be described by the Maxwell–Wagner–Sillars relaxation. The analysis of the complex electric modulus in the frequency range from 20 Hz to 1 MHz revealed that the interfacial relaxation followed the Cole–Davidson distribution of relaxation times. The universal power-law of ac conductivity was observed in the epoxy/carbon fiber composites. The calculated power exponent (near unity) is physically acceptable within this applied model.

Relaxation dynamics of the conductive processes in BaTiO3 ceramics at high temperature

Abstract The temperature and frequency dependences of the undoped BaTiO 3 ceramics dielectric properties were measured between 25 °C and 700 °C and 100 Hz to 10 MHz, respectively. A dielectric anomaly was observed at low frequencies in the temperature range of 400–700 °C. This anomaly was associated to a low frequency dispersion process taking place at high temperature. The relaxation dynamics of the conductive process in BaTiO 3 ceramics was investigated. A relaxation function in the time domain ( Φ ( t )) was determined from the frequency dependence of the dielectric modulus, using a relaxation function in the frequency domain ( F* ( ω )). In BaTiO 3 ceramics context, the best relaxation functions ( F *( ω )), in the temperature ranges of 220–400 °C and 425 °C and 630 °C, were found to be a Cole–Cole and Davidson–Cole distribution functions, respectively. The relaxation function ( f ( t )) obtained by the time domain method was found to be a Kohlrausch–Williams–Watts (KWW) function type. The activation energy values (0.72 eV and 0.8 eV) reveal a mechanism correlated with the movement of single ionized oxygen vacancies and electrons of the second level of ionization, probably due to the formation of a titanium liquid phase during the sintering process.

Cole–Davidson dynamics of simple chain models

Rotational relaxation functions of the end-to-end vector of short, freely jointed and freely rotating chains were determined from molecular dynamics simulations. The associated response functions were obtained from the one-sided Fourier transform of the relaxation functions. The Cole-Davidson function was used to fit the response functions with extensive use being made of Cole-Cole plots in the fitting procedure. For the systems studied, the Cole-Davidson function provided remarkably accurate fits [as compared to the transform of the Kohlrausch-Williams-Watts (KWW) function]. The only appreciable deviations from the simulation results were in the high frequency limit and were due to ballistic or free rotation effects. The accuracy of the Cole-Davidson function appears to be the result of the transition in the time domain from stretched exponential behavior at intermediate time to single exponential behavior at long time. Such a transition can be explained in terms of a distribution of relaxation times with a well-defined longest relaxation time. Since the Cole-Davidson distribution has a sharp cutoff in relaxation time (while the KWW function does not), it makes sense that the Cole-Davidson would provide a better frequency-domain description of the associated response function than the KWW function does.

Dielectric spectroscopy on aqueous electrolytic solutions

In this work, detailed dielectric measurements are presented on aqueous electrolytic solutions of NaCl and KCl in a broad frequency range, typical for modern telecommunication techniques. The complex dielectric permittivity or equivalently the complex conductivity are systematically studied as function of frequency (100 MHz-40 GHz), temperature (10-60 degrees C) and molar concentration (0.001-1 mol/l). By a detailed analysis of the dielectric results using an asymmetrically broadened Cole-Davidson distribution of relaxation times, in addition to dc conductivity, the dielectric response as function of frequency, temperature, and molar concentration was fully parameterized by a total of 13 parameters. This model ansatz and the 13 parameters include an enormous predictive power, allowing a reasonable estimation of the dielectric constant, loss, and the conductivity for any set of external variables frequency, temperature and concentration. The proposed method is not only useful for rather simple electrolytic solutions, but also for cell suspensions and biological matter, if additional processes, especially at low frequencies, are adequately taken into account.

Dielectric effects on positive temperature coefficient composites of polyethylene and short carbon fibers

By adding the short carbon fibers to the polyethylene matrix, excellent positive temperature coefficient (PTC) effect was achieved. Alternating current (AC) electrical properties of this PTC composite were studied as a function of frequency. The analysis of AC electrical conductivity and dielectric permittivity was done by using a micro-morphology model, which included conductive carbon fiber-aggregates in series with an equivalent circuit of resistor–capacitor parallel that represent the blends at these contact regions. The observed electrical properties of PTC composites were due to the breakage of the conduction networks caused by thermal expansion. The dielectric behaviors of the interfacial polarization between polyethylene matrix and carbon fibers could be described by Maxwell–Wagner–Sillars relaxation when the composite was heated above 116 °C. The analysis of the electric modulus in the frequency range from 100 Hz to 10 MHz revealed that the interfacial relaxation followed the Cole–Davidson distribution of relaxation time.

Dielectric relaxation time of bulk water at 136–140K, background loss and crystallization effects

Dielectric relaxation time, tau, of ultraviscous bulk water has been determined by analyzing its loss tangent, tan delta, data, which had been measured on heating the vapor-deposited amorphous solid water and hyperquenched glassy water in our earlier studies. [Johari, Hallbrucker, and Mayer, J. Chem. Phys. 95, 2955 (1991); 97, 5851 (1992)]. As for glasses and liquids generally, the measured tan delta of water is the sum of a frequency-independent background loss and a frequency-dependent relaxational loss. A two-frequency method is provided for determining the background loss and used for obtaining the relaxational part of tan delta. After considering the structural relaxation and crystal-nuclei growth effects, tau for water has been determined. At 136+/-1 K, it is 2.5+/-0.6 s when a single relaxation time is (untenably) assumed, and 42+/-14 s when a distribution of relaxation times, a characteristic of viscous liquids, is assumed, with Davidson-Cole distribution parameter of 0.75. Structural relaxation time of approximately 70 s for water at 136 K, which was originally estimated from the DSC endotherm [Johari, Hallbrucker, and Mayer, Nature (London) 330, 552 (1987)], has been revised to approximately 33 s. Temperature dependence of tau could not be determined because ultraviscous water crystallizes too rapidly to cubic ice containing stacking faults and intergranular water. The study demonstrates that water is a liquid over the 136-155 K range, thus removing the basis for a recent contention on its state.

Detection of percolation phenomena in binary polar liquids by broadband dielectric spectroscopy

In the previous papers [Stengele, A., Rey, St., Leuenberger, H., 2001. A novel approach to the characterization of polar liquids. Part 1: pure liquids. Int. J. Pharm. 225, 123–134; Stengele, A., Rey, St., Leuenberger, H., 2002. A novel approach to the characterization of polar liquids. Part 2: binary mixtures. Int. J. Pharm. 241, 231–240], it was shown that the Clausius–Mossotti–Debye equation for the quasi-static dielectric constant ( ɛ) can be extended to liquids if the parameter E i/ E is introduced. E i corresponds to the local mean field due to close molecule–molecule interactions after the application of an external electric field E. In the present paper it is demonstrated that the E i/ E parameter and the relaxation behavior of the dipole moment of the polar molecule in binary mixtures of water, respectively, methanol or benzylalcohol with 1,4-dioxane can be used for the detection of percolation phenomena. As 1,4-dioxane has no intrinsic dipole moment but can form hydrogen bonds and is completely miscible with water, respectively, methanol or benzylalcohol, percolation phenomena can be related to the relaxation behavior of the dipole moment of the polar co-solvent. The relaxation behavior of the binary mixtures can be modeled by applying the Debye equation, and the Cole–Davidson distribution function. Superpositions such as the Debye equation and the Cole–Davidson distribution function or a sum ( ∑ i ) of Debye equations are also considered.

Far infrared spectroscopy of water at different temperatures: GHz to THz dielectric spectroscopy of water

The power absorption coefficient and refractive index of water at temperatures of 4, 8.9, 30 and 50 °C have been measured in the wavenumber range 50–220 cm −1. The power absorption coefficient profile is observed to increase with increase in temperatures from 4 to 8.9 and then to 30 °C. This is followed by a decrease in the profile at 50 °C. These results show that the absorbance arises from both the dipolar orientational and the translational stretch of the hydrogen bond. The dielectric loss spectrum in the frequency range 1 GHz to 7 THz wavenumber range (0.03 to 233.3 cm −1) at 25 °C, with those compiled from the literature and combined with these results, is reanalysed in terms of the three relaxation processes and the fourth resonant process. The relaxation times at a temperature of 25 °C are found to be 8.31, 1.0 and 0.10 ps. For the resonant process, the dielectric loss peak is centred at 175 cm −1. The lowest frequency process is pure Debye and interpreted as arising from the activation of the water molecule, from one of the four sites surrounding a central molecule, to a neighbouring unoccupied site. The relaxation process is perfect Debye because the activation has the same barrier for each of the four sites. The second process follows Davidson–Cole distribution and is interpreted as arising from the rotation of single water molecules that are not hydrogen-bonded at a given instant of time. The fastest relaxation process is associated with the intermolecular energy transfer or the energy dissipation through the interaction between the O–H stretch modes. This process may well arise partly from the weak 60 cm −1 band due the hydrogen bond bending and/or from a weak 30 cm −1 band reported in the literature. The resonant process centred at 175 cm −1 arises from the translational modes arising from the stretching of the hydrogen bonds and this involves fluctuations both in the dipole moment and the polarizability as the band is seen in the Raman spectra too.

Microwave dielectric relaxation, electrical conductance and ultrasonic relaxation of LiClO4in polyethylene oxide dimethyl ether–500 (PEO-500)

The relevance of establishing the structure of electrolyte solutions of low dielectric permittivity is briefly recalled. Microwave complex dielectric permittivities e* = e′ − Je″ in the frequency range ∼1 to 130 GHz for LiClO4 dissolved in poly(ethylene oxide)dimethyl ether of average molar mass 500 (PEO-500), in the concentration range 0.40 to 1 mol dm−3, at 25 °C, are reported. The data are interpreted by a Cole–Davidson distribution function. Molar electrical conductance for the same system, in the concentration range 7 × 10−4 to ∼1 mol dm−3, shows a minimum and a maximum as a function of the electrolyte concentration. The data, in the diluted range (⩽0.017 mol/dm3) are interpreted by the Fuoss–Onsager equation, in terms of free ions and ion-pair-species. Ultrasonic relaxation spectra in the frequency range 1 to 300 MHz and concentration range 0.1 to ∼0.4 mol dm−3, are interpreted by the ion-pair dimerization equilibrium 2LiClO4 ↔ (LiClO4)2, the dimer being an apolar antiparallel species , thus rationalizing the depolarization of the solution with the consequent appearance of a maximum in the dielectric permittivity of the solution at high electrolyte concentration. A parallel with previous work on LiPF6 dissolved in the same solvent is drawn.

Molecular dynamics in supercooled di-isobutyl phthalateclose to the glass transition

The dynamics of the α-relaxation in a glass-forming low-molecular-weight system, di-isobutyl phthalate (DIBP), has beenstudied by means of dielectric, nuclear magnetic and mechanicalspectroscopies. By using these techniques we have covered a widetimescale ranging from 10-11 to 102 s. Vogel-Fulchercorrelation time behaviour and the Davidson-Cole distribution functionmake possible a simultaneous fitting of T1, the nuclear Overhauser enhancement (NOE) factor and dielectricdata over the temperature range (Tg-4 K)<T<(Tg + 155 K). These results together with the viscosity data havebeen analysed in terms of the Debye diffusion.

Molecular Dynamics in Solid Riboflavin as Studied by 1H NMR

Spin–lattice relaxation times T1 and T1d as well as NMR second moment were employed to study the molecular dynamics of riboflavin (vitamin B2) in the temperature range 55–350 K. The broad and flat T1 minimum observed at low temperatures is attributed to the motion of two nonequivalent methyl groups. The motion of the methyl groups is interpreted in terms of Haupt's theory, which takes into account the tunneling assisted relaxation. An additional mechanism of relaxation in the high temperature region is provided by the motion of a proton in one of the hydroxyl groups. The Davidson–Cole distribution of correlation times for this motion is assumed.

Frequency dependent study of the correlation functions in EPR spectroscopy — the Cole–Davidson approach: 1. Perdeuterated 2,2,6,6-tetramethyl-4-piperidone N-oxide in toluene

EPR linewidth measurements of PD-Tempone in toluene at 1 (L-Band), 4 (S-Band), 9 (X-Band) and 34 GHz (Q-Band) microwave frequencies indicate the presence of a distribution of relaxation times. The empirical response parameter introduced by Cole–Davidson for the analysis of dielectric relaxation in liquids has been used for the analysis of EPR relaxation data in the low frequency region. The Cole–Davidson parameter can assume values in the range 0< β≤1 When β=1, one obtains the Debye-type spectral density. The calculated linewidth data at 1 GHz agrees with a Cole–Davidson distribution function with a width parameter 0.83±0.04 for a spherical solute. β<1 at L-band suggests the presence of an asymmetrical distribution of relaxation times associated with different modes of relaxation mechanisms or internal molecular motions. This study shows EPR experiments at low microwave frequencies are more sensitive to the shape of the correlation function.

Characterization of Doped Silicon from 0.1 to 2.5 THz Using Multiple Reflection

Via THz Time domain spectroscopy, the characterization of high conductive n-type, 1.31Ω cm silicon can be measured by directly analyzing the multiple reflections using Fabry-Perot theory. The magnitude and phase difference of total transmission show good agreement between theoretical and experimental values over a 2.5 THz frequency range with complex index of refraction and power absorption. The measured absorption and dispersion are strongly frequency-dependent, and all of the results are well fit by a Cole-Davidson type distribution

Universality of NMR Results in LISICON Systems and Other Solid Lithium Conductors

Abstract The temperature evolution of the 7Li NMR spectra and relaxation rates in many investigated solid lithium conductors shows more or less the same behavior. These compounds are characterized by a disordered lithium sublattice with a surplus of cation sites in cavities and channels of the anionic network. At low temperature, the spectra consist of a central line and a distributed satellite base structure. Upon increasing temperature, both components narrow until a reduced constant width with a well resolved quadrupole structure is reached. The mean nuclear quadrupole coupling parameters reduce by either about 5 or by a factor of 15 in all the compounds. The spin-lattice relaxation rates 1/T1 are asymmetric as a function of reciprocal temperature and of quadrupolar origin. The activation energy of the main process of ionic motion may best be obtained from the temperature dependence of the dipolar spin-spin-relaxation rate 1/T2 . The spectral densities of the relaxation dependences correspond to those for inhomogeneous motions; they may be described by modification of the BPP equation, a Cole-Davidson distribution or a Kohlrausch-Williams-Watts function. Within this study three LISICON systems, Li4-3x Gax GeO4 , and two phosphates Li3M2 (PO4)3 (M = Sc, In) were investigated or re-investigated which fit well into this scheme. Activation energies of 39-43 kJ/mol (Li4-3x Gax GeO4 with x = 0.06,0.14, 0.24), 53 kJ/mol (Li 3 Sc 2 (PO4)3) and 75 kJ/mol (Li3 In2 (PO4)3) were obtained.