Dependence Of Kt Research Articles

Young’s Modulus-Independent Determination of Fibre Parameters for Rayleigh-Based Optical Frequency Domain Reflectometry from Cryogenic Temperatures up to 353 K

The magnetic spectrometer AMS-100, which includes a superconducting coil, is designed to measure cosmic rays and detect cosmic antimatter in space. This extreme environment requires a suitable sensing solution to monitor critical changes in the structure such as the beginning of a quench in the superconducting coil. Rayleigh-scattering-based distributed optical fibre sensors (DOFS) fulfil the high requirements for these extreme conditions but require precise calibration of the temperature and strain coefficients of the optical fibre. Therefore, the fibre-dependent strain and temperature coefficients KT and Kϵ for the temperature range from 77 K to 353 K were investigated in this study. The fibre was integrated into an aluminium tensile test sample with well-calibrated strain gauges to determine the fibre’s Kϵ independently of its Young’s modulus. Simulations were used to validate that the strain caused by changes in temperature or mechanical conditions was the same in the optical fibre as in the aluminium test sample. The results indicated a linear temperature dependence of Kϵ and a non-linear temperature dependence of KT. With the parameters presented in this work, it was possible to accurately determine the strain or temperature of an aluminium structure over the entire temperature range from 77 K to 353 K using the DOFS.

Analytical treatment of the critical properties of a generalized van der Waals equation

Abstract The two-parameter van der Waals (vdW) equation of state is generalized, by adding another two parameters to the attractive term. General relations between thermodynamic functions of the generalized vdW equation and the hard sphere gas are derived. The cubic equation of the generalized vdW is solved and the critical points (P c , V c , T c ) are obtained for general k. The critical properties of the vdW real gas such as the isothermal compressibility K T , the isobaric expansion coefficient α and the isobaric heat capacity C P are calculated exactly. The temperature dependence of K T , α and C P is investigated close to the critical point on the critical isobar path P r = 1(P = P c ). Numerical calculations for K T and C P are presented above and below P r .

Kinetic modelling of the reversible addition–fragmentation chain transfer polymerisation of N-isopropylacrylamide

Reversible addition–fragmentation chain transfer (RAFT) polymerisation is a complex process involving multiple fundamental reaction steps that makes estimation of rate coefficients and exact kinetic simulation problematical. In this paper, we show that the homopolymerisation of N-isopropylacrylamide (NIPAM) can be kinetically modelled using a much-simplified method involving only a few kinetic parameters. The experimental polymerisation of NIPAM was performed with 4-cyano-4-(phenylcarbonothioylthio)pentanoic acid as the RAFT agent and 4,4′-azobis(4-cyanopentanoic acid) as the initiator in dioxane at three different temperatures (60, 70 and 80 °C) with the initial [RAFT]:[initiator] ratio kept at 5:1 and different initial [monomer]:[RAFT] ranging from 100:0.2 to 100:1. Our method for the kinetic simulation makes use of a pseudo-concentration of polymer, in which all chains are assumed to have the same molar mass equal to the average molar mass Mn. With these approximations, the apparent rate coefficients for propagation, termination and initiator decomposition (kp,app, kt,app and kd,app respectively) and three fitting parameters, αp (ratio of propagating radicals to macroRAFT agent), αkp (ratio krein,app/kp,app), and αkt (exponent describing the chain length dependence of kt) were optimised. Our modelling results adequately simulated the experimental polymerization kinetics. Values of kt,app, kd,app, αp, αkp, αkt appeared essentially independent of [monomer]:[RAFT] and the values kt,app, kd,app and αkt were consistent in magnitude with literature values of the corresponding kt, α and kdf. Retardation associated with the RAFT process is encompassed in the value of kp,app, which was strongly dependent on [monomer]:[RAFT], with higher kp,app and Rp being observed for higher [monomer]:[RAFT].

Water reduction into hydrogen using Rh-doped SrTiO3 photoelectrodes surface-modified by minute amounts of Pt: Insights from heterogeneous kinetic analysis

In this paper, we report on the photoelectrochemical hydrogen evolution reaction (HER) from water using Rhodium-doped strontium titanate, surface-modified by the addition of minute amounts of platinum nanoparticles. Platinum has been used in an attempt to increase the charge transfer efficiency of the photoelectrode but also to investigate the balancing effect of surface co-catalysis on charge transfer versus recombination processes. The dynamics of the HER in pH-neutral aqueous solutions has been investigated by Photoelectrochemical Impedance Spectroscopy. Experimental impedance data has been analyzed using a model derived from considerations on heterogeneous kinetics. From the best experimental vs. calculation PEIS fits, microscopic rate parameters have been determined over an extended range of electrical potential. The charge transfer (kt) and recombination (kr) rate constants have been found to vary exponentially with the applied potential. The analysis of the potential dependence of kt led to a Butler-Volmer transfer coefficient of 0.21. The analysis of the potential dependence of the recombination rate constant kr led to the determination of a non-ideality factor β = 0.04 (β = 1 for ideal interfaces), demonstrating the strong non-ideal behavior of the interface. Such non-ideal behavior has been attributed to the presence of a high density of surface states. The space charge capacitance under inversion conditions, i.e. when a strong reversed bias is applied and when the surface of this p-semiconductor exhibits a n-type conductivity, has also been measured and analyzed.

Chain-Length-Dependent Termination of Sodium Methacrylate Polymerization in Aqueous Solution Studied by SP-PLP-EPR

Via the single pulse–pulsed laser polymerization–electron paramagnetic resonance (SP-PLP-EPR) technique, the chain-length-dependent termination of 5 and 10 wt % sodium methacrylate (NaMAA) in aqueous solution was measured from 5 to 60 °C. The rate coefficients kt(i,i) for termination of two ionized radicals of identical size i were analyzed by the composite model. Three out of the four composite-model parameters behave similarly to nonionized monomers, whereas the fourth parameter, the rate coefficient for termination of two radicals of chain length unity, kt(1,1), exhibits a distinctly different behavior. The temperature dependence of kt(1,1) is significantly below the one of fluidity (inverse solution viscosity). Moreover, absolute kt(1,1) increases with NaMAA concentration, i.e., toward higher viscosity. Both observations indicate that the termination kinetics of ionized radicals largely differs from the Smoluchowski-type behavior.

Growth and electrical properties of Na0.5Bi0.5TiO3–K0.5Bi0.5TiO3 lead-free single crystals by the TSSG method

A series of NBT-KBT lead-free crystals with dimensions of Φ 35×10mm were successfully grown by the TSSG method. The as-grown crystals possess rhombohedral perovskite structure at room temperature. The curves ε(T) for all crystals show two abnormal dielectric peaks. The depolarization temperatures Td derived from the first peak of curves tanδ(T) vary with the KBT content, which are 130, 150, 140, and 115°C respectively, for (100−x)NBT−xKBT (x=5, 8, 12, 15) crystals, being well consistent with the Td obtained from the temperature dependence of kt. A notable thermal hysteresis, ΔT≈35°C, for ferroelectric-antiferroelectric phase transition was also disclosed for 92NBT-8KBT crystal. The investigation of orientation dependence for electrical properties disclosed the dielectric parameters show weak anisotropy. The piezoelectric constants (d33) are 147, 175, 205, 238 pC/N and the values of kt are 38%, 52%, 52%, 54%, respectively for (100−x)NBT−xKBT (x=5, 8, 12, 15) crystals.

Propagation and Chain-Length-Dependent Termination Rate Coefficients Deduced from a Single SP–PLP–EPR Experiment

The laser single pulse (SP)–pulsed laser polymerization (PLP)–electron paramagnetic resonance (EPR) technique allows for deducing propagation (kp) and termination (kt) rate coefficients, including the chain-length dependence of kt, from a single pulsed-laser experiment. The method, which is particularly well suited for slowly terminating radicals, e.g., sterically hindered and ionic radicals, is illustrated for di(n-butyl) itaconate in bulk at temperatures from 30 to 60 °C. The time evolution of the DBI radical concentration is measured with a high time resolution at constant magnetic field. Propagation is associated with a relatively low pre-exponential A(kp), which is responsible for the small kp value of 6.8 L mol–1 s–1 at 30 °C. The chain-length dependence (CLD) of kt, deduced from the same SP–PLP–EPR signal as is kp, turns out to be adequately represented by the composite model. Whereas typical numbers are found for the power-law exponents for short and long radicals and for the crossover chain lengt...

A stress-concentration-formula generating equation for arbitrary shallow surfaces

Analytical understanding of how stress concentrates is invaluable. An equation that generates stress concentration formulas is derived and shown to apply very well to a number of shallow irregularities on surfaces, for the plane stress conditions and to a first-order approximation. Under shallow conditions, for any first-order Hölder-continuous surface function f(x), the derived equation is: kt(x)=1-2Hf′(x), where H is the Hilbert transform and f′(x) is the spatial derivative of f with respect to the independent variable. It is shown that using this generating equation, well-known traditional results can be easily derived. Also, a number of other stress concentration formulas for various cases are generated. Furthermore, a second-order approximation is introduced, which shows the dependence of kt on not only the slope but also on the concavity of the surface. The approach used herein can be extended to finding closed-form solutions to other integral equations possessing similar kernels for applications such as the variation of the stress intensity factor due to an arbitrary crack front profile (work in progress).

Dynamic and Double-Diffusive Instabilities in a Weak Pycnocline. Part I: Observations of Heat Flux and Diffusivity in the Vicinity of Maud Rise, Weddell Sea

Abstract An expedition to study the stability of the weakly stratified water column in the eastern Weddell Sea was undertaken in the austral winter of 2005. A regional CTD survey around Maud Rise delineated water mass boundaries associated with flow around the seamount and identified areas most susceptible to overturning. A downstream region of the seamount Taylor column was found least stable, with a potential density difference across the pycnocline less than 0.018 kg m−3. Intensive water column measurements, including 1300 profiles of temperature, conductivity, and fast-response microconductivity, were made during a series of 13 drift stations to investigate vertical turbulent transports and the evolution of water column stability. The dependence of pycnocline turbulent diffusivity kT on Froude number Fr (turbulence generated by internal wave shear) and density ratio Rρ (turbulence generated by diffusive layering and possibly diapycnal cabbeling) is investigated. The Fr alone cannot explain completely the observed kT variability. Instead, there is also a strong dependence on Rρ. Turbulent diffusivity is an order of magnitude larger in the weakly stratified Taylor cap over Maud Rise (where Rρ approaches one) than in the surrounding water column that is unaffected by flow around Maud Rise. In terms of water column stability, diffusive heat flux across the pycnocline inhibits winter ice growth and densification of the surface layer. The observed Rρ dependence of kT thus provides a strong negative feedback on the winter evolution of the Maud Rise area water column toward overturning instability.

Growth and electrical properties of 0.88Na0.5Bi0.5TiO3–0.12K0.5Bi0.5TiO3 lead-free piezoelectric single crystal

A large-size 0.88Na0.5Bi0.5TiO3–0.12K0.5Bi0.5TiO3 lead-free piezoelectric single crystal with dimensions of ∅30 mm ×10 mm was successfully grown by the top-seeded solution growth method. X-ray powder diffraction measurement showed that the as-grown crystal possesses a rhombohedral perovskite structure at room temperature. The temperature dependence of dielectric constant and loss, polarization hysteresis loop, electric-field-induced strain, and the electromechanical coupling property were investigated for the -oriented as-grown crystal. The remanent polarization Pr, the piezoelectric constant d33, and the electromechanical coupling coefficient kt at room temperature are 13.8 pC/cm2, 205 pC/N, and 51 %, respectively. The maximum strain of 0.23 % was obtained under 30 kV/cm. The temperature dependence of kt demonstrated a relatively stable thermal electromechanical coupling property over the temperature range of 20–150 °C.

Pressure-volume-temperature equation of state of tungsten carbide to 32 GPa and 1673 K

We have obtained pressure-volume-temperature (P-V-T) equation of state for hexagonal tungsten carbide (α-WC) up to 32 GPa and 1673 K using synchrotron x-ray diffraction in a multianvil apparatus at the SPring-8 facility. MgO and Au were used as pressure calibrants. A least-squares fit of the P-V-T-data to a high-temperature Birch–Murnaghan equation of state yielded V0=20.750±0.002 Å3, KT=384±4 GPa, K′=4.65±0.32, temperature derivative of the bulk modulus (∂KT/∂T)P=−0.014±0.002 GPa/K, and thermal expansion α=a0+a1T with a0=0.96(±0.05)×10−5 K−1 and a1=0.48(±0.05)×10−8 K−2. The data showed an anisotropic nature of compressibility, with the a-axis (KTa=341±6 GPa) more compressible than the c-the axis (KTc=506±12 GPa) as well as an anisotropic temperature dependence of KT. The estimated thermal Grüneisen parameters are 1.44–1.64 and the Debye temperature is calculated to be 1220 K, which is different from previous estimates.

Compartmentalization in Atom Transfer Radical Polymerization to High Conversion in Dispersed Systems: Effects of Diffusion-Controlled Reactions

Effects of diffusion control in a compartmentalized atom transfer radical polymerization (ATRP) system have been investigated by modeling and simulations employing modified Smith−Ewart equations in connection with conversion-dependent rate coefficients for the system styrene/polystyrene-Cl/CuCl/4,4′-di(5-nonyl)-2,2′-bipyridine (dNbpy) at 75 °C. The effects of conversion-dependent rate coefficients for deactivation (kdeact) and termination (kt) were investigated in detail for the particle diameter 30 nm, i.e., a strongly compartmentalized system. The decrease in kdeact at high conversion results in an increase in the number of monomer units added per activation−deactivation cycle, thereby causing slight loss of control relative to when kdeact remains constant. The conversion dependence of kt results in a minor increase in livingness at high conversion relative to when kt remains constant. The individual compartmentalized rates of deactivation and termination have been analyzed in detail based on the partic...

Termination kinetics of free-radical polymerization in ionic liquids

Termination rate coefficients, kt, for free-radical polymerization of 15vol.-% methyl methacrylate (MMA) dissolved in 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl)-imide and in 1-butyl-3-methylimidazolium tetrafluoroborate were measured at 10°C via the single pulse − pulsed laser polymerization − electron paramagnetic resonance (SP–PLP–EPR) technique. Whereas absolute kt in ionic liquid solution is by about one order of magnitude below the associated bulk MMA value, the chain-length dependence of kt is very similar in both liquid environments.

Free-radical propagation and termination kinetics of the butyl acrylate dimer studied by pulsed laser polymerization techniques

The propagation and termination rate coefficients for bulk polymerization of the butyl acrylate dimer (BA dimer) are determined by pulsed laser techniques. The rate coefficient for propagation, kp, is deduced for temperatures from 20 to 90°C via the pulsed laser polymerization–size exclusion chromatography (PLP–SEC) method at pulse repetition rates between 1 and 10Hz. The Arrhenius parameters were found to be: EA(kp)=(34.2±1.0)kJmol−1 and A(kp)/Lmol−1s−1=(1.08±0.49)×107Lmol−1s−1. The termination rate coefficient, kt, has been measured via SP-PLP–ESR, single pulse-pulsed laser polymerization in conjunction with time-resolved electron spin resonance detection of radical concentration. The resulting Arrhenius parameters as deduced from the temperature range −15 to +30°C are: EA(〈kt〉)=(22.8±3.7)kJmol−1 and log(A/Lmol−1s−1)=10.6±1. The chain-length dependence of kt was studied at 30°C. For short chains a significant dependence was found which may be represented by an exponent α=0.79 in the power-law expression kt(i)=kt0i−α.

Chain Length Dependent Termination Rate Coefficients of Methyl Methacrylate (MMA) in the Gel Regime: Accessing kti,i Using Reversible Addition-Fragmentation Chain Transfer (RAFT) Polymerization

The termination rate coefficient, kti,i, for propagating chains of near equal length, i, was evaluated using the RAFT−CLD−T method over a wide range of chain lengths and up to a conversion of 70% for MMA polymerizations carried out in the presence of the RAFT agent, CPDB, at 80 °C. We found that the conversion for the gel onset corresponded to the conversion at which polymer chains begin to overlap (i.e., c*), and was found to range from 15 to 30% conversion depending on the Mn. It was further shown that c* also corresponded with the gel onset conversions for vinyl acetate and methyl acrylate. The chain length dependence of kt in the gel regime scaled as αgel(x) = 1.8x + 0.056, suggesting that reptation alone does not play a role in our system. A composite model was then derived to accurately describe kti,i for chain lengths up to 3200 and conversions up to 70%. The kti,i profiles for well-known termination models were tested and most gave unsatisfactory agreement with our experiments. Our model can be readily applied to any monomer provided accurate kti,i(x) data can be determined.

Termination Kinetics of tert‐Butyl Methacrylate and of n‐Butyl Methacrylate Free‐Radical Bulk Homopolymerizations

AbstractSummary: Termination kinetics in tert‐butyl methacrylate (tert‐BMA) and n‐butyl methacrylate (n‐BMA) bulk homopolymerizations has been studied via the single pulse‐pulsed laser polymerization‐near infrared (SP‐PLP‐NIR) method between 40 and 80 °C at pressures from 500 to 2 250 bar. Toward increasing monomer conversion, the chain‐length averaged termination rate coefficient, 〈kt〉, for both monomers exhibits the methacrylate‐specific sequence of an initial plateau region, assigned to control by segmental diffusion, followed by a steep decrease of 〈kt〉 at intermediate conversion, which is assigned to translational diffusion control, and a weaker decrease of 〈kt〉, associated with reaction‐diffusion control, at still higher degrees of monomer conversion. Despite this similarity, the two isomeric monomers clearly differ in absolute size of 〈kt〉 and in the monomer concentration ranges where the transitions between the different types of diffusion control occur. The differences are assigned to effects of chain mobility which is hindered to a larger extent in tert‐BMA than in n‐BMA. As a consequence, the 〈kt〉 behavior of tert‐BMA at 80 °C is close to the one of n‐BMA at 40 °C. Investigations into the chain‐length dependence of kt, in particular into kt(i,i), the rate coefficient for termination of two radicals of identical size, support the evidence on the different types of diffusion control that operate as a function of monomer conversion. In the initial conversion range, the power‐law exponent which characterizes the chain‐length dependence of larger (entangled) radicals, is found for both monomers to be close to the theoretical value of α = 0.16.Dependence of log(〈kt〉/kp) on monomer conversion, X, for n‐BMA and tert‐BMA bulk homopolymerizations at 2 000 bar and 70 °C. Circles and triangles represent independent data sets obtained from separate experiments.magnified imageDependence of log(〈kt〉/kp) on monomer conversion, X, for n‐BMA and tert‐BMA bulk homopolymerizations at 2 000 bar and 70 °C. Circles and triangles represent independent data sets obtained from separate experiments.

A Random-sequential Mechanism for Nitrite Binding and Active Site Reduction in Copper-containing Nitrite Reductase

The homotrimeric copper-containing nitrite reductase (NiR) contains one type-1 and one type-2 copper center per monomer. Electrons enter through the type-1 site and are shuttled to the type-2 site where nitrite is reduced to nitric oxide. To investigate the catalytic mechanism of NiR the effects of pH and nitrite on the turnover rate in the presence of three different electron donors at saturating concentrations were measured. The activity of NiR was also measured electrochemically by exploiting direct electron transfer to the enzyme immobilized on a graphite rotating disk electrode. In all cases, the steady-state kinetics fitted excellently to a random-sequential mechanism in which electron transfer from the type-1 to the type-2 site is rate-limiting. At low [NO(-)(2)] reduction of the type-2 site precedes nitrite binding, at high [NO(-)(2)] the reverse occurs. Below pH 6.5, the catalytic activity diminished at higher nitrite concentrations, in agreement with electron transfer being slower to the nitrite-bound type-2 site than to the water-bound type-2 site. Above pH 6.5, substrate activation is observed, in agreement with electron transfer to the nitrite-bound type-2 site being faster than electron transfer to the hydroxyl-bound type-2 site. To study the effect of slower electron transfer between the type-1 and type-2 site, NiR M150T was used. It has a type-1 site with a 125-mV higher midpoint potential and a 0.3-eV higher reorganization energy leading to an approximately 50-fold slower intramolecular electron transfer to the type-2 site. The results confirm that NiR employs a random-sequential mechanism.

Chain Length Dependent Termination in Butyl Acrylate Free-Radical Polymerization Studied via Stationary and Pulsed Laser Initiated RAFT Polymerization

The chain-length dependence of the termination rate coefficient, kt, in butyl acrylate free-radical polymerization has been determined by two independent methods, RAFT-SP-PLP and RAFT-CLD-T, both employing control of radical chain length by reversible addition fragmentation chain transfer (RAFT) polymerization. Within RAFT-SP-PLP, the polymerization induced by a laser single pulse is monitored via near-IR spectroscopy with a time resolution of microseconds. In RAFT-CLD-T, isothermal reaction rate measurements are carried out via DSC under stationary polymerization conditions. The resulting kt data refer to the situation of living/controlled radical polymerization, where both radical chain length and monomer conversion increase during the course of the reaction. The RAFT-SP-PLP measurements were carried out at 60 °C and two pressures, 5 and 1000 bar. The RAFT-CLD-T experiments were run at ambient pressure and at two temperatures, 60 and 80 °C, respectively. In absolute value, the termination rate coefficients for identical pressure and temperature deduced from the two methods differ by less than a factor of 2. For the dependence of kt on chain length, i, almost identical information is provided by the two techniques. The chain-length dependence of kt may be described by the power-law expression kt(i) = kt(1,1)i-α with, however, α being different for short-chain and long-chain radicals. RAFT-SP-PLP yields α1 = 1.25 for the short-chain regime from 1 < i < 30, and α2 = 0.22 for chain lengths above i = 50. RAFT-CLD-T results in α1 = 1.04 and α2 = 0.20 in identical chain length regimes. kt(1,1) values are found to be close to 1 × 109 L mol-1 s-1.

Termination Kinetics of Dibutyl Itaconate Free‐Radical Polymerization Studied via the SP–PLP–ESR Technique

AbstractSummary: The termination kinetics of dibutyl itaconate (DBI) bulk polymerization was studied via SP–PLP–ESR single pulse–pulsed laser polymerization with time‐resolved detection of free‐radical concentration by electron‐spin resonance, at temperatures between 0 and 60 °C. As is characteristic of PLP experiments, termination rate coefficients, kt(i,i), are measured for radicals of (almost) identical chain length (CL) i. CL‐averaged 〈kt〉, for chain lengths up to 200 monomer units, and also k referring to termination of very small‐size radicals are directly deduced from measured DBI radical concentration vs time traces. At 45 °C, 〈kt〉 is (3.4 ± 0.6) · 105 L · mol−1 · s−1 and k is (7.2 ± 1.0) · 105 L · mol−1 · s−1. Both rate coefficients are independent of monomer conversion up to the highest experimental conversion of 18%. The associated activation energies are EA(〈kt〉) = 23.0 ± 3.2 kJ · mol−1 and EA(k) = 27.6 ± 2.8 kJ · mol−1, respectively. “Model‐dependent” and “model‐free” analyses of radical concentration vs time profiles indicate a pronounced CL dependence of kt(i,i) for DBI radicals of moderate size, 5 &lt; i &lt; 50. The lowering of kt(i,i) with CL corresponds to an exponent α close to 0.5 in a power‐law expression kt(i,i) = k · i−a. At higher chain lengths, the variation of kt(i,i) with CL becomes weaker and may be represented by an α value of 0.16 or even below. These results are consistent with models according to which α varies to a larger extent at low CL and to a smaller extent at high CL with the crossover region between the two ranges being located somewhere around i = 100.Conversion‐dependence of 〈kt〉 and k from laser‐induced photopolymerization of DBI.magnified imageConversion‐dependence of 〈kt〉 and k from laser‐induced photopolymerization of DBI.

Linear Chain Termination in the Kinetics of Postpolymerization of Dimethacrylates

The kinetics of postpolymerization (after turning off UV irradiation) of various dimethacrylates differing in their nature and molecular weight was studied over a wide range of temperatures. For each temperature, a series of kinetic curves, varying in the initial conversion during a dark period, was obtained. The proposed kinetic model is based on the following assumptions. The process in the interphase layer at the liquid monomer-solid polymer boundary has the most significant contribution to the kinetics of postpolymerization. Chain termination in the interphase layer occurs by the unimolecular reaction, is controlled by the chain growth rate, and presents the act of “self-burying” of an active radical in a conformational trap. A wide spectrum of characteristic times is inherent in unimolecular chain termination, and the relaxation function is described by the Kohlraush' stretched exponential law. The rate law obtained agrees well with experimental data. This fact made it possible to estimate the rate constants (kt) and the activation energies of chain termination and to establish the scale dependence of kt on the molar concentration [M0] of the monomer in a block. It is suggested that both the stretched exponential law and the scale kt-[M0] dependence are due to a wide spectrum of characteristic times of relaxation exhibiting the properties of a fractal set.