Stretched Exponential Research Articles

Probing Gelation and Rheological Behavior of a Self-Assembled Molecular Gel.

Molecular gels have been investigated over the last few decades; however, mechanical behavior of these self-assembled gels is not well understood, particularly how these materials fail at large strain. Here, we report the gelation and rheological behavior of a molecular gel formed by self-assembly of a low molecular weight gelator (LMWG), di-Fmoc-l-lysine, in 1-propanol/water mixture. Gels were prepared by solvent-triggered technique, and gelation was tracked using Fourier transform infrared (FTIR) spectroscopy and shear rheology. FTIR spectroscopy captures the formation of hydrogen bonding between the gelator molecules, and the change in IR spectra during the gelation process correlates with the gelation kinetics results captured by rheology. Self-assembly of gelator molecules leads to a fiber-like structure, and these long fibers topologically interact to form a gel-like material. Stretched-exponential function can capture the stress-relaxation data. Stress-relaxation time for these gels have been found to be long owing to long fiber dimensions, and the stretching exponent value of 1/3 indicates polydispersity in fiber dimensions. Cavitation rheology captures fracture-like behavior of these gels, and critical energy release rate has been estimated to be of the order 0.1 J/m2. Our results provide new understanding of the rheological behavior of molecular gels and their structural origin.

Origin of stretched-exponential photoluminescence relaxation in size-separated silicon nanocrystals

A detailed understanding of the photoluminescence (PL) from silicon nanocrystals (SiNCs) is convoluted by the complexity of the decay mechanism, including a stretched-exponential relaxation and the presence of both nanosecond and microsecond time scales. In this publication, we analyze the microsecond PL decay of size-resolved SiNC fractions in both full-spectrum (FS) and spectrally resolved (SR) configurations, where the stretching exponent and lifetime are used to deduce a probability distribution function (PDF) of decay rates. For the PL decay measured at peak emission, we find a systematic shift and narrowing of the PDF in comparison to the FS measurements. In a similar fashion, we resolve the PL lifetime of the ‘blue’, ‘peak’, and ‘red’ regions of the spectrum and map PL decays of different photon energy onto their corresponding location in the PDF. A general trend is observed where higher and lower photon energies are correlated with shorter and longer lifetimes, respectively, which we relate to the PL line width and electron-phonon coupling.

Ageing of Ge24.9Sb11.6S63.5 thin films under various conditions

Amorphous Ge24.9Sb11.6S63.5 thin films prepared by thermal evaporation were aged for more than 1 year under various conditions: in the dark in a desiccator, in a dark and humid atmosphere and under laboratory conditions under daylight. The bleaching of thin films (the blue shift of the optical band gap) was observed with the magnitude depending on the storage conditions. The overall kinetics of ageing (bleaching) follows well the stretch exponential function with the formal rate depending on the storage conditions and the stretch exponent slightly varying in the region of 0.54–0.6. The photo-sensitivity of the aged thin films was significantly affected by storage conditions and was found to decrease with the increasing time of ageing. The changes in thin films during the ageing were monitored with respect to the chemical composition, the structure, the surface wettability, the surface topology and information about the depth and optical parameters of the formed over-layer and were finally correlated with the storage conditions.

Lithium ion conductivity and dielectric relaxation in dendritic nanostructured LiTaO3 glass–nanocrystal composites

Lithium tantalate in single-crystalline and coarse-grained configurations is a poor ionic conductor and does not qualify as a solid electrolyte for lithium-based batteries. In this work, ionic conductivity was sought to be enhanced by the use of nanocrystals of LiTaO3 embedded in a borate-based glass matrix. Glasses of composition 3Li2O–4B2O3–Ta2O5 were formed by melt-quenching. The crystallization process was described by using isothermal crystallization kinetics, invoking the Johnson–Mehl–Avrami–Kolmogorov equation, which indicated a three-dimensional growth with an Avrami exponent of 3.5 and an effective activation energy for crystallization of 735 ± 65 kJ mol–1. Heat treatment of the as-quenched glasses was performed between 530 and 560 °C, and the evolution of LiTaO3 phase was studied by X-ray powder diffraction, scanning electron microscopy, and transmission electron microscopy. The heat treatment yielded coalesced LiTaO3 nanocrystals of 18–32 nm size, forming dendritic structures in the glass matrix. Impedance analyses of the as-quenched and heat-treated glasses showed a dramatic improvement in dc conductivity (σdc), with a maximum around 3 × 10–3 S m−1 at 200 °C (σdcT = 1.5 S m−1 K) and activation energy of 0.54 eV for 530 °C/3 h heat-treated glasses. The values of σdc of the as-quenched glasses and of the 530 °C/3 h and 540 °C/3 h heat-treated glasses were about seven orders of magnitude higher than those of the single crystalline LiTaO3. Furthermore, the effect of heat treatment on lithium ion dynamics in the 40–200 °C temperature range was investigated by modulus formalism invoking the stretched exponential Kohlrausch–Williams–Watts function. The 7Li magic angle spinning NMR was used to investigate lithium self-diffusion in the nanostructured glass nanocrystal composites as a function of temperature between −10 °C and 60 °C.

Application of the stretched exponential equation to sorption of mine gases and sorption induced swelling of bituminous coal

An evaluation of the ability of the stretched exponential (SE) equation to describe the sorption kinetics and the rate of expansion/contraction of hard coal in coal–carbon dioxide, coal–methane and coal–CO2/CH4 mixture systems was performed. In order to address this issue the adequate sorption experiments were carried out at high pressure by means of the volumetric method on a cubicoid solid samples. Simultaneously the kinetics of coal sorption-induced swelling were monitored. For two coals under investigation a linear and a non-linear relation between coal expansion and the amount of sorbed gas was observed. The SE equation shows a very good agreement with the sorption experimental data obtained for both coals, it is also able to accurately describe the sorption-induced-swelling of lower rank coal and the CH4-induced swelling of higher rank coal however it does not give a reasonable fit to swelling kinetics induced by sorption of CO2 and CO2\\CH4 mixtures on higher rank coal. The highest characteristic rate parameter k was found for CO2 sorption on lower rank coal, the lowest was calculated for CH4 sorption on the same coal and the values of k for higher rank coal are in between. The values of stretching parameter b for higher rank coal are greater than for lower rank coal, i.e. between 0.55 and 0.75 and <0.5, indicating a relatively narrow and a relatively broad distribution of characteristic relaxation times of sorption processes, respectively. Generally, the values of parameters k and b calculated for CO2, CH4 and CO2\\CH4 mixture sorption-induced swelling are lower for lower rank coal. The values of parameter b for swelling phenomena are not compatible with these of sorption processes.

The Stretched Exponential Behavior and Its Underlying Dynamics. The Phenomenological Approach

AbstractWe show that the anomalous diffusion equations with a fractional spatial derivative in the Caputo or Riesz sense are strictly related to the special convolution properties of the Lévy stable distributions which stem from the evolution properties of stretched or compressed exponential function. The formal solutions of these fractional differential equations are found by using the evolution operator method where the evolution is conceived as integral transform whose kernel is the Green function. Exact and explicit examples of the solutions are reported and studied for various fractional order of derivatives and for different initial conditions.

Experimental decomposition of the positive bias temperature stress-induced instability in self-aligned coplanar InGaZnO thin-film transistors and its modeling based on the multiple stretched-exponential functions

Decomposition of the positive gate-bias temperature stress (PBTS)-induced instability into contributions of distinct mechanisms is experimentally demonstrated at several temperatures in top-gate self-aligned coplanar amorphous InGaZnO thin-film transistors by combining the stress-time-divided measurements and the subgap density-of-states (DOS) extraction. It is found that the PBTS-induced threshold voltage shift (ΔVT) consists of three mechanisms: (1) increase of DOS due to excess oxygen in the active region; (2) shallow; and (3) deep charge trapping in the gate insulator components. Corresponding activation energy is 0.75, 0.4, and 0.9 eV, respectively. The increase of DOS is physically identified as the electron-capture by peroxide. Proposed decomposition is validated by reproducing the PBTS time-evolution of I–V characteristics through the technology computer-aided design simulation into which the extracted DOS and charge trapping are incorporated. It is also found that the quantitative decomposition of PBTS-induce ΔVT accompanied with the multiple stretched-exponential models enables an effective assessment of the complex degradation nature of multiple PBTS physical processes occurring simultaneously. Our results can be easily applied universally to any device with any stress conditions, along with guidelines for process optimization efforts toward ultimate PBTS stability.

A Model for Approximately Stretched-Exponential Relaxation with Continuously Varying Stretching Exponents

Relaxation in glasses is often approximated by a stretched-exponential form: $f(t) = A \exp [-(t/\tau)^{\beta}]$. Here, we show that the relaxation in a model of sheared non-Brownian suspensions developed by Cort\'e et al. [Nature Phys. 4, 420 (2008)] can be well approximated by a stretched exponential with an exponent $\beta$ that depends on the strain amplitude: $0.25 < \beta < 1$. In a one-dimensional version of the model, we show how the relaxation originates from density fluctuations in the initial particle configurations. Our analysis is in good agreement with numerical simulations and reveals a functional form for the relaxation that is distinct from, but well approximated by, a stretched-exponential function.

The relation between stretched-exponential relaxation and the vibrational density of states in glassy disordered systems

Amorphous solids or glasses are known to exhibit stretched-exponential decay over broad time intervals in several of their macroscopic observables: intermediate scattering function, dielectric relaxation modulus, time-dependent elastic modulus, etc. This behaviour is prominent especially near the glass transition. In this Letter we show, on the example of dielectric relaxation, that stretched-exponential relaxation is intimately related to the peculiar lattice dynamics of glasses. By reformulating the Lorentz model of dielectric matter in a more general form, we express the dielectric response as a function of the vibrational density of states (DOS) for a random assembly of spherical particles interacting harmonically with their nearest-neighbours. Surprisingly we find that near the glass transition for this system (which coincides with the Maxwell rigidity transition in this model), the dielectric relaxation is perfectly consistent with stretched-exponential behaviour with Kohlrausch exponents 0.56<β<0.65, which is the range where exponents are measured in most experimental systems. Crucially, the root cause of stretched-exponential relaxation can be traced back to soft modes (boson-peak) in the DOS.

Dependence of the photobleaching on laser radiation wavelength in bismuth-doped germanosilicate fibers

Photobleaching of IR luminescence in bismuth-doped fibers was studied using laser radiation with different wavelengths: 407, 532, 639, 975, 1460nm. For each wavelength the bleaching time was determined by fitting a stretched-exponential function to the data measured. It was revealed that the bleaching time increases with the decrease of the photon energy. The measured dependence of the bleaching rate on the launched power at 532nm is also presented. The effect of temperature on the bleaching rate was also investigated. The role of the oxygen-deficient centers in the bleaching process is discussed.

Dielecric relaxation and conduction mechanism of dysprosium doped barium bismuth titanate Aurivillius ceramics

In this report, we present the structure, electrical properties and optical study of dysprosium doped bismuth layer-structured BaBi4Ti4O15 (BBT) with a general formula Ba(Bi1−xDyx)4Ti4O15 (BDBT), prepared by solid state reaction route. The X-ray diffraction study shows a single phase orthorhombic structure with a space group A21 am in all compositions. The relaxation and conduction mechanism was studied by frequency and temperature dependent impedance spectroscopy. Impedance studies show a non-Debye type relaxation, and relaxation frequency shift to the higher side with an increase in temperature indicating conduction in material and favouring the long range motion of mobile charge carriers. The Nyquist plot shows overlapping semicircles, for grain and grain boundary of Dy doped BBT ceramics. The Kohlrausch–Williams–Watts function is used to explain the modulus analysis and confirms the coexistence of grain and grain boundary effects. The frequency dependent AC conductivity at different temperatures indicates that the conduction process is thermally activated and the spectra follow the universal power law. The variation of DC conductivity confirms that the BDBT ceramics exhibit negative temperature coefficient of resistance behaviour in high temperature. Activation energies are obtained from the Arrhenius fitting of impedance and conductivity plots.

Production Patterns of Eagle Ford Shale Gas: Decline Curve Analysis Using 1084 Wells

This paper analyzes and quantifies characteristic production behavior using historical data from 1084 shale gas wells in the Eagle Ford shale play from 2010 to 2014. Decline curve analysis, using Hyperbolic and Stretched Exponential models, are used to derive average decline rates and other characteristic parameters for shale gas wells. Both Hyperbolic and Stretched Exponential models fit well to aggregated and individual well production data. The hyperbolic model is found to perform slightly better than the Stretched Exponential model in this study. In the Eagle Ford shale play, about 77% of wells reach the peak production of 1644–4932 mil cubic feet per day; the production decline rate of the first year is around 70%, and over the first two years it is around 80%; shale gas wells were estimated to yield estimated ultimate recoverable total resources of 1.41–2.03 billion cubic feet (20 years as life span), which is in line with other studies.

Stretched-Exponential Trends in a-Igzo Tfts

A bias temperature stress (BTS) investigation of amorphous InGaZnO thin-film transistors (a-IGZO TFTs) is undertaken using the stretched-exponential (SE) relation to model the threshold voltage shift (ΔVTH) as a function of time, temperature, and applied gate voltage. Explicitly, the SE equation employed herein is described as ΔVTH (t, T, VG) = (VG-VTH0) (1-exp(-(t/τ)β)), where VTH0 is the initial unstressed TFT threshold voltage, τ is a trapping time, τ = τ0exp(Eτ/kBT), where τ0 is a trapping time prefactor, Eτ is an energy barrier, and β is an energy barrier parameter [1, 2]. Simulation indicates that a larger τ and/or β results in less ΔVTH, i.e., a more stable a-IGZO TFT. An alternate form of the SE voltage prefactor (VG-VTH0) involves the use of V0, which is identified as a saturation voltage [3, 4]. Either prefactor formulation can be used to accurately fit ΔVTH trends for stress times less than ~3 hours. However, as shown in Fig. 1, fitting of a sample data set restricted to stress times less than 3 hours leads to dramatically different saturated threshold voltage shifts, even though both prefactors precisely fit the initial 2 hours of data (see insert of Fig. 1). Physically, this is associated with the fact that the (VG-VTH0) prefactor corresponds to the applied overvoltage which explicitly depends on VG, while V0 is independent of VG. Simulation is useful for revealing various kinds of ΔVTH (t, T, VG) trends. For example, Fig. 2(a) shows that the relevant energy barrier, Eτ, associated with ΔVTH (10-5 s, 300 K, VG = 30 V) is very narrow for β = 1 and broadens dramatically as β → 0. The Eτ-β relationship is better represented by taking the derivative of ΔVTH with respect to Eτ, as shown in Fig. 2(b), where the magnitude of the peak increases and the distribution of Eτ narrows as β increases. As a larger β is indicative of a more stable TFT, it is reasonable to theorize that a narrow Eτ distribution is meaningful in predicting stable long-term TFT reliability. [1] M. Mativenga, et al., “High stable amorphous indium-gallium-zinc-oxide thin-film transistor using an etch-stopper and a via-hole structure,” Journal of Information Display, Vol. 12, No. 1, 47-50 (2011)[2] T. Y. Hsieh, et al., “Investigating degradation behavior of InGaZnO thin-film transistors induced by charge-trapping effect under DC and AC gate bias stress,” Thin Solid Films, 528, 53-56 (2013)[3] J. M. Lee, et al., “Bias-stress-induced stretched-exponential time dependence of threshold voltage shift in InGaZnO thin film transistors,” Applied Physics Letters, 93, 093304 (2008)[4] S. Park, et al., “Threshold voltage shift prediction for gate bias stress on amorphous InGaZnO thin film transistors,” Microelectronics Reliability, 52, 2215-2219 (2012) Figure 1

Reply to “Comment on ‘Revisiting the 1894 Omori Aftershock Dataset with the Stretched Exponential Function’ by A. Mignan” by S. Hainzl and A. Christophersen

ABSTRACT Hainzl and Christophersen (2016) recently suggested that the approach used by Mignan (2016) to visualize the Omori law in a log–log plot using the complementary cumulative density function (CCDF) was misleading. They showed that one should use a temporal upper bound t max =max( t obs ), with t obs the occurrence time of a set of observed aftershocks, instead of t max →∞, as used by Mignan. They found that both the Omori law and stretched exponential function (SEF) are undistinguishable on a corrected CCDF log–log plot but that the Omori law is preferred based on maximum‐likelihood estimations (MLEs). I first clarify the rationale for using the CCDF log–log plot for comparison of the SEF with a power law but verify that the comparison indeed becomes misleading when the data sequence is incomplete. However, I then show that MLEs obtained for the Omori law are ambiguous, because this function may win versus the SEF, even if the true relaxation process follows an SEF (subject to early data incompleteness). This debate highlights that statistics alone may not be enough to choose between the stretched exponential and the Omori law and that physical considerations should be added to the discussion.

Comment on “Revisiting the 1894 Omori Aftershock Dataset with the Stretched Exponential Function” by A. Mignan

ABSTRACT Mignan (2016) claimed that a stretched exponential function describes Omori’s original aftershock data of the 1891 Great Nobi earthquake better than the well‐known Omori law, which is a power law. Besides his preference for the stretched exponential function based on general physical reasoning, he proposed that the Omori law does not hold when the proper visualization method is used; that is, the complementary cumulative density function (CCDF) in a log–log plot. However, his proposed plot is misleading, because it compares data of a finite observation interval with functions integrated over infinite periods. Using the same data as Mignan (2016), we find that an appropriate comparison leads to visually indistinguishable fits for this dataset. The Omori law is preferred based on maximum‐likelihood values. Moreover, the extrapolation shows that it also fits the long‐term data until the centennial anniversary of the Nobi event significantly better than the stretched exponential.

Static and dynamic properties of supercooled water in small nanotubes.

The static and dynamic properties of water in small silicon-carbide and carbon nanotubes have been studied over the temperature range 100 K-298 K, using extensive molecular dynamics simulations. The computed properties include the radial distribution function, the cage correlation function, the space-time autocorrelation function, the velocity autocorrelation function, and the self-diffusivity. They all indicate that, under the conditions that we study, water does not freeze in small nanotubes; the Stokes-Einstein relation breaks down, and the self-diffusivity exhibits a transition around 230 K, very close to 228 K, the temperature at which a fragile-to-strong dynamic crossover is supposed to happen. The cage correlation function C(t) decays according to a stretched-exponential function, C(t) ∼ exp[ - (t/τ)(β)], where τ is a relaxation time and β is a topological exponent.

Monoexponential, biexponential and stretched-exponential models based diffusion weighted imaging: a comparative study in the differential diagnosis of benign and malignant breast lesions

Objective To investgate the value of various parameters obtained from monoexponential, biexponential, and stretched exponential diffusion-weighted imaging models in the differential diagnosis of breast lesions. Methods A retrospective study performed in 54 patients with pathologically confirmed malignant tumors(n=30), benign lesions(n=34) and normal fibroglandular tissues (n=30). All patients underwent T1WI, T2WI, dynamic enhancement and diffusion weighted MRI with multi-b values at a 3.0 T magnetic resonance imaging unit. All parameters were measured at a workstation. ADC was calculated by using monoexponential analysis(b=0, 800 s/mm2). Slow apparent diffusion coefficient (ADC- slow), fast apparent diffusion coefficient (ADC- fast), and perfusion fraction (f) were calculated using the biexponential model. Distributed diffusion coefficient (DDC) and water molecular diffusion heterogeneity index (α) were obtained from diffusion-weighted images using the stretched exponential model. All parameters were statistically compared among normal fibroglandular tissues, benign lesions and malignant tumors using Kruskal- Wallis rank sum test. Mann- Whitney U test were used for further comparisons between specific group pairs. ADC values were compared with ADC-slow and DDC for different groups by paired Wilcoxon test. Correlations between ADC-value, ADC-slow and DDC were assessed by using Spearman rank correlation coefficient. Receiver operating characteristic curve was used to analyze and compare the ability of these parameters in differentiation of benign and malignant breast lesions. Results The ADC, ADC-slow, f, DDC and α values were significantly different among the normal fibroglandular tissues, benign tumors and malignant tumors (P<0.05). Further differential comparisons of the four parameters between each pair showed that the ADC, ADC-slow, f, DDC and α values were significantly lower in malignant tumors than both in normal fibroglandular tissues and benign tumors(P<0.016 7). The ADC values of normal fibroglandular tissues, benign tumors and malignant tumors were significantly higher than ADC- slow and DDC (P<0.05). The f and DDC had higher area under the receiver operating characteristic curve (0.688 and 0.657 respectively). The optimal cutoff values for ADC, ADC- slow, f, DDC and α were 1.235 × 10-3mm2/s, 0.428×10-3mm2/s, 57.8%, 1.175×10-3mm2/s and 0.721, respectively. The α value showed higher specificity (65.5% ) and the f value had higher sensitivity (82.9%). Conclusion The parameters derived from biexponential and stretched exponential DWI could be helpful for differentiation between benign and malignant breast tumors. Key words: Breast neoplasms; Magnetic resonance imaging; Comparative study

Dynamics of Users Activity on Web-Blogs

Activity of users on Internet discussion forums is analyzed. The rank of users is shown to be approximated better by stretched-exponential function than by Zipfs law. Cumulative distribution function is found as an excellent tool in analysis of the dynamics of the collective social phenomena. We are able to approximate the number of blog comments with time by simple functions that resemble FermiDirac distribution function: probability of posting a comment is given by P(t) = P0(t)/(1+P0(t), where P0(t) = exp(a·log(t/t0)). It is argued that dynamics of blog entries by a specific user ought to be related to personality of each user.

NMR Evidence for Inhomogeneous Nematic Fluctuations in BaFe_{2}(As_{1-x}P_{x})_{2}.

We present evidence for nuclear spin-lattice relaxation driven by glassy nematic fluctuations in isovalent P-doped BaFe_{2}As_{2} single crystals. Both the ^{75}As and ^{31}P sites exhibit a stretched-exponential relaxation similar to the electron-doped systems. By comparing the hyperfine fields and the relaxation rates at these sites we find that the As relaxation cannot be explained solely in terms of magnetic spin fluctuations. We demonstrate that nematic fluctuations couple to the As nuclear quadrupolar moment and can explain the excess relaxation. These results suggest that glassy nematic dynamics are a common phenomenon in the iron-based superconductors.

Revisiting the 1894 Omori Aftershock Dataset with the Stretched Exponential Function

In 1894, Fusakichi Omori published his landmark article on aftershocks in which he described the temporal decay rate of aftershocks as a power law. In this article, I show that the expression proposed by Omori does not hold when the proper visualization method is used (i.e., complementary cumulative density function in log–log plot) and that the choice of a stretched exponential function should have been preferred on physical grounds. I reproduce the emblematic aftershock sequence of the 1891 Great Nobi earthquake used by Omori and show that a stretched exponential expression describes the data better than a power law.