Watts Function Research Articles

Physical aging of a glassy polymer in cultural heritage conservation

AbstractArtworks, particularly easel paintings, are multi‐component materials intended to last indefinitely. The protective coatings applied to these artworks often consist of amorphous glass‐like polymers, which undergo slow physical aging and structural recovery below their glass transition temperature. This process alters key physical properties such as mechanical strength, optical clarity, and thermal stability over extended periods. This study investigates the time‐dependent evolution of these properties in Laropal A81, a widely used synthetic resin for cultural heritage conservation, particularly as a replacement for ancient varnishes. The investigation involves characterization of enthalpy recovery via differential scanning calorimetry, refractive index evolution via refractometry, and creep compliance evolution via rheological measurements. The aging behavior of Laropal A81 is further analyzed using the Kohlrausch–Williams–Watts function and the Tool–Narayanaswamy–Moynihan model, enabling the quantification of critical dynamic parameters such as activation energy, nonlinearity, partition coefficients, and non‐exponentiality. The gained insights into the long‐term behavior of this coating's properties can be valuable for improving preservation and restoration strategies for cultural heritage.

PREFACE — SPECIAL ISSUE ON FRACTALS AND LOCAL FRACTIONAL CALCULUS: RECENT ADVANCES AND FUTURE CHALLENGES

Fractal geometry plays an important role in the description of the characteristics of nature. Local fractional calculus, a new branch of mathematics, is used to handle the non-differentiable problems in mathematical physics and engineering sciences. The local fractional inequalities, local fractional ODEs and local fractional PDEs via local fractional calculus are studied. Fractional calculus is also considered to express the fractal behaviors of the functions, which have fractal dimensions. The interesting problems from fractional calculus and fractals are reported. With the scaling law, the scaling-law vector calculus via scaling-law calculus is suggested in detail. Some special functions related to the classical, fractional, and power-law calculus are also presented to express the Kohlrausch–Williams–Watts function, Mittag-Leffler function and Weierstrass–Mandelbrot function. They have a relation to the ODEs, PDEs, fractional ODEs and fractional PDEs in real-world problems. Theory of the scaling-law series via Kohlrausch–Williams–Watts function is suggested to handle real-world problems. The hypothesis for the tempered Xi function is proposed as the Fractals Challenge, which is a new challenge in the field of mathematics. The typical applications of fractal geometry are proposed in real-world problems.

ANOMALOUS DIFFUSION MODELS AND MANDELBROT SCALING-LAW SOLUTIONS

In this paper, the anomalous diffusion models are studied in the framework of the scaling-law calculus with the Mandelbrot scaling law. A analytical technology analogous to the Fourier transform is proposed to deal with the one-dimensional scaling-law diffusion equation. The scaling-law series formula via Kohlrausch–Williams–Watts function is efficient and accurate for finding exact solutions for the scaling-law PDEs arising in the Mandelbrot scaling-law phenomena.

CALCULUS OPERATORS AND SPECIAL FUNCTIONS ASSOCIATED WITH KOHLRAUSCH–WILLIAMS–WATTS AND MITTAG-LEFFLER FUNCTIONS

In this paper, many important formulas of the subtrigonometric, subhyperbolic, pretrigonometric, prehyperbolic, supertrigonometric, and superhyperbolic functions sin Wiman class are developed for the first time. The subsine, subcosine, subhyperbolic sine, and subhyperbolic cosine associated with Kohlrausch–Williams–Watts (KWW) function and their scaling-law ODEs are proposed. The supersine, supercosine, superhyperbolic sine, and superhyperbolic cosine functions associated with Mittag-Leffler (ML) function and their fractional ODEs are obtained. The conjectures for the supercosine functions containing ML function are presented in detail.

Exploring the comparison of optical, dielectric and photocatalytic performance of Yb3+ and Gd3+ half-doped DyCrO3 nanostructures

The auto combustion sol-gel method was utilized to prepare Dy0.5Gd0.5CrO3 (DGCO) and Dy0.5Yb0.5CrO3 (DYCO) nanomaterials. The single-phase formation of materials was confirmed with the X-ray diffraction technique. Utilizing Kubelka-Munk function, the optical bandgap energy values were found to be 3.39 eV (DGCO) and 2.61 eV (DYCO). The dielectric properties were analyzed in frequency range from 10 kHz to 2 MHz and a temperature range from 25 °C to 225 °C. The conduction mechanism followed the universal Jonscher's power law, which revealed that conduction process was primarily governed by the small polaron hopping. The CBH, NSPT, and QMT models were applicable for the a.c. conduction in the DGCO nanomaterial whereas the CBH and NSPT models were more appropriate for the DYCO nanomaterial. The Kohlrausch, Williams and Watts function was used to correlate the imaginary electrical modulus spectrum, which confirmed that the relaxation process in the materials was non Debye-type. An impedance spectroscopic measurement revealed the influence of grain and grain boundaries and it followed negative temperature coefficient resistance behaviour at higher temperatures. Both DGCO and DYCO nanomaterials were also investigated for their photocatalytic degradation of Rhodamine B (RhB) and Methylene Blue (MB) dyes. The degradation, under direct solar irradiation, was 84 % (RhB) and 63 % (MB) for DYCO nanomaterial and 54 % (RhB) and 44 % (MB) for DGCO nanomaterial, indicating their potential in water treatment.

Effect of thermal modification on the stress relaxation behavior and microstructure of the cell wall

The stress relaxation behavior and cell wall microstructure of sugi were evaluated after thermal modification. Stress relaxation is observed and has a broad relaxation spectrum, implying various relaxation mechanisms. The relaxation was analyzed using a stretched exponential function, namely, the Kohlrausch–Williams–Watts (KWW) function, which contains two parameters. Moreover, the structure of the amorphous phase in the cell wall was examined by small-angle X-ray scattering (SAXS) analysis using the mass fractal dimension. The variation in the relaxation spectrum reduced, and the specific relaxation time increased by thermal modification at 220 °C. The mass fractal dimension in SAXS increased owing to modification, indicating that the structure of the cell wall includes some defects between cellulose microfibrils. The mass fractal dimension was related to the relaxation parameter of the KWW function. Considering the change in crystallinity, the amorphous phase in the cell wall decomposed and condensed by thermal decomposition, which caused a longer relaxation time. Thus, the KWW function may be used to evaluate the stress relaxation behavior of wood, and the mass fractal dimension in SAXS can indicate the amorphous structure in the cell wall.

Low-temperature dielectric relaxation mechanism and correlated barrier hopping transport in neodymium perovskite chromite

Herein, neodymium chromite (NdCrO3) has been synthesized by the sol–gel auto-combustion method. The Rietveld analysis of the X-ray diffraction pattern has shown that NdCrO3 crystalizes in the perovskite structure with orthorhombic unit cell and Pbnm space group symmetry. The electrical transport properties have been investigated by complex impedance spectroscopy over the experimental frequency and temperature ranging from 40 to 1 × 106 Hz and 158–283 K, respectively. The equivalent circuit modeling of Nyquist plots revealed the contribution of grains and grain boundaries to the charge conduction and the relaxation mechanism of NdCrO3, respectively. The AC conductivity analysis divulged the correlated barrier hopping transport mechanism in the NdCrO3. The imaginary electrical modulus spectra fitted by the Kohlrausch, Williams, and Watts function presented the non-Debye type nature of the relaxation mechanism in the material. The dielectric permittivity has exhibited the Maxwell–Wagner type polarization of the material, and the loss factor has exhibited an increasing trend with temperature.

Investigation on electrical transport and dielectric relaxation mechanism in TbCrO3 perovskite orthochromite

Herein, the TbCrO3 (TCO) chromite was synthesized by the sol-gel auto-combustion route and sintered at 900 °C. The Rietveld analysis of the room temperature powder X-ray diffraction profile confirmed the single-phase formation of the material crystallized in the orthorhombic phase of the perovskite structure with Pbnm space group symmetry. The surface morphology, the elemental composition, and the chemical states of the compound were investigated by field emission scanning electron microscopy, energy dispersive X-ray analysis, and X-ray photoelectron spectroscopy, respectively. The electronic density of states, band structure, and electron density were analyzed by Density Functional Theory. The electrical transport and dielectric relaxation mechanism were investigated using complex impedance spectroscopy over the experimental frequency and temperature ranging from 40-1.2x106Hz and 138−273 K, respectively. Two semicircular arcs in the impedance complex plane plots were modeled to an equivalent circuit configured by RG∥QG+RGB∥QGB which were attributed to the contributions of the highly resistive grain boundaries and relatively conducting grains in the material. The electrical conductivity as a function of frequency followed Jonscher’s universal power law behavior and an increasing nature of the temperature-dependent frequency exponent suggested small polaron hopping as the electrical conduction mechanism of the material. The least capacitive response of the TCO system was investigated by the complex electrical modulus formalism and the frequency-dependent imaginary modulus spectra were modeled by the Kohlrausch–Williams–Watts function, which resolved the thermally activated peaks of bulk and interface effects to the dielectric relaxation mechanism. The dielectric constant exhibited Maxwell-Wagner interfacial polarization effects and the dielectric loss factor showed frequency dispersion at all temperatures.

Volume relaxation of soda-lime silicate glasses below glass transition temperature

Structural relaxation is a widely known phenomenon that occurs in glassy systems, which still attracts strong industrial and research interest. Although the volume change associated with structural relaxation is well described by the Kohlrausch–Williams–Watts function, its origin, particularly from a glass structural viewpoint, is not clearly defined. To understand the behavior of structural relaxation, in this study, we performed volume relaxation evaluations, Raman spectroscopy assessments, and surface resistivity measurements before and after annealing at 50 K below the glass transition temperature Tg of soda-lime silicate glasses with the same Tg but different fragilities. The combined results indicated that the following changes in the glass structure occurred during the structural relaxation: (1) reorganization of the SiO2 network; (2) transfer of Na ions from the ion channel region into the SiO2 network region; and (3) segmentation of the ion channel region in the modified random network model.

Mitigating of Thin-Film Composite PTMSP Membrane Aging by Introduction of Porous Rigid and Soft Branched Polymeric Additives.

This work was focused on the mitigation of physical aging in thin-film composite (TFC) membranes (selective layer ~1 μm) based on polymer intrinsic microporosity (PTMSP) by the introduction of both soft, branched polyethyleneimine (PEI), and rigid, porous aromatic framework PAF-11, polymer additives. Self-standing mixed-matrix membranes of thicknesses in the range of 20-30 μm were also prepared with the same polymer and fillers. Based on 450 days of monitoring, it was observed that the neat PTMSP composite membrane underwent a severe decline of its gas transport properties, and the resultant CO2 permeance was 14% (5.2 m3 (STP)/(m2·h·bar)) from the initial value measured for the freshly cast sample (75 m3 (STP)/(m2·h·bar)). The introduction of branched polyethyleneimine followed by its cross-linking allowed to us to improve the TFC performance maintaining CO2 permeance at the level of 30% comparing with day zero. However, the best results were achieved by the combination of porous, rigid and soft, branched polymeric additives that enabled us to preserve the transport characteristics of TFC membrane as 43% (47 m3 (STP)/(m2·h·bar) after 450 days) from its initial values (110 m3 (STP)/(m2·h·bar)). Experimental data were fitted using the Kohlrausch-Williams-Watts function, and the limiting (equilibrium) values of the CO2 and N2 permeances of the TFC membranes were estimated. The limit value of CO2 permeance for neat PTMSP TFC membrane was found to be 5.2 m3 (STP)/(m2·h·bar), while the value of 34 m3(STP)/(m2·h·bar) or 12,600 GPU was achieved for TFC membrane containing 4 wt% cross-linked PEI, and 30 wt% PAF-11. Based on the N2 adsorption isotherms data, it was calculated that the reduction of the free volume was 1.5-3 times higher in neat PTMSP compared to the modified one. Bearing in mind the pronounced mitigation of physical aging by the introduction of both types of fillers, the developed high-performance membranes have great potential as support for the coating of an ultrathin, selective layer for gas separation.

Influence of Microalloying Process on Dynamic Mechanical Relaxation of ZrCo-Based Amorphous Alloy

In this study, the dynamic mechanical spectroscopy was used to characterize the effects of minor Ni addition on the relaxation behavior of ZrCoAl bulk metallic glass (BMG). For this purpose, the Kohlrausch–Williams–Watts (KWW) function and quasi-point defect (QPD) model were used for evaluation of relaxation under the different aging temperatures. The results indicated that the Ni addition shifted the relaxation process to the higher temperatures. Moreover, the estimations showed that the activation energy of relaxation was 5.951 eV and 6.205 eV for ZrCoAl and ZrCoAlNi, respectively. It was also revealed that the microalloying process enhanced the structural defects in the system and led to the improvement of dynamic heterogeneity in the BMG. Comparing the physical models, it is suggested that a small change in the structural defects intensifies the dynamic heterogeneity in the material.

Accelerated relaxation behavior during water desorption in the mechano-sorptive creep of wood: modeling and analysis based on the free volume concept and Kohlausch–Williams–Watts function

Abstract The accelerated creep behavior during water desorption of wood was modeled on the Kohlausch–Williams–Watts (KWW) function and free volume concept. Assuming that the water desorption rate dmc/dln t $\text{dmc/dln\,t}$ is involved in the creation of a new free volume, the net creep compliance during desorption J du(t) was obtained as ln J du ( t ) = β c ln t − β c ( ln τ c 0 + k f dmc / d ln t ) + ln J c ∞ $\mathrm{ln} {J}_{\text{du}}\left(t\right)={\beta }_{\text{c}} \mathrm{ln} t-{\beta }_{\text{c}}\left(\mathrm{ln} {\tau }_{\text{c}0}+{k}_{\text{f}}\,\text{dmc}/\text{d}\mathrm{ln} t\right)+\mathrm{ln} {J}_{\text{c}\infty }$ where β c, τ c0 and J c∞ are the stretching parameter, characteristic time of the relaxation, and long-time asymptote compliance at a constant moisture content (mc) equal to the initial mc of the accelerated creep, respectively, and k f is the coefficient of the water desorption rate. These parameters were determined by simulation using experimental data and literature results and were found to be reasonable. The simulation results agreed well with the experimental ones. This verifies the validity of the model and its derived equations.

Kinetics of viscous flow sintering of glass–zircon composites. Effect of zircon volume fraction and particle size distribution

Viscous flow sintering of glass–zircon composites was studied by isothermal and constant-rate heating experiments. The sintering curves fitted very well to the developed kinetic model, based on the Kohlrausch–Williams–Watts (KWW) function. Increasing the zircon volume fraction and decreasing the zircon–glass particle size ratio was verified to slow down the sintering rate. The combined effect of these variables on sintering was interpreted on the basis of percolation theory. The influence of temperature on the composite sintering rate corresponded to the influence of this variable on glass matrix flowability. Dimensionless relationships between the kinetic parameters and zircon volume fraction and the particle size ratio were obtained, which were independent of the thermal treatment used.

Structural origin of thermal shrinkage in soda-lime silicate glass below the glass transition temperature: A theoretical investigation by microsecond timescale molecular dynamics simulations.

Microscopic dynamical features in the relaxation of glass structures are one of the most important unsolved problems in condensed matter physics. Although the structural relaxation processes in the vicinity of glass transition temperature are phenomenologically expressed by the Kohlrausch-Williams-Watts function and the relaxation time can be successfully interpreted by Adam-Gibbs theory and/or Narayanaswamy's model, the atomic rearrangement, which is the origin of the volume change, and its driving force have not been elucidated. Using the microsecond time-scale molecular dynamics simulations, this study provides insights to quantitatively determine the origin of the thermal shrinkage below Tg in a soda-lime silicate glass. We found that during annealing below Tg, Na ions penetrate into the six-membered silicate rings, which remedies the acute O-O-O angles of the energetically unstable rings. The ring structure change makes the space to possess the cation inside the rings, but the ring volume is eventually reduced, which results in thermal shrinkage of the soda-lime silica glass. In conclusion, the dynamical structural relaxation due to the cation displacement evokes the overall volume relaxation at low temperature in the glassy material.

Molecular dynamics in the supercooled liquid and glassy states of bezafibrate and binary mixture of fenofibrate

Two solubility limited pharmaceuticals bezafibrate and fenofibrate, belonging to BCS class II drugs, were compared, by investigating molecular dynamics of the former and compared with the neat and as a probe in an apolar host polystyrene of mole. wt. 800 (PS800) of the latter by broadband dielectric spectroscopy (BDS), other supporting thermal and spectroscopic experimental means and density functional theory (DFT) calculations. The BDS experiments from frozen glassy state to vibrant molten state of the title drugs were done to explore the role of parameters describing their complex molecular dynamics and the stabiliy of amorphous phase, such as glass transition temperature (Tg), temperature dependence of α-relaxation times (τα), width of structural dispersion and its role on the occurrence of Johari-Goldstein (JG) β-relaxation, fragility index (m) and apparent activation energy of secondary processes. Realizing that most of the pharmaceuticals have narrow α-dispersion which limits the visibility of JG β-relaxation as a separate process in the dielectric window, to resolve the conundrum, we have further explored the amorphous dynamics of fenofibrate as a probein an apolar host polystyrene of molecular weight 800 (PS800) having higher Tg than the polar probe. The Tg of bezafibrate and PS800-fenofibrate mixture were determined to be 298 K and 275 K respectively, whereas the fractional exponent of Kohlrausch–Williams–Watts function βKWW of bezafibrate was determined to be 0.68 which is closer to that of fenofibrate, whereas that of PS800-fenofibrate mixture was found to be much broader with a value of 0.39, all in the vicinity of Tg. Even though βKWW decreased in binary mixture of fenofibrate when compared to its pure form due to concentration fluctuations and other aspects, but cold crystallization tendency was absent and Tg is elevated than that of fenofibrate. Due to higher Tg than that of fenofibrate, amorphous bezafibrate is more stable than fenofibrate. Finally, using quantum computational calculations using DFT, the relevant flexible parts of the two drug molecules, whose motions manifests the secondary γ-relaxation was identified by calculating potential energy barriers and dipole moments on relaxed dihedral angle scans, as no experimental technique is available for this purpose.

Stress relaxation and humidity dependence in sodium alginate-glycerol films

The control of viscoelastic properties of alginate biopolymer that mimics the matrix properties of biological substrates plays an important role for the success of its biomedical applications. For this purpose stress relaxation behavior of glycerol plasticized sodium alginate films is characterized at room temperature as a function of the glycerol concentration ranging up to 40%. A series of experiments are thus conducted at relative humidity levels of 38 ± 1% and 51 ± 1%. The glycerol content is demonstrated to amplify the effect of humidity on relaxation profiles. In the case of 30% glycerol, normalized stress at the level of 65% is recorded at the end of the first 30 min at the low humidity level, whereas the corresponding value drops to 8% with increased humidity. Alginate films with no glycerol content exhibit much higher normalized stresses of 89% and 73% at low and high humidity levels, respectively. In addition, stress relaxation is observed to continue well beyond the first 30 min, especially for glycerol concentrations lower than 30%, where 9-hour parameters for a stretched exponential Kohlrausch – Williams – Watts function are reported demonstrating the importance of relaxation time for successful modeling.

Non-overlapping small polaron tunneling conduction coupled dielectric relaxation in weak ferromagnetic NiAl2O4

The inverse spinel nickel aluminate micro-particles was successfully synthesized via solid-state synthesis techniques. The single phase cubic structure with space group Fd3m of the as-prepared sample was entrenched from Rietveld refinement of x-ray diffraction pattern. In addition, the photoluminescence (PL) and FTIR spectra were also performed to give strong evidence of its pure phase formation. The narrow hysteresis m–h loop and UV-DRS spectra at room temperature demonstrate its weak ferromagnetic and semiconducting nature with saturation magnetization 64.96 × 10−3 emu gm−1 and direct optical band gap 2.03 eV respectively. The high-resolution FESEM micrographs and EDS elemental analysis exhibit its grain growth in µm range (217 µm) and presence of elemental compound Ni, Al and O respectively. The electrical transport properties were accomplished by complex impedance spectroscopy as a function of frequency (100 Hz–1 MHz) with the evolution of high temperature (300 °C–500 °C). The Nyquist plots (Z″ versus Z′) were well fitted with an equivalent circuit model (QR) (QR) consisting of a series combination of intra and inter-granular contribution. Furthermore, the imaginary modulus spectra were also fitted with Kohlrausch–Williams–Watts (KWW) function, which represents two thermally activated peaks of grain and grain boundary effects. The low-frequency dispersive ac conductivity was elucidated using the following equation: . The increasing nature of temperature dependent frequency exponent (n) suggests the quantum mechanical tunneling: non-overlapping small polaron tunneling (NSPT) concept for conduction mechanism. In low-frequency region, the lattice and charge carrier polarization simultaneously contribute in dielectric permittivity.

Nanofiller-Induced Ionic Conductivity Enhancement and Relaxation Property Analysis of the Blend Polymer Electrolyte Using Non-Debye Electric Field Relaxation Function

The enhancement of conductivity of a composite polymer as a dielectric material is an essential requirement for electrostatic storage devices. We have modified the microstructure of the polymer matrix by introducing an insulating nanofiller SiO2. The effect of such a filler on the ionic conductivity of the composite polymer electrolyte has been investigated using a variety of experimental techniques along with the non-Debye type of relaxation functions. We have achieved optimum conductivity enhancement at a threshold filler concentration of 0.7 wt % in the blend polymer matrix composed of poly(ethylene oxide), poly(vinylidene fluoride) (80:20), and salt NH4 I (35 wt %). Such an enhancement of conductivity is a result of formation of a highly conducting interphase region around the nanofiller surface. The mobility of the conducting species is found to increase enormously in the presence of a filler. As a consequence, the ionic conductivity of the filler-induced blend polymer electrolyte increases 3 times of its magnitude (3.02 × 10–3 S/cm) compared to that without a filler. The occurrence of two different activation energies which decrease with increasing filler concentrations, as determined from temperature-dependent conductivity, has been well explained from the dynamics of free and contact ions. A non-Debye behavior of relaxation properties has been analyzed using a newly approached one-parameter Mittag-Leffler function. The experimental decay function fits very well using the Mittag-Leffler function as compared to the conventional non-Debye Kohlrausch–Williams–Watts function used in the literature.

Electrical and room temperature multiferroic properties of polyvinylidene fluoride nanocomposites doped with nickel ferrite nanoparticles

The higher values of magneto-dielectric coupling is observed in flexible multiferroic polyvinylidene fluoride (PVDF) nanocomposites doped with nickel ferrite (NFO) nanoparticles.

Characteristic Exponent of Normal and Oblique Rolls in Homeotropically Aligned Nematic Liquid Crystal

Soft-mode turbulence (SMT) is one of an experimental example of spatiotemporal chaos, observed in electroconvection system of homeotropically aligned nematic liquid crystal (NLC), due to a non-linear interaction between Nambu-Goldstone mode denoted by the C(r)- director and the convective mode q(r). There are two types of stripe patterns in the SMT, namely normal rolls (NR) and oblique rolls (OR) which separated by a point of applied frequency, called the Lifshitz frequency (fL). We report a study of the phase transition from normal to oblique rolls by observing the patterns with an applied frequency below and beyond of fL. The temporal fluctuations of the pattern images had been analyzed using autocorrelation function. It fits with Kohlrausch Williams Watts (KWW) function, showing there is a dynamical glass-forming liquid in the transition of NR-OR regime. Also, we found a new type of defect in the NR regime which never been reported before, a dynamic defect which takes the shape of a ring first to a spot in the end.