Williams Watts 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.

Structural properties, dielectric relaxation, and impedance spectroscopy of NASICON type Na3+xZr2−xPrxSi2PO12${\rm Na}_{3+x} {\rm Zr}_{2-x} {\rm Pr}_{x} {\rm Si}_2 {\rm PO}_{12}$ ceramics

AbstractWe investigate the dielectric and impedance spectroscopic investigation of Pr‐doped NASICON type () samples as a function of temperature and frequency. The Rietveld refinement of X‐ray diffraction patterns confirms the monoclinic phase having C2/c space groups for all the samples. The scanning electron microscopy shows the granular‐like structure and energy dispersive X‐ray analysis confirms the desired compositions. The temperature (90–400 K) and frequency (20 Hz–2 MHz) dependence of electric permittivity are explained using Maxwell–Wagner–Sillars (MWS) polarization and space charge polarization mechanisms. The dielectric relaxation shows nearly equal activation energy for all the samples with a non‐Debye type of relaxation in the measured temperature range. The complex impedance analysis shows the presence of broad grain and grain boundary relaxation peaks. The stretched exponent analysis of electric modulus using the Kohlrausch–Williams–Watts (KWW) function further confirms the non‐Debye type of relaxation. Moreover, scaling analysis of the electric modulus shows a similar type of relaxation for all the samples. The ac conductivity data are fitted using modified power law, where the temperature dependence of exponent () confirms the correlated barrier hopping (CBH) type conduction for all the samples. Our results indicate that the Pr‐doped NASICON samples are potential candidates for charge storage devices due to their large electric permittivity.

Study of temperature-dependent interface with viscoelastic behavior through shear testing and modeling

In this work, the effect of temperature on interfaces with viscoelastic behavior was investigated using experimental and theoretical means. In particular, rubber/woven fabrics with thickness of 0.5 mm were subjected to self-designed shear tests at temperatures ranging from 263.15 to 383.15 K. Derived interface cyclic regeneration was observed through shear tests. Subsequently, a bi-linear cohesive zone model (CZM) was used to describe the shear test data. A novel image processing algorithm was developed to analyze the damaged morphology of the rubber/weave interface. It was found that hairniness amount decreased by 68.4 % when the temperature increased from 263.15 to 383.15 K. In addition, by combining the bi-linear CZM with the Kohlrausch–William–Watts law, Wiliams-Landel-Ferry equation and BK criterion, a 3D temperature-dependent constitutive model (KBW) was developed. The model successfully simulated the viscoelasticity of interfaces at different temperatures with margin of error less than 9 %. Thus, this study provides a strategy for elucidating the temperature dependence of interfaces with viscoelastic behavior.

Magnetic exchange interactions and non-Debye relaxation in spin-3/2 frustrated Kagomé magnet Co3V2O8

We report the experimental determination of the magnetic exchange parameter ( J/kB = 2.88 ± 0.02 K) for the Spin-3/2 ferromagnetic (FM) Kagomé lattice system: Co3V2O8 using the temperature dependence of dc-magnetic susceptibility χ(T) data by employing the fundamental Heisenberg linear chain model. Our results are quite consistent with the theoretically reported nearest neighbor dominant FM exchange coupling strength Jex−NN∼ 2.45 K. Five different magnetic phase transitions (6.2–11.2 K) and spin-flip transitions (9.6–7.7 kOe) have been probed using the ∂ ( χT )/ ∂T vs. T, heat capacity (CP -T) and differential isothermal magnetization curves. Among such sequence of transitions, the prominent ones being incommensurate antiferromagnetic (AFM) state at 11.2 K, commensurate AFM state at 8.8 K, and commensurate FM state across 6.2 K. All the successive magnetic phase transitions have been mapped onto a single H-T plane through which one can easily distinguish the above-mentioned different phases. The magnetic contribution of the CP -T near TN (11.2 K) has been analyzed using the power-law expression CM = A|T — TN|−α resulting in the critical exponent α = 0.18 ± 0.01 (0.15 ± 0.003) for T < TN (T > TN ), respectively for the Co3V2O8. It is interesting to note that non-Debye type dipole relaxation is quite prominent in Co3V2O8 and was evident from the Kohlrausch–Williams–Watts analysis of complex modulus and impedance spectra (0 ⩽β⩽ 1). Mott’s variable-range hopping of charge carriers process is evident through the resistivity analysis ( ρac -T −1/4 ) in the temperature range 275 ∘C–350 ∘C. Moreover, the frequency-dependent analysis of σac (ω) follows Jonscher’s power law yielding two distinct activation energies ( Ea∼ 0.37 and 2.29 eV) between the temperature range 39 ∘C–99 ∘C and 240 ∘C–321 ∘C.

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.

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.

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.

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.

Coupling between hydrolysis and mechanical relaxation in an ester‐based polyurethane network

AbstractIn this study, we first investigate the impact of a chain scission process due to hydrolysis on the relaxation behavior of an elastomer. We then explore the possible influence of a deformed state on the chemical reactions associated with hydrolysis. To perform the experiments coupling mechanical relaxation and hydrolysis, we select not only specific conditions such as the type of elastomer but also severe conditions such as immersions at high temperatures. Here, an ester‐based polyurethane network is chosen to easily hydrolyze the ester function, thus leading to a chain scission process. In addition, relaxation tests are performed in water at several temperatures (i.e., 40, 60, and 80 °C) to accelerate the hydrolysis rate. Results show that a deformed state does not accelerate the hydrolysis rate. However, a chain scission process clearly modifies the relaxation behavior. Finally, this behavior is modeled by coupling the chain scission kinetics with the relaxation time based on the Kohlrausch–Williams–Watts approach.

Temperature dependence of the aging effect of permittivity in Pb(Zn1/3Nb2/3)O3–PbTiO3 and BaTiO3 single crystals

The temperature dependence of the aging effect of permittivity in the paraelectric phase of 95.5%Pb(Zn1/3Nb2/3)O3–4.5%PbTiO3 (PZN–4.5%PT) and BaTiO3 (BT) single crystals has been investigated. The time dependence of the Re of permittivity at constant temperatures has been found in the paraelectric phase not only in the relaxor ferroelectric PZN–4.5%PT, but also in the typical oxide ferroelectric BT. Such time dependence of permittivity can be empirically analyzed using the Kohlausch–Williams–Watts relaxation function. We discuss the relationship between the aging effect and the relaxation process of the collective motion of polar nanoregions.

Analyzing the strong influence of the cationic environment on the charge carrier’s jumping in a superionic conductor

In this work the relationship between the structure of a solid ionic material and its ionic dynamics has been studied by impedance spectroscopy, differential scanning calorimetry and molecular dynamics simulation. Nanometric samples of AgI were synthesized to study the cation dynamics when it reaches its superionic behavior. The observed thermal events corresponded to the changes obtained on the dc conductivity behavior. This correspondence was microscopically explained by the ion dynamics revealed in the molecular dynamics simulation. The comprehensive analysis made it possible to confirm the weak short-range interactions between silver cations and iodide anions and exposed that the most mobile cation jumps are constrained by their neighbor cations. This finding explains the Kohlrausch–Williams–Watts β parameter value of 0.9 in the α phase where the AgI becomes a superionic conductor, an aspect that becomes essential to design a superionic conductor by doping ionic conductor materials.

Relaxation Functions Interpolating the Cole–Cole and Kohlrausch–Williams–Watts Dielectric Relaxation Models

To describe non-Debye relaxation phenomena observed in dielectric materials, the Cole–Cole (CC) relaxation model in the frequency domain and the Kohlrausch–Williams–Watts (KWW) relaxation model in the time domain were introduced in the physics of dielectrics. In this paper, we propose a new relaxation model with two parameters besides a relaxation time by expressing the relaxation function in the time domain in terms of the Mittag–Leffler functions. The proposed model represents a group of non-Debye relaxation phenomena and shows a transition between the CC and the KWW models. The relaxation properties described by the new model are analyzed, including the response function, the normalized complex dielectric permittivity, dielectric storage and loss factors as well as the relaxation frequency and time spectral functions. The presented relaxation function has a concise form and is expected to be applied to more complex relaxation phenomena.

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.

Rate-dependent properties of resistance-welded carbon fiber-reinforced polyetheretherketone joints: Simulations and experiments

The rate-dependent characteristics of the resistance-welded joints of thermoplastic composites are investigated via simulations and experiments. A rate-dependent cohesive zone model based on the Kohlrausch–William–Watts formulation is proposed to capture the mechanical response of the joints at different loading rates, which is validated by the experimental results in terms of stiffness, strength, and failure. The proposed model is equivalent to the generalized Maxwell model with only eight input parameters to describe the welding interface's time- and history-dependent mechanical response. The stress distribution, crack emergence, and damage evolution of welded joints at different loading rates are revealed through numerical calculations. Additionally, the experimental results indicate that the strength of the joint is enhanced with the increase in the loading speed, as the strengthening effect caused by the high strain rate and the shorter loading time results in a tightly bonded stainless-steel/matrix interface and a rougher fracture surface.

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.

Effect of TiO2 Nanoparticles and Extrusion Process on the Physicochemical Properties of Biodegradable and Active Cassava Starch Nanocomposites.

Biodegradable polymers have been strongly recognized as an alternative to replace traditional petrochemical plastics, which have become a global problem due to their long persistence in the environment. In this work, the effect of the addition of titanium dioxide nanoparticles (TiO2NP) on the morphology, physicochemical properties and biodegradation under industrial composting conditions of cassava starch-based nanocomposites obtained by extrusion at different screw speeds (80 and 120 rpm) were investigated. Films performed at 120 rpm (S120 and S120-TiO2NP) showed completely processed starch and homogeneously distributed nanoparticles, leading to much more flexible nanocomposites than those obtained at 80 rpm. The incorporation of TiO2NP led to an increase in storage modulus of all films and, in the case of S120-TiO2NP, to higher strain at break values. From the Kohlrausch-Williams-Watts theoretical model (KWW), an increase in the relaxation time of the nanocomposites was observed due to a decrease in the number of polymer chains involved in the relaxation process. Additionally, S120-TiO2NP showed effective protection against UV light, greater hydrophobicity and faster biodegradation in compost, resulting in a promising material for food packaging applications.

Effect of the physical aging on the secondary β relaxation process in a La-based metallic glass

Dynamic mechanical relaxation processes, i.e., main (α) relaxation and secondary (β) relaxation, are important issues to understand mechanical deformation, atomic diffusion as well as glass transition phenomenon of metallic glasses. In current work, La68Ni15Al15Cu2 metallic glass was selected as a protocol glass system. Mechanical relaxation processes were probed by dynamic mechanical analysis. The effects of annealing at different temperatures were analyzed by Kohlrausch–Williams–Watts (KWW)-type equation. The Kohlrausch exponent βKWW reflects the deviation from a single Debye relaxation, indicating the fact that dynamics in metallic glass are actually heterogeneous originating from the structural heterogeneity. The effects of thermal treatments were also discussed, which provides a potential solution to tune the relaxation behaviors in metallic glasses.

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.

High Ionic Conduction and Polarity-Induced Piezoresponse in Layered Bimetallic Rb4Ag2BiBr9 Single Crystals

The emergence of lead-free metal-halide perovskites in modern-day energy research is primarily due to their competent semiconducting and optoelectronic properties, as well as their nontoxicity and improved stability in ambient operating conditions. However, a detailed understanding of ion transport dynamics and the relaxation behavior of these materials is still elusive. In this article, we report on the single-crystal (SC) growth of a bimetallic two-dimensional layered Rb4Ag2BiBr9 and explore its dielectric, piezoelectric, and ferroelectric properties. The dielectric attributes are investigated by studying the temperature-dependent complex impedance, complex electric modulus, and AC conductivity over an extensive frequency range from 4 Hz to 8 MHz. The role of grain and grain boundaries in the effective impedance is established using the Maxwell–Wagner equivalent circuit model from the Nyquist plots. The observed electric modulus spectra are studied using the Havriliak–Nigami and the Kohlrausch–Williams–Watts formalisms to understand the ionic transport and relaxation mechanisms in Rb4Ag2BiBr9. The DC conductivity and relaxation time show Arrhenius-like behavior with the inverse temperature validating the hopping motion of ions. The values of the activation energy required for ion hopping are calculated independently from the relaxation time, hopping frequency, and DC conductivity Arrhenius plots and are in good agreement with a value of 0.47 eV. The scaling of the temperature-dependent conductivity and electric modulus spectra into a single master curve demonstrates the congruence of the ionic conduction and the relaxation phenomena at different temperatures for Rb4Ag2BiBr9. This SC demonstrates decent thermal and ambient stability up to 500 °C and 12 months, respectively. The pristine orthorhombic Rb4Ag2BiBr9 SC shows a piezoelectric amplitude value of ∼564 pm at the maximum applied bias (±10 V), and a saturation polarization of ∼0.14 nC/cm2 estimated from the piezoelectric force microscopy and polarization hysteresis loop measurement, respectively. The ferroelectric and semiconducting attributes of this material can be harnessed for prospective applications as a thin film in designing mechanical energy harvesters as well as functional photoferroelectrics, such as optical switches and ferroelectric photovoltaics.

Electrical conductivity and relaxation in lithium-doped barium vanadate glasses investigated by impedance spectroscopy

We present herein a detailed study of the electric and dielectric properties of glasses with the composition 60V2O5–(40–x)BaO–xLi2O, where x = 10, 15, 20, or 25 mol%. The conduction and relaxation mechanisms have been studied by impedance spectroscopy over the frequency range from 40 Hz to 10 MHz and at various temperatures. With constant transition metal oxide content, the conductivity variation in the glasses is assumed to be mainly due to an ionic contribution. The dependence of electrical conductivity on frequency has been analysed using Jonscher's power law. Properties of the glasses, such as electric modulus loss and relaxation mechanism, have been analysed in terms of electric modulus using the Kohlrausch–William–Watts (KWW) equation, which describes the relaxation behaviour of the mobile ions as non-Debye in nature. Scaling of ac conductivity and electric modulus data has shown that conductivity relaxation is independent of temperature. The non-monotonic variation in the stretching parameter β has been found to show a positive correlation with various activation energies, namely electrical activation energy, relaxation activation energy, and the activation energy from the Anderson–Stuart model. The anomalous behaviour of activation energy with respect to conductivity has been substantiated through structural insight. The conductivity was found to be highest in glasses with 25 mol% Li2O, while coupling between the mobile ions was best for glasses with 20 mol% Li2O.