Debye Law Research Articles

Emergence of Debye Scaling in the Density of States of Liquids under Nanoconfinement.

In the realm of nanoscience, the dynamic behaviors of liquids at scales beyond the conventional structural relaxation time, τ, unfold a fascinating blend of solid-like characteristics, including the propagation of collective shear waves and the emergence of elasticity. However, in classical bulk liquids, where τ is typically of the order of 1 ps or less, this solid-like behavior remains elusive in the low-frequency region of the density of states (DOS). Here, we provide evidence for the emergent solid-like nature of liquids at short distances through inelastic neutron scattering measurements of the low-frequency DOS in liquid water and glycerol confined within graphene oxide membranes. In particular, upon increasing the strength of confinement, we observe a transition from a liquid-like DOS (linear in the frequency ω) to a solid-like behavior (Debye law, ∼ω2) in the range of 1-4 meV. Molecular dynamics simulations confirm these findings and reveal additional solid-like features, including propagating collective shear waves and a reduction in the self-diffusion constant. Finally, we show that the onset of solid-like dynamics is pushed toward low frequency along with the slowing-down of the relaxation processes upon confinement. This nanoconfinement-induced transition, aligning with k-gap theory, underscores the potential of leveraging liquid nanoconfinement in advancing nanoscale science and technology, building more connections between fluid dynamics and materials engineering.

Effect of doping and preparation parameter on AC conductivity and dielectric modulus formalism of Al doped DLC thin films

Here, we report the composite effect of aluminum doping on the dielectric response of Diamond-Like Carbon (DLC) thin films. In addition, the contribution from preparation parameters like acetylene flow rate is also considered to understand sample behavior. AC conductivity study shows the existence of a nonlinear increase of conductivity with frequency leading toward the Jump Relaxation Model (JRM). The transition frequency (ft) is observed, which increases with conductivity. The study reveals that dielectric response is influenced by space charge conductivity, prompting us to adopt the modified Debye law. Deviation from ideal Debye behavior is also confirmed in dielectric modulus analysis. The deviation is frequency-dependent, and a higher deviation is observed in the high-frequency region. Plotting of the Master curve shows the presence of different relaxation dynamics in different samples. A current-voltage (I-V) study of different samples is also undertaken, showing higher values (greater than unity) of the ideality factor.

Effect of substitution on the structural, electrical properties, and dielectric relaxor behavior in lead-free BiFeO3-based ceramics.

BiFeO3-based ceramics have recently garnered much interest among researchers owing to their valuable and outstanding characteristics. For this reason, the 0.7(Na0.5Bi0.5)TiO3-0.3(Bi0.7Sm0.3FeO3) ceramic was successfully synthesized by a solid-state route. The central purpose of this research is to investigate the substitution influence of Na, Ti, and Sm on the structural, dielectric, and electric properties of 0.7(Na0.5Bi0.5)TiO3-0.3(Bi0.7Sm0.3FeO3), as well as to explore its potential applications as it exhibits multiple novel functions. Notably, a structural transition from rhombohedral R3c to orthorhombic P4mm occurred within this material. In this respect, a suitable equivalent electrical circuit was invested to assess the contributions of grains and grain boundaries to the complex impedance results. Electrical conductivity was attributed to the correlated barrier hopping (CBH) motion of the oxygen vacancies in the sample. The temperature dependence of the dielectric constants revealed the presence of a phase transition. The local disorder provides a dependence of the real part of the permittivity on the frequency which characterizes a relaxor ferroelectric-type behavior of a lead-free material. The modified Curie-Weiss law, in addition to the Vogel Fulcher and Debye law relationships, was utilized to analyze the diffuse transition phase. Furthermore, the studied compound displayed promising electrical properties and chemical stability and proved to be a good relaxor. In this regard, a correlation between dielectric and electric behavior near the ferro-paraelectric phase transition was established.

Fractal character of coal nanopore and effect of deviation corrected, coal rank, and gas adsorption

Pore structure in coal is fundamental to determining the flowability properties and mechanical functions. Various insights into pore structure, including pore size distribution, specific area, and pore volume, have been obtained in experiments with many methods. Systematic research is lacking on how to characterize pore structure with more detailed information using fractal theory. Here, using the small-angle X-ray scattering (SAXS) method, we studied the fractal characters of four coal samples and showed how the deviation corrected, coal rank, and gas adsorption affected the fractal dimension of the pore structure. The average pore diameters of four samples obtained by the gyration radius using Guinier curves were 50.25–52.95 nm. The surface fractal of coal samples was the dominant feature, and the fractal features altered with the q value. Scattering information of simple bodies, such as pores and fractures, could superimpose the non-integer nature of the fractal dimension. The Porod and Debye laws guided the deviation correction process in the high q region. Pore fractals might be changed into surface fractals through the fractal dimension after the deviation correction, and the SAXS data obtained by Debye law correction maintained the originality and validity better than Porod law correction. The irregular shape of M was observed on the curve of fractal dimension as the maximum reflectance of vitrinite increases, resulting in three stages. More adsorption sites per unit projection area could be found in a larger fractal dimension of the pore surface, which results in an increase in Langmuir adsorption volume. The increase in the adsorption pressure reduced the fractal dimension. Our findings enhance the experimental support for comprehending the structure of coal nanopores.

A fresh look at the vibrational and thermodynamic properties of liquids within the soft potential model.

Contrary to the case of solids and gases, where Debye theory and kinetic theory offer a good description for most of the physical properties, a complete theoretical understanding of the vibrational and thermodynamic properties of liquids is still missing. Liquids exhibit a vibrational density of states (VDOS) which does not obey Debye law, and a heat capacity which decreases monotonically with temperature, rather than growing as in solids. Despite many attempts, a simple, complete and widely accepted theoretical framework able to formally derive the aforementioned properties has not been found yet. Here, we revisit one of the theoretical proposals, and in particular we re-analyze the properties of liquids within the soft-potential model, originally formulated for glasses. We confirm that, at least at a qualitative level, many characteristic properties of liquids can be rationalized within this model. We discuss the validity of several phenomenological expressions proposed in the literature for the density of unstable modes, and in particular for its temperature and frequency dependence. We discuss the role of negative curvature regions and unstable modes as fundamental ingredients to have a linear in frequency VDOS. Finally, we compute the heat capacity within the soft potential model for liquids and we show that it decreases with temperature, in agreement with experimental and simulation data.

Vibrational Phenomena in Glasses at Low Temperatures Captured by Field Theory of Disordered Harmonic Oscillators.

We investigate the vibrational properties of topologically disordered materials by analytically studying particles that harmonically oscillate around random positions. Exploiting classical field theory in the thermodynamic limit at T=0, we build up a self-consistent model by analyzing the Hessian utilizing Euclidean random matrix theory. In accordance with earlier findings [T. S. Grigera etal.J. Stat. Mech. (2011) P02015.JSMTC61742-546810.1088/1742-5468/2011/02/P02015], we take nonplanar diagrams into account to correctly address multiple local scattering events. By doing so, we end up with a first principles theory that can predict the main anomalies of athermal disordered materials, including the boson peak, sound softening, and Rayleigh damping of sound. In the vibrational density of states, the sound modes lead to Debye's law for small frequencies. Additionally, an excess appears in the density of states starting as ω^{4} in the low frequency limit, which is attributed to (quasi-) localized modes.

Experimental Confirmation of the Universal Law for the Vibrational Density of States of Liquids.

An analytical model describing the vibrational density of states (VDOS) of liquids has long been elusive, owing to the complexities of liquid dynamics. Nevertheless, Zaccone and Baggioli have recently developed such a model which was proposed to be the universal law for the vibrational density of states of liquids. Distinct from the Debye law, g(ω) ∝ ω2, for solids, the universal law for liquids reveals a linear relationship, g(ω) ∝ ω, in the low-energy region. We have confirmed this universal law with experimental VDOS measured by inelastic neutron scattering on real liquid systems including water, liquid metal, and polymer liquids, and have applied this model to extract the effective relaxation rate for the short time dynamics for each liquid. The model has also been further evaluated in the prediction of the specific heat with comparison to existing experimental data as well as with values obtained by different approaches.

DFT study of structural, electronic, magnetic, elastic, vibrational, thermodynamic and thermoelectric properties of XCrP (X= Ni, Pd and Pt)

This study of physical properties of the half-Heuslers compounds of XCrP (X = Ni, Pd, Pt) is performed in the framework of density functional theory (DFT). The results show that all compounds are stable in the ferromagnetic state. The DOSs reveal that NiCrP is half-metallic, whereas, PdCrP and PtCrP are nearly half-metallic. The elastic properties indicate that these compounds are ductile and anisotropic. The phonons dispersions show that NiCrP is dynamically stable, while the two others are unstable. The variation of specific heat at constant volume Cv with temperature obeys the Debye's law at low-temperature, and it obeys the classical Dulong–Petit's law at high temperature for NiCrP, while Cv of PdCrP and PtCrP don't follow the Debye law due to the negative phonons frequencies. The thermoelectric properties (TE) indicate that these compounds are n-type materials and suitable for TE applications.

Unifying Description of the Vibrational Anomalies of Amorphous Materials.

The vibrational density of states D(ω) of solids controls their thermal and transport properties. In crystals, the low-frequency modes are extended phonons distributed in frequency according to Debye's law, D(ω)∝ω^{2}. In amorphous solids, phonons are damped, and at low frequency D(ω) comprises extended modes in excess over Debye's prediction, leading to the so-called boson peak in D(ω)/ω^{2} at ω_{bp}, and quasilocalized ones. Here we show that boson peak and phonon attenuation in the Rayleigh scattering regime are related, as suggested by correlated fluctuating elasticity theory, and that amorphous materials can be described as homogeneous isotropic elastic media punctuated by quasilocalized modes acting as elastic heterogeneities. Our numerical results resolve the conflict between theoretical approaches attributing amorphous solids' vibrational anomalies to elastic disorder and localized defects.

Temperature-Dependent Dielectric Relaxation in Ionic Acetamide Deep Eutectics: Partial Viscosity Decoupling and Explanations from the Simulated Single-Particle Reorientation Dynamics and Hydrogen-Bond Fluctuations.

We report here temperature-dependent (293 ≤ T (K) ≤ 336) dielectric relaxation (DR) measurements of (acetamide + LiBr/NO3-/ClO4-) deep eutectic solvents (DESs) in the frequency window of 0.2 ≤ ν (GHz) ≤ 50 and explore, via molecular dynamics simulations, the relative roles for the collective single-particle reorientational relaxations and the H-bond dynamics of acetamide in the measured DR response. In addition, DR measurements of neat molten acetamide were performed. Recorded DR spectra of these DESs require multi-Debye fits and produce well-separated DR time scales that are spread over several picoseconds to ∼1 ns. Simulations suggest DR time scales derive contributions from both the collective reorientational () relaxation and structural H-bond (CHB(t)) dynamics of acetamide. A good correlation between the measured and simulated activation energies further reveals a strong connection between the measured DR and the simulated and CHB(t). Average DR times exhibit a strong fractional viscosity dependence, suggesting substantial microheterogeneity in these media. Simulations of and CHB(t) reveal strong stretched exponential relaxations with a stretching exponent, 0.4 ≤ β ≤ 0.7. The ratio between the average reorientational correlation times of first and second ranks, , deviates appreciably from Debye's law for homogeneous media. Importantly, a pronounced translation-rotation decoupling between the simulated reorientation and center-of-mass diffusion times was observed.

Physics of phonon-polaritons in amorphous materials.

The nature of bosonic excitations in disordered materials has remained elusive due to the difficulties in defining key concepts such as quasi-particles in the presence of disorder. We report on an experimental observation of phonon-polaritons in glasses, including a prominent boson peak (BP), i.e., excess of THz modes over the Debye law. A theoretical framework based on the concept of diffusons is developed to describe the broadening linewidth of the polariton due to disorder-induced scattering. It is shown here for the first time that the BP frequency and the Ioffe-Regel (IR) crossover frequency of the polariton collapse onto the same power-law decay with the diffusivity of the bosonic excitation. This analysis dismisses the hypothesis of the BP being caused by a relic of the van Hove singularity. The presented framework establishes a new methodology to analyze bosonic excitations in amorphous media, well beyond the traditional case of acoustic phonons, and establishes the IR crossover as the fundamental physical mechanism behind the BP.

Role of Anharmonicity in Specific Heat of Metals and Insulators at Low Temperatures

In this paper we discuss the role of anharmonicity of lattice waves in electronic specific heat of metals and insulators at very low temperatures. Previous researchers have revealed the anharmonicity of the lattice waves to cause deviation in specific heat from Debye T3law. Different researchers have given different concepts of anharmonicity in electronic specific heat of metals at very low temperatures. The energy spectrum of phonons can not be fully described by the Debye law. On the whole all reports dilute the contention of the electronic contribution to be responsible for linear temperature dependent component of specific heat of metals at very low temperatures. For anharmonicity of lattice waves this contribution must be expected for metals and insulators also.

Random Matrix Theory and the Boson Peak in Two-Dimensional Systems

The random matrix theory is used to describe the vibrational properties of two-dimensional disordered systems with a large number of degrees of freedom. It is shown that the correlated Wishart ensemble allows one to take into account the most significant mechanical properties of amorphous solids. In this ensemble, an excess density of vibrational states in comparison with the Debye law is observed, which is expressed in a peak in the reduced density of states g(ω)/ω. It is known as the boson peak, observed in many experiments and numerical calculations concerning two-dimensional and three-dimensional disordered systems. It is shown that the asymptotic behavior of the boson peak in two-dimensional systems has a number of features.

Применение теории случайных матриц для описания бозонного пика в двумерных системах

The random matrix theory is applied to describe the vibrational properties of two-dimensional disordered systems with a large number of degrees of freedom. It is shown that the most significant mechanical properties of amorphous solids can be taken into account using the correlated Wishart ensemble. In this ensemble, an excess vibrational density of states over the Debye law is observed as a peak in the reduced density of states g(ω)/ω. Such a peak is known as the boson peak, which was observed in many experiments and numerical simulations for two-dimensional and three-dimensional disordered systems. It is shown that two-dimensional systems have a number of differences in the asymptotic behavior of the boson peak.

A Simple Model for Stretched Exponential Relaxation in Three‐Level Jumping Process

In this study, operator formalism is briefly introduced which is used to model Debye‐type relaxation with three‐level jumping based on Markovian framework. The author generalizes this formalism to the non‐Markovian process to model the non‐exponential relaxation and internal friction of real bcc metals or systems like Snoek‐type. By using this formalism, stretched exponential relaxation and the frequency and temperature dependence of internal friction depending upon β parameter are obtained. For β = 1, it is shown that the relaxation is exponential which obeys Debye law, and the frequency and temperature dependence of internal peak are represented by a single Debye peak. However, for 0 < β < 1 the author shows that relaxation is given by stretched exponential form, and the internal friction deviates from single Debye peak. It is concluded that β represents the memory, aging, and coupling effects of stochastic dynamics in complex materials.

Effect of water adsorption on positron–electron annihilation kinetics in the MgO–Al2O3 ceramics in the frequency domain

Positron–electron annihilation kinetics in the MgO–Al2O3 ceramics sintered at different temperatures (1100, 1200 and 1400 °C) with the following water adsorbtion procedure has been calculated and analyzed in a frequency domain. The spectra of real (in-phase) $$\chi_{ 1} \left( \omega \right)$$ and imaginary (quadrature) $$\chi_{2} \left( \omega \right)$$ components of modulated positron–electron annihilation response have been obtained numerically from temporal kinetic characteristics using integral Fourier transform. The obtained complex spectra of positron-electron annihilation in MgO–Al2O3 ceramics in the frequency domain obey a sum of two Debye law components denying a correlation between elementary positron annihilation processes. Strong increasing of amplitude of lower frequency Debye component caused by water adsorbtion on the frequency spectra has been observed. Characteristic frequencies of Debye-type components of water-immersed samples show weaker dependencies on sintering temperature than in just-sintered ones. It is shown that position of large maxima on the frequency dependencies of imaginary part corresponds to the fastest average relaxation lifetime representing the most intensive interaction process of positrons with small cavity traps in solids.

Glassy anomalies in the heat capacity of an ordered 2-bromobenzophenone single crystal

The heat capacity $C(T)$ of a 2-bromobenzophenone (2-BrBP) single crystal measured at temperatures from 0.4 to 30 K demonstrates anomalies inherent in disordered solids: Instead of the Debye law ${C}_{\mathrm{D}}\ensuremath{\propto}{T}^{3}, C(T)$ shows a linear temperature dependence and a boson peak, i.e., peculiarities typical of solids with disorder. Computations for a pair of interacting 2-BrBP molecules revealed a few low-frequency states which are expected to actively couple to long-wave phonons. Raman scattering spectra demonstrate two strong spikes at energies close to the boson peak center at $26\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$. We relate the Raman spikes to low-energy intramolecular modes, which are the rotational oscillations of the substituted phenyl ring around the C-C link to the ketone. Thus, we have found that a completely ordered crystal made up of molecules with low-energy intramolecular modes can show low-$T$ properties, which are inherent in irregular solids such as various glasses, etc.

Frequency domain kinetic of positron–electron annihilation in the MgO–Al2O3 spinel-type ceramics

In this work, the kinetic of positron–electron annihilation in the MgO–Al2O3 spinel-type ceramics sintered at different temperatures (1100, 1200 and 1400 °C) has been calculated and analyzed in a frequency domain. The spectra of real (in-phase) and imaginary (quadrature) components of positron–electron annihilation kinetic have been obtained numerically from usual temporal characteristics using integral Fourier transform. The numerical calculations were carried out using cubic spline interpolation of the pulse characteristics of MgO–Al2O3 ceramics in time domain with following analytical calculations of integrals. The obtained spectra as real so imaginary part of MgO–Al2O3 ceramics in frequency domain almost good obey a Debye law denying correlation between elementary positron annihilation processes. Complex diagrams of frequency domain responses of as-prepared samples have a shape of semicircles with close characteristic frequencies. Some deviation on low-frequency side of the semicircles is observed confirming an availability of longer time kinetic processes. Sintering temperature dependencies of the relaxation times and characteristic frequencies of positron–electron annihilation processes have been obtained. It is shown that position of large maxima on the frequency dependencies of imaginary part corresponds to fast average relaxation lifetime representing the most intensive interaction process of positrons with small cavity traps in solids.

Continuum limit of the vibrational properties of amorphous solids

The low-frequency vibrational and low-temperature thermal properties of amorphous solids are markedly different from those of crystalline solids. This situation is counterintuitive because all solid materials are expected to behave as a homogeneous elastic body in the continuum limit, in which vibrational modes are phonons that follow the Debye law. A number of phenomenological explanations for this situation have been proposed, which assume elastic heterogeneities, soft localized vibrations, and so on. Microscopic mean-field theories have recently been developed to predict the universal non-Debye scaling law. Considering these theoretical arguments, it is absolutely necessary to directly observe the nature of the low-frequency vibrations of amorphous solids and determine the laws that such vibrations obey. Herein, we perform an extremely large-scale vibrational mode analysis of a model amorphous solid. We find that the scaling law predicted by the mean-field theory is violated at low frequency, and in the continuum limit, the vibrational modes converge to a mixture of phonon modes that follow the Debye law and soft localized modes that follow another universal non-Debye scaling law.

Thermodynamic properties of fullerite C70

A new expression for the isochoric heat capacity and the equation of state of fullerite C70 are obtained in the framework of a quantum-statistical method. Analogs of the Debye law and Dulong–Petit law for this fullerite are formulated. Fullerene C70 molecules are modeled by isotropic quantum oscillators under the assumption that their nonsphericity weakly influences the thermodynamic properties of the condensed phase. The intramolecular oscillations of carbon atoms are described using the Debye theory and the cold contribution to the free energy of fullerite is calculated using the Lennard-Jones pair potential for fullerene molecules. A comparison of the proposed theory to experiment shows good agreement.