Low-frequency Density Of States Research Articles

Low Frequency Vibrations and Diffusion in Disordered Polymers Bearing an Intrinsic Microporosity as Revealed by Neutron Scattering

The microscopic diffusion and the low frequency density of states (VDOS) of PIM-EA-TB(CH3) are investigated by inelastic and quasi-elastic neutron scattering where also the demethylated counterpart of PIM-EA-TB(H2) is considered. These intrinsic microporous polymers are characterized by large BET surface area values of several hundred m2/g and pore sizes between 0.5 and 2 nm. Detailed comparison is made to the archetype of polymers of intrinsic microporosity, PIM-1, and polynorbornenes also bearing a microporosity. Due to the wavelength of neutrons, the diffusion and vibrations can be addressed on microscopic length and time scales. From the inelastic neutron scattering experiments the low frequency density of states (VDOS) is estimated which shows excess contributions to the Debye-type VDOS known as Boson peak. It was found that the maximum frequency of the Boson peak decreases with increasing microporosity characterized by the BET surface area. However, besides the BET surface area, additional factors such as the backbone stiffness govern the maximum frequency of the Boson peak. Further the mean squared displacement related to microscopic motions was estimated from elastic fixed window scans. At temperatures above 175 K, the mean squared displacement PIM-EA-TB(CH3) is higher than that for the demethylated counterpart PIM-EA-TB(H2). The additional contribution found for PIM-EA-TB(CH3) is ascribed to the rotation of the methyl group in this polymer because the only difference between the two structures is that PIM-EA-TB(CH3) has methyl groups where PIM-EA-TB(H2) has none. A detailed comparison of the molecular dynamics is also made to that of PIM-1 and the microporous polynorbornene PTCNSi1. The manuscript focuses on the importance of vibrations and the localized molecular mobility characterized by the microscopic diffusion on the gas transport in polymeric separation membranes. In the frame of the random gate model localized fluctuations can open or close bottlenecks between pores to enable the diffusion of gas molecules.

Low frequency vibrational density of state of highly permeable super glassy polynorbornenes - the Boson peak.

Inelastic incoherent neutron time-of-flight scattering was employed to measure the low frequency density of states for a series of addition polynorbornenes with bulky side groups. The rigid main chain in combination with the bulky side groups give rise to a microporosity of these polymers in the solid state. The microporosity characterized by the BET surfaces area varies systematically in the considered series. Such materials have some possible application as active separation layer in gas separation membranes. All investigated materials show excess contributions to the Debye type density of states characteristic for glasses known as Boson peak. The maximum position of the Boson peak shifts to lower frequency values with increasing microporosity. Data for PIM-1 and Matrimid included for comparison are in good agreement to this dependency. This result supports the sound wave interpretation of the Boson peak.

Disorder and excess modes in hard-sphere colloidal systems

The anomalous thermodynamic properties of glasses remain incompletely understood, notably the anomalous peak in the heat capacity at low temperatures; it is believed to be due to an excess of low-frequency vibrational modes and a manifestation of the structural disorder in these systems. We study the thermodynamics and vibrational dynamics of colloidal glasses and (defected) crystals. The experimental determination of the vibrational density of states allows us to directly observe a strong enhancement of low-frequency modes. Using a novel method (Zargar R. et al., Phys. Rev. Lett. 110 (2013) 258301) to determine the free energy, we also determine the entropy and the specific heat experimentally. It follows that the emergence of the excess modes and high values of the specific heat are directly related and are specific to the glass: even for solids containing a very large amount of defects, both the low-frequency density of states and the specific heat are significantly smaller than for the glass.

Vibrational behavior of metal nanowires under tensile stress

We have investigated the vibrational density of states (VDOS) of a thin Cu nanowire with $<100>$ axial orientation and considered the effect of axial strain. The VDOS are calculated using a real space Green's function approach with the force constant matrices extracted from interaction potential based on the embedded atom method. Results for the vibrational density of states of a strain-free nanowire show quite distinctive characteristics compared to that of a bulk atom, the most striking feature of which is the existence of high frequency modes above the top of the bulk spectrum. We further predict that the low frequency characteristics of the VDOS reveal the quasi-1 dimensional (Q1D) behavior only when the wire is extremely thin. Through decomposition of VDOS into local atoms we also exhibit that while the anomalous increase in low frequency density of states is mainly due to the corner atoms, the enhancement in high frequency modes is primarily moderated by core atoms. We, additionally, find that while the high frequency band above the top of the bulk phonon shifts to higher frequencies, the characteristics at low frequencies remains almost the same upon stretching the nanowire along the axial direction.

Vibrational Density of States of Hydration Water at Biomolecular Sites: Hydrophobicity Promotes Low Density Amorphous Ice Behavior

Inelastic neutron scattering experiments and molecular dynamics simulations have been used to investigate the low frequency modes, in the region between 0 and 100 meV, of hydration water in selected hydrophilic and hydrophobic biomolecules. The results show changes in the plasticity of the hydrogen-bond network of hydration water molecules depending on the biomolecular site. At 200 K, the measured low frequency density of states of hydration water molecules of hydrophilic peptides is remarkably similar to that of high density amorphous ice, whereas, for hydrophobic biomolecules, it is comparable to that of low density amorphous ice behavior. In both hydrophilic and hydrophobic biomolecules, the high frequency modes show a blue shift of the libration mode as compared to the room temperature data. These results can be related to the density of water molecules around the biological interface, suggesting that the apparent local density of water is larger in a hydrophilic environment.

Ioffe–Regel’ crossover and boson peaks in disordered solid solutions and similar anomalies in heterogeneous crystalline structures

Low-frequency features of the phonon spectra of disordered solid solutions and heterogeneous crystalline structures are analyzed at the microscopic level. It is shown that boson-peak type excitations can arise in disordered solid solutions whose sites have only translational degrees of freedom. Thus it is established that such excitations appear mainly because of the additional positional dispersion of sound waves which is due to the disordering. The influence of boson-peak excitations on the low-temperature specific heat is investigated. It is found that in a number of cases the specific heat is more sensitive to excitations of this kind than the low-frequency density of states is. It is shown that anomalies similar to Ioffe–Regel’ crossover and boson peaks can also arise in disordered heterogeneous crystalline structures with a complicated lattice.

Relative pair dynamics in simple supercooled liquids: longitudinal contributions.

The pair dynamics of simulated argon samples is investigated at the melting (85 K), supercooled (55 K), and quenched (20 K) liquid states, and in the crystal (20 K) state. Tagged pairs, initially lying in a given shell, were divided into incoming and outgoing groups and followed along simulated trajectories. Over them, specific correlation functions deltaB(r(0);t), involving the pair separation vector projected along its initial value r(0) (longitudinal dynamics), have been evaluated. More or less pronounced oscillations are detected according to the temperature of the thermodynamic states (and, obviously, their solid or liquid nature); for each state, they depend on the initial pair distance r(0), too. The oscillations vanish after few picoseconds (fast dynamics) in the case of crystal, whereas in the supercooled liquid they decay towards a plateau, whose height increases with the temperature. It is shown that the power spectrum of deltaB(r(0);t) practically yields the same density of states (DOS) produced by the pair velocity correlation function. The deltaB(r(0);t) functions obtained from the argon crystal at 20 K produce DOS curves dominated by two main frequency contributions, at about 40 and 60 cm(-1) (Einstein and Debye frequency, respectively). Their shape is quite well reproduced by damped harmonic oscillator-like (DHO) functions vibrating at that frequencies. In liquid states, the deltaB(r(0);t) plateau, that forms after the fast DHO dynamics, accounts for the system diffusivity. The relaxation towards the plateau is modeled by an exponential function whose decay time is comparable with the average vibration period. Evidence that the liquid states conserve a certain memory of the vibrational modes of the crystal is obtained. In these states, the DHO functions at the Einstein and Debye frequencies plus an exponential function cannot reproduce the deltaB(r(0);t) shape. A pronounced shoulder, that forms around 0.5 ps, requires the contribution of a third DHO. In the DOS, it yields a band centered below 20 cm(-1) that produces low frequency DOS excess in comparison with the DOS of the crystal. This contribution is present in liquid and supercooled high temperature states and survives near the temperature of the glass transition whereas the diffusion practically vanishes.

The low-frequency density of states for amorphous and crystalline ices

Incoherent inelastic neutron scattering studies have been performed on two amorphous polymorphs and two crystalline polymorphs of ice, including high-density amorphous (hda) ice. Subsequent annealing cycles allowed data to be collected from low-density amorphous ice (lda), and crystalline ices Ic and Ih. Data were collected between 5 and 80 K for hda, and between 5 and 120 K for the other phases. The low-frequency (ν<1 THz) densities of states of the four phases are compared. The data illustrate clearly an excess number of modes in the hda density of states at 5 K centered at 0.65 THz. The magnitude of the excess g(ν) is considerably reduced upon heating the sample from 5 to 20 K. Above 60 K the density of states of hda was found to exhibit predominantly a Debye behavior, g(ν)∝Aν2, as ν0 THz. No such excess modes were observed in lda or the two crystalline phases studied.

Nature of the Boson peak of silica glasses from hyper-Raman scattering

Hyper-Raman scattering from the Boson peak has been performed in vitreous silica and in a silica-based lead glass. A comparison with Raman scattering and infrared measurements suggests that in v-SiO 2 the low- ω excitations are associated with rotational motions of SiO 4 tetrahedra. These modes dominate the vibrational density of states in that frequency range. In the lead glass, despite a strong modification of the SiO 4 network as compared to pure silica, the results indicate that the low-frequency density of states might also be dominated by hyper-Raman active modes.

Diffusion with critically correlated traps and the slow relaxation of the longest-wavelength mode.

We study diffusion on a substrate with permanent traps distributed with critical positional correlation, modeled by their placement on the perimeters of a critical percolation cluster. We perform a numerical analysis of the vibrational density of states and the largest eigenvalue of the equivalent scalar elasticity problem using the method of Arnoldi and Saad. We show that the critical trap correlation increases the exponent appearing in the stretched exponential behavior of the low frequency density of states by approximately a factor of two as compared to the case of no correlations. A finite size scaling hypothesis of the largest eigenvalue is proposed and its relation to the density of states is given. The numerical analysis of this scaling postulate leads to the estimation of the stretch exponent in good agreement with the density of states result.

The low frequency density of states and vibrational population dynamics of polyatomic molecules in liquids

Instantaneous normal mode calculations of the low frequency solvent modes of carbon tetrachloride (CCl4) and chloroform (CHCl3), and experiments on the vibrational population dynamics of the T1u CO stretching mode (∼1980 cm−1) of tungsten hexacarbonyl in CCl4 and CHCl3 are used to understand factors affecting the temperature dependence of the vibrational lifetime. Picosecond infrared pump–probe experiments measuring the vibrational lifetime of the T1u mode from the melting points to the boiling points of the two solvents show a dramatic solvent dependence. In CCl4, the vibrational lifetime decreases as the temperature is increased; however, in CHCl3, the vibrational lifetime actually becomes longer as the temperature is increased. The change in thermal occupation numbers of the modes in the solute/solvent systems cannot account for this difference. Changes in the density of states of the instantaneous normal modes and changes in the magnitude of the anharmonic coupling matrix elements are considered. The calculated differences in the temperature dependences of the densities of states appear too small to account for the observed difference in trends of the temperature dependent lifetimes. This suggests that the temperature dependence of the liquid density causes significant changes in the magnitude of the anharmonic coupling matrix elements responsible for vibrational relaxation.

Vibrational properties of hierarchical systems

The vibrational properties of one-dimensional hierarchical systems are investigated and results are obtained for both their eigenvalues and eigenvectors. Two cases are considered, the first one with a hierarchy of spring constants and the latter with a hierarchy in the masses. In both cases the eigenspectrum is found to be a zero-measure, two-scale Cantor set with a fractal dimension between 0 and 1. The scaling properties of the spectra are calculated using renormalization group techniques and are verified by extensive numerical work. The low-frequency density of states and low-temperature specific heat are calculated and a singularity is found in the scaling behavior. The eigenvectors are found to be either extended or critical and self-similar. A transfer matrix formalism is introduced to calculate the scaling properties of the envelope of the critical eigenvectors. Furthermore, a connection is established between the hierarchical vibration and diffusion problems, as well as to the same problems in random systems, thus showing the universality of the observed features.

Role of the Coulomb interaction in the low-frequency density of states of DNA double helices.

The complete vibrational frequency spectrum of several DNA double-helical oligomers is calculated using established pair potentials. Various cutoff values are used for the range of the Coulomb interactions. At very low frequency the integrated density of states shows a noninteger exponent with values ranging from 0.75 to 1.55, depending on the cutoff value for the Coulomb interactions. We conclude that the cumulative densities of states in those molecules depend more on competing interactions than on various proposed universal laws.

Density of states and specific heat of elastic vibrations in layer structures

On the basis of a determination of normal modes in materials consisting of periodic arrangements of macroscopic layers (of period d), the low frequency density of states and the corresponding low temperature specific heat were calculated numerically. The average temperature dependence of the latter changes in the vicinity of a characteristic temperature T 0 (proportional to 1/d), from a low temperature ( α 0T 3)- to a higher temperature ( αT 3+ βT 2)- law. Depending on the material parameters, β may be positive (especially if Stonely waves are present) or negative. The coefficients α and α 0 can differ by a factor larger than 2. Characteristics of thick and thin layers and the implications of the results on the interpretation of experimental data are discussed.

Memory function parametrization in the GLE-Ghost atom formalism: A microscopic approach

The memory function used in the GLE approach to solid atom motion is modeled as a generalization of the position autocorrelation function for a multidimensional Brownian oscillator. The exact microscopic memory function and the assumed form are required to agree in two limits: at short times and for the low frequency density of states. This suffices to specify all the parameters, including those which control the memory of correlated motion between primary zone atoms as mediated by the rest of the solid. Determination of these parameters utilizes as the fundamental input the frequency matrix for the solid. An example is presented for various fcc solid surfaces assuming Lennard-Jones (12, 6) forces between the solid atoms. The approach is capable, in principle, of treating such interesting structures as high Miller index surfaces, molecular solids and adsorbed species while retaining only a small number of primary zone atoms.

The specific heat of bulk amorphous arsenic

The specific heat of bulk amorphous arsenic deposited from the vapor at 415°K has been measured between 1.8 and 20°K. A comparison with compacted amorphous As films deposited at room temperature reveals appreciable differences in the form of C/T 3. The results indicate a relative increase in the density of low frequency vibrational modes in the bulk amorphous As. The specific heat measurements support x-ray and Raman scattering studies that suggest variations in the local structure of amorphous As prepared under different deposition conditions. A comparison with neutron scattering measurements in bulk amorphous As also indicates an underestimate of the low frequency density of states.

Mössbauer effect studies on 2-MeV proton irradiatedNb3Sn

Temperature-dependent recoil-free fraction M\ossbauer studies on virigin, 1.65-MeV $^{1}\mathrm{H}$- and 2.0-MeV $^{1}\mathrm{H}$- irradiated, and 185-Mrad $\ensuremath{\gamma}$-irradiated ${\mathrm{Nb}}_{3}$Sn have been carried out in the range $78\ensuremath{\le}T\ensuremath{\le}320$ K, using the 23.8-keV radiation of $^{119}\mathrm{Sn}$. These data permit the calculation of an effective lattice temperature, ${\ensuremath{\Theta}}_{M}$, as probed by the motion of the tin atom. The data indicate that a proton dose rate of \ensuremath{\sim}1.2 \ifmmode\times\else\texttimes\fi{} ${10}^{19}$ charges ${\mathrm{cm}}^{\ensuremath{-}2}$ lowers ${\ensuremath{\Theta}}_{M}$ by \ensuremath{\sim}50\ifmmode^\circ\else\textdegree\fi{} owing to the creation of a statistical distribution of site defects which increase the low-frequency density of states of the phonon spectru. These site defects can be annealed out at \ensuremath{\sim}750\ifmmode^\circ\else\textdegree\fi{}C. ${\mathrm{Nb}}_{3}$Sn is exceptionally stable to $\ensuremath{\gamma}$ radiation damage effects, a dose of 185 Mrad incident on a foil sample apparently has no effect on ${\ensuremath{\Theta}}_{M}$ as probed by the tin atom.