Floppy Modes Research Articles

Floppy Modes and Nonaffine Deformations in Random Fiber Networks

We study the elasticity of random fiber networks. Starting from a microscopic picture of the nonaffine deformation fields, we calculate the macroscopic elastic moduli both in a scaling theory and a self-consistent effective medium theory. By relating nonaffinity to the low-energy excitations of the network ("floppy modes"), we achieve a detailed characterization of the nonaffine deformations present in fibrous networks.

Glass transition phenomenology and flexibility: An approach using the energy landscape formalism

The thermodynamic phenomenology of the glass transition in chalcogenide glasses is studied by using rigidity theory, which treats covalent bonding as mechanical constraints. Since flexible systems have a certain number of nearly zero frequency modes (called floppy modes), these modes provide channels in the energy landscape of the glass, and as a consequence, the entropy and fragility depend upon the number of constraints, even for the supercooled melt. Using this approach, the variation of the glass transition temperature with the chemical composition can be obtained from the number of floppy modes, since low frequencies enhance in a considerable way the average quadratic displacement of atomic vibrations. The result reproduces the observed experimental variation of the glass transition temperature with chemical composition.

Variation of the glass transition temperature with rigidity and chemical composition

The effects of flexibility and chemical composition in the variation of the glass transition temperature are obtained by using the Lindemann criteria, which relates melting temperature with atomic vibrations, and rigidity theory. Using this criteria and that floppy modes produce an excess of vibrational states at low frequencies which enhance in a considerable way the average quadratic displacement, we show that the consequence is a modified glass transition temperature. This approach allows us to obtain in a simple way the empirically modified Gibbs-DiMarzio law, which has been widely used in chalcogenide glasses to fit the changes in the glass transition temperature with the chemical composition. The method predicts that the constant that appears in the law depends upon the ratio of two characteristic frequencies (or temperatures). This constant is estimated for the ${\mathrm{Se}}_{1\ensuremath{-}x\ensuremath{-}y}{({\mathrm{Ge}}_{y}{\mathrm{As}}_{1\ensuremath{-}y})}_{x}$ glass by using the experimental density of vibrational states, and the result shows a good agreement with the experimental fit from glass transition temperature variation.

The boson peak in Raman spectra of AsxS1−x glasses

The Raman spectra of AsxS1−x glasses with x < 40 at % (Z < 2.4) have been studied in a wide temperature range (20–300 K). A well resolved boson peak is observed in the low-frequency portion of the spectrum, not withstanding the appearance of floppy modes in the glasses under study. It is shown that the boson peak is characterized by two parameters: intensity and the peak position. A comparison of the intensity variation for the boson peaks indicates that the degree of disorder increases as x decreases. This effect is caused by floppy modes in the glass network and by sulfur phase separation. Studies of reduced boson peaks in the Raman spectra of AsxS1−x glasses confirms the theoretical assumption that the shape of the peaks is independent of composition and temperature.

Energy landscape and rigidity

The effects of floppy modes in the thermodynamical properties of a system are studied. From thermodynamical arguments, we deduce that floppy modes are not at zero frequency and thus a modified Debye model is used to take into account this effect. The model predicts a deviation from the Debye law at low temperatures. Then, the connection between the topography of the energy landscape, the topology of the phase space, and the rigidity of a glass is explored. As a result, we relate the number of constraints and floppy modes to the statistics of the landscape. We apply these ideas to a simple model for which we provide an approximate expression for the number of energy basins as a function of the rigidity. This helps to understand certain features of the glass transition, like the jump in the specific heat or the reversible window observed in chalcogenide glasses.

Boson peaks and floppy modes: some relations between constraint and excitationphenomenology, and interpretation, of glasses and the glass transition

We make here an effort to find commonality between the athermal constraint approach toglassiness due to Phillips and Thorpe, in which composition is the key variable, and themore conventional temperature-induced softening approaches to the glass transition atconstant composition. A starting point for our discussion is the parallel in behaviour of theboson peak, derived from the vibrational density of states, which is enhanced both byincreasing glass fictive temperature (potential energy), and by decreasing glass coordinationnumber (through a rigidity threshold), e.g. in chalcogenide glasses. We relate the potentialenergy of the glass to a topological defect concentration, and see defect formation as ameans of lifting constraints, and hence promoting flow in formally overconstrained glasses.This viewpoint is supported by observations on irradiation of glasses, in whichthe athermal introduction of defects, or lifting of barriers, may induce flow, orrelaxation/annealing. These considerations emphasize the importance of takingtemperature, and fictive (‘structural’) temperature, considerations into account inevaluating the properties of laboratory glasses for comparison with constraint theorypredictions.

Quantum corrections to classical time-correlation functions: hydrogen bonding and anharmonic floppy modes.

Several simple quantum correction factors for classical line shapes, connecting dipole autocorrelation functions to infrared spectra, are compared to exact quantum data in both the frequency and time domain. In addition, the performance of the centroid molecular dynamics approach to line shapes and time-correlation functions is compared to that of these a posteriori correction schemes. The focus is on a tunable model that is able to describe typical hydrogen bonding scenarios covering continuously phenomena from tunneling via low-barrier hydrogen bonds to centered hydrogen bonds with an emphasis on floppy modes and anharmonicities. For these classes of problems, the so-called "harmonic approximation" is found to perform best in most cases, being, however, outperformed by explicit centroid molecular dynamics calculations. In addition, a theoretical analysis of quantum correction factors is carried out within the framework of the fluctuation-dissipation theorem. It can be shown that the harmonic approximation not only restores the detailed balance condition like all other correction factors, but that it is the only one that also satisfies the fluctuation-dissipation theorem. Based on this analysis, it is proposed that quantum corrections of response functions in general should be based on the underlying Kubo-transformed correlation functions.

Low-frequency vibrational excitations in zeolite ZSM-5 and its partially crystalline derivatives

Simulations of zeolite ZSM-5 based partially amorphous materials show that for higher crystallinity the intensity of the boson peak decreases upon amorphization, whereas the opposite behavior is observed for forms with lower crystallinity. The former effect is associated with a decrease of the concentration of tenfold rings and a general lowering of symmetry by puckering of large rings. The latter behavior is related to an increasing participation of undeicoordinated centers in the relevant low-frequency motions. Booth observations can he explained in the framework of Maxwell counting of floppy modes. The modes associated with the hoson peak for these materials are found to he mainly optic in nature.

Floppy modes and relaxation in GexSe1−x glasses

Floppy modes and relaxation processes in the GexSe1−x glasses system are studied by vibrational methods. We report Raman spectral line shapes over a temperature range from 100 to 300 K. Normalized frequency shifts with temperature of the GeSe4/2 breathing mode and the Se–Se stretching mode depend on the Ge composition. The results are associated with the stiffness transition window in which we suggest a radical decrease of Sen (n>2) segments which could be regarded as the microscopic origin of floppy modes in the Se-containing network glasses.

Compressibility, kinetics, and phase transition in pressurized amorphous silica

We model the process of densification of silica glass using molecular dynamics simulation in order to resolve the current controversy regarding the existence of the first-order phase transition in this material. We propose the picture in which the structural changes start to take place in the pressure window between 3 and 5 GPa, after which significant modifications take place with the structural breakdown in the medium range. We also study microscopic processes behind temperature-induced volume decrease of pressurized glass, seen experimentally. We simulate this process and observe similar negative thermal swelling, accompanied by considerable rebonding and relaxations processes. Global nature of rebonding, resulting from the extended character of floppy modes present in silica glass, yields a large value of temperature-induced densification. The densified structure shows broadening of the rings distribution, and we identify the microscopical changes that lead to the breakdown of the medium-range structure. The interesting observation from the long annealing of pressuried glass is the large-amplitude cooperative flow of atoms, which takes place as the structure relaxes through continuous rebonding and relaxation events.

Structural relaxation and self-organization in network glasses at stiffness transition

We have investigated the changes in quasi-elastic (QE) relaxation behavior in a low-frequency range 10–80 cm −1 near stiffness transition in Ge x Se 1− x glasses using temperature-programmed Raman scattering technique. We confirm that the QE contribution originates from the fast (β) process of relaxation. Since the QE intensity depends on glass composition, the origin of the relaxational modes is traced to floppy modes. We attribute the relaxation modes mainly to the damping or jumping motions of the rotating Se n segments, which could be regarded as the microscopic origin of floppy modes in the Se-contained network glasses.

Protein unfolding: rigidity lost.

We relate the unfolding of a protein to its loss of structural stability or rigidity. Rigidity and flexibility are well defined concepts in mathematics and physics, with a body of theorems and algorithms that have been applied successfully to materials, allowing the constraints in a network to be related to its deformability. Here we simulate the weakening or dilution of the noncovalent bonds during protein unfolding, and identify the emergence of flexible regions as unfolding proceeds. The transition state is determined from the inflection point in the change in the number of independent bond-rotational degrees of freedom (floppy modes) of the protein as its mean atomic coordination decreases. The first derivative of the fraction of floppy modes as a function of mean coordination is similar to the fraction-folded curve for a protein as a function of denaturant concentration or temperature. The second derivative, a specific heat-like quantity, shows a peak around a mean coordination of <r> = 2.41 for the 26 diverse proteins we have studied. As the protein denatures, it loses rigidity at the transition state, proceeds to a state where just the initial folding core remains stable, then becomes entirely denatured or flexible. This universal behavior for proteins of diverse architecture, including monomers and oligomers, is analogous to the rigid to floppy phase transition in network glasses. This approach provides a unifying view of the phase transitions of proteins and glasses, and identifies the mean coordination as the relevant structural variable, or reaction coordinate, along the unfolding pathway.

Floppy modesin silica glass under pressure

We study the effects of pressure on the existence of floppymodes in silica glass. We find that small values of pressure donot alter the ability of silica glass to sustain floppy modes,as long as they are not accompanied by changes in the topology of the glass, and thus do not change the balance betweenthe number of constraints and degrees of freedom of the system.As increased pressure leads to the appearance of increasedcoordination numbers, this balance is changed, and the structureloses its ability to support floppy modes. We find that whileat low values of pressure floppy modes are spatially extended,as they are in uncompressed structures, high pressure leads tothe localization of low-ω (albeit non-zero) floppy modes.

Disorder in tetrahedrally bonded amorphous solids

Abstract We have recently found that the density of the low-energy excitations in amorphous silicon films and, in particular, in hydrogenated amorphous silicon (a-Si: H) films can decrease by over three orders of magnitude, to a level of complete disappearance in certain cases from that observed in all other amorphous solids. This observation breaks down the universality of these excitations in amorphous solids. Extending this study, we have measured the low-temperature internal friction of a variety of a-Si: H films, hydrogenated amorphous germanium films and amorphous carbon films, under different deposition conditions or post-annealing treatments. We find that the density of the low-energy excitations in these systems is smaller than in all other amorphous solids, and that they vary considerably between different films. We attribute this property to be a uniaue feature of their tetrahedral bonding, where the overconstrained structure restricts the tunnelling motion of atoms. We conclude that the low-energy excitations in tetrahedrally bonded amorphous solids are sensitive to the existence of local floppy modes caused by structural inhomogeneity.

Protein flexibility and dynamics using constraint theory

A new approach is presented for determining the rigid regions in proteins and the flexible joints between them. The short-range forces in proteins are modeled as constraints and we use a recently developed formalism from graph theory to analyze flexibility in the bond network. Forces included in the analysis are the covalent bond-stretching and bond-bending forces, salt bridges, and hydrogen bonds. We use a local function to associate an energy with individual hydrogen bonds, which then can be included or excluded depending on the bond strength. Colored maps of the rigid and flexible regions provide a direct visualization of where the motion of the protein can take place, consistent with these distance constraints. We also define a flexibility index that quantifies the local density of flexible or floppy modes, in terms of the dihedral angles that remain free to rotate in each flexible region. A negative flexibility index provides a measure of the density of redundant bonds in rigid regions. A new application of this approach is to simulate the maximal range of possible motions of the flexible regions by introducing Monte Carlo changes in the free dihedral angles, subject to the distance constraints. This is done using a method that maintains closure of the rings formed by covalent and hydrogen bonds in the flexible parts of the protein, and van der Waals overlaps between atoms are avoided. We use the locus of the possible motions of HIV protease as an example; movies of its motion can be seen at http://www.pa.msu.edu/∼lei. © 2001 by Elsevier Science Inc.

Dynamics of silica glass:two-level tunnelling states and low-energy floppy modes

We present the results of a computational study of the low-energydynamics of silica glass. Molecular dynamics simulation resultsshow that parts of the glass structure can undergo large cooperativereorientations of SiO4 tetrahedra. These motions involvereorientations of about 30 tetrahedra with an energy barrier ofabout 0.06 eV. We relate these motions to the presence ofdouble-well potentials which give rise to two-level tunnellingstates in the model, thereby providing the mechanism for theanomalous low-temperature thermal properties of glasses. Simulationof larger structures of silica glass shows that jump events becomemore frequent and uncorrelated with each other. In addition tostudying the flexibility of silica glass in terms of the largetetrahedral rearrangements, we also address the flexibility ofsilica glass in terms of its ability to sustain low-ωfloppy modes. The latter part of the study is supported byinelastic neutron scattering data, and we compare experimental andcalculated dynamic structure factors in the energy range 0-10 meVand scattering vector range 0-8 Å-1. By applying theanalysis of the rigid-unit-mode model as initially developed forcrystalline silicates to structures of silica glass we find thatsilica glass is surprisingly similar to its correspondingcrystalline phases in its ability to support low-ω floppymodes. The same conclusion follows from the comparison ofcalculated vibrational densities of states of silica glass and itscrystalline phases, and is borne out in the inelastic neutronscattering data.

Floppy modes and the Boson peak in crystalline and amorphous silicates: an inelastic neutron scattering study

AbstractWe present the results of an inelastic neutron scattering study of a crystalline polymorph of SiO2 (α-quartz), and a number of silicate glasses (pure silica, SiO2, and three glasses in the alkali series (K,Li)2Si2O5). We demonstrate the presence of the Boson peak in the inelastic spectra of the crystalline and amorphous materials and argue that it arises simply from the dispersion characteristics of transverse acoustic modes. Furthermore, we investigate the role of the lower-energy floppy modes in the inelastic spectra, and show that they are controlled by chemical effects.

Contribution of floppy modes to the heat capacity jump and fragility in chalcogenide glasses

The jump of the heat capacity in chalcogenide glasses during glass transition is estimated using the number of floppy modes as a free energy. A comparison with experimental data shows good agreement, without using any free parameter. This result allows a determination of the change of the glass fragility and excess thermal expansivity as a function of the average coordination number.

Orientational Disordering in CsCo(ND3)6(ClO4)2Cl2 Crystals Studied by Single Crystal Neutron Diffraction between 20 and 290 K

A member of the series M(I)M(III)(ND3)6(ClO4)2X2, CsCo(ND3)6(ClO4)2Cl2, has been examined by single crystal time-of-flight neutron diffraction at 20, Tc−10, Tc+10, and 290 K, adding to previous X-ray diffraction studies of some compounds in this series. Tc is the phase transition temperature of 250(1) K observed by DSC between closely related simple trigonal structures—twinned ordered R3 and disordered R3m, both with Z=3 and the former with a=7.210(2) Å, c=26.819 Å at 20 K. The twin populations, orientational order parameter, and deuterium disordering are much more clearly observed by neutron than by X-ray diffraction. A simple model involving both twinning and order–disorder with differing ranges of ionic orientational correlations is appropriate for this crystal at all temperatures. This type of subtle transition, sometimes involving anharmonicity, floppy modes, and substantial disorder below the transition temperature, is easily missed by diffraction studies and yet the thermal and electrical consequences of such disorder may be severe.

RIGIDITY IN GLASSES AND PROTEINS

We review recent progress in applying the theory of rigidity to glassy networks and to proteins. These three dimensional systems require a generalization of Laman's theorem, which we have used to develop a technique called the Pebble Game which allows the rigid regions (containing both isostatic and overconstrained parts) and the flexible joints between them, to be found. We show that a flexibility index, which measures the local density of floppy modes, is useful in characterizing the network. A sampling of recent results is given for network glasses, where we show how the glass structure can self-organize to produce an intermediate phase that is stress-free and contains a percolating isostatic cluster. In proteins, we show how maps of the rigid regions and flexible joints, as well as maps of the flexibility index, can help to elucidate the connection between structure and function.