Local Elastic Constants Research Articles

Soft dynamics of the excited state: Lambda-shaped optical spectra

Anomalous optical spectra of centres with strongly reduced local elastic constants in the excited electronic state are studied. In this case, transitions take place to the vicinity of a flat minimum of the potential energy in the configurational coordinate space. The existence of a flat minimum results in the appearance of the resonant (pseudo-local) mode of a low frequency. This leads to a strong enhancement of the low-frequency part of the phonon wing, causing a formation of the possibly modulated lambda-shaped optical spectra; the modulation comes from the Airy oscillations of the quantum accelerated motion.

Renormalization group analysis and numerical simulation of propagation and localization of acoustic waves in heterogeneous media

Propagation of acoustic waves in strongly heterogeneous elastic media is studied using renormalization group analysis and extensive numerical simulations. The heterogeneities are characterized by a broad distribution of the local elastic constants. We consider both Gaussian-white distributed elastic constants, as well as those with long-range correlations with a nondecaying power-law correlation function. The study is motivated in part by recent analysis of experimental data for the spatial distribution of the elastic moduli of rock at large length scales, which indicated that the distribution contains the same type of long-range correlations as what we consider in the present paper. The problem that we formulate and the results are, however, applicable to acoustic wave propagation in any disordered elastic material that contains the types of heterogeneities that we consider in the present paper. Using the Martin-Siggia-Rose method, we analyze the problem analytically and find that, depending on the type of disorder, the renormalization group (RG) flows exhibit a transition to a localized or extended regime in any dimension. We also carry out extensive numerical simulations of acoustic wave propagation in one-, two-, and three-dimensional systems. Both isotropic and anisotropic media (with anisotropy being due to stratified) are considered. The results for the isotropic media are consistent with the RG predictions. While the RG analysis, in its present form, does not make any prediction for the anisotropic media, the results of our numerical simulations indicate the possibility of the existence of a regime of superlocalization in which the waves' amplitudes decay as $\mathrm{exp}[\ensuremath{-}{(\ensuremath{\mid}\mathbf{x}\ensuremath{\mid}∕\ensuremath{\xi})}^{\ensuremath{\gamma}}]$, with $\ensuremath{\gamma}>1$, where $\ensuremath{\xi}$ is the localization length. However, further investigations may be necessary in order to establish the possible existence of such a localization regime.

Local Elastic Constants For Epoxy-Nanotube Composites From Molecular Dynamics Simulation

ABSTRACTA method from molecular dynamics simulation is developed for determining local elastic constants of an epoxy/nanotube composite. The local values of C11, C33, K12, and K13 elastic constants are calculated for an epoxy/nanotube composite as a function of radial distance from the nanotube. While the results possess a significant amount of statistical uncertainty resulting from both the numerical analysis and the molecular fluctuations during the simulation, the following observations can be made. If the size of the region around the nanotube is increased from shells of 1Å to 6Å in thickness, then the scatter in the data reduces enough to observe trends. All the elastic constants determined are at a minimum 20Å from the center of the nanotube. The C11, C33, and K12 follow similar trends as a function of radial distance from the nanotube. The K13 decreases greater distances from the nanotube and becomes negative which may be a symptom of the statistical averaging.

Expressions for the stress and elasticity tensors for angle-dependent potentials

The stress and elasticity tensors for interatomic potentials that depend explicitly on bond bending and dihedral angles are derived by taking strain derivatives of the free energy. The resulting expressions can be used in Monte Carlo and molecular dynamics simulations in the canonical and microcanonical ensembles. These expressions are particularly useful at low temperatures where it is difficult to obtain results using the fluctuation formula of Parrinello and Rahman [J. Chem. Phys. 76, 2662 (1982)]. Local elastic constants within heterogeneous and composite materials can also be calculated as a function of temperature using this method. As an example, the stress and elasticity tensors are derived for the second-generation reactive empirical bond-order potential. This potential energy function was used because it has been used extensively in computer simulations of hydrocarbon materials, including carbon nanotubes, and because it is one of the few potential energy functions that can model chemical reactions. To validate the accuracy of the derived expressions, the elastic constants for diamond and graphite and the Young's Modulus of a (10,10) single-wall carbon nanotube are all calculated at T = 0 K using this potential and compared with previously published data and results obtained using other potentials.

Local mechanical properties of polymeric nanocomposites

The inclusion of a nanoparticle into a polymer matrix is studied by efficient Monte Carlo simulations. The resulting structural changes in the melt and glass exhibit a strong dependence on the strength of the polymer attraction to the surface of the filler. The mechanical properties of the nanocomposite are analyzed in detail through a formalism that permits calculation of local elastic constants. The average shear and Young's modulus of the nanocomposite are higher than those of the pure polymer for neutral or attractive nanoparticles. For repulsive particles, these moduli are lower. Simulation of local properties reveals that a glassy layer is formed in the vicinity of the attractive filler, contributing to the increased strength of the composite material. In contrast, a region of negative moduli emerges around repulsive fillers, which provides a mechanism for frustration relief and a lowering of the glass transition temperature.

Local stresses and elastic constants at the atomic scale

In this paper, we examine the problem of deriving local stress measures and associated elastic constants for a region containing a fixed number of atoms that is part of a larger body. By means of an expansion of the local interatomic potential energy, we derive expressions for the local second Piola–Kirchoff stress and the associated elastic constants. We relate these to other previously derived atomic scale stress measures. Numerical calculations are presented for the case of an oversized inclusion embedded in an otherwise perfect face-centred cubic crystal.

Local surface elastic constants by resonant-ultrasound microscopy

We report a method—resonant-ultrasound microscopy—for measuring elastic-constant distribution over a solid’s surface. Applying an oscillating electric field to a rectangular-parallelepiped oscillator of langasite (La3Ga5SiO14) crystal by a surrounding solenoid coil, we generated and detected vibrations of the crystal without electrodes and without wires. Acoustic coupling of the specimen to the oscillator is only made at an antinodal vibration point on the crystal’s bottom surface. The crystal’s resonance-frequency shift reflects elastic constants of the specimen in the contacting area. Point-contact measurement permits sensitive, quantitative evaluation of a material’s local elastic constants. As an illustrating example, we measured the elastic-stiffness distribution of a Nb–Ti/Cu resin superconductive wire. We compared our measurements with both static-contact and dynamic-contact models.

Method for determining the elastic modulus at grain boundaries for polycrystalline materials

On the basis of the model of non-equilibrium grain boundary segregation induced by tensile stress, a set of kinetic equations is derived to formulate this process. These kinetic equations allow excellent simulation of the grain boundary segregation of phosphorus and sulphur observed in steels subjected to low tensile stresses. In the present paper, based on such a widely approved model, a new approach is proposed to quantify the elastic modulus at grain boundaries for polycrystalline materials. Using the observation of Misra, the grain boundary elastic modulus Egb = 2.03 × 109 Pa at 883 K for Cr-Mo-V-2.6Ni steel is obtained for the first time. This result shows excellent agreement with the local elastic constants simulated theoretically by Kluge et al., and indicates that the grain boundary elastic modulus for a polycrystalline material is much lower than the commonly assumed value.

How is the indentation modulus of bone tissue related to its macroscopic elastic response? A validation study

This work consists of the validation of a novel approach to estimate the local anisotropic elastic constants of the bone extracellular matrix using nanoindentation. For this purpose, nanoindentation on two planes of material symmetry were analyzed and the resulting longitudinal elastic moduli compared to the moduli measured with a macroscopic tensile test. A combined lathe and tensile system was designed that allows machining and testing of cylindrical microspecimens of approximately 4 mm in length and 300 μm in diameter. Three bovine specimens were tested in tension and their outer geometry and porosity assessed by synchrotron radiation microtomography. Based on the results of the traction test and the precise outer geometry, an apparent longitudinal Young's modulus was calculated. Results between 20.3 and 27.6 GPa were found that match with previously reported values for bovine compact bone. The same specimens were then characterized by nanoindentation on a transverse and longitudinal plane. A longitudinal Young's modulus for the bone matrix was then derived using the numerical scheme proposed by Swadener and Pharr and the fabric–elasticity relationship by Zysset and Curnier. Based on the matrix modulus and a power law effective volume fraction, an apparent longitudinal Young's modulus was predicted for each microspecimen. This alternative approach provided values between 19.9 and 30.0 GPa, demonstrating differences between 2% and 13% to the values provided by the initial tensile test. This study therefore raises confidence in our nanoindentation protocol of the bone extracellular matrix and supports the underlying hypotheses used to extract the anisotropic elastic constants.

Local elastic constants in thin films of an fcc crystal.

In this work we present a formalism for the calculation of the local elastic constants in inhomogeneous systems based on a method of planes. Unlike previous work, this formalism does not require the partitioning of the system into a set of finite volumes over which average elastic constants are calculated. Results for the calculation of the local elastic constants of a nearest-neighbor Lennard-Jones fcc crystal in the bulk and in a thin film are presented. The local constants are calculated at exact planes of the (001) face of the crystal. The average elastic constants of the bulk system are also computed and are consistent with the local constants. Additionally we present the local stress profiles in the thin film when a small uniaxial strain is applied. The resulting stress profile compares favorably with the stress profile predicted via the local elastic constants. The surface melting of a model for argon for which experimental and simulation data are available is also studied within the framework of this formalism.

Imaging and measurement of local mechanical material properties by atomic force acoustic microscopy

AbstractIn atomic force acoustic microscopy (AFAM) the cantilever of an atomic force microscope is vibrated at ultrasonic frequencies while a sample surface is scanned with the sensor tip contacting the sample. As a consequence, the amplitude and phase of the cantilever vibration as well as the shift of the cantilever resonance frequencies contain information about local tip–sample contact stiffness and can be used as imaging quantities. An appropriate theoretical description of the transfer of ultrasound in an atomic force microscope enables the measurement of the local mechanical material parameters of the sample surface by evaluating experimental cantilever vibration spectra. In the experiments presented here, we examine the sensitivity of the technique using silicon single crystals. Furthermore, we show that the ferroelectric domains of lead zirconate–titanate ceramics can be imaged by AFAM and that local elastic constants of the sample surface can be determined quantitatively. The lateral resolution of the technique is given by the contact area formed by the sensor tip and the sample surface, which can have a diameter of <10 nm. Copyright © 2002 John Wiley & Sons, Ltd.

Tunable local polariton modes in semiconductors

We study the local states within the polariton bandgap that arise due to deep defect centers with strong electron-phonon coupling. Electron transitions involving deep levels may result in alteration of local elastic constants. In this case, substantial reversible transformations of the impurity polariton density of states occur, which include the appearance/disappearance of the polariton impurity band, its shift and/or the modification of its shape. These changes can be induced by thermo- and photo-excitation of the localized electron states or by trapping of injected charge carriers. We develop a simple model, which is applied to the $O_P$ center in $GaP$. Further possible experimental realizations of the effect are discussed.

Nanometer-scale resolution and depth discrimination in near-field optical microscopy studies of electric-field-induced molecular reorientation dynamics

Electric-field-induced molecular reorientation dynamics in polymer-dispersed liquid crystal (PDLC) films are characterized in detail using near-field scanning optical microscopy (NSOM) methods developed previously [Mei and Higgins, J. Phys. Chem. A 102, 7558 (1998)]. In these experiments, a modulated electric field is applied between the aluminum-coated NSOM probe and an indium-tin-oxide (ITO) substrate. The field causes reorientation of the liquid crystal within the ITO-supported PDLC film. The reorientation process is observed by near-field optical means. In this paper, it is conclusively shown that under appropriate conditions the dynamics observed occur in extremely small volumes, and are substantially confined within the near-field optical regime. The volume in which the dynamics are probed may be controlled by varying the experimental parameters (i.e., field strength and modulation frequency) employed. Conclusive evidence for confinement is obtained from both theoretical arguments and experimental results. Calculations of the electric fields in a model dielectric medium show that the largest fields occur very near the NSOM probe. Experimental observation of spatial variations in the threshold (i.e., the “Frederiks transition”) for liquid crystal reorientation provide further evidence. The most direct evidence is provided by the observation of sub-diffraction-limited resolution in dynamics images of ≈ 1 μm thick samples. Spatial variations in the observed dynamics are interpreted to reflect the energetics of local liquid crystal organization, the details of the reorientation process, and also polymer/liquid-crystal interfacial interactions. Finally, important information on the local rotational viscosity and elastic force constants within individual liquid-crystal droplets is obtained.

Structure, stresses, and local dynamics in glasses

The interrelations between short range structural and elastic aspects in glasses and glass forming liquids pose important and yet unresolved questions. In this paper these relations are analyzed for mono-atomic glasses and stressed liquids with a short range repulsive-attractive pair potentials. Strong variations of the local pressure are found even in a zero temperature glass, whereas the largest values of pressure are the same in both glasses and liquids. The coordination number z(J) and the effective first peak radius depend on the local pressures J's. A linear relation was found between components of site stress tensor and the local elastic constants. A linear relation was also found between the trace of the squares of the local frequencies and the local pressures. Those relations hold for glasses at zero temperature and for liquids. We explain this by a relation between the structure and the potential terms. A structural similarity between liquids and solids is manifested by similar dependencies of the coordination number on the pressures.

Low-frequency Raman scattering in glasses: Thermal treatments and viscosity

Abstract We compare the changes with thermal treatments of the first-order Raman spectra of different silicate glasses. The low-frequcncy Raman scattering of pure silica glass is shown to have an anomalous behaviour with fictive temperature (like other macroscopic properties) compared with E glass. This behaviour is correlated with the viscosity of silica glass. It is further shown that the boson peak is not correlated quantitatively with the macroscopic sound velocity but more probably to the heterogeneous disorder of the glass through local elastic constants or a characteristic length.

Optical detection for scanning microdeformation microscopy

Scanning microdeformation microscopy is a kind of ac contact force microscopy sensitive to the variations of the local elastic constants of the investigated material. In this letter a new optical interferometer designed for detecting the small displacement of the cantilever is reported. The setup is described and some applications of the laser probe to scanning microdeformation microscopy are demonstrated.

Relation between bridging conformation and rheology of suspensions

The rheological properties of suspensions flocculated by polymer bridging are studied as a function of adsorption affinity of polymer for the particle surfaces. Polymer adsorption is irreversible under ordinary conditions, because the polymer chain may attach to the surface at many points and not be able to desorb simultaneously from all sites. When both the particle and polymer concentrations exceed some critical levels, the suspensions elastically respond to small deformation at very low frequencies. Since the flocs are considered to consist of sites (particles) connected by bonds (bridges), the elastic responses due to the three-dimensional network of unbounded flocs are analyzed in terms of site—bond percolation. With decreasing adsorption affinity of polymer chain, the fraction of loops increases at the expense of trains which are in direct contact with surface. The decrease in the fraction of segments in trains causes the adsorption—desorption process to reversibly occur by Brownian motion. As a result, the polymer bridges are constantly forming, breaking and reforming in a quiescent state. When the adsorption affinity of polymer is decreased by the addition of surfactant, the viscosity decreases at low shear rates and increases at high shear rates. Therefore, the viscosity behavior changes from shear-thinning to Newtonian flow. The suspensions in which the particles are bridged by a flexible coil show viscosity profiles consisting of low-shear-rate Newtonian flow, shear-thickening flow at moderate shear rates, and shear-thinning flow at high shear rates. Although the extension of polymer coil connecting particles in shear fields is an intrinsic mechanism, the network structure is essential for shear-thickening flow of suspensions. The elastic properties of highly deformable network are also analyzed by percolation theory. The scaling analysis shows that the critical exponent is 4.0 for network constructed by irreversible bridges and 1.8 for highly elastic network by reversible bridges. The difference in critical behavior for elastic percolation may be attributed to the difference in local elastic constants.

Investigation of the stress dependences of optical and EPR spectra for a KMgF 3:V 2+ crystal

For a V 2+ ion in a KMgF 3 crystal, by assuming that the local elastic stiffness constants c ij in the vicinity of the impurity ion are greater than those of the host due to the size-misfitting substitution, the stress dependences of optical and EPR spectra, and also the optical spectra and EPR g factor, can be reasonably explained in a unified away. It appears that the assumption is reasonable and that the optical and EPR measurements may be suitable methods of studying these local properties for the crystals containing transition-metal ion impurities.

線集束型超音波顕微鏡によるジルコニア/ステンレス鋼焼結複合材料の弾性定数の評価

The elastic constants of the sintered composites of zirconia/stainless steel were measured by a LFB acoustic microscope as a function of volume mixing ratio. Specimens from various fabrication routes were used to see the applicability of the LFB acoustic microscope to the composites having specific sintered microstructures. It has been shown that the present method enables us to measure the local elastic constants, such as Young’s modulus and Poisson’s ratio, of the sintered metal/ceramic composites with sufficient accuracy. It has also been emphasized that the LFB acoustic microscope would be one of the most viable methods to evaluate the elastic constant distribution of the recently developed functionally gradient materials, which have definite compositional gradients and the pertaining microstructural transitions.

Monte Carlo study of the shear modulus at the surface of a Lennard-Jones crystal

In this paper, we give a microscopic definition of local elastic constants. We apply this to the numerical evaluation of the shear modulus of an interface which is sharp as compared with the interaction range. The algorithm is applied to a study of the (001) face of a face-centered-cubic (fcc) Lennard-Jones crystal. The vanishing of the shear modulus gives an estimate of the melting temperature of the first layer which is well below the bulk triple point. Some theoretical aspects of surface melting are briefly discussed.