Total Mean Square Displacement Research Articles

Change in the Phase Composition and Lattice Parameters of the Solid Solution Based on α-Ti in the Surface Layers of the Ti–6Al–4V Alloy Subjected to Electron-Beam Treatment

The microstructure, phase composition, and lattice parameters of the α-Ti- based solid solution in the Ti–6Al–4V alloy treated by pulsed and continuous electron beams with the energy density of 18–24 and 450 J/cm2, respectively, have been investigated using the methods of X-ray diffraction analysis and transmission and scanning electron microscopy. In the initial state, the two-phase (α + β) alloy had a polycrystalline structure with the equiaxed α-phase grains and β-phase grains located at the junctions or along the boundaries of the α-phase grains. After the electron-beam treatment, α' martensite with a lamellar structure is formed in the molten surface layer, which then experiences an α' → α + α'' + β phase transformation. In the α phase, the lamellar structure inherited from the α' martensite is retained; the β phase is located along the boundaries of lamellar grains of the α phase; the α'' phase is located both in the β phase and inside the lamellar grains of the α phase. It has been revealed that the greater the total volume fraction of the β and α'' phases, the greater the lattice parameters of α-Ti and their axial ratio c/a, and the less the total mean-square displacements of atoms in the 101 direction in the α-Ti phase. The decrease in the total mean-square displacements in the α-Ti phase is due to the diffusion of the vanadium atoms into the β phase. With an increase in the energy density of the electron beam and with a decrease in the rate of cooling of the molten layer, the total volume fraction of the β and α'' phases increases and reaches 6%.

Effect of Aluminum Concentration on the Lattice Parameters and Mean-Square Displacements of Atoms in Cu–Al and Ti–6Al–4V Alloys

It is shown that the fcc lattice parameter and the total mean-square displacements of atoms increase when copper is doped with aluminum. Parameters а and с of the hcp α-Ti lattice decrease when titanium is doped with aluminum and vanadium. This effect is attributed to the size factor of radii of alloy elements. The lattice parameter of Ti–6Al–4V alloy samples subjected to electron-beam processing may vary either way compared to α-Ti prior to processing. It is demonstrated that the observed variation of the lattice parameter is induced by the redistribution of aluminum and silicon in surface layers. The mean-square displacement of atoms and the lattice parameters in α-Ti and the Ti–6Al–4V alloy are directly proportional.

Molecular dynamics simulation of cross-linked epoxy resin and its interaction energy with graphene under two typical force fields

A model of cross-linked epoxy system composed of bisphenol-A resin, 2,3,6-tetrahydro-3-methylphthalic anhydride curing agent, and 2,4,6-tris(dimethylaminomethyl)phenol accelerator was established and molecular dynamics simulations were performed to calculate the properties of the epoxy and its composites. The results show that the mean square displacement (MSD) and glass transition temperature (Tg) calculated by Dreiding force field are always lower than that by PCFF force field, and the simulation results of Dreiding force field are better consistent with experiments. With the increasing simulation size, total MSD increases while Tg decreases slightly. The simulated systems with DGEBA more than 12 have Tg values similar to experiments. The molecular motion of epoxy system is also influenced by the cross-linking degree, and the presence of uncross-linked particles increases the total MSD. In the graphene/epoxy composites, interaction energy between modified monolayer graphene (MMG) sheets is larger than that between epoxy and graphene, indicating that MMG sheets tend to agglomerate when mixing with epoxy. Functionalization of graphene can reduce interaction energy between MMG sheets and increase that between epoxy resin and graphene, which is beneficial to the dispersion of graphene.

Resolution Effects on the Mean Square Displacement as Obtained by the Self-Distribution-Function Procedure

In the present contribution, a procedure for molecular motion characterization based on the evaluation of the Mean Square Displacement (MSD), through the Self-Distribution Function (SDF), is presented. It is shown how MSD, which represents an important observable for the characterization of dynamical properties, can be decomposed into different partial contributions associated to system dynamical processes within a specific spatial scale. It is also shown how the SDF procedure allows us to evaluate both total MSD and partial MSDs through total and partial SDFs. As a result, total MSD is the weighed sum of partial MSDs in which the weights are obtained by the fitting procedure of measured Elastic Incoherent Neutron Scattering (EINS) intensity. We apply SDF procedure to data collected,by IN13, IN10 and IN4 spectrometers (Institute Laue Langevin), on aqueous mixtures of two homologous disaccharides (sucrose and trehalose) and on dry and hydrated (H2O and D2O) lysozyme with and without disaccharides. It emerges that the hydrogen bond imposed network of the water-trehalose mixture appears to be stronger with respect to that of the water-sucrose mixture. This result can justify the higher bioprotectant effectiveness of trehalose. Furthermore, it emerges that partial MSDs of sucrose and trehalose are equivalent in the low Q domain (0÷1.7) Å−1 whereas they are different in the high Q domain (1.7÷4) Å−1. This suggests that the higher structure sensitivity of sucrose should be related to the small spatial observation windows. Moreover, the role of the instrumental resolution in EINS is considered. The nature of the dynamical transition is highlighted and it is shown that it occurs when the system relaxation time becomes shorter than the instrumental energy time. Finally, the bioprotectants effect on protein dynamics and the amplitude of vibrations in lysozyme are presented.

Motion characterization by self-distribution–function procedure

In the present paper a procedure for the biomolecular motion characterization based on the evaluation of the Mean Square Displacement (MSD), through the Self Distribution Function (SDF), is presented. In particular it will be shown how the MSD, which represents a good observable for the characterization of the dynamical properties in disordered systems, can be decomposed into partial contributions associated to the system dynamical processes within a specific spatial scale. It will be shown how the SDF procedure allows to evaluate both the total MSD and the partial MSDs through the total SFD and the partial SDFs. As a result, the total MSD is the weighed sum of the partial MSD contributions in which the weights are obtained by the fitting procedure of measured EINS intensity data. We apply the SDF procedure at EINS data collected, by the IN13 backscattering spectrometer at the Institute Laue-Langevin, Grenoble, on aqueous mixtures of two homologous disaccharides (sucrose and trehalose) and on dry myoglobin in trehalose environment. It emerges that the hydrogen bond imposed network of the water–trehalose mixture appears to be stronger with respect to that of the water–sucrose mixture and this result can justify the highest bioprotectant effectiveness of trehalose in comparison with sucrose. Furthermore it emerges that, the partial MSD behaviours of sucrose and trehalose are equivalent in the low Q domain (0–1.7) Å−1 whereas they are different in the high Q domain (1.7–4) Å-1. This circumstance suggests that the higher structure sensitivity of sucrose in respect to trehalose should be related to the small spatial observation windows.

Mean square displacement evaluation by elastic neutron scattering self-distribution function

In the present work an operational recipe for the mean square displacement (MSD) determination, highlighting the connection between elastic incoherent neutron scattering (EINS) intensity profiles and the associated self-distribution function, is presented. The determination of the thermal behavior of the total MSD and of its partial contributions is tested on EINS data collected by the backscattering spectrometer IN13 (ILL, Grenoble) on a model system such as PolyEthylene Glycol with a mean molecular weight of 400 Dalton (PEG 400).

Elastic neutron scattering study of proton dynamics in glycerol

A recent neutron scattering investigation on lysozyme embedded in glycerol–water mixtures has shown that solvent dynamics is crucial in determining the dynamical properties of the biomolecule itself (Biophys. J. 83 (2002) 1157). To better understand the role played by solvent, we have performed an elastic incoherent neutron scattering (INS) experiment as a function of the temperature on pure glycerol. To directly compare the dynamics of the solvated protein and that of the pure solvent, we settled the same experimental conditions and applied the same data analysis procedure as in Paciaroni et al. (Biophys. J. 83 (2002) 1157). By using a double-well model and taking into account for the global molecular diffusion, we exploited the measured intensity to estimate the mean square displacements (MSD) of glycerol hydrogen atoms. We found that the total MSD deviate from the low-temperature vibrational harmonic trend at approximately T=235 K, consistent with the value of the critical temperature reported in literature (Phys. Rev. E 55 (1997) 3183). The present investigation suggests that the internal dynamics of glycerol molecules, i.e. the vibrations and reorientations of hydrogen atoms relative to the molecular centre of mass (c.o.m.), can be put in relationship with the protein dynamics.

Possible origin of the non-linear long-term autocorrelations within the Gaussian regime

In this work we extend the recently considered toy model of Weierstrass or Lévy walks with varying velocity of the walker (Quantitative Finance 3 (2003) 201; Chem. Phys. 284 (2002) 481; Comp. Phys. Comm. 147 (2002) 565; Phys. A 264 (1999) 84; Phys. A 264 (1999) 107) by introducing a more realistic possibility that the walk can be occasionally intermitted by its momentary localization; the localizations themselves are again described by the Weierstrass or Lévy process. The direct empirical motivation for developing this combined model is, for example, the dynamics of financial high-frequency time series or hydrological and even meteorological ones where variations of the index are randomly intermitted by flat intervals of different length exhibiting no changes in the activity of the system. This combined Weierstrass walks was developed in the framework of the non-separable generalized continuous-time random walk formalism developed very recently (Lecture Notes Comput. Sci. 2657 (2003) 407; Eur. Phys. J. B 33 (2003) 495). This non-Markovian two-state (walking-localization) model makes possible to cover by the unified treatment a broad band of known up to now types of non-biased diffusion from the dispersive one over the normal, enhanced, ballistic, and hyperdiffusion up to the Richardson law of diffusion which defines here a part of the borderline which separates the latter from the ‘Lévy ocean’ where the total mean-square displacement of the walker diverges. We observed that anomalous diffusion is characterized here by three fractional exponents: one (temporal) characterizing the localized state and two (temporal and spatial) characterizing the walking one. By considering successive dynamic (even) exponents we constructed a series of different diffusion phase diagrams on the plane defined by the temporal and spatial (partial) fractional (dynamic) exponents characterizing the walking state. To adapt the model to the description of empirical data (the discrete time series), which are collected with a discrete time step, we used in the continuous-time series produced by the model a discretization procedure. We observed that such a procedure generates, in general, long-range non-linear autocorrelations even in the Gaussian regime, which appear to be similar to those observed, e.g., in the financial time series (Phys. A 287 (2000) 396; Phys. A 299 (2001) 1; Phys. A 299 (2001) 16; Phys. A 299 (2001) 16), although single steps of the walker within continuous time are, by definition, uncorrelated. This suggests a suprising origin of long-range non-linear autocorrelations alternative to the one proposed very recently (cf. Mosaliver et al. (Phys. Rev. E 67 (2003) 021112) and refs. therein) although both approaches involve related variants of the well-known continuous-time random walk formalism applied yet in many different branches of knowledge (Phys. Rep. 158 (1987) 263; Phys. Rep. 195 (1990) 127; in: A. Bunde, S. Havlin (Eds.), Fractals in Science, Springer, Berlin, 1995, p. 1).

Study of the non-linear autocorrelations within the Gaussian regime

In this work we extend the recently considered toy model of Weierstrass or Levy walks with varying velocity of the walker [1] by introducing a more realistic possibility that the walk can be occasionally intermitted by its momentary localization; the localizations themselves are again described by the Weierstrass or Levy process. The direct empirical motivation for developing this combined model is, for example, the dynamics of financial high-frequency time series or hydrological and even meteorological ones where variations of the index are randomly intermitted by flat intervals of different length exhibiting no changes in the activity of the system. This combined Weierstrass walks was developed in the framework of the non-separable generalized continuous-time random walk (GCTRW) formalism developed recently [2]. This approach makes it possible to study by stochastic simulations the whole spatial-temporal range while analytically we can study only the initial, pre-asymptotic and asymptotic regions (but not the intermediate one). Our approach is possible since the Weierstrass walks is a geometric superposition of regular random walks each of which can be simply treated by stochastic simulations. This non-Markovian two-state (walking-localization) model makes possible to cover by the unified treatment a broad band of known up to now types of non-biased diffusion from the dispersive one over the normal, enhanced, ballistic, and hyperdiffusion up to the Richardson law of diffusion which defines here a part of the borderline which separates the latter from the `Levy ocean' where the total mean-square displacement of the walker diverges. We observed that anomalous diffusion is characterized here by three fractional exponents: temporal one characterizing the localized state and two, temporal and spatial ones, characterizing the walking state. By considering successive dynamic (even) exponents we constructed a series of different diffusion phase diagrams on the plane defined by the spatial and temporal fractional dimensions of the walking state. To adapt the model to the description of empirical data (or discrete time series) which are collected with a discrete time-step we used in the continuous-time series produced by the model a discretization procedure. We observed that such a procedure generates, in general, long-range non-linear autocorrelations even in the Gaussian regime, which appear to be similar to those observed, e.g., in the financial time series [3–6], although single steps of the walker within continuous time are, by definition, uncorrelated. This suggests a surprising explanation alternative to the one proposed very recently (cf. [7] and Refs. therein) although both approaches involve related variants of the well-known CTRW formalism applied yet in many different branches of knowledge [8–10].

Vibrations of H8Si8O12, D8Si8O12, and H10Si10O15 As Determined by INS, IR, and Raman Experiments

A detailed study of the vibrational structure of the silasesquioxanes H8Si8O12 and H10Si10O15 based on INS, IR, and Raman spectra and on a normal-coordinate analysis is reported. The inelastic neutron scattering (INS) spectrum of crystalline H8Si8O12 is in good agreement with the optical data and allowed the assignment of optically forbidden transitions. The previously published force field, determined from IR and Raman data of H8Si8O12 and D8Si8O12, provided a good fit to the INS frequencies and intensities except in the O−Si−H bending region, which is sensitive to intermolecular interactions. The INS spectrum, a comparison of the Raman spectra of dissolved and solid H8Si8O12, and published diffraction data were used to analyse the influence of crystal packing. A modified force field was developed which is adapted to crystalline H8Si8O12 and distinguishes between O−Si−H bending force constants for the axial and the equatorial hydrogen atoms. Excellent agreement with all spectra was thus obtained. This force field was also applied to the INS spectra of H10Si10O15 and D8Si8O12. Based on a normal-coordinate analysis of the INS data, the total mean-square displacements (msd) of the hydrogen atoms of H8Si8O12 were extracted as well as limits for the total msd of the silicon atoms. The msd's and the neutron diffraction values are in good agreement, showing that no or very little static disorder is present in crystalline H8Si8O12. Based on the msd's, the lowest internal torsional frequency was estimated to be 41 ± 7 cm-1.

Green's function and atomic mean-square displacement: Phonon shift and width contributions

Abstract We present a derivation of the finite-temperature expressions for the lowest-order anharmonic (λ2) contributions to the atomic mean-square displacement (MSD) from the Green's function (GF) method and show that there is no contribution to MSD from the polarization mixing in the self-energy of the GF. These contributions arise from the cubic- and quartic-phonon shifts and the cubic-phonon width for a phonon mode qj where q is the wave-vector and j is the polarization index. Since the present MSD expressions are valid for all temperatures they can be used in numerical evaluation of the λ2 contributions for Ne and other quantum crystals where the high-temperature expressions are not valid. In the high-temperature limit the total MSD from the cubic shift and width agrees with the cubic contribution to MSD derived in the classical limit by Maradudin and Flinn (MF) and the cubic MSD derived in the ω → 0 limit of the self-energy of the GF by Shukla and Hubschle (SH). The quartic contribution to MSD in th...

Static mean-square displacement as an indicator of the crystal-to-amorphous transformation

Catastrophic amorphization of crystalline alloys supersaturated beyond a critical concentration has been postulated by Fecht, Desré and Johnson [Philos. Mag. B59 (1989) 577]. NbPd is a good candidate system for testing their ideas because of its plunging T 0 line on the Nb-rich side of the phase diagram. Signs of instability in bcc NbPd solutions were looked for by elastic neutron diffraction measurements of atomic mean-square displacements (MSDs) between 12 K and room temperature. Thermally induced MSD values have been estimated from low-temperature heat capacity measurements and subtracted from the total MSD. The remaining static MSD increases rapidly with Pd concentration and is much larger than expected from the size mismatch of Nb and Pd atoms. At 42 at.% Pd, the rms static disorder reaches approximately half the value at which the Lindemann criterion predicts that the lattice should melt. Supersaturation also causes the Debye temperature to decrease, suggesting that an elastic modulus softens. In light of these results, the usefulness of static atomic disorder as a measure of the stability of a crystal against amorphization is discussed.

Volume-dependence of the melting temperature with Debye's model for Si and Ge

Lindeman's criterion for melting of covalent crystals is formulated using Debye's model. The contribution of the acoustical, especially transverse-like acoustical phonon modes to the total mean-square displacement is predominant at higher temperatures and essential to the melting scheme. Using the volume-dependence of the transverse-like Grüneisen constant at higher temperatures, the volume effect on the melting temperature of Si and Ge is studied by Lindeman's melting law. The obtained melting curve decreases as function of the compressed volume, and is qualitatively in agreement with the observed tendency for Si and Ge.

Measurement and analysis of the response field of turbulent boundary layer excited panels

Reported here are experimental results 0f the wall pressure covariance on a rigid wall, the dynamic response, and acoustic radiation characteristics of rigidly clamped panels excited by turbulent flow. Statistical techniques are employed. A duct 0f 7·3 × 3·5 in. crosssection was used, with flow speeds up t0 700 ft/sec. Two functional representations 0f the wall pressure covariance in the broad frequency band have been obtained. One representation uses a probability density distribution which fits the experimental cross-correlation distribution but is of limited range of frequency components. The other covariance is constructed from the auto-correlation obtained from the power spectral density with temporal decay in a moving frame of reference, corresponding to a broader range 0f frequency components. The panels tested were 7 × 12 in. made 0f aluminium flush mounted in the wall of a turbulent flow duct. Panel thicknesses were 0·020 0·040 0·060 and 0·080 in. It is found that the modal mean-square displacement, along the flow direction, varies with Mach number and panel thickness. The higher order modes show the effect 0f aerodynamic coincidence corresponding to the maximum displacement amplitude. A similar maximum is indicated by the bending stresses. The broad-band longitudinal space-time correlation 0f the displacement 0f the 0·040 in. panel clearly shows pronounced oblique ridges and valleys which are interpreted as moving waves having the same phase velocity as the convected wall pressure fluctuations. The correlation 0f the 0·080 in. panel shows a more irregular wave pattern with n0 evidence of coincidence. This difference in response between the thickest and the thinnest panel is also manifested in the total mean-square displacement, the total acoustic power radiated, and the modal bending stresses; indicating a transition in power laws for conditions above and below coincidence. The degree of coupling between the structural field and the acoustic field is discussed. It is shown that the modal acoustic impedance is a measurable quantity which increases with modal number, and less significantly with Mach number.