Thermal Density Fluctuations Research Articles

Thermal density fluctuations in polymer glasses

Journal of Polymer Science: Polymer Physics EditionVolume 19, Issue 8 p. 1277-1279 Note Thermal density fluctuations in polymer glasses Richard E. Robertson, Richard E. Robertson Engineering and Research Staff, Metallurgy Department, Ford Motor Company, Dearborn, Michigan 48121Search for more papers by this author Richard E. Robertson, Richard E. Robertson Engineering and Research Staff, Metallurgy Department, Ford Motor Company, Dearborn, Michigan 48121Search for more papers by this author First published: August 1981 https://doi.org/10.1002/pol.1981.180190812Citations: 1AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Citing Literature Volume19, Issue8August 1981Pages 1277-1279 RelatedInformation

Fluctuations in the Ion-Cyclotron-Frequency Range: Possible Measurement ofTeTiin Tokamaks

The thermal density fluctuations in a typical tokamak plasma, for a wave vector perpendicular to the magnetic field and for a wavelength of the order of the ion gyroradius, are shown to be maximum at the frequency of the zero-group-velocity ion cyclotron waves. This frequency is sensitive to the ion- to electron-temperature ratio and the spectrum remains peaked despite a poor scattering-vector definition and destabilizing effects such as presence of a current. Possible applications to tokamak diagnosis are emphasized.

SAXS study of the structure of glasses containing no modifying oxides

The method of X-ray scattering at small angles (SAXS) was used to study the structure of onecomponent glasses — vitreous B 2O 3, GeO 2 and quartz glasses as well as two-component borosilicate and borogermanate glasses. No submicroheterogeneous structure is present in onecomponent glasses and SAXS intensity is due to only thermal density fluctuations. In case of vitreous B 2O 3 no process of structure relaxation was observed at temperatures above and below T g. At temperatures below T g the relaxation time grows apparently infinitely long and the structure is frozen. The temperature dependence of SAXS intensity can be used to determine T g. In two-component glasses a submicroheterogeneous structure appears due to the non-uniform distribution of structural elements of one network in the volume of the other. The degree of inhomogeneity increases with increasing concentration of either of components and reaches a maximum at approximately equal components concentration.

The determination of diffuse-boundary thicknesses of polymers by small-angle X-ray scattering

Thermal density fluctuations within phases and finite interphase widths lead to systematic deviations from Porod's law. The validity of present methods used to analyze these deviations and determine diffuse-boundary widths is determined. In view of the inadequacies found in these methods, a simple yet accurate method is proposed to determine the diffuse-boundary width from direct graphical analysis of slit-smeared intensity data. The diffuse interface is modelled by a sigmoidal-gradient model which is justified on thermodynamic grounds, with the interphase thickness as a function of the Flory–Huggins interaction parameter.

Structure of glassy polymers. V. Small‐angle x‐ray scattering from polystyrene

AbstractSmall‐angle x‐ray scattering (SAXS) from glassy atactic polystyrene has been measured using a Bonse–Hart system. After correcting for absorption, background, and beam divergence, the scattering has been placed on an absolute basis using a standard silica suspension as a reference.The desmeared absolute intensity decreases strongly with increasing scattering angle over the range between 20 sec and 20 min. At larger angles, the intensity decreases much more slowly with increasing angle and approaches the value expected for density fluctuations frozen‐in at the glass transition. The angular variation of intensity is well described by the scattering from heterogeneities of various sizes and concentrations superimposed on the scattering from thermal density fluctuations. These heterogeneities range in radius from 10 to 4000 Å. The present SAXS results on glassy polystyrene seem inconsistent with the presence of nodular features as representative of the bulk polymer.

Role of density fluctuations in bimolecular reaction kinetics

The influence of initial thermal density fluctuations of reactants on the bimolecular reaction kinetics (A + B→C) is considered. It is shown that the

Fluctuations in the diffusion of particles in a non-uniform medium

We study the thermal fluctuations of density and flow velocity of particles diffusing in a non-uniform medium with memory character. The fluctuation spectra are derived from the macroscopic equations with the aid of the generalized Nyquist theorem. A comparison is made with other methods.

Enhanced scattering at the upper-hybrid resonance

Discussed the problem of e.m. wave scattering near the upper hybrid resonance. The scattering flux is calculated using cold collisional plasma theory and retaining ponderomotive force terms in the non-linear current. Two cases of interest are considered: (a) incoherent scattering by thermal density fluctuations; (b) coherent scattering by an ion-acoustic wave. A good agreement is found with experiments.

On the structure of glassy polymers. III. Small‐angle x‐ray scattering from amorphous polyethylene terephthalate

AbstractThe small‐angle x‐ray scattering (SAXS) from glassy polyethylene terephthalate has been measured using a Bonse–Hart system. The data cover the angular range (2θ) between 20 sec and 2 deg. After correcting for absorption, background, and beam divergence, the data have been placed on an absolute basis by comparison with the scattering from a standard silica suspension. The corrected absolute intensity decreases strongly with increasing angle over the range between 20 sec and 15 min, decreases more gradually in the range between 15 min and 45 min, and reaches a nearly constant asymptotic value over the range between 45 min and 2 deg. The magnitude of the scattering in the constant range, about 0.4 (electrons)2 Å−3, is very close to the value predicted by the thermodynamic fluctuation theory for fluids applied at the glass‐transition temperature [0.34 (electrons)2 Å−3]. The increase in intensity at angles smaller than about 45 min cannot be described by structures on the scale and volume fraction of the nodules reported in amorphous PET (50–100 Å), but can be well represented by small concentrations of heterogeneities, ranging in size from 100 to 2000 Angstroms, superimposed on the thermal density fluctuations frozen‐in at the glass transition. The bulk structure of this material seems well described as a random amorphous solid, containing simple thermal fluctuations and a small concentration (<1 vol‐%) of heterogeneities covering a range of sizes. The heterogeneities in the small end of the range may well be crystallites which formed on cooling.

On the structure of glassy polymers. IV. Small‐angle x‐ray scattering from rigid polyvinyl chloride

AbstractThe small‐angle x‐ray scattering (SAXS) from quenched and annealed rigid polyvinyl chloride (PVC) has been measured using a Bonse–Hart system. After correcting for absorption, background, and beam divergence, the scattering has been placed on an absolute basis using a standard silica suspension as a reference. The scattering from annealed (6 days at 75°C) and unannealed PVC was identical within experimental error, varying with scattering angle in a manner similar to the SAXS from other amorphous polymers. The intensity decreases rapidly with increasing scattering angle over the ranges from 20 sec to 20 min. Beyond 20 min the intensity is fairly constant, decreasing only slightly with increasing angle. At the largest angles of measurement (in the range of 120 min), the measured intensity is close to the value calculated for thermal density fluctuations frozen‐in at the glass transition.The angular variation of intensity is well described by the scattering from heterogeneities of various sizes and concentrations superimposed on the scattering from thermal density fluctuations. These heterogeneities range in radius from 50 to 4500 Å and, assuming the crystalline excess density, the total concentration of heterogeneities is less than 0.5%. The mean‐square fluctuation in density, determined from the measured intensity invariant, is also consistent with such a distribution of heterogeneities.The present SAXS results on rigid PVC are inconsistent with the presence of nodular features as representative of the bulk polymer. Rather, it is suggested that they are associated with surface effects. It is further suggested that previously indicated volume fractions of crystallinity in rigid PVC (generally in the range of 5–12%) are incorrect, and that the model of a three‐dimensional network of crystallites used to explain the rheological behavior of this material should be re‐examined.

Electron microscopy and small angle x-ray studies of amorphous polymers

Abstract The small-angle X-ray scattering (SAXS) from polycarbonate, polymethyl methacrylate, and polyethylene terephthalate seem inconsistent, both in the magnitude of the SAXS intensity and in its variation with scattering angle, with nodular features as representative of the bulk polymers. Rather than nodules, the SAXS data indicate the presence of thermal density fluctuations frozen-in at the glass transitions, together with a small concentration of large heterogeneities (hundreds to thousands of angstroms in size). Electron microscopy (EM) of amorphous polycarbonate provides further support for the suggestion that nodular features are not representative of the bulk material. The only structure seen in bright field is the “salt and pepper” background common to all materials. In dark field, very small structures (<30 Å in size) occupying small volume fractions were observed. These results are at variance with previously reported EM observations of nodular features in polycarbonate. It is suggested that the structure of these polymers be regarded as that of a random amorphous solid, and that the nodular structures may reflect surface features of the materials.

On the structure of glassy polymers. II. Small‐angle x‐ray scattering from poly(methyl methacrylate)

AbstractThe small‐angle x‐ray scattering (SAXS) from glassy poly(methyl methacrylate) has been measured using a Bonse–Hart system. The data cover the angular range (2θ) between 20 sec and 2 deg. After correcting for absorption, background, and beam divergence, the data have been placed on an absolute basis by comparison with the scattering from a standard silica suspension. The corrected absolute intensity decreases strongly with increasing angle over the range between 20 sec and 30 min, and is nearly constant between 30 min and 2 deg. The magnitude of the scattering in the constant range, 0.6 (electrons)2 Å−3, is within a factor of 1.5 of the value predicted by the thermodynamic fluctuation theory for fluids applied at the glass transition temperature. The increase in intensity at smaller angles cannot be described by structures on the scale of the nodules reported in highly isotactic PMMA (150–200 Å), but can be well represented by small concentrations of heterogeneities, several thousand angstrom units in size, superimposed on the thermal density fluctuations frozen‐in at the glass transition. The bulk structure of this material is well described as a random amorphous solid, containing simple thermal fluctuations and a small concentration of relatively large heterogeneities.

On the structure of glassy polymers. I. Small‐angle X‐ray scattering from polycarbonate

AbstractThe small‐angle x‐ray scattering (SAXS) from glassy bisphenol‐A polycarbonate has been measured using a Bonse–Hart system. The data cover the angular range (2θ) between 20 sec and 2 deg. After correcting for absorption, background, and vertical beam divergence, they have been placed on an absolute basis by comparison with the scattering from a standard silica suspension. The corrected absolute intensity decreases strongly with increasing angle over the range between 20 sec and 30 min, and is nearly constant between 30 min and 2 deg. The magnitude of the scattering in the constant range, 0.44 (electrons)2 Å−3, is well represented by the thermodynamic theory for fluids applied at the glass‐transition temperature. The increase in intensity at smaller angles cannot be described by structures on the scale of the nodules reported in this material (50–200 Å), but can be well represented by a small concentration of heterogeneities (0.04% by volume or less), several thousand angstrom units in size, superimposed on the thermal density fluctuations frozen in at the glass transition.It is suggested that the nodular features reported for this material are not representative of bulk material but should be associated with surface effects. The bulk structure can—as far as the SAXS is concerned—be well described as a random amorphous solid, containing simple thermal fluctuations and a small concentration of relatively large heterogeneities.

Thermal density fluctuations in amorphous polymers as revealed by small angle X-ray diffraction

Thermal density fluctuations in amorphous polymers as revealed by small angle X-ray diffraction

CO 2 laser scattering measurements of turbulence in a high-β collisionless shock wave

Long wavelength ( γ = 0.01 cm) suprathermal (3 × 10 4 times thermal) density fluctuations are observed in a collisionless shock wave by scattering of CO 2 laser light. The fluctuations first propagate parallel to the electron drift, 20 ns later in a perpendicular direction as well.

Measurement of Plasma Properties from the Cross Spectrum of Thermal Density Fluctuations

The relation between the density cross-correlation function and the cross spectrum of fluctuations in the electron saturation currents drawn by two Langmuir probes is used to measure basic plasma properties. The measured quantities using this approach are the ambipolar diffusion coefficient and the electron density. It is found that the method has sufficient sensitivity to determine changes in the ionic composition. These measurements are compared with plasma parameters obtained by using conventional probe techniques. Discrepancies are shown to exist and the source of these discrepancies is discussed.

A Theory of Collision Dominated Electron Density Fluctuations in a Plasma with Applications to Incoherent Scattering

The problem of thermal electron density fluctuations in a weakly ionized gas is considered from the point of view of particle diffusion in single particle phase space (μ space). When ion-neutral and electron-neutral two-body collisions constitute the only important force on the charged particles apart from an electrostatic self-consistent field, it is shown how the fluctuation theory may be developed from a transition probability describing the random walk of a single charged particle in the absence of the interaction field. Specific calculations of thermally excited low frequency electron density fluctuation spectra are carried out on the assumption that the random walk may be described in terms of classical Brownian motion or in terms of a combination of free flight and classical Brownian motion. The latter model is shown to be a first approximation to a more exact solution which may be obtained by solving an integral equation for the transition probability. First-order calculations are carried out for the case where the ion-neutral collisions may be treated as hard sphere elastic ones. The computational results are compared with those of earlier theories of fluctuations in a weakly ionized gas.

Spectrum of Light Doubly Scattered by an Opalescent Fluid

An expression is derived for the frequency spectrum of light which is scattered twice by thermal diffusion-type density fluctuations in a fluid. In the two limiting cases of forward and backward scattering, the shape of the spectrum is insensitive to the presence of a double-scattering component.

Nonlinear Incoherent Light Scattering

Classical equations are used to calculate nonlinear incoherent light scattered from two coherent beams of electromagnetic radiation of different frequencies propagating in a homogeneous plasma without magnetic field. With a suitable choice of the difference frequency, enhanced light scattering occurs near the plasma frequency. In this case the frequency spectrum is mainly given by the thermal ion density fluctuations.

An application of slowing down kernels to thermal neutron density fluctuations in nuclear reactors

The backward equation for the space and energy-dependent probability generating function of the density fluctuations in a nuclear reactor, has been reduced to a more tractable form by the application of a slowing down kernel technique. Simple working expressions have been obtained for the variance, the correlation function and the power spectral density (P.S.D.) in terms of the infinite medium slowing down kernel. The method includes the effects of detectors, delayed neutrons and slowing down time. In addition, the frequency characteristics of the detection apparatus have been accounted for. It is shown that the dominant break frequency, of the fundamental mode component in the power spectral density, is markedly dependent on the slowing down lengths of the delayed neutrons, a result that can lead to appreciable corrections to the experimentally calculated neutron lifetime in certain systems. A general formalism accounting for slowing down, delayed neutrons and detector geometry is developed, and is applied to detectors of various shapes. Marked differences in the P.S.D. are noted for the different detectors in an infinite medium; however, for the finite medium, the effect of detector geometry is not so important and it is shown that, in certain circumstances, experiments can be analysed according to the simple, infinite medium-infinite detector formula.