Finite Shear Modulus Research Articles

Acoustic radiation force on gas bubbles and soft elastic scatterers in tissue

Acoustic radiation force on a scatterer in tissue depends on the compressibility and shear modulus of both the tissue and the scatterer. This force is related to the monopole and dipole scattering coefficients. The finite shear modulus of the tissue decreases the radiation force in comparison with the force exerted on the same scatterer surrounded by liquid. Shear moduli for soft tissue range from several kilopascals (breast, liver) to tens of kilopascals and higher for cornea, cartilage, and cancerous tissue. As reported previously, the radiation force on a bubble in tissue having 100 kPa shear modulus is 50% less than if the bubble is in water. This difference decreases for scatterers with finite shear moduli, examples of which are reported here. Additionally, displacement of a scatterer due to radiation force is inversely proportional to the shear modulus of the tissue, which permits measurement of the latter. Experiments demonstrating this technique are reviewed. In these experiments, the radiation force is applied to a gas microbubble produced by laser-induced optical breakdown, while displacement of the microbubble is measured by high-frequency ultrasound as a function of time. Results are reported for tissue-mimicking phantoms and animal crystalline lenses in vitro.

Glass Elasticity from Particle Trajectories

Using positional data from video microscopy of a two-dimensional colloidal system and from simulations of hard disks, we determine the wave-vector-dependent elastic dispersion relations in glass. The emergence of rigidity based on the existence of a well defined displacement field in amorphous solids is demonstrated. Continuum elastic theory is recovered in the limit of long wavelengths which provides the glass elastic shear and bulk modulus as a function of temperature. The onset of a finite static shear modulus upon cooling marks the fluid-glass transition in an intuitive and unique way.

Random networks of cross-linked directed polymers

We explore the effect of random permanent cross-links on a system of directed polymers confined between two planes with their end points free to slide on them. We treat the cross-links as quenched disorder and we use a semimicroscopic replica field theory to study the structure and elasticity of this system. Upon increasing the cross-link density, we get a continuous gelation transition signaled by the emergence of a finite in-plane localization length. The distribution of localization length turns out to depend on the height along the preferred direction of the directed polymers. The gelation transition also gives rise to a finite in-plane shear modulus which we calculate and turns out to be universal, i.e., independent of the energy and length scales of the polymers and the cross-links. Using a symmetry argument, we show that cross-links of negligible extent along the preferred axis of the directed polymers do not cause any renormalization to the tilt modulus of the uncross-linked system.

Gravitational radiation from crystalline color-superconducting hybrid stars

The interiors of high mass compact (neutron) stars may contain deconfined quark matter in a crystalline color-superconducting (CCS) state. On a basis of microscopic nuclear and quark matter equations of states we explore the internal structure of such stars in general relativity. We find that their stable sequence harbors CCS quark cores with masses ${M}_{\mathrm{core}}\ensuremath{\le}(0.78--0.82){M}_{\ensuremath{\bigodot}}$ and radii ${R}_{\mathrm{core}}\ensuremath{\le}7\text{ }\text{ }\mathrm{km}$. The CCS quark matter can support nonaxisymmetric deformations, because of its finite shear modulus, and can generate gravitational radiation at twice the rotation frequency of the star. Assuming that the CCS core is maximally strained we compute the maximal quadrupole moment it can sustain. The characteristic strain of gravitational wave emission ${h}_{0}$ predicted by our models are compared to the upper limits obtained by the LIGO and GEO 600 detectors. The upper limits are consistent with the breaking strain of CCS matter $\ensuremath{\sigma}\ensuremath{\le}{10}^{\ensuremath{-}4}$ and large pairing gaps $\ensuremath{\Delta}\ensuremath{\sim}50\text{ }\text{ }\mathrm{MeV}$, or, alternatively, with $\ensuremath{\sigma}\ensuremath{\sim}{10}^{\ensuremath{-}3}$ and small pairing gaps $\ensuremath{\Delta}\ensuremath{\sim}15\text{ }\text{ }\mathrm{MeV}$. An observationally determined value of the characteristic strain ${h}_{0}$ can pin down the product $\ensuremath{\sigma}{\ensuremath{\Delta}}^{2}$. On the theoretical side a better understanding of the breaking strain of CCS matter will be needed to predict reliably the level of the deformation of CCS quark core from first principles.

Compression and shear surface rheology in spread layers of β-casein and β-lactoglobulin

We investigate the surface viscoelasticity of β-lactoglobulin and β-casein spread surface monolayers using a recently discovered method. Step compressions are performed, and the surface pressure is measured as a function of time. This is a common experiment for surface monolayers. However in our experiments the pressure is recorded by two perpendicular sensors, parallel and perpendicular to the compression direction. This enables us to clearly measure the time relaxation of both the compression and shear moduli, at the same time, in a single experiment, and with a standard apparatus. β-Lactoglobulin and β-casein monolayers are interesting because of their importance in food science and because they exhibit universally slow dynamical behavior that is still not fully understood. Our results confirm that the compressional modulus dominates the total viscoelastic response in both proteins. Indeed for β-casein we confirm that the shear modulus is always negligible, i.e., the layer is in a fluid state. In β-lactoglobulin a finite shear modulus emerges above a critical concentration. We emphasize that in Langmuir trough dynamic experiments the surface pressure should be measured in both the compression and the perpendicular directions.

Viscoelastic dynamics of spherical composite vesicles

A micromechanical model for the low-frequency dynamics of spherical composite vesicles (CVs) is proposed. Solid-like viscoelastic properties of the CVs are taken into account. The equations of motion of a CV surrounded by a viscous liquid are derived. They have discrete solutions which describe linearly coupled stretching and bending relaxation modes and an independent shear mode. The qualitative difference between the bending modes excited in a spherical vesicle and that in a flat membrane is demonstrated. The shear elasticity of the CVs gives an essential contribution to the relaxation rate of the bending mode at small wave numbers. It is also shown that even in an incompressible spherical vesicle with a finite shear modulus, the bending mode involves both radial and tangent displacements. These reasons make both in-plane and out-of-plane low-frequency responses of the CV quite different with respect to those of the flat membrane. To compare our theoretical results with published experimental data, the power spectra of the actin-coated CV are calculated.

A study on the homogeneous nucleation kinetics of model charged spheresuspensions

We investigated the kinetics of homogeneous nucleation from a shear melt for amodel system of deionized aqueous suspensions of charged spheres. With the aidof nearly index-matched perfluorinated particles of low polydispersity,we extended previous studies into the regime of large metastability andnucleation-dominated solidification. With increasing particle number densityn,the solidification time determined by the appearance of a finite shearmodulus decreased from minutes to milliseconds. Nucleation rate densitiesJ werederived from the width of the principal peak in the static structure factoras measured by means of light scattering after complete solidification.Jwas observed to increase approximately exponentially withn asexpected from classical nucleation theory in the absence of a kinetic glasstransition. Additional measurements of the elastic and dynamic behaviour,however, show that for the largest concentrations, the sample properties areglass-like.

Frustration of structural fluctuations upon equilibration of shear melts

We report on the formation of amorphous solids from aquaeous suspensions of charged colloidal spheres. Comprehensive light scattering and microscopic studies show that in these systems the nucleation rate density continuously increases to very high values. At the highest particle densities of 47.5 μm −3 (packing fraction Φ=0.146) an amorphous state is observed of only short range order, finite static shear modulus and frozen long time dynamics. This state is composed of a piling of––as we propose pre-critical––nuclei. Differences from the Hard Sphere case are discussed in some detail. There the arrest of density fluctuations is observed and described by Mode Coupling scenarios. In the charged system a pronounced pre-ordering of the metastable melt is present which leads to structural fluctuations i.e. the formation of bond orientational order with more or less retained density. In both scenarios, however, mutual frustration of fluctuations, which evolve during the equilibration of the metastable (shear-) melt, is the main source of vitrification.

Vortex matter phase transitions in Bi 2Sr 2CaCu 2O 8

Vortex-lattice phase transitions in Bi 2Sr 2CaCu 2O 8 crystals are studied using local magnetization measurements. The vortex matter is found to exhibit at least three distinct phases. A rather ordered quasilattice phase is present at low fields. At elevated temperatures the quasilattice melts (or sublimates) through a first-order phase transition, whereas at lower temperatures a transition to a highly disordered vortex solid occurs at the second magnetization peak. A very low density of columnar defects results in significant pinning of the quasilattice, demonstrating the finite shear modulus of this phase. Local ac measurements reveal that in the presence of columns the pinned quasilattice still melts through a first-order phase transition with an observable step in equilibrium magnetization at elevated temperatures.

Vortex-matter phase transitions inBi2Sr2CaCu2O8:Effects of weak disorder

The vortex matter phase diagram in Bi{sub 2}Sr{sub 2}CaCu{sub 2}O{sub 8} crystals is investigated by introducing very low doses of point and correlated disorder. We conclude that at least three distinct phases are present. The ordered low-field quasilattice phase has a finite shear modulus which vanishes at the first-order melting or sublimation transition at elevated temperatures. At lower temperatures the quasilattice transforms into a disordered solid as field is increased above the second magnetization peak. This disorder-driven transition shifts to lower fields with increased point disorder. The first-order transition displays corresponding downward curvature in the vicinity of the critical point. {copyright} {ital 1997} {ital The American Physical Society}

Nonlinear vibroacoustic properties of rocks

This is a summary of recent theoretical and experimental work performed by the Nonlinear Acoustic Group (L. Ostrovsky, A. Sutin, V. Nazarov, I. Belyaeva, Yu. Zaitsev) at the Institute of Applied Physics, N. Novgorod, Russia. Experiments show that Earth materials can have anomalously strong mechanical nonlinearity. However, in many cases there is a lack of clear understanding of specific mechanisms responsible for these nonlinear effects. Some theoretical models of structurally inhomogeneous media have been constructed and experimentally verified. These models include: (1) porous waterlike media characterized by small (but finite) shear modulus; (2) grainy media where nonlinearity is caused by the intergrain contacts (Hertz model); (3) some nonlinear models of cracks. Laboratory experiments have demonstrated a good agreement with theoretical results, and in all three cases the values of the nonlinearity parameter (defined similarly to that used in nonlinear acoustics for gases and liquids) could achieve the values of the order of 103–104 and even more. Some estimates for real rocks are also given. They show that nonlinear parameters may prove to be much more sensitive to the details of the material structure than the usual linear ones (e.g., sound velocity). This can serve as a base for developing new methods of seismic surveillance.

Phase coherence in two-dimensional type-II superconductors.

We show that in two-dimensional type-II superconductors, the vortex liquid-solid transition is not accompanied by a simultaneous divergence of the phase correlation length. Even in the ``vortex-solid'' phase, despite a finite shear modulus and quasi-long-range density correlations, the phase coherence remains short ranged, with a correlation length of order one magnetic length. We also find that, in the solid phase, there is no ``Bose condensation,'' in the sense that any particular Abrikosov lattice state has zero statistical weight in the thermodynamic limit. The vortex-solid phase in two dimensions is therefore best described as a Cooper-pair charge-density wave with no superconducting phase order.

Helicity modulus and Meissner effect in a fluctuating type-II superconductor.

The helicity modulus for a fluctuating type-II superconductor is computed within the elastic medium approximation, as a probe of superconducting phase coherence and the Meissner effect in the mixed state. We argue that at the vortex line lattice melting transition, there remains superconducting coherence parallel to the applied magnetic field, provided the vortex line liquid retains a finite shear modulus at finite wave vector.

Surface instability of viscoelastic thin films

The surfaces of thin, liquid films can be unstable due to thinning van der Waals interactions, leading to the formation of holes in the initially uniform film. These instabilities can be greatly retarded in viscoelastic materials (and completely inhibited in elastic materials) even when the finite frequency shear modulus, E, is small compared to the infinite frequency modulus, G. This occurs when E/G≫(a/h0)5≪1 where a is a molecular size and h0 is the film thickness. We relate the growth rate of the instability to the dynamic viscosity, $\eta~(\omega)$, with examples for the cases of a polymer brush, an elastic fluid (gel), and a transient polymer network, described by reptation dynamics.

Paradox in phase transitions with volume change.

In gels, a critical point at which swelling transition becomes continuous, is characterized by vanishing of the bulk modulus $K$. At this point, however, an instability is triggered by a single mode representing homogeneous expansion while fluctuations with finite wave numbers are still suppressed by the finite shear modulus $\ensuremath{\mu}$. Critical divergence of the fluctuation spectrum is expected not at this critical point but only in an unstable region. This unusual feature has been overlooked because of small values of $\ensuremath{\mu}$ in previous experiments.

Fluctuating Interfaces: a Series of Synchrotron X-Ray Scattering Studies of Interacting Stacked Membranes

AbstractIn this paper we concentrate on fluctuation phenomena encountered in interacting multilayered fluid membranes using synchrotron x-ray scattering as the primary tool. These systems consisting of surfactant or lipidic surfaces, are prototype models for understanding the statistical behavior of fluctuating surfaces embedded in three-dimensions. In elucidating the nature of fluid membranes we stress a unique intrinsic property: in contrast to usual surfaces with finite shear moduli, where surface tension plays a central role, the free energy of these essentially tensionless surfaces is governed by their geometrical shape and its fluctuations. We present data in a new regime of stability for very dilute and flexible membranes with interlayer separations of order hundreds of Angstroms. This is in contrast to most rigid membranes where the interlayer interactions are dominated by detailed microscopic interactions such as hydration and van der Waals. We show that the stability of these dilute lamellar phases is associated with violent out-of-plane fluctuations of the membranes giving rise to an effectively large long-range repulsive interaction theoretically elucidated by Helfrich. Because of its entropic origin, this interaction is universal and we present data for two different surfactant systems. Finally, we show that this new regime is distinct from the classical regime in which largely separated membrane sheets are stabilized because of their mutual electrostatic repulsion.

Shear modulus effect on acoustically transparent materials

Materials have been developed that match density <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> and dilatational sound speed <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> to the values of water as closely as possible while maintaining sufficient rigidity to serve for structural purposes. Matching of density and speed results in transparency for fluids only; the shear modulus in a solid admits the presence of a shear wave which causes deviation from ideal <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pc</tex> behavior. The effect of a finite shear modulus on the reflection of plane waves by an infinite plane is analyzed. The shear modulus of the material was measured following a method developed by R. L. Adkins in 1966. Examples are given of the reflection coefficient as a function of incidence angle for values of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</tex> close to those of the medium, and various ratios of plate thickness to dilatational wavelength. The condition of total reflection is shown to be related to the occurrence of thickness-shear resonances. Insertion-loss measurements on a structurally rigid <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">pc</tex> panel are compared with results of the analysis.

Influence of shear modulus on the behavior of acoustically transparent materials

Materials are under development that match density (ρ) and dilatational sound speed (c) to the values of seawater as closely as possible while maintaining sufficient rigidity to serve for structural purposes. Another paper in this meeting by C. M. Thompson and J. R. Griffith (preceding abstract) describes the composition of the material and some of its properties. Matching of density and speed results in transparency for fluids only; the shear modulus in a solid admits the presence of a shear wave which causes deviation from ideal ρ‐c behavior. In this paper, the effect of a finite shear modulus on the reflection of plane waves by an infinite plane is analyzed. The shear modulus of the material was measured following a method developed by R. L. Adkins in 1966. Examples are given of the reflection coefficient as a function of incidence angle for values of ρ and c close to those of the medium, and various ratios of plate thickness to dilatational wavelength. The effect of a finite loss tangent in the shear ...

SmecticsAandCAre Always Glasses

Smectics A and C, tilted discotics, and the smectic-C˜ phase are shown to have finite shear moduli for shears applied in directions which have no translational order (liquid directions): This is the signature of a glass. This modulus is generated purely by dynamical fluctuation effects. It leads in smectics A and C to a violation of the basic rule of conventional hydrodynamics which asserts the equality of the numbers of hydrodynamic modes and fields.

SmecticsAandCAre Always Glasses

Smectics $A$ and $C$, tilted discotics, and the smectic-$\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{C}$ phase are shown to have finite shear moduli for shears applied in directions which have no translational order (liquid directions): This is the signature of a glass. This modulus is generated purely by dynamical fluctuation effects. It leads in smectics $A$ and $C$ to a violation of the basic rule of conventional hydrodynamics which asserts the equality of the numbers of hydrodynamic modes and fields.