Complex Longitudinal Modulus Research Articles

Thorough ultrasonic rheology of soft, visco-elastic materials: Example of crosslinked Polyurethane elastomer

We present a thorough procedure for measuring the rheological properties of soft, highly attenuating, visco-elastic materials at ultrasonic frequencies. The material chosen for this illustration is a crosslinked Polyurethane (PU) elastomer (Sika UR3440 type), which is widely used in the field of underwater acoustics. We determine its complex longitudinal modulus M and shear modulus G as function of frequency in the range 1–5 MHz and of temperature in the range 5–40 °C. M is determined from the measurement of the transmission of longitudinal, plane waves by a slab of PU immersed in water. G is determined by contact measurements from the transmission and reflection of transverse, plane waves by a slab of PU. This determination of G for such a soft and viscous material as PU is made possible by the use of thin slabs and the implementation of an original signal analysis. M, G and bulk modulus K are found to obey the time–temperature superposition principle and to be accurately described by a fractional derivative rheological model. This allows us to propose analytic formulas for the frequency and temperature dependence of M, G and K valid above PU glass transition temperature.

High-sensitivity and high-specificity biomechanical imaging by stimulated Brillouin scattering microscopy.

Label-free, non-contact imaging with mechanical contrast and optical sectioning is a substantial challenge in microscopy. Spontaneous Brillouin scattering microscopy meets this challenge, but encounters a trade-off between acquisition speed and the specificity for biomechanical constituents with overlapping Brillouin bands. Stimulated Brillouin scattering microscopy overcomes this trade-off and enables the cross-sectional imaging of live Caenorhabditis elegans at the organ and subcellular levels, with both elasticity and viscosity contrasts at high specificity and with practical recording times.

Mechanical Properties of cellulose fibers measured by Brillouin spectroscopy

We investigate the potential of Brillouin Light Scattering (BLS) Microspectroscopy for fast non-invasive all-optical assessment of the mechanical properties of viscose fibers and bleached softwood pulp. Using an optimized Brillouin spectrometer, we demonstrate fast spatial mapping of the complex longitudinal modulus over extended areas (> 100 µm). Our results reveal that while the softwood pulp has a relatively uniform moduli, the viscous fibers have significant spatial heterogeneous in the moduli. Specifically, the viscose fibers exhibited a regular pattern of increasing and decreasing modulus normal to the fiber axis. The potential influence of a locally changing refractive index is investigated by holographic phase microscopy and ruled out. We discuss our results in light of the anisotropic mechanical properties of the fibers and are able to estimate the relative difference between the modulus along the fiber axis and that perpendicular to it. Results are presented alongside reference measurements of the quasi-static mechanical properties transverse to the fiber axes obtained using AFM-nanoindentation which reveal a similar trend, hinting at the potential usefulness of BLS for mechanical characterization applications. However, more detailed investigations are called for to uncover all the factors influencing the measured high-frequency BLS modulus and its significance in relation to physical properties of the fiber that may be of practical interest.

The longitudinal modulus of bitumen: Pressure and temperature dependencies

Bitumen retains significant solid-like behavior even in temperatures in excess of 50°C. Traditional ultrasonic wave-propagation studies have, however, largely ignored the existence of the shear modulus in such materials, and they have mostly assumed that the observed longitudinal (P) wave speeds solely depend on the fluid’s bulk modulus. To further study this, we have measured ultrasonic longitudinal (P) wave transmission speeds through viscous bitumen at different pressures (0.1–15 MPa) and temperatures (7–132°C) using an adapted version of the technique that consists of two piezoelectric receivers placed at unequal lengths from the transmitter. As such, we are able to calculate the P-wave attenuation and velocity that is used to derive the material’s complex longitudinal modulus. Using parallel measurements of the bitumen’s complex shear modulus, we find that the bulk modulus differs from the longitudinal modulus particularly at lower (reservoirs) temperatures. The results, together with the realization that bitumen experiences a sequence of various compositional and thermophysical phase that is primarily temperature-dependent, can be implemented to improve the fluid-substitution analyses of rock-physics studies of bitumen-saturated reservoirs.

Modeling nonlinear viscoelasticity in dynamic acoustoelasticity

Dynamic Acoustoelastic Testing (DAET) has been developed to non-invasively assess the nonlinear viscoelastic properties of fluids and solids. We propose a phenomenologically motivated model for harmonic regime to identify nonlinear viscoelastic parameters from DAET experiments. The nonlinear elastic and viscous parameters are derived from the real and imaginary components of the Taylor series expansion of the complex longitudinal modulus. The model is applied to Newtonian fluids that exhibit classical elastic nonlinearities and glass beads saturated with water that exhibit nonclassical viscoelastic nonlinearities. Hysteresis, asymmetry, and DC offset are well accounted for in the model.

Ultrasonic Dynamic Mechanical Analysis of Polymers

Abstract The propagation of ultrasonic waves in polymers depends on their viscoelastic behaviour and density, resulting significantly affected by phase transitions occurring with changing temperature and pressure or during chemical reactions. Therefore, the application of low intensity ultrasound, acting as a high frequency dynamic mechanical deformation applied to a polymer, can monitor the changes of viscoelastic properties associated with the glass transition, the crystallization, the physical or chemical gelation, the crosslinking. Thanks to the non-destructive character (due to the very small deformation amplitude), low intensity ultrasound can be successfully used for polymer characterization. Moreover, this technique has a big potential as a sensor for on-line and in-situ monitoring of production processes for polymers or polymer matrix composites. Recently, in the laboratory of Polymeric Materials of Lecce University a custom made ultrasonic set-up for the characterization of polymeric material, even at high temperatures, has been developed. The ultrasonic equipment is coupled with a rotational rheometer. Ultrasonic waves and shear oscillations at low frequency can be applied simultaneously on the sample, getting information on its viscoelastic behaviour over a wide frequency range. The aim of this paper is to present the potential and reliability of the ultrasonic equipment for the ultrasonic dynamic mechanical analysis (UDMA) of both thermosetting and thermoplastic polymers. Three applications of UDMA to different polymeric systems will be reviewed, concerning the cross-linking of a thermosetting resin, the crystallisation from melt of a semicrystalline polymer and the water sorption in a dry hydrogel film. From the ultrasonic velocity and attenuation measurements, the viscoelastic properties of the tested polymers are evaluated in terms of complex longitudinal modulus and compared with the results of conventional dynamic mechanical analysis, carried out at low frequency.

Ultrasonic investigation of wheat starch retrogradation

A recently developed technique, based on ultrasonic wave propagation, was applied to study the effect of storage time on the retrogradation process of wheat starch extrudates during storage at room temperature. A specifically designed experimental set-up enabled to propagate in situ ultrasonic waves continuously for several days at constant water content and storage temperature (∼34% water and 25 °C). The velocity and the attenuation of the ultrasonic waves changed as a result of the recrystallisation process of amylopectin molecules of wheat starch samples during storage. The propagation of ultrasonic waves, acting as a dynamic mechanical deformation at high frequency (10 MHz), gave access to the complex longitudinal modulus that, compared to the complex Young modulus obtained from low frequency DMA, enabled a better insight of the changes in viscoelastic behaviour of starch extrudates during retrogradation. The results of the ultrasonic monitoring of starch retrogradation were correlated with those obtained from X-ray diffraction and differential scanning calorimetry. The present study demonstrated the potential of the ultrasonic technique in detecting the changes in physical properties of concentrated starch systems.

Determination of the elastic moduli of polymer films by a new ultrasonic reflection method

The development and first applications of a new ultrasonic reflection method for determination of the viscoelastic properties of polymer films are reported. The complex shear modulus G* and the complex longitudinal modulus L*=K*+4/3 G* of the samples are derived from the measured complex reflection coefficients of an ultrasonic shear and longitudinal wave, respectively. From G and L the Young's modulus E, the compression modulus K and the Poisson ratio v can be calculated for isotropic materials. A LiNbO 3 -transducer (10° rotated Y-cut) is used for the simultaneous excitation of longitudinal and transversal ultrasonic waves, which allows to determine different elastic constants by one measurement. A measuring cell with normal incidence of the ultrasonic waves is used. The equipment has been applied to study the time dependence of the moduli during film formation from an aqueous polymer dispersion and the isothermal curing of an epoxy resin. Furthermore, the temperature dependence of the elastic constants of a carbon-black filled rubber and during non-isothermal crystallization of a semi-crystalline polymer has been studied.

On phase connectivity descriptions in modeling viscoelasticity of fiber or sphere reinforced composites

AbstractWe consider composites exhibiting different scales of heterogeneities and some hierarchical phase connectivities, referred to as “composites of composites.” Following literature suggestions, and as an alternative to the use of complicated pattern‐based models, effective elastic or viscoelastic moduli of such structures, with spherical (isotropic) or fiber‐like (transverse isotropic) symmetry are estimated following a stepwise homogenization procedure. The steps are chosen according to the successive identified heterogeneity scales, making use, at each step, of the simplest n‐layered (n=3 or 4) sphere or fiber inclusion model accounting for the scale dominant connectivity features. We quantitatively check the sensitivity of such an estimate process to the chosen simplifications for the morphology description, on comparing different choices for morphologies typical of nonuniformly reinforced matrices. The comparison first investigates, in elasticity, the effects of volume fractions of phases and of stiffnesses ranging over several orders, as well as phase connectivity permutations for 3‐phase arrangements. One then compares several estimates obtained for the effective complex longitudinal modulus of a typical fiber reinforced polymer morphology using the correspondence principle for its viscoelasticity. The material description is compared to experiment, which for the morphological characteristics is the most important.

Relaxations in thermosets. XVIII. Ultrasonic studies of curing kinetics of ethylene‐diamine‐cured epoxide

AbstractVelocity and attenuation of longitudinal acoustic waves in a frequency range of 5–16 MHz have been measured during the curing of diglycidyl ether of bisphenol‐A with ethylene diamine. The velocity monotonically increases and reaches a limiting value and the attenuation reaches a maximum and thereafter decreases as curing proceeds. Both kinds of data have been transformed into a complex‐longitudinal‐modulus formalism. The complex‐plane plots of the longitudinal modulus show an arc‐skewed shape at the long (curing) time end. It is fitted, for the initial phase of curing, to a stretched exponential decay function with an exponent γ = 0.22 to 0.28. These results then are considered in terms of a process with a negative feedback between molecular diffusion and chemical reactions to obtain the increase in the relaxation time with the curing time. This increase is sigmoidal. Calculations of the complex longitudinal modulus show the consistency of our formalism.

Short time structural relaxation processes in liquids: Comparison of experimental and computer simulation glass transitions on picosecond time scales

Recent Brillouin scattering studies of relaxation in the simple ion system Ca++–K+/NO−3 (two particles of argon structure and a small anion) provide the experimental basis for a direct comparison of glass transformation phenomenology in computer simulated LJ argon ‘‘glass,’’ with that in laboratory substances studied on the same time scale. Most of the attenuated glass transition characteristics observed for LJ argon are found in the ionic system, and the same relation of the dispersion midpoint to the so-called ‘‘ideal’’ glass transition temperature is observed. Analysis of the real and imaginary parts of the complex longitudinal modulus shows that at high temperatures the relaxation function for Ca++–K+–NO−3, in strong contrast with that at normal low temperature behavior, closely approaches a simple exponential decay with Arrhenius form for the relaxation time. Furthermore, the high temperature Arrhenius plot extrapolates naturally to the reciprocal quasilattice vibration frequency determined by far infrared absorption studies on the glassy solid. Because of the low activation energy in the high temperature regime the glass ‘‘transition’’ observed with decreasing temperature is smeared out almost beyond recognition. It is argued that this will be a rather general phenomenon for hyperquenched glasses and that for such cases, the fictive temperature concept which associates the glass structure with an internally equilibrated liquid structure (that at Tfictive) must collapse. Parallels between LJ argon and Ca+–K+–NO3–glasses are discussed.

Control ranges and the optimal solutions of the minimal reflectance problem; homogeneous materials, normal incidence, and narrow bandwidths

We study the optimal solutions of the minimal sound reflection problem for planar inhomogeneous sound absorbing layers at normal incidence and narrow bandwidths as a function of their controls. Homogeneous materials are considered. The controls are the real and imaginary parts of the inverse complex longitudinal modulus, K* = K(1 + iη) and the density, ρ. The optimal solutions correspond to multilayered systems, with piecewise constant, extremal control values on the sublayers. The dependence of layer structures, i.e., numbers of sublayers, sublayer thicknesses, and parameter distributions, on the controls can be studied as a function of a certain polyhedron in control space. The shape of the polyhedron, which is determined by the boundary values of the controls, will be correlated with desirable characteristics of the resulting layered structures.

Studying the longitudinal flow of polymers with periodic deformation

1. At low rates of elongation the state of a polymer melt depends neither on the strain nor on the strain rate. The frequency dependence of the components of the complex longitudinal modulus, measured by superposition of a periodic deformation on a steady longitudinal flow, resemble, at low elongation rates, the frequency characteristics of the components of the complex shear modulus. 2. Relaxation processes due to steady longitudinal flow affect the components of the complex longitudinal modulus measured by periodic deformation. 3. A melt of a polydisperse polymer under conditions of longitudinal flow begins, at some definite strain rate, to respond to periodic deformation as a rubber-like body and continues to do so as the amount of elongation increases, which is manifested by the trend of the frequency dependence of the modulus component E', this trend becoming similar to that for a solid body. 4. As the strain increases during elongation, the upper boundary of the flow range, which is determined by the periodic deformation, shifts toward low frequencies. 5. The viscoelasticity characteristics of a material subject to elongation can be accurately enough described by nonlinear equations with the aid of the linear shear-relaxation spectrum.

A study of viscoelastic properties of butanediol-1,3 using optical digital correlation spectroscopy

A temperature study of viscoelastic relaxation in butanediol-1,3 was made using digital correlation and Brillouin scattering spectroscopic methods. The digital correlator provided measurement of the relaxation function from which average relaxation times were calculated. The Brillouin scattering method allows a more accurate determination of the high frequency longitudinal modulus than older ultrasonic methods that require extrapolation of data over a large temperature range for the determination of this modulus. From the light scattering data, it has been concluded that, contrary to previous studies, the relaxations observed in the real and imaginary parts of the complex longitudinal modulus are the same. Under these circumstances the use of a ’’hysteresis’’ effect to interpret the nature of the relaxation at high frequencies is unnecessary. A theoretical derivation has been given that allows the average relaxation time obtained via correlator studies to be compared directly to the measured shear viscosity.

Calculation of the Dynamic Moduli of Linear Viscoelastic Materials from Vibrating Reed Measurements

The direct relation between the complex longitudinal modulus E* and the experimentally measured quantities—the amplitude ratio A and phase angle φ—has been derived for the forced vibrating reed instrument and is given by: exp (iφ)/A=1+∑K=1∞(−ω2ρ/E*)K[AK+BKρ−1+CKρ−2+….], where terms in ρ with negative exponents are not admitted. E* is equal to E′+iE″, where E′ is the longitudinal storage modulus and E″ is the longitudinal loss modulus. A is the ratio of the amplitude of the free end of the reed—either loaded with a pin or unloaded—to that of the clamped sinusoidally driven end and φ is the angle by which the former lags the latter. In practice, the sample dimensions and frequency range are chosen such that the sum in the above equation is rapidly convergent. The dynamic moduli then can be calculated by including only up to the first two or three terms in the sum. Where phase measurements are relatively inconvenient to make and when all but the first term under the sum can be neglected, the equation can be used to calculate dynamic moduli from two amplitude measurements at the same frequency for reeds of two different lengths; or from amplitude at resonance alone, provided E″≪E′. Illustrative examples are given. The procedures for calculating the dynamic moduli are applicable to the data obtained in all experiments where a relation of the form given above is obtained.