Storage Compliance Research Articles

Dynamic Viscoelastic and Steady Flow Properties of High Density Polyethylene Melts

Shear storage (G′) and loss moduli (G″), and steady shear (σ) and first normal stress difference (N1) of a commercial high density polyethylene (HDPE) melts and the fractions have been measured with a Weissenberg Rheogoniometer. Pseudo-equilibrium modulus (GeN0) and average molecular weight between entanglement points (Me) were found to be 1.32 × 106 Pa and 2.18 × 103, respectively. The value of Me is approximately half of the literature value reported for Mc, the critical molecular weight for entanglement coupling estimated from molecular weight dependence of viscosity. The value of zero shear viscosity (η0) obtained from both dynamic and steady shear measurements are in good agreement and found to be proportional to 3.8 power of the weight- average molecular weight. Cox- Merz rule does not hold so well at high rate of shear for both the whole polymer and its fraction. Steady state compliance (Je0) of the fractions is much higher than expected value and is remarkably dependent on molecular weight. Storage compliance (J′) obtained from dynamic measurement and steady shear compliance (Js) from steady flow measurement agree with each other in the value and the behavior for each fraction. It was found that J′ (and JS) of the fractions at low frequencies (low shear stresses) increase as increasing molecular weight and molecular weight distribution.On the contrary, J′ at higher angular frequencies decreases with increasing molecular weight and molecular weight distribution. It is suggested that Js, at high rates of shear decreases with increasing molecular weight and molecular weight distribution supporting Mendelson's results.

Determination of Dynamic Storage and Loss Compliances from Creep Data

A pair of empirical equations are proposed for the purpose of approximating the dynamic mechanical viscoelasticity functions, the storage compliance, J′(ω), and the nonviscous loss compliance, J″(ω)−1/ωη, from the value of the recoverable compliance, Jr(t), and its logarithmic slope, m(t)=d log Jr(t)/d log t. The angular frequency is ω and the shear viscosity is η. Three examples representing various kinds of polymeric, viscoelastic response are presented as tests of the effectiveness of the proposed expressions. Most of the calculated points are in error by less than 10%.

Properties of amorphous and crystallizable hydrocarbon polymers. I. Melt rheology of fractions of linear polyethylene

AbstractThe dynamic moduli G′(ω) and G″(ω) for two groups of linear polyethylene fractions (reported Mw/Mn < 1.2) were measured in the melt state using the eccentric rotating disk method. Values of zero shear viscosity η0 were obtained and compared with published results on similar fractions. Molecular weight data were converted to a common basis through intrinsic viscosities in trichlorobenzene (TCB) at 135°C. With recent data on Mw (light scattering) vs. [η]TCB, for linear polyethylene, the relationship at 190°C, η0 = 3.40 × 10−14(Mw)3.60, was obtained. The flow activation energy Ea was 6.4 kcal (T = 140–195°C). The plateau modulus G at 190°C was determined from the area under the loss modulus peak in one high‐molecular‐weight sample. The value obtained, G = 1.58 × 107 dyn/cm2, corresponds to an apparent molecular weight between entanglements of 1850. The storage compliance J′(ω) becomes anomalously large at low frequencies. The recoverable compliance J could not be determined for any of the fractions.

Approximate relations among compliance functions of linear viscoelasticity for amorphous polymers

AbstractIn the glass–rubber transition region of viscoelastic behavior of amorphous polymers, it is found that is a good first approximation. Here J(t) is the value of the shear creep compliance at time t and J″(1/t) is the value of the shear loss compliance at an angular frequency of 1/t. Previous approximations related the creep compliance to the storage compliance. When J(t) is proportional to tm, where m is a positive constant, J″(1/t) is within 30% of J(t) for m > 0.6. Over the entire range, is a better approximation than either of the other two. The relaxation modulus G(t) is hardly ever a good approximation for the loss modulus G″(1/t).

Dynamic viscoelastic behavior of polybutadiene networks

AbstractDynamic viscoelastic properties of polybutadiene networks, prepared by conventional sulfur vulcanization of polymers differing in primary molecular weight, molecular weight distribution, long‐chain branch content, and branching pattern, were measured in the region of the viscoelastic spectrum of principal interest in dynamic applications of rubbers. At lower frequencies, the storage compliance was found to correlate closely with the fraction of free (terminal) chains in the network, while the loss compliance could be correlated with either the free end fraction or with the fraction of chain entanglements not trapped into the network by primary valence crosslinks. At higher frequencies, approaching the plateau region of the viscoelastic spectrum, effects of network structural features diminished as contributions of untrapped entanglements became more pronounced. Networks prepared in presence of a diluent exhibited the same trends as pure gum vulcanizates. Carbon black produced a leveling effect, ascribed to a second entanglement entrapment mechanism by adsorption sites on the filler surface.

The Effects of High-Molecular-Weight Components on the Viscoelastic Properties of Polystyrene

The effects of molecular weight distribution (MWD) on the steady-state shear and the dynamic viscoelastic properties of high-molecular-weight (MW>97000) polystyrene have been examined with a Weissenberg rheogoniometer. Synthetic MWDs were produced by blending polystyrene components of defined MWs and narrow MWDs. Experiments were designed to examine the effects of small amounts of a high-MW component on the viscous and elastic properties of the melt.The frequency response of the storage and loss moduli, G′ and G″, and the storage compliance J′, are compared with the shear rate dependence of the shear stress τ12, the primary normal stress difference P11−P22, and the shear modulus G, as a function of the composition of the blend. At low deformation rates the dynamic and the steady-state shear properties are equal; G″=τ12, G′=(P11−P22/2, and G=1/J′. At higher deformation rates the steady-state shear properties are primarily controlled by the high-MW component while the dynamic properties reflect the response of each component. The zero shear viscosities of the blends and the narrow-MWD components exhibit the same dependence on the weight-average MW. Four predictions for the MWD dependence of the steady-state shear compliance Je° are compared with the experimental results. Although overestimating the magnitude in the maximum in Je°, only Graessley’s theory accounts for the observed behavior on either side of the maximum.The data is used to discuss the previously reported dependence of the characteristic relaxation time, τ=η°Je°, on the shape of the non-Newtonian flow response. The τ is found to be a measure of the time scale of the relaxation processes of the high-MW component.

Dynamic mechanical behavior of lightly crosslinked networks of poly(n‐butyl methacrylate) and poly(n‐butyl acrylate) in the rubberlike region

AbstractAn investigation was made of the dynamic mechanical behavior in the rubberlike region of poly(n‐butyl methacrylate) (PBuMA) and poly(n‐butyl acrylate) (PBuA) networks lightly crosslinked with ethylene dimethacrylate to concentrations from 10−6 to 10−4 mole/cm3. The measurements were carried out by use of an apparatus for low‐frequency forced vibrations working in the frequency range 2.5 × 10−4 to 1 Hz. With parameters c1 and c2 of the Williams‐Landel‐Ferry equation, obtained from data in the main transition region, the data did not reduce in the rubberlike region for the poly(butyl methacrylate) networks; the spread of the deviations decreases with increasing concentration of the crosslinking agent. Superposition could be achieved in all cases when a shift factor was used on the vertical axis. At sufficiently low reduced frequencies and at high temperatures the storage compliance decreases in both series of polymers with increasing concentration of the crosslinking agent as expected. At higher reduced frequencies and at higher temperatures of measurement, however, anomalous behavior was observed with uncrosslinked samples having a lower compliance than those crosslinked to a very low degree. This finding was explained as due to very long relaxation times of the untrapped entanglements present in the noncrosslinked polymer, which are absent in the same polymer crosslinked already to very low degrees. The retardation spectra of both PBuMA and PBuA exhibited secondary relaxation mechanisms which were shifted by four logarithmic decades toward higher retardation times in comparison with the primary retardation maximum.

Viscoelastic properities of cross-linked solutions of poly (β-hydroxyethyl methacrylate) in diethylene glycol. V. Influence of secondary low-temperature transitions

Abstract The influence of secondary low-temperature dispersions on the storage and loss compliances of four poly(β-hydroxyethyl methacry-late) solutions in diethylene glycol was studied by comparing previously published double-transducer data with isochronal torsion pendulum data measured starting from the temperature of liquid nitrogen. It was found that the contribution of the main transition as observed in the storage compliance J'p decreased with increasing swelling degree of the samples. For a sample swollen to volume fraction of polymer v2 = 0.168, log Jp changes only about one decade, from −6 to −7; the further decrease of log J′p from −7 to −10.5 corresponds to the secondary low-temperature dispersion characteristic for polar side chains—polar diluent systems. For the latter dispersion the Arrhenius plot obtained using the double-transducer data gave δH = 24 kcal/mole. It was concluded that in some special cases the role of low-temperature secondary dispersions can be of primary importance in infl...

Numerical calculation of creep compliance from dynamic data for linear viscoelastic materials

Numerical formulae are given for calculation of creep compliance from the known course of the storage and loss compliance with frequency for linear viscoelastic materials. These formulae involve values of the storage compliance and/or loss compliance at frequencies which are equally spaced on a logarithmic frequency scale. The ratio between successive frequencies corresponds to a factor of two. A method is introduced by which bounds for the relative error of those formulae can be derived. These bounds depend on the value of the damping, tanδ, at the angular frequency,ω 0, at which the calculation is performed. The lower this damping, the easier is the calculation of the creep compliance. This calculation involves either the value of the storage compliance at a frequencyω 0 = 1/t, and the values of the loss compliance in a rather narrow frequency region aroundω 0; or the value of the storage compliance at frequencyω 0, the value of the loss compliance at frequencyω 0/2, and the derivative of the storage compliance with respect to the logarithm of frequency in a frequency region aroundω 0.

The numerical calculation of storage and loss compliance from creep data for linear viscoelastic materials

Numerical formulae are given for calculation of storage and loss compliance from the course of the creep compliance for linear viscoelastic materials. These formulae involve values of the creep compliance at times which are equally spaced on a logarithmic time scale. The ratio between succeeding times corresponds to a factor of two. A method is introduced by which bounds for the relative error of those formulae can be derived. These bounds depend on the value of the damping, tan δ. The calculation of the storage compliance is easier with the lower damping values. This calculation involves the value of the creep compliance at time t0=1/ω, and that of its derivative with respect to the logarithm of time in a rather narrow region around t0. In contrast the calculation of the loss compliance is more difficult with the lower damping values. This calculation involves the value of the derivative of the creep compliance with respect to the logarithm of time in a broad interval around t0. © 1969 Dr. Dietrich Steinkopff Verlag.

Dynamic mechanical properties of cross‐linked rubbers. V. An approximate analysis of the modulus contributions of trapped and untrapped entanglements

AbstractUsing the criterion that a coupling entanglement in a polymer network is trapped if all four strands radiating from it terminate in chemical cross‐links (an approximation to a more rigorous treatment of Langley), an equation is derived relating the equilibrium modulus to the magnitude of the compliance in the frequency region where all coupling entanglements, whether trapped or not, contribute to the elasticity. The latter value is estimated from the storage compliance at the frequency where the storage compliance of the uncross‐linked polymer corresponds to the entanglement compliance derived from integration over the loss compliance. The theory agrees rather well with data on vulcanizates of natural rubber. For 1,4‐polybutadiene and styrene–butadiene rubbers, the agreement is somewhat less satisfactory, but the results support the hypothesis that the low‐frequency losses observed in lightly cross‐linked rubbers are due to relaxation of untrapped entanglements.

Viscoelastic properties of polybutadienes—linear and lightly crosslinked near the gel point

AbstractViscoelastic properties of four linear and three very lightly crosslinked polybutadienes (microstructure about 50% trans) were studied. Of the latter, two had not reached the gel point, and their molecular weight distributions were determined by sedimentation velocity analysis; the third was crosslinked just past the gel point, with only 32% gel fraction present. The crosslinking agent was sulfur. Complex shear compliances were measured over a frequency range from 0.1 to 1000 cps at temperatures from −70 to 30°C. with a Fitzgerald transducer and a Plazek torsion pendulum; and torsional creep measurements were made over time periods up to about three days. The creep data were converted to the corresponding dynamic viscoelastic functions at very low frequencies by conventional approximation methods. All data were reduced to 25°C. by shift factors calculated from a previously adopted equation of the WLF form. In the transition zone, the viscoelastic properties of linear samples were almost independent of molecular weight. The entanglement spacing, derived from the minimum in the loss tangent and the inflection in the storage compliance, was 130 to 160 chain atoms. The maximum in the retardation spectrum attributable to motions of individual network strands was closely similar to the corresponding maxima for more highly crosslinked vulcanizates previously studied, showing that even in the latter it is associated with entanglement network strands rather than strands between chemical crosslinks. For a linear sample with molecular weight 180,000, the retardation processes disappear at times beyond about 10 sec. at 25°C. With crosslinking short of the gel point (i.e., branching) the slow retardation processes are enormously increased and prolonged to longer times. With further crosslinking through the gel point and beyond, the slow retardation processes decrease progressively in magnitude. Qualitatively, this behavior resembles the sharp maximum in content of highly branched and aggregated molecular species which is predicted at the gel point by crosslinking statistics; but the slow processes (or low‐frequency losses) persist farther past the gel point than would be expected on this basis. The steady‐state compliances of the linear samples were smaller, but for a sample crosslinked short of the gel point were much larger, than the prediction of the Rouse theory modified for molecular weight distribution.

Dynamic Compressibility of Poly (Vinyl Acetate) and Its Relation to Free Volume.

The complex bulk compliance (dynamic compressibility) of a commercial sample of poly (vinyl acetate), AYAT, was measured at frequencies from 50 to 1,000 cycles per second, temperatures from 0 to 100 °C, and static hydrostatic stresses from 0 to 981 bars (gage pressure) using an alternating hydrostatic stress generated and detected by piezoelectric transducers mounted in an essentially noncompliant cavity with dimensions small in comparison to a wavelength. The above temperature range was more than sufficient to cover the dispersion region in which an inflection in the storage compliance and a maximum in the loss compliance were observed. The data were reduced to functions of reduced variables using the WLF Equations extended to include static pressure with the “universal” WLF Constants and dT/dP = 0.020 °C/bar. The difference in limiting compliances at zero and infinite frequencies was larger than that predicted from the dT/dP shift using the free volume concept. A discussion is presented on possible processes that might contribute to an excessive value between limiting compliances.

Dynamic Mechanical Properties of Stretched Natural Rubber

Abstract Dynamic shear compliance measurements on a natural rubber vulcanizate have been carried out from 50 to 5000 cps as a function of static elongation in the range 0–400%. Above 300% elongation several very large resonance dispersions appear in the region 1000–2000 cps, but at lower elongations no such resonances are observed. The appearance of mechanical resonance dispersion thus coincides with the development of oriented crystallinity in the stretched rubber as shown from x-ray diffraction data. On the other hand, the magnitudes of the dispersions decrease with increasing elongation above 300% whereas x-ray measurements indicate the amount or extent of crystallinity increases with stretching above 300% elongation. The resonances thus clearly decrease as the crystallinity becomes more perfect or more extensive and hence appear to reflect the imperfections of the crystalline regions formed by elongation rather than the crystallinity itself. Observed changes in the compliance spectrum with time at each elongation also favor this view. Broad retardation dispersions were observed when the rubber was allowed to contract back to elongations below 300% after long times above 300%. These are tentatively ascribed to an extended amorphous state in which the rubber molecules are kept partially extended by very small randomly oriented crystalline regions which persist for some time after retraction. Another interesting feature of the results is an apparent correlation between the percent crystallinity determined from x-ray diffraction and the reciprocal of the general level of storage compliance 1/J′ at frequencies well below or well above the dispersion region.

Dynamic Mechanical Properties of Polyethylene from 25° to 150°C

Measurements of the complex shear compliance (J* = J′ − iJ″) have been carried out for three polyethylene samples of increasing weight average to number average molecular weight ratios (M̄w/M̄n) of 19, 70, and 110. The measurements extended to temperatures well above the crystalline melting points of the polyethylenes and over a frequency range from 25 to 2500 cps. A sudden increase in compliance occurred at all frequencies at the melting points of each type of polyethylene; there were no marked differences in dynamic mechanical behavior below the melting points. Above the melting points values of storage compliance (J′) were very nearly the same for all the polyethylenes but differences in the loss compliance (J″) were found. In particular at higher frequencies (1300 cps) values of J″ diverged with temperature in ascending order as M̄w/Mn increased. Values of mechanical loss tangent (J″/J′) showed sudden jumps at the melting points but them was no evidence of maxima or peaks at these temperatures. Attempts to superpose the data in the melt were unsuccessful as values of J″ at various temperatures could not be shifted into coincidence even though values of J′ superposed fairly well. The superposition anomalies are tentatively ascribed to the presence of a secondary maximum in the loss compliance vs frequency characteristics of the materials while the secondary maximum, in turn, is attributed to either (1) a split molecular weight distribution with two separate maxima, or (2) a β mechanism involving the motion of very long branches.

Mechanical Resonance Dispersion in Metals at Audio-Frequencies

Measurements of the complex shear compliance (${\mathrm{J}}^{*}={J}^{\ensuremath{'}}\ensuremath{-}i{J}^{\ensuremath{'}\ensuremath{'}}$) of pure, polycrystalline lead, indium, and aluminum at closely spaced frequencies in the audio-frequency range have resulted in the discovery of multiple frequency dispersions of the resonance type. Values of the loss compliance (${J}^{\ensuremath{'}\ensuremath{'}}$) are found to have sharply defined maxima in the range from 100 to 5000 cps; values of the storage compliance (${J}^{\ensuremath{'}}$) rise to a maximum and then drop to pass through a minimum as a narrow (50 to 100 cps wide) dispersion region is traversed in the direction of increasing frequency. The shapes of the complex compliance vs frequency curves are thus similar to those found for the variation of the complex index of refraction with frequency (in absorption bands) at infrared and optical frequencies. The results indicate the presence of a dispersion mechanism different from those leading to the relaxation dispersions previously observed in connection with the mechanical properties of metals. Results on lead indicate that some of the resonances can be eliminated by high-temperature annealing of the sample material. The possibility that dislocation vibrations are responsible for the observed phenomena is considered, but no specific mechanism is found to explain the results. A number of dislocation theories are examined but these fail to predict any mechanical resonance dispersion at audio-frequencies. In addition to their theoretical implications, the results are believed to be of considerable practical importance since they indicate that certain materials when subject to mechanical vibrations of relatively low frequencies may undergo large changes in modulus at a particular frequency or frequencies.

Mechanical Resonance Dispersion in Crystalline Polymers at Audio-Frequencies

Measurements of the complex shear compliance (J*=J′—iJ″) of crystalline polymers (polytetrafluorethylene, linear polyethylene, and polyvinyl stearate) at closely spaced frequencies in the audio-frequency range have resulted in the discovery of multiple frequency dispersions of the resonance type. Values of the loss compliance (J″) are found to have sharply defined maxima in the range from 100 to 5000 cps; values of the storage compliance (J′) rise to a maximum and then drop to a minimum as a narrow (50 to 100 cps wide) dispersion region is traversed in the direction of increasing frequency. The results indicate the presence of a dispersion mechanism different from those leading to the usual relaxation dispersions observed in connection with the mechanical properties of polymers. The magnitudes of the mechanical loss maxima and the frequencies at which the effect is observed depend on the static stress history and state of the sample material as well as temperature. Similar results obtained for pure polycrystalline metals (lead, indium, and aluminum) lead to speculation that stress-induced motions of dislocations or other imperfections in the crystalline structure may be important in producing this type of dispersion. At present, however, there is no satisfactory explanation for these phenomena.