John D. Ferry

Extensions of the Rouse Theory of Viscoelastic Properties to Undiluted Linear Polymers

The Rouse theory for viscoelastic properties of very dilute solutions is modified for application to undiluted linear polymers. With the effective segment mobility expressed in terms of steady‐flow viscosity, the theory is applied to polymers of rather low molecular weight essentially without further change. In high molecular weight polymers, it is assumed that for modes of motion with relaxation times above a critical value the effective segment mobility drops abruptly, in accordance with the effect of entanglement coupling on steadyflow viscosity as described by Bueche. Properties in both the transition region between glasslike and rubberlike consistency and the rubberlike or plateau region are predicted semiquantitatively with no arbitrary parameters. In an alternative application to the transition region, the average effective friction coefficient per monomer unit can be calculated for both linear and lightly cross‐linked systems.

The Relaxation Distribution Function of Polyisobutylene in the Transition from Rubber‐Like to Glass‐Like Behavior

The steady flow viscosity of a sample of polyisobutylene of viscosity‐average molecular weight 1.35 million, distributed by the National Bureau of Standards, has been measured from 15° to 100°C. Its logarithm is a linear function of 1/T2. Application of the method of reduced variables to dynamic mechanical data from −45° to 100°, previously reported for this polyisobutylene, yields composite curves reduced to 25°C for the real and imaginary parts of the complex compliance and complex shear modulus; the real part of the complex dynamic viscosity; and the mechanical loss tangent. The latter exhibits a broad and peculiarly asymmetric maximum. The reduced time scale extends from 1 to 10−9 sec. The reduction factors aT obtained in this way are slightly higher than those derived either from the viscosity or from stress relaxation measurements of Tobolsky and associates. The distribution functions of relaxation and retardation times have been calculated by second approximation methods and their detailed shapes a...

The free volume interpretation of the dependence of viscosities and viscoelastic relaxation times on concentration, pressure, and tensile strain

SummaryThe free volume interpretation of the dependence of relaxation times and viscosities on temperature can be extended to their dependence on concentration, pressure, and tensile strain. The coefficient which in theWilliams/Landel/Ferry equation represents the thermal expansion of the relative free volume has the following analogs, respectively: for concentration dependence, a constant closely related to the fractional free volume of the diluent liquid; for pressure dependence, the part of the compressibility attributable to collapse of free volume; for tensile strain, a constant proportional to 1–2μ, whereμ isPoissons ratio. The treatment is probably limited to conditions where the fractional free volume is not greater than 0.08. In the case of pressure dependence, some viscosity data on hydrocarbon and silicone liquids at high pressures are found to agree closely with the predicted form with reasonable values of the associated parameters.ZusammenfassungDie Interpretation der Abhängigkeit von Relaxationszeiten und Viskositäten von der Temperatur auf Grund des freien Volumens läßt sich in bezug auf Konzentrations-, Druck- und Zugkraftbereich erweitern. Der Koeffizient, der in derWilliams/Landel/Ferry -Gleichung die thermische Expansion des relativen freien Volumens repräsentiert, hat die folgenden bezüglichen Analoga: Für die Konzentrationsabhängigkeit eine Konstante, die eng mit dem fraktionellen freien Volumen der Verdünnungsflüssigkeit verkoppelt ist; für die Druckabhängigkeit den Anteil an Kompressibilität, der dem Verschwinden an freiem Volumen zugeschrieben werden muß; für die Zugkraft eine Konstante proportional zu 1–2μ, wobeiμ dasPoissonsche Verhältnis ist. Die Behandlung ist wahrscheinlich auf Bedingungen begrenzt, in denen das freie Volumen nicht größer als 0.08 wird. Im Fall der Druckabhängigkeit findet sich, daß einige Viskositätsdaten an Kohlenwasserstoff- und Silikonflüssigkeiten bei hohen Drucken sehr gut mit der vorausgesagten Form und mit vernünftigen Werten der assoziierten Parameter übereinstimmen.

Some approximate equations useful in the phenomenological treatment of linear viscoelastic data

Abstract Two new approximation methods for treatment of viscoelastic data are introduced. In the first, the relaxation (or retardation) spectrum is obtained from several values of the real (or imaginary) component of the complex modulus (or compliance) without differentiation. In the second, the relaxation (or creep) function is obtained by a very simple calculation from the real and imaginary parts of the complex modulus (or compliance)

Dynamic Mechanical Properties of Polyisobutylene

Values of complex shear compliance (and rigidity) have been obtained for a sample of polyisobutylene of viscosity‐average molecular weight 1.35×106 at 22 temperatures from −45 to 100°C and at about 20 frequencies from 30 to 5000 cps. Measurements were made by means of the transducer method of Fitzgerald and Ferry with a precision of ±2 percent and are estimated to be accurate to within ±3 percent as evidenced by agreement obtained between 7 different samples of widely varying dimensions. Values of the real part of the complex shear compliance J′ vary from 3.1×10−7 cm2/dyne at 99.9°C to 1.0×10−10 cm2/dyne at −44.6°C. The frequency dependence of the loss tangent J″/J′ indicates the presence of a low, broad maximum of 1.7 at −10 to −5°C and a second, smaller maximum at lower temperatures. This second maximum is also evident in a plot of J″/J′ at a fixed frequency against temperature. The wide temperature and frequency ranges of the measurements have provided an essentially complete experimental description o...

Dynamic mechanical properties of poly-n-butyl methacrylate☆

Abstract The real (J′) and imaginary (J″) components of the complex compliance have been measured between 15 and 3600 cycles/sec. in the temperature range from −14° to 130°C. for a fractionated poly-n-octyl methacrylate with molecular weight 3.62 × 106. The method of reduced variables gave superposed curves for J′ and J″ except for an anomaly at the lowest temperatures where J″ passes through a minimum and fails to superpose, suggesting the onset of a β mechanism. The temperature dependence of relaxation times follows the WLF form with fg (relative free volume at the glass transition temperature, Tg) = 0.027 and αf (thermal expansion coefficient of free volume) = 2.5 × 10−4 deg.−1. The relaxation and retardation spectra, reduced to 100°C., resemble in shape those for the other methacrylates studied except for minor features. The logarithm of the monomeric friction coefficient (in dynes-sec./cm.) is −5.78 at 100°C. and 2.54 at −20°C. (Tg). The relaxation and retardation spectra for the ethyl, n-butyl, n-hexyl, and n-octyl polymers at 100°C. can be brought into corresponding states (choosing the butyl as reference) by adjusting the magnitudes with a dilution factor representing the number of monomoles per unit volume and the time scales with a factor derived from the friction coefficients. Their positions then agree very closely at short times and moderately well in the transition zone, but near the maximum of the retardation spectrum there are residual differences, probably associated with the Bueche entanglements. The time scale adjustments can be fairly well accounted for by the differences in free volume at 100°C. as deduced by Rogers and Mandelkern.

Electron microscopy of fine fibrin clots and fine and coarse fibrin films: Observations of fibers in cross-section and in deformed states☆

Fine fibrin clots and coarse and fine fibrin films (both ligated and unligated), formed by shrinkage of clots in one dimension, were examined by electron microscopy. Specimens of clots were prepared by critical point drying and by embedding and sectioning; specimens of films were prepared by embedding and sectioning only. In the fine clots, network junctions appeared to be formed by fiber segments in which two or more protofibrils were gently twisted around each other for distances of the order of 200 nm and then diverged to give trifunctional branch points. This topology appeared to be preserved in the fine films. It is proposed that the strength of the junctions is primarily provided by the twisting topology, though reinforced by non-covalent bonding involving the B sites uncovered by thrombin. In coarse films, bundles of protofibrils, lying primarily in the film plane, had diameters of 40 to 200 nm and were gently twisted around each other to form thicker cables. Uniaxial stretching, up to 100%, of either fine or coarse film before fixing caused suprisingly extensive orientation of the protofibrils or bundles. However, random orientation was recovered if a stretched ligated film was allowed to retract to its original dimensions before fixing. In a stretched coarse film sectioned perpendicular to the stretch direction, fiber bundles could be seen in cross-section; these were roughly circular with scalloped edges. The changes with stretching and recovery are discussed in relation to possible mechanisms of deformation and elastic energy storage.

Second approximation methods for determining the relaxation time spectrum of a viscoelastic material

Abstract Second-approximation methods are given for deriving the distribution function of relaxation times of a viscoelastic material from experimental measurements of stress relaxation after sudden strain, stress relaxation after cessation of steady-state flow, the real and imaginary parts of the complex dynamic rigidity, and the real part of the complex dynamic viscosity. The numerical coefficients required for the calculations are tabulated, and the methods are illustrated by application to experimental data on solutions of polyvinyl acetate.

Method for determining the dynamic mechanical behavior of gels and solids at audio-frequencies; comparison of mechanical and electrical properties

A method is described for measurements of complex shear modulus or compliance on samples ranging from soft gels to stiff solids at −50 to +150°C. over the frequency range 25 to 5000 cycles/sec. By a proper choice of sample dimensions a precision of ±2% is obtained for values of compliance varying from about 10−5 to 10−10 cm.2/dyne although a single sample cannot be measured with this precision over the entire frequency-temperature range. Results are given for a sample of polyisobutylene from the National Bureau of Standards at 25.0°C., for a polyvinyl chloride-dimethylthianthrene gel (10% polymer by volume) at ten temperatures between −25 and 25°C., arid for polyvinyl chloride plasticized with dimethylthianthrene (40% polymer by volume) at ten temperatures between 5 and 60°C. Comparison of the dynamic mechanical results with measurements of complex dielectric constant previously reported for the polyvinyl chloride-dimethylthianthrene compositions shows that the dielectric constant and compliance give roughly the same type of temperature-frequency dispersion. While the frequency of maximum electrical loss tangent (ϵ″/ϵ′) and that of maximum mechanical loss tangent (J″/J′) are not the same at a given temperature, these maxima do appear to shift the same amount with temperature. Thus for each concentration the slopes of curves obtained by plotting the logarithm of the frequency of maximum loss tangent against the reciprocal of the absolute temperature are identical for the electrical and mechanical cases. This indicates that the activation energies for electrical and mechanical responses are the same. The maximum loss tangent for the mechanical case is much larger than the electrical maximum for both the 10 and 40% polyvinyl chloride concentrations.

Rheology of fibrin clots II: Linear viscoelastic behavior in shear creep

Abstract Creep of human fibrin clots in small shearing deformations has been studied over a time scale from 20 to 10 4 s. Most clots were prepared by the action of thrombin on purified fibrinogen, under various conditions of pH and ionic strength to produce networks with either coarse or line structure, and with or without ligation by calcium ion and fibrinoligase. Combination of data with previous results on dynamic viscoelastic properties provided a description of viscoelastic behavior over seven logarithmic decades of time or frequency scale. Fine, unligated clots showed very little creep over most of this range. Coarse, unligated clots showed substantial creep but this was suppressed by ligation. It is postulated that the basic fibril element formed by polymerization of fibrin units remains intact under stress but that, in coarse clots, some slippage of these elements within aggregated bundles occurs unless they are secured by α-α ligation. The elastic modulus (in a time scale of seconds) is proportional to the 1.5 power of fibrin concentration in unligated clots (the magnitude for fine clots being somewhat smaller than for coarse) and to the 2.3 power for coarse, ligated clots. Creep behavior with different shear strain magnitudes (up to 0.2) and tests of creep recovery by the Boltzmann superposition principle show strict adherence to linear viscoelastic behavior, indicating no basic structural changes during the experiments, provided the structure is fully developed before strain is imposed. If progressive ligation continues in the strained state, subsequent creep recovery is incomplete. Clots formed with ancrod instead of thrombin show a smaller modulus and much more extensive creep.