Dynamic shear oscillation

A torsional pendulum (Figure 5.80) is often used to determine dynamic properties. The lower end of the specimen is clamped rigidly and the upper clamp is attached to the inertia arm. By moving the masses of the inertia arm, the rotational momentum of inertia can be adjusted so as to obtain the required frequency of rotational oscillation. The dynamic shear modulus, G, can be measured in this manner. A related device is the dynamic mechanical analyzer (DMA), which is commonly used to evaluate the dynamic mechanical properties of polymers at temperatures down to cryogenic temperatures. 

Figure 11. Schematic diagram of a torsion pendulum for measuring dynamic shear modulus and damping. A typical damped oscillation curve is illustrated at the bottom of the figure (25)...

For measurements on polymer melts, an apparatus of the concentric cylinder type can be used. The internal cylinder of such an apparatus is preferably suspended between two torsion wires. One of them is fixed with its lower end in the bottom of the unit, the other is twisted sinusoidally with the prescribed angular frequency at its upper end. Phase difference and ratio of amplitudes of the oscillations of the upper wire end and of the internal cylinder are measured. From these measurements the dynamic shear moduli, as defined above, can be deduced, when the inertia of the system is taken into account. Such an apparatus has been developed by Den Otter (26) at this Institute, making use of earlier experiences, as made by several other authors. (See e.g. ref. [27, 28 and 29).]... 

To obtain measurements during oscillatory shear, the drive motor causes the fixture to oscillate from high to low shear rates deforming the sample. The transducer detects the periodic stress which is generated by the deformation. The magnitude of the stress is converted into dynamic shear moduli. 

Dynamic shear rheology involves measuring the resistance to dynamic oscillatory flows. Dynamic moduli such as the storage (or solid-like) modulus (G ), the loss (or fluid-like) modulus (G"), the loss tangent (tan 8 = G"IG ) and the complex viscosity ( / ) can all be used to characterize deformation resistance to dynamic oscillation of a sinusoidally imposed deformation with a characteristic frequency of oscillation (o). 

In dynamic-shear rheology an oscillating or dynamic shearing deformation is applied to the material, and dynamic-shear properties are measured. For the simplest case of an infinite parallel-plate system, Figure 3.68, one can define relevant properties as follows. 

Dynamic-shear measurements are of the complex viscosity rj ) as a function of the dynamic oscillation rate (o), at constant temperature. These tests are defined as isothermal dynamic frequency sweeps. Since the dynamic frequency sweeps are conducted at a given amplitude of motion, or strain, it is necessary to ensure that the sweeps are conducted in the region where the response is strain-independent, which is defined as the linear viscoelastic region. This region of strain independence is determined by an isothermal strain sweep, which measures the complex viscosity as a function of applied strain at a given frequency. This ensures that a strain at which the dynamic frequency sweep may be conducted in the linear viscoelastic region is selected. 

Torsional Pendulum Analysis (TPA). A freely oscillating torsional pendulum (7) operating at ca. 1 Hz was used for the determination of dynamic shear modulus of all cured samples as a function of temperature. The procedure recommended in ASTM-D-2236-70 was followed. 

Blends of PI with PB were dynamically sheared at large amplitude (y = 0.8) and frequency CO = 0.63 and 6.3 rad/s [Matsuzaka et al., 1997]. After a temperature jump, the spinodal decomposition (SD) was in-situ observed at the lower frequency, but not at the higher. In the latter case, after stopping the oscillation, a modified SD pattern emerged. The authors postulated that the dynamic flow induced a structure in miscible system, quite different from that that exists in the non-sheared specimens. 

Fig. 4.10. Frequency-dependence of the dynamic shear modulus G

The storage and loss shear moduli, G and G", vs. oscillation frequency ta, and the creep compliance J vs. time t, measured at each concentration and tanperature, were temperature shifted with respect to frequency or time. These temperature master curves at each concentration were then shifted to overlap one another along the frequency or time axis. The dynamic shear moduli master curves as a function of reduced frequency (oa ac are shown in fig. 4.4, and the shear creep compliance master curves as a function of reduced time tlajUc are shown in fig. 4.5. Master curves... 

The polymeric materials which show the nonlinear viscoelastic properties exhibit dynamic shear stress containing higher-order odd harmonics even under small-amplitude oscillation. These viscoelastic functions can accurately be determined only by many experiments of various strain amplitude, Vq. 

Small-amplitude oscillatory flow is often referred to as dynamic shear flow. Fluid deformation under d)mamic simple shear flow can be described by considering ttie fluid wiflun a small gap dX2 between two large parallel plates of which the upper one undergoes small amplitude oscillations in its own plane with a frequency velocity field within the gap can be given by d , = ydxj but y is not a constant as in steady simple shear. Instead it varies sinusoidally and is given by... 

Differential dynamic measurements have also been made with other kinds of oscillating deformations. In studies by Painter of small dynamic shear deformations superimposed on large static shear strains in the same direction, the dynamic storage modulus G of cross-linked natural rubber and poly(dimethyl sil-oxane) at 24 Hz was found to increase with increasing static strains in excess of 721 = 0.2. Here 721 is defined as u jxt in the notation of Chapter 1. After a history of large static strain, however, the change in G with static strain in a second (and subsequent) sequence of experiments was much smaller. These history-dependent effects are no doubt related to the behavior in repeated stress-strain cycles in extension mentioned in Section 3 above. Some experiments on torsional deformations of stretched rubber strips have been reported. " ... 

The modification of the surface force apparatus (see Fig. VI-4) to measure viscosities between crossed mica cylinders has alleviated concerns about surface roughness. In dynamic mode, a slow, small-amplitude periodic oscillation was imposed on one of the cylinders such that the separation x varied by approximately 10% or less. In the limit of low shear rates, a simple equation defines the viscosity as a function of separation... 

The Weissenbeig Rheogoniometer (49) is a complex dynamic viscometer that can measure elastic behavior as well as viscosity. It was the first rheometer designed to measure both shear and normal stresses and can be used for complete characterization of viscoelastic materials. Its capabilities include measurement of steady-state rotational shear within a viscosity range of 10-1 —13 mPa-s at shear rates of 10-4 — 104 s-1, of normal forces (elastic effect) exhibited by the material being sheared, and of an oscillatory shear range of 5 x 10-6 to 50 Hz, from which the elastic modulus and dynamic viscosity can be determined. A unique feature is its ability to superimpose oscillation on steady shear to provide dynamic measurements under flow conditions all measurements can be made over a wide range of temperatures (—50 to 400°C). 

Dynamic mechanical analysers, as discussed in chapter 9, can be constructed so that they can be used with unvulcanised materials and, hence, the in phase and out of phase components of modulus and the loss angle measured. The usual test piece geometries for cured rubbers are not convenient for the uncured materials where some form of oscillating shear is probably the best approach. This is the geometry used in cure meters discussed in the next section and such instruments have formed the basis for apparatus which measures dynamic properties from before and through the curing process. 

The Yerzley oscillograph is specified in ASTM D94519 and is shown schematically in Figure 9.7. It consists of a horizontal beam pivoted so as to oscillate vertically and in so doing deform the test piece mounted between the beam and a fixed support. A pen attached to one end of the beam records the decaying train of oscillations on a revolving drum chart. The dynamic deformation of the test piece can be superimposed on a static strain and the mode of deformation can be either shear or compression. The mass and, hence, the inertia of the beam can be varied by attached weights. 

This is the dynamic viscosity in small amplitude oscillatory shear which is the real component of the complex shear viscosity which is a function of the angular frequency of oscillation. 

The bulk rheological properties of the PFPEs, including the melt viscosity (p), storage modulus (G ), and loss modulus (G"), were measured at several different temperatures via steady shear and dynamic oscillation tests. Note that we denoted p as melt viscosity and r as solution viscosity. An excellent description of the rheology is available in Ferry [99]. 

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