Oscillatory frequency

The RMS-800 provides steady-shear rotational rates from 10 to 100 rad/s and oscillatory frequencies from 10 to 100 rad/s. An autotension device compensates for expansion or contraction. With the standard 25- and 50-mm parallel plates, the viscosity range is 50-10 mPa-s, and the shear modulus range is 8 x 10 to 10 N/m. These ranges can be expanded with nonstandard plates, cones, and a Couette system. The temperature range is 20-350°C (-150 0 optional). 

In the previous subsection, the forcing frequency was exactly twice the natural oscillatory frequency. Thus the motion around one oscillation gives exactly two circuits of the forcing cycle for one revolution of the natural limit cycle. The full oscillation of the forced system has the same period as the autonomous cycle and twice the forcing period. The concentrations 0p and 6r return to exactly the same point at the top of the cycle, and subsequent oscillatory cycles follow the same close path across the toroidal surface. This is known as phase locking or resonance. We can expect such locking, with a closed loop on the torus, whenever the ratio of the natural and forcing... 

The behaviour and magnitude of the storage and loss moduli and yield stress as a function of applied stress or oscillatory frequency and concentration can be modelled mathematically and leads to conclusions about the structure of the material.3 For supramolecular gels, for example, their structure is not simple and may be described as cellular solids, fractal/colloidal systems or soft glassy materials. In order to be considered as gels (which are solid-like) the elastic modulus (O ) should be invariant with frequency up to a particular yield point, and should exceed G" by at least an order of magnitude (Figure 14.2). 

Several companies supply density equipment which was considered suitable for automatic, continuous operation with sufficient precision for calculation of polymerization conversion. These break down into three classes based on mode of operation y-ray absorption, oscillatory frequency of a sample filled tube, and mass measurement at fixed volume. Only one of these, an oscillator-based system distributed by Mettler Instrument Corp. (representing Anton Paar Ag.) has models with dead volumes small enough for laboratory scale experimentation. The other units generally also suffered from narrow density spans when the precision was sufficient for conversion studies. Table... 

Figure 4.17. Linear melt-state rheological properties as a function of oscillatory frequency (a) storage modulus, G and (b) loss modulus, G" (c) Dependence of complex viscosity on temperature for ABS nanocomposites. Reprinted with permission from ref (68).

While the dynamic experiments described above are to be conducted in the linear viscoelastic range, another experiment can be conducted in which the results obtained in the non-linear range are useful. With a controlled-stress rheometer, one can conduct an experiment in which the stress is increased continuously at a constant oscillatory frequency, say 1 Hz. Results obtained in such an experiment are shown schematically in Figure 3-40. As the stress is increased continuously, initially, G and G" remain relatively constant until at a critical value of stress, Oc, the magnitude of G decreases sharply and that of G" also decreases not as sharply after a slight inerease. One may also use the value of the applied stress at which the curves of G and G" intersect... 

Lopes da Silva et al. (1993) obtained the flow data by increasing the shear stress from zero to a maximum value, that depended on the concentration of the sample, and reducing it over the same range in a continuous manner. The time taken for the up and down curves was standardized. For examining the correlation between dynamic and steady shear properties, the down flow curves (decreasing shear stress) and the dynamic data obtained also by decreasing the oscillatory frequency (from 10 to 0.01 Hz) were used. 

Typically, in dynamic oscillatory testing, a sinusoidal (oscillatory) small-amplitude stress is applied to the sample and the mechanical response measured as functions of both oscillatory frequency and, in some instances, temperature. 

Generally, the operative frequency range for the torsional pendulum method is 0.01 to 50 Hz, the upper limit of the frequency defined by the dimensions of the oscillatory frequency relative to the dimensions of the sample. At higher investigative temperatures, the polymeric materials may undergo extensional deformation (creep) due to the weight of the inertia bar. Under such circumstances, a modified torsion pendulum apparatus may be used in which an inertia disk is attached directly onto the end of the sample (12). In addition, this method is frequently employed to measure the torsion modulus at low frequencies. 

In the second category, the viscoelastic properties of polymeric materials, particularly orientated polymers and composites, may be determined by the application of oscillatory frequencies in the megahertz frequency range. These methods are more complicated in design, and the interpretation of results is more complex. Further information on these methods is given in the excellent texts by Ward (11) and Ward and Hadley (12). 

If V and the form of

repulsive curve Vi can be determined by an RKR-like method that computes individual turning points (Child, 1973, 1974). This method is useful for obtaining an initial approximation for the repulsive potential. However, if only a few experimental r -values are known, it is difficult to identify unambiguously the oscillatory frequency of T versus v. For example, in Fig. 7.20 the number of vibrational levels sampled is insufficient to determine the actual shape of r . 

The result expressed as a function of the oscillatory frequency w is often referred to as the viscoelastic spectrum. Because the polymeric liquid has both the viscous and elastic properties, the time dependence of the induced stress will not be totally either in phase or out of phase with the oscillatory rate-of-strain. Substituting Eq. (4.31) into Eq. (4.22), we obtain the stress... 

Quartz Crystal Microbalance. This experiment depends on the manner in which the resonant frequency of an oscillating quartz crystal piezoelectric element responds to a change in mass on the Interface perpendicular to the oscillatory motion. Typically the oscillation is excited between facing metal film electrodes on opposite side of the crystal as shown in Figure 2(top). A change in mass at one of the Interfaces produces a proportional oscillatory frequency decrease. 

Fig. 10.45 Transition temperature as a function of oscillatory frequency during temperature ramp with a rate 1 °C/min. The triangle point at zero frequency is the transition temperature determined from isothermal frequency sweep (Wei et al. 2011)...

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