Sinusoidal oscillations Viscosity

Dynamic melt viscosity studies on the star blocks and a similar triblock were carried out using a Rheometric Mechanical Spectrometer (RMS) (Rheometrics 800). Circular molded samples with -1.5 mm thickness and 2 cm diameter were subjected to forced sinusoidal oscillations (2% strain) between two parallel plates. The experiment was set in the frequency sweep mode. Data were collected at 180 and 210 °C. 

The dynamic melt viscosity measurements of select star blocks and a similar triblock were carried out on a rheometric mechanical spectrometer, RMS. Circular molded samples of 2 cm diameter and -1.5 mm thickness were subjected to forced sinusoidal oscillations. Dynamic viscosities were recorded in the frequency range of 0.01-100 rad/s at 180 °C. Figure 10 shows the complex viscosities of two select star blocks and a similar linear triblock. The plots showed characteristic behavior of thermoplastic elastomers, i.e., absence of Newtonian behavior even in the low frequency region. The complex viscosity of the star block... 

The viscosity of a macromolecular solution can undergo changes when subjected to a periodic shear wave of frequency, w, instead of a steady-state shearing stress. The response of the particles to such a sinusoidally oscillating shear can be expressed in terms of a complex viscosity, rt -. 

In reality, neither the viscosity p (= G"/(q) nor the storage modulus G correctly defines the relationship between tensions and deformations except in the case of pure sinusoidal oscillations (9). In fact, what is measured in free oscillations is the dynamic modulus not at a real frequency but at a complex frequency. 

For fluids of low viscosity such as dilute polymer solutions, it is customary to describe the viscoelasticity with the complex modulus and the experiments have been performed in simple shear flow. The sinusoidally oscillating simple shear flow employed in this paper is illustrated in Fig. 1.1. The velocity of the fluid in the cartesian coordinate... 

Pig. 1. (a) When a sample is subjected to a sinusoidal oscillating stress, it responds in a similar strain wave, provided the material stays within its elastic limits. When the material responds to the applied wave perfectly elastically, an in-phase, storage, or elastic response is seen (b), while a viscous response gives an out-of-phase, loss, or viscous response (c). Viscoelastic materials fall in between these two extremes as shown in (d). For the real sample in (d), the phase angle S and the amplitude at peak k are the values used for the calculation of modulus, viscosity, damping, and other properties. 

In order to get to higher shear rates, it is customary to change over to a dynamic mode, in which the steady rotation is replaced by a sinusoidal oscillation, performed at a constant strain but varying frequencies. Viscosities at frequencies of as high as 500 rad/s are achievable with this... 

Most adsorbed surfactant and polymer coils at the oil-water (0/W) interface show non-Newtonian rheological behavior. The surface shear viscosity Pg depends on the applied shear rate, showing shear thinning at high shear rates. Some films also show Bingham plastic behavior with a measurable yield stress. Many adsorbed polymers and proteins show viscoelastic behavior and one can measure viscous and elastic components using sinusoidally oscillating surface dilation. For example the complex dilational modulus c obtained can be split into an in-phase (the elastic component e ) and an out-of-phase (the viscous component e") components. Creep and stress relaxation methods can be applied to study viscoelasticity. 

Viscosity versus shear rate for polystyrene at 200°C, using capillary, cone and plate, and sinusoidal oscillations with... 

First normal stress coe i-cient by steady cone and plate by sinusoidal oscillation, (2G /w. eq. 4.2.5), and integrating capillary viscosity data using eq. 6.5.1. 

Errors introduced in transient viscosity measurements (broken lines) as a result of instrument inertia can be eliminated by an active control loop (—). (a) Stress ramp 0-W seconds for a 5 mPa S Newtonian standard. Adapted from Franck (1992). (b) Sinusoidal oscillations on a 100 mPa-s standard. 

Rotor inertia affects all transient measurements on low viscosity fluids. Figure 8.2.1 lb illustrates the problem in sinusoidal oscillation testing. Even for this relatively high viscosity standard, it was not possible to obtain accurate data above about 0.2 rad/s without compensating for inertia. The most difficult corrections are for step changes in torque like start-up and recovery. Even for these cases, Franck (1992) reports that his correction software yields true values less than 2 seconds after the step for a very low 5 mPa s, viscosity standard. 

Viscosity versus shear rate and shear stress can also be determined at various temperatures with automated cone-plate or parallel plate analyzers, such as supplied by Rheometrics and others. Typically, a sample is subjected to a sinusoidal oscillating strain and the resultant torsional force measured. Strain gauges lead to output of elastic modulus, loss modulus and tan A. These can be obtained over a wide frequency range at a given temperature, or over a broad temperatore range, showing first- and second-order transitions. Typically, the equipment is quite expensive. 

The TA Instruments CSL2 rheometer can perform low frequency oscillatory measurements as well as steady-state viscosity determinations, even though it has a simple mechanical system. The sinusoidal wave form is generated mathematically in the computer rather than with an electromechanical drive system. The stress is controlled, and the resulting strain is determined and stored in memory. The computer analyzes the wave form and calculates the viscosity and elasticity of the specimen at the frequency of the test. As of this writing (1996), the oscillation software covers a frequency range of 10-4 -40 Hz. This range could be increased as faster software and computers become available. 

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). 

Problem 3-27. Oscillating Cylinder as a Viscometer. Consider a cylinder immersed in a large bath of fluid with kinematic viscosity v that rotates sinusoidally about its axis with angular velocity Qz = sin( >t). The cylinder has a radius R, length L, and L/R p> 1. It is... 

Normally pressure-driven rheometers are used only to measure steady shear viscosi. However, several devices have been developed that oscillate the flow rate sinusoidally (Thurston, 1961 Brokate and Cast, 1992). Typically oscillations are large amplitude and the strain field is nonhomogeneous, so G and G" cannot be measured directly. However, such rheometers have been shown to be sensitive to structure in low viscosity liquids ( filastic, 1992). 

The cone-n-plate viscometer can be used for oscillatory shear measurements as well. In this case, the sample is deformed by an oscillating driver which may be mechanical or electromagnetic. The amplitude of the sinusoidal deformation is measured by a strain transducer. The force deforming the sample is measured by the small deformation of a relatively rigid spring or tension bar to which a stress transducer is attached. Because of the energy dissipated by the viscoelastic polymer melt, a phase difference develops between the stress and the strain. The complex viscosity behavior is determined from the amplitudes of stress and strain and the phase angle between them. The results are usually interpreted in terms of the material functions t, G, G", and others [21-28]. 

Figure 6.8 shows the variation in rotational viscosity ( y ) with time t. The most remarkable feature of the result in (6.56) is that, as a function of time, the square of the amplitude of the rotational viscosity y q ) oscillates sinusoidally (Fig. 6.8). After rising to a maximum of [ (t-Tc)-kK ]Y 2, it drops back to zero. Since, (6.56) is obtained by comparing (6.55) with (6.45), the maximum value allotted to ( y ) must not exceed one, else the perturbative method will be invalid and hence, (6.56). This result is highly surprising. At times t = where n = 1, 2, 3... the FLC molecules certain to have almost infinite rotational mobility resulting into zero rotational viscosity as evident from Fig. 6.8. Therefore, the applied field should not keep for a longer period of time but it should turn off after an interval of time for maximizing the chances to produce finite rotational torque on the FLC director and as a result to obtain the system with finite rotational viscosity. Once the field is turned off, this reduced viscosity seems to produce a finite but weaker torque at the memory state and thereby, promotes multistability and resolution in the memory states as proposed earlier.

Many time-dependent flows have been studied by polymer chemists. A particularly important one is the small-amplitude sinusoidal shear flow (Figure 2b) where = Re(y e " y, Vy = 0, v = 0, in which is in general a complex quantity and co is the frequency of oscillation the notation Re means the real part of . Then because the amplitude of the oscillation is vanishingly small, the shear stress is also sinusoidal Ty = Reltyj e, where Zy is complex. We now define a complex viscosity by... 

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