Modulus dynamic shear

Dynamic shear moduli are conveniently determined with automated equipment, for instance, with the torsion pendulum. However, moduli derived from dynamic tests are often higher than the results from static tests for lack of relaxation. Examples are shown in Table 3.3. Young s moduli of the polymers A, B, C, D, derived from tensile tests (frequency 0.01 Hz) are compared with shear moduli S determined with the torsion pendulum (frequency > 1 Hz). For rubberlike materials is 3S/E = 1, according to Eq. 

Recently, an electrorheological effect, i.e., an increase in the viscosity and dynamic shear moduli of lecithin/n-decane solutions in the presence of small amounts of polar additives (water or glycerol) when an external electric field is applied to the system, has been observed [65]. 

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).]... 

For the dynamic shear moduli one obtains the well-known expressions for a linear elastico-viscous material, viz. ... 

With the aid of these relaxation times other rheological properties like normal stresses, flow birefringence and dynamic shear moduli can be calculated. A more detailed discussion of this procedure wiE be given in Chapter 4. 

Experimentally, the dynamic shear moduli are usually measured by applying sinusoidal oscillatory shear in constant stress or constant strain rheometers. This can be in parallel plate, cone-and-plate or concentric cylinder (Couette) geometries. An excellent monograph on rheology, including its application to polymers, is provided by Macosko (1994). 

Fig. 2.17 Frequency dependence of dynamic shear moduli computed using a model for the linear viscoelasticity of a cubic phase based on slip planes, introduced by Jones and McLeish (1995). Dashed line G, solid line G".The bulk modulus is chosen to be G — 105 (arb. units). The calculation is for a slip plane density AT1 - 10 5 and a viscosity ratio rh = = 1, where rjs is the slip plane viscosity and t] is the bulk viscosity. The strain...

Fig. 4.28 Dynamic shear moduli as a function of temperature for a PS-PI diblock (Afw = 60kgmoL1, 17wt% PS) (points) in dibutyl phthalate (

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. 

FIG. 13.13 The dynamic shear moduli, G and G", for a high molecular weight polymer, as functions of the angular frequency. Classification is also indicated. Analogous to Ferry (1970) and Te Nijenhuis (1980, 2006). 

FIG. 15.12 Double logarithmic plot of the dynamic shear moduli G and G" vs. angular frequency reference temperature of 170 °C. After Gortemaker (1976) and Gortemaker et al. (1976). 

In Fig. 15.27, the transient extensional viscosity of a low-density polyethylene, measured at 150 °C for various extensional rates of strain, is plotted against time (Munstedt and Laun, 1979). Qualitatively this figure resembles the results of the Lodge model for a Maxwell model in Fig. 15.26. For small extensional rates of strain (qe < 0.001 s ) 77+(f) is almost three times rj+ t). For qe > 0. 01 s 1 r/+ (f) increases fast, but not to infinite values, as is the case in the Lodge model. The drawn line was estimated by substitution of a spectrum of relaxation times of the polymer (calculated from the dynamic shear moduli, G and G") in Lodge s constitutive equation. The resulting viscosities are shown in Fig. 15.28 after a constant value at small extensional rates of strain the viscosity increases to a maximum value, followed by a decrease to values below the zero extension viscosity. 

Interfacial shear properties can be determined either by steady state or by oscillatory measurements (see sec. 3.6f). In the latter case dynamic shear moduli are obtained in the former the interfacial shear modulus or the interfacial shear viscosity is obtained. 

Still another relationship between experimental parameters is a direct consequence of the Boltzmann superposition principle. We will derive the equations relating the shear stress relaxation modulus G(t) to the in-phase and out-of-phase dynamic shear moduli G oi) and G"(co) starting from equation (2-46)... 

Figure 9.5 Master curves of dynamic shear moduli at 463K for (a) iPP/EHR33 (70/30) and (b) iPP/ EHRS 1 (70/30). (From Reference 24 with permission from John Wiley Sons, Inc.)...

Fig. 245. Dynamic shear moduli vs angular frequency for a 1% gellan solution at 25 °C without and with 50mmol/l NaCl circles without NaCl squares with 50 mmol/1 NaCl (O) and ( ) G ( ) and ( ) G" (constructed from data presented in [590])...

Figure 2.19 Storage and loss dynamic shear moduli (G and G ) for the reaction mixture BFPA, plotted logarithmically as a function of total reaction time at 30 "C. Reproduced with permission from A. Zlatanic, B. Dunjic and J. Djonlagic, Macromolecular Chemistry and Physics, 1999,200, 2048 1999. 1999,...

Samples of H-H and H-T PS were also subjected to the measurements of the dynamic shear complex viscosity and dynamic shear moduli at 160° and 190°C (53). At lower shear stress the behavior of the H-T is essentially Newtonian. The departure from the Newtonian behavior occurs above 10 dyn/cm. On the other hand, the behavior of the H-H PS is non-Newtonian even at 160°C. and at low shear stresses of 10 dyn/cm. The melt viscosity of H-H PS decreases more rapidly with stress as does the melt viscosity of the H-T polymer. As temperature and stress is increased, the rheological behavior of the two polymers are the same (as can be seen at 190°C.). The dynamic shear storage modulus reveals also a small but significant difference in the rheological behavior of H-T and H-H PS as the G with u for the H-H PS is smaller than for the H-T polymer. Results from the melt rheology studies also indicate as does solution behavior that the polymer chain in H-H PS is stiffer than is H-T PS (53). 

Figure 7 Dynamic shear moduli recovery for a 19.8 VB wt.% composite with even fabric distribution heated to 250°C at heating/cooling rates of 3.6 C/min (upper plot) and 11.1 °C/min (lower plot). Shown are data for G (— —) and G (— —) during heating and G (— —) and G (—O—) during cooling. Frequency = 10 rad/s. Strain =1%.

Rheological measurements were performed in a stress rheometer fixture with a 2-cm cone and plate having a 1° cone angle and gap of 27 pm. Dynamic shear moduli were measured at 0.5% strain between 0.1 and 100 rad/s. Creep compliance was measured with a constant applied stress in the range of 0.1 to 5 kPa. Both measurements were performed over a series of temperatures to obtain data for time-temperature superposition. 

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... 

FIGURE 4.4 Dynamic shear moduli temperature-concentration shifted master curves for PFOM with dissolved PFPE Zdol (a) storage modulus and (b) loss modulus. 

Figure 2. Dynamic shear moduli vs. time for the cure of B2 with at 85°C, r=1.0. Measurements at 5% strain at ti3 10 rad sec.

In this work [91], the effects of the quality of carbon black dispersion on the dynamic shear moduli of uncured, carbon-black-filled, natural rubber compositions were investigated. Effects due to changes in temperature,... 

B. Dynamic Shear Moduli of Unvulcanized Samples as Functions of Temperature, Frequency, Strain Amplitude, and Degree of Carbon Black Dispersion... 

Fig, 21 Effect of a large range of changes in shear-strain amplitude on dynamic shear moduli. (Adapted from Ref. 58.)... 

Several theories based on elongated rigid axisymmetric models for a macromolecule predict frequency dependence of the dynamic shear moduli and intrinsic viscosities which can be expressed in the following form ... 

Dense grafting of side chains onto linear backbones, and homopolymerization of macromonomers, are both used to synthesize macromolecular brushes. Steric repulsion of the side chains forces the main chain into an extended wormlike conformation, resulting in liquid-crystalline phases, and lower dynamic shear moduli than linear flexible coils in concentrated solutions [93, 94]. Densely grafted polymeric brushes on sliding surfaces have been found to reduce friction, and therefore have potential for providing biolubrication for artificial implants [95]. 

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