Rubbery shear modulus

The moduli and Tg s of the networks formed from the bulk reactions of the five systems of Figure 9 are shown in Table IV(29). The first five columns define the systems, the next two give the experimental values of G(at 298K) and Tg, and the last three give the values of pr,c, Mc, and G/G°. The last quantity is the reduction in rubbery shear modulus on the basis of that expected for the perfect network(G°). G/G° is in fact equal to M /Mc. 

The network from system 3 is distinct from the rest, being a glass at room temperature and also having a rubbery shear modulus near the value expected on the basis of G°. Possible reasons for this high value of G/G° follow those discussed previously with reference to Mc/Mc and Figure 9. The more flexible chains of the aliphatic systems give lower values of Tg, resulting in elastomers at room temperature. 

The rubbery shear modulus is higher when hard segment crystallinity is present because of the filler effect. 

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

We conclude that high internal stresses are generated by simple shear of a long incompressible rectangular rubber block, if the end surfaces are stress-free. These internal stresses are due to restraints at the bonded plates. One consequence is that a high hydrostatic tension may be set up in the interior of the sheared block. For example, at an imposed shear strain of 3, the negative pressure in the interior is predicted to be about three times the shear modulus p. This is sufficiently high to cause internal fracture in a soft rubbery solid [5]. 

Note 4 Loose ends and ring structures reduce the concentration of elastically active network chains and result in the shear modulus and Young s modulus of the rubbery networks being less than the values expected for a perfect network structure. 

Filler-filler interaction (Payne effect) - The introduction of reinforcing fillers into rubbery matrices strongly modifies the viscoelastic behavior of the materials. In dynamic mechanical measurements, with increasing strain amplitude, reinforced samples display a decrease of the storage shear modulus G. This phenomenon is commonly known as the Payne effect and is due to progressive destruction of the filler-filler interaction [46, 47]. The AG values calculated from the difference in the G values measured at 0.56% strain and at 100% strain in the unvulcanized state are used to quantify the Payne effect. 

In addition to knowing the temperature shift factors, it is also necessary to know the actual value of ( t ) at some temperature. Dielectric relaxation studies often have the advantage that a frequency of maximum loss can be determined for both the primary and secondary process at the same temperature because e" can be measured over at least 10 decades. For PEMA there is not enough dielectric relaxation strength associated with the a process and the fi process has a maximum too near in frequency to accurately resolve both processes. Only a very broad peak is observed near Tg. Studies of the frequency dependence of the shear modulus in the rubbery state could be carried out, but there... 

This approach can be illustrated by unsaturated polyesters based on an almost equimolar combination of maleate and phthalate of propylene glycol, crosslinked by styrene (45 wt%) (Mortaigne el al., 1992). Six samples differing by the prepolymer molar mass were analyzed. The chain-ends concentration, b, was determined by volumetric analysis of alcohols and acids in the initial reactive mixture. Then, the system was cured, elastic measurements were made in the rubbery state at Tg + 30° C, and the shear modulus G was plotted against chain-ends concentration (Fig. 14.7). The following relationship was obtained ... 

In some cases, network structure is modified by aminolysis reactions25. An example is the polymer formed from diglycidylic ester of o-phthalic acid and diaminodiphenilmethane. Aminolysis makes the chain between crosslinks shorter and influences the properties of the polymer (dynamic shear modulus in a rubbery... 

For the films and conditions we have used, the transmission line and lumped element models give indistinguishable results. Fitting of the data of Fig. 13.7 yields G as a function of time. These values increase at short times (due to nucleation phenomena) to long time limiting values of G = 1.9 x 106 dyne cm-2 and G" = 3.0 x 108 dyne cm-2. These values of the shear modulus components show that, in dichloromethane, the PVF film is a very rubbery polymer in which there is considerable viscoelastic loss when the film thickness exceeds 1 p.m. 

Isothermal measurements of the dynamic mechanical behavior as a function of frequency were carried out on the five materials listed in Table I. Numerous isotherms were obtained in order to describe the behavior in the rubbery plateau and in the terminal zone of the viscoelastic response curves. An example of such data is shown in Figure 6 where the storage shear modulus for copolymer 2148 (1/2) is plotted against frequency at 10 different temperatures. 

Chain entanglements are the cause of rubber-elastic properties in the liquid. Below the "critical" molecular mass (Mc) there are no indications of a rubbery plateau. The length of the latter is strongly dependent on the length of the molecular chains, i.e. on the molar mass of the polymer. From the shear modulus of the pseudo rubber plateau the molecular weight between entanglements may be calculated ... 

The number of resonant peaks that can be measured is dependent on the loss factor of the material, but, typically, there are three to five peaks. As expected, the resonant peaks appear at higher frequencies in the glassy state than in the rubbery state. From the amplitude and frequency of each measured resonant peak. Young s modulus and loss factor are determined at the corresponding frequency and temperature. By assuming Poisson s ratio of 0.5, Young s modulus is converted to shear modulus. The loss factor in extension is assumed to equal the loss factor in shear. 

The data shows the shift in soft segment T from -48 C (100 percent 1,4-BDO) to 6 C (100 percent dMPD) as expected due to phase mixing. As anticipated, hard segment crystallization occurs when the DMPD content is less than 50 percent. The shear modulus data (Figure 9) show a steady decrease in the rubbery modulus as the DMPD content of the blend increases because the degree of hard segment crystallinity is decreasing, and these results... 

Instantaneously deformed high molar mass polymer melts (long polymer chains in their liquid state) behave at intermediate times as networks with well-defined values of shear modulus, called the plateau modulus Ge, which is independent of molar mass for long-chain polymers. This rubbery plateau is seen for all polymer melts with... 

The use of rubbers (particularly epoxy-terminated butadiene nitrile, ETBN, rubber or carboxy-termi-nated butadiene acrylonitrile, CTBN, rubber) to toughen thermoset polymers is perhaps the most widely explored method and has been applied with some measure of success in epoxy resins. Phase separation of the second rubbery phase occurs during cure and its incorporation in the epoxy matrix can significantly enhance the fracture toughness of the thermoset. Although the rubber has a low shear modulus, its bulk modulus is comparable to the value measured for the epoxy, ensuring that the rubber inclusions introduced... 

Goodier (42) showed that for a particle which possesses a considerably lower shear modulus than the matrix, the maximum stress concentration occurs at the equator of the particle. Rubbers are commonly found to undergo cavitation quite readily under the action of a triaxial tensile stress field. Thus, the microvoids are produced by cavitation around the rubbery particles during fatigue crack propagation, together with localized plastic deformation due to interaction between the stress field ahead of the crack and the rubber particles. 

The major, and most important, advantage of being able to use high values of the Poisson ratio is that this implies that the value of K of the rubber particle is high. Indeed, input values for of 1 MPa and v of 0.49992 imply a value of K of about 2 GPa, which is of the order expected for a rubbery polymer (19). Thus, the values of the bulk modulus, K, and of the shear modulus,... 

If the rubbery equilibrium shear modulus does not show evidence of crosslink mobility for some other family of thermosets, then the factor (fav-2)/fav would drop out of Equation 6.18, so that the dependence of Tg on network architecture would be expressed more simply, just in tenns of Mc. It could, in that case, be expressed equivalently as in Equations 6.16 and 6.17. For thermosets known to or expected to manifest crosslink mobility, it should, then, generally be possible to combine the functional form of the dependence on fav shown in Equation 6.18 with Equation 6.16, to obtain Equation 6.19 which is an alternative form for the relationship for the Tg of thermosets manifesting crosslink mobility. 

B.3. Structure-Property Relationships for Rubbery Polymers ll.B.3.a. Shear Modulus... 

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