Rubber modulus, cross-link density

Many of the most floppy polymers have half-melted in this way at room temperature. The temperature at which this happens is called the glass temperature, Tq, for the polymer. Some polymers, which have no cross-links, melt completely at temperatures above T, becoming viscous liquids. Others, containing cross-links, become leathery (like PVC) or rubbery (as polystyrene butadiene does). Some typical values for Tg are polymethylmethacrylate (PMMA, or perspex), 100°C polystyrene (PS), 90°C polyethylene (low-density form), -20°C natural rubber, -40°C. To summarise, above Tc. the polymer is leathery, rubbery or molten below, it is a true solid with a modulus of at least 2GNm . This behaviour is shown in Fig. 6.2 which also shows how the stiffness of polymers increases as the covalent cross-link density increases, towards the value for diamond (which is simply a polymer with 100% of its bonds cross-linked. Fig. 4.7). Stiff polymers, then, are possible the stiffest now available have moduli comparable with that of aluminium. 

Compo- sition number Resin type/parts per 100 parts of mbber (phr) Sulfur (phr) Y Method of preparation Cross- link density, r/2 (moles X 10 per ml of mbber) Rubber particle size (pm) Young s modulus (MPa) Stress at 100% strain (MPa) Tens. str. (MPa) UlL elong. (%) Tens. set (%)... 

The underlying nonlinearity function m (A). which is independent of the type of deformation, is very similar for different amorphous rubbers. For SBR, it is independent of the cross-link" density over moderate changes in cross-link density (62) and independent of the temperature down to —40°C, a temperature where the modulus has increased by a factor of 2 to 3 over the room-temperature value (61). The function A) is insensitive to the presence of moderate amounts of carbon black filler for strains up to about 100% (63). 

Most elastomers are amorphous, but those with regular structures can crystallize when cooled to extremely low temperatures. Vulcanized soft rubber, which has a low cross-link density, when stretched crystallizes in a reversible process, and the oriented polymer has a high modulus (high stress for small strains, i.e., stiffness) and high tensile strength. 

The shear modulus of the rubber network is related to the molecular weight between cross-link points or M. The lower the molecular weight of chains between cross-links (network chains), the higher the cross-link density and the higher the modulus. This is shown in the following expression ... 

Elastomers are cross-linked macromolecules above the glass transition temperature. The cross-link density is the fundamental engineering quantity which, for instance, determines the modulus of elasticity. Usually, it is measured during vulcanization of well-defined rubber samples in a vulcameter by the moment necessary to perform a given torsional shear of the test sample. Swelling experiments can be performed alternatively, but are problematic for filled elastomers. Such measurements are based on the assumption that the measured quantity does not vary over the sample volume. Inhomogeneous cross-link densities can be determined from the surface hardness, but volumetric resolution is achieved by conventional methods only after cutting the sample. 

Cross-link densities were characterized above T by equilibrium modulus measurements employing rubber elasticity theory. The results indicate that this method yields surprisingly reasonable values. 

Based on studies of an homologous, endlihked, epoxy/amine network series, the simple theory of rubber elasticity has proved effective for determining reasonable cross-link densities from equilibria modulus measurements in the rubbery state. 

A typical entropy-elastic material is cross-linked natural rubber, ds-poly(l-methyl-1-butenylene) or cts-l,4-polyisoprene, as summarized in Fig. 5.166 (see also Fig. 1.15). Its extensibility is 500 to 1,000%, in contrast to the 1% of typical energy-elastic sohds. Natural rubber has a molar mass of perhaps 350,000 Da (about 5,000 isoprene monomers or 20,000 carbon backbone bonds) and is then vulcanized to have about 1% cross-links (see Fig. 3.50). A rubber with a Young s modulus of 10 Pa (depending on cross-link density) must be compared to its bulk modulus (= 1/p,... 

A mechanism for formation of chromium ion cross-linked anionic polyacrylamide gels has been discussed by Prud homme, et al Theories of rubber elasticity have shown that storage modulus G is proportional to the cross-link density v as shown below ... 

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