Guth-Smallwood Relationship

In the mid-1940 s Guth (1945) and Smallwood (1944) developed a widely employed (Cohan, 1947 Kraus, 1965c, Chapter 4) equation expressing rubber reinforcement directly in terms of filler concentration. An important form of the equation can be written as 

Many other equations have been applied to rubber reinforcement. Perhaps the most important include the rule of mixtures laws, the Mullins equation, and the upper- and lower-bound relations (Broutman and Krock, 1967). Since these relationships often apply more satisfactorily to plastic rather than elastomeric composites, they will be broadly developed in Section 12.1. 

While the melting point of undeformed natural rubber is just over room temperature, the melting point of crystallites in stretched samples is much higher. At the highest elongations possible, natural rubber melts at about 100 C (Greensmith et a/., 1963, esp. p. 264), placing a temperature limitation on this mode of reinforcement. 

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The very presence of the high-modulus filler imparts a certain stiffness to the system, as given by the lower-bound-type Guth-Smallwood relationship. In normally reinforced systems, however, the thermodynamic reinforcement mode, by way of internal energy increases, is much more important. Thermodynamic reinforcement acts in addition to the Guth-Smallwood mode. While the presence of the filler tends to increase the crosslink density slightly, this is apparently not a major cause of elastomer modulus increases. 

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