Ideal Rubber Elasticity

The first law of thermodynamics states that the change in internal energy of an isolated system A is equal to the difference of the heat added to the system Q and the work done by the system W [23]  

From the second law, we know that an increment of entropy d5 is equal to an increment of heat d(2 added reversibly at temperature T  

If we stretch a piece of rubber with force/a distance (dL) we also do work involving pressure and volume  

Combining all these contribntions, we have (differentiating Equation 9.55) 

With the introduction of several assumptions, the chief one being that V does not change greatly on stretching (a good assumption for rubber), the thermodynamic equation of state for rubber is 

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For an ideal network T is in the numerator of the formula for E, so that E is proportional to the absolute temperature. The log E - T curve thus shows a positive slope (not a straight line because of the log-scale but slightly curved upward). In reality this simple picture is often disturbed by deviations from ideal rubber-elastic behaviour. 

The average molecular weight between crosslinking points can be estimated, based on Ideal rubber elasticity theory In which... 

For a hair, when it is 2 < 1.25, the value of fgjf is independent of I and is 0.11. Also, for the hair where the SS bond is decreased by 14% by a reduction reaction, the fgjf value is 0.08 for k < 1.55 and the hair shows gel-like entropy elasticity [54]. The fgjf value of natural rubber is 0.18-0.25 [55]. In the diluted system using this special mixed solvent, because even a fiber rich with SS bonds such as hair shows almost ideal rubber elasticity at a large deformation, quantitative analysis of the crosslinks is done using this phenomenon. 

For the higher-order structure formation and properties of hair, the results on the crosslinking and property development by SS bond obtained by both chemical and physical analysis have been described. Fibrous proteins such as hair are found to show nearly ideal rubber elasticity in 8M LiBr/BC dilute solution. Starting finm the elucidation of the number of crosslink points and crosslink pattern from high elongational curves, the... 

From the DLTMA curves, the authors reported that besides the likely relaxation effect observed at -50°C (see Fig. 8), the silicone rubber behaves as an ideal rubber-elastic solid over the whole temperature range. The polysulfide and polyurethane sealants, on the other hand, exhibit viscous flow at temperatures higher than 150°C. The polyacrylate sealant will soften above 50°C indicating a pseudoplastic behavior. 

Eqnation 9.51 differs from the result of ideal rubber elasticity presented below by a factor of 4/5. The reasons for this factor are complexities in the model [19] that are beyond the scope of this text, but one would expect a factor of less than unity becanse of chain ends that do not contribute to the elasticity of the entangled polymer network. Given that the plateau modulus is independent of molecular weights, M, for M > Me, the value of in Equation 9.51 is a constant for a given melt but will vary from polymer to polymer depending on the structural properties of the polymer such... 

Flory, who developed the theory of ideal rubber elasticity presented above, introduced early on a correction to account for the fact that in the random cross-linking of a melt of polymer chains with a number-average molecular weight there will be pendant inelastic strands at the ends of the original uncross-linked chains [23], The number of moles of original chains is given by... 

For small deformations, oc/n 1, the above equation reduces to the ideal rubber elasticity formula of Equation 9.70 when one keeps only the lowest term in Equation 9.77. For higher deformations, Equation 9.78 gives much better agreanent with experimental results as it has the additional parameter... 

Note the analogy to Equation 9.70 where the phenomenological coefficient 2Cj can be equated to RTN, the elastic modulus of the ideal rubber elasticity model. If we also keep the next term in the series in Equation 9.85, for i = 0 and j = 1, we get... 

Coarse-grained molecular models Bead-spring model Generalized Maxwell model Terminal relaxation Hydrodynamic interaction Ideal rubber elasticity Packing length... 

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