Elastically Effective Chains and Entanglements

The number of network chains active in the elastic behaviour of networks (elastically effective chains) can be obtained from the equilibrium behaviour of networks subjected to various types of stress as described in Chapters III and IV. Unfortunately, (i) the statistical 

Free chain ends (unreacted functionalities) reduce the number of active network chains in a network compared with the same network without free ends. Disregarding possible presence of loops and entanglements, C — 1 C crosslinks are necessary according to Flory (55) to connect C chains into one giant macromolecule. Additional crosslinks will be elastically effective. Their number is given by 

Tobolsky et al. (170) arrived at another expression by a consideration of the severance of the network chains in a network originally without chain ends. Every cut produces two free ends and two three-functional units so that there is a loss of one network chain per cut  

The discrepancy was explained by Scanlan (755) as well as by Mullins and Thomas (129). Because of the chain ends the network contains elastically effective four- and threefunctional units (in addition to ineffective two- and unifunctional units), which can be equally effective in constraints on four and three chains so that the number of elastically effective chains is given by 

The probability that a randomly chosen functionality is not a part of a tie borne by its parent unit, at a degree of conversion a, is 

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