Entanglements to the modulus

Equation (29) shows that the modulus is proportional to the cycle rank , and that no other structural parameters contribute to the modulus. The number of entanglements trapped in the network structure does not change the cycle rank. Possible contributions of these trapped entanglements to the modulus therefore cannot originate from the topology of the phantom network. 

The different forms of the modulus are still widely disputed and different opinions are expressed in the literature. Hsu and Mark " " prepared networks of polybutadiene by endlinking. According to dynamic measurements polybutadiene has a high plateau modulus in viscoelasticity and one can expect a strong contribution to the modulus from entanglements. In this study experiments have been fitted to the stress-strain relations with the Flory constraint-fluctuation model as in ref. 245. The authors concluded that there is no contribution from entanglements to the modulus. The same conclusion was drawn in refs. 245-248. 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

Data obtained by various investigators (13,14,18,19) indicate that trapped entanglements commonly contribute to the modulus. To represent the modulus, an equation has been used which, if all junctions are trifunctional, becomes ... 

Starting from simple statistical considerations concerning the coupling between similar or dissimilar chains, a model was constructed for the architecture of a class of IPN where interaction among phases occurs on a molecular level. It was shown(Equation 13) that the contributions to the modulus due to crosslinks are subject to linear additivity, while a square root additivity rule holds for the contributions due to entanglements. 

Through use of classical network theories of macromolecules, G has been shown to be proportional to crosslink density by G = nKT -i- Gen, where n is the nnmber density of crosslinkers, K is the Boltzmann s constant, T is the absolnte temperature, and Gen is the contribution to the modulus because of polymer chain entanglement (Knoll and Prud Homme, 1987). The loss modulus (G") gives information abont the viscous properties of the fluid. The stress response for a viscous Newtonian fluid would be 90 degrees out-of-phase with the displacement but in-phase with the shear rate. So, for an elastic material, all the information is in the storage modulus, G, and for a viscous material, aU the information is in the loss modulus, G". Refer to Eigure 6.2, the dynamic viscosities p and iT are defined as... 

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