Entanglements trapped

The cycle rank completely defines the connectivity of a network and is the only parameter that contributes to the elasticity of a network, as will be discussed further in the following section on elementary molecular theories. In several other studies, contributions from entanglements that are trapped during cross-linking are considered in addition to the chemical cross-links [23,24]. The trapped entanglement model is also discussed below. 

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. 

Experimental determinations of the contributions above those predicted by the reference phantom network model have been controversial. Experiments of Rennar and Oppermann [45] on end-linked PDMS networks, indicate that contributions from trapped entanglements are significant for low degrees of endlinking but are not important when the network chains are shorter. Experimental results of Erman et al. [46] on randomly cross-linked poly(ethyl acrylate)... 

The deformation of polymer chains in stretched and swollen networks can be investigated by SANS, A few such studies have been carried out, and some theoretical results based on Gaussian models of networks have been presented. The possible defects in network formation may invalidate an otherwise well planned experiment, and because of this uncertainty, conclusions based on current experiments must be viewed as tentative. It is also true that theoretical calculations have been restricted thus far to only a few simple models of an elastomeric network. An appropriate method of calculation for trapped entanglements has not been constructed, nor has any calculation of the SANS pattern of a network which is constrained according to the reptation models of de Gennes (24) or Doi-Edwards (25,26) appeared. 

Molecular theories presuming a contribution from trapped entanglements in small strain gave good agreement with the data and offered reasonable explanation of the trends observed. 

Data Interpretation. Often the same stress-strain data is utilized by two different research groups as evidence to both support and deny the existence of a trapped entanglement contri-... 

For imperfect epoxy-amine or polyoxypropylene-urethane networks (Mc=103-10 ), the front factor, A, in the rubber elasticity theories was always higher than the phantom value which may be due to a contribution by trapped entanglements. The crosslinking density of the networks was controlled by excess amine or hydroxyl groups, respectively, or by addition of monoepoxide. The reduced equilibrium moduli (equal to the concentration of elastically active network chains) of epoxy networks were the same in dry and swollen states and fitted equally well the theory with chemical contribution and A 1 or the phantom network value of A and a trapped entanglement contribution due to the similar shape of both contributions. For polyurethane networks from polyoxypro-pylene triol (M=2700), A 2 if only the chemical contribution was considered which could be explained by a trapped entanglement contribution. 

According to the rubber elasticity theory ( 1, 2), the equilibrium shear modulus, Ge, is proportional to the concentration of EANC s and an additional contribution due to trapped entanglements may also be considered ... 

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. 

Background. Consider that pairwise interactions between active network chains (13), commonly termed trapped entanglements, do not significantly affect the stress in a specimen deformed in simple tension or compression. Then, according to recent theory (16,17), the shear modulus for a network in which all junctions are trifunctional is given by an equation which can be written in the form (13) ... 

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 ... 

Since the excellent work of Moore and Watson (6, who cross-linked natural rubber with t-butylperoxide, most workers have assumed that physical cross-links contribute to the equilibrium elastic properties of cross-linked elastomers. This idea seems to be fully confirmed in work by Graessley and co-workers who used the Langley method on radiation cross-linked polybutadiene (.7) and ethylene-propylene copolymer (8) to study trapped entanglements. Two-network results on 1,2-polybutadiene (9.10) also indicate that the equilibrium elastic contribution from chain entangling at high degrees of cross-linking is quantitatively equal to the pseudoequilibrium rubber plateau modulus (1 1.) of the uncross-linked polymer. 

Ronca and Allegra (12) and Flory ( 1, 2) assume explicitly in their new rubber elasticity theory that trapped entanglements make no contribution to the equilibrium elastic modulus. It is proposed that chain entangling merely serves to suppress junction fluctuations at small deformations, thereby making the network deform affinely at small deformations. This means that the limiting value of the front factor is one for complete suppression of junction fluctuations. 

Thus, the simplified Two-Network experiment shows by a direct comparison of forces at constant length that the trapped entangled structure of a well cross-linked elastomer contributes to the equilibrium modulus by an amount that is approximately equal to the rubber plateau modulus. The modulus contribution from the trapped entangled structure will be less for lower molecular weights and especially at low degrees of cross-linking (14). 

This is a theoretical study on the structure and modulus of a composite polymeric network formed by two intermeshing co-continuous networks of different chemistry, which interact on a molecular level. The rigidity of this elastomer is assumed to increase with the number density of chemical crosslinks and trapped entanglements in the system. The latter quantity is estimated from the relative concentration of the individual components and their ability to entangle in the unmixed state. The equilibrium elasticity modulus is then calculated for both the cases of a simultaneous and sequential interpenetrating polymer network. 

It is also assumed that the fraction of trapped entanglements along a chain of the one species caused by associations with chains of the other species is equal to the fractional participation of the steps of the second species in the total step population ... 

Because v, is a fractional quantity. Equation 18 always predicts modulus values larger than the corresponding expression for a simultaneous IPN(Equation 13). For the special case of a network with no defects or trapped entanglements (i)) 1, "), an earlier... 

A "trapped" entanglements between crystals - essentially infinite lifetime... 

---