Links, and entanglements

This paper presents some results obtained with the polarizing microscope and the scanning electron microscope. The importance of cross-linking and entanglements is discussed, and results of ESR measurements on popcorn polymer systems are presented. 

Molecular weights of the polymer and of the monomer unit Molecular weight of chains between entanglement points Number of random links between cross-links and entanglement points... 

Forgan RS, Sauvage JP, Stoddart JF (2011) Chcanical topology complex molecular knots, links, and entanglements. Chem Rev 111 5434—5464... 

As a result, the concentration of the elastically effective subchains (i.e., subchains contributing to the elasticity modulus) differs from the concentration of subchains Vch calculated from the amount of cross-linker used in the gel synthesis under the assumption of the formation of an ideal network, where all the cross-linker molecules are incorporated in such a way that they connect elastically effective network subchains. To describe this difference, the elasticity modulus G is represented as a sum of two components Gch and Gg associated with chemical cross-links and entanglements, respectively. This approach was called a two-network model. " °... 

For NR based IPNs, Mathew et al. had investigated the sorption and diffusion of three aromatic solvents (benzene, toluene, and xylene) through NR/polystyrene semi-, and full IPNs, and reported that as the polystyrene content increased, the solvent uptake value decreased. This was attributed to the fact that the introduction of plastic phase decreased the chain flexibility of the network. The study further showed that the nature and size of the penetrant molecule affected the transport behaviour. Temperature, it was observed, affected the transport properties. The solvent uptake was found to inerease with temperature up to 65 °C, and at 70 °C, a deerease in uptake was observed (Figure 22.4). The values of sorption and permeation eoeffieients obtained showed a direct dependency on sample characteristics, blend composition, and crosslink level. The study further showed that as the number of crosslinks increased in the blends, the resistance offered to solvent uptake increased since the solvent molecules have to overcome the dense barrier of polymer cross-linking and entanglements to diffuse into the blend. 

This formula shows how the cross links and entanglements combine to enclose a chain. The strain energy has now become... 

Equations 22.3-22.14 represent the simplest formulation of filled phantom polymer networks. Clearly, specific features of the fractal filler structures of carbon black, etc., are totally neglected. However, the model uses chain variables R(i) directly. It assumes the chains are Gaussian the cross-links and filler particles are placed in position randomly and instantaneously and are thereafter permanent. Additionally, constraints arising from entanglements and packing effects can be introduced using the mean field approach of harmonic tube constraints [15]. 

The elastic free energy of the constrained-junction model, similar to that of the slip-link model, is the sum of the phantom network free energy and that due to the constraints. Both the slip-link and the constrained-junction model free energies reduce to that of the phantom network model when the effect of entanglements diminishes to zero. One important difference between the two models, however, is that the constrained-junction model free energy equates to that of the affine network model in the limit of infinitely strong constraints, whereas the slip-link model free energy may exceed that for an affine deformation, as may be observed from Equation (41). 

Networks with tri- and tetra-functional cross-links produced by end-linking of short strands give moduli which are more in accord with the new theory if quantitative reaction can be assumed (3...13) However, the data on polydimethylsiloxane networks, may equally well be analyzed in terms of modulus contributions from chemical cross-links and chain entangling, both, if imperfect reaction is taken into account (J 4). Absence of a modulus contribution from chain entangling has therefore not been demonstrated by end-linked networks. 

The relationship between degree of cross-linking and trapping of the entangled structure (the trapping is nearly complete in the present work). 

Since the two effects work in parallel in ordinary networks, it is necessary to know the concentration of effective cross-links and to have a molecular theory which correctly relates the modulus to the concentration of cross-links. The contribution from chain entangling is then found as the difference between the observed and the calculated modulus. This seems to be an almost hopeless task unless the network structure is very simple and the contribution from chain entangling is large. 

In 1944, Flory (3) noted that the moduli of cross-linked butyl rubbers generally differ somewhat from values calculated from the crosslink density according to the kinetic theory of rubber elasticity. In many cases, the modulus also depends on the primary (uncross-linked) molecular weight distribution of the polymer. He attributed both observations to three kinds of network defects chain ends, loops, and chain entanglements. The latter are latent in the system prior to cross-linking and become permanent features of the network when cross-links are added. 

---