Effects of Entanglements

Model networks may also be used to test of molecular predictions for the modulus of a network of known degree of cross linking. Some experiments on model networks have given values of the elastic modulus in 

Aspects of greatest importance appear to be (i) investigating the effects of cross linking in solution, (ii) studying the effects of swelling on networks cross linked in the bulk (dry) state, (iii) building on the demonstration by 

Vilgis and Erman that the constraint models and slip-link models have much in common, (iv) elucidating the effects of cross-link functionality and degree of cross linking, (v) exploring a variety of elastomeric polymers, particularly those having very different values of the plateau modulus, and (vi) generalizing rubber-elasticity models to include viscoelastic effects as well. 

For purposes of illustration, it is useful to consider some studies of Op-permann and coworkers on end-linked PDMS networks of different functionalities, since some of the aforementioned issues arose in the analysis of their experimental data. Here, studies of the small-strain shear modulus of end-linked PDMS networks with pentafunctional junctions were carried out as a function of network chain density. The data agree with predictions of the molecular theories at larger values of the chain or junction densities. At lower junction densities however the measured 

Stress-strain measurements on bimodal PDMS networks exhibited upturns in modulus that were much less pronounced than those in crystal-lizable polymer networks. The upturns are independent of temperature and are not diminished by incorporation of solvent. These characteristics are expected for the case of limited chain extensibility. - 

---

Since the slippage factor is a fraction, Eq. (2.59) states in mathematical terms something we realize must be the case, namely, that the effects of entanglements on the neighbors of the original molecule must diminish as we move away from that molecule to prevent the coupling from producing an infinite viscosity. 

In the rubbery plateau, a new impediment to movement must be overcome entanglements along the polymer chain. In discussing the effects of entanglements in Chap. 2, we compared them to crosslinks. Is it any surprise, then, that rubbery behavior similar to that shown by cross-linked elastomers characterizes this region ... 

Chemical crosslinks and entanglements differ, however. The former is permanent, the latter transient. Given sufficient time, even the effects of entanglements can be overcome and stationary-state flow is achieved. An... 

Since, in contrast to experiment, the simulation knows in detail what the connectivity looks hke, how long the strands are, and how the network loops are distributed, one can attribute this behavior to the non-crossability of the chains. Actually, one can even go further by allowing the chains to cross each other but still keep the excluded volume. Such a technical trick, which is only possible in simulations, allows one to isolate the effect of entanglement and non-crossability in such a case. As one would expect, if one allows chains to cross through each other one recovers the so-called phantom network result. 

The empirical frictional factor (T fric) is independent of shear rate but increases in poor solvent this permits to account for the dependence of the scission rate constant on solvent quality. The entanglement part (r enl), as given by Graessley s theory which considers the effect of entanglement and disentanglement processes, is a complex function of shear rate ... 

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). 

The slope of versus c is greater as M is higher (at lower concentrations more effect of entanglements). 

We may have to consider that the segment distribution fluctuates in the cylindrical domain in order to formulate the effects of entanglement and jamming in a solution as illustrated in Fig. 13b. In other words, we may no longer be permitted to consider the fuzzy cylinder a hard-core cylinder of the geometry specified by Eq. (43), but have to make its periphery fluctuate. 

Edwards SF, Grant JWV (1973) The effect of entanglements on diffusion in a polymer melt. J Phys A Math Nucl Gen 6 1169-1185... 

Higgins JS, Roots JE (1985) Effect of entanglement on the single-chain motion of polymer molecules in melt samples observed by neutron scattering. J Chem Soc Faraday Trans II 81 757-767... 

Pokrovskii VN, Kokorin YuK (1985) Effect of entanglement on the macromolecule mobility. 

DeGennes, P., Kinetics of diffusion-controlled processes in dense polymer systems II. Effects of entanglements, Journal of Chemical Physics, Vol. 76, No. 6, 1982, pp. 3322-3326. 

The principle of the NMR approach to semi-local properties of polymeric melts is considered in Section 2 it is shown how the existence of a temporary network structure is detected from the relaxation of the transverse magnetisation of protons attached to chains. The observation of segmental motions from the longitudinal relaxation of proton magnetisation is described in Section 3 it is also shown how local motions in concentrated polymeric solutions can be probed from the diffusion process of small molecules. Section 4 is devoted to the analysis of the effect of entanglement relaxation on NMR properties. 

Note that the velocity of craze tip advance is very sensitive to the energy of the surface being created (Vq decreases strongly as F increases since 10 < n < 20 for most polymer glasses ). This feature will be invoked later to explain the important effect of entanglements on the stresses required for crazing. 

Much less attention has been given to the effects of entanglements on the mechanical properties of crazes and on the competition between crazing and other. 

Having set out a detailed model of craze growth, we now compare its predictions with recent experiments in two principal areas 1) the effects of entanglement density and 2) the effects of temperature. 

The effect of entanglements on the relaxation of polymer chains is illustrated in Fig. 3-22, which shows the storage modulus G for a series of polystyrene melts of differing... 

A simple way to think about the effect of entanglements is written in the spirit of Eq. (7.48) ... 

---