Gaussian chain entangled

The overall result is that in the melt the polymer molecules adopt Gaussian configurations and behave as thermodynamically ideal entities. This combination of ideality and chain entanglement has been confirmed by neutron scattering experiments and is well established despite the apparent paradox. 

The role of chain entangling in cross-linked elastomers is an old issue which has not yet been settled. The success of Flory s new rubber elasticity theory 0-5) in describing some of the departures from the simple Gaussian theory has acted as a strong catalyst for new work in this area. 

The two-network method has been carefully examined. All the previous two-network results were obtained in simple extension for which the Gaussian composite network theory was found to be inadequate. Results obtained on composite networks of 1,2-polybutadiene for three different types of strain, namely equibiaxial extension, pure shear, and simple extension, are discussed in the present paper. The Gaussian composite network elastic free energy relation is found to be adequate in equibiaxial extension and possibly pure shear. Extrapolation to zero strain gives the same result for all three types of strain The contribution from chain entangling at elastic equilibrium is found to be approximately equal to the pseudo-equilibrium rubber plateau modulus and about three times larger than the contribution from chemical cross-links. 

In particular it has been conjectured that the terminal relaxation of star polymers might be the most sensitive test of the dilution exponent P in Go theta solvents suggest a mean value of nearer 2.3 [32]. A physically reasonable scahng assumption for the density of topological entanglements in a melt of Gaussian chains leads to a value of 7/3 [31]. 

Strictly speaking. Equation 23 pertains to networks of Gaussian chains which interact only at junctions chain entanglements and intrachain molecular interactions have been neglected. Note that G is the zero (low) frequency storage modulus. 

Note that p is independent of molecular weight, since for Gaussian chains M/ [R )q is a constant. At 140°C, the entanglement molecular weight Mg, plateau modulus G, and the tube diameter a (defined below in Section 3.7.1) are related to p by (Fetters et al. 1994)... 

Due to the dual filler and crosslinking nature of the hard domains in TPEs, the molecular deformation process is entirely different than the Gaussian network theories used in the description of conventional rubbers. Chain entanglements, which serve as effective crosslinks, play an important role in governing TPE behavior. The stress-strain results of most TPEs have been described by the empirical Mooney-Rivlin equation ... 

In Chapter 3, we used the Rouse model for a polymer chain to study the diffusion motion and the time-correlation function of the end-to-end vector. The Rouse model was first developed to describe polymer viscoelastic behavior in a dilute solution. In spite of its original intention, the theory successfully interprets the viscoelastic behavior of the entanglement-free poljuner melt or blend-solution system. The Rouse theory, developed on the Gaussian chain model, effectively simplifies the complexity associated with the large number of intra-molecular degrees of freedom and describes the slow dynamic viscoelastic behavior — slower than the motion of a single Rouse segment. 

This equation also implies that each entanglement strand is sufficiently long to be described by a Gaussian chain model, i.e. 

Direct measurement of the cross-link density of thermoset polymers including those from epoxy resins remains one of the most difficult analytical challenges in the field. A far too common approach simply relates the rubbery modulus (Gr), the thermoset modulus above Tg, to the molecular weight between cross-links (Me) using the theory of rubbery elasticity (133,134). Unfortunately thermoset networks have much more complex features than do true elastomers, including non-Gaussian chain behavior, interchain interactions, and entanglements (172). 

Assume that we are dealing with polymers in good solvents and in the semidilute solution. If r is a scale to measure, then the chain entanglement shows the following properties. At r >, that is, outside the blob, the repulsive interactions between monomers are screened out by other chains in the solution so that the whole chain is composed of blobs connected in an ordinary random walk without excluded volume effect. Overall, the chain follows Gaussian statistics. At r <, that is, within the blob, the chain does not interact with other chains, but there is a strong excluded volume effect. 

A first hint of this complication is found in dilute solutions. The theoretical models (Zimm or Rouse) that we introduced both consider phantom chains, they ignore the topological constraints due to the connections between polymer chains, and thus the effects of entanglements this turns out to be a reasonable assumption if the number of entanglements or knots is small and is actually the case for an isolated polymer in a good solvent. However, in a Gaussian chain of N monomers (in a 0 solvent), the number of knots scales as in a 0 solvent, polymer chains are self-entangled. 

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