Entangled system elasticity

Initially, the elasticity of concentrated polymer systems was ascribed to the existence of a network in the system formed by long macromolecules with junction sites (Ferry 1980). The sites were assumed to exist for an appreciable time, so that, for observable times which are less than the lifetime of the site, the entangled system appears to be elastic. Equation (1.44) was used to estimate the number density of sites in the system. The number of entanglements for a single macromolecule Z = M/Me can be calculated according to the modified formula... 

Considering the entangled systems in the coarse-grained approximation, we forget about segments the theory contains the effective elastic forces between the fictious adjacent particles, and the stress tensor (equation (6.7)) can be expressed through the variables and u"k in the form... 

In this form the expression for the stresses is valid for any dynamics of the chain. For the macromolecules in an entangled system, the elastic and internal viscosity forces according to equations (26) and (40) - (42) have the form... 

The consequences for elasticity of this model are discussed next but note, by intuition, that the Wall-Flory model now begins to look more plausible for entangled systems than for phantom systems. 

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. 

This is a theoretical study on the entanglement architecture and mechanical properties of an ideal two-component interpenetrating polymer network (IPN) composed of flexible chains (Fig. la). In this system molecular interaction between different polymer species is accomplished by the simultaneous or sequential polymerization of the polymeric precursors [1 ]. Chains which are thermodynamically incompatible are permanently interlocked in a composite network due to the presence of chemical crosslinks. The network structure is thus reinforced by chain entanglements trapped between permanent junctions [2,3]. It is evident that, entanglements between identical chains lie further apart in an IPN than in a one-component network (Fig. lb) and entanglements associating heterogeneous polymers are formed in between homopolymer junctions. In the present study the density of the various interchain associations in the composite network is evaluated as a function of the properties of the pure network components. This information is used to estimate the equilibrium rubber elasticity modulus of the IPN. 

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. 

To calculate the characteristics of viscoelasticity in the framework of mesoscopic approach, one can start with the system of entangled macromolecules, considered as a dilute suspension of chains with internal viscoelasticity moving in viscoelastic medium, while the elastic and internal viscosity forces, according to equations (3.4)-(3.6) and (3.8), have the form... 

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

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