Entangled rubber elasticity

Langley, N.R. and Polmanteer, K.E., Role of chain entanglements in rubber elasticity. Polym. Prep. Am. Chem. Soc. Div. Polym. Chem., 13(1), 235-240 (1972). 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

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 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. 

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. 

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. 

It is clearly shown that chain entangling plays a major role in networks of 1,2-polybutadiene produced by cross-linking of long linear chains. The two-network method should provide critical tests for new molecular theories of rubber elasticity which take chain entangling into account. 

Equation (2.53) is stating that the network modulus is the product of the thermal energy and the number of springs trapped by the entanglements. This is the result that is predicted for covalently crosslinked elastomers from the theory of rubber elasticity that will be discussed in a little more detail below. However, what we should focus on here is that there is a range of frequencies over which a polymer melt behaves as a crosslinked three-dimensional mesh. At low frequencies entanglements... 

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. 

Each of the viscoelastic parameters G°, rj0, and Je° has associated with it a characteristic molecular weight which either measures an equivalent spacing of entanglement couples along the chain (Me, deduced from G with the kinetic theory of rubber elasticity), or marks the onset of behavior attributed to the presence of entanglements (Mc and AT, deduced from r/0 and Je° as functions of molecular weight). Table 5.2 lists Me, Mc, and M c for several polymers. Aside from certain difficulties in their evaluation, each is a rather direct and independent reflection of experimental fact. 

Analysis of networks in terms of molecular structure relies heavily on the kinetic theory of rubber elasticity. Although the theory is very well established in broad outline, there remain some troublesome questions that plague its use in quantitative applications of the kind required here. The following section reviews these problems as they relate to the subject of entanglement. 

Thurston GB, Peterlin A (1967) Influence of finite number of chain segments, hydrodynamic interaction, and internal viscosity on intrinsic birefringence and viscosity of polymer solutions in an oscillating laminar flow field. J Chem Phys 46(12) 4881 4884 Treloar LRG (1958) The physics of rubber elasticity. Oxford University Press, London Tsenoglou C (2001) Non-Newtonian rheology of entangled polymer solutions and melts. Macromolecules 34 2148-2155... 

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 ... 

Chain entanglements are the cause of rubber-elastic properties in the liquid. Below the "critical" molecular mass (Mc) there are no indications of a rubbery plateau. The length of the latter is strongly dependent on the length of the molecular chains, i.e. on the molar mass of the polymer. From the shear modulus of the pseudo rubber plateau the molecular weight between entanglements may be calculated ... 

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