Rubber elasticity, theory

From Eq. (3-13), polymer stresses in a flow or deformation can be calculated if the distribution function for that flow can be predicted. To make such predictions, molecular 

The theory of rubber elasticity was developed by Wall (1942), Flory and Rehner (1943), James and Guth (1943), and Treloar (1943). Suppose that cross-link points are evenly spaced along each polymer chain a distance random-walk steps apart. The portion of polymer 

We now consider an extensional deformation of an incompressible rubber network (Fig. 3-7), where the stretch axes are oriented along the coordinate axes apd where the stretch ratios X, k2, and I3 are in directions 1, 2, and 3, respectively. For the example in Fig. 3-7, the deformation is a uniaxial extension that increases the length of the cylinder by a factor of X1 over its initial length. By volume conservation, the radius of the cylinder then shrinks to times the original radius. If the cross-link points are convected with 

For a volume-conserving deformation, the strands that before the deformation had end-to-end vectors between R and R -f dR have end-to-end vectors between R and R - - ifR after the deformation. Therefore ffo R )dR = (R)dR. Since R = Ri/h and i/ = X k2 3dR = dR (since I1A2A3 = 1), we have 

Each surface of equal probability is therefore deformed from a sphere to an ellipsoid with principal axes Ai, A2, and X3 times the diameter of the undeformed sphere. 

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An increase in the swelling degree usually results in lowering elastic modulus. According to the rubber elasticity theory [116-118] the shear modulus of the gel G can be expressed as ... 

The above equations gave reasonably reliable M value of SBS. Another approach to modeling the elastic behavior of SBS triblock copolymer has been developed [202]. The first one, the simple model, is obtained by a modification of classical rubber elasticity theory to account for the filler effect of the domain. The major objection was the simple application of mbber elasticity theory to block copolymers without considering the effect of the domain on the distribution function of the mbber matrix chain. In the derivation of classical equation of rabber elasticity, it is assumed that the chain has Gaussian distribution function. The use of this distribution function considers that aU spaces are accessible to a given chain. However, that is not the case of TPEs because the domain also takes up space in block copolymers. 

The formal thermodynamic analogy existing between an ideal rubber and an ideal gas carries over to the statistical derivation of the force of retraction of stretched rubber, which we undertake in this section. This derivation parallels so closely the statistical-thermodynamic deduction of the pressure of a perfect gas that it seems worth while to set forth the latter briefly here for the purpose of illustrating clearly the subsequent derivation of the basic relations of rubber elasticity theory. 

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 chains of typical networks are of sufficient length and flexibility to justify representation of the distribution of their end-to-end lengths by the most tractable of all distribution functions, the Gaussian. This facet of the problem being so summarily dealt with, the burden of rubber elasticity theory centers on the connections between the end-to-end lengths of the chains comprising the network and the macroscopic strain. 

Here, only the form of rubber elasticity theory used to analyze the structure of hydrogels prepared in the presence of a solvent is presented... 

Figure 5.11 Dependence of the reduced equilibrium shear modulus, Ge/wg// 7" on the molar ratio of [OH]/[NCO] groups, ah, for poly(oxypropylene)triol (Niax LG 56)-4,4 -diisocyanatodiphenylmethane system (—-) limits of the Flory-Erman junction fluctuation rubber elasticity theory. The dependence has been reconstructed from data of ref. [78]...

There exist a number of experimental methods for determination of structure sensitive parameters of a system undergoing branching and crosslinking. However, evaluation of some of the results requires application of a theoretical approach to the phenomenon the measurement is concerned with. Then, we may be testing two theories at once. The equilibrium elasticity is one example, since there exist alternative rubber elasticity theories. However, certain conclusions can always be made. 

However, in doing so one tests two theories the network formation theory and the rubber elasticity theory and there are at present deeper uncertainties in the latter than in the former. Many attempts to analyze the validity of the rubber elasticity theories were in the past based on the assumption of ideality of networks prepared usually by endllnklng. The ideal state can be approached but never reached experimentally and small deviations may have a considerable effect on the concentration of elastically active chains (EANC) and thus on the equilibrium modulus. The main issue of the rubber elasticity studies is to find which theory fits the experimental data best. This problem goes far beyond the network... 

Newer rubber elasticity theories based on the tube model (35) consider special constraint release mechanisms which allow a physi-... 

Here, v is Poisson s ratio which is equal to 0.5 for elastic materials such as hydrogels. Rubber elasticity theory describes the shear modulus in terms of structural parameters such as the molecular weight between crosslinks. In the rubber elasticity theory, the crosslink junctions are considered fixed in space [19]. Also, the network is considered ideal in that it contained no structural defects. Known as the affine network theory, it describes the shear modulus as... 

If the network is given a tensile deformation in the 2 direction and the comers move affinely, the four chains in the tetrahedron contribute independently to the modulus, with the well known result obtained from the rubber elasticity theory ... 

Front factor in modulus equation from rubber elasticity theory (Part 7). Fraction of configurations of free chains which are consistent with specified end-to-end coordinates (Part 7). 

It is evident that composite or heterogeneous networks, which result from macro- and micro-syneresis respectively, are not suitable for the verification of basic rubber-elasticity theories. The interpretation of their behaviour in the light of existing network elasticity theories should be quite complicated. Especially for heterogeneous networks, additional... 

Despite the preceding remarks and the fact that thermosets are far from being ideal rubbers, the basic rubber elasticity theory works surprisingly well in most practical cases, as illustrated by the data of Table 10.9. 

The rubber elasticity theory in its simplest version always predicts a good order of magnitude for the modulus, with 0.4 si 0 sc 1.6 for every system under consideration. It seems difficult to go beyond, predicting for instance 0 from the network structure, for many reasons ... 

J.-P. Jarry and L. Monnerie, Effects of a nematic-like interaction in rubber elasticity theory, Macromolecules, 12, 316 (1979). 

The mechanical properties of single hydrated dextran microcapsules (< 10 pm in diameter) with an embedded model protein drug have also been measured by the micromanipulation technique, and the information obtained (such as the Young s modulus) was used to derive their average pore size based on a statistical rubber elasticity theory (Ward and Hadley, 1993) and furthermore to predict the protein release rate (Stenekes et al., 2000). 

In Ref. 26 34) the mechanical behaviour of several epoxy-aromatic amine networks was analyzed in the rubbery state at temperatures of about T + 40 °C. Figure 14 gives some experimental results. If the rubber elasticity theory is obeyed, the following relation holds for uniaxial deformation ... 

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