Rubber elasticity Gaussian theory

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. 

According to the importance of the cross-links, various models have been used to develop a microscopic theory of rubber elasticity [78-83], These models mainly differ with respect to the space accessible for the junctions to fluctuate around their average positions. Maximum spatial freedom is warranted in the so-called phantom network model [78,79,83], Here, freely intersecting chains and forces acting only on pairs of junctions are assumed. Under stress the average positions of the junctions are affinely deformed without changing the extent of the spatial fluctuations. The width of their Gaussian distribution is predicted to be... 

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 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. 

In other statistical theories of rubber elasticity (see e.g. reviews 29,34)) the Gaussian statistics is not valid even at small deformations and the intramolecular energy component is dependent on deformation. 

The non-Gaussian theories of rubber elasticity have the disadvantage of containing parameters which generally can be determined only by experiment. Recently,... 

Curro and Mark 38) have proposed a new non-Gaussian theory of rubber elasticity based on rotational isomeric state simulations of network chain configurations. Specifically, Monte Carlo calculations were used to determine the distribution functions for end-to-end dimensions of the network chains. The utilization of these distribution functions instead of the Gaussian function yields a large decreases in the entropy of the network chains. 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

In conclusion, it can be said that the theory can well describe the development of the gel structure. The correlation between the equilibrium modulus and sol fraction is very good so that the sol fraction can alternatively be used for determination of the concentration of EANC s if an accurate and precise determination of conversion meets with difficulties. It is to be recalled here that the Gaussian rubber elasticity theory does not apply to highly crosslinked networks of usual stoidiiometric systems. When a good theory is available, the calculated value of taking possibly into account the topological limit of the reaction will be ne ed. 

For a network of uniform length chains. Lake and Thomas substituted for L with an Equation predicted from rubber elasticity theory. They also derived an alternate expression for L for a network of random Gaussian chains. The two expressions differ only by a small numerical constant. Making either substitution, and rearranging terms, it can be shown that... 

C. Menduina, C. McBride, and C. Vega (2001) Correctly averaged Non-Gaussian theory of rubber-like elasticity - application to the description of the behavior of poly(dimethylsiloxane) bimodal networks. Phys. Chem,. Chem. Phys. 3, p. 1289... 

Using only the first term is essentially equivalent to the Gaussian approximation of rubber elasticity theory developed in section 6.4.4. The form of (P2(cos9) versus X for the rubber model is then as shown... 

The shear modulus of a rubber is inversely proportional to the average network chain length according to the Gaussian theory of rubber elasticity [19]. Therefore, examination of the storage modulus above the Tg provides the evidence for network hydrolytic resistance. Figure 12.13 shows that the storage modulus (Eg)... 

Consider a strand of polymer chain between two cross-links. The vector R between the positions of the two cross-links changes with deformation. Any molecular theory on rubber elasticity is based on the probability distribution function for R. As seen in Chapter 1, if the number of segments N on the strand is large, the probability distribution (R, N) of the end-to-end vector R is a Gaussian function... 

The above equation has found high applicability by combining it with the relationship for the true stress, as derived from the Gaussian theory of rubber elasticity. Specifically this theory leads to... 

From the statistical theory of rubber elasticity follows for the case of uniaxial deformation of a network of independent Gaussian chains the relation ... 

Several techniques are available for the mechanical characterization of cryogels in swollen and dried states. Uniaxial compression tests are conducted on cylindrical cryogel samples to determine the Young s modulus E or shear modulus G from the slope of stress-strain curves at low compressions, while the stress at 3 or 5 % compression is reported as the compressive stress uniaxial compression of a cylindrical gel sample, the statistical theories of rubber elasticity yield for Gaussian chains an equation of the form [77, 78] ... 

James and Guth developed a theory of rubber elasticity without the assumption of affine deformation [18,19,20]. They introduced the macroscopic deformation as the boundary conditions applied to the surface of the samples. Junctions are assumed to move freely under such fixed boundary conditions. The network chains (assumed to be Gaussian) act only to deliver forces at the junctions they attach to. They are allowed to pass through one another freely, and they are not subject to the volume exclusion requirements of real molecular systems. Therefore, the theory is called the phantom network theory. 

The state of the ideal rubber can be specified by the locations of all the junction points, ij, and by fce end-to-end vectors for all tire chains connecting the junction points,. The first postulate of the statistical theory of rubber elasticity is that, in the rest state with no external constraints, the distribution fimction for the set of chain end-to-end vectors is a Gaussian distribution witii a mean-squared end-to-end distance that is proportional to the molecular weight of the chains between jimcnons ... 

The phenomenon of rubber elasticity fascinated scientists of all kinds during the 19th century. The explanation of the temperature dependence of the force of retraction in terms of the Gaussian theory of chain statistics was one of the first great triumphs of the macromolecular paradigm. One of the recurrent themes of this book is the clear explanation of actual phenomena in terms of the macromolecular nature of the system. 

Figure 3.9 Theoretical non-Gaussian free extension curve obtained by fitting experimental data (0) to the James and Guth theory, with NkT = 0.273 MPa, n — 75. (Reproduced with permission from Treloar, The Physics of Rubber Elasticity, 3rd edn, Oxford University Press, Oxford, 1975)...

The theory of Bernstein, Kearsley and Zapas [20] and developments of it (e.g. Zapas and Craft [21]) - so-called BKZ theories - are aimed in particular at large deformation behaviour. The Gaussian model of rubber elasticity tells us that in uniaxial stretching the true stress o is in the form... 

In the development of rubber elasticity theory (Section 9.7.1), it will be shown that the restoring force,/, on a chain or chain portion large enough to be Gaussian, is given by... 

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