Gaussian theory of rubber elasticity

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. 

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)... 

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... 

The value of Me may also be determined by equilibrium swelling methods. In Chapter 4 it was shown that, assuming the validity of the Flory-Huggins equation and the Gaussian theory of rubber elasticity as embodied in eqn (3.46), the relationship between Me and the equilibrium swelling (llvr) was given by the expression ... 

The contribution of the elastic term in lightly cross-linked networks can be described by the Gaussian theory of rubber elasticity (2,3). In fully neutralized polyelectrolytes, in the presence of added salt, the ionic term is not expected to play an exphcit role. Ionic interactions, however, may modify the mixing free energy contribution. In neutral polymer solutions the Flory-Huggins theory (I), based on the lattice model of solutions, expresses the mixing pressure as... 

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... 

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. 

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. 

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... 

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 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... 

Further deviations from eqn (4.13) are to be expected where segments are highly extended and Gaussian statistics no longer apply, because of neglect of the experimentally observed C2 term and by all the other imperfections of the Gaussian theory of high elasticity. In spite of this eqn (4.13) (or rather its simplifications (4.18) and (4.19)) has been of importance in the elucidation of the structure of vulcanized rubbers (see Chapter 8). 

We shall now develop a statistical theory of rubber elasticity using the Gaussian formula to represent the most probable distribution of molecular end-to-end distance. The mass of rubber is assumed to consist of N chains each end of which is joined to the ends of other chains. Suppose that the mbber is extended by a factor a. 

In summary, the anomalous upturn in modulus observed for crystallizable polymers such as natural rubber and cw-1,4-polybutadiene is largely, if not entirely, due to strain-induced crystallization. In the case of the noncrystaUizable PDMS model networks it is clearly due to the limited chain extensibility, and thus the results on this system will be extremely useM for reliable evaluation of the various non-Gaussian theories of rubber-like elasticity. 

This Gaussian expression is fundamental to the statistical mechanical theory of rubber elasticity. 

The rubber in a blown-up balloon is stretched in a biaxial fashion. Derive the force-strain relationship under the assumption that the rubber follows the Gaussian statistical theory of rubber elasticity. 

STATISTICAL THEORY OF RUBBER ELASTICITY 9.6.1 Affine and Phantom Gaussian Models... 

PET with the draw ratio X can be expressed by a relationship based on the Gaussian form of the theory of rubber elasticity... 

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