Rubber elasticity, classical

Substituting Eq. (12) into Eq. (11) permits us to derive the Hookean spring force law, well-known in the classical theory of rubber elasticity ... 

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. 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

Figure 1. Stress relaxation curves for three different extension ratios. Uncross-linked high-vinyl polybutadiene with a weight average molecular weight of 2 million and a reference temperature of 283 K. G is the apparent rubber elasticity modulus calculated from classical affine theory. (Solid line is data from Ref. 1).

For the free energy of elastic deformation of network chains, Fel, the following expression of the classical theory of rubber elasticity can be used [21] ... 

During the last decade, the classical theory of rubber elasticity has been reconsidered significantly. It has been demonstrated (see, e.g. Ref.53>) that, for the phantom noninteracting network whose chains move freely one through the other, the equations of state of Eqs. (28) and (29) for simple deformation as well as for W, Q and AIJ [Eqs. (30)-(32) and (35)—(37)] are proportional not to v but to q, which is the cycle rank of the network, i.e. the number of independent circuits it contains. For a perfect phantom network of uniform functionality cp( > 2)... 

It is not necessary to be expert in the theory of rubber elasticity to test rubbers but it is a distinct advantage to be conversant with the main principles. A classic account of the development of the basic theories is given by Treloar1 which, if not digested from cover to cover, should be compulsory reading for those concerned with physical testing. 

This work is based on the molecular dynamic simulation of a monomer scale model corresponding to the affine network model of rubber elasticity.3 However, whereas the classic model has no nonbonded interactions, our model... 

The problem of determination of the partition function Z(k, N) for the iV-link chain having the fc-step primitive path was at first solved in Ref. [17] for the case a = c by application of rather complicated combinatorial methods. The generalization of the method proposed in Ref. [17] for the case c> a was performed in Refs. [19,23] by means of matrix methods which allow one to determine the value Z(k,N) numerically for the isotropic lattice of obstacles. The basic ideas of the paper [17] were used in Ref. [19] for investigation of the influence of topological effects in the problem of rubber elasticity of polymer networks. The dependence of the strain x on the relative deformation A for the uniaxial tension Ax = Xy = 1/Va, kz = A calculated in this paper is presented in Fig. 6 in Moon-ey-Rivlin coordinates (t/t0, A ), where r0 = vT/V0(k — 1/A2) represents the classical elasticity law [13]. (The direct Edwards approach to this problem was used in Ref. [26].) Within the framework of the theory proposed, the swelling properties of polymer networks were investigated in Refs. [19, 23] and the t(A)-dependence for the partially swollen gels was obtained [23]. In these papers, it was shown that the theory presented can be applied to a quantitative description of the experimental data. 

Usually the deformation behaviour of rubbery crosslinked polymers with long chains between crosslinks obeys the classical theory of rubber elasticity. However, the situation for densely crosslinked polymers is not so simple. 

These theories are based on the classical theories of rubber elasticity of macromolecular solids, wherein permanent chemical crosslinks connect segments of molecules, forcing them to move together. This central idea can be applied to polymeric liquids. However in this case, the interactions between molecules are assumed to be localized at junctions and are supposed to be temporary. Whatever their nature, physical or topological, these crosslinks are continually created and destroyed but, at any time, they ensure sufficient connectivity between the molecules to give rise to a certain level of cooperative motion. 

In this work we used polystyrene-based ionomers.-Since there is no crystallinity in this type of ionomer, only the effect of ionic interactions has been observed. Eisenberg et al. reported that for styrene-methacrylic acid ionomers, the position of the high inflection point in the stress relaxation master curve could be approximately predicted from the classical theory of rubber elasticity, assuming that each ion pah-acts as a crosslink up to ca. 6 mol %. Above 6 mol %, the deviation of data points from the calculated curve is very large. For sulfonated polystyrene ionomers, the inflection point in stress relaxation master curves and the rubbery plateau region in dynamic mechanical data seemed to follow the classical rubber theory at low ion content. Therefore, it is generally concluded that polystyrene-based ionomers with low ion content show a crosslinking effect due to multiplet formation. More... 

For the free energy Fel of elastic deformation of the polymer gel the expression of classical theory of rubber elasticity modified by Birshtein is... 

In this chapter, we first discuss the thermodynamics of rubber elasticity. The classical thermodynamic approach, as is well known, is only concerned with the macroscopic behavior of the material under investigation and has nothing to do with its molecular structure. The latter belongs to the realm of statistical mechanics, which is the subject of the second section, and has as its... 

This demonstrates that liquid crystallinity is not exhibited by many polymer networks because their liquid crystal to isotropic transitions are too low to be observed. But the residual liquid crystallinity affects the stress-strain relationship at higher temperatures. These results form a deviation from classical rubber elasticity theory. 

The deformation ability of networks strongly swollen with benzene and those slightly swollen in cyclohexane was unexpectedly found to be the same. What is surprising here is the absence of any correlation between the volume increase of model networks on swelling and their deformation under compression or elongation [130], as it would have to foUow from the classic theory of rubber elasticity. This theory does not predict any difference between the extensional modulus and the shear modulus that controls the swelling. Nevertheless, the experimental ratio of Ce(CH)/ Ce(BZ) = 6 is twice as large as the ratio of E(CH)/E(BZ) = 3 (irrespective ofp) [123]. 

Samulski [90, 91] described these interactions in the mean field approximation by an additional intermolecular potential from the classical theory of rubber elasticity. A similar expression is proposed for the elastic free energy. 

From mechanical measurements, classical rubber elasticity theory gives, for low uniaxial deformations and polymer chains that are long compared to M, the following relation [104]... 

We refer the reader to the classic survey of rubber elasticity by Treloar (1975) and to three recent reviews that give fuller accounts of the molecular theory (Graessley, 2004 Mark and Erman, 1988,1992). The author thanks Mr. R.A. Paden for drawing several of the figures. 

Models of rubber elasticity have been reviewed for finite deformation and compared with experimental data by Boyce and Arruda (2000). A hybrid model of the Flory-Erman model for low stretch deformation and the Arruda-Boyce model for large stretch deformation is shown to give an accurate predictive description of Treloar s classic data over the entire stretch range for all deformation states. 

An old point of controversy in rubber elasticity theory deals with the value of the so-called front factor g = Ap which was introduced first in the phantom chain models to connect the number of elastically effective network chains per unit volume and the shear modulus by G = Ar kTv. We use the notation of Rehage who clearly distinguishes between A andp. The factor A is often called the microstructure factor. One obtains A = 1 in the case of affine networks and A = 1 — 2/f (f = functionality) in the opposite case of free-fluctuation networks. The quantity is called the memory factor and is equal to the ratio of the mean square end-to-end distance of chains in the undeformed network to the same quantity for the system with junction points removed. The concept of the memory factor permits proper allowance for changes of the modulus caused by changes of experimental conditions (e.g. temperature, solvent) and the reduction of the modulus to a reference state However, in a number of cases a clear distinction between the two contributions to the front factor is not unambiguous. Contradictory results were obtained even in the classical studies. 

Affine deformation This model assumes that the deformation of each configuration of the chains is affine in the macroscopic deformation. It is not compatible with known classical theories of rubber elasticity. 

Thermodynamics, both classical [Appendix 3.A] and statistical [Appendix 2A], have been applied to many topics in polymer science. The results have provided insights into the origin of rubber elasticity, the nature of polymer crystalline, polymeric heat capacities and the miscibility of polyblends. 

Equations describing rubber elasticity can be derived in a straightforward fashion from classical thermodynamics based on free energy considerations. Free energy in turn can be related to experimentally accessible quantities as shown in the derivation below. 

Historically, the question of mechanism of elasticity has been one of evaluating the relative contributions of three different proposed mechanisms (1) the random chain network (classic rubber elasticity) theory, - (2) the solvent entropy theory, and (3) the damping of internal chain dynamics on extension. Ttie first is due to the Flory school the second was initiated by Weis-Fogh and Andersen, and the third is due to the present author and coworkers of the last quarter century. 

The presence of positive cooperativity, explained by the interconversion between different conformational states, also demonstrates that these entropic elastic model proteins are not properly described as random chain networks, as the adherents of the relevance of the classical theory of rubber elasticity to protein elasticity are compelled to argue. 

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