Rubber theory, statistical

It is also possible to estimate the cross-link density from the stress-strain data, using the statistical theory of rubber-like elasticity [47,58]. For a swollen rubber the relationship is... 

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. 

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. 

Simultaneous IPN. According to the statistical theory of rubber elasticity, the elasticity modulus (Eg), a measure of the material rigidity, is proportional to the concentration of elastically active segments (Vg) in the network [3,4]. For negligible perturbation of the strand length at rest due to crosslinking (a reasonable assumption for the case of a simultaneous IPN), the modulus is given by ... 

The classical statistical theory of rubber elasticity1) for a Gaussian polymer network which took into account not only the change of conformational entropy of elastically active chains in the network but also the change of the conformation energy, led to the following equation of state for simple elongation or compression 19-2,1... 

In the current statistical theory of rubber elasticity, it is suggested that the front-factor molecular forces. They have proposed a semiempirical equation of state taking into account the dependence... 

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 statistical theory of rubber elasticity predicts that isothermal simple elongation and compression at constant pressure must be accompanied by interchain effects resulting from the volume change on deformation. The correct experimental determination of these effects is difficult because of very small absolute values of the volume changes. These studies are, however, important for understanding the molecular mechanisms of rubber elasticity and checking the validity of the postulates of statistical theory. 

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. 

The relationships resulting from the statistical theory fall well short of fully describing the stress strain curves of filled rubbers. From the alternative phenomenological approach a general relation for W is given by ... 

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). 

The average length (or molecular weight) of network chains in a crosslinked polymer can be experimentally determined from the equilibrium rubbery modulus. This relationship is a direct result of the statistical theory of rubber-like elasticity . In the last decade or so, modem theories of rubber-like elasticity 2127) further refined this relationship but have not altered its basic foundation. In essence, it is... 

Stress-strain properties for unfilled and filled silicon rubbers are studied in the temperature range 150-473 K. In this range, the increase of the modulus with temperature is significantly lower than predicted by the simple statistical theory of rubber elasticity. A moderate increase of the modulus with increasing temperature can be explained by the decrease of the number of adsorption junctions in the elastomer matrix as well as by the decrease of the ability of filler particles to share deformation caused by a weakening of PDMS-Aerosil interactions at higher temperatures. 

The presence of filler in the rubber as well as the increase of the surface ability of the Aerosil surface causes an increase in the modulus. The temperature dependence of the modulus is often used to analyze the network density in cured elastomers. According to the simple statistical theory of rubber elasticity, the modulus should increase twice for the double increase of the absolute temperature [35]. This behavior is observed for a cured xmfilled sample as shown in Fig. 15. However, for rubber filled with hydrophilic and hydrophobic Aerosil, the modulus increases by a factor of 1.3 and 1.6, respectively, as a function of temperature in the range of 225-450 K. It appears that less mobile chain units in the adsorption layer do not contribute directly to the rubber modulus, since the fraction of this layer is only a few percent [7, 8, 12, 21]. Since the influence of the secondary structure of fillers and filler-filler interaction is of importance only at moderate strain [43, 47], it is assumed that the change of the modulus with temperature is mainly caused by the properties of the elastomer matrix and the adsorption layer which cause the filler particles to share deformation. Therefore, the moderate decrease of the rubber modulus with increasing temperature, as compared to the value expected from the statistical theory, can be explained by the following reasons a decrease of the density of adsorption junctions as well as their strength, and a decrease of the ability of filler particles to share deformation due to a decrease of elastomer-filler interactions. 

In contrast to the filled samples, the deformation energy for the unfilled ones increases proportionally to the increase in the absolute temperature according to the prediction of the simple statistical theory of rubber elasticity. Thus, it appears that the change of the modulus and the deformation energy with increasing temperature reveals a decrease of the density of adsorption junctions in the elastomer matrix, as well as a decrease of the ability of filler particles to share deformation, resulting from a weakening of elastomer-filler interactions. 

An equation for the modulus of ideal rubber was derived from statistical theory that can be credited to several scientists, including Flory, and Guth and James (Sperling, 1986). A key assumption in derivation of the eqmtion is that the networks are Gaussian. 

This, model contradicts with the statistical theory of rubber elasticity, and the artificial assumption oia = 3k TJb has been used. 

Filled Rubbers and the Statistical Theory of Rubber Elasticity. . . 186... 

There exists as yet no rigorous extension of the statistical theory of rubber elasticity to a filled elastomer. Nevertheless, many attempts have been made to apply the theory to data on filled rubbers, usually with the objective of obtaining at least an approximate estimate of the number of filler-rubber attachments. These attempts have been discussed in earlier reviews (17, 126) and will not be considered here in full detail. We only restate briefly some of the experimental and theoretical difficulties inherent in this approach. 

If the objective of the experiment is to estimate the network chain density by the statistical theory, and if swelling is resorted to as a means for alleviating the above experimental difficulties, then the measurement of stresses becomes redundant the theory of equilibrium swelling of Flory and Rehner (127,128) makes an estimate of the desired network chain density possible with only the knowledge of the equilibrium swelling volume. The usual procedure is to calculate the volume fraction of polymer in the swollen rubber (ar) under the assumption that the filler does not swell. The Flory-Rehner theory then yields, for a network containing no sol fraction ... 

An interesting, indirect application of the statistical theory to filled rubbers has been published by Wolff (/33). In recent years increasing use has been made of oscillating disc rheometers to follow the course of vulcanization. These instruments operate at low frequencies and at vulcanization temperatures, so that torque recorded at full cure is a good measure of an equilibrium shear modulus and hence proportional to v. Wolff finds for polymers filled with furnace blacks and cross-linked with dieumyl peroxide ... 

In the formulation of the statistical theory of rubber elasticity,5 11 the following simplifying assumptions are made ... 

The statistical theory of rubber elasticity discussed in the preceding section was arrived at through considerations of the underlying molecular structure. The equation of state was obtained directly from the Helmholtz free energy of deformation (or simply conformational entropy of deformation, since the energy effects were assumed to be absent), which we can recast with the aid of equations (6-45) and (6-59) as... 

According to the statistical theory of rubber elasticity, the elastic stress of an elastomer under uniaxial extension is directly proportional to the concentration... 

In analogy to the kinetic theory of ideal gases, the statistical theory of rubber elasticity is often called the kinetic theory of rubber elasticity. Reflect upon the similarities and differences between the basic philosophies of these two theories. 

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