Rubber elasticity, statistical mechanical theory

Kaliske, M., Heinrich, G., 1999. An extended tube-model for rubber elasticity Statistical-mechanical theory and finite element implementation. Rubber Chem. Technol. 72 (4), 602-632. Khokhlov, A.R., 1992. In Dusek, K. (Ed.), Responsive Gels Volume Transitions I. Springer, Verlag Berlin, p. 125. 

Kaliske, M. and Heinrich, G. (1999) An extended tube-model for rubber elasticity statistical-mechanical theory and finite element implementation. Rubber Chem. 

According to the statistical-mechanical theory of rubber elasticity, it is possible to obtain the temperature coefficient of the unperturbed dimensions, d InsjdT, from measurements of elastic moduli as a function of temperature for lightly cross-linked amorphous networks [Volken-stein and Ptitsyn (258 ) Flory, Hoeve and Ciferri (103a)]. This possibility, which rests on the reasonable assumption that the chains in undiluted amorphous polymer have essentially their unperturbed mean dimensions [see Flory (5)j, has been realized experimentally for polyethylene, polyisobutylene, natural rubber and poly(dimethylsiloxane) [Ciferri, Hoeve and Flory (66") and Ciferri (66 )] and the results have been confirmed by observations of intrinsic viscosities in athermal (but not theta ) solvents for polyethylene and poly(dimethylsiloxane). In all these cases, the derivative d In sjdT is no greater than about 10-3 per degree, and is actually positive for natural rubber and for the siloxane polymer. 

The mean-squared, end-to-end distance in Equation [9] is the simplest average property of interest for a polymer chain. Among other physical properties, this quantity appears in the equations of statistical mechanical theories of rubber-like elasticity. In Equation [9], the angle brackets denote the ensemble (or time) average over all possible conformations. The subscript 0 indicates that the average pertains to an unperturbed chain (theta conditions no excluded volume effects are present). (See Figure... 

From this rough outline of some examples of current problems in the physics of rubber elasticity, it is clear that it is important to have a well-founded statistical-mechanical theory of equilibrium properties of rubber-elastic networks. Consequently, first junction and entanglement topology are described and discussed. Then a section briefly reviews the theory of the phantom network. In the following two sections, theories of equilibrium properties of networks and a comparison of theoretical results with experimental data are presented. 

In other theories of rubber elasticity, the network structure is explicitly considered. However, the polymer on the surface is taken to be fixed (according to an affine deformation) upon deformation. - A truly statistical mechanical theory would also treat the surface statistically. More fundame ntally, however, in these theories the fixed point character of the surface i hen completely determines the behavior of the bulk material. This would appear to be nonsense in the thermodynamic limit of infinite volume, unless the fixed surface were of finite extent. In this case, the theory is no longer statistical in nature. 

The statistical mechanical theory for rubber elasticity was first qualitatively formulated by Werner Kuhn, Eugene Guth and Herman Mark. The entropy-driven elasticity was explained on the basis of conformational states. The initial theory dealt only with single molecules, but later development by these pioneers and by other scientists formulated the theory also for polymer networks. The first stress—strain equation based on statistical mechanics was formulated by Eugene Guth and Hubert James in 1941. 

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

These statistical mechanical theories of rubber elasticity are based on two fundamental postulates, supported by thermoelastic and neutron scattering experiments (i) molecular chain configurations are random in undeformed amorphous polymers and (ii) the elastic response of the network originates within the chains and not, to a significant extent, from interactions between them. 

The bracket (1 — 2/f) was introduced into the theory of rubber elasticity by Graessley [23], following an idea of Duiser and Staverman [28]. Graessley discussed the statistical mechanics of random coil networks, which he had divided into an ensemble of micronetworks. 

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine 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. 

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). 

Weakly crosslinked epoxy-amine networks above their Tg exhibit rubbery behaviour like vulcanized rubbers and the theory of rubber elasticity can be applied to their mechanical behaviour. The equilibrium stress-strain data can be correlated with the concentration of elastically active network chains (EANC) and other statistical characteristics of the gel. This correlation is important not only for verification of the theory but also for application of crosslinked epoxies above their Tg. 

The Epons 828,1001,1002,1004, and 1007 fully cured with stoichiometeric amounts of DDS are examples of well-characterized networks. Therefore, mechanical measurements on them offer insight into the viscoelastic properties of rubber networks. The shear creep compliance J t) of these Epons were measured above their glass temperatures [11, 12, 14]. From the statistical theory of rubber elasticity [1-5, 29-33] the equilibrium modulus Ge is proportional to the product Tp, where p is the density at temperature T, and hence the equilibrium compliance is proportional to (Tpy Thus J t) is expected to be proportional to and J(t)Tp is the quantity which should be compared at different temperatures. Actually the reduced creep compliance... 

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] ... 

There is an extensive body of literature describing the stress-strain response of rubberlike materials that is based upon the concepts of Finite Elasticity Theory which was originally developed by Rivlin and others [58,59]. The reader is referred to this literature for further details of the relevant developments. For the purposes of this paper, we will discuss the developments of the so-called Valanis-Landel strain energy density function, [60] because it is of the form that most commonly results from the statistical mechanical models of rubber networks and has been very successful in describing the mechanical response of cross-linked rubber. It is resultingly very useful in understanding the behavior of swollen networks. 

High moduli, memory effects, and SANS results which are inconsistent with classical theories of rubber elasticity provoke the need for a new theory. The ideas of junction rearrangement, if correct, require that none of the models of affine deformation should be expected to apply. A statistical mechanical partition function, properly formulated for a polymeric elastomer, should yield predictions of chain deformation, and additional assinnptions relating macroscopic and molecular geometry are superfluous. 

It is apparent from considerations of the structure in Section 4.2 that semi-crystalline polymers are two-phase materials and that the increase in modulus is due to the presence of the crystals. Traditional ideas of the stiffening effect due to the presence of crystals were based upon the statistical theory of rubber elasticity (Section 5.3.2). It was thought that the crystals in the amorphous rubber behaved like cross-links and produced the stiffening through an increase in cross-link density rather than through their own inherent stiffness. Although this mechanism may be relevant at very low degrees of crystallinity it is clear that most semi-crystalline... 

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