The Statistical Theory of Rubber Elasticity

The essential concept involved in the statistical theory of rubber elasticity is that a macroscopic deformation of the whole sample leads to a microscopic deformation of individual polymer chains. The microscopic model of an ideal rubber consists of a three-dimensional network with junction points of known functionality greater than 2. An ideal rubber consists of fully covalent junctions between polymer chains. At short times, high-molecular-weight polymer liquids behave like rubber, but the length of the chains needed to describe the observed elastic behavior is independent of molecular weight and is much shorter than the whole chain. The concept of intrinsic entanglements in uncrosslinked polymer liquids is now well established, but the nature of these restrictions to flow is still unresolved. The following discussion focuses on ideal covalent networks. 

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  

There is no preferred direction for the end-to-end vectors in the rest state, and the full set of lengths is represented by the ensemble of chains that constitute the sample of rubber. 

The second postulate of the ideal-rubber theory is that, after deformation, the distribution of chain end-to-end vectors is perturbed in exactly the ratio determined by the macroscopic deformation. This assumption is called the principle of affine deformation. The distribution of chain end-to-end vectors is now given by  

In the rest state, the distribution of chain end-to-end vectors is spherically symmetric. In an ideal uniaxial deformation, the distribution of x-, y-, and z-components is cylindrically symmetric. The normalized distribution for the x-component (the stretch direction) is  

Section 6.3 deals with purely phenomenological theories. In this section the predictions of a theory based on the microstructure of a rubber are considered. By 1788 at the latest the term rubber was being applied to the material obtained from the latex of the tree Hevea braziliensis because of its ability to remove pencil marks from paper. The first printed account of this use for wiping off from paper the marks of black lead pencil was given by Joseph Priestley as early as 1770. This material is now called natural rubber and its chemical structure is shown in fig. 6.9. 

Many synthetic materials have similar physical properties. These are the synthetic rubbers, a subgroup of polymers often called elastomers. The repeat units of some important natural and synthetic rubbers are shown in fig. 6.10. 

Natural rubber in its raw state will flow under continuously applied stress and is therefore not very useful. Cross-linking the molecules prevents flow and this can be done by heating natural rubber with sulphur, in the process called vulcanisation, which was discovered by Goodyear in 1839. Other chemical reagents are used more generally to cross-link the mole- 

10 Repeat units of some important natural and synthetic rubbers. In the copolymers marked with an asterisk the respective monomer units occur in a random sequence along the chain (but see also section 12.3.3). 

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 quantity to be calculated for the gas in a container of volume Vo is the probability 0 that all of the gas molecules will spontaneously move to a portion of the container having a volume V, The probability that any given molecule occurs in this volume is U/Fo, and the probability that all of them, v in number, are there simultaneously is 

According to the Boltzmann relation, the entropy change AS for the process of compression is given by 

After the formation of the network structure has been completed, let the sample be subjected to any type of homogeneous strain (including swelling, to be treated in Chap. XIII) which may be described as an alteration of its dimensions X, Y, and Z by factors and az, 

for example, the sample is stretched in the o -direction, its volume remaining constant, the ir-components of all vectors will be increased by the factor a, while the y- and 2-components will be decreased as is 

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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 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. 

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. 

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. 

We return now to the difference in behavior between the two types of PMMA with molecular weights of 1.5 x 10 and 3.6 x 10 . We note that the rubbery modulus of the type with the higher molecular weight reaches a plateau at /iR = 3.4 MPa. As we discuss in Chapter 6 on rubber elasticity, the entanglement molecular weight Me can be determined from this modulus through the statistical theory of rubber elasticity (Ferry, 1980) as... 

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] ... 

It will thus be obvious that both covalent and polar crosslinks contribute to the crosslink network and hence modulus of polyurethane elastomers. The dependence of this modulus or temperature can be divided into contributions from a covalently linked network conforming to the statistical theory of rubber elasticity, and contributions from secondary crosslinks which are assumed to have a temperature dependence governed by the Arrhenius law in which the modulus of elasticity is represented by the equation ... 

The statistical theory of rubber elasticity is based on the concepts of random chain motion and the restraining power of cross-links that is, it is a molecular... 

Equation (9.63), with two additional terms over the statistical theory of rubber elasticity, fits the data quite well (see Figure 9.18) (64). 

The statistical theory of rubber elasticity has undergone significant and continuous refinement, resulting in a series of correction terms. These are sometimes omitted and sometimes included in scientific and engineering research, as the need for them arises. In this section we briefly consider some of these. 

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