Statistical Thermodynamics of Rubber Elasticity

It is convenient to divide the derivation of equations such as (9.4) into two parts. First, the equation of state for a single chain in space is derived. Then we show how a network of such chains behaves. 

It is convenient to start again with the general equation for the Helmholtz free energy, equation (9.5)  

This can be rewritten in statistical thermodynamic notation F = constant -kT In Q(r, T) 

From a quantitative point of view, at each particular end-to-end distance, all possible conformations of the chain need to be counted, holding the ends 

As before, the quantity U, assumed to be constant (or zero), drops out of the calculation, leaving only the entropic contribution. In this case, for a single chain, the quantity / for force must be used. The cross section of the individual chain, necessary for a determination of the stress, remains undefined. 

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James HM, Guth E (1953) Statistical thermodynamics of rubber elasticity. J Chem Phys 21 1039-1049... 

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... 

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 molecular models of rubber elasticity relate chain statistics and chain deformation to the deformation of the macroscopic material. The thermodynamic changes, including stress are derived from chain deformation. In this sense, the measurement of geometric changes is fundamental to the theory, constitutes a direct check of the model, and is an unambiguous measure of the mutual consistency of theory and experiment. 

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. 

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 equation of state of rubber elasticity will now be calculated via statistical thermodynamics, rather than the classical thermodynamics of Section 9.5. Statistical thermodynamics makes use of the probability of finding an atom, segment, or molecule in any one place as a means of computing the entropy. Thus tremendous insight is obtained into the molecular processes of entropic phenomena, although classical thermodynamics illustrates energetic phenomena adequately. 

Polymer networks are conveniently characterized in the elastomeric state, which is exhibited at temperatures above the glass-to-rubber transition temperature T. In this state, the large ensemble of configurations accessible to flexible chain molecules by Brownian motion is very amenable to statistical mechanical analysis. Polymers with relatively high values of such as polystyrene or elastin are generally studied in the swollen state to lower their values of to below the temperature of investigation. It is also advantageous to study network behavior in the swollen state since this facilitates the approach to elastic equilibrium, which is required for application of rubber elasticity theories based on statistical thermodynamics. ... 

Finally, it is interesting and helpful to make a comparison between rubber elasticity and gas pressure from the view point of statistical thermodynamics. A gas particle (atom or molecule) has more space to move about in a large container than in a small one. In other words, the total number of the states available for the gas particle to occupy, all having the same potential energy, is proportional to the volume V of the container. Thus, corresponding to Eq. (2.9), the entropy of the gas particle can be... 

Guth E, James HM (1941) Elastic and thermodynamic properties of rubber-like materials a statistical theory. Ind Eng Chem 33 624... 

In Section 9.5 some of the basic classical thermodynamic relationships for rubber elasticity were examined. Now the classical and statistical formulations are combined (108,109). 

The swelling of vulcanized crosslinked rubber by solvents has long been observed. This behavior was first modeled in the 1940s by Flory and Refiner [47]. They combine the Meyer-Flory-Huggins statistical thermodynamic theory of polymer solutions (Section 3.3) with the molecular theory of crosslinked rubber elasticity [48]. The molecular weight between crosslinks of the vulcanizates was predicted from the swelling to be... 

The properties of elastomeric materials are controlled by their molecular structure which has been discussed earlier (Section 4.4.5). They are basically all amorphous polymers above their glass transition and normally cross-linked. Their unique deformation behaviour has fascinated scientists for many years and there are even reports of investigations into the deformation of natural rubber from the beginning of the last century. Rubber elasticity is particularly amenable to analysis using thermodynamics, as an elastomer behaves essentially as an entropy spring . It is even possible to derive the form of the basic stress-strain relationship from first principles by considering the statistical thermodynamic behaviour of the molecular network. 

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