Thermodynamics, rubbers statistical

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. 

D 2. Doty, P. M. Diskussionsbeitrag zu G. Gee — The interaction between rubber and liquids. X. Some new experimental tests of a statistical thermodynamic theory of rubber-liquid systems. Trans. Faraday Soc. 42 B, 46 (1946). 

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... 

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

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. 

However, most students not broadly exposed to statistical thermodynamics find such calculations difficult to follow at first. For this reason we first derive the ideal gas law via the very same principles that are employed in calculating the stress-elongation relationships in rubber elasticity. 

Fluids flow, boil, freeze, and evaporate. Solids melt and deform. Oil and water don t mix. Metals and semiconductors conduct electricity. Crystals grow. Chemicals react and rearrange, take up heat and gi e it off. Rubber stretches and retracts. Proteins catalyze biological reactions. What forces drive these processes This question is addressed by statistical thermodynamics, a set of tools for modeling molecular forces and behavior, and a language for interpreting experiments. 

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 4.5). They are basically all amorphous polymers above their glass transition and normally crosslinked. 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 nineteeth century. Elastomer deformation 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. 

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. ... 

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. 

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... 

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