On Rubberlike Elasticity

Mark, J.E., Some recent theory, experiments and simulations on rubberlike elasticity, J. Phys. Chem. B, 107, 903-9013, 2003. 

Mark, J. E., Some Unusual Elastomers and Experiments on Rubberlike Elasticity. Prog. Polym. Sci. 2003,28,1205-1221. 

Multimodal networks represent a method to determine the effect of network chain-length distribution - on rubberlike elasticity. Chain-length distribution has not received much attention even though manipulation of the chain-length distribution can give large improvements in mechanical properties. There are two primary reasons for this... 

Mark JE. Some unusual elastomers and experiments on rubberlike elasticity. Prog Polym Sci 2003 28 1205-21. 

Many other parepistemes were stimulated by the new habits of precision in theory. Two important ones are the entropic theory of rubberlike elasticity in polymers, which again reached a degree of maturity in the middle of the century (Treloar 1951), and the calculation of phase diagrams (CALPHAD) on the basis of measurements of thermochemical quantities (heats of reaction, activity coefficients, etc.) here the first serious attempt, for the Ni-Cr Cu system, was done in the Netherlands by Meijering (1957). The early history of CALPHAD has recently been... 

The statistical distribution of r values for long polymer chains and the influence of chain structure and hindrance to rotation about chain bonds on its root-mean-square value will be the topics of primary concern in the present chapter. We thus enter upon the second major application of statistical methods to polymer problems, the first of these having been discussed in the two chapters preceding. Quite apart from whatever intrinsic interest may be attached to the polymer chain configuration problem, its analysis is essential for the interpretation of rubberlike elasticity and of dilute solution properties, both hydrodynamic and thermodynamic, of polymers. These problems will be dealt with in following chapters. The content of the present... 

Before proceeding further it is desirable to point out that dH/dL)T,p will differ indiscernibly from dE/dL)T,p in any likely application to rubberlike elastic phenomena. This may be seen by observing that the second term on the right-hand side of the relation... 

J.E. Mark, Illustrative modeling studies on elastomers and rubberlike elasticity. In M. Laudon, and B. Romanowiczs (Eds.), International Conference on Computational Nanoscience. Computational Publications, Boston, Hilton Head Island, SC, 2001, p. 53. 

Distribution functions for the end-to-end separation of polymeric sulfur and selenium are obtained from Monte-Carlo simulations which take into account the chains geometric characteristics and conformational preferences. Comparisons with the corresponding information on PE demonstrate the remarkable equilibrium flexibility or compactness of these two molecules. Use of the S and Se distribution functions in the three-chain model for rubberlike elasticity in the affine limit gives elastomeric properties very close to those of non-Gaussian networks, even though their distribution functions appear to be significantly non-Gaussian. 

The present theoretical approach to rubberlike elasticity is novel in that it utilizes the wealth of information which RiS theory provides on the spatial configurations of chain molecules. Specifically, Monte Carlo calculations based on the RIS approximation are used to simulate spatial configurations, and thus distribution functions for end-to-end separation r of the chains. Results are presented for polyethylene and polydimethylsiloxane chains most of which are quite short, in order to elucidate non-Gaussian effects due to limited chain extensibility. 

In most real systems, energy and entropy changes can occur. The elasticity of an ideal network is entropy controlled. In this picture stresses are caused by the chain orientation. From the theory of rubberlike elasticity it can be shown that the shear modulus of an ideal network depends on the number of elastically effective cahins between the crosslinks (19) Gq = v k-T where v means the number of elastically effective chains in unit volume. 

The molecular theory of rubberlike elasticity predicts that the first coefficient, Ci, is proportional to the number N of molecular strands that make up the three-dimensional network. The second coefficient, C, appears to reflect physical restraints on molecular strands like those represented in the tube model (Graessley, 2004) and is in principle amenable to calculation. The third parameter,, is not really independent. When the strands are long and flexible, it will be given approximately by 3X, where Xm is the maximum stretch ratio of an average strand. But is inversely proportional to N for strands that are randomly arranged in the unstretched state (Treloar, 1975). Jm is therefore expected to be inversely proportional to Ci. Thus the entire range of elastic behavior arises from only two fundamental molecular parameters. 

The ratios of mean-squared dimensions appearing in Eq. (4.13) are microscopic quantities. To express the elastic free energy of a network in terms of the macroscopic state of deformation, an assumption has to be made relating microscopic chain dimensions to macroscopic deformation. Their relation to macroscopic deformations imposed on the network has been a main area of research in the area of rubberlike elasticity. Several models have been proposed for this purpose, which are discussed in the following sections. Before, we describe the macroscopic deformation, stress, and the modulus of a network. 

It can seen from Figure 5.4 that the equilibrium compliance Je decreases uniformly from the 1007/DDS to the 828/DDS and is expected on the basis of the kinetic theory of rubberlike elasticity, since the concentration of network chains increases and the molecular weight per crosslinked unit, Mx, decreases in the same order. The Mx values calculated as /o/JT/g are remarkably close to the molecular weight values of the starting epoxy resins. 

L. R. G. Treloar, The Physics of Rubber Elasticity, third ed.. Clarendon Press, Oxford, 1975. J. E. Mark, Thermoelastic results on rubberlike networks and their bearing on the foundations of elasticity theory, Macromol. Revs. 11, 135 (1976). 

The most important property of elastomers and elastoplasts is their accentuated high, hard or soft rubberlike elasticity. The commercially interesting property values in these cases are generally only reached after formulating or compounding with fillers, plasticizers, etc. The subsequent cross-linking depends on the type of rubber, that is, on the nature of the cross-linkable or vulcanizable groups. 

The goals of the new edition of this book are much the same as those described in the Preface to the First Edition, namely a broad overview of elastomers and rubberlike elasticity. Again, the emphasis is on a unified treatment, ranging from chemical aspects such as elastomer synthesis and curing, through theoretical developments and characterization of equilibrium and dynamic properties, to final applications (including tire manufacture and engineering). 

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