Guth and James

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. 

An equation for the modulus of ideal rubber was derived from statistical theory that can be credited to several scientists, including Flory, and Guth and James (Sperling, 1986). A key assumption in derivation of the eqmtion is that the networks are Gaussian. 

Of fundamental importance is the fact that it has proved possible to calculate the stress-strain diagram for a material like rubber starting from some simple assumptions concerning the free rotation, and flexibility of the coiled molecules. Thus, Guth and James derived the formula ... 

Equation (3.31) is called the equation of state of the rubber. This equation was firstly derived by Guth and James in 1941 (Guth and James 1941). We conventirai-ally make an ideal-chain approximation with C = 1. 

Through the research of Guth and James (31-35), Treloar (36), Wall (37), and Flory (38), the quantitative relations between chain extension and entropy reduction were clarified. In brief, the number of conformations that a polymer chain can assume in space were calculated. As the chain is extended, the number of such conformations diminishes. (A fully extended chain, in the shape of a rod, has only one conformation, and its conformational entropy is zero.)... 

Since most rheologists would be unfamiliar with the microscopic theories of mbber elasticity, Alfrey presented a thorough discussion of the development of these equations. The key names include Kuhn, Guth and Mark, Wall, Treloar, Flory, and Guth and James. Separate treatments of the work of Wall and of Treloar will be found in another chapter. Alfrey is especially interested in the large strain limit, where the Gaussian theories are not appropriate. The inverse Langevin function appears here as a better representation of the chain when it is near full extension. 

Alftey includes a synopsis of the work of Hory on the formation of gels by polycondensation reactions [23]. He also recapitulates the discussion of Guth and James [24] on the explanation of the observed stress-extension curve. 

Guth E, James HM and Mark IT, "The Kinetic Theory of Rubber Elasticity", in Mark H and Whitby GS (Eds), "Scientific Progress in the Field of Rubber and Synthetic Elastomers" Interscience Publishers, New York, Vol. II, pp 253-299,1946. 

The mean values of the chain end-to-end vectors are displaced affinely with the macroscopic extension in the James-Guth theory (James and Guth 1943). The fluctuations of the junctions are independent of the deformation of the sample. As a consequence, the end-to-end vectors are deformed not affinely. The free energy of elasticity of the free-fluctuation limit is... 

For a tetrafunctional (/ = 4) random crosslink, g =1/2 - the James-Guth and Edwards-Freed result. In the limit of high / values, g approaches one, the value for the affine transformation theories. Flory (1), using cycle rank theory, has obtained the same result as Graessley. Mark [19] gives an introduction to cycle rank theory. Table 7.1 lists the various models and values of g obtained from each. 

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

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. 

In discussing more realistic models we consider first the modulus of the constrained fluctuation theory of Flory. Flory s assumption, that entanglements only restrict the fluctuations of the crosslinks, gives at once the result that the modulus is between the extremes — affine and James and Guth. The constraint parameter k interpolates between both models. This is revealed by the following expression " ... 

Figure 3.5 Comparison of experiment (points) and theory [Eq. (3.49)] for the entropy elasticity of the same sample shown in Fig. 3.3. [Reprinted with permission from H. M. James and E. Guth, J. Chem. Phys. 11 455 (1943).]...

Early theories of Guth, Kuhn, Wall and others proceeded on the assumption that the microscopic distribution of end-to-end vectors of the chains should reflect the macroscopic dimensions of the specimen, i.e., that the chain vectors should be affine in the strain. The pivotal theory of James and Guth (1947), put forward subsequently, addressed a network of Gaussian chains free of all interactions with one another, the integrity of the chains which precludes one from the space occupied by another being deliberately left out of account. Hypothetical networks of this kind came to be known later as phantom networks (Flory, 1964,... 

James and Guth showed rigorously that the mean chain vectors in a Gaussian phantom network are affine in the strain. They showed also that the fluctuations about the mean vectors in such a network would be independent of the strain. Hence, the instantaneous distribution of chain vectors, being the convolution of the distribution of mean vectors and their fluctuations, is not affine in the strain. Nearly twenty years elapsed before his fact and its significance came to be recognized (Flory, 1976,... 

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