Entropic Theory

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... 

Above the amorphous transition, the modulus of the tie molecules increases with temperature ( is proportional to T) according to entropic theory of rubber elasticity. n is nearly constant so... 

The experimental data about rubbers deformation are usually interpreted within the frameworks of the high-elastieity entropic theory [1-3], elaborated on the basis of assumptions about high-elastic polymers incompressibility (Poisson s ratio V = 0.5) and polymer chains Gaussian statistics. As it is known [4], the Gaussian statistic is characteristic only for the networks, prepared by chains concentrated solution curing, in the case of their compression or weak (draw ratio A, < 1.2) tension. For such stmctures the fractal dimension d = 2 and in case of v = 0.5 the following classical expression was obtained [3] ... 

As a rule, at present crosslinked polymer networks are characterised within the frameworks of entropic rubber high-elasticity concepts [2, 3]. However, in recent years works indicating a more complex structure of crosslinked rubbers have appeared. Flory [4] demonstrated the existence of dynamic local order in rubbers. Balankin [5] showed principal inaccuracy of the entropic high-elasticity theory and proposed a high-elasticity fractal theory of polymers. These observations suppose that more complete characterisation of these materials is necessary for the correct description of the structure of rubbers and their behaviour at deformation. In paper [6] this was carried out by the combined use of a number of theoretical physical concepts, namely the rubber high-elasticity entropic theory, the cluster model of the amorphous state structure of polymers [7, 8] and fractal analysis [9]. 

Therefore, the complete methods of calculation of the characteristics of crosslinked networks was proposed, which combines the ruhher high-elasticity entropic theory, the cluster model of amorphous state structure of polymers and fractal analysis methods. The proposed method has shown that growth in statistical segment length is observed as the drawing ratio increases. This snpposes that the chain statistical flexibility depends not only on its chemical constitntion, but also on the network deformed state. The considered method can be nsed for computer simulation and prediction of the structure of crosslinked polymer networks [6]. 

The co-ordination of theoretical and experimental results at higher e is realised either within the frameworks of phenomenological modifications of entropic theory... 

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

By following the standard procedures of statistical mechanics, one arrives at an equation which can be converted into the BET equation (2.12) by the simple substitution 9i/9i , ki = c. Thus parameter c acquires a significance different from that in the BET theory in essence it now involves entropic terms as well as energetic terms. 

The natiue of the rate constants k, can be discussed in terms of transition-state theory. This is a general theory for analyzing the energetic and entropic components of a reaction process. In transition-state theory, a reaction is assumed to involve the formation of an activated complex that goes on to product at an extremely rapid rate. The rate of deconposition of the activated con lex has been calculated from the assumptions of the theory to be 6 x 10 s at room temperature and is given by the expression ... 

For more than two decades researchers have attempted to overcome the inadequacies of Flory s treatment in order to establish a model that will provide accurate predictions. Most of these research efforts can be grouped into two categories, i.e., attempts at corrections to the enthalpic or noncombinatorial part, and modifications to the entropic or combinatorial part of the Flory-Huggins theory. The more complex relationships derived by Huggins, Guggenheim, Stavermans, and others [53] required so many additional and poorly determined parameters that these approaches lack practical applications. A review of the more serious deficiencies... 

Crosslinked polymers are rather peculiar materials in that they never melt and they exhibit entropic elasticity at elevated temperatures. The present review on the influence of crosslink density is structured around model polymers of uniform composition but with widely varying numbers of crosslinks. The degree of crosslinking in the polymers was verified by use of the theory of rubber elasticity. 

Although the basic concept of macromolecular networks and entropic elasticity [18] were expressed more then 50 years ago, work on the physics of rubber elasticity [8, 19, 20, 21] is still active. Moreover, the molecular theories of rubber elasticity are advancing to give increasingly realistic models for polymer networks [7, 22]. 

The large deformability as shown in Figure 21.2, one of the main features of rubber, can be discussed in the category of continuum mechanics, which itself is complete theoretical framework. However, in the textbooks on rubber, we have to explain this feature with molecular theory. This would be the statistical mechanics of network structure where we encounter another serious pitfall and this is what we are concerned with in this chapter the assumption of affine deformation. The assumption is the core idea that appeared both in Gaussian network that treats infinitesimal deformation and in Mooney-Rivlin equation that treats large deformation. The microscopic deformation of a single polymer chain must be proportional to the macroscopic rubber deformation. However, the assumption is merely hypothesis and there is no experimental support. In summary, the theory of rubbery materials is built like a two-storied house of cards, without any experimental evidence on a single polymer chain entropic elasticity and affine deformation. 

In this chapter, AFM palpation was introduced to verify the entropic elasticity of a single polymer chain and affine deformation hypothesis, both of which are the fundamental subject of mbber physics. The method was also applied to CB-reinforced NR which is one of the most important product from the industrial viewpoint. The current status of arts for the method is still unsophisticated. It would be rather said that we are now in the same stage as the ancients who acquired fire. However, we believe that here is the clue for the conversion of rubber science from theory-guided science into experiment-guided science. AFM is not merely high-resolution microscopy, but a doctor in the twenty-first century who can palpate materials at nanometer scale. 

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