Cycle rank theory

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. 

The cycle rank completely defines the connectivity of a network and is the only parameter that contributes to the elasticity of a network, as will be discussed further in the following section on elementary molecular theories. In several other studies, contributions from entanglements that are trapped during cross-linking are considered in addition to the chemical cross-links [23,24]. The trapped entanglement model is also discussed below. 

During the last decade, the classical theory of rubber elasticity has been reconsidered significantly. It has been demonstrated (see, e.g. Ref.53>) that, for the phantom noninteracting network whose chains move freely one through the other, the equations of state of Eqs. (28) and (29) for simple deformation as well as for W, Q and AIJ [Eqs. (30)-(32) and (35)—(37)] are proportional not to v but to q, which is the cycle rank of the network, i.e. the number of independent circuits it contains. For a perfect phantom network of uniform functionality cp( > 2)... 

It is possible to calculate a number of other structural parameters, for instance those listed in Table 1. In Ref. relations were derived for the average lunctionality of active branch points, fe, a quantity which is important in the rubber elasticity theory for conversion of into the (effective) cycle rank. In terms of pgf T (z), f is defined by... 

This chapter studies the local and global structures of polymer networks. For the local structure, we focus on the internal structure of cross-Unk junctions, and study how they affect the sol-gel transition. For the global structure, we focus on the topological connectivity of the network, such as cycle ranks, elastically effective chains, etc., and study how they affect the elastic properties of the networks. We then move to the self-similarity of the structures near the gel point, and derive some important scaling laws on the basis of percolation theory. Finally, we refer to the percolation in continuum media, focusing on the coexistence of gelation and phase separation in spherical coUoid particles interacting with the adhesive square well potential. 

This article presents uniaxial extension measurements on cis-1,4-polybutadiene networks of known junction functionality. The observed values of the reduced force from uniaxial extension measurements conform to the constrained junction theory of Flory. The reduced force intercept at 1/cX = 0 is fully comprehensible in terms of the cycle rank of the network, and can be calculated from chemical considerations. This holds even though the polybutadiene melt has a high plateau modulus. Therefore, discrete topological entanglements do not contribute perceptibly to the equilibrium modulus of polybutadiene networks. 

Reduced force as functions of for Network A. Data for bulk samples (+) and samples swollen with decane (triangles) are given and the polymer volume fractions are indicated with each isotherm. The solid curves were calculated according to theory with parameters given in Table III. The lower dashed line represents the cycle rank density calculated from the chemical constitution of Newtwork A with corrections due to dangling chain ends included. The upperdashed line represents the affine limit for a perfect network obtained with respect to the corrected chemical value of Network A. 

The number of elastic degrees of freedom is three times the cycle rank. The elasdc behaviour of real networks in particular in the unswollen suite, deviates from that of phantom networks. A recent theory to explain this deviation is based upon the assumption of a difference between pofyfimctiorud and bijwictiorml junctions which is alien to the concept of phantom networks and is pkysicedly not plausible. An alternative theory is presented based upon die concept of constrained elastic degrees of freedom instead of constrained junctions. 

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