Cycle rank

The structure of a perfect network may be defined by two variables, the cycle rank and the average junction functionality (f>. Cycle rank is defined as the number of chains that must be cut to reduce the network to a tree. The three other parameters used often in defining a network are (i) the number of network chains (chains between junctions) v, (ii) the number of junctions p, and (iii) the molecular weight Mc of chains between two junctions. They may be obtained from and using the relations... 

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. 

Equation (29) shows that the modulus is proportional to the cycle rank , and that no other structural parameters contribute to the modulus. The number of entanglements trapped in the network structure does not change the cycle rank. Possible contributions of these trapped entanglements to the modulus therefore cannot originate from the topology of the phantom network. 

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... 

A network is formed by network chains (N number of net chains) which are connected in the cross-links (junction points). The functionality / of cross-links (number of chains connected in a junction point) depends on their chemical nature, e.g., on the cross-linker. In a perfect network each network chain starts in one junction point and ends in another one. A real network is imperfect, but it can be described by two quantities, the network cycle rank and the number of junction points p/. 

The cycle rank or number of independent circuits, characterizes the network with greater generality, regardless of the nature of its imperfection. is the minimum number of scissions required to reduce the network to a spanning tree. 

The cycle rank is given by the difference of the number of net chains Ac and the number of junction point p,j... 

The cycle rank of a network is defined as the number of cuts required to reduce the network to a tree ( ). It is a structural factor characteristic of the perfection of a network ( ) for a perfect network, the cycle rsuik per chain ( is given by C>l-2/f. The ( values for various networks are listed in Table 3. The gels produced at very high extents of reaction still eidiibit various kinds of structural imperfections I for example, the cycle rank of the... 

Table III. Cycle Rank per Chain for Various Networks...

The last two models have been investigated extensively by Flory He considered networks forming part of a larger network with fixed jimctions outside the network under consideration. The crosslinking points of the inner network were assumed to fluctuate freely, or to have fixed positions in space. In both cases, the inner network can be considered as localized by boundary conditions. In particular, Flory considered a network formed in two hypothetical steps. First, a giant acyclic molecule is formed by joining all chains via the available multifunctional junctions such a tree-like molecule can be characterized by v -H 1 v junctions plus chain ends (i.e. number of labelled points). Figure 3 shows an example of the network after the first step. In the second step additional connections are formed by the reaction of 2 unreacted functionalities, which reduces the number of labelled points to approximately The number is called the cycle rank, which can be defined... 

In the perfect network, the cycle rank % is connected to the number of junctions M and the number of chains by the relation... 

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... 

FIGURE 29.3. Schematic representation of a network structure with r ei = 12, /He, = 9, and = 4 (a). Note that the cycle rank is the number of cuts needed to reduce the network to a tree (b). 

To characterize the strueture of gels, parameters such as the number of cross-links /X, the number of subchains v, and their average molecular weight M, the branching index of the cross-links <, the number of free ends Vend, and the cycle rank should be specified [1]. 

Because of the topological relations of the networks, v(l — 2/< ) is identified to be of cycle rank f, and hence the free energy is given 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. 

The quantities X, are the principal extension ratios, which specify the strain relative to an isotropic state of Reference Chapt, 5. The cycle rank 4 was first introduced by Flory and is the number of independent circuits in the network or the number of chains which have to be cut to reduce the network to an acyclic structure or tree. Subsidiary quantities called the number of effective chains and junctions noted, respectively, and Pe can be defined by the relationship... 

As already mentioned, a network is characterized by its cycle rank density 4/Vo-Scanlan and Case have defined an active junction as one joined by at least three paths to the gel network and an active chain as one terminated by active junctions at both its ends. Pearson and Graessley have shown that for a randomly interconnected network whose junctions are of even functionality... 

Another way to obtain networks with irregularities is to mix chains end and cross-linking agent in a non-stoichiometric ratio of reactive groups The cycle rank and the topology of the resulting networks can be predicted by Monte-Carlo simulation... 

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