Scanlan-Case criterion

Scanlan-Case criterion for elastically effective chains. Chains with i >3 and i > 3 are effective. 

Fig. 8.4 (a) Scanlan-Case criterion to find the elastically effective chains. A chain with both ends... 

The Scanlan-Case (SC) criterion states that a subchain is elastically effective if both its ends are connected to the elastically active junctions, i.e., whose path numbers / and i are larger than or equal to 3 (Figure 4.11). The criterion leads to... 

In Section 4.2, we introduced the Scanlan-Case (SC) criterion to find the elastically effective junctions and chains. SC proposes that a chain with both ends connected to elastically effective junctions (junctions whose path number is larger or equal to three) is elastically effective (Figure 8.4(a)) [9,10],... 

There remain many inactive subchains after such an end correction is made. For instance, a group of chains, such as shown in Figure 4.10, may be dangling from one junction as a whole. They are inactive. Scanlan and Case applied the graph theory and introduced a criterion forjudging the activity of a given subchain [15,16]. It is stated as follows. 

We next employ the criterion of Scanlan [12] and Case [13] that only subchains connected at both ends to junctions with at least three paths to the gel are elastically effective. We, thus, have i,i > 3 for an effective chain. A junction with one path (i = 1) to the gel unites a group of subchains dangling from the network matrix whose conformations are not affected by an applied stress. A junction with two paths (i = 2) to the gel merely extends the length of an effective subchain. We may call the junctions with i > 3 elastically effective junctions. An effective chain is defined as a chain connecting two effective junctions at its both ends. We, thus, find... 

Scanlan and Case have used another criterion to identify the chains and junctions which should control the elastic properties of networks. In their scheme, a junction is elastically active if three or more of its arms are independently attached to the network. A strand is elastically active if it is attached at both ends to an active junction. However, it can be shown that equation (55) still holds if Vg and Pg are replaced by the numbers of active chains and junctions Pa, respectively. This is stated in equation (56). In the phantom network model, the elastic modulus depends only on the cycle rank and therefore is not influenced by the choice of criterion. Furthermore, there is no difference between the two criteria when applied to perfect networks. All the equations derived above by graph-theoretical arguments hold for odd as well as even values of and may be adapted to networks of variable functionality merely by replacing (j) by its average 0.124,126,127... 

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