Real networks

Staverman, A.J. Properties of Phantom Networks and Real Networks. Vol. 44, pp. 73-102. 

Any real network must contain terminal chains bound at one end to a cross-linkage and terminated at the other by the end ( free end O of a primary molecule. One of these is indicated by chain AB in Fig. 92, a. Terminal chains, unlike the internal chains discussed above, are subject to no permanent restraint by deformation their configurations may be temporarily altered during the deformation process, but rearrangements proceeding from the unattached chain end will in time re-... 

Number (or number of moles) of cross-linked or branched units (Chaps. IX and XI). Hence, also the number of chains in a perfect network structure (Chap. XI). Effective number (or number of moles) of chains in a real network (Chaps. XI and XIII). 

Neurons are not used alone, but in networks in which they constitute layers. In Fig. 33.21 a two-layer network is shown. In the first layer two neurons are linked each to two inputs, x, and X2- The upper one is the one we already described, the lower one has w, = 2, W2 = 1 and also 7= 1. It is easy to understand that for this neuron, the output )>2 is 1 on and above line b in Fig. 33.22a and 0 below it. The outputs of the neurons now serve as inputs to a third neuron, constituting a second layer. Both have weight 0.5 and 7 for this neuron is 0.75. The output yfi j, of this neuron is 1 if E = 0.5 y, + 0.5 y2 > 0.75 and 0 otherwise. Since y, and y2 have as possible values 0 and 1, the condition for 7 > 0.75 is fulfilled only when both are equal to 1, i.e. in the dashed area of Fig. 33.22b. The boundary obtained is now no longer straight, but consists of two pieces. This network is only a simple demonstration network. Real networks have many more nodes and transfer functions are usually non-linear and it will be intuitively clear that boundaries of a very complex nature can be developed. How to do this, and applications of supervised pattern recognition are described in detail in Chapter 44 but it should be stated here that excellent results can be obtained. 

Two conditions must be met if this conclusion is to be revealed by the analysis. First, appropriate experimental procedures must be adopted to assure establishment of elastic equilibrium. Second, the contribution to the stress from restrictions on fluctuations in real networks must be properly taken into account, with due regard for the variation of this contribution with deformation and with degree of cross-linking. Otherwise, the analysis of experimental data may yield results that are quite misleading. 

Most probable positions of the chains are determined by the use of a characteristic vector r. This vector is representative of an average network chain of N links (the average links per chain). It deforms affinely whereas the actual network chains might not, and its value depends only upon network deformation. Crystallization leaves r essentially unaltered since the miniscule volume contraction brought about by crystallization can be ignored. But real network chains are severely displaced by crystallization. These displacements, however, must be compatible with the immutability of r. So in a sense, the characteristic vector r limits the configurational variations of the chains to those consistent with a fixed network shape and size at a given deformation. 

Classical molecular theories of rubber elasticity (7, 8) lead to an elastic equation of state which predicts the reduced stress to be constant over the entire range of uniaxial deformation. To explain this deviation between the classical theories and reality. Flory (9) and Ronca and Allegra (10) have separately proposed a new model based on the hypothesis that in a real network, the fluctuations of a junction about its mean position may may be significantly impeded by interactions with chains emanating from spatially, but not topologically, neighboring junctions. Thus, the junctions in a real network are more constrained than those in a phantom network. The elastic force is taken to be the sum of two contributions (9) ... 

Inhomogeneities in a real network may occur either because of a continuous distribution of molecular weight between crosslinks or due to the regions of different average molecular weights (as may be the case in randomly crosslinked networks). 

It is shown that model, end-linked networks cannot be perfect networks. Simply from the mechanism of formation, post-gel intramolecular reaction must occur and some of this leads to the formation of inelastic loops. Data on the small-strain, shear moduli of trifunctional and tetrafunctional polyurethane networks from polyols of various molar masses, and the extents of reaction at gelation occurring during their formation are considered in more detail than hitherto. The networks, prepared in bulk and at various dilutions in solvent, show extents of reaction at gelation which indicate pre-gel intramolecular reaction and small-strain moduli which are lower than those expected for perfect network structures. From the systematic variations of moduli and gel points with dilution of preparation, it is deduced that the networks follow affine behaviour at small strains and that even in the limit of no pre-gel intramolecular reaction, the occurrence of post-gel intramolecular reaction means that network defects still occur. In addition, from the variation of defects with polyol molar mass it is demonstrated that defects will still persist in the limit of infinite molar mass. In this limit, theoretical arguments are used to define the minimal significant structures which must be considered for the definition of the properties and structures of real networks. 

We see that, only if the parameter I is temperature independent, the entropic and energetic components of real networks with the sterical restrictions are identical to that of the phantom or affine network. 

At the very beginning, it seems worthwhile to put forward a definition of an ideal network, so that we can treat any real network by reference to this definition. An ideal network then, is defined to be a collection of Gaussian chains between /-functional junction points (crosslinks) under the condition that all functionalities of the junction points have reacted with the ends of all and different chains. Furthermore, neither the grouping of chain-ends into crosslinks, nor any external effect, such as interaction with a surrounding diluent, should change the Gaussian statistics of the individual chains. 

Even if completely homogeneous and disordered in the relaxed state, a real network differs from the ideal network, defined in Chapter I. Three types of network defects are commonly considered to be present in polymer networks unreacted functionalities, closed loops, and permanent chain entanglements. Within each group there are several possibilities dependent on the arrangement of chains the effect of defects on the elastic properties of the network is thus by no means simple, as has been stressed e.g. by Case (28). Several possible arrangements are shown in Fig. 1, where only nearest neighbour defect structures have been drawn. 

It has been pointed out repeatedly that the elastic behaviour of virtually all real networks in the unswollen state deviates appreciably from Gaussian behaviour. Often these deviations depend on the history... 

One of the key discoveries was the realization that very many real networks in nature, technology (e.g., the Internet and WWW) and human relations have similar structure and growth patterns, and can be described by the same mathematical formulas. All of them share similar properties and behavior. This discovery and the new theory have created an unprecedented opportunity for investigating resilience and vulnerabilities of the Internet and the WWW. For this reason we consider the scale-free network theory and related empirical results as being a significant development in the Cyberspace Security and Defense. 

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