Network chains definition

The next step in the development of a model is to postulate a perfect network. By definition, a perfect network has no free chain ends. An actual network will contain dangling ends, but it is easier to begin with the perfect case and subsequently correct it to a more realistic picture. We define v as the number of subchains contained in this perfect network, a subchain being the portion of chain between the crosslink points. The molecular weight and degree of polymerization of the chain between crosslinks are defined to be Mj, and n, respectively. Note that these same symbols were used in the last chapter with different definitions. 

The general conclusion of the mentioned works was that the appearance of the jump on the dependences of network volume on the composition of the solvent or on temperature is reached only at some definite content of ionic groups in the network chains. For neutral networks with flexible chains, the collapse is usually not observed. Exceptions to this rule were reported for poly(isopropylacrylamide) (PIPAA) [16], poly(vinylcaprolactam) and poly-(2-vinylpyrrolidone) [17] gels. The specific feature of these systems is that the transition takes place in structured solvents water or concentrated aqueous solutions of aluminium sulfate. 

In this Chapter we will consider the elasticity and swelling of networks which deviate from our definition of ideal networks only because of a certain number of network defects (see Chapter II, Section 2). We will designate by v the number of elastically effective network chains, which in an ideal network equals the number of chains because of the absence of defects. 

According to our definition the end-to-end distances of the network chains have an a priori probability distribution which is Gaussian. The effect of finite extensibility of the chains will be postponed to Chapter IV, because it is a special aspect of non-Gaussian behaviour. 

The theoretical approach for determining the deformation behaviour of a network due to swelling or due to a mechanical force (stress-strain measurements, compression experiment) is based on a hypothetical phantom network. A phantom network is, by definition, a network with the fictitious property that chains and junctions can move freely through one another without destroying the cormectivity of the network. Usually, the network chains behave as Gaussian chains. Within the phantom network model, three network types can be distinguished ... 

Three-dimensional cross-linked networks are, according to definition, considered to be infinite in size. It is therefore pointless to consider their molecular weights. Such cross-link networks are classified according to the network chain lengths, branch type, and branch density. 

If both sides are divided by the original cross-sectional area Aq = VJIq, then, with the definition of the tensile stress a, = F/Aq, the gas constant R = /cVl, and the molar concentration [Mj = Ni/VqNi of the network chains, equation (11-35) becomes... 

Whea there are reactants with three or more functionahties participating ia the polymerization, branching and the formation of iatermolecular linkages, ie, cross-linking of the polymer chains, become definite possibiUties. If extensive cross-linking occurs in a polymer system to form network stmctures, the mobiUty of the polymer chains is greatiy restricted. Then the system loses its fluidity and transforms from a moderately viscous Hquid to a gelled material with infinite viscosity. The experimental results of several such reaction systems are collected in Table 6. 

Model Networks. Constmction of model networks allows development of quantitative stmcture property relationships and provide the abiUty to test the accuracy of the theories of mbber elasticity (251—254). By definition, model networks have controlled molecular weight between cross-links, controlled cross-link functionahty, and controlled molecular weight distribution of cross-linked chains. Sihcones cross-linked by either condensation or addition reactions are ideally suited for these studies because all of the above parameters can be controlled. A typical condensation-cure model network consists of an a, CO-polydimethylsiloxanediol, tetraethoxysilane (or alkyltrimethoxysilane), and a tin-cure catalyst (255). A typical addition-cure model is composed of a, ffl-vinylpolydimethylsiloxane, tetrakis(dimethylsiloxy)silane, and a platinum-cure catalyst (256—258). 

In the derivation of eqn. (7) it was assumed that n (number of equivalent random links) is the same for all chains. For our samples (B2 system, Mw/Mn=1.45), this assumption is definitely not correct. Therefore, it is desirable to obtain birefringence results on networks prepared from monodisperse polymer (in that n is constant), before the validity of n itself is questioned. 

Porter s value chain is one basis for the development of the supply chain. The term supply chain was created by consultant Keith Oliver in 1982 according to Heckmann et al. (2003). Compared to the company-internal focus of Porter s value chain, the supply chain extends the scope towards intra-company material and information flows from raw materials to the end-consumer reflected in the definition of Christopher (1992) a supply chain is a network of organizations that are involved through upstream and downstream linkages in different processes and activities that product value in the form of products and services in the hand of the ultimate consumer . Core ideas of the supply chain concept are ... 

When characterizing polymer networks, the following definitions are typically applied [150] and are illustrated in Fig. 7. When a radical on a polymer chain propagates through a pendant double (i.e. a double bond from a monomer with one double bond already reacted), a crosslink, secondary cycle, or primary cycle can be formed. A crosslink forms when the radical reacts with a pendant... 

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. 

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