Networks, infinite definition

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. 

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 define hybrid inorganic-organic framework materials as compounds that contain both inorganic and organic moieties as integral parts of a network with infinite bonding connectivity in at least one dimension. This definition excludes systems that are molecular or oligomeric, such as the... 

The origins of the present three-dimensional molecular-level branching concepts can be traced back to the initial introduction of infinite network theory by Flory [62-65] and Stockmayer [66, 67], In 1943, Flory introduced the term network cell, which he defined as the most fundamental unit in a molecular network structure [68]. To paraphrase the original definition, it is the recurring branch juncture in a network system as well as the excluded volume associated with this branch juncture. Graessley [69, 70] took the notion one step further by describing... 

During emulsion polymoization proccesses crosslinking can occur due to the presence of radicals and unsaturation in the polymer chain. Chain transfer agents are often used to decrease the level of crosslinking during polymerization. The gel content of a crosslinkable polymo is d ned as the fraction of material of infinite MM experimental criteria are usually more aibitraiy, and strongly depend on the experimental procedure. The same holds for the gel point, Le. the conversion at which the first insoluble polymer network makes its appearance. For example, in ABS gel content can be based on the toluene extraction of the soluble fraction of the polybutadioie. Hie insoluble residue is the gel fraction by definition. 

In DDD, systems are modeled as networks of streams, which are infinite sequences over a type. Streams and other nonfinite structures raise issues in model theory that are only now being ironed out (Barwise Moss 1996). These problems are intrinsic to logics based in well-founded set theory, which must deal with streams indirectly. The typical representation is mapping from natural numbers to values. The problem is that proofs about streams under this representation reduce to inductions over the naturals, instead of "structural coinductions that reflecting the definition style. 

As the gel point is the point where at least one molecule just has an infinite molecular weight, it is the point where the system is just not completely soluble any more. Consequently, extraction of the system is a method to determine the gel point. It will be dear that this method is in general not an appropriate one and definitely not suitable for physical networks. 

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. 

When one molecnle grows to span the entire network, the system is at the gel point. The gel point is best determined from measurements of flow behaviour. It is the point at which the zero-shear viscosity of the system becomes infinite and the system develops a shear modulus. However, these definitions are not very helpfnl practically, becanse it is difficult to measure very large values of viscosity or a very small shear modulus. A more useful definition relies on measnrements of the dynamic shear modnli, the gel point being defined by the condition G = G". The gel point is also signalled by the onset of insolnbility of the three-dimensional network. 

---