One-Step Networks

In more detail, the crudest version of the affine deformation problem of flexible junction-point networks will lead to an energy 

These are standard forms with Vi = 0 when one is deforming an equilibrium system. To improve on such expressions one must review the physical circumstances. The problem arises that it is far easier to compress a rod by the Euler 

Crude as this model is, it shows that one can derive an effective modulus, and such models can be refined if experimental situations warrant it. However, these models depend on a uniformity which does not in general occur unless some agency (like a bridge building engineer) forces it. In general, more uniform structures result from tree type growth as we will see below. 

More realistic calculations need to include features which are intuitively obvious, but not easy to handle. The affine assumption can be extended to rigid junction-points, so that if there is an angular energy cos 0 where 

If the rods buckle there has to be a recalculation of n allowing for this, and if the junction point is truly rigid, i.e., cos 0 remains the same and extra buckling has 

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Unlike one-step networks, where all stiff segments projecting from each branchpoint are completely or almost completely indistinguishable from each other, in... 

Figures 42 and 43 represent two extreme cases of one-step networks, one formed in a concentrated environment at Co S> C, and the other in a dilute solution at Co < C and subsequently concentrated. In Fig. 42 we draw one tree which, for simplicity, is in the plane. The other trees represented by dots where they cross the plane. These dots do not represent unreacted monomers.

Fig. 42. One-step network prepared at high concentration, C, > C. Each tree (fractal polymer) is deeply interpenetrated with others resulting in rather homogeneous network stmcture and low probability of large voids...

Fig. 43. One-step network prepared by first polymerizing trees (fractal polymers) at high dilution, Co < C. and then concentrating the solution to C to bring the trees in contact with each other. A small number of interfractal covalent bonds may be formed if polycondensation is allowed to proceed, and the network is inhomogeneous with high probability of large interfractal voids...

Fig. 44. An example of intrafractal drcuit. In the case of rigid one-step networks such circuits are rare. More common are small and large circuits created during and after gelation, where each contains segments and branchpoints belonging to more than one precursor FP...

Reaction measurement studies also show that the chemistry is often not a simple one-step reaction process (37). There are usually several key intermediates, and the reaction is better thought of as a network of series and parallel steps. Kinetic parameters for each of the steps can be derived from the data. The appearance of these intermediates can add to the time required to achieve a desired level of total breakdown to the simple, thermodynamically stable products, eg, CO2, H2O, or N2. 

It may also be possible to crosslink the acrylic PSA with the help of multifunctional acrylates or methacrylates [87], These monomers can simply be copolymerized with the balance of the other monomers to form a covalently crosslinked network in one step. Since the resulting polymer is no longer soluble, this typ)e of crosslinking is typically limited to bulk reactions carried out as an adhesive coating directly on the article or in emulsion polymerizations where the crosslinked particles can be dried to a PSA film. 

Tricyclohexaprenol, a possible forerunner of sterols in the evolution of biomembranes, was synthesized by construction of the cyclic network in one step using cation-olefin tricyclization and subsequent stereocontroUed attachment of the Cio appendage to ring C. 

An example for a synthesis of a poly(siloxane) network is shown in Fig. 37b. In a one-step reaction the mesogenic moieties as well as the crosslinking agent are coupled via an addition reaction to the reactive linear poly(methylhydrogensiloxane) backbone 92). Because of similar reactivity of the crosslinking agent and mesogenic molecules, a statistical, disordered addition to the backbone has to be expected. 

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... 

Applying mass-balance and thermodynamic constraints typically leaves one without a precisely defined (unique) solution for reaction fluxes and reactant concentration, but instead with a mathematically constrained feasible space for these variables. Exploration of this feasible space is the purview of constraint-based analysis. It has so far been left unstated that any application in this area starts with the determination of the reactions in a system, from which the stoichiometric matrix arises. This first step, network reconstruction, integrates genomic and proteomic data to determine carefully the enzymes present in an organism, cell, or subcellular compartment. The network reconstruction process is described elsewhere [107]. 

For the purpose of this rule, the forward and reverse directions of a reversible step are counted as two steps. Each arrowhead in the network, except if pointing at a co-product or co-reactant, thus corresponds to one step. Steps in which a species i does not participate as reactant or product do not contribute to the respective r. For example, in the network... 

The term competing steps is used if one and the same component participates as reactant in more than one step of the pathway or network. [The idea is that such steps "compete" with one another for the reactant they have in common an alternative term is series-parallel steps.] The simplest such case is [21]... 

In practice, many reaction systems involve non-trace intermediates, but no obvious non-simple reactions of intermediates. A good strategy in such situations is to cut the overall reaction network into portions at the non-trace intermediate or intermediates (see Section 6.5), then reduce the portions as described for simple networks in Section 6.4.1. Network reduction makes it unnecessary to keep track of trace intermediates (except those reacting in a non-simple manner) and so obviates much of the hard work Trace intermediates are the more troublesome ones in network elucidation because they are difficult or impossible to detect, identify, analyze for, or synthesize, tasks that usually do not pose problems with intermediates that rise above trace level. Often, the network portions will turn out to be "piecewise simple" (see Section 6.5). If not, further cutting at additional nonsimple steps is called for when these become apparent. 

A catalyst may exist in two or more forms with different catalytic activities. In the simplest systems of this type, two such forms interconvert in a quasi-equilibrium step. The conversion may or may not involve other species. It may, for example, be a ligand exchange. The two catalyst species may catalyze the same reaction or different ones. A network of this type, with ligand exchange and different reactions A — P and B — Q, is... 

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