Reaction, extent networks

The task now is to select the linear combinations that will most probably correspond to independent parts of the reaction network with easily interpretable stoichiometry. A simplification of the data in the matrix can be achieved by such a rotation that the axes go through the points in Fig. A-2 (this is equivalent to some zero-stoichiometric coefficients) and that the points of Fig. A-3 are in the first quadrant (this corresponds to positive reaction extents) if possible. Rotations of the abscissa through 220° and the ordinate through 240° lead to attaining both objectives. The associated rotation matrix is ... 

At the gel point, when in the course of a three-dimensional polymerization reaction a network first appears, its mass is negligible practically all of the polymerizing material still exists as monomers, dimers, trimers, and larger groups which are not bound to the network. As the reaction proceeds after the gel point, more and more of these loose groups become attached to the network, so that its relative mass (the gel fraction ) gradually increases and approaches unity. Approximate equations for the increase in gel fraction with extent of reaction have been given by Flory (1941) and Stockmayer (1943). 

Fig. 2). If the excitation frequency (oj) is much faster than tu , then the molecules do not have time to respond and there is no flow if to < o , then the molecules slide easily past one another (i.e., the viscosity is low) and little energy is dissipated. Similarly, if the frequency is fixed, but the degree of reaction (extent of cross-linking) is low, the sol is fluid and W (or G") is small as the reaction proceeds p increases), the viscosity increases and G" passes through a maximum before the gel becomes purely elastic (when the network is too stiff to flow) and dissipation is arrested. The storage modulus, G, is the familiar elastic (static) shear modulus when (o> u>o, but G (

The question is now Which reaction pathways arc Followed, and to what extent This asks for a detailed modeling of the kinetics of the individual reaction steps of this network. This can be achieved on the basis of the half-lives of four s-triazinc herbicides in soil [17]. Figure 10.3-13 shows the four compounds For which data were Found in the literature. 

We noted above that the presence of monomer with a functionality greater than 2 results in branched polymer chains. This in turn produces a three-dimensional network of polymer under certain circumstances. The solubility and mechanical behavior of such materials depend critically on whether the extent of polymerization is above or below the threshold for the formation of this network. The threshold is described as the gel point, since the reaction mixture sets up or gels at this point. We have previously introduced the term thermosetting to describe these cross-linked polymeric materials. Because their mechanical properties are largely unaffected by temperature variations-in contrast to thermoplastic materials which become more fluid on heating-step-growth polymers that exceed the gel point are widely used as engineering materials. 

Equation (5.47) is of considerable practical utility in view of the commercial importance of three-dimensional polymer networks. Some reactions of the sort we have considered are carried out on a very large scale Imagine the consequences of having a polymer preparation solidify in a large and expensive reaction vessel because the polymerization reaction went a little too far Considering this kind of application, we might actually be relieved to know that Eq. (5.47) errs in the direction of underestimating the extent of reaction at... 

Ammonium salts of the zeolites differ from most of the compounds containing this cation discussed above, in that the anion is a stable network of A104 and Si04 tetrahedra with acid groups situated within the regular channels and pore structure. The removal of ammonia (and water) from such structures has been of interest owing to the catalytic activity of the decomposition product. It is believed [1006] that the first step in deammination is proton transfer (as in the decomposition of many other ammonium salts) from NH4 to the (Al, Si)04 network with —OH production. This reaction is 90% complete by 673 K [1007] and water is lost by condensation of the —OH groups (773—1173 K). The rate of ammonia evolution and the nature of the residual product depend to some extent on reactant disposition [1006,1008]. 

Baekeland had to make important discoveries before he could bridge the gap between the initial concept and final products. In particular, he found that the base-catalysed condensation of phenol and formaldehyde can be carried out in two parts. If the process is carefully controlled, an intermediate product can be isolated, either as a liquid or a solid, depending on the extent of reaction. At this stage, the material consists of essentially linear molecules and is both fusible and soluble in appropriate solvents. When heated under pressure to 150 °C, this intermediate is converted to the hard, infusible solid known as bakelite . This second stage is the one at which the three-dimensional cross-linked network develops. 

Hydroxy-terminated PDMS, however, has disadvantages. The monofunctional ends limit the number of connections between the polymer (or oligomer) molecule and the glass network to two. This limitation raises the possibility that some PDMS molecules are not tied at both ends to the glass network if the polycondensation does not go to completion i.e. there may be "dangling" or loose PDMS chains in the final sol-gel material. This occurance of free ends would indeed be anticipated since the extent of reaction most likely is not 100%. Hence, the physical properties, specifically the mechanical behavior of the overall material, would be expected to suffer as a result of loose PDMS chains in the system. Disregarding this potential problem, the mechanical behavior of the sol-gel hybrids are, ultimately, influenced by the mechanical behavior of the modifying elastomer ... 

Although the reaction scheme shows a complete hydrolysis before condensation begins, this is likely not correct as stated earlier. The relative rates and extents of these two reactions will particularly depend on the amount of water added and the acidity of the system (10,11). The high functionality of the triethoxysilane endcapped PTMO oligomer should enhance the incorporation of PTMO molecules into the TEOS network. It was also assumed that the reactivities would be the same between silanol groups from silicic acid and endcapped PTMO. Therefore, no preferential condensation was expected and the deciding factors for which type of condensation (self- or co-) took place would be the diffusivities and local concentrations. 

Observed monomer concentrations are presented by Figure 2 as a function of cure time and temperature (see Equation 20). At high monomer conversions, the data appear to approach an asymptote. As the extent of network development within the resin advances, the rate of reaction diminishes. Molecular diffusion of macromolecules, initially, and of monomeric molecules, ultimately, becomes severely restricted, resulting in diffusion-controlled reactions (20). The material ultimately becomes a glass. Monomer concentration dynamics are no longer exponential decays. The rate constants become time dependent. For the cure at 60°C, monomer concentration can be described by an exponential function. 

These excesses diminished for those networks with diluent present during network formation or with low extents of the network formation reaction. 

To compare the predictions of the various molecular theories of rubber elasticity, three sets of high functionality networks were prepared and tested In this Investigation. The first set of networks tested were formed In bulk and attained a high extent of the endllnklng reaction, i.e., eX).9 where e Is the extent of reaction of the terminal vinyl groups. The second set of networks studied were formed In the presence of diluent and also achieved a high extent of reaction (e>0.9). The final group of experiments were performed on networks formed In bulk at low extents of reaction (0.4 [Pg.333]

Figure 9. Molar mass between elastically effective junction points (Mc) relative to that for the perfect network (Mc°) versus extent of intramolecular reaction at gelatin (pr,c) for polyurethane networks (29).

The factors which influence pre-gel intramolecular reaction in random polymerisations are shown to influence strongly the moduli of the networks formed at complete reaction. For the polyurethane and polyester networks studied, the moduli are always lower than those expected for no pre-gel intramolecular reaction, indicating the importance of such reaction in determining the number of elastically ineffective loops in the networks. In the limit of the ideal gel point, perfect networks are predicted to be formed. Perfect networks are not realised with bulk reaction systems. At a given extent of pre-gel intramolecular... 

To obtain accurate values of the sol, thin specimens (1 mm) in one study (13) were kept in the solvent for six weeks in another study (14), thin specimens were extracted for more than 18 days in Soxhlet extractors. When the present experimental data were obtained (6), there was little interest in knowing the sol fraction accurately. However, as discussed subsequently, to compute the extent of the curing reactions and the concentration of elastically active network chains, the sol fraction must be known accurately. 

According to the equations derived by Miller and Macosko (24), outlined in the Appendix, the probability that a short segment selected at random in the network is part of an elastically active chain is (P /p)2 where p is the final extent of the curing reaction. If an entanglement results from pairwise interactions between chains, as proposed (13), then Te in eq 2 can be equated to (Pxi/p)4, which is the probability that two interacting chains are active. 

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