Network properties

As it is seen from Eq. (23), the thermodynamic opportunity of the reaction initiation (AG = 0) is defined by network properties (Ccon, fix, Tiim), as well as by the conditions of production, storage, and exploitation (/iph, a), and by external influence (T, A). As mentioned previously, the Ccon value is the function of chemical structure of the network and the solvent (in a number of cases the solvent amount disposed in the network may depend on ph and intermolecular bonds distribution). 

Network properties and microscopic structures of various epoxy resins cross-linked by phenolic novolacs were investigated by Suzuki et al.97 Positron annihilation spectroscopy (PAS) was utilized to characterize intermolecular spacing of networks and the results were compared to bulk polymer properties. The lifetimes (t3) and intensities (/3) of the active species (positronium ions) correspond to volume and number of holes which constitute the free volume in the network. Networks cured with flexible epoxies had more holes throughout the temperature range, and the space increased with temperature increases. Glass transition temperatures and thermal expansion coefficients (a) were calculated from plots of t3 versus temperature. The Tgs and thermal expansion coefficients obtained from PAS were lower titan those obtained from thermomechanical analysis. These differences were attributed to micro-Brownian motions determined by PAS versus macroscopic polymer properties determined by thermomechanical analysis. 

The post-gel Miller-Macosko derivation determines network properties by first calculating the probability that looking out from a A group is a finite chain, P(F ° ). This probability is equal to the probability that A has not reacted (1-a) plus the probability that A has reacted times the probability that looking in to a B group is finite ... 

The use of this analytical expression greatly simplifies the calculation of network properties for random copolymers. 

The paper first considers the factors affecting intramolecular reaction, the importance of intramolecular reaction in non-linear random polymerisations, and the effects of intramolecular reaction on the gel point. The correlation of gel points through approximate theories of gelation is discussed, and reference is made to the determination of effective functionalities from gel-point data. Results are then presented showing that a close correlation exists between the amount of pre-gel intramolecular reaction that has occurred and the shear modulus of the network formed at complete reaction. Similarly, the Tg of a network is shown to be related to amount of pre-gel intramolecular reaction. In addition, materials formed from bulk reaction systems are compared to illustrate the inherent influences of molar masses, functionalities and chain structures of reactants on network properties. Finally, the non-Gaussian behaviour of networks in compression is discussed. 

The properties of a polymer network depend not only on the molar masses, functionalities, chain structures, and proportions of reactants used to prepare the network but also on the conditions (concentration and temperature) of preparation. In the Gaussian sense, the perfect network can never be obtained in practice, but, through random or condensation polymerisations(T) of polyfunctional monomers and prepolymers, networks with imperfections which are to some extent quantifiable can be prepared, and the importance of such imperfections on network properties can be ascertained. In this context, the use of well-characterised random polymerisations for network preparation may be contrasted with the more traditional method of cross-linking polymer chains. With the latter, uncertainties can exist with regard to the... 

The experimental determination of RBA, however, is difficult but some attempts have been made and these include direct observation, measurements of electrical conductivity, shrinkage energy, gas adsorption and light scattering. The linear elastic response of paper has been explained in terms of various micromechanical models which take into account both fibre and network properties, including RBA. An example of one which predicts the sheet modulus, Es is given below ... 

Equation (99) implies that it is often possible to specify intervals or approximate values for the scaled elasticities in terms of relative saturation, even when detailed kinetic information is not available. For example, as a rule of thumb, the substrate concentration can often be considered to be on the order of the Km value. As the scaled elasticities, by means of the control coefficients, can be directly translated into a systemic response, it is possible to utilize such heuristic arguments to acquire an initial approximation of global network properties. 

Finally, and more profoundly, not all properties require explicit knowledge of the functional form of the rate equations. In particular, many network properties, such as control coefficients or the Jacobian matrix, only depend on the elasticities. As all rate equations discussed above yield, by definition, the assigned elasticities, a discussion which functional form is a better approximation is not necessary. In Section VIII we propose to use (variants of) the elasticities as bona fide parameters, without going the loop way via explicit auxiliary functions. 

R. Steuer and G. Zamora Lopez, Global network properties. Analysis of Biological Networks, Wiley Series on Bioinformatics Computational Techniques and Engineering. B. H. Junker and F. Schreiber, eds., John Wiley Sons, Inc. 2008. 

Incorporation into the network structure of specific groupings that affect network properties. 

Dendrimers represent a model for compact multifunctional precursor of polymer networks. Polymer networks prepared by crosslinking of dendrimers were suggested several years ago [64]. Since then, some experimental work has been performed, but there are still many points in structural interpretation of network formation and network properties that are not well understood. 

Several applications of hyperbranched polymers as precursors for synthesis of crosslinked materials have been reported [91-97] but systematic studies of crosslinking kinetics, gelation, network formation and network properties are still missing. These studies include application of hyperbranched aliphatic polyesters as hydroxy group containing precursors in alkyd resins by which the hardness of alkyd films was improved [94], Several studies involved the modification of hyperbranched polyesters to introduce polymerizable unsaturated C=C double bonds (maleate or acrylic groups). A crosslinked network was formed by free-radical homopolymerization or copolymerization. 

The statistical approach to chemical kinetics was developed by Li et al. (2001, 2002), and high-dimensional model representations (HDMR) were proposed as efficient tools to provide a fully global statistical analysis of a model. The work of Feng et al. (2004) was focused on how the network properties are affected by random rate constant changes. The rate constants were transformed to a logarithmic scale to ensure an even distribution over the large space. 

It is the opinion of the present authors that frequently too much emphasis is placed in network studies on the influence of network defects (Chapter II, Section 2), while in reality pre-existing order, inhomogeneous crosslinking, composite network formation, or microsyneresis may play an important role in the mechanical behaviour, as well as in most other network properties. Examples of this will be given in Chapter IV. 

Because the duration for one measurement is very short (e.g., with a 1-Hz input, a cycle is completed in 1 sec), a dynamic test is suitable for gaining information in a short time frame or for monitoring time-dependent changes in gel network properties. When monitoring the gelation process at a fixed frequency, it usually takes a few hours for G to become approximately constant. The constancy can be judged by a constant value of G at a fixed frequency during a subsequent frequency or strain sweep test, which usually takes several minutes. 

As in linear polymers, the relative influence of the molecular structure (scale of nanometers and monomers), and the macromolecular structure (crosslink density), on network properties, depends on temperature, as shown in Fig. 10.9. In the glassy state, the physical behavior is essentially controlled by cohesion and local molecular mobility, both properties being mainly under the dependence of the molecular scale structure. As expected, there are only second-order differences between linear and network polymers. Here, most of the results of polymer physics, established on linear polymers, can be used to predict the properties of thermosets. Open questions in this domain concern the local mobility (location and amplitude of the (3 transition). 

In this example only one half of the chain parts are effectively contributing to the network properties the remaining part, the loose ends, are not being deformed when the material is strained. 

Networks comprised of linear, classically synthesized polymers that are cross-linked have been reviewed1 -41 and will not be treated in this Chapter. Further, mathematical treatments of network properties will not be discussed herein. 

Modolo J., Garenne A., Henry J., Beuter A. Probabilistic model of the subthalamic nucleus with small-world networks properties (XXVIIIth Symposium of Computational Neuroscience, Montreal (Canada), 2006). 

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