Modeling of Network Formation

Introduction of new precursor architectures brings about new challenges in description and modeling of network formation because the apparent reactivities of functional groups become dependent on the size and shape of the precursors... 

The mathematical model of network formation in the pregel stage will focus on the prediction of the gel conversion and the evolution of number-and mass-average molar masses, Mn and Mw, respectively. For chainwise polymerizations, calculations will be restricted to the limit of a very low concentration of the polyfunctional monomer (A4 in the previous example). Thus, homogeneous systems will always be considered. 

To provide a rough estimation of the gel conversion arising from such a mechanism, let us assume a very simple model of network formation in which a first stage involves the hydrolytic condensation of the monomer to produce octahedra as shown in Fig. 3.23. In the formation of octahedra,... 

The description of a network structure is based on such parameters as chemical crosslink density and functionality, average chain length between crosslinks and length distribution of these chains, concentration of elastically active chains and structural defects like unreacted ends and elastically inactive cycles. However, many properties of a network depend not only on the above-mentioned characteristics but also on the order of the chemical crosslink connection — the network topology. So, the complete description of a network structure should include all these parameters. It is difficult to measure many of these characteristics experimentally and we must have an appropriate theory which could describe all these structural parameters on the basis of a physical model of network formation. At present, there are only two types of theoretical approaches which can describe the growth of network structures up to late post-gel stages of cure. One is based on tree-like models as developed by Dusek7 I0-26,1 The other uses computer-simulation of network structure on a lattice this model was developed by Topolkaraev, Berlin, Oshmyan 9,3l) (a review of the theoretical models may be found in Ref.7) and in this volume by Dusek). Both approaches are statistical and correlate well with experiments 6,7 9 10 13,26,31). They differ mainly mathematically. However, each of them emphasizes some different details of a network structure. 

Charmot D, GuUlot J. Kinetic modelling of network formation in styrene-butadiene emulsion copolymers a comparative study with the generahzed form of Flory s theory of gelation. Polymer 1990 31 352-360. 

Recently the polymeric network (gel) has become a very attractive research area combining at the same time fundamental and applied topics of great interest. Since the physical properties of polymeric networks strongly depend on the polymerization kinetics, an understanding of the kinetics of network formation is indispensable for designing network structure. Various models have been proposed for the kinetics of network formation since the pioneering work of Flory (1 ) and Stockmayer (2), but their predictions are, quite often unsatisfactory, especially for a free radical polymerization system. These systems are of significant conmercial interest. In order to account for the specific reaction scheme of free radical polymerization, it will be necessary to consider all of the important elementary reactions. 

There are yet many problems to be solved to build a more realistic model for network formation, and more experimental information will be necessary in order to clarify these complicated phenomena. We do however believe these kinetic models will provide greater insight into the phenomenon of network formation. 

The deformation of polymer chains in stretched and swollen networks can be investigated by SANS, A few such studies have been carried out, and some theoretical results based on Gaussian models of networks have been presented. The possible defects in network formation may invalidate an otherwise well planned experiment, and because of this uncertainty, conclusions based on current experiments must be viewed as tentative. It is also true that theoretical calculations have been restricted thus far to only a few simple models of an elastomeric network. An appropriate method of calculation for trapped entanglements has not been constructed, nor has any calculation of the SANS pattern of a network which is constrained according to the reptation models of de Gennes (24) or Doi-Edwards (25,26) appeared. 

A model of lamellae formation In stretched networks is proposed. Approximately one-half of the chains do not fold. Formation of such lamellae Is accompanied by declining stress. Highly folded systems (high crystallinity), however, can cause a stress Increase. In the calculations crosslinks are assigned to their most probable positions through the use of a characteristic vector. A contingent of amorphous chains Is also Included. The calculations suggest that the concept of fibrillar-lamellar transformations may be unnecessary to explain observed stress-temperature profiles In some cases. 

Throughout the chapter, the importance of network formation theories in understanding and predicting structural development is stressed. Therefore, a short expose on network formation theories is given in this chapter. Although the use of theoretical modeling of network build-up and comparison with experiments play a central role in this chapter, most mathematical relations and their derivation are avoided and only basic postulates of the theories are stated. The reader can always find references to literature sources where such mathematical relations are derived. 

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. 

Low-molecular-weight model compounds such as phenylglycidyl or other mono-glycidyl ethers as well as primary, secondary and tertiary amines have been used for the study of the kinetics, thermodynamics and mechanism of curing. To reveal the kinetic features of network formation, results of studies of the real epoxy-amine systems have also been considered. Another problem under discussion is the effect of the kinetic peculiarities of formation of the epoxy-amine polymers on their structure and properties. 

In building mathematical models of product formation in a mold it is possible to treat a polymeric material as motionless (or quasi-solid), because the viscosity grows very rapidly with the formation of a linear or network polymer thus, hydrodynamic phenomena can be neglected. In this situation, the polymerization process itself becomes the most important factor, and it is worth noting that the process occurs in nonisothermal conditions. 

A major focus of this chapter is the effect of network formation and network density on the relaxation times of the H and 13C nuclei in the NMR experiment. The changes in relaxation times can be exploited to obtain information about crosslink density and chain motion, and must be taken into account in the design of experiments to determine changes in chemical structure. In this section we examine how crosslinking changes the transverse (T2) relaxation times of nuclei, and how this information can be of use. Two different approaches have been taken in the literature, namely changes in T2 can be used to estimate crosslink density, or used to develop and verify models of chain motion. 

Many dairy emulsions destabilize by flocculation and networking. Hib-berd et al, (1997a, b) obtained the ultrasonic response to flocculation showing that floe size increased during the experiment. The experiment was repeated with a higher level of hydroxyethyl cellulose (0.1%, v/v) with the result that flocculation occurred more rapidly with the formation of a densely connected network of particulate material. The residual root-mean-square error associated with fitting the ECAH model to the ultrasonic data at various points in the flocculation reaction increased rapidly at the onset of network formation and could, in principle, be used to detect such phenomena in a process context. 

Although the major interest in experimental and theoretical studies of network formation has been devoted to elastomer networks, the epoxy resins keep apparently first place among typical thermosets. Almost exclusively, the statistical theory based on the tree-like model has been used. The problem of curing was first attacked by Japanese authors (Yamabe and Fukui, Kakurai and Noguchi, Tanaka and Kakiuchi) who used the combinatorial approach of Flory and Stockmayer. Their work has been reviewed in Chapter IV of May s and Tanaka s monograph Their experimental studies included molecular weights and gel points. However, their conclusions were somewhat invalidated by the fact that the assumed reaction schemes were too simplified or even incorrect. It is to be stressed, however, that Yamabe and Fukui were the first who took into account the initiated mechanism of polymerization of epoxy groups (polyetherification). They used, however, the statistical treatment which is incorrect as was shown in Section 3.3. 

It should be noted that the above results were obtained on endlinked systems [103, 104] yielding model networks. In this case the value of M, at the end of network formation is close to the number-average MM of the initial sample [105]. 

An identical mathematical description of the kinetics of curing of reactants different in chemical nature and that obtained on the basis of fundamentally different experimental methods allows us to assume that this apparent selfacceleration course of some rheokinetic parameters is common to the processes of formation of materials with a crosslinked structure. It should be emphasized once more that the self-acceleration" effect must not be identified with the self-catalysis of the reaction of interaction between epoxy monomers and diamines which is studied in detail on model compounds [116, 117]. For each particular curing process the self-acceleration effect is influenced by the mechanism of network formatic, namely, chemical self catalysis [118], the appearance of local inhomogeneities [120], the manifestation of gel eff t [78], parallel course of catalytic and noncatalytic reactions [68]. It is probably true that the phenomena listed above may in one form or another show up in specific processes and make their contribution into self-acceleration of a curing reaction. 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

Macosko and Miller (1976) and Scranton and Peppas (1990) also developed a recursive statistical theory of network formation whereby polymer structures evolve through the probability of bond formation between monomer units this theory includes substitution effects of adjacent monomer groups. These statistical models have been used successfully in step-growth polymerizations of amine-cured epoxies (Dusek, 1986a) and urethanes (Dusek et al, 1990). This method enables calculation of the molar mass and mechanical properties, but appears to predict heterogeneous and chain-growth polymerization poorly. 

Data on the fractal forms of macromolecules, the existence of which is predetermined by thermodynamic nonequilibrium and by the presence of deterministic order, are considered. The limitations of the concept of polymer fractal (macromolecular coil), of the Vilgis concept and of the possibility of modelling in terms of the percolation theory and diffusion-limited irreversible aggregation are discussed. It is noted that not only macromolecular coils but also the segments of macromolecules between topological fixing points (crosslinks, entanglements) are stochastic fractals this is confirmed by the model of structure formation in a network polymer. 

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