Step growth models

In this section we examine some examples of cross-linked step-growth polymers. The systems we shall describe are thermosetting polymers of considerable industrial importance. The chemistry of these polymerization reactions is more complex than the hypothetical AB reactions of our models. We choose to describe these commercial polymers rather than model systems which might conform better to the theoretical developments of the last section both because of the importance of these materials and because the theoretical concepts provide a framework for understanding more complex systems, even if they are not quantitatively successful. 

Fig. 8.2 Strain-generated active path mechanisms, (a) Often referred to as the film rupture model and (b) the slip step dissolution model. In both cases growth is by dissolution film rupture is the rate controlling step, not the mechanism of crack growth...

Appendix 1 Toward a Model for Step-Growth Copolymerization w ith Phase Separation... 

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

In the first step, lipid model membranes have been generated (Fig. 15) on the air/liquid interface, on a glass micropipette (see Section VIII.A.1), and on an aperture that separates two cells filled with subphase (see Section VIII.A.2). Further, amphiphilic lipid molecules have been self-assembled in an aqueous medium surrounding unilamellar vesicles (see Section VIII.A.3). Subsequently, the S-layer protein of B. coagulans E38/vl, B. stearother-mophilus PV72/p2, or B. sphaericus CCM 2177 have been injected into the aqueous subphase (Fig. 15). As on solid supports, crystal growth of S-layer lattices on planar or vesicular lipid films is initiated simultaneously at many randomly distributed nucleation... 

The current research shows that the model describing this step-growth polymerization is valid at relatively low conversions. Experimental monomer concentrations and the moments of the distribution are adequately fit, yielding estimates of the model parameters. The simulation demonstrates that fitting molar concentrations of polymeric species is substantially more demanding. 

The smoothing terms have a thermodynamic basis, because they are related to surface gradients in chemical potential, and they are based on linear rate equations. The magnitude of the smoothing terms vary with different powers of a characteristic length, so that at large scales, the EW term should predominate, while at small scales, diffusion becomes important. The literature also contains non-linear models, with terms that may represent the lattice potential or account for step growth or diffusion bias, for example. 

The book is thus a series of steps, from the multiplier and its role in the reproduction schema in Chapter 2 to the Kalecki principle in Chapter 3 and a detailed consideration of the circuit of money in Chapter 4. Having built up a macro monetary model of the reproduction schema, in which both money and aggregate demand are featured, Chapter 5 derives the Domar growth model from these analytical foundations. The relevance of this growth model to Marxian theories of crisis is explored and further developed in Chapter 6. 

LaMer, Monomer Addition Growth Model. Most of the recent publications (13,18,37,43-45) concerning the Stober silica precipitation describe a first-order hydrolysis of TEOS as the rate-limiting process in the silica particle precipitation. The second reaction step, the condensation reaction, was found to be faster by at least a... 

Ultrahigh-vacuum (UHV) surface spectroscopy has been used with molecular beams of SiH4 and mass spectroscopy to elucidate the Si growth mechanism (67, 143). Joyce et al. (67) found that Si growth is preceded by an induction period when surface oxide was removed as SiO. The subsequent film growth proceeds by growth and coalescence of adjacent nuclei with no apparent formation of defects. Henderson and Helm (144) proposed a step-flow model in which adatoms from SiH4 surface reactions difluse to kink sites. 

Comparison of the measured peak shape with simulations based on Eqs. (2-5) and (2-6) reveals that a nucleation and growth model describes the reduction of Fe304 to Fe best. Thus, the formation of metallic iron nuclei at the surface of the particles is the difficult step. Once these nuclei have formed, they provide the site where molecular hydrogen dissociates to yield atomic hydrogen, which takes care of further reduction. The studies of Wimmers and co-workers [8] show nicely that TPR allows for detailed conclusions on reduction mechanisms, albeit in favorable cases only. 

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