Pseudo-network model

Equation (32a) has been very successful in modelling the development of birefringence with extension ratio (or equivalently draw ratio) in a rubber, and this is of a different shape from the predictions of the pseudo-affine deformation scheme (Eq. (30a)). There are also very significant differences between the predictions of the two schemes for P400- In particular, the development of P400 with extension ratio is much slower for the network model than for the pseudo-affine scheme. 

Such considerations appear to be very relevant to the deformation of polymethylmethacrylate (PMMA) in the glassy state. At first sight, the development of P200 with draw ratio appears to follow the pseudo-affine deformation scheme rather than the rubber network model. It is, however, not possible to reconcile this conclusion with the temperature dependence of the behaviour where the development of orientation reduces in absolute magnitude with increasing temperature of deformation. It was proposed by Raha and Bowden 25) that an alternative deformation scheme, which fits the data well, is to assume that the deformation is akin to a rubber network, where the number of cross-links systematically reduces as the draw ratio is increased. It is assumed that the reduction in the number of cross-links per unit volume N i.e. molecular entanglements is proportional to the degree of deformation. 

Fig.2 compares the predictions of the pseudo-affine model and the affine model with different values of the nvunber n of links per chain. It is clearly seen that the pseudo-affine scheme gives a much more rapid initial orientation than the affine network model. 

A pseudo-affine model predicts the variation of 2 with the deformation of a semi-crystalline polymer. It assumes that the distribution of crystal c axes is the same as the distribution of network chain end-to-end vectors r, in a rubber that has undergone the same macroscopic strain. Figure 3.12 showed the affine deformation of an r vector with that of a rubber block. 

In a stretched rubber, the molecules elongate, and the r vectors move towards the tensile axis. Fience the variation of Pi with extension ratio will differ from the pseudo-affine model. For moderate strains the increase of Pi with extension ratio is linear, but at high extensions the approximation used in Eq. (3.12), that both q and q are large, breaks down. Treloar (1975) described models which consider the limited number of links in the network chains. Figure 3.33 shows that the orientation function abruptly approaches 1 as the extension ratio of the rubber exceeds v. Although the model is successful for rubbers, it fails for the amorphous phase in polypropylene (Fig. 3.32), presumably because the crystals deform and reduce the strain in the amorphous phase. 

It is also to be expected on the above model that an S—B diblock or a B—S—B triblock would not behave as a pseudo-network rubber unless it was separately vulcanized. This has also been found to be true in practice. 

Actually, we can apply this to stracture elements and ions in all the models of solutions available in thermodynamics and in particular those deduced Irom statistical thermodynamics such as the model of strictly regular solutioa The basic assumptions of these models apply without reservation to stracture elements and even, in this case, the assumption of a pseudo-network, which it is necessary to admit in the case of liquid phases, does not obviously present arty difficulty for crystallized solids. 

The general reaction occurring in hydrodesulfurization has been described in Section 2.1.1. The most studied model compound is DBT. The reactivity towards hydrogenation of the phenyl substituents already mentioned (Section 2.1.1) is also observed in the hydroprocessing of sulfur compounds. The reactivity towards hydrogenolysis of the C-S bond masks the effects associated to aromatics hydrogenation. The DBT reaction network is sketched in Fig. 8 the pseudo-first-order reaction constants measured by Houalla [68] have been included. 

However, if the atoms are not related by symmetry, the normal rules break down. The homoionic N-N bond in the hydrazinium ion is an electron pair bond, but one in which N1 contributes 1.25 and N2 0.75 electrons. How can we apply the bond valence model in such cases where no solution to the network equations is possible One approach is to isolate the non-bipartite portion of the graph into a complex pseudo-atom. Thus in the hydrazinium ion the homoionic bond and its two terminating N atoms are treated as a single pseudo-anion which forms six bonds with a valence sum equal to the formal charge of —4. 

The enolate ions are unstable intermediates, hence the pseudo steady state approximation can be applied to these intermediates, resulting in a kinetic model in which only stable components figure. It also can be proven (ref.5) that such a model will be mathematically equivalent to the one as follows from the network presented in figure 1. 

In this section we will focus on addressing the difficulties arising in the first two tasks of sequential HEN synthesis and we will discuss simultaneous approaches developed in the early 90s. More specifically, in section 8.5.1 we will discuss the simultaneous consideration of minimum number of matches and minimum investment cost network derivation. In section 8.5.2 we will discuss the pseudo-pinch concept and its associated simultaneous synthesis approach. In section 8.5.3, we will present an approach that involves no decomposition and treats HRAT as an explicit optimization variable. Finally, in section 8.5.4, we will discuss the development of alternative simultaneous optimization models for heat integration which address the same single-task HEN problem as the approach of section 8.5.3. 

Figure 15.2 2H NMR relaxation function obtained in a PDMS model network in the relaxed state, showing the two components the fast relaxing component (the curve with alternatively long and short dashes) is pseudo-solid and belongs to elastic chains (which have restricted motions), the slowly relaxing, exponential component (dashed line), is liquid-like and belongs to dangling chains. The total signal (black points) is a superposition of both contributions, which gives the fraction of dangling chains...

Dynamics of the Reaction Network. Information on the concentration changes of 4-chlorophenol and its intermediates (4-chlorocatechol, 4-chlororesorcinol, hydroquinone, and hydroxyhydroquinone) allows us to perform some dynamic analysis of the photocatalytic oxidation of 4-chlorophenol. Scheme IV shows a reaction network that can be established to describe the overall mineralization of 4-chlorophenol. According to this reaction network, 4-chlorophenol (4-CP) first decomposes to 4-chlororesorcinol (4-CRE), 4-chlorocatechol (4-CCA), or hydroquinone (HQ) 4-chlorocatechol is the major primary product. Further oxidation of the primary intermediates yields hydroxyhydroquinone (HHQ), as the secondary intermediate, which is readily mineralized to carbon dioxide. For simplicity, a pseudo-first-order expression was used to model the dynamics of the reaction network (eqs 9-14). 

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