Fast Propagation

As the size of the crack increases due to its propagation, Kj may reach the fracture toughness of the material (i. e. a threshold value Kjc) and fast propagation takes place (the propagation rate is of the same order as the sound rate through the material), which leads to brittle fracture. From Eq. (1) the critical size of the crack can be calculated as  

For a material of given fracture toughness (Kjc) nd for a given nominal stress (o ), a critical crack size can be calculated above which failure occurs. For instance. 

The critical conditions with regard to the stress are shown by two horizontal lines a = and a = a, those with regard to fCj can be obtained from Eq. (1) by replacing Kj with Kjscc and fCjc, respectively 

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Third generation initiators are based on the NHC system of second generation initiators, but do not contain any phosphine ligand. Instead, one or two pyridine ligands are weakly bound to the ruthenium centre (c/. Fig. 3.28, complexes 73 and 74c). Pyridine dissociates very easily and hardly competes with the olefin for the coordination site. As a result, complete initiation and fast propagation are enabled, therefore living polymerisation is rendered possible. 

The polymerization of styrene by Zr (benzyl) 4 has the characteristics of producing high molecular weight polymers by a very slow polymerization process. This can be reasonably explained by a slow initiation process followed by a fast propagation reaction. These two processes are probably chemically similar but differ significantly in rate for structural reasons. 

The central question at issue is still whether pseudocationic polymerisation, propagated by a large concentration of activated ester, is a real phenomenon sui generis or whether all these reactions can be explained in terms of very fast propagation by a very small concentration of conventional carbonium ions. 

Abstract. Auto-accelerated polymerization is known to occur in viscous reaction media ("gel-effect") and also when the polymer precipitates as it forms. It is generally assumed that the cause of auto-acceleration is the arising of non-steady-state kinetics created by a diffusion controlled termination step. Recent work has shown that the polymerization of acrylic acid in bulk and in solution proceeds under steady or auto-accelered conditions irrespective of the precipitation of the polymer. On the other hand, a close correlation is established between auto-acceleration and the type of H-bonded molecular association involving acrylic acid in the system. On the basis of numerous data it is concluded that auto-acceleration is determined by the formation of an oriented monomer-polymer association complex which favors an ultra-fast propagation process. Similar conclusions are derived for the polymerization of methacrylic acid and acrylonitrile based on studies of polymerization kinetics in bulk and in solution and on evidence of molecular associations. In the case of acrylonitrile a dipole-dipole complex involving the nitrile groups is assumed to be responsible for the observed auto-acceleration. 

For most collagens, the folding of the triple helical domain proceeds from the carboxyl end toward the amino end of the trimeric molecule in a zipper-like fashion with a rate that is limited by cis—trans isomerization of peptidyl prolyl bonds." The fast propagation of the triple helix formation is followed by a slower folding... 

The combustion index is applicable to solids and gives a qualitative indication about combustibility, ranging from one to six. Index 1 corresponds to no combustion and Index 6 to a violent combustion with fast propagation. From Index 4, the combustion propagates through to the solid. 

In the former case [Eq. (20)], the rate does depend on [M], Thus, kinetics alone may erroneously indicate that the covalent species D reacts directly with M (monomer), although it first ionizes to C. The latter case [Eq. (21)] results in zero-order kinetics in monomer. However, this may not happen in the polymerization process. If k-, < A 2[M], then once the ions C are generated, they can react many times with monomer before deactivation. Thus, the kinetics may be first order with respect to monomer and resemble a system with slow initiation and fast propagation. 

Avvakumov E.G., Kosova N.V. Fast propagating sohd-state mechanochemical reactions. Sib. Khim. Zhurn. 1991 5 62-6. 

Viting L.M., Avvakumov E.G. About correlation between Gibbs energies and average orbital electronegativities of complex oxides. Zhum. Fis. Khim. 2001 75 1120-22. Avvakumov E.G., Kosova N.V. Fast propagating solid-state mechanochemical reactions. Sibir. Khim. Zhum. 1991 5 62-. ... 

The high reactivity of ionic active centers which yields fast propagation rates also results in a greater propensity toward side reactions and interference from trace impurities. Low polymerization temperatures favor propagation over competing reactions, and ionic polymerizations are often performed at colder temperatures than those used in free-radical processes, which would be impossibly slow under the same conditions. 

The reaction medium in cationic polymerizations is usually a moderately polar chlorinated hydrocarbon like CHjCI (dielectric constant = 12.6 at —2(TC). A greater proportion of the macroions are free of their counterions in cationic than in anionic polymerizations in the usual solvents for the latter processes. Cationic polymerizations are characterized by extremely fast propagation rates. 

The fast initiation step is followed by an equally fast propagation reaction. While the rate constant of the former has been measured by Kunitake and Takarabe the dimerisation kinetics have not been measured for this particular system. The following slow reactions consist of the proton transfer between the dimeric cation and the monomer to gjve alternatively the unsaturated dimer or the indanylic one. Finally, the protonated dimer can isomerise to a more stable configuration due to the direct interaction of the two phenyl groups through space polarisation effects Thus ... 

In 1969 Penczek and Kubisa [59] reported an exhaustive study of the kinetics and mechanism of BCMO polymerization initiated by the (i-C4H9 )j AI/H2O system and carried out in chlorobenzene solution at 55—95°C. This system produced homogeneous conditions for the polymerization and the whole process could be described as a non-stationary reaction with slow initiation, fast propagation, and slow degradative chain transfer to polymer. 

Due to its high speed, the cationic initiation reaction contributes to the over-all lactam consumption, in contrast to the anionic polymerization in which the disproportionation is usually very slow as compared with the fast propagation reactions involving acyllactam growth centres. Neutral and protonated polymer amide groups can also take part in the dispro-... 

Figure 12. Average degrees of polymerization An and polydispersity indices as functions of conversion for a fast propagating monomer in the absence and in the presence of an initial reduced excess concentration r/o of the persistent species. Parameters as used for Figure 11, and kp = 20000 M 1 s 1, [M]o = 10 M. Circles are added according to the analytical equations.

The initiation rate was low in this system, kinetic curves showed marked acceleration periods. Analysis of the initial periods of kinetic curves, considering the slow initiation-fast propagation process 19), yielded of an apparent rate constant of initiation kj = 1.6 1(T3 mol-1 1 s 1 (C6H5C1, 70 °C) 18> (k = 8.5 mol"1 I s-1 for the same conditions cf. next paragraph). 

The radical chain mechanism outlined here avoids the ineffective direct reaction of molecular oxygen with the substrate hydrocarbon. The fast propagation reactions produce ROOH that in turn can initiate new radical chains. As the primary product of the reaction initiates new reactions, one ends up with an autocatalytic acceleration. The propagating peroxyl radicals can also mutually terminate and yield one molecule of alcohol and ketone in a one-to-one stoichiometry. The ratio between the rate of propagation and the rate of termination is referred to as the chain length and is of the order of 50-1000. As the desired chain products are more susceptible to oxidation, autoxidations are normally carried out at low conversions in order to keep the selectivity to an economically acceptable level. 

According to traditional textbooks, the hydroperoxide is produced in the fast propagation reaction, whereas alcohol and ketone are formed in the slow termination step. As such, one would expect the hydroperoxide selectivity to be an order of magnitude higher than the alcohol and ketone selectivity. This is, however, not in agreement with the experimental observations, indicating that (an) important source(s) of alcohol and ketone is (are) missing in this model. Moreover, the... 

Digestion procedures do have some common characteristics, all of which are certainly far from ideal for a fast propagation into analytical laboratories. They are time consuming,... 

Note that Zvmax>0, if Kuexp(l), we have 2vmax<0, tvmax<0, and the velocity has no maximum. In that case, v decreases with the distance, which corresponds to the onset of the fast propagation phase at f = 0. 

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