Chain scission rate equation

Here a(r) = random scission rate to end-chain scission rate. Now, for a distribution with initially large degree of polymerization under isothermal conditions, it may be confirmed by direct substitution into Equation 18.33 that the solutions of these equations for / > I arc... 

For semi-crystalline polymers, the governing equation for chain scission rate and short chain diffusion are... 

Although primary and secondary alkyl hydroperoxides are attacked by free radicals, as in equations 8 and 9, such reactions are not chain scission reactions since the alkylperoxy radicals terminate by disproportionation without forming the new radicals needed to continue the chain (53). Overall decomposition rates are faster than the tme first-order rates if radical-induced decompositions are not suppressed. 

In flow-induced degradation, K is strongly dependent on the chain length and on the fluid strain-rate (e). According to the rate theory of molecular fracture (Eqs. 70 and 73), the scission rate constant K can be described by the following equation [155]... 

By using the kinetic equations developed in Sect. 5.2, the degradation yield as a function of strain rate and temperature can be calculated. The results, with different values of the temperature and preexponential factor, are reported in Fig. 51 where it can be seen that increasing the reaction temperature from 280 K to 413 K merely shifts the critical strain rate for chain scission by <6%. 

Equation (7.20) is general, although the expression for Te itself depends on the primary distribution, the gel fraction, and the relative rates of random cross-linking and chain scission. 

The condensed phase mechanism was explained taking into account the decrease of the pyrolysis rate of polypropylene BiCl3 could catalyze the condensation between chloroparaffin and polypropylene by addition to chain end double bonds (Equation 4.25) formed either in reaction (Equation 4.22) or in chain scission occurring during volatilization of polypropylene 31... 

If random scission occurs together with end-chain scission (but at a different rate) then defining x by dx/dt = k(i the population balance equations (PBEs) for these combined processes are... 

The factor a is a measure of the accessibility of the bonds (13), and k is the specific rate constant for the rate of breaking of normal IC-O-4 C bonds in the anhydroglucose chain (k in Equations 1-3 equals ak). The expression 100[1 /(DPn)t — 1/(DP )0] gives the percentage of the initial number of bonds that have been broken. Equation 4 is also the equation that would apply if an equal number of bonds were broken in equal periods of time, that is, at a constant rate of chain scissioning (zero-order kinetics). Conformance to Equation 4, therefore, cannot be taken as proof of first-order kinetic behavior (12). 

These results provide additional confirmation for the mechanism of pyrolysis of simple polyolefins. The absence of monomer in the volatile products, the maxima in the rate curves, and the sharp decrease in the intrinsic viscosity for linear polymethylene (29) and polypropylene (2, 6, 13, 30) all point to an essentially random scission, due to pronounced intermolecular chain transfer, Equation 2. However, deviations appear when a, the fraction of bonds broken, or, what amounts to the same, the number average DP is examined as a function of time. For small a, the former relation should be one of simple proportionality and hnearity in 1/P. Instead, for both polypropylene (6) and polymethylene [see Figure 5, in (29)] curvature appears, indicating a reduction of the scission rate after an initial period of rapid degradation. For polypropylene this has been interpreted as a breaking of weak and normal bonds. Between 250° and 280° C., one weak link per 2.4 X 10 is found (6). At 295° C., the existence of more than two types of bonds would have to be postulated. 

This method was further simplified by Ozawa [13] and applied to the random degradation of polymers where the proportion of sample remaining is defined in terms of the fractional number of broken bonds. Random degradation is observed in certain polymers where main-chain scission occurs at random points with equal probability. This method assumes that the degree of conversion is constant at the DTG peak temperature, for all heating rates. For a given fractional mass the left-hand side of equation 5.20 is constant and therefore... 

Let us consider a chain scission process occurring at a constant rate r = dSIdt. Assuming that PI Pig == 2, Equation [12.6] can be rewritten as ... 

According to the authors, many literature data agree with this expression. As the rate equation is second order with a limit Ainu not first order in molecular weight, they consider that chain scission is related to the chain length of adjacent polymer molecules. The same equation was applied by Fujii [30] to mastication of poly(methyl methacrylate). For poly(vinyl chloride) the exponent was 1 instead of 2, indicating a first-order reaction. 

The rate constant for random chain scission (fc ) is given by equation [1292] ... 

The Equations 4.1-4.6 show that oxygen promotes recombination of the macro radicals and, consequently, the rate of chain scission increases in its presence. This would enable structurisation of the involatile residue from PTFE. However, Equations 4.4-4.6 show that the recombination of the secondary peroxide radicals results in the formation of radicals which readily isomerise with cleavage of the macromolecules. For this reason, restructuring of the final residue of PTFE is observed only at the late stages of its thermal oxidation. 

This section provides a rate equation for polymer chain scission in which the effect of carboxylic end groups of the long and short chains is separated using a set of partitioning parameters. 

It is often claimed that biodegradable polymers degrade through random scissions of the polymer chains. However the ester bonds next to chain ends may hydrolyse faster than those inside the chains, which is known as end scission. To account for the effect of end scission, GleadaU et al. (2014) used separate equations for random scission rate, / dt, and end scission rate, / dt, respectively, such that... 

The rate Equation [3.5] for chain scission, as presented in Chapter 3, can be applied to the amorphous phase of a semi-crystalline polymer. Using superscript amp to represent the amorphous phase, the rate equation can be rewritten as ... 

If diffusion of short chains is prohibited, the rate equation for chain scission of sani-... 

Rate equation for chain scission in presence of buffering reactions... 

The rate equation for polymer chain scission was presented in Chapter 3 for amorphous polymers and Chapter 4 for semi-crystaUine polymers. These equations are summarised in this section. The terms in these equations that are affected by the diffusion of short chains are highlighted. 

For semi-crystalline polymers, following Section 4.2 the rate Equation [4.6] for chain scission can be rewritten as... 

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