Propagation, chain-length

Determine the rate of propagation, chain length and Ext for the reaction. 

Schilow recognized that induced reactions fall into two classes. The first class now is called an induced chain reaction, which can be described in terms of an initiation step, a propagation sequence, and a termination step. The other class is the coupled reaction, which can be distinguished from an induced chain reaction by the behavior of the induction factor defined by the ratio equivalents of induced reaction/ equivalents of primary reaction. In an induced chain reaction the induction factor increases without limit as the propagation chain length is increased. In a coupled reaction the induction factor approaches some definite small value such as 1, 2, or 1/2 as the induction reaction is favored. 

Keywords chain-length dependent propagation chain-length dependent termination ... 

It has been shown experimentally that the propagating chain length has no effect on the reaction rate. Each propagation step proceeds at the same rate for a homopolymeiisation. 

The kinetic chain length has a slightly different definition in the presence of chain transfer. Instead of being simply the ratio Rp/R, it is redefined to be the rate of propagation relative to the rates of all other steps that compete with propagation specifically, termination and transfer (subscript tr) ... 

We shall consider these points below. The mechanism for cationic polymerization continues to include initiation, propagation, transfer, and termination steps, and the rate of polymerization and the kinetic chain length are the principal quantities of interest. 

In writing Eqs. (7.1)-(7.4) we make the customary assumption that the kinetic constants are independent of the size of the radical and we indicate the concentration of all radicals, whatever their chain length, ending with the Mj repeat unit by the notation [Mj ], This formalism therefore assumes that only the nature of the radical chain end influences the rate constant for propagation. We refer to this as the terminal control mechanism. If we wished to consider the effect of the next-to-last repeat unit in the radical, each of these reactions and the associated rate laws would be replaced by two alternatives. Thus reaction (7. A) becomes... 

An important descriptor of a chain reaction is the kinetic chain length, ie, the number of cycles of the propagation steps (eqs. 2 and 3) for each new radical introduced into the system. The chain length for a hydroperoxide reaction is given by equation (10) where HPE = efficiency to hydroperoxide, %, and 2/ = number of effective radicals generated per mol of hydroperoxide decomposed. For 100% radical generation efficiency, / = 1. For 90% efficiency to hydroperoxide, the minimum chain length (/ = 1) is 14. 

If the initiation reaction is much faster than the propagation reaction, then all chains start to grow at the same time. Because there is no inherent termination step, the statistical distribution of chain lengths is very narrow. The average molecular weight is calculated from the mole ratio of monomer-to-initiator sites. Chain termination is usually accompHshed by adding proton donors, eg, water or alcohols, or electrophiles such as carbon dioxide. 

Copolymers with butadiene, ie, those containing at least 60 wt % butadiene, are an important family of mbbers. In addition to synthetic mbber, these compositions have extensive uses as paper coatings, water-based paints, and carpet backing. Because of unfavorable reaction kinetics in a mass system, these copolymers are made in an emulsion polymerization system, which favors chain propagation but not termination (199). The result is economically acceptable rates with desirable chain lengths. Usually such processes are mn batchwise in order to achieve satisfactory particle size distribution. 

Termination. The conversion of peroxy and alkyl radicals to nonradical species terminates the propagation reactions, thus decreasing the kinetic chain length. Termination reactions (eqs. 7 and 8) are significant when the oxygen concentration is very low, as in polymers with thick cross-sections where the oxidation rate is controlled by the diffusion of oxygen, or in a closed extmder. The combination of alkyl radicals (eq. 7) leads to cross-linking, which causes an undesirable increase in melt viscosity. 

Radical Scavengers Hydrogen-donating antioxidants (AH), such as hindered phenols and secondary aromatic amines, inhibit oxidation by competing with the organic substrate (RH) for peroxy radicals. This shortens the kinetic chain length of the propagation reactions. 

The step in which the reactive intermediate, in this case A-, is generated is called the initiation step. In the next four equations in the example above, a sequence of two reactions is repeated this is the propagation phase. Chain reactions are characterized by a chain length, which is the number of propagation steps that take place per initiation step. Finally, there are termination steps, which include any reactions that destroy one of the reactive intermediates necessary for the propagation of the chain. Clearly, the greater the frequency of termination steps, the lower the chain length will be. 

In the literature on radical polymerization, the rate constant for propagation, ( is often taken to have a single value (i.e. kp( I) - kv(2) - kvQ) - kp(n) - refer Scheme 4.45). However, there is now good evidence that the value of k is dependent on chain length, at least for the first few propagation steps (Section 4.5.1), and on the reaction conditions (Section 8.3). 

Chain Length Dependence of Propagation Rate Constants... 

It is usually assumed that propagation rate constants in homopolymerization ( p) arc independent of chain length and, for longer chains (length >20), there is experimental evidence to support this assumption.356 6 However, there is now a body of indirect evidence to suggest that the rate constants for the first few propagation steps p(l), kp(2), etc. can be substantially different from (overall) (refer Scheme 4.45). The effect can be seen as a special ease of a penultimate unit effect (Section 7.3,1.2). Evidence comes from a number of sources, for example ... 

Before any chemistry can take place the radical centers of the propagating species must conic into appropriate proximity and it is now generally accepted that the self-reaction of propagating radicals- is a diffusion-controlled process. For this reason there is no single rate constant for termination in radical polymerization. The average rate constant usually quoted is a composite term that depends on the nature of the medium and the chain lengths of the two propagating species. Diffusion mechanisms and other factors that affect the absolute rate constants for termination are discussed in Section 5.2.1.4. 

A given chain will undergo i-1 propagation steps (each with probability 0) before terminating (with probability 1-0). Thus, if termination is wholly by chain transfer or disproportionation, the chain length distribution is given by cq. 17 (Figure 5.1) ... 

In conventional radical polymerization, the chain length distribution of propagating species is broad and new short chains are formed continually by initiation. As has been stated above, the population balance means that, termination, most frequently, involves the reaction of a shorter, more mobile, chain with a longer, less mobile, chain. In living radical polymerizations, the chain lengths of most propagating species are similar (i.e. i j) and increase with conversion. Ideally, in ATRP and NMP no new chains are fonned. In practice,... 

The parameters indexed with a are connected with the nucleation step or other effects occurring only once per triple helix. Parameters denoted by s are related with the equilibrium constants of the propagation steps and are ordered to be independent of the position of the reacting chain segment. This implies that end effects are neglected. Since the same dependences are valid for AH° and AS, with the help of their chain length dependence we can determine AG by extrapolation up to 3 n - 2 = 0, and thus, a can be estimated it depends neither on temperature nor on the chain length. 

As already mentioned, the enthalpy change A//° involved in an elementary propagation step corresponds to the equilibrium constant S. The parameter a, however, is purely entropically influenced mainly due to the steric restrictions during the formation of a helical nucleus. The determination of a, since it is related to the same power (3n - 2) of s, requires the consideration of the dependence of the thermodynamic parameters on the chain length (Eq. (9 a)). 

The solvent influence, calculated with the Huron-Claverie method, reverses the qualitative graduation of the reaction enthalpies of the propagation steps as the chain length increases, in comparison to the gas phase. The same results were obtained using the same model system by Basilevski et al.125), while using a fundamentally different model for the solvent influence. 

Largest polymer chain length in polymer distribution Smallest polymer chain length in seed distribution and reactor effluent Initiation rate constant Propagation rate constant... 

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