Rate of Polymerization Expression

It should be noted that Eq. (6.26) represents the instantaneous rate of polymerization corresponding to [M] and [I] values at any instant. Since these values change with conversion, Eq. (6.26) must be integrated over a period of time to determine the overall extent of polymerization. 

If the initiator decomposes in a unimolecular reaction [cf. Eq. (6.3)], the corresponding rate expression is first order in initiator  

The half-life is thus independent of the initial concentration. It is a convenient criterion for initiator activities. Table 6.1 lists the half-lives for several common initiators [3,4] at various temperatures.] 

This gives the amount of polymer produced (in terms of the moles of monomer converted) in time t at a given temperature. It is also useful for 

Problem 6.3 The decomposition of benzoyl peroxide is characterized by a half-life of 7.3 h at 70°C and an activation energy of 29.7 kcal/mol. What concentration (mol/L) of this peroxide is needed to convert 50% of the original charge of a vinyl monomer to polymer in 6 hours at 60°C (Data / = 0.4 fcp/fe = 1.04x10 L/mbl-s at 60°C.) 

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In order to gain some insight into actual rather than relative flow rates, it is necessary to know the polymerization rate. Let r represent the fractional rate of polymerization expressed as the fraction of the reactor capacity converted to polymer per hour. 

The core of the derivation for the rate of polymerization expression is the assumption that the rate of initiation equals the rate of termination, equation 46 in its simplified form. This assumption is mandatory for the establishment of a constant free radical concentration (eq. 54). 

Acrylamide polymerization by radiation proceeds via free radical addition mechanism [37,38,40,45,50]. This involves three major processes, namely, initiation, propagation, and termination. Apart from the many subprocesses involved in each step at the stationary state the rates of formation and destruction of radicals are equal. The overall rate of polymerization (/ p) is so expressed by Chapiro [51] as ... 

The synthetic methods of macromolecules having an active pendant group include (1) the transformation reactions of polymer and copolymers, and (2) polymerization and copolymerization of functional monomers having active pendant groups. The macromolecules, either in the shape of film or microbeads, can be used as the substrate. As we have mentioned previously, the rate of polymerization initiated with the Ce(IV) ion redox system is much faster than that initiated by Ce(l V) ion alone, as expressed in / r 1. Therefore, the graft... 

Reaction rate functions expressing rate of polymerization R generally depend upon the molar concentrations of monomer and Initiator, and temperature. 

Derive an expression for the rate of polymerization of stoichiometric amounts of adipic acid and hexamethylene diamine. Indicate the assumptions inherent in the derivation. Derive an expression for the rate of polymerization of nonstoichiometric amounts of the two reactants. 

The rate of propagation, and therefore the rate of polymerization, is the sum of many individual propagation steps. Since the rate constants for all the propagation steps are the same, one can express the polymerization rate by... 

The rate expression Eq. 3-32 requires a first-order dependence of the polymerization rate on the monomer concentration and is observed for many polymerizations [Kamachi et al., 1978], Figure 3-2 shows the first-order relationship for the polymerization of methyl methacrylate [Sugimura and Minoura, 1966], However, there are many polymerizations where Rp shows a higher than first-order dependence on [M], Thus the rate of polymerization depends on the -power of the monomer concentration in the polymerization of styrene in chlorobenzene solution at 120°C initiated by t-butyl peresters [Misra and Mathiu, 1967]. The benzoyl peroxide initiated polymerization of styrene in toluene at 80°C shows an increasing order of dependence of Rp on [M] as [M] decreases [Horikx and Hermans, 1953], The dependence is 1.18-order at [M] = 1.8 and increases to 1.36-order at [M] = 0.4. These effects may be caused by a dependence of the initiation rate on the monomer concentration. Equation 3-28 was derived on the assumption that Rt is independent of [M], The initiation rate can be monomer-dependent in several ways. The initiator efficiency / may vary directly with the monomer concentration... 

The rate of polymerization in nonterminating systems can be expressed as the rate of propagation... 

Tibitt et al. were also able to obtain an analytical expression for the rate of polymerization by assuming that hydrogen atoms are equivalent to free radicals. The final form of rp is given by... 

In classical free radical polymerizations, the rate of polymerization, Rp, is often expressed as [92]... 

The above expression indicates that the rate of polymerization is proportional to the square root of both the incident light intensity and the photoinitiator concentration. This expression is accurate to within 2.5% for thin films where e A]b < 0.2. As the thickness of the sample is increased, the light intensity can decrease appreciably across the sample, and Eq. (4) must be used. 

The rate of polymerization (/ p) in emulsion polymerization is expressed using n and N as... 

Thus the growing anionic chain can assume at least two identities the free anion and the anion-cation ion pair (several types of solvated ion-pairs can also be considered). Furthermore, the kinetics of these propagation reactions, which generally show a fractional dependency on chain-end concentration ranging from one-half to unity, can best be explained by assuming that the monomer can react with both the free anion and the ion-pair (4, 5, 60, but at different rates. Thus, for example, in the polymerization of styrene by organosodium, the rate of polymerization (Rp) can be expressed as... 

For an accurate evaluation of the efficiency of UV photons in initiating the polymerization of acrylate photoresists, it is necessary to determine the polymerization quantum yield, , that corresponds to the number of acrylate functions which have polymerized per photon absorbed. can be expressed as the ratio of the rate of polymerization to the absorbed light-intensity and calculated from the following expression, after introducing Equations 1 and 2 ... 

Based on the assumption that termination of the growing polymer chains occurs by both mono- and bimolecular pathways, the overall rate of polymerization can then be expressed, at a first approximate, as the sum of two terms which depend on the first power and on the square root of the absorbed light-intensity, respectively ... 

A conventional description of the polymerization process is given in Fig. 2, in which I represent the initiator, M is the monomer, and R is a free radical. By application of the steady state assumption to the concentrations of R- and the growing chain radical RMn-, and assuming that the only important loss of monomer is through the propagation step, one can derive an expression for the overall rate of polymerization, (eq. 1). 

The initial rate of polymerization of the inverse emulsion polymerization using sodium di-2-ethyl-hexyl sulfosuccinate (AOT) and sorbitan monooleat (SMO) as emulsifiers and an oil soluble azo initiator can be expressed by [13] ... 

As an example, for styrene at 0°C. the following data are typical (30) rate of polymerization = 14.5%/hour, dose rate = 2.2 X 103 rads/hour, average DP 103. Since 1 rad corresponds to the absorption of 100 ergs/gram of material, it can be demonstrated, on the basis of the above values, that approximately 6 X 105 monomer units are incorporated into polymer for each 100 e.v. of energy absorbed. In conventional radiation chemistry terminology this would be expressed as G(.m0nomer)... 

The rate of polymerization is expressed as yield per liter per hour—i.e., as an average value since the actual rate of polymerization depends on time, as a rule decreasing gradually the time of polymerization was standardized to 1 or 2 hours in order to facilitate comparisons. 

With all the equations (hat have now been thrown at you, you re probably a bit confused about where we are and where we re going. So, first of all, we ve summarized our three kinetic equations, for initiation, propagation and termination, in Figure 4-16. Now, the first thing we want to obtain is an expression for the rate of polymerization, R. This is simply the rate of formation of polymer, which, if you think about it, is the same as the rate of disappearance of... 

Making this type of assumption may seem a bit dubious to you, but it s often the type of thing you have to do in science. In the end, the validity of any assumption you make must stand the test of experimental confirmation. So let s see how well this and also Flory s assumption, that reactivity is independent of chain length, works. The equation for the rate of polymerization is obtained by substituting the expression for the concentration of radical species into the equation for the rate of polymerization (Equations 4-29). 

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