Rate and Degree of Polymerization

The rate of living cationic polymerization is expressed as the rate of propagation of ion pairs 

Some living polymerization show a greater than first-order dependence of Rp on [M] indicating that the initiation rate is slow and dependent on monomer. 

The number-average degree of polymerization for a living cationic polymerization is defined as the concentration of monomer consumed divided by the total concentration of all propagating chains (dormant and active) 

Poisson distribution [Flory, 1940 Peebles 1971 Szwarc, 1968], the same for ATRP and other living polymerizations 

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The kinetic expressions which describe the rate and degree of polymerization in cationic polymerizations are derived in a manner analogous to that for radical polymerization. The results are similar with the main difference being that the direct and inverse dependencies of the rate and degree of polymerization, respectively, on the initiator concentration or initiation rate are both first-order, not half-order as in radical polymerization. The difference arises from cationic termination being mono-molecular in the propagating species instead of bimolecular as in radical polymerization. 

The effect of temperature on the rate and degree of polymerization is of prime importance in determining the manner of performing a polymerization. Increasing the reaction temperature usually increases the polymerization rate and decreases the polymer molecular weight. Figure 3-13 shows this effect for the thermal, self-initiated polymerization of styrene. However, the quantitative effect of temperature is complex since Rp and X depend on a combination of three rate constants—kd, kp, and kt. Each of the rate constants for initiation, propagation, and termination can be expressed by an Arrhenius-type relationship... 

Consider the bulk polymerization of neat styrene by ultraviolet irradiation. The initial polymerization rate and degree of polymerization are 1.0 x 10 3 mol L 1 s 1 and 200, respectively, at 27°C. What will be the corresponding values for polymerization at 77°C ... 

The same initial polymerization rate and degree of polymerization as in Problem 3-15 are obtained at 27°C for a particular AIBN thermal-initiated polymerization of styrene. Calculate the Rp and X values at 77°C. 

The number of polymer particles is the prime determinant of the rate and degree of polymerization since it appears as the first power in both Eqs. 4-5 and 4-7. The formation (and stabilization) of polymer particles by both micellar nucleation and homogeneous nucleation involves the adsorption of surfactant from the micelles, solution, and monomer droplets. The number of polymer particles that can be stabilized is dependent on the total surface area of surfactant present in the system asS, where as is the interfacial surface area occupied by a surfactant molecule and S is the total concentration of surfactant in the system (micelles, solution, monomer droplets). However, N is also directly dependent on the rate of radical generation. The quantitative dependence of N on asS and R,- has been derived as... 

Quantitatively compare the rate and degree of polymerization of styrene polymerized in bulk at 60°C with an emulsion polymerization (case 2 behavior h — 0.5) containing 1.0 x 1015 polymer particles per milhhter. Assume that [M] = 5.0 molar, R, = 5.0 x 1012 radicals per milliliter per second, and all rate constants are the same for both systems. For each polymerization system, indicate the various ways (if any) by which the polymerization rate can be affected without affecting the degree of polymerization. 

From a consideration of Eqs. 5-34 and 5-35, the composite activation energies Er and Ej for the rate and degree of polymerization, respectively, are obtained as... 

The number-average molecular weight of the initially formed polymer is 20,000. A 1.00-g sample of the polymer contains 3.0 x 10-5 moles of OH groups it does not contain chlorine. Show the reaction sequence of initiation, propagation, and termination steps for this polymerization and derive the appropriate expressions for the rate and degree of polymerization. Indicate clearly any assumptions made in the derivations. 

A 1.5 M solution of styrene in tetrahydrofuran is polymerized at 25°C by sodium naphthalene at a concentration of 3.2 x 10 5 M. Calculate the polymerization rate and degree of polymerization using appropriate data from Table 5-11. What fractions of the polymerization rate are due to free ions and ion pairs, respectively Repeat the calculations for 3.2 x 10-2 M sodium naphthalene. 

The rate of polymerization (at constant initiator concentration) depends on the number of micelles and therefore on the emulsifier concentration. The rate and degree of polymerization can be increased simultaneously. 

Exact temperature control is very important in polymerization reactions, since, among other things, the rate and degree of polymerization are strongly dependent on temperature. For accurate work, for example, for kinetic analysis with a dilatometer, a thermostat filled with water or paraffin oil may be used instead of thermostatting in the normal way with the aid of a contact thermometer and an immersion heater. 

The initiation of a radical polymerization of a monomer can be achieved with practically every peroxo or azo compound. This means that in these cases the type of initiator influences only the rate and degree of polymerization, the nature of the end groups and branching but not the polymerizability of the monomer as such. This is not the case with redox systems as radical initiators. As a... 

Photoinitiation is limited to thin layers due to the low penetration of light, nevertheless it possesses several advantages compared to the common techniques. Thus, it is possible to control the rate and degree of polymerization and also the number and length of crosslinks, by the intensity of light. 

The rates and degrees of polymerizations in radical copolymerizations conform essentially to the same laws as for radical homopolymerization (see Sect. 3.1). Raising the initiator concentration causes an increase in the rate of polymerization and at the same time a decrease in the molecular weight a temperature rise has the same effect. However, these assertions are valid only for a given... 

So far our attention has been paid to the polymerization rate and degree of polymerization. There exists no reason why future study should not be directed to other quantities. One example is the steric... 

A The expressions for the rate and degree of polymerization when the major termination step is step 5 are given in (8) and (9) respectively. The solution is based on the steady state assumption, k3(R )(M) = ki+(M )(M), which implies that the rates of Steps 3 and 4 are at least five times faster than termination. 

Fifl. 4. Temperature dependence of the rates and degrees of polymerization for the radiation induced polymerization of styrene. Dose rate OJ093 Mrad/hr (Oarreau er al. 1979 reproduced with permission of Journal of Colloid and Inlerfoct Science.)... 

These relations for rate and degree of polymerization, although based on a relatively simple chemical picture, agree well with experimental data obtained in solution and bulk polymerizations. In emulsion polymerization, however, a complication arises from the fact that the volume of the reaction mixture is subdivided into a very large number of very small volume elements, the particles, which are suspended in water. These particles are so small that they can accommodate only a limited number of polymer radicals at any one time. The peculiarities of emulsion polymerization stem from this limitation. Polymer radicals, being insoluble in water, are confined to the particle in which they are generated. Thus, a radical in one particle cannot be terminated by a radical in another particle. Therefore, the rate of termination in emulsion is much lower than that given by Equation 1. 

For values of a between 1 and 10, the character of the polymerization kinetics is intermediate between that of emulsion and that of solution polymerization. This is the region of suspension or pearl polymerization, where the rate and degree of polymerization are somewhat higher than for reaction in solution. The value... 

In the derivation of the kinetic relations it was assumed that free radicals enter the particles one by one the initiation process just described satisfies this condition. This is not the case when radicals are formed by thermal decomposition of an oil-soluble initiator. Such decomposition produces pairs of radicals in the hydrocarbon phase. One would expect a pair of radicals, confined to the extremely small volume of a latex particle, to recombine rapidly. The kinetics of this type of polymerization have been described above. It is recalled here that the subdivision factor, z, and hence rate and degree of polymerization are smaller than 1 and decrease with a. These predictions from kinetic theory are in contradiction to experimental observations. Although some oil-soluble initiators, which are good catalysts in solution systems, are poor initiators in emulsion polymerizations—e.g., benzoyl peroxide—other thermally decomposing peroxides and azo compounds produce polymer in emulsion at rates comparable to those observed in polymerization initiated by water-soluble catalysts, where the radicals enter the particles one by one. Such is the case for cumene hydroperoxide, which at low concentrations yields a rate of polymerization per particle equal to that of a persulfate-initiated reaction. It must therefore be concluded that, although oil-soluble initiators may decompose into radical pairs within the particles, polymer radicals are formed one by one. The following mechanisms are consistent with formation of polymer radicals singly. 

On the other hand, for polyesterification (an example of a stepwise reaction), the rate and degree of polymerization depend on the number of functional groups in the system, regardless of whether they are on the monomer, the high polymer, or any species intermediate in molecular weight (Equations 11, 12, and 13) ... 

Most epoxide polymerizations have the characteristics of living polymerizations, that is, the ability to polymerize successive monomer charges forming block copolymers. The expressions for the rate and degree of polymerizations are essentially those used in living chain polymerizations (see Chapter 8). The polymerization rate is given by... 

Under certain conditions, polymerizations of cationic cyclic ethers show the characteristics of living polymerizations in that the propagating species are long-lived and narrow MWDs are obtained. The rate and degree of polymerizations are then given by expressions previously described [Eqs. (10.15) and (10.16)]. Living polymerizations occur when initiation is fast relative to propagation and there is an absence of termination processes. Such conditions are found for polymerizations initiated with acylium (I) and... 

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