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for MMA polymerization due to the presence of intramolecular solvation. Intramolecular solvation involves electron donation from the carbonyl oxygen of the penultimate unit (i.e., the unit just before the end unit of the propagating chain) as shown in XXVIII [Kraft et al., 1978, 1980]. This additional binding of the counterion to the polymer accounts for the low dissociation constant (K < 10 9 ) for polymethyl methacrylate and also poly(2-vinylpyridine) ion
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pairs [Tardi and Sigwalt, 1972; Van Beylen et al., 1988]. Intramolecular solvation in poly (2-vinylpyridine) involves electron donation from the nitrogen of a penultimate pyridine ring. This effect may also be responsible for the decrease in dissociation constant of the poly(2vinylpyridine) ion pair with decreasing size of counterion in THF. Thus K is 1:1 10 9 , 2:5 10 9 , and 8:3 10 10 for Cs , K , and Na , respectively [Szwarc, 1983]. This is the reverse of the order observed for polystyryl ion pairs in THF. Apparently, smaller counterions t better or more tightly into the intramolecular solvation sphere. 5-3d-3 Degree of Polymerization
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The number-average degree of polymerization for a living anionic polymerization is the ratio of the concentration of reacted monomer to the concentration of living ends (Eq. 5-58). For the usual situation where all the initiator I is converted into propagating chain ends, Eq. 5-58 becomes
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depending on the mode of initiation. Equation 5-98 applies to polymerizations initiated by electron transfer since each nal polymer molecule originates from two initiator molecules (e.g., one dianionic propagating species is formed from two sodium naphthalenes). Initiation processes other than electron transfer (e.g., alkyllithium) involve one polymer molecule per initiator molecule, and Eq. 5-99 is applicable. For polymerization involving dormant species in equilibrium with ions and ion pairs (e.g., GTP), [I]0 is the sum of the concentrations of ion, ion pair, and dormant species. Narrow molecular weight distributions are obtained for systems with fast initiation and ef cient mixing in the absence of depropagation, termination, and transfer reaction (Eq. 3-230). PDI values below 1.1 1.2 are found for many living polymerizations. The living polymer technique offers a unique method of synthesizing standard polymer samples of known and wellde ned molecular weights. Commercially available molecular weight standards are now
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available for a number of polymers polystyrene, polyisoprene, poly(a-methylstyrene), poly(2-vinylpyridine), poly(methyl methacrylate), polyisobutylene, and poly(tetrahydrofuran). All except the last two polymers are synthesized by living anionic polymerization. The last two are obtained by living cationic polymerization with ring-opening polymerization (Chap. 7) used for poly(tetrahydrofuran). These polymer standards are useful as calibration standards in molecular weight measurements by size exclusion chromatography, membrane and vapor pressure osmometry, and viscometry. The occurrence of any termination, transfer or side reactions result in broadening of the molecular weight distribution. The termination reactions in methacrylate polymerizations (at other than low temperatures in polar solvents) and depropagation in a-methylstyrene polymerizations broaden PDI [Chaplin and Yaddehige, 1980; Malhotra et al., 1977; Malhotra, 1978]. Although the bulk of propagation is carried by a small fraction of the propagating species (i.e., the free ions), this does not signi cantly broaden the molecular weight distribution since the free ions and ion pairs are in rapid equilibrium. Each polymer chain propagates as both free ions and ion pairs over its lifetime and the average fractions of its lifetime spent as free ions and ion pairs are not too different than for any other propagating chain.
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The data available on the temperature dependence of the rates of living polymerization show the experimental activation energy ER is generally relatively low and positive. One should note that ER for living polymerization is the activation energy for propagation. The polymerization rates are relatively insensitive to temperature but increase with increasing temperature. Furthermore, the activation energy varies considerably depending on the solvent employed in the polymerization as was the case for cationic polymerization. Thus the activation energy for propagation in the system styrene sodium naphthalene is 37.6 kJ mol 1 in dioxane and only 4.2 kJ mol 1 in tetrahydrofuran [Allen et al., 1960; Geacintov et al., 1962; Stretch and Allen, 1961]. The molecular weight of the polymer produced in a nonterminating polymerization is unaffected by temperature if transfer agents are absent. The situation can be different if transfer agents are initially present. Most of the activation energy data reported in the literature are apparent activation energies corresponding to the values for the apparent propagation rate constant. The effect of temperature on propagation is complex temperature simultaneously affects the relative concentrations of free ions and ion pairs and the individual rate constants for the free ions and ion pairs. Temperature affects the rate constants kp and kp in the manner all rate con stants are affected, increasing temperature increases the values of kp and kp . However, the effect of temperature on the concentration of free ions relative to ion pairs is in the opposite direction. The change with temperature of the equilibrium constant for dissociation of ion pairs into free ions is given by the relationship
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where H is negative and K increases with decreasing temperature [Schmitt and Schultz, 1975; Shimomura et al., 1976b]. For example, H is about 37 kJ mol 1 for polystyryl sodium in tetrahydrofuran, which corresponds to an increase in K by a factor of about 300 as the temperature changes from 25 to 70 C. The free-ion concentration is higher by a factor of about 20 at 70 C compared to 25 C for a living end concentration of 10 3 M. The
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