Institut für Physikalische Chemie der Universität Wien, Währinger Straße 42, A-1090 Wien, Austria

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1 Chain Length-Dependent Termination in Pulsed-Laser Polymerization. VIII. The Temperature Dependence of the Rate Coefficient of Bimolecular Termination in the Bulk Polymerization of Styrene OSKAR FRIEDRICH OLAJ, PHILIPP VANA Institut für Physikalische Chemie der Universität Wien, Währinger Straße 42, A-1090 Wien, Austria Received 30 August 1999; accepted 23 November 1999 ABSTRACT: The photosensitized polymerization of styrene in bulk was investigated in the temperature range of C with respect to the average rate coefficient of bimolecular chain termination k t, especially its chain length dependence at low conversions, by means of pulsed laser polymerization (PLP). Three methods were applied: two of them were based on equations originally derived for chain length independent termination taking the quantity k t contained therein as an average k t, while the third one consisted in a nonlinear fit of the experimental chain length distribution (CLD) obtained at very low pulse frequencies (LF-PLP) to a theoretical equation. The exponent b characterizing the extent of chain length dependence was unanimously found to decrease from about at 25 C to at 70 C, slightly depending on which of the three methods was chosen. This trend toward more ideal polymerization kinetics with rise of polymerization temperature is tentatively ascribed to a quite general type of polymer solution behavior that consists in a (slow) approach to a lower critical solution temperature (LCST), which is associated with a decrease of the solvent quality of the monomer toward the polymer, an effect that should be accompanied with a decrease of the parameter b John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: , 2000 Keywords: styrene; pulsed laser polymerization; (chain-length dependent) termination rate coefficient INTRODUCTION Among the various mechanisms of vinyl polymerization, the free radical mechanism certainly is the most investigated one. It is no wonder, therefore, that the overwhelming majority of problems concerning radical polymerization is solved nowadays. There are only very few aspects where the scientific community is not (yet) in full command Correspondence to: O. F. Olaj ( oskar.friedrich. olaj@univie.ac.at) Journal of Polymer Science: Part A: Polymer Chemistry, Vol. 38, (2000) 2000 John Wiley & Sons, Inc. of all details. One of these rare examples is the bimolecular termination process between two radical chains. This is mainly so for two reasons. On the one hand, there is no really reliable method at hand of direct measurement of the rate constant of chain termination under realistic polymerization conditions. On the other, the process is not a direct one, but involves several steps 1 : the radical chains have to approach each other by (translational) diffusion, they have to interpenetrate and the radical chain ends have to come into positions (by segmental diffusion) in which they may annihilate their spins. Because of the complexity of this overall process the rate constant of 697

2 698 OLAJ AND VANA bimolecular chain termination k t because of diffusion control 2 5 must be considered to be a coefficient rather than a constant inasmuch because it depends on chain length, as at least one out of these steps is a noncompensated function of chain length. Finally, entropic factors also subject to control by chain length may play an additional role. 6,7 It is not at all surprising that the data reported for k t exhibit an enormous scatter, 8 even with respect to the best-investigated and wellbehaved monomers like styrene and methylmethacrylate. It was only in the last decade (a relatively short time span in the history of radical polymerization) that some progress was achieved. The merit goes mainly to the introduction of laser methods into polymerization kinetics such as single-pulse methods with subsequent time-resolved analysis of conversion 9,10 or pseudostationary methods followed by an analysis of the chain length distribution (CLD) of the polymer formed. 11 Especially the latter technique, which is meanwhile accepted as the favorite and recommended technique of determining k p, the rate constant of chain propagation, has been important in this context: being in command of k p values it was no longer necessary to resort to k 2 p /k t and k p /k t data (the latter ones taken from pseudostationary methods) and to resolve these two different combinations of k p and k t into their individual components, but it was sufficient to insert the k p data into either of the two combinations to arrive at reasonable k t data. This represented a remarkable progress with respect to accuracy because the solution of the two equations relating k p and k t (represented by k 2 p /k t and k p /k t data), due to inevitable experimental errors, is a very diffuse one that could be shown to be responsible of a good deal of the scatter encountered in k p and especially k t values evaluated in this way. 16 On this basis we were able to develop two different methods to evaluate k t as a function of chain length. The first one 17 (in the following denoted as Method A) makes use of the invariance of the second moment (represented by the product of the weight average degree of polymerization P w and the rate of polymerization p, strictly speaking a second moment per time) of the CLD toward initiation conditions 18 (stationary or pseudostationary) in case of chain length-independent termination, which allows calculation of an average k t (denoted k m t ) for chain length-dependent termination according to P w p k 2 p m k M 2 3 (1) t [M] being the concentration of monomer and the quantity representing the contribution of disproportionation to overall termination. The second one (Method B) is derived from the expression for the rate of polymerization p in pulsed laser polymerization (PLP), 19 equally developed originally for chain length-independent polymerization, 20 yielding a different average k t (denoted as k* t ) if termination is chain length dependent p k p M ln 1 k * t t 0 k* t t /2 k* t t 0 (2) with being the concentration of radicals produced in each pulse and t 0 the reciprocal laser frequency. As could be shown by computer simulations, 19 the latter quantity k* t is very close to the true (event weighted) average k t while k m t underestimates k t by about 15 20%. 16 The two methods were successfully applied to the polymerization of styrene 21 and methylmethacrylate 22 at 25 C in bulk as benchmark systems. In both cases, a moderate dependence of the average k t was found when k m t or k* t, respectively, were plotted vs. the (average) length of the radical chains in the moment of undergoing termination, being related to the number average degree of polymerization P n of the polymer by a constant factor according to P n 1 /2 (3) Written in a power law of the form k t A b (4) the exponent b was found to be in the range of Because it is reasonable to assume that either of the lengths of the two chains involved in termination will influence k t the law governing the termination between two radicals of lengths x and y will read k t x,y k t 1,1 x, y b (5)

3 CHAIN LENGTH-DEPENDENT TERMINATION 699 with the same exponent b, and k t (1,1) of the same order of magnitude as A, where x, y is some average of the two chain lengths x and y, k t ( x,y) being the rate constant of bimolecular termination between two radical chains of length x and y. k t (1,1) does not necessarily represent the true rate constant of termination between two (hypothetical) radicals of unity chain lengths, but rather describes the bimolecular reaction between two radical chains (!) extrapolated to unity degree of polymerization. The values found for the exponent b were afterwards confirmed by a third method 23 (Method C), which, in essence, is based on the analysis of the CLD of a polymer prepared by PLP at a very low frequency (low frequency PLP, LF-PLP). In this case, nearly all radicals have died out when the next pulse arrives so that practically all radicals in the system have the same chain length at a given time (within the very narrow Poisson distribution). Under these conditions the CLD can be calculated and the experimental CLD fitted to the theoretical one. Values close to 0.20 have been obtained by this procedure for styrene as well as for methylmethacrylate at 25 C. Admittedly, 25 C is a temperature that is rather convenient for photochemical polymerization (because thermal initiation can be neglected), but is by no means typical for normal thermal radical polymerization. The primary aim of this communication, therefore, is to extend the methods to higher temperatures and to evaluate k t and its chain length dependence for those temperatures that are of real practical interest for free radical polymerization of styrene, and at which the majority of experiments concerning thermally initiated styrene polymerization experiments has been carried out. Furthermore, because the methods so far have yielded fairly consistent results, the prospects of obtaining some information on whether, and (if so) how the chain length dependence of termination varies with temperature may not be too bad. Of course, such findings may also cast a new light on the mechanism of bimolecular termination. EXPERIMENTAL The PLP experiments for Methods A and B were carried out at the temperatures indicated (40, 55, and 70 C, respectively) as described in our recent communication 21 from which the results obtained for 25 C also were taken. The overall procedure (sample preparation, polymerization, analysis of the CLD by size-exclusion chromatography (SEC), and calculational procedure) was exactly the same as had been outlined in that place. 21 The absence of thermal initiation during the time necessary for performing the PLP experiments was carefully checked by dummy experiments in the dark. In essence, all the necessary data with the exception of and t 0 (rate of polymerization p, number- and weight-average degree of polymerization, rate constant of propagation k p ) were extracted from the CLD, the quantity required for Method B was calculated from 2 p t 0 / P n 1 (6) To obtain reliable values for the parameter b, care was taken to cover a sufficiently broad range of values by a proper variation of experimental conditions, mainly t 0 and (the latter by variation of sensitizer concentration and/or laser intensity). Our experimentally determined data of the rate constant of propagation k p exhibited a slight dependence on chain length, strictly speaking on the quantity L 0, which in itself provides the access to k p according to the basic relationship 11 k p L 0 / t 0 M (7) L 0 being taken as the position of the point of inflection preceding the maximum in the CLD. This variation consisted in a slight decrease of k p with increasing L 0 up to L , fairly independent of polymerization temperature. For higher L 0 (or multiples of L 0 if higher order points of inflection were considered) this decrease came to an end, the k p values exhibiting a plateau or even a very slight increase for L For these reasons the data analysis was not only carried out using our k p data obtained from the CLD curves (referring to L , denoted as k p (1000) ) but also using the benchmark data reported in ref. 14 (denoted as k p (b) ). All the k p data entering the calculations are compiled in Table I. With respect to the quantity, a twofold procedure was adopted also: on the one hand, in accordance with earlier results, 24,25 a constant value of fix 0.1 was used for all temperatures; on the other, the calculations were carried out with a temperature-dependent var as given by Berger and Meyerhoff. 26

4 700 OLAJ AND VANA Table I. Propagation Rate Constants of Styrene in Bulk Obtained From the CLD Curves Referring to L , k (1000) p and Benchmark Data, 14 k (b) p,as Well as According to The Contribution of Disproportionation to Overall Termination fix and var by Berger and Meyerhoff 26 T C k p (1000) L mol 1 s 1 (b) k p L mol 1 s 1 fix var The procedure on which Method C is based has been in extenso described quite recently. 23 It is completely independent of all the quantities entering into the two other methods such as k p,, p, P n, P w, and is nearly insensitive toward the choice of. Essentially, it consists in carrying out a three-parameter nonlinear regression, fitting the experimental CLD (likewise determined by SEC) to that derived for an ideal single-pulse polymerization ignoring the Poissonian character of chain growth, 23 by means of Prism 3.00 by Graphpad Software. Two out of the three parameters evaluated in this way are of minor interest only, and moreover, are interrelated through a normalization constant that, unfortunately, cannot be determined independently. The third parameter, on which all the concern is concentrated, corresponds to the exponent b in eqs. 4 and 5, characterizing the chain length dependence of termination. Unfortunately, the absolute value of k t cannot be obtained in this way. The only critical point is that the experimental parameters (especially t 0 ) have to be suitably modified to account for the faster propagation at higher temperatures. RESULTS In Table I, the k p data (k p (1000) derived from our experiments, together with the benchmark k p values calculated from ref. 14, k p (b) ) are summarized. Table I also includes the values, var, which have been calculated according to Berger and Meyerhoff. 26 Table II gives a compilation of the parameters of the log log plots of k t m and k* t vs. according to eq. 4. Four different combinations of and k p values are tabled: (i) k p (1000) and fix, (ii) k p (b) and fix, (iii) k p (1000) and var, (iv) k p (b) and var. The influence of temperature on the overall behavior is best demonstrated by comparing the Table II. Parameters A and b of eq. 4 Derived From the Log Log Plots of k m t and k* t vs. for the Parameter Sets (i) fix, k (1000) p (ii) fix, k (b) p (iii) var, k (1000) (b) p, and (iv) var, k p From k m t (Method A) From k* t (Method B) T C b A 10 8 L mol 1 s 1 b A 10 8 L mol 1 s 1 i ii iii iv

5 CHAIN LENGTH-DEPENDENT TERMINATION 701 log log plots for 40 and 70 C (Fig. 1); the corresponding figure for 25 C has been already presented earlier. 21 Apart from the (trivial) increase of k t m or k* t with temperature the decrease of the slope ( b) corresponding to a marked reduction of chain-length dependence of k t is clearly visible for both types of averages. A complete survey on the overall situation comprising all data is given for a different combination of and k p values in Figure 2, for k t m and k* t separately. Paying tribute to the importance of parameter b, Figure 3 gives a graphical representation of b as a function of temperature for all the four combinations of and k p values used. Due to the variation of chain-length dependence with temperature the evaluation of Arrhenius parameters is not trivial. As an example, the Arrhenius parameters were evaluated for four different (average) chain lengths of the reacting radicals corresponding to ln 0, 3, 5, and 7 ( 1, 20, 148, and 1097, respectively) using the data obtained with the help of parameter combination (i). These are summarized in Tables III (for k t m ) and IV (for k* t ), respectively. Finally, the LF-PLP experiments carried out at 25 C 23 were extended to 70 C. Due to the higher rate of propagation much lower t 0 values (10 20 s) were sufficient than at the lower temperature. On average, b was obtained as (compared to 0.20 for 25 C 23 ). Figure 1. Double logarithmic plots of k m t (F) and k* t (E) calculated from experimental data of styrene polymerization at 40 and 70 C according to eq. 1 and eq. 2 vs. the number-average chain length of the radicals at the moment of their termination, for contribution of disproportionation to overall termination fix 0.1 and k (1000) p 140 L mol 1 s 1 and 400 L mol 1 s 1, respectively; ( ) best linear fit. Figure 2. Double logarithmic plots of k m t (filled symbols) and k* t (open symbols) calculated from experimental data of styrene polymerization at 25 C (F,E), 40 C (,ƒ), 55 C (Œ, ), and 70 C (, ) according to eq. 1 and eq. 2 vs. the number-average chain length of the radicals at the moment of their termination, for a constant contribution of disproportionation to overall termination fix 0.1 and k (1000) p (see Table I); ( ) best linear fit.

6 702 OLAJ AND VANA Figure 3. Temperature dependence of the parameter b of eq 4 calculated from k m t ( ) and k* t ( ) using the parameter set (i) fix and k (1000) p, (ii) fix and k (b) p, (iii) var and k (1000) p, (iv) var and k (b) p. Error bars correspond to twice the standard deviation. DISCUSSION The data presented in Table II demonstrate that the choice of the set of quantities and k p does influence the results; this influence, however, is of quantitative rather than qualitative nature. This observation is consistent with the sensitivity analysis carried out in an earlier publication, 21 which proved the changes to be of limited extent only. Again, k* t exceeds k t m for all the temperatures by about 15 20%, as it has been predicted theoretically, where k* t is supposed to be the more correct estimate of the true average k t. 17 The importance of k t m, however, lies in the fact that it is influenced by a possible chain transfer to a different extent in comparison to k* t. Thus, as long as the ratio of these two averages is not seriously altered, this may be taken as an indication that a Table III. k m t Calculated Using k (1000) p and fix 0.1 at Four Different Degrees of Polymerization and Evaluated Arrhenius Parameters ( Preexponential Factor, E A Energy of Activation) T C 1/T 10 3 K 1 ln(k t m /10 8 L mol 1 s 1 ) ln /10 9 L mol 1 s E A /J mol

7 CHAIN LENGTH-DEPENDENT TERMINATION 703 Table IV. k* t Calculated Using k (1000) p and fix 0.1 at Four Different Degrees of Polymerization and Evaluated Arrhenius Parameters ( Preexponential Factor, E A Energy of Activation) T C 1/T 10 3 K 1 ln(k* t /10 8 L mol 1 s 1 ) ln /10 9 L mol 1 s E A /J mol falsification of the results by chain transfer can be excluded. Actually, eq 2 is completely insensitive to chain transfer; it is only the determination of according to eq 6 that might be influenced by the presence of chain transfer. By the way, it should be noted that chain transfer in bulk styrene polymerization is mainly due to the Diels Alder intermediate, 27 which, however, seems to be destroyed under the influence of light 28 (or is not formed at all under the conditions of photopolymerization). As a consequence, chain transfer should not present a notable limitation to the accuracy of these methods. The main result is the decrease of the exponent b, with increasing temperature from about 0.17 (at 25 C) to (at 70 C) which, at least in tendency, is also well corroborated by the LF-PLP investigations. This means that the bulk polymerization approaches the conditions of ideal polymerization with increasing temperature. At the moment, the reasons for this effect only can be guessed. It is one point that the chain-length dependence of reactions between chain ends, as found by computer simulation 29 which is characterized by an exponent b 0.17 in a good solvent is decreased to b 0.06 in a solvent. For instance, in the polymerization of 1: 1 mixtures of styrene and cyclohexane at 25 C, a solvent for polystyrene at 34 C, in fact b parameters smaller than 0.17 (ca. 0.12) were found. 30 Monomeric styrene, however, will remain to be what we should like to call a good solvent for polystyrene at higher temperatures. The other point is that first discovered by Freeman and Rowlinson 31 polymer solutions, due to the disparity in density and thermal expansivity between solvent (monomer) and polymer, for entropic reasons quite generally will be subject to a high-temperature phase separation exhibiting a lower critical solution temperature (LCST) if the temperature is sufficiently elevated. 32 An example of this type is represented by the polystyrene butanone system. 33 It is well conceivable that such a tendency might also be operative in the polystyrene monostyrene system, leading to a decrease of the parameter b. The Arrhenius parameters (energy of activation E A and preexponential factor ) presented in Tables III and IV reveal some interesting features, too. Again, with respect to the general tendency and independent of the choice of the set of k p and data, the (formal) energies of activation exhibit a pronounced increase with increasing chain length from rather low values for the hypothetical chain of unity length 1(2 E A /(kj mol 1 ) 7) to 8 E A /(kj mol 1 ) 12 for 20, 10 E A /(kj mol 1 ) 15 for 148 and finally 13 E A /(kj mol 1 ) 19 for The most important point is that a good deal of this increase already takes place in the low molecular weight region and already for 20 reaches the activation energy of flow reported for styrene 34 (E A 10 kj mol 1 ). Admittedly, and not unexpectedly, the precision of the parameters is not very high. The overall picture, however, which is a direct consequence of the decrease of b with chain length, does not look unreasonable on the whole, and might be fairly compatible with the general view on radical chain behavior. Two points, however, should be borne in mind: (i) the values quoted above are average chain lengths only, and (ii) most of the increase of E A falls into a range that corresponds to extrapolated (average) values only, and thus refer to hypothetical and not real radical chains. The energies of activation furthermore compare fairly well with some

8 704 OLAJ AND VANA literature data such as those reported by Buback and Kuchta 35 (E A kj mol 1, chainlength dependence not considered), Mahabadi and O Driscoll 36 (E A 10.0 kj mol 1 ), which seems to be a little low in view of the high degree of polymerization, P n 2400 corresponding to 1320 given, Berger 26 (E A 9.96 kj mol 1, chain-length dependence not considered). Less agreement exists with Yamada s data 37 (E A 15.6 kj mol 1, resulting from ESR measurements) and absolutely no agreement is found with the energy of activation derived from the data given in the Polymer Handbook 8 (E A 25.2 kj mol 1 ), which admittedly is based on the agglomerated and unselected mass of data reported there. Concerning the absolute values of k t, a comparison with literature data is meaningful only for the more modern ones, where k t has been determined from some combination of k t with k p and a separately evaluated k p. As an example, the work of Buback et al. should be quoted. 35 Unfortunately, neither the chain-length dependence of k t was considered nor the average chain length given to which the data referred. Thus, only a very gross comparison can be made. This, however, shows an absolutely fair accordance with the data presented in this communication if reasonable average radical chain lengths are assumed. CONCLUSIONS At the moment, the observation that the deviation from ideal polymerization behavior decreases with increasing reaction temperature is restricted to the bulk polymerization of styrene. It will be a rewarding task to find out whether this appears to be a phenomenon of some generality. If so, this will give relief and satisfaction to the older generations of polymer kineticists whose reputation otherwise would be compromised because of having overlooked a substantial chain length dependence of k t in their work concentrating on more elevated polymerization temperatures. Support of this work by the Austrian Science Fund (Project #13114) is gratefully acknowledged. REFERENCES AND NOTES 1. Horie, K.; Mita, I.; Kambe, H. Polym J (Tokyo) 1973, 4, Schulz, G. V. Z Phys Chem 1956, 8, Benson, S. W.; North, A. M. J Am Chem Soc 1959, 81, North, A. M.; Reed, G. A. Trans Faraday Soc 1961, 57, Allen, P. E.; Patrick, C. Makromol Chem 1961, 47, 154; 1964, 72, Olaj, O. F.; Zifferer, G. Makromol Chem Rapid Commun 1982, 3, Olaj, O. F.; Zifferer, G. Makromol Chem 1988, 189, Berger, K. C.; Meyerhoff, G. In Polymer Handbook; Brandup, J.; Immergut, E. H., Eds.; J. Wiley & Sons: New York, 1989, p II/67ff, 3rd ed. 9. Buback, M.; Hippler, J.; Schweer, J.; Vögele, H.-P. Makromol Chem Rapid Commun 1986, 7, Buback, M.; Schweer, J. Makromol Chem Rapid Commun 1988, 9, Olaj, O. F.; Bitai, I.; Hinkelmann, F. Makromol Chem 1987, 188, Buback, M.; Garcia Rubio, L. H.; Gilbert, R. G.; Napper, D. H.; Guillot, J.; Hamielec, A. E.; Hill, D.; O Driscoll, K. F.; Olaj, O. F.; Shen, J.; Solomon, D.; Moad, G.; Stickler, M.; Tirrell, M.; Winnik, M. A. J Polym Sci Part C Polym Lett 1988, 26, Buback, M.; Gilbert, R. G.; Russell, G. T.; Hill, D. J. T.; Moad, G.; O Driscoll, K. F.; Shen, J.; Winnik, M. A. J Polym Sci Part A Polym Chem 1992, 30, Buback, M.; Gilbert, R. G.; Hutchinson, R. A.; Klumperman, B.; Kuchta, F.-D.; Manders, B. G.; O Driscoll, K. F.; Russell, G. T.; Schweer, J. Macromol Chem Phys 1995, 196, Beuermann, S.; Buback, M.; Davis, T. P.; Gilbert, R. G.; Hutchinson, R. A.; Olaj, O. F.; Russell, G. T.; Schweer, J.; van Herk, A. M. Macromol Chem Phys 1997, 198, Olaj, O. F.; Kornherr, A.; Zifferer, G. Macromol Theory Simul, to appear. 17. Olaj, O. F.; Kornherr, A.; Zifferer, G. Macromol Rapid Commun 1997, 18, Olaj, O. F.; Zifferer, G. Eur Polym J 1989, 25, Olaj, O. F.; Kornherr, A.; Zifferer, G. Macromol Rapid Commun 1998, 19, Olaj, O. F.; Bitai, I.; Gleixner, G. Makromol Chem 1985, 186, Olaj, O. F.; Vana, P. Macromol Rapid Commun 1998, 19, Olaj, O. F.; Vana, P. Macromol Rapid Commun 1998, 19, Olaj, O. F.; Vana, P.; Kornherr, A.; Zifferer, G. Macromol Chem Phys 1999, 200, Olaj, O. F.; Breitenbach, J. W.; Wolf, B. Monatsh Chem 1964, 95, Gleixner, G.; Olaj, O. F.; Breitenbach, J. W. Makromol Chem 1979, 180, 2581.

9 CHAIN LENGTH-DEPENDENT TERMINATION Berger, K. C.; Meyerhoff, G. Makromol Chem 1975, 176, 1983; Berger, K. C. Makromol Chem 1975, 176, Olaj, O. F.; Kauffmann, H. F.; Breitenbach, J. W. Makromol Chem 1976, 177, Olaj, O. F.; Kauffmann, H. F.; Breitenbach, J. W.; Bieringer, H. J Polym Sci Polym Lett Ed 1977, 15, Olaj, O. F.; Zifferer, G. Makromol Chem 1988, 189, Olaj, O. F.; Kolar, M. hitherto unpublished results; see Kolar, M. M.Sc. Thesis, University of Vienna (1999). 31. Freeman, P. I.; Rowlinson, J. S. Polymer 1960, 1, Patterson, D. Rubber Chem Technol 1967, 40, Saeki, S.; Kuwara, N.; Konno, S.; Kaneko, M. Macromolecules 1973, 6, Visnawath, D. S. Data Book on the Viscosity of Liquids, Hemisphere Publ. Co.: New York, Buback, M.; Kuchta, F.-D. Macromol Chem Phys 1997, 198, Mahabadi, H. K.; O Driscoll, K. F. J Macromol Sci Chem 1977, 11, Yamada, B.; Kageoka, M.; Otsu, T. Polym Bull 1992, 29, 385.