- Email: [email protected]
The importance of the molecular weight distribution for the direct determination of the kinetic constants of free radical polymerization
Radiation Physics and Chemistry 67 (2003) 335–339
The importance of the molecular weight distribution for the direct determination of the kinetic constants of free radical polymerization . Irene Schnoll-Bitai Inst. fur Physikalische Chemie, Wahringer Str. 42, A-1090 Vienna, Austria .
Abstract The distribution curves obtained by stationary, pseudostationary and instationary techniques are surveyed in rough outlines. Special emphasis is laid on the possibilities to determine directly kinetic quantities from both theoretical and experimental point of view. A simple method for the direct determination of axial dispersion is also presented based on the comparison of ideal and experimental absolute peak widths. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Radical polymerization; Molecular weight distribution; Kinetics
1. Introduction In principle, a molecular weight distribution bears the imprint of the genesis of the polymer as each event (initiation, propagation, termination, transfer) influences the number and/or chain length of the synthesized polymer. The choice of initiation conditions is of utmost importance for the modeling of chain length distributions (CLDs). The wealth of distributions and information contained therein direct the attention to the following questions: (1) How can the kinetic information be extracted? (2) How can the knowledge of these constants be used for the intentional modeling of distributions. For the nexus of theoretical distribution curves with experimental data obtained by size exclusion chromatography (SEC) conversion of the measured signal as a function of elution volume is necessary. Both axes need to be transformed (Trathnigg, 1998). The elution volume is converted to log (M) by the use of a calibration curve which is in most cases linear log ðMÞ ¼ a bVe :
ð1Þ
The change from a logarithmic to a linear scale necessitates the proper conversion of the ordinate. The
height of the raw data hðlogðMÞÞ from SEC for a masssensitive detector is converted to the number nðMÞ; molar mass wðMÞ or hyper distribution hðMÞ by hðlog MÞpM 2 nðMÞpMwðMÞphðMÞ:
ð2Þ
Simple multiplication of a number distribution with i (degree of polymerization) or i2 lead to the not normalized molar mass or hyper distribution.
2. Schulz-Zimm distributions (Elias, 1999) nðiÞ ¼
kk ik1 eki=Pn Pkn Gðk þ 1Þ
ð3Þ
Continuous initiation is necessary for the development of a stationary state and leads to monomodal Schulz-Zimm distributions which is a time-invariant distribution for low conversion. The degree of coupling k (1,2) and the number average degree of polymerization Pn are the parameters influencing the shape (Fig. 1). The extrema (maximum, points of inflection) are obtained by . analysis of the curves (Schnoll-Bitai, 2002a) ( rffiffiffiffiffiffiffiffiffiffiffi) k1 1 Pn ; ipi;n ¼ imax;n 17 k > 1; imax;n ¼ k k1
E-mail address: [email protected] . (I. Schnoll-Bitai). 0969-806X/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0969-806X(03)00063-X
ð4Þ
I. Schnoll-Bitai / Radiation Physics and Chemistry 67 (2003) 335–339 .
336
Fig. 2. PLP kp ¼ 100 l mol1 s1, ½M ¼ 10 mol l1, t0 ¼ 100 ms, kt r ¼ 100 s1 (Schn.oll-Bitai et al., in preparation).
3. Distributions from pseudostationary polymerization
Fig. 1. The conversion of Schulz-Zimm distributions from number to molar mass, and hyper distribution and finally to chromatographic dimension for a mass sensitive detector.
( imax;w ¼ Pn
ipi;w ¼ Pn
( rffiffiffi) rffiffiffi) 1 1 17 ¼ imax;w 17 ; k k ð5Þ
kþ1 kþ1 Pn ¼ Pw ipi;h ¼ Pn k k ( rffiffiffiffiffiffiffiffiffiffiffi) 1 : ¼ imax;h 17 kþ1
imax;h ¼
(
rffiffiffiffiffiffiffiffiffiffiffi) 1 17 kþ1 ð6Þ
Without additional information only the type of termination and the number and weight average degrees of polymerization can theoretically be determined by Eqs. (4)–(6); individual rate constants are not accessible. Gilbert et al. (1999) developed the CLD method for the determination of CM ¼ ktr;M =kp (ratio of the chain transfer and propagation rate coefficients) as the slope of the linear part of the CLD in the high molecular weight region in the limit of low initiation.
Pulsed laser polymerization (Olaj and Bitai, 1987; van Herk, 2000; Olaj and Zifferer, 1996; Buback and Beuermann, 2002) (=pseudostationary technique) lead to the first direct determination of the rate constant of propagation, kp : The periodic modulation of the total concentration between a minimum and maximum value gives rise to molecular weight distributions, which are characterized by the appearance of the so-called additional peaks (c.f. Fig. 2). For the peaks the location of the point of inflection on the low molecular weight side (or maximum), L0 ; is directly related to the monomer concentration ½M; the duration of the dark period separating two laser pulses t0 and kp L0 ¼ nkp ½Mt0
n ¼ 1; 2; 3; y :
ð7Þ
According to theoretical calculations the point of inflection is the better choice for the direct determination of kp in the intermediate and low termination regime whereas the maximum should be used in the high termination limit (Hutchinson et al., 1996; van Herk et al., 1995). The accuracy of the data depends strongly on the quality of the measured and converted molecular weight distributions. It is best for polymers where narrow standards (of the same type) are used for calibration either directly or via universal calibration. The analysis of copolymer systems is a more delicate task as neither narrow standards nor reliable Mark-Houwink constants exist for the different copolymer compositions but are, in principle, possible with multidetector systems. Even under ideal conditions there remain two questions: (1) At the moment there exists no experimental criterion for the decision whether the point of inflection or the maximum is the better choice for the deduction of kp : (2) It is always assumed that the influence of axial dispersion is negligible but Buback et al. (1996) demonstrated that the location of the extrema changes
I. Schnoll-Bitai / Radiation Physics and Chemistry 67 (2003) 335–339 .
with the primary radical concentration and the axial dispersion. An average value of the rate constant of termination kt can also be deduced from the chromatographic data (Olaj et al., 1987) by making use of the rate of polymerization vp ; the mass average degree of polymerization Pw and the relative contribution of termination by disproportionation to the overall termination d kt ¼
kp2 ½M2 ð3 dÞ : Pw vp
ð8Þ
The validity of an assumed power law dependence kt;i ¼ kt1;1 ib was shown by a three parameter fit ða; b; cÞ of experimentally measured distribution for poly(methyl methacrylate) samples in the low frequency (singlepulse) limit at 25 C (Olaj et al., 1999) according to ( ib wp ¼ ai2 d 2 1 þ c=1 b i1b ) 1d ði=2Þb 2 : ð9Þ þ 4 1 þ c=1 bði=2Þ1b 2 1 þ d Buback and Beuermann, 2002 combined the PLPwith the CLD-technique for the direct determination of ktr from the measured molecular weight distribution.
4. Distributions from instationary polymerization (Schn.oll-Bitai, 2002b)
sffiffiffiffiffiffiffiffiffi! 1 1 ; kp ½MtD ¼ Llow exp þ 2Llow Llow sffiffiffiffiffiffiffiffiffiffi! 1 1 kp ½MtD ¼ Lhigh exp : 2Lhigh Lhigh
For a finite continuos initiation period tIni the distribution is characterized by a change in slope; the location of the corresponding point of inflection can be described by Eq. (10) when tD and Lmax are replaced by tIni and Lpi : For low values of the rate of initiation and/or the rate constant of termination almost rectangular distributions are obtained. Even the combination of several periods differing in their initiation characteristics can be used for the direct determination of kp from the points of inflection (Fig. 3) i1ext;j Ekp ½M
j1 X
ð11Þ
ð12Þ
tnm :
ð13Þ
m¼0
Besides the determination of kp from the peak maxima Davis et al. (2002) also extracted kt -values by a procedure that separated the peak areas of the RCLD and PCLD from multipulse quenched instationary polymerization. The absolute peak width D is defined as the difference between two successive points of inflection belonging to a peak D ¼ ihigh ilow
ð14Þ
and is an invariant quantity with respect to the number, molar mass and hyper distribution. The absolute peak width depends again on the initiation conditions pffiffiffiffiffiffiffiffi D ¼ 2 imax for d-initiation; ð15Þ D ¼ kp ½MtIni
Under instationary polymerization conditions the radical (RCLD) and polymer (PCLD) CLDs depend strongly on the total polymerization time. The usually transient radical distributions become feasible by a fast and efficient quenching process which deactivates almost instantaneously all radicals at a given time. All distributions are a superposition of the RCLD and PCLD. The shape depends on the chosen initiation conditions and can vary from a monomodal Poissonlike distribution to multimodal or even step-like functions. For single d-pulse initiation the RCLD is a Poisson distribution and the extrema (maximum Lmax ; points of inflection Llow ; Lhigh ) depend on the duration of the dark period tD
1 kp ½MtD ¼ Lmax exp ð10Þ ELmax ; 2Lmax
337
for continuous initiation:
ð16Þ
The invariance of the absolute peak widths is a necessary criterion that the axial dispersion can be determined from experimental data. The relative peak width introduced as the ratio of two . successive points of inflection (Schnoll-Bitai, 2002a, c) (always>1) is a valuable mean for the direct determination of axial dispersion in SEC as it is directly related to the peak variance ihigh d¼ ; ð17Þ ilow pffiffiffiffiffiffiffiffi imax þ imax pffiffiffiffiffiffiffiffi for d-initiation; dE ð17aÞ imax imax d¼
ik ik1
Pk
i¼1 ¼ Pk1 i¼1
ti ti
for continuous initiation;
2sSEC ¼ DSEC ¼ vlow vhigh 1 ipi;high 1 1 ¼ ¼ d: Mpi;high Mpi;low ¼ b b ipi;low b
ð17bÞ
ð18Þ
The validity of the additive contribution of axial . dispersion to the peak variance (Elias, 1999; SchnollBitai, 2002c) s2SEC ¼ s2peak þ s2ad
ð19Þ
338
I. Schnoll-Bitai / Radiation Physics and Chemistry 67 (2003) 335–339 .
Fig. 3. The influence of different initiation conditions on the resulting RCLD and PCLD for d-pulse or continuous initiation and the combination of different initiation periods; the corresponding points of inflection are included; parameters as in Fig. 2.
heterogeneous systems to elucidate still unsolved problems. A simple method for the direct determination of axial dispersion in SEC was developed and should be used to increase the accuracy of the kinetic data extracted from experimentally measured distribution curves. For the sake of compactness only the latest publications or review articles were chosen and use can be made of the articles cited therein:
References Fig. 4. Comparison of measured peak widths for narrow . polymer standards from Polymer Standard Service (SchnollBitai, 2002a).
was demonstrated for different narrow polymer standards (Fig. 4) as all standards had comparable peak widths for the same value of the peak maximum.
5. Conclusion Pseudostationary and instationary techniques are suitable for direct determination of kp and sometimes even kt : This enables the investigation of copolymerization systems, polymerization in solution as well as in
Buback, M., Beuermann, S., 2002. Rate coefficients of freeradical polymerization deduced from pulsed laser experiments. Prog. Polym. Sci. 27, 191–254. Buback, M., Busch, M., L.ammel, R.A., 1996. Modeling of molecular weight distribution in pulsed laser free-radical homopolymerizations. Macromol. Theory Simul. 5, 845–861. Davis, T.P., Vana, P., Yee, L.H., 2002. Multipulse initiation in pulsed laser and quenched instationary polymerization: determination of the propagation and termination rate coefficients for dicyclohexyl itaconate polymerization. Macromolecules 35 (8), 3008–3016. Elias, H.-G., 1999. In: Elias, H.G. (Ed.), Makromolekule . B 1: Chemische Struktur und Synthese, 6th Edition. Wiley-VCH, Weinheim, pp. 71–75.
I. Schnoll-Bitai / Radiation Physics and Chemistry 67 (2003) 335–339 . Gilbert, R.G., Davis, T.P., Kukulj, D., 1999. Chain transfer to monomer in the free radical polymerization of methyl methacrylate, styrene, and a-methylstyrene. Macromolecules 31, 994–999. Hutchinson, R.A., Beuermann, S., Paquet, D.A., McMinn, J.H., 1996. Determination of free-radical propagation rate coefficients of butyl, 2-ethylhexyl, and dodecyl acrylates by pulsed laser polymerization. Macromolecules 29, 4206–4215. Olaj, O.F., Bitai, I., 1987. The laser flash-initiated polymerization as a tool of evaluating (individual) kinetic constants of free radical polymerization, 3. Information from degrees of polymerization. Angew. Makromol. Chem. 155, 177–190. Olaj, O.F., Zifferer, G., 1996. The incorporation of side reactions into pseudostationary polymerization, 1. Macromol. Theory Simul. 5, 901–921. Olaj, O.F., Bitai, I., Hinkelmann, F., 1987. The laser-flashinitiated polymerization as a tool of evaluating (individual) kinetic constants of free-radical polymerization, 2. Macromol. Chem. 188, 1689–1702. Olaj, O.F., Vana, P., Kornherr, A., Zifferer, G., 1999. Chain-length dependent termination in pulsed-laser polymerization, 7. The evaluation of the power-law exponent b from the chainlength distribution in the low frequency (single-pulse) limit for the reference systems styrene and methyl methacrylate in bulk at 25 C. Macromol. Chem. Phys. 200, 2031–2039.
339
. Schnoll-Bitai, I., 2002a. A comparative study of the properties of different distribution curves. Macromol. Phys. Chem. 203, 1754–1762. . Schnoll-Bitai, I., 2002b. The quenched instationary polymerization system, 5. The peak broadness as an invariant quantity of the number, molar mass and hyper distributions. Macromol. Theory Simul. 11, 199–212. . Schnoll-Bitai, I., 2002c. Quenched instationary polymerization systems, 6. The direct determination of axial dispersion in size exclusion chromatography by comparison of theoretical and ideal peak widths. Macromol. Theory Simul. 11, 770–776. . Schnoll-Bitai, I., Kornherr, A., Olaj, O.F., Zifferer, G., in preparation. Trathnigg, B., 1998. Equipment and materials. In: Pasch, H., Trathnigg, B. (Eds.), HPLC of Polymers. Springer, Berlin, pp. 33–38. van Herk, A.M., 2000. Pulsed initiation polymerization as a means of obtaining propagation rate coefficients in freeradical polymerizations. II review up to 2000. Macromol. Theory Simul. 9, 433–441. van Herk, A.M., Manders, B.G., German, A.L., 1995. Monte carlo simulation of the chain length distribution in pulsedlaser polymerization in microemulsion. Makromol. Theory Simul. 4, 325–333.
The importance of the molecular weight distribution for the direct determination of the kinetic constants of free radical polymerization . Irene Schnoll-Bitai Inst. fur Physikalische Chemie, Wahringer Str. 42, A-1090 Vienna, Austria .
Abstract The distribution curves obtained by stationary, pseudostationary and instationary techniques are surveyed in rough outlines. Special emphasis is laid on the possibilities to determine directly kinetic quantities from both theoretical and experimental point of view. A simple method for the direct determination of axial dispersion is also presented based on the comparison of ideal and experimental absolute peak widths. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Radical polymerization; Molecular weight distribution; Kinetics
1. Introduction In principle, a molecular weight distribution bears the imprint of the genesis of the polymer as each event (initiation, propagation, termination, transfer) influences the number and/or chain length of the synthesized polymer. The choice of initiation conditions is of utmost importance for the modeling of chain length distributions (CLDs). The wealth of distributions and information contained therein direct the attention to the following questions: (1) How can the kinetic information be extracted? (2) How can the knowledge of these constants be used for the intentional modeling of distributions. For the nexus of theoretical distribution curves with experimental data obtained by size exclusion chromatography (SEC) conversion of the measured signal as a function of elution volume is necessary. Both axes need to be transformed (Trathnigg, 1998). The elution volume is converted to log (M) by the use of a calibration curve which is in most cases linear log ðMÞ ¼ a bVe :
ð1Þ
The change from a logarithmic to a linear scale necessitates the proper conversion of the ordinate. The
height of the raw data hðlogðMÞÞ from SEC for a masssensitive detector is converted to the number nðMÞ; molar mass wðMÞ or hyper distribution hðMÞ by hðlog MÞpM 2 nðMÞpMwðMÞphðMÞ:
ð2Þ
Simple multiplication of a number distribution with i (degree of polymerization) or i2 lead to the not normalized molar mass or hyper distribution.
2. Schulz-Zimm distributions (Elias, 1999) nðiÞ ¼
kk ik1 eki=Pn Pkn Gðk þ 1Þ
ð3Þ
Continuous initiation is necessary for the development of a stationary state and leads to monomodal Schulz-Zimm distributions which is a time-invariant distribution for low conversion. The degree of coupling k (1,2) and the number average degree of polymerization Pn are the parameters influencing the shape (Fig. 1). The extrema (maximum, points of inflection) are obtained by . analysis of the curves (Schnoll-Bitai, 2002a) ( rffiffiffiffiffiffiffiffiffiffiffi) k1 1 Pn ; ipi;n ¼ imax;n 17 k > 1; imax;n ¼ k k1
E-mail address: [email protected] . (I. Schnoll-Bitai). 0969-806X/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0969-806X(03)00063-X
ð4Þ
I. Schnoll-Bitai / Radiation Physics and Chemistry 67 (2003) 335–339 .
336
Fig. 2. PLP kp ¼ 100 l mol1 s1, ½M ¼ 10 mol l1, t0 ¼ 100 ms, kt r ¼ 100 s1 (Schn.oll-Bitai et al., in preparation).
3. Distributions from pseudostationary polymerization
Fig. 1. The conversion of Schulz-Zimm distributions from number to molar mass, and hyper distribution and finally to chromatographic dimension for a mass sensitive detector.
( imax;w ¼ Pn
ipi;w ¼ Pn
( rffiffiffi) rffiffiffi) 1 1 17 ¼ imax;w 17 ; k k ð5Þ
kþ1 kþ1 Pn ¼ Pw ipi;h ¼ Pn k k ( rffiffiffiffiffiffiffiffiffiffiffi) 1 : ¼ imax;h 17 kþ1
imax;h ¼
(
rffiffiffiffiffiffiffiffiffiffiffi) 1 17 kþ1 ð6Þ
Without additional information only the type of termination and the number and weight average degrees of polymerization can theoretically be determined by Eqs. (4)–(6); individual rate constants are not accessible. Gilbert et al. (1999) developed the CLD method for the determination of CM ¼ ktr;M =kp (ratio of the chain transfer and propagation rate coefficients) as the slope of the linear part of the CLD in the high molecular weight region in the limit of low initiation.
Pulsed laser polymerization (Olaj and Bitai, 1987; van Herk, 2000; Olaj and Zifferer, 1996; Buback and Beuermann, 2002) (=pseudostationary technique) lead to the first direct determination of the rate constant of propagation, kp : The periodic modulation of the total concentration between a minimum and maximum value gives rise to molecular weight distributions, which are characterized by the appearance of the so-called additional peaks (c.f. Fig. 2). For the peaks the location of the point of inflection on the low molecular weight side (or maximum), L0 ; is directly related to the monomer concentration ½M; the duration of the dark period separating two laser pulses t0 and kp L0 ¼ nkp ½Mt0
n ¼ 1; 2; 3; y :
ð7Þ
According to theoretical calculations the point of inflection is the better choice for the direct determination of kp in the intermediate and low termination regime whereas the maximum should be used in the high termination limit (Hutchinson et al., 1996; van Herk et al., 1995). The accuracy of the data depends strongly on the quality of the measured and converted molecular weight distributions. It is best for polymers where narrow standards (of the same type) are used for calibration either directly or via universal calibration. The analysis of copolymer systems is a more delicate task as neither narrow standards nor reliable Mark-Houwink constants exist for the different copolymer compositions but are, in principle, possible with multidetector systems. Even under ideal conditions there remain two questions: (1) At the moment there exists no experimental criterion for the decision whether the point of inflection or the maximum is the better choice for the deduction of kp : (2) It is always assumed that the influence of axial dispersion is negligible but Buback et al. (1996) demonstrated that the location of the extrema changes
I. Schnoll-Bitai / Radiation Physics and Chemistry 67 (2003) 335–339 .
with the primary radical concentration and the axial dispersion. An average value of the rate constant of termination kt can also be deduced from the chromatographic data (Olaj et al., 1987) by making use of the rate of polymerization vp ; the mass average degree of polymerization Pw and the relative contribution of termination by disproportionation to the overall termination d kt ¼
kp2 ½M2 ð3 dÞ : Pw vp
ð8Þ
The validity of an assumed power law dependence kt;i ¼ kt1;1 ib was shown by a three parameter fit ða; b; cÞ of experimentally measured distribution for poly(methyl methacrylate) samples in the low frequency (singlepulse) limit at 25 C (Olaj et al., 1999) according to ( ib wp ¼ ai2 d 2 1 þ c=1 b i1b ) 1d ði=2Þb 2 : ð9Þ þ 4 1 þ c=1 bði=2Þ1b 2 1 þ d Buback and Beuermann, 2002 combined the PLPwith the CLD-technique for the direct determination of ktr from the measured molecular weight distribution.
4. Distributions from instationary polymerization (Schn.oll-Bitai, 2002b)
sffiffiffiffiffiffiffiffiffi! 1 1 ; kp ½MtD ¼ Llow exp þ 2Llow Llow sffiffiffiffiffiffiffiffiffiffi! 1 1 kp ½MtD ¼ Lhigh exp : 2Lhigh Lhigh
For a finite continuos initiation period tIni the distribution is characterized by a change in slope; the location of the corresponding point of inflection can be described by Eq. (10) when tD and Lmax are replaced by tIni and Lpi : For low values of the rate of initiation and/or the rate constant of termination almost rectangular distributions are obtained. Even the combination of several periods differing in their initiation characteristics can be used for the direct determination of kp from the points of inflection (Fig. 3) i1ext;j Ekp ½M
j1 X
ð11Þ
ð12Þ
tnm :
ð13Þ
m¼0
Besides the determination of kp from the peak maxima Davis et al. (2002) also extracted kt -values by a procedure that separated the peak areas of the RCLD and PCLD from multipulse quenched instationary polymerization. The absolute peak width D is defined as the difference between two successive points of inflection belonging to a peak D ¼ ihigh ilow
ð14Þ
and is an invariant quantity with respect to the number, molar mass and hyper distribution. The absolute peak width depends again on the initiation conditions pffiffiffiffiffiffiffiffi D ¼ 2 imax for d-initiation; ð15Þ D ¼ kp ½MtIni
Under instationary polymerization conditions the radical (RCLD) and polymer (PCLD) CLDs depend strongly on the total polymerization time. The usually transient radical distributions become feasible by a fast and efficient quenching process which deactivates almost instantaneously all radicals at a given time. All distributions are a superposition of the RCLD and PCLD. The shape depends on the chosen initiation conditions and can vary from a monomodal Poissonlike distribution to multimodal or even step-like functions. For single d-pulse initiation the RCLD is a Poisson distribution and the extrema (maximum Lmax ; points of inflection Llow ; Lhigh ) depend on the duration of the dark period tD
1 kp ½MtD ¼ Lmax exp ð10Þ ELmax ; 2Lmax
337
for continuous initiation:
ð16Þ
The invariance of the absolute peak widths is a necessary criterion that the axial dispersion can be determined from experimental data. The relative peak width introduced as the ratio of two . successive points of inflection (Schnoll-Bitai, 2002a, c) (always>1) is a valuable mean for the direct determination of axial dispersion in SEC as it is directly related to the peak variance ihigh d¼ ; ð17Þ ilow pffiffiffiffiffiffiffiffi imax þ imax pffiffiffiffiffiffiffiffi for d-initiation; dE ð17aÞ imax imax d¼
ik ik1
Pk
i¼1 ¼ Pk1 i¼1
ti ti
for continuous initiation;
2sSEC ¼ DSEC ¼ vlow vhigh 1 ipi;high 1 1 ¼ ¼ d: Mpi;high Mpi;low ¼ b b ipi;low b
ð17bÞ
ð18Þ
The validity of the additive contribution of axial . dispersion to the peak variance (Elias, 1999; SchnollBitai, 2002c) s2SEC ¼ s2peak þ s2ad
ð19Þ
338
I. Schnoll-Bitai / Radiation Physics and Chemistry 67 (2003) 335–339 .
Fig. 3. The influence of different initiation conditions on the resulting RCLD and PCLD for d-pulse or continuous initiation and the combination of different initiation periods; the corresponding points of inflection are included; parameters as in Fig. 2.
heterogeneous systems to elucidate still unsolved problems. A simple method for the direct determination of axial dispersion in SEC was developed and should be used to increase the accuracy of the kinetic data extracted from experimentally measured distribution curves. For the sake of compactness only the latest publications or review articles were chosen and use can be made of the articles cited therein:
References Fig. 4. Comparison of measured peak widths for narrow . polymer standards from Polymer Standard Service (SchnollBitai, 2002a).
was demonstrated for different narrow polymer standards (Fig. 4) as all standards had comparable peak widths for the same value of the peak maximum.
5. Conclusion Pseudostationary and instationary techniques are suitable for direct determination of kp and sometimes even kt : This enables the investigation of copolymerization systems, polymerization in solution as well as in
Buback, M., Beuermann, S., 2002. Rate coefficients of freeradical polymerization deduced from pulsed laser experiments. Prog. Polym. Sci. 27, 191–254. Buback, M., Busch, M., L.ammel, R.A., 1996. Modeling of molecular weight distribution in pulsed laser free-radical homopolymerizations. Macromol. Theory Simul. 5, 845–861. Davis, T.P., Vana, P., Yee, L.H., 2002. Multipulse initiation in pulsed laser and quenched instationary polymerization: determination of the propagation and termination rate coefficients for dicyclohexyl itaconate polymerization. Macromolecules 35 (8), 3008–3016. Elias, H.-G., 1999. In: Elias, H.G. (Ed.), Makromolekule . B 1: Chemische Struktur und Synthese, 6th Edition. Wiley-VCH, Weinheim, pp. 71–75.
I. Schnoll-Bitai / Radiation Physics and Chemistry 67 (2003) 335–339 . Gilbert, R.G., Davis, T.P., Kukulj, D., 1999. Chain transfer to monomer in the free radical polymerization of methyl methacrylate, styrene, and a-methylstyrene. Macromolecules 31, 994–999. Hutchinson, R.A., Beuermann, S., Paquet, D.A., McMinn, J.H., 1996. Determination of free-radical propagation rate coefficients of butyl, 2-ethylhexyl, and dodecyl acrylates by pulsed laser polymerization. Macromolecules 29, 4206–4215. Olaj, O.F., Bitai, I., 1987. The laser flash-initiated polymerization as a tool of evaluating (individual) kinetic constants of free radical polymerization, 3. Information from degrees of polymerization. Angew. Makromol. Chem. 155, 177–190. Olaj, O.F., Zifferer, G., 1996. The incorporation of side reactions into pseudostationary polymerization, 1. Macromol. Theory Simul. 5, 901–921. Olaj, O.F., Bitai, I., Hinkelmann, F., 1987. The laser-flashinitiated polymerization as a tool of evaluating (individual) kinetic constants of free-radical polymerization, 2. Macromol. Chem. 188, 1689–1702. Olaj, O.F., Vana, P., Kornherr, A., Zifferer, G., 1999. Chain-length dependent termination in pulsed-laser polymerization, 7. The evaluation of the power-law exponent b from the chainlength distribution in the low frequency (single-pulse) limit for the reference systems styrene and methyl methacrylate in bulk at 25 C. Macromol. Chem. Phys. 200, 2031–2039.
339
. Schnoll-Bitai, I., 2002a. A comparative study of the properties of different distribution curves. Macromol. Phys. Chem. 203, 1754–1762. . Schnoll-Bitai, I., 2002b. The quenched instationary polymerization system, 5. The peak broadness as an invariant quantity of the number, molar mass and hyper distributions. Macromol. Theory Simul. 11, 199–212. . Schnoll-Bitai, I., 2002c. Quenched instationary polymerization systems, 6. The direct determination of axial dispersion in size exclusion chromatography by comparison of theoretical and ideal peak widths. Macromol. Theory Simul. 11, 770–776. . Schnoll-Bitai, I., Kornherr, A., Olaj, O.F., Zifferer, G., in preparation. Trathnigg, B., 1998. Equipment and materials. In: Pasch, H., Trathnigg, B. (Eds.), HPLC of Polymers. Springer, Berlin, pp. 33–38. van Herk, A.M., 2000. Pulsed initiation polymerization as a means of obtaining propagation rate coefficients in freeradical polymerizations. II review up to 2000. Macromol. Theory Simul. 9, 433–441. van Herk, A.M., Manders, B.G., German, A.L., 1995. Monte carlo simulation of the chain length distribution in pulsedlaser polymerization in microemulsion. Makromol. Theory Simul. 4, 325–333.