Anal. Chem. 1996, 68, 1303-1308
Comparison of Most Probable Peak Values As Measured for Polymer Distributions by MALDI Mass Spectrometry and by Size Exclusion Chromatography Christian Jackson,* Barbara Larsen, and Charles McEwen
Corporate Center for Analytical Sciences, DuPont Central Research & Development, P.O. Box 80228, Wilmington, Delaware 19880-0228
Matrix-assisted laser desorption/ionization mass spectrometry (MS) has been shown to provide most probable peak (Mp) values for poly(methyl methacrylate) polymers that are low relative to manufacturers, Mp values measured by size exclusion chromatography (SEC). Comparison of theoretical Mp values determined by MS or SEC is shown to be a function of how the data are displayed. For narrow polymer distributions, the theoretical Mp value determined by MS will be 2 monomer units smaller than the Mp determined by SEC. For wide polydispersity, the Mp value determined by MS will be considerably lower than those obtained from SEC. The Mp value reported should be reserved for weight fraction vs log mass plots as in SEC. The modal molecular mass, Mm, is recommended for data represented as number fraction vs linear mass as with MS data. Matrix-assisted laser desorption/ionization (MALDI) mass spectrometry has generated considerable interest as a new method for characterizing polymers.1-10 The method is especially valuable for structural characterization of individual oligomers of low-mass polymers.10-14 However, considerable attention has been focused on the ability of MALDI to provide accurate molecular weight distributions.6-10,14 Because polymer chemists have developed property/function correlations based on measurable values such (1) Bahr, U.; Deppe, A.; Karas, M.; Hillenkamp, F.; Giessmann, U. Anal. Chem. 1992, 64, 2866. (2) Danis, P. O.; Karr, D. E.; Mayer, F.; Holle, A.; Watson, C. H. Org. Mass Spectrom. 1992, 27, 843. (3) Danis, P. O.; Karr, D. E. Org. Mass Spectrom. 1993, 28, 923. (4) Eggert, M.; Freitag, R. J. Polym. Sci., Polym. Chem. Ed. 1994, 32, 803. (5) Freitag, R.; Bates, T.; Eggert, M. J. Poly. Sci., Polym. Chem. Ed. 1994, 32, 3019. (6) Montaudo, G.; Montaudo, M. S.; Puglisi, C.; Samperi, F. Rapid Commun. Mass Spectrom. 1994, 8, 1011. (7) Montaudo, G.; Montaudo, M. S.; Puglisi, C.; Samperi, F. Rapid Commun. Mass Spectrom. 1995, 9 453. (8) Burger, H. M.; Muller, H.-M.; Seebach, D.; Bornsen, K. O.; Schan, M.; Widmer, H. M. Macromolecules 1993, 26, 4783. (9) Montaudo, G.; Montaudo, M. S.; Puglisi, C.; Samperi, F. Rapid Commun. Mass Spectrom. 1994, 8, 981. (10) Danis, P. O.; Karr, D. E.; Simonsick, W. J.; Wu, D. T. Macromolecules 1995, 28, 1229. (11) Weidner, St.; Kuhn, G.; Just, U. Rapid Commun. Mass Spectrom. 1995, 9, 697. (12) Abate, R.; Ballistreri, A.; Montaudo, G.; Garozzo, D.; Impaliomeni, G.; Critchley, G.; Tanaka, K. Rapid Commun. Mass Spectrom. 1993, 7, 1033. (13) Pasch, H.; Gores, F. Polymer 1995, 36, 1999. (14) Larsen, B. S.; Simonsick, W. J., Jr.; McEwen, C. N. J. Am. Soc. Mass Spectrom. 1996, 7, 287. 0003-2700/96/0368-1303$12.00/0
© 1996 American Chemical Society
as number average (Mn), weight average (Mw), and polydispersity (D), accurate determination of these values is important for characterizing polymers. Mass spectrometry (MS) can provide absolute molecular weights and has the potential to provide molecular weight distributions that are more accurate than those determined by methods such as size exclusion chromatography (SEC), light scattering, and osmometry. However, because synthetic polymers are a mixture of oligomers that differ in molecular weight, end groups, branching, etc., accurate molecular weight analysis requires that both the mass and abundance of each oligomer species be correctly measured over a rather wide mass range. In mass spectrometry, an accurate abundance measurement for polymers requires that the ionization process, ion transmission, and ion detection be independent of mass (or at least a known function of mass) and oligomer structure (end groups, number of repeat units, etc.). Results obtained from different laboratories using MALDI for polymer analysis have provided different conclusions concerning the applicability of MALDI for polymer molecular weight analysis.1,7,8,10 This is not unexpected because MALDI is an ionization method which itself has considerable variation, especially in sample preparations as well as instrumental effects. Significant variation of polymer molecular weight distributions is possible even with time-of-flight mass analyzers because of differences in ion acceleration, ion focusing, ion detection, and so forth. Thus, instrumentation or sample preparation may be the cause of different conclusions concerning polymer molecular weight analyses rather than the MALDI ionization process. If this is the case, it should be possible with an instrument properly designed for polymer analysis to find a matrix and sample preparation that will provide accurate molecular weight results for a given polymer type. Knowing how to evaluate when MALDI provides accurate molecular weight distributions is important. The common method for determining the accuracy of mass spectral molecular weight distributions for synthetic polymers is by comparison with values provided by the manufacturer for polymer standards. These values are frequently obtained using SEC. The polymer standards are often identified by the Mp values. Mp is the mass value for the most probable (abundant) point on the SEC chromatogram. Because the mass of the most abundant peak is easy to determine by mass spectrometry, it is common to compare mass spectral generated Mp values with the manufacturer’s Mp values.1,3,6,7 The Mp values thus become one measure Analytical Chemistry, Vol. 68, No. 8, April 15, 1996 1303
Figure 1. MALDI mass spectrum of poly(methyl methacrylate) 2400 obtained in the reflectron mode with 20 kV postacceleration. The doublet observed for each oligomer species is a function of ion detection.14
of the accuracy of MALDI for polymer analysis. This paper discusses the applicability of comparing Mp values generated by SEC and MS and evaluates the accuracy of MALDI generated Mp values. EXPERIMENTAL SECTION Poly(methyl methacrylate) (PMMA) standards were obtained from Polymer Laboratories (Church Stretton, U.K.). The poly(tetramethylene ethylene glycol) used is a commercial product of Hodogaya Chemical Co. The sample preparation for MALDI involved dissolving the polymers in tetrahydrofuran to make 10100 µmol solutions. 2,5-Dihydroxybenzoic acid (DHB) (Sigma) was dissolved in THF at ∼10 mg/mL. Either the two solutions were mixed 3:1 matrix to polymer and 0.5 µL was spotted on the target or 0.7 µL of matrix solution was dried on the target and 0.4 µL of polymer solution was added to the dried matrix. The polymer solution was prepared in soft glass vials as a means of increasing the sodium ion content of the solution so that [M + Na]+ ions dominate. The MALDI mass spectrometer used for these experiments was a Finnigan MAT Vision 2000 (Hemel Hempstead, England) fitted with discrete dynode electron multipliers for both the linear and reflectron modes. The reflectron mode was operated using 5 kV ion acceleration and 20 kV postacceleration. Spectra obtained in the linear mode were acquired using 30 kV ion acceleration with no postacceleration. Spectra were acquired by summing spectra from ∼100 selected laser shots. A 337 nm nitrogen laser (Laser Science, Inc., Newton, MA) was used with the beam attenuated to just above threshold for polymer ions. The chromatograph and detectors used in these experiments consists of a Waters 150C size exclusion chromatograph (Waters Associates, Milford, MA) and a DAWN DSPF laser photometer (Wyatt Technology, Santa Barbara, CA) equipped with a 5 mW helium neon laser operating at 632.8 nm. The detectors are connected in series with the photometer first after the columns and the refractometer last. Four 300 mm × 7.5 mm i.d. columns packed with Mixed-C5 micron PLGel packing (Polymer Laboratories, Amherst, MA) were 1304
Analytical Chemistry, Vol. 68, No. 8, April 15, 1996
Table 1. Mp Values for Narrow Distribution PMMA Standardsa SEC
Mp (MALDI)
SEC
Mp (MALDI)
1270 2400 3100 3800
1102, 1102b 2100, 2100*c 2700*c 3400
4700 5200 6950 15100
4200*c 4600, 4600d 6500 14750
a Spectra acquired using DHB as matrix. b Acquired in linear mode. Results obtained from ref 7. d Spectrum acquired using indole acrylic acid as matrix.
c
used in tandem. The mobile phase was HPLC grade THF, and the operating temperature was 30 °C. Polymer solutions were prepared at a concentration of 2 mg/mL, and the injected volume was typically 100 µL. Data were collected and analyzed using commercial (TriSEC, Viscotek Corp.) and in-house software. RESULTS AND DISCUSSION Narrow Distribution Case. Figure 1 shows the MALDI mass spectrum for a narrow distribution PMMA standard. The Mp value for the oligomer as determined by MALDI-TOF mass spectrometry operated in the reflectron mode is ∼2100 while that given by the manufacturer of the standard using carefully obtained SEC values is 2400. An Mp value of 2100 was also reported for the same standard using a different manufacturer’s MALDI-TOF instrument operated in the linear mode.7 For narrow distribution PMMA standards, MALDI-MS consistently produces Mp values that are low compared to SEC and has been a point of discussion concerning the accuracy of MALDI results.7 Table 1 compares Mp values provided by the manufacturer for a number of PMMA standards with values obtained by mass spectrometry. Interestingly, even though the mass spectral values are uniformly lower than those determined by SEC, the mass spectral data agree between laboratories even when acquired on different instruments using different solvent and matrix protocols. Furthermore, results from our laboratory14 demonstrate that isolated MMA oligomers
Table 2. Values of the Peak Molecular Weight for the Poisson Molecular Weight Distribution distribution
peak mol wt Mp
P(x) W(x) W(log x)
(v + 1/2) M0 (v + 11/2) M0 (v + 21/2) M0
having molecular weights of about 2500 and 5000 when mixed in equimolar amounts and run by MALDI result in molecular ion ([M + Na+]) peaks of equal area providing the ions impact the detector with sufficient velocity to eliminate detector mass discrimination. Thus, unless an oligomer distribution produces ion suppression that is not observed with a mixture of two oligomers, then MALDI-MS, at least within the mass range studied,14 should provide accurate Mp values. This is especially true for narrow distributions where Mp is determined over a narrow mass range. While the discrepancy between MALDI and SEC Mp values has been cited as an error in the MALDI measurement,7 it can be shown that, even for narrow distributions, a difference between the MALDI and SEC Mp values is expected. The weight fraction or concentration of each species in the distribution is measured by SEC using either a differential refractometer or a UV photometer. A narrow polymer distribution can be simulated by a Poisson distribution. Displaying the data from a Poisson distribution using the normal SEC format (weight-average intensity vs a log mass scale) produces Mp values that are ∼2 repeat unit masses higher than when the same data are displayed using the mass spectral format (number-average intensity vs linear mass scale). Theoretical Description of a Narrow Molecular Weight Distribution: Possion Distribution. A polymer formed by monomer addition without termination, such as in anionic “living” polymerizations, has a number fraction distribution P of molecules of degree of polymerization x described by a Poisson distribution15,16
P(x) )
e-vvx-1 (x - 1)!
(1)
where v is the number of monomers reacted per initiator. The molecular weight is given by x(M0), where M0 is the monomer molecular weight and x is the number of monomers. This function describes the distribution measured in the MS experiment. An alternative description of the distribution is as a weight fraction distribution W given by
W(x) )
1 xe-vvx-1 xn (x - 1)!
(2)
where the factor 1/xn is a normalization factor and xn is the number-average degree of polymerization, xn ) v + 1. In size exclusion chromatography, the separation of the molecules depends on the logarithm of the molecular weight and so the calibration curve of log molecular weight is a linear function of the elution volume. The shape of the weight fraction distribu(15) Flory, P. J. J. Am. Chem. Soc. 1940, 62, 1561. (16) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (17) Yau, W. W.; Kirkland, J. J.; Bly, D. D. Modern Size Exclusion Chromatography; John Wiley and Sons: New York, 1979. (18) Jackson, C.; Yau, W. W. J. Chromatogr. 1993, 645, 209.
Figure 2. Simulated narrow distribution of oligomers for v ) 20.5 (a, top) plotted on a linear and (b, bottom) on a logarithmic scale.
tion, W(log x), from such a separation is given by17,18
W(log x) )
x2 e-vvx-1 xnxw (x - 1)!
(3)
where xw is the weight average degree of polymerization,
xw )
v2 + 3v + 1 v+1
(4)
The degree of polymerization at the peaks of these three distribution, P(x), W(x), and W(log x), can be found by calculation and are listed in Table 2. The peak molecular weight measured by SEC is found to be always 2 monomer units higher than that measured by MS for a polymer with a Poisson molecular weight distribution (v > 3). Parts a and b of Figure 2 are simulated distributions for PMMA (M0 ) 100), for v ) 20.5, demonstrating the difference between MS and SEC data, respectively. In Figure 2a the peak molecular weight is 2100 whereas in Figure 2b it is 2300. In practice, the molecular weight distributions are slightly broader then theoretically predicted due to impurities and a (19) Peebles, L. H. Molecular Weight Distributions in Polymers; Wiley-Interscience: New York, 1971.
Analytical Chemistry, Vol. 68, No. 8, April 15, 1996
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W(x) )
1 xP(x) xn
(7)
The weight-fraction distribution measured by SEC is given by
W(log x) )
1 2 x P(X) xnxw
(8)
The difference between the peak degree of polymerization on the number distribution and that measured by SEC is given by
1 ∆x ) xpeak SEC - xpeak MS ) (xxn + 8xn - 8 - xn) (9) 2
Figure 3. Variance between SEC and MS data for the narrow oligomer distribution.
difference of 3-4 monomer units in molecular weight would be expected. At low molecular weights, this difference in peak molecular weights will be apparent. However, at higher molecular weights, the difference will become insignificant compared to errors in the SEC and MALDI measurements. A similar conclusion can be derived by approximating the Poisson distribution by a normal distribution19 in which the variance of the distribution is determined by the degree of polymerization. This restriction makes the normal distribution a one-parameter distribution like the Poisson distribution. If we set the relationship between the number- and weight-average degrees of polymerization to be the same as for the Poisson distribution then the variance of the distribution is given by
s2 ) v
(5)
and the number-fraction distribution is given by
P(x) )
1
x2πσ
(
exp -
)
(x - xn)2 2σ2
The weight-fraction distribution is given by
(6)
This function is shown in Figure 3. It can be seen that the difference is greater than 1.5 for xn > 8 and that the asymptotic limit is 2, in agreement with the calculated values found for the Poisson distribution. The above discussion demonstrates that, for a narrow distribution (polydispersity is approaching 1), Mp measured using a number-fraction abundance scale and a linear mass scale, as is the case with MS, will be ∼2 repeat units lower in mass (for a homopolymer) than when measured using a weight-fraction abundance scale and a log mass scale, as is the case with SEC. Thus, in the case of the PMMA 2400 standard shown in Figure 1, the comparable Mp value measured by MS and graphed like SEC would be 2300. The 1 repeat unit difference (4% mass error) is well within the accuracy of the measurements. The Mn and Mw values determined for PMMA 2400 by MALDI-MS and SEC differ by 100 Da (1 repeat unit). Therefore, small differences in Mp values are expected for narrow distributions and are a result of how the data are plotted. However, as can be seen from Table 1, Mp values frequently differ by more than a 4% mass error even after accounting for the 2 repeat mass difference. The discrepancy of Mp values is especially large for wide distribution standards. Wide Distribution Case. For wide polydisperse polymers, large discrepancies in Mp as well as polydispersity measurements are observed between MALDI and SEC measurements. Montaudo et al.7 plotted a relationship between polydispersity and the “error” in measuring Mp by MALDI. The error increased with increasing polydispersity. This can also be seen in results from our laboratory.
Figure 4. MALDI mass spectrum of poly(tetramethylene ethylene glycol). 1306
Analytical Chemistry, Vol. 68, No. 8, April 15, 1996
Table 3. Values of the Peak Molecular Weight for the Schulz Molecular Weight Distribution Mp distribution P(x) W(x) W(log x)
peak mol wt Mp
k)1
Mn(k - 1)/k Mn Mw
M0 Mn Mw
k)2 1/
2Mn Mn Mw
Flory most probable distribution. For addition polymerization, p is the probability of propagation steps among the combined total of propagation and termination steps. In general, k ) 1 for termination by disproportionation and k ) 2 for termination by second-order combination. The corresponding weight fraction distribution is given by
W(x) )
(-ln p)k+1xkpx Γ(k + 1)
(11)
and the weight-fraction distribution measured by SEC is given by
W(log x) )
(-ln p)k+2xk+1px Γ(k + 2)
(12)
The number- and weight-average degrees of polymerization are given by
k ln p
(13)
k+1 ln p
(14)
xn ) Figure 5. SEC chromatogram of poly(tetramethylene ethylene glycol) (Figure 4) (a, top) plotted on a logarithmic scale and (b, bottom) number fraction plotted on a linear scale against the square root of the molecular weight.
Figure 4 shows the MALDI distribution measured for a commercial poly(tetramethylene ethylene glycol). The measured Mp by MS is ∼1400 but by SEC is ∼4000 (Figure 5a). However, plotting the SEC data as a number fraction vs a square root mass scale (Figure 5b) gives an Mp value of 1450 (382), in close agreement with the MALDI results. Note the close similarity of the shape of the molecular weight distribution when the mass spectral data (Figure 4) and SEC data (Figure 5b) are plotted in the same manner. Clearly, plotting the mass spectral data as a number-fraction intensity vs a linear (or square root) mass scale has a considerable influence on the measured values of Mp. We suggest that the peak molecular weight, as measured by MALDIMS, be referred to as Mm for modal molecular weight so that it is not confused with Mp as measured by SEC. The theoretical description of Mp values related to wide distributions is given below. Theoretical Description of a Wide Molecular Weight Polymer Distribution. The molecular weight distribution produced by linear condensation polymerization and by many addition polymerizations can be described by the Schulz distribution.19-21
F(x) )
(-ln p)kxk-1px Γ(k)
(10)
For linear condensation polymerization, p is the extent of the reaction and k is equal to 1. In this case, eq 10 reduces to the (20) Schulz, G. V. Z. Phys. Chem. 1939, B43, 25. (21) Zimm, B. H. J. Chem. Phys. 1948, 16, 1099.
xw ) -
As in the case of the Poisson distribution, the molecular weights at the peaks of these distributions are different. In addition, in the case of polydisperse molecular weight distributions, the peak in the MS spectra depends upon the value of k in eq 10. The peak molecular weights are summarized in Table 3. The peak molecular weight measured by SEC is always the weight-average molecular weight. However, measured by MS the peak molecular weight varies from the monomer mass (k ) 1) to half the number-average molecular weight, depending on the type of termination. Parts a and b of Figure 6 show calculated molecular weight distributions with p ) 0.90 and k ) 1 measured by MS and SEC, respectively. In Figure 6a the monomer is clearly the most abundant species by number fraction and in Figure 6b the peak molecular weight is the weight-average value of 1900 g/mol (eq 14). Figure 7 shows calculated molecular weight distributions with the same value of p, 0.90, but with k ) 2. The peak from the MS distribution (Figure 7a) is between 900 and 1000, which is half the number-average molecular weight (eq 14) and the SEC peak (Figure 7b) is at 2850, the weight-average molecular weight. CONCLUSION Comparison of Mp values as measured by MS vs SEC is shown not to be valid. Mp values are a function of how the data are displayed. In SEC, the data are displayed as weight-fraction intensity vs a log mass scale. MS uses a number-fraction abundance axis and a linear mass axis. Changing from a linear to a log mass scale or from weight-fraction to number-fraction intensity will change the mass at which the most probable peak appears. Analytical Chemistry, Vol. 68, No. 8, April 15, 1996
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Figure 6. Theoretical distribution for a wide distribution of oligomers (k ) 1, p ) 0.9): (a, top) simulated mass spectrum plotted on a linear scale and (b, bottom) simulated SEC chromatogram plotted on a logarithmic scale.
For narrow distributions, the difference between Mp values as determined by MS and SEC will be small (∼2 repeat units) but increases with increasing polydispersity. For wide polydisperse samples, Mp, as measured by SEC, represents Mw but as plotted from mass spectral data is dependent on the type of polymerization and can vary from the mononer mass in condensation polymerization to 1/2Mn for addition polymerization. It is therefore recommended that Mp be reserved for weight fraction vs log mass plots. Because the most abundant peak in a number-fraction plot is related to the conditions used for polymerization, it can be an
1308 Analytical Chemistry, Vol. 68, No. 8, April 15, 1996
Figure 7. Theoretical distribution for a wide distribution of oligomers where k ) 2 and p ) 0.9: (a, top) simulated mass spectrum plotted in linear scale and (b, bottom) simulated SEC chromatogram plotted on a logarithmic scale.
important parameter and should have its own identity. We suggest the acronym Mm for modal molecular weight, which happens to be easily obtained by MALDI-MS. Received for review November 7, 1995. Accepted January 26, 1996.X AC951103G X
Abstract published in Advance ACS Abstracts, March 1, 1996.
