Article pubs.acs.org/Macromolecules
On the Origin of the Major Peak Cluster Spacing in the Mass Spectra of Copolymers Michael Petr,† Eugenia Kharlampieva,‡ Donald Cropek,† and Stephen Grimme†,* †
Engineer Research and Development Center, Construction Engineering Research Laboratory, U.S. Army Corps of Engineers, Champaign, Illinois 61826, United States ‡ Department of Chemistry, University of Alabama at Birmingham, Birmingham, Alabama 35294, United States S Supporting Information *
ABSTRACT: Mass spectrometry (MS) is a uniquely informative technique in the characterization of copolymers, where spectra prominently feature peak clustering. The spacing of these clusters, in general, is dominated by the spacing of one repeat unit, and contained herein is the theory to explain this observation. Extension of this theory also explains the more subtle observation that, even though the spacing is generally that of one unit, occasionally, the spacing between the maxima of adjacent clusters shifts by that of the other unit. Furthermore, the theory predicts that, in the low molecular weight region of the spectrum, there is a total switch to the spacing of the other unit along with asymmetric peak clusters that have a “sawtooth” shape. The analysis uses the Gaussian, log−normal, and Schulz−Zimm models as well as the random coupling hypothesis to explicitly demonstrate that (1) the major peak cluster spacing naturally arises from the unit in the copolymer with the widest distribution, as measured by the scaled standard deviation, (2) the spacing shift naturally occurs due to the marginal probabilities away from the spectrum maximum, and (3) the low molecular weight switch is a natural consequence of the tail of the distribution of the unit with the widest distribution. Results are provided to predict which unit in the copolymer will govern the major peak cluster spacing, how often the spacing will shift to that of another unit in the middle and high molecular weight regions of the spectrum, the molecular weight and composition of the maximum peak in every cluster, and the molecular weight below which the spacing will be that of the another unit. We believe that our results are the first to provide tangible theory to explain the previously unknown origins of these empirically observed phenomena. predicted by Tobita,18,19 confirm the random coupling hypothesis,7−9 measure the extent of functionalization,20,21 measure the relative activity of catalysts,22 and identify the constituent amino acids or nucleic acids in biopolymers.23 MS is a crucial tool to characterize the variety of copolymers in all of these scenarios because it is a direct measure of molecular weight and the MWD, unlike most other techniques which suffer from the error associated with measuring another property correlated with molecular weight. For example, copolymer molecular weight measurements with GPC have intrinsic error because GPC measures hydrodynamic radius, which is correlated with molecular weight using homopolymer standards; however, there can be error in the final molecular weight determination because interactions between the solvent and each unit in a copolymer and the homopolymer standards can be different and, thus, influence the apparent molecular weight. On the other hand, with the improved copolymer characterization capability of MS comes the difficulty that the spectra are more complicated than those of homopolymers because they consist of the discrete superposition
1. INTRODUCTION Mass spectrometry (MS) has become an important method in the characterization of polymers of all types.1,2 Unlike other methods which observe some other property correlated to molecular weight, such as gel permeation chromatography (GPC), laser light scattering (LLS), osmometry, and viscosimetry, MS is used to observe the exact molecular weights and occurrence frequencies of individual polymer chains. This quality is critical for polymer characterization because it provides a large amount of information, including the polymer’s identity, molecular weight, molecular weight distribution (MWD), and end groups. Traditionally, soft ionization techniques, such as matrix-assisted laser desorption/ionization (MALDI), electrospray ionization (ESI), and secondary ion mass spectrometry (SIMS), that do not fragment the polymers are used because the resulting spectra are direct observations of the MWDs.3 Furthermore, MS is becoming an especially useful method in the characterization of copolymers because of the unique information it provides.4 In particular, MS has been used to determine the MWDs of individual blocks,5−9 determine the exact composition and occurrence frequency of each set of copolymer chains with the same molecular weight,7−15 observe the chemical heterogeneity15−17 and compositional distribution © XXXX American Chemical Society
Received: March 18, 2013 Revised: July 9, 2013
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cluster spacing in the mass spectra of copolymers. Our analysis demonstrates that the major peak cluster spacing naturally arises from the repeat unit in the copolymer with the widest distribution, as measured by the scaled standard deviation. Next, we demonstrate that there is a shift in the spacing at regular intervals due to marginal probabilities away from the spectrum maximum. Finally, our analysis predicts an outright shift in the low molecular weight region of the spectrum to the spacing of the unit with the narrower distribution along with a shift to asymmetric, “sawtooth” shaped peak clusters, both due to the tail of the MWD. We believe our investigation is significant because, despite the observation of some of these particular features both experimentally5−15,17,20−22,27 and numerically,24−26 until now, no explanation has been provided for their origins. Furthermore, the results of this analysis can be applied broadly to copolymers and provide simple expressions to examine mass spectra of copolymers of any identity, composition, length, distribution, or architecture. Most importantly, this analysis provides physical insight and fundamental theory to explain the various features of the major peak cluster spacing over the entire molecular weight ranges of copolymer mass spectra.
of the MWDs of multiple units. Nonetheless, if sufficiently analyzed, MS can be a powerful tool to characterize copolymers, both analytically7 or in conjunction with more sophisticated simulations.24−26 The most prominent feature in the mass spectrum of a copolymer is that the peaks corresponding to individual copolymer chains are grouped into clusters.5−15,17,20−22,27 For simplicity, the following discussion describes a two-component copolymer; however, the ideas and definitions extend to copolymers of more than two components, as will be demonstrated later. Therefore, in a copolymer of A and B units, each cluster consists of copolymer chains all with the same degree of polymerization but with different compositions of A and B. Each cluster has a low intensity tail at low molecular weight, a low intensity tail at high molecular weight, and a maximum intensity in between, and, because the domains of the clusters span the entire mass spectrum, the tails of the clusters overlap. In addition to peak clusters, there is a characteristic spacing between the clusters themselves, defined as the molecular weight between the maximum intensity peaks in adjacent clusters and referred to herein as the “major peak cluster spacing,” and finally, there is a characteristic spacing between the individual peaks within a given cluster, referred to herein as the “minor peak cluster spacing.” All of these featuresthe clusters, the major spacing, the minor spacing, and the overlap points are illustrated in Figure 1, which is a simulation of the mass
2. MODEL DEVELOPMENT To determine the origin of the major peak cluster spacing, a general copolymer of A and B units is assumed, and the MWD of each unit is represented by a series of common, two-parameter models: Gaussian, log−normal, and Schulz−Zimm.28 Presented first is the full analysis using the Gaussian model, which produces the simplest and most straightforward results, and presented second are the results for the other two models, the analysis for which is in the Supporting Information. The MWD of the copolymer is represented in eq 1 by P, which is the probability distribution function for the entire copolymer. Also, eq 1 is the mathematical statement of the random coupling hypothesis,7−9 which is the assumption that the A and B distributions are independent. Substitution of the Gaussian model28 for each MWD into eq 1 produces eq 2, which condenses to eq 3. Thus, eq 3 is the final form of the probability distribution function, which corresponds to the intensity in a mass spectrum: P(nA , nB) = P(nA )P(nB)
P(nA , nB) =
Figure 1. MS simulation using a Gaussian model of an example polystyrene-poly(α-methylstyrene) (A-B) copolymer with MWD parameters of nA = nB = 20, PDIA =1.5, and PDIB = 1.05. The clusters are labeled and separated with dotted lines to mark the points at which the cluster tails overlap, and the major and minor peak cluster spacings have been marked at the top.
1 σA ×
(1)
2⎤ ⎡ 1 ⎛ n − nA̅ ⎞ ⎥ exp⎢ − ⎜ A ⎟ ⎢⎣ 2 ⎝ σA ⎠ ⎥⎦ 2π
1 σB
2⎤ ⎡ 1 ⎛ nB − nB̅ ⎞ ⎥ ⎢ exp − ⎜ ⎟ ⎢⎣ 2 ⎝ σB ⎠ ⎥⎦ 2π
(2)
P(nA , nB)
spectrum of an example polystyrene−poly(α-methylstyrene) copolymer. As for the significance of these features, the minor peak cluster spacing is a consequence of the difference in the molecular weight of the A and B repeat units,7 and the major peak cluster spacing is usually that of the A or the B repeat unit. This observation leads to the questions: what is the origin of the major peak cluster spacing in copolymers, is it uniformly that of the A unit or that of the B unit, under what conditions is it the spacing of the A unit, and under what conditions is it the spacing of the B unit? We report on a novel but simple approach to MS peak analysis using three standard polymer models to explain the major peak
=
2 ⎡ ⎛ ⎛ n − nB ⎞2 ⎞⎤ 1 1 ⎛ n − nA̅ ⎞ ̅ ⎟ ⎟⎥ exp⎢ − ⎜⎜⎜ A ⎟ +⎜ B ⎢ 2 ⎝ σA ⎠ 2πσAσB σ ⎝ ⎠ ⎟⎠⎥⎦ B ⎝ ⎣
(3)
where P is the probability of an individual polymer chain with nA units of A and nB units of B, nA̅ and n̅B are the number-average units or the degrees of polymerization of A and B, respectively, and σA and σB are the standard deviations of the A and B MWDs, respectively, defined by eqs 4 and 5:28,29 σA = nA̅ PDIA − 1 B
(4)
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σB = nB̅ PDIB − 1
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predicted with the model is the outright switch to the repeat of the unit with the narrower MWD along with a switch to asymmetric, “sawtooth” shaped peak clusters in the low molecular weight region of the spectrum. Major Peak Cluster Spacing. To determine the origin of the major peak cluster spacing in general, as observed in the MS data of Crecelius et al.,5 Wilczek-Vera et al.,7−9 Cox et al.,10 Montaudo et al.,13 Willemse et al.,15 and Sevaty et al.,17 the relative influence of the A MWD is compared to that of the B MWD using the developed probability distribution function. This can be done by defining the probability ratio shown in eq 7:
(5)
where PDIA and PDIB are the polydispersity indices of the MWDs of A and B, respectively. Although these probability distributions are continuous, the actual copolymers have discrete molecular weights because the indices nA and nB can only take positive integer values. Accordingly, the molecular weight of an individual copolymer chain is defined by eq 6: MW = nA MA + nBMB
(6)
where MW is the molecular weight of the copolymer and MA and MB are the molecular weights of the A and B repeat units, respectively. This model for the molecular weight distribution is quite general because it assumes only that the MWDs of the polymers are independent and that they can be represented with a Gaussian distribution (or log−normal or Schulz−Zimm distributions in the Supporting Information). Therefore, this analysis is broadly relevant to a copolymer of any identity, composition, length, distribution, or architecture that fits these two basic assumptions.
P(̃ kA , kB) ≡
P(nA̅ + kA , nB̅ ) P(nA̅ , nB̅ + kB)
(7)
where P̃ is the ratio of the probability of the average plus a scale factor kA units of A to the probability of the average plus a scale factor kB units of B. Both kA and kB are small (usually k ≤ 3), positive integers that represent a small number of repeat units, and they arise because MA and MB are not necessarily close but can instead have approximate ratios that are simple rational numbers other than 1, such as 1.5, 2, or 3. Subsequently, the scale factors kA and kB are determined by minimizing the minor peak cluster spacing within the resolution of the spectrum, as illustrated by eq 8, while maximizing the number of peaks in each cluster, as illustrated in eq 9:
3. RESULTS AND DISCUSSION Once the model for the copolymer is described, it can be used to determine the origin of all the features of the major peak cluster spacing. Figure 2 is the full spectrum of a simulation of the mass
δ ≡ |kAMA − kBMB| > δmin
(8)
⎧ MA , A spacing ⎪ ⎪ δkB N=⎨ ⎪ MB ⎪ δk , B spacing ⎩ A
(9)
where δ is the minor peak cluster spacing, δmin is the minimum observable spacing determined by the sample and instrument resolution, and N is the number of minor peak spacings between major peak spacings or the number of individual peaks (excluding the overlapping tails) in a peak cluster. It should be noted that both expressions for A and B unit spacing in eq 9 are analogous and nominally produce the same value for the number of peaks in a cluster; thus, eqs 8 and 9 determine the same scale factors for a given MA and MB, regardless of whether there is A or B unit spacing. More importantly, it should be emphasized that the minor peak spacing is not the absolute calculable minimum of eq 8 alone. Instead, it is the minimum of eq 8 that is above the minimum observable spacing, where the scale factors are integers and where the number of peaks in a clusterdetermined with eq 9is maximized. For example, if the A and B polymers are polystyrene (104 g/mol) and poly(α-methylstyrene) (118 g/mol), then kA and kB are both 1.7,8 If, however, they are polyisobutylene (56 g/mol) and poly(α-methylstyrene) (118 g/mol), then kA is 2 and kB is 1,10 and, in an extreme example, if they are polystyrene (104 g/mol) and polyisoprene (68 g/mol), then kA is 2 and kB is 3.15 Of course, if the terms in the expression in eq 8 are exactly equal within the resolution of the sample and the MS instrument (say within a few g/mol or less), such as with polyethylene (28 g/mol) and poly(carbon monoxide) (28 g/mol),30 then δ is 0, there will be no minor peak spacing, and there will be no peak clusters because all the peaks in a given potential cluster will overlap. To establish an explicit form for the probability ratio, substitution of the probability expressions from eq 3 into eq 7
Figure 2. Simulation using a Gaussian model of the full mass spectrum of an example polystyrene−poly(α-methylstyrene) (A-B) copolymer with MWD parameters of nA = nB = 20, PDIA =1.5, and PDIB = 1.05. Each perceived peak is actually a peak cluster, and the various molecular weight ranges and features are marked, including the medium and high molecular weight region, the locations of the spacing shifts, and the low molecular weight region.
spectrum of an example polystyrene−poly(α-methylstyrene) copolymer. Because Figure 2 shows such a wide range of molecular weights, individual peaks are not visible; instead, each perceived peak is actually a peak cluster. More importantly, in Figure 2, all the relevant regions and features are identified. In particular, peak clusters in the mass spectrum of a copolymer normally have the spacing of the unit with the wider MWD in the medium and high molecular weight regions of the spectrum, but, at regular cluster intervals, this spacing shifts by that of the unit with the narrower MWD. Furthermore, the maximum peak in any given cluster in the medium and high molecular weight regions is determined with a simple expression. The last feature C
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produces eq 10, which reduces to the final form of the probability ratio in eq 11:
P(̃ kA , kB) =
1 2πσAσB 1 2πσAσB
⎡ 1⎛ exp⎢ − 2 ⎜ ⎣ ⎝ ⎡ 1⎛ exp⎢ − 2 ⎜ ⎣ ⎝
(
nA̅ + kA − nA̅ σA
(
nA̅ − nA̅ σA
2
2
) +(
) +(
copolymer meet the condition in eq 16 (i.e., the B unit has a wider MWD), then the major peak cluster spacing will be that of the B unit. However, when the properties of the copolymer meet or nearly meet the condition in eq 15, then there will be no clear major peak cluster spacing. Instead, the spacing between peak clusters will be that of both the A and the B units, which manifests itself as an average of the molecular weights of the A and B repeat units, if the mass resolution is low, or as an alternation in the spacing between peak cluster maxima between that of the A unit and that of the B unit, if the resolution is high. These various scenarios are illustrated in Figure 3, which is a series of MS simulations using the Gaussian model for an example polystyrene− poly(α-methylstyrene) (A−B) copolymer,7,8 where kA and kB are both 1 as mentioned above. Figure 3a is the full spectrum with a base set of parameters, where the MWD for both A and B are identical, Figure 3b is zoomed in around the center of the spectrum, and Figure 3c-e contain spectra with perturbations from the base set of parameters. In Figure 3b, P̃ is 1, so the major peak cluster spacing is that of both the A and the B units. In Figure 3c and Figure 3d, P̃ is greater than 1 by increasing nA̅ and PDIA, respectively, so the major peak cluster spacing is that of the A unit. In Figure 3e and Figure 3f, P̃ is less than 1 by increasing nB̅ and PDIB, respectively, so the major peak cluster spacing is that of the B unit. To illustrate the effect of the scale factors, Figure 4 is another MS simulation but for an example poly(isobutylene)− poly(α-methylstyrene) (A−B) copolymer. Even though both units have the same number averages, polydispersities, and standard deviations, the spacing is that of the B unit because kA is 2 and kB is 1, as mentioned above.10 Also in Figure 4, there appear to be intermediate peak clusters with the A unit spacing. In actuality, there are two series of peak clusters as a consequence of the fact that the overall spectrum has the spacing of the larger unit, as predicted by eq 16. Both series have the spacing of the B unit, but one has an odd number of A units, and the other has an even number of A units. Therefore, these intermediate peaks comprise one of the series, and the two series are offset by the A unit spacing. Similar analyses using the log−normal and Schulz−Zimm models28 are performed in the Supporting Information, and the results of these analyses are shown in eqs 17 and 18, respectively:
2 ⎞⎤
nB̅ − nB̅ σB
) ⎟⎠⎥⎦
nB̅ + kB − nB̅ σB
) ⎟⎠⎥⎦
2 ⎞⎤
(10)
⎡ ⎛⎛ ⎞ 2 ⎛ ⎞ 2 ⎞⎤ k 1 k P(̃ kA , kB) = exp⎢ ⎜⎜⎜ B ⎟ − ⎜ A ⎟ ⎟⎟⎥ ⎢ 2 ⎝ σB ⎠ ⎝ σA ⎠ ⎠⎥⎦ ⎣ ⎝
(11)
The simplified form of the probability ratio in eq 11 can be used to determine the major peak cluster spacing by comparing P̃ to 1, as in eq 12. If P̃ is greater than 1, then A spacing dominates because the probability of having different numbers of A units is higher than the probability of having different numbers of B units. If P̃ is less than 1, then B spacing dominates because the probability of having different numbers of B units is higher than the probability of having different numbers of A units. To solve eq 12, the natural log is taken, and then the terms are rearranged to give the final result in eq 13: ⎡ ⎛⎛ ⎞ 2 ⎛ ⎞ 2 ⎞⎤ k 1 k P(̃ kA , kB) = exp⎢ ⎜⎜⎜ B ⎟ − ⎜ A ⎟ ⎟⎟⎥ {>, = , , = , , =, 1;
nB̅ PDIB − 1 nA̅ PDIA − 1 σ σA > B; > kA kB kA kB
P ̃ {> , = , , = , , = , , = , , =,
