Data Analysis Methods for Synthetic Polymer Mass Spectrometry: Autocorrelation.

Autocorrelation is shown to be useful in describing the periodic patterns found in high- resolution mass spectra of synthetic polymers. Examples of this usefulness are described for a simple linear homopolymer to demonstrate the method fundamentals, a condensation polymer to demonstrate its utility in understanding complex spectra with multiple repeating patterns on different mass scales, and a condensation copolymer to demonstrate how it can elegantly and efficiently reveal unexpected phenomena. It is shown that using autocorrelation to determine where the signal devolves into noise can be useful in determining molecular mass distributions of synthetic polymers, a primary focus of the NIST synthetic polymer mass spectrometry effort. The appendices describe some of the effects of transformation from time to mass space when time-of-flight mass separation is used, as well as the effects of non-trivial baselines on the autocorrelation function.

1. Introduction

The advent of rapid, high-resolution, broadmass-range mass spectrometry has revolutionized synthetic polymer single-chain characterization [1]. Along with this new measurement technology has come a flood of high-quality mass spectral data of an exceedingly complex nature. It is not unusual for synthetic polymer mass spectra to contain hundreds of separate peaks even when excluding those simply derived from naturally-occurring isotope distributions. Automated data analysis methods are needed in order to make full and timely use of the data.

Time series analysis, which first came to fore with the publication of Norbert Wiener’s seminal text Extrapolation, Interpolation, and Smoothing of Stationary Time Series with Engineering Applications [2] in 1949, has proved invaluable in many fields of data analysis. Weiner’s text represents the first complete exposition of the study of operations on time series, including autocorrelation and cross-correlation. In the intervening years these correlation methods have been applied to many types of mass spectral data for many purposes [3-5]. Owens has reviewed the use of correlation functions in mass spectroscopy, in particular, the use of autocorrelation and cross-correlation as applied to ion fragments in order to identify small organic molecules in standard libraries [6]. Hercules and coworkers have used autocorrelation of isotope distributions as a method to optimize automated data collection [7]. Here we discuss the application to synthetic polymer mass spectra for the purpose of efficiently extracting information from complex data.

First we define the mass autocorrelation and show how to treat the data properly for its use. Then we present autocorrelation for a spectrum of a simple polyethylene oxide homopolymer to establish the fundamentals. Following that we present data on two more complicated structures, specifically two silsesquioxanes produced by condensation polymerization [8] in which the mass spectra can be related directly to the polymer architecture. Finally, we apply autocorrelation to the issue of quantitation in polymer mass spectrometry using the example of polybutadiene.

2. The Mass Autocorrelation Function

We define the mass autocorrelation function as where S(m) is signal at mass m taken on equal intervals of mass, δm. Equal intervals of mass are used because most correlation algorithms, and the closely related field of fast Fourier transforms (FFT), require the signal to be evenly spaced points on the scale of interest.

Time-of-flight (TOF) mass separation [9] is the technique most often applied to synthetic polymers due to their high molecular masses, typically in excess of 1000 u and often much greater (into the 100 000 u range and beyond). No other mass separation technique can reach such high masses. The TOF signal, s(t), is collected on equal intervals of time. The transformation from this time-base signal s(t) to a mass-base signal S(m) involves both an interpolation and a change of the signal itself by a Jacobean transform. The mathematics to affect this transformation is discussed in Appendix A.

3. Example 1: A Simple Linear Homopolymer

The most obvious use of mass autocorrelation function is to get an accurate representation of the repeat unit of the polymer. This can be difficult in a spectrum with noise where identification of peak position will inevitably lack precision and lead to inaccuracies in calculating the repeat unit mass. Figure 1 shows the mass spectrum for a low-molecular-mass polyethylene oxide (repeat unit: [–CH2–CH2–O–]); while Fig. 2 is its autocorrelation function with different values of δm. Data were obtained by matrix-assisted laser desorption/ionization (MALDI) TOF mass spectrometry [10, 11]. Before autocorrelation a baseline was pulled off the data in time space and the data was subsequently transformed from time space to mass space by the partial integration method described in Appendix A. The autocorrelation clearly shows the 44.03 u repeat unit of polyethylene oxide with a precision difficult to match by simply picking adjacent peaks and calculating a mass difference.

Fig. 1

Matrix-assisted laser desorption/ionization time-of-flight mass spectrum from a polyethylene oxide of a molecular mass centered around 1440 u.

Fig. 2

Mass autocorrelation of the data in Fig. 1. The effect of various coarse graining parameters on the representation of the data is seen. Notice as δm increases above 0.5 u the isotope resolution is lost.

Now consider the effect of varying the δm for partial integration or interpolation. The spectrum and its autocorrelation function with δm chosen to be from 0.1 u to 2.0 u are also shown in Fig. 2. It is clear we get a varying representation of the repeat unit and its isotope effect depending on the choice of δm. By increasing δm, that is, by integrating over a wider window of the data for each point, we obtain less sensitivity to the isotopes, that is, a greater smoothing effect on the data but less accuracy in peak position.

4. Example 2: A Complex Homopolymer

Polysilsesquioxanes are three-dimensional polymers with a tri-functional repeat unit of the form [RSiO3/2] where each silicon atom is coordinated with three oxygen atoms. They are most often produced by a low temperature sol-gel hydrolysis-condensation reaction from silicon alkoxides [12]. One important unknown in the processing of silsesquioxanes is the “degree-of-condensation” as a function of molecular mass. That is, how many of the silicon atoms are three-fold coordinated with bridging oxygen atoms and how many have terminal silanol (≡SiOH) groups?

The mass spectrum of methacrylpropyl silsesquioxane (R = (CH2)3–O–CO–CCH2–CH3) is seen in Fig. 3 [13, 14]. Each major cluster of peaks corresponds to a single oligomer with a given number of repeat units n. Since the monomer contains one silicon atom the value of n also corresponds to the number of silicon atoms in that oligomer. For this material this average mass of the basic repeat unit is 188.25 u. (The average is taken over all isotopes of each element present using their natural abundances.) This is the value of the mass difference between groups of peaks seen in Fig. 3. Knowing that ionization occurs via the attachment of adventitious Na+, and by including the mass of the two O1/2 H end groups, an exact identification of each oligomer present in the sample can be made.

Fig. 3

The full mass spectrum of the methacrylpropyl silsesquioxane showing the characteristic shape of a condensation polymer. Estimated standard uncertainty (Type A) of the peak position from calibration and repeatability studies is 0.2 u, and the estimated standard uncertainty in overall signal intensity from repeatability studies is 15 %.

Figure 4 shows the detail of a single low-mass oligomer from Fig. 3. The maximum possible mass of an oligomer with n repeat units occurs when every silicon atom has one silanol group in addition to one R-group and two bridging oxygen atoms. Two bridging oxygen atoms are the minimum number necessary for the formation of a polymer, that is, conceptually polymerization requires difunctionality at a minimum. Thus, the repeat unit in this case can be given as [RSi(O1/2)2OH]. For an oligomer with n repeat units the mass of the heaviest oligomer is n times the mass of this “difunctional” oligomer (plus the mass of the Na+ ion and the end groups). This heaviest oligomer is the linear or branched structure. However, the highest intensity peak generally does not appear at the maximum possible mass. Instead, lower mass peaks are more intense. These peaks correspond to the loss of water as pairs of Si-OH groups react. This in turn immediately indicates that intramolecular reactions are occurring during polymerization. If intermolecular reactions were occurring the value of n would change and a new, higher mass, oligomer would be formed. In Fig. 4, n = 10 and the number of closed loops t is given across the top of the figure. The value of t ranges from 0 to 6 with 3 being the most likely value. Note that each peak is separated by 18 u indicating the loss of water.

Fig. 4

Detail around a single oligomer of the methacrylpropyl silsesquioxane from Fig. 3 for n=10. Across the top of the figure is given the number of closed loops t indicated by the loss of water (18u). The maximum value for t was 3 with the lowest t being 0 and the highest being 6.

For the condensation polymer derived from the silsesquioxane monomer considered here, the mass m of the linear oligomer having n repeat groups is given in units of u by the equation: where n is the number of repeat groups whose mass is 188.25 u, p is the mass of the cation (either 23 u for sodium, or 39 u for potassium), and 18 u is for the two O1/2H end groups. It is easy to show that either a strictly linear or a branched-linear polymer, which does not have one of the branches forming a closed loop with the oligomer itself, follows the above formula for mass. This formula would explain a single peak for each oligomer but cannot explain the major clusters that were observed and ascribed to intramolecular ring formation.

This suggests a modified version of Eq. (2) that includes intramolecular closed loop formation: where again n is the number of repeat units, p is the mass of the cation, t is equal to the number of closed loops in the molecule (i.e., the number of lost water molecules), and 18 u in the last term is for the added end groups.

Applying these concepts to the full mass spectrum, Fig. 5 gives the number of closed loops t per oligomer with n repeat units, that is t vs n. The solid circles give the number of closed loops for the most intense minor cluster of each major cluster. (Recall that a major cluster corresponds to an oligomer with n repeat units.) The points marked with an x are for the least intense peaks observed in each major cluster, that is, the weakest peaks found before the baseline noise overtakes the signal. The regression fit of the solid circles given by the solid line in Fig. 5 has a slope of 0.273 with a standard uncertainty of 0.006, an intercept of 0.226 with a standard uncertainty of 0.192, and a correlation coefficient of 0.998. (The “standard uncertainty” is the estimated standard deviation of the fitted parameter.) The first observation is that the ratio of t/n remains roughly constant for all n with a value of about 1/4. This suggests that the molecule is no more or less likely to interact with itself based solely on its size. Stated another way, the molecule may be fractal-like with its closed-ring topology independent of molecular size [15]. A fully-condensed polyhedral structure with an even number of repeat units will follow the equation t = 1/2n + 1, while for an odd number of repeat units the governing equation is t = 1/2(n – 1) + 1. This is shown as a dashed line in Fig. 5 on the other hand, a branched linear chain with no closed loops will have t = 0 (by definition), and thus t/n = 0 which is merely the abscissa of the graph. Therefore, in general it appears as if the specific silsesquioxane studied has on the average an assortment of closed loops and linear branches in each molecule. No fully-condensed polyhedra were observed except at very low mass (n<10) because the experimentally-observed t/n ratio was on the order of 1/2 well below the fully-condensed-polyhedron value of (for large n). The analysis of this data requires analysis of each peak and identifying it with each species. This can be very laborious if one wishes to screen a large number of compounds.

Fig. 5

plot of the number of intramolecular closed loops, t, versus the number of repeat units in a given oligomer, n. The solid circles represent the maximum intensity peak for each oligomer, and points marked with an x give the maximum and minimum number of observed loops. The solid line is a linear regression fit to the solid circles, while the dashed line is the expected value t = 1/2n + 1 (for n even) for the fully-condensed polyhedral structure. The sample showed an intermediate behavior between a branched linear structure and a fully-condensed structure.

The mass autocorrelation function was applied to the data in Fig. 3 with the lag, L, in the range from (0 to 1000) u and with δm = 1 u and is shown in Fig. 6. It largely replicates the original mass spectrum without much of the baseline noise. In this way it can be roughly thought of as a kind of “averaging.” The peaks at 188.25 u are for correlations of Δn = 1, those at 376.5 u are for Δn = 2, etc. Figure 7 is the low mass region of the autocorrelation function expanded. There are a series of five low mass peaks, marked with stars in the figure, starting at 18 u and each 18 u apart. This indicates that the number of closed loops per oligomer should be about five, that is, there should be five peaks in each major cluster. Recall that this was shown in Fig. 4 where the difference for each oligomer between the maximum and minimum number of closed loops observed, t, is about five. Likewise, in Fig. 7 the number of peaks in the autocorrelation function around mass 188.25 u should be about 10, marked with the symbol x in the figure, that is, correlations of the five peaks of two adjacent major clusters. Lastly, since the spacing used in this autocorrelation function is 1 u, the isotopic resolution that should be apparent at 1 u is not seen, instead autocorrelation within the minor peaks is simply smeared out.

Fig. 6

The full autocorrelation function with a lag, L, from (1 to1000) u for the mass spectrum shown in Fig. 3. The autocorrelation coefficient is plotted versus L in units of u.

Fig. 7

Low-mass-region detail of autocorrelation function shown in Fig. 6. The stars show the 5 peaks shifted by 18 u found in each major cluster. The positions marked with an x are the 10 peaks found by correlations between major clusters 188.25 u apart.

Figure 8 shows the autocorrelation function centered at 941.25 u (corresponding to correlations over five repeat unit masses, i.e., 5×188.25 u = 941.25 u) superimposed over the autocorrelation function at 188.25 u. This was done simply by subtracting 753.0 u = 4 × 188.25 u from the autocorrelation function centered at 941.25 u. Notice that the maximum peak for 941.25 u group is 18 u to the left of the maximum peak for the 188.25 u group. This indicates that as four repeat units are added to an oligomer (n = n + 4) one added closed loop is formed per molecule on average (t = t + 1). Once again this can be seen from the slope of the line in Fig. 5: for each step of n equal to four, t is increased by about one. (Strictly, since experimentally t/n = 0.273, an increase in n of 4 should yield an increase in t of 1.1. This is hinted at in the peak heights of Fig. 8). This provides another view that the molecule is self-affine in that adding additional repeat groups changes proportionately the number of closed loops. In contrast, a strictly linear polymer undergoing a random walk crosses itself in proportion to the square root of the number of repeat units, i.e., . This behavior is clearly not seen in this material. Each of these trends is revealed rapidly by mass autocorrelation and would not be as readily apparent in a peak-to-peak indexing of the data.

Fig. 8

Shift of the five-repeat-unit correlation function (dashed line) onto the single-repeat-unit correlation function (solid line) showing the 18 u offset between the maximum peak for each group.

5. Example 3: A Copolymer

MALDI-TOF mass spectrometry was performed on a low molecular mass fraction of a copolymer of methyl silsesquioxane (repeat unit: [CH3SiO3/2]) and dimethysiloxane (repeat unit: [(CH3)2SiO]) monomers (Dow-Corning Metflex™)1. This fraction had a nominal mass of 3400 u by size-exclusion chromatography.

Figure 9 shows the full spectrum of sample while Fig. 10 shows a detailed region of this spectrum highlighting individual oligomers. The traditional way to analyze this data is to take knowledge of the mass of the two monomers along with the polymerization reaction involved and assign each individual peak in the spectrum to a particular composition, typically several hundred peaks for a condensation-hydrolysis resin such as this. Although this may be the most thorough method to analyze the data it requires very high precision data and may not reveal significant trends in the data. Typically it is discovery of these trends and not accounting of each peak in the spectrum that is desired, especially in production quality control situations.

Fig. 9

The full mass spectrum of the methylsilsesquioxane/dimethylsiloxane copolymer showing the characteristic shape of a condensation polymer. Estimated standard uncertainty (Type A) of the peak position from calibration and repeatability studies is 0.2 u, and the estimated standard uncertainty in overall signal intensity from repeatability studies is 15 %.

Fig. 10

Detail of the mass spectrum shown in Fig. 9 showing the complexity of the signal.

Figure 11 shows the autocorrelation of the data in Fig. 9. Peaks appear in the autocorrelation at each of the repeat distances of the main spectrum. There is a large peak at 74 u indicative of the dimethyl siloxane unit (D). That is to say that there frequently occurs pairs of oligomer separated in mass by 74 u, i.e., that the higher mass oligomer has grown by one D unit. Interestingly there is no peak at 67 u, which is the mass of the methyl silsesquioxane unit (T). However, there is a peak at 134 u that is twice the mass the silsesquioxane unit (2T). This immediately indicates that each oligomer present has an even number of T units. (Actually, to show this you also need to observe that there are also peaks at 4T, 6T, 8T, etc., but not at 3T, 5T, 7T, etc.) Each of the other peaks in the autocorrelation can be shown to be linear differences of 2T and D units forming the general function n2T–mD. Table 1 shows some of these combinations at lower mass. Notice that for every combination there is a peak in the autocorrelation and there are no peaks in the autocorrelation that are not in Table 1. Since the interpolation was done at 1 u intervals there are uncertainties of about 1 u between the table and Fig. 11.

Fig. 11

Mass autocorrelation of the data in Fig. 9. Labels indicate that the silsesquioxane repeat unit only appears as a dimer (134 u) and not as a monomer (67 u) while the siloxane repeat unit does appear as a monomer (74 u).

Table 1

Identification of peaks in the mass autocorrelation of the copolymer resin using |n2T−mD|

m	n
	0	1	2	3	4
0	0 u	134 u	268 u	402 u	536 u
1	74 u	60 u	194 u	328 u	463 u
2	148 u	14 u	120 u	254 u	389 u
3	222 u	88 u	46 u	180 u	315 u
4	296 u	162 u	28 u	106 u	240 u

The next observation to be made is that there is no peak at 18 u in the autocorrelation. The hydrolysis-condensation reaction gives off water when two silanols combine to form a bridging oxygen. As discussed previously, in incompletely-condensed silsesquioxanes a strong autocorrelation peak is seen at 18 u indicative of oligomers with the same number of repeat units but different degrees of condensation. The lack of a peak at 18 u immediately indicates that either full intra molecular condensation of silanols has taken place, or no intra molecular condensation of silanols has taken place. Only an exact indexing of peaks in the mass spectrum (which can be quite time consuming) can answer this question, however, it seems unlikely that condensation can occur to polymerize the material (intermolecular condensation) with some concomitant intramolecular condensation also occurring [16]. Additionally, an even number of T units is a strong indication of complete condensation since an odd number of T units would always leave at least one silanol in the material leading to further condensation reactions.

6. Autocorrelation in Signal-to-Noise Determinations

Up to now the autocorrelation function has been applied over the whole range of the polymer spectrum to understand polymer structure. However, in addition to polymer structure it is also often used to calculate moments of the molecular mass distribution (see Appendix C). To do so it is important to find the low-intensity oligomer peaks at the extrema of the molecular mass distribution. To accomplish this consider the use of the autocorrelation over only a part of the polymer spectrum. (This is not the “partial” autocorrelation function often discussed in time series analysis.) This “windowed” autocorrelation, analogous to a windowed FFT, is useful to determine where the signal has returned to baseline, that is, where does the signal devolve into the noise. This is crucial in the calculation of molecular mass distributions (MMD) from mass spectral data as the low and high mass oligomers at the extremes of the distribution have a disproportionate effect on the calculation. Since the thrust of the NIST polymer mass spectrometry effort is to make such determinations of MMD from mass spectral data it is of primary importance to us.

Figure 12 shows such a situation for polybutadiene (PBD, repeat unit: [–CH2–CH = CH–CH2–]). We propose to use the autocorrelation function to tell us more about where there is no signal in the noise. Let us say we use an integration window of a width 8 to 10 times the mass of the repeat unit and a maximum lag one half of the window length. Then we can move the integration window with increasing initial masses, mi, to higher and higher values. There will be a mass mi where the correlation coefficient at the repeat unit mass will not rise above background. At this mi, we assume we have no signal while below it, we take it that we have signal. However, we must be careful about the baseline. If we have not taken the baseline off correctly, we will still see positive signal for the autocorrelation function not at the repeat unit. In fact, the baseline alone should be smooth signal between the repeat units with no peaks. Peaks should only appear at the repeat units. If they appear at other places at these high masses, we may suspect significant loss of an important signal (or perhaps a repeat unit present only at high mass).

Fig. 12

Matrix-assisted laser desorption/ionization time-of-flight mass spectrum from a polybutadiene of a molecular mass centered around 4100 u.

In Fig. 13 we apply our window choice on real polybutadiene data of Fig. 12 for about 10 repeat units (a range about 500 u wide) for lags out to nearly 3 repeat units starting each new window at 250 u increments with windows moving from 4877 u to the high molecular mass tail of the distribution. We notice a repeat unit in the window from the middle of the MMD at 54 u. This is the polybutadiene repeat unit mass. Additionally there are much weaker peaks at about 20 u and 34 u that are due to fragments along the chain backbone. For windows above mi of 5377 u, we see no repeat unit signal at all. We then take our cut off of signal at 5627 u, the start of the next window. One might expect the autocorrelation function of a baseline of pure noise to be zero but it is not. If the noisy baseline were offset by a constant, the autocorrelation function would be unity. The linear autocorrelation function indicates an essentially constant baseline in time (see Appendix B).

Fig. 13

Mass autocorrelation of the data in Fig. 12 with windowing at the high mass edge of the mass spectra. The mass in the legend refers to the mass at the low edge of the window.

In Fig. 14 we apply the same window width on the same data with windows moving toward the low tail of the distribution. Again, we notice a repeat unit in the window taken from the middle of the MMD at 54 u as well as much weaker peaks at about 20 u and 34 u. For windows with masses above 2127 u, we see only a repeat unit signal. Below this we may see some signal. Clearly here, the baseline signal is causing difficulty so we have redrawn the baseline for this data and the autocorrelation functions for windows starting at mass 1636 u are shown in Fig. 15. Once we draw a more correct baseline (i.e., through the noise in the spectra), the balancing of noise and the signal become clearer. For the peaks at mass 54 u on window 1636 u to 2386 u there are clearly peaks and some new peaks appear, apparently the appearance of another repeating species perhaps matrix clusters or silver cation clusters [17]. In this particular polymer, the average mass of silver (107.88 u), introduced as a cationizing agent, is about the same as two polybutadiene repeat units, confusing the issue somewhat.

Fig. 14

Autocorrelation function windowing at the low mass edge of the mass spectra. The mass in the legend refers to the mass at the low edge of the window.

Fig. 15

Autocorrelation function windowing at the low mass edge of the mass spectra after redrawing of the baseline. Compare to Fig. 12 where little signal is noticeable between the 1877 u window and the 2377 u window. Even the lowest window starting at 1636 u going to 2136 u shows mass signal at the repeat unit. It is mixed in with other repeats not identified yet. The mass in the legend refers to the mass at the low edge of the window.

7. Conclusion

We have shown that the autocorrelation function applied to the mass spectra of synthetic polymers allows one to more easily gain insight into the polymer single-chain structure. This offers a tool for looking at homopolymers with architectural changes like the silsesquioxanes and at the structure of complex polymers like the siloxane-silsesquioxane copolymer presented. Finally, we have shown how the windowed autocorrelation function can be used to separate signal from noise.

Table 2

Computed MMD moments for various baselines with A = 100 in Eqs. (6), (7), and (8)

	Mn(u)	Mw(u)	Mz(u)
Without baseline	1424	1447	1468
Constant baseline	1420	1504	1580
Linear baseline	1229	1290	1353
Exponential baseline	1174	1239	1313

Table 3

Computed MMD moments for various baselines with A = 10 in Eqs. (6), (7), and (8)

	Mn(u)	Mw(u)	Mz(u)
Without baseline	1424	1447	1468
Constant baseline	1422	1486	1546
Linear baseline	1289	1343	1395
Exponential baseline	1254	1315	1375