Error Analysis for NMR Polymer Microstructure Measurement without

Anal. Chem. 2009, 81, 8585–8589

Error Analysis for NMR Polymer Microstructure Measurement without Calibration Standards XiaoHua Qiu,* Zhe Zhou, Gian Gobbi, and Oscar D. Redwine The Dow Chemical Company, Building 1897, Midland, Michigan 48674 We report an error analysis method for primary analytical methods in the absence of calibration standards. Quantitative 13C NMR analysis of ethylene/1-octene (E/O) copolymers is given as an example. Because the method is based on a self-calibration scheme established by counting, it is a measure of accuracy rather than precision. We demonstrate it is self-consistent and neither underestimate nor excessively overestimate the experimental errors. We also show the method identified previously unknown systematic biases in a NMR instrument. The method can eliminate unnecessary data averaging to save valuable NMR resources. The accuracy estimate proposed is not unique to 13C NMR spectroscopy of E/O but should be applicable to all other measurement systems where the accuracy of a subset of the measured responses can be established. Measurement, by its nature, is inexact; the magnitude of that “inexactness” is the error. In this definition, error is different from a mistake, which is the introduction of an error that can be traced to its source and can be detected, quantified, and corrected. Error is inherent in measurement, and incorporates such things as the precision of the measuring tools, their proper adjustment, and competent application. Because accuracy is the degree of conformity with a standard (the “truth”), the accuracy of a measurement is indeterminate if such standards do not exist. In a generic chemical reaction A + B f A′xB′y, it is desirable to determine the exact ratio of x to y. Analytical methods to determine this ratio need to be validated against well established and widely accepted standards. Gravimetric standards can be obtained only if the amount of A and B reacted can be determined accurately. The task becomes more difficult if A and/or B can get physically incorporated into A′xB′y and become gradually separated over time. In the case of polymerization reactions, even if standards with exact ratios of x to y do exist, absolutely accurate measurements of sequence distributions for A′xB′y are difficult because it is virtually impossible to obtain standards for the infinite number of possible sequence distributions. The lack of standards described above typifies the situation for all ethylene/R-olefin copolymers. This study focuses on the example of ethylene/1-octene (E/O) copolymers. During the production of E/O, a mixture of ethylene and 1-octene is vented. * Corresponding author. E-mail: [email protected]. 10.1021/ac901565u CCC: $40.75  2009 American Chemical Society Published on Web 09/18/2009

Another mixture of ethylene and 1-octene is absorbed onto the resin and vaporizes during the devolatilization process. Therefore, it is difficult to accurately calculate the composition of the resin from mass balance. The sequence distribution standards for E/O are even harder to acquire. Yet compositions and sequence distributions of E/O are essential to their material properties. They also provide critical information about the polymerization chemistry and have been measured extensively by both academic and industrial scientists.1,2 13C NMR spectroscopy has been accepted as a primary method for composition and sequence distribution analyses. However, critical questions remain: How accurate are these measurements? If one measurement disagrees with another, are the differences large compared to the errors? If yes, which one is correct? In NMR spectroscopy, multiple resonance lines from the same repeat unit can usually be observed in the same experiment. We propose a “self calibration” scheme to estimate errors in a primary method where calibration standards do not exist. In this case, a subset of the system being measured (resonances from octene repeat units) can be related to each other through counting. Because counting is inherently accurate, this subsystem can be treated as the “truth” or the calibration standards. We use the measured values of this subsystem to calibrate against the “truth” and estimate the errors in the measurement. Once experimental errors are quantified using the “self calibration”, Monte Carlo (MC) simulation is an effective way to investigate their effect on the final results. Random fluctuations of the size determined by the “self calibration” can be introduced to the experimental data using MC. Multiple results generated from analyzing these simulated data can be statistically evaluated for the accuracy of the analyses. One complication is that experimental random noise and systematic biases cannot be easily separated in the “self calibration”. They are all treated as random noise in MC. If the accuracy analysis results and reproducibility results differ too much (such as one decade), one should investigate the experimental setup carefully and try to identify and eliminate any dominating systematic biases. Compared to the conventional precision analysis, the proposed “self calibration” has the following advantages. First, it is fundamentally a calibration study and not a reproducibility study. The results of the error analyses include both accuracy and precision effects. Second, it can be conducted for each single measurement instead of multiple experiments required for precision analysis. Third, it can be conducted simultaneously with data acquisition, (1) Maddams, W. F.; Parker, S. F. Markromol. Chem. 1988, 189, 333. (2) Randall, J. C. Rev. Macromol. Chem. Phys. 1989, C29 (2, 3), 201.

Analytical Chemistry, Vol. 81, No. 20, October 15, 2009

8585

Figure 1. 13C NMR spectrum of a commercial poly(ethylene-co-1-octene) with approximately 12 mol % octene. The chemical shift range and structure assignments for the integrals are listed in Table 1. Table 1. Peak Names, Integral Regions, and Assignments for E/O

start (ppm) end (ppm) assignment

A

B

C

D

E

F

G

H

I

J

K

39.8 41.6 RR

37.9 38.7 EOE CH

35.7 36.2 OOE CH

34.3 35.6 Rγ+, 6B6

33.4 33.8 OOO CH

32.0 32.4 3B6

29.0 31.6 γ+γ+

25.9 28.2 βγ+, 5B6

24.0 25.1 ββ

22.5 23.2 2B6

13.9 14.4 1B6

which eliminates uncertainties associated with the instrumental deterioration during the time between the calibration process and actual measurement. Fourth, it is a data analysis step, and no extra data acquisitions are needed. Thus, it can be applied retroactively to all historical measurements, making data comparison more meaningful. Fifth, it can be incorporated into instrumentation and be conducted at the end of each data averaging step, eliminating excessive data averaging that has marginal value in terms of desired accuracy level. EXPERIMENTAL SECTION Polymer samples were prepared as 6 wt % solutions. The solvent was a 5/95 (w/w) mixture of p-dichlorobenzene-d4 and o-dichlororbenzene with 0.025 M chromium actetylacetonate added as the relaxation agent. Typically, 0.2 g of polymer was dissolved in 2.5 g of the solvent mixture in a 10 mm NMR tube. After N2 purge, the NMR tube was capped and heated in a heating block set at 150 °C to dissolve the polymer. Samples were vortexed during heating to facilitate sample homogenization. Once the sample/solvent achieved the appearance of a single phase and flowed consistently, the sample tube was left in the heating block for more than 24 h for homogenization purposes. Both Varian and Bruker 400 MHz systems were used to acquire data. The Varian system is the “default” (when the instrument on which data were collected is not identified) system, and results from the Bruker system are mentioned specifically in this report. The following parameters are the default set up on both instruments: temperature at 400 K, 25 000 Hz spectral width, 1.3 s acquisition time, 90° pulse, 6 s relaxation delay, 8000 scans, and inverse gated decoupling with Waltz modulation. The decoupler frequency was set to the 1H resonance of the main chain methylene peak. The number of scans does change for some 8586

Analytical Chemistry, Vol. 81, No. 20, October 15, 2009

experiments and is specified when it does. The inverse gated decoupling was implemented on Varian with “s2pul” as the pulse sequence and the decoupler mode (dm) set to ‘nny’. It was first implemented on Bruker with “zgpg/bi_Waltz65_256” as the pulse sequence and set PL13 to 120 db. The pulse sequence was later changed to “zgig/bi_Waltz65_64pl”. The details of the two decoupling pulse sequences are included in the Supporting Information. The reason for this change will be discussed later in the report. The majority of the free induction decay (FID) files on the Varian system were processed using Felix95, and the Bruker FIDs were processed using NUTS. A limited number of Varian files were processed using NUTS as well. There is no noticeable change in either the integrals or the calculated results between files processed with either software. The FID was apodized with a cosine function. It was then zero filled once and Fourier transformed. The spectrum was phased and baseline corrected manually. Baseline correction was performed on the spectral region between 54 and 4 ppm, a straight line fitting was used for Felix95, and the “fb” routine was used for NUTS. Figure 1 shows a typical spectrum of E/O with a defined integral range. The chemical shift ranges are specified in Table 1. Scheme 1 illustrates the numbering system used in this report. The assignment for these peaks can be found in an earlier report.3 The “linear leastsquares analysis with the constraint” in Seger and Maciel4 was used to analyze the integral list for composition and triad distribution. Once the population standard deviation was calculated, the Excel random number generator for normal distribution was used (3) Qiu, X. H.; Redwine, O. D.; Gobbi, G.; Nuamthanom, A.; Rinaldi, P. L. Macromolecules 2007, 40, 6879. (4) Seger, M. R.; Maceil, G. E. Anal. Chem. 2004, 76, 5734.

Scheme 1. Diagram and Carbon Nomenclature for E/O

to generate a random noise for each integral in Table 1. The simulated random noises were added to the experimental spectrum to generate a simulated spectrum. The “linear least-squares analysis with the constraint” method was then applied to this simulated spectrum, and results were recorded. Thirty simulated spectra were generated for each experiment. Averages and standard deviations from the thirty sets of results were calculated using existing Excel functions. RESULTS AND DISCUSSION There are seven unique carbons that belong to an octene repeat unit in E/O. The methine carbon, 1B6, 2B6, and 3B6 have resolved peaks (capital letters) in the 13C NMR spectrum. 1B6 ) K; 2B6 ) J; 3B6 ) F;

∑ CH ) B + C + E

(1)

R and β carbons only exist where an octene is incorporated. They can be calculated from the following triads (5B6 is treated as a β carbon and 6B6 is treated as an R carbon).

∑ R ) A + D ) EOE × 3 + (OOE + EOO) × 2.5 + OOO × 2

(2)

∑ β ) H + I ) EOE × 3 + (OOE + EOO) × 2 + OOO-OEO

(3)

When counting R and β carbons in a triad, we only count methylenes that are R and β to the CH of the central octene unit. Equation 2 can be derived from the following analysis. Each octene repeat unit has three R carbons. However the RR carbons are shared with another octene repeat unit. So each RR carbon is counted as only half for a given triad. Equation 3 can be derived from the following analysis. Each octene in the center of EOE triad has three β carbons, each octene in the center of OOE/ EOO triad has two β carbons, and each octene in the center of OOO triad has only one β carbon. The ββ carbon is counted twice since it is β to two octenes. The quantity of ββ carbons equals the quantity of the OEO triads and is subtracted from the sums of three EOE, two OOE/EOO, and one OOO to correct for the double counting of ββ carbons. Using the assignment in Table 1, we can rewrite eqs 2 and 3 into A + D ) 3 × B + 2.5 × C + 2 × E

(4)

H+2×I)3×B+2×C+E

(5)

Equations 1, 4, and 5 all have integrals B, C, and E on the right side of the equation. A relative error sampled by R carbon is defined as the ratio of the left side of eq 4 to the right side of the same equation. This ratio captures the inconsistency from the A and D integrals relative to the B, C, and E integrals. An R equivalent CH, ∑CHR, is defined by multiplying the above ratio with ∑CH.

∑ CHR ) (A + D)/(3 × B + 2.5 × C + 2 × E) × (B + C + E)

(6)

Similarly we can calculate a β equivalent CH, ∑CHβ,

∑ CHβ ) (H + 2 × I)/(3 × B + 2 × C + E) × (B + C + E)

(7)

The known systematic biases are corrected in the following way. The long spin-lattice relaxation time for the 1B6, 2B6, and 3B6 carbons was corrected by comparing to a run with 30 s relaxation delay. The 1B6 peak integral is 1.05 times, the 2B6 peak integral is 1.02 times, and the 3B6 peak integral is 1.005 times bigger in the spectrum with 30 s relaxation delay compared to the 6 s relaxation delay used for all the spectra collected in this report. Therefore, for the spectra collected in this report, the corresponding peak integrals were multiplied by the factors identified in the previous sentence. The complications from real chain ends 1s, 2s, and 3s were corrected by subtracting the integral for the 4s peak from the integrals for 1B6, 2B6, and 3B6. Since the 4s peak is usually on the right tail of the main chain methylene carbon peak, a subspectral region between 29.8 and 29.4 ppm was baseline fitted again after all the other peak integrals were recorded. After the second baseline correction, the integral for the 4s peak at 29.6 ppm was recorded for the real chain ends correction. After the corrections, the integrals for peak 1B6, 2B6, 3B6, ∑CH, ∑CHR, and ∑CHβ (F, J, K, B + C + E, (A + D)/(3*B + 2.5*C + 2*E)*(B + C + E), and (H + 2*I)/(3*B + 2*C + E)*(B + C + E)) should be the same if no experimental errors are present in the spectrum. The sample standard deviation was calculated for the six numbers. A t test factor of 2.57 for 5 degrees of freedom was applied for 95% confidence level. It was converted back to an estimated population standard deviation by dividing it by 1.96. This number was used in MC to evaluate experimental uncertainties in the results. In the absence of standards, we use two examples to demonstrate that error analysis is both self-consistent and very sensitive to previously unknown instrumental mistakes. Analytical Chemistry, Vol. 81, No. 20, October 15, 2009

8587

Table 2. Composition and Triad Analysis Results for Two E/O and Their Blends octene mol % sample id octene rich octene poor 10.52 wt % 21.85 wt % 49.07 wt %

EEE

“σ” 15.01 0.40 12.98 12.90 10.81 10.83 6.48 6.53

measured predicted measured predicted measured predicted

EEO “σ”

0.17 0.01 0.11 0.18 0.11 0.18 0.03 0.18

0.598 0.988 0.652 0.654 0.710 0.710 0.825 0.824

OOE

EOE

“σ”

0.004 0.000 0.002 0.004 0.001 0.004 0.001 0.004

0.225 0.008 0.195 0.194 0.163 0.163 0.099 0.099

“σ”

0.005 0.000 0.006 0.005 0.002 0.005 0.001 0.005

0.021 0.000 0.019 0.018 0.015 0.015 0.009 0.009

0.002 0.000 0.002 0.002 0.001 0.002 0.001 0.002

“σ” 0.129 0.004 0.111 0.111 0.093 0.093 0.056 0.057

0.002 0.000 0.001 0.002 0.001 0.002 0.001 0.002

Table 3. Composition and Triad Analysis Results for an E/O on Bruker and Varian Instruments octene mol % run no. Bruker1 Bruker2 Bruker3 Bruker4 Varian Bruker5

EEE

“σ” 20.28 20.28 20.29 20.32 19.72 19.48

0.53 0.91 0.58 0.63 0.18 0.36

EEO “σ”

0.476 0.476 0.476 0.475 0.493 0.496

0.015 0.010 0.007 0.010 0.002 0.006

Two E/O copolymers, one octene rich and one octene poor, were used to make three gravimetric blends to demonstrate the self-consistency of the error analysis. Table 2 lists composition and triad analyses for the two starting materials and the blends, as well as the expected results calculated from weight fractions. The corresponding spectra are included in the Supporting Information. Because we stated absolute standards are very difficult to obtain for copolymers in the Introduction, this test only serves to demonstrate that the data analysis and the error analysis are self-consistent. The tabulated “σ” is from MC. The concentrations of the blends are listed as weight percentage of the octene poor sample in the blend. The errors involved in weighting are negligible compared to the experimental errors in NMR spectroscopy. All the NMR results from the gravimetric blends agree with the predicted results, based on weight, within one standard deviation. This demonstrates that the error analysis introduced in this report is self-consistent. Table 3 shows the NMR composition and triad analyses of E/O using the same sample (different from the two standards used in Table 2) but analyzed on two different instruments. Run Bruker1-4 showed good reproducibility, typical of a single operator repeated run on a state-of-art instrument. However, the implied precision conflicts with the results obtained on a Varian spectrometer. The results would normally evolve into a passionate debate about the two major NMR vendors. However, MC results can be used to estimate how accurate each measurement is. First, all the measured numbers agree with each other within one standard deviation. This demonstrates that the error analysis does not underestimate errors. Second, the difference between the Bruker1-4 results and the Varian result is close to the sum of the standard deviations for each system. This demonstrates that the error analysis does not excessively overestimate errors. The standard deviations for Bruker1-4 are surprisingly large when compared to the reproducibility. This observation points to the possibility that a large systematic error exists for the Bruker system. After extensive testing, we identified the pulse sequence employed for the Bruker system was the culprit for the systematic 8588

Analytical Chemistry, Vol. 81, No. 20, October 15, 2009

OOE “σ”

0.270 0.270 0.269 0.270 0.265 0.266

0.050 0.025 0.029 0.026 0.004 0.008

EOE “σ”

0.033 0.033 0.033 0.034 0.037 0.036

0.025 0.030 0.024 0.028 0.004 0.008

“σ” 0.170 0.170 0.170 0.170 0.159 0.158

0.017 0.023 0.019 0.013 0.002 0.004

error. The “zgpg/bi_Waltz65_256” pulse sequence turns the decoupler on 100 ms before the 13C 90° pulse. Some nuclear Overhauser enhancement (NOE) was built up during the 100 ms. Since NOE buildup rates are different for the different carbons in E/O, it causes a systematic error that was captured in the “self-calibration”. NMR details about the steps taken to correct it were reported elsewhere.5 Eventually, “zgig/ bi_Waltz65_64pl” pulse sequence was developed to eliminate NOE build up before acquisition. The resulting Bruker results are shown in Table 3 as Bruker5. The agreement between the two different systems is much better. After the differences between the instruments were resolved, the individual normalized (the 2B6 integral is normalized to 100) integral for each region was compared between the spectra taken with the “zgpg/bi_Waltz65_256” and with the “zgig/ bi_Waltz65_64pl” pulse sequences. Indeed the integrals in the former show increases that inversely correlated with the T1 of the carbon. The carbons associated with the hexyl branch such as the methine carbon and the R and β methylene carbons have shorter T1s and were overcounted with the “zgpg/ bi_Waltz65_256” pulse sequence, and that is why the octene contents were systematically high in the Bruker1-4. Table 2 in ref 5 gave the integral increasing factors between the two Bruker pulse sequences. Because the increasing factors are not uniform for all the peaks, there are more inconsistencies between the integrals in the “self-calibration”, and that is why the standard deviations reported for Bruker1-4 are much larger than the reproducibility. Using these factors and the Varian spectrum, one can generate the same results as those from Bruker 1-4. Therefore, the systematic bias in Table 2 can be completely explained by the first pulse sequence. The acquisition time for quantitative 13C NMR of E/O is on the order of hours because of the low sensitivity of NMR. Table 4 lists the results from experiments on an E/O sample. (The (5) Zhou, Z.; Kummerle, R.; Qiu, X. H.; Redwine, O. D.; Cong, R.; Taha, A.; Baugh, D.; Winniford, W. J. Magn. Reson. 2007, 187, 225.

Table 4. Composition and Triad Analysis of an E/O with a Different Number of Scans octene mol % no. of scans

time (min)

128 384 1536 6144 24576

16 47 187 748 2990

EEE

“σ” 16.93 16.78 16.80 16.87 16.85

0.30 0.20 0.14 0.12 0.10

EEO “σ”

0.561 0.564 0.563 0.562 0.563

spectra are included in the Supporting Information.) The only difference is the number of scans in each experiment. The results with 384 scans already have the desired accuracy for most E/O that we encounter. Therefore, 1 h acquisition time is all that is required to achieve the accuracy requirements of ±0.2 mol %, compared to at least 6-7 h employed today. The 4-fold reduction in experimental time is significant with respect to valuable NMR resources. Since accuracy can now be estimated from a single experiment, it is possible to incorporate the estimation into the vendor software and be conducted at the end of each scan. The required accuracy can be set at the beginning of an experiment and the spectrometer can stop the acquisition once this requirement is satisfied. CONCLUSIONS In the absence of standards, the “self-calibration” method proposed in this report is an effective, and currently the only, way to estimate the accuracy for 13C NMR spectra of E/O. Accuracy estimates can be used to compare results from different

OOE “σ”

0.004 0.004 0.003 0.003 0.002

0.231 0.231 0.231 0.231 0.230

0.007 0.004 0.004 0.004 0.003

EOE “σ”

0.029 0.029 0.030 0.030 0.030

0.005 0.003 0.002 0.002 0.002

“σ” 0.140 0.138 0.139 0.139 0.139

0.003 0.002 0.002 0.001 0.002

spectrometers. It can be used to identify unsuspected large systematic biases, and eliminate unnecessary acquisition time, to make valuable NMR resources available to other problems. The method developed in this report for accuracy estimates is not unique to E/O. It can be applied to other systems where the accuracy of a subsystem can be established, and used to “self-calibrate” the experimental observations. ACKNOWLEDGMENT We thank Anne Leugers and Jenny Birk for their constructive feedback. SUPPORTING INFORMATION AVAILABLE Extra information as noted in the text. This material is available free of charge via the Internet at http://pubs.acs.org. Received for review July 15, 2009. Accepted September 1, 2009. AC901565U

Analytical Chemistry, Vol. 81, No. 20, October 15, 2009

8589