Quantitative 13C NMR Analysis of Sequence Distributions in Poly

Anal. Chem. 2004, 76, 5734-5747

Quantitative 13C NMR Analysis of Sequence Distributions in Poly(ethylene-co-1-hexene) Mark R. Seger† and Gary E. Maciel*

Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523-1872

Different 13C NMR methods of determining triad distributions in two poly(ethylene-co-1-hexene) copolymers are examined using high signal-to-noise ratio 13C NMR spectra of the copolymers dissolved in deuterated 1,2,4trichlorobenzene at 398 K. This examination includes a comparison of three integration techniques. The experimental impact of decoupler sidebands and significantly nonequal 13C NOE values are examined. A least-squares regression analysis technique for solving for triad mole fractions is tested and appears to be more reliable than two published algebraic expressions (and other expressions examined in the work reported here). The resultant triad mole fractions are compared to sequence distribution parameters expected by Bernoullian and first-order Markovian statistical models. On the basis of 13C NMRdetermined average reactivity ratios, the copolymer designated sample B (5.3 mol % 1-hexene) appears to be a Bernoullian copolymer resulting from a single-site catalytic system. The copolymer designated sample S (3.6 mol % 1-hexene overall) is better described as a mixture of polyethylene and a Bernoullian copolymer with 6.4 mol % 1-hexene content, and thus appears to result from a multisite catalytic system. General Information. Recent advances in transition-metal catalysis of ethylene polymerization at low pressure can provide a high degree of control of branching and narrow molecular weight ranges.1 Polyethylene polymers with controlled branching can be prepared by (1) ethylene homopolymerization using group10 diazadiene complexes as catalysts or (2) copolymerization of ethylene and a 1-alkene using group-4 metallocenes. In contrast to the complex mixtures of branched polyethylenes obtained with conventional multisite catalysts, these metallocene catalyst systems are examples of single-site catalysts (i.e., only one type of catalytic site) and produce very uniform polymers with narrow molar mass distributions (Mw/Mn ≈ 2). The ethylene-only homopolymerization mechanism can produce alkyl side chains via what is alternatively known as “2,ωpolymerization”2 or “chain-walking”.3 Group-10 metal alkyls can migrate along a growing polyethylene polymer chain by repeated * To whom correspondence should be addressed. E-mail: maciel@lamar. colostate.edu. † Current address: Department of Chemistry and Geochemistry, Colorado School of Mines, Golden, CO 80501. (1) Maeder, D.; Heinemann, J.; Walter, P.; Muelhaupt, R. Macromolecules 2000, 33, 1254-1261. (2) Moerhring, V. M.; Fink, G. Angew. Chem. 1985, 97, 982-984.

5734 Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

β-eliminations followed by reinsertions. When chain growth occurs exclusively at the chain end, this mechanism produces methyl branched polyethylene polymers. In the copolymerization of ethylene with 1-alkenes by group-4 metallocene catalysts, control of branching is provided by properly choosing the molecular architecture of the metallocene catalysts. Of particular importance in this regard are the improved processing characteristics of linear low-density polyethylene (LLDPE), as compared to high-density polyethylene (mostly unbranched) and low-density polyethylene (a high degree of uncontrolled branching, including long branches). The physical characteristics of LLDPE, such as melting temperature and crystallization behavior, can be “fine-tuned” by changing the 1-olefin monomer identity or concentration or by altering the metallocene catalyst, yielding products with a wide range of applications, from plastic grocery bags to agricultural weed control. In the study reported here, the branching of two poly(ethylene-co-1-hexene) copolymers was investigated by liquid sample 13C nuclear magnetic resonance (NMR). Isomers of Poly(ethylene-co-1-hexene). Several types of poly(ethylene-co-1-hexene) isomers are possible, including (1) conformational/configurational and (2) substitutional.4-8 In this work, we are primarily concerned with substitutional isomerisms of the sequence type, which are the main types of structural isomers detectable by liquid sample 13C NMR. These types of isomers are the result of adding either an ethylene or 1-hexene monomer unit to the end of a growing polymer chain, whereas the previous monomer unit added may also be either ethylene or 1-hexene. The common way to describe these sequences is by means of n-ads, namely, monads (one monomer unit), diads (two monomer units), triads, tetrads, etc. The sum of the mole fractions of the two monads in the poly(ethylene-co-1-hexene) copolymer system, ethylene (E) and 1-hexene (H) is unity, i.e., [E] + [H] ) 1. Just as there are three possible diads in poly(ethylene-co-1hexene) copolymers (EE, EH, HH), six triads are possible (EEE, (3) (a) Jurkiewicz, A.; Eilerts, N. W.; Hsieh, E. T. Macromolecules 1999, 32, 5471-5476. (b) Johnson, L. K.; Killian, C. M.; Brookhart, M. J. Am. Chem. Soc. 1995, 117, 6414-6415. (4) Randall, J. C. J. Macromol. Sci., Rev. Macromol. Chem. Phys. 1989, C29, 201-317. (5) Bovey, F. A. High-Resolution NMR of Macromolecules; Academic Press: New York, 1972. (6) (a) Bovey, F. A. Polymer Conformation and Configuration; Academic Press: NY, 1969. (b) Bovey, F. A. High-Resolution NMR of Macromolecules; Academic Press: New York, 1979. (7) Randall, J. C. Polymer Sequence Determination; Academic Press: New York, 1977. (8) Axelson, D. E.; Levy, G. C.; Mandelkern, L. Macromolecules 1979, 12, 4152. 10.1021/ac040104i CCC: $27.50

© 2004 American Chemical Society Published on Web 09/03/2004

Table 1. Necessary Relationships for the Poly(ethylene-co-1-hexene) System4 type

necessary relationship

monad-monad diad-diad monad-diad

[E] + [H] ) 1 [EE] + [EH] + [HH] ) 1 [E] ) [EE] + 1/2[EH] [H] ) [HH] + 1/2[EH] [EEE] + [EEH] + [HEH] + [EHE] + [EHH] + [HHH] ) 1 [EEH] + 2[HEH] ) [EHH] + 2[EHE] [E] ) [EEE] + [EEH] + [HEH] [H] ) [EHE] + [EHH] + [HHH] [EE] ) [EEE] + 1/2[EEH] [EH] ) [EEH] + 2[HEH] ) [EHH] + 2[EHE] [HH] ) [HHH] + 1/2[EHH] [EEEE] + [EEEH] + [HEEH] + [EEHE] + [EEHH] + [HEHH] + + [EHEH] + [EHHE] + [EHHH] + [HHHH] ) 1 2[HEEH] + [EEEH] ) [EEHE] + [EEHH] 2[EHHE] + [EHHH] ) [HEHH] + [EEHH] [EE] ) [EEEE] + [EEEH] + [HEEH] [EH] ) [EEHE] + [EEHH] + [HEHH] + [EHEH] [HH] ) [EHHE] + [EHHH] + [HHHH] [EEE] ) [EEEE] + 1/2[EEEH] [EEH] ) 2[HEEH] + [EEEH] ) [EEHE] + [EEHH] [HEH] ) 1/2[EHEH] + 1/2[HEHH] [EHE] ) 1/2[EHEH] + 1/2[EEHE] [EHH] ) 2[EHHE] + [EHHH] ) [HEHH] + [EEHH] [HHH] ) [HHHH] + 1/2[EHHH]

triad-triad monad-triad diad-triad tetrad-tetrad

diad-tetrad

triad-tetrad

EEH, HEH, EHE, EHH, HHH), with EEH and HEE being equivalent, as are EHH and HHE, and the sum of their mole fractions is unity. It is easily seen that

[EEE] + [EEH] + [HEH] ) [E]

(1)

[EHE] + [EHH] + [HHH] ) [H]

(2)

and

Necessary relationships are expressions that algebraically relate various n-ad mole fractions (e.g., relate diad to triads), independent of polymerization mechanism or statistics. Table 1 lists necessary relationships, which were advanced by Randall,4 for the EH copolymer system. A more detailed discussion of isomers and the necessary relationships of EH copolymers is given in the Supporting Information. Nomenclature and 13C Chemical Shift Assignments. The nomenclature used in this study is that introduced by Randall9 and by Carman and Wilkes10 and later extended by others. This system is represented in Figure 1, where it is seen that Greek letters are used to denote the positions of a given backbone carbon site relative to methine carbons and side-chain carbons (“B” for branch) are labeled using the format nBm, where m represents the length of the side chain and n refers to the position of the carbon in question, as counted from the end of the side chain.5 Additional details on nomenclature are included in the Supporting Information. (9) Randall, J. C. J. Polym. Sci., Polym. Phys. Ed. 1973, 11, 275-287. (10) Carman, C. J.; Wilkes, C. E. Rubber Chem. Technol. 1971, 44, 781-804.

Figure 1. Nomenclature examples for poly(ethylene-co-1-hexene) substructures.

The chemical shift assignments of the 13C spectrum of poly(ethylene-co-1-hexene) (dissolved in suitable solvents, e.g., dichloro- or trichlorobenzenes) are well reported in the literature,6,11 with essentially full agreement among independent reports, except for very small differences in reported chemical shifts. A set of chemical shifts and their assignments, reported by Hsieh and Randall,11 is shown in Table 2. These assignments were made by a variety of techniques, including the use of both low and high molecular weight model compounds and a variety of NMR methods (e.g., off-resonance decoupling, spectral editing techniques such as DEPT3 and APT,12 and two-dimensional methods, such as J-resolved spectroscopy13 and heteronuclear shift correlation spectroscopy (HETCOR) in conjunction with proton COSY results.13 The necessary relationships can be often be used to select between alternative assignments, or confirm tentative assignments. Randall and Hsieh14 reported the detection of two types of polymer chain ends: an n-alkyl chain end (sCH2sCH2sCH2s CH3, labeled as “s” for saturated) and a terminal olefin (sCH2s CHdCH2). Table 2 includes the chemical shifts they reported for these chain ends, where 1s, 2s, and 3s are the carbons counting from the end of the saturated chain end, 1v and 2v are the olefinic carbons, again counting from the chain end (“v” for vinylic), and “a” refers to the allylic carbon. As most commercial poly(ethyleneco-1-hexene) materials are of high molecular weight, these chain ends contribute only very small 13C NMR peaks and are often not detectable. (11) Hsieh, E. T.; Randall, J. C. Macromolecules 1982, 15, 1402-1406. (12) Galland, G. B.; deSouza, R. F.; Mauler, R. S.; Nunes, F. F. Macromolecules 1999, 32, 1620-1625. (13) Bruch, M. D.; Payne, W. G. Macromolecules 1986, 19, 2712-2721.

Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

5735

Table 2. 13C Chemical Shift Assignments for the Poly(ethylene-co-1-hexene) System, as Determined by Hsieh and Randall11 carbon site

sequence

obsd 13C chem shift (ppm)

obsd 13C carbon chem shift site sequence (ppm)

RR RR RR CH(EHE) CH(EHH) 4B4 Rγ Rγ Rδ+

HHHH EHHH EHHE EHE EHH HHH HEHH EHEH EEHH

41.40 40.86 40.18 38.13 35.85 35.37 35.00-34.90 35.00-34.90 35.00-34.90

4B4

EHH

35.00-34.90 1B4

Rδ+ 4B4 CH(HHH) γγ γδ+ δ+δ+

EEHE EHE HHH HEEH EEEH (EEE)n

34.54 34.13 33.47 30.94 30.47 29.98

3B4 3B4 3B4 βδ+ βδ+ ββ ββ ββ 2B4

2v 1v a 3s 2s 1s

EHE EHH HHH EEHE EEHH EHEHE EHEHH HHEHH EHE + EHH +HHH EHE + EHH + HHH

29.51 29.34 29.18 27.28 27.09 24.53 24.39 24.25 23.37 14.12 139.46 114.34 33.91 32.18 22.86 14.15

EXPERIMENTAL SECTION Polymers. Poly(ethylene-co-1-hexene) samples were obtained from ExxonMobil Chemical Co. (sample B) and Eastman Chemical Co. (sample S). The 1-hexene incorporation levels were determined to be 5.3 (B, bigger) and 3.6 mol % (S, smaller). Sample Preparation. NMR samples were prepared for both 5- and 10-mm (o.d.) tubes as follows: weighed amounts of polymer and solvent, 1,2,4-trichlorobenzene-d3 (TCB-d3) or 1,4-dichlorobenzene-d4 (DCB-d4), and hexamethyldisiloxane (HMDS, for both chemical shift referencing and monitoring magnetic field homogeneity), were placed in the NMR tube at room temperature, with approximate weight percentages of 15, 83, and 2%, respectively. The total solution weight was typically 1.1 g for 5-mm NMR tubes or 5.5 g for 10-mm NMR tubes, resulting in sample heights of 50 and 70 mm, respectively. The sample mixtures in NMR tubes were heated to 155 °C in a stirred silicone oil bath for 8-12 h. The polymer/solvent/HMDS mixtures became quickly transparent at 155 °C but required at least 4 h of heating to appear visibly uniform with respect to refractive index and homogeneous with respect to viscosity, when probed by a quartz rod (samples stirred three times with a thin quartz rod during the first 4 h of dissolution). NMR Spectroscopy. Liquid-state 1H and 13C NMR experiments were performed using Varian Inova-400 and Varian Inova500 NMR spectrometers, with static magnetic field strengths of 9.4 and 11.7 T, respectively. The former was equipped with a 5-mm triple-resonance indirect-detection probe, while the latter employed a 5-mm triple-resonance probe and a 10-mm double-resonance probe. Since sample spinning was found not to affect significantly the 13C NMR line shapes in the liquid sample polymer spectra of this study, such spectra were obtained without sample spinning. The NMR tubes were heated to 125 °C in the NMR probes. Dry nitrogen gas (obtained from liquid nitrogen boil off) was used for the variable-temperature gas and to flush the probes. Temperature calibration was done by replacing the NMR sample tubes with a tube containing neat ethylene glycol at the same sample height as the polymer solutions.5 Actual temperatures within the 5736 Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

neat ethylene glycol were determined using the Varian tempcal routine, inputting the observed 1H chemical shifts of ethylene glycol’s methylene and hydroxyl protons. Temperatures were stable to within (0.1 °C over the course of the NMR observations. A thermal equilibration delay period of at least 1 h was required with the 5-mm probe, whereas the 10-mm probe required at least 3 h to thermally equilibrate, as judged by constancy in probe shimming and probe tuning. The 2H NMR signals of either TCB-d3 or DCB-d4 solvent provided a signal for field-frequency lock. Magnetic field shimming was based on the deuterium lock signal intensity; shimming on the time domain signal (free induction decay (FID) shimming, based on the signal ring-down characteristics) did not provide better 13C NMR line shapes. Typical HMDS line widths (at halfheight) were 0.5 Hz for 5-mm copolymer solution samples and 1.0 Hz for 10-mm samples. 1H and 13C NMR chemical shifts were referenced using the methyl proton signal of HMDS (determined in this study to be at 0.02 ppm) and the largest 13C copolymer peak (the polyethylenelike polymer backbone, with a reported chemical shift of 29.98 ppm),11 respectively. Unless stated otherwise, all 13C NMR spectra were obtained with a spectral width covering (at least) the chemical shift range from 50 to -10 ppm, taking care that any aliased aromatic carbon signals from the solvent do not overlap with the copolymer peaks. A 90° 13C pulse width of ∼14 µs (corresponding to a 17.6-kHz 13C radio frequency field) was used, with WALTZ-16 proton decoupling13 (1H power level of ∼2 W, corresponding to a 2.3-kHz 1H field, centered on the most intense proton resonance at 1.25 ppm). At least 32 000 data points were acquired for each FID; spectral processing included zero-filling the time-domain signal to 256k points and 2 Hz exponential apodization prior to Fourier transformation. Signal-to-noise ratios were determined using the Varian dsnmax algorithm, utilizing the most intense 13C peak (29.98 ppm) and finding the lowest rootmean-squared noise region (2 ppm wide) in the spectral region between 45 and 10 ppm. All 13C NMR spectra were obtained in the double precision digitization mode. For the purpose of determining triad mole fractions, quantitative 13C NMR spectra were obtained using either 25- or 30-s relaxation delays; the latter value is more than 5T1 for all copolymer carbons in both samples (vide infra),15,16 except methyl carbons. Some quantitative 13C NMR spectra were obtained using continuous wave (CW) proton decoupling (centered on the most intense proton resonance) to eliminate the small decoupler sidebands observed with WALTZ-16 decoupling (vide infra).17 13C T values were determined at 125 °C by the inversion1 recovery method, using repetition delays of 50 s. No statistically significant systematic (vs random) deviations were found in the computer fits of the inversion-recovery data for a singleexponential decay. 13C nuclear Overhauser enhancement (NOE) factors were determined from comparison of signal intensities or integrals in the presence or absence of proton decoupling during the repetition delay (both 25- and 50-s repetition delays were (14) Randall, J. C.; Hsieh, E. T. ACS Symp. Ser. 1984, No. 247, 131-151. (15) Schaefer, J.; Natusch, D. F. S. Macromolecules 1972, 5, 416-427. (16) (a) Farrar, T. C.; Becker, E. D. Pulse and Fourier Transform NMR; Academic Press: New York, 1969. (b) Slichter, C. P. Principles of Magnetic Resonance, 2nd ed.; Springer-Verlag: Berlin 1978. (17) Waugh, J. S. J. Magn. Reson. 1982, 50, 30-49.

Figure 2. Liquid-state 13C NMR spectrum (126 MHz) of sample B dissolved in 1,2,4-trichlorobenzene-d3 at 125 °C (with WALTZ-16 decoupling). (A) ×1; all peaks on scale. (B) ×100; most intense peaks are off scale.

used).15 For many NOE experiments, WALTZ-16 proton decoupling was utilized during data acquisition (whether or not decoupling is applied during the repetition delays). Some NOE experiments used other proton decoupling techniques, including CW proton decoupling or even no proton decoupling (resulting in proton-coupled 13C NMR spectra). For spectral subregions that lack complete baseline peak separation, start and end points for integration were visually placed at the minimum intensity “valley” between overlapping peaks. Once integral regions were selected, baseline correction was performed using the bc command of the Varian software, involving a polynomial fit that uses anchor points that are set as points of zero intensity. The absolute intensity of each integral region was determined in some instances by using a single set of baseline correction parameters for the entire aliphatic region (45-10 ppm); no zero-order and first-order integral corrections were used in these cases. The resultant integral values are termed machine integrals here, as they do not depend on human decisions or evaluation. Baseline correction of each integral region separately by the operator provides what we refer to as manual integral values. Manual adjustment of the zero- and first-order integral correction parameters were made visually to flatten the integration plot over noise regions without peaks or decoupler sidebands. Regions 3s, D, and E were baseline-corrected together because of apparent peak overlap of the broad wings of the very intense D region. Likewise, the integral of each C subregion was not baselinecorrected individually but as a group, and regions F, G, and 2s were baseline-corrected together. Another method used for peak integrations in this study is deconvolution derived integration. In this method, each region is fitted by deconvolution, and the resulting deconvoluted contributions are taken as the integrated intensities. RESULTS AND DISCUSSION NMR Approaches. Figure 2 shows a liquid sample 13C NMR spectrum of a polymer-B/TCB-d3 solution that is typical of the spectra on which this study is based. As detailed in the Supporting Information, both liquid sample 1H NMR and solid-state 13C NMR suffer dramatically by comparison, in terms of structural resolution, and were not pursued in depth in this study.

Collective Assignments. Although the most advanced 13C NMR techniques and equipment available today can achieve a higher level of peak separation than was available to Randall and Hsieh at the time of their experimental work (early 1980s),11,14 we nevertheless found it useful to employ an approach that they introduced, collective assignments, which provides a highly robust methodology that can be applied straightforwardly by any competent NMR spectroscopist on any modern NMR spectrometer. In practice, separate peaks are not observed for all the 13C chemical shifts listed in Table 2. Many of the 13C resonances are partially or completely overlapped. Hsieh and Randall dealt with resonance overlap by means of the concept of collective assignments.11 Instead of trying to integrate overlapping peaks separately, the spectrum is divided into baseline-resolved regions and then integrated region by region. The total integral of each region is then compared algebraically to the n-ads with 13C chemical shifts contributing to that region. As Hsieh and Randall showed,11 this procedure not only simplifies quantitative analysis of a spectrum but also simplifies the shift assignments, since configurational splittings or long-range sequence effects (which often give rise to line broadening or poorly resolved peak splittings) can be neglected. In this way, the spectrum need be assigned only in terms of triad and tetrad contributions, and stereochemical effects (such as meso vs racemic relative configurations) can be ignored. An in-depth analysis of the tetrads and triads contributing to each spectral region is not included here; instead, a summary, according to Hsieh and Randall,11 is presented in Table 3. The indicated collective assignments have been confirmed in this work and by other researchers.18 Triad Expressions. Hsieh and Randall11 showed that the integrated intensities of the various collective assignment regions (A, B, ..., G; defined in Table 3) can be related algebraically, in many ways, to the concentrations of the six different triads in EH copolymer. They chose to work with the following six equations:

k[EHE] ) B

(3)

k[EHH] ) 2(G - B - A)

(4)

k[HHH] ) 2A + B - G

(5)

k[HEH] ) F

(6)

k[EEH] ) 2(G - A - F)

(7)

k[EEE] ) 1/2(A + D + F - 2G)

(8)

Hseih and Randall11 chose this particular set of triad expressions for several reasons: (1) to eliminate expressions involving regions E and H, the former because of the potential for a slight overlap of spectral intensity between regions D and E and the latter because the long T1 value of the H methyl carbons requires long repetition delays (sufficiently long repetition delay for quantitative integration of the methyl region H would adversely affect overall spectral sensitivity, an important issue, since many of the important regions, such as A and F, have very small but important signals); (2) because of the potential presence of allylic (18) DePooter, M.; Smith, P. B.; Dohrer, K. K.; Bennett, K. F.; Meadows, M. D.; Smith, C. G.; Schouwennaars, H. P.; Geerhards, R. A. J. Appl. Polym. Sci. 1991, 42, 399-408.


5737

Table 3. Collective Assignment Regions for Poly(ethylene-co-1-hexene)11

region

13C chem shift range (ppm)

A B C

42.0-39.5 38.1 36.0-33.0

D

31.0-28.5

E F G H

27.5-26.5 25.0-24.0 23.4 14.1

carbon type(s)

contributing n-ads

RR CH(EHE) CH(EHH) CH(HHH) 4B4 Rγ Rδ+ γγ γδ+ δ+δ+ 3B4 βδ+ ββ 2B4 1B4

HHHH + EHHH + EHHE EHE EHH HHH EHE + EHH + HHH EHEH + HEHH EEHE + EEHH HEEH EEEH (EEE)n EHE + EHH + HHH EEHE + EEHH EHEHE + EHEHH + HEHEH EHE + EHH + HHH EHE + EHH + HHH

carbons from terminal olefin polymer chain ends, it is advantageous to eliminate region C from the triad analysis; and (3) the apparent simplicity of these six equations. Many other possible triad expressions are possible, especially if some of Hsieh and Randall’s choices for region elimination are

altered. Further possibilities involve the creation of spectral subregions; Randall has pointed out this possibility, especially with region C.7,9 Hsieh and Randall chose to ignore the potential presence of saturated-polymer chain-end signals, which can affect the values of the integrals D, G, and H.11 These issues are examined below. Spectral Regions. As discussed above, Hsieh and Randall11 used the collective assignment regions shown in Table 3. Some new subregions are proposed in this work. Randall7,9 previously suggested the C1 and C2 subregions, such that C ) C1 + C2. In the work reported here, the C region was separated into four subregions for integration: C ) C1 + C2a + C2b + C2c. (The C2 integral value then equals C2a + C2b + C2c). The D region was likewise divided in this study into three subregions, such that D ) D1 + D2 + D3. The proposed new subregions appear to lack complete baseline separation from neighboring peaks, although better resolution can now be achieved than was possible in the work of Hsieh and Randall about 20 years ago. These more detailed spectral regions were included in the current work (Tables 4 and 5) in order to examine alternate algorithms for triad distributions and sequence parameters. Decoupler Sidebands. During the process of confirming 13C chemical shift assignments for the spectrum of poly(ethylene-co-

Table 4. 13C Liquid Sample NMR Relaxation Parameters for Sample B at 125 °Ca-c

region A B C1 C2a C2b C2c D1 D2 D3 E F G H d

peak chem shift (ppm)

T1 (s) at 101 MHz

T1 (s) at 126 MHz

T2 (ms) at 126 MHz

pred natural line width at 126 MHz (fwhh in Hz)d

line width obsd at 126 MHz (fwhh in Hz)

40.2 38.1 35.8 35.0 34.5 34.1 30.4 29.98 29.5 27.2 24.5 23.3 14.1

1.2 ( 0.6 1.71 ( 0.06 0.83 ( 0.28 0.79 ( 0.20 1.13 ( 0.02 1.36 ( 0.06 1.64 ( 0.02 2.11 ( 0.01 2.40 ( 0.03 1.44 ( 0.02 0.64 ( 0.26 4.57 ( 0.04 8.3 ( 0.2

1.2 ( 0.4 1.62 ( 0.01 0.94 ( 0.12 0.75 ( 0.06 1.09 ( 0.01 1.30 ( 0.02 1.62 ( 0.01 2.12 ( 0.01 2.27 ( 0.02 1.38 ( 0.01 0.75 ( 0.18 4.48 ( 0.02 8.2 ( 0.1

165 ( 79 200 ( 78 438 ( 60 292 ( 113 191 ( 57 162 ( 59 290 ( 33 81 ( 21 172 ( 81 251 ( 67 52 ( 24 761 ( 330 200 ( 127

1.9 ( 0.9 1.6 ( 0.6 0.7 ( 0.1 1.1 ( 0.4 1.7 ( 0.5 2.0 ( 0.7 1.1 ( 0.1 3.9 ( 1.0 1.9 ( 0.9 1.3 ( 0.3 6(3 0.4 ( 0.2 1.6 ( 1.0

9.19 3.89 3.53 5.74 5.08 4.86 5.06 4.64 3.65 4.12 4.02 3.40 ∼5

a Data obtained with CW decoupling. b Intensities obtained as peak heights. c Indicated uncertainties correspond to two standard deviations. Predicted “natural line width”, (πT2)-1.

Table 5. 13C T1 and NOE Values Measured for Samples B and S at 126 MHz and 125 °Ca-c region A B C1 C2a C2b C2c D1 D2 D3 E F G H

peak chem shift (ppm)

sample B T1 (s)c

sample B NOE factord,e

sample S T1 (s)c

sample S NOE factord

40.2 38.1 35.8 35.0 34.5 34.1 30.4 29.98 29.5 27.2 24.5 23.3 14.1

1.2 ( 0.4 1.62 ( 0.01 0.94 ( 0.12 0.75 ( 0.06 1.09 ( 0.01 1.30 ( 0.02 1.62 ( 0.01 2.12 ( 0.01 2.27 ( 0.02 1.38 ( 0.01 0.75 ( 0.18 4.48 ( 0.02 8.23 ( 0.10

2.8 ( 2.0 2.71 ( 0.10 2.54 ( 0.52 2.66 ( 0.10 2.64 ( 0.09 2.60 ( 0.11 2.26 ( 0.11 2.43 ( 0.16 2.81 ( 0.16 2.57 ( 0.10 2.41 ( 0.71 2.65 ( 0.28 2.40 ( 0.11e

? 2.01 ( 0.04 1.09 ( 0.22 0.92 ( 0.10 1.34 ( 0.01 1.55 ( 0.03 1.94 ( 0.02 2.59 ( 0.005 2.75 ( 0.04 1.67 ( 0.01 1.01 ( 0.23 5.28 ( 0.07 9.58 ( 0.27

2.8 ( 3.0 2.70 ( 0.25 2.71 ( 0.76 2.61 ( 0.35 2.77 ( 0.11 2.77 ( 0.24 2.52 ( 0.27 2.59 ( 0.06 2.69 ( 0.34 2.73 ( 0.15 2.42 ( 1.05 2.61 ( 0.09 2.29 ( 0.09e

a Indicated uncertainties correspond to two standard deviations. b Data obtained with CW decoupling. c Intensities taken as peak heights. d Integrals obtained by deconvolution (sample B) or manual integration (sample S). e Accuracy of NOE values for region H is affected by insufficient relaxation between scans.


13C

Figure 3. Comparison of the 126-MHz liquid-state NMR spectra of sample B dissolved in 1,2,4-trichlorobenzene-d3 at 125 °C: (A) with WALTZ-16 decoupling; (B) with CW decoupling and the same decoupler power setting.

1-hexene), small peaks were observed that were never previously reported for similar samples. For example, a small peak was reproducibly observed at ∼26 ppm, a region that was previously reported to be lacking any significant resonances arising from the polymer; similarly, a small peak at ∼37 ppm is seen in this study, but not previously reported. After extensive analysis of the problem, including enormously valuable discussions with scientific colleagues, it appeared likely that the anomalous peaks were “decoupler sidebands” of the very large δ+δ+ peak at 29.98 ppm (due to polyethylene-like carbons). Decoupler sidebands result from periodic modulations of the 13C magnetization due to the cyclic nature of most proton decoupling methods that are based on pulse sequences and are generated at levels above the noise from only the most intense resonance(s). A comparison of the 13C spectra of the poly(ethylene-co-1hexene) sample B, acquired with two different decoupler modes, is shown in Figure 3. The top spectrum was acquired using standard WALTZ-16 decoupling17 (2.5 kHz decoupler field strength), while the bottom spectrum was obtained using CW decoupling, using the same decoupler field strength. The peaks at ∼26 and ∼37 ppm are not generated with CW decoupling, indicating that these peaks are decoupler sidebands occurring during the WALTZ sequence. The concern then became whether there are other decoupler sidebands in the WALTZ-decoupled spectrum that may overlap with peaks whose integrals are used in the quantitative triad analysis. If so, these decoupler sidebands could present a source of systematic error in the analysis, by slightly distorting the true intensities of the various peaks and regions. For determining the locations and intensities of these decoupler sidebands, a spectrum containing only the δ+δ+ peak would be ideal for analysis. For this purpose, three different polyethylene samples were prepared, but upon 13C NMR analysis, all showed additional 13C peaks due to branching, which complicates detection of the decoupler sidebands. Instead, a model compound (DMSO) was analyzed at the same temperature and instrumental conditions as used for sample B. Figure 4 shows the 126-MHz liquid sample 13C NMR spectrum of neat DMSO at 125 °C, obtained with either WALTZ-16 decoupling (spectrum A) or with CW decoupling (spectrum B).

Figure 4. 126-MHz liquid-state 13C NMR spectrum of neat DMSO at 125 °C: (A) with WALTZ-16 decoupling; (B) with CW decoupling and the same decoupler power setting. Note: the 13C peak of DMSO was arbitrarily set to a position of 29.98 ppm (to facilitate comparison to the copolymer spectrum).

The 13C peak of DMSO was arbitrarily set to a chemical shift of 29.98 ppm (instead of the correct chemical shift of 39.5 ppm relative to TMS) to facilitate comparison to the copolymer spectrum; thus, Figure 4 should show where the decoupler sidebands are expected that arise from the large δ+δ+ peak at 29.98 ppm in the copolymer spectrum. Figure 4A (with WALTZ16 decoupling) shows the presence of six large decoupler sidebands, symmetrically disposed about the central peak, and possibly eight smaller decoupler sidebands. Peaks arising from impurities in the DMSO are labeled by “i” in Figure 4; these two known impurity peaks are observed in both spectra, whereas the decoupler sidebands are seen in the WALTZ-16 spectrum but are absent during CW decoupling. With this information, reliable 13C NMR intensities could be obtained either with CW decoupling or with WALTZ-16 decoupling (by taking proper account of decoupler sidebands). Additional details on decoupler sidebands are presented in the Supporting Information. 13C Relaxation Times. 13C T and T values were measured 1 2 at 101 or 126 MHz, or both, for samples B and S at 125 °C, using the Varian T1 and T2 routines, which analyze peak heights from inversion-recovery and Carr-Purcell-Meiboom-Gill data sets, respectively.16 This involves a nonlinear least-squares fit of peak heights to the following equations:

Iz ) I0[1 - 2 exp(-τ/T1)]

(9)

Iz ) I0 exp(-τ/T2)

(10)

and

The results are summarized in Tables 4 and 5; the uncertainties shown are (2 standard deviations. T1 values were needed in order to obtain quantitatively significant 13C integrals, since for full 90° 13C excitation pulses, one needs to wait at least five T1’s between 13C pulses.16 In Table 4 we see that the 13C T1 values are as small as 0.75 s for regions C2a and F. The C2a region contains Rγ methylenes from HEHH and EHEH and Rδ+ methylenes from EEHH and 4B4 methylenes Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

5739

from EHH, while the F region is derived from ββ methylene carbons in the HEH triad. In general, the smallest T1 values are for methylene carbons that are R or β to a branch point (methine carbon). Presumably, methylenes close to branch points may have more restricted motions than isolated methylenes (such as δ+δ+), which can undergo a ”crankshaft”-type motion.6 The largest T1 values belong to the n-butyl side chains resulting from the incorporation of 1-hexene into the copolymer. Except for the methylene groups closest to the branch point (4B4), the T1 values of the branch carbons are greater than 2 s. The largest T1 is for the methyl carbon (1B4; region H); this reflects the influence of more rapid (and more isotropic) motion for branch carbons than for backbone carbons. Table 5 shows a comparison of the 13C T1 values for samples S and B at 126 MHz. In all cases, the T1 values for sample S are larger than the values for the corresponding sites in sample B, although the two samples were prepared and analyzed as similarly as possible. Assuming that the greater the correlation time for the molecular motion, the smaller will be the T1 values (i.e., correlation times are smaller than ωo-1), then copolymer B appears to have slower molecular motion. Table 4 indicates that 13C T1 values are, except for cases of large experimental uncertainties, slightly smaller at 126 MHz than at 101 MHz (corresponding to 400- and 500-MHz 1H frequencies, respectively). These differences are close to being within the observed confidence limits of the T1C measurements. In terms of well-known relationships16 between relevant spectral densities and molecular reorientation correlation time, τc (as represented in Figure SI-7A in the Supporting Information), these small (or zero) T1C differences imply, at least qualitatively, that the relevant correlation time is much smaller than the inverse of the Larmor frequency, ωo-1; i.e., τc is smaller than 101-1 and 126-1 MHz-1. This situation is commonly known as the “extreme narrowing” condition.6 However, comparison of 13C T and T values in Table 4 shows that T C , T C, a situation 1 2 2 1 that is commonly understood as manifesting the condition τc . ωo-1 (as represented in Figure SI-7B). The 13C T2 values for sample B were determined in order to compare the “natural” 13C line widths to those observed in the spectrum. The former are the inherent line widths of a single carbon signal (as determined by T2C), while the observed line widths may be the result of chemical shift dispersion (the extreme overlap of individual 13C signals due to slightly different structural environments) and static magnetic field inhomogeneities. Table 4 compares the observed full width at half-height (fwhh) values obtained from the spectrum of sample B to the natural line widths calculated from the measured T2C values. For most peak regions listed in Table 4, the predicted line widths are significantly smaller than the experimentally observed line widths, indicating that the largest line width contributions arise from isotropic chemical shift dispersion. The T1C and T2C behavior outlined above can be reconciled, as has been done previously for various organic polymers, by recognizing the dependences of T1 and T2 on various spectral density terms, J(ω), that describe the relevant molecular dynamics.19 In large polymer molecules, T2C is dominated by a zerofrequency term, J(0), for the dipole-dipole interaction between 13C and directly attached (or nearby) protons. This J(0) term is large because it depends on the overall reorientation of the main 5740


polymer chain, which is very slow. Since there is no contribution of a J(0) term to (T1C)-1, spin-lattice relaxation is much slower (than spin-spin relaxation) and is dominated by much faster motions that do not involve the reorientation of the entire polymer molecule and that have much smaller correlation times; i.e., τc < ωo-1. Of course, this simple, qualitative interpretation should be qualified by realization that (a) the motions involved may not be isotropic (an assumption behind the simplest formulations), (b) there is likely to be a distribution of correlation times, and (c) relaxation other than that due to the dipolar mechanism may contribute. Nuclear Overhauser Effect Enhancements. The NOE20 is a key concern whenever liquid-state 13C NMR is performed with the aim of quantitation. Proton decoupling can cause saturation of the proton spins coupled to 13C, resulting in enhanced 13C signals. The resulting 13C resonances may be as much as 2.99 times larger than the intensities obtained without NOE (i.e., without saturation of the proton resonances). If the NOE ratio is identical for all 13C nuclei in the sample, the peak integrals obtained are all scaled by the same factor, and no corrections need be applied in order to achieve a quantitative triad analysis. If the NOE ratios are different for different carbons, they must be determined so that the appropriate factors can be applied. If one wishes to detect 13C resonances without NOE enhancement, a technique known as gated decoupling, in which the decoupler is turned off except during data acquisition, may be used.16 Typically, for 13C/1H spin pairs in compounds such as the copolymers examined here, it may take several seconds for the NOE to build to its steady-state value, because of the 13C/1H relaxation mechanisms involved. By turning the decoupler on only during data acquisition (off during the long repetition delay period after data acquisition), 13C signals are acquired without NOE enhancement. Not all systems give the full NOE enhancement possible. The NOE effect operates via 1H-13C (usually dipolar) coupling; maximum 13C NOE values are obtained when the heteronuclear dipolar interaction with protons is the overwhelmingly dominant relaxation mechanism of the observed carbons. Alternate 13C relaxation mechanisms, such as chemical shift anisotropy, 1H13C scalar coupling, or dipolar interactions with the unpaired electrons of a paramagnetic impurity, can compete with protonmediated dipolar 13C relaxation and lead to less than full NOE enhancement. The NOE ratio, f, as used here, is defined as the ratio of 13C signal integrals with (Iz) and without (I0) the nuclear Overhauser enhancement: or

f ) Iz/I0 ) η + 1

(11)

η ) (Iz - I0)/I0

(12)

where the symbol η is called the nuclear Overhauser enhancement factor. When NOE factors were measured in the study reported here, the data were acquired in blocks of roughly 1 or 2 h apiece, (19) (a) McCall, D. W. Acc. Chem Res. 1971, 4, 223-232. (b) Sefcik, M. D.; Schaefer, J.; Stejskal, E. O.; McKay, R. A. Macromolecules 1980, 13, 11321137. (20) Freeman, R. Double Resonance. In Encyclopedia of Nuclear Magnetic Resonance; Grant, D. M., Harris, R. K., Eds.; Wiley: New York, 1999; pp 1740-1749.

Table 6. Some Alternate Methods for Determining Triad Mole Fractions from Collective Assignment Region Integrals method

triads

Hsieh-Randall

k[EHE] ) B k[EHH] ) 2(G - B - A) k[HHH] ) 2A + B - G k[HEH] ) F k[EEH] ) 2(G - A - F) k[EEE] ) (1/2)(A + D + F - 2G) k[EHE] ) B k[EHH] ) C1 k[HHH] ) A - (1/2)G k[HEH] ) F k[EEH] ) E k[EEE] ) (1/2)D - (1/2)G - (1/4)E k[EHE] ) B k[EHH] ) C1 k[HHH] ) A1 - (1/2)A2 (where A1 + A2 + A3 ) A) k[HEH] ) F k[EEH] ) E k[EEE] ) (1/2)D - (1/2)G - (1/4) E linear least-squares analysis with the constraint [EEH] +2[HEH] ) [EHH] +2[EHE]

Randall

Figure 5. 126-MHz liquid-state 13C NMR spectrum of sample S: (A) ×1; all peaks on scale. (B) ×100; most intense peaks are off scale.

Seger

interleaving continuous decoupling with gated decoupling (i.e., with and without NOE), although NOE values obtained without interleaving did not differ significantly from those obtained with interleaving. Numerous NOE determinations were made on the B and S polymer solutions, and substantial experimental errors were encountered, as detailed in the Supporting Information. The “best” NOE values were chosen as those NOE values determined immediately before or after acquisition of the high-S/N spectrum being analyzed quantitatively (to minimize unintended variations in temperature or other experimental parameters). The best values, obtained from high signal-to-noise 126-MHz 13C NMR spectra of samples B and S, are shown in Table 5. These “best” NOE factors were reproducible only to within about (0.2 ((0.1 for the most intense peaks, and > (0.2 for the smallest peaks) and appear to be often significantly smaller than NOE factors reported in the literature for similar polymer structures. For example, Hansen et al.21 reported an observed NOE factor of 2.67 for the large δ+δ+ peak (region D2, 29.98 ppm) in a low-density polyethylene sample containing butyl side chains (in addition to ethyl and hexyl branches), as compared to the NOE factors of 2.43 and 2.59 determined here for samples B and S, respectively. Dissolved oxygen may be responsible in part for the differences, as samples B and S were not intentionally degassed, whereas Hansen et al. purged their sample with nitrogen gas before analysis. It should be noted, however, that Hansen et al. did report that most of the NOE factors that they determined are significantly below the theoretical maximum of 2.99, as is also seen in the study reported here (Table 5). Additional details on the determination and interpretation of NOE values in this study are given in the Supporting Information. Chain Ends and Head-to-Head, Head-to-Tail, and Tailto-Tail Linkages. Figure 5 shows the 126-MHz 13C NMR spectrum obtained for sample S (with CW decoupling and NOE enhancement), which can be compared to the corresponding spectrum for sample B shown in Figure 2. Considering the chainend group assignments made by Hsieh and Randall11 shown in Table 2, both samples B and S contain easily detected saturated end groups, as indicated by the 3s and 2s peaks at 32.2 and 22.9 ppm, respectively (the 1s peak overlaps with the 1B4 peak, the methyl at the end of each butyl branch). No peaks were observed for the 2v and 1v carbons (139.5 and 114.3 ppm, respectively) of

an unsaturated (vinylic) chain end; the corresponding allylic carbon peak (labeled “a” in Table 2) at 33.9 ppm, if present, would be obscured by the 4B4 peak at 34.1 ppm. Assuming the chainend groups are all saturated, the number-average molecular weight is determined (by 13C NMR) to be 5.1 × 104 g/mol for sample B and 3.0 × 104 g/mol for sample S. On average, the polymer chains in sample B contain 1565 ethylene monomer units and 88 1-hexene monomer units, while polymer chains in sample S (on average) contain 964 ethylene monomer units and 36 1-hexene monomer units, as determined by these 13C NMR results. An interesting issue is the possibility of head-to-head or tailto-tail polymerization of 1-hexene monomers. Randall7,9 and other authors reported no signs of such linkages between consecutive 1-hexene monomers. The Lindemann and Adams method22 predicts 13C chemical shifts of 37.05 and 39.52 ppm for the methine carbons of the head-to-head and tail-to-tail cases, respectively. Examination of Figures 2 and 5 indicates that very small amounts of these linkages might be present in both samples S and B, as there appear to be very small peaks at 37.5 and 39.6 ppm that are barely discernible above the noise in these high-sensitivity spectra. The 4B4 carbon in the tail-to-tail linkage may also be resolvable, with a chemical shift of 31.78 ppm, as predicted by the Lindemann and Adams method; there is a very small peak at 31.5 ppm in both Figures 2 and 5 which may be due to this carbon site. In any case, if head-to-head or tail-to-tail linkages exist in these samples, they represent less than 1% of HH diads. A more precise value cannot be determined, as these very small peaks are very difficult to integrate meaningfully. Comparison of Integration Methods and Triad Expressions. Table 6 shows some alternate methods (algorithms) for determining triad mole fractions from integrals of collective assignment regions, including methods developed in this study. The Hsieh-Randall triad method11 is shown in eqs 3-8; the

(21) Hansen, E. W.; Blom, R.; Bade, O. M. Polymer 1997, 38, 4295-4304.

(22) Lindemann, L. P.; Adams, J. Q. Anal. Chem. 1971, 43, 1245-1252.

Solver


5741

Table 7. Triad Mole Fractions Determined with NOE for Poly(ethylene-co-1-hexene) Sample B by Various Triad Methods Defined in Table 6, Using Various Methods of Peak Integrationa,b peak integration method triad method Hsieh-Randall

Randall

Seger

machine integral (mol %)b

manual integral (mol %)a

deconvolution integral (mol %)a

[EHE] ) 4.76 ( 0.11 [EHH] ) 0.89 ( 0.41 [HHH] ) - 0.25 ( 0.29 [HEH] ) 0.37 ( 0.36 [EEH] ) 9.67 ( 0.29 [EEE] ) 84.55 ( 0.31 [EHE] ) 4.78 ( 0.11 [EHH] ) 0.44 ( 0.06 [HHH] ) - 0.02 ( 0.10 [HEH] ) 0.37 ( 0.07 [EEH] ) 9.47 ( 0.19 [EEE] ) 84.95 ( 0.27 [EHE] ) 4.78 ( 0.11 [EHH] ) 0.44 ( 0.06 [HHH] ) 0.02 ( 0.03 [HEH] ) 0.37 ( 0.07 [EEH] ) 9.47 ( 0.20 [EEE] ) 84.92 ( 0.22

[EHE] ) 4.74 ( 0.16 [EHH] ) 0.84 ( 0.35 [HHH] ) - 0.20 ( 0.24 [HEH] ) 0.34 ( 0.04 [EEH] ) 9.64 ( 0.16 [EEE] ) 84.63 ( 0.15 [EHE] ) 4.76 ( 0.16 [EHH] ) 0.42 ( 0.09 [HHH] ) 0.01 ( 0.09 [HEH] ) 0.35 ( 0.04 [EEH] ) 9.42 ( 0.15 [EEE] ) 85.03 ( 0.25 [EHE] ) 4.76 ( 0.16 [EHH] ) 0.42 ( 0.09 [HHH] ) 0.02 ( 0.02 [HEH] ) 0.35 ( 0.04 [EEH] ) 9.42 ( 0.15 [EEE] ) 85.03 ( 0.26

[EHE] ) 4.76 ( 0.02 [EHH] ) 0.96 ( 0.09 [HHH] ) - 0.19 ( 0.09 [HEH] ) 0.35 ( 0.03 [EEH] ) 9.78 ( 0.13 [EEE] ) 84.34 ( 0.14 [EHE] ) 4.79 ( 0.02 [EHH] ) 0.43 ( 0.04 [HHH] ) 0.07 ( 0.07 [HEH] ) 0.35 ( 0.03 [EEH] ) 9.47 ( 0.05 [EEE] ) 84.89 ( 0.09 [EHE] ) 4.79 ( 0.02 [EHH] ) 0.43 ( 0.04 [HHH] ) 0.04 ( 0.02 [HEH] ) 0.35 ( 0.03 [EEH] ) 9.47 ( 0.05 [EEE] ) 84.92 ( 0.08 [EHE] ) 4.80 ( 0.03 [EHH] ) 0.47 ( 0.03 [HHH] ) 0.10 ( 0.04 [HEH] ) 0.32 ( 0.03 [EEH] ) 9.44 ( 0.05 [EEE] ) 84.87 ( 0.06

Solver

a

Corrected using NOE factors given in Table 5. b Indicated uncertainties equal three standard deviations.

Randall method is given in ref 4; and the Seger method is the same as the Randall method, except that [HHH] was determined using the expression, k[HHH] ) A1 + 1/2A2. In this analysis, the A region is subdivided into three subregions: A1 represents the HHHH RR carbon and is located at 41.4 ppm, A2 represents the EHHH RR carbon at 40.9 ppm, and A3 represents the EHHE RR carbon at 40.2 ppm. Not surprisingly, no discernible peak is observed in the A1 region of any of the spectra obtained in this study since the HHHH content of these polymers is extremely low. Using the number molecular weight estimates indicated earlier, it can be estimated that only one in eighty polymer chains of sample B contains a single HHHH tetrad, while only one in six hundred polymer chains of sample S contains a single HHHH tetrad. Analysis of the indicated random errors (given as two standard deviations) indicates that in general the reproducibility is significantly better using deconvolution-derived integrals than those obtained by either the “machine” or “manual” integration methods. Similarly, the Seger triad method has better reproducibility than either the Hsieh-Randall or Randall triad methods. Included in Table 6 is a fourth triad method denoted the Solver method. This linear regression method uses the Solver subroutine of Microsoft Excel to determine the best least-squares fit of the “deconvolution” integral data to the region assignment relationships (as shown in the Supporting Information, eqs SI-13-SI-22), requiring that the necessary relationship, [EEH] + 2[HEH] ) [EHH] + 2[EHE], be maintained. As discussed below, the Solver linear regression method appears to be superior, as far as reproducibility, to the other triad methods shown, which all involve analytical formulas to determine triad mole fractions. The Solver method may be expected to be affected less by systematic errors than the other methods, as all the spectral integral information is used simulta5742 Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

neously. A similar conclusion was reached by Cheng,23 who used a simplex routine to determine the best triad mole fraction values in ethylene/propylene copolymers. Table 7 shows triad mole fractions, based on NMR intensities that have been corrected for T1 and NOE, determined for sample B, using three different methods of integration (machine, manual, and deconvolution integrals, as discussed above) and using three or four different sets of formulas to determine triad mole fractions. A conspicuous problem with the results from the Hsieh-Randall method, as shown in Table 7, is the negative value determined for [HHH], a clearly unrealistic result. It is not clear why this method gives a [HHH] value significantly below zero, but negative [HHH] values were obtained in almost every instance in which the Hsieh-Randall method was utilized in this study. Systematic errors appear to be involved but have not been identified. As is apparent in Table 6, the algebraically overdetermined system of interest here allows for many possible expressions for [HHH]. Table 8 has a somewhat expanded set of algorithms for [HHH], but still includes only most of the simpler algebraic expressions for [HHH] that are possible; more complicated algorithms (i.e., with more terms) were avoided, as the cumulative uncertainty increases as more integral terms are included in the [HHH] expression. Note that [HHH] is more susceptible to random error than the other triad mole fractions, owing to the low level of 1-hexene incorporation in samples B and S, which results in very small [HHH] values. The negative [HHH] values in Table 7 are in most cases within two standard deviations of zero. Evaluation of uncertainties reveals that the Solver method provides the most reproducible [HHH] values. One finds in Table (23) Cheng, H. N. Anal. Chem. 1982, 54, 1828-1833.

Table 8. Some Alternate Expressions for Determining [HHH], Using Manual Integrals of Spectra Obtained with NOE11,14,18,a [HHH] mol % expression

sample B

sample S

k[HHH] ) 2A + B - G Hsieh-Randall method11 k[HHH] ) A - (1/2)C1 Randall method4 k[HHH] ) 3G - C k[HHH] ) C2c - B k[HHH] ) A1 + (1/2)A2 Seger method k[HHH] ) (1/2) (4A + E + 2F - C1-2G) k[HHH] ) G - B - C1 k[HHH] ) 2G - C2A - C2B k[HHH] ) (1/2) (2G - E - 2F - C1) k[HHH] ) 2A - C + G + 2F + E Randall4 k[HHH] ) (1/2) (2C2c - E - 2F + C1)

-0.21

-0.31

-0.02

-0.02

0.14 0.25 0.02

0.87 -0.06 0.02

-0.08 0.17 0.22 0.04 0.03

-0.31 0.28 0.53 0.27 0.28

0.12

-0.07

0.06 0.14 0.01

0.13 0.36 0.02

mean of above values standard deviation of above values Solver linear regression method a

Corrected using NOE factors given in Table 5.

8 that the value determined by the Solver method for [HHH] is smaller for sample B than for sample S, although sample B has greater 1-hexene incorporation. However, taking the estimated (0.02 mol % uncertainty into account, the [HHH] values of both sample B and sample S are not significantly nonzero. We know from Figures 2 and 5 that the RR(EHHH) carbon peak is clearly present at 40.9 ppm in the 13C NMR spectra of both samples, so polymers B and S must contain some HHH triads, although we cannot quantify the HHH mole fraction meaningfully, even in the highest sensitivity spectra obtained. Table 9 shows triad mole fractions obtained for samples B and S, from 13C spectra acquired with CW decoupling (i.e., with NOE enhancement and with NOE correction applied), integrated by the manual method and analyzed using the published HsiehRandall, Randall, and Solver triad methods. Based on the reproducibility considerations mentioned above, the Solver values are thought to be most reliable. Summing the Solver values for [EHE], [EHH], and [HHH] of sample B gives a value of 5.32 mol % for [H]; i.e., 1-hexene is incorporated into the copolymer at a level of 5.32 mol %. The corresponding value for sample S is 3.62 mol %. Use of the Hsieh-Randall triad method yields the results, [H] ) 5.35 mol % for sample B and 3.84 mol % for sample S; similarly, the Randall triad method gives values of 5.17 and 3.56 mol % for samples B and S, respectively. The left three columns of Table 10 show the effect of NOE correction on triad and monad mole percentages, obtained in one set of experiments (run a) for polymer B. Each of the spectra was obtained with 500 transients accumulated with CW decoupling, and each was analyzed by the deconvolution integral method. The NOE correction factors used to determine the values in the right three columns of Table 10 are taken from Table 5; these factors were obtained from careful analysis of high signalto-noise spectra of the same sample. The left three columns allow one to compare triad results obtained under NOE-assisted conditions (with or without NOE corrections applied) with triad results

obtained without NOE assistance (gated decoupling). The resulting triad mole percentages shown in Table 10 are consistent to the nearest 0.2 mol %, except for [EEH] and [EEE]; the latter is consistent only to ∼1 mol %. Values of the 1-hexene monomer contents, [H], shown in Table 10 differ significantly (whether NOE corrections are applied) for the two NOE approaches, considering the random error (as indicated by the 90% confidence limits shown). By comparing triad mole percentages obtained with and without NOE corrections, it appears that the use of NOE-uncorrected values from NOEassisted experiments somewhat overestimates the contribution of H-containing triads and underestimates the contribution of [EEE]. A direct comparison of the triad mole percentages obtained from polymer B data obtained without NOE with those derived from data obtained with NOE (whether corrected for NOE differences; see Table 10) is made difficult by the much larger random errors obtained from the lower signal-to-noise spectra obtained by gated decoupling (i.e., without Overhauser enhancement). Assuming that the results derived with gated decoupling are the most accurate (although less precise) values in Table 10, since no systematic error due to NOE variations or enhancement corrections is present, it cannot be determined whether correcting the different NOE values among the various regions actually provides more accurate triad mole fractions than using NOE-uncorrected data obtained with NOE assistance. Hence, it appears that, for determining triad fractions to within ∼0.2 mol % (at least for the polymer composition of sample B), if one obtains data with the time-saving assistance of the NOE, one cannot readily justify spending the extra time of determining NOE correction factors. The right two columns of Table 10 show NOE-corrected results determined using a higher signal-to-noise spectrum (run b; 5740 transients), obtained in a separate experiment using the same sample of polymer B, utilizing both manual and deconvolution integration methods. Random errors (90% confidence limits) in the run b triad mole fractions are estimated to be less than (0.02 mol %, based on the standard deviations of the run a triad mole fractions and the greater number of transients involved in the run b spectrum. One sees a strong correspondence between the run a (left three columns) and run b (right two columns) triad mole fractions; the greatest relative difference appears to be in the [HHH] value. On the basis of numerous repetitive integrations, it appears that for weak peaks, such as those of the A region, one can place greater confidence in integrals obtained by the manual method than those obtained by the deconvolution method. It appears that the deconvolution method overestimates the [HHH] triad content, which is very sensitive to the Al region integral. For this reason, taking into account both the precision (apparently better for the deconvolution method) and suspected systematic errors of the various integration methods, we conclude that the “best” triad mole fractions experimentally obtained for this set of spectra (run b) are those shown for run b with manual integration, presented as the right-most column in Table 10. These triad mole fractions are used in subsequent sequence parameter analyses, along with similarly obtained results for polymer S. Sequence Distributions. Accurate triad mole fractions provide information about the sequence distribution of monomers within the copolymer, which in turn is related to the physical properties of the polymer and the mechanism of polymerization. Analytical Chemistry, Vol. 76, No. 19, October 1, 2004

5743

Table 9. Triad Mole Fractions Determined for Poly(ethylene-co-1-hexene) Samples by Various Methods,4,11,18 Using Spectra Obtained with NOE and NOE Corrections (from Manual Integration Data) method

triads

Hsieh-Randall

k[EHE] ) B k[EHH] ) 2(G - B - A) k[HHH] ) 2A + B - G k[HEH] ) F k[EEH] ) 2(G - A - F) k[EEE] ) (1/2)(A + D + F - 2G) k[EHE] ) B k[EHH] ) C1 k[HHH] ) A - (1/2)G k[HEH] ) F k[EEH] ) E k[EEE] ) (1/2)D - (1/2)G - (1/4)E linear least-squares analysis with the constraint [EEH] + 2[HEH] ) [EHH] + 2[EHE]

Randall

Solver

sample B results (mol %)

sample S results (mol %)

[EHE] ) 4.67 [EHH] ) 0.89 [HHH] ) - 0.21 [HEH] ) 0.34 [EEH] ) 9.55 [EEE] ) 84.76 [EHE] ) 4.68 [EHH] ) 0.51 [HHH] ) -0.02 [HEH] ) 0.34 [EEH] ) 9.46 [EEE] ) 85.03 [EHE] ) 4.76

[EHE] ) 3.18 [EHH] ) 0.97 [HHH] ) -0.31 [HEH] ) 0.49 [EEH] ) 6.34 [EEE] ) 89.33 [EHE] ) 3.20 [EHH] ) 0.38 [HHH] ) -0.02 [HEH] ) 0.50 [EEH] ) 5.80 [EEE] ) 90.13 [EHE] ) 3.18

[EHH] ) 0.55 [HHH] ) 0.01 [HEH] ) 0.31 [EEH] ) 9.44 [EEE] ) 84.93

[EHH] ) 0.42 [HHH] ) 0.02 [HEH] ) 0.49 [EEH] ) 5.79 [EEE] ) 90.10

Table 10. Comparison of NOE-Corrected Triad Mole Fractions Determined for Sample B by the Solver Analysis Method, Using the Deconvolution and Manual Integration Methodsa mole fractions (as mol %) by various NOE and integration methods

n-ad mol fractn [EHE] [EHH] [HHH] [HEH] [EEH] [EEE] [E] [H]

run a gated decoupling (no NOE) deconvolution integrals

run a CW decoupling (uncorrected for NOE) deconvolution integrals

run a CW decoupling (corrected for NOE)b deconvolution integrals

run b CW decoupling (corrected for NOE)b deconvolution integrals

run b CW decoupling (corrected for NOE)b manual integrals

4.91 ( 0.29 0.50 ( 0.08 0.19 ( 0.17 0.29 ( 0.16 9.74 ( 0.62 84.38 ( 0.30 94.40 ( 0.30 5.60 ( 0.30

5.15 ( 0.02 0.51 ( 0.03 0.16 ( 0.03 0.41 ( 0.02 10.00 ( 0.03 83.77 ( 0.05 94.18 ( 0.06 5.82 ( 0.05

4.80 ( 0.03 0.47 ( 0.03 0.10 ( 0.04 0.32 ( 0.03 9.44 ( 0.05 84.87 ( 0.06 94.62 ( 0.08 5.38 ( 0.06

5.13 0.51 0.06 0.41 9.95 83.93 94.29 5.70

4.76 0.55 0.01 0.31 9.44 84.93 94.68 5.32

a Run a involves six spectra of 500 transients each, while run b involves a single spectrum of 5740 transients; both runs used continuous CW decoupling. Values shown for run a represent the mean values. b NOE correction factors given in Table 5.

In commercial poly(ethylene-co-1-hexene) with less than 6 mol % 1-hexene, the presence of butyl branches disrupts the crystal packing of polyethylene chains and lowers the polymer density and melting point, improving processability.24 It is thought that widely dispersed butyl branches are more effective in this regard than would be an equal number of butyl branches in mutual proximity. As ethylene and 1-hexene are polymerized, a variety of monomer sequences may be created, including ethylene-only portions of the polymer chain, 1-hexene-only segments, and sequences with both ethylene and 1-hexene units included. A knowledge of the concentrations of diads and higher order n-ads provides information about the number and types of sequences present. In practice, the resolution achievable in the 13C NMR spectrum of poly(ethylene-co-1-hexene) makes triad analysis practical; although tetrad-level descriptions may be achievable, pentad and higher n-ad analyses do not appear to be practical with commonly available 13C NMR capabilities. (24) Wignall, G. D.; Alamo, R. G.; Laondono, J. D.; Mandelkern, L.; Kim, M. H.; Lin, J. S.; Brown, G. M. Macromolecules 2000, 33, 551-561.


Sequence distribution information can provide insight regarding the mechanism of polymer catalysis. If the addition of either ethylene or 1-hexene to the growing polymer chain occurs after the rate-determining step of the catalytic cycle, the probability of adding 1-hexene instead of ethylene (PH) may not depend on the identity of the previous monomer added; in this case, a growing polymer chain ending in 1-hexene would be as likely to add another 1-hexene as would a chain ending in ethylene. Such a polymer is known as a Bernoullian copolymer6 and is characterized by a single probability for adding a 1-hexene monomer, independent of the identity of the last monomer unit added. For Bernoullian copolymers, a specific random distribution of 1-hexene is obtained, with a predictable set of diad and triad mole fractions (as shown in Table SI-4 in the Supporting Information). Clearly, block copolymers or alternating copolymers do not correspond to a Bernoullian triad distribution. It should be noted, however, that not all random copolymers give the same triad distributions. Consider a situation in which 1-hexene is twice as likely to add to a growing polymer chain to which another

1-hexene was the last monomer unit added, relative to its adding to a chain with an ethylene added last. The triad distribution expected would be quite different from a Bernoullian polymer, with larger values for [EHH] and [HHH]. A first-order Markov sequence is a random sequence distribution of a type that can be generated on the basis of two independent probability parameters, e.g., PE/H and PH/E, where PE/H represents the probability of adding a 1-hexene monomer to a growing chain in which ethylene was the last monomer unit added.6 Markovian statistics do not demand a chemically specific type of mechanism.25 The diad and triad mole fractions expected for a first-order Markov copolymer (given in Table SI-5 of the Supporting Information) can be different from those expected for a Bernoullian copolymer, but if PE/H ) PH/H and PH/E ) PE/E, the first-order Markov model becomes the Bernoullian model.6 A number of parameters have been developed to characterize the sequence distribution in copolymers, based on diad and triad mole fraction values. Harwood and Ritchey26 developed the run number (sometimes also know as the sequence number), the average number of times the monomer switches from E to H (or H to E, but not both, according to the updated definition given by Randall4) per 100 monomer units. In terms of triads,

run number ) 100{[EHE] + 1

PE/H ) [EH]/2[E]

(19)

PH/E ) [EH]/2[H]

(20)

and

It follows that F ) (PE/H + PH/E)-1. Another parameter defined by Coleman and Fox28 is Ω:

ΩE ) [E][EE]/[EEE]

(21)

ΩH ) [H] [HH]/[HHH]

(22)

or

An Ω value of unity is expected for both Bernoullian and firstorder Markov statistics, and thus, this parameter can serve as a test for consistency to the latter model. The cluster index was defined by Randall9 as follows:

cluster index ) 10{[H] - [EHE]}/{2[H]2 - [H]3} (23)

1

/2[EHH]} ) 100{[HEH] + /2[EEH]} (13)

The average sequence length is the average number of monomers in an E-only or H-only monomer sequence (or “run”), and was given by Harwood and Ritchey26 in terms of triads as

nE ) {[EEE] + [EEH] + [HEH]}/{[HEH] + 1/2[EEH]} (14) nH ) {[EHE] + [EHH] + [HHH]}/{[EHE] + 1/2[EHH]} (15) The persistence ratio, F, was defined by Coleman and Fox27 as

F ) 2[E] [H]/[EH]

(16)

Using the necessary relationships in Table 1, this can be re-expressed as

F ) 2{[EE] + 1/2[EH]} {[HH] + 1/2[EH]}/[EH] (17) F ) 2{[EEE] + [EEH] + [HEH]}{[EHE] + [EHH] + [HHH]}/{[EEH] + 2[HEH]} (18) The persistence ratio provides a simple test for Bernoullian character, as F ) 1 for the Bernoullian case. A polymer exhibiting a persistence ratio greater than unity would have monomer clustering greater than expected by Bernoullian statistics. Since, for first-order Markov statistics,7 (25) (a) Price, F. P. J. Chem. Phys. 1961, 35, 1884-1892. (b) Price, F. P. J. Chem. Phys. 1962, 36, 209-218. (26) Harwood: H. J.; Ritchey, W. M. J. Polym. Sci., Part B 1964, 2, 601-607. (27) Coleman, B. D.; Fox, T. G. J. Chem. Phys. 1963, 38, 1065-1075.

A cluster index of 0 would indicate that all H monomer units are separated by at least one E unit (i.e., no clustering of H), while a cluster index of 10 is consistent with Bernoullian statistics. A value greater than 10 would indicate more clustering than expected by the Bernoullian model. The average reactivity ratio product, 〈r1r2〉, is also a useful measure (or predictor) of sequence distribution.29 The parameters, r1 and r2, represent the reactivities of monomers 1 and 2 in a copolymerization reaction. Randall7 and others30 have shown that, for a copolymer made by a single-site catalyst at constant comonomer concentrations, and ignoring diffusion or mixing effects, reactivity ratios can be used to relate the relative molar monomer concentration in the feedstock, M ) M1/M2, to the relative molar monomer concentration incorporated into the copolymer, m ) m1/m2 (e.g., [E]/[H]), as follows:

m ) M(r1M + 1)/(r2 + M)

(24)

If the copolymer follows first-order Markov statistics, then

r1M ) (1 - P12)/P12 ≡ (1 - PE/H)/PE/H

(25)

r2/M ) (1 - P21)/P21 ≡ (1 - PH/E)/PH/E

(26)

and the reactivity ratio product, r1r2, does not depend on M. By multiplying eqs 25 and 26, one can see that

r1r2 ) (1 - PE/H)(1 - PH/E)/PE/HPH/E

(27)

(28) Coleman, B. D.; Fox, T. G. J. Am. Chem. Soc. 1963, 85, 1241-1244. (29) Kakugo, M.; Naito, Y.; Mizunuma, K.; Miyatake, T. Macromolecules 1982, 15, 150-1152. (30) Cozewith, C. Macromolecules 1987, 20, 1237-1244.


5745

Table 11. Comparison of Sequence Distribution Parameter Values Obtained for Poly(ethylene-co-1-hexene) Samples B and Sa

parameter name persistence ratio cluster index Ω (E-based) Ω (H-based) reactivity ratio productb r1r2 av reactivity ratio productc 〈r1r2〉

Bernoullian value

first-order Markovian value

sample B value

sample S value 1.031 17.10 0.995 0.416 1.907 1.867

1 10 1 1 1

any 1 1 1

1.001 10.16 0.999 1.516 1.028

1

1

1.010

a Based on Solver results from Table 9. b Computed using eqs 19, 20, and 27. c Computed using eq 28.

The average reactivity ratio product, 〈r1r2〉, is defined on the basis of experimental diad mole fractions, as follows:29

〈r1r2〉 ) 4[EE] [HH]/[EH]2

(28)

Although this definition is not model-specific, 〈r1r2〉 can nevertheless serve as a useful parameter, since 〈r1r2〉 ) 1 if Bernoullian or first-order Markov statistics are followed during polymerization, and single-site catalysis is involved. Significant deviations of 〈r1r2〉 from unity indicate that either multiple catalytic sites (with different reactivities) are involved or non-Markovian (second-order or higher-order Markovian) distributions are involved.30 It is typical for metallocene-catalyzed poly(ethylene-co-1-alkene) polymers to yield NMR-derived r1r2 values close to (or slightly less than) unity and for corresponding polymers prepared with Ziegler-Natta catalysts to yield higher r1r2 values.30 Table 11 shows the values determined for various sequence distribution parameters for samples B and S, based on the Solver triad mole fractions given in Table 9 and the parameter definitions given above. Also included in Table 11 are values of the sequence distribution parameters expected on the basis of either Bernoullian or first-order Markov kinetics models. By all parameters, except the H-based Ω value, sample B appears to follow Bernoullian statistics very closely; the deviation of the H-based Ω value from unity is not statistically significant, since this parameter is extremely sensitive to random errors in the integral values. Sample S, on the other hand, shows greater deviations from the parameter values than are expected on the basis of Bernoullian or first-order Markov statistics (Table 11). The large deviation from unity in the average reactivity ratio product (< r1r2>) for sample S in Table 11 is consistent with the fact that this polymer was prepared from a catalyst of the ZieglerNatta type (supported titanium chloride with an aluminum alkyl cocatalyst).31 In contrast, polymer B, with a 〈r1r2〉 value near unity, was prepared using an unsupported metallocene catalyst32 and appears to result from single-site Bernoullian catalysis.23,33-35 While other mechanistic models may be capable of rationalizing the (31) J. Vanderbilt, private communication. (32) R. R. Chance, private communication. (33) Randall, J. C. J. Polym. Sci., A: Polym. Chem. 1998, 36, 1527-1542. (34) Alamo, R. G.; Blanco, J. A.; Agarwal, P. K.; Randall, J. C. Macromolecules 2003, 36, 1559-1571.

5746


deviation from unity in the average reactivity ratio for sample S, it may still be informative to examine whether a simple two-site Bernoullian model fits well to the experimental triad mole fractions. A comparison (Table SI-6) of the best (Solver) fits of one-site and two-site Bernoullian models to the experimental triad values for samples B and S shows that for the latter there is a fairly good fit of a two-site Bernoullian model to the experimentally determined triad mole fractions. This two-site Bernoullian model catalyst system would produce a mixture of polymer chains: site I produces a Bernoullian ethylene/1-hexene copolymer with 6.4 mol % 1-hexene content and contributes 57% of the monomer units in sample S, while site II appears to polymerize only ethylene and contributes 43% of sample S. The production of both ethylene homopolymer and copolymer, as evidenced by TREF experiments, has been reported previously.36 The two-site Bernoullian model reproduces the experimentally determined triads for sample S with a residual sum of squares that is 3 times smaller than that of the one-site Bernoullian model, a statistically meaningful improvement, even in view of the additional statistical degrees of freedom in the two-site model.32 The two-site Bernoullian model also produces a calculated 〈r1r2〉 value that is in excellent agreement with the experimental value. Examination of the residuals (shown in Table SI-7) indicates, however, that the quality of fit to the E-centered triad mole fractions is substantially poorer than the quality of fit to the H-centered triad mole fractions. This suggests that more complex mechanistic models (such as a two-site first-order Markovian model or a threesite Bernoullian model) may provide a significantly better fit and, thus, a more realistic model of polymerization kinetics. A similar comparison for sample B shows no statistically significant improvement in the quality of fit (only 24% reduction in the triad residual sum of squares) when a two-site Bernoullian model is used instead of the one-site model; the latter provides an excellent fit to the experimental data. Thus, we conclude that sample B is well modeled as resulting from a simple, single-site Bernoullian catalyst system. Several other poly(ethylene-co-1alkene) copolymers have been shown23,30-32 to result from singlesite or multisite catalysis models, depending on the catalyst system used to prepare the copolymer. CONCLUSIONS Two samples of poly(ethylene-co-1-hexene), samples B and S, were found to contain 5.3 and 3.6 mol % 1-hexene incorporation, respectively. Integrals obtained by spectral deconvolution were found to provide more reproducible results than by either machine or manual integration methods, although the manual method may provide more accurate [HHH] values. The accurate determination of NOE factors can be problematic, but sufficient sensitivity can be obtained with such copolymer samples that unenhanced (nonNOE) integrals can be used. Furthermore, it appears that, at least for poly(ethylene-co-1-hexene) samples with composition similar to that of sample B, triad mole fractions (with the possible exception of [EEE]) that are accurate to within (0.002 can be derived from data obtained with NOE enhancements, without having to determine/make NOE corrections. (35) Randall, J. C.; Alamo, R. G.; Agarwal, P. D.; Ruff, C. J. Macromolecules 2003, 36, 1572-1584. (36) Yau, W. W. Houston Polyolefin Symp., February 25, 2001.

Triad mole fractions were determined most reproducibly (and presumably most accurately) using a least-squares Microsoft Excel Solver linear regression. Polymer B is well described as a simple Bernoullian triad distribution, resulting from a single-site Bernoullian catalyst system of the metallocene type. On the basis of a value of the average reactivity ratio product of 1.87, polymer S does not appear to follow very closely either Bernoullian or first-order Markov statistics for a single-site catalyst, consistent with a polymer produced by a multiple-site catalyst system of the Ziegler-Natta type. The experimentally determined triad mole fractions and 〈r1r2〉 value determined for sample S fit fairly well to a two-site Bernoullian catalyst model, producing a mixture of polymer chains: site I produces a Bernoullian ethylene/1-hexene copolymer with 6.4 mol % 1-hexene content and contributes 57% of polymer S, while site II appears to polymerize only ethylene and contributes 43% of polymer S. Significant residuals in the E-centered triad mole fractions for the best fit of a two-site Bernoullian catalyst model for sample S suggest that more complex mechanistic models may be more appropriate.

ACKNOWLEDGMENT This work was supported in part by AFOSR Grant F49620-950192 and DOE Grant DE-FG03-95ER14558. We thank Drs. R. R. Chance, S. F. Rucker, and J. C. Randall of ExxonMobil and Dr. D. W. Lowman of Eastman Chemical Co. for helpful discussions. SUPPORTING INFORMATION AVAILABLE Includes additional details on isomerism in EH copolymers, “necessary relationships”, nomenclature, collective assignments for triad analysis, other NMR techniques (including 13C CP-MAS and 1H CRAMPS), and decoupler sidebands, NOE, sequence distribution and polymerization mechanisms, with five tables and seven figures. This material is available free of charge via the Internet at http://pubs.acs.org.

Received for review May 24, 2004. Accepted July 16, 2004. AC040104I


5747

Quantitative 13C NMR Analysis of Sequence Distributions in Poly

Recommend Documents