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Macromolecules 2011, 44, 208–214 DOI: 10.1021/ma101803h
Detecting Structural Polydispersity in Branched Polybutadienes Si Wan Li,† Heon E. Park, John M. Dealy,* and Milan Maric Department of Chemical Engineering, McGill University, Montreal, QC, Canada H3A 2B2
Hyojoon Lee,† Kyuhyun Im,‡ Heungyeal Choi, and Taihyun Chang Department of Chemistry and Division of Advanced Materials Science, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea
M. Shahinur Rahman and Jimmy Mays Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, United States. † Equal contribution. ‡Current address: Samsung Advanced Institute of Technology, Korea Received September 27, 2010; Revised Manuscript Received November 9, 2010
ABSTRACT: The structural details of a set of highly entangled H-shaped polybutadienes (PBDs) prepared by anionic polymerization were examined in detail by three reputable laboratories using size exclusion chromatography (SEC) and temperature gradient interaction chromatography (TGIC). While SEC data indicated that samples having the desired structures (i.e., nearly monodisperse H-shaped polymer) had been produced, additional SEC data from other laboratories showed that the samples were structurally more complex than originally thought. TGIC data revealed that while the samples did not contain high molecular weight byproducts, they did contain low molecular weight byproducts. To discern these structural details of the branched PBDs, small amounts of sample were fractionated by TGIC. By combining knowledge of the polymerization process with the TGIC data of fractionated samples, it was possible to work out the detailed compositions of the samples and the branching structures of each component.
Introduction The polymers described here were synthesized for a study of the effects of long-chain branching and polydispersity on the rheological behavior of entangled polymers, the results of which were to be used for the evaluation of molecular models of viscoelasticity. The desired structural details (arm and crossbar lengths) of these polymers were estimated prior to synthesis so that the polymers would be highly entangled but with terminal behavior accessible using available rheometric techniques. In order to facilitate the evaluation of molecular models as predictors of rheological behavior, a series of molecularly uniform H-shaped polybutadienes (H-PBDs) with a high concentration of 1,4-addition were required, and polydispersity was to be generated by blending. In this way the detailed structures of the samples would be known. H-PBD was chosen because H is the simplest structure containing two branch points, and PBD has a relatively low molecular weight between entanglements (Me = 2 kg/mol1) among polymers that can be synthesized by anionic polymerization. The preference for high 1,4-addition is due to the fact that 1,2-addition results in a vinyl side group (a short branch), and the double bond on the vinyl side group is vulnerable to cross-linking. A novel anionic polymerization method2 was used to prepare samples having the desired characteristics. While size exclusion chromatography (SEC) data indicated that this synthesis method had produced the desired materials, later analyses using SEC with different chromatographic detectors and temperature gradient interaction chromatography (TGIC) revealed that while the production of high molecular byproducts *Corresponding author. E-mail:
[email protected]. pubs.acs.org/Macromolecules
Published on Web 12/20/2010
had been obviated, the samples were considerably more complex than originally thought. This was discouraging from the point of view of the intended rheological study, but if the detailed composition of the samples could somehow be discerned, they could still be of use. We explain below how this was accomplished. The H-PBD samples are named according to the intended molecular weights of the arms and crossbars; e.g., “HA12B40” denotes a sample with four arms of equal molar mass of 12 kg/mol and a crossbar with a molar mass of 40 kg/mol. Experimental Section Synthesis of H-PBDs. A novel strategy for preparing H-PBDs by anionic polymerization was developed by Rahman et al.2 in order to minimize the formation of high molecular weight byproducts like those reported by Perny et al.3 It is based on the idea that the coupling of two side arms with trichloromethylsilane prior to any reaction with the crossbar should significantly reduce the production of high molecular weight byproducts.4 This method requires the use of high-vacuum and ampoulization techniques.5 Details of the synthesis procedures and dilute solution properties have been described previously.2 In general, the synthesis involves a difunctional diphenylethylene-based coupling agent 4-(dichloromethylsilyl)diphenylethylene (DCMSDPE), sec-BuLi, butadiene, and benzene. As shown in Figure 1, the synthesis consists of five main steps, and this reaction scheme will be used to infer the molecular structures of the resulting polymers in a later section. Determination of Isomeric Composition. To elucidate microstructures, 1H nuclear magnetic resonance (NMR) was used. As the sensitivity and resolution of NMR spectra increase with magnetic field strength, the characteristic peaks for cis and trans r 2010 American Chemical Society
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Alliance 2695 separations module and a Viscotek TDA model 302 consisting of a RI detector at a wavelength of 660 nm, a 4-capillary viscometer, and a 7 light scattering detector (LS) in 3 mW power at a wavelength of 670 nm were used for these samples. Two Polymer Laboratories Polypore columns (300 7.5 mm, 5 μm) were used, and the detectors were temperature controlled in the same oven as the Viscotek TDA. Measurements were done at 40 C with THF as the mobile phase at a flow rate of 1.0 mL/min; dn/dc was allowed to float between 0.133 and 0.138 mL/g. Each sample was injected three times, and the data were mathematically averaged before processing. The third SEC characterization (SEC-3) was performed on the final product, using a Wyatt miniDAWN LS detector in 60 mW power at a wavelength of 658 nm, and a Shodex RI-101 RI detector uses white light as the light source with two PLgel mixed-C columns (300 7.5 mm, 5 μm). This characterization was performed at 40 C with THF as the mobile phase at a flow rate of 0.8 mL/min; dn/dc was 0.128 mL/g. Determination of Molecular Weights by TGIC. Unlike SEC, TGIC6,7 is an interaction chromatographic technique in which the separation is driven by enthalpic interactions between the sample and the stationary phase. The interaction strength is controlled by varying the column temperature7,8 and eluent composition,9,10 and the molecular weight resolution is little affected by chain architecture. Therefore, TGIC is believed to be more sensitive to the chemical nature of a polymer7-9 and to have a higher resolution than SEC.3,11-13 The TGIC separations were carried out using a standard high-performance liquid chromatography (HPLC) system equipped with a C18 bonded silica column (Phenomenex, Kromasil, 300 A˚ pore, 1504.6 mm, 5 μm particle size); the mobile phase was 1,4-dioxane at a flow rate of 0.5 mL/min, and dn/dc was 0.095 mL/g. The system was equipped on line with a Wyatt miniDAWN LS detector and a Shodex RI-101 RI detector. The column temperature profile is indicated by solid lines in Figure 5 which were adjusted in a way to cover the range of molar mass of each sample.
Results and Discussion Isomeric Composition. Characteristic peaks determined by H NMR for the cis/trans 1,4 and 1,2 vinyl configurations were identified,14,15 and the corresponding areas under the peaks were calculated by the operating software. As proposed by Tanaka et al.,15 the mole fractions of 1,2 and 1,4 units can be calculated by eqs 1 and 2 where the bracket denotes the mole fraction of the respective isomeric unit indicated and I(δ) is the relative integrated intensity of the signal with chemical shift δ. The ratio of the integrated intensity of the signal of the trans type protons to that of the cis ones in the 2 ppm (aliphatic) or 5 ppm (olefinic) region indicates the ratio of trans to cis in the polymer.14,16 As noted by Santee et al., the aliphatic region is not as distinct as the olefinic region due to spin-spin coupling effects.14 Thus, the trans-to-cis ratio was calculated by eq 3 using the olefinic region. 1
Figure 1. Synthesis steps for the anionic polymerization of H-PBD.
configurations can be resolved by use of 500 MHz field. Our NMR studies were carried out on a 500 MHz Varian unity spectrometer at room temperature. A sample of 10 mg was dissolved in 0.7 mL of deuterated chloroform (Cambridge Isotope Laboratories Inc.) in 5 mm 508 Up NMR tubes. Samples were injected into the probe, shimmed, and scanned 32 times. Determination of Molecular Weights by SEC. Molecular weights of the H-PBDs were determined in three laboratories under the supervision of qualified scientists. The first characterization (SEC-1) was performed using SEC equipped with twoangle laser light scattering (TALLS) in 30 mW power at a wavelength of 685 nm, a refractive index (RI) detector at a wavelength of 680 nm, and a Viscotek differential viscometer to determine the precursors and final products. The columns were Waters Ultrastyragel HR series, HR-2, HR-4, HR-5E, and HR6E, with pore sizes 103, 104, and 105 A˚. Measurements were done at 40 C with tetrahydrofuran (THF) as the mobile phase at a flow rate of 1.0 mL/min. An average value of the refractive index increment (dn/dc) 0.130 mL/g was used in determining the molar mass of the intermediate products. The characterization was performed in parallel with the synthesis2 and thus provided information on the product of each stage of the polymerization (i.e., step 2 and step 4 in Figure 1) in addition to the final product. “Final product” refers to the result of step 5 in Figure 1 after fractionation with toluene/methanol as the solvent/nonsolvent pair.2 The second SEC characterization (SEC-2) was performed on the final product using a SEC system of Waters
Ið1:3Þ 2½1, 2 ¼ Ið2:0- 2:1Þ ½1, 2 þ 4½1, 4
ð1Þ
½1, 2 þ ½1, 4 ¼ 1
ð2Þ
Ið5:4Þ Ið5:3Þ : Ið5:4Þ þ Ið5:3Þ Ið5:4Þ þ Ið5:3Þ
ð3Þ
The mole fractions of cis 1,4 and trans 1,4 are then calculated by multiplying the above ratio by the overall 1,4 content given by eq 2. Results of isomeric content are summarized in Table 1, which shows that all samples had high levels of 1,4-addition (g94%) as had been desired.
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Characterization by SEC. It is well-known that the molar mass averages and molar mass distributions (MMDs) determined by SEC can vary significantly from one laboratory to another, and a comparison of data from 15 sources for a branched polyethylene revealed an interlaboratory variability for Mw of about 18% regardless of the detectors used.17 Possible causes of these variations in molar mass average and MMD among different laboratories have been examined in detail by Trathnigg.18 And in our study, the molar mass averages and MMDs determined in three laboratories were quite different from each other. The average molar masses determined by SEC are listed in Table 2. An “arm” is the product of step 2 in Figure 1, “1/2-H” refers to that of step 4 in Figure 1, and “final product” refers to that of step 5, although this may contain byproducts other than an H molecule, even after fractionation using toluene/methanol as the solvent/nonsolvent pair.2 Since the molar mass characterization by SEC-1 was performed in parallel with the synthesis,2 it is the only one that provides information on precursors at various stages of polymerization. In other words, it contains molar mass information on arms and 1/2-Hs. Complete elution curves at various stages of polymerizations can be found in a previous publication.2 Generally, the SEC-1 elution profiles of final products, shown in Figure 2, had single peaks and narrow distributions. Moreover, a previous study on the branching parameter g for our samples showed very good to fair agreement with literature values on H-shaped polystyrene.2 However, the RI elution profiles for the same final products characterized by SEC-2, as shown in Figure 3, had double peaks except for HA12B40. The molecular weights at the two peaks are in a ratio of about 1:2, indicating the presence of 1/2-H molecules, which are equivalent to asymmetric 3-arm stars, implying that most of the samples were actually mixtures of H molecules and asymmetric 3-arm stars (1/2-H). The weight fraction calculated from the area under the peaks after deconvolution of the RI signal by a Gaussian distribution indicates that the amount of 1/2-H material is significant, ranging from 0 to 47 wt %, as shown in Table 2. We note that the MMD obtained from SEC-2 was the broadest of the distributions from the three laboratories. The elution profiles obtained by SEC-3 shown in Figure 4 also indicate the presence of low molecular weight byproducts in most of the samples. In most cases, MMDs obtained
by SEC-3 were broader than those found by SEC-1, likely due to better separation in SEC-3. For HA30B40 and HA40B40, the molecular weights found by SEC-3 were comparable to those found by SEC-1. However, lower molecular weights were indicated for the other samples using SEC-3
Figure 2. SEC-1 (LS response at 15) elution profiles on final products; samples were analyzed in THF at 40 C at a flow rate of 1.0 mL/min.
Table 1. Microstructural Characteristic of H-PBDs microstructure sample code
cis 1,4 (%)
trans 1,4 (%)
vinyl 1,2 (%)
HA12B40 HA30B40 HA40B40 HA12B100
51 52 52 53
43 42 42 41
6 6 6 6
Figure 3. SEC-2 (RI response) elution profiles on final products; samples were analyzed in THF at 40 C at a flow rate of 1.0 mL/min.
Table 2. Summary of SEC Characterizations in Three Laboratories (Molar Mass in kg/mol) SEC-1 1
arm
/2-H
SEC-2 final product
SEC-3
final product
final product
sample code
Mn
PDI
Mn
PDI
Mn
PDI
Mn
wt %
PDI
Mpa
PDI
HA12B40 HA30B40
10.6 29.6
1.01 1.01
41 80.2
1.11 1.05
82.3 161
1.03 1.06
70.6 164
N.A.b N.A.
41.6
1.01
1.06
212
1.05
1.22
216
N.A.
HA12B100
15.3
1.03
1.07
158
1.04
100 8 92 23 77 47 53
1.25 1.32
HA40B40
70.8 93.3 174 109.6 218.8 81.3 170
1.28
114 196
N.A.
a
106 82.4
Peak molar mass. b N.A.: not available.
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Figure 4. SEC-3 (RI and LS response at 90) elution profiles on final products; samples were analyzed in THF at 40 C at a flow rate of 0.8 mL/min.
Figure 5. TGIC chromatograms for final products; samples were processed in 1,4-dioxane at a flow rate of 0.5 mL/min. Corresponding peak molar mass is shown in Table 3. Column temperature is also shown in the plot. Table 3. Peak Molar Mass Measured by TGIC peak molar mass Mp (kg/mol) sample code
peak 1
peak 2
peak 3
peak 4
peak 5
peak 6
HA12B40 HA30B40 HA40B40 HA12B100
20 65 110 103
28 117 121 170
43 131 205 198
57 164
70
83
compared to SEC-1, despite the fact that both techniques are supposed to yield absolute molecular weights. Although it is well-known that molar mass averages and MMDs vary from one laboratory to another, especially since the light scattered intensity is proportional to the square of dn/dc, using different dn/dc values would cause deviation in molar mass measurement by LS; it should not be confused with our case in which one SEC result shows a single peak while others show double peaks. Thus, we suspected that our
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samples were structurally more complex than originally thought. Characterization by TGIC. The SEC profiles shown in Figures 2-4 suggested that some of the samples were binary blends (H and 3-arm stars), but the TGIC chromatograms shown in Figure 5 indicated that all the samples were mixtures of three or more components, even those fractionated with toluene/methanol as the solvent/nonsolvent pair.2 All samples contained multiple, low molecular weight byproducts as revealed by the molecular weight corresponding to each peak, i.e., Mp shown in Table 3. Although TGIC showed higher resolution than SEC by resolving more peaks, further information is needed to identify the branching structures of these extra components. It is noted that the column temperature profile of HA12B40 is different from the other samples. A lower starting temperature is used on HA12B40 because the sample contains relatively low molecular weight components (e.g., peaks 1, 2, and 3 with Mp < 45 kg/mol). If the starting temperature of HA12B40 is chosen at 23 C, i.e., same starting temperature as the other samples, these low molecular weight components cannot be resolved. On the other hand, an increasing temperature is needed to elute any high molecular weight components beyond peak 6 of HA12B40. Inference of Structures. Currently our ability to identify branching structures is quite limited.19 A commonly used characterizing parameter is the branching contraction parameter g=ÆRg2æb/ÆRg2æl, where ÆRg2æ is the mean-square radius of gyration and the subscripts b and l denote branched and linear polymer of identical molar mass, respectively. This parameter reflects the effect of branching on ÆRg2æ at a given molar mass. Several quantitative relationships have been proposed to relate g to branching structure.20,21 Recently, van Ruymbeke et al.22 developed a method to obtain structural information for systems containing linear and star molecules. The MMD obtained from SEC was split into contributions of various structures using a statistical approach. They found that the calculated structural compositions depended strongly on the MMD used; i.e., the amounts of linear, 3-arm star, and 4-arm star material found by SEC with a RI detector (SEC/RI) were very different from those found by SEC with a multiangle laser light scattering detector (SEC/MALLS). To avoid such discrepancies, they used MMD data from the SEC/RI for low and SEC/MALLS for high molar mass material. However, the switching point from one MMD to the other is arbitrary, and we concluded that for our samples more reliable structural composition information could be obtained by taking advantage of the higher resolution of TGIC. First, a small amount of material was fractionated using TGIC separation columns, and eluates over given time intervals were collected into separate vials. Average molar masses (Mn and Mw) for each collected fraction of eluates were then determined by SEC-3. We found that the Mn of the last collected fraction Mn,highest molecular component (highest molecular weight fraction in Table 4) of each sample was close to 2 times that of 1/2-H molecules (Table 1) with an average difference of 4% between the two Mn values. Taking into account that the uncertainty of molecular weights measured by SEC/TALLS is typically (5%,23 we concluded that the highest molecular weight fraction of each sample consisted of H molecules. The lower molecular weight materials were hypothesized to be the byproducts shown in Figure 6 which was based on the synthesis mechanism shown in Figure 1, and the molecular weights of these components were estimated.
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Table 4. Estimated Structures and Compositions of H-PBDs samples
fraction name
Mn (kg/mol)
PDI
10 22 1.01 29 1.00 20 0 43 1.01 3 55 1.00 40 69 1.01 50 79 1.01 60 0 103 1.02 HA12B100 1 168 1.01 20 187 1.02 30 83 1.01 HA30B40 10 0 108 1.01 2 138 1.00 30 167 1.00 40 113 1.01 HA40B40 10 129 1.01 20 0 205 1.00 3 a Number refers to the structure identification number shown in Figure 7. HA12B40
inferred structurea
wt % 3 6 15 21 25 30 33 15 52 18 21 27 34 6 17 77
2 3 4 or 6 8 9 4 or 6 9 4 or 6 8 9 4 or 6 7
linear linear linear or 3-arm star 4-arm star 3-arm star H linear or 3-arm star 3-arm star H linear or 3-arm star 4-arm star 3-arm star H linear or 3-arm star 3-arm star H
Figure 6. The most possible byproducts formed in the H-PBD synthesis. The number shown in brackets corresponds to the structure identification number shown in Figure 7, while the length of each segment is shown in parentheses with a and b equal to the arm length shown in Table 2 and the crossbar length calculated by eq 4, respectively.
We estimated Mn,crossbar (Mn of the crossbar) by use of eq 4: Mn, crossbar ¼ Mn, highest molecular component - 4Mn, arm
ð4Þ
where Mn,arm is Mn of the arm given in Table 2. Using Mn,crossbar and Mn,arm, we calculated the Mn values of the byproducts
most likely to be produced during synthesis according to the composition of arms and crossbars indicated in parentheses in Figure 7. The formation of byproducts other than those shown in Figure 7, with higher molar masses or more complicated structures (e.g., branch-on-branch) is possible but very unlikely due to steric hindrance of the bulky aromatic groups in the DCMSDPE and the poor mobility of a heavy molecule.
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Figure 7. Schematic representations of the most probable structures of low molecular weight byproducts; the length of each segment is shown in parentheses with a and b equal to the arm length shown in Table 2 and the crossbar length calculated by eq 4, respectively.
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The TGIC data shown in Figure 8 also indicate negligible amounts of higher molecular weights byproducts compared to those shown in Figure 7. We then assigned structures to the remaining fractions by matching their Mn with those of the byproducts shown in Figure 7. Using this method, we found that the measured Mn values of the fractions of eluate shown in Table 4 were very close to the calculated Mn values of the most probable byproducts for all samples, with an average difference of 4%. Finally, to obtain the weight fraction of each structure, the RI signals shown in Figure 8 were deconvoluted using Gaussian fitting, and the results are shown in Table 4. We use HA12B40 data to illustrate this procedure. The vertical solid lines in the TGIC curve of Figure 8 indicate the time intervals to collect various fractions of the eluate, while the dashed lines are the Gaussian fits of each peak obtained from deconvolution. The measured Mn of fraction 60 (79 kg/mol) is very close to 2 times that of a 1/2-H molecule (82 kg/mol), implying that fraction 60 consists of H molecules. Taking Mn,arm to be 10.6 kg/mol, as measured on the product of step 2 in Figure 1, allows us to assign fraction 50 to H molecules with a missing arm (i.e., structure 9). Using eq 4, Mn,crossbar was found to be 36.7 kg/mol. With this information, fraction 40 (Mn=55 kg/mol) is best described as an asymmetric 4-arm star (i.e., structure 8). The same analysis was applied to the remaining fractions, whereas fractions 10 and 20 are best described as linear chains (structures 2 and 3, respectively), and fraction 30 can be identified as either a linear or an asymmetric 3-arm star (structure 4 or 6). Conclusions Size exclusion chromatography (SEC) techniques were unable to resolve the structural compositions of our long chain branched polymers containing various byproducts due to its inherent low resolution. Furthermore, data depending on sample treatment, quality of the separation columns, data acquisition, and processing revealed significant variation in SEC results among three reputable laboratories. Nonetheless, the SEC results implied that samples originally thought to be monodisperse H-shaped polymers were more complex, and this was confirmed by temperature gradient interaction chromatography (TGIC) analysis. We combined knowledge of the synthesis mechanism with TGIC data to infer the detailed structural composition of the samples. The results showed that while the synthesis method had successfully obviated the production of high-molecular-weight byproducts, our polymers were mixtures of the intended H molecules with lowmolecular-weight linear, 3-arm star, and 4-arm star byproducts. This made it possible to use these samples for further rheological and molecular modeling studies. At the same time, the results clearly demonstrated the limitation in the SECbased characterization of branched polymers and the necessity of employing complementary tools such as TGIC for a rigorous characterization.
Figure 8. Fractionation of final products by TGIC and the SEC-3 elution profile (RI response) of each collected fraction. The rectangles indicate the time intervals to collect various fractions of the eluate for SEC characterization.
Acknowledgment. The H-PBDs were produced under the National Science Foundation (NSF) grant-DMR 0604965. The Natural Sciences and Engineering Research Council of Canada (NSERC) provided financial support for the structure study. T.C. acknowledges the support from NRF via NRL (R0A-2007-00020125-0), SRC (R11-2008-052-03002), and WCU (R31-2008000-10059-0) programs. S.W.L. gratefully acknowledges the assistance at Drs. Willem deGroot, Ray Brown, and Tianzi Huang at the Dow Chemical Company for the SEC-2 data and Dr. Jacques Roovers for his valuable advice regarding anionic polymerization.
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References and Notes (1) Fetters, L. J.; Lohse, D. J.; Colby, R. H. Chain dimensions and entanglement spacings. In Physical Properties of Polymers Handbook; 2nd Ed., Springer: New York, 2006. (2) Rahman, M. S.; Aggarwal, R.; Larson, R. G.; Dealy, J. M.; Mays, J. Macromolecules 2008, 41, 8225–8230. (3) Perny, S.; Allgaier, J.; Cho, D.; Lee, W.; Chang, T. Macromolecules 2001, 34, 5408–5415. (4) Roovers, J.; Toporowski, P. M. Macromolecules 1981, 14, 1174–1178. (5) Uhrig, D.; Mays, J. W. J. Polym. Sci., Part A: Polym. Chem. 2005, 43, 6179–6222. (6) Lee, H. C.; Chang, T. Polymer 1996, 37, 5747–5749. (7) Chang, T. J. Polym. Sci., Part B: Polym. Phys. 2005, 43, 1591–1607. (8) Chang, T. Adv. Polym. Sci. 2003, 163, 1–60. (9) Gl€ ockner, G. Gradient HPLC of Copolymers and Chromatographic Cross-Fractionation; Springer Verlag: New York, 1992. (10) Pasch, H.; Trathnigg, B. HPLC of Polymers; Springer Verlag: Berlin, 1998. (11) Lee, W.; Lee, H. C.; Chang, T.; Kim, S. B. Macromolecules 1998, 31, 344–348. (12) Lee, W.; Lee, H.; Cha, J.; Chang, T.; Hanley, K. J.; Lodge, T. P. Macromolecules 2000, 33, 5111–5115.
Li et al. (13) Lee, H. C.; Chang, T.; Harville, S.; Mays, J. Macromolecules 1998, 31, 690–694. (14) Santee, E. R.; Chang, R.; Morton, M. J. Polym. Sci., Polym. Lett. Ed. 1973, 11, 449–452. (15) Tanaka, Y.; Takeuchi, Y. J. Polym. Sci., Part A-2 1971, 9, 43–57. (16) Sadeghi, G. M.; Barikani, M.; Morshedian, J.; Taromi, F. A. Iran. Polym. J. 2003, 12, 515–521. (17) D’Agnillo, L.; Soares, J. B. P.; Penlidis, A. J. Polym. Sci., Part B: Polym. Phys. 2002, 40, 902–921. (18) Trathnigg, B. Size-exclusion chromatography of polymers. In Encyclopedia of Analytical Chemistry; John Wiley & Sons Ltd.: Chichester, 2000. (19) Dealy, J. M.; Larson, R. G. Structure and Rheology of Molten Polymers: From Structure to Flow Behavior and Back Again; Hanser-Gardner Publications: Cincinnati, OH, 2006. (20) Bonchev, D.; Markel, E. J.; Dekmezian, A. H. Polymer 2002, 43, 203–222. (21) Zimm, B. H.; Stockmayer, W. H. J. Chem. Phys. 1949, 17, 1301– 1314. (22) van Ruymbeke, E.; Coppola, S.; Balacca, L.; Righi, S.; Vlassopoulos, D. J. Rheol. 2010, 54, 507–538. (23) Terao, K.; Mays, J. W. Eur. Polym. J. 2004, 40, 1623–1627.
