J. Phys. Chem. 1996, 100, 19939-19944
19939
Carbon-13 and Proton Relaxation Study of Backbone Dynamics of Poly(vinyl acetate) in Solution Sapna Ravindranathan Department of Inorganic and Physical Chemistry, Indian Institute of Science, Bangalore 560 012, India ReceiVed: April 30, 1996; In Final Form: August 20, 1996X
The dynamics of poly(vinyl acetate) in toluene solution has been examined by 13C and proton relaxation. 13C spin-lattice relaxation time and nuclear Overhauser enhancement measurements were carried out as a function of temperature at 50.3 and 100.6 MHz. The spin-lattice relaxation times for backbone protons were measured at different temperatures at 200 MHz. The relaxation data have been analyzed using the Hall-Weber-Helfand (HWH) model, which describes backbone dynamics in terms of conformational transitions and the Dejean-Laupretre-Monnerie (DLM) model, which includes bond librations in addition to conformational transitions. The parameters obtained from the analysis of 13C relaxation data were utilized to predict the proton relaxation data. The DLM model was found to be more successful in reproducing the experimental results. To study the influence of libration further, proton relaxation data for poly(vinyl acetate) over a wider range of temperature reported in the literature were analyzed by these two models. The DLM model could reproduce the experimental data at all temperatures whereas the HWH model was found to be successful only in accounting for the experimental data at high temperatures. The results demonstrate the importance of including the librational mode in the description of the backbone dynamics in polymers.
Introduction Nuclear magnetic resonance techniques have been used extensively in the study of the dynamical processes occurring in synthetic polymers. Most of these studies are based on proton and 13C nuclei, since these nuclei are present in almost all polymers. Nuclear magnetic relaxation parameters such as spin-lattice relaxation time, spin-spin relaxation time, etc., provide information about the molecular motion through their relationship to the spectral density function which is the Fourier transform of the orientational time correlation function of the relaxing dipoles. The time correlation function embodies the mechanisms and rates of the molecular motions, and obtaining information on this function is the objective of the relaxation studies. Since the 13C nucleus is of low natural abundance, the relaxation is dominated by interactions with the directly bonded hydrogens, and the interpretation of relaxation data is relatively simple. On the other hand, interpretation of proton relaxation data is complicated mainly by the interactions between several neighboring protons and the strong dependence on interproton distances, which makes it necessary to have a knowledge of the local conformation of the polymer. However, measurements of proton relaxation data in addition to 13C data are significant in testing the various models of polymer motion since proton relaxation data offer a means of probing the spectral density over a wider frequency range. There are several models which describe backbone motion in terms of the conformational transitions which occur in short segments of the chain.1-3 However, recent studies of 13C relaxation in several polymers have demonstrated that more localized motions, namely librations of the C-H vectors of the backbone, should also be considered in addition to the conformational transitions, in order to understand the differences in the dynamics of the C-H vectors at different sites of the backbone.4-6 The importance of including the librational mode has been noticed in particular at the T1 minimum and on the higher correlation time side of the T1 minimum. In the present study, 13C spin-lattice relaxation times (T1 ) and nuclear Overhauser enhancements (NOEs) of poly(vinyl X
Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01226-9 CCC: $12.00
acetate) (PVA) have been measured in toluene solution as a function of temperature at 50 and 100 MHz. Proton spin-lattice relaxation times have also been measured at different temperatures at 200 MHz. The 13C and proton relaxation data have been analyzed in terms of the Hall-Weber-Helfand (HWH) model,7 which describes dynamics in terms of conformational transitions along the backbone, and the Dejean-LaupretreMonnerie (DLM) model,4 which includes bond librations in addition to conformational transitions. In order to study the influence of the librational mode in proton relaxation, we have compared the performance of the two models in accounting for the observed proton relaxation data, since the parameters for backbone motion determined from the 13C relaxation data should also be able to account for the proton relaxation data. It should be noted, however, that computer simulation studies indicate that the orienatation correlation functions associated with the proton and 13C relaxation could be different.8 The influence of the librational mode is more at the T1 minimum, and for PVA it occurs at low temperature. We have therefore analyzed the proton relaxation data on PVA reported by Heatley and Cox,9 which includes measurements at low temperatures. In this case also, we have compared the performance of the HWH and DLM models in order to understand the influence of libration in proton relaxation, especially in the region of the T1 minimum. The same experimental data had been analyzed earlier in terms of several models describing backbone dynamics in terms of conformational transitions.10,11 It was found that none of these models could account for the experimental data at low temperatures. We show here that the DLM model successfuly accounts for the experimental observations at all temperatures whereas the HWH model is satisfactory only at temperatures well above the T1 minimum. Experimental Section The PVA sample used in the present study was obtained from Aldrich and has a weight average molecular weight 124 800. The sample used for NMR experiments were 10% (w/v) in toluene-d8. The 13C nuclear magnetic resonance experiments were carried out on Bruker ACF-200 and Bruker AMX-400 spectrometers © 1996 American Chemical Society
19940 J. Phys. Chem., Vol. 100, No. 51, 1996
Ravindranathan
operating at 50.3 and 100.6 MHz, respectively, for the 13C nucleus. The sample temperature was regulated to (1 K. The spin-lattice relaxation times were measured by the inversion recovery technique, which uses a 180°-τ-90° pulse sequence. The delay times between two sequences were 5 times longer than the highest T1, among those which were to be determined simultaneously. An initial estimate of the T1 values was obtained from preliminary experiments. A total of 200-300 acquisitions were accumulated for each set of 14 arrayed τ values. The T1 values were determined by fitting the signal intensities as a function of delay time to a three-parameter exponential function.12 The nuclear Overhauser enhancements (NOEs) were measured by comparing 13C signal intensities of spectra acquired by continuous 1H decoupling and inverse gated decoupling. In the NOE experiments, delays of 10T1 were used between the acquisitions. Experiments were performed on undegassed samples, since the relaxation times of interest are less than 4 s, and in such cases the presence of dissolved oxygen is not expected to contribute significantly to relaxation.13 The proton relaxation experiments were carried out on the Bruker ACF-200 spectrometer. The spin-lattice relaxation time measurements were carried out by the inversion recovery technique with homonuclear decoupling.9 For the measurement of the relaxation time of the methine proton (A), the methylene signal is saturated by a decoupling field. Similarly, for the methylene proton (X), the relaxation time is measured by saturating the methine proton resonance. Samples for proton experiments were degassed and sealed. The measured T1’s are accurate to about 8% or better, and the NOEs are accurate to about 15%. The accuracy of the measured relaxation time for the methylene proton is only about 10%, since the signal is close to that of the methyl proton of the side chain.
[ ] [( [( [ ] [( [(
2 1 1 µ0pγH ) TAA 10 4π
+ rAA6 4J(ωA - ωX) + 12J(ωA) + 24J(ωA + ωX)
2 1 µ0pγH 1 ) TXX 10 4π
]
1 N µ0pγHγC 2 ) [J(ωH - ωC) + 3J(ωC) + 6J(ωH + ωC)] T1 10 4πr3 (1)
[
]
6J(ωH + ωC) - J(ωH - ωC) γH (2) γC J(ωH - ωC) + 3J(ωC) + 6J(ωH + ωC)
where γC and γH are the magnetogyric ratio of the 13C and 1H nuclei, respectively, µ0 is the permeability of vacuum, N is the number of directly bonded protons, r is the C-H internuclear distance, p ) h/2π where h is the Planck’s constant, and ωH and ωC are the Larmor frequencies of 1H and 13C, respectively. We have used an r value of 1.09 obtained from quantum chemistry for all the C-H distances.15 The phenomenological equations governing the time evolution of the longitudinal magnetization of the methine and methylene protons involve cross-relaxation terms.16 In the inversion recovery experiment with decoupling, the cross-term is held constant by the perturbing field. The relaxation of the unperturbed nucleus will therefore be exponential with time constant TAA in the case of the A{X} experiments or TXX in the case of X{A} experiments. Assuming only intramolecular contributions to the relaxation, the time constants are given by9 where ωA is
)]
)]
(3)
3J(ωX) + 12J(ωX)
rAX6
)]
(4)
the proton Larmor frequency and ωA - ωX is the observed chemical shift difference between the methine and methylene protons. The expressions for the time constants involve interproton distances. Heatley and Cox9 have calculated the effective internuclear distances by averaging over several conformations and tacticities. We have employed these reported values in the calculations, namely, rAA ) 2.726 Å, rAX ) 2.415 Å, and rXX ) 1.66 Å. The spectral density function J(ω) is obtained by the Fourier transformation of the orientation correlation function. The correlation function is derived on the basis of specific models for polymer motion. The models employed in this study are described below. (i) Hall-Weber-Helfand (HWH) Model. The HWH model takes into account pair transitions involving simultaneous rotations about two bonds and isolated transitions which involve rotations about one bond. The spectral density is given by4,7
J(ω) ) Re
[
1 (R + iβ)1/2
]
(5)
where
R)
Assuming a purely 13C-1H dipolar relaxation mechanism, the expressions for the the spin-lattice relaxation time, T1, and the nuclear Overhauser enhancement, NOE, under conditions of complete proton decoupling are given by14
NOE ) 1 +
2
rAX6
+ rXX6 2J(ωA - ωX) + 6J(ωX) + 12J(ωA + ωX)
Relaxation Equations and Motional Models
[
)]
6J(ωA) + 24J(2ωA)
2
2 1 + - ω2 2 τ τ τ0 0 1
(6)
and
β ) -2ω
[
1 1 + τ0 τ1
]
(7)
Here, τ1 and τ0 are the correlation times for pair and isolated transitions, respectively. This model considers only a single motional mode, namely conformational transitions of the backbone for describing the dynamical process. (ii) Dejean-Laupretre-Monnerie (DLM) Model. Recently there has been considerable emphasis in the literature on the necessity to consider two classes of motions occurring on well-separated time scales for a better description of the backbone motion, and this was first proposed by Dejean et al.4 The HWH model underestimates the value of T1 at the minimum, and it fails to account for the different local dynamics observed at different carbons of the polymer backbone. The DLM model overcomes these deficiencies by superimposing an additional independent motion on the backbone rearrangement of the HWH model. The additional motion involves librations of the C-H vectors of the backbone. The DLM spectral density function is given by
J(ω) )
f τl 1-f + 1/2 (R + iβ) 1 + ω2 τl2
(8)
Inclusion of the librational motion introduces two additional
Dynamics of Poly(vinyl acetate) in Solution
J. Phys. Chem., Vol. 100, No. 51, 1996 19941
parameters, namely the correlation time for libration, τl, and f, the relative weight of the librational component.
TABLE 1: Experimental Carbon-13 Spin-Lattice Relaxation Times (ms) and NOEa for BackBone Carbons of PVA as a Function of Temperature at Two Magnetic Fields
Numerical Calculations
CH
Relaxation data were analyzed by using the spectral density functions described in the previous section. In order to obtain a fit of T1 as a function of temperature, we have introduced the temperature dependence of correlation times in the spectral density expression. Helfand17 has applied Kramers’18 theory for the diffusion of a particle over a potential barrier to conformational transitions in a polymer. According to this theory, the correlation time τ, for a conformational transition involving an energy barrier Ea, is written as
τ ) AeEa/RT
(9)
where A ) ηc, η being the viscosity and c a molecular constant. In the calculations, the activation energy Ea and the preexponential factor, A, were taken as adjustable parameters. The temperature-dependent 13C T1 and NOE values at both the frequencies were fitted simultaneously. Experimental relaxation data at 263 and 283 K were not included. For the HWH model, the experimental T1 and NOE values for the methine carbon at different temperatures were used as input data. The parameter τ0/τ1 was varied manually, and the values of A and Ea which provided the best fit were obtained. Fitting was achieved by minimizing the sum of the squares of the deviations between calculated and experimental values using a simplex routine. The parameters thus obtained were used to predict the relaxation data of the methylene carbon and backbone protons. In the case of the DLM model parameters A, Ea, and f were included in the fitting procedure, and the remaining parameters, namely τ0/τ1 and τ1/τl, were adjusted manually. The experimental data corresponding to the methine carbon were utilized to obtain the model parameters. These parameters were held constant while the values of f giving the best fit to the methylene carbon data were determined. The parameters for the DLM model which were obtained from the experimental 13C relaxation data were then employed to predict the relaxation data of the backbone protons. In the calculation of the proton relaxation data using the DLM model, we assume that the CH2 or CHR (where R is the side group) unit is rigid so that the interproton distances can be taken as constant even in the presence of the librational mode. The deviations between the calculated and experimental quantities are expressed in terms of mean-square deviations, denoted as d. Results and Discussion PVA has been studied by NMR, and the spectral features are well-documented in the literature.19,20 In the proton spectrum, the methine and methylene protons are well-resolved, and the spectrum indicated random tacticity for the sample under investigation. The signals from the backbone carbons in the 13C spectrum also show splittings arising from sensitivity to stereochemical configuration along the chain. In Table 1, we summarize the 13C T1 and NOE values for the backbone carbons of PVA. On examining the relaxation of the different components of the backbone carbon resonances, the differences in the T1 values were found to be within experimental error. The T1 and NOE values reported here are the average of the values for the different components. The T1 values increase with temperature and are higher at the higher magnetic field. The NOE values decrease with increasing magnetic field and are below the limiting value of 3.0 in the
temp, K
50.3 MHz
263
121 (1.60) 132 (1.95) 150 (2.05) 209 (2.55) 264 (2.57) 323 (2.59)
283 296 313 323 333 a
CH2
100.6 MHz
50.3 MHz
100.6 MHz
266 (1.90) 281 (1.97) 318 (2.30) 376 (2.40)
71 (1.58) 75 (1.92) 85 (2.20) 122 (2.44) 152 (2.55) 177 (2.61)
150 (1.92) 157 (1.99) 176 (2.34) 214 (2.45)
Values in parentheses.
TABLE 2: Experimental Proton Spin-Lattice Relaxation Times (ms) of Backbone Protons of PVA as a Function of Temperature at 200 MHz temp, K
TAA
TXX
303 313 323 333
366 394 476 523
167 191 206 243
entire temperature range at both the fields. These observations suggest that the motional characteristics are away from the extreme narrowing limit. Another important observation is that the ratio of the T1 values of the CH group to the CH2 group, that is, T1(CH)/T1(CH2 ), is approximately 1.8 throughout the temperature range, at both the fields. This value is different from 2, which is expected from the number of directly bonded protons. This suggests that the C-H internuclear vectors associated with the CH and CH2 groups may be subjected to different local motions. The spin-lattice relaxation times for the backbone protons are given in Table 2. The measurements were made by the inversion recovery experiment in which one nucleus is saturated by a decoupling field. The relaxation time measured in the case of the A{X} experiment is denoted TAA, and that in the case of the X{A} experiment is denoted TXX. In the following we discuss the analysis of the experimental relaxation data using the HWH and DLM correlation functions. The contribution of overall tumbling of the polymer chain to relaxation is not considered here since it becomes insignificant for molecular weights over 10 000 or so in vinyl polymers.2,3 The best fit T1 and NOE versus temperature curves for the methine carbon calculated on the basis of the HWH and DLM models are shown in Figure 1, and the corresponding model parameters are summarized in Table 3. It is clear that the HWH model does not give a good fit to the experimental T1 and NOE values simultaneously. The calculated T1 values are below the experimental values at both the fields in the entire temperature range. If the reorientation of the backbone C-H vectors is solely due to the conformational transitions along the backbone, the parameters obtained from the methine carbon data should also be able to account for the relaxation data of the methylene carbon. Figure 2 shows the T1 and NOE curves for the methylene carbon predicted bythe HWH model. The calculated T1 values deviate significantly from the experimental values. This indicates that the reorientation of the backbone C-H vectors is not entirely determined by the segmental motions. On the other hand, the DLM model, which considers conformational transitions as well as C-H bond librations, gives
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Ravindranathan
Figure 1. Temperature dependence of (a) T1 and (b) NOE for the methine carbon of PVA in toluene-d8 at field strengths of 100.6 (square) and 50.3 MHz (triangle). Solid and dashed lines represent best fit values calculated by the DLM and HWH models, respectively.
TABLE 3: Simulation Parameters for Backbone Carbon Relaxation Data of PVA model
τ0/τ1
HWH
5
DLM
4
a
τ1/τl
200
f
0.233a 0.327b
Ea, kJ/mol
104A, s
103d
27.6
0.674
26.8
1.157
7.43a 3.39b 4.03a 2.63b
Methine carbon. b Methylene carbon.
a good fit to the T1 and NOE values of the backbone carbons in the entire temperature range as seen from Figures 1 and 2. From the parameters given in Table 2 it is seen that the librational motion is about 200 times faster than the segmental motion. The librational mode accounts for about 23% of the decay of the correlation function in the case of the methine carbon and for about 33% in the case of the methylene carbon. The amplitude of the librational motion can be estimated from the parameter f in conjunction with Howarth’s restricted rotation model using the expression21
1-f)
[
]
cos θ - cos3 θ 2(1 - cos θ)
2
(10)
The librational motion is described as the motion of the C-H vector inside a cone of half-angle θ, the axis of the cone being the rest position of the C-H bond. In this case, the conic halfangles estimated for the C-H vectors of the methine and the
Figure 2. Temperature dependence of (a) T1 and (b) NOE for the methylene carbon of PVA in toluene-d8 at field strengths of 100.6 (square) and 50.3 MHz (triangle). Solid and dashed lines represent best fit values calculated by the DLM and HWH models, respectively.
methylene groups are 23.8°and 28.9°, respectively. The smaller librational amplitude observed in the case of the methine group indicates greater steric hindrance to the librational motion of the C-H vector of the methine group relative to that of the methylene group. This is because the presence of a side group on the methine carbon can physically hinder the librational motion. This observation agrees with the earlier findings for other polymers.22-26 Guillermo et al.27 have proposed a method for superimposing NT1 data obtained at different Larmor frequencies. This superposition is possible only if the molecular motions that are considered in the correlation function are governed by a single basic correlation time τ(T). A plot of log (NT1/ωC) vs log(ωCτ1) shown in Figure 3 indicates that the 13C relaxation data at the two Larmor frequencies superpose fairly well. The success of the superposition implies that the mechanisms of local motions as reflected in the correlation function are independent of temperature, and it justifies the attempts to fit the data to a correlation function whose shape is independent of temperature. The validity of a motional model can be tested further by its ability to predict other experimental quantities. The parameters which fit the 13C relaxation data are employed to predict the proton relaxation data of PVA. Prediction of proton relaxation data is probably a more stringent test for a dynamic model because in this case the spectral density is probed over a rather wide range of frequencies. Figure 4 shows the predicted spin-
Dynamics of Poly(vinyl acetate) in Solution
J. Phys. Chem., Vol. 100, No. 51, 1996 19943
Figure 3. Frequency-temperature superposition of 13C NT1 values for the backbone methine carbon of PVA in toluene-d8. τ1 values are the correlation times obtained from the analysis of the relaxation data by using the DLM model. Larmor frequencies are 100.6 (square) and 50.3 MHz (triangle).
Figure 5. Temperature dependence of the relaxation data of the backbone methine (square) and methylene (triangle) protons of PVA in toluene-d8: (a) TAA, TXX and (b) ηA, ηX. Solid and dashed lines represent values calculated by the DLM and HWH models. The points are experimental data from ref 8.
Figure 4. Comparison of the experimental spin-lattice relaxation times of backbone protons of PVA in toluene-d8: (a) TAA and (b) TXX with the predictions of the DLM and HWH models.
lattice relaxation times for the backbone protons of PVA. The prediction of the DLM model is close to the experimental values whereas the values predicted by the HWH model show larger deviations from experiment. This observation further demonstrates the importance of including the librational mode in addition to conformational transitions, in modeling backbone motion. The inclusion of the librational mode in the correlation function influences the calculated relaxation data through two factors, namely, the correlation time for libration and the quantity f, which is related to the amplitude of libration. Dejean et al.4 have examined the effect of these two factors and have shown that the librational correlation time shows a noticeable influence only on the higher correlation time side of the T1 minimum. The factor f, on the other hand, has a direct influence on the height of the T1 minimum. It would therefore be interesting to test the performance of the DLM model in accounting for proton relaxation data on the higher correlation time side, that is, at low temperature. For this purpose, we have
analyzed the experimental proton relaxation rates measured by Heatley and Cox9 for PVA in toluene-d8 as a function of temperature at 300 MHz, using the HWH and DLM models. In the analysis using the the HWH model, the spin-lattice relaxation time for the methine proton (TAA) and the Overhauser enhancement factor (ηA ) are utilized to obtain the parameters in the model. The spin-lattice relaxation times of the methylene proton (TXX ) and the Overhauser enhancement factor (ηX ) are then predicted by these parameters. The results from the HWH model are shown in Figure 5, and the parameters giving the best fit to experimental data are given in Table 4. It is clear that there is a large deviation from experimental values toward lower temperatures. It is important to note that in an earlier analysis of the same experimental data by Skolnick and Yaris,10,11 in which several models based on conformational transitions were tested, no model could account satisfactorily for the measured relaxation data at the lowest temperature, namely 228 K. A reasonably good fit could be obtained only at the cost of a nonphysical parametrization. In the analysis using the DLM model, the experimental quantities TAA and ηA are utilized to obtain the parameters. All parameters except f are then held constant while obtaining the best fit to TXX. The Overhauser enhancement ηX is predicted by the model. The results from the DLM model are included in Figure 5, and the corresponding best fit parameters are given in Table 4. The DLM model reproduces the experimental observations successfully. Comparing the performance of the HWH and DLM models, one finds that at higher temperatures the results from both models are similar. The differences are observed in the region of the T1 minimum and beyond. A similar situation arises in 13C relaxation. At temperatures much higher than the T1 minimum, that is, in the lower correlation time side, both HWH and DLM models have been found to be successful.23,28
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Ravindranathan
TABLE 4: Simulation Parameters for the Backbone Proton Relaxation Data of PVAa model
τ0/τ1
HWH
25
DLM
6
a
τ1/τl
200
f
0.325b 0.407c
Ea, kJ/mol
1013A, s
103d
18.4
1.104
17.7
3.781
4.58b 5.82c 0.13b 0.77c
Reference 9. b Methine proton. c Methylene proton.
The parameters estimated by fitting the models to the 13C data in the present study and the proton data reported earlier are different. For example, the differences in the apparent activation energies lead to estimates of shorter correlation times from proton data compared to that from 13C. Interestingly, Weber and Helfand8 have concluded from computer simulations that the vector correlation functions for vectors oriented at different angles relative to the backbone decay at different rates. By fitting the simulation data to a functional form which is not unlike the DLM correlation function, they have obtained different parameters corresponding to different orientations of the vectors. Even though their results are not strictly applicable to the second harmonic functions arising in NMR relaxation, it may be reasonably inferred that dipole-dipole interactions oriented at different angles relative to the backbone will perhaps have different correlation functions. Proton and 13C relaxation experiments probe the reorientation of vectors having different orientations relative to the backbone, and it is possible that this leads to the differences in the estimated values of the parameters. However, more experimental data on several polymers are necessary before confirming this possibility. Conclusions The 13C and proton relaxation data for the backbone nuclei of PVA have been analyzed by the HWH and DLM models. Only the DLM model gives a good fit to both the T1 and NOE of the backbone carbons simultaneously. The parameters derived from a fit of the 13C relaxation data were used to predict the proton relaxation data. Results from the DLM model are closer to the experimental values, implying that the librational mode is important in proton relaxation. For a further understanding of the influence of librational mode, the proton relaxation data reported by Heatley and Cox, which cover a wider range of temperature, were also analyzed by using these two models. As observed in the case of other models which describe backbone dynamics only on the basis of conformational transitions, the HWH model also fails to account for the low-
temperature data. The DLM model, on the other hand, reproduces the experimental data at all temperatures, thereby showing the importance of including the the librational mode in backbone dynamics. Acknowledgment. The 400 MHz NMR experiments were carried out at the Sophisticated Instruments Facility, Indian Institute of Science. The award of a fellowship by the Council of Scientific and Industrial Research (CSIR), India, is appreciated. References and Notes (1) Heatley, F. In Dynamics of Chain in Solution by NMR Spectroscopy; Booth, C., Price, C., Eds.; Comprehensive Polymer Science; Pergamon Press: New York, 1990; Vol. 18, p 377. (2) Heatley, F. Prog. Nucl. Magn. Reson. Spectrosc. 1979, 13, 47. (3) Heatley, F. Annu. Rep. NMR Spectrosc. 1986, 17, 179. (4) Dejean de la batie, R.; Laupretre, F.; Monnerie, L. Macromolecules 1988, 21, 2045. (5) Gisser, D. J.; Glowinkowski, S.; Ediger, M. D. Macromolecules 1991, 24, 4270. (6) Ravindranathan, S.; Sathyanarayana, D. N. Macromolecules 1995, 28, 2396. (7) Hall, C. K.; Helfand, E. J. Chem. Phys. 1982, 77, 3275. (8) Weber, T. A.; Helfand, E. J. Phys. Chem. 1983, 87, 2881. (9) Heatley,F.; Cox, M. K. Polymer 1977, 18, 225. (10) Skolnick, J.; Yaris, R. Macromolecules 1983, 15, 1046. (11) Skolnick, J.; Yaris, R. Macromolecules 1983, 16, 492. (12) Sass, M.; Ziessow, D. J. Magn. Reson. 1977, 25, 263. (13) Levy, G. C.; Peat, I. R. J. Magn. Reson. 1975, 18, 500. (14) Doddrell, D.; Glushko, V.; Allerhand, A. J. Chem. Phys. 1972, 56, 3683. (15) Pople, J. A.; Gordan, M. S. J. Am. Chem. Soc. 1967, 89, 4253. (16) Noggle, J. H.; Schirmer, R. E. The Nuclear OVerhauser Effect; Academic Press: New York, 1971. (17) Helfand, E. J. Chem. Phys. 1971, 54, 4651. (18) Kramers, H. A. Physica 1940, 7, 284. (19) Wu, T. K.; Ovenall, D. W. Macromolecules 1974, 7, 776. (20) Ramey, K. C.; Lini, D. C. J. Polym. Lett. Ed. 1967, 5, 39. (21) Howarth, O. W. J. Chem. Soc., Faraday Trans. 2 1979, 75, 863. (22) Dejean de la Batie, R.; Laupretre, F.; Monnerie, L. Macromolecules 1988, 21, 2052. (23) Dejean de la Batie, R.; Laupretre, F.; Monnerie, L. Macromolecules 1989, 22, 122. (24) Radiotis, T.; Brown, G. R.; Dais, P. Macromolecules 1993, 26, 1445. (25) Dais, P.; Nedea, M. E.; Morin, F. G.; Marchessault, R. H. Macromolecules 1990, 23, 3387. (26) Dais, P.; Nedea, M. E.; Marchesssault, R. H. Polymer 1992, 33, 4288. (27) Guillermo, A.; Dupeyre, R.; Cohen-Addad, J. P. Macromolecules 1990, 23, 1291. (28) Dais, P.; Nedea, M. E.; Morin, F. G.; Marchessault, R. H. Macromolecules 1989, 22, 4208.
JP961226K
