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Mode Analysis of NMR Relaxation in a Polymer Melt Marius Hofmann*
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Department of Chemistry, Louisiana State University, Baton Rouge, Louisiana 70803, United States ABSTRACT: Spin−lattice relaxation rates of unentangled poly(dimethylsiloxane) (PDMS) melts of different M measured by field-cycling (FC) 1H NMR are analyzed using a new approach. By fitting the time domain mode distribution of the Rouse model to the experimental data, interpolation of the latter is achieved and a mode separation is performed. The evolution of the Rouse relaxation spectrum with increasing M is studied. From the model parameters, the diffusion coefficient D, the Rouse time τR, and the statistical length are calculated, all in good agreement with literature values and the Rouse model. The new approach allows for a more accurate calculation of the segmental mean square displacement, removes previous discrepancies between FC and field-gradient NMR and verifies the relevant relaxation theory. Furthermore, for the first time, the analysis provides the radius of gyration RG, the values of which are in agreement with literature values obtained by small-angle neutron scattering. The Mdependence RG ∝ M0.53±0.04 indicates ideal chain statistics down to M = 860 g/mol.
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curves in the susceptibility representation12−14 χ′′(ωaT) ≡ ωR1(ωaT) are constructed, which extend the effective frequency range to up to 10 decades. Provided that the αrelaxation is covered, the shift factors aT yield the temperature dependence of the corresponding time constant τα. In polymer melts, FTS enables the study of segmental motion from the αrelaxation to Rouse and entanglement dynamics and eventually up to the center-of-mass diffusion.13−16 The widely accepted tube-reptation (TR) model17 together with refinements18 provides a satisfactory description of the segmental dynamics in entangled polymer melts. It incorporates the Rouse model19,20 and predicts from short to long times four power-law relaxation regimes reflecting Rouse, constrained Rouse, reptation, and diffusion dynamics, as well as the corresponding crossover times. Previous FC studies on polymer melts scrutinized model predictions on a rather phenomenological basis, e.g., by identifying such power-law regimes.6,13−16,21 The modeling of 1H relaxation data is usually complicated by the existence of both an intra- and an intermolecular relaxation pathway which are related to reorientational and translational dynamics, respectively.2,3,8 A separation of the total relaxation rate along R1(ω) = R1,inter(ω) + R1,intra(ω) is achieved by isotope dilution experiments.6,22−25 In simple liquids and polymer melts it was revealed that already at frequencies/times ω−1,t ≲ τα the frequency dependence of the total relaxation, R1(ω), is largely determined by intermolecular relaxation. It was demonstrated that FC 1H NMR relaxometry can access the time dependence of the mean square displacement (msd) of segments in polymer melts which is obtained from
INTRODUCTION The molecular dynamics in liquids can be studied by a multitude of NMR methods. Already shortly after the discovery of NMR, Bloembergen, Purcell, and Pound related molecular motion to NMR relaxation.1 Ever since the pioneering work by Noack2 and Kimmich,3 NMR relaxometry based on the field-cycling (FC) method has established itself as a powerful tool for measuring the slow dynamics occurring in complex liquids. The FC technique is predominantly applied to measure the spin−lattice relaxation rate R1(ω) in dependence on the Larmor frequency ω ∝ B, with B denoting the magnetic flux density. In contemporary, commercially available FC relaxometers which are based on electromagnets B is varied from 1 T down to about 200 μT. Advanced home-built devices even unlock significantly lower fields Me. In the Rouse regime, D ∝ M−1.5±0.2 is found, whereas the Rouse model predicts an exponent of −1. Yet, the iso-frictional quantity DτS which is also shown in Figure 4b, follows DτS ∝ M−1.1±0.2, again in accordance with the Rouse model. For M > Me, entanglement effects entail the well-known power law D ∝ M−2.2.29,50,54
Figure 4a displays the M-dependence of the Rouse time. For M ≤ 11k, τR follows a power law τR ∝ M2.5±0.2. The absolute values of τR are consistent with literature values measured by dynamic light scattering32 (DLS) after temperature rescaling (cf. Appendix A) and with a value extracted from NS.30 Specifically, from the values on PDMS 6500, ⟨R G 2⟩ = 21.3Å and D = 2.7 × 10−11 m2/s, τR was calculated and reported using the relationship48 τR =
2⟨R G 2⟩ ζl 2N 2 = 3π 2kbT π 2D
(10)
In refs 49 and 50, further values of τR or τt, respectively, based on rheology and FG NMR are reported. They are also consistent with the ones determined by FC NMR. The scaling exponent of τR ∝ Mγ exceeds the predicted one for Rouse dynamics (γ = 2) slightly. However, quantities which depend on the friction coefficient ξ need to be adjusted for potential molar mass dependence.29,51 In fact, the values of ξ (cf. Table 1) vary with M within a factor of about 3, not only due to the weak M-dependence of the α-relaxation (cf. Results),14,45,51 but also because the apparent values of ξ absorb all imperfections in master-curve construction and temperature inaccuracy. The apparent segmental relaxation time τS ∝ ξ which follows from eq 4 was chosen to calculate the isoF
DOI: 10.1021/acs.macromol.8b01189 Macromolecules XXXX, XXX, XXX−XXX
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Macromolecules In Figure 5a the apparent exponent α(M) of the msd ⟨r2(t)⟩ ∝ tα in the Rouse regime extracted from the data shown in Figure 3 is addressed. It was determined from the derivative of the msd data on log−log scale. The procedure is described in a previous study.55 The exponent decreases with M and reaches a value of 0.5, the prediction of the Rouse model, only asymptotically. In Figure 5b the reduced msd which results from subtracting the diffusion term is shown. This quantity follows (⟨r2(t)⟩ − 6Dt) ∝ t0.5 almost up to the Rouse time τR, as expected. As also suggested by the simulations of eq 5 (cf. Figure 1a), the diffusion term foreshadows the subdiffusive regime of internal chain relaxation, leading to an overestimation of the Rouse exponent, particularly at small M. A similar trend was reported for the exponent of the dynamic susceptibility of PDMS measured by depolarized DLS,56 for which an exponent of 0.5 is expected likewise. The exponent values of that work are included in Figure 5a, yet, the asymptotic approach toward 0.5 was explained by the establishment of Gaussian chain statistics. Alternatively, it may be speculated that rotational diffusion foreshadows the actual Rouse regime also in case of depolarized DLS, a technique which is sensitive to reorientational motion. The long-time plateau value of the reduced msd, which is read off from Figure 5b, is related to the rms radius of gyration (cf. eq 6): 1 lim (⟨r 2(t )⟩ − 6Dt )= ⟨R G 2⟩ = R G 2 t →∞
given in ref 39 as an approximation was also tested using the diffusion coefficient resulting from the fit (cf. Table 1). The result is included in Figure 6. In comparison to the NS based reference values, eq 12, underestimates RG slightly, at least at higher M. The statistical length l obtained from FC NMR varies within 11.5−14.5 Å (cf. Table 1), in fair agreement with literature values reported for the Kuhn length lK = 13−15 Å.10,42,58,61 As shown in Figure 7a, the M-dependence of the number of modes, N, obeys an almost linear relationship, N ∝ M1.1, as expected. For M = Me, a number of Ne ≈ 18 Rouse modes is projected. The number of bonds is given by n0 = M/m0 (also indicated in Figure 7a), with m0 = 38 g/mol denoting the average mass per bond, following from the molecular structure. From the number of statistical segments, which is equated with N = M/mS, it can be estimated that a statistical segment comprises mS/m0 ≈ 20 bonds and therefore a molar mass of mS = 20m0 ≈ 760 g/mol. This value is twice as high as the Kuhn molar mass, 381 g/mol, reported in ref 61, but is in better agreement to the DLS-based critical value mR ≈ 560 g/mol reported by Ding et al.56 for the establishment of ideal chain statistics. However, the lowest M studied, that for PDMS 860, already exceeds this value, and the question “when does a molecule become a polymer?” cannot be answered, based on the preceding data. Dividing the apparent friction coefficient ξ of the statistical segment (cf. Table 1) by mS/(2m0) ≈ 10 yields an estimate for the friction coefficient per monomer ξm. The values span over a range of 2 ≲ ξm × 10−12 mN−1 s−1 ≲ 7 with a tendency to increase with M and saturate at M ≥ 6k. This trend appears to follow the increase of τα with M (cf. Results). The absolute values of ξm at high M are in tune with the value 5.4 × 10−12 Ns/m following from ξm(T)=ξ0 exp[κ/(T−TVFT)] using the parameters ξ0 = 2.4 × 10−13 Ns/m, κ = 0.00144/K and TVFT = 100 K reported in ref 42. Another comment on the segmental relaxation time τS is worthwhile. The values, which were calculated using eq 4, span from 270 to 860 ps (cf. Table 1) and are by almost up to two orders larger than τα ≈ 10 ps (cf. Appendix A). The segmental relaxation time is proportional to the square of the statistical length, τs ∝ l2 and proportional to the α-relaxation time τα, which controls the temperature dependence of relaxation, τs ∝ ταl2. This motivates the definition of a length scale lα associated with the α-relaxation as probed by NMR from τS/τα =(l/lα)2. The calculation yields an estimate 1.5 Å ≲ lα ≲ 2.7 Å, which is 1−2 times the bond length in the backbone of PDMS (1.65 Å, ref 42). This indicates that at least in PDMS, equating the α- with the segmental relaxation time is inappropriate, a fact which was already mentioned in a previous work.26 The rather large discrepancy between τα and τS would imply that the local (α-) and the polymer relaxation spectra are well-separated in the case of PDMS. In fact, this would explain peculiarities in the relaxation spectra of PDMS reported in previous studies.14,62 As mentioned, the entanglement time of PDMS at 323 K is τe = 1.2μs and the number of Rouse modes for M ≈ Me was estimated to Ne ≈ 18. The TR model17 predicts a value of τTR e = τsNe2 ≈ 0.3 μs, to which the experimental value agrees fairly well. Figure 7b shows the quantity R1(ω → 0)ns/τS plotted vs N. The factor ns/τS removes the influence of the varying density (cf. eq 7 and Table 1) and makes it an iso-frictional quantity. The experimental values yield a power-law dependence with an exponent 0.72 ± 0.09, which is larger than 0.5 predicted by the
(11)
Figure 6 displays the values of RG, which were calculated from NMR relaxation data for the first time, in comparison to
Figure 6. Radius of gyration of PDMS melts determined by FC NMR in comparison to literature values obtained by SANS.30,57−60 A power law with exponent 0.53 is indicated (dashed line). The result of the estimate eq 12 is also included (open triangles).
values reported in different small-angle neutron scattering (SANS) works.30,57−60 A very good agreement is found and a power-law relationship RG ∝ M0.53±0.04, in accordance with ideal chain statistics (exponent 0.5), is revealed down to the lowest M PDMS 860. The relationship iμ y R G ≈ jjj 0 zzz k 4π {
2
2 4 π ℏ γH ns 6 DR1, inter(0)
(12) G
DOI: 10.1021/acs.macromol.8b01189 Macromolecules XXXX, XXX, XXX−XXX
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Figure 7. (a) Number of Rouse modes N vs M. The solid line reflects the number of bonds given by M/m0. (b) Iso-frictional zero-field intermolecular relaxation rate R1(ω → 0)ns/τS. For comparison, the prediction of the Rouse model for purely intramolecular relaxation, R1,intra(ω → 0, N), is also provided (dotted line). In both figures, power-law behavior and Me are indicated.
field-cycling and field-gradient NMR are now consistent. For the first time, the radius of gyration, RG, of polymer chains was calculated from NMR relaxation data. It follows RG ∝ M0.53±0.04, in accordance with ideal chain statistics and agrees with numerous literature values obtained by neutron scattering. Furthermore, the analysis suggests that the statistical segment comprises 10 monomer units with a molar mass of 760 g/mol and a length scale of 11.5−14.5 Å. Also, the segmental relaxation time is much longer than the one associated with the α-relaxation, which would provide an explanation for the peculiar NMR relaxation spectra of PDMS discovered previously.
simulations of an ideal, long Rouse chain (cf. Inset of Figure 1b). The deviation can only partially be explained by the Mdependence of the diffusion coefficient, which is somewhat stronger than predicted by the Rouse model, as mentioned. Finally, for comparison, the prediction of the Rouse model for purely intramolecular relaxation, R1,intra(ω → 0,N), is also illustrated in Figure 7b. As derived in ref 13, the quantity features a logarithmic N-dependence, R1,intra(0,N) ∝ ln(N), which differs from the found power-law trend significantly. Possible traces of intramolecular relaxation can therefore not explain the deviation between the apparent exponent 0.72 and the predicted one, 0.5. The real nature of the chains, as well as polydispersity effects, might provide an explanation.
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CONCLUSION Focusing on the Rouse regime, the polymer dynamics of nonentangled PDMS melts of different M was studied by fieldcycling 1H NMR relaxometry. Master curves of spin−lattice relaxation rates R1(ω) were taken from the literature and scaled to a reference temperature ideal for observing Rouse dynamics. The rates R1(ω) are governed by intermolecular relaxation in case of PDMS, and therefore reflect translational segmental motion over a broad frequency range. Assuming that the relaxation rate can be additively decomposed into a local and a polymer relaxation component, the relaxation rate due to pure polymer dynamics, Rpolymer (ω), was obtained. Performing 1 numerical cosine transformation during fitting, the timedomain mode distribution of the Rouse model, which only depends on three free parameters, was fitted to the data, yielding a perfect description of the polymer relaxation. In contrast to previous studies which only provided a phenomenological description of the NMR relaxation dispersion in polymers, a quantitative mode analysis was performed and the spectrum of relaxation times from the segmental (τS) to the Rouse time (τR) was revealed. After correcting for the M-dependence of the α-relaxation, the latter follows τR/τS ∝ M2.1±0.2, as predicted by the Rouse model. In addition to the mode spectrum, the refined analysis yields the diffusion coefficient, the M-dependence of which follows DτS ∝ M−1.1±0.2, also in agreement with the Rouse model and literature values. The new approach allows determining the segmental mean square displacement ⟨r2(t)⟩ more accurately than before and with less assumptions. Concerning PDMS 22k, the data from
APPENDIX A
Temperature Dependence of the α-Relaxation
In order to rescale the data from FC NMR and from the literature to the reference temperature 323 K, the data need to τα(T ) be multiplied by factors . As demonstrated τα(Tref = 323K )
elsewhere,63 the temperature dependence of τα(T) can be decomposed into an Arrhenius-like high-T part and in a cooperative part along τα(T ) = τ∞exp[T −1(E∞ + Ecoop(T ))] with Ecoop(T ) ÄÅ ÉÑ ÅÅ μ ÑÑ Å Å = E∞expÅÅ− (T − TA)ÑÑÑÑ ÅÅÇ E∞ ÑÑÖ
(13)
The temperature dependence of τα of PDMS was measured by dielectric spectroscopy and NMR.14 The fit parameters used for calculating τα(T = 323K) of different M are found in ref46.
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APPENDIX B
Mode Decomposition of the Intermolecular 1H Relaxation Rate
Figure 8 displays the decomposition of the intermolecular relaxation dispersion of PDMS 11k into Rouse modes (cf. Figure 2b). The fit parameters (ξ and l) given in Table 1 were used to calculate the partial sums of eq 5 in consecutive order for N = 1,2,···,16. The diffusion term 6Dt (0th mode) is also included in every partial sum and is indicated for the case N = 16. H
DOI: 10.1021/acs.macromol.8b01189 Macromolecules XXXX, XXX, XXX−XXX
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(9) Kresse, B.; Becher, M.; Privalov, A. F.; Hofmann, M.; Rössler, E. A.; Vogel, M.; Fujara, F. 1H NMR at Larmor frequencies down to 3 Hz by means of Field-Cycling techniques. J. Magn. Reson. 2017, 277, 79−85. (10) Ferry, J. D. Viscoelastic Properties of Polymers; John Wiley & Sons: New York, 1980. (11) Kremer, F.; Schönhals, A. Broadband Dielectric Spectroscopy; Springer: Berlin, 2003. (12) Cohen-Addad, J. P.; Messa, J. P. J. Concentration frequency dependence of short range chain fluctuations observed in concentrated polymer solutions from nuclear magnetic relaxation rates. J. Phys., Lett. 1976, 37, 193−196. (13) Kariyo, S.; Brodin, A.; Gainaru, C.; Herrmann, A.; Hintermeyer, J.; Schick, H.; Novikov, V. N.; Rössler, E. A. From Simple Liquid to Polymer Melt. Glassy and Polymer Dynamics Studied by Fast Field Cycling NMR Relaxometry: Rouse Regime. Macromolecules 2008, 41, 5322−5332. (14) Hofmann, M.; Herrmann, A.; Abou Elfadl, A.; Kruk, D.; Wohlfahrt, M.; Rössler, E. A. Glassy, Rouse, and Entanglement Dynamics As Revealed by Field Cycling 1H NMR Relaxometry. Macromolecules 2012, 45, 2390−2401. (15) Herrmann, A.; Kresse, B.; Gmeiner, J.; Privalov, A. F.; Kruk, D.; Fujara, F.; Rö ssler, E. A. Protracted Crossover to Reptation Dynamics: A Field Cycling 1H NMR Study Including Extremely Low Frequencies. Macromolecules 2012, 45, 1408−1416. (16) Hofmann, M.; Kresse, B.; Privalov, A. F.; Heymann, L.; Willner, L.; Aksel, N.; Fatkullin, N.; Fujara, F.; Rössler, E. A. Segmental Mean Square Displacement: Field-Cycling 1H Relaxometry vs Neutron Scattering. Macromolecules 2016, 49, 7945−7951. (17) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Oxford University Press: New York, 1986. (18) Likhtman, A. E.; McLeish, T. C. B. Quantitative Theory for Linear Dynamics of Linear Entangled Polymers. Macromolecules 2002, 35, 6332−6343. (19) Rouse, P. E., Jr. A Theory of the Linear Viscoelastic Properties of Dilute Solutions of Coiling Polymers. J. Chem. Phys. 1953, 21, 1272−1280. (20) Verdier, P. H. Monte Carlo Studies of Lattice-Model Polymer Chauns. I. Correlation Functions in the Statistical-Bead Model. J. Chem. Phys. 1966, 45, 2118−2121. (21) Kimmich, R.; Fatkullin, N. Polymer Chain Dynamics and NMR. Adv. Polym. Sci. 2004, 170, 1−113. (22) Morita, A.; Ando, I.; Nishioka, A.; et al. 1H-NMR Relaxation and Molecular Motion of Molten Polyethylene by Isotopic Dilution Method. J. Polym. Sci., Polym. Lett. Ed. 1980, 18, 109−113. (23) Kehr, M.; Fatkullin, N.; Kimmich, R. Deuteron and proton spin-lattice relaxation dispersion of polymer melts: Intrasegment, intrachain, and interchain contributions. J. Chem. Phys. 2007, 127, 084911. (24) Kehr, M.; Fatkullin, N.; Kimmich, R. Molecular diffusion on a time scale between nano- and milliseconds probed by field-cycling NMR relaxometry of intermolecular dipolar interactions: Application to polymer melts. J. Chem. Phys. 2007, 126, 094903. (25) Herrmann, A.; Kresse, B.; Wohlfahrt, M.; Bauer, I.; Privalov, A. F.; Kruk, D.; Fatkullin, N.; Fujara, F.; Rössler, E. A. Mean Square Displacement and Reorientational Correlation Function in Entangled Polymer Melts Revealed by Field Cycling 1H and 2H NMR Relaxometry. Macromolecules 2012, 45, 6516−6525. (26) Kresse, B.; Hofmann, M.; Privalov, A. F.; Fatkullin, N.; Fujara, F.; Rössler, E. A. All Polymer Diffusion Regimes Covered by Combining Field-Cycling and Field-Gradient 1H NMR. Macromolecules 2015, 48, 4491−4502. (27) Lozovoi, A.; Mattea, C.; Hofmann, M.; Saalwächter, K.; Fatkullin, N.; Stapf, S. Segmental dynamics of polyethylene-altpropylene studied by NMR spin echo techniques. J. Chem. Phys. 2017, 146, 224901. (28) Meier, R.; Kruk, D.; Bourdick, A.; Schneider, E.; Rössler, E. A. Inter- and Intramolecular Relaxation in Molecular Liquids by Field
Figure 8. Spin−lattice rate R1,inter(ω) associated with polymer relaxation of PDMS 11k at 323 K separated into Rouse modes. The red line represents the sum over all internal modes including the contribution by the diffusion term (blue dashed line). The gray lines reflect the partial sums of the internal modes in consecutive order.
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AUTHOR INFORMATION
Corresponding Author
*(M.H.) E-mail:
[email protected]. ORCID
Marius Hofmann: 0000-0001-9330-2742 Notes
The author declares no competing financial interest.
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ACKNOWLEDGMENTS The work was financially supported by a postdoc fellowship of the German Academic Exchange Service (DAAD). The author thanks Ernst Rössler, Franz Fujara, Nail Fatkullin, and Benjamin Kresse for fruitful discussions and many years of collaboration.
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REFERENCES
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2⟨R G 2⟩ 2
π D
, instead of τR =
⟨R G 2⟩ 18π 2D
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DOI: 10.1021/acs.macromol.8b01189 Macromolecules XXXX, XXX, XXX−XXX
