From Simple Liquid to Polymer Melt. Glassy and Polymer Dynamics Studied by Fast Field Cycling NMR Relaxometry: Low and High Molecular Weight Limit

Glass spectrum, excess wing phenomenon, and master curves in molecular glass formers: A multi-method approach

The relaxation spectra of glass formers solely displaying an α-peak and excess wing contribution collected by various methods are reanalyzed to pin down their different spectral evolution. We show that master curve construction encompassing both α-peak and emerging excess wing works for depolarized light scattering (DLS) and nuclear magnetic resonance (NMR) relaxometry. It reveals the self-part of the slow dynamics' spectrum. Master curves are to be understood as a result of a more extensive scaling covering all temperatures instead of strict frequency-temperature superposition. DLS and NMR display identical relaxation spectra; yet, comparing different systems, we do not find a generic structural relaxation at variance with recent claims. Dielectric spectroscopy (DS) spectra show particularities, which render master curve construction obsolete. The DS α-peak is enhanced or suppressed with respect to that of DLS or NMR, yet, not correlated to the polarity of the liquid. Attempting to single out the excess wing from the overall spectrum discloses a stronger exponential temperature dependence of its amplitude compared to that below Tg and a link between its exponent and that of the fast dynamics' spectrum. Yet, such a decomposition of α-peak and excess wing appears to be unphysical. Among many different glasses, the amplitude of the excess wing power-law spectrum is found to be identical at Tg, interpreted as a relaxation analog to the Lindemann criterion.

Glassy and Polymer Dynamics of Elastomers by 1H-Field-Cycling NMR Relaxometry: Effects of Fillers

1H spin-lattice relaxation rate (R1) dispersions were acquired by field-cycling (FC) NMR relaxometry between 0.01 and 35 MHz over a wide temperature range on polyisoprene rubber (IR), either unfilled or filled with different amounts of carbon black, silica, or a combination of both, and sulfur cured. By exploiting the frequency-temperature superposition principle and constructing master curves for the total FC NMR susceptibility, χ″(ω) = ωR1(ω), the correlation times for glassy dynamics, τs, were determined. Moreover, the contribution of polymer dynamics, χpol″(ω), to χ″(ω) was singled out by subtracting the contribution of glassy dynamics, χglass″(ω), well represented by the Cole-Davidson spectral density. Glassy dynamics resulted moderately modified by the presence of fillers, τs values determined for the filled rubbers being slightly different from those of the unfilled one. Polymer dynamics was affected by the presence of fillers in the Rouse regime. A change in the frequency dependence of χpol″(ω) at low frequencies was observed for all filled rubbers, more pronounced for those reinforced with silica, which suggests that the presence of the filler particles can affect chain conformations, resulting in a different Rouse mode distribution, and/or interchain interactions modulated by translational motions.

Glassy and Polymer Dynamics of Elastomers by 1H Field-Cycling NMR Relaxometry: Effects of Cross-Linking

¹H spin lattice relaxation rate (R₁) dispersions were acquired by field-cycling (FC) NMR relaxometry between 0.01 and 35 MHz over a wide temperature range on polyisoprene (IR), polybutadiene (BR), and poly(styrene-co-butadiene) (SBR) rubbers, obtained by vulcanization under different conditions, and on the corresponding uncured elastomers. By exploiting the frequency–temperature superposition principle, χ″(ωτ_s) master curves were constructed by shifting the total FC NMR susceptibility, χ″(ω) = ωR₁(ω), curves along the frequency axis by the correlation times for glassy dynamics, τ_s. Longer τ_s values and, correspondingly, higher glass transition temperatures were determined for the sulfur-cured elastomers with respect to the uncured ones, which increased by increasing the cross-link density, whereas no significant changes were found for fragility. The contribution of polymer dynamics, χ_pol^″(ω), to χ″(ω) was singled out by subtracting the contribution of glassy dynamics, χ_glass^″(ω), well represented using a Cole–Davidson spectral density. For all elastomers, χ_pol^″(ω) was found to represent a small fraction, on the order of 0.05–0.14, of the total χ″(ω), which did not show a significant dependence on cross-link density. In the investigated temperature and frequency ranges, polymer dynamics was found to encompass regimes I (Rouse dynamics) and II (constrained Rouse dynamics) of the tube reptation model for the uncured elastomers and only regime I for the vulcanized ones. This is clear evidence that chemical cross-links impose constraints on chain dynamics on a larger space and time scale than free Rouse modes.

The dynamics of the plastically crystalline phase of cyanoadamantane revisited by NMR line shape analysis and field-cycling relaxometry

The dynamics of cyanoadamantane (CN-ADA) in its plastically crystalline phase encompasses three processes: overall tumbling of the rigid molecule, rotation around the molecular symmetry axis, and vacancy diffusion. This makes CN-ADA a prototypical case to be studied by field-cycling as well as by conventional NMR relaxometry. Data are collected from 430 K down to about 4 K and frequencies in the range of 10 kHz-56 MHz are covered. The overall tumbling is interpreted as a cooperative jump process preceding along the orthogonal axis of the cubic lattice and exhibiting a temperature independent non-Lorentzian spectral density. Consequently, a master curve is constructed, which yields model-independent correlation times, which agree well with those reported in the literature. It can be interpolated by a Cole-Davidson function with a width parameter β_CD = 0.83. The uniaxial rotation persisting in the glassy crystal (T < T_g = 170 K) is governed by a broad distribution of activation energies, g(E). In this case, the standard master curve construction applied for the overall tumbling, for example, fails, as the actually probed distribution of correlation times G(ln τ) strongly changes with temperature. We suggest a scaling method that generally applies for the case that a relaxation process is determined by a distribution of thermally activated processes. Frequency as well as temperature dependence of the relaxation rate can be used to reconstruct g(E). In addition, g(E) is extracted from the proton line-shape, which was measured down to 4 K. Vacancy diffusion governs the relaxation dispersion at highest temperatures; yet, a quantitative analysis is not possible due to instrumental limitations.

Application of proton field-cycling NMR relaxometry for studying translational diffusion in simple liquids and polymer melts

With the availability of commercial field-cycling relaxometers together with progress of home-built instruments nuclear magnetic resonance relaxometry has gained new momentum as a method of investigating the dynamics in viscous liquids and polymer melts. The method provides the frequency dependence of the spin-lattice relaxation rate. In the case of protons, one distinguishes intramolecular and intermolecular relaxation pathways. Whereas the intramolecular contribution prevails at high frequencies and reflects rotational dynamics, the often ignored intermolecular relaxation contribution dominates at low-frequency and provides access to translational dynamics. A universal low-frequencies dispersion law holds which in pure systems allows determining the diffusion coefficient in a straightforward way. In addition, the rotational time constant is extracted from the high-frequency relaxation contribution. This is demonstrated for simple and ionic liquids and for polymer melts.

Time Domain NMR in Polymer Science: From the Laboratory to the Industry

Highly controlled polymers and nanostructures are increasingly translated from the lab to the industry. Together with the industrialization of complex systems from renewable sources, a paradigm change in the processing of plastics and rubbers is underway, requiring a new generation of analytical tools. Here, we present the recent developments in time domain NMR (TD-NMR), starting with an introduction of the methods. Several examples illustrate the new take on traditional issues like the measurement of crosslink density in vulcanized rubber or the monitoring of crystallization kinetics, as well as the unique information that can be extracted from multiphase, nanophase and composite materials. Generally, TD-NMR is capable of determining structural parameters that are in agreement with other techniques and with the final macroscopic properties of industrial interest, as well as reveal details on the local homogeneity that are difficult to obtain otherwise. Considering its moderate technical and space requirements of performing, TD-NMR is a good candidate for assisting product and process development in several applications throughout the rubber, plastics, composites and adhesives industry.

Scaling analysis of the viscoelastic response of linear polymers

Viscoelastic response in terms of the complex shear modulus G^*(ω) of the linear polymers poly(ethylene-alt-propylene), poly(isoprene), and poly(butadiene) is studied for molar masses (M) from 3k up to 1000k and over a wide temperature range starting from the glass transition temperature T_g (174 K-373 K). Master curves G'(ωτ_α) and G″(ωτ_α) are constructed for the polymer-specific relaxation. Segmental relaxation occurring close to T_g is independently addressed by single spectra. Altogether, viscoelastic response is effectively studied over 14 decades in frequency. The structural relaxation time τ_α used for scaling is taken from dielectric spectra. We suggest a derivative method for identifying the different power-law regimes and their exponents along G″(ωτ_α) ∝ ω^ε″. The exponent ε″ = ε″(ωτ_α) ≡ d ln G″(ωτ_α)/d ln(ωτ_α) reveals more details compared to conventional analyses and displays high similarity among the polymers. Within a simple scaling model, the original tube-reptation model is extended to include contour length fluctuations (CLFs). The model reproduces all signatures of the quantitative theory by Likhtman and McLeish. The characteristic times and power-law exponents are rediscovered in ε″(ωτ_α). The high-frequency flank of the terminal relaxation closely follows the prediction for CLF (ε″ = -0.25), i.e., G″(ω) ∝ ω^-0.21±0.02. At lower frequencies, a second regime with lower exponent ε″ is observed signaling the crossover to coherent reptation. Application of the full Likhtman-McLeish calculation provides a quantitative interpolation of ε″(ωτ_α) at frequencies below those of the Rouse regime. The derivative method also allows identifying the entanglement time τ_e. However, as the exponent in the Rouse regime (ωτ_e > 1) varies along ε_eRouse = 0.66 ± 0.04 (off the Rouse prediction ε_Rouse = 0.5) and that at ωτ_e < 1 is similar, only a weak manifestation of the crossover at τ_e is found at highest M. Yet, calculating τ_e/τ_α= (M/M_o)², we find good agreement among the polymers when discussing ε″(ωτ_e). The terminal relaxation time τ_t is directly read off from ε″(ωτ_α). Plotting τ_t/τ_e as a function of Z = M/M_e, we find universal behavior as predicted by the TR model. The M dependence crosses over from an exponent significantly larger than 3.0 at intermediate M to an exponent approaching 3.0 at highest M in agreement with previous reports. The frequency of the minimum in G″(ωτ_α) scales as τ_min ∝ M^1.0±0.1. An M-independent frequency marks the crossover to glassy relaxation at the highest frequencies. Independent of the amplitude of G″(ω), which may be related to sample-to-sample differences, the derivative method is a versatile tool to provide a detailed phenomenological analysis of the viscoelastic response of complex liquids.

Segmental dynamics of polyethylene-alt-propylene studied by NMR spin echo techniques

Segmental dynamics of a highly entangled melt of linear polyethylene-alt-propylene with a molecular weight of 200 kDa was studied with a novel proton nuclear magnetic resonance (NMR) approach based upon ¹H → ²H isotope dilution as applied to a solid-echo build-up function I^SE(t), which is constructed from the NMR spin echo signals arising from the Hahn echo (HE) and two variations of the solid-echo pulse sequence. The isotope dilution enables the separation of inter- and intramolecular contributions to this function and allows one to extract the segmental mean-squared displacements in the millisecond time range, which is hardly accessible by other experimental methods. The proposed technique in combination with time-temperature superposition yields information about segmental translation in polyethylene-alt-propylene over 6 decades in time from 10^-6 s up to 1 s. The time dependence of the mean-squared displacement obtained in this time range clearly shows three regimes of power law with exponents, which are in good agreement with the tube-reptation model predictions for the Rouse model, incoherent reptation and coherent reptation regimes. The results at short times coincide with the fast-field cycling relaxometry and neutron spin echo data, yet, significantly extending the probed time range. Furthermore, the obtained data are verified as well by the use of the dipolar-correlation effect on the Hahn echo, which was developed before by the co-authors. At the same time, the amplitude ratio of the intermolecular part of the proton dynamic dipole-dipole correlation function over the intramolecular part obtained from the experimental data is not in agreement with the predictions of the tube-reptation model for the regimes of incoherent and coherent reptation.

Cooperativity at the glass transition: A perspective from facilitation on the analysis of relaxation in modulated calorimetry

The glass transition region in nonconfined polymeric and low-molecular-weight supercooled liquids is probed by temperature-modulated calorimetry at a frequency of 3.3 mHz. From the distribution of relaxation times derived by analyzing the complex heat capacity, the number N_{α} of cooperatively rearranging units is estimated. This is done by resorting to a method in which cooperative motion is viewed as a result of a spontaneous regression of energy fluctuations. After a first, local, structural transition occurs, the energy threshold for the rearrangement of adjacent molecular units decreases progressively. This facilitation process is associated to a corresponding evolution of the density of states in a canonical representation and may be considered as a continuous spanning through different dynamic states toward a condition in which configurational constraints disappear. A good agreement is found with the N_{α} values obtained from the same calorimetric data within the framework of Donth's fluctuation theory. It is shown that, at variance from previous treatments, N_{α} can be estimated from just the relaxation function, without resorting to the knowledge of the configurational entropy. Examples point to a modest dependence of the N_{α} estimates on the experimental method used to derive the relaxation function.

Intermolecular spin relaxation and translation diffusion in liquids and polymer melts: insight from field-cycling 1H NMR relaxometry

With the advent of commercial field-cycling (FC) spectrometers, NMR relaxometry has gained new momentum as a method of investigating dynamics in liquids and polymers. The outcome of FC NMR experiments is spin-lattice relaxation time versus frequency (relaxation dispersion). In the case of protons, due to the intra- and intermolecular origin of dipolar interactions, the relaxation dispersion reflects rotational as well as translational dynamics. The latter shows a universal dispersion law at low frequencies, which allows determination of the diffusion coefficient D(T) in addition to the rotational correlation time τ(rot)(T), that is, FC (1)H NMR becomes an alternative to field-gradient NMR spectroscopy. Subdiffusive translation found in polymers can be accessed by singling out the intermolecular relaxation through isotope dilution experiments, and the mean square displacement can then be revealed as a function of time, thus complementing neutron scattering experiments. Likewise, information on reorientational dynamics is provided by the intramolecular relaxation. Assuming frequency-temperature superposition the corresponding correlation functions can be monitored up to eight decades in amplitude and time, which allows thorough testing of current polymer theories.

Probing α-relaxation with nuclear magnetic resonance echo decay and relaxation: a study on nitrile butadiene rubber

One dimensional (1)H NMR measurements have been performed to probe slow molecular motions in nitrile butadiene rubber (NBR) around its calorimetric glass transition temperature Tg. The purpose is to show how software aided data analysis can extract meaningful dynamical data from these measurements. Spin-lattice relaxation time, free induction decay (FID) and magic sandwich echo (MSE) measurements have been carried out at different values of the static field, as a function of temperature. It has been evidenced how the efficiency of the MSE signal in reconstructing the original FID exhibits a sudden minimum at a given temperature, with a slight dependence from the measuring frequency. Computer simulations performed with the software SPINEVOLUTION have shown that the minimum in the efficiency reconstruction of the MSE signal corresponds to the average motional frequency taking a value around the inter-proton coupling. The FID signals have been fitted with a truncated form of a newly derived exact correlation function for the transverse magnetization of a dipolar interacting spin pair, which allows one to avoid the restriction of the stationary and Gaussian approximations. A direct estimate of the conformational dynamics on approaching the Tg is obtained, and the results are in agreement with the analysis performed via the MSE reconstruction efficiency. The occurrence of a wide distribution of correlation frequencies for the chains motion, with a Vogel-Fulcher type temperature dependence, is addressed. A route for a fruitful study of the dynamics accompanying the glass transition by a variety of NMR measurements is thus proposed.

Long-Time Diffusion in Polymer Melts Revealed by 1H NMR Relaxometry

We demonstrate that field-cycling ¹H NMR relaxometry can be used as a straightforward method of determining translational diffusion coefficient D = D(M) in polymer systems. The ¹H spin-lattice relaxation dispersion for polybutadiene of different molecular masses M (446 < M/(g mol^-1) < 9470) is measured at several temperatures (233 < T/K < 408) in a broad frequency range. The diffusion coefficient D(T) is determined from the intermolecular contribution to the overall spin-lattice relaxation rate R₁(ω), which dominates in the low-frequency range and follows a universal dispersion law linear in √ω. The extracted diffusion coefficients are in good agreement with the values obtained previously by field gradient NMR. The molecular mass dependence D = D(M) exhibits two power laws: D ∝ M^-1.3±0.1 and ∝M^-2.3±0.1. They show a crossover for M = 2300, a value that is close to the entanglement molecular mass M_e of polybutadiene. The corresponding power-law exponents are close to the prediction of the tube-reptation model.

On the theory of the proton free induction decay and Hahn echo in polymer systems: the role of intermolecular magnetic dipole-dipole interactions and the modified Anderson-Weiss approximation

The influence of the intermolecular magnetic dipole-dipole interaction on the free induction decay (FID) as well as on the Hahn-echo of proton spins in polymer melts is investigated. It is shown that for isotropic models of polymer dynamics, when polymer segment displacements do not correlate with an initial chain conformation, the influence of the intermolecular magnetic dipole-dipole interactions to the FID and Hahn echo is increasing more rapidly with evolution time than the corresponding influence of the intramolecular magnetic dipole-dipole interactions. On the other hand, the situation is inverted for the tube-reptation model: here the influence of the intramolecular magnetic dipole-dipole interactions to the FID and Hahn echo is increasing faster with time than the contribution from intermolecular interactions. A simple expression for the relative mean squared displacements of polymer segments from different chains is obtained from the intermolecular contribution to the FID. A modified Anderson-Weiss approximation, taking into account flip-flop transitions between different spins, is proposed and on that basis, the conditions for extracting the relative intermolecular mean squared displacements of polymer segments from the intermolecular contribution to the proton FID is established. Systematic investigations of intermolecular contributions, which were considered as an unimportant factor for FID and Hahn echo in polymer systems by most previous works, actually cannot be considered as negligible and opens a new dimension for obtaining information about polymer dynamics in the millisecond regime.