Protracted Crossover to Reptation Dynamics: A Field Cycling 1H NMR Study Including Extremely Low Frequencies

Effects of end groups and entanglements on crystallization and melting behaviors of poly(ε-caprolactone)

The topology including end groups, entanglement loops, and tie molecules has a significant impact on the rheological and crystallization behavior and consequently on the functionality of a polymer. Unentangled, weakly entangled, and strongly entangled poly(ε-caprolactone)s (PCLs) with end groups and various molecular weights were synthesized. POM and DSC were used to observe spherulite growth and characterize thermal properties during crystallization and melting. The viscosity and structure of the samples were probed by rheology and X-ray analysis, respectively. The crossover of the scaling relationship of viscosity vs molecular weight demonstrates that the samples cover a wide range of entanglement density, and the bulky end groups cause deviations from the classical scaling laws. In situ simultaneous SAXS/WAXS investigations showed that the crystal structure of PCLs did not change with end groups and heating. The results of POM and DSC imply that the end groups and entanglements affect the crystallization rate and the spherulite morphology. The melting of PCLs containing end groups was found to be a multi-step process involving various nanoscale crystalline structures. The evolution of nanoscale crystalline structures of isothermally crystallized PCLs during heating was analyzed by fitting 1D SAXS profiles, and the continuous structural evolution was found to be a process influenced by end groups and entanglements. The results show that end groups and entanglements affect the chain dynamics and lead to constrained crystallization behavior and the formation of metastable structures, ultimately affecting the structure evolution during melting.

NMR Relaxometry Accessing the Relaxation Spectrum in Molecular Glass Formers

It is a longstanding question whether universality or specificity characterize the molecular dynamics underlying the glass transition of liquids. In particular, there is an ongoing debate to what degree the shape of dynamical susceptibilities is common to various molecular glass formers. Traditionally, results from dielectric spectroscopy and light scattering have dominated the discussion. Here, we show that nuclear magnetic resonance (NMR), primarily field-cycling relaxometry, has evolved into a valuable method, which provides access to both translational and rotational motions, depending on the probe nucleus. A comparison of ¹H NMR results indicates that translation is more retarded with respect to rotation for liquids with fully established hydrogen-bond networks; however, the effect is not related to the slow Debye process of, for example, monohydroxy alcohols. As for the reorientation dynamics, the NMR susceptibilities of the structural (α) relaxation usually resemble those of light scattering, while the dielectric spectra of especially polar liquids have a different broadening, likely due to contributions from cross correlations between different molecules. Moreover, NMR relaxometry confirms that the excess wing on the high-frequency flank of the α-process is a generic relaxation feature of liquids approaching the glass transition. However, the relevance of this feature generally differs between various methods, possibly because of their different sensitivities to small-amplitude motions. As a major advantage, NMR is isotope specific; hence, it enables selective studies on a particular molecular entity or a particular component of a liquid mixture. Exploiting these possibilities, we show that the characteristic Cole-Davidson shape of the α-relaxation is retained in various ionic liquids and salt solutions, but the width parameter may differ for the components. In contrast, the low-frequency flank of the α-relaxation can be notably broadened for liquids in nanoscopic confinements. This effect also occurs in liquid mixtures with a prominent dynamical disparity in their components.

In Situ RheoNMR Correlation of Polymer Segmental Mobility with Mechanical Properties during Hydrogel Synthesis

Understanding polymer gelation over multiple length-scales is crucial to develop advanced materials. An experimental setup is developed that combines rheological measurements with simultaneous time-domain ¹ H NMR relaxometry (TD-NMR) techniques, which are used to study molecular motion (<10 nm) in soft matter. This so-called low-field RheoNMR setup is used to study the impact of varying degrees of crosslinking (DC) on the gelation kinetics of acrylic acid (AAc) and N,N'-methylene bisacrylamide (MBA) free radical crosslinking copolymerization. A stretched exponential function describes the T₂ relaxation curves throughout the gelation process. The stretching exponent β decreases from 0.90 to 0.67 as a function of increasing DC, suggesting an increase in network heterogeneity with a broad T₂ distribution at higher DC. The inverse correlation of the elastic modulus G' with T₂ relaxation times reveals a pronounced molecular rigidity for higher DC at early gelation times, indicating the formation of inelastic, rigid domains such as crosslinking clusters. The authors further correlate G' with the polymer concentration during gelation using a T₁ filter for solvent suppression. A characteristic scaling exponent of 2.3 is found, which is in agreement with theoretical predictions of G' based on the confining tube model in semi-dilute entangled polymer solutions.

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.

Coarse-grained molecular dynamics simulations of poly(ethylene terephthalate)

We have constructed efficient coarse-grained (CG) models of poly(ethylene terephthalate) (PET), using three mapping schemes, in which a repeat unit is lumped into either three or four beads. The CG potentials are parameterized to reproduce target distributions of an underlying accurate atomistic model [H. Eslami and F. Müller-Plathe, Macromolecules 42, 8241-8250 (2009)]. The CG simulations allow equilibration of long PET chains at all length scales. The CG results on the density of PET in melt and glassy states, chain dimension, local packing, and structure factor are in good agreement with experiment. We have established a link between the glass transition temperature and the local movements including conformational transitions and mean-square displacements of chain segments. Temperature transferabilities of the three proposed models were studied by comparing CG results on the static and thermodynamic properties of a polymer with atomistic and experimental findings. One of the three CG models has a good degree of transferability, following all inter- and intra-structural rearrangements of the atomistic model, over a broad range of temperature. Furthermore, as a distinct point of strength of CG, over atomistic, simulations, we have examined the dynamics of PET long chains, consisting of 100 repeat units, over a regime where entanglements dominate the dynamics. Performing long-time (550 ns) CG simulations, we have noticed the signature of a crossover from Rouse to reptation dynamics. However, a clear separation between the Rouse and the reptation dynamics needs much longer time simulations, confirming the experimental findings that the crossover to full reptation dynamics is very protracted.

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.

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.

Correction of environmental magnetic fields for the acquisition of Nuclear magnetic relaxation dispersion profiles below Earth's field

T₁ relaxation times can be measured at a range of magnetic field strengths by Fast Field-Cycling (FFC) NMR relaxometry to provide T₁-dispersion curves. These are valuable tools for the investigation of material properties as they provide information about molecular dynamics non-invasively. However, accessing information at fields below 230 μT (10kHz proton Larmor frequency) requires careful correction of unwanted environmental magnetic fields. In this work a novel method is proposed that compensates for the environmental fields on a FFC-NMR relaxometer and extends the acquisition of Nuclear Magnetic Relaxation Dispersion profiles to 2.3μT (extremely low field region), with direct application in the study of slow molecular motions. Our method is an improvement of an existing technique, reported by Anoardo and Ferrante in 2003, which exploits the non-adiabatic behaviour of the magnetisation in rapidly-varying magnetic fields and makes use of the oscillation of the signal amplitude to estimate the field strength. This increases the accuracy in measuring the environmental fields and allows predicting the optimal correction values by applying simple equations to fit the data acquired. Validation of the method is performed by comparisons with well-known dispersion curves obtained from polymers and benzene.

Solid state field-cycling NMR relaxometry: instrumental improvements and new applications

The paper reviews recent progress in field cycling (FC) NMR instrumentation and its application to solid state physics. Special emphasis is put on our own work during the last 15years on instrumentation, theory and applications. As far as instrumentation is concerned we report on our development of two types of electronical FC relaxometers, a mechanical FC relaxometer and a combination of FC and one-dimensional microimaging. Progress has been achieved with respect to several parameters such as the accessible field and temperature range as well as the incorporation of sample spinning. Since an appropriate analysis of FC data requires a careful consideration of relaxation theory, we include a theory section discussing the most relevant aspects of relaxation in solids which are related to residual dipolar and quadrupolar interactions. The most important limitations of relaxation theory are also discussed. With improved instrumentation and with the help of relaxation theory we get access to interesting new applications such as ionic motion in solid electrolytes, structure determination in molecular crystals, ultraslow polymer dynamics and rotational resonance phenomena.

Simultaneous measurement of very small magnetic fields and spin-lattice relaxation

A field cycling (FC) NMR experiment is presented which allows for the simultaneous determination of very small magnetic fields down to about 3 μT and the concomitant measurement of nuclear spin-lattice relaxation times in these fields. The technique will enable broadband spin-lattice relaxation dispersion experiments down to about 100 Hz (1)H Larmor frequency. Limitations of its applicability are discussed.

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.

Interpretation of 1H and 2H spin-lattice relaxation dispersions: insights from molecular dynamics simulations of polymer melts

We demonstrate that molecular dynamics simulations are a versatile tool to ascertain the interpretation of spin-lattice relaxation data. For (1)H, our simulation approach allows us to separate and to compare intra- and inter-molecular contributions to spin-lattice relaxation dispersions. Dealing with the important example of polymer melts, we show that the intramolecular parts of (1)H spectral densities and correlation functions are governed by rotational motion, while their inter-molecular counterparts provide access to translational motion, in particular, to mean-squared displacements and self-diffusion coefficients. Exploiting that the full microscopic information is available from molecular dynamics simulations, we determine the range of validity of experimental approaches, which often assume Gaussian dynamics, and we provide guidelines for the determination of free parameters required in experimental analyses. For (2)H, we examine the traditional methodology to extract correlation times of complex dynamics from relaxation data. Furthermore, based on knowledge from our computational study, it is shown that measurement of (2)H spin-lattice relaxation dispersions allows one to disentangle the intra- and inter-molecular contributions to the corresponding (1)H data in experimental work. Altogether, our simulation results yield a solid basis for future (1)H and (2)H spin-lattice relaxation analysis.

Combining 7Li NMR field-cycling relaxometry and stimulated-echo experiments: a powerful approach to lithium ion dynamics in solid-state electrolytes

We use (7)Li NMR to study lithium ion dynamics in a (Li2S)-(P2S5) glass. In particular, it is shown that a combination of (7)Li field-cycling relaxometry and (7)Li stimulated-echo experiments allows us to cover a time window extending over 10 orders of magnitude without any gaps. While the (7)Li stimulated-echo method proved suitable to measure correlation functions F2(t) of lithium ion dynamics in solids in recent years, we establish the (7)Li field-cycling technique as a versatile tool to ascertain the spectral density J2(ω) of the lithium ionic motion in this contribution. It is found that the dynamic range of (7)Li field-cycling relaxometry is 10(-9)-10(-5)s and, hence, it complements in an ideal way that of (7)Li stimulated-echo experiments, which amounts to 10(-5)-10(1)s. Transformations between time and frequency domains reveal that the field-cycling and stimulated-echo approaches yield results for the translational motion of the lithium ions that are consistent both with each other and with findings for the motional narrowing of (7)Li NMR spectra of the studied (Li2S)-(P2S5) glass. In the (7)Li field-cycling studies of the (Li2S)-(P2S5) glass, we observe the translational ionic motion at higher temperatures and the nearly constant loss at lower temperatures. For the former motion, the frequency dependence of the measured spectral density is well described by a Cole-Davidson function. For the latter phenomenon, which was considered as an universal phenomenon of disordered solids in the literature, we find an exponential temperature dependence.

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.

Dynamics of Linear Polybutadienes in Solution Studied by Field Cycling 1H NMR

Field cycling ¹H NMR relaxometry is utilized to investigate dynamics in solutions of monodisperse polybutadienes of different molecular mass (M) and deuterated toluene. Broad temperature and polymer mass fraction ranges (c = 5-100%) are studied. By applying frequency-temperature superposition, susceptibility master curves χ″_DD(ωτ_s) are constructed. They cover the segmental relaxation and polymer chain dynamics, and provide the concentration dependence of the segmental time constant τ_s(T). The relaxation strength of polymer dynamics is reduced similarly for all M with decreasing c; for the lowest c, almost no polymer dynamics shows up, that is, the dipolar correlation function obtained via Fourier transform decays almost completely due to segmental dynamics. The dipolar correlation function is decomposed into contributions of segmental and polymer dynamics. Its long-time power-law exponent associated with entanglement dynamics is increased from its bulk value with reduced c. This is interpreted as a continuous increase of the effective entanglement molecular mass.