All Polymer Diffusion Regimes Covered by Combining Field-Cycling and Field-Gradient 1H NMR

Complex Monte Carlo Light-Driven Dynamics of Monomers in Functionalized Bond Fluctuation Model Polymer Chains

We study Monte Carlo dynamics of the monomers and center of mass of a model polymer chain functionalized with azobenzene molecules in the presence of an inhomogeneous linearly polarized laser light. The simulations use a generalized Bond Fluctuation Model. The mean squared displacements of the monomers and the center of mass are analyzed in a period of Monte Carlo time typical for a build-up of Surface Relief Grating. Approximate scaling laws for mean squared displacements are found and interpreted in terms of sub- and superdiffusive dynamics for the monomers and center of mass. A counterintuitive effect is observed, where the monomers perform subdiffusive motion but the resulting motion of the center of mass is superdiffusive. This result disparages theoretical approaches based on an assumption that the dynamics of single monomers in a chain can be characterized in terms of independent identically distributed random variables.

Predicting Solute Diffusivity in Polymers Using Time-Temperature Superposition

We demonstrate a novel application of the time-temperature superposition (TTS) principle to predict solute diffusivity D in glassy polymers using atomistic molecular dynamics simulations. Our TTS approach incorporates the Debye-Waller factor ⟨u²⟩, a measure of solute caging, along with concepts from thermodynamic scaling methods, allowing us to balance contributions to the dynamics from temperature and ⟨u²⟩ using adjustable parameters. Our approach rescales the solute mean-squared displacement curves at several temperatures into a master curve that approximates the diffusive dynamics at a reference temperature, effectively extending the simulation time scale from nanoseconds to seconds and beyond. With a set of "universal" parameters, this TTS approach predicts D with reasonable accuracy in a broad range of polymer/solute systems. Using TTS greatly reduces the computational cost compared to standard MD simulations. Thus, our method offers a means to rapidly and routinely provide order-of-magnitude estimates of D using simulations.

Lattice Boltzmann framework for accurate NMR simulation in porous media

Nuclear magnetic resonance (NMR) responses of fluids saturating porous media arise from complex relaxation-diffusion dynamics of polarized spins. These constitute a sensitive probe of the microstructure and are described by the Bloch-Torrey equations. An NMR simulation framework based on an augmented lattice Boltzmann method aimed at the fine-scale resolution of nuclear polarization density is presented. The approach encapsulates the time evolution of the full magnetization vector and naturally incorporates the mechanisms of diffusional transport. Spin dephasing mechanisms are fully resolved at tomogram voxel scale to account for magnetic field inhomogeneity. The approach is validated against analytical solutions of spin-echo decays for simple pore geometries. An application to a nano-computed-tomography image of chalk with inhomogeneous internal fields yields T_{2} spectral measures in good agreement with experiment and illustrates the spatial pore-scale dynamics of net magnetization. Findings establish the feasibility of the framework for pure diffusion and present an approach vector to modeling the evolution of magnetization under flow conditions.

Dynamical Comparison of Different Polymer Architectures-Bottlebrush vs Linear Polymer

Different polymer architectures behave differently regarding their dynamics. We have used a combination of dielectric spectroscopy, and fast field cycling nuclear magnetic resonance (NMR) to compare the dynamical behavior of two different polymer architectures, with similar overall molecular weight. The systems of interest are a bottlebrush polymer and a linear one, both based on poly(dimethylsiloxane) (PDMS). To verify the structure of the PDMS-g-PDMS bottlebrush in the melt, small-angle neutron scattering was used, yielding a spherical shape. Information about the segmental dynamics was revealed by dielectric spectroscopy and extended to higher temperatures by fast field cycling NMR. One advantage of fast field cycling NMR is the detection of large-scale chain dynamics, which dielectric spectroscopy cannot probe for PDMS. While segmental relaxation seems to be independent of the architecture, the large-scale chain dynamics show substantial differences, as represented by the mean square displacement. Here, two regions are detected for each polymer. The linear polymer shows the Rouse regime, followed by reptation. In contrast, the bottlebrush polymer performs Rouse dynamics and diffusion in the available time window, and entanglement effects are completely missing.

Pulsed field gradient NMR diffusion measurement in nanoporous materials

Labeling in diffusion measurements by pulsed field gradient (PFG) NMR is based on the observation of the phase of nuclear spins acquired in a constant magnetic field with purposefully superimposed field gradients. This labeling does in no way affect microdynamics and provides information about the probability distribution of molecular displacements as a function of time. An introduction of the measuring principle is followed by a detailed description of the ranges of measurements and their limitation. Particular emphasis is given to an explanation of possible pitfalls in the measurements and the ways to circumvent them. Showcases presented for illustrating the wealth of information provided by PFG NMR include a survey on the various patterns of concentration dependence of intra-particle diffusion and examples of transport inhibition by additional transport resistances within the nanoporous particles and on their external surface. The latter information is attained by combination with the outcome of tracer exchange experiments, which are shown to become possible via a special formalism of PFG NMR data analysis. Further evidence provided by PFG NMR concerns diffusion enhancement in pore hierarchies, diffusion anisotropy and the impact of diffusion on chemical conversion in porous catalysts. A compilation of the specifics of PFG NMR and of the parallels with other measurement techniques concludes the paper.

Numerical calculation of free-energy barriers for entangled polymer nucleation

The crystallization of entangled polymers from their melt is investigated using computer simulation with a coarse-grained model. Using hybrid Monte Carlo simulations enables us to probe the behavior of long polymer chains. We identify solid-like beads with a centrosymmetry local order parameter and compute the nucleation free-energy barrier at relatively high supercooling with adaptive-bias windowed umbrella sampling. Our results demonstrate that the critical nucleus sizes and the heights of free-energy barriers do not significantly depend on the molecular weight of the polymer; however, the nucleation rate decreases with the increase in molecular weight. Moreover, an analysis of the composition of the critical nucleus suggests that intra-molecular growth of the nucleated cluster does not contribute significantly to crystallization for this system.

One dimensional magnetic resonance microscopy with micrometer resolution in static field gradients

Magnetic resonance microscopy (MRM) is a valuable tool for spatially resolved studies. While it is desirable to address voxels in the general case, it is sufficient to resolve slices of the sample in many cases of practical importance, e.g., for layered structures or at planar surfaces. We demonstrate that use of high static field gradients of 73 T/m in combination with a specially designed probe head enable MRM with an ultrahigh resolution of ∼2 μm in one dimension. The key feature of the built probe head is a precise computer controlled adjustment of the sample position and orientation, which allows for an accurate alignment of the samples with respect to the gradient of the magnetic field. Since slice-wise scanning of extended samples with this high spatial resolution is time-consuming, we introduce a methodology to reduce the experimental time significantly. Unlike the usual approach, which involves elaborate hardware and software correction, experimental imperfections are removed by stepwise moving the sample in our case. We demonstrate the capabilities of high-resolution 1D MRM for a solid sample with a layered structure and a liquid droplet on a planar solid substrate.

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.

Molecular Dynamics in the Lyophases of Copolymer P123 Investigated with FFC NMR Relaxometry

Associative block copolymers of the type (EO) _x(PO) _y(EO) _x (where EO and PO represent ethylene and propylene oxides, respectively) in aqueous solution have far reaching commercial applications such as solubilization, controlled-drug delivery, etc. The molecular dynamics of a self-associating triblock copolymer (EO)₂₀(PO)₇₀(EO)₂₀ (known as P123 with a molecular weight of ∼5800), in aqueous solution (D₂O), consisting of various lyotropic liquid crystalline phases such as isotropic micellar, cubic, hexagonal, and lamellar phases, is investigated using the fast field cycling nuclear magnetic resonance (FFC NMR) relaxometry technique in the Larmor frequency range from 5 kHz to 30 MHz. A nuclear spin-relaxation model consisting of chain modes (Rouse modes) and order fluctuation (OF) modes typical for polymers and liquid crystals, respectively, is considered to explain the observed proton magnetic relaxation dispersion (PMRD) data in the lyophases under investigation. The PMRD analysis in both isotropic micellar and cubic phases revealed a Rouse frequency dependence of spin-lattice relaxation rate ( R₁), i.e., R₁ ∝ -τ_s ln(ωτ_s), in the entire frequency range of study. Hexagonal and lamellar phase data show Rouse modes as well as OF modes, leaving the signature of the latter as R₁ ∝ ω^{- p}, where p ∼ 0.5 is typical for nematic mesogens. The activation energies were also determined from segmental correlation times in the lyophases of study. To the best of our knowledge, the present FFC NMR relaxometry study is unique and quantitative in unraveling molecular dynamics of the associative copolymer P123 in aqueous solution.

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.

Self-diffusion studies by intra- and inter-molecular spin-lattice relaxometry using field-cycling: Liquids, plastic crystals, porous media, and polymer segments

Field-cycling NMR relaxometry is a well-established technique for probing molecular dynamics in a frequency range from typically a few kHz up to several tens of MHz. For the interpretation of relaxometry data, it is quite often assumed that the spin-lattice relaxation process is of an intra-molecular nature so that rotational fluctuations dominate. However, dipolar interactions as the main type of couplings between protons and other dipolar species without quadrupole moments can imply appreciable inter-molecular contributions. These fluctuate due to translational displacements and to a lesser degree also by rotational reorientations in the short-range limit. The analysis of the inter-molecular proton spin-lattice relaxation rate thus permits one to evaluate self-diffusion variables such as the diffusion coefficient or the mean square displacement on a time scale from nanoseconds to several hundreds of microseconds. Numerous applications to solvents, plastic crystals and polymers will be reviewed. The technique is of particular interest for polymer dynamics since inter-molecular spin-lattice relaxation diffusometry bridges the time scales of quasi-elastic neutron scattering and field-gradient NMR diffusometry. This is just the range where model-specific intra-coil mechanisms are assumed to occur. They are expected to reveal themselves by characteristic power laws for the time-dependence of the mean-square segment displacement. These can be favorably tested on this basis. Results reported in the literature will be compared with theoretical predictions. On the other hand, there is a second way for translational diffusion phenomena to affect the spin-lattice relaxation dispersion. If rotational diffusion of molecules is restricted, translational diffusion properties can be deduced even from molecular reorientation dynamics detected by intra-molecular spin-lattice relaxation. This sort of scenario will be relevant for adsorbates on surfaces or polymer segments under entanglement and chain connectivity constraints. Under such conditions, reorientations will be correlated with translational displacements leading to the so-called RMTD relaxation process (reorientation mediated by translational displacements). Applications to porous glasses, protein solutions, lipid bilayers, and clays will be discussed. Finally, we will address the intriguing fact that the various time limits of the segment mean-square displacement of polymers in some cases perfectly reproduce predictions of the tube/reptation model whereas the reorientation dynamics suggests strongly deviating power laws.

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.

1H NMR at Larmor frequencies down to 3Hz by means of Field-Cycling techniques

Field-Cycling (FC) NMR experiments were carried out at ¹H Larmor frequencies down to about 3Hz. This could be achieved by fast switching a high polarizing magnetic field down to a low evolution field which is tilted with respect to the polarization field. Then, the low frequency Larmor precession of the nuclear spin magnetization about this evolution field is registered by means of FIDs in a high detection field. The crucial technical point of the experiment is the stabilization of the evolution field, which is achieved by compensating for temporal magnetic field fluctuations of all three spatial components. The paper reports on some other basic low field experiments such as the simultaneous measurement of the Larmor frequency and the spin-lattice relaxation time in such small fields as well as the irradiation of oscillating transversal magnetic field pulses at very low frequencies as a novel method for field calibration in low field FC NMR. The potential of low field FC is exemplified by the ¹H relaxation dispersion of water at frequencies below about 2kHz stemming from the slow proton exchange process.