Nail Fatkullin

Polymer Chain Dynamics and NMR

The universal features of polymer dynamics are specifically represented by laws for (anomalous) segment diffusion and chain relaxation modes. Nuclear magnetic resonance (NMR)-based techniques provide direct access to these phenomena. This in particular refers to NMR relaxation and diffusion studies. Methods suitable for this purpose are described in detail. Three basic classes of polymer dynamics models, namely the Rouse model, the tube/reptation model, and the renormalized Rouse models are outlined and discussed with respect to predictions for NMR measurands. A wealth of experimental NMR data are reviewed and compared with predictions of the model theories. It is shown that characteristic features of all three types of models can be verified in great detail provided that the model premisses are suitably mimicked in the experiments. Rouse dynamics is shown to be relevant for polymer melts with molecular weights below the critical value and for solutions of diminished entanglement effect. Features specific for the renormalized Rouse model reveal themselves in the form of high- and low-mode-number limits of the spin–lattice relaxation dispersion. These results are considered to mirror the analytical structure of the Generalized Langevin Equation. Finally, anomalous-diffusion and relaxation laws characteristic for the tube/reptation model can be perfectly reproduced in experiment if the polymer chains are confined in a nanoporous, solid matrix whereas bulk melts are not in accord with these predictions. The dynamics of chains confined in artificial tubes can be treated analytically assuming a harmonic radial potential for the polymer/wall interaction. These results derived for a real tube closely render the characteristic features of the original Doi/Edwards model predicted for a fictitious tube.

Chain dynamics in entangled polymers: Power laws of the proton and deuteron spin-lattice relaxation dispersions

Chain modes of entangled polymer melts can directly be probed in a frequency range 102 Hz<ν<108 Hz with the aid of field-cycling proton or deuteron relaxometry. The frequency dispersion of proton spin-lattice relaxation universally shows crossovers between the power laws T1∝ν0.5±0.05 (region I), T1∝ν0.25±0.05 (region II), and T1∝ν0.45±0.05 (region III) from high to low frequencies. Regions I and II are identified as limits of a theory based on the renormalized Rouse model assuming intrasegment dipolar interactions. Region III does not appear in distinct form in the deuteron T1 dispersion of perdeuterated chains. It is inferred that proton relaxation in region III is influenced by intersegment interactions which are negligible with deuterons. A corresponding formalism is given. The comparison with the experimental data suggests some multi-chain correlation of the displacement dynamics.

Molecular diffusion on a time scale between nano- and milliseconds probed by field-cycling NMR relaxometry of intermolecular dipolar interactions: Application to polymer melts

A formalism is presented permitting the evaluation of the relative mean-squared displacement of molecules from the intermolecular contribution to spin-lattice relaxation dispersion of dipolar coupled spins. The only condition for the applicability is the subdiffusive power law character of the time dependence of the mean-squared displacement as it is typical for the chain mode regime in polymer liquids. Using field-cycling NMR relaxometry, an effective diffusion time range from nano- to almost milliseconds can be probed. The intermolecular spin-lattice relaxation contribution can be determined with the aid of isotopic dilution, that is, mixtures of undeuterated and deuterated molecules. Experiments have been performed with melts of polyethyleneoxide and polybutadiene. The mean-squared segment displacements have been evaluated as a function of time over five decades. The data can be described by a power law. The extrapolation to the much longer time scale of ordinary field-gradient NMR diffusometry gives good coincidence with literature data. The total time range thus covers nine decades.

Field-cycling NMR relaxometry of polymers confined to artificial tubes: verification of the exponent 3/4 in the spin–lattice relaxation dispersion predicted by the reptation model

Abstract 2 H field-cycling NMR relaxometry was applied to deuterated linear polyethyleneoxide (PEO) of different chain lengths confined in a porous matrix of cross-linked polyhydroxyethylmethacrylate (PHEMA). The PHEMA pore diameter was of the order of 10 nm, i.e. smaller than or similar to the dimension which the PEO coils would have in the unconfined melt. The frequency and molecular weight dependences of the spin–lattice relaxation time predicted by de Gennes as T 1 ∝ M 0 ω 3/4 on a timescale short compared with the Rouse relaxation time are well reproduced experimentally.

Deuteron and proton spin-lattice relaxation dispersion of polymer melts: Intrasegment, intrachain, and interchain contributions

Proton and deuteron field-cycling NMR relaxometry was applied to deuterated and undeuterated bulk polyethyleneoxide and polybutadiene melts and mixtures thereof with molecular weights above the critical value. Spin-lattice relaxation data due to intrasegment (quadrupolar) couplings and intra- and interchain (dipolar) interactions were evaluated. Diverse dynamic limits are identified both with the proton and deuteron frequency dispersion data. The comparison between the intrachain and the interchain contributions leads to the conclusion that only model theories based on largely isotropic chain dynamics can account for the experimental findings. The extremely anisotropic character of the well-known tube/reptation model is too restrictive in this respect.

Spin diffusion in melts of entangled polymers

Based on theoretical considerations [N. F. Fatkullin, Sov. Phys. JETP 72, 563 (1991)], immaterial spin diffusion mediated by flip–flop transitions of dipolar coupled spins on different macromolecules was predicted to influence the diffusion coefficient measured in nuclear magnetic resonance field-gradient experiments. In order to test this hypothesis, we have carried out supercon fringe field proton magnetic resonance diffusometry experiments with polyethylene oxide melts (Mw=438 000) using field gradients of up to 60 T/m. The polymer chains were dispersed in a matrix of deuterated chains of an equivalent molecular mass. The time-dependent segment diffusion coefficients measured in the diluted and undiluted polymer coincided for diffusion times below about 200 ms. However, increasing the diffusion time up to 1 s leads to a reduction of the diffusion coefficient in the deuterated matrix by a factor of about 2 relative to the undeuterated system. The long-time diffusion coefficient measured with long polyme...

Segment diffusion and nuclear magnetic resonance spin-lattice relaxation of polymer chains confined in tubes: Analytical treatment and Monte Carlo simulation of the crossover from Rouse to reptation dynamics

The frequency and molecular mass dependences of nuclear magnetic resonance spin-lattice relaxation and the time dependence of the mean-squared segment displacement of Kuhn segment chains confined in static straight and randomly coiled tubes with “soft” and “hard” walls were studied. “Soft” walls were modeled in the form of a cylindrical distribution of a harmonic radial potential. This scenario is analytically solvable in contrast to the situation of “hard” (reflecting) walls corresponding to an infinitely deep square-well radial potential. In the latter case, we have therefore employed Monte Carlo simulations using a modified Stockmayer chain model. In both situations, qualitatively equivalent results were obtained. Depending on the effective tube diameter (or width of the potential well) a crossover from Rouse to reptation behavior occurs which sets on already far beyond the Flory radius of the polymer. In terms of the spin-lattice relaxation dispersion, reptation reveals itself by T1∝M0ω3/4 in the chai...

The “corset effect” of spin-lattice relaxation in polymer melts confined in nanoporous media

Linear polyethylene oxides with molecular weightsMw of 1665 and 10170 confined in pores with variable diameters in a solid methacrylate matrix were studied by proton field-cycling nuclear magnetic resonance relaxometry. The pore diameter was varied in the range of 9–57 nm. In all cases, the spin-lattice relaxation time shows a frequency dependence close toT1∞ v3/4 in the range ofv=3·10−1-2·101 MHz as predicted by the tube-reptation model. This protonT1 dispersion essentially reproduces that found in a previous deuteron study (R. Kimmich, R.-O. Seitter, U. Beginn, M. Möller, N. Fatkullin: Chem. Phys. Lett. 307, 147, 1999). As a feature particularly characteristic for reptation, this finding suggests that reptation is the dominating chain dynamics mechanism under pore confinement in the corresponding time range. The absolute values of the spin-lattice relaxation times indicate that the diameter of the effective tubes in which reptation occurs is much smaller than the pore diameters on the time scale of spin-lattice relaxation experimens. An estimation leads to a valued*∼0.5 nm. The impenetrability of the solid pore walls, the uncrossability of polymer chains (“excluded volume”) and the low value of the compressibility in polymer melts create the “corset effect” which reduces the lateral motions of polymer chains to a microscopic scale of only a few tenths of a nanometer.

NMR field gradient diffusometry of segment displacements in melts of entangled polymers

Segment diffusion in a polyethyleneoxide melt (Mw = 5 000 000) was studied with the aid of the supercon fringe field version of field‐gradient NMR diffusometry. The evaluation based on the second moment of the probability density function, i.e., the mean squared displacement, shows reasonable agreement with the predictions of the tube/reptation model. However, taking into account the whole probability density function, leads to substantial discrepancies.

Confinement effect of chain dynamics in micrometer thick layers of a polymer melt below the critical molecular weight

Polymer melts confined in micrometer thick layers were examined with the aid of field-cycling NMR relaxometry. It is shown that chain dynamics under such moderate confinement conditions are perceptibly different from those observed in the bulk material. This is considered to be a consequence of the corset effect, which predicts a crossover between Rouse and reptationlike dynamics for molecular weights below the critical value at confinement length scales much larger than 10RF, where RF is the Flory radius of the bulk polymer coil [Fatkullin et al., New J. Phys. 6, 46 (2004)]. For the polymer species studied, a perfluoropolyether with a molecular weight of 11 000, the Flory radius is of the order 10 nm, so that the experiment refers to the far end of the predicted crossover region from confined to bulk chain dynamics. Remarkably the confinement effect is shown to reach polymer-wall distances of the order 100 Flory radii.