Rainer Kimmich

Strange kinetics, porous media, and NMR

Abstract Nuclear magnetic resonance (NMR) techniques cover a broad range of length and time scales on which dynamic properties of fluids confined in porous media can be investigated. This report refers to field-cycling NMR relaxometry, field gradient NMR diffusometry and NMR microscopy. The objective was to examine diffusion, hydrodynamic dispersion, flow, and thermal convection under the influence of geometrical confinements and surface interactions in porous media. The anomalous character of these phenomena will be demonstrated and discussed in comparison with computer simulations and theoretical concepts. The first part of this presentation is devoted to nanoporous samples. It is shown that molecular Levy walks along inner surfaces occur under certain conditions. Mutual “obstruction” of molecules in molecular sieves and zeolites is another source of diffusion anomaly known as single-file diffusion which can be described by Gaussian propagators with a diffusion coefficient depending on time in a certain limit. In the case of polymers confined in narrow artificial tubes of a porous solid matrix, the characteristics of reptation were experimentally verified. The second part mainly refers to “trapping” effects as a source of anomalous transport characterised by non-Gaussian propagators. Model objects fabricated on the basis of percolation cluster models were examined with respect to flow, diffusion, thermal convection and hydrodynamic dispersion. The elucidation of transport laws in model systems of well defined and mathematically describable geometries is considered to be a promising way for the exploration of the structure/dynamics relationship in porous media as a long-term objective.

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.

Molecular dynamics in confined monomolecular layers. A field-cycling nuclear magnetic resonance relaxometry study of liquids in porous glass

Liquids filled in porous media such as porous glass do not freeze at the bulk freezing temperatures. Two phases must be distinguished. A one to at most two monolayer thick film adsorbed on the inner surfaces does not freeze at all, whereas free liquid within the pores freezes at reduced temperatures relative to the bulk values as predicted by the Gibbs/Thompson equation. The fraction of non‐freezing liquid can be evaluated from the reduction factor of the low‐frequency spin‐lattice relaxation time upon freezing of the free liquid. A method for the determination of the pore size may be established on this basis. Water and tetradecane, i.e., a polar and a nonpolar adsorbate, filled in porous glass have been studied with the aid of field‐cycling nuclear magnetic resonance (NMR) relaxometry. Above the freezing range the frequency dependences of the spin‐lattice relaxation time T1 of the two liquids strongly deviate from each other owing to the different adsorption properties. On the other hand, with frozen sa...

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.

The anomalous adsorbate dynamics at surfaces in porous media studied by nuclear magnetic resonance methods. The orientational structure factor and Lévy walks

Diffusion of adsorbate molecules along surfaces of porous media was examined with respect to ordinary and Levy walk diffusion mechanisms. The orientational structure factor formalism of the “reorientation mediated by translational displacements” (RMTD) mechanism originally derived for ordinary diffusion is generalized to Levy walks. The two cases can be distinguished experimentally using field-cycling NMR relaxometry. The low-frequency spin-lattice relaxation dispersion is influenced by the dynamics on the surfaces as well as by the surface geometry. The experiments were carried out with polar and nonpolar liquids filled into porous glasses and fine particle agglomerates (ZnO, TiO2). The spin-lattice relaxation dispersion of polar and nonpolar adsorbate species shows dramatic differences, and reflects the limits of “strong” and “weak” adsorption, respectively. The low-frequency behavior is explained by RMTD along the surfaces. At temperatures below the freezing point of the confined liquids, one or two mo...

Geometrical restrictions of water diffusion in aqueous protein systems. A study using NMR field-gradient techniques

Geometrical restrictions of water diffusion in different aqueous protein systems were studied using two versions of the NMR field gradient technique. The samples were aqueous systems of bovine serum albumin, gelatin and horse myoglobin at concentrations ranging from diluted solutions to almost dry powders being only partly hydrated. Hydrated protein aerogels were produced by the aid of a special preparation procedure and studied in addition. The experiments referred to the, temperature and concentration dependences of the water diffusion coefficient above and below the free-water freezing temperature. The diffusion coefficient within clusters of overlapping hydration shells is reduced by one order of magnitude compared with that of bulk water. Geometrical restrictions manifest themselves (a) by the obstruction effect observed at low protein concentrations, (b) by the topologically two-dimensional diffusion in the network of overlapping hydration shells, (c) by the percolation threshold appearing at about 15%b.w. water and (d) by the anomalous diffusion behaviour concluded from the protein aerogel study.

Diffusion measurements with the pulsed gradient nonlinear spin echo method

Spin echo signal attenuation by diffusion is examined for coherence evolution in the course of ordinary pulsed gradient spin echoes and for nonlinear evolution in the presence of a spatially modulated demagnetizing field. It is shown, that, for given field gradient pulse widths (or equivalently for a given gradient strength), echo attenuation by diffusion is much more efficient for nonlinear echoes than for Hahn echoes. Remarkably, in the case of nonlinear echoes the refocusing process is spoiled by diffusion not only during the gradient intervals but also thereafter. The effect of displacements occurring in the gradient intervals is enhanced according to the order of the nonlinear echo the pulse sequence is adjusted for. A second attenuation mechanism takes place after the gradient pulses due to displacements in the presence of the spatially modulated demagnetizing field. This effect even occurs when the gradient intervals are too short to contribute. A complete formalism is presented describing all feat...

NMR imaging of solids by Jeener-Broekaert phase encoding

Abstract A method for imaging of spectral parameters of solid samples is presented. The detected NMR signals are Jeener-Broekaert echoes. No read field gradient is applied during the acquisition, so that wide-line spectral parameters can be evaluated and be transferred to image contrasts. On the other hand, multipulse line-narrowing sequences can be applied during the echoes in order to obtain high-resolution spectra. The imaging principle is a pure phase-encoding Fourier technique in two or three dimensions. The phase-encoding gradients are active in the interval between the first two pulses of the Jeener-Broekaert three-pulse sequence. Between the second and the third pulse, the information is conserved in the dipolar (or quadrupolar) order state which is insensitive to field gradients and governed by the relatively slow dipolar (or quadrupolar) relaxation. This interval therefore can be chosen to be long enough to switch the gradients off. The third pulse “reads” the information of the spin-state order and produces an echo under homogeneous field conditions. In the case of two-dimensional imaging, a slice is preselected prior to the whole Jeener-Broekaert sequence by the aid of a LOSY slice-selection pulse. Test experiments are reported, and applications to polymer and biological materials are discussed.

Diffusion on random-site percolation clusters: Theory and NMR microscopy experiments with model objects

Quasi-two-dimensional random-site percolation model objects were fabricated based on computer-generated templates. Samples consisting of two compartments, a reservoir of H2O gel attached to a percolation model object, which was initially filled with D2O, were examined with nuclear magnetic resonance microscopy for rendering proton spin density maps. The propagating proton/deuteron interdiffusion profiles were recorded and evaluated with respect to anomalous diffusion parameters. The deviation of the concentration profiles from those expected for unobstructed diffusion directly reflects the anomaly of the propagator for diffusion on a percolation cluster. The fractal dimension of the random walk d(w) evaluated from the diffusion measurements on the one hand and the fractal dimension d(f) deduced from the spin density map of the percolation object on the other permits one to experimentally compare dynamical and static exponents. Approximate calculations of the propagator are given on the basis of the fractional diffusion equation. Furthermore, the ordinary diffusion equation was solved numerically for the corresponding initial and boundary conditions for comparison. The anomalous diffusion constant was evaluated and is compared to the Brownian case. Some ad hoc correction of the propagator is shown to pay tribute to the finiteness of the system. In this way, anomalous solutions of the fractional diffusion equation could experimentally be verified.