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.

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.

The heterogeneous distribution of the liquid phase in partially filled porous glasses and its effect on self-diffusion.

The spatial distribution of the liquid phase in a typical, partially filled, porous glass (VitraPor #5) has been examined with the aid of magnetic resonance microscopy and field gradient nuclear magnetic resonance diffusometry techniques. The correlation length of the material turned out to be long enough to permit the visualization of the microscopic heterogeneity of the material by magnetic resonance imaging. Contrasts are dominated by transverse relaxation depending on local filling degree, which in turn depends on local microstructure. The bimodal heterogeneity of the latter was also visualized by scanning electron microscopy. The effect of heterogeneity on an effective diffusion coefficient has been examined for polar (water) and nonpolar (cyclohexane) molecules.

Investigations of polymer dynamics in nanoporous media by field cycling NMR relaxometry and the dipolar correlation effect.

The chain dynamics of short-chain perfluoropolyether melts confined in Vycor nanoporous media has been characterized by field cycling nuclear magnetic resonance relaxometry and the dipolar correlation effect. The slowdown of motions under confinement, leading to larger residual dipolar couplings, has been probed by looking at the quotient of stimulated and primary echoes. Using field cycling relaxometry, it has been shown that there is strong evidence of reptation-like motion, even for such short-chain polymers as shown by the frequency and molecular weight dependences of the spin-lattice relaxation time.

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.

Cooperative polymer dynamics under nanoscopic pore confinements probed by field-cycling NMR relaxometry.

Reptational dynamics of bulk polymer chains on a time scale between the Rouse mode relaxation time and the so-called disengagement time is not compatible with the basic thermodynamic law of fluctuations of the number of segments in a given volume. On the other hand, experimental field-cycling NMR relaxometry data of perfluoropolyether melts confined in Vycor, a porous silica glass of nominal pore dimension of 4 nm, closely display the predicted signatures for the molecular weight and frequency dependences of the spin-lattice relaxation time in this particular limit, namely T1 proportional M-1/2nu1/2. It is shown that this contradiction is an apparent one. In this paper a formalism is developed suggesting cooperative chain dynamics under nanoscopic pore confinements. The result is a cooperative reptational displacement phenomenon reducing the root-mean-squared displacement rate correspondingly but showing the same characteristic dependences as the ordinary reptation model. The tube diameter effective for cooperative reptation is estimated on this basis for the sample system under consideration and is found to be of the same order of magnitude as the nominal pore diameter of Vycor.

Probing four orders of magnitude of the diffusion time in porous silica glass with unconventional NMR techniques.

The combined use of two unconventional NMR diffusometry techniques permits measurements of the self-diffusion coefficient of fluids confined in porous media in the time range from 100 microseconds to seconds. The fringe field stimulated echo technique (FFStE) exploits the strong steady gradient in the fringe field of a superconducting magnet. Using a standard 9.4 T (400 MHz) wide-bore magnet, for example, the gradient is 22 T/m at 375 MHz proton resonance and reaches 60 T/m at 200 MHz. Extremely short diffusion times can be probed on this basis. The magnetization grid rotating frame imaging technique (MAGROFI) is based on gradients of the radio frequency (RF) field. The RF gradients not necessarily need be constant since the data are acquired with spatial resolution along the RF gradient direction. MAGROFI is also well suited for unilateral NMR applications where all fields are intrinsically inhomogeneous. The RF gradients reached depend largely on the RF coil diameter and geometry. Using a conic shape, a value of at least 0.3 T/m can be reached which is suitable for long-time diffusion measurements. Both techniques do not require any special hardware and can be implemented on standard high RF power NMR spectrometers. As an application, the influence of the tortuosity increasing with the diffusion time is examined in a saturated porous silica glass.

Dissimilar electro-osmotic flow and ionic current recirculation patterns in porous media detected by NMR mapping experiments.

Random-site percolation clusters were milled into ceramic (polar) and polystyrene (nonpolar) plates as a paradigm for porous media or complex microsystem channel networks. The pore space was filled with electrolyte solutions. Using NMR microscopy techniques, maps of the following quantities were recorded: (i) flow velocity driven by external pressure gradient, (ii) electro-osmotic flow (EOF) velocity, (iii) ionic current density in the presence of EOF, (iv) ionic current density in the absence of EOF. As far as possible, the experiments were supplemented by computational fluid dynamics simulations. It is shown that electro-osmotic flow as well as the electric current density include vortices and recirculation patterns. Remarkably, all transport patterns turned out to be dissimilar, and the occurrence and positions of vortices do not coincide in the different maps.

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.

Sub- and superdiffusive molecular displacement laws in disordered porous media probed by nuclear magnetic resonance.

Hydrodynamic dispersion of water flowing through porous glasses with nominal pore sizes in the range 40 to 160 micrometers was studied with the aid of a pulsed gradient nuclear magnetic resonance technique compensating for coherent flow velocities. The crossover from effectively subdiffusive mean square displacement, proportional variant t0.84, in the absence of hydrodynamic flow to a superdiffusive, almost ballistic power law, proportional, variant t1.95, at the highest flow rates was observed. At intermediate flow rates, a gradual conversion between these two limiting power laws occurs. As a function of the Péclet number, the effective dispersion coefficient is in accordance with a power law with an exponent 1.2.

CYCLCROP indirect 13C mapping for long-time and chemical-shift selective diffusion measurements in complex hydrocarbon systems.

Cyclic cross-polarization from a proton magnetization to 13C and from there back to proton coherences permits the indirect, 13C chemical shift selective detection of hydrocarbon compounds in the proton NMR channel. This excitation technique can be combined with elements of one-, two- or three-dimensional magnetic resonance imaging permitting the measurement of time-resolved spatial distributions of hydrocarbon components. Beginning this sort of CYCLCROP mapping experiment with a non-equilibrium distribution of the constituents in the system allows one to study the time evolution of the concentrations of all components that can be identified by characteristic 13C resonance lines. As applications, studies of ingress, mixing, gel formation, transport and metabolism in living plants, long-time inter- and self-diffusion in complex hydrocarbon systems are suggested. As a test experiment, the diffusion of methanol in swollen polymethylmethacrylate was examined.

Effect of hydrodynamic flow on low-field spin-lattice relaxation in liquids in the nanoscopic vicinity of solid surfaces: theory and Monte Carlo simulations of model pore spaces.

It is shown that slow hydrodynamic flow with velocities of a few millimeters per second reduces the spin-lattice relaxation rate of fluids confined to pores of a diamagnetic, polar, solid material. The effect is predicted by an analytical theory and Monte Carlo simulations of model pore spaces. Adsorbate molecules diffusing in the vicinity of pore surfaces can perform adsorption, desorption, and readsorption cycles, effectively leading to displacements along the surface (also termed "bulk mediated surface diffusion" or BMSD). Since the surface determines the orientation of the adsorbed molecule relative to the external magnetic field, desorption at one site and readsorption at another site of a nonplanar surface will cause molecular reorientation. This is the basis of the "reorientation mediated by translational displacements" (RMTD) relaxation mechanism. If hydrodynamic flow is superimposed on diffusion, the RMTD process will be accelerated in a sort of rotational analog to translational hydrodynamic (or Taylor-Aris) dispersion. This reveals itself by a prolongation of spin-lattice relaxation times at low frequencies. The flow-relaxation effect takes place in the vicinity of the pore surfaces on the order of nanometers. The conclusions are (i) the BMSD and RMTD relaxation mechanism of fluids in porous materials is corroborated, (ii) hydrodynamic dispersion affects molecular displacements at surfaces, and (iii) interfacial slip in the sense of a molecular hopping, i.e., a desorption-readsorption process takes place.

Polymer chain dynamics under nanoscopic confinements.

It is shown that the confinement of polymer melts in nanopores leads to chain dynamics dramatically different from bulk behavior. This so-called corset effect occurs both above and below the critical molecular mass and induces the dynamic features predicted for reptation. A spinodal demixing technique was employed for the preparation of linear poly(ethylene oxide) (PEO) confined to nanoscopic strands that are in turn embedded in a quasi-solid and impenetrable methacrylate matrix. Both the molecular weight of the PEO and the mean diameter of the strands were varied to a certain degree. The chain dynamics of the PEO in the molten state was examined with the aid of field-gradient NMR diffusometry (time scale, 10(-2)-10(0) s) and field-cycling NMR relaxometry (time scale, 10(-9)-10(-4) s). The dominating mechanism for translational displacements probed in the nanoscopic strands by either technique is shown to be reptation. On the time scale of spin-lattice relaxation time measurements, the frequency dependence signature of reptation (i.e., T1 approximately nu(3/4)) showed up in all samples. A "tube" diameter of only 0.6 nm was concluded to be effective on this time scale even when the strand diameter was larger than the radius of gyration of the PEO random coils. This corset effect is traced back to the lack of the local fluctuation capacity of the free volume in nanoscopic confinements. The confinement dimension is estimated at which the crossover from confined to bulk chain dynamics is expected.

NMR study of the vapor phase contribution to diffusion in partially filled silica glasses with nanometer and micrometer pores.

The contribution of the vapor phase to molecular diffusion in porous silica glasses with nanometer (Vycor) and micrometer (VitraPor#5) pores partially filled with water (polar) or cyclohexane (nonpolar) was investigated with the aid of field-gradient NMR diffusometry. Due to the vapor phase, the effective diffusion coefficient of cyclohexane filling micrometer pores (VitraPor#5) increased up to 10 times relative to the value in bulk liquid upon reduction of the pore space filling factor. On the other hand, the effective diffusion coefficient of water first decreases and then increases when the liquid content is reduced. The dependence of the effective diffusion coefficient on the pore filling factor is strongly related to the pore dimension. A general two-phase exchange model is presented that is well accounting for all experimental diffusion features.

Field-gradient NMR diffusometry in poly(ethylene oxide) melts confined to nanoscopic pores of solid methacrylate matrices.

Depending on the choice of matrix constituents, the diameters of strands of linear, monodisperse poly(ethylene oxide) confined to nanoscopic pores of cross-linked methacrylate matrices can be varied considerably. The samples were characterized by DSC, TEM, SEM and fringe field-gradient NMR diffusometry with respect to the strand diameter. A formalism evaluating diffusive spin echo attenuation curves based on the tube/reptation model allows the determination of the strand diameter. Values in the range 8-58 nm were found in accordance with TEM and SEM micrographs of shadow-cast freeze-fractured surfaces of the samples.

Flow measurements below 50 mum: NMR microscopy experiments in lithographic model pore spaces.

Quasi two-dimensional random site percolation model objects have been prepared using a synchrotron radiation lithography technique with a spatial resolution better than 50 microm and an aspect ratio of up to 17. Flow of water through the pore space was studied with the aid of an NMR velocity mapping method and compared with a computational fluid dynamics simulation. In order to be able to measure and map widely distributed flow velocities with microscopic resolution (typically 40 x 40 microm), an experimental protocol that permits one to cover an effectively very wide velocity field of view (0.6-10 mm/s) had to be developed.

Flow-enhanced molecular reorientations and interfacial slip probed by field-cycling NMR relaxometry in microscopic pores.

It is shown that hydrodynamic flow has an effect on spin-lattice relaxation in water filled into a porous monolithic silica material. This is a rotational analogue of translational hydrodynamic (or Taylor-Aris) dispersion arising from incoherent Brownian motion in combination with coherent flow. The effect is demonstrated with the aid of field-cycling NMR relaxometry and confirmed by theoretical considerations. The results directly verify bulk mediated surface diffusion and reveal interfacial slip at fluid-solid interfaces.

Nuclear magnetic resonance study of the vapor contribution to diffusion in silica glasses with micrometer pores partially filled with liquid cyclohexane or water.

The contribution of the vapor phase to molecular diffusion in porous silica glass (Vitrapor#5; mean pore diameter 1 micrometer) partially filled with cyclohexane (nonpolar) or water (polar) was investigated with the aid of field-gradient NMR diffusometry. Due to the vapor phase, the effective diffusion coefficient of cyclohexane increased up to ten times relative to the value in bulk liquid upon reduction of the pore space filling factor. On the other hand, the effective diffusion coefficient of water first decreases and then increases when the liquid content is reduced. A two-phase exchange theory is presented accounting well for all experimental diffusion features. The diffusion behavior in the samples with micrometer pores under investigation here is in contrast to previous findings for the same solvents in a material with nanometer pores (Vycor; mean pore diameter 4 nm) where the fast-exchange limit had to be assumed [Ardelean et al., J. Chem. Phys. 119, 10358 (2003)]. It is concluded that the pore size plays a crucial role for the relevance of molecular exchange limits relative to the experimental diffusion/exchange time.

Polymer dynamics under nanoscopic constraints: the "corset effect" as revealed by NMR relaxometry and diffusometry.

A spinodal demixing technique was employed for the preparation of linear poly(ethylene oxide) (PEO) confined in nanoscopic strands, which in turn are embedded in a quasi-solid methacrylate matrix impenetrable to PEO. Both the molecular weight of the PEO and the mean diameter of the strands are variable to a certain degree. Chain dynamics of the PEO in the molten state were examined with the aid of field-gradient NMR diffusometry and field-cycling NMR relaxometry. The dominating mechanism for translational displacements in the nanoscopic strands is shown to be reptation. A formalism for the evaluation of NMR diffusometry is presented, which permits the estimation of the mean PEO strand diameter. Samples of different composition revealed diameters in the range 9-58 nm, in reasonable agreement with electron micrographs. The time scale of the diffusion measurements was 10-300 ms. On the much shorter time scale of field-cycling NMR relaxometry, 10(-9)-10(-4)s, a frequency dispersion of the spin-lattice relaxation time characteristic for reptation clearly showed up in all samples. An effective tube diameter of only 0.6 nm was found even when the strand diameter was larger than the radius of gyration of the PEO chain random coils. The finding that the tube diameter effective on the short time scale of field-cycling NMR relaxometry is much smaller than the diameter of the confining structure is termed the "corset effect", and is traced back to the lack of local free-volume fluctuation capacity under nanoscale confinements. The order of magnitude of the 'pore' diameter, at which the cross-over from confined to bulk chain dynamics is expected, is estimated.