Nuclear magnetic resonance (NMR) has provided us with many beneficial opportunities for science and technology. Its continued use in novel fields has yielded impressive strength and attractiveness for nearly a century. This is particularly true with regards to the topic of this book, the exploration of “Fluid Transport in Porous Solids and Heterogeneous Materials”.

Here, the benefit of NMR in being able to look “from the outside” into a system becomes particularly evident. NMR operates as an “ideal spy”, providing information without interfering with internally occurring phenomena. NMR is able to give information on pore spaces as well as anything that might happen within them. This wide-range of information that is accessible is illustrated by the examples in this book. The origin of some of these developments can, most remarkably, be traced back over many decades, to the very beginning of NMR research. In this chapter we will recollect some of the roots of the challenges we face today with applying NMR to studying “Fluid Transport in Porous Solids and Heterogeneous Materials”—albeit with some bias by personal experiences and impressions.

The output of nuclear magnetic relaxation on pore space architecture and guest dynamics in porous materials is, generally, based on model assumptions. These assumptions are, as a rule, well established and supported by experimental evidence. In its early years, however, NMR was used for studying molecular diffusion. The information gained stands on its own. Hahn's seminal paper in 1950¹  provided us with an opportunity that, in subsequent years, has been extensively exploited for diffusion measurements with liquids.²  With the application of pulsed field gradients by Stejskal and Tanner,³  the gradient intensity could be chosen large enough so that, eventually, diffusion measurements with porous materials have become possible. In his seminal paper of 1965⁴  John Tanner introduced the technique under the title “Pulsed Field Gradients for NMR Spin-Echo Diffusion Measurements”. Since then, the method has found application in quite a number of different communities. Its widespread use might have contributed to a diversification in nomenclature, with currently two names in common use: pulsed field gradient (PFG)^5–9  and pulsed gradient spin echo (PGSE)^10–12  NMR. In either case, Tanner's original wording is easily recognized.

The development of NMR was, essentially from its very beginning, closely related with the search for its application to petrophysical studies. The oil industry became aware of the potential of this novel source of information and vigorously promoted research on logging projects. The data in Table 1.1, taken from the paper of Kleinberg and Jackson,¹³  illustrate this intense and most rewarding partnership from its beginning until 2000.

In 1948, two years after the discovery of NMR in condensed matter by Bloch¹⁴  and Purcell,¹⁵  the thesis of Bloembergen and the classical paper by Bloembergen, Purcell and Pound (BPP theory¹⁶ ) explained many features of the relaxation of NMR signals in bulk liquids by interpreting the dependence of the relaxation times on parameters related to molecular motion, including temperature, viscosity and distance between spins. A retrospective article by Bloembergen gives a review of NMR attempts before 1946 and of early work on relaxation.¹⁷  It has also been recognized that fluid molecules can be adsorbed near a solid surface, resulting in a decrease in their mobility. The existence and influence of pore walls were later found to appear in the relaxation patterns of NMR. The application of the BPP theory to the adsorbed layers could have caused researchers to think that the relaxation times of molecules in the adsorbed layers could have been decreased and so decreasing the relaxation times of fluids inside the pore space of porous media; but it seems that nobody had that intuition.

However, the idea was raised to build a device to be lowered inside the wells to get the signals of oil and water from the porous rock formation outside the borehole at depths of thousands of meters. Russel Varian had demonstrated that it was possible to observe NMR by free precession (at about 2 kHz) in the Earth's field. Numerous studies about the feasibility of the application of Nuclear Magnetic resonance for well Logging (NML) by Varian Associates^18–20  followed. In those pioneering researches, the now widespread use and importance of the NMR single-sided NMR devices,²¹  that led to the evolution of the concept of compact and mobile devices,²²  able to detect NMR signal outside the magnet, outside the laboratory, in a non-destructive way, regardless of the sample sizes emerged. The key feature of NML was intended to be the possibility to exploit the different relaxation times of bulk oil and water (10 times larger for water than for oil) to distinguish their signals. Since water and oil have about the same ¹H nuclei density, the fraction of water and oil could have been determined by their signal ratio and the porosity of the rock formation by the total signal, and all this at depths of thousands of meters. Three NML research projects started at that time: Varian with Byron-Jackson, Schlumberger in Ridgefield, and what is now Chevron; nuclear Magnetic Resonance studies for fluids in Porous Media (MRPM) also started at Shell and Magnolia (later Mobil) to understand the properties of fluids in porous media for the purpose of characterizing reservoir rocks.

When it was found that surface effects shortened water relaxation times to where water could not be distinguished from oil or even to where it could not be observed, it appeared that NML might not be very useful. However, it was soon realized that relaxation times inversely proportional to pore surface areas gave information on pore size distributions, thereby giving information on the permeability of the rock to the flow of pore fluids, even more important than the original objectives.^23–25 

In the 1950s many kinds of data were interpreted to suggest thick reduced-mobility liquid layers of water or other fluids adsorbed on surfaces including those of rock grains. Field dependent relaxation measurements at Chevron (from a micro-Tesla to a Tesla) did not support this and even showed that the postulated ice-like layers in DNA did not exist.²⁶  The enhanced pore fluid relaxation comes mainly from a single adsorbed liquid layer at the surface.

In the late 1950s it was well understood that the local relaxation times for fluids were greatly shortened in not much more than one molecular layer at the solid surface. It was shown²³  that if a pore is small enough that diffusion maintains nuclear magnetization uniform inside the pore, the rate of the observed relaxation time of the fluid in the pore is 1/T=1/T_b+(V_s/V)/(T_s+τ), where T_b is the relaxation time of the bulk fluid, V_s the volume of the surface layer, V the pore volume, T_s the relaxation time of the surface layer, and τ the residence time of a molecule in the surface layer. In 1956 Henry Torrey, Jan Korringa and Bob Brown wrote a U.S. Patent where many of the most basic features of MRPM were summarized, including relaxation for water and oils of different viscosity, and their behavior inside porous media at different temperatures. The first experimental NML was run in 1960 and limited commercial earth's-field NML service became available, and useful applications were found.

The effects of pore sizes and surface properties on relaxation were investigated at Shell.^27,28  Wettability effects had already been noted by Brown and Fatt.²⁹  Most of the papers used the longitudinal relaxation time T₁, but also the transverse relaxation time T₂ started to be studied.^30,31 

The porosity of a water- or oil-saturated porous material can be determined from the NMR signal, with proper calibration. Other properties can be related to T₁ or T₂ relaxation curves. It was assumed that signal with T₁ less than some “cutoff” time was “irreducible water” and that only fluid with longer relaxation times would be produced. Timur³²  found a cutoff time of about 12 ms. Studies on the determination of water and oil when both phases are present in the pore space led also to the proposals of NMR estimates of the residual oil saturation.³³  Permeability estimates from relaxation data were developed by Seevers³⁴  and by Timur³²  and later by Kenyon et al.³⁵  The coming of MRI contributed to the understanding of oil industry applications.³⁶ 

Many porous media have a wide distribution of pore sizes, the distribution of classes of fluids with a distribution of relaxation times, possibly in different regimes of exchange, can determine a multi-exponential relaxation. A stochastic theory for the relaxation in heterogeneous systems with many exchanging water phases was proposed in 1957 by Zimmerman and Brittin.³⁷  It started in an oil industry laboratory, to justify the behavior of T₁ and T₂ in water systems adsorbed on silica gel, and had great success in the study of systems also of biological interest, that for many aspects can be considered as porous media. An example is given by the study on DNA water reported by Brown.²⁶  After some examples of sporadic interest for NMR relaxation in biological systems,³⁸  interest grew significantly with the appearance of an NMR study on HeLa cells (the first “immortal” human cells grown in a lab³⁹ ), and the Damadian paper⁴⁰  that indicated the possibility to detect tumors by increased relaxation times for the first time. Let's not forget that the papers, posing the basis for Magnetic Resonance Imaging (MRI), appeared around the same time^41,42  with a clear focus on biological systems.

The two fields of petrophysical and biological studies enjoyed reciprocal advantages by exchange of experiences, methods, and theories, given by the MRPM studies. A clear example of this is given by the seminal work of Brownstein and Tarr⁴³  that gave the interpretation of the multi-exponential behavior by classical diffusion in the presence of relaxation sinks on the confining surfaces, without the need of the assumption of different water phases. Written for cell water, later this theory influenced the interpretation of multi-exponential relaxation for porous media of any nature, including rocks.⁴⁴  In any case, it became clear that the observed multi-exponential relaxation, giving rise to distributions of relaxation times strongly depended on the diffusion regimes in the complex network of the pore space. In a real porous medium, with the same surface properties, in the case of a fast diffusion regime, one would observe a single exponential decay only if the diffusion is fast enough inside each pore and among the pores to make the magnetization uniform inside the whole pore space, or, of course if the diffusion is fast and the pores are all the same. However, if the diffusion is fast inside each pore, but slow among pores, the relaxation will be multi-exponential.^45,46  Algorithms have been developed to invert multi-exponential curves to distributions of relaxation times.⁴⁷  An algorithm was proposed with a smoothing coefficient varying along the relaxation time distribution, in order to maintain uniform the penalty.⁴⁸ 

Over time these methods have been exploited to study the pore size distributions of porous media of different nature and for different applications. Simple methods to separate solid and liquid components on the free induction decay, combined with quasi-continuous analysis of the two data sets, have been exploited to follow the kinetics of Portlandite and liquid component formation in hydrate cements.⁴⁹  In coral skeletons, the pore-sizes can be analyzed, with a single NMR measurement, at multiple length scales. The effect of increasing acidity on increasing the macro-scale porosity, whilst the linear extension rate remained the same, revealed the acclimation of the corals in a warming acidifying ocean.⁵⁰ 

Later, algorithms were developed for two-dimensional (2D) inversion of experimental 2D data,⁵¹  in order to obtain Relaxation–Relaxation and Diffusion–Relaxation correlation functions or pore-to-pore exchange parameters (for more details see Chapter 4).^52–54  It is clear that caution is needed to interpret multi-exponential relaxation in terms of pore-size distributions, especially for T₂, for water subjected to diffusion inside field gradients. Not only large scale gradients can be present, but also internal gradients inside the pore due to the susceptibility difference between water and the solid material. For water diffusing inside a constant gradient, for unrestricted diffusion, the dependence of 1/T₂ on the half-echo time in a CPMG sequence is expected to be quadratic.⁵⁵  In many porous media it was found to be linear instead of quadratic, and this was interpreted as due to a distribution of correlation times for molecular diffusion.⁵⁶ 

Shortly after, NMR with pulsed field gradients enabled diffusion measurements of water in zeolites,⁵⁷  probably the most important representative of “microporous” materials.⁵⁸  Pore sizes of such substances are known to be of molecular dimensions. Transverse nuclear magnetic relaxation times of guest molecules in such host materials are generally very small so that, as a rule, NMR diffusion studies necessitate the use of “pulsed” field gradients. As a most astonishing outcome of these studies, water diffusion in zeolites was found to be only slightly exceeded by that in the neat liquid.

This puzzling result gave rise to an in-depth study of molecular diffusion in zeolites in the very place where Felix Bloch was working as the first PhD student of Werner Heisenberg. Owing to the activities of Artur Lösche and Harry Pfeifer and their groups,⁵⁹  Leipzig was now on the way to becoming a place which Richard Ernst, during a talk in Leipzig in 1992, referred to as the “East Pole of Magnetic Resonance”.⁶⁰  Benefitting from being part of the Eastern hemisphere, researchers in Leipzig had access to probably the largest zeolite crystals available at this time, synthesized in the famous laboratory of Sergey Petrovitch Zhdanov in Leningrad. In this way, by a purposeful variation of the diffusion path lengths in relation to the crystal sizes, the high diffusivities reported in ref. 57 could be attributed to “long-range” diffusion, i.e. to mass transfer in free space between the individual zeolite crystallites. Water diffusivities in the micropores, however, was determined to be notably smaller, in complete agreement with the expected behavior.⁶¹  The application of diffusion NMR to beds of zeolite crystals gave rise to the development of two concepts of data analysis,⁶²  which have become part of the general tool box of NMR, namely the formalism of two-range diffusion^63,64  for taking account of mass exchange between different compartments⁶⁵  such as biological cells⁶⁶  and the introduction of the “mean” propagator.⁶⁷ 

The diffusion of guest molecules in zeolites was (so far) mainly based on the measurement of transient uptake and release curves initiated by a pressure step in the surrounding atmosphere. Diffusivities were determined with the understanding that these phenomena were controlled by the guest diffusivity within the zeolite pore space. It came as a great surprise, therefore, when in many cases the intracrystalline diffusivities—now directly measured owing to the potentials of NMR—proved to exceed the so far generally accepted values by several orders of magnitude.⁶⁸  As the only solution of the problem, mass transfer in such crystals had to be required to be controlled by additional transport resistances on the external crystal surface or in intracrystalline space rather than by exclusively the diffusional resistance of the genuine pore space as so far generally implied.⁶⁹  NMR diffusion studies did thus provide evidence of these barriers long before they became an object of high-resolution electron microscopy⁷⁰  and initiated a paradigm shift in the understanding of mass transfer in nanoporous materials.⁷¹ 

In a sense, the invention of MRI brought about a revolution of thinking in the NMR community. Being used to associate spatial resolution with phenomena like optical holography or scattering of particles or electromagnetic waves, the pioneers in the field soon realized striking analogies. In particular, the signal patterns recorded with the standard MRI technique, i.e. “spin-warp imaging”,⁷²  were identified as representations in reciprocal-space just as known with optical holograms. That is, real-space images can be rendered by two- (or three-) dimensional Fourier transformations of the NMR signal intensities as functions of the respective wave-vector components. Reciprocal-space variables (wavenumbers) are converted in this way to conjugate real-space coordinates.

This analogy can be extended to diffusion NMR. It was mainly Paul Callaghan and his group who put this idea forward.⁷³  Again, a real-space/reciprocal-space notion was employed. The spin-echo attenuation function due to translational diffusion is actually the Fourier conjugate to the real-space distribution of molecular displacements, where one is normally dealing with Gauss functions for both space-representations. By definition, the reciprocal-space variable, the ‘wavenumber’, is determined by the strength and duration of the effective field gradient pulse acting on the evolution of the precessing transverse magnetization.

As a matter of course, the effect of field gradients in MRI or NMR diffusometry experiments has little to do with travelling electromagnetic or matter ‘waves’. One is therefore tempted to discount the analogy as something entirely formal. However, the resemblance between real wave scenarios and magnetization-evolution effects in the presence of field-gradient goes much further. The wavelength defining the wavenumber turns up again as the pitch of the so-called helix of the spatial distribution of transverse magnetization after a gradient pulse. In the stimulated-echo variant of NMR diffusometry,^74,75  this helix is temporarily and partially converted to a sinusoidal modulation of the longitudinal magnetization along the gradient axis with just the ‘wavelength’ attributed to the relevant wavenumber.

There is even more to it than that. The phenomena most characteristic for waves are scattering and diffraction. As shown in ref. 76, there is a formal coincidence of the diffusive echo-attenuation function with the incoherent dynamic structure factor defined for scattering experiments. That is, field-gradient NMR diffusometry extends the accessible wavenumber range of quasi-elastic neutron scattering, for instance. Entirely distinct techniques for studies of molecular dynamics can thus be combined in a most favorable way as extensively outlined in ref. 77.

The second and most striking finding in this context is that diffraction patterns can be observed in diffusion experiments. Callaghan et al.⁷⁸  demonstrated that exactly the same intensity patterns as in optical diffraction experiments appear for echo attenuation when the field-gradient induced ‘wavelength’ matches structural lengths of a confining matrix such as a porous medium. The relevant length scale is usually in the order of a few micrometers. Striking applications of this kind of ‘NMR diffusion diffraction’ to aqueous erythrocyte suspensions have been reported by Kuchel et al.⁷⁹ 

In terms of wavenumbers, the determination of very low diffusion coefficients with field-gradient techniques requires as large values as possible. An optimal efficiency in this respect can be achieved by using firstly steady instead of pulsed gradients, secondly the stimulated-echo variant, and thirdly the particularly strong and stable gradients of the fringe field of superconducting magnets⁸⁰  (see also Chapter 8). With such a set-up it was even possible to reach the ultimate physical low-end limit of any field-gradient NMR diffusometry measurement, namely immaterial spin-diffusion mediated by flip-flop spin transitions. The direct assessment of this limiting spin-diffusion coefficient was reported in ref. 81 for polymers.

A further result obtained with the fringe-field NMR technique is the evaluation of the before mentioned incoherent dynamic structure factor for polymers diffusing in nanoporous matrices. Referring to the well-known tube/reptation concept,⁸²  it was shown that the echo attenuation function is non-Gaussian in this case.⁸³  It depends on the pore diameter in a characteristic way. That is, pore diameters have consistently been determined with the aid of diffusion experiments.⁸⁴  Remarkably, these results exceed the diffusion time and wavenumber ranges accessible by quasi-elastic neutron scattering by far.

Fluid transport through porous media can be mediated by molecular diffusion, hydrodynamic flow, and—between these two extremes—by hydrodynamic (or Taylor/Aris) dispersion. The latter contains elements both of potentially tortuous but coherent flow, and diffusive displacements in a superimposed manner.⁸⁵ 

It has long been predicted that the incoherent part of displacements should be sub-diffusive if diffusion prevails and super-diffusive in cases where tortuous flow dominates in random porous media. The crossover between such mean square displacement laws deviating from Einstein's linear relationship has indeed been observed with the aid of pulsed field-gradient NMR for the first time.⁸⁶  Moreover, a rotational analogue of Taylor/Aris dispersion has been concluded in a field-cycling NMR relaxometry study of flow along inner surfaces of porous media.⁸⁷  This suggests a kind of interfacial slip including intermittent adsorption/desorption hopping cycles along the pore walls instead of the no-slip boundary condition frequently anticipated for viscous flow.

Studies of anomalous diffusion, i.e. of the sub- or super-diffusive mean squared displacement laws for disordered porous media, require experimental access to a time scale as wide as possible. The incoherent dynamic structure factors determined with the aid of quasi-elastic neutron scattering on the one hand, and—in the form of echo-attenuation functions—by steady- or pulsed-field gradient NMR techniques on the other leave a diffusion time gap from about 100 ns to about 100 µs.⁷⁷  This gap can be bridged by evaluating the inter-molecular contribution to spin-lattice relaxation as detected by field-cycling NMR relaxometry experiments. Intermolecular spin-lattice relaxation directly reflects translational diffusion. It can be distinguished from the intramolecular counterpart by isotopic dilution. That is in particular by partially replacing proton containing molecules by perdeuterated species. In combination, an enormous, hitherto unprecedented range of seven orders of magnitude of the diffusion time becomes accessible in this way.^88,89 

As outlined above, a very important information source exploited with well-logging NMR is based on nuclear magnetic relaxation of the liquids confined in porous rocks. Porous rock and building materials such as cement usually contain electron-paramagnetic impurities that can act as efficient relaxation sinks in addition to diamagnetic mechanisms.⁹⁰  Irrespective of the relevant spin interactions dominating spin-lattice relaxation, the strong increase of relaxation rates in pores suggests correlation times longer by orders of magnitude than in the bulk-liquid state.

The first idea to explain this was that a kind of immobilization takes place at the pore walls. One even spoke of “irrotationally bound” molecules. However, quite amazingly it turned out that the translational mobility for example in hydration layers of adsorbing surfaces is only moderately reduced. Geometrical constraints may even be more restrictive than binding forces. Not unlike the findings for zeolites,^62,68  field-gradient NMR studies reveal that translational diffusion of adsorbate molecules at surfaces is unexpectedly fast. This was demonstrated for instance in so-called “non-freezing” surface layers,⁹¹  with crystal water in myoglobin single crystals⁹²  and at silica surfaces.⁹³ 

Before this background, the most intriguing question arose, how could the retardation of molecular reorientations in porous media by up to eight orders of magnitude relative to bulk conditions be reconciled with the almost bulk-like translational diffusivities at short displacement length scales? The answer is that adsorbate molecules probe diamagnetic surfaces via translations from adsorption site to adsorption site. The mechanism is called bulk mediated surface diffusion (BMSD) and was identified as a kind of Lévy walk along pore walls.^94,95  That is, molecular reorientation at surfaces is a matter of the topology. The correlation of molecular orientations is maintained as long as the sites probed by the molecules are correlated with respect to surface orientations. Thus, reorientation is mediated by translational displacements along the surface (RMTD process).⁹⁶ 

Apart from porous rock, hydrated cement or mortar at its various processing stages turned out to be one of the media where electron-paramagnetic “impurities” can act as relaxation sinks (see Chapter 10). This especially refers to iron ions that are known to be incorporated in the respective pore walls at random. Again, surface diffusion is an essential element of the relaxation process. Korb et al.^90,97  developed a model of 2D adsorbate diffusion providing the interaction contacts to the paramagnetic relaxation centers. Characteristic material properties such as the specific surface area, for instance, have been determined on this basis. Relaxation-based findings can be compared with results from field-gradient NMR diffusion studies referring to much longer time and displacement length scales.⁹⁸  In this context, it should also be noted that electron paramagnetic centers such as nitroxide free radicals can also be incorporated artificially in otherwise diamagnetic materials for just the purpose to determine local diffusivities of solvents.^99,100 

The diffusion studies so far referred to, concern porous media saturated with liquids. Mainly as a consequence of the geometrical restrictions in the pore network, the self-diffusion coefficient in the pore network tends to be reduced relative to the bulk fluid. However, in pores only partially filled with liquids, a quite puzzling phenomenon arises. The effective diffusivities measured under these circumstances can exceed the values for bulk liquids by up to an order of magnitude.^101,102  Depending on the vapor pressure, this is a consequence of the contribution of the vapor phase in the pores. It reveals itself in spite of the three orders of magnitude lower density of the vapor. The diffusion coefficient in the vapor is four orders of magnitude larger than in the respective liquid phase. Molecular exchange between the two phases on the time scale of the experiment thus leads to more or less averaged diffusivities. These are largely determined by the vapor contribution, whereas—notabene—the NMR signal is still dominated by the liquid. Whether molecular exchange between the two phases is fast or slow relative to the diffusion time, that is, how the vapor-diffusion enhancement effect discloses itself in experiments, was shown to be a question of the pore size.^103,104  Furthermore, and provided the vapor pressure is high enough, vapor-enhanced diffusion along surfaces also contributes to the RMTD process mentioned above for spin-lattice relaxation as demonstrated in ref. 105.

A retrospective view on science over the last few decades would be incomplete without reference to the particular conditions of a world divided into two power blocks, and to the beneficial influence of science ensuring mutual contacts even across the “iron curtain”. It was the existence of these contacts which, eventually, accelerated and facilitated the re-establishment of unlimited international exchange after the fall of the Berlin Wall. This is particularly true with the field of MRPM. At the end of the 1980s the number of groups working on NMR for porous media independently and, as a rule, by very different routes attained such a level that the time was ripe for a direct exchange of their experiments and ideas, as stated by the organizers in the proceedings of the first conference on MRPM.¹⁰⁶  In 1988 the idea came out to organize an international meeting on NMR in porous media. The meeting took place in 1990, devoted to the progress in magnetic resonance in porous media and in understanding porous media themselves and on the behavior of fluids inside. It was also an opportunity to stimulate contacts among researchers from various parts of academia and industry. The Bologna meeting was the first of a long series that continues today, as summarized in the Table 1.2.

The seventh conference was the first one without Giulio Cesare Borgia, the co-founder and promoter of these conferences, who died unexpectedly in September 2002. The seventh conference was an important event from the point of view of the future of the MRPM community and of this conference series. The growing vitality and interest in these conferences and the consideration that the community, which the conference series served had grown considerably covering all continents, discussed how the future of these meetings would be assured; it was decided that the community would join the Groupement Ampere as the MRPM Division. The conferences, which are now explicitly called the “Bologna MRPM Conferences,” became Ampere Events and, in commemoration of the co-founder and promoter, the “Giulio Cesare Borgia Award for Young Researchers” was established. Thus, it is by far not incidental to find some of these awardees, notably Denis S. Grebenkov, Jonathan Mitchell and Rustem Valiullin, among the contributors to this book.

Conference proceedings appeared as special volumes of “Magnetic Resonance Imaging”, of “Conference Proceedings Series of the American Institute of Physics” and, since 2013, of “Microporous and Mesoporous Materials” (see http://mrpm.org for more details). Since 2007 these editions are accompanied by the presentation of supplementary conference communications in the free-access “Diffusion Fundamentals” online journal (http://diffusion-fundamentals.org). The present volume does, in a certain way, refer to the highlights of the conferences and conference reports with, hopefully, ample potentials for paving the way to further florescence and success of NMR on exploring fluid transport in porous solids and heterogeneous materials.