Translational motion in solution (e.g., diffusion, flow or advection) plays a central role in science. Self-diffusion can be rightfully considered as being the most fundamental form of transport at the molecular level and, consequently, it lies at the heart of many chemical reactions and can even govern the kinetics. Diffusion, due to its very ubiquity, is encountered in a myriad of scientific studies ranging from diseases to separation science and nanotechnology. Further, the translational motion of a species not only reflects intrinsic properties of the species itself (e.g., hydrodynamics), but can also shed light on the surrounding environment (e.g., intermolecular dynamics or motional restriction). Consequently, being able to study and ultimately understand the translational motion of molecules and molecular systems in their native environment is of inestimable scientific value.

Measuring translational motion at the molecular level presents special difficulties since labelling (e.g., radiotracers) or the introduction of thermodynamic gradients (which leads to mutual diffusion and consequently irreversible thermodynamics) in the measurement process can have deleterious effects on the outcome. Also, in many instances it is of interest to measure the diffusion of species at quite high concentrations. Fortunately, nuclear magnetic resonance (NMR) provides a means of unparalleled utility and convenience for performing non-invasive measurements of translational motion. Of particular significance is that, in general, the species of interest inherently contain NMR-sensitive nuclei and thus sample preparation generally requires nothing more than placing the sample into the NMR spectrometer.

Introduction

This chapter details the instrumentation for generating magnetic gradients and related technical issues. A basic understanding of gradient pulse generation provides insight into spectrometer limitations and related problems. The basic considerations and components of NMR probes and of the generation of high-intensity pulsed field gradients have been reviewed elsewhere. Many of the complications that affect PGSE measurements also apply to imaging experiments, consequently some of the solutions to the technical problems were developed with imaging in mind. Indeed, the design of a B0 gradient probe for diffusion measurements is essentially similar to that of an NMR imaging or microscopy probe except that the gradients used for the B0 gradient probe are often larger and greater precision is required in gradient pulse generation (i.e., pairs of gradient pulses need to be matched to the ppm level). Many high-resolution NMR probes come equipped with gradient coils capable of generating magnetic gradients in the range of 0.5 T m−1, whereas modern high-gradient diffusion probes are capable of generating gradients in excess of 20 T m−1 (Figure 5.1). There is also an interest in making probes capable of performing measurements on samples at high temperature and pressure and for use in solid-state studies.

To perform PGSE measurements, the spectrometer must be equipped with a current amplifier under the control of the acquisition computer which can send current pulses to a gradient coil placed around the sample. The hardware aspects of pulsed field gradient NMR have been discussed by numerous authors.

Introduction and reviews

The applications of NMR techniques to the study of translational motion is enormous and it is impossible to give anything approaching a comprehensive review. Consequently, only a smattering of papers from the different areas of application is presented and, in general, instead of citing the first paper with respect to each application, more recent papers have been chosen and the interested reader should consult the references listed therein. The classification of different studies is complicated since many studies have significance in more than one area. Numerous reviews on PGSE NMR have already appeared in the literature including ones of a general nature. Similarly, there are many books and review articles devoted entirely or in part to the use and applications of MRI techniques to study translational motion and mass transfer including clinical applications and rheological studies.

There are also a large number of more specialised reviews (or reviews on specialised areas including sections on gradient-based NMR techniques) dealing with NMR measurements of translational motion on diffusion-weighted spectroscopy for studying intact mammalian tissues, drug binding, exchange and combinatorial chemistry, flow, heterogeneous systems, liquid crystals, membranes and surfactants, organometallics, polymers, porous systems including zeolites, and solids.

Reviews have also been presented on the complementarity of the structural information that can be obtained from NMR diffusion measurements with that obtained from NOE experiments, the use of PGSE NMR in the studies of physicochemical processes in molecular systems, applications to environmental science, ENMR, the spectral editing of complex mixtures with particular emphasis on techniques involving diffusion, and B1 gradient-based measurements.

Introduction

A requirement in measuring transport (e.g., transmembrane) or exchange (e.g., ligand binding) is to be able to identify a measurable NMR parameter that has a different value in each state. Modulation of this parameter by the transport or exchange process is examined to characterise the process. Traditionally, NMR chemical shifts or relaxation times have been used for this purpose. With the advent of PGSE methods, a difference in diffusion properties (i.e., a difference in diffusion coefficient between sites or a difference in motional restriction) becomes another measurable NMR parameter that can be used to probe transport or exchange.

In the simplest case the exchange will occur between two freely diffusing sites (e.g., a ligand binding to a macromolecule; Figure 4.1); however, in many real systems (e.g., a suspension of biological cells) one site, or both sites if at higher cellular volume fractions, may be restricted. In contrast to the previous chapter where only simple restricting systems with reflecting boundary conditions were considered and the diffusing species did not interact with other restricting geometries, in real systems (e.g., biological cells, porous systems) it may also be necessary to consider the effects of a combination of exchange, restriction, obstruction and polydispersity in addition to surface and bulk relaxation as well as different bulk diffusion coefficients in each medium (e.g., Figure 4.2). As a consequence, modelling such systems can be very complicated and various approximations are necessarily used.

Introduction

This chapter introduces the concept of diffusion and other associated forms of translational motion such as flow, together with their physical significance. Measurements of translational motion and their interpretation are necessarily tied to a mathematical framework. Consequently, a detailed coverage of the mathematics, including the partial differential equation known as the diffusion equation, is presented. Finally, the common techniques for measuring diffusion are discussed.

Types of translational motion – physical interpretation and significance

‘Diffusion’ is used in the scientific literature with imprecision and ambiguity as there are a number of types of diffusion. With respect to molecular motion, diffusion is used to denote self-diffusion, mutual diffusion and ‘distinct’ (not in the sense of individual to a species) diffusion coefficients. Confusion arises since, although related and having the same units (i.e., m2s−1), these phenomena are physically distinct. The confusion is exacerbated in the NMR literature with the term ‘spin-diffusion’ which is a distinct NMR cross relaxation – based phenomenon involving the random migration of magnetisation via mutual spin flips in neighbouring nuclei, even though it can be measured using techniques related to those outlined in this book. In this book ‘diffusion’ signifies self-diffusion, which will also be referred to as translational diffusion, although some consideration will be given to mutual diffusion since many of the alternative methods for measuring diffusion, especially those based on scattering, provide information on mutual diffusion which is often compared with the results of NMR measurements of translational diffusion.

Introduction

In the previous chapter we considered the various methods for relating echo attenuation with diffusion in the case of free isotropic diffusion for a single diffusing species. It was observed that the echo signal attenuation was single exponential with respect to q2 and the correct value of the diffusion coefficient was determined irrespective of the measuring time (i.e., Δ). Due to the relatively long timescale of the diffusion measurement (i.e., Δ), gradient-based measurements are sensitive to the enclosing geometry (or pore) in which the diffusion occurs (i.e., restricted diffusion) and an appropriate model must be used to account for the effects of restricted diffusion when analysing the data. The effects of the restriction can be used to provide structural information for pores with characterisitc distances (a) in the range of 0.01–100 μm. Thus, gradient methods are especially suited to studying the physics of restricted diffusion and transport in porous materials.

Non-single-exponential decays can arise in a number of ways including multicomponent systems, anisotropic or restricted diffusion. These effects are the subject of the next two chapters (more complex models are studied in Chapter 4). The relevant analytical formulae for diffusion between planes and inside spheres are presented (diffusion in cylinders is presented in the following chapter). It is remarked that these are the commonly used models for benchmarking numerical approaches. We also mention that Grebenkov has recently presented a review of NMR studies of restricted Brownian motion.

Introduction

Most simplistically, mutual diffusion can be probed by imaging diffusion profiles (e.g., the ingress of a solvent into a material). However, the integration of MRI techniques with the gradient-based measurements of translational motion that we have discussed in previous chapters allows for potentially more information to be obtained – especially from spatially inhomogeneous samples. It also provides additional techniques for measuring such motions. Diffusion is extremely important in MRI, and, amongst other effects, at very high resolutions it determines the ultimate resolution limit when the distance moved by a molecule is comparable to voxel dimensions. Further, since motion is more restricted near a boundary, the spins near the boundary are less dephased (attenuated) during the application of imaging gradients in high resolution imaging, consequently a stronger signal is obtained near the boundary and this has become known as diffusive edge enhancement. Relatedly, since the length scales that can be probed with NMR diffusion measurements encompass those that restrict diffusion in cellular systems, the combination of PGSE with imaging techniques can result in MRI contrasts. Whilst there can be diffusion anisotropy at the microscopic level (e.g., diffusion in a biological cell), the MRI sampling is coarse and thus if there is too much inhomogeneity of the ordering of the microscopic anisotropic systems, the information obtained from the voxel will appear isotropic.

Introduction

As soon as the spin-echo was discovered by Hahn in 1950 it was realised that it could form the basis of self-diffusion measurements. Indeed, certainly within the next decade the concept of spin-echo-based diffusion measurements using static magnetic gradients (i.e., Steady Gradient Spin-Echo or SGSE NMR) had become widespread and used in quite sophisticated measurements such as on water and 3He. Many of the experimental limitations of static gradient measurements were removed with the suggestion in 1963 by McCall, Douglass and Anderson and experimental introduction in 1965 by Stejskal and Tanner of applying the magnetic gradients as pulses in the spin-echo sequence (i.e., Pulsed Gradient Spin-Echo NMR or PGSE NMR). Carr and Purcell were the first to discuss NMR flow measurements and in 1960 NMR flow measurements were considered for the purpose of measuring sea-water motion.

Virtually all contemporary NMR diffusion (and flow experiments) are based on some form of spin-echo. Indeed, for all but the simplest cases the dependence of the observed echo amplitudes on diffusion rapidly becomes very complicated and this can be exacerbated in pulse sequences where the magnetisation is kept in a steady state. However, in the following discussions we will assume, unless otherwise noted, that all pulse sequences start with the spin system being in thermal equilibrium (i.e., M0). As the diffusing species necessarily contains a nuclear spin, the terms spin and particle will henceforth become synonymous.

Introduction

There are a number of potential problems that must be addressed in PGSE measurements if high quality data is to be obtained. The problems include: (i) rf interference, (ii) radiation damping and long-range dipolar interactions, (iii) convection, (iv) homogeneity of the applied magnetic field gradient, (v) background magnetic gradients, (vi) eddy currents and static magnetic field disturbances generated by the gradient pulses, and lastly (vii) gradient pulse mismatch and sample movement. Almost invariably these problems lead to increased signal attenuation and thus overestimates of the diffusion coefficient and misinterpretation of the experimental data, and it has been noted that all PGSE systems have thresholds below which artefactual attenuation exceeds diffusive attenuation. Here, we consider the origins of these problems, their symptoms and some methods to alleviate them.

RF problems

The addition of gradient coils to an NMR probe can have deleterious effects on probe performance. Due to the proximity of the gradient coils to the sample region, the gradient coils and leads can, without appropriate precautions, act as antennae and introduce rf interference. A related problem is the possible strong mutual inductance between the gradient and the rf coils. Thus, the quality factor Q (= ωL/R where ω, L and R are the resonance frequency, inductance and resistance, respectively) of the rf coil(s) are diminished resulting in longer pulses for the same flip angle, poorer decoupling efficiency and S/N.

The effects of non-ideal B1 pulses and B1 inhomogeneity are well-known on spin-echoes, but have not been widely considered with respect to NMR diffusion measurements.

The preceding chapters developed the basic principles needed to describe selforganization and self-assembly in a variety of systems in either initially prepared unstable and metastable states or far-from-equilibrium states. The underlying mesoscopic description involved order parameter fields whose evolution was given in terms of either free energy functionals or amplitude equations. The latter approach is used for systems for which the free-energy-based description is not applicable. However, in many applications to physical and biological problems there is no clear-cut distinction between these two approaches. Often physical systems operate far from equilibrium, and the dynamics may involve both a free energy functional component and a component that cannot be expressed in this form. In this and the following chapters we describe several applications that illustrate how the methods developed in the body of the book may be used to construct models that capture the important aspects of the dynamics. We begin with a discussion of laser-induced melting in this chapter. In the following chapter we consider reactive physical and biological systems where phase segregation and reaction–diffusion dynamics are combined. The last chapter considers active materials where the constituent elements undergo driven or self-propelled motion. The analysis involves the combination of liquid crystal free energy formulations with order parameter dynamics to account for the active motion.

Laser-induced melting

When a laser with an appropriate intensity is focused onto a solid semiconductor film it can create a variety of ordered and disordered lamellar patterns of coexisting solid and melt regions (Fig. 29.1). The periodicity of the ordered patterns is commensurate with the wavelength of the incident laser radiation (van Driel et al., 1982).

Growth of order from disorder is a natural phenomenon which is seen in a variety of systems. An important class of such phenomena involves the kinetics of phase ordering and phase separation. The examples of such growth processes that were described in Chapter 2 had common characteristics. Now, we discuss the experimental results that point to common features of the kinetics of phase separation processes. A combination of techniques from nonequilibrium statistical mechanics and nonlinear dynamics is used to study the formation and evolution of spatial structures. Substantial progress in our understanding of the kinetics of domain growth during a first-order phase transition has been made over the past few decades. The knowledge gained in these studies forms the underpinning of the descriptions of many such processes which create order from disorder.

Kinetics of phase ordering and phase separation

Phase separation is usually initiated by a rapid change or quench in a thermodynamic variable (often temperature and sometimes pressure), which places a disordered system in a post-quench initial nonequilibrium state. The system then evolves towards an inhomogeneous ordered state of coexisting phases, which is its final equilibrium state. Depending on the nature of the quench, the post-quench state may be either thermodynamically unstable or metastable (see Fig. 2.1). In the former case, the onset of separation is spontaneous, and the kinetics that follows is known as spinodal decomposition. For the metastable case, nonlinear fluctuations are needed to initiate the separation process. The system is said to undergo phase separation through homogeneous nucleation if the system is pure. Phase separation occurs by heterogeneous nucleation if the system has impurities or surfaces which initiate nucleation events.

The idea for this book arose from the observation that similar-looking patterns occur in widely different systems under a variety of conditions. In many cases the patterns are familiar and have been studied for many years. This is true for phase-segregating mixtures where domains of two phases form and coarsen in time. A large spectrum of liquid crystal phases is known to arise from the organization of rod-like molecules to form spatial patterns. The self-assembly of molecular groups into complex structures is the basis for many of the developments in nanomaterial technology. If systems are studied in far-from-equilibrium conditions, in addition to spatial structures that are similar to those in equilibrium systems, new structures with distinctive properties are seen. Since systems driven out of equilibrium by flows of matter or energy are commonly encountered in nature, the study of these systems takes on added importance. Many biological systems fall into this far-from-equilibrium category.

In an attempt to understand physical phenomena or design materials with new properties, researchers often combine elements from the descriptions of equilibrium and nonequilibrium systems. Typically, pattern formation in equilibrium systems is studied through evolution equations that involve a free energy functional. In far-from-equilibrium conditions such a description is often not possible. However, amplitude equations for the time evolution of the slow modes of the system play the role that free-energy-based equations take in equilibrium systems. Many systems can be modeled by utilizing both equilibrium and nonequilibrium concepts.

Currently, a wide variety of methods is being used to analyze self-organization and self-assembly. In particular, microscopic and mesoscopic approaches are being developed to study complex self-assembly in considerable detail.