Introduction

This chapter is concerned with the practical issues and key considerations involved in setting up PGSE experiments and the subsequent data analysis. Selection of PGSE parameters is discussed in Section 6.2 and sample preparation is discussed in Section 6.3. The various methods of gradient calibration are considered in Section 6.4. Finally, PGSE data analysis and display are considered in Section 6.5. Under favourable conditions it is possible to measure diffusion coefficients with greater than 99% accuracy. Indeed simple PGSE experiments have been shown to be reasonably robust with respect to experimental parameters (e.g., rf pulse flip angle). It cannot be overemphasised that the overall accuracy of a diffusion measurement is intimately connected to the accuracy of the gradient calibration. It is too easy to confuse the apparent precision of a diffusion measurement obtained from analysing the PGSE data with the true overall accuracy. For example, the PGSE data obtained from an experiment may be highly single exponential, but the gradient calibration or temperature control may have been inaccurate such that the analysis of the PGSE data leads to a highly precise but unfortunately a highly inaccurate diffusion coefficient.

Irrespective of the aim of the PGSE experiment, the analysis is always simplified by starting with a distortion-free data set with good signal-to-noise and, especially when the system has multiple components, good resolution.

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.

Single-particle Green's functions, density response functions and other correlation functions are calculated in many different ways in the literature on Bose-condensed gases. An in-depth comparison and classification of different approaches was first given in the classic paper by Hohenberg and Martin (1965), with an emphasis on the various exact identities (conservation laws) that are satisfied. A key feature to be included in any theory is that a Bose broken symmetry leads to a hybridization of the single-particle excitations and the collective density fluctuations in such a way that the two excitation spectra become identical. This key feature is demonstrated in Chapter 5 of Griffin (1993).

How to relate and assess various approximations for correlation functions in a Bose superfluid has been a topic of continual interest (and some controversy) since the late 1950s. These questions were largely resolved by the early 1960s at a conceptual level but the detailed applications of the theory were limited to dilute Bose-condensed gases at T = 0. Since it was difficult to relate the theory to the properties of superfluid. He at a quantitative level, this formalism based on a Bose broken symmetry was of little interest to experimentalists. The creation of superfluid Bose condensed gases in 1995 changed all this and has given new life to the many body theory of dilute weakly interacting Bose condensed gases.

Various approximations for the Beliaev single-particle self-energies Σαβ were derived in Sections 4.3 and 4.4. The discussion in Chapter 4 was somewhat abstract.

Since the dramatic discovery of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995 (Anderson et al., 1995), there has been an explosion of theoretical and experimental research on the properties of Bose-condensed dilute gases. The first phase of this research was discussed in the influential review article by Dalfovo et al. (1999) and in the proceedings of the 1998 Varenna Summer School on BEC (Inguscio et al., 1999). More recently, this research has been well documented in two monographs, by Pethick and Smith (2008, second edition) and by Pitaevskii and Stringari (2003). Most of this research, both experimental and theoretical, has concentrated on the case of low temperatures (well below the BEC transition temperature, TBEC), where one is effectively dealing with a pure Bose condensate. The total fraction of noncondensate atoms in such experiments can be as small as 10% of the total number of atoms and, equally importantly, this low-density cloud of thermally excited atoms is spread over a much larger spatial region compared with the high-density condensate, which is localized at the centre of the trapping potential. Thus most studies of Bose-condensed gases at low temperatures have concentrated entirely on the condensate degree of freedom and its response to various perturbations. This region is well described by the famous Gross–Pitaevskii (GP) equation of motion for the condensate order parameter Φ(r, t). As shown by research since 1995, this pure condensate domain is very rich in physics.

The main goal of the present book, in contrast, is to describe the dynamics of dilute trapped atomic gases at finite temperatures such that the noncondensate atoms also play an important role.

Since the creation of Bose–Einstein condensation (BEC) in trapped atomic gases in 1995, there has been an enormous amount of research on ultracold quantum gases. However, most theoretical studies have ignored the dynamical effect of the thermally excited atoms. In this book, we try to give a clear development of the key ideas and theoretical techniques needed to deal with the dynamics and nonequilibrium behaviour of trapped Bose gases at finite temperatures. By limiting ourselves from the beginning to a relatively simple microscopic model, we can concentrate on the new physics which arises when dealing with the correlated motions of both the condensate and noncondensate degrees of freedom. This book also sets the stage for the future generalizations that will be needed to understand the coupled dynamics of the superfluid and normal fluid components in strongly interacting Bose gases, where there is significant depletion of the condensate even at T = 0.

The core of this book is based on a long paper published by the authors (Zaremba, Nikuni and Griffin, 1999). In the last decade, together with our coworkers, we have extended and applied this work in many additional papers. The starting point of our approach is not original, in that it consists of combining the Gross–Pitaevskii equation for the condensate with a Boltzmann equation for the noncondensate atoms. The kinetic equation for trapped superfluid Bose gases we use was first developed and studied in a pioneering series of papers by Kirkpatrick and Dorfman in 1985 on a uniform Bose gas at finite temperatures.

In Chapter 3, we introduced a simple but reasonable approximation for the nonequilibrium dynamics of a Bose-condensed gas based on a generalized GP equation coupled to a kinetic equation. In Chapters 4–7, we turn to the question of how to derive such coupled equations for the condensate and noncondensate components in a way that gives a deeper understanding of the ZNG theory. Chapters 4–7 involve an introduction to Green's functions and field theoretic techniques for nonequilibrium problems. These provide the natural language and formalism to deal with the many subtle aspects of a Bose-condensed gas at finite temperatures. These four chapters are fairly technical. This chapter is mainly based on Kadanoff and Baym (1962) and Imamović-Tomasović (2001). Readers who are not interested in these questions can go straight to Chapter 8, which begins the discussion of applications of the ZNG coupled equations given in Chapter 3.

Overview of Green's function approach

To derive a microscopic theory of the nonequilibrium behaviour of a dilute weakly interacting Bose-condensed gas at finite temperatures, there are several different approaches available in the literature. We will use the wellknown Kadanoff–Baym (KB) nonequilibrium Green's function method. The generalization of this formalism to a Bose-condensed system was first considered by Kane and Kadanoff (1965), whose goal was to derive the Landau two-fluid hydrodynamic equations for a system with a Bose broken symmetry.

The general problem consists of how to calculate the nonequilibrium response of a system induced by an external (space- and time-dependent) perturbation. In response to such an external perturbation, many interesting physical phenomena appear, including the excitation of collective modes and various transport processes.