1. Power of Experiment Progress in science is intimately connected with our insights into the processes and

phenomena occurring in our surroundings. Since ancient times, people were puzzled by sophisticated apparatuses designed by the elite of the time. The transition from the Middle Ages into modern time, the period of Renaissance, is characterized by the purposeful application of such devices for a better and deeper understanding of our world. This fundamental issue of science remains unchanged even today and diffusion research is no exception. The first contribution, authored by Nikolaus Nestle, introduces us to the phenomenon of diffusion in general and to the options for its observation, by which diffusion turns out to be a "macroscale dwarf and nanoscale giant". Nuclear magnetic resonance (NMR) is among the most powerful techniques of diffusion research. Its arsenal includes options to trace diffusion phenomena over essentially all space and time scales, starting from the elementary steps of diffusion over fractions of nanometers up to tens of meters (for example in the study of concentration profiles in geological systems). We start with two contributions by William S. Price and Philip W. Kuchel, highlighting two aspects of the measuring technique related to sample-polydispersity and to the realization of extra-short observation times. These general studies are complemented by examples of various applications of NMR to special tasks of diffusion research which are presented in the subsequent chapters. The papers mainly dedicated to experimental aspects of diffusion research are completed by Stefano Brandani's presentation of the "Challenges in Macroscopic Measurement of Diffusion in Zeolites", i.e. in one of these prominent nanoporous materials to which we shall return later on in a separate chapter dedicated to "holes and channels". This paper provides a nice example of the mutual benefit of the simultaneous application of different experimental techniques to one and the same system. With particular pleasure and gratitude we refer to the first poster abstract of this chapter, authored by Daniel Monceau and Jean Philibert, which introduces us to a web-site with a special chapter on diffusion and its relevance in material sciences and technology, as an initiative of DIMAT 2000.

As two local examples of apparatuses which have attracted people's interest and admiration since ages, this chapter is graced with two fountains. Both are located in the immediate neighbourhood of churches, San Pietro (XIII century) and San Domenico (early XIV century), respectively, providing excellent places for contemplating the wonders of nature and experiment.

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14

15

Diffusion: Macroscale Dwarf and Nanoscale Giant

Nikolaus Nestle1

1 BASF Aktiengesellschaft, Ludwigshafen, Germany Corresponding author: Nikolaus Nestle BASF AG Ludwigshafen GKP/P, G201 D-67056 Ludwigshafen E-Mail: nikolaus.nestle@basf.com

Abstract Diffusion processes of particles or energy exhibit a characteristic scaling behavior

with space and time. In this contribution, the background of this scaling behavior will be shortly described and its consequences in physics, chemistry, biology and various areas of technology shall be explored. Furthermore, some demonstration experiments for different aspects of diffusion and its scaling behavior will be discussed in the course of the article.

Keywords: Diffusion, scaling-behaviour, chemical technology, food science,

demonstration experiments

1. Introduction and a short look into usual demonstration experiments A standard demonstration of diffusion processes in German experimental physics

lectures involves a large (up to 1 m high) glass cylinder the lower part of which is filled with a blue, concentrated solution of copper sulfate which then is carefully overlaid with colourless demineralized water. If done by a skilful experimentalist, the interface between both solutions is initially quite sharp (with a transition zone of maybe 1 to 2 millimeters and with the density difference between the water and the solution additionally stabilizing the original stratification. Over the course of a two-hour lecture, one can already recognize some widening of the interface region due to interdiffusion of blue copper ions (and their colorless sulfate counterions) into the demineralized water region. The interface zone may grow to about 1 to 2 cm in this time. In some places, especially in the State of Saxony, the cylinder is then carefully positioned into a wooden rack and kept on public display for the remaining term or even longer (see fig. 1). There is even a special name for this rack with the cylinders in it: It is called “Semesteruhr”, i.e. “term clock”. This name illustrates quite well that major changes in the distribution of the copper ions over the cylinder may take several terms, coming close to equilibration even years.

The appearance of a “term clock” a few weeks after filling perfectly represents the typical perception of diffusion on the macroscale: It may lead to some blurring of formerly sharp interfaces, but is slow and it does not reach very far, and in order to

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, N. Nestle 16

prevent it, one must carefully pay attention to avoid other more effective mixing processes such as convection. However, the initial hours of the experiment provide already some clue that this may not be the complete picture: In this phase of the experiment, a notable change in the interface region between copper sulfate solution and free water can be observed as a matter of one or two hours. This suggests that diffusion may initially be “faster” than at later stages.

Not working inside academia right now and therefore lacking the means to set up a real “term clock” for closer inspection of the initial hours of the process, I was looking for a simpler demonstration experiment for interdiffusion in order to provide some experiment-based illustrations for this article. Searching on internet, I first bumped into the suggestion to use just a few crystallites of potassium permanganate in a small water-filled dish. The example pictures coming along with the experiment description looked quite similar to those shown in fig. 2. Indeed one can see “diffuse” color fronts inside the dish, however the asymmetric appea-rance of the colored region clearly indi-cates some asym-metry which should not be observable for a purely diffu-sive process.

Fig. 1: “Term clock” at the University of Leipzig grand phy-sics lecture theatre, spring 2000.

Fu

Fig. 2: Some KMnO4 crystallites in a slightly tilted water-filled plate (directly after application and ca. 5 min later).

rthermore, we see the KMnO4 ven-turing away from

the crystallite to distances much bigger than one should expect on the basis of the mean diffusive shift of water (1.2 mm using eq. 2.12 assuming D = 2.3×10-9m2/s). Actually, the propagation of the dissolved permanganate ions is strongly affected by density-driven flow (the density of KMnO4 solution is higher than that of water). Only in the longer run diffusion effects will soften up the concentration gradient between the heavy KMnO4 solution creeping along the bottom of the dish and the lighter overstanding water. The dominating role of density-driven flow is not just unique for the dissolution of KMnO4 but happens in a similar way for other water-

17

soluble substances. It can be suppressed by adding appropriate thickening agents to the water. Some bacterial exopolysaccharide materials are known to have only a minor impact on the diffusion behaviour of water and dissolved molecules while at the same time dramatically increasing the viscosity of the water (see section 4.4) By adding such agents to the water, we can suppress density-driven flow processes almost completely while not affecting the diffusion too much. Fig. 3 shows some snapshots from an experiment performed with eosin (C20H6Br4Na2O5) as a dye in a dish filled just with water and the other with a watery solution of 0.3 g/l xanthan gum [1] as a thickening agent. (Doing the experiment with KMnO4 is not possible as xanthan gum and similar thickeners chemically react with KMnO4.) As can be seen from the figure, the purely diffusive transport of eosin in the xanthan-doped water is orders of magnitude slower than the density-driven flow in the pure water.

Fig. 3: Distribution of dissolving eosin in a water-filled dish (left) and a dish with 0.3 % xanthan gum solution (right): In the water-filled dish, the eosin has already covered a substantial part of the dish while the crystallites are still applied (A, B). After 5 min, more than half of the dish is occupied by the eosin solution while the propagation of the eosin in the xanthan gum solution is just about 1 mm (C). After 30 min, the eosin in the xanthan gum solution has propagated by a few mm while the eosin in the water is homogeneously distributed (D). After 90 min, the eosin in the xanthan gum has spread around each crystallite by about 5 mm (E), and after 625 min the diffusive spread is about 15 mm (F). Note also the different intensities of the individual eosin spots which are due to different amounts of eosin in the original crystallites.

As the gravity-driven flow tends to stabilize the interface between two miscible liquid phases of different density, it can be used to prepare a demonstration experiment for purely diffusive transport: Fig. 4 shows a water-filled vial where an intensively coloured, sugar-containing syrup was injected locally near the bottom. As can be seen from the figure, this procedure leads, after a few minutes, to a sharp and flat interface between the syrup and the water – a perfect starting point for a 1D diffusion experiment. Some snapshots of this are given in the figure. For a semiquantitative observation of the diffusion process, a simple digital camera with remote control from a PC or a long-term

18

serial image function is sufficient. Alternatively, the experiment can be run completely qualitatively with stuff completely available at any bar.

Fig. 4: Diffusive mixing between a lower layer of sweet red fruit syrup and the overstanding water. A: directly after the injection of the lingonberry syrup, B: about 6 h later, C: about 12 h later and D: about 24 h later. Again, the pictures in fig. 4 show the characteristic behavior with the diffusive broadening of the interface front starting quite fast and becoming subsequently slower.

The length scale of the experiment in figure 4 is a few mm, and the time scale is on the order of hours, and the slowing down on a time scale of days is obvious. So this again fits very well with the idea of diffusion on macroscopic scales being a slow process. Looking back at the density-driven flow of KMnO4 and eosin in water, we get an indication that diffusion on short length scales must be a much more effective process: In order to feed the long-range propagation by the density driven flow, the molecules from the dissolving crystallites must first diffuse through an almost stagnant laminary liquid layer at the surface of the crystallite. Obviously the diffusive flow through this layer must be quite fast as it can sustain the fast long-range flow of considerable amounts of material.

Common to all the diffusion experiments described up to now is the observation of a particle current against a concentration gradient in absence of any other driving force except this gradient. Furthermore, all those processes were instationary diffusion processes: Due to the diffusive current, the concentration gradient itself is beginning to smoothen out. In a stationary diffusion process, by contrast, the concentration gradient against which the diffusion is occurring remains constant.

Visualizing a stationary diffusion process in condensed matter experimentally is rather difficult [2]. A macroscopic scale experiment in liquid phase is additionally complicated by the fact that the stationary diffusion will be preceded by a long period of instationary diffusion until a concentration equilibrium is reached in the diffusion bridge between the two reservoirs of constant concentration. Therefore, it is typically not demonstrated in lecture hall experiments.

As diffusion of energy is much faster than diffusive material in condensed matter, it is much easier to demonstrate stationary diffusion of energy instead of matter. The diffusion of energy is more commonly known as “heat conduction”. The analog to the concentration of a substance in the heat diffusion experiment is the temperature. An everyday example of (almost) stationary heat conduction is the temperature gradient along a wall between a heated (or chilled) building and the outside world (see fig. 5). Demonstrating stationary heat diffusion experimentally is much easier, and a heat current can for example be determined by the evaporation losses of a liquid phase or by electronic heat flow meters. Alternatively, the temperature gradient along the heat bridge

19

may be observed via thermochromic dyes, multisite temperature sampling with thermoelements or thermographic imaging via IR cameras.

Fig. 5: Stationary heat diffusion and temperature profile through a wall separating a cold and a hot reservoir (e.g. the air-conditioned inside of a house from the hot outside).

2. From the labtable to the blackboard: Fick’s laws and a phenomenological description of diffusion

2.1 Stationary diffusion and Fick’s first law The phenomenological description of stationary diffusion is given by Fick’s first law

which is also known as Fourier’s law in the context of heat diffusion (actually, Fick developed his phenomenological description of diffusion on the basis of Fourier’s work on heat conduction [2]). It relates the diffusive current jD to the concentration gradient:

xcDjD d

d−= in the 1D case, (2.1)

cDjD ∇−= in the 3D case, respectively. (2.2) Fourier’s law is written completely analog and relates the heat current jQ to the

concentration gradient :

xTjQ d

dλ−= in the 1D case, (2.3)

TjQ ∇−= λ in the 3D case, respectively. (2.4)

The diffusion coefficient D and the heat conductivity λ are scalar material constants in the standard textbook treatment of these transport phenomena. In anisotropic materials, this is not any more the case, and a diffusion or heat conductivity tensor is needed to describe the transport properties of the materials. Anisotropic diffusion properties are quite common in biological systems, anisotropic heat conduction is typical for anisotropically structured materials of biological or technical origin (e.g. wood: about 1.8 times higher thermal conductivity in fiber direction [3]).

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2.2 Instationary diffusion and Fick’s second law In the description of instationary diffusion, we will consider the diffusion balance

inside a given differential element under the assumption that Fick’s first law is valid on differential time and length scales. For the sake of simplicity, we will provide the detailed calculation just for the 1D diffusion case.

If we consider the concentration change dc in a volume element dx localized at the site x during the time dt we find on the basis of a simple material balance for the concentration change:

( )d()(ddd xxjxjx

tc

DD +−= )

i.e. the difference between inflow and outflow leads to a corresponding concentration change. This equation can be rewritten as

( )x

xxjxjtc DD

d)d()(

dd +−

=

and inserting Fick’s law for the diffusion current, we arrive at

2

2

dd

dd

)d(dd

)(d

dd

xcD

xx

xxcxxcD

tc

=⎟⎠⎞

⎜⎝⎛ +

−−= . (2.5)

The same argument in 3D leads to

cDtc

Δ=dd

(2.6)

and in heat conduction to

TTCt

TΔ=Δ= αλ

dd

(2.7)

with C denoting the heat capacity of the material and Cλα = being known under the

terms “temperature conductivity” or “heat diffusion coefficient”. Despite their innocent looks which are quite similar to a wave equation

uctu

p Δ=∂∂ 2

2

2

(2.8)

or Schrödinger’s equation

ψψΔ−=

∂∂

mi

t 2 (2.9)

equations (2.5) to (2.7) prove mathematically and numerically quite hard to treat, and whole books [4,5] have been devoted to their solution, and it has been only very recently that solutions of the diffusion equation have been incorporated into the repertoire of computer algebra programs such as Maple or Mathematica. Only some good-natured

special cases are easier to solve. One of them is the stationary case (i.e. 0=dtdc ). This

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corresponds to Poisson’s equation which leads to the well-known linear concentration or temperature gradient in the 1D case.

The most simple case of an instationary solution of the diffusion equation is the spreading of an “instantaneous” or Delta-source which results in a Gaussian of increasing width

Dttw 2)( = (2.10) and decreasing amplitude:

)4

exp(2

)),(2

Dtx

DtCtxc −=π

(2.11)

In addition to the purely theoretical case of the “instantaneous” source, this solution is also valid for a Gaussian initial concentration profile of the width w (which is equivalent

to a shift in the starting time by D

wto 2

2

= ). Other initial concentration profiles such as a

step-function will lead to mathematically more complicated concentration profiles involving error functions or series of error functions [5].

The corresponding solutions in the 3D case are

Dttw D 6)(3 = (2.12) and

)4

)(exp()(8

)),,,(222

3 Dtzyx

DtCtzyxc ++

−=π

(2.13)

2.3 The “velocity” of diffusive transport The common feature of all these solutions in the 1D case is that the width of the

diffuse concentration profile will increase according to a square-root-of-time law as in the simple Gaussian case. This square-root-of-time behavior is the reason for the perceived slowdown of diffusive transport over macroscopic distances: Calculating a “velocity” vD

from the mean diffusive shift over a length scale LDtL 2= , we arrive at

LDL

tLv

DL

LD

2

22 === , (2.14)

i.e. the “velocity” of diffusive transport is inversely proportional to the length scale L! The ratio between velocities of directed movement (flow, advection) and the

“velocity” of diffusive transport in a system is often characterized by the Péclet number Pe which was initially introduced in the analysis of heat transfer [6].

The Péclet number Pe for diffusion is given as the ratio

DLv

=Pe , (2.15)

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for heat transfer as αLv

=Pe . (2.16)

In the macroscopic world, Péclet numbers tend to be very high, i.e. directed transport processes tend to be much faster than diffusive processes. The propagation of permanganate ions by gravity-driven flow in figure 2 is an example.

On the microscale, by contrast, we find many processes taking place at very low Péclet numbers. The tendency to lower Péclet numbers at short length scales is not only due to smaller values for L but also due to the decreasing flow velocities in smaller

structures (at constant pressure difference essentially proportional to 21

L ). In the

further course of this contribution, we will explore the consequences of this phenomenon in natural and technical systems. But before doing so, we will take a very short look at the molecular mechanism of diffusion and the idea of self-diffusion.

3. Diffusion in statistical mechanics, self-diffusion

3.1 The statistical picture of diffusion The description of diffusion provided by Fick’s law is phenomenological and

continuous, and quantities like concentrations and temperature are average quantities that become meaningless on the molecular length scale. In order to understand the nature of diffusion on the sub-nm scale, one therefore has to resort to statistical mechanics. First statistical calculations on diffusion in gases were made by James Clerk Maxwell [7] in 1867. A more general statistical description of diffusion processes by statistical mechanics was suggested in 1905 independently by Albert Einstein [8] and Marian von Smoluchowski [9].

In the statistical picture, each molecule is undergoing a random walk which leads to a mean square displacement increasing linear with time – which in turn gives rise to the square-root-of-time behaviour already discussed in the last chapter. If different species of molecules are non-uniformely distributed over a system (which is an unlikely stage in the absence of barriers corralling the molecules in their uneven distribution), we will see a leveling off of these concentration gradients resulting from the random walks of all molecules together: The system moves from an unlikely stage with concentration gradients to a more likely stage with a stochastically uniform distribution of each species of molecules. The change into this more likely stage leads to an increase in the entropy of the system, and it is obvious that this interdiffusion process is irreversible.

While changes in the local chemical composition of the system and increases in entropy only take place in the case of interdiffusion processes along concentration gradients, the molecular motions leading to each molecule’s random walk are also there in thermal equilibrium.

3.2 Experimental approaches to self-diffusion Due to the indistinguishable nature of molecules, this self-diffusion process cannot be

observed directly without manipulations trying to mark individual molecules. The classical way to mark the molecules is the use of (radioactive or stable) isotope tracers: If

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the usually tiny differences between molecules containing different isotopes are neglected, following the changing distribution of the isotope tracer in the system provides an experimental access to the self-diffusion process in the system. A key player in developing the radiotracer approach to diffusion studies was von Hevesy [10] who developed the radiotracer approach both for studies of self-diffusion and interdiffusion processes. Other experimental approaches to self-diffusion studies are quasielastic neutron scattering (where essentially the momentum transfer between individual molecules and an incident neutron beam is probed) and nuclear magnetic resonance (which allows the transient encoding of position information onto the nuclear spins by appropriate manipulations of their precession phase in field gradient NMR [11,12] or by selective excitation of very thin slices [13]). Special features of field gradient NMR are the options of working at well-defined diffusion times [14,12] and well-defined diffusive shift lengths [15]. With such experimental options available, field gradient NMR allows the study of porous systems from the perspective of the molecules diffusing in them. Depending on the diffusion coefficient of the liquid phase and its NMR behavior inside the porous material of interest, the diffusion length scales accessible in the experiments may range from a few 10 nm to several 100 µm. The dependence of the measured diffusion coefficient on diffusion times and/or reciprocal shift vectors therefore provides unique information on the internal structure of the material’s pore system. The field-gradient NMR experiment is non-destructive and can be continuously repeated. Like that, it is also possible to study changes in the structure and transport properties of a developing pore system such as a hydrating cement stone matrix [16]. NMR effects are not only sensitive to the well-defined magnetic field gradients used in field gradient NMR but also to statistical gradients present in many micro-and nano-structured materials [17,18]. The way how internal magnetic field variations inside a sample manifest themselves in its NMR behavior is dependent on the ratio of the mean diffusive shift during the time scale of the NMR experiment and the length scale of the magnetic field variation inside the sample: If the length scale of the variation is much bigger than that of diffusion, a distribution of well-defined NMR frequencies can be observed. If the diffusion length scale is larger than the length scale of field variation, it manifests itself mainly by a characteristic attenuation of the NMR signals. This difference in the NMR behavior is the result of the decreasing effectiveness of diffusive mixing processes: if the length scale of the magnetic field gradients is comparable or shorter than the mean diffusive shift during the NMR time scale, each diffusing molecule experiences its own stochastic averaging of NMR frequencies. This in turn leads to the macroscopic attenuation of the NMR signal. If the time scale of an NMR experiment is varied, the joint action of diffusion and NMR can be different, too (see fig. 6).

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Fig. 6: NMR signal attenuation due to diffusion in internal magnetic field gradients: turbo spin echo MRI scan (A; echo time: 96 ms, 7 refocussing pulses during echo time) and spin echo MRI scan (B; echo time 96 ms, no refocusing during echo time) of a bottle filled with water and sand layers of different grain size (written in mm next to image A). The stronger echo attenuation in the fine-grained sand layers near the ends of the bottle in image B is due to diffusion effects in the internal magnetic field gradients produced by the sand grains.

4. Diffusive transport in technical and natural systems; the role of geometry

4.1 Diffusion limited reactions with a migrating reaction zone; the Stefan model In the absence of convective processes, any reaction in an extended medium needs

diffusive transport between an external reservoir and the reaction zone (see fig. 7). The same holds true for transport of heat to a phase transition zone. If the reaction process is sufficiently fast and irreversible, we can assume that the reactant is used up as soon as it meets unreacted material. In this case, it can be shown that the reaction zone travels into the medium according to the following relation [19,5,20] in the 1D case:

bind

res

ctDc

tx2

)( = (4.1)

with cbind denoting the concentration of available binding sites in the material and cres denoting the concentration of the reactant in the reservoir.

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Fig. 7: Reaction front propagating by purely diffusive transport into an extended medium: With increasing intrusion of the front into the medium, the zone of diffusion limited transport becomes wider. This in turn leads to an increasing diffusion resistance which has to be overcome by the reactant. The same model can be applied for the migration of a phase transition zone [19]. Both the reaction zone in this case and the mean square displacement in the case of pure diffusion propagate according to a square-root of time-law into the medium. The difference between the two cases is the shape of the intruding front: in the Stefan case, the concentration profile goes to exactly 0 after the reaction zone while the concentration profile in the purely diffusive case exhibits a Gaussian “tail” extending far into the medium. It should be noted that the diffusion length may actually be longer than the intrusion depth of the reaction front into the medium [21].

The reaction rate in a Stefan-type process is limited by the diffusive current between the reservoir and the reaction zone and therefore decreases like the “velocity” of a pure diffusion process (see eqn. 2.14) with increasing intrusion depth x(t) of the reaction front:

)(dd

txc

Dt

N resprod = (4.2)

As the Stefan case is mathematically more simple and exhibits a well-defined reaction zone, some typical aspects of diffusion-limited processes shall be discussed on the basis of this model.

The initially very pronounced decrease of the reaction rate is typical for a diffusion-limited process. If high reaction-rates are intended, it is necessary to keep the length scale of diffusive transport as small as possible. This can be achieved by minimizing diffusion barriers inside the reservoir (e.g. by stirring). The diffusion barrier inside the medium can be minimized by working with small particles of the reactive medium. It must however be noted that the transport geometry in the case of small particles is different from that in the case of an extended medium. The same holds true for reaction geometries in which the reactant is provided from a spherical or an extended cylindrical source into the surrounding medium (inside-out-case). In fig. 8, the propagation speeds of the reaction zone in the 1D case and in the spherical and cylindrical cases are plotted as a function of the distance to the reaction zone both for “inside out” and “outside in” geometries. The most noteworthy difference between the different geometries is the asymmetry between the “inside-out” and “outside-in” case for the cylindrical and spherical geometries: In the first case, the reaction rates tend to decrease much faster in the case of the cylinder and even stronger for the sphere, while the reaction rate in the inside-out case is fastest for the sphere. Furthermore, the outside-in cases exhibit an increasing reaction speed shortly

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before the innermost core of the sphere or the cylinder is reached and the material is completely saturated.

Fig. 8: Propagation speed of the reaction zone in a Stefan process. The “relative intrusion depth” in the case of the 1D geometry is normalized to the same length as the radius of the cylinder and the sphere. While the intrusion of the reaction zone in the 1D case is symmetrical, the intrusion of the reaction zone in the cylindrical and spherical geometries is different for the “outside in” and the “inside out” cases. The reason for this is the dependence of the radial volume element on the radius.

The reason for the differences between the geometries and the asymmetry between “outside-in” and “inside-out” diffusion cases is the volume element in the different geometries: In the cylindrical and in the spherical geometry, the volume element is given as

rrrVcyl d2)(d π= or . rrrVsph d4)(d 2π=This corresponds to a decreasing size of the volume element in the case of the outside-in geometry and an increasing size in the case of the inside-out geometry. As can be seen from figure 8, the decreasing volume element leads to an almost constant and finally even increasing traveling speed of the reaction zone inside the particle. Correspondingly, the growing volume element in an inside-out scenario leads to a progressive slowdown of the reaction front compared to the 1D case. Consequences of these geometry effects shall be discussed in sections 4.3, 4.4 and 4.8. Despite the constant or even increasing propagation speed of the reaction front, one must be aware that the overall reaction rate nevertheless is decreasing with the intrusion of the reactant into the particles.

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4.2 Technical compromises in the design of particle-bed processes In order to achieve minimal losses in reactivity due to the diffusion-limitations just

described, very small particles seem optimal for outside-in reactions such as ion exchange. However, handling very small particles often is inconvenient in the implementation of technical processes:

• The permeability of a particle bed dramatically decreases along with the particle size, furthermore, fine particle beds are more susceptible to clogging due to filtration effects,

• Keeping a fine particle bed together requires special filters at the exit of the column, otherwise particles might escape the column in suspension (and possibly do some mischief downstream),

• Fine particulate materials will require more precautions with respect to appropriate packaging and possibly also with respect to occupational health issues.

Fig. 9 Rough sketch of diffusive transport limitations (A) in a monolithic reactant bead (B) in a bead with internal porosity.

For a monolithic material, the best compromise will be choosing the smallest particle size which still can be reasonably handled. If the diffusion coefficient inside the particle material is much smaller than that in the surrounding medium, introducing additional porosity is a good option (see fig. 9): In this case, the actual diffusion length inside the active reaction medium is much shorter, and it is going to occur from channels with much faster diffusion inside. If the channels are filled by a free liquid phase such as water (D = 2.3×10-9m2/s), the mean diffusive shift inside the channels will be on the order of a 1mm in a minute. Like that, a substantial diffusive material transfer from and to the active reaction medium can be achieved for a 2 mm diameter bead within a minute. If we assume a diffusion coefficient of D = 2.3×10-13m2/s for water inside the active material itself, a comparable diffusive material exchange would take 100 minutes, i.e. more than 1 hour.

While the porosity trick provides some remedy, it still doesn’t solve the problem of diffusion limitation in general: increasing the particle diameter will still lead to increasingly strong diffusion limitations, and furthermore one must be aware that the

28

packing density of active material is decreased by introducing porosity. Nevertheless, particles with diameters on the mm scale and with some internal (macro-)porosity are a very common approach in the design of columns for applications as diverse as ion exchange and heterogeneous catalysis.

Basically, the design of optimal particles for column processes follows the rules: • Minimize diffusive transport at low diffusion coefficient by introducing higher-

diffusivity macroporosity, • Keep diffusion-limited transport lengths at any diffusion coefficient as short as

possible in order to fulfill other design requirements. It must of course be stressed that the design of reactive columns is a very complex

issue in research and technology and that diffusion effects are really just one issue among a multitude of others.

4.3 Inside-out diffusion processes and controlled release While reaction columns typically are designed in order to provide a high reaction rate,

there are other fields of technology where one is interested in a long-term and possibly constant release of some active agent. Examples are the release of active agents in pharmaceutical or agrochemical applications.

Applied in dissolved form or as a soluble, fine-grained material (such as the eosin dye in fig. 3), the active agent will be initially present at a very high concentration (which might actually be too high and poisonous) but readily be washed away or wasted by some decay reaction. This is not what we are interested in: Rather, we want to provide a moderate (and effective but not poisonous) concentration of the active agent over a long time. In this scenario (called controlled release), the slowing-down-effect of diffusion is actually a friend: However, the square-root-of-time effect and the geometry effects are again to be taken into account: Depending on the requirements of the application, the insertion of additional diffusion barriers (which might even be designed in a way that they slowly degrade along with the decreasing concentration of active agent), the use of slow-decaying precursor formulations or the observation of appropriate application schemes may be used in order to optimize the time-concentration profile achieved by the controlled release. Typical length scales in controlled release applications range from µm to mm. Nanoscale formulations are not of great interest for diffusion-controlled release as no sufficient delay by diffusion can be achieved in this case.

Nevertheless, nanoparticulate formulations are of great interest when it comes to poorly soluble active agents: here, the large surface area and the faster Brownian motion of smaller particles increase the distribution and the contact area of the agent in the system (leading to shorter distances the dissolved agent molecules have to travel in the water phase before reaching their sites of action). Furthermore, nanoparticles are capable of penetrating into biological cells. This latter aspect is presently a major challenge in research both for pharmaceutical technology and the study of possible health risks arising from nanoparticulate materials [22].

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4.4 Diffusive transport in organisms and biofilms The dimensions of biological cells are on a similar order of magnitude as mean

diffusive shift of small molecules in aqueous media such as cytoplasm during a time of a second or less:

The length of individual bacteria is typically around 1 µm. At these dimensions, they do not need any active circulation systems (and circulation with Péclet numbers > 1 would require unrealistically fast flow inside the cells). Recently, giant bacteria with diameters up to 750 µm have been described [23]. In these bacteria, however, most of the cell volume is taken up by a large storage vacuole, and the metabolically active part of the bacterium is limited to a thickness of a few µm.

Large eukaryotic single-cell organisms such as Paramecium or Noctiluca very often measure several 100 µm or even more. This is already much larger than the mean diffusive shift of water during 1s. Such large single-cell organisms typically are equipped with at least some active intracellular transport mechanism for nutrients (for example, in the case of Paramecium it is possible to observe nutrient-filled vacuoles circulating through the organism).

In polycellular organisms, the need for active transport systems is even more obvious. Therefore, we find a wide range of different types of circulation and/or peristaltic systems inside all higher organisms. The tracheal system of insects combines the much faster diffusion of oxygen in the gas phase (with the finest tracheal openings penetrating into individual cells), some ventilation and an active pumping system for the nutrient-containing hemolymph. In vertebrates, both oxygen and nutrients are transported via the blood circulation. The vascular system again extends down to the cellular length scale so that diffusive transport is mainly limited to the sub-mm scale.

Understanding the mechanisms of vascular growth inside the human body is considered a key to novel therapeutic approaches to conditions such as coronary heart disease or cancer: Studies into the growth mechanisms of tumor tissues have indicated that solid tumors without an own vascular system can only reach sizes on the order of 1 mm [24]. In order to grow larger, the tumor needs to be connected to the blood stream by angiogenesis. Angiogenesis inhibitors such as bevacizumab [25] are in the progress of being introduced into the clinical therapy of cancer.

While polycellular organisms typically have at least some form of active transport systems that complements diffusive transport to larger length scales, some bacteria and other microorganisms go actually the other way and build some diffusion barriers around themselves. (In the case of a polycellular organism, the cells inside live in a cell-friendly environment so that their main concern is sufficient feeding. The environment for individual single-cell-organisms, by contrast tends to be a much rougher place so that protection against external hazards is at least as important as sufficient supply of nutrients.) The most frequent forms of diffusion barriers are gels consisting mainly of polysaccharides (so-called exopolysaccharides such as Xanthan or Gellan). While the exopolysaccharide layer forms only a minor diffusion barrier for dissolved oxygen or small nutrient molecules, the access of colloidal particles or predator microorganisms to the bacterium is strongly reduced.

Large and strong exopolysaccharide layers require a high metabolic investment from the respective microorganims. Due to the unfavourable inside-out-geometry, the amount

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of material needed to form an exopolysaccharide layer around an individual bacterium is especially high. At the same time, the risk of damage such as partial dissolution of the gel barriers is also rather high for a free-floating microorganism.

A much more effective use of the exopolysaccharide barrier can be made by forming a biofilm: Sharing the gel barrier among many organisms will reduce the individual cell’s metabolic investment in forming the barrier. Furthermore, the biofilm will lead to a quasi-one-dimensional diffusion geometry (with no-outside-in effect and thus with a smaller diffusive flow from the solution phase to the microorganism). Like that, the diffusion barrier formed by the exopolysaccharide has a mitigating effect on transient concentration highs of poisonous substances which will hit the microorganisms later and with a restricted peak value of its concentration.

Another aspect of moving together into a biofilm is the possibility of the microorganisms to form a “Verbund”-like community in which metabolites produced by one species may serve as nutrients for another microorganism present in the biofilm. Depending on their location and the organisms involved, biofilms can present a major problem (in medicine, hygiene and with respect to biofouling) or they are very useful (in some areas of biotechnology, e.g. for waste water treatment).

4.5 Diffusion in polymer technology Due to their large molecular weight, the diffusion coefficients of polymers are

generally much lower than those of small molecules. This is even the case in dilute solutions. In more concentrated solutions or even more so in polymer melts, the translational movements of each polymer molecule are further restricted by the geometrical constraints exerted by the surrounding molecules. This effect is described by the reptation model of polymers and various refinements to it [26].

4.5.1 Diffusion-limitations in polymerization processes: bulk, solution and emulsion polymerizations

In any case, diffusion coefficients in polymer melts tend to be quite low (well below 10-13 m2/s) and also the viscosities of polymer melts are quite high. As a result, stirring concentrated solutions or melts of polymers is quite difficult, and diffusive mixing is very slow. While the diffusion coefficients for residual monomers are orders of magnitude higher than those of the polymer molecules, also their diffusion is strongly slowed down compared to a free liquid phase. Therefore, the diffusion limitation inside a polymerization batch becomes more and more of a problem with increasing degree of polymerization. In addition to material transport, also heat transport may be a serious issue as also the heat conductivity of polymer melts tends to be quite low and the high viscosities successfully suppress convective heat transfer, too.

Despite all these problems, many polymers still are produced in concentrated solutions (dilute solutions are unattractive as large quantities of solvent would have to be removed in order to recover the polymerization product) or in melts. For other polymers, however, emulsion polymerizations [27] have become more and more popular. In this process, the polymerization process is initiated inside amphiphilic micelles in a water phase. The monomers are provided in solution or as an emulsion . Monomer molecules diffusing from the water phase into the micelles with the growing polymer chains are

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incorporated into the polymer chains. In the end, aqueous polymer dispersions [28] with solid contents of over 50% can be produced like that. In contrast to solution polymers and polymer melts, these dispersions exhibit moderate viscosities, and the final size of the polymer particles is about several 50 to several 100 nm in diameter. Over the length scales of such particles, diffusive transport is reasonably fast even for very low diffusion coefficients. The short length scales for intraparticle diffusion and the low viscosity of the dispersion allow easy mixing and heat transfer inside the polymerization batch. In contrast to bulk polymerization products, the resulting polymer dispersion contains large quantities of surfactants and other adjuvants. If dried above their respective minimum film formation temperature (MFFT), such dispersions will coalesce into continuous films. Depending on the properties of the polymer particles, the resulting films can be used for applications in paints, adhesives or paper and textile coating materials. In the film formation process, diffusion of polymer chains between individual particles may play some role; the main mechanism for the stabilization of the films is due to electrostatic and steric interaction forces between individual particles. Emulsion polymerizates can also be used for compounding into melts of other polymers (e.g. ABS where a synthetical rubber from an emulsion polymerisation process is mixed into an SAN melt).

4.5.2 Diffusive migration of small molecules in polymer materials

Other issues on diffusion in polymers include the diffusion of solvents, plasticizers, residual monomers or other small molecules through polymer matrices. Especially the possible diffusion of plasticizers and residual monomers from the polymer into surrounding phases creates great concern, especially in applications where the polymers are in contact with body fluids or food [29,30]. Diffusive migration of low-molecular weight species through polymers is not restricted to plasticizers or residual monomers: Small molecules may migrate through polymeric packaging materials. A demonstration for this effect can be seen from fig. 10 where two 10 ml aliquots of concentrated hydrochloric acid in a 10 ml polyethylene bottle (wall thickness 2.5 mm) and sealed into a 40 µm plastic bag were stored in closed tin containers for 1 day: While the container with the bottle is still intact, the container with the bag is already heavily corroded by HCl diffused through the polyethylene. In the case of the bottle, a comparable corrosion of the container wall was observed after about 100 days. The migration of oxygen through polymeric packaging materials is a major challenge in food packaging applications, for example polymeric bottles for beer (which is the common beverage most sensitive against oxygen diffusion [31]). A range of different approaches such as barrier layers [31,32], special coatings or oxygen-scavenging additives [33] in the polymers are used to overcome the oxygen-diffusion problem.

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Fig. 10. Corrosion of a tin container by HCl diffusion through a PE bottle and a PE plastic bag: (A) both containers in intact state (B) situation after a day: while the walls of the container with the bottle are still clear, the container with the plastic bag is already heavily corroded.

Minimizing the diffusive ingress of oxygen and water vapour is an even more challenging task in the development of polymeric housings for polymeric or organic-molecule-based electronic devices such as Organic LEDs (OLEDs) which are highly sensitive to degradation by oxygen and water. Presently, only housings containing inorganic layers such as glass are capable of providing a sufficient diffusion barrier that allows a reasonable shelf and service life of such devices. Flexible devices may be realized by polymer-reinforced ultrathin glass layers [34].

4.6 Diffusion in functional materials for electronics Diffusion is not just a challenge in organic electronics, but also in the production and

performance of classical devices such as semiconductor electronics or batteries. With some exceptions such as ion-conducting materials, (chemical) diffusion coefficients in crystalline solids tend to be very low1. For example, the diffusion coefficients of doping atoms such as P or As in Si are about 10-18m2/s at temperatures around 1000°C. This corresponds to mean diffusive shifts in the sub-µm range during a 30-min tempering episode at 1000°C [35]. Extrapolating empirical findings on the temperature dependence of dopant diffusion in Si [36] down to room temperature, we arrive at diffusion coefficients around 10-48m2/s which would correspond to a mean diffusive shift of 1 nm after 1022 years! However, it should be noted that other atomic species such as for example Cu exhibit much higher diffusion coefficients in solid Si or SiO2 barrier materials and therefore may already lead to rapid diffusive degradation of microelectronic devices at temperatures around 300 °C or lower [37].

1 It should be noted that there are also charge carrier diffusion processes taking place in solids. For example, diffusion currents of electrons and holes are responsible for the specific current transport properties of semiconductor p-n-junctions (diodes). The diffusion of the charged quasiparticles can be described by means of the Nernst-Planck-Equation (see 4.6.2).

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4.6.1 Semiconductor nanostructures: stability and growth The high activation energies of diffusion in semiconductors explain why it is possible

to produce devices with dopant profiles at nm-resolution and nevertheless have completely stable structures at room temperature for “infinitely” long times. It also explains why the development of high-temperature electronics presents special material challenges.

By appropriate growing techniques such as molecular beam epitaxy (MBE) it is possible to realize high-quality multilayer structures of different semiconductor materials at “atomic” resolution. A major reason why these techniques actually work is the dramatic difference between the diffusion coefficient of ad-atoms on the surface and the diffusion of atoms in already complete layers [38]. Due to the temperature dependence of both the inter-layer diffusion and the diffusion of ad-atoms, the choice of the substrate temperature plays a key role in controlling MBE growth. Further improvements in controlling diffusion between the layers can be achieved by introducing special diffusion barrier layers [39]. 4.6.2 Battery materials: long-term stability despite high diffusion coefficients

In charged primary and secondary batteries, the two electrodes are separated by an ion-conducting liquid, polymeric or solid layer. In fig. 11, a schematic representation of a cell for a lithium battery is given. In commercial lithium batteries, the thickness of the individual cells is on the order of a few 100 µm, and they are stacked or scrolled together to squeeze as much as possible storage capacity inside the battery housing. The space left for the separator inside such cells is on the order of 100 µm or less.

Fig. 11. Layout of film electrodes and separator in a Li ion battery. The overall layers structures are just a few 100 µm thick and are scrolled together into the battery housing.

Present separators consist of lithium-conducting polymers, gels or porous materials soaked with a non-aqueous solution of a Li salt such as LiPF6. In order to be good ion conductors for Li+ (which is needed in order to achieve a low internal resistance of the battery), these systems are characterized by high mobilities of the Li+ ions. This Li+ mobility is not only reflected in the ionic conductivities but also in self-diffusion coefficients of Li+ ions that may range around several 10-10 m2/s [40]. Such diffusion coefficients correspond to a mean diffusive shift of 100 µm in roughly a minute. Therefore it may seem likely that such a battery should self-discharge as a matter of minutes. However, state-of-the-art lithium batteries have self-discharge times of several months. The reason for this seemingly paradox stability is the charged nature of the Li+

34

ions: the diffusive propagation of Li+ ions without appropriate counterions is suppressed by the buildup of strong local electric fields. The description of the diffusion of ions therefore needs a modified form of Fick’s first law which is known as the Nernst-Planck equation:

)dd

dd(

xRTFcz

xcDj ii

iii

Φ+−= . (4.3)

The diffusion of different ionic species i with electrical charge zi is coupled together by the derivative of the electric potential Φ. (R denotes the general gas constant, F the Faraday constant and T the temperature.)

In the case of the charged battery, the absence of counterions capable of diffusion through the separator leads to a stabilization of the non-equilibrium distribution of the Li+ ions despite the absence of diffusion barriers.

4.7 Diffusive mixing in microfluidics Microfluidic systems are increasingly used in chemical research labs (“lab on a chip”,

high-throughput screening, …) and the first industrial applications of microreactors for production purposes have also been presented [41,42,43]. Due to the short distances between the active surfaces and the “bulk liquid” in typical channels (5 to 100 µm diameter), diffusion often provides sufficiently good mixing of the reactants so that no further active mixing measures have to be taken [41]. In systems with high flow velocities (several mm/s or higher), diffusive mixing still can take several seconds which correspond to channel lengths of several cm [44]. If such long channels are not desired or if fast reaction kinetics dictate faster mixing time, mixing has to be sped up. A range of active mixing devices such as miniaturized magnetic stir bars [45], paramagnetic flagellae driven by rotating magnetic fields [46], miniaturized peristaltic pumps [47] or particles operated by optical traps [48] have been suggested and demonstrated. The essential effect of all these devices is that they are supposed to reduce the length scale for diffusive mixing even further down as mixing times (and such needed flow path lengths) will decrease with the square of the remaining length scale. Active mixing devices are often difficult to produce by established microstructuring techniques and they need additional infrastructure such as magnetic fields or laser beams for operation. Therefore, passive mixing techniques aiming at creating cross-flows between the individual laminae of the liquid inside the channels are considered a much easier way to enhance mixing performance in microfluidic devices [49] or the introduction of porous structures with well-defined tortuosity properties inside the microfluidic channels [44].

4.8 Heat transfer at micro- and macroscales Heat transfer processes on the macroscopic scale are present in many aspects of

everyday life: Heat conduction through house walls, cooking and heat management in chemical process technology are just a few examples. In most of these situations, we’re dealing with instationary heat transfer processes. In this case, not just the heat conductivity λ but also the heat diffusion coefficient α is relevant for the heat propagation process: While the heat conductivity describes the heat current through a wall during stationary conditions, the heat diffusion coefficient is relevant for the propagation of a

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temperature variation through the wall. Two walls with the same heat current under stationary conditions (e.g. a wooden wall with λ= 0.13 Wm-1K-1 and 5 cm thick and a stone wall with λ= 2.6 Wm-1K-1 and 1 m thick) will not necessarily take the same time to heat up: In the case of the wooden wall (heat diffusion coefficient 1.1×10-7m2/s), it takes roughly 3 hours for half of the temperature difference to arrive at the inner side of the wall. In the case of the stone wall (heat diffusion coefficient 1.1×10-6m2/s), it takes roughly 6 hours for half of the temperature difference to arrive at the inner side of the wall. This difference is mirrored in traditional house construction schemes in different parts of the world: In arctic regions throughout the word, we find relatively thin wooden walls, while thick stone or clay walls are common in regions with dry and hot summers and high temperature variations during the day such as Mediterranean or the Middle East. In such houses, daytime high temperatures will take a very long time to migrate through the wall, and it is possible to keep the house much cooler than the outside high temperatures over quite long times. In the arctic, by contrast, warm daytime temperatures are rather welcome in the house and so the rather fast heat diffusion through the wall actually proves advantageous.

In cooking, a uniform heating of the pot’s content is desired. In low-viscosity liquids such as water or oil this can be achieved without further measures due to the formation of density-driven convection rolls. In higher-viscosity food systems such as thickened sauces or for interstitial water between solid foodstuffs such as noodles, the formation of convection rolls is suppressed and active stirring is necessary to avoid overheating (and subsequent sticking to the pot) near the bottom of the pot at high heating rates. At lower heating rates, no sticking will occur but heating up all the food by just heat conduction may take very long time and the cooking times in the pot will be unevenly distributed. For extended solid materials such as pieces of meat, heating the inside to the required cooking temperature is either realized by frying (i.e. a high heating rate and much higher temperatures reached at the outside) or by boiling at constant temperature in water. The frying time needed for pieces of different thickness increases again with the square of the pieces’ thickness. Furthermore, the resulting texture of the fried food will vary much stronger with depth inside a thick piece than inside a thin one. Thermally instable (yet nutrionally valuable) components of the food may be degraded due to long periods at elevated temperatures during heating and cooling of large batches of food. A common example for this is ultrahightemperature (UHT) processing of milk (several seconds at temperatures up to 150 °C) which is nearly as effective as sterilization (several minutes at 121 °C, heating inside the whole bottles) for achieving long shelf life at room temperature. Heating and cooling a small quantity of the milk to 150 °C leads to much smaller losses in micronutrients than the sterilization process in which long heating and cooling times have to be considered along with the duration of the actual high-temperature treatment.

Similar effects are also found in the handling of biotechnological products which may need very fast cooling as a first step to product recovery. When just mixing a cell slurry with ice water for cooling, heat diffusion processes inside the material may lead to partial degradation of the desired products. Cooling the cell suspension in microfluidic heat exchanger systems can avoid much of these problems [50].

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Finally, heat management in chemical reactions depends a lot on the dimensions of the reactor system: In a sufficiently thick structure, even a moderate exothermic effect such as the heat release by hydrating cement may lead to substantial heating (massive concrete structures may easily reach temperatures of 80°C inside during the first days of hydration; in order to minimize this problem, special, slow-hydrating concrete mixtures may be used for such structures). In the case of more exothermic reactions, much more dramatic heating may occur and often must be avoided in order to prevent a catastrophic further heating of the reaction system. The most simple approach to active heat management is again stirring which might be complemented by the introduction of heat exchanger systems inside the reactor. Adjusting heating and cooling in a reactor system is one of the challenges that need to be addressed in scaling up a process from conventional lab size or production plant size. In the chemical industry, this is often done with experiments in miniplants as an intermediate stage.

With the advent of microreactor systems, another option for heat management has become available: here, only very short distances need to be traveled by heat conduction through the reacting mixture, and furthermore, the conductive heat transfer through the reactor housing roughly corresponds to a cylindrical “inside-out” heat conduction scenario. Stacks of microreactors instead of one large-scale reactor will provide an option for easier heat management even at production scale in the future (at least for some products), and the problematic scaling up process with changing conditions for heat-management may come down to a simple “numbering up” [51].

5 Conclusion As the different examples in the last chapter show, the square-root-of-time scaling

behaviour of both energy and particle diffusion leads to considerable differences in transport processes at different length scales. Understanding and controlling the role of diffusion processes for transport at small length scales is a fascinating challenge in many fields of science ranging from basic physics to chemical engineering, medicine and even food science and civil engineering. Research into diffusion therefore provides an important basis for sustainable technological development.

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Effects of Polydispersity on PGSE NMR Coherence Features

Nirbhay N. Yadav and William S. Price

College of Health and Science, University of Western Sydney, Australia Corresponding author: William S. Price College of Health and Science University of Western Sydney Locked Bag 1797, Penrith South DC, NSW 1797, Australia E-Mail: w.price@uws.edu.au

Abstract Real systems always contain some degree of polydispersity and yet the effects of this

real and very important problem have not been studied in great detail in NMR diffusion experiments. The effects of polydispersity become even less clear when we are outside the short gradient pulse (SGP) limit (which we generally are). Here we investigate the effects of polydispersity, in the form of a Gaussian distribution of characteristic distances, on the coherence features of PGSE NMR experiments of a model system. Characteristic pore sizes were determined from the coherence features and compared to characteristic distances determined from Fourier transforms of the second derivative.

Keywords PGSE; restricted diffusion; diffusion diffraction; polydispersity; feature enhancement.

1. Introduction Pulsed gradient spin-echo (PGSE) NMR is a powerful tool ideally suited for measuring molecular dynamics in porous systems. PGSE NMR is unique in that it is non-invasive, it can simultaneously study multiple species independently in a mixed sample, and it can probe the internal structure of samples at length scales much smaller than conventional NMR imaging. Consequently, PGSE NMR is routinely used in a wide range of fields and disciplines such as oil exploration [1-3], drug delivery systems [4], remediation of contaminated waste-water [5,6], and analytical techniques involving chromatographic processes [7-9]. PGSE NMR measures molecular motion via the attenuation of a spin-echo signal. Under certain conditions, diffraction-like effects appear on the echo attenuation curve at regular intervals of the wave vector q (=γδg/2π). These coherence features can subsequently be used to probe morphological characteristics of restricted systems (i.e., pore size, tortuosity, and connectivity). For example, polystyrene spheres [10], red blood cells [11], yeast cells [12], brain white matter [12], water-in-oil emulsions [13], and molecules between glass plates [14,15] have been studied using PGSE NMR.

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, W. Price 40

Despite the successful characterisation of the systems mentioned above, experimental factors such as background gradients, wall relaxation, and polydispersity can distort or even remove the coherence features in the signal attenuation curve. While bipolar gradients in PGSE pulse sequences have been reasonably successful in removing the effects of background gradients [16-18] and extra terms in the analytical expressions can account for the additional decay due to wall relaxation [19], the mechanisms for dealing with polydispersity have not been investigated in detail. The systems mentioned above (like all experimental systems) contain some degree of polydispersity (i.e., a distribution of characteristic distances). The subsequent coherence features in polydisperse samples are “damped” because the average weighted signal used in PGSE NMR superimposes nodes at slightly different locations on q. In extreme cases the coherence features can completely disappear and the signal attenuation curve may only show Gaussian diffusion and hence may lead to a mis-characterisation of the sample. Models which assume pores with slanted walls [15,20], a Gaussian distribution of characteristic distances [15], and a combination of smooth and zigzag surfaces [20] have all been fitted successfully to experimental data. Nevertheless, the authors are unaware of any experimental studies which have tested the validity of these models against a range of experimental conditions. A method for delineating the underlying coherence features in poorly defined PGSE data was proposed by Kuchel et al. [21]. The method involves weighting the data using the Blackman-Harris window function [22], interpolating with a shifting cubic spline, and then the second derivative is taken prior to Fourier transformation. The ability of this numerical tool to identify instances where more than one set of coherence features are present may allow for a more accurate determination of pore structure. Subsequently, the second derivative method can be an important tool for interpreting PGSE diffraction profiles of polydisperse samples.

In this study we use a well defined model system (parallel planes) prepared to give different amounts of polydispersity to test the validity of the Gaussian distribution of characteristic distances model describing polydispersity. Fourier transforms of the second derivative were used in the analysis of the PGSE data.

2. Theory PGSE NMR spatially labels the

position of the spins in a system by recording the phase of the transverse magnetisation via spin-echoes (Fig. 1). The position of the spins is recorded at two instants in time which correspond to two positions (say r and r’). If r and r’ are identical (i.e., the spins remain stationary or return to their starting position), the net phase change, γδg·(r - r’) = 0, and the spins create a refocussed echo. However, if the spins

Fig. 1: Schematic representation of the Tanner NMR pulsed field gradient stimulated spin-echo pulse sequence for the PGSE experiment.

41

move to a different position during the time between the gradient pulses (Δ), γδg·(r - r’) ≠ 0, their contribution to the echo will be phase shifted. The degree of the dephasing due the applied gradient is proportional to the displacement in the direction of the gradient during Δ. In the case of translational diffusion, each spin in a system will have a random phase change causing the spin-echo signal (Eq. (1)) to attenuate due to averaging of the incoherent phases over the entire sample. The short gradient pulse (SGP) approximation gives the normalised echo attenuation for a two impulse scheme, such as a Hahn spin-echo based sequence or stimulated spin-echo based sequence (Fig. 1), by [23] ( ) ( ) ( ) ( )| ', exp ' 'E P iρ γδΔ = Δ ⋅ −⎡ ⎤⎣ ⎦∫ ∫g r r r g r r rd dr (1)

where ρ(r) is the equilibrium spin density and P (r|r’, Δ), the diffusion propagator, is given by the solution to the diffusion equation. By ensuring δ << Δ, mathematically tractable equations which describe diffusion within symmetrical pores (e.g., planar [23,24], cylindrical [19,25], and spherical [26,27]) can be derived from Eq. (1). A significant advantage of using the SGP approximation over some other methods is that it can model diffraction-like effects. Diffraction effects in PGSE NMR derive from the average propagator, which is the probability ( ),P ΔR that a spin at any starting position will displace by R during period Δ, is given by ( ) ( ) ( ), ,P PρΔ = + Δ∫R r r r R, dr (2) where the diffusion propagator is multiplied by the equilibrium spin density and integrated across the whole sample. In terms of the average propagator Eq. (1) reduces to ( ) ( ) 2, , iE P e π ⋅Δ = Δ∫ q Rq R dR (3) where q is introduced to include the effects of the gradient into the analysis. In the long time limit (Δ→∞), all species trapped within a pore become independent of their starting positions and, therefore, diffusional processes, so ( ) ( ), ,P R ρ+ ∞ = +r r r R . (4) Consequently, the average propagator becomes ( ) ( ) ( ),P ρ ρ∞ = +∫R r r R dr . (5) In the limit where the diffusing molecules have sampled the entire pore, the propagator is an autocorrelation function of ρ (r). Using the Wiener-Kintchine theorem [28] and Eq. (3) we find that E (q, ∞) is the power spectrum of ρ (r), ( ) ( ) 2

,E S∞ =q q (6)

where S (q) is the Fourier transform of ρ ( r’). The signal obtained has no phase information, therefore a Fourier transformation cannot be carried out to obtain structural information from a “conventional” image. Instead, structural information is obtained by plotting the signal attenuation vs. the reciprocal space of acquisition (q). In the case of parallel plane pores,

42

( )( )

( )( ) ( ) ( )

( ) ( )

2 22

2 2 2 21

2 1 cos 2 1 1 cos 24 2 exp

2 2

n

n

a an DE q aaa a n

π ππππ π π

∞

Δ=

−⎡ ⎤ − −⎛ ⎞− Δ⎣ ⎦= + ⎜ ⎟ 2⎡ ⎤⎝ ⎠ −⎣ ⎦

∑q q

qq q

(7)

where D the diffusion coefficient and the gradient (g) is perpendicular to the planes. The second term in Eq. (7) disappears at long Δ and diffractive minima appear at q = n/a (n = 1, 2, 3, …) (see Fig. 2). The accuracy of the SGP approximation is however limited by its reliance on (i) gradient pulses which have durations much shorter than their separations (δ << Δ), and (ii) the distance diffused during the gradient pulse is small compared to the characteristic dimensions of the system (Eq. (8)),

2a

Dδ (8)

where δ is the length of the gradient pulse and a is the characteristic distance of the pore. In the case of large molecules in small pores, meeting the condition in Eq. (8) can pose problems for the NMR hardware subsequently leading to errors including an underestimation of the pore size [29,30]. As mentioned earlier, the porous nature of the sample may be lost when these conditions are not met. Subsequently, some authors have developed methods which approximate finite-width gradient pulses by discretising the gradient pulses into intervals of infinitesimally narrow pulses [30,31]. The matrix formalism [30] is particularly successful in significantly reducing errors associated with diffusion during the gradient pulse leading to an underestimation of the pore dimensions and has been experimentally verified [15]. Using this discretisation, the PGSE pulse sequence (Fig. 1) is subdivided into 2N+1 intervals of length τ such that the total length of the sequence is (2N+1)τ with

Fig. 2: Simulation of a PGSE coherence feature modelled using the SGP approximation and matrix formalism. Δwas set to 2 s, δ was 2 ms, and the planar separation (a) was 128 μm. The effects of the finite gradient pulses are evident in the matrix-based simulations with the diffractive minima moving to higher q.

12

N τ⎛ ⎞Δ = +⎜ ⎟⎝ ⎠

, (9)

12

Mδ τ⎛ ⎞= +⎜ ⎟⎝ ⎠

, (10)

hence the total effective scattering wave vector amplitude is ( ) ( )( ) 1

net 1 1 2q M q M gτ π γ τ−= + = + . (11)

43

Finally, the matrix equation for the attenuation is (12) ( ) ( ) ( ) ( )† †MM N ME S q RA q R RA q RS q− ⎡ ⎤= ⎡ ⎤⎣ ⎦ ⎣ ⎦

where component matrices are given by ′=S BS , (13)

† ′=A C A C , (14)

and ( )2 2 2exp /R k Dπ τ= − a . (15)

B and C are diagonal matrices defined by

1

2

2

a

a

a

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

B , (16)

1

2

2

a

a

a

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

C , (17)

( )( ) ( )( ) ( )

( )( ) ( )( ) ( )

2 2

2 2

2 exp 2 cosodd

2

2 exp 2 sineven

2

k

i a i a a ak

a kS

a i a a ak

a k

π π π

π π

π π π

π π

⎧⎪

−⎪′ = ⎨⎪⎪ −⎩

q q q

q

q q q

q

(18)

and

12kk k kk kA S S′ ′+′−

⎡ ⎤′ ′ ′= +⎣ ⎦ . (19)

44

3. Experimental The model experimental system used in this study was a single pore with parallel boundaries. A water solution (20% H2O and 80% D2O) was placed between the tube and plunger of susceptibility-matched microtubes (BMS-3; Shigemi, Tokyo) with the plunger positioned to give a separation of about 150 μm (a) with the bottom of the tube with the gradient direction being perpendicular to the planes (Fig. 3a). The end of the plunger was polished with different grades (FEPA system) of sand paper (600 – 120) to obtain surfaces with different degrees of rugosity. After the plunger was polished with the 120 grade sandpaper it was extremely difficult to achieve a small pore size (due to removal of the taper on the plunger as seen in Fig. 3b) therefore experiments carried out had a characteristic distance of approximately 250 μm. To ensure the polished surface was perpendicular the sides of the plunger, a hole (diameter 4 mm) was drilled in an aluminium block that had been machined with high precision to give perpendicular surfaces. The polishing system used only allowed polishing the end of the plunger and not the plane surface within the tube. 1H NMR experiments were performed at 298 K on a Bruker Avance 500 wb spectrometer equipped with a 5 mm broadband inverse probe equipped with a single (i.e., z) shielded gradient coil connected to a GREAT 3/10 current amplifier. The strength of the gradient was calibrated using the known diffusion coefficient of water whilst the temperature in the NMR probe was calibrated using methanol. The stimulated echo pulse sequence with “rectangular” gradient pulses was used to obtain the characteristic diffusion-diffraction pattern. The tube arrangement gave two resonances: a narrow resonance for water between the planes and a broader resonance from water on the sides of the plunger. The two resonances overlapped at low q values and could not be separated without significantly degrading the field homogeneity. Simulations of PGSE coherence features modelled using the matrix formalism were carried out using Mathcad 13 (Mathsoft, Cambridge, MA) and Fourier transforms of the second derivative were performed in Mathematica 5.0.

Fig. 3: (a) Shigemi tube arrangement showing the pore of characteristic distance a created between the plunger and tube, (b) parallel boundary pore with idealised walls, and (c) parallel boundary pore with rough walls which is approximated by a Gaussian distribution of characteristic distances. The arrow indicates the direction of the magnetic field gradient (g).

45

4. Results and Discussion 4. Results and Discussion Firstly, the rugosity in the parallel pore was modelled using a Gaussian distribution about the spacing in the middle of the tube (a0) where a weighting factor was introduced to reflect that the NMR signal will be larger from regions of larger separation

Firstly, the rugosity in the parallel pore was modelled using a Gaussian distribution about the spacing in the middle of the tube (a0) where a weighting factor was introduced to reflect that the NMR signal will be larger from regions of larger separation

( )

( )( )

Fig. 4: PGSE coherence data simulated using the matrix formalism of a Gaussian distribution of characteristic distances where σ was varied between 0.0a0 to 0.4a0. The inset displays the probability of finding a spin at a particular value of z.

2022

2

z azz e σρσ π

−−

= [20]

where ρ(z) is probability of finding a spin at some value of z and σ is the standard deviation. The effect of a Gaussian distribution of characteristic distances can clearly be seen in (Fig. 4). A standard deviation of 0.2a0 is enough to remove all but the first diffraction minima and a standard deviation of 0.4a0 removes all coherence features.

Secondly, the simulations above are supported by experimental data in Fig. 5 where the samples with a higher degree of rugosity clearly show a damping of the coherence features. The absolute attenuation of the experimental data in Fig. 5 is greater than that in the simulations because the free diffusing spins, which attenuate more rapidly, contribute to 1H peak at low qa values. Consequently, the σ values used to damp the modelled diffraction minima and fit the model to the experimental data may be slightly lower than what would be expected if the experimental data only consisted of spins undergoing restricted diffusion.

The shifting of experimental minima to higher qa than the models predict which was seen in previous studies [15] is not observed. Also in Fig. 5a – Fig. 5c, the amount of rugosity in the system was insufficient to decrease the number of maxima/minima cycles in our attenuation curves.

The only other factors which could cause such a damping of the coherence features as seen in Fig. 5 are wall relaxation [24], background gradients [15], and neglecting the motion of spins to adjacent regions with different characteristic dimensions during Δ. Firstly, it has been proven in a number of studies that the surface relaxivity of glass is such that the perfectly reflecting wall approximation is reasonable [15,20,32,33]. Secondly, background gradients still form at the glass-water interface despite using susceptibility matched tubes [34]. Two glass-water interfaces within 100 μm to 300 μm, like those used in this study, result in substantial background gradients which cannot be removed even when using bipolar gradient pulses because the long Δ used enable the spins to travel into regions of different magnetic field strength.

46

Fig. 5: PGSE signal attenuation profile for water diffusing between planes in a Shigemi tube after (a) no polishing, (b) the plunger polished using 600 grade paper, (c) 400 grade paper, and (d) 120 grade paper. The experimental data in (a-c) was obtained with Δ = 4 s and δ = 2 ms whereas experiments for (d) has Δ = 7 s and δ = 2 ms in order to maintain the DΔ/a2 ratio. Each spectrum was the average of 32 transients with a recycle delay of typically 36 s which was sufficient to allow for full relaxation (i.e., > 5 x T1) between each transient. The experimental data was fitted to the model for a Gaussian distribution of characteristic distances (dotted line). The characteristic distances stated on each figure were determined by fitting to the matrix formalism and confirmed using an optical travelling microscope (PTI, Hampshire, England).

Background gradients can be modelled using a cosine profile [35] which has recently been solved [36]. Finally, coherence features only become apparent when a significant portion of the spins sample the boundaries of the confining geometry of a particular characteristic distance. If a spin moves into an adjacent region with a different characteristic distance during Δ, coherence features become less distinct. Pore hopping theory has successfully characterised random walk simulations within regularly spaced rectangular barriers [37] and could be applicable in our model where the interconnection size would approach zero. Presently, the authors are working on more sophisticated models which include background gradients and investigating the applicability of pore hopping models and stochastic boundary conditions. The probability of finding a spin at each characteristic distance (weighted propagator P(z)) was obtained by interpolating the experimental data in Fig. 5, applying the Blackman-Harris window function, taking the second derivative of the interpolations, and finally numerically Fourier transforming the derived data. The weighted propagators in Fig. 6a and Fig. 6c both show characteristic distances very similar to those determined

47

directly from the coherence features. The characteristic dimensions determined from Fig. 6b and Fig. 6d are also similar to values determined directly from the coherence features but give two distinct characteristic distances between the parallel planes. Two characteristic distances from samples polished using the 600 and 120 grade sandpaper could explain why there is a misalignment of some nodes from the matrix formalism to nodes in the experimental data.

Fig. 6: Weighted propagator P(z) for PGSE NMR q-space data after the application of a Fourier transform of the second derivative. The data in (a) is from a pore of characteristic distance of 150 μm, (b) 165 μm pore after sanding the plunger with 600 grade sandpaper, (c) 140 μm pore after sanding with 400 grade sandpaper, and (d) a 250 μm pore after sanding with 120 grade sandpaper.

The potency of the second derivative method for interpreting damped coherence features is further demonstrated in Fig. 7 which contains experimental data and Fourier transforms of the second derivative of the PGSE data obtained from the sample polished using the 120 grade sandpaper at different diffusion intervals (Δ). Even though the coherence features are much less distinct in Fig. 7a compared to Fig. 7b, Fourier transforms of the second derivative gives the same result. Subsequently the second derivative method could be used in more polydisperse systems where the coherence features are less distinct. The Gaussian distribution model of characteristic distances used in this study was successful in simulating the damping effect of polydispersity but some anomalies exist. For example, the damped coherence features in Fig. 5a also correspond to parallel planes with a wall slanted by 0.4 degrees. For our system a model which incorporates both

48

Fig. 7: PGSE attenuation curves (top) and corresponding weighted propagators (bottom) after application of the second derivative method for a pore after sanding with 120 grade sandpaper. The data in (a) and (c) was obtained with Δ = 4 s whist (b) and (d) was obtained with Δ = 7 s. Despite the coherence features in (a) being significantly less distant than those in (b), the weighted propagators give the same characteristic dimensions.

slanted pore walls and a Gaussian distribution of characteristic distances would provide a better fit but will still only be approximations for a stochastic boundary condition. Given a set of possibly polydisperse PGSE data and with currently available models and methods, one should apply the Fourier transform of the second derivative to determine if there are a few characteristic distances within the sample. Using either the Gaussian distribution of characteristic distances or slanted pore wall models, one can then quantify the polydispersity in the system by fitting the data to simulations of PGSE coherence features using either the matrix formalism or SGP approximation. The authors are currently working on improved techniques for characterising polydisperse PGSE data.

5. Conclusion The Gaussian distribution of characteristic distances is reasonably successful in describing the polydispersity between parallel planes although a model which considers stochastic boundary conditions would be more ideal. A Fourier transform of the second derivative was however shown to be extremely useful in characterising our model system and explaining possible inconsistencies in experimental results.

49

6. Acknowledgements The financial support of the NSW State Government BioFirst award and the UWS postgraduate award is gratefully acknowledged. The authors thank Prof. Philip Kuchel for providing Mathematica worksheets on the second derivative method and Mr. Phillip Whitton for his technical support.

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5950-5953. [9] R. Bujalski, and F.F. Cantwell, Anal. Chem. 78 (2006) 1593-1605. [10] P.T. Callaghan, A. Coy, D. MacGowan, K.J. Packer, and F.O. Zelaya, Nature 351

(1991) 467-469. [11] A.M. Torres, R.J. Michniewicz, B.E. Chapman, G.A.R. Young, and P.W. Kuchel,

Magn. Reson. Imaging 16 (1998) 423-434. [12] C. Malmborg, M. Sjobeck, S. Brockstedt, E. Englund, O. Söderman, and D.

Topgaard, J. Magn. Reson. 180 (2006) 280-285. [13] B. Hakansson, R. Pons, and O. Söderman, Magn. Reson. Imaging 16 (1998) 643-

646. [14] W.S. Price, and O. Söderman, Israel J. Chem. 43 (2003) 25-32. [15] W.S. Price, P. Stilbs, and O. Söderman, J. Magn. Reson. 160 (2003) 139-143. [16] P.Z. Sun, J.G. Seland, and D. Cory, J. Magn. Reson. 161 (2003) 168-173. [17] P. Galvosas, F. Stallmach, and J. Kärger, J. Magn. Reson. 166 (2004) 164-173. [18] J.G. Seland, G.H. Sørland, K. Zick, and B. Hafskjold, J. Magn. Reson. 146 (2000)

14-19. [19] P.T. Callaghan, J. Magn. Reson. A 113 (1995) 53-59. [20] D. Topgaard, and O. Söderman, Magn. Reson. Imaging 21 (2003) 69-76. [21] P.W. Kuchel, T.R. Eykyn, and D.G. Regan, Magnet. Reson. Med. 52 (2004) 907-

912. [22] F.J. Harris, P. IEEE 66 (1978) 51-83. [23] J.E. Tanner, and E.O. Stejskal, J. Chem. Phys. 49 (1968) 1768-1777. [24] A. Coy, and P.T. Callaghan, J. Chem. Phys. 101 (1994) 4599-4609. [25] O. Söderman, and B. Jönsson, J. Magn. Reson. A 117 (1995) 94-97. [26] J.S. Murday, and R.M. Cotts, J. Chem. Phys. 48 (1968) 4938-4945. [27] B. Balinov, B. Jönsson, P. Linse, and O. Söderman, J. Magn. Reson. A 104 (1993)

17-25. [28] D.C. Champeney, Fourier transforms and their physical applications, Academic

Press, London ; New York, 1973. [29] P.P. Mitra, and B.I. Halperin, J. Magn. Reson. A 113 (1995) 94-101.

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[30] P.T. Callaghan, J. Magn. Reson. 129 (1997) 74-84. [31] A. Caprihan, L.Z. Wang, and E. Fukushima, J. Magn. Reson. A 118 (1996) 94-

102. [32] M. Appel, G. Fleischer, D. Geschke, J. Kärger, and M. Winkler, J. Magn. Reson.

A 122 (1996) 248-250. [33] S.L. Codd, and P.T. Callaghan, J. Magn. Reson. 137 (1999) 358-372. [34] W.S. Price, P. Stilbs, B. Jönsson, and O. Söderman, J. Magn. Reson. 150 (2001)

49-56. [35] L.J. Zielinski, and P.N. Sen, J. Magn. Reson. 147 (2000) 95-103. [36] D.S. Grebenkov, J. Chem. Phys. 126 (2007) 104706-104715. [37] P.T. Callaghan, A. Coy, T.P.J. Halpin, D. MacGowan, K.J. Packer, and F.O.

Zelaya, J. Chem. Phys. 97 (1992) 651-662.

51

NMR Methods for the Fast Recording of Diffusion

Guilhem Pages,1 Philip W. Kuchel,1

1 University of Sydney, School of Molecular and Microbial Biosciences, Sydney 2006, Australia

Corresponding author: Guilhem Pages School of Molecular and Microbial Biosciences University of Sydney NSW 2006, Australia E-Mail: g.pages@mmb.usyd.edu.au

Abstract The time taken to complete an NMR diffusion experiment is typically ~10 minutes.

For systems that rapidly evolve conventional pulsed field gradient spin-echo (PGSE) experiments cannot be used to obtain reliable estimates of molecular mobility. Modern ‘fast-diffusion’ experiments provide a means of obtaining this information and thus open up new vistas for the application of PGSE NMR. In this paper we review the various advantages and disadvantages of these methods.

Keywords: Diffusion experiment; diffusion ordered spectroscopy; DOSY; echo train; fast diffusion; lineshape fitting; multiple coherence pathways; single scan.

Abbreviations: BPPSTE, bipolar stimulated echo; CPMG, Carr-Purcell-Meiboom-Gill; DDF, distant dipolar field; DOSY, diffusion ordered spectroscopy; FID, free induction decay; MMME, multiple modulation multiple echoes; NMR, nuclear magnetic resonance; PFG, pulsed field gradient; PFGMSE, pulsed-field gradient multiple spin-echo; S/N, signal to noise ratio.

1. Introduction Pulsed field gradient spin-echo nuclear magnetic resonance (PGSE NMR)

experiments are a method of choice for studying diffusional mobility of molecules in living and inanimate systems. The method is non-invasive and is able to measure in a few minutes the diffusion coefficients of solutes and solvents in complex mixtures [1-5]. In PGSE NMR the signal intensity S(g) of the molecule of interest is measured as a function of the magnitude of the magnetic field gradient pulses that are used. The diffusion coefficient D is estimated from the data by regressing onto them the following Stejskal-Tanner equation [6]:

2 2 2 '0( ) g DS g S e γ δ− Δ= (1)

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, G. Pages 52

where S0 denotes the signal intensity without gradients, γ is the nuclear magnetogyric ratio, g is the gradient amplitude, δ is the gradient pulse duration, and Δ’ is the ‘corrected’ diffusion time. In the simplest pulse sequences Δ’ is simply the interval between the two field gradient pulses, or more generally Δ’ = (Δ - δ/3). One well-known extension to data acquisition and processing using PGSE NMR is diffusion ordered spectroscopy (DOSY) [7]. The final output is a 2-dimensional contour map in which one axis records diffusion coefficients and the other chemical shifts. Thus one dimension indicates a physical property and not, as is usually the case, information on magnetic interactions between nuclei.

New pulse sequences and processing methods have been developed to study a diverse variety of samples from: discrete to polydisperse samples; chemical to biological ones; unrestricted to restricted ones or those with anisotropic diffusion; and stagnant to rapidly flowing mixtures. Several reviews have been written describing these applications [1,3,8-10].

Even if the total time of the experiment is short, it may be too long to interrogate systems in which diffusion coefficients change rapidly. Such systems include biological ones with enzyme-catalyzed reactions, or those in rapidly flowing fluids in industrial processes. The first fast-diffusion NMR pulse sequence was described in 1969 [11]; and recently a few more methods have been published that can measure a diffusion coefficient in less than 1 minute. We review these techniques in this paper.

Pulse sequences to measure diffusion coefficients very rapidly have also been developed to handle the specific requirements of grossly inhomogeneous magnetic fields [12], and for use with laser-excited hyper-polarized gas of relatively short half-life [13]. Because these methods are highly specialized we will not discuss with them in any detail. We have classified the remainder of the methods into two classes: (1) those from which the diffusion coefficient is extracted from data obtained by using only one or two gradient pulses; and (2) those in which diffusion is elicited by a train of gradient pulses. The latter we analyze and discuss first.

2. Diffusion measured using a train of gradient pulses Saving time by decreasing phase cycling By keeping to the traditional approach with PGSE NMR experiments the only

recourse to saving time is to decrease the number of transients in each phase cycle. In the bipolar-pulse stimulated-echo (BPPSTE) experiment [14], the phase cycling of the 180° pulses requires a EXORCYCLE [15] with 16 (or optimally 64) transients in order to record only the spatially encoded magnetization. This EXORCYCLE enforces a zero-order coherence during the delay Δ.

The approach developed by Morris et al. [16] is based on suppression of unwanted coherence-transfer pathways using deliberately unbalanced pulse pairs (Fig. 1). The magnitude of the two pulses in a pair have the ratio 1:1+α. During the experiment, α is held constant and the appropriately transformed Stejskal-Tanner equation is:

2 2

2 2 2 2 1( )6 2( )

(0)gS g e

S

α αγ δ δ τ− −− Δ+ +

= (2)

53

where τ denotes the delay between the two components of the bipolar magnetic field gradient pulses. The choice of the value of α depends on BB1 inhomogeneity and signal-to-noise ratio and is usually ~ 0.2.

Fig. 1: PGSE pulse sequences that use composite pulses. A, classical BPPLED sequence B, the pulse sequence designed by Morris et al. [16] that uses unequal gradient pulses in each bipolar pair. Additional gradient pulses can be used in B to refocus the lock signal and to dephase unwanted coherences. Spoil gradients can also be added to the BPPLED pulse sequence. RF denotes the radio-frequency time train and Gz the gradient pulse time train. δ is the gradient duration, g is the gradient amplitude, Δ is the diffusion time (midpoints of the two diffusion encoding periods), τ is the time between the midpoints of the antiphase field gradients, and α is the unbalancing factor of the gradient pulses.

Series of gradient echoes Another way to minimize the acquisition time of a PGSE experiment is to record

within a single transient the echoes that are modulated by diffusion. There are four pulse sequences that are based on this principle.

The first two pulse sequences are based on the Carr–Purcell–Meiboom–Gill (CPMG) experiment. In one, each 180° pulse is sandwiched between two gradient pulses of identical magnitude to encode and then decode spatial information (Fig. 2A); the pulse sequence is called the Pulsed Field-Gradient Multiple Spin-Echo (PFGMSE) experiment [17]. The delay between the initial 90° pulse and the first 180° is τ and the delay between each 180° is twice this value, 2τ, and each echo corresponds to a different gradient magnitude. The number of echoes that are recorded is the number of points used in the diffusion decay analysis, so a prolonged series of echoes provides a better estimate of D. To extract the value of D, the ratios between two consecutive echoes must be calculated (Eq. 3). It is then a simple matter to determine D from the slope of the graph:

2 2 2 2

2

( 1) 2ln '( )

S n n g DS n T

τ γ δ+= − − Δ (3)

54

where n denotes to the nth echo that is recorded.

The second pulse sequence, that is based on a CPMG RF-pulse train, uses magnetic field gradient pulses that are of constant amplitude (Fig. 2B) [18]. The diffusion coefficient is extracted from the linewidths of the Fourier transform of the echo train. The width of the resonance at half of the peak height, Δν1/2, of the adsorption signal in the frequency domain, for a Lorentzian lineshape is:

2 2 21

2 2

1 12 6

g DT

δυ γ δπ τ τ

⎡ ⎤Δ⎛Δ = + − ⎞⎢ ⎥⎜

⎝ ⎠⎟

⎣ ⎦ (4)

To fit the data, the value of the first term in parentheses of Eq. (4) is required. Hence an experiment is run with the gradient pulse-amplitudes set to 0 thus eliminating the second term of the equation. To extend this method to the case of non-Lorentzian lineshapes and/or overlapping peaks in the echo spectrum, the first 90° hard pulse is substituted by a 90° selective one. Fig. 2 emphasizes the similarity between both of the CPMG-based pulse sequences. They differ only in the variation or not of the gradient magnitudes.

Fig. 2: CPMG-based PGSE methods for measuring D in a single multiple-echo transient. A, the PGMSE method [17]; and B, the method of Chandrakumar et al. [18]. For A the gradient amplitude is increased systematically, while for B it is constant. The notation is identical to that in Fig. 1 except that τ is the delay between the 90° and 180° RF pulses. The subscripted square-brackets around the pulse sequence indicates that it is repeated N times.

The third method in the present class involves recording a series of echoes that are

created by alternation of the diffusion gradient [19]; these are called gradient echoes. Each echo is attenuated further by an additional diffusion-gradient pulse. The expression for the signal amplitude differs between even (Eq. 5) and odd (Eq. 6) echoes:

2 2 3

2 30

2 2 g g3Dg n n D

even n nS A e eγ δ γ δ−

− = i (5)

55

(2 2 3 2 202 2 g g3

Dg n D nodd n nS A e e

γ δ )3γ τδ δ− −

− =i

(6) where An is the amplitude without diffusion attenuation, and the vector dot product g0●g accounts for the interaction between the gradient pulses and a static magnetic field. Fitting the signal amplitude using both previous equations is used to estimate D.

Finally, a recent approach called Difftrain has been described [20]. This is like the previous one because it exploits attenuation of an echo by an additional gradient pulse. The position of a nucleus is encoded and then it is decoded multiple times. This yields all the information necessary to fit the Stejskal-Tannner equation (Eq. 1). To conserve chemical shift information, the pulse sequence uses the inversion recovery method that is routinely used for measuring longitudinal relaxation times [21]. Positions of the spins are encoded in the transverse plane and then promoted to the longitudinal direction with an RF pulse; then a small proportion of the spin population is returned to the transverse plane for the detection via a spin echo. After detection, the residual magnetization is destroyed by a spoil gradient, before applying another gradient and RF pulse to detect the next echo.

Difftrain can be used to measure the distribution of droplet sizes in an oil-water emulsion that is undergoing evolution during phase separation [22]. A fast-diffusion experiment (taking less than 4 s) was needed to characterize the system because the emulsion was thermodynamically unstable. In contrast, the acquisition time for a classical PGSE NMR experiment is too long to capture the evolving droplet sizes in the system. The authors also describe the possibility of measuring the apparent velocity of water flow through an ion-exchange (desalting) column.

All the abovementioned experiments take significantly less time than the classical

PGSE NMR diffusion experiments to yield their data. They are still based on the evolution of a signal whose variation is described by an expression that contains D. Comparing the methods, the one described by Morris et al. [16] appears to encompass the best compromise between the difficulty of setting up the experiment on the NMR spectrometer and processing the resulting data. Other approaches have been developed to measure D in a single-scan, but the major difference is that they do not require a change in gradient magnitude during the experiment.

3. Pulse sequences with a small number of gradients Burst pulse sequence The Burst pulse sequence [23], was initially developed to rapidly acquire magnetic

resonance images (MRI) under conditions of high spectral resolution [24]. A train of echoes is created from a series of low flip-angle RF pulses. To enable image formation, the pulses are applied in the presence of a persistent magnetic field gradient; and the echoes are detected by using a magnetic ‘read’ gradient (Fig. 3).

56

Fig. 3: The Burst pulse sequence modified to measure D [24]. On the upper right of the diagram each arrow represents an echo; d1 is the delay between the end of the last pulse and switching off the gradient; d2 is the period between the last RF pulse and switching the read gradient on; τ is the period between the midpoints of two consecutive RF pulses. For other notation, see Fig. 1.

To analyze the effect of this pulse sequence on a spin system we note that the first

series of RF pulses is simply a DANTE pulse train [25]. This realization suggests the use of the ‘small-angle approximation’ [26] that is used in predicting the excitation envelope of the pulse train. The analysis is then extended to include the effects of diffusion. Accordingly the signal intensity for each echo is described by the equation:

2 2 2

3n

n ng D

nS eδ

γ δ ⎛ ⎞− Δ −⎜⎝=

'⎟⎠ (7)

where Δn = nτ+d2 and δn = nτ+d1. D is estimated by fitting Eq. (7) to a graph of the intensity of the NMR signal versus the grouped parameters in the exponent of the expression.

Multiple Modulation Multiple Echoes (MMME)

A further method to rapidly estimate D of a molecular species is to record the spin-echo from a series of different coherence pathways. This approach increases the information content over that of a single-transient [27]. For spin-1/2 nuclei such as 1H, we can define three states of magnetization: M0, M- and M+. For a train of RF pulses, a coherence pathway is characterized by a series of N + 1 numbers from q0 to qN where qN is the magnetization after the Nth pulse and q0 is the magnetization-state before the first pulse. Each coherence pathway can be written as a product of three terms:

MQ = AQ BBQ CQ (8) where Q represents one coherence pathway between 0 and N; AQ is the frequency spectrum; BBQ is a term that describes diffusion; and CQ is the relaxation-attenuation factor. To write an expression for BQB , it is necessary to introduce the instantaneous wave-vector k(t) [28,29], the magnitude of the constant field-gradient g and the instantaneous value of q, q(t’):

57

(9) 0

( ) ( ') 't

k t g q t dtγ= ∫ In the case of diffusion in an isotropic unbounded medium the attenuation of the

signal is given by,

2

0

( )

B

t

D k t dt

Q e− ∫

= (10)

where t = 0 specifies the beginning of the pulse sequence and T is the time at the maximum of the spin echo.

Now consider a persistent magnetic field gradient and a train of four RF pulses with tipping angles α1,…,α4 and a time spacing between them of τ1 to τ3, respectively. Fig. 4 shows this sequence using α1 = α2 = α3 = 90° and α4 = 180°.

Fig. 4: MMME4 pulse sequence [27] used to measure D in a single transient. RF, Gz and g are defined in Fig. 1. τ1 is chosen and then τ2 and τ3 are calculated using Eq. (15).

During the τ periods in the pulse sequence of Fig. 4, transverse magnetization (q =

±1) acquires a phase of qgτι (i = 1, 2 or 3) and the spin echo appears when the total phase is zero:

1 1 2 2 3 3 4 0q q qτ τ τ τ+ + − = (11)

where τ4 is the delay after the last pulse and prior to acquisition of the free induction decay. For a ratio τ1 = τ2/3 = τ3/9, Eq. 11 is expressed by the ratio τ4/τ1 which represents the phase index: τ4/τ1 = q1 + 3q2 + 9q3 (12)

Therefore, fourteen different phase indices are obtained, and by using the convention, q0 = 0 and q4 = -1 to detect the signal it is seen that an echo is detected every τ4 = nτ1

58

where n is the phase index. Thus we detect a new spin-echo signal every τ1 seconds as indicated in Table 1.

Table 1: Different coherence pathways in a MMME4 pulse sequence. One free induction decay (coherence pathway 0,0,0,0,-1) with 13 echoes are recorded. Phase index q0 q1 q2 q3 q4

0 0 0 0 0 -1 1 0 1 0 0 -1 2 0 -1 1 0 -1 3 0 0 1 0 -1 4 0 1 1 0 -1 5 0 -1 -1 1 -1 6 0 0 -1 1 -1 7 0 1 -1 1 -1 8 0 -1 0 1 -1 9 0 0 0 1 -1

10 0 1 0 1 -1 11 0 -1 1 1 -1 12 0 0 1 1 -1 13 0 1 1 1 -1

For each coherence pathway, molecular diffusion is responsible for signal attenuation

which is described by the expression:

2 3

Qb D gQ e γ τ−=B (13)

where bQ is calculated for each coherence pathway. The CQ term contains the longitudinal and transverse relaxation times, T1 and T2, respectively.

The previous experiment can be extended to a sequence of N pulses whereupon it is called MMMEN. For N pulses, the number of coherence pathways that are generated is given by [30]:

[3(N-1) – 1]/2 + 1 (14)

Table 2 shows the number of coherence pathways that are recorded as a function of the number of RF pulses. Table 2: Number of coherence pathways recorded for a pulse sequence incorporating from 2 to 7 RF pulses [26]. Number of RF pulses 2 3 4 5 6 7 Number of coherence pathways 2 5 14 41 122 365

59

In order to record spin echoes at particular times we set the period τi to be a power of 3:

τi = 3i-1 τ1 (15) Because the experiment is performed with a field gradient that is on all the time, and the sample extends beyond the region of the RF coils, each pulse must be slice-selective so that off-resonance signals are avoided. Using these conditions, the spin-echo signals and line shapes are able to be rigorously defined. Thus a single-transient experiment is sufficient to measure D. If the sample size is not sufficiently large spin-echoes may not have uniquely determined shapes and/or areas. In this case, two transients must be acquired using two different τ values (τ and τ’); then the diffusion coefficient is obtained from the ratio of the intensities of both echoes, as follows:

2 2 3 3( '( )

( ')Qb D gQ

Q

Se

Sγ τ ττ

τ− −= ) (16)

This method has been successfully implemented in the SMART imaging pulse

sequence [31,32]. The pulse sequence [31] measures both D and T2 , and for unspecified reasons it requires a phase cycle of four transients. MMME has also been applied to measuring the velocity of flowing fluid in a single transient [33]. By adding magnetic field gradient pulses in multiple directions, it is possible to record a diffusion tensor that characterizes diffusion in all three directions of a Cartesian coordinate system [34]. Thus the values of Dxx, Dyy, and Dzz can be deduced for each dimension in anisotropic media using a single-transient.

Fast-CRAZED

In this experiment RF pulses are used to excite spin systems in the sample so that a dipolar magnetic field is transiently established. This so called ‘distant dipolar field’ (DDF) is then used to measure simultaneously both D and T2. Thus the CRAZED pulse sequence led to fast-CRAZED [35] due to its ability to rapidly measure the NMR parameters [36] (Fig. 5). In this pulse sequence only one gradient pulse is used. The DDF created by the modulated magnetization performs the role of field gradient pulses by refocussing dephased magnetization. To select intermolecular zero-quantum transitions mediated by the DDF, a two-step phase cycle is used with a transient recorded with a β RF pulse (Fig. 5) of 45° subtracted from one recorded with β = 135°. A train of 180° pulses is used as a sandwich of 90°- 180°- 90° pulses [37] to minimize the loss of magnitude of the DDF. The first 90° RF pulse can be an adiabatic one thus selecting particular coherence pathways. The total time for an experiment using this sequence is ~10 seconds; the slightly longer time than perhaps expected is due to the long train of 90°- 180°- 90° pulses.

60

Fig. 5: Fast-CRAZED pulse sequence [38]. β is a 45o or 135° RF pulse. t1 is the period between the end of the adiabatic pulse and the middle of the β pulse and τ is the delay between the midpoints of the β pulse and the sandwiched 180° pulse. The sandwiched train is repeated N times. For other items of notation see Fig. 1.

The theoretical underpinning of this experiment relies on the non-linear term that is

added to the Bloch-Torrey equations to account for the presence of the DDF [39-41]; the parameter D is incorporated into this term. An analytical solution of this equation is available only if diffusion-mediated signal-attenuation is much stronger than the rephasing of the magnetization due to the DDF. Thus diffusion attenuation is encapsulated in the relationship 1/(D(γgδ)2) while the dipolar characteristic time is given by:

0 0

1d M

τμ γ

= (17)

where μ0 is the magnetic permeability of a vacuum, and M0 is the equilibrium magnetization. The signal from the pulse sequence represented in Fig. 5 is given by Eq. (18) that includes the DDF in its local form [42] assuming that (see Fig. 5) t1 << T1.

(2 )1 2

22 2 211

122 ( 2 )2 ( )0 31 2

2

1

(3cos 1)( , ) sin2 (1 )

14 2

t t

T Dk tDk t T

d

M eM t t e eDk

T

δθβ

τ

− +

− +− −+ −= × ×

+− (18)

where θ is the angle between G and BB0, and k is the wave vector of spatial modulation of the magnetization caused by the magnetic field gradient; it is equal to γgG. A log-normal plot of the signal amplitude versus t2 allows the extraction of estimates of both D and T2 as follows:

61

(i) if 22

1

112

tDk

T+

, T2 is estimated from the slope of the function that is

fitted to the data.

(ii) if 2

1

12DkT

, D is obtained from the data by considering the value of T1 to

be effectively infinite and fitting the resulting expression.

One-dimensional DOSY As noted above, a well-known data representation from PGSE NMR is called DOSY

[7,9]; the experiment and subsequent data analysis generate a 2-dimensional contour map with the estimates of D along one axis and chemical shift along the other. In the realm of experiments used to rapidly measure D the name of a new method is derived from the fact that information on both diffusion and chemical shift are recorded using a single-transient; hence it is called one-dimensional DOSY [43,44].

The principle of the method is that information on diffusion is encoded in the lineshape of each resonance. To achieve this outcome there must be a spatial dependence of the chemical shift, and diffusion will broaden peaks during signal acquisition. There are two reported methods based on this principle that record data in a single transient, and from which the data are then analyzed to yield an estimate of D. The two methods differ in the way they create the spatial dependence of chemical shift. In the first method [43], this dependence is created by using a non-uniform magnetic field gradient that is generated by an additional current in the z2 shim coils, after a hardware modification that enables its switching on and off during the pulse sequence.

In the second method [44], spatial encoding is generated using an adiabatic frequency-swept 180° RF pulse that is applied while the diffusion-detecting magnetic field gradients are applied. For both pulse sequences, the peak broadening that is invoked during signal acquisition is brought about with a weak linear ‘read’ gradient. But because the second method does not require hardware modification it is easier to implement than the first one. Therefore henceforth we focus on this pulse sequence (Fig. 6).

Fig. 6: One-dimensional DOSY pulse sequence [44]. Two negative gradient pulses are added to reduce eddy currents. tp is the adiabatic pulse duration, Gd and Gr are, respectively, the intensity of the diffusion and read gradients, and Δ is the diffusion time.

62

The diffusion gradients Gd are chosen to bring about different chemical shifts (off-set frequencies) as a function of spatial position. During the first of the magnetic field gradient pulses an adiabatic RF pulse of duration tp, is swept through a range of off-set frequencies. The spins are nutated through the x’,y’-plane by this pulse at different times during the application of the magnetic field gradient; thus they are exposed to diffusion sensing over different times depending on their spatial location in the sample. A schematic representation of this effect is shown in Fig. 7. For spins located at one end of the sample, the adiabatic RF 180° pulse exerts its effect immediately and their magnetization is dephased by the gradient field for a time tp. On the other hand, magnetization in the middle of the sample is dephased during tp/2, then the adiabatic swept-frequency RF 180° pulse is applied; finally the magnetization is dephased by the gradient pulse during the last tp/2 period. This last period leads to refocusing of the spin magnetization vectors. The combination of gradient and sweep pulse allows the acquisition of a net signal that depends on the spatial position of the spins. To record the signal attenuation along the sample, signal acquisition occurs in the presence of a read-gradient Gr. Therefore spectral peaks are obtained which are diffusion-weighted images of the sample, but the weighting varies along the sample.

Fig. 7: One-dimensional DOSY. Effect of the first 90o hard pulse and then the adiabatic pulse and diffusion gradient, as a function of the position of spins (spin isochromats) in the sample. The time scale is that of the adiabatic RF pulse of duration tp.

To analyze the data, the net phase acquired by spin isochromats at the end of the

adiabatic pulse needs to be expressed as a function of the experimental parameters and D. Hence, if a spin at position z experiences RF-induced 180o flipping at time α(z)tp, where 1 ≥ α(z) ≥ 0, the following relationship holds:

63

[ ] [ ]( ) 1 ( ) 1 2 ( )d p d p dG z z t G z z t z G ztγ α γ α α γ− − = − − p (19)

During the frequency-sweep pulse, the spin isochromats precess as if they were under

an effective gradient of strength Geff:

[ ]( ) 1 2 ( )eff dG z z Gα= − (20)

In this simplified analysis it is assumed that the spin isochromats experience a 180° nutation at the instant the sweep is on-resonance. A more exact approach is described in the Appendix of the original paper [44].

After calculating the effective magnetic field gradient associated with each off-set frequency (chemical shift position, hence z) a predicted lineshape is calculated for a value of Dfit using a slightly modified form of Eq. (1):

2 2 ( )0( ) eff fitA z DS z S e γ− Δ= (21)

where S0 is a scaling factor, and Aeff corresponds to Geff(z)tp.

4. Limitations of fast-recording methods General

None of the methods described above have been developed to estimate two different values of D from the one spectral peak. The systems that have been studied have been well defined ones, and applications to more complex heterogeneous samples are awaited. In the particular case of biological samples where the temperature is relatively high convection can introduce a systematic overestimate of the value of D, and yet none of the fast-diffusion methods are convection compensated [3,45]. In addition, the pulse sequences have not yet been used with systems that have restricted diffusion, such as water in red blood cells [46].

Another general limitation is the available signal-to-noise (S/N) ratio in the NMR spectra. The gain in time efficiency is off-set by a lower S/N ratio; only in systems in which the detection is facile, i.e., a high solute concentration, can the experiments be used.

The particular fast-diffusion methods that are based on the recording of spin-echoes [17-20,24,27,36] also have the same drawbacks related to S/N. To avoid the overlap of spin echoes, the intensity of each subsequent echo must be significantly less than the previous one. The second drawback is the loss of chemical shift information in these latter methods, making them inappropriate for studies on mixtures of solutes [18-20].

Limitations of experiments with trains of gradients

A common observation with respect to the estimates of Ds obtained with fast-diffusion experiments is that they are high relative to those obtained with standard PGSE methods. The coefficient of variation on the estimates of D made from a peak that is

64

broad in the diffusion dimension of a 2-dimensional (DOSY) map, is routinely higher from a fast-diffusion pulse sequence. In the method of Morris et al. [16] errors arise from the presence of a small extra amount of signal that survives the pulse train due to an insufficient absolute difference in gradient area between the two last gradient pulses. This undesired residual signal increases the signal intensities above what is expected at the higher magnitudes of the field gradient pulses. This results in a smaller estimate of the apparent D. On the other hand, for the pulse sequences that use a CPMG RF-pulse train [17,18], an imperfect 180° refocusing pulse gives an error in the signal intensity that manifests itself as a faster decay and hence an artifactually high estimate of D. For both of the CPMG-based methods, obviously the total acquisition time must be shorter than the T2 of the nuclei of interest. Unfortunately, in most cases, this condition can not be satisfied. The method of Van Gelderen et al. [19] allows the capture of chemical shift information when a large number of points are sampled in each gradient echo but the subsequent data processing is long and complex: both the chemical shift (off-set frequency) and gradient-induced evolution evolve in the opposite way. And, the data set from each echo must be processed separately. In the CPMG methods the magnetic field gradient pulses must be switched on and off very rapidly, thus promoting eddy currents in conducting parts of the NMR probe. These give rise to distortion of spectral line shapes and lead to artifacts in the estimates of D. The Difftrain pulse sequence has two drawbacks: The first is the requirement that the sample has long relaxation times, and there must be plenty of samples with a high S/N. The second, is the need to acquire two transients to estimate D: the first transient is recorded without the magnetic field gradient on to record the signal decay due to T1; while the second one is recorded with gradients to measure both diffusion and relaxation effects.

Limited gradient pulse methods

The estimates of D obtained with the one-dimensional DOSY methods are often lower than those estimated with the more conventional PGSE methods. In performing the experiments the spectroscopist should have this consideration in mind. In order to have a coefficient of variation in the estimate of D of ~1.5%, the S/N ratio should be greater than ~100. Because the CPMG-diffusion methods are based on analyzing diffusion-broadened spectral peaks, the likelihood of peak-overlap is increased; therefore fitting of the requisite lineshape function to the data can be problematical. To apply the methods to estimate the D of solutes in mixtures requires well-resolved peaks as a precondition.

For both the MMME and Burst pulse sequences, a long period is used for the evolution of the magnetization of the system, and this occurs in the presence of magnetic field gradients. This is especially true of MMME (Fig. 4) in which the gradients are on for the full duration of the pulse sequence. Even if the gradient magnitudes are only a few gauss per centimeter, the total experimental time is critical. The manufacturers recommend using gradient pulses that are on for less than 10 ms at the maximum current. If we consider a linear relationship between current and g, the maximum experimental time obtained by using only 1% of the maximum current is 1 second. Hence, this duration determines the maximum duration of the MMME pulse sequence.

65

The final inconvenience lies in the restricted domain of parameter values in which the equation that is used for estimating D from the data remains valid. For the Fast-CRAZED method, Eq. (18) is valid only in regimes of highly diffusion-attenuated signals. This is generally found in systems that contain small molecules in solution or in the gas phase. Furthermore, the number of transients that are needed to retrieve the parameters to fit with Eq. (18) increases with the number of components. Thus it appears that fast-diffusion experiments have only been applied, at most, with binary mixtures with each component being ‘non dilute’. And with the Burst pulse sequence the total DANTE angle is made less than 30° to improve the veracity of the estimate of D. With the latter, and indeed all the pulse sequences discussed here, a significant amount of effort must be expended on trial experiments that are used to optimize performance on each particular NMR spectrometer.

5. Conclusions A knowledge of the diffusion coefficients of solvent and solutes in mixtures of

various origins is valuable for predicting the physical and chemically-reactive properties of the system. If systems of interest evolve rapidly on the time scale of conventional PGSE NMR diffusion experiments then the pulse sequences described here might be of value.

There are principally two different classes of NMR fast-diffusion experiments. The first records the signal as a function of the magnitude of the magnetic field gradients that are used in the experiment. To speed up signal acquisition various pulse sequences have been designed to minimize the extent of phase cycling; or to record a series of spin-echo signals from a range of magnitudes of the magnetic field gradients.

The second class of experiments uses only one or two magnetic field gradient magnitudes. To estimate D, spatial information is encoded by using a different approach from that commonly used in, say, the Stejskal-Tanner experiment [6]. Each method is based on a different physical phenomenon. The first uses a DANTE RF-pulse train, while the second pulse sequence achieves in one transient the sampling of many different magnetization-coherence pathways. The third method uses a distant dipolar field (DDF) and both adiabatic and gradient pulses are combined to give spectra from which D is estimated. Each of the methods described presents its own limitations. None of the pulse sequences have been used to study restricted diffusion for which long diffusion times are usually needed. Systems in which there is convection pose another problem, as none of the methods are convection compensated. Some of the methods record a train of spin-echoes, and chemical shift information is lost or is barely accessible, thus limiting the application of the methods to only simple mixtures of solutes.

Overall the one-dimensional DOSY method appears to be the method of choice for the (biological) systems which we study [3]. Currently, this method is being developed in our laboratory to study rapidly evolving systems such as the morphological changes in the human red blood cell, its membrane flickering, and the variation of D of guest molecules inside an extended support such as gelatin that undergoes changes in length [47,48].

66

Acknowledgements The work was supported by a Discovery Grant from the Australian Research Council

to PWK.

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68

Challenges in Macroscopic Measurement of Diffusion in Zeolites

Stefano Brandani

University of Edinburgh, School of Engineering and Electronics, UK Corresponding author: Stefano Brandani IMP-SEE University of Edinburgh The King’s Buildings Edinburgh EH9 3JL UK E-Mail: s.brandani@ed.ac.uk

Abstract The paper presents a brief historical overview of the field of the measurement of

diffusion in zeolites. The focus will be on macroscopic measurements, i.e. when diffusion through entire crystals is studied. While this may appear to be a simple task, we will see that there are difficulties in obtaining reliable results. Two case studies, the volumetric/piezometric experiment and the Transient Analysis of Products (TAP) apparatus are discussed and ways to improve either the experiments or the way in which the results are analyzed are presented. The author’s perspective into what are the current challenges in the field concludes the paper.

Keywords: diffusion measurements; zeolites; adsorption.

1. Introduction Zeolites are both naturally occurring and synthetic crystals that have very well

characterized nanopores [1]. This feature makes these materials extremely useful in several applications in catalysis and adsorption separation processes. These applications are controlled by both adsorption equilibrium and mass transfer kinetics and as a result many different experimental techniques have been developed to measure these quantities.

Measurement of diffusion in zeolites can be divided into two differing approaches: microscopic studies, which typically measure the movement of molecules and relate the root mean square displacement over time to the diffusion coefficient using Einstein’s equation, and macroscopic studies, which measure the mass transfer between the gas and the solid adsorbent over the entire crystal and solve Fick’s law of diffusion to obtain the diffusional time constant.

The fact that the crystal structure of zeolites is apparently well characterized makes them a suitable candidate for the development of theories of diffusion in confined geometries and thus of great importance also for a purely fundamental study.

Figure 1 shows a condensed historical overview of the development of the field. While the specific techniques are very interesting on their own merits, what is important

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, S. Brandani 69

to note is the fact that, apart from the pioneering work of Tiselius [2], who observed by microscopy the adsorption front of water in a very large crystal of natural heulandite, for a considerable period of time uptake rate experiments, introduced originally by Barrer [3], were considered to be the means of carrying out these measurements. To follow the mass of the adsorbent as it changes in time, in the presence of adsorbate molecules, seemed to be the obvious way to measure diffusivities in zeolites. This can be achieved either through a direct measurement of the mass (gravimetric methods) or the measurement of gas pressure in a closed system (volumetric/piezometric methods).

Year1930 1940 1950 1960 1970 1980 1990 2000

Year1930 1940 1950 1960 1970 1980 1990 2000

Non-equilibriumEquilibrium

DirectVisualObservatTiselius(1934)

TransientUptakeBarrer (1938)

TracerExchangeBarrer(1941)

DirectVisualObservatTiselius(1934)

TransientUptakeBarrer (1938)

TracerExchangeBarrer(1941)

NMRRelaxationResing, Pfeifer, Michel(1967)

PFG-NMRPfeifer, Kärger(1971)

NMRRelaxationResing, Pfeifer, Michel(1967)

PFG-NMRPfeifer, Kärger(1971)

TFRGrenier,Meunier (1998)

Chromatography Haynes,Ruthven(1973)

ZLCEic, Ruthven(1988)

CoherentQENSJobic (1999)

Effectiveness Factor. Haag, Post(1981)

FRYasudaRees (1982)

TAPNijhuis et al., Baerns Keipert(1997)

MembranePermeationHayhurst,Weernick(1983)

FTIRKarge(1991)

PEPvan Santen(2000)

IRMicroscopyKarge(1974)

IFMKärger,Schemmert(1999)

IncoherentQENSCohen deLara, Jobic(1983)

Tracer ZLCHufton, Brandani,Ruthven(1994)

Exchange NMRChmelka(1998)

TFRGrenier,Meunier (1998)

Chromatography Haynes,Ruthven(1973)

ZLCEic, Ruthven(1988)

CoherentQENSJobic (1999)

Effectiveness Factor. Haag, Post(1981)

FRYasudaRees (1982)

TAPNijhuis et al., Baerns Keipert(1997)

MembranePermeationHayhurst,Weernick(1983)

FTIRKarge(1991)

PEPvan Santen(2000)

IRMicroscopyKarge(1974)

IFMKärger,Schemmert(1999)

IncoherentQENSCohen deLara, Jobic(1983)

Tracer ZLCHufton, Brandani,Ruthven(1994)

Exchange NMRChmelka(1998)

…

Fig. 1: Development of the measurement of diffusion in zeolites [2-20]. Adapted from [21].

Figure 1 clearly shows a “dramatic event” at the end of the 1960s, early ‘70s, with the

introduction of the first NMR measurements [6, 7]. With the advent of the PFG-NMR technique, pioneered by Pfeifer and Kärger, a number of inconsistencies in the results reported in the literature came to light. This led to a re-examination of many “fast diffusing” systems for which gravimetric measurements where affected by spurious kinetic resistances. To overcome these limitations many new techniques were proposed. In many cases, the earlier results were shown to be incorrect and by the ’80s and early ‘90s most researchers in the field came to the conclusion that only microscopically determined diffusivities were reliable and that macroscopic methods, when not in agreement with these measurements, reflected the kinetic limitations due to heat transfer, external film, bed effects and surface barriers. For a detailed discussion one should refer to the monograph [22].

70

The story does not end here. Figure 2 shows results published only in the last 15 years for the system n-hexane/silicalite. These measurements were carried out by groups who knew of the earlier difficulties and ensured that their respective systems were not affected. For particular adsorbents, in this case silicalite, it is apparent that the “measured” diffusivity is dependent on the length-scale and time of the observation. The current general consensus is that all these measurements are in fact correct, within their respective uncertainties, and the results reflect the fact that large silicalite crystals have internal imperfections that dominate the macroscopic measurements. The mass transfer constant measured by macroscopic techniques differs by up to 3 orders of magnitude from what it would be in an ideal crystal, based on the results from neutron scattering and PFG-NMR which will be discussed by Hervè Jobic in a separate paper in this book.

n-Hexane in Silicalite after 1989

1.E-14

1.E-13

1.E-12

1.E-11

1.E-10

1.E-09

1.E-08

1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5

1000/T (K)

D (m

2 /s)

MBRN-PRM TEOscMbal QENS SingCrysMemPiezo FR PFG-NMR ChromatGrav-Uptk (P.Voogd) ZLC Grav-Uptk (SU.Kulkarni) Grav-Uptk (J.Caro et al.)Grav-Uptk (M.Jama & Ruthven)

3 orders of magnitude!3 orders of magnitude!

Fig. 2: Recent diffusion in zeolites data reported in the literature [23-31].

The interesting conclusion that can be derived from all this is that, to date, we do not

have a valid theory that will be able to predict a-priori the mass transfer kinetics in non-ideal zeolite crystals. As a consequence it will be necessary to carry out macroscopic measurements for the foreseeable future.

This field has relatively few groups who specialize in macroscopic diffusion measurements in nanoporous materials. The majority of the data that is being published is mainly from those who investigate mass transfer kinetics as part of a wider project, who sometimes do not fully understand the underlying difficulties in making such measurements. The problem is also that there are no commercially available instruments designed for this purpose, so researchers often modify or use equipment designed for equilibrium purposes or have to develop their own systems. One of the real challenges is therefore that of providing simple guidelines that can be used to identify when the experiments are likely to be successful. We will discuss this in detail with the aid of two case studies.

2. Volumetric/Piezometric method: a closed system. We begin the discussion with a very simple system, which is shown schematically in

Figure 3. This is the volumetric or more accurately the piezometric (pressure is

71

measured) experiment. This system was developed for equilibrium measurements. For this purpose the dosing and uptake (where the zeolite is placed) cells are connected to vacuum and the valve is then closed. The dosing volume is charged to a known pressure, i.e. a finite known amount of gas is introduced into the system. By opening the valve and measuring the final pressure, one can calculate the amount of gas adsorbed. For this measurement one needs only one pressure transducer in the dosing volume.

In principle this system can be used also for kinetic measurements, if the pressure vs time curve is recorded. Consider the following example:

1) Measurements are carried out without the zeolite present to investigate the flow characteristics of the valve.

2) Measurements are carried out with the zeolite present. If a significant difference is observed from the previous experiments, one concludes that mass transfer kinetics is being measured.

An example of this is shown in Figure 4.

Fig. 3: Volumetric/Piezometric system.

Fig. 4: Qualitative difference between experiments with and without adsorbent present in the uptake cell.

Is there something obviously wrong with proposition 1 and 2? Not at first sight. An experimentalist would comment that flow through a valve is non-linear and therefore proposition 1 would establish the characteristics of the valve in a region that is not representative of the adsorption experiments. When adsorption is taking place the gas flowing through the valve can be much larger than that in the non adsorbing case, if fast diffusing and strongly adsorbed components are being investigated. The second issue is that adsorption is a phase change phenomenon and heat is being generated, so one should consider also the effect of heat transfer resistances. If we limit the discussion to single component systems, one has to be careful also with the way the solid is arranged as this will change the heat transfer characteristics. Finally if the pressure step is relatively large, the equilibrium isotherm is not necessarily linear, introducing additional

complications in the analysis of the results.

PP

Dosing cell Uptake cell

T controlT control

0

0.2

0.4

0.6

0.8

1

0.00 0.01 0.02 0.03 0.04 0.05Time

Dos

ing

volu

me

pres

sure

Equilibrium

We can conclude that an accurate measurement is more complicated than originally thought. The student with a passion for modeling may suggest to include these effects in the model equations and to determine additional parameters from the experiments. After all, we observe a difference between the adsorbing and non-adsorbing case, so we should be able to measure kinetics.

Empty

With adsorbent What we have missed, so far, by focusing on the experimental details is the more general question: can we use the comparison with a non adsorbing system to confirm that mass transfer kinetics is being measured?

72

The answer to this question is no! This appears to be an obvious conclusion to very few researchers. I will try to justify this answer by looking at Figure 3 and proposing a thought experiment: replace the valve with a very small capillary tube so that the flow between the dosing and uptake volume is slow. The pressure is changing, but the change is so slow that the adsorbed phase equilibrates with the gas immediately. In this case the experiment with and without the adsorbent will be quite different, but the dynamics can be described entirely using only the equilibrium properties and the flow characteristics of the capillary. By looking at Figure 3 we can therefore identify two characteristic time constants, one linked to the valve conductance and the other due to the diffusion in the zeolite. It is the ratio of these time constants that dictates in which regime we are and if we can measure diffusion. Figure 4 also shows the limiting case of equilibrium control where the adsorbed phase is always at equilibrium with the gas (see section 4).

Fig. 5: Pressure transient in the uptake cell.

Qualitatively the equilibrium control model has a shape that is very similar to that of the full curve, so how can we be sure that we are indeed measuring diffusion? The answer to this question can be the use of accurate models coupled to dynamic sensitivity analysis to distinguish between them, but this solution may lead to cases where one is not certain of the result. Is there an unequivocal way to prove directly from the experiments that mass transfer kinetics is being measured? Again look at Figure 3. If we add a second pressure transducer and measure the pressure in the uptake cell we now have unequivocal

experimental proof of what we are measuring. This is because, while the pressure in the dosing cell, whichever the controlling mechanism, will be a monotonically decreasing function in time, the pressure in the uptake volume should initially increase, go through a maximum and then decrease to the final equilibrium value if mass transfer or heat transfer limits the dynamics of this system (see Figure 5). The pressure in the uptake cell will be increasing monotonically if the adsorbed phase is always at equilibrium with the gas. Therefore, it is possible to distinguish between these two cases from a direct observation of the shape of the pressure vs time curve of the uptake cell.

0.0

0.1

0.2

0.3

0.4

0.5

0.00 0.01 0.02 0.03 0.04 0.05Time

Upt

ake

cell

pres

sure

Equilibrium

With adsorbent

Empty

The second pressure measurement offers an additional advantage. It allows the calculation of the amount of gas present in the adsorbed phase at any time, i.e. the average adsorbed amounts are known. The knowledge of the average solid concentration and the boundary condition on the solid, given by the measured pressure in the uptake cell, allows the interpretation of the uptake data, without the need for an empirical valve model.

Based on this discussion we have shown that it is possible to establish experimentally if diffusion is being measured and that a modification to an equilibrium apparatus can lead to a much improved system as well as ease in the analysis of the results. For a detailed discussion of this system, one should refer to [32], which reports also an analysis of the data for the system benzene-NaX measured using this technique [33]. One still

73

finds in the recent literature data measured using a volumetric/piezometric system with very fast diffusion and only the pressure on the dosing cell being recorded [34].

3. Transient Analysis of Products (TAP) method: a flow system. The TAP system is another apparatus that has been used to measure diffusivities. A TAP bed consists of a tube in which the solids are loaded. The column is then exposed to a high vacuum chamber at the outlet. Once the system is equilibrated, a very small pulse of gas is introduced at the inlet of the column. The flux coming out of the adsorbent bed is measured with time and diffusion in the solid is thus calculated. A schematic diagram of a TAP bed is shown in Figure 6.

Fig. 6: TAP system.

The fact that the entire system is under vacuum ensures that the experiment is carried out in the region where the adsorption isotherm is linear. The system can be assumed to be isothermal because as the pulse is traveling along the column an approximately equal amount of molecules are being adsorbed and desorbed. One should also consider that there is a very large heat capacity of the bed and metal components in comparison with the heat generated by the relatively small amount of adsorbate molecules. Finally the gas is introduced pure and there are no external mass transfer limitations due to the presence of a second carrier gas, i.e. diffusion through an external film. This seems to be an ideal system to measure the diffusivity at low concentrations. It is not surprising that with this type of system Nijhuis et al. [16] appear to be the only researchers to date who claim to have measured macroscopically diffusivities in large silicalite crystals in agreement with PFG-NMR results.

Fig. 7: Qualitative results with/without adsorption.

Nijuis adopted the following approach: 1) The column was loaded with

silicalite that had the template molecule needed for the sythesis still present. In this case there are no empty nanopores. Experiments were carried out on these crystals, thus showing the response without adsorption.

2) The column was exposed to air at high temperature and the template was burned off, thus opening the nanopores. Experiments were repeated under these conditions and a large difference was observed.

Nijhuis used a detailed model to interpret his results and claimed that, under the conditions in which he carried out his experiments, statistical analysis of his parameter fitting procedure indicated that he obtained reliable results.

z-axisHigh

vacuum to Mass Spec

Inlet pulse Bed of adsorbent particles

0

1

2

3

4

5

6

7

8

9

10

0.00E+00 2.00E-01 4.00E-01 6.00E-01 8.00E-01 1.00E+00 1.20E+00 1.40E+00 1.60E+00 1.80E+00 2.00E+00

Time

Sign

al

Not adsorbing

Adsorbing higher temperature

Adsorbing lower temperature

74

This second example gives the opportunity to explain why I believe that one of the challenges in this field is that of determining directly from the experiments if diffusion is being measured. I will explain this without the use of any mathematical model, but observe that:

1) This system, neglecting any other complicating detail such as premixing and postmixing, has at least two time constants: one linked to diffusion in the zeolites; the second due to Knudsen diffusion in the bed.

2) If we are able to normalize the experimental response using only one time constant, then the system must be controlled by Knudsen transport which is always present.

Nijhuis et al. [16] presented complete experimental curves at 527 K and 623 K for the system n-butane/silicalite-1. They reported the signal as measured by the mass spectrometer and the procedure that one should carry out on their data is as follows:

i. Calculate the integral of the signal as a function of time and divide the signal by this amount. An advantage of this is that now the curves do not depend on the actual amount injected into the system.

ii. Divide the time by a characteristic time, such as the time corresponding to the maximum of the curve, or any constant time.

iii. Multiply the signal/integral by this characteristic time, so that the area under the curve is 1 and the plotted curve is dimensionless.

If different curves are made dimensionless in both measured quantity and time in this way and it is not possible to distinguish between them, then only one time constant can be extracted from the experiments. This has to be that related to Knudsen flow as explained above. Figure 8 shows the result of the procedure outlined above when applied to the data of Nijhuis et al. [16].

Fig. 8: Normalized plot of experimental data.

The procedure is extremely simple and can unambiguously prove that the claims of Nijhuis are not correct. Using a very similar apparatus, Keipert and Baerns [35] also studied the same systems, but concluded that they were not able to

measure the fast diffusivities. In fact the TAP can be used only in a very small window of conditions, because if the diffusivity in the solid is too low, the adsorbate will exit the column without interacting with the solid, while if the diffusivity is too fast the solid and gas will be at equilibrium.

0.0

0.5

1.0

1.5

2.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0Dimensionless time

Dim

ensi

onle

ss fl

ux

623 K - 1.08

527 K - 0.48

Theoretical - Equilibrium Control

4. Equilibrium control. What should be apparent by now is that for closed systems there is a need to test the dynamic response of the apparatus in the absence of adsorption, to establish an order of magnitude for its response time. In the case of a flow system, one needs to test the non-

75

adsorbing case to be sure that a significant portion of molecules enters the adsorbent during the experiment. These checks are usually carried out by experimentalists. For both systems it is also true that there is a second limit that has to be considered but does not appear to be as obvious. We term this limit equilibrium control. To understand what is meant we return to the data of Nijhuis and ask: what did he measure? Nijhuis argues that 22 RDLDK >>τ , where DK is the Knudsen diffusion in the column, L is the length of the column, τ is the tortuosity in the packed bed (typically 2 to 3), D is the diffusivity in the zeolite and R is the equivalent radius of the crystals. So how can the Knudsen flow be slow compared to the diffusion in the zeolite? The problem in this argument is that we should be comparing the flux in the gas, which is proportional to 2LDK τ , with the flux in the zeolite, which is proportional to 2RHD , where H is the equilibrium Henry constant. In zeolites this is often very large. The best way to explain this is that in the equilibrium control limit the model of the TAP column reduces to Fick’s equation. In the TAP bed we can write the mass balance in the gas phase as

012

2

=∂∂

−∂∂−

+∂∂

zcD

tq

tc K

τεε (1)

where the three terms in the equation are the accumulation in the fluid, the accumulation in the solid and the dispersion due to diffusion. The mass balance in the solid phase is

02 =∇−∂∂ qD

tq (2)

where q is the average adsorbed phase concentration. These two equations are coupled with the relevant boundary conditions. If we assume that diffusion in the solid is the controlling mass transfer resistance, then the concentration at the solid surface will be at equilibrium with the gas: . HCqS =This general model has a limit corresponding to the non adsorbing case if one takes either D = 0 or H = 0. The second limit is that of equilibrium control, which is obtained when D = ∞. In this second limit, there is no internal gradient in the solid particle (otherwise there would be an infinite flux), therefore the average concentration is equal to the concentration at the surface and

012

2

=∂∂

−∂∂−

+∂∂

zcD

tcH

tc K

τεε (3)

This can be rewritten in the form of the diffusion equation

01 2

2

=∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−∂∂

zc

HD

tc K

εε

τ (4)

Therefore Nijhuis in his experiments measured an effective Knudsen diffusivity (the term in parenthesis), which is only a function of the equilibrium constant!

76

5. Conclusion I have tried to present very simple examples to show that the macroscopic

measurement of diffusion in zeolites is very challenging. I hope that the general approach of comparing kinetic experiments to the equilibrium control limit may be applicable to other fields where diffusion is measured. In general, as shown in both case studies, it should be possible to devise experiments where one can have direct evidence that the conditions are far from the equilibrium control limit. Since equilibrium can be measured through independent experiments or from the final result in closed systems, even if experimental validation cannot be achieved, one should always compare the dynamic response to that which would be predicted under equilibrium control. Figure 8 includes the curve which corresponds to this limit, showing that the simple model can describe the experimental results. Only if a clear difference in the predicted equilibrium control curve and the measured dynamics can be observed, one can conclude that mass or heat transfer kinetics are quantifiable.

There are no commercially available apparatuses that can be used for the measurement of diffusion in nanoporous materials. Even if these become available, it is unlikely that a single system will be able to provide accurate results for the wide range of time constants that may be encountered in all the materials that need to be tested. This is especially true if one considers also the extension to “formed” materials, such as pellets and monoliths, in which diffusion in the macropores through molecular, Knudsen or viscous flow may be relevant.

In order to have reliable equipment that can be used by the non-expert, the main challenges that will have to be resolved are:

a) The need for experimental protocols that unequivocally confirm the validity of the results.

b) A systematic approach to establish the experimental conditions in which physical parameters may be measured in different apparatuses.

c) The development of automated systems, which are particularly important in industrial applications.

d) The development of software tools that allow the analysis of multiple experiments to extract the diffusion coefficient, combined with dynamic sensitivity analysis.

References [1] Ch. Baerlocher, W.M. Meier and D.H. Olson, Atlas of Zeolite Framework Types,

5th revised edition, Elsevier, Amsterdam, 2001. [2] A.W.K. Tiselius, Z. Phys. Chem. A169 (1934) 425-458 . [3] R.M. Barrer Proc. Roy. Soc. London A167 (1938) 392-420. [4] R.M. Barrer, Trans. Faraday Soc. 37 (1941) 590-599. [5] H.A. Resing, J.H. Thompson, J. Phys. Chem. 46 (1967) 2876-2880. [6] H. Pfeifer, in NMR – Basic Principles and Progress 7, Springer, Berlin, 1972,

pp.53-153. [7] H.W. Haynes, P.N. Sarma Adv. Chem. 133 (1974) 205-217.

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[8] H.G. Karge, K. Klose, Ber. Bunsen. Ges. Phys. Chem. 78 (1974) 1263. [9] W.O. Haag, R.M. Lago, P.B. Weisz, Faraday Discuss. Chem. Soc. 72 (1981) 317-

330. [10] Y. Yasuda, J. Phys. Chem. 86 (1982), 1913-1917. [11] A. Paravar, D.T. Hayhurst, in Proc. 6th Int. Zeolite Conf., Reno 1983, D. Olson, A.

Bisio Eds, Butterworth, Guildford, 1984, pp. 217-224. [12] E. Cohen de Lara, R. Kahn, F. Mezei, J. Chem. Soc. Faraday Trans. 1, 79 (1983)

1911-1920. [13] M. Eic, D.M. Ruthven, Zeolites, 8 (1988) 40-45. [14] S. Brandani, J.R. Hufton, D.M. Ruthven, Zeolites, 15 (1995) 624-631. [15] V. Bourdin, Ph. Grenier, F. Meunier, L.M. Sun, AIChE J. 42 (1996) 700-712. [16] T. A. Nijhuis, L. J. P. van den Broeke, J. M. van de Graaf, F. Kapteijn, M. Makkee,

J. A. Moulijn, Chem. Eng. Sci. 52 (1997) 3401–3404. [17] D. J. Schaefer, D. E. Favre, M. Wilhelm, S. J. Weigel, B. F. Chmelka, J. Am. Chem.

Soc. 119 (1997) 9252-9267. [18] U. Schemmert, J. Kärger, J. Weitkamp, Microporous and Mesoporous Materials 32

(1999) 101-110. [19] H. Jobic, J. Kärger, M. Bée, Phys. ReV. Lett. 82 (1999) 4260-4263. [20] R.R. Schumacher, B.G. Anderson, N.J. Noordhoek, F.J.M.M. de Gauw, A.M. de

Jong, M.J.A. de Voigt, R.A. van Santen, Microporous and Mesoporous Materials 35-6 (2000) 315-326.

[21] D.M. Ruthven, in Fluid Transport in Nanoporous Materials, W.C. Conner, J. Fraissard Eds, Springer, Dordrecht, 2006, pp. 151-186.

[22] J. Kärger, D.M. Ruthven, Diffusion in Zeolites and Other Microporous Solids, John Wiley, New York, 1992.

[23] P. Voogd, H. van Bekkum, D. Shavit, H.W. Kouwenhoven J. Chem. Soc. Faraday Trans. 87 (1991) 3575-3580.

[24] W. Heink, J. Kärger, H. Pfeifer, K.P. Datema, A.K. Nowak J. Chem. Soc. Faraday Trans. 88 (1992) 3505-3509.

[25] J. Caro, M. Noack, J. Richtermendau, F. Marlow, D. Peterson, M. Griepentrog, J. Kornatowski, J. Phys. Chem. 97 (1993) 13685-13690.

[26] A. Micke, M. Bülow, M. Kocirik, P. Struve, J. Phys. Chem. 98 (1994) 12337-12344 [27] M.A. Jama, M.P.F. Delmas, D.M. Ruthven, Zeolites 18 (1997) 200-204. [28] L.J. Song, L.V.C. Rees J. Chem. Soc. Faraday Trans. 93 (1997) 649-657. [29] O. Talu, M.S. Sun, D.B. Shah, AIChE J. 44 (1998) 681-694. [30] H. Jobic, Journal of Molecular Catalysis A: Chemical, 158 (2000)135-142. [31] W. Zhu, F. Kapteijn, J.A. Moulijn, Microporous and Mesoporous Materials 47

(2001) 157-171. [32] S. Brandani Adsorption 4 (1998) 17-24. [33] M. Bülow, W. Mietk, P. Struve, P. Lorenz J. Chem. Soc. Faraday Trans. I, 79

(1983) 2457-2466. [34] R.S. Todd, P.A. Webley, R.D. Whitley, M. J. Labuda, Adsorption 11 (2005) 427-

432. [35] O.P. Keipert, M. Baerns, Chem. Eng. Sci. 53 (1998) 3623-3634.

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A Web Site Dedicated to Materials Science Education, Specially Diffusion

Daniel Monceau, Jean Philibert

Former professor Université Paris-Sud, Orsay France

jean.philibert@u-psud.fr This site, created a few years ago, following a round table organised by the international conference DIMAT 2000, has just been completely revamped. Its purpose is to gather information, references, books, visuals, multimedia documents, numerical simulations, softwares, links to other websites, in order to help teachers and students alike to find, share, exchange,… resources in Materials Science. It is (and will be) a source of documents and valuable sites links to complement their lectures and courses as well as a collections of exercises to illustrate their courses. Fifteen scientific domains should be covered from – in alphabetic order – Corrosion to Thermodynamics, plus a chapter on distant learning. To-day the chapters “Diffusion“ and “Crystallography” are the most extensive. Some others are still under development. All data served on this site are carefully checked by an Editorial Committee for the relevant scientific domain, who select them for their pertinence and quality. Any one interested can submit some content to be placed on this site. Their proposal will follow the usual procedure of refereeing under the control of the relevant Editorial Committee. All languages are acceptable as far as a member of the Editorial Committee is able to read and review it. The Chapter “Diffusion” covers many topics from Brownian motion to applications on structural transformations. Up to now, most of its contents is of “classical” relevance. It could be enriched with less conventional aspects, such as several of those discussed in the present Conference. The editors should appreciate receiving such proposals from speakers/authors of “Diffusion Fundamentals” in order to help teachers to enrich their courses with documents, examples, softwares, animations, … on topics and applications which are not found in usual text-books and could be very attractive to students. For more information, please visit: http://e-materials.ensiacet.fr

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, J. Philibert 79

Methodical Aspects of 2D NMR Correlation Spectroscopy under Conditions of Ultra High Pulsed Field Gradients

Marcel Gratz, Petrik Galvosas

Universität Leipzig, Fakultät für Physik und Geowissenschaften, Linnéstr. 5, 04103 Leipzig, Germany, E-Mail: m.gratz@physik.uni-leipzig.de

1. Introduction Multidimensional NMR methods have become an important tool in the 1970s. In the

past years a new set of experiments has developed whose postprocessing is based on a multidimensional inverse Laplace transformation rather than a Fourier transformation which provides the capabilities to find correlations between various parameters such as diffusion coefficients D and NMR relaxation times T1 and T2 [1,2].

The aim of the present work is to implement these techniques to our spectrometers in order to combine the concept of ultra high pulsed field gradients of up to 35 T/m [3] with modern 2D NMR correlation experiments.

2. Experimental The main idea of 2D NMR experiments is to execute two suitable pulse sequences in

succession, thus, the signal of the second subsequence depends on the NMR signal of the first one. This signal dependency enables one to design NMR experiments, which reveal the correlation between measured parameters such as T1 and T2 or D and T2, respectively.

The D-T2 correlation experiment (DRCOSY), as introduced by Hürlimann [4] for the stray field of a superconducting magnet, combines an echo experiment (D Domain) and a CPMG echo train (T2 Domain). As opposite to this experiment, we use pulsed magnetic field gradients within the diffusion part of the pulse sequence (see, e.g. [5]) which may require a matching procedure for the gradient pulses. This holds in particular when using ultra high pulsed field gradients as outlined in [3]: If due to the finite experimental accuracy the aforementioned pair of gradients does not match exactly, the echo displaces and may even have a smaller magnitude which may lead to a wrong value of the diffusion coefficient determined. Therefore, a gradient adjustment is essential during data acquisition in order to correct for the mismatch. Here, we follow the concept suggested in [3], record the first echo completely and use it for the matching procedure. While this enables one to match the pulsed field gradients, it would exceed the size of the acquisition memory if one acquires each subsequent echo in the CPMG part with the same number of data points per echo. Therefore, the remaining echo train is acquired with only a few points per echo.

Another challenge is to find the highest gradient value necessary for a given PFG NMR experiment as it is crucial for the inverse Laplace transformation, as a model-free approach, to find a complete attenuated signal decay, regardless of whether the source of this (multi-exponential) decay is a relaxation or diffusion process. On the other hand, the determination of the echo shift, used to match the magnetic field gradients, fails without

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, M. Gratz 80

any distinct signal. To solve this conflict, a termination condition based on the signal-to-noise ratio and/or a pre-determined absolute echo shift is introduced.

For the simple case of bulk water, as a proof of working, the results of the DRCOSY are in good agreement with the expected diffusion and relaxation behaviour of water (Fig. 2).

Fig. 2: inverse Laplace transformation of Fig. 1 with single peak at T

Fig. 1: Acquired map of NMR signals, CPMG on t-axis and echo attenuation on g

Similar, but less complicated is the implementation of the T1-T2 correlation experiment [1] as no gradient matching procedure is required. The results obtained for water agree with the expected relaxation time values.

3. Conclusion While the implementation of RRCOSY experiments into our spectrometers was

straightforward, a couple of issues had to be addressed for the set up of DRCOSY experiments. Due to the use of ultra high pulsed field gradients and its ramifications we had to introduce an adjustment loop, as well as an automatic termination of the measurement based on the signal-to-noise ratio. We obtained expected behaviour of water for both T1-T2 correlation as well as D-T2 correlation.

References[1] Song, Y.Q., Venkataramanan, L., Hurlimann, M.D., Flaum, M., Frulla, P. & Straley,

C., J. Magn. Reson., 154 (2002) 261-268. [2] Callaghan, P.T., Godefroy, S. & Ryland, B.N., J. Magn. Reson., 162 (2003) 320-327. [3] Galvosas, P., Stallmach, F., Seiffert, G., Kärger, J., Kaess, U. & Majer, G., J. Magn.

Reson., 151 (2001) 260-268. [4] Hürlimann, M.D., Venkataramanan, L. & Flaum, C., J. Chem. Phys., 117 (2002)

10223-10232. [5] Godefroy, S. & Callaghan, P.T., Magn. Reson. Imaging, 21 (2003) 381-383.

2-axis 2 = 8ms and D = 2.3x10-9m2 -1 s

81

Combined Use of Pulsed Gradient Spin Echo and High Resolution Magic Angle Spinning to Investigate Solutes Diffusion in Presence of a

Chromatographic Stationary Phase

Stéphane Viel,1 Grégory Excoffier,1 Guilhem Pagès,2 Fabio Ziarelli,3 Corinne Delaurent,1 and Stefano Caldarelli1

1 Aix-Marseille Université, JE2421 TRACES, service 512, av. Escadrille Normandie Niémen, 13397 Marseille cedex 20, France; 2 University of Sydney, School of Molecular

and Microbial Biosciences, Sydney 2006, Australia; 3 CNRS, Fédération des Sciences Chimiques de Marseille, Spectropole, service 511, av. Escadrille Normandie Niémen,

13397 Marseille cedex 20, France. E-Mail: s.viel@univ-cezanne.fr

1. Introduction We recently introduced a novel analytical method based on the Pulsed Gradient Spin Echo (PGSE) experiment and High Resolution Magic Angle Spinning (HRMAS), which combines the advantages of column chromatography separation and NMR structural analysis [1]. Specifically, we showed that, in a PGSE experiment, the separation of the NMR spectra of the components of a mixture could be enhanced by several orders of magnitude upon addition of a typical stationary phase used in HPLC, such as either a normal (silica gel) or a reversed (functionalized ODS silica gel) phase. HRMAS is then required to recover high resolution NMR spectra by removing magnetic susceptibility broadenings caused by the presence in solution of the solid support. This combined PGSE and HRMAS technique was subsequently applied to investigate indirectly crucial steps of reversed phase liquid chromatography, such as the partitioning of the analyte between different phases [2]. Interestingly, with respect to HPLC, this technique offers surprising perspectives for mixture analysis [3]. Indeed, while reversed phase liquid chromatography (RPLC) is required to separate the components of a model mixture (benzene, naphthalene, and anthracene dissolved in an acetonitrile/water mix), both normal and reversed stationary phases give excellent discrimination in NMR. Because of its unexpected potential for mixture analysis, this technique was hence referred to as ‘Chromatographic NMR’.

To investigate the discrimination process evidenced in Chromatographic NMR, the apparent diffusion coefficients (D) of the above mentioned 3-component mixture were measured in the presence of a constant weight of bare silica as a function of the amount of liquid solution in the MAS rotor.

2. Results and Discussion The discrimination obtained for benzene, naphthalene, and anthracene, in an acetonitrile/water mix, is best illustrated in the form of a DOSY map (Fig. 1). Basically, Diffusion Ordered NMR Spectroscopy (DOSY) is a convenient way of displaying PGSE data showing a pseudo two-dimensional spectrum with chemical shifts and D values on the horizontal and vertical axes, respectively [4]. In this way, the orders of magnitude difference in the diffusion coefficients of the three solutes is clearly apparent.

S. Brandani, C. Chmelik, J. Kärger, R. Volpe (Editors)Diffusion Fundamentals II, Leipziger Universitätsverlag, Leipzig 2007

© 2007, S. Viel 82

Solution volume (μL)5 10 15 20 25

Dbe

nzen

e (x 1

0-9 m

2 s-1

)

3

4

5

6

7

8

Fig. 1. DOSY map recorded at 303 K ona mixture of benzene, naphthalene, andanthracene, at 90.0, 9.0, and 1.6 g L-1, respectively, in an acetonitrile/water mix(90/10, v/v).

Fig. 2. Diffusion coefficient of benzene in CDCl3 (1 g L-1) at 303 K with the bare silica. The dashed line shows the diffusion coefficient of benzene measured in the same conditions without stationary phase.

Initially, this particular system was selected as a model because it is commonly employed in RPLC for testing the properties of the chromatographic support used under reversed phase conditions. However, due to differences in solubility, the solutes concentrations are very different. Therefore, another solvent was required and CDCl3 was first tried because it is a good solvent for aromatics and compatible with bare silica. Surprisingly, a clear discrimination could also be observed in this case.

Subsequently, the D values of the three components analyzed separately in CDCl3 were measured as a function of the solution volume. While relatively little variation was observed for both naphthalene and anthracene, the D value of benzene clearly increased as the solution amount was reduced (Fig. 2). This may be related to the work of D’Orazio et al. [5] who described a vapor phase contribution to the diffusion of water confined in porous glass with different filling fractions. As a consequence of this specific molecular behavior, the differences in D can be modulated.

3. Conclusion These preliminary results show that, in loose analogy to HPLC, several

physicochemical parameters can be fruitfully used to optimize the resolution in Chromatographic NMR.

References [1] S. Viel, F. Ziarelli, S. Caldarelli, Proc. Natl. Aca. Sci. USA 100 (2003) 9696-9698. [2] G. Pagès, C. Delaurent, S. Caldarelli, Anal. Chem. 78 (2006) 561-566. [3] G. Pagès, C. Delaurent, S. Caldarelli, Angew. Chem. Int. Ed. 45 (2006) 5950-5953. [4] K. F. Morris, C. S. Johnson, Jr. J. Am. Chem. Soc. 114 (1992) 3139-3141. [5] F. D’Orazio, S. Bhattacharja, W. P. Halperin, R. Gerhardt Phys. Rev. Lett. 63

(1989) 43-46.

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