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10 PFG NMR Studies of Anomalous Diﬀusion

J¨org K¨arger and Frank Stallmach

10.1 Introduction

Depending on the system under study, particle diﬀusivities cover a huge range

of values from ∼ 10

−20

−1

for solids (see Chaps. 1-9) up to ∼ 1m

−1

for

dilute gases. However, the nature of a system does not only aﬀect the rate

of particle propagation, it is also the pattern of its time dependence,which

may be a function of the system. In general, the mean square displacement of

the migrating particles increases in proportion to the observation time. This

is the case of normal diﬀusion, which is commonly described by Fick’s laws.

The fundamental relations of normal diﬀusion are summarized in Chap. 1 of

this textbook, and most of the other chapters are also devoted to this case.

However, under the inﬂuence of internal structures, particle propagation may

deviate from this pattern. Owing to the intimate relation between structure

and mobility, in the last few years the study of “anomalous diﬀusion” has

become a most attractive subject of both fundamental and applied research.

Since the time exponent of the mean square displacement is generally found

to be smaller than 1, anomalous diﬀusion is often referred to as subdiﬀusion.

Chapters 13, 16 - 19, 21 and 22 deal with experimental and theoretical aspects

of this case. The present chapter is devoted to the inﬂuence of interfaces on

molecular propagation. It is based on the corresponding chapter of the ﬁrst

edition, which one of us has co-authored with G. Fleischer and U. Roland.

With great regret we have to announce that Gerald Fleischer has passed away

in 1999. His oeuvre as one of the pioneers in applying PFG NMR to diﬀusion

in macromolecular systems remains of lasting value.

In the present chapter, the term interface is meant in a rather broad

sense. It refers to ﬂuid/solid interaction in microporous zeolitic adsorbate-

adsorbent systems (Sect. 10.4) and in meso- and macroporous amorphous

materials (Sect. 10.5) as well as to ﬂuid/ﬂuid interaction in internally struc-

tured macromolecular melts and solutions (Sect. 10.6). In all these cases,

deviations from ordinary diﬀusion occur, and the method of pulsed ﬁeld gra-

dient (PFG) NMR (Sect. 10.3) turns out to be a most helpful technique for

their investigation.

418 J¨org K¨arger and Frank Stallmach

10.2 The Origin of Anomalous Diﬀusion

The diﬀusion path of a particle consists of numerous displacements caused by

the stochastically changing interaction between the particle and its surround-

ings. In many cases, in particular in homogeneous systems and for suﬃciently

large observation times, it is possible to divide the diﬀusion time into inter-

vals, which are large enough so that the displacements in any of these time

intervals are independent of each other. In this case, the mean square dis-

placement r

(t) of the particles may be shown to depend in a simple way

on the observation time t. For this purpose we consider the vector sum of the

displacements ∆r

during the individual time intervals, which for simplicity

are assumed to be of equal duration ∆t.Itholds

r

(t) =





∆r









(∆r

)







i=j

∆r





(∆r)



· t/∆t (10.1)

where the last equality results as a consequence of the missing correlation

between subsequent displacements ∆r

.With



(∆r)



as the mean square

value of the displacement during the considered time interval ∆t, (10.1) re-

ﬂects the expected proportionality between the mean square displacement

r

(t) and the observation time t in the case of ordinary diﬀusion. The ab-

sence of correlation between the displacements during diﬀerent time intervals,

i.e. the disappearance of the cross term in (10.1), turns out to be a necessary

and suﬃcient condition for normal diﬀusion. This statement does clearly not

imply that under the condition of normal diﬀusion any correlation between

subsequent displacements is excluded. A famous example is the correlation

eﬀect of solid state diﬀusion (see Sect. 1.6.1 in Chap. 1 and Sect. 18.4.3 in

Chap. 18), where subsequent displacements are preferentially directed into

opposite directions. However, summing the times for the individual steps to

longer time intervals with correspondingly larger displacements (being the

vector sum of the individual displacements), these larger displacements are

soon found to be independent of each other. Hence, the correlation eﬀect

in solid state diﬀusion is found to aﬀect the absolute value of the diﬀusiv-

ity (by the “correlation factor” f – see Sect. 1.6.1 and Sect. 18.4.3) without

leading to deviations from normal diﬀusion. Anomalous diﬀusion can only be

observed if the correlation between the displacements is preserved over the

total duration of the experiment. It is therefore one of the crucial questions to

be answered in the following sections, what is the mechanism, which ensures

the correlation between the displacements over the whole observation time.

Deviations from normal diﬀusion may follow various patterns. Using the

conception of a fractal structure (see Chap. 19), the correlation between the

mean square displacement and the observation time may be represented in

a quite general way. For deriving this correlation we have only to imply the

10 PFG NMR Studies of Anomalous Diﬀusion 419

self-similarity of the system. Owing to the stochastic nature of diﬀusion, it

is suﬃcient to imply the self-similarity of the system in a statistical sense.

The eﬀect of this self-similarity of the structure on molecular propagation is

assumed to be reﬂected by a scaling law, which implies that by enhancing

the elementary structure length from l to µl the corresponding time step has

to be enhanced from t to νt. The actual values of the scaling parameters µ

and ν depend on the system under study. The eﬀorts to be exerted on their

determination (see Chap. 19 and [1, 2]) are generally not negligibly small.

However, when these parameters are known, the relation between the mean

square displacement and the observation time t may be deduced very easily:

As a consequence of the scaling law one may note the correspondence

r

 = f(t) ↔



(µr)



= f (νt), (10.2)

where f(t) is an (at ﬁrst) unknown function representing the time dependence

of the mean square displacement. The right-hand side of this correspondence

may be rearranged leading to the equation

r

 = f(t)=

f(νt) (10.3)

The second equality is obviously obeyed by

r

(t) = f (t) ∝ t

(10.4)

with

κ =2

log µ

log ν

. (10.5)

The mean square displacement is thus quite generally found to obey a power-

law time-dependence (cf. (10.4)) where exponent κ is given by (10.5). In the

case of ordinary diﬀusion due to one-dimensional random walk with con-

stant step size, e.g., µ and ν are found to be 2 and 4, respectively ([3], p.

61). This leads to the familiar result of proportionality between r

 and t,

which has been veriﬁed, already, by means of (10.1). In the case of anomalous

diﬀusion, independent of the considered time interval, subsequent displace-

ments are more likely directed into opposite directions, making the cross

term



i=j

∆r

in (10.1) negative. The mean square displacement is thus

expected to increase less than linearly with the observation time. One has

therefore to expect that κ ≤ 1, with κ = 1 characterizing the case of normal

diﬀusion.

In several places of this book (Chaps. 2, 3, 18, 19 and 23), the phenom-

enon of diﬀusion is considered on the basis of the so-called propagator or (as

generally referred to in neutron scattering and M¨oßbauer spectroscopy) Van

Hove self-correlation function (with the notation G

(r, t)). It represents the

probability density of particle propagation and contains the maximum infor-

mation accessible about the diﬀusion properties of the system. Like in the

420 J¨org K¨arger and Frank Stallmach

case of quasielastic neutron scattering (QENS, see chapters 2, 3 and 13), the

propagator is directly related to the experimental data of pulsed ﬁeld gradi-

ent (PFG) NMR, which shall be explicitly shown in the subsequent section.

In contrast to QENS, which traces molecular motion over not more than a

few tens of nanometers maximum, typical displacements monitored by PFG

NMR are of the order of micrometers. Therefore, the propagator generally

results as the superposition of many individual displacements. If – as in the

case of normal diﬀusion – these displacements are independent of each other,

on the basis of the central limit theorem [4] the resulting probability distri-

bution may be shown to be a Gaussian ( [3], pp. 63–67). In PFG NMR one

is able to monitor molecular displacements in the direction of the magnetic

ﬁeld gradient, which is commonly assumed to coincide with the z-direction.

In the case of normal diﬀusion the propagator thus results to be

P (z, t)=

(

2πz

(t)

exp



−z

2z

(t)



, (10.6)

where P (z, t)dz denotes the probability that during a time interval t the

z coordinate of a particle is shifted by a value between z and z +dz.

Equation(10.6) may be interpreted as an immediate consequence of the cen-

tral limit theorem. However, it also follows by applying Fick’s second law

(see, e.g., (1.8) of Chap. 1) with the initial condition

P (z, 0) = δ(z) (10.7)

to the propagator. Moreover, in doing so, the mean square displacement is

found to be

z

(t) =2Dt (10.8)

in complete agreement with (10.1) and with the requirement of the central

limit theorem that the square width of the distribution increases in propor-

tion with the number of elements (time intervals) considered for the total

distribution. Equation (10.8) is commonly referred to as Einstein’s relation

with D denoting the coeﬃcient of self-diﬀusion (tracer diﬀusion) as appearing

in Fick’s ﬁrst law for labelled particles.

Even under the conditions of anomalous diﬀusion, the propagator is often

found to be satisfactorily well approached by a Gaussian (cf. (10.6)). In some

cases, e.g. in the long-time limit of single-ﬁle diﬀusion (see Sect. 10.4.5), (10.6)

holds even strictly. It is therefore common practice in PFG NMR to analyse

diﬀusion phenomena exclusively on the basis of an eﬀective diﬀusivity

eﬀ

(t) ≡

z

(t)

, (10.9)

which results by inverting (10.8). In the case of normal diﬀusion, D

eﬀ

is clearly

independent of the observation time and coincides with the self-diﬀusivity.

10 PFG NMR Studies of Anomalous Diﬀusion 421

Under the conditions of anomalous diﬀusion, D

eﬀ

becomes time dependent.

If (10.4) is valid, the time dependence of D

eﬀ

becomes

eﬀ

(t) ∝ t

κ−1

. (10.10)

Since κ ≤ 1, the eﬀective diﬀusivity is found to decrease with increasing ob-

servation time. Anomalous diﬀusion is therefore (like in the cases considered

in Chaps. 12, 13, 16 and 19) often referred to as sublinear diﬀusion or even

subdiﬀusion. In the case of completely restricted diﬀusion, e.g. diﬀusion in a

space conﬁned by impenetrable barriers, z

 remains constant for suﬃciently

long observation times, and we have D

eﬀ

∝ t

−1

for t →∞.

10.3 Fundamentals of PFG NMR

10.3.1 The Measuring Principle

As in the case of NMR tomography [5], the application of NMR for studying

molecular transport is based on the Larmor condition

ω = γB (10.11)

between the magnetic ﬁeld and the resonance frequency (multiplied by 2π)for

transitions between the Zeeman levels of the nuclear spins in the magnetic

ﬁeld. The magnetogyric ratio γ is a characteristic quantity of the nucleus

under study (e.g. 2.67 · 10

−1

for

H). The Larmor frequency ω may

be intuitively understood as the precessional frequency of the nuclear spins

(and hence of the nuclear magnetization) about the direction of the magnetic

ﬁeld. Superimposing the constant magnetic ﬁeld B

by an inhomogeneous

ﬁeld B

add

= gz (the “ﬁeld gradient”), the Larmor frequency becomes space

dependent:

ω = ω(z)=γ(B

+ gz)=ω

+ γgz , (10.12)

where the z-coordinate is assumed to be aligned along the direction of the

applied ﬁeld gradient. In the PFG NMR technique (see, e.g., [3, 6–10]), the

inhomogeneous ﬁeld is applied over two short time intervals of duration δ

separated by the “observation time” ∆. These two ﬁeld gradient pulses are

applied either with opposite signs (i.e. the amplitudes +g and −g)orwith

a radiofrequency (rf) pulse of suitable duration (a “π-pulse”) in between. In

both cases, the eﬀect of the second ﬁeld gradient pulse has to be subtracted

from that of the ﬁrst one. The phase shift ϕ of a nuclear spin, which during

the observation time ∆ has been displaced over a distance (z

− z

)inz-

direction, in comparison with a spin, which has remained at the same position,

is therefore

ϕ = γgδ(z

− z

) . (10.13)

The quantity monitored in PFG NMR is the amplitude of the “spin echo”

as generated, e.g., in the

– τ – π – τ – echo sequence (see also Sect. 9.3

422 J¨org K¨arger and Frank Stallmach

in Chap. 9 (Fig. 9.10), Sect. 13.2.1 in Chap. 13, Sect. 17.2.2 in Chap. 17

(Fig. 17.1)). Instead of PFG NMR, therefore, sometimes also the term PGSE

(pulsed gradient spin-echo) NMR is used. The amplitude of this spin echo

is proportional to the total magnetization, i.e., to the vector sum of the

contributions of the individual spins. The application of ﬁeld gradient pulses

thus leads to a signal (spin-echo) attenuation

Ψ =



p(z

)P (z

,∆)cos(γgδ(z

− z

)) dz



P (z, ∆)cos(γgδz)dz, (10.14)

with

P (z, ∆)=



p(z

)P (z

+ z,z

,∆)dz

. (10.15)

p(z

)isthea priori probability (density) to ﬁnd a spin at position z

for

t =0andP (z

,∆) the probability (density) that the spin has moved

from z

to position z

in the time interval t = ∆. P (z, ∆) is the so-called

mean propagator, i.e., the propagator averaged over all starting positions

. It is the probability density that an arbitrarily selected particle in the

sample is shifted over a distance z in z-direction (which is the direction of

the magnetic ﬁeld gradient) during the time interval between the two ﬁeld

gradient pulses. Since in a heterogeneous sample the probability function of

molecular displacement may depend on the starting point, in this case the

propagator as used in (10.15) is understood as a mean value over the sample.

10.3.2 The Mean Propagator

By Fourier inversion of (10.14) the mean propagator may be directly deduced

from the primary data of the PFG NMR experiment, yielding

P (z, ∆)=

2π



Ψ(δg, ∆)cos(γδgz)d(γδg) . (10.16)

As an example, Fig. 10.1 displays the ﬁrst application of this possibility show-

ing the propagation patterns of ethane in beds of zeolite NaCaA with two

diﬀerent crystallite sizes [11]. Zeolite NaCaA consists of microporous crys-

tallites whose diﬀusion properties are discussed in more detail in Sect. 10.4

and in Sect. 23.4 of Chap. 23 [12–14]. Being symmetric in z, for simplicity

the propagator is only represented for z ≥ 0. For the lowest temperature

(153 K), the distribution widths of molecular displacement during the con-

sidered time intervals (5 ...45 ms) are found to be small in comparison with

the mean radius of the larger crystallites (8 µm). In this case, the observed

mean square displacement increases in proportion with the observation time

as required for normal diﬀusion so that PFG NMR is able to monitor genuine

10 PFG NMR Studies of Anomalous Diﬀusion 423

Fig. 10.1. Propagator representation of the self-diﬀusion of ethane in NaCaA

zeolite: (a) loading 40 mg/g, mean crystallite radius R =8µm; (b) 58 mg/g,

R =0.4 µm [11].

intracrystalline self-diﬀusion. In the smaller crystallites, obviously, molecu-

lar propagation is terminated by the surface of the crystallites so that PFG

NMR provides information on the size of the crystallites rather than the in-

trinsic mobility. This way of tracing the extension of microscopic regions has

become popular under the name “dynamic imaging” [7, 15]. With increasing

temperature, however, the thermal energy of the diﬀusing molecules becomes

large enough so that a substantial fraction of the ethane molecules are able

to surpass the step in the potential energy from the intracrystalline space

into the surrounding gas phase (intercrystalline space). Consequently, distri-

bution widths of molecular propagation much larger than the crystallite radii

become possible. Eventually, the conditions for the application of the central

limit theorem are again obeyed and molecular propagation is described by a

Gaussian with an eﬀective diﬀusivity D

. D

may be shown to be the prod-

424 J¨org K¨arger and Frank Stallmach

uct of the diﬀusivity of the molecules in the intercrystalline space and their

relative amount [16].

10.3.3 PFG NMR as a Generalized Scattering Experiment

Using complex notation and introducing the magnetic ﬁeld gradient vector

g, (10.14) may be written in the form

Ψ(γδg,∆)=



P (r,∆)exp(iγδgr)dr . (10.17)

The structure of this relation coincides with that of the self-term of the inter-

mediate dynamical structure factor I

(Q, t) of QENS as introduced by (3.7)

in Sect. 3.2 of Chap. 3 and as S(Q, t) in Sect. 13.2 of Chap. 13. Similar

relations occur in M¨oßbauer spectroscopy (Chap. 2) and dynamic light scat-

tering (Chaps. 15 and 16). They may be derived by Fourier transformation of

(2.1) in Chap. 2 and of (15.3) together with (15.1) in Chap. 15, respectively,

from frequency to time domain. PFG NMR may therefore be considered as a

scattering experiment with a generalized scattering vector q = γδg [17–20].

With typical maximum values of g (50 T/m) and δ (2 ms; cf., e.g., [21–25]),

for protons the generalized scattering vector is found to be of the order of

3 ·10

−1

. This corresponds to minimum displacements observable by PFG

NMR of 10 ...100 nm.

Inserting (10.6) and (10.8) into (10.14) (or the equivalent three-dimensional

expressions into (10.17)), the spin-echo attenuation is found to be a simple

exponential

Ψ =exp(−γ

D∆) . (10.18)

Identifying γδg as the generalized scattering vector, (10.18) coincides with the

intermediate scattering function in the so-called diﬀusion limit in QENS and

light scattering. Obviously, in this case the self-diﬀusivity (and via (10.8) the

mean square displacement) results in a straightforward way from the slope of

the spin-echo attenuation in a semilogarithmic representation versus γ

As long as the propagator is a Gaussian, (10.18) is rigorously correct and

implies the cases of normal and anomalous diﬀusion, if D is understood in

the more general sense as an eﬀective diﬀusivity as deﬁned by (10.9).

If molecular diﬀusion is conﬁned to a certain range, for suﬃciently large

observation times the propagator approaches a time-independent function

∞

(r), which is determined by the distribution of the molecules under study

in the conﬁning volume. Therefore, the spin-echo attenuation attains a time-

independent value

∞

(γδg) ≡ lim

∆→∞

Ψ(γδg,∆)



p(z

) P (z

, ∞)exp(iγgδ(z

− z

)) dz

(10.19)





p(z)exp(iγgδz)dz



= |F (q)|