Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

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10 PFG NMR Studies of Anomalous Diﬀusion 455

the observation time. In fact, an estimate of the domain size of the separated

phases by small angle neutron scattering yields values of about 75 nm [98,

100], which are notably exceeded by the diﬀusion path lengths during the

observation times. Studying such eﬀects of molecular restrictions by choosing

experimental conditions, where the observed molecular displacements may be

of the order of the phase extensions, are among the most challenging tasks

of future research.

10.7 Conclusion

PFG NMR has been introduced as a sensitive technique for studying the time

dependence of molecular propagation. In homogeneous systems the probabil-

ity distribution of molecular displacements (the “propagator”) is described by

a Gaussian, which is completely determined by its mean square width (which

is most appropriately visualized by the “mean square displacement”). In ho-

mogeneous systems, the mean square displacement increases in proportion

to the observation time, with the self-diﬀusivity being the factor of propor-

tionality. Deviations from homogeneity may lead to characteristic deviations

from this proportionality as well as from the Gaussian shape of the prop-

agator. Such deviations are most likely to be encountered for molecules in

contact with interfaces. We have highlighted a wide spectrum of possibilities

for deviations from homogeneity including micro- and mesoporous adsorbate-

adsorbent systems, ﬂuid-saturated natural sediments as well as the case of

self-structurization in macromolecular systems. PFG NMR could be shown

to provide unique information about both structural and dynamical proper-

ties, thus complementing the information from other techniques about these

systems.

Notation

a unit cell dimension in x-direction

b unit cell dimension in y-direction

B magnetic ﬂux density

c sorbate concentration; unit cell dimension in z-direction

grain (particle) diameter

D coeﬃcient of self-diﬀusion (self-diﬀusivity)

coeﬃcient of long-range diﬀusion

coeﬃcient of transport diﬀusion

par/perp

main components of axisymetrical diﬀusion tensor

fractal dimension of the grain surface

F mobility factor of single-ﬁle diﬀusion

g magnetic ﬁeld gradient

l elementary length; mean free distance between adjacent molecules

456 J¨org K¨arger and Frank Stallmach

N number of diﬀusants

p pressure

P (z, t) (mean) propagator in z-direction

q generalized scattering vector

r vector of displacement or position

∆r displacement during time interval ∆t

R radius of crystallites

root mean square end-to-end distance of polymer chains

, S

speciﬁc surface area, surface-to-volume ratio

t time, time step

∆t time interval

longitudinal nuclear magnetic relaxation time

transverse nuclear magnetic relaxation time

(t) velocity of particle i at time t

z coordinate

γ magnetogyric ratio

δ duration (width) of the ﬁeld gradient pulses

∆ separation of the ﬁeld gradient pulses (“observation time”)

θ occupation probability

κ time exponent of anomalous diﬀusion

λ jump distance

µ scaling parameter; chemical potential

ν scaling parameter

τ distance between the π/2- and π-pulse in the spin-echo experiment;

mean residence time between subsequent jumps

,τ

distances between the ﬁrst and second π/2-pulse, and the second

and third π/2-pulse in the stimulated echo sequence, respectively

ϕ phase shift in the precessional motion

Ψ attenuation of the NMR signal (spin echo) in PFG NMR experi-

ments

ω Larmor frequency

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11 Diﬀusion Measurements by Ultrasonics

Roger Biel, Martin Schubert, Karl Ullrich W¨urz, and Wolfgang Grill

11.1 Introduction

The velocity of acoustic waves depends on the elastic properties and the den-

sity of the transport medium. The variation of these properties can therefore

be observed with ultrasonic waves. The velocity of sound can sensitively be

detected by observation of the phases of transit signals. Resolutions of 1 part

in 10

with respect to changes of the velocity of ultrasound can be achieved

for ultrasonic waves in the frequency regime of typically 10 to 100 MHz. Spa-

tial resolution can be obtained with narrow beams of plane waves or with

focused ultrasonic waves.

Ultrasonic beams generated and detected by planar transducers are rou-

tinely used to observe the temperature dependence of the velocity of sound

in liquid or solid materials. Variations of the temperature lead to changes

of the elastic properties as well as to thermal expansion. The temperature

dependence of the velocity of acoustic waves is often used to observe phase

transitions or to gain information on other thermodynamic properties of the

samples. The observation of the velocity of sound can also be used as an ex-

tremely sensitive thermometer, capable to resolve temperature changes even

below 1 mK. It is therefore almost surprising that only little eﬀort has been

paid so far to observe thermal diﬀusion processes or other transport processes

by diﬀusion with ultrasonic detection schemes. An exception are somewhat

related photoacoustic excitation schemes including the so-called thermal wave

methods which have even been applied with microscopic resolution.

To demonstrate the feasibility of ultrasonic detection schemes with phase

resolution, applications are presented for the observation of the diﬀusion of

hydrogen in single-crystalline tantalum and the diﬀusion of heavy water in

gels and living cells. Acoustic interferometric detection schemes and acoustic

imaging techniques have so far only been used in these applications. They

are demonstrated here to inform about these novel developments and to sup-

ply the information needed to decide if such methods can be used advanta-

geously for measurements which cannot easily be performed by established

techniques.

462 Roger Biel et al.

11.2 Diﬀusion of Hydrogen in Single-Crystalline

Tantalum

The transport properties of hydrogen in metals have an obvious technical

relevance for hydrogen storage devices where metals are employed, capable

to store large amounts of hydrogen dissolved in the metal matrix. Single-

crystalline samples are normally used to study the basic physical properties

under idealized conditions.

A two-beam interferometric setup has been employed (Fig. 11.1) to ob-

serve the temperature dependence of the transport properties of hydrogen in

metals [1]. The ultrasonic interferometric device is comparable to an opti-

cal Mach-Zehnder interferometer. Acoustic waves travel along two paths at

diﬀerent positions in the sample, monitoring integrally the hydrogen concen-

tration along the paths. Piezoelectric transducers (quartz) mounted on the

sample convert electric signals into acoustic signals and vice versa. Diﬀerent

to the optical equivalent the remaining part of the interferometer involves

only electrical signals. This is possible since ultrasonic transducers are linear

devices converting the amplitudes and not simply the power of the respective

electric or acoustic signals.

The dependence of the velocity of transverse polarized ultrasonic waves

on the hydrogen content c with respect to a small variation of this content

∆c is given by [2]:

+(∆c

/∆c)c

ρ +(∆ρ/∆c)c

, (11.1)

where ρ is the density and c

the element i, i of the tensor of the elastic

stiﬀness constants of tantalum. The relative change of c

per variation of the

hydrogen content of one atomic percent is 1.4 ·10

−3

[2]. The respective value

for c

is only 8.9 · 10

−5

[2]. Therefore transverse polarized acoustic waves,

travelling along the [100] direction of the tantalum crystal, are employed for

Fig. 11.1. Tantalum sample with gradient in the hydrogen distribution and four

transducers for generation and detection of ultrasonic waves travelling along two

separated paths.

11 Diﬀusion Measurements by Ultrasonics 463

Fig. 11.2. Schematic of the ultrasonic two-path interferometer with vector phase

detection. Channel 1 is used for phase-locking of the oscillator. Channels 2 and 3

detect relative changes of the phase of the ultrasonic signal with respect to chan-

nel 1. Changes in the velocity in path 1 can be determined from the frequency

variation of the locked RF-oscillator.

optimum sensitivity to the hydrogen content which is about 16 times higher

than for longitudinal waves.

In the experimental setup phase detection is achieved by multiplicative

mixing techniques (Fig. 11.2). The frequency used for excitation is varied in

such a way, that the phase observed in one of the paths with respect to an

electronic reference derived from the oscillator used for excitation remains at

a constant value. The amplitude is stabilized to achieve optimum resolution.

The phase observed for the other path is monitored with respect to the same

electronic reference by vector detection schemes. Pulses are used to discrim-

inate echoes in the acoustic paths. Some of these features are introduced to

achieve optimum sensitivity and long-term stability. This is necessary since

rather slow changes have to be observed. Equilibration times for samples of

typical sizes of 1 cm can reach several days or even months, depending on

the temperature of the sample.

464 Roger Biel et al.

Fig. 11.3. Time dependence of the phase signals (channels 2 and 3 in Fig. 11.2)

referenced with a shift of 0

◦

and 90

◦

with respect to the oscillator signal during

the equilibration of the graded hydrogen distribution in tantalum by diﬀusion at a

temperature of 273 K (frequency of the ultrasound: 10.6 MHz).

The data are collected and stored with a computer, controlling also the

temperature of the sample.

The hydrogen distribution in the tantalum sample is varied by applying

unidirectional electric currents at temperatures of about 400 K. This creates a

gradient in the hydrogen content across the sample. Subsequently the crystals

are cooled to the temperature where the transport processes are studied. The

equilibration of the hydrogen distribution is observed with one path of the

acoustic interferometer in the region of elevated hydrogen content and the

other path in the region where the content has been reduced.

The data collected over time spans up to several days (Fig. 11.3) may

be used to determine the diﬀusion coeﬃcient D taking into account that

the relaxation of the hydrogen distribution is determined by a quasi-one di-

mensional diﬀusion process following Fick’s second law (see, e.g., (1.8) in

Chap. 1):

∂c

∂t

= D

∂

∂x

. (11.2)

The diﬀusivities obtained for diﬀerent temperatures are presented in Fig. 11.4.

The temperature dependence of the diﬀusion coeﬃcient is found to follow an

Arrhenius relation

D = D

−

·T

(11.3)

with E

=0.14 eV, where D

is the diﬀusivity at temperature T →∞.

Results presented in the literature [3] which have been obtained by con-

ventional methods show errors of up to almost 100%, even for data from the

11 Diﬀusion Measurements by Ultrasonics 465

Fig. 11.4. Arrhenius plot of the diﬀusion coeﬃcient D derived from measurements

as represented in Fig. 11.3 (crosses) at diﬀerent temperatures and line ﬁtted to the

data.

same author. The results obtained here show deviations of only up to 23%

from the expected linear behaviour. The extrapolation to the value of the dif-

fusion coeﬃcient for T →∞leads to D

=4.5 · 10

−8

/s, which compares

well with the value of D

=4.4 ·10

−8

/s obtained by V¨olkl and Alefeld [3].

The ultrasonic detection scheme has the advantage, that surface eﬀects

play a negligible role, since the hydrogen content is monitored along a path

in the volume of the sample. Even extremely slow processes can be studied.

As unwillingly demonstrated in the data (Fig. 11.3), the measurement can

be continued in the case of temporal failure of the electronics, caused here by

a failure of the current supply. Such events are otherwise disastrous for long-

time observations. Furthermore the rather low power level of the acoustic

signals does not inﬂuence the observed hydrogen distribution in the sample.

The method presented here can be used for any sample with two plane

parallel surfaces. Due to the integrated detection along the paths of the ul-

trasonic waves, transport properties can be studied for a direction normal to

these surfaces. Lateral resolution is restricted by the fact that planar acoustic

waves are employed. Resolutions down to typically 2 mm are feasible with

available transducers. Temporal resolution is determined by the repetition

rate at which the ultrasonic pulses are generated. The maximum repetition

rate depends on the absorption in the sample, since ultrasonic echoes must

reach a negligible level prior to the following excitation. Repetition rates of