Grain-Growth Kinetics of Nanocrystalline Iron Studied in Situ by Synchrotron Real-Time X-ray Diffraction

Article abstract of DOI:10.1021/jp991622d

Pulsed electrodeposition (PED) is used to prepare nanocrystalline iron with an average grain size of 19 nm and thermal stability up to 550 K. At 663 K ≤ T ≤ 783 K the kinetics of grain growth, with respect to size and size distribution, is studied in situ by means of real-time synchrotron X-ray diffraction. The Bragg peak line shapes of the large number of diffractograms are analyzed using a Warren/Averbach procedure improved with respect to reliability and efficiency. We observe two regimes of grain growth: at less elevated temperatures grain growth is smooth and moderate up to limiting size values between 50 and 100 nm, depending on temperature. The initially rather narrow width of the size distribution increases slightly, and the activation energy of grain growth, about 100 kJ/mol, corresponds to the literature value for grain boundary self-diffusion in nanocrystalline Fe. At higher temperatures the grains grow first rapidly and then slowly up to limiting values between 200 and 400 nm, depending on temperature. The size distribution becomes rather broad, and the activation energy for grain growth, about 175 kJ/mol, corresponds to the literature value for grain boundary self-diffusion in coarse-grained polycrystalline Fe. We do not find evidence for a change of the type of distribution which indicates normal grain growth. The quality of our diffraction data allows a critical evaluation of different kinetic models of grain growth.

Full text of DOI:10.1021/jp991622d

J. Phys. Chem. B 2000, 104, 2467-2476
2467
Grain-Growth Kinetics of Nanocrystalline Iron Studied In Situ by Synchrotron Real-Time
X-ray Diffraction
H. Natter,†,§ M. Schmelzer,†,§ M.-S Lo1ffler,†,§ C. E. Krill,† A. Fitch,‡ and R. Hempelmann*,†,§
Physikalische Chemie, Technische Physik, UniVersita¨t des Saarlandes, D-66123 Saarbru¨cken, Germany, and
ESRF, Grenoble, France
ReceiVed: May 19, 1999
Pulsed electrodeposition (PED) is used to prepare nanocrystalline iron with an average grain size of 19 nm
and thermal stability up to 550 K. At 663 K e T e 783 K the kinetics of grain growth, with respect to size
and size distribution, is studied in situ by means of real-time synchrotron X-ray diffraction. The Bragg peak
line shapes of the large number of diffractograms are analyzed using a Warren/Averbach procedure improved
with respect to reliability and efficiency. We observe two regimes of grain growth: at less elevated temperatures
grain growth is smooth and moderate up to limiting size values between 50 and 100 nm, depending on
temperature. The initially rather narrow width of the size distribution increases slightly, and the activation
energy of grain growth, about 100 kJ/mol, corresponds to the literature value for grain boundary self-diffusion
in nanocrystalline Fe. At higher temperatures the grains grow first rapidly and then slowly up to limiting
values between 200 and 400 nm, depending on temperature. The size distribution becomes rather broad, and
the activation energy for grain growth, about 175 kJ/mol, corresponds to the literature value for grain boundary
self-diffusion in coarse-grained polycrystalline Fe. We do not find evidence for a change of the type of
distribution which indicates normal grain growth. The quality of our diffraction data allows a critical evaluation
of different kinetic models of grain growth.
1. Introduction
doping), the excess energy of the interface is reduced to such
an extent that there is no longer a driving force for grain growth.9
On the other hand, thermal stability can be of kinetic origin,
such that by appropriate measures (solute doping atoms or
segregates in the grain boundary), the mobility of the grain
boundaries is sufficiently reduced.10 These latter phenomena are
often called solute drag or Zener drag, respectively.11 In any
case, for a fundamental understanding of thermal stability,
accurate experimental studies of the kinetics of grain growth
are necessary, and this is the aim of the present contribution.
The grain growth kinetics of conventional coarse-grained
materials is a well-established field,11 as outlined in section 2.
The grain sizes are commonly determined by metallographic
techniques or by electron microscopy, i.e., ex situ. From grain
size histograms mostly comprising only rather small numbers
of grains, average grain sizes and sometimes even grain size
distributions are deduced, with correspondingly low statistical
significance. The new feature of kinetic grain growth studies
on nanocrystalline materials is the possibility to deduce the size
and the size distribution and, additionally, the concomitant
microstrain from a line-shape analysis of appropriate Bragg
peaks in X-ray diffractograms recorded in situ during the grain
growth. The advantage of in situ, as compared to ex situ,
experiments is obvious: exactly the same part of the sample is
studied at all times, and problems of heating-up and cooling-
down times do not exist. To exploit this possibility, we have
used an X-ray diffractometer based at a synchrotron source, for
these reasons: (1) the extreme angular resolution enables us to
determine the grain-size-induced line broadening up to grain
sizes of about 500 nm, whereas the resolution of a standard
laboratory X-ray diffractometer sets an upper limit at about 100
nm, and (2) the high intensity available at a synchrotron source
enables us to record a diffractogram in a few minutes with
Nanocrystalline solids consist of crystalline grains with sizes
between 5 and 100 nm separated by disordered grain boundary
regions.1,2 Because of the large volume fraction of interfaces
and/or the small crystallite/grain sizes, nanocrystalline solids
differ in certain physical properties from coarse-grained poly-
crystalline solids of the same chemical composition.3 In materi-
als science, this is a novel feature: because certain physical or
chemical properties dependsin the nanometer regimeson the
grain size or on the interface volume fraction, respectively, the
fascinating possibility arises that materials properties can be
tuned to desired values simply by adjusting the grain-size
appropriately during the production process. Examples of grain
size dependent properties are the coercivity and permeability
of ferromagnetic transition metals,4,5 the hardness of WC/Co
hard-metal composites,6 enhanced corrosion resistance in nano-
Ni,7 and increased hardness in nano-Au.8
Unquestionably, the grain-size dependence of materials
properties is promising for quite a number of applications.
During these applications, at the operating temperatures, grain
growth must be excluded or at least restricted to an acceptable
low level, because otherwise the relevant properties of the
corresponding devices change with time during operation.
Therefore, sufficient thermal stability with respect to grain
growth is the prerequisite of every application of nanocrystalline
solids. On one hand, thermal stability can be of thermodynamic
origin; this is the case when by appropriate measures (mostly
* To whom correspondence should be addressed. E-mail: hempelmann@
rz.uni-sb.de.
† Universita¨t des Saarlandes.
‡ ESRF.
§ Dedicated to Prof. Dr. Dr. h.c. mult. E. Wicke on the occasion of his
85th birthday.
10.1021/jp991622d CCC: $19.00 © 2000 American Chemical Society
Published on Web 02/24/2000

2468 J. Phys. Chem. B, Vol. 104, No. 11, 2000
Natter et al.
sufficient count rates; such a time resolution for a real-time
experiment of the grain-growth kinetics cannot be reached with
a standard laboratory diffractometer.
certain stage; consequently, eq 3 has to be supplemented by a
growth-retarding term b2:
a2
dD(t)
dt
Taking advantage of these two features of synchrotron X-ray
diffractometry, we present in this contribution a detailed
experimental study, isothermally at different temperatures, of
the temporal evolution of grain size, grain-size distribution, and
microstrain of a nanocrystalline metal. Thereby, to our knowl-
edge for the first time, we observe in situ the transition from
the nanocrystalline to the polycrystalline grain-growth kinetics.
Of essential importance for the present study is the reliability
and efficiency of the Bragg peak line shape analyses, which
will therefore be described in some detail.
)
- b2
(6)
D(t)
With the initial grain size D0, the solution of eq 6 is
D0 - D(t)
a
b22
a2 - b2D0
+
2 ln
) t
(7)
(
)
b2
a2 - b2D(t)
At t f ∞ the grains do not grow anymore, i.e., dD(t)/dt ) 0;
hence D∞ ) a2/b2 and eq 7 can be expressed as
2. Kinetics of Grain Growth
D0 - D(t)
D∞ - D0
k2t )
+ ln
(8)
(
)
D∞
The theoretical basis for understanding grain growth was set
forth about 50 years ago in the classic papers of Smith,12
Burke,13 and Burke and Turnbull,14 who deduced grain growth
from the movement of grain boundaries. They assume that the
driving pressure on a grain boundary arises from its curvature,
which is identified with the mean radius (or diameter, D,
respectively) of the grains; then, according to the Kelvin
equation, the driving pressure is proportional to the reciprocal
radius (or diameter) of the grain:
D∞ - D(t)
with k2 ) b2/a2 ) a2/D2 . An expression for D(t) can only
2
∞
numerically be obtained from eq 8, and therefore, in this grain
growth model with impediment, for the data evaluation it is
convenient to fit the reverse function t(D) to the experimental
data instead of D(t).
Michels et al.17 recently suggested that the retardation
constant, b2, should be a function of the grain size in grain-
growth processes of nanocrystalline metals. They argue that in
these systems, the impurities in the grain boundaries are more
and more enriched when, in the process of grain growth, the
grain-boundary volume fraction decreases. The simplest as-
sumption is b2 D, such that
4γ
D
Pdrive ) R
(1)
where R is a small geometric constant. Furthermore, the specific
interphacial energy γ is assumed to be the same for all
boundaries and to remain unchanged in the course of grain
growth. Then the radial velocity of the grain boundary, dD/dt,
is proportional to the product of grain boundary mobility, M,
and driving pressure:
a3
dD(t)
dt
)
- b3D(t)
(9)
D(t)
With the same initial condition as above, the solution of this
differential equation is
dD
dt
MPdrive
(2)
a
b
a3
b3
3 + D0
-
exp(-2b3t)
(10)
2
D(t) )
Hence, combining eqs 1 and 2, for the average grain diameter
D(t), one obtains the differential equation
(
)
x
3
Using D∞ ) a /b this can be expressed as
x
3
3
a1
dD(t)
dt
)
(3)
2
2 - (D∞2 - D0 ) exp(-k3t)
(11)
D(t)
D(t) )
D
x
∞
where the constant a1 comprises all the above-mentioned
quantities; with D(t) ) D0 for t ) 0, the solution of the
differential equation is
with the rate constant k3 ) 2b3 ) 2a3/D∞2. In the present
contribution we shall call this the grain-growth model with size-
dependent impediment.
D(t)2 - Do2 ) k1t
(4)
As outlined above, the constants a1,2,3 contain the product of
specific interface energy, γ, and grain boundary mobility, M.
In compliance with the fluctuation-dissipation theorem, we
relate the mobility of the grain boundary to the self-diffusion
coefficient, Ds, in the grain boundary:
D0 is the initial grain size and k1 ) 2a1, the temperature-
dependent rate constant. The last equation is often written in a
more general form as
Ds
D(t)n - D0n ) k1t
(5)
M )
(12)
kBT
which we henceforth shall call the generalized parabolic grain-
growth model. The empirical grain-growth exponent, n, is in
the above analysis equal to 2; such a process is called parabolic
grain growth. Experimentally, n varies between values of 2 and
4, sometimes even up to 10,11 and for a given system n usually
varies with temperature.15 In any case, in this model the grains
continue to grow at long times.
where kB is the Boltzmann constant. In general form eq 12 is
known as the Einstein equation. The diffusivity obeys the
Arrhenius law with the activation energy, Q, but according to
eq 12 the mobility and thus the kinetic rate constants derived
above exhibit an additional T-1 temperature dependence, such
that
Burke13 and later Grey and Higgins16 took account of the
experimental observation of a limiting grain size: in their model,
as a result of grain growth, the driving pressure vanishes at a
Q2,3
2
Tk2,3D∞
exp -
(13)
(
)
RT

Grain Growth Kinetics of Nanocrystalline Iron
J. Phys. Chem. B, Vol. 104, No. 11, 2000 2469
7/2
-5/2
Hence, the activation energies of the grain boundary diffusion
processes can be obtained from the slopes of plots of Tk1 or
Tk2,3D∞2, respectively, versus T-1; thereby γ is assumed to be
temperature-independent. Concerning the temperature depen-
dence of the retarding terms b, we are not aware of theoretical
predictions.
µ ) D
D
(17)
(18)
area
vol
σ ) exp( ln( D vol/ D area))
x
3.2 Column Length. It is well-known that small grain sizes
cause a broadening of X-ray diffraction peaks. The primary
experimental quantity, however, is the volume-weighted average,
L
vol, of so-called column lengths L, i.e., the lengths of columns
3. X-ray Line-Shape Analysis
of cubic cells aligned perpendicular to the set of diffracting
lattice planes. The interrelation between grain size and column
lengths is visualized in Figure 1 for a spherical particle: the
sphere consists of columns with different lengths
Elastic scatteringsor, synonymously, diffraction of radiation
on mattersmathematically corresponds to a Fourier transforma-
tion from real space to momentum space; hence the X-ray
diffraction pattern of a sample represents a complete mapping
of its structure and its microstructure in momentum space, which
is often called reciprocal space or, in the case of periodic
structures, reciprocal lattice. In most investigations structural
information is extracted from the diffraction pattern, namely
from the angular positions and intensities of the Bragg peaks.
Here, however, we are interested in the microstructure, or, more
strictly speaking, the nanostructure of our sample. Very roughly,
the nanostructure can be quoted by just one quantity, the
volume-weighted average size; this information can, in a certain
approximation, be derived from the line widths of Bragg peaks.
Generally, the line width of a Bragg peak of a nanocrystalline
material is due to the instrumental width (resolution) of the
diffractometer, a broadening due to the small grain size and a
broadening due to microstrain. But, as will be described below,
more detailed information is accessible in the line shapes of
Bragg peaks; their evaluation allows us to characterize the
nanostructure more thoroughly, in terms of a grain-size distribu-
tion and in terms of an isotropic mean-squared microstrain.
Thereby “size”, strictly speaking, denotes the size of coherently
scattering regions. Stacking faults, for example, cause an
apparently reduced grain size if the grain size is calculated from
diffraction peaks; because stacking faults are not essential for
bcc metals, they are not considered here, and we refer to recent
literature.18
L ) nd*
(19)
where d* is the side length of the cubes; d* is a fictitious lattice
spacing to be explained later, in eq 26.
Even for an ensemble of monodisperse spheres, the column
lengths exhibit a distribution; only the central column length
equals the particle’s diameter, and all other columns are smaller.
The primarily obtained average column lengths of an ensemble
of particles can be transformed into average grain sizes if all
the crystallites in the sample have roughly the same shape;25,26
the standard assumption is a spherical shape, then
3
2
D
)
)
L
L
(20)
(21)
area
vol
area
vol
4
3
D
3.3 Scherrer Formula and Williamson/Hall Plots. For
completeness, we briefly mention here the two most-known
formalisms for the evaluation of line widths of Bragg peaks.
Scherrer27 attributed the total width to grain-size effects and
obtained for Γ, the full width at half-maximum:
Kλ
cos θ L
Γ )
(22)
vol
3.1 Size Distribution Function. For nanocrystalline materials,
the (normalized) log-normal distribution function has turned out
to be appropriate1 for describing the grain-size distribution:
The Scherrer constant K equals 0.9 for spherical particles.
Additionally, taking into account eq 21 (which is not always
done in the literature), eq 22 allows, in a very simple way, an
estimate of the volume-weighted average grain size, which for
small grain sizes (large broadening) and small microstrain is
quite reasonable.
2
ln(D/µ)
ln σ
1
1
2
g(D) )
exp -
(14)
(
)
[
]
x
2πD ln σ
The Williamson/Hall technique28 takes into account both size
and strain effects. For the separation, the different θ dependences
are utilized: The size broadening is proportional to cos-1 θ (see
eq 22), and the strain broadening is proportional to tan θ; hence,
according to
It is characterized by the median diameter, µ, and the
geometrical standard deviation, σ. Typical σ-values are: larger
than 2 for materials nanoscaled by ball milling,19 1.7-1.8 for
nanocrystalline metals prepared via inert-gas condensation,20
1.3-1.4 for nanocrystalline metals prepared by pulsed elec-
trodeposition,21,22 1.1-1.2 for nanocrystalline oxide particles
prepared by a sol-gel-type hydrolysis in microemulsion-
provided nanoreactors23 and as low as 1.03 for SiO2 particles
via Sto¨ber synthesis.24 σ ) 1 corresponds to a δ-function. Log-
normal distributions are later displayed in Figure 5; characteristic
Kλ
Γ cos θ )
+ 4ꢀ sin θ
(23)
L
vol
a plot of Γ cos θ versus sin θ enables the separation of
microstrain (slope) from the size (ordinate intersection) effects.
The additive combination of the two line-broadening contribu-
tions in eq 23 implies a Lorentzian line shape for both terms,
which, in general, is not the case; therefore, some size average
is obtained which is not well defined. In general, the microstrain
is anisotropic; therefore, these so-called Williamson/Hall plots
have to be done separately for different directions in the
reciprocal lattice. An apparently anisotropic grain size might
indicate lattice faults. Thus, Williamson/Hall plots exhibit quite
direct nanostructural information. In most cases both the
quantities are the area-weighted average grain size D
and
area
the volume-weighted average grain size D vol, as marked in
Figure 5, which can be expressed in terms of µ and σ:
D
D
area ) µ exp(5/2 ln2 σ)
vol ) µ exp(7/2 ln2 σ)
(15)
(16)
or, vice versa:

2470 J. Phys. Chem. B, Vol. 104, No. 11, 2000
Natter et al.
(1) It represents a very efficient smoothing procedure; data
with a certain amount of statistical fluctuations can still be
handled reliably, i.e., comparatively short measuring times are
acceptable.
(2) The fit functions are known from -∞ to +∞, and so
truncation effects do not exist.
(3) Overlapping of Bragg peakssup to a certain degreesis
no problem.
(4) The determination of the background level is straightfor-
ward.
The disadvantagesinherent in all data fittingsis that the
evaluation is biased: if a given Bragg peak is fitted with one
Pearson VII function, implicitly a monomodal size distribution
is postulated. The fitsapart from smoothing statistical fluctu-
ationsscould, however, also erase real physical features in the
data. One has to be aware of this risk and carefully check the
quality of every fit, for instance by using the ør2 test: ør2 for a
certain peak should be only slightly above 1 over a whole series
of diffractograms. If ør2 deviates appreciably from 1, then the
presumption of monomodality is questionable and a more
complex model should be tried, for instance a bimodal distribu-
tion, i.e., a fit with two Pearson VII functions.
The fit functions R(2θ) of the artificial resolution function
and I(2θ) of the experimental intensity of the sample are
transformed into functions of the variable s ) (2 sin θ)/λ, where
the respective Bragg peak is assumed to be centered at s0. For
the Fourier transformation, both R(s) and I(s) are expressed as
Fourier cosine series (the sine terms vanish for even functions):
Figure 1. Relation between column length and diameter for a spherical
particle.
Scherrer and the Williamson/Hall methods are applied with an
approximate deconvolution of the instrumental width. This can
be accomplished in a very crude way by simply subtracting the
instrumental line width from the experimental one.
3.4 Resolution Correction. Information about the grain-size
distribution is contained in the line shape of Bragg peaks. The
experimentally observed line shape is the convolution of the
intrinsic line shape (scattering function) of the sample with the
line shape of the diffractometer.29 This so-called resolution
function is determined using a standard LaB6 sample which does
not cause any line broadening. Every Bragg peak of LaB6 is
fitted with a Pearson VII function:
2
-m
500
2θ - P0
HWHM
R0
I(θ)
1 + (21/m - 1)
(24)
(
(
) )
R(s) )
+
R cos[nπ(s - s )2d*]
(25a)
(25b)
∑
n
0
2
n)1
Subsequently, the resulting m and HWHM values are empirically
500
I0
parametrized:
Ihkl(s) )
+
I cos[nπ(s - s )2d*]
n 0
∑
2
n)1
HWHM2 ) A tan P02 + B tan P0 + C
m ) DP02 + EP0 + F
The fictitious interplanar distance, d*, is given by
λ
1
2d* )
)
(26)
s2 - s0
2(sin θ2 - sin θ0)
In the case of asymmetric instrumental line shapes, a common
feature of laboratory X-ray diffractometers, it is advantageous
to employ split Pearson VII functions. Using the resulting
empirical parameters A, B, C, D, E, and F, an artificial resolution
function R(2θ) can be calculated at any desired 2θ value. For
obvious reasons (i.e., Fourier transformation of the data has to
be done anyway), the resolution correction is performed by
deconvolution in Fourier space, as is explained below.
3.5 Warren/Averbach Method. The Warren/Averbach30
method is a Fourier method; it allows the separation of size
and strain effects and enables us to determine the grain-size
distribution. The critical step is the Fourier transformation, which
is very sensitive to statistical fluctuations in the data, to
truncation effects, and to the background level. Thereforesfor
the original Warren/Averbach methodsextremely good counting
statistics and a wide angular measuring range are necessary;
these requirements, however, contradict those of a real-time
diffraction experiment. To reach a good time resolution, only
the relevant part of the diffractogramsas narrow as possibles
has to be recorded with counting statistics that are just sufficient.
As a solution to this dilemma we have developed the following,
meanwhile optimized procedure (early versions have been
briefly mentioned before):31,21,22 every relevant diffraction peak
is fitted with a Pearson VII function, and then the resulting fit
function is Fourier transformed. The prior fitting step has a
number of advantages:
where θ2 is the upper limit of the evaluation interval; if the
lower and upper limits, s1 and s2 in terms of the variable s,
were chosen strictly according to the periodicity of the lattice,
i.e., s1 ) 0.5‚s0 and s2 ) 1.5‚s0, then d* would be equal to dhkl
.
The Fourier coefficients are determined from the integrations
R ) 2d* s +1/2d*R(s) cos(nπ(s - s0)2d*)
n ) 0, 1, 2, ...
0
∫
n
s0-1/2d*
(27a)
I ) 2d* s +1/2d*I(s) cos(nπ(s - s0)2d*)
n ) 0, 1, 2, ...
0
∫
n
s0-1/2d*
(27b)
For numerical reasons, to minimize truncation effects, we
usually chose θ2 ) θ0 + 20‚HWHM, because beyond 20 line
widths the Pearson VII function vanishes. Because the experi-
mental intensity, I(s), is the convolution of the scattering
function, A(s) with the resolution function, R(s), deconvolution
is possible by a division of the Fourier coefficients:
In
Ahnkl
)
(28)
Rn
Ahnkl are the cosine Fourier coefficients of the (resolution-
corrected) scattering function of the hkl-Bragg peak. According

Grain Growth Kinetics of Nanocrystalline Iron
J. Phys. Chem. B, Vol. 104, No. 11, 2000 2471
Figure 3. Determination of area- and volume-weighted average column
lengths in the framework of the Warren/Averbach method.
Figure 2. Separation of size and strain effects in the Fourier coefficients
in the framework of the Warren/Averbach method.
to Bertaut,32 n‚d*, the real-space conjugate variable of s-s0, is
identical with the column length, L, discussed in Section 2.2
(see eq 19).
The scattering function, A(s), describes the line-broadening
effects of small-particle size and microstrain (macroscopic strain
would lead to a change of the lattice constant, i.e., to a shift of
the Bragg peak). If size and strain effects are uncorrelated, A(s)
is the convolution of both effects. Correspondingly, then, the
Fourier coefficients of the scattering function are the product
of both terms:
According to eqs 20 and 21, average column lengths are
transformed into average diameters; then, according to eqs 17
and 18, under the assumption of a log-normal size distribution,
the parameters µ and σ, and thus the desired grain size
distribution are obtained. Grain sizes below 5 nm cause Bragg
peak line widths that are too broad for a reliable evaluation.
For standard laboratory diffractometers, the upper grain size
that can be evaluated is about 100 nm; otherwise, the line
broadening is too small. But with 5 nm e D
e 100 nm,
vol
just the nanocrystalline size regime is covered; this explains
the importance of X-ray line shape analyses for nanocrystalline
materials. A synchrotron-based diffractometer exhibits a much
better angular resolution, which extends the measuring range
up to 500 nm, as will be shown below.
Ahnkl ) Ansize Asntrain(hkl)
(29)
The size term does not depend on hkl, whereas the strain
term does; this allows a separation. Usually, one assumes a
Gaussian distributed strain. For small strain, an expansion yields
4. Details of the Synchrotron Experiment
We used the beamline “BM16-Powder Diffraction” of the
ESRF in Grenoble (France). This diffractometer is equipped
with a primary Si(311) double monochromator, thus providing
a variable incident wavelength. We have chosen λ ) 0.4898
Å; at this small wavelength, X-ray absorption in the sample is
low. The measuring range was 10° e 2θ e 30°, which
corresponds to 0.356 Å-1 e s e 1.057 Å-1. For bcc iron, this
range comprises the (110) up to the (220) Bragg reflection. The
measuring time per diffractogram was 1 min; in total, 800
diffractograms have been recorded, but for reason of count
statistics we have assembled them into groups of five; hence
the “time resolution” of our real-time diffraction experiment
was 5 min. The detector system consists of 9 detectors, each
with a Si(111) secondary detector in front of it. In this way an
angular resolution in 2θ of 0.006° FWHM is reached, more than
an order of magnitude better than a common laboratory X-ray
diffractometer. The instrumental resolution was determined
using a NIST-LaB6 standard sample in the manner described
above.
2π2 ꢀ2(L) L2
Ahnkl ) Asnize 1 -
(30)
2
(
)
dhkl
where ꢀ2(L) is the mean-square inhomogeneous strain of a
column of lattice cells of length L, which in most cases is
anisotropic, i.e., different for different sets of reflections (hkl).
In the next step, the Fourier coefficient Ahnkl is normalized such
that Ah0kl ) 1 for all hkl. In this step, all the information
contained in the intensity, including the structural information,
is lost, but simultaneously all the problems connected with
integrated intensities (Lorentz polarization factor, Debye-Waller
factor, absorption, etc.) are eliminated. A plot of the normalized
-2
hkl
Fourier coefficients versus d
yields the strain-corrected
coefficients Asnize as ordinate intersection, as demonstrated in
Figure 2. In the case of cubic symmetry one can, according to
d2hkl ) a2/(h2 + k2 + k2) ) a2/m2 also plot versus m2. In
general, only Bragg reflections of the same (hkl) set can be used
(in the present contribution (110)-(220)), but if in a Williamson/
Hall plot the strain has turned out to be small and isotropic, the
choice of Bragg reflections is not restricted. Subsequently, the
normalized strain-corrected Fourier coefficients are plotted
versus the column length L ) nd* as shown in Figure 3. From
the initial slope at L f 0 and from the integration, the area-
weighted average column length and the volume-weighted
column length, respectively, are obtained:29
The nanocrystalline iron sample, which turned out to be
brittle, was crushed into a powder using an agate mortar in an
Ar glovebox, filled into quartz capillaries (L 0.7 mm) and
sealed. The diffractograms were recorded in transmission in
θ-2θ mode, at the following temperatures: 663, 683, 703, 739,
753, and 783 K. Sample heating was done by means of a hot-
air blower, and the temperature accuracy was ( 2 K.
5. Preparation of Nano-Fe by Pulsed Electrodeposition
dAsize
dL
1
Since several years PED is applied for the preparation of
nanocrystalline metals and alloys, for instance nano-Ni,33,22
nano-Pd,31 nano-Cu,21 nano-Co5 or nano-Ni1-xCux.22 For the
electrochemical deposition of metals with nanocrystalline micro-
structure, very small crystallites have to be formed on the
) -
(31)
(32)
|
L
L
Lf0
area
∞Asize dL )
vol
∫
0
2

2472 J. Phys. Chem. B, Vol. 104, No. 11, 2000
Natter et al.
cathode. For that purpose, use is made of the electrochemical
version of the Kelvin equation:34
2γVm
rc )
(33)
ze0|η|
rc is the critical nucleation radius (i.e., grains with r > rc are
stable), γ the specific surface energy, Vm the atomic volume in
the crystal, z the number of elementary charges e0, and η the
overvoltage.
According to eq 33, the higher the overvoltage, the smaller
the crystallites. With the high voltage (about 10 V), a high
current (up to 2 A/cm2) is connected, i.e., a very high deposition
rate of the metal cations from the electrolyte. The control of
this process via eq 33 is possible only for a few milliseconds;
after that period, the concentration of metal cations in the
vicinity of the cathode is reduced so much that the process
becomes diffusion-controlled. To maintain control of the grain
size, one has to apply a pulsed current, with correspondingly
short pulse widths (a few milliseconds). In the break between
two pulses (several tens of milliseconds), metal cations diffuse,
as desired, from the bulk of the electrolyte into the vicinity of
the cathode. But simultaneouslysundesiredlyson the cathode,
due to exchange currents, Ostwald ripening sets in: the
somewhat larger grains grow at the expense of the smaller ones.
To suppress this exchange process, appropriate organic complex
formers are added to the electrolyte bath: the nanocrystals are
protected during the break, but complexation is not so strong
that metal deposition in the next pulse is impeded.
Figure 4. Set of diffractograms of nano-Fe recorded at T ) 753 K;
only the selected angular range is displayed, which comprises the (110)
Bragg peak.
TABLE 1: Results of Evaluation of Grain-Size Distributions
for Nano-Fe in Initial State and in Final States at the End of
the Different Isothermal Real-Time Experiments
T (K)
D
area (nm)
D
vol (nm)
µ (nm)
σ
strain (%)
initial
663
683
703
739
753
783
16
37
70
19
46
92
103
150
234
395
9.8
22.0
36.6
36.0
42.6
44.5
47.2
1.54
1.58
1.67
1.73
1.82
1.99
2.18
0.684
0.303
0.201
0.179
0.103
0.076
0.023
76
104
145
215
For the electrochemical preparation of nano-Fe, we use a
sacrificial Fe anode and an inert Ti cathode. The electrolyte
(citrate bath) contains 50 g/L (NH4)2 Fe(SO4)2, 20 g/L citric
acid trisodium salt, 10 g/L citric acid, and 40 g/L boric acid.
The bath temperature is 303 K. With the pulse parameters ton
) 2 ms, toff ) 8 ms, and Ipulse ) 0.1 A/cm2, we obtain nano-Fe
with D vol ) 19 nm and σ ) 1.54. The comparatively moderate
pulse parameters are necessary to keep the H2 evolution, the
competing electrochemical reaction, at an acceptable low level.
Generally, with very large overvoltage and correspondingly large
pulse current density, most of the metallic impurities of the
starting materials are co-deposited together with the main
constituent; in the present case, however, under the compara-
tively mild electrochemical conditions described above, a certain
electrolytical refinery effect can be expected such that the
resulting nanocrystalline iron should contain fewer metallic
impurities than the starting materials.
Figure 5. Initial and final grain-size distributions of the different
isothermal grain growths experiments on nano-Fe; the dashed/dotted/
solid marks indicate the µ/ D area/ D values, respectively.
vol
6. Results
Figure 4 shows a selected set of diffractograms recorded at
753 K. We have taken data at five additional temperatures; about
20 diffractograms at each temperature cover a time period of
up to 150 min. It can be clearly recognized in Figure 4 that the
(110) Bragg peak is broad at the beginning of the real-time
isothermal grain-growth experiment, becomes narrow rather
quickly, and then, at later times, does not change significantly
with time. For the line shape analyses, we used the (110) and
the (220) Bragg reflections. The quality of the fits of the profiles
with Pearson VII functions is confirmed by the low ø2r values,
which exhibit a slight but not significant increase with temper-
ature and time, without any sudden change in between.
The foil for the anode was 99.5% Fe purchased from
Goodfellow, with 3000 wt ppm Mn, 1000 ppm Si, <800 ppm
C, and <500 ppm S. The Fe-salt (NH4)2 Fe(SO4)2 contained
<500 ppm Co, <500 ppm Mn, and <500 ppm Ni as main
impurities, according to the manufacturer. Chemical analysis
of the resulting nano-Fe by means of ICP-MS yields the
following (in wt ppm): B 538 ppm, Cr 8.3 ppm, Mn 9.9 ppm,
Co 190 ppm, Ni 541 ppm, Cu 33 ppm, Mo 4.4 ppm, Ag 1.0
ppm, Cd 1.2 ppm, and Sn 1.8 ppm; the boron content obviously
originates from boric acid in the electrolyte. Chemical analysis
of the interstitial impurities in the resulting nano-Fe, by means
of hot extraction, gave the following results (in wt %): hydrogen
0.09%, carbon 0.87%, oxygen 1.5%, and nitrogen 0.20%. The
closed porosity of our PED-nanometal samples is very low;
typical values for the relative bulk density are 97% to 99%,
depending on the grain size.22
In Table 1, the complete results of the evaluation of our
primary X-ray diffraction data evaluation are listed for the initial
state of our nano-Fe and for the final states at the different
temperatures of our experiment. From the primary quantities,
D
and D area, the other quantities have been derived as
vol
described in section 2. The median diameter, µ, and the
geometrical standard deviation, σ, are used to calculate and
visualize the distribution functions in Figure 5: the final medians
increase with temperature by a factor of only 2.1, whereas the

Grain Growth Kinetics of Nanocrystalline Iron
J. Phys. Chem. B, Vol. 104, No. 11, 2000 2473
Figure 6. Temperature dependence of the geometrical standard
deviations σ of the log-normal distributions of nano-Fe at the end of
the different real-time experiments.
Figure 8. Temperature and time evolution of the volume-weighted
average crystallite diameters of nano-Fe; the lines represent fits with
different kinetic grain-growth models. Dashed lines represent fits with
the generalized parabolic grain-growth model; solid lines represent fits
with the grain-growth model with impediment; dashed-dotted lines
represent fits with size-dependent impediment.
Figure 7. Temperature and time evolution of the microstrain of nano-
Fe.
final volume-weighted diameters increase by a factor of 8.6,
i.e., the distribution becomes very broad at high temperatures.
At T ) 663 K, isothermally, the median diameter, µ, increases
moderately with time, from its initial value of 9.8 nm to, finally,
22 nm. The width parameter, σ, changes, slightly from initially
1.54 to finally 1.58; at T ) 783 K, the final values are µ )
47.2 nm and σ ) 2.18, indicating a considerable grain growth
and a very broad distribution. The final σ values of the different
temperature runs are displayed in Figure 6; they exhibit a kind
of kink at about 740 K. At the low temperatures, the major
changes of both µ and σ occur within the first 10 min; after
about 30 min, both µ and σ become more or less time-
independent.
In Figure 7, the evolution of the microstrain with time and
temperature is displayed. For nano-Fe as prepared, the micro-
strain amounts to 0.68%; in the isothermal grain growth
experiments, this value decreases extremely rapidly right at the
beginning and then remains constant at limiting values between
0.3% at 663 K and 0.1% at 753 K. At the highest temperature,
783 K, the microstrain essentially disappears.
Figure 9. Temperature dependence of the impediment factors. Solid
line: from fits with the grain-growth model with impediment; a high
temperature and a low temperature regime can clearly be distinguished.
Dashed-dotted line: from fits with size-dependent impediment.
(2) The solid lines are the results of the fits with the growth
model with impediment, eq 8. The resulting growth-retarding
parameter, b2, is temperature-independent for 663 K e T e 793
K, but then, for 753 K and 783 K, b2 clearly increases with
temperature as displayed in Figure 9; thus, the two temperature
regimes are apparently distinguishable.
(3) The dashed-dotted lines are the results of fits with the
grain-growth model with size-dependent impediment, eq 11. The
resulting b3 values amount to about 0.02 s-1 for all temperatures
except the lowest one, where it is about twice that value (see
Figure 9).
The resulting volume-weighted average diameters, D vol, are
displayed as data points in Figure 8; at high temperatures they
show a fast increase in the grain size at the beginning of each
experiment, and evidently a limiting grain size value at long
times, which exhibits a pronounced temperature dependence.
At less elevated temperatures, only a moderate and compara-
tively smooth grain growth is observed, which stops after a short
time period.
All three models contain two adjustable parameters (D0 is
kept fixed at D0 ) 19 nm). The normalized mean squared
residuals
N
1
2
øn
)
(Dexp - Dfiit)2
(34)
∑
i
2
i)1
ND∞
Fits of the grain sizes with the three kinetic models outlined
in section 2 yield the lines in Figure 8 as follows:
(1) The dashed lines are the results of fits with the generalized
parabolic grain-growth model, eq 5. The resulting grain-growth
exponents n decrease from about 12 at the lowest temperature
to about 3 at the highest temperature of our experiments.
are plotted in Figure 10 as a function of temperature; at the
lowest two temperatures, the generalized parabolic model gives
the best fits, but with very high grain growth exponents n; at
medium and high temperatures, the best fits are obtained for

2474 J. Phys. Chem. B, Vol. 104, No. 11, 2000
Natter et al.
7. Discussion
Nanocrystalline iron has been prepared by pulsed elec-
trodeposition (PED). By this preparation technique, noble metals
with positive standard electrode potential such as Pd31 or Cu21
can easily be prepared in the nanocrystalline state with low
metallic impurity content, low porosity, controlled and adjustable
grain size, and comparatively narrow (1.3 e σ e 1.4) grain-
size distribution. The preparation of nano-Fe by PED, however,
is delicate because the Fe standard electrode potential, EL(Fe2+
/
Fe) ) -0.409 V, is considerably below the standard hydrogen
electrode potential, and so H2 evolution would be the preferential
electrochemical process. However, formation of H2 is strongly
impeded at the (growing) Fe cathode due to the large reaction
overvoltage of the reaction 2Hads f H2. To keep H2 evolution
at a low level, compromises in the pulse parameters are
necessary, and therefore we have reached only a σ value of
1.54 for our nano-Fe as starting material for the kinetic studies.
Still, this means a narrower initial size distribution than can be
achieved by other methods, in particularly by ball milling
(BM),35 the most common method of nano-preparation. Fur-
thermore, PED samples exhibit only a comparatively small
amount of lattice strain, in contrast to BM samples, which
experience severe plastic deformation during milling.
Figure 10. Normalized mean-squared residuals of the fits of Figure
8. Dashed line represent fits with the generalized parabolic grain growth
model; solid line represent fits with the grain growth model with
impediment; dashed-dotted line represents fits with size-dependent
impediment.
The content of metallic impurities of our PED nano-Fe is
low, but interstitial impurities such as O, N, and H cannot be
avoided, because the preparation takes place in aqueous phase
and N2 gas is bubbled through the electrolyte to drive out O2.
Bulk Fe is intrinsically ductile, but for our nano-Fe we notice
a pronounced and (in our view) intergranular brittleness as a
result of strong segregation, during the formation process, of
interstitial impurities in the grain boundaries. Here these
impurities are supposed to enhance the thermal stability of nano-
Fe with respect to grain growth; actually, at room temperature
and up to 550 K, grain growth is not detectable. For all these
reasons we consider our PED-nano-Fe as a well-suited material
for a detailed study of grain growth of nanocrystalline metals.
The experimental method of choice is in situ synchrotron
X-ray diffraction, which to our knowledge has not been used
before for investigations of this kind. The novel feature is the
possibility to observe the evolution of the (volume-weighted
average) grain size over more than 1 order of magnitude in
size in one experiment; thus, as will be discussed below, the
transition from the nanocrystalline regime (5-100 nm) to the
polycrystalline regime can be observed; the latter regime is
accessible only because of the extreme angular resolution of
synchrotron X-ray diffractometers.
Altogether about 800 raw diffractograms have been recorded
and for further treatment grouped into 160 diffractograms, each
containing about 2000 data points. Only because of the efficiency
of the data eValuation scheme outlined in section 3 could this
huge amount of data be handled and could volume-weighted
and area-weighted average diameters be calculated for each
diffractogram. From these primarily obtained quantities the size
distribution has been calculated for each diffractogram. In this
waysand this is also a new feature for a kinetic grain-growth
investigationswe could study the temporal evolution of the size
distribution at different temperatures. We observe a slight
increase in the width of the distribution at low temperatures
and a rather broad final distribution at high temperatures. Thus
we note indications of two different temperature regimes, most
clearly from the kink in Figure 6. All data are compatible with
a monomodal log-normal distribution, and we find no indication
that this type changes in the course of our real-time experiments.
We interpret this as evidence for normal grain growth. We would
Figure 11. Temperature dependence of the rates a2 (solid lines) and
a3 (dashed-dotted lines) in extended Arrhenius representations according
to eq 13; a high temperature and a low temperature regime can clearly
be distinguished.
the model with (constant) impediment. The new model with
size-dependent impediment appears to be no improvement;
visual inspection of Figure 8 gives this indication, too.
Conventional Arrhenius evaluations of the rates a1 ) k1/2
2
and a2,3 ) k2,3D∞ yield activation energies of Q1 ) 221 kJ/
mol, Q2 ) 121 kJ/mol, and Q3 ) 118 kJ/mol. There is, however,
evidence for deviations from straight-line behavior of a2 and
a3. The extended Arrhenius evaluations of these rates, according
to eq 13, are displayed in Figure 11. In mutual agreement both
model 2, with constant impediment, and model 3, with the
retarding term proportional to the size, clearly exhibit different
activation energies in the low- and high-temperature regimes,
respectively. For the kinetic model 2, we obtain in the high-
temperature regime QH2 T ) (173 ( 10) kJ/mol, and in the low-
temperature regime QL2T ) (98 ( 8) kJ/mol. The correspond-
ing values for model 3 are QH3 T ) (165 ( 10) kJ/mol and QL3T
) (84 ( 8) kJ/mol. Conventional, but split, Arrhenius evalu-
ations yield values that are smaller by 6 kJ/mol in the high-
temperature regime and by 5 kJ/mol in the low-temperature
regime, respectively.
Since the microstrain in n-Fe in the as-prepared state
decreases rapidly in the initial stage of grain growth at each
temperature (see Figure 7), one could argue that the kinetic
modeling should disregard the earliest data point in Figure 8
and treat D0 as a variable parameter. Then the growth models
with impediment and with size-dependent impediment fit the
data equally well over the entire temperature range. An
Arrhenius evaluation of the resulting rate constants yields
activation energies of about 150 kJ/mol for both models without
evidence for distinct low and high-temperature regimes.

Grain Growth Kinetics of Nanocrystalline Iron
J. Phys. Chem. B, Vol. 104, No. 11, 2000 2475
like to emphasize that with our data evaluation scheme it is
possible to detect anomalous grain growth, as we have
demonstrated for nano-Ni in a previous ex situ X-ray study.22
Why nano-Fe exhibits normal grain growth and nano-Ni
anomalous grain growth is not clear to us at present.
temperature regime, the resulting activation energy of the rate
constant, QH2 T ) (173 ( 10) kJ/mol, agrees perfectly with the
activation energy of the grain-boundary self-diffusion coefficient
of polycrystalline Fe, Q ) 174.5 kJ/mol.40,41 Actually, apart
from the first and second data points, the grain sizes observed
in the real-time experiments are outside the nanocrystalline size
regime, i.e., they are in the sub-µ-regime. The diffusivity and
growth kinetics in this regime correspond to those of polycrys-
talline materials, and the basic assumption, that grain growth is
connected with the mobility of the grain boundaries and thus
with the grain-boundary self-diffusion, is quantitatively con-
firmed. The low-temperature regime of our real-time experi-
ments corresponds to the nanocrystalline size regime. Here the
activation energy of the rate constant, QL2T ) (98 ( 10) kJ/
mol, is considerably lower, and the absolute value of the rate
constant at the lowest temperature of our experiment, T ) 663
K, is nearly an order of magnitude larger than the extrapolated
high-temperature value. This is in full accordance with Fe tracer
diffusion studies of Tanimoto et al.42 on nano-Fe. They observe
a considerably enhanced diffusion in the nanocrystalline state
with an activation energy of about 109 kJ/mol; the authors
interpret their self-diffusion coefficient in terms of grain
boundary type-C diffusion kinetics40,43 where the volume
diffusion in the crystallites is expected to be negligible. The
resulting retardation term, b2, does not depend on the temper-
ature in the nanocrystalline size regime. Because a higher
temperature also means a larger time-averaged size during the
run, we conclude that the retardation term seems to be
independent of size in the nanocrystalline size regime.
The microstrain, ꢀ2(L) for L ) Dmax, i.e., for the most
frequent diameter, denotes the relative mean-squared deviation
of the lattice spacing from its average value. Our PED-nano-Fe
in the initial state, as prepared, contains a moderate amount of
microstrain, 0.68%. It can be separated into two parts: one part
relaxes very rapidly right at the beginning of each temperature
run, and the other one obviously does not decay (see Figure 7).
We attribute the fast relaxation process to irreversible, local,
single-atomic jumps in the grain boundaries; in this way, with
one or two jumps, local energetic minima are reached, and the
interphacial pressure (see eq 1) on the grains is substantially
reduced. The nonrelaxing part of the microstrain apparently
depends on the temperature of the experiment. In our view,
however, it basically scales with the size of the nanocrystallites
(see Table 1) and, as a further experimental observation, it scales
with the lattice constant of the nanocrystallites. These findings
and their interpretation are the content of a forthcoming
publication and are beyond the scope of the present one. Because
of the very low porosity in our PED-nanometal samples (see
section 5) the pores in our view do not have a significant
influence on the grain-growth kinetics.
The generalized parabolic growth model fits the data well at
low temperatures but with unrealistic, unphysically large grain-
growth exponents n > 10; the model clearly yields the worst
fits at the elevated temperatures (see Figures 8 and 10), although
it is mathematically extremely flexible because of the variability
of n. For the empirical growth parameters n we obtain values
that agree satisfactorily with literature values for BM-nano-Fe
of Malow and Koch,15 Moelle and Fecht,36 and Holzer et al.;37
all these authors used the same model for their data evaluation.
The empirical parameters n change from about 12 at low
temperatures to about 3 at high temperatures. This could be
and has been interpreted as a gradual change of the growth
mechanism, but we consider this very indirect hint not as very
convincing. The determination of an activation energy for the
rate constant k1 is a problem, because with decreasing n the
dimension of k1 changes dramatically, in our case from nm12/s
to nm3/s, a physically not very appealing situation. As an
admittedly questionable measure, we keep n ) 5 constant in
the fit and then obtain an activation energy of 221 kJ/mol, which
is in striking disagreement with the values obtained below for
the same quantity in terms of the other kinetic models. This
value, however, is in accordance with the result of Vandermeer
and Huo,38 Q ) 249 kJ/mol, for grain growth in polycrystalline
iron in the temperature range 883 K e T e 1253 K. The
activation energy for self-diffusion in iron is 251 kJ/mol39,40
and for grain boundary diffusion, 174.5 kJ/mol.40,41 In our view,
the disagreement of the resulting activation energy with the grain
boundary diffusion value and the physically unrealistically large,
experimentally determined growth parameters indicate a failure
of the generalized parabolic grain growth model to describe grain
growth at least in nanocrystalline iron; presumably this statement
holds for nanocrystalline metals in general.
This conclusion is in contradiction with the assumption of
Michels et al.17 in their growth model with size-dependent
impediment. The parameter b3 resulting from a fit of our data
with this model decreases with temperature at the lowest
temperatures of our experiment (see Figure 9) and, as we
suppose, in this way in the nanocrystalline state somehow the
postulated size dependence is compensated. On the other hand,
at higher temperatures the parameter b3 remains roughly
constant, whereas the parameter b2 increases significantly. The
latter behavior seems unphysical, given that the drag on grain-
boundary migration exerted by segregated impurities would be
expected to remain constant or decrease with increasing tem-
perature.
For both kinetic models with impediment discussed above,
disregarding the first data point in Figure 8 leads to an Arrhenius
plot with no clear separation into low and high-temperature
regimes, primarily because the uncertainties in the low-
temperature rate constants are large if D0 is allowed to vary. A
single activation energy is found that is somewhat lower than,
but still compatible with, the activation energy for grain-
boundary self-diffusion in polycrystalline Fe.
Malow and Koch15 have performed a kinetic grain growth
study on BM-nano-Fe in the temperature range 625 K e T e
875 K by means of ex situ standard laboratory X-ray diffraction.
For evaluation of the line profiles of the Bragg peaks, they use
simply the Scherrer equation (eq 22) without taking proper
account of the broadening due to lattice strain, which is large
in BM samples. The resulting grain sizes approach limiting
values at long times, but these values are in striking disagree-
ment with ours (ours are about 1 order of magnitude larger).
Also, the resulting kinetic parameters are in such strong
disagreement with ours that a comparison does not make sense.
We are convinced that Malow and Koch BM- nano-Fe exhibits
a kinetic behavior different from our PED-nano-Fe, but in view
The growth model with impediment yields the best fits of
the three considered models (see Figures 8 and 10). The resulting
fit parameters, the retardation term b2 (Figure 9) and the rate
constant a2 ) k2D∞2 (Figure 11), exhibitsin mutual agreement
and in accordance with the width of the grain-size distribution
as mentioned abovestwo temperature regimes. In the high-

2476 J. Phys. Chem. B, Vol. 104, No. 11, 2000
Natter et al.
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hardly possible.
Summary
By combining a chemically and nanostructurally well-defined
sample (PED-nano-Fe) with a high-performance physical mea-
suring technique (synchrotron X-ray diffraction), we have been
able to produce a large amount of high-quality, real-time
diffraction data on the grain-growth kinetics of nanocrystalline
metals. Using a (primary) data evaluation scheme optimized
with respect to reliability and efficiency, we could deduce from
these many diffractograms kinetic (secondary) data with such
accuracy that a critical comparison with existing kinetic models
of grain growth becomes meaningful.
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of the grain size with time; furthermore, it enables us to observe
a change in the kinetics at the transition from the nanocrystalline
to the polycrystalline regime; indicators are both the rate
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model, we consider the agreement between the high-temperature
activation energy for grain growth and the literature value for
the activation energy of the Fe grain-boundary self-diffusion
coefficient.
A further new feature of our present study is the determination
of grain-size distributions in a real-time experiment. All our
data are compatible with a monomodal log-normal grain-size
distribution. We observe a slight and gradual increase with time
of the median and of the width at low temperatures; at high
temperatures, both parameters change rapidly right at the
beginning of each real-time experiment and then slowly
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Acknowledgment. The present study was initiated and
performed at the University of Saarbru¨cken in the framework
of the “Sonderforschungsbereich 277 Grenzfla¨chenbestimmte
Materialien”, and we gratefully acknowledge the financial
support by the Deutsche Forschungsgemeinschaft. We thank our
colleagues in the SFB, L. Helfen for assistance with the
synchrotron measurements, Prof. Dr. K.-H. Ehses for stimulating
discussions and Prof. Dr. H. Ph. Beck and Dr. K. Ziegler for
the chemical analyses. R. H. thanks the Fonds der Chemischen
Industrie for generous financial support.
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