Full text of DOI:10.1002/1439-7641(20010119)2:1<55::AID-CPHC55>3.0.CO;2-S

[12] J. Hofkens, M. Maus, T. Gensch, T. Vosch, M. Cotlet, F. Köhn, A. Herrmann, K.
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1999, 304, 1.
[14] Y. Karni, S. Jordens, G. De Belder, G. Schweitzer, J. Hofkens, T. Gensch, M.
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[16] T. Weil, U. M. Wiesler, A. Herrmann, K. Mullen, J. Am. Chem. Soc. 2000,
submitted.
Effects of Oxidation on the Nanoscale
Mechanisms of Crack Formation in
Aluminum**
Emily A. A. Jarvis,[a] Robin L. Hayes,[a] and Emily A. Carter*[a]
KEYWORDS:
aluminum
´ cracking ´ density functional calculations ´
[17] F. Morgenroth, E. Reuther, K. Mullen, Angew. Chem. 1997, 109, 647; Angew.
Chem. Int. Ed. Engl. 1997, 36, 631.
materials science ´ surfaces
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[19] F. Morgenroth, C. Kubel, M. Muller, U. M. Wiesler, A. J. Berresheim, M.
Wagner, K. Mullen, Carbon 1998, 36, 833.
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Materials failure, in the form of cracking, is a phenomenon of
fundamental scientific interest and one that impacts a variety of
applications spanning a wide range of fields, particularly those of
materials science and engineering. Experiments have investi-
gated extensively the macroscopic properties associated with
cracking within homogeneous materials as well as at interfaces
between dissimilar materials. Likewise, theoretical modeling via
engineering finite-element approaches[1] and molecular dynam-
ics simulations with empirical embedded-atom potentials[2, 3]
have provided some insight into cracking mechanisms. Never-
theless, these models rely on inherent assumptions concerning
interatomic and/or bulk behavior, a drawback in instances where
fundamental atomic interactions are poorly understood or
improperly characterized by overly simplified model potentials.
A comprehensive study incorporating a first principles approach
at the atomic scale and effectively linking this information to the
macroscopic scale poses an array of challenges, implementa-
tional and otherwise. To date, these difficulties and the computa-
tional expense associated with first principles calculations have
generally motivated employing empirical assumptions to treat
mechanics of the smallest length-scale regime. Resorting to
empirical models limits the chemistry that may be accounted for.
Accordingly, despite widespread scientific interest in the crack-
ing phenomenon, aspects of the microscopic failure mecha-
nisms remain largely a mystery. Herein, we investigate some
aspects of the atomic-level properties which lead to chemically
induced crack formation within a simple model. Finding
methods to smoothly and effectively couple microscopic to
macroscopic modeling is an active area of research[4, 5] and will
provide much-needed insight into a complete mechanism for
chemically induced cracking.
[34] G. Schweitzer, G. De Belder, S. Jordens, Y. Karni, F. C. De Schryver,
unpublished results.
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8186.
Aluminum is an important engineering material used in a
variety of applications; to understand its behavior under stress is
essential. Under ambient conditions, a self-limiting oxide layer
forms on the aluminum surface and protects the underlying
metal from further oxidation. This, in addition to their light
[42] P. Changenet, H. Zhang, M. J. van der Meer, K. J. Hellingwerf, M. Glasbeek,
Chem. Phys. Lett. 1998, 282, 276.
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283.
[a] Prof. E. A. Carter, E. A. A. Jarvis, R. L. Hayes
Department of Chemistry and Biochemistry
University of California, Los Angeles
Los Angeles, CA 90095-1569 (USA)
Fax: (‡1)310-267-0319
Received: July 31, 2000 [Z80]
E-mail: eac@chem.ucla.edu
[**] We are grateful to the US Air Force Office of Scientific Research through the
US Department of Defense MURI program for support of this work.
CHEMPHYSCHEM 2001, No. 1 ꢀ WILEY-VCH-Verlag GmbH, D-69451 Weinheim, 2001 1439-4235/01/02/01 $ 17.50+.50/0
55

weight and strength, makes aluminum alloys popular materials
from which to fabricate, for example, airplane components.[6]
Since the oxidation of aluminum affects its mechanical strength,
it may prove critical to account for such effects in a simulation of
crack growth. Accordingly, we consider a simple model of the
effect of oxidation on crack formation by examining, at the
atomic level, how cracks form in aluminum and its fully oxidized,
stable partner a-Al2O3. More specifically, we calculate the
dependence of the crystal energies upon tensile stress applied
perpendicular to the face-centered cubic (fcc) Al (111) planes and
hexagonal a-Al2O3 (0001) planes, also known as the basal plane
of alumina. These planes were chosen since the most energeti-
cally favorable, low-index surfaces of aluminum are those parallel
to the (111) planes[7] and Al2O3 formed from the oxidized metal
has been shown to grow with surfaces parallel to the (0001)
planes.[8]
Figure 1a shows the area-normalized energy increase relative
to the bulk crystal with respect to width of vacuum introduced,
namely, the initial crack separation. The initial steep slope
represents a region where relaxed atoms of the material
uniformly expand to heal the crack. Conversely, the region with
small slope corresponds to the situation where the vacuum is
Using periodic slab density functional theory (DFT)[9, 10] within
the generalized gradient approximation (GGA),[11, 12] we inves-
tigate crack formation in a-Al2O3 and aluminum.[13] To simulate
crack formation, we introduce a region of vacuum parallel to the
bulk (0001) planes for a-Al2O3 and the bulk (111) planes for Al
and allow the atomic coordinates of the system to relax from
that starting point. We perform a series of such calculations,
introducing successively wider regions of vacuum until the
relaxed system energetics are equivalent to those of isolated
surfaces.
Editorial Advisory Board Member:[*]
Emily Ann Carter is a Professor of
Chemistry at the University of Califor-
nia, Los Angeles. One of those rare
native Californians, she received her
B.S. in Chemistry at UC Berkeley in
1982 and Ph.D. in Chemistry at Cal-
tech in 1987. She ventured eastward
for a one year postdoctoral fellowship
at the University of Colorado, Boulder,
before returning to Southern Califor-
nia as an assistant professor at UCLA in 1988. She was promoted to
tenure in 1992 and to her current position in 1994. She is known for
her work combining ab initio quantum chemistry with dynamics
and kinetics, especially as applied to surface chemistry. More
recently, her interests have moved toward bridging chemistry, solid
state physics, materials science, and mechanical engineering, with
her work on linear scaling, orbital-free density functional methods
that afford treatment of thousands of atoms from first principles
and her embedding theory that combines quantum chemistry with
band structure calculations. These techniques are now being
combined with finite element approaches to undertake multi-
length-scale simulations of materials. This work has garnered
Figure 1. a) Energy versus separation relative to the bulk for a-Al2O3 and Al with
values for both initial (unrelaxed) atomic coordinates and final (relaxed) atomic
coordinates. Separation refers to the width of crack introduced (zero separation
represents the bulk equilibrium structure). b) Relaxation versus initial separation
relative to the bulk for a-Al2O3 and Al. Zero relaxation represents the bulk
equilibrium interlayer spacing. c) Comparison of UBER curves and our values (the
same as (a)) obtained for Al crack formation. The fits from UBER for both the
relaxed and unrelaxed structures involved separate optimization of the UBER
parameters.
her
a number of awards, including Fellowships of the
American Vacuum Society, the American Physical Society, and
the AAAS. More details of her work can be found at her website:
http://www.chem.ucla.edu/carter.
[*] Members of the Editorial Advisory Board will be introduced to the readers
with their first manuscript.
56
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too large for crack healing to occur upon relaxation; instead, the
system is forced to form a surface on either side of the crack. The
calculated surface energy of a-Al2O3 is much higher than that of
aluminum; it is greatly affected by relaxation of atomic
coordinates due to unfavorable ªbareº Alꢀ3‡ on the initially
cleaved surface. Upon relaxation, the Al cations relax toward the
plane of oxygen atoms, as has been noted previously.[14±18] From
this plot, we can estimate the cleavage energy, which is the
energy required to form a crack in each material, from the energy
increase, relative to bulk, of the relaxed structure immediately
after a nonhealing crack is formed. For Al2O3, this occurs at a
separation of 1.2 Š with a cleavage energy of 2.6 Jmꢁ2 relative to
the bulk, while the cleavage energy of aluminum is much lower,
1.5 Jmꢁ2 at a separation of 2.9 Š. Although the significantly
higher value of a-Al2O3 compared to Al may seem counter-
intuitive, recall the present discussion is limited to ideal cleavage
energetics that do not account for effects such as dislocation
nucleation, inherent to ductile cracking. Creating such transient
defects permits macroscopic aluminum to appear much ªstron-
gerº due to its more efficient dissipation of crack-tip energy via
such mechanisms; the inclusion of macroscopic energy dissipa-
tion is essential for a macroscopic strength estimate. On the
other hand, dislocations are less likely to form in Al2O3, due to the
rigidity of ceramic materials, and hence the cleavage energy
calculated here for Al2O3 may be realistic. While Al2O3 appears
ªstrongerº than Al when no dissipative mechanisms are allowed,
its brittleness results in crack formation on a much shorter
separation length scale, as we now describe.
In Figure 1b, we plot the relaxation of surface atomic
coordinates from their bulk value with respect to the width of
the vacuum region introduced. This figure dramatically displays
the greater extent of the separation region over which Al,
relative to Al2O3, is able to heal the introduced crack. Such
behavior is indicative of ductile versus brittle materials failure.
The a-Al2O3 forms cracks when only 1.2 Š of vacuum is
introduced, whereas Al is able to heal cracks over 2.3 Š wide!
The interlayer spacing in bulk aluminum is also 2.3 Š, so this
introduced vacuum results in an interlayer separation of 4.6 Š
over which the Al crystal can heal. The interlayer spacing
between the Al in a-Al2O3 is 0.5 Š, hence the crystal is only able
to heal over an interlayer separation of ꢀ1.6 Š. The Al(111)
surface expands slightly (1.5%) upon relaxation for large separa-
tions, consistent with experimental observations.[19, 20] As explained
for Figure 1a, the Al2O3 (0001) surface experiences dramatic inward
relaxation. A surface is created on both sides of the crack;
accordingly, this plot indicates twice the surface relaxation.
Figure 2. Valence electron-density plots for a-Al2O3 and aluminum. Dark values
indicate regions of highest density. (To facilitate ease of visualization, due to the
high values of the localized electron density on oxygen, the a-Al2O3 plots show ten
equivalent contours whereas the corresponding aluminum plots only have six
contours.) a) Al2O3 unrelaxed structure with 1.0 Š vacuum introduced. Note the
continuity of electron density across the crack. This crack heals upon relaxation.
b) Al2O3 unrelaxed structure with 1.2 Š vacuum introduced. Note the lack of
electron density bridging the crack. This crack further separates (does not heal)
upon relaxation. c) Aluminum unrelaxed structure with 2.1 Š vacuum introduced.
Note the continuous shaded area of low electron density bridging the crack. This
crack heals upon relaxation. d) Aluminum unrelaxed structure with 2.9 Š vacuum
introduced. Note the lack of electron density connecting the two surfaces. This
crack does not heal upon relaxation.
region in Al2O3 shows a dramatic decline going from 1.0 Š to
1.2 Š of introduced vacuum (Figures 2a and b). By contrast, such
a decline is observed for Al only beyond 2.0 Š vacuum. The
behavior associated with relaxing metal surfaces in close
proximity, namely the ªavalancheº effect or ªjump to con-
tactº,[21±24] is related to the extent of the electron density into the
vacuum. The contrast in decay rate of the electron density
between insulators and metals is central to the atomic-level
reasons for brittle versus ductile behavior. The long range of the
density falloff in the metal allows atomic-scale preference for
alternate modes of energy dissipation, namely, atomic ªstretch-
ingº to bridge the void. Naturally, at the macroscopic level,
further evidence for ductile behavior is shown by the preference
for forming dislocation planes over crack-tip propagation.
Although such behavior of the electron density has not been
previously compared for the specific systems of Al2O3 and Al, it is
a result anticipated from understanding electron-density decay
While the aluminum crack can heal even with introduction of a
void larger than 2.3 Š, the Al2O3 can only heal across crack widths
of ꢂ1.1 Š. Figure 2 provides an explanation for this observation
with plots of valence electron density. The highly localized
electron density of the ionic crystal a-Al2O3 appears in sharp
contrast to the delocalized (metallic) electron density of
aluminum.[*] The electron density extending across the vacuum
as a function of the relative band gap in the crystal.[25±27]
A
[*] The lack of electron density at the atomic positions in the Al crystal is due to
the pseudopotential which replaces and mimics the effect of the core
electrons on the valence electrons. Recall the ABC stacking of the fcc (111) cell.
previous study designed to model the (111) surfaces of the
semiconductor Si showed a decline of electron density roughly
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57

intermediate to the void widths required for ªcrackingº of our
insulating and metallic systems.[28] A much earlier series of
studies on metal adhesion, including that of aluminum, dis-
played a ªstrong bondingº range of roughly 2 Š.[29] Recently,
Ismail-Beigi et al. derived a simple empirical approximation to
the electron-density decay for metals and crystals with relatively
small bandgaps. However, this same study concluded that for
tightly bound insulators,[**] a simple analytical form to describe
the spatial decay of the electron density is not readily
obtainable; rather, it strongly depends on specifics of the atomic
potentials in the crystal.[30] This highlights the importance of a
high quality materials description at the atomic scale, even when
one is interested in macroscopic phenomena.
Finally, this work implies that the oxidation of aluminum
enhances both crack initiation at Al surfacesÐdue to the brittle
behavior of Al2O3Ðand perhaps crack propagation, since
ambient air will oxidize both the crack tip leading edge and its
surfaces, to render cracking into the bulk more facile.
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The so-called universal binding energy ± distance relationship
(UBER) is a popular empirical relation, purported to describe
cohesion and adhesion of metals, that has been shown to
capture the essential features of the energy versus separation
curve for a variety of covalent diatomic species and unrelaxed
metal surfaces.[31] An attempt to describe ionic crystals, related in
spirit to the UBER, has been proposed but has been limited to
very simple crystals in application.[32] Just as generic analytical
forms for the decay of the electron density are inadequate for
tightly bound insulators, so too the UBER is unable to capture
the features of relaxed energy versus separation curves.
Although the UBER works fairly well for rigidly separated cracks
in metals, obtaining a good UBER fit is not possible when surface
relaxation is allowed. For example, Figure 1c displays our UBER
fits for both the unrelaxed and relaxed aluminum data sets. The
two parameters in the UBER were varied to obtain optimal fits.
Given the small surface relaxation in aluminum compared to
Al2O3 and that the UBER was only claimed to be applicable to
metals and covalently bonded diatomic species,[31] this should
have been the optimal case for the UBER to be successful. The
large relaxation of the Al2O3 surface and the ionicity of the crystal
make it a very poor candidate for the UBER approximation.
Nevertheless, despite the relatively ªidealº properties of alumi-
num for application of the UBER, the approximation breaks
down with inclusion of surface relaxation. Similar limitations in
the UBER approximation apply to previous theoretical studies of
the adhesive ªavalancheº[21±24, 28] and in attempts to model
experimentally observed behavior between an STM tip and an Al
surface.[33, 34] This provides further incentive for use of ab initio
data as input to continuum models, since surface relaxation will
occur on the time scales for most mechanisms of crack
propagation.[35]
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electrons; nonlinear core corrections to exchange and correlation were
included for Al. We tested for convergence with respect to k-point
sampling density and kinetic energy cutoff of the plane wave basis for
bulk a-Al2O3 and fcc Al structures. As a result of these convergence tests,
we employed a k-point sampling of 3 Â 3 Â 1 for the hexagonal unit cell
of Al2O3 and 8 Â 8 Â 8 for the bulk fcc cell of Al. For both sets of
calculations, we used the same kinetic energy cutoff of 338 eV for the
planewave basis and 554 eV for the augmentation charge basis needed
for our ultrasoft pseudopotentials. We optimized the volume of the
primitive crystals by performing
a series of single point energy
calculations uniformly scaling the lattice vectors within ꢀ5% of the
equilibrium value and fitting to the Murnaghan equation of state. The
optimum bulk cell lattice vectors (which are within ꢀ1% of experiment
[C. Rodriguez, O. Cappannini, E. Peltzer y Blanca, R. Casali, Phys. Status
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construction of the cells used in crack simulation, with vacuum added
beyond these values in the direction perpendicular to the Al (111) and
Al2O3 (0001) planes, respectively. Test cases for Al with 6, 9, 12, and 15
atomic layers were performed. For Al2O3, test cases of 6, 12, and 18 Al2O3
units were also investigated. From these calculations, we determined that
nine atomic layers for Al and six Al2O3 units (ꢀ18 atomic ªlayersº) for Al2O3
were sufficient to minimize the interaction of the two surfaces across the
periodic slab. In the Al calculations, only three atomic layers on either side
of the crack were allowed to relax, that is, the atomic coordinates of the
center three layers were kept fixed to their bulk values. Tests have shown
that this constraint has little effect on the results.
In sum, we find that the sharp falloff in electron density at
Al2O3 surfaces provides a nanoscale explanation for the brittle-
ness of alumina, since cracks form after the lattice experiences
small expansions that correspond to loss of ªbridgingº electron
density across the crack. Moreover, the large structural relaxation
of Al2O3 surfaces inhibits healing of small cracks, once they are
formed. This structural relaxation diminishes the acceptability of
simple interaction models in macroscale simulations and
suggests such relaxation should be accounted for in the future.
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58
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[35] Hypersonic cracks may be an exception: F. F. Abraham, H. Gao, Phys. Rev.
Lett. 2000, 84, 3113 ± 3116.
The compound under investigation is a bent-core or so called
banana molecule, with the structure given by 1.
Received: August 9, 2000 [Z88]
Fractal Growth Patterns in Liquid
Crystals**
Its phase-transition temperatures, as determined in this study
by polarising microscopy on cooling, are Iso. 169.6 B2 150.3 BX
145.5 Cryst. (all temperatures in degrees Celsius). These vary
from the originally reported transition temperatures,[15] which is,
however, of no relevance to the present investigations. In this
study, we are only concerned with the transition from the
isotropic phase to the liquid crystalline B2 phase. The transition
enthalpy for Iso. !B2 was determined by differential scanning
calorimetry to be DH ˆ 5.7 kJmolꢁ1.[15] The B2 phase is a fluid
smectic phase of C2v symmetry. It has recently attracted much
attention, due to its polar order and antiferroelectric switching
behavior, despite the fact that the constituent molecules are
achiral.[16] For further details on banana molecules, phase
structure and properties, refer to a recent review by Pelzl et al.[17]
The compound does show a rather broad two-phase region
on the formation of the liquid crystalline phase of about 3 K,
which is quite common for bent-core mesogens. As the sample
is not a mixture of different components, this is attributed to
impurities contained within the LC. Here, the transition temper-
ature was taken as the one where first germs of the B2 phase
were observed within the isotropic melt on very slow cooling.
For the determination of the fractal dimension D, several
different methods were employed, as listed below.[1, 2, 18]
1) The box-dimension method, where the fractal dimension Db
is given by the exponent in the proportionality in Equa-
tion (1), with N(d) representing the number of boxes of size d
occupied by the data set.
Ingo Dierking*[a]
KEYWORDS:
crystal growth ´ fractals ´ liquid crystals ´ phase transitions
Fractal geometry is nowadays employed in many branches of
science, from physics, chemistry and biology to engineering and
geology. It can be used in the description of percolation systems,
growth models, material fracture or surfaces and interfaces, to
name just a few of its applications. For an introduction on
general topics of fractals and disordered systems, see references
[1, 2]. In the area of liquid crystal (LC) research, fractal analysis
has been employed in the study of the Saffman ± Taylor
instability in Hele-Shaw cells.[3±6] Patterns formed in viscous
fingering were found not to be fractal, while their perimeter was.
An estimation of the fractal dimension of a dendritic-type texture
of a discotic columnar ± hexagonal ordered phase was reported
in reference [7]. Other investigations were related to the
[a] Dr. I. Dierking[‡]
Department of Physics
Chalmers University of Technology
41296 Göteborg (Sweden)
‡
[ ] Present address:
Institut für Physikalische Chemie
Technische Universität Darmstadt
Petersenstrasse 20, 64287 Darmstadt (Germany)
Fax: (‡49)6151-16-4924
1
dD
N(d) ꢀ
(1)
b
E-mail: dierking@hrz2.hrz.tu-darmstadt.de
2) The information-dimension method, which defines the fractal
[**] The compound used in this study was kindly provided by G. Pelzl (Halle,
Germany). I would like to acknowledge the financial support of the Fonds
der Chemischen Industrie, the Swedish Foundation for Strategic Research
and the European network LC-PHOTONET.
dimension Di from the proportionality in Equation (2).
I(d) ꢀ ꢁ Di log(d)
(2)
CHEMPHYSCHEM 2001, No. 1 ꢀ WILEY-VCH-Verlag GmbH, D-69451 Weinheim, 2001 1439-4235/01/02/01 $ 17.50+.50/0
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