A maximum in the strength of nanocrystalline copper.

Abstract

We used molecular dynamics simulations with system sizes up to 100 million atoms to simulate plastic deformation of nanocrystalline copper. By varying the grain size between 5 and 50 nanometers, we show that the flow stress and thus the strength exhibit a maximum at a grain size of 10 to 15 nanometers. This maximum is because of a shift in the microscopic deformation mechanism from dislocation-mediated plasticity in the coarse-grained material to grain boundary sliding in the nanocrystalline region. The simulations allow us to observe the mechanisms behind the grain-size dependence of the strength of polycrystalline metals.

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A Maximum in the Strength of
Nanocrystalline Copper
Jakob Schiøtz* and Karsten W. Jacobsen
We used molecular dynamics simulations with system sizes up to 100 million
atomstosimulateplasticdeformationofnanocrystallinecopper.Byvaryingthe
grain size between 5 and 50 nanometers, we show that the flow stress and thus
the strength exhibit a maximum at a grain size of 10 to 15 nanometers. This
maximumisbecauseofashiftinthemicroscopicdeformationmechanismfrom
dislocation-mediated plasticity in the coarse-grained material to grain bound-
ary sliding in the nanocrystalline region. The simulations allow us to observe
the mechanisms behind the grain-size dependence of the strength of poly-
crystalline metals.
The yield stress, flow stress, and hardness of
metals and alloys usually increase when the
grain size of the material is reduced (1, 2). For
a range of materials, the increase in yield stress
is inversely proportional to the square root of
the grain size, and the other quantities follow a
similar relationship. This relation, known as the
Hall-Petch relation, is well established experi-
mentally from millimeter-sized grains down to
the submicron regime and has led to sugges-
tions of nanocrystalline metals as candidates for
high-hardness materials. It is thus relevant to
ask how deep into the nanoscale regime this
behavior can be expected to continue. Experi-
ments seem to indicate that the hardness levels
off at small grain sizes (3, 4) and in some cases
even decreases (5, 6), but the situation is still
unclear because of difficulties in preparing
high-quality samples (3, 7).
The plastic behavior of crystalline mate-
rials is usually controlled by the motion of
dislocations, which are line defects in the
crystalline lattice. The Hall-Petch relation is
believed to be because of the grain bound-
aries hindering dislocation activity, thereby
making plastic deformation more difficult at
small grain sizes. However, the detailed
mechanisms behind the relation are not fully
resolved (8, 9). For grains smaller than 12
nm, molecular dynamics (MD) simulations
for copper and nickel have shown a decreas-
ing flow stress as the grain size is reduced,
i.e., a reverse Hall-Petch effect (10–14). The
simulations indicate that the deformation
mechanism is different in this regime. The
plastic deformation is no longer dominated
by dislocation motion but is instead carried
by atomic sliding in the grain boundaries,
which for very small grains constitute a sub-
stantial fraction of the material. When the
grain size is reduced, the fraction of grain
boundary atoms increases, leading to the ob-
served softening. It has thus been anticipated
(15–17) that a maximum in flow stress and
hardness should appear in the crossover re-
gime between larger grains with dislocation-
mediated plastic deformation and smaller
grains with grain-boundary sliding. Here, this
picture is put on a much firmer basis by direct
atomistic simulations of the crossover re-
gime. Furthermore, the simulations provide
insights into which dislocation mechanisms
are active in the crossover regime and thus
facilitate accurate modeling of plasticity at
longer length scales.
The simulations are carried out with the use
of standard MD techniques, with the atomic
interactions in copper treated by an effective
medium theory potential (18). The large num-
berofatomsnecessarytodescribeagrainstruc-
ture with about 50-nm grains realistically re-
quires a very efficient and massively parallel
implementation of the dynamics and also puts
high demands on hardware stability during the
simulations. After the setup of the nanocrystal-
line structure (19), the system is uniaxially
strained with a strain rate of 5 ? 108s?1.
During the elongation, the stress is calculated,
and the resulting stress-strain curves from 10
simulations with varying grain sizes are shown
in Fig. 1A. In the systems with the smallest
grain size, grain boundary sliding gradually sets
in as the stress builds up, but only very limited
dislocation activity is seen in the grains. In the
more coarse-grained systems, grain boundary
sliding is less important, and numerous dislo-
cations are nucleated at 2.5 to 3% strain. No
dislocations are seen at lower strains, because
the simulations start from perfect dislocation-
free structures and a large stress is required to
nucleate dislocations from grain boundary
sources. Once dislocations appear in the grains,
massive plastic deformation occurs, and the
stress drops somewhat. The initial peak in flow
stress before and during the nucleation of the
dislocations is amplified by the large strain rate:
Lowering the strain rate by an order of magni-
tude almost completely removes the overshoot
in the stress-strain curves (fig. S1).
After about 6 to 7% of deformation, a
steady-state-likesituation
where the stress is nearly independent of
strain. This flow stress, however, depends
strongly on the grain size, as shown in Fig.
1B. At grain sizes below about 10 nm, the
flow stress increases sharply with grain
size, reaching a maximum value of about
2.25 GPa. For grains larger than 15 nm, the
flow stress decreases again.
isobtained
Center for Atomic-Scale Materials Physics (CAMP),
Department of Physics, Technical University of Den-
mark, DK-2800 Lyngby, Denmark.
*To whom correspondence should be addressed. E-
mail: schiotz@fysik.dtu.dk
Fig. 1. The grain-size depen-
dence of the flow stress. (A)
Stress-strain curves for 10 sim-
ulations with varying grain siz-
es. (B) The flow stress, defined
as the average stress in the
strain interval from 7 to 10%
deformation. The error bars in-
dicate the fluctuations in this
strain interval (1 standard devi-
ation). A maximum in the flow
stressis seen
sizesof10to15nm,causedby
a shift from grain boundary–
mediatedto
mediated plasticity.
forgrain
dislocation-
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Because of the short time scale accessible
with molecular dynamics, the sample must be
deformed rapidly. The calculated flow stresses
are thus obtained at a very high strain rate as
comparedtoexperiments,andnotunexpectedly
they are therefore somewhat higher than the
flow stresses observed experimentally. For
grain sizes above 15 nm where experiments are
readily available, the difference is a factor of 2
to 3 (3). However, one should also note that
microvoids and flaws are experimentally un-
avoidable and may tend to reduce the measured
values(3,7).Ifthestrainratesensitivitiesofthe
two deformation mechanisms are different, the
balance between them could be different at
experimentally accessible strain rates. In that
case, the optimal grain size would shift slightly
compared to what we report here.
The mechanisms behind the plastic defor-
mation can be disclosed through an atomic-
scale analysis of the simulation results. Fig-
ure 2 shows atomic-scale structures obtained
after 10% deformation and compares them
with the atomically resolved strain that re-
sults from an additional 1% deformation. For
the small grain sizes (Fig. 2, C and D), the
strain is localized in the grain boundaries,
indicating that the main deformation mecha-
nism operating here is grain boundary slid-
ing. For the larger grains (Fig. 2, A and B),
the strain lies mainly at glide planes in the
interior of the grains, although some strain is
still seen in the grain boundaries. This is an
indication that the deformation has occurred
mainly through dislocations moving through
the grains. A change in deformation mecha-
nism has therefore occurred. Of course, this is
not a sharp transition, and, even in the case of
the 49-nm grains, some deformation is seen
to occur in the grain boundaries. Earlier sim-
ulations with grain sizes up to 12 nm have
shown that for small grains the dislocation
activity increases with grain size as a precur-
sor to the transition (20). Recent simulations
of nanocrystalline aluminum for large grains
with a quasi-two-dimensional grain structure
also demonstrated dislocation-mediated plas-
tic deformation (21, 22). There is experimen-
tal evidence for a change in mechanism, too.
Nanocrystalline gold with grain sizes in the
10- to 20-nm range deforms without disloca-
tion activity (23), whereas dislocations are
observed in the deformation of nanocrystal-
line nickel with a grain size down to 30 nm
(20, 24). Dislocations may have been seen in
nickel grains as small as 10 nm, but because
of the difficulties of controlling the image
conditions in such small grains they could not
be identified unambiguously (20).
Atomic-scale analysis of the simulations for
large grains reveals more detailed information
about the dislocation processes in the Hall-
Petch regime, consistent with the behavior ex-
pected in coarse-grained materials. In the flow
regime, a great variety of dislocation processes
are observed. Throughout the simulation, new
dislocations are generated at the grain bound-
aries, glide through grains, and are absorbed at
other grain boundaries. At these grain sizes, the
main sources of new dislocations are the grain
boundaries rather than, for example, Frank-
Read sources in the grain interiors. The dislo-
cations sometimes intersect or get trapped by
stacking faults. However, the interactions do
not lead to dislocation tangles and more perma-
nent immobilization of the dislocations, al-
though rarely a few Lomer-Cottrell locks are
created and soon after destroyed. This is in
good agreement with the stress-strain curves,
which do not exhibit strain hardening; i.e., the
flow stress does not increase with strain. This
Fig. 2. The deformation mode at
different grain sizes. (A) The struc-
ture that appears in the simulation
with average grain diameter d ? 49
nm after 10% deformation. Blue at-
oms are in a perfect face-centered
cubic crystalline environment; yel-
low atoms are at stacking faults and
twin boundaries; and red atoms are
in grain boundaries and dislocation
cores. (B) The additional strain ac-
cumulated when the deformation is
increased from 10 to 11%. The
main deformation is seen to occur
on slip planes in the interior of the
grains. (C and D) show the same for
a system with d ? 7 nm. Here, the
majority of the deformation occurs
in the grain boundaries. Because the
system in (A) and (B) contains 102
million atoms, the circles indicating
the individual atoms have been omitted. The scale bars in (B) and (D) are 5 nm.
Fig. 3. A look inside
the large grain in the
upper-left corner of
Fig. 2A. Only atoms in
grain boundaries and
dislocation cores are
shown; atoms in dislo-
cation cores are light-
er colored than grain
boundary
the right side, a full
dislocation split into
two partial
tions is seen (A); most
other dislocations are
single Shockley partial
dislocations. Some of
the dislocations pile
up against the grain
boundary in the back-
ground;the
are indicated by the
arrows (B) and (C). By
watching an anima-
tion of the simulation,
it can be seen that
these are indeed pile-
ups. The dislocation
activity leading to this
configuration is shown in Movie S1.
atoms.In
disloca-
pileups
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lack of strain hardening is believed to contrib-
utetothelowtensileductilityofnanocrystalline
metals (25). The most notable feature observed
in the simulations is the formation of disloca-
tion pileups: a series of dislocations being
pushed toward a grain boundary by the stress,
as can be seen in Fig. 3. A pileup consists of
multiple dislocations queued up on the same (or
nearby) slip planes, pressed toward a grain
boundary by the applied stress, but held apart
by their mutual repulsion. As new dislocations
arrive in the rear of the pileup, their stress field
helps to press the front-most dislocations into
the grain boundary (Movie S1). Arrays of dis-
locations resembling pileups have been seen
moving through copper grains as small as 50
nm (26). The pileup labeled “B” in Fig. 3
consists of five to six dislocations in a 35- to
40-nm-long pileup, consistent with the expect-
ed size of such a pileup at a tensile stress near 2
GPa (9), although the elastic fields from other
nearby dislocations clearly perturb the pileup.
Several models for the Hall-Petch behavior
have been suggested (8, 9), and it is difficult if
not impossible to distinguish between them ex-
perimentally, but these simulations provide
someinsight.Thepresenceofpileupsabovethe
hardness maximum gives some credibility to
the original suggestion of the Hall-Petch effect
being mainly caused by pileups (1): For small
grains, the extent of a pileup is limited and
therefore the local buildup of stress in a pileup
is reduced. A higher applied stress is thus re-
quired before plastic flow occurs. Estimates for
grain size where the breakdown of the Hall-
Petch effect occurs depend somewhat on the
detailed assumptions of this model, but they fall
in the range from 11 nm (15, 17) to 50 nm (27)
for copper, in agreement with the 10 to 15 nm
found for the hardness maximum in the simu-
lations. Other models focus on grain boundary
sources (28), which certainly also play a crucial
role in the simulations. Deformation twins have
been suggested as dislocation obstacles in the
smallest grains (22, 29). Only a few twins are
observed in our simulations, but a large number
of stacking faults might play the same role.
However, if they contributed substantially to
the Hall-Petch effect, it would be seen in Fig.
1A as work hardening, because the number of
stacking faults increases during the simulations.
Furthermore, on the basis of the simulations,
models explaining the Hall-Petch effect as
stemming from increased work hardening in
small grains (30, 31) can certainly be excluded
for nanocrystalline copper.
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Supporting Online Material
www.sciencemag.org/cgi/content/full/301/5638/1357/
DC1
Materials and Methods
Fig. S1
Movie S1
9 May 2003; accepted 23 July 2003
Christopher W. Schadt,1* Andrew P. Martin,1David A. Lipson,2
Steven K. Schmidt1†
The finding that microbial communities are active under snow has changed the esti-
matedglobalratesofbiogeochemicalprocessesbeneathseasonalsnowpacks.Weused
microbiologicalandmoleculartechniquestoelucidatethephylogeneticcompositionof
undersnowmicrobialcommunitiesinColorado,theUnitedStates.Here,weshowthat
tundra soil microbial biomass reaches its annual peak under snow, and that fungi
account for most of the biomass. Phylogenetic analysis of tundra soil fungi revealed a
highdiversityoffungiandthreenovelcladesthatconstitutemajornewgroupsoffungi
(divergentatthesubphylumorclasslevel).Anabundanceofpreviouslyunknownfungi
thatareactivebeneaththesnowsubstantiallybroadensourunderstandingofboththe
diversity and biogeochemical functioning of fungi in cold environments.
About40%oftheterrestrialenvironmentconsists
of biomes that are covered by snow for varying
lengths of time in the winter (1). Soils in these
environments contain a large reservoir of organic
carbon (2, 3). Until recently, it was assumed that
soil microorganisms were inactive during the
snow-covered period. However, measurements
of a high efflux of CO2and other greenhouse
gases through the snow suggest that microbial
populations can be active under the snow (4–6).
These results prompted a reevaluation of whether
some seasonally snow-covered environments are
sinks of atmospheric CO2(7). In addition, under-
snow microbial metabolism is an important bio-
geochemical sink for nitrogen (8), and the sub-
sequent release of microbial nitrogen during
snowmelt is a major contributor to high primary
productivity during the short growing season in
the tundra (8, 9). Despite this progress, we know
little about the identity or seasonal dynamics of
the microbes involved.
We used standard methods (10) to estimate
microbialbiomassincoldsoils(8,11).Theresults
showthatmicrobialbiomassvariesseasonallyand
reachesmaximumannuallevelsduringlatewinter
under the snow in tundra soils (Table 1) (P ?
0.001). This observation parallels several recent
studies that have found peaks in microbial bio-
mass in the late winter (8, 11, 12). Most of the
1Department of Ecology and Evolutionary Biology,
University of Colorado, Boulder, CO 80309–0334,
USA.2Department of Biology, San Diego State Uni-
versity, San Diego, CA 92182–4614, USA.
*Present address: Environmental Sciences Division,
Oak Ridge National Laboratory, Oak Ridge, TN
37831–6038, USA.
†To whom correspondence should be addressed. E-
mail: schmidts@spot.colorado.edu
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