Pseudoelasticity of Single Crystalline Cu Nanowires Through

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Pseudoelasticity of Single
Crystalline Cu Nanowires
Through Reversible Lattice
Reorientations
Wuwei Liang
Min Zhou1
The George W. Woodruff School of Mechanical
Engineering Georgia Institute of Technology,
Atlanta, GA 30332-0405 USA
1
Molecular dynamics simulations are carried out to analyze the structure and mechanical
behavior of Cu nanowires with lateral dimensions of 1.45–2.89 nm. The calculations
simulate the formation of nanowires through a “top-down” fabrication process by “slicing” square columns of atoms from single-crystalline bulk Cu along the [001], [010], and
[100] directions and by allowing them to undergo controlled relaxation which involves
the reorientation of the initial configuration with a 具001典 axis and {001} surfaces into a
new configuration with a 具110典 axis and {111} lateral surfaces. The propagation of twin
planes is primarily responsible for the lattice rotation. The transformed structure is the
same as what has been observed experimentally in Cu nanowires. A pseudoelastic behavior driven by the high surface-to-volume ratio and surface stress at the nanoscale is
observed for the transformed wires. Specifically, the relaxed wires undergo a reverse
transformation to recover the configuration it possessed as part of the bulk crystal prior
to relaxation when tensile loading with sufficient magnitude is applied. The reverse transformation progresses with the propagation of a single twin boundary in reverse to that
observed during relaxation. This process has the diffusionless nature and the invariantplane strain of a martensitic transformation and is similar to those in shape memory
alloys in phenomenology. The reversibility of the relaxation and loading processes endows the nanowires with the ability for pseudoelastic elongations of up to 41% in reversible axial strain which is well beyond the yield strain of the approximately 0.25% of bulk
Cu and the recoverable strains on the order of 8% of most bulk shape memory materials.
The existence of the pseudoelasticity observed in the single-crystalline, metallic nanowires here is size and temperature dependent. At 300 K, this effect is observed in wires
with lateral dimensions equal to or smaller than 1.81⫻ 1.81 nm. As temperature increases, the critical wire size for observing this effect increases. This temperature dependence gives rise to a novel shape memory effect to Cu nanowires not seen in bulk
Cu. 关DOI: 10.1115/1.1928915兴
Introduction
Nanocomponents are of great technical importance because of
their unique structures, properties, and potential applications in
nanoelectronics, nano-optics, and nanocomposites. The unique
properties are primarily derived from novel structures associated
with the nanosize scales. For example, Yakobson et al. 关1兴 and
Falvo et al. 关2兴 observed in experiments and atomistic simulations
that carbon nanotubes can completely recover from sever deformations with strains up to 15% without inducing residual defects.
Bilalbegovic 关3兴 reported that Au nanowires can recover their initial lengths and radii after very large compressive strains. However, irreversible defects are nucleated under even small compressive strains, in contrast to the defect-free processes seen for
carbon nanotubes. Diao et al. 关5兴 and Gall et al. 关4兴 analyzed the
strength and the significant asymmetry in the yield behavior of Au
nanowires arising from surface stress and the activation of different slip systems in tension and compression.
Metallic nanowires are generally fabricated via “bottom-up” or
“top-down” approaches. The bottom-up approach is templatedirected and involves electrochemical deposition and the reduc1
Author to whom correspondence should be addressed; Email:
[email protected]
Contributed by the Materials Division of ASME for publication in the OURNAL OF
ENGINEERING MATERIALS AND TECHNOLOGY. Manuscript received: January 2, 2005. Final manuscript received: March 7, 2005. Review conducted by: Kenneth Gall.
tion of compounds in the channels of the templates 关6兴. On the
other hand, the top-down approach involves the fabrication of
nanocomponents from bulk materials by thinning through opticalbeam, electron-beam, ion-beam, or scanning-probe lithography.
Compared with the bottom-up approach, the top-down approach
possesses great flexibilities in producing nanowires of nearly any
shape 关7,8兴. One issue associated with the top-down approach is
that nanowires may undergo morphological and dimensional
changes upon separation from the bulk due to surface or structural
reconstructions. The reason is that the surface-to-volume ratios of
nanowires are extremely high compared with those of bulk materials. To put this in perspective, consider a wire with length l and
a square cross section whose side length is d. If the area of the end
surfaces is omitted, the surface-to-volume ratio is
RS/V =
4dl 4
= .
d 2l d
共1兲
RS/V is independent of the wire length and is inversely proportional to the lateral dimension. For a nanowire with d = 4 nm,
RS/V = 1.0⫻ 109 m−1; while for a typical macroscopic tensile
specimen with d = 4 mm, RS/V = 1.0⫻ 103 m−1. The nanowire has a
surface-to-volume ratio 106 times that of the macroscopic specimen.
Because of such high surface-to-volume ratios and the resulting
high surface-stress-induced bulk stress, the effect of the creation
of surfaces and the rearrangement of atoms on surfaces at the
Journal of Engineering Materials and Technology
Copyright © 2005 by ASME
OCTOBER 2005, Vol. 127 / 423
nanoscale during fabrication can extend well into the interior of
nanowires. For example, Kondo et al. 关9兴 discovered that the surfaces of Au nanofilms change from 共100兲 into 共111兲 when film
thickness is decreased to less than eight atomic layers. In addition,
Kondo and Takayanagi 关10兴 found that Au nanowires with 具001典
axes cut from 共100兲 films whose thickness is smaller than 2 nm
reconstruct into hexagonal prism shapes with 具110典 axes and 兵111其
lateral surfaces. Such reconstructions do not occur when the thickness of the Au films is larger than the above values. Diao et al.
关11兴 observed a similar process in atomistic simulations. Specifically, when the cross-sectional areas are equal to or less than
1.83⫻ 1.83 nm, Au nanowires with initial face-centered-cubic
共fcc兲 structures and 具100典 axes reorganize spontaneously into
body-centered-tetragonal 共BCT兲 structures 关11兴.
On one hand, structural changes present a challenge for controlling the morphologies and dimensions of nanowires during the
fabrication process. On the other hand, they also provide an important mechanism behind many attractive properties of nanowires. Obviously, in order to obtain nanowires with desired properties, morphologies, and dimensions, it is important to
understand the mechanisms and quantify the effects of the structural changes in the top-down fabrication process. The characterization of the structural changes also allows the behavior of the
nanowires to be quantified. This paper concerns the structure and
mechanical behavior of Cu nanowires formed through a top-down
fabrication process. The computational framework entails the
“slicing” of square columns of atoms from single-crystalline bulk
Cu and relaxing the wires under controlled conditions. The analysis focuses on the mechanisms for the structural reconstruction
from the sliced atomic columns to relaxed, free-standing nanowires. The conditions under which the transformation occurs in
terms of size and temperature are outlined. The behavior of the
relaxed wires under tensile loading is also analyzed. The relaxed
structures obtained through the atomistic simulations here are in
accord with the structures of laboratory Cu nanowires fabricated
recently by Liu and Bando 关12兴 and Liu et al. 关6兴. The computations show that these wires exhibit a pseudoelastic behavior under
tensile loading and unloading with the ability to recover elongation strains of up to 41% 共relative to the relaxed wire length兲. This
pseudoelastic behavior is the manifestation of a structural transformation in reverse to the structural reorganization seen during
the relaxation process. Specifically, the relaxed 具110典-axis/
兵111其-surface nanowires reassume their original bulk structure
during tensile loading. The complete transformation requires a
tensile strain of approximately 41% which corresponds to the
strain associated with the original relaxation of approximately
−29.3% 共relative to the original, unrelaxed wire length兲. Upon
subsequent unloading, the nanowires revert to the relaxed structure through the same lattice reorientation process observed during the original relaxation. The reversibility of the relaxation and
loading processes endows the nanowires with the ability for pseudoelastic elongations well beyond the yield strain of approximately 0.25% of bulk Cu and the recoverable strains on the order
of 8% of most bulk shape memory materials 关13兴. In the following
sections, the morphological changes associated with the lattice
reorientation during relaxation, the reverse transformation during
tensile loading, and the recovery of the tensile strain during unloading are analyzed. The emphasis is on the identification of the
factors influencing the processes and on the quantification of the
conditions under which this pseudoelastic behavior occurs.
2
Framework of Analysis
Molecular dynamics 共MD兲 simulations with the embeddedatom-method 共EAM兲 potential of Daw and Baskes 关14兴 and Foiles
et al. 关15兴 for Cu are carried out to analyze the structure and
mechanical behavior of free standing Cu nanowires. The wires are
created by “slicing” square columns of atoms from single-crystal
bulk Cu along the 关001兴, 关010兴, and 关100兴 directions and by sub424 / Vol. 127, OCTOBER 2005
Fig. 1 The structure of a 1.81Ã 1.81 nm „5 Ã 5 lattice constants… Cu nanowire before relaxation: „a… ˆ001‰ square lattice
on square cross sections; „b… external view of the nanowire; „c…
ˆ001‰ square lattice on lateral surfaces
sequent computational relaxation following the slicing. The calculations simulate a top-down fabrication process for nanowires in
the spirit of what is described in Kondo and Takayanagi 关10兴. The
nanocolumns initially isolated from bulk have the perfect fcc crystal structure of single-crystalline bulk Cu at 300 K with a lattice
constant of 0.3615 nm. The initial length of the columns is 21.69
nm 共or 60 lattice constants兲. The lateral dimensions of the columns vary from 1.45⫻ 1.45 nm 共or 4 ⫻ 4 lattice constants兲 to
2.89⫻ 2.89 nm 共or 8 ⫻ 8 lattice constants兲. The axes of the nanocolumns are in the 关001兴 direction and both their cross sections
and lateral surfaces are 兵001其 planes with the same square lattice,
as illustrated in Fig. 1.
Because higher-energy free surfaces are created during the topdown fabrication process, the wires are not at equilibrium when
they are initially sliced out of the bulk lattice. In order to obtain
more stable structures, the nanowires are then allowed to relax,
with one end fixed in the axial direction and the other end free of
constraints. Temperature is kept constant at 300 K during relaxation by rescaling atomic velocities. The time step for the relaxation is 1 fs. The structural transformation during relaxation and
the structure after an equilibrium state is reached are analyzed.
After relaxation, the free-standing wires are subjected to
uniaxial tensile loading with one end fixed and the other end
pulled at a constant velocity. To minimize the generation of a
shock wave in the wire and to achieve relatively uniform deformation, an initial linear distribution of structural deformation velocity in the tensile direction is superimposed on to the random
thermal velocities of the atoms, as shown in Fig. 2. The initial
structural velocity varies from 0 at the fixed end to the prescribed
pulling velocity of V = 3.615 ms−1 at the other end. After the first
time step, only velocities on boundary atoms are maintained at the
pulling velocity, providing the tensile loading necessary for the
wire to deform at a constant nominal strain rate of 1.67
⫻ 108 s−1. Although this high strain rate is difficult to reproduce in
a laboratory setting, it is partly necessitated by the timeconsuming nature of MD calculations without artificial constraints. More importantly, it allows the behavior of the nanowires
to be analyzed under fully dynamic conditions. Simulations approximating conditions with lower strain rates and quasistatic
loading are also important and are being carried out. The results
will be reported separately in the future.
Transactions of the ASME
Fig. 3 „Color… The progression of the structural transformation in a 1.81Ã 1.81 nm „5 Ã 5 lattice constants… Cu nanowire at
300 K
Fig. 2 Computational model for the tensile deformation of Cu
nanowires, boundary atoms are colored blue
3
Behavior of Nanowires
3.1 Structural Transformation During Relaxation: Formation of Free-Standing Wires. The 兵001其 surfaces of some fcc
transition and noble metals are known to reconstruct into 兵111其
planes with a close-packed hexagonal lattice to reduce their energy 共Hove et al. 关16兴, Binnig et al. 关17兴兲. At the nanoscale, such
a surface change can extend into the substrate. In addition, the
transformation shows a strong dependence on size. For example,
atomistic simulations carried out by Diao et al. 关11兴 show surfacestress-induced phase transformations in Au nanowires from a fcc
structure to a body-centered-tetragonal 共BCT兲 structure when the
cross section is equal to or smaller than 1.83⫻ 1.83 nm. Diao et
al. 关18兴 reported the formation of low-energy 具110典 Au wires with
rhombic cross sections from 具100典 wires with square cross sections equal to or smaller than 2.85⫻ 2.85 nm. This transformation
is a surface stress driven lattice reorientation process. Diao et al.
关11,18兴 quantified the effect of the surface stress. For wires with a
square cross-section, the axial compressive stress due to surface
stress is
␴=
4fd 4f
= ,
d2
d
共2兲
where f is the axial component of the surface stress as calculated
by Streitz et al. 关19兴 and d is the side length of the cross section.
Obviously, ␴ increases as the wire size decreases and can be very
high in small wires. For example, ␴ = 5.95 GPa in a 1.83
⫻ 1.83 nm Au nanowire with the BCT crystalline structure 关11兴.
Such a high stress level can induce structural transformations,
leading to a reduction in the system energy.
A similar transformation is observed in Cu nanowires here. The
process involves the complete wire, from the surfaces to the interior. It is a structural reorientation process that transforms the
wires from a 具100典-axis/ 兵001其-surfaces configuration to a
具110典-axis configuration with 兵111其 side surfaces and 兵110其 crosssectional planes. Both configurations have the same fcc structure
albeit different lattice orientations and different shapes. Moreover,
a strong size dependence of the transformation is also observed.
Specifically, wires smaller than 2.17⫻ 2.17 nm 共6 ⫻ 6 lattice constants兲 undergo a spontaneous lattice reconstruction to lower their
energy. Figure 3 shows the progression of the transformation of a
1.81⫻ 1.81 nm 共5 ⫻ 5 lattice constants兲 wire. The process initiates
from one end and propagates to the other end, resulting in a reJournal of Engineering Materials and Technology
duction in the wire length and an increase in its lateral dimensions. The complete transformation yields an axial strain of approximately −29% and an increase in the cross-sectional area of
approximately 13.3%. Wires with cross sections larger than 2.17
⫻ 2.17 nm 共6 ⫻ 6 lattice constants兲 do not undergo such a spontaneous lattice reconstruction at 300 K without external stimuli. The
morphologies of these larger wires remain very similar to their
state as part of a bulk crystal. The most noticeable change is in the
length which shows a contraction of less than 1% due to elastic
lattice straining. Figure 4 summarizes the length changes of wires
of different sizes, clearly identifying the critical size for transformation as 2.17⫻ 2.17 nm 共6 ⫻ 6 lattice constants兲. A more detailed analysis of the conditions for the initiation of the spontaneous transformation will be given shortly. For now, the discussion
focuses on the mechanisms of the transformation at 300 K without
external stimuli.
The structural characteristics of the wires before and after the
transformation can be analyzed by comparing their pair correlation functions 共PCF兲 关20兴 with that of bulk Cu crystals. Figure 5
shows the three PCFs at 300 K. The peaks in the curves indicate
that the first, second, and third nearest neighbor distances are RA
= 2.547 Å, RB = 3.615 Å, and RC = 4.426 Å, respectively. The
peaks in all three curves have essentially the same positions, indicating that the relaxed nanowires have the same fcc crystalline
structure as that for bulk Cu crystals and the unrelaxed wires. The
curves also show that the lattice constant of the transformed nanowires is approximately the same as that of bulk Cu crystals 共3.615
Å兲. The peaks for the nanowires are lower than those for bulk Cu
Fig. 4 Axial strain ␧1 after relaxation as a function of wire size
OCTOBER 2005, Vol. 127 / 425
Fig. 5 Pair correlation functions for Cu nanowires before and
after reconstruction and bulk Cu crystal at 300 K
crystals because the wires have a large proportion of surface atoms which have fewer neighbors. The structure of the reconstructed nanowires has also been analyzed using the centrosymmetry parameter which is a measure for inhomogeneous
distortions of symmetric lattices 关21兴. The results are consistent
with the conclusions from Fig. 5, showing that the transformed
wires have a defect-free, single-crystalline fcc structure.
While maintaining the fcc crystalline structure, the reconstructed wires have clearly different morphologies with their axes
and surfaces coincide with different crystalline directions and
planes from those of the unreconstructed wires. Specifically, the
initially 具001典-oriented nanowires 共with 具001典 axes and all around
兵100其 surfaces with square lattices兲 reorient to assume 具110典 axes,
兵111其 lateral surfaces, and 兵110其 cross sections, as illustrated in
Fig. 6. While the new side surfaces have perfect hexagonal lattices
typical of 兵111其 crystalline planes, the reconstructed cross sections
are characterized by elongated hexagonal lattices indicative of
兵110其 planes as illustrated in Fig. 7 关22兴. Also, as shown in Figs.
1共a兲 and 6共a兲, the shape of the cross sections changes from square
to rhombic. This shape change is associated with the reorganization of the side surfaces into 兵111其 atomic planes, resulting in a
reduction in the surface energy. A quantification of the morphological changes is listed in Table 1. Note that the data for the
transformed configurations of wires with different sizes are for
Fig. 7 A schematic illustration of the projections of two „110…
planes „black and gray atoms… in a fcc structure observed
along the †110‡ directions; the „110… plane is characterized by
elongated hexagonal lattice with ␣ = 70.5deg and ␤ = 109.5deg
„Ref. †21‡…
different temperatures. Discussions on the temperature required
for the transformation as a function of wire size will be given in
Sec. 3.2. For the wire sizes analyzed here, the measured angles of
the rhombic cross-sections 关␣ and ␤ in Fig. 6共a兲兴 are consistent
with the angles between the 共1̄11兲 and 共1̄11̄兲 crystalline planes
viewed from the 关110兴 axis, as illustrated in Fig. 7. Specifically,
the angles are approximately 109.5 and 70.5 deg, respectively.
Also, the side length of the cross sections increases 21.0% from
1.81 to 2.19 nm for a 1.81⫻ 1.81 nm 共5 ⫻ 5 lattice constants兲
wire. The lateral surface area decreases 12.3% from 156.82 to
137.06 nm2, resulting in an increase in the atomic density of
14.5% 共from 15.2 to 17.4 atoms/ nm2兲 on the surfaces. Overall,
the number of surface atoms remains the same 共2380兲 after the
transformation, indicating that there is no atomic diffusion between the surfaces and the interior. This observation is consistent
with the characteristics of a diffusionless martensitic transformation 关13,23兴 and is in contrast to surface reconstructions in bulk
materials which involve atomic migration from substrate to surfaces 关24兴.
To quantify the deformation associated with the transformation,
the deformation function is evaluated. This analysis is a phenomenological quantification by considering the wire as a continuum
and by focusing on the overall deformation outcome. As illustrated in Fig. 8, the relations between the coordinates before the
transformation 共x, y, and z兲 and the coordinates after the transformation 共x̄, ȳ, and z̄兲 can be written as
冦
Fig. 6 The reconstructed structure of a Cu nanowire with initial dimensions of 1.81Ã 1.81 nm „5 Ã 5 lattice constants…: „a…
Š110‹ elongated hexagonal lattice on a rhombic cross section;
„b… external view of the reconstructed nanowire; „c… ˆ111‰ hexagonal lattice on lateral surfaces
426 / Vol. 127, OCTOBER 2005
d
共x + y cos ␣兲,
d0
d
ȳ = y sin ␣, and
d0
l1
z̄ = z;
l0
x̄ =
冧
共3兲
where d0 and d are the side lengths of the cross sections before
and after the transformation, respectively, and l0 and l1 are the
wire lengths before and after the transformation, respectively. The
deformation gradient corresponding to Eq. 共3兲 is
Transactions of the ASME
Table 1 Morphological change of nanowires associated with reconstruction
d0 共lattice
constant兲
d0共nm兲
d共nm兲
␣
共deg兲
␤
共deg兲
Wire
length
l共nm兲
4
5
6
7
8
1.45
1.81
2.17
2.53
2.89
1.76
2.19
2.65
3.07
3.39
70.5
70.0
69.7
70.7
68.3
109.5
110.0
110.3
109.3
111.3
15.33
15.61
15.87
16.04
16.41
冤 冥
共4兲
For the 1.45⫻ 1.45 nm 共4 ⫻ 4 lattice constants兲 wire in Table 1
F=
冤
0
0
1.150
0
0
0
0.707
冥
Lateral
surface
area A
共nm2兲
Temperature
共K兲
125.45
156.82
188.18
219.55
250.91
110.46
137.06
168.42
196.67
222.28
100
300
450
600
900
⍀
= det共F兲 = 0.992
⍀0
d d
cos ␣ 0
d0 d0
d
sin ␣ 0 .
F= 0
d0
l1
0
0
l0
1.220 0.407
Lateral
surface
area A0
共nm2兲
,
共5兲
yielding a volume ratio of
共6兲
for the transformation. In the above expression, ⍀0 and ⍀ are,
respectively, the volumes before and after the transformation and
det共F兲 denotes the determinant of F. The volume change associated with the transformation is small because the wire maintains
the same fcc structure after the transformation. The Lagrangian
strain tensor for the deformation from the initial configuration to
the transformed configuration is
冤
0.244
1
E = 共FT · F − I兲 = 0.250
2
0
0.250
0
0.244
0
0
− 0.250
冥
共7兲
.
Here, I is the identity tensor. The corresponding Eulerian strain
tensor or the negative of the Lagrangian strain for the inverse
deformation from the transformed configuration to the initial 共or
stretched兲 configuration is
冤
0.164
1
E* = 共I − F−T · F−1兲 = 0.119
2
0
−1
Fig. 8 A schematic illustration of the deformation of a wire
associated with the lattice transformation; the gray dash lines
indicate the wire configuration before the transformation, the
solid lines denote the configuration after the transformation
Journal of Engineering Materials and Technology
0.119
0
0.080
0
0
− 0.500
冥
共8兲
−T
where, F denotes the inverse and F denotes the inverse transpose of F.
E and E* allow the forward and reverse deformations to be
fully quantified. In particular, the axial strains associated with the
relaxation is Ezz = −0.250. This corresponds to an engineering
strain of ␧1 = 共l1 − l0兲 / l0 = Fzz − 1 = 冑2Ezz + 1 − 1 = −0.293 共relative to
the original unrelaxed length l0兲. For the reverse 共tensile兲 deformation from the relaxed state which will be discussed in Sec. 3.3,
*
= 0.500 when the wire recovers its original
the axial strain is −Ezz
length prior to relaxation, corresponding to an engineering strain
*
of ␧2 = 共l0 − l1兲 / l1 = 1 / Fzz − 1 = 冑−2Ezz
+ 1 − 1 = 0.414.
A comparison of the 共110兲 planes containing the wire axes before and after the transformation provides another quantification
of the deformation associated with the reconstruction. Figure 9共a兲
illustrates the 共110兲 atomic plane before the reconstruction. This is
an interior plane in the wire which contains the original wire axis
共关001兴兲 and a diagonal 共关1̄10兴兲 of the square cross section 共001兲,
see Fig. 1共a兲. Figure 9共c兲 illustrates the same 共110兲 plane after the
reconstruction. This transformed plane contains the new wire axis
关11̄0兴 and the long diagonal 共关001兴兲 of the transformed 共110兲 cross
section, see Fig. 6共a兲. The shaded rectangles in Fig. 9 correspond
to the shaded rectangle in Fig. 7. The length and width of the
rectangle are, respectively, a and 共冑2 / 2兲a ⬇ 0.707a, where a is the
lattice constant. Before reconstruction, the long side of the rectangle is aligned with the nanowire axis. After reconstruction, the
long side of the rectangle is rotated by 90 deg and is aligned with
one diagonal of the cross section of the wire. Obviously, the axial
strain associated with the transformation is
OCTOBER 2005, Vol. 127 / 427
Fig. 9 Lattice rotation associated with the structural transformation during relaxation, „a… a „110… atomic plane containing
the original wire axis „†001‡… and a diagonal „†1̄10‡… of the „001…
square cross section of the original wire before reconstruction
†Sec. 1-1 in Fig. 1„a…‡, „c… a „110… atomic plane containing the
new wires axis †11̄0‡ and the long diagonal „†001‡… of the transformed cross section „110… plane †Sec. 2-2 in Fig. 6„a…‡
␧1 =
冉冑 冊冒
2
a−a
2
a = ␧relaxation = 共l1 − l0兲/l0 = − 0.293.
共9兲
This value is consistent with the axial strain associated with the
length reduction during the full reconstruction of wires smaller
than 2.17⫻ 2.17 nm 共6 ⫻ 6 lattice constants兲 as shown in Fig. 4
and discussed above. It is interesting to note that the length reduction is only 19.3% for a 2.17⫻ 2.17 nm nanowire at 300 K 共Fig.
4兲. This is because such a wire only undergoes a partial reconstruction upon relaxation at the room temperature, as shown in
Fig. 10. Clearly, part of this wire shows reconstruction into a
具110典 axis and 兵111其 surfaces while the remaining part retains the
original 关001兴 axis and 兵001其 surfaces. Defects in the form of
twins can be seen between the transformed and untransformed
Fig. 10 „Color… The configuration of a Cu nanowire with initial
lateral dimensions of 2.17Ã 2.17 nm „6 Ã 6 lattice constants…:
„a… external view; „b… section view; „c… defects „twin boundaries… only. Atoms are colored according to their centrosymmetry values
428 / Vol. 127, OCTOBER 2005
Fig. 11 Comparison of the potential energy of the wires with
Š001‹ / ˆ001‰ and Š110‹ / ˆ111‰ configurations at 300 K
segments. The partially reconstructed nanowire exhibits an irregular shape because of the interfaces between the two different lattice orientations. Since the interfaces divide the wire into regions
with different crystal orientations, the wire can be regarded as
polycrystalline with each region resembling a nanosized grain and
the lattice interfaces serving as grain boundaries.
The partially reconstructed configuration in Fig. 10 provides an
illustration of the twinning process responsible for the transformation. Although at the bulk level twinning is usually observed in
bcc and hcp materials which have fewer slip systems, it has been
observed in nanostructured fcc materials such as Cu 关25,26兴, silver 关27兴, and Au 关26兴. For example, Wang et al. 关26兴 observed
microtwinning in electrochemically deposited Cu nanowires with
diameters of 30–50 nm. Molares et al. 关25兴 found twin structures
in Cu nanowires with a diameter of 70 nm. In the simulations
here, twin boundaries initially nucleate at the fixed end and propagate toward the free end via the gliding of 兵111其 具112典 partial
dislocations under the compressive stress induced by surface
stress. This twinning process causes the 90 deg lattice rotation
illustrated in Fig. 9 and transforms the wire from the 具001典 / 兵001其
configuration to the 具110典 / 兵111其 configuration as the twin boundaries sweep through the wire length.
The structures of the fully reconstructed Cu nanowires reported
above are in good agreement with those of the laboratory Cu
nanowires fabricated recently by vacuum vapor deposition 关12兴
and by a complex surfactant-assisted hydrothermal reduction process 关6兴. Just like what is seen above in the simulations, the laboratory nanowires also have defect-free single-crystalline fcc structures with 具110典 axes and 兵111其 surfaces, indicating that 具110典 is
the preferred growth orientation and 兵111其 planes are preferred
lateral surfaces.
3.2 Size and Temperature Dependence of Reconstruction.
Even though the reconstructed nanowires retains the same defectfree, single-crystalline fcc structures as the wires before the transformation, the energy of the systems changes due to the increase
in atomic density on surfaces when the initial 兵001其 surfaces reorganize into closely packed 兵111其 surfaces 关28兴. Specifically, the
surface energy per unit area for 兵001其 and 兵111其 planes is 1.28 and
1.17 Jm−2, respectively 关15兴. Moreover, the surface areas are reduced after reorientation for wires of all sizes, as shown in Table
1. A quantification of the difference in potential energy is given in
Fig. 11. The change of the average potential energy per atom is
evaluated at a temperature of 300 K, accounting for reductions in
surface areas 共see Table 1兲, the surface energy density, and volume. This evaluation is achieved by “slicing” 具001典 wires with
兵001其 surfaces and 具110典 wires with 兵111其 surfaces out of bulk Cu
crystals at 300 K with appropriate orientations and dimensions.
The potential energy of the unreconstructed wires with the
具001典 / 兵001其 configuration is calculated immediately after they are
Transactions of the ASME
“sliced” out. The potential energy of the wires with the
具110典 / 兵111其 configuration is computed after they reach their equilibrium states through relaxation at 300 K. Clearly, the potential
energy per atom decreases with the wire size for each configuration because smaller wires have larger surface-to-volume ratios, as
shown in Eq. 共1兲. On the other hand, regardless of size, wires with
the 具110典 / 兵111其 configuration always have lower energy levels
compared with their unreconstructed counterparts with the
具001典 / 兵001其 configuration. The reorientation from the 具001典 / 兵001其
configuration to the 具110典 / 兵111其 configuration is always energetically favored because there is always an energy reduction in the
system associated with the increase in atomic density on the surfaces at all sizes. Therefore, when fabricated through a “bottomup” approach, Cu nanowires are likely to adopt the configuration
with 具110典 axes and 兵111其 surfaces since 具110典 / 兵111其 wires are
more stable than the 具001典 / 兵001其 wires. Also, all 具001典 / 兵001其
wires should have a natural tendency for spontaneous transformation into 具110典 / 兵111其 wires.
It is illustrative to point out that the lateral dimensions of laboratory Cu nanowires reported in the literature range from 50–100
nm, significantly larger than the critical size for the lattice reorientation observed in our simulations. One reason for this superficial contradiction has to do with the energetic barrier and driving
force of the transformation. While the transformed configuration
always has a lower energy level than the untransformed configuration at the room temperature 共300 K, see Fig. 11兲, the lower
energy state must be reached through structural reconstruction in a
top-down manufacturing process. Such a transformation not only
requires a driving force provided by the energy difference of the
configurations but also requires the driving force to be sufficient
to overcome the barrier for initiating and completing the reconstruction process 关24兴. Sufficient driving force for overcoming the
size-dependent energy barrier for the transformation can be
achieved by either reducing the wire size or by increasing the
temperature. These effects are discussed here.
Consider the surface stress-induced stress as a measure for the
driving force of the structural reconstruction. When the nanowire
is first “sliced” out from bulk Cu, surface atoms lose the symmetry of forces from neighboring atoms and experience only forces
exerted by atoms in the wire, giving rise to surface stress which
induces a compressive stress in the wire during relaxation. This
average stress at this time is
␴=
兺
1
rij 丢 fij ,
2⍀0 j共⫽i兲
共10兲
where ⍀0 is the initial volume of the wire, rij = r j − ri with ri being
the position of atom i, and 丢 denotes the tensor product of two
vectors with 共a 丢 b兲␣␤ = a␣b␤ 共a␣ and b␤ are Cartesian components
of a and b, respectively, ␣, ␤ = 1,2,3兲, and fij is the interatomic
forces applied on atom i by atom j.
Figure 12 shows the average stress in the axial direction ¯␴
关component ␴zz of ␴ in Eq. 共10兲兴 at different temperatures as a
function of wire size. Also indicated in this figure are the values of
the critical stress ␴cr at which wires of different sizes begin a full
reconstruction from the 具001典 / 兵001其 configuration to the lower
energy 具110典 / 兵111其 configuration. The transformation occurs
when ¯␴ is higher than ␴cr. The value of ␴cr for each wire size is
obtained by carrying out relaxation calculations at slowly increasing temperatures. Specifically, relaxation starts at 0 K. If reconstruction does not occur within 150 ps, the temperature is increased slightly 共50 K兲 and the relaxation calculation is carried out
for another 150 ps. If reconstruction is not observed at the new
temperature, the temperature is increased again. This process is
repeated until full structural reconstruction is observed at some
temperature Tcr.
Clearly, ¯␴ decreases with wire size as larger wires have smaller
surface-to-volume ratios. Hence, reconstruction only occurs in
Journal of Engineering Materials and Technology
Fig. 12 Surface-stress-induced stress in the axial direction
when wires are just “sliced” out of a bulk Cu crystal, the solid
curve indicates the critical stress ␴cr and temperature Tcr for a
wire of a certain size to fully reconstruct
wires smaller than a critical size at a given temperature. A similar
size effect has also been observed in Au nanofilms and nanowires
experimentally fabricated with a top-down method 关9兴. Specifically, 兵100其 Au nanofilms are found to reconstruct into 兵111其 films
at a temperature lower than 429 K when their thickness is less
than eight atomic layers. Also, Au nanowires cut from 共001兲 Au
films have been observed to reconstruct into hexagonal prism
structures with 具110典 axes and 兵111其 lateral surfaces when the
thickness of the films is less than 2 nm 关10兴.
In addition to the size dependence, the reconstruction is also
temperature dependent as ¯␴ increases with temperature for any
given wire size, as seen in Fig. 12. The critical size for reconstruction increases with increasing temperature. For example, 2.53
⫻ 2.53 nm 共7 ⫻ 7 lattice constants兲 nanowires do not reconstruct
at all at 300 K but fully reconstruct at 600 K.
It is important to point out that, like in all MD models, the
accuracy of the simulations carried out here depends on the
atomic potential used. Since the critical temperatures and sizes for
transformation are closely related to 兵111其 and 兵001其 surface energies and surface stresses and different atomic potentials for Cu
predict different surface energies for 兵111其 and 兵001其 surfaces
关15,29,30兴, the predicted critical condition in terms of temperature
and size may vary from potential to potential. However, the overall trend and dependence are expected to be the same regardless of
which potential is used. More importantly, it must be noted that it
is the difference in the surface energies for 兵111其 and 兵001其 surfaces, not the absolute energy values, that determines the nature of
the observed transformation. Since the difference is found to be of
the same order in all the Cu potentials 关15,29–31兴, it is expected
that different potentials would still lead to predictions of similar
structural behaviors. Indeed, the same transformation mechanism
and behavior have been observed in simulations carried out using
the EAM potential for Cu in Ref. 关32兴. In addition, similar transformation mechanisms are observed by Diao et al. for surface
stress driven lattice reorientation in Au nanowires when different
atomistic potentials, including an EAM, a modified embedded
atom method, and a surface embedded atom potential, are used
关18兴.
3.3 Pseudoelasticity in Transformed Cu Nanowires. The
shape-memory effect is normally associated with martensitic 共displacive or diffusionless兲 transformations which involve a shearlike
deformation mechanism through cooperative displacement of atoms. The transformation results in an overall change in shape and
size, with the motion of all atoms associated with each shear
variant being in the same direction 关13兴. Although most martensitic transformations occur in alloys, similar transformations have
also been observed in other materials, such as A15 superconducting compounds and Ar-N2 solid solutions 共Nakanishi 关33兴 and
OCTOBER 2005, Vol. 127 / 429
Fig. 13 The stress-strain relation of a Š110‹ / ˆ111‰ reconstructed wire with an initial dimension of 1.81Ã 1.81 nm „5 Ã 5
lattice constants… during tensile loading and unloading at 300 K
Barrett 关34兴兲. The fundamental requirement for martensitic transformations is that the deformation has an invariant-plane strain
关23兴. Specifically, the operation of an invariant-plane strain always
leaves one plane of the parent crystal completely undistorted and
unrotated. This plane is referred to as the invariant plane. One
example of an invariant-plane strain is mechanical twinning in
which the parent lattice is homogeneously sheared into the twin
orientation, with the twinning plane unaffected by the deformation
and forming a coherent boundary between the parent and product
crystals.
Such a mechanical twinning process is observed in the loading
and unloading processes of the relaxed or transformed
具110典 / 兵111其 Cu nanowires 共e.g., wires smaller than 2.17
⫻ 2.17 nm or 6 ⫻ 6 lattice constants at 300 K兲 discussed above.
The reversible lattice reorientation associated with this twinning
process has the diffusionless nature of a martensitic transformation, endowing the nanowires with a pseudoelastic behavior similar to that of shape-memory alloys which is not seen in bulk Cu.
The following discussion focuses on the behavior of reconstructed wires. The interest is on the mechanical response to axial
tensile loading and unloading. Uniaxial tensile loading is applied
to free-standing, fully reconstructed wires. Figure 13 shows the
stress-strain relation of a reconstructed wire with an initial dimension of 1.81⫻ 1.81 nm prior to relaxation. Initially, the wire has a
defect-free, uniform fcc structure with 具110典 axis and 兵111其 surfaces. The side length of the rhombic cross sections is 2.19 nm,
and the angles between the sides are 70.0 and 110 deg, respectively, as shown Fig. 6. Under the tensile loading, the wire first
deforms elastically with stress increasing linearly with strain. This
stage of uniform stretching of the wire lattice ends at a strain of
approximately 2% 共point A in Fig. 13兲.
The linear elastic stage of deformation is followed by a lattice
reorientation process in the direction opposite to that for the transformation during relaxation. While the transformation during relaxation is driven by surface stress-induced stress and is from the
具001典 / 兵001其 configuration to the 具110典 / 兵111其 configuration, the
reverse transformation is driven by the externally applied stress
and is from the 具110典 / 兵111其 configuration back to the 具001典 / 兵001其
configuration. A phase 共twin兲 boundary separating the 具001典 / 兵001其
region and the 具110典 / 兵111其 region is seen 共Fig. 14兲 propagating
from the former to the later. When the strain rate for the tensile
deformation is 1.67⫻ 108 s−1, the average propagation speed is
12.6 ms−1, a small fraction of the stress wave speed in the 具001典
direction of single crystalline Cu which is 2092 ms−1 for uniaxial
stress and 4198 ms−1 for uniaxial strain. The wire recovers its
430 / Vol. 127, OCTOBER 2005
Fig. 14 „Color… Lattice orientations on the cross sections of a
Cu nanowire with initial dimensions of 1.81Ã 1.81 nm „5 Ã 5 lattice constants… at a strain of 0.24; „a… a sectional view along the
wire axis and the Š110‹ diagonal of the cross section, „b… elongated hexagonal lattice in the Š110‹ / ˆ111‰ region, „c… a cross
section in the transition region containing both the Š001‹ / ˆ001‰
and the Š110‹ / ˆ111‰ configurations, „d… square lattice on the
cross section in the Š001‹ / ˆ001‰ region. Atoms are colored according to their centrosymmetry values
original structure and orientation prior to relaxation when the
phase 共twin兲 boundary has propagated through the whole wire
length. The reorientations in both directions are stress-induced
processes. For the 1.81⫻ 1.81 nm 共5 ⫻ 5 lattice constants兲 nanowire in Figs. 13 and 14, the transformation during relaxation occurs with a critical compressive stress of 5.9 GPa at 300 K 共see
Fig. 12兲. On the other hand, the reverse reorientation initiates
when the externally applied tensile stress reaches 2.0 GPa 共Fig.
13兲. The stress increases gradually throughout the reorientation
process while the wire undergoes significant elongation. Note that
the reverse transformation is completed at point B in Fig. 13 with
a strain of approximately ␧2 = 0.414, bringing the wire to its original length before relaxation 共l0兲. As discussed in Sec. 3.1, the
axial strain upon full relaxation is ␧1 = ␧relaxation = −0.293, where
the reference state is the original wire with the 具001典 / 兵001其 configuration. On the other hand, the reference state for the tensile
deformation is the reconstructed wire with the 具110典 / 兵111其 configuration, the strain upon full reverse transformation is
␧2 = ␧loading = 共l0 − l1兲/l1 = − ␧relaxation/共1 + ␧relaxation兲
= 0.293/共1 − 0.293兲 = 0.414.
共11兲
The reverse lattice reorientation progresses with the propagation
of a single twin boundary. The twin boundary nucleates at one end
and propagates to the other end until the whole nanowire recovers
its 具001典 orientation. Figure 14 shows a sectional view of the
1.81⫻ 1.81 nm nanowire in Fig. 13 while the twin boundary is in
the middle of the transforming wire. The twin boundary separates
the transformed 共具001典 / 兵001其兲 region from the untransformed
共具110典 / 兵111其兲 region. Clearly, the transformed region has square
lattices typical of the original unrelaxed wire on both crosssectional and lateral surfaces. The untransformed region retains
the 具110典 elongated hexagonal lattice on cross sections and 兵111其
hexagonal lattice on lateral surfaces typical of the relaxed wire.
The cross section at the interface clearly shows the transition from
the 具110典 orientation to the 具001典 orientation, as shown in Fig.
Transactions of the ASME
Fig. 16 An illustration of the shape-memory effect in Cu nanowires, Tcr is the critical temperature for a Cu nanowire of a
certain size to fully reconstruct or show the effect
Fig. 15 „Color… Twin boundary between the Š001‹ and Š110‹
lattices during the reorientation process, the details of a „110…
lattice section which is part of Sec. A-A in Fig. 14„a… is shown. ␥
denotes the misorientation angle between the lattices on both
sides of the twin boundary. The atoms are colored according to
their centrosymmetry values
14共c兲.
The mobile twin boundary is formed by mismatch defects between the 具110典 / 兵111其 lattice and the 具001典 / 兵001其 lattice, as illustrated in Fig. 15. The stacking sequence within each region is
ABCABC in the direction perpendicular to the boundary. At the
interface, the stacking sequence is ABC兩A兩CBA where the middle
A is the mirror plane. Crystallographically, this is a 兵111其 twin
plane with its misorientation axis aligned in the 关1̄12̄兴 direction.
The misorientation angle ␥ between the lattices is measured to be
110.0 deg. Thus, the twin plane can be considered as a ⌺3 兵111其
109.5 deg grain boundary since it separates two regions of perfect
lattices 关35兴. Under loading, atoms on both sides of the twin plane
shear homogeneously, leading to the axis reorientation from the
关11̄0兴 direction to the 关001兴 direction while keeping the twin plane
unaffected by the deformation. This is a case of twinning as an
operation of invariant-plane strain with the twin plane being the
invariant plane. The twin plane moves in the axial direction via
the gliding of partial dislocations, as shown in Fig. 15.
Although the lattice reorientation during loading involves defect activities, the associated deformation can be fully recovered
upon unloading. This is similar to the reconstruction during relaxation discussed in Sec. 3.1. While the propagation of a single twin
boundary is primarily responsible for the lattice reorientation during loading, multiple twin boundaries propagate along the wire
axis and eventually disappear during unloading, leading to the full
recovery of the deformation. The dash line in Fig. 13 shows the
unloading path from a strain of ␧2 = 0.414 共measured relative to
the relaxed free length of the wire兲. At this strain, the wire has its
original length l0 it possessed prior to relaxation 共or ␧1 = 0.0兲.
When the tensile stress is reduced to zero, the wires returns to its
fully relaxed state with ␧1 = −0.293 or ␧2 = 0. The loading and unloading paths together form a hysterisis loop typical of materials
Journal of Engineering Materials and Technology
with the shape-memory effect 关36兴.
During the unloading process, surface stress drives the nanowire toward the 具110典 / 兵111其 orientation and the free-standing
configuration without external loading. This spontaneous lattice
reorientation from the 具001典 / 兵001其 configuration to the
具110典 / 兵111其 configuration and the opposite process during loading
provide the mechanisms for fully reversible tensile deformations.
From the perspective of the 共110兲 plane containing the wire axis
共关11̄0兴兲 and the longer diagonal of the cross-section 共关001兴兲, the
lattice orientations are essentially 90 deg rotations in opposite
directions which occur, respectively, during relaxation/unloading
and during tensile loading, as illustrated in Fig. 9. As shown earlier, the maximum recoverable strain is ␧loading = 0.414 共measured
relative to the relaxed free length of the wire兲 during the loading
process and is ␧relaxation = −0.293 共measured relative to the original
unrelaxed length兲 during the relaxation or unloading process.
These values do not account for the elastic stretching of the crystal lattice which only has a very small effect on such large strains.
The reversibility of these strains endows the nanowires with the
ability for pseudoelastic elongations of up to 41.4% in reversible
axial strain, well beyond both the yield strain of approximately
0.25% of bulk Cu and the recoverable strains on the order of 8%
of most bulk shape memory materials 关13兴.
In summary, wires with the initial 具001典 / 兵001其 configuration
spontaneously transform into the 具110典 / 兵111其 configuration upon
relaxation at a temperature higher than Tcr 共Fig. 12兲. On the other
hand, wires with the 具110典 / 兵111其 configuration recover their original 具001典 / 兵001其 configuration they possessed prior to relaxation
when tensile loading is applied to them. These stretched wires
with the 具001典 / 兵001其 configuration can spontaneously return to
their 具110典 / 兵111其 configuration from strains up to 0.414, as illustrated in Fig. 16. This shape memory effect at the nanoscale directly results from the high surface-to-volume rations of the Cu
nanowires analyzed here. It is a unique property at the nanoscale
which does not exist in bulk Cu. This unique behavior may lead to
important applications at the nanoscale, including sensors, transducers, and actuators in nanoelectromechanical systems 共NEMS兲
关37,38兴. Because of the temperature dependence of the pseudoelastic behavior discussed here, such nanowires may also have
pyromechanical applications such as thermally activated sensors
or actuators 关39兴.
4
Conclusions
1. The structure and mechanical behavior of Cu nanowires fabricated via a top-down approach are analyzed. The calculations use MD simulations to model the relaxation process
OCTOBER 2005, Vol. 127 / 431
during the fabrication and the tensile deformation process of
fully relaxed nanowires. At a temperature of 300 K, spontaneous lattice reorientation is observed in wires smaller than
2.17⫻ 2.17 nm 共6 ⫻ 6 lattice constants兲 but not in wires with
larger sizes. The wires change from the 具001典 / 兵001其 configuration to the 具110典 / 兵111其 configuration through a 90 deg
lattice reorientation, with the cross-sectional shape changing
from square to rhombic and the wire length decreasing approximately 29.3%. This transformation progresses through
the nucleation and propagation of twin boundaries driven by
surface stress-induced stress. The critical wire size for the
transformation is found to increase with increasing temperature. The calculated structure of the reconstructed wires is in
good agreement with what has been observed experimentally in laboratory Cu nanowires.
2. The analysis shows that the 具110典 / 兵111其 configuration is always energetically favored over the 具001典 / 兵001其 configuration regardless of the size of the nanowire. The existence of
a critical size for transformation at a given temperature
points to the existence of an energy barrier for the initiation
of the transformation. A quantification of this barrier in
terms of the average stress induced by surface stress as a
function of temperature and wire size is obtained. The critical stress can be regarded as the stress needed to nucleate
and drive twin boundaries across the nanowire.
3. A pseudoelastic behavior and a novel shape memory effect
have been observed in the relaxed nanowires with the
具110典 / 兵111其 configuration during tensile loading and unloading. The mechanism responsible for this pseudoelasticity is a reversible transformation opposite to that observed
during relaxation. This transformation occurs through the
propagation of a twin boundary separating the 具110典 / 兵111其
and the 具001典 / 兵001其 region. Driven by the externally applied
tensile loading, the twin boundary sweeps through the wire
axis and leads to the recovery of the 具001典 / 兵001其 configuration which the wire assumed prior to relaxation when the
elongation reaches approximately ␧loading = 0.414. This process possesses the diffusionless nature and the typical
invariant-plane strain of martensitic transformations. The reversibility of these processes during loading and unloading
endows the nanowires with the ability for pseudoelastic
elongations of up to 41.4% in reversible axial strain which is
well beyond both the yield strain of approximately 0.25% of
bulk Cu and the recoverable strains of most bulk shape
memory materials. This pseudoelastic behavior and the
shape memory effect of Cu nanowires observed here are
similar to those of many shape memory alloys in bulk. However, the mechanism responsible for this behavior at the
nanoscale does not exist for bulk Cu. The size dependence
and temperature dependence of this behavior suggest possible applications of these nanowires as sensors, transducers,
and pyromechanical actuators in NEMS.
Acknowledgments
This research is supported by NASA Langley Research Center
through Grant No. NAG-1-02054. Computations are carried out at
the NAVO and ERDC MSRCs through AFOSR MURI number
D49620-02-1-0382 at Georgia Tech. We wish to thank S.
Plimpton for sharing his MD code WARP. W.L. thanks Douglas.
E. Spearot for helpful discussions. The images of deformation in
this paper are created with the graphics package visual molecular
dynamics 共VMD兲 共Ref. 关40兴兲.
关3兴
关4兴
关5兴
关6兴
关7兴
关8兴
关9兴
关10兴
关11兴
关12兴
关13兴
关14兴
关15兴
关16兴
关17兴
关18兴
关19兴
关20兴
关21兴
关22兴
关23兴
关24兴
关25兴
关26兴
关27兴
关28兴
关29兴
关30兴
关31兴
关32兴
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OCTOBER 2005, Vol. 127 / 433
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