RESEARCH ARTICLE | JANUARY 10 2013
Mechanical properties and scaling laws of nanoporous gold
Xiao-Yu Sun; Guang-Kui Xu; Xiaoyan Li; ... et. al
Journal of Applied Physics 113, 023505 (2013)
https://doi.org/10.1063/1.4774246
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Ligament and joint sizes govern softening in nanoporous aluminum
JOURNAL OF APPLIED PHYSICS 113, 023505 (2013)
Mechanical properties and scaling laws of nanoporous gold
Xiao-Yu Sun,1 Guang-Kui Xu,1 Xiaoyan Li,2 Xi-Qiao Feng,1,a) and Huajian Gao2
1
AML and CMM, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
School of Engineering, Brown University, Providence, Rhode Island 02912, USA
2
(Received 21 June 2012; accepted 17 December 2012; published online 10 January 2013)
I. INTRODUCTION
Biological materials such as feathers, antlers, cortical
bones, and diatom algae have evolved nanoporous structures
that are crucial for their functions.1–5 Recent years have also
witnessed biomimetic synthesis of organic, metallic, and
composite nanoporous materials.6–9 The nanoporous microstructures of these materials can endow them with unusual
physical, chemical, and mechanical properties, including
huge surface-to-volume ratio, superior electrocatalytic performance, surface-enhanced Raman scattering, strain reversibility, and high yield strength.10–13 These properties make
them promising candidates for potential applications in a variety of fields such as biosensing, actuation, tissue engineering, and proton exchange membrane fuel cells.14–17
Much experimental and theoretical efforts have been
devoted toward exploring the properties of nanoporous materials. In particular, nanoporous gold (np-Au) is an ideal candidate for studying the mechanical behavior of nanoporous
metals due to its easy fabrication and remarkable stability
against corrosion and oxidation.8 The yield strength and elastic modulus of np-Au have been experimentally measured by
various methods including microtension, nanoindentation,
and buckling-based metrology.7,8,18 It was found that the mechanical properties of np-Au depend sensitively on microstructural parameters such as the ligament size and relative
mass density (i.e., the ratio between the density of the nanoporous material and that of its bulk counterpart). Interestingly, the experiments of Biener et al.8 showed that even for
relative mass density as low as 0.25, the load-carrying
capacity (e.g., yield strength) of np-Au is still much larger
than that of the bulk Au.
So far, most theoretical studies on the mechanical
properties of nanoporous materials have been based on the
a)
Author to whom correspondence should be addressed. Electronic mail:
fengxq@tsinghua.edu.cn.
0021-8979/2013/113(2)/023505/9/$30.00
Gibson-Ashby model,19 which allows to estimate the effective Young’s modulus, yield strength, and ultimate tensile
strength of a porous material in terms of its relative mass
density. While the Gibson-Ashby model has been widely
used for porous materials with characteristic microstructural
sizes on the order of micrometers and above, it may be inaccurate or even invalid for nanoporous materials due to the
large surface-to-volume ratio. For example, for np-Au samples with an average ligament size of 100 nm, the yield
strength can be 10–100 times higher than that predicted from
the Gibson-Ashby model.6,13 Therefore, the Gibson-Ashby
model should be modified for nanoporous materials according to their deformation mechanisms at the nanoscale. Hodge
et al.20 presented a modified scaling equation for the yield
stress as a function of the relative mass density and ligament
size of np-Au, in which they incorporated a Hall-Petch-type
relation between the average yield strength and average diameter of ligaments. Feng et al.21 proposed a unit-cell micromechanics model to predict the size-dependent elastic
property of nanoporous materials by considering the effects
of surface energy and residual surface stresses. Xia et al.22,23
theoretically investigated the mechanical properties of nanoporous materials with hierarchical structures observed in
experiments.
Molecular dynamics (MD) simulations can be a potentially powerful tool to investigate the mechanical properties of
nanoporous materials with atomic resolution. Using a tetrakaidecahedron simulation model, Cantrell24 investigated the mechanical behavior of closed-cell nanoporous copper foams. To
et al.25 constructed a simulation model from randomly generated spheres that partially overlap with one another, and performed MD simulations to examine the softening behavior of
nanoporous aluminum under tension. They found that the softening slows down significantly at sufficiently small ligament
sizes. These existing MD simulations are typically based on
highly simplified geometrical models, with microstructures
distinctly different from those of real nanoporous materials.
113, 023505-1
C 2013 American Institute of Physics
V
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Nanoporous metals are a class of novel nanomaterials with potential applications in many fields
such as sensing, catalysis, and fuel cells. The present paper is aimed to investigate atomic
mechanisms associated with the uniaxial tensile deformation behavior of nanoporous gold. A phase
field method is adopted to generate the bicontinuous open-cell porous microstructure of the
material. Molecular dynamics simulations then reveal that the uniaxial tensile deformation in such
porous materials is accompanied by an accumulation of stacking faults in ligaments along the
loading direction and their junctions with neighboring ligaments, as well as the formation of
Lomer–Cottrell locks at such junctions. The tensile strain leads to progressive necking and rupture
of some ligaments, ultimately resulting in failure of the material. The simulation results also
suggest scaling laws for the effective Young’s modulus, yield stress, and ultimate strength as
C 2013 American
functions of the relative mass density and average ligament size in the material. V
Institute of Physics. [http://dx.doi.org/10.1063/1.4774246]
023505-2
Sun et al.
As a result, the underlying physical mechanisms behind the
unique deformation behaviors of nanoporous materials are
still open to further investigations. In the present paper, we
will introduce a phase field method to generate the bicontinuous microstructures that closely mimic those in real nanoporous metallic materials. MD simulations are then performed to
study the detailed deformation mechanisms of np-Au under
tension, and insights from these simulations are used to develop scaling laws for the mechanical properties of np-Au in
terms of microstructural parameters.
II. METHODS
A. Phase field method
where uðx; y; z; tÞ (1 u 1) is the difference in concentration of the two phases, t is the evolutionary time of the
system, f ðuÞ is the free energy function, and h is the width
of transition region between the two phases. In our simulations, we adopt the double-well potential function f ðuÞ
¼ 14 ðu2 1Þ2 and set h ¼ 0:01; the latter value does not
interfere with the results qualitatively.33
mation of spatial derivatives results in the following discrete
equation:
m
umþ1
ijk uijk
s
h
i
3
2 2 m
m
¼ r2 ðum
ijk Þ uijk h r uijk ;
(2)
m
m
m
m
m
where r2 um
ijk ¼ ½uðiþ1Þjk þ uði1Þjk þ uiðjþ1Þk þ uiðj1Þk þ uijðkþ1Þ
m
m
2
þ uijðk1Þ 6uijk =b is used to approximate the continuous
Laplacian on the discrete lattices. To numerically solve Eq.
(2), an adaptive integration time step s should be properly
selected. Too large an integration time step might induce
divergence or instability of the numerical solution, while too
small a time step increases the computational cost unnecessarily. Numerical tests suggest that s ¼ 0:015 is an appropriate choice that ensures high accuracy, good stability, and
reasonable computational cost. A set of initial values
uðib; jb; kb; 0Þ 2 ½5; 5 104 are randomly generated, and
periodic boundary conditions are adopted in all three directions of the box.
During the spinodal simulation, the two phases gradually separate, which is subsequently followed by a coarsening process. Figure 1 shows a series of representative
distribution patterns of u in the middle region of the simulation box from t ¼ 1 104 s to t ¼ 6 104 s. A cutoff um
c is
set to distinguish the gold phase from the pores. The phase
of point ðib; jb; kbÞ at time t ¼ ms is defined as
m
m
(3)
Gm
ijk ¼ H uijk uc ;
m
where Hðum
ijk uc Þ is the standard step function. When
m
m
um
>
u
,
Hðu
um
ijk
c
ijk
c Þ ¼ 1, indicating that the point is occum
pied by gold; otherwise, Hðum
ijk uc Þ ¼ 0, corresponding to
a pore space. To achieve a specified relative mass density q,
we adjust the value of the cutoff as um
cs such that the statistisatisfies
the
condition
cal distribution of um
ijk
Gm
c ¼
N X
N X
N
N X
N X
N
1 X
1 X
m
m
G
¼
Hðum
ijk ucs Þ ¼q:
N 3 i¼1 j¼1 k¼1 ijk N 3 i¼1 j¼1 k¼1
(4)
After the specified relative mass density has been achieved,
the average ligament diameter is measured by a statistical
method described below.
B. Control of the relative mass density
In this subsection, we propose a method to generate
atomic models of nanoporous materials with controlled relative mass density, q ¼ qnp =qb , where qnp and qb denote the
mass densities of a nanoporous material and its bulk phase,
respectively. A three-dimensional bicontinuous structure is
generated by solving Eq. (1) in a cubic box which is discretized into a N N N lattice, where N is a sufficiently large
number taken to be N ¼ 100. Let um
ijk denote the discrete
value of uðib; jb; kb; msÞ at lattice node point ðib; jb; kbÞ,
where b and s are the mesh size and the integration time
step, respectively. Discretizing Eq. (1) with central approxi-
FIG. 1. Distribution of differential alloy concentration u in the middle section of the phase field simulation box from t ¼ 1 104 s to t ¼ 6 104 s.
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Establishing an appropriate atomic model of nanoporous
metals is a crucial step in the present study. Most nanoporous
materials produced by fabrication techniques such as dealloying9 exhibit bicontinuous microstructures consisting of
random-sized open cells.26 Newman et al.27 showed that the
bicontinuous structures of np-Au are similar to those associated with spinodal decomposition, a process that can be
adequately described by phase field methods.28–31 Here, we
adopt such a method to generate atomic models of nanoporous metals with controlled relative mass density and average ligament size.
In the phase field method, we mimic spinodal decomposition of a binary fluid mixture of two components, A and B,
with initially uniform spatial distributions but different volume fractions. Upon cooling below a critical temperature,
the mixture spontaneously decomposes into two phases with
bicontinuous microstructures,28 each phase characterized by
an individual component. After the phase separation, one
phase A is removed to create pores, while the remaining
phase B forms a nanoporous microstructure. This spinodal
evolution can be described by the Cahn-Hilliard equation32
@u
df ðuÞ
¼ r2
h2 r2 u ;
(1)
@t
du
J. Appl. Phys. 113, 023505 (2013)
023505-3
Sun et al.
C. Control of the average ligament diameter
where u ¼ a b, a and b being the angles measured from
the x-axis to O1A and O2B, respectively; d is the normalized
diameter of the cylinder; h is the normalized distance
between the two cross sections. By randomly selecting u
2 ½0; 2p and h 2 ½0; l, where l is the length of the cylinder,
the statistical distribution of the secant line length can be
determined. For illustration, Fig. 2(b) shows the distribution
of secant line lengths for a cylinder with d ¼ 20b and
l ¼ 100b. It is seen that the distribution has the maximum frequency around 20b, corresponding to the value of d. We
FIG. 2. A statistical method for measuring the diameter of a ligament. (a)
Schematic of a cylinder with the geometrical parameters involved, (b) distribution of the secant line length, and (c) comparison of the calculated result
and exact value.
have tested the efficiency and accuracy of this method by a
number of examples with different diameters and lengths, as
shown in Fig. 2(c).
Figure 3(a) shows the distribution of secant line lengths
for a np-Au sample using the above method. For each sample, we measure the diameters of 20 different ligaments,
It is found that the average ligament diwith mean value d.
ameter in the phase field model increases with the evolution
time, implying coarsening of the microstructure (Fig. 3(b)).
In the early stage, the two phases separate quickly and the
coarsening rate of microstructure is high. The coarsening
rate will slow down gradually with further evolution. Once
the ligament diameter reaches a specified value, the phase
field simulation is terminated, and the recorded phase of all
discrete points Gm
ijk is used to generate the MD simulation
model. This method is demonstrated to be highly efficient in
obtaining a three-dimensional bicontinuous nanoporous material model with similar microstructures as real np-Au
samples.
D. Molecular dynamics method
For the sake of simplicity, each simulated sample
obtained from the phase field method described above is
used to filter a face-centered-cubic (fcc) single crystalline
Au cube with f100g orientations in all three directions into a
np-Au sample. Each discrete point with Gm
ijk ¼ 1 represents a
FIG. 3. Phase field method for generating the molecular dynamics simulation model of nanoporous materials with bicontinuous microstructure. (a)
Distribution of the secant line lengths, where the value with the maximal frequency corresponds to the average ligament diameter. (b) Evolution of the
average ligament diameter with the phase field simulation time.
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In real nanoporous materials, most ligaments have irregular cross sections and non-uniform dimensions along their
length directions. In our study, each ligament is approximately regarded as a cylinder (Fig. 2(a)). The ligament diameter is determined by the following method. First, a point in
the ligament, e.g., point T in Fig. 2(a), is randomly selected.
A secant line with a random orientation (i.e., AB in Fig.
2(a)) is created through this point, and then its length is calculated using the coordinates of intersection points. Such
procedure is repeated for a sufficiently large number of times
(e.g., ten thousand times). Statistically, it is known that the
secant line length with the maximum frequency corresponds
to the ligament diameter.
To validate the above procedure, let A and B denote the
two points where the random secant line passing through T
intersects the ligament surface, as shown in Fig. 2. The two
cross sections passing A and B have geometrical centers at
O1 and O2, respectively. The length of the secant line AB can
be calculated by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
h ui2
2
þh ;
(5)
AB ¼
d sin
2
J. Appl. Phys. 113, 023505 (2013)
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Sun et al.
FIG. 4. Atomic configuration of a relaxed np-Au sample with the average
ligament diameter of d ¼ 3.26 nm and relative mass density of q ¼ 0.30.
III. RESULTS AND DISCUSSIONS
A. Stress-strain curves
The uniaxial tensile stress-strain curves for the np-Au
samples of types I and II are shown in Figs. 5(a) and 5(b),
respectively. At the initial deformation stage, the stress
increases linearly with strain until the yield stress rs is
reached. After reaching the maximum stress value (defined
as the ultimate tensile strength ru ), the stress decreases as
the strain is further increased. The Young’s modulus Es , the
yield strength rs , and the ultimate strength ru of a given npAu sample are measured from the stress-strain curves and
summarized in Table I. As indicated in Table I, the effective
Young’s modulus Es , the yield strength rs , and the ultimate
strength ru all increase with increasing relative mass density
q. For example, ru increases by 71.5%, from 154.6 MPa to
265.2 MPa as q changes from 0.24 to 0.36. Therefore, for a
fixed ligament diameter, the mechanical properties of np-Au
strongly depend on the relative mass density.
For type II np-Au samples with the same mass density
but different average ligament diameters, the ultimate
strength ru varies significantly while the effective Young’s
modulus Es and the yield stress rs of samples exhibit little
changes. When the average ligament diameter d increases
from 2.45 nm to 4.08 nm, the mean ultimate strength
decreases from 221.1 MPa to 176.9 MPa, corresponding to a
FIG. 5. The stress-strain curves of np-Au samples for (a) type I samples
with fixed ligament diameter of d ¼ 3.26 nm but different relative densities
of q ¼ 0.24, 0.27, 0.30, 0.33, and 0.36, and (b) type II samples with fixed relative mass density of q ¼ 0.30 but different ligament diameters d ¼ 2.45,
2.86, 3.26, 3.67, and 4.08 nm, respectively.
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crystal cell, while all atoms with Gm
ijk ¼ 0 are removed to
create pore space.
To investigate the dependence of mechanical properties
of the constructed np-Au on its relative mass density q and
ligament diameter d, two types of samples (I and II) are created using the phase field method. For type I, all samples are
made to have the same ligament diameter of d ¼ 3.26 nm but
different relative mass densities of q ¼0.24, 0.27, 0.30, 0.33,
and 0.36. For type II, the samples have the same relative
mass density of q ¼0.30 but different ligament diameters of
d ¼ 2.45, 2.86, 3.26, 3.67, and 4.08 nm. For each combination of q and d, we generate three different models with
bicontinuous nanoporous structures and perform simulations
under uniaxial tension. The largest simulated system contains 1 465 852 atoms.
We performed MD simulations using LAMMPS.34 The embedded atom method (EAM) potential,35 which has been
widely used in the simulations of metals and alloys,36,37 is
utilized to describe the interatomic interactions. Prior to
loading, each sample is relaxed to the minimum energy configuration using conjugate gradient method and then thermally equilibrated at 300 K for 60 ps using a Nose-Hoover
thermostat. Figure 4 shows a typical atomic configuration of
the np-Au sample with d ¼ 3.26 nm and q ¼ 0.30 after equilibration. Periodic boundary conditions are imposed in all
three directions, and the time step is set as 1 fs. Uniaxial tension is applied by stretching the cubic box along the y-direction. Specifically, in each loading step, a prescribed uniform
strain increment of 0.1% is applied, and then the system is
relaxed for 1 ps at a constant temperature of 300 K, corresponding to a constant strain rate of 109/s. This stepwise
straining method is commonly used in MD simulations of
various materials.25,38,39
The common neighbor analysis (CNA)40 is adopted to
identify defects generated during deformation. Four types of
atoms are painted in different colors: red for fcc atoms, green
for body-centered-cubic (bcc) atoms, yellow for hexagonalclose- packed (hcp) atoms, and blue for non-12-coordinated
atoms. The atomic stresses are calculated using Virial theorem,41 while the macroscopic stress is obtained by averaging
all atomic stresses over the last 200 time steps of the relaxation period at each loading step.
J. Appl. Phys. 113, 023505 (2013)
023505-5
Sun et al.
J. Appl. Phys. 113, 023505 (2013)
TABLE I. Mechanical properties of np-Au samples from molecular dynamics simulations.
Sample
Relative mass density q
Ligament diameter d (nm)
Young’s modulus Es (GPa)
Yield strength rs (MPa)
Ultimate strength ru (MPa)
Type I
0.24
0.27
0.30
0.33
0.36
3.26
2.83 6 0.13
3.03 6 0.13
3.54 6 0.05
4.04 6 0.02
4.88 6 0.09
130.3 6 7.2
136.8 6 7.5
161.2 6 11.3
174.4 6 18.4
234.8 6 11.9
154.6 6 3.8
165.6 6 5.7
190.4 6 4.0
219.1 6 2.9
265.2 6 4.6
Type II
0.30
2.45
2.86
3.26
3.67
4.08
3.41 6 0.09
3.45 6 0.11
3.54 6 0.05
3.64 6 0.11
3.62 6 0.16
163.9 6 2.0
154.0 6 7.4
161.2 6 11.3
166.9 6 9.1
168.6 6 4.1
221.1 6 7.0
198.0 6 6.5
190.4 6 4.0
183.6 6 1.6
176.9 6 10.4
B. Deformation behaviors and physical mechanisms
To investigate the deformation features and size effects
of np-Au, we first study the sample with a ligament diameter
of d ¼ 3.26 nm and relative mass density of q ¼ 0.30. The
microstructural evolution and defect density levels in this
sample are quantitatively characterized by relative variations
in the fractions of hcp atoms and surface atoms during loading. By convention, an atom is thought to be on surface if its
coordination number is smaller than 9.42 The relative variation in the number of surface atoms is defined as
n ¼ ðNe N0 Þ=N0 , where Ne is the number of surface atoms
in the sample under tensile strain e and N0 is that at the initial
undeformed state (i.e., e ¼ 0). During loading, the value of n
tends to increase as a result of the deformation of free surfaces and the generation of new surfaces in the sample. The
fraction of hcp atoms is defined as g ¼ Nhcp =Nall , where Nhcp
and Nall are the number of hcp atoms and the total number of
atoms in the sample, respectively. The increase in g is
mainly attributed to the formation of stacking faults and twin
planes.43
Figure 6(a) shows a typical tensile stress-strain curve for
the sample with q ¼ 0.30 and d ¼ 3.26 nm. The corresponding variation of the surface atom fraction n and the hcp atom
fraction g with strain are displayed in Figs. 6(a) and 6(b),
respectively. These curves can be characterized by the following four regimes of distinct deformation behaviors.
(i)
(ii)
In the first regime (AB), most ligaments undergo elastic deformation and the stress increases linearly with
the applied strain. The Young’s modulus from the
stress-strain curve is about 3.54 GPa, and the plastic
yield stress at point B is measured to be 170.8 MPa.
Because the surface deformation is negligible and
there is no dislocation activity in this stage, no evident
changes are observed in either g or n.
In the second regime (BC), the stress increases further
but shows a nonlinear relation with the increasing
strain. The ultimate strength is reached at point C,
with ru ¼ 198:1 MPa. Meanwhile, the stacking faults
(iii)
gradually accumulate, resulting in a pronounced
increase in the fraction of hcp atoms g, as shown in
Fig. 6(b). The normalized surface atom number n also
increases (see Fig. 6(a)) because of the incremental
plastic deformation in some ligaments and the generation of new surfaces.
As the applied strain further increases, the stressstrain curve enters the third regime (CD), which
exhibits a negative slope. In this regime, both g and n
continue to rise, indicating a combination of complicated microstructural evolution, large deformation,
and successive rupture of ligaments.
FIG. 6. Deformation characteristics of np-Au with relative mass density
q ¼ 0.30 and average ligament diameter d ¼ 3.26 nm. (a) Variations of stress
r (left ordinate, blue) and surface atom number n (right ordinate, red) with
the applied strain e. (b) The hcp atom fraction g as a function of the strain e.
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relative change of about 25.0%, whereas Es and rs only
show slight increases by 6.2% and 2.9%, respectively. The
deformation mechanisms and size effects related to these
observations will be investigated in detail shortly.
023505-6
(iv)
Sun et al.
In the fourth regime, in which the applied strain
exceeds 22:8% (DE) in the simulation, the fraction of
hcp atoms g shows a slight increase while that of surface atoms n remains nearly the same. The specimen
approaches its final breaking point but the surface
area change is small.
FIG. 7. Evolution of dislocations and microstructure in a np-Au sample
under different applied strains: (a) e ¼ 0, (b) e ¼ 0:080, (c) e ¼ 0:192, and
(d) e ¼ 0:274. A slab between planes z ¼ 30 nm and z ¼ 40 nm is shown.
The loading direction is in the vertical axis, and atoms are colored according
to the CNA method.
planes and travel through a whole ligament or junction (see
Fig. 7(c)), to be finally absorbed by the opposite free surface
while leaving behind stacking faults in the interior. As the
applied strain rises to 22.8% (point D), the fraction of hcp
atoms has increased to almost twice of that at the initial state
(point A). The stacking faults are preferentially found in the
junctions and vertical ligaments, while the horizontal ligaments remain nearly free of such defects. The ligaments in
this cross section exhibit successive necking and rupture,
somewhat similar to crack propagation. Apparently, with
increasing strain, some of the weakest vertical ligaments
begin to fail, resulting in a net reduction in the real crosssectional area of the np-Au sample. Rupture of one ligament
induces redistribution and increase of stresses in other ligaments, leading to successive deformation and failure of the
vertical ligaments. Meanwhile, those ligaments initially not
perfectly aligned with the applied load rotate gradually toward the tensile direction. The superior deformability of ligaments is evidenced by the long strain-softening stage shown
in Fig. 6(a). As the total strain exceeds 22.8%, almost all vertical ligaments are broken, as circled in Fig. 7(d), and the deformation is dominated by necking at this stage.
C. Deformation mechanisms of ligaments
To further understand the deformation and fracture
mechanisms of np-Au, we now focus on a representative ligament in the tensile direction and its junctions with neighboring ligaments, where most stacking faults are located. Figure
8 illustrates the deformation process of such a ligament with
increasing strains. It is observed that partial dislocations initially nucleate from surface steps (see Fig. 8(a)) and travel
through the interior of the ligament. When the externally
applied strain increases to 15%, a large number of stacking
faults are generated in the middle of the ligament (see Fig.
8(b)). As the strain increases further, the ligament starts to
show necking (see Figs. 8(c) and 8(d)) and eventually ruptures (see Fig. 8(e)). It appears that the deformation and fracture mechanisms of an individual nanometer-sized ligament
are essentially the same as those of a single crystalline gold
nanowire.44
Figure 9 illustrates the atomic structures around a junction of a vertical ligament and its neighbors at the applied
strain of 15.4%. Note that many stacking faults appear in the
junction (see blue circle in Fig. 9(a)) but the horizontal ligaments connected to this junction are nearly free of defects
due to much lower stresses in them. These simulation results
FIG. 8. A sequence of snapshots for deformation and rupture of a typical ligament under different strains: (a) e ¼ 0:098, (b) e ¼ 0:150, (c) e ¼ 0:206,
(d) e ¼ 0:244, and (e) e ¼ 0:256.
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To further reveal physical mechanisms behind the
observed deformation behaviors, we investigated the microstructural evolution during plastic deformation. Figure 7 captures a sequence of snapshots in the slab between the planes
of z ¼ 30 nm and z ¼ 40 nm.
Figure 7(a) shows the atomic configuration of the sample at the initial state. It is observed that there are few intrinsic stacking faults across the ligaments. These stacking faults
form during equilibration due to the relaxation of residual
stress. In the elastic regime (AB), the deformation of ligaments is expected to be fully reversible. However, there are
still a few partial dislocations emitted from free surfaces,
which is responsible for the serrations in the stress-strain
curve in Fig. 6(a).
As the applied strain exceeds a critical value of 5:2%
(point B), more and more partial dislocations and stacking
faults are generated in ligaments and their junctions, as
shown in Fig. 7(b). As a result, the average radius of nanopores increases and some ligaments show plastic necking.
However, most ligaments can still bear the increasing load
so that the material exhibits a strain-hardening behavior until
the ultimate strength is reached, as shown in Fig. 6(a).
In the third (CD) regime, many partial dislocations are
nucleated from the free surface, then glide on f111g slip
J. Appl. Phys. 113, 023505 (2013)
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Sun et al.
J. Appl. Phys. 113, 023505 (2013)
FIG. 9. (a) Nucleation of dislocations at a junction (circle) and (b) formation
of a Lomer–Cottrell dislocation lock due to the reaction of two extended
dislocations.
D. Scaling law of Young’s modulus
In this section, we will analyze the scaling laws for the
mechanical properties of np-Au, including the Young’s modulus Es , the yield stress rs , and the ultimate strength ru .
Gibson and Ashby have systematically investigated the
mechanical properties of open-cell foam materials with characteristic microstructural sizes on the order of microns, millimeters, or even larger. Based on the Timoshenko beam
theory, they derived the following scaling law for the macroscopically effective Young’s modulus Es in terms of the relative mass density q ¼ qnp =qb :19
Es
¼ C1 q2 ;
Eb
(6)
where Eb is the Young’s modulus of the bulk phase, and C1
is a constant to be determined from experiments. Figure
10(a) shows the Young’s modulus Es of np-Au type I samples as a function of the relative mass density q. To avoid
the empirical constant C1 , we have normalized Es and q by
the corresponding parameters of the sample with the relative
mass density q ¼ 0:24. In Fig. 10(a), an apparent deviation
is observed between the results of our MD simulations and
the Gibson-Ashby’s scaling law in Eq. (6). Such difference
could be attributed to a strong size effect in mechanical properties of nanosized ligaments.
To account for the simulation results, we propose the
following modified relationship between the effective
Young’s modulus and the relative mass density of nanoporous materials,
FIG. 10. Average properties of np-Au as a function of the relative mass density: (a) the effective Young’s modulus; (b) the yield strength and the ultimate strength. Here, Es , rs , and ru are normalized by their corresponding
values Es0 ¼ Es jq¼0:24 , rs0 ¼ rs jq¼0:24 , and ru0 ¼ ru jq¼0:24 at q ¼ 0:24,
respectively.
Es
¼ C2 ðq2 þ C3 qÞ;
Eb
(7)
where C2 and C3 are dimensionless constants to be determined by experiments or theoretical calculations. The two
terms on the right-hand side of Eq. (7) correspond mainly
to the bending and tensile deformations of ligaments,
respectively. The first term originates from GibsonAshby’s scaling law of Young’s modulus, as given in
Eq. (6). To account for the contribution of the tensile deformation of ligaments, one can invoke the unit cell model
established by Gibson and Ashby,19,21 whereby we can
derive Es =Eb ¼ ðd=lÞ2 and relate the relative density q to
the sizes of the ligaments by q / ðd=lÞ2 . Thus, for the
contribution of the tensile deformation of ligaments, one
has Es =Eb / q.
For open-cell porous materials with a low relative density, the deformation is dominated by the bending of ligaments, while for those with a higher relative density, the
tensile deformation of ligaments is controlling. Therefore, in
the case of conventional open-cell porous materials with a
low mass density, Eq. (7) is reduced to the classical GibsonAshby relation in Eq. (6). For np-Au samples in this
study, fitting our MD simulation results by Eq. (7) leads to
C2 ¼ 0:14 and C3 ¼ 0:97.
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are in agreement with the experimental observation that dislocations preferentially assemble near the junctions of vertical ligaments of np-Au under uniaxial compression.45 The
detailed dislocation structures in this junction are shown in
Fig. 9(b). Due to the complicate stress state in the junctions,
different slip systems can be activated, leading to the formation of diverse dislocation configurations. For example, Fig.
9(b) shows two extended dislocations of Burgers vectors e1
and e2 slipping on ð1 11Þ and ð111Þ planes, respectively. The
reaction of the two dislocations leads to the formation of a
Lomer–Cottrell lock,46 which may help resisting fracture at
the junction.
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Sun et al.
J. Appl. Phys. 113, 023505 (2013)
E. Scaling law of yield strength
Gibson and Ashby19 pointed out that the plastic yielding
of open-cell foams under tension involves two typical mechanisms, namely, plastic collapse at junctions and axial yielding of ligaments. When the former mechanism dominates,
Gibson and Ashby derived the following relation between
the yield strength and relative mass density:19
rs
¼ C4 q3=2 ð1 þ q1=2 Þ;
rb
(8)
where rs and rb denote the yield strengths of the nanoporous
material and its bulk phase, respectively, and C4 is a constant. On the other hand, if the axial yielding of ligaments
dominates, the Gibson-Ashby model gives19
(9)
where C5 is also a constant.
The yield strength rs of type I samples of np-Au is plotted against the relative mass density q in Fig. 10(b), where
rs and q are normalized by the corresponding parameters of
the sample with q ¼ 0:24, as in Subsection III D. Our numerical results are compared with the two Gibson-Ashby’s scaling laws in Eqs. (8) and (9), i.e., rs =rb / q3=2 ð1 þ q1=2 Þ and
rs =rb / q, as shown in Fig. 10(b). Note that our data match
the scaling law rs =rb / q very well, indicating that the
yielding behavior of np-Au is dominated by the axial yielding of ligaments. This finding is consistent with the observed
deformation mechanisms in Subsection III B.
The variations of rs with respect to q can be understood
as follow. The plastic properties of np-Au mainly depend on
those of the ligaments. For type I samples with variable relative mass density q but fixed ligament diameter d, the larger
the q, the shorter the ligament length l. The increased yield
stress of np-Au with q simply reflects the fact that shorter
ligaments tend to have fewer defects.
F. Scaling law of ultimate strength
For the np-Au samples with different relative mass densities q, the normalized ultimate strengths ru obtained from
MD simulations are shown in Fig. 10(b). It is found that the
FIG. 11. Relationship between the ultimate strength and the average ligament diameter of np-Au.
ru
¼ C6 q:
rb
(10)
In addition, the ultimate strength ru of np-Au shows a
strong dependence on the average ligament diameter d, with
an exponential relation ru =rb / d0:44 , as indicated by our
simulation results in Table I and Fig. 11. Such distinct size
effect of the ultimate strength of nanoporous metals can be
understood from the size-dependent strength of single crystalline nanowires, which often exhibits a similar exponential
relation.47,48
IV. CONCLUSIONS
In summary, the tensile deformation behaviors and size
effects of open-cell nanoporous gold have been investigated
using MD simulations. It is found that np-Au primarily fails
through plastic necking and rupture of those ligaments in
alignment with the loading direction. Partial dislocations can
form Lomer–Cottrell locks at junctions, hindering subsequent dislocation activities and enhancing the strength of
junctions. The ultimate tensile strength of np-Au exhibits a
distinct size effect on the average ligament diameter due to a
number of reasons, including suppressed nucleation, as well
as emission and reabsorption of dislocations by free surfaces.
In addition, a number of scaling laws have been found for
the effective Young’s modulus, yield stress, and ultimate
strength of nanoporous metals. The present study provides
not only new insights into the mechanical behaviors and
associated mechanisms of np-Au but also a practical guide
for their applications.
ACKNOWLEDGMENTS
Supports from the National Natural Science Foundation
of China (Grant Nos. 10972121 and 107320503) and the 973
program of MOST (2010CB631005) are acknowledged.
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