Nanoporous structures
DOI: 10.1002/smll.200800129
Nanoporous Structures: Smaller is Stronger**
Gang Ouyang, Guowei Yang,* Changqing Sun, and
Weiguang Zhu
Nanoporous structures have become the focus of intensive
research in recent years owing to their unique applications in
mesoscopic physics and chemistry, and potential technological
applications, including sensing, catalysis, DNA translocation,
and as templates for nanostructure self-assembly.[1–6] The
study of nanoporous structures such as nanopore-, nanocavity-,
and nanochannel-array materials can afford a deep understanding of the new scientific results in such good systems with
negative curvature surface. As the number of atoms near the
inner surface of nanoporous structures is very large relative to
the total number of atoms, the surface effects on the physical
and chemical properties can be dominant. It is well known that
many physical properties of nanomaterials and nanostructures
such as melting temperature, surface free energy, elastic
modulus, and cohesive energy show strong size effects.[7–9]
Nanoporous materials with large internal surface area have
been more extensively employed than other nanomaterials as
good host materials in nanotechnology.[10] For example,
nanoporous structures and nanocavities show a novel sink
effect to capture molecules, and the effect can be controlled by
tuning the pore size and porosity.[2]
Since the lower coordination of atoms of nanoporous
structures can lead to the redistribution of electronic charge
and change the cohesive energy of single atoms in a matrix, the
mechanical responses differ from those of atoms in the bulk
counterpart. Additionally, the mechanical applications of
nanoporous structures are currently an important subject of
much research. Thus, numerous expressions about the porosity dependence of the isotropic elastic modulus have been
developed by effective medium theory.[11–12] The effective
elastic modulus is suitable for describing the mechanical
[] Prof. G. W. Yang
State Key Laboratory of Optoelectronic Materials and Technologies
Institute of Optoelectronic and Functional Composite Materials
School of Physics, Science, and Engineering, Zhongshan University
Guangzhou, 510275 (P.R. China)
E-mail: [email protected]
Prof. C. Q. Sun, Prof. W. G. Zhu, Dr. G. Ouyang
School of Electrical & Electronic Engineering
Nanyang Technological University
Singapore, 639798 (Singapore)
Dr. G. Ouyang
College of Physics and Information Science
Hunan Normal University
Changsha, 410081 (P.R. China)
[] This work was supported by the National Natural Science Foundation of China (10747129, 50525206, and U0734004), the Ministry
of Education (106126), and the Hunan Normal University
(070622).
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properties of nanoporous structures. In general, the effective
elastic modulus of porous material is expressed as the elastic
module of the matrix and the porosity inclusion in the
materials.[13] However, the surface elasticity of nanostructures
is different from the bulk, which is generalized by the Young–
Laplace equation based on the mechanical equilibrium
principle.[14] Dual et al. pointed out that the stiffness of
nanoporous materials may be made to exceed those of the
nonporous counterpart bulk by satisfying certain surface
modifications.[15] In fact, nanoporous structures are similar
with nanocavities with negative curvature of inner surface in
matrix.Sentence ambiguous. Please rephrase. The large innersurface energy can lead to the effective elastic modulus of the
surface with a negative curvature being larger than that of the
plane case.[16] Importantly, the inner skin of nanocavities will
undergo local hardening owing to the local bond stiffening
around nanocavities when the void size becomes small.[16–17]
However, there are not any quantitative theories to predict the
mechanical responses of nanoporous structures when the
cylindrical pore size is in the range of several nanometers.[2]
Herein, we propose a quantitative theory to investigate the
stiffening of nanoporous structures and show that the effective
bulk elastic modulus of nanoporous structures is determined
by the cylindrical pore size and the porosity.
A constructed model of nanoporous structures is shown in
Figure 1. In fact, the physical properties of the inner surface of
nanoporous structures are the same as those of nanotubes. The
inner skin of nanoporous structures differs from that of the
bulk counterpart as a result of the negative curvature.
Thermodynamically, the inner-surface free energy originates
from the atomic-bonding energy and the elastic strain energy
of the inner surface of the nanoporous structures.[18] The
negative curvature of nanoporous structures can induce a
steady increase in the density of the atomic bonds with a
decrease in size. Similarly, the density of the elastic strain
energy increases as the cylindrical pore size decreases. Thus,
according to our previous considerations,[8,17] the innersurface free energy (g) of a cylindrical pore of nanoporous
structures can be written as Equation (1):
g ¼ g struc þ g chem
(1)
In Equation (1) g struc is the structural contribution induced
by the elastic strain energy of the inner-surface atoms and
g chem is that from the cohesive energy of the surface atoms
based on the broken-bond rule,[8] which is written as Equation
(2):
g ðdÞ ¼ g 0 ð1 þ 4h=dÞ þ a "2
(2)
In Equation (2) g o, h, d, and a are the value in the plane
surface with zero curvature, the atomic diameter, the diameter
of a cylindrical nanopore in the matrix, and the average spring
constant of every pair of atoms in the deformation lattice e is
the lattice strain of the inner skin. Owing to the relaxation and
reconstruction of the inner-skin atoms of nanoporous
structures, the inner shell can undergo more stiffening than
before. On the basis of the bond-order–length–strength
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In Equations (6) and (7), t is the thickness of the surface
layer in the nanoporous structures and n is the Poisson ratio.
Furthermore, the surface constitutive equation can be written
based on Hooke’s law as Equation (8):
s ¼ ls ðtr"s Þ1 þ 2ms "s
Figure 1. Schematic illustration of nanoporous structures (d ¼ diameter
of a cylindrical pore, t ¼ thickness of the inner surface). Inset: arbitrary
cylindrical nanopore in nanoporous structures.
(BOLS) correlation mechanism,[18] the bond broken at the
inner surface of nanoporous structures will cause the
remaining bonds of the undercoordinated atoms to contract
spontaneously. Furthermore, the bond strength is stronger
than that in the bulk. Thus, the surface Young modulus (Esnp )
in the inner shell can be calculated as shown in Equation
(3):[18]
s
DE=E0 ¼ ðEnp
E0 Þ=E0 ¼ ðas =a0 Þm 3as =a0 þ 2
(3)
In Equation (3) m is a characterization parameter of the
bond nature (for alloys m ¼ 4 and for metals m ¼ 1), E0 is the
Young modulus in bulk, and as and ao are the lattice constants
in the surface and in bulk, respectively. The surface atomic strain
in the self-equilibrium state can be derived as @U=V0 @eje¼^e ¼ 0
(U is the total strain energy per unit volume (V0) of the
nanoporous structures). If we assume that in our case we are
dealing with the isotropic state, the relationship between
the surface lattice constant and the surface free energy in
nanoporous structures in the self-equilibrium state can be
deduced as shown in Equation (4):
4gðdÞ
J
as ¼ a0 1 d
(4)
In Equation (4) J is the orientation-dependent constant.
Therefore, the inner-surface Young modulus can be expressed
as the size function Equation (5):
4g ðd Þ
s
J
Enp
¼ E0 3 2 1 d
(5)
If we consider that the matrix and the inner surface of
cylindrical pores are both isotropic,[19] the Lame elastic
parameters at the surface can be calculated as:
s
l ¼ E0
s
m ¼ E0
4g ðdÞ
nt
J
32 1
d
ð1 nÞð1 2nÞ
4g ðdÞ
t
J
32 1
d
2ð 1 þ n Þ
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(6)
(7)
(8)
In Equation (8) s is the surface stress tensor, 1 is the
second-order unit tensor in two-dimensional space, and l5 and
m5 are the surface elastic constants for the isotropic surface.
According to the generalized self-consistent method
(GSAM),[15,20] we employ the nonvanishing strain components with a cylindrical coordinate system. The transverse bulk
elastic modulus ke can be solved by the boundary conditions
and the displacement solutions, which are expressed as
u0r ¼ "0m r, u0f ¼ 0, u0z ¼ 0 and uir ¼ ai r þ bri , uiz ¼ 0, uif ¼ 0 (r
is the radius of the nanopore, and i (i ¼ m, e) denotes the
matrix and the effective medium, respectively. It is found that
the ke for nanoporous structures can be expressed in
Equation (9):
ke ð1 2nÞ½2ð1 pÞ þ ð1 þ p 2pnÞðð2ls þ 4ms Þ=ðdmm ÞÞ
¼
2ð1 þ p 2nÞ þ ð1 pÞð1 2nÞðð2ls þ 4ms Þ=ðdmm ÞÞ
k
(9)
In Equation (9) k is the the bulk elastic modulus, p and mm
denote the porosity and shear modulus, respectively. By
combining Equations (6), (7), and (9), we can obtain the
porosity- and size-dependent effective bulk elastic modulus of
nanoporous structures.
On the basis of the established model mentioned above, we
calculate the inner-surface free energy, the Young modulus of
the surface, and the effective bulk elastic modulus of Au
nanoporous structures. Clearly, the inner-surface free energy
of Au nanoporous structures becomes large with decreasing
diameter as shown in Figure 2a. Interestingly, when d > 5 nm,
the surface free energy of the inner surface smoothly
approaches to that of the bulk. The physical reasons are
attributed to the density of the elastic strain energy of the
lattice relaxation and the density of the atomic dangling bond
become large as the size decreases. Additionally, the
variations of Esnp of Au nanoporous structures are plotted
in Figure 2b. It can be seen that the Young modulus of the
inner surface of Au nanoporous structures increases with
decreasing diameter. Similarly, when d > 5 nm, Esnp smoothly
approaches that of the bulk. Thus, 5 nm seems a threshold
value for the size dependence of Au nanoporous structures.
Indeed, the state of undercoordinated atoms in the inner skin
in nanoporous structures would be different from that in the
bulk, which is similar to the case of nanowires.[21] The surface
layers of nanowires can be approximated as a composite
wire with a core–shell structure that is composed of a cylinder
core in bulk and a shell in the surface layer coaxial with the
core. Thus, the surface Young modulus would be higher than
that of the core bulk. In our case, nanoporous structures can be
treated to be the inner shell–outer core structure.
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small 2008, 4, No. 9, 1359–1362
Figure 3. Dependence of the effective bulk elastic modulus of Au
nanoporous structures on the diameter and the porosity. The necessary
parameters for Au were obtained from References [31, 32]:
h ¼ 0.2884 nm, Eo ¼ 78 GPa, n ¼ 0.44, and g o ¼ 1.59 J m2.
Figure 2. a) Size-dependent inner surface free energy and b) surface
Young modulus of Au nanoporous structures.
According to Equation (9), the relationships between the
cylindrical pore size, the porosity, and the ratio of the effective
bulk elastic modulus of nanoporous structures and bulk at
constant porosities of 0.1, 0.2, and 0.3 is shown in Figure 3.
Clearly, the value of ke/k smoothly increases with decreasing
diameter of the nanoporous structures. The effective bulk
elastic modulus of nanoporous structures with small cylindrical pore size is larger than that of nanoporous structures with
large cylindrical pore size. Very recently, Mathur et al.[22]
experimentally showed that the effective Young modulus of
Au nanoporous leaf shows a strong size-effect, that is, it
increases with decreasing ligament size in the range 3–40 nm.
Thus, our theoretical predictions can depict the trend of
mechanical responses. When p ! 1, the ratio ðke =kÞ
approaches 0. Furthermore, at constant cylindrical pore sizes
of 1, 2, 5, and 8 nm, we can obtain the effective bulk elastic
modulus of Au nanoporous structures with various porosities
as shown in Figure 4. The variation trend in the Au
nanoporous structures with small pores becomes higher than
those with large pore size. Surprisingly, the bulk elastic
modulus of nanoporous structures with a cylindrical pore size
less than 2 nm exhibits a stiffening effect in Figure 4. In other
words, the effective bulk elastic modulus of nanoporous
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structures with pore size less than 2 nm is larger than that of the
bulk counterpart. These theoretical results are consistent with
the recent predictions.[15]
Interestingly, a certain amount of defects such as atomic
vacancies or point defects of solid specimens can enhance the
mechanical strength.[23] Wu et al.[24] indicated that hollow
polymer nanofibers show a great axial stiffening effect and
found that the fiber diameter has an evident effect on the
mechanical response of nanofibers. Similarly, the hardness of
FeAlN and WAlC correlates directly with the concentration
of nitrogen and carbon vacancies.[25,26] In particular, Biener
et al.[27] demonstrated that the enhancement of the hardness of
metal foams can reduce the length scale of ligaments and
pores. Moreover, the strength of nanoporous Au has a value
about ten times higher than that of micrometer-sized porous
structures.[28] Accordingly, the mechanical strength of solid
specimens could be enhanced by the introduction of atomic
vacancies, nanocavities, or nanoporous structures. Impor-
Figure 4. The relationship between the effective bulk elastic modulus,
the porosity, and the diameter.
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1361
communications
tantly, these experimental results are consistent with the
theoretical predictions described herein.
In fact, the surface energy of nanoporous structures plays a
significant role in their mechanical properties. The elastic
response of materials is a fundamental physical property,
which allows the understanding and prediction of the size
effect in the mechanical properties of the nanomaterials.
Generally, for nanomaterials such as nanoparticles, nanowires
and nanofilms, the effect of surface energy on the mechanical
responses can be enhanced when the characteristic sizes are
below 10 nm.[29] By combining Equations (5), (6), and (7), we
can anticipate that local stiffening will take place around
the cylindrical pore skin owing to the size-dependent surface
energy of the inner skin. In other words, the inner-surface
energy of nanoporous structures increases monotonically with
decreasing diameter of the pores in the size range > 2 nm. In
contrast, stiffening of nanoporous structures is less than that of
the nonporous materials when the size and porosity are
beyond the critical size.
There are other reasons that may induce the stiffening
effect in nanoporous structures. Parida et al.[30] reported a
macroscopic reduction by up to 30% in volume of Au
nanoporous leaf during dealloying. A large number of
dislocations and defects appear during the Au nanoporous
leaf formation. According to the effective medium theory, the
effective elastic modulus increases owing to the relative
density increasing rapidly.
In summary, to understand the mechanical responses of
nanoporous structures better, we developed an analytical
approach to shed light on the effective bulk elastic modulus of
nanoporous structures on the basis of nanothermodynamics
and continuum mechanics. Our theoretical results show that
the effective bulk elastic modulus of nanoporous structures
with pore size less than 2 nm is higher than that of nonporous
materials. The porosity and the size of cylindrical pores play a
significant role in the stiffening of nanoporous structures. We
expect that our theory can be applied to the mechanical
responses of nanomaterials with negative curvature structures,
such as nanoporous structures and nanocavities.
mechanical properties . nanoporous structures . surface
free energy
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Received: January 25, 2008
Revised: June 3, 2008
Published online: August 8, 2008
Keywords:
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