Noname manuscript No.
(will be inserted by the editor)
Enhanced mechanical properties of ZnO
nanowire-reinforced nanocomposites: a size-scale effect
Kasra Momeni
Received: Aug 24, 2013
Abstract A multiscale approach is pursued to develop a shear-lag model in
combination with core-surface and core-shell models for capturing size-scale
effect on mechanical properties of ZnO nanowire (NW)-reinforced nanocomposites. In the core-surface (core-shell) model surface effects are represented
by a zero-thickness (finite-thickness) surface with different elastic modulus as
of the central part of NW. Molecular dynamics (MD) technique is utilized
for calculating thickness of the shell, in the core-shell model. Closed form of
governing equations are derived considering linear elasticity for axisymmetric problem and cylindrical coordinate system. Effect of different parameters
including diameter and aspect ratio of NWs are studied to demonstrate application of the developed model. Numerical results disclose that NWs with
larger aspect ratio and smaller diameter can carry larger portion of applied
stress and are preferred in designing high performance nanocomposites. This
result is in agreement with reported computational and experimental data.
Keywords Nano composites · Polymer-matrix composites (PMCs) · Smart
materials · Electrical properties · Multiscale modeling · Size-scale effect
1 Introduction
Recently, ZnO nanostructures and in particular nanowires have attracted the
attention of research community for their mechanical [1, 2] and piezoelectric
properties [3, 4]. Zinc oxide NWs have superior mechanical properties relative to their bulk form [1, 2], large aspect ratio, and low production costs.
K. Momeni
Aerospace Engineering Department, Iowa State University, Ames, IA, USA 50011
Tel.: +1 (515) 294-8387
Fax: +1 (515) 294-3262
E-mail: kmomeni@iastate.edu
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Kasra Momeni
Generating electric potential by bending ZnO NWs, or using multiscale devices constructed of an array of ZnO NWs have been reported [5, 6]. Finite
element modeling technique followed by experimental studies were utilized to
demonstrate the generation of electric potential in ZnO NWs [5, 7].
The atoms that are located at the surface/interface experience a
different environment than their bulk counterparts. They have a different coordination number and consequently a different energy. Reducing the size of materials and structures to inter-atomic distances,
as in the case of nanomaterials and nanostructures, places a larger
portion of the atoms at the surface/interface. Therefore, total energy
of the structure and consequently its properties will be size dependent. Effect of size-scale on mechanical properties of ZnO NWs was captured
using both experimental studies [1, 2] and theoretical calculations [2, 8]. Experimental studies show the elastic modulus depends on diameter, d, of ZnO
NWs, and increases from 147.3GPa for thick NWs (200nm ≤ d ≤ 400nm)
to 249.3GPa for thin (d ' 40nm) NWs [9]. This behavior was interpreted in
the context of a core-surface [10] and a core-shell [2] model. Commonly, using
individual ZnO NWs can not provide sufficient power for micro/nano devices
[5, 6] and utilizing multi-scale devices is inevitable. Various energy harvesting
nanodevices were fabricated based on connecting individual ZnO NWs [6, 11].
Multiscale modeling assists in designing nanodevices which utilize ZnO NWs
by predicting the overall property of the device as a function of its structural
geometry [12].
Nanocomposite electrical generators (NCEGs) are one of the multiscale
devices successfully built experimentally [11] and studied theoretically [12],
where the load transfer mechanism between polymer matrix and ZnO NWs
plays a key role in their performance. Efforts have been made to assess the load
transfer between matrix and nanoscale reinforcement phase of nanocomposites, which includes experimental studies [13], molecular dynamics simulations
[14, 15], and continuum based models [16, 17]. Extensive studies has also been
performed on the stress transfer problem for traditional fiber-reinforced composites, e.g. the shear-lag model proposed by Cox [18]. Although this model
provides a good estimate for the stress transfered to the matrix, it neither accounts for the stress transfer from matrix at the ends of fiber, nor the size-scale
effects.
The objective of this study is developing a multiphysics analytical model
to predict size-scale effects on load transfer between the matrix material and
embedded ZnO NWs. This is a step toward understanding the strengthening
mechanism of NWs in nanocomposites, and also the first step toward capturing
size-scale effect on electric potential generation in nanocomposite electrical
generators. The remainder of this paper is organized as follows: in the first
following section a core-shell and a core-surface model are described. Then a
shear-lag model is elaborated, followed by describing case studies of ZnO NWs
with different diameters. Finally, results of this study is summarized in the last
section.
Title Suppressed Due to Excessive Length
3
2 Size-scale effect on mechanical properties of ZnO NWs
Mechanical properties of different nanowires have been studied experimentally, which shows a wide range of behavior for nanowires of different materials [19, 20]. In addition to the experimental studies, mechanical properties of
nanowires were also studied theoretically by adding corrections to the continuum mechanics theory [21, 22] and atomistic simulation techniques [23], where
they are explained in terms of surface stresses [22] and using direct atomic
simulations [24]. Difference between the elastic modulus of nanosized materials calculated using atomistic simulaitons and continuum mechanics technique
was attributed to the geometry, a characteristic length, surface and bulk elastic
constants of these structures.
In the case of ZnO NWs, their mechanical properties were reported experimentally [1, 2, 25], theoretically[26] and using computational simulations
[23]. The elastic modulus of ZnO NWs shows a strong dependence on the
utilized experimental technique. The reported elastic modulus of ZnO NWs
are: 29(±8)GPa using a single clamped NW bending experiment [27], and
31(±2)GPa by three-point bending test [28]. Experimental results of size-scale
effect on the elastic modulus of ZnO NWs show different trends – i.e., increase
in the elastic modulus of ZnO NWs as a function of diameter [2] as well as
diameter-independent [25] are reported.
Different mechanisms have been proposed in order to demonstrate the sizedependence of mechanical properties of ZnO nanostructures. Theoretical studies [29] show (101̄0) free surfaces relax by contraction of bond length, which
extend at least five to six layers below the film surface [29]. Although in some
studies [29] nonlinear elastic response of NW core was considered to be a major
factor in determining the elastic modulus, surface bond saturation [30], and
surface bond reconstruction [2] are also reported as the source of size-scale
effects.
In this study, size-dependent elastic modulus of ZnO NWs is described
in the context of a core-surface [2] and a core-shell [10] models. Although
different models exists to capture the surface and size-scale effects
such as semi-continuum models [31] and non-local elasticity [32],
the core-surface and core-shell models have advantages. Their main
advantage is capturing the size-dependent properties of the nanostructures with an acceptable accuracy while keeping the formulation
analytically tractable. This is a key point in deriving the governing
equations of complex structures such as nanocomposites. However,
these models have limitations on the size of nanostructures that
they can be applied to. Their main assumption–i.e., presence of a
core structure with properties of the bulk material, is not valid for
materials with extremely small dimensions such as complexions.[33]
Subsequently, a multi-scale continuum model is developed to capture the sizescale effect on the stress transfer in ZnO NW reinforced nanocomposites. The
developed model is then utilized for analyzing effect of different parameters
on the performance and mechanical characteristics of NCEGs.
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Kasra Momeni
Fig. 1 (a) In the real NW the size-scale effect causes the bonds to relax as we move toward
the surface which results in higher elastic modulus of the surface atoms. (b) In the Core-Shell
model, surface effect is modeled by a shell of thickness t with an effective constant elastic
modulus which is different than the elastic modulus of the core part with radius rc .
2.1 Core-Surface model
Effect of surface on elastic properties of materials has been extensively studied
by Gurtin and Morduch [34] using force and moment balance laws. Cammarata
[35] has modeled surface and interface stresses as work per unit area necessary to stretch the surface. A core-surface model was introduced by Miller
and Shenoy [36] which introduces a size-scale factor to calculate deviation in
mechanical behavior of nanomaterial from what was predicted by continuum
theory.
The core-surface model considers the NW as a composite structure of a core
with elastic modulus of bulk and a zero-thickness surface of elastic modulus,
Esr ,(with units of P a · m). Elastic modulus of a NW under tensile loading is
Esr
,
(1)
E = Ec + 2
r
where r is the radius of NW [10]. Although classical core-surface models consider the effect of surface tension [34, 36, 37], this effect has been neglected
here [10]. Considering elastic isotropic materials–i.e. G = E/2(1 + υ), and assuming same υ for core and surface, the core-surface model for shear modulus
is
Gsr
G = Gc + 2
,
(2)
r
where Gc is the shear modulus of core and Gsr is the shear modulus of surface.
2.2 Core-Shell model
As stated above, surface bond relaxation results in the change of mechanical
properties of nanostructured materials [38]. In a core-shell model surface effects
were captured by considering the NW as a composite structure, Fig. 1, with
a core having elastic modulus of bulk material, Ec , and an outer surface shell
coaxial with the core having constant surface elastic modulus of Es . All surface
effects including the change in elastic modulus as a result of bond contraction
is captured in surface elastic modulus Es .
Using the relation for axial stiffness of NW:
EA = Ec Ac + Es As ,
(3)
Title Suppressed Due to Excessive Length
5
Fig. 2 Effect of diameter on thickness of the shell. Different ZnO NWs are simulated using
MD technique to find thickness of the shell in the core-shell model. Simulations are performed
for infinitely long NWs using periodic boundary conditions and details can be found in [39].
Shell-thickness is defined as the thickness at which radial atomic strain is larger than 0.01–
i.e., the atomic strain for smallest NW, ro ∼ 2nm, which core-shell model is still
applicable. Nanowires with diameters (1) 4.245 nm, (2) 4.9 nm, (3) 5.55 nm, (4) 6.2 nm,
(5) 6.857 nm, and (6) 19.31 nm are simulated, which show an approximately constant shell
thickness of < 2.5 nm.
where E is effective elastic modulus, A is the cross-section area, Ac is surface
area of core, and As is surface area of shell, respectively. Approximating NW
as a perfect cylindrical object followed by considering rc as radius of the core,
and t as shell thickness, the relation between elastic modulus of NW and elastic
properties of core and shell are:
Ec (r − t)2 + Es (2r − t)t
.
(4)
E=
r2
Here r = rc + t is the radius of NW. Considering the relation between shear
stress, G, and elastic modulus, E, for elastic isotropic materials – i.e. G =
E/2(1 + υ), and assuming same υ for core and shell, the core-shell model for
shear modulus is
Gc (r − t)2 + Gs (2r − t)t
G=
,
(5)
r2
where Gc is the shear modulus of core and Gs is shear modulus of shell.
Using the core-shell model requires the knowledge of the shell
thicknesses, which needs to be obtained from experimental measurements or atomistic calculations. Although inventing the in situ
characterization techniques made the experimental measurement of
this thickness feasible, utilizing such techniques are still challenging.
Here, the latter approach–i.e., MD simulation technique, is pursued
to calculate the thickness of the shell.
The ZnO NWs with wurtzite crystal structure and diameters ranging from
4.25 nm to 19.31 nm (Fig. 2) are studied. Rigid ion approximation [40] is
used and interactions between the Zn and O ions are modeled with Binks
inter-atomic potential [41],
qi qj
−rij
C
E (rij ) =
+ A exp
(6)
− 6,
rij
ρ
rij
where A, C, and ρ are material parameters and their corresponding values are
listed in table 1; rij is the distance between two ions; and qi is the charge of
i’th ion. The terms on the right-hand side of equation (6) are the long-range
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Kasra Momeni
Table 1 Parameters of the Buckingham potential of ZnO, introduced by Binks et al. [41]
Ions
O2− − O2−
Zn2+ − Zn2+
Zn2+ − O2−
A (eV)
9547.096
0.0
529.70
ρ (Å)
0.21916
0.0
0.3581
C(eV −6 )
32.0
0.0
0.0
Coulombic, and short-rage repulsive and attractive interactions, respectively.
The long-range Coulombic interactions for ZnO NWs were calculated using the
Wolf summation technique, which correctly treats the long-range interactions
at the surfaces of finite size structures. [42] The damping coefficient in the
Wolf summation technique is chosen to be 0.4 and a cut-off radius of 1 nm are
considered [39, 43].
The molecular models of ZnO NWs were simulated for 20ps in the NVT
ensemble at 0.01K without applying any external load. The structure of ZnO
NWs before and after equilibrium are compared to calculate the shell thickness.
It was revealed that a nonzero core forms only for NWs with a diameter greater
than ∼ 4.9 nm, which has a radial strain of ∼ 0.01. This strain value is used
to determine the shell thickness of NWs with larger diameters. It was revealed
that this thickness remains constant as the diameter of the NW increases.
It is shown that the relaxed surface thickness–i.e., thickness with radial
strain larger than 0.01, is < 2.5 nm and independent of diameter. The value
of 0.01 is chosen as it is the radial atomic strain for smallest NW,
ro ∼ 2nm, which core-shell model is still applicable. For the core-shell
model, shell thickness was considered to be fixed and equal to 2.42 nm, which
is in consistent with reported experimental values [10] and interatomic bond
length reported using atomistic calculations [44]. The minimum diameter of a
NW that core-shell model can be applied to is the one with zero core radius–
i.e., a radius of 2.42 nm. Therefore the proposed core-shell model is only
applicable for NW with diameter larger than 4.84 nm. Except this limitation
of core-shell model, there is no limitation on size of NW that can be modeled
in this context. Size-scale effect on elastic modulus of ZnO NWs is shown in
Fig. 3, using both a core-surface and a core-shell model.
3 Problem configuration and shear-lag model
Despite significant challenges, fabrication of nanocomposites reinforced by
well-aligned and evenly disseminated nanofibers have been reported [45]. For
the purpose of analytical modeling of such nanocomposites a hypothetical
model composed of perfectly aligned and uniformly distributed NWs in an
epoxy matrix is considered, as shown in Fig. 4. A cylindrical representative
volume element (RVE) (depicted by a shaded cylinder in Fig. 4 and sketched
separately in Fig. 5) composed of a center-aligned NW perfectly bonded to
an epoxy matrix, is considered. Although a prismatic RVE with square
Title Suppressed Due to Excessive Length
7
Fig. 3 Elastic modulus of ZnO NWs as a function of radius modeled with the core-surface,
black solid line, and the core-shell, red dotted line is shown.
Fig. 4 Schematic representation of the nanocomposites model. The corresponding RVE is
shown by a shaded cylinder enclosing a single NW.
cross-section is ideal, considering a cylindrical RVE is justified by
the assumption of low NW concentration and vanishing interaction
between NWs in neighboring unit cells. The chosen RVE configuration makes the solution to be analytically tractable, while keeping
the results within an acceptable range of error [46, 47]. Here, the
ZnO NW and epoxy matrix are treated as isotropic materials. Although ZnO
NW is an orthotropic material, previous studies revealed that using isotropic
assumption leads to results with acceptable accuracy [48].
A shear-lag model was originally introduced by Cox[18] for analyzing the
load transfer between a matrix material and embedded fiber. Later on this
model was elaborated by Nairn [49] using equations of elasticity for axisymmetric stress states in transversely isotropic materials. However, in these studies
effect of stress transfer from both ends of the fiber is neglected. An improved
shear-lag model which also considers the stress transfer between the matrix
and fiber was introduced by Gao and Li [46]. Despite advantages of latter
model, it considers a constant elastic modulus for the nanofibers.
In this research study, the shear-lag model obtained in Ref [46] is expanded
for ZnO NWs with diameter-dependent elastic coefficient. Two different mod-
8
Kasra Momeni
Fig. 5 Schematic representation of the composite’s RVE which has an outer radius of R.
The radius of NW is ro , and origin of the coordinate system is considered to be at the
center of NW which has a length 2Lf . Reprinted with permission from [12]. Copyright 2010,
American Institute of Physics.
els, core-surface [10] and core-shell [2], are used for describing relation between elastic modulus of a NW and its diameter. There are three major assumptions in deriving governing equations [49] which are i ) bonding between
matrix and fiber is perfect and variation of axial stress along axis in fiber
f
and matrix are functions of the location along axis – i.e. ∂σzz
/∂z = f (z),
m
∂σzz /∂z = g(z); ii ) axial deformation is much larger than radial deformation
–i.e. |∂w/∂r| |∂u/∂z|; iii ) axial stress is much larger than hoop stresses–i.e.
σrr + σθθ σzz . Simulation results indicate that maximum stress in NWs is
a function of diameter and aspect ratio of NWs, and increases by decreasing
(increasing) the diameter (aspect ratio) of NWs.
3.1 Governing equations
Considering the axisymmetric configuration of this problem, governing continuum equations were expressed in cylindrical coordinate – i.e. (r, θ, z). Properties of NW and matrix are specified by superscripts f and m, respectively. The
conservation of linear momentum in absence of body forces and acceleration
is
∂σrr
∂σrz
σrr − σθθ
+
+
= 0,
(7a)
∂r
∂z
r
∂σrz
∂σzz
σrz
+
+
= 0.
(7b)
∂r
∂z
r
Defining u and w as displacement along r and z direction, geometrical equations for ZnO NW can be written as
εrr = ∂u/∂r, (8a)
γrz = ∂u/∂z + ∂w/∂r.
εθθ = u/r,
(8d)
(8b)
εzz = ∂w/∂z, (8c)
Title Suppressed Due to Excessive Length
9
Considering the ZnO NW and epoxy matrix as isotropic materials under
axisymmetric loading, using infinitesimal strain formulation the constitutive
equations for NW and matrix are
1
1
[σrr − ν (σθθ + σzz )] , (9a)
εθθ =
[σθθ − ν (σzz + σrr )] ,(9b)
εrr =
E
E
σrz
1
γrz =
.
(9d)
εzz =
[σzz − ν (σrr + σθθ )] , (9c)
G
E
f
It should be noted that elastic modulus of NW, E , is a function of radius.
The governing boundary condition for axial loading on RVE is
T m |r=R = 0,
(10a)
T m |z=±Lf = ±σ êz ,
(10b)
where T is traction vector, R is radius of RVE, L is length of RVE, and êz is
the unit vector along z-axis. Force balance at the interface of NW and matrix
results in
(11a)
T f −L <z<L ,r=ro = T m |−Lf <z<Lf ,r=ro ,
f
f
m
f
(11b)
T z=±L ,0<r<ro = T |z=±Lf ,0<r<ro .
f
In order to derive the governing equations, RVE is divided into two distinct
parts one that is reinforced by NW, and the other contains only the matrix.
Solution of governing equations is elaborated in next two sections.
3.2 Solution in reinforced region
Integrating equation (7b) with respect to r from 0 to ro for a NW is
Z ro
Z ro
f
∂σzz
1
1 ∂
1
f
(2πr)
dr
+
rσrz
(2πr) dr =
2
2
πro 0
∂z
πro 0 r ∂r
Z ro
∂
1
2
f
f r0
= 0,
(12)
σ
(2πr)
dr
+ 2 rσrz
zz
2
0
∂z πro 0
ro
where the first term in equation (12) shows average axial normal stress over
cross section of NW and can be expressed as follows
Z ro
1
f
f
σ̄zz = 2
σzz
(r, z) · (2πr) dr.
(13)
πro 0
Taking derivative of equation (13) with respect to z and substituting to (12)
f
f
at r = ro is represented by τof .
/dz = −2τof /ro , where σrz
results in dσ̄zz
f
Assuming ∂σzz /∂z = f (z) and using equation (7b),
f
∂σrz
σf
+ f (z) + rz = 0,
(14)
∂r
r
f
with a solution as, σrz
= −f (z)·r/2+c1 /r; Considering axisymmetric geometry
f
of this problem, σrz vanishes on the axis, and solution of equation (14) reduces
to,
1
f
σrz
= − f (z) · r.
(15)
2
Therefore equation (15) simplifies as
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Kasra Momeni
r
2 f
f
σrz
= τof (z) .
(16b)
τo (z),
(16a)
ro
ro
Integrating (7b) with respect to r from ro to R and using (10a) gives,
RR m
m
m
(r, z) (2πr) dr. Asdσ̄zz
/dz = 2ro /(R2 − ro2 )τof where σ̄zz
(z) = π(R21−r2 ) ro σzz
o
m
suming ∂σzz /∂z = g(z), where g(z) is an unknown function which must be
determined. Utilizing equation (10a) and integrating with respect to r from r
to R leads to
f (z) = −
m
σrz
=
g(z) R2 − r2
·
.
2
r
(17)
Substituting (17) to (11b),
2 · ro f
τ .
R2 − ro2 o
Combining equations (17) and (18) leads to,
2
R − r2
ro
m
σrz (r) = 2
τof .
R − ro2
r
Assuming |∂u/∂z| |∂w/∂r|, and using equations (8d) and (9d),
g(z) =
(18)
(19)
∂wm
∂wf
m
σrz
= Gm
.
(20b)
,
(20a)
∂r
∂r
Substituting equation (20b) in (19),τof = Gm (R2 −ro2 )/(R2 −r2 )·r/ro ·∂wm /∂r
which by integrating from ro to R results in,
m
wR
− wrmo
R2 − ro2
τof = Gm
.
(21)
2
ro
R ln (R/ro ) − 1/2 (R2 − ro2 )
Combining equations (21) and (19), result in
m
2
2
− wrmo
wR
m
mR − r
σrz (r) = G
.
(22)
r
R2 ln (R/ro ) − 1/2 (R2 − ro2 )
Substituting equation (22) to (20b) and integrating from ro to r results in
"
#
m
wR
− wrmo · R2 ln (r/ro ) − 1/2 r2 − ro2
m
m
.
(23)
wr (r, z) = wro +
R2 ln (R/ro ) − 1/2 (R2 − ro2 )
f
σrz
= Gf
Using equations (8c) and (9c), and assuming σrr + σθθ σzz for both NW
and matrix material gives the constitutive equations of NW and matrix along
z-axis as
∂wm
∂wf
m
f
σzz
= Em
.
(25)
σzz
= E f (r)
,
(24)
∂z
∂z
Substituting equation (23) to (25),
R2 ln (r/ro ) − 1/2 r2 − ro2
m
m
m
σzz =σzz |r=ro + 2
[σ m |r=R − σzz
|r=ro ] . (26)
R ln (R/ro ) − 1/2 (R2 − ro2 ) zz
Using the conservation of linear momentum along the z-axis,
Z ro
Z R
f
m
πR2 σ =
σzz
· (2πr)dr +
σzz
(2πr)dr,
(27)
0
ro
Title Suppressed Due to Excessive Length
11
equations (13), (26), and (7b) results in,
m
m
f
m
σzz
|r=R = σzz
|r=ro + R2 σ − σ̄zz
ro2 + σzz
|r=ro ro2 − R2
R2 ln (R/ro ) − 1/2 R2 − ro2
× 4
. (28)
R ln (R/ro ) − 1/4 (R2 − ro2 ) · (3R2 − ro2 )
Combining equations (21), (25) and (28) gives the governing differential
equations for NW’s stress along z-axis as
f
m
σ̄zz
ro2 − R2 σ + σzz
|ro R2 − ro2
R2 − ro2
1
d2 σ fzz
=
.
(29)
dz 2
ro2
1 + νm R4 ln (R/ro ) − 1/4 (R2 − ro2 ) (3R2 − ro2 )
Assuming perfect bonding between NW and matrix material – i.e. εm
zz |r=ro =
εfzz |r=ro , results in
Em f
m
σ |r=ro .
(30)
σzz
|r=ro = f
E (r) zz
f
f
Assuming low volume fraction of fiber, σzz
≈ σ̄zz
, and using equation (30)
gives
f
d2 σ̄zz
f
− η 2 α2 (r)σ̄zz
= −R2 η 2 σ,
(31)
dz 2
which is an ordinary differential equation with constant coefficients, and η 2 =
R2 −ro2
1
Em
1
2
2
2
2
ro2
1+νm × R4 ln(R/ro )−1/4(R2 −ro2 )(3R2 −ro2 ) and α (r) = ro + E f (r) R − ro .
Solving equation (31) results in
R2 σ
f
(32)
σ̄zz
= A(r) sinh (η α(r)z) + B(r) cosh (η α(r)z) + 2 .
α (r)
Substituting in equation (13), gives
2τ f
− o = η α(r) [A(r) cosh (η α(r) z) + B(r) sinh (η α(r) z)] .
(33)
ro
Substituting equations (28), (30), and (32) in (26) results in
R2 ln (r/ro ) − 1/2 r2 − ro2
m
R2 σ +
σzz = 2
R ln (R/ro ) − 1/4 (R2 − ro2 ) · (3R2 − ro2 )

2
h
i 
m
 Em
R ln (R/ro ) − 1/2 R2 − ro2 · ro2 + EEf (r) R2 − ro2 
−
×
 E f (r)

R4 ln (R/ro ) − 1/4 (R2 − ro2 ) · (3R2 − ro2 )
R2 σ
A(r) sinh (η α(r)z) + B(r) cosh (η α(r)z) + 2
.
(34)
α (r)
Using (33) in (16b) results in,
r
f
σrz
= − ηα(r) {A(r) cosh (η α(r)z) + B(r) sinh (η α(r)z)} .
(35)
2
Finally, the use of (33) in (19) gives,
ro2 η α(r)
R2
m
σrz =
r−
{A(r) cosh (η α(r)z) + B(r) sinh (η α(r)z)} .
2 (R2 − ro2 )
r
(36)
f
f
m
m
Equations (32) and (34) to (36) represent σ̄zz
, σrz
, σzz
and σrz
, respectively. The pure matrix region must be considered in order to calculate two
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Kasra Momeni
constant coefficients A and B. Equations that are derived in the reinforced region will be used in the pure matrix region by considering a virtual NW with
mechanical properties the same as mechanical properties of matrix material.
3.3 Solution in the pure matrix region
Solution of reinforced matrix region is used for pure matrix region by substituting Ef = Em in equation (32) that results in
q 2 2
R −ro
R2
1
m
z
+
σ̄zz
= A0 (r) sinh
2
2
2
2
2
4
q 2 ro2 1+νm R ln(R/ro )−1/4(R −ro )(3R −ro )
(37)
2
R −ro
1
R
B 0 (r) cosh
z
+
σ,
2
4
2
2
2
2
r
1+νm R ln(R/ro )−1/4(R −r )(3R −r )
o
o
o
where A0 and B 0 are the same as A and B in equation (32) and functions of
r. A0 and B 0 are determined to be zero using boundary conditions (10a).
In the next step, using equation (32), (10a) and (11b), A and B are calculated as
(38b)
B = −R2 + α2 σ sech[Lf α η]/α2 .
A = 0, (38a)
Substituting A and B into equations (32) to (36) gives,
−R2 + α2 σ cosh[z α η]sech[Lf α η]
R2 σ
f
σ̄zz
= 2 +
,
(39a)
α
α2
ro −R2 + α2 ησ sech[Lf α η]sinh[z α η]
,
(39b)
σof = −
2α
R2 ln (r/ro ) − 1/2 r2 − ro2
m
R2 σ +
σzz
= 2
R ln (R/ro ) − 1/4 (R2 − ro2 ) · (3R2 − ro2 )

i 
h
2
m
 Em
R ln (R/ro ) − 1/2 R2 − ro2 · ro2 + EEf (r) R2 − ro2 
−
×
 E f (r)

R4 ln (R/ro ) − 1/4 (R2 − ro2 ) · (3R2 − ro2 )
#
"
−R2 + α2 σ sech[Lf α η]
R2 σ
cosh (η α(r)z) + 2
(39c)
α2
α (r)
(
)
−R2 + α2 σ sech[Lf α η]
r
f
σrz = − ηα(r)
sinh (η α(r)z) ,
(39d)
2
α2
(
2
2
−R2 + α2 σ sech[Lf α η]
r
η
α(r)
R
o
m
σrz =
r−
2 (R2 − ro2 )
r
α2
)
sinh (η α(r)z) ,
(39e)
where size-scale effect is captured through elastic modulus of the fiber, E f ,
and is a function of r.
Title Suppressed Due to Excessive Length
13
4 Case study
Parameter studies are used for illustrating the developed model. Here, effect
of aspect ratio, Lf /r, and radius, r, on axial and interfacial shear stresses
and their maximum values are studied for three different NWs with aspect
ratios of 7.8, 10.1, and 12.8, and three radii of 10 nm. Elastic modulus of
core, surface, shell, and matrix are Ec = 115 GPa [10], Esr = 267 Pa · m
[10], Es = 244.4 GPa [10], Em = 2.41 GPa [46], respectively. Poisson’s ratio
of matrix is νm = 0.35 and thickness of the shell is t = 2.42 nm [10] that is
also in consistence with MD simulations. Distribution of normalized average
f
f
axial normal stress, σ̄zz
/σ, and normalized shear stress, σrz
/σ, are studied
when elastic modulus is size-dependent utilizing the core-surface and core-shell
models (Figs. 6, 8). Size independent model is obtained by assuming zero
surface elastic modulus in the core-surface model. The same goal is achieved
for the core-shell model using the same elastic modulus for the shell material
as used for the core material – i.e. Ec = Es . Distribution of normalized shear
stress and normalized average axial normal stress calculated using this model
without surface effects is the same as previously proposed shear-lag models
[46, 47], which verifies the feasibility of the proposed model.
f
/σ along the length of three NWs with aspect ratios of 12.8,
Variation of σ̄zz
10.1, and 7.8 is illustrated in Fig. 6 using the core-surface and the core-shell
model. The normalized average normal axial stress vanishes at both ends of
NW and it reaches a maximum value at the center of NW as shown in Fig. 6.
f
/σ for a given point along a NW is higher for
Furthermore, it is seen that σ̄zz
the ones with larger aspect ratio. Considering the equilibrium conditions at
each cross-section of RVE, it is expected that σ̄zz /σ follows an opposite trend
m
/σ in matrix is inversely proportional
in matrix material. In other words, σ̄zz
f
to the aspect ratio of NW. Maximum value of σ̄zz
/σ as a function of radius
of NWs with aspect ratios of 12.8, 10.1, and 7.8 is shown in Fig. 7 for the
f
/σ
core-surface and the core-shell models. It is seen that maximum value of σ̄zz
increases by increasing the aspect ratio of NWs. Furthermore, it reduces as
diameter of NWs increase and reaches a value predicted by the corresponding
model with neglected surface effects.
f
/σ, is shown in Fig.
The change in normalized interfacial shear stress, σrz
8 along the length of three NWs with aspect ratios of 12.8, 10.1, and 7.8
using the core-surface and the core-shell models. The normalized interfacial
shear stress vanishes at center the of NW and reaches an extremum value at
both ends of NW, which are shown in Fig. 8. Furthermore, it is seen that
f
/σ for a given point along NW is smaller for NWs with larger aspect ratio.
σrz
f
/σ as a function of radius of NWs with aspect ratio of
Maximum value of σrz
12.8, 10.1, and 7.8 for a core-surface and a core-shell model is shown in Fig.
9. It is illustrated that maximum normalized interfacial shear stress increases
by increasing aspect ratio of NWs and reduces by increasing the diameter
of NWs to a value predicted by the corresponding model with no surface
effects. Comparing the results for a core-surface and a core-shell model, it
14
Kasra Momeni
r = 10 nm
r = 10 nm
20
20
L /r = 12.8 nm
L /r = 12.8 nm
f
f
L /r = 10.1 nm
15
15
f
L /r = 10.1 nm
f
L /r = 7.8 nm
L /r = 7.8 nm
f
f
5
zz
10
f
zz
f
10
5
0
0
-5
-5
-100
-50
(a)
0
50
100
-100
-50
(b)
z (nm)
0
50
100
z (nm)
f
Fig. 6 Distribution of normalized average axial normal stress, σ̄zz
/σ, which is calculated
from (a) Core-Surface model and (b) Core-Shell model, is shown along NWs with three
different aspect ratios (I) Lf /r = 12.8, (II) Lf /r = 10.1, and (III) Lf /r = 7.8.
Core-Surface Model
Core-Shell Model
18
18
W ith
Size-Effect
W ith
W ithout Size-Effect
16
16
L /r = 12.8
f
L /r = 12.8
12
14
L /r = 10.1
f
zz
f
zz
f, max
L /r = 10.1
f, max
f
14
12
L /r = 7.8
f
L /r = 7.8
f
10
(a)
Size-Effect
W ithout Size-Effect
10
0
20
40
60
r (nm)
80
100 120
0
(b)
20
40
60
80
100 120
r (nm)
f,max
Fig. 7 Maximum normalized average normal axial stress, σ̄zz
/σ, as a function of NW’s
radius, r, calculated from (a) the Core-Surface, and (b) the Core-Shell model, is shown.
Here, NW’s with three aspect ratios (I) Lf /r = 12.8 (blue line), (II) Lf /r = 10.1 (red line),
(III) Lf /r = 7.8 (black line) are considered for a case including surface effects (solid lines)
and the one without surface effects (dotted lines).
can be concluded that the maximum interfacial shear stress predicted by the
core-surface and the core-shell models are almost the same.
5 Summary
A multiscale approach was pursued for modeling interfacial stress transfer in
a polymer matrix nanocomposite which reinforced with ZnO NWs. Size scale
effect is implemented using a core-surface and a core-shell model for elastic
modulus of NWs. A cylindrical shaped RVE of a ZnO NW embedded in an
epoxy matrix is considered. Although ZnO NWs have hexagonal cross-section,
previous studies show that modeling ZnO NWs as cylinder is an acceptable
approximation [48]. The problem was formulated in cylindrical coordinate using elasticity theory and shear-lag analysis. Using this approach, closed form
formulas were derived for normal and shear stress components in NW and
matrix.
Title Suppressed Due to Excessive Length
15
r = 10 nm
r = 10 nm
2
2
L /r = 12.8
1
L /r = 12.8
1
f
f
L /r = 10.1
L /r = 10.1
L /r = 7.8
L /r = 7.8
f
f
f
0
rz
f
0
rz
f
f
-1
-1
-2
-2
-150
-100
-50
0
50
100
150
-150
z (nm)
(a)
-100
-50
0
50
100
150
z (nm)
(b)
f
Fig. 8 Distribution of normalized shear stress, σrz
/σ, is calculated using (a) Core-Surface
model and (b) Core-Shell, along NWs with three different aspect ratios (I) Lf /r = 12.8
(blue line), (II) Lf /r = 10.1 (red line), and (III) Lf /r = 7.8 (black line).
Core-Shell
Core-Surface
1.9
1.9
W ith
W ith
Size-Effect
1.8
f
1.7
L /r = 10.1
rz
f
L /r = 7.8
1.6
f
1.5
L /r = 12.8
f
1.7
rz
L /r = 12.8
f, max
1.8
f, max
Size-Effect
W ithout Size-Effect
W ithout Size-Effect
L /r = 10.1
f
L /r = 7.8
1.6
f
1.5
0
20
40
60
r (nm)
80
100
120
0
20
40
60
80
100
120
r (nm)
f,max
Fig. 9 Maximum normalized interfacial shear stress, σrz
/σ, calculated as a function
of NW’s radius from (a) the Core-Surface, and (b) the Core-Shell model is shown. Three
different aspect ratios (I) Lf /r = 12.8 (blue line), (II) Lf /r = 10.1 (red line), (III) Lf /r =
7.8 (black line) are considered when surface effects are included (solid line) and without
surface effects (dotted line).
It was shown that newly developed model is consistent with classical shearlag formulation when surface effects are neglected [47]. A parameter study on
the effect of aspect ratio and diameter of NWs was performed to demonstrate
this model. Numerical results show that aspect ratio and diameter of NWs
play a key role in mechanical performance of nanocomposites, and NWs with
smaller diameter and larger aspect ratios should be used to achieve higher
performance. This is in agreement with previously reported theoretical and
experimental studies.
In order to simplify the model and make it analytically tractable, effect
of piezoelectric response of ZnO NWs on their mechanical properties [50] has
been neglected. However, neglecting the coupling of electrical and elastic properties of NW results in overestimating the elastic modulus of ZnO NWs and
subsequently overestimating stress in NW and underestimating it in matrix,
which must be considered when designing a nanocomposite. Nevertheless, surface effects were included in our model using a core-surface, equation (1), and
a core-shell, equation (4), model through the elastic modulus of NWs, Ef , and
subsequently in calculation of stress through equations (39e) to (39a). It has
16
Kasra Momeni
f
also been shown that normalized average normal axial stress,σ̄zz
/σ, is inversely
proportional to diameter of NWs, which is crucial in designing nanocomposite
electrical generators. Considering the fact that generated electric potential in
ZnO NWs is proportional to the stress, using NW with smaller diameter is
recommended for achieving higher electric output in nanocomposite electrical
generators.
Acknowledgments
The author gratefully acknowledges the Iowa State department of Aerospace
Engineering, Dr. M. Momeni, and Dr. A. Soghrati for providing the financial
support.
References
1. A. V. Desai and M. A. Haque. Mechanical properties of ZnO nanowires.
Sensor. Actuat. A-Phys., 134(1):169–176, 2007.
2. C. Q Chen, Y Shi, Y S Zhang, J Zhu, and Y J Yan. Size dependence of
Young’s modulus in ZnO nanowires. Phys. Rev. Lett., 96(7):075505, 2006.
3. Jun Zhou, Peng Fei, Yifan Gao, Yudong Gu, Jin Liu, Gang Bao, and
Zhong Lin Wang. Mechanical-Electrical Triggers and Sensors Using Piezoelectric Micowires/Nanowires. Nano Lett., 8(9):2725–2730, 2008.
4. P. X Gao, J Song, J Liu, and Z. L Wang. Nanowire Piezoelectric Nanogenerators on Plastic Substrates as Flexible Power Sources for Nanodevices.
Adv. Mater., 19(1):67–72, 2007.
5. Z. L Wang. Piezoelectric Nanogenerators Based on Zinc Oxide Nanowire
Arrays. Science, 312(5771):242–246, 2006.
6. X Wang, J Song, J Liu, and Z. L Wang. Direct-Current Nanogenerator
Driven by Ultrasonic Waves. Science, 316(5821):102–105, 2007.
7. Yifan Gao and Zhong Lin Wang. Electrostatic Potential in a Bent
Piezoelectric Nanowire. The Fundamental Theory of Nanogenerator and
Nanopiezotronics. Nano Lett., 7(8):2499–2505, 2007.
8. J Hu. Surface effect on the size-and orientation-dependent elastic properties of single-crystal ZnO nanostructures. J. Appl. Phys., 105(3):034302–
034302, 2009.
9. A. Asthana, K Momeni, A. Prasad, Y. K. Yap, and R. S Yassar. In
situ observation of size-scale effects on the mechanical properties of ZnO
nanowires. Nanotechnology, 22(26):265712, 2011.
10. Feng Xu, Qingqun Qin, Ashish Mishra, Yi Gu, and Yong Zhu. Mechanical
properties of ZnO nanowires under different loading modes. Nano Res.,
3(4):271–280, 2010.
11. Sheng Xu, Yong Qin, Chen Xu, Yaguang Wei, Rusen Yang, and Zhong Lin
Wang. Self-powered nanowire devices. Nat. Nanotechnol., 5(5):366–373,
2010.
Title Suppressed Due to Excessive Length
17
12. K Momeni, G. M Odegard, and R. S Yassar. Nanocomposite electrical
generator based on piezoelectric zinc oxide nanowires. J. Appl. Phys.,
108(11):114303, 2010.
13. D Qian, EC Dickey, R Andrews, and T Rantell. Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites. Appl.
Phys. Lett., 76(20):2868–2870, 2000.
14. SJV Frankland and VM Harik. Analysis of carbon nanotube pull-out from
a polymer matrix. Surf. Sci., 525:L103–L108, 2003.
15. K Liao and S Li. Interfacial characteristics of a carbon nanotubepolystyrene composite system. Appl. Phys. Lett., 79(25):4225–4227, 2001.
16. H Daniel Wagner. Nanotube–polymer adhesion: a mechanics approach.
Chem. Phys. Lett., 361(1–2):57–61, 2002.
17. S A Meguid, J M Wernik, and Z Q Cheng. Atomistic-based continuum
representation of the effective properties of nano-reinforced epoxies. Int.
J. Solids Struct., 47(13):1723–1736, 2010.
18. H L Cox. The elasticity and strength of paper and other fibrous materials.
British J. Appl. Phys., 3(3):72, 1952.
19. S. G Nilsson, X Borrise, and L Montelius. Size effect on Young’s modulus
of thin chromium cantilevers. Appl. Phys. Lett., 85(16):3555–3557, 2004.
20. Xinxin Li, Takahito Ono, Yuelin Wang, and Masayoshi Esashi. Ultrathin
single-crystalline-silicon cantilever resonators: Fabrication technology and
significant specimen size effect on Young’s modulus. Appl. Phys. Lett.,
83(15):3081–3083, 2003.
21. D. E. Segall, Sohrab Ismail-Beigi, and T. A. Arias. Elasticity of nanometersized objects. Phys. Rev. B, 65:214109, 2002.
22. Vijay B. Shenoy. Atomistic calculations of elastic properties of metallic
fcc crystal surfaces. Phys. Rev. B, 71:094104, 2005.
23. A. J. Kulkarni, M. Zhou, and F J Ke. Orientation and size dependence of the elastic properties of zinc oxide nanobelts. Nanotechnology,
16(12):2749–2756, 2005.
24. Jeremy Q. Broughton, Christopher A. Meli, Priya Vashishta, and Rajiv K.
Kalia. Direct atomistic simulation of quartz crystal oscillators: Bulk properties and nanoscale devices. Phys. Rev. B, 56:611–618, 1997.
25. Baomei Wen, John E. Sader, and John J. Boland. Mechanical properties
of zno nanowires. Phys. Rev. Lett., 101:175502, 2008.
26. M. S. Majdoub, P. Sharma, and T. Cagin. Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect.
Phys. Rev. B, 77:125424, 2008.
27. Jinhui Song, Xudong Wang, Elisa Riedo, and Zhong L Wang. Elastic
Property of Vertically Aligned Nanowires. Nano Lett., 5(10):1954–1958,
2005.
28. Hai Ni and Xiaodong Li. Young’s modulus of ZnO nanobelts measured
using atomic force microscopy and nanoindentation techniques. Nanotechnology, 17(14):3591–3597, 2006.
29. B. Meyer and Dominik Marx. Density-functional study of the structure
and stability of zno surfaces. Phys. Rev. B, 67:035403, 2003.
18
Kasra Momeni
30. Lixin Zhang and Hanchen Huang. Young’s moduli of ZnO nanoplates: Ab
initio determinations. Appl. Phys. Lett., 89(18):183111, 2006.
31. C.T. Sun and Haitao Zhang. Size-dependent elastic moduli of platelike
nanomaterials. Journal of Applied Physics, 93(2):1212–1218, 2003.
32. A Cemal Eringen. On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of Applied Physics,
54(9):4703–4710, 1983.
33. Ming Tang, W Craig Carter, and Rowland M. Cannon. Grain Boundary
Transitions in Binary Alloys. Phys. Rev. Lett., 97:075502, August 2006.
34. Morton E Gurtin and A Ian Murdoch. A continuum theory of elastic
material surfaces. Arch. Ration. Mech. An., 57(4):291–323, 1975.
35. R C Cammarata. Surface and interface stress effects on the growth of thin
films. J. Electron. Mater., 26(9):966–968, 1997.
36. Ronald E Miller and Vijay B Shenoy. Size-dependent elastic properties of
nanosized structural elements. Nanotechnology, 11(3):139, 2000.
37. R C Cammarata. Surface and Interface Stress Effects in Thin-Films.
Progress in Surface Science, 46(1):1–38, 1994.
38. Robert Everest Newnham. Structure-property relations. Springer, 1975.
39. K Momeni, Gregory M. Odegard, and Reza S. Yassar. Finite size effect
on the piezoelectric properties of ZnO nanobelts: A molecular dynamics
approach. Acta materialia, 60(13–14):5117–5124, 2012.
40. Shuangxing Dai, Martin L Dunn, and Harold S Park. Piezoelectric constants for ZnO calculated using classical polarizable core–shell potentials.
Nanotechnology, 21(44):445707, 2010.
41. D Jason Binks and Robin W Grimes. The non-stoichiometry of zinc and
chromium excess zinc chromite. Solid State Communications, 89(11):921–
924, 1994.
42. D Wolf, P. Keblinski, S R Phillpot, and J Eggebrecht. Exact method for
the simulation of Coulombic systems by spherically truncated, pairwise
r[sup -1] summation. The Journal of Chemical Physics, 110(17):8254–
8282, 1999.
43. Kasra Momeni and Hamed Attariani. Electromechanical properties of 1D
ZnO nanostructures: Nanopiezotronics building blocks, surface and sizescale effects. Physical Chemistry Chemical Physics, pages –, 2013.
44. Chang Q Sun, B K Tay, X T Zeng, S Li, T P Chen, Ji Zhou, H L Bai,
and E Y Jiang. Bond-order–bond-length–bond-strength (bond-OLS) correlation mechanism for the shape-and-size dependence of a nanosolid. J.
Phys.-Condens. Mat., 14(34):7781, 2002.
45. R. Haggenmueller, H.H. Gommans, A.G. Rinzler, J.E. Fischer, and K.I.
Winey. Aligned single-wall carbon nanotubes in composites by melt processing methods. Chem. Phys. Lett., 330(3–4):219–225.
46. X Gao and K Li. A shear-lag model for carbon nanotube-reinforced polymer composites. Int. J. Solids Struct., 42(5-6):1649–1667, 2005.
47. CY Li and TW Chou. Multiscale Modeling of carbon nanotube reinforced
polymer composites. J. Nanosci. Nanotechnol., 3(5):423–430, 2003.
Title Suppressed Due to Excessive Length
19
48. Yifan Gao and Zhong Lin Wang. Electrostatic Potential in a Bent
Piezoelectric Nanowire. The Fundamental Theory of Nanogenerator and
Nanopiezotronics. Nano Lett., 7(8):2499–2505, 2007.
49. John A Nairn. On the use of shear-lag methods for analysis of stress
transfer in unidirectional composites. Mech. Mater., 26(2):63–80, 1997.
50. A. V. Desai and M. A. Haque. Influence of electromechanical boundary conditions on elasticity of zinc oxide nanowires. Appl. Phys. Lett.,
91(18):183106–183106, 2007.