- Email: [email protected]
polyethylene composites: A molecular dynamics simulation study
Accepted Manuscript Title: Interfacial properties of carboxylic acid functionalized CNT/Polyethylene composites: A molecular dynamics simulation study Author: Zeshuai Yuan Zixing Lu Mingyang Chen Zhenyu Yang Fan Xie PII: DOI: Reference:
S0169-4332(15)01373-2 http://dx.doi.org/doi:10.1016/j.apsusc.2015.06.039 APSUSC 30558
To appear in:
APSUSC
Received date: Revised date: Accepted date:
7-4-2015 28-5-2015 7-6-2015
Please cite this article as: Z. Yuan, Z. Lu, M. Chen, Z. Yang, F. Xie, Interfacial properties of carboxylic acid functionalized CNT/Polyethylene composites: A molecular dynamics simulation study, Applied Surface Science (2015), http://dx.doi.org/10.1016/j.apsusc.2015.06.039 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
A periodic molecular dynamics model is proposed to eliminate end effect of CNT. Load transfer mechanism in the interphase of functionalized SWNT reinforced
ip t
polymer. The effect of functionalization and sliding velocity on interfacial shear strength.
Ac ce p
te
d
M
an
us
cr
A criterion is proposed to characterize the strength of the interface.
Page 1 of 47
2
Interfacial properties of carboxylic acid functionalized CNT/Polyethylene composites: A molecular dynamics simulation study
ip t
Zeshuai Yuan, Zixing Lu *, Mingyang Chen, Zhenyu Yang, Fan Xie Institute of Solid Mechanics, Beihang University, Beijing, 100191, China
an
E-mail: [email protected]
us
TEL: +86 10 82315707; FAX: +86 10 82328501
cr
* Corresponding author
Abstract
M
A periodic molecular dynamics (MD) model is proposed to investigate the
d
mechanical properties of the interface between a functionalized single-walled carbon
te
nanotube (SWNT) and matrix, with “end effect” eliminated. The load transfer
Ac ce p
mechanism in the layer of interphase surrounding SWNT is investigated as the SWNT is sliding in the polymer. It is indicated that the deformation of the interphase is mainly driven by the functional groups, which determines the interfacial properties that is found to be sensitive to the sliding velocity. The effective interfacial shear strengths in different sliding stages and variable sliding velocities are evaluated, and an empirical model is proposed to describe the effect of the sliding velocity and the amount of functionalization on the interfacial strength. Based on the calculation of shear properties of polyethylene (PE) bulk and an equivalent shear load transfer model, we did an original investigation of the interfacial properties determined by the
Page 2 of 47
3
amount of functionalization of SWNT and the sliding velocity. In addition, a criterion is proposed to characterize the strength of the interface. The results show that the
ip t
assumption of perfect bonding is reasonable for the modeling of these functionalized SWNT/PE composites at a higher scale.
us
properties of interface; Molecular dynamics simulation.
cr
Keywords: Functionalized carbon nanotube; Polymer composites; Mechanical
an
1.Introduction
M
Discovered in 1991 [1], carbon nanotubes (CNTs) have stimulated huge interests in both academics and industry due to their unprecedented mechanical, thermal and
d
electrical properties. These excellent properties, especially the impressive mechanical
te
behavior with Young’s modulus as high as 1TPa and strength higher than 1GPa, give
Ac ce p
CNTs the potential to serve as ideal fillers in polymer-matrix composites [2]. Generally, the effective properties of the composites are determined by the CNTs, the polymer matrix and the interface between them. Among these, the interfacial bonding between the CNTs and the polymer matrix is an important aspect, as it is crucial for the load transfer between the CNTs and the polymer matrix [3,4]. Since CNTs usually agglomerate due to Van der Waals force, a significant challenge in developing high performance CNT/polymer composite is to introduce the individual CNTs in a polymer matrix in order to achieve better dispersion and strong interfacial interactions [5]. In this regard, the surface functionalization of CNTs is an
Page 3 of 47
4
effective way to prevent agglomeration, which helps to better disperse and stabilize the CNTs within a polymer [6]. Up to now, several approaches have been proposed to
ip t
functionalize the CNT surface, including defect functionalization, non-covalent functionalization and covalent functionalization [7]. Among them, adding carboxylic
cr
acid functional groups, -COOH, is more convenient than other approaches, and can be
us
readily used for further covalent and non-covalent functionalization of CNT [5]. Thus carboxylic acid functionalization is very attractive and has been widely employed for
an
the preparation of CNT/polymer composites [8]. Kanagaraj et al. [8] studied the
M
mechanical and tribological properties of high density polyethylene (HDPE) reinforced with acid treated CNTs. A considerable improvement on mechanical
d
properties of the material was observed when the volume fraction of CNT was
te
increased, which showed a good load transfer effect and interface link between the
Ac ce p
functionalized CNT and HDPE.
Several researchers have investigated the interaction between CNTs and polymer
matrix through pull-out experiments[9-11]. Wagner et al. [9] reported that the stress transfer ability of MWNT-polymer (urethane/diacrylate oligomer EBECRYL 4858) interfaces is of the order of 500 MPa under compression and tension. Cooper et al. [10] measured the interfacial strength by pulling out individual single walled carbon nanotube (SWNT) ropes and MWNTs from an epoxy matrix using a scanning probe microscope tip, and evaluated the interfacial strength in the range of 35-376 MPa. It is remarkable that most of the SWNT ropes were fractured instead of being pulled-out,
Page 4 of 47
5
which implied that substantial adhesion exists between the CNTs and the epoxy resin matrix in some of their specimen. Barber et al. [11] have shown that the average
ip t
interfacial stress required to remove a MWNT from the polyethylene-butene matrix is 47 MPa by performing reproducible nanopullout experiments using atomic force
cr
microscopy. The scattered results demonstrate that the interaction mechanism between
us
CNTs and polymer matrix is complex on the nanoscale. However, nano-experiments [12] are expensive and can only evaluate the performance of a fabricated
an
CNT/polymer composite, rather than revealing the mechanisms of reinforcement or
M
providing a way to optimize the mechanical properties.
Molecular dynamics (MD) simulations can provide alternative methods to obtain
d
detailed information of the interfacial interaction and sliding at the molecular level
te
[13,14], especially the interaction between CNTs and the surrounding polymer matrix
Ac ce p
[15,16]. Frankland et al. [17] studied the influence of chemical cross-links between a SWNT and a polyethylene (PE) matrix on the interfacial strength by using MD simulation. The simulation predicted that the strength can be enhanced by over an order of magnitude with the formation of cross-links involving less than 1% of the CNT atoms. Gou et al. [4] simulated the pull-out process of a CNT from a cured epoxy resin by MD method. By estimating the interaction energy, Gou et al. were able to determine that the interfacial strength between the CNT and the epoxy resin was 75 MPa. Zheng et al. [18] investigated the interfacial bonding characteristics between the SWNT, on which –COOH, –CONH2, –C6H11, or –C6H5 groups have been chemically
Page 5 of 47
6
attached, and the PE matrix by performing pullout simulations. The results showed that appropriate functionalization of nanotubes at low densities of functionalized
ip t
carbon atoms drastically increase their interfacial bonding and shear stress between the nanotubes and the polymer matrix. Recently, Haghighatpanah et al. [19] employed
cr
MD and molecular mechanics (MM) methods to investigate the interfacial properties
us
of SWNT/polyethylene and SWNT/polyacrylonitrile composites using pull-out simulations. They investigated how the percentage and the location of –COOH groups
an
functionalized carbon atoms have influenced the interfacial strength.
M
A great majority of literatures on this topic investigated the interaction between CNT and surrounding polymer matrix through a pull-out simulation. However, Li and
d
his co-workers [20] found that the interfacial shear stress (ISS) between CNT and
te
polymer matrix distributes at each end of the embedded CNT within the range of 1nm
Ac ce p
during the pull-out process. By molecular mechanics simulation, they also found that a capped CNT presents a higher pull-out force than that of the corresponding open-ended CNT [21]. As the MD simulation is mostly limited to systems with several nanometers and tens of particles [22,23], a large proportion of atoms, whether CNT atoms or polymer ones, have to be located in the region where the “end effect” is intense. Thus the results of the pull-out simulation are sensitive to the CNT “end effect” because in the pull-out process the periodic boundary condition is difficult to be applied in the axial direction of the CNT. Nevertheless, CNTs with several micrometers in length are widely produced by catalytic chemical vapour deposition
Page 6 of 47
7
(CCVD), and are dispersed in the nanocomposite as reinforcements [24]. Considering the large aspect ratio of CNTs, the “end effect” were usually ignored when modeling
ip t
and analyzing the CNT/polymer composites [25-27]. Therefore, it is important to investigate the mechanism of the “pure interfacial shear” between CNTs and the
cr
surrounding polymer matrix through a molecular-level model that ignores the “end
us
effect”. However, few literatures reports relevant works. Based on the authors’ knowledge, among the many literatures, only Zhang et al. [28] simulated the sliding
an
process of a non-functionalized SWNT in PE matrix by using a periodic MD model.
M
Their simulation indicated that the curve of the ISS as a function of CNT sliding displacement is shaped like that of sliding friction.
d
In addition, according to the observation of the densification and crystallization
te
from MD simulation [29,30] and experimental characterization [31], the structural
Ac ce p
arrangement of matrix polymer in the vicinity of the CNTs is altered as a layer of interphase by the interaction with the CNTs. Several researchers have tried to introduce the effect of the interphase on the properties of non-functionalized CNT/polymer composites in finite element model [32], MD model [33] or multi-scale model [34]. These modeling works indicated that ignoring the interphase could bring about misleading results for the effective properties of the composite. Thus, it is important to investigate the load transfer mechanism of the interphase, as it will have a great influence on the mechanical properties of CNTs/matrix interface. In this work, a periodic MD model has been constructed to investigate the
Page 7 of 47
8
interfacial properties of functionalized SWNT/polymer composite, which avoids the “end effect” of SWNTs and is more convenient to obtain the load transfer mechanism.
ip t
The SWNT is randomly coated with carboxylic acid functional groups –COOH and then is embedded into PE matrix. The emphasis of our study was laid on the load
cr
transfer mechanism of the interphase. Moreover, the effective interfacial shear
us
strengths in different sliding stages and variable sliding velocities are evaluated. Based on the calculation of shear properties of PE bulk and an equivalent shear load
an
transfer model, we investigated the effect of the amount of functionalization of SWNT
M
and the sliding velocity on the interfacial properties. In conclusion, a criterion was proposed to distinguish the strong interface and the weak one.
te
2.1 Atomistic models
d
2. Method
Ac ce p
In MD simulation, an armchair type SWNT with a chiral vector of (10, 10) is chosen as reinforcement, and an amorphous linear PE is used as polymer matrix. A single chain of the amorphous PE consists of 200 –CH2–CH2– unit monomers, since
in chains longer than that, the variation of the interfacial properties can be neglected [28]. The number of PE chains in the model varies according to the size of the unit cell. Although the SWNT chirality may affect the SWNT-polymer interaction [17], the influences are expected to be minor compared to the effect of the functionalization we force on here [19]. Thus the trends obtained with carboxylic acid functionalization of the (10, 10) SWNT are expected to be valid for all chiralities. In total, three kinds of
Page 8 of 47
9
atomistic models are constructed for the investigation and comparison, including functionalized SWNT-polymer systems, a non-functionalized SWNT-polymer system
procedures, we only detail the first one in details here.
ip t
and a pure polymer bulk. As the three kinds of models have similar constructing
cr
Firstly, periodic unit cells are constructed with sufficiently large dimensions of
us
49.19 A 70.00 A 70.00 A in order to prevent interaction between atoms and their
periodic images during the whole MD simulation. The SWNT functionalized with
an
carboxylic acid groups, which are randomly distributed on the SWNT carbon atoms,
M
is added at the center of the unit cell aligned to the X direction as is depicted in Fig. 1(a). After that, random-walk PE chains are generated around the SWNT to fill the
d
unit cell at a target density of 0.7 g/cm3 by using Accelyrs Amorphous Cell package,
te
which is based on the method proposed by Meirovitch [35] and Theodorou and Suter
Ac ce p
[36]. Then, the total potential energy of the unit cell is minimized by the conjugate gradient method with a target convergence threshold for the specified maximum energy change of 0.001 kcal/mol [27,34] (Fig. 1(b)). All the cell construction and the subsequent energy minimization processes are performed using a commercial molecular dynamics simulation package Material Studio 7.0 [37], with the polymer consistent force field (PCFF) [38,39] to describe inter- and intra-atomic interactions. For non-bond calculations, an atom-based cutoff of 11 Å was used in the van der Waals (vdW) interaction, and the Ewald summation method was used to treat the long-range characteristics of the Coulomb interaction [40].
Page 9 of 47
10
When initial structures are sufficiently minimized, they are then inserted into the MD code where an equilibration sequence with four different steps will be performed
ip t
to relax any high energy configurations that are artificially created. Initially, the unit cells are equilibrated for 50 ps with a time step of 1fs under the NVT ensemble at the
cr
temperature of 500 K followed by a relaxation for 150 ps with a time step of 0.5 fs
us
under the NPT ensemble at the temperature of 500 K and the pressure of 1 atm. The next relaxation cooled the structure down to the desired temperature with a step of 1
an
K for every 1 ps, followed by a further relaxation of 300 ps at the desired temperature.
M
The latter two equilibrium processes are both performed under the NPT ensemble at the pressure of 1 atm with a time step of 0.5 fs. The target temperature is set to 100 K,
d
where the PE molecules can be regarded as a glassy and rigid polymer [41,42]. A
te
parallel MD code, LAMMPS [43], has been used to perform the equilibration and the
Ac ce p
following simulations using the same force field (PCFF) mentioned above. The system temperature and pressure are regulated by the Nosé-Hoover methods[44-46]. Similar procedures are performed to construct the atomic models for the
non-functionalized SWNT-polymer system and the pure polymer bulk (Fig. 1(c)). The equilibration sequence relaxes the pure PE bulk to a final density of 0.90 g/cm3, which
is within the range of experimental values 0.89-0.925 g/cm3 for linear low-density PE [47]. The densities of the stable SWNT-PE systems are in the range of 0.93-0.97 g/cm3, which is also in agreement with the experimentally determined densities for the system [47].
Page 10 of 47
11
2.2 MD simulation for SWNT sliding and PE bulk shear After the equilibration, production runs are performed to obtain the ISS of the
ip t
SWNT-PE systems and the shear stress-strain curve of the pure PE bulk. For the former case, the PE atoms outside the interphase are fixed to prevent the
cr
rigid body motion of the PE matrix, shown as the green region in Fig. 2. Uniform
us
velocities ranging from 1m/s to 100m/s are applied to the carbon atoms of the SWNT, to simulate the sliding process. In the simulations, periodic boundary conditions are
an
applied in three directions, and hence the “end effect” of SWNTs can be avoided. As
M
shown in Fig. 2, the interphase for load transferring is defined as a concentric cylinder within the yellow circle with a radius of R1 while beyond the white circle which
d
represents the radius of the SWNT R0 . The conventional definition of the interphase
te
is a layer of dense polymer atoms surrounding the CNT. The local density change is
Ac ce p
used as a criterion to determine the boundary of the interphase region and the bulk matrix [33]. Different from that, in this work, we define the interphase as a layer of atoms for load transferring from the bulk matrix to the CNT. The thickness of the interphase T should be big enough to eliminate the direct interaction between the CNT and the bulk matrix outside. At the same time, it needs to be small to improve computational efficiency. Obviously, the interphase defined here will cover the conventional one. Based on detailed discussions of the density change and the effective interfacial shear strength, we will determine the thickness of the interphase in the following section.
Page 11 of 47
12
As a reference for studying the interfacial shear strength, the shear stress-strain curves of the pure PE bulk are obtained under different strain rates, by applying a
ip t
continuous external shear strain to the unit cell as shown in Fig. 3. Based on the stress-strain curves, a 1-D constitutive model is established to determine the critical
cr
strength of the interface under the theoretical framework of continuum mechanics.
us
The corresponding load transfer model and the comparison will be discussed in details
3. Results and discussion
M
3.1 The interphase for load transferring
an
in Section 3.4.
The radius density distribution (RRD), which has been widely used in
d
CNT/polymer systems [27,48], of the PE matrix has been calculated to reflect the
te
effect of SWNT on the surrounding PE matrix. The local density at different radius R
Ac ce p
is determined by dividing the atomic mass of cylindrical shell by the corresponding volume. Specifically, the inner boundary is defined as the (10,10) SWNT radius R0=6.78 Å, and the thickness of the shell is set to 0.3 Å. The RRDs of non-, 1%, and
5% functionalized SWNT composites are respectively shown in Fig. 4 with the reference density of pure PE bulk. It can be seen from Fig. 4 that whether the SWNT is functionalized or not, the RRD oscillates around the mean value of PE bulk density, which can be attributed to the fact that the linear PE chains are easy to form regular lamellar-type structures on the SWNT surface [49]. In contrast, the fluctuation of the RRD of the pure PE, corresponding to an amorphous structure, is much smaller.
Page 12 of 47
13
Furthermore, for the functionalized SWNTs, the variations of the RRDs are relatively slight, and all have a peak value at around T=4 Å. On the other hand, the amplitude of
ip t
the RRD decreases as T increases, until the density is almost equal to the PE bulk at around 17 Å. However, as the amount of functionalization goes up, the amplitude of
cr
the RRD shows a distinct drop, which may be attributed to the fact that the regular
us
structure of the surrounding PE chains are disturbed by the –COOH groups.
To give a precise evaluation of the thickness of the interphase for load
an
transferring analysis, eight simulations with different thickness of interphase T have
M
been performed for 1% and 5% functionalized SWNT composites respectively. Similar to the definition in Ref. [28], two key parameters for the ISS of the sliding
d
process have been calculated, i.e. the critical interfacial shear stress (CISS) and the
te
steady sliding interfacial shear stress (SISS), which respectively refer to the peak
Ac ce p
value and the mean value of the steady sliding stage, as shown in Sub-figure 5(b). The details of these parameters and the sliding process will be further discussed in the following section.
Then, the CISS and the SISS are calculated for the 1% and 5% functionalized
cases as the thickness of the interphase T increases. As shown in Fig. 5(a), both CISS and SISS decrease as the thickness of the interphase increases and tend to converge to constant values when the interphase reaches a sufficient thickness. This means that the influence of the SWNT on the PE atoms can be neglected at a distance large enough. To determine the appropriate value for the thickness, the relative error between the
Page 13 of 47
14
neighboring data points in Fig. 5(a) is evaluated. We find that the relative error of the CISS and SISS between models with the thickness of 17 Å and 19 Å is smaller than
ip t
5% in both cases. This means that the influence of the SWNT is small enough on the PE atoms at the distance of 17 Å from the SWNT surface, hence an interphase
cr
thickness of 17 Å will be used in all following simulations.
us
3.2 Sliding process of CNT
The ISS is studied by examining the energy changes during the sliding of the
Fi 2R0 L
(1),
M
an
SWNT embedded in the PE matrix, which is calculated as
where Fi is the axial component (x-direction) of the total force on the SWNT
te
d
imposed by PE molecules, R0 is the SWNT radius, and L is the embedded length of SWNT. In the calculation the resultant force on the SWNT is evaluated by
Ac ce p
summing the force interaction between the group of PE molecules and that of atoms of SWNT which consists the atoms of the pristine SWNT and the –COOH groups. Fig. 6 shows the typical ISS as a function of SWNT sliding displacement d CNT for different amounts of functionalization under the sliding velocity of 100 m/s. Similar to the ISS of pristine SWNT sliding [28], three distinct stages can be observed during the sliding process for the functionalized SWNTs, including (1) a linear increase before a critical stress point, (2) a linear decrease, and (3) a steady sliding stage. The peak value after the first stage is defined as the CISS c . During the steady sliding stage, the ISS oscillates irregularly because of the random distribution of functional
Page 14 of 47
15
groups, which is different from the periodic oscillation of the pristine SWNTs [28]. The steady sliding interfacial shear stress (SISS) s is defined by averaging the ISS
ip t
over the whole steady sliding time steps. As shown in Fig. 6, the shape of the ISS curve is similar to the curve of sliding friction, and the SWNT sliding becomes easier
cr
in the steady sliding stage (as kinetic friction) after reaching the CISS (as static
us
friction). Furthermore, Fig. 6 indicates that both the CISS and the SISS go up as the amount of the functionalization increases, while the corresponding sliding
an
displacement for CISS is around 4 Å for all cases.
M
The CISS and SISS are calculated respectively for the nanocomposite with the amount of SWNT functionalization from 0.5% to 5.0% based on the results of MD
d
simulations. Table 1 shows a comparison of the CISS/SISS between the
te
functionalized and non-functionalized SWNT under the sliding velocity of 100 m/s.
Ac ce p
It is obvious that when –COOH groups are added to the SWNT surface, CISS and SISS increase dramatically. As the present periodic model avoids the “end effect”, the values of CISS and SISS for the non-functionalized SWNT-PE system are much smaller than those calculated by the pull-out simulation, but have the same order of magnitude as those for Ref. [28] (~8MPa of CISS and ~1.5MPa of SISS in corresponding PE chain length and sliding velocity). The differences of our results from those of Ref. [28] may be attributed to the temperature difference and the united-atom model in their simulation. Table 1 also shows the values of ISS reported in Ref. [19], in which a SWNT pull-out simulation was implemented. In order to
Page 15 of 47
16
avoid the “end effect” for a valid comparison with our results, the enhancements of the ISS for the functionalized SWNT cases are calculated as presented in Table 1. It
ip t
shows that enhancement values of the SISS in our work are very close to those of the ISS in the pull-out simulation, which indicates that the “end effect” has a minor
cr
influence on the reinforcement of the functionalization. In other word, the –COOH
us
groups may mainly enhance the sliding resistance between the SWNT and the surrounding polymer matrix; this mechanism will be discussed in detail. Meanwhile,
an
the comparison also verifies the validity of our model for reflecting the effects of the
M
–COOH groups.
In order to analyze the loading transfer mechanism, the detailed deformations of
d
the interphase region in the cross-section perpendicular to the y-direction and
te
through the axis of the SWNT are presented in Fig. 7, demonstrating the SWNT with
Ac ce p
the amount of functionalization f 5% under the sliding velocity of 100 m/s, corresponding to the case of black point in Fig. 6. In Fig. 7(a), the atoms are colored according to their molecule ID; a typical –COOH group in the yellow circle marks the sliding of the SWNT. Initially, the –COOH group lies on the left side of the red PE chain. When the SWNT moves to dCNT=6 Å, PE chains undergo an obvious
deformation along with the SWNT translational motion, but the distance between the marked –COOH group and the red chain remains almost unchanged, which indicates that little relative slide happens between the SWNT and the surrounding PE chains. This phenomenon occurs at the end of the first sliding stage, where the ISS reaches
Page 16 of 47
17
approximately the CISS. When dCNT=12 Å, the overlap of the –COOH group and the red PE chain can be observed, apart from the further deformation of the PE matrix.
ip t
This indicates the existence of distinct sliding displacement between the SWNT and the PE matrix. When dCNT=24 Å, the –COOH group moves to the right side of the
cr
red chain and no further deformation of the PE chains is observed comparing to the
us
case of when dCNT=12 Å. This process corresponds to the steady sliding stage indicated in Fig. 6.
an
The SWNT axial displacement of each PE atom is presented in Fig. 7(b) for
M
these four cases. It is indicated that the atomic displacement significantly goes up as dCNT increases from 0 Å to 12 Å, but the variation of the atomic displacement can be
d
neglected for a bigger dCNT. Fig. 7(b) more obviously reflected the deformation of the
te
PE chains surrounding the SWNT. Similar to the analysis above, the primary load
Ac ce p
transfer mechanism of the interphase is the deformation of the PE chains at the first stage of the SWNT sliding and, in contrast, the relative sliding between the SWNT and the PE chains in the steady sliding stage. Furthermore, the atomic displacement follows a similar distribution along the SWNT radial direction for each case, i.e. the closer the PE atom is to SWNT, the bigger the atomic displacement is, which implies a shear deformation in the interphase for load transferring. Fig. 8 shows a clearer variation of the atomic deformation as a function of the radial thickness and the SWNT displacement. The value of the atomic deformation is the average value for atoms in a specific radius. The saddle-shaped surface obviously indicates the shear
Page 17 of 47
18
deformation in the interphase. As the SWNT slides, the interphase undergoes a large deformation at first, and then stays steady. On the other hand, the PE molecules closer
ip t
to the SWNT undergo larger deformation as discussed above. To investigate the effect of the amount of the functionalization of SWNT on the
cr
load transferring in the interphase, a comparison is presented in Fig. 9 between the
us
cases of f 5% and f 1% . An obvious difference of the atomic displacement distribution is indicated between the two cases in Fig. 9(a). When f 5% , the
in contrast,
only the PE atoms below the SWNT clearly move, and the
M
f 1% ,
an
atomic displacement distribution is near symmetrical between the SWNT axis. When
atoms above have small movement, where no –COOH group exists. Fig. 9(b) shows a
d
comparison of the averaged atomic displacements as a function of the radial thickness.
te
The curves of the 1% case are obviously lower than those of the 5% one. This implies
Ac ce p
that the shear deformation of the interphase mainly depends on the functional groups on the SWNT surface, regardless of the position and the amount. Fig. 9(b) also shows that the variation of the atomic displacement can be neglected when the radial thickness reaches around 15 Å, which means the interphase thickness of 17 Å is big enough.
3.3 The interfacial shear strength and criterion As the inertia effect and the viscosity should be cautiously considered in the polymer system [50], we employ an extrapolation to infer the properties of the interface and the matrix in quasi-static. A power-law model has been employed to
Page 18 of 47
19
approximately describe the variation of CISS and SISS respectively, with the amount of functional groups f and the SWNT sliding velocity u CNT as follows: nc ncf u CNT f 0 c c f u 0 CNT 0 nsu nsf u CNT 0 f s s f u 0 CNT 0
ip t
u
cr
(2),
us
where c0 , s0 , ncf , ncu , nsf , and nsu are the constants determined by fitting with the MD results, presented in Table 2. c0 and s0 are reference stress; f 0 and
an
u CNT 0 are respectively the reference amount of functional groups and the SWNT
M
sliding velocity, which we set to f 0 1% and u CNT 0 1m / s in the present work. It is indicated that the exponents ncf and nsf are both approximate to 1.0, in other
d
words, the relationship of the interfacial shear strength and the amount of
te
functionalization is approximately linear, and the load transferring in the interphase
Ac ce p
mainly depends on the interaction between the functional groups and the surrounding PE chains.
The MD results and the fitting curves are presented in Fig. 10 for the CISS and
SISS as a function of the SWNT sliding velocity. As the sliding velocity increases, both CISS and SISS increase obviously for each amount of functionalization. As the analysis in Section 3.2 has shown, the value of ISS depends on the shear deformation of PE chains in the interphase, which is mainly driven by the –COOH groups. Since the mechanical properties of PE is sensitive to the strain rate [41], a higher value of ISS can be obtained for a faster SWNT sliding which corresponds to a higher strain
Page 19 of 47
20
rate of the interphase. In the following, we attempt to establish a criterion to determine whether the
ip t
interface is a strong one or a weak one under the theoretical framework of continuum mechanics. As a reference, the shear strain-stress relationship of the pure PE has been
cr
obtained by MD simulation. The stress-strain curves under different strain rates are
us
shown in Fig. 11(a). It shows that a 1-D elastic-perfectly plastic model is able to describe the stress-strain response at a certain strain rate. The two material parameters,
an
elastic modulus G and yield strength Y , are assumed as functions of the strain rate
M
, which can be obtained by fitting with the MD results. As shown in Fig. 11(b), a power-law model is used to express the relationship
(3),
Ac ce p
te
d
nG G G0 0 n Y Y0 0
where we set to 0 1s 1 in the present paper; nG and n are respectively the stiffness relation exponent and the strength relation exponent. Next, an equivalent shear load transfer model, detailed in Appendix A, is
employed. Fig. 12 shows a sketch for the load transferring in the interphase based on the shear deformation with an assumption that the PE interphase matrix is perfectly bonded to the SWNT. Under this assumption, the critical interfacial yield strength is determined by the equivalent yield strength of the surrounding interphase matrix layer, which is given by Eq. (A7). Combining Eq. (2) and Eq. (A7), a relationship of the
Page 20 of 47
21
amount of functional groups and the sliding velocity can be obtained to distinguish
(4),
us
cr
1 n nsf u u c nG 1 f upper f 0 Y00 0 n u 0 ns u n ns s R0 1 n ncf n 1 ncu n ncu n Y0 c G f lower f 0 0 0 u 0 u c R0
ip t
the strong interface and the weak one, i.e.
which is presented in Fig. 13. The upper and the lower bounds are calculated from the
an
SISS and the CISS, respectively. When the value of f is higher than the upper
M
bound, the interfacial strength stays bigger than the interphase matrix layer, whichever stage the sliding process lies on. This interface can be defined as a strong one,
d
because the yield occurs earlier in the interphase matrix. In contrast, a value of f
Ac ce p
than that of the matrix.
te
lower than the red dash line means a weak interface of which strength stays smaller
Different from the absolute boundary between the strong and weak interface, Fig.
13 shows that a translation state exists between the upper and the lower bounds. As we have discussed above, the ISS curve of the SWNT sliding has a shape like that of the conventional sliding friction, in other words, the value of CISS is generally higher than SISS. It means that a bigger load is required to activate the motion of the SWNT from a static state, but a smaller load is enough to maintain the steady sliding state. Therefore, if a load smaller than CISS but bigger than SISS is applied, the SWNT can not begin to slide from a static state. However, this load could keep the SWNT sliding
Page 21 of 47
22
once the motion of the SWNT is activated by, say, an impact. The existence of the translation state between the strong and weak interface may bring a more notable
ip t
influence on the dynamical properties of the composite rather than the quasi-static nanotube pulling-out behavior. Moreover, even in the quasi-static experiments for
cr
testing the macro-properties of the nanocomposite, the local disturbance can not be
us
ignored, hence it is meaningful to discuss the interface condition by considering the existence of the translation state between the strong and weak interface.
an
Furthermore, it is worth noting that the values of the two bounds decrease
M
nonlinearly as the sliding velocity slows down. When the sliding velocity is around 10-8 m/s, corresponding to the pull-out rate in experiment [51], the value of the upper
d
bound is 0.62%, which can be easily reached for functionalized CNTs [5]. Therefore,
te
it is reasonable to assume a perfect bonding between the functionalized CNT and the
Ac ce p
PE matrix in most cases of modeling at a higher scale. 4. Conclusion
In this paper, a periodic MD model is constructed to investigate the mechanical
properties of the interface between the functionalized SWNT and the polymer matrix. The present model avoids the “end effect” of SWNTs and is more convenient to obtain the load transfer mechanism. Carboxylic acid groups are randomly coated onto the SWNT to study the effect of functionalization, and an amorphous linear PE is used as polymer matrix. Based on the analysis of the RRD and the interfacial strength, an effective interfacial thickness of 17 Å is obtained. During the sliding process of the
Page 22 of 47
23
SWNT, the shear deformation occurs in the layer of interphase to transfer the load from the matrix to the SWNT. The deformation behavior is similar to an
ip t
“elastic-plastic” solid as the SWNT moves, which increases initially and then stays steady thereafter. In addition, the shear deformation is mainly driven by the functional
cr
groups.
us
The ISS is calculated as a function of the SWNT sliding displacement, which shows a similar shape of the curve of sliding friction, i.e. a lower steady sliding stage
an
(SISS) exists following a peak value (CISS). A power-law model is employed to
M
describe the variation of CISS and SISS respectively, with considering the effects of the amount of functionalization and the sliding velocity. Finally, we propose an
d
interfacial strength criterion by considering the amount of functionalization of SWNT
te
and the sliding velocity. The results imply that an assumption of perfect bonding
Ac ce p
between the functionalized SWNT and the PE matrix is reasonable for the modeling at a higher scale under the experimental velocity. It may be noted that the load transfer ability of the interface with more
functionalization, such as 2%, is much higher than the yield strength of the matrix. However, the excessive load can still be transferred to the SWNT because of the interaction between the functional groups and the chains in the interphase whose properties have been modified by the SWNT and the functional groups. The existence of the interphase may further improve the overall properties of the nanocomposite.
Page 23 of 47
24
Acknowledgments The authors thank the support from the National Natural Science Foundation of
ip t
China (11472025, 11272030 and 11272300).
cr
Appendix A. Equivalent shear load transfer model
us
By assuming a perfect interfacial bonding and the decoupling of tensile and shear deformations, the shear strain in the interphase layer can be written as:
G
0 R0
an
nG
G0 R 0
(A1),
M
where 0 is the shear stress at the interface, R0 is the radius of the SWNT.
d
In the elastic stage, the deformation of the surrounding matrix layer satisfies the
te
superposition principle, thus 0. The shear strain rate can be expressed by taking
Ac ce p
the derivative of Eq. (A1) with respect to time as follows: R nG 0 0 0 G0 R
1
nG 1
(A2).
Then the SWNT sliding velocity u CNT can be expressed as:
u CNT
1 dR 1 R0 nG R1
0 R00 nG G0
1
nG nG 1 nG nG 1 R nG 1 R 1 0
(A3).
Therefore, 0 can be written as a function of the sliding velocity,
G u G 0 c 0 CNTnG R00
n 1
where c
1 1 1 n G
nG nG R1 nG 1 R0 nG 1
nG 1
(A4),
.
Page 24 of 47
25
Further, 0 is evaluated based on the time integral for Eq. (A4), G 0 c 0 R0
u CNT 0
nG
d CNT
(A5).
ip t
The strain rate at the interface can be also obtained, 1
(A6).
cr
c nG 1 ( R R0 ) u CNT R0
us
Moreover, the yield occurs at the position near the CNT, and hence the equivalent yield strength of the surrounding matrix layer can be evaluated by
an
introducing Eq. (A6) into the second term of Eq. (3), n
n
(A7).
d
M
c nG 1 u Y Y0 CNT R0 0
te
References:
[1]S. Iijima, Helical microtubules of graphitic carbon, Nature, 354(1991) 56-58.
Ac ce p
[2]J.N. Coleman, U. Khan, W.J. Blau, Y.K. Gun Ko, Small but strong: a review of the mechanical properties of carbon nanotube–polymer composites, Carbon, 44(2006) 1624-1652.
[3]D. Qian, E.C. Dickey, R. Andrews, T. Rantell, Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites, Appl. Phys. Lett., 76(2000) 2868.
[4]J. Gou, B. Minaie, B. Wang, Z. Liang, C. Zhang, Computational and experimental study of interfacial bonding of single-walled nanotube reinforced composites, Comp. Mater. Sci., 31(2004) 225-236.
[5]N.G. Sahoo, S. Rana, J.W. Cho, L. Li, S.H. Chan, Polymer nanocomposites based on functionalized carbon nanotubes, Prog. Polym. Sci., 35(2010) 837-867.
[6]J. Yu, B. Tonpheng, G. Gröbner, O. Andersson, Thermal properties and transition studies of multi-wall carbon nanotube/nylon-6 composites, Carbon, 49(2011) 4858-4866. [7]J.L. Bahr, J.M. Tour, Covalent chemistry of single-wall carbon nanotubes, J. Mater. Chem., 12(2002) 1952-1958. [8]S. Kanagaraj, F.R. Varanda, T.V. Zhil Tsova, M.S. Oliveira, J.A. Simões, Mechanical properties of high density polyethylene/carbon nanotube composites, Compos. Sci. Technol., 67(2007) 3071-3077. [9]H.D. Wagner, O. Lourie, Y. Feldman, R. Tenne, Stress-induced fragmentation of multiwall carbon
Page 25 of 47
26
nanotubes in a polymer matrix, Appl. Phys. Lett., 72(1998) 188. [10]C.A. Cooper, S.R. Cohen, A.H. Barber, H.D. Wagner, Detachment of nanotubes from a polymer matrix, Appl. Phys. Lett., 81(2002) 3873. [11]A.H. Barber, S.R. Cohen, H.D. Wagner, Measurement of carbon nanotube–polymer interfacial strength, Appl. Phys. Lett., 82(2003) 4140. ‐Nanotube/Polymer Interface, Adv. Mater., 18(2006) 83-87.
ip t
[12]A.H. Barber, S.R. Cohen, A. Eitan, L.S. Schadler, H.D. Wagner, Fracture Transitions at a Carbon [13]Z. Yang, Y. Zhao, Adsorption of His-tagged peptide to Ni, Cu and Au (100) surfaces: Molecular dynamics simulation, Eng. Anal. Bound. Elem., 31(2007) 402-409.
cr
[14]Z. Yang, Y. Zhao, Multi-scale simulation of an immobilizing F1-ATPase molecular motor, Journal of Korean Physical Society, 52(2008) 1211.
us
[15]Q. Li, G. He, R. Zhao, Y. Li, Investigation of the influence factors of polyethylene molecule encapsulated into carbon nanotubes by molecular dynamics simulation, Appl. Surf. Sci., 257(2011) 10022-10030.
an
[16]S. Rouhi, Y. Alizadeh, R. Ansari, On the interfacial characteristics of polyethylene/single-walled carbon nanotubes using molecular dynamics simulations, Appl. Surf. Sci., 292(2014) 958-970. [17]S.J.V. Frankland, A. Caglar, D.W. Brenner, M. Griebel, Molecular Simulation of the Influence of Chemical Cross-Links on the Shear Strength of Carbon Nanotube−Polymer Interfaces, The
M
Journal of Physical Chemistry B, 106(2002) 3046-3048.
[18]Q. Zheng, D. Xia, Q. Xue, K. Yan, X. Gao, Q. Li, Computational analysis of effect of modification on the interfacial characteristics of a carbon nanotube–polyethylene composite system, Appl.
d
Surf. Sci., 255(2009) 3534-3543. [19]S. Haghighatpanah, M. Bohlà N, K. Bolton, Molecular level computational studies of
te
polyethylene and polyacrylonitrile composites containing single walled carbon nanotubes: effect of carboxylic acid functionalization on nanotube-polymer interfacial properties, Frontiers in
Ac ce p
Chemistry, 2(2014).
[20]Y. Li, Y. Liu, X. Peng, C. Yan, S. Liu, N. Hu, Pull-out simulations on interfacial properties of carbon nanotube-reinforced polymer nanocomposites, Comp. Mater. Sci., 50(2011) 1854-1860.
[21]Y. Li, S. Liu, N. Hu, X. Han, L. Zhou, H. Ning, L. Wu, G. Yamamoto, C. Chang, T. Hashida, Pull-out simulations of a capped carbon nanotube in carbon nanotube-reinforced nanocomposites, J. Appl. Phys., 113(2013) 144304.
[22]Q. Liu, Z. Lu, M. Zhu, Z. Yuan, Z. Yang, Z. Hu, J. Li, Simulation of the tensile properties of silica aerogels: the effects of cluster structure and primary particle size, Soft matter, 10(2014) 6266-6277.
[23]Q. Yuan, Y. Zhao, Multiscale dynamic wetting of a droplet on a lyophilic pillar-arrayed surface, J. Fluid Mech., 716(2013) 171-188. [24]S. Bal, S.S. Samal, Carbon nanotube reinforced polymer composites—a state of the art, B. Mater. Sci., 30(2007) 379-386. [25]F.T. Fisher, R.D. Bradshaw, L.C. Brinson, Effects of nanotube waviness on the modulus of nanotube-reinforced polymers, Appl. Phys. Lett., 80(2002) 4647-4649. [26]Z. Yuan, Z. Lu, Numerical analysis of elastic–plastic properties of polymer composite reinforced by wavy and random CNTs, Comp. Mater. Sci., 95(2014) 610-619.
Page 26 of 47
27
[27]S. Yang, S. Yu, J. Ryu, J. Cho, W. Kyoung, D. Han, M. Cho, Nonlinear multiscale modeling approach to characterize elastoplastic behavior of CNT/polymer nanocomposites considering the interphase and interfacial imperfection, Int. J. Plasticity, 41(2013) 124-146. [28]Z.Q. Zhang, D.K. Ward, Y. Xue, H.W. Zhang, M.F. Horstemeyer, Interfacial characteristics of carbon nanotube-polyethylene composites using molecular dynamics simulations, ISRN Materials Science, 2011(2011).
ip t
[29]H. Yang, Y. Chen, Y. Liu, W.S. Cai, Z.S. Li, Molecular dynamics simulation of polyethylene on single wall carbon nanotube, The Journal of Chemical Physics, 127(2007) 94902.
[30]S. Kumar, S.K. Pattanayek, G.G. Pereira, Organization of polymer chains onto long, single-wall
cr
carbon nano-tubes: Effect of tube diameter and cooling method, The Journal of Chemical Physics, 140(2014) 24904.
us
[31]L. Li, C.Y. Li, C. Ni, L. Rong, B. Hsiao, Structure and crystallization behavior of Nylon 66/multi-walled carbon nanotube nanocomposites at low carbon nanotube contents, Polymer, 48(2007) 3452-3460.
an
[32]F. Han, Y. Azdoud, G. Lubineau, Computational modeling of elastic properties of carbon nanotube/polymer composites with interphase regions. Part I: Micro-structural characterization and geometric modeling, Comp. Mater. Sci., 81(2014) 641-651.
[33]S. Herasati, L.C. Zhang, H.H. Ruan, A new method for characterizing the interphase regions of
M
carbon nanotube composites, Int. J. Solids Struct., 51(2014) 1781-1791. [34]S. Yang, S. Yu, W. Kyoung, D. Han, M. Cho, Multiscale modeling of size-dependent elastic properties of carbon nanotube/polymer nanocomposites with interfacial imperfections, Polymer,
d
53(2012) 623-633. [35]H. Meirovitch, Computer simulation of self-avoiding walks: Testing the scanning method, The
te
Journal of Chemical Physics, 79(1983) 502.
[36]D.N. Theodorou, U.W. Suter, Detailed molecular structure of a vinyl polymer glass,
Ac ce p
Macromolecules, 18(1985) 1467-1478. [37]Accelrys Inc., San Francisco
[38]H. Sun, S.J. Mumby, J.R. Maple, A.T. Hagler, An ab Initio CFF93 All-Atom Force Field for Polycarbonates, J. Am. Chem. Soc., 116(1994) 2978-2987.
[39]T.L. Chantawansri, T.W. Sirk, E.F. Byrd, J.W. Andzelm, B.M. Rice, Shock Hugoniot calculations of polymers using quantum mechanics and molecular dynamics, The Journal of chemical physics, 137(2012) 204901.
[40]H. Sun, COMPASS: an ab initio force-field optimized for condensed-phase applications overview with details on alkane and benzene compounds, The Journal of Physical Chemistry B, 102(1998) 7338-7364.
[41]D. Hossain, M.A. Tschopp, D.K. Ward, J.L. Bouvard, P. Wang, M.F. Horstemeyer, Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene, 2010, pp.6071-6083. [42]Y. Li, G.D. Seidel, Multiscale modeling of the effects of nanoscale load transfer on the effective elastic properties of unfunctionalized carbon nanotube – polyethylene nanocomposites, Model. Simul. Mater. Sc., 22(2014) 25023. [43]S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys.,
Page 27 of 47
28
117(1995) 1-19. [44]S. Nosé , A unified formulation of the constant temperature molecular dynamics methods, The Journal of Chemical Physics, 81(1984) 511. [45]W.G. Hoover, Canonical dynamics: equilibrium phase-space distributions, Phys. Rev. A, 31(1985) 1695. [46]W.G. Hoover, Constant-pressure equations of motion, Phys. Rev. A, 34(1986) 2499-2500.
ip t
[47]C. Vasile, M. Pascu, Practical guide to polyethylene, iSmithers Rapra Publishing, 2005.
[48]Y. Kuang, B. Huang, Effects of covalent functionalization on the thermal transport in carbon nanotube/polymer composites: A multi-scale investigation, Polymer, 56(2015) 563-571.
cr
[49]A. Minoia, L. Chen, D. Beljonne, R. Lazzaroni, Molecular modeling study of the structure and stability of polymer/carbon nanotube interfaces, Polymer, 53(2012) 5480-5490.
us
[50]J. Yin, Y. Zhao, R. Zhu, Molecular dynamics simulation of barnacle cement, Materials Science and Engineering: A, 409(2005) 160-166.
[51]Y. Ganesan, C. Peng, Y. Lu, P.E. Loya, P. Moloney, E. Barrera, B.I. Yakobson, J.M. Tour, R.
an
Ballarini, J. Lou, Interface Toughness of Carbon Nanotube Reinforced Epoxy Composites, ACS
Ac ce p
te
d
M
Applied Materials & Interfaces, 3(2011) 129-134.
Page 28 of 47
29
Figure captions Fig. 1. The unit cells constructed before equilibration: (a) the functionalized SWNT;
ip t
(b) the functionalized SWNT/PE system; (c) the pure PE bulk. Fig. 2. Atomic model for the SWNT/PE composite.
cr
Fig. 3. Pure shear simulation of the PE bulk.
us
Fig. 4. The variation of the local density in the matrix with the radial thickness of the matrix, the X-axis is the interphase thickness T=R1-R0.
an
Fig. 5. The variation of the CISS/SISS with the interphase thickness.
M
Fig. 6. The typical curve of ISS as a function of the SWNT sliding displacement under the sliding velocity of 100m/s.
d
Fig. 7. The specific distribution of the movement of PE atoms with the SWNT sliding.
te
The SWNT atoms are indicated by pink color.
Ac ce p
Fig. 8. The atomic displacement of the surrounding PE chains as a function of SWNT sliding displacement and radial thickness. Fig. 9. A comparison of the deformation in the interphase between the functionalization of 5% and 1%. Fig. 10. The effect of the SWNT sliding velocity on (a) CISS and (b) SISS. Fig. 11. Shear deformation of the pure PE bulk: (a) the stress-strain curves obtained by MD simulation; (b) the power-law model expressing the effect of strain rate on the elastic modulus and yield strength. Fig. 12. The interphase transferring the load from the matrix to the SWNT through
Page 29 of 47
30
shear deformation. Fig. 13. The interfacial condition determined by the amount of functionalization and
Ac ce p
te
d
M
an
us
cr
ip t
the sliding velocity.
Page 30 of 47
31
Functionalization
0.0%
0.5%
1.0%
1.5%
2.0%
5.0%
CISS (MPa)
6.35
47.45
137.29
202.66
262.84
415.83
Enhancement (MPa)
-
41.1
130.9
196.3
256.5
409.5
SISS (MPa)
3.48
34.83
75.27
106.79
142.29
290.72
Enhancement (MPa)
-
31.3
71.8
103.3
ip t
Table 1 Comparison of ISS with respect to the amount of functionalization
Corresponding results of the pull-out simulation in Ref. [18] 150
-
230
-
Enhancement (MPa)
-
-
80
-
287.2
-
450
-
300
us
cr
ISS (MPa)
138.8
Table 2 Summary of the constants in Eq. (2) by fitting with the MD results
c0 (MPa) 73.10
s0 (MPa)
0.12
nsf
nsu
0.99
0.13
Ac ce p
te
d
38.81
ncu
1.10
M
SISS
ncf
an
CISS
Page 31 of 47
Ac ce p
te
d
M
an
us
cr
ip t
Figure 1
Page 32 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 2
Page 33 of 47
Ac ce p
te
d
M
an
us
cr
ip t
Figure 3
Page 34 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 4
Page 35 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 5
Page 36 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 6
Page 37 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 7
Page 38 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 8
Page 39 of 47
Ac ce p
te
d
M
an
us
cr
ip t
Figure 9(a)
Page 40 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 9(b)
Page 41 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 10(a)
Page 42 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 10(b)
Page 43 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 11(a)
Page 44 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 11(b)
Page 45 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 12
Page 46 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 13
Page 47 of 47
S0169-4332(15)01373-2 http://dx.doi.org/doi:10.1016/j.apsusc.2015.06.039 APSUSC 30558
To appear in:
APSUSC
Received date: Revised date: Accepted date:
7-4-2015 28-5-2015 7-6-2015
Please cite this article as: Z. Yuan, Z. Lu, M. Chen, Z. Yang, F. Xie, Interfacial properties of carboxylic acid functionalized CNT/Polyethylene composites: A molecular dynamics simulation study, Applied Surface Science (2015), http://dx.doi.org/10.1016/j.apsusc.2015.06.039 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
1
A periodic molecular dynamics model is proposed to eliminate end effect of CNT. Load transfer mechanism in the interphase of functionalized SWNT reinforced
ip t
polymer. The effect of functionalization and sliding velocity on interfacial shear strength.
Ac ce p
te
d
M
an
us
cr
A criterion is proposed to characterize the strength of the interface.
Page 1 of 47
2
Interfacial properties of carboxylic acid functionalized CNT/Polyethylene composites: A molecular dynamics simulation study
ip t
Zeshuai Yuan, Zixing Lu *, Mingyang Chen, Zhenyu Yang, Fan Xie Institute of Solid Mechanics, Beihang University, Beijing, 100191, China
an
E-mail: [email protected]
us
TEL: +86 10 82315707; FAX: +86 10 82328501
cr
* Corresponding author
Abstract
M
A periodic molecular dynamics (MD) model is proposed to investigate the
d
mechanical properties of the interface between a functionalized single-walled carbon
te
nanotube (SWNT) and matrix, with “end effect” eliminated. The load transfer
Ac ce p
mechanism in the layer of interphase surrounding SWNT is investigated as the SWNT is sliding in the polymer. It is indicated that the deformation of the interphase is mainly driven by the functional groups, which determines the interfacial properties that is found to be sensitive to the sliding velocity. The effective interfacial shear strengths in different sliding stages and variable sliding velocities are evaluated, and an empirical model is proposed to describe the effect of the sliding velocity and the amount of functionalization on the interfacial strength. Based on the calculation of shear properties of polyethylene (PE) bulk and an equivalent shear load transfer model, we did an original investigation of the interfacial properties determined by the
Page 2 of 47
3
amount of functionalization of SWNT and the sliding velocity. In addition, a criterion is proposed to characterize the strength of the interface. The results show that the
ip t
assumption of perfect bonding is reasonable for the modeling of these functionalized SWNT/PE composites at a higher scale.
us
properties of interface; Molecular dynamics simulation.
cr
Keywords: Functionalized carbon nanotube; Polymer composites; Mechanical
an
1.Introduction
M
Discovered in 1991 [1], carbon nanotubes (CNTs) have stimulated huge interests in both academics and industry due to their unprecedented mechanical, thermal and
d
electrical properties. These excellent properties, especially the impressive mechanical
te
behavior with Young’s modulus as high as 1TPa and strength higher than 1GPa, give
Ac ce p
CNTs the potential to serve as ideal fillers in polymer-matrix composites [2]. Generally, the effective properties of the composites are determined by the CNTs, the polymer matrix and the interface between them. Among these, the interfacial bonding between the CNTs and the polymer matrix is an important aspect, as it is crucial for the load transfer between the CNTs and the polymer matrix [3,4]. Since CNTs usually agglomerate due to Van der Waals force, a significant challenge in developing high performance CNT/polymer composite is to introduce the individual CNTs in a polymer matrix in order to achieve better dispersion and strong interfacial interactions [5]. In this regard, the surface functionalization of CNTs is an
Page 3 of 47
4
effective way to prevent agglomeration, which helps to better disperse and stabilize the CNTs within a polymer [6]. Up to now, several approaches have been proposed to
ip t
functionalize the CNT surface, including defect functionalization, non-covalent functionalization and covalent functionalization [7]. Among them, adding carboxylic
cr
acid functional groups, -COOH, is more convenient than other approaches, and can be
us
readily used for further covalent and non-covalent functionalization of CNT [5]. Thus carboxylic acid functionalization is very attractive and has been widely employed for
an
the preparation of CNT/polymer composites [8]. Kanagaraj et al. [8] studied the
M
mechanical and tribological properties of high density polyethylene (HDPE) reinforced with acid treated CNTs. A considerable improvement on mechanical
d
properties of the material was observed when the volume fraction of CNT was
te
increased, which showed a good load transfer effect and interface link between the
Ac ce p
functionalized CNT and HDPE.
Several researchers have investigated the interaction between CNTs and polymer
matrix through pull-out experiments[9-11]. Wagner et al. [9] reported that the stress transfer ability of MWNT-polymer (urethane/diacrylate oligomer EBECRYL 4858) interfaces is of the order of 500 MPa under compression and tension. Cooper et al. [10] measured the interfacial strength by pulling out individual single walled carbon nanotube (SWNT) ropes and MWNTs from an epoxy matrix using a scanning probe microscope tip, and evaluated the interfacial strength in the range of 35-376 MPa. It is remarkable that most of the SWNT ropes were fractured instead of being pulled-out,
Page 4 of 47
5
which implied that substantial adhesion exists between the CNTs and the epoxy resin matrix in some of their specimen. Barber et al. [11] have shown that the average
ip t
interfacial stress required to remove a MWNT from the polyethylene-butene matrix is 47 MPa by performing reproducible nanopullout experiments using atomic force
cr
microscopy. The scattered results demonstrate that the interaction mechanism between
us
CNTs and polymer matrix is complex on the nanoscale. However, nano-experiments [12] are expensive and can only evaluate the performance of a fabricated
an
CNT/polymer composite, rather than revealing the mechanisms of reinforcement or
M
providing a way to optimize the mechanical properties.
Molecular dynamics (MD) simulations can provide alternative methods to obtain
d
detailed information of the interfacial interaction and sliding at the molecular level
te
[13,14], especially the interaction between CNTs and the surrounding polymer matrix
Ac ce p
[15,16]. Frankland et al. [17] studied the influence of chemical cross-links between a SWNT and a polyethylene (PE) matrix on the interfacial strength by using MD simulation. The simulation predicted that the strength can be enhanced by over an order of magnitude with the formation of cross-links involving less than 1% of the CNT atoms. Gou et al. [4] simulated the pull-out process of a CNT from a cured epoxy resin by MD method. By estimating the interaction energy, Gou et al. were able to determine that the interfacial strength between the CNT and the epoxy resin was 75 MPa. Zheng et al. [18] investigated the interfacial bonding characteristics between the SWNT, on which –COOH, –CONH2, –C6H11, or –C6H5 groups have been chemically
Page 5 of 47
6
attached, and the PE matrix by performing pullout simulations. The results showed that appropriate functionalization of nanotubes at low densities of functionalized
ip t
carbon atoms drastically increase their interfacial bonding and shear stress between the nanotubes and the polymer matrix. Recently, Haghighatpanah et al. [19] employed
cr
MD and molecular mechanics (MM) methods to investigate the interfacial properties
us
of SWNT/polyethylene and SWNT/polyacrylonitrile composites using pull-out simulations. They investigated how the percentage and the location of –COOH groups
an
functionalized carbon atoms have influenced the interfacial strength.
M
A great majority of literatures on this topic investigated the interaction between CNT and surrounding polymer matrix through a pull-out simulation. However, Li and
d
his co-workers [20] found that the interfacial shear stress (ISS) between CNT and
te
polymer matrix distributes at each end of the embedded CNT within the range of 1nm
Ac ce p
during the pull-out process. By molecular mechanics simulation, they also found that a capped CNT presents a higher pull-out force than that of the corresponding open-ended CNT [21]. As the MD simulation is mostly limited to systems with several nanometers and tens of particles [22,23], a large proportion of atoms, whether CNT atoms or polymer ones, have to be located in the region where the “end effect” is intense. Thus the results of the pull-out simulation are sensitive to the CNT “end effect” because in the pull-out process the periodic boundary condition is difficult to be applied in the axial direction of the CNT. Nevertheless, CNTs with several micrometers in length are widely produced by catalytic chemical vapour deposition
Page 6 of 47
7
(CCVD), and are dispersed in the nanocomposite as reinforcements [24]. Considering the large aspect ratio of CNTs, the “end effect” were usually ignored when modeling
ip t
and analyzing the CNT/polymer composites [25-27]. Therefore, it is important to investigate the mechanism of the “pure interfacial shear” between CNTs and the
cr
surrounding polymer matrix through a molecular-level model that ignores the “end
us
effect”. However, few literatures reports relevant works. Based on the authors’ knowledge, among the many literatures, only Zhang et al. [28] simulated the sliding
an
process of a non-functionalized SWNT in PE matrix by using a periodic MD model.
M
Their simulation indicated that the curve of the ISS as a function of CNT sliding displacement is shaped like that of sliding friction.
d
In addition, according to the observation of the densification and crystallization
te
from MD simulation [29,30] and experimental characterization [31], the structural
Ac ce p
arrangement of matrix polymer in the vicinity of the CNTs is altered as a layer of interphase by the interaction with the CNTs. Several researchers have tried to introduce the effect of the interphase on the properties of non-functionalized CNT/polymer composites in finite element model [32], MD model [33] or multi-scale model [34]. These modeling works indicated that ignoring the interphase could bring about misleading results for the effective properties of the composite. Thus, it is important to investigate the load transfer mechanism of the interphase, as it will have a great influence on the mechanical properties of CNTs/matrix interface. In this work, a periodic MD model has been constructed to investigate the
Page 7 of 47
8
interfacial properties of functionalized SWNT/polymer composite, which avoids the “end effect” of SWNTs and is more convenient to obtain the load transfer mechanism.
ip t
The SWNT is randomly coated with carboxylic acid functional groups –COOH and then is embedded into PE matrix. The emphasis of our study was laid on the load
cr
transfer mechanism of the interphase. Moreover, the effective interfacial shear
us
strengths in different sliding stages and variable sliding velocities are evaluated. Based on the calculation of shear properties of PE bulk and an equivalent shear load
an
transfer model, we investigated the effect of the amount of functionalization of SWNT
M
and the sliding velocity on the interfacial properties. In conclusion, a criterion was proposed to distinguish the strong interface and the weak one.
te
2.1 Atomistic models
d
2. Method
Ac ce p
In MD simulation, an armchair type SWNT with a chiral vector of (10, 10) is chosen as reinforcement, and an amorphous linear PE is used as polymer matrix. A single chain of the amorphous PE consists of 200 –CH2–CH2– unit monomers, since
in chains longer than that, the variation of the interfacial properties can be neglected [28]. The number of PE chains in the model varies according to the size of the unit cell. Although the SWNT chirality may affect the SWNT-polymer interaction [17], the influences are expected to be minor compared to the effect of the functionalization we force on here [19]. Thus the trends obtained with carboxylic acid functionalization of the (10, 10) SWNT are expected to be valid for all chiralities. In total, three kinds of
Page 8 of 47
9
atomistic models are constructed for the investigation and comparison, including functionalized SWNT-polymer systems, a non-functionalized SWNT-polymer system
procedures, we only detail the first one in details here.
ip t
and a pure polymer bulk. As the three kinds of models have similar constructing
cr
Firstly, periodic unit cells are constructed with sufficiently large dimensions of
us
49.19 A 70.00 A 70.00 A in order to prevent interaction between atoms and their
periodic images during the whole MD simulation. The SWNT functionalized with
an
carboxylic acid groups, which are randomly distributed on the SWNT carbon atoms,
M
is added at the center of the unit cell aligned to the X direction as is depicted in Fig. 1(a). After that, random-walk PE chains are generated around the SWNT to fill the
d
unit cell at a target density of 0.7 g/cm3 by using Accelyrs Amorphous Cell package,
te
which is based on the method proposed by Meirovitch [35] and Theodorou and Suter
Ac ce p
[36]. Then, the total potential energy of the unit cell is minimized by the conjugate gradient method with a target convergence threshold for the specified maximum energy change of 0.001 kcal/mol [27,34] (Fig. 1(b)). All the cell construction and the subsequent energy minimization processes are performed using a commercial molecular dynamics simulation package Material Studio 7.0 [37], with the polymer consistent force field (PCFF) [38,39] to describe inter- and intra-atomic interactions. For non-bond calculations, an atom-based cutoff of 11 Å was used in the van der Waals (vdW) interaction, and the Ewald summation method was used to treat the long-range characteristics of the Coulomb interaction [40].
Page 9 of 47
10
When initial structures are sufficiently minimized, they are then inserted into the MD code where an equilibration sequence with four different steps will be performed
ip t
to relax any high energy configurations that are artificially created. Initially, the unit cells are equilibrated for 50 ps with a time step of 1fs under the NVT ensemble at the
cr
temperature of 500 K followed by a relaxation for 150 ps with a time step of 0.5 fs
us
under the NPT ensemble at the temperature of 500 K and the pressure of 1 atm. The next relaxation cooled the structure down to the desired temperature with a step of 1
an
K for every 1 ps, followed by a further relaxation of 300 ps at the desired temperature.
M
The latter two equilibrium processes are both performed under the NPT ensemble at the pressure of 1 atm with a time step of 0.5 fs. The target temperature is set to 100 K,
d
where the PE molecules can be regarded as a glassy and rigid polymer [41,42]. A
te
parallel MD code, LAMMPS [43], has been used to perform the equilibration and the
Ac ce p
following simulations using the same force field (PCFF) mentioned above. The system temperature and pressure are regulated by the Nosé-Hoover methods[44-46]. Similar procedures are performed to construct the atomic models for the
non-functionalized SWNT-polymer system and the pure polymer bulk (Fig. 1(c)). The equilibration sequence relaxes the pure PE bulk to a final density of 0.90 g/cm3, which
is within the range of experimental values 0.89-0.925 g/cm3 for linear low-density PE [47]. The densities of the stable SWNT-PE systems are in the range of 0.93-0.97 g/cm3, which is also in agreement with the experimentally determined densities for the system [47].
Page 10 of 47
11
2.2 MD simulation for SWNT sliding and PE bulk shear After the equilibration, production runs are performed to obtain the ISS of the
ip t
SWNT-PE systems and the shear stress-strain curve of the pure PE bulk. For the former case, the PE atoms outside the interphase are fixed to prevent the
cr
rigid body motion of the PE matrix, shown as the green region in Fig. 2. Uniform
us
velocities ranging from 1m/s to 100m/s are applied to the carbon atoms of the SWNT, to simulate the sliding process. In the simulations, periodic boundary conditions are
an
applied in three directions, and hence the “end effect” of SWNTs can be avoided. As
M
shown in Fig. 2, the interphase for load transferring is defined as a concentric cylinder within the yellow circle with a radius of R1 while beyond the white circle which
d
represents the radius of the SWNT R0 . The conventional definition of the interphase
te
is a layer of dense polymer atoms surrounding the CNT. The local density change is
Ac ce p
used as a criterion to determine the boundary of the interphase region and the bulk matrix [33]. Different from that, in this work, we define the interphase as a layer of atoms for load transferring from the bulk matrix to the CNT. The thickness of the interphase T should be big enough to eliminate the direct interaction between the CNT and the bulk matrix outside. At the same time, it needs to be small to improve computational efficiency. Obviously, the interphase defined here will cover the conventional one. Based on detailed discussions of the density change and the effective interfacial shear strength, we will determine the thickness of the interphase in the following section.
Page 11 of 47
12
As a reference for studying the interfacial shear strength, the shear stress-strain curves of the pure PE bulk are obtained under different strain rates, by applying a
ip t
continuous external shear strain to the unit cell as shown in Fig. 3. Based on the stress-strain curves, a 1-D constitutive model is established to determine the critical
cr
strength of the interface under the theoretical framework of continuum mechanics.
us
The corresponding load transfer model and the comparison will be discussed in details
3. Results and discussion
M
3.1 The interphase for load transferring
an
in Section 3.4.
The radius density distribution (RRD), which has been widely used in
d
CNT/polymer systems [27,48], of the PE matrix has been calculated to reflect the
te
effect of SWNT on the surrounding PE matrix. The local density at different radius R
Ac ce p
is determined by dividing the atomic mass of cylindrical shell by the corresponding volume. Specifically, the inner boundary is defined as the (10,10) SWNT radius R0=6.78 Å, and the thickness of the shell is set to 0.3 Å. The RRDs of non-, 1%, and
5% functionalized SWNT composites are respectively shown in Fig. 4 with the reference density of pure PE bulk. It can be seen from Fig. 4 that whether the SWNT is functionalized or not, the RRD oscillates around the mean value of PE bulk density, which can be attributed to the fact that the linear PE chains are easy to form regular lamellar-type structures on the SWNT surface [49]. In contrast, the fluctuation of the RRD of the pure PE, corresponding to an amorphous structure, is much smaller.
Page 12 of 47
13
Furthermore, for the functionalized SWNTs, the variations of the RRDs are relatively slight, and all have a peak value at around T=4 Å. On the other hand, the amplitude of
ip t
the RRD decreases as T increases, until the density is almost equal to the PE bulk at around 17 Å. However, as the amount of functionalization goes up, the amplitude of
cr
the RRD shows a distinct drop, which may be attributed to the fact that the regular
us
structure of the surrounding PE chains are disturbed by the –COOH groups.
To give a precise evaluation of the thickness of the interphase for load
an
transferring analysis, eight simulations with different thickness of interphase T have
M
been performed for 1% and 5% functionalized SWNT composites respectively. Similar to the definition in Ref. [28], two key parameters for the ISS of the sliding
d
process have been calculated, i.e. the critical interfacial shear stress (CISS) and the
te
steady sliding interfacial shear stress (SISS), which respectively refer to the peak
Ac ce p
value and the mean value of the steady sliding stage, as shown in Sub-figure 5(b). The details of these parameters and the sliding process will be further discussed in the following section.
Then, the CISS and the SISS are calculated for the 1% and 5% functionalized
cases as the thickness of the interphase T increases. As shown in Fig. 5(a), both CISS and SISS decrease as the thickness of the interphase increases and tend to converge to constant values when the interphase reaches a sufficient thickness. This means that the influence of the SWNT on the PE atoms can be neglected at a distance large enough. To determine the appropriate value for the thickness, the relative error between the
Page 13 of 47
14
neighboring data points in Fig. 5(a) is evaluated. We find that the relative error of the CISS and SISS between models with the thickness of 17 Å and 19 Å is smaller than
ip t
5% in both cases. This means that the influence of the SWNT is small enough on the PE atoms at the distance of 17 Å from the SWNT surface, hence an interphase
cr
thickness of 17 Å will be used in all following simulations.
us
3.2 Sliding process of CNT
The ISS is studied by examining the energy changes during the sliding of the
Fi 2R0 L
(1),
M
an
SWNT embedded in the PE matrix, which is calculated as
where Fi is the axial component (x-direction) of the total force on the SWNT
te
d
imposed by PE molecules, R0 is the SWNT radius, and L is the embedded length of SWNT. In the calculation the resultant force on the SWNT is evaluated by
Ac ce p
summing the force interaction between the group of PE molecules and that of atoms of SWNT which consists the atoms of the pristine SWNT and the –COOH groups. Fig. 6 shows the typical ISS as a function of SWNT sliding displacement d CNT for different amounts of functionalization under the sliding velocity of 100 m/s. Similar to the ISS of pristine SWNT sliding [28], three distinct stages can be observed during the sliding process for the functionalized SWNTs, including (1) a linear increase before a critical stress point, (2) a linear decrease, and (3) a steady sliding stage. The peak value after the first stage is defined as the CISS c . During the steady sliding stage, the ISS oscillates irregularly because of the random distribution of functional
Page 14 of 47
15
groups, which is different from the periodic oscillation of the pristine SWNTs [28]. The steady sliding interfacial shear stress (SISS) s is defined by averaging the ISS
ip t
over the whole steady sliding time steps. As shown in Fig. 6, the shape of the ISS curve is similar to the curve of sliding friction, and the SWNT sliding becomes easier
cr
in the steady sliding stage (as kinetic friction) after reaching the CISS (as static
us
friction). Furthermore, Fig. 6 indicates that both the CISS and the SISS go up as the amount of the functionalization increases, while the corresponding sliding
an
displacement for CISS is around 4 Å for all cases.
M
The CISS and SISS are calculated respectively for the nanocomposite with the amount of SWNT functionalization from 0.5% to 5.0% based on the results of MD
d
simulations. Table 1 shows a comparison of the CISS/SISS between the
te
functionalized and non-functionalized SWNT under the sliding velocity of 100 m/s.
Ac ce p
It is obvious that when –COOH groups are added to the SWNT surface, CISS and SISS increase dramatically. As the present periodic model avoids the “end effect”, the values of CISS and SISS for the non-functionalized SWNT-PE system are much smaller than those calculated by the pull-out simulation, but have the same order of magnitude as those for Ref. [28] (~8MPa of CISS and ~1.5MPa of SISS in corresponding PE chain length and sliding velocity). The differences of our results from those of Ref. [28] may be attributed to the temperature difference and the united-atom model in their simulation. Table 1 also shows the values of ISS reported in Ref. [19], in which a SWNT pull-out simulation was implemented. In order to
Page 15 of 47
16
avoid the “end effect” for a valid comparison with our results, the enhancements of the ISS for the functionalized SWNT cases are calculated as presented in Table 1. It
ip t
shows that enhancement values of the SISS in our work are very close to those of the ISS in the pull-out simulation, which indicates that the “end effect” has a minor
cr
influence on the reinforcement of the functionalization. In other word, the –COOH
us
groups may mainly enhance the sliding resistance between the SWNT and the surrounding polymer matrix; this mechanism will be discussed in detail. Meanwhile,
an
the comparison also verifies the validity of our model for reflecting the effects of the
M
–COOH groups.
In order to analyze the loading transfer mechanism, the detailed deformations of
d
the interphase region in the cross-section perpendicular to the y-direction and
te
through the axis of the SWNT are presented in Fig. 7, demonstrating the SWNT with
Ac ce p
the amount of functionalization f 5% under the sliding velocity of 100 m/s, corresponding to the case of black point in Fig. 6. In Fig. 7(a), the atoms are colored according to their molecule ID; a typical –COOH group in the yellow circle marks the sliding of the SWNT. Initially, the –COOH group lies on the left side of the red PE chain. When the SWNT moves to dCNT=6 Å, PE chains undergo an obvious
deformation along with the SWNT translational motion, but the distance between the marked –COOH group and the red chain remains almost unchanged, which indicates that little relative slide happens between the SWNT and the surrounding PE chains. This phenomenon occurs at the end of the first sliding stage, where the ISS reaches
Page 16 of 47
17
approximately the CISS. When dCNT=12 Å, the overlap of the –COOH group and the red PE chain can be observed, apart from the further deformation of the PE matrix.
ip t
This indicates the existence of distinct sliding displacement between the SWNT and the PE matrix. When dCNT=24 Å, the –COOH group moves to the right side of the
cr
red chain and no further deformation of the PE chains is observed comparing to the
us
case of when dCNT=12 Å. This process corresponds to the steady sliding stage indicated in Fig. 6.
an
The SWNT axial displacement of each PE atom is presented in Fig. 7(b) for
M
these four cases. It is indicated that the atomic displacement significantly goes up as dCNT increases from 0 Å to 12 Å, but the variation of the atomic displacement can be
d
neglected for a bigger dCNT. Fig. 7(b) more obviously reflected the deformation of the
te
PE chains surrounding the SWNT. Similar to the analysis above, the primary load
Ac ce p
transfer mechanism of the interphase is the deformation of the PE chains at the first stage of the SWNT sliding and, in contrast, the relative sliding between the SWNT and the PE chains in the steady sliding stage. Furthermore, the atomic displacement follows a similar distribution along the SWNT radial direction for each case, i.e. the closer the PE atom is to SWNT, the bigger the atomic displacement is, which implies a shear deformation in the interphase for load transferring. Fig. 8 shows a clearer variation of the atomic deformation as a function of the radial thickness and the SWNT displacement. The value of the atomic deformation is the average value for atoms in a specific radius. The saddle-shaped surface obviously indicates the shear
Page 17 of 47
18
deformation in the interphase. As the SWNT slides, the interphase undergoes a large deformation at first, and then stays steady. On the other hand, the PE molecules closer
ip t
to the SWNT undergo larger deformation as discussed above. To investigate the effect of the amount of the functionalization of SWNT on the
cr
load transferring in the interphase, a comparison is presented in Fig. 9 between the
us
cases of f 5% and f 1% . An obvious difference of the atomic displacement distribution is indicated between the two cases in Fig. 9(a). When f 5% , the
in contrast,
only the PE atoms below the SWNT clearly move, and the
M
f 1% ,
an
atomic displacement distribution is near symmetrical between the SWNT axis. When
atoms above have small movement, where no –COOH group exists. Fig. 9(b) shows a
d
comparison of the averaged atomic displacements as a function of the radial thickness.
te
The curves of the 1% case are obviously lower than those of the 5% one. This implies
Ac ce p
that the shear deformation of the interphase mainly depends on the functional groups on the SWNT surface, regardless of the position and the amount. Fig. 9(b) also shows that the variation of the atomic displacement can be neglected when the radial thickness reaches around 15 Å, which means the interphase thickness of 17 Å is big enough.
3.3 The interfacial shear strength and criterion As the inertia effect and the viscosity should be cautiously considered in the polymer system [50], we employ an extrapolation to infer the properties of the interface and the matrix in quasi-static. A power-law model has been employed to
Page 18 of 47
19
approximately describe the variation of CISS and SISS respectively, with the amount of functional groups f and the SWNT sliding velocity u CNT as follows: nc ncf u CNT f 0 c c f u 0 CNT 0 nsu nsf u CNT 0 f s s f u 0 CNT 0
ip t
u
cr
(2),
us
where c0 , s0 , ncf , ncu , nsf , and nsu are the constants determined by fitting with the MD results, presented in Table 2. c0 and s0 are reference stress; f 0 and
an
u CNT 0 are respectively the reference amount of functional groups and the SWNT
M
sliding velocity, which we set to f 0 1% and u CNT 0 1m / s in the present work. It is indicated that the exponents ncf and nsf are both approximate to 1.0, in other
d
words, the relationship of the interfacial shear strength and the amount of
te
functionalization is approximately linear, and the load transferring in the interphase
Ac ce p
mainly depends on the interaction between the functional groups and the surrounding PE chains.
The MD results and the fitting curves are presented in Fig. 10 for the CISS and
SISS as a function of the SWNT sliding velocity. As the sliding velocity increases, both CISS and SISS increase obviously for each amount of functionalization. As the analysis in Section 3.2 has shown, the value of ISS depends on the shear deformation of PE chains in the interphase, which is mainly driven by the –COOH groups. Since the mechanical properties of PE is sensitive to the strain rate [41], a higher value of ISS can be obtained for a faster SWNT sliding which corresponds to a higher strain
Page 19 of 47
20
rate of the interphase. In the following, we attempt to establish a criterion to determine whether the
ip t
interface is a strong one or a weak one under the theoretical framework of continuum mechanics. As a reference, the shear strain-stress relationship of the pure PE has been
cr
obtained by MD simulation. The stress-strain curves under different strain rates are
us
shown in Fig. 11(a). It shows that a 1-D elastic-perfectly plastic model is able to describe the stress-strain response at a certain strain rate. The two material parameters,
an
elastic modulus G and yield strength Y , are assumed as functions of the strain rate
M
, which can be obtained by fitting with the MD results. As shown in Fig. 11(b), a power-law model is used to express the relationship
(3),
Ac ce p
te
d
nG G G0 0 n Y Y0 0
where we set to 0 1s 1 in the present paper; nG and n are respectively the stiffness relation exponent and the strength relation exponent. Next, an equivalent shear load transfer model, detailed in Appendix A, is
employed. Fig. 12 shows a sketch for the load transferring in the interphase based on the shear deformation with an assumption that the PE interphase matrix is perfectly bonded to the SWNT. Under this assumption, the critical interfacial yield strength is determined by the equivalent yield strength of the surrounding interphase matrix layer, which is given by Eq. (A7). Combining Eq. (2) and Eq. (A7), a relationship of the
Page 20 of 47
21
amount of functional groups and the sliding velocity can be obtained to distinguish
(4),
us
cr
1 n nsf u u c nG 1 f upper f 0 Y00 0 n u 0 ns u n ns s R0 1 n ncf n 1 ncu n ncu n Y0 c G f lower f 0 0 0 u 0 u c R0
ip t
the strong interface and the weak one, i.e.
which is presented in Fig. 13. The upper and the lower bounds are calculated from the
an
SISS and the CISS, respectively. When the value of f is higher than the upper
M
bound, the interfacial strength stays bigger than the interphase matrix layer, whichever stage the sliding process lies on. This interface can be defined as a strong one,
d
because the yield occurs earlier in the interphase matrix. In contrast, a value of f
Ac ce p
than that of the matrix.
te
lower than the red dash line means a weak interface of which strength stays smaller
Different from the absolute boundary between the strong and weak interface, Fig.
13 shows that a translation state exists between the upper and the lower bounds. As we have discussed above, the ISS curve of the SWNT sliding has a shape like that of the conventional sliding friction, in other words, the value of CISS is generally higher than SISS. It means that a bigger load is required to activate the motion of the SWNT from a static state, but a smaller load is enough to maintain the steady sliding state. Therefore, if a load smaller than CISS but bigger than SISS is applied, the SWNT can not begin to slide from a static state. However, this load could keep the SWNT sliding
Page 21 of 47
22
once the motion of the SWNT is activated by, say, an impact. The existence of the translation state between the strong and weak interface may bring a more notable
ip t
influence on the dynamical properties of the composite rather than the quasi-static nanotube pulling-out behavior. Moreover, even in the quasi-static experiments for
cr
testing the macro-properties of the nanocomposite, the local disturbance can not be
us
ignored, hence it is meaningful to discuss the interface condition by considering the existence of the translation state between the strong and weak interface.
an
Furthermore, it is worth noting that the values of the two bounds decrease
M
nonlinearly as the sliding velocity slows down. When the sliding velocity is around 10-8 m/s, corresponding to the pull-out rate in experiment [51], the value of the upper
d
bound is 0.62%, which can be easily reached for functionalized CNTs [5]. Therefore,
te
it is reasonable to assume a perfect bonding between the functionalized CNT and the
Ac ce p
PE matrix in most cases of modeling at a higher scale. 4. Conclusion
In this paper, a periodic MD model is constructed to investigate the mechanical
properties of the interface between the functionalized SWNT and the polymer matrix. The present model avoids the “end effect” of SWNTs and is more convenient to obtain the load transfer mechanism. Carboxylic acid groups are randomly coated onto the SWNT to study the effect of functionalization, and an amorphous linear PE is used as polymer matrix. Based on the analysis of the RRD and the interfacial strength, an effective interfacial thickness of 17 Å is obtained. During the sliding process of the
Page 22 of 47
23
SWNT, the shear deformation occurs in the layer of interphase to transfer the load from the matrix to the SWNT. The deformation behavior is similar to an
ip t
“elastic-plastic” solid as the SWNT moves, which increases initially and then stays steady thereafter. In addition, the shear deformation is mainly driven by the functional
cr
groups.
us
The ISS is calculated as a function of the SWNT sliding displacement, which shows a similar shape of the curve of sliding friction, i.e. a lower steady sliding stage
an
(SISS) exists following a peak value (CISS). A power-law model is employed to
M
describe the variation of CISS and SISS respectively, with considering the effects of the amount of functionalization and the sliding velocity. Finally, we propose an
d
interfacial strength criterion by considering the amount of functionalization of SWNT
te
and the sliding velocity. The results imply that an assumption of perfect bonding
Ac ce p
between the functionalized SWNT and the PE matrix is reasonable for the modeling at a higher scale under the experimental velocity. It may be noted that the load transfer ability of the interface with more
functionalization, such as 2%, is much higher than the yield strength of the matrix. However, the excessive load can still be transferred to the SWNT because of the interaction between the functional groups and the chains in the interphase whose properties have been modified by the SWNT and the functional groups. The existence of the interphase may further improve the overall properties of the nanocomposite.
Page 23 of 47
24
Acknowledgments The authors thank the support from the National Natural Science Foundation of
ip t
China (11472025, 11272030 and 11272300).
cr
Appendix A. Equivalent shear load transfer model
us
By assuming a perfect interfacial bonding and the decoupling of tensile and shear deformations, the shear strain in the interphase layer can be written as:
G
0 R0
an
nG
G0 R 0
(A1),
M
where 0 is the shear stress at the interface, R0 is the radius of the SWNT.
d
In the elastic stage, the deformation of the surrounding matrix layer satisfies the
te
superposition principle, thus 0. The shear strain rate can be expressed by taking
Ac ce p
the derivative of Eq. (A1) with respect to time as follows: R nG 0 0 0 G0 R
1
nG 1
(A2).
Then the SWNT sliding velocity u CNT can be expressed as:
u CNT
1 dR 1 R0 nG R1
0 R00 nG G0
1
nG nG 1 nG nG 1 R nG 1 R 1 0
(A3).
Therefore, 0 can be written as a function of the sliding velocity,
G u G 0 c 0 CNTnG R00
n 1
where c
1 1 1 n G
nG nG R1 nG 1 R0 nG 1
nG 1
(A4),
.
Page 24 of 47
25
Further, 0 is evaluated based on the time integral for Eq. (A4), G 0 c 0 R0
u CNT 0
nG
d CNT
(A5).
ip t
The strain rate at the interface can be also obtained, 1
(A6).
cr
c nG 1 ( R R0 ) u CNT R0
us
Moreover, the yield occurs at the position near the CNT, and hence the equivalent yield strength of the surrounding matrix layer can be evaluated by
an
introducing Eq. (A6) into the second term of Eq. (3), n
n
(A7).
d
M
c nG 1 u Y Y0 CNT R0 0
te
References:
[1]S. Iijima, Helical microtubules of graphitic carbon, Nature, 354(1991) 56-58.
Ac ce p
[2]J.N. Coleman, U. Khan, W.J. Blau, Y.K. Gun Ko, Small but strong: a review of the mechanical properties of carbon nanotube–polymer composites, Carbon, 44(2006) 1624-1652.
[3]D. Qian, E.C. Dickey, R. Andrews, T. Rantell, Load transfer and deformation mechanisms in carbon nanotube-polystyrene composites, Appl. Phys. Lett., 76(2000) 2868.
[4]J. Gou, B. Minaie, B. Wang, Z. Liang, C. Zhang, Computational and experimental study of interfacial bonding of single-walled nanotube reinforced composites, Comp. Mater. Sci., 31(2004) 225-236.
[5]N.G. Sahoo, S. Rana, J.W. Cho, L. Li, S.H. Chan, Polymer nanocomposites based on functionalized carbon nanotubes, Prog. Polym. Sci., 35(2010) 837-867.
[6]J. Yu, B. Tonpheng, G. Gröbner, O. Andersson, Thermal properties and transition studies of multi-wall carbon nanotube/nylon-6 composites, Carbon, 49(2011) 4858-4866. [7]J.L. Bahr, J.M. Tour, Covalent chemistry of single-wall carbon nanotubes, J. Mater. Chem., 12(2002) 1952-1958. [8]S. Kanagaraj, F.R. Varanda, T.V. Zhil Tsova, M.S. Oliveira, J.A. Simões, Mechanical properties of high density polyethylene/carbon nanotube composites, Compos. Sci. Technol., 67(2007) 3071-3077. [9]H.D. Wagner, O. Lourie, Y. Feldman, R. Tenne, Stress-induced fragmentation of multiwall carbon
Page 25 of 47
26
nanotubes in a polymer matrix, Appl. Phys. Lett., 72(1998) 188. [10]C.A. Cooper, S.R. Cohen, A.H. Barber, H.D. Wagner, Detachment of nanotubes from a polymer matrix, Appl. Phys. Lett., 81(2002) 3873. [11]A.H. Barber, S.R. Cohen, H.D. Wagner, Measurement of carbon nanotube–polymer interfacial strength, Appl. Phys. Lett., 82(2003) 4140. ‐Nanotube/Polymer Interface, Adv. Mater., 18(2006) 83-87.
ip t
[12]A.H. Barber, S.R. Cohen, A. Eitan, L.S. Schadler, H.D. Wagner, Fracture Transitions at a Carbon [13]Z. Yang, Y. Zhao, Adsorption of His-tagged peptide to Ni, Cu and Au (100) surfaces: Molecular dynamics simulation, Eng. Anal. Bound. Elem., 31(2007) 402-409.
cr
[14]Z. Yang, Y. Zhao, Multi-scale simulation of an immobilizing F1-ATPase molecular motor, Journal of Korean Physical Society, 52(2008) 1211.
us
[15]Q. Li, G. He, R. Zhao, Y. Li, Investigation of the influence factors of polyethylene molecule encapsulated into carbon nanotubes by molecular dynamics simulation, Appl. Surf. Sci., 257(2011) 10022-10030.
an
[16]S. Rouhi, Y. Alizadeh, R. Ansari, On the interfacial characteristics of polyethylene/single-walled carbon nanotubes using molecular dynamics simulations, Appl. Surf. Sci., 292(2014) 958-970. [17]S.J.V. Frankland, A. Caglar, D.W. Brenner, M. Griebel, Molecular Simulation of the Influence of Chemical Cross-Links on the Shear Strength of Carbon Nanotube−Polymer Interfaces, The
M
Journal of Physical Chemistry B, 106(2002) 3046-3048.
[18]Q. Zheng, D. Xia, Q. Xue, K. Yan, X. Gao, Q. Li, Computational analysis of effect of modification on the interfacial characteristics of a carbon nanotube–polyethylene composite system, Appl.
d
Surf. Sci., 255(2009) 3534-3543. [19]S. Haghighatpanah, M. Bohlà N, K. Bolton, Molecular level computational studies of
te
polyethylene and polyacrylonitrile composites containing single walled carbon nanotubes: effect of carboxylic acid functionalization on nanotube-polymer interfacial properties, Frontiers in
Ac ce p
Chemistry, 2(2014).
[20]Y. Li, Y. Liu, X. Peng, C. Yan, S. Liu, N. Hu, Pull-out simulations on interfacial properties of carbon nanotube-reinforced polymer nanocomposites, Comp. Mater. Sci., 50(2011) 1854-1860.
[21]Y. Li, S. Liu, N. Hu, X. Han, L. Zhou, H. Ning, L. Wu, G. Yamamoto, C. Chang, T. Hashida, Pull-out simulations of a capped carbon nanotube in carbon nanotube-reinforced nanocomposites, J. Appl. Phys., 113(2013) 144304.
[22]Q. Liu, Z. Lu, M. Zhu, Z. Yuan, Z. Yang, Z. Hu, J. Li, Simulation of the tensile properties of silica aerogels: the effects of cluster structure and primary particle size, Soft matter, 10(2014) 6266-6277.
[23]Q. Yuan, Y. Zhao, Multiscale dynamic wetting of a droplet on a lyophilic pillar-arrayed surface, J. Fluid Mech., 716(2013) 171-188. [24]S. Bal, S.S. Samal, Carbon nanotube reinforced polymer composites—a state of the art, B. Mater. Sci., 30(2007) 379-386. [25]F.T. Fisher, R.D. Bradshaw, L.C. Brinson, Effects of nanotube waviness on the modulus of nanotube-reinforced polymers, Appl. Phys. Lett., 80(2002) 4647-4649. [26]Z. Yuan, Z. Lu, Numerical analysis of elastic–plastic properties of polymer composite reinforced by wavy and random CNTs, Comp. Mater. Sci., 95(2014) 610-619.
Page 26 of 47
27
[27]S. Yang, S. Yu, J. Ryu, J. Cho, W. Kyoung, D. Han, M. Cho, Nonlinear multiscale modeling approach to characterize elastoplastic behavior of CNT/polymer nanocomposites considering the interphase and interfacial imperfection, Int. J. Plasticity, 41(2013) 124-146. [28]Z.Q. Zhang, D.K. Ward, Y. Xue, H.W. Zhang, M.F. Horstemeyer, Interfacial characteristics of carbon nanotube-polyethylene composites using molecular dynamics simulations, ISRN Materials Science, 2011(2011).
ip t
[29]H. Yang, Y. Chen, Y. Liu, W.S. Cai, Z.S. Li, Molecular dynamics simulation of polyethylene on single wall carbon nanotube, The Journal of Chemical Physics, 127(2007) 94902.
[30]S. Kumar, S.K. Pattanayek, G.G. Pereira, Organization of polymer chains onto long, single-wall
cr
carbon nano-tubes: Effect of tube diameter and cooling method, The Journal of Chemical Physics, 140(2014) 24904.
us
[31]L. Li, C.Y. Li, C. Ni, L. Rong, B. Hsiao, Structure and crystallization behavior of Nylon 66/multi-walled carbon nanotube nanocomposites at low carbon nanotube contents, Polymer, 48(2007) 3452-3460.
an
[32]F. Han, Y. Azdoud, G. Lubineau, Computational modeling of elastic properties of carbon nanotube/polymer composites with interphase regions. Part I: Micro-structural characterization and geometric modeling, Comp. Mater. Sci., 81(2014) 641-651.
[33]S. Herasati, L.C. Zhang, H.H. Ruan, A new method for characterizing the interphase regions of
M
carbon nanotube composites, Int. J. Solids Struct., 51(2014) 1781-1791. [34]S. Yang, S. Yu, W. Kyoung, D. Han, M. Cho, Multiscale modeling of size-dependent elastic properties of carbon nanotube/polymer nanocomposites with interfacial imperfections, Polymer,
d
53(2012) 623-633. [35]H. Meirovitch, Computer simulation of self-avoiding walks: Testing the scanning method, The
te
Journal of Chemical Physics, 79(1983) 502.
[36]D.N. Theodorou, U.W. Suter, Detailed molecular structure of a vinyl polymer glass,
Ac ce p
Macromolecules, 18(1985) 1467-1478. [37]Accelrys Inc., San Francisco
[38]H. Sun, S.J. Mumby, J.R. Maple, A.T. Hagler, An ab Initio CFF93 All-Atom Force Field for Polycarbonates, J. Am. Chem. Soc., 116(1994) 2978-2987.
[39]T.L. Chantawansri, T.W. Sirk, E.F. Byrd, J.W. Andzelm, B.M. Rice, Shock Hugoniot calculations of polymers using quantum mechanics and molecular dynamics, The Journal of chemical physics, 137(2012) 204901.
[40]H. Sun, COMPASS: an ab initio force-field optimized for condensed-phase applications overview with details on alkane and benzene compounds, The Journal of Physical Chemistry B, 102(1998) 7338-7364.
[41]D. Hossain, M.A. Tschopp, D.K. Ward, J.L. Bouvard, P. Wang, M.F. Horstemeyer, Molecular dynamics simulations of deformation mechanisms of amorphous polyethylene, 2010, pp.6071-6083. [42]Y. Li, G.D. Seidel, Multiscale modeling of the effects of nanoscale load transfer on the effective elastic properties of unfunctionalized carbon nanotube – polyethylene nanocomposites, Model. Simul. Mater. Sc., 22(2014) 25023. [43]S. Plimpton, Fast parallel algorithms for short-range molecular dynamics, J. Comput. Phys.,
Page 27 of 47
28
117(1995) 1-19. [44]S. Nosé , A unified formulation of the constant temperature molecular dynamics methods, The Journal of Chemical Physics, 81(1984) 511. [45]W.G. Hoover, Canonical dynamics: equilibrium phase-space distributions, Phys. Rev. A, 31(1985) 1695. [46]W.G. Hoover, Constant-pressure equations of motion, Phys. Rev. A, 34(1986) 2499-2500.
ip t
[47]C. Vasile, M. Pascu, Practical guide to polyethylene, iSmithers Rapra Publishing, 2005.
[48]Y. Kuang, B. Huang, Effects of covalent functionalization on the thermal transport in carbon nanotube/polymer composites: A multi-scale investigation, Polymer, 56(2015) 563-571.
cr
[49]A. Minoia, L. Chen, D. Beljonne, R. Lazzaroni, Molecular modeling study of the structure and stability of polymer/carbon nanotube interfaces, Polymer, 53(2012) 5480-5490.
us
[50]J. Yin, Y. Zhao, R. Zhu, Molecular dynamics simulation of barnacle cement, Materials Science and Engineering: A, 409(2005) 160-166.
[51]Y. Ganesan, C. Peng, Y. Lu, P.E. Loya, P. Moloney, E. Barrera, B.I. Yakobson, J.M. Tour, R.
an
Ballarini, J. Lou, Interface Toughness of Carbon Nanotube Reinforced Epoxy Composites, ACS
Ac ce p
te
d
M
Applied Materials & Interfaces, 3(2011) 129-134.
Page 28 of 47
29
Figure captions Fig. 1. The unit cells constructed before equilibration: (a) the functionalized SWNT;
ip t
(b) the functionalized SWNT/PE system; (c) the pure PE bulk. Fig. 2. Atomic model for the SWNT/PE composite.
cr
Fig. 3. Pure shear simulation of the PE bulk.
us
Fig. 4. The variation of the local density in the matrix with the radial thickness of the matrix, the X-axis is the interphase thickness T=R1-R0.
an
Fig. 5. The variation of the CISS/SISS with the interphase thickness.
M
Fig. 6. The typical curve of ISS as a function of the SWNT sliding displacement under the sliding velocity of 100m/s.
d
Fig. 7. The specific distribution of the movement of PE atoms with the SWNT sliding.
te
The SWNT atoms are indicated by pink color.
Ac ce p
Fig. 8. The atomic displacement of the surrounding PE chains as a function of SWNT sliding displacement and radial thickness. Fig. 9. A comparison of the deformation in the interphase between the functionalization of 5% and 1%. Fig. 10. The effect of the SWNT sliding velocity on (a) CISS and (b) SISS. Fig. 11. Shear deformation of the pure PE bulk: (a) the stress-strain curves obtained by MD simulation; (b) the power-law model expressing the effect of strain rate on the elastic modulus and yield strength. Fig. 12. The interphase transferring the load from the matrix to the SWNT through
Page 29 of 47
30
shear deformation. Fig. 13. The interfacial condition determined by the amount of functionalization and
Ac ce p
te
d
M
an
us
cr
ip t
the sliding velocity.
Page 30 of 47
31
Functionalization
0.0%
0.5%
1.0%
1.5%
2.0%
5.0%
CISS (MPa)
6.35
47.45
137.29
202.66
262.84
415.83
Enhancement (MPa)
-
41.1
130.9
196.3
256.5
409.5
SISS (MPa)
3.48
34.83
75.27
106.79
142.29
290.72
Enhancement (MPa)
-
31.3
71.8
103.3
ip t
Table 1 Comparison of ISS with respect to the amount of functionalization
Corresponding results of the pull-out simulation in Ref. [18] 150
-
230
-
Enhancement (MPa)
-
-
80
-
287.2
-
450
-
300
us
cr
ISS (MPa)
138.8
Table 2 Summary of the constants in Eq. (2) by fitting with the MD results
c0 (MPa) 73.10
s0 (MPa)
0.12
nsf
nsu
0.99
0.13
Ac ce p
te
d
38.81
ncu
1.10
M
SISS
ncf
an
CISS
Page 31 of 47
Ac ce p
te
d
M
an
us
cr
ip t
Figure 1
Page 32 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 2
Page 33 of 47
Ac ce p
te
d
M
an
us
cr
ip t
Figure 3
Page 34 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 4
Page 35 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 5
Page 36 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 6
Page 37 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 7
Page 38 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 8
Page 39 of 47
Ac ce p
te
d
M
an
us
cr
ip t
Figure 9(a)
Page 40 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 9(b)
Page 41 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 10(a)
Page 42 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 10(b)
Page 43 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 11(a)
Page 44 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 11(b)
Page 45 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 12
Page 46 of 47
Ac
ce
pt
ed
M
an
us
cr
i
Figure 13
Page 47 of 47