Multimodal and self-healable interfaces enable strong and ... - CiteSeerX

Journal of the Mechanics and Physics of Solids 70 (2014) 30–41

Contents lists available at ScienceDirect

Journal of the Mechanics and Physics of Solids journal homepage: www.elsevier.com/locate/jmps

Multimodal and self-healable interfaces enable strong and tough graphene-derived materials Yilun Liu a,n, Zhiping Xu b,n a

International Center for Applied Mechanics, SV Lab, School of Aerospace, Xi’an Jiaotong University, Xi'an 710049, China Applied Mechanics Laboratory, Department of Engineering Mechanics and Center for Nano and Micro Mechanics, Tsinghua University, Beijing 100084, China b

a r t i c l e in f o

abstract

Article history: Received 8 September 2013 Received in revised form 15 February 2014 Accepted 8 May 2014 Available online 20 May 2014

Recent studies have shown that graphene-derived materials not only feature outstanding multifunctional properties, but also act as model materials to implant nanoscale structural engineering insights into their macroscopic performance optimization. In this work, we explore strengthening and toughening strategies of this class of materials by introducing multimodal crosslinks, including long, strong and short, self-healable ones. We identify two failure modes by fracturing functionalized graphene sheets or their crosslinks, and the role of brick-and-mortar hierarchy in mechanical enhancement. Theoretical analysis and atomistic simulation results show that multimodal crosslinks synergistically transfer tensile load to enhance the strength, whereas reversible rupture and formation of healable crosslinks improve the toughness. These findings lay the ground for future development of high-performance paper-, fiber- or film-like macroscopic materials from lowdimensional structures with engineerable interfaces. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Graphene-derived materials Mechanical enhancement Crosslink Self-healing Multimodality

1. Introduction The synergetic excellence of mechanical, thermal and electronic properties of graphene has attracted immersed interests in, but not limited to, the materials community (Geim and Novoselov, 2007). Focus has been placed not only on utilization of its high intrinsic stiffness and strength up to 1 TPa and 120 GPa, ultimate strain to failure of 20%, but also how to transfer the strong and tough performances of graphene monolayer into macroscopic applications (Liu et al., 2012). Within this scenario, nanostructures such as graphene, graphite nanoplatelets and carbon nanotubes have been widely used as reinforcing phases in high-performance composites. Major advantages based on this approach include well-enhanced stiffness, strength, resilience, as well as the multifunctionality (Stankovich et al., 2006; Young et al., 2012). However, it is also widely recognized that the native interface between these nanostructures and matrices creates weak points in the mechanical sense. This issue critically prevents successful transfer of outstanding performance of graphene across multiple length scales up to the macroscopic level, in addition to other difficulties such as lack of efficient technique to uniformly disperse nanostructures into the matrix at a high volume fraction (Gong et al., 2010; Young et al., 2012). Graphene-derived materials (papers, fibers, films etc.) usually feature layer-by-layer microstructures with single sheet spanning over hundreds of micrometers (An et al., 2011; Compton et al., 2011; Kotov et al., 1996; Qiu et al., 2012; Stankovich et al., 2006;

n

Corresponding authors. E-mail addresses: [email protected] (Y. Liu), [email protected] (Z. Xu).

http://dx.doi.org/10.1016/j.jmps.2014.05.006 0022-5096/& 2014 Elsevier Ltd. All rights reserved.

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

31

Xu et al., 2013; Zhao et al., 2013). In comparison with particle or fiber-based reinforced composites, they demonstrate many advantages, such as the extreme exposure of graphene sheets to the environment for functionalization, enabling rich and tunable crosslinking mechanisms between them. Recently, a number of theoretical and experimental efforts have been made in predicting and optimizing mechanical performance of this class of materials. Similar structural hierarchy broadly appears in biological materials, such as bones, teeth and nacre, where brittle minerals and soft proteins are integrated for their superior strength and toughness (Cheng et al., 2013; Dimas et al., 2013; Finnemore et al., 2012; Gao, 2006; Yao et al., 2013; Zhang et al., 2010, 2011; Zhong et al., 2013; Zuo and Wei, 2007). Mechanical enhancement is achieved by the staggered arrangement of mineral platelets that distributes the tensile load, which is transferred across the material by shear in soft protein interphases. In this manner the high stiffness, strength of mineral platelets and toughness of proteins are integrated, yielding superior overall mechanical properties, as captured in the tension–shear chain model proposed by Ji and Gao (2004). Later works have also focused on the topological optimization and stabilization of the staggered layer-by-layer structures (Zhang et al., 2010, 2011). Besides of these elegant insights into rational design of high-performance materials, a whole new dimension is added in graphene-derived materials by engineering crosslinks in the interlayer gallery, through covalent, dative, ionic, hydrogen bonds or van der Waals interactions (Compton et al., 2011; Gao et al., 2011; Park et al., 2008). As the aspect ratio of a functionalized graphene sheet (the lateral size divided by its thickness) is usually very high ( 4104), there is enough space between the sheets for nanoscale structural engineering. By further considering the intralayer deformation of functionalized graphene sheets, a deformable tension–shear (DTS) model was proposed to describe the mechanical properties of graphenederived materials (Liu et al., 2012), which predicts that giant graphene oxide (GO) sheets can be used to build ultrastrong materials, as achieved in recent experimental work (Xu et al., 2013). Beyond these points, recent progresses in functionalizing graphene sheets with various chemical groups (An et al., 2011; Bekyarova et al., 2013; Compton et al., 2011; Gao et al., 2011; Liu et al., 2011, 2012, 2013; Park et al., 2008; Xu et al., 2013), and improved understanding of crosslinking mechanisms in complex materials, such as multimodality, sacrificial bonds and self-healing behaviors (Hartmann and Fratzl, 2009; Liu et al., 2011; Mark and Erman, 2007; Wojtecki et al., 2011; Xu, 2013), could further enable optimal design in a new direction by tuning material interfaces at the molecular level. For example, gels with enhanced toughness were synthesized with multimodal crosslinks (Sun et al., 2012, 2013). The synergy of crack bridging by covalent crosslinks and hysteresis by unzipping ionic crosslinks allows synthesized polymer hydrogels to be stretched beyond 20 times their initial length, and have fracture energies of
9000 J/m2. In this regime, covalent crosslinks preserves the memory of the initial state, so that much of the large deformation is removed on unloading, while unzipped ionic crosslinks cause internal damage that is healed by re-zipping (Sun et al., 2012). Polyampholytes, polymers bearing randomly dispersed cationic and anionic repeat groups, form tough and viscoelastic hydrogels, where strong ionic bonds serve as permanent links holding the structure integrity under loading, while weak and reversible ionic bonds play a sacrificial role in mechanical energy dissipation, offering toughening and mechanical recovery, to heal the damage and fatigue thus being created (Sun et al., 2013). It is worth noting that, mechanical responses of polymeric gels are usually weak and delayed due to their entropic elasticity and random network structures, while layered materials derived from functionalized graphene sheets allow more intensive and prompt responses to external cues, and thus become ideal candidates to implement ideas of mechanical enhancement by introducing multimodal and self-healable crosslinks. In this article, we first clarify failure modes of graphene-derived materials, and develop an analytical model to describe the failure propagation in Section 2. In Section 3, we introduce the concepts of multimodal and self-healable crosslinks based on atomistic simulation results, followed by discussions on strengthening and toughening effects of the material in Section 4. 2. Failure of graphene-derived materials Graphene-derived materials with layer-by-layer microstructures may experience two distinct failure modes when subjecting to in-plane tensile loads, i.e. fracture of functionalized graphene sheets (by breaking covalent bonds) or interlayer crosslinks (also known as the ‘pull-out’ mode). Their mechanical performance is critically defined by the selection of failure modes. Thus in order to achieve enhancement by engineering their interlayer crosslinks, we need to identify the failure criteria of these two contrastive modes first. 2.1. Characterization of the failure mode The mechanical behaviors of graphene-derived materials under tension can be captured in the DTS model by considering both intralayer elasticity of the functionalized graphene sheet and interlayer crosslinks as continuum phases (Liu et al., 2012). The unidirectional load applied is transferred along a path consisting of both the sheets under tension and interlayer crosslinks under shear. Based on a representative volume element (RVE) illustrated in Fig. 1a, we consider a tension force F0 applied on the RVE in the DTS model. The tensile strain ε in the sheet along its in-plane direction and shear strain γ in crosslinks are (Liu et al., 2012)
 F 0 1 coshðx=l0 Þ 1 þc  þ sinhðx=l0 Þ εðxÞ ¼ ∂u1 =∂x ¼ ð1aÞ 2 2s D 2

32

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

Fig. 1. (a) A representative volume element (RVE) of the deformable tensile–shear (DTS) model consisting of two functionalized graphene half-sheets in regularly staggered arrangement. (b) Illustration of the two failure modes of graphene-derived materials with layer-by-layer microstructures, i.e. failure of sheets or crosslinks. (c) The spatial distributions of tensile strain in the sheet (for sheet 1 in panel b) and shear strain in crosslinks for l/l0 ¼ 10. (d) Failure modes characterized by two parameters k1 and k2 as defined in the text.

γðxÞ ¼


 u1  u2 F 0 l0 ð1 þ cÞcoshðx=l0 Þ  s sinhðx=l0 Þ ¼ s h0 h0 D

ð1bÞ

Here u1 and u2 are the displacement fields in neighboring sheets 1 and 2 as a function of position x measured from the left side of RVE (x ¼0), i.e. the free end of the lower sheet 1 in Fig. 1a. According to the staggered arrangement of sheets in the RVE, the free end of one sheet has the same coordinate x as the center of its neighboring sheets. h0 is the interlayer distance, s ¼sinh(l/l0), c¼cosh(l/l0), and l is the size of RVE or the half-sheet length. We define here a parameter l0 ¼(Dh0/4G)1/2 to characterize the length scale of interlayer load transfer through parameters including interlayer space h0, the effective shear modulus of interlayer crosslinks G, and tensile stiffness of the functionalized graphene sheet D¼Yh, where Y and h are its Young's modulus and thickness. Eqs. (1a) and (1b) indicate that the interlayer shear strain γ(x) localizes near the end/center of sheets within a distance of
l0, offering efficient interlayer load transfer, while tensile strain ε(x) increases from zero at the free end to the maximum at the sheet center. We illustrate these spatial distributions in Fig. 1c for l/l0 ¼10. Under tensile loading, the material could fail in two distinct modes: fracture in the functionalized graphene sheet above a critical tensile strain εcr (denoted as mode G), or failure of interlayer crosslinks when the shear strain exceeds γcr (mode I). The selection of failure modes is determined by both structural and mechanical properties of the sheet and crosslinks. As both tensile and shear strain maximize at or near the end/center of sheets (in the staggered arrangement shown in Fig. 1a), fracture will nucleate there. The strain-based failure criteria are thus εðlÞ ¼

F0 rεcr D

ð2aÞ

γðlÞ ¼

F 0 l0 1 þ c r γ cr h0 D s

ð2bÞ

By defining two controlling parameters k1 ¼l/l0 and k2 ¼γcr(4Gh0/D)1/2/εcr, the diagram of failure mode is illustrated in Fig. 1d, and tensile strength of the RVE is 8 cr 1 < Dε ; k2 Z 1 þsinhk 2h0 coshk1 ss ¼ ð3Þ cr 1 : 2l sDγ ; k2 o 1 þsinhk coshk1 0 ð1 þ cÞ

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

33

As indicated in Fig. 1b, in failure mode G, the sheet is fractured at the center and the entire tension–shear chain is broken. Crack nucleates across the materials transversely and therefore no tensile load can be borne further as crosslinks cannot bear tensile loads. While in mode I the crosslinks fail from regions near the ends and centers of the sheets. The remaining crosslinks, however, could still carry the load and thus no catastrophic failure occurs. The damaged material with partial failure of interlayer crosslinks can be considered as a material with reduced overlapping area. With large-scale functionalized graphene sheets synthesized, significant enhancement in load transfer capability and thus the toughness can be achieved during progressive failure of interlayer crosslinks. This mechanism has been identified in aforementioned biological materials where the high toughness is attributed to the protein interphase that bears shear (Ji and Gao, 2004; Zhang et al., 2011). From these understandings, one would expect to maximize the material strength when mode G and mode I are achieved simultaneously so load capacities of both the sheets and interlayer crosslinks are fully utilized, or k2 ¼sinh k1/(1þcosh k1) according to Eq. (3). However to benefit from the progressive failure of interlayer crosslinks, where rupture and reformation of healable crosslinks dissipate significant mechanical energy, mode I failure should be assured with k2 osinh k1/(1þcosh k1). 2.2. Failure propagation and toughness We now discuss propagation of failure in the material. In failure mode G, crack nucleates and propagates through the center of the functionalized graphene sheet where tensile stress maximizes. However, as interlayer crosslinking phase cannot bear tensile load, the crack will advance abruptly by a distance of 2h0 after fracturing one sheet (see the staggered arrangement of sheets in Fig. 1a). Therefore the fracture toughness ΓG is Γ G ¼ Γ g h=2h0

ð4Þ

Here h ¼0.34 nm and h0 are the thickness of graphene and interlayer space of graphene-derived materials, respectively. For graphene, the fracture surface energy Γg is 10 J/m2 (Zhang et al., 2012), while in functionalized graphene sheets such as GO, the value could change significantly due to the presence of crosslinks, as well as defects and lattice distortion induced by chemical functionalization. In mode I, interlayer crosslinks fracture progressively from regions near the ends and centers of the sheets, while unbroken interlayer crosslinks can still transfer the load applied. Based on the DTS model (Liu et al., 2012), the maximum attainable tensile stress ssa and strain εsa with failed crosslinks of length 2a (Fig. 1b) are   γ cr D sinh l1 =l0   ð5aÞ ssa ¼ 2l0 ð1 þ cosh l1 =l0 Þ εsa ¼

u1 ðlÞ  u2 ð0Þ h0 ssa γ cr h0 ¼ ðl þ 2aÞ þ l Dl l

ð5bÞ

Here l1 ¼l  2a is the length of unbroken crosslinks in a RVE. The results based on Eqs. (5a) and (5b) are plotted as functions of a in Fig. 2a for different sheet sizes ranging from l¼ 5l0 to 100l0. We have shown earlier in Eqs. (1a) and (1b) and Fig. 1c that efficient tensile–shear load transfer is limited within a distance l0 from the ends and centers of sheets. Thus if part of interlayer crosslinks fails while the length of remaining crosslinks l1 is still larger than 2l0, the weakening effect will not be significant. For example, ssa is almost a constant for 2a well below the half-sheet length l, as shown in Fig. 2a for l ¼100l0. The maximum attainable strain εsa increases with a first below a critical failure length acr, beyond which the value drops immediately. This reduction in εsa indicates that the RVE cannot bear higher tensile strain with the staggered layer-by-layer structure, and the remaining crosslinks will fail catastrophically. The strain εsa includes both in-plane sheet elongation that increases with a (the first term in Eq. 5b), as well as interlayer shear (the second term in Eq. 5b) remains as a constant, as the failure of interlayer crosslinks advances. However, as ssa decreases rapidly as l1 approaches zero, the first term of Eq. (5b)

Fig. 2. (a) Relations between maximum attainable tensile stress ssa (solid lines), strain εsa (dashed lines) and the failure length a of interlayer crosslinks. (b) The critical failure length acr of interlayer crosslinks (solid line, black), its theoretical prediction 2acr/l ¼ (l  6l0)/l (dashed line, black) and toughness (solid line, black) of the material plotted against the relatively sheet size l/l0.

34

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

increases at the beginning of the interlayer crosslinks failure and decreases afterwards. The maximum value of εsa can be determined by letting dεsa/da ¼0 as sinh

l1 l1 l1 l þ2a l1 l þ2a þ sinh cosh  cosh  ¼0 l0 l0 l0 l0 l0 l0

ð6Þ

By numerically solving Eq. (6) we obtain the critical failure length acr at the stationary point in the strain curve of Fig. 2a, which is plotted in Fig. 2b as a function of the sheet size. For relatively large sheet (2l 4150l0), acr can be predicted by 2acr/l¼(l 6l0)/l. Beyond acr, the entire graphene-derived materials are damaged as the interlayer crosslinks fail in a pull-out mode abruptly. Failure of interlayer crosslinks can then propagate to this point, and the toughness calculated from this RVE approach is Z acr ss ð0Þεs ð0Þ þ T cr ¼ ssa ðaÞdεsa ðaÞ 2 0 Z acr 2 h0 γ cr D s½s=2 þð1 þ cÞl0 =l þ ssa ðaÞdεsa ðaÞ ð7Þ ¼ 2 2 ð1 þ cÞ 0 4l0 where the first term is the toughness before interlayer crosslinks fail (corresponding to the linear part of stress–strain curve) and the second term describes energy absorption during their failure propagation. The overall toughness Tcr is about 3 times of the value without this additional contribution, which increases with the sheet size further (Fig. 2b). In contrast, for failure mode G, the sheets fail prior to interlayer crosslinks, thus the toughness is limited by the first term in Eq. (7). In failure mode I, the failure advances along the interlayer crosslinks between functionalized graphene sheets, and the development of crack corresponds to pull-out of sheets. The effective crack length in perpendicular to the tensile loading direction increases by 2h0 after all interlayer crosslinks fail in a RVE, and the fracture toughness is Γ I ¼ 2h0 lT cr =2h0 ¼ lT cr

ð8Þ

Although failure mode I is beneficial to enhance the toughness of graphene-derived materials, the load-carrying capacity cannot be improved after the failure of interlayer crosslinks, which lacks a failure arrest mechanism. Moreover, structures of broken crosslinks could become unstable under complicated load conditions. To solve this problem, we propose in this work to utilize bimodal crosslinks, consisting of both long, strong and short, self-healable ones, to improve the strength, toughness, trap material failure and simultaneously maintain the structural stability, as will be discussed below. 3. Atomistic simulations of multimodal and self-healable crosslinking mechanisms As discussed earlier, interlayer crosslinks and their failure play critical roles in defining mechanical performance of graphene-derived materials. Here we explore mechanical and failure behaviors of hydrogen bond (H-bond) and covalent bond (with glutaraldehyde, GA) crosslinked GO papers as illustrative examples for the multimodal and self-healable concepts (Compton et al., 2011; Gao et al., 2011). We perform molecular dynamics (MD) simulations to investigate their effects on the tensile strength and toughness. In order to capture bond breaking and (re)forming processes of crosslinks, we use the reactive force field (ReaxFF) in the simulations where chemical reaction and charge redistribution are included (Chenoweth et al., 2008). 3.1. Self-healable interlayer crosslinks We first explore the self-healable mechanism in H-bond interlayer crosslinks. Our models consist of two GO layers with length lGO ¼16 nm and width wGO ¼4 nm. Periodic boundary conditions are applied in the width direction only. Tensile loads on the system are applied by constraining the left end of the supporting (bottom) layer, and pulling the right end of the top layer at a constant speed of v ¼2 m/s along the length direction, as illustrated in Fig. 3a. The interlayer shear stress is calculated by dividing this tensile force f by the overlapping area between two GO sheets. Two model materials are explored. One consists of graphene sheets randomly functionalized with epoxy and hydroxyl groups (Kim et al., 2012) (Fig. 3a and c). The chemical composition of GO is nC:nO:nH ¼1:0.25:0.125, within the typical range characterized in experiments (Gao et al., 2009; Kim et al., 2012). Here n is the number density of atoms. The equilibrium distance between GO sheets at room temperature is 0.56 nm in the absence of interstitial water, which is close to the value obtained in our previous firstprinciples calculations (Liu et al., 2011). In a second model, 10% vacancies are further introduced by removing carbon atoms from the GO sheet (Fig. 3b and d). The relaxed atomic structures of both models are shown in Fig. 3a and b, clearly showing wrinkles due to the presence of defects and lattice distortion. In the first model, these wrinkled sheets are firstly flattened under tensile loading before the interlayer crosslinks start to break. For the relatively small sizes of GO sheets used in our simulations, all crosslinks contribute to interlayer load transfer after the sheets being straightened. There exists a plateau in the shear stress– displacement curve (Fig. 3e), following a reduction in the stress amplitude. This can be explained as follows: (1) functional groups are randomly distributed in the graphene sheet, so the equilibrium lengths of interlayer H-bonds differ. Due to this multimodality as will be discussed later in more detail, interlayer crosslinks will break progressively under shear between

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

35

Fig. 3. Failure of interlayer crosslinks between neighboring graphene oxide (GO) sheets without (a) and with (b) 10% in-plane vacancies. (c) and (d) Atomic structures of graphene oxide sheets corresponding to (a) and (b). (e) Relationship between the interlayer shear stress and displacement. The straight lines indicate the sliding resistances measured as the average shear stresses during sliding failure.

functionalized graphene sheets, transfer the load, and thus yield the plateau in shear stress amplitude. (2) Moreover, the H-bonds could break and reform reversibly during the relative sliding between GO sheets at the interface, and thus load transfer could still be maintained after breakage of H-bonds, offering a self-healable interface. As confined in the layer-bylayer structure, the H-bonds can reform immediately after breaking. The reforming rate of H-bonds, which is the key parameter defining the self-healing behavior, depends on the density and distribution of functional groups in the new position after failure, as well as the loading rate. Once the H-bond interface cannot be fully healed, the maximum tensile force will decrease to a lower value. For the second model with
10% vacancies in the functionalized graphene sheet, the shear stress–displacement relation is similar, with H-bonds also reforming after breaking during the sliding failure of interface (Fig. 3e). However, the wrinkling is more significant, and thus more stretch is required to straighten the sheets. According to the simulation results, the effective shear strengths (the peak stress) τs in these two models are obtained as 279 and 321 MPa respectively, and sliding resistances at the self-healing stage are τr ¼101 MPa and 140 MPa, which are defined as the average shear stress during the sliding failure (see Fig. 3e). The shear strength and sliding resistance in GO sheets with defects are higher due to the enhanced interlayer interaction at wrinkles and defective sites. However it should be noticed that this enhancement of shear performance is achieved by reducing in-plane tensile strength in the functionalized graphene sheet. Our simulation results show that the tensile strength is reduced by
28% with 10% oxygen-containing functional groups, consistent with recent reported value for carbon nanotubes that shows 30% reduction in strength when the coverage of functional groups exceeds 10% (Zhang et al., 2008). 3.2. Multimodal interlayer crosslinks In order to highlight the multimodal nature of crosslinks and their effects in mechanical enhancement, we further explore GO bilayers crosslinked by covalent bonds with glutaraldehyde (GA), water molecules, as well as their combination. According to recent experimental work, the strength and ultimate tensile strain of GO papers can be improved by intercalating GA (Gao et al., 2011), and the H-bond network formed between water and oxygen containing groups in the sheets enhances the load transfer capacity (Compton et al., 2011; Medhekar et al., 2010). In our models,
5% carbon atoms randomly selected in GO are covalently crosslinked by GA with an areal density of 1.9 nm  2. The interlayer distance h0 between GO sheets is 1.2 nm that is closed to experimental measurements (Gao et al., 2011). The modulation through H-bond network is investigated by intercalating water at a weight content of 25%, with h0 ¼0.8 nm also consistent with experiments (Liu et al., 2013). Other settings of the MD simulations are the same as those introduced in Section 3.1. Their atomic structures are illustrated in Fig. 4a, including the combination of GA and water intercalation with h0 ¼1.3 nm. The shear stress–displacement relations measured from MD simulations are summarized in Fig. 4b. The results show that for GA crosslinks, the shear stress first increases to 490 MPa, followed by abrupt decrease to a lower level of 140 MPa, which is closed to the sliding resistance τr of H-bonds crosslinks of defective GO as presented in Section 3.1 as the GA crosslinks fail and only H-bonds between the oxygen groups in GO contribute to the load transfer.

36

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

Fig. 4. (a) Atomic structures of the interlayer gallery between GO sheets crosslinked by glutaraldehyde (GA), water and their combination. (b) Shear stress–distance relations for models illustrated in panel (a), obtained by MD simulations.

It should be remarked here that the shear strength τs ¼490 MPa measured here increases with the areal density of GA crosslinks. For GO with water intercalation, the shear strengths τs ¼400 MPa is higher than the values 279 and 321 MPa for GO and defective GO without water molecules participating the H-bond network for mechanical enhancement. In comparison with GA crosslinks, the stress curve does not display an abrupt drop and the sliding resistance after peak stress is significantly higher. This is because the mobility of water molecules assists the reconstruction of H-bonds. With these benefits from GA and water intercalations, the bimodally crosslinked GO sheets demonstrate combined high shear strength from GA and after-failure performance from the reversibility of H-bond network between water and oxygen containing groups in GO. 4. Graphene-derived materials with multimodal and self-healable interfaces From the two sets of MD simulations presented in the previous section, two key features of interfacial crosslinks are elucidated in governing the load transfer process between functionalized graphene sheets, i.e. the multimodal crosslinks and self-healable behavior during interfacial failure. We now include these effects into an analytical model in order to predict the overall mechanical properties of graphene-derived materials. 4.1. Multimodal and self-healable crosslinks Our previous work (Liu et al., 2012) have shown that mechanical properties of graphene-derived papers can be finely tuned by adjusting the structure and distribution of graphene sheets and crosslinks. However, observations from our MD simulations here further suggest that introducing multimodal crosslinks can simultaneously enhances mechanical properties of strength and toughness for graphene-derived materials. For example, bimodal crosslinks with both short, reversible crosslinks (SCs, e.g. dative, ionic, hydrogen bonds, and van der Waals interactions (Compton et al., 2011; Park et al., 2008)) and long, resilient crosslinks (LCs, e.g. covalent bonds through polymer intercalation (Gao et al., 2011)) are expected to introduce effective strengthening and toughening effects. In recent experiments, it has been reported that both stiffness and strength of GO papers and fibers are improved by introducing ions (e.g. calcium, magnesium, boron) into the gallery regions (An et al., 2011; Park et al., 2008; Xu et al., 2013), yielding a record tensile strength
0.5 GPa. On the other hand, covalent crosslinks using GA or poly-vinyl alcohol (PVA) also improve their load-bearing capabilities (Gao et al., 2011; Liu et al., 2013). The role of multimodal networks has been studied in rubberlike materials consisting of both long and short polymer chains in a crosslinked network (Mark and Erman, 2007). Despite of the similarity in concept, synergetic enhancement by LCs and SCs could be established in graphene-derived materials more effectively due to their layer-by-layer microstructures and the outstanding mechanical properties of single graphene sheet. Thus the stiffness, strength and toughness can be improved by combining LCs and SCs. Several merits arise distinctly by introducing multimodal and self-healable interlayer crosslinks: (1) self-healable SCs act as a sacrificial interface to continuously dissipate the mechanical energy by rupturing and reconstructing of the crosslinks, while the LCs maintain the structural integrity, (2) the self-healable feature of the crosslinks enables robust performance for cyclic loading, (3) SCs keep the compact layer-by-layer structure of the graphenederived materials, which is critical for their high mechanical performance, and (4) LCs could also prevent structural failure of graphene-derived materials under complicated load conditions such as bending, torsion, and moisture-induced swelling. As illustrated in Fig. 5a, a two-dimensional RVE is constructed by considering functionalized graphene sheets with a uniform lateral size 2l, where adjacent layers are regularly staggered and crosslinked by bimodal agents of covalent LCs and self-healable SCs. Here the mechanical response of the LCs is assumed to be hyperelastic. The constitutive relation between

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

37

Fig. 5. Functionalized graphene sheets crosslinked bimodally in a layer-by-layer structure: (a) an illustration of the model and loading condition for a RVE consisting of both short crosslinks (SCs) and long crosslinks (LCs) between functionalized graphene sheets. F0 is the tensile force applied to the RVE, under which LCs start to fail in regions near the edges and centers of sheets where shear strain maximizes, while SCs break and reconstruct as failure proceeds. (b) Schematic shear stress–strain relation by including bimodal and self-healing crosslinking effects.

shear stress and strain is τL ¼GL((λ  1)/λ2)/3 (Mark and Erman, 2007). We assume the deformation of interlayer crosslinks to be pure shear and the longitudinal elongation λ (the ratio between the extended and rest length of LCs) is expressed in the interlayer shear strain γ as λ ¼1þ γ (Mark and Erman, 2007). The shear strain to failure for LCs is assumed to be γL. For deformation in the small strain regime the relation degenerates to the linear relation of τL ¼GLγ and we define GL as the effective shear modulus of the LCs. The mechanical response of SCs is simplified into a linear relation with self-healable feature, where the interlayer shear stress contributed by SCs is τS ¼GSγR. GS is the effective shear modulus of SCs and γR is the interlayer shear strain of SCs after their last reconstruction, which is defined as γR ¼ [u1(x  1, t  1-t) u2(x  1, t  1-t)]/h0. We assume the SCs heal instantaneously after the rupture. Here t  1 and x  1 are the time and position of SCs for the last reconstruction, and u1(2)(x  1, t  1-t) is the displacement of sheet 1(2) from time t  1 to t at the point x  1 (Fig. 5a). The critical failure shear strain of the SCs is assumed as γS. By including the combined effects of LCs and SCs, the constitutive relation of interlayer crosslinks is schematically shown in Fig. 5b. As the shear strain increases beyond γS, SCs start to fail so the shear stress drops down. After the reconstruction at a new equilibrium position, the shear stress gradually increases again as the shear deformation further proceeds. During the 2 repeatingly breaking and recovery processes of the SCs, mechanical energy ED ¼nGSγS corresponding to the shaded area in Fig. 5b is dissipated, which depends on the number n of self-healed SCs, their effective shear modulus and shear strain to failure. 4.2. Analysis based on the DTS model We now incorporate the effects of multimodal and self-healable interlayer crosslinks into the DTS model following the RVE approach. According to our previous study (Liu et al., 2012), we assume that the interlayer crosslinks are uniformly distributed. By substituting the expression of shear stress into the DTS model one obtains the governing equations: ∂2 u1 ðx; tÞ D ¼ 2ðτL þτS Þ ∂x2

ð9aÞ

∂2 u2 ðx; tÞ ¼  2ðτL þ τS Þ D ∂x2

ð9bÞ

As SCs are usually stiffer and more brittle than LCs (where entropic elasticity governs), here we assume GS ¼10GL and γS ¼ 0.1γL in the following discussion. The interlayer distance h0 varies for different crosslinks. According our simulation

38

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

results, h0 for covalent GA-crosslinks is 1.3 nm. For comparison, h0 measured experimentally for PVA crosslinks is 1.32 nm, while for neat GO films h0 ¼0.83 nm (Liu et al., 2013). Therefore, in the bimodal model, we assume the interlayer distance is hL ¼1.32 nm for LCs and hS ¼0.83 nm for SCs. The length of functionalized graphene sheet is set to 2l ¼4l0, where l0 ¼(DhL/ 4GL)1/2 is the span of crosslinking region with efficient shear load transfer by LCs, as we have introduced earlier. We consider three representative systems with (1) both LCs and self-healable SCs, (2) only LCs, and (3) only self-healable SCs. In the following analysis, we will focus on the loading range where LCs are not broken, and thus the maximum shear strain in a RVE is determined by the critical failure shear strain of LCs γL. We also assume that SCs are instantaneously reconstructed at the new position. Predicted stress–strain relations for these three model systems are obtained by numerically solving Eqs. (9a) and (9b) using the conjugate gradient method. The results are summarized in Fig. 6. According to the combined effects from LCs and SCs, virtues such as the high stiffness and self-healability of SCs, high extension and toughness of LCs are fully utilized and transferred into the macroscopic material properties for system 1. As a result, tensile stiffness in the small deformation regime (the linear region in Fig. 6a), strength and ultimate tensile strain are higher than those of system 2 with LCs only (see solid and dashed lines in Fig. 6a). This is consistent with experimental results that the ultimate tensile strain can be significantly improved by adding the water to GA-crosslinked GO papers (Gao et al., 2011). Due to the self-healability of SCs, mechanical energy is dissipated by fracturing and reforming SCs, which is indicated by the hysteresis between loading and unloading curves. It should also be noticed that during the unloading process, the deformation of graphene-derived materials – by rupturing and reconstructing the SCs – is neither elastic nor reversible. In contrast, for interlayer crosslinks with SCs only (system 3), interlayer load transfer is maintained and enhanced after reconstruction of SCs (see the nonlinear region in Fig. 6a). The strength here is
65% of the value in system 1 where the synergistic effect of LCs and SCs works, but the stiffness is higher because the interlayer distance of SCs (0.83 nm) is smaller than that of LCs (1.32 nm). It should be remarked here that the effect of interlayer distance h0 on the mechanical properties of graphene-derived materials is two-fold as shown in Eq. (3). Firstly it defines the density of functionalized graphene sheets and thus reduced stiffness and strength at a larger value of h0. Secondly, the length scale of efficient load transfer l0 scales 1/2 with h0 . Hysteresis is also observed in the loading and unloading cycles for system 3, and the dissipated energy in one cycle is
55% of the value for system 1 (Fig. 6a). However, one should be noticed that due to the absence of LCs, the integrity of the layer-by-layer structure could be destroyed by rupturing SCs, especially when complicated loading conditions, such as bending and torsion, are applied. In system 1, the maximum tensile stress increases with the rupture and reconstruction of SCs, which could arrest the failure of crosslinks. A diagram for the tensile strength sM of graphene-derived materials with bimodal crosslinks (both LCs and self-healable SCs) normalized by that with LCs only, i.e. sL, is plotted in Fig. 6b. Here the sheet size is 4l0, and we consider SCs with parameters of GS/GL from 2 to 20 and γL/γS from 2 to 20. The strength enhancement by including self-healable SCs increases with GS and γS, which is described using a single parameter GSγS/GLγL. In small deformation regime, as GS is usually much higher than GL, the stiffness of model system 1 is mainly attributed to SCs. The breakage of SCs leads to kinks in the tensile stress curves (Fig. 6a), and a lower value of γS yields smoother response in the stress and more stabilized mechanical performance of the material. Although our results in Fig. 6c suggest that the strength enhancement of graphene-derived materials decreases with the sheet size, the absolute value of the tensile strength still increases. According to the results obtained from the DTS model, the enhancement by enlarging the sheet size converges for l 48l0. In the current study, we focus on the regime before LCs are broken. However, as the failure of LCs develops in mode I, energy dissipation and toughness are further enhanced till the abrupt pull-out failure occurs. According to our previous analysis, interlayer shear strain localizes at the regions near edges and centers of the functionalized graphene sheets for large sheet size (measured by l0 ¼(DhL/4GL)1/2 from the edges and centers), where the rupture and reconstruction of SCs nucleate. The tensile load in the sheet is transferred by interlayer shear, where both LCs and SCs contribute. Because of the rupture and reconstruction of SCs, we can assume that the average shear stress (over different states from the newly reconstruction to nearly failure) contributed by SCs is GSγS/2 within a distance l0 from the edges and centers. For each sheet, the effective load transfer length of SCs is thus 4l0 by considering its two sides and two edges (see the illustration of RVE in Fig. 5a). As a result the tensile force transferred by the interlayer shear of SCs is 2GSγSl0 and the effective tensile stress is 2GSγSl0/2hL. Based on our previous work, the tensile strength of solely LCs is sL ¼

sγ L D 2ð1 þcÞl0

ð10Þ

Here in order to obtain an analytical expression for the strength enhancement by both LCs and SCs, we adopt a linear hyperelastic constitutive relation, which is expected to work well when the failure shear strain γL is less than 1. Thus the enhancement by LCs and SCs can be predicted as sM sS 2GS γ S l0 =2hL ð1 þ cÞ GS γ S ¼ 1þ ¼ 1þ ¼ 1þ 2s GL γ L sL sL sγ L D=2ð1 þ cÞl0

ð11Þ

where sM ¼sL þ sS. The prediction from Eq. (11) works well for large sheet size and small values of GSγS/GLγL (Fig. 6c), because that the shear strain localizes near the edges and centers in large-size sheets, and the effect of SCs on shear strain distribution becomes not significant for relatively small GSγS/GLγL.

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

39

Fig. 6. (a) Tensile stress–strain relation of the RVE with three types of interlayer crosslinks. With LCs only, the stress–strain relation is elastic and there is no hysteresis between loading and unloading curves, in contrast to those where SCs are introduced. (b) The strength enhancement sM/sL with bimodal crosslinks and self-healable interfaces calculated for different values of GS and γS, where sL is strength with LCs only. (c) The relation between the strength enhancement and GSγS/GLγL for different sheet sizes, i.e. l ¼ 2l0 (□), l¼ 4l0 (○) and l ¼ 8l0 (△). The solid line is the theoretical prediction from Eq. (11).

The stiffness, strength and toughness of graphene-derived materials are simultaneously enhanced by introducing both LCs and self-healable SCs. The strength is about the summation of their individual contributions. Besides, the tensile strain

40

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

to failure can be higher than the system with LCs only. If only LCs are available, the stiffness and strength are usually low. This could be improved by increasing the density of LCs that is equivalent to increase the effective shear modulus GL of LCs (Liu et al., 2013). However, the density of LCs is limited by the number density of anchoring sites in the functionalized graphene sheets and the layer-by-layer microstructure (one LCs may entangle with others at a high density). On the other hand, if there are only non-self-healable SCs at the interface (see the linear region in Fig. 6a), the strength and toughness is usually low due to the limited deformability of SCs. The self-healing SCs can improve partly the toughness and the strength, but the staggered structure could become unstable during rupturing SCs (An et al., 2011; Xu et al., 2013). We finally note here that the discussion above is based on the failure mode I, while if the failure is in mode G then the toughening effect by interlayer crosslinks will not be available.

4.3. Additional remarks Our discussion in this section is limited by the DTS model and RVE approach we follow. There are a few open issues that may be critical in practical applications of graphene-derived materials, and thus we add some comments here. In our theoretical model, the usage of RVE excludes the localized nature of stress concentration at fracture front as it propagates in the material. In order to take into account this effect, the constraint in RVE must be released. Moreover, functionalized graphene sheets are assumed to be flat and in regularly staggered arrangement. However, in graphenederived materials, non-flat structures and irregular arrangement are typical due to the presence of defects and lattice distortion in the sheet, as well as the non-uniform distribution of sheet size and shape. According to these defects, fracture in graphene-derived materials could be localized and the assumption made by using the RVE should be adjusted. The nonplanar geometry and spatial arrangement of stiff platelets in nacre have been studied in recent work (Tang et al., 2007; Zhang et al., 2010), and could be extended to the material we have explored here. Specifically, Zhang et al. concluded that stairwise staggering (including regular staggering as focused in this work) could achieve overall excellent performance, namely high stiffness and strength comparable to those of the reinforcing platelets as well as high failure strain and energy storage capacity comparable to those of the soft matrix (Zhang et al., 2010, 2011). Our discussions on the mechanical behaviors here are based on the assumption that interlayer crosslinks responses, including their rupture and reformation, are instantaneous. However, the reconstruction of SCs is complicated and ratedependent. As reported recently, factors including the loading rate, amplitude, and temperature are critical for the timedependent mechanical response of H-bond networks (Qian et al., 2008; Ruiz and Keten, 2014; Ruiz et al., 2013). The reforming rate of crosslinks may further depend on the distribution of the binding sites in functionalized graphene sheets and crosslinking agents in the interlayer gallery. This could be, for example, modulated by the water content between GO sheets. In our MD simulation results, as healable behavior of the H-bond network is observed at a loading rate of 2 m/s, we estimate a reforming rate higher than
0.01 ps  1. Thus the instantaneous assumption we made is reasonable for typical realistic loading conditions. However, rigorous consideration of these effects could improve the current model and enable predictions for time-dependent mechanical properties of graphene-derived materials beyond the quasi-static limit in this work, including the creeping effect and their viscoelastic properties.

5. Conclusion In this study we have explored the mechanical properties of graphene-derived materials by introducing multimodal selfhealable interlayer crosslinks into their regularly staggered layer-by-layer microstructures. The mechanisms of mechanical enhancement are elucidated by both atomistic simulations and theoretical model analysis. The strengthening effect originates from the synergy of multimodal crosslinks that almost adds up contributions of individual ones, whereas toughening roots from the continuous rupture and reconstruction of reversible crosslinks. Besides, the increase in stress during interlayer crosslinks fracture offers a failure arrest mechanism. Nanoscale confinement between functionalized graphene sheets enables efficient load transfer that is critically defined by the crosslinks. Optimized hierarchical materials consisting of large-size sheets and multimodal crosslinks that include both long, strong and short self-healable ones offer the opportunity to make use of the outstanding mechanical properties of graphene to the largest extent. The models and results provide key concepts in optimal design of graphene materials in their macroscopic forms. The rationales identified here can be extended straightforward to other materials made of lowdimensional nanostructures that could be functionalized with specified crosslinks.

Acknowledgments Y.L. acknowledges the support by the National Natural Science Foundation of China through Grant 11302163 and 11321062. Z.X. acknowledges supports by the National Natural Science Foundation of China through Grant 11222217, 11002079, and Tsinghua University Initiative Scientific Research Program 2011Z02174. The simulations were performed on the Explorer 100 cluster system of Tsinghua National Laboratory for Information Science and Technology.

Y. Liu, Z. Xu / J. Mech. Phys. Solids 70 (2014) 30–41

41

References An, Z., Compton, O.C., Putz, K.W., Brinson, L.C., Nguyen, S.T., 2011. Bio-inspired borate cross-linking in ultra-stiff graphene oxide thin films. Adv. Mater. 23, 3842–3846. Bekyarova, E., Sarkar, S., Wang, F., Itkis, M.E., Kalinina, I., Tian, X., Haddon, R.C., 2013. Effect of covalent chemistry on the electronic structure and properties of carbon nanotubes and graphene. Acc. Chem. Res. 46, 65–76. Cheng, Q., Wu, M., Li, M., Jiang, L., Tang, Z., 2013. Ultratough artificial nacre based on conjugated cross-linked graphene oxide. Angew. Chem. Int. Ed. 52, 3750–3755. Chenoweth, K., van Duin, A.C.T., Goddard, W.A., 2008. ReaxFF reactive force field for molecular dynamics simulations of hydrocarbon oxidation. J. Phys. Chem. A 112, 1040–1053. Compton, O.C., Cranford, S.W., Putz, K.W., An, Z., Brinson, L.C., Buehler, M.J., Nguyen, S.T., 2011. Tuning the mechanical properties of graphene oxide paper and its associated polymer nanocomposites by controlling cooperative intersheet hydrogen bonding. ACS Nano 6, 2008–2019. Dimas, L.S., Bratzel, G.H., Eylon, I., Buehler, M.J., 2013. Tough composites inspired by mineralized natural materials: computation, 3D printing, and testing. Adv. Funct. Mater. 23, 4629–4638. Finnemore, A., Cunha, P., Shean, T., Vignolini, S., Guldin, S., Oyen, M., Steiner, U., 2012. Biomimetic layer-by-layer assembly of artificial nacre. Nat. Commun. 3, 966. Gao, H., 2006. Application of fracture mechanics concepts to hierarchical biomechanics of bone and bone-like materials. Int. J. Fract. 138, 101–137. Gao, W., Alemany, L.B., Ci, L., Ajayan, P.M., 2009. New insights into the structure and reduction of graphite oxide. Nat. Chem. 1, 403–408. Gao, Y., Liu, L.-Q., Zu, S.-Z., Peng, K., Zhou, D., Han, B.-H., Zhang, Z., 2011. The effect of interlayer adhesion on the mechanical behaviors of macroscopic graphene oxide papers. ACS Nano 5, 2134–2141. Geim, A.K., Novoselov, K.S., 2007. The rise of graphene. Nat. Mater. 6, 183–191. Gong, L., Kinloch, I.A., Young, R.J., Riaz, I., Jalil, R., Novoselov, K.S., 2010. Interfacial stress transfer in a graphene monolayer nanocomposite. Adv. Mater. 22, 2694–2697. Hartmann, M.A., Fratzl, P., 2009. Sacrificial ionic bonds need to be randomly distributed to provide shear deformability. Nano Lett. 9, 3603–3607. Ji, B., Gao, H., 2004. Mechanical properties of nanostructure of biological materials. J. Mech. Phys. Solids 52, 1963–1990. Kim, S., Zhou, S., Hu, Y., Acik, M., Chabal, Y.J., Berger, C., de Heer, W., Bongiorno, A., Riedo, E., 2012. Room-temperature metastability of multilayer graphene oxide films. Nat. Mater. 11, 544–549. Kotov, N.A., Dékány, I., Fendler, J.H., 1996. Ultrathin graphite oxide–polyelectrolyte composites prepared by self-assembly: transition between conductive and non-conductive states. Adv. Mater. 8, 637–641. Liu, L., Gao, Y., Liu, Q., Kuang, J., Zhou, D., Ju, S., Han, B., Zhang, Z., 2013. High mechanical performance of layered graphene oxide/poly(vinyl alcohol) nanocomposite films. Small 9, 2466–2472. Liu, Y., Xie, B., Xu, Z., 2011. Mechanics of coordinative crosslinks in graphene nanocomposites: a first-principles study. J. Mater. Chem. 21, 6707–6712. Liu, Y., Xie, B., Zhang, Z., Zheng, Q., Xu, Z., 2012. Mechanical properties of graphene papers. J. Mech. Phys. Solids 60, 591–605. Mark, J.E., Erman, B., 2007. Rubberlike Elasticity: A Molecular Primer. Cambridge University Press, Cambridge. Medhekar, N.V., Ramasubramaniam, A., Ruoff, R.S., Shenoy, V.B., 2010. Hydrogen bond networks in graphene oxide composite paper: structure and mechanical properties. ACS Nano 4, 2300–2306. Park, S., Lee, K.-S., Bozoklu, G., Cai, W., Nguyen, S.T., Ruoff, R.S., 2008. Graphene oxide papers modified by divalent ions: enhancing mechanical properties via chemical cross-linking. ACS Nano 2, 572–578. Qian, J., Wang, J., Gao, H., 2008. Lifetime and strength of adhesive molecular bond clusters between elastic media. Langmuir 24, 1262–1270. Qiu, L., Liu, J.Z., Chang, S.L.Y., Wu, Y., Li, D., 2012. Biomimetic superelastic graphene-based cellular monoliths. Nat. Commun. 3, 1241. Ruiz, L., Keten, S., 2014. Directing the self-assembly of supra-biomolecular nanotubes using entropic forces. Soft Matter 10, 851–861. Ruiz, L., VonAchen, P., Lazzara, T.D., Xu, T., Keten, S., 2013. Persistence length and stochastic fragmentation of supramolecular nanotubes under mechanical force. Nanotechnology 24, 195103. Stankovich, S., Dikin, D.A., Dommett, G.H.B., Kohlhaas, K.M., Zimney, E.J., Stach, E.A., Piner, R.D., Nguyen, S.T., Ruoff, R.S., 2006. Graphene-based composite materials. Nature 442, 282–286. Sun, J.-Y., Zhao, X., Illeperuma, W.R.K., Chaudhuri, O., Oh, K.H., Mooney, D.J., Vlassak, J.J., Suo, Z., 2012. Highly stretchable and tough hydrogels. Nature 489, 133–136. Sun, T.L., Kurokawa, T., Kuroda, S., Ihsan, A.B., Akasaki, T., Sato, K., Haque, M.A., Nakajima, T., Gong, J.P., 2013. Physical hydrogels composed of polyampholytes demonstrate high toughness and viscoelasticity. Nat. Mater. 12, 932–937. Tang, H., Barthelat, F., Espinosa, H.D., 2007. An elasto-viscoplastic interface model for investigating the constitutive behavior of nacre. J. Mech. Phys. Solids 55, 1410–1438. Wojtecki, R.J., Meador, M.A., Rowan, S.J., 2011. Using the dynamic bond to access macroscopically responsive structurally dynamic polymers. Nat. Mater. 10, 14–27. Xu, Z., 2013. Mechanics of metal–catecholate complexes: the roles of coordination state and metal types. Sci. Rep. 3, 2914. Xu, Z., Sun, H., Zhao, X., Gao, C., 2013. Ultrastrong fibers assembled from giant graphene oxide sheets. Adv. Mater. 25, 188–193. Yao, H., Song, Z., Xu, Z., Gao, H., 2013. Cracks fail to intensify stress in nacreous composites. Compos. Sci. Technol. 81, 24–29. Young, R.J., Kinloch, I.A., Gong, L., Novoselov, K.S., 2012. The mechanics of graphene nanocomposites: a review. Compos. Sci. Technol. 72, 1459–1476. Zhang, T., Li, X., Kadkhodaei, S., Gao, H., 2012. Flaw insensitive fracture in nanocrystalline graphene. Nano Lett. 12, 4605–4610. Zhang, Z., Zhang, Y.-W., Gao, H., 2011. On optimal hierarchy of load-bearing biological materials. Proc. R. Soc. B: Biol. Sci. 278, 519–525. Zhang, Z.Q., Liu, B., Chen, Y.L., Jiang, H., Hwang, K.C., Huang, Y., 2008. Mechanical properties of functionalized carbon nanotubes. Nanotechnology 19, 395702. Zhang, Z.Q., Liu, B., Huang, Y., Hwang, K.C., Gao, H., 2010. Mechanical properties of unidirectional nanocomposites with non-uniformly or randomly staggered platelet distribution. J. Mech. Phys. Solids 58, 1646–1660. Zhao, X., Xu, Z., Zheng, B., Gao, C., 2013. Macroscopic assembled, ultrastrong and H2SO4-resistant fibres of polymer-grafted graphene oxide. Sci. Rep. 3, 3164. Zhong, D., Yang, Q., Guo, L., Dou, S., Liu, K., Jiang, L., 2013. Fusion of nacre, mussel, and lotus leaf: bio-inspired graphene composite paper with multifunctional integration. Nanoscale 5, 5758–5764. Zuo, S., Wei, Y., 2007. Effective elastic modulus of bone-like hierarchical materials. Acta Mech. Solida Sin. 20, 198–205.