Multiscale mechanics of noncovalent interface in graphene oxide layered nanocomposites

ZeZhou He, YinBo Zhu, HengAn Wu

CAS Key Laboratory of Mechanical Behavior and Design of Materials, Department of Modern Mechanics, CAS Center for Excellence in Complex System Mechanics, University of Science and Technology of China, Hefei 230027, China

Keywords:Multiscale mechanics Noncovalent interface Commensurate and incommensurate Shear-lag model Layered nanocomposites

ABSTRACT Noncovalent interfaces play a vital role in inelastic deformation and toughening mechanisms in layered nanocomposites due to their dynamical recoverability. When interfacial engineering is applied to design layered nanocomposites, shear-lag analysis is usually implemented to evaluate the capability of interfacial loading transfer. Here, we introduce a multiscale shear-lag model that correlates macroscale mechanical properties with the molecular mechanisms to quantify the effects of interfacial configuration in graphene oxide (GO) layered nanocomposites. By investigating the mechanical responses of commensurate and incommensurate interfaces, we identify that the commensurate interface exhibits a pronounced size effect due to the nucleation and propagation of interfacial defects, whereas the incommensurate interface displays uniform deformation. Our predictions are further validated through large-scale molecular dynamics simulations for GO layered nanocomposites. This work demonstrates how size effects and interfacial configurations can be exploited to fabricate layered nanocomposites with superior mechanical properties despite relying on weak noncovalent interfaces.

Nanomaterials such as carbon nanotube, graphene, nanocellulose, and so on, which possess excellent physical, chemical, and mechanical properties [1–3] , are primary building blocks for designing high-performance structural and functional nanocomposites [2–4] . They are assembled sequentially through interface engineering and bottom-up design, by which the prominent properties of functional motifs at the nanoscale are transferred into the macroscale materials [ 2 , 5-8 ]. For instance, with typical layered structures, graphene and graphene oxide (GO) have been widely applied as building blocks assembled layer-by-layer into nacrelike microstructures [9–11] , known as layered nanocomposites. The key to success of these nanocomposites is their hierarchical interfaces, which not only provide adequate cohesion between adjacent building blocks and maintain the structural integrity but also distribute the overall load onto each functional nano-motifs[ 2 , 12-14 ]. Numerous effort s have been made to explore the complexity and synergism behind the interplay of building blocks and interfaces in layered nanocomposites with the goal of preparing high-performance nanocomposites [ 15 , 16 ]. From this perspective,the shear-lag model has been demonstrated to be essential in theoretically advancing knowledge of loading transfer in nacre [17] ,GO paper [ 13 , 18 , 19 ], nanocellulose paper [20] , and fiber-reinforced composites [21] . Generally, the shear-lag analysis mainly focuses on the mismatch between building block elasticity and interface shear deformation in brick-interface systems [ 18 , 22-24 ], where the deformation behavior depends on the competition between the overlap length of brick and shear-lag characteristic length [ 17 , 25 ].The interfacial shear stress distribution is uniform when the overlap length is below the characteristic length and localized when the overlap length exceeds the characteristic length, so there are critical overlap length scales governing the balance of mechanical properties [ 17 , 18 , 24 ].

Fig. 1. Schematics of the multiscale shear-lag model for noncovalent interface in layered nanocomposites: a Sketch map of regular staggered structure and the RVE highlighted by the blue box. b The shear-lag model of the staggered structure. c Deformation process of noncovalent interface, involving initial configuration, interfacial sliding,and the breaking and reforming of noncovalent interactions. d Profile of the effective interface constitutive relation. Γ1 and δp are the interfacial toughness and period of commensurate interface, respectively.

Though layered nanocomposites promise high mechanical properties, challenges remain in simultaneously improving strength and toughness as these two properties are generally mutually exclusive [ 26 , 27 ]. To overcome this conflict, recent material designs increasingly seek to incorporate weak interfaces and mimic the mechanical behavior of biological interfaces in biocomposites that exhibit an excellent combination of mechanical properties far beyond their components, such as nacre, bone, spider silk, and wood[ 12 , 16 , 27 , 28 ]. In these biocomposites, interfaces are usually weaker than the building blocks to channel nonlinear deformations and deflect cracks into configurations [ 12 , 29 ]. Inspired by that, recently,we proposed that incorporating noncovalent interfaces, which are inherently weak but reversible, within materials is a powerful approach to balance their mechanical properties and regulate toughness mechanisms [ 13 , 23 , 30 ]. Unlike traditional unrecoverable interfaces that possess a failure strain, noncovalent interfaces can dynamically break and re-form under large sliding displacement and maintain loading transfer capability until bricks are completely pulled out [31–34] . As a consequence, the classical shear-lag model fails to predict the mechanical responses of noncovalent interfaces and the characteristic overlap lengths that govern the mechanical properties of layered nanocomposites [ 23 , 35 , 36 ]. Besides, except for van der Waals interfaces in multilayer two-dimensional materials and double-walled carbon nanotubes [ 36 , 37 ], interface stacking configurations also plays a vital role in other noncovalent interfaces, such as hydrogen bonding in silkβ-sheet nanocrystallites and sacrificial ionic bonding in bone, which exhibit unusual deformation behaviors with respect to varied overlap length under different interfacial configurations [ 35 , 38 ]. They can be classified by their structural characteristics into two types, i.e., commensurate interface for regular structure and incommensurate interface for random structure, related to the spatial distribution of functional groups and molecular configurations [23] . However, there is still a lack of multiscale analysis that decodes the correlation between macroscale mechanical performances of GO layered nanocomposites and molecular mechanisms in noncovalent interfaces over several hierarchical length scales.

In this Letter, by including the interfacial configurations with commensurate and incommensurate types, we introduce a multiscale shear-lag model to delineate the nonlinear deformation of noncovalent interface and to decode the molecular mechanisms underlying macroscopic mechanical responses through two characteristic overlap lengths. This model is then applied to GO layered nanocomposites combined with molecular dynamics (MD) simulations to reveal the mechanisms of size effect in noncovalent interface and relevant strengthening and toughening design for layered nanocomposites.

As shown in Fig. 1 a, the layered nanocomposites dominated by noncovalent interfaces are assumed to be regularly staggered in a “brick-and-mortar” pattern for simplicity. The overlap length isL/2, half the length of the brick. The mechanical responses under uniaxial load, such as displacement field, tensile strengthσe, and effective toughnessWecan be obtained by the shear-lag analysis based on the representative volume element (RVE) [13] , as indicated in Fig. 1 b. The shear-lag analysis focuses on the mismatched deformation between brick elasticity and interfacial sliding, while interfacial mechanical response related to the molecular mechanisms can be described by the constitutive relation [23] . Hence functional nano-motifs (bricks) are assumed to be elastic and brittle with Young’s modulusEband tensile strengthσcr. The interfacial constitutive relation can be expressed as a function of interlayer relative sliding distance, written asτ=τcrχ(Δ), whereτcris shear strength, andχ(Δ) is the shape function associated with the interfacial configuration [23] .

For noncovalent interface, atomic interactions, such as van der Waals interaction, hydrogen bond, andπ-πinteraction, can be expressed as a function of the distance between two atoms and be neglected beyond a cutoff distance [39] . The most significant difference from covalent interaction is that noncovalent interaction can dynamically break and re-form so that the noncovalent interface is self-healable and can maintain the capability of loading transfer under large interlayer sliding, as illustrated by Fig. 1 c. According to the interfacial shear response, the noncovalent interface can be summarized in two configurations, i.e., commensurate and incommensurate interfaces. Commensurate interface exhibits periodic shear stress and high shear strength, while the shear strength of incommensurate interface is much less than that of commensurate one and is a constant under large relative sliding. Here, we assume that the constitutive relation of incommensurate interface is linear-sliding form, and that of commensurate one is a triangular wave, where the positive amplitude isτcr, the negative amplitude is -kτcr, andkreflects the interfacial roughness ( Fig. 1 d). Smallerkimplies a rougher interface. Based on these interfacial constitutive relations, we can correlate interfacial sliding with molecular mechanisms of noncovalent interactions, where shear strength of commensurate interface is related to interaction strength, and the integral of shear stress–displacement curve within one period reflects the energy dissipation by the breaking and reforming of noncovalent bonds [23] .

As shown in Fig. 1 b, when a displacementΔis applied at the right end of the top brick, the mechanical equilibrium of the RVE is governed by [23]

Fig. 2. The mechanical responses of RVE under different overlap lengths L /2 and interfacial roughness k . a Typical shear stress distribution under different overlap lengths with k = 0.5. b Normalized tensile strength as a function of overlap length under different interfacial roughness. c Typical effective stress-strain curves under different overlap lengths with k = 0.5. Dashed lines are the normalized effective stress-strain curves of incommensurate interface. d Normalized effective toughness as a function of the ratio between two lengths L and l 1cr under different interfacial roughness. L ?is the length of effective toughness reaching its extremum.

To apply our theoretical predictions in functional materials, we implement large-scale MD simulation for GO layered nanocomposites by using the LAMMPS package [41] . As illustrated in Fig. 3 a,the commensurate interface is constructed through the regular distribution of hydroxyl groups on graphene with a carbon-oxygen ratio of 4:1, while the incommensurate interface is accomplished by distributing hydroxyl groups randomly. Water molecules are introduced to regulate the interfacial configuration and mechanical responses with a constant linear density of 20 per nanometer, as indicated in Fig. 3 b. The width of graphene sheet is set to be 5.1 nm. The CHARMM force field, which has been proved to describe the interfacial noncovalent bonding well, is employed to characterize the interactions of GO and water [ 42 , 43 ]. The mechanical responses of GO layered nanocomposites are obtained through the uniaxial tension of RVE with a strain rate of 2 ×108s?1 under isothermal–isobaric (NPT) ensemble at the temperature of 300 K.

Figure 3 c exhibits the stress-strain curves of 100 nm RVE under different interfacial configurations. As we expect, the maximum achievable tensile stress of commensurate interface is higher than that of incommensurate and wet interfaces, demonstrating that commensurate configuration possesses superior advantages in interfacial loading transfer for short overlap length. Comparing the mechanical properties of dry and wet interfaces, we find that water molecules decline the effective modulus but slightly improve the effective strength of wet-incommensurate interface. These results can be attributed to that water molecules increase interlayer distances and shear strength of incommensurate interface. Previous experiments and simulation work have also reported these phenomena and revealed that water molecules improve the density or strength of interlayer hydrogen bonds (H-bonds) [30] . However,the wet-commensurate interface displays a contrary result, where the tensile strength is much lower than that of commensurate one.This anomalous phenomenon has not been presented before and cannot be explained by previous analysis. Here, we demonstrate that shear strength depends not only on the strength and density of H-bonds but also on the interfacial configuration. Water molecules can disrupt the commensurate configuration, resulting in a mechanical response similar to that of the incommensurate interface and pronounced reduction of shear strength. Besides, the wet-commensurate interface shows higher tensile strength compared with dry and wet incommensurate ones. It is possible that the wet-commensurate interface is rougher than dry and wet incommensurate configurations.

Fig. 3. The structural configurations and mechanical responses of GO staggered structure:. a The RVE of dry commensurate and incommensurate configurations. b The RVE of wet commensurate and incommensurate configurations. Water molecules colored by green and blue atoms are randomly filled in the interlayer. The GO flakes are assembled by five layers of unit sheets. The mechanical responses of RVE under different interfacial configurations with brick length of c L = 100 nm and d L = 400 nm. The dashed line in c and d is the value calculated by the Dugdale model.

For the mechanical response of 400 nm RVE, as shown in Fig. 3 d, both dry and wet incommensurate interfaces exhibit a similar trend to the short RVE, while the tensile strength of wetcommensurate interface exceeds that of commensurate one. This significant transformation can be ascribed to the differences in deformation modes between dry and wet commensurate interfaces.Our recent study revealed that there are multiple interface defects nucleating and stacking due to the deformation mismatch between interfacial sliding and GO elasticity, dividing the commensurate interface into small segments [23] . Hence, the effective length of loading transfer is the sum of these small segments for the long commensurate interface. On the contrary, the effective length for incommensurate interface is the entire overlap length because the interfacial shear deformation is uniform. As a result, the effective length of incommensurate interface is larger than that of commensurate one as the overlap length is ultralong, and proper interface regulation can make the tensile strength of incommensurate interface surpasses that of commensurate one.Furthermore, it should be noted that the mechanical response of commensurate interface exhibits stress drop as the tensile stress reaches the value determined by the Dugdale model. This behavior corresponds with our theoretical prediction illustrated in Fig. 2 c,which further validates our multiscale shear-lag model. Then the stress-strain curve is followed by strain hardening, resulting in the tensile stress exceeding the upper limit governed by the Dugdale model.

Through a series of MD simulations, we obtain the tensile strength of RVE as a function of GO flake length under different interfacial configurations, as indicated in Fig. 4 a. The tensile strength of incommensurate interface increases linearly with the brick length due to the uniform shear stress distribution. The tensile strength of commensurate interface obeys the Dugdale model whenL< 300 nm and then follows linear growth whenL> 300 nm, similar to the tendency anticipated in Fig. 2 b. As the commensurate interface is intercalated by water molecules, commensurate configuration transforms into an incommensurate one, so the tensile strength of wet-commensurate interface becomes a linear function of brick length. Besides, as elucidated by our theoretical model discussed above, the tensile strength of incommensurate configurations (incommensurate and wet interfaces) is far less than that of commensurate one whenL< 100 nm, while their tensile strength approaches or even exceeds that of commensurate configuration whenL> 300 nm. Contrasting dry and wet interfaces, we find that water molecules can slightly enhance the tensile strength of incommensurate interface for a wide range of flake lengths. For commensurate configuration, the tensile strength of wet interface is lower than that of dry one asL< 300 nm but higher than that asL> 300 nm. This result implies that the loading transfer of commensurate interface is efficient at short overlap length, while incommensurate interface possesses high efficiency at long overlap length, agreeing to the results obtained by the modified Frenkel–Kontorova model in our recent work [23] .

To illustrate the efficiency of loading transfer under different interfacial configurations, we define the slope of linear fitting in Fig. 4 a as the loading transfer capability. Combining Eq. (3) , we can acquire the effective shear strength of different interfacial configurations. The value of commensurate interface is attained by linear fitting the data ofL≥300 nm. As indicated in Fig. 4 b, the effective shear strength of commensurate interface (0.116 GPa) is smaller than that of incommensurate one (0.128 GPa). A certain amount of water molecules can reinforce the effective shear strength of both the incommensurate (0.14 GPa) and commensurate interfaces(0.137 GPa). Additionally, both dry and wet incommensurate configurations possess higher effective shear strength than commensurate one due to the fact that the incommensurate interface is rougher than commensurate one, which endows incommensurate interface with high loading transfer capability and energy dissipation under long overlap length.

Fig. 4. a The tensile strength of RVE as a function of GO flake length under varied interfacial configurations. The dashed lines are obtained from linear fitting. b The effective shear strength of different interfacial configurations.

From the example of GO layered nanocomposites, we analyze their tensile strength varied with interfacial configurations, shearing properties, and overlap length, simply expressed asσm=σm(φ,τ,Γ1,L,hb). For staggered structure, however, there are generally two failure modes that govern the mechanical properties, i.e.,the failure of noncovalent interface when the brick is totally pulled out (denoted as mode I) and the brick fracture when the maximum stress in brick is beyond its strengthσcr(denoted as mode F) [13] .When the mechanical performances are dominated by the interfacial failure, the maximum tensile stress appears before global sliding occurs, while the maximum toughness is usually obtained as the overlap length is reduced to zero. Therefore, the full expression of tensile strengthσecan be described as

In summary, by including the interfacial configuration at the atomic scale, we introduce a multiscale shear-lag model to correlate the macroscale mechanical responses of GO layered nanocomposites with the molecular mechanisms of noncovalent interface.The noncovalent interfaces can be classified by their shear responses into two types, i.e., commensurate interface for periodic shear stress-displacement curve and incommensurate interface for the linear-sliding curve. The shear-lag analysis reveals that the shear stress distribution of incommensurate interface is uniform,while that of commensurate one exhibits a pronounced size effect due to the periodicity of shear response. These distinguished deformation behaviors lead to unusual mechanical responses of the staggered structure, where that of commensurate interface shows send peak and strain hardening with the increase of overlap length due to the nucleation and propagation of interfacial defects. Then,our predictions are further validated through large-scale MD simulations for GO layered nanocomposites. Counterintuitively, at long overlap lengths, the incommensurate interface can reach extremely high strength and even exceed that of commensurate one because interfacial defects in commensurate interface extremely reduce the effective length of loading transfer. Furthermore, when the interface is intercalated by water molecules, the mechanical response of commensurate configuration transforms into the incommensurate one, which improves the loading transfer capability for long overlap length. Our results reveal that layered nanocomposites can simultaneously achieve ultrahigh strength and toughness through proper configuration design of weak noncovalent interfaces.

Our findings are general for other types of nanomaterials dominated by noncovalent interfaces, such as nanocellulose, aramid nanofiber, and collagen nanofiber, to name a few. The application of our results to the design of hierarchical nanocomposites can present us with new strategies in balancing mechanical properties and toughness mechanisms, leading toward a novel class of functional nanocomposites that are both strong and tough.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

AcknowledgmentsThis work was jointly supported by the National Natural Science Foundation of China (Nos. 11872063 and 12172346), the University of Science and Technology of China (USTC) Research Funds of the Double First-Class Initiative (No. YD2480 0 020 02), and China Postdoctoral Science Foundation (No. 2021TQ0323). The numerical calculations have been done on the supercomputing system in the Supercomputing Center of University of Science and Technology of China.