Formation and Mechanical Behavior of Nanocomposite

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Functional Nanostructured Materials (including low-D carbon)

Formation and Mechanical Behavior of Nanocomposite Superstructures from Interlayer Bonding in Twisted Bilayer Graphene Mengxi Chen, Andre Rodrigues Muniz, and Dimitrios Maroudas ACS Appl. Mater. Interfaces, Just Accepted Manuscript • DOI: 10.1021/acsami.8b09741 • Publication Date (Web): 08 Aug 2018 Downloaded from http://pubs.acs.org on August 8, 2018

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ACS Applied Materials & Interfaces

Formation and Mechanical Behavior of Nanocomposite Superstructures from Interlayer Bonding in Twisted Bilayer Graphene Mengxi Chen,(1) Andre R. Muniz,(2) and Dimitrios Maroudas(1),* (1)

Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 01003, United States

(2)

Department of Chemical Engineering, Federal University of Rio Grande do Sul, Porto Alegre, Brazil

KEYWORDS. Graphene; twisted bilayer graphene; nanodiamond superstructures; mechanical properties; molecular dynamics.

ABSTRACT

We report a comprehensive study on the design of two-dimensional graphene-diamond nanocomposite superstructures formed through interlayer covalent bonding of twisted bilayer graphene with commensurate bilayers.

The interlayer bonding is induced by patterned

hydrogenation that leads to formation of superlattices of 2D nanodiamond domains embedded 1 ACS Paragon Plus Environment

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between the two graphene layers. We generalize a rigorous algorithm for the formation of all possible classes of these superstructures: the structural parameters employed to design such carbon nanocomposites include the commensurate bilayer’s twist angle, the stacking type of the nanodomains where the interlayer bonds are formed, the interlayer bond pattern, and the interlayer C-C bond density that is proportional to the concentration of sp3-hybridized interlayerbonded C atoms.

We also analyze systematically the mechanical behavior of these

nanocomposite superstructures based on molecular-dynamics simulations of uniaxial tensile straining tests according to a reliable interatomic bond-order potential. We identify a range of structural parameters over which the fracture of these superstructures is ductile, mediated by void formation, growth, and coalescence, contrary to the typical brittle fracture of graphene. We introduce a ductility metric as an order parameter for the brittle-to-ductile transition, demonstrate its direct dependence on the fraction of sp3-hybridized interlayer-bonded C atoms, and show that increasing the fraction of interlayer-bonded C atoms beyond a critical level in certain classes of these superstructures induces their ductile mechanical response.

1. Introduction Graphene and graphene-based nanomaterials have great potential toward enabling nextgeneration technologies and have attracted significant attention over the past several years, due to their intriguing set of electronic, thermal, and mechanical properties.1–5 For example, graphene flakes have been used as “filler” material toward reinforcement of mechanical properties in polymer-matrix nanocomposites.6–8 Structural modification of single-layer and few-layer graphene, including modification through chemical functionalization,4,9–11 laser treatment,3,12 and ion/electron

irradiation,13,14

has

been

studied

extensively

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both

experimentally

and

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theoretically.8,15–18 Such structural modification allows for fine tuning of physical properties, through the controlled insertion of defects, impurities, and passivating agents in the graphene lattice structure, including tilt grain boundaries in graphene sheets,19 sp3 C-C bonds in hydrogenated or fluorinated graphene,4,11,20 pores in graphene nanomeshes,21,22 monovacancies in single-layer graphene,23 flaws in nanocrystalline graphene,24 as well as vacancies and voids in irradiated amorphous graphene.14,25,26 In few-layer graphene, structural modifications also can be achieved by inducing interlayer covalent bonding between atoms from adjacent layers in the material, through a sp2-sp3 C-C bonding transition facilitated by selective surface functionalization. In previous studies,27–29 we have introduced a class of carbon-based nanomaterials achieved by the insertion of such bonds in twisted bilayer graphene (TBG), namely, formation of interlayer-bonded twisted bilayer graphene (IB-TBG). The structure of these materials is characterized by superlattices of diamond-like nanocrystals embedded within the TBG layers and having the symmetry and periodicity of the underlying Moiré pattern of the TBG. In other words, these graphene bilayer materials constitute ordered nanocomposite superstructures, with the nanodiamond domains playing the role of “filler” in the graphene “matrix”. Through first-principles density functional theory (DFT) calculations, we have demonstrated the stability of such configurations and established that their electronic, mechanical, and thermal properties can be controlled by varying the density and spatial distribution of interlayer bonds. For example, the electronic behavior of these materials range from semi-metallic to semiconducting27,29,30 with band gaps ranging from a few meV to ~1.2 eV, and their thermal expansion coefficient changes from negative to positive by increasing the concentration of interlayer bonds.31 Several experimental studies have demonstrated the feasibility of interlayer bonding in few-layer graphene employing various 3 ACS Paragon Plus Environment


techniques, including femtosecond-laser excitation of graphite,12 ion/electron beam irradiation of few-layer graphene,13,32 and functionalization and high-pressure treatment of few-layer graphene.9,32–34 A fundamental analysis of the mechanical behavior of such IB-TBG superstructures was conducted in previous studies, which revealed the superior mechanical properties of these nanocomposite materials28,35 and indicated the possibility of ductile failure upon loading in addition to the typical brittle fracture of graphene. In spite of the above studies, a comprehensive and rigorous approach to designing IB-TBG superstructures with optimal properties and function, such as mechanical behavior and structural response under uniaxial straining, remains elusive. The purpose of this article is to propose a systematic approach toward formation of such IB-TBG superstructures, providing a general rule (algorithm) for creating such 2D materials over a broad range of twist angles, as well as to investigate the mechanical response exhibited by these materials under tensile straining. Achieving this goal will set the stage for establishing rigorous quantitative relationships between the structure of these materials, through well-defined structural parameters, and their mechanical behavior aiming at development of mechanically superior graphene-diamond nanocomposites. The rest of this article is structured as follows. The computational methods employed in this study are described in Section 2. The atomic structure of the IB-TBG nanocomposite superstructures is analyzed in Section 3.1, resulting in their classification into three superstructure types and generalizing the concept of IB-TBG superstructure formation. The structural characterization of these superstructures is carried out in Section 3.2, with emphasis placed on how the structural parameters that determine the superstructure type control the internal stress in the superstructures due to the presence of distorted interlayer bonds. The mechanical behavior of the superstructures in response to uniaxial tensile straining is analyzed 4 ACS Paragon Plus Environment

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and discussed in Section 3.3, leading to a criterion for determining which of these structures are ductile. Finally, the main conclusions of our study are summarized in Section 4.

2. Computational Methods The main computational effort in this study consisted of molecular-dynamics (MD) simulations of mechanical tests on the IB-TBG superstructures examined as described below. For the characterization of the superstructures in Section 3.2, fully relaxed configurations were employed that were generated also by MD simulation as described below. The formation of unrelaxed configurations of these superstructures is a mainly analytical geometrical task and is detailed in Section 3.1. The MD simulations of uniaxial tensile straining tests on

IB-TBG nanocomposite

superstructures were conducted using the LAMMPS software package.36 The interatomic interactions were described according to the adaptive interatomic reactive empirical bond-order (AIREBO) potential, widely used for carbon allotropes and nanostructures.19,23,24,37,38 To avoid spurious high forces at higher strains, the C-C cutoff distance in the potential was set to 2.0 Å, as done in various previous studies.28,35 Prior to tensile straining, the IB-TBG superstructures were fully relaxed and equilibrated at constant temperature (300 K) and zero pressure for sufficient time periods. The Berendsen thermostat and barostat39 were used for temperature and pressure control in all simulations. Subsequently, the relaxed superstructures were strained along either the armchair (y) or the zigzag (x) direction at 300 K with a constant applied engineering strain rate of 1×10-4 ps-1, and the classical equations of motion were integrated numerically using a velocity Verlet algorithm with a time step of 0.15 fs. We have confirmed that the reported mechanical properties are not affected (to within statistical error) by varying the applied strain 5 ACS Paragon Plus Environment


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rate over a two-orders-of-magnitude interval (strain rates lower/higher by a factor of 10 than the value reported above). Atomic-level stresses on the C atoms were calculated along the trajectories according to the virial theorem by averaging over a period of 150 fs. Periodic boundary conditions were applied in all directions and a vacuum layer of thickness equal to 40 Å was used in the z direction of the supercell (i.e., normal to the graphene planes) to avoid interactions between periodic images. The thickness of the IB-TBGs was assumed to be 6.8 Å to compute the atomic volume28,35, consistent with the thickness of 3.4 Å typically employed for single-layer graphene.11,28,35,38 From each MD simulation, we generated stress-strain curves and monitored the distribution of the atomic-level von Mises stress, τvM, i.e., the second scalar invariant of the atomic-level stress tensor,35 on the material prior to (i.e., during equilibration) and during the tensile straining test. The ultimate tensile strength, , fracture strain, , and Young’s modulus of each configuration tested were determined directly from the computed stress-strain curve, whereas the corresponding fracture toughness refers to the area under the stress-strain curve and was

calculated from the integral of the stress over the resulting strain interval, ≡ , which provides a metric of the energy dissipated during the fracture process. The mechanical response to uniaxial straining of all the IB-TBGs examined was analyzed as a function of the respective interlayer bond density. As a metric of this interlayer bond density, we used the fraction of sp3-hybridized interlayer-bonded C atoms (with respect to the total number of C atoms) in the superstructure, fsp3. The range of the fsp3 metric can be effectively controlled by the commensurate bilayer’s twist angle, the stacking type of the nanodomains where the interlayer bonds are formed, and the interlayer bond pattern that controls the maximum interlayer bond


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density through the maximum number of interlayer C-C bonds per unit cell, NIB,

max,

that

structurally stable IB-TBGs can reach.29,30 For each IB-TBG configuration examined, at least five independent MD simulations of uniaxial tensile straining were conducted and statistical analysis of the results was performed. The employed supercell sizes varied depending on the structural parameters of the IB-TBGs; in general, the supercell sizes on the xy-plane ranged from approximately 105 Å × 40 Å to 360 Å × 140 Å, with the corresponding number of total atoms ranging from 1440 to 15858.

3. Results and Discussion Here, we generalize the structure formation procedure that was introduced in our previous studies28–30,35to a rigorous algorithm for the generation of the IB-TBG superstructures. This comprehensive algorithm reveals that IB-TBG superstructures can be classified into three distinct classes (types) depending on their structural parameters. These superstructure types are then characterized focusing on their internal stress distribution that arises due to the interlayer CC bonding and correlates strongly with their mechanical response to uniaxial tensile straining. The analysis of the mechanical behavior of these IB-TBG superstructures culminates with the definition of a ductility metric for each superstructure, thus establishing a rigorous, quantitative relationship between their atomic structure and their response to mechanical loading, focusing here on uniaxial straining.

3.1. Structure Formation There have been several previous studies regarding the introduction of sp3-hybridized bonds into graphene and graphene-based structures.27,29,30,40–42 In the present study, we generated 7 ACS Paragon Plus Environment


interlayer-bonded bilayer graphene structures with an ordered pattern of diamond-like nanodomains. Starting with perfectly aligned bilayer graphene and specifying an axis of rotation normal to the graphene planes, we rotated the two graphene layers with respect to each other by a twist angle, θ. As a result of this rotation, two different types of subdomains with distinct stacking type were formed at different localized regions of a superlattice pattern with different well-known local alignments, namely, AA (or hexagonal) and AB (or Bernal) stacking arrangements. The bilayer is perfectly AA-stacked when θ = 0 and it is perfectly AB-stacked when θ = 60o according to the symmetry of graphene’s honeycomb lattice. When the twist angle

θ ranges over the interval 0° < < ~16° (equivalently ~44° < < 60°) the existence of local AA or AB stacking is ensured;15,43,44 such domain superlattices characterized by local AA- or AB-stacked domains have the same periodicity and symmetry with the underlying Moiré pattern characteristic of the rotational stacking of two graphene layers; the size of the stacked nanodomains in such bilayers decreases with increasing θ. Within these AA- or AB-stacked nanodomains, sp3-hybridized interlayer C-C bonds were then created by displacing pairs of relatively aligned C atoms from each of the two graphene layers toward each other. Within these nanodomains, every other C atom of each layer is paired with a C atom of the opposite layer and one interlayer C-C bond is formed between them, with the remaining neighboring C atoms that do not form interlayer bonds requiring chemical functionalization to passivate C dangling bonds and stabilize this bonding environment. In this study, hydrogenation is used to passivate the adjacent unpaired C atoms, thus forming the embedded

diamond-like nanodomain as depicted in Figure 1; we mention that other means of chemical functionalization, such as fluorination,29 also can be used to accomplish such interlayer C-C bonding in twisted bilayer graphene. The resulting configurations were then fully relaxed to 8 ACS Paragon Plus Environment

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ensure the structural stability of the superstructures before undergoing a uniaxial straining test. The embedded diamond-like sp3-bonded domains are termed as nanodiamond domains throughout this article. Figure 1 shows various configurations of these nanodiamond domain superstructures created in the present study. In each part of Figure 1, the first frame on the left shows the location in the commensurate graphene bilayer where nanodiamond domains were created; in Figures 1(a-b) these domains are formed within AA-stacked regions of the superlattice pattern of the rotational stacking, while in Figures 1(c-d) the domains are formed within AB-stacked regions.

The remaining three frames of each part of Figure 1 depict

nanodiamond domains with increasing size (more interlayer C-C bonds) from left to right. There are two types of such nanodiamonds that can be created depending on the exact location of the center of rotation in forming the twisted bilayer. In Figures 1(a) and 1(c), the center of rotation of the twisted bilayer is located at the geometrical center of a hexagonal cell of a perfectly aligned AA- or AB- stacked bilayer, respectively; as a result of such symmetry, the minimum number of interlayer bonds required to form the smallest closed-shell nanodiamond domain is equal to 3. Such nanodiamonds and their corresponding superstructures are termed as Type I. On the other hand, in Figures 1(b) and 1(d), the center of rotation of the twisted bilayer is located at a C atom of a perfectly aligned AA- or AB- stacked bilayer, respectively; in this case, the minimum number of interlayer bonds required to form the smallest closed-shell nanodiamond domain is equal to 1. Such nanodiamonds and their corresponding superstructures are termed as Type II. The algorithm to form type I and type II nanodiamonds starts from the minimum number of interlayer bonds in the unit cell of the superlattice and proceeds with the formation of sequences of fully closed shells of interlayer bonds that surround the minimum-size nanodiamond cluster; 9 ACS Paragon Plus Environment


we choose to proceed with sequences of closed shells of interlayer bonds in order to avoid imperfections at the nanodiamond domain boundary, which plays the role of a graphenenanodiamond interface in these superstructures, and have the nanodiamond domains grow in a radially outward fashion from the center of rotation. The resulting nanodiamond domain has truncated triangular and hexagonal shapes for type I and type II domains, respectively. The corresponding number of C-C interlayer bonds per nanodiamond domain, is expressed as = 3 for type I nanodiamonds and = 3 − 1 + 1 for type II nanodiamonds, where the positive integer n = 1, 2, 3, … is the number of closed shells of interlayer bonds involved in the nanodiamond domain formation. The maximum NIB that can be reached in this structure formation process, NIB,max, depends on the twist angle, θ. Larger twist angles lead to smaller AA- or AB-stacked domain sizes, since neighboring (in z) atoms from the two graphene layers in the commensurate bilayer become increasingly misaligned from the perfect alignment (AA or AB) with increasing distance from the center of rotation and more misaligned for larger twist angles. As a result of this misalignment and the ensuing distortion and straining of the formed interlayer C-C bonds, the structural stability of the formed nanodiamond superstructures is lost beyond a maximum value of n that determines NIB,max for each twist angle and nanodiamond type. It should be mentioned that a higher number of stable interlayer bonds can be created by forming type I nanodiamonds rather than forming type II nanodiamonds as a result of the nanodiamond type geometry detailed above. This is an important observation for the purpose of developing strategies for introducing large nanodiamonds in fabricating such superstructures, aiming at optimizing certain desired properties.


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Figure 1. Top views of atomic configurations of representative nanodiamond domain superstructures generated by interlayer C-C bonding in twisted bilayer graphene. sp2-bonded C atoms and the respective C-C bonds, interlayer-bonded C atoms, and H atoms are colored yellow, blue, and red, respectively. (a) AA-stacked type I embedded nanodiamond domain with truncated triangular shape, (b) AA-stacked type II embedded nanodiamond domain with hexagonal shape, (c) AB-stacked type I embedded nanodiamond domain with truncated triangular shape, and (d) AB-stacked type II embedded nanodiamond domain with hexagonal shape. In each case, the corresponding Moiré pattern of the commensurate twisted bilayer is shown on the left, with a magnified view of the region marked by the dashed black square shown on the next 3 panels of each row with increasing nanodiamond domain size (n = 1, 2, and 3, respectively) from left to right.



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Considering that these nanodiamond clusters can be created, i.e., embedded, within AA- or AB-stacked domains of the twisted bilayer, there are, in general, four possible configurations of interlayer-bonded twisted bilayer graphene (IB-TBG) that can be formed; such representative configurations are depicted in Figure 2. In Figures 2(a) and 2(b), nanodiamonds are created only within a single AA-stacked or a single AB-stacked domain of the unit cell of the superlattice, respectively. In Figure 2(c), nanodiamonds are created within both the AA- and AB-stacked regions of the unit cell of the superlattice. In Figure 2(d), nanodiamonds are formed within ABstacked regions only, but at both of the possible AB-stacked domains of the superlattice unit cell, i.e., both AB-stacked domains in the unit cell are interlayer-bonded.

Consistent with our

previous studies,30,35 the triplet , $%, is used in Figure 2 to denote each configuration, where the stacking type, st, denotes the local layer stacking in the embedded nanodiamond clusters, and can be one of four types (namely, AA, AB, AAAB, and ABAB).


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Figure 2. Atomic configurations of representative interlayer-bonded twisted bilayer graphene superstructures (IB-TBGs) investigated. Each frame shows different (top and side) views of IBTBGs. (a) (7.341o, AA, NIB = 27), (b) (6.009o, AB, NIB = 19), (c) (9.430o, AAAB, NIB = 10), and (d) (9.430o, ABAB, NIB = 14) nanodiamond domain superstructures. Yellow, blue, and red spheres denote sp2-hybridized C atoms, interlayer-bonded C atoms, and H atoms, respectively. In the interval 0° & < ~16°, there are countably infinite values of the twist angle, θ, given by the equation15,43 ,- . )/.

= '($ )* +,-. 0/. 1, 1 that lead to commensurate twisted bilayers for positive integer values of p and q. We examined IB-TBG configurations with several such twist angles, based on various combinations of p and q for nanodiamond clusters embedded in the generated IB-TBG configurations, which can be of type I or type II. Based on our superstructure formation algorithm with a twist angle determined by various combinations of p and q, we can identify the IB-TBG superstructures to be either pure 13 ACS Paragon Plus Environment


or hybrid configurations. As pure IB-TBG superstructures, we denote all the superstructures consisting of embedded nanodiamonds that are of either type I or type II, while IB-TBG superstructures characterized by coexistence of both type I and type II nanodiamonds we term as hybrid. In particular, pure type I and pure type II IB-TBG superstructures contain only type I and only type II nanodiamonds, respectively, arranged in a hexagonal pattern. On the contrary, hybrid IBTBG superstructures contain both type I and type II nanodiamonds. The classification of an IB-

TBG superstructure as pure or hybrid depends exclusively on the value of . Specifically, our structure formation algorithm in conjunction with Equation (1) generate pure-type IB-TBG superstructures if and only if 2 = 1 and 3 is odd. This observation is consistent with previous studies on commensurate twisted bilayers without interlayer bonding,15,43 which also reported super-periodicities for the underlying Moiré pattern, based on the supersymmetry of the resulting local stacked subdomains.

Various values of twist angle for creating pure type IB-TBG

superstructures, including θ = 13.174°, 9.430°, 7.341°, 6.009°, and 5.086°, have been observed experimentally in Moiré patterns in graphite and/or used in previous theoretical studies.28–30,35,44 Any combination of p and q other than p = 1 and q = odd that leads to pure-type IB-TBG superstructures, as detailed above, yield twist angles calculated through Equation (1) that generate hybrid IB-TBG superstructures. Figures 3(a-c) show representative pure and hybrid AA-stacked IB-TBG configurations. The superstructures in Figures 3(a) and 3(b) are pure type I and pure type II configurations, respectively, and magnifications of their unit cells also are shown: each unit cell contains a single type I or a single type II nanodiamond, respectively. Figure 3(c) shows a representative configuration of a hybrid IB-TBG superstructure and its unit cell, where the embedded nanodiamond in the center of the unit cell is of type I, while the nanodiamonds at the edge of the 14 ACS Paragon Plus Environment

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unit cell are of type II, with each unit cell of this hybrid superstructure containing one type I nanodiamond and three type II nanodiamonds. The number of carbon atoms per unit cell of these superstructures, denoted by NC, can be calculated analytically and is equal to the number of C atoms in the unit cell of the superlattice of the twisted bilayer without interlayer bonding, which is given by15,43 4 =

31 33 + 2 . 2 56

In Equation (2), 5 = 3/ :' 2, 3 and 6 = :' 2 + 33, 2 − 33 , where gcd denotes the greatest common divider; clearly, interlayer bonding in the twisted bilayers does not affect NC. In Figure 3(d), NC is plotted as a function of θ. The unit cell size decreases with increasing twist angle consistently with AA- and AB-stacked domain sizes in the unit cell. The interlayer-bonded configurations of hybrid type have larger unit cell sizes compared to those of the pure-type configurations as a result of supersymmetry breaking through the p and q choices that lead to hybrid superstructure formation; this is demonstrated in Figure 3 through the superstructure unit cells that contain only one nanodiamond in the pure-type cases of Figures 3(a) and 3(b), but four nanodiamonds in the hybrid-type case of Figure 3(c). The parameters in the triplet , $%, together with the designation of the pure/hybrid type of the superstructure are sufficient to comprehensively identify the atomic structure of any IB-TBG configuration.



Figure 3. Atomic configurations of representative AA-stacked only interlayer-bonded superstructures investigated. Frames (a), (b), and (c) show the top views of the interlayer-bonded bilayers and different (top and side) views of the corresponding unit cells. (a) (5.086o, AA, NIB = 27) with type I (truncated triangular), (b) (5.086o, AA, NIB = 19) with type II (hexagonal), and (c) (5.509o, AA, NIB = 65) with both type I (truncated triangular) and type II (hexagonal), i.e., hybrid, nanodiamond domain superstructures. Dashed black lines are used to mark the edges of the unit cells of the superstructures. Yellow, blue, and red spheres denote sp2-hybridized C atoms, interlayer-bonded C atoms, and H atoms, respectively. (d) Number of C atoms in each unit cell, NC, of the nanodiamond domain superstructures as a function of the twist angle θ for pure (blue open diamonds) and hybrid (red open triangles) type AA-stacked interlayer-bonded configurations.

3.2. Structural Characterization Prior to being subjected to uniaxial tensile straining, all IB-TBG superstructures were fully relaxed.

Figure 4 shows some representative relaxed atomic configurations, generated as

described in Section 3.1 prior to their relaxation, and corresponding to the configurations depicted in Figures 2 and 3. To characterize these relaxed configurations, we focused on the distribution of the internal or residual stress in these structures as a result of the atomic displacements due to the interlayer C-C bonding in the commensurate twisted bilayers. The ensuing out-of-plane C-C bond distortions induce internal strains and, thus, development of 16 ACS Paragon Plus Environment

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residual stress. This is highlighted in Figure 4 where the atoms are colored according to their atomic-level von Mises stress. The formation of the sp3-hybridized interlayer bonds requires displacements in the z-direction of C atoms that were not stressed with respect to their equilibrium state in the commensurate bilayer prior to interlayer bonding. The varying distortion of these C-C bonds in the nanodiamond domains due to the misalignment of the interlayerbonded C atoms, in addition to the atomic displacements perpendicular to the graphene planes, is responsible for the residual stress variation and the stress gradients within each superstructure’s unit cell. The nanodiamond domains can be easily visualized through the stress maps of Figure 4 with highly stressed atoms (colored red) marking the nanodiamond clusters’ boundaries. The C atoms at the center of these sp3-bonded nanodiamond domains are perfectly aligned and are characterized by low residual stresses. C atoms in twisted bilayer graphene located farther away from each AA- or AB-stacked subdomain center become gradually misaligned and, consequently, the interlayer bonds formed between these C atoms become increasingly more strained, thus increasing the residual stress and changing the color of these C atoms in Figure 4. The C atoms at the edge of these interlayer-bonded nanodiamond domains, which also are the atoms at the interfaces between sp2- and sp3-hybridized C atoms, i.e., the graphene-nanodiamond interfaces, always sustain the highest levels of residual stress. These residual stresses are higher for IB-TBG configurations with relatively larger nanodiamonds (higher fsp3), implying that these structures will be weaker under tensile straining.



Figure 4. Top views of relaxed atomic configurations of the IB-TBG superstructures shown in Figs. 2 and 3, (a) (7.341o, AA, NIB = 27) with type I nanodiamonds, (b) (6.009o, AB, NIB = 19) with type II nanodiamonds, (c) (9.430o, AAAB, NIB = 10), (d) (9.430o, ABAB, NIB = 14), (e) (5.086o, AA, NIB = 27) with type I nanodiamonds, (f) (5.086o, AA, NIB = 19) with type II nanodiamonds, and (g) (5.509o, AA, NIB = 65) with hybrid (of type I and type II) nanodiamonds, characterized according to their internal stress due to the strained C-C bonds present in each such superstructure. In each configuration, the C atoms are colored according to their atomic-level von Mises stress. In most structures, Figures 4(b-g), the most relaxed regions (colored darkest blue) are located in the sp2-bonded graphene matrix between nanodiamond domains, with very low or zero residual stress. For configurations with high interlayer bond density, such as that of Figure 4(a), the most relaxed sites are located at the centers of the nanodiamond domains. Such structural relaxation and residual stress distributions have important implications for the mechanical


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response of the corresponding superstructures to uniaxial straining, which is discussed in the next section.

3.3. Mechanical Response Under Uniaxial Tensile Strain To analyze the mechanical behavior of the nanocomposite superstructures that we generated, we subjected IB-TBG superstructures of the three types discussed in Sections 3.1 and 3.2 to uniaxial tensile straining tests, varying the interlayer bond density metric, fsp3, in all cases. From the MD simulations of these straining tests, we obtained stress-strain curves for each structure examined and calculated the corresponding mechanical properties in addition to monitoring their structural response. Figure 5 depicts representative stress-strain curves accompanied with atomic configurations of the respective superstructures near failure, with the atoms in each configuration again colored according to their atomic-level von Mises stress; the IB-TBG superstructures, whose response is shown in Figure 5, are the same superstructures that were characterized in Figure 4 undergoing straining along the zigzag (x) direction. Figures 5(b-g) demonstrate typical brittle fracture for the corresponding structures, with the crack in the inserted configurations propagating along the y direction, i.e., perpendicular to the straining direction, causing the brittle cleavage of the superstructure. The stress-strain curves of Figures 5(b-g) also are typical of such brittle fracture, with the stress increasing until crack initiation and then dropping abruptly to zero, indicating the complete failure of the respective structure. In all of these cases, Figures 5(b-g), crack initiation occurs at the edge of a nanodiamond domain, i.e., at sites where the highest residual stresses develop as discussed in Section 3.2 (Figure 4); in all such cases, subsequent to their initiation, cracks propagate at a very fast rate facilitating brittle fracture.



Figure 5. Stress-strain curves obtained by MD simulations of uniaxial tensile straining tests along the zigzag (x) direction for the superstructures characterized in Figs. 2, 3 and 4: (a) (7.341o, AA, NIB = 27) with type I nanodiamonds, (b) (6.009o, AB, NIB = 19) with type II nanodiamonds, (c) (9.430o, AAAB, NIB = 10), (d) (9.430o, ABAB, NIB = 14), (e) (5.086o, AA, NIB = 27) with type I nanodiamonds, (f) (5.086o, AA, NIB = 19) with type II nanodiamonds, and (g) (5.509o, AA, NIB = 65) with hybrid (of type I and type II) nanodiamonds. In all cases, the insets show atomic configurations near failure with the atoms colored according to their atomic-level von Mises stress. The IB-TBG superstructure in (a) undergoes ductile failure, while all the other superstructures undergo brittle failure. Contrary to the responses depicted in Figures 5(b-g), the superstructure with the higher interlayer bond density whose mechanical response to uniaxial tensile straining is shown in Figure 5(a), exhibited a completely different type of mechanical behavior and fracture, which is typical of ductile failure. Instead of brittle cleavage immediately following crack initiation, in the IB-TBG configuration of Figure 5(a), voids were formed in sequence and the failure of this 20 ACS Paragon Plus Environment

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material was the outcome of such void formation, growth, and coalescence as the straining progressed. This type of ductile fracture was originally reported in our prior study.35 Similar to the brittle fracture cases, crack initiation occurs at a site located on the nanodiamond domain boundary, where the interlayer C-C bonds are distorted and strained, but at a lower strain compared to those corresponding to crack initiation in the cases of Figures 5(b-g). Considering that the applied stress at this low strain is not sufficiently high to break ; bonds in the surrounding graphene matrix, instead of fast crack propagation in a certain direction and brittle cleavage, only the weaker sp3 bonds inside the nanodiamond domain break in this case, facilitating void formation. The existence of the C-H bonds in the non-interlayer-bonded C atoms of the nanodiamond domains does not restrict the rotation of C-C bonds in these IB-TBG configurations, thus enabling out-of-plane vibrational and rotational motion, which aids in the formation and growth of the voids. This role of H atoms in the fracture of graphene derivatives also was observed and discussed in previous studies of mechanical behavior of H-functionalized graphene38 and hydrogenated electron-irradiated graphene.45 In the stress-strain curve of Figure 5(a), each new void formation in this ductile fracture mechanism causes a local stress drop, which also indicates local stress relief and partial energy dissipation. Such void formation occurs sequentially through random crack initiation at sites of high stress concentration throughout the superstructure. To quantitatively characterize the mechanical behavior of the IB-TBG superstructures, we determined from the stress-strain curves their mechanical properties, including their ultimate tensile strength (equal to the maximum stress on the structure reached in the straining test), fracture strain, Young modulus, and fracture toughness. We have already established that the tensile strength, fracture strain, and Young modulus of the brittle IB-TBG superstructures 21 ACS Paragon Plus Environment


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decrease monotonically with increasing fsp3, since the insertion of sp3-hybridized C-C interlayer bonds

and

the

formation

of

graphene-nanodiamond

interfaces

weaken

the

whole

superstructure.28,35 Here, we place special emphasis on the dependence of fracture toughness on interlayer bond density (i.e., on fsp3) and on the brittle-to-ductile transition in these superstructures. In Figure 6(a), the computed fracture toughness, Kf as defined in Section 2, is plotted as a function of fsp3 for all the IB-TBG superstructures with AA stacking type examined in this study. It is evident from Figure 6(a) that the fracture toughness decreases with fsp3 until this interlayer-bonded C fraction reaches a value fsp3 ≈ 15%, and then it increases with increasing fsp3, indicating a typical ductile response and the onset of a brittle-to-ductile transition at the critical value fsp3,c ≈ 0.15. The above characterization refers to all the IB-TBG superstructures examined; however, it is clear from Figure 6(a) that fsp3,c is reached only in pure IB-TBG superstructures and is exceeded substantially only in such pure superstructures of type I. In addition, it should be mentioned that although the ultimate tensile strength of the superstructures decreases with increasing fsp3, the lowest strength computed was 8.8 GPa. Furthermore, the lowest fracture toughness in such superstructures computed in this study was 0.71 GPa. These computed mechanical properties indicate that the IB-TBG superstructures that we generated remain remarkably strong and tough even at very high concentrations of sp3-hybridized interlayer-bonded C atoms.


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Figure 6. (a) Fracture toughness and (b) ductility metric, Φ, as a function of the fraction of sp3hybridized interlayer-bonded C atoms, fsp3, for IB-TBG superstructures (AA-stacked only) from MD simulations of uniaxial tensile straining along the zigzag (x) direction. Blue open circles, red open diamonds, and green open squares denote pure type I, pure type II, and hybrid IB-TBG superstructures, respectively. Each data point shown is obtained by averaging over five independent MD simulations. In (a), the dashed lines represent optimal polynomial fits of the computed fracture toughness as a function of fsp3. In (b), the inset shows the ductility metric, Φ, as a function of fsp3, for IB-TBG superstructures with the same twist angle, θ = 5.086o, and increasing nanodiamond size (increasing n). The black horizontal dot-dashed line in (b) is used as a marker for the onset of ductile failure at Φ = ½. To formulate a rigorous, quantitative criterion for the onset of the brittle-to-ductile transition in the IB-TBG superstructures under uniaxial tensile straining, we define a ductility metric, Φ, as Φ≡ 1−

= , 3 23 ACS Paragon Plus Environment


where is the fracture toughness of the superstructure and = is the corresponding integral at

crack initiation, i.e., = ≡ > , where εi is the applied strain at crack initiation. For brittle fracture, εf = εi and Kf = Ki yielding Φ = 0, while for ductile fracture, such as in the case of the superstructure of Figure 4(a) with the mechanical response of Figure 5(a), = ≪ , implying according to Equation (3) that Φ approaches the value of 1; consequently, 0 & Φ & 1, and Φ constitutes an order parameter for analyzing the brittle-to-ductile transition. In the inset of Figure 6(b), we have plotted Φ as a function of fsp3 for IB-TBG superstructures with identical twist angle of = 5.086°. We see that with increasing fsp3 (through increasing the number of closed shells of interlayer-bonded carbon atoms, n, and altering nanocomposite superstructure type, from pure type I to pure type II), a brittle-to-ductile transition is observed to occur accompanied by the abrupt increase in Φ from ~ 0 at fsp3 ≈ 0.10 to ~ 0.8 at fsp3 ≈ 0.20. In Figure 6(b), the ductility metric Φ is plotted as a function of fsp3 for the IB-TBG superstructures examined with AA stacking type. At low values of fsp3, Φ is practically equal to zero (to within statistical fluctuations), with an abrupt increase in ductility Φ as fsp3 reaches values in the vicinity of 0.15, indicating the occurrence of a brittle-to-ductile transition consistent with the appearance of a minimum in the fracture toughness as a function of fsp3 in Figure 6(a). Given the abrupt change in Φ as fsp3 exceeds its critical value, we use Φ = ½ as the value of the ductility metric that marks the onset of the brittle-to-ductile transition; in the plot of Figure 6(b) this yields fsp3,c ≈ 0.15. Every data point in Figure 6 was obtained by averaging over at least five independent MD simulations of uniaxial straining and the resulting stress-strain curves and the error bars from the statistical analysis (corresponding to 95% confidence intervals) are displayed. The largest statistical error in the calculation of Φ for the IB-TBG superstructures is exhibited in 24 ACS Paragon Plus Environment

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the vicinity of the critical point for the brittle-to-ductile transition; the reason for this was that among the various MD runs we performed for these superstructures with fraction of interlayerbonded C atoms in the vicinity of fsp3,c, ductile response was not observed for every single MD simulation, with the same superstructure configuration exhibiting brittle response sometimes. Such uncertainty in mechanical response and large fluctuations in ductility Φ provided strong evidence of proximity to the critical point for the brittle-to-ductile transition in this class of carbon nanocomposite superstructures. It should be mentioned that for ductile IB-TBG superstructures (with the largest values of Φ computed), the statistical error in the calculation of the ductility metric Φ was very low (below 5%) although voids formed randomly within different nanodiamond domains and produced different stress-strain curves in every single test (among the various independent MD runs for the same structure, i.e., the same set of structural parameters). We also mention that, throughout our extensive parametric study, ductile response was not observed in any AB-stacked IB-TBG superstructures, but it was exclusively observed in superstructures with AA stacking type (st = AA).

In addition, more AA-stacked IB-TBG

superstructures of pure type I were found to exhibit ductile mechanical response than those of pure type II and no superstructures of hybrid type were found to be ductile. We conclude that pure type I AA-stacked IB-TBG superstructures are the most promising such nanocomposite superstructures regarding their ductility; this is an important finding in terms of designing mechanically superior graphene-based nanomaterials, given the brittle nature of pristine single-layer graphene. The different ductility of the different superstructure types can be understood through the structural stability of the various superstructures prior to uniaxial straining, namely, in that sufficiently large nanodiamond domains that substantially exceed the critical fsp3 value can be formed only in IB-TBG superstructures of pure type I. Finally, we 25 ACS Paragon Plus Environment


mention that such a brittle-to-ductile transition also has been observed to occur in other graphene derivatives and metamaterials. Specifically, uniaxial tensile straining tests on computer models of electron-irradiated single-layer graphene revealed a brittle-to-ductile transition, with this mechanical transition onset correlating with the onset of the irradiation-induced amorphization transition.26 Moreover, in MD simulations of uniaxial tensile straining of graphene nanomeshes with circular pores, the ductility of the nanomeshes was found to increase for porosities higher than ~15%.46

4. Conclusions In summary, we have investigated systematically and comprehensively the formation of all possible classes of interlayer-bonded twisted bilayer graphene (IB-TBG) superstructures, characterized by the presence of ordered arrays of nanodiamond domains consisting of sp3hybridized interlayer-bonded C atoms. We characterized these embedded nanodiamond clusters and the resulting IB-TBG superstructures emphasizing on the residual stress distribution in such structures as a function of their structural parameters, namely, the twist angle of the commensurate graphene bilayers, the stacking type of the interlayer-bonded domains, and the size and type of the embedded nanodiamond clusters with the size expressed as the fraction of interlayer-bonded C atoms in the superstructure. Following the formation and characterization of the IB-TBG superstructures, we conducted MD simulations of uniaxial tensile straining tests that we subjected these superstructures to, and systematically probed a brittle-to-ductile transition that they undergo as the fraction of interlayerbonded C atoms in the superstructures increases. IB-TBGs with relatively low interlayer bond density were found to be brittle with an abrupt decrease in stress during their fracture by brittle 26 ACS Paragon Plus Environment

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cleavage. On the contrary, certain superstructure classes, namely, with pure type nanodiamond domains of AA stacking type, at higher-than-critical interlayer bond density were found to undergo ductile failure mediated by the nucleation, growth, and coalescence of voids forming in the nanodiamond domains of these superstructures. To characterize the ductility of the IB-TBG superstructures, we introduced a new ductility metric, Φ, based on the superstructures’ fracture toughness; for the superstructures that undergo ductile failure, this ductility metric was found to increase abruptly when the fraction of interlayer-bonded C atoms (fsp3) in the superstructure exceeds a critical value of approximately 15%. Although these ductile IB-TBG superstructures have substantially lower strength and toughness compared to those of pristine single-layer graphene, they are very strong and tough, and generally mechanically superior compared to most conventional engineering materials, with significantly improved deformability under tension. Our study establishes rigorous, quantitative structure-properties-function relationships in a very promising class of graphene nanocomposite superstructures and demonstrates how their structural characteristics can be tailored to control their mechanical response.

Although

experimental synthesis of such superstructures is well beyond the scope of this article, the findings of our study provide the fundamental knowledge and understanding of mechanical behavior required for designing experimental protocols to fabricate such nanocomposite superstructures with superior mechanical properties and functionality. In closing, we mention that the worst mechanical attribute of graphene is that it is intrinsically a brittle material. Therefore, development of materials engineering strategies which result in ductile graphene derivatives, metamaterials, or nanocomposites, as accomplished here with certain classes of IBTBG superstructures, is particularly significant in optimizing the function of these materials in technological applications. 27 ACS Paragon Plus Environment


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AUTHOR INFORMATION Corresponding Author * E-mail address: [email protected] Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS M.C. and D.M. acknowledge financial support by the Army Research Laboratory under Grant No. W911NF-10-2-0098, Subaward No. 14-215454-02, and the usage of the facilities of the Massachusetts Green High-Performance Computing Center (MGHPCC). A.R.M. acknowledges financial

support

by

CNPq/Brazil

through

Grant

No.

449824/2014-4

(MCTI/CNPQ/UNIVERSAL 14/2014) and computational support by LNCC/SDUMONT. REFERENCES (1)

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ABSTRACT GRAPHIC


Formation and Mechanical Behavior of Nanocomposite

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