Structural and mechanical properties of monolayer amorphous carbon and boron nitride

Amorphous materials exhibit various characteristics that are not featured by crystals and can sometimes be tuned by their degree of disorder (DOD). Here, we report results on the mechanical properties of monolayer amorphous carbon (MAC) and monolayer amorphous boron nitride (maBN) with different DOD. The pertinent structures are obtained by kinetic Monte Carlo (kMC) simulations using machine-learning potentials with density-functional theory-level accuracy. An intuitive order parameter, namely, the areal fraction $F_{\mathrm{x}}$ occupied by crystallites within the continuous random network, is proposed to describe the DOD. We find that $F_{\mathrm{x}}$ captures the essence of the DOD: Samples with the same $F_{\mathrm{x}}$ but different sizes and arrangements of crystallites, obtained using two distinct kMC procedures, have virtually identical radial distribution functions as well as bond-length and bond-angle distributions. Furthermore, by simulating the fracture process with molecular dynamics, we found that the mechanical responses of MAC and maBN before fracture are mainly determined by $F_{\mathrm{x}}$ and are insensitive to the sizes and specific arrangements and to some extent the numbers and area distributions of the crystallites. The behavior of cracks in the two materials is analyzed and found to mainly propagate in meandering paths in the continuous random network region and to be influenced by crystallites in distinct ways that toughen the material. The present results reveal the relation between structure and mechanical properties in amorphous monolayers and may provide a universal toughening strategy for two-dimensional materials.

Figure 1

Validation of MLP. (a),(d) Energy differences caused by Stone-Wales transformations or bond exchanges (only in BN systems) at randomly selected steps in the kMC algorithm. kMC-accepted Δ$E$'s are shown as red triangles while kMC-denied Δ$E$'s are shown as blue reverse triangles; (a) MAC and (d) maBN. (b),(e) Comparison of atomic forces in fractured (b) MAC and (e) maBN between DFT results and MLP predicted values. The inset is a schematic of fractured configuration, where red, blue, and black represent atoms at crack edges/tip, strings, and other regions, respectively. (c),(f) Phonon dispersion comparisons for (c) graphene and (f) h-BN, calculated using DFT (black solid), MLP (red dashed), and a state-of-the-art empirical potential (blue dashed).

Figure 2

Atomic structures of monolayer carbon and BN in kMC simulation. (a),(b) Atomic structures of monolayer carbon and BN from different kMC steps, respectively. The hexagons in crystallite regions are colored in green (canonical hexagons) and blue (noncanonical hexagons). The percentage of canonical hexagons (only the green region) and corresponding DOD order parameter $F_{\mathrm{x}}$ are listed. (c),(d) RDF of monolayer carbon and BN with five different $F_{\mathrm{x}}$, respectively.

Figure 3

Relation of $F_{\mathrm{x}}$ to different manifestations of the DOD. (a),(b) Atomic structures of four different MAC and maBN samples with identical $F_{\mathrm{x}}=0.50$ but generated from different kMC procedures (one initiates from h-BN, the other three from different random configurations). (c) RDFs of three MAC (upper panel) and maBN (lower panel) samples in (a) and (b). (d) Bond-length and (e) bond-angle distributions of the three MAC (upper panel) and three maBN (lower panel) samples in (a) and (b).

Figure 4

Mechanical response of MAC and maBN. (a) Strain-stress curves of MAC and with three different $F_{\mathrm{x}}$ comparing with the precrack graphene along a zigzag direction. The blue star represents a snapshot containing two arrested cracks. (b) Stress-strain curves of maBN with different $F_{\mathrm{x}}$ compared with the precrack monolayer h-BN along a zigzag direction. (c) Schematic of stretching simulations. (d),(e) Strain-stress curves for three groups of (d) MAC and (e) maBN samples, respectively. Samples in each group have similar $F_{\mathrm{x}}$ values, but are generated from different procedures: one initiates from crystal, and three initiate from different random configurations, which are labeled by Gr/hBN and R1–R3, respectively, and have completely different atomic structures. Samples labeled by R1 in the legend correspond to the sample in (a) for MAC and (b) for maBN.

Figure 5

Crack propagation. (a) Crack path in crystallite-maBN. The main crack interacts with crystallites in three main ways: (b),(e) crystallites stop the propagation of the main crack; (c),(f) the main crack is deflected by crystallites; and (d),(g) while the main crack is stopped by crystallite, another crack initiates near the crystallite and a bridge is formed between the two cracks.