A review on mechanics and mechanical properties of 2D materials—Graphene and beyond
Introduction
The isolation of monolayer graphene flakes by mechanical exfoliation of bulk graphite opened the field of two-dimensional (2D) materials [1]. Since then, many other 2D materials have been discovered, such as transition metal-dichalcogenides (TMDs, e.g., MoS2), hexagonal boron-nitride (h-BN), and black phosphorous or phosphorene. The family of 2D materials offers a full spectrum of physical properties, from conducting graphene to semiconducting MoS2 and to insulating h-BN. Moreover, the 2D crystal structures render a unique combination of mechanical properties, with high in-plane stiffness and strength but extremely low flexural rigidity. Together, the 2D materials are promising for a wide range of applications [2], [3].
Here we review recent theoretical and experimental studies related to mechanics and mechanical properties of 2D materials. We emphasize how mechanics is indispensable in the study of mechanical properties including interfacial properties and the coupling between the mechanical and other physical properties. The review is divided into five self-contained sections. As the most basic mechanical properties, elastic properties of 2D materials are discussed in Section 2. Experimental methods to measure the in-plane elastic properties of graphene have been developed [4] and extended to other 2D materials [5], [6]. A recent experiment reported surprising results for the in-plane stiffness of graphene [7], opening a question on the effects of defects and statistical rippling. Direct measurement of the elastic bending modulus is more challenging for 2D materials [8]. Here again, a recent experiment [9] reported orders of magnitude higher values than theoretical predictions, raising a question on the fundamental mechanics of bending an ultrathin membrane with the effect of thermal fluctuations [10]. Theoretically, density functional theory based first-principles calculations have been used to predict linear and nonlinear elastic properties of graphene and other 2D materials. On the other hand, the accuracy of empirical potentials for molecular dynamics (MD) simulations remains to be improved. The effects of thermal rippling on the elastic properties of graphene are discussed based on MD simulations and statistical mechanics of elastic membranes [11].
Section 3 focuses on inelastic properties of 2D materials, starting with a description of defects such as vacancies, dislocations, and grain boundaries [12]. The strength and toughness are then discussed. The strength of a pristine 2D material is usually high [4], [6], but it could vary substantially due to the presence of topological defects and out-of-plane deformation [13], [14]. The fracture toughness of graphene obtained from experiments is relatively low [15], for which potential toughening mechanisms have been explored [16]. Fundamental questions have also been raised on the appropriate definition of fracture toughness for 2D materials based on fracture mechanics.
Section 4 deals with the coupling between mechanical deformation and other physical properties of 2D materials. Recent theoretical and experimental work has shown unprecedented effects of strain on many physical properties of graphene and other 2D materials [17], making “strain engineering” a viable approach for a wide range of potential applications involving 2D materials. As a sampling of the vast literature on this subject, we focus here on pseudomagnetic fields (PMFs) in deformed graphene [18], [19], phase transitions of TMDs under different mechanical constraints [20], [21], phonon and electronic structures of TMDs under hydrostatic pressure and strain [22], [23], and piezo- and flexoelectricity of 2D materials that couple strains and strain gradients (curvature) to polarization [24], [25].
Section 5 is devoted to interfacial properties of 2D materials. Adhesion and friction experiments have been developed to measure the mechanical interactions between graphene and other materials as its substrate or probing tips. In addition to the measurement of adhesion energy [26], more detailed measurements and analysis revealed the strength and range of the interactions in form of traction–separation relations [27], which provided further insight into the underlying mechanisms of the mechanical interactions. While van der Waals interactions have been commonly assumed to be the primary mechanism, experimental evidence suggests that other mechanisms may also have to be considered, such as the effects of water capillary, reactive defects, and surface roughness. Theoretically, it is also possible to unify the adhesion and friction properties of the interface within the framework of mixed-mode, nonlinear fracture mechanics.
In Section 6, a brief account of potential applications related to the mechanics and mechanical properties of 2D materials is presented, including synthesis and transfer for large-scale manufacturing, graphene origami and kirigami, flexible electronics and biomedical applications.
In the final section of this review, we provide an outlook for further studies related to mechanics and mechanical properties of 2D materials including and beyond graphene.
Section snippets
Elastic properties
Like thin membranes, 2D materials may be deformed by in-plane stretching or by bending out-of-plane. As a result, the elastic properties of 2D materials include both in-plane and bending moduli. With combined stretching and bending, a set of coupling moduli may also be defined theoretically [28], as noted for graphene being rolled into carbon nanotubes [29]. This section reviews recent experiments for measuring the elastic properties of 2D materials as well as theoretical predictions from first
Inelastic properties: Defects, strength and toughness
Mechanical properties of 2D materials beyond elasticity depend sensitively on the presence of defects (e.g., vacancies, dislocations, grain boundaries, and crack-like flaws) and their evolution during deformation. Strength and toughness are two distinct mechanical properties describing the onset of failure in terms of stress and energy, respectively. While the toughness is defined as the energy per unit volume of the material with a unit of J/m3 in elementary mechanics, it is defined more
Electromechanics of graphene and strain engineering
Many of the properties of graphene are strongly tied to its lattice structure. The large elastic deformability of graphene (e.g., 20%) allows for substantial change of the graphene lattice structure, therefore opening up fertile opportunities to tailor the electronic properties (e.g., charge carrier dynamics) of graphene through mechanical strain. For example, despite its many exceptional physical properties, graphene suffers from one key drawback in potential usage in electronics — it is
Interfacial properties: Adhesion and friction
Graphene and other 2D materials have the highest surface to volume ratios of any class of materials. As a result, surface forces are expected to play a significant role as these materials are being integrated into microelectronics, MEMS and NEMS devices and composite materials. Surface forces are also important when the 2D materials have to be transferred from one substrate to another via selective delamination and adhesion. This section presents a brief review of recent developments in
Applications
This section briefly reviews a few applications of 2D materials where mechanics and mechanical properties play important roles, including synthesis and transfer, graphene origami and kirigami, flexible and biomedical applications.
Summary and outlook
The family of 2D materials has grown beyond graphene, and together they hold great promise for a wide range of applications. The mechanics and mechanical properties of 2D materials play important roles in many applications including large-scale manufacturing and integration. Fundamental research on the mechanics and mechanical properties of 2D materials has made significant progress over the last decade, both theoretically and experimentally. As an outlook for future studies, a few topics of
Acknowledgments
This review results from discussions at the AmeriMech Symposium on Mechanical Behavior of 2D Materials–Graphene and Beyond, which was held at the University of Texas at Austin on April 4–6, 2016. We gratefully acknowledge financial support of the symposium by the National Science Foundation through Grant No. 1625862 and the National Academy of Sciences through the US National Committee of Theoretical and Applied Mechanics (USNCTAM) with Grant No. 2000006243.
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