Moiré superlattice effects on interfacial mechanical behavior: A concise review

The moiré superlattice, arising from the interface of mismatched single crystals, intricately regulates the physical and mechanical properties of materials, giving rise to phenomena such as superconductivity and superlubricity. This study delves into the profound impact of moiré superlattices on the interfacial mechanical behavior of van der Waals (vdW) layered materials, with a particular focus on tribological properties. A comprehensive review of continuum modeling approaches for vdW layered materials is presented, accentuating the incorporation of moiré superlattice effects in theoretical models to unravel their distinctive interfacial frictional behavior and thermodynamic properties. The exploration of moiré superlattices has significantly advanced our fundamental understanding of interface phenomena in vdW layered materials. This progress provides crucial theoretical insights that can inform the design of multifunctional devices based on the unique properties of twisted layered materials.


| INTRODUCTION
Since the successful isolation of graphene via mechanical exfoliation in 2004, [1] two-dimensional (2D) van der Waals (vdW) layered materials have attracted significant interest.A plethora of 2D vdW layered materials, such as hexagonal boron nitride (h-BN), [2] molybdenum disulfide (MoS 2 ), [3] and phosphorene, [4] have emerged in recent years.One of the primary characteristics of vdW layered materials is their extreme anisotropy: atoms form strong covalent bonds within the layers, while the vdW interaction between layers is comparatively weak. [5,6]The exceptional characteristics of 2D vdW layered materials position them as popular candidates for addressing energy and environmental challenges, as well as for advanced devices.27] It is noteworthy that although the building blocks of vdW layered materials, that is, monolayer materials, inherently possess excellent mechanical, [28,29] thermal, [30,31] and electrical properties, [1,[32][33][34][35] there are limitations in finetuning their physical properties owing to the inherent atomic structure of monolayer systems. [36,37]Interestingly, the assembly of vdW stacking structures has proven to be an effective method for substantially adjusting their material properties. [38]These stacking structures introduce a competitive mechanism between interlayer potential energy and elastic deformation energy.[46] Specifically, weak interlayer vdW binding enables interlayer sliding and twisting, providing a distinctive method to modulate the electrical, mechanical, and tribological properties of 2D vdW layered materials.
Illustrative results of interlayer twisting are depicted in Figure 1.When a bilayer graphene is twisted from its commensurate configuration (Figure 1A) with an angle θ, the lattice mismatch between the atomic layers induces the creation of well-defined moiré superlattices (Figure 1B). [64]For a general vdW heterostructure, the unit cell size ( a | | m ) of the moiré superlattices [65,66] (red hexagon in Figure 1F [63] ) can be described by [67,68] a δ a where a is the lattice constant of the top layer material, δ a a = / − 1 sub is the lattice misfit between interfacing layers (a sub is the lattice constant of the substrate), and θ is the twist angle between the top layer and the substrate.The angle between a m and the zigzag direction of the substrate lattice reads as F I G U R E 1 Interlayer twisting induced moiré superlattice in van der Waals (vdW) layered materials.(A, B) The commensurate and incommensurate stacking states in bilayer graphene.l 1 and l 2 are the armchair directions of the substrate and the top layer graphene, respectively.The angle between the two vectors gives the twist angle θ. a m is one of the orientations of the moiré superlattice, and the angle between a m and l 1 (or l 2 ) is θ/2.(C) The twist-angle dependent period of twisted bilayer graphene (tBLG) moiré superlattice.(D-F) Scanning tunneling microscopy (STM) images of tBLG with different moiré period (D) 2.4, (E) 6.0, and (F) 11.5 nm; the hexagonal shape marked by the red dashed line in (F) represents a moiré superlattice.The scale bars in (D-F) are 5 nm.Reproduced with permission. [63]Copyright 2012, Springer Nature.
For twisted bilayer graphene, δ = 0, the period of the superlattice a | | m and the angle ψ degenerate to a θ 2 sin( / 2) gr and + θ π 2 2 , respectively.Typical twist angle-dependent period of the superlattice in bilayer graphene is shown in Figure 1C.
The moiré superlattices formed between 2D vdW layered materials significantly influence their deformation modes.In the case of two vertically stacked graphene or h-BN monolayers with a small twist angle, spontaneous atomic reconstruction occurs, driven by the interplay between interlayer vdW interaction and intralayer elasticity (Figure 2A). [41,76]This process results in the rearrangement of atoms, forming domains with locally commensurate stacking and strained soliton boundaries (Figure 2B). [70,76]Upon relaxation, the AB regions undergo a notable transformation from circular (Figure 2K) to quasi-triangular shapes (Figure 2L).Conversely, at large twist angles, the shapes of these regions remain essentially unchanged (Figure 2M,N). [74,77]In some cases, the small twist angle can even result in the formation of a helical structure with broken symmetry (Figure 2E-G). [70,72,75,78]imultaneously, the in-plane atomic reconstruction significantly affects the distribution of the out-of-plane deformation field (Figure 2H-J). [73]It is evident that at twist angles larger than ~1.1°, the out-of-plane deformation can be approximated by a trigonometric function (denoted by the black dashed line in Figure 2J).As the twist angle further decreases, intensified in-plane reconstruction increases the proportion of the AB stacking regions, making the deformation field no longer analytically approximate (e.g., 0.6°in Figure 2J). [73]tomic reconstruction has been identified as a significant factor influencing the interfacial mechanical behavior of vdW layered materials.A compelling study has elucidated that atomic reconstruction induces moiré distortion during the interlayer sliding of graphene/h-BN heterostructures, resulting in a moiré-level stick-slip behavior that aligns well with experimental observations. [46]In these heterostructures, moiré-induced atomic reconstruction at the heterogeneous interface profoundly modulates their mechanical behavior.Notably, the penetration length of moiré-induced atomic reconstruction undergoes a distinct reduction with increasing twist angle, indicating the twist angle-dependence of the atomic reconstruction effect. [79]Furthermore, the distinctive patterns stemming from atomic reconstruction can bring about significant alterations in the Young's modulus and the Poisson's ratio of the graphene/h-BN heterostructure, as demonstrated in a recent study. [43]Importantly, this effect extends beyond graphene-based systems to encompass transition metal dichalcogenide heterostructures, and only aligned heterostructures with small twist angles (≤5°) exhibit high efficiency in interlayer strain transfer, as detailed in a recent publication. [80]I G U R E 2 Moiré superlattice dependent deformation field.(A) Schematic of the local atomic reconstruction in two-dimensional (2D) materials. [69](B-G) The in-plane deformation field.[71] Broken-symmetry structure (E) such as the bending model of tBLG with a twist angle of 0.83°(F) and the MoSe 2 /WSe 2 heterostructure with a twist angle of 0.3°(G). [70,72](H-J) Scanning tunneling microscopy characterizations of moiré structure. [73]The scale bars in (C, D, F, G) are 200, 20, 5, and 10 nm, respectively.(K-N) The interlayer potential energy of tBLG system with a twist angle of 1.12°(K, L) and 5.08°(M, N).And (K, M) are calculated based on the rigid system, (L, N) are the corresponding interlayer potential energy based on their optimized configurations.The color bar represents the range of −31.5 to −17.3 meV/atom.(A, D) Reproduced with permission. [69]Copyright 2021, Springer Nature.(B, E, G) Reproduced under the terms of the CC-BY Creative Commons Attribution 4.0 International License. [74]Copyright 2023, American Chemical Society.(C) Reproduced with permission. [71]Copyright 2021, Springer Nature.(F) Reproduced with permission. [75]Copyright 2016, American Chemical Society.(H-J) Reproduced under the terms of the CC-BY-NC Creative Commons Attribution NonCommercial License 4.0. [73]Copyright 2020, The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science.
It is evident that, for a thorough theoretical exploration of the mechanical properties of large-scale twisted vdW layered materials, incorporating the moiré superlattice effect into continuum mechanical models is imperative.The objective of this review is to offer a comprehensive and in-depth examination of the moiré superlattice effect on the mechanical and tribological properties of vdW layered materials.In Section 2, we delve into the impact of the moiré superlattice on the tribological properties of layered materials, showcasing cases of atomic force microscope (AFM) experimental results and related models.Additionally, a novel friction mechanism arising from the compensation of the moiré superlattice at the edges of a slider is introduced.Section 3 provides a review of the development of theoretical models that consider the elastic effects of the moiré superlattice through thermodynamic approaches.Finally, in Section 4, a concise summary is provided, accompanied by valuable insights into future prospects.

| Moiré-dependent mechanical and tribological properties of vdW layered materials
Taking hexagonal lattice type layered materials as an example, when there is interlayer twisting or lattice mismatch between adjacent layers (Figure 3A,B), hexagonal moiré superlattices will form at the interface, with their unit cell shown as the white hexagonal in Figure 3A,B. [81]It is observed that the deformation field is closely related to the moiré superlattice (Figure 3A,B), [7,16,26,54] and the morphology of the unit cell includes six AA-stacking corners (blue dots in Figure 3B) surrounding a nearly circular AB-stacking region (the black circular denoted region in Figure 3B).The connecting parts between the AA-stacking regions are termed domain walls (white dashed lines between blue dots in Figure 3B). [41,48,57,61,82,83]It has been reported that the measured Young's modulus from nanoindentation experiments in AB regions of graphene/h-BN is significantly smaller than that measured in AA regions (Figure 3C,D). [81,84]This observation is predicted because the AA-stacking regions exhibit local protrusions, which lead to increased interlayer spacings.During the nanoindentation process, these protrusions will generate in-plane deformation in the graphene layer.As a result, the ultra-high in-plane Young's modulus of graphene assist in resisting the indentation deformation, resulting in a higher measured Young's modulus in the AA regions of graphene/h-BN heterostructure (Figure 3C,D). [81,84]These microscopic mechanisms are evidenced by the observations in the nanoindentation experiments (Figure 3A-D). [81]Obviously, the vertical undulations of the system will be notably influenced by the configuration of the superlattice, leading to a pronounced sensitivity to its mechanical properties based on parity. [43]On the other hand, changes in the moiré superlattice also significantly affect the in-plane mechanical properties, as seen in the transition of the moiré superlattice from hexagonal (Figure 3E) to flamingo-shaped (Figure 3F) as the number of layers increases.Consequently, the inplane Young's modulus and Poisson's ratio exhibit a parity dependence on the number of layers (Figure 3G,H). [43]emarkably, when scanning an AFM tip on a surface with moiré superlattices, a dual-scale stick-slip behavior has been observed from the molecular dynamics (MD) simulations employing state-of-the-art force fields.[46] The small stick-slip period arises from the interaction between the AFM tip and atomic lattices, while the larger modulation (i.e., the moirélevel stick-slip) is attributed to the effect of moiré superlattice (Figure 3I-O), [44][45][46] originating from the strong coupling between in-plane deformation and out-of-plane distortion of the moiré superlattice.As the interlayer twist angle increases, the periodicity of the long-range modulation decreases.[46] A deformation coupled (DC) PT model has been proposed to describe the moiré-level stick-slip behavior, [46,85,86] x U x q q t q x y U x q q t q y tip ( , , , ) tip q ( , , , ) q q q ( , , , ) q q x y x y where m tip and m q are the mass of the AFM tip and the graphene substrate in the contact region, respectively, γ x and γ q are their corresponding damping coefficients.The total potential energy of the model system reads ( ) By introducing the moiré superlattice-induced potential energy surface U moiréa nd the tip-induced out-of-plane deformation into the DC-PT model, and solving the governing equations, the moiré-level stick-slip behavior has been theoretically repeated. [46]Alternatively, by introducing an extra contact spring between the top layer and the fixed end in the DC-PT model to describe the total potential energy of the system, the dual-scale stick-slip frictional behavior in h-BN/graphene system is also reproduced. [44]6]

| Frictional scaling behavior of twisted vdW layered materials: Moiré compensation
In addition to the stick-slip behavior observed during sliding AFM tips on surfaces with moiré superlattices, another interesting phenomenon, that is, structural superlubricity (SSL), [97,98] can occur when there is a lattice F I G U R E 3 Moiré effect on mechanical behaviors and AFM friction test.(A-D) Moiré-dependent interlayer contact stiffness. [81]E, F) Moiré-dependent mechanical properties. [43](G, H) The parity dependence of the mechanical properties of alternating graphene/h-BN heterostructures. [43][46] (J) The theoretical model of improved Prandtl-Tomlinson (PT) model. [44][46] The scale bars in (A, B) are 10 nm and in (I, L) are 20 nm.(A-D) Reproduced with permission. [81]Copyright 2021, Springer Nature.(E-H) Reproduced with permission. [43]Copyright 2021, American Physical Society.(I-K) Reproduced with permission. [46]Copyright 2022, John Wiley and Sons.(L, M) Reproduced with permission. [44]Copyright 2022, American Physical Society.(N, O) Reproduced with permission. [45]Copyright 2022, American Chemical Society.mismatch at the interface.In this case, the frictional force is ultra-low and the wear is nearly zero during relative motion between incommensurate crystal interfaces. [99],100] One key issue of SSL is still the size dependence, although there have been numerous studies on the size effect of interfacial friction (F A γ  ).[103][104][105][106][107][108] Based on MD simulations, it was found that for incommensurate and commensurate circular contact interfaces, γ takes values of 0.25 and 1, respectively (Figure 4A), and the oscillation period of the sliding force is approximately equal to the size of the moiré superlattice. [109]Considering the different contributions of geometric edge, complete moiré contact region, and moiré rim to sliding friction, a general theoretical model for interlayer sliding friction is proposed: [102] F where A is the contact area, γ e and γ MR represent the contributions of the geometric edge and moiré rim to the frictional force, respectively.α describes the scaling character of the moiré rim, and σ reflects the effects of sliding direction and edge shape.The theory predicts a characteristic size of D ~10 2 nm, above which the scaling transits from sublinear to linear (Figure 4B).The scaling behavior can also be tuned by applying strain, [67] where the incomplete moiré superlattice at the edges can result in different compensation under different strains (Figure 4D).Similar conclusions are drawn by developing a misfit interval statistical method to quantitatively identify the geometrical characteristics of moiré patterns, [111] in which the distribution of lattice misfits was observed following the evolution of moiré superlattices.It is also a good indicator to determine (non-)superlubricity at the interfaces (Figure 4E,F).By identifying incomplete moiré superlattices with atomiclevel precision through MD simulations, it was observed that the moiré boundary dynamically changes with the F I G U R E 4 Moiré effect on frictional scaling.(A) The friction scaling of circular sliders.The exponents of γ = 0.25 and γ = 1 correspond to incommensurate and commensurate contact interfaces, respectively. [109](B) The scaling behavior of moiré-modulated friction changes from sublinear to linear behaviors with increasing size. [102](C) The friction scaling of square twisted bilayer graphene (tBLG), where the twist angle is 5°.The open circles were obtained by molecular dynamics (MD) simulations, and the black dashed lines and the blue solid line were predicted from theory. [110](D) Moiré compensation during sliding.The rim regions 1, 3, and 5 in panel (iv-3) form a complete moiré superlattice as revealed by the translational stitching method shown in panel (iv-4). [67](E, F) The moiré patterns and corresponding misfit interval statistical method distribution. [111](G) The evolution of the moiré boundary (the top panel) and interlayer potential energy (the bottom panel) with sliding distance. [77](H) Moiré compensation mechanism underlying the dual-period oscillation behavior in (C), where the color map reflects the atomic potential energy with a twist angle of 5°. [110](A) Reproduced with permission. [109]Copyright 2016, American Physical Society.(B) Reproduced with permission. [102]Copyright 2019, American Chemical Society.(C, H) Reproduced with permission. [110]Copyright 2024, Elsevier.(D) Reproduced with permission. [67]Copyright 2019, American Chemical Society.(E, F) Reproduced with permission. [111]Copyright 2022, Elsevier.(G) Reproduced with permission. [77]Copyright 2023, Elsevier.
interlayer sliding process, for example, the 'W ' shaped moiré boundary can transform into the 'Z' shape during sliding, and the energy barrier is predominantly contributed by the edge atoms determined by the moiré boundary (Figure 4G). [77]t is noteworthy that a recent study shows that the friction scaling significantly depends on the slider shape. [110]For sliders of polygons, the maximum static friction between twisted interface exhibits a unique dualperiod oscillation behavior, as well as size-independent scaling (F A s max 0  , the blue line in Figure 4C).The mechanism underlying the dual-period oscillation behavior is revealed by the moiré compensation mechanism.Specifically, the shorter period (a s in Figure 4H or the period of oscillation of the black dashed line in Figure 4C) is attributed to the compensation of moiré superlattice at opposite edges (e.g., the two blue arcs enclosed regions in the left panel of Figure 4H), while the longer period modulation (a L in Figure 4H or the period of oscillation of the blue line in Figure 4C) originates from the of moiré superlattice at the same edge (e.g., the two blue arcs enclosed regions in the right panel of Figure 4H).In addition, there is a special twisting angle for vdW layered materials with a hexagonal lattice.Due to the 60°r otational symmetry of the hexagonal lattice, when the interlayer twisting angle is 30°, the moiré superlattice is no longer a triangular lattice but exhibits a quasi-crystalline structure (Figure 5A). [112]This unique twelve-fold symmetric structure (Figure 5C) can lead to changes in interface electrical [114][115][116] and tribological [113] properties.Specifically, the frictional force will exhibit a linear dependence on the slider size (with an exponent of γ= 0.5) (Figure 5B), and the long-period modulation disappears.The oscillation period of the frictional force is ~a 3.58 m , consistent with the minimum characteristic size of the quasi-crystalline a θ a (2 + 3 ) = cos( /2) (2 + m a 3 ) ~3.6 m (Figure 5D). [113]

| Analytical frictional scaling laws for vdW layered materials
Considering that moiré superlattices have a significant effect on the mechanical and physical properties of vdW layered materials, incorporating the moiré effect into continuum modeling is highly desired.Based on the observed one-to-one correspondence between the atomic potential energy distribution and the moiré superlattice configuration (Figure 4H), [77,110] the moiré superlattice-modulated interlayer potential energy can be approximated by (under the rigid assumption), [46,77,[117][118][119] where U 0 is the amplitude of the potential energy landscape corrugation per unit area and a m is the size of the moiré superlattice.Integrating Equation ( 6) over the entire contact area of a slider (S slider ) yields the shape and positiondependent interaction energy between the slider and the substrate, slider 0 0 , where x y ( , ) 0 0 is the geometric center of the slider.By taking the derivative of the energy with respect to the sliding direction, the force trace during sliding can be obtained by: [110] F where a s is the lattice constant of the substrate.Based on Equation ( 7), the size-dependent friction of sliders can be obtained.Typical results for regular polygonal (square) graphene flakes sliding along the armchair direction of graphene substrate are shown in Figure 4C, which can be predicted by: [110]  and the envelope of the friction (Equation [8]) can be described by (blue line in Figure 4C), where w is determined by the shape of the slider and the twist angle. [110]Similarly, when the slider is circular, the relationship between the sliding friction force (F ) and the twist angle (θ) and size (R) can be derived as, [77,120] where is the Bessel function of the first kind.The theoretical predictions based on Equation (9) (blue lines in Figure 6K,L) are in excellent agreement with the results obtained from MD simulations (red diamonds in Figure 6K,L).By simplifying the Bessel functions in Equation ( 9), the maximum energy barrier is found proportional to θ and R as 6K,L). [109,127]t is worth noting that the above derivation is based on the rigid body assumption.[133] Therefore, the influence of flake flexibility, temperature perturbation, and substrate should be considered.By treating graphene as a thin film, the system free energy of graphene on a rigid substrate was derived using the thermodynamic method (Figure 6D). [123]It is found that the interface adhesion can be significantly modulated by thermal-induced out-of-plane deformation for flexible layered materials atop rigid substrates.For instance, the adhesive strength decreases linearly with increasing temperature (Figure 6F-H). [124]A recent study shows that the thermal fluctuation gradient can induce tangential entropic forces in vdW layered materials, originating from the variation of the thermodynamic potential under the condition of flexible substrates. [132]36][137] F I G U R E 5 The unique quasi-crystalline structure formed in the twisted interface.(A) The quasi-crystalline structure in twisted bilayer graphene (tBLG) with a twist angle of 30°. [112](B) The frictional scaling of tBLG with a twist angle of 30°.(C) Corresponding 12-fold symmetric structure of quasi-crystalline in (A). [113](D) The green, red, blue, and yellow colors represent dodecagonal structures with radii of a (2 + 3 ) n , n = 1, 2, 3, 4, respectively. [113](A) Reproduced with permission. [112]Copyright 2020, American Chemical Society.(B) Reproduced with permission. [113]Copyright 2016, American Physical Society.
F I G U R E 6 Continuum model for two-dimensional (2D) van der Waals (vdW) layered materials.(A) The thermal induced ripples in monolayer graphene. [121](B) Snapshots of thermal rippling of a graphene with size of L = 20 nm at T = 300 K. [122] (C) Top view of thermal rippling of graphene on a rigid substrate with T = 1000 K. [123] (D) The height and deflection profile along the white line in (C). [123]E) Illustration of the effects of substrate and temperature on the surface morphology of an ultra-soft membrane. [124](F-H) The probability distribution of the normal distance w of carbon atoms above the copper substrate with different interaction strengths and the average interlayer distance at different temperatures. [124](I) The moiré superlattice-dependent out-of-plane deformation field at different temperatures (θ = 2°, T = 0, 300 K). [125] (J) The continuum approximation of the deformation fields in (I). [125](K, L) Comparison of the interlayer sliding barrier obtained from the continuum model (blue and black lines obtained by Equation [8]) and the MD simulation results (open red diamonds). [77](M) The twist angle and size dependent free energy of bilayer graphene system. [125](N) Experimentally observed spontaneously rotation of graphene flakes from the incommensurate to the commensurate stacks atop of a graphite substrate. [126]A) Reproduced with permission. [121]Copyright 2007, Springer Nature.(B) Reproduced with permission. [122]Copyright 2014, Elsevier.(C, D) Reproduced with permission. [123]Copyright 2016, AIP Publishing.(E-H) Reproduced with permission. [124]Copyright 2019, Elsevier.(I, J, M) Reproduced with permission. [125]Copyright 2022, Elsevier.(K, L) Reproduced with permission. [77]Copyright 2023, Elsevier.(N) Reproduced with permission. [126]Copyright 2013, American Chemical Society.

| Thermal dynamical model of twisted vdW layered materials
One drawback of treating 2D vdW layered materials (e.g., graphene) as thin films is the neglect of lattice information, which is the physical origin of moiré superlattices.In the meantime, the lattice mismatch, proven to play a crucial role in the interfacial mechanics of vdW layered materials, [43][44][45][46] will also be neglected.It is noted that the out-of-plane deformation field at finite temperature follows the evolution of moiré superlattices (Figure 6I).Therefore, the out-of-plane deformation can be approximated as (with the assumption of small deformation) [125] h h h py py px + Δ cos (cos + cos 3 ), eq  (10)   where h eq and h Δ are the equilibrium interlayer distance and amplitude of the out-of-plane deformation field, respectively.p reflects the size of the moiré superlattice.Equation (10) shows excellent agreement with MD simulation results at 0 K (Figure 6J).But there is some deviation from the simulation results at finite temperatures (e.g., 300 K, Figure 6J) since the thermal-induced fluctuation is ignored in Equation (10).Therefore, the out-of-plane deformation at finite temperature should be replaced with , where W x y ( , ) represents the fluctuation induced by temperature, which can be generally expressed using a Fourier series. [122,124,129,131,132]By incorporating the deformation field h T in the thermodynamic model, the free energy of the system with different twist angles can be obtained (Figure 6M). [125]It is observed that there exists a free energy gradient pointing towards the commensurate stack (with a twist angle of 0°), which provides a good explanation for the experimental observed spontaneous rotation of graphene flakes from incommensurate to commensurate stacking atop the graphite substrate (Figure 6N). [126]n the other hand, even in the state of superlubricity, the observed dynamic evolution of the moiré superlatticedependent deformation fields during sliding could also introduce an additional dissipative channel. [125,138,139]It is reported that by introducing a normal load to suppress the moiré superlattice-dependent out-of-plane deformation, interlayer friction can be reduced, and it even leads to an intriguing negative friction behavior. [138]

| CONCLUSIONS AND PROSPECTS
The formation of moiré superlattices in 2D vdW layered materials through interlayer twisting presents a unique avenue for manipulating material properties, giving rise to burgeoning research fields like twistronics, [140,141] moiré physics, [65] and superlubricity. [55,140]Furthermore, the distinctive physical and mechanical behaviors exhibited by layered materials due to the moiré superlattice effect find extensive applications in diverse areas.
][10] However, the practical utilization of these novel physical phenomena relies on a comprehensive understanding and accurate description of the moiré superlattice effect.In our review, we have provided a concise overview of the impact of moiré superlattices on interfacial mechanical behaviors, encompassing moirédependent mechanical and tribological properties, with a particular emphasis on theoretical models.Despite notable progress in recent years, certain aspects still warrant clarification, and key experiments should be devised and conducted to address these outstanding questions.
First, atomic reconstruction in a large moiré superlattice (e.g., for very small twist angles) can lead to significant local deformation (Figure 7B), [73] the current trigonometric deformation field approximation is only applicable to 2D vdW layered materials with large twist angles (Figure 7A  and 2J). [74]However, the most intriguing physical phenomena, such as the superconductivity, occur within a very small range of twist angles.Therefore, highly localized strain (Figure 7B) should play an important role in these physical phenomena.Unfortunately, modeling and simulating the effect of atomic reconstruction on the mechanical F I G U R E 7 The challenges of theoretical modeling the moiré superlattice effect in two-dimensional (2D) van der Waals (vdW) layered materials.(A, B) The deformation mode can significantly change with the twist angle. [74](C) The importance of the edge effect in the tribological behavior. [146](D, E) Complex moiré superlattices in multilayer structures [147] and multiscale moiré superlattices. [148]F) Dependent of moiré superlattice on the crystal unit cell.(A, B) Reproduced with permission. [74]Copyright 2015, IOP Publishing.(C) Reproduced with permission. [146]Copyright 2021, Springer Nature.(D) Reproduced with permission. [147]Copyright 2017, American Chemical Society.(E) Reproduced with permission. [148]Copyright 2022, American Chemical Society.
behavior of vdW layered materials remain significant challenges.Theoretically, the interlayer potential energy and the in-plane or out-of-plane deformation fields cannot be easily approximated by simple trigonometric functions (Equation [10]).Therefore, accurately describing the changes of the deformation field caused by this atomic reconstruction needs further investigation.From a simulation perspective, since the atomic reconstruction typically occurs at small twist angles, the number of atoms within a moiré superlattice can reach tens of thousands, far exceeding the computational capabilities of high-precision atomic calculation methods such as density functional theory (DFT).For MD simulations, although it can capture the deformation transition behavior of moiré superlattices (Figure 2L), performing calculations at the experimental scales (typically at the micrometer level) remains a substantial challenge. [54]n addition, various lattice types exist, including rectangular lattice, and face-centered cubic lattices [149] (Figure 7F), along with thousands of combinations involving multilayers of homogeneous and heterogeneous interfaces (Figure 7D). [38,147]There is also the potential for the formation of multiscale moiré superlattices within these structures (Figure 7E). [64,88,89,148,150]hese variations pose a key challenge for theoretical modeling of the moiré superlattice effect in different systems.Moreover, the geometric edge of 2D vdW layered materials can also play a crucial role in the tribological behavior. [101,146]Therefore, incorporating the effect of the edge into the continuum modeling of moiré superlattices and quantifying the compensation of incomplete moiré superlattices at the edge when considering elastic deformation are worth further study.
Last but not least, the theoretical predictions on the effects of size, shape, and twist angle of a slider on the friction between the slider and vdW layered materials are important for the guidance of superlubricity application.However, this also poses experimental challenges in decoupling the aforementioned factors and effects, such as edge pinning and sliding direction.