Anisotropic Mechanical Properties of 2-D Materials

*Qiang Li*

## **Abstract**

While prior reviews and research articles focused on the various synthetic routes and microstructural controls of 2D nanomaterials as well as their functional applications, this chapter discloses the anisotropic behaviors of 2D materials and puts emphasis on the mechanical anisotropy of three distinct 2D materials, namely graphene, MoS2 and Al alloy coating, representative of carbon, inorganic and metallic 2D crystalline materials. Except for the relatively low interlayer cohesive stress, the in-plane anisotropy of the former two materials classes is subjected primarily to the hexagonal structure of the unit cells of the graphene and MoS2. The anisotropy of metallic thin films with high-density grain boundaries with preferential directionality, rendered by the non-equilibrium synthetic methods, results from both the conventional Taylor factor and the directionality of the grain boundaries. Despite 2D materials' wide spectrum of applications, such as electronics, energy devices, sensors, coating etc., the mechanical anisotropy could be critical for certain mechanical applications, such as friction, and provide instructions on the durability, reliability and property optimization in the various applications of different 2D materials.

**Keywords:** anisotropy, mechanical behaviors, 2D materials, metallic materials, non-metallic materials

## **1. Introduction**

As time passes, the advancement of nanotechnology has spread to all fronts. The concept states that at least two dimensions that construct nanomaterials fall between 1 and 100 nm. Nanomaterials are classified as zero- (0D), one- (1D), two- (2D), and three-dimensional (3D) nanostructures. The nanoscale has unprecedented attributes that fundamentally alter materials' properties. Since K. Novoselov et al. successfully mechanically exfoliated a single layer of graphene off the graphite in 2004 [1], extensive efforts and progress have been made on the synthesis and applications of graphene and various 2D nanomaterials in resemblance to graphene nanostructure, including transition metal dichalcogenides (TMDs), hexagonal boron nitride (BN), and perovskites, just to name a few. They have lateral extension but their individual layer is merely a single or few atoms thick. Hence, they have characteristics like electron confinement and anisotropy in various properties

manifested in two dimensions, while they possess extended interlayer spacing for active kinetic and physicochemical events, which has attracted broad research interests on their physicochemical, electrochemical, electronic and mechanical properties. For metallic materials, metallic thin films/coatings have 2D extension but limited thickness. The 2D materials selected to represent each materials family are crystalline materials and, in general, possess crystal anisotropy in their mechanical behaviors and even functional property. The single crystal face-centered cubic (FCC) structure is taken as an example. It is well known that the FCC single crystals have crystal anisotropy determined by the Schmid factor that associates the loading direction to the load resolved on the specific slip system [2]. Because the glide of the dislocations is favored on the slip systems subjected to a larger Schmid factor, plastic anisotropy manifests in the form of different cellular substructure made of dislocation walls. As we alleged, the non-metallic 2D nanomaterials have extended interlayer spacing and metallic thin films, fabricated primarily by ultrahigh vacuum techniques and electrodeposition, feature high-density directional grain boundaries (GBs) and preferential texture. As a result, anisotropy in 2D materials is prone to deviate from that of the bulk crystals, and plays substantial roles in their mechanical applications and the reliability of the apparatuses and devices with 2D materials as components or building blocks, but has not been put emphasis on as much as their functional properties and synthesis. Prior to comprehension toward the mechanical anisotropy of 2D materials, their general microstructural features and applications are first set forth so as to better grasp the anisotropy in their mechanical response to external stimuli and its importance in their functional and engineering applications.

S-Mo-S layered structure [4]. Hence, it has drawn enormous attention for its potent applications in energy storage and conversion, such as photocatalysis for the pollutant degradation and biosensors, just to name a few. In addition, MoS2 has a good tunability toward its band gap, which offers high flexibility in property customization and optimization. At the same time, MoS2 manifests comparable physical attributes when compared to graphene, including high charge carrier mobility and superb wear properties. Compared to hexagonal structure, other structurally complex 2D materials, such as arsenic trisulfide (As2S3), also showed mechanical anisotropy [5]. A unit cell of As2S3 consists of two layers inverted with respect to a symmetry center and is defined by 20 atoms in contrast to two for graphene and

Now, we turn our attention to a different 2D materials family, i.e. metallic coating and thin films. The protective coatings, from an engineering point of view, are essential as to apparatus maintenance and the enhanced equipment safety and lifespan. One application of metallic or their composite coating is to prevent corrosion. Coatings should render compatibility with parental materials and operate at extreme atmospheres, such as high temperature and corrosive conditions. Metallic coatings either provide passive protection by forming a barrier of oxides or offer active protection obtained through the adsorption of chemical inhibitors [6]. Metallic coatings as biomaterials are potent components in body implants and they ought to possess superb mechanical behaviors and biocompatibility, and high corrosion resistance, while they are required to release minimal metallic ions to avoid the toxicity. Ti, NiTi, Pt and 316 L austenitic stainless steel are often implemented. Furthermore, various metals with unique characteristics are used in the applications of thin film optics, such as surface plasmon generation and optoelectronics. Metallic thin films are frequently applied onto ceramic matrices, rendering high-quality broadband reflective finishing highly desired to control over the directionality of the laser beam. In addition, Cu is commonly utilized as an interconnect material and serves as thin conductive layers ensuring the adhesion to dielectrics and inhibiting diffusion into silicon or dielectrics, and provide capability of electrodeposition of Cu [7]. From the aforementioned applications of metallic coating and thin films, it is realized that their fabrication often relies on non-equilibrium routes, such as a variety of ultrahigh vacuum techniques and electrodeposition/electroplating. The energetic adatoms landing on the substrate often first form nanometric epitaxial zone and then become 3D clusters during the growth process [8]. Moreover, the sophisticated compositions or the interaction between matrix atoms and impurity atoms as dopants and alloying elements often exert pinning effects. These factors result in the formation of abundant GBs among the columnar nanograins. Many thin films have been grown homo- or heteroepitaxially on single crystal templates or locally on polycrystalline templates, giving rise to the preferential texture in the films. Both the GB directionality and the preferential texture in the coatings or thin films should lead to mechanical and crystal anisotropies which greatly affect their performance in practical applications and hint at the property optimization along

In FCC single crystals, the Schmid factors mainly explains the crystal anisotropy. For a given crystal, different loading directions result in different sets of Schmid factors on the 24 slip systems FCC structure intrinsically has, eventually tailoring

three for MoS2.

each direction.

**55**

**1.2 Basics of anisotropy**

*1.1.2 Metallic thin films*

*Anisotropic Mechanical Properties of 2-D Materials DOI: http://dx.doi.org/10.5772/intechopen.96598*

## **1.1 The general microstructural features and applications of 2D materials**

### *1.1.1 Non-metallic 2D materials*

Graphene is a typical 2D carbon allotrope and a monolayer of graphene, with a thickness of 0.335 nm, has a hexagonal honeycomb structure. This 2D nanomaterial is remarkably electric and thermal conductive and is equipped with the promising quantum Hall effect. Moreover, the pristine graphene possesses an elastic modulus of �1 TPa and a mechanical quality factor of 104 at an elevated temperature of 5 K. Despite similar sheet-like nanostructures, 2D nanomaterials made of inorganic compounds can render intriguing properties and versatility due to their more complex compositions. In contrary to chemically inert graphene with no intrinsic bandgap, MoS2 with layered structure is one transition metal dichalcogenide. MoS2 has been often synthesized using chemical vapor deposition (CVD) and its reaction principle involves first the transformation of solid-state MoO3 and sublimed surfur to gas state and then the mixed gases driven by argon caused the formation of gasphase MoO3-x and MoS2 and eventually the formation of solid-state MoS2, the reactions of which is expressed as [3]:

$$\rm{MoO\_3(g)} + \left(\frac{\infty}{2}\right) \rm{S(g)} \leftrightharpoons \rm{MoO\_{3-x}(g)} + \left(\frac{\infty}{2}\right) \rm{SO\_2} \tag{1}$$

$$\mathrm{MoO}\_{3-x}(\mathrm{g}) + (7-\varkappa)\mathrm{S}(\mathrm{g}) \rightleftharpoons \mathrm{MoS}\_{2}(\mathrm{g}) + \left(\frac{3-\varkappa}{2}\right)\mathrm{SO}\_{2}\,\mathrm{.}\tag{2}$$

MoS2 structure is comprised of two layers of closely packed S atoms layers sandwiching a layer of Mo atoms and it features strong covalent bonding as a result of the Mo-S interaction and the Van der Waals force between S layers. This leads to the comparably facile kinetic transportation of ions and even molecules through

*Anisotropic Mechanical Properties of 2-D Materials DOI: http://dx.doi.org/10.5772/intechopen.96598*

S-Mo-S layered structure [4]. Hence, it has drawn enormous attention for its potent applications in energy storage and conversion, such as photocatalysis for the pollutant degradation and biosensors, just to name a few. In addition, MoS2 has a good tunability toward its band gap, which offers high flexibility in property customization and optimization. At the same time, MoS2 manifests comparable physical attributes when compared to graphene, including high charge carrier mobility and superb wear properties. Compared to hexagonal structure, other structurally complex 2D materials, such as arsenic trisulfide (As2S3), also showed mechanical anisotropy [5]. A unit cell of As2S3 consists of two layers inverted with respect to a symmetry center and is defined by 20 atoms in contrast to two for graphene and three for MoS2.

#### *1.1.2 Metallic thin films*

manifested in two dimensions, while they possess extended interlayer spacing for active kinetic and physicochemical events, which has attracted broad research interests on their physicochemical, electrochemical, electronic and mechanical properties. For metallic materials, metallic thin films/coatings have 2D extension but limited thickness. The 2D materials selected to represent each materials family are crystalline materials and, in general, possess crystal anisotropy in their mechanical behaviors and even functional property. The single crystal face-centered cubic (FCC) structure is taken as an example. It is well known that the FCC single crystals have crystal anisotropy determined by the Schmid factor that associates the loading direction to the load resolved on the specific slip system [2]. Because the glide of the dislocations is favored on the slip systems subjected to a larger Schmid factor, plastic anisotropy manifests in the form of different cellular substructure made of dislocation walls. As we alleged, the non-metallic 2D nanomaterials have extended interlayer spacing and metallic thin films, fabricated primarily by ultrahigh vacuum techniques and electrodeposition, feature high-density directional grain boundaries (GBs) and preferential texture. As a result, anisotropy in 2D materials is prone to deviate from that of the bulk crystals, and plays substantial roles in their mechanical applications and the reliability of the apparatuses and devices with 2D materials as components or building blocks, but has not been put emphasis on as much as their functional properties and synthesis. Prior to comprehension toward the mechanical anisotropy of 2D materials, their general microstructural features and applications are first set forth so as to better grasp the anisotropy in their mechanical response to external stimuli and its importance in their functional and engineering applications.

**1.1 The general microstructural features and applications of 2D materials**

Graphene is a typical 2D carbon allotrope and a monolayer of graphene, with a thickness of 0.335 nm, has a hexagonal honeycomb structure. This 2D nanomaterial is remarkably electric and thermal conductive and is equipped with the promising quantum Hall effect. Moreover, the pristine graphene possesses an elastic modulus of �1 TPa and a mechanical quality factor of 104 at an elevated temperature of 5 K. Despite similar sheet-like nanostructures, 2D nanomaterials made of inorganic compounds can render intriguing properties and versatility due to their more complex compositions. In contrary to chemically inert graphene with no intrinsic bandgap, MoS2 with layered structure is one transition metal dichalcogenide. MoS2 has been often synthesized using chemical vapor deposition (CVD) and its reaction principle involves first the transformation of solid-state MoO3 and sublimed surfur to gas state and then the mixed gases driven by argon caused the formation of gasphase MoO3-x and MoS2 and eventually the formation of solid-state MoS2, the

*S g*ð Þ ⇋ *MoO*<sup>3</sup>�*<sup>x</sup>*ð Þþ *g*

*x* 2 

3 � *x* 2 

*SO*<sup>2</sup> (1)

*SO*2*:* (2)

*1.1.1 Non-metallic 2D materials*

*Material Flow Analysis*

reactions of which is expressed as [3]:

**54**

*MoO*3ð Þþ *g*

*x* 2 

*MoO*<sup>3</sup>�*<sup>x</sup>*ð Þþ *g* ð Þ 7 � *x S g*ð Þ ⇋ *MoS*2ð Þþ *g*

MoS2 structure is comprised of two layers of closely packed S atoms layers sandwiching a layer of Mo atoms and it features strong covalent bonding as a result of the Mo-S interaction and the Van der Waals force between S layers. This leads to the comparably facile kinetic transportation of ions and even molecules through

Now, we turn our attention to a different 2D materials family, i.e. metallic coating and thin films. The protective coatings, from an engineering point of view, are essential as to apparatus maintenance and the enhanced equipment safety and lifespan. One application of metallic or their composite coating is to prevent corrosion. Coatings should render compatibility with parental materials and operate at extreme atmospheres, such as high temperature and corrosive conditions. Metallic coatings either provide passive protection by forming a barrier of oxides or offer active protection obtained through the adsorption of chemical inhibitors [6]. Metallic coatings as biomaterials are potent components in body implants and they ought to possess superb mechanical behaviors and biocompatibility, and high corrosion resistance, while they are required to release minimal metallic ions to avoid the toxicity. Ti, NiTi, Pt and 316 L austenitic stainless steel are often implemented. Furthermore, various metals with unique characteristics are used in the applications of thin film optics, such as surface plasmon generation and optoelectronics. Metallic thin films are frequently applied onto ceramic matrices, rendering high-quality broadband reflective finishing highly desired to control over the directionality of the laser beam. In addition, Cu is commonly utilized as an interconnect material and serves as thin conductive layers ensuring the adhesion to dielectrics and inhibiting diffusion into silicon or dielectrics, and provide capability of electrodeposition of Cu [7]. From the aforementioned applications of metallic coating and thin films, it is realized that their fabrication often relies on non-equilibrium routes, such as a variety of ultrahigh vacuum techniques and electrodeposition/electroplating. The energetic adatoms landing on the substrate often first form nanometric epitaxial zone and then become 3D clusters during the growth process [8]. Moreover, the sophisticated compositions or the interaction between matrix atoms and impurity atoms as dopants and alloying elements often exert pinning effects. These factors result in the formation of abundant GBs among the columnar nanograins. Many thin films have been grown homo- or heteroepitaxially on single crystal templates or locally on polycrystalline templates, giving rise to the preferential texture in the films. Both the GB directionality and the preferential texture in the coatings or thin films should lead to mechanical and crystal anisotropies which greatly affect their performance in practical applications and hint at the property optimization along each direction.

#### **1.2 Basics of anisotropy**

In FCC single crystals, the Schmid factors mainly explains the crystal anisotropy. For a given crystal, different loading directions result in different sets of Schmid factors on the 24 slip systems FCC structure intrinsically has, eventually tailoring

dislocations on different slip systems. Along with the dislocation populations, the dislocations would self-organize into certain low energy substructures with cellular shape. For instance, [111]-loading generally leads to planar-shaped cell substructures, whereas [100] direction renders spherical-shaped ones. To be specific, the Schmid factors of the four slip planes for three different loading directions are present in **Table 1**. Z. Q. Wang et al. introduced an H-factor based on Schmid factors to comprehend the deformation heterogeneity and the formula is expressed as [2].

$$H = \frac{\tilde{m}\_{i,\max} - \tilde{m}\_{i,\min}}{\sum\_{i} \tilde{m}\_{i}} \tag{3}$$

**1.3 Anisotropy in non-metallic 2D nanomaterials**

*Anisotropic Mechanical Properties of 2-D Materials DOI: http://dx.doi.org/10.5772/intechopen.96598*

properties but also functional properties.

**1.4 Anisotropy in metallic 2D thin films**

*pressure. Reprinted with permission from reference [14].*

**Figure 1.**

**57**

The 2D non-metallic materials display remarked structural anisotropy due to the large interlayer spacing and comparably low interlayer cohesion and interaction, which causes that a monolayer of graphene could be readily mechanically exfoliated and hexagonal MoS2 (h-MoS2) with lamellar structure can be used as a solid lubricant owing to its superlubricity causing the facile glide among MoS2 nanosheets. However, in the in-plane direction, the assumption of mechanical isotropy in 2D materials is premature just based on the six-fold symmetry in their hexagonal lattice when the isotropy has been assumed for some estimations of the elastic behaviors in carbon nanotubes. Prior researches unveiled that friction force exerted on both graphene and MoS2 along in-plane 'zigzag' and 'armchair' directions of the hexagonal lattice gave rise to different results and friction tests along armchair direction resulted in larger friction forces. M. Dienwiebel et al. found the angular interval between two friction peak force being approximate 60° upon friction tests on graphite [12]. This suggests that the 2D materials with hexagonal lattice manifest a sixfold anisotropy with a 60° periodicity. Meanwhile, studies showed that the anisotropy in both graphene and MoS2 has a thickness dependence [13]. 2D nonmetallic nanomaterials have been often used as building blocks or components for micro/nano-electromechanical systems (M/NEMSs) and nanoelectronics. The anisotropy of those 2D materials have great influence on not only mechanical

Metallic coating and thin films have been largely fabricated adopting nonequilibrium ultrahigh vacuum techniques and electrodeposition. When the nuclei heterogeneously grow and then 3D clusters collide amid the coalescence process, forming intercrystalline interface. This process generally gives rise to nanocolumnar grains whose grain size is small, even in monolithic metals, in contrast to other equilibrium processes. **Figure 1** shows the structure zone diagram after energetic deposition of a thin film on a substrate, indicating that columnar grains preferentially being generated at different generalized temperature *T/Tm* and argon pressure [14]. The columnar structure could even exist in amorphous Al-Cr thin films prepared by the sputtering technique as a result of chemical segregation [15]. These 2D

*Structure zone diagram after energetic deposition at different generalized temperature* T/Tm *and argon*

where *m*~ *<sup>i</sup>* is the sum of all three values of the Schmid factors for each slip plane under one loading orientation, and *m*~ *<sup>i</sup>*, *max* and *m*~ *<sup>i</sup>*, *min* are the maximum and minimum of *<sup>m</sup>*<sup>~</sup> *<sup>i</sup>* out of four different planes, respectively. This causes the <sup>211</sup> � �-, 111 ½ � and 100 ½ �-loadings to render the respective H-factor of 0.376, 0.333 and 0. Moreover, this suggests that 211 � �-loading brings about highest flow stress and most heterogeneous cellular structure, where 100 ½ �-loading the lowest flow stress and most homogeneous structure out of three loading conditions.

For polycrystalline cubic metallic materials, the Yield strength is associated to a Taylor factor, *M*, based on the critical resolved shear stress (CRSS) and the Taylor factor relies on the active slip systems at the grain level of a polycrystalline metal with crystallographic preferential texture in the aggregate. According to the classic dislocation mechanics [9], the Yield strength can be defined as

$$
\Delta \sigma\_{\mathcal{Y}} = \mathcal{M} \left[ \tau\_0 + \left( \frac{\pi^\* \mu b}{\pi (1 - v) L} \right)^2 \right] \tag{4}
$$

where *τ*<sup>0</sup> and L are the lattice friction and the mean spacing between a dislocation source to obstacle, respectively. *τ* <sup>∗</sup> denotes the barrier shear stress for a single dislocation transmission across a GB and *v*, *b* and *μ* are the Poisson's ratio, Burgers vector and the shear modulus, respectively. The Taylor factor is intimately associated with the stacking fault energy (SFE) as well as the ratio, ξ, of CRSS for twinning and slip for FCC metals. Besides FCC structure, various crystalline materials are anisotropic. For example, various carbides hexagonal M7C3 (M = Fe, Cr, W, Mo) exhibited anisotropy and difference in chemical bonding along different crystallographic orientation determined the anisotropy and their elastic anisotropy could be further tailored by the different atomic arrangement controlled by multialloying [10]. Despite different mechanisms, some liquid crystals can be defined as the anisotropic fluids whose form is between the isotropic liquid and the anisotropic crystalline phase [11].


**Table 1.** *Schmid factors of the four slip systems for three different loading directions. Reconstructed from reference [2].* dislocations on different slip systems. Along with the dislocation populations, the dislocations would self-organize into certain low energy substructures with cellular shape. For instance, [111]-loading generally leads to planar-shaped cell substructures, whereas [100] direction renders spherical-shaped ones. To be specific, the Schmid factors of the four slip planes for three different loading directions are present in **Table 1**. Z. Q. Wang et al. introduced an H-factor based on Schmid factors to comprehend the deformation heterogeneity and the formula is expressed

> *<sup>H</sup>* <sup>¼</sup> *<sup>m</sup>*<sup>~</sup> *<sup>i</sup>*, *max*P� *<sup>m</sup>*<sup>~</sup> *<sup>i</sup>*, *min i m*~ *i*

where *m*~ *<sup>i</sup>* is the sum of all three values of the Schmid factors for each slip plane under one loading orientation, and *m*~ *<sup>i</sup>*, *max* and *m*~ *<sup>i</sup>*, *min* are the maximum and mini-

� �-loading brings about highest flow stress and most

*π*ð Þ 1 � *v L*

� �<sup>2</sup> " #

where *τ*<sup>0</sup> and L are the lattice friction and the mean spacing between a dislocation source to obstacle, respectively. *τ* <sup>∗</sup> denotes the barrier shear stress for a single dislocation transmission across a GB and *v*, *b* and *μ* are the Poisson's ratio, Burgers vector and the shear modulus, respectively. The Taylor factor is intimately associated with the stacking fault energy (SFE) as well as the ratio, ξ, of CRSS for twinning and slip for FCC metals. Besides FCC structure, various crystalline materials are anisotropic. For example, various carbides hexagonal M7C3 (M = Fe, Cr, W, Mo) exhibited anisotropy and difference in chemical bonding along different crystallographic orientation determined the anisotropy and their elastic anisotropy could be further tailored by the different atomic arrangement controlled by multialloying [10]. Despite different mechanisms, some liquid crystals can be defined as the anisotropic fluids whose form is between the isotropic liquid and the

½ � **100 -loading** ½ � **111 -loading 211**

ð Þ 111 0.41, 0.41, 0.0 0.0, 0.0, 0.0 0.0, 0.0, 0.0 111 � � 0.41, 0.41, 0.0 0.27, 0.27, 0.0 0.41, 0.27, 0.14 111 � � 0.41, 0.41, 0.0 0.27, 0.27, 0.0 0.41, 0.27, 0.14

� � 0.41, 0.41, 0.0 0.27, 0.27, 0.0 0.27, 0.27, 0.0

*Schmid factors of the four slip systems for three different loading directions. Reconstructed from reference [2].*

and 100 ½ �-loadings to render the respective H-factor of 0.376, 0.333 and 0. More-

heterogeneous cellular structure, where 100 ½ �-loading the lowest flow stress and

<sup>Δ</sup>*σ<sup>y</sup>* <sup>¼</sup> *<sup>M</sup> <sup>τ</sup>*<sup>0</sup> <sup>þ</sup> *<sup>τ</sup>* <sup>∗</sup> *<sup>μ</sup><sup>b</sup>*

For polycrystalline cubic metallic materials, the Yield strength is associated to a Taylor factor, *M*, based on the critical resolved shear stress (CRSS) and the Taylor factor relies on the active slip systems at the grain level of a polycrystalline metal with crystallographic preferential texture in the aggregate. According to the classic

mum of *m*~ *<sup>i</sup>* out of four different planes, respectively. This causes the 211

most homogeneous structure out of three loading conditions.

dislocation mechanics [9], the Yield strength can be defined as

(3)

(4)

� �**-loading**

� �-, 111 ½ �-

as [2].

*Material Flow Analysis*

over, this suggests that 211

anisotropic crystalline phase [11].

111

**Table 1.**

**56**
