Friction properties of black phosphorus: a first-principles study

Based on the first-principle, the friction anisotropy, structural super-lubricity and oxidation induced ultra-low friction of black phosphorus at atomic scale under different loads have been studied. The results show that the interface friction of black phosphorus is anisotropic, that is, the friction along the armchair direction is greater than that along the zigzag direction. Moreover, the friction between the black phosphorus interfaces shows a structural superlubricity property, and the incommensurate interface friction is approximately one thousandth of the commensurate interface friction, which is mainly due to the less electronic charge and the smaller amplitude of electronic charge change between the incommensurate interfaces during the friction process. In addition, the oxidation of black phosphorus is beneficial for lubrication between interfaces.


Introduction
Friction is an ancient and mysterious mechanical phenomenon in daily life. When two objects slide relative to each other, friction can block the relative sliding of objects, dissipate the energy of mechanical systems, and wear the objects in contact with each other. In engineering applications, friction and wear often lead to huge energy losses and machine failures [1,2]. Therefore, effective control and reduction of friction and wear between mechanical components can improve their efficiency and life, and has become one of the greatest challenges of mobile mechanical components today [3].
Among all the charming two-dimensional layered materials, BP has received extensive attention in recent years due to its unique mechanical, electrical, optical and tribological properties [30][31][32][33][34][35][36][37]. As an emerging two-dimensional material, BP has shown more and more excellent performance and broad application prospects in the field of tribology. For instance, the friction characteristics of few-layer BP flakes, including the influence of BP layers on friction and the friction anisotropy at atomic scale, have been studied by using atomic force microscopy (AFM) [38]. Robust superlubricity of BP modified by NaOH as lubricant additive to water, which attribute to the very low shear resistance of the water layer retained by BP-OH nanosheets, has been observed [39]. In addition, the AFM experiment shows that, besides the layered structure, the environmental degradation of BP is also conducive to its lubrication performance [40]. The experimental research using CSM friction and wear tester shows that BP-GO mixed nano-materials as water-based additives can significantly reduce friction and wear [41]. A macro scale superlubricity has been achieved under a high contact pressure via the lubrication of BP quantum dots in aqueous ethylene glycol aqueous suspension at the Si 3 N 4 /sapphire frictional interface [42]. In addition, a macro-scale superlubricity at the steel/steel contact under high pressure has been observed in the presence of BP used as an oil-based nano-additive [43]. Recently, the atomic-scale friction of black and violet phosphorus crystals has also been investigated by friction force microscopy [44]. As a water-based lubricating additive, a BP nanosheet also exhibits excellent lubrication performance [45].
Theoretically, anisotropic friction and superlubricity behaviors of BP have been investigated by molecular dynamics simulations [46][47][48][49][50]. Moreover, density functional theory (DFT) is an indispensable tool to study the mechanism of friction, wear and lubrication, and has made great contributions to the tribology field by accurately calculating the potential energy surface, shear strength, charge transfer between friction pairs, etc [51]. Recently, structural superlubricity in phosphorene has been identified by means of DFT calculations [52]. Insensitivity of friction of BP under the high load, which is closely related to its negative Poisson's ratio, has been also studied from first-principles calculations [53]. Phosphorene shows good performance as a solid lubricant for steel [54], which attributes to the strong chemical adsorption of phosphorene on the iron/steel surface. In adition, by the first-principles method, a BP/graphene heterostructure with oxide functionalization, which shows ultra-low frictional properties under high pressure conditions, has been designed [55]. However, as far as we know, the friction characteristics of oxidized and hydroxyl containing BP have not been studied theoretically. The surface of degraded BP often contains oxygen and hydroxyl functional groups [56]. Therefore, it is necessary to carry out first-principles calculations to study the atomic scale friction characteristics of oxidized and hydroxyl containing BP in detail.
In this paper, friction of BP at atomic scale under different loads has been investigated through the first-principle method.
Our computed results show that the friction between BP interfaces shows anisotropy, that is, the friction along the armchair direction is greater than that along the zigzag direction. Structural superlubricity of BP has also studied here. The friction of incommensurate interface is far less than that of the commensurate one. In addition, the oxidation of BP is beneficial to the lubrication between interfaces. However, hydroxyl groups are not conducive to the lubrication of BP interfaces.

Calculation methods and models
All the DFT calculations in this work have been carried out by the Vienna Ab initio Simulation Package [57,58]. The electron wave functions are extended by the projector augmented wave method [59] to describe the interaction between ion and valence electrons. We utilize the generalized gradient approximation with the Perdew-Burke-Ernzerhof exchange correlation functional [60]. In all calculations, the DFT-D2 scheme of Grimme [61] is used to consider the van der Waals long-range interaction. Cut off energy of 550 eV and 13 × 15 × 1 Monkhorst-Pack mesh k-points are utilized in the DFT simulations. The calculation convergence criterions for energy and force are 10 −5 eV and 0.005 eV Å, respectively. The vacuum layer perpendicular to the surface is greater than 15 Å to ensure that there is no interaction between adjacent supercells.
Our calculated lattice constants of BP are a = 4.41 Å and b = 3.33 Å, which are good agreement with previous theoretical [52] and experimental [62] values. Two layers of BP sheet are selected to simulate the friction between layers (figure 1). As shown in the figures 1(a) and (b), when the two layers are stacked in the same direction (armchair-armchair and zigzag-zigzag), we call the two layers of BP tablets commensurate (BP-R0). When the upper layer rotates 90°( armchair versus zigzag), we call these two layers of BP tablets incommensurate (BP-R90). A 4 × 3 supercell with only 0.7% lattice mismatch is constructed to model the incommensurate interface (figures 1(c) and (d)). In order to understand the friction behaviors of degraded BP, we studied the friction characteristics of BP interface containing two functional groups, that is, oxidized (figure 1(e)) and hydroxyl passivated (figure 1(f)) interfaces.

Friction anisotropy of black phosphorus
Friction anisotropy has been studied with commensurate double-layer BP ( figure 1(a)). First, under 0 load, potential energy and frictional force of the relative sliding of the bilayer commensurate black phosphorus (BP-R0) along three representative directions have been calculated in accordance with the Zhong's theory [63] , n is the potential energy at position x along the given sliding direction under a certain load f , is the binding energy of the bilayer BP, z is the interlayer distance between two layers of BP, ( ) is the minimum value of potential energy along the given sliding direction, that is the selected zero potential energy point, E isolated is the energy of an isolated layer of BP, and ( ) E x z , t represents the total energy of double BP.
Because the ratio of BP lattice constants is about a: b = 4:3, the 37°direction is the diagonal direction of the unit cell. It can be seen from figure 2(a) that the sliding potential barrier along the armchair direction is greater than the zigzag    [52,53]. The corresponding frictional force along these three directions is shown in figure 2(b). The frictional force along the x sliding direction under a certain load f , , n is the potential energy at position x along the given sliding direction under a certain load f . n The maximum value of ( ) f x f , x n is the static friction governing the onset of stick-slip motion. From the figure 2(b), we can also see that the maximal frictional force along the armchair direction is greater than that along the other two directions.
Potential energy barrier and maximal frictional force as a function of load is shown in figure 3. For a given sliding direction, such as the armchair direction, the frictional force generally meets the classical Amontons-Coulomb law of friction, and increases linearly with the increase of load. Under the same load, the frictional force along the armchair direction is greater than that along the zigzag and 37°directions. In other words, the friction between the double-layer BP sheets is anisotropic. The friction anisotropy of BP has also been reported in previous works by the atomic force microscope (AFM) experiment [38] and molecular dynamics simulations [46,47,49]. It is very important to understand the friction anisotropy of BP for strain engineering devices and micro/nano electromechanical systems.

Structural superlubricity of black phosphorus
Structural superlubricity is a kind of superlubricate state caused by incommensurate interfaces or lattice mismatch [3,4,18,23,25]. Among all kinds of methods proposed to achieve superlubricity, structural superlubricity is considered to be one of the most promising methods. Potential energy surfaces (PES) under 0 load of bilayer phosphorene for the commensurate black phosphorus (BP-R0) and the incommensurate black phosphorus (BP-R90) have been shown in figures 4(a) and (b), respectively. It can be seen that the friction is significantly anisotropic for the commensurate bilayer black phosphorus (BP-R0). The potential barrier is about 175 meV cell −1 . However, for the incommensurate double-layer black phosphorus (BP-R90), it is almost indistinguishable from the energy point of view, and the friction anisotropy disappears. The potential barrier is only 1.1 meV cell −1 and is about 0.6% of that of the commensurate bilayer black phosphorus (BP-R0).
According to Zhong's theory [63], we can calculate the coefficient of friction by the formula Here, DV, f n and Dx represent the potential energy barrier , load and sliding distance along the x sliding direction, respectively. Coefficient of friction as a function of load has been shown in figure 5. It can be seen from the figure 5 that for the incommensurate bilayer black phosphorus (BP-R90), the friction coefficient is two orders of magnitude smaller than the commensurate one. Superlubricity can be realized between the incommensurable double-layers of black phosphorus (BP-R90). Our results also confirm that BP can be used as a promising solid lubricant.
In order to find out the reason behind the structural superlubricity, we calculated the electronic charge density of the bilayer BP during sliding along the armchair direction. Figure 6(a) shows the percentage of the electronic charge between the bilayer BP to the total electronic charge of the system during sliding along the armchair direction. It can be seen from the figure 6(a) that the electronic charge between commensurate interfaces during sliding is significantly more than that between incommensurate interfaces. For the commensurate bilayer black phosphorus (BP-R0), the more electronic charges between interfaces, the stronger the interaction between them, and the greater the friction during sliding. On the contrary, for the incommensurate double-layer black phosphorus (BP-R90), the less the electronic charge between the interfaces, the weaker the interaction between them, and the smaller the friction in the sliding process. Moreover, the amplitude of electronic charge change at the commensurate interface is obviously larger than that at the incommensurate interface.
Electronic charge density difference of the commensurate and incommensurate bilayer BP have been shown in figures 6(b) and (c), respectively. Electronic charge density difference r r r r D = --, bilayer upper bottom r , bilayer r upper and r bottom represent the electronic charge density of bilayer, upper layer and bottom layer BP, respectively. It can reflect the electronic charge transfer at the interface. Compared with the commensurate interface, less electronic charge has been transferred from the upper and lower layers to the incommensurate interface. Therefore, in the sliding process, the incommensurate interface friction is small, mainly because there is less electronic charge transfer from the upper and lower layers of BP at the interface. Moreover, the incommensurate interface spacing, 3.31 Å, is greater than the commensurate interface spacing, 3.01 Å.

Oxidation induced ultra-low friction of black phosphorus
One of the biggest challenges of BP research and application is its instability under environmental conditions. The oxidation and degradation of BP leads to rapid loss of performance, which limits the potential application of BP in the field of electronics and optoelectronics. However, BP is often used as a liquid lubricating additive to show good lubricating performance [36,39,45]. Moreover, the experimental studies show that the oxidation and degradation of BP are beneficial to its lubrication [15][16][17]40]. Molecular dynamics simulations show that even under ultra-high contact pressure, due to the formation of abundant P=O and P-OH bonds on the surface of oxidized BP, water molecules can also be retained at the friction interface, which is conducive to achieving stable super lubrication [16].
As far as we know, there is no first-principles calculation study on the friction properties of oxidized BP. Here, we use the models (e) and (f) shown in the figure 1 to investigate the effects of oxygen and hydroxyl functional groups on the friction properties of BP, respectively. Figures 7(b) and (c) show the PES of BP with oxygen and hydroxyl functional groups sliding relatively under zero load, respectively. To facilitate comparison, the potential energy surface of the commensurate BP with relative sliding under zero load is also shown in figure 7(a). It can be seen from the potential energy surfaces that the friction between the interfaces of BP containing hydroxyl is obviously higher than that of pristine and oxidized BP. Figures 7(d) and (e) show the potential energy as a function of the sliding distance along the armchair and zigzag direction under zero load respectively. For the armchair direction, due to the introduction of hydroxyl, the friction between BP interfaces increases. However, for the zigzag   direction, the introduction of hydroxyl can reduce friction. In order to consider the effect of load on friction, frictional force as a function of load is also shown in the figures 7(f) and (g). In general, the introduction of oxygen functional groups can reduce the friction between BP interfaces. However, the introduction of hydroxyl increases friction between them.
In order to explain the friction changes caused by the introduction of functional groups, charge density difference shown in figure 8 have been calculated. It can be seen from the figure that for oxidized BP, due to the introduction of oxygen functional groups, the charge transfer between the BP interfaces is significantly reduced and the interaction is weakened, so the friction is reduced. However, due to the introduction of hydroxyl, the charge transfer between the BP interfaces is significantly increased and the interaction is enhanced, so the friction is greater. It is worth noting that due to the introduction of hydroxyl, the formation of P-H bond (bond length 2.35 Å) between the relatively sliding BP interfaces may be the main reason for increasing the interface friction.

Conclusions
The friction characteristics of black phosphorus were systematically studied by the first-principle method. The conclusions are as follows: (1) the friction between the black phosphorus interfaces is anisotropy. (2) The friction between the black phosphorus interfaces shows a structural superlubricity property. (3) Oxidation is beneficial to the lubrication of black phosphorus interface. (4) Due to the P-H bond between interfaces, hydroxyl groups are not conducive to the lubrication of black phosphorus interfaces.