Friction law for atomic-scale contact assisted by atomistic simulations

Non-empirical law depicting how atomic-scale friction behaves is crucial to facilitate the practical design of tribosystems. However, progress in developing a practically usable friction law has stagnated because atomic-scale friction arises from the continuous forming and rupturing of interfacial chemical bonds and such interfacial chemical reactions are difficult to measure precisely in experiments. Here, we propose a usable friction law for atomic-scale contact by using atomistic simulations to correctly measure the interfacial chemical reactions of a realistic rough surface, and confirm its applicability to predicting how atomic-scale friction varies with temperature, sliding velocity, and load.


3
Friction is particularly important for reducing energy consumption because around 30% of the world's energy is lost to friction 1,2 . Generally, friction between two solids sliding relative to each other depends on not only the types of contacting materials but also external conditions such as temperature, applied load, and sliding velocity. Practically applicable friction laws that predict how these external conditions influence the friction force are crucial because they would benefit greatly the practical design of lubricating systems.
Conventionally, friction force ( ) between two surfaces is proportional to the applied normal force ( ). This linear relation between and is explained by the well-known Amontons' law, which states that the real contact area ( ) between two macroscale surfaces is generally proportional to 3,4 ; thus, if the shear stress on is reasonably assumed to be constant, then the linear relations ∝ ∝ appear. Recently, however, with the advances in nanotribological experiments, it has been found that frictional behavior at nanoscale contact does not generally obey Amontons' law 5,6 . Instead, nanoscale friction behaves like a thermally activated process 7 , and various analytical models have been established to explain the nanoscale friction data. For example, by treating nanoscale friction as a point-contact scheme in which a contacting atom is thermally activated to scan over a corrugated potential surface, successful predictions have been made regarding how the friction force of this contacting atom varies with temperature and scanning velocity, such as the Prandtl-Tomlinson (PT) model and its extended theories [8][9][10][11] . validation of a practically applicable friction law. In this paper, we propose that the interfacial 5 chemical reactions and real surface temperature could be monitored very effectively and precisely using the large-scale reactive molecular dynamics (MD) simulations with our previously developed high-accuracy reactive potentials [20][21][22] . We aim to establish a practically and easily applicable friction law that can predict how the friction force varies with the external conditions at the atomic scale, based on the precisely measured , , and surface temperature from MD simulations.

Results
We use diamond-like carbon (DLC) 23,24 as the test material because DLC is a widely used and studied solid lubricant. Fig. 1A shows the friction simulation model, in which two hydrogenpassivated DLC substrates with self-affine roughness 25  distributed homogeneously and is around 300 K; however, during friction, temperature increases with proximity to the friction interface. Here, is defined as the mean of the highest 6 temperature in the distribution profile, which is about 500 K higher than . Using this definition, we observe that the obtained increases almost linearly with increasing ( Fig. 1C). See the Supplementary Information for detailed simulation setups and discussions on . The next task is to know how friction force ( ) and interfacial chemical reactions are affected by in the MD simulations. Open squares in Fig. 2A   friction process is rather than or the environmental temperature. Meanwhile, the lateral force of rupturing an interfacial bond, , could be correctly obtained by the multi-bond model 9 proposed by Urbakh's group 5,[16][17][18][19] . However, Urbakh's multi-bond model is too complex to be applied simply, because its analytical expression involves multiple hard-to-determine empirical parameters. Instead, can be estimated by the simpler PT model because the thermally activated process of rupturing an interfacial bond can be regarded as the process of climbing a potential curve. Using a previous form 9 , we have where is potential curve corrugation of contact surface and is a function of the critical force ( ).
is a material-and system-dependent parameter as discussed exhaustively in the Supplementary   Information. Here, only and as fitting parameters should be determined by comparing with the experimental/simulation results. By fitting Eq. 3 to the MD-measured using = , we obtain = 2.006 nN and = 1.000. Thus, with the values obtained above for Δ , , and , Eq. 1 succeeds to predict a mountain-type temperature dependence of the friction force, which matches the MD-measured perfectly ( Fig. 2A).
It is also essential to precisely understand the frictional behaviors as well as the interfacial chemical reactions affected by different and , and we expect Eqs. 1-3 (with the above extracted , , and ) to be valid for describing the load and velocity effects. According to Eqs. 2 and 3, it is obvious that, if temperature is unchanged, and would only have influences on and , respectively. Thus, to check the relationship between and and assess the validity of Eq. 2, we perform friction simulations using the same model as that in Fig. 1A respectively. Focusing on ( ) firstly, the measured increases monotonically with , and then we should confirm whether this relationship obeys Eq. 2 or not. Note that will increase with even though is kept unchanged because larger applied load brings more frictional heat (Fig. S2A in the Supplementary Information). Obviously, to predict ( ), the exact values of must be taken into consideration; otherwise, for instance, if = is substituted into Eq. 2, the prediction (red line in Fig. 3A)   Here, as a trial application of Eqs. 1-3, we predict how and affect the temperature dependence of . In Fig. 2C Fig. 4B shows the predicted under a constant of 675 nN but different . 12 Herein we observe the entire shift-up of the curve with the increasing again, but interestingly the increases with increasing , differing from the load effect. The above predicted trends, namely, that the sliding velocity changes whereas the applied load does not, agree qualitatively with previous experiments 19,29 . This result helps the reduction of friction by choosing the proper temperature and sliding velocity, thereby benefiting greatly the practical design of lubricating systems. We must also point out the limitations of Eqs. 1-3 for friction prediction. First, they are not available for the systems in which non-bonding interactions such as van der Waals are dominant. 13 Such systems include (i) atomic-scale contact of ultra-smooth surfaces (because surface adhesion becomes a dominant factor with decreasing surface roughness 30,31 ) and (ii) friction of twodimensional materials such as graphene and MoS2 (because friction of these materials stems mostly from the van der Waals interactions between neighboring layers). Second, Eqs. 1-3 may fail under extremely high or low temperatures, loads, and sliding velocities, as discussed exhaustively in the Supplementary Information.

Discussion
In summary, we succeeded in using large-scale reactive MD simulations to precisely monitor the real surface temperature and interfacial chemical reactions of DLC in a realistic rough-surface contact state. Based on the MD simulation results, we established a proper theoretical friction law to describe the atomic-scale frictional behavior and further predict how the friction force varies with temperature, sliding velocity, and applied load. Furthermore, we showed that the actual temperature of the contacting surface must be used in the proposed friction model to practically and correctly predict the friction force; otherwise, the predictive results may deviate greatly if the substrate or environmental temperature is used in the proposed friction law. This work could contribute greatly to understanding the fundamentals of friction and the practical design of lubricating systems.

Data availability
All data needed to evaluate the conclusions are present in the paper and/or the Supplementary Information. Additional data are available from the authors upon request.