Yi-Wei Li(李毅偉) Peng-Fei Xu(許鵬飛) and Yong-Ge Yang(楊勇歌)

1Department of Mathematics,Shanxi Agricultural University,Jinzhong 030801,China

2School of Applied Mathematics,Guangdong University of Technology,Guangzhou 510520,China

Keywords: Frenkel–Kontorova model,Gaussian colored noise,hysteresis,super-lubricity

1. Introduction

Nano-friction is becoming more and more often a central issue in the emerging field of nanoscale science and technology, which is related to the atom processes occurring at the interface of two interacting systems in relative motion.[1–3]Recently, remarkable achievements have been made in nanofriction testing technology (atomic and friction force microscope,surface-force,and quartz-crystal microbalance)and the computing power(realistic molecular dynamics simulations).And these advances have largely encouraged to search for simple mathematical models capable of describing the essential physics in friction processes.[3]In particular, the driven Frenkel–Kontorova (FK)[4]models have recently received an increasing amount of attention and have been extensively investigated both theoretically and experimentally in the nanofriction areas.[5,6]The standard FK[3,4]model describes the dissipative motion of a chain of harmonically interacting particles that slides over a rigid substrate potential due to the application of an external driving force. Moreover, the classic FK model has been used to study the dynamic phenomena in nano-tribology, and an increasing interest as a possible interpretative key has been found to understand the atomic processes occurring at the interface of two materials in relative motion.[1–21]So far,the FK models have become an extremely important theoretical tool in the field of nano-tribology. Relevant generalizations of the FK models have also been proposed in the literature to cover a large class of physically interesting nano-friction phenomena,[7–13]such as hysteresis, superlubricity, and stick-slip,etc. Furthermore, both the dynamics process of friction and the mechanism of energy dissipation on the nanoscale are explained, according to the FK models and nano-friction.

However, most studies on FK models focus on deterministic systems,[7–13]and only a few involves the stochastic systems.[15–20]In fact,noise is usually inevitable as modeling a real system at the nanoscale level. Teki′cet al.[15–18]studied the noise effects on the dynamical mode-locking phenomena in the overdamped FK model. The effects of the temperature and substrate disorder on the FK model are investigated by Guerraet al.[19]The noise,[20,22]as a stochastic excitation,can not only make deterministic systems random but also change their dynamical behaviors. The interference of random factors is an important factor that cannot be ignored in the study of nonlinear dynamics. The random factor refers to Gaussian white noise or some associated colored noise. As the spectrum of Gaussian white noise[20]is unbounded and there is no Gaussian white noise in nature, the introduction of Gaussian white noise may go against the essential characteristics of some systems in nature.So the introduction of colored noise is very necessary. Therefore,the study of stochastic FK models under colored noise excitation has important practical significance for the understanding of nano-friction mechanism. This paper mainly studies the FK model under the Gaussian colored noise excitation and studies the variation of some nano-friction phenomena,such as hysteresis,maximum static friction force,and super-lubricity. The Langevin molecular dynamics approach allows us to introduce Gaussian colored noise via the inclusion of a stochastic force.[19,20]Working in the dissipative regime, we analyze the display of hysteretic behavior in theB(F) characteristics[19]for the variation of the external driving force and the relationship between the maximum static friction force and the noise parameters.

The rest of the paper is organized as follows.Section 2 introduces the driven FK model with the Gaussian colored noise excitation. In Section 3, the simulation results are presented.The effects of different parameters (such as the noise intensity and the correlation time) on the chain mobility and the maximum static friction force have been investigated in detail for incommensurate case and commensurate case,respectively. Finally,the main conclusions are drawn in Section 4.

2. Model

The system here describes the dynamics of a driven FK model, whoseNparticle positionsxisatisfy the following equation of motion:[3,19,20]

whereDis the noise intensity and the correlation timeτ=λ-1.

The ratioB=VCM/Fof the time-averaged CM velocity to the external applied force(the chain mobility).[19]Observation of finite static friction implies that the contacting solids have locked into a local energy minimum,andFsis maximum static friction force which represents the force needed to lift them out of it.

The researches of Vanossi[3]and Braun[9]on the FK models show that when strengthKis small[3,10]and the system is in an underdamped state,the hysteresis of the system is obvious which is also convenient for the explanation of some problems in this article. In this paper, we focus on the effects of Gaussian colored noise in the underdamped[2,20]dynamics of a onedimensional chain of interacting atoms sliding over a substrate potential. If not stated differently,the valuem=1,K=1,andγ=0.7 define our system units.Bf=(mγ)-1represents the maximum asymptotic value of the chain mobility.[19,20]

3. Results and disussion

The length scale competition between the substrate and interatomic potentials controls the static and dynamic behavior of the system,[3,4,17]resulting in a rich complexity of spatially modulated structures for the chain particles. Following the previous studies in the FK models,[1–3]we have studied the incommensurate case(b/a=144/233)and the commensurate case (b/a=1). For different values of the model parameters noise intensityDand the correlation timeτ, we explore the behavior of the chain mobility as a function of variations ofF.And the combined effects of noise intensityDand correlation timeτon the maximum static friction forceFshave also been investigated in detail.

3.1. Noise intensity effects

3.1.1. Incommensurate case (b/a = 144/233)

The behavior of the chain mobilityBas a function of variations of the driving forceF,for three different values of noise intensityDand fixed correlation times (τ=1,τ=10, andτ=100), is shown in Fig. 1, to investigate the effect of the noise intensityD. In this section, we consider the case of the incommensurate, and we choosea=1,b/a=144/233,c=a/β=144/89,so that the system size isL=144 and the chain is made up ofN=233 particles.[7,15]

Fig.1. Noise effects on static friction and hysteresis of the B(F)characteristics for the incommensurate case when D=0,0.005,0.05 for three cases:(a)τ =1,(b)τ =10,and(c)τ =100. Triangles and circles denote,respectively,the increasing and decreasing processes of F.

In Fig.1(a),when noise intensityD=0,a large hysteresis is clear. The larger value of noise intensity contributes to the disappearance of hysteretic and the decrease of the maximum static friction forceFs(whenD=0,Fs≈0.18;D=0.005,Fs≈0.15;D= 0.05,Fs＜0.017), which indicates that the noise makes the atoms move along the chain much easier than themselves. Moreover,as the value of theFrises,the increase ofDcomes with a greater chain mobilityB(whenF=0.6,D=0,B ≈0.58;D=0.005,B ≈0.62;D=0.05,B ≈0.65).Recent studies[19,20]of the one-dimensional FK models have demonstrated that noise excitation makes it easier for atoms to escape from the substrate potential. The results concluded from Fig.1 also verify this view.

Figures 1(a) and 1(b) have a similar conclusion that is the hysteresis and maximum static frictionFsof the system decline as the noise intensity grows. The difference between Figs. 1(a) and 1(b) can be concluded that the increase of the correlation timeτrenders a slower decline of the hysteresis andFs. As shown in Fig. 1(c) (τ=100), the comparatively large value of the correlation timeτmakes the hysteresis andFsalmost insusceptible to noise intensity. We can obtain from Fig. 1 that the greater the correlation timeτ, the slower the decrease of the hysteresis andFs. The result means that noise intensity has a positive effect on reducing hysteresis andFs,whereas a change in correlation time will hinder this process.

Fig.2. Noise effects on maximum static friction force for the incommensurate case.

Figure 2 displays the detailed behavior of the maximum static friction forceFswith growing noise intensityD. The maximum static frictionFsdrops asDincreases. However,this trend varies depending on the correlation time. The variation is dramatic whenτ=1 and invisible whenτ=10.In particular, with appropriate parameters (whenD ≈0.15,τ=1),the system will give rise to super-lubricity. And the region,in which super-lubricity can happen, is marked in Fig. 3. However, there is no super-lubricity while the correlation time is too large(τ=100)which indicates that super-lubricity is possible for us if we choose appropriate noise parameters.

Fig.3. Super-lubricity region for the incommensurate case.

Figures 1 and 2 show that for the incommensurate case,noise excitation makes it easier for atoms to escape from the substrate potential and accelerate the motion of the system. With suitable noise intensity and correlation time,superlubricity happens. To verify the relevant conclusions indepth,we have also carried out simulations for other irrational choices of the three characteristic lengthsa,b, andcof the model.

3.1.2. Commensurate case (b/a = 1)

In this section, we shall focus on the hysteretic behavior ofBas a function ofFand the effect of noise intensity imposed onFsfor the case of a commensurate choice among the three model length scales already considered in a previous study.[19,20]Therefore, the numerical results refer to a substrate potential characterized by the parametersa=1 andc=a/β=30/24. The simulations are performed for a sizeL=140 and a chain made up ofN=140 particles.[2,10]

Fig.4. Noise effects on static friction and hysteresis of the B(F)characteristics for the commensurate case when D=0, 0.005, 0.05 for three cases:τ =1,τ =10,and τ =100. Triangles and circles denote the increasing and decreasing process of F respectively.

Figure 4 shows the noise intensityDeffects on the mobility-driving characteristics for a commensurate interface.In Fig. 4, when the correlation time is fixed, the width of the hysteretic region and the maximum static friction forceFsdecrease to varying degrees due to the increase ofD.This trend is less pronounced when the value of correlation time is large. Moreover, as shown in Fig.4(a), the introduction of the noise accelerates the chain motion of the system as the forceFrises (whenF= 0.6,D= 0,B ≈0.6;D= 0.005,B ≈0.63;D=0.05,B ≈0.8), the greater the noise intensity,the greater the chain mobility. The relevant change diminishes with the increase of correlation time which can be observed from Figs.4(b)and 4(c). As a result,the noise intensity tends to reduce the hysteretic region at finite correlation time, destroy the occurrence of parametric resonances inside the chain,and favor sliding with higher mobility values.

Conclude from Figs. 1(a) and 4(a): for a commensurate interface,the chain length isL=144,and the number of atoms isN=233,whilst an incommensurate case has the followingL=140 andN=140.Albeit similar in chain length, despite having fewer atoms, when the commensurate interface has a noise intensity ofD=0,its hysteresis region andFsare significantly larger in comparison.Figure 1(a)shows that the hysteresis andFsvary by noise. The distinction between the two circumstances is that for the commensurate interface (Figs. 4 and 5), the atoms are more strongly coupled and entrapped by the substrate potential[5,10,19]so that the introduction of the noise make this coupling unstable and makes the atoms move along the chain much easier than before. Therefore,the influence is more vigorous for the commensurate case.

Fig.5.Noise effects on maximum static friction force for the commensurate case.

The variation of the maximum static friction forceFswith respect to noise intensityDfor the commensurate interface is plotted in Fig.5. It is found that the increase ofDcontributes to the decrease ofFs. The shorter the correlation timeτ, the more obviouslyFsvaries.Figure 6 depicts the region,in which the super-lubricity can appear. It is noteworthy that the system is much easier to give rise to super-lubricity for the incommensurate interface (τ ≈0.01,D ≈0.1) than the commensurate interface(τ ≈0.01,D ≈0.1).

Fig.6. Super-lubricity region for the commensurate case.

3.2. Correlation time effects

The above simulation results indicate that the hysteretic and maximum static friction forceFsdepends notably on the correlation timeτ.In order to study the effect of the correlation time on the nano-friction phenomena in more detail, we plot the variation of theB(F)for different values of correlation timeτwith different noise intensitiesD.Meanwhile,the maximum static friction forceFswith respect to correlation timeτis given. This section is divided into the incommensurate case and the commensurate case.

3.2.1. Incommensurate case (b/a = 144/233)

In this section,we have considered the case of the incommensurate, so we choosea=1,b/a=144/233,c=a/β=144/89,so that the system size isL=144 and the chain is made up ofN=233 particles.[2,7]

As depicted in Fig. 7, for the incommensurate case, the correlation time has a clear effect on the hysteresis and maximum static friction forceFs.And the hysteretic region tends to increase with increasing values of correlation timeτ, and the correlation time also affects the maximum static friction forceFs(in Fig. 7(a),τ=0.1,Fs≈0.11;τ=1,Fs≈0.15;τ=10,F ≈0.17). Figures 7(a) and 7(b) demonstrate similar conclusion,however,under the fixed noise intensityD,the trend of increasing the hysteretic region andFsare different due to different correlation time.

Fig. 7. Correlation time on static friction forces as well as the hysteresis for the incommensurate case, in which triangle and circle denote the processes of increasing and decreasing external forces depicting the hysteresis behavior for two cases: (a)D=0.005,(b)D=0.05.

Figure 8 displays the detailed behavior of the maximum static friction forceFswith the change of correlation timeτat different noise intensityD.Fsis increased asτgrows,and eventually reaches a relatively stable value (D=0.005,Fs≈0.18;D= 0.05,Fs≈0.17). The difference between the two situations (D=0.005,D=0.05) is that the greater the noise intensity,the more sharply the increase. The results mean that the correlation time encourages the increase of the hysteresis and the maximum static friction forceFs,however,the noise intensity plays the opposite role. The findings summarized in Figs. 7 and 8 verify some of the conclusions in Subsection 3.1.

Fig.8.Correlation time on maximum static friction force for the incommensurate case.

3.2.2. Commensurate case (b/a = 1)

In order to further study the effect of the correlation time on the nano-friction phenomena, this section investigates the commensurate case. Therefore, the numerical results refer to the substrate potential characterized by the parametersa=1 andc=a/β=30/24.In this case, the chain lengthL=140 and the number of atomsN=140.

Fig.9. Correlation time effects on static friction and hysteresis of the B(F)characteristics for the incommensurate case, when τ =0.1, 1, 10 for two cases: (a)D=0.005,(b)D=0.05.

Figure 9 gives the process of the hysteretic behavior in the characteristicsB versus Ffor the commensurate interface.It is shown that for given values of noise intensityD(whereD=0.005, 0.05 respectively), with the increase of the correlation timeτ,the region of the hysteretic is increased. Meanwhile, the maximum static friction forceFsgoes up with increasing the correlation times(in Fig.9(a),τ=0.1,Fs≈0.25;τ= 10,Fs≈0.3). To sum up, the difference between the Figs.9(a)and 9(b)is the increased tendencies of the hysteresis region which is related to noise intensityD.

For the commensurate case,the variation of the maximum static friction forceFswith respect to correlation timeτfor different noise intensitiesDis plotted in Fig. 10. The maximum static friction forceFsincreases to a relatively stable value(D=0.005,τ ≈0.33;D=0.05,τ ≈0.31)whileτincreases. The greater the noise intensity, the more evident the increase.

Fig.10. Correlation time on maximum static friction force for the commensurate case.

It is summarized from Figs.7–10 that for commensurate interface,the influence of colored noise on hysteresis andFsis more evident. From what we have mentioned above,noise intensityD=0.05,correlation time rises fromτ=0 toτ=10,hystersis region (Figs. 7(b) and 9(b)) and the maximum friction (Figs. 6 and 8) both experienced the processes from appearing to growing. However, difference also exists. Given the larger increase ofFsfrom 0 to 0.31 for commensurate interface(Fig.7(b),τ=10)and from 0 to 0.17 for incommensurate case(Fig.5(b),τ=10)and wider hysteresis area for the former interface,it is easy to conclude that noise poses a more significant effect on hysteresis andFsfor commensurate case.

4. Conclusions

In this paper,we have investigated the effects of Gaussian colored noise on a one-dimensional chain of interacting atoms driven by an external force. Starting from the two geometrically opposite ideal cases of commensurate and incommensurate interface. In particular,we have focused our study on the variation regularity of the hysteretic behavior and the maximum static friction force which are affected by the Gaussian colored noise.

The results indicate that the noise intensity has a positive effect on reducing hysteresis andFs,whereas the change in correlation time hinders this process. In particular,suitable correlation time and noise intensity give rise to super-lubricity.The difference between the two circumstances is that for the commensurate mating contacts, the influence of the noise is much stronger in terms of triggering the motion of the FK model than for the incommensurate interface since the atoms in the former case are coupled and entrapped more strongly by the substrate potential. We hope that the results presented in this work may be relevant to future theoretical and experimental studies concerning microscopic tribology of the real physical systems,where the geometrical features of the interfaces in relative motion could play a major role.

Acknowledgements

Project supported by the National Natural Science Foundation of China(Grant No.11902081),the Science and Technology Innovation Foundation of Higher Education Institutions of Shanxi Province, China (Grant No. 2020L0172), the Natural Science Foundation for Young Scientists of Shanxi Agricultural University,China(Grant No.2020QC04),and the Research Fund of Shanxi Agriculture University,China(Grant No.2021BQ12).