Multiscale simulation on shearing transitions of thin-film lubrication with multi-layer molecules

TL;DR

This paper uses a multiscale simulation to study shearing transitions in thin-film lubrication, revealing how coupling strength and load affect slip behaviors and film thickness, aligning with experimental observations.

Contribution

It introduces a multiscale method to analyze shearing transitions in multi-layer molecular thin films, highlighting the effects of coupling strength and load on tribological behavior.

Findings

Three types of interlayer slip match experimental results.

Tribological behaviors are insensitive to small coupling strength.

Effective film thickness increases with larger coupling strength, decreases with load.

Abstract

Shearing transitions of multi-layer molecularly thin-film lubrication systems in variations of the film-substrate coupling strength and the load are studied by using a multiscale method. Three kinds of the interlayer slips found in decreasing the coupling strength are in qualitative agreement with experimental results. Although tribological behaviors are almost insensitive to the smaller coupling strength, they and the effective film thickness are enlarged more and more as the larger one increases. When the load increases, the tribological behaviors are similar to those in increasing coupling strength, but the effective film thickness is opposite.

Equations

$$x_i^t=x_i^b+\alpha a,$$

$$V_{eff}({\mathbf{R}}^{N_n})=\sum _{e=1}^{N_e}(N_a^e{\tilde{u}}_e+N_s^e{f}_e),$$

$${\tilde{u}}_e=\frac{1}{2}\sum _{j\ne i}{u}_{ss}(r_{ij}),$$

$$u_{ab}=\left\{\begin{array}[]{ll}\phi_{ab}(r)-\phi_{ab}(r_{c}),&\mbox{if}~{}~{}~{}r<r_{c};\\ 0,&\mbox{if}~{}~{}~{}r\geq r_{c},\end{array}\right.$$

$${\varphi }_{ab}(r)=4{ϵ}_{ab}[(\sigma /r{)}^{12}-(\sigma /r{)}^6],ab=ff,fs,ss.$$

$$f_e=2k_{B}T\ln [h(\det D{)}^{1/4}/k_{B}T],$$

$$\begin{array}[]{ll}U_{cf}({\bf r}^{N_{f}},{\bf r}^{N^{\prime}_{s}},{\bf R}^{N_{n}})=&\frac{1}{2}\sum_{i=1}^{N_{f}}\sum_{j\neq i}^{N_{f}}u_{ff}(r_{ij})+\sum_{i=1}^{N_{f}}\sum_{j=1}^{N^{\prime}_{s}}u_{fs}(r_{ij})\\ &+\frac{1}{2}\sum_{i=1}^{N^{\prime}_{s}}\sum_{j\neq i}^{N^{\prime}_{s}}u_{ss}(r_{ij})+\frac{1}{2}\sum_{i=1}^{N^{\prime}_{s}}\sum_{j=1}^{N^{\prime\prime}_{s}}u_{ss}(r_{ij})\\ &+\frac{1}{2}\sum_{i=1}^{N_{w}}\sum_{j=1}^{N^{\prime\prime}_{s}}u_{ss}(r_{ij})+\sum_{e=1}^{N_{e}}N_{a}^{e}[\frac{1}{2}\sum_{j\neq i}u_{ss}(r_{ij})]\\ &+\sum_{e=1}^{N_{e}}N_{s}^{e}2k_{B}T{\rm ln}[\hbar({\rm det}D)^{1/4}/k_{B}T]\end{array}$$

$$\begin{array}[]{ll}T_{yx,cf}=&\frac{1}{2L_{x}}\sum_{i=1}^{N_{f}}\sum_{j\neq i}^{N_{f}}<u^{\prime}_{ff}(r_{ij})x_{ij}y_{ij}/(r_{ij}L_{y})>\\ &+\frac{1}{L_{x}}\sum_{i=1}^{N_{f}}\sum_{j=1}^{N^{\prime}_{s}}<u^{\prime}_{fs}(r_{ij})x_{ij}y_{ij}/(r_{ij}L_{y})>\\ &+\frac{1}{2L_{x}}\sum_{i=1}^{N^{\prime}_{s}}\sum_{j\neq i}^{N^{\prime}_{s}}<u^{\prime}_{ss}(r_{ij})x_{ij}y_{ij}/(r_{ij}L_{y})>\\ &+\frac{1}{2L_{x}}\sum_{i=1}^{N^{\prime}_{s}}\sum_{j=1}^{N^{\prime\prime}_{s}}<u^{\prime}_{ss}(r_{ij})x_{ij}y_{ij}/(r_{ij}L_{y})>\\ &+\frac{1}{2L_{x}}\sum_{i=1}^{N_{w}}\sum_{j=1}^{N^{\prime\prime}_{s}}<u^{\prime}_{ss}(r_{ij})x_{ij}y_{ij}/(r_{ij}L_{y})>\\ &+\frac{1}{L_{x}}\sum_{e=1}^{N_{e}}N_{a}^{e}[\frac{1}{2}\sum_{j\neq i}<u^{\prime}_{ss}(r_{ij})x_{ij}y_{ij}/(r_{ij}L_{y})>]\\ &+\frac{1}{L_{x}}\sum_{e=1}^{N_{e}}N_{s}^{e}[\frac{k_{B}T}{2}<\frac{1}{|D|}\frac{\partial|D|}{\partial{\alpha a}}>],\end{array}$$

Full text

Keywords  Thin-film lubrication; Coarse-graining model; Shear transitions; Tribology

1 Introduction

A thin fluid film confined between two solid substrates may prevent surfaces of the substrates coming into molecular contact and becoming damaged in the dry friction, which is termed as thin-film lubrication. The systematic friction force is reduced three orders of magnitudes when the dry friction is replaced by the thin-film lubrication. The thin-film lubrication is a very interesting topic on both fundamental theory and engineering application, such as viscous dampers, magnetic storage devices and designs of MEMS/NEMS[1, 2, 3]. A stick-slip phenomenon (the freezing-melting transition) appears in the molecularly thin film lubrication when the substrates are assumed as rigid. The critical friction force, which is quantized with the number of fluid molecular layers separating the surfaces, decreases as the number of fluid molecular layers increases[4, 5, 6, 7]. For the thin film lubrication with multi-layer fluid molecules, the slipping process may have the partial and whole film melting due to the different interaction potentials between the solid and fluid molecules[8, 9]. Although the surfaces of the substrates are smooth on macroscopic scale, they actually make molecular contact through the medium of the thin-film. In particular, the solid substrates may have multiple spatial scales and influence the microscopic motions of both the solid and fluid molecules near/within the film[10, 11, 12]. It is therefore necessary to include the details of molecular motions in the substrates to treat the shearing transitions of the thin-film lubrication with multi-layer fluid molecules. To fully understand the physical mechanism of the thin-film lubrication system, the heterostructure interface is described by an atomistic model, while the elastic behavior of the much thicker substrate is modeled by the finite element method. Such multiscale problem of the thin-film lubrication system must be reflected in a theoretical model.

During the past three decades a series of effective methods combining atomistic and continuum descriptions of fluids and solids are developed[13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 35]. On the one hand, several effects on fluid flow near the walls have been made by applying the molecular dynamics to follow the trajectories of the fluid atoms in the layers near the walls and extend the non-sliding boundary conditions near the walls to the sliding ones. Beyond that the hydrodynamic equations are employed, the molecular dynamics and hydrodynamic treatments are self-consistently joined in the transition region that overlaps the two regimes of the atomistic and continuum descriptions. On the other hand, within the context of solid materials, two kinds of the effects to combine atomistic and continuum descriptions of the behaviors of materials have been completed at zero temperature. One is the FEAt procedure, where a core region described at the atomistic scale is surrounded by a transition region connected with the continuum regions covered by finite elements(FE). The other is a quasicontinuum treatment, where the whole lattices of material are overlayed with a FE mesh. The energy density in each FE is evaluated with the movements of its nodes. By including effects of thermal motions on the dynamics of certain processes, the FEAt and quasicontinuum methods are extended to finite temperature. Moreover, in a treatment of crack propagation in pure silicon, to make a multiscale modeling approach that dynamically couples different length scales and accounts for thermal effects, the macroscopic atomistic ab-initial dynamics(MAAD) is developed. The molecular dynamics(MD) with the tight-binding approximation, the MD with the semi-empirical potential energy and the FE method are applied in the different length scales ranging from the atomic scale through the microscale and finally to the mesoscale/macroscale. In particular, the soft matter as a subfield of condensed matter comprising a variety of biological systems such as macromolecular assemblies has both fluid and solid properties. To investigate its long time- and length-scale behaviors, the multiscale approaches with quasi-equilibrium assumptions[36, 37], the hybrid continuum method mixing the classical force field and coarse grained model[38, 39] and the multiscale algorithm with the order parameters [40, 41] have been developed. The multiscale analysis as a powerful methodology plays a critical role in the biological processes.

It is noted that for the thin-film lubrication system, the evolution of the system is slower than the molecular motions. In other words, on the time scale of the relevant process the system remains thermodynamic equilibrium. Such reversible or quasi-static processes of the thin-film lubrication can be described by means of Monte Carlo methods. In previous articles[42, 43] we developed the multiscale description of fluid-solid interfacial systems and applied it to treat the monolayer film lubrication with elastic substrates. The far-region solid substrates are coarse-grained by local and nonlocal elements. The systematic free-energy related to the elements is corrected by the local harmonic approximation. The principal conclusion of papers is that the hybrid atomistic-coarse-graining (HACG) scheme yields the shear-stress profiles and the mean separation curves in good agreement with those in the fully atomistic description of the system over a wide variety of conditions. However, since the simulations were performed only for a monolayer molecularly film lubrication, an extension to multi-layer molecularly film lubrication systems and an investigation of microscopic structures during the shearing slips are the purpose of the present article. In particular, the tribological behaviors in the multi-layer molecularly film lubrication system not only depend on the elasticity of substrates, but also on the adhesion strength between the film and substrates and the cohesive strength of the film. It was shown that an optimal choice of the interaction within fluids relative to the interaction between the fluids and substrates may reduce the kinetic friction force of the system[44]. Therefore, it is important to investigate the effects of the adhesion strength between the film and substrates on the shearing transitions in the system when the cohesive strength of the film is fixed.

The paper is organized as follows. In section 2, a short description of the coarse-graining model and the computational procedure is provided. The numerical results for two cases of three-layer and four-layer molecularly film lubrication are analyzed in section 3. Finally, in section 4, some conclusions and discussions are given.

2 Coarse-graining model and computational procedure

The idealized 2D contact consists of two identical hexagonal close-packed crystalline substrates separated by a multi-layer molecularly thin-film at an atomically flat interface, as shown schematically in Fig. 1. The top/bottom wall is taken to be rigid with the nearest-neighbor distance, i.e., the lattice constant ${\it a}$. The bottom wall remains stationary in the “laboratory” reference frame; the top wall can be translated in the $x$- and $y$- directions, but remains parallel with the bottom wall. The walls serve as handles by which the substrates can be manipulated. The lateral alignment of the walls is specified by the register $\alpha$, which is defined by

[TABLE]

where $x^{t}_{i}$ u and $x^{b}_{i}$ denote the corresponding atomistic positions in the top and bottom walls, respectively. $\alpha$ is the fraction of the lattice constant by which the top wall is displaced laterally with respect to the bottom wall.

The tribological system only comprises the solid substrates plus the fluid film, but does not cover the walls. The solid substrates are divided into the near- and far-regions, which are depicted by using the atomistic and coarse-grained descriptions, respectively. The coarse-grained far regions of substrates are covered with a mesh of triangular elements. This coarse-graining partitions the original substrate atoms into two subsets: $N_{n}$ nodal atoms and $N_{s}$ non-nodal atoms. If one integrates the Boltzmann factor over the 2$N_{s}$ degrees of freedom of the non-nodal atoms, one obtains an effective potential energy governing the motion of just the nodal atoms

[TABLE]

where ${\bf R}^{N_{n}}$ stands for the nodal configuration, $N_{e}$ for the number of elements, $N_{a}^{e}$ the number of atoms underlying element $e$. The configurational energy per atom $\tilde{u}_{e}$ is expressed as

[TABLE]

where $i$ denotes the “centroid” atom (i.e., the atom nearest the centroid of $e$) and $j$ labels atoms that lie within the circle of radius $r_{c}$ that is centered on $i$. The following shifted Lennard-Jones (12,6) potentials are taken as the pair interactions of the atoms

[TABLE]

where

[TABLE]

The effective diameter $\sigma$, $\epsilon_{ab}$ and range $r_{c}$ is the same for all pairs. $f_{e}$ is the Helmholtz energy per atom. By using the local harmonic approximation[45], $f_{e}$ can be estimated as

[TABLE]

where the elements of the $2\times 2$ dynamical matrix are given by $D_{kl}=m^{-1}(\partial^{2}\tilde{u}_{e}/\partial x_{k}\partial x_{l})_{0}(k,l=1,2)$, $k$ and $l$ label Cartesian components $(x_{1}=x,x_{2}=y)$ of the position of the reference atom, and the subscript 0 signifies that the partial derivative is evaluated at the equilibrium configuration. To show the dynamic properties of molecularly thin-film lubrication, it was found that the numerical results based on the Lennard-Jones potentials are in qualitative agreement with the experimental ones[4].

The total configurational energy $U_{cf}$ and the shear stress $T_{yx,cf}$ of the free-energy corrected HACG system are respectively given

[TABLE]

and

[TABLE]

where $N^{\prime}_{s}$ stands for the number of atoms in the near regions of the substrates, $N^{\prime\prime}_{s}$ for the number of underlying atoms in the far regions, $N_{e}$ for the number of elements, $N_{a}^{e}$ ($N_{s}^{e}$) the number of atoms (non-nodal atoms) underlying element $e$, $k_{B}$ is Boltzmann s constant, $\hbar$ is Planck s constant and $D$ is the local dynamical matrix associated with the centroid atom in a local element. The sliding with the register $\alpha$ is viewed as a quasistatic (or reversible) process. To compute the thermomechanical properties in the forward and backward processes we perform isothermal-isobaric Monte Carlo (MC) simulations. More details of the numerical method were presented in [42, 43].

For the simulations, Table I lists the values of the various parameters in the system. Numerical values are expressed in dimensionless units based on the Lennard-Jones parameters for the solid-solid interaction: distance is expressed in units of $\sigma$; energy in units of $\epsilon_{ss}$; stress in units of $\epsilon_{ss}/\sigma$; temperature in units of $\epsilon_{ss}/k_{B}$. The film-substrate attractive well depth is supposed to obey a modified combining rule $\epsilon_{fs}=\kappa\sqrt{\epsilon_{ff}\epsilon_{ss}}=\kappa/3$, where $\kappa$ is an adjustable constant. $\epsilon_{fs}$ and $\epsilon_{ff}$ represent approximately the adhesion strength between the film and substrates and the cohesive strength of the film in the system, respectively. In the following, we take two cases of (three-layer and four-layer) molecularly thin-film to investigate effects of the film-substrate coupling strength and loads on the tribological dynamical processes of the thin-film lubrication system at fixed $\epsilon_{ss}$ and $\epsilon_{ff}$.

3 Results

3.1 Three-layer molecularly thin-film lubrication

3.1.1 Effects of film-substrate coupling strength at a fixed load

Figure 2 displays plots of the shear stress $T_{yx}$ and the mean separation $L_{y}$ between walls for the three-layer molecularly thin-film lubrication system at the load $T_{yy}$=-1.0 as a function of register $\alpha$ for different $\kappa(\epsilon_{fs})$. Since all shear-stress and mean separation profiles possess similar behaviors, we take $\kappa$=1 ($\epsilon_{fs}$=1/3) as an example to summarize their common features. The profiles display a forward periodic process in $\alpha$ with a period length 1, which composites two elastic (stick) regions ($0\leq\alpha<0.6$, $0.7<\alpha\leq 1$) and a transition (slip) region ($0.6\leq\alpha\leq 0.7$). At $\alpha=0$, fluid atoms with an initial random distribution converge to a stable close-packed structure with solid atoms of the upper and lower substrates as shown in Fig. 3(a). The perfect symmetric atomistic structure exhibits the system in an equilibrium state and leads to $T_{yx,cf}$=0. As $\alpha$ increases gradually in the range $0<\alpha<0.6$, the upper and lower substrates adsorbing the film just like a whole crystal to generate an elastic shear behavior in the $x$ direction. The film atoms as in a solid state hold the shear stress in the system. A net force provided to the upper substrate/the film by the film/the lower substrate varies stronger and leads to the increase of $T_{yx,cf}$. Meanwhile, to hold the shear stress in the thin film, the effective film thickness (or the mean separation $L_{y}$) also increases. As $\alpha$ varies in the range $0.6\leq\alpha\leq 0.7$, the systematic shear strain reaches a critical value, which corresponds to the largest shear stress. For the lubrication system, the interaction between the film and the substrates or among the interlayers in the film cannot hold the strong shear stress. Under the action of the shear stress, the interlayer atoms in the film or the surface atoms of the upper substrate abruptly stride over the atoms in the layer touching with them as barriers, so that the upper and lower substrates make a relative slippage to restrain the increase of the systematic shear stress. After the sliding process, the system performs again an elastic shear behavior. Since the net force provided to the upper substrate/the film by the thin film atoms/the lower substrate has changed its direction, the shear stress is negative. In this process, the effective film thickness (or the mean separation $L_{y}$) reaches the largest value and then drops gradually. When $\alpha$ increases continuously in the range $0.7<\alpha<1$, the systematic behavior is asymmetric with that at $1-\alpha$. When the register finally reaches $\alpha=1$, the system is precisely in the same state as it is at $\alpha=0$. As $\alpha$ increases from 1 to 1.2, the above profiles in the range $0<\alpha\leq 0.2$ are repeated. Shearing from $\alpha=1.2$ to $\alpha=0$ carries the system in reverse through the same states as shearing in the forward direction. In the forward and backward processes, the slip process deviates from the center $\alpha=0.5$ but appears at its both sides duo to the longer relaxation time required to reach the quasi-equilibrium state[46, 47]. So the paths for the stick and the slip processes constitute a hysteresis loop.

In Fig. 2(a), the thermal elastic coefficient ($c_{\kappa}=\frac{\Delta T_{yx,cf}}{\Delta\alpha}$) in the stick process decreases when $\kappa$ decreases from 3 to 1/6. It is almost a constant when $\kappa$ decreases over $1/6\leq\kappa\leq 1/20$. The thermal elastic coefficient reflects the ability of the system resisting the shear strain. The larger $\kappa(\epsilon_{fs})$ has influence on $c_{\kappa}$ while the smaller $\kappa(\epsilon_{fs})$ is insensitive to it. So only the larger one between $\epsilon_{fs}$ and $\epsilon_{ff}$ is more significant to determine the thermal elastic coefficient of the system. Meanwhile, the width of the hysteresis loop reveals the deviation of the system from the quasi-equilibrium state. The larger $\kappa(\epsilon_{fs})$ corresponds to the more width of the hysteresis loop while the smaller $\kappa(\epsilon_{fs})$ is insensitive to it. So the width of hysteresis loop has the above similar features with the thermal elastic coefficient. Moreover, in Fig. 2(b), the mean separation $L_{y}$ (or the effective film thickness) monotonically decreases with decreasing $\kappa$ except $\kappa$=3. In the interaction between the film and the substrates at $\kappa$=3 ($\epsilon_{fs}=\epsilon_{ss}$), the film atoms adsorbed by the substrates take effects as the surface atoms of the substrates. So the effective film thickness is reduced approximately from three-layer to monolayer. This leads to the decrease of $L_{y}$. In order to display the effects of $\kappa$ on the shearing transitions, three kinds of the atomistic configuration of the film and the near substrates at $\alpha=0$ and $\alpha=1.0$ for the different $\kappa(\epsilon_{fs})$ are shown in Fig. 3.

(1) The slip process appearing in the internal layers of the film atoms is termed as the internal slip. (i)For $\kappa=3$ and 1, $\epsilon_{fs}$(=1 and 1/3) is larger than $\epsilon_{ff}$. In the thin-film lubrication system, the weakest interaction among atoms appears in the film. The slip process happens between the first and second layers in the film as shown in Fig. 3(b). The fluid atoms of the first layer are moved a lattice distance to the right with respect to the second layer in the film. At the same time, the upper substrate adsorbing the fluid atoms of the first layer is moved together with the first layer in the film. (ii) For $\kappa=1/3$, $\epsilon_{fs}$(=1/9) is equal to $\epsilon_{ff}$. In the thin-film lubrication system, the weaker interaction of the atoms appears in the film and between the substrates and the film. The slip process happens between the second and the third layers in the film as shown in Fig. 3(c). The fluid atoms of the second layer is moved a lattice distance to the right with respect to the third layer in the film. In this case, the first layer in the film adsorbed by the upper substrate is moved together with the second layer in the film.

(2) The slip process appearing between the film and substrates is termed as the interfacial slip. (iii) For $\kappa=1/6$ and 1/9, $\epsilon_{fs}$(=1/18 and 1/27) is smaller than $\epsilon_{ff}$. In the thin-film lubrication system, the weakest interaction among atoms appears between the substrates and the film. The slip process happens between the upper substrate and the first layer in the film as shown in Fig. 3(d). The surface atoms of the upper substrate are moved a lattice distance to the right with respect to the first layer in the film. (iv) For $\kappa=1/12$, $\epsilon_{fs}$(=1/36) is smaller than $\epsilon_{ff}$. In the thin-film lubrication system, the weakest interaction among atoms also appears between the substrates and the film. The slip process happens between the third layer in the film and the lower substrate as shown in Fig. 3(e). The fluid atoms in the third layer are moved a lattice distance to the right with respect to the lower substrate.

(3) The slip processes appearing not only between the internal layers of the film atoms but also between the film and substrates are termed as the mixing slip, i.e., mixing of the internal and the interfacial slips. (v) For $\kappa=1/16$ and 1/20, $\epsilon_{fs}$(=1/48 and 1/60) is smaller than $\epsilon_{ff}$. In the thin-film lubrication system, the weakest interaction among atoms appears between the substrates and the film. The slip processes happen simultaneously between the upper substrate and the first layer in the film, between the first and second layers in the film and between the third layer in the film and the lower substrate as shown in Fig. 3(f). Only the second and the third layers in the film move together. The surface atoms in the upper substrate are moved a lattice distance to the right with respect to the first layer in the film. The fluid atoms in the second and third layers are moved a lattice distance to the left with respect to the first layer in the film and the lower substrate.

3.1.2 Effects of loads at a fixed film-substrate coupling strength

Figure 4 displays the shear-stress and the mean separation profiles of the three-layer molecularly thin-film lubrication at $\kappa=1$ ($\epsilon_{sf}=1/3$) for different load $-T_{yy}$ on the system. As the load $-T_{yy}$ increases, the thermal elastic constant $c_{1}$ increases and the width of the hysteresis loop enlarges. Meanwhile, the vertical movement of the atoms in the system is more restrained. It leads to the decrease of the effective film thickness (or the mean separation $L_{y}$). For $T_{yy}=-0.5$, the slip process happens between the second and third layers in the film. The fluid atoms of the second layer are moved a lattice distance to the right with respect to those of the third layer. At the same time, the first layer in the film adsorbed by the upper substrate is moved together with the second layer in the film. For $T_{yy}=-1.0$, the slip process happens between the first and second layers in the film. The fluid atoms of the first layer are moved a lattice distance to the right with respect to those of the second layer. For $T_{yy}=-1.5$, the slip process happens between the second and third layers in the film. The fluid atoms of the second layer are moved a lattice distance to the right with respect to those of the third layer. So the internal slip at the fixed $\kappa=1$ ($\epsilon_{sf}=1/3$) is kept as the load increases. In comparing with the effects of the film-substrate coupling strength on the system, it is found that the effects of the load $-T_{yy}$ on the system have similar tribological dynamical behaviors (the thermal elastic constant and the width of the hysteresis loop), but are opposite on the effective film thickness(or the mean separation $L_{y}$) in system.

3.2 Four-layer molecularly thin-film lubrication

3.2.1 Effects of film-substrate coupling strength at a fixed load

Figure 5 displays plots of the shear stress $T_{yx}$ and the mean separation $L_{y}$ for the four-layer molecularly thin-film lubrication system at the load $T_{yy}$=-1.0 as a function of register $\alpha$ for different $\kappa(\epsilon_{fs})$. Since all shear-stress profiles possess similar behaviors, we take $\kappa$=1 ($\epsilon_{fs}$=1/3) as an example to summarize their common features. The profiles display a forward periodic process in $\alpha$ with a period length 1. The periodic process composites two elastic (stick) regions ($-0.5\leq\alpha<0.1$, $0.15<\alpha\leq 0.5$) and a transition (slip) region ($0.1\leq\alpha\leq 0.15$). At $\alpha=-0.5$, the film atoms with an initial random distribution converge to a stable close-packed structure with the surface atoms of the upper and lower substrates as shown in Fig. 6(a). It exhibits the system in an equilibrium state with $T_{yx,cf}=0$. As $\alpha$ increases gradually from -0.5 to 0.1, the elastic shear behavior to the $x$ direction in the combination of the upper and lower substrates connected by the film is enhanced. This leads directly to the increase of $T_{yx,cf}$. As $\alpha$ increases in the range $0.1<\alpha<0.15$, the fluid atoms in the film abruptly stride over the atoms in the layer touching with them as barriers, so that the shear strain between the upper and lower substrates are varied from a positive value to a negative one. As $\alpha$ increases gradually from 0.15 to 0.5, the systematic behavior is asymmetric with that at $-\alpha$, i.e., the shear strain of the system is varied from the negative value to zero. When the register finally reaches $\alpha=0.5$, the system is in precisely the same state as it is at $\alpha=-0.5$. As $\alpha$ increases from 0.5 to 0.7, the above scenario ($-0.5<\alpha<-0.3$) is repeated. Shearing from $\alpha=0.7$ to $\alpha=-0.5$ carries the system in reverse through precisely the same states as shearing in the forward direction.

In Fig. 5, the thermal elastic coefficient $c_{\kappa}$ decreases when $\kappa$ decreases from 3 to 1/6. It is almost a constant when $\kappa$ decreases over $1/6\leq\kappa\leq 1/20$. Meanwhile, as $\kappa(\epsilon_{fs})$ decreases, the width of hysteresis loop has the above similar features with the thermal elastic coefficient. Moreover, the mean separation $L_{y}$ (or effective film thickness) monotonically decreases with decreasing $\kappa$ except $\kappa$=3. In the interaction between the film and the substrates at $\kappa$=3 ($\epsilon_{fs}=\epsilon_{ss}$), the film atoms adsorbed by the substrates take effects as the surface atoms of the substrates, so that the effective film thickness is reduced approximately from four-layer to two-layer. This leads to the decrease of $L_{y}$. In order to display the effects of $\kappa$ on the shearing transitions, two kinds of the atomistic configuration of the film and the near substrates at $\alpha=-0.5$ and $\alpha=0.5$ for the different $\epsilon_{fs}$ are shown in Fig. 6.

(1) The internal slip: (i) for $\kappa=$ 3, 1 and 1/3, $\epsilon_{fs}$(=1, 1/3 and 1/9) is not smaller than $\epsilon_{ff}$. The slip process happens between the second and third layers in the film as shown in Fig. 6(b). The fluid atoms of the second layer are moved a lattice distance to the right with respect to those of the third layer. In this case, the first layer in the film adsorbed by the upper substrate is moved together with the second layer in the film. (ii) For $\kappa=1/6$, $\epsilon_{fs}$(=1/18) is smaller than $\epsilon_{ff}$. The slip process happens between the third and fourth layers in the film as shown in Fig. 6(c). The fluid atoms of the third layer is moved a lattice distance to the right with respect to those of the fourth layer. At the same time, the second layer in the film is moved together with the third layer in the film.

(2) The interfacial slip: (iii) for $\kappa=1/9$ and 1/12, $\epsilon_{fs}$(=1/27 and 1/36) is smaller than $\epsilon_{ff}$. The slip process happens between the upper substrate and the first layer in the film as shown in Fig. 6(d). The surface atoms of the upper substrate are moved a lattice distance to the right with respect to the first layer in the film. (iv) For $\kappa=1/12$ and 1/16, $\epsilon_{fs}$(=1/36 and 1/48) is smaller than $\epsilon_{ff}$, the slip process happens between fourth layer in the film and the lower substrate as shown in Fig. 6(e). The fluid atoms in the fourth layer are moved a lattice distance to the right with respect to to the lower substrate.

(3) The slip processes appearing not only between the film and the upper substrate but also between the film and the lower substrate are also termed as the mixing slip, i.e., mixing of the upper and lower the interfacial slips. (v) For $\kappa=1/20$, $\epsilon_{fs}$(=1/60) is smaller than $\epsilon_{ff}$. The slip processes appear not only between the upper substrate and the first layer in the film but also between the fourth layer in the film and the lower substrate as shown in Fig. 6(f). The fluid atoms of the first, second, third and fourth layers are moved a lattice distance to the right with respect to the lower substrate. The surface atoms of the upper substrate are moved two lattice distance to the right with respect to the first layer in the film. So the upper and lower substrates have only the relative slippage with a lattice distance.

3.2.2 Effects of loads at a fixed film-substrate coupling strength

Figure 7 displays the shear-stress and the mean separation profiles of the four-layer molecularly thin-film lubrication at $\kappa=1$ ($\epsilon_{sf}=1/3$) for different load $-T_{yy}$ on the system. As the load $-T_{yy}$ increases, the thermal elastic constant increases and the width of the hysteresis loop enlarges. Meanwhile, $L_{y}$ (or the effective film thickness) becomes thinner. For $T_{yy}=-0.5$, the slip processes happen not only between the first and second layers, between the second and third atomistic layers in the film, but also between the third and fourth layers in the film. The fluid atoms of the third layer are moved a lattice distance to the right with respect to those of the fourth layer. The fluid atoms of the second layer are moved a lattice distance to the right with respect to those of the third layer. The fluid atoms of the first layer are moved a lattice distance to the left with respect to those of the second layer. So the surface atoms of the upper substrate and the fluid atoms of the first layer are moved together a lattice distance to the right with respect to those of the fourth layer adsorbed by the lower substrate. The slip processes with the multiple internal slips are also termed as the mixing slip. For $T_{yy}=-1.0$, the slip process happens between the second and third layers in the film. The fluid atoms of the second layer are moved a lattice distance to the right with respect to those of the third layer. For $T_{yy}=-1.5$, the slip process happens between the third and fourth layers in the film. The fluid atoms of the third layer are moved a lattice distance to the right with respect to those of the fourth layer. So in the four-layer thin-film lubrication at the fixed $\kappa=1$ ($\epsilon_{sf}=1/3$), the mixing slip is changed to the internal slip as the load increases. In comparing with the effects of the film-substrate coupling strength on the system, it is found that the effects of the load $-T_{yy}$ on the system have similar tribological dynamical behaviors (the thermal elastic constant and the width of the hysteresis loop), but are opposite on the effective film thickness (or the mean separation $L_{y}$) in system.

4 Conclusions and discussions

Shearing transitions of multi-layer molecularly film lubrication systems in variations of the film-substrate coupling strength and the load have been studied by using a hybrid atomistic-coarse-grained method. For the thin-film lubrication with multi-layer molecules at a fixed load, three kinds of the interlayer slips (the internal, the interfacial and the mixing slips) in the system are found as the film-substrate coupling strength decreases. The internal and the interfacial slips have one relative slippage in the interlayers of the systems, but the mixing slip has more relative slippages in them. For the three-layer molecularly thin-film lubrication, the mixing slip includes the internal slip and the upper and lower interfacial slips. For the four-layer molecularly thin-film lubrication, the mixing slip includes the upper and lower interfacial slips or the multiple internal slips. These phenomena are in qualitative agreement with the experimental results[8, 9]. Meanwhile, the tribological dynamical behaviors (the thermal elastic constant and the width of the hysteresis loop) are almost insensitive to the smaller film-substrate coupling strength. However, they and the effective film thickness are enlarged more and more as the larger film-substrate coupling strength increases. When the load increases, the tribological dynamical behaviors are similar to those in increasing film-substrate coupling strength, but the effective film thickness is opposite.

Acknowledgments This research is supported by the National Natural Science Foundation of China through the Grants No. 11172310 and No. 11472284, and the CAS Strategic Priority Research Program XDB22040403. The author thanks the IMECH research computing facility for assisting in the computation.