Nanotribology: Simulation Results

Soft and hard lubricants
Soft lubricant
Hard lubricant
Role of substrate geometry
Conclusion

Soft and hard lubricants

Constant-force algorithm: hysteresis

Simulation results for V_ss=3, V_sl=1/3, lattice=12×11, N_l=1 to 5:

"soft" lubricant: V_ll <<V_sl (V_ll = 1/9) two lubricant layers are “glued” to the substrates sliding occurs at a middle of the lubricant lubricant melts at the onset of sliding stick-slip motion is due to melting/freezing mechanism	"hard" lubricant: V_ll >> V_sl (V_ll = 1) lubricant film remains solid at sliding sliding – at the lubricant/substrate interface(s) stick-slip motion is due to inertia mechanism crystalline structure → perfect sliding (superlubricity)

N_l=5 (enlarged view)	N_l=1 & 5 (enlarged view)

Spring algorithm: stick-slip

Simulation results for k_spring= 3·10^-4

Soft lubricant: V_ll = 1/9 melting/freezing mechanism (v_spring= 0.03)	Hard lubricant: V_ll = 1 (ideal structure) inertia mechanism (v_spring= 0.01)

N_l=3 movie (avi 2 Mb)	N_l=3 movie (avi 1.6 Mb)

The melting temperature T_m of a thin lubricant film confined between two solid surfaces is higher than the corresponding bulk value, because the confinement decreases the entropy of the film and shifts the bulk melting transition to higher temperatures.

The melting temperature T_m as a function of the number of lubricant layers N_l for f_load= 0.1.
The filled markers correspond to MD simulation results, while the open markers are the theoretical values.

The solid curves describe the fits
T_m= 0.405 + 0.165/N_l for the hard lubricant and
T_m= 0.045 + 0.350/N_l for the soft lubricant.

The dependence T_m(N_l) can be calculated analytically using the empirical Lindemann criterion: the melting starts when the amplitude of mutual displacement of nearest neighboring atoms reaches some threshold value of order 10% of the lattice constant. In the case of a thin film, mutual displacements decrease for the confined film, where the oscillations of the boundary atoms are suppressed. The mutual displacements are maximal at the lubricant/substrate interface for the hard-lubricant case, and at the middle of the soft-lubricant film where the internal interactions are weaker than the interactions with the substrates. Therefore, in the hard-lubricant system the melting starts at the interface, while in the soft-lubricant case the melting starts in the middle of the lubricant.
Thus, a very thin confined film is typically in a solid state. This results in a nonzero static frictional force f_s.

The dynamics of the film is significantly affected by the substrate, both in the solid and in the molten phases. The solid phase, able to sustain shear stress, shows large diffusional motions of the atoms; the molten phase shows a layered structure.
For more details, see O.M. Braun and M. Peyrard, Phys. Rev. E 68 (2003) 011506 "Dynamics and melting of a thin confined film" (pdf files may be found here).

Soft lubricant: melting/freezing mechanism

First, the soft lubricant freezes in a metastable configuration (which is different from stick to stick), where the two utmost lubricant layers are glued to the bottom and top substrates correspondingly and are characterized by a larger atomic concentration (e.g., for <10%) than the middle layers. The structure of the lubricant film has many defects (e.g., see two configurations below). The static threshold f_s grows with the time of stationary contact (aging of the contact). Because the substrate-lubricant interaction (V_sl=1/3) is much larger than the interaction inside the lubricant (V_ll=1/9), the sliding should begin somewhere within the lubricant, by depinning of the defects that stick the substrates.

Configurations of the N_l=5 lubricant just prior the beginning of sliding

The annealed configuration at f = 0.087	The stick configuration obtained during stick-slip motion with v_spring= 0.01 at f = 0.017

Second, there are two different steady-state sliding regimes, the "layer-over-layer sliding" (LoLS) regime, when the lubricant remains in the solid state, and the "liquid-sliding" (LS) regime, when the lubricant is melted due to sliding. These two regimes exist for different intervals of the driving force or velocity.

Third, the behavior of a very thin lubricant film with two or less atomic layers, differs qualitatively from the behavior of thicker films.

Two sliding regimes (LoLS and LS) for N_l>2

Figure soft5:

(a) the kinetic frictional force,

(b) the lubricant effective temperature, and

as functions of the driving velocity for the N_l=5 system.

Stars show the data obtained with the spring algorithm, while circles and dash curves show the results obtained with the constant-force algorithm (open circles are for the layer-over-layer sliding regime, and solid circles are for the liquid-sliding regime).

Layer-over-layer sliding. In the LoLS state the lubricant takes the well-defined layered structure; the utmost lubricant layers are glued to the corresponding substrates and move together with them, while the other (middle) layers slide (creep) one over another.

Figure soft7: Distribution of the x components of atomic velocities across the film in the LoLS regime at f = 0.008; see also the corresponding movie (animated gif, 3 Mb).

Bottom panel shows the distribution of atomic concentration across the lubricant.

The lubricant effective temperature grows with the velocity but remains lower than the melting one (T_melt≈ 0.11 for the five-layer soft flat system). Therefore, the lubricant is in a "solid" state; however, there is exchange of atoms between different lubricant layers in this state. Due to atomic exchange, the 2D atomic concentrations in the layers may vary during sliding and, when two adjacent layers occur to have close concentrations, they become commensurate, lock together and move with the same velocity, while the main sliding occurs at the most "incommensurate" interface (e.g., see Fig.soft7).

The layer-over-layer sliding is stable for the velocities 0.05 < v_top< 0.3 and forces 0.003 < f < 0.011 (for the five-layer film at f_load= 0.1; these intervals of velocities/forces shift to higher values as the load increases or N_l decreases). The system locks at lower forces/velocities, while at larger ones the lubricant melts and the system goes to the liquid-sliding regime.

Liquid sliding. The LS regime appears with the increase of the driving velocity (v_spring> 0.3) or force (f > 0.01 for N_l=5). At lower values of the forces/velocities, e.g., for f = 0.011 and v_top< 1 or ~1, the system exhibits smooth sliding with two utmost layers approximately ordered and glued to the corresponding substrates (the "glued" lubricant layer may exhibit a slow creep motion relative the substrate), while the middle layers are melted. When the force/velocity increases to f = 0.012 or v_top>> 0.3, the glued layers melt as well. In the LS regime the x-component of the atomic velocities exhibits a linear gradient across the melted film. The sliding heats the lubricant to a temperature larger than the melting one, so that the driving itself maintains the lubricant in the molten state.

The transition from stick-slip to smooth sliding with the increase of driving velocity:

Figure soft26: Dynamics of the N_l=3 flat system with the attached spring (k_spring= 3×10^–4) for three driving velocities: v=0.03 (left column), v=0.1 (middle column), and v=0.3 (right column).

The top row shows the spring force,

the second row shows the velocity of the top substrate,

the third row shows the lubricant thickness, and

the bottom row shows the effective lubricant temperature (note that T_melt≈ 0.15 for the three-layer soft flat system).

The corresponding configurations are shown below


stick configuration for stick-slip with v_spring= 0.03	LoLS, smooth sliding at v_spring= 0.3	LS, smooth sliding at v_spring= 1

Onset of sliding

The threshold force f_s as well as the system dynamics at the onset of sliding first of all depend on the starting configuration. Namely, if the starting configuration contains a small concentration of defects (dislocations) that pin the substrates so that f_s only slightly exceeds the kinetic frictional force, then the interstitial defective atoms are pulled back into the lubricant layers, the lubricant film is ordered, and the sliding begins with the layer-over-layer mechanism. On the other hand, when the starting configuration is far from the ordered configuration (e.g., as the annealed configuration shown above), then at the onset of sliding the lubricant has to be reorganized by plastic deformation, its effective temperature grows above the melting one, and the lubricant film melts almost immediately with the beginning of sliding. See also an example (*.mov 12 Mb, needs QuickTime installed).

Very thin lubricant films (N_l=2 or 1)

The system behavior changes qualitatively for very thin lubricant films, N_l≤ 2, when all lubricant atoms interact directly with the substrates. The lubricant films with two or less atomic layers are characterized by much larger values of the static threshold f_s (which now does not change with time, i.e., there are no aging of the contact) and the critical velocity v_c, than those for more thick films described above.

Two-layer lubricant film. Here, the two lubricant layers are glued each to the corresponding substrate, and the lubricant takes the two-layer crystalline structure at stick configurations (e.g., see the figure below). Because both layers are made of the same material, they are almost commensurate, thus the sliding should start from the breaking of bonds between these layers. For the two-layer soft flat system, the lubricant melts at T_melt≈ 0.23 and freezes (when the temperature decreases) at T_freeze≈ 0.19. Because the static threshold force is large for the two-layer system, at the onset of sliding the effective lubricant temperature increases well above T_melt. Therefore, the sliding always corresponds to the LS regime, and the hysteresis, as well as stick-slip, occurs due to the melting-freezing mechanism. The film melts at slips, although it remains well layered in three layers (see typical configurations below).


stick configuration for stick-slip with v_spring= 0.1	LS, smooth sliding at v_spring= 1

In the LS state, the bottom lubricant layer is almost glued to the immobile bottom substrate, moving with a very small average velocity corresponded to creep motion with respect the bottom substrate, the top lubricant layer in the same manner is almost glued to the top substrate, while the middle layer moves with the v_top/2 velocity (in average) as demonstrated in Fig.soft32 below.

Figure soft32: The x-component of atomic velocities v_x versus z for all lubricant and substrate atoms in the smooth sliding regime for N_l=2 and v_spring= 1.

One-layer lubricant film. On the other hand, the N_l=1 system behaves essentially similar to the hard-lubricant system described below. Although the lubricant is heated due to sliding up to temperatures T_lubr~ 0.1 to 0.4, this temperature is still well below the melting temperature T_melt≈ 0.57 of the one-layer film. Therefore, the hysteresis, as well as the stick-slip, exist due to inertia mechanism. Analyzing MD trajectories, at the onset of sliding we first observe the motion of two domain walls, which soon transform into a channel of moving lubricant atoms, and then to the "running" state of all lubricant atoms.

Figure soft37:

(a) the velocity of the top substrate,

(b) the effective lubricant temperature, and

as functions of the applied force for the one-layer lubricant film.

Solid symbols are for flat surfaces, and

open symbols, for the curved geometry.

Indicate the huge hysteresis: when the driving grows, the sliding starts at f > 0.1, but the backward transition to the locked state, when the force decreases, occurs at f ~ 0.02

The same remains true for the case of the curved top substrate. In this case we observed at the onset of sliding a solitonic mechanism, which operates in the narrow region due to a bump ahead of it, so that the concentration in the narrow region increases and becomes incommensurate with the substrates.

stick configuration of the N_l=1 lubricant with the curved geometry just prior the beginning of sliding at f = 0.102 < f_s≈ 0.105

LoLS, smooth sliding at f = 0.06 when v_top= 1.99

Hard lubricant: Perfect sliding

Flat surfaces and solid lubricant with an ideal crystalline structure: the universal dependence.

In the solid-sliding regime, when the top rigid substrate with one attached s-layer moves as a whole with the average velocity ‹v_top›, the bottom rigid substrate with one attached s-layer does not move at all, and the lubricant film moves as a whole with the velocity v_lub= (1/2)‹v_top›, the washboard frequency is equal to ω_wash= 2πv_lub/a = π‹v_top›/a, where a=a_s=3. The balance of forces for the top substrate + one attached s-layer is F ≡ N_S f = N_al m_l η^*(v_top) v_lub, where we have introduced the total viscous damping coefficient η^*(v_top) for an atom in the utmost lubricant layer. In the "perfect-sliding" approximation the atoms in the utmost lubricant layers feel only the external damping η_ext(v_top) ≈ η₁(z_lubr) [η_ph(ω_wash) + η_eh] due to energy exchange with the rigid substrates. Assuming that all the damping within the lubricant is due to the external one, η^*(v_top) = η_ext(v_top), we obtain the "universal" ("perfect-sliding") dependence

v_top^(uni)(f) = 2N_S

N_al f

m_lη_ext(v_top)

which depends neither on the number of lubricant layers nor on the substrate masses m_s and m_S, because it corresponds to the steady state (while a delay of response of v_top when f varies non-adiabatically, has to depend on the masses).

üat small forces the effective friction is very small, η_eff ~ v⁴

üplateau at v_top~ 1, when ω_wash is within the phonon zone of the substrates

üf_max≈ 7 and v_max≈ 10.8 (these values depend on ω_m) – further increase of f leads to unstable motion, because the energy injected into the system, cannot be dissipated

The lubricant is heated due to driving.

The distributions of velocities in all cases can be well approximated by Gaussian curves if we use different "temperatures" for the lubricant and the s-atomic substrate layers as well as for different degrees of freedom.
Definition: T_a= m‹( v_α– ‹v_α›)²›, where ‹...› is for the averaging over time and over all atoms in a given layer, etc.

Simulation results (see figure to the right):

the lubricant temperature increases as f grows until the lubricant finally melts at f = f_f
T_z >> T_x > T_y → the driven system is strongly out of equilibrium.

Hard lubricant: Energy losses

When one part of a system moves with respect to another part with a relative velocity v, then the rate at which work is done is equal to ε = vf, where f is the total force acting on the former. This may be used to define the energy losses of a given atom as

ε_i = –

∑
i′

(v_i – v_i′) f_ii′

where the sum is over all atoms of the system including the s-atoms of the rigid substrates, and f_ii′ is the force acting on the i-th atoms from the i′-th one.

Illustration: let us consider an ideal case, when the bottom substrate is immobile, the top substrate moves with a velocity v in the x direction, the lubricant film moves as a whole with the velocity v/2, and the force F is applied to the top substrate. In this case we have

F_lub/top= –F_top/lub= F_bot/lub= –F_lub/bot= F, and we obtain for the losses

ε_top = –(1/2)(–F)·( v – (1/2)v ) = (1/4) Fv,

ε_lub = –(1/2)[ F·( (1/2)v – v ) – F·(1/2)v ] = (1/2) Fv,

ε_bot = –(1/2)F·( 0 – (1/2)v ) = (1/4) Fv,
thus the total losses ε = Fv = the work done by the external force as expected.

Simulation results (see figure to the right): the anharmonicity is large, and main energy losses are at the interfaces.

Hard lubricant: melting of the confined film

thermal melting of the hard lubricant (N_l= 5, T = T_m, f = 0): movie for T = 0.45 (avi 1 Mb)

force-induced melting of the hard lubricant (N_l= 5, T = 0, f = f_f): movie for f = 0.5 >> f_f (avi 1.6 Mb)

If T >T_m, then the lubricant is liquid, the static friction is zero, f_s= 0, there are no hysteresis, and the kinetic friction is much higher than for the perfect-sliding case.

Figure: distribution of x-velocity across the lubricant for N_l= 5 (f = 0.01) in

the solid state (T = 0.4, perfect sliding), see movie (avi 0.46 Mb)
the molten state (T = 0.5, liquid sliding), see movie (avi 0.43 Mb)

Hard lubricant: Role of temperature


enlarged view	enlarged view

The solid lubricant with the ideal structure provides the smallest frictional force (superlubricity). If the lubricant film is in a liquid state, the frictional force is much larger. When the lubricant is frozen again, it takes an amorphous structure because, due to the contact with the substrates, the energy is removed from the lubricant very fast, and the lubricant film has no time to order. The frictional force in the case of an amorphous lubricant is typically larger than for the molten one.

Ideal crystalline configuration (provides the perfect sliding)	Configuration after melting/freezing (“amorphous” lubricant)

Hard lubricant: Transition from stick-slip to smooth sliding

Perfect sliding: System dynamics for the N_l=3 hard system (k_spring= 3·10^–4) for three values of the driving velocity:

v=0.01 (stick-slip) →

v=0.03 (irregular/chaotic) →

v=0.1 (smooth perfect sliding)

"Amorphous" lubricant: the frictional force vs time for three values of the spring velocity for N_l=2 and N_l=5 with flat geometry

Hard lubricant: Phenomenological theory of kinetic friction

The results obtained with MD simulations, however, still remain on an empirical level. Being quite time consuming, MD provides the value of friction only for a given set of model parameters and hardly may explain general trends and laws, e.g., the dependence of friction on the temperature, sliding velocity, shape of lubricant molecules, parameters of the interaction between the lubricant and substrates, etc. The kinetic friction appears due to energy flow from the sliding interface into the bulk of substrates, which finally is to be converted to heat. The energy losses emerge mainly because of creation of phonons; the rate of this process depends on many factors, such as the density of phonon states, positions and velocities of the lubricant atoms relative the surfaces, which in turn depend on the sliding velocity, the heating of the interface due to sliding, etc. Below we outline a phenomenological approach which allows us to explain the simulation results presented above, as well as to predict the friction properties of similar systems without (or before) MD simulations.

The energy balance approach. The most natural way of f_k calculation is through the energy balance arguments: the energy dE_in/dt = Fv_top = N_subfv_top pumped into the system per time unit due to external driving, must be equal to the energy dE_diss/dt dissipated in the substrates. The only way of energy dissipation in the model is through the viscous damping term m_lη(z,v)v in Langevin equations of motion. When the lubricant has an effective temperature T_lub due to heating by the substrate as well as by the external driving, then its atoms move with a thermal velocity ‹v_th›=(k_BT_lub/m_l)^1/2 ~ 0.3−- 0.7 at temperatures T_lub~ 0.1 − 0.5. When a lubricant atom is near a substrate at a distance z_l from the nearest surface and moves with an average velocity v_l with respect to it, then it losses per time unit the energy

ε(v_l; T_lub) = m_lη₁(z_l)

⌠
⌡

+∞

-∞

dv η₂(v) v² P(v−v_l; T_lub) ,

(ph3)

where P(v;T) = (m_l /2πk_BT)^1/2exp(−m_lv²/2k_BT) is the Maxwell distribution. In the thermodynamic equilibrium state, v_l= 0 and T_lub= T_sub, this loss must be compensated by "energy gain" due to the stochastic force in Langevin equation, which acts on the mobile atoms from the substrate thermostat. Let N'_at be the number of atoms in the lubricant layer just adjoined to the substrate surface, and v_lx the average x-velocity of atoms in this layer relative the substrate, while v_ly= v_lz= 0 for the motion along y and z. The total energy loss due to friction can be estimated as

dE_diss

≈ sN'_atE₁(v_lx) , where

(ph4)

E₁(v_lx) = ε(v_lx; T_lub) + 2ε(0; T_lub) − 3ε(0; T_sub) .

(ph5)

The first contribution in E₁ comes from the x-motion, and the second one, from the motion along y and z. The last term in Eq.(ph5) describes the energy gain coming from the thermostat to mobile atoms due to action of the stochastic force; this contribution has to be subtracted from the frictional losses. In the thermodynamic equilibrium state the energy gain must be equal to the energy loss due to the thermostat, ε(0;T_sub). We assume that this contribution (from the stochastic force emerged due to the thermostat) remains the same in the nonequilibrium driven state. The factor s=2 in front of the r.h.s. of Eq.(ph4) appears in the case of symmetric sliding, where there are two sliding lubricant/substrate interfaces. In the asymmetric case, when there is only one sliding interface, one has to put s=1 and

dE_diss

≈ N'_at [ E₁(v_lx) + E₁(0) ] ,

(ph6)

where the first contribution comes from the sliding interface and the second one, from the stick lubricant/substrate interface.

From the equality dE_in/dt = dE_diss/dt we finally obtain for the kinetic friction force

f_k ≈ m_l G η₁(z_l) F(v_top) ,

(ph7)

where G = sN'_at/N_sub and η₁(z_l) are "geometrical" factors which only indirectly depend on the velocity and temperature through a change of the lubricant structure due to sliding (typically G < 1.2 and η₁~ 0.1 in our model). The last factor in Eq.(ph7) is the main one that determines the dependence of the kinetic friction on the driving velocity and temperature, and is defined (for the symmetric sliding, s=2) as

F(v_top) = v_top^-1

⌠
⌡

+∞

-∞

dv η₂(v) v²[ P(v−v_lx; T_lub) + 2P(v; T_lub) − 3P(v; T_sub) ].

(ph8)

If we take into account only the minimal contribution η_min in Eq.(ph2), then the factor F becomes equal to

F_min(v_top) = η_min

é
ë

v_lx² +

3k_B

m_l

(T_lub − T_sub)

ù
û

v_top,

(ph9)

which grows linearly with the velocity (at T_lub= T_sub) as well as with the lubricant temperature (at fixed v_top and T_sub).

In a general case, the factor F grows with the driving velocity and temperature too. Its value is determined by the velocity v_lx and the lubricant temperature T_lub. It is easy to find v_lx in the case of solid lubricant: v_lx= v_top/2 for symmetric sliding and v_lx= v_top for asymmetric sliding. In the case of a liquid lubricant (the LS regime), one has 0 < v_lx< v_top/2. As for the lubricant temperature, its value depends on the sliding velocity due to sliding-induced heating.

Thus, we have to take into account two issues: first, the lubricant is always heated due to driving, and second, the lubricant structure depends on the sliding steady state, so that the kinetic friction force is determined by a rather delicate balance of these factors.

The geometrical factor. While the factor F in Eq.(ph7) increases with the driving velocity and temperature, the geometrical factors G and η₁ decrease thus compensating the growth. First, the lubricant thickness d increases with the lubricant temperature due to thermal expansion. As a result, the distance of lubricant atoms from the nearest surface will increase with temperature too, e.g., as z_l≈ z_l0+ β_zlT_lub. The increase of z_l will result in the decrease of the coefficient η₁. Second, the geometrical factor G is directly proportional to N'_at, i.e., to the number of atoms in the lubricant layer just adjoined to the substrate surface. If N'_l is the effective number of layers in a given configuration, then N'_at can be found from the conservation of the total number of atoms, N'_atN'_l= N_atN_l. For the perfect crystalline structure of the lubricant we have N'_at= N_at. In the case of hard lubricant with "amorphous" structure N_l< N'_l≤ N_l+1 so that N'_at< N_at, or N'_at= α_NN_at, where α_N< 1. The case of liquid lubricant, when the lubricant structure is changed with the driving velocity and temperature, is more involved: the number of atoms that interact with the substrate, N'_at, should decrease when the film thickness grows. For T_lub> T_melt we found the dependence

N'_at = α_N N_at / [1+ β_N (T_lub− T_melt)]^2/3.

(ph11)

The geometrical factor G, which is directly proportional to N'_at, may strongly decrease with temperature and driving velocity, especially in the LS regime.

The lubricant temperature. The very important issue is that the lubricant temperature increases with the driving velocity. It can be found analytically with the help of energy balance arguments similarly to the approach used above: an energy R₊ pumped into the lubricant (per atom and per time unit), must be equal to the dissipated energy R₋. The energy flux from the lubricant to the substrates, which emerges due to the difference of their temperatures, may be defined as

R₋ = 6m_l

⌠
⌡

+∞

-∞

dv η(z,v) v²[ P(v,T_lub) − P(v,T_sub) ].

(ph15)

The pumped energy R₊ emerges due to the "shaking" of the lubricant during sliding. This rate can be estimated with the help of linear response theory: Let the lubricant be perturbed by an oscillating force of an amplitude f₀ and a frequency ω₀. When one lubricant layer slides over another one with a relative velocity v₀, it is disturbed with a "lubricant washboard frequency" ω₀=2πv₀/r, where r is an average interatomic distance in the lubricant. Then, the rate R₊ can be found as

R₊ =

πs f₀²

4m_l

⌠
⌡

+∞

-∞

dv ρ_lub(2πv/r) [ P(v−v₀,T_lub) − P(v,T_lub) ],

(ph16)

where ρ_lub(ω) is the (local) density of phonon states in the lubricant. The value v₀ in Eq.(ph16) is directly proportional to the driving velocity v_top, although it has to be defined in different ways for different types of sliding. In the limit v_top→ 0, Eq.(ph16) gives R₊∝ v₀², which leads to the correct low-velocity relation ΔT ∝ v_top². At higher velocities, the value of R₊ depends on the lubricant phonon spectrum. Extracting all necessary parameters from the simulation data, we obtained the following dependences:

The perfect sliding.

Figure ph12: Phenomenological dependences for the perfect sliding of the N_l=5 solid lubricant for f₀= 0.2. The symbols correspond to simulation data.

(a) Dependence of the kinetic friction force on the sliding velocity at T_sub= 0.1 (open diamonds) and T_sub= 0.4 (red solid circles).

(b) Dependence of the kinetic friction on the substrate temperature for three values of the sliding velocity v_top= 0.1 (black), 0.3 (red), and 1 (blue).

Here v_lx= v_top/2, s=2, G=1.212, and from the simulation data we extracted other parameters: β_z≈ 3.6, z_l0≈ 5.21, β_zl≈ 0.3, and z_s≈ 2.12.

Note that if we ignore all temperature dependences, we obtain in the T_sub= 0 case the "universal" dependence described above.

The lubricant with the "amorphous" structure.

Figure ph13: Phenomenological dependences of the kinetic friction force on the sliding velocity at T_sub= 0 (blue curve and open diamonds, the symbols correspond to simulation data) and T_sub= 0.3 (red curve and solid circles) for the "amorphous" lubricant.

Here α_N= N_l/(N_l+1) ≈ 0.833 (simulation suggests α_N≈ 0.81), and from the simulation data we extracted other parameters: β_z≈ 2.33, z_l0≈ 5.22, β_zl≈ 0.333, and z_s≈ 2.112. Note, however, that we had to take different values for the fitting parameter f₀, namely, f₀ = 1.4 for T_sub= 0 but f₀= 1.8 for T_sub= 0.3.

The sliding becomes asymmetric at low driving velocities v_top< 0.3; in this case we put s=1 and v_lx= v_top. Unfortunately, we were unable to obtain a good quantitative agreement with simulation using the procedure described above. Instead, a good agreement was achieved with an (artificial?) dependence T_lub≈ 2v_top³.

The T=0 movies can be found here (animated gif, 8.6 Mb).

The LS (liquid sliding) regime. When the confined film is molten, its thickness changes with the temperature and driving velocity, and the factor G strongly decreases with temperature. To find N'_at, we calculated the distribution of atomic concentration around the first lubricant layer P₁(z) (see Fig.ph10) and then integrated it, N'_at=C ∫ dz P₁(z); the dependence N'_at(T_lub) can be fitted with a good accuracy by Eq.(ph11). Then, we have to know the average velocity v_lx of the first lubricant layer with respect to the substrate. Using the distribution of x-velocities across the lubricant obtained in simulation, we schematized it as shown in Fig.ph16. The dependence v_x(z) is linear within the lubricant, but undergoes jumps at the substrate/lubricant interfaces. Using this picture, we assumed that v_lx can be described by the expression

v_lx = αv_top (z_l − z_s)/

(ph20)

which includes a phenomenological parameter α. The values of α extracted from the simulation data, may be fitted by the dependence α = 2.53 − 0.71T_lub.


Figure ph10: Distribution of atomic concentration across the lubricant in the liquid-sliding regime (schematically).	Figure ph16: Schematic presentation of distribution of the x-velocity of lubricant atoms across the lubricant in the LS regime.

In the result, we obtained the following dependences for the LS regime:

Figure ph18: Dependence of the kinetic friction force

(a) on the driving velocity at a fixed substrate temperature, T = 0.3 (black), 0.4 (blue), 0.5 (red), and 0.6 (magenta), and

(b) on the substrate temperature at a fixed driving velocity v_drive= 0.75 (red) and 0.25 (black) for the LS regime.

Symbols correspond to simulation data, while solid curves describe the phenomenological dependences.

Here β_z≈ 36.4, z_s≈ 2.09, β_N≈ 2.84, T_melt= 0.44, α_N= 0.65, z_l0= 5.41, β_zl≈ 0.313, and f₀≈ 4.3.

Discussion and conclusion. Thus, the simulation results can be satisfactorily explained with the help of the phenomenological approach. Almost all parameters that we extracted from simulation data, may in principle be calculated from first principles or estimated at least, except the two fitting parameters f₀ and α. The first parameter, the coefficient f₀ in Eq.(ph16), describes the amplitude of the oscillating ("shaking") force responsible for the sliding-induced heating of the lubricant. Note that the value of f₀ correlates with the structure of the lubricant: the smallest value (f₀= 0.2) was found for the perfect sliding regime, while much larger values (f₀= 1.4 to 1.8), for the "amorphous" lubricant structure. The largest value (f₀= 4.3) was obtained for the liquid lubricant. We could suggest that the value of f₀ is proportional to "defectivity" of the lubricant film structure, although we have no clear understanding of this problem, and it requires further investigation. Second, the friction force in the LS regime strongly depends on the velocity of the first lubricant layer relatively to the substrate. According to Eq.(ph20), v_lx is determined by the parameter α, i.e., by the gradient of x-velocity at the substrate/lubricant interface. This question is closely related to the "slip" or "no-slip" behavior of a liquid flow near a solid surface. The latter is characterized by the so-called slip length defined as

L_s = v_lx

dv_x

|
|

z=z_l

In turn, the slip length is determined by the liquid-surface interaction, i.e., either the surfaces are wetting or non-wetting by the lubricant (the latter situation corresponds to the hard-lubricant system described here).

Although the phenomenological theory presented above still uses some parameters, it allows us to explain the simulation results. In particular, it naturally explains the increase of kinetic friction with the driving velocity. Then, it explains the temperature dependence of the friction, which emerges due to interplay of two factors F and Gη₁: the first factor increases with temperature, while the geometrical one, Gη₁, decreases with T. Thus, the phenomenological theory allows us to predict, at least qualitatively, the behavior of other tribological systems in a general framework.

For more details of the phenomenological approach, see O.M. Braun, Phys. Scr. 78 (2008) 015802 "Phenomenological theory of kinetic friction for the solid lubricant film" (pdf files may be found here).

Role of substrate geometry

In the results presented above, the top and bottom substrates were identical. Now let us consider the tribosystem, where the substrates are not identical. In this case, the sliding becomes asymmetric, and essential changes of sliding mechanisms occur in the case of soft lubricants.

Curved surface

Note: magnitude of frictional forces, as well as the critical velocity v_c, for the hard lubricant are in about ten times smaller, than for the soft lubricant.

low driving velocity: stick-slip motion	high driving velocity: smooth sliding	N_l=3 V_sl=1/3
		soft lubricant: V_ll << V_sl
		hard lubricant: V_ll >> V_sl
configurations / movies (mpeg)
soft lubricant (V_ll=0.2): melting/freezing stick-slip motion (v=0.3) movie (6.6 Mb)	soft lubricant (V_ll=0.2) smooth sliding (v=1) movie (3.4 Mb)
hard lubricant (V_ll=1): inertia mechanism stick-slip motion (v=0.03) movie (3.4 Mb)	hard lubricant (V_ll=1): smooth sliding (v=0.1) movie (4.5 Mb)

Stepped surface


n_steps=3, smooth LoLS, v_spring=0.1	n_steps=3, smooth LS, v_spring=1

Rotated surface

Figure rot14: Friction force as a function of the misfit angle φ (i.e., when the substrates are rotated one with respect to another on the angle φ).

Diamonds and solid curve describe the static friction,

dotted curves show the kinetic friction force at different driving velocities (from v = 0.1 to 3) as indicated in legend.

Star symbols correspond to the regime of "solid sliding".

Panel (a) is for T = 0, and (b) is for the "room" temperature T = 0.025.

As expected, in the LS regime (e.g., for v = 1 and 3 in Fig.rot14) the friction is almost independent of φ. However, the static friction, as well as the kinetic friction in the LoLS regime, may change on more than one order in magnitude, when the misfit angle varies. The highest friction is for φ=0. For all other angles studied, the sliding always corresponds to the LoLS regime at low driving velocities. Contrary to the φ=0 case, now the sliding is typically asymmetric, the sliding takes place at a single interface only, between the middle and one of the attached lubricant layers, i.e., the middle layers move either with the top or bottom substrate. The middle layers may remain ordered during sliding or, for other values of φ, they may be 2D melted (in the former case, the friction is lower).

For few special misfit angles, namely, for φ = 15.4, 27.2 and 42.8 indicated by stars in Fig.rot14, we observed the "superlubricity" characterized by a very low friction. In these cases, the lubricant film is solid and ordered during sliding, and moves together with one of the substrates. This is the "solid sliding" (SS) regime. However, the lubricant is not rigid during motion, which now corresponds to the "solitonic" mechanism.

Conclusion

Soft lubricant film (3 ≤ N_l≤ 5):
- for the static frictional force f_s, Amontons law is explained by the interactions within the lubricant, that leads to μ ~ 0.5; the value of f_s is determined by a particular structure of the frozen metastable lubricant film (i.e., by defects that should be broken at the sliding onset);
- f_s slowly grows with the time of stationary contact (up to in four times of magnitude) and reaches its maximum for the annealed configuration (aging of the contact);
- the hysteresis of v_top(f) (when the driving force increases/decreases adiabatically) as well as the stick-slip motion at low driving velocities, is due to the melting-freezing mechanism; the transition from stick-slip to smooth sliding takes place at v_c~ 0.1;
- there are two steady-state sliding regimes, the LoLS sliding at low driving forces/velocities and the LS regime at larger ones. In the LoLS regime the sliding stops when the force/velocity decrease and v becomes lower than v_top~ 0.1; when the force/velocity increases, the lubricant melts and the LS regime begins to operate. The LoLS regime is more stable for narrower films, than for thicker ones;
- at smooth sliding, the kinetic frictional force is approximately independent on the driving velocity, and is lower (or much lower) than the static frictional force. However, the lubricant effective temperature (the heating of the lubricant due to sliding) as well as the thickness of the film are proportional to the driving velocity;
Very thin soft lubricant films (N_l=2 and 1):
- for two-layer lubricant film, both its layers are glued to the corresponding substrates in the annealed and stick configurations, and the film structure is crystalline. As a result, the static frictional force is high, the lubricant always melts at the onset of sliding, so that only the LS regime operates. The hysteresis of v_top(f), as well as the stick-slip, always correspond to the melting-freezing mechanism, and the threshold velocity of the transition from stick-slip to the smooth sliding (v_c~ 0.6) is also higher than for thicker films;
- for one-layer lubricant film, the static friction is the largest, f_s~ 0.1. The film typically is not melted during sliding, and the stick-slip motion is due to inertia mechanism similarly to the hard lubricant system (because of too high melting temperature of the one-layer film);
Perfect sliding (hard lubricant):
- the solid lubricant film with ideal crystalline structure provides the lowest frictional forces, μ ~ 10^-3 to 10^-2;
- the kinetic frictional force depends on the sliding velocity v (Amonton’s laws does not operate for f_kinetic);
- the transition from stick-slip motion to smooth sliding occurs at v_c~ 0.03 to 0.1;
Lubricant with amorphous structure (hard lubricant):
- the solid lubricant film with a metastable (amorphous) structure leads to a rather high frictional forces, μ ~ 0.1 to 1;
- the kinetic frictional force depends on v as f_kinetic≈ Bv with B ~ 0.015 to 0.035;
- the transition from stick-slip motion to smooth sliding occurs at v_c~ 0.1 to 0.5;
Role of substrate temperature (hard lubricant):
- if one starts from the perfect-sliding state, then the velocity decreases as T grows until the film melts. After that, the mobility grows with T, although it remains lower than the perfect-sliding value. For amorphous lubricant, the mobility increases with T ;
- in the molten state, the kinetic frictional force is proportional to the sliding velocity v, e.g., as f_kinetic≈ 0.017v_top;
- f_kinetic has peculiarities at T = T_m and T = T_freeze, where the mobility slows down.

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Last updated on November 26, 2009 by O.Braun. Copyright © by O.Braun.

Translated from L^AT_EX by T_TH, configurations produced by RasTop, movies – by VMD

Two sliding regimes (LoLS and LS) for Nl>2

Very thin lubricant films (Nl=2 or 1)

Very thin soft lubricant films (Nl=2 and 1):

Next: Self-ordering of the lubricant

Back to main page or tribology page

Two sliding regimes (LoLS and LS) for N_l>2

Very thin lubricant films (N_l=2 or 1)

Very thin soft lubricant films (N_l=2 and 1):