Tribology: An Introduction


Definition: The word tribology is derived from the Greek word tribos, meaning rubbing ("to rub"); thus, a direct translation of "tribology" is "the science of rubbing". However, now tribology incorporates also the studies of adhesion, contact formation, wear, fracture, lubrication, nanoindentation and, first of all, friction.

Historical remarks: The macroscopic laws of the classical (macroscopic) friction were found by da Vinci, Amontons, Euler and Coulomb.

Why friction is important goto top

Hydrodynamic and boundary lubrication goto top

At stop/start, (almost) always the regime of boundary lubrication occurs. In what follows we consider the boundary lubrication only.


Experimental techniques goto top


(1) Standard technique (Bowden and Tabor 1950; Rabinowicz 1965; surface forces apparatus by Bhusham, Israelachvili, Landman, Granick; experiments with paper on paper by Heslot et al)

  • typically in air

  • the solids may be well defined as, e.g., for atomically flat mica plates

  • the separation between the surfaces may be controlled using optical interferometry (it can be measured to within ~by studying multiple beam interference finger)

  • measured: spring force ("tribological" friction  μ Ffriction/Fload)

  • control parameters:

    • vspring (pulling velocity)

    • kspring (machine stiffness)

    • Fload (loading mass)

    • T  (temperature)

(2) STM/AFM/FFM: scanning tunneling microscope (Gerd Binnig & Heinrich Rohrer 1981, Nobel Price 1986) for conducting surfaces, and atomic-force microscope (Binnig et al 1985) for dielectric surfaces measure surface topography;  friction-force microscope (Mate et al 1987) measures forces transverse to the surface

(3) Quartz-crystal microbalance (QCM) technique (Krim and Widom 1986): Gas atoms, such as Kr, Xe, or Ar, condense onto the surface of a quartz-crystal oscillator covered by a (111) oriented noble-metal substrate such as Au or Ag. The added mass of the adsorbate and the dissipation due to slip of the layer over the substrate shift and broaden the microbalance resonance peak. By measuring these changing, information about the form and magnitude of the frictional force can be obtained

(4) Levitating tribometer (O.A. Marchenko & V.S. Kulik, Institute of Physics NASU; see movie)

  • up to submonolayer lubricant films
  • both surfaces and film structure may be controlled with STM/AFM before and after measurments!
  • low loads

(5) Molecular dynamics (MD)

Results and laws for solid-on-solid friction goto top

  1. μ Ffriction/Fload ≈ constant, and  μ  is less than or about of 1 (Amontons 1699) →  real (actual) contact area  Areal~Fload  (Philip Bowden & David Tabor 1950). Indeed, real surfaces are rough. Thus,  Areal  will grow until the external loading force will be balanced by the contact pressure integrated over  Areal.

    Let  Pload(real) = PloadA/Areal  be the real pressure at the contact. Then:


  2. μkineticor << μstatic,  and  μkinetic  is approximately independent of the driving velocity  v


  3. Stick-slip motion ↔ smooth sliding


  4. "Memory/age" effects: frictional forces depend on the previous dynamical history of the solid-solid contact, e.g.,  μstatic(t) ≈ as + bs ln(t)  and  μkinetic(v) ≈ μstatic(aφ/v)  with the characteristic length  aφ~1 μm  (due to plasticity of the system)


  5. Full (large-scale) MD simulation showed that the lubricant structure may be

Important notes (what is known already) goto top

  1. Contacts / junctions / asperities:

    An example (from Persson's book): Put a steel cube of 10 × 10 × 10 cm3  on a steel table. Taking  Pload(real) ~ Pyield ≈ 109 N/m2,  we obtain  Areal~ 0.1 mm2,  so that ~103 to 105 junctions are expected at the interface. Note: STM/AFM/FFM techniques just study a single contact.


  2. Forces are of atomic-scale value close to the plasticity threshold. Estimation:


  3. (Almost) always there is a  lubricant between the solids (called "third bodies" by tribologists): either a (specially chosen) lubricant film, or grease (oil), or dust, or wear debris produced by sliding, or water or/and a thin layer of hydrocarbon etc. adsorbed from air. Thus, the frictional force is almost entirely determined by the force required to shear the lubricant film itself.


  4. (Almost) always the lubricant and the solids are "incommensurate":


  5. A thin film (less than 10 molecular diameters) is (almost) always layered, because the plates induce crystalline order in the film (solidify / freeze the lubricant; Thompson et al 1995). Moreover, when the lubricant thickness is less than three layers, most films behave like solid (glass-like in most cases, although may be ordered for relatively simple lubricant molecules).

See also an interesting tutorial with movies at Nano-World - a Swiss Virtual Campus Project


Macroscopic theory (Bowder & Tabor 1950) goto top

Assume that the yield stress τs at the contact is linearly coupled with the local pressure p,

τs(p) = τ0 + αp.


Then, integrating over the area of real contact, we obtain

Ffriction= τ0Areal + αFload



μ = α + τ0/<p>,


where <p> = Fload/Areal. Amontons' law operates if either  <p> >> τ0, or if  <p>  is constant, the latter holds for the ideally plastic surfaces (when <p> = Pyield) as well as for elastic surfaces (Volmer and Natterman 1997: real surfaces are self-affine fractals, thus as  Fload  increase, the old contacts are expanding and new contacts are created). Finally, the real area of contact must become equal to the apparent area, and Amontons' law will not operate. It is so for rubber but not for steel a machine would be destroyed earlier.


Stick-slip motion and smooth sliding goto top

In a typical experiment as well as in real machines, a spring is attached to the slider (even if there is no spring, the elasticity of the block plays the same role). The free end of the spring moves with a velocity vspring, and the force in the spring is registered. draw02.gif
Let initially the system is in rest and the spring has its natural length. Then when the base moves, the spring stretches,  f  increases until the threshold value  fs (the static frictional force), and the block starts to move. Then due to inertia, the slider accelerates to catch the base. If  vspring  is small,  f decreases until the "backward" threshold force  fb,  and the slider stops. Then the process repeats, so the stick-slip motion occurs. Otherwise, when  vspring is large, the smooth sliding takes place.

     low velocity &/or soft system                high velocity &/or stiff system


Experiment: the behavior depends on the values  vspring  and  kspring. The smooth sliding is observed if the spring is stiff enough, and/or if the velocity is high enough. Otherwise, the stick-slip motion is observed. During stick, the elastic energy is pumped into the system by the driving device; during slip, this elastic energy is released into kinetic energy, and eventually dissipated as heat.


Note: if one assumes that the frictional force depends on the instantaneous velocity only,  ffric= f(v),  then the boundary separating two regimes corresponds to the vertical line in the (vspring, kspring)  plane (i.e., the critical velocity is independent of kspring), that is wrong. Thus, "memory" effects must be included.


(a nice applet written by F.-J. Elmer clearly demonstrates stick-slip and smooth motion)


Phenomenological theory goto top

The phenomenological model of the transition "stick-slip motion « smooth sliding" has been developed by Heslot et al 1994, Baumberger et al 1994, and Persson 1997. The model is based on the assumption that the static frictional force depends on the time of stationary contact. For example, if the sliding-to-locked transition has occur at  t=0, then the static frictional force will evolve as

fs(t) = fs1 + (fs2fs1)(1–et),
so that just after the locking the force is  fs(0) = fs1,  but later, at t, it approaches the value  fs() = fs2 > fs1.

Qualitatively (Persson): In case (a) the spring force increases faster with  t  than the initial linear increase of the static frictional force; hence the motion of the slider will not stop and no stick-slip motion will occur; In case (b) the spring force will be smaller than the static frictional force until  t  reaches the value  t1 at which time slip starts; hence stick-slip motion will occur. Thus, the critical velocity is determined by the initial slope of the dependence  fs(t),

vc ~ kspring–1 dfs(t)/dt|t=0


The phenomenological model includes two differential equations: The motion of the top block is described by the equation
(t) + M η

(t) + ffric[x(t)] = fdrive[x(t)] = kspring [vspringt x(t)],

where x(t) is the coordinate of the sliding block, M is its mass, and η is a phenomenological coefficient describing the viscous damping when the block slides over the bottom block. The second equation has to describe the frictional force ffric[x(t)]. The idea is to introduce some artificial variable called the "contact-age function" φ(t) which depends on the prehistory of the system, and is defined by the following differential equation,
(t) = 1 –
(t) φ(t)/aφ
where  aφ is some characteristic distance of microscopic-scale order, e.g., the substrate lattice constant  a. Then one assumes that  ffric[x(t)]  in equation for the top block is determined by Eq.(*) where, however, instead of time one has to substitute the contact-age function  φ,  instead of  fs1,  the backward force  fb,  and instead of  fs2,  the static frictional force  fs,  so that
ffric[x(t)] = fb + (fs fb)( 1 – exp{–φ[x(t)]/τ} ).
Thus for the stationary contact we have φ(t) = t and we recover the dependence Eq.(*), while for the steady sliding regime  φ = aφ /vspring= τv(vspring)  does not depend on time (τv is the average time a junction survives before being broken by the sliding motion), and the force  ffric(v) = fb + (fs fb)[1 – exp(–aφ/vspringτ)]  is the constant dependent on vspring,  it is large at low velocities,  ffric= fs  for  vspring→0, and small at high velocities,  ffric= fb  for  vspring. This set of equations leads to smooth sliding for large  vspring  and to stick-slip motion at low velocities. Using the parameters Mkspring  and  vspring  corresponded to an experimental setup, taking some reasonable values for the forces  fb  and  fs,  and playing with the phenomenological parameters  τη  and  aφ,  one can achieve excellent agreement with experimental results. However, while the dependence Eq.(*) can be explained with physical models, the phenomenological dependence Eq.(**) has no good physical background, it does not follow from simulation.


Low-dimensional models goto top

Zero-dimensional model one atom in the periodic substrate potential: the rigid bottom substrate (external sinusoidal potential) + one atom + the dc force is applied directly to the atom + Langevin motion equation + underdamped external friction


One-dimensional models:

Tomlinson model

Frenkel-Kontorova model



(nice applets for the Tomlinson model and the dc driven FK model were written by F.-J. Elmer)


More complicated models:

Two-dimensional models

Next: 3D model goto top

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Last updated on October 15, 2008 by Oleg Braun.                    Translated from LATEX by TTH