nanotribology
Since their
emergence, the scanning microscopes offer an ability to study
micro/nanotribology. Below we present some essential principles of
tribology and discuss the technique of lateral forces investigation.
The frictional force
is an aggregate effect arising from various physical phenomena:
elasticity, adhesion, viscosity, capillary forces, surface chemistry,
phononic and electrostatic interaction, etc. Any of them can dominate
depending on conditions.
Each tribology
field investigates friction at its own scale. Macrotribology deals with
large objects and do not take into consideration a matter structure. On
the contrary, nanotribology explains friction on the level of
individual atoms interaction. Microtribology is an intermediate field.
The macrotribology basic law is the Amontons-Coulomb law stating that the friction force is proportional to the loading (normal) force:
 |
, |
(1) |
where k
– dimensionless coefficient of friction containing all the tribology
information. It depends on many factors, such as temperature, humidity,
sliding velocity, etc.
In macrotribology,
it is considered that geometrical contact area of two bodies is equal
to (or slightly differs from) the real contact area on the atomic
scale. This is certainly an approximation because in fact even very
flat surfaces seem rough at lower scale, so the true contact area is
much less – only asperities are in perfect contact. At the macroscale,
the contact is a great number of microcontacts (Fig.2).
In this case, the macroscopic frictional force is an averaged
microscopic frictional force of individual microcontacts that can vary
greatly.
 |
| Fig. 2. Ball-silicon wafer contact on the macroscopic scale [1]. |
Microtribology
studies such individual contacts. As a rule, it is implied that a small
single asperity interacts with a surface. Namely this model made the
AFM an attractive experimental tool for microtribological studies.
As it is well
known, friction is the dissipative force. When the two surfaces in
contact slide against each other, mechanical energy dissipation occurs.
For example, to maintain constant sliding velocity, an internal force
must produce work, therefore, every factor giving rise to friction has
a mechanism of the energy dissipation. Considering microtribology, let
us list some of them.
Friction can be
either dry or wet. It is considered to be wet even when an extremely
thin (few atomic layers) film of liquid occurs on the surface. Due to
the adsorption this is always the case except for the following:
- hydrophobic surfaces of tip and sample;
- friction in vacuum;
- large normal load resulting in squeezing out of the liquid
from the interface, true contact of surfaces establishing and actual
realization of the dry friction mechanismа.
It is considered
that in case of dry friction surface asperities hit against each other.
While overcoming obstacles, atomic-lattice vibrations are generated and
dissipated as phonons which carry away the energy. Moreover, when
adhesion links between hills of surfaces in contact are broken,
electron-hole pairs are created in metallic samples and this process
also requires energy (this effect is much weaker than phononic
dissipation). In case of soft samples, microasperities can be destroyed
(the so called "plowing") and mechanical energy is spent on atomic
links break.
Wet friction
depends much on liquid layer thickness. If a film is monomolecular,
friction is dry-like. If a film is two-three monolayers thick, the
energy dissipation in a phonon channel is blocked and liquid layer
viscosity is of major importance. For more thick films capillary
effects predominate which results in contacting surfaces asperities
attraction upon shear.
What is the relation between frictional force and loading force in microtribology? The Amontons-Coulomb low analogue here is the Bowden-Tabor relation (model) written as:
This area depends on degree of both contacting surface hills mutual
indenting. As it is known, the area of such contact is given by the
Hertz theory:
 |
(3) |
where R – tip radius of curvature, N – normal loading force, K – reduced Young's modulus, given by
 |
, |
(4) |
with E, E' – Young's moduli and m, m' – Poisson's ratios of tip and sample, respectively. For the silicon probe and sample 
, 
, 
.
 |
, |
(2) |
where 
– shear stress, 
– true area of elementary contact (in contrast to geometrical contact area in macrotribology).
As it can be seen, dependence of the frictional force on normal load N is nonlinear. If a film of liquid exists, it is necessary to add toN an adhesion term arising from the capillary force. Using the DMT model, it can be written as:
 |
(5) |
where 
– surface tension coefficient. This is the additional attractive force between the contacting surfaces.
The Bowden-Tabor model is verified well in experiments. In Fig. 3 are shown experimental data [1] acquired in vacuum (lack of liquid film and capillary effect) and in air; theoretical curve (3) is presented for comparison.
 |
| Fig. 3. Experimental frictional force as a
function of normal load in air and in vacuum. Thick line shows
theoretical Bowden-Tabor relation [1]. |
In microtribology, the phenomenon such as "stick-slip" motion is frequently observed. The frictional force between the two sliding surfaces is irregular and is of saw-tooth character (Fig. 4).
If a hill of one surface is stuck to a "site" of the other surface via
adhesion and capillary forces, it will hardly be unstuck without
predominant force. Once it is separated, it jumps (slips) into another
"site" where sticks for a while and so on.
The stick-slip behaviour depends much on scan speed (Fig. 5). To investigate the dependence between frictional force and slip speed, the experiment was carried out [1].
In this experiment the frictional force was measured between silicon
ball having radius 0.5 mm and flat silicon surface. Both bodies were
hydrophilic. The roughness was 0.2 nm and 0.17 nm for the ball and
surface respectively. At low speed the stick-slip phenomenon is
pronounced, the jumps frequency is small and amplitude is large.
Increasing the speed results in rising the frequency and lowering the
amplitude. At some maximum critical sliding velocity the effect
vanishes and frictional force becomes regular. In experiment the
critical speed 0.4 mm/s was reached at normal load equal 70 mN.
 |
 |
| Fig. 4. Frictional force vs. sliding velocity [1]. Frictional force behavior at sliding velocities above and under critical is shown in boxes. |
Fig. 5. Frictional force amplitude and frequency at stick-slip motion as a function of scan speed [1]. |
On the same samples the dependence between frictional force and temperature/humidity was studied[1].
From the beginning both solids were hydrophilic. Then, in order to
remove the oxide film and make them hydrophobic, they were etched in
hydrofluoric acid for two minutes.
Thus, the
frictional force was measured versus relative humidity (RH) at various
temperatures for hydrophilic and hydrophobic samples. The measuring
system was placed into a camera with adjustable humidity and
temperature. RH was in the range from 85% to 20%. The normal load was
maintained constant and amounted N = 2000 mN. The results for high and low temperatures are shown in Fig. 6 [1].
 |
| Fig. 6. Friction vs. humidity at various temperatures for hydrophilic and hydrophobic systems [1]. |
The hydrophilic
sample can adsorb ample quantity of water. Thus, the more the
environment humidity is, the more liquid can be adsorbed and the higher
the frictional force is. As the temperature grows, desorption starts to
prevail over adsorption and friction decreases. The more the
temperature is, the more energetic water molecules are and the easier
they leave the surface and return to it. That is why friction slightly
depends on humidity.
Hydrophobic
silicon, in contrast to the hydrophilic one, reveals weak dependence of
friction on humidity at any temperature. With temperature growth, the
friction slightly rises. This means that, as a result of desorption,
solid surfaces are in tighter contact, Van der Waals forces start to
act between them and chemical bonds arise.
Nanotribology
deals with individual atoms interaction. Imagine a surface atom of one
body (AFM tip) that slides in a periodic potential of surface atoms of
the other body (sample) (Fig. 7) and the energy is not dissipated.
 |
Fig. 7. Left: Potential energy and tip path.
Right: Instantaneous and average frictional force [3]. |
Nonconservatism is
introduced as follows. Reaching the point of potential maximum, the
atom, which can be modeled as a spring suspended ball, loses contact
with the surface and "falls" to the point of potential minimum (or its
neighborhood). The atom passes into the site with another energy, i.e.
the potential becomes "nonpotential". The instantaneous frictional
force in this case is
 |
(6) |
 |
It may be thought
that thanks to the spring suspension, the energy is transmitted deep
into the body, i.e. from the nanoscopic point of view it is dissipated.
This model leads to a nonconservative (on average) force shown in Fig.
7 which is the frictional force. This averaged nonconservative force is
considered to be the frictional force in microtribology.
As an example, we present experimental data [2], [4] (Fig. 8) obtained for highly oriented pyrolytic graphite (HOPG).
 |
| Fig. 8. (a) Topography and frictional force
images for the HOPG sample sized 1 nm x 1 nm. (b) Schematic overlay of
the two images. Symbols mark corresponding maximums. Spatial shift
between images is clearly seen [2], [4]. |
It is seen that the
surface topography and frictional force image are of the same
periodicity and shifted relative each other in accordance with the
above presented theory.
Notice that the
HOPG surface should be dry. Water adsorption in this case is of greater
significance than on the microscopic scale. Capillary forces damp
stick-slip motion so the acquired image becomes fuzzy.
Summary.
- The science of friction – tribology –
is subdivided into macrotribology, microtribology, and nanotribology.
To describe friction at different scales, various models are used.
- The friction depends sufficiently on humidity, temperature, adsorption, etc. and can be of dry or wet type.
- The basic equation of
macrotribology is the Amontons-Coulomb law. The macroscopic area of
contacting bodies is considered to be multiple elementary contacts
whose total area is much less than the gross contact area.
- Dry friction in the elementary
contact is described by the Bowden-Tabor model. It employs the Hertzian
theory on the elastic deformation in place of contact, the friction
parameter being the shear stress.
- Capillary forces are of major importance in the wet friction.
- In microtribology, "stick-slip" phenomenon results in the frictional force irregularity and its saw-toothed variation.
- The nanotribology describes
friction in terms of atoms interaction. Considering the motion of one
body atoms in the potential of the other body atoms, it is possible to
introduce the nonconservative force describing friction.
References.
- Scherge Matthias, Biological micro- and nanotribology: Nature's solutions. Springer, 2001
- N.P. D'Costa, J.H. Hoh, Rev. Sci.Instrum. 66 (1995) 5096-5097
- Wiesendanger R., Guentherodt H.-J. (eds.), Scanning tunneling
microscopy. - 2d ed. 3 : Theory of STM and related scanning probe
methods. 1996
- Bhushan B., Wear 225-229 (1999) 465-492.
