Institute of Physics, National Academy of Sciences of Ukraine, 46 Science Avenue, 03028 Kiev, Ukraine
presented. The main attention is devoted to the results obtained by the Molecular Dynamics
PACS numbers: 539.8
Science about friction, or tribology (from Greek tribos, that is translated as “grind”) is extraordinarily important
both from scientiﬁc and practical points of view. From the beginning we emphasize that if in one situations it is
desirable to lower friction as much as possible, in others – vice versa, to attain maximally large friction. We remind
also that friction may be static and kinetic. The static friction force f
is the force which is necessary to apply to
motion of a car are possible, as well as stick of constructions with screw-bolts and nuts. The kinetic friction force f
the force necessary for maintenance of the smooth sliding with a given speed v. Therefore, every time unit, the energy
v is pumped into the system, which is converted into a heat and ﬁnally goes to heating of atmosphere. According to
developed countries. Therefore even a small reduction of friction promises an enormous economic eﬀect. In typical
, for example, f
, and the ratio of the friction force to the loading force f
known as the friction coeﬃcient in tribology, in order of magnitude usually takes values of µ ∼ 0.1.
Because of importance of friction, its study began more than three centuries ago . The ﬁrst advanced study of
friction, reaching us, belongs to Leonardo da Vinchi (1452-1519), who discovered that the friction coeﬃcient does not
depend on the area of contact. Later Giyom Amontons (1663-1705) showed that the friction is directly proportional
to the load, i.e., to the weight of sliding block. Leonard Eyler (1707-1783) noted that it is necessary to distinguish the
static friction studied by Vinchi, and the kinetic friction explored by Amontons. Finally, Charles Coulomb (1736-1806)
discovered that the kinetic friction does not depend on the speed of sliding.
These laws, getting the name of the Amontons laws, remained purely empiric up to a middle of past century, when
Bowder and Tabor  made the ﬁrst attempt of their explanation from the physical point of view. They paid attention
to the fact that the contacting surfaces are practically always rough. Therefore, the real contact is attained only on
“tubercles”, or asperities. A simple estimation  shows that the real area of contact A
makes only ∼ 10
– forces in the contacts are close to the limit of plasticity of materials that form the contact. This explains the
Amontons laws: with the increase of the load f
, the real contact area grows either due to the increase of the number
result, the ratio µ = f
Later, more careful experiments showed that the Amontons laws are valid approximately only, and the problem
of friction is essentially more involved. Firstly, friction depends on the speed nevertheless. Secondly, it depends on
the prehistory of contact, i.e., friction occurs to be diﬀerent for the “newborn” contact and for the contact which
already undergone some sliding. A new era in the study of friction began only about 15-20 years ago, thanks to
development of new experimental methods (ﬁrst of all, the “tip-based technologies” coming from the surface physics
– the scanning tunnel microscope (STM)  and its subsequent improvements – the atomic force microscope (AFM)
 and the friction force microscope (FFM) ), and also due to enormous progress in computer power allowing to
make simulations by the Molecular Dynamics (MD) method for real tribosystems.
In this brief review we try to present a modern look on the problem of friction from the physical point of view,
making the main accent on the study of kinetic friction by the MD method. We consider the regime of boundary
friction only, when the surfaces are separated by a very thin, of few monomolecular layers, lubricant ﬁlm. We note
that such a ﬁlm is almost always present: it may be either a specially chosen lubricant, or it may correspond to a
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fat (oil), dust, wreckages (scales) of the surface material appearing as a result of sliding, or to water or hydrocarbon
molecules adsorbed from air, etc. – all this is called the “third bodies” in tribology. Moreover, even if the lubricant
ﬁlm is thick, at the moments of the onset of motion or at its stop, the lubricant is squeezed out from the contact area,
and the system turns out into the regime of boundary lubrication.
In physics, a very large role is played by simple models, which, from one side, correctly describe the basic aspects of
a problem and, from the other side, allow either the exact solution or at least a well grounded solution with a predicted
accuracy. In tribology, there are two such models – the Tomlinson model  (see Fig. 1) and the Frenkel-Kontorova
(FK) model , schematically shown in Fig. 2. Later, a large number of generalized and combined models was proposed
as well, description of which can be found, e.g., in the review  and in the monograph . However, already the
simplest one-dimensional (1D) model – an atom in the external periodic potential – allows us to understand the
important aspects of friction. Let us assume that the periodic potential of surface can be described by the sinusoidal
function with the period a
= 2π and the amplitude ε
= 1. If we apply to the atom a constant force f , it will remain
in a rest state (in the local minimum of the potential V (x) = sin x − f x) whilst f < f
= 1; thus, the force f
the atom will begin to slide over the potential relief. However, if we
M (here η is the damping
coeﬃcient and M is the atomic mass), as the atom can overcome the maxima of the potential relief due to its inertia.
The force f
is the analog of the kinetic friction force. The important result is that the minimal speed v
, at which
the atom can slide due to inertia, is of the “atomic-scale” order, v
∼ 50 ˚
FIG. 2: The Frenkel-Kontorova model.
In a real tribosystem, the periodic potential corresponds, for example, to the surface potential of the lower (immo-
bile) substrate, while the “atom”, to the moving top substrate. As v
, one may speculatively assume that
in a macroscopically large system, when M → ∞, we will get v
→ 0. This assumption, however, is wrong [12, 13].
contact) is stopped, and this takes place at a speed of atomic-scale order. At the same moment, a stopping wave is
created, and then the second, third, etc. atomic layers of the top substrate are stopped successively one after another.
The stopping wave takes away the accumulated kinetic energy of motion to the bulk of substrates.
If now we will move the atom not directly, but through a spring (which describes, e.g., the elasticity of the top
substrate), the end of the spring moves at a speed of v, we come to the Tomlinson model. At v > v
– the so-called stick-slip motion, well known as door
until it reaches the static threshold f
. At this moment the system begins to move with the increasing speed, until it
undergoes the end of the spring; thus the spring is weakened again, and the driving force falls down. As a result, the
system slows down up to the complete stop, and the whole process repeats itself. In the stick-slip regime, the friction
force does not depend on speed; however, if the system temperature is nonzero, T > 0, there is a weak (logarithmic)
dependence f (v) because of thermally activated jumps of lubricant atoms .
The second important model widely used in tribology is the Frenkel-Kontorova model. Firstly it was proposed for
description of dislocations in solids, and then it was widely used in surface physics for description of commensurate
and incommensurate structures of ﬁlms adsorbed on a surface . Generally, a breach in understanding of friction
problems, attained lately, above all things is connected with the progress in surface physics, a large contribution
to which was brought by Ukrainian scientists from the Institute of Physics, Institute of Physics of Semiconductors,
Institute of Chemistry of Surfaces, etc. However, the problems of tribology are more diﬃcult, than in the surface
physics: if in the latter is studying the “opened ﬁlms” adsorbed on a solid surface, in tribology systems the lubricant
ﬁlm is “clutched” by surfaces from both sides and therefore it is less accessible to direct study.
The FK model describes a chain of interacting atoms (e.g., adatoms or lubricant atoms), placed in the external
periodic potential created by surface atoms of the substrate. A success in the use of the FK model is connected with
that in the continuum limit (valid at a strong interaction between the atoms) its equations of motion are reduced to
the exactly integrable sine-Gordon equation, the solutions of which, besides the linear waves (phonons), include the
topological solitons (so-called “kinks”) and dynamical solitons (“breathers”). The kink describes a spatially localized
compression of the chain (or its extension in the case of antikink), and is characterized by extremely high mobility.
Namely kinks are responsible for the rapid transfer of mass along the chain, i.e., for mobility of the chain (the adlayer
or the lubricant ﬁlm). In two-dimensional (2D) or three-dimensional (3D) systems instead of kinks, conceptions of
domain walls or dislocation are used, but the physics of processes remains qualitatively the same. For example, a
mechanism of motion of a ﬁnite chain (or an island in the 2D system) is the following: a kink is created at one (free)
end of the chain, then it rapidly moves along the chain and annihilates on the other chain’s end; as a result, the whole
chain is displaced on the distance of one lattice spacing [15, 16].
Notion of “incommensurability” is other extremely important conception of the FK model. Namely, if the lattice
constants of the chain a
and the substrate a
in the inﬁnite system are incommensurate (i.e., their ratio χ = a
irrational), there always exists a critical value of the elastic constant of the chain g
, such that for a higher rigidity the
extremely small. This phenomenon (known in physics from the beginning of 1970th as the Aubry transition, or “the
transition with destruction of analyticity” [17–21]) acquired an extreme actuality in tribology in connection with the
prediction of “superlubricity” , i.e., the existence of lubricants providing extremely low friction. In the FK model
the best conditions for appearance of the state with f
= 0 are carried out at incommensurability proper to the “gold
5 − 1)/2. If the chain is placed between two one-dimensional “surfaces”, the so-called ”spiral ratio”
Thus, the simple models already provide several answers for the basic questions of tribology, at least on a qualita-
tively level. For example, it is clear that a solid lubricant could be the most eﬀective: it should provide the maximal
friction in the case of commensurate surface/lubricant interface (the so-called “cold welding of contacts”) and minimal
(up to zero) friction – for an incommensurable interface; in the case of a liquid lubricant the friction coeﬃcient should
take on intermediate values.
MOLECULAR DYNAMICS SIMULATION OF FRICTION
system must be three-dimensional. It is connected with the fact that the basic mechanism of energy losses at sliding
is excitation of phonons [1, 10]. The rate of this process is directly proportional to the density of phonon states
in the substrates which cannot be correctly modelled with the help of one- or two-dimensional systems. In other,
the modelling of tribosystems is carried out by standard MD methods. The bottom and top substrates are usually
modelled by a single or few atomic layers each, and the lubricant atoms (or molecules) are placed between the
substrates (Fig. 3). It is assumed that all atoms interact between themselves. The interaction is described, for
example, by the Lennard-Jones potential or by a more realistic for the given system potential. In the longitudinal
directions x and y the periodic boundary conditions are used. The bottom substrate is usually ﬁxed (immobile),
and to the top substrate, a load force (which corresponds, e.g., to its weight) and the driving force are applied,
usually through a spring, the end of which moves with a given speed of v. During simulation the spring force, which
corresponds to the friction force, and also a large number of other parameters, such as the thickness of the lubricant
ﬁlm, its structure, distribution of temperature and atomic velocities through the contact, etc., are saved.
The modelling of tribosystems has, however, two important features. Firstly, as the number of lubricant atoms
is ﬁxed (and up to now the accessible for MD simulation number of atoms is still relatively small), the results of
simulation may be sensible to the number of lubricant atoms N – for example, whether the lubricant atoms form
exactly two atomic layers or two layers with a half. To reduce related errors, one may make one or both surfaces
“corrugated” as shown in Fig. 3 (that, by the way, is closer to a real situation, where surfaces are rough almost
always). Besides, it is desirable to make MD simulations with a diﬀerent numbers N .
The second problem in modelling of friction is more serious. We remind that any tribological system is a “machine”
on transformation of energy of forward motion in a heat. Namely, the driving force constantly pumps energy into
the system, and if we will not remove it, the system soon simply will evaporate or burst. Therefore, using of solely
Newtonian equations of motion is impossible; it is also impossible to use artiﬁcial methods of removing energy (such
as, e.g., the widely used method of renormalization of atomic velocities at every or few MD steps), as the rate of
removing of energy in the end will determine the friction force. Ordinary reception used in such situations is to model
the substrates by a large number of layers, and then for layers distant from the interface, to use Langevin equations
with damping, which smoothly increases with the distance from the contact, modelling in such a way an eﬀectively
“inﬁnite” substrate . However, this method leads to a catastrophic increase of the system size and the necessary
computer power, the more so unjustiﬁed, that in the end only the trajectories of lubricant atoms are of real interest.
A solution of this problem was proposed in Ref. . It consists in the use of Langevin equations for all lubricant
and substrate atoms, but with a “realistic” damping coeﬃcient, which depends on the coordinate r
and velocity v
of the given atom relatively the surfaces in contact, and correctly describes the energy exchange between the moving
atom and the substrates. For the dependence η(r
), it was proposed to use the expression found earlier for an
adatom which vibrates near the crystal surface [25, 26]. Of course, the use of the dependence obtained for vibration
of a single atom, for the case of the system of interacting moving atoms, can result in some errors, but in any case
this approach is much better, than to use as the damping coeﬃcient η some “taken from a ceiling” constant, as in
majority of MD simulation of friction . The use of the velocity-dependent damping coeﬃcient requires in turn a
substantial development of the method of stochastic equations, as was done in Ref. .
The use of the described MD method showed [10, 24] that the basic factor which determines the behavior of a
tribosystem, is the relation between the amplitude of interatomic interaction in the lubricant V
and the interaction
. In the case of traditional (e.g., oil) lubricants, the inequality V
holds, i.e., the lubricant atoms are coupled to the surfaces much stronger, than between themselves; it is the so-called
themselves is strong, V
, and as a result, the lubricant remains in the solid state even at sliding.
MELTING OF A THIN LUBRICANT FILM
surface, essentially diﬀer from those in a bulk, and are characterized by a large variety . The same is true for the
lubricant ﬁlm conﬁned between two surfaces. The ﬁrst and obvious fact is that the temperature of melting of the
lubricant ﬁlm T
is essentially higher, than the bulk melting temperature T
; for example, for a monolayer ﬁlm
the ratio T
may take values around 3. The value of T
monotonically decreases with the increase of the number
in the lubricant ﬁlm and approaches the bulk value only for N
> 5. T
grows also with the increase of
pressure. Such a behavior is related to the limitation of motion of lubricant atoms in the transverse direction because
of the contact with the surfaces.
The mechanisms of melting of the hard and soft lubricants are also diﬀerent . In the hard tribosystem, the
lubricant atoms in contact with the substrates, can vibrate with a larger amplitude, than in the middle of the ﬁlm;
therefore its melting begins from the boundary layers. In the opposite case of the soft lubricant, where the boundary
layers are strongly coupled with the substrates, the melting begins from a middle of the ﬁlm. The T
obtained with the help of the MD method, can be well explained by the known Lindeman criterion . In both cases,
however, the melting is related to the increase of the speciﬁc volume, that in the given system, as the MD simulation
shows, is expressed in a sharp increase of the ﬁlm thickness and formation of an additional atomic layer.
Properties of the molten lubricant ﬁlm diﬀer from those of the bulk liquid – the former demonstrates a well expressed
layered structure, which is saved at sliding as well. Although the discovery of this fact caused the surprise in tribology
community, from point of surface physics this phenomenon is natural: the crystalline structure of the surfaces imposes
a structure to the near-by layers of the liquid lubricant both in the transverse and, in less degree, in the longitudinal
directions x and y (the latter, however, is destroyed at sliding).
It is interesting that in the solid state at T < T
the ﬁlm structure also substantially diﬀers from that in the bulk:
in the bulk, as characteristic for the same temperature. It is related to the presence of a large number of defects (in
particular, vacancies) in the conﬁned ﬁlm, the state of which is closer to a glasslike, than to the ideally crystalline.
Finally, we note that the lubricant ﬁlm can be melted not only because of the rise of temperature, but also due to its
sliding (the sliding-induced melting). The mechanism of this melting, however, diﬀers from that described above .
kinetic friction. For example, theory predicts  that for the contact of two elastic surfaces the static friction
practically always should be zero, that totally conﬂicts with all known experiments. Complication of calculation of
and badly deﬁned (for example, it is assumed that it rather corresponds to a glasslike structure), diﬀers from contact
to contact, and also changes with time (f
grows with the time of stationary contact – the so-called aging of the
kinetic friction only, i.e., the smooth sliding regime, when the system is in the well deﬁnite steady state. We remind
that such a regime exists only at very high speeds of sliding, v > 1 − 10 m/s.
In the case of a traditional (oil-based) lubricant, or the soft tribosystem, the boundary layers of the lubricant
ﬁlm are strongly coupled to the surfaces and, therefore, sliding must begin with a break of bonds somewhere in the
middle of ﬁlm. As a result, the ﬁlm is melted with the onset of motion, and remains liquid both in the smooth
sliding regime (at high speeds v > v
), and in the sliding phase of the stick-slip regime at speeds v < v
. In the
melting-freezing mechanism [34, 35]. In the smooth sliding regime, the liquid state of the ﬁlm is supported due to its
heating because of sliding. The friction coeﬃcient in this system takes on intermediate values of order µ ∼ 0.1, and µ
is directly proportional to the viscosity of lubricant, which for a thin ﬁlm may be in 2 − 3 times higher than the bulk
viscosity [10, 24].
On the other hand, in the case of solid lubricant, or the hard tribosystem, the sliding is carried out at the sur-
face/lubricant interface (usually only at one of the two boundaries, as the system is rarely fully symmetric). And if
the surface and the hard lubricant have an ideal crystalline structure, we get the system with extremely low friction.
The reason of this consists that the substrate and lubricant are, as a rule, rigid enough, so that their elasticities are
higher than the Aubry threshold, i.e., the sliding mode is realized. In addition, the substrates and the lubricant are
made of diﬀerent materials as a rule and, therefore, they have diﬀerent lattice periods, incommensurable in a general
case. But even if the periods coincide or are commensurate, for formation of the commensurable interface in the
two-dimensional contact it is necessary that the axes of these two lattices be strictly aligned, as any, even smallest
dismiss of the axes will result in incommensurability of the lattices. Thus, the regime of extremely low friction should
be carried out practically always if, we emphasize, the substrates and the solid lubricant have the ideal crystalline
structure [10, 24]. Namely this fact explains the very good tribological characteristics of the graphite-based lubricants
as well as other layered materials such as MoS
. Extremely low friction is indeed observed experimentally,
for example, at scanning of the W(011) tip on the Si(001) surface , or at sliding of a graphite scale on the graphite
surface . We note that a large progress in development of hard lubricants is achieved at the Institute of Material
Problems NASU [38, 39].
However, the described above dignities of the solid lubricants disappear totally, if the contacting surfaces are not
ideal, for example, if there are steps, asperities or other defects on the surfaces, where the pinning (hooking) of the
surfaces takes place. At depinning from these defects, the lubricant may be melted, and then, during the stop, it will
be solidiﬁed again, but already with a structure close to amorphous, as the freezing process is very fast due to good
thermal contact with the substrates. In the case of imperfect (amorphous or glasslike) structure of the solid lubricant
ﬁlm, the friction becomes quite large – larger than for the liquid lubricants characteristic for soft tribosystems [10, 24].
Nevertheless, by the careful choice of parameters of the solid lubricant it is possible to recover its good tribological
characteristics. We remind that at sliding the lubricant is heated to some temperature T
, and also that its
is proportional to the amplitude of the interatomic interaction V
. If we will pick up the V
parameter so that at sliding T
be close to T
(but do not exceed it), the defects of the ﬁlm may be removed, and
an imperfect ﬁlm, we observe the stick-slip motion. The ﬁlm temperature sharply rises during the phase of sliding,
the ﬁlm self-orders remaining in the solid state and, after a few stages of sliding, the system passes to the smooth
sliding regime with a very low friction. For realization of the self-ordering mechanism, it is necessary to choose the
amplitude of the V
interaction large enough, so that the ﬁlm is not melted during sliding, but not too large, so that
The results of the MD modelling allowed us also to build the phenomenological theory of kinetic friction , by
which it is possible to predict analytically the behavior of a tribosystem with the change of its parameters.
The described above microscopic mechanisms of friction are for sure important for constructing of nano-mechanical
devices. However there is a question, are they in any relation with the processes of friction in a macrocosm? Foremost
divergence in the value of the critical velocity of the transition from stick-slip to smooth sliding causes suspicion: at
experiment the transition is observed at speeds about 1−10 µm/s [1, 41], while the MD simulation gives v
∼ 1−10 m/s
is done, the higher value of v
is observed . The second problem is related to the viscosity of the thin lubricant
experiment shows their diﬀerence on many orders of magnitude. However, these two problems are linked together.
Indeed, viscosity of the thin ﬁlm is deﬁned as f d/va
is the lattice constant
taken from the experiment, we
between the simulation and experiment for other values (forces, ﬁlm thickness, etc.) .
FIG. 4: The earthquake model.
This contradiction can be resolved with the help of the earthquakes (EQ) model – third from the basic models used in
tribology. The name of this model appeared because the same type of models is used for modelling of earthquakes .
Physics of both processes is qualitatively identical, but diﬀers by the spatio-temporal scale – nanometers and seconds
or hours in tribology on comparison with kilometers and years or centuries in geology. In the EQ model the sliding
interface is treated as a set of contacts bound by springs with the moving top base (the springs model the elasticity of
the top substrate), as shown in Fig. 4. A single contact behaves in accordance with the results of STM experiments
or MD simulations: it is immobile until the total force f
acting on it, does not exceed the static threshold f
(usually for simpliﬁcation
it is supposed f
= 0). Also it is taken into account that the contacts elastically interact with each other, thus a
relaxation as well or even cause an avalanche of relaxations.
The basic issue in the EQ model for description of friction is consideration of aging of contacts, i.e., the threshold
dependence of system dynamics on the sliding velocity . In addition, distribution of contacts should be chaotic, and
the system should be two-dimensional (the 1D model does not succeeded to reproduce the experimentally observed
dependences ). Then at a small speed of sliding, when all contacts have enough time to “grow old” and attain
approximately the same value f
, depinning of contacts takes place simultaneously over the whole system, i.e., their
the threshold values f
for diﬀerent contacts are diﬀerent, therefore they move asynchronically, and as a result, the
at which, we emphasize, the contacts themselves are still in the regime of (microscopic) stick-slip. For the contact of
rough surfaces, a typical distance between the contacts is a ∼ 10
m, and the aging time of contacts is of order
τ ∼ 1 − 10
; thus the change of sliding regimes should take place at the speed v ∼ a/τ , as is observed experimentally.
the sliding surfaces are made of mica, which may have the ideal structure of macroscopic area (up to mm
e.g., with diﬀerent orientation, because this will lower the free energy of the system due to the increase of entropy.
Domains of diﬀerent orientations have diﬀerent values for the thresholds f
, i.e., they play the same role as the
Further development of researches in this direction  should allow us to describe friction on the mesoscopic level,
and that is the basic approach in modern material science.
Majority from the results described above, as well as many others not included in the given review because of lack
of space, was obtained just in the last 5-10 years, that indicates the swift progress in tribology. However, there still is
extraordinarily actual the further improvement of experimental methods, able to ﬁx not only the average friction force,
but also to provide a detailed information about processes inside the lubricant ﬁlm. In this plan, there is perspective
to use the methods, that are already well developed in surface physics. For example, an important information on
the energy exchange in adsorbed ﬁlms can be obtained by the QCM (quartz crystal microbalance) method . From
other interesting new experimental methods, one may note the methods used in works [49–51], and also a recently
developed at the Institute of Physics (NASU) technique of “ﬂoating substrate”, where a sliding block holds out above
the surface by the magnetic ﬁeld .
From the problems not considered in the given review, ﬁrst of all we have to mention the problem of search of
methods of control and purposeful operation of friction both by chemical methods by addition of specially chosen
molecules to the base lubricant , and mechanical methods, for example, using special nanopatterned surfaces 
or applying to the system an external oscillating force .
We also did not touch the important question of the form of lubricant molecules. As was shown above, the minimal
friction is achieved in the case of contact of two ideal crystalline surfaces. The form of molecules of the hard lubricant
in this case is unimportant, as the main role plays the surface structure. In the case of traditional, or soft tribosystems,
the kinetic friction force is directly proportional to viscosity of the liquid lubricant; therefore, a lower is the viscosity,
the smaller must be the friction. As a speculative example one may mention the use of air as a “lubricant” between
the rotating disk and the reading head in computer disks, where the head “levitates” over the disk like an airplane.
Another example known from times of ancient Egypt, but recently acquired the special actuality in connection with
development of nano-mechanical devices, is the use of usual water [56, 57] or water solutions  as a lubricant.
Everybody knows how slippery is the surface of ice covered by a thin water ﬁlm. One has to note also that in
the process of evolution, the nature chose namely water solutions as lubricants in living organisms. However, daily
experience says just about reverse: if to smear hands by a butter, they will be far more slippery, than if it is simple
to get wet of them, i.e., the experience prompts that often a liquid with a high viscosity is a better lubricant. This
is related to squeezing of the lubricant out from the contact area: a higher is the viscosity, the slower is the process
of squeezing out. More rigorously, in the case of boundary lubrication, the important is not the lubricant viscosity,
but the length L of lubricant molecules: a longer is the molecule, the by the greater number of the atoms it holds
on the surface, and the more diﬃcult is to remove it out from the contact area . In some systems, however, the
dependence µ(L) may be nonmonotonic . Another interesting idea is to use advantages of rolling friction, e.g.,
to use as a lubricant the spherical molecules C
(fullerenes) [60, 61] – as is well known, in macroscopic systems the
times lower than the frictions of sliding. Lately, development of nano-
and micro-mechanical elements and machines became actual, for example, micro-bearing using carbon nanotubes or
fullerenes as rollers or marbles, and also “microcars” able to transport “loads” on the crystal surface .
In conclusion we emphasize that the problem of friction is many-branched and requires the coordinated intra-
disciplinary eﬀorts – from the side of physicists, chemists, material scientists and mechanics, and then one may expect
a great progress in tribology in nearest future.
This article is dedicated to the 90-th anniversary of the National Academy of Sciences of Ukraine and of its president
– the Academician Boris Paton. The content of the article is based on the results of researches and numerous
discussions with coauthors and colleagues — Alan Bishop, Thierry Dauxois, Yuri Kivshar, Maxim Paliy, Bo Persson,
Michel Peyrard, Erio Tosatti and Mikhael Urbakh — to which I would like to express my sincere gratitude. I thank
also I.K. Pohodnya, the Editor of the collection of papers devoted to the 90-th anniversary of NASU, for the invitation
to write this article, and A.G. Naumovets for a support and numerous useful comments.
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