Boundary friction on molecular lubricants: rolling mode?
A theoretical model is proposed for low-temperature friction between two smooth rigid solid surfaces separated by lubricant molecules, admitting their deformations and rotations. The appearance of different modes of energy dissipation (by «rocking» or «rolling» of lubricants) at slow relative dis...
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
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irk-123456789-1195122017-06-08T03:02:26Z Boundary friction on molecular lubricants: rolling mode? Loktev, V.M. Pogorelov, Yu.G. Низкоразмерные и неупорядоченные системы A theoretical model is proposed for low-temperature friction between two smooth rigid solid surfaces separated by lubricant molecules, admitting their deformations and rotations. The appearance of different modes of energy dissipation (by «rocking» or «rolling» of lubricants) at slow relative displacement of the surfaces is shown to be accompanied by stick-and-slip features and reveals a nonmonotonic (mean) friction force vs external load. 2004 Article Boundary friction on molecular lubricants: rolling mode? / V.M. Loktev, Yu.G. Pogorelov // Физика низких температур. — 2004. — Т. 30, № 4. — С. 426-430. — Бібліогр.: 15 назв. — англ. 0132-6414 PACS: 46.55.+d, 81.40.Pq http://dspace.nbuv.gov.ua/handle/123456789/119512 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы |
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Низкоразмерные и неупорядоченные системы Низкоразмерные и неупорядоченные системы Loktev, V.M. Pogorelov, Yu.G. Boundary friction on molecular lubricants: rolling mode? Физика низких температур |
| description |
A theoretical model is proposed for low-temperature friction between two smooth rigid solid
surfaces separated by lubricant molecules, admitting their deformations and rotations. The appearance
of different modes of energy dissipation (by «rocking» or «rolling» of lubricants) at slow relative
displacement of the surfaces is shown to be accompanied by stick-and-slip features and reveals
a nonmonotonic (mean) friction force vs external load. |
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Article |
| author |
Loktev, V.M. Pogorelov, Yu.G. |
| author_facet |
Loktev, V.M. Pogorelov, Yu.G. |
| author_sort |
Loktev, V.M. |
| title |
Boundary friction on molecular lubricants: rolling mode? |
| title_short |
Boundary friction on molecular lubricants: rolling mode? |
| title_full |
Boundary friction on molecular lubricants: rolling mode? |
| title_fullStr |
Boundary friction on molecular lubricants: rolling mode? |
| title_full_unstemmed |
Boundary friction on molecular lubricants: rolling mode? |
| title_sort |
boundary friction on molecular lubricants: rolling mode? |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2004 |
| topic_facet |
Низкоразмерные и неупорядоченные системы |
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http://dspace.nbuv.gov.ua/handle/123456789/119512 |
| citation_txt |
Boundary friction on molecular lubricants: rolling mode? / V.M. Loktev, Yu.G. Pogorelov // Физика низких температур. — 2004. — Т. 30, № 4. — С. 426-430. — Бібліогр.: 15 назв. — англ. |
| series |
Физика низких температур |
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AT loktevvm boundaryfrictiononmolecularlubricantsrollingmode AT pogorelovyug boundaryfrictiononmolecularlubricantsrollingmode |
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2025-07-08T16:00:30Z |
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2025-07-08T16:00:30Z |
| _version_ |
1837095120651943936 |
| fulltext |
Fizika Nizkikh Temperatur, 2004, v. 30, No. 4, p. 426–430
Boundary friction on molecular lubricants:
rolling mode?
V.M. Loktev
Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine,
Metrologichna Str. 14-b, Kiev, 03143 Ukraine
Yu.G. Pogorelov
CFP/Departamento de F�sica, Universidade do Porto, 4169-007 Porto, Portugal
E-mail: ypogorel@fc.up.pt
Received September 23, 2003, revised November 13, 2003
A theoretical model is proposed for low-temperature friction between two smooth rigid solid
surfaces separated by lubricant molecules, admitting their deformations and rotations. The appear-
ance of different modes of energy dissipation (by «rocking» or «rolling» of lubricants) at slow re-
lative displacement of the surfaces is shown to be accompanied by stick-and-slip features and re-
veals a nonmonotonic (mean) friction force vs external load.
PACS: 46.55.+d, 81.40.Pq
Introduction
In modern tribology there is a still increasing inter-
est in studies of wearless friction on atomically smooth
surfaces [1,2] as a possibility to provide information
about the basic processes of energy losses on the micro-
scopic level, important for the purposes of optimization
in many technological applications. This is also con-
nected with the search for the best coating and lubri-
cant materials. The principal physical picture usually
considered in relation to boundary friction on few mo-
lecular layers of lubricant liquid is a sequence of «freez-
ing—melting» processes of the lubricant, giving rise to
discontinuous (stick-and-slip) displacement of sliding
surfaces [3]. Recently a new theoretical approach was
proposed for microscopic sliding processes at extremely
low velocities of motion and upon a monolayer of lubri-
cant atoms [4], as can be the case for friction force mi-
croscopy (FFM). Based on the adiabatic formation of
metastable states (similar to dislocations in usual de-
formed crystals or defects in the Frenkel—Kontorova
model [5,6]) and their following relaxation, this treat-
ment shows how the (average) microscopic friction co-
efficient depends upon the material parameters of the
contacting solids and lubricants and also how
stick-and-slip jumps with atomic periodicity can de-
velop in the microscopic friction force. It should be
noted that a similar microscopic mechanism of dislo-
cation-assisted sliding was recently proposed for
contacting asperities in dry friction [7].
Ultimately, with the discovery of the almost spher-
ical fullerene C60 molecule [8] (and/or cylindrical
carbon nanotubes [9]) a hope had arosen to use such
closed molecular structures as «free rotating» lubri-
cants, for considerable reduction of the friction coeffi-
cient. For instance, a reduction of the sliding friction
coefficient was already discussed, due to involvement
of spinning motion of surfaces in contact [10].
However, the FFM experiments with use of C60
monolayers deposited over atomically smooth solid
surfaces brought some contradictory results [11–13].
To get a better understanding of them, a further theo-
retical insight is desirable on the elementary processes
of boundary friction.
The present communication is aimed at extention of
the above-mentioned adiabatic approach to the pro-
cesses of boundary friction which include the internal
degrees of freedom of the lubricant molecules. Within a
simplest model, we make an attempt to show that, due
to the discrete atomic structure of such a molecule,
there are possible qualitatively different modes of slow
motion, either dissipative or nondissipative, depending
on the applied external load on the contact. For
dissipative modes, there are energy losses resulting
© V.M. Loktev and Yu.G. Pogorelov, 2004
from stick-and-slip discontinuities, but these losses
turn out much lower than for similar processes at slid-
ing solid surfaces upon atomic lubricants. Besides, the
mean value of friction force is found to be a non-
monotonic function of the external load and can be sen-
sitive to temperature in the cryogenic range.
The model
Let us consider a two-dimensional model for the
boundary friction system which includes two semi-in-
finite atomic arrays, «solids», with identical triangu-
lar lattice structure and a spatial separation d between
their surfaces, and clusters of four atoms, the «molec-
ular lubricants», confined between the solids (Fig. 1).
The distances between nearest neighbor atoms in both
solids and in clusters are supposed invariable* (that
means, corresponding to «absolutely rigid» bonds),
and the bond length for the molecule is equal to the
lattice parameter a for the solid surfaces.
The model includes certain «soft links», which de-
termine the energy variation vs relative displacement
of the solids. In the first turn, this is a weak
Lenard-Jones interaction between an atom of solid and
a lubricant molecule at distance r:
V r
r
r
r
rL J� �
�
�
�
�
�
� �
�
�
�
�
�
�
�
�
�
�
�
( ) �0
0
12
0
6
2 , (1)
characterized by the adhesion energy �0 and equilib-
rium distance r0. Besides, a lubricant molecule can
experience rhombic deformation which changes its en-
ergy as ksin2 �, where k and � are the elastic constant
and rhombic angle, respectively. We neglect the in-
teraction between different lubricants, that is, we
omit their possible collective modes. Thus the molec-
ular lubricants are supposed to contribute independ-
ently to the total friction force. Then we choose one
particular molecule and set its center of mass as the
origin of the reference frame, so its configuration is
determined by the «internal» rhombic angle � and the
«external» orientation angle � (say, with respect to
the normal to the interface). In such a frame, the two
solids are supposed to be displaced symmetrically
with respect to the lubricant molecule, and their con-
figuration is fully defined by the displacement param-
eter p chosen, e.g., as the smallest positive longitudi-
nal coordinate of the atoms of the upper solid.
All this permits us to write the total energy (per
lubricant) in the simple form:
E V kL J n m
n m
� � ��� (| | ) sin
,
r R 2 �, (2)
where r Rn mand are respectively the positions of the
atoms of solids and the atoms in the lubricant mole-
cule. Obviously, this extremely simplified model does
not pretend to give a quantitative description, and ex-
planation of friction experiments in real systems with
complex and typically incommensurate solid and mo-
lecular structures. It serves mainly to illustrate some
new qualitative possibilities for weakly dissipative
processes associated with the internal (rolling or rota-
tional) degrees of freedom of molecular (including
nanotubes) lubricants, in contrast to the dissipation by
only translational motions of the «point-like»
(atomic) lubricants.
The adiabatic dynamics
The adiabatic treatment of the system, correspond-
ing to the expression (2), follows the lines suggested in
Ref. 4. The equilibrium distance for Lenard-Jones in-
teraction (1) is taken equal to the interatomic distance
in the solids: r a0 � . At a given separation d between
the surfaces, we calculate numerically the total energy
profile E p( , , )� � as a function of the displacement
parameter p, and also of the angles � and �. Next this
function is optimized with respect to the deformation
angle � to result in the profiles E p( , ),� such as those
Boundary Friction on Molecular Lubricants: Rolling Mode?
Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 427
* Here mean distances are meant, as the only relevant ones for adiabatic equilibrium states, while small thermal
fluctuations become important for activation of relatively fast transitions between these equilibrium states.
d
p a
x
y
��
Fig. 1. Schematic of the friction system: two solid arrays
(grey circles) with identical lattice structures are sepa-
rated by a four-atom lubricant molecule (white circles).
The molecule center rests at the origin, and its configura-
tion is determined by the tilt angle � and rhombic angle
�, while the symmetric displacements of the solids are de-
scribed by the parameter p (see the text).
displayed in Figs. 2–5 (for different values of p,
through the whole displacement period from 0 to a, and
at different separations d).
The primary optimization in � refers to the stronger
elastic deformation constant (we took k � 0 5 0. � , while
the amplitude of relevant energy oscillations in � is
~ .01 0� ) and hence to a faster relaxation in � than in
�*. Then the system behavior at very slow uniform
variation of the parameter p with time (that is, the
slow dynamics) is obtained from the analysis of the
profile E p( , )� . Below we analyze how this profile
changes with growing external load, which is here
simulated by a gradual decrease of the separation
distance d.
i) At the greatest separations (evidently correspon-
ding to the lowest loads), it is seen from Fig. 2 that the
energy profile has a single �/2 periodicity in �. This
means that, for any given p, there is a single equilib-
rium state for the molecule, characterized by its orien-
tation �( )p (up to aC4 rotation) and energy E p( ( ))� ,
such that ( ) ( )� �� �E/ p � 0, ( ) ( )� ���
�
2 0E/ p � . In this
case, the phase trajectory E( )� (shown by the sequence
of bold dots linked by arrows, for growing displace-
ment) is closed and continious. Thus, the system energy
changes in a fully reversible way and, though some
forces are exerted in the process, their mean value over
the cycle and thus the mean friction force are exactly
zero. This reversible variation of the angle � around its
median position (such that the long diagonal of the
rhomb points vertically, inset to Fig. 2), corresponds to
a «rocking mode» of the molecular motion.
ii) As the two solid surfaces draw closer (at increas-
ing load), the first qualitative change that appears in
the system behavior is the doubling of its energy mini-
ma. Now there are two split minima at the initial con-
figuration p � 0 (Fig. 3), which correspond to a two-
fold degenerate equilibrium state of the deformed
lubricant molecule: the long diagonal of the rhomb
can deviate by a finite angle � �/2 to both sides from
the vertical (inset to Fig. 3). Since the molecule is
considered a classic object, it initially occupies only
one of the minima (the left one is chosen in Fig. 3).
However, the degeneracy gets lifted for p � 0, so that
one of the split minima turns into a stable (s) and the
other a metastable (m) equilibrium state. As is seen from
the consecutive curves in Fig. 3, the energy barrier h be-
tween the m state and the nearest s state (here that to
the left of m) decreases by many times with growing p,
as a certain function h p( ). Since the adiabatic lifetime
for the m state is: � � �m a h� exp( ) (where � a ~ 10 12� s
is the atomic oscillation time and � is the inverse tempe-
rature), it decreases in this course by many orders of
428 Fizika Nizkikh Temperatur, 2004, v. 30, No. 4
V.M. Loktev and Yu.G. Pogorelov
* A similar hierarchy of relaxation rates between two degrees of freedom was also reported for point-like lubricants,
corresponding to their normal (transverse) and longitudinal displacements [4].
– /4� 0 �/4
0.6 0.4
0.8 0.2
p = 0
d = 2.6a
0
–0.9
–1.0
–1.1
–1.2
�, rad
P
o
te
n
tia
l e
n
e
rg
y
(
u
n
its
)
� 0
Fig. 2. Series of profiles of potential energy vs orientation
(at optimal deformation, see the text) and the related
equilibrium states (bold circles) of a lubricant molecule at
growing displacement of the solids (the numbers indicate
p/a values) and at the highest separation between them.
The phase trajectory (arrows) shows that the molecule
performs continuous «rocking» motions and after the full
cycle (at p/a � 1) returns to its initial state with no en-
ergy dissipation. Inset: the rhombic configuration at p � 0.
0.2
0.4
0.4 0.6
0.8
0
d = 2.32a
p = 0
– /4� 0 �/4 �/2
–1.6
–1.7
–1.8
–1.9
P
o
te
n
tia
l e
n
e
rg
y
(
u
n
its
)
� 0
�, rad
Fig. 3. The same as in Fig. 2 but at closer separation be-
tween the solids (higher external load). The lubricant
molecule after «rocking» returns to its initial state, but in
this course it experiences a thermally activated jump (the
dotted arrow over the potential barrier for p/a � 04. ) be-
tween the metastable (0.4) and stable ( . )04� minima, and
the corresponding energy difference is irreversibly lost. In-
set: the configuration at p � 0 corresponds to one of the
split minima.
magnitude. Eventually, this lifetime gets comparable to
the characteristic time �0 of slow displacement by an
atomic period, at a very sharply defined instant when
p p� 0 so that h p h / a( ) ln( )0 0
1
0� � �� � � . Hence it is
almost exactly at this instant that a thermally activated
jump from the m to the s state is realized. After the
jump, the energy difference between the m and s states
(to the moment of transition) is irreversibly lost,
through the creation of quasiparticles (phonons, for in-
sulating solids) which are finally thermalized in the
bulk. For typical displacement rates in FFM~ 102
Å/s,
one has �0
210~ � s, so that the barrier to the transition
moment is still as high as h0
123~ � � . For the situation
presented in Fig. 4, the transition corresponds to
p a0 0 4� . (the potential profile shown by the solid line).
It is also seen that, with further increase of p, up to a,
the system returns to its initial state; thus the phase
trajectory is still closed («rocking mode» again) though
discontinuous. This discontinuity produces stick-and-slip
features in the microscopic behavior of the force vs dis-
placement and it is the only source of irreversible losses in
our model system, in full similarity to the model of slid-
ing solids upon atomic lubricants.
There are, however, some distinctions between the
two systems. First, the stick-and-slip profile for mo-
lecular lubricants is more complicated than the simple
triangular sawtooth for atomic lubricants. But espe-
cially important is the fact that the irreversible forces
for molecular lubricants are smaller than the elastic,
reversible forces (the jump heights in Figs. 3–5 are no-
ticeably lower than the amplitudes of the smooth os-
cillations), while the irreversible forces for atomic lu-
bricants are orders of magnitude higher than the
reversible ones. Since the reversible forces in both
cases are characterized by the same energy scale �0,
this indicates a possibility to substantially reduce the
dissipation by molecular (rotating) lubricants vs that
by atomic lubricants. Note, at least, that if the mole-
cule occurs initially (at p � 0) in the right minimum,
it simply spends a period, 0 � �p a, in the s state, but
then passes to the m state during the next period and
eventually comes to the same closed discontinuous
rocking.
iii) Now let us bring the surfaces yet a little bit
closer, as by a very small change of the distance d be-
tween the solids (Figs. 3 and 4). Then the system evo-
lution acquires yet a new quality: now at the transi-
tion moment the nearest s state is to the right of the
given m state. As a results, after the transition, at
p p� 0, the molecule does not return to its initial state
but will eventually be rotated by an angle �/2 (the
dashed circles and arrow in the inset to Fig. 4), and
this rotation will be repeated in each successive pe-
riod. Thus in this case we have both unclosed (corre-
sponding to the «rolling mode») and discontinuous re-
gimes of lubricant motion.
iv) With still increasing loads, the situation of a
single minimum for E( , )0 � will be restored again
(though displaced by the angle �/4), corresponding
to a «square» molecule (inset to Fig. 5). But now, un-
like the case of low loads, this minimum gets split
with increasing displacement p (here at p � 0 3. ). The
molecule returns from the «rolling» to «rocking» re-
gime («rock-n-roll dance»), but with considerably
reduced energy dissipation: the energy loss at a jump
for d a� 21. (Fig. 5) is about 2.5 times smaller than for
d a� 2 28. (Fig. 4), i.e., it decreases with increasing
load. We do not present here detailed results of nu-
merical simulations for even higher loads, noting only
Boundary Friction on Molecular Lubricants: Rolling Mode?
Fizika Nizkikh Temperatur, 2004, v. 30, No. 4 429
P
o
te
n
tia
l e
n
e
rg
y
(
u
n
its
)
� 0
�, rad
–1.6
–1.7
–1.8
–1.9
–2.0
–2.1
0 �/4 �/2 3 /4�
0
0.2
0.4
0.4 0.6
0.8
1.0
d = 2.28a
Fig. 4. The same as in Figs. 2 and 3 but at still higher load.
The thermally activated jump changes its direction and the
lubricant molecule rotates by the angle �/2 after every pe-
riod in p («rolling mode»). The energy loss and hence the
friction force is somewhat larger than that in Fig. 3.
P
o
te
n
tia
l e
n
e
rg
y
(
u
n
its
)
� 0
�, rad
–1.2
–1.6
–2.0
–2.4
–2.8
0 �/4 �/2
0.6 0.6 0.4
00.8
d = 2.1a
0.2
Fig. 5. The same as in Figs. 2–4 at futher growing load.
The molecule returns from «rolling» to «rocking» regime
with an intermediate jump, but at much smaller energy
loss than in Figs. 3 and 4.
that they reveal a number of subsequent dissipative re-
gimes, either with growing and falling friction forces.
Finally, it should be noticed that the irreversible
processes considered above are triggered by the actual
temperature and this parameter can be as important as
the applied pressure for changing the dissipated en-
ergy, for changing the dissipation regime, and for the
very presence of energy losses. Thus, using typical LJ
energy for an C–C pair � ~ 1eV, the critical condition
p a0 0 4� . for the rolling process iii) in Fig. 4 is ex-
pected at � �1 ~ 15 K, but it will change to p a0 0 2� .
with about a twofold decrease of the dissipated energy
at � �1 ~ 40 K.
Conclusions
The above simple analysis demonstrates that
boundary friction with participation of molecular
(spheroidal or cylindrical) lubricants can possess
quite unusual properties, such as the existence of vari-
ous regimes of molecular motion, either nondissipative
or dissipative, with abrupt transitions from one regime
to another upon continuous variation of the external
load. In the sequence of regimes («rocking» and «roll-
ing») the obtained friction force vs load displays
nonmonotonic and hence nonlinear behavior. This
model (of course, with due improvements to be
incorporated) may provide a mechanism for explain-
ing the data of real experiments [14,15] with the
fullerenesC60 as lubricants and give indications for an
optimal regime of their practical applications.
V.M.L. acknowledges partial support of this inves-
tigation by Science and Technology Center of Ukraine
(grant No 2669).
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