The effective viscosity temperature dependence of the lubricant is taken into consideration within the thermodynamic model of boundary friction. Transitions between the friction kinetic modes are described as the second-order phase transitions. For the system kinetic investigation the mechanical analogue of the simple tribological system has been considered. The time dependences of block coordinate, velocity, friction force and shear stress have been build. Therefore, the stick-slip motion has been described for a wide range of the system parameters. A comparison of simulation results with experimental data in sequence increasing shear rate has been made

Liashenko, Yakiv Oleksandrovych

Zaskoka, Anton Mykolaiovych

Заскока, Антон Миколайович

Заскока, Антон Николаевич

Ляшенко, Яков Александрович

Ляшенко, Яків Олександрович

Electronic Sumy State University Institutional Repository

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE  NANOMATERIALS: APPLICATIONS AND PROPERTIES Vol. 3 No 1, 01NTF18(3pp) (2014)   2304-1862/2014/3(1)01NTF18(3) 01NTF18-1  2014 Sumy State University The Viscosity Temperature Dependence in the Boundary Friction Model under a Periodic External Action  A.M. Zaskoka*, I.A. Lyashenko  Sumy State University, 2, Rymsky Korsakov Str., 40007 Sumy, Ukraine  (Received 15 July 2014; published online 29 August 2014)  The effective viscosity temperature dependence of the lubricant is taken into consideration within the thermodynamic model of boundary friction. Transitions between the friction kinetic modes are described as the second-order phase transitions. For the system kinetic investigation the mechanical analogue of the simple tribological system has been considered. The time dependences of block coordinate, velocity, friction force and shear stress have been build. Therefore, the stick-slip motion has been described for a wide range of the system parameters. A comparison of simulation results with experimental data in sequence increas-ing shear rate has been made.  Keywords: Boundary friction, Phase transition, Elastic and viscous stress, Effective viscosity.   PACS numbers: 47.15.gm, 47.50. – d, 62.20.Qp                                                    * zaskoka23@ukr.net 1. INTRODUCTION  The dynamic properties of an ultrathin film which is confined between two atomically flat solid surfaces are significantly different from a bulk lubricant [1,2]. The boundary friction mode set in, when the lubricant thickness is less than several molecular layers. In this case, the lubricant film does not form stable thermody-namic solid or liquid phase. Instead, the regions with solid-like or liquid-like structures appear [1,3]. After a prolonged rubbing mechanisms operation at high pres-sures, the lubricant squeezes out of the surfaces and the boundary mode is established [3]. The periodical phase transitions of the first-order [3] and the second-order [4] between structural states lead to the inter-rupted motion. One of the most common experimental techniques is based on the use of surface force appa-ratus (SFA) [3]. It is used for investigation of quasi-spherical and chain molecules behavior which are con-fined between two atomically flat mica surfaces [3,5]. The detailed experimental investigation of the effec-tive viscosity dependence on the temperature and the velocity gradient has been done in the work [6]. The main aim of this research is to investigate the tribolog-ical system behavior with help of the thermodynamic model, which takes into consideration the dependence of the effective viscosity on the temperature and the shear rate.  2. TRIBOLOGICAL SYSTEM  Let us consider the mechanical analogue of the tribological system [7] which shown in Fig. 1.    Fig. 1 – The scheme of the tribological system.  A block of mass M is located on rollers. The rolling friction for the block can be neglected. The block is confined between two springs with stiffness K each. On the surface of the block another one is placed, which is forced to move by external forces. In the presence of interaction forces between the two blocks, the motion of the upper block causes the motion of the lower one. However, their trajectories do not coincide and depend on the friction mode established in the system. Let us denote the upper block coordinate and veloci-ty by X and V = Ẋ, respectively. Moreover, x, v = ẋ are a coordinate and a velocity of the lower block. The upper block moves according to the cyclic law:   cosmX X t , (2.1)  sinmV X t , (2.2)  where Xm is the amplitude and ω is the cyclic frequency. In this case motion equation takes form [8]:   2 0Mx Kx F . (2.3)  3. VISCOSITY AND FRICTION FORCE DE-PENDENCE ON  TEMPERATURE  The stresses arising in the lubricant layer during the movement contain elastic σel and viscous σv compo-nents. The friction force F is determined as a multipli-cation of the total stresses and the rubbing surfaces contact area A:   el vF A  (3.1)  The viscous stresses is found according to the equa-tion [6]:   v eff , (3.2)  where ηeff is the effective viscosity. In the work [6] the effective viscosity ηeff depend-ence on the temperature T and on the deformation rate of the layers consisting of hydrocarbon polymeric  A.M. ZASKOKA, I.A. LYASHENKO PROC. NAP 3, 01NTF18 (2014)   01NTF18-2 chain molecules has been investigated by molecular dynamics methods. The resultant dependence has the form [6]:   10nCeff , (3.3)  where C(n) and n are the temperature dependant pa-rameters. It is experimentally found, that the parameter n varies from 1 in the case of low temperatures (solid-like state) to n = 0 (liquid-like state) in the case of high temperatures. In order to take this peculiarity into account, authors of article [6] introduce relationship between T and n in form:   11 ( )knT T, (3.4)  where constants Tk and β depend on the polymer used. Particularly, for eicosan C20H42 those parameters equal Tk = 353 К, β = 4.09 [6]. For all lubricants kinds, the dependence C(n) can be approximately expressed by equality [6]:   10 9 3 8C n . (3.5)  4. FREE ENERGY  Let’s write down the free energy density in the form [4,9]:    2 2 2 4( )2 4c ela bf T T , (4.1)  where T is the lubricant temperature, Tc is the critical temperature, ɛ el is the shear component of elastic strain, α, a, b are positive constants, ϕ is the order parameter which in the solid-like state takes nonzero values and equals to zero, when lubricant is liquid-like [4,9]. The elastic stresses arising in the lubricant are de-fined as σel = ∂f/∂ɛ el [4]:   2el el ela , (4.2)  where μ is the shear modulus which takes zero values in liquid-like state. Nonzero values of the shear modu-lus which correspond to solid-like state, can be found from condition ∂f/∂ϕ = 0 [4]:    22 ( )c elaT T ab, (4.3)  From the expression (4.3) let us found the elastic strain critical value [4]    2 ( )cel cT Ta. (4.4)  The lubricant melts when it exceeds this value. The elastic stresses (4.4) stationary value depends on the film thickness h, the internal stresses relaxation time τε and the surfaces relative velocity (V−v) [10,11]:    0( )elV vh. (4.5)  From the Eq. (4.4) using expressions (4.5) we can evaluate the critical velocity Vc0:   02 ( )cch T TVa. (4.6)  The friction force F can be written as combination of (3.2) – (3.5), (4.2) and (4.5):  110 9 3 8sgn( )10nnelV vF V v Ah. (4.7)  5. STICK-SLIP FRICTION  The typical time dependences of the system param-eters are shown in Fig. 2. Here the dashed curves cor-respond to the upper block coordinates X and velocities V and the solid curves to the bottom block coordinates x and velocities v.    Fig. 2 – The dependences of coordinates X and x, velocities V and v, elastic stresses σel and friction force F on the time t at the parameters α = 0.95 J K−1/m3, Tc = 290 K, a =4∙1012 Pa, b = 230 J/m3, Tk = 353 K, β = 4.09, A = 0.6∙10−9 m2, τɛ  = 10−8 s, h = 10−9 m. M = 0.4 kg, δ = 100 J−1∙m3/s, K = 3000 N/m, T = 235 K, Xm = 0.9∙10−6 m, ω = 5 rad/s.   THE VISCOSITY TEMPERATURE DEPENDENCE IN THE BOUNDARY… PROC. NAP 3, 01NTF18 (2014)   01NTF18-3 At the initial time moment, the lubricant is solid-like. At time t > 0 the upper block starts motion in accordance with Eq. (2.1). The bottom blocks moves together with the upper one, because of a friction force F, arising between the surfaces. However, the bottom block motion is pre-vented by stretching out and compressing of the springs. Therefore, in general case the velocities V and v do not coincide. Consequently, the relative shear velocity |V−v| increases. After exceeding critical value |V−v| > Vc0 (4.6), the lubricant melts. The friction force |F| abruptly de-creases (Fig. 2, c), and the block slips on a significant distance (Fig. 2, a). The bottom block velocity sharply grows and as a result the value |V−v| increases. At this moment in Fig.2, b an increase in the friction force abso-lute value can be observed due to the stress viscous com-ponent increase according to Eq. (4.2), (4.5) and (4.7). Whereupon |F| decreases since the relative velocity re-ducesеs and F reverses sign. Here the block slips so much, so stretched spring compresses and vice versa. After ful-fillment of the condition |V−v| < Vc0 the lubricant solidi-fies.    Fig. 3 – The friction force F dependences on the time t for the same parameters as in Fig. 2 and Xm1 = 0.67∙10−6, Xm2 = 0.9∙10−6, Xm3 = 1.31∙10−6, Xm4 = 5∙10−6 m. The panel a corresponds to the lubricant temperature T1 = 235 K, and the panel b to the T2 = 280 K one.  Lets consider the shear amplitude influence on the system behavior. Fig. 3, a demonstrates the friction force time dependence for 4 different shear amplitudes of the upper block. It is shown that the increase in amplitude Xm leads to the rise of phase transition amount over period (Fig. 3, a, amplitudes Xm1, Xm2, Xm3). It is explained by the fact, that the block velocity Xmωsin(ωt) increases. Therefore the phase transition frequency grows [3,5,8,12]. As the amplitude is elevat-ed to Xm = 5 μm the lubricant solidifies twice over a period. In this situation the lubricant solidifies when the block changes its direction. It is also worth noting that amplitudes were chosen so that their subsequent changing leads to stationary state at minimal time. For instance, for intermediate value Xm = 0.75 μm, which is between values Xm1 and Xm2, the stick-slip motion is observed. In this case, the blocks twice “stick” and twice “slip” over a period. However, significant time is required to establish the stationary state. Fig. 3, b was built for the higher lubricant tempera-ture T2 = 280 K. The increase in temperature causes the decrease in the phase transition frequency. At the same time, the maximum friction force value reduces [3,5,7,8,11-13]. The figure first section determines the stick-slip motion, where lubricant melts twice over a period. This case is similar to that one described above for Xm4. For the subsequent amplitude value Xm2, the lubricant does not solidify after melting. Therefore, the further amplitude growing does not influence the fric-tion mode and slide friction is established in the system [13, 14].  6. CONCLUSION  In the work the thermodynamic model of ultrathin lubricant melting has been further developed. The novelty of the investigation consists in consideration of the de-pendence of the polymeric lubricant viscosity on the tem-perature and velocity gradient within the second-order phase transition model. The dependence evidences, that the effective viscosity logarithm is proportional to the logarithm of the shear rate. The proportionality factor value varies from 1 in the solid-like state to 0 in the liq-uid-like state. It has been shown that stick-slip motion is a result of second-order phase transition between liquid-like and solid-like lubricant structures. The time dependencies of coordinate, velocity, fric-tion force and elastic stress have been built. The com-puter simulation results have been presented, which predicts stick-slip disappearance and further system functioning in the sliding friction mode.  AKNOWLEDGEMENTS  The work was executed at support of the Ministry of Education and Science of Ukraine (MESU) within framework of the project “Modeling of friction of metal nonparties and boundary liquid films which interact with atomically flat surfaces” (No. 0112U001380).  REFERENCES  1. J. Israelachvili, Surf. Sci. Rep. 14, 109 (1992). 2. B.N.J. Persson Sliding friction. Physical principles and applications (New York: Springer-Verlag: 2000) 3. H. Yoshizawa, J. Israelachvili, J. Phys. Chem. 97, 11300 (1993). 4. V.L. Popov, Tech. Phys. 46, 605 (2001). 5. A.D. Berman, W.A. Ducker, I.N. Israelachvili, Langmuir 12, 4559 (1996). 6. I.M. Sivebaek, V.N. Samoilov, B.N.J. Persson, Phys. Rev. Lett. 108, 036102 (2012). 7. C.-R. Yang, R.-T. Lee, Y.-C. Chiou, Tr. Int. 30, 719 (1997). 8. C.-R. Yang, Y.-C. Chiou, R.-T. Lee, Tr. Int. 32, 443 (1999). 9. V.L. Popov, Sol. State Commun. 115, 369 (2000). 10. I.A. Lyashenko, Tech. Phys. 56, 701 (2011). 11. I.A. Lyashenko, A.M. Zaskoka, Ukr. J. Phys. 58, 91 (2013). 12. P.D. Baets, J. Degrieck, F.V.D. Velde, A.P.V. Peteghem, Wear 243, 48 (2000). 13. I.A. Lyashenko, A.M. Zaskoka, J. Phys. Stud. 17, 1002 (2013). 14. I.A. Lyashenko, A.V. Khomenko, A.M. Zaskoka, Trib. Trans. 56, 1019 (2013). 

The Viscosity Temperature Dependence in the Boundary Friction Model under a Periodic External Action

SocioEconomic Challenges Journal (Sumy State University)

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE  
NANOMATERIALS: APPLICATIONS AND PROPERTIES 
Vol. 3 No 1, 01NTF18(3pp) (2014) 
 
 
2304-1862/2014/3(1)01NTF18(3) 01NTF18-1  2014 Sumy State University 
The Viscosity Temperature Dependence in the Boundary Friction Model under a Periodic 
External Action 
 
A.M. Zaskoka*, I.A. Lyashenko 
 
Sumy State University, 2, Rymsky Korsakov Str., 40007 Sumy, Ukraine 
 
(Received 15 July 2014; published online 29 August 2014) 
 
The effective viscosity temperature dependence of the lubricant is taken into consideration within the 
thermodynamic model of boundary friction. Transitions between the friction kinetic modes are described as 
the second-order phase transitions. For the system kinetic investigation the mechanical analogue of the 
simple tribological system has been considered. The time dependences of block coordinate, velocity, friction 
force and shear stress have been build. Therefore, the stick-slip motion has been described for a wide range 
of the system parameters. A comparison of simulation results with experimental data in sequence increas-
ing shear rate has been made. 
 
Keywords: Boundary friction, Phase transition, Elastic and viscous stress, Effective viscosity. 
 
 PACS numbers: 47.15.gm, 47.50. – d, 62.20.Qp 
 
 
                                                 
* zaskoka23@ukr.net 
1. INTRODUCTION 
 
The dynamic properties of an ultrathin film which 
is confined between two atomically flat solid surfaces 
are significantly different from a bulk lubricant [1,2]. 
The boundary friction mode set in, when the lubricant 
thickness is less than several molecular layers. In this 
case, the lubricant film does not form stable thermody-
namic solid or liquid phase. Instead, the regions with 
solid-like or liquid-like structures appear [1,3]. After a 
prolonged rubbing mechanisms operation at high pres-
sures, the lubricant squeezes out of the surfaces and 
the boundary mode is established [3]. The periodical 
phase transitions of the first-order [3] and the second-
order [4] between structural states lead to the inter-
rupted motion. One of the most common experimental 
techniques is based on the use of surface force appa-
ratus (SFA) [3]. It is used for investigation of quasi-
spherical and chain molecules behavior which are con-
fined between two atomically flat mica surfaces [3,5]. 
The detailed experimental investigation of the effec-
tive viscosity dependence on the temperature and the 
velocity gradient has been done in the work [6]. The 
main aim of this research is to investigate the tribolog-
ical system behavior with help of the thermodynamic 
model, which takes into consideration the dependence 
of the effective viscosity on the temperature and the 
shear rate. 
 
2. TRIBOLOGICAL SYSTEM 
 
Let us consider the mechanical analogue of the 
tribological system [7] which shown in Fig. 1. 
 
 
 
Fig. 1 – The scheme of the tribological system. 
 
A block of mass M is located on rollers. The rolling 
friction for the block can be neglected. The block is 
confined between two springs with stiffness K each. On 
the surface of the block another one is placed, which is 
forced to move by external forces. In the presence of 
interaction forces between the two blocks, the motion of 
the upper block causes the motion of the lower one. 
However, their trajectories do not coincide and depend 
on the friction mode established in the system. 
Let us denote the upper block coordinate and veloci-
ty by X and V = Ẋ, respectively. Moreover, x, v = ẋ are a 
coordinate and a velocity of the lower block. The upper 
block moves according to the cyclic law: 
 
 cosmX X t , (2.1) 
 sinmV X t , (2.2) 
 
where Xm is the amplitude and ω is the cyclic 
frequency. In this case motion equation takes form [8]: 
 
 2 0Mx Kx F . (2.3) 
 
3. VISCOSITY AND FRICTION FORCE DE-
PENDENCE ON  TEMPERATURE 
 
The stresses arising in the lubricant layer during 
the movement contain elastic σel and viscous σv compo-
nents. The friction force F is determined as a multipli-
cation of the total stresses and the rubbing surfaces 
contact area A: 
 
 el vF A  (3.1) 
 
The viscous stresses is found according to the equa-
tion [6]: 
 
 v eff , (3.2) 
 
where ηeff is the effective viscosity. 
In the work [6] the effective viscosity ηeff depend-
ence on the temperature T and on the deformation rate 
of the layers consisting of hydrocarbon polymeric 
 A.M. ZASKOKA, I.A. LYASHENKO PROC. NAP 3, 01NTF18 (2014) 
 
 
01NTF18-2 
chain molecules has been investigated by molecular 
dynamics methods. The resultant dependence has the 
form [6]: 
 
 10
nC
eff , (3.3) 
 
where C(n) and n are the temperature dependant pa-
rameters. 
It is experimentally found, that the parameter n 
varies from 1 in the case of low temperatures (solid-like 
state) to n = 0 (liquid-like state) in the case of high 
temperatures. In order to take this peculiarity into 
account, authors of article [6] introduce relationship 
between T and n in form: 
 
 
1
1 ( )k
n
T T
, (3.4) 
 
where constants Tk and β depend on the polymer used. 
Particularly, for eicosan C20H42 those parameters equal 
Tk = 353 К, β = 4.09 [6]. 
For all lubricants kinds, the dependence C(n) can be 
approximately expressed by equality [6]: 
 
 10 9 3 8C n . (3.5) 
 
4. FREE ENERGY 
 
Let’s write down the free energy density in the 
form [4,9]:  
 
 2 2 2 4( )
2 4
c el
a b
f T T , (4.1) 
 
where T is the lubricant temperature, Tc is the critical 
temperature, ɛ el is the shear component of elastic 
strain, α, a, b are positive constants, ϕ is the order 
parameter which in the solid-like state takes nonzero 
values and equals to zero, when lubricant is liquid-
like [4,9]. 
The elastic stresses arising in the lubricant are de-
fined as σel = ∂f/∂ɛ el [4]: 
 
 2el el ela , (4.2) 
 
where μ is the shear modulus which takes zero values 
in liquid-like state. Nonzero values of the shear modu-
lus which correspond to solid-like state, can be found 
from condition ∂f/∂ϕ = 0 [4]:  
 
 22 ( )c el
a
T T a
b
, (4.3) 
 
From the expression (4.3) let us found the elastic 
strain critical value [4]  
 
 
2 ( )c
el c
T T
a
. (4.4) 
 
The lubricant melts when it exceeds this value. 
The elastic stresses (4.4) stationary value depends 
on the film thickness h, the internal stresses relaxation 
time τε and the surfaces relative velocity (V−v) [10,11]:  
 
 0
( )
el
V v
h
. (4.5) 
 
From the Eq. (4.4) using expressions (4.5) we can 
evaluate the critical velocity Vc0: 
 
 0
2 ( )c
c
h T T
V
a
. (4.6) 
 
The friction force F can be written as combination of 
(3.2) – (3.5), (4.2) and (4.5): 
 
1
10 9 3 8sgn( )10
n
n
el
V v
F V v A
h
. (4.7) 
 
5. STICK-SLIP FRICTION 
 
The typical time dependences of the system param-
eters are shown in Fig. 2. Here the dashed curves cor-
respond to the upper block coordinates X and velocities 
V and the solid curves to the bottom block coordinates x 
and velocities v. 
 
 
 
Fig. 2 – The dependences of coordinates X and x, velocities V 
and v, elastic stresses σel and friction force F on the time t at 
the parameters α = 0.95 J K−1/m3, Tc = 290 K, a =4∙1012 Pa, 
b = 230 J/m3, Tk = 353 K, β = 4.09, A = 0.6∙10−9 m2, τɛ  = 10−8 s, 
h = 10−9 m. M = 0.4 kg, δ = 100 J−1∙m3/s, K = 3000 N/m, 
T = 235 K, Xm = 0.9∙10−6 m, ω = 5 rad/s. 
 
 THE VISCOSITY TEMPERATURE DEPENDENCE IN THE BOUNDARY… PROC. NAP 3, 01NTF18 (2014) 
 
 
01NTF18-3 
At the initial time moment, the lubricant is solid-like. 
At time t > 0 the upper block starts motion in accordance 
with Eq. (2.1). The bottom blocks moves together with the 
upper one, because of a friction force F, arising between 
the surfaces. However, the bottom block motion is pre-
vented by stretching out and compressing of the springs. 
Therefore, in general case the velocities V and v do not 
coincide. Consequently, the relative shear velocity |V−v| 
increases. After exceeding critical value |V−v| > Vc0 (4.6), 
the lubricant melts. The friction force |F| abruptly de-
creases (Fig. 2, c), and the block slips on a significant 
distance (Fig. 2, a). The bottom block velocity sharply 
grows and as a result the value |V−v| increases. At this 
moment in Fig.2, b an increase in the friction force abso-
lute value can be observed due to the stress viscous com-
ponent increase according to Eq. (4.2), (4.5) and (4.7). 
Whereupon |F| decreases since the relative velocity re-
ducesеs and F reverses sign. Here the block slips so much, 
so stretched spring compresses and vice versa. After ful-
fillment of the condition |V−v| < Vc0 the lubricant solidi-
fies. 
 
 
 
Fig. 3 – The friction force F dependences on the time t for the 
same parameters as in Fig. 2 and Xm1 = 0.67∙10−6, 
Xm2 = 0.9∙10−6, Xm3 = 1.31∙10−6, Xm4 = 5∙10−6 m. The panel a 
corresponds to the lubricant temperature T1 = 235 K, and the 
panel b to the T2 = 280 K one. 
 
Lets consider the shear amplitude influence on the 
system behavior. Fig. 3, a demonstrates the friction 
force time dependence for 4 different shear amplitudes 
of the upper block. It is shown that the increase in 
amplitude Xm leads to the rise of phase transition 
amount over period (Fig. 3, a, amplitudes Xm1, Xm2, 
Xm3). It is explained by the fact, that the block velocity 
Xmωsin(ωt) increases. Therefore the phase transition 
frequency grows [3,5,8,12]. As the amplitude is elevat-
ed to Xm = 5 μm the lubricant solidifies twice over a 
period. In this situation the lubricant solidifies when 
the block changes its direction. It is also worth noting 
that amplitudes were chosen so that their subsequent 
changing leads to stationary state at minimal time. For 
instance, for intermediate value Xm = 0.75 μm, which is 
between values Xm1 and Xm2, the stick-slip motion is 
observed. In this case, the blocks twice “stick” and 
twice “slip” over a period. However, significant time is 
required to establish the stationary state. 
Fig. 3, b was built for the higher lubricant tempera-
ture T2 = 280 K. The increase in temperature causes 
the decrease in the phase transition frequency. At the 
same time, the maximum friction force value reduces 
[3,5,7,8,11-13]. The figure first section determines the 
stick-slip motion, where lubricant melts twice over a 
period. This case is similar to that one described above 
for Xm4. For the subsequent amplitude value Xm2, the 
lubricant does not solidify after melting. Therefore, the 
further amplitude growing does not influence the fric-
tion mode and slide friction is established in the system 
[13, 14]. 
 
6. CONCLUSION 
 
In the work the thermodynamic model of ultrathin 
lubricant melting has been further developed. The novelty 
of the investigation consists in consideration of the de-
pendence of the polymeric lubricant viscosity on the tem-
perature and velocity gradient within the second-order 
phase transition model. The dependence evidences, that 
the effective viscosity logarithm is proportional to the 
logarithm of the shear rate. The proportionality factor 
value varies from 1 in the solid-like state to 0 in the liq-
uid-like state. It has been shown that stick-slip motion is 
a result of second-order phase transition between liquid-
like and solid-like lubricant structures. 
The time dependencies of coordinate, velocity, fric-
tion force and elastic stress have been built. The com-
puter simulation results have been presented, which 
predicts stick-slip disappearance and further system 
functioning in the sliding friction mode. 
 
AKNOWLEDGEMENTS 
 
The work was executed at support of the Ministry of 
Education and Science of Ukraine (MESU) within 
framework of the project “Modeling of friction of metal 
nonparties and boundary liquid films which interact 
with atomically flat surfaces” (No. 0112U001380). 
 
REFERENCES 
 
1. J. Israelachvili, Surf. Sci. Rep. 14, 109 (1992). 
2. B.N.J. Persson Sliding friction. Physical principles and 
applications (New York: Springer-Verlag: 2000) 
3. H. Yoshizawa, J. Israelachvili, J. Phys. Chem. 97, 11300 
(1993). 
4. V.L. Popov, Tech. Phys. 46, 605 (2001). 
5. A.D. Berman, W.A. Ducker, I.N. Israelachvili, Langmuir 
12, 4559 (1996). 
6. I.M. Sivebaek, V.N. Samoilov, B.N.J. Persson, Phys. Rev. 
Lett. 108, 036102 (2012). 
7. C.-R. Yang, R.-T. Lee, Y.-C. Chiou, Tr. Int. 30, 719 (1997). 
8. C.-R. Yang, Y.-C. Chiou, R.-T. Lee, Tr. Int. 32, 443 (1999). 
9. V.L. Popov, Sol. State Commun. 115, 369 (2000). 
10. I.A. Lyashenko, Tech. Phys. 56, 701 (2011). 
11. I.A. Lyashenko, A.M. Zaskoka, Ukr. J. Phys. 58, 91 (2013). 
12. P.D. Baets, J. Degrieck, F.V.D. Velde, A.P.V. Peteghem, 
Wear 243, 48 (2000). 
13. I.A. Lyashenko, A.M. Zaskoka, J. Phys. Stud. 17, 1002 (2013). 
14. I.A. Lyashenko, A.V. Khomenko, A.M. Zaskoka, Trib. 
Trans. 56, 1019 (2013). 


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The Viscosity Temperature Dependence in the Boundary Friction Model under a Periodic External Action

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