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Influence of external load on the frictional characteristics of rotary model using a molecular dynamics approach
Computational Materials Science 122 (2016) 201–209
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Influence of external load on the frictional characteristics of rotary model using a molecular dynamics approach Weijie Shi, Xiaohui Luo ⇑, Zuti Zhang, Yinshui Liu, Wenlong Lu School of Mechanical Science and Engineering, Huazhong University of Science and Technology, No. 1037 Luoyu Road, Wuhan, Hubei 430074, China
a r t i c l e
i n f o
Article history: Received 15 January 2016 Received in revised form 22 May 2016 Accepted 24 May 2016 Available online 1 June 2016 Keywords: Wear modeling Nanotribology Frictional force Coefficient of friction Molecular dynamics
a b s t r a c t Based on the swash plate–slipper in water hydraulic axial piston pump, a rotary model is built using the molecular dynamics approach to study the frictional characteristics in the perspective of atoms. The model consists of a diamond rotator and a copper substrate and the diamond rotator rotates around the cooper substrate at different external loads. The frictional force is calculated by summing all the atomic forces of diamond rotator along the opposite direction of linear velocity and the real contact area is defined by the number of atoms that interact chemically across the contact interface. Influences of external load on the frictional force and coefficient of friction are analyzed. The simulation results show that the total atomic force varies with rotary cycle according to the sine law with a periodic cycle of 2p in the rotary model. However, the frictional force is basically not a periodic signal. It fluctuates around the average frictional force which is linearly dependent on the real contact area. Besides, it is demonstrated that the coefficient of friction decreases with the increase of external load, which is due to the nonlinear increase of real contact area. As a result, the present work will have an important impact on the fundamental understanding of the wear mechanism of rotary model. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Water hydraulics which uses water directly as the working media has many advantages, such as environmental friendliness, cleanliness, low operation cost and easy disposal [1–3]. As a crucial component in the hydraulic system, the axial piston pump has been used for many years to transmit power hydraulically. Swash plate–slipper is one of the most important friction pairs in the hydraulic axial piston pump and it plays a key role in the pump’s performance [4–6]. However, the low viscosity and poor lubrication of water make it difficult to form the water lubrication film, which can produce boundary friction or even unlubrication friction [7,8]. Therefore, it is essential to study the frictional characteristics of the swash plate–slipper friction pair. Nowadays, nanomaterials have become one of the hottest topics in the scientific research field and they are regarded as another industrial revolution in the 21th century. They have abroad scope in water hydraulic because of their excellent frictional characteristics [9–12]. Compared with traditional experimental methods, molecular dynamics (MD) simulation has become a useful tool to study the frictional characteristics of
⇑ Corresponding author. E-mail address: [email protected] (X. Luo). http://dx.doi.org/10.1016/j.commatsci.2016.05.031 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.
nanomaterials. Besides, many studies have showed that working conditions which affect atomic friction, such as normal load, temperature and velocity, can be studied well by using MD simulation [13–28]. For example, Zhang and Tanaka [13] investigated a diamond-copper sliding system using MD simulation, and revealed that key sliding parameters, such as indentation depth and sliding speed have huge influence on the deformation. Yan [18] conducted the MD simulation to understand AFM-based nano-scratching process deeply. They carried out two adjacent scratching tests at different feeds and found that the feed has an important influence on the deformation of the scratching forces. Zhang and Tang [19] carried out a MD analysis to understand how slider surface roughness influences the friction and wear behavior of a nano-system. They showed that water molecules could alter the friction characteristics by reducing the contact area and surface-surface interactions were important to friction. Mo et al. [26] performed the MD simulations to study the friction in nanoscale single asperity contacts. The results showed that friction is controlled by the mean number of atoms that interact chemically across the contact interface. However, similar studies focus on the MD simulation of the sliding but not the rotary system. The rotary velocity in the MD simulation is larger than the typical velocity in experiments, but some studies suggest that the low speed in the MD simulation can be scaled up. For example, Nair et al. [29] adopted the velocity
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10 m/s to conduct a MD simulation, while the velocity in the indentation experiment was relatively slower (106–109 m/s). Xu et al. [24] performed the MD simulations of nano-indentation and the indentation velocity in MD simulations was larger than the typical velocity in experiments. They found that loading speed has no effect on the simulations results. Therefore, the method of scaling the rotary velocity in the rotary model is adopted to conduct the MD simulation. Also, calculation of frictional force is another problem in the rotary model with MD simulation. At present, most researchers often use the method that summing all the atomic forces along the sliding direction to calculate the frictional force, and they also average the atomic forces with some time steps as the average frictional force [16,19–25]. For the rotary sliding friction in the macroscale, the frictional force is distributed along the opposite direction of linear velocity and its direction varies all the time. Based on the previous studies, the frictional force in the rotary model is calculated by summing all the atomic forces along the opposite direction of linear velocity. In this paper, MD simulation was used to study the frictional characteristics of rotary model which was conducted based on the swash plate–slipper friction pair of the axial piston pump. The simulation model was realized by reducing the scale size of swash plate–slipper and increasing the rotary speed of the slipper. Additionally, influence of external load FL on the frictional characteristics was investigated to provide theoretical basis for the wide application of MD in the water hydraulics field.
2. Computational scheme 2.1. Geometry modeling The rotary model is built based on the swash plate–slipper friction pair. As shown in Fig. 1, it consists of a rotator and a substrate, which represent the slipper and the swash plate, respectively. The rotator is a combination of a partial sphere and a cylinder. Upper pffiffiffi part of the rotator is a cylinder, with diameter of D1 ¼ 3 nm and height of H1 = 1.2 nm. Bottom part of the rotator is a partial pffiffiffi sphere, with radius of R ¼ 3 nm. The center of the partial sphere is located at H2 = 0.5 nm above the bottom of the cylinder. The substrate is a cylinder and its diameter and height are equal to
D2 = 20 nm and H4 = 5 nm, respectively. The substrate is below H3 = 1 nm the rotator and the distance of central axes between the rotator and substrate is L = 5 nm. 2.2. Molecular dynamics The MD simulation with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [30] is adopted to perform the MD simulation and AtomEye [31] is used for the visualization of atomic data. Based on the geometric model in Fig. 1, the model in MD simulation is built, which is shown in Fig. 2. The model consists of a diamond rotator and a copper substrate. The copper substrate contains 134,750 copper atoms, including 14,450 boundary atoms, 28,875 thermostat atoms and 91,425 Newtonian atoms. The atoms distributed along the z axis from bottom to top are boundary atoms, thermostat atoms and Newtonian atoms, respectively. In the simulation, the boundary atoms fix the substrate and the thermostat atoms achieve a stable system by controlling the temperature. Besides, the Newtonian atoms and thermostat atoms obey the Newton’s second law, which can simulate the process of friction. Periodic boundary conditions are adopted in the x and y directions. The diamond rotator contains 2574 carbon atoms and it rotates counter clockwise around the central axis of copper substrate. There are three different atomic interactions in the simulation, including the C–C atoms interaction in the diamond rotator, the Cu–Cu atoms interaction in the copper substrate and the C–Cu atoms interaction between the diamond rotator and copper substrate. In the simulation, an EAM potential written by Foiles et al. [32,33], is adopted to illustrate the Cu–Cu atoms interaction. As the diamond rotator is treated as the rigid in the simulation, C–C interaction can be ignored. The Morse potential is widely adopted for C–Cu interaction in MD simulation, which is calculated by
E ¼ D0 exp2aðrr0 Þ 2expaðrr0 Þ
ð1Þ
where D0, a, r and r0 is the cohesive energy, the elastic modulus, dynamic distances and equilibrium distance between C atoms and Cu atoms, respectively. The Morse potential parameters are adopted as D0 = 0.087 eV, a = 5.14, r0 = 2.05 Å, from Ref. [34]. To acquire a steady conformation, an energy minimization with a steepest descent algorithm is used. The MD system is relaxed by
Fig. 1. Schematic description of rotary model.
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the x and y directions, respectively. As illustrated in Fig. 3, the total atomic forces in the x and y directions are divided along the opposite direction of linear velocity. Meanwhile, the component frictional forces (fx, fy) are generated. The summation of component frictional forces is the frictional force. Therefore, the frictional force can be illustrated by Eq. (2), which can explain how the total atomic forces in the x and y directions make contributions to the frictional force.
8 > < f x ¼ F x cos h f y ¼ F y sin h > : f ¼ fx þ fy
Fig. 2. Rotary model using the MD simulation.
constant number of particles, volume, and energy (NVE) and used a Langevin thermostat to keep the temperature of thermostat atoms at 298 K. Pressure is an important parameter in the field of water hydraulics, which is treated as the external load in the rotary model. In order to investigate the influence of external load on the frictional characteristics, different external loads (2.5, 12.5 and 37.5 nN) are imposed on the diamond rotator in the negative direction of z axis at different rotary cycles (15, 20 and 30 ps). After full relaxation, the external load is kept constant and the diamond rotator is made a circuit of the central axis of copper substrate. 2.3. Computational method The atomic forces of diamond rotator, including attractive force and repulsive force caused by the MORSE potential energy between the atoms of diamond rotator and every atom of copper substrate, can be acquired by using the LAMMPS software. A force analysis is carried out for further calculation of frictional force, which is shown in Fig. 3. Fx and Fy is the total atomic forces acquired by summing atomic forces of all the carbon atoms in
Fig. 3. Schematic diagram of frictional force analysis in the rotary model.
ð2Þ
① is the initial position of diamond rotator. h is the rotary angle where the diamond rotator revolves around the central axis of copper substrate. In the rotary process of diamond rotator, rotary angle has close relationship with the time steps, step number and rotary cycle in the simulation and it is estimated by
h ¼ ðDt n=TÞ 2p
ð3Þ
where Dt, n, T is the time steps, step number and rotary cycle, respectively. Hence, the frictional force in the rotary model can be calculated by
f ¼ F x cosð2p Dt n=TÞ þ F y sinð2p Dt n=TÞ
ð4Þ
Defining contact area is another question to understand the friction in nanoscale contacts. A real contact area defined by Mo and Szlufarska [26–28] is Areal = NatAat, where Nat is the mean number of atoms in contact. An atom from the diamond rotator is defined as the atom in contact when this atom is within the range of chemical interactions of any atom of the copper substrate. The value of Aat is calculated by dividing the total surface area of diamond rotator by the total number of atoms in contact. Additionally, an asperity contact area Aasp is also defined using the concept of the edge of the contact zone. The schematic diagram of the asperity contact area is shown in Fig. 4. The asperity contact area is the maximum cross sectional area where the rotator meets the substrate. The furrow depths are calculated by calculating the distance between the atoms of the upper of cooper substrate and the atoms of the bottom of furrow. When the depth of furrow is higher than the height of sphere, the asperity contact area is acquired by calculating the cross sectional area of the cylinder. Therefore, the asperity contact area can be calculated with Eq. (5).
Fig. 4. Schematic diagram of nominal contact area.
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(
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S ¼ pr 2 ¼ pðR2 ðR hÞ Þ h 6 hmax D21 =4
S¼p
h > hmax
ð5Þ
where R is the radius of the partial sphere, r is the radius of junction of diamond rotator and copper substrate, D1 is the diameter of cylinder part, h is the depth of furrow and hmax is the height of semi-sphere. According to the rotary model in Fig. 1, the value pffiffiffi of R, D and hmax is equal to 1 nm, 3 nm and 0.5 nm, respectively. It can be deduced that the asperity contact area will increase when the depth of furrow is increased. 3. Results and discussions 3.1. Analysis of the frictional force
negative maximum value at position ①. Similarly, at position ②, ③ and ④, the total atomic forces in the x direction are positive maximum value, 0, negative maximum value, respectively, and the total atomic forces in the y direction are 0, positive maximum value, 0, respectively. It can be seen that the periodic cycle of total atomic force is 2p. Therefore, the total atomic forces are sine curve and cosine curve with periodic cycle of Tcycle = 2p. The above analysis is carried out in the cases when the rotary model is a symmetric system and no wear occurs. Actually, the model experiences wear, so the debris and plough are produced, which makes the symmetric system become asymmetric. The phase angle and amplitude of total atomic forces deviate from zero point slightly. As a result, the total atomic forces in the x and y directions can be illustrated by Eq. (6).
F x ¼ a1 þ b1 sinðh þ u1 Þ
ð6Þ
As Eq. (2) illustrates, the frictional force contains component frictional forces in the x and y directions. Meanwhile, the component frictional forces are made up of total atomic forces in the x and y directions. It can be deduced that the total atomic forces and component frictional forces have enormous influence on the frictional force. Therefore, it is necessary to study how the total atomic forces and component frictional forces vary with the rotary angle. Fig. 5 shows the total atomic forces in the x and y directions with external load of 12.5 nN and rotary cycle of 30 ps. ①, ②, ③ and ④ corresponds to the positions in Fig. 3. The total atomic force fluctuates with the rotary angle. Besides, there is a sine relationship between the total atomic force and the rotary angle. The periodic cycle of total atomic force is 2p, which is related to the structure of rotary model. Assuming that the diamond rotator rotates on the condition when no wear happens, it will do uniform circular motion. According to the feature of uniform circular motion, the total atomic forces in the normal direction can be considered as a constant, which is called the centripetal force Fn in the paper. Based on the force analysis in Fig. 3, the centripetal force is equal to Fn = Fxsin h Fycos h. It can be deduced that the waveforms of total atomic forces are sine and cosine curves, which can make the centripetal become a constant. Moreover, the rotary model is a model of symmetrical distribution. For example, when the rotator is at position ①, as shown in Fig. 3, the total atomic force in the x direction can be cancelled out since the number of atoms is equal on both sides of y axis. Also, the atomic force equals to the centripetal force in the y direction and it points to the negative direction of x axis, so the atomic force in the y direction is at its
where c1 + c2 is the positive value of the curve offsets zero. Fig. 7 shows that the frictional force varies with the rotary angle. The frictional force fluctuates around a positive value which is called the average frictional force. It is considered that the frictional force fluctuation is very common in micro scale [35]. The fluctuations are due
Fig. 5. Total atomic forces of diamond rotator in the x and y directions.
Fig. 6. Relationship between component frictional force and rotary angle.
F y ¼ a2 b2 cosðh þ u2 Þ
Combining Eq. (2) with Eq. (6), the component frictional forces can be acquired with Eq. (7).
(
f x ¼ a1 cosðhÞ þ b1 sinðh þ u1 Þ cosðhÞ f y ¼ a2 sinðhÞ b2 cosðh þ u2 Þ sinðhÞ
ð7Þ
It can be deduced from the wave in Fig. 5 that b1/a1 1, b2/ a2 1. Thus, the first items of Eq. (7) (a1cos(h) and a2sin(h)) have little influence on the periodic cycle of the component frictional forces, while the second items of Eq. (7) can make the cycle reduce by half. As shown in Fig. 6, the periodic cycle of component frictional forces is half of the periodic cycle of total atomic forces. The curve fitting for the component frictional forces in Fig. 6 is made by using sine formula and Eq. (8) is acquired.
(
f x ¼ c1 þ d1 sinð2h þ d1 Þ f y ¼ c2 d2 sinð2h þ d2 Þ
ð8Þ
where c1 c2, c1 > 0, c2 > 0, d1 d2, d1 d2. Synthesizing Eqs. (2) and (8), Eq. (9) is gotten.
f ¼ c 1 þ c2
ð9Þ
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Fig. 7. Relationship between frictional force and rotary angle.
to the stick–slip behavior between the diamond rotator and the cooper substrate. In conclusion, the periodic cycle of total atomic forces is 2p, while the periodic cycle of component frictional forces is p. It is coincident with the force analysis in the macro. The formula of calculating the frictional force is proved to be correct from the force analysis and mathematical deduction. 3.2. Influence of external load on the frictional force Fig. 8 shows the frictional forces at different external loads (2.5, 12.5 and 37.5 nN) with different rotary cycles (15, 20 and 30 ps), where a force promotes the motivation is defined as the negative frictional force. Eq. (4) is applied to calculate the frictional forces. Meanwhile, the frictional forces are averaged over blocks of consecutive time steps, with each lasting for 500 time steps. It can clearly be seen that they fluctuate around the average frictional forces, no matter what the external loads are. The changes in these curves are not significantly different for different rotary cycles. In other words, the rotary cycles adopted in the simulation have little effect on the frictional forces. The average frictional forces fave are calculated by averaging the frictional forces within one rotary cycle. With increasing external load, the average frictional force becomes larger and it is almost proportional to the external load. Moreover, oscillation amplitudes of the frictional forces increase with the increase of external loads. Furthermore, the frictional forces sometimes become negative at low external loads (2.5 and 12.5 nN), which indicates that the frictional forces promote the motivation of diamond rotator. However, it does not appear with the increasing external load, such as at external load of 37.5 nN. The frictional forces in a macroscopic friction are acquired by averaging the atomic forces on the basis of huge number of atoms involved, while only a limited number of atoms play an important role in the nanometer scale, so individual atomic interactions can lead to the instant variation of the frictional force [19]. When the external load is at 2.5 nN or 12.5 nN, the number of atoms behind the diamond rotator can be more than that in front of diamond rotator at certain instant of the rotary motion. As a result, the attraction from the atoms behind the diamond rotator may be larger than the resistance from the atoms in front of the diamond rotator. The direction of the resultant force is changed and the frictional force becomes negative. Contrary to the condition at low external load, a large number of copper atoms pile up in front of
diamond rotator and the resistance plays a leading role almost all the time at high external load of 37.5 nN. Thus, the frictional forces are basically all positive at external load of 37.5 nN. The MD simulation is performed to determine the dependence of average frictional force on the contact area and external load. Based on the data acquired from the MD simulation, fitting curves are given by using power function. Fig. 9(a) illustrates the relationship between average frictional force and external load at different rotary cycles. No matter how long the rotary cycle is, it is predicted 2=3
that the average frictional force f av e / F L . Similarly, the simula2=3 tion results also show that the real contact area Areal / F L , which is shown in Fig. 9(b). Fig. 9(c) shows the asperity contact area using the concept of the edge of the contact zone. Compared with Fig. 9 (b), it can be seen that the real contact area is slightly higher than the asperity contact area at external load below 12.5 nN. When the external load is increased, the asperity contact area does not increase any more, but the real contact area increases slowly. Combined with the depth of furrow in Fig. 9(d), it can be deduced that when the external load is above 12.5 nN, the partial sphere of rotator has completely entered into the substrate and part of the cylinder of the diamond rotator begins to take part in friction. As a consequence, the asperity contact area cannot illustrate the friction law in the rotary model and the average frictional force should be defined in terms of the real contact area. The dependence of the average frictional force on the external load is shown in Fig. 10. The simulation data reveals that the average frictional force is proportional to the real contact area.
3.3. Influence of external load on the coefficient of friction The coefficient of friction is the ratio of the average frictional force to the normal force which is the summation of the atomic forces in the z direction. Fig. 11 shows the coefficient of friction varying with the external loads at different rotary cycles. By comparing Fig. 9(b) with Fig. 11, it can be seen that the coefficient of friction has close relationship with the real contact area. When the external load is raised, the real contact increases nonlinearly and the coefficient of friction shows a downward trend. But the decreasing rate is different at different external load. For example, when the external load is lower than 12.5 nN, the coefficient of friction declines rapidly, which is the period when the partial sphere part of diamond rotator is pressing into the copper
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Fig. 8. Frictional force varying with rotary angle at different rotary cycles: (a) 15 ps, (b) 20 ps, and (c) 30 ps. The black, pink and blue lines correspond to the external loads of 2.5, 12.5 and 37.5 nN, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
substrate. A good description of the frictional morphology is illustrated in the Fig. 12(a). At this moment, the real contact area increases rapidly, which makes the frictional coefficient fall abruptly. However, when the external load is larger than 12.5 nN, the partial sphere part has entirely pressed into the substrate. As Fig. 12(b) and (c) shows, the cylinder part of the diamond rotator
begins to take part in friction and the real contact area increases steadily. Meanwhile, the coefficient of friction tends towards stability. So it can be concluded that the increasing rate of real contact area determines the decreasing rate of the coefficient of friction. The results indicate that the external load is significantly important to the coefficient of friction by changing the real contact area.
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Fig. 9. Influence of external load on the frictional characteristics of rotary model. (a) Average frictional force, (b) real contact area, (c) asperity contact area, and (d) depth of furrow.
Fig. 10. Dependence of average frictional force on the real contact area at different rotary cycles.
Fig. 11. Coefficient of friction varying with the external loads at different rotary cycles. The black, pink and blue lines correspond to the rotary cycles of 15, 20 and 30 ps, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4. Conclusion In conclusion, MD simulation was used to investigate the influence of external load on the frictional characteristics of swash
plate–slipper friction pair in the perspective of atom. The formula of frictional force was proposed to analyze the frictional characteristics of rotary model. The conclusions were drawn as follows:
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Fig. 12. Snapshots of friction morphology at rotary cycle of 20 ps with different external loads (a) 2.5 nN, (b) 12.5 nN, and (c) 37.5 nN.
(1) In the rotary model, the total atomic forces in the x and y directions vary with rotary cycles according to the sine law with a periodic cycle of one circle (2p). However, the periodic cycle of the component frictional force decreases one half and the frictional force is basically not a periodic signal. (2) The frictional force fluctuates with rotary angle and it even promotes the motivation of diamond rotator when the external load is low. The average frictional force depends linearly on the real contact area and it shows approximately the same 2/3 power-law dependence on the external load. Also, the real contact area scales with external load as F 2=3 L . (3) The external load has enormous influence on the coefficient of friction by changing the real contact area. When the external load is raised, the coefficient of friction declines abruptly at low external load, which is due to the rapid increase of real contact area. But when the external load is above 12.5 nN, the real contact area increase steadily and the coefficient of friction tends towards stability.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Influence of external load on the frictional characteristics of rotary model using a molecular dynamics approach Weijie Shi, Xiaohui Luo ⇑, Zuti Zhang, Yinshui Liu, Wenlong Lu School of Mechanical Science and Engineering, Huazhong University of Science and Technology, No. 1037 Luoyu Road, Wuhan, Hubei 430074, China
a r t i c l e
i n f o
Article history: Received 15 January 2016 Received in revised form 22 May 2016 Accepted 24 May 2016 Available online 1 June 2016 Keywords: Wear modeling Nanotribology Frictional force Coefficient of friction Molecular dynamics
a b s t r a c t Based on the swash plate–slipper in water hydraulic axial piston pump, a rotary model is built using the molecular dynamics approach to study the frictional characteristics in the perspective of atoms. The model consists of a diamond rotator and a copper substrate and the diamond rotator rotates around the cooper substrate at different external loads. The frictional force is calculated by summing all the atomic forces of diamond rotator along the opposite direction of linear velocity and the real contact area is defined by the number of atoms that interact chemically across the contact interface. Influences of external load on the frictional force and coefficient of friction are analyzed. The simulation results show that the total atomic force varies with rotary cycle according to the sine law with a periodic cycle of 2p in the rotary model. However, the frictional force is basically not a periodic signal. It fluctuates around the average frictional force which is linearly dependent on the real contact area. Besides, it is demonstrated that the coefficient of friction decreases with the increase of external load, which is due to the nonlinear increase of real contact area. As a result, the present work will have an important impact on the fundamental understanding of the wear mechanism of rotary model. Ó 2016 Elsevier B.V. All rights reserved.
1. Introduction Water hydraulics which uses water directly as the working media has many advantages, such as environmental friendliness, cleanliness, low operation cost and easy disposal [1–3]. As a crucial component in the hydraulic system, the axial piston pump has been used for many years to transmit power hydraulically. Swash plate–slipper is one of the most important friction pairs in the hydraulic axial piston pump and it plays a key role in the pump’s performance [4–6]. However, the low viscosity and poor lubrication of water make it difficult to form the water lubrication film, which can produce boundary friction or even unlubrication friction [7,8]. Therefore, it is essential to study the frictional characteristics of the swash plate–slipper friction pair. Nowadays, nanomaterials have become one of the hottest topics in the scientific research field and they are regarded as another industrial revolution in the 21th century. They have abroad scope in water hydraulic because of their excellent frictional characteristics [9–12]. Compared with traditional experimental methods, molecular dynamics (MD) simulation has become a useful tool to study the frictional characteristics of
⇑ Corresponding author. E-mail address: [email protected] (X. Luo). http://dx.doi.org/10.1016/j.commatsci.2016.05.031 0927-0256/Ó 2016 Elsevier B.V. All rights reserved.
nanomaterials. Besides, many studies have showed that working conditions which affect atomic friction, such as normal load, temperature and velocity, can be studied well by using MD simulation [13–28]. For example, Zhang and Tanaka [13] investigated a diamond-copper sliding system using MD simulation, and revealed that key sliding parameters, such as indentation depth and sliding speed have huge influence on the deformation. Yan [18] conducted the MD simulation to understand AFM-based nano-scratching process deeply. They carried out two adjacent scratching tests at different feeds and found that the feed has an important influence on the deformation of the scratching forces. Zhang and Tang [19] carried out a MD analysis to understand how slider surface roughness influences the friction and wear behavior of a nano-system. They showed that water molecules could alter the friction characteristics by reducing the contact area and surface-surface interactions were important to friction. Mo et al. [26] performed the MD simulations to study the friction in nanoscale single asperity contacts. The results showed that friction is controlled by the mean number of atoms that interact chemically across the contact interface. However, similar studies focus on the MD simulation of the sliding but not the rotary system. The rotary velocity in the MD simulation is larger than the typical velocity in experiments, but some studies suggest that the low speed in the MD simulation can be scaled up. For example, Nair et al. [29] adopted the velocity
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10 m/s to conduct a MD simulation, while the velocity in the indentation experiment was relatively slower (106–109 m/s). Xu et al. [24] performed the MD simulations of nano-indentation and the indentation velocity in MD simulations was larger than the typical velocity in experiments. They found that loading speed has no effect on the simulations results. Therefore, the method of scaling the rotary velocity in the rotary model is adopted to conduct the MD simulation. Also, calculation of frictional force is another problem in the rotary model with MD simulation. At present, most researchers often use the method that summing all the atomic forces along the sliding direction to calculate the frictional force, and they also average the atomic forces with some time steps as the average frictional force [16,19–25]. For the rotary sliding friction in the macroscale, the frictional force is distributed along the opposite direction of linear velocity and its direction varies all the time. Based on the previous studies, the frictional force in the rotary model is calculated by summing all the atomic forces along the opposite direction of linear velocity. In this paper, MD simulation was used to study the frictional characteristics of rotary model which was conducted based on the swash plate–slipper friction pair of the axial piston pump. The simulation model was realized by reducing the scale size of swash plate–slipper and increasing the rotary speed of the slipper. Additionally, influence of external load FL on the frictional characteristics was investigated to provide theoretical basis for the wide application of MD in the water hydraulics field.
2. Computational scheme 2.1. Geometry modeling The rotary model is built based on the swash plate–slipper friction pair. As shown in Fig. 1, it consists of a rotator and a substrate, which represent the slipper and the swash plate, respectively. The rotator is a combination of a partial sphere and a cylinder. Upper pffiffiffi part of the rotator is a cylinder, with diameter of D1 ¼ 3 nm and height of H1 = 1.2 nm. Bottom part of the rotator is a partial pffiffiffi sphere, with radius of R ¼ 3 nm. The center of the partial sphere is located at H2 = 0.5 nm above the bottom of the cylinder. The substrate is a cylinder and its diameter and height are equal to
D2 = 20 nm and H4 = 5 nm, respectively. The substrate is below H3 = 1 nm the rotator and the distance of central axes between the rotator and substrate is L = 5 nm. 2.2. Molecular dynamics The MD simulation with the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) [30] is adopted to perform the MD simulation and AtomEye [31] is used for the visualization of atomic data. Based on the geometric model in Fig. 1, the model in MD simulation is built, which is shown in Fig. 2. The model consists of a diamond rotator and a copper substrate. The copper substrate contains 134,750 copper atoms, including 14,450 boundary atoms, 28,875 thermostat atoms and 91,425 Newtonian atoms. The atoms distributed along the z axis from bottom to top are boundary atoms, thermostat atoms and Newtonian atoms, respectively. In the simulation, the boundary atoms fix the substrate and the thermostat atoms achieve a stable system by controlling the temperature. Besides, the Newtonian atoms and thermostat atoms obey the Newton’s second law, which can simulate the process of friction. Periodic boundary conditions are adopted in the x and y directions. The diamond rotator contains 2574 carbon atoms and it rotates counter clockwise around the central axis of copper substrate. There are three different atomic interactions in the simulation, including the C–C atoms interaction in the diamond rotator, the Cu–Cu atoms interaction in the copper substrate and the C–Cu atoms interaction between the diamond rotator and copper substrate. In the simulation, an EAM potential written by Foiles et al. [32,33], is adopted to illustrate the Cu–Cu atoms interaction. As the diamond rotator is treated as the rigid in the simulation, C–C interaction can be ignored. The Morse potential is widely adopted for C–Cu interaction in MD simulation, which is calculated by
E ¼ D0 exp2aðrr0 Þ 2expaðrr0 Þ
ð1Þ
where D0, a, r and r0 is the cohesive energy, the elastic modulus, dynamic distances and equilibrium distance between C atoms and Cu atoms, respectively. The Morse potential parameters are adopted as D0 = 0.087 eV, a = 5.14, r0 = 2.05 Å, from Ref. [34]. To acquire a steady conformation, an energy minimization with a steepest descent algorithm is used. The MD system is relaxed by
Fig. 1. Schematic description of rotary model.
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the x and y directions, respectively. As illustrated in Fig. 3, the total atomic forces in the x and y directions are divided along the opposite direction of linear velocity. Meanwhile, the component frictional forces (fx, fy) are generated. The summation of component frictional forces is the frictional force. Therefore, the frictional force can be illustrated by Eq. (2), which can explain how the total atomic forces in the x and y directions make contributions to the frictional force.
8 > < f x ¼ F x cos h f y ¼ F y sin h > : f ¼ fx þ fy
Fig. 2. Rotary model using the MD simulation.
constant number of particles, volume, and energy (NVE) and used a Langevin thermostat to keep the temperature of thermostat atoms at 298 K. Pressure is an important parameter in the field of water hydraulics, which is treated as the external load in the rotary model. In order to investigate the influence of external load on the frictional characteristics, different external loads (2.5, 12.5 and 37.5 nN) are imposed on the diamond rotator in the negative direction of z axis at different rotary cycles (15, 20 and 30 ps). After full relaxation, the external load is kept constant and the diamond rotator is made a circuit of the central axis of copper substrate. 2.3. Computational method The atomic forces of diamond rotator, including attractive force and repulsive force caused by the MORSE potential energy between the atoms of diamond rotator and every atom of copper substrate, can be acquired by using the LAMMPS software. A force analysis is carried out for further calculation of frictional force, which is shown in Fig. 3. Fx and Fy is the total atomic forces acquired by summing atomic forces of all the carbon atoms in
Fig. 3. Schematic diagram of frictional force analysis in the rotary model.
ð2Þ
① is the initial position of diamond rotator. h is the rotary angle where the diamond rotator revolves around the central axis of copper substrate. In the rotary process of diamond rotator, rotary angle has close relationship with the time steps, step number and rotary cycle in the simulation and it is estimated by
h ¼ ðDt n=TÞ 2p
ð3Þ
where Dt, n, T is the time steps, step number and rotary cycle, respectively. Hence, the frictional force in the rotary model can be calculated by
f ¼ F x cosð2p Dt n=TÞ þ F y sinð2p Dt n=TÞ
ð4Þ
Defining contact area is another question to understand the friction in nanoscale contacts. A real contact area defined by Mo and Szlufarska [26–28] is Areal = NatAat, where Nat is the mean number of atoms in contact. An atom from the diamond rotator is defined as the atom in contact when this atom is within the range of chemical interactions of any atom of the copper substrate. The value of Aat is calculated by dividing the total surface area of diamond rotator by the total number of atoms in contact. Additionally, an asperity contact area Aasp is also defined using the concept of the edge of the contact zone. The schematic diagram of the asperity contact area is shown in Fig. 4. The asperity contact area is the maximum cross sectional area where the rotator meets the substrate. The furrow depths are calculated by calculating the distance between the atoms of the upper of cooper substrate and the atoms of the bottom of furrow. When the depth of furrow is higher than the height of sphere, the asperity contact area is acquired by calculating the cross sectional area of the cylinder. Therefore, the asperity contact area can be calculated with Eq. (5).
Fig. 4. Schematic diagram of nominal contact area.
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(
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S ¼ pr 2 ¼ pðR2 ðR hÞ Þ h 6 hmax D21 =4
S¼p
h > hmax
ð5Þ
where R is the radius of the partial sphere, r is the radius of junction of diamond rotator and copper substrate, D1 is the diameter of cylinder part, h is the depth of furrow and hmax is the height of semi-sphere. According to the rotary model in Fig. 1, the value pffiffiffi of R, D and hmax is equal to 1 nm, 3 nm and 0.5 nm, respectively. It can be deduced that the asperity contact area will increase when the depth of furrow is increased. 3. Results and discussions 3.1. Analysis of the frictional force
negative maximum value at position ①. Similarly, at position ②, ③ and ④, the total atomic forces in the x direction are positive maximum value, 0, negative maximum value, respectively, and the total atomic forces in the y direction are 0, positive maximum value, 0, respectively. It can be seen that the periodic cycle of total atomic force is 2p. Therefore, the total atomic forces are sine curve and cosine curve with periodic cycle of Tcycle = 2p. The above analysis is carried out in the cases when the rotary model is a symmetric system and no wear occurs. Actually, the model experiences wear, so the debris and plough are produced, which makes the symmetric system become asymmetric. The phase angle and amplitude of total atomic forces deviate from zero point slightly. As a result, the total atomic forces in the x and y directions can be illustrated by Eq. (6).
F x ¼ a1 þ b1 sinðh þ u1 Þ
ð6Þ
As Eq. (2) illustrates, the frictional force contains component frictional forces in the x and y directions. Meanwhile, the component frictional forces are made up of total atomic forces in the x and y directions. It can be deduced that the total atomic forces and component frictional forces have enormous influence on the frictional force. Therefore, it is necessary to study how the total atomic forces and component frictional forces vary with the rotary angle. Fig. 5 shows the total atomic forces in the x and y directions with external load of 12.5 nN and rotary cycle of 30 ps. ①, ②, ③ and ④ corresponds to the positions in Fig. 3. The total atomic force fluctuates with the rotary angle. Besides, there is a sine relationship between the total atomic force and the rotary angle. The periodic cycle of total atomic force is 2p, which is related to the structure of rotary model. Assuming that the diamond rotator rotates on the condition when no wear happens, it will do uniform circular motion. According to the feature of uniform circular motion, the total atomic forces in the normal direction can be considered as a constant, which is called the centripetal force Fn in the paper. Based on the force analysis in Fig. 3, the centripetal force is equal to Fn = Fxsin h Fycos h. It can be deduced that the waveforms of total atomic forces are sine and cosine curves, which can make the centripetal become a constant. Moreover, the rotary model is a model of symmetrical distribution. For example, when the rotator is at position ①, as shown in Fig. 3, the total atomic force in the x direction can be cancelled out since the number of atoms is equal on both sides of y axis. Also, the atomic force equals to the centripetal force in the y direction and it points to the negative direction of x axis, so the atomic force in the y direction is at its
where c1 + c2 is the positive value of the curve offsets zero. Fig. 7 shows that the frictional force varies with the rotary angle. The frictional force fluctuates around a positive value which is called the average frictional force. It is considered that the frictional force fluctuation is very common in micro scale [35]. The fluctuations are due
Fig. 5. Total atomic forces of diamond rotator in the x and y directions.
Fig. 6. Relationship between component frictional force and rotary angle.
F y ¼ a2 b2 cosðh þ u2 Þ
Combining Eq. (2) with Eq. (6), the component frictional forces can be acquired with Eq. (7).
(
f x ¼ a1 cosðhÞ þ b1 sinðh þ u1 Þ cosðhÞ f y ¼ a2 sinðhÞ b2 cosðh þ u2 Þ sinðhÞ
ð7Þ
It can be deduced from the wave in Fig. 5 that b1/a1 1, b2/ a2 1. Thus, the first items of Eq. (7) (a1cos(h) and a2sin(h)) have little influence on the periodic cycle of the component frictional forces, while the second items of Eq. (7) can make the cycle reduce by half. As shown in Fig. 6, the periodic cycle of component frictional forces is half of the periodic cycle of total atomic forces. The curve fitting for the component frictional forces in Fig. 6 is made by using sine formula and Eq. (8) is acquired.
(
f x ¼ c1 þ d1 sinð2h þ d1 Þ f y ¼ c2 d2 sinð2h þ d2 Þ
ð8Þ
where c1 c2, c1 > 0, c2 > 0, d1 d2, d1 d2. Synthesizing Eqs. (2) and (8), Eq. (9) is gotten.
f ¼ c 1 þ c2
ð9Þ
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Fig. 7. Relationship between frictional force and rotary angle.
to the stick–slip behavior between the diamond rotator and the cooper substrate. In conclusion, the periodic cycle of total atomic forces is 2p, while the periodic cycle of component frictional forces is p. It is coincident with the force analysis in the macro. The formula of calculating the frictional force is proved to be correct from the force analysis and mathematical deduction. 3.2. Influence of external load on the frictional force Fig. 8 shows the frictional forces at different external loads (2.5, 12.5 and 37.5 nN) with different rotary cycles (15, 20 and 30 ps), where a force promotes the motivation is defined as the negative frictional force. Eq. (4) is applied to calculate the frictional forces. Meanwhile, the frictional forces are averaged over blocks of consecutive time steps, with each lasting for 500 time steps. It can clearly be seen that they fluctuate around the average frictional forces, no matter what the external loads are. The changes in these curves are not significantly different for different rotary cycles. In other words, the rotary cycles adopted in the simulation have little effect on the frictional forces. The average frictional forces fave are calculated by averaging the frictional forces within one rotary cycle. With increasing external load, the average frictional force becomes larger and it is almost proportional to the external load. Moreover, oscillation amplitudes of the frictional forces increase with the increase of external loads. Furthermore, the frictional forces sometimes become negative at low external loads (2.5 and 12.5 nN), which indicates that the frictional forces promote the motivation of diamond rotator. However, it does not appear with the increasing external load, such as at external load of 37.5 nN. The frictional forces in a macroscopic friction are acquired by averaging the atomic forces on the basis of huge number of atoms involved, while only a limited number of atoms play an important role in the nanometer scale, so individual atomic interactions can lead to the instant variation of the frictional force [19]. When the external load is at 2.5 nN or 12.5 nN, the number of atoms behind the diamond rotator can be more than that in front of diamond rotator at certain instant of the rotary motion. As a result, the attraction from the atoms behind the diamond rotator may be larger than the resistance from the atoms in front of the diamond rotator. The direction of the resultant force is changed and the frictional force becomes negative. Contrary to the condition at low external load, a large number of copper atoms pile up in front of
diamond rotator and the resistance plays a leading role almost all the time at high external load of 37.5 nN. Thus, the frictional forces are basically all positive at external load of 37.5 nN. The MD simulation is performed to determine the dependence of average frictional force on the contact area and external load. Based on the data acquired from the MD simulation, fitting curves are given by using power function. Fig. 9(a) illustrates the relationship between average frictional force and external load at different rotary cycles. No matter how long the rotary cycle is, it is predicted 2=3
that the average frictional force f av e / F L . Similarly, the simula2=3 tion results also show that the real contact area Areal / F L , which is shown in Fig. 9(b). Fig. 9(c) shows the asperity contact area using the concept of the edge of the contact zone. Compared with Fig. 9 (b), it can be seen that the real contact area is slightly higher than the asperity contact area at external load below 12.5 nN. When the external load is increased, the asperity contact area does not increase any more, but the real contact area increases slowly. Combined with the depth of furrow in Fig. 9(d), it can be deduced that when the external load is above 12.5 nN, the partial sphere of rotator has completely entered into the substrate and part of the cylinder of the diamond rotator begins to take part in friction. As a consequence, the asperity contact area cannot illustrate the friction law in the rotary model and the average frictional force should be defined in terms of the real contact area. The dependence of the average frictional force on the external load is shown in Fig. 10. The simulation data reveals that the average frictional force is proportional to the real contact area.
3.3. Influence of external load on the coefficient of friction The coefficient of friction is the ratio of the average frictional force to the normal force which is the summation of the atomic forces in the z direction. Fig. 11 shows the coefficient of friction varying with the external loads at different rotary cycles. By comparing Fig. 9(b) with Fig. 11, it can be seen that the coefficient of friction has close relationship with the real contact area. When the external load is raised, the real contact increases nonlinearly and the coefficient of friction shows a downward trend. But the decreasing rate is different at different external load. For example, when the external load is lower than 12.5 nN, the coefficient of friction declines rapidly, which is the period when the partial sphere part of diamond rotator is pressing into the copper
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Fig. 8. Frictional force varying with rotary angle at different rotary cycles: (a) 15 ps, (b) 20 ps, and (c) 30 ps. The black, pink and blue lines correspond to the external loads of 2.5, 12.5 and 37.5 nN, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
substrate. A good description of the frictional morphology is illustrated in the Fig. 12(a). At this moment, the real contact area increases rapidly, which makes the frictional coefficient fall abruptly. However, when the external load is larger than 12.5 nN, the partial sphere part has entirely pressed into the substrate. As Fig. 12(b) and (c) shows, the cylinder part of the diamond rotator
begins to take part in friction and the real contact area increases steadily. Meanwhile, the coefficient of friction tends towards stability. So it can be concluded that the increasing rate of real contact area determines the decreasing rate of the coefficient of friction. The results indicate that the external load is significantly important to the coefficient of friction by changing the real contact area.
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Fig. 9. Influence of external load on the frictional characteristics of rotary model. (a) Average frictional force, (b) real contact area, (c) asperity contact area, and (d) depth of furrow.
Fig. 10. Dependence of average frictional force on the real contact area at different rotary cycles.
Fig. 11. Coefficient of friction varying with the external loads at different rotary cycles. The black, pink and blue lines correspond to the rotary cycles of 15, 20 and 30 ps, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
4. Conclusion In conclusion, MD simulation was used to investigate the influence of external load on the frictional characteristics of swash
plate–slipper friction pair in the perspective of atom. The formula of frictional force was proposed to analyze the frictional characteristics of rotary model. The conclusions were drawn as follows:
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Fig. 12. Snapshots of friction morphology at rotary cycle of 20 ps with different external loads (a) 2.5 nN, (b) 12.5 nN, and (c) 37.5 nN.
(1) In the rotary model, the total atomic forces in the x and y directions vary with rotary cycles according to the sine law with a periodic cycle of one circle (2p). However, the periodic cycle of the component frictional force decreases one half and the frictional force is basically not a periodic signal. (2) The frictional force fluctuates with rotary angle and it even promotes the motivation of diamond rotator when the external load is low. The average frictional force depends linearly on the real contact area and it shows approximately the same 2/3 power-law dependence on the external load. Also, the real contact area scales with external load as F 2=3 L . (3) The external load has enormous influence on the coefficient of friction by changing the real contact area. When the external load is raised, the coefficient of friction declines abruptly at low external load, which is due to the rapid increase of real contact area. But when the external load is above 12.5 nN, the real contact area increase steadily and the coefficient of friction tends towards stability.
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