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Journal of Tribology | Volume 140 | Issue 3 | research-article
Research Papers: Friction and Wear

Effect of Substrate Shape on Friction Regimes and on Tip Jump Probability in Atomic Scale Friction

[+] Author and Article Information
E. Djiha Tchaptchet

Laboratory of Mechanics,
Department of Physics,
Faculty of Science,
University of Yaoundé 1,
P.O. Box 812,
Yaoundé 337, Cameroon
e-mail: djihae@yahoo.fr

G. Djuidje Kenmoe

Laboratory of Mechanics,
Department of Physics,
Faculty of Science,
University of Yaoundé 1,
P.O. Box 812,
Yaoundé 337, Cameroon;
Centre d'Excellence en Technologies de
l'Information et de la Communication (CETIC),
University of Yaoundé 1,
Yaoundé 337, Cameroon
e-mail: kdjuidje@yahoo.fr

T. C. Kofane

Laboratory of Mechanics,
Department of Physics,
Faculty of Science,
University of Yaoundé 1,
P.O. Box 812,
Yaoundé 337, Cameroon;
Centre d'Excellence en Technologies de
l'Information et de la Communication (CETIC),
University of Yaoundé 1,
Yaoundé 337, Cameroon
e-mail: tckofane@yahoo.com

1Corresponding author.

Contributed by the Tribology Division of ASME for publication in the JOURNAL OF TRIBOLOGY. Manuscript received April 22, 2017; final manuscript received October 5, 2017; published online December 20, 2017. Assoc. Editor: Min Zou.

J. Tribol 140(3), 031606 (Dec 20, 2017) (8 pages) Paper No: TRIB-17-1145; doi: 10.1115/1.4038409 History: Received April 22, 2017; Revised October 05, 2017
Article
References
Figures

We investigate the effect of the shape potential on the frictional behavior transitions. The Tomlinson parameter for the deformable substrate potential is calculated theoretically and its influence on friction force is studied. Futhermore, effects of temperature and substrate shape on the tip jump probability are presented. We find two critical times, which characterize the tip dynamics. The first critical time is the time spent by the tip to reach next potential minimum and the second is the time at which the tip exhibits an equiprobability of forward and backward jump. We show that these critical times depend strongly on the substrate shape as well as on the temperature.
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References

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Figures

Grahic Jump Location
Fig. 1

Schematic of model. The tip interacts with the substrate, which is modeled by a nonsinusoidal potential (Remoissenet–Peyrard potential). The system is driven by a rigid support that moves with a constant velocity v. The shape of the corrugation potential depends on the parameter r.

+Schematic of model. The tip interacts with the substrate, which is modeled by a nonsinusoidal potential (Remoissenet–Peyrard potential). The system is driven by a rigid support that moves with a constant velocity v. The shape of the corrugation potential depends on the parameter r.
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Schematic of model. The tip interacts with the substrate, which is modeled by a nonsinusoidal potential (Remoissenet–Peyrard potential). The system is driven by a rigid support that moves with a constant velocity v. The shape of the corrugation potential depends on the parameter r.
Grahic Jump Location
Fig. 2

(a) Numerical result showing the variation of friction force as a function of spring stiffness. The vertical dotted line marks the transition between stick-slip regime and smooth sliding regime. (b) Friction force versus spring stiffness for different values of r. One can observe that increasing the shape parameter r leads to later transition to smooth sliding. (c) Comparison of the value of the critical stiffness kr obtained numerically and the theoretical expression. One can observe that the critical value of spring stiffness increases with r; moreover, the theoretical expression of kr is in good agreement with numerical data. Other model parameters: U0 = 1.1 ev; a = 0.52 nm; v = 1 μm/s.

+(a) Numerical result showing the variation of friction force as a function of spring stiffness. The vertical dotted line marks the transition between stick-slip regime and smooth sliding regime. (b) Friction force versus spring stiffness for different values of r. One can observe that increasing the shape parameter r leads to later transition to smooth sliding. (c) Comparison of the value of the critical stiffness kr obtained numerically and the theoretical expression. One can observe that the critical value of spring stiffness increases with r; moreover, the theoretical expression of kr is in good agreement with numerical data. Other model parameters: U0 = 1.1 ev; a = 0.52 nm; v = 1 μm/s.
View Large  |  Save Figure  |  Download Slide (.ppt)
(a) Numerical result showing the variation of friction force as a function of spring stiffness. The vertical dotted line marks the transition between stick-slip regime and smooth sliding regime. (b) Friction force versus spring stiffness for different values of r. One can observe that increasing the shape parameter r leads to later transition to smooth sliding. (c) Comparison of the value of the critical stiffness kr obtained numerically and the theoretical expression. One can observe that the critical value of spring stiffness increases with r; moreover, the theoretical expression of kr is in good agreement with numerical data. Other model parameters: U0 = 1.1 ev; a = 0.52 nm; v = 1 μm/s.
Grahic Jump Location
Fig. 3

(a) Total energy versus tip displacement for different values of the relative corrugation ηr, (b) friction force versus relative corrugation ηr for different values of shape parameter r, and (c) the effect of shape parameter r on transitions between slip regimes.

+(a) Total energy versus tip displacement for different values of the relative corrugation ηr, (b) friction force versus relative corrugation ηr for different values of shape parameter r, and (c) the effect of shape parameter r on transitions between slip regimes.
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(a) Total energy versus tip displacement for different values of the relative corrugation ηr, (b) friction force versus relative corrugation ηr for different values of shape parameter r, and (c) the effect of shape parameter r on transitions between slip regimes.
Grahic Jump Location
Fig. 4

Illustration of slip between two adjacent energy minima. Pi is the probability of the tip residing in the current potential well, i, where the energy barrier is ΔVi→j and Pj is the probability of the tip residing at the next minima, j, where the corresponding energy barrier is ΔVj→i.

+Illustration of slip between two adjacent energy minima. Pi is the probability of the tip residing in the current potential well, i, where the energy barrier is ΔVi→j and Pj is the probability of the tip residing at the next minima, j, where the corresponding energy barrier is ΔVj→i.
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Illustration of slip between two adjacent energy minima. Pi is the probability of the tip residing in the current potential well, i, where the energy barrier is ΔVi→j and Pj is the probability of the tip residing at the next minima, j, where the corresponding energy barrier is ΔVj→i.
Grahic Jump Location
Fig. 5

Illustration of transition rate for three values of shape parameter (r = 0.3 (a), r = 0.6, (b) and r = −0.4, and (c)) at different temperatures. One can observe that transition rate increases with lateral friction force and reaches a maximum value around the transition point. This maximum value increases with the temperature T and depends on the shape parameter r.

+Illustration of transition rate for three values of shape parameter (r = 0.3 (a), r = 0.6, (b) and r = −0.4, and (c)) at different temperatures. One can observe that transition rate increases with lateral friction force and reaches a maximum value around the transition point. This maximum value increases with the temperature T and depends on the shape parameter r.
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Illustration of transition rate for three values of shape parameter (r = 0.3 (a), r = 0.6, (b) and r = −0.4, and (c)) at different temperatures. One can observe that transition rate increases with lateral friction force and reaches a maximum value around the transition point. This maximum value increases with the temperature T and depends on the shape parameter r.
Grahic Jump Location
Fig. 6

Friction force versus temperature for three values of shape parameter: r = 0.2, r = 0.5, and r = −0.6. The numerical data are in good agreement with the analytical expression.

+Friction force versus temperature for three values of shape parameter: r = 0.2, r = 0.5, and r = −0.6. The numerical data are in good agreement with the analytical expression.
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Friction force versus temperature for three values of shape parameter: r = 0.2, r = 0.5, and r = −0.6. The numerical data are in good agreement with the analytical expression.
Grahic Jump Location
Fig. 7

Probability Pi as a function of time for five values of shape parameter r (a) and probability Pj as a function of time for five values of shape parameter r (b). Other model parameters: U0 = 1.1ev; a = 0.52 nm; and k = 1.2 N/m. The probability of the tip residing in the current well i decreases when the support moves since it reaches the zero value (a). At the same time, the probability of the tip residing at the next minima j increases since it reaches the values of 1 (b). One can observe that the evolution of the probabilities Pi and Pj depends on the shape parameter r and on the temperature T (c). So, the time to reach the next potential minimum decreases for negative values of r and increases for positive values of r as we can see in (d). Note that the increase of temperature leads to a significant reduction of the time to reach the next potential as we can observe in (d).

+Probability Pi as a function of time for five values of shape parameter r (a) and probability Pj as a function of time for five values of shape parameter r (b). Other model parameters: U0 = 1.1ev; a = 0.52 nm; and k = 1.2 N/m. The probability of the tip residing in the current well i decreases when the support moves since it reaches the zero value (a). At the same time, the probability of the tip residing at the next minima j increases since it reaches the values of 1 (b). One can observe that the evolution of the probabilities Pi and Pj depends on the shape parameter r and on the temperature T (c). So, the time to reach the next potential minimum decreases for negative values of r and increases for positive values of r as we can see in (d). Note that the increase of temperature leads to a significant reduction of the time to reach the next potential as we can observe in (d).
View Large  |  Save Figure  |  Download Slide (.ppt)
Probability Pi as a function of time for five values of shape parameter r (a) and probability Pj as a function of time for five values of shape parameter r (b). Other model parameters: U0 = 1.1ev; a = 0.52 nm; and k = 1.2 N/m. The probability of the tip residing in the current well i decreases when the support moves since it reaches the zero value (a). At the same time, the probability of the tip residing at the next minima j increases since it reaches the values of 1 (b). One can observe that the evolution of the probabilities Pi and Pj depends on the shape parameter r and on the temperature T (c). So, the time to reach the next potential minimum decreases for negative values of r and increases for positive values of r as we can see in (d). Note that the increase of temperature leads to a significant reduction of the time to reach the next potential as we can observe in (d).
Grahic Jump Location
Fig. 8

Illustration of the critical time tc2 for two values of shape parameter r (r = 0.4 (a) and r = 0.6 (b)). The increase of temperature induces a reduction of tc2 (c). One can observe that tc2 decreases for negative values of r and increases for positive values of r (d). Other model parameters: U0 = 1.1 ev; a = 0.52 nm; k = 1.2 N/m.

+Illustration of the critical time tc2 for two values of shape parameter r (r = 0.4 (a) and r = 0.6 (b)). The increase of temperature induces a reduction of tc2 (c). One can observe that tc2 decreases for negative values of r and increases for positive values of r (d). Other model parameters: U0 = 1.1 ev; a = 0.52 nm; k = 1.2 N/m.
View Large  |  Save Figure  |  Download Slide (.ppt)
Illustration of the critical time tc2 for two values of shape parameter r (r = 0.4 (a) and r = 0.6 (b)). The increase of temperature induces a reduction of tc2 (c). One can observe that tc2 decreases for negative values of r and increases for positive values of r (d). Other model parameters: U0 = 1.1 ev; a = 0.52 nm; k = 1.2 N/m.

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