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A new measure of local temperature distribution in non-equilibrium molecular dynamics simulation of friction
ELSEVIER
Surface and Coatings Technology 83 (1996) 3 3-316
A new measureof local temperature distribution in non-equilibrium molecular dynamics simulation of friction Keiji Hayashi aa*,Noriyuki Sakudoa,Toshio Kawai b a Electron Device Systems Core, Kanazawa Institute of Technology, 7-1, Ohgigaoka, Nonoichi, Ishikawa 921, Japan b Department of Physics, Keio University, 14-1, Hiyoshi 3 chome, Kohoku-ku, Yokohama 223, Japan
Abstract Analysis methods of energy dissipation mechanisms in the non-equilibrium molecular dynamics simulation of friction were studied. A spatial distribution of local quasi-temperature, defined by averaging over time, was found to be a potentially convenient measure of the heat generation and transport accompanying friction or, more generally, in microscopic non-equilibrium systems characterized by strong spatial and slight temporal inhomogeneities. Moreover, the local quasi-temperature is applied to the visualization of modes in which a steady spatial distribution is formed by wearless friction and then collapses owing to wear with the passing of time. Keywords: Friction; Molecular dynamics simulation; Energy dissipation; Complexity
1. Introduction In developing low friction, durable surfaces, a better understanding of the atomistic mechanisms of sliding friction is indispensable [l-3]. The goal of our investigation is to clarify the origin of wearless friction, and to establish the guiding principles of process design to improve the quality of material surfaces.As a first step, non-equilibrium molecular dynamics (MD) simulation [4] of wearless friction was carried out in this study. In this paper, we propose the use of the local quasitemperature, defined later, as a potentially convenient measure of the heat generation and heat transport in microscopic non-equilibrium systems characterized by strong spatial and slight temporal inhomogeneities. The definition and applications of the local quasi-temperature are described.
2. Simulation methods MD simulations of friction between the plane surfaces of two closest-packed lattices were carried out using a simplified two-dimensional model illustrated in Fig. 1. Particles belonging to the bottom layer of lattice A, * Corresponding author. Tel.: +81 762 48 1100 (ext. 2424); fax: + 81 762 94 6707; e-mail: [email protected]. 0257-8972/96/$15.000 1996Elsevier ScienceS.A. Ml rights reserved
*X
. fix :ed Fig. 1. Simulation model of surfaces placed in sliding contact. The initial configuration is shown. A periodic boundary condition was imposed on the lateral boundaries.
particles belonging to the top layer of lattice B and the other particles are referred to as bottom particles, top particles and passive particles respectively in this paper.
314
K. Hayashi et nl./Surface atzdCoatbzgsTechnology83 (1996) 313-316
The total number of passive particles was in the range 100-1600 in this study. Lennard-Jones potentials [4] ~AA(l’ij), SBB(I.ij)and ~AB(I’ij), where ~ij represents the distance between the i th and jth particles, were assumed between two A particles, between two B particles and between A and B particles respectively
local quasi-temperature derived by averaging over time as a rough measure. In the following equations, a timeaveraged value around time t is denoted by angular brackets with a subscript t. For example, the timeaveraged value of an arbitrary function f( t’) around time t is t+At
L
(1)
where the subscript ,?,uis AA or BB or AB. The existence of the interface was taken into account by assuming that EABis smaller than EAAand cBB. The initial load was controlled by adjusting the distance L, between the bottom and top layers. The positions of the bottom particles were fixed and the distance L, was kept constant throughout each simulation run. The positions of the top particles were also fixed at first, when the initial temperature of the passive particles was controlled. The simulation of friction was started at time t = 0. Friction occurred by moving the top particles along the interface at a Constant velocity $troke much slower than the sound velocities of the lattices. The constant velocity &,k,$ was also added to the thermal velocity of each passive B particle at t = 0. The motions of the passive particles were traced by solving equations of motion numerically. A periodic boundary condition was imposed on the lateral boundaries. The phenomenological control of the mean temperature or total kinetic energy of the passive particles was not performed during t > 0. and In this paper, CAA,?nA(mass Of an A particle) DAA,which corresponds to the lattice constant of crystal A, are adopted as units of energy, mass and length respectivm]. MD slmulatlons were carried out for a variety of values of the simulation parameters: g’BB (lattice constant of crystal B), 0.526-1.0; Ed, 0.01-0.2; cBB, 1.0-5.0; nzg (mass of a B particle), 1.0-8.0; !&r&e, 0.01-0.5.
3. Results and discussion 3.1. Local quasi-temperntwe A difficulty encountered in the MD analysis of the mechanisms of energy dissipation accompanying wearless friction involves the strong spatial inhomogeneity of the thermal properties, originating inherently from the existence of an abrupt interface between the two materials. In practical cases,however, the relative sliding speed of the two surfaces is usually much slower than the thermal motion of the particles. Thus in order to clarify the processesof energy dissipation in microscopic non-equilibrium systemscharacterized by strong spatial and slight temporal inhomogeneities, we defined the
t
t-At
= &
s
f(f)
dt’
(2)
The time-averaged kinetic energy of each passive particle can be separated into two components imj(Fj(t’)“)t
+
=$~~~j(~j(t’))f
(3)
K,(t)
where lllj is the mass of the particle. The second term on the right-hand side of Eq. (3) is a component which originates from the fluctuation of the velocity Uj(t’) of the passive particle K,(t)
K,(t)
-
=i?nj({Cj(t’)
(4)
(Cj(t’))t)‘)t
In these equations, the subscript j stands for a physical quantity of the jth particle. The local quasi-temperature of the D-dimensional system (for example, D = 2 for the model described in Section 2) is defined by q(t)
q(t)
= $
Kj(t)
(5)
where k is the Boltzmann constant, In the following figures, the unit of local quasi-temperature is taken so that the value of k is unity. It should be noted that, in this study, the value of the averaging time At in Eq. (2) was selected to be of the same order as ~~~~~~~~~~~ so that At is sufficiently long compared with the periods of almost all modes of lattice vibration. A typical example of the steady spatial distribution of the local quasi-temperature observed in an MD simulaI
s 3 Q, z.E e !
I
OB XA
N
0 0.1 0.2 Local Quasi-Temperature
(a. u.)
Fig. 2. A typical example of the steady spatial distribution of the local quasi-temperature originating from wearlessfriction. The z coordinate is plotted as the ordinate and the local quasi-temperature as the abscissa for each passive particle. The z coordinates of the bottom particles, top particles and interface are labelled L, U and I respectively.
315
K. Hayashi et al./Sulface and Coatings Technology 83 (1996) 313-316
Furthermore, another common feature of the steady spatial distribution of the local quasi-temperature is that, on the whole, the gradient of the local quasitemperature as a function of the z coordinate is steepest in the vicinity of the interface and is approximately zero at the top and the bottom layers, which are assumed to be adiabatic. Considering that heat fluxes beyond these boundary layers should be zero, these features of spatial distribution may imply that the heat flux is, roughly speaking, proportional to the gradient of the local quasitemperature. Therefore the local quasi-temperature is expected to be helpful for discussing the mechanisms of energy dissipation in spatially inhomogeneous non-equilibrium systems on the basis of MD simulations.
tion of wearless friction is shown in Fig. 2. In this figure, the z coordinate is plotted as the ordinate and the local quasi-temperature as the abscissa for each passive particle. Although the quantitative details depend on the selection of the value of the averaging time At, the following qualitative features of this plot are commonly observed in this study over fairly wide ranges of At, time t and other parameters mentioned in Section 2. Firstly, the local quasi-temperature is highest at the interface between lattices A and B, and tends to decrease on moving to the bottom layer of lattice A or to the top layer of lattice B. This suggests that heat is mainly generated at the interface at which the particles frequently collide with each other at high relative velocities.
I
(a) J
I
I
” 300 a s. ij 200 5 3 2
I
400 800 Time (a. u.)
Time
(a. u.)
100
I 800
400 Time (a. u.)
400 800 Time (a. u.)
Fig. 3. The frictional work WfI done by the top layer of lattice B (a), the total energy Etotnlobtained by summing the kinetic and potential energies of the passive particles (b), the total kinetic energy Ek of the passive particles (c) and the load F, (d) as a function of the time t.
K. Hayashi et al./Surfuce nnd Coatings Technology83 (1996) 313-316
316
3.2. Someapplications
Fig. 3 shows the frictional work W, done by the top layer of lattice B (a), the total energy Etotalobtained by summingthe kinetic and potential energiesof the passive particles (b), the total kinetic energy Ek of the passive particles (c) and the load F, (d) as a function of the time L.The law of energyconservationis satisfied,as observed by comparing Fig. 3(b) with Fig. 3(a) where W, was obtained by integrating the product of the frictional force F, and the constant velocity !.&,kewith respectto time t. Fig. 3(a) indicates that the frictional force F,, which is proportional to the slope of this figure, initially increasesin region (I) and then decreasesin region (II). The first increasecan be accounted for by considering that the frictional force F, is a function of both the load F, and the mean temperature which is roughly proportional to the total kinetic energy Ek in this case.Since the volumeof the model illustrated in Fig. 1 was assumed to remain constant during the MD simulation, the increase in the mean temperature caused by friction (Fig. 3(c)) results in an increasein the pressureor load F, (Fig. 3(d)), thus leading to an increasein the frictional force F,. Fig. 4 shows the spatial distributions of the local quasi-temperatureat times t of -25 (i), 125 (ii), 275 (iii) and 425 (iv). The formation processof a steady distribution can be clearly seen from this figure. Moreover, the spatial distributions of the local quasitemperature at times t of 425 and 925 are plotted in Fig. 5, where the data for B particles and A particles are shown separately in (a) and (b). As can be seen from this figure, wear does not take place until t = 425. In region (II) of Fig. 3(a), however, the high temperature causesthe wear or surfacemelting of materials A and B. The reduction in the frictional force F, in region (II) is due to the lubrication effect of this interfacial liquid
(I)(ii)
(iii)
’
(id
Local Quasi-Temperature (a. u,) Fig. 5. Spatial distributions of the local quasi-temperature T, at times t of 425 and 925. Data for B particles and A particles are plotted in (a) and (b) respectively. The z coordinates of the bottom particles, top particles and interface are labelled L, U and I respectively.
layer. Fig. 5 shows that the steady spatial distribution of the local quasi-temperature observed at t=425 is destroyed by t= 925 due to the disappearanceof the abrupt interfacebetweenmaterials A and B. 4. Conclusions
Analysis methods of energy dissipation mechanisms in the non-equilibrium MD simulation of wearlessfriction were studied. The local quasi-temperaturedefined in Section 3.1 was found to be a potentially useful measure of the heat generation and heat transport in microscopic non-equilibrium systemscharacterized by strong spatial and slight temporal inhomogeneitiesas is often the casewith practical friction. Moreover, the local quasi-temperaturewas applied to the visualization of modesin which a steadyspatial distribution is gradually formed by wearlessfriction and then collapsesowing to wear with the passingof time. Acknowledgement One of the authors (K.W.) wishes to express his gratitude to Professor Hiroshi Kuwano of Keio University for his encouragement. References
Local Quasi-Temperature (a. u.) Fig. 4. Spatial distributions of the local quasi-temperature q at times t of -25 (i), 125 (ii), 275 (iii) and 425 (iv). The z coordinates of the bottom particles, top particles and interface are labelled L, U and I respectively.
[1] LL. Singer, J. Vm. Sci. Techrlol. A, 12 (1994) 2605. [2] J.A. Harrison, CT. White, R.J. Colton and D.W. Brenner, J. Phps. Chern.,97 (1993) 6573. [3] J.N. Glosli and G.M. McClelland, P/~J’s,Reo.Letf,, 70 (1993) 1960. [4] Wm. G. Hoover, Cor,lputatiorral Statistical Mcchaaics, Elsevier, Amsterdam, 1991,p. 288.
Surface and Coatings Technology 83 (1996) 3 3-316
A new measureof local temperature distribution in non-equilibrium molecular dynamics simulation of friction Keiji Hayashi aa*,Noriyuki Sakudoa,Toshio Kawai b a Electron Device Systems Core, Kanazawa Institute of Technology, 7-1, Ohgigaoka, Nonoichi, Ishikawa 921, Japan b Department of Physics, Keio University, 14-1, Hiyoshi 3 chome, Kohoku-ku, Yokohama 223, Japan
Abstract Analysis methods of energy dissipation mechanisms in the non-equilibrium molecular dynamics simulation of friction were studied. A spatial distribution of local quasi-temperature, defined by averaging over time, was found to be a potentially convenient measure of the heat generation and transport accompanying friction or, more generally, in microscopic non-equilibrium systems characterized by strong spatial and slight temporal inhomogeneities. Moreover, the local quasi-temperature is applied to the visualization of modes in which a steady spatial distribution is formed by wearless friction and then collapses owing to wear with the passing of time. Keywords: Friction; Molecular dynamics simulation; Energy dissipation; Complexity
1. Introduction In developing low friction, durable surfaces, a better understanding of the atomistic mechanisms of sliding friction is indispensable [l-3]. The goal of our investigation is to clarify the origin of wearless friction, and to establish the guiding principles of process design to improve the quality of material surfaces.As a first step, non-equilibrium molecular dynamics (MD) simulation [4] of wearless friction was carried out in this study. In this paper, we propose the use of the local quasitemperature, defined later, as a potentially convenient measure of the heat generation and heat transport in microscopic non-equilibrium systems characterized by strong spatial and slight temporal inhomogeneities. The definition and applications of the local quasi-temperature are described.
2. Simulation methods MD simulations of friction between the plane surfaces of two closest-packed lattices were carried out using a simplified two-dimensional model illustrated in Fig. 1. Particles belonging to the bottom layer of lattice A, * Corresponding author. Tel.: +81 762 48 1100 (ext. 2424); fax: + 81 762 94 6707; e-mail: [email protected]. 0257-8972/96/$15.000 1996Elsevier ScienceS.A. Ml rights reserved
*X
. fix :ed Fig. 1. Simulation model of surfaces placed in sliding contact. The initial configuration is shown. A periodic boundary condition was imposed on the lateral boundaries.
particles belonging to the top layer of lattice B and the other particles are referred to as bottom particles, top particles and passive particles respectively in this paper.
314
K. Hayashi et nl./Surface atzdCoatbzgsTechnology83 (1996) 313-316
The total number of passive particles was in the range 100-1600 in this study. Lennard-Jones potentials [4] ~AA(l’ij), SBB(I.ij)and ~AB(I’ij), where ~ij represents the distance between the i th and jth particles, were assumed between two A particles, between two B particles and between A and B particles respectively
local quasi-temperature derived by averaging over time as a rough measure. In the following equations, a timeaveraged value around time t is denoted by angular brackets with a subscript t. For example, the timeaveraged value of an arbitrary function f( t’) around time t is t+At
L
(1)
where the subscript ,?,uis AA or BB or AB. The existence of the interface was taken into account by assuming that EABis smaller than EAAand cBB. The initial load was controlled by adjusting the distance L, between the bottom and top layers. The positions of the bottom particles were fixed and the distance L, was kept constant throughout each simulation run. The positions of the top particles were also fixed at first, when the initial temperature of the passive particles was controlled. The simulation of friction was started at time t = 0. Friction occurred by moving the top particles along the interface at a Constant velocity $troke much slower than the sound velocities of the lattices. The constant velocity &,k,$ was also added to the thermal velocity of each passive B particle at t = 0. The motions of the passive particles were traced by solving equations of motion numerically. A periodic boundary condition was imposed on the lateral boundaries. The phenomenological control of the mean temperature or total kinetic energy of the passive particles was not performed during t > 0. and In this paper, CAA,?nA(mass Of an A particle) DAA,which corresponds to the lattice constant of crystal A, are adopted as units of energy, mass and length respectivm]. MD slmulatlons were carried out for a variety of values of the simulation parameters: g’BB (lattice constant of crystal B), 0.526-1.0; Ed, 0.01-0.2; cBB, 1.0-5.0; nzg (mass of a B particle), 1.0-8.0; !&r&e, 0.01-0.5.
3. Results and discussion 3.1. Local quasi-temperntwe A difficulty encountered in the MD analysis of the mechanisms of energy dissipation accompanying wearless friction involves the strong spatial inhomogeneity of the thermal properties, originating inherently from the existence of an abrupt interface between the two materials. In practical cases,however, the relative sliding speed of the two surfaces is usually much slower than the thermal motion of the particles. Thus in order to clarify the processesof energy dissipation in microscopic non-equilibrium systemscharacterized by strong spatial and slight temporal inhomogeneities, we defined the
t-At
= &
s
f(f)
dt’
(2)
The time-averaged kinetic energy of each passive particle can be separated into two components imj(Fj(t’)“)t
+
=$~~~j(~j(t’))f
(3)
K,(t)
where lllj is the mass of the particle. The second term on the right-hand side of Eq. (3) is a component which originates from the fluctuation of the velocity Uj(t’) of the passive particle K,(t)
K,(t)
-
=i?nj({Cj(t’)
(4)
(Cj(t’))t)‘)t
In these equations, the subscript j stands for a physical quantity of the jth particle. The local quasi-temperature of the D-dimensional system (for example, D = 2 for the model described in Section 2) is defined by q(t)
q(t)
= $
Kj(t)
(5)
where k is the Boltzmann constant, In the following figures, the unit of local quasi-temperature is taken so that the value of k is unity. It should be noted that, in this study, the value of the averaging time At in Eq. (2) was selected to be of the same order as ~~~~~~~~~~~ so that At is sufficiently long compared with the periods of almost all modes of lattice vibration. A typical example of the steady spatial distribution of the local quasi-temperature observed in an MD simulaI
s 3 Q, z.E e !
I
OB XA
N
0 0.1 0.2 Local Quasi-Temperature
(a. u.)
Fig. 2. A typical example of the steady spatial distribution of the local quasi-temperature originating from wearlessfriction. The z coordinate is plotted as the ordinate and the local quasi-temperature as the abscissa for each passive particle. The z coordinates of the bottom particles, top particles and interface are labelled L, U and I respectively.
315
K. Hayashi et al./Sulface and Coatings Technology 83 (1996) 313-316
Furthermore, another common feature of the steady spatial distribution of the local quasi-temperature is that, on the whole, the gradient of the local quasitemperature as a function of the z coordinate is steepest in the vicinity of the interface and is approximately zero at the top and the bottom layers, which are assumed to be adiabatic. Considering that heat fluxes beyond these boundary layers should be zero, these features of spatial distribution may imply that the heat flux is, roughly speaking, proportional to the gradient of the local quasitemperature. Therefore the local quasi-temperature is expected to be helpful for discussing the mechanisms of energy dissipation in spatially inhomogeneous non-equilibrium systems on the basis of MD simulations.
tion of wearless friction is shown in Fig. 2. In this figure, the z coordinate is plotted as the ordinate and the local quasi-temperature as the abscissa for each passive particle. Although the quantitative details depend on the selection of the value of the averaging time At, the following qualitative features of this plot are commonly observed in this study over fairly wide ranges of At, time t and other parameters mentioned in Section 2. Firstly, the local quasi-temperature is highest at the interface between lattices A and B, and tends to decrease on moving to the bottom layer of lattice A or to the top layer of lattice B. This suggests that heat is mainly generated at the interface at which the particles frequently collide with each other at high relative velocities.
I
(a) J
I
I
” 300 a s. ij 200 5 3 2
I
400 800 Time (a. u.)
Time
(a. u.)
100
I 800
400 Time (a. u.)
400 800 Time (a. u.)
Fig. 3. The frictional work WfI done by the top layer of lattice B (a), the total energy Etotnlobtained by summing the kinetic and potential energies of the passive particles (b), the total kinetic energy Ek of the passive particles (c) and the load F, (d) as a function of the time t.
K. Hayashi et al./Surfuce nnd Coatings Technology83 (1996) 313-316
316
3.2. Someapplications
Fig. 3 shows the frictional work W, done by the top layer of lattice B (a), the total energy Etotalobtained by summingthe kinetic and potential energiesof the passive particles (b), the total kinetic energy Ek of the passive particles (c) and the load F, (d) as a function of the time L.The law of energyconservationis satisfied,as observed by comparing Fig. 3(b) with Fig. 3(a) where W, was obtained by integrating the product of the frictional force F, and the constant velocity !.&,kewith respectto time t. Fig. 3(a) indicates that the frictional force F,, which is proportional to the slope of this figure, initially increasesin region (I) and then decreasesin region (II). The first increasecan be accounted for by considering that the frictional force F, is a function of both the load F, and the mean temperature which is roughly proportional to the total kinetic energy Ek in this case.Since the volumeof the model illustrated in Fig. 1 was assumed to remain constant during the MD simulation, the increase in the mean temperature caused by friction (Fig. 3(c)) results in an increasein the pressureor load F, (Fig. 3(d)), thus leading to an increasein the frictional force F,. Fig. 4 shows the spatial distributions of the local quasi-temperatureat times t of -25 (i), 125 (ii), 275 (iii) and 425 (iv). The formation processof a steady distribution can be clearly seen from this figure. Moreover, the spatial distributions of the local quasitemperature at times t of 425 and 925 are plotted in Fig. 5, where the data for B particles and A particles are shown separately in (a) and (b). As can be seen from this figure, wear does not take place until t = 425. In region (II) of Fig. 3(a), however, the high temperature causesthe wear or surfacemelting of materials A and B. The reduction in the frictional force F, in region (II) is due to the lubrication effect of this interfacial liquid
(I)(ii)
(iii)
’
(id
Local Quasi-Temperature (a. u,) Fig. 5. Spatial distributions of the local quasi-temperature T, at times t of 425 and 925. Data for B particles and A particles are plotted in (a) and (b) respectively. The z coordinates of the bottom particles, top particles and interface are labelled L, U and I respectively.
layer. Fig. 5 shows that the steady spatial distribution of the local quasi-temperature observed at t=425 is destroyed by t= 925 due to the disappearanceof the abrupt interfacebetweenmaterials A and B. 4. Conclusions
Analysis methods of energy dissipation mechanisms in the non-equilibrium MD simulation of wearlessfriction were studied. The local quasi-temperaturedefined in Section 3.1 was found to be a potentially useful measure of the heat generation and heat transport in microscopic non-equilibrium systemscharacterized by strong spatial and slight temporal inhomogeneitiesas is often the casewith practical friction. Moreover, the local quasi-temperaturewas applied to the visualization of modesin which a steadyspatial distribution is gradually formed by wearlessfriction and then collapsesowing to wear with the passingof time. Acknowledgement One of the authors (K.W.) wishes to express his gratitude to Professor Hiroshi Kuwano of Keio University for his encouragement. References
Local Quasi-Temperature (a. u.) Fig. 4. Spatial distributions of the local quasi-temperature q at times t of -25 (i), 125 (ii), 275 (iii) and 425 (iv). The z coordinates of the bottom particles, top particles and interface are labelled L, U and I respectively.
[1] LL. Singer, J. Vm. Sci. Techrlol. A, 12 (1994) 2605. [2] J.A. Harrison, CT. White, R.J. Colton and D.W. Brenner, J. Phps. Chern.,97 (1993) 6573. [3] J.N. Glosli and G.M. McClelland, P/~J’s,Reo.Letf,, 70 (1993) 1960. [4] Wm. G. Hoover, Cor,lputatiorral Statistical Mcchaaics, Elsevier, Amsterdam, 1991,p. 288.