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The relationship between static and kinetic friction and atomic level “stick-slip” motion
The relationship between “stick- slip” motion
static and kinetic
friction
and atomic
level
Abstract It
is shown
velcoity “stick
that
the fact that
approaches slip”
motion
the force
of static
7ero) for solid-on-solid that
occurs
during
sliding
the sliding
Friction
is larger
of two elastic
I. Introduction In recent theoretical work [I] it was proposed that the atomic level kinetic friction ,fl, observed in atomic force microscopy (AFM) experiments [2] is equal to the mean interfacial force, for a monolayer of palladium sliding on a graphite substrate. This is based on the assumption that, at low speed sliding. the net work done against the interfacial force to move a surface atom to a potential maximum is all dissipated and therefore is equal to the work done by the force of kinetic friction. In this article. it is shown that, unlike the monolayer case considered in ref. I, for a multilayer solid (a more appropriate model for modeling the AFM experiments [2]) sliding on a similar substrate, one must study the strain needed to overcome the force of static frictionfl, all or part of which is released in the slip part of the “stick-slip” motion which occurs at low speed sliding. This model accounts naturally for the accepted notion that,/, >,fi. It is interesting to note that other work by the present author [3] shows that, for any three-dimensional solid sliding system, ,fk would approach zero in the limit as the sliding velocity approaches zero if the motion in this limit were smooth rather than stick-slip in nature, implying that such non-smooth motion at low speeds is essential for having ,I, approach a non-zero value in the limit of zero sliding velocity.
2. Stick-slip
motion and kinetic friction
Consider the following model. Two macroscopic elastic bodies in contact slide relative to each other, one on top of the other, and the surface of each body at the interface moves in a potential due to the other body. In order for sliding to take place, a critical force,f; must be
0040-6090/9
I /S3.50
than
can be accounted
the force
of kinetic
for by considering
bodies
relative
friction
the strain
(in the limit released during
as the siding atomic
level
to each other.
applied to the upper surface of the upper body (the lower surface of the lower body is assumed to be held in place) sufficiently large to overcome the maximum force due to the potential of interaction at the interface between the bodies. Since the interaction force between the bodies is position dependent, the motion in the slow speed limit will be stick-slip in nature because strain will be built up as the upper surface of the upper body is pushed until the resulting stress is able to overcome the maximum pinning force due to the lower body. At this point the upper body slips forward suddenly. A fraction of the built-up strain in the bodies is released in each slip. If this fraction is very small, the force that must be applied to the upper surface to maintain the motion ,fl, will remain nearly equal to j; because the strain determining the force on the upper surface remains nearly constant, but, if it is large, as the strain is released, ,/1, will drop well below ,f\ between slips, making the average ,fi smaller. (The sliding speed is assumed to be sufficiently slow to allow this energy to spread throughout the system, if the system is sufficiently large, or to be released into space.) This correlation between the relative sizes of,f( andj, and the strain released in each slip should be able to be tested experimentally. Although in the remainder of this article we shall consider the simple case of a homogeneous crystal which is commensurate with the potential, the result described in this paragraph, however, that ,f; <,f, is related to atomic level stick-slip motion is quite general and would apply to inhomogeneous crystals as well as crystals which are incommensurate with the potential. The one difference would be that the abrupt slips described above that occur during low speed sliding would now be local, rather than involving the crystal as a whole. Thus. this atomic level stick-slip motion for a large inhomogeneous system might average out on a microscopic scale, and therefore, not be observed as
c
1991
Elsevier
Sequoia.
Lausanne
J. B. Sokolqf
1 Friction
untl utomic
stick-slip motion of the system as a whole. This case will be studied in a future publication. For relatively small surfaces (i.e. with dimensions not too large compared with the interatomic spacing), however, we would expect that the atomic level stick-slip motion would not average out. The present model includes only stick-slip motion due to potential energy sticking occurring at a microscopic level.
3. Models for the slip motion In order to calculate the strain released in one consider the following equations of motion, which resents the motion of one of the sliding solids periodic potential (assumed to be caused by the ond) : .Y,,= - c((2.X,,- s,, _ , - s,, + , ) - ii ,,.,J(_Y/.$)
slip, repin a sec-
(1)
with n running from 0 to N - 1, wheref(x,) is a force of period a (the lattice constant), x,, is the displacement of the nth layer of atoms (each layer will move rigidly if the periodic force is commensurate with the crystal plane) and c( is the interlayer force constant. The model is illustrated in Fig. 1. Equation (1) is subject to the boundary condmon that xN = .xNi , c-uN+, is introduced formally in order to allow us to apply eqn. ( 1) to the Nth atomic layer. but there really is no N + 1 layer. This boundary condition eliminates the harmonic interaction between levels N and N + 1.) The mass has been absorbed into a, and the coordinate x,, is set equal to a value which creates a stress in the system just large enough to overcome the forcef(.u,) in order to study the stick-slip motion. All the other x are at the equilibrium values corresponding to this value of _Y”. (In principle, this model can be applied to any size layers, from one atom up to an infinite number of atoms, although all layers are assumed to be of equal size. As such, it can in principle be applied to asperities which are only a few atoms wide as well as to infinitely wide flat surfaces). In the motion that follows, the lowest plane moves one or more periods of the periodic force
209
level “srick -slip”
before becoming stuck, as all the translational motion of the system is converted into vibrational. The assumption is that between slips the vibrational energy given up to the system either spreads out throughout the entire system or is dissipated to the environment. (The interaction with the environment was not included in the present model.) We shall see that how far the lower plane slides depends on the form off: For example, let us consider two models which can be solved exactly. In one model, the “saw-tooth potential model”, f is equal to a constant f. for .xN between 0 and u. When s, passes the value a (i.e. reaches the end of the sawtooth), each atom in the lower layer is given a pulse of kinetic energy OSmz;’ = j& where m is the atomic mass as a result of dropping a lower potential energy. (This energy is the potential energy that was built up as the atomic level climbed the saw-tooth and which is now given back as it slides off the end.) This force repeats over each succeeding distance of length U. In order to solve this model, let us set x,, equal to x,,O + u,,, where x ’ is the equilibrium value of x,,, which can be done bicause the values of .Y” and X, have been chosen so that the resulting stress is just balanced by the force of the saw-tooth potential. In order to solve for the motion which occurs after the lower layer falls off the edge of the saw-tooth potential, let us take the Laplace transform of eqn. (1). We have s’u;, + x(2ui, -
u;,_, -u;,+
where ui, is the Laplace
u:,=
,) = ti,y(t = 0)6,, N transform
of u,,, defined
(2) by
J
u,, em.” dt
0
and s is the Laplace (2) we have
transform
variable.
Solving
(3) where
N is the
number of layers, m is an integer, + 1) and (c)Oz(q,,l)=2c([ 1 -cos(q,,,)]. In solving this equation, we have fixed u0 at the value 0, and this condition is satisfied for the above solution. The above choice of q,,, guarantees that uN is always condition. Taking equal to uN+, , the second boundary the inverse Laplace transform of eqn. (3) gives q,,, = (2m + l)n/(2N
un: =
N-'C,(O) 1
sin’(q,,,N)sin[~dq,,,)tl (00(q,,,)
I),
z (4x) -‘C,,,(O)
dq sin[to,(q)t]/tu,(q) s
,v ti,v(h/c)(47r) Fig. I. The model is shown. Dots connected by straight lines represent a rigid layers of atoms, and curly lines represent harmonic forces.
eqn.
’
dx sin X/.X = (8 &)
_ ‘ti,,,(h/c)
210
.I. B. Sokolqf
/ Friction
a constant in the long time limit, because in this limit the integral over q is dominated by small q where w,(q) z (c/b)q, where c is the speed of sound perpendicular to the surface and b is the lattice constant perpendicular to the surface. This approximation was used in eqn. (4). Thus, as long as the velocity given by the saw-tooth potential is small compared with the speed of sound, the lowest atomic layer slips less than one period of the potential, and thus is not able to release all the strain. Let us consider a model in whichf(x,,,) in eqn. (1) is a periodic array of parabolic potentials. Then, it is given by ,f(X/v) = -&)[x,
-(I
+ 0.5)a]
for (I- 0.5)~ < xN < (1+ 0.5)~ where I is an integer and i,, is the strength of the force. As we did for the saw-tooth potential, we set x,’ =x,,” + u,, with xN chosen to be at the maximum of the left-hand side of one of the parabolic potentials so that the force on it from the potential is towards the right and the x,,” are chosen so that the layer N - 1 exerts a force equal to 0.5&a also towards the right. The reason for this choice is that this is the force necessary to put the system in equilibrium if the Nth layer were near the maximum of the parabolic potential to the left of the present potential. We are assuming that the force on layer N was just barely large enough to push it into the potential well in which it at present finds itself, causing it to undergo a slip, which we will now study to determine how far it will slip before it dissipates all its kinetic energy. With the above substitution for x,,, we obtain for the Laplace-transformed equation of motion s’u;, +‘Y(2U;, -u;,_, whose
solution
u;, = (2N + 1)
-uI,+,)
= -6,,.,~&(u:,-a/s)
level “stick
-slip”
where 6 is chosen to be to the right of any singularities in the complex s plane. Since we are interested in the maximum distance that the lower layer of atoms can displace, we want to calculate u,, in the large t limit. In this limit, eqn. (7) is dominated by the contribution for small s (i.e. IsI2 < 4~). Thus, to a good approximation, we can replace (s2 + 4~) ‘12 by (4~) “‘. Then the integral in eqn. (7) gives am
=a{1
-exp[
-(&/cl”‘)t])
(8)
which implies that the lower layer displaces only one lattice constant (i.e. one period of the periodic potential) before being stopped, implying that it just falls short of getting out of the first potential, and thus will not continue to slip. The approximation made in eqn. (6b) of replacing the sum over m by an integral is valid only if the system is sufficiently thick so that t ~’ is always much larger than the spacing between successive values of w”(q,,,). This condition will already begin to fail for times longer than about 10 _ 5 s for the force constant z”’ = 10” (a typical lattice vibrational frequency) and N z 10’. For longer times, the discreteness of the phonon spectrum must be taken into account. In order to accomplish this, let us first replace wo(q,,?) by its small q,,, approxito the approximamation, x”2q,,l, which is equivalent tion of replacing (s2 + 4a) ‘I2 by (4~) ‘P made in eqn. (7) in order to obtain eqn. (8). The sum over m in eqn. (6b) may be expressed as the following contour integral [4] G =(2yN))’
dw tanh( o/2y)(s’
- cu’)
’
(9)
where j’ = (2c() ‘j2(2N + 1) - ’ and where the contour is shown in Fig. 2 (since the hyperbolic tangent has poles at o/2;’ = i(2m + 1)7r/2). This is shown in ref. 4 to be
is ‘A”(&
- u/s)
w2
c
(64 0
I)?
where qJPzand o,,(q,,,) are given under G=N-’
(5)
unri atomic
,,Z
eqn. (3). We use
sin*(q,,,N) c 1>1.G + w”2(q,ll)
z(27c)
’
dq [s2 + o,‘(q,,,)]
’
s = [OSs(S
’
+ 4c() “‘I
(6b)
where we have used the fact that, in the large N limit, sin2(q,,, N) z 1. Then, the solution for u,,,(t) is
ds e”‘{s[s(s2+4c() “2 -t 2&]} - ’
UN(t) = jt”Q(27Ci) - ’ s h
I
-7
(7)
Fig. 2. The contour
in the complex
w plane used in eqn. (9) is shown.
equivalent
to
d(c) tanh( to/2;‘)s - ‘[ cS(.s- W) - fi( w + s)]
= ( 2;,N.s) Substituting u:,,
=
unio
’ tanh(s/2:!)
(10)
in eqn. (6) we obtain tanh(s/2;,)
(.s[2;*N.s + rri,, tanh(s/2;,)])
’ (11)
which can easily be shown to have poles at s = 0 and at s = is, (.s? positive and real) which is a solution to the equation 2;‘N.s, + rri,, tan(s,/2;,) = 0. Using these poles to perform the contour integration for finding the inverse Laplace transform, as in eqn. (7) one obtains u,(t)
= u
r ,I
I - 1 4;q sin( (I),,/;*)(!I,,
twice s,, and then to swing back. For a parabolic potential force constant equal to 0.024 of the interlayer force constant (which would give a value of .f; comparable with the Peierls force for a material containing dislocations), .vV only moved about one period of the potential. Let us now consider the slip distance for a more realistic model of the interface potential. Zhong and Tomanek [I] have shown that the appropriate potential for palladium on graphite is to a good approximation a sinusoidal potential. Thus let us now study the mode1 of eqn. ( 1) with such a potential. Consider ,f’(s,,) = i.,,cos[( 27[/(1).v,v] with j.,,/x = 0.158~ (corresponding to equal intersurface and interatomic layer force constants and comparable with the value of i.,,/sr used in ref. I). For N = 1000, the lower layer was found to slide by about 16lr, implying that all the strain is released in the slip and that ,fi z 0.5f;. whereas. for ;.,,/x = 0.003&r, it slid by only 0.89a. For the latter value of i,,/r, and for N = 5000 and N = 10000, the lower level slid by about 5a and IOtr respectively, implying that about l/4 of the built-up strain is released in each slip. This implies an average ,f; which is about 88% of ,f;. In Fig. 3, .vv is plotted as a function of time for the case of equal intersurface and interlayer force constants considered above. As can be seen, .Y,\, overshoots the value of _Q, and then settles back to a value near x~,. The slipping distances when there are defects in the crystal planes which are in contact will probably be smaller than these values because of the excitation of intralayer vibrations (which are not included in this model). Their inclusion would make ,f; closer to ,C. If the solid were only IO-100 atoms thick, as occurs for the case of asperity contact, we would therefore conclude that the slippage would be less than the 0.89~ calculated above for 1000 atoms and i,,/r = 0.0038rr. For j_,,/x = 0.158a and only 100 atoms thick. the slippage should be more like l.6u,
’cos( co,, t)1 (12)
where w,, is the n th pole of eqn. ( I I ). (The higher-order poles have the form w,, z (2n + l)rc;‘.) In the thick solid limit, ix. large N or small ;I, the sum in eqn. ( 12) can be replaced by an integral of the form
0
where f = i,,/2;lN = i,,/r’!‘. This integral gives the e -” term in eqn. (8). For longer times, if N is kept finite, the continuum approximation will not be valid and the resulting discrete sum will give for the summation in eqn. (12) a term which oscillates between both positive and negative values, which indicates that LIP will pass into the next parabolic well, which occurs at 11,~= u, implying that there will be slippage past the first well. How far the slippage goes cannot be determined easily by this analytic solution and at this point a numerical calculation would be more appropriate. The reason that this mode1 allows slips of more than one period whereas the triangular potential did not is that here there will be a force from the potential on the lower layer during the slip, whereas for the triangular potential which produces a constant force, exactly cancelled by the applied stress, there is not. Both of the two models described above were also tested by computer simulations (ix. the equations of motion were solved numerically). Identical results were found for the triangular potential. For a periodic array of parabolic potentials, if the force constant of the parabolic potentials were taken to be equal to the interlayer force constant for 1000 layers, .Y,\. was found to increase to a value equal to about
I”“l”“I’~“l’~“I”20 -
15 -
;
lo-
5-
0
0
“‘,“““,,“‘,“““,“‘I 1000
2000
3000
4000
5000
t (‘0
Fig. 3. The position .s, of the ,zth layer (in units of the period of the potential) period.
as a function
i.c,. x”~).
of time (in units of an interlayer
vibrational
212
J. B. Sokolo#‘/
as compared with the 16~ reported ness of 1000 atoms.
above
F~Yctiotl cd
for a thick-
crtomic lrcd “stick
References
1
Acknowledgment 3 I would like to thank for their support during carried out.
the Office of Naval Research the time that this work was
slip”
4
W. Zhong and D. Tomanek, PIz~~.v. Rec. h/t.. 64 ( 1990) 3054. C. M. Mate, G. M. McClelland. R. Erlandsson and S. Chiang. P/z_w Rec. Lrtt., 59 ( 1987) 1942. J. B. Sokoloff, P/!I..F. Rw. B.31 ( 1985)2270: Suf. Sci.. 144 (1984) 267: unpublished, 1991. A. A. Abrikosov. L. P. Gorkov and 1. E. Dzyaloshinski. 1I4prkorl.v of’ Qutmfurn Firld T/wo~,t UT Stcrtisriurl Ph~xic~.v. Prentice-Hall. Englewood Cliffs, NJ, 1963, p. 178.
static and kinetic
friction
and atomic
level
Abstract It
is shown
velcoity “stick
that
the fact that
approaches slip”
motion
the force
of static
7ero) for solid-on-solid that
occurs
during
sliding
the sliding
Friction
is larger
of two elastic
I. Introduction In recent theoretical work [I] it was proposed that the atomic level kinetic friction ,fl, observed in atomic force microscopy (AFM) experiments [2] is equal to the mean interfacial force, for a monolayer of palladium sliding on a graphite substrate. This is based on the assumption that, at low speed sliding. the net work done against the interfacial force to move a surface atom to a potential maximum is all dissipated and therefore is equal to the work done by the force of kinetic friction. In this article. it is shown that, unlike the monolayer case considered in ref. I, for a multilayer solid (a more appropriate model for modeling the AFM experiments [2]) sliding on a similar substrate, one must study the strain needed to overcome the force of static frictionfl, all or part of which is released in the slip part of the “stick-slip” motion which occurs at low speed sliding. This model accounts naturally for the accepted notion that,/, >,fi. It is interesting to note that other work by the present author [3] shows that, for any three-dimensional solid sliding system, ,fk would approach zero in the limit as the sliding velocity approaches zero if the motion in this limit were smooth rather than stick-slip in nature, implying that such non-smooth motion at low speeds is essential for having ,I, approach a non-zero value in the limit of zero sliding velocity.
2. Stick-slip
motion and kinetic friction
Consider the following model. Two macroscopic elastic bodies in contact slide relative to each other, one on top of the other, and the surface of each body at the interface moves in a potential due to the other body. In order for sliding to take place, a critical force,f; must be
0040-6090/9
I /S3.50
than
can be accounted
the force
of kinetic
for by considering
bodies
relative
friction
the strain
(in the limit released during
as the siding atomic
level
to each other.
applied to the upper surface of the upper body (the lower surface of the lower body is assumed to be held in place) sufficiently large to overcome the maximum force due to the potential of interaction at the interface between the bodies. Since the interaction force between the bodies is position dependent, the motion in the slow speed limit will be stick-slip in nature because strain will be built up as the upper surface of the upper body is pushed until the resulting stress is able to overcome the maximum pinning force due to the lower body. At this point the upper body slips forward suddenly. A fraction of the built-up strain in the bodies is released in each slip. If this fraction is very small, the force that must be applied to the upper surface to maintain the motion ,fl, will remain nearly equal to j; because the strain determining the force on the upper surface remains nearly constant, but, if it is large, as the strain is released, ,/1, will drop well below ,f\ between slips, making the average ,fi smaller. (The sliding speed is assumed to be sufficiently slow to allow this energy to spread throughout the system, if the system is sufficiently large, or to be released into space.) This correlation between the relative sizes of,f( andj, and the strain released in each slip should be able to be tested experimentally. Although in the remainder of this article we shall consider the simple case of a homogeneous crystal which is commensurate with the potential, the result described in this paragraph, however, that ,f; <,f, is related to atomic level stick-slip motion is quite general and would apply to inhomogeneous crystals as well as crystals which are incommensurate with the potential. The one difference would be that the abrupt slips described above that occur during low speed sliding would now be local, rather than involving the crystal as a whole. Thus. this atomic level stick-slip motion for a large inhomogeneous system might average out on a microscopic scale, and therefore, not be observed as
c
1991
Elsevier
Sequoia.
Lausanne
J. B. Sokolqf
1 Friction
untl utomic
stick-slip motion of the system as a whole. This case will be studied in a future publication. For relatively small surfaces (i.e. with dimensions not too large compared with the interatomic spacing), however, we would expect that the atomic level stick-slip motion would not average out. The present model includes only stick-slip motion due to potential energy sticking occurring at a microscopic level.
3. Models for the slip motion In order to calculate the strain released in one consider the following equations of motion, which resents the motion of one of the sliding solids periodic potential (assumed to be caused by the ond) : .Y,,= - c((2.X,,- s,, _ , - s,, + , ) - ii ,,.,J(_Y/.$)
slip, repin a sec-
(1)
with n running from 0 to N - 1, wheref(x,) is a force of period a (the lattice constant), x,, is the displacement of the nth layer of atoms (each layer will move rigidly if the periodic force is commensurate with the crystal plane) and c( is the interlayer force constant. The model is illustrated in Fig. 1. Equation (1) is subject to the boundary condmon that xN = .xNi , c-uN+, is introduced formally in order to allow us to apply eqn. ( 1) to the Nth atomic layer. but there really is no N + 1 layer. This boundary condition eliminates the harmonic interaction between levels N and N + 1.) The mass has been absorbed into a, and the coordinate x,, is set equal to a value which creates a stress in the system just large enough to overcome the forcef(.u,) in order to study the stick-slip motion. All the other x are at the equilibrium values corresponding to this value of _Y”. (In principle, this model can be applied to any size layers, from one atom up to an infinite number of atoms, although all layers are assumed to be of equal size. As such, it can in principle be applied to asperities which are only a few atoms wide as well as to infinitely wide flat surfaces). In the motion that follows, the lowest plane moves one or more periods of the periodic force
209
level “srick -slip”
before becoming stuck, as all the translational motion of the system is converted into vibrational. The assumption is that between slips the vibrational energy given up to the system either spreads out throughout the entire system or is dissipated to the environment. (The interaction with the environment was not included in the present model.) We shall see that how far the lower plane slides depends on the form off: For example, let us consider two models which can be solved exactly. In one model, the “saw-tooth potential model”, f is equal to a constant f. for .xN between 0 and u. When s, passes the value a (i.e. reaches the end of the sawtooth), each atom in the lower layer is given a pulse of kinetic energy OSmz;’ = j& where m is the atomic mass as a result of dropping a lower potential energy. (This energy is the potential energy that was built up as the atomic level climbed the saw-tooth and which is now given back as it slides off the end.) This force repeats over each succeeding distance of length U. In order to solve this model, let us set x,, equal to x,,O + u,,, where x ’ is the equilibrium value of x,,, which can be done bicause the values of .Y” and X, have been chosen so that the resulting stress is just balanced by the force of the saw-tooth potential. In order to solve for the motion which occurs after the lower layer falls off the edge of the saw-tooth potential, let us take the Laplace transform of eqn. (1). We have s’u;, + x(2ui, -
u;,_, -u;,+
where ui, is the Laplace
u:,=
,) = ti,y(t = 0)6,, N transform
of u,,, defined
(2) by
J
u,, em.” dt
0
and s is the Laplace (2) we have
transform
variable.
Solving
(3) where
N is the
number of layers, m is an integer, + 1) and (c)Oz(q,,l)=2c([ 1 -cos(q,,,)]. In solving this equation, we have fixed u0 at the value 0, and this condition is satisfied for the above solution. The above choice of q,,, guarantees that uN is always condition. Taking equal to uN+, , the second boundary the inverse Laplace transform of eqn. (3) gives q,,, = (2m + l)n/(2N
un: =
N-'C,(O) 1
sin’(q,,,N)sin[~dq,,,)tl (00(q,,,)
I),
z (4x) -‘C,,,(O)
dq sin[to,(q)t]/tu,(q) s
,v ti,v(h/c)(47r) Fig. I. The model is shown. Dots connected by straight lines represent a rigid layers of atoms, and curly lines represent harmonic forces.
eqn.
’
dx sin X/.X = (8 &)
_ ‘ti,,,(h/c)
210
.I. B. Sokolqf
/ Friction
a constant in the long time limit, because in this limit the integral over q is dominated by small q where w,(q) z (c/b)q, where c is the speed of sound perpendicular to the surface and b is the lattice constant perpendicular to the surface. This approximation was used in eqn. (4). Thus, as long as the velocity given by the saw-tooth potential is small compared with the speed of sound, the lowest atomic layer slips less than one period of the potential, and thus is not able to release all the strain. Let us consider a model in whichf(x,,,) in eqn. (1) is a periodic array of parabolic potentials. Then, it is given by ,f(X/v) = -&)[x,
-(I
+ 0.5)a]
for (I- 0.5)~ < xN < (1+ 0.5)~ where I is an integer and i,, is the strength of the force. As we did for the saw-tooth potential, we set x,’ =x,,” + u,, with xN chosen to be at the maximum of the left-hand side of one of the parabolic potentials so that the force on it from the potential is towards the right and the x,,” are chosen so that the layer N - 1 exerts a force equal to 0.5&a also towards the right. The reason for this choice is that this is the force necessary to put the system in equilibrium if the Nth layer were near the maximum of the parabolic potential to the left of the present potential. We are assuming that the force on layer N was just barely large enough to push it into the potential well in which it at present finds itself, causing it to undergo a slip, which we will now study to determine how far it will slip before it dissipates all its kinetic energy. With the above substitution for x,,, we obtain for the Laplace-transformed equation of motion s’u;, +‘Y(2U;, -u;,_, whose
solution
u;, = (2N + 1)
-uI,+,)
= -6,,.,~&(u:,-a/s)
level “stick
-slip”
where 6 is chosen to be to the right of any singularities in the complex s plane. Since we are interested in the maximum distance that the lower layer of atoms can displace, we want to calculate u,, in the large t limit. In this limit, eqn. (7) is dominated by the contribution for small s (i.e. IsI2 < 4~). Thus, to a good approximation, we can replace (s2 + 4~) ‘12 by (4~) “‘. Then the integral in eqn. (7) gives am
=a{1
-exp[
-(&/cl”‘)t])
(8)
which implies that the lower layer displaces only one lattice constant (i.e. one period of the periodic potential) before being stopped, implying that it just falls short of getting out of the first potential, and thus will not continue to slip. The approximation made in eqn. (6b) of replacing the sum over m by an integral is valid only if the system is sufficiently thick so that t ~’ is always much larger than the spacing between successive values of w”(q,,,). This condition will already begin to fail for times longer than about 10 _ 5 s for the force constant z”’ = 10” (a typical lattice vibrational frequency) and N z 10’. For longer times, the discreteness of the phonon spectrum must be taken into account. In order to accomplish this, let us first replace wo(q,,?) by its small q,,, approxito the approximamation, x”2q,,l, which is equivalent tion of replacing (s2 + 4a) ‘I2 by (4~) ‘P made in eqn. (7) in order to obtain eqn. (8). The sum over m in eqn. (6b) may be expressed as the following contour integral [4] G =(2yN))’
dw tanh( o/2y)(s’
- cu’)
’
(9)
where j’ = (2c() ‘j2(2N + 1) - ’ and where the contour is shown in Fig. 2 (since the hyperbolic tangent has poles at o/2;’ = i(2m + 1)7r/2). This is shown in ref. 4 to be
is ‘A”(&
- u/s)
w2
c
(64 0
I)?
where qJPzand o,,(q,,,) are given under G=N-’
(5)
unri atomic
,,Z
eqn. (3). We use
sin*(q,,,N) c 1>1.G + w”2(q,ll)
z(27c)
’
dq [s2 + o,‘(q,,,)]
’
s = [OSs(S
’
+ 4c() “‘I
(6b)
where we have used the fact that, in the large N limit, sin2(q,,, N) z 1. Then, the solution for u,,,(t) is
ds e”‘{s[s(s2+4c() “2 -t 2&]} - ’
UN(t) = jt”Q(27Ci) - ’ s h
I
-7
(7)
Fig. 2. The contour
in the complex
w plane used in eqn. (9) is shown.
equivalent
to
d(c) tanh( to/2;‘)s - ‘[ cS(.s- W) - fi( w + s)]
= ( 2;,N.s) Substituting u:,,
=
unio
’ tanh(s/2:!)
(10)
in eqn. (6) we obtain tanh(s/2;,)
(.s[2;*N.s + rri,, tanh(s/2;,)])
’ (11)
which can easily be shown to have poles at s = 0 and at s = is, (.s? positive and real) which is a solution to the equation 2;‘N.s, + rri,, tan(s,/2;,) = 0. Using these poles to perform the contour integration for finding the inverse Laplace transform, as in eqn. (7) one obtains u,(t)
= u
r ,I
I - 1 4;q sin( (I),,/;*)(!I,,
twice s,, and then to swing back. For a parabolic potential force constant equal to 0.024 of the interlayer force constant (which would give a value of .f; comparable with the Peierls force for a material containing dislocations), .vV only moved about one period of the potential. Let us now consider the slip distance for a more realistic model of the interface potential. Zhong and Tomanek [I] have shown that the appropriate potential for palladium on graphite is to a good approximation a sinusoidal potential. Thus let us now study the mode1 of eqn. ( 1) with such a potential. Consider ,f’(s,,) = i.,,cos[( 27[/(1).v,v] with j.,,/x = 0.158~ (corresponding to equal intersurface and interatomic layer force constants and comparable with the value of i.,,/sr used in ref. I). For N = 1000, the lower layer was found to slide by about 16lr, implying that all the strain is released in the slip and that ,fi z 0.5f;. whereas. for ;.,,/x = 0.003&r, it slid by only 0.89a. For the latter value of i,,/r, and for N = 5000 and N = 10000, the lower level slid by about 5a and IOtr respectively, implying that about l/4 of the built-up strain is released in each slip. This implies an average ,f; which is about 88% of ,f;. In Fig. 3, .vv is plotted as a function of time for the case of equal intersurface and interlayer force constants considered above. As can be seen, .Y,\, overshoots the value of _Q, and then settles back to a value near x~,. The slipping distances when there are defects in the crystal planes which are in contact will probably be smaller than these values because of the excitation of intralayer vibrations (which are not included in this model). Their inclusion would make ,f; closer to ,C. If the solid were only IO-100 atoms thick, as occurs for the case of asperity contact, we would therefore conclude that the slippage would be less than the 0.89~ calculated above for 1000 atoms and i,,/r = 0.0038rr. For j_,,/x = 0.158a and only 100 atoms thick. the slippage should be more like l.6u,
’cos( co,, t)1 (12)
where w,, is the n th pole of eqn. ( I I ). (The higher-order poles have the form w,, z (2n + l)rc;‘.) In the thick solid limit, ix. large N or small ;I, the sum in eqn. ( 12) can be replaced by an integral of the form
0
where f = i,,/2;lN = i,,/r’!‘. This integral gives the e -” term in eqn. (8). For longer times, if N is kept finite, the continuum approximation will not be valid and the resulting discrete sum will give for the summation in eqn. (12) a term which oscillates between both positive and negative values, which indicates that LIP will pass into the next parabolic well, which occurs at 11,~= u, implying that there will be slippage past the first well. How far the slippage goes cannot be determined easily by this analytic solution and at this point a numerical calculation would be more appropriate. The reason that this mode1 allows slips of more than one period whereas the triangular potential did not is that here there will be a force from the potential on the lower layer during the slip, whereas for the triangular potential which produces a constant force, exactly cancelled by the applied stress, there is not. Both of the two models described above were also tested by computer simulations (ix. the equations of motion were solved numerically). Identical results were found for the triangular potential. For a periodic array of parabolic potentials, if the force constant of the parabolic potentials were taken to be equal to the interlayer force constant for 1000 layers, .Y,\. was found to increase to a value equal to about
I”“l”“I’~“l’~“I”20 -
15 -
;
lo-
5-
0
0
“‘,“““,,“‘,“““,“‘I 1000
2000
3000
4000
5000
t (‘0
Fig. 3. The position .s, of the ,zth layer (in units of the period of the potential) period.
as a function
i.c,. x”~).
of time (in units of an interlayer
vibrational
212
J. B. Sokolo#‘/
as compared with the 16~ reported ness of 1000 atoms.
above
F~Yctiotl cd
for a thick-
crtomic lrcd “stick
References
1
Acknowledgment 3 I would like to thank for their support during carried out.
the Office of Naval Research the time that this work was
slip”
4
W. Zhong and D. Tomanek, PIz~~.v. Rec. h/t.. 64 ( 1990) 3054. C. M. Mate, G. M. McClelland. R. Erlandsson and S. Chiang. P/z_w Rec. Lrtt., 59 ( 1987) 1942. J. B. Sokoloff, P/!I..F. Rw. B.31 ( 1985)2270: Suf. Sci.. 144 (1984) 267: unpublished, 1991. A. A. Abrikosov. L. P. Gorkov and 1. E. Dzyaloshinski. 1I4prkorl.v of’ Qutmfurn Firld T/wo~,t UT Stcrtisriurl Ph~xic~.v. Prentice-Hall. Englewood Cliffs, NJ, 1963, p. 178.