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In-situ measurement of the visco-elastic properties of a sliding lubricated contact
The Third Body Concept / D. Dowson et al. (Editors) 0 1996 Elsevier Science B.V. All rights reserved.
185
In-situ measurement of the visco-elastic properties of a sliding lubricated contact A. Tonck, D. Mazuyer, J.-M. Georges
Laboratoire de Tribologie et Dynamique des SystBmes, URA CNRS 855, Ecole Centrale de Lyons B.P. 163, F-69131 Ecully Cedex, France In this paper, a new type of experiment is presented which was enable by the recent development of the surface force apparatus and of the molecular tribometer used in our Laboratory. In a single test, we obtain the pertinent parameters involved in boundary lubrication and about third body behaviour. To the now classical stage of characterisation using surface force procedure (bulk viscosity measurements, thickness of immobile layer, adhesion and confined layer under load), is added a frictional test with continuous measurement of the viscoelastic properties of the interface layer under shear. This in-situ characterisation is made in both the normal and the tangential directions using very low superimposed vibrations and gives both the compressive and shear modulus. The appropriate choice of the amplitudes and frequencies used allows these extra measurements to be performed without any perturbance towards the friction process and also with a good accuracy. Thanks to this procedure, i t is now possible t o know precisely the rheological behaviour of the interface during the friction process and not only after the sliding, where the interface can be modified by molecular relaxation process of the lubricant. The method is applied to a sphereplane contact lubricated with self-assembled stearic acid monolayers whose thickness remains almost constant during the loading. 1. INTRODUCTION
To understand t h e mechanism of lubrication in a boundary regime, the lubricants a r e generally tested on tribometers measuring the evolution of the frictional force as a function of various external parameters such as sliding speed, normal load, temperature, that can be easily obtainedll-31. Yet, t h e friction force is governed, in t he absence of ploughing friction due t o the solid asperities, by adhesive bonding and hence from the mechanical shear strength of the sliding interface 141. Furthermore, a molecularly thin film is present in the interface whose shearing behaviour and structure control the sliding process that can give viscous or
friction forces [51. This behaviour strongly depends on the variations during the sliding, of internal properties of the film such as thickness, its viscosity, its elastic shear modulus [61. So, the knowledge of the evolution of these properties under shearing is necessary for a better understanding of the dissipative process i n boundary lubrication. For example, in a sphere-plane contact lubricated with two monolayers of stearic acid which is the lubricant also used in the present work, it has been shown [61 that the shear process seems to be related to : 1-the elastic shear modulus of the bilayer, 2- the limit of the shear elastic deformation of the bi-layer before sliding characterised by a critical length X*,
186
3- small variations of the thickness of the bi-layer. Variations of the order of O.Olnm are observed at each changes of the state of the film : beginning of sliding, changes of speed, stops, reverse sliding (dilatance phenomena observed when the speed is lower than about lOnm/s). These variations are interpreted as to be due to the interdigitation processes between the two mono 1ay er s . After sliding, the interface thickness is also modified by a molecular relaxation process of the lubricant. The great problem then encountered is that the rheological behaviours of the interface (point 1 and 2) are measured only before or after the friction test, whereas they change during friction, like the thickness does. The precise measurement of these internal parameters, at a molecular scale and under shear, leads t o two main difficulties. The first problem arises from a
structural effect : the compliance of the interface is very often much lower than the compliance of the solids or of the experimental device used, so the elastic behaviour is dominated by these external parts. The second is related t o the accuracy needed for the measurement of very low compliances in a linear regime response. To minimise the structural effect, the only possible way is a reduction of the ratio 2db (diameter of the contact (2aYthickness of the film B) either using of a very low radius as with AFM technique [7-91 or using a low normal load, with an apparatus especially designed to measure such compliances. Our device, first used as surface force apparatus and then as molecular tribometer [61, has been developed to work at low loads, as a surface viscoelasticimeter in order to obtain valid i n - s i t u rheological measurements during all the friction test.
Figure 1. Schematic diagram of the molecular tribometer derived from Surface Force Apparatus
187 2. EXPERIMENTAL DEVICE
Nanofriction experiments and in-situ viscoelastic characterisations a r e performed with a three-axial instrument derived from a Surface Force Apparatus, developed in our laboratory [SI. Figure 1. shows a schematic diagram of the apparatus. Three piezoelectric elements are used to move the sphere in the three directions x, y and z. They make possible a maximum displacement of 10mm when they are supplied with an high voltage of 300V.Three sensitive capacitive transducers measure the relative displacements of the samples holders in the three directions. These three transducers and the very low compliance of the samples as well as their mountings (lower than 2.10-7 m/N) allow the relative displacements to be measured without any further displacements. Bending due to load frame compliances is negligible. The resolutions reached are better than 0. lnm. Two force transducers specially built and based on the capacitive measurement of elastic bending of two double cantilevers are used t o measure the normal and tangential forces : Fz and Fx. They have a high resolution in spite of very low compliances, 10-8 N and 25.10m6m / N respectively. Three closed loops are used to feed the high voltage amplifiers via PI controllers and then supply the three piezoelectric elements. Two displacement closed loops control the tangential displacements x and y, while the operation in the normal direction z can be selected either in displacement or normal force control. The standard set-up used in the tests obviously includes the continuous quasistatic measurements of the displacements and the resulting normal and tangential forces. But it also includes the simultaneous measurements of t h e rheological behaviours of the contact. In order to do that, small sinusoidal motions can be added to normal and tangential displacements.
Extremely light dynamic motions such as 0.1 nm can be used. The resulting displacements and forces are measured using double phase synchronous analysers which give the in-phase and out of phase signals of both the normal and tangential mechanical transfer functions of the contact between the sphere and the sample. The out of phase signals are related to the dissipative phenomena a s viscous or friction effect and the in-phase signals are related t o the conservative or elastic contribution. For dynamical measurements, the frequency response of the complete measurement chain is carefully recorded as the apparatus transfer functions. These transfer functions are then used to calculate from the measured dynamical forces and displacements, the behaviour of the interface or of the contact itself. We are thus able to obtain a rheological analysis in the two axis, on a frequency range of 0.01 up to 500Hz. The normal behaviour either for tip or sphere-plane indentation is obviously a non-linear process, first by the about square law for tip behaviour or Hertzian law for sphere-plane behaviour of the repulsive force. Thanks to the low level of dynamical motion, the behaviour can be considered as locally linear. The surfaces used in this work are cobalt coatings on fused boro-silicate glass for the sphere (radius R = 2.45 mm) and on silicon wafer for the plane. This cobalt layer is deposited under a low argon pressure using cathodic sputtering. Before deposition, the chamber is pumped for 8 hours at a pressure of Pa. Atomic force microscopy observations show that the surfaces consist of irregular connected clusters leading to a corrugated "blackberry"-like roughness (0.8 nm peak to valley with a wave length of 70 nm). A droplet of 1 m M solution of stearic acid (Aldrich) in anhydrous n-dodecane (pure grade from Aldrich Sure Seal : grade 99 96) is deposited between the two cobalt surfaces. This droplet forms a meniscus of radius r = 1-2 mm in the sphere/plane interface. A monolayer of stearic acid is
188 obtained after one hour adsorption 110-113. All the experiments were performed in dry air at a temperature of 24.3 f 0.2 "C.
3.EXPERIMENTAL PROCEDURE The aim is t o obtain, in a single friction test, rheological information about the contact on a wide range of normal loads (from 1 pN to 10 mN), including in-situ normal and tangential viscoelastic characterisation. The experiment is carried out by making a normal approach of the sphere t o the plane while a tangential displacement is simultaneously applied between the two solids (figure 2.).
t
x
Z
fhPBa
The evolution of the frictional force can then be continuously measured as function of the normal load, The speed of the motion of the sphere along the Z direction is sufficiently slow to consider that the friction of the interface is in its steady state phase. The superimposed vibrating motions both in the X and 2 directions are also used to characterise the surface stearic acid layer before contact and then to perform the insitu viscoelastic measurements of the interface under shearing. The simultaneous t a n g e n t i a l stiffness measurement of a sliding contact is not obvious and precautions must be taken t o insure the validity of the rheological measurement. In order to successfully do that, we have to carefully take into account some problems listed below, that will lead to the choice of the range of the suited amplitudes and frequencies. Tangential visco-elastic characterhtion d a static contact Even for a static contact, the measurement of the tangential visco-elastic properties may be af'fected by the sliding or the micro-sliding of the contact, resulting of the dynamical tangential sollicitation. This well known and already studied phenomenon [121 leads t o the formation of hysteresis cycles in the tangential forcedisplacement curves. The effect on the measured transfer function is a reduction of the visible measured stiffness and the occurrence of a frictional dissipative part which may be mistaken as a viscous effect. In order to avoid this problem, the only possible way is the use of a very low tangential amplitude, t o stay in a linear regime. At low loads, classical mechanical behaviour of a contact between two solids [121 predicts extremely low tangential sollicitation. But fortunately, at this scale of load, the behaviour appears t o be often dominated by t h e interface layers themselves, for which we have already shown that the sliding occurs only after a critical distance X* having a molecular 3.1.
g
stearic acid molecule
dodecane molecule
Figure 2. Schematic description of the experimental procedure of the friction test. The sphere is simultaneously displaced in the directions X and Z at different speed (Vx=O.l nm/s and Vz=O.Ol nm/s). A sinusoidal vibration with different a m p l i t u d e s a n d frequencies is superimposed t o the continuous motion along these 2 directions. Then, tangential and normal forces, tangential and normal stiffnesses can be continuously measured during the experiment.
189
scale (figure 3.). This distance is almost independent of the normal load and the tangential sollicitation selected is taken t o be its tenth part. The estimation of the value of X* 0.3 nm) in a previous work [61 leads to an amplitude of 0.03nm. (5
Fx
f x* (I
FXP
X
Figure 3. Characteristic evolution of the frictional force with the sliding distance when the stearic acid bi-layer is sheared. After an linear period characterised by an elastic stiffness Kx, the force Fx becomes non-linear and reaches a limiting value Fxc. The transition between these 2 periods occurs a t the critical distance X*=Fxe/Kx. The superimposed vibration (amplitude Ax) to the sliding motion causes small linear returns of the friction force with a slope Kx. This stiffness is continuously measured during sliding thanks t o the appropriate amplitude Ax and frequency Rx.
3.2. Friction and superimposed viscoelastic characterisation In addition to the previous points, the intrinsic non linear behaviour of a sliding
contact appears clearly in the case of friction experiment superimposed with tangential visco-elastic characterisation and must be well considered. The dynamic displacement of amplitude A x and frequency Rx results in little cycles such as those shown in figure 3., assuming there is no significant speed effect and that the friction behaviour is well described by a
tangential stiffness Kx (elastic reversible part) and a plateau (non-linear part)(figure 3.). These highly non linear responses have needed a peculiar analysis based on a first harmonic method that has been already used for nano-indentation experiment (case of superimposed elastic measurements) 1131 and extended to a friction test. This method exhibits a criterion t h a t ensures an accuracy on the viscoelastic measurements better than 1% when the following relation is checked :
where Vx is the tangential speed. This method shows the non-linearity causes a phase of 0.36" that does not affect the measurement of the damping of the film. According to relation (l),it is seen that the higher the frequency is, the higher the average tangential speed might be. So the maximum speed that can be applied for a frequency of 220 Hz and an amplitude of 0.03 nm is 0.7 nm/s. This value could appear very low for a conventional friction test, but preceding tribology experiments at a molecular scale, show that the physical phenomena in the sliding process occur for this order of magnitude of speed and even lower [61. In a liquid medium, when the dynamic behaviour of the interface is measured under a normal solicitation, the in-phase component of the transfer function is not only due t o the elasticity of the compressed layer but also to the elastic deformation of the solid surfaces. Therefore in order to minimise this contribution and to obtain directly the normal contact stiffness, a low frequency is used for the normal vibrating motion. On the opposite, the use of high frequency is very convenient for the measurement of the tangential stiffness of a squeezed layer, because in a tangential solicitation, no similar flow pressure is created and the measured tangential stiffness represents the real stiffness of the contact.
190
Table 1 Experimental conditions used for the quasi-static motion and the superimposed vibrating solicitations both in the normal and tangential directions
Quasi-static speed (rids) Vibration frequency (Hz) Vibration amplitude (nm)
Normal motion
Tangential motion
0.01 (VZ)
0.1 (VX) 220 (ax) 0.03 (Ax)
37 ( n z )
0.1 (Az)
The experimental conditions that have been chosen, after accounting for the main sources of disturbance, caused by dynamic measurements both in the normal and tangential directions, on the friction process are given in table 1 . 4. RESULTEl AND DISCUSSION
Before the friction experiment, the lubricant is squeezed in a normal approach in order to measure the evolution of the static force versus displacement and the transfer function of the contact (with the superimposed sinusoidal vibration). The difference between the measured inverse damping function and the measured inverse of the derivative of the contact capacitance [141 show that the thickness of the hydrodynamic layer is 4.9 nm (twice the length of a stearic acid molecule). The attractive part of the normal force is very small (adhesion force : -2.5 pN) is well described by a Van der Waals law whose value suggest that the contact between the layers is made through methyl groups [61, [151, [161. The evolution of the repulsive normal force versus the sphere/plane distance shows the contact behaves as if the interface were a rigid wall 6 nm thick. These measurements in a simple squeeze experiment mean that the interface can be considered as the contact between one monolayer adsorbed onto each surface. The variations of the average thickness is
deduced from the evolution of the normal force, assuming t h a t the contact is equivalent to a Hertzian sphere/plane contact separated by a rigid layer of thickness D and is given by the relation :
a2 D=Z+R
where a is the Hertzian contact radius and R is the sphere radius. In relation (2), the a2 term - represents the total elastic
R
deformation of the solids according t o Hertz's law. The measurement of the electrical spherdplane contact capacitance during the experiment is also a useful control of the value the thickness D given by relation (2). During a friction experiment in which the loading is increased in a quasi-static way, the normal and tangential forces, Fz and Fx, respectively , the normal and tangential elastic stiffnesses, Kz and Kx, respectively are simultaneously measured versus time, In a classical nanotribology experiment at constant normal load Fzo, the evolution of the tangential force F x versus the sliding distance is characterised by two periods 161: -an linear period described by a tangential stiffness Kx, -a non-linear period, where the tangential force increases until an equilibrium value
Fx~.
191
The ratio Fxt/Kx is a length noted X * (figure 3.) that is characteristic of the transition between the linear and the nonlinear period of friction process and can be viewed as the critical distance above which sliding occurs. Then, the two parameters Kx (elastic parameter) and length X * (plasticity onset) can be used to describe the frictional behaviour of the monolayers. Thanks t o the experimental procedure detailed in this paper, it is possible t o continuously measure the stiffness Kx and the characteristic length during the sliding process by the evolution of the ratio F a x . Therefore, the tangential force Fx can be written as : FX = FX! = KX x X*
(3)
According to the relation (81,the friction coefficient is made of two components :
E
-the first term (-)is a conservative part P given by the average tangential elasticity of the monolayers in contact relative to the pressure, -the second term is a dissipative part described by the molecular length X * adimensionnalized by the thickness of the interface and defined as the minimum sliding distance from which sliding starts.
So, the friction coefficient p is given by :
0.02
(4) L
I
0.015
The previous relation may be regarded in terms of elastic shear modulus/contact pressure rather t h a n in terms of stiffness/normal force using the following relationship :
- KXXB a=----;;xxaz
0
where -d is the mean elastic shear modulus of the interface. So, relation ( 5 ) implies : (6)
therefore,
where is the mean contact pressure. Combining relations (4) and (71,the friction coefficient can be expressed as :
1
2 3 4 NORMAL FORCE Fz ( mN )
5
Figure 4. Continuous evolution of the friction coefficient with normal load . The friction coefficient becomes independent of the load when the two monolayers are in contact (at loads greater than 50 pN) which corresponds t o constant sphere/plane distance of 4.9 nm. Then, the friction coefficient reaches a very low value of 0.0075.
The simultaneous measurements during the sliding, of tangential force Fx, thickness of the interface and tangential
192
stiffness Kx allows us, according t o relations (3) and (71,t o determine these two components. Figure 4. shows the continuous evolution of the friction coefficient as a function of the loading force. This curve exhibits a very low friction coefficient (about 0.0075) that is independent of load in the range 50 pN5000pN (pressure range from 4 MPa t o 5OMPa). In this pressure range, these selfassembled monolayers of stearic acid obey the Amontons law but give only half the friction coefficient than that previously measured on the same contact [61. This difference may be due to an increase of the density of the monolayers in the experiments presented in this work. L
31.5
'
to another evolution of the length X* and of
the ratio
B P
with Fz, as shown in figure 5.
These curves are both derived from tangential stiffness measurement t h a t includes not only the stiffness of the squeezed monolayers but also the stiffness Kc of the Hertzian spherdplane contact [121. This contribution becomes important especially at high loads and the measured stiffness is then given by : (9)
The curves plotted in figure 5. are based on corrected values of the stiffness K x by the relation (9). It is noted that length X * increases with the loading, from 0.06 nm and 0.1 nm. This slight increase in X* is
E
balanced by a simultaneous decrease of = P
from 1to 0.5 leading to a friction coefficient independent of normal load - Fz. The G measured values of X* and = are quite P
1 2 3 4 NORMAL FORCE Fz ( mN )
0
5
Figure 5 . Simultaneous evolutions of the G ratio and the critical length X* versus
P
normal load.
-G P
is experimentally obtained
by the measurements of tangential Stiffness Kx, normal force Fz and distance Band
-
=&xD.X*is P Fz the ratio of tangential force Fx to tangential stiffness Kx. X* increases from 0.05 to 0.1
derived from the relation
nm while
-G P
. I
decreases from 1 to 0.5.
This different molecular structure leads
different from those that have been obtained in other experiments for only three states of loading [61 (in that case, X*=0.5 nm and G -=r=O.l). As for the reduction of the friction P
coefficient, these significant quantitative differences are attributed t o a higher ordering of the monolayers and do not modify the qualitative analysis of our results. These observations show t h a t the intrinsic length X* is not only related to the molecular nature of the lubricant but also to the structure of the molecular layers it forms at the neighbourhood of the surfaces, depending on the interactions lubricantlsurface. These results are confirmed by friction experiments in which the normal force is maintained constant (figure 6.).
193
0 0 v
U
4
-0.5 150 ~.
".l
0
50
TIME (s)
100
Figure 6. Correlated evolutions of the frictional force and tangential stiffness in a friction test at constant load (Fz=500pN), for different sliding speed Vx. When the sliding speed is increased the stiffness decreases while the tangential force slightly decreases (after a small increase). When the sliding motion is stopped the friction force suddenly relaxes to a nonzero value while the stiffness Kx increases on a much longer time. These variations are associated with slight changes (0.5 m) in the thickness of the interface that cannot totally explain the important variation of Kx, especially when the sliding is stopped. This effect might be due t o the interdigitation between the monolayers. In this type of experiment, it is possible to check that the value of the measured tangential stiffness Kx is not far from the value of the static stiffness given by the slope
(%)=o x
= Kx.
As shown in figure 6.,
the evolution of the friction force (for a normal load equal to 500p.N), is associated with variations of both the tangential stiffness and the electrical capacitance. This effect is sensitive to the change in the tangential speed and is particularly exhibited when the sliding is stopped suddenly resulting in a relaxation
experiment : the frictional force drops steeply to a minimum non-zero value in a few seconds whilst tangential stiffness Kx increases by 40% over a longer period. A slight increase in the electrical contact capacitance is simultaneously observed. If we assume that the capacitance variation is only due to a decrease of the interface thickness, this latter would be about 0.0004 nm. This very small value cannot lead to the significant increase of stiffness measured in this experiment. This gap suggests that stiffness Kx is certainly related not to the elasticity of the whole interface but to the behaviour of the shear plane (small interdigitating zone between the sliding monolayers [171). 5. CONCLUSIONS
The original experimental procedure used in this work associated with an appropriate choice of the amplitude and the frequencies of the vibrating motions (limited by microsliding effects and the non-linearity of the sliding process) both in the normal and in the tangential directions make now possible the in-situ measurement of the visco-elastic behaviour of a sliding contact at a molecular level, in a single test. The molecular tribometer is used a s a viscoelasticimeter of surfaces and interfaces that can be used in a spherdplane contact as well as in tip/plane geometry. The preliminary experiments carried out on sliding stearic acid monolayers adsorbed on cobalt surfaces show the interest of measuring the rheological properties of the interface under shearing and not only after or before the friction test and their evolution with the normal load. Thanks t o the measurements of the thickness of the interface, the tangential stiffness and the tangential force during the shearing, i t is possible t o completely describe the friction coefficient. It has been shown t h a t the
194
friction coefficient can then be written as :
G , directly obtained from P Kx is related to the experimental data - x 6 Fz the elastic behaviour of the interface or of a p a r t of the interface. The length X* experimentally obtained by the ratio Fx/Kx is a critical length over which sliding starts and strongly depends on the structure of the layer. The mean thickness D of the interface changes during sliding in the case of these stearic acid boundary layers while X* increases from 0.06 nm to 0.1 nm
where the term
and
7
a decreases from 1 to 0.5 with the load
-c
P leading to a low constant friction coefficient. The variations of may be more important for polymeric lubricant 1181, for example. So, t h e simultaneous evolutions of
D, X* and ti
during the P friction process govern the evolution of the friction coefficient. Last, the amplitudes of the tangential vibrating motion necessary to obtain precisely the dynamic response of the sliding contact have initially been chosen to be much lower than the critical length X* in a ratio 1 t o 10. Actually, in the experiments presented in this paper and for the layer we have characterised, X* is found to be 0.06 nm while the amplitude of the tangential vibration is 0.03 nm (half of X*). These particular experimental conditions do not disturb the measurement of the frictional force nor the tangential stiffness. This means that the value of the amplitude to make precise measurements of the viscoelastic response of an interface under shearing can be high and is only limited by the intrinsic length X*. -e
REFERENCES 1. F. P. Bowden, D. Tabor, The Friction and Lubrication of Solids, Clarendon, Oxford, 1964. 2. R. C. Bowers, W. A. Zisman, J. Applied Physics, 39,12 (1968)5385. 3. B. J. Briscoe, D. Tabor, J. of Adhesion, 9 (1978)145. 4. A. M. Homola, J. N. Israelachvili, M. L. Gee and P. J. Mac Guiggan, Tribology, 111 (1989)675. 5 . H. Yoshizawa, Y. L. Chen, J. N. Israelachvili, J. Chem. Phys., 97 (1993) 4128. 6. J. M. Georges, A. Tonck, D. Mazuyer, Wear, 175 (1994)59. 7. C. Mate, Phys. Rev. Lett., 68 (1992)3323. 8. R.Overney, E. Meyer, J. Frommer, H. J. Giintherodt, M. Fujihira, H. Takano, Y. Gotoh, Langmuir, 10 (1994)1281. 9. S. A. Joyce, R. C. Thomas, J. E. Houston, T. A. Michalske, R. Crooks, Phys. Rev. Lett.. 68. 18 (1992)2790. 10. M.-Jacquet , J: M. Georges, J. Chimie Physique (Paris), 11 (1974)1529. 11. E. Smith, C. A. Alves, J. W. Anderegg, F. Porter a n d M. D. Siperko, Langmuir, 9 (1993). 12. R. D. Mindlin, ASME J. Applied Mechanics, 16 (1949)259. 13. S. Bec, These de doctorat no 92.62,Ecole Centrale de Lyon (1992). 14. J. M. Georges, S. Millot, J. L. Loubet and A. Tonck, J. Chem. Phys., 98, 8 (1993)7345. 15. B. V. Derjaguin, V.M. Muller and Y. P. Toporov, J. of Coll. and Inter. Sci., 53 (1975)314. 16. J. N. Israelachvili, Intermolecular and Surface Forces, 2nd Edition, Academic Press, 1992. 17. J. F.Joanny, Langmuir, 8 (1992)989. 18. J. M. Georges, A.Tonck, D. Mazuyer, J. L. Loubet, E. Georges, Proceedings of the 21st Leeds-Lyon Symposium on Tribology : Lubricants and Lubrication, T. Childs, D. Dowson, C. Taylor, G. Dalmaz (Eds), Elsevier Science Publishers B. V., Amsterdam, 1995.
185
In-situ measurement of the visco-elastic properties of a sliding lubricated contact A. Tonck, D. Mazuyer, J.-M. Georges
Laboratoire de Tribologie et Dynamique des SystBmes, URA CNRS 855, Ecole Centrale de Lyons B.P. 163, F-69131 Ecully Cedex, France In this paper, a new type of experiment is presented which was enable by the recent development of the surface force apparatus and of the molecular tribometer used in our Laboratory. In a single test, we obtain the pertinent parameters involved in boundary lubrication and about third body behaviour. To the now classical stage of characterisation using surface force procedure (bulk viscosity measurements, thickness of immobile layer, adhesion and confined layer under load), is added a frictional test with continuous measurement of the viscoelastic properties of the interface layer under shear. This in-situ characterisation is made in both the normal and the tangential directions using very low superimposed vibrations and gives both the compressive and shear modulus. The appropriate choice of the amplitudes and frequencies used allows these extra measurements to be performed without any perturbance towards the friction process and also with a good accuracy. Thanks to this procedure, i t is now possible t o know precisely the rheological behaviour of the interface during the friction process and not only after the sliding, where the interface can be modified by molecular relaxation process of the lubricant. The method is applied to a sphereplane contact lubricated with self-assembled stearic acid monolayers whose thickness remains almost constant during the loading. 1. INTRODUCTION
To understand t h e mechanism of lubrication in a boundary regime, the lubricants a r e generally tested on tribometers measuring the evolution of the frictional force as a function of various external parameters such as sliding speed, normal load, temperature, that can be easily obtainedll-31. Yet, t h e friction force is governed, in t he absence of ploughing friction due t o the solid asperities, by adhesive bonding and hence from the mechanical shear strength of the sliding interface 141. Furthermore, a molecularly thin film is present in the interface whose shearing behaviour and structure control the sliding process that can give viscous or
friction forces [51. This behaviour strongly depends on the variations during the sliding, of internal properties of the film such as thickness, its viscosity, its elastic shear modulus [61. So, the knowledge of the evolution of these properties under shearing is necessary for a better understanding of the dissipative process i n boundary lubrication. For example, in a sphere-plane contact lubricated with two monolayers of stearic acid which is the lubricant also used in the present work, it has been shown [61 that the shear process seems to be related to : 1-the elastic shear modulus of the bilayer, 2- the limit of the shear elastic deformation of the bi-layer before sliding characterised by a critical length X*,
186
3- small variations of the thickness of the bi-layer. Variations of the order of O.Olnm are observed at each changes of the state of the film : beginning of sliding, changes of speed, stops, reverse sliding (dilatance phenomena observed when the speed is lower than about lOnm/s). These variations are interpreted as to be due to the interdigitation processes between the two mono 1ay er s . After sliding, the interface thickness is also modified by a molecular relaxation process of the lubricant. The great problem then encountered is that the rheological behaviours of the interface (point 1 and 2) are measured only before or after the friction test, whereas they change during friction, like the thickness does. The precise measurement of these internal parameters, at a molecular scale and under shear, leads t o two main difficulties. The first problem arises from a
structural effect : the compliance of the interface is very often much lower than the compliance of the solids or of the experimental device used, so the elastic behaviour is dominated by these external parts. The second is related t o the accuracy needed for the measurement of very low compliances in a linear regime response. To minimise the structural effect, the only possible way is a reduction of the ratio 2db (diameter of the contact (2aYthickness of the film B) either using of a very low radius as with AFM technique [7-91 or using a low normal load, with an apparatus especially designed to measure such compliances. Our device, first used as surface force apparatus and then as molecular tribometer [61, has been developed to work at low loads, as a surface viscoelasticimeter in order to obtain valid i n - s i t u rheological measurements during all the friction test.
Figure 1. Schematic diagram of the molecular tribometer derived from Surface Force Apparatus
187 2. EXPERIMENTAL DEVICE
Nanofriction experiments and in-situ viscoelastic characterisations a r e performed with a three-axial instrument derived from a Surface Force Apparatus, developed in our laboratory [SI. Figure 1. shows a schematic diagram of the apparatus. Three piezoelectric elements are used to move the sphere in the three directions x, y and z. They make possible a maximum displacement of 10mm when they are supplied with an high voltage of 300V.Three sensitive capacitive transducers measure the relative displacements of the samples holders in the three directions. These three transducers and the very low compliance of the samples as well as their mountings (lower than 2.10-7 m/N) allow the relative displacements to be measured without any further displacements. Bending due to load frame compliances is negligible. The resolutions reached are better than 0. lnm. Two force transducers specially built and based on the capacitive measurement of elastic bending of two double cantilevers are used t o measure the normal and tangential forces : Fz and Fx. They have a high resolution in spite of very low compliances, 10-8 N and 25.10m6m / N respectively. Three closed loops are used to feed the high voltage amplifiers via PI controllers and then supply the three piezoelectric elements. Two displacement closed loops control the tangential displacements x and y, while the operation in the normal direction z can be selected either in displacement or normal force control. The standard set-up used in the tests obviously includes the continuous quasistatic measurements of the displacements and the resulting normal and tangential forces. But it also includes the simultaneous measurements of t h e rheological behaviours of the contact. In order to do that, small sinusoidal motions can be added to normal and tangential displacements.
Extremely light dynamic motions such as 0.1 nm can be used. The resulting displacements and forces are measured using double phase synchronous analysers which give the in-phase and out of phase signals of both the normal and tangential mechanical transfer functions of the contact between the sphere and the sample. The out of phase signals are related to the dissipative phenomena a s viscous or friction effect and the in-phase signals are related t o the conservative or elastic contribution. For dynamical measurements, the frequency response of the complete measurement chain is carefully recorded as the apparatus transfer functions. These transfer functions are then used to calculate from the measured dynamical forces and displacements, the behaviour of the interface or of the contact itself. We are thus able to obtain a rheological analysis in the two axis, on a frequency range of 0.01 up to 500Hz. The normal behaviour either for tip or sphere-plane indentation is obviously a non-linear process, first by the about square law for tip behaviour or Hertzian law for sphere-plane behaviour of the repulsive force. Thanks to the low level of dynamical motion, the behaviour can be considered as locally linear. The surfaces used in this work are cobalt coatings on fused boro-silicate glass for the sphere (radius R = 2.45 mm) and on silicon wafer for the plane. This cobalt layer is deposited under a low argon pressure using cathodic sputtering. Before deposition, the chamber is pumped for 8 hours at a pressure of Pa. Atomic force microscopy observations show that the surfaces consist of irregular connected clusters leading to a corrugated "blackberry"-like roughness (0.8 nm peak to valley with a wave length of 70 nm). A droplet of 1 m M solution of stearic acid (Aldrich) in anhydrous n-dodecane (pure grade from Aldrich Sure Seal : grade 99 96) is deposited between the two cobalt surfaces. This droplet forms a meniscus of radius r = 1-2 mm in the sphere/plane interface. A monolayer of stearic acid is
188 obtained after one hour adsorption 110-113. All the experiments were performed in dry air at a temperature of 24.3 f 0.2 "C.
3.EXPERIMENTAL PROCEDURE The aim is t o obtain, in a single friction test, rheological information about the contact on a wide range of normal loads (from 1 pN to 10 mN), including in-situ normal and tangential viscoelastic characterisation. The experiment is carried out by making a normal approach of the sphere t o the plane while a tangential displacement is simultaneously applied between the two solids (figure 2.).
t
x
Z
fhPBa
The evolution of the frictional force can then be continuously measured as function of the normal load, The speed of the motion of the sphere along the Z direction is sufficiently slow to consider that the friction of the interface is in its steady state phase. The superimposed vibrating motions both in the X and 2 directions are also used to characterise the surface stearic acid layer before contact and then to perform the insitu viscoelastic measurements of the interface under shearing. The simultaneous t a n g e n t i a l stiffness measurement of a sliding contact is not obvious and precautions must be taken t o insure the validity of the rheological measurement. In order to successfully do that, we have to carefully take into account some problems listed below, that will lead to the choice of the range of the suited amplitudes and frequencies. Tangential visco-elastic characterhtion d a static contact Even for a static contact, the measurement of the tangential visco-elastic properties may be af'fected by the sliding or the micro-sliding of the contact, resulting of the dynamical tangential sollicitation. This well known and already studied phenomenon [121 leads t o the formation of hysteresis cycles in the tangential forcedisplacement curves. The effect on the measured transfer function is a reduction of the visible measured stiffness and the occurrence of a frictional dissipative part which may be mistaken as a viscous effect. In order to avoid this problem, the only possible way is the use of a very low tangential amplitude, t o stay in a linear regime. At low loads, classical mechanical behaviour of a contact between two solids [121 predicts extremely low tangential sollicitation. But fortunately, at this scale of load, the behaviour appears t o be often dominated by t h e interface layers themselves, for which we have already shown that the sliding occurs only after a critical distance X* having a molecular 3.1.
g
stearic acid molecule
dodecane molecule
Figure 2. Schematic description of the experimental procedure of the friction test. The sphere is simultaneously displaced in the directions X and Z at different speed (Vx=O.l nm/s and Vz=O.Ol nm/s). A sinusoidal vibration with different a m p l i t u d e s a n d frequencies is superimposed t o the continuous motion along these 2 directions. Then, tangential and normal forces, tangential and normal stiffnesses can be continuously measured during the experiment.
189
scale (figure 3.). This distance is almost independent of the normal load and the tangential sollicitation selected is taken t o be its tenth part. The estimation of the value of X* 0.3 nm) in a previous work [61 leads to an amplitude of 0.03nm. (5
Fx
f x* (I
FXP
X
Figure 3. Characteristic evolution of the frictional force with the sliding distance when the stearic acid bi-layer is sheared. After an linear period characterised by an elastic stiffness Kx, the force Fx becomes non-linear and reaches a limiting value Fxc. The transition between these 2 periods occurs a t the critical distance X*=Fxe/Kx. The superimposed vibration (amplitude Ax) to the sliding motion causes small linear returns of the friction force with a slope Kx. This stiffness is continuously measured during sliding thanks t o the appropriate amplitude Ax and frequency Rx.
3.2. Friction and superimposed viscoelastic characterisation In addition to the previous points, the intrinsic non linear behaviour of a sliding
contact appears clearly in the case of friction experiment superimposed with tangential visco-elastic characterisation and must be well considered. The dynamic displacement of amplitude A x and frequency Rx results in little cycles such as those shown in figure 3., assuming there is no significant speed effect and that the friction behaviour is well described by a
tangential stiffness Kx (elastic reversible part) and a plateau (non-linear part)(figure 3.). These highly non linear responses have needed a peculiar analysis based on a first harmonic method that has been already used for nano-indentation experiment (case of superimposed elastic measurements) 1131 and extended to a friction test. This method exhibits a criterion t h a t ensures an accuracy on the viscoelastic measurements better than 1% when the following relation is checked :
where Vx is the tangential speed. This method shows the non-linearity causes a phase of 0.36" that does not affect the measurement of the damping of the film. According to relation (l),it is seen that the higher the frequency is, the higher the average tangential speed might be. So the maximum speed that can be applied for a frequency of 220 Hz and an amplitude of 0.03 nm is 0.7 nm/s. This value could appear very low for a conventional friction test, but preceding tribology experiments at a molecular scale, show that the physical phenomena in the sliding process occur for this order of magnitude of speed and even lower [61. In a liquid medium, when the dynamic behaviour of the interface is measured under a normal solicitation, the in-phase component of the transfer function is not only due t o the elasticity of the compressed layer but also to the elastic deformation of the solid surfaces. Therefore in order to minimise this contribution and to obtain directly the normal contact stiffness, a low frequency is used for the normal vibrating motion. On the opposite, the use of high frequency is very convenient for the measurement of the tangential stiffness of a squeezed layer, because in a tangential solicitation, no similar flow pressure is created and the measured tangential stiffness represents the real stiffness of the contact.
190
Table 1 Experimental conditions used for the quasi-static motion and the superimposed vibrating solicitations both in the normal and tangential directions
Quasi-static speed (rids) Vibration frequency (Hz) Vibration amplitude (nm)
Normal motion
Tangential motion
0.01 (VZ)
0.1 (VX) 220 (ax) 0.03 (Ax)
37 ( n z )
0.1 (Az)
The experimental conditions that have been chosen, after accounting for the main sources of disturbance, caused by dynamic measurements both in the normal and tangential directions, on the friction process are given in table 1 . 4. RESULTEl AND DISCUSSION
Before the friction experiment, the lubricant is squeezed in a normal approach in order to measure the evolution of the static force versus displacement and the transfer function of the contact (with the superimposed sinusoidal vibration). The difference between the measured inverse damping function and the measured inverse of the derivative of the contact capacitance [141 show that the thickness of the hydrodynamic layer is 4.9 nm (twice the length of a stearic acid molecule). The attractive part of the normal force is very small (adhesion force : -2.5 pN) is well described by a Van der Waals law whose value suggest that the contact between the layers is made through methyl groups [61, [151, [161. The evolution of the repulsive normal force versus the sphere/plane distance shows the contact behaves as if the interface were a rigid wall 6 nm thick. These measurements in a simple squeeze experiment mean that the interface can be considered as the contact between one monolayer adsorbed onto each surface. The variations of the average thickness is
deduced from the evolution of the normal force, assuming t h a t the contact is equivalent to a Hertzian sphere/plane contact separated by a rigid layer of thickness D and is given by the relation :
a2 D=Z+R
where a is the Hertzian contact radius and R is the sphere radius. In relation (2), the a2 term - represents the total elastic
R
deformation of the solids according t o Hertz's law. The measurement of the electrical spherdplane contact capacitance during the experiment is also a useful control of the value the thickness D given by relation (2). During a friction experiment in which the loading is increased in a quasi-static way, the normal and tangential forces, Fz and Fx, respectively , the normal and tangential elastic stiffnesses, Kz and Kx, respectively are simultaneously measured versus time, In a classical nanotribology experiment at constant normal load Fzo, the evolution of the tangential force F x versus the sliding distance is characterised by two periods 161: -an linear period described by a tangential stiffness Kx, -a non-linear period, where the tangential force increases until an equilibrium value
Fx~.
191
The ratio Fxt/Kx is a length noted X * (figure 3.) that is characteristic of the transition between the linear and the nonlinear period of friction process and can be viewed as the critical distance above which sliding occurs. Then, the two parameters Kx (elastic parameter) and length X * (plasticity onset) can be used to describe the frictional behaviour of the monolayers. Thanks t o the experimental procedure detailed in this paper, it is possible t o continuously measure the stiffness Kx and the characteristic length during the sliding process by the evolution of the ratio F a x . Therefore, the tangential force Fx can be written as : FX = FX! = KX x X*
(3)
According to the relation (81,the friction coefficient is made of two components :
E
-the first term (-)is a conservative part P given by the average tangential elasticity of the monolayers in contact relative to the pressure, -the second term is a dissipative part described by the molecular length X * adimensionnalized by the thickness of the interface and defined as the minimum sliding distance from which sliding starts.
So, the friction coefficient p is given by :
0.02
(4) L
I
0.015
The previous relation may be regarded in terms of elastic shear modulus/contact pressure rather t h a n in terms of stiffness/normal force using the following relationship :
- KXXB a=----;;xxaz
0
where -d is the mean elastic shear modulus of the interface. So, relation ( 5 ) implies : (6)
therefore,
where is the mean contact pressure. Combining relations (4) and (71,the friction coefficient can be expressed as :
1
2 3 4 NORMAL FORCE Fz ( mN )
5
Figure 4. Continuous evolution of the friction coefficient with normal load . The friction coefficient becomes independent of the load when the two monolayers are in contact (at loads greater than 50 pN) which corresponds t o constant sphere/plane distance of 4.9 nm. Then, the friction coefficient reaches a very low value of 0.0075.
The simultaneous measurements during the sliding, of tangential force Fx, thickness of the interface and tangential
192
stiffness Kx allows us, according t o relations (3) and (71,t o determine these two components. Figure 4. shows the continuous evolution of the friction coefficient as a function of the loading force. This curve exhibits a very low friction coefficient (about 0.0075) that is independent of load in the range 50 pN5000pN (pressure range from 4 MPa t o 5OMPa). In this pressure range, these selfassembled monolayers of stearic acid obey the Amontons law but give only half the friction coefficient than that previously measured on the same contact [61. This difference may be due to an increase of the density of the monolayers in the experiments presented in this work. L
31.5
'
to another evolution of the length X* and of
the ratio
B P
with Fz, as shown in figure 5.
These curves are both derived from tangential stiffness measurement t h a t includes not only the stiffness of the squeezed monolayers but also the stiffness Kc of the Hertzian spherdplane contact [121. This contribution becomes important especially at high loads and the measured stiffness is then given by : (9)
The curves plotted in figure 5. are based on corrected values of the stiffness K x by the relation (9). It is noted that length X * increases with the loading, from 0.06 nm and 0.1 nm. This slight increase in X* is
E
balanced by a simultaneous decrease of = P
from 1to 0.5 leading to a friction coefficient independent of normal load - Fz. The G measured values of X* and = are quite P
1 2 3 4 NORMAL FORCE Fz ( mN )
0
5
Figure 5 . Simultaneous evolutions of the G ratio and the critical length X* versus
P
normal load.
-G P
is experimentally obtained
by the measurements of tangential Stiffness Kx, normal force Fz and distance Band
-
=&xD.X*is P Fz the ratio of tangential force Fx to tangential stiffness Kx. X* increases from 0.05 to 0.1
derived from the relation
nm while
-G P
. I
decreases from 1 to 0.5.
This different molecular structure leads
different from those that have been obtained in other experiments for only three states of loading [61 (in that case, X*=0.5 nm and G -=r=O.l). As for the reduction of the friction P
coefficient, these significant quantitative differences are attributed t o a higher ordering of the monolayers and do not modify the qualitative analysis of our results. These observations show t h a t the intrinsic length X* is not only related to the molecular nature of the lubricant but also to the structure of the molecular layers it forms at the neighbourhood of the surfaces, depending on the interactions lubricantlsurface. These results are confirmed by friction experiments in which the normal force is maintained constant (figure 6.).
193
0 0 v
U
4
-0.5 150 ~.
".l
0
50
TIME (s)
100
Figure 6. Correlated evolutions of the frictional force and tangential stiffness in a friction test at constant load (Fz=500pN), for different sliding speed Vx. When the sliding speed is increased the stiffness decreases while the tangential force slightly decreases (after a small increase). When the sliding motion is stopped the friction force suddenly relaxes to a nonzero value while the stiffness Kx increases on a much longer time. These variations are associated with slight changes (0.5 m) in the thickness of the interface that cannot totally explain the important variation of Kx, especially when the sliding is stopped. This effect might be due t o the interdigitation between the monolayers. In this type of experiment, it is possible to check that the value of the measured tangential stiffness Kx is not far from the value of the static stiffness given by the slope
(%)=o x
= Kx.
As shown in figure 6.,
the evolution of the friction force (for a normal load equal to 500p.N), is associated with variations of both the tangential stiffness and the electrical capacitance. This effect is sensitive to the change in the tangential speed and is particularly exhibited when the sliding is stopped suddenly resulting in a relaxation
experiment : the frictional force drops steeply to a minimum non-zero value in a few seconds whilst tangential stiffness Kx increases by 40% over a longer period. A slight increase in the electrical contact capacitance is simultaneously observed. If we assume that the capacitance variation is only due to a decrease of the interface thickness, this latter would be about 0.0004 nm. This very small value cannot lead to the significant increase of stiffness measured in this experiment. This gap suggests that stiffness Kx is certainly related not to the elasticity of the whole interface but to the behaviour of the shear plane (small interdigitating zone between the sliding monolayers [171). 5. CONCLUSIONS
The original experimental procedure used in this work associated with an appropriate choice of the amplitude and the frequencies of the vibrating motions (limited by microsliding effects and the non-linearity of the sliding process) both in the normal and in the tangential directions make now possible the in-situ measurement of the visco-elastic behaviour of a sliding contact at a molecular level, in a single test. The molecular tribometer is used a s a viscoelasticimeter of surfaces and interfaces that can be used in a spherdplane contact as well as in tip/plane geometry. The preliminary experiments carried out on sliding stearic acid monolayers adsorbed on cobalt surfaces show the interest of measuring the rheological properties of the interface under shearing and not only after or before the friction test and their evolution with the normal load. Thanks t o the measurements of the thickness of the interface, the tangential stiffness and the tangential force during the shearing, i t is possible t o completely describe the friction coefficient. It has been shown t h a t the
194
friction coefficient can then be written as :
G , directly obtained from P Kx is related to the experimental data - x 6 Fz the elastic behaviour of the interface or of a p a r t of the interface. The length X* experimentally obtained by the ratio Fx/Kx is a critical length over which sliding starts and strongly depends on the structure of the layer. The mean thickness D of the interface changes during sliding in the case of these stearic acid boundary layers while X* increases from 0.06 nm to 0.1 nm
where the term
and
7
a decreases from 1 to 0.5 with the load
-c
P leading to a low constant friction coefficient. The variations of may be more important for polymeric lubricant 1181, for example. So, t h e simultaneous evolutions of
D, X* and ti
during the P friction process govern the evolution of the friction coefficient. Last, the amplitudes of the tangential vibrating motion necessary to obtain precisely the dynamic response of the sliding contact have initially been chosen to be much lower than the critical length X* in a ratio 1 t o 10. Actually, in the experiments presented in this paper and for the layer we have characterised, X* is found to be 0.06 nm while the amplitude of the tangential vibration is 0.03 nm (half of X*). These particular experimental conditions do not disturb the measurement of the frictional force nor the tangential stiffness. This means that the value of the amplitude to make precise measurements of the viscoelastic response of an interface under shearing can be high and is only limited by the intrinsic length X*. -e
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