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Langmuir 1997, 13, 3454-3463
Contact of Two Carbon Surfaces Covered with a Dispersant Polymer E. Georges, J.-M. Georges,*,† and S. Hollinger Laboratoire de Tribologie et Dynamiques des Syste` mes, Ecole Centrale de Lyon, 69131 Ecully, France Received January 21, 1997. In Final Form: April 7, 1997X Using a surface force apparatus (SFA), we have studied, first the adhesive force between two smooth amorphous carbon surfaces in air, and second, the repulsive force due to the contact of two brush polymer layers adsorbed on carbon surfaces. Adhesion in dry air is compared to that due to the effect of a small meniscus formed in humid air and the partial wetting of a large water meniscus. Dispersive and acid-base contributions of carbon surfaces are also determined, using Fowkes analysis. The adsorption of a polyisobutenesuccinimide (PIB) in dilute solution with hydrocarbon solvent at 25 °C, and the resulting interactions between carbon surfaces are also characterised with the SFA. Our results show that this dispersant polymer on a carbon surface forms a 5.5 nm thick anisotropic brush layer. This anisotropy is detected mechanically and compared with that of other polymer or surfactant. The ability of PIB molecules to adsorb on carbon particles and thus stabilize them in the hydrocarbon solution, is of great implication in lubrication technology. The results clarify that this additive function by steric stabilization mechanism and not by electrostatic repulsion The results provide a basis for understanding the mechanism of action of of polyisobutenesuccinimides widely used as dispersant additives in lubricants.
1. Introduction Colloidal stability and aggregation in nonaqueous media of polymer-coated particles are of interest in many technological applications.1 Polymer stabilization is also one of the means to prevent aggregation and sedimentation of particles such as wear particles, or soot, which are essentially carbon particles.2 Polymers having such properties are called dispersants; one example is polyisobutenesuccinimide (PIB).3,4 It is a polyisobutene chain having about 70 carbons and is very soluble in nonaqueous media. The polymer chains are so firmly anchored to particles and extend far enough into the medium that whenever particles collide, they are kept far enough apart. Steric and or electrostatic repulsives forces are invoked to explain this property.4 Moreover, the mechanical properties of surfaces coated with adsorbed polymers control not only phenomena such as colloidal stability but also the tribological properties of solids. Thus, dispersants have been used since the 1950s in lubricant technology.5 Concerning the adsorption, various techniques have been useful in determining the polymer adsorption: the carbon black sedimentation test, micelle formation test, zeta potential measurement, and heat adsorption measurement.3,4,12 More recently, the surface force apparatus (SFA)6 and atomic force microscope have been used to study surfactant adsorption and aggregation at the solidliquid interface.31 Over the last 10 years, surface force apparatus (SFA) have been used to ascertain the steric and electrostatic effects of polymers near solid surfaces. Generally the surfaces are mica or metallic.6-8 We are presenting here the first experiments with carbon surfaces. † X
Member of Institut Universitaire de France. Abstract published in Advance ACS Abstracts, June 1, 1997.
(1) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1992. (2) Pugh, R. J.; Matsunaga, T. Colloids Surf. 1993, 7, 183-207. (3) Forbes, E. S.; Neustadter, E. L. Tribology 1972, 5, 72-77. (4) Fowkes, F. M.; Pugh, R. J. In Colloid Properties of Nonaqueous Dispersion; American Chemical Society: Washington, DC, 1984. (5) Spikes, H. A.; Cann, P. M.; Coy, R. C.; Wardle, R. W. N. Lubr. Sci. 1990, 3, 45-62. Cann, P. M.; Spikes, H. A. STLE/ASME, 93-TC-1B-1, 1993. (6) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd Ed.; Academic Press: London, 1992. (7) Luckham, P.; Klein, J. Macromolecules 1985, 18, 721-728.
S0743-7463(97)00065-6 CCC: $14.00
Therefore the aim of this study is to investigate the carbon surface adhesive force and the repulsive forces of polyisobutenesuccinimide layers adsorbed from a dilute solution. The paper is organized as follows. First, principles of the experiments with the surface force apparatus are described. Then the nature of the materials is given. The carbon surfaces and liquid solutions are described (Section 2). The adhesion and the wetting properties of carbon surfaces are presented (Section 3). The nanorheology results with the brush polymer are presented in Section 4; they include two subsections, one on the interactions and one on the isotropic elasticity of the polymer layer. 2. Experimental Section (a) Surface Force Apparatus (SFA). The Ecole Centrale de Lyon surface force apparatus (SFA) used in these experiments has been described in previous publications.9-11 The general principle is that a macroscopic spherical body can be moved toward and away from, in the three directions 0XYZ, a planar one using the expansion and the vibration of a piezoelectric crystal (Figure 1). A sphere of radius R ) 2.5 mm is firmly fixed to the three axial piezoelectric translator. The plane specimen is supported by double sensors, measuring normal and tangential forces (F,T). Each of these is equipped with a capacitive sensor and a double cantilever spring. The sensor’s high resolution allows a very low compliance to be used for the force measurement (2 × 10-6 m/N). Three capacitive sensors are designed to measure relative displacements in the three directions between the supports of the two solids, with a displacement resolution of 0.01 nm in each direction. Each sensor capacitance is determined by incorporating it in an LC oscillator acting in the range 5-12 MHz.11 Specimens are conductors, and the electrical capacitance C of the sphere-plane interface is also measured; thus the sphere-plane interface with a closest distance, D, is detected (Figure 1). The normal approach of the sphere to the plane is mainly used in this study. The normal speed is a superposition of two (8) Klein, J.; Kumacheva, E.; Mahalu, D.; Perahia, D.; Fetters, L. J. Nature 1994, 370, 634. (9) Tonck, A.; Georges, J. M.; Loubet, J. L. J. of Colloid Interface Sci. 1988, 126 (1),1540-1563. (10) Georges, J. M.; Millot, S.; Loubet, J. L.; Tonck, A. J. Chem. Phys. 1993, 98 (8), 7345-7360. (11) Georges, J.-M.; Tonck, A.; Mazuyer, D.; Georges, E.; Loubet, J.-L.; Sidoroff, F. J. Phys. II, France 1996, 6, 57-76.
© 1997 American Chemical Society
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Langmuir, Vol. 13, No. 13, 1997 3455
Figure 1. (a) Principles of the experiments in normal approach. The sphere (radius R ≈ 2.5 mm) is moved toward (or away) from a plane (OZ direction). As the specimens are conductors, the electrical capacitance, C, of the sphere-plane interface is measured and then the sphere-plane interface’s closest distance, D, is detected. (b) The normal speed is a superposition of two components; one is a steady ramp, which gives a constant approach speed of D˙ ) Z˙ ) 0.2 nm/s, and the other one is a small amplitude oscillatory motion D ˜ of about 0.1 nm RMS, with a pulsation of ω ) 4.2 × 10-3 rad/s. components. The first one is a steady ramp, which gives a constant approach speed of D˙ ) Z˙ ) 0.2 nm/s (Figure 1). The other one is a small amplitude oscillatory motion D ˜ of about 0.1 nm RMS, with a pulsation of ω ) 4.2 × 10-3 rad/s. This combination allows two different kinds of measurements to be simultaneously recorded with the sphere-plane capacitance C: they are the “quasi-static” normal force, Fs, and the mechanical transfer function. From this complex transfer function, only the imaginary part will be presented in this paper. Experiments are performed in an isolated room where the temperature variations are negligible with an experimental duration time of a few hours. The humidity rate is measured. For almost all experiments, humidity is kept under 0.1% by placing the experimental device under a bell jar with the drying agent P2O5. A laminar flow bench is used to reduce dust during the fabrication and manipulation of the surfaces. All these precautions are necessary to ensure the success of the experiments. (b) Carbon Surfaces. Both the sphere and the plane used consist of fused borosilicate glass, whose Poisson’s ratio is 0.22:1 and Young’s modulus is 65 GPa, (glass 732-01, Sovirel Corporation). Each surface is annealed in order to destroy and remove dust, and to obtain a polished substrate. These surfaces are coated with metallic cobalt and then carbon and deposited under low argon pressure (5 × 10-2 Pa), using cathodic sputtering. The cobalt is used as a conductor; for good conduction a 60 nm layer is deposited. The carbon deposit thickness is around 3 nm. Atomic force microscopy and scanning tunneling microscopy examination of the sputtered surfaces show that the surfaces consist of irregularly connected clusters, producing a slightly bumpy corrugation with a “blackberrylike” roughness (peak to valley 1.2 nm, measured with a scan length of 1 µm) (Figure 2a). The corrugation diameter is about 50 nm. The low amplitude of the surface roughness is therefore negligible compared with the thickness of polymer layers considered in this study. X-ray photoelectron spectroscopy gives the composition of the deposited layer, confirming the presence of a metallic cobalt unoxidized layer (Figure 2b). Furthermore a C-C bond (nongraphitic) for the carbon layer is characterized as well as a residual presence of oxygen and argon adsorbed during the deposition and the transfer process.
Figure 2. (a) Atomic force microscopy (AFM) analysis of the sputtered carbon surface. The image size is 1 µm2, and the roughness is 1.2 nm peak to valley measured with a scan length of 1 µm. (b) X-ray photoelectron spectroscopy (XPS) analysis of the sputtered layer of carbon on the cobalt-borosilicate glass substrate. Detection of a C-C nongraphitic bond and residual presence of argon and oxygen. Wetting experiments conducted on carbon surfaces with the SFA are presented later. (c) Solutions Tested. Experiments were carried out with pure solvent and a polymer solution. The solvent is a 175 neutral solvent (175NS) base stock (from Esso Port Je´roˆme, France) whose kinematic viscosities are 33.60 cSt at 40 °C and 5.65 cSt at 100 °C. Density at 15 °C is 0.8707, and the dielectric constant is 2.188 at 20 °C. The molecular weight of 175NS is 416 g/mol, and it is composed of 4.30% of aromatic carbon, 68.85% of paraffinic carbon, and 26.85% of naphthenic carbon. The sulphur level is 0.55% by weight. The polymer dispersant used in this study is a polyisobutenesuccinimide (PIB) (Oloa 1200) made by Lubrizol (USA). Its solubility in nonaqueous media such as 175NS comes from a liquid polyisobutene chain having about 70 carbons. The anchoring group is diethylenetriamine attached to the polyisobutene chain by a succinimide group. Oloa 1200 is used extensively as an automotive crankcase dispersant. It is, in particular, used to control and stabilize soot in engine oils.3,4,12,13 PIB is tested with a 175NS solution containing 2.5% of active polymer. The molecular weight of the polymer is 1200 g/mol. (d) SFA Experimental Procedure. The experimental procedure used in this study is as follows (Figure 3): first, the dry contact between the carbon surfaces is tested. The adhesive force versus the sphere-plane distance is detected. Second, a (12) Pugh, R. J.; Matsunaga, T.; Fowkes, F. M. Colloid Surf. 1983, 7, 183-207. (13) Pugh, R. J.; Fowkes, F. M. Colloids Surf. 1984, 11, 423-427.
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Figure 3. Experimental procedure. First, the dry contact between the carbon surfaces is tested and the adhesive forces versus the sphere-plane distance detected. Second, a drop of solution is placed at the interface to create a meniscus, and the wetting force is measured. Third, the nanorheology of the solution is tested. The simultaneous measurements of static and dynamic forces versus the sphere-plane distance leads principally to two values. First, the thickness of the polymer layer adsorbed on each carbon surface, L, is obtained from the quasi-static force profile. Second, the hydrodynamic thickness, LH, which is the thickness of the adsorbed layer which does not participate in the flow of the bulk solution, is determined from the dynamic forces measurements. drop of solution is placed at the interface to create a meniscus of radii r1 ) 1 mm and r2 ≈ 1 µm. A stabilization time of 1 h is required for the polymer solution. The addition of a solution drop at the interface produces a jump in the quasi-static force. This jump is due to the wetting force, FW, created by the Laplace pressure, which can be written as, for D , r2,6
FW ) 4πRγLV cos θ
(1)
where R is the sphere radius, γLV is the surface tension of the test solution, and θ is the contact angle of the solution with the carbon surface. Relation 1 provides the value of γLV cos θ. The surface tension γLV is measured independently with a Wilhemy balance.14 Due to the small variations of r during the experiments, the surface of the meniscus is constant; therefore there is no variation in the wetting force. Third, the nanorheology of the solution is tested (Appendix, part c).
3. Adhesion and Wetting Properties of Carbon Surfaces Physico chemical properties of carbon surfaces are tested with SFA experiments. (a) Carbon Surfaces in Dry Air. Figure 4a shows the variations of the quasi-static force/radius, Fs/R, measurements during downward and upward displacement, Z. This distance, Z, is equal to the sphere-plane (14) Chappuis, J.; Georges, J.-M. J. Chim. Phys.-Chim. Biol. 1974, 71 (4), 567-575. (15) Johnson, K. L. Contact Mechanics; Cambridge: Cambridge, U.K., 1985. (16) (a) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. Roy. Soc. London, Ser. A, 1971, 345, 301-313. (b) Derjaguin, B. V.; Muller, V. M.; Toporov, Y. Colloids Surf. 1983, 7, 251-259. (c) Pashley, M. D.; Pethica, J. B.; Tabor, D. Wear 1984, 100, 7-31. (17) Maugis, D. J. Colloid Interface Sci. 1992, 150, 243.
Figure 4. (a) The variations of the quasi-static force/radius measurements during the downward and upward displacement, Z, for two carbon surfaces in dry air (relative humidity, RH, of air less than 0.1%). Strong adhesion is detected. During the outward approach a pull-off force is measured: Fpo S /R ) 436 ( 2 mN/m. (b) Same experiment as in part a. The force/radius values, during the upward displacement, are reported using the sphere-plane distance, D, as first coordinate. D is calculated with the JKR method16 and leads to a typical representation with a barrier, corresponding to the mechanical origin. This mechanical origin differs from the electrical zero and will be taken as a reference in the following measurements.
distance, D, plus the solid deformation, δ. The value Z ) 0 is given by the electrical measurement.10 For the experiment in dry air, with a relative humidity, RH, of less than 0.1%, strong adhesion is detected. In particular, during the outward approach a “pull-off” force is measured: Fpo S /R ) 436 ( 2 mN/m. In the case of adhesive contact and in dry air, the pulloff force is related to the solid vapor surface energy, γSV, by the following relation (as explained in Appendix, part a):
Fpo S /R ) -3πγSV
(2)
Equation 2 leads, for the experiment presented in Figure 4, to γSV ) 46.3 ( 0.2 mN/m. This value is less than that found for out-gassed graphite (γSV ) 123-130 mN/m),21 but it is in agreement with the values obtained for carbon fibers (γSV ) 40-60 mN/m).20 Values for soots are not known. Figure 4b shows the same force/radius values presented in Figure 4a, but against the sphere-plane distance, D, where
D )Z - δ
(3)
Two Carbon Surfaces with a Dispersant Polymer
Langmuir, Vol. 13, No. 13, 1997 3457 Table 1. Comparison of the Measurements of the “Pull-Off” Force for the Three Cases Presented: Dry Air, Humid Air RH ) 50%, and with a Large Water Meniscus (r ≈ 1 mm)
interface
in mN/m
Fpo S /πR
relevant interfacial energy in mN/m
dry air RH < 0.1% water meniscus humid air RH ) 50%
436 304 472
3γSV dry air 4γLV cos θ 4(γLV cos θ + 0.75 γSL)
γSV dry air ) 46.3 γLV cos θ ) 24.2 γSV humid air ) 41.8, γSL ) 17.6
Fpo S /R
and White analysis,18 the sum of the wetting force, FW, due to the meniscus and the surface force of carbon-carbon inside of the liquid, FSS. Figure 5. Variations of the quasi-static force/radius measurement during upward displacement, Z, for two carbon surfaces in wet air and in dry air; visualization of the effects of a micromeniscus.
δ is calculated with the Johnson, Kendall, and Roberts (JKR) method16a (Appendix, part a). The plotting presents a different characteristic for the Fs/R vs D representation, with a barrier at roughly 1 nm from the origin. This origin comes from an electrical measurement and corresponds to the point where the two conductive parts of the sphere and the plane are in contact. The presence of a barrier indicates a shift in the contacting point, which is at 1 nm, from the electrical origin and leads to define the real origin as the mechanical origin. This shift can be explained by considering the electrical properties of the carbon layer. The deposit is not a conductor; otherwise the barrier would be on the electrical zero. It is not either an insulator; that would put the barrier at 6 nm from the electrical zero, corresponding to twice the thickness of the carbon deposit. Therefore the carbon deposit is a partial conductor, and the mechanical contact does not occur at the electrical origin, but at a “mechanical” origin. The distance measurements have thus to be corrected to get the real origin for Z or D. Surface force measurements conducted in dry air showed that the carbon deposit was a partial conductor and lead to the solid vapor surface energy that we will use in the next stages. (b) Micro- and MacroMenisci. The humidity of the air appears as a key problem in the dry contact. Figure 5 shows the quasi-static force/radius, Fs/R, measurements during upward displacement, Z, for two different manipulations. One is the dry contact test already presented (Figure 4a). The other one is performed without P2O5: the relative humidity is 50%. It clearly appears that something on the carbon surfaces is interfering with the force profile of the carbon meniscus. The fact that this product is neutralized by a drying agent and has a viscosity of 1-2 mPa‚s (see later for the principle of this measurement) suggests that water condenses on the carbon surfaces. It is thus possible to detect the presence of a micromeniscus. The pull off forces correspond to different solid-solid or solid-liquid-solid interactions. (i) In dry air, the pull off force is given by eq 2. (ii) In wet air, inside of a small liquid annulus, the pull off force is, according to the Fogden (18) Fogden, A.; White, L. R. J. Colloid Interface Sci. 1990, 138, 414430. (19) Schwarz, U. D.; Bluhm, H.; Holscher, H.; Allers, W.; Wiesndanger, R. In Physics of sliding friction; Person, B. N. J., Tosatti, E., Eds.; Nato ASI, Series E, Applied Sciences 311; Kluwer Academic Publishers: Dordrecht, 1995; pp 369-402. (20) Nardin, M.; Schultz, J. J. Composite Interface 1988, 1, 124. (21) Fowkes, F. M. In Advances in Chemistry Series 43: Contact Angle, Wettability and Adhesion; Gould, R. F., Ed.; American Chemical Society Applied Publications: Washington, DC, 1964; p 99.
Fpo S ) FW + FSS ) 4πRγLV cos θ + 1.5πRWSLS ) 4πR(γLV cos θ + 0.75γSL) (4) (iii) In the case of a large water meniscus and when the distance, D, is also large, the surface force of carboncarbon, FSS is negligible; consequently Fpo S is written as
Fpo S ) FW ) 4πRγLV cos θ
(1)
Table 1 gives the comparison of the measurements of the pull off forces for the three cases presented: dry air, humid air with RH ) 50%, and a large water meniscus (r1 ≈ 1 mm). For the experiment in humid air, the solidliquid surface energy, γSL, is obtained by using the value of γLV cos θ measured with a large meniscus of water. From these results, it is clear that γSV dry air > γSV humid air. Figure 5 also shows that, in dry air, formation and rupture of adhesive contact are made with a very small hysteresis. Adhesive contact occurs for Z ) 3.6 nm and rupture for 3.6-4.0 nm. In wet air (RH ) 50%), adhesion occurs at Z ) 4.1 nm and rupture at Z ) 16 nm. The force law, during the outward displacement, is, in this case, related to the volume of condensed liquid in the contact zone, as shown recently.29 It is concluded that with a drying agent a micromeniscus cannot easily be formed on the carbon surfaces; its thickness is less than 0.5 nm. (c) Characterization of the Carbon Deposit through the Wetting. The aim of this subsection is to give the components of the carbon surface energy obtained with different wetting measurements in the SFA apparatus. The surface energy of the carbon deposit can be written as the sum of two components:21
γSV ) γSVLW + γSVAB
(5)
LW where γSV is the dispersion part containing the London AB and van der Waals contributions and γSV is a measure the acid-base character of the solid. It is assumed that, in a dry contact, only the nondispersive forces play a role; LW ) 46.3 mN/m. consequently γSV ) γSV Wetting experiments are conducted by introducing a meniscus of pure liquids, in the sphere-plane interface, when the sphere-plane distance is in the range of D ≈ 1-2 µm (step two of the procedure, Section 2). The difference in the static force, between the sphere and the plane, with and without a large meniscus, is related to the wetting force, FW, (eq B4).22 Therefore values of
(22) Good, R. J.; van Oss, C. J. In Modern Approaches of wettability. Theory and Applications; Schrader, M. E., Loeb, G. I., eds.; Plenum Press: New York, 1992.
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Table 2. Liquid-Vapor Surface Energy, γLV, and Its Components (the Dispersion Part Containing London and van der AB a Waals Contributions γLW LV and a Measure of the Acid-Base Character of the Liquid γLV ) pure liquids with carbon surfaces
γLV mN‚m-1
LW γLV mN‚m-1
AB γLV mN‚m-1
γLV cos θ mN‚m-1
cos θ
θa
72.8b
21.8b
51.0b
72.8b 25.5b 28.9c 58.2b
21.8b 25.5b 28.9c 30.5b
51.0b 0.0b 0.0 27.7b
24.2 18.6 25.5 28.9 28.4
0.333 0.255 1.000 1.000 0.480
70.0° (receding angle) 75.2° (advancing angle) 0° (advancing angle) 0° (advancing angle) 61.3° (advancing angle)
water water n-dodecane 175NS formamide
a The wetting force F gives the value of γ W LV cos θ and that of the contact angle θ of the liquid with the carbon surfaces. When the wetting force is measured for a large inward displacement, D, an advancing contact angle is obtained. When the wetting force is measured, for a large outward displacement D, a receding contact angle is obtained. b Referenced in the literature.21 c Measured with a Wilhemy balance.14
Figure 6. Evolutions of cos θ + 1 versus the values of LW 1/2 (γLV ) /γLV for various liquids on a dry carbon surface. Two different types of behavior can be distinguished. First, for nonpolar liquid, such as n-dodecane and 175NS, spread out on the carbon surface and cos θ + 1 ) 2. A is the nonpolar liquid, with the highest surface energy, which wetted the carbon surface. The line OA thus represents the first term of the Fowkes equation. Second, a polar compound like water and formamide does not wet the surface, so both first and second terms are LW is the discalculable from the Fowkes representation. γSV persing part gathering the London and van der Waals contriAB is the nondispersing part representing the butions and γSV acid-base character of the solid, which are determined.
γLV cos θ and that of the contact angle θ of pure liquids with the carbon surfaces are determined, using the liquid surface tension values of the literature. Experiments are conducted with four pure liquids: water, n-dodecane, formamide, and pure 175NS (Table 2). The high contact angle values obtained for the carbon surface with pure water (advancing angle ) 75.2°; receding angle ) 70.0°) reveal the hydrophobic character of carbon surfaces. LW AB and γSV for a dry carbon The determination of γSV surface is obtained following the Fowkes procedure (Appendix, part b), using eq B2.
cos θ + 1 ) 2x
γLW SV
γAB xγLW LV AB x LV + 2xγSV γLV γLV
first term
(B2)
second term
Figure 6 shows the evolution of cos θ + 1 versus the LW 1/2 ) for dry carbon surfaces. For a nonpolar values of (γLV liquid, such as n-dodecane and 175NS, cos θ + 1 ) 2. But it can be noted that these two liquids spread instantaneously; therefore the Young-Dupre´ relation is γSV - γSL > γLV. For such liquids, Fowkes equation is reduced to LW LW 1/2 LW LW 1/2 2 < 2[(γSV γLV ) /γLV] or (1/γSV ) < [(γLV ) /γLV]. Point A LW 1/2 (Figure 6) has for an ordinate [γLV ) /γLV] ) 1/(46.3)1/2. A corresponds to the upper limit of purely dispersive liquids
that wet the carbon surface. The line OA represents nonpolar compounds that give incomplete wetting (θ > 0 AB and γLV ) 0): it represents the values of the first term of the Fowkes equation (B2). Experiments conducted with water and formamide show that the second term of the equation is not equal to zero, indicating that there are AB * 0. Idenpolar sites on carbon surfaces and that γSV AB tification with water experiments gives γSV ) 5.5 ( 0.2 LW AB + γSV ) 46.3 + 5.5 ) 51.8 mN/m. Therefore γSV ) γSV mN/m. The results for carbon surfaces in air can be summarized as follows. (i) A homogeneous and thin layer of amorphous carbon is deposited on cobalt smooth surfaces. The layer is a partial conductor. Therefore the mechanical contact between the sphere and the plane is not the electrical one. But, for a dry contact, a residual film evaluated to D0 ) 1 nm, is found. (ii) The carbon surfaces are hydrophobic. Their advancing contact angle with water, θa, is 75°. (iii) The carbon surfaces have polar sites as revealed by the Fowkes analysis of wetting. (iv) An open question that remains is whether there will always be a formation of a micromeniscus on the carbon surfaces, even when a drying agent is used. 4. Nanorheology of the Brush Polymer (a) Interactions with the Polymer Solution. Behavior of the Pure Solvent. Before studying the adsorbed layers on carbon surfaces with polymer solution, it is first necessary to present the results observed for the hydrocarbon solvent (175NS). Figure 7 shows the damping function and the quasi-static force against the sphereplane displacement, D, using for D ) 0, the mechanical origin. The inverted damping function, 1/A, increases linearly with D, following eq C2. The slope leads to the dynamic viscosity η0 ) 59.0 ( 1.5 mPa‚s, which is in good agreement with the viscosity measured on a capillary viscometer (η0 ) 60 ( 0.3 mPa‚s) at 25 °C. The extrapolation of the damping function intercepts the surface separation axis at 2LH ) 4.0 nm. The hydrodynamic layer will thus be of the same order as the roughness (1.2 nm). The static force profile is a barrier with a small hysteresis. A close look at the downward displacement corresponding to the approach will reveal a small attraction due to the van der Waals force just before the beginning of the repulsion starting at 2L ) 4.4 nm. The distance L is close to LH: L ≈ LH ) 2.1 nm. The static force measured here corresponds to the surfaces forces and can be written as the sum of three components: the van der Waals contribution, FvdW, the entropically driven force, Fst, and the elastic contribution, Fel:
FS ) FvdW + Fst + Fel
(6)
where the first two forces are attractive and only the third
Two Carbon Surfaces with a Dispersant Polymer
Langmuir, Vol. 13, No. 13, 1997 3459
Figure 7. (a) Variations of the damping function 6πR2/A with the sphere-plane separation, D, for carbon surfaces and for the hydrocarbon solvent (base oil 175NS) and the succinimide solution in hydrocarbon (175NS + PIB). LH is the hydrodynamic layer thickness. η is the bulk viscosity of the test liquid. (b) Quasi static force/radius versus the sphere-plane separation, D, for the hydrocarbon solvent (base oil 175NS) and a succinimide solution in hydrocarbon (175NS + PIB). 175NS presents adhesion of the carbon surfaces. PIB solution presents a repulsion process between the two carbon surfaces.
is repulsive. The van der Waals force for undistorted surfaces is
FvdW ) -
A131R 6D2
(7)
where A131 is the Hamaker constant for the carbon175NS-carbon interface. If it is considered that A131 is given by (6), A131 ≈ [(A11)1/2 - (A33)1/2]2 where A11 is the Hamaker constant of carbon and where A33 is the Hamaker constant of the hydrocarbon (A33 ) 5 × 10-20 J).6 Hamaker constant value of carbon (A11) of 9.5 × 10-20 J is obtained using the equation A11 ) 24πγSVD20, where D0, the separation contact, is set as 0.165 nm,6 and where γSV ) 46.3 mN/m. Therefore A131 ) 7.2 × 1021 J. This value is in the range of the Hamaker constant expected for such an interface and is comparable to the reported value of (8 - 20) × 1021 J.4,23 According to eq 7, the van der Waals force for D ) 4.4 nm, is FvdW/R ) 0.7 × 10-3 N/m; this value is in agreement with the measurement presented in Figure 7. The contribution of the entropically driven force is negligible, in comparison to the measured values of the force. The repulsive elastic force starts to dominate when the first repulsive layers begin to interact at D ) 2L ) 4.4 nm. The repulsive layer is not squeezed out during the indentation; in fact it has been found that it will resist the highest pressure allowed by our experimental device: 0.1 GPa. It appears that a relatively thick immobile layer is present at the solid surface; this layer (23) Lyklema, J. Fundamentals of interface and Colloid Science. 1: Fundamentals; Academic Press: London, 1991.
is detectable during the squeezing process and resists high pressures. Such results have already been encountered with another base oil by Christenson and Israelachvili on mica surfaces.24 Polyisobutenesuccinimide Solution. Figure 7 also shows the damping function and the quasi-static force, against D, for the solution 2.5% PIB in 175NS. Experiments were carried out after 5 h of solution adsorption at 25 °C. The first repulsive effect begins at 11.5 ( 0.3 nm. This proves that polymer molecules adsorb on the solid surface, creating an immobile film responsible for the increase of the first repulsion distance. The dynamic measurements confirm these results: the damping function 6πR2/A varies linearly with distance D. The slope gives a viscosity of 78.5 mPa‚s, close to the value obtained on a capillary viscometer (76.4 mPa‚s), and the extrapolation of the linear part conducts to an immobile layer thickness of 2LH ) 10.5 ( 0.25 nm. The fact that L = LH ) 5.3-5.8 nm indicates that the film covering the surfaces is homogeneous. Furthermore it is important to note that the adsorbed layer thickness corresponds to the length of the PIB molecule (5 nm). The adsorbed layer can thus be imagined as a brush structure made of PIB molecules anchored on the surface through their polyamide moiety. The contact of two brush-bearing surfaces is analyzed by the Alexander-de Gennes theory.27 Chains are attached to the solid surface. The force p(D), per unit surface between two parallel plates separated by a distance D, and bearing surface layers are given by eq 8, where kB is
p(D) ≈
kBT 2L s3 D
9/4
([ ]
[2LD ] ) 3/4
-
(8)
Bolzmann’s constant, T is the absolute temperature, s is the average distance between the attachment points, and 2L is twice the layer thickness for the adsorbed polymer as shown on Figure 8. The first term of eq 8 is an osmotic repulsion, while the second term is an elastic restoring force. This pressure can be obtained for the experiments presented in Figure 7. According to the Derjaguin approximation,6 the pressure, p(D), or force per unit surface between two parallel plates separated distance D is related to the force, F, on the sphere-plane interface eq 9.
p(D) ≈
1 dF 2πR dD
(9)
Figure 8 shows the evolution of pressure p(D) in the sphere-plane contact versus the sphere-plane distance D, for PIB solution and for the hydrocarbon solvent. On the same Figure 8, eq 8 is plotted for 2L ) 11.5 nm and three values of s (1.4, 1.6, and 1.8 nm). The best fit between theory and experiment is obtained for L ) 5.7 nm and s ) 1.6 nm or s2 ) 2.56 nm2. For D < 7 nm, the Alexanderde Gennes theory is not completely followed for all the compressions of the polymer layers. This is due to the elastic deformation of the two substrates, which is not (24) Christenson, H. K.; Israelachvili, J. N. J. Colloid Interface Sci. 1992, 119, 1. (25) de Gennes, P. G. Macromolecules 1981, 14 (6), 1639-1644. (26) de Gennes, P. G. Adv. Colloid Interface Sci. 1987, 27, 189-194. (27) Kornbrekke, R. E.; Morrison, I. D.; Oja, T. Langmuir 1992, 8, 1211-1217. (28) Bec, S.; Tonck, A., Georges, J.-M. In The third body concept; Dowson, D. et al., Eds.; Elsevier Science: Amsterdam, 1996; pp 173184. (29) Tonck, A.; Mazuyer, D.; Georges, J.-M. In The third body concept; Dowson, D. et al., Eds.; Elsevier Science: Amsterdam, 1996; pp 185194. (30) Crassous, J.; Charlaix, E.; Loubet, J.-L. Europhys. Lett. 1994, 28, 37-42.
3460 Langmuir, Vol. 13, No. 13, 1997
Georges et al.
Figures 7 and 8 have shown that the adsorption of PIB on carbon surfaces gives repulsion forces when the two surfaces are in contact. Two mechanisms can be used to explain this repulsion: steric effects and the double-layer effect. The first effect has been explained. The nature of the double-layer effects in nonaqueous media such as the sign and amplitude of electrical charges has yet to be explained, but measurements on a PIB solution (Oloa 1200) have recently been published by Kornbrekke et al.27 The Zeta potential was measured and found to be around 30 mV. Looking at this result, it is possible to calculate the thickness of the double layer through the Debye length 1/κ:
(1κ)
0kT
2
Figure 8. The evolution of the pressure Π(D) in the sphereplane contact versus the sphere-plane distance, D. For the hydrocarbon solvent (base oil 175NS) and for the succinimide solution in hydrocarbon (175NS + PIB), data are obtained by using the Derjaguin relation. Results are compared with the Alexander-de Gennes theory.27 s is the average distance between the attachment points of the PIB brush, and 2L is twice the layer thickness. Best fit between theory and experiment is obtained for L ) 5.7 nm and s ) 1.6 nm.
taken into account in this theory. At the beginning of the bilayer indentation, the compressive elastic Young’s modulus can be evaluated as
Ef ≈ D dp(D)/dD It is found that Ef ≈ (5-10) MPa. This value is in accordance with the estimated Young’s modulus of a Langmuir monolayer of stearic acid molecules (STA) in liquid condensed state as recently measured by scanning force microscopy.31 This data is in agreement with the adsorption data of PIB (Oloa 1200), on carbon black studied by Pugh and Fowkes.13 They have observed a strong adsorption in the first hours which decreases rapidly until an equilibrium value is reached. Such an evolution with time is characteristic for the adsorption of large molecules. Measurements of adsorption isotherms were also completed at different temperatures; they show a linear evolution of the adsorbed quantity against the equilibrium concentration, revealing that molecules are in pseudoequilibrium. Knowing the specific surface of the carbon black and the quantities adsorbed, it was found that the area occupied per molecule was s2 ) 1.8 nm2. The SFA data (s2 ) 2.56 nm2) suggest that, after 5 h, equilibrium adsorption of PIB on carbon surfaces is not reached. Wetting measurements on carbon surfaces, similar to those conducted on hydrocarbon solvents, were obtained from the PIB solution. The experiments give γLV cos θ ) 19.9 mN/m and γLV 30.1 mN/m, respectively. These values can be compared to the results of the experiments on 175NS. They give γLV cos θ ) 28.9 mN/m and γLV ) 28.9 mN/m. Therefore using the Young-Dupre equation, and the fact that γSV has the same value in the two cases, the following relation is obtained:
(γSL)175NS - (γSL)OLOA 1200 ) (γLV cos θ)175NS (γLV cos θ)OLOA 1200 ) ∆γ ) 9.6 mN‚m-1 (10) ∆γ is positive, which is another indication of the presence of an adsorbed layer. (31) Tsukruk, V. V.; Bliznyuk, V. N.; Hazel, J.; Visser, D.; Everson, M. P. Langmuir 1996, 12, 4840-4849.
) e
2
r 2Nz2
(11)
where 0 is the vacuum permitivity, e is the electron charge, r is the media relative constant (r ) 2.2), z is the ion valence, and N is the ion number by unit volume. For a concentration c ) 28 × 10-3 g/cm3 we have N ) 9.3 × 1018 ions per cm3. Numerical application of relation 11, gives for the double-layer width 1/κ ) 532 nm. If the double layer plays a role in the repulsive force, the pressure law will be p(D) ∝ exp(-D/ψ),6 with ψ ) 1/κ ) 532 nm. Experimentally it is found ψ ≈ 1-2 nm, a value much lower than 1/κ. Therefore, it appears that the repulsive aspect can be considered as purely steric, as suggested by Forbes and Neustadter.3 (b) Anisotropic Elasticity of the Succinimide Layer. Experiments presented in Figure 8 suggest that the succinimide molecules organize themselves in a relatively dense layer, with a brush structure. If that is the case, the layer is structured though lateral cohesive forces as for a fatty acid monolayer deposited with the Langmuir-Blodgett method. The contact of two brushes will have an isotropic mechanical behavior. In particular, the normal elastic compressibility modulus (OZ) will not be related to the elastic tangential modulus (OX). Experiments were conducted in order to simultaneously measure the normal and the tangential properties of polymer films during the indentation procedure (Appendix, part d). Figure 9 presents the normal stiffness of the contact KZ versus the tangential stiffness of the contact KX, for three different layers: PIB brush layer, stearic acid (STA) monolayer, and polyisoprene (PI) layer. The stearic acid monolayer is obtained by adsorption on a smooth cobalt surface from a dilute solution of stearic acid in ndodecane.29 The polyisoprene layer is obtained by adsorption of the polymer from a semidilute solution of cis1,4-polyisoprene in 2,4-dicyclohexyl-2-methylpentane, which is a small hydrocarbon molecule and a good solvent for the polyisoprene at 23 °C.11 For each experiment, the evolution of the KZ function of KX is obtained by changing the indentation depth Z (sphere-plane distance). When the indentation depth of the double film is zero, KZ ) KX ) 0. During the indentation KZ and KX increase simultaneously. For the PIB and STA layers, when the indentation depths are small, KZ ≈ 25KX. For more important indentation depths, dKZ/dKX is 2.5. In opposition for PI layers, KZ ≈ 3KX. On the same figure are reported the typical values of the stiffness ratio KZ/KX for the case of a nonadhesive contact between a sphere and a plane and for the case of the indentation of a isotropic layer elastically indented by a rigid sphere (Appendix d). A comparison between these KZ versus KX evolutions clearly shows that at the beginning of the indentation PIB and STA layers, the tangential stiffness value is much
Two Carbon Surfaces with a Dispersant Polymer
Langmuir, Vol. 13, No. 13, 1997 3461
Acknowledgment. The authors thank Andre´ Tonck for his technical skills and B. Cambou, C Diraison, D. Faure, and H. Mathais for helpful discussions. We also thank Elf-Antar-France for making this work possible. Appendices (a) Adhesive Contacts. Hertz15 has already shown a hundred years ago that, in the case of a nonadhesive contact, solids deformations, δ, and the radius of the contact area, a, are related to the applied load and given by
δ) with
E* )
Figure 9. Normal stiffness of the contact KZ versus the tangential stiffness of the contact KX, for three different layers: PIB, stearic acid STA monolayer, and polyisoprene films. The anisotropic character of the PIB and STA films are clearly detected.
less than the normal stiffness values. This is due to the anisotropic structure of the layer which induces anisotropic elastic behavior. That is not the case for the PI layer. The brush stucture of the PIB layer is clearly proven.
2/3
[
(A1)
]
2 1 - ν22 4 1 - ν1 + 3 E1 E2
-1
(A2)
vi is the Poisson’s ratio, Ei the Young’s modulus of sphere and flat, respectively (i ) 1 and 2). Relation A1 is only valid for a nonadhesive contact. Therefore, Johnson, Kendall, and Roberts16a (henceforth referred to as JKR) extended the Hertzian theory to two solids which adhere due to their finite surface energy. They recognized that the contribution of the additional adhesive forces can be considered by the introduction of an apparent Hertzian load FS,JKR which is somewhat larger than the externally applied load Fs.
FS,JKR ) FS + 3πRWSS + x6πRFSWSS + (3πRWSS)2 (A3)
5. Conclusions Experiments with a surface force apparatus are made, using a homogeneous, thin layer of amorphous carbon deposited on smooth cobalt surfaces. The contact of carbon surfaces is tested in dry and wet air. Adhesion forces are measured and compared with those obtained in water. The carbon surfaces present polar sites revealed by the Fowkes analysis of wetting. Using these carbon surfaces, the repulsive forces created by the adsorption of a brush polymer are studied. It is a dilute solution of polyisobutenesuccinimide (PIB) (Oloa 1200), in a hydrocarbon solvent at 25 °C. The PIB molecules organize themselves in a homogeneous, relatively dense layer with a brush structure. The layer thickness corresponds to the molecule length. The repulsion effect is obtained by a purely steric effect of the two brushes. The mechanical properties of the double film are anisotropic, and the normal compression modulus is much greater than the shearing modulus. The results provide a basis for understanding the mechanism of action of polyisobutenesuccinimides widely used as dispersant additives in lubricants. The criterion for stabilization carbon colloidal suspension in which only van der Waals and steric mechanisms take place is a critical separation distance, D0, during the particles collisions. D0 is twice the effective adsorbed film thickness on each carbon particle, where the attraction dispersive energy is κT. For two spherical particles of same radius a, D0 depends on radius a and the Hamaker constant A131 , with the relation D0 ) RA131/12κT. Substitution of the data shows that the largest particles that can be stabilized by the PIB shell of L ) 5.5 nm would have a radius a ) 0.07 µm. This relatively low value explains why it is interesting to consider other polymer molecules for the stabilization of the carbon particles in automotive lubricants.3
[ ]
a2 1 FS ) 1/3 R R E*
where R is the radius of the sphere and WSS is the solidsolid interaction energy. In this case, the solid deformations δ and contact radius a are
δ)
2Fs a2 + 3R 3aE*
(A4)
with
a)
[E*R ]
1/3
[FS + 3πRWSS +
x6πRWSSFS + (3πRWSS)2]1/3
(A5)
Note that at small negative loads (Fs < 0) eq A5 predicts a contact area between the sphere and plane (i.e., sphere still adheres). The negative load reaches the pull-off force po FS,JKR . po Fs,JKR ) -1.5πRWSS
(A6)
Although the JKR theory was shown to be excellently applicable to adhesive contact with the SFA, it fails for spheres with small radii since it does not take into account attractive forces outside of the contact area and predicts an infinite stress at the edge of the contact circle. This infinite stress disappears as soon as attractive interactions outside of the contact area are considered, as in the theory of Derjaguin, Muller, and Toporov (DMT).16b In this case the solids only separate when the contact area has reduced to zero. The pull-off force is po ) -2πRWSS FS,DMT
(A7)
A dimensionless quantity R was proposed to select between the JKR and DMT theories:16a,b
3462 Langmuir, Vol. 13, No. 13, 1997
R ) 6π
Georges et al.
WSSR1/2
(A8)
E*De3/2
R must be large compared with unity for the JKR theory to apply.16c Numerical application, taking into account the results of Figure 4, gives
R)
6π 69.8 × 10-3 (2.5 × 10-3)1/2 ) 34.5 10 -9 3/2 6 × 10 (1 × 10 ) (A8a)
R is large compared with unity; therefore the JKR theory is applied. The nature of the transition from the JKR theory to the DMT theory was most extensively studied by Maugis.17 Their theories offer the possibility to consider the effect of capillary forces, which are the dominant attractive forces under ambient conditions. Fogden and White (FW)18 called their theory “generalised Hertzian theory”; the contact itself was considered to be nonadhesive and the capillary forces were assumed to be the only attractive forces acting outside of the contact area. In the FW theory, the load-contact area curve is determined by a set of two equations which depend on a fundamental parameter, β. 1/2
β)
3π 23/2
1 E* rK - DS 2 2γLVR1/2
(
3/2
)
(
1-
)
DS 2rK
-1
(A9)
Here, rK represents the Kelvin radius of the meniscus, γLV the surface tension of the liquid-vapor interface per unit area (here, water-air), and DS the separation of the surfaces. The two equations cannot be solved analytically. In the limits of small or large β, however, analytic approximations can be obtained. A small value of β corresponds to the JKR type limit. A large value corresponds to the DMT type limit. The main result is that for a given separation, DS, between the sphere and the plane a stable meniscus (radius rK) forms from the capillary condensation of water vapor even if its pressure, P0, is inferior to the saturated vapor pressure, PSAT.6 The expressions for rK and DS are the following:
rK )
0.54 P0 ln PSAT
[ ]
DS ) 0.54
2 cos θ P0 ln PSAT
[ ]
6 × 1010[(3 - 0.5) × 10-9]3/2
(A10)
[1 - 61]
-3 1/2
2(72.8 × 10 )[2.5 × 10 ] -3
relative humidity P0/PSAT Kelvin radius rK (nm) Ds (nm)
1 ∞ ∞
0.9 -5 3.3
-1
) 2.4 (A9)
In conclusion the JKR model is applied. (b) Fowkes’ Analysis. The solid-liquid-solid interaction energy, WSLS, is related to the interfacial energy, γSL, by WSLS ) 2γSL. The interfacial energy or interfacial tension, γSL, is given by the Dupre´ equation:6
γSL ) γSV + γLV - WSL
(B1)
0.5 -0.8 0.5
0.1 -0.3 0.2
The solid-liquid adhesion work, WSL, can be written as LW AB LW WSL ) WSL + WSL , where WSL corresponds to the contribution of long distance interaction forces and AB WSL corresponds to the Lewis acid-base contribution: LW LW LW 1/2 AB AB AB 1/2 WSL ) 2(γSV γLV ) and WSL ) 2(γSV γLV ) . When these equations are taken into account, the following relation, proposed by Fowkes, is obtained:21 LW AB AB γSL ) γSV + γLV - 2xγLW SV γLV - 2xγSV γLV
(B2)
When the contact angle is θ, equilibrium between the different surface tensions is given by the Young-Dupre´ relation:
γSV - γSL ) γLV cos θ
(B3)
This relation is verified if θ > 0. Fowkes relation combined with the Young-Dupre´ relation leads to
cos θ + 1 ) 2x
γLW SV
γAB xγLW LV ABx LV + 2xγSV γLV γLV
(B4)
Equation proposed also by Fowkes. (c) “Hydrodynamic” Layer Thickness. The experiments are conducted in three steps as shown in Figure 3. First, the dry contact of carbon surfaces is tested. The presence of surface forces between the elastic sphere (radius R) and plane is revealed by the pull-off force Fpo S , which is related to the surface energy of the solid surfaces γSV. Here, the pull-off force is given by the following relation:6 Fpo S ) -3πRγSV. Second, a drop of the solution is placed between the sphere and the plane, and a meniscus (of radius r1 ≈ 1 mm) forms. The stabilization time for adsorption is about a few minutes for the pure solvent to some hours for the polymer solution on the carbon surface. The total force measured, FT, present between the sphere and the plane has three origins: a wetting force, FW, a surface force, FS, and a hydrodynamic force, FH:
FT ) FW + FS + FH
with DS and rK in nanometers. Taking θ ) 70° as the receding contact angle for a drop of water on a carbon surface, the numerical applications of these relations are given in Table 3. Numerical application of eq A9 leads to
β ) 1.879
Table 3. Kelvin Radius and Critical Separation, Ds, Versus the Relative Humidity
(C1)
The presence of a meniscus of liquid confined between sphere and plane (distance D) produces a wetting force, FW, due to the Laplace pressure which is, when D , r:
FW ) 4πRγLV cos θ
(1)
where γLV is the surface tension of the solution and θ the contact angle of the solution with the surface. The experiments are conducted in such a manner that the surface of the meniscus stays constant during the experiment. Third, to test the solution, the sphere-plane displacement, D, is monitored with two speed components.7 One is a steady ramp, which gives a constant approach speed of 0.1 nm‚s-1. Superimposed on this is a small amplitude oscillatory motion of about 0.1 nm RMS, with a vibration period of 2.5 × 10-2 s. The low speed permits the measurement of the quasi-static normal force because the contribution of the hydrodynamic force, FH to the total force, FT (C1) is negligible. But, the oscillatory motion permits the study of the mechanical transfer function of
Two Carbon Surfaces with a Dispersant Polymer
Langmuir, Vol. 13, No. 13, 1997 3463
the sphere-plane interface. From this complex transfer function, only the imaginary part will be used in this paper. This component, which is the dynamic force in quadrature with respect to the oscillatory motion, gives the damping function 1/A:
D - 2LH 1 iωD ˜ ) ) A FH 6πηR2
with
(D1)
[
]
2 1 - ν22 4 1 - ν1 + 3 E1 E2
-1
(A2)
where a is the radius of indentation. The tangential stiffness, KX, is given by the Mindlin relation:10
4G KX ) a 2 - νS
(C2)
where η is the viscosity of the fluid. LH is the “hydrodynamic” thickness, which is the thickness of the adsorbed layer which does not participate in the flow of the “bulk” solution. (d) Measurement of the Layer Anisotropy. The purpose of the experiment is to measure simultaneously normal and tangential properties of polymer films during the indentation procedure. The method has already been presented in the literature.28 The normal approach (OZ) is realized with two speed components; one D˙ ) Z˙ ) 0.2nm/s is a steady ramp, and the second one is a small amplitude oscillatory motion of about D ˜ ) 0.1 nm RMS, with a pulsation of ω ) 4.2 × 10-3 rad/s. A tangential motion (OX) is added with a small amplitude oscillatory motion of about 0.2 nm RMS, with a pulsation of ω ) 10 × 10-3 rad/s. Forces in the two directions are simultaneously recorded, with an accuracy of 10-8 N. Such measurements give for each sphere-plane displacement the tangential and normal stiffness of the contact KX and KZ. The theoretical results of the contact stiffness are the following: (1) In the case of a nonadhesive contact between a sphere and a plane, the normal Hertzian stiffness, KZ, is given by11
3 KZ ) aE* 2
E* )
(D2)
where GS is the shear elastic modulus of the solid and vi is the Poisson’s ratio. For isotropic material,the two elastic moduli are related, by the relation:
GS )
ES 2(1 + νS)
(D3)
Therefore, eqs D1, D2, and D3 give
2 - νS KZ ) KX 2(1 - νS)
(D4)
For the typical values of νS (between 0 and 0.5), KZ/KX vary between 1 and 1.5. (2) In the case of a layer elastically indented by a rigid sphere, the normal stiffness is KZ ) Efπa2/D, where a is the radius of indentation, D is the thickness of the double film, and Ec is the compressive modulus of the double film. The tangential stiffness is given by
KX )
Gfπa2 D
If the layer is isotropic, Ef ) 2(1 + νf)Gf and relations D4 and D5 lead to (D6):
KZ ) 2(1 + νf) KX
(D6)
For the typical values of νf , KZ/KX varies between 2 and 3. LA970065O
