Stick Slip Contact Mechanics between Dissimilar Materials: Effect of

Mar 15, 2008 - Measurements of the contact radius as a function of applied force between a mica surface and a silica surface (mica/silica) in air are ...
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Stick Slip Contact Mechanics between Dissimilar Materials: Effect of Charging and Large Friction Patricia M. McGuiggan Department of Materials Science and Engineering, Johns Hopkins UniVersity, Baltimore, Maryland 21218 ReceiVed December 12, 2007. In Final Form: February 2, 2008 Measurements of the contact radius as a function of applied force between a mica surface and a silica surface (mica/silica) in air are reported. The load/unload results show that the contact radius generally increases with applied force. Because of the presence of charging due to contact electrification, both a short-range van der Waals adhesion force and longer-range electrostatic adhesive interaction contribute to the measured force. The results indicate that approximately 20% of the pull-off force is due to van der Waals forces. The contact radius versus applied force results can be fit to Johnson-Kendall-Roberts (JKR) theory by considering that only the short-range van der Waals forces contribute to the work of adhesion and subtracting a constant longer-range electrostatic force. Also, an additional and unexpected step function is superimposed on the contact radius versus applied force curve. Thus, the contact diameter increases in a stepped dependence with increasing force. The stepped contact behavior is seen only for increasing force and is not observed when symmetric mica/mica or silica/silica contacts are measured. In humid conditions, the contact diameter of the mica/silica contact increases monotonically with applied force. Friction forces between the surfaces are also measured and the shear stress of a mica/silica interface is 100 times greater than the shear stress of a mica/mica interface. This large shear stress retards the increase in contact area as the force is increased and leads to the observed stepped contact mechanics behavior.

Introduction The contact behavior between two smooth, nonadhesive homogeneous elastic solids is well described by Hertz theory.1 Two elastic solids in contact will deform under an applied force F, which is equal to the elastic restoring force of the solids. No interaction is assumed to occur between the solids except for a repulsion at contact. At zero applied force, the surfaces separate. More recent continuum mechanics theories have included adhesive surface forces in the model of the contact of elastic spheres.2,3 Johnson, Kendall, and Roberts4 (JKR) modified the Hertz interaction to include surface energy in the interaction. Only the short-range adhesive forces inside the contact area are included; any interaction forces outside the contact area are excluded in JKR theory. JKR theory leads to a greater contact diameter at a fixed force than found from Hertz theory and is applicable when the elastic deformation is large compared to the range of the surface forces. The contact radius has a finite value just before separation. Another contact theory used to model the deformation between adhesive surfaces is Derjaguin, Muller, and Toporov (DMT) theory.5,6 In this theory, long-range surface forces are assumed to act on the region outside the contact area. The region inside the contact is assumed to follow Hertz theory. Furthermore, the solids separate when the contact radius falls to zero. The contact theories model the adhesive interaction between ideal surfaces: perfectly elastic, homogeneous, and smooth. Many materials are composites of solids where the solid materials are often dissimilar and rough. Roughness can significantly decrease the adhesion between materials by decreasing the area of (1) Johnson, K. L. Proc. Inst. Mech. Eng. 1982, 196, 363-378. (2) Adams, G. G.; Nosonovsky, M. Tribol. Int. 2000, 33, 431-442. (3) Johnson, K. L. Contact Mechanics; Cambridge University Press: Cambridge, U.K., 1985. (4) Johnson, K.; Kendall, K.; Roberts, A. Proc. R. Soc. London, Ser. A 1971, 324 (1558), 301-313. (5) Derjaguin, B. V.; Muller, V. M.; Toporov, Y. P. J. Colloid Interface Sci. 1975, 53, 314-326. (6) Pashley, M. D. Colloids Surf. 1984, 12, 69-77.

contact.7-9 Dissimilar surfaces may also develop charging as the two surfaces adhere.10 This may give rise to a large adhesion energy as well as hysteresis in the loading/unloading cycles. Previous studies of the adhesion between a mica surface and a silica surface in dry nitrogen atmosphere have measured a work of adhesion that is at least 50 times greater than the adhesion of either material to itself.10 Thus, the work of adhesion between mica/mica surfaces, silica/silica surfaces, and mica/silica surfaces has been measured to be 110 mJ/m2, 80 mJ/m2, and >6000 mJ/m2, respectively.10-12 The unexpectedly large work of adhesion between the dissimilar mica/silica materials is due to contact electrification.13 When the surfaces are in contact, silica acquires a negative charge and mica a positive charge.10 The force needed to pull charged surfaces apart is greater than the force required to separate uncharged surfaces, leading to the high adhesion between the mica/silica surfaces. With such a difference between the work of adhesion for charged and uncharged interfaces, one might expect the contact mechanics between charged and uncharged surfaces to also be quite different. All the contact mechanics theories work within certain limits. DMT and JKR theories are extreme limits of an adhesive contact mechanics problem, and a continuous transition between JKR and DMT behavior has been shown.14,15 DMT theory generally applies to relatively rigid systems with low adhesion and small radii of curvature, whereas JKR theory is best for high adhesion, large radii, and compliant materials. Neither JKR nor DMT theory applies to charged interfaces. Numerous theoretical and experi(7) Fuller, K. N. G.; Tabor, D. Proc. R. Soc. London, Ser. A 1975, 345, 327342. (8) Schaefer, D. M.; Carpenter, M; Gady, B.; Reifenberger, R.; Demejo, L. P.; Rimai, D. S. J. Adhes. Sci. Technol. 1995, 9, 1049-1062. (9) Maugis, D. J. Adhes. Sci. Technol. 1996, 10, 161-175. (10) Horn, R. G.; Smith, D. T. Science 1992, 256, 362-364. (11) Smith, D. T.; Horn, R. G. Mater. Res. Soc. Symp. Proc. 1990, 170. (12) Horn, R. G.; Israelachvili, J. N.; Pribac, F. J. Colloid Interface Sci. 1987, 115, 480-492. (13) Lowell, J.; Roseinnes, A. C. AdVa. Phys. 1980, 29, 947-1023. (14) Maugis, D. J. Colloid Interface Sci. 1992, 150, 243-269. (15) Tabor, D. J. Colloid Interface Sci. 1977, 58, 2-13.

10.1021/la703882h CCC: $40.75 © 2008 American Chemical Society Published on Web 03/15/2008

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mental papers have investigated the applicability and limits of JKR theory.16-18 Recently, theoretical and experimental measurements have shown how JKR theory can be applied to layered materials as well as homogeneous materials.19 In addition, viscoelastic effects have been studied.20 However, questions remain regarding the effect of surface charge on adhesion and contact behavior. In addition, the JKR assumption of full slip at the interface has not been explored. In this paper, the effect of surface charge on contact behavior is investigated by measuring the contact radius as a function of force between a mica surface and a silica surface. Interfacial slip is also studied by measuring the friction between the surfaces. The results are compared to measurements of symmetric silica/ silica interfaces and mica/mica interfaces. Experimental Technique The surface forces apparatus (SFA) was used to measure contact radius a, applied force F, and friction force between the surfaces.21,22 The apparatus uses two molecularly smooth sheets bonded to cylindrical silica lenses with Shell Epon 1004 epoxy hot-melt adhesive. The cylindrical lenses are mounted with their cylindrical axes at right angles. The resulting contact between the two crossed cylinders is geometrically equivalent to a sphere on a flat. The back surface of each sheet is coated with 500 Å of a reflective silver film. Multiple-beam optical interference between the two silver layers produces fringes of equal chromatic order (FECO). Measurements of the shape of the fringes when the surfaces are not in contact are used to compute the local radius of curvature of the surfaces. Deformation of the surfaces, hence the fringes, occurs once the surfaces are in contact due to the deformation of the relatively soft epoxy adhesive beneath the thin mica and silica sheets. The radius of the flattened contact is directly measured from the amount of flattening of the deformed FECO fringe pattern. The bottom cylindrical lens is mounted on a double-cantilever spring. The double-spring design minimizes shearing at the contact as the spring deflects under load. The applied force is measured by noting the deflection of the spring, which in this work had a spring constant k ) 1 × 105 (( 0.1 × 105) N/m. Unless otherwise noted, the ( refers to the standard uncertainty in the measurements and is taken as 1 standard deviation of the observed values. This SFA experiment uses two different surfaces. For the mica surface, muscovite mica was cleaved to give a molecularly smooth mica surface. For the silica surface, a thin silica sheet was prepared from Hypersil silica by a blown-bubble technique.10 After a bubble was blown, freshly cleaved mica patches were placed on the silica to protect the silica surface from contamination. The silica surface was silvered and the sheet was glued, silver side down, to the support disks. The mica sheet was then peeled from the silica surface, exposing a smooth amorphous surface. The procedure produces amorphous silica surfaces with less than 5 Å surface roughness.23 The surfaces were placed in a UV cleaner for 10 min and then mounted in the SFA. The chamber was then purged with dry N2 for at least 20 min. Solid P2O5 was placed in the chamber to further ensure a dry atmosphere. The measurements were made at 23 °C. The SFA experiments were performed by bringing the surfaces into van der Waals contact at zero applied force. The force was increased and the contact radius was measured. The measurements continued until a compressive force of approximately 200 mN was (16) Chaudhury, M. K.; Whitesides, G. M. Langmuir 1991, 7, 1013-1025. (17) Deruelle, M.; Hervet, H.; Jandeau, G.; Leger, L. J. Adhes. Sci. Technol. 1998, 12, 225-247. (18) Shull, K. R. Mater. Sci. Eng., R 2002, 36, 1-45. (19) McGuiggan, P. M.; Wallace, J. S.; Smith, D. T.; Sridhar, I.; Zheng, Z. W.; Johnson, K. L. J. Phys. D: Appl. Phys. 2007, 40, 5984-5994. (20) Greenwood, J. A.; Johnson, K. L. Philos. Mag. A 1981, 43, 697-711. (21) Israelachvili, J. N.; Adams, G. E. Nature 1976, 262, 773-776. (22) Israelachvili, J. N.; McGuiggan, P. M.; Homola, A. M. Science 1988, 240, 189-191. (23) Horn, R. G.; Smith, D. T.; Haller, W. Chem. Phys. Lett. 1989, 162, 404408.

Figure 1. Measurements of the load (b) and unload (O) curve for one mica surface and one silica surface interacting in dry nitrogen. The lower line is the JKR analysis from eq 1 with W12 as measured from the pull-off force, Fp ) -73 mN. The upper line is the JKR analysis, where W12 is obtained from the integrated force-distance curve. Both curves are fit by use of E* ) 22 GPa.

Figure 2. Schematic diagram of the interaction between a mica surface and a silica surface in dry nitrogen. The surfaces are (1) separated and initially uncharged, (2) in contact where charging spontaneously occurs, (3) in contact with increased charge due to larger contact area, or (4) separated and the surfaces remain charged until the charge dissipates. applied. The force was then decreased with contact diameters being measured at each force until the surfaces spontaneously separated (“pull-off”) at a negative (tensile) force, Fp. The time between loading steps was approximately 1 min, for a total loading/unloading time of approximately 40 min. Generally, the surfaces were separated by distances of at least 5 mm for 4 h before the contact was remeasured. This was done to allow complete neutralization of any charges that might be on the surfaces. The silica and mica surfaces were between 2 and 5 µm thick and the epoxy layers were 10-30 µm thick. The mean radius of the cylinders was 1.5 cm (R1 ) 1.3 cm, R2 ) 1.8 cm). The Young’s moduli E for the mica, blown silica, epoxy adhesive, and silica support surfaces are 62 ( 2, 71 ( 1, 3.4 ( 0.04, and 71 ( 1 GPa, respectively.19,24

Results In a dry nitrogen atmosphere, the mica/silica interaction is adhesive and the surfaces jump together once the attractive forces overcome the spring force and the spring force gradient equals the attractive force gradient. At this point the surfaces are in adhesive contact and have a finite contact radius, a, but the external applied force is approximately zero. Figure 1 shows the measured contact diameter versus the applied force for one mica surface and one silica surface (mica/silica) interacting in dry (24) Oliver, W. C.; Pharr, G. M. J. Mater. Res. 1992, 7, 1564-1583.

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nitrogen. For this experiment, with a mean cylinder radius of 1.5 ( 0.2 cm, the contact radius on loading at zero applied force is 35 ( 3 µm. The contact radius increases with increasing applied force and then decreases as the applied force is reduced from a compressive (positive) to a tensile (negative) value until pull-off occurs at a force Fp. Since the surfaces are adhesive, the pull-off force is negative and pull-off occurs when there is still a finite contact radius. The surface profile is significantly distorted when the two surfaces are pulled apart, indicating that there is still an adhesive interaction occurring between the now charged surfaces even though the surfaces are separated. According to JKR theory for homogeneous materials, the dependence of the contact radius a with applied force F is given by4

a3 )

3R [F + 3πRW12 + x6πRW12F + (3πRW12)2] (1) 4E*

where a is the contact radius, R is the mean radius of the cylinders, E* is the effective modulus of the system, F is the applied force, and W12 is the work of adhesion of the surfaces. For the interaction of materials 1 and 2, the work of adhesion is given by W12 ) γ1 + γ2 - γ12. The homogeneous JKR equation predicts that the surfaces separate when the force, given as the pull-off force Fp, is given by Fp ) (3/2)πRW12. For layered surfaces, the pull-off force will also depend on the layer thicknesses, but this will not be considered.19 The solid and dashed curves in Figure 1 are the fit to JKR theory (eq 1) using E* ) 22 GPa as a fitting parameter since E* was not directly measured in the experiment. For layered surfaces, E* is dependent on the modulus and thickness of the mica, silica, and epoxy layers as well as the contact diameter.19 The value of E* is expected to be between E of the epoxy (Eepoxy ) 3.4 GPa) and E of the surfaces (Emica ) 62 GPa and Esilica ∼ 72 GPa).19 In Figure 1, two solutions to eq 1 are obtained. The dashed curve was calculated with W12 ) Fp/(3/2)πR ) 1000 mJ/m2 where Fp ) -73 mN was the measured pull-off force. The solid line was calculated with W12 ) 6000 mJ/m2, which was obtained from previous measurements by integrating the entire force-distance curve to take into account the long-range electrostatic interaction.10 Neither curve accurately predicts the measured forces, especially at low applied forces. This discrepancy is expected since JKR theory does not take into account long-range interactions such as electrostatic interactions that occur during charging. When the surfaces are in contact, previous measurements have shown that silica acquires a negative charge and mica a positive charge.10 Figure 2 shows a schematic diagram of the surface charge during a load/unload measurement. Contact first occurs between two initially uncharged surfaces. Once contact occurs, the contact area becomes charged. If the position of the charge groups on the surface is assumed to be localized to the area within the contact zone, increasing the contact radius increases the total amount of charge at the interface although the charge density is likely to remain constant. However, on unloading, the surfaces are now charged and separation occurs between charged areas. For the charged mica/silica surfaces, there are two contributions to the adhesive force: a short-range van der Waals interaction and a longer-range electrostatic force. The nonretarded van der Waals force, FVDW, between two crossed cylinders varies with separation distance, D, according to25 (25) Israelachvili, J. N. Intermolecular & Surface Forces; Academic Press: New York, 1992.

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FVDW )

AH R 6D2

(2)

where AH is the Hamaker constant, R is the mean radius of the cylinders, and D is the separation distance. For surfaces with R ∼ 2 cm, most of the interaction occurs within the last 2 nm of contact. A simplified expression of the electrostatic interaction is to model the charged surfaces as a fixed voltage parallel plate capacitor.10 The electrostatic force Fel per unit area A is given by

Fel σ2 )A 20

(3)

where σ is the surface charge density and 0 is the permittivity of free space. The electrostatic force is due to surface charging and is independent of the separation for small distances. In dry atmosphere, it has been shown that the surfaces maintain a constant charge after separation until the surface separation exceeds 500800 nm.10 At this separation, discharge begins to occur. For surfaces of similar radii (R ∼ 2 cm), the range of the electrostatic interaction has been measured to exceed 3 µm.10 In order to determine if the assumption of constant surface charge during unloading is justified, the surface separation of the entire charged area should be less than 0.5 µm prior to pull-off. Consider the force curve shown in Figure 1. The contact radius at the highest applied force is 65 µm. Since the contact radius decreases to 25 µm before pull-off, geometric analysis assuming Hertz interaction gives the separation distance D at the edge of the maximum contact radius as less than 0.2 µm (D ∼ δ ) a2/2R, where δ is the indentation). This is shown schematically in Figure 3. Since no discharge occurs for surface separations less than 0.5 µm and the maximum separation of the charged edges before pull-off occurs is 0.2 µm, constant surface charge can be assumed during the measurement. The above calculation is an oversimplification since discrete charge interactions and image charges are not included, and therefore the electrostatic interaction at and near contact might be quite different than the interaction when the surfaces are separated by a few nanometers. There is currently no theory that predicts the contact behavior of adhesive, charged surfaces. Since JKR and DMT theories consider a single component of the adhesive force, these theories cannot accurately describe the interaction force for the mica/ silica interaction, as was shown in Figure 1. Barthel26,27 has modeled the interaction of two surfaces when the interaction is complex. He proposes that when the interaction is composed of two forces with very different decay lengths, the interaction can be split into the specific short- and long-range components. Since the electrostatic force is assumed to be independent of separation for small distances, the electrostatic force can be considered to be a constant adhesive force, as long as the surface charge density remains constant during the measurement. This electrostatic force can then be added to the short-range van der Waals interaction to give the total interaction force. A similar analysis has been proposed by Rimai et al.28 If the adhesive components can be split, then adhesion theories such as JKR theory (eq 1) may still be valid since the effect of charging may be simply a constant added to the short-range (26) Barthel, E. “Modelling the adhesion of spheres, when the form of the interaction is complex”, Colloids Surf., A 1999, 149, 99-105. (27) Huguet, A. S.; Barthel, E. J. Adhes. 2000, 74, 143-175. (28) Rimai, D. S.; Ezenyilimba, M. C.; Quesnel, D. J. J. Adhes. 2005, 81, 245-269.

Contact Mechanics between Dissimilar Materials

Figure 3. Schematic diagram of the surface separation during unloading, with the assumption of Hertz interaction. At maximum applied force, the contact radius am gives the maximum surface charge radius. As the surfaces are separated, the outer edges of the charged surfaces will separate until pull-off occurs. D is the surface separation at the edge of the maximum contact, δ is the indentation, a is the contact radius, and R is the mean radius of the cylinders.

adhesive component of the force. With the long- and short-range interactions separated, only the short-range van der Waals forces will be included in JKR theory. The expected van der Waals pull-off force for crossed cylinders can be calculated from eq 2. For cylinders with a mean radius of 1.5 cm and the assumptions that D ≈ 0.2 nm for a separation distance at contact and AH ≈ 9 × 10-20 J (the mean Hamaker constant of mica and silica surfaces),25 eq 2 gives FVDW ≈ -5.6 mN. The pull-off force measured between mica/mica and silica/ silica surfaces of similar radii was found to be -12 mN and -(9-17) mN, respectively.11,12 This shows that the measured pull-off force for uncharged surfaces is consistent with the FVDW calculation.29 The pull-off force measured between mica/silica surfaces was -70 ( 25 mN. This value is about 6 times higher than the pull-off force measured between mica/mica and silica/silica surfaces of similar radii. Since charge transfer is not observed between mica/mica and silica/silica contact, it can be estimated that approximately -12 mN of the pull-off force between mica/ silica is due to the van der Waals interaction and the remaining -58 mN is coming from the electrostatic force. In other words, approximately 20% of the adhesion is due to van der Waals forces and 80% is due to electrostatic forces. For the charged mica/silica interface, a range of pull-off force values, which depended upon the number of previous contacts, was measured. Previous experiments have found that increasing the number of contacts increases the charge density on a surface; thus, the greater the number of contacts, the higher the pull-off force.10 For this experiment, -110 mN < Fp < -40 mN. If enough time was given to allow complete dissipation of the charge that had formed on contact due to contact electrification, the pull-off force returned to the original measurement. Generally, separating the surfaces by about 2 mm for at least 4 h was sufficient to dissipate the charge. Note, however, that for the mica/silica interface, the surfaces do not jump apart beyond the range of the interaction forces. Even though the surfaces have moved ∼0.2 µm apart, a substantial attractive force is still measured. In the presence of long-range forces, there is an additional work of adhesion that comes from integrating the attractive force-distance curve out to infinite surface separation. Horn and Smith10 measured the range of the attractive force between similar radii mica/silica surfaces to be 3 µm. The integrated force-distance curve is divided by the maximum area of contact to obtain W12 ) (6600 - 8800) mJ/ (29) Homola, A. M.; Israelachvili, J. N.; Gee, M. L.; McGuiggan, P. M. J. Tribol. 1989, 111, 675-682.

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Figure 4. Measurements of the load (b) and unload (O) curve for one mica surface and one silica surface interacting in dry nitrogen (reproduced from Figure 1). The dashed line is the JKR analysis from eq 1 using the expected work of adhesion for uncharged surfaces. The solid line shows the JKR prediction (dashed line) where the force is shifted by -73 mN to account for the long-range electrostatic forces. The exact value of the contact diameter is measured to (5%.

m2.10 In the JKR equation, the long-range force is not reflected in the usual relationship between the pull-off force and the work of adhesion according to W12 ) Fp/(3/2)πR. The JKR relationship assumes that there is no long-range force, or in other words, the force-distance integral has a value of 0. Therefore, if a value of W12 is calculated from the pull-off force, one finds the much smaller value of W12. Specifically, for the mica/silica interface, the pull-off force was measured to be Fp ) -73 mN (R ) 1.5 cm), giving W12 ) 1000 mJ/m2. This value is only 10-15% of the work of adhesion obtained from integrating the force-distance curve. In order to investigate the specific short- and long-range interactions, the data in Figure 1 are replotted in Figure 4. The dashed curve in Figure 4 shows the theoretical predictions of JKR theory (eq 1) using the work of adhesion for uncharged surfaces (W12 ∼ 110 mJ/m2) and again using E* as a fitting parameter. For a best fit, the value of E* was found to be 16 GPa. As shown by the dashed line in Figure 4, the JKR prediction using the short-range work of adhesion expected for uncharged surfaces (W12 ∼ 110 mJ/m2) underestimates the dependence of the contact radius with applied force. If an arbitrary constant electrostatic force of -73 mN is subtracted from the applied force, a better fit to the data is obtained. This is shown as the solid curve in Figure 4. According to eq 3, if a contact area based on the maximum contact radius of 65 µm is assumed, an electrostatic force of -73 mN gives a surface charge density of 7 mC/m2. The high surface charge density measured between the mica and silica surfaces reflects the intimate contact achieved over the molecularly smooth contact area. A surface charge density of 7 mC/m2 agrees with previous measurements of the electrostatic charge.10 The data in Figure 4 shows hysteresis, that is, a difference between the loading and unloading curves. This difference is seen most clearly at low forces. Because there is hysteresis, two values of the contact radius are measured at the same force, one on the loading curve and the other on the unloading curve. Because the pull-off force value is taken from the unload curve, the JKR theory was fitted to the unload curve. JKR theory does not consider hysteresis and so one might argue whether the load or unload curve should be fitted to the theory. Hysteresis is observed even

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Figure 5. Measurements of the load curve for one mica surface and one silica surface interacting in dry nitrogen. The contact diameter increases in a stepped behavior with applied force. The relative contact diameter is measured to (1 µm.

for uncharged surfaces.19,30 The causes of hysteresis have been reviewed by Israelachvili.31 Surprisingly, the load curve shows that the measured contact diameter for mica/silica contact does not increase smoothly with force but rather increases incrementally. A typical loading curve is shown in Figure 5. The stepped contact radius versus applied force behavior can also be seen in the loading curve in Figure 1. The increase in the diameter of contact corresponds to approximately 5 µm/step and, correspondingly, an increase in the area of contact by about 650 µm2 or about 7-10% of the previous contact area. The steps observed on the loading curve were generally not evident on the unloading curve. In spite of the differences in the pull-off force, the stepped contact diameter versus load response was present for both the first and subsequent contact runs on loading. In addition, the steps were clearly visible from the FECO fringes when a constant loading was applied and when a force was applied incrementally. The magnitude of the interaction varied with charge, but the stepped behavior was present regardless of the number of contacts and the time between contacts. Since the pull-off force varied with the number of contacts, the presence of the stepped contact dependence was independent of the adhesion force. The interaction between a silica surface and a mica surface was also measured in 100% humidity as shown in Figure 6. At 100% humidity, the interaction is dominated by capillary forces due to a liquid meniscus surrounding the contact. The liquid meniscus formed as the surfaces came into contact and slowly evaporated once the surfaces jumped apart at a pull-off force Fp ) -110 ( 20 mN. The contact diameter increased monotonically with applied force. Thus the stepped contact diameter dependence on applied force disappeared in humid conditions. The loading/unloading contact behavior between two silica surfaces in dry air was also measured, and the results are shown in Figure 7. The solid line is calculated from JKR theory (eq 3) with a measured pull-off force of Fp ) -17 ( 2 mN (E*/R ) 4 × 1012 N/m3). Hysteresis between the loading and unloading curves was measured, but no steps were observed on loading. At large applied loads there is a deviation from JKR theory due to the effects of the silica substrate under the epoxy layer. For the silica/silica contact, multiple contacts damaged the surfaces. (30) Horn, R. G. Measurement of Surface Forces and Adhesion. In ASM Handbook; Henry, S. D., Ed.; -ASM International: Herndon, VA, 1992; pp 399-405. (31) Israelachvili, J. J. Vac. Sci. Technol., A 1992, 10, 2961-2971.

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Figure 6. Measurements of the load (b) and unload (O) contact radius for one mica surface and one silica surface interacting in 100% humidity. A liquid meniscus forms between the surfaces at contact.

Figure 7. Measurements of the load (b) and unload (O) contact radius for two silica surfaces in dry N2. The solid line is the JKR analysis from eq 1 using the measured Fp ) -17 mN.

The friction force was also measured between mica/silica surfaces and silica/silica surfaces in dry air.22 The surfaces were first brought together at no externally applied load before a shear force was applied. For the mica/silica and silica/silica surfaces, the interaction was adhesive and the surfaces remained pinned prior to the initiation of sliding. This was followed by a large slip, which tended to damage the surfaces. Hence, only the maximum shear stress of the static friction before the slip was recorded. The shear stress S (friction/area of contact) was measured to be S ) 2.3 × 108 (( 0.5 × 108) N/m2 for silica/silica surfaces and S ) 3 × 109 (( 0.5 × 109) N/m2 for mica/silica surfaces. The shear stress between mica/mica surfaces has been previously measured to be 2.5 × 107 N/m2.29 Thus, in dry air, Smica/silica ≈ 10 times Ssilica/silica ≈ 100 times Smica/mica.

Discussion Industrial applications such as substrate cleaning processes, powder processing, and electrophotography involve the adhesion and contact mechanics of charged particles and surfaces.32-34 Despite the technological importance, the adhesion and contact (32) Krupp, H. AdV. Colloid Interface Sci. 1967, 1, 111-239. (33) Rimai, D. S.; Quesnel, D. J. J. Adhes. 2002, 78, 413-429. (34) Rimai, D. S.; Ezenyilimba, M.; Goebel, W. K.; Cormier, S.; Quesnel, D. J. J. Imaging Sci. Technol. 2002, 46, 200-207.

Contact Mechanics between Dissimilar Materials

mechanics of charged interfaces, and in particular charged particle adhesion, are not well understood. For example, consider the studies of charged toner particles adhering to substrates. It is accepted that both van der Waals and electrostatic interactions are present in toner particles. However, a consistent explanation of the interaction has not emerged. In particular, it has been found that van der Waals interactions dominate,35,36 electrostatic interactions dominate,28,37 and both electrostatics and van der Waals contribute equally.38 One reason for the discrepancy is that the results have been found to depend on surface roughness, particle shape, and particle diameter.33,34 Since the electrostatic interaction is a weakly decaying function, it does not scale with particle radius. Even between ideal mica surfaces, the adhesion is complex. For example, the adhesion between mica/mica sheets depends on the relative crystallographic orientation of the mica sheets,39,40 the thickness of the mica and subsurface layers,19 and the presence of condensates and intervening films.41 The additional longrange force present between the mica and silica surfaces makes the interaction more complex. The contact radius-force dependence between symmetric mica/mica and silica/silica surfaces can be fitted by JKR theory if a constant modulus is assumed, as expected for homogeneous systems, and if the measured pull-off force is used to determine the work of adhesion. The mica/silica load/unload curve can be fitted to JKR theory by using a work of adhesion as expected for uncharged surfaces and then subtracting a constant longrange adhesive interaction from the applied load. In this fit, the work of adhesion for uncharged surfaces is about 90% less than the work of adhesion calculated from the measured pull-off force. This shows that the pull-off force is measuring both short- and long-range adhesive component. The contact radius versus applied force for the mica/silica interaction could not be fitted to JKR theory by use of either the measured pull-off force or the work of adhesion, measured by integrating the force-distance curve. This is not surprising since JKR theory includes only shortrange interaction forces and both the pull-off force and the work of adhesion, obtained by integrating the force-distance curve, include long-range forces. Although only the contact radius is measured in this experiment, it is likely that the shape of the surfaces near the contact region is also affected by the long-range electrostatic force. The interferometry used to measure the contact radius can also be used to measure the entire surface profile. Previous measurements between mica/mica surfaces have shown a surface profile that is consistent with JKR theory.12 It is possible that the surface profile of the mica/silica surfaces would better fit the DMT profile since long-range forces are included in the theory. Future experiments will measure the surface profiles of the mica/silica surfaces. In these measurements, the contact radius for the mica/mica and silica/silica interaction increases monotonically with applied force. However, small jumps have been observed on unloading mica/mica contacts in dry air, and these jumps have been attributed to dynamic instabilities and the presence of charged domains.42 (35) Mastrangelo, C. J. Photogr. Sci. Eng. 1982, 26, 194-197. (36) Sounilhac, S.; Barthel, E.; Creuzet, F. J. Appl. Phys. 1999, 85, 222-227. (37) Donald, D. K. J. Appl. Phys. 1969, 40, 3013-3019. (38) Hays, D. A.; Wayman, W. H. Inst. Phys. Conf. Ser. 1983, 237-242. (39) McGuiggan, P. M.; Israelachvili, J. N. J. Mater. Res. 1990, 5, 22322243. (40) McGuiggan, P. M.; Israelachvili, J. N. Chem. Phys. Lett. 1988, 149, 469472. (41) Hough, D. B.; White, L. R. AdV. Colloid Interface Sci. 1980, 14, 3-41. (42) Frantz, P.; Artsyukhovich, A.; Carpick, R. W.; Salmeron, M. Langmuir 1997, 13, 5957-5961.

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In contrast, the interaction between a mica and a silica surface in air increases in a steplike manner on loading and thus has an additional dependence superimposed on the contact radius versus load curve. A number of physical phenomena could account for this unusual step behavior including (1) charging, (2) differences in the moduli of the two surfaces, and (3) friction. The role of each of these will be considered. 1. Contact Electrification. No charging is expected to occur between symmetric mica/mica and silica/silica surfaces. Between mica/silica surfaces, the loading curve involves contact of initially uncharged surfaces whereas the unloading curve involves separation of charged surfaces. As the surfaces initially jump into contact giving rise to a finite contact radius, charging occurs spontaneously in the contact region. The force is then increased and the change in the contact diameter is measured. The increased force brings about new contact area between the mica and silica surfaces at the periphery of the previous contact. The new area, which was uncharged, now becomes charged. On unloading, the contact area decreases from a large charged area. However, the charged area remains charged on separation since no charge dissipation occurs at small surface separations. This charging difference could give rise to the adhesion hysteresis that was measured, but it is unlikely that it is the cause of the stepped dependence because the stepped dependence is independent of the amount of charge on the surface. If the charging were directly related to the stepped dependence, then the behavior should change with surface charge. However, since the stepped dependence was present on the first and subsequent contacts, it appears that the stepped dependence is independent of the surface charge. Nonuniform surface charge might also affect the interaction. Recent pull-off force measurements have used a patch charge model to explain variation in the pull-off force observed in atomic force microscopy.43 In addition, nonuniformly charged patches have also been observed to control adhesion of polydisperse toner particles.44 The possibility that the surfaces do not have a uniform charge, and that this variation might account for the stepped contact radius dependence, cannot be ruled out. 2. Difference in the Modulus of Each Surface. The same epoxy was used in all the experiments; however, the thickness was not controlled. The thickness of the epoxy is typically 1030 µm thick. In addition, the mica sheet is thinner than the silica sheet. Although the mica and silica have similar moduli, the mica and silica thickness differences and epoxy thickness differences imply that there is a mismatch in the mechanical properties. This may lead to an unbalanced stress at the interface that may be velocity-dependent. However, the mechanical properties should not change significantly with humidity. Since the steps disappeared with humidity, the mechanical property mismatch is likely not the reason for the stepped dependence. 3. Friction. The sliding of two surfaces past each other can occur in a stick-slip manner.45,46 Even with no externally applied tangential force, two contacting surfaces can have a shear stress. For contact between a sphere and a flat surface, Bowden and Tabor47 described experimental evidence for a locked inner region (43) Pollock, H. M.; Burnham, N. A.; Colton, R. J. J. Adhes. 1995, 51, 71-86. (44) Eklund, E. A.; Wayman, W. H.; Brillson, L. J.; Hays, D. A. Electrostatics 1995, 143, 85-92. (45) Gee, M. L.; McGuiggan, P. M.; Israelachvili, J. N.; Homola, A. M. J. Chem. Phys. 1990, 93, 1895-1906. (46) Israelachvili, J.; McGuiggan, P.; Gee, M.; Homola, A.; Robbins, M.; Thompson, P. J. Phys.: Condens. Matter 1990, 2, SA89-SA98.

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surrounded by a region of slip, all within the contact region. For Hertz contact, finite friction effects have been theoretically investigated; however, a stepped contact mechanics behavior was not predicted.48 In dry nitrogen, the shear stress at the mica/silica interface is quite high, S ) 3 × 109 (( 0.5 × 109) N/m2. This large shear stress may not allow full slip at the interface, as is assumed in JKR theory. The presence of humidity can change the interaction of the surfaces. At high humidity, the surfaces are interacting across a thin water film formed between the surfaces due to capillary condensation. This thin water film changes the adhesion and friction properties.49 In the case of mica/mica surfaces, the shear stress decreases by an order of magnitude in 100% humid conditions.50 Under these decreased shear stress conditions, interfacial slip can easily occur and a monotonic increase in the contact diameter versus force is found. Therefore, it is likely that the large friction between the mica and silica surface in dry air retards the slip and gives rise to the unexpected step behavior. (47) Bowden, F. P.; Tabor, D. The Friction and Lubrication of Solids; Clarendon Press: Oxford, U.K., 1986. (48) Spence, D. A. J. Elasticity 1975, 5, 297-319. (49) Kim, D. I.; Ahn, H. S.; Choi, D. H. Appl. Phys. Lett. 2004, 84, 19191921. (50) Homola, A. M.; Israelachvili, J. N.; McGuiggan, P. M.; Gee, M. L. Wear 1990, 136, 65-83.

McGuiggan

Conclusions The contact mechanics behavior between a positively charged mica surface and a negatively charged silica surface is measured. The results indicate that approximately 20% of the pull-off force is due to van der Waals forces and the remaining 80% due to electrostatic forces for crossed cylinders of R ≈ 2 cm. The enhanced adhesion between the mica and silica surfaces due to charge transfer can generally be accounted for theoretically by assuming only short-range van der Waals forces contribute to the adhesive force in JKR theory and subtracting a longer-range electrostatic force. However, an unexpected stepped behavior is superimposed on the contact diameter versus force curve for the mica/silica interface. This behavior is believed to be due to retardation of the interfacial slip due to the high interfacial shear stress. Thus, the full-slip condition necessary in JKR theory cannot be realized. Acknowledgment. I thank Doug Smith for preparation of the silica surfaces and Lee White, Jacob Israelachvili, Etienne Barthel, Mark Robbins, Jay Wallace, and Don Rimai for helpful discussions. This material is based upon work supported by the National Science Foundation under Grant 0709187. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation (NSF). LA703882H