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Simulations of Nanotribology with Realistic Probe Tip Models† Michael Chandross,*,‡ Christian D. Lorenz,§ Mark J. Stevens,‡ and Gary S. Grest‡ Sandia National Laboratories, Albuquerque, New Mexico 87185, and Materials Research Group, Kings College London, Strand, London WC2R 2LS, England, U.K. ReceiVed July 31, 2007. In Final Form: NoVember 8, 2007 We present the results of massively parallel molecular dynamics simulations aimed at understanding the nanotribological properties of alkylsilane self-assembled monolayers (SAMs) on amorphous silica. In contrast to studies with opposing flat plates, as found in the bulk of the simulation literature, we use a model system with a realistic AFM tip (radius of curvature ranging from 3 to 30 nm) in contact with a SAM-coated silica substrate. We compare the differences in response between systems in which chains are fully physisorbed, fully chemisorbed, and systems with a mixture of the two. Our results demonstrate that the ubiquitous JKR and DMT models do not accurately describe the contact mechanics of these systems. In shear simulations, we find that the chain length has minimal effects on both the friction force and coefficient. The tip radius affects the friction force only (i.e., the coefficient is unchanged) by a constant shift in magnitude due to the increase in pull-off force with increasing radius. We also find that at extremely low loads, on the order of 10 nN, shearing from the tip causes damage to the physisorbed monolayers by removal of molecules.
1. Introduction Although microelectromechanical systems (MEMS) can be manufactured from a host of different materials, the most prevalent by far is silicon. This is due to not only the massive Si manufacturing infrastructure already in place but also the existing expertise on the properties and processing of silicon, as well as the desirability of achieving actuation and logic on the same substrate. Silicon, however, does not come without issues, with a major obstacle being its tendency to rapidly oxidize, forming a highly reactive, hydrophilic layer.1 Without treatment, etched parts can easily adhere, rendering devices useless. This active layer has been successfully passivated through the use of self-assembled monolayers (SAMs) that reduce the surface energy and render the surface itself hydrophobic while greatly reducing adhesion and friction.2 Although this process has been in use for a number of years, there still exist a number of open questions regarding the properties of SAM coatings, ranging from fundamental questions such as the substrate attachment mechanism and the dependence of friction coefficients on chain length and coverage to engineering issues such as the stability and lifetime of applied coatings, including wear resistance and the long-time effects of water. Some of the earliest work on the tribological properties of SAMs as boundary lubricants investigated the chain-length dependence of the friction between an atomic force microscope (AFM) tip and alkylsilane chains on mica3 or alkanethiols on gold.4 The latter work demonstrated that much of the frictional response is dominated by disorder in the monolayer systems, with shorter chains having more disorder than longer chains (n > 8, where n is the number of carbons in the backbone Cn). In these studies, the magnitude of the friction force decreases with increasing chain length, and the slopes of †
Part of the Molecular and Surface Forces special issue. * Corresponding author. E-mail:
[email protected]. ‡ Sandia National Laboratories. § Kings College London. (1) Iler, R. K. The Chemistry of Silica: Solubility, Polymerization, Colloid and Surface Properties, and Biochemistry; Wiley: New York, 1979; p 866. (2) Srinivasan, U.; Huston, M. R.; Howe, R. T.; Maboudian, R. J. Microelectromech. Syst. 1998, 7, 252-260. (3) Xiao, X.; Hu, J.; Charych, D. H.; Salmeron, M. Langmuir 1996, 12, 235237. (4) Lio, A.; Charych, D. H.; Salmeron, M. J. Phys. Chem. B. 1997, 101, 38003805.
the friction force versus applied load curves (i.e., friction coefficient) either monotonically decrease with chain length or show some nonmonotonic behavior. Similar results were seen in interfacial force microscopy (IFM) experiments on alkoxyl monolayers on treated SiO2 surfaces.5 The effects of disorder on the frictional response was systematically probed with a series of experiments on mixtures of alkanethiols with spiroalkanedithiols.6,7 The latter molecule has two tail groups of variable length bonded to a single head group. Combinations of C17 alkanethiols and spiroalkanedithiols where one chain was of length 17 and one was of length 1, 2, 12, or 17 demonstrated that any amount of disorder (i.e., when the second chain has a length other than 17) increased friction and caused the friction versus load curves to be identical to each other.6,7 Similar results were found for mixures of methyl- and trifluoromethyl-terminated films8 in which the addition of a small percentage of fluorinated chains increased friction. Recent work by Flater et al. demonstrated the effects of increasing disorder on frictional properties by separately measuring the friction of C18 chains in both liquid expanded (LE, lower density) and liquid condensed (LC, higher density) phases.9 Although the friction coefficient remained unchanged, the LE phase consistently showed a higher friction force as compared to the LC phase. Recent NEXAFS and FTIR measurements investigating the chain-length dependence of SAM structure found a high degree of molecular orientation at all chain lengths between C12 and C18.10 The authors interpret this as indicating highly ordered monolayers. This group found a nonmonotonic dependence of the friction coefficient on chain length, with lower values for intermediate chain lengths (C12) and higher values for both shorter (C10, which was found to be only partially ordered) and longer (5) Major, R. C.; Kim, H. I.; Huston, J. E.; Zhu, X. Y. Tribol. Lett. 2003, 14, 237-244. (6) Perry, S. S.; Lee, S.; Shon, Y.-S.; Colorado, R.; Lee, T. R. Tribol. Lett. 2001, 10, 81-87. (7) Lee, S.; Shon, Y. S.; Colorado, R.; Guenard, R. L.; Lee, T. R.; Perry, S. S. Langmuir 2000, 16, 2220-2224. (8) Kim, H. I.; Graupe, M.; Oloba, O.; Koini, T.; Imaduddin, S.; Lee, T. R.; Perry, S. S. Langmuir 1999, 15, 3179-3185. (9) Flater, E. E.; Ashurst, W. R.; Carpick, R. W. Langmuir 2007, 23, 92429252. (10) Sambasivan, S.; Hsieh, S.; Fischer, D. A.; Hsu, S. M. In Effect of SelfAssembled Monolayer Film Order on Nanofriction; AVS: 2006; pp 1484-1488.
10.1021/la702323y CCC: $40.75 © 2008 American Chemical Society Published on Web 01/10/2008
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(C16 and C18) chains. Much shorter and longer chains (C5 and C30) were found to be disordered and showed the highest friction coefficients. Modeling work on these systems has generally relied on molecular simulations in which opposing flat plates are compressed and sheared in order to study the tribological response11-17 Although these systems do not accurately reflect the geometry of the experiments, they benefit from the removal of complications of the tip/substrate geometry such as load-dependent contact areas. Effectively, the area dependence of the friction force is removed, and the friction depends purely on the applied load. Our previous work15 systematically studied the effects of chain length and coverage by holding one variable fixed while varying the other. The major results were that for well-ordered, fully packed SAMs on a crystalline substrate the friction coefficient varies nonmonotonically with chain length, in agreement with experiments on both alkylsilanes on silica10 and alkanethiols on gold.4 When defects were introduced into the monolayers, either by removing chains from the crystalline substrate or by using an amorphous substrate that had been annealed for the desired coverage of active sites on the surface, the chain length dependence disappeared. However, this work also demonstrated that friction force (more specifically, measured shear stress) increased with increasing disorder as seen in recent experiments.9 This effect has been seen in previous simulations by Mikulski and Harrison that examined the differences in friction between tightly and loosely packed hydrocarbon chains on diamond.13 These authors found that removing 30% of the chains leads to higher friction as a result of energy dissipation through fluctuations in the bond lengths. A similar result was later found in simulations of sliding of an infinite, flat amorphous carbon tip against pure C14 monolayers as compared to mixed C12/C16 monolayers.18 By comparing the forces that contribute both positively and negatively to the friction, it was shown that higher friction in the mixed system is due to an increase in energy dissipation. Flat plate simulations such as these, however, do not correctly represent the contact mechanics of the experimental systems, which makes direct comparison to experiment difficult. The major differences between flat plates and the tip/substrate geometry lie in the load-dependent contact area in the latter, as well as the possibility of tip-induced damage to the substrate and material transfer from the substrate to the tip. Because a major goal of our work is to understand the wear and degradation of MEMS coatings, the ability to induce damage to the substrates is a strong motivator for tip-based simulations. Previous nanotribological simulations with model tips do exist but have been affected by major issues. The most common approximation is the use of extremely sharp tips19-24 whereas (11) Glosli, J. N.; McClelland, G. M. Phys. ReV. Lett. 1993, 70, 1960-1963. (12) Tupper, K. J.; Brenner, D. W. Thin Solid Films 1994, 253, 185-189. (13) Mikulski, P. T.; Harrison, J. A. J. Am. Chem. Soc. 2001, 123, 6873-6881. (14) Chandross, M.; Grest, G. S.; Stevens, M. J. Langmuir 2002, 18, 83928399. (15) Chandross, M.; Webb, E. B.; Stevens, M. J.; Grest, G. S.; Garofalini, S. H. Phys. ReV. Lett. 2004, 93, 166103. (16) Mikulski, P. T.; Herman, L. A.; Harrison, J. A. Langmuir 2005, 21, 1219712206. (17) Lorenz, C. D.; Chandross, M.; Grest, G. S.; Stevens, M. J.; Webb, E. B. Langmuir 2005, 21, 11744-11748. (18) Mikulski, P. T.; Gao, G.; Chateauneuf, G. M.; Harrison, J. A. J. Chem. Phys. 2005, 122, 024701-024709. (19) Sorensen, M. R.; Jacobsen, K. W.; Stoltze, P. Phys. ReV. B: Condens. Matter 1996, 53, 2101-2113. (20) Bonner, T.; Baratoff, A. Surf. Sci. 1997, 377-379, 1082-1086. (21) Abdurixit, A.; Baratoff, A.; Meyer, E. Appl. Surf. Sci. 2000, 157, 355360. (22) Foster, A. S.; Shluger, A. L.; Nieminen, R. M. Nanotechnology 2004, 15, S60-S64. (23) Trevethan, T.; Kantorovich, L. Surf. Sci. 2003, 540, 497-503.
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other work has used rigid tips,25 nonatomistic tips,26 or infinitely flat tips18 or lacked true dynamics.27,28 These approximations are, in general, made to lower the computational burden of the simulations. Part of our motivation, therefore, is to apply large computational resources to overcome these issues by simulating fully atomistic tips with realistic dimensions (i.e., radii of curvature (F) in the tens of nanometers). Like all previous nanotribological simulations both with two flat substrates and those with a tip, we cannot approach the low shear rates in experiments because this is more an issue of computing time rather than resources. However, the shear rates in MEMS are only about 1 order of magnitude lower than the simulation range. We present here results from massively parallel molecular dynamics (MD) simulations of compression and shear of explicit AFM tips on amorphous silica substrates coated with alkylsilane SAMs. We study a range of tip sizes (F ) 0.3 to 30 nm, representing extremely sharp and blunted tips, respectively) and chain lengths (n ) 8 to 18) in order to compare both to experiment as well as previous simulations, both from our group as well as others.
2. Computational Model and Methodology Our simulations are fully atomistic, using the OPLS force field29-31 with the LAMMPS MD code.32 Nonbonded interactions are cut off at 10 Å except for long-range Coulomb interactions, which are calculated using a 2D particle-particle-particle mesh Ewald summation.33 The rRESPA multiple time step algorithm34 is used with time steps of 0.5, 1.0, and 2.0 fs for bonds, angles/ dihedrals, and nonbonded interactions, respectively. Temperature is constrained to 300 K with a Langevin thermostat in the direction perpendicular to compression and shear only. The silica tips and substrate were created from a large, bulk amorphous SiO2 sample created by the melting and quenching of a bulk R-quartz silica crystal.15 Tips with F ) 3, 10, and 30 nm (as shown in Figure 1) were carved out of the bulk to model an extremely sharp AFM tip, a reasonable size for a tip in use, and a blunted tip, respectively. Most data presented here are for the F ) 10 nm tip. The tips were annealed to achieve the desired coverage of active sites on the exposed surface, which for our purposes here means predominantly hydrophobic tips with only a few active sites remaining. These sites are capped with H or OH groups for undercoordinated O or Si, respectively. The two larger tips are hollow, leaving a shell of 1.5 nm thickness, to reduce the computational burden. Oxygen atoms on the inner surface of the tip are held rigid to control tip motion during both compression and shear simulations. Although experimental tips are rarely made of SiO2, we consider this model to be an approximation of an oxide layer on a Si or Si3N4 tip (modeled by the fixed oxygens). Although it has not been conclusively (24) Gao, G.; Cannara, R. J.; Carpick, R. W.; Harrison, J. A. Langmuir 2007, 23, 5394-5405. (25) Ohzono, T.; Fujihira, M. Physical ReV. B: Condens. Matter 2000, 62, 17055-17071. (26) Jun, S.; Lee, Y.; Kim, S. Y.; Im, S. Nanotechnology 2004, 15, 11691174. (27) Leng, Y.; Jiang, S. J. Chem. Phys. 2000, 113, 8800-8806. (28) Zhang, L.; Leng, Y.; Jiang, S. Langmuir 2003, 19, 9742-9747. (29) Jorgensen, W. L.; Maxwell, D. S.; Tirado-Rives, J. J. Am. Chem. Soc. 1996, 118, 11225. (30) Watkins, E. K.; Jorgensen, W. L. J. Phys. Chem. A 2001, 105, 4118. (31) Jorgensen, W. L. Private communication, 2003. (32) Plimpton, S. J. Comput. Phys. 1995, 117, 1-19. (33) Crozier, P. S.; Rowley, R. L.; Henderson, D. J. Chem. Phys. 2001, 114, 7513-7517. (34) Plimpton, S.; Pollock, R.; Stevens, M. Proceedings of the Eighth SIAM Conference on Parallel Processing for Scientific Computing; SIAM Activity Group on Supercomputing: Philadelphia, PA, 1997; p 8.
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Figure 1. Tips used in the simulations with radii of curvature of (a) 3, (b) 10, and (c) 30 nm. Note that the two larger tips are hollow to reduce the computational burden. Silicon atoms are shown in yellow, oxygen in red, and hydrogen in white.
demonstrated that Si3N4 forms an oxide layer, it has been shown that cantilevers made of the same material contain at least some oxygen.35 The substrates have a thickness of 1.5 nm and are annealed until the desired coverage of active surface groups is achieved (4 to 5 OH/nm2), with OH groups on the bottom of the substrate held fixed to control substrate motion. The size of the substrate varied from 11.1 × 11.1 nm2 to 47.7 × 48.2 nm2 depending on F. Experimentally, alkylsilane molecules adsorb onto the silica surface to form the SAMs. We study systems in which chains are both fully physisorbed (i.e., hydrogen bonded), which we refer to as alkylsilanes, and fully chemisorbed, which is a more appropriate model for alkoxylsilanes.5 We will use this nomenclature to distinguish between the two cases. The alkylsilane molecule has a silicon head group that is attached to three hydroxyl groups with a hydrocarbon backbone, Si(OH)3(CH2)n - 1CH3. The hydrocarbon part is hereafter referred to as Cn, including the methyl group. Here we present results for n ) 8-18. Initially, the alkylsilane chains are aligned perpendicularly to and randomly placed within 3 Å of the amorphous substrate. Upon equilibration, the chains quickly move to bring their headgroups closer to the hydroxyls on the substrate. The alkoxylsilane SiO(CH2)n - 1CH3 has no hydroxyl groups because of its formation from an alcohol precursor and the chemical bond between its silicon atom and the substrate. The alkoxylsilane SAMs are generated in the same manner as discussed in ref 15. Because there is some controversy over the relative amounts of physi- and chemisorption36 and in order to see the effect of noncovalently bonded molecules, we have also studied mixed systems where either 10 or 50% of chains are physisorbed while the remaining are chemisorbed. The SAMs are placed at coverages consistent with the experimentally observed coverage of 25.0 Å2/chain. During a compression simulation, the tip and substrate, initially separated by approximately 1.5 nm, are brought together at a constant rate of 5 or 0.5 m/s. During compression, the contact forces and areas are monitored. Although these strain rates are high compared to experiment, our previous work demonstrated that for two flat substrates the velocity does not have a strong effect on the contact force.15 However, this is no longer true in (35) Carpick, R. W. Private communication, 2007. (36) Onclin, S.; Ravoo, B. J.; Reinhoudt, D. N. Angew. Chem., Int. Ed. 2005, 44, 6282-6304.
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the case of tip simulations, in which case chains in the SAM can move to regions of lower applied force as discussed in detail below. Although the applied force can, in general, be quite large, we limit our studies to systems in which the maximum compressive force is approximately 100 nN in order to keep the tip from buckling and also to avoid measuring properties of SiO2 in the tip or substrate. To measure the contact area, a precise definition of contact at an atomic scale becomes necessary. This is particularly difficult given the nature of the van der Waals interaction between atoms. Here we define tip and substrate atoms as being in contact when one of the former is within 0.5 nm of one of the latter. The positions of all contacting atoms are then projected onto a plane where a circle that encompasses approximately 95% of the contacting points is then found. The radius of this circle is then the contact radius, which can be used to determine the applicability of various contact mechanics models as discussed in the next section. Shear measurements are performed by applying constant opposite velocities to the fixed oxygen atoms in both the tip and the substrate. In general, we study a shear velocity of 2 m/s, which is several orders of magnitude faster than velocities reached in AFM experiments. As stated above, this is a computational limitation that is presently not possible to overcome. We have previously studied the effects of shear velocity on the frictional response of the flat-plate simulations,15,17 but were unable to complete a statistically significant number of simulations. The strongest effect seen in these simulations appeared to be a reduction in the friction coefficient with decreasing velocity similar to experimental results.37,38
3. Results A. Compression Simulations. In simulations with a curved tip, we see many novel features that are not possible in flat-plate simulations. Snapshots of a compression simulation are shown in Figure 2a for C11 chains physisorbed on silica in contact with the 10 nm tip. A 4-nm-thick slice from the center of the simulation box is shown for clarity. The important features of this Figure are (1) evidence of chains splaying away from the tip, (2) a burm of material pushed up from the substrate accumulating around the edge of the tip, and (3) material that has been forced out of the SAM by the increased lateral pressure from excluded volume effects. Some of the material from the SAM remains on the tip after removal, as shown in Figure 2b. This transfer of material to the tip is discussed in detail in a separate publication.39 Compression curves for the three tip sizes are shown in Figure 3. As in the flat-plate simulations, the initial interaction between the tip and substrate is zero when the separation is greater than the cutoff. As the tip approaches the substrate, there is a small attractive region as the interactions between the tip and SAM turn on. Shown in the inset of Figure 3 is an enlargement of the attractive region for the different tips. The difference in pull-off force expected for different size tips is evident here as different maximum forces of attraction. Although these forces are small, we have divided them by the actual contact area and determined that the pressures are comparable to those in our previous work15 and nearly independent of tip radius. This calculation also serves to partially justify our definition of contact. As the tip and SAM are moved closer together, they come into contact, and the (37) Gnecco, E.; Bennewitz, R.; Gyalog, T.; Loppacher, C.; Bammerlin, M.; Meyer, E.; Guntherodt, H. J. Phys. ReV. Lett. 2000, 84, 1172-1175. (38) Hild, W.; Ahmed, S.; Hungenbach, G.; Scherge, M.; Schaefer, J. Tribol. Lett. 2007, 25, 1-7. (39) Chandross, M.; Lorenz, C. D.; Stevens, M.; Grest, G. S. To be submitted for publication.
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Figure 5. Plots of 100-point running averages of normal force vs separation for (]) chemisorbed, (×) 10% chemisorbed, and (4) 50% chemisorbed for a compression rate of 5 m/s and physisorbed at compressions rates of (O) 5 m/s, (0) 0.5 m/s, and (X) after equilibration. Results are for C11 SAMs with the 10 nm tip.
Figure 2. Snapshots for the 10 nm tip (a) during compression and (b) after decompression for C11 alkylsilane SAM. Silicon atoms are shown in yellow, oxygen in red, carbon in blue, and hydrogen in white. For clarity, only a 4-nm-thick slice from the center of the simulation box is shown.
Figure 3. Normal force as a function of compression for a tip in contact with a C11 alkylsilane SAM for the (0) 3, (]) 10, and (O) 30 nm tips. The inset shows an enlargement of the attractive regime with the 3, 10, and 30 nm tips shown as a solid line, a dashed line, and a solid line with points, respectively (where lines here simply connect points to guide the eye). The zero of the x axis is a tip-SAM separation of approximately 1.5 nm. The compression rate is 5 m/s.
Figure 4. Normal force as a function of separation for (×) C8, (b) C11, and (]) C18 alkylsilane SAMS in contact with the F ) 10 nm tip. The x axis has been shifted so that 0 corresponds to the beginning of repulsion. The compression rate is 5 m/s.
repulsive interaction quickly takes over, causing the force to rise. Because of the chain splay, it is easier to insert a sharper tip into the SAM than a more blunt tip and thus, in the repulsive region, the tips with larger F see the SAM as being stiffer. Figure 4 shows the compressive force for various chain lengths with the F ) 10 nm tip. As seen previously,14,40 shorter chains are stiffer than longer chains, resulting in the different curves seen in the Figure. The attractive region is independent of chain
length, as would be expected because the attraction is primarily between the terminal CH3 groups and the tip. Figure 5 shows the difference in behavior during compression between physisorption and chemisorption at the same chain length, n ) 11. As found in our previous work,17 chemisorbed chains are stiffer than physisorbed chains. This is a consequence of the fact that the latter cannot move out of the compression region, which removes a mechanism for force dissipation. Also shown in Figure 5 are systems in which either 10 or 50% of the chains are chemisorbed, with the remaining being physisorbed. As expected, the behavior of these systems is intermediate between the two extremes of 0 and 100% chemisorbed. In this Figure, we have shifted the x axis so that the zero of separation coincides with the first contact, as defined above, between tip and substrate atoms. The effect of compression speed on the simulation results is also shown in Figure 5. Compressing at 0.5 m/s, compared to 5 m/s, results in a SAM that appears to be more compliant. This is not surprising because the slower compression rate gives the chains more time to move out of the way of the tip. Also shown are the results of simulations similar to creep experiments in which we compress to a given separation and then equilibrate. The difference between these equilibrated points and the dynamic compression is quite large. Approximately 1 ns of simulation time is required for the force at a given separation to equilibrate. Although even experimental nanoindentation studies are not equivalent to creep measurements, these points are an indication that our compression rate is likely too fast to compare directly to experimental measurements. Because these simulations allow for the simultaneous measurement of both applied forces and true contact areas, we are able to test the predictions of various contact mechanics models without any fitting parameters. To this end, we show in Figure 6 the comparison of two standard contact mechanics models, the JKR41 and DMT42 models, as well as the relatively new thin coating contacts mechanics (TCCM) model developed by Reedy43 to our data. In this case, we show results in the elastic regime only for C11 chains with the 10 nm tip. In all cases, all material parameters were extracted from simulations, including the compression simulations shown in Figure 4 as well as simulations in which a flat SiO2 surface is brought into contact with a SAMcoated SiO2 surface (not shown). Both the standard JKR and DMT models are seen to underpredict the contact radius as a (40) Tutein, A. B.; Stuart, S. J.; Harrison, J. A. J. Phys. Chem. B 1999, 103, 11357-11365. (41) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301-313. (42) Derjaguin, B. V.; Muller, V. M.; Toporov, Y. P. J. Colloid Interface Sci. 1975, 53, 314-326. (43) Reedy, E. D. J. Mater. Res. 2006, 21.
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Figure 6. Contact radius a as a function of applied load P. Simulation data for physisorbed C11 chains with the 10 nm tip are shown as points. Predictions from JKR, DMT, and TCCM models are shown as dashed, dotted, and solid lines, respectively.
function of applied load, indicating the importance of the contribution of the thin compliant layer. In particular, note that the functional form of the dependence of contact radius on applied load is incorrect for these models. The TCCM model, however, shows excellent quantitative agreement with our measured contact areas, indicating that this model alone is appropriate for the study of these systems. It is not entirely unexpected that the JKR and DMT models provide poor descriptions of the contact mechanics because these models were not designed for systems in which the deformation is small outside of the contact region. However, these methods are commonly used to interpret experimental data, and the data here indicate that more caution should be exercised. We have not performed fits within these models to determine what aspects of the contact mechanics are incorrectly predicted; therefore, we are currently unable to comment on the possible implications for work in which they are used. The problems seen in application of the JKR and DMT models to tip/SAM contacts has been seen previously in experiments by Major et al., who attempted to use these models to explain the friction versus load curves obtained by the IFM.5 The authors found that whereas good fits for C12 and C18 were possible with the DMT model, the JKR model was needed to fit the C6 results. Other simulations have also noted issues with the application of these models to nanoscale contacts. Wenning and Muser first applied simple models to study the effects of tip curvature on the friction as a function of load L.44 This work found that friction is linearly dependent on L for commensurate surfaces and proportional to L2/3 for incommensurate interfaces. More recently, Gao, et al. found discrepancies between continuum predictions of contact radius and MD simulations of small diamond tips in contact with (001) and (111) diamond surfaces. These authors also found a linear dependence of friction on applied load for both commensurate and incommensurate interfaces. Detailed simulations to test the validity of continuum contact mechanics on the nanoscale have been performed by Luan and Robbins, first without45 and later with46 adhesion. For contact between rigid cylindrical or spherical surfaces (including a bent crystalline slab and cuts from amorphous and crystalline slabs) and an elastic substrate, these authors found that contact mechanics models can underpredict the contact areas by as much as 100% at small loads,45,46 with similar problems in the work of adhesion and stresses. One contribution to the discrepancy between the continuum calculations and the simulations is the roughness of the contact, which is not considered in the former. Another factor, (44) Wenning, L.; Muser, M. H. Europhys. Lett. 2001, 54, 693-699. (45) Luan, B. Q.; Robbins, M. O. Nature 2005, 435, 929-932. (46) Luan, B. Q.; Robbins, M. O. Phys. ReV. E 2006, 74, 026111.
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Figure 7. Snapshot for C11 alkylsilane SAMS in contact with the 10 nm tip after 2 ns of shear at a velocity of 2 m/s and a normal load of 10 nN. A 4 nm slice from the center of the SAM is shown with all atoms in the tip.
which may be more relevant here, is the compliance of the contact, which is generally neglected in continuum mechanics. However, the compliance is taken into account in the TCCM model, which may explain its more accurate predictions. The largest deviation from the TCCM model in Figure 6 is in the adhesive regime (i.e., P < 0) where the predicted slope of a versus P seems to be incorrect. This is likely an effect of the definition of contact, which is different in the simulations and the TCCM model where only tip and substrate atoms with repulsive interactions contribute to the contact area.43 The contact area used by Luan and Robbins is also different in that it includes contributions from any atomic pair with nonzero interactions (i.e., all pairs within the cutoff of the interactions).45,46 This definition would be inappropriate for our simulations because of the long-ranged Coulomb interactions in the atomistic force field. We therefore decided upon a small cutoff for contact interactions to approximate the conditions leading to overlap of the Lennard-Jones radii of tip and substrate atoms. We have verified that small changes in our definition of contact (i.e., separations different from 0.5 nm) do not lead to qualitative differences in our results. B. Shear Simulations. A snapshot from a shear simulation of C11 alkylsilane SAMS in contact with the 10 nm tip is shown in Figure 7. The general features seen in this snapshot are similar to those for other chain lengths and tip F. Shear proceeds with the tip moving to the left at 1 m/s in the Figure while the substrate moves to the right at the same speed for a relative velocity of 2 m/s. It has been shown in flat-plate simulations that chains tilt in the direction of shear.15 Similarly, chains in the SAM at the leading edge of the tip are forced to tilt in the direction of shear. An additional feature of the tip simulations is the material plowed up from the substrate by the tip that is visible above the SAM. We have found that the removal of material from the substrate occurs at loads below 10 nN. The effects of this material on the measured friction will be discussed below. In Figure 8, we show the accumulated results for the measured friction force as a function of applied normal force for four different chain lengths with the 10 nm tip. Again, the general features from the flat-plate simulations are seen in that the friction is linear as a function of load and there is little to no chain-length dependence. Following the derivation in Brukman et al.,47 we consider that the friction force depends primarily on the contact area and the applied load as F ) τoA(L) + µL, where F is the friction force, A is the contact area, L is the applied load, and τo and µ are constants. The linearity implies that the dependence on contact area is low and the second term dominates. We can (47) Brukman, M. J.; Marco, G. O.; Dunbar, T. D.; Boarman, L. D.; Carpick, R. W. Langmuir 2006, 22, 3988-3998.
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Figure 8. Friction force as a function of normal load for (b) C8, (9) C11, ([) C16, and (2) C18 alkylsilane SAMS in contact with the 10 nm F tip. Lines are fits from linear regression.
Figure 9. Friction vs normal force for C11 SAMs with the 10 nm tip. Systems shown are (b) fully physisorbed, (2) fully chemisorbed, (]) 10% chemisorbed, and (9) in-plane physisorbed (see text). Lines are fits from linear regression. Some points for the fully chemisorbed systems lie outside the range of this graph.
therefore reasonably compare friction between systems by comparing the friction coefficient µ. Although µ varies little with chain length, as found previously, the values of µ here are higher than those from the flat-plate simulations by a factor of 3 to 4. To understand this difference, we first note that the major new features of this work as compared to our previous work are (a) the physisorption of chains, this is different from our previous work on physisorbed chains because of the potential for the tip to penetrate the monolayer and cause chains to move, (b) the removal of material from the substrate, and (c) the tip/substrate geometry. In Figure 9, we show the results of a number of different shear simulations of C11 chains with the 10 nm tip aimed at trying to resolve the differences in the friction coefficient. The comparison between the data for fully physisorbed and fully chemisorbed systems is seen in the Figure to account for a reduction in the friction coefficient by a factor of 2. In fact, with only 10% of the chains chemisorbed (with the remaining 90% physisorbed) the same effect is seen. This alone, however, does not distinguish between points a and b above because it is unclear whether it is the physisorption of chains itself that contributes to the higher friction coefficient or the removal of material. To resolve this problem, we created a system in which the chains in the SAM are fully physisorbed to the substrate but the height of the head group is fixed. In other words, chains are able to move freely throughout the plane but are unable to be removed from the SAM. We refer to this situation as in-plane physisorption. The data in Figure 9 demonstrates that the measured friction for chains that are physisorbed in plane is identical to that for fully physisorbed chains. Therefore, the removal of material does not contribute strongly to the friction coefficient. The major factors in the higher friction coefficient seen here are likely the following: (1) The physisorption of chains: the ability of chains to move laterally throughout the monolayer leads to tighter packing in the monolayer (i.e., less area per chain) upon tip insertion (see below), which increases energy dissipation.
Langmuir, Vol. 24, No. 4, 2008 1245
Figure 10. Friction vs load for C11 alkylsilane SAMS in contact with the (O) 3, (0) 10, and (4) 30 nm tips. The difference in pull-off force is evident at the y-axis crossing. Lines are fits from linear regression.
(2) Tip-on-substrate geometry: flat-on-flat geometry enables the movement of the SAM as a whole, whereas a tip will expend more energy moving chains that are outside its local deformation zone. Separate simulations of flat, bare silica substrates shearing on a flat SAM demonstrate a lower friction coefficient. (3) Hydrophobic tip coating: the same simulations discussed in point 2 still show higher friction than SAM-on-SAM shear simulations, indicating that SiO2 sliding on the SAM leads to a higher friction coefficient. (4) Shear velocity: previous simulations15,17and experiments37,38 tend to indicate that the high shear rates in these simulations lead to higher friction coefficients, but computational restrictions make this difficult to test. Compiled results for shear simulations on C11 SAMs with different tip sizes are shown in Figure 10. As predicted from the plowing mechanism proposed by Carpick and co-workers,9 a change in tip radius should have two effects. First, the contact area will change, but this has been shown above to have a minimal impact on friction in comparison to the load dependence. Second, the pull-off force will change. Therefore it is expected that different tip sizes will have friction versus load curves that are parallel lines offset by the pull-off force. This is exactly what is seen in Figure 10. The difference in pull-off force is evident in different y-axis crossings as well as in the inset of Figure 3, where we show the adhesive regime of the compression simulations for C11 chains with the different tips. Note that we do not expect the linear relationship between friction force and normal load to apply throughout the adhesive regime, as implied in Figure 10. We have shown in our previous work on flat-plate systems that the nonmonotonic (and therefore nonlinear) dependence of friction on normal load in the adhesive regime seen in IFM experiments5,48 can also be seen in simulations at constant separation.17 The physisorbed systems respond to the insertion of a tip through tighter packing within the monolayer and a corresponding increase in the lateral pressure in the SAM. These effects can be seen in the snapshots shown in Figure 12a,b. In both Figures, the same C18 monolayer is shown during a shear simulation, but in Figure 11a the applied load is 5.7 nN whereas in Figure 11b it is 17.6 nN. At the lower load, the tip has only slightly penetrated the monolayer and is predominantly sliding on top of the chains. The lower pressure is clear from the defect in the SAM to the left of the tip. Note that even without penetration the tip is still clearly pushing over chains at the leading edge. With only approximately 10 nN more of applied force, however, the increase in lateral pressure and tighter chain packing is obvious in Figure 11b. Not only has the defect hole (a consequence of the procedure used to physisorb the chains in our model) to the left of the tip (48) Kim, H. I.; Boiadjiev, V.; Houston, J. E.; Zhu, X. Y.; Kiely, J. D. Tribol. Lett. 2001, 10, 97-101.
1246 Langmuir, Vol. 24, No. 4, 2008
Figure 11. Snapshots of the center 4 nm of a shear simulation with C18 alkylsilane SAMS and a 10 nm tip at applied loads of (a) 5.6 and (b) 17.6 nN. The increase in packing and lateral pressure with increased applied force is discussed in the text.
Figure 12. Top down view of the monolayer slice of the C11 alkylsilane SAMS from Figure 11 with the tip and substrate removed for clarity. The alignment of chains on the left side of the image due to the shearing of the tip can be seen.
closed but chains to the right of the tip can be seen being forced out of the SAM. The pushing over of chains is again evident, as is the plowing up of material from the substrate. Further qualitative confirmation of the plowing mechanism is visible in Figure 12, where we show a top-down view of the 4 nm slice of the SAM from Figure 7 with the tip removed. On the right side of the image, outside of the region affected by the tip, the chains are randomly aligned. In the center, under the tip, the compression and deformation of the SAM are obvious, whereas on the left side of the Figure the alignment of chains in the direction of shear is clear.
Conclusions Tip-based nanotribology simulations bring about two major changes as compared to flat-plate simulations, namely, loaddependent contact areas and the possibility of the removal of material from the substrate. In the case of material removal, our
Chandross et al.
simulations demonstrate that, at least for fully physisorbed monolayers, even extremely low loads can lead to damage to the monolayer. Although the effect on the measured friction is small, this has strong implications for the wear and degradation of SAM coatings, particularly in the presence of water. We will discuss these effects at length in a different publication.49 It is, however, unlikely that a fully physisorbed monolayer is an accurate description of experimental systems.36 We have found that adding covalent bonds to 10% of the chains in the systems, which has a significant effect on the frictional response, does not prevent such damage. The dependence of the contact area A on the applied load L has been shown to be correctly described by the thin coating contact mechanics theory,43 which describes the relationship as A∝L1/2, in contrast to the ubiquitous JKR and DMT theories that use an exponent of 2/3. However, the contact area appears to have little effect on the frictional response of the monolayer. The more dominant effect on the friction appears to be the disorder present in the system, as has been claimed previously both from experiments4 and simulations.50,15 Both our flat-plate and tip-based simulations show that at a given coverage the friction is independent of chain length. Experimentally, it is difficult to decouple the chain length and the coverage (and therefore disorder), resulting in the often-quoted result that friction decreases with increasing chain length. In our flat-plate simulations on highly ordered SAMs as well as some experimental work on more ordered systems,10 the native-chain-length dependence is revealed to be nonmonotonic. A majority of experiments are therefore likely focused on the effects of coverage rather than chain length. This conclusion is supported by both simulations13,16,15 and experiments6-9 that have clearly demonstrated that the addition of disorder to a system increases the friction. Recent experimental work9 has ascribed the measured frictional response to “plowing” of the chains, in which the dominant energy-dissipation mechanism is the tilting of chains at the leading edge of the tip. Our simulations verify this mechanism, at least in a qualitative fashion at this point. Future work will address a quantitative demonstration of the plowing mechanism. Acknowledgment. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC04-94AL85000. LA702323Y (49) Lane, J. M. D.; Chandross, M.; Lorenz, C. D.; Stevens, M.; Grest, G. S. Langmuir, submitted for publication, 2007. (50) Tutein, A. B.; Stuart, S. J.; Harrison, J. A. Langmuir 2000, 16, 291-296.
