Langmuir 2002, 18, 1453-1456
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Measurement of Retarded Dispersion Forces of Mica A. M. Stewart, V. V. Yaminsky,* and S. Ohnishi Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT, 0200, Australia Received October 15, 2001. In Final Form: December 28, 2001 We report direct high-resolution capacitor measurements of the force against distance between crossed cylindrical mica surfaces in air. The dispersion interaction in the range of distances between 20 and 100 nm occurs in the retarded regime. The force is found to vary as the inverse cube of the distance of nearest separation in good agreement with the early measurements of Tabor and Winterton. The measured deformation on contact is in agreement with the theory of Johnson, Kendall, and Roberts. The plane of origin of the retarded van der Waals force is shifted outward from the equilibrium contact position at zero external load by the magnitude of contact deformation.
Introduction The problem of the interaction between solid surfaces in air (vacuum) is at the heart of modern colloid science. In the early 20th century, Tomlinson1 followed by Bradley2 observed large, reproducible adhesion and friction forces between fused silica surfaces. The experiments, carried out with freshly molten filaments and spheres, have shown that the surface smoothness of the elastic macroscopic glass bodies enables a quantitative study of van der Waals forces to be carried out. Such measurements and their later extension to the study of interaction of hydrophobic surfaces in liquid media3 have led to the development of new experimental techniques of surface preparation and force measurement. Fundamental advances that followed the pioneering observations include the Derjaguin approximation,4 the exact accounting of the surface geometry, theories of adhesion and contact elasticity5 (the JKR-Maugis), and the Hamaker-Lifshitz theory of surface forces.6-8 By pointing to the long-range nature of the van der Waals attraction, the theory inspired a new round of search for experimental evidence of the effect.9-12 The experiments inspired new refinements of the theory.13,14 The research continues15-17 with extensions toward thin films, colloid * Corresponding author. E-mail:
[email protected]. (1) Tomlinson, G. Philos. Mag. 1928, 6, 695. (2) Bradley, R. S. Philos. Mag. 1932, 13, 853. (3) Yaminsky, V. V.; Yusupov, R. K.; Amelina, E. A.; Pchelin, V. A.; Shchukin, E. D. Kolloidn. Zh. 1975, 37, 918 [Colloid J. USSR 1975, 37, 824]. (4) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155. (5) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301. (6) Hamaker, H. C. Physica 1937, 4, 1058. (7) Lifshitz, E. M. Zh. Eksp. Teor. Fiz. 1956, 29, 94 [Sov. Phys. JETP 1956, 2, 73]. Dzyaloshinskii, I. E.; Lifshitz, E. M.; Pitaevskii, L. P. Adv. Phys. 1961, 10, 165. (8) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: London, 1976. (9) Derjaguin, B. V.; Abrikosova, I. I. Zh. Eksp. Teor. Fiz. 1951, 21, 945. Derjaguin, B. V.; Abrikosova, I. I.; Lifshitz, E. M. Q. Rev., Chem. Soc. 1956, 10, 295. (10) Sparnaay, M. J. Physica (Amsterdam) 1958, 24, 751. (11) Tabor, D.; Winterton, R. H. S. Proc. R. Soc. London, Ser. A 1969, 312, 435. (12) Israelachvili, J. N.; Tabor, D. Proc. R. Soc. London, Ser. A 1972, 331, 19. (13) White, L. R.; Israelachvili, J. N.; Ninham, B. W. J. Chem. Soc., Faraday Trans. 1 1976, 72, 2526. (14) Chan, D.; Richmond, P. Proc. R. Soc. London, Ser. A 1977, 353, 163. (15) Ederth, T. Phys. Rev. A 2000, 62, 062104-1. (16) Harris, B. W.; Chen, F.; Modiheen, U. Phys. Rev. A 2000, 62, 052109.
stability, and other areas of the modern physics of condensed states of matter.18 In the early work, relatively little attention was paid to surface characterization. Distance resolution remained limited. The effort culminated with the work of Tabor and Winterton,11 the most advanced and up-to-date measurement of van der Waals forces between mica surfaces in air, done with a prototype of the modern surface force apparatus (SFA).19 It is essentially the macroscopic radius of the mica surfaces that enables accurate measurements to be made. Like those of vitreous silica, the crystalline mica surfaces are smooth and homogeneous. On an atomic scale, molecular alignment for liquids and glasses is due to the effect of surface tension; in the case of mica, it is due to crystallographic regularity at the plane of cleavage, but with the inevitable introduction of defects.20 Multiple beam interferometry is at the basis of the traditional SFA technique. The instrument was subsequently adapted to study forces between surfaces in a liquid.19 Although advantageous in historical retrospective, in the perspective of quantitative measurements the SFA suffers significant limitations, hindering progress in the area.21 Indeed, the distance resolution of surface interferometry is not sufficient to detect submonolayer effects of adsorption and forces acting at relatively short distances. Complexity of surface preparation and data analysis and associated difficulties of automation of surface force measurements stand in the way of easy acquisition of accurate results. The proliferation of commercial atomic force microscopes (AFMs) as a substitute tool for quantitative surface force studies22 did not solve the problem. This method of surface force measurement is also lacking the desired simplicity, in view of the laborious procedure of attachment of a “particle” to a microscopic “cantilever” that inevitably leads to surface contamination. Compared to the properties of freshly molten glass or freshly cleaved mica, the surface chemistry and geometry of the microscopic particles are even more difficult to control. Accurate mechanical control is not easy (17) Bevan, M. A.; Prieve, D. C. Langmuir 1999, 15, 7925. (18) Israelachvili, J. N. Intermolecular and Surface Forces; Academic Press: New York, 1985; 2nd ed. 1992. (19) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (20) Ohnishi, S.; Hato, M.; Tamada, K.; Christenson, H. Langmuir 1999, 15, 3312. (21) Yaminsky, V. V.; Ninham, B. W. Adv. Colloid Interface Sci. 1999, 83, 227. (22) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239.
10.1021/la0156311 CCC: $22.00 © 2002 American Chemical Society Published on Web 02/01/2002
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to achieve either, due to effects such as cantilever twist. Calibration is among unresolved issues.23 Because of the microscopic particle radius, the force, according to the Derjaguin approximation, is orders of magnitude smaller than in the SFA case. Stray mechanical noise typically is not a problem for SFA measurements in a liquid due to viscous damping of the large radii surfaces but is a reason force measurements in the vapor phase are scarce. With an AFM, attachment of a relatively massive sphere to a low-stiffness cantilever takes the originally high resonance frequency down, into the range of external excitation. The problem is overcome in the AFM case by use of inertial suspensions. The advantages of the different methods have been combined in a new technique.24 In brief, the ordinary SFA has been equipped with a capacitor, the most accurate physical sensor of small displacements. Earlier attempts to develop alternative means of surface force measurements25,26 included the use of capacitors27,28 and piezoelectric sensors.29-32 The instrument we use is simple, reliable, versatile, and most accurate. Measurements can be carried out in manual and automatic modes, allowing extended dynamic control, as in the AFM, and with a greater range of possibilities. The instrument retains all the SFA benefits of a macroscopic device, including the simplicity of surface mounting and high surface energy resolution. The F/R gain that follows by use of large curvature surfaces is amplified by higher sensitivity of the capacitor sensor. In combination with traditional optical SFA microscopy/interferometry, the method allows simultaneous measurement of normal and lateral contact deformations, as well as “surface to surface” distances. The technique is not restricted to semitransparent mica surfaces silvered at the back and glued onto silica disks before use, following the laborious cleavage procedure that is needed to form macroscopic areas of atomically uniform planes of micron thicknesses. The Tomlinson-Bradley spheres/cylinders still remain an attractive alternative. These geometrically ideal surfaces form under conditions that are more easily controlled. They can be used in the new device in extending our previous studies3 later taken with capacitor27 and bimorph detectors,29-32 along with nontransparent objects and nonSFA applications. The method can be readily adapted to a wide range of physical problems that require simultaneous measurement of small forces and displacements. This is crucial in the study of confined materials, molecular rheology and tribology, wetting, nanoscale devices, and so forth. Experimental Technique The measurements were carried out with the capacitance dilatometer attachment for the surface force apparatus described (23) Maeda, N.; Senden, T. J. Langmuir 2000, 16, 9282. (24) Stewart, A. M. Meas. Sci. Technol. 2000, 11, 298. (25) Peschel, G.; Adlfinger, K. H. Bunsen-Ges. Phys. Chem. 1970, 74, 351. Peschel, G.; Belouschek, P.; Muller, M. M.; Muller, M. R.; Konig, R. Colloid Polym. Sci. 1982, 260, 444. (26) Bailey, A. I.; Daniels, H. J. Phys. Chem. 1973, 77, 501. (27) Yaminskii, V. V.; Steblin, V. N.; Shchukin, E. D. Pure Appl. Chem. 1992, 64, 1725. (28) Tonck, A.; Georges, J. M.; Loubet, J. L. J. Colloid Interface Sci. 1988, 126, 150. Crassous, J.; Charlaix, E.; Gayvallet, H.; Loubet, J. L. Langmuir 1993, 9, 1995. Frantz, P.; Agrait, N.; Salmeron, M. Langmuir 1996, 12, 3289. (29) Stewart, A. M.; Parker, J. L. Rev. Sci. Instrum. 1992, 63, 5626. Parker, J. L.; Stewart, A. M. Prog. Colloid Polym. Sci. 1992, 88, 162. (30) Parker, J. L.; Yaminsky, V. V.; Claesson, P. M. J. Phys. Chem. 1993, 97, 7706. (31) Stewart, A. M. Meas. Sci. Technol. 1995, 6, 114. (32) Yaminsky, V. V.; Ninham, B. W.; Stewart, A. M. Langmuir 1996, 12, 836. Yaminsky, V.; Jones, C.; Yaminsky, F.; Ninham, B. Langmuir 1996, 12, 3531.
Letters previously.24 The planar capacitor, with an adjustable gap of around 100 microns over an area of about 1 cm2, mounted on the spring carrying shaft, senses a displacement of the spring under the effect of a force. This can be the surface force that varies by changing the distance or for example the external magnetic load that provides an alternative mode of force control.3,27,33 In the measurements of long-range van der Waals forces reported here, a magnet, particularly beneficial for studies of contact deformations, was not used in order to reduce the mass and susceptibility to vibration of the moving surface assembly. The data were acquired by driving the rigidly mounted surface with a piezoelectric transducer, the standard SFA/AFM mode of displacement control. We use a double cantilever spring to eliminate shear tilt effects that might interfere with measurements of adhesion in this mode. To detect the small forces caused by dispersion interaction, a relatively weak spring of spring constant 171 N/m was used. The use of variable stiffness and electromechanical feedback springs is possible.29 In this way, the capacitance bridge out-of-balance signal is directly proportional to the deflection of the spring from its equilibrium position and hence to the surface force. Accurate force and displacement calibration is straightforward and can be readily done in several independent ways ultimately traceable to interferometry. Nonlinearity is negligible for typical surface force measurements over ranges less than a micron. For applications requiring the monitoring of much larger spring deflections, the full nonlinear analysis can be carried out, as described previously,24 and in fact was used in these measurements. Signal instability, whenever it might occur, is entirely due to thermal drift; this is insignificant under our experimental conditions, a temperature-controlled room with temperature fluctuations of less than 0.1 °C. The apparatus was suspended by three rubber cords in order to reduce ambient vibration. The resonant frequency of the suspension, of the order of 1 Hz, is sufficiently low to cut off the mechanical noise of the building effectively. Although the mechanical resonant frequency of the double cantilever spring itself was low, in our case around 20 Hz, the amplitude of the stray vibration of the moving surface was reduced to around 0.1 nm. This efficient vibration isolation proved essential for the high-resolution measurements in air. A large drive voltage of 25 V amplitude compared to the 10 V amplitude used previously24 was used to increase the electrical signal-to-noise ratio. Van der Waals forces were measured in a dry atmosphere, using CaH2 as the drying reagent. Some measurements were done under controlled humidity, by use of saturated aqueous salt solutions, and in vapors of organic liquids. The local radii of curvature R of the surfaces were measured in two perpendicular directions35 by interferometry. The silica spheres used in some experiments were formed by the standard procedure2,3 of melting an end of a 2 mm diameter Suprasil rod in an oxygen-gas flame before the experiment. Unlike the Tabor-Israelachvili interferometer in which the thin glued mica is rippled, leading to a variation of the local radius, the Bradley spheres are geometrically uniform at all scales down to atomic, as is obvious from imaging and force measurement. Local radii that enter the DerjaguinJKR equations are equal in this case to the macroscopic radii of curvature. The latter can be readily measured to a great precision. Pull-off forces are reproducible and are found to be the same at different contact positions. As for mica, the magnitude of adhesion, as expected from the surface energy estimates based on wetting, adhesion, and adsorption, is the ultimate proof of surface smoothness.
Results and Discussion Figure 1 shows the results obtained with mica surfaces when they approach each other at a constant speed of the piezoelectric drive; the speed was slow enough to have no effect on the results. The distance of separation D was calculated by taking into account the measured deflection (33) Stewart, A. M.; Christenson, H. K. Meas. Sci. Technol. 1990, 1, 1301. (34) Ohnishi, S.; Yaminsky, V. V.; Christenson, H. K. Langmuir 2000, 16, 8360. (35) Stewart, A. M. J. Colloid Interface Sci. 1995, 170, 287.
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Figure 1. Two mica surfaces interacting across air: the force (F/R) versus displacement D measurement with the capacitor. The long-range attraction is essentially the same in a dry atmosphere and at low ambient humidity (below 50%). The force-distance profile in this range does not depend on the speed of approach between the surfaces. The retarded Lifshitz form holds over the entire range of spring stability down to 20 nm. The inset shows the data on a double logarithmic scale; the slope of the experimental line corresponds to the inverse third power of the distance. The estimated (refs 11 and 12) nonretarded form is shown as a separate line. The straight dashed line shows the asymptotic form of the jump. The plane of origin of the van der Waals force occurs at a distance of 14.2 nm from the measured central position of the interacting bodies at zero load. The corresponding value of the normal contact deformation that defines the zero distance between undistorted surfaces, as calculated by the JKR theory from the radius of contact measured by interferometry, is also 14.2 nm. The experimental result shows an excellent match between the Lifshitz and JKR theories. Taken together, they provide a complete description of the interaction over the entire range of physical displacements. Experimental results obey the JKR relation between the radii of the undistorted surfaces, the maximum attraction (the pull-off force), and the elastic modulus. The contact JKR regime that appears on the scale of the plot almost as “hard wall” is indeed shifted from the zero distance position of undistorted surfaces by the value of the elastic compliance. The two complementary theories provide a self-consistent description of the interaction based on a complete set of experimental data, with no adjustable parameters involved.
of the spring carrying the other surface. With the present setup, the resolution is limited only by the electrical noise, effectively less than 0.1 nm in amplitude in the mechanical equivalent by using the relatively weak spring. This corresponds to a force amplitude of less than 20 nN. For the given spring stiffness, the detection limit, and the sphere radii R (2.34 cm), the van der Waals force shows up from distances below 100 nm (Figure 1). With decreasing distance, the force increases by orders of magnitude and can be directly measured down to a distance of about 20 nm. Below this point, the force gradient of the van der Waals attraction exceeds the stiffness of the spring. At this critical instability condition, mechanical equilibrium is lost, and the surfaces “jump” with acceleration into contact; the straight dashed line in the figure shows the asymptote of the jump. Use of stiffer springs reduces the range of the jump-in instability. Larger and steeper attractive forces acting at shorter distances then can be measured. In our case, the forces are resolved at a high accuracy over a wide dynamic interval with a single spring. The hitherto most advanced measurements by Tabor and coworkers11,12 were based on an indirect approach, by using
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several springs of different stiffnesses, and comparison of the jump-in distances from a number of separate experiments plotted together. In our case, the entire curve is recorded in one run. The measurement typically takes less than a minute. Over the detection range, the experimental curve scales with the limiting form for the retarded van der Waals attraction. As can be seen from the inset to the figure, the force varies as F/R ) -(2π/3)B/D3 from the jump distance up to the largest distance of measurement. The continuous line through data points in the figure shows the theoretical equation, and the broken line shows the nonretarded form with the Hamaker constant according to the measurements at shorter distances.11,12 The best fit value of the retardation constant B ) 0.93 × 10-28 J m found by us agrees with the value of 0.81 × 10-28 J m reported by Tabor and Winterton11 and that of 0.97 × 10-28 J m measured later by Israelachvili and Tabor.12 Error, of the order of (5%, is mostly associated with the measurement of R. The data of Tabor and co-workers were discussed, based on theoretical calculations from the dielectric spectrum of mica, by Chan and Richmond.14 In contact, the surfaces deform elastically. During approach, the mechanical repulsion shows up by deceleration of the surfaces over the range of negative displacements. Accurate solution of the equation of motion allows studying forces over the range of the jump.34 According to the Johnson-Kendall-Roberts (JKR) theory5 of adhesion, the contact diameter at zero force a(0) corresponds to normal displacement of the surfaces by δ ) a(0)2/3R giving δ ) 14.2 nm with R ) 2.34 cm and a(0) ) 31.6 microns, measured by interferometry. The plane of origin of the van der Waals force, as obtained from the best fit of the measured force-distance profile to the retardation form, is situated at precisely the same distance of 14.2 nm, as theoretically expected. Actually, the fit in the contact region in Figure 1 is to a full JKR compliance curve, but the negative slope of the curve is too large to be seen in the figure. With experimental errors of the two independent δ estimates of several percent, our result is in excellent agreement with the two independent theories, the Lifshitz theory of long-range forces, rigorous in the limit of large distances, and the theory of contact mechanics that assumes effectively zero range of adhesion forces. The result further shows that the JKR normalto-lateral displacement relationship does indeed hold accurately, although the surface substrates are not uniform bodies but layered structures. The effective elastic modulus K calculated from the measured pull-off force of 12.1 mN by JKR theory5 is quite high (K ) 3.6 × 1010 J/m3), in accordance with the relatively large thickness of the mica sheets (about 8.5 microns). Essentially, the experiment provides a complete set of data allowing a self-consistent description of the interaction over the entire range distances, from physical infinity (over 100 nm) to essentially negative displacements that correspond to squashing of the bodies in the adhesive contact. The results enable testing the theories of surface forces and contact deformations with which they comply. Measurements using fused silica spheres of an order of magnitude smaller radius show similar effects, but the forces and contact deformations are accordingly smaller, as follows from the Derjaguin and JKR equations. The Lifshitz theory of van der Waals forces, the geometrical Derjaguin approximation, the JKR theory of adhesion of elastic bodies, and related surface energy parameters can be unambiguously verified also in this case. Extended
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analysis of these results will be provided in a forthcoming publication.36 From the results presented in this paper, it is clear that the technique offers great potential for the quantitative study of surface forces. To test further possibilities, we have carried out some of the measurements in undersaturated and saturated vapors of water and organic liquids. Adsorption of vapors on the capacitor plates does not change the position of the baseline to a significant extent because the adsorbed thicknesses remain many orders of magnitude smaller than the widths of the capacitor gap. Capillary forces and volumes of the bridging menisci37 can be measured accurately along with other surface phenomena. The instrument can be used to study forces acting across liquids and solutions, as the capacitor sensor is operable in the vapor phase. More detailed measurements of retarded and nonretarded forces, with account for changing surface geometry, material elasticity, surface chemistry, and medium effects of capillary condensation and adsorption, are currently under way. Conclusions Hitherto, SFA measurements showed forces of the van der Waals attraction in vapor indirectly, mainly by (36) Yaminsky, V. V.; Stewart, A. M. Langmuir, submitted. (37) Ohnishi, S.; Yaminsky, V. V. Langmuir, submitted.
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measurement of the distance of jump into contact. In many cases, the force curves before the onset of the mechanical instability are not clearly resolved. This makes it difficult to discriminate between retarded and nonretarded forces acting at the instant of the jump. Some AFM measurements with much lower stiffness springs show the curves explicitly, but effects of surface roughness and precise calibration remain an issue.16 The effects of contact deformation are invariably disregarded in AFM experiments. Our results show that with mica the retardation form is effectively exact and holds accurately at distances larger than 20 nm. Fits of the jump-in data to the nonretarded Hamaker equation, as is often done in the SFA and AFM literature, suffer from ambiguity. Our results show that a self-consistent account of long-range surface forces and contact deformations can be achieved by measurements with a capacitor. Acknowledgment. We thank Professor Richard Pashley for sharing reminiscences of earlier measurements of the van der Waals attraction and Professor Derek Chan for a literature update on the AFM studies of Casimir forces. LA0156311
