Langmuir 2003, 19, 4037-4039
Interaction of Glass Surfaces in Air: Dispersion Forces in the Retarded Regime V. V. Yaminsky and A. M. Stewart* Department of Applied Mathematics, Research School of Physical Sciences and Engineering, The Australian National University, Canberra, ACT, 0200, Australia Received December 2, 2002. In Final Form: February 19, 2003
Introduction In a previous paper,1 we reported measurements of the interaction between mica surfaces in air carried out with a capacitor dilatometer attachment to a surface force apparatus (SFA).2 The device works by measuring, with a capacitance sensor, the distance that a double cantilever spring moves under the influence of a force. The sensor has distance and force (F/R) resolutions higher than those of current SFA and atomic force microscopy (AFM) techniques, and the instrument permits simultaneous measurement of long-range dispersion forces and elastic contact deformations. The method allows the study of new surfaces and materials as well as the traditional SFA substrate, mica, provided that the surfaces are smooth on a nanometer scale. The surfaces need not transmit light because the methods of optical interferometry are not used. In this paper, we report measurements on Pyrex glass whose surface can be made smooth by melting. In earlier years, the van der Waals forces predicted by the Lifshitz theory3-5 were measured between macroscopic glass lenses in air.6,7 In these pioneering experiments, the roughness of the mechanically polished surfaces and archaic force measurement techniques did not allow useful results to be obtained. The first credible measurements of dispersion forces of mica in air, by Tabor and Winterton,8 were carried out with the precursor of the modern SFA. Surface force measurements in air are made difficult by ambient vibration that is not damped by viscosity, an important factor that allows more accurate measurements of forces in liquids. Even in the absence of vibration, the distance resolution of the optical SFA technique remains limited. Given this, the forces were measured indirectly through the “flip” point of mechanical instability by using springs of different stiffnesses. The data from the separate experiments, each covering a single distance point, were combined on one plot.8 Hitherto, in air, only mica surfaces have been investigated with the SFA in the retarded regime. More recently, the interaction of commercial powder particles of silica and other materials was studied * Corresponding author. E-mail:
[email protected]. (1) Stewart, A. M.; Yaminsky, V. V.; Ohnishi, S. Langmuir 2002, 18 (5), 1453-1456. (2) Stewart, A. M. Meas. Sci. Technol. 2000, 11, 298-304. (3) Hamaker, H. C. Physica 1937, 4, 1058-1072. (4) 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. (5) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: London, 1976. (6) Derjaguin, B. V.; Abrikosova, I. I. Zh. Eksp. Teor. Fiz. 1951, 21, 945-946. Derjaguin, B. V.; Abrikosova, I. I.; Lifshitz, E. M. Q. Rev., Chem. Soc. 1956, 10, 295. (7) Sparnaay, M. J. Physica (Amsterdam) 1958, 24, 751. (8) Tabor, D.; Winterton, R. H. S. Proc. R. Soc. London, Ser. A 1969, 312, 435.
4037
by AFM,9 the imaging method adapted for measurement of surface forces. The results suffer from ill-defined surface geometry, calibration difficulties, stray mechanical distortions, and problems of contamination. Recently, we reported the first direct measurement of retarded van der Waals forces between mica surfaces in air with a macroscopic SFA device using a single spring.1 The enhanced resolution enabled us to record both the long-range attraction and the contact repulsion in a single run. In our present measurements, we have utilized the property of molecular smoothness of molten glass surfaces, as suggested by the old10,11 and more recent12 studies of adhesion. Our experimental setup does not inherit the intrinsic inaccuracies and uncertainties of surface force measurements based on studying the interaction of microscopic particles affixed to microscopic cantilevers. We verify the consistency of our fitting procedure by calculating the same parameter, the contact displacement of origin of the dispersion force, from different physical properties measured in the experiment. Materials and Methods Surface smoothness, homogeneity, and absence of plasticity are the desirable requirements for the substrates used in surface force measurements. Glasses are particularly suited for this role. The vitreous state retains the structural evenness of liquid surfaces aligned by surface tension. Silica glasses are chemically inert and insoluble in most liquids. Fresh surfaces, smooth and clean, are easy to produce by flaming. The elimination of the laborious preparation and mounting procedures needed for the optical measurements with mica reduces the risks of contamination. Glasses are hard, high-strength, nonplastic materials. Their response to strain is purely elastic; deformations are reversible to the point of brittle fracture. Because of their isotropic physical properties, the exact equations of contact mechanics are applicable in this case. The theory is simpler to apply than for crystalline bodies and particularly for the case of the elastically complex mica-glue-silica structures.13 Compared to pure silica that softens at high temperatures but does not melt into a liquid, one can readily form a glass droplet of up to a centimeter in diameter just by melting the end of a glass rod in an oxygen-propane flame. The semispheroids so obtained were installed in the SFA straight after cooling to room temperature without further treatment or were washed with distilled Millipore water and dried with nitrogen before installation. Because of gravitational squashing in the course of the bulk melting, the solidified macroscopic droplets of glass are not exactly spherical. Figure 1 shows a microscope image of one of the surfaces used in these experiments. The curvatures of two orthogonal cross sections were measured by geometrical fitting of a computercaptured image. The exact expression relating the four principal curvatures was used to calculate the effective radius R.14,15 Unlike with the mica-SFA-interferometer, where the locally varying effective curvature has to be measured at each contact position, the glass surfaces are closer to the geometrically ideal, having the same microscopic curvature at any point as the macroscopic curvature around the contact zone. The fact that the force measurement results do not change by changing the contact position confirms that effective values of curvature, as (9) Senden, T. J. Curr. Opin. Colloid Interface Sci. 2001, 6 (2), 95101. (10) Tomlinson, G. Philos. Mag. 1928, 6, 695. (11) Bradley, R. S. Philos. Mag. 1932, 13, 853. (12) Yaminsky, V. V.; Yusupov, R. K.; Amelina, E. A.; Pchelin, V. A.; Shchukin, E. D. Kolloidn. Zh. 1975, 37, 918-925 [Colloid J. USSR 1975, 37, 824-829]. (13) Christenson, H. K. Langmuir 1996, 12, 1404-1405. (14) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155. (15) Stewart, A. M. J. Colloid Interface Sci. B 1995 170, 287-289.
10.1021/la0269412 CCC: $25.00 © 2003 American Chemical Society Published on Web 03/25/2003
4038
Langmuir, Vol. 19, No. 9, 2003
Notes
Figure 1. The image of a quasi-spherical glass surface and the geometrical fit to its profile. The point where the surfaces touch at contact is at the bottom, and the curvature is matched at that point. Because of the deformation under gravity during melting, the surface is not exactly spherical and is flattened in the axial (vertical) direction. The meridian radius of curvature at the pole on the contact point on the bottom is larger by 10% than the equatorial radius in the horizontal cross section. The axial z symmetry of the surface follows from observation in two orthogonal x and y directions. The second main cross section (the y profile is not shown) gives the same radius of curvature, 0.483 cm. Averaged over the two surfaces as R ) R1R2/(R1 + R2) in the sphere-sphere approximation, the geometrical factor R in the Derjaguin and JKR equations corresponds to R ) 0.242 cm with both surfaces having almost identical curvatures in this case. well as the surface energy related to the surface chemical properties, do not vary over the surfaces. The surfaces are chemically and geometrically uniform. The surface force apparatus with capacitive detection of displacement was that used previously,1,2 with a double cantilever spring of stiffness 144 N/m. The surfaces, which were rigidly clamped to their holders, were moved together with a piezoelectric transducer. Simultaneous measurement of pull-off forces and force-displacement profiles provides a complete set of accurate surface force data. The measurements were carried out in air of low humidity (relative humidity of about 30%).
Results and Discussion Experimental values of force F (in µN) between the surfaces against separation between the surfaces D, measured on the approach between two Pyrex surfaces with R ) 0.242 cm, are shown in Figure 2. The data, obtained by bringing the surfaces together by means of the piezoelectric transducer sufficiently slowly that inertial effects were unimportant up to the point of mechanical instability, when the surfaces jumped together, took about 1 min to obtain. The data points to the right of D ) 0 are in the dispersion regime; the almost vertical column of data points to the left of D ) 0 are in the contact or Johnson-Kendall-Roberts (JKR) regime. The results at different contact positions, obtained by slightly offsetting the surfaces horizontally, are similar, indicating the absence of local radius variations. Averaging over several curves enables us to enhance the accuracy beyond the level of the noise (∼0.1 nm amplitude). When the surfaces are not in contact, as in the right side of the figure down to the point of inward jump, which is about 8 nm from contact, they retain their essentially undistorted shape. However, when they snap into contact, as in the left-hand side, they squash together and distort in the way described by the theory of Johnson, Kendall, and Roberts.16 Accordingly, because the axial contact deformation on contact cannot be obtained directly, the origin of the D coordinate is not apparent from the bare experimental data and has to be deduced from analysis of it. The criterion to be met is that any fit satisfies, on theoretical grounds, both the appropriate form of the (16) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Ser. A 1971, 324, 301.
Figure 2. Force F (in µN) between two freshly molten glass spheres in air against surface separation D (nm). The data are plotted on linear (a) and logarithmic (b) scales. The mean radius R of the surfaces is 0.242 cm; the spring constant is 144 N/m. The approach is at a constant speed (10 nm/s) of the piezoelectric drive. The dashed line corresponds to the retarded form of the van der Waals force F ) -(2π/3)RB/D3 with B ) 0.39 × 10-28 J m. The plane of zero separation is positioned δ ) 7.5 nm from the position of contact with zero external force.
dispersion force and also the axial contact distortion expected from JKR theory. The limiting forms of the van der Waals interaction are in the nonretarded regime F ) -AR/(6D2) and in the retarded regime F ) -(2π/3)BR/D3, where A and B are constants. The quantity A is called the Hamaker constant. Following our experience with the dispersion forces of mica,1 we fit our data to the retarded limiting form, adjusting both B and the origin of D. The best fit to the retarded form is shown as the dashed line in Figure 2. It is seen from Figure 2b that F scales as D-3 with parameters B ) 0.39 × 10-28 J m and a surface distortion of 7.5 nm. In other words, after the surfaces make contact we predict that they squash together by a distance δ ) 7.5 nm. The remaining step is to confirm that this surface distortion is consistent with JKR theory and the known materials parameters of glass. JKR theory predicts that δ is related to the radius of contact at zero external load a(0) by
δ ) a(0)2/3R
(1)
By using this relation in our work on mica,1 in conjunction with optically measured a(0), we avoided invoking the bulk elastic modulus which is not well-defined for the mechanically complex mica-glue-silica sandwich. We cannot do this in the present measurements, because a(0) cannot be obtained with the experimental system, so instead we use the measured pull-off force Fp, the force
Notes
Langmuir, Vol. 19, No. 9, 2003 4039
needed to separate the surfaces once they have adhered, which we measure to be 0.0021 N. According to JKR theory, this gives
a(0) ) (4RFp/K)1/3
(2)
and hence a(0) and δ may be obtained. The quantity K is an elastic bulk modulus given by
K ) (2/3)E/(1 - ν2)
(3)
where E is Young’s modulus and ν is Poisson’s ratio. The accepted mechanical properties of fused silica17 are given as E ) 6.9 × 1010 J/m3 and ν ) 0.2, leading to K ) 4.8 × 1010 J/m3. Taking this value of K, we get a(0) ) 7.5 µm and δ ) 7.7 nm, the latter in good agreement with the experiment result of 7.5 nm. The small mismatch, barely above experimental uncertainty, between our experimental and calculated values of δ is likely to be associated with friction-related adhesion hysteresis that the JKR theory does not consider, the difference of the bulk parameters of Pyrex from those of silica and the wellknown complexity of the gel structure of surface layers of silica glasses which depends on the conditions of flame treatment, prewashing, and humidity.18 We discuss our measured value of the dispersion force of glass in the retarded regime of F ) -(2π/3)BR/D3 with Bglass ) 0.39 × 10-28 J m. Our measurements appear to be the first ones done for this material on the distance dependence of the dispersion force, and we are not aware of any theoretical calculations for the retarded region. We compare our results with those we obtained earlier for mica.1 For that material, we found Bmica ) 0.93 × 10-28 J m. For mica, also, the value of the Hamaker coefficient A in the nonretarded regime, based on the measurements of Tabor and Winterton, was deduced to be Amica ) 13.5 × 10-20 J.8 Theoretical calculations19 confirm this magnitude and indicate that the actual distance dependence is complex in the transition between the nonretarded and retarded regimes. The precise distance dependence is difficult to resolve because of experimental error. A typical calculated value of the Hamaker constant for silica-airsilica is Asilica ) 6.5 × 10-20 J.20 We see that the Hamaker (17) Handbook of Chemistry and Physics, 67th ed.; CRC Press: Boca Raton, FL, 1986. (18) Yaminsky, V. V. Langmuir 1977, 13, 2-7. (19) Chan, D.; Richmond, P. Proc. R. Soc. London, Ser. A 1977, 353, 163-176. (20) Senden, T. J.; Drummond, C. J. Colloids Surf., A 1995, 94, 2951.
constant of silica is 48% that of mica. Our results show that the ratio of the retarded dispersion constant B of glass to that of mica is Bglass/Bmica ) 42%, a comparable ratio. When freshly cleaved mica21 or freshly molten silica surfaces12 are left for some time in a laboratory atmosphere or in an SFA box filled with dry nitrogen, their surface properties change gradually. Without going into detail, the problem is attributed to adsorption and possibly absorption of monolayers and larger quantities of environmental contaminants segregating at the surface.18,21,22 The effect manifests itself in a reduction of adhesion, measured by the pull-off force, with time. In a series of measurements that we have made over a week following preparation of fresh glass surfaces, we have found that although the pull-off force decreased substantially over this period, there is no change of the dispersion force with time. Force-distance profiles are similar to that in Figure 2 except that δ decreases with exposure time, consistent with JKR theory. The presence of monolayers reduces adhesion without significantly influencing van der Waals interactions at large distances. The original adhesion of the aged surfaces can be restored by flame polishing, changing δ back to its pristine value without altering the radii of the surfaces. Summary The molecular smoothness of fused glass surfaces allows simultaneous measurement of long-range forces and contact deformations. Our measurements were done with a novel SFA that utilizes a capacitor to measure displacement. Local curvature variations and elastic inhomogeneity, which complicated measurements with mica surfaces, do not occur in the case of glass. This allows prediction of the contact deformation from the measured pull-off force, the macroscopic radius of the surfaces, and material elastic properties. The results of the measurements are consistent with rigorous theories, the macroscopic Lifshitz theory of the long-range van der Waals interaction and the JKR theory of adhesion. These two theories, accurate in limits of their validity at the two opposite extremes of the interaction range, match each other, with the values of experimental parameters providing a self-consistent theoretical description of the interaction from effectively infinite separation into contact. As with mica, the retardation form of the van der Waals force is valid at large separations. LA0269412 (21) Christenson, H. K. J. Phys. Chem. 1993, 97, 12034-12041. (22) Christenson, H. K.; Israelachvili, J. N. J. Colloid Interface Sci. 1987, 117 (2), 576-577.
