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Langmuir 1996, 12, 1404-1405
Surface Deformations in Direct Force Measurements H. K. Christenson* Experimental Surface Physics, Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National University, Canberra, ACT 0200, Australia Received October 18, 1994. In Final Form: November 7, 1995
Recent years have seen an increase in the number of devices available for the direct measurement of surface forces. This Note aims to summarize what is known about surface deformations in the surface force apparatus (SFA) and to compare with what can be expected with some other techniques. Some misleading ideas about surface deformations and limiting rigidities in the SFA may have been obtained by casual readers of recent publications.1-3 The SFA4-6 uses mica surfaces, of a radius of curvature R ≈ 2 cm, glued to polished silica disks with an epoxy resin (Epon 1004 from Shell Chemical Co.). One surface is mounted at the end of a double-cantilever spring,7 the force-measuring spring. The surface separation is determined by multiple-beam interferometry, which also allows surface deformations to be monitored in situ. In particular, the zero of separation is measured directly by bringing the surfaces into contact in nitrogen. In the atomic force microscopes (AFM) most commonly used, the surfaces are often so-called colloid probes (of varying nature) of R ≈ 3-10 µm, glued with epoxy resin to the end of a single cantilever of silicon nitride.8 The absolute surface separation cannot be directly determined but is inferred from the force vs distance behavior, and contact is assumed to be at “constant compliance”, i.e., when the deflection of the cantilever becomes linear with respect to sample displacement. The observed hysteresis in the position of this “contact line” has been the subject of recent discussion.9 In a third device3 fused silica surfaces (R ≈ 2 mm) are mounted at the end of a piezoelectric bimorph, which permits the spring bending to be measured electronically. The surface separation is then calculated from this bending using a constant compliance region as in the AFM. Clearly, absolute distances relative to a predetermined contact (e.g., contact in nitrogen) can only be determined using interferometry, as in the SFA. In order to summarize what is known about the effects of surface deformation on the measured forces it is helpful to recall some equations from the theory of elastic deformations. According to the JKR (Johnson-KendallRoberts) theory10 the elastic constant (bulk modulus) K * FAX: 61-6-249 0732. Tel: 61-6-249 3357. E-mail: hkc110@ rsphysse.anu.edu.au. (1) Parker, J. L.; Stewart, A. M. Prog. Colloid Polym. Sci. 1992, 88, 162. (2) Parker, J. L.; Attard, P. J. Phys. Chem. 1992, 96, 10405. (3) Parker, J. L.; Claesson, P. M. Langmuir 1994, 10, 635. (4) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1 1978, 74, 975. (5) Parker, J. L.; Christenson, H. K.; Ninham, B. W. Rev. Sci. Instrum. 1989, 60, 3135. (6) Israelachvili, J. N.; McGuiggan, P. M. J. Mater. Res. 1990, 5, 2223. (7) Christenson, H. K. J. Colloid Interface Sci. 1988, 121, 170. (8) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239. (9) Hoh, J. H.; Engel, A. Langmuir 1993, 9, 3310. (10) Johnson, K. L.; Kendall, K; Roberts, A. D. Proc. R. Soc. London 1971, A324, 301. (11) Christenson, H. K.; Claesson, P. M. J. Colloid Interface Sci. 1990, 139, 589.
is related to the radius of the flattened contact zone under zero load a0 by
a03 )
12πγR2 K
(1)
where γ is the surface energy. With a knowledge of R, γ, and a0 it is thus possible to calculate K, on the assumption that the layered mica-glue-silica system obeys JKR theory. For an isotropic solid sphere on an isotropic solid plane (equivalent to the crossed cylinders of the SFA) K is related to Young’s modulus E and Poisson’s ratio ν (the ratio of longitudinal extension to transverse contraction; for most materials ν2 is small, e0.1) by
K)
E 2 3 (1 - ν2)
(2)
In JKR theory the surface energy is related to the pull-off force F by
γ)
F 3πR
(3)
The corresponding relationship for undeformed surfaces is
γ)
F 4πR
(4)
Despite early controversy7,11 it has been shown that for mica surfaces in strongly adhesive contact JKR theory provides a good description of the relationship between surface energy and pull-off force12 as well as the profiles of the deformed surfaces.13 The effective elastic constant of the mica-glue system may be easily calculated from measured values of a0 and F/R, using eqs 1 and 3. The results of a large number of experiments are shown in Figure 1, where the measured pull-off force in nitrogen is plotted as a function of the elastic constant E/(1 - ν2). (For a description of the experimental procedure, see ref 12.) The measured values of the elastic constant are large, of the same order as that of silica, 7 × 1010 N m-2,14 and only a factor of 3 or 4 less than that of mica itself, 2 × 1011 N m-2.15,16 The average of all the measurements is (5 ( 2) × 1010 N m-2, which is the same as that found in another recent study (2-7 × 1010 N m-2).17,18 There is no correlation between the measured values of F/R and the elastic constant, which shows that the variations in mica and glue thickness that occur from experiment to experiment do not affect the validity of eq 3. There is a weak correlation between mica thickness and E/(1 - ν2), with thicker mica sheets showing on average slightly higher values of E/(1 - ν2). This is to (12) Christenson, H. K. J. Phys. Chem. 1994, 97, 12034. (13) Maugis, D.; Gauthier-Manuel, B. J. Adhesion Sci. Technol. 1994, 8, 1311. (14) Handbook of Chemistry and Physics, 57th ed.; CRC Press: Cleveland, OH, 1976. (15) Gaines, G. L.; Tabor, D. Nature 1956, 178, 1304. (16) Simmons, G.; Wang, H. Single Crystal Elastic Constants and Calculated Aggregate Properties, 2nd ed.; M.I.T. Press: Cambridge, MA, 1971. (17) Chen, Y. L.; Helm, C. A.; Israelachvili, J. N. J. Phys. Chem. 1991, 95, 10736. (18) Using the Hertz theory of elastic deformations and very high applied loads, much lower values of E, in the range (0.4-1.8) × 1010 N -2 m , were obtained for surfaces immersed in KCl solution, where the forces should be net repulsive; see: Horn, R. G.; Israelachvili, J. N.; Pribac, F. J. Colloid Interface Sci. 1987, 115, 480. The reason for the difference is not clear; perhaps it is because a different regime of loads was investigated.
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Langmuir, Vol. 12, No. 5, 1996 1405
Figure 1. Measured pull-off force F, normalized by radius of curvature R, for mica surfaces in dry nitrogen as a function of the elastic constant E/(1 - ν2), for 20 different contact positions of 16 different pairs of mica sheets (from a series of measurements published in ref 12). E is Young’s modulus, and ν is Poisson’s ratio. The elastic constant is calculated from the measured contact area under zero load and the measured pulloff force using JKR theory (eqs 1 and 3). The dashed lines are least squares fits to the data. The near-zero slope of the horizontal line shows the lack of correlation between the elastic constant and the pull-off force. The vertical line consequently gives the average of the elastic constant. The points with the highest (>9 × 1010) and lowest ( 10, and for µ < 1 the measured adhesion is essentially unaffected by surface deformation and the relationship is the same as for the undeformed case (eq 4). For intermediate µ values there is a transition region, shown in Figure 3 of ref 19. With typical values for an SFA adhesion (in nitrogen) experiment12 (γ ) 150 mJ m-2, R ) 2 cm, E/(1 - ν2) ) 5 × 1010 N m-2, De ) 0.2 nm) one obtains µ ) 82, which is very far into the regime of JKR theory, as expected. (19) Muller, V. M.; Yushchenko, V. S.; Derjaguin, B. V. J. Colloid Interface Sci. 1980, 77, 91. (20) Maugis, D. J. Colloid Interface Sci. 1992, 150, 243. (21) Christenson, H. K. J. Chem. Phys. 1983, 78, 6906. (22) Shubin, V. E.; Ke´kicheff, P. J. Colloid Interface Sci. 1993, 155, 108. (23) Christenson, H. K. Phys. Rev. Lett. 1995, 74, 4675.
From eq 5 it can be seen that surface deformation in the SFA is due almost entirely to the large radius of curvature of the surfaces. Even the hardest substance known, diamond (E ≈ 9 × 1011 N m-2 16) would have µ ) 12 (other quantities kept equal) and still be in the JKR regime. Only for γ e 0.2 mJ m-2 (|F/R| e 2 mN m-1, using eq 3 or 4) will surface deformation not appreciably affect the measured interactions in the mica-glue system for R ≈ 2 cm. For colloidal silica particles of R ≈ 5 µm, deformation should be unimportant for γ e 20 mJ m-2. Although the above examples are strictly valid only for adhesive forces, and the exact numerical values should obviously not be taken too seriously, it is nevertheless clear that surface deformation is largely unavoidable when dealing with macroscopic surfaces and large forces. Note that the onset of deformation (flattening) during measurements of solvation forces (where the local gradient of the force is also very steep) is observed to occur for applied loads of about 2 mN m-1,21 or for the same magnitude of interaction as in the adhesive example above. Flattening will only occur for larger magnitudes with less steeply varying force laws (which can be related to the parameter De in eq 5), typically at about F/R ≈ 10 mN/m for doublelayer forces.22 Reducing the radius of curvature by an order of magnitude, or using silica instead of mica and glue, as in ref 3, can only have a very marginal effect on surface deformation. The only realistic way of avoiding surface deformation in direct force measurements with macroscopic surfaces is to limit the measurements to weak forces. Even measurements on micron-size particles, as in an AFM, may not avoid the problem of surface deformation for very large forces, such as adhesion in gas or between hydrophobic surfaces in water. Thanks to the interferometric technique, however, one can at least monitor the deformation with the SFA. The elasticity of the mica and glue system obviously constitutes the limit of stiffness of the SFA, and this sets an effective upper bound to the stiffness of the forcemeasuring spring. This limiting spring constant (normalized to k/R) is not, however, 2 × 105 N m-2 1 as appears to follow from published measurements with a forcefeedback system of limited response (see Figure 5B of ref 1; note that the distance scale is in error by a factor of 10). Reference 2 easily makes the reader conclude that the limiting stiffness k of the SFA is 5.5 × 104 N m-1 (see Figure 10). The quoted spring constant k/R of a “rigid support” 12 (3 × 106 N m-2) does not imply that this is some limiting rigidity caused by the glue. If good rigidity of the mechanical components is ensured, it is a simple matter to obtain k ) 3 × 105 N m-1, which with R ) 2 cm corresponds to k/R ) 1.5 × 107 N m-2.6,23 In summary, surface deformation in the surface force apparatus is not caused by the glue but is an unavoidable consequence of the use of macroscopic surfaces. Surface deformation can only be completely avoided by using surfaces of colloidal dimensions (R ≈ 5 µm), or by limiting measurements to weak forces (F/R e 2 mN/m for strongly adhesive or repulsive forces, F/R e 10 mN/m for doublelayer forces, with R ≈ 2 cm). Acknowledgment. I am grateful for helpful discussions with and comments by P. M. Claesson, R. G. Horn, T. Sawkins, T. Senden, and V. V. Yaminsky. I thank B. Gauthier-Manuel for sending me a reprint of ref 13. LA9408127
