Langmuir 1996, 12, 3289-3294
3289
Use of Capacitance to Measure Surface Forces. 1. Measuring Distance of Separation with Enhanced Spatial and Time Resolution P. Frantz, N. Agrait,† and M. Salmeron* Materials Sciences Division, Lawrence Berkeley National Laboratory, University of California, Berkeley, California 94720 Received January 2, 1996X We describe a simple method to measure the separation and forces between two macroscopic surfaces mounted in a crossed cylinder arrangement. It is shown that by measuring the capacitance between silver layers deposited on the backside of thin mica substrates, it is possible to achieve spatial resolution of less than 0.1 nm and time resolution of 1 ms. The growth of capacitance with decreasing distance of separation follows the predicted dependence for an ideal sphere and plate capacitor. To demonstrate the capability of obtaining force-distance profiles with this technique, octamethylcyclotetrasiloxane (OMCTS) has been confined between the mica sheets. We find that the process of collapse of the confined liquid, from four molecular layers to three, occurs more rapidly than 1 ms. This technique is also shown to give excellent results for opaque surfaces used in the surface forces apparatus (SFA), and we show that the adhesion energy between two surfaces of oxidized silicon is 3 mJ/m2. Applications of this technique to the study of contact mechanics and dielectric properties of confined materials are discussed.
Introduction The surface forces apparatus (SFA) has demonstrated extensive utility as a tool to study static and dynamic forces1,2 between smooth, macroscopic surfaces and the behavior of molecules confined between such surfaces.3 With this instrument, two freshly cleaved mica surfaces oppose each other in a crossed cylinder geometry to produce a circular contact that is nearly ideal; mechanical and chemical properties may be manipulated and characterized with great precision. In its most common form, the instrument relies on interferometry to determine the distance of separation between the surfaces.4 Interference of white light occurs in a Fabry-Perot interference cavity formed by two opposing layers of silver deposited on the backside of the mica substrates. As the surfaces are separated, the discrete wavelengths of light selected by the cavity increase with the distance between the silver layers, with a spatial resolution approaching 0.1 nm. These interference fringes of equal chromatic order (FECO) are then recorded and treated individually to yield information about the distances and forces in the normal direction, as well as contact area. Although this technique is elegant in its conceptual simplicity, we find that meticulous care and patience must be exercised to obtain useful data. The preparation of a pair of surfaces which produce interference fringes yielding resolution of less than a few angstroms is not trivial, requiring a convergence of delicately thin mica (roughly 1 µm), silver layers with a small window of useful reflectivity, and a contaminant free surface. Interference fringes are often diffuse enough to allow for human error in the judgment of the fringe position. Due to the tedium of collecting information from these fringes, approach curves of static forces vs distance are rarely composed of † Permanent address: Dept. Fisica de la Materia Condensada, Facultad de Ciencias, Universidad Autonoma de Madrid, 28049 Madrid, Spain. X Abstract published in Advance ACS Abstracts, June 1, 1996.
(1) Israelachvili, J. N. Handbook of Micro/Nano Tribology; CRC Press: New York, 1995. (2) Israelachvili, J. N. Chemtracts: Anal. Phys. Chem. 1989, 1, 1. (3) Granick, S. Science 1991, 253, 1374. (4) Israelachvili, J. N. J. Colloid Interface Sci. 1973, 44, 259.
S0743-7463(96)00014-5 CCC: $12.00
more than a few tens of data points. Furthermore, as the understanding of the behavior of confined materials grows, so does the need to perform rapid measurements of dynamic forces in the normal direction. Using piezoelectrics5,6 and strain gauges,7 it has long been possible to sample the lateral forces at a rate faster than many confined liquids can respond, yet it takes several seconds to collect a single data point in the normal direction using the FECO fringes. Interest has grown in the pursuit of obtaining dynamic force measurements in the normal direction. Horn8 and Israelachvilli9 have independently employed videocameras to record rapidly changing fringe positions, with a time resolution of about 0.5 s, during the drainage of fluid from the contact region. Results were used to infer changes in the viscosity as a function of film thickness, and the rate of drainage was seen to violate continuum theories of hydrodynamics.8 Georges and co-workers10,11 have developed experimental and analytical tools to determine various rheological properties of polymer films confined between a rigid metal sphere and plate. In this work, the distance between the surfaces was oscillated to induce flow into and away from the contact, and capacitance between the surfaces was used to monitor the distance of separation. Although these rough surfaces are suitable for the study of polymers, they cannot allow confinementinduced ordering effects in molecules which are smaller than the characteristic roughness of the surfaces.3 Even when ideally smooth surfaces are used, interpretation of dynamic force measurements in the normal direction may not be as straightforward as in the case of static forces due to the lack of a dynamic force analog to the Derjaguin approximation.12 (5) van Alsten, J.; Granick, S. Phys. Rev. Lett. 1988, 6, 2570. (6) van Alsten, J.; Granick, S. Tribol. Trans. 1990, 33, 436. (7) Israelachvili, J. N.; Homola, A. M.; Mcguiggan, P. M. Science 1988, 240, 189. (8) Chan, D. Y. C.; Horn, R. G. J. Chem. Phys. 1985, 83, 5311. (9) Israelachvili, J. N. Colloid Polym. Sci. 1986, 264, 1060. (10) Georges, J. M.; Millot, S.; Loubet, J. L.; Tonck, A. J. Chem. Phys. 1993, 98, 7345. (11) Pelletier, E.; Montfort, J. P.; Loubet, J. L.; Tonck, A.; Georges, J. M. Macromolecules 1995, 28, 1990. (12) van Alsten, J.; Granick, S.; Israelachvili, J. N. J. Colloid Interface Sci. 1988, 125, 739.
© 1996 American Chemical Society
3290 Langmuir, Vol. 12, No. 13, 1996
Figure 1. Schematic of the instrumental arrangement used to measure thecapacitance between the silver layers on the backside of thin mica substrates in the surface forces apparatus.
Recent efforts have been made to measure forces between crossed mica cylinders without the reliance on interference fringes as the primary source of information. Piezoelectric bimorphs have been employed as strain gauge leaf springs to monitor changes in the relative vertical positions.13,14 Displacement of the surface from its initial position induces a charge on the bimorph which may be calibrated to yield the distance of vertical translation with resolution surpassing that of FECO fringes. In our laboratory, a fiber-optic interferometer has been employed in a geometry reminiscent of a scanning probe microscope to measure relative changes in distance between the surfaces with resolution of ∼0.1 nm.15 Although excellent results have been obtained from these techniques, the conclusions may be uncertain; the motion that is detected is spatially removed from the actual motion of the mica surfaces at the contact. By measuring the capacitance between the surfaces, one is probing directly the area of contact between the surfaces, since the measured signal is heavily weighted by the area of least separation. In this article we describe how this technique is used to measure the distance of separation when the two surfaces are brought together in a gaseous or liquid medium. When used in conjunction with the FECO fringes (for purposes of calibration), measurements of normal force can be made with ease and accuracy; they are much less demanding of the quality of mica and the silver coating, and rapid processes at the molecular level may be probed at a rate limited only by the electronics used (in our case, 1 ms, though this may be overcome by use of a faster technique for detection of capacitance changes). We also show that electrically conductive opaque surfaces, such as oxidized silicon, may be used as substrates for this apparatus. In a forthcoming publication we describe in detail the use of this technique to study contact mechanics,16 including rapid measurements of adhesion and adhesion hysteresis, and to investigate changes in dielectric properties of confined materials.17
Frantz et al. exists between points A and B at the beginning of an experiment. When the relative position of the crossed cylinders is changed, a difference in potential proportional to the change in capacitance arises between A and B. The in-phase and out-of-phase components of this high-frequency signal are then collected by a two-channel lock-in amplifier and sent off to a 486 PC for analysis. A home-made surface forces apparatus, described in detail elsewhere,15 was used for these experiments. This apparatus has several features which distinguish it from the common SFA. First of all, as with the apparatus of Stewart and Christenson,18 the position of one surface is controlled through magnetic force deflection. A permanent magnet, mounted beside the surface held by leaf springs, feels a force due to a magnetic field gradient created by two coils of copper wire located outside the apparatus. This method of manipulation has the advantage of large range (1 mm), smooth computer controlled operation, and absolute lack of hysteresis.18 Second, in addition to the use of FECO fringes, a fiber-optic interferometer has been constructed to measure the distance of separation.15 Although similar instruments are commercially available, the capacitance bridge was easily constructed from spare components; commercial capacitance bridges are very expensive. These components include two identical variable resistors (0-200 kΩ), two identical fixed capacitors (mica, 2.2 nf), and a variable capacitor. The capacitance of the fixed capacitors was chosen to be large with respect to that of the SFA capacitor (commonly 10 pf), to improve the spatial resolution. We chose to construct our own variable capacitor due to uncertainties in the stability of commercially produced capacitors. The design for this component was taken directly from that of Steinitz et al.,19 although the dimensions were adjusted to provide a capacitance range of 0-50 pf. All wires and components have been shielded to avoid noise from stray capacitances. The high-frequency signal generator is contained within a twophase lock-in amplifier (Stanford Research Systems). The signals used to manipulate the relative positions of the crossed cylinders are controlled by an electronic unit constructed for use with a scanning probe microscope (STM 100, RHK Technology). This affords the option of feedback capability; the signal from the capacitance or the fiber-optic interferometer may be used as a reference to avoid the unstable jump-in to contact when the surfaces are brought together within an attractive potential. It also provides for the more mundane task of supplying simple analog signals for control of a DC current source (HP 6267B) to create the magnetic field for deflection of the leaf springs, and it simultaneously collects signals from multiple sources for processing by the PC.
Performance The resolution of the capacitance measurement device is given by
VA - VB ) V[Z3/(Z1 + Z3) - Z4/(Z2 + Z4)] where Z1, Z2, Z3, and Z4 are the impedances of the SFA, the variable capacitor, and the two large fixed capacitors, respectively. Since we are close to balance,
Z1 + Z3 ) Z2 + Z4 + dZ Therefore,
Design
VA - VB ) VδZ/Z4 ) VδC/C4
A schematic of the apparatus is shown in Figure 1. The signal generator sends a high-frequency sine wave of amplitude 1 V to the capacitance bridge. This signal is passed along to the SFA surfaces through wires connected to the silver on the backside of the mica substrates. The capacitance of the crossed cylinder capacitor, and the resistance of the capacitor circuit, is balanced on the other side of the bridge so that no potential difference
An imbalance of 200 nV, the highest sensitivity attainable with the current setup, corresponds to ∆C ) 0.44 fF. The behavior of a sphere and plate capacitor has been reviewed by Georges and co-workers.20 Since this analysis has been presented in detail, we refer the reader to this work and write only the final result. For distances of separation much less than
(13) Parker, J. L. Langmuir 1992, 8, 551. (14) Frantz, P.; Salmeron, M. Unpublished data. (15) Frantz, P.; Wolf, F.; Xiao, X.-D.; Du, Q.; Salmeron, M. Submitted for publication. (16) Frantz, P.; Carpick, R.; Salmeron, M. Submitted for publication. (17) Frantz, P.; Agrait, N.; Salmeron, M. Submitted for publication.
(18) (a) Stewart, A. M.; Christenson, H. K. Meas. Sci. Technol. 1990, 1, 1301. (b) Stewart, A. M.; Parker, J. L. Rev. Sci. Instrum. 1992, 63, 5626. (19) Steinitz, M. O.; Gonossar, J.; Schnepf, W.; Tindall, D. A. Rev. Sci. Instrum. 1986, 57, 297. (20) Boyer, C.; Houze, F.; Tonck, A.; Loubet, J.-C.; Georges, J. M. J. Phys. D 1994, 27, 1504.
Use of Capacitance to Measure Surface Forces
Langmuir, Vol. 12, No. 13, 1996 3291
Figure 2. Signal intensity (VA - VB) vs relative distance of separation between the conductive layers of the sphere-plate capacitor as the surfaces are brought into contact from 16 µm separation in dry air. Inset: the inverse of the derivative of the data shown in the main body of this figure vs distance of separation. A straight line in this figure is predicted by the expression for the capacitance of an ideal sphere-plate capacitor.
data shown in the inset of Figure 2 may be used to find the absolute thickness of the mica and to designate the origin of contact. However, since the slope of this line is proportional to the dielectric constant of the medium that is expelled as the silver surfaces approach one another, the slope after contact must be increased by the ratio of the dielectric constant of mica to that of the interposing medium. This is, however, an approximation, since the silver is separated by mica only at the contact and by mica and air outside the contact. The thickness of mica is then the distance between the point of mica-mica contact and the x-intercept of this line. The approximation is good, because the capacitance is dominated by the point of closest approach.16 Thus one sees that, in the case of Figure 2, the silver surfaces are separated by 0.44 fF, we can, in principle, expect to do better than 0.1 nm resolution. As explained below, this resolution improves rapidly with decreasing thickness of mica. The response of the apparatus during approach of the surfaces from 16 um separation in dry air is shown in Figure 2. The measured signal, the difference in potential between A and B (Figure 1), is plotted as a function of relative distance. The absolute distance between silver layers may be determined by measuring the wavelength between FECO fringes when the surfaces are in contact4 or by the method indicated below. Relative changes in distance are determined by using FECO fringes or the fiber-optic interferometer to calibrate the vertical deflection induced by a given change in the current through the electromagnetic coils. In this case, a distance of 12 um corresponded to ∼1 A. The spatial resolution for this pair of surfaces may be determined from the slope of this line at closest approach (18 000 V/m) and the resolution in voltage provided by the lockin amplifier and capacitance balance (500 nV), giving reliable changes in distance down to ∼0.03 nm. The inset of Figure 2 shows the inverse of the derivative of this data set. The linearity of this figure shows that the capacitance can well be described by the above expression for a sphere and plate capacitor. This figure also demonstrates the advantage gained by minimizing the thickness of the mica surfaces; signal intensity increases rapidly with decreasing distance, with no concomitant increase in noise. Determination of the origin deserves a special note. Since the rate of change of capacitance grows to infinity as the distance between the silver layers approaches zero, the x-intercept of the
F ) -V2π(R/H) Thus, for a sphere of radius 1 cm, the force is ∼100 nN. In the present case, we have used a spring constant of 200N/m, so a 100 nN force should induce a detectable motion of ∼0.5 nm when the surfaces are out of contact. The frequency of this oscillation is much higher than the resonance frequency of our apparatus (300 Hz), so we expect this high-frequency force to be equivalent to a dc force of half the amplitude, or ∼0.25 nm. This constant deflection has not yet been observed.
Results To demonstrate the capabilities of this technique, octamethylcyclotetrasiloxane (OMCTS) has been confined between the mica surfaces. This simple, nonpolar, Newtonian, low molecular weight relative of the silicone oils has been extensively studied in previous surface force measurements.21 Previous results show ordering of the liquid into discrete layers when the surfaces are separated by less than eight times the molecular radius.22 In the present experiment, the surfaces began separated by 20 nm and a linearly increasing current was applied to the coils to bring them into contact. To convert the resulting change in the capacitance signal into units of distance, two corrections must first be considered. First, we have seen that there is a nonlinear dependence of the capacitance on the distance of separation; the growth of capacitance with distance is faster when the surfaces are closer together. Applying eq 1 to the present case where the silver surfaces are separated by 2 µm of mica, we find that the change in distance calibration across an approach curve spanning 20 nm is ∼1% of our noise level. Therefore, we may treat such small changes in capacitance as varying linearly with distance. Secondly, as the liquid confined between the mica sheets becomes ordered, it will sustain a normal force capable of deforming the surfaces. As the sphere undergoes Hertzian deformation, the measured capacitance increases. This effect, which causes a capacitance change of nearly 10% over the range of load applied in a typical approach curve (30 µN), is easily accounted for by fitting the early deviation from linearity (21) Christenson, H. K. J. Chem. Phys. 1983, 78, 6906. (22) Klein, J.; Kumacheva, E. Science 1995, 269, 816.
3292 Langmuir, Vol. 12, No. 13, 1996
Figure 3. Distance of separation between the mica surfaces vs the magnetic deflection force applied to a permanent magnet mounted on the lower surface support. Steps in this figure show the layering of this liquid into discrete strata. As the pressure is increased, individual layers are squeezed out. The total thickness of mica between the silver layers was 2.9 µm, and the response time set by the lock-in was 10 ms.
Frantz et al.
Figure 4. More steps appear at smaller applied loads as the size of each increment of force is reduced. In this case, the total thickness of mica was 1.7 µm, and the response time of the lock-in was 300 ms. As a result, the spatial resolution has improved in comparison with the experiment of Figure 3, but the discrete steps in distance are smeared by the decay time of the lock-in.
in the capacitance vs distance curve with a function describing the change in area with load at a hertzian contact:23
a2 ) (RF/K)2/3 Here, the radius R of the sphere is 6 mm, F is the force applied to the substrate support by the magnetic lever deflection, starting from the point at which the liquid supports a normal force (where the slope of the capacitance curve begins to deviate from the value of the leaf spring constant), and K, the elastic modulus, is an adjustable parameter. This least squares fitted curve is then subtracted from the raw data. To justify the validity of this approach, it is necessary to show that the area of contact is proportional to the measured capacitance. The contribution to the capacitance from the area of contact between the surfaces should simply scale linearly with the area; however, the contribution from outside the area of contact is not trivial. Here we state only that this contribution is approximately constant and small compared to the contact capacitance, and we reserve detailed analysis for a treatise on the use of capacitance to study the mechanics of contact between two macroscopic objects.16 Figure 3 shows the change in separation with magnetic force applied to the permanent magnet mounted on the lower surface support with a drop of OMCTS placed between the mica sheets. The raw data has been corrected for substrate deformation as described above and converted into units of distance by scaling with a calibration from the FECO fringes.4 Interference fringes were also used to set an absolute zero to the distance of separation. We find that the liquid was able to support a measurable normal force at a separation of 4.5 nm, roughly equal to 5 molecular diameters, and individual layers were squeezed out of the contact area as the load was increased. In this case, where the total thickness of mica between the silver layers was slightly
