Computer-Controlled Experiments in the Surface Forces Apparatus

Langmuir 1996, 12, 3885-3890

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Computer-Controlled Experiments in the Surface Forces Apparatus with a CCD-Spectrograph T. Gru¨newald and C. A. Helm* Institut fu¨ r Physikalische Chemie, Johannes Gutenberg-Universita¨ t, Jakob-Welder Weg 11, D-55099 Mainz, Germany Received January 2, 1996X We present a computer-controlled technique to measure the distance-dependent forces in the surface forces apparatus. The power of our setup is shown by a measurement of the repulsive forces between mica surfaces immersed in a 0.01 M NaCl solution. At close distances we find an oscillatory force with a periodicity of 0.30 nm, which roughly corresponds to the diameter of a water molecule. For the distance determination we use the standard interferometric technique: the interferometer consists of a medium sandwiched between two mica sheets of equal thickness silvered on the backside. The surface separation is measured by comparing the resonance wavelength to the one obtained from contacting mica sheets. By substitution of the eye by a CCD-camera we improve the reproducibility of the method and we can determine the cavity parameters quantitatively.

Introduction The surface forces apparatus (SFA) has become an established tool to measure quantitatively and directly surface forces as a function of their separation.1,2 The fundamental importance of the SFA is mainly due to the intimate relationship between intermolecular and surface forces. Thus, the physical origin of the surface interaction is inferred from the distance dependence of the force. However, in the case of large forces, surface deformation and rearranging surface molecules are common features, and it is essential to observe these effects. In the classic SFA1,2 this is done optically. Indeed, from the size of the contact area at a given load, one can calculate the adhesion and the adhesion hysteresis.3,4 Also it is possible to correlate attractive hydrophobic forces between lipid membranes with their propensity to thin continually or even to fuse.5-7 Capillary condensation at the edges of the contact area in vapors can be observed directly.8 Also, lateral movement of adsorbed surface molecules in vapor was observed for a surfactant monolayer, which partially dissolved into the water meniscus.9 An even more intriguing example of surface wetting is given by an adsorbed polyelectrolyte monolayer which (even at r.h. ) 0%) can be pushed out of the contact area to form a polyelectrolyte meniscus; after separation of the surfaces, the monolayer anneals again.10 Recently, we started to study dyes in the SFA. There we demonstrated that the shape of the transmission peaks gives additional infor* To whom correspondence should be addressed. Fax: 00496131-393768. Phone: 0049-6131-392470. E-mail: chelm@ mzdmza.zdv.uni-mainz.de. X Abstract published in Advance ACS Abstracts, July 1, 1996. (1) Israelachvili, J. N.; Tabor, D. Proc. R. Soc. London A Ser, 1972, 331, 19. (2) Israelachvili, J. N. J. Chem. Soc., Faraday Trans. 2 1973, 69, 1729. (3) Horn, R. G.; Israelachvili, J. N.; Pribac, F. JCIS 1986, 115, 480. (4) Chen, Y. L.; Helm, C. A.; Israelachvili, J. N. J. Phys. Chem. 1991, 95, 10736. (5) Horn, R. G. Biochim. Biophys. Acta 1984, 778, 224. (6) Helm, C. A.; Israelachvili, J. N.; McGuiggan, P. M. Science 1989, 246, 919. (7) Helm, C. A.; Israelachvili, J. N.; McGuiggan, P. M. Biochemistry 1992, 31, 1794. (8) Fisher, L. R.; Israelachvili, J. N. Colloids Surf. 1981, 3, 303. (9) Chen, Y. L.; Helm, C. A.; Israelachvili, J. N. J. Phys. Chem. 1991, 95, 10736. (10) Lowack, K.; Helm, C. A. Macromolecules 1995, 28, 2912.

S0743-7463(96)00002-9 CCC: $12.00

mation about the dye concentration and orientation at the surfaces.11,12 Since it is possible to obtain so much and so varied information from the careful analysis of the fringes, it is very useful to improve the interferometric technique. One can measure the surface separation more accurately and quickly than with the eyepiece. This is very useful for many applications, such as the one shown below. Furthermore, in order to control the interface kinetics, we want to measure the force-distance curves automatically. The surfaces are to approach (to separate from) each other at a constant and known speed, interrupting the approach (the separation) for a well defined time interval to monitor the distance between the surfaces. In the classical paper2 the distance was measured interferometrically with an eyepiece and a little scale, which were attached to a spectrograph. This method is tedious (i.e. unpopular with graduate students), and the reproducibility depends a bit on personal skills. Therefore, various apparatus with automatic distance determination were built. One approach is to measure the distance electronically with a capacitance device.13,14 While this can be extremely accurate, as shown by Loubet and co-workers, one can no longer observe the fringes and thus has no information about surface deformation or rearrangement of the surface molecules. There are also some automatic distance measurements described in the literature, which are based on interferometry. The main idea can be found in the thesis of Deitrick.15 It is necessary to use either a professional video system (u-matic) or a CCD-camera. Both techniques allow one to look at a movie frame by frame in a reproducible way (which is impossible with a VHS video system; with VHS each frame is shifted an arbitrary amount relative to the next). For each frame, the FECO fringe (fringes of equal chromatic order) is fitted to a fourthorder polynomial, from the fit, the wavelength at the center and eventually the distance between the surfaces were calculated. A disadvantage of this ansatz is the rather (11) Ma¨chtle, P.; Mu¨ller, C.; Helm, C. A. J. Phys. II France 1994, 4, 481. (12) Mu¨ller, C.; Ma¨chtle, P.; Helm, C. A. J. Phys. Chem. 1994, 98, 11119. (13) Parker, J. L.; Christenson, H. K.; Ninham, B. W. Rev. Sci. Instrum. 1989, 60, 3135. (14) Tonck, A.; Georges, J. M.; Loubet, J. L. JCIS 1988, 126, 150. (15) Deitrick, G. L. Ph.D. Thesis, University of Minnesota, 1990.

© 1996 American Chemical Society

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Figure 2. Cartoon of the experimental setup. The sample is irradiated with polarized white light. The transmitted light is focused on the entrance slit of an imaging spectrograph. Depending on the mirror position at the exit of the spectrograph, the fringes are either viewed by a light sensitive camera (which is connected to an image amplifier and a video recorder) or by a computer-controlled CCD-camera.

Figure 1. Schematic view of the optical setup. Top row, left: Crossed cylinders are used, which correspond to the geometry of a sphere on a flat. Top row, right: If they are irradiated by monochromatic light, Newton’s rings are observed. Center row, left: With a hardware slit the central area of the Newton’s rings is focused on the entrance slit of an imaging spectrograph; at a given wavelength one observes a point pattern in the y-direction. When the surfaces are irradiated with white light, fringes of equal chromatic order are observed. Their shape reflects the surface geometry. Center row, right: The smallest wavelength of a given fringe is used to determine the surface separation. Bottom row: The central area is selected by a software slit, and transmission spectra are measured.

clumsy data analysis; from each two-dimensional picture only one number is deduced:16 the surface separation. To find a convenient way to measure force runs quickly and simply, we take a look at the way the classic SFA works: Because of the experimental convenience, often mica or modified mica surfaces are investigated. Mica can be cleaved in micrometer-thin platelets, which are atomically smooth. The thin platelets can be easily cut to obtain two sheets of equal thickness. They are rather flexible and can be glued onto silica half cylinders (a crossed cylinder geometry corresponds to the sphere-flat geometry; the measured forces can be correlated via the Derjaguin-relation to the interaction energy between two flats.17,18 Mica is transparent to visible light, and therefore, after silvering the mica sheets on the backside, it is possible to construct an optical cavity and to measure the distance between the surfaces interferometrically. Figure 1 gives a cartoon of such an alignment. If the surfaces are irradiated with monochromatic light, Newton’s rings are observed (Figure 1, top row, right), which are circular due to the sphere-flat geometry. With the entrance slit of an imaging spectrograph the central (16) Quon, R. A.; Levins, J. M.; Vanderlick, T. K. JCIS 1994, 171, 474. (17) Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (18) Evans, D. F.; Wennerstro¨m, H. The Colloidal Domain; Verlag Chemie: Weinheim, 1994.

cross section of Newton’s rings is selected and can be observed as space-resolved dots in the spectrograph. If the surfaces are irradiated with white light, fringes of equal chromatic order are observed (Figure 1, center row). The shape of the fringes reflects the shape of the surfaces. At the smallest wavelength of a given fringe the surfaces are closest together; there the distance between the tip of the sphere and the flat can be calculated. Now, we take an additional (software) slit, to obtain the wavelength dependent intensity at the center of the fringes (Figure 1, bottom row). As expected from a cavity, we obtain transmission peaks with Lorentzian shapes11,19 and calculate from the peak position the surface separation. In the range of surface forces, the distance between the surfaces is controlled with a piezoelectrical crystal. One of the surfaces is mounted on a spring of known spring constant, if a force acts between the surfaces, the spring is deflected. The voltage applied to the piezoelectrical crystal is computer-controlled and varied step-by-step. The possibilities and limitations to measure force runs this way will be described in this paper. Materials and Methods NaCl was from Merck, Darmstadt, Germany and used as received. The water was filtered in a Milli-Q unit and subsequently destilled. The glue, EPOXI 1004, was from Shell. The cavity for the distance measurements consists of two mica sheets of equal thickness, which are silvered on their backsides (about 500 Å of silver) (BAE 250 coating system with a QSG 301 quartz crystal thickness monitor and a QRG 301 rate meter, all Balzers, Liechtenstein). The sheets are glued on cylindrical glass disks and mounted in the SFA (Mark IV, Anutech, Australia). Small angle X-ray scattering (SAXS) experiments to measure the silver thickness were performed with a Siemens D-500 powder diffractometer using copper KR radiation with a wavelength of 1.54 Å and data acquisition via a DACO-MP interface connected to a personal computer. Figure 2 gives a sketch of the experimental setup to measure the transmission properties of the cavity. Briefly, light from a 100 W halogen lamp (Mu¨ller Instruments, Grasbrunn, Germany) (19) Yeh, P. Optical Waves in Layered Media; John Wiley & Sons: New York, 1988.

Computer-Controlled Force Measurements after passing through an infrared filter and a polarizer is focused on the cavity. Behind the interferometer a microscope objective (magnification 5) focuses the transmitted light on the entrance slight of a 75 cm long imaging spectrograph (SpectraPro-750, Acton Research Corporation, Acton, MA), after it passes through a second polarizer. The overall magnification is a factor of ≈25 and is determined by the distance between the spectrograph and the SFA objective. The spectrograph has a table with three different gratings; by rotation of the table a suitable grating and the desired wavelength interval are selected. The gratings and their respective resolutions in our setup are as follows: 150 grooves/mm, 0.193 nm/pixel; 600 grooves/mm, 0.047 nm/pixel; 1800 grooves/mm, 0.013 nm/pixel. The spectrograph has two exits; one is connected to a SIT-camera (Proxitronic, HL5, Bensheim, Germany), an image-intensifier (Argus-10, Hamamatsu, Herrsching, Germany), and a video recorder (VHS), and the other is connected to a slow-scan CCD-camera (22 µm × 22 µm) with 576 × 384 pixels (EEV CCD 02, Spectroscopy Instruments, Gilching, Germany). The SIT-camera is quick (60 frames per second), but it exhibits a nonlinear relationship between the intensity and the brightness. Yet it is very useful for alignment, as well as for the observation of fast moving surface molecules. The linear intensity readout of the CCD is helpful for automatic force-distance measurements, since one obtains reliable transmission peaks, and is essential for our work with dyes.11,12 The image of the CCD is read via a controller (ST 138, Spectroscopy Instruments) in a computer (IBM-PC 4-86, 66 MHz); the build-up of a picture takes 1 s. The limiting time factor is the readout time of the CCD-chip. To obtain the surface separation, we need the transmission spectrum at the center of the image (cf. Figure 1, center row right and bottom row). The height of the irradiated area is selected with a software slit (which is defined by the vertical readout pixels of the CCD-chip), no longer with a hardware slit as in our earlier work.11 For distance measurements, we do not use the spacial resolution of the CCD-camera but average over a height of 5-10 pixels and just measure intensity vs wavelength. These measurements may be performed fairly quickly; one can obtain a spectrum in 200 ms (limited by the shutter, which can be changed easily), the speed being independent of the amount of pixels over which we average. For improved statistics, and thus more accurate peaks, of course one can measure for a longer time interval (provided there are no drifts). The peak shape can be described by a Lorentzian;11,12,19 the position of the peak is the resonance wavelength from which the surface separation is calculated. Yet, mica is birefringent. Depending on the relative orientation of the mica sheets, two peaks of the same order may be very close to each other; i.e., the intensity between the peaks is not zero, and one has to fit a double-peak. This may lead to a rather large error regarding the peak position. The two peaks exhibit different polarization directions. To suppress one of the peaks, we use two polarizers, one between the lamp and the SFA, and the other between the SFA and the spectrograph to minimize the background. Before an experiment, they are set in such a way that only one peak is observed. Before starting a force run, the background b0 is determined. To fit the remaining transmission peak to a Lorentzian is not very convenient for on-line analysis of peaks. The peaks measured during a force run are often obtained quickly (i.e. with mediocre statistics) or are distorted due to drifts. Because of the large error in the data points, one will gain nothing from a careful data analysis. Therefore, we determine the peak maximum for each spectrum obtained during a force run in the following way: (i) Find the value of maximum intensity ymax(λmax). (ii) Determine the fwhm (full width at half maximum) of the peak according to y(λ+) - b0 ) y(λ-) - b0 ) (ymax - b0)/2 and fwhm ) λ+ - λ-. (iii) Calculate the center of mass of the peak; the summation limits are N- ) λmax - 2(fwhm) and N+ ) λmax + 2(fwhm). So we have N+

λcenter )

∑ N-

N+

λy(λ)/

∑y(λ) N-

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Figure 3. A transmission spectrum of the interferometer measured by superimposing ten high resolution spectra (measured with a 600 g/mm grating). In the inset a peak measured at the overlap of two spectra is shown, the overlap is good within 0.1 nm. The straight line was obtained by a simulation of the transmission; the parameters are given in the text. λcenter is the resonance wavelength, from which the surface separation is deduced. We find that the accuracy of the center is as good as the one obtained from a fit to a Lorentzian. For a force run, the voltage as well as the voltage increments and additionally the time intervals (both for changing the voltage and for measuring the distance) are computer-controlled. Besides that, the spectrograph grating and the wavelength window are selected with the computer. If the wavelength window is rather small, and a longrange force is to be measured, either one has to readjust the wavelength window during the force run or one has to measure with fringes of decreasing chromatic order, which pass through the wavelength window during the approach of the surfaces (which we usually do). With ≈2 µm thick mica and the 600 grooves/mm grating we see two fringes in the field of view, which is sufficient. With a constant wavelength window we avoid any offsets which may be caused by the movement of the grating.

Results and Discussion A transmission spectrum of the Fabry-Pe´rot interferometer is shown in Figure 3. It is obtained from two mica sheets in contact (in air). To obtain an overview spectrum with high resolution, ten spectra, all of them measured with a fine grating, (600 grooves/mm) are superimposed. The intensities of the peaks are a function of the spectrum of the excitation lamp, the reflectivity of the grating, and the spectral response of the CCD-camera. In the inset a peak measured at the overlap of two spectra is shown, the overlap is good within 0.1 nm. The straight line is the result of the simulation of the transmission of a FabryPe´rot interferometer. We used indices of refraction from the literature: mica,20 nGl ) 1.56 + 5890/λ2; silver,19 nAg ) 0.3816 - 0.001177λ + (1.041 × 10-6)λ2 + i(1.21 0.00818λ). The calculated parameters were the silver thickness (50.5 nm, which was confirmed by SAXS) and the thickness of the mica sheets (2.362 µm) (we assumed that the absolute intensity of transmitted light increased linearly with the wavelength), which were determined by least-square techniques. The quality of the simulation as shown in the inset is typical. To demonstrate how the setup works, we show fairly standard experiment, which still is rather challenging in terms of distance resolution. The system was mica in a 10 mM NaCl solution, where we expect a long-range repulsive electrostatic force21 and at small distances an (20) Bailey, A. I.; Kaye, S. M. Br. J. Appl. Phys. 1965, 16, 39. (21) Pashley, R. JCIS 1980, 83, 531.

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Figure 4. (top) The force (on a logarithmic scale) between two mica surfaces in NaCl solution at the concentrations indicated. A long range electrostatic repulsion and for the low ion concentration, a short range van der Waals attraction can be distinguished. The dotted lines are fits to an exponential decaying force; the thus obtained decay lengths are indicated. (bottom) The same measurement as above (on a linear scale) in 0.01 M NaCl solution at surface separtations of the order of a few molecular diameters. The broken line is the same as in the top figure. The repulsive force increases stepwise. The right inset is a histogram of all the distances measured on approaching surfaces. Six peaks can be observed; their maximum position was determined by fits to a Gauss function. The left inset gives the subsequent positions of the maxima, from which a step size of 0.30 nm is deduced. (Due to alignment problems, we lost the absolute zero of the distance measurements; so the relative changes of the distance are reliable, but the absolute values are not.)

oscillatory force law due to water- and ion-layering effects.17,22 Each mica sheet was 4.27 µm thick; measurements close to contact were performed with the fringe of order 47 and the 1800 grooves/mm grating (wavelength resolution 0.013 nm/pixel or distance resolution ≈ 0.17 nm/pixel). Figure 4 (top) gives the force on a logarithmic scale as a function of the distance; the exponential decay with a decay length of 3.04 nm agrees amazingly well with the 3.04 nm expected from electrostatics (the average time for a distance measurement was 1 s. The applied voltage was increased in steps of 0.1-1 V. If no forces act, a step of 0.1 V corresponds roughly to a change in separation of 1 Å. For large separations, small voltage changes were chosen; when repulsive forces act, the voltage steps were increased.). Surface flattening can be clearly seen if the repulsive force exceeds 30 mN/m. According to the literature,21,22 the repulsive exponentially decaying elec(22) Israelachvili, J. N.; Pashley, R. M. Nature 1983, 306, 249.

Gru¨ newald and Helm

trostatic force should be superimposed by an oscillatory force, while the van der Waals attraction is completely shielded. Such an attraction can only be observed when the salt concentration is at least one order of magnitude smaller. Figure 4 (bottom) shows the force on a linear scale for very small surface separations. We observe a stepwise increase of the force, which is typical for an oscillatory force. In the right inset a histogram of small surface separations is given. Six maxima can be observed which represent the respective distances at which a steep increase of the force occurs. The position of each maximum was determined by a fit to a Gauss function. The size of the steps represents the diameter of the more or less spherical molecules confined between the surfaces. The left inset of Figure 4 (bottom) gives the number of oscillations as a function of surface separation. From the slope, one deduces a periodicity of 0.30 ( 0.01 nm. This agrees reasonably well with the 0.25 ( 0.03 nm observed with mica surfaces in KCl solutions and corresponds to the diameter of a water molecule.17,22 In the original paper a periodicity of 0.26 nm was suggested,22 which was deduced from the molecular dimensions of water and gives a volume of about 0.01 nm3. However, if one calculates the volume in the bulk, one finds an average volume for a water molecule of 0.03 nm3. The same result is obtained by X-ray measurements, where the electron density is 10e-/0.03 nm3 ((ref 23 ), note that water has 10 electrons per molecule). Frozen water, i.e. ice, crystallizes in the diamond structure, where the excluded volume amounts to about 2/3.24 Apparently, in fluid water, the excluded volume is of the same order of magnitude. Therefore, we expect for the periodicity of the oscillatory force a number between 0.26 and 0.31 nm, depending on the orientation and mobility of the water molecules between the mica sheets. There is no evidence for a strong influence of the Na+ ions on the structural force. Since mica is negatively charged, the Na+ concentration close to the surfaces is increased by about two orders of magnitude. Yet, the diameter of the bare ion is 0.19 nm and thus well below the step size measured whereas the diameter of the hydrated ion (0.66 nm) is well above the periodicity observed.17 The peak position can be measured more accurately the more narrow the peak is. To find ways to influence the peak shape, we take a quick look at the optics of a Fabry-Pe´rot interferometer.11,12 The transmission |T|2 of an interferometer consisting of two mica sheets (thickness, DGl; index of refraction, nGl) is given by

|T|2 )

(1 - r2)2 (1 - r2)2 + 4r2 sin2 φGl

(1)

r is the reflection coefficient of the mirror, i.e. the mica/ silver/glue interface, λ is the wavelength of the light, and

φGl ) (4π/λ)nGlDGl is the phase shift the light experiences traveling once between the mirrors. The resonance condition of the interferometer is

mλ0m ) 4nGlDGl, m is an integer

(2)

(23) Braslau, A.; Deutsch, M.; Pershan, P. S.; Weiss, A.; Als-Nielsen, J.; Bohr, J. Phys. Rev. Lett. 1985, 54, 114. (24) Ashcroft, N. W.; Mermin, N. D. Solid State Physics; Holt, Rinehart; Winston: London, 1976.

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Then a peak in the transmission occurs. The fwhm of the peak is given by

fwhm(λ0m)

λ0m(1 - r2) ) mπr

(3)

It is obvious that the peak is more narrow the more r approaches 1. It is important to note, that a separation of the surfaces does shift the resonance wavelength of the cavity, but the shape of the transmission peak is unchanged. The reflection coefficient of the three-layer system mica/silver/glue (note that glue/silver/mica is different) is given by

|

r ) rGl,Ag +

|

tAg,GltGl,AgrAg,Su exp(-4πinAgDAg/λ) 1 + rGl,AgrAg,Su exp(-4πinAgDAg/λ)

(4)

Here, the subscripts Ag and Su stand for silver and substrate (i.e. the glue), respectively. The reflection and transmission coefficients are defined as usual:19

rGl,Ag )

nGl - nAg nAg - nSu , rAg,Su ) , nGl + nAg nAg + nSu tAg,GltGl,Ag )

4nGlnAg (nAg + nGl)2

(5)

Because of the large imaginary part of the silver index of refraction, wavelength-dependent phase shifts at the interfaces mica/silver and silver/glue occur. Therefore, the mica thickness according to eq 2 is more a first approximation than exact. Figure 5 (top) shows the square of the reflection coefficient as a function of the wavelength for various silver thicknesses together with experimental data. The wavelength-dependent indices of refraction for silver and mica are the same as in Figure 3, nSu ) 1.594. Due to the small imaginary part of nAg at low wavelengths, r is rather small, too. With increasing wavelength, the imaginary part of nAg increases, leading to an increase in r. The diamonds represent experimentally determined values of r from transmission spectra similar to the one shown in Figure 3. One observes the theoretically expected increase of r with the wavelength. To investigate the influence of the silver thickness, various amounts of silver were evaporated onto the mica sheets. The thickness of the silver was determined by SAXS.25 A representative curve of the reflected intensity is given in Figure 5 (bottom) the thickness was calculated according to

∆qmindAg ) 2π

(6)

where ∆qmin is the difference between two neighboring wave vectors at the minimum of the reflected intensity.25 In Figure 5 (bottom) nine minima can be observed, from which one obtains a silver thickness of 539 Å. Also, r was measured in the SFA; the result is shown in Figure 5 (top). One finds the theoretically expected trend that with growing silver thickness the reflection coefficient increases. However, the absolute value is somewhat lower, which may be due to scattering at the silver/glue interface, inhomogeneities in the glue, and/or misalignment. Also, our evaporator produces silver layers of laterally varying silver thickness; we tried to minimize this error, yet cannot exclude it completely. However, the increasing silver thickness leads to a decrease in transmitted intensity, which leads to worse (25) Kiessig, H. Ann. Phys. 1931, 10, 769.

Figure 5. (top) The square of the reflection coefficient calculated according to eq 4 as a function of the wavelength for various silver thicknesses as indicated. The experimental data were obtained by determining the line width of a transmission peak and deducing the reflection coefficient according to eq 3. The silver thickness given for the experimental data was determined by SAXS. (bottom) Typical SAXS experiment to determine the silver thickness from the minima in the reflected intensity. The inset gives the slope of the minima, from which the silver thickness is deduced.

counting statistics of the peak. At each round trip between the mirrors the light somewhat penetrates into the silver. Due to the pronounced imaginary part of the silver refractive index, some light gets absorbed. The thicker the silver layer, the more light gets absorbed. Therefore, experimentally, one compromises between a narrow peak (high silver thickness) and high transmitted intensity (low silver thickness). For our experiments, we find that DAg ) 50 ( 3 nm is a very convenient silver thickness. We would like to point out that experimental mishaps may broaden the transmission peak, for instance bad alignment or water condensation on some glass parts in the SFA. Worse is either evaporating from a dirty evaporator or melting the silver during gluing (leading to yellow silver mirrors), because it both reduces the transmitted intensity and increases the peak width. Finally, we would like to discuss systematic errors in the distance determination which may arise with a somewhat sloppy alignment. We work with a sphere on a flat geometry. Depending on the radius of the sphere, the chosen software slit is between 5 and 10 pixels high. It is chosen by looking for maximum peak intensity, without broadening the peak. (Thus, it is chosen according to the same criteria as the hardware slit, i.e. the entrance slit of the spectrograph.) With our magnification (≈ 25; i.e. 0.95 µm/pixel in the y-direction) and an entrance slit width between 50 and 200 µm, we measure the distance

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between the flat and a small cap of the sphere (radius 2.5-5 µm). If the distance between the tip of the cap and the flat is D, then the distance at the rim of the cap is D + h. Assuming a sphere radius R and a circular cap with radius r, using the chord theorem (R - h)2 + r2 ) R2 and neglecting h2, we obtain h ) r2/2R. Choosing representative values, i.e. R ) 1.5 cm and r ) 3 µm, we find h ) 3 Å. Now, it is interesting to calculate the mean distance between the circular cap and the tip, which is actually measured. We deduce 2

r ∫0r(D + h)F ∂F ∫02π∂φ)/(πr2) ) D + 4R

D* ) (

(7)

Inserting the same numbers as before, we obtain D* ) D + 1.5 Å. Obviously, the offset can be reduced if the entrance slits are smaller; i.e., a horizontal hardware slit of 100 µm (experimentally, the fringes look more narrow) and a perpendicular software slit of 4 pixels correspond to r ≈ 2 µm, which leads to D* ) D + 0.7 Å, a number which is well below the resolution. The systematic offset is independent of the surface separation; i.e., as long as no surface flattening occurs, the change of the surface separation can be measured accurately. If the surfaces flatten, we have D* ) D. However, one has to be careful because the compressive pressure does not only increase the contact area but also thins the mica platelets by a few angstrom;9 i.e., now we may have D* < D. Both effects tend to overestimate changes of the surface separation. We may conclude that the onset of surface flattening is rather likely to be accompanied by a few angstrom apparent compression due to a systematic error in surface separation.

Conclusion and Implications We have shown force measurements in a 0.01 M NaCl solution, which exhibit at short separations oscillations with a periodicity of 0.30 nm. The oscillations could be reliably observed, since we were able to obtain many experimental data points. High distance resolution was obtained by using a spectrograph with a CCD-camera with a suitable wavelength resolution. The quality of the cavity was checked by a controlled variation of the silver mirrors; the results were consistent with X-ray measurements of the silver thickness and simulations of the cavity transmission. The experimental setup described here opens up new possibilities for measuring the distance-dependent force of systems with slow kinetics. By analyzing the transmission peaks of absorbing media, we can measure the optical density of the medium together with the force.

Acknowledgment. We enjoyed helpful discussions with Christel Mu¨ller, Klaus Lowack, and Peter Ma¨chtle. Diethelm Johannsmann and Harald Bock measured the index of refraction of the glue. Helmuth Mo¨hwald put a BMBF-grant in our way and encouraged the whole project. Thus we have now the good luck to acknowledge the financial support of both the Bundesministerium fu¨r Bildung, Wissenschaft, Forschung und Technologie (Nr. 13N6284) and the Sonderforschungsbereich 262 (Project D13/D19). LA960002Q