10398
J. Phys. Chem. 1992,96, 10398-10405
Deformatlon of Surfaces Due to Surface Forces John L.Parker**+and Phil Attard Department of Applied Maths, Research School of Physical Sciences and Engineering, Australian National University, G.P.O.Box 4, Canberra A.C.T.2601, Australia (Received: June 23, 1992; In Final Form: September 8, 1992)
Surface deformations are often observedduring the course of surface force experiments. We have calculated these deformations from continuum elastic theory for three fundamental types of surface force: attractive van der Waals interactions, electrical double layer repulsions, and oscillatory solvation forces. Strong repulsive surface interactions cause the surfaces to flatten, whereas the surfaces bulge out toward one another under the influence of strong attractive forces. The Derjaguin approximation, which relates the force scaled by the undeformed radius of curvature of the surfaces to the interaction free energy between planar plates, cannot be used when the surface deformations are large. The error in the free energy when the Derjaguin approximation is used is shown for the different force laws.
I. Introduction Advances in both theory and in experiment during the last two decades have brought about an enormous increase in understanding of the origin and nature of surface forces. One significant experimental technique is the surface force apparatus (SFA).l,z These instruments provided the first measurements of the force and separation between two surfaces and allowed the most direct comparison between theory and experiment. The SFA, principally p i o n d by Israelachvili,' employs multiple beam interferometry3 to measure the separation between two crossed cylinders. The total force is detected by the deflection of a weak spring on which one of the surfaces is mounted. Theory and experiment can be linked by invoking the Derjaguin approximation which relates the force (F)normalized by the radius ( R ) of curvature to the interaction free energy (E) between two planar half spaces.c6 F / R = 2rE (1) The interaction free energy is a universal quantity characteristic of the surfaces and the fluid. The Derjaguin approximation allows the free energy to be derived from a measurement of the total load which depends on the size and geometry of the surfaces. For spherical surfaces the Derjaguin approximationworb even at very small radii.57 When the measurement deviates from theory, an extra attraction or repulsion is inferred to be present. For instance, the observation of an extra repulsive force in addition to that predicted by DLVO theory has led to the discovery of hydration f0rces.89~ The application of eq 1 when the surfaces are deformed must be queetioned. All materials have a finite elasticity and will deform under the influence of surface forces. Muscovite mica is the favored material for surface force measurement because it is easily cleaved into thin transparent sheets suitable for the interferometry measurements. The mica sheets are silvered on one side and glued to silica disca. Figure 1 shows a schematic diagram of the surfaces. The glue adhering the mica to the discs is rather compliant, and the surfacesdeform when they are placed under load. For instance when the surfaces are brought into a strong adhesive contact a "flattened" region forms and is easily observed in the FECO fringe pattern. With a large repulsive interaction the surfaces still "flatten", but they adopt a different shape. The surface force apparatus provides a precise measurement of the force between the surfaces even when the surfaces are highly deformed, but obviously the Derjaguin approximation is no longer valid and the free energy per unit area between planes cannot be obtained directly. It has recently become possible to calculate self-consistently the surface shape between two elastic spheres interacting with an arbitrary surface force." The normal displacement of the surface 'On leave at The Surface Force Group, Depertmcnt of Physical Chemistry, The Royal Institute of Technology, S100.44 Stockholm and The Institute for Surface Chemistry, Box 5607, S-114.86 Stockholm.
0022-3654/92/2096- 10398$03.Oo/O
at a point r off the central axis under a given pressure distribution is
ivz
u(r) = - -X p ( t ) k ( r , t ) tdt where
K(m) is the complete elliptic integral of the first kind, p ( t ) = p,(h(t)) is the value of the pressure at t which depends on the deformed separation, h(t), E is the Young's modulus, and Y is Poissons's ratio. Early solutions of the elastic equations include the asymptotic results for repulsive interactions of Hughes and WhiteI2J3and numerical results for attractive Lennard-Jones interactions by Muller, Yushchenko, and Derjaguin.14 Complete numerical 80lution to eqs 2 and 3 has only recently been achieved, and in a previous publication" we have explicitly dealt with the deformations which m r when two elastic bodies are in contact. The results obtained were compared with the classical theories of H e a l s Johnson, Kendall, and Roberts (JKR),I6 and Derjaguin, Muller, and Toporov (DMT)I7 for simple model potentials. We found, in agreement with earlier c a l ~ ~ l a t i o nthat s ' ~Hertz ~ ~ theory is applicable when the range of the surface force is short, DMT theory is valid for rigid surfaces and attractive interactions, and JKR theory is more accurate (than DMT) for soft bodies. An interesting and novel prediction was that hysteresis occurs for soft adhesive bodies in contact." This paper is concerned with the surface deformations which occur prior to the surfaces contacting each other. We have found that the deformations that occur are very often significant and that great care must be taken in applying the Derjaguin a p proximation to compare measurement and theory. Often deviations from theory occur since only part of the applied force is attributable to the surface force between the bodies, and the rest goes into their elastic deformation. The types of "additional" forces which are observed depends critically on how the measurements are made and the types of forces present in the system.
II. Force Measurement Techdques There are a number of different techniques for measuring surface forces. In the simplest technique one surface is mounted at the end of a weak spring and the other surface is rigidly attached to a piezo electric crystal; the surface separation is measured by interferometry. Force measurement is begun with the surfaces at large separation, and the piem crystal is used to step one surface toward the other. The spring deflection can be either measured (8
1992 American Chemical Society
The Journal of Physical Chemistry, Vol. 96, No. 25, 1992 10399
Deformation of Surfaces Due to Surface Forces
\
silica+
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Light
Figure 1. Schematic diagram of the surfaces used in the surface forces apparatus. The two surfaces consist of a silica-glucxnica composite, and the type of deformations the surface experience under the influence of a repulsive surface force is shown to the right. At a separation h(O), the surfaces are pushed away from each other by an amount ho(0)- h(0) = 2240). The undeformed surface separation is ho(0).
directly, with a force sensor, or it can be inferred from a calibration of the piezo motion control. This technique will henceforth be referred to as the deflection technique. The deflection technique is not suitable for the measurement of strongly attractive surface interactions because of the mechanical instability which occurs when the slope of the attractive interaction exceeds the spring constant of the force measuring spring. In order to overcome this restriction a device is often used which allows the spring constant to be varied during the course of an experiment. The device is used to measure the separation at which the spring instability occurs (the jump position) and this is recorded along with the spring constant. In this way a plot of the derivative of the force, which in the Derjaguin approximation is just the pressure between planes, as a function of separation can be constructed. From here on this will be referred to as the jump method. Another technique has recently been developed which partly overcomes the spring instability problem and allows continual recording of attractive surface forces.'* In this technique a force sensorI9is used to regulate a magnetic force applied to the end of the cantilever in a feedback loop. With high feedback gains the sensor is maintained at a constant nominal null deflection regardless of the presence or not of surface forces. The voltage used to control the strength of the magnetic field is then directly proportional to the strength of the surface forces. Because of the null deflection of the bimorph, the nominal or undeformed separation can be inferred from the movement of the supporting piezo without the need to use optical interferometry.20(This is also the case for the atomic force microscope.) This technique will be referred to as the force feedback method. Obviously the effect of surface deformations on measurements will depend upon which one of the three procedures outlined above is used,as will be explicitly shown in what follows. First we briefly describe the theoretical analysis and then present model results for three fundamental types of surface force forces, namely exponentially repulsive, power law (or van der Waals) attractive, and oscillatory (or solvation) forces.
III. Analysis
where is the undeformed surface shape. Details of the numerical techniques have been given previously. In order to calculate the surface deformation, the pressure between two plates p,(h(r)) and the elastic constant need to be specified. The surface profile h(r) and the total force can be calculated as a function of ho. For a converged, self-consistent p(r), the total force is obtained by numerical integration over the surface profile. With these data it is possible to generate force curves which would be obtained with the different measurement techniques. The interferometrictechnique measures the separation between the backs of the mica surfaces. If we ignore compression of the thin mica sheet, then h is equivalent to this measurement. The glue and glass are much thicker and more compliant than the mica sheets so we expect most of the deformations to occur here. The variation in stress is rapidly damped, and so the major contribution comes from the glue, and the composite structure of the sample can be ignored. A plot of h versus the total FIR corresponds to the data which would be obtained with the deflection method. The force feedback method can produce the same data if FECO is used to measure the surface separation, but it also gives FIR as a function of ho, which is how we have chosen to represent the data. A plot of h versus dF/dho corresponds to the jump method. Note that no force measuring spring is included in the calculation, and so the results represent measurements made with an effectively infinite spring constant force measuring spring. B. Interaction Forces. DLVO and repulsive hydration forces are often measured with the surface force apparatus. The repulsive component in DLVO forces is electrostatic in origin, and for simplicity we have modeled this with a repulsive exponential pressure
P,(h) = P&-Kh
(5)
where Po = 2cw2@2,and K is the inverse debye screening length, \k0 electrostatic is surface potential, and €eo is the dielectric constant. Hydration forces can also be a model with a single exponential decay with much shorter decay lengths. The pressure for the attractive van der Waals profiles is taken from a Lennard-Jones continuum model of the solids.
Typical measured values for Hamaker constant ( A ) are 1.35 J for mica across air," 2.2 X lWN J for mica across water', J for hydrocarbon across water. We have chosen and 6 X values of A = 1.0 X 1.0 X and 1.0 X J. A repulsive component is included in the Lennard-Jones model, and the zero of the pressure occurs at zo = O h m . The addition of the attractive van der Waals interaction eq 6 and the exponential repulsion eq 5 leads to the full DLVO model. Finally the oscillatory force law is modeled with a simple damped sinusoid X
E(h) = A cos ( 2 ~ $ J Z ) e - ~ l ~
(7)
and the corresponding pressure
A. Numerical Model. Equation 2 relates the deformation of one body to the pressure exerted by another. The pressure is simply that due to the presence of a surface force. Figure 1 shows schematic profiles of two surfaces as they might appear if pushed toward each other with a repulsive surface interaction. The actual separation between the surfaces at a distance r from the central axis is denoted by h(r). The undeformed separation is denoted by ho(r). As one might expect, the effect of a repulsive surface force is to flatten the surfaces and to increase their separation compared to the undeformed surface shape. The pressure at a given radial separation from the center p(r) is taken to equal the pressure between two planes at that separation p(r) = ps(h(r)). The pressure determines the deformation and the deformation in tum determines the surface separation:
h(r) = ho(r) - 2u(r)
(4)
p,(h) = Ae-h/A(27rf sin (27r$JZ)+ cos (27r$JZ)/X) (8) with values of A = 0.02 N/m, l / f = 0.70 nm, and X = 0.65 nm. It has recently been shown that both experiment and theory for Oscillatory forces in the liquid OMCTS are well described by such a damped sinusoid with these parameter^.^ C. Elastic Constants. The values reported for the elastic constant E/( 1 - 3)for the composite mica-glueglass system are quite varied. White et a1.6 chose values between 5 X 1Olo and 10'' Nm-2, whereas Horn et a1.21found experimentally that the elastic constant can vary between 2.5 X 1 O9 and 1.16 X 1 0loNm-2. These values were obtained by measuring the contact area between two surfaces as a function of load at high electrolyte concentrations. In much more recent experimentson coated surfaces, Helm et a1.22reported values between 2 X 1Olo and 7 X 1Olo Nm-*. One
lo400 The Journal of Physical Chemistry, Vol. 96, No. 25, 1992
Parker and Attard
Repulsive
......._..... Oscillatory
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Figure 2. A plot the separation between two spheres at the point of closest approach against the undeformed separation ho for three fundamental types of surface interaction. The lower line on the figure indicates the results obtained with an attractive van der Waals interaction (q6 in the text) with the parameters A = J, zo = 0.5 nm. A repulsive surface force of the form of q 5 with parameters K-' = 1 nm and q0= 85 mV. In the absence of a surface force there is no deformation and as a consequence h = ho up until contact. Departures from nonlinearity occur in the presence of a force. For strong attractive interactions the surfaces are not stable at every separation. The surfaces jump spontaneously into contact when the slope of interaction exceeds an effective spring constant for the system and this point is indicated by the arrow. Instabilities occur for oscillatory force profiles at each oscillation (indicated by arrows).
2 1
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reason for the variation may be that the high adhesions imply that JKR theory, which was used to analyze the data, is not applicable." For fused quartz E/(1- 3)= 1.1 X 10" Nm-2 this value would represent an upper bound. We have c h a m elastic constants of 1 X lo9, 1 X lolo, and 1 X 10" Nm-2. With these values we are confident that the real measurements cannot lie outside of our predictions.
IV. Results At large surface separations, beyond the range of any surface force the change in surface Separation, h, is equal to how far the surfaces have bem moved &. When the surfaces begin to interact via surface forces this is no longer the case. Figure 2 illustrates this effect for the three fundamental types of surface forces. For an attractive interaction the surfaces are puled toward each other, and h d c c r w a mort rapidly than 4. For a repulsive interaction the surface separation remains larger than h,,. For an oscillatory force law the surface separation begins to oscillate with respect to the surface displacement at low forces, but as the force is increased h decreasca much less rapidly than does ho except for jumps from one oscillation to the next (full details follow). In the absence of surface deformations h = h,,. Negative values of 4 would correspond to interpenetration of the surface if they were not deformed. These results were computed for E/(1 - 3) = 1Olo Nm-2, a value between the Youngs moduli reported in the literature for the mica-glutglass system. The most startling result from inspection of Figure 2 is the fact that there is a substantial effect of surface compression well before contact between the surfaces is reached. All the CUNCBbegin to deviate from the undeformed case (h = ho) at a surface separation h = 5 nm. A. Attractive Forcer. The cross sectional surface separation for two surfaces interacting with an attractive surface force are shown in Figure 3A (A = J, E/(1 3) = 1OloNm-z). The profiles appear parabolic rather than circular because of the asymmetric axes (the Y axis is in units of m,whereas the X axis is in units of nm). The data is plotted in this manner 80 that the profdea repnsmt the shape of the fringes which would be observaed during experiments. The surface remains roughly spherical until the surfaces jump together at which point a large flattened contact
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r (Pm) Figure 3. (A) Surface profiles at varying separation for two surfaces interacting with an attractive force law R = 1.5 cm, A = J, and E/(1- 9)= 1O'O Nm-*. Profiles are also drawn after the surfaces have been pushed beyond the contact position, and this results in an increase in the contact area. (B) A plot of the surface profile at the point of maximum deformation (h = 1.40nm and ho = 1.95 nm, R = 1.5 cm, A = lWm J, and E/(1 - 9)= 1O'O N&). Both the surface shape and undeformed surface shape (dashed line) are shown. The amount of deformation as a function of the distance from the central axis is shown in the inset. region forms. This region, although it appears flat, is in fact curved as a result of the nonuniform pressure distribution across the area.11-13 The surfaces are deformed by the influence of the surface force prior to reaching contact, and this is illustrated in Figure 3B. The surface shape is plotted at a separation of h = 1.40 nm, and the undeformed surface shape is shown for comparison (dashed line). The inset shows the amount of deformation as a function of the radial distance from the central axis. The deformation at the center at this position is nearly 0.6 nm. Computed force w e a for the deflection method and the force feedback method are shown in Figure 4A for E/( 1 - v2) = 1Olo
The Journal of Physical Chemistry, Vol. 96, No. 25, 1992 10401
Deformation of Surfaces Due to Surface Forces
becomes more notable. In general, measurements made using the deflection method are slightly less attractive than the interaction free energy. This is due to the fact that the attractive force distorts the surfaces and pulls them in toward each other and this leads to a smaller radius of curvature. The feedback technique provides precisely the same values of force; however, we are free to choose whether to plot the data as a function of h (in which case an identical curve to the deflection method would be obtained) or as a function of ho which is how it is represented in Figure 4A. One would choose to present data in this way if optical interferometry is not being or cannot be used. In this case the force as a function of & lies to the more attractive side of the interaction free energy. Thii is due to the fact that the surfaces are distorted or pulled toward each other by an mount equal to & - h. Another important point to note is that the last points plotted in Figure 4A represent the last stable positions before the surfaces jump into contact. As seen in Figure 2 the surfaces are not stable below a particular surface separation. In a previous publication we have shown that for the force law of eq 6 the surfaces jump into contact at the position h given by
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-2 1’ Figure 4. (A) The computed force (symbols) and the interaction free energy (curves) for three different Hamaker constants (A = lO-I9, lo-*’ J with an elastic constant E/(l- 9) = 1OIoNm-2 and R = 1.5 cm). The separation, D = h or ho corresponds to the deflection measurement method (left m a t symbol) or to the force feedback method (symbol to the right), respectively. The interaction free energy lies between the computed force for the different measurement methods. The insct shows the force plotted against ho on a reduced scale. The finite slope of the curves when the surface come to contact is due to compliance of the system. (B)The effect of changing the sample elasticity on the computed results for the deflection (open symbols) and feedback technique (closed E/(1 - 9) = 1o’O symbols) is illustrated for attractive forces A = Nm-2 (data from Figure 4 open squares and filled diamonds) and E/( 1 - 9) = 10” Nm-2, R = 1.5 cm (open diamonds and triangles). Note that it is not possible to measure fotces beyond the position of the final point.
Nm-2. For small Hamaker constants (A = J) there is good agreement between forces plotted against both h and ho and the corresponding interaction free energy. This simply means that for small attractive forces there is little deformation. As the Hamaker constant is increased, the effect of surface deformations
Whether or not the surface is stable at a particular position depends on the derivative of the pressure and the ratio of the deformation to the pressure. The inset to Figure 4A shows the computed h,, vs E curves on an expanded scale. When the surfaces are in contact (ho = 2.5 nm indicated by an arrow), the bodies can be further pushed together (equivalent to further decreasing the undeformed surface separation ho), and the resulting slope in force is due to the compliance of the system. Experimental data qualitatively similar to this has been observed with the constant deflection feedback system.’* The effect of varying the elastic constant on the compound force curve is illustrated in Figure 4B. The data from Figure 4A (A = J) is replotted with calculations with an elastic modulus E/(1- 3)= lo9Nm-2. The effect of lowering the elastic COIIstant is to shift the jump positions to larger surface separations and obviously the amount of distortion prior to the jump becomes larger. Results computed for the measurement of attractive interactions with the jump method are shown in Figure 5. At large separations the jump positions agree relatively well with the pressure due to the surface force and there is little deformation. Accurate results are obtained for small forces but only up to a point. The last point on the upper curve (squares in Figure 5 ) represents the last stable position before the surfaces jump into contact. The consequences of elastic instability for the measurement of attractive forces are rather drastic. The first and most important consequence is that there are separations at which direct surface force measurements are impossible. The inset to Figure 5 shows the jump position as a function of the strength of the force and the elasticity of the surface calculated from eq 9. For small elasticities and strong attractive forces corresponding to the left top most region of the plot the jump position can be of the order of nanometers. Even with stiffer surfaces (Le., higher E/( 1 - 3)) the surfaces still jump from a reasonably large surface separation. In a realistic measurement A is of the order of J and E / ( 1 - v2) 5 Nm-2; under such circumstances it would be impossible to make measurements at separations less than 1.5 nm even with an infinite spring constant force measuring spring. Force measurements by the jump method are made by gradually increasing the spring constant of the force measuring spring and measuring the jump position. The measured jump @tion cannot change after the elastic yield for the system has been reached regardless of how stiff the measuring spring is made. Under such circumstancesif measurements are continued to still higher spring stiffness, then the results would follow the dashed line indicated in Figure 5. These results when compared with van der Waals or even a more elaborate theory would display an apparent extra
10402 The Journal of Physical Chemistry, Vol. 96, No. 25, 1992
Parker and Attard
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Fcp,5. Results computed for the spring jump method D = h, A = (upper) and A = (lower) for E/(1 - u2) = 1Olo Nm-2. The solid line is the analytic ~ u r between e planar walls (of unit area), and the points am obtained from numerical derivative of the F/2rR with respect to from tho data in F w e 4. The dashed linea indicate the separations at which the elastic instability occurs. The inset shows the position of the elastic hutability (from cq 9) as a function of the Hamaker constant: (a) R = 1.5 cm,E/(1 - 2 ) = lo9 Nm-2, (b) R = 1.5 cm, E/(1 - 2 ) = 1OloNm-*, (c) R = 1.5 cm, E/(1 - u2) = 10" Nm-2, (d) R = 0.1 cm, E/(1-9)= 109 Nm-2, (e) R = 0.1 cm, E/(1 - uz) = 1O1ONm-2, (0 R = 0.1 cm, E (1 2) = 10" Nm-2, and (8) R = 3 pm and E/(1 - u2) = 1010Nm-l -
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F i e 6. Surface profiles for two surfaces interacting with a repulsive surface. force.: K-I = 1 nm, q0= 85 mV, R = 1.5 cm, E/(1 - 2 ) = 1O'O
Nm-2.
calculations of surface deformation effects at these conditions using the nonlinear Poisson-Boltzmann law show little evidence of surface flattening. Figure 7C shows the computed fonx for E/( 1 - uz) = 1O1O Nm-2, I' = 13.8 nm, q0= 93 mV, and A = lWzo J which are the parameters obtained from fitting experimental data in ref 8. Surface deformation effects become extremely important when trying to reach an estimate for the decay length of a hydration attraction. Obviously it is important not to confuse a surface force from surface force measurements or even the decay length elastic instability with the instability due to the force measuring of double layer forces at high electrolyte concentrations. Indeed, spring. at very high electrolyte and high surface charge, surface deforB. acpldre Forcea The crow section of the surface separation mation can prevent the surfaces from coming to an adhesive between two surfaces interacting with an exponential repulsion contact even though there is a minima present in the interaction is shown in Figure 6 at various surface separations. The surface free energy. Certainly the measurementsof hydration force are profile is very different to that for an attractive interaction. The influenced by surface deformations but it is difficult to predict ~ ~ 1 section 8 0 never appears flat even when the surfaces are pushed by how much, and this is now the subject of closer scrutiny by together under high loads. In fact for short decay lengths both calculation and experiment. enormous forces need to be applied to reach contact. This is so C. osdlhtory Forces. Oscillatory surface forces, as the name because as the surfaces approach each other they begin to flatten implies, vary from attractive to repulsive as a function of the and consequently the forces increase because the pressure is inreparation between the surfaces. The effect of these forces on tegrated over a larger area. Of course, eventually the surfaces the surface profile is to contort the surface into a stepped surface must cinnc into contact but this may occur at tremendously high shape. This is most clearly seen in the inset to Figure 8. This forca (and wen passibly beyond the range which can be applied is a result of the fact that at one radial distance from the central with the instrument). This effect is best illustrated by the force axis the part of the surface at that separation is experiencing an curves shown in F i r e 7B. These are forces computed with K-I attractive interaction, whereas at slightly wider radial separation = 1 nm, q0= 40,60, and 80 mV, and with E/(1 - 9)= 1O1O the surface is experiencing a repulsive surface force. Nm-z and a Lennard-Jones type attraction A = 1WmJ. At low The results of forces calculated for the deflection method (in potentials the computed measurement have a somewhat shorter figure 9 squares represent the surfaces being pushed togetha and 1triangla signify unloading) begin to deviate from the interaction decay lcn#th than does the interaction free energy. However as thc~~isinaaased,largcrandlargerdeviatsonsare~ed. free energy ( d i d line) at the fourth Osciuation. At the very next An apparent extra repulsionis present in the force measurements. maxima the force is extremely high due to the fact that the The size of tho deviation due to the surface flattening depends data have become quite flat at this eepetation. This is precisely also on the elasticity of the surfaces. Larger flattening occurs what is observed experimentally. The force maxima for the infor softer surfaces,and hence there is a larger departure from the nermdt oscillations are well above the values one would expect free-. Fi7Ashowsthecompltedfo~xcurws for three for a simple exponentially decaying oscillatory fora. Intuwingly Werent clasticconstants, E/(1- 9)= 1O1O,loll, and 10I2NmV2, the valuea for the minima obtained arc very close to the interaction i1 = 1 nm,and qo= 80 mV. free energy. Hydration f o m obecrved between mica surfaces in aqueous Another rather interesting observation is that the radius of the e~ectrolytcsolutions are roughly exponential repulsions which flattened region is about 9 pm at a surface separation of 2.4 nm, appear as forced in addition to double layer foras. White et al.13 whereas at a similar force the radius for two surfaces i n t e r a w 1with an attractive interaction is about 4 pm (A = lWm J Figure found that surface flattening did not account for the observation of hydration forces between mica at low salt concentrations. 3A). The depth of the attractive intawtion, or the surface energy iin this case, is very much larger than the depth of the minima Hydration forces are said to appear at 5 X lo4 M NaCl,* and
Deformation of Surfaces Due to Surface Forces
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The Journal of Physical Chemistry, Vol. 96, No.25, 1992 10403
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r (Pm) Figure 8. Surface profiles for two surfaces interacting with an oscillatory surface force (cq 7), A = 0.02 N/m, l/f= 0.70 nm, A = 0.65 nm, R = 1.5 cm, and E/(1 - 9)= 1Olo Nm-*. The solid lines show the surface shape on moving the surfaces together and the dashed lines indicate the surface s h a m on separating. The inset shows an expansion of the data, and steps with the same periodicity as the force law arc clearly evident. 0
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F i v 7. (A) Computed force curvcs (deflection method D = h) for surfaces interacting with the same parameters as Figure 6 with three different clastic constantsE/(1- 2)= lo9(dotted line), 1Olo (solid lime), and 10'' (dashed line) Nm-2. (B)Computed force curves (D= h) for surface? interacting with three different surface potentials: q0= 40 (dashes and dots), 60 (long dashes) and 80 mV (short dashes) (I(-' = 1 nm). The interaction free energy for the three potentials is shown as solid lines. An attractive van der Waals interaction has been added to the repulsive force law A = 1 X lom, J, E/(1 - 9)= 1Olo Nm-$ R = 1.5 cm (i.e., a combination of cqn 5 and 6). (C) Calculated forces curves for the full nonlinear Poisson-Boltzman description of the electrical double layer. The quam are computed with constant charge boundary conditions and the circles for constant potential K-I = 13.8 nm and Yo = 93 mV. E/(1- 2)= lO'O, R = 1.5 cm. The Derjaguin approximation would be indistinguishable from these results on this scale.
in the oscillation, and so one may expect that the size of the flattened region to be very much larger. The force gradient for an oscillatory force varies much more rapidly with surface s e p aration than for a purely attractive force. This means that large
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D (nm> Figure 9. Computed force curve8 for the same oscillatory force law as in Figure 8. The solid symbols display the results for the force f d b a c k method ( D = ho), the open symbols for the force deflection method (squares for loading, triangles for unloading in both cases), and the d i d line is the specified interaction free energy, which one is attempting to measure. The inset shows ho vs F / 2 r R on a wider scale.
deformations do OCCUT even for relatively small forces if the gradient of force changes rapidly. The force as a function of ho is also plotted in Figure 9 (solid squares and triangles and over an expanded scale in the inset). The slope of the repulsive parts of the d a t i o n s is quite shallow and is again due to compression of the surfaces. The similarity
10404 The Journal of Physical Chemistry, Vol. 96, No. 25, 1992
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1.5
Force ( N x I O . ~ )
Figure 10. The surface deformation along the central axis (h - h,) against the applied force F(N) for both attractive and repulsive surface interactions (R = 1.5 cm and E/( 1 - v2) = 1Olo Nm-2). The line r e p resents the best fit to the data and it has a slope of 5.5 X 104 Nm-l. This number represents the upper limit for the stiffness of the apparatus.
between the measurements of ref 18 and the data shown in Figure 9 are quite remarkable.
v.
Discussioo We have calculated the shapes of surfaces and force between them as they interact with three fundamental types of surface forces. The type of surface deformations which occur depena in a complex way on the nature of the interaction between the surfaces. Large forces and rapidly changing force gradients cause large enough deformations to make the application of the Derjaguin approximation with the undeformed radius of curvature inappropriate. One way to try to unify the behavior for the different forces is consider the deformation along the central axis. The difference between h and ho provides the surface deformation at this point. A plot of this value against the force for attractive and repulsive forces are shown in Figure 10. The data lie roughly on a straight line, and the behavior can to a very rough approximation be described by Hookes law (f = K x ) where K is an effective spring ' Nm-Zthis effective spring constant amstant. For E/(I- 2) = 1OO is about 5 X 104 Nm-I. Approximate solutions for the elastic equations can be obtained when the deformation varies slowly with the curvature." Under this approximation analytic expressions can be obtained relating the force and the central deformation for both the simple attractive eq 6 and the repulsive eq 5 surface force. The effective spring constant for the system for a repulsive interaction is
and for an attractive interaction is
where d = h - ho. For values of E/(1 - v2) = 1OloNm-2, R = 1.5 cm,and 6'= 1 nm a value of 4.8 X lo4 Nm-' is obtained, and for an attractive interaction at h = 1 nm F / b = 3.6 X 104 Nm-'. The value obtained from the plot in Figure 10 is close to w 10 is roughly linear these values. The fact that the data in F justifies the approach taken in refs 18 and 5 in the analysis of the experimental results where the surface elasticity was removed with a linear approximation. Accounting for elasticity with this effective spring constant provides an approximate relation between h and &, valid for low applied loads. It thus allows thoee methods which specify ho (the force feedback technique and the atomic force microwope) to be corrected for the separation. It does not correct for error in the force due to the deformation. (Incidently, the common practice
Parker and Attard of dividing the total repulsive load by the area of flattened contact is not a valid method for determining the p u r e W e e n planes; see the pressure distributions given in ref 11.) This effective spring is in series with the force measuring spring, and it follows that the jump method will fail when the force maswing spring amstant exceeds the effective spring constant due to the surface elasticity (cf. Figure 5 ) . Up until recently only mica and a few other surfaces have been found to be suitable for study with the surface force apparatus. Optical interferometry can be used to obtain the surface separation only when the materials used are relatively thin (1-3 pm) and transparent. One way to overcome this problem is to measure the force and separation between surfaces using nonoptical techniques, and recently this has become possible using a bimorph force sensor.I9 With this sensor it is possible to measure the deflection of the spring on which one surface is mounted directly. The surface separation can be calculated from the deflection and the position (usually set by the voltage applied to the piezo) of the other surface. It must be remembered that the spring is mounted on the back of the surface, and if the distance is calculated in this way then the effect of deformations is completely ignored. For an attractive interaction the point of closest approach between the surfaces will be at a smaller distance than that deduced from the calculation outlined above because the surfaces are deformed toward each other. Conversely larger surface separations are deduced from calculations using this procedure for repulsive surface interactions. The value obtained for the force is identical to that obtained by interferometry but it is assigned to a different surface separation. Obviously the differences in the force curves obtained will become larger for stronger forces. It is possible to measure the force between a colloidal particle and a macroscopic surface with an atomic force microscope.a The separation between the surface and the colloidal probe can be deduced from the cantilever deflection and the surface position using the same procedure as outlined above. An optical light lever is used to detect the deflection of the end of the cantilever, and the position of the macroscopic surface is controlled with a piezo electric actuator. The sensitivity of the light lever is determined by pushing the surfaces into contact and then measuring the voltage produced by a differential light sensitive diode and using the known piezo calibration to calculate the lever deflection. With this calibration the spring deflection and hence the force and surface separation can be calculated. There are two potential problems with this procedure. Firstly the calibration for the light lever obtained may be incorrect due to deformation of the surface and the probe,and, secondly, the surface separation obtained from the deflection is not equal to the point of closest approach. The radius of colloidal probes used for such experiments are of the order of 10 pm. The effective elasticity of the system can be estimated from eqs 10 and 11, and for E/(1 - 9)= 10" Nm-2, R = 10 pm, and 6'= 1 nm a value of 1.3 X 104 Nm-' is obtained and for an attractive interaction at h = 1 nm 9.4 X IO3Nm-' is obtained. These values are lower by 4-5 than the effective elasticity for the surface force apparatus with E/( 1 - 2)one order of magnitude higher for the colloidal probe. The total force exerted by surface forces is about IO00 times smaller for the case of the colloidal probe but for similar sample rigidities the effective elasticity is ten times lower. One would then expect that for colloidal particles the size of surface deformations dong the central axis would be 20 times smaller than for the surface apparatus. The forces applied in the measurement of the region of constant compliance are usually much stronger (relative to the radius than those applied in the surface force apparatus), and so it is difficult to estimate the effects of surface deformation on these measurements without full analysis and this will be the subject of a future publication. The force feedback technique allows the simultaneous determination of the force, the undeformed separation hO and, if interferometry is being used, the separation h. Thus it is possible to determine experimentally all of the quantities which define the surface deformation (including the surface shape from the interferometricprofiles), and it is possible to determine the validity
J. Phys. Chem. 1992,96, 10405-10411 of eqns 2 and 3 and the results described here. A detailed experimental investigation is currently under way, and the results will be reported elsewhere.
References and Notes (1) Israelachvili, J. N.; Adams, G. E. J . Chem.Soc., Furuduy Trans. 1 1978, 74, 975. (2) Parker, J. L.; Christenson, H. K.; Ninham, B. W. Rev.Sci. Instrum. 1989, 60, 3135. (3) Israelachvili, J. N. J . Colloid Interface Sci. 1973, 44, 259. (4) Derjaguin, B. V. Kolloid Z . 1934, 69, 155. (5) Attard, P.; Parker, J. L. J . Phys. Chem.1992, 96, 5086. (6) White, L. R. J . Colloid Interjace Sci. 1983. 95, 286. (7) Attard, P.; B&ard, D. R.; Ursenbach, C. P.; Patey, G. N. Phys. Reu. A 1991,44,8224. (8) Pashley, R. M. J . Colloid Interfuce Sci. 1981, 80,153. (9) Pashley, R. M. J. Colloid Interfuce Sci. 1981, 83, 531. (10) Pashley, R. M.; Israelachvili,J. N. J. Colloid InterfaceSci. 1984,101, 511.
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(11) Attard, P.; Parker, J. L. Phys. Rev.A, in press. (12) Hughes, B. D.; White, L. R. Quart. J . Mech.Appl. Muth. 1979,32, 445. (13) Hughes, B. D.; White, L. R. J. Chem.Soc., Faruday Trans. 11980, 76, 963. (14) Muller, V. M.; Yushchenko, V. S.; Derjaguin, B. V. J . Colloid Interfuce Sci. 1983, 92, 92. (15) Hertz, H.; Reine, J. Angew. Murh. 1881. 92, 156. (16) Johnson. K. L.: Kendall, K.; Roberts, A. D. Proc. R.Soc. London A 1971,324. 301 (17) Derjaguin, 8.V.; Muller, V. M.; Toporov Yu, J. Colloid Interfuce Sci. 1975, 53, 314. (18) Parker, J. L.; Stewart, A. M. Prog. Colloid Polym. Sci., in press. (19) Parker, J. L. Lungmuir 1992,8, 551. (20) Israelachvili, J. N.; Tabor, D. Nature (Phys. Sci.) 1972, 236, 106. (21) Horn, R. G.; Israelachvili, J. N.; F'ribac, F. J. Colloid Interface Sei. 1987, 115,480. (22) Chen, Y. L.; Helm, C. A.; Israelachvili, J. N. J . Phys. Chem.1991, 95, 10736. (23) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991,353,239.
Reduction of the Roughness of Silver Films by the Controlled Application of Surface Forces John M. Levins and T.Kyle Vanderlick* Department of Chemical Engineering, University of Pennsylvania, Philadelphia, Pennsylvania I91 04-6393 (Received: June 24, 1992; In Final Form: September 17, 1992)
Using the surface forces apparatus, we have brought into contact a thermally evaporated silver film with a smooth mica sheet and have measured the time-dependent changes in the wavelengths of fringes of equal chromatic order generated using multiple beam interferometry. Based on a theoretical analysis of the interferencefringes produced for a silver surface with prescribed roughness, the wavelength shifts that we observe can be explained by an irreversible reduction in the roughness of the silver surface. Assuming a sinusoidal profile for the silver surface, our measured wavelength shifts correspond to a decrease in root-mean-squared roughness from between 10 and 30 A. Concurrent with the reduction of silver roughness, we observe an increase in both the area of deformed contact and the pull-off force between mica and silver. We also measured the thicknesses of alkanethiol self-assembled monolayers chemisorbed on the silver surface. Although our thicknesses are lower than those obtained by others using ellipsometry, our monolayer thicknesses increase as expected with the length of the alkanethiol hydrocarbon chain.
Introduction
Developed into its present form by Israelachvili,' the surface forces apparatus (SFA) has become a mainstay experimental technique of colloid and interface science. The apparatus allows the measurement of the force acting between two opposed surfaces as a function of their separation. In nearly all applications of the SFA, mica surfaces have been employed, primarily because mica can be cleaved molecularly smooth. Within the past few years, however, other surfaces-such as silica? poly(ethy1ene terephthalate),' sapphire,' and platinum fh5-have been used, with varying degrees of succcss, in the SFA. In this paper, we demonstrate that silver can be used as one of the two surfaces, mica being the other, without significant loss of resolution in the separation measurement. Moreover, we find that the roughness of thermally evaporated silver films can be r e d u d by mmpnaping the silver between two molecularly smooth mica sheets. Our interest in silver films is based on their use as substrates for alkanethiol self-assembled monolayer formation.61 I Self-assembled monolayers form via the spontaneouschemisorption from solution of functionalized molecules onto a host solid surface. We have formed self-assembled monolayers on silver surfaces in the SFA and demonstrated that angstrom-level resolution in monolayer thickness measurements can be obtained. The ultimate goal of our research is to use self-assembled monolayers to prepare surfaces with well-defined chemical and *To whom correspondence should be addressed.
structural properties for use in the SFA. We intend to employ these surfaces to study several colloidal interactions that are poorly understood-specifically, hydration f ~ r c e s , ' ~steric - ' ~ undulation forces,IsJ6 and hydrophobic forces.17-19 The first step toward achieving this goal is the focus of this paper: demonstrating that a surface suitable for the formation of self-assembledmonolayers (silver) can be used in the SFA. Although alltanethiol self-assembled monolayers are known to form on the other noble metals: silver has the advantage that it is ideally suited for the application of multiple beam interferometry, which is used to determine surface separation in the SFA. White light is directed through an interferometer consisting of two highly reflective metallic filar separated by a layer, or layers, of dielectric material with an overall thickness greater than the wavelength of light in the visible spectrum. The light undergoes multiple reflections between the reflective layers and emerges as a series of fringa of equal chromatic order (FECO).MThe FECO occur at discrete wavelengths which depend on the distance between the two reflective films. Silver is generally used as the metallic coating because of its high reflectivity. Israelachvili2' derived analytical expresSons relating the distance between the silver films to the wavelength, A, of any particular FECO. Rather than account explicitly for the optical properties of the reflective films, Israelachvili treated the silver layers as planes with reflectivity approaching unity. Using the multilayer matrix method-which accounts for the optical properties of each layer in the interferometer-Clarknz2examined the accuracy
0022-36541921209610405S03.0010 Q 1992 American Chemical Societv
