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Direct Measurement of Interaction Forces between Colloidal Particles Using the Scanning Force Microscope Y. Q. Li,? N. J. Tao,+J. Pan,? A. A. Garcia,t and S.M. Lindsay*rt Department of Physics and Astronomy and Department of Chemical, Biological and Materials Engineering, Arizona State University, Tempe, Arizona 85287 Received May 4,1992. In Final Form: January 26,1993 We have measured the interaction forces versus distance between pairs of 2-pm diameter polystyrene spheres immersed in electrolytes containing up to 1 M KC1. A reproducible long-range repulsion was observed for concentrationsbelow 0.01M. Adhesion varied considerablyfrom measurementto measurement (values ranged from 3.5 to 140 nN) and showed no systematic trend as a function of electrolyte strength. Data for repulsion in 0.001 M KC1are well fitted by a simple result based on the Derjaguin approximation and the linearized Poisson-Boltzmann equation. This fit yields a surface potential ($0) of 87 mV and a Debye length of 7.7 f 0.1 nm, in reasonable agreement with the expected Debye length (9.5 nm) and the measured { potential (83 f 2 mV). A full numerical solution of the Poisson-Boltzmann equation shows that the simpletheory works reasonably in these conditions,despitethe fact that $dkgT> 1. Measurements made in pure water and measurementsof the interaction between a sphere and a mice surface gave results that varied from run to run, none of which agreed with theoretical calculations.
Introduction The interaction forces (or interaction potentials) between two solid surfaces, especially for small particles, have been regarded as fundamental to the understanding of various colloidal and surface phenomena such as flocculation, stability, ordering, shear flow, adsorption, etc. Direct measurements of surface forces have been carried out by many groups using a number of methods.14 The surface force apparatus (SFA) is the most prevalent amongthemS2Most of these methods employmacroscopic surfaces. Recently there have been reports of measurements of surface forces with the scanning force microscope (SFM)."-lO In most of these experiments,the force-sensing tip has been used as one of the surfaces. Both long-range repulsion" and local adhesion1*have been studied, yielding data (at least in one case) that may show the breaking of individual hydrogen bonds.13 Both Ducker et ale1*and Buttll have replaced the normal force-sensing tip with a smallglass sphere, to obtain the first measurements of the interaction between a colloidal particle and a glass or mica11J4surface. We have taken this process one step further, and in this paper, we report measurements of the interaction between + Department of Physics and Astronomy.
* Department of Chemical, Biological and Materials Engineering.
(1)Derjaguin, B. V.;Rabinovich, Y. I.; Churaev, N. M. Nature 1978, 272,313-318. (2)Israelachvili, J. N.;Adams, G. E. J. Chem. SOC.,Faraday Trans. 1 1978,1,975-1001. (3)Israelachvili, J. N.Adu. Colloid Interface Sci. 1982,16, 31-47. (4)Israelachvili, J. N. Surface Sci. Resp. 1992,14,109-159. (5)Lodge, K.B. Adv. Colloid Interj. Sci. 1982,16,27-73. (6)Ninham, B. W. Adu. Colloid Interj. Sci. 1982,16,3-15. (7)Tang, S. L.; Bokor, J.; Storz, R. H. Appl. Phys. Lett. 1988,52, 188-190. (8) Weisenhorn, A. L.; Hansma, P. K.; Albrecht, T. R.; Quate, C. F. Appl. Phys. Lett. 1989,54,2651-2653. (9)Burnham, N.A.; Dominguez, D. D.; Mowery, R. L.; Colton, R. J. Phys. Rev. Lett. 1990,64,1931-1934. (10)Blackman, G. S.;Mate, C. M.; Philpott, M. R. Phys. Reu. Lett. 1990,65,2270-2273. (11)Butt, H.-J. Biophys. J. 1991,60, 777-785. (12)Mizes, H.A.; Loh, K.-G.; Miller, R. J. D.; Ahuja, S. K.; Grabowski, E.F. Appl. Phys. Lett. 1991,59,2901-2903. (13)Hoh, J. H.; Cleveland, J. P.; Prater, C. B.; Revel, J.-P.;Hansma, P. K.J. Am. Chem. SOC.1992,114,4917-4918. (14)Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991,353, 239-241.
pairs of spheres in electrolyte solution. We have chosen to study polystyrene latex spheres because they are ubiquitous colloidalvehiclesused both as the basis of many emulsions and as analytical and preparative components in chemistry and biochemistry. They are available with a range of functional groups on their surfaces, permitting biomolecules (such as antibodies) to be attached to their surfaces. We have exploited the ability of the AFM to position the tip in the xy plane in order to align a sphere on the force-sensing cantilever coaxially with a sphere chemically "glued" to a mica substrate. For surface potentials, $0, smaller than ~ B and T for low salt concentrations, one obtains, from the linearized Poisson-Boltzmann equation, the Derjaguin approximation, and a sum of van der Waals interactions,16 the following simple result (often referred to as the DLVO theory):
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F(D) = 2?ra2Re-& tEoK
AR 12D2
In (l), R is the radius of one of the pair of spheres (separated by a distance, D), u is its surface charge density, e is the dielectricconstant of the medium, 1 / is~the Debye length, and A is the Hamaker constant. This result is based on the assumption that the surface charge remains constant as the spheres approach one another. Since the surface charge is probably a function of salt concentration (see below), this is not a completely valid approach. Nonetheless, numerical simulations show that it is better than the other simplealternative of holdingthe surface potential constant because there are large changes of surface potential as the two double layers interpenetrate. In any event, surface potentials are too high for the assumptions under which (1)was derived to hold, and a full calculation (described below) is required. It is difficult to determine the surface charge directly by other means. Measurementsof electrophoreticmobility are usually interpreted in terms of the potential at the hydrodynamicshear plane (the {potential, which is usually nearly equal to the surface potential) because this datum can be obtained without the use of a model for the double~
~~~~
(15)Israelachvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, 1992; p 450.
0743-7463/93/2409-0637$04.00/00 1993 American Chemical Society
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638 Langmuir, Vol. 9, No. 3, 1993
layer capacitance of the sphere.16 However, in order to make a comparison with the surface charge that appears in (l),we will use the Grahame equation for a plane: u = 0.117
[KCl]'/2 sinh(+,-J51.4)
(2)
where u is in coulombs per meter squared, [KCl] is molar concentration,and $0 is in millivolts. Numerical solutions (see below) show that this is reasonably accurate at 0.001 M electrolyte (despite the fact that jl/~/tZgT> 1). For comparison,we also examinethe interactionbetween a sphere and a charged plane (mica surface). In this case, the result of simple theory id5
where u1 and u2 are the charge densities on the sphere and plane, respectively. For identical charge densities, this yields twice the force experienced by a pair of spheres. The adhesion is characterized by the hysteresis in the force distance curve, the tip remaining in contact with the surface after the tip is retracted for the extra distance needed to generate a force in the cantilever equal to the adhesion force. However, in contrast to data for the longrange repulsion, curves were not reproducible from scan to scan between a given pair of spheres. Experimental Section The scanning force microscope used in our experiment is a commercially available machine, the NanoScope I1 (Digital Instruments, Santa Barbara, CA). The sample and the tip are immersed in a liquid cell supplied by the manufacturer. This cell has ports which allow the solution in the cell to be exchanged without disturbing the experimental setup. Data were obtained using the NanoScope software with a modification that allows the height to be scanned slowly. Raw data (cantilever deflection vs voltage applied to the z scanner) are displayed on the NanoScope screen. The Nanoscope I1 does not allow these data to be extracted, so these displays are saved as screen dumps. We have written code which extracts the data from these files. They were turned into force vs distance curves by the following procedure. The point of contact was identified as the point where rapid downward displacementof the cantilever changesinto an upward displacement (the tip having jumped into contact). The deflection of the cantilever from its position at large distances was then subtracted from the raw height scan values point by point. In principle, these curves should also be corrected for compression of the sample after contact, but these effects are small (see discussion below). The vertical sensitivity of the height scanner was calibrated as described e1~ewhere.l~The force display was calibrated by pushing the tip with a rigid (steel) substrate. The spring constant of the cantilever was obtained by pushing the cantilever against a glass fiber and monitoring the relative deflection of the cantilever under an optical microscope. The glass fibers had been calibrated by measuring their deflection when loaded with a small weight. The cantilevers used in this work had a spring constant of 0.58 f 0.09 N/m. A typical SFM tip is a Si3N4pyramid about 5 pm in size grown on a cantilever. We modified the tip by gluing a polystyrene sphere to the end of a cantilever using Torr Seal (Varian Vacuum Products), a low vapor pressure resin sealant. This does not appear to cause significant contamination. We have verified, using optical microscopy, that all the balls used in this work do protrude beyond any other part of the cantilever structure, so we are sure that the ball dominates the interaction. A scanning electron micrograph (SEM) of a 16-pm ball on a cantilever is shown in Figure 1. (16) Vold, R.D.; Vold, M. J. Colloid and Interface Chemistry;AddisonWesley Publishing Co., Inc.: Reading, MA, 1983; p 694. (17) Li, Y.; Lindsay, S. M. Reu. Sci. Instrum. 1991,62,2630-2633.
Figure 1. Scanning electron microscope image of a 16-pmdiameter sphere glued to an SFM cantilever. The substrates were either freshly cleaved muscovite mica or thoroughly cleaned glass cover slips which were treated with a silane in order to bond the balls. (3-Aminopropy1)triethoxysilane (AldrichChemical Co., Inc.) was made into a solution containing 98% (v/v) ethanol and 2% (v/v) acetic acid (pH 4.9) for a final silane concentration of 3.5% (v/v). Substrates were first rinsed in ethanol and then immersed in this silane solution for about 15 min. They were then rinsed with ethanol and heated in air a t 110 "C for 30 min in order to volatilize weakly bound silane. After heat treatment, 100 mL of 0.25% (v/v) polystyrene microsphere suspension was placed on the substrate surface. The colloidal suspension was allowed to dry thoroughly in air before rinsing with triply distilled 18-MQdeionizedwater. Suspensions were obtained from Dow Diagnostic (2-pm-diameter spheres) and PolySciences(16-pm-diameterspheres). The smallerspheres were prepared by free radical emulsion polymerization, and the larger spheres were prepared by suspension polymerization.The smaller particles are free of surfactant. The manufacturers state that trace amounts of surfactant may be found on the larger spheres. The smaller spheres formed uniform arrays, allowing periodicities (and hence the sphere diameter) to be checked directly by imaging.I7 The 16-pm-diameter spheres were too polydisperseto permit measurement of size by imaging (because of the uncertainty in the tip radius), so we have used the manufacturer's value of 15.8 f 2.8 pm diameter. We first imaged the spheres under an aqueous solution with the modified tip. Figure 2 showsa series of imagesof polystyrene spheres 16 pm in diameter on a mica surface obtained with the same size sphere tip. The tip was then positioned on top of a sphere on the surface by a succession of zooms as indicated in Figure 2. We found that force-distance curves did depend on the degree of alignment (asexpected) but that careful centering of the two balls resulted in reproducible curves. We then obtained the force vs distance curves by ramping the sample stage in the z direction while the xy scan was effectively disabled. We have found that hysteresis is reduced to an insignificant level when the ramp speed is reduced to below 200 nm/s. All the force-distance curves were recorded a t this speed or lower. Initial data were obtained in 18MQwater. Salt solution was then flushed through the cell, the balls were realigned, and the data were recorded again. We found that data in a given run were very reproducible. We did not average curves (20 were recorded over each ball) because each curve was essentially identical. However, we did observe large variations from preparation to preparation. Somepreparations gave no indication of long-range interactions. We believe that this may reflect contamination of the spheres during the drying process. We have selected the data we show here because they are representative of the longest range interactions we have seen (contamination appears to destroy this long-range interaction). Experle1,and 1 M KCl iments were carried out using solutions. We also monitored adhesion by recording the "pullofF force. The results are listed in Table I. Adhesion forces
Langmuir, Vol. 9, No. 3, 1993 639
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Figure 2. Polystyrene spheres deposited on a mica surface imaged with a same size sphere as the tip: (a) 65-pm scan, (b) 40-pm scan, (c) 20-pm scan, (d) 2.7-pm scan. The mean diameter of the spheres is 15.8 pm. The sphere indicated by the letter A is the one of interest. Table I. Values of Adhesive Force between 2-"-Diameter Polystyrene Latex Spheres
mean mean adhesive std dev adhesive std dev ion concn force (nN) (nN) ion concn force (nN) (nN) 18 MQ HzO 90.3 35.0 lO-'M KC1 13.9 8.8 M KC1 M KC1
30.0 93.0
5.0 22.5
1 M KC1
82.3
28.8
varied from 3.5 to 140 nN, and we could not discern a systematic trend as a function of electrolyte strength. The charge on the spheres was characterized experimentally using measurements of their electrophoretic mobility, from which the {potential was extracted.I8 These measurements were kindly carried out by Professor Fuerstenau's group a t the University of California, Berkeley. They yielded the following data: For the 2-pm spheres, V, was -83 f 2 mV in 0.001 M KCl and -62 i 3 mV in 0.01 M KC1. For the 16-pm-diameter spheres, Vc was -74 f 2 mV in 0.001 M KCl and -60 f 2 mV in 0.01 M KCl.
Results and Discussion Figure 3a shows the force-distance curves between two polystyrene spheres, each 2 pm in diameter, for various ion concentrations. All the force-distance curves are corrected as described above, but are not averaged or smoothed. Note that, in this case, the point of contact is quite unambiguous as the points above "0" on the horizontal axis of Figure 3a. The gradient of the forcedistance curve after contact (i.e., to the left of the distance origin in Figure 3a) is essentiallyidentical to that obtained when the same tip is scanned over a clean stainless steel disk so that the elastic deformation of the polystyrene (18) Hunter, R.J. Zeta Potential in Colloid Science;Academic Press: London, 1981; p 386.
cannot be measured with this particular cantilever.lg At low ion concentration, a long-range repulsive force is evident. A t the distance marked by arrows (Figure 3a) the tip jumps into contact as a result of mechanical instability. This is evident as an almost vertical portion of the raw deflection data (corrected here as described earlier). The jump-into contact was found to occur at the same point (within experimental uncertainty) for each scan. This is not evident in the corrected data (Figure3a) where the sharp jump has been smeared out by the correction that has been applied for cantilever displacement. Data in the region between the onset of the jump and the point of contact are subject to considerable uncertainty because of the rapid motion of the cantilever in this region. We had considerable difficulty measuring a long-range interaction between a ball on the tip and freshly cleaved mica. We could not obtain any data with the 2-pmdiameter spheres, only observing a repulsion with the 16pm-diameter spheres. These data are shown in Figure 4. In this case, a point of contact was not obvious. As can be seen from the jump-in regions (marked by arrows) on Figure 4, the point of contact appears to change with salt concentration. The origin of the discrepancy lies in the "soft" tail that has appeared in the data at the lower salt concentrations. We have kept the origin at the point measured at high salt where contact is more obvious. We do not understand the origin of this tail, but the above procedure yields a Debye length for the 0.001 M KC1 of 6.9 f 0.4 nm, which is in agreement with the fit in the ball-ball experiment. On the other hand, the forces are very low. If the mica were charged as much as the (19) Tao, N.J.; Lindsay, S. M.;Lees, S. Biophys. J. 1992,63,1-5.
Let tera
640 Langmuir, Vol. 9, No. 3, 1993 .....................................................
".........................................................
Distance (nm)
0
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0
40 60 Distance (nm)
20
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BO
100
Figure 3. (a,top) Experimental force-distance curvesfor various electrolyte strengths as marked. The bar on the left indicates 5 mN/m, and curves are arbitrarily displaced for clarity. (b, bottom) Experimental data in 0.001 M KC1 (diamonds) plotted with the fit to the DLVO theory (eq 1,solid line) and the result of the numerical solution of the Poisson-Boltzmann equation. ......l... ................."............................
~
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, 10
30
50
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, , 90
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Figure 4. Forve vs distance curves for a polystyrene sphere 16 pm in diameter and a mica plane surface in various ion concentrations.
polystyrene surface, then these data should yield a force that is 16times that measured in the previous experiment. In fact,the forcesare similar,suggestinga very small charge on the mica (whichwould explain why we could not observe the electrostatic interaction with the 2-pm spheres where the force would be 8 times smaller). Data for the f potential indicate that the surface . potential of the spheres is many times ~ B TTherefore, we have to fit our data with a full numerical solution of the Poisson-Boltzmann equation. We began with a Pascal Butt, and described program kindly sent to us by Dr. H.-J. elsewhere." The experimental geometry leads to a large
calculation because of the large sphere diameter (pm) relative to the length scale (nm) on which forces were measured. We therefore rewrote the code in C and ran it on a UNIX workstation (Silicon Graphics Personal Iris). The approach is similar to that described by Butt,ll but we note the following points: Due to geometrical symmetry, we remapped the two-sphereproblem from a whole space into a quadrant, thus imposing boundary conditions of vanishing surface normal derivative at the midplane between two spheres and at the plane containing the line joining the centers of the two spheres. We used the following boundary conditions at the sphere-electrolyte interface: The mesh points at the boundary fall into 16 different situations, with each one yielding a different linear equation relating neighboring potential values. For example, consider mesh point M inside the electrolyte near the lower half of the polystyrenesphere and therefore closer to the midplane. One of its nearest neighboring mesh points is inside the polystyrene sphere. The surface normal passing through mesh point Fvl allows us to extrapolate the surface normal derivativesof the potential for both sides of the boundary by relating it to the potentials at M and at its four nearest and two second nearest neighbormesh points. The expressions€orsurface normal derivatives are substituted into the boundary condition such that the change of the surface normal derivative (weightedby their dielectricconstants) matches the surface charge density. Thus, the equation relating the potentials at M and its four nearest and two second nearest neighbors is established. These equations were combined with those of the finite differenceversion of the nonlinear Poisson-Boltzmann equation inside the electrolyte and Laplace equation inside the sphere. They were solved by the Gauss-Seidel method. The force between the two spheres was then calculated using the potential obtained at the midplane. At low surface charges we obtained results which agreed with (1)and (3). Figure 3b shows a comparison of the experimental data for 10 mM KC1 (points) with the results of fitting the simple theory (eq 1) and the results of the numerical solutionof the Poisson-Boltzmann equation (dashed line). The numerical solution is for a surface charge density of 2X C/m2. The fit to (1)is for u = 9.6 X 103 C/m2 with 1 / =~7.7 f 0.1 nm. This is in reasonable agreement with the calculated Debye length15which is 9.5 nm in a M 1:l electrolyte. Equation 1was fitted using a leastsquares procedure in Kaleidagraph. The numberical calculation was repeated with different values of u until the calculated force at 6-nm separation matched the observed force. This gave a somewhatworse overall fit to the data than the simple theory, but the two approaches do not differ considerably, Numerical calculations show that (2) gives reasonable M KC1 provided that the radius of the values for $0 in sphere greatly exceeds the Debye length. The calculated (eq 1)and 2 X le2 C/m2(numerical) charges of 9.6 X correspond to surface potentials of 87 and 123 mV, respectively. These are reasonably close to the measured { potential (83 f 2 mV). The long-range tail is more obvious in pure water (see Figure 3a), but the data do not agree with either of the results of a simple theory or a full calculation. Since the surface potential is much higher in this case, we expect that a full calculation is essential. The results of this calculation (not shown) yield a very rapid initial decay with a long tail. The decay of the long tail is similar to the simple DLVO prediction, but if this tail alone were fitted, the correspondingvalues of the surfacecharge would
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Langmuir, Vol. 9, No. 3, 1993 641
be far too low. Hone et aL20 have shown that a Yukawa potential is a reasonable fit to the interaction in this case. Thus, we expect to see a long-range decay with a decay length similar to the simple Debye result. This would be 1pm in pure water (the calculated initial decay is very rapid with a decay length of a few nanometers). The tail seen in the data for pure water has a decay length of 10 nm, in complete disagreementwith theory. The solution is unbuffered, but its measured pH (-6.0) would not account for this discrepancy. We believe that contamination must play a role in this case, and understand that salt can be leached from the interior of the spheres when they are immersed in pure water.21 We can also fit the le3M data for the ball-plane experiment (Figure 4) quite well with (3). We obtain a Debye length of 6.9 f 0.4 nm. Measurements of the potential yield similar values for the two spheres, so if we assume that the surface charge density of the 16" sphere is also -1 X 10-2 C/m2, we can extract a value for the surface charge density of the mica using (3). We obtain 4.1 X 1W C/m2. Equation 3 then yields $0 = 5.7 mV. The surface potential of mica in 10-3M 1:l electrolyte has been measured by Israelachvilli and Adams who report a value of 75 mV. We do not understand the origin of this discrepancy. It may be that mica is more easily contaminated than the surface of the polystyrene spheres.
Summary
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(20) Hone,D.;Alexander, S.;Chaikin,P. M.;Pincus,P.J.Chem.Phys. 1983,79,1474-1479. (21) Clarke, N.A. Personal communication.
Long-range interactions and adhesion have been measuted between pairs of polystyrene latex particles. Data obtained in 103 M KC1 are in reasonable agreement with calculations of the long-range electrostatic interaction based on exact solution of the Poisson-Boltzmann equation. This long-rangeforce shows a systematic dependence on electrolyte concentration, decreasing as the concentration is increased. Adhesion forces vary considerably (3.5-140 nN) and show no consistent dependence on electrolyte concentration. Experiments in pure water showed a long-rangerepulsive interaction. The data could not be fitted by calculations,probablyas a consequence of contamination. Measurementsof the interaction between a sphere and a mica plane yield a reasonable value for the Debye length in 10-3 M KC1, but the estimated surface potential of the mica is very low.
Acknowledgment. We are grateful to Dr. Fuemtenau and his graduate student M. Cholic for the potential measurements. We thank Dr. J.-H. Butt for insightful discussionsand his help on the computercodes. We thank J. P. Carrejo for taking the SEM image of the modified AFM tip. Also, we would like to thank Rick Oden, Larry Nagahara, and James D e b s e for many helpfuldiscuseions. The work has been supported by the National Science Foundation (Grant Dir 89-20053) and the Office of Naval Research (Grant "14-90-5-1455).
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