Probing the Limits of the Derjaguin Approximation with Scanning

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Probing the Limits of the Derjaguin Approximation with Scanning Force Microscopy Brian A. Todd and Steven J. Eppell* Department of Biomedical Engineering, Case Western Reserve University, Cleveland, Ohio 44106-7207 Received July 8, 2003. In Final Form: February 26, 2004 We have measured the interaction force between a silicon nitride scanning force microscopy (SFM) probe and the basal plane of highly oriented pyrolitic graphite as a function of pH and ionic concentration in aqueous solutions. Forces in the range (50 pN were reconstructed from measured signals using dynamical analysis of the cantilever. We modeled the force-separation data using a flat plate electric double-layer interaction and assumed the Derjaguin approximation to adapt the flat plate geometry for the SFM probe shape. Measured forces were well modeled by the theory at high ionic concentrations (10 and 100 mM), where Debye lengths were 3.0 and 0.96 nm, respectively. The theory failed to model forces at a lower ionic concentration (1 mM), where the Debye length was 9.6 nm. To investigate this, we calibrated the SFM probe geometry using blind reconstruction and obtained an apex radius of 7 nm. This value suggested that failure of the theory was due to an invalidation of the Derjaguin approximation at long Debye lengths, where the characteristic length scale for the interaction was larger than the size of the SFM probe. The errors were reduced by replacing the Derjaguin approximation with a surface element integration. The result experimentally demonstrates the limitations of the Derjaguin approximation for predicting interactions of nanoscale colloids.

I. Introduction In many colloidal systems, the characteristic size of the colloid is smaller than the characteristic length scale for interparticle interactions. For example, most polymers and biomolecules have characteristic dimensions on the order of a nanometer, comparable to the decay length for van der Waals forces. Likewise Debye lengths and even solvent forces can extend over many nanometers, larger than the size of some colloids. Yet, nearly all of the experimental investigations of colloidal interactions to date involve the opposite case: systems where the characteristic size of the object is much larger than the interaction length scale. This is partly an historical artifact since the leading instrument in intermolecular and surface force measurements, the surface forces apparatus, measures the interaction between two large mica sheets (radius of curvature ∼1 cm).1 Scanning force microscopy (SFM)2 can measure the interactions between small colloids and even single atoms.3-5 However, micrometer-sized particles are often glued to the SFM cantilever (so-called “colloid probe microscopy”)6 to provide well-characterized surface geometry and larger, more easily measured signals. Unfortunately, this abolishes the high lateral resolution afforded by the sharp tip (a strong point of making force measurements using SFM), and interaction forces on small colloids remain less understood. From a theoretical standpoint, the interactions of small colloids are expected to be significantly different from those * To whom correspondence should be addressed. E-mail: sje@ cwru.edu. (1) Israelachvili, J. Intermolecular and Surface Forces; 2nd ed.; Academic Press: San Diego, 1992. (2) Binnig, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930-933. (3) Ohnesorge, F.; Binnig, G. Science 1993, 260, 1451-1456. (4) Giessibl, F. J.; Herz, M.; Mannhart, J. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 12006-12010. (5) Hoffmann, P. M.; Oral, A.; Grimble, R. A.; Ozer, H. O.; Jeffery, S.; Pethica, J. B. Proc. R. Soc. London, Ser. A 2001, 457, 1161-1174. (6) Ducker, W. A.; Senden, T. J.; Pashley, R. M. Nature 1991, 353, 239-241.

of large colloids. In particular, for large colloids, the Derjaguin approximation applies.7 In this limiting case, the separation dependence of the force-separation relationship is identical for all large colloids. In other words, using either a ∼1 cm surface with the surface forces apparatus or a ∼10 µm particle with a colloidal probe microscope, the functional form of the measured forceseparation curve is identical; only the scaling of the forces is different (1000 times smaller for a 10 µm surface). As the size of the colloid approaches the length scale for the interaction, however, a new behavior emerges and the functional form for the force-separation relationship is expected to change.8 This behavior is more complicated to model, however, it has recently been shown that the case of a sphere interacting with a surface can be calculated using a relatively straightforward numerical quadrature dubbed surface element integration (SEI).9 Since this geometry is similar to the SFM probe interacting with a surface, the SEI should be useful for interpreting SFM force data where a nanoscale probe interacts with a smooth substrate. The SEI predicts forces smaller than the Derjaguin approximation.9 In this study, we investigate the interactions of an SFM probe with a flat substrate and compare the experimental data with those of the Derjaguin approximation and SEI. To determine the size of the probe apex, we reconstructed its geometry using a modified blind reconstruction algorithm.10 At the nanometer length scales relevant to the tip-surface interaction, this apex size is the “characteristic radius” for the SFM probe colloid. We then carried out high-bandwidth force experiments11 in aqueous buffers, controlling the length scale for the interaction (Debye (7) Derjaguin, B. V. Kolloid-Z. 1934, 69, 155-164. (8) Adamczyk, Z.; Weronski, P. Adv. Colloid Interface Sci. 1999, 83, 137-226. (9) Bhattacharjee, S.; Elimelech, M. J. Colloid Interface Sci. 1997, 193, 273-285. (10) Todd, B. A.; Eppell, S. J. Surf. Sci. 2001, 491, 473-483. (11) Todd, B. A.; Eppell, S. J.; Zypman, F. R. Appl. Phys. Lett. 2001, 79, 1888-1890.

10.1021/la035235d CCC: $27.50 © 2004 American Chemical Society Published on Web 05/12/2004

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length) using the electrolyte concentration. With this setup, we were able to measure forces both when the characteristic radius of the SFM probe was larger than the Debye length and when it was smaller. The results confirm that the Derjaguin approximation overestimates forces for small particles and long Debye lengths. Using the SEI provided a better fit to the experimental data. The result experimentally demonstrates the limitations of the Derjaguin approximation in predicting interactions of nanoscale colloids. II. Materials and Methods a. Force-Distance Measurements. Forces between a silicon nitride SFM probe and the basal plane of highly orientated pyrolytic graphite (HOPG(bp)) were measured as a function of pH and ionic concentration using a multimode scanning force microscope with a Nanoscope III controller and liquid cell (Digital Instruments, Santa Barbara, CA). All solutions were freshly prepared within 2 days of the experiment using 18.2 MΩ water from a Millipore UV plus system (Millipore, Bedford, MA). The concentration of buffering species was minimized so that the pH and ionic concentration could be controlled independently. A minimum buffer concentration that controlled the pH of solutions to (0.1 for 2 days was determined to be 0.1 mM. Solutions of pH 4.0, 6.7, and 9.7 were prepared at this concentration using potassium hydrogen phthalate, potassium phosphate monobasic, and potassium borate pentahydrate as the buffering species, respectively. The ionic concentration was varied at 1, 10, and 100 mM using NaCl. The cantilever and solution cell were rinsed with acetone prior to the experiment. All buffer contacting surfaces were then rinsed 5× each with methanol, ethanol, and then an excess of 18.2 MΩ water. For each solution, several milliliters was flowed through the liquid cell, flow was stopped, and the system was allowed to equilibrate for 15 min. The SFM probe was then engaged, and the surface was imaged with a scan range of 500 nm to ensure smoothness. Finally, 50 force-distance curves were measured with a z-range of 1 µm and repeat rate of 10 Hz. This rate is expected to be very slow with respect to the development of electric double-layer (EDL) forces, so only one rate was used and not a “spectrum” as is often done in investigations of slow biological interactions.12 The cantilever deflection signal was digitized via the signal access module using a high-speed digitizer at 10 MHz (Compuscope CS1100, Gage Applied Sciences, Montreal, Canada). This experimental setup is depicted in Figure 1. b. Force-Separation Curve Reconstruction. Prior to reconstruction of force-separation curves, each measurement was preprocessed to remove photodetector drift and an additional artifact resulting from interference between the laser reflection off the cantilever and the reflection off the HOPG(bp) surface. The artifact was removed by fitting a portion of each forcedistance curve that was far from the tip-surface contact point to an interferometer equation plus a line

martifact(t) ) A sin2(ωt + φ) + c0 + c1t

(1)

and then subtracting the fit from the raw measured curve to obtain a more accurate representation of the cantilever deflection response. The frequency, ω, was consistently close to the value expected on the basis of the velocity of the surface (10 µm/s) and the wavelength of the laser (670 nm). Forces were then calculated as previously described.11 Briefly, a dynamical model of the cantilever is used to reconstruct the probe force from the measured cantilever signal. The reconstruction involves carrying out a regularized inversion of the linear system of equations

Kf ) m

(2)

where m is vector containing the digitized cantilever deflection measurements, f is a vector of forces, and K is a matrix (12) Merkel, R.; Nassoy, P.; Leung, A.; Ritchie, K.; Evans, E. Nature 1997, 397, 50-53.

Figure 1. Experimental setup. Interactions between a silicon nitride SFM probe and an HOPG(bp) surface are measured by monitoring the deflection of the cantilever with an optical lever while cycling the height of the surface. The photodetector signal of the optical lever is digitized at 10 MHz using a computerbased data acquisition system. The experiment is performed using a liquid cell in nine different solution compositions, three different pH values by three different NaCl concentrations. representing the cantilever instrument response. Values for the three kinematic parameters of the cantilever are required to calculate K and were calibrated using Sader’s method13 to obtain k ) 0.017 N/m, f1 ) 3100 Hz, and Q ) 1.5 for the spring constant, first-mode resonant frequency, and first-mode quality factor, respectively. Fits obtained for the calibration confirmed that measurement noise was appropriate for thermally limited fluctuations of the coupled cantilever-buffer system. The inversion of eq 2 was carried out in MATLAB (Mathworks, Natick, MA) using the L-curve analysis14 and Tikhonov regularization15 implementations provided by Hansen’s regularization tools.16 A force-separation curve was then obtained by plotting force versus the difference between the SFM tip and HOPG(bp) surface positions. The force curves reconstructed in this fashion are accurate but have a large amount of noise.17 The noise was reduced by registering each of the 50 force-separation curves at “contact” (defined by the steep portion of the force-separation curve at the smallest separation) and then binning and averaging with a separation bin size of 0.25 nm. This yielded a single average force-separation curve representing the average probe-surface interaction for each of the nine solution concentrations (three pH values by three ionic concentrations). c. Regression Analysis. The nine average force-separation curves were analyzed using an EDL model (first term) plus a hard-core repulsion model (second term) for the potential energy per unit area between between two infinite planes1

G(D) )

2π B (σ σ )e-κ(D-D0) + 0κ 1 2 72π(D - D0)8

(3)

In this equation, G is the potential energy per area and D is the measured tip-surface separation. The parameters are associated with hard-core repulsion B, the product of the surface and tip charge densities σ1σ2, the inverse Debye length κ, the dielectric constant of the medium , the permittivity of free space 0, and the unknown absolute tip-sample separation at contact D0. The product of the surface charge densities is enclosed in parentheses to emphasize that it is a single indistinguishable degree of freedom. The specific form chosen for the EDL interaction is strictly valid only for interactions between ionpermeable surfaces.18 However, the general exponential form, (13) Sader, J. E.; Chon, J. W. M.; Mulvaney, P. Rev. Sci. Instrum. 1999, 70, 3967-3969. (14) Hansen, P. SIAM Rev. 1992, 34, 561-580. (15) Tikhonov, A. N. Dokl. Akad. Nauk. SSSR 1963, 151, 501-504. (16) Hansen, P. C. Numer. Algorithms 1999, 20, 195-196. (17) Todd, B. A.; Eppell, S. J. J. Appl. Phys. 2003, 94, 3563-3572.

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differing only by small factors, is valid for a variety of configurations. This includes any configuration of permeable or impermeable surfaces at large separations18 and the permeable/ impermeable configuration when the impermeable surface has a much larger charge than the permeable surface. The form that we use is often simply given as the electrostatic free energy between charged surfaces in liquids without reference to the permeability of the boundaries.1 To use the flat plate theory to model the interaction between an SFM probe and a surface, it is necessary to adapt it to the SFM probe/surface geometry. Most commonly the Derjaguin approximation is made, whereby it is assumed that the characteristic radius of the probe is much larger than the characteristic length scale for the interaction. For this case, the measured force is proportional to the interaction potential according to1

Fda(D) ≈ 2πRG(D)

(4)

where R is the characteristic radius of the SFM probe. More recently, it has been shown that the case of a small spherical particle interacting with an infinite plane can be calculated using SEI. This requires a numerical quadrature of the flat plate potential9

∫[

FSEI(D) ) -

R

0

∂G(D + R - R[1 - (r/R)2]1/2) ∂D ∂G(D + R + R[1 - (r/R)2]1/2) r dr (5) ∂D

]

We carried out the numerical integration using Simpson’s rule with a tolerance of 1 × 10-3 pN. Fda(D) and FSEI(D) were used to fit the force-separation data. The number of free parameters in the fits was minimized by using theoretical values for parameters where they could reasonably be assumed. The value for B was taken from Israelachvili1 at 10-72 J m6, and this single value was used for all nine curves. D0 values were determined manually by overlapping the steep portions of the measured and model forceseparation curves at contact. The Debye lengths are known from the solution concentrations1 and are 9.6, 3.0, and 0.96 nm for salt concentrations 1.0, 10, and 100 mM, respectively. The dielectric constant of the medium  was taken as that of water (78.5), and the permittivity of free space 0 is 8.85 × 10-12 C2/(J m). The only value allowed to vary in the fit was the product of the surface charge densities σ1σ2. A different value of σ1σ2 was allowed for each solution pH to account for titratable acidic and basic surface groups on the silicon nitride probe surface. In total, there were three free parameters for the nine force-separation curves (each with 200 points). The fits were initially performed using all nine force-separation curves. However, from this initial fit, it became clear that the Derjaguin model could not accommodate the data at the longest Debye length. Hence, in the final analysis, the fit was restricted to the six force-separation curves at 10 and 100 mM, where the Derjaguin approximation is most likely to apply. Fits were also restricted to separations between 3 and 25 nm, where solvation forces are negligible.19 The fitting procedure was exactly the same for the SEI model so that the Derjaguin and SEI approximations could be directly compared. Agreement between the model and the data was quantified in terms of their root-mean-square (RMS) difference. d. SFM Probe Geometry Calibration. The geometry of the probe used in the experiment was calibrated as previously described.10 Briefly, five scans of a Nioprobe (General Micro, Edmonton, Canada) sample were obtained using contact mode SFM with a scan range of 128 nm at 256 × 256 pixels. Both the “trace” and the “retrace” images were captured. For each image, the number of pixels was reduced to 128 × 128 by averaging 2 × 2 clusters of pixels and then averaging the trace and retrace images for each scan to obtain five reduced noise images at 1 nm/pixel resolution. The averaged images were then used as the (18) Parsegian, V. A.; Gingell, D. Biophys. J. 1972, 12, 1192-1204. (19) Chin, C. J.; Yiacoumi, S.; Tsouris, C. Environ. Sci. Technol. 2002, 36, 343-348.

Figure 2. One example of a force-distance measurement. (a) The raw photodetector signal (dashed line) contains an optical interference artifact with a characteristic 33 ms period. The artifact is removed by fitting eq 1 to the data at large separations/ early times (bold solid line) and then subtracting out the artifact. (b) The last 2 ms prior to contact is isolated and used as the input to reconstruct the tip-sample force (c). (d) The forceseparation is obtained by plotting force versus separation (tip minus sample position) as parametric functions of time. input to a modified blind reconstruction algorithm10 to reconstruct five independent representations of the probe tip. These were in turn averaged to obtain the calibrated probe geometry. Finally, the reconstructed tip was fit to a sphere to determine the probe apex radius R.

III. Results a. Experimental Force-Separation Curves. An example of one force-separation curve reconstruction from data collected at pH 4.6 and 1 mM NaCl concentration is shown in Figure 2. We began with the raw digitized cantilever deflection signal (dotted line in Figure 2a). The signal contains an interference artifact that is readily

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Figure 3. Average force-separation curves at the nine different solution concentrations analyzed using the Derjaguin approximation. Each curve (dotted line) was obtained by binning and averaging 50 individual force-separation curves with a separation bin size of 0.25 nm. Each row is obtained at constant pH and each column at constant NaCl concentration. A fit to the flat plate potential using the Derjaguin approximation (solid line) closely models the experimental data at the two highest salt concentrations (middle and right-hand columns) but overestimates the interaction force at the lowest salt concentration (left-hand column).

identified from its characteristic ∼33 ms period. The artifact was removed by fitting to eq 1 (solid line) over the time range of 0-50 ms and then subtracting the fit from the measured signal. With this time range, the last 300400 nm prior to contact is not used in the zeroing. From the processed signal, the last 2 ms prior to contact was isolated (Figure 2b). This served as input to the inverse problem of reconstructing forces from the measured deflections. The Tikhonov regularized solutions to the inverse problem yields the output forces in Figure 2c. The force-separation curve is then obtained by plotting force versus the difference between the probe position and the surface position (Figure 2d). Because the reconstruction is ill-conditioned, noise in the reconstructed forces is significantly larger (in this case about 5×) than estimated by equipartition (∼10 pN).20 We reduced the noise through ensemble averaging. The result of averaging 50 curves with a separation bin size of 0.25 nm at each solution composition is shown in Figure 3 (experimental data are the dotted lines). The graphs are arranged so that rows are all obtained at constant pH and columns are obtained at constant NaCl concentration. The upper left-hand plot corresponds to the average curve obtained from the force-separation curve in Figure 2d and 49 similar curves. (20) Gittes, F.; Schmidt, C. F. Eur. Biophys. J. Biophys. Lett. 1998, 27, 75-81.

Table 1. Regression Parameters for the Force-Separation Models model Derjaguin SEI, R fixed SEI, R varied

σ1σ2 × 107 (C2/m4) pH 4.6 pH 6.7 pH 9.7 -1.5 -2.0 -2.2

1.9 2.2 2.3

2.0 2.5 2.2

R (nm)

RMS errorb (pN)

7a 7a 6.8

15 1.9 1.5

a Measured value; fixed in the regression. b RMS error calculated using the fits to data at all nine pH and ionic concentration combinations.

b. Derjaguin Approximation. We modeled each of the averaged curves according to the EDL theory (eq 3) using the Derjaguin approximation (eq 4). The model predictions are shown as the solid lines in Figure 3, and the values obtained for the three free parameters in the fit representing the product of the tip and sample surface charge densities are given in Table 1. The experimental data closely follow the theory at the higher salt concentrations, namely, 100 and 10 mM, but deviate by as much as 300% at 1.0 mM. To investigate the cause for differences between experiment and the Derjaguin approximation we calibrated the SFM probe. The reconstructed probe is shown in Figure 4. Note that the vertical dimensions in the rendering are exaggerated by a factor of 4 to facilitate visualization. We quantified the size of the probe by fitting to a model of a spherical surface. The radius from the fit was 7 nm. This

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Figure 4. Calibrated shape of the SFM probe used in the experiment. Vertical dimensions are exaggerated by a factor of 4 to facilitate contrast. The probe apex radius obtained from fitting a spherical surface to the probe shape is 7 nm.

is significantly larger than the Debye lengths where the theory closely matched experiment (10 and 100 mM) but smaller than the Debye length where the theory overestimated the interaction force (1 mM). This suggested that differences between the measured forces and the theoretical predictions were due to a violation of a basic assumption in the Derjaguin approximation that requires each of the interacting surfaces to have a characteristic radius that is much larger than the characteristic range of the interaction. c. SEI. To account for the small size of the SFM probe, we modeled the experimental force-separation curves using EDL theory (eq 3) but replaced the usual Derjaguin approximation with SEI (eq 5). We applied SEI first using the measured probe apex radius of 7 nm (solid line in Figure 5). In this case, there are exactly the same number of free parameters (fitted values in Table 1) as with the Derjaguin approximation and the RMS difference between theory and experiment is reduced by ∼87% (see Table 1). The predicted interaction forces at the longest Debye length (1.0 mM), in particular, are greatly improved (compare left-hand columns in Figures 3 and 5). Furthermore, if we allow the radius of the SEI sphere to vary in the fit, it converges to 6.8 nm (curves similar to the solid line in Figure 5 and not shown, fitted values in Table 1). This fitted radius is not significantly different from the 7 nm tip apex radius determined by blind reconstruction. This shows that independent measurements of force and tip shape are consistent and supports our assertion that the Derjaguin approximation was violated. IV. Discussion We have collected and analyzed a total of 450 forcedistance curves for the interaction between an SFM probe and an HOPG(bp) surface, 50 each at 3 different pH values and 3 different sodium chloride concentrations. The measurements contained a substantial artifact due to interference between the laser light reflected off the cantilever and the HOPG(bp) surface. We subtracted out the artifact using an interferometer equation obtained by fitting the signal only at large separations (>300 nm or 30 times larger than the largest Debye lengths in the experiment). This ensured that we did not zero out contributions to the signal arising from tip-sample forces. We reconstructed complete force-separation curves over the entire range of solution concentrations using a dynamical analysis of the measured cantilever signal.11 This model was essential at pH 4.6 because the snap-tocontact instability was present and the traditional Hooke’s law model (F ) kz) is known to calculate inaccurate forces in that case.21 Individual force-separation curves had ∼50 pN RMS noise, about a factor of 5 greater than expected (21) Todd, B. A.; Eppell, S. J.; Zypman, F. R. J. Appl. Phys. 2000, 88, 7321-7327.

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on the basis of equipartition.20 This is a result of the illposedness that exists when using a dynamical cantilever model.17 We are able to overcome this problem and reduce the noise by ensemble averaging to obtain smooth average force-separation curves. These average force-separation curves have an RMS noise level of only ∼2 pN, a factor of 5 smaller than expected on the basis of equipartition.20 The forces present were sensitive to ionic concentration in the manner characteristic of EDL interactions. This behavior was somewhat surprising since HOPG(bp) is often thought of as a nonpolar uncharged surface. Initially assuming this, we expected the charged silicon nitride tip to induce an image charge of opposite sign in the nonpolar and uncharged HOPG(bp). However, this would have resulted in forces that were attractive at all pH values.22 Instead, we observed a change from attractive to repulsive interaction in going from pH 4.6 to pH > 6.7. Taking into account the expected pH dependence for the silicon nitride tip surface charge (positive below pH ≈ 6 and negative above),23 our experimental data are consistent with HOPG(bp) bearing a static negative charge at all pH values. Possible scenarios leading to this are strong physisorption and/or anion intercalation or oxidation of the graphite surface to form carboxyl groups.24 Either of these cases would result in negative charges and a force-separation behavior with silicon nitride that is consistent with our experimental observations. The products of the silicon nitride and HOPG(bp) surface charge densities σ1σ2 were around 2 × 10-7 C/m2. This is smaller but within an order of magnitude of surface charge densities measured by SFM for the symmetric silicon nitride system by Senden and Drummond23 and Raiteri et al.25 Reconstructed force-separation curves were modeled by the EDL theory and Derjaguin approximation. There was good agreement between theory and experiment at higher ionic concentration, where the Debye lengths were 0.96 and 3.0 nm. At the lowest ionic concentration, where the Debye length was 9.6 nm, however, the theory predicted a much larger force than was present in the experiment. This behavior is consistent with errors expected when the Derjaguin approximation is applied outside its range of validity; it overestimates forces for small particle sizes and long Debye lengths.8,9 We confirmed that this was the case by experimentally measuring the probe apex radius of the scanning probe microscope at 7 nm. We then tried a different model, SEI,9 that reduced the restrictions on particle size and calculated the EDL forces between a spherical particle and an infinite flat plane. Although this geometry is not entirely appropriate for the SFM probe (the probe’s shape in Figure 4 is not entirely spherical), it is capable of modeling a behavior whereby the size of a particle influences the separation dependence of the interaction. In particular, for small particle sizes at small separations, the flat plate interactions are attenuated more strongly compared to those at large separations. The Derjaguin approximation cannot model this behavior because the size of the particle scales the interaction uniformly at all separations (see that R multiplies G(D) in eq 4). Using the calibrated value for the probe apex radius and precisely the same regression (22) Zypman, F. R.; Eppell, S. J. J. Vac. Sci. Technol., B 1997, 15, 1853-1860. (23) Senden, T. J.; Drummond, C. J. Colloids Surf., A 1995, 94, 2951. (24) Kinoshita, K. Carbon Electrochemical and Physiochemical Properties; John Wiley & Sons: New York, 1988. (25) Raiteri, R.; Margesin, B.; Grattarola, M. Sens. Actuators, B 1998, 46, 126-132.

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Figure 5. Average force-separation curves at the nine different solution concentrations analyzed using SEI with the measured tip radius. The experimental data (dotted line) are the same as plotted in Figure 3. A fit to the flat plate potential using SEI with the calibrated probe apex radius, 7 nm (solid line), more closely matches the experimental data than the Derjaguin approximation (Figure 3). The fits at the longest Debye length (left-hand column), in particular, are improved.

conditions as with the Derjaguin approximation, we obtained a fit for SEI that had 8 times less discrepancy between experiment and theory (RMS difference 1.9 pN compared to 15 pN). Furthermore, when the radius in the SEI fit was allowed to vary, it converged to a value not significantly different from the radius measured by blind reconstruction (6.8 nm as compared to 7 nm). This shows that independent measurements of force and probe shape indicate that the experiment includes a regime where the Derjaguin approximation was violated. V. Conclusions We measured interaction forces between a silicon nitride SFM probe and an HOPG(bp) surface as a function of pH and ionic concentration. The usual EDL potential and Derjaguin approximation matched the measured forces at Debye lengths below 3 nm but overestimated forces at a Debye length of 9.6 nm. By calibrating the SFM probe,

we were able to understand this discrepancy as a violation of the Derjaguin approximation, which states that the characteristic probe radius (R ≈ 7 nm) must be larger than the Debye length. Replacing the Derjaguin approximation with an SEI provided a significantly better fit between theory and experiment. Hence, we have used SFM to experimentally demonstrate an altered forceseparation relationship governing colloids that violate the Derjaguin approximation. Interactions in this regime have been less studied and are relevant to polymers, biological molecules, and other nanoscale colloids. The general methodology outlined in this paper should be useful in further study of these systems. Acknowledgment. we thank the Whitaker Foundation and the National Institutes of Health, Grant No. AR45664-01, for generous financial support. LA035235D