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Determination of Single-Bond Forces from Contact Force Variances in Atomic Force Microscopy John M. Williams, Taejoon Han, and Thomas P. Beebe, Jr.* Department of Chemistry, University of Utah, Salt Lake City, Utah 84112 Received June 22, 1995. In Final Form: October 13, 1995X Atomic force microscopy contact forces are shown to obey a Poisson distribution, so that the ratio of their variance to mean gives the force of a single chemical bond between the tip and sample. The derived single-bond force was able to distinguish nominal van der Waals interactions (60 ( 3 pN) from hydrogenbond interactions (181 ( 35 pN) between atomic force microscope tips and gold and mica surfaces, respectively. This technique greatly reduced sampling time and sample wear, allowed quantitative use of low-resolution force data from a commercially available instrument, and detected important chemical differences between surface functional groups on the samples. These experiments constitute an important step in obtaining chemically specific information in atomic force microscopy.
I. Introduction In the years since its invention,1 the atomic force microscope (AFM) has probed the structure of both metallic and nonmetallic surfaces at scales reaching atomic resolution.2 Although its original use was for topographic measurements, the chemical forces between the AFM probe tip and the surface can also produce unique localized chemical information about the sample.3-6 In addition, the AFM can operate in liquid environments, and so it is well suited for microscopy of some biological samples, for example.7-9 When the AFM tip withdraws from contact with a sample surface, an adhesion force develops between tip and sample. In the absence of capillary forces (for example, in liquid media), the adhesion force arises from chemical bonds between the tip and sample surfaces. Analysis of the adhesion forces, or measurement of a single bond force, can thus produce localized chemical information about the sample-liquid interface, such as the nature of surface functional groups10 or the local effects of the liquid medium. Several reports have presented measurements of singlebond forces with the AFM.11-14 These have relied upon resolution of discrete single-bond contact forces. Another approach, presented herein, does not rely upon such resolution and derives from Poisson statistics.15,16 The * Author to whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, January 1, 1996. (1) Binnig, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930. (2) Albrecht, T. R.; Quate, C. F. J. Appl. Phys. 1987, 62, 2599. Albrecht, T. R.; Quate, C. F. J. Vac. Sci. Technol. A 1988, 6, 271. (3) Burnham, N. A.; Colton, R. J.; Pollock, H. M. J. Vac. Sci. Technol. A 1991, 9, 2548 and references therein. (4) Overney, R. M.; Meyer, E.; Frommer, J.; Brodbeck, D.; Lu¨thi, R.; Howald, L.; Gu¨ntherodt, H.-J.; Fujihira, M.; Takano, H.; Gotoh, Y. Nature 1992, 359, 133. (5) Frisbie, C. D.; Rozsnyai, L. F.; Noy, A.; Wrighton, M. S.; Lieber, C. M. Science 1994, 265, 2071. (6) Wilbur, J. L.; Biebuyck, H. A.; MacDonald, J. C.; Whitesides, G. M. Langmuir 1995, 11, 825. (7) Edstrom, R. D.; Yang, X.; Lee, G.; Evans, D. F. FASEB J. 1990, 4, 3144 and references therein. (8) Marti, O., Amrein, M., Eds. STM and SFM in Biology; Academic Press: San Diego, 1993, and references therein. (9) Leavitt, A. J.; Wenzler, L. A.; Williams, J. M.; Beebe, T. P., Jr. J. Phys. Chem. 1994, 98, 8742. (10) Thomas, R. C.; Houston, J. E.; Crooks, R. M.; Kim, T.; Michalske, T. A. J. Am. Chem. Soc. 1995, 117, 3830. (11) Hoh, J. H.; Cleveland, J. P.; Prater, C. B.; Revel, J.-P.; Hansma, P. K. J. Am. Chem. Soc. 1992, 114, 4917. (12) Ohnesorge, F.; Binnig, G. Science 1993, 260, 1451. (13) Lee, G. U.; Chrisey, L. A.; Colton, R. J. Science 1994, 266, 771. Lee,G. U.; Kidwell, D. A.; Colton, R. J. Langmuir 1994, 10, 354. (14) Moy, V. T.; Florin, E.-L.; Gaub, H. E. Science 1994, 266, 257. Florin, E.-L.; Moy, V. T.; Gaub, H. E. Science 1994, 264, 415.
Poisson distribution describes the probability of observing a random sample of r discrete events when the mean number is m events (in this case m chemical bonds); the variance of this Poisson distribution is also m. If the total contact force between tip and sample surfaces is the sum of discrete bond forces F, then the total force should vary as a Poisson distribution with mean µ ) mF and variance σ2 ) mF2. Thus, the ratio of the variance and mean of the total force gives the individual bond force: F ) σ2/µ. A plot of σ2 against µ should therefore be linear with slope F and zero intercept. The mean number m of discrete bonds may also be determined and used to estimate the tip-sample contact area. This analysis also applies to high-resolution discrete-force data (as from refs 11-14) to provide independent support for the force estimates, and to confirm that the total force is the sum of discrete forces, as is chemically reasonable. Preliminary results from this method have already been published elsewhere.17 The assumptions underlying this model may be summarized as follows: The total adhesion force develops as the sum of a finite number of discrete, independent chemical bonds, as is reasonable for a microscopic interface under a liquid medium. Also, these bonds form randomly, and all have similar force values; this assumption will be tested and supported below in the Discussion section. An unphysical feature of the Poisson model is that there would be no upper limit to the possible number of interfacial bonds. A more realistic model would include a finite upper limit as an extra parameter, which produces a binomial model. The binomial model approximates to the Poisson when the upper force limit is relatively large compared to the mean (i.e., by a factor of 20 or greater). At present, the data do not clearly indicate which is the appropriate model, so the Poisson model was chosen for simplicity. There are several advantages to this analysis of contact force data. It required fewer than 100 force curves, compared to more than 4000 by Hoh et al.11 or several hundred by Florin et al.;14 this reduces measurement time and sample damage due to repetitive contact. Also, it allows use of force data resolutions that are too low to (15) A different application of related principles may be found in the following: Stone, T. A.; Merkel, R.; Ritchie, K.; Lauffenburger, D. A.; Zukoski, C. F.; Evans, E. A. Probing the Rupture Force of a Receptor/ Ligand Pair with a Membrane Force Transducer. Presented at the 69th Colloid & Surface Science Symposium, Salt Lake City, UT, 1995. Evans, E.; Berk, D.; Leung, A. Biophys. J. 1991, 59, 838. (16) A brief overview of this distribution is found in the following: Barlow, R. J. Statistics; Wiley: New York, 1989; pp 28 ff. (17) Han, T.; Williams, J. M.; Beebe, T. P., Jr. Anal. Chim. Acta 1995, 307, 365.
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Figure 1. Simplified schematic diagram of the surface layers of Si3N4 AFM tips in aqueous medium. Equilibrium hydrolysis of siloxane bonds creates surface silanols; the surface silanols can participate in hydrogen bonding at the solid-liquid interface. See also ref 22.
resolve discrete force differences (such as commercial AFM systems routinely attain). In order for this analysis to succeed, it is necessary to ensure that other errors contributing to the variance are negligible, because their contribution will tend to exaggerate F and produce a more Gaussian distribution.18 In this series of experiments the noise on the force measurements, mainly due to thermal noise, was negligible compared to the mean forces and to the pull-off force variances, so that source of error was not significant, as will be discussed below.
Figure 2. Plot of σ2 vs µ for data in Table 1 (Si3N4-gold forces), with least-squares linear regression. Slope ) 60 ( 3 pN; the origin is plotted but not included in the regression. Table 1. Results for van der Waals Forces between Bare AFM Tips and Au(111) Films in Water
II. Experimental Section All force measurements employed a Topometrix Explorer AFM and Digital Instruments pyramidal silicon nitride (Si3N4) tips, with spring constant k ) 0.114 N/m. The cantilever spring constants were calculated from their resonance frequencies according to the method of Cleveland et al.19 The piezoceramic response was linearized by the manufacturer.20 The digitization resolution of less than 1.5 pN/bit was negligible compared to the experimental noise, which will be discussed in the next section. To avoid capillary forces, force measurements were made under a fresh film of distilled, doubly-deionized water (Barnstead purification system) held between the sample and scan head surfaces, with an atmospheric interface only at a narrow perimeter. Control experiments ensured that carbonate absorption from the air was negligible over the experimental time scale.21 The medium was unbuffered to avoid electrolyte adsorption effects. In this medium silanol groups cover the tip surface, as shown schematically in Figure 1 , and can participate in hydrogen bonding.11,22 Multiple sets of 46-80 force-distance curves were collected for two tip-surface systems to investigate the possibility of using contact forces to distinguish different surface functional groups. Each set was collected at a different sample area, and the maximum applied force was kept as constant and small as possible (less than 7 nN) to minimize sample wear. This also kept the tip indentation depth and tip-sample contact area relatively constant. The measurements were performed at a separation rate of 5 µm/s; this is comparable to the rates used by Ohnesorge and Binnig,12 Florin et al.,14 and Evans et al.,15 but is faster than the rates reported by other groups.11,13 It was noted that contact forces decreased after prolonged exposure of the mica to water, possibly because potassium cations (dissolved from the mica) had adsorbed to the contact areas,23 so only data obtained in fresh liquid medium were accepted to ensure wellcontrolled surface chemistry. (18) Barlow, R. J., op. cit., p 49. (19) Cleveland, J. P.; Manne, S.; Bocek, D.; Hansma, P. K. Rev. Sci. Instrum. 1993, 64, 403. (20) Topometrix User’s Manual, v. 3.05, pp 6-8. (21) After exposure to air, the pH change of a fresh aliquot of water was found to be no greater than 0.02 pH unit over the time scale of a force experiment, which corresponds to a relative carbonate increase of about 5%. (22) Bousse, L.; Mostarshed, S. J. Electroanal. Chem. 1991, 302, 269. Although the silicon oxynitride surface can also express amino groups, in aqueous media these are only about 1% as frequent as surface silanols, and can be neglected. (23) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: San Diego, 1992; Section 13.5.
set
mean force µ (nN)
variance σ2 (nN2)
mean no. of bonds m ) µ/(60 pN)
size of set N
A B C
16.7 17.1 1.90
1.00 1.08 0.143
278 285 31.7
52 46 80
Table 2. Results for Hydrogen-Bond Forces between Bare AFM Tips and Mica in Water set
mean force µ (nN)
variance σ2 (nN2)
mean number bonds m ) µ/(181 pN)
size of set N
D E F G
15.6 8.00 4.73 3.59
2.87 0.984 0.726 0.805
86.2 44.2 26.1 19.8
80 52 58 68
Hydrogen-bonding interactions were examined with bare Si3N4 tips and mica surfaces, designated as data sets D, E, F, G. In order to compare a different chemical interaction, van der Waals forces were examined with bare Si3N4 tips and Au(111) surfaces (data sets A, B, C). Au(111) films 2000 Å thick were prepared by evaporation onto mica substrates under high vacuum (approximately 8 × 10-7 Torr) and then annealed at 500 °C for 1 h.24,25 Both of these sample surfaces, mica and Au(111), furnish atomically flat (zero radius of curvature) terraces as large as microns; thus, all contact sites are virtually identical and effects of local surface curvature differences are avoided.
III. Results and Discussion Table 1 lists the results for each set of pull-off force measurements on the Au(111) surface. Figure 2 shows a plot of the force variance σ2 vs the mean force µ for each set. These results for interactions with Au(111) form a line with slope ) 60 ( 3 pN and insignificant intercept, as expected from the theory above. This indicates a nominal van der Waals force of FvdW ) 60 pN. Table 2 lists the results for measurements on mica; the plot in Figure 3 also forms a line (r ) 0.981), with slope ) 181 ( 35 pN and insignificant intercept. The results from this system are also consistent with the Poisson model and indicate a nominal hydrogen-bond force of FH-bond ) (24) DeRose, J. A.; Thundat, T.; Nagahara, L. A.; Lindsay, S. M. Surf. Sci. 1991, 256, 102. (25) Clemmer, C. R.; Beebe, T. P., Jr. Scanning Microscopy 1992, 6, 319.
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Figure 3. Plot of σ2 versus µ for data in Table 2 (Si3N4-mica forces), with least-squares linear regression. Slope ) 181 ( 35 pN; the origin is plotted but not included in the regression.
181 pN. The hydrogen-bond force in this system was about 3 times greater than the van der Waals force obtained above, as expected from recent literature.10 The results from both types of chemical systems support the analysis of contact force data according to Poisson statistics, and the calculated bond forces agree, within an order of magnitude, with similar bond forces reported previously.11-14 (Extrapolated bond energies derived from this method, and reported elsewhere,17 also have been comparable to accepted values.) It has been noted that the probe-surface separation rate affects the observed bond forces (kinetic effect);13,15 until the dynamics of this effect have been determined, the use of widely varying rates by different researchers may prevent the achievement of agreement to better than an order of magnitude. The 1σ experimental errors on force measurements (about 46 pN for data A, B, and C, and 30 pN for data D, E, F, and G) were about 2 orders of magnitude smaller than the pull-off mean forces, and contributed variances (about 0.002 nN2 for data A, B, and C, and 0.001 nN2 for data D, E, F, and G) that were 2-3 orders of magnitude smaller than the force variances. Thus, the experimental errors were negligible in this analysis for single-bond forces. However, the errors were not negligible compared to the F values themselves, especially in the case of van der Waals forces; it will be shown below that this would interfere with attempts to directly detect quantized bond forces (as in refs 11-14). This indicates the utility of the Poisson analysis for AFM systems that cannot resolve single-bond forces, as is typically the case with commercial instruments at present. We must also consider the possiblity of a nonspecific long-range force26 F0 acting between the surface and tip, as often occurs in colloid chemistry. The action of a nonspecific force would significantly alter the σ2 vs µ plots:
µ ) mF + F0 σ2 ) mF2 ) µF - FF0. The intercept of this line would be -FF0, not 0 as in our results. Inasmuch as our experimental intercepts were not significantly different from zero, the action of such a (26) Suggested by reviewer.
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Figure 4. Histograms of force differences for data set A. Set A shows some force quantization at the expected multiples of 60 pN.
nonspecific force was not significant in our experiments. The use of an aqueous medium, which often reduces Hamaker constants by an order of magnitude,27 may have suppressed any long-range forces relative to adhesion forces. The contact force histograms may also reveal underlying Poisson statistics. Some tentative results of this kind seemed to approximate Poisson distributions qualitatively and are presented in the Appendix. In an elegant experiment, Hoh et al. have reported the resolution of individual bond forces from discrete contact force differences.11 However, under unfavorable conditions the random errors on the force data may be so large as to obscure the discrete differences or force quantization and thus prevent that type of analysis. The Poisson technique, however, can determine individual bond forces even when the data are not resolved well enough to discriminate them. Histograms of force differences from well-resolved data11 would show data clusters at integral multiples of the individual bond force F. A histogram of force differences for data A, plotted in Figure 4, only shows indistinct quantization at the expected multiples of 60 pN. Furthermore, results from the combined sets for forces on Au(111) and mica, Figure 5, show a lack of structure or quantization of the measured forces about a discrete bond force. Instead, the overall distributions of force differences have an exponential decay, which is typical of the distribution of intervals between occurrences of uniform probability. This indicates that random errors in our measurements have leveled out and obscured the discreteness of the contact bond forces, and demonstrates the utility of the Poisson method, where resolution of individual force differences is unnecessary. In private communications, our colleagues have questioned the underlying assumption that contact forces have discrete values, rather than vary continuously depending on bond position and orientation. However, the literature11-14 shows that discrete, quantized bond forces can be measured, and that could only be the case if (to a good approximation) each bond could be treated as being either “on” or “off”, a quantized model. In other words, the quantized results already in the literature support the (27) Israelachvili, J., op. cit., Sections 11.1-11.5.
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Figure 6. Comparison of the histograms for pseudo-jump-in (sliding friction) force sets, plotted as bars, to the corresponding Poisson distributions (lines). Note the close qualitative similarities, although the experimental histograms are more strongly peaked: (upper) set C, m ) 2.87; (middle) set F, m ) 1.35; (lower) set G, m ) 1.13.
Figure 5. Histograms of force differences for combined sets A, B, C (upper); and D, E, F, G (lower). The combined sets show no quantization, only exponential decays that are typical for data without distinctly quantized structure. Random measurement errors have leveled out the discreteness of the bond forces.
model of (approximately) quantized bond forces. Of course, a more refined model would include force variations due to other effects, but they may well be minor, according to the cited results. These private communications have not raised any other plausible explanation for our observations of Poisson statistics in these systems. If the discrete bond force F is not constant, but can vary, then this would introduce nonlinearity into the plots of σ2 vs µ. The effect can be calculated as follows:
σ2 ) F2m + 〈n2〉σ2F ) F2m + (σ2n + 〈n〉2)σ2F ) F2m + (m + m2)σ2F ) Fµ +
(
) F+
(
)
µ µ2 2 + σ F F2 F
)
σ2F σ2F µ + 2 µ2 F F
where µ, m, F, and σ2 are defined as in the Introduction; n denotes the number of bonds ruptured in a given experiment, angle brackets (〈 〉) denote expectation values, and σ2F is the hypothetical variance of the discrete bond force. This equation implies that any variance σ2F in the bond force F would be manifest as a curvature in the plots of σ2 vs µ, by introducing a nonzero second-order coefficient in the last term above. However, such curvature was not observed. Quadratic fits to the data of Figures 2 and 3, along with supplementary data (not shown), had second-
order coefficients that were not significantly different from zero. This confirms that variations in F were not significant in these systems. Thus, the observed bond forces appear to be approximately quantized, within the resolution of this method, as has been reported previously.11-14 IV. Conclusions Individual bond forces can be determined from sets of contact force data by the ratio of pull-off force variance to the mean force. This is due to familiar properties of the Poisson distribution for the sum of discrete bond forces. This method works well even under unfavorable conditions where the force data are not resolved well enough to show quantized force differences. Also, the Poisson analysis requires much fewer force measurements, and so is less time-consuming and less damaging to the sample. The results were able to distinguish interactions on Au(111) from interactions on mica, which we attributed tentatively to van der Waals and hydrogen-bond interactions, respectively. The ability to distinguish these surfaces adds a dimension of chemical specificity to atomic force microscopy with unmodified silicon nitride tips. Acknowledgment. The authors thank Topometrix for support under a collaborative arrangement and Drs. Vladimir Hlady, Henry White, and Joel Harris for useful suggestions. This material is based upon work supported in part under a National Science Foundation Graduate Research Fellowship and by NSF National Young Investigator (CHE9357188) and Camille Dreyfus TeacherScholar Awards (TPB). Appendix The contact force histograms (frequency distributions) also supported this Poisson statistical method. Such distributions sort the total force data in each set according to the number of discrete adhesion bonds, using the F values from Figures 2 and 3. The Poisson distribution for m > 20, where m is the mean number of individual bonds,
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is practically indistinguishable from a Gaussian curve; in this case the analysis of distribution shape is inconclusive. All sets of pull-off force results listed in Tables 1 and 2, with m g 20, had roughly a Gaussian distribution. However, when the AFM tip approaches contact with the surface, an apparent jump-in can occur due to frictional forces if the tip slides on the surface. This effect has been described in the literature.28 In this case, we attribute our pseudo-jump-in forces to tip sliding and frictional adhesion, which might also exhibit quantization effects because the sliding friction arises from interfacial forces.29,30 The pseudo-jump-in force results of sets C, F, and G had (28) Hoh, J. H.; Engel, A. Langmuir 1993, 9, 3310. (29) Stuart, J. K.; Hlady, V. Langmuir 1995, 11, 1368. (30) Israelachvili, J. N.; Chen, Y.-L.; Yoshizawa, H. J. Adhesion Sci. Technol. 1994, 8, 1231.
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much smaller m, and indeed Figure 6 shows that their histograms seem to approximate Poisson distributions31 qualitatively for m ) 2.87, 1.35, and 1.13, respectively, although the experimental histograms are more sharply peaked. These histograms provide further support for the validity of this technique. It is possible that local forces at tip asperities are responsible for quantized jumpin effects here, but it is more likely that the quantization in the pseudo-jump-in force arises from tip-sliding effects. This evidence may be tentative, but the experimental histograms of Figure 6 compare well with the theoretical Poisson distributions and are compelling enough to warrant notice. LA950500J (31) The probability function is P(r) ) N[(e-mmr)r!].
