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Reconstruction of the Tip-Surface Interaction Potential by Analysis of the Brownian Motion of an Atomic Force Microscope Tip Oscar H. Willemsen, Laurens Kuipers, Kees O. van der Werf, Bart G. de Grooth, and Jan Greve* Department of Applied Physics, Applied Optics Group, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands Received October 15, 1999. In Final Form: February 14, 2000 The thermal movement of an atomic force microscope (AFM) tip is used to reconstruct the tip-surface interaction potential. If a tip is brought into the vicinity of a surface, its movement is governed by the sum of the harmonic cantilever potential and the tip-surface interaction potential. By simulation of the movement of a tip in a model potential, it was demonstrated that a potential can be reconstructed from the probability distribution of the tip position. By application of the reconstruction technique to an experimentally obtained distribution function, it was demonstrated that the method is very sensitive to drifts in the AFM setup. In addition to this, the tip-surface interaction potential cannot be derived because the cantilever potential adds an undetermined term to the measured potential. By use of the force-distance curves to carefully control the movement of the cantilever, the position of the cantilever is determined at all times. This enables the determination of the tip-surface interaction potential. Because a force-distance curve has an internal calibration of the position of the cantilever potential, individual curves can be averaged to improve the accuracy of the method. The novel method is tested on a model system of a Si3N4 tip that interacts with mica. In 100 mM KCl buffer, the tip-surface interaction potential can be determined with an accuracy below the thermal energy kbT. The interaction potential has a minimum of 22 kbT because of the combination of van der Waals attraction and Born repulsion. At 3 mM KCl, the tip-surface interaction is dominated by the electrostatic interaction.
Introduction The measurement of surface-surface interaction forces has been pioneered by Israelachvili, who developed the surface force apparatus (SFA).1-3 The instrument consists of two crossed cylinders, with a radius of 1 cm, that are moved toward each other while the force on the cylinders is measured. With the SFA it became possible to experimentally verify relationships that had already been formulated more than 15 years earlier.1,2,4 The use of crossed cylinders causes the detected forces to be relatively high because the contact area of two cylinders in adhesive contact is approximately 6 µm2,5 which is rather large when compared to molecular dimensions. The atomic force microscope6 (AFM) is well-known for its ability to image surfaces with submolecular resolution. In addition to the imaging capacity, it can be used to measure surface forces. The advantage of the AFM over the SFA is that the AFM can probe a surface force more locally because a tip with a radius of only 20 nm is brought into contact with a surface, creating a contact area of ≈12 nm2. Although the small size of the probe causes the forces on the probe to be much lower, the spring constants of the AFM cantilevers are smaller than those used in the SFA, implying that the intrinsic force sensitivity is higher. In the so-called force-distance mode of an AFM, the tip is brought to the surface and subsequently retracted while * To whom correspondence should be addressed. Phone: +31 53 4893157. Fax: +31 53 4891105. E-mail:
[email protected]. (1) Tabor, D.; Winterton, R. H. S. Proc. R. Soc. London A 1969, 312, 435. (2) Israelachvili, J. N.; Tabor, D. Proc. R. Soc. London A 1972, 331, 19. (3) Israelachvili, J. N.; Tabor, D. Prog. Surf. Membr. Sci. 1973, 7, 1. (4) Israelachvili, J. N. Surf. Sci. Rep. 1992, 14, 109. (5) Israelachvili, J. N. Intermolecular & Surface Forces, 2nd ed.; Academic Press: New York, 1991. (6) Binnig, G.; Quate, C. F.; Gerber, C. Phys. Rev. Lett. 1986, 56, 930.
the force is measured. The most emphasis is put on interpreting the forces measured while the tip is retracted. From this part of the curve, it is possible to detect quantized adhesion effects7 and even single molecular bonds.8,9 From the approach part, continuous surface forces, like van der Waals and DLVO forces, have been measured.10,11 Interesting effects that occur during the approach of the tip to the surface, like those due to the discreteness of solvents, are very difficult to detect. To facilitate the detection of these effects, different modes of operation of the AFM have been developed. O’Shea and co-workers, for instance, have shown that when the tip movement is modulated, it is indeed possible to see discrete layering effects.12,13 Measurements of adhesion forces do not reveal the shape of the tip-surface interaction potential. The disruption force measured for a ligand-receptor binding, for instance, does not give any information on the dissociation rate of this weak bond, unless this force is measured at multiple force loading rates.14-16 Evans and co-workers have formulated a theory to describe the dependence of the disruption force on the loading rate. Their model is based (7) Hoh, J. H.; Cleveland, J. P.; Pratter, C. B.; Revel, J.-P.; Hansma, P. K. J. Am. Chem. Soc. 1992, 114, 4917. (8) Florin, E.-L.; Moy, V. T.; Gaub, H. E. Science 1994, 264, 415. (9) Lee, G. U.; Kidwell, D. A.; Colton, R. J. Langmuir 1994, 10, 354. (10) Burnham, N. A.; Colton, R. J. J. Vac. Sci. Technol. A 1989, 7, 2906. (11) Butt, H.-J. Biophys. J. 1991, 60, 1438. (12) O’Shea, S. J.; Welland, M. E.; Pethica, J. B. Chem Phys. Lett. 1994, 223, 336. (13) O’Shea, S. J.; Welland, M. E. Langmuir 1998, 14, 4186. (14) Rief, M.; Gautel, M.; Oesterhelt, F.; Fernandez, J. M.; Gaub, H. E. Science 1997, 276, 1109. (15) Fritz, J.; Katopodis, A. G.; Kolbinger, F.; Anselmetti, D. Proc. Natl. Acad. Sci. U.S.A. 1998, 95, 12283. (16) Merkel, R.; Nassoy, P.; Leung, A.; Ritchie, K.; Evans, E. Nature 1999, 397, 50.
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on Kramers theory17 for bond dissociation rates in a liquid.18,19 This theory shows that both the association and dissociation rates of a bond depend only on the shape of the tip-surface interaction potential, from which both rates can be calculated. Therefore, it is important to find a method to determine tip-surface interaction potentials with an AFM. The conventional way to determine tip-surface interaction potentials from force-distance curves is to integrate the force times the distance traveled by the tip while approaching the surface. Following this method, not all tip-surface interaction potentials can be obtained. If the force gradient due to the tip-surface interaction is higher than the spring constant of the cantilever used (d2U/dr2 > kcant; with U the potential energy, r the distance, and kcant the spring constant of the cantilever), the integration of a force-distance curve will break down. The tip will come in an energetically unstable regime and the tip snaps into contact.10,20 The force is therefore ill-defined over the distance of snapping and, as a consequence, it is impossible to use the integration method to obtain the potential. Alternative approaches are presented to circumvent the problem of snapping into contact. When magnetism21 or capacitance22 is used to exert force on the cantilever, the spring constant of the cantilever can actively be modified with a force feedback. In these approaches, the problems of snapping into contact are postponed until the force gradient of the tip-surface interaction potential reaches a higher value than this new, actively modified spring constant. The disadvantage of snapping into contact is inherent in force measurements using springs and thus is also present in the measurement with the SFA.4 A new approach for measuring interaction potentials was proposed recently by Cleveland and co-workers23 for AFM experiments. In this approach, they used the Boltzmann distribution to calculate the Helmholtz free energy of the tip and surface from the probability distribution of the position of the tip, while the cantilever was maintained at a fixed position. The method was used to demonstrate the presence of multiple potential energy wells in the vicinity of a calcite surface. A similar approach is used in optical trapping experiments.24,25 The approach is very elegant because it directly determines the energy without the need for an integration step. Another advantage of using the approach of Cleveland and coworkers21 is that the energy can be determined for interactions where d2U/dr2 > kcant. In the approach of Cleveland and co-workers23 the total potential of the tip, which is the sum of the interaction potential and the harmonic cantilever potential, is measured. The relative position of these two potentials depends on the distance between the cantilever base and surface. If the cantilever is brought to the surface in an uncontrolled way, this relative position is not known, unfortunately. In addition to this, the method requires the tip to explore a considerable part of the interaction potential, which makes the measurement inherently slow. Because of this (17) Kramers, H. A. Physica (Utrecht) 1940, 7, 284. (18) Evans, E.; Ritchie, K. Biophys. J. 1997, 72, 1541. (19) Evans, E.; Ritchie, K. Biophys. J. 1999, 76, 2439. (20) Willemsen, O. H.; Snel, M. M. E.; Kuipers, L.; Figdor, C. G.; Greve, J.; de Grooth, B. G. Biophys. J. 1999, 76, 716. (21) Jarvis, S. P.; Yamada, H.; Yamamoto, S.-I.; Tokumoto, H.; Pethica, J. B. Nature 1996, 384, 247. (22) Joyce, S. A.; Houston, J. E. Rev. Sci. Instrum. 1991, 62, 710. (23) Cleveland, J. P.; Scha¨ffer, T. E.; Hansma, P. K. Phys. Rev. B. 1995, 52, R8692. (24) Florin, E.-L.; Pralle, A.; Stelzer, E. H. K.; Ho¨rber, J. K. H. Appl. Phys. A 1998, 66, S75. (25) Crocker, J. C.; Matteo, J. A.; Dinsmore, A. D.; Yodh, A. G. Phys. Rev. Lett. 1999, 82, 4352.
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lack of speed, drifts in the AFM setup may offset the shape of the measured interaction potential. In this study, we present a new approach to measure the tip-surface interaction potential with the help of the probability distribution of tip position. We generate forcedistance curves at such a low approach rate that the tip has sufficient opportunity to explore a large part of the continuously varying potential energy curve. Because the tip is brought into contact with the surface during each approach, we accurately know the absolute position of the cantilever potential. This method can be used to obtain the tip-surface interaction potential and at the same time lead to a decreased sensitivity for drift in the z-direction of the AFM. Moreover, it allows averaging over several force-distance curves to improve the statistical accuracy of measured probability distributions of the tip position. The new approach is applied on a model system of a Si3N4 tip in the vicinity of a mica surface. In a buffer solution, containing 100 mM KCl, the contributions of van der Waals attraction and Born repulsion to the tipsurface interaction potential are demonstrated. At 3 mM KCl, an electrostatic contribution is detected, which is dominant over the van der Waals attraction. Because the method even works for these weak interactions, it will certainly be sensitive enough to measure the association and dissociation rates of single molecular bonds in the future. Materials and Methods All experiments were done with silicon nitride tips. The spring constant of the cantilevers (Sharpened Microlevers; cantilever B; Park Scientific Instruments, Sunnyvale, CA) was calibrated as described elsewhere.26,27 The tips were all from the same wafer and had a low adhesion force on freshly cleaved mica (lower than 100 pN in 100 mM KCl buffer). Mica surfaces were prepared by cleavage. The mica was immediately submersed in the buffer solution to prevent the adhesion of contaminants to the surface. Buffer solutions were prepared from HPLC grade water (Sigma-Aldrich, U.K.) because AFM images demonstrated that Milli-Q water still contained hydrocarbons that could form a monolayer on the freshly cleaved mica. Buffers contained 100 or 3 mM KCl at a pH ) 6. Measurements of the tip-surface interaction potentials were made by operating the AFM in force-distance mode. The stabilization procedures and way of operation have been described elsewhere.28,29 The scan size was set to zero to maintain the tip at the same lateral position during the generation of the forcedistance curves. The rate of generating force-distance curves was 0.5 Hz. The piezo travel was set to 15 or 48 nm for the measurements at 100 and 3 mM KCl, respectively. Each measurement contained a set of 256 force-distance curves. The curves were analyzed with a software routine that is described below. The measured curves represented the deflection signal, sampled with a frequency of 65 kHz. According to the Nyquist criterion, tip movements with a frequency up to 32 kHz could therefore be measured. This maximal frequency is high enough to detect the movement of a tip in liquid (resonance frequency of cantilever B is 2-3 kHz). The software routine to convert a set of force-distance curves into probability distributions of tip position was written in interactive data language (IDL; RSI, Boulder, CO). A typical force-distance curve that is analyzed with the software routine is depicted in Figure 4a. Only the part of the force-distance curve in which the cantilever base is approaching the surface is analyzed (line A-E). Analysis of the retraction part of the force(26) Hutter, J. L.; Bechhoefer, J. Rev. Sci. Instrum. 1993, 64, 1868. (27) Butt, H.-J.; Jaschke, M. Nanotechnology. 1995, 6, 1. (28) van der Werf, K. O.; Putman, C. A. J.; de Grooth, B. G.; Greve, J. Appl. Phys. Lett. 1994, 65, 1195. (29) Willemsen, O. H.; Snel, M. M. E.; van Noort, S. J. T.; van der Werf, K. O.; de Grooth, B. G.; Figdor, C. G.; Greve, J. Ultramicroscopy 1999, 80, 133.
Analysis of the Brownian Motion of an AFM Tip distance curve (line E-G) can be performed in a similar way. The raw data consisted of voltages, linearly proportional to the deflection of the cantilever, that were plotted against the cantilever base-surface distance. The part of the force-distance curve where the tip is in contact (line C-E) was used to calibrate the deflection signal and convert it to nanometers. Because a tip in contact with an uncompressible surface follows the movement of the cantilever base, the slope of section C-E in a calibrated curve is unity. Taking this into account, calibration of the forcedistance curves is performed by fitting a straight line to section C-E for each force-distance curve of the data set and dividing the scale of the y-axis of the curves by the average slope. At 3 mM KCl, no snap into contact was observed, implying that the deflection cannot be calibrated from the force-distance curve as described above. Instead, we used the sensitivity of the cantilever signal, as obtained from the experiments at 100 mM KCl, for the slope of a straight line that was fitted through the last 5000 points ()3.5 nm) of trace C-E. The calibration of the zero point of cantilever base-surface distance D is obtained from the intersection of the line fitted from section C-E and the baseline. This calibration will eliminate the effect of drift in the z-direction. The baseline is fitted from sections A-B and F-G. The fitted line immediately shows the influence of low-frequency noise in the deflection signal. For less than 50 of the 256 acquired curves the noise was too large, rendering these force-distance curves unusable. The vertical axis of the calibrated force curve contains deflection of the cantilever instead of the position of the tip with respect to the surface. To transform the vertical axis, the line that is fitted to line C-E is subtracted from the graph. The resulting curves that show the position of the tip as a function of cantilever base-surface distance are used to construct histograms of position. The approach part of the force-distance curve (line A-E) is divided into 262 equal portions, which means that the total potential is evaluated at 262 intervals of cantilever base-surface distance D. The size of the interval is 0.06 or 0.18 nm for piezo travels of 15 and 48 nm, respectively. Each interval contains 250 measured tip positions. The probability distribution was created by making a histogram of the tip positions. A minimum of -2 nm and a maximum of 40 nm was used with a bin size of 0.025 nm. The procedure was repeated for all forcedistance curves of the data set and all histograms were added. The resulting histograms were normalized to 1. The procedure provides us with an array of data, representing the probability distribution of the tip position as a function of cantilever basesurface distance. The procedure to obtain the tip-surface interaction potential by the integration of force over tip-surface distance was also written in IDL. It calibrates the raw data into graphs of deflection versus cantilever base-surface as has been described above. The deflection is then multiplied with the spring constant to obtain the force on the cantilever. The tip-surface distance is obtained by subtraction of the calibrated deflection signal from the cantilever base-surface distance. All force-distance curves that were used in the stochastic approach were also used in the force integration method to have equal statistical information. All curves were averaged and the resulting curve contained 65 536 data points. The points were compressed to 1064 data points before integration. The resulting force versus tip-surface distance curve was integrated, using a stepwise integration procedure. The obtained energy was scaled to the thermal energy kbT (kb is the Boltzmann constant and T a temperature of 300 K).
Results We will begin with a detailed model analysis of the movement of an AFM tip near a mica surface. Using a model for the total potential energy of a tip in the vicinity of the surface, we will simulate the thermal movement of the tip. Then, it is shown that, from the experimentally observed movement of the tip, the total potential energy can be reconstructed. The influence of drift on a real measurement is discussed. Modeling of Nonspecific Interactions. In a previous study we used potential energy curves to describe the
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Figure 1. (a) Schematic diagram (not to scale) of the model used to describe tip-surface interactions. The AFM tip is conically shaped with a spherical end face. It is attached to the cantilever base by a harmonic spring, the cantilever. The cantilever base is moved with respect to the surface and the deflection of the cantilever is determined. (b) Calculated forcedistance curve that resulted from the model of the tip-surface interaction in a buffer, containing 100 mM KCl. In this curve the point where the cantilever base-surface distance is defined as zero is indicated with a dot.
nonspecific interaction between the tip and surface.20 Because this model predicted tip behavior near a surface that could be verified experimentally, it will also be used in this study. The model describes the potential energy contributions that are present in the interaction of a Si3N4 tip with a mica surface The schematic representation of all the objects present in the model is shown in Figure 1a. The geometry of tip and surface, a conical tip with a spherical end face30 near a semi-infinite mica surface, is used to calculate the tip-surface energy as a function of tipsurface distance (d). The contributions of the Born repulsion and van der Waals attraction to the tip-surface interaction potential are described by a Lennard-Jones potential. A harmonic spring potential is added to account for the potential energy of the cantilever. In addition to these contributions, an exponentially decaying energy contribution can be added to the tip-surface interaction potential to account for electrostatic interactions. The sum of all these contributions is defined as the total potential and this is the potential that will be measured in an AFM experiment. The shape of the total potential depends on the relative position of the tip-surface interaction potential with respect to the position of the cantilever potential. In the model of the tip-surface interaction, the relative position is determined by the cantilever base-surface distance D. To construct a force-distance curve using this model, the cantilever base-surface distance is varied and the position of the tip is calculated without the presence of thermal fluctuations. When an electrostatic energy contribution (30) Butt, H.-J. Biophys. J. 1991, 60, 777.
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simulation step is calculated by taking the current position of the tip and adding the product of the velocity and time step of the simulation ∆t. The velocity is calculated by taking the current velocity and adding the contributions derived from the force as follows from the shape of the potential at that particular position, the damping of the cantilever, and the stochastic fluctuations. The latter two contributions were chosen in such a way that they fulfill the Langevin equation. A detailed description of the simulation will be given elsewhere.31 The simulation was carried out with the potential that is plotted in Figure 2a that results from the model of nonspecific interaction with electrostatic contribution. The simulation was used to obtain a time series of tip positions, as is shown in Figure 2b. The graph shows that the stochastic fluctuations in the tip energy effectively allow it to explore the potential over a range of tip-surface distances of almost 2 nm. Because the energy barrier to jump from the left to the right well is only 0.5 kbT, the tip will only rarely remain in the left well. Most of the time, it directly jumps back to the other well, where it is more likely to be. To validate the simulation, the average potential and kinetic energy of the tip are calculated. As expected, they both have a value of 0.5 kbT after sufficient simulation steps. To find out whether the movement of the tip is indeed governed by stochastic fluctuations and the shape of the total potential, the distribution of the tip position is used to reconstruct this potential with the method of Cleveland and co-workers.23 They used the Boltzmann distribution, which gives the relation between energy difference and probability,
P(d) ) Ce-U(d)/kbT Figure 2. (a) Potential energy as a function of the tip-surface distance d. The tip-surface interaction potential contains a Lennard-Jones contribution and an electrostatic contribution for a buffer, containing 100 mM monovalent salt. The LennardJones potential used a Hamaker constant of 3.1 × 10-21 J for the Si3N4-water-mica system. The van der Waals interaction term was adapted for the shape of the tip, as is described elsewhere.20 The electrostatic repulsion term had a decay length of 1.0 nm and an energy at 0 nm of 20 kbT. The harmonic cantilever with a spring constant of 0.02 N/m has been added at a cantilever base-surface distance of -0.2 nm. (b) The simulated position of a tip in the potential of (a) plotted as a function of time. The two dotted lines are added to indicate the position of the energy wells. The lower line indicates the left minimum of (a) and the higher line indicates the right minimum. (c) The potentials that are reconstructed from the probability distribution of tip position. The thin line has been reconstructed by analyzing 1 800 simulation steps and the thick line by analyzing 58 000 simulation steps. The y-axis has been calibrated in units of kbT.
corresponding to a buffer containing 100 mM KCl is used, the force-distance curve of Figure 1b is obtained. One of the total potential energy curves that is used in the construction of the force-distance curve of Figure 1a is shown in Figure 2a. A noticeable feature of this potential is that it has two energy minimums that are separated by a few kbT only (1 kbT ) 4.1 × 10-21 J). As has already been shown in our previous study, the tip can “hop” between the energy wells. To understand the exact behavior of a tip in such a potential, we performed a simulation, which is discussed below. Simulation of Tip Movement in a Potential. The simulation that has been performed uses an iterative process to calculate the position of a particle in a potential energy well. The position that the tip will have in the next
(1)
in which P(d) is the probability of the tip to be at tipsurface distance d, C is a constant, U(d) is the total potential at tip-surface distance d, kb is Boltzmann’s constant, and T is the temperature. Equation 1 implies that from the measured energy probability distribution of the tip position the total potential can be obtained:
U(d) ) -kbT ln(P(d)) + const.
(2)
The simulation of tip movement used before was adapted for the new approach. The position of the tip is used to calculate the probability distribution of the tip position. When the probability distribution is inserted into eq 2, a total potential is obtained, which is plotted in Figure 2c (thin line). The curve clearly shows that even though it has been constructed after ≈1800 simulation steps, the overall shape is already visible. When the simulation is carried out for almost 60 000 simulation steps, the shape almost perfectly resembles the original potential. This is shown in Figure 2c (thick line). It was verified that the reconstruction of the potential energy curve was independent of damping and temperature. Interestingly, at critical damping it took the least simulation steps to reconstruct the total potential. Reconstruction of the Total Potential from Experimentally Observed Tip Position. To find out whether the reconstruction of the total potential can be used for experimentally obtained data, the position of a tip near a surface in a buffer of 100 mM KCl was measured. The cantilever base is brought so close to the surface that the tip makes discrete jumps between positions in the (31) De Grooth, B. G. Am. J. Phys. 1999, 67, 1248.
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Figure 3. (a) and (b) Two sections of a time trace of the deflection signal of a cantilever. The cantilever and tip are brought close to the surface with a piezo spindle and by manual addition of a voltage to the z-stage of the scan tube. The deflection trace was recorded by sampling the unfiltered deflection signal with a frequency of 50 kHz. Part (a) shows the time trace during the time interval [0.11, 0.21] s and (b) shows it during the interval [2.10, 2.20] s. (c) Total potential energy that has been reconstructed from (a), using eq 2. (d) Total potential that has been reconstructed from (b). The y-axes of (c) and (d) have been calibrated in units of kbT.
potential wells. The deflection, representing the position of the tip, was recorded for 3.0 s. Two sections of the time trace have been plotted in Figure 3a,b. The time trace of Figure 3a shows the position of the tip during the interval [0.11, 0.21] s of the total time trace and Figure 3b shows the time trace during the interval [2.10, 2.20] s. Comparing these two time traces, it is very clear that the statistical behavior of the tip is quite different. For instance, the probability of the tip to be in contact with the surface has decreased in the second interval. Also, other statistical properties, like the hopping frequency and the average dwell time in both wells, have changed. Because we know that the stability of our AFM setup in the z-direction is at least 0.07 nm/s,29 we realize that the cantilever basesurface distance is changing during a measurement period of 3.0 s. When the model for tip-surface interactions is used, it becomes clear that this variation of D will change the shape of the total potential. The tip behavior will change accordingly. To demonstrate the effect of the drift on the total potential, the time traces are used to construct a probability distribution, which in turn is used to calculate the total potential. Parts c and d of Figure 3 show the total potential calculated from the time traces of parts a and b, respectively. It is clear that the different tip movement is caused by a different shape of the potential. There is a second reason which makes it impossible to calculate the tip-surface interaction potential from parts c or d of Figure 3. The position of the cantilever potential is unknown and it can therefore not be subtracted from the total potential to get the tip-surface interaction potential. This problem was already noted in the study of Cleveland and co-workers21 because only plots of the total potential instead of the tip-surface interaction potential are shown. In the next section we will suggest a new approach, which allows measurement of the tip-surface interaction potential. In our approach we use force-distance curves
to move the cantilever potential in a controlled way. We will explain how it is possible to determine the absolute value of the cantilever base-surface distance and why the method is less sensitive to drift in the AFM setup. Reconstruction of the Tip-Surface Interaction Potential from Force-Distance Curves. The modeling of tip-surface interaction showed clearly that the tip is forced to move in a continuously varying total potential,20 resulting in a changing behavior of the cantilever. To measure the reason for this behavior, we divide the forcedistance curve into very small sections, where we assume the total potential to be constant. In our new approach, the force-distance curve is generated at a sufficiently low cantilever velocity and the position of the tip is determined with a high sampling rate, making it possible to calculate the probability distribution for each of the sections. The main advantage over the method where the tip is simply brought near the surface is that, with the new approach, the tip is gradually forced to come into adhesive contact in a controlled way. This means that there is an internal calibration of the zero point of the cantilever base-surface distance D in each force-distance curve. Because the velocity by which D is changed is equal to the cantilever velocity, the position of the cantilever potential is known for each section of the force-distance curve. If it is assumed that the tip-surface interaction potential is the same for all force-distance curves, it is possible to average the probability distributions of tip position from different force-distance curves. Averaging of the probability distribution not only improves the accuracy but also allows the rate of generating forcedistance curves to be increased, provided that the tip is still able to explore the total potential. This can be done because two force-distance curves, generated at twice the original rate, give the same result as one curve, recorded at the original rate. The generation of force-
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Figure 4. (a) Typical force-distance curve that is recorded with a cantilever in 100 mM KCl buffer (pH 6). The characters have been added for easy reference. The zero-point of the x-axis has been calibrated as described in the Materials and Methods section. This is done for the approach and retraction part of the curve. The inset shows a detailed view on the approach part of the force-distance curve where the tip snaps in and out of contact. (b) Total potential obtained from the probability distribution of the tip position at a cantilever base-surface distance (D) of 8.2 nm. The curve is fitted with a harmonic function, which is shown with the dotted line. (c) and (d) The total potential at a cantilever base-surface distance of 2.9 and 2.3 nm, respectively. The dotted lines denote the position of the cantilever potential. (e) Parts of the tip-surface interaction potential that have been obtained by subtraction of the total potential curves of (b)-(d). The y-axes of (b)-(e) have been calibrated in units of kbT.
distance curves at a higher rate relaxes the demands for the z-stability of the AFM. A typical force-distance curve obtained in 100 mM KCl is plotted in Figure 4a. From comparison of this curve with the theoretical curve of Figure 1b, it is clear that the snapping of the tip to the surface is not a single event. Instead, the tip hops between two potential wells. Evidently, the cantilever is moved slowly enough so that the tip is really able to explore a considerable part of the potential. It can also be seen that, once D is lowered, the probability to be in contact with the surface increases, and that the distance between the potential wells becomes smaller. These observations confirm our earlier results.20 For analysis, the force-distance curves are transformed into probability distributions of tip position, as is described in the Materials and Methods section. Most importantly, the force-distance curves are calibrated in such a way that the average position of a tip ()the tip-surface distance) in contact is 0 nm. Because the tip is fluctuating around this position, also negative tip-surface distances are possible. As a result, we obtain a set of histograms containing the probability distributions of tip position for each value of D, which are inserted in eq 2 to calculate the total potential. The tip-surface interaction potential is now constructed by subtracting the cantilever potential at the known cantilever base-surface distance. To this
end, the proper shape of the cantilever potential and the correct cantilever base-surface distance are determined. First, the cantilever base is brought far from the surface. The movement of the tip is governed by stochastic fluctuations and cantilever potential only. The total potential for this situation is a harmonic potential, as plotted in Figure 4b, and can be adequately described by a second-order polynomial function, which is used for further analysis. Because the cantilever movement is controlled, the position of the cantilever potential is known for each of the 262 sections of the force-distance curve. These sections are used to reconstruct the tip-surface interaction potential by subtracting the cantilever potential from the total potential. This is illustrated in Figure 4c-e. The total potential, obtained at a cantilever basesurface distance of 2.9 nm, is plotted in Figure 4c. The interaction with the surface causes a second minimum that is located to the left of the original minimum. Thereafter, the cantilever base is brought 0.6 nm closer to the surface. The total potential measured is plotted in Figure 4d. It is clear that the relative depths of the energy wells have changed, indicating that the probability to be in adhesive contact with the surface has increased. Hereafter, the cantilever energy is subtracted at the proper position and the results for three sections are plotted in
Analysis of the Brownian Motion of an AFM Tip
Figure 4e. For each section, only a small part of the tipsurface interaction potential is obtained as a result of the limited time a tip spends in each part of the potential. Therefore, only a limited range of tip-surface distances can be explored. The part of the tip-surface interaction potential that has been obtained by subtraction of the cantilever potential from the total potential of Figure 4b is a flat line, as expected. Noise at the edges is higher because the total potential at these tip-surface distances was obtained from those bins of the histogram that were filled with only a few events. The results show a positive slope in the tipsurface interaction potential, indicating attraction between surface and tip. Bringing the tip closer to the surface, it is able to explore smaller tip-surface distances and a different part of the tip-surface energy can be calculated. The result of the subtraction of the cantilever potential from the potential of Figure 4d shows an effect that should be noticed. In the overlapping region of the tip-surface interaction potential curves of Figure 4c,d, the curves have similar shapes but different offsets. This originates from the fact that we use histograms to calculate energy differences instead of absolute energy levels. Because it is physically obvious that different analysis steps should result in the same energy at the same tip-surface distance, the effect has to be compensated for. The compensation is accomplished by maximizing the overlap of the curves, obtained from adjacent sections of the force-distance curves. After the overlap has been maximized, a set of values representing the value of the tip-surface energy at that particular tip-surface distance is obtained. These values, which all have their own error margins, have to be added with proper weight factors. The adding procedure is described in the Appendix. Measurement of the Tip-Surface Interaction Potential at 3 mM KCl. The new method of reconstruction of the tip-surface interaction potential was compared with the conventional method of force integration. A repulsive tip-surface interaction is a suitable test system because the tip will not snap into contact, implying that both methods should work. To construct a repulsive potential that ranges over several nanometers, a salt concentration of 3 mM KCl was used, creating an electrostatic energy contribution with a Debeye decay length of 5.5 nm.11 As initial measurements demonstrated that a piezo travel of 15 nm was insufficient to reconstruct the tip-surface potential, it was increased to 48 nm. A typical force-distance curve that was recorded at a salt concentration of 3 mM KCl is plotted in Figure 5a. As predicted, the tip never snaps into contact, indicating that the electrostatic repulsion is dominant during the whole measurement. Therefore, the two methods for determining the tip-surface interaction potential can be used. The tip-surface energy, calculated with the stochastic fluctuation technique, is plotted in Figure 5b. Apparently, the tip-surface energy is repulsive in the whole region measured. Because the tip never snaps into contact, the definition of the zero point of tip-surface distance is not an absolute one. Yet it is possible to calculate whether the energy decays exponentially, as is shown in Figure 5b. The Debye decay length derived from the exponential decay is 5.0 nm. This is in reasonably good agreement with the theoretical value of 5.5 nm. The tip-surface energy, calculated with the force integration method, is plotted in Figure 5c. The decay length of the electrostatic energy contribution is 5.2 nm, which is almost the same as that obtained with the
Langmuir, Vol. 16, No. 9, 2000 4345
Figure 5. (a) Typical force-distance curve that is recorded with a cantilever in 3 mM KCl buffer (pH 6). The zero-point of the x-axis has been calibrated for the approach and attraction part of the curve as described in the Materials and Methods section. (b) Tip-surface interaction potential that has been obtained with the analysis, using the stochastic fluctuations of the tip position. The tip-surface interaction potential is fitted with a single exponentially decaying function with a decay length of 5.0 nm (dashed line). (c) The tip-surface interaction potential that has been obtained by using the integration of the force on the tip over tip-surface distance. The tip-surface interaction potential is fitted with a single exponentially decaying function with a decay length of 5.2 nm (dashed line). The y-axes of (b) and (c) have been calibrated in units of kbT.
statistical fluctuation method. The differences between parts b and c of Figure 5 are due to the lack of absolute calibration of the zero point, which causes a horizontal shift between the two potentials. This results in a proportional change of the vertical scale for an exponentially decaying curve. Second, in the force integration method a value for the spring constant has to be estimated, while in the stochastic method the spring constant is obtained in the measurement itself. For the former approach, the spring constant is determined using the thermal noise spectrum of the cantilever and thus effectively eliminating 1/f noise.26 In the latter approach, this noise can cause erroneous overlay of the data from the individual force-distance curves, resulting in a broadening of the histograms. The estimated spring constant will be too low, resulting in a lower value of the tip-surface energy. Measurement of the Tip-Surface Interaction Potential at 100 mM KCl. To show that the statistical fluctuation method can also be applied to the measurement of attractive tip-surface interaction potentials, the salt concentration is increased to 100 mM KCl. As has already been shown in Figure 4a, the tip snaps into contact under
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Langmuir, Vol. 16, No. 9, 2000
Figure 6. The figure shows the tip-surface interaction potential that has been obtained at 100 mM KCl. The y-axis has been calibrated in units of kbT.
these conditions and only the statistical fluctuation method can be applied. The total potential obtained from the analysis is plotted in Figure 6. In this graph the potential energy is plotted against tip-surface distance d. For positive values of d the potential has a positive slope, which is due to van der Waals attraction between the tip and surface. At zero distance, there is a minimum in the potential with a value of -22 kbT and below zero nanometer the slope of the potential becomes positive because of the Born repulsion. Experiments performed under the same conditions but with a different tip showed that the depth of the well is not constant (up to 50 kbT; data not shown), which is expected because the exact shape of the tip apex varies from tip to tip. The resolution of the tip-surface interaction potential is high enough to detect energy differences lower than the thermal energy kbT. It should be noted that, for tip-surface distances between 0.7 and 1.8 nm, the second derivative of the tip-surface interaction potential is higher that the spring constant of the cantilever. This observation demonstrates that the method using stochastic fluctuation can be used to calculate the tip-surface interaction potential, even in regions where the conventional way of integrating a force-distance curve over the distance cannot be applied. Discussion and Conclusions In this study the Brownian movement of an AFM tip in a potential well is used to calculate the total potential from the derived probability distribution of the tip position. Then, the tip-surface interaction potential is obtained by subtraction of the cantilever potential of the total potential. This could be accomplished by the use of the force-distance mode AFM, where the cantilever potential is moved in a controlled way and the correct position of the cantilever is determined. The approach was tested on a model system of a tip interacting with mica in 100 mM monovalent salt buffer. The tip-surface interaction potential could be obtained with a resolution that was better than the thermal energy kbT and it showed a well depth of 22 kbT. Thus, the novel approach, using forcedistance curves that move the tip in a controlled way through the tip-surface interaction potential, can detect energy differences of several tens of kbT. This implies that, in principle, it is possible to determine the interaction potential of ligand-receptor pairs, which require energy differences of 50 kbT.18 The use of force-distance curves to reconstruct the tipsurface interaction potential has many advantages over the method where a tip is brought close to a surface. We have shown that, without the new approach, the shape of the total potential may change considerably (Figure 3c,d) in only 2 s of the measurement time. This is attributed to the drift in the AFM setup, which will induce a
Willemsen et al.
displacement of the cantilever potential relative to the tip-surface interaction potential. When the forcedistance curves are effectively divided into very small sections where the total potential energy may be considered constant, the influence of drift is reduced. In addition to this advantage, the controlled movement of the cantilever potential allows the deduction of the tip-surface interaction potential from the measured total potential. Finally, averaging over several force-distance curves to increase the accuracy of the method is possible because the position of the cantilever potential is calibrated for each separate force-distance curve. When the force-distance curves are measured, the range of tip-surface distances over which the tip-surface interaction potential can be determined is much higher than that when the cantilever is held at a fixed position. In the calculation of the total potential, it is assumed that the potential is stable during a small section of a forcedistance curve only. During such a section, the tip should be able to explore the potential long enough that most parts of the energy states have been visited many times. In the simulation (Figure 2c) it already appeared that 1800 simulation steps (in the experiment this corresponds to 28 ms of measurement time) were enough to obtain the coarse shape of potential. The experiment (Figures 4b-e) demonstrated that 200 force-distance curves, containing 50 000 data points, are sufficient to reconstruct a section of the potential with energy differences of 8 kbT and a range of 4.5 nm. The range of tip-sample distances that is explored should be so large that it overlaps with the range of distances that is explored in an adjacent section of the force-distance curves. For weak bonds, this condition is easily met, as is seen from parts c and d of Figure 4, but for strong bonds the tip-surface interaction potential is highly curved and the method may require more force-distance curves to increase the number of detected tip positions per section. For tip-surface distances below 0 nm, the shape of the curve is not correct. Comparing Figures 4e and 6, it is observed that, in this region, the shape is not the same. Because the latter figure is the result of adding all sections that have been obtained from the individual parts of the force-distance curve, different shapes of the potential must have been used. This is physically not correct. The problem is attributed to the limited accuracy of the beam deflection system that is used to detect the position of the tip. This inaccuracy can cause the tip to be detected at an unphysical position and therefore will give the wrong total potential at these distances. This lack of detection accuracy also causes another problem that is discussed below. Comparing the simulated time trace of Figure 2b with the measured time trace of Figure 3c, it is evident that the average dwell time in the wells in the simulation is much shorter than that in the experiment. In the simulation, the low-energy barrier between the wells in combination with the narrow well makes the probability of hopping from the left well to the right well high and therefore the average dwell time low. In the experiment, the detected accuracy of the tip position is limited, resulting in erroneous apparent positions. This limitation results in a broadening of the reconstructed potential, especially for highly curved ones, and in a decrease of depth of potential wells. The two consequences of limited position detection affect the predicted statistical behavior of the tip. On one hand, the increased width of the potential well decreases the attempt frequency to hop between the wells. On the other hand, the lowering of the potential barrier results in a higher probability to hop. Analyzing Figures 3a and c with this knowledge, it is unlikely that
Analysis of the Brownian Motion of an AFM Tip
Langmuir, Vol. 16, No. 9, 2000 4347
the potential energy of Figure 3c can account for the actual behavior of the tip, observed in Figure 3a. Because hopping of the tip, which is due to a double-well potential, is still observed, the uncertainty in tip position detection causes the right potential well to disappear. These considerations have to be taken into account when the statistical behavior of a tip is predicted from a reconstructed potential. The lack of accuracy in detecting the tip position is attributed to three sources. The first two sources are due to the laser diode that is focused on the back of the cantilever and which has shot noise and a limited beam pointing stability.23,29 These two sources have been investigated by replacing the cantilever by a fixed mirror to exclude all additional movements. From the acquired spectra it is estimated that 30% is caused by shot noise and 70% by beam-pointing stability. The total uncertainty in tip position, caused by the laser diode, would give a broadening of the total energy curves of 0.008 nm (fwhm). The third source is the method of beam deflection to detect the position of the tip. Butt and Jaschke27 have already calculated that higher order bending modes cause a finite deflection signal, even when the tip is fixed on a surface. This is caused by the fact that beam deflection is sensitive to bending of the cantilever instead of absolute position. With the combination of the three sources, the tip in contact can only be detected with an accuracy of 0.3 nm p-p. Appendix To obtain the complete tip-surface interaction potential, we have to add the different sections of this potential. For each value of the tip-surface distance the energy is the sum of the number of contributions, which each have different error margins. The contributions have to be added with the proper weight factor. At a fixed tip-surface distance dk we have obtained a set of n independent measurements of the energy U with values uik and with error margins ∆uik:
( )
∆Uk Uk )0 ∂wi
∂
for all i ) 1...n
We can use eq A3 to write one of the weight factors, for instance, w1, as a function of all the others. Then, all the remaining wi, i*1, are independent. Using this and eq A4, we can rewrite eq A5 as
∂ ∂wi
((
(1 -
∑ l)2
)
∆u1k
wl)‚
Uk
( ) ( ))
2
∆uik
+ ... + Uk ∆unk 2 wn‚ ) 0 (A6) Uk
-2w1‚∆u1k2 + 2wi‚∆uik2 ) 0
1
wi ) ∆uik‚
Uk ) w1‚u1k + w2‚u2k + ... + wn‚unk
∆uik )
where the sum of all weight factors is unity:
(A3)
From eqs A1 and A2 the relative error of U is defined as
∆Uk ) Uk
x(
)
∆u1k Uk
w1‚
2
(
)
∆uik Uk
+ ... + wi‚
2
(
To obtain the best possible value of U, the weight factors have to be chosen so that the relative error of U is minimized. This is accomplished by solving the following set of equations:
Nik
(A9)
Nim ∑ m
( )
-kbT -kbT ‚∆Nik ) Nik xN
(A10)
ik
In this equation we have used the fact that the error of bin of a histogram is equal to the square root of the number of events in this bin. Substitution of eq A10 into eq A8, we obtain
wi )
)
∆unk 2 Uk (A4)
+ ... + wn‚
∆ulk2
where the summation is performed over every tip-sample distance of the ith measurement. The summation is performed on every tip-sample distance dl where the tip is detected. Using eqs 2 and A9, we can calculate the error in the total potential:
(A2)
∑l wl ) 1
∑l
(A8)
1
Hereafter, we will use eq A8 to find the proper weight factors in the determination of the tip-surface interaction potential. The total potential is calculated by inserting the probability distribution of tip position in eq 2. The probability distribution of tip position is obtained by making a histogram of the tip-surface distances. Let Nik be the number of events at tip-surface distance dk, obtained in the ith independent measurement. Then, we obtain for the probability pik,
uik ( ∆uik, ...
To obtain the weighted average Uk of the measurements, we add the measured values, with weight factors w1, w2, ..., wn.
(A7)
Combining eqs A3 and A7, we obtain
pik )
(A1)
2
+ ... + wi‚
Using the independence of wi and eq A6, we obtain
u1k ( ∆u1k, ...
unk ( ∆unk
(A5)
Nik n
(A11)
Nik ∑ l)1 From eq A11 we learn that the proper weight factors in the adding procedure, which is used to obtain the tipsurface interaction potential, is the probability of the tip to be at that particular tip-surface distance. LA991368G
