Scanning probe acceleration microscopy (SPAM) in fluids: Mapping mechanical properties of surfaces at the nanoscale
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Edited by Francisco de la Cruz, Centro Atómico Bariloche, Rio Negro, Argentina, and approved January 19, 2006 (received for review July 12, 2005)
Abstract
One of the major thrusts in proximal probe techniques is combination of imaging capabilities with simultaneous measurements of physical properties. In tapping mode atomic force microscopy (TMAFM), the most straightforward way to accomplish this goal is to reconstruct the timeresolved force interaction between the tip and surface. These tip–sample forces can be used to detect interactions (e.g., binding sites) and map material properties with nanoscale spatial resolution. Here, we describe a previously unreported approach, which we refer to as scanning probe acceleration microscopy (SPAM), in which the TMAFM cantilever acts as an accelerometer to extract tip–sample forces during imaging. This method utilizes the second derivative of the deflection signal to recover the tip acceleration trajectory. The challenge in such an approach is that with real, noisy data, the second derivative of the signal is strongly dominated by the noise. This problem is solved by taking advantage of the fact that most of the information about the deflection trajectory is contained in the higher harmonics, making it possible to filter the signal by “comb” filtering, i.e., by taking its Fourier transform and inverting it while selectively retaining only the intensities at integer harmonic frequencies. Such a comb filtering method works particularly well in fluid TMAFM because of the highly distorted character of the deflection signal. Numerical simulations and in situ TMAFM experiments on supported lipid bilayer patches on mica are reported to demonstrate the validity of this approach.
Tapping mode atomic force microscopy (TMAFM) (1) is a widely used dynamic imaging technique that maps surface topography by monitoring the oscillation amplitude of a cantilever integrated with an ultrasharp tip probe, driven by a piezeolectric bimorph element mounted at the cantilever root. In this imaging mode, the cantilever is commonly driven near its resonance frequency ω_{ο}, and the intermittent tip–sample contacts lead to the decrease of cantilever oscillation amplitude from the “free” amplitude A _{o} to tapping amplitude A. The sample surface acts as a repulsive barrier that limits the tapping amplitude of the cantilever (2–4). For a rigid surface, this decrease of cantilever oscillation amplitude is linear with the decrease of the distance between the tip and the sample D _{o}. Thus, the surface topography can be tracked by rastering the tip in the xy plane and using a feedback loop to continuously adjust the vertical (z) extension of the piezoelectric scanner to maintain the constant setpoint s = A/A _{o}.
There is considerable interest in using TMAFM to study elastic and viscoelastic mechanical properties of surfaces, which would be beneficial in enhancing the ability to characterize materials and map mechanical and/or chemical variations of surfaces at the nanoscale in a much gentler fashion. Much of this information can be ascertained from the timeresolved force interaction between the surface and the tip, but currently there is no straightforward manner to obtain the value of this force in tapping mode imaging. In the absence of such a straightforward technique, the phase of the cantilever in tapping mode is commonly used to obtain some information about the mechanical properties of surfaces (5, 6); however, multiple sources of energy dissipation [i.e., capillary forces (7), viscoelasticity of the sample (8), cross talk with topography (9), frictional forces associated with the tilt of the cantilever and/or surface (10), etc.] make it difficult to interpret phase images.
A more complete insight into the mechanical properties of the sample can be obtained by deeper analysis of the cantilever deflection trajectory involving especially its higher harmonic content (11–13). When a harmonic drive signal is applied to the cantilever, the resulting oscillation is also harmonic. The harmonic motion of the cantilever is distorted at the bottom of each tapping oscillation cycle, and as a consequence, a certain amount of power is shifted to higher harmonics. In traditional TMAFM, which monitors cantilever deflection only at the oscillation frequency, information about this distortion is lost. The easiest way to retain it is to digitize the entire cantilever trajectory at sufficiently high frequency (at least twice the frequency of the highest harmonic) and high bit resolution. Recent developments in highspeed analog/digital converters make this task entirely possible, as demonstrated by Stark et al. (12), who proposed to reconstruct tip/sample force by taking the inverse Fourier transform of the Fouriertransformed cantilever trajectory divided by its transfer function. Currently, the most straightforward method of measuring the transfer function of a cantilever involves using the oscillation decay of a cantilever subjected to an initial deflection (12). The initial deflection may be imposed by performing a force curve experiment on a strongly adhesive surface. It also must be noted that there are several sources of distortion in cantilever deflection signals, including nonlinearities of the detector and electronics of the atomic force microscope (AFM). Also, higher eigenmodes of the cantilever can complicate analysis of higher harmonics. Conversely, cantilevers that couple with higher eigenmodes can enhance the higher harmonic content in TMAFM signals (14). Another common problem in analyzing higher harmonics is the rapid decay of the harmonic envelope, which can effectively place the signal below the noise level.
In an effort to more fully understand the tip–sample forces in TMAFM, numeric simulations are often used that describe the motion of a cantilever as a driven, damped harmonic oscillator (15) where m _{eff} is the effective mass of a cantilever, b is the damping coefficient, k is the cantilever spring constant, a _{o} is the drive amplitude, ω is the drive frequency, D _{0} is the resting position of the cantilever base, F _{ext} is the tip–sample force, and z is the position of the cantilever with respect to the surface. It has to be noted that in practice, AFM monitors the deflection (y) of the cantilever given by rather than its position, z. For systems characterized by high values of quality factor (Q), such as tapping mode operation in air, the difference between y and z (not counting the constant factor D _{o}) is minimal, because the cantilever oscillation amplitude exceeds the drive oscillation amplitude a _{o} by several hundred times. In operation in liquids, however, the cantilever motion undergoes strong viscous damping (Q factor is low), and its oscillation amplitude is comparable with a _{o}. From Eq. 2 , it is clear that under such circumstances one has to make the distinction between cantilever position and deflection, and to make a connection with experiment, Eq. 1 should be rewritten in terms of y as Rearranging Eq. 3 , it can be shown that the tip acceleration can be decomposed into a pulselike tip–sample force (F _{ext}) and other slowly varying factors oscillating at a frequency ω Thus, the contribution to the acceleration because of the timeresolved tip/sample force may be easily distinguished from other terms in the second derivative of the cantilever deflection signal, providing a simple method to reconstruct the tapping force. This finding is the basis of a previously undescribed technique, referred to hereafter as scanning probe acceleration microscopy (SPAM), which makes use of the entire cantilever deflection trajectory to map local forces in a timeresolved manner. The major challenge in this method of reconstructing the force is the noise in the deflection signal. Such noise has already been shown to complicate analysis of force–distance curves obtained by scanning force microscopes (16). It will be shown that the noise can be effectively suppressed through “comb filtering” based on higher harmonics. This approach is made possible by the fact that higher harmonics contain virtually complete information on the distortion of cantilever deflection trajectory. The comb filter is used to extract the harmonics from the Fourier transform of the deflection signal, and an inverse Fourier transform is calculated to give the filtered deflection trajectory.
The effectiveness of the use of the comb filter to accurately reconstruct the deflection trajectory depends on the number of harmonics that appear above the noise level in the Fourier transform. TMAFM under fluids is characterized by largely distorted character of the cantilever trajectory (17–19), as shown in Fig. 1. This large distortion of the trajectory in fluid TMAFM can be directly related to the low Q of the cantilever because of the viscous damping in the fluid (20, 21). Here, we show that for highly distorted signals inherent in fluid TMAFM, the large number of higher harmonics facilitates comb filtering, and reconstruction of the tip/sample tapping force from the second derivative (or acceleration) of the cantilever deflection signal. The magnitude of tapping force can be directly related to material properties of the surface, in particular to the surface Young's modulus. In the following, the use of this approach is illustrated through both numerical simulations and experiments with supported lipid bilayers on mica. Supported lipid bilayer patches on mica were chosen as a model system because the change in surface modulus between the bilayer patch and mica offers an opportunity to compare tip–surface force interactions on different surfaces.
Results
Numerical Simulations.
Single degreeoffreedom simulations of TMAFM in fluids were performed with parameters based on actual AFM experiments to be described later. Typical conditions used in simulations of fluid TMAFM were as follows: resonance frequency of 8 kHz (with operating frequencies slightly above resonance), k of 0.5 N/m, Q of 2, and cantilever free amplitude of 40–75 nm. Although 8 kHz is not a true resonant frequency of cantilevers used in the experiment, this frequency was used in simulations because operation of fluid TMAFM experiments near this frequency is known to produce superior results (22–24). The model was equipped with a feedback loop (integral gain) that allowed simulation of an actual imaging operation. The free cantilever oscillation amplitude was set to A _{o} = 75 nm to closely correspond to actual experiments; the feedback loop maintained the tapping amplitude at 75% of the free amplitude (setpoint ratio of A/A _{o} = 0.75). As discussed earlier, fluid TMAFM deflection trajectories are characterized by a pronounced distortion (Fig. 1). The simulated AFM experiments reproduced this characteristic shape (Fig. 2 a), and comparison with the tip–sample force plots (Fig. 2 b) showed that this distortion coincided with intermittent contact between the tip and sample surface. The distortion became less pronounced with the decrease of surface Young's modulus (Fig. 2 a). Fig. 2 b also shows the well known change in tapping force pulse shape with the change in surface modulus: at a constant setpoint, the area under the peak remains constant, whereas its width (contact time) increases and its height decreases with decreasing E. The latter dependence of maximum tapping force on sample modulus provides a basis for SPAM analysis. Based on Eq. 4 , the second derivative of the deflection signal properly scaled by the effective mass of the cantilever gives the total force acting on the cantilever (Fig. 2 c). Note that in addition to the initial sharp “tapping” pulse, the total force contains components varying at ω, originating from damping and cantilever deflection.
The effect of noise in the deflection signal on reconstruction of tip–sample force from cantilever acceleration also was explored through simulations by “corrupting” the signal with evenly distributed random noise with signaltonoise (S/N) ratios of 1,000:1, 100:1, and 10:1 (Fig. 3). Whereas the characteristic distortion associated with TMAFM in fluids was still discernible even with S/N ratios reaching 10:1, the derivative of the signal was completely obscured by noise at S/N ratios as low as 100:1. Because S/N levels in typical AFM deflection signals may be of this order (or worse), this result indicates that this method of force reconstruction would require filtering or signal processing of noisy deflection signals for use with real systems. Below, we demonstrate that the use of a “harmonic comb filter” satisfies this demand (Fig. 4). In this process, the Fourier transform of the deflection signal is combfiltered, i.e., only intensities corresponding to integer harmonic frequencies are kept, and these intensities are used to reconstruct a deflection signal, y _{rec}(t), by inverse Fourier transform based on the following equation: where ω_{oper} is the operating frequency, δ is Dirac's delta function (Fig. 4 c), and N is the highest harmonic still distinguishable above the noise level. Because of the suppression of some harmonics, a portion of the force magnitude is lost in reconstruction as may be seen, for example, by comparing the peaks in Fig. 2 b (the 100GPa sample) and their reconstructed equivalents shown in Fig. 4 d. Despite this loss of information, the reconstructed signal makes it possible to differentiate between surfaces with different elasticity. By excluding the constant offset and first harmonic in the comb filter, this underlying oscillation can be suppressed.
Fluid TMAFM experiments involving imaging of a rectangular 5nm step were explored in another set of simulations (Fig. 5; see also Movie 1, which is published as supporting information on the PNAS web site). Young's modulus of the surface was 60 GPa before and after the step, but it was lower on the step, where it could range from 1 to 59 GPa. The simulation parameters were chosen to correspond to imaging a 2.5μm line with a scan rate of 5 Hz. As shown in Fig. 5 a, the AFM model was able to track the surface step, although the height of the trace over the step was smaller than the actual step height. This difference was caused by the different compressibility of the step in comparison with its surroundings and is the origin of well known compliancebased contrast in AFM. Importantly, because the feedback loop maintained constant cantilever amplitude along the whole trace (with the exception of edges, where transients appeared), the average force per cycle remained constant. In contrast, the maximum value of tapping force and the width of the force pulse varied when the cantilever passed over the regions of different Young's moduli (Fig. 5 b). These changes in force also were faithfully reproduced in the second derivative of the combfiltered deflection signal (Fig. 5 c). Fig. 5 d shows the relationship between the ratio of maximum force in the softer and more rigid regions and the ratio of the respective values of Young's moduli. Such a curve can be used as a calibration to extract relative values of the surface moduli. It is important to note that the use of a sliding window Fourier transform was needed to accurately reproduce local changes in the deflection signals, such as transients associated with step edges. The window was five oscillation cycles wide and was advanced each time by one oscillation cycle. With the application of harmonic combfiltering, it was possible to reproduce the relationship between the ratio of maximum forces for different ratios of Young's moduli from deflection trajectories with S/N ratios as low as 20:1 (Fig. 5 d).
Implementation of SPAM.
Bilayer patches on mica were imaged using fluid TMAFM while the entire cantilever deflection trajectory was digitized. Height and amplitude (error signal) images of a typical bilayer patch are shown in Fig. 6 a and b. The acquired cantilever deflection trajectories were combfiltered by using a sliding window Fourier transform as described above, and the entire maximum force map of the surface was reconstructed (Fig. 6 c; for a demonstration of one reconstructed line from the image, see Movie 2, which is published as supporting information on the PNAS web site), with darker colors corresponding to lower values of maximum tapping force. Because of limitations in the memory of the data acquisition card, the deflection trajectory was collected in timestamped portions that were merged, interpolated, and reshaped into an appropriately sized matrix corresponding to the AFM image. Because of the latency associated with flushing the card memory, small portions of the cantilever deflection trajectory were missed, and the resulting gaps are manifested as horizontal lines in the maximum force image. A histogram of maximum tapping force over the entire reconstructed force image (Fig. 6 d) was distinctly bimodal, with modes corresponding to the mica and bilayer surfaces. Consistent with the simulation results, the mode corresponding to the bilayer was lower in comparison with that corresponding to mica. With the assumption that Young's modulus of mica is 60 GPa, an estimate of the effective Young's modulus of a supported bilayer can be made based on the ratio of maximum forces and the simulated calibration curve in Fig. 5 d, resulting in a value of 1–3 GPa for Young's modulus of a bilayer. Individual reconstructed tapping pulses for the regions corresponding to mica and bilayer are shown, respectively, in Fig. 6 e and f. Because the effective modulus of the bilayer on mica is much lower than that of bare mica, the tapping force pulses on mica (Fig. 6 e) were taller and narrower than force pulses on the bilayer (Fig. 6 f).
Discussion
Simultaneous mapping of sample topography and properties is one of the particularly attractive features of proximal probebased microscopies. Accomplishing this feat, however, is quite challenging. In the very popular TMAFM, it is further complicated by the fact that, in an attempt to minimize the invasiveness of the imaging process, the tip/sample interaction is limited to a very brief encounter when the probe strikes the surface near the bottom of each oscillation cycle. Here, we have demonstrated that despite this complication, information about the tip/sample force interaction can be obtained from cantilever deflection trajectories by taking advantage of the fact that much of the information concerning this interaction is stored in higher harmonics, making it possible to filter and analyze noisy deflection signals to reconstruct the timeresolved tip/sample tapping force. With this ability, spatially resolved force maps can be constructed. It should be noted that the described harmonic comb filter is applicable only to 1periodic motion of the kind commonly observed in TMAFM and cannot be used in the case of motion exhibiting period doubling or aperiodic/chaotic character. Furthermore, the reconstructed trajectory obtained from harmonic comb filtering is missing some information contained in harmonics below the noise level, and the accuracy of measured transients is limited because of the loss of information not contained in the harmonics.
Spatially resolved force maps of a surface can be directly correlated to material properties such as modulus and adhesion. Such force maps can be obtained in socalled force volume imaging, which takes a force curve at every point in an AFM image; however, this method is limited by slow scan rates because it can take as long as several hours to obtain one image. In contrast, the force map shown in Fig. 6 c was obtained in <1 min. Studying elastic properties of surfaces by force mapping will allow for the unambiguous assignment of observed surface domains of samples such as phaseseparated polymer films. This ability could allow for the timeresolved monitoring of changes of nanoscale surface properties under various conditions such as temperature and pH. Because this technique is especially useful in fluid TMAFM due to the particularly pronounced distortion of cantilever trajectory resulting in a large number of harmonics needed for combfiltering, it has significant potential in biological applications. For example, changes in the modulus of bilayers, cells, and other biological surfaces under the influence of external factors (cholesterol content, structuremodifying drugs, etc.) could be easily studied.
Materials and Methods
Numerical Model of Fluid TMAFM.
Numerical simulations were performed with simulink and matlab (MathWorks, Natick, MA) by using a single degreeoffreedom model of TMAFM cantilever based on Eq. 1 . In TMAFM the tip–sample force varies according to the tip–sample distance D. Throughout the portion of the cycle where the tip does not contact the surface, the tip–sample force was modeled by using the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory, and in the portion where the tip is in contact with the surface, the tip/sample force was described by the Derjaguin–Muller–Toporov (DMT) theory for the interaction between a sphere and a surface (25) where σ_{s} and σ_{tip} are the surface charge densities of sample and tip, ε_{o} is the permittivity of vacuum, ε_{e} is the dielectric constant of the medium, λ_{D} is the Debye length (λ_{D} = 0.304/e _{c} for monovalent electrolytes with e _{c} being the electrolyte concentration), R _{tip} is the radius of the tip, H is the Hamaker constant (26, 27), a _{DMT} is the intermolecular distance parameter of DMT potential (25), and where E _{1}, ν_{1} and E _{2}, ν_{2} are Young's modulus and the Poisson coefficient of the tip and the sample, respectively. The surface charge densities for mica and silicon nitride used in the simulation were equal to −0.0025 and −0.032 C/m^{2}, respectively (26, 27). A feedback loop equipped with an integral gain that adjusts the sample height to the desired set–point ratio was implemented to simulate the process of imaging model steps, which were 5 nm tall, with Young's moduli ranging from 1 to 59 GPa. Young's modulus of the region surrounding the step was set to 60 GPa.
Preparation of Bilayer Patches.
Total brain lipid extract was purchased from Avanti Polar Lipids, dried under a stream of nitrogen, lyophilized, and resuspended in PBS (pH 7.3) at a concentration of 1 mg/ml. By using an acetone/dryice bath, bilayers and multilayer lipid sheets were formed by five sequential freeze–thaw cycles (28). The lipid suspensions then were sonicated for 15 min to promote vesicle formation. Then 40 μl of the suspended vesicle solution diluted five times was added directly to the AFM fluid cell by using the hanging drop method and placed on freshly cleaved mica, allowing the vesicles to flatten and fuse in situ.
AFM Imaging Conditions.
In situ AFM experiments were performed with a Nanoscope III MultiMode scanning probe microscope (Digital Instruments, Santa Barbara, CA) by using a tapping fluid cell equipped with an Oring and a Vshaped oxidesharpened silicon nitride cantilever with a nominal spring constant of 0.5 N/m. Images were acquired with a “vertical engage” Jscanner. Scan rates were set at 1–2 Hz with cantilever drive frequencies ranging from ≈8 to 10 kHz, and 5 × 1.25μm images were captured at 256 × 64pixel resolution. Cantilever deflection trajectories were simultaneously captured through an AFM signal access module (Digital Instruments) by using a CompuScope 14100 data acquisition card (Gage Applied Technologies, Lachine, QB, Canada) and customwritten software. Trajectories were captured at 5–10 MS/s and 14bit resolution.
Acknowledgments
We thank Prof. Guy C. Berry for suggestions in manuscript preparation. This work was supported by National Science Foundation Grants CTS0304568 and DMR9974457 and the Howard Hughes Medical Institute (fellowship for M.P.).
Footnotes
 *To whom correspondence should be addressed. Email: tomek{at}andrew.cmu.edu

Author contributions: T.K. designed research; J.L., B.C., and T.K. performed research; M.P. contributed new reagents/analytic tools; J.L. analyzed data; and J.L. and T.K. wrote the paper.

Conflict of interest statement: No conflicts declared.

This paper was submitted directly (Track II) to the PNAS office.
 Abbreviations:
 AFM,
 atomic force microscope/microscopy;
 TMAFM,
 tapping mode AFM;
 SPAM,
 scanning probe acceleration microscopy;
 S/N,
 signaltonoise
 © 2006 by The National Academy of Sciences of the USA
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