Prototype cantilevers for quantitative lateral force microscopy

Prototype cantilevers are presented that enable quantitative surface force measurements using contact-mode atomic force microscopy (AFM). The "hammerhead" cantilevers facilitate precise optical lever system calibrations for cantilever flexure and torsion, enabling quantifiable adhesion measurements and friction measurements by lateral force microscopy (LFM). Critically, a single hammerhead cantilever of known flexural stiffness and probe length dimension can be used to perform both a system calibration as well as surface force measurements in situ, which greatly increases force measurement precision and accuracy. During LFM calibration mode, a hammerhead cantilever allows an optical lever "torque sensitivity" to be generated for the quantification of LFM friction forces. Precise calibrations were performed on two different AFM instruments, in which torque sensitivity values were specified with sub-percent relative uncertainty. To examine the potential for accurate lateral force measurements using the prototype cantilevers, finite element analysis predicted measurement errors of a few percent or less, which could be reduced via refinement of calibration methodology or cantilever design. The cantilevers are compatible with commercial AFM instrumentation and can be used for other AFM techniques such as contact imaging and dynamic mode measurements.     

Influence of Poisson's ratio variation on lateral spring constant of atomic force microscopy cantilevers

Atomic force microscopy (AFM) can be used to measure the surface morphologies and the mechanical properties of nanostructures. The force acting on the AFM cantilever can be obtained by multiplying the spring constant of AFM cantilever and the corresponding deformation. To improve the accuracy of force experiments, the spring constant of AFM cantilever must be calibrated carefully. Many methods, such as theoretical equations, the finite element method, and the use of reference cantilever, were reported to obtain the spring constant of AFM cantilevers. For the cantilever made of single crystal, the Poisson's ratio varies with different cantilever-crystal angles. In this paper, the influences of Poisson's ratio variation on the lateral spring constant and axial spring constant of rectangular and V-shaped AFM cantilevers, with different tilt angles and normal forces, were investigated by the finite element analysis. When the cantilever's tilt angle is 20 degrees and the Poisson's ratio varies from 0.02 to 0.4, the finite element results show that the lateral spring constants decrease 11.75% for the rectangular cantilever with 1muN landing force and decrease 18.60% for the V-shaped cantilever with 50nN landing force, respectively. The influence of Poisson's ratio variation on axial spring constant is less than 3% for both rectangular and V-shaped cantilevers. As the tilt angle increases, the axial spring constants for rectangular and V-shaped cantilevers decrease substantially. The results obtained can be used to improve the accuracy of the lateral force measurement when using atomic force microscopy.     

Finite-element vibration analysis of tapping-mode atomic force microscopy in liquid

Investigation of morphology and mechanical properties of biological specimens using atomic force microscopy (AFM) often requires its operation in liquid environment. Due to the hydrodynamic force, the vibration of AFM cantilevers in liquid shows dramatically different dynamic characteristics from that in air. A good understanding of the dynamics of AFM cantilevers vibrating in liquid is needed for the interpretation of scanning images, selection of AFM operating conditions, and evaluation of sample's mechanical properties. In this study, a finite element (FE) model is used for frequency and transient response analysis of AFM cantilevers in tapping mode (TM) operated in air or liquid. Hydrodynamic force exerted by the fluid on AFM cantilevers is approximated by additional mass and hydrodynamic damping. The additional mass and hydrodynamic damping matrices corresponding to beam elements are derived. With this model, numerical simulations are performed for an AFM cantilever to obtain the frequency and transient responses of the cantilever in air and liquid. The comparison between our simulated results and the experimentally obtained ones shows good agreement. Based on the simulations, different characteristics of cantilever dynamics in air and liquid are discussed.     

An interferometric platform for studying AFM probe deflection

This paper describes an interferometric platform for measuring the full-field deflection of atomic force microscope (AFM) probes and generic cantilevers during quasi-static loading. The platform consists of a scanning white light interferometer (SWLI), holders for the cantilevers, a translation stage, a rotation (tip-tilt) stage, and an adapter plate to connect these items to the SWLI table. Visualization of cantilever bending behavior is demonstrated for snap-in against a rigid surface, cantilever-on-cantilever tests, and a damaged AFM probe. A new approach to normal force calculation using a polynomial fit to the cantilever deflection profile is also presented and verified experimentally. The method requires only the coefficient for the third order (cubic) term from the fit to the deflection profile, the elastic modulus, and the area moment of inertia for the cantilever under test.     

Improved parallel scan method for nanofriction force measurement with atomic force microscopy

Based on Ruan and Bhushan's study [J. Ruan and B. Bhushan, J. Tribol. 116, 378 (1994)], an improved method for quantitative nano/microfriction force measurements with the atomic force microscope (AFM) is presented. The related theoretical derivation is given in detail. The coefficient of friction can be calculated by scanning in the direction parallel to the long axis of the AFM cantilever. Then conversion factor, which can convert the lateral deflection response of the photodetector into corresponding friction force, is identified with the Meyer and Amer method [G. Meyer and N. M. Ame, Appl. Phys. Lett. 57, 2089 (1990)]. Like Ruan and Bhushan method, the advantage of this approach is that the coefficient of friction can be obtained with the plan-view geometry of AFM cantilevers and some common uncertainties, such as thickness, coating, and material properties, are not necessary. The result of the experiments performed utilizing rectangular cantilevers of different lengths shows that this improved method produces an accurate agreement for cantilevers of different lengths, thus the method can be used to measure nano/microfriction force.     

Scanning electron microscopy studies of protein-functionalized atomic force microscopy cantilever tips

Protein-functionalized atomic force microscopy (AFM) tips have been used to investigate the interaction of individual ligand-receptor complexes. Herein we present results from scanning electron microscopy (SEM) studies of protein-functionalized AFM cantilever tips. The goals of this study were (1) to examine the surface morphology of protein-coated AFM tips and (2) to determine the stability of the coated tips. Based on SEM images, we found that bovine serum albumin (BSA) in solution spontaneously adsorbed onto the surface of silicon nitride cantilevers, forming a uniform protein layer over the surface. Additional protein layers deposited over the initial BSA-coated surface did not significantly alter the surface morphology. However, we found that avidin-functionalized tips were contaminated with debris after a series of force measurements with biotinylated agarose beads. The bound debris presumably originated from the transfer of material from the agarose bead. This observation is consistent with the observed deterioration of functional activity as measured in ligand-receptor binding force experiments.     

Improved lateral force calibration based on the angle conversion factor in atomic force microscopy

A novel calibration method is proposed for determining lateral forces in atomic force microscopy (AFM), by introducing an angle conversion factor, which is defined as the ratio of the twist angle of a cantilever to the corresponding lateral signal. This factor greatly simplifies the calibration procedures. Once the angle conversion factor is determined in AFM, the lateral force calibration factors of any rectangular cantilever can be obtained by simple computation without further experiments. To determine the angle conversion factor, this study focuses on the determination of the twist angle of a cantilever during lateral force calibration in AFM. Since the twist angle of a cantilever cannot be directly measured in AFM, the angles are obtained by means of the moment balance equations between a rectangular AFM cantilever and a simple commercially available step grating. To eliminate the effect of the adhesive force, the gradients of the lateral signals and the twist angles as a function of normal force are used in calculating the angle conversion factor. To verify reliability and reproducibility of the method, two step gratings with different heights and two different rectangular cantilevers were used in lateral force calibration in AFM. The results showed good agreement, to within 10%. This method was validated by comparing the coefficient of friction of mica so determined with values in the literature.     

High-resolution constant-height imaging with apertured silicon cantilever probes

We present high-resolution aperture probes based on non-contact silicon atomic force microscopy (AFM) cantilevers for simultaneous AFM and near-infrared scanning near-field optical microscopy (SNOM). For use in near-field optical microscopy, conventional AFM cantilevers are modified by covering their tip side with an opaque aluminium layer. To fabricate an aperture, this metal layer is opened at the end of the polyhedral probe using focused ion beams (FIB). Here we show that apertures of less than 50 nm can be obtained using this technique, which actually yield a resolution of about 50 nm, corresponding to λ/20 at the wavelength used. To exclude artefacts induced by distance control, we work in constant-height mode. Our attention is particularly focused on the distance dependence of resolution and to the influence of slight cantilever bending on the optical images when scanning at such low scan heights, where first small attractive forces exerted on the cantilever become detectable.     

Dynamic spring constants for higher flexural modes of cantilever plates with applications to atomic force microscopy

In atomic force microscopy (AFM) a sharp tip fixed close to the free end of a cantilever beam interacts with a surface. The interaction can be described by a point-mass model of an equivalent oscillator with a single spring located at the position of the tip. However, other spring constants have to be used to describe the oscillation behavior correctly if forces are acting on the cantilever over an extended lateral range. A point-mass model is then no longer valid. In the present study we derive expressions for the spring constants of cantilevers that can interact with any part of their plan view area along the beam and for all flexural modes. The equations describe the oscillation behavior in the corresponding mass model and are based on the eigenfrequencies and modal shapes of the free cantilever. The results are of high practical relevance, for example if an AFM is operated in a higher flexural mode, if the tip is not located at the free end of the cantilever beam, or if the external conservative forces affecting cantilever movement are not restricted to a single point. The limitations of the approach are discussed.     

Precise atomic force microscope cantilever spring constant calibration using a reference cantilever array

A method for calibrating the stiffness of atomic force microscope (AFM) cantilevers is demonstrated using an array of uniform microfabricated reference cantilevers. A series of force-displacement curves was obtained using a commercial AFM test cantilever on the reference cantilever array, and the data were analyzed using an implied Euler-Bernoulli model to extract the test cantilever spring constant from linear regression fitting. The method offers a factor of 5 improvement over the precision of the usual reference cantilever calibration method and, when combined with the Systeme International traceability potential of the cantilever array, can provide very accurate spring constant calibrations.     

Comparative study of the conditions required to image live human epithelial and fibroblast cells using atomic force microscopy

Successful imaging of living human cells using atomic force microscopy (AFM) is influenced by many variables including cell culture conditions, cell morphology, surface topography, scan parameters, and cantilever choice. In this study, these variables were investigated while imaging two morphologically distinct human cell lines, namely LL24 (fibroblasts) and NCI H727 (epithelial) cells. The cell types used in this study were found to require different parameter settings to produce images showing the greatest detail. In contact mode, optimal loading forces ranged between 2-2.8 x 10(-9) and 0.1-0.7 x 10(-9) (N) for LL24 and NCI H727 cells respectively. In tapping (AC) mode, images of LL24 cells were obtained using cantilevers with a spring constant of at least 0.32 N/m, while NCI H727 cells required a greater spring constant of at least 0.58 N/m. To obtain tapping mode images, cantilevers needed to be tuned to resonate at higher frequencies than their resonance frequencies to obtain images. For NCI H727 cells, contact mode imaging produced the clearest images. For LL24 cells, contact and tapping mode AFM produced images of comparable quality. Overall, this study shows that cells with different morphologies and surface topography require different scanning approaches and optimal conditions must be determined empirically to achieve images of high quality.     

Quantitative comparison of two independent lateral force calibration techniques for the atomic force microscope

Two independent lateral-force calibration methods for the atomic force microscope (AFM)--the hammerhead (HH) technique and the diamagnetic lateral force calibrator (D-LFC)--are systematically compared and found to agree to within 5?% or less, but with precision limited to about 15?%, using four different tee-shaped HH reference probes. The limitations of each method, both of which offer independent yet feasible paths toward traceable accuracy, are discussed and investigated. We find that stiff cantilevers may produce inconsistent D-LFC values through the application of excessively high normal loads. In addition, D-LFC results vary when the method is implemented using different modes of AFM feedback control, constant height and constant force modes, where the latter is more consistent with the HH method and closer to typical experimental conditions. Specifically, for the D-LFC apparatus used here, calibration in constant height mode introduced errors up to 14?%. In constant force mode using a relatively stiff cantilever, we observed an ≈?4?% systematic error per μN of applied load for loads ≤?1 μN. The issue of excessive load typically emerges for cantilevers whose flexural spring constant is large compared with the normal spring constant of the D-LFC setup (such that relatively small cantilever flexural displacements produce relatively large loads). Overall, the HH method carries a larger uncertainty, which is dominated by uncertainty in measurement of the flexural spring constant of the HH cantilever as well as in the effective length dimension of the cantilever probe. The D-LFC method relies on fewer parameters and thus has fewer uncertainties associated with it. We thus show that it is the preferred method of the two, as long as care is taken to perform the calibration in constant force mode with low applied loads.     

Analysis of the effect of mechanical properties of liquid and geometrical parameters of cantilever on the frequency response function of AFM

The dynamic behavior of atomic force microscopy (AFM) cantilevers in liquid is completely different from its behavior in air due to the applied hydrodynamic force. Exciting cantilever with frequencies close to resonance frequency and primary position alignment are two critical issues that should be considered in deriving frequency response function (FRF). In this paper, the hydrodynamic force has been modeled with string of spheres and the effect of the damping and the added mass on the model has been analyzed. Afterward, this force is applied to the dynamic equation so that the dynamic behavior of AFM cantilevers is studied in liquids by analyzing the effect of some important parameters such as added mass, internal, and fluid damping. By simulations of the dynamic equations for a silicon cantilever, FRF is determined in both air and liquid. In addition, the effects of two significant parameters of liquid mechanical properties (liquid viscosity and density) and geometrical parameters of cantilever on FRF are studied. The results for string of spheres model are compared with the other hydrodynamic model and the experimental data. When length/width ratio decreases, it is found that string of spheres model has a better agreement than the other hydrodynamic model with experimental data.     

The effect of off-end tip distance on the nanomanipulation based on rectangular and V-shape cantilevered AFMs

The AFM system, which is used as a nanomanipulator, includes a probe consistent of a cantilever and a tapered tip. In cantilevers, the tip can be located in different distances from the cantilever free end. This causes to change in stiffness of the cantilever and therefore changing in pushing force of the nanomanipulation. In this paper, the effect of the tip distance on the cantilever stiffness is studied using the equations of Hazel, and Neumeister and Ducker (ND), and a new equation to correct the torsional stiffness of V-shaped cantilevers (VSC) is proposed, which is based on the ND equation. Then, the effect of distance on pushing force of AFM-based nanomanipulations with rectangular cantilevered (RC) and VSC AFMs is simulated. The obtained results using proposed equation show that increasing of distance causes to non-linear increment of torsional stiffness of VSC. Error of the proposed equation is achieved less than 3% in comparison with result of torsional stiffness equation of ND. Moreover, it is observed that the torsional stiffness of VSC predicted by Hazel’s equation is considerably inaccurate. In nanomanipulation studies, the necessary pushing forces of nanoparticle motion are increased by increment of distance, for both types of cantilevers (RC and VSC). Moreover, critical time for RC AFM increases, but in the case of VSC AFM, the critical time decreases at first, then it is almost constant at a limited range of d, and finally it starts to increase by increasing the distance.     

Computational models of a nano probe tip for static behaviors

It is difficult to predict the measurement bias arising from the compliance of the atomic force microscope (AFM) probe. The issue becomes particularly important in this situation where nanometer uncertainties are sought for measurements with dimensional probes composed of flexible carbon nanotubes mounted on AFM cantilevers. We have developed a finite element model for simulating the mechanical behavior of AFM cantilevers with carbon nanotubes attached. Spring constants of both the nanotube and cantilever in two directions are calculated using the finite element method with known Young's moduli of both silicon and multiwall nanotube as input data. Compliance of the nanotube-attached AFM probe tip may be calculated from the set of spring constants. This paper presents static models that together provide a basis to estimate uncertainties in linewidth measurement using nanotubes. In particular, the interaction between a multiwall nanotube tip and a silicon sample is modeled using the Lennard-Jones theory. Snap-in and snap-out of the probe tip in a scanning mode are calculated by integrating the compliance of the probe and the sample-tip interacting force model. Cantilever and probe tip deflections and points of contact are derived for both horizontal scanning of a plateau and vertically scanning of a wall. The finite element method and the Lennard-Jones model provide a means to analyze the interaction of the probe and sample and measurement uncertainty, including actual deflection and the gap between the probe tip and the measured sample surface.     

Interlaboratory round robin on cantilever calibration for AFM force spectroscopy

Single-molecule force spectroscopy studies performed by Atomic Force Microscopes (AFMs) strongly rely on accurately determined cantilever spring constants. Hence, to calibrate cantilevers, a reliable calibration protocol is essential. Although the thermal noise method and the direct Sader method are frequently used for cantilever calibration, there is no consensus on the optimal calibration of soft and V-shaped cantilevers, especially those used in force spectroscopy. Therefore, in this study we aimed at establishing a commonly accepted approach to accurately calibrate compliant and V-shaped cantilevers. In a round robin experiment involving eight different laboratories we compared the thermal noise and the Sader method on ten commercial and custom-built AFMs. We found that spring constants of both rectangular and V-shaped cantilevers can accurately be determined with both methods, although the Sader method proved to be superior. Furthermore, we observed that simultaneous application of both methods on an AFM proved an accurate consistency check of the instrument and thus provides optimal and highly reproducible calibration. To illustrate the importance of optimal calibration, we show that for biological force spectroscopy studies, an erroneously calibrated cantilever can significantly affect the derived (bio)physical parameters. Taken together, our findings demonstrated that with the pre-established protocol described reliable spring constants can be obtained for different types of cantilevers.     

Developments for inverted atomic force microscopy

Atomic force microscopy (AFM) has been used to study a wide range of systems. Chemically and biologically modified probes have extended AFM by coupling chemical and biological information with the physical measurements. In an effort to further expand the capabilities of modified AFM probes, previous studies investigated the use of an inverted AFM design (i-AFM), wherein a microfabricated tip array is used to image a cantilever-supported sample. This report details developments in cantilever and tip array fabrication which are aimed at improving the applicability and performance of this i-AFM design. Using an epoxy-based procedure, commercial cantilevers were modified with a series of standard substrates, including template-stripped gold, highly oriented pyrolytic graphite, and mica. The samples on these cantilevers were imaged with i-AFM, and lateral force images are obtained. This paper demonstrates the first use of i-AFM for measuring friction.     

Resonance-mode effect on microcantilever mass-sensing performance in air

This research investigates the air drag damping effect of the micromachined cantilevers in different resonance modes on the quality factor, which are operated in ambient air. Based on a simplified dish-string model for air drag force acting on the resonant cantilever, the air drag damping properties of the cantilevers vibrating in different modes are analyzed with theoretic vibration mechanics, which is complemented and further confirmed with finite-element simulation. Four kinds of integrated cantilevers, which resonate in the first flexural mode, the second flexural mode, the first torsional mode, and the second torsional mode, respectively, are designed and fabricated by using micromachining techniques. Finally, biomolecular sensing experiments are carried out to verify the theoretical results obtained before. From both the modeling and experimental results, it can be seen that damping characteristics of the torsional cantilever resonators are generally better than that of the flexural ones, and quality factor of the cantilever resonator in a higher-frequency mode is always superior to that in a lower-frequency one. Among the four kinds of microcantilever resonators operated in our experiments, the one operated in the second flexural modes exhibits the highest Q factor and the best biomass sensing performance.     

Study of the tip mass and interaction force effects on the frequency response and mode shapes of the AFM cantilever

The vibrational characteristics of an atomic force microscope (AFM) cantilever beam play a key role in dynamic mode of the atomic force microscope. As the oscillating AFM cantilever tip approaches the sample, the tip–sample interaction force influences the cantilever dynamics. In this paper, we present a detailed theoretical analysis of the frequency response and mode shape behavior of a cantilever beam in the dynamic mode subject to changes in the tip mass and the interaction regime between the AFM cantilever system and the sample. We consider a distributed parameter model for AFM and use Euler–Bernoulli method to derive an expression for AFM characteristics equation contains tip mass and interaction force terms. We study the frequency response of AFM cantilever under variations of interaction force between AFM tip and sample. Also, we investigate the effect of tip mass on the frequency response and also the quality factor and spring constant of each eigenmodes of AFM micro-cantilever. In addition, the mode shape analysis of AFM cantilever under variations of tip mass and interaction force is investigated. This will incorporate the presentation of explicit analytical expressions and numerical analysis. The results show that by considering the tip mass, the resonance frequencies of the cantilever are decreased. Also, the tip mass has a significant effect on the mode shape of the higher eigenmodes of the AFM cantilever. Moreover, tip mass affects the quality factor and spring constant of each modes.     

Probing material properties of polymeric surface layers with tapping mode AFM: Which cantilever spring constant,tapping amplitude and amplitude set point gives good image contrast and minimal surface damage?

A phase shift between the oscillatory motion and drive motion of an AFM-cantilever used for tapping mode AFM imaging can be related to adhesive and elastic properties of surface layers. In this study it was studied how optimal contrast between hard and soft surface layers can be achieved while minimizing the surface damage. This was investigated by performing classical force-distance measurements while driving the cantilever as in tapping mode imaging. The amplitude and phase response as a function of the average tip-surface separation was recorded. Five different cantilevers with a wide range of spring constants and four different tapping amplitudes were investigated and compared. Based on these experiments it is concluded that too stiff cantilever, high free tapping amplitude and low amplitude set point value often lead to surface damage, while too low spring constant and low free tapping amplitude result in poor phase image contrast. Intermediate values where little surface damage and significant image contrast are obtained were identified. In all cases it was observed that the best image contrast was obtained when the amplitude set point was chosen such that the amplitude during imaging was reduced to approximately 50% of the free amplitude.