Brian A. Todd

A method to improve the quantitative analysis of SFM images at the nanoscale

Abstract Quantitative analysis of scanned force microscope (SFM) images at the nanoscale requires removing the contributions of tip size and shape from the images. Mathematical morphology provides tools for doing this but determination of the probe tip geometry is required first. Blind reconstruction is among the most popular methods for determining tip geometry. We show that, at the nanoscale, spatially anisotropic noise generally present in SFM data results in artificially asymmetric tip geometries as determined by blind reconstruction. We present an easily implemented improvement to the publicly available computer code for blind reconstruction that alleviates this problem. Experimental evidence is presented to show that the method results in tip geometries that are consistent with expected shapes based on self-imaging using very sharp surface features.

Squeezing out hidden force information from scanning force microscopes

A method to measure force-separation curves with a scanning force microscope is presented. Forces within the “snap to contact” are obtained by high-speed (MHz) measurement of cantilever deflection signals analyzed using the generalized beam theory. Numerical simulation is used to demonstrate the effectiveness of the method. Experimental results show that the method yields complete continuous force-separation curves with flimsy cantilevers in fluids allowing for sensitive force measurements in nonvacuum environments.

Connecting Nanoscale Images of Proteins with Their Genetic Sequences

We present a technique for reconstructing biomolecular structures from scanning force microscope data. The technique works by iteratively refining model molecules by comparison of simulated and experimental images. It can remove instrument artifacts to yield accurate dimensional measurements from tip-broadened data. The result of the reconstruction is a model that can be chosen to include the physically significant parameters for the system at hand. We demonstrate this by reconstructing scanning force microscope images of the cartilage proteoglycan aggrecan. By explicitly including the protein backbone in the model, we are able to associate measured three-dimensional structures with sites in the protein primary structure. The distribution of aggrecan core protein lengths that we measure suggests that 48% of aggrecan molecules found in vivo have been partially catabolized at either the E(1480)-(1481)G or E(1667)-(1668)G aggrecanase cleavage site.

Improved analysis of the time domain response of scanning force microscope cantilevers

The snap-to-contact instability encountered in scanning force microscopy-force spectroscopy (SFMFS) limits the range of forces measurable in SFM force–distance experiments. We have generalized the flexural beam theory for SFM cantilevers to include tip interactions that are present in the snap-to-contact region. We compare solutions for the beam theory with the simple harmonic oscillator (SHO) that is often used to approximate SFM cantilevers. The limitations of the SHO model are encountered when large force gradients are present. This causes the beam shape to change leading to error when the SHO is used to reconstruct force curves collected in the snap-to-contact region. We quantify the error introduced into a force–separation curve reconstructed using the SHO approximation by numerical simulation. The force–separation curve reconstructed by the SHO was significantly inaccurate and had distorted separation dependence. This makes physical interpretation of force curves reconstructed using the SHO approxim...

Inverse problem of scanning force microscope force measurements

The Hooke’s Law model, traditionally used to determine forces from the deflection of scanning force microscope (SFM) cantilevers, restricts the bandwidth to well below the cantilever resonant frequency. The limitation imposed on measurements by Hooke’s Law can be overcome by modeling the multimodal and dynamical response of the cantilever (i.e., accounting for the viscous and inertial response) using a beam model. However, when calculating the forces on the cantilever from deflection measurements, this “inverse problem” is ill posed (i.e., it amplifies noise in the measurement, so that simple inversion produces useless solutions). Regularization techniques provide mathematical solutions to this ill-posed problem but introduce nontrivial mechanisms by which inputs to the system are propagated to outputs. In this article, we investigate the propagation of errors in the inverse problem of SFM force measurements. We first develop a noise model and validate it using experimental measurements. This is then applied to simulate a force–distance experiment with a relevant amount of noise. This provides a cantilever deflection signal based on known forces with which to evaluate the accuracy and precision of a force reconstruction algorithm. We show that Tikhonov regularized solutions obtained with an L-curve analysis reconstruct forces with an accuracy of better than 90% on average but with poor precision, yielding a signal-to-noise ratio of ∼2 for a 0.02 N/m cantilever. Ensemble averaging is recommended to improve the precision. This allows both accurate and precise force curves to be reconstructed with a high bandwidth.