Reconstructing nonlinearities with intermodulation spectroscopy

Intermodal coupling spectroscopy of mechanical modes in microcantilevers

Atomic force microscopy (AFM) is highly regarded as a lens peering into the next discoveries of nanotechnology. Fundamental research in atomic interactions, molecular reactions, and biological cell behaviour are key focal points, demanding a continuous increase in resolution and sensitivity. While renowned fields such as optomechanics have marched towards outstanding signal-to-noise ratios, these improvements have yet to find a practical way to AFM. As a solution, we investigate here a mechanism in which individual mechanical eigenmodes of a microcantilever couple to one another, mimicking optomechanical techniques to reduce thermal noise. We have a look at the most commonly used modes in AFM, starting with the first two flexural modes of cantilevers and asses the impact of an amplified coupling between them. In the following, we expand our investigation to the sea of eigenmodes available in the same structure and find a maximum coupling of 9.38 × 10³ Hz/nm between two torsional modes. Through such findings we aim to expand the field of multifrequency AFM with innumerable possibilities leading to improved signal-to-noise ratios, all accessible with no additional hardware.

Square-Wave Electrogravimetry Combined with Voltammetry Reveals Reversible Submonolayer Adsorption of Redox-Active Ions

Square-wave voltammetry on electrolytes containing reversible redox pairs in solution was complemented by acoustic microgravimetry, where multifrequency lock-in amplification provides for a time resolution of 2.5 ms and a frequency resolution after accumulation of 12 mHz. The instrument operates similar to a quartz crystal microbalance with dissipation monitoring (QCM-D). The use of square-waves rather than linear ramps makes the analysis more transparent because it reduces the contribution of non-Faraday currents. Also, square-wave electrogravimetry determines the rates of mass transfer with much better sensitivity than its counterpart based on linear voltage ramps. The shifts of frequency and bandwidth are in agreement with the Sauerbrey prediction, meaning that the overtone-normalized frequency shifts, Δf/n, are similar on the different overtones and that the shifts in half bandwidth, ΔΓ, are smaller than the shifts in frequency. Small deviations from the Sauerbrey prediction presumably result from the softness of the adsorbed layer. Because the response time of the QCM signals is much longer than the RC time of double layer recharging as determined with electrochemical impedance spectroscopy (EIS), interpretation in terms of adsorption and desorption is more plausible than interpretation in terms of changed viscosity in the diffuse double layer. Ions of methyl viologen (MV) were found to adsorb to the electrode surface more strongly in the state with a single charge than in the fully oxidized state carrying two charges. The difference in apparent thickness between the oxidized and the reduced state was up to 2 nm, depending on concentration. The gravimetric results obtained on flavin adenine dinucleotide (FAD) depended on pH. At neutral pH, adsorption was largest close to the redox potential. Presumably, the adsorbed molecules are semiquinones, that is, are the intermediates of the underlying two-electron process.

Tuning of strong nonlinearity in radio-frequency superconducting-quantum-interference-device meta-atoms

Strong nonlinearity of a self-resonant radio-frequency (rf) superconducting-quantum-interference-device (SQUID) meta-atom is explored via intermodulation (IM) measurements. Previous work in zero dc magnetic flux showed a sharp onset of IM response as the frequency sweeps through the resonance. A second onset at higher frequency was also observed, creating a prominent gap in the IM response. By extending those measurements to nonzero dc flux, different dynamics are revealed, including dc flux tunability of the aforementioned gaps and enhanced IM response near geometric resonance of the rf SQUID. These features observed experimentally are understood and analyzed theoretically through a combination of a steady-state analytical modeling and a full numerical treatment of the rf SQUID dynamics. The latter in addition predicts the presence of chaos in narrow parameter regimes. The understanding of intermodulation in rf SQUID metamaterials is important for producing low-noise amplification of microwave signals and tunable filters.

Universal Theory of Dynamic Force Microscopy for Exact and Robust Force Reconstruction Using Multiharmonic Signal Analysis

Force reconstruction in dynamic force microscopy (DFM) is a nontrivial problem that requires the deconvolution of integrals. However, conventional reconstruction methods, which recover forces from single-frequency motion of the cantilever at its resonance, exhibit non-negligible error and reconstruction instability in the highly nonlinear force regime when the tip oscillates with its amplitude comparable to the decay length of the interaction. Here, we develop a theoretical platform of DFM based on multiharmonic signal analysis for exact and robust reconstruction of conservative and dissipative forces, valid for all oscillation amplitudes and entire tip-sample distances in both amplitude- and frequency-modulation atomic force microscopy. We achieve accuracy improvement by an order of magnitude for oscillation amplitudes comparable to or larger than the decay length, and by 2 orders of magnitude for smaller amplitudes at the force minimum, even in cases where conventional methods show poor accuracy (≳5%). Moreover, we obtain greater robustness with respect to the oscillation amplitude error, resulting in a fivefold increase in reconstruction precision. Our results demonstrate a fast and versatile reconstruction scheme for nanomechanical force characterization, with higher harmonics measured with sufficient signal-to-noise ratio, which provides unprecedented accuracy and stability beyond conventional methods.

Kinetics of viscoelasticity in the electric double layer following steps in the electrode potential studied by a fast electrochemical quartz crystal microbalance (EQCM)

A fast EQCM measures the kinetics of the viscosity changes inside the double layer following voltage jumps.

Nanomechanical mapping in air or vacuum using multi-harmonic signals in tapping mode atomic force microscopy

We present a method by which multi-harmonic signals acquired during a normal tapping mode (amplitude modulated) AFM scan of a sample in air or vacuum with standard microcantilevers can be used to map quantitatively the local mechanical properties of the sample such as elastic modulus, adhesion, and indentation. The approach is based on the observation that during the tapping mode operation in air or vacuum, the 0th and 2nd harmonic signals of microcantilever vibration are encountered under most operating conditions and can be mapped with sufficient signal to noise ratio. By measuring the amplitude and phase of the driven harmonic as well as the 0th and 2nd harmonic observables, we find analytical/semi-analytical formulas that relate these multi-harmonic observables to local mechanical properties for several classical tip-sample interaction models. Least squares estimation of the local mechanical properties is performed pixel by pixel. The method is validated through computations as well as experimental data acquired on a polymer blend made of Polystyrene and Polyolefin elastomer.

A Quartz Crystal Microbalance, Which Tracks Four Overtones in Parallel with a Time Resolution of 10 Milliseconds: Application to Inkjet Printing

A quartz crystal microbalance (QCM) is described, which simultaneously determines resonance frequency and bandwidth on four different overtones. The time resolution is 10 milliseconds. This fast, multi-overtone QCM is based on multi-frequency lockin amplification. Synchronous interrogation of overtones is needed, when the sample changes quickly and when information on the sample is to be extracted from the comparison between overtones. The application example is thermal inkjet-printing. At impact, the resonance frequencies change over a time shorter than 10 milliseconds. There is a further increase in the contact area, evidenced by an increasing common prefactor to the shifts in frequency, Δf, and half-bandwidth, ΔΓ. The ratio ΔΓ/(-Δf), which quantifies the energy dissipated per time and unit area, decreases with time. Often, there is a fast initial decrease, lasting for about 100 milliseconds, followed by a slower decrease, persisting over the entire drying time (a few seconds). Fitting the overtone dependence of Δf(n) and ΔΓ(n) with power laws, one finds power-law exponents of about 1/2, characteristic of semi-infinite Newtonian liquids. The power-law exponents corresponding to Δf(n) slightly increase with time. The decrease of ΔΓ/(-Δf) and the increase of the exponents are explained by evaporation and formation of a solid film at the resonator surface.

An ultrafast quartz crystal microbalance based on a frequency comb approach delivers sub-millisecond time resolution

Quartz crystal microbalance with dissipation monitoring (QCMD) is a simple and versatile sensing technique with applications in a wide variety of academic and industrial fields, most notably electrochemistry, biophysics, quality control, and environmental monitoring. QCMD is limited by a relatively poor time resolution, which is of the order of seconds with conventional instrument designs at the noise level usually required. In this work, we present a design of an ultrafast QCMD with submillisecond time resolution. It is based on a frequency comb approach applied to a high-fundamental-frequency (HFF) resonator through a multifrequency lock-in amplifier. The combination allows us to reach data acquisition rates >10 kHz. We illustrate the method using a toy model of a glass sphere dropped on the resonator surfaces, bare or coated with liposomes, in liquid. We discuss some interesting features of the results obtained with the dropped spheres, such as bending of the HFF resonators due to the impact, sphere bouncing (or the absence of it), and contact aging.

High-veracity functional imaging in scanning probe microscopy via Graph-Bootstrapping

The key objective of scanning probe microscopy (SPM) techniques is the optimal representation of the nanoscale surface structure and functionality inferred from the dynamics of the cantilever. This is particularly pertinent today, as the SPM community has seen a rapidly growing trend towards simultaneous capture of multiple imaging channels and complex modes of operation involving high-dimensional information-rich datasets, bringing forward the challenges of visualization and analysis, particularly for cases where the underlying dynamic model is poorly understood. To meet this challenge, we present a data-driven approach, Graph-Bootstrapping, based on low-dimensional manifold learning of the full SPM spectra and demonstrate its successes for high-veracity mechanical mapping on a mixed polymer thin film and resolving irregular hydration structure of calcite at atomic resolution. Using the proposed methodology, we can efficiently reveal and hierarchically represent salient material features with rich local details, further enabling denoising, classification, and high-resolution functional imaging.

A Novel Method to Reconstruct the Force Curve by Higher Harmonics of the First Two Flexural Modes in Frequency Modulation Atomic Force Microscope (FM-AFM)

Atomic force microscope (AFM) is an idealized tool to measure the physical and chemical properties of the sample surfaces by reconstructing the force curve, which is of great significance to materials science, biology, and medicine science. Frequency modulation atomic force microscope (FM-AFM) collects the frequency shift as feedback thus having high force sensitivity and it accomplishes a true noncontact mode, which means great potential in biological sample detection field. However, it is a challenge to establish the relationship between the cantilever properties observed in practice and the tip-sample interaction theoretically. Moreover, there is no existing method to reconstruct the force curve in FM-AFM combining the higher harmonics and the higher flexural modes. This paper proposes a novel method that a full force curve can be reconstructed by any order higher harmonics of the first two flexural modes under any vibration amplitude in FM-AFM. Moreover, in the small amplitude regime, short range forces are reconstructed more accurately by higher harmonics analysis compared with fundamental harmonics using the Sader-Jarvis formula.

Big, Deep, and Smart Data in Scanning Probe Microscopy

Scanning probe microscopy (SPM) techniques have opened the door to nanoscience and nanotechnology by enabling imaging and manipulation of the structure and functionality of matter at nanometer and atomic scales. Here, we analyze the scientific discovery process in SPM by following the information flow from the tip-surface junction, to knowledge adoption by the wider scientific community. We further discuss the challenges and opportunities offered by merging SPM with advanced data mining, visual analytics, and knowledge discovery technologies.

Dynamic force microscopy simulator (dForce): A tool for planning and understanding tapping and bimodal AFM experiments

We present a simulation environment, dForce, which can be used for a better understanding of dynamic force microscopy experiments. The simulator presents the cantilever-tip dynamics for two dynamic AFM methods, tapping mode AFM and bimodal AFM. It can be applied for a wide variety of experimental situations in air or liquid. The code provides all the variables and parameters relevant in those modes, for example, the instantaneous deflection and tip-surface force, velocity, virial, dissipated energy, sample deformation and peak force as a function of time or distance. The simulator includes a variety of interactions and contact mechanics models to describe AFM experiments including: van der Waals, Hertz, DMT, JKR, bottom effect cone correction, linear viscoelastic forces or the standard linear solid viscoelastic model. We have compared two numerical integration methods to select the one that offers optimal accuracy and speed. The graphical user interface has been designed to facilitate the navigation of non-experts in simulations. Finally, the accuracy of dForce has been tested against numerical simulations performed during the last 18 years.

Dynamic calibration of higher eigenmode parameters of a cantilever in atomic force microscopy by using tip-surface interactions

We present a theoretical framework for the dynamic calibration of the higher eigenmode parameters (stiffness and optical lever inverse responsivity) of a cantilever. The method is based on the tip-surface force reconstruction technique and does not require any prior knowledge of the eigenmode shape or the particular form of the tip-surface interaction. The calibration method proposed requires a single-point force measurement by using a multimodal drive and its accuracy is independent of the unknown physical amplitude of a higher eigenmode.

Unlocking higher harmonics in atomic force microscopy with gentle interactions

In dynamic atomic force microscopy, nanoscale properties are encoded in the higher harmonics. Nevertheless, when gentle interactions and minimal invasiveness are required, these harmonics are typically undetectable. Here, we propose to externally drive an arbitrary number of exact higher harmonics above the noise level. In this way, multiple contrast channels that are sensitive to compositional variations are made accessible. Numerical integration of the equation of motion shows that the external introduction of exact harmonic frequencies does not compromise the fundamental frequency. Thermal fluctuations are also considered within the detection bandwidth of interest and discussed in terms of higher-harmonic phase contrast in the presence and absence of an external excitation of higher harmonics. Higher harmonic phase shifts further provide the means to directly decouple the true topography from that induced by compositional heterogeneity.

Polynomial force approximations and multifrequency atomic force microscopy

We present polynomial force reconstruction from experimental intermodulation atomic force microscopy (ImAFM) data. We study the tip-surface force during a slow surface approach and compare the results with amplitude-dependence force spectroscopy (ADFS). Based on polynomial force reconstruction we generate high-resolution surface-property maps of polymer blend samples. The polynomial method is described as a special example of a more general approximative force reconstruction, where the aim is to determine model parameters that best approximate the measured force spectrum. This approximative approach is not limited to spectral data, and we demonstrate how it can be adapted to a force quadrature picture.

High-resolution nanomechanical analysis of suspended electrospun silk fibers with the torsional harmonic atomic force microscope

Atomic force microscopes have become indispensable tools for mechanical characterization of nanoscale and submicron structures. However, materials with complex geometries, such as electrospun fiber networks used for tissue scaffolds, still pose challenges due to the influence of tension and bending modulus on the response of the suspended structures. Here we report mechanical measurements on electrospun silk fibers with various treatments that allow discriminating among the different mechanisms that determine the mechanical behavior of these complex structures. In particular we were able to identify the role of tension and boundary conditions (pinned versus clamped) in determining the mechanical response of electrospun silk fibers. Our findings show that high-resolution mechanical imaging with torsional harmonic atomic force microscopy provides a reliable method to investigate the mechanics of materials with complex geometries.

Interpreting motion and force for narrow-band intermodulation atomic force microscopy

Intermodulation atomic force microscopy (ImAFM) is a mode of dynamic atomic force microscopy that probes the nonlinear tip-surface force by measurement of the mixing of multiple modes in a frequency comb. A high-quality factor cantilever resonance and a suitable drive comb will result in tip motion described by a narrow-band frequency comb. We show, by a separation of time scales, that such motion is equivalent to rapid oscillations at the cantilever resonance with a slow amplitude and phase or frequency modulation. With this time-domain perspective, we analyze single oscillation cycles in ImAFM to extract the Fourier components of the tip-surface force that are in-phase with the tip motion (F(I)) and quadrature to the motion (F(Q)). Traditionally, these force components have been considered as a function of the static-probe height only. Here we show that F(I) and F(Q) actually depend on both static-probe height and oscillation amplitude. We demonstrate on simulated data how to reconstruct the amplitude dependence of F(I) and F(Q) from a single ImAFM measurement. Furthermore, we introduce ImAFM approach measurements with which we reconstruct the full amplitude and probe-height dependence of the force components F(I) and F(Q), providing deeper insight into the tip-surface interaction. We demonstrate the capabilities of ImAFM approach measurements on a polystyrene polymer surface.

The role of nonlinear dynamics in quantitative atomic force microscopy

Various methods of force measurement with the atomic force microscope are compared for their ability to accurately determine the tip-surface force from analysis of the nonlinear cantilever motion. It is explained how intermodulation, or the frequency mixing of multiple drive tones by the nonlinear tip-surface force, can be used to concentrate the nonlinear motion in a narrow band of frequency near the cantilever's fundamental resonance, where accuracy and sensitivity of force measurement are greatest. Two different methods for reconstructing tip-surface forces from intermodulation spectra are explained. The reconstruction of both conservative and dissipative tip-surface interactions from intermodulation spectra are demonstrated on simulated data.

Repulsive bimodal atomic force microscopy on polymers

Bimodal atomic force microscopy can provide high-resolution images of polymers. In the bimodal operation mode, two eigenmodes of the cantilever are driven simultaneously. When examining polymers, an effective mechanical contact is often required between the tip and the sample to obtain compositional contrast, so particular emphasis was placed on the repulsive regime of dynamic force microscopy. We thus investigated bimodal imaging on a polystyrene-block-polybutadiene diblock copolymer surface and on polystyrene. The attractive operation regime was only stable when the amplitude of the second eigenmode was kept small compared to the amplitude of the fundamental mode. To clarify the influence of the higher eigenmode oscillation on the image quality, the amplitude ratio of both modes was systematically varied. Fourier analysis of the time series recorded during imaging showed frequency mixing. However, these spurious signals were at least two orders of magnitude smaller than the first two fundamental eigenmodes. Thus, repulsive bimodal imaging of polymer surfaces yields a good signal quality for amplitude ratios smaller than A(01)/A(02) = 10:1 without affecting the topography feedback.

Multifrequency imaging in the intermittent contact mode of atomic force microscopy: beyond phase imaging

The cantilever dynamics in single-frequency scanning probe microscopy (SPM) are undefined due to having only two output variables, which leads to poorly understood image contrast. To address this shortcoming, generalized phase imaging scanning probe microscopy (GP-SPM), based on broad band detection and multi-eigenmode operation, is developed and demonstrated on diamond nanoparticles with different functionalization layers. It is shown that rich information on tip-surface interactions can be acquired by separating the response amplitude, instant resonance frequency, and quality factor. The obtained data allow high-resolution imaging even in the ambient environment. By tuning the strength of tip-surface interaction, different surface functionalizations can be discerned.

The emergence of multifrequency force microscopy

In atomic force microscopy a cantilever with a sharp tip attached to it is scanned over the surface of a sample, and information about the surface is extracted by measuring how the deflection of the cantilever - which is caused by interactions between the tip and the surface - varies with position. In the most common form of atomic force microscopy, dynamic force microscopy, the cantilever is made to vibrate at a specific frequency, and the deflection of the tip is measured at this frequency. But the motion of the cantilever is highly nonlinear, and in conventional dynamic force microscopy, information about the sample that is encoded in the deflection at frequencies other than the excitation frequency is irreversibly lost. Multifrequency force microscopy involves the excitation and/or detection of the deflection at two or more frequencies, and it has the potential to overcome limitations in the spatial resolution and acquisition times of conventional force microscopes. Here we review the development of five different modes of multifrequency force microscopy and examine its application in studies of proteins, the imaging of vibrating nanostructures, measurements of ion diffusion and subsurface imaging in cells.

Theoretical study of the frequency shift in bimodal FM-AFM by fractional calculus

Bimodal atomic force microscopy is a force-microscopy method that requires the simultaneous excitation of two eigenmodes of the cantilever. This method enables the simultaneous recording of several material properties and, at the same time, it also increases the sensitivity of the microscope. Here we apply fractional calculus to express the frequency shift of the second eigenmode in terms of the fractional derivative of the interaction force. We show that this approximation is valid for situations in which the amplitude of the first mode is larger than the length of scale of the force, corresponding to the most common experimental case. We also show that this approximation is valid for very different types of tip-surface forces such as the Lennard-Jones and Derjaguin-Muller-Toporov forces.

Dual frequency open-loop electric potential microscopy for local potential measurements in electrolyte solution with high ionic strength

Recent development of open-loop electric potential microscopy (OL-EPM) has enabled to measure local potential distribution at a solid/liquid interface. However, the operating environment of OL-EPM has been limited to a weak electrolyte solution (<1 mM). This has significantly limited its application range in biology and chemistry. To overcome this limitation, we have developed dual frequency (DF) mode OL-EPM. In the method, an ac bias voltage consisting of two frequency components at f(1) and f(2) is applied between a tip and sample. The local potential is calculated from the amplitudes of the f(1) and |f(1) - f(2)| components of the electrostatic force. In contrast to the conventional single frequency (SF) mode OL-EPM, the detection of the 2f(1) component is not required in DF mode. Thus, the maximum bias modulation frequency in DF mode is twice as high as that in SF mode. The high bias modulation frequency used in DF mode prevents the generation of electrochemical reactions and redistribution of ions and water, which enables to operate OL-EPM even in a strong electrolyte solution. In this study, we have performed potential measurements of nanoparticles on a graphite surface in 1 and 10 mM NaCl solution. The results demonstrate that DF mode OL-EPM allows measurements of local potential distribution in 10 mM electrolyte solution.

Noninvasive protein structural flexibility mapping by bimodal dynamic force microscopy

Mapping of the protein structural flexibility with sub-2-nm spatial resolution in liquid is achieved by combining bimodal excitation and frequency modulation force microscopy. The excitation of two cantilever eigenmodes in dynamic force microscopy enables the separation between topography and flexibility mapping. We have measured variations of the elastic modulus in a single antibody pentamer from 8 to 18 MPa when the probe is moved from the end of the protein arm to the central protrusion. Bimodal dynamic force microscopy enables us to perform the measurements under very small repulsive loads (30-40 pN).

Nanomechanical coupling enables detection and imaging of 5 nm superparamagnetic particles in liquid

We demonstrate that a force microscope operated in a bimodal mode enables the imaging and detection of superparamagnetic particles down to 5 nm. The bimodal method exploits the nanomechanical coupling of the excited modes to enhance the sensitivity of the higher mode to detect changes in material properties. The coupling requires the presence of nonlinear forces. Remarkably, bimodal operation enables us to identify changes of slowly varying forces (quasi-linear) in the presence of a stronger nonlinear force. Thus, unambiguous identification of single apoferritin (non-magnetic) and ferritin (magnetic) molecules in air and liquid is accomplished.

Note: The intermodulation lockin analyzer

Nonlinear systems can be probed by driving them with two or more pure tones while measuring the intermodulation products of the drive tones in the response. We describe a digital lockin analyzer which is designed explicitly for this purpose. The analyzer is implemented on a field-programmable gate array, providing speed in analysis, real-time feedback, and stability in operation. The use of the analyzer is demonstrated for intermodulation atomic force microscopy. A generalization of the intermodulation spectral technique to arbitrary drive waveforms is discussed.

Resolution theory, and static and frequency-dependent cross-talk in piezoresponse force microscopy

Probing the functionality of materials locally by means of scanning probe microscopy (SPM) requires a reliable framework for identifying the target signal and separating it from the effects of surface morphology and instrument non-idealities, e.g. instrumental and topographical cross-talk. Here we develop a linear resolution theory framework in order to describe the cross-talk effects, and apply it for elucidation of frequency-dependent cross-talk mechanisms in piezoresponse force microscopy. The use of a band excitation method allows electromechanical/electrical and mechanical/topographic signals to be unambiguously separated. The applicability of a functional fit approach and multivariate statistical analysis methods for identification of data in band excitation SPM is explored.