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1.  Multiple regimes of operation in bimodal AFM: understanding the energy of cantilever eigenmodes 
Summary
One of the key goals in atomic force microscopy (AFM) imaging is to enhance material property contrast with high resolution. Bimodal AFM, where two eigenmodes are simultaneously excited, confers significant advantages over conventional single-frequency tapping mode AFM due to its ability to provide contrast between regions with different material properties under gentle imaging conditions. Bimodal AFM traditionally uses the first two eigenmodes of the AFM cantilever. In this work, the authors explore the use of higher eigenmodes in bimodal AFM (e.g., exciting the first and fourth eigenmodes). It is found that such operation leads to interesting contrast reversals compared to traditional bimodal AFM. A series of experiments and numerical simulations shows that the primary cause of the contrast reversals is not the choice of eigenmode itself (e.g., second versus fourth), but rather the relative kinetic energy between the higher eigenmode and the first eigenmode. This leads to the identification of three distinct imaging regimes in bimodal AFM. This result, which is applicable even to traditional bimodal AFM, should allow researchers to choose cantilever and operating parameters in a more rational manner in order to optimize resolution and contrast during nanoscale imaging of materials.
doi:10.3762/bjnano.4.45
PMCID: PMC3701429  PMID: 23844344
atomic force microscopy; bimodal AFM; cantilever eigenmodes; polymer characterization
2.  Trade-offs in sensitivity and sampling depth in bimodal atomic force microscopy and comparison to the trimodal case 
Summary
This paper presents experiments on Nafion® proton exchange membranes and numerical simulations illustrating the trade-offs between the optimization of compositional contrast and the modulation of tip indentation depth in bimodal atomic force microscopy (AFM). We focus on the original bimodal AFM method, which uses amplitude modulation to acquire the topography through the first cantilever eigenmode, and drives a higher eigenmode in open-loop to perform compositional mapping. This method is attractive due to its relative simplicity, robustness and commercial availability. We show that this technique offers the capability to modulate tip indentation depth, in addition to providing sample topography and material property contrast, although there are important competing effects between the optimization of sensitivity and the control of indentation depth, both of which strongly influence the contrast quality. Furthermore, we demonstrate that the two eigenmodes can be highly coupled in practice, especially when highly repulsive imaging conditions are used. Finally, we also offer a comparison with a previously reported trimodal AFM method, where the above competing effects are minimized.
doi:10.3762/bjnano.5.125
PMCID: PMC4142983  PMID: 25161847
amplitude modulation; bimodal; multifrequency atomic force microscopy; indentation depth modulation; Nafion; open loop; proton exchange membranes; trimodal
3.  High-bandwidth multimode self-sensing in bimodal atomic force microscopy 
Summary
Using standard microelectromechanical system (MEMS) processes to coat a microcantilever with a piezoelectric layer results in a versatile transducer with inherent self-sensing capabilities. For applications in multifrequency atomic force microscopy (MF-AFM), we illustrate that a single piezoelectric layer can be simultaneously used for multimode excitation and detection of the cantilever deflection. This is achieved by a charge sensor with a bandwidth of 10 MHz and dual feedthrough cancellation to recover the resonant modes that are heavily buried in feedthrough originating from the piezoelectric capacitance. The setup enables the omission of the commonly used piezoelectric stack actuator and optical beam deflection sensor, alleviating limitations due to distorted frequency responses and instrumentation cost, respectively. The proposed method benefits from a more than two orders of magnitude increase in deflection to strain sensitivity on the fifth eigenmode leading to a remarkable signal-to-noise ratio. Experimental results using bimodal AFM imaging on a two component polymer sample validate that the self-sensing scheme can therefore be used to provide both the feedback signal, for topography imaging on the fundamental mode, and phase imaging on the higher eigenmode.
doi:10.3762/bjnano.7.26
PMCID: PMC4778537  PMID: 26977385
atomic force microscopy; charge sensing; feedthrough cancellation; multimode sensor; piezoelectric cantilever; self-sensing
4.  Improving image contrast and material discrimination with nonlinear response in bimodal atomic force microscopy 
Nature Communications  2015;6:6270.
Atomic force microscopy has recently been extented to bimodal operation, where increased image contrast is achieved through excitation and measurement of two cantilever eigenmodes. This enhanced material contrast is advantageous in analysis of complex heterogeneous materials with phase separation on the micro or nanometre scale. Here we show that much greater image contrast results from analysis of nonlinear response to the bimodal drive, at harmonics and mixing frequencies. The amplitude and phase of up to 17 frequencies are simultaneously measured in a single scan. Using a machine-learning algorithm we demonstrate almost threefold improvement in the ability to separate material components of a polymer blend when including this nonlinear response. Beyond the statistical analysis performed here, analysis of nonlinear response could be used to obtain quantitative material properties at high speeds and with enhanced resolution.
Bimodal atomic force microscopy is a promising approach in obtaining high-quality image contrast and material property mapping. Here, the authors show that by considering nonlinear response in bimodal atomic force microscopy, significant improvements in image contrast and material discrimination can be achieved.
doi:10.1038/ncomms7270
PMCID: PMC4346977  PMID: 25665933
5.  Multi-frequency tapping-mode atomic force microscopy beyond three eigenmodes in ambient air 
Summary
We present an exploratory study of multimodal tapping-mode atomic force microscopy driving more than three cantilever eigenmodes. We present tetramodal (4-eigenmode) imaging experiments conducted on a thin polytetrafluoroethylene (PTFE) film and computational simulations of pentamodal (5-eigenmode) cantilever dynamics and spectroscopy, focusing on the case of large amplitude ratios between the fundamental eigenmode and the higher eigenmodes. We discuss the dynamic complexities of the tip response in time and frequency space, as well as the average amplitude and phase response. We also illustrate typical images and spectroscopy curves and provide a very brief description of the observed contrast. Overall, our findings are promising in that they help to open the door to increasing sophistication and greater versatility in multi-frequency AFM through the incorporation of a larger number of driven eigenmodes, and in highlighting specific future research opportunities.
doi:10.3762/bjnano.5.175
PMCID: PMC4222484  PMID: 25383276
amplitude-modulation; bimodal; frequency-modulation; multi-frequency atomic force microscopy; multimodal; open loop; trimodal
6.  Bimodal atomic force microscopy driving the higher eigenmode in frequency-modulation mode: Implementation, advantages, disadvantages and comparison to the open-loop case 
Summary
We present an overview of the bimodal amplitude–frequency-modulation (AM-FM) imaging mode of atomic force microscopy (AFM), whereby the fundamental eigenmode is driven by using the amplitude-modulation technique (AM-AFM) while a higher eigenmode is driven by using either the constant-excitation or the constant-amplitude variant of the frequency-modulation (FM-AFM) technique. We also offer a comparison to the original bimodal AFM method, in which the higher eigenmode is driven with constant frequency and constant excitation amplitude. General as well as particular characteristics of the different driving schemes are highlighted from theoretical and experimental points of view, revealing the advantages and disadvantages of each. This study provides information and guidelines that can be useful in selecting the most appropriate operation mode to characterize different samples in the most efficient and reliable way.
doi:10.3762/bjnano.4.20
PMCID: PMC3628543  PMID: 23616939
amplitude-modulation; atomic force microscopy; frequency-modulation; phase-locked loop; spectroscopy
7.  Theoretical study of the frequency shift in bimodal FM-AFM by fractional calculus 
Summary
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.
doi:10.3762/bjnano.3.22
PMCID: PMC3323908  PMID: 22496992
AFM; atomic force microscopy; bimodal AFM; frequency shift; integral calculus applications
8.  Challenges and complexities of multifrequency atomic force microscopy in liquid environments 
Summary
This paper illustrates through numerical simulation the complexities encountered in high-damping AFM imaging, as in liquid enviroments, within the specific context of multifrequency atomic force microscopy (AFM). The focus is primarily on (i) the amplitude and phase relaxation of driven higher eigenmodes between successive tip–sample impacts, (ii) the momentary excitation of non-driven higher eigenmodes and (iii) base excitation artifacts. The results and discussion are mostly applicable to the cases where higher eigenmodes are driven in open loop and frequency modulation within bimodal schemes, but some concepts are also applicable to other types of multifrequency operations and to single-eigenmode amplitude and frequency modulation methods.
doi:10.3762/bjnano.5.33
PMCID: PMC3999742  PMID: 24778952
amplitude-modulation; bimodal; frequency-modulation; liquids; multifrequency atomic force microscopy
9.  Optimization of phase contrast in bimodal amplitude modulation AFM 
Summary
Bimodal force microscopy has expanded the capabilities of atomic force microscopy (AFM) by providing high spatial resolution images, compositional contrast and quantitative mapping of material properties without compromising the data acquisition speed. In the first bimodal AFM configuration, an amplitude feedback loop keeps constant the amplitude of the first mode while the observables of the second mode have not feedback restrictions (bimodal AM). Here we study the conditions to enhance the compositional contrast in bimodal AM while imaging heterogeneous materials. The contrast has a maximum by decreasing the amplitude of the second mode. We demonstrate that the roles of the excited modes are asymmetric. The operational range of bimodal AM is maximized when the second mode is free to follow changes in the force. We also study the contrast in trimodal AFM by analyzing the kinetic energy ratios. The phase contrast improves by decreasing the energy of second mode relative to those of the first and third modes.
doi:10.3762/bjnano.6.108
PMCID: PMC4463493  PMID: 26114079
bimodal AFM; dynamic AFM; tapping mode
10.  A Unique Self-Sensing, Self-Actuating AFM Probe at Higher Eigenmodes 
Sensors (Basel, Switzerland)  2015;15(11):28764-28771.
With its unique structure, the Akiyama probe is a type of tuning fork atomic force microscope probe. The long, soft cantilever makes it possible to measure soft samples in tapping mode. In this article, some characteristics of the probe at its second eigenmode are revealed by use of finite element analysis (FEA) and experiments in a standard atmosphere. Although the signal-to-noise ratio in this environment is not good enough, the 2 nm resolution and 0.09 Hz/nm sensitivity prove that the Akiyama probe can be used at its second eigenmode under FM non-contact mode or low amplitude FM tapping mode, which means that it is easy to change the measuring method from normal tapping to small amplitude tapping or non-contact mode with the same probe and equipment.
doi:10.3390/s151128764
PMCID: PMC4701306  PMID: 26580619
AFM; quartz tuning fork; higher eigenmode; non-contact; finite element analysis
11.  Interpreting motion and force for narrow-band intermodulation atomic force microscopy 
Summary
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.
doi:10.3762/bjnano.4.5
PMCID: PMC3566785  PMID: 23400552
atomic force microscopy; AFM; frequency combs; force spectroscopy; high-quality-factor resonators; intermodulation; multifrequency
12.  Frequency, amplitude, and phase measurements in contact resonance atomic force microscopies 
Summary
The resonance frequency, amplitude, and phase response of the first two eigenmodes of two contact-resonance atomic force microscopy (CR-AFM) configurations, which differ in the method used to excite the system (cantilever base vs sample excitation), are analyzed in this work. Similarities and differences in the observables of the cantilever dynamics, as well as the different effect of the tip–sample contact properties on those observables in each configuration are discussed. Finally, the expected accuracy of CR-AFM using phase-locked loop detection is investigated and quantification of the typical errors incurred during measurements is provided.
doi:10.3762/bjnano.5.30
PMCID: PMC3999761  PMID: 24778949
contact-resonance AFM; dynamic AFM; frequency modulation; phase-locked loop; viscoelasticity
13.  Dynamic calibration of higher eigenmode parameters of a cantilever in atomic force microscopy by using tip–surface interactions 
Summary
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.
doi:10.3762/bjnano.5.200
PMCID: PMC4222386  PMID: 25383301
atomic force microscopy; calibration; multimodal AFM; multifrequency AFM
14.  Atomic Force Microscopy and Wettability Study of Oxidized Patterns at the Surface of Polystyrene 
The surface properties of patterned surfaces made by a combination of photolithography and oxygen plasma treatment of polystyrene (PS) were investigated. PS and plasma-treated PS (PSox) were first characterized using X-ray photoelectron spectroscopy and the study of wetting dynamics (Wilhelmy plate method) in water and in solutions of different pH. The results indicated that the PSox surface may be viewed as covered with a polyelectrolyte-like gel, which swells depending on pH. It was then shown, using atomic force microscopy (AFM), that the adhesion force measured on PS with a silicon tip in water was higher compared with that measured on PSox. This feature allowed imaging of the oxidation patterns using the adhesion mapping mode. The origin of the pulloff force contrast, which could not be explained by combining Johnson–Kendall–Roberts theory and thermodynamic considerations, was attributed to repulsion between the tip and hydrated polymer chains present on the oxidized surface. Imaging was also performed in the lateral force mode, a higher friction being recorded On PS than On PSOX.
doi:10.1006/jcis.1999.6524
PMCID: PMC3261235  PMID: 10550254
atomic force microscopy; plasma treatment; polystyrene; Wilhelmy plate method; Johnson–Kendall–Roberts model; surface forces
15.  Measurement of Mechanical Properties of Cantilever Shaped Materials 
Sensors (Basel, Switzerland)  2008;8(5):3497-3541.
Microcantilevers were first introduced as imaging probes in Atomic Force Microscopy (AFM) due to their extremely high sensitivity in measuring surface forces. The versatility of these probes, however, allows the sensing and measurement of a host of mechanical properties of various materials. Sensor parameters such as resonance frequency, quality factor, amplitude of vibration and bending due to a differential stress can all be simultaneously determined for a cantilever. When measuring the mechanical properties of materials, identifying and discerning the most influential parameters responsible for the observed changes in the cantilever response are important. We will, therefore, discuss the effects of various force fields such as those induced by mass loading, residual stress, internal friction of the material, and other changes in the mechanical properties of the microcantilevers. Methods to measure variations in temperature, pressure, or molecular adsorption of water molecules are also discussed. Often these effects occur simultaneously, increasing the number of parameters that need to be concurrently measured to ensure the reliability of the sensors. We therefore systematically investigate the geometric and environmental effects on cantilever measurements including the chemical nature of the underlying interactions. To address the geometric effects we have considered cantilevers with a rectangular or circular cross section. The chemical nature is addressed by using cantilevers fabricated with metals and/or dielectrics. Selective chemical etching, swelling or changes in Young's modulus of the surface were investigated by means of polymeric and inorganic coatings. Finally to address the effect of the environment in which the cantilever operates, the Knudsen number was determined to characterize the molecule-cantilever collisions. Also bimaterial cantilevers with high thermal sensitivity were used to discern the effect of temperature variations. When appropriate, we use continuum mechanics, which is justified according to the ratio between the cantilever thickness and the grain size of the materials. We will also address other potential applications such as the ageing process of nuclear materials, building materials, and optical fibers, which can be investigated by monitoring their mechanical changes with time. In summary, by virtue of the dynamic response of a miniaturized cantilever shaped material, we present useful measurements of the associated elastic properties.
doi:10.3390/s8053497
PMCID: PMC3675557
Microcantilever; mechanics; ageing; environment; stress; gas; materials; sensor; pressure; temperature
16.  Acute Degradation of Surface-Bound Unsaturated Polyolefins in Common Solvents under Ambient Conditions 
We describe the detachment of covalently grafted polybutadiene and polynorbornenechains – which were prepared by surface-initiated ring-opening metathesis polymerization (SiROMP) – from Si/SiO2 substrates upon brief exposure to common solvents in air. Degradation and disappearance of grafted polybutadiene films after successive rinses with dichloromethane was monitored by ellipsometry. Changes in surface topography were analyzed by atomic force microscopy. The rapid auto-oxidation of allylic carbon-hydrogen bonds renders these thin films extremely susceptible to degradation under ambient conditions. Polymers in the tethered state suffer more acute degradation (on the time scale of seconds) compared to those dissolved in solution (not detectable after days). To prevent degradation, unsaturated polymersprepared by SiROMP and the subsequent conversion (of unsaturated groups) need to be carriedout under inert atmosphere. For example, smooth polybutadiene thin films of ~ 100 Å thick were covalently attached to silicon substrates via SiROMP of cyclooctadiene in the vapor phase. Solvent rinsing to remove unreacted monomers and free oligomers/polymers was carried out prior to the conversion of double bonds to epoxide groups. When these steps were carried out under nitrogen, negligible film loss was observed and surface topography of the thin film was preserved. Once the unsaturation was removed from polybutadiene, the epoxidized and hydroxylated polybutadiene films were stable toward solvent exposure in air.
doi:10.1021/la902340z
PMCID: PMC2756074  PMID: 19788207
17.  The nanoscale phase distinguishing of PCL-PB-PCL blended in epoxy resin by tapping mode atomic force microscopy 
Nanoscale Research Letters  2012;7(1):153.
In this work, we investigated the bulk phase distinguishing of the poly(ε-caprolactone)-polybutadiene-poly(ε-caprolactone) (PCL-PB-PCL) triblock copolymer blended in epoxy resin by tapping mode atomic force microscopy (TM-AFM). We found that at a set-point amplitude ratio (rsp) less than or equal to 0.85, a clear phase contrast could be obtained using a probe with a force constant of 40 N/m. When rsp was decreased to 0.1 or less, the measured size of the PB-rich domain relatively shrank; however, the height images of the PB-rich domain would take reverse (translating from the original light to dark) at rsp = 0.85. Force-probe measurements were carried out on the phase-separated regions by TM-AFM. According to the phase shift angle vs. rsp curve, it could be concluded that the different force exerting on the epoxy matrix or on the PB-rich domain might result in the height and phase image reversion. Furthermore, the indentation depth vs. rsp plot showed that with large tapping force (lower rsp), the indentation depth for the PB-rich domain was nearly identical for the epoxy resin matrix.
doi:10.1186/1556-276X-7-153
PMCID: PMC3311068  PMID: 22360980
tapping mode AFM; PCL-PB-PCL; phase image; force-probe
18.  Spectroscopic Investigation of Local Mechanical Impedance of Living Cells 
PLoS ONE  2014;9(7):e101687.
We studied nanoscale mechanical properties of PC12 living cells with a Force Feedback Microscope using two experimental approaches. The first one consists in measuring the local mechanical impedance of the cell membrane while simultaneously mapping the cell morphology at constant force. As the interaction force is increased, we observe the appearance of the sub-membrane cytoskeleton. We compare our findings with the outcome of other techniques. The second experimental approach consists in a spectroscopic investigation of the cell while varying the tip indentation into the membrane and consequently the applied force. At variance with conventional dynamic Atomic Force Microscopy techniques, here it is not mandatory to work at the first oscillation eigenmode of the cantilever: the excitation frequency of the tip can be chosen arbitrary leading then to new spectroscopic AFM techniques. We found in this way that the mechanical response of the PC12 cell membrane is found to be frequency dependent in the 1 kHz - 10 kHz range. In particular, we observe that the damping coefficient consistently decreases when the excitation frequency is increased.
doi:10.1371/journal.pone.0101687
PMCID: PMC4084948  PMID: 24999625
19.  Tuning Core vs. Shell Dimensions to Adjust the Performance of Nanoscopic Containers for the Loading and Release of Doxorubicin 
Detailed studies were performed to probe the effects of the core and shell dimensions of amphiphilic, shell crosslinked, knedel-like polymer nanoparticles (SCKs) on the loading and release of doxorubicin (DOX), a widely-used chemotherapy agent, in aqueous buffer, as a function of the solution pH. Effects of the nanoparticle composition were held constant, by employing SCKs constructed from a single type of amphiphilic diblock copolymer, poly(acrylic acid)-b-polystyrene (PAA-b-PS). A series of four SCK nanoparticle samples, ranging in number-average hydrodynamic diameter from 14–30 nm, was prepared from four block copolymers having different relative block lengths and absolute degrees of polymerization. The ratios of acrylic acid to styrene block lengths ranged from 0.65 to 3.0, giving SCKs with ratios of shell to core volumes ranging from 0.44 to 2.1. Although the shell thicknesses were calculated to be similar (1.5–3.1 nm by transmission electron microscopy (TEM) calculations and 3.5–4.9 nm by small angle neutron scattering (SANS) analyses), two of the SCK nanoparticles had relatively large core diameters (19 ± 2 and 20 ± 2 nm by TEM; 17.4 and 15.3 nm by SANS), while two had similar, smaller core diameters (11 ± 2 and 13 ± 2 nm by TEM; 9.0 and 8.9 nm by SANS). The SCKs were capable of being loaded with 1500–9700 DOX molecules per each particle, with larger numbers of DOX molecules packaged within the larger core SCKs. Their shell-to-core volume ratio showed impact on the rates and extents of release of DOX, with the volume occupied by the poly(acrylic acid) shell relative to the volume occupied by the polystyrene core correlating inversely with the diffusion-based release of DOX. Given that the same amount of polymer was used to construct each SCK sample, SCKs having smaller cores and higher acrylic acid vs. styrene volume ratios were present at higher concentrations than were the larger core SCKs, and gave lower final extents of release., Higher final extents of release and faster rates of release were observed for all DOX-loaded particle samples at pH 5.0 vs. pH 7.4, respectively, ca. 60% vs. 40% at 60 h, suggesting promise for enhanced delivery within tumors and cells. By fitting the data to the Higuchi model, quantitative determination of the kinetics of release was made, giving rate constants ranging from 0.0431 to 0.0540 h−½ at pH 7.4 and 0.106 to 0.136 h−½ at pH 5.0. In comparison, the non-crosslinked polymer micelle analogs exhibited rate constants for release of DOX of 0.245 and 0.278 h−½ at pH 7.4 and 5.0, respectively. These studies point to future directions to craft sophisticated devices for controlled drug release.
doi:10.1016/j.jconrel.2011.01.009
PMCID: PMC3119510  PMID: 21241750
core-shell nanoparticles; block copolymers; doxorubicin; drug delivery; nanoparticle dimensions; release kinetics
20.  Structural Evolution of Low-Molecular-Weight Poly(ethylene oxide)-block-polystyrene Diblock Copolymer Thin Film 
The Scientific World Journal  2013;2013:539457.
The structural evolution of low-molecular-weight poly(ethylene oxide)-block-polystyrene (PEO-b-PS) diblock copolymer thin film with various initial film thicknesses on silicon substrate under thermal annealing was investigated by atomic force microscopy, optical microscopy, and contact angle measurement. At film thickness below half of the interlamellar spacing of the diblock copolymer (6.2 nm), the entire silicon is covered by a polymer brush with PEO blocks anchored on the Si substrate due to the substrate-induced effect. When the film is thicker than 6.2 nm, a dense polymer brush which is equal to half of an interlamellar layer was formed on the silicon, while the excess material dewet this layer to form droplets. The droplet surface was rich with PS block and the PEO block crystallized inside the bigger droplet to form spherulite.
doi:10.1155/2013/539457
PMCID: PMC3835911  PMID: 24302862
21.  A simple and efficient quasi 3-dimensional viscoelastic model and software for simulation of tapping-mode atomic force microscopy 
Summary
This paper introduces a quasi-3-dimensional (Q3D) viscoelastic model and software tool for use in atomic force microscopy (AFM) simulations. The model is based on a 2-dimensional array of standard linear solid (SLS) model elements. The well-known 1-dimensional SLS model is a textbook example in viscoelastic theory but is relatively new in AFM simulation. It is the simplest model that offers a qualitatively correct description of the most fundamental viscoelastic behaviors, namely stress relaxation and creep. However, this simple model does not reflect the correct curvature in the repulsive portion of the force curve, so its application in the quantitative interpretation of AFM experiments is relatively limited. In the proposed Q3D model the use of an array of SLS elements leads to force curves that have the typical upward curvature in the repulsive region, while still offering a very low computational cost. Furthermore, the use of a multidimensional model allows for the study of AFM tips having non-ideal geometries, which can be extremely useful in practice. Examples of typical force curves are provided for single- and multifrequency tapping-mode imaging, for both of which the force curves exhibit the expected features. Finally, a software tool to simulate amplitude and phase spectroscopy curves is provided, which can be easily modified to implement other controls schemes in order to aid in the interpretation of AFM experiments.
doi:10.3762/bjnano.6.229
PMCID: PMC4685832  PMID: 26734515
atomic force microscopy (AFM); modeling; multifrequency; multimodal; polymers; simulation; spectroscopy; standard linear solid; tapping-mode AFM; viscoelasticity
22.  Modeling viscoelasticity through spring–dashpot models in intermittent-contact atomic force microscopy 
Summary
We examine different approaches to model viscoelasticity within atomic force microscopy (AFM) simulation. Our study ranges from very simple linear spring–dashpot models to more sophisticated nonlinear systems that are able to reproduce fundamental properties of viscoelastic surfaces, including creep, stress relaxation and the presence of multiple relaxation times. Some of the models examined have been previously used in AFM simulation, but their applicability to different situations has not yet been examined in detail. The behavior of each model is analyzed here in terms of force–distance curves, dissipated energy and any inherent unphysical artifacts. We focus in this paper on single-eigenmode tip–sample impacts, but the models and results can also be useful in the context of multifrequency AFM, in which the tip trajectories are very complex and there is a wider range of sample deformation frequencies (descriptions of tip–sample model behaviors in the context of multifrequency AFM require detailed studies and are beyond the scope of this work).
doi:10.3762/bjnano.5.224
PMCID: PMC4273292  PMID: 25551043
atomic force microscopy; creep; dissipated energy; multifrequency; stress relaxation; tapping mode; viscoelasticity
23.  Bimodal Analysis of Mammary Epithelial Cell Migration in Two Dimensions 
Annals of biomedical engineering  2008;37(1):230-245.
Cell migration paths of mammary epithelial cells (expressing different versions of the promigratory tyrosine kinase receptor Her2/Neu) were analyzed within a bimodal framework that is a generalization of the run-and-tumble description applicable to bacterial migration. The mammalian cell trajectories were segregated into two types of alternating modes, namely, the “directional-mode” (mode I, the more persistent mode, analogous to the bacterial run phase) and the “re-orientation-mode” (mode II, the less persistent mode, analogous to the bacterial tumble phase). Higher resolution (more pixel information, relative to cell size) and smaller sampling intervals (time between images) were found to give a better estimate of the deduced single cell dynamics (such as directional-mode time and turn angle distribution) of the various cell types from the bimodal analysis. The bimodal analysis tool permits the deduction of short-time dynamics of cell motion such as the turn angle distributions and turn frequencies during the course of cell migration compared to standard methods of cell migration analysis. We find that the two-hour mammalian cell tracking data do not fall into the diffusive regime implying that the often-used random motility expressions for mammalian cell motion (based on assuming diffusive motion) are invalid over the time steps (fraction of minute) typically used in modeling mammalian cell migration.
doi:10.1007/s10439-008-9592-y
PMCID: PMC3586332  PMID: 18982450
Bimodal framework; mammalian cell; turn angle distribution; random motility coefficient
24.  Probing viscoelastic surfaces with bimodal tapping-mode atomic force microscopy: Underlying physics and observables for a standard linear solid model 
Summary
This paper presents computational simulations of single-mode and bimodal atomic force microscopy (AFM) with particular focus on the viscoelastic interactions occurring during tip–sample impact. The surface is modeled by using a standard linear solid model, which is the simplest system that can reproduce creep compliance and stress relaxation, which are fundamental behaviors exhibited by viscoelastic surfaces. The relaxation of the surface in combination with the complexities of bimodal tip–sample impacts gives rise to unique dynamic behaviors that have important consequences with regards to the acquisition of quantitative relationships between the sample properties and the AFM observables. The physics of the tip–sample interactions and its effect on the observables are illustrated and discussed, and a brief research outlook on viscoelasticity measurement with intermittent-contact AFM is provided.
doi:10.3762/bjnano.5.176
PMCID: PMC4222404  PMID: 25383277
amplitude-modulation; bimodal; dissipation; frequency modulation; multi-frequency atomic force microscopy; viscoelasticity; standard linear solid
25.  Optimization of Functionalized Polymer Layers for Specific Targeting of Mobile Receptors on Cell Surfaces 
The reversible binding between a planar polymer layer functionalized by targeting groups and a planar cell surface containing different densities of mobile receptors has been studied by Monte Carlo simulations. Using the acceptance-ratio method the distance-dependent profiles for the average number of ligands bound to receptors, the total free energy for the polymer layer-cell surface interaction and the interaction force were obtained. Four main design parameters for the polymer layer were considered: the degree of functionalization, chain degree of polymerization, polymer density of grafting and the binding energy for the targeting group-receptor interaction. We found that an increase in the degree of functionalization or in the absolute energy of ligand-receptor binding results in a larger number of ligands bound to the receptors, lower free energy and stronger attractive force. Polymer layers composed of shorter chains were found to exhibit a deeper and narrower free energy profile and a larger attractive force, while longer tethers can interact with the cell surface at a larger and broader range of separation distances, in agreement with experimental observations. Our simulation results show that the increase in polymer grafting density from the mushroom to brush regime enhances the ligand availability and results in a stronger attractive force, increases the maximum binding distance, but exhibits a shallower free energy minimum due to the smaller tolerance to compression for polymer layers with high grafting density. We used two measures of the polymer layer binding affinity to the cell surface: the free energy minimum, related to the equilibrium binding constant and the fraction of bound ligands. We found that the polymer layers with a smaller chain length and grafting density, larger degree of functionalization and larger absolute binding energy exhibit both a larger equilibrium binding constant to the cell surface and a larger average number of bound ligands, except for high binding energies when the maximum level of binding is reached independently of polymer length and grafting density. We showed that high binding specificity can be achieved by the polymer layers with intermediate ligand-receptor binding energies or an intermediate number of ligands, as a larger binding energy or number of ligands ensures a high binding affinity but lacks specificity while a smaller binding energy or number of ligands provides inadequate affinity. We found that the results for polymer layers with different properties follow a similar pattern when both high binding affinity to cells with high receptor density and high binding specificity are considered. As a result, the optimal design of the polymer layers can be achieved by using several different strategies, which are discussed.
doi:10.1021/la801935h
PMCID: PMC2731659  PMID: 18834163

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