Atomic force microscopy is highly suited for characterizing morphology and physical properties of nanoscale objects. The application of this technique to nanomechanical studies is, therefore, exploited in a wide range of fields from life sciences to materials science and from miniature devices to sensors. Although performing a mechanical measurement can be straightforward and accessible to novice users, obtaining meaningful results requires knowledge and experience not always evident in standard instrumental software modules. In this paper, we provide a basic guide to proper protocols for the measurement and analysis of force curves and related atomic force microscopic techniques. Looking forward, we also survey the budding application of machine learning in this discipline.
I. INTRODUCTION
Since its development in the 1980s, atomic force microscopy (AFM) has served as the engine driving progress in many areas of nanoscale science and technology. As its name implies, the technique measures and delivers forces, allowing correlation of nanomechanical and morphological information at the nanoscale. Following its early implementations, application of AFM to force measurements has undergone enormous development and proven to be an invaluable tool in many areas of nanoscience. A few reviews give some idea of the impact of AFM force studies in fields of life sciences and zoology, polymers and polymer composites, 1 and 2D materials, and smart and other materials of current interest such as molecular solids and metal-organic frameworks.1–6
There are some nuances in AFM operation that make quantification of such measurements difficult relative to standard measurements made at larger scales such as tensile testing, microindentation, and even instrumented nanoindentation. In AFM, the interaction with the surface is with a small probe that is typically a consumable, and thus, its geometry needs to be carefully characterized for each experiment. The same holds for the commonly used cantilevered beams whose mechanical properties (stiffness, possibly resonance frequency) must be known. These matters are certainly surmountable through a number of established procedures. Another issue of concern is that due to the small interaction area/volume, small changes in the tip can profoundly influence the results obtained. This could be due to contamination or wear that can be continuously changing factors during the course of measurement. When modeling the mechanical system, it is common practice to apply analytical expressions arising from contact mechanics. Frequently, this approach ignores the fact that some of the basic assumptions underlying those equations are not met. This could be due to the nature of the sample (inhomogeneous, anisotropic, and rough). Other factors relate to the probe itself—for instance, the commonly used cantilever beam is not constrained to the plane perpendicular to the surface but has additional degrees of lateral freedom. Also, the probe geometry is ill-determined as noted above. Despite these limitations and uncertainties, which may lead to a wide range of reported values for a sample measured in different labs, and/or by different techniques, AFM nanoindentation is quite reliable for comparative studies using the same probe and conditions in which case many of the errors cancel out.
Many papers and reviews have discussed specific and general procedures and concerns for AFM-based nanomechanical measurements.7–10 The purpose of this paper, in line with this collection, is twofold—first, to provide some clear guidelines for novice experimenters on proper procedures in common nanomechanical measurements while stressing pitfalls that can lead to erroneous results, and second, to present and discuss some recent developments from machine learning and how they can be applied to the force curve analysis. For convenience, a list of abbreviations used in this work is shown in Table I.
Abbreviations used in this work.
Abbreviation . | Full term . |
---|---|
a | Contact radius |
A | Contact area |
AFM | Atomic force microscopy |
DMT | Derjaguin Müller Toporov |
ΔD | Cantilever deflection |
EV | Extracellular vesicle |
E | Young's modulus |
E* | Reduced Young's modulus |
F | Applied force |
G | Shear modulus |
GPa | gigapascal |
JKR | Jackson Kendal Roberts |
k | Cantilever spring constant |
kPa | kilopascal |
mJ | milliJoule |
ML | Machine learning |
MPa | megapascal |
OBD | Optical beam deflection |
R | Tip radius |
Rm | Membrane radius |
r | Nanotube/wire radius |
S | Deflection sensitivity |
Sc | Contact stiffness |
t | Sample thickness |
t-SNE | t-distributed stochastic neighbor embedding analysis |
w | Work of adhesion |
Z | Piezo-position in direction perpendicular to the sample plane |
δ | Sample deformation or deflection |
λ | Parameter distinguishing between bending or stretching of membrane |
λo | Elasticity parameter |
σo | Film pretension |
ν | Poisson ratio |
Abbreviation . | Full term . |
---|---|
a | Contact radius |
A | Contact area |
AFM | Atomic force microscopy |
DMT | Derjaguin Müller Toporov |
ΔD | Cantilever deflection |
EV | Extracellular vesicle |
E | Young's modulus |
E* | Reduced Young's modulus |
F | Applied force |
G | Shear modulus |
GPa | gigapascal |
JKR | Jackson Kendal Roberts |
k | Cantilever spring constant |
kPa | kilopascal |
mJ | milliJoule |
ML | Machine learning |
MPa | megapascal |
OBD | Optical beam deflection |
R | Tip radius |
Rm | Membrane radius |
r | Nanotube/wire radius |
S | Deflection sensitivity |
Sc | Contact stiffness |
t | Sample thickness |
t-SNE | t-distributed stochastic neighbor embedding analysis |
w | Work of adhesion |
Z | Piezo-position in direction perpendicular to the sample plane |
δ | Sample deformation or deflection |
λ | Parameter distinguishing between bending or stretching of membrane |
λo | Elasticity parameter |
σo | Film pretension |
ν | Poisson ratio |
II. BRIEF DESCRIPTION OF METHODS
A. General considerations
In AFM nanoindentation, the movement of the probe and/or sample is controlled by a piezoelectric actuator. The probe deflection due to contact with the surface is most typically measured by a laser beam reflected onto a photodiode termed optical beam deflection (OBD). There are other optical means11,12 and some instruments employ self-sensing piezoresistive cantilevers,13 and we will limit our discussion here to the more prevalent OBD technique illustrated in Fig. 1(a). The forces between the tip and the surface can be measured knowing the cantilever spring constant k, combined with the deflection sensitivity S (ΔD/ΔV). The latter represents the ratio between the extent of deflection of the tip in nm, ΔD, and the corresponding change in the controller voltage output, while a controlled displacement is applied between the probe and the sample using the piezoelectric actuator controlling the z-motion.14 The AFM is capable of monitoring processes in natural environments as it can be operated in vacuum, air, or under liquid.
(a) Schematic representation of a typical AFM working principle showing the tip indented into the sample. The z-motion (ΔZ) is controlled by the piezo-scanner and bend at the end of cantilever (caused by tip movement ΔD) is measured on a quadrant photodiode giving a voltage change (ΔV), which can be converted to nanometers through the sensitivity S, while ΔD is converted to force units through multiplication by the cantilever spring constant k. ΔZ–ΔD is the deformation of tip into sample, δ. (b) Model force vs distance curve for an indentation experiment. The thinner line with left-pointing arrows represents approach and thicker line with right-pointing arrows retract. Zo is the point of contact. (c) Attractive and adhesive forces between the tip and sample are seen as dips in the approaching and retracting curves, respectively.
(a) Schematic representation of a typical AFM working principle showing the tip indented into the sample. The z-motion (ΔZ) is controlled by the piezo-scanner and bend at the end of cantilever (caused by tip movement ΔD) is measured on a quadrant photodiode giving a voltage change (ΔV), which can be converted to nanometers through the sensitivity S, while ΔD is converted to force units through multiplication by the cantilever spring constant k. ΔZ–ΔD is the deformation of tip into sample, δ. (b) Model force vs distance curve for an indentation experiment. The thinner line with left-pointing arrows represents approach and thicker line with right-pointing arrows retract. Zo is the point of contact. (c) Attractive and adhesive forces between the tip and sample are seen as dips in the approaching and retracting curves, respectively.
The positioning capabilities of the AFM allow performing such measurements at specific locations on the surface with nanometer precision. In some cases, a surface map of these measurements is desired. Although historically this was done with reduced pixel resolution due to long times involved in acquiring force versus distance curves, newer methods, using subresonance modulation modes15 allows the accumulation of force-distance curves at every pixel of the scanned images. These modes, based on “pulsed-force mode,”16 are now able to acquire such images without reduction in scanning speed. Thus, simultaneous topography and nanomechanical maps can be made over a fully resolved image; these maps containing information such as adhesion, deformation, and elastic properties.15
Using the measured cantilever deflection and z-transducer motion under tip-surface contact, the recorded ΔV versus distance curves can be transformed into force versus indentation (deformation) F-δ curves. Fitting these curves to an analytical model, with input of tip geometry, yields the sample's indentation modulus.17,18 Although the values obtained are frequently reported as Young's modulus or just modulus of elasticity, these are not strictly identical. Differences can arise from the stress and strain occurring in different directions and depend on the anisotropy of the material under study.19,20
These equations do not account for plastic (irreversible) deformation, which is treated for instrumented nanoindentation using appropriate analysis of the unloading curve.28 The tendency to plastic deformation is difficult to avoid in AFM since high-resolution dictates the use of sharp tips. This leads to a small contact area and subsequent high stress in the contact zone. The criterion for maintaining the indentation conditions in the linear elastic regime is that the tip radius should be much larger than the contact radius. Tip radius is also a factor determining the regime of contact mechanical behavior as seen in Eq. (1). The importance of choosing appropriately large tip radii has been carefully evaluated previously29 and is discussed below in Sec. III (see Fig. 4 and the accompanying text).
As mentioned above, to calculate the modulus of the material, it is necessary to provide and measure the instrumental and probe parameters, i.e., spring constant, sensitivity, and probe shape. In addition, fitting the force versus indentation curves to a model will also necessitate a few assumptions. A recently published review has summarized the possible uncertainties associated with AFM nanoindentation measurements.26 A basic point to consider, which is inherent to nearly all AFMs, is the finite angle between the cantilever long axis and the surface, typically 10°–15°. The cantilever is almost always orientated with an angle relative to the sample surface, which can induce lateral forces and affect the measured mechanical properties.30 This can be largely corrected for by adding an x motion component [see Fig. 1(a)] to the z approach to compensate for the tilt.31 The tilt can also be corrected mathematically through a model that can be used for a wide range of tilt angles.32 Table II summarizes universally present components of the measurement and analysis that require care, as detailed in the following. The main considerations are briefly summarized below.
Key considerations in force-curve evaluation and analysis.
Relevant factor . | Physical characteristic . | Contribution to analysis . | Quantification . | Underlying causes for errors . |
---|---|---|---|---|
Tip | Geometry/shape, hardness | Parameter for fit to the analytical model | Tip characterizer, blind-reconstruction, indentation on material of known modulus, SEM | Wear/contamination |
Cantilever | Spring constant | Converting cantilever bend to the applied force | Thermal noise, Sader, calibrated cantilever | Method of determining k does not correspond to that felt at (unknown) tip position, cantilever too stiff to allow accurate thermal tune |
Sensitivity | Conversion volts to nm | Converting (optical) signal to deflection/deformation | F-ΔD curve on stiff surface, contactless methods | Deformation of surface/tip during calibration, poor positioning of laser on the cantilever, calibrating or measuring on sloped surface |
Sample dimensions | Lateral and vertical dimensions | Determines suitability to analytical model/influence of substrate | Topographical image/preknowledge of sample | Sample thickness too small, Sample geometry leads to poorly defined contact |
Relevant factor . | Physical characteristic . | Contribution to analysis . | Quantification . | Underlying causes for errors . |
---|---|---|---|---|
Tip | Geometry/shape, hardness | Parameter for fit to the analytical model | Tip characterizer, blind-reconstruction, indentation on material of known modulus, SEM | Wear/contamination |
Cantilever | Spring constant | Converting cantilever bend to the applied force | Thermal noise, Sader, calibrated cantilever | Method of determining k does not correspond to that felt at (unknown) tip position, cantilever too stiff to allow accurate thermal tune |
Sensitivity | Conversion volts to nm | Converting (optical) signal to deflection/deformation | F-ΔD curve on stiff surface, contactless methods | Deformation of surface/tip during calibration, poor positioning of laser on the cantilever, calibrating or measuring on sloped surface |
Sample dimensions | Lateral and vertical dimensions | Determines suitability to analytical model/influence of substrate | Topographical image/preknowledge of sample | Sample thickness too small, Sample geometry leads to poorly defined contact |
B. Cantilever deflection sensitivity (S)
The first fundamental step for quantitative measurements is accurately estimating the cantilever deflection sensitivity. The deflection output from the AFM controller electronics is in volts, and this must be converted into nm. The deflection sensitivity S is the ratio between a known cantilever deflection (usually obtained by pressing the tip against a hard surface not deformed by the tip, over a known distance) and the change in voltage measured on the photodiode (ΔV). Uncertainties using this method can arise from inconsistent placement of the detection laser on the cantilever, contamination of tip by compressible impurities, or for very stiff cantilevers, unquantified indentation into the surface.
The deflection sensitivity is also related to specific optical configuration, light source alignment, cantilever coating, and detector and can be influenced by an unavoidable drift over time. The latter can be corrected by recalibration during the experiment. The commonly used OBD method measures a deflection angle rather than the absolute z displacement, making it very sensitive to the (somewhat unknown) position of the tip relative to the cantilever end.
Determining S requires exact knowledge of the z-piezo displacement, thus problematic for systems operating under an open-loop z position control.33 Piezoelectric displacement is known to experience artifacts like nonlinearity, hysteresis, and creep in the motion response to change in voltage.34 Uncertainty in piezo-positioning can reach 30% error for large movements.26,35 Most modern AFMs incorporate a sensor that independently measures the z piezo-motion, thus eliminating or greatly reducing this uncertainty.
Once S is calculated, the spring constant k can be calculated as described in Sec. II C below. Depending on the calibration method chosen, errors in calculating S will also affect the calculation of k, which can also be affected by other factors like the position of the laser spot on the cantilever, an additional source of error.36,37
C. Spring constant of cantilever
Two common methods to calculate k are the Sader method40 and the thermal noise method.41 Combining them allows determination also of S without contact between tip and surface, simplifying the calibration process and eliminates the probability of change in tip shape due to the potentially destructive contact calibration method. The Sader method is based on the hydrodynamic response of cantilevers in the measured medium, cantilever length and width, resonant frequency, and quality factor. Although first derived and most commonly implemented for rectangular cantilevers, corrections for different geometries are also available.42 k is also often calculated using thermal noise and the equipartition theorem. Here, the thermal noise derived for the normal mode is inversely related to the square root of the spring constant. This method requires knowledge of S, and due to the square relationship between S and k, a small 10% error in S results in 20% error in k.43 By using both the Sader and thermal noise methods in tandem, the result can be inverted to solve for S that allows complete noncontact conversion of changes in the photodiode voltage signal to force.42 The Sader method can be applied with good accuracy for many types of commercially available cantilevers listed in Refs. 12, 26, 44, and 45.
The inverse relation between the cantilever stiffness k and thermal noise means that stiff cantilevers undergo smaller thermal fluctuations. This makes the thermal calibration limited and less accurate for very stiff cantilevers. Therefore, in some cases, the cantilever must be calibrated using another, precalibrated cantilever. Here, the stiffness of the unknown cantilever can be calculated by pressing on a precalibrated lever and measuring the deflection of both beams under the applied displacement. This method offers accuracies in the range of 5%–10% and is suitable for very stiff cantilevers that cannot be calibrated accurately (due to low thermal noise) by the common thermal method.46,47 Accuracy of better than 0.5% was achieved using large-scale precalibrated (dozens of mm in length) cantilevers.48 In closing, we note that in principle the spring constant of the cantilever can be computed from standard beam equations, knowing length, width, thickness, density, and modulus of the cantilever beam. The calibration methods described above are necessary since the dimensions (primarily the thickness) of the beam are not well controlled and can vary from the manufacturer’s specifications. Also, for lithographic processes that fashion the beam using a deposited material (such as silicon nitride), the modulus and density can vary from batch to batch, as well as the type and thickness of the reflective coating on the cantilever that further contribute to this uncertainty. It is also important to note that the relevant k is felt at the position of the tip, and when the tip is offset along the axis away from its end, the cantilever is effectively shorter, and hence, k is larger.
D. Tip shape
Accurate knowledge of the tip shape is crucial for proper modulus calculation when using contact mechanics models. Several methods can be used to obtain this information. Commercial vendors provide tips with a known shape (typically spherical) with defined radius. These could be a colloid particle glued to the cantilever or a spherical shape formed directly on the integrated silicon probe. Tip shape can also be measured by a tip characterizer—a sample with sharp spikes much smaller than the tip radius so that they provide inverse imaging of the tip. Sometimes electron microscopy is applied to measure the tip shape. Finally, “Blind reconstruction” can also be used—this is an algorithm that can be simply explained by finding the sharpest features on the surface to define the maximum tip size; this new proposed tip size is used to erode the image to account for tip shape, and the process is continued until it converges.49 A related method is to make indentations on a material of known modulus (similar to that anticipated for the material under study), applying the relevant contact model to fit the curve, now solving for the tip radius rather than the modulus. It was suggested that this latter method leads to the underestimation of tip size at small depths.50 After characterization, it is important to keep in mind that the tip can change due to contamination or by wear—particularly on hard surfaces, see Sec. IV below. Although sharp tips can improve spatial resolution, their size is harder to quantify. Significant errors could arise from large relative uncertainty of the tip shape, which propagate to sizable errors in the calculated modulus. For example, a factor of 2 error in probe radius will result in a 40% error in Young's modulus for thick samples and even larger errors for thin samples, when the substrate influence is ignored (see below, Sec. II F).51
Clearly, the contact area determination is critical. The standard contact-mechanics relations used for deriving the modulus presume that the surface geometry is a flat plane. Topographical features with dimensions on the order of tip radius will lead to smaller or larger actual contact areas depending on the location on the sample. For instance, as seen in Fig. 2, a mapping of the modulus computed in this way “on the fly” shows that the grain boundaries always have apparently higher modulus. This can arise because the true contact area is larger when the tip contacts the sloped boundaries between neighboring grains, so the area used in calculation (which appears in the denominator) is too small leading to overestimation of E. It can also arise from the changing vector of the force interaction with the surface. Therefore, when computing the modulus from such a mapping, only areas where the surface is relatively flat should be used. This effect is shown in Figs. 2(d) and 2(e). These data show that on a sloped surface, the surface normal has a component both along the tip axis and perpendicular to it. The latter causes an additional bending of the cantilever, which can augment or reduce the cantilever bend leading to over- or under-estimation of the modulus, respectively. Sometimes, the roughness is uniformly large so that there is no meaning to the modulus value obtained.
(a) Topography, (b) modulus, and (c) error signal mappings of the grain structure showing enhanced apparent stiffness at grain boundaries (see the text). (d) Perspective image of calibration grid with a 70° angle between the upward and downward slope. Cartoon of the cantilever and the tip shows the influence of the slope on the cantilever bend. On the right slope, the tip is pushed to the right, leading to a downward bend of the cantilever that reduces the recorded voltage response, thus increasing the detector sensitivity S. On the left slope, the tip is pushed to left, augmenting the voltage signal and thus diminishing S. (e) Cross-sectional profile of image in (d), showing the sensitivity values measured on upward and downward slope of this grid.
(a) Topography, (b) modulus, and (c) error signal mappings of the grain structure showing enhanced apparent stiffness at grain boundaries (see the text). (d) Perspective image of calibration grid with a 70° angle between the upward and downward slope. Cartoon of the cantilever and the tip shows the influence of the slope on the cantilever bend. On the right slope, the tip is pushed to the right, leading to a downward bend of the cantilever that reduces the recorded voltage response, thus increasing the detector sensitivity S. On the left slope, the tip is pushed to left, augmenting the voltage signal and thus diminishing S. (e) Cross-sectional profile of image in (d), showing the sensitivity values measured on upward and downward slope of this grid.
E. Force curve alignment before modeling
Prior to modeling, the force indentation curve should be aligned as follows, The baseline {part of the force curve where the tip and surface are not in contact [see Fig. 1(b)]} should be aligned and assigned to zero force. Though we expect no change in the force when the tip is not in contact with the surface, the baseline is often sloped. Also, sometimes an offset is observed between incoming and retracting curves. The latter effect could occur due to compression of air between the cantilever and surface or hydrodynamic drag when measuring in fluid. The former could occur due to imperfect alignment of the optical detection system. The tilt in the baseline should be corrected, commonly by using a linear or polynomial fit and subtracting the offset to align the base line.52,53
It is also important to determine the point of contact of the tip with the surface. This point is used to define the relative z position so that values below zero give the indentation depth into the sample, and values above zero the baseline before contact [see Fig. 1(b)]. The calculated modulus values are very sensitive to the location of the contact point.26,51
F. Substrate effect
The indenting probe is sensitive to the material properties in a volume under and around the tip position, which extends beyond the actual volume defined by the contact. This effect should be considered carefully when indenting thin and soft layers on a hard substrate. A rough rule of thumb to avoid the substrate effect is that the indentation depth should not exceed 10% of the total film thickness (“Buckle's rule”). Nonetheless, some works suggest that the value may be much smaller than this, and the reliability of this rule depends on the difference in elastic modulus between the substrate and film as well as the contact area in relation to the film thickness.9,26,54,55 Finite element calculations showed much higher substrate sensitivity for hard on soft films than for soft on hard films.56 For very thin films, this limitation may prove difficult to achieve, and corrections to eliminate the substrate effect are available.51,57,58
The models and techniques applied should account for the relative stiffness of the film and substrate. For compliant films of sufficient thickness, deformation will be governed by film properties. For the opposite case, which sometimes arises for a thin film of large modulus on a thicker, compliant substrate, the thin film actually serves as a secondary indenter deforming the substrate. This situation can be treated by analytical approximations59 or by finite element analysis.60
Very thin films and samples with small lateral dimensions sometimes require additional preparation for meaningful measurement. Figure 3 describes force measurements made on two different Malaria-derived extracellular vesicle (EV) subpopulations, each population having a very different mean size. Due to their small size, the standard force curve analysis could not be applied because the smaller EV populations would return “stiffer” behavior only due to a greater substrate influence. Therefore, the EVs were disassembled into supported bilayers [Fig. 3(b)], and the mechanical properties of two subpopulation fractions 3 and 4 were compared by performing puncture assays to determine the force required to penetrate the bilayer, as illustrated in Fig. 3(c).61 This method gave a clear difference in the cohesive strengths of the bilayers originating from different EV populations as seen in Fig. 3(d). Additional nuances of thin films are discussed further in Sec. V.
Conversion of small EVs to a supported lipid bilayer membrane for AFM nanomechanical measurements and machine learning. (a) High-resolution AFM images (scale bar—50 nm) of extracellular vesicles (EVs) from two different sized EV fractions. These EVs were then ruptured by washing with pure water to form supported lipid bilayers, giving the images shown in (b) (scale bar: 1 μm). (c) Force-separation curves measured on these bilayers to determine the puncture force. Definition of puncture force is shown as Fmax indicated on the curve of fraction 3. (d) Statistics of puncture forces for the two fractions presented as box plots. (e) and (f) Machine-learning grouping based on features engineered from the puncture data by t-distributed stochastic neighbor embedding analysis (t-SNE) and K-means (see the text). (e) In the t-SNE analysis, two clusters are found representing two size fractions, F3 (stars) and F4 (circles). The Xs mark the center of the clusters determined by K-means. (f) K-means loss function for the analysis is shown in (e), displaying loss function as a function of the number of clusters, shown as the 's. The dashed line serves as a guide. The elbow point indicates the most probable number of clusters that best represent the data. Reproduced with permission from Abou Karam et al. EMBO Rep. 23, e54755 (2022). Copyright 2022, John Wiley and Sons.
Conversion of small EVs to a supported lipid bilayer membrane for AFM nanomechanical measurements and machine learning. (a) High-resolution AFM images (scale bar—50 nm) of extracellular vesicles (EVs) from two different sized EV fractions. These EVs were then ruptured by washing with pure water to form supported lipid bilayers, giving the images shown in (b) (scale bar: 1 μm). (c) Force-separation curves measured on these bilayers to determine the puncture force. Definition of puncture force is shown as Fmax indicated on the curve of fraction 3. (d) Statistics of puncture forces for the two fractions presented as box plots. (e) and (f) Machine-learning grouping based on features engineered from the puncture data by t-distributed stochastic neighbor embedding analysis (t-SNE) and K-means (see the text). (e) In the t-SNE analysis, two clusters are found representing two size fractions, F3 (stars) and F4 (circles). The Xs mark the center of the clusters determined by K-means. (f) K-means loss function for the analysis is shown in (e), displaying loss function as a function of the number of clusters, shown as the 's. The dashed line serves as a guide. The elbow point indicates the most probable number of clusters that best represent the data. Reproduced with permission from Abou Karam et al. EMBO Rep. 23, e54755 (2022). Copyright 2022, John Wiley and Sons.
III. MEASUREMENTS ON SOFT MATTER
By soft matter, we include both fragile materials with low hardness and very compliant materials with low modulus. Some polymers and many biological specimens fall in this category. Force measurements on the latter must take into account the often extensive steps required to obtain a meaningful measurement. Thus, in addition to the precautions used due to their fragile nature, they are sensitive to the environment and must be measured in the correct buffer and environmental conditions. In addition, they must be anchored to a substrate firmly enough to allow probing with the AFM tip but not too strongly to alter their mechanical response.
The first and often most trying step for a successful bio-AFM measurement is immobilization of the sample to a surface. A cell or a tissue slice that is not securely bound may detach and drift toward the tip, making the deformation measurement unreliable. Cells are commonly adsorbed to the Petri dish or wells made from glass or plastic by coating them with polylysine.43 Due to the fact that many biological specimens carry a negative charge, the positively charged polylysine creates an electrostatic attraction. The efficacy of attachment of cells can be visually checked through an optical microscope by gently shaking the dish and verifying that they do not become dislodged. Furthermore, if an imaging mode providing simultaneous topographical and mechanical data is utilized, suspicious motion of the sample can be detected directly from the AFM scan. Other common methods to facilitate electrostatic adsorption of negatively charged biological sample are coating the substrate (glass/plastic/mica) surface with (3-aminopropyl)triethoxy silane,62 which binds to the substrate through the silane moiety and to the biosample through the amine, or by using commercially available adhesives.63 For adsorbing small molecules such as proteins, DNA, or EVs, the negatively charged atomically smooth mica surface can be modified by divalent ion adsorption using Mg2+ or Ni2+ salts.64,65
In addition to the considerations of tip choice provided in Sec. II, measurements on the soft material require special attention. Although, in general, sharp tips are advantageous for AFM work due to enhanced spatial resolution, the small contact area leads to higher local stress, which could cause irreversible local damage to the soft sample.66 For example, in Fig. 4, the comparison between EVs scanned with a sharp or a rounded tip. The soft, round vesicles are easily damaged with the sharp tip. Such damage would render the F–Z curves invalid for the analysis. In addition, for thin samples, it is critical to control the indentation depth to avoid contribution from the substrate, and this can be harder to control when using a sharp tip to measure a soft sample. Here, it is important to recall the requirement (Sec. II F) to keep the contact area small relative to sample thickness, which highlights the special attention to details required in these measurements.
Atomic force microscopy (AFM) imaging of a Plasmodium falciparum-derived EV sample for similar regions on the same sample scanned with two different tips (on similar cantilevers). (a) Image scanned with a sharp tip (tip apex R < 10 nm). Even at the lowest force that can applied for imaging with this probe, less than 100 pN, the EVs are damaged and swept from the surface as can be seen by comparison with (b) where the EVs are imaged gently without displacing or damaging them, when using a probe with the same cantilever spring constant as in (a) but with a rounded tip, apex R ∼ 30 nm. Reproduced with permission from Rosenhek-Goldian et al., Imaging of Extracellular Vesicles Derived from Plasmodium falciparum–Infected Red Blood Cells Using Atomic Force Microscopy (Springer US, New York, 2022), pp. 133–145. Copyright 2022, Springer Nature.
Atomic force microscopy (AFM) imaging of a Plasmodium falciparum-derived EV sample for similar regions on the same sample scanned with two different tips (on similar cantilevers). (a) Image scanned with a sharp tip (tip apex R < 10 nm). Even at the lowest force that can applied for imaging with this probe, less than 100 pN, the EVs are damaged and swept from the surface as can be seen by comparison with (b) where the EVs are imaged gently without displacing or damaging them, when using a probe with the same cantilever spring constant as in (a) but with a rounded tip, apex R ∼ 30 nm. Reproduced with permission from Rosenhek-Goldian et al., Imaging of Extracellular Vesicles Derived from Plasmodium falciparum–Infected Red Blood Cells Using Atomic Force Microscopy (Springer US, New York, 2022), pp. 133–145. Copyright 2022, Springer Nature.
For the reasons enumerated above, mechanical measurements on very soft materials such as cells, gels, and tissue often employ a colloidal probe.51 Indeed, indentation measurements with a sharp tip resulted in higher modulus values on cells67 or poly(vinyl-alcohol) gels51 compared to a colloidal or flat tip. The biophysical cause for the higher modulus values measured using shallow indentations with a sharp probe as compared to larger spherical probes has been assigned to the sensitivity of sharper tip probes primarily to the outer stiffer cell cortex, whereas larger spherical tips are influenced by the stiffness of both the cortex and the underlying (more compliant) cytoskeleton.68 Studies have shown that similar modulus values can be obtained by using either colloidal or somewhat sharper conical/pyramidal tips (with rounded tip apex ∼ 30–100 nm).69,70 When using much sharper tips, with the radius of curvature lower than 20 nm, puncture events are harder to avoid. In addition, for inhomogeneous samples as the cell, using a sharper probe and deeper penetration depths enables the monitoring of changes in intracellular elements.71,72
For fragile cells, several precautions are recommended to ensure their stability. In order to verify that a cell remains intact following the indentation measurement, it is important to image it after completing the indentation experiment. The integrity of the cell is crucial for obtaining reliable modulus measurements. Keeping the biological specimen viable during measurement is another consideration. Culture medium conditions (buffer type, strength, and pH), operation temperature, and culture density can have a major influence on the measured modulus values.67,73 Cell viability may be judged by morphological changes, sometimes seen optically (for example, RBCs with abnormal membrane, termed echinocytes, can be observed at elevated pH conditions74–76), or assessed by fluorescent stains like DAPI,77,78 whereby the dead cell membrane is more permeable to the dye leading to stronger cell staining.79 A color change due to phenol red dye of the culture medium can indicate pH changes due to CO2 concentration in the environment. A possible solution to avoid this effect is to maintain controlled CO2 concentration in the measurement chamber80,81 or by ensuring that the measurement time is short enough to avoid loss of viability.
As mentioned in Sec. II C, the cantilever stiffness should be chosen to be on the order of contact stiffness. Thus, frequently very soft cantilevers are employed for biological work, with spring constant on the order of a few hundredths of N/m. Nonetheless, in some cases, a stiffer cantilever is appropriate, in order to minimize noise-induced fluctuations associated with very soft cantilevers,82 to prevent sticking of the probe to an adhesive sample, or to perform faster measurements without influence of the hydrodynamics.83
IV. MEASUREMENTS ON HARD/STIFF MATTER
Hard matter presents some very different issues than soft matter. Although typically these are stable materials imaged in ambient air with no issues of sample attachment or tendency to tear, the choice of tip, cantilever, and mode of operation are often more limited here.
For such robust materials, very stiff cantilever springs are used to apply high forces. This can raise issues with the tip radius that contributes in a major way to the computed area. There are two concerns here. One is that the tip wear can lead to a changing tip radius during the course of measurement so that a value measured at the start is not representative. Although this was mentioned in passing above, this problem is particularly acute for hard and stiff samples. It is always advisable to use a tip made of or coated with a hard material such as diamond or refractory metal such as metal carbides, but even for hard tips such as hard, diamondlike carbon, wear has been reported.84
These problems make measurements of very high modulus materials such as many ceramics somewhat out of reach for quasistatic force measurements. Although probes using diamond shards for the tip and cantilevers with k of hundreds of N/m can be found and applied for AFM nanoindentation, it is harder to justify the application of the simplistic contact models in this case. This due to the fact that forces transverse to the surface normal will be significant, leading to friction and possibly slip at the surface and/or bend of the cantilever beyond the confines of Hooke's law. The former is due to higher normal forces leading to larger friction as well as the larger transverse force due to the larger moment arm of these long shards. The latter is because larger forces require larger deflection of the cantilever. Using lower forces, hence, lower deformation can alleviate these obstacles, but computing the small deformation can lead to large experimental uncertainty. Therefore, multifrequency techniques are employed. An example is the contact resonance (CR) technique.85 In this method, the tip is brought into hard contact with the surface, and the sample (or sometimes tip) is excited by high-frequency (ultrasonic) excitation. The dynamics are evaluated using Euler–Bernoulli beam relations, whereby the contact resonance frequency is proportional to the 1/3 power of the sample modulus.86 Contact resonance methods have other advantages compared to quasistatic force curves. One is the range of response:
A cantilever excited at the second contact resonance will have a much higher effective stiffness due to nodes forming along the length of the cantilever that effectively shorten the cantilever.87 Recent work has also shown that the technique can probe the modulus and even thickness of subsurface layers.88 In addition to its clear advantage for very stiff materials, CR has additionally been applied to some more compliant ones.5,86 The cantilever oscillations in CR have small amplitude; however, these higher forces can still damage the tip. Common practice is to locate a region of interest in the regular imaging mode and then drastically reduce the number of pixels for the modulus determination using CR in order to minimize indent cycles and thus tip damage. The CR technique and other oscillatory modes hold great promise and have provided valuable insights into a number of fields but cannot be covered within the scope of this elementary survey. Interested readers can find more in-depth information in a recent review.5 Nonetheless, the general issues discussed here such as sample geometry, careful calibration, and choice of probe are relevant for their proper application.
V. CHALLENGING SAMPLE GEOMETRIES
A. General considerations
The continuum contact-mechanics models described above in Sec. II A are applicable for an axisymmetric tip indenting into a homogeneous, infinite half-plane. AFM is frequently used to probe properties of samples for which one or more or the dimensions are on the scale of the tip radius. This largely invalidates simplistic applications of those aforementioned models and requires special considerations for measurement and/or evaluation. 1D and 2D materials have attracted a great deal of attention due to their special properties, of note their mechanical properties. Some works showed that carbon nanotubes and graphene have elastic moduli ranking among the highest of all known materials.89,90 Other materials with similar geometry such as nanorods and nanowires also exhibit exceptional mechanical properties.91,92 Measuring the elastic properties of these structures, though, requires protocols that account for their geometry. For instance, an indentation test on a substrate-supported single layer graphene film is not amenable to determine the modulus from simple F–Z curves due to its small thickness and proximity of the substrate. Furthermore, nanoindentation on nanotubes would need to contend with two consequences of their geometry—(1) they are not a solid cylinder but rather hollow, and (2) the contact shape will depend on the tip radius that could be on the order of the nanotube radius.
B. 2D materials and ultrathin films
Scheme of the suspended nanotube for modulus measurement showing the suspended length L, and deformation δ for lateral displacement leading to torsional twist of the cantilever and combination of bending, stretching, and radial deformation of the nanotube depending on the geometry and force applied. The inset shows the undeformed radius r of nanotube and its wall thickness t, as well as radial deformation that could result from such measurements.
Scheme of the suspended nanotube for modulus measurement showing the suspended length L, and deformation δ for lateral displacement leading to torsional twist of the cantilever and combination of bending, stretching, and radial deformation of the nanotube depending on the geometry and force applied. The inset shows the undeformed radius r of nanotube and its wall thickness t, as well as radial deformation that could result from such measurements.
For small values of λ, the structure bends like a plate and for large values stretches as a membrane. The critical values appear in Ref. 94. An advantage of this method is that it allows an accurate modulus measurement of very thin films of high modulus.89
Here, the superscript “2D” refers to the 2D values, and σo and E are the film pretension and modulus that are obtained from fitting the F versus δ data to Eq. (5).
Another useful AFM technique for determining the modulus of ultrathin films is the bulge test, whereby the membrane is formed over a pressurizable cavity so that application of pressure causes the membrane to bulge upward, and AFM is used to determine the height of the bulge.96 This method of course requires that a pressure-tight seal be formed between the membrane and the framework of the substrate.
C. Nanowires and nanotubes
Additional approaches for determination of modulus of thin and small objects exist. These are useful, for instance, when the thickness and small lateral dimensions of the object studied are incompatible with any sort of standard analytical analysis of force curves, and the sample cannot easily be prepared in the suspended configurations as described above, or when the three-point configuration does not allow access to the desired mode. This is the case for measuring the radial stiffness of a nanotube because the radial modulus of nanotubes is often much smaller than the longitudinal one and can best be detected by suppressing any stretching. Therefore, a protocol is required for indenting a nanotube supported on a hard substrate. We can consider two limits, the first when the indenting tip has a radius larger than that of the nanotube. In this case, a Hertzian model was applied, which considered the geometric contact between the sphere, i.e., tip, and the cylinder, i.e., nanotube, while ignoring the internal structure.100 On the other extreme, a model considering a shell-like structure deformed by a point load was proposed.101 This could be relevant for larger, multiwalled nanotubes, indented by a sharp tip. Due to the complex geometry, different regimes developed during the indentation process, the first being direct deformation of the shell structure and the last collapse of the tube so that subsequent deformation is of the flat, layered structure. While providing important insights, these models cannot fully consider the deformation of a hollow tube with multiple layers that are also capable of shear within the layers of the shell. The additional complexity can be treated through modeling, which includes full tubule geometry and boundary conditions, either by continuum models such as finite element analysis or by atomistic computations.102,103
Small fibrils also exhibit small cylindrical geometry but may not be amenable to suspension over a gap. It has been shown that imaging the distribution of angular bends along the length of the fibril (if they form bent configurations on the surface) allows estimation of the elastic modulus using a wormlike chain model without performing force curves.104,105
VI. MACHINE LEARNING
Machine learning (ML) has become an important asset to improve and speed up collection, interpretation, and presentation of microscopy data. While in its infancy, the forays into this field show great promise for deciphering mechanical information from AFM force curves. This section will be less explicit than previous ones in defining proper measurement protocols, with a primary goal of showing the overall benefit of the approach, relevant applications, and general suggestions for implementation.
Although ML is developing very rapidly for the image analysis, it also provides many benefits for the force-curve analysis. Although the applications in SPM are still somewhat limited and not yet widespread, their efficacy is well established and applications are quickly expanding.106 In supervised learning, a dataset that is labelled according to different classes of objects is used to teach the system how to distinguish between these classes. Frequently, machine-learning algorithms are trained using a portion of the dataset set aside for that purpose. In order to make the algorithm reliable, a large dataset is required. Acquiring large numbers of images can be a bottleneck. However, force curves can be acquired rapidly, and it is more feasible to acquire a sufficient experimental set in relatively short time.
The utility of relying on ML methods opens new opportunities for the analysis of force curves. For instance, in many practical cases, there is no need to measure a precise elastic modulus value for the sample but rather to compare the modulus across different samples or treatments, sample position, etc. In these cases, rather than performing a fit of F–Z curves to an analytical model, it is sufficient to show the corresponding differences in curve shape characteristic of different sample populations. This approach was applied to distinguish healthy from cancerous brain tissue more efficiently than the standard Hertzian analysis (Sneddon), which compares analytically derived moduli values.107 Furthermore, the probability of incorrect assignment could be computed as a function of the number of curves acquired, allowing an objective standard for the minimum number of curves necessary to obtain a valid conclusion. Here, the entire curve shape was used in the analysis rather than just a portion of it (i.e., approaching or receding) as is done with the classical analysis.
Another method to classify force data without deriving a specific quantity is feature extraction.108 Here, several different characteristic features of the curves are chosen (often relying on prior, expert knowledge of the physics of interaction), and machine learning is used to find similarities between different curves based on such features. We have recently used this to distinguish different EV populations by AFM membrane puncture experiments.61 In this work, it was also possible to determine the number of different categories that the EVs grouped into. Although frequently the distinction is binary (e.g., healthy or diseased), there could sometimes be intermediate or more extreme divisions. Using t-distributed stochastic neighbor embedding analysis (t-SNE) and K-means analyses, two distinct populations characterized by similar features were identified as seen in Figs. 3(e) and 3(f). The t-SNE allows grouping the copious datasets into distinct clusters that showed two prominent groups. The K-means plots a loss-function (i.e., calculated error for each choice) versus the number of clusters to find the most probable number of such groups that best describes the data. A technique for simultaneously acquiring multiple features including those related to force events (adhesion) as well as topography (roughness) was used to distinguish between healthy and cancerous cells.109 Here, the adhesion obtained from the AFM measurements was the useful physical characteristic, and the distinction between healthy and cancerous cells was made using feature engineering, which extracts deeper information from the raw data. Here, the analysis used differences in the distribution of the set of adhesion values (i.e., average and root mean square, skewness, peak to peak values, kurtosis, and others). Of course, these analyses are not limited to binary situations. A deep learning neural network approach allowed classifying F–Z curves into three different categories distinguished by differing adhesive properties for cells of the common pathogen Candida albicans.110 Over 16 000 such full curves were classified within 90 s after supervised training.
In cases where a numerical value for modulus is desired, automation is still possible, but it is then necessary to choose an analytical model to fit the data. This has been demonstrated using a Python-based program “nanoite.”111 Several of the steps mentioned above (Sec. II F) were performed automatically, such as baseline removal and determination of the contact point. Of equal practical importance is the need to remove curves that are invalid due to poor contact, insufficient retraction, instabilities, and other features. This requires defining a set of criteria for separating valid and invalid data. This step, which must often be performed manually by an expert operator, was efficiently performed here by machine learning. Although in this work, some of the criteria derived from the magnitude of residuals in the fit, in principle only those criteria divorced from the fit quality could be used. This step could then be separated from the actual fit to a model, allowing model-free curve comparison, which also filters out artifactual curves.
An intermediate approach considered that it is often not known a priori which contact model best fits the data. To address this problem, three different contact models were considered, distinguished by whether or not they allow tip-surface adhesion and whether the tip is more spherical or spheroconical.112 These were used to train the ML model by finding the most appropriate model for the curve and subsequently the associated elastic modulus. These fit values were associated with seven different characteristic (dimensionless) features of each force curve. By using dimensionless features (obtained by division of characteristic maximum/minimum force values, slope/curvature of curve to other force/displacement values, etc.), the influence of variations in tip geometry and material variation could be nullified. This initial training identified characteristic features of the force-distance curves. These characteristic features are, therefore, associated with a quantitative modulus without requiring knowledge of these potentially unknown factors. Training by a supervised machine learning regressor model then allowed the assignment of modulus based on these seven dimensionless features of the force curves. Once trained, this supervised model provided high accuracy for modulus determination. This approach not only obviated the need to predict the contact mechanics model but in addition removed the necessity to provide the tip geometry, which otherwise requires a separate calibration. It also removes concern about changing tip shape during the course of an experiment due to wear.
Another important but somewhat elusive characteristic to obtain from F–δ curves is the viscoelasticity. Although often ignored in the force-curve analysis, this can have significant influence in the force-distance behavior, reflected in different shape curves dependent on the speed they are acquired. There are a number of ways to evaluate the viscoelasticity using nanoindentation.113 A recent attempt to obtain this information using a nonsupervised neural network shows some promise, although still requires more development.114 The experiment recorded stress relaxation curves, obtained by pushing the tip by a constant depth into the surface and recording the change in force with time. The results were fed into a Maxwellian model (consisting of springs and dashpots) to fit the data from those curves. Nine viscoelastic parameters were extracted from these data, which were then put into a neural network to produce self-organized maps (SOMs). These SOMs showed how the nine parameters obtained from the Maxwellian analysis could be specifically correlated with different therapeutic treatments of breast cancer cells.
VII. CONCLUSIONS
We have presented here some fundamental considerations that should be applied for meaningful measurement of surface elasticity by AFM. Although the measurement itself can be quite straightforward, and most AFM vendors provide dedicated software to perform such measurements, without proper attention to all the details, it is easy to produce useless results. Fortunately, proper methodology allows successful application of the technique. We have additionally presented here the exciting and blooming discipline of machine learning as applied to the force curve analysis. This field is still in its infancy but has already provided impressive results.
ACKNOWLEDGMENTS
The authors thank Ido Azuri for helpful discussions.
AUTHOR DECLARATIONS
Conflict of Interest
The authors have no conflicts to disclose.
Author Contributions
Irit Rosenhek-Goldian: Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). Sidney R. Cohen: Conceptualization (lead); Data curation (lead); Visualization (supporting); Writing – original draft (equal); Writing – review & editing (equal).
DATA AVAILABILITY
The data that support the findings of this study are available within the article.