Generalized Hertz model for bimodal nanomechanical mapping

Introduction

Over the decades since its invention [1] the atomic force microscope (AFM) has been used in a variety of modes to characterize micro- and nanoscale heterogeneous structures in composites and other advanced materials. The AFM can provide high resolution topographic and mechanical properties mapping using techniques such as force curves [2,3], contact resonance [4,5], force modulation [6,7], phase imaging [8,9], loss tangent imaging [10], friction force microscopy [11], creep compliance [12], shear modulation force microscopy [13], pulsed force microscopy [14] and torsional approaches [15]. These techniques can be broadly classified as either “parametric” or “spectroscopic” techniques.

In parametric nanomechanical techniques, the sample properties are deduced from changes in the parameters of a driven cantilever that is oscillating in a (quasi) steady state while interacting with the sample surface. For example, tapping-mode AFM [16,17] (also known as amplitude-modulation (AM) AFM [18-20]), is one of the most commonly used parametric techniques, where the cantilever is driven on resonance and the cantilever–sample distance is adjusted by a feedback loop to maintain a constant oscillation amplitude at every image pixel. The time required for the cantilever to reach a steady state defines the acquisition speed, allowing tapping-mode imaging to achieve very high speeds ultimately only limited by the cantilever bandwidth. However, the small number of tapping-mode observables (amplitude and phase) limits the extraction of absolute storage and loss moduli, as they cannot be distinguished from changes in indentation depth. In tapping mode, only the ratio of the storage to loss modulus can be measured [10,21,22]. The same limitation applies to many other parametric techniques, such as force modulation [6,7] and other single-frequency imaging modes, such as frequency-modulation (FM) AFM [23]. Separating the storage and loss moduli, and quantifying them, requires either additional independent observables or the use of spectroscopic methods.

Spectroscopic techniques rely on changing the operating conditions of the cantilever to provide the necessary information to extract nanomechanical properties of the sample for a given image pixel. This can be achieved by changing the cantilever–sample distance [24] or sweeping the drive frequency [25], amongst others [26]. Examples of well-established spectroscopic techniques are nanoindentation [27] and force curves as well as dynamic force curves performed with an oscillated cantilever. The time-varying cantilever response serves as input to a model for extracting nanomechanical properties of the sample at any location. These techniques are by nature slow for imaging, as they measure time-varying changes of the cantilever at every pixel.

Recently, parametric techniques have been extended by driving two or more cantilever resonances simultaneously in order to increase the number of observables, which is required to quantify the storage and loss moduli. Advances in this direction include bimodal [28-33], trimodal [34] or more generally multimodal/multifrequency [35] techniques, and have demonstrated quantitative mapping without compromising on the high speeds that define parametric imaging techniques. Currently, state-of-the-art bimodal methodologies are mostly based on FM-AFM techniques that rely on elaborate mathematical theories [36-41], involving fractional calculus and Laplace transforms for relating AFM observables to nanomechanical properties. The mathematical complexity of these techniques can obscure physically intuitive understanding of the cantilever dynamics in bimodal AFM experiments.

Here, we present a simplified theory for bimodal AFM with a large fundamental resonance oscillation amplitude and small higher resonance amplitude. The theory is based on a binomial approximation of the weight function for extracting the interaction stiffness for both resonant modes, each yielding a simple analytical expression. These two independent pieces of information are refactored to provide information about modulus and indentation depth. While the theory is generally applicable to a wide range of tip–sample interaction models, the derivation here begins in the context of a Hertzian contact with a paraboloidal tip and is then generalized to any tip shape described by a power-law profile. This theory is then experimentally applied to three variations of bimodal AFM involving different dynamic AFM modes of operation [42,43], namely amplitude modulation (AM) [1,18,19], phase modulation (PM) [43-46] and frequency modulation (FM) [23,36,47]. Finally, a method for extracting the tip size and tip shape from bimodal AFM approach curves is presented and demonstrated on a polystyrene sample. Figure 1 provides a diagram of the theory presented in the following three sections.

Methods

Hertzian contact mechanics

The Hertzian contact model involves the interaction stiffness k_int versus indentation depth δ between a paraboloidal tip of radius R and a flat sample as

where the effective Young’s modulus E_eff combines deformation of the tip and sample [48]. Similar expressions can be derived for a tip in the shape of a punch or a cone [49].

Here, these three special cases are generalized to any axisymmetric tip shape whose cross-sectional radius r, as a function of height z, is governed by the power law

where the characteristic length scale L and the exponent parameter fully define the tip size and shape, respectively. Figure 2a illustrates Equation 2 for five values of m, including the three special cases of punch, paraboloid, and cone. Note that “sphere” is used as a shorthand for “paraboloid” in the scientific vernacular.

Upon purely elastic indentation of such a power-law indenter into a sample surface to an indentation depth δ, the true contact radius r_c is smaller than the cross-sectional radius r(δ) because of deformation of the surface, as can be understood from Figure 2b. The exception is the punch model for m = 1, where no reduction in radius occurs. The contact radius correction factor α_c quantifies the reduction in contact radius r_c with respect to r(δ) by

As expected, α_c = 1 for m = 1 (punch), and then α_c monotonically decreases to α_c = 2/π ≈ 0.64 for m = 2 (cone). The mathematical form of α_c is presented in the Appendix (b).

Importantly, it is the true contact radius r_c that defines the tip–sample interaction stiffness, as derived by Oliver and Pharr [27]:

Substituting in the expression for r_c leads to the general form

The interaction stiffness in Equation 5 is plotted for three special cases in Figure 3. For the punch model with m = 1, the length scale parameter L is equal to the punch radius R. For the paraboloidal indenter model with m = 3/2, L is the effective “sphere” diameter 2R. For the conical indenter with m = 2, L drops out and the cone half-angle θ = 45°. (Given the loss of the length scale parameter L in the degenerate case m = 2 (cone), a half-angle parameter θ may be introduced by multiplying k_int by (tan θ)⁻¹ to fully define the geometry of the conical indenter, if necessary.)

Bimodal interaction theory

This section first describes how the tip–sample interaction stiffness of a paraboloidal (“spherical”) tip affects the changes in effective stiffness of the first and second eigenmodes of the cantilever. Then, the results are generalized to a power-law tip shape.

Paraboloidal tip

For the first eigenmode driven with a large oscillation amplitude A₁, the change in stiffness of the interacting cantilever Δk₁ averaged over one cycle can be computed by integrating k_int(δ) with a semi-circular weight function that spans the peak-to-peak cantilever oscillation [50], as represented in Figure 3. Mathematically,

where δ_max is the maximum indentation depth and the normalized distance u = (δ_max − δ)/A₁. It is worth noting the distinction between k_int(δ), which is the instantaneous stiffness profile experienced by the oscillating cantilever tip, and Δk₁, which is the time-averaged effective change in stiffness of the cantilever–tip–sample system that is experimentally measurable by the AFM user. Measuring Δk₁ is the topic of Section ‘Simple harmonic oscillator theory‘, presented later.

In the limit that the fundamental amplitude A₁ is much larger than the interaction length scale, the stiffness profile k_int(δ) only affects a small portion of the cantilever sinusoidal oscillation where the tip indents the sample, as shown in Figure 3a. Therefore, the integration limits in Equation 6 reduce to [0, δ_max/A₁] such that u << 1 throughout the integration. Consequently, the weight function can be approximated very accurately with the first term in the binomial expansion, This approximation, used previously [51], is graphically illustrated in Figure 3e,f. Applying it to Equation 6 results in

which can be easily integrated for a paraboloidal indenter to give

The relative error in Δk₁ introduced by the binomial approximation is quantified in Figure 4; it typically falls on the order of 1% in large-amplitude dynamic AFM, where the interaction amplitude A₁ greatly exceeds δ_max.

Meanwhile, the second (or higher) eigenmode is deliberately driven at a small amplitude A₂, such that the interaction stiffness it experiences is roughly constant throughout one of the higher eigenmode oscillation cycles. However, the instantaneous interaction stiffness experienced by the second eigenmode slowly changes along the trajectory of the first eigenmode. Because the second mode rides along the slower sinusoidal motion of the first mode, its time-averaged change in interaction stiffness Δk₂ can be calculated by

where T is the oscillation period of the first eigenmode. A rigorous derivation of Equation 9 is provided in Appendix (a). Parametrizing this time integral with respect to distance by the substitution t = cos⁻¹(u)/ω₁ results in

Similarly to the approach taken with the first eigenmode, this integral can be solved analytically for a paraboloidal indenter after applying the binomial approximation such that

Note that the second-mode change in stiffness scales linearly with indentation depth, Δk₂δ_max, while the first mode stiffness scales with the square of the indentation depth, . These scaling laws have been verified experimentally for a paraboloidal indenter [39]. The system of two equations (Equation 8 and Equation 11) can be solved for two unknowns, namely

and

This operation is central to bimodal imaging, as it separates the changes in modulus from changes in indentation depth. This distinction cannot be made in single-mode dynamic AFM imaging.

The key to bimodal nanomechanical imaging is that the same stiffness profile is measured simultaneously by two different eigenmodes with different weight functions: and The fact that these weight functions are related by a derivative operation results in independent measures of nanomechanical properties by both eigenmodes while imaging. Notably, the fact that the weight function of the second eigenmode increases drastically as the tip approaches the sample, as seen in Figure 3f, explains the high sensitivity and increased spatial resolution of bimodal imaging noted in the past [52].

Power-law tip

The derivation so far revolved about a paraboloidal indenter. Applying the same approach used in Equations 6–13 to the generalized stiffness profile of Equation 5 result in the generalized indentation and modulus equations

and

where β is a scaling constant that depends only on m and is mathematically defined in the Appendix (c). As expected, choosing m = 3/2 in Equation 14 and Equation 15 recovers Eqution 12 and Equation 13.

The tip shape exponent m is a free parameter in the generalized stiffness profile. It can be inferred experimentally by measuring changes in Δk₁ and Δk₂ for a fixed A₁ and relating them to Equation 15 by the scaling law

as will be demonstrated later.

Simple harmonic oscillator theory

Measurements of Δk₁ and Δk₂ are necessary for calculating δ_max and E_eff in Equation 12 and Equation 13 or Equation 14 and Equation 15.

A single equation that extracts Δk from a driven eigenmode (on and off resonance) with any number of changing AFM observables is presented in the Appendix (d). The derivation assumes the eigenmode in question can be modeled as a simple harmonic oscillator (SHO) within the range of interactions explored in the experiment. Here, three special cases (AM, PM, FM) for starting an experiment on-resonance were extracted from the general SHO solution for simplicity. These special cases are presented in Table 1, and described in the following subsections. The constant-excitation FM mode (CEFM) is also included in Table 1 for completeness.

The last subsection extends the SHO theory to measuring two eigenmodes simultaneously, as required for bimodal AFM.

Amplitude modulation (AM) mode

For an eigenmode driven in AM mode, the corresponding time-averaged interaction stiffness can be calculated from the measured interaction amplitude A and interaction phase as in

while the cantilever quality factor Q_c, the cantilever stiffness k_c, and the reference amplitude A_r are all measured in the absence of tip–sample interactions (A_r is often referred to as the “free amplitude”).

Phase modulation (PM) mode

Alternatively, PM mode uses an automatic-gain-controller (AGC) to maintain a constant cantilever amplitude A by varying the drive amplitude D; meanwhile, the drive frequency remains fixed just as in AM mode. In this mode,

Note that an AGC is especially beneficial for use on the second eigenmode, because A₂ is often chosen close to the detection limit and may drop substantially in AM mode when the resonance frequency shifts upon interaction with the sample. The AGC in the PM mode assures that A₂ remains above the detection limit throughout the experiment.

Frequency modulation (FM) mode

If FM mode is employed for any of the eigenmodes, where the resonance frequency f_c is tracked with a phase-locked-loop (PLL), the measured frequency shift Δf can be used to estimate the interaction stiffness by the approximation

In this mode, the oscillation amplitude is held constant with an AGC. The use of an AGC will be assumed for “FM mode” henceforth.

Alternatively, constant-excitation FM mode (CEFM) employs a PLL but does not use an AGC [53,54]; therefore, the oscillation amplitude is not necessarily constant throughout the tip–sample interaction. This less-common technique is not explored in this article.

Bimodal configurations

In bimodal AFM, both eigenmodes behave like independent SHO’s because the first mode is driven in the large-amplitude limit and the second is driven in the small-amplitude limit. An empirical verification of this independence will be performed and discussed later.

In principle, both Δk₁ and Δk₂ can be measured with any of the four measurement modes described so far, leading to 16 possible configurations: AM-AM, AM-PM, AM-FM, FM-FM, etc. As an example, the first experimental bimodal measurements were taken in the AM-AM configuration [33]. In their pioneering work, Heruzzo and Garcia demonstrated FM-FM measurements [39,55]. Here, the AM-AM, AM-PM and AM-FM configurations are explored in detail. Practical considerations for these choices will be discussed later.

Results

A polystyrene surface was probed with bimodal approach curves (also known as “force–distance” curves), where the bimodally oscillating cantilever approaches the sample with a constant velocity. The experiment is described in the first subsection, where the AM-AM, AM-PM, and AM-FM configurations are compared to investigate the validity of the SHO model assumed by the bimodal theory. Then, multiple approach curves are used to extract the shape and size of the tip. Finally, the newly defined tip geometry is used to extract the modulus from all the approach curves to assess consistency of the results and their dependence on imaging parameters.

Experiment

The experiment was performed on a Cypher AFM with an ASYELEC-02 cantilever, which has a Ti/Ir-coated tip of nominal tip radius R = 28 ± 10 nm. Photothermal excitation [56] was used, which ensures stable imaging [57] and accurate FM tracking [58-60]. An automated calibration method [61] was used to obtain the stiffness of the first eigenmode (k_c1 = 43.2 N/m), which was then used as a basis of calibration for the second eigenmode stiffness k_c2 = 818 N/m by a recently established calibration protocol [62]. Next, a thermal power spectral density [63] of the cantilever was acquired close to the surface (within a few hundred nanometers) to determine the remaining cantilever properties (f_c1, f_c2, Q_c1, Q_c2). For this calibration step, the proximity to the surface is important. Performing it far from the surface incorrectly introduces long-range cantilever–sample interactions into the final measurement. Finally, the equipartition theorem was used to convert the amplitudes of both eigenmodes into nanometers [64-66].

Figure 5 shows approach curves acquired in the AM-AM, AM-PM, and AM-FM configurations. Equations 17–19 were used to recover values of Δk₁ and Δk₂ where appropriate. The excellent agreement between all three bimodal configurations across the entire approach curve suggests that the simple harmonic oscillator is a valid model to describe the second eigenmode of the cantilever in this experiment.

Calibration of the tip shape and size

When driving the first eigenmode in the AM mode, large variations in indentation depth can be achieved by varying the reference amplitude A_r1 ("free amplitude") while keeping A₁ fixed. Varying the reference amplitude is more effective than changing A₁ which carries a weak dependence on indentation for a setpoint around A₁/A_r1 ≈ 0.5. Consecutive bimodal approach curves on a polystyrene sample were performed while varying the reference amplitude A_r1 between 50 nm and 91 nm in 21 increments. The AM-AM, AM-PM, and AM-FM configurations were alternated. A total of 63 approach curves were performed within 45 min.

The values of Δk₁ and Δk₂ at a fixed amplitude (A₁ = 40 nm) were used to determine the tip shape using the scaling law in Equation 16. The results are shown in Figure 6, where the exponent m values for each bimodal configuration were extracted from a power-law fit and shown to be equal within error; they averaged to m = 1.24 ± 0.04. This measurement of the tip shape (m) can now be used to calculate the tip size (L), which is uniquely determined if the sample modulus is known. Fitting Equation 15 to the data and setting E_eff = 3 GPa (as approximately expected for polystyrene [67]) results in a value for the tip length scale L = 32 nm. This tip characterization suggests a tip shaped as a rounded punch, somewhere between a punch of radius R = 32 nm and a paraboloid of radius R = 16 nm.

With a defined exponent m value, the absolute indentation depth at the various A_r1 can be calculated by Equation 15, as shown in the inset of Figure 6. The calibration of the tip shape and size is expected to be valid for the explored range of indentations (between 0.6 nm and 1.2 nm in this case).

Measurement of modulus

With a modeled tip shape and size (m = 1.24, L = 32 nm), the modulus E_eff was extracted from all 63 approach curves, as shown in Figure 7. At A₁ = 40 nm, the average modulus must be 3 GPa because this was the assumed value from which the tip size was extracted. Notably, the modulus appears to be independent of both A₁ and bimodal configuration, as will be discussed in the next section.

Discussion

Independence of imaging parameters

The approach curves in Figure 7 span indentation depths from 0.6 nm to 1.2 nm, interaction amplitudes from 25 nm to 70 nm, and reference amplitudes between 50 nm and 91 nm. The modulus values at A₁ = 40 nm are constrained to match 3 GPa by the tip shape and size calibration. However, throughout this range of imaging parameters, the modulus also remains within a 0.15 GPa standard deviation (5%) of 3 GPa, with no obvious trend. In addition, the three bimodal configurations agree to better than this standard deviation.

The independence on imaging parameters provides confidence that the system was accurately modeled by a rounded punch of shape m = 1.24 and size L = 32 nm. In contrast, assuming a paraboloidal tip (m = 1.5) results in increased variability in the modulus, as well as noticeable dependence on imaging parameters. Also, assuming a paraboloidal tip results in a tip radius R = 93 nm for these experiments, which is far from the nominal radius for this cantilever tip (R = 28 ± 10 nm).

These results lead to the conclusion that calibrating the tip shape and size provided a more accurate picture of the tip–sample contact mechanics, which led in turn to improved accuracy in extracting the modulus of polystyrene over a wider range of imaging parameters.

Given that the bimodal AFM theory was derived in the context of Hertzian contact mechanics where only repulsive interactions are considered, only data where A₁/A_r1 < 0.75 were analyzed to ensure that repulsive forces dominate over attractive and adhesive forces. This criterion is somewhat arbitrary and specific to the current dataset; other samples and tip geometries may require different threshold values of A₁/A_r1 or may require more elaborate models.

AM/PM/FM equivalence

The agreement between AM, PM and FM modes of operation for Δk₂ shown in Figure 5 validates the mathematical framework leading up to Equations 17–19 that assumes SHO behavior of the cantilever. This agreement is reassuring, as a sinusoidally-driven SHO model was used for both eigenmodes to derive the bimodal interaction theory that relates the cantilever parameters to the interaction profile. Furthermore, the agreement between the modes of operation suggests that the feedback loops used for the PM and FM modes tracked changes in the cantilever eigenmode appropriately.

Large and small amplitude approximations

The amplitude of the first mode was modeled in the large limit, while the amplitude of the second mode was modeled in the small limit. The validity of these assumptions was implicitly verified in the dataset acquired for this experiment, as described herein.

The analysis in Figure 7 suggests that the variations in interaction amplitude (25 nm < A₁ < 70 nm) do not affect the outcome of the measured E_eff. This suggests that the large-amplitude limit for the first mode was fulfilled across the range of explored amplitudes.

The small-amplitude limit for A₂ is tested by the comparison of experimental configurations. Whereas the second mode amplitude A₂ for AM-AM drops significantly (from 500 pm to 200 pm) during a single approach curve, A₂ for AM-PM and AM-FM is held fixed by the AGC. The fact that all three configurations lead to the same extracted E_eff suggests that the small-amplitude limit for the second mode was fulfilled.

Sensitivity of bimodal AFM

The high sensitivity of bimodal AFM was predicted in the theory described in Section ‘Bimodal interaction theory’ and attributed to the introduction of an additional eigenmode driven at a small amplitude. The argument was based on the fact that the weight function of the second eigenmode rises quickly as the tip approach the sample, as shown in Figure 3f.

This fact is demonstrated experimentally in the data of Figure 5, where Δk₂ is 20× to 50× larger than Δk₁. Whereas both Δk₁ and Δk₂ are weighted integrals of the identical stiffness profile k_int(δ), the weight function of the second mode captures much more of the interaction stiffness for small indentations.

This illustrates the benefit of introducing a small-amplitude second eigenmode into the measurement: it provides a high-sensitivity channel of information, without disrupting the measurement as explained in the previous section.

Error from binomial approximation

The error introduced by the approximation that allowed the analytical closed-form in Equation 14 and Equation 15 is now calculated. In the worst case scenario for this experiment with δ_max = 1.2 nm and A₁ = 40 nm, a relative error of 0.5% is caused by the approximation In turn, this results in an overestimation of 0.9% in indentation depth, and an underestimation of 1.0% in modulus as far as Equation 14 and Equation 15 are concerned. These errors from approximation are negligible compared with experimental variability and absolute calibration error.

Importantly, applying Equation 19 to both eigenmodes results in FM-FM equations that are identical to those derived by Herruzo et al. [38,39]. This leads to the conclusion that the approximation is sufficient for deriving analytical equations for bimodal FM-FM and other bimodal configurations. In other words, the use of fractional calculus, Laplace transforms, Padé approximants, and Bessel functions can be avoided with no loss in accuracy when deriving bimodal AFM theory.

Merits of AM-FM

Although the three configurations of bimodal methods tested in Figure 5 were equally accurate, the AM-AM method has the disadvantage of potentially having more noise relative to AM-PM, which in turn may have more noise relative to AM-FM. In AM-AM and AM-PM, the second eigenmode drive frequency is held fixed. In that case, when the second eigenmode resonance frequency shifts by more than its bandwidth (f_c/2Q_c), the signal-to-noise ratio degrades because the signal drops relative to the noise floor. Furthermore, in AM-AM operation, A₂ decreases upon interaction with the surface – further aggravating the drop in signal-to-noise ratio. It is therefore often advisable to operate the second eigenmode in FM mode in practical situations.

On the other hand, the first eigenmode typically undergoes more modest changes in resonance frequency upon interaction with the surface. Also, these changes are controllable by the AFM user by the setting of an amplitude setpoint. For this reason, AM operation on the first eigenmode does not cause the same decrease in signal-to-noise ratio often observed on the second eigenmode. More importantly, dynamic AFM imaging in ambient conditions has relied on the robustness of AM (tapping-mode) imaging, mostly attributable to the monotonicity of the amplitude versus distance relationship, for stable operation and topography tracking.

For these two reasons, the authors generally recommend the use of AM-FM over the other configurations (AM-PM, FM-FM, etc.), as it is the most robust, versatile and sensitive configuration for bimodal nanomechanical mapping in a wide range of imaging conditions.

Conclusion

An analysis framework for bimodal AFM with a power-law tip and Hertzian contact was presented. The derived theory was used to extract the tip shape and size for an experiment on a polystyrene sample. For a wide range of imaging parameters, the experimental data returned a nearly constant modulus of the material when analyzed with this model. Three configurations (AM-AM, AM-PM, AM-FM) were tested and shown to provide equally accurate results, thereby supporting our assumption that the cantilever second eigenmode can be modeled as a simple harmonic oscillator for the range of interactions explored in the experiment.

The approximations used for deriving an analytical closed-form solution for bimodal AFM were also investigated. Both the first mode large-amplitude approximation and the second mode small-amplitude approximation were verified to be accurate for typical imaging conditions. Notably, the fact that the weight function of the second eigenmode increases drastically as the tip approaches the sample explains the high sensitivity and spatial resolution of bimodal imaging. The binomial approximation of the stiffness weight function was shown to introduce negligible error (<1%), yet it can be used to derive bimodal AFM theory without invoking the use of fractional calculus.

The experimental in situ determination of the tip shape and size is a pivotal step towards absolute quantitative nanomechanical measurements in a variety of techniques. This work demonstrates the benefits of tip characterization in the context of bimodal nanomechanical mapping, which improves the accuracy of fast parametric techniques such as AM-FM nanomechanical mapping.

Appendix

a. Correction factor for contact radius: α_c

The correction factor that accounts for deformation of the sample that leads to a true contact radius different from the nominal contact radius [68,69] is given by

where Γ is the gamma function. Figure 1 graphs α_c for values between m = 1 and m = 2. Note that obtaining α_c = 1 at the value at m = 1 requires a taking a limit.

b. Weighted interaction stiffness of the higher eigenmode

The weighted integral used to average the interaction stiffness across one oscillation cycle of the first eigenmode (Equation 6) also applies to higher eigenmodes; it can be rewritten as

where u’ is the normalized deflection of the higher eigenmode in this context. The major distinction between Equation 6 and Equation 21 is the substitution δ_max → δ(t) which accounts for the non-zero amplitude of the first eigenmode A₁ that introduces a time-varying component to the k_int experienced by the higher mode. For a first-mode period T = 1/f_c1, this effect can be averaged over a full cycle by

In the limit that A₂ → 0 and f_c2/f_c1 → ∞, k_int remains constant throughout any full oscillation cycle of the second mode, such that Equation 22 solves to

Notably, this integral that determines Δk₂ only depends on the trajectory of the first eigenmode because A₂ is assumed small.

c. Correction factor for power-law force model: β

When integrating k_int(δ) in Equation 6 and Equation 10 to obtain Δk₁ and Δk₂, respectively, for a power-law model as defined by Equation 5, gamma functions emerge due to the integration of non-integer powers. They are summarized here as

Note that the gamma functions that compose α_c in Appendix a relate to deformation of the sample and have no direct relationship to the gamma functions of β, which relate to the effects of power-law stiffness profiles affecting the cantilever parameters.

d. Generalized SHO equation

The on-resonance Equations 17–19 are special cases of the general solution to a simple harmonic oscillator model, which can be driven on or off resonance with concurrent changes in drive frequency, phase, oscillation amplitude and drive amplitude. This section derives a general equation that makes no assumption about which of these variables is held fixed upon interaction with the sample.

The response of a freely vibrating cantilever with effective stiffness k_c, mass m_c, and damping b_c can be described by the complex-valued cantilever impedance [70]

where the angular frequency is defined as ω = 2πf. The subscript “c” reminds that k_c, m_c, b_c are properties of the cantilever (prior to the tip–sample interaction) that do not change throughout the experiment.

A “reference” measurement of this cantilever impedance can be made prior to tip–sample interaction by exciting the cantilever sinusoidally with some reference driving force F_r at some reference drive frequency ω_r, which results in a reference oscillation amplitude A_r and a reference phase :

The reference driving force can be defined by isolating the imaginary components of both equations and solving for the driving force:

In the presence of some interaction, the cantilever impedance is subject to a time-averaged change in stiffness Δk and a time-averaged change in damping Δb, such that the interaction impedance

The interaction impedance is inferred during the experiment by

where the drive force F, the drive frequency ω, the oscillation amplitude A and the phase response are measured during tip–sample interaction.

Subtracting the reference measurement from the interaction measurement , then isolating the real components and solving for Δk leads to

Assuming that the drive force F is proportional to the drive amplitude D and that their relationship is frequency-independent implies that

Substituting Equation 27 and Equation 31 into Equation 30 results in

With the substitutions

and

that introduce the cantilever resonance frequency ω_c and the quality factor Q_c, Equation 32 can be rewritten in a more experimentally friendly form as

Lastly, driving the cantilever on resonance prior to tip sample interactions (ω_r = ω_c; = 90°) simplifies the result to

For AM operation, setting D = D_r and ω = ω_c results in Equation 17.

For PM operation, setting A = A_r and ω = ω_c results in Equation 18.

For FM operation, setting A = A_r, enforcing = 90°, and applying a binomial approximation results in Equation 19.