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.


Appendix 1: Tip-sample interaction model
In [1] a tip-sample interaction model is suggested, which includes surface energy hysteresis. The model was based on a DMT model. In the original formulation, the attractive and repulsive parts are handled separately (the attractive part was called "long-range dissipative interfacial interactions"). Using a slightly different notation, they can be combined into one expression as follows where H, R, a 0 , γ are the Hamaker constant, tip radius, intermolecular distance, and surface energy, respectively, E * is the reduced elasticity E * = (1 − ν 2 tip )/E tip + (1 − ν 2 sample )/E sample −1 and ν and E are the Poisson's ratio and Young's modulus of the tip and the sample. We note that this model includes only a single parameter γ, which is the change in surface energy (J/m 2 ) between approach and retract, to describe the strength of the hysteresis. Two parameters are given in [1], one for d < a 0 and a separate one for d > a 0 . Although that is a more general case, it means that the force described by that model is not necessarily continuous at d = a 0 . We have restricted ourselves only to the case where the force is continuous.
This model has been shown to match several features of experimental energy dissipation mea-S2 surements, and it is well suited to analysis (e.g., the method of averaging). However, for numerical simulation it can present a problem. The switch between the approach and retract forces happens instantaneously. This means that the force is discontinuous at the switch, which is clearly nonphysical.
This may cause difficulties for the differential equation solver, and it can also introduce nonphysical high-frequency oscillation of the cantilever into the simulation.
Therefore, we suggest a modification that allows the force to be continuous everywhere.
Initially,ḋ < 0 and F ts (d) = F ts,app (d). If, at time t = t 0 and d = d 0 , the velocity switches sign fromḋ < 0 toḋ > 0, then the force for time t > t 0 is defined as F ts (d) = F ts,ret (d) + where λ is a decay length (we use 0.1 nm typically). In other words, when the velocity switches, the current trajectory is smoothly transitioned into the new trajectory. The difference between the current and new trajectories decays exponentially. This is illustrated in Figure S1. The above definition is sufficient for single-frequency AFM (e.g., AM-AFM). However, for bimodal AFM, it is possible that the velocity might reverse two (or more) times, and the second reversal might happen before the transition to the new trajectory is complete. Therefore, we instead use this definition: If the velocity reverses at time t = t 0 and d = d 0 , then let F * be the force at time t 0 . Then the force for time t > t 0 is defined as F ts = F ts,m (d) + (F * − F ts,m (d 0 )) e (−d−d 0 )/λ , and "m" corresponds to either "app" or "ret", depending on which direction the velocity has switched to. This allows an arbitrary number of reversals at arbitrary distances, while still always maintaining continuity of the forces.

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Appendix 2: Derivation for cantilever energy By definition, energy is force times displacement. The inertia force on the cantilever is given by m iqi and the displacement is q i . Therefore the kinetic energy at time t is E i (t) = m iqi dq i = t 0 m iqiqi dt. Assuming a harmonic response at the natural frequency, q i = A i cos(ω i t + φ ), theṅ The spring force on the cantilever is given by k i q i . Therefore the potential energy at time t is E i (t) = k i q i dq i = t 0 k i q iqi dt. Using a similar derivation, it can be shown that the maximum kinetic energy over the cycle is equal to the maximum potential energy over the cycle (the maximums happen at different times). Therefore, the body of the paper could have been formulated in terms of the potential energy stored in each eigenmode with no change in the conclusions.