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**NONLINEAR dynamicS of atomic force microscopY**

GIUSEPPE REGA Sapienza University of Rome, Italy Department of Structural and Geotechnical Engineering Co-workers: VALERIA SETTIMI, Ugo ANDREAUS, LUCA PLACIDI

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**OUTLINE 1. INTRODUCTION A. NONCONTACT AFM 2. ModelING**

3. bifurcations/response scenarios 4. global Dynamics and integrity B. TAPPING AFM 3. NONLINEAR hysteresis 4. INFLUENCE OF modal DAMPING (Q-FACTORS) C. control OF AFM RESPONSE 5. external feedback control OF Noncontact afm 6. Weakly nonlinear dynamics 7. Strongly nonlinear dynamics 8. CONCLUSIONS NONLINEAR dynamicS of atomic force MicroscopY GIUSEPPE REGA

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**1. complementary AFM TOPICS for NONCONTACT and TAPPING**

AIM: Noncontact: Investigating conditions for occurrence of unwanted jump to contact, under beam vertical (scan horizontal) excitation, realizing conditions of external (parametric) forcing at primary (fundamental), subharmonic (principal), and superharmonic resonances Tapping: Discussing criticality of intermittent contact, highlighting effects of higher order eigenmodes, with relevant damping ratios, on overall system dynamics at various resonances ATOMIC INTERACTION POTENTIAL: Noncontact: solely attractive Tapping: attractive-repulsive CONTINUOUS MODELING: Noncontact: geometrically nonlinear beam, realizing a general platform for refined investigations Tapping: simple linear beam, to highlight involved effects of atomic interaction and modal dampings REDUCED-ORDER MODELING: Noncontact: minimal-order (single-mode) allowing systematic bifurcation analyses Tapping: multi-mode, with Rayleigh-based modal damping evaluation PHENOMENOLOGICAL FEATURES OF INTEREST: Noncontact: attractors robustness, basins erosion, dynamic integrity, system practical safety with respect to escape Tapping: nonlinear hysteresis, higher harmonics contribution, approach/retract separation, impact velocity, contact force, patterns/ranges of modal Q-factors (damping) NONLINEAR dynamicS of atomic force MicroscopY GIUSEPPE REGA

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**b. tapping AFM - AIM OF INVESTIGATION - 1 -**

TAPPING-AFM: the tip operates in the ATTRACTIVE and REPULSIVE FORCE region, and TOUCHES the surface only FOR SHORT PERIODS, in order to reduce damages to potentially fragile samples. Recently, higher order bending modes gained significant interest (mostly for AFM in liquids) because of their very high Q-factors (low attenuation) and dynamic stiffnesses (Raman et al, 2008): Allow to drive a tip with very small amplitudes, which enables atomic-scale resolution. Shorter cycle time period of higher eigenmode considerably shorter time response (at least one cycle of oscillation) needed to capture data Non-trivially different responses in soft-impact dynamics of a cantilever beam when considering equivalent single-mode model more reliable multi-mode model Of major importance to reliably characterize velocities and forces at contact For TAPPING AFMs in AIR at various resonances: Extent of BISTABLE BEHAVIOR and IMPACT VELOCITY/CONTACT FORCE Effect of APPROACH/RETRACT SEPARATION Importance of HIGHER-ORDER EIGENMODES Influence of PATTERN/RANGE of DAMPING RATIOS NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 2. Modeling Extended Hamilton principle initial boundary value problem for transverse vibration: d Attractive-repulsive tip-sample interaction: van der Waals and Derjaguin-Muller-Toporov contact forces NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 2.1. modelING - MULTI-mode model Nondimensionalization, assumed mode technique system of ODEs Damping matrix Cij : Rayleigh assumption with and constants with respect to modes Alternative expression in terms of Q-factors of j-th mode (inverse of damping ratio), usually referred to in experimental AFM dynamics Relation between Q-factor and Rayleigh damping with =0 and evaluated with the first mode NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 2.2. modelING - QUASI-STATIC BEHAVIOR - 1 - The inclination of the green line represents the stiffness of the micro-cantilever d Sample its intersection with the interaction force represents the equilibrium tip position NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 2.2. modelING - QUASI-STATIC BEHAVIOR - 2 - As the probe specimen separation (distance) is reduced, the cantilever tip experiences an increasing attractive force toward the sample But when the distance falls below a critical value, there is a change in the interaction between tip and sample, and the tip snaps into contact When the distance is decreased and increased again, the tip snaps off on retraction This hysteretic and bistable behavior may meaningfully affect sample imaging, making the interpretation of the signal produced by the microscope quite difficult. NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 3. NONLINEAR Hysteresis Same hysteretic and bistable behavior in dynamics Sample AFM tip y0 Sin Ωt Vibrations of cantilever tip induced through oscillations of a dither piezo at cantilever support. Z(t) Typical single-mode frequency-response diagram (primary resonance) For decreasing tip-sample separation, amplitude- saturation and bistability different maximum distances at fixed frequency and separation values NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**Single-mode vs three-mode (enough for converging response)**

07/06/2013 3.1. NONLINEAR hysteresis - FREQUENCY SWEEP Single-mode vs three-mode (enough for converging response) Maximum distance Phase difference Frequency range of bistable solution with three modes non-trivially smaller Significant variation on saturated branch Higher harmonic contributions NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 3.2. NONLINEAR hysteresis - approach/retract Separation Sweep - 1 - Single-mode vs three-mode: resonance of 1st mode Response in nominally monostable region: range of “high” separation values (24 26 nm) Maximum distance Phase difference Hysteresis phenomena also occur !! NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 3.2. NONLINEAR hysteresis - approach/retract Separation Sweep - 2 - Three-mode model: resonance of 2nd mode Bistable region: range of very low separation values (- 4 2 nm) various harmonics Steep decrease in fundamental and 1/3 subharmonic correspond to steep increases of 1/4 and 1/2 subharmonics meaningful transfer of energy from excited second mode to lower harmonics. Hysteretic behaviour due to 1/3 subharmonic hints for filtering the component source of hysteresis, permitting to eliminate coexisting solutions and improve image resolution NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 3.2. NONLINEAR RESPONSE - Separation Sweep - CONTACT FORCE How response features at nominal impact/contact depend on the cantilever being excited at FIRST or SECOND resonance ? FIRST resonance: Hysteresis in the range of both high and very low separation high separation low separation NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**SECOND resonance: range of very low separation values**

07/06/2013 3.2. NONLINEAR RESPONSE - Separation Sweep - IMPACT VELOCITY/CONTACT FORCE SECOND resonance: range of very low separation values Intersection with zero distance line and chaotic response Impact velocity and contact force: ONE ORDER OF MAGNITUDE LARGER than at first resonance (high and low separation values) importance of exciting second mode for harmful tapping effects NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4. INFLUENCE OF modal DAMPING (Q-FACTORS)**

07/06/2013 4. INFLUENCE OF modal DAMPING (Q-FACTORS) Single-mode model: characterizing damping through equivalent Q-factor (as in experimental AFM) Q1 = 33.3 Three-mode model with Rayleigh criterion (structural damping, system behavior in air, uncommon in AFM community): damping increasing (Q-factor decreasing) with increasing number of mode Q1 = Q2 = 5.31, Q1 = 1.9 Literature patterns of quality factors not always understandable (or internally consistent), with the relevant range varying from ten to a few hundred in air up to several thousands in vacuum conditions. Exploring INFLUENCE OF HIGHER MODES on nonlinear hysteresis for nominal values of MODAL DAMPING (Q-FACTOR) IN HIGHER RANGE, or with INCREASING Q-FACTOR values, more typical of experimental AFM IN LIQUIDS NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 4.1. INFLUENCE OF Q-FACTORS - aiR vs LIQUID-LIKE values - 1 - First resonance HIGHER RANGE of Q-factor values: Q1 = Q2 = 333, Q3 = 3330 Nominal FLUID with VERY LOW VISCOSITY phase difference Low modal dampings, staying far away from sample no significant change in nonlinear hysteresis NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**LOW RANGE of Q-factor values: Q1 = 3.33. Q2 = 5.31, Q3 = 19 **

07/06/2013 4.1. INFLUENCE OF Q-FACTORS - aiR vs LIQUID-LIKE values - 2 - First resonance LOW RANGE of Q-factor values: Q1 = Q2 = 5.31, Q3 = 19 Nominal FLUID with RELATIVELY HIGH VISCOSITY phase difference High modal damping, getting closer to sample no meaningful differences with respect to low damping range, nor with respect to reference case (increasing modal damping) Contribution of HIGHER MODES POORLY DEPENDS on (INCREASING OR DECREASING) PATTERN of MODAL DAMPINGS, IRRESPECTIVE of CONSIDERED RANGE NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)**

07/06/2013 4.1. INFLUENCE OF Q-FACTORS - aiR vs LIQUID-LIKE values - 3 - 1st reson. meaningful differences with APPROACH VS RETRACT within hysteresis High range of Q-factor values approach orbit “contained” within attractive branch due to repulsive effect below a certain distance in the approach stage retract orbit encompassing both repulsive and attractive branches Low range of Q-factor values approach orbit still “contained” within attractive retract orbit trapped close to repulsive branch NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**Importance of EXCITING A HIGHER ORDER MODE ! ! !**

07/06/2013 4.1. INFLUENCE OF Q-FACTORS - LIQUID-LIKE values - multiperiodic orbit Second resonance: high viscosity liquid-like Three coexisting attractors two periodic with comparably low amplitudes one MULTIPERIODIC of large amplitude in large separation region: encompasses branches penetrates sample very large contact force Importance of EXCITING A HIGHER ORDER MODE ! ! ! NONLINEAR dynamicS of atomic force MicroscopY B. tapping AFM GIUSEPPE REGA 23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013)

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**NONCONTACT AFM TAPPING AFM 8. Conclusions**

GLOBAL DYNAMICS OF A SINGLE MODE MODEL OF NONCONTACT AFM very rich scenario DYNAMIC INTEGRITY practical escape thresholds associated with a priori safe design targets EXTERNAL FEEDBACK CONTROL modified involved scenario effective local technique to be supported by comprehensive analysis of global dynamics aimed at detecting proper operation ranges TAPPING AFM MULTIMODE MODEL importance of exciting a higher order mode limited influence of pattern and range of air- or liquid-like Q-factors NONLINEAR HYSTERETIC BEHAVIOR also multistability of response, affecting quality/robustness of scan process hints to filtering components which are source of hysteresis, to eliminate coexisting solutions and improve image resolution NONLINEAR dynamicS of atomic force MicroscopY GIUSEPPE REGA

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