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NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY GIUSEPPE REGA S APIENZA U NIVERSITY OF R OME, I TALY D EPARTMENT OF S TRUCTURAL AND G EOTECHNICAL E NGINEERING 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 2. MODELING 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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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) 1. COMPLEMENTARY AFM TOPICS for NONCONTACT and TAPPING GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY

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GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM B. TAPPING AFM - AIM OF INVESTIGATION 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

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 5/31 d Attractive-repulsive tip-sample interaction: van der Waals and Derjaguin-Muller-Toporov contact forces GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 2. MODELING Extended Hamilton principle initial boundary value problem for transverse vibration:

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 6/31 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM Nondimensionalization, assumed mode technique system of ODEs Damping matrix C ij : 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 2.1. MODELING - MULTI-MODE MODEL

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 7/31 The inclination of the green line represents the stiffness of the micro-cantilever Sample its intersection with the interaction force represents the equilibrium tip position d d GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 2.2. MODELING - QUASI-STATIC BEHAVIOR - 1 -

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 8/31 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. GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 2.2. MODELING - QUASI-STATIC BEHAVIOR - 2 -

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 9/31 Vibrations of cantilever tip induced through oscillations of a dither piezo at cantilever support. Sample AFM tip y 0 Sin Ωt Same hysteretic and bistable behavior in dynamics Z(t) GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 3. NONLINEAR HYSTERESIS 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

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 10/31 Maximum distance Single-mode vs three-mode (enough for converging response) GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 3.1. NONLINEAR HYSTERESIS - FREQUENCY SWEEP Phase difference Frequency range of bistable solution with three modes non-trivially smaller Significant variation on saturated branch Higher harmonic contributions

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 11/31 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 3.2. NONLINEAR HYSTERESIS - APPROACH/RETRACT SEPARATION SWEEP Single-mode vs three-mode: resonance of 1 st mode Maximum distancePhase difference Hysteresis phenomena also occur !! Response in nominally monostable region : range of high separation values (24 26 nm)

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 12/31 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 3.2. NONLINEAR HYSTERESIS - APPROACH/RETRACT SEPARATION SWEEP Three-mode model: resonance of 2 nd 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

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 13/31 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 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

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 14/31 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 3.2. NONLINEAR RESPONSE - SEPARATION SWEEP - IMPACT VELOCITY/CONTACT FORCE SECOND resonance: range of very low separation values 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 Intersection with zero distance line and chaotic response

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 15/31 Single-mode model: characterizing damping through equivalent Q-factor (as in experimental AFM) Q 1 = 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 Q 1 = Q 2 = 5.31, Q 1 = 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 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 4. INFLUENCE OF MODAL DAMPING (Q-FACTORS)

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 16/31 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM First resonance HIGHER RANGE of Q-factor values : Q 1 = Q 2 = 333, Q 3 = 3330 Nominal FLUID with VERY LOW VISCOSITY 4.1. INFLUENCE OF Q-FACTORS - AIR vs LIQUID-LIKE VALUES Low modal dampings, staying far away from sample no significant change in nonlinear hysteresis phase difference

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 17/31 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 4.1. INFLUENCE OF Q-FACTORS - AIR vs LIQUID-LIKE VALUES 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 First resonance LOW RANGE of Q-factor values : Q 1 = Q 2 = 5.31, Q 3 = 19 Nominal FLUID with RELATIVELY HIGH VISCOSITY

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 18/31 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 4.1. INFLUENCE OF Q-FACTORS - AIR vs LIQUID-LIKE VALUES st 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

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23/07/2013 4th Canadian Conference on Nonlinear Solid Mechanics (CanCNSM2013) Pagina 19/31 GIUSEPPE REGA NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY B. TAPPING AFM 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 ! ! !

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8. CONCLUSIONS NONCONTACT AFM 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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