NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY GIUSEPPE REGA S APIENZA U NIVERSITY OF R OME, I TALY D...

NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY

GIUSEPPE REGASAPIENZA UNIVERSITY OF ROME, ITALY

DEPARTMENT OF STRUCTURAL AND GEOTECHNICAL ENGINEERING

Co-workers: VALERIA SETTIMI, UGO ANDREAUS, LUCA PLACIDI

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 MICROSCOPYGIUSEPPE REGA

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 REGANONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY


B. TAPPING AFM

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

23/07/20134th 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


B. TAPPING AFM

2. MODELING

• Extended Hamilton principle initial boundary value problem for transverse vibration:




B. TAPPING AFM

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

2.1. MODELING - MULTI-MODE MODEL



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


B. TAPPING AFM

2.2. MODELING - QUASI-STATIC BEHAVIOR - 1 -



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.


B. TAPPING AFM

2.2. MODELING - QUASI-STATIC BEHAVIOR - 2 -



Maximum distance

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


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




B. TAPPING AFM

3.2. NONLINEAR HYSTERESIS - APPROACH/RETRACT SEPARATION SWEEP - 1 -

Single-mode vs three-mode: resonance of 1st mode

Maximum distance Phase difference

Hysteresis phenomena also occur !!

Response in nominally monostable region: range of “high” separation values (24 26 nm)




B. TAPPING AFM

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 harmonicsSteep 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




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




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



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 = 33.3. 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


B. TAPPING AFM

4. INFLUENCE OF MODAL DAMPING (Q-FACTORS)




B. TAPPING AFM

First resonance

HIGHER RANGE of Q-factor values: Q1 = 33.3. Q2 = 333, Q3 = 3330

Nominal FLUID with VERY LOW VISCOSITY

4.1. INFLUENCE OF Q-FACTORS - AIR vs LIQUID-LIKE VALUES - 1 -

Low modal dampings, staying far away from sample no significant change in nonlinear hysteresis

phase difference




B. TAPPING AFM


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: Q1 = 3.33. Q2 = 5.31, Q3 = 19

Nominal FLUID with RELATIVELY HIGH VISCOSITY




B. TAPPING AFM


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




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 ! ! !

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 MODElimited 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

NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY GIUSEPPE REGA S APIENZA U NIVERSITY OF R OME, I TALY D...

Documents

Transcript of NONLINEAR DYNAMICS OF ATOMIC FORCE MICROSCOPY GIUSEPPE REGA S APIENZA U NIVERSITY OF R OME, I TALY D...