AFM Tip–Sample Force Curve: Snap-In & Pull-Off

Piezo & Cantilever

Z-piezo position 8.0 nm

Auto approach / retract sweep

Spring constant k 2.0 N/m

Hamaker constant A 1.0 ×10⁻¹⁹ J

Imaging Mode

Drive frequency 1.00 × f₀

Live Readout

Tip–sample force

0.0 nN

Gap distance d

8.0 nm

Deflection

0.0 nm

Regime

—

Osc. amplitude

— nm

Phase lag

— °

blue = F(d) · orange dashed = cantilever load line

How it works

The tip is modelled as a sphere of radius R = 20 nm approaching a flat sample. Below the contact separation d₀ = 0.3 nm the attractive van der Waals force dominates; below it, elastic repulsion (DMT contact model) takes over:

d ≥ d₀:  F(d) = − A·R / (6 d²)                          (van der Waals)
d < d₀:  F(d) = − A·R/(6 d₀²) + (4/3) E*·√R · (d₀−d)^1.5   (DMT elastic contact)

The cantilever is a damped spring driven by the piezo: m·d̈ = −k(d − z) − c·ḋ + F(d), where z is the commanded piezo height (plus a sinusoidal dither in tapping mode). Mechanical snap-in happens when the local slope of the attractive force, dF/dd, exceeds the cantilever stiffness k — the spring can no longer hold a stable equilibrium and the tip jumps into contact; pulling back out shows the mirror-image pull-off jump at a different piezo height, an adhesion hysteresis loop visible as a loop on the F–d chart during an auto-sweep.

In tapping mode the piezo base is oscillated at a set fraction of the cantilever's natural frequency f₀ while the tip periodically taps the surface. Amplitude and phase are recovered the way a real AFM's lock-in amplifier does it: running in-phase/quadrature averages X = ⟨d·cos ωt⟩, Y = ⟨d·sin ωt⟩ give amplitude = 2√(X²+Y²) and phase = atan2(Y,X) — amplitude damping and phase lag relative to the free-air drive are exactly the signals real tapping-mode AFMs use to image topography and material contrast.

Note: real cantilevers oscillate at kHz–MHz; the dynamics here run on a slowed, illustrative timescale (mass is scaled so the natural frequency reads a few hertz) so the snap-in, pull-off and ring-down are visible in real time — the force law and equilibrium condition above are unscaled and exact.

CONTACT MODE

Cantilever & tip Sample surface

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