This is the Prandtl-Tomlinson model of atomic friction — the physics behind friction/lateral-force microscopy (FFM). A tip is held by a lateral spring of stiffness k whose free end (the "support") is dragged at constant velocity. The tip itself feels a periodic lattice potential from the atoms below, of period a and corrugation force amplitude F₀:
γ·ẋ = k·(X_s − x) − F₀·sin(2πx/a) (overdamped tip motion)
η = 2πF₀ / k (stability parameter)
- When η > 1 (soft spring / strong corrugation), the force-balance equation has multiple solutions: the tip stays pinned ("sticks") in one lattice well while the support keeps moving, building up spring tension — until the well's restoring force is overcome and the tip suddenly snaps ("slips") into the next well. This produces the sawtooth lateral-force signal and a burst of kinetic energy that is dissipated as heat (phonons) at every slip — exactly what a real FFM cantilever's torsional signal records, atom row by atom row.
- When η < 1 (stiff spring / weak corrugation), the tip can always find a smooth equilibrium as the support advances — motion is continuous, the lateral force trace is a smooth sinusoid, and almost no energy is dissipated. This is the microscopic origin of superlubricity.
- The forward (trace) and backward (retrace) sweeps trace out a hysteresis loop whose area equals the energy dissipated per cycle; half the vertical gap between the two branches is the friction force reported by a real FFM measurement.