Confined-Film Stick-Slip: Prandtl–Tomlinson Simulator (2D)

This 2D companion to the nanotribology squeeze-film simulator keeps the same real solvation-force layering physics (confined layer count n = round(D/σ), oscillatory force F(D) ≈ e^(−D/ξ)·cos(2πD/σ)) but computes friction a genuinely different way. Instead of scripting a sawtooth or sine-wave slide, it integrates the actual overdamped Prandtl-Tomlinson equation of motion — a spring-driven tip riding a periodic lattice potential whose corrugation is screened by the number of confined lubricant layers — with 4th-order Runge-Kutta at every frame. Stick-slip is not switched on by a threshold: it emerges on its own once the dimensionless Tomlinson parameter η = 4π²U₀/(kσ²) crosses 1, exactly the bifurcation that produces the atomic-scale stick-slip friction traces measured by real AFM and Surface Forces Apparatus experiments. Squeeze the gap down to thin the film and watch the live hysteresis-loop panel switch from a smooth, single-valued curve to a jagged sawtooth as the confined layers stop screening the substrate's atomic corrugation.