A direct method for an elusive parameter

The parameter nobody could measure directly

Friction at the atomic scale, as probed by friction-force microscopy (FFM), produces characteristic stick-slip force traces. Standard analysis of those traces extracts two things from the stick phase: the effective tip stiffness and the maximum lateral (slip) force. The damping coefficient γ — the quantity that governs how mechanical energy is dissipated into heat and lattice vibrations at the contact — has no direct method. It gets inferred indirectly, through friction-loop hysteresis, through barrier-crossing rate theory, or by correlation with other observables. In the thesis I called it the “holy grail” of nanotribology: an elusive parameter with no direct method I’m aware of, because the tools that are good at stick phases are blind to the thing you want.

Flip the analysis onto the slip phase

The move in the thesis is to throw away the stick-phase obsession and look at what happens during and just after the slip — specifically, the ringdown. A slip is a rapid, damped oscillation in a harmonic-like potential well; the analytic solution to that oscillation (under-, critically-, or over-damped) has γ as a parameter. Fit the post-slip ringdown, read off γ directly.

To verify this works, I needed a system where the ground-truth damping was known. That meant a full Prandtl–Tomlinson (PT) Langevin simulator, from scratch, in Python:

• Fourth-order stochastic Runge–Kutta — chosen because Runge–Kutta-type methods don’t require derivatives of the drift or diffusion coefficients.
• Physically-calibrated parameters: dt = 2×10−8 s, m = 10−12 kg, tip spring KS = 1–10 N/m, scan velocity V ≈ 103 nm/s, thermal T = 300 K, PT corrugation parameter η = 12 and η = 15.
• 1-D and 2-D systems, Numba JIT compilation — single-trajectory integration from ~40 s to ~80 ms so parameter sweeps finished in minutes instead of hours. Dataset ended at 100 trajectories × ~158 K points each, 16M+ points total.

Repository: github.com/Zuzutus/sde-solver.

Detecting slips without handwaving

Before you can fit a slip you have to find one. The signature I used is that the force trace changes value slowly during stick and rapidly during slip — so a 20-point windowed moving standard deviation of the trace lights up exactly where a slip is happening. Threshold-crossing plus a local-max finder gives a slip midpoint; a dynamic forward/backward segmentation grows the window until the point-to-point delta falls below a fraction of the local peak. A Gaussian moving average smooths first; a fit-window sweep keeps the best R² sub-segment and explicitly excludes the jump initiation and the stick re-entry, since the linear oscillator solution doesn’t describe those. 100 slip events per parameter condition, statistics reported as medians and IQRs.

Result: fitted γ tracks the simulator γ

Because γ_sim is set explicitly in the simulator, the validation is a direct comparison against ground truth. The method recovers γ in 1-D independent of load and in 2-D independent of scan angle on a hexagonal lattice. The fitted angular frequency ω tracks the effective stiffness K_eff with Pearson correlation 0.953–0.995 in 1-D and 0.931–0.979 in 2-D.

At the unphysical extremes the method has bounds: γ is overestimated by up to 19% for heavy underdamping and underestimated by up to 15% for heavy overdamping. These extremes aren’t what real systems do — but quantifying the degradation there was how I figured out the method’s practical edges.

One finding that surprised me: critically-damped simulations come out slightly underdamped in the fits. Rather than hand-wave, I derived why — the effective stiffness K_eff picked up by the ringdown is strictly smaller than the bare tip stiffness K_S — and ended the thesis with a practical recommendation: use the underdamped expression for both underdamped and critically damped systems. Experimentalists get a direct tool; theoreticians get a little more to argue about.

Ongoing — applying the method to MD data

The thesis validated the method against a simulator where the answer was known. The natural next step is to run it on molecular-dynamics trajectories of tip–surface sliding, where the answer isn’t known. I’m currently porting the pipeline to MD force traces: the fitting and segmentation carry over; the open problem is robust, physically-grounded filtering, since real MD output has baseline drift and noise structure the PT simulator doesn’t. That’s the active blocker — not the math, the pre-processing.