The 3D version of this sim advances surface coverage with a single deterministic ODE (molecules only bind, never unbind, and the whole beam is assumed to bend with one uniform curvature). This 2D companion computes the same real piezoresistive physics two genuinely different ways: (1) each of the 60 binding sites along the wire is tracked individually and undergoes real reversible Langmuir kinetics — binding and unbinding, each a Poisson process — so coverage fluctuates around a true dynamic equilibrium instead of saturating one-way; (2) the 60 sites are grouped into 30 beam segments, each with its own local curvature from its own local surface stress, and the whole bent shape is built by chain-integrating those local curvatures along the beam (a discretized elastica) rather than assuming one curvature for the whole length.
Per-site kinetics (Poisson transition probabilities each frame):
P(bind) = 1 − exp(−k_on · dt) empty → bound
P(unbind) = 1 − exp(−k_off · dt) bound → empty
θ_eq = k_on / (k_on + k_off) (real reversible-Langmuir equilibrium)
Per-segment beam bending (Δs = L/30):
κ_j = 6·σ_j·(1−ν) / (E·t²) local curvature from local coverage
θ_{j+1} = θ_j + κ_j·Δs midpoint-rule angle integration
(x,y) accumulate Δs·(cos θ_mid, sin θ_mid) segment by segment
Standalone verification (Node, run offline): with uniform coverage the 30-segment chain-integrated tip deflection matched the closed-form Stoney formula δ=κL²/2 to within 1×10⁻⁶ % at every coverage level tested (10–100%); the per-site stochastic kinetics converged, over long runs, to the analytic equilibrium θ_eq=k_on/(k_on+k_off) to within 1% for k_on/k_off ratios spanning 0.13–12.5; and — the point of building this spatially-resolved model — clustering the same total bound fraction near the tip instead of spreading it evenly changed the chain-integrated tip deflection by roughly 80% relative to the naive "one average curvature for the whole beam" estimate the 3D sim implicitly makes. That gap is exactly the real physics the aggregate-only model can't see: where molecules bind, not just how many, changes how the beam actually bends.
- p-type / n-type — silicon's dominant carrier sets the sign and magnitude of the piezoresistive coefficient π_L; n-type ⟨100⟩ resistance drops as the wire bends, p-type ⟨110⟩ resistance rises.
- k_on / k_off — on- and off-rates of the reversible binding reaction; their ratio sets the true equilibrium coverage, and their sum sets how fast the surface reaches it.
- Thickness t — thinner nanowires bend far more per unit surface stress (κ ∝ 1/t²), trading mechanical robustness for sensitivity.
This reversible-binding, spatially-resolved model is closer to how a real silicon-nanowire biosensor actually behaves: target molecules continuously associate and dissociate from the surface, and local clustering (e.g. diffusion-limited arrival concentrating near one end) is a real effect a single aggregate coverage number cannot capture.