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Use a freely rotating chain: fixed bond length $b$, fixed angle $\vartheta$ between forward bond vectors, and independently random torsion angles. Then $⟨{\hat{t}}_0\cdot {\hat{t}}_n⟩=(\cos \vartheta {)}^n=e^{-nb/{\ell }_p}$, giving the orientational persistence length of a freely rotating chainThe equivalent long-chain Kuhn length divided by two, $b(1+\cos \vartheta )/[2(1-\cos \vartheta )]\simeq 3.48\mathrm{nm}$, is nearly equal here but is a distinct discrete-chain convention. A chemical internal bond angle is supplementary to the forward-vector angle used above.

Synthetic examples are relatively stiff poly(p-phenylene terephthalamide) and flexible polydimethylsiloxane. Light scattering and viscometry established rodlike solution conformations for the former; single-chain mechanical measurements support the latter's flexibility. Persistence is solvent- and model-dependent, so no universal numerical value is inferred for either example.^[ref][ref]

single-molecule force spectroscopy measures force and extension of an individual tether while controlling position or load. In an atomic-force arrangement, a synthetic chain connects a cantilever probe to a translated substrate. For the cantilever compliance correction, a deflection sensor gives $d$, a calibrated spring constant gives $F=k_{c}d$, and substrate displacement $Z$ gives chain extension $x=Z-d$ after geometric offsets are removed. Additional linkers and instrument compliance must be included if appreciable. Thermal deflection calibration uses $k_c⟨d^2⟩=k_{B}T$ only after isolating the cantilever's calibrated thermal contribution.

An optical-tweezers implementation detects a tethered bead's displacement from a calibrated trap centre, with $F=k_{\mathrm{trap}}\Delta x$ in the harmonic regime. Detector calibration, drag and trap linearity determine the usable force and frequency range. These instruments measure individual force–extension curves and distributions; averaging many curves can hide multiple tether populations.

For a synthetic freely jointed chain with Kuhn length $b$,A strongly stretched worm-like chain instead has $1-x/L\simeq \sqrt{k_{B}T/(4F{\ell }_p)}$ before backbone stretching becomes important. Fits can estimate contour and persistence lengths only in their appropriate regimes. Hysteresis and pulling-rate dependence test whether an apparent mechanical transition is quasistatic or activated.

Control single versus multiple attachments, subtract nonspecific adhesion, vary pulling speed, calibrate temperature and stiffness, and correct drift. A rupture peak is not a universal bond strength independent of loading history. A primary nonbiological example measured individual polydimethylsiloxane chains by atomic-force methods; its flexibility interpretation combines those measurements with molecular calculations.^[ref]