Superparamagnetic Fe₃O₄ nanoparticles, coated with antibodies, tumble freely in an oscillating magnetic field and their moment tracks the field almost instantly. When a biomarker is present it cross-links several nanoparticles into a cluster — its hydrodynamic volume jumps, so its rotation (and its moment) lags the field far more. That extra phase lag, read out as the AC-susceptibility signal, is the whole assay. The chamber panel shows the population directly; the phasor panel is the same lag drawn as two rotating clock hands; the Debye panel shows where that lag sits on the universal χ″(ωτ) response curve.
Brownian relaxation: τ_B = 3 η V_h / (k_B T)
Néel relaxation: τ_N = τ_0 exp(K_a V_core / k_B T)
Effective: 1/τ = 1/τ_B + 1/τ_N
Phase lag: δ = atan(ω τ), ω = 2π f
Out-of-phase (Debye): χ″ ∝ ωτ / (1 + (ωτ)²)
Binding (Langmuir): θ = [A] / ([A] + K_d)
Cluster hydrodynamics: V_h,bound = V_h,free · n̄ with n̄ = 1 + θ(n_max−1)
- Biomarker concentration — sets the bound fraction θ via a Langmuir binding curve; more target means more nanoparticles are pulled into clusters (orange) in the chamber panel.
- AC frequency — the drive frequency f of the rotating field; the phase lag δ = atan(ωτ) grows once ωτ approaches 1, visible as the orange hand falling behind the yellow hand on the phasor panel.
- Core diameter — below ~12 nm the Néel time τ_N is far shorter than τ_B, so the internal moment flips freely inside a stationary particle and binding barely changes the signal — too small to sense. Above ~18 nm τ_N is effectively frozen (blocked) and only the physical rotation (τ_B) — which binding slows — carries the signal. That is why real assays pick particles in the 20–30 nm range.
- Viscosity — a thicker matrix (serum vs. buffer) slows Brownian rotation for everyone, shifting the whole assay's optimal frequency window; watch both markers slide along the Debye curve together.
This is the physical principle behind magnetic-relaxation / frequency-mixing immunoassays used in point-of-care diagnostics: unlike optical assays, they work directly in turbid or opaque samples (whole blood, serum) because the readout is magnetic, not optical.