Magnetic Nanoparticle Relaxation Biosensor

Assay Conditions

Biomarker concentration 10 nM

AC drive frequency 16 kHz

Nanoparticle core diameter 20 nm

Solution viscosity 1.0 cP (buffer)

Live Readout

Bound fraction θ

Cluster size n̄

1.0

τ_Brownian (bound)

— μs

Phase lag δ

0.0°

Out-of-phase signal χ″ (a.u.)

0.00

Legend

Free nanoparticle (unbound)

Bound cluster (on biomarker)

Rotating AC field vector

How it works

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.

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^(1)  with  n̄ = 1 + θ(n_max−1)

Biomarker concentration — sets the bound fraction θ via a Langmuir binding curve; more target means more nanoparticles get pulled into clusters (orange).
AC frequency — the drive frequency f of the rotating field; the phase lag δ = atan(ωτ) grows once ωτ approaches 1.
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.

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.

Free particles Bound clusters

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