Magnetic relaxation time

How long does a suspended particle keep its magnetic orientation?

The formula

τB = \frac{3 η0 Vh}{kT}, \frac{1}{τ} = \frac{1}{τB} + \frac{1}{τN}

τB: Brownian relaxation time, s
η0: Carrier viscosity, Pa·s
Vh: Hydrodynamic volume, π(d + 2δ)³ ÷ 6, m³
kT: Thermal energy, k_B × T in kelvin, J
τ: Effective relaxation time, s
τN: Néel relaxation time, τ₀ exp(KV ÷ k_BT), s

LaTeX

τB = \frac{3 \cdot η0 \cdot Vh}{kT}, \qquad \frac{1}{τ} = \frac{1}{τB} + \frac{1}{τN}

Work it out

m Particulate

The magnetic solid. Sets the anisotropy constant that holds the moment inside the grain, and the single-domain limit.

d Particle diameter

µm

The magnetic core, treated as a sphere. Both relaxation times grow with its volume — one linearly, one exponentially.

c Carrier fluid

The liquid the particle turns in. Its viscosity is what resists the Brownian rotation.

T Temperature

°C

Thermal energy is what randomises the orientation, by both routes.

δ Surfactant shell thickness

The coating that keeps the particles apart. It turns with the particle, so it adds to the hydrodynamic volume but not to the magnetic one.

or see a worked example with typical numbers

Method

Convert the temperature to kelvin and multiply by Boltzmann's constant, 1.38 × 10⁻²³ J/K, for the thermal energy k_BT — about 4 × 10⁻²¹ J at room temperature.
Find the hydrodynamic volume: a sphere of diameter d + 2δ, the core plus the surfactant shell on both sides, π(d + 2δ)³ ÷ 6. The shell turns with the particle, so it counts here.
The Brownian time is three times the carrier's viscosity times that volume, divided by k_BT. Water at 1 mPa·s and a micron particle give about 0.4 s; a hundred-nanometre particle, a thousand times less volume, gives 0.4 ms.
Find the magnetic volume, πd³ ÷ 6 of the core alone, and multiply by the material's anisotropy constant K for the barrier the moment must hop over. The Néel time is an attempt time of one nanosecond times e raised to that barrier over k_BT.
If the barrier exceeds a hundred k_BT the exponential is astronomically large and the number means nothing more than "blocked" — it is capped there and said so.
If the particle is larger than the material's single-domain limit it is many domains, and the Néel picture does not describe it at all: the effective time is the Brownian one.
Otherwise add the two rates: 1/τ = 1/τ_B + 1/τ_N. The shorter time dominates, and its name is the regime.

Assumptions

A rigid sphere in a Newtonian liquid, rotating freely — Debye's rotational diffusion. A slurry dense enough that the particles touch or chain rotates as a structure, not as spheres, and holds its orientation longer than this says.
A single-domain particle with one uniaxial easy axis and a barrier of KV — the Néel–Brown picture with a nanosecond attempt time. Real τ₀ is anywhere from 10⁻⁹ to 10⁻¹¹ s, and the anisotropy of a small particle is often larger than the bulk constant used here; the exponent is what matters, and a factor of a few in it moves the time by orders of magnitude.
Above the single-domain limit the moment is not one moment, and "Néel relaxation" is the wrong question: a multi-domain particle's magnetisation changes by domain walls moving, which takes a field above the coercivity. In a field-free stretch of tube the only relaxation is the Brownian one.
No applied field. The moment relaxes toward random; in a field it relaxes toward the field instead, on times of the same order.
The carrier's viscosity is its 20 °C handbook value regardless of the temperature entered — the temperature affects the thermal energy here, not the viscosity table. A warm carrier is thinner and the Brownian time shorter than shown.
Nothing here is Meyer's. The estate material gives a speed for the medium and no particle size, no carrier and no surfactant; the calculation exists because whether a magnetised slug survives the trip to a pickup coil turns entirely on those three things.