Chain formation onset

Will the particles chain up in the field?

λ = \frac{Ec}{kT}, \qquad Ec = \frac{\mu_0 \cdot mp^2}{4 \pi \cdot d^3}

Method

1. Find the particle's moment: the saturation magnetisation times the sphere's volume, πd³ ÷ 6. The particle is taken as fully magnetised, which in any field worth applying it is.
2. The energy of two such moments touching head to tail is μ₀m² ÷ 4πd³ — the dipole–dipole energy at a centre separation of one diameter.
3. Divide by k_BT for λ. Because m² goes with d⁶ and the denominator with d³, λ goes with d³: doubling the diameter multiplies it by eight.
4. Read the regime off the thresholds: below 1 the particles stay separate; between 1 and 3 they chain along the field and disperse when it is removed; above 3 the chains persist and the medium is a structured fluid.

Assumptions

• Identical spheres, fully magnetised, touching — the contact separation is one diameter and any surfactant shell is ignored. A shell of thickness δ raises the separation to d + 2δ and lowers the coupling by (d ÷ (d + 2δ))³, which for a ten-nanometre grain with a two-nanometre coat is a factor of 2.7; a coated ferrofluid is more stable than the bare figure says.
• The thresholds at λ ≈ 1 and 3 are from simulation and experiment on monodisperse dipolar hard spheres in the dilute limit. Real powders are polydisperse, and the largest particles chain first; the loading also matters — at a few per cent the chains are sparse, and at tens of per cent they span the tube — and this page does not take the loading as an input, because λ is a threshold, not a structure.
• What the page does not say: chain length, yield stress, or the viscosity of the chained medium. It says only when the separate-spheres assumption of the magnetoviscous calculation has failed, and above λ of a few it has.
• Nothing here is Meyer's. The estate material puts a magnetic slurry through a coil; the calculation exists because a slurry of micron iron in a field is not a liquid of separate particles at all — the coupling is in the millions — and the medium in the coil is a very different thing from the medium in the pump.