Ferrofluid Spikes
Raise the field past a threshold and a flat pool erupts into a lattice of black spikes, each holding its distance from its neighbours. Drop the field and it is a puddle again at once.
Magnetic fluid in a vertical field is pulled upward by the field and held down by gravity and surface tension. Below a critical field the flat surface wins. Above it, the flat surface is unstable to a particular wavelength, and the pool breaks into spikes at that spacing, arranged six around one. The threshold, the spacing and the shape of the lattice are not free: they follow from the density, the surface tension and the permeability of the fluid. None of which the fluid supplies. The field is handed in from outside, and the whole structure lives exactly as long as the field does.
A Pool in a Vertical Field
Where the Numbers Come From
The fluid is the one Gollwitzer, Rehberg and Richter measured in their hexagon-to-square experiments, Ferrotec APG 512 A: density 1236 kg/m³, surface tension 30.6 mN/m, susceptibility χ = 1.17. Everything below follows from those three numbers and gravity.
λc = 2π/kc
Λ = μ0(μr−1)² / (1 + 1/μr)
Hc² = 2√(ρgσ) / Λ
Bc = μ0μrHc
a = (2/√3) λc
| quantity | computed | measured |
|---|---|---|
| capillary wavenumber kc | 0.630 /mm | 0.63 /mm |
| critical wavelength λc | 9.98 mm | about 1 cm |
| spike spacing a | 11.53 mm | not reported |
| critical induction Bc | 15.60 mT | 16.7 mT |
The last row is the honest one. Linear theory lands 6.6% below the measured onset, and it should: it assumes the magnetisation rises in proportion to the field, and in a real ferrofluid it does not. The same authors, doing the quantitative version of this comparison, feed their measured magnetisation curve into a finite-element calculation for exactly that reason. Six and a half percent is what the straight-line assumption costs.
The spacing row needs one remark. The lattice spacing is not λc. A hexagonal pattern is built from three wave vectors of length kc at 120° to one another, and the peaks those three make sit (2/√3) λc apart, about 15% further than the wavelength. The panel measures the spacing and prints the prediction beside it.
Why This Is an Attractlet
There are two things on this page and they are classified differently. The panel
is a simulation, and the kernel’s Attractlet Model class (ATTRACTLET,
§ 7.4) covers simulations and numerical solvers outright. That one is not a judgement call.
The physical pool is an authored reading, which is why the card tag carries no citation. The reading is this: the spike lattice is bounded, it is stable, and it returns when you disturb it, so on surface measurement it passes for an attractor. What it never does is make the thing holding it up. The field comes from a coil somebody switched on. Press Cut the field: the lattice does not decay, wind down, or leave a remnant. It stops being held, and the pool is flat inside a tenth of a second of model time.
The Shape of the Basin
At any field above threshold, almost every starting pool arrives at the same lattice: same spacing, same six neighbours, from a flat surface or from a disturbed one. Press Knock a spike down and the gap fills in within a frame or two. The lattice is a strong attracting state, and that is exactly what makes it easy to mistake.
What the basin does not give back is which lattice. The spacing returns; the positions do not have to. Where the spikes stand is set by whichever disturbance grew first, and the panel is deliberately built in a box that the predicted lattice tiles exactly, so the reader sees a clean pattern rather than one fighting its own boundaries. A pool in a dish has no such courtesy and is full of defects.
There is a second thing worth watching. Raise the field until spikes appear, then bring it back down: they do not vanish at the field where they appeared. They persist well below it and then collapse all at once. That is a subcritical bifurcation, and it is the same shape of behaviour, an abrupt jump with a loop, that the experiments report for the later hexagon-to-square change too.
What This Model Is Not
It is not a fluid solve. Between frames the panel advances an amplitude equation, the standard weakly-nonlinear description of pattern formation, whose linear operator was built to peak at the real kc and cross zero at the real threshold. That is why the threshold and the spacing come out right. Nothing else about it is a claim.
The clock is arbitrary. Growth rates are set by a relaxation constant, not by the viscosity of the fluid. Only ratios of times mean anything, and one of those ratios is real: near the threshold the pattern takes far longer to appear than well above it, which is critical slowing down and is not a defect of the panel.
The spike height is arbitrary. The vertical scale in the view is the amplitude equation’s, stretched for legibility, not a height in millimetres. The horizontal scale is real, which is why the readout measures spacing and says nothing about how tall a spike is. The surface is also smoother than the real thing: an amplitude equation gives rounded peaks, and a real Rosensweig spike comes to a much sharper point.
The hysteresis loop is half a prediction. A single quadratic coefficient in the amplitude equation does three jobs at once: it makes the spikes point up rather than down, it picks hexagons over stripes, and it makes the bifurcation subcritical. That the loop exists follows from the pattern being hexagonal. How wide it is follows from that coefficient, which is not measured from the fluid. Take the width as illustrative.
The slider stops at 17.1 mT on purpose. Above that this model stops holding a hexagonal lattice and goes to a nearly up-down symmetric state. Real ferrofluid does change pattern above threshold, from hexagons to squares, starting near 20.7 mT in the experiment cited below, and the two are not the same event: squares are still spikes, and what this model produces is not. Rather than show a model failing and let it be read as physics, the range ends where the lattice was verified to hold.
Cowley, M. D. & Rosensweig, R. E. (1967). The interfacial stability of a
ferromagnetic fluid. Journal of Fluid Mechanics, 30(4), 671–688.
Rosensweig, R. E. (1985). Ferrohydrodynamics. Cambridge University Press.
Gollwitzer, C., Rehberg, I. & Richter, R. (2006). Via hexagons to squares in ferrofluids:
experiments on hysteretic surface transformations under variation of the normal magnetic field.
arXiv:nlin/0607059. Source of the fluid parameters,
of Bc = 16.7 mT, of kc = 0.63 /mm, and of the hexagon-to-square fields.
Cross, M. C. & Hohenberg, P. C. (1993). Pattern formation outside of equilibrium.
Reviews of Modern Physics, 65(3), 851–1112.