Chain formation onset, two ways

The same calculation run twice, side by side. Change anything on either side and the difference is shown output by output. This page is a link: both sets of numbers are in the address bar.

Side A open alone

m Particulate

The magnetic solid. Its saturation magnetisation sets each particle's moment, taken as fully magnetised.

d Particle diameter

µm

The coupling goes with the cube of this: ten times the diameter, a thousand times the coupling. The default is a ten-nanometre ferrofluid grain, which is the only size range where the answer is in doubt.

T Temperature

°C

Thermal energy is what keeps the particles apart.

Side B open alone

m Particulate

The magnetic solid. Its saturation magnetisation sets each particle's moment, taken as fully magnetised.

d Particle diameter

µm

T Temperature

°C

Thermal energy is what keeps the particles apart.

Swap sides

What changed

The two sides are the same. Change something on either.

Output	A	B	B against A
λ Dipolar coupling constant	19.58	19.58	same
Ec Contact energy	7.923e-11 nJ	7.923e-11 nJ	same

Side A, with its numbers

19.58 = \frac{7.923e-11 nJ}{4.047e-12 nJ}, 7.923e-11 nJ = \frac{μ_{0} {8.901e-19}^{2}}{4 π 10^{nm 3}}

LaTeX

19.58 = \frac{7.923e-11\,\mathrm{nJ}}{4.047e-12\,\mathrm{nJ}}, \qquad 7.923e-11\,\mathrm{nJ} = \frac{\mu_0 \cdot 8.901e-19^2}{4 \pi \cdot 10\,\mathrm{nm}^3}

λ is 19.6, above 3: the dipole attraction is well beyond thermal energy and the particles chain along the field into structures that persist. The medium in the field is non-Newtonian — it has a yield stress and a viscosity that depends on the shear — and the magnetoviscous calculation's assumption of separate spheres has failed; its figure is not a floor but the wrong picture. Thermal energy is 4.047e-12 nJ against a contact energy of 7.923e-11 nJ.
This is a threshold, not a structure: λ says whether chains form, and nothing about how long they are or what the chained medium's yield stress is. The thresholds are from simulation and experiment on ideal spheres, and the archive holds this page at lower confidence than the rest of the group.

Side B, with its numbers

19.58 = \frac{7.923e-11 nJ}{4.047e-12 nJ}, 7.923e-11 nJ = \frac{μ_{0} {8.901e-19}^{2}}{4 π 10^{nm 3}}

LaTeX

19.58 = \frac{7.923e-11\,\mathrm{nJ}}{4.047e-12\,\mathrm{nJ}}, \qquad 7.923e-11\,\mathrm{nJ} = \frac{\mu_0 \cdot 8.901e-19^2}{4 \pi \cdot 10\,\mathrm{nm}^3}

λ is 19.6, above 3: the dipole attraction is well beyond thermal energy and the particles chain along the field into structures that persist. The medium in the field is non-Newtonian — it has a yield stress and a viscosity that depends on the shear — and the magnetoviscous calculation's assumption of separate spheres has failed; its figure is not a floor but the wrong picture. Thermal energy is 4.047e-12 nJ against a contact energy of 7.923e-11 nJ.
This is a threshold, not a structure: λ says whether chains form, and nothing about how long they are or what the chained medium's yield stress is. The thresholds are from simulation and experiment on ideal spheres, and the archive holds this page at lower confidence than the rest of the group.