Put a suspension of magnetic colloids under a microscope and you get one of two films. In the first, particles wander, meet, drift apart again, and nothing survives the encounter. In the second they find each other and stay found, and within a few seconds the frame fills with chains and rings that were not there at the start. Same particles, same temperature, same solvent. What separates the two films is a single number, and in our simulations we call it lambda.
Two energies, one division
A magnetic colloid in water is pulled on by two things. Its neighbour acts on it through a magnetic dipole field, and the water kicks it at random. Both of those are energies, so comparing them is division rather than anything deeper:
λ = μ0m2 / (4πa3kT)
The numerator is the magnetic energy of two dipoles sitting roughly a particle apart, with m the dipole moment of one particle and μ0 the permeability of vacuum. The denominator is thermal: k is Boltzmann's constant, T the absolute temperature, and a the particle radius, which enters cubed because the dipolar energy falls off with the cube of separation.
When λ is large the magnetic term dominates and a bond, once formed, survives the next thermal kick. When λ is small the kicks win and the suspension stays dispersed. That is the whole idea, and it is the reason we never report a magnetic colloid result without saying which λ it belongs to.
Where the threshold actually sits
That comparison needs numbers, so here are ours. In the quasi-two-dimensional study we published in 2019, at a surface fraction of 0.01, we saw no aggregation at all below roughly λ = 15. From 15 up to 45 was the working window where structures formed and could be characterised. Above about 75 the answer began to depend on how large we made the simulation box rather than on the physics, which is a warning sign rather than a result.
Those thresholds belong to that geometry and that concentration, and it is worth being blunt about it. In the three-dimensional, field-free runs we published this year we swept λ from 10 to 100 at a volume fraction of 0.001 and used λ = 45 as the representative case. A threshold is a property of a system, not of the symbol.
Lambda sets the clock, not the shape
The part that surprises people is what λ does not control. Raising it mostly makes assembly faster rather than different. In our 2019 results the long-time structures were set by where the dipole sits inside the particle, a displacement we write as s, while λ governed how quickly the system got there. At λ = 45 with a laterally shifted dipole, the long-time average cluster size fell from about 20 particles at s = 0 to about 3 at s = 0.6. The shift did that, not the coupling.
There is one place where λ does reach into structure. At λ of 80 and above we found a coexistence of loops and island-like clusters around s = 0.3 to 0.4 that does not appear at lower coupling. One exception in a clean division of labour is a good sign that the division is real.
So if you want chains to form at all, λ has to clear the threshold for your geometry. If you want them to form sooner, raise λ. If you want a different shape, λ is the wrong knob, and the two posts on moving the magnet off-centre and on what happens to those clusters under shear are about the knobs that are.
The trap in the definition
Here is the practical warning. Not every paper writes λ the same way. Some build it on the particle radius and a 4π, as above. Others build it on the diameter and a 2π. The underlying idea is identical in both, dipolar energy over thermal energy, but the prefactors are not, so the numbers are not interchangeable.
Go back to each paper's printed definition and put both on one convention before comparing. A reported λ of 20 is not the same physical state as another reported λ of 20 if one of them was defined on diameter. We flag this in our own notes because we have had to do the conversion more than once.
What the number does not cover
One last boundary. λ compares magnetism with temperature and nothing else. It is silent about flow, which is the Péclet number's job, and silent about an applied field, which brings in a separate Langevin parameter. It also says nothing about hydrodynamic interactions, which the dilute simulations above neglect outright and name as the next thing to add.
A good dimensionless number is narrow. λ answers one question well: will a magnetic bond outlast a thermal kick? Everything else needs its own ratio.
Vega-Bellido, G. I., DeLaCruz-Araujo, R. A., Kretzschmar, I., & Córdova-Figueroa, U. M. (2019). Self-assembly of magnetic colloids with shifted dipoles. Soft Matter, 15(20), 4078–4086. DOI: 10.1039/c8sm02591f