Most of what we know about magnetic colloids with off-centre dipoles comes from simulations that squeeze the particles into a plane, the way a real sample sits between two glass slides. That geometry is honest about many experiments and it has been productive for us. It also quietly removes one degree of freedom from every particle, and the question of what that removal costs had been sitting in our own outlook sections for years.
Our new paper in Physical Review E, by Wilmer A. Martinez-Valle and me, answers it by taking the walls away.
What was simulated
The particles carry a magnetic moment displaced along the radius, pointing out toward the surface rather than sitting at the centre. We call that displacement s, scaled by the particle radius. We simulated it from 0 up to 0.7, and beyond that the figures show only a schematic continuation, because the model stops being trustworthy there and we would rather say so than extrapolate.
Everything happens by Brownian dynamics in a periodic cubic box with no applied field, at a volume fraction of 0.001, with the dipolar coupling constant λ swept from 10 to 100 and λ = 45 as the representative case. If λ is unfamiliar, it is the ratio of magnetic to thermal energy, and the companion explainer published earlier this month walks through it term by term. Two particles count as bonded when their centres sit closer than 1.2 diameters. The integration steps at one ten-thousandth of the diffusive time and runs out past a thousand diffusive times, which is long enough for the slow cases to stop being transient.
The dilute limit lets us drop hydrodynamic interactions, and we do. That is a simplification, not a discovery, and the paper lists it first among the things to add next.
Confinement was hiding structures
Sweep the shift upward in three dimensions and the suspension walks through a sequence. At small s the particles build chains and loops, which is what the planar work also gives. Past that, rings and mixed loops take over. Keep going and the system produces compact globules, multiring structures and vesicle-like shells.
The last group is the point. Those are not variations on a planar structure; they are assemblies that the quasi-two-dimensional geometry suppresses outright, because closing a shell needs particles to come at each other from directions the plane forbids. Our 2020 planar study of the same radially shifted particles mapped eight cluster phases across the coupling and the shift. None of them was a shell.
So dimensionality is not a background detail of the simulation setup. It acts as a control parameter in its own right, on the same footing as the coupling constant and the dipole shift, and a phase map drawn in a plane is answering a narrower question than it appears to.
Growth has two stages and four regions
The kinetics came out less tidy than the structures, in a useful way. Aggregation does not start when the clock does. There is a delay while the system nucleates, and only then does the mean cluster size climb as a power of time, with the exponent depending on both the shift and the coupling.
Plotting that exponent across the coupling and the shift gives four regions rather than one law. In one corner, at low coupling and large shift, the exponent sits near zero and the suspension essentially never aggregates. A slow region follows. Then a broad region where the exponent lands between about 0.4 and 0.5, which is the behaviour most of the literature reports. And at high coupling with small shift, a fourth region where growth runs faster still.
A single growth exponent quoted for this class of system is therefore a regional claim, not a universal one, and the same holds for the rings-versus-shells question above. Strong coupling and a nearly centred dipole put you in the fast corner. Weak coupling and a strongly displaced dipole put you somewhere that looks, on an experimental timescale, like nothing happening at all.
What we would want next
The model is deliberately plain: no hydrodynamic coupling, a finite cutoff on the dipolar interaction, identical particles, no external field. Each of those is a real limit and each is named in the paper as the next step, which is the honest order to do this in. The field case is the one we are most curious about, because an applied field competes with the dipole geometry rather than simply reinforcing it, and that competition is where the planar work found its sharpest boundaries.
The simulation data are deposited on Zenodo; the paper cites a specific version of that record.
Martinez-Valle, W. A., & Córdova-Figueroa, U. M. (2026). Three-dimensional aggregation dynamics of magnetic Janus colloids with radially shifted dipoles. Physical Review E, 114(2), 025414. DOI: 10.1103/c4zw-zwrn