# Colloidal test bed for universal dynamics of phase transitions

- Department of Physics, University of Massachusetts, Boston, MA 02125

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Early insight on the critical dynamics of phase transitions arose in a cosmological setting in an effort to understand the origin of structure formation in the early Universe. Kibble pointed out that in a spontaneous symmetry breaking scenario, when a system is driven across a phase transition from a high-symmetry phase to a topologically nontrivial vacuum manifold, causally disconnected regions of the system choose independently the broken symmetry (1, 2). These conflicting choices result in the formation of topological defects, such as domain walls in a ferromagnet and vortices in a superfluid, to name a couple of familiar examples.

Soon after, Zurek indicated that signatures of universality in the dynamics of a phase transition could be tested in condensed matter systems, e.g., superfluid helium (3, 4). Further, he improved the estimate for the average size of the domains and predicted a universal power law for the density of topological defects as a function of the rate at which the phase transition is crossed. The combination of these ideas is known as the Kibble−Zurek mechanism (KZM) and has been a lively subject of both theoretical and experimental research during the last decades. The abundant attempts to verify the KZM in the laboratory have, however, faced a variety of shortcomings, and, while different aspects of the mechanism have been confirmed, a definite test is still missing (5). A remarkable step forward is reported by Deutschländer et al., who used colloidal monolayers as a test bed for universal critical dyamics with unprecedented accuracy (6).

In a nutshell, the paradigmatic KZM provides a framework to describe the dynamics across a continuous phase transition. At equilibrium, the correlation length diverges as a universal power law in the thermodynamic limit when the critical point *A*, the KZM exploits the adiabatic impulse approximation to “chop” the evolution through the phase transition in three sequential stages, where the dynamics is quasi-adiabatic, frozen, and quasi-adiabatic again, as the control parameter takes values

The quest for a conclusive test verifying the KZM scaling faces the following major challenges. Given that the mechanism uses equilibrium properties of the system to account for the nonequilibrium dynamics, measurements of the equilibrium correlation length and relaxation time and the associated critical exponents (

Deutschländer et al. studied the universal nonequilibrium dynamics induced by cooling, at a tunable finite rate, a colloidal monolayer. The phase transition under consideration is made clear by the Kosterlitz−Thouless−Halperin−Nelson−Young (KTHNY) theory (9⇓–11). At high temperatures, an isotropic phase is found with short-range orientational order and isolated disclinations. As the system is cooled down below a critical temperature *B*. This assumes that the hexatic phase, being narrow in parameter space, can be ignored and that the dynamics in the crystalline phase is down.

Apart from involving a two-step process, the main peculiarity of the KTHNY universality class for the purpose of testing the KZM is that correlation length and relaxation time do not follow an algebraic divergence at equilibrium but, rather, an exponential one, e.g., *b* > 0, and ^{3}, within a sample with over a hundred thousand particles with an extension about a hundred times the interparticle distance. These parameters compare favorably to other tests of the KZM (5). Moreover, density gradients in the initial state were suppressed by an exquisite control of the horizontal inclination, and the authors assessed that temperature gradients were absent as well.

The beads forming the colloid are superparamagnetic, and repulsive dipole−dipole Deutschländer et al. show that colloids are an ideal platform to advance our understanding of universal dynamics in critical systems.

interactions can be induced by applying an external magnetic field. The control parameter is the ratio of the magnetic energy to the thermal energy, and its rate of change can be tuned by nearly three orders of magnitude. The ensuing overdamped nonequilibrium dynamics was analyzed by video microscopy with single-particle resolution that allows unprecedented access to all stages of the phase transition dynamics. The authors recorded the evolution of the density of defects and the domain size. The characteristic size of the domains was studied as a function of the quench rate, and remarkable agreement was found with the KZM prediction

Overall, Deutschländer et al. (6) show that colloids are an ideal platform to advance our understanding of universal dynamics in critical systems. A variety of exciting prospects for future research can be envisioned. Examples include the critical dynamics of confined colloids across a (zero-temperature) second-order phase transition (16), under inhomogeneous driving (17⇓–19), and in the presence of quenched disorder (20).

## References

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*Advances in Chemical Physics*, eds Rice SA, Dinner AR (Wiley, Hoboken, NJ), Vol 156 - ↵
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