Students of thermodynamics learn that closed systems tend toward states of increasing entropy, which is often considered synonymous with decreasing order. But in some systems, entropy and order can be allies, not opponents: The system tends toward greater order as—and precisely because—its entropy increases.
The phenomenon isn’t as paradoxical as it sounds. The secret is to partition the system’s degrees of freedom into two subsets so that ordering in one subset increases the entropy of the other—and thus of the system as a whole. The trick is well known in soft-matter physics, where entropy-driven order shows up in contexts such as colloidal crystallization: When an ensemble of particles assembles into an ordered lattice, each one has more room to move around.
The mechanical motion of colloidal particles involves continuous degrees of freedom, which can be complicated to model and difficult to precisely measure. Now Yale University’s Peter Schiffer, Los Alamos National Laboratory’s Cristiano Nisoli, and their colleagues have shown that entropy-driven order can also occur in an array of nanomagnets called an artificial spin ice—a system whose degrees of freedom are solely discrete.
The figure shows a representation of the array they looked at, which they dubbed “tetris ice.” Each arrow represents a nanomagnet whose position is fixed and whose magnetization is free to point in either direction along its length. The system is frustrated: There’s no way for the magnetic moments to arrange themselves so that every vertex sits at its lowest-energy configuration. As a result, tetris ice has a lot of available ground-state configurations that are nearly degenerate in energy. (For more on frustration in artificial spin ices, see the article by Ian Gilbert, Cristiano Nisoli, and Peter Schiffer, Physics Today, July 2016, page 54.)
The colors in the figure denote the partitioning of the magnets into two sets: the blue “backbones” and the red “staircases.” As illustrated, the backbone configuration has long-range order—a repeating pattern not just within each backbone but also from backbone to backbone. The staircases, on the other hand, are nearly random, because a staircase between two ordered backbones can adopt many nearly degenerate configurations.
Any deviation from backbone order wouldn’t necessarily impose an energy penalty on the system. But it would impose an entropy penalty. To keep the overall energy low, the staircases in the disordered-backbone region would no longer have access to many random configurations. Instead, they’d be locked into just one.
It follows that of all the low-energy configurations the lattice can access, the vast majority have backbone order. And indeed, when the researchers actually constructed the lattice, they observed a high degree of long-range correlation among the backbone moments. Because the system’s exact state is easy to measure using x-ray circular-dichroism photoemission electron microscopy and easy to model using Monte Carlo simulations, tetris ice is a convenient new platform for investigating the real-time dynamics of an entropically ordering system. (H. Saglam et al., Nat. Phys., 2022, doi:10.1038/s41567-022-01555-6.)
