Active particles push the boundaries of two-dimensional solids

Physics 16, 146

Active particles can form 2D solids that are different from those formed by immobile particles, and exhibit long-range crystalline ordering with giant spontaneous deformations.

Figure 1: (Left) At equilibrium, a two-dimensional system of particles settles into an ordered, crystal lattice-like structure, although this arrangement becomes less pronounced at longer scales. (middle) These active systems settle into an ordered phase similar to a solid but with very large fluctuations around a perfect crystal lattice, as shown by Shi et al. (1). (Right) The contour lines trace the massive displacements around the lattice derived from the team’s numerical simulations. The offsets can be as large as the many grid locations shown in the diagram. In the figure, brightness refers to the magnitude of the offset and the direction of the color.<

__truncato_root__> (left) At equilibrium, a two-dimensional system of particles settles into an ordered crystal lattice-like structure, although this arrangement becomes less pronounced at longer scales. (middle) These active systems…

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If you compress a liquid slowly enough at low temperatures, it will freeze into an ordered solid: a crystal. Or at least that’s what we’re used to seeing in three dimensions. If you look instead at particles confined to a two-dimensional (2D) plane, the result is quite different. For equilibrium systems, the two-dimensional solid settles into a structure that lacks long-range order, as it becomes less organized away from the central lattice site. However, the behavior of systems far from equilibrium, such as self-propelled particles, remains an open question. In a numerical study of bacteria-like particles, Xia Qingxi and colleagues from Soochow University in China show that active crystals follow a slightly different set of rules than their immobile counterparts (1). Like 2D equilibrium crystals, 2D active systems settle into a solid-state-like phase but with very large particle fluctuations around forming a perfect crystal lattice. This discovery could help guide the design of future materials based on energetic particles.

A distinctive feature of crystal structures is their periodicity. They exhibit a long-range order, which means a regular arrangement of particles that repeats cyclically over the entire crystal. At equilibrium, two-dimensional solids cannot display true long-range order (2). Instead, the correlation function of the particle’s position decays with distance, according to a force law captured by the famous Berezinsky-Kosterlitz-Tholes-Halperin-Nielson-Young (BKTHNY) theorem (3). Interestingly, despite their lack of long-range positional order, these 2D solids still feature true long-range directional order: the particles are perfectly aligned throughout the system. However, for any system that breaks inverse time symmetry, established theories cannot say whether real two-dimensional crystals can be stabilized at all, much less what such stabilization might look like. This theoretical challenge is particularly relevant to active matter systems (those that convert energy into mechanical work at microscopic scales, such as chemically driven populations of bacteria or colloidal particles) when they transform from a liquid state into a dense, solid-like phase (4–7).

Shi and his team now show, numerically and analytically, how to implement positional and directional ordering in a two-dimensional active system. Their study investigates a common game model of active matter, which is a system of mutually repulsive particles. Particles are self-moving in a direction defined by their inner polar axes, which are aligned in the short term with the axes of their neighbors but also fluctuate. The authors first show that this system exhibits a dense, ordered phase, much like an equilibrium solid, for a sufficiently small self-propelling velocity, degree of alignment, and rotational propagation. Next, they focused their attention on how this dense phase changes when one adjusts various system parameters.

The team showed numerically that the system exhibits a quasi long-range positional order and a true long-range directional order throughout the volume, just like a equilibrium solid. The twist, however, is that the force law advocates describing the localized system cover a very wide range, predicting values ​​of up to 20, which might seem shockingly large to someone familiar with equilibrium physics. Not only does the usual BKTHNY theory set an upper limit of 1/3 to the power-law power in two-dimensional equilibrium solids (3), but it is also very unusual to see a critical power take on such large values ​​in general. In fact, typical critical exponents of correlation length versus temperature or magnetic order versus temperature are of order 1.

To support their numerical observations, Shi and colleagues devised a simple linear elastic theory that describes the displacement of particles with respect to a perfectly ordered lattice. To do this, they replaced the term equilibrium, which describes fluctuations in white noise (the variance of which is related to temperature), with an active term, which describes those fluctuations in terms of changing self-propulsion directions. This leads to the definition of an “effective active temperature” that replaces that used in the BKTHNY power law expression, allowing the accepted theory to escape the equilibrium constraints. This effective temperature also helps explain an interesting feature of 2D active solids: the positions of their particles display very large spontaneous fluctuations around positions that would be present in an ideal crystal. These gigantic spontaneous distortions are much larger than those that would occur in any equilibrium position. But, as in a two-dimensional equilibrium crystal, the particles retain true long-range directional order.

Shi’s study highlights how moving out of balance allows systems to break the rules we often take for granted. At the most basic level, the use of the theory of minimum elasticity and effective temperature elegantly connects active systems to equilibrium systems in a manner reminiscent of effective temperatures in glasses (8). Beyond game models, this concept can help explain the behavior of dense and confined biological systems, such as tissues (9).

On a more practical basis, a better understanding of the interaction between system and fluctuations in active solids is essential to begin incorporating actives into materials and manufacturing processes. For example, macroscopic mechanical materials can use active elements to intermittently generate larger amounts of deformations without affecting the integrity of the material (10). Similarly, in the context of colloidal materials (11), introducing activity at critical steps during preparation can facilitate self-assembly of a wide variety of structures.


  1. X. -q.shi et al.“Extreme spontaneous deformations of active crystals” Phys. Rev. Litt. 131108301 (2023).
  2. ND Mermin, “The Crystal System in Two Dimensions” Phys. pastor. 176250 (1968).
  3. RR Nelson and P. Halperin, “Dislocation-mediated fusion in two dimensions,” Phys. Rev. B 192457 (1979); AP Young, “Melting and Vector Coulomb Gas in Two Dimensions,” 191855 (1979).
  4. G. Briand and O. Dauchot, “Self-propelled hard drive crystal,” Phys. Rev. Litt. 117098004 (2016).
  5. LF Cugliandolo and G. Gonnella, “Phases of Active Matter in Two Dimensions”, in Active Matter and Unequilibrium Statistical Physics, Les Houches Summer School Lecture NotesEdited by J. Taylor et al. (Oxford University Press, Oxford, 2022) (Amazon) (WorldCat).
  6. El Cabrini et al.“Entropones as collective excitations in active solids” J Kim. Phys. 159 (2023).
  7. B. baconier et al.Selective and group machining in active solids. nat. Phys. 181234 (2022).
  8. L.F. Cogliandolo, “Effective Temperature”, G Phys. 44483001 (2011).
  9. skim et al.“Embryonic tissue as active foams” nat. Phys. 17859 (2021).
  10. Joe Surjadi et al.Mechanical materials and their engineering applications. case. M. rainy. 211,800,864 (2019).
  11. M. he et al.“colloidal diamond” nature 585524 (2020).

About the author

Photo by Matthias Cassiolis

Mathias Kasiolis is a soft materials physicist and is currently a postdoctoral assistant in the Simmons Center for Computational Physical Chemistry and in the Soft Materials Research Center, both at New York University (NYU). Matthias holds two master’s degrees in physics from ESPCI Paris and the École Normale Supérieure and a PhD in physics from the Sorbonne. Mathias was a postdoctoral fellow at the Technion Israel Institute of Technology before transferring to New York University. Their research focuses on the numerical study of emerging properties in soft and energetic matter, particularly on unconventional system shapes.

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