Active Points push the limits of two-dimensional Solids

September 5, 2023& bullet; Physics 16, 146

Active particles can form two-dimensional solids that differ from those formed by non-motile particles, exhibiting long-range crystalline arrangements accompanied by giant spontaneous deformations.

A statue — **Figure 1:** (Left) At equilibrium, the two-dimensional system of particles is stable in an ordered structure similar to a crystal lattice, although that order becomes less clearly defined at longer length scales. (Middle) These active systems are stabilized in a planar phase similar to that of a solid but with much greater variability around a perfect crystal, as shown by Shi and co-workers. [1]. (Right) The contour lines trace the maximum displacement around the lattice taken from the numerical simulation of the group. The displacement can be as large as many of the lattice points shown in the diagram. In the drawing, the light shows the amount of displacement and the shape of the color.

to enlarge the picture — **Figure 1:** (Left) At equilibrium, the two-dimensional system of particles is stable in an ordered structure similar to a crystal lattice, although that order becomes less clearly defined at longer length scales. (Middle) These active systems are stabilized in a planar phase similar to that of a solid but with much greater variability around a perfect crystal, as shown by Shi and co-workers. [1]. (Right) The contour lines trace the maximum displacement around the lattice taken from the numerical simulation of the group. The displacement can be as large as many of the lattice points shown in the diagram. In the drawing, the light shows the amount of displacement and the shape of the color.

If you compress a liquid slowly enough at low temperatures, it will freeze into an ordered solid: a crystal. Or that’s what we usually see in three dimensions. If you instead consider particles confined to a two-dimensional (2D) plane, the result is quite different. In parallel systems, the 2D solid is stable in a structure with no long-range order—the smaller the order away from the central lattice site. The behavior of systems far from equilibrium, such as autonomous particles, remains, however, an open question. In a numerical study of bacteria-like particles, Xia-qing Shi of Soochow University in China and his colleagues have now shown that active crystals follow a slightly different set of rules than others. [1]. Like symmetrical 2D crystals, 2D active systems are stable in a planar phase similar to that of a solid but with very large fluctuations around the configuration of a perfect crystal lattice. The discovery could help guide the design of future materials based on active particles.

A defining characteristic of crystal structures is their periodicity. They exhibit long-term ordering, meaning a regular arrangement of particles that repeats itself throughout the crystal. Equally, 2D solids cannot exhibit true long-range order [2]. Instead, the coordinate function of the particle’s position decays with distance, according to the power law captured by the well-known theory of Berezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young (BKTHNY) [3]. Interestingly, despite the lack of long-range spatial order, these 2D solids still exhibit true long-range order: the particles are perfectly aligned throughout the system. For any system that breaks the reversal symmetry, however, established theories cannot say whether true 2D crystals can be stabilized at all, much less what this stabilization would look like. This theoretical challenge is particularly related to active matter systems (those that convert energy into mechanical work on small scales, such as a collection of bacteria or chemically dissolved colloidal particles) when they move from a liquid to a dense phase, such as a solid phase. [4–7].

Shi and team now show, numerically and analytically, how positional and orientational order play out in a functional 2D system. Their study examines a common toy model of active matter, a system of opposing particles. Particles moving along a path are defined by their internal polar axes, which align at short range with the axes of their neighbors but also change. The researchers first showed that this system exhibits a dense, ordered phase, similar to that of a homogeneous solid, at a small enough speed for self-propulsion, a degree of coordination, and a circular distribution. Then, they focused their attention on how this dense phase changes as a single parameter of the system.

The team showed numerically that the system exhibits long-range spatial alignment and real long-range order throughout the volume, similar to equilibrium reinforcement. However, the twist is that the power-law exponents that describe the positional order cover a very wide range, estimating values as high as 20, which may sound surprising to someone familiar with equilibrium physics. Not only does the general BKTHNY theory establish an upper limit of 1/3 of the law-law exponent for 2D equilibrium solids. [3], but it is not very common to see the critical exponent take large values in general. Well, the most important exponents of the order length versus temperature or the magnetic order versus temperature are of order 1.

To support their numerical observations, Shi and his colleagues developed a simple linear elastic theory that describes the displacement of particles with respect to a perfectly ordered lattice. To do so, they replaced the term equilibrium which describes the variable as white noise (the variance of which is correlated with temperature) and the term functional which describes the variable in different directions of self-propulsion. This leads to the definition of “effective temperature” that replaces the one used in the expression of the BKTHNY law, which allows the accepted theory to escape the measurement limits. This effective temperature also helps to explain an interesting feature of 2D active solids: their particle positions exhibit large spontaneous fluctuations about their positions in a perfect crystal. These spontaneous defects are much larger than those occurring in any case of alignment. But, as in a 2D equilibrium crystal, the particles retain real long-range order.

Shi’s study highlights how being away from equilibrium allows systems to break rules we often take for granted. At a fundamental level, the application of the theory of small elasticities and effective heat perfectly combines active and equilibrium processes in a manner reminiscent of effective temperatures in glasses. [8]. Apart from toy models, this concept can help explain the behavior of dense, confined biological systems such as tissues. [9].

In more active fields, a better understanding of the interplay between the order and dynamics of active solids is necessary to begin incorporating active factors into materials and manufacturing processes. For example, macroscopic mechanical metamaterials can use active materials to produce large amplitudes of deformation without affecting the integrity of the material. [10]. Similarly, in the context of colloidal substances [11]informing the worker of important steps during preparation can facilitate the assembly of different types of structures.

References

X.-q. Shi continuous.“Spontaneous Degradation of Active Crystals,” Phys. Pastor Lett. 131108301 (2023).
ND Mermin, “Crystalline Order in Two Dimensions,” Phys. Teacher. 176250 (1968).
RR Nelson and BI Halperin, “Dislocation-mediated melting in two dimensions,” Phys. Pastor B 192457 (1979); AP Young, “Dissolution and the coulomb vector gas in two dimensions,” 191855 (1979).
G. Briand and O. Dauchot, “Crystallization of self-propelled hard disks,” Phys. Pastor Lett. 117098004 (2016).
LF Cugliandolo and G. Gonnella, “Phases of active matter in two dimensions,” in Practical Matter and the Physics of Incoherent Information, Les Houches Summer School Lecture Notesedited by J. Tailor continuous. (Oxford University Press, Oxford, 2022)[Amazon][WorldCat].
L. Caprini continuous.“Entropons as coupled excitations in active solids,” J. Chem. Phys. 159 (2023).
P. Baconnier continuous.“Selective and Combined Reactions in Active Solids,” Nat. Phys. 181234 (2022).
LF Cugliandolo, “Effective Heat,” J. Phys. 44483001 (2011).
S. Kim continuous.“Embryonic tissues as functional foams,” Nat. Phys. 17859 (2021).
JU Surjadi continuous.“Mechanical Metamaterials and Their Engineering Applications,” Adv. Eng. Mater. 211800864 (2019).
M. He continuous.“Colloidal Diamond,” Nature 585524 (2020).

About the Author

Mathias Casiulis is a soft physicist currently working as a postdoctoral assistant at the Simons Center for Computational Physical Chemistry and the Center for Soft Matter Research, both at New York University (NYU). Mathias holds two masters degrees in physics from ESPCI Paris and École Normale Supérieure and a PhD in physics from Sorbonne University. Mathias was a post-doctoral student at the Technion Israel Institute of Technology before transferring to NYU. Their research focuses on the numerical study of the emergent properties of soft and reactive materials and in particular on irregular patterns of order.