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# Hydrodynamics of random-organizing hyperuniform fluids

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved October 7, 2019 (received for review July 7, 2019)

## Significance

Recently, an exotic hyperuniform fluid state was found in a number of nonequilibrium systems, which shows the promise for fabrication of novel lifelike functional materials capable of self-healing and self-adapting. However, the general mechanism of fluidic hyperuniformity (HU) and its fundamental difference from equilibrium fluids remain unclear, which hinders the rational design of nonequilibrium hyperuniform fluids. In this work, we propose a nonequilibrium hard-sphere model of hyperuniform fluids and formulate a full hydrodynamic description based on the generalized Navier–Stokes equations. Our theory reveals the mechanism of fluidic HU in a simple and intuitive way, which is confirmed by simulating a realistic active spinner system.

## Abstract

Disordered hyperuniform structures are locally random while uniform like crystals at large length scales. Recently, an exotic hyperuniform fluid state was found in several nonequilibrium systems, while the underlying physics remains unknown. In this work, we propose a nonequilibrium (driven-dissipative) hard-sphere model and formulate a hydrodynamic theory based on Navier–Stokes equations to uncover the general mechanism of the fluidic hyperuniformity (HU). At a fixed density, this model system undergoes a smooth transition from an absorbing state to an active hyperuniform fluid and then, to the equilibrium fluid by changing the dissipation strength. We study the criticality of the absorbing-phase transition. We find that the origin of fluidic HU can be understood as the damping of a stochastic harmonic oscillator in q space, which indicates that the suppressed long-wavelength density fluctuation in the hyperuniform fluid can exhibit as either acoustic (resonance) mode or diffusive (overdamped) mode. Importantly, our theory reveals that the damping dissipation and active reciprocal interaction (driving) are the two ingredients for fluidic HU. Based on this principle, we further demonstrate how to realize the fluidic HU in an experimentally accessible active spinner system and discuss the possible realization in other systems.

- fluidic hyperuniformity
- nonequilibrium fluids
- Navier–Stokes equations
- absorbing-phase transition
- active spinners

Hyperuniform structures are characterized by vanishing long-wavelength density fluctuations with the structure factor

## Results

### Model.

We first consider a minimal model consisting of N hard spheres (Fig. 1*A*). Each particle has a mass m, a diameter σ, and a random initial velocity. Particles undergo active collision with reciprocal center-to-center impulsions, in which an additional kinetic energy *Materials and Methods* has simulation details). As the conclusion that we obtain is independent of dimensionality, in the following, we focus on the 2D systems and leave the results of 3-dimensional (3D) systems in *SI Appendix*.

### Absorbing-Phase Transition.

In our model with random initial particle velocities, when *B*, we plot the phase diagram of the 2D system in the representation of *B* shows the phase boundary of the system obtained from simulations, and the dashed line in Fig. 1*B* shows *SI Appendix*, Figs. S1 and S2, by using the finite-size scaling analysis (33, 34), we investigate the criticality of this transition in 2D systems with weak inertia (*SI Appendix*, Figs. S3 and S4 and Table S1.

### From Equilibrium Simple Fluids to Nonequilibrium Hyperuniform Fluids.

Our model is a driven-dissipative system. In the active state, the power of energy injection and dissipation per particle at the mean-field level can be written as*A*, we plot *A*) and the systems with different *A*). We find the 2 sets of data collapsing onto a single curve when *B*. We find that *SI Appendix*, Fig. S5). In Fig. 2*C*, we plot the structure factor *C*). With decreasing

### Hydrodynamic Theory for Hyperuniform Fluids.

The random-organizing hard-sphere fluid has a well-defined kinetic temperature in the large *SI Appendix*. With the additional damping term **5** and **6** describe a nonequilibrium fluid, which violates the fluctuation–dissipation theorem (37). Using the standard hydrodynamic linearization procedure, we find the density fluctuation **7**, in fact, describes a damped stochastic harmonic oscillator if we map **9** can be also directly obtained using our hydrodynamic theory (*SI Appendix*). From Eqs. **8** and **9**, we find *E*). For nonequilibrium fluids (**7** can be neglected compared with *F*). This mechanism of fluidic HU can be also understood as that the hyperuniform fluid has a q-dependent temperature, which vanishes at *C*, the results of Eq. **9** are shown as dashed lines, which well agree with the simulation results at small q. In *SI Appendix*, Fig. S6, we show the insensitivity of *SI Appendix*, Fig. S7, we also show how

Dynamically, the oscillator represented by Eq. **7** is in the resonance state in regime**7** can be written as a simple diffusion equation:*G*, we plot the schematic diagram of density modes in the representation of *G*), while for *G*) and acoustic modes (green region in Fig. 2*G*) at relative small q and nonacoustic “fast-decay mode” at large q. The upper boundary of the acoustic regime is determined by Eq. **10**, which becomes less accurate (dashed line in Fig. 2*G*) when the system approaches the critical point

Generally, different dynamic modes can be characterized by different shapes of dynamic structure *A*) and systems with fixed q but different *B*) at the same density *A* (or Fig. 3*B*). The theoretical predictions of Eq. **12** are plotted as dashed lines in Fig. 3 by using the accurate expression of sound speed *C*. We find that the theoretical predictions of Brillouin peaks match well not only with data from the equilibrium fluid but also, with nonequilibrium hyperuniform fluids. This indicates that the sound speed and the compressibility in hyperuniform fluids with kinetic temperature *SI Appendix* has a discussion), it predicts the bimodal acoustic modes and diffusive modes faithfully for hyperuniform fluids with small *B* indicate

### Power Spectrum of Local Density Fluctuations.

The aforementioned dynamic regimes in hyperuniform fluids can also be distinguished by the noise color reflected by the power spectrum of local density fluctuations (32, 43): i.e.,*SI Appendix*. Here, we only summarize the main results. We first introduce 3 typical frequencies *D*, we plot *C* but distinguish them by using *D*. We find that, when *C* with Fig. 2*D*, we find that the development of hyperuniform scaling *G*, while the white noise regime with upper-bound frequency *G*, respectively. These special noise features provide us with an additional tool to identify and study the hyperuniform fluids experimentally in the frequency domain. We emphasize that all of these structural and dynamic properties of our hyperuniform fluids are independent of dimensionality, and we show the simulation data for the 3D system in *SI Appendix*, Figs. S8 and S9 and Movie S4.

### Hyperuniform Active Spinner Fluids.

The hydrodynamic theory above suggests 2 important conditions for forming nonequilibrium hyperuniform fluids: 1) the damping dissipation characterized by *A*) as an example to realize the hyperuniform fluid. Active spinners systems exhibit many interesting phenomena and have been extensively studied both theoretically (44⇓⇓–47) and experimentally (48⇓⇓⇓⇓–53). We use the active spinner model from ref. 45 (*SI Appendix* has details), in which each spinner is a dimer consisting of 2 spherical hard monomers. Driven by a constant torque Ω, spinners perform self-rotating motion with the same chirality. The dynamics of both translational and rotational freedom degrees of spinners are underdamped. Thus, when 2 fast-rotating spinners collide, some part of their rotational kinetic energy is transferred to their translational freedom degree, inducing the active collision similar to the previous hard-sphere model. With the fixed substrate friction coefficient, we find that Ω in the spinners system plays the similar role of *B* and *C*, we calculate *B* and *C*) compared with the equilibrium dimer fluid (open symbols in Fig. 4 *B* and *C*). When Ω is small, we find the same hyperuniform scaling

## Discussion and Conclusion

In conclusion, a general hydrodynamic mechanism of fluidic HU is discovered by investigating a random-organizing hard-sphere fluid model and formulating a fluctuating hydrodynamic theory based on the generalized NS equations. We find that the state of the system is determined by 2 characteristic lengths: the dissipation length

## Materials and Methods

In our simulations, we use square (2D) or cubic (3D) simulation boxes to calculate the

Details about the event-driven simulations, finite-size scaling analysis of absorbing-phase transitions, hydrodynamic theory, theoretical derivations of *SI Appendix*, *Supplementary Text*. A list of symbols used in this work can be found in *SI Appendix*, Table S2. The original data for Figs. 1 to 4 are provided in Datasets S1–S9.

## Acknowledgments

We thank Profs. Dov Levine and Hao Hu for helpful discussions. This work is supported by Nanyang Technological University Start-Up Grant M4081781.120; Academic Research Fund from Singapore Ministry of Education Grant M4011873.120; and Advanced Manufacturing and Engineering Young Individual Research Grant A1784C0018 by the Science and Engineering Research Council of Agency for Science, Technology and Research Singapore.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: r.ni{at}ntu.edu.sg.

Author contributions: Q.-L.L. and R.N. designed research; Q.-L.L. performed research; Q.-L.L. analyzed data; and Q.-L.L. and R.N. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1911596116/-/DCSupplemental.

Published under the PNAS license.

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