How does a hyperuniform fluid freeze?

Significance

Most discontinuous phase transitions in equilibrium can be triggered by localized fluctuations, of which the kinetic pathway is nucleation and growth. Here, in a system of reactive particles forming nonequilibrium hyperuniform fluids, we find that the discontinuous freezing transition of the metastable nonequilibrium hyperuniform fluid into an absorbing state does not have the kinetic pathway of nucleation and growth, and it can only be triggered by long-wavelength fluctuations starting from the largest length-scale of the system. The rate of the phase transition decreases with increasing the system size. This challenges the common understanding of metastability in discontinuous phase transitions and suggests that the phase transformations in nonequilibrium systems can be fundamentally different from the ones in equilibrium.

Abstract

All phase transitions can be categorized into two different types: continuous and discontinuous phase transitions. Discontinuous phase transitions are normally accompanied with significant structural changes, and nearly all of them have the kinetic pathway of nucleation and growth, if the system does not suffer from glassy dynamics. Here, in a system of barrier-controlled reactive particles, we find that the discontinuous freezing transition of a nonequilibrium hyperuniform fluid into an absorbing state does not have the kinetic pathway of nucleation and growth, and the transition is triggered by long-wavelength fluctuations. The transition rate decreases with increasing the system size, which suggests that the metastable hyperuniform fluid could be kinetically stable in an infinitely large system. This challenges the common understanding of metastability in discontinuous phase transitions. Moreover, we find that the “metastable yet kinetically stable” hyperuniform fluid features a scaling in the structure factor $S(k\to 0)\sim k^{1.2}$ in 2D, which is the third dynamic hyperuniform state in addition to the critical hyperuniform state with $S(k\to 0)\sim k^{0.45}$ and the nonequilibrium hyperuniform fluid with $S(k\to 0)\sim k^2$.

The concept of hyperuniformity was introduced by Torquato and Stillinger (1), in which a hyperuniform structure is defined if the structure factor $S(|\mathbf{k}|\to 0)=0$ with $\mathbf{k}$ the wavevector. Besides the “ordered” hyperuniform structures, like crystals and quasicrystals (2), over the past decades, a number of disordered hyperuniform structures have been found in various systems, including perturbed lattices (3, 4), perfect glasses (5), jammed structures (6, 7), avian photoreceptor patterns (8), biological tissues (9), early universe fluctuations (10), etc. These disordered hyperuniform structures have shown even better properties than the ordered hyperuniform structures like isotropic photonic bandgaps opened at low dielectric contrast (11–13) and abnormal transparency (14, 15), which suggests a direction in design and fabrication of disordered hyperuniform functional materials. Recently, emergent dynamic hyperuniform states were also found in nonequilibrium systems theoretically (16, 17), numerically (16–23) and experimentally (24–27), and most of them can be categorized into the critical hyperuniform state and the nonequilibrium hyperuniform fluid. It is found that the nonequilibrium hyperuniform fluid originates from the interplay between the reciprocal active excitation between particles and the frictional dissipation of the background. With decreasing the active excitation or increasing the friction of the solvent, the system undergoes an absorbing transition into an immobile state (17). While most absorbing transitions are continuous phase transitions, it was recently found that in systems of barrier-controlled reactive particles, by increasing the reaction barrier, the absorbing transition can be discontinuous (28). Here, using mean field theory and computer simulation, we investigate the kinetic pathway for the nonequilibrium hyperuniform fluid of barrier-controlled reactive particles undergoing the discontinuous phase transition into an absorbing state. Intriguingly, we find that the discontinuous absorbing transition of nonequilibrium hyperuniform fluid does not have the kinetic pathway of nucleation and growth, which suggests that the metastable hyperuniform fluid could be kinetically stable in the thermodynamic limit with a hyperuniform scaling in the structure factor $S(k\to 0)\sim k^{1.2}$ belonging to the strongest category of hyperuniformity (Class I) (29).

Results

Model.

We consider a generalized reactive particle system in $d$-dimension consisting of $N$ hard spheres with mass $m$, diameter $\sigma$ and random initial velocities $\mathbf{v}$ (28) as shown in Fig. 1A, and we focus on $d=2$. Particles can undergo active or passive collisions depending on their colliding velocities. For two particles $({\mathbf{r}}_i,{\mathbf{v}}_i)$ and $({\mathbf{r}}_j,{\mathbf{v}}_j)$ colliding at time $t$, if the relative kinetic energy between them is larger than the reaction barrier $E_b$, i.e., $\frac{1}{2}m{(\mathrm{\Delta }{v}_{i,j}^{\perp })}^2>E_b$, they undergo an active collision, and an extra energy $ϵ$ is injected reciprocally to the kinetic energy of the two particles in the colliding direction; otherwise, the two particles undergo an elastic passive collision. Here, $\mathrm{\Delta }{v}_{i,j}^{\perp }$ is the relative velocity along the center-to-center direction, and $\mathrm{\Delta }{v}_{i,j}^{\perp }=\mathrm{\Delta }{\mathbf{v}}_{i,j}·\frac{\mathrm{\Delta }{\mathbf{r}}_{i,j}}{\sigma }$ with $\mathrm{\Delta }{\mathbf{r}}_{i,j}={\mathbf{r}}_j-{\mathbf{r}}_i$. The equation of motion for particle $i$ between two consecutive collisions is described by the underdamped Langevin equation

$$\begin{array}{c}\hfill m\frac{d{\mathbf{v}}_i(t)}{\mathit{dt}}=-\gamma {\mathbf{v}}_i(t),\end{array}$$ [1]

Fig. 1.

Barrier-controlled reactive particles. (A) Schematic of the barrier-controlled reactive hard spheres, where the green, red, and drak blue solid arrows indicate the friction of solvent, active, and passive collisions, respectively, with the dashed arrows the velocities of particles. (B) Schematic of the mean field prediction on the phase behavior of barrier-controlled reactive particles, in which depending on the parameter $b$, $T_k$ changes with parameter $a$ either continuously or discontinuously to zero. Here, $V(T_k)$ is the effective potential constructed in the mean field theory. (C) Steady-state kinetic temperature $T_k$ as a function of reaction barrier $E_b$ for systems of $N=$10,000 particles at various density $\tilde{\rho }$ with $l_d=\sqrt{5}\sigma$.

with the friction coefficient $\gamma$. The system is simulated using an event-driven algorithm (17, 28). The dimensionless particle density of the system is $\tilde{\rho }\equiv N{\sigma }^d/V$, where $V=L^d$ is the volume of the system, and $L$ is the box length with periodic boundary conditions in all directions. The dissipation length $l_d\equiv \sqrt{mϵ}/\gamma$, and $l_d$ is set as $\sqrt{5}\sigma$ in this work. Here, the time unit is ${\tau }_0\equiv \sigma /v_0$ with the typical excitation speed defined as $v_0\equiv \sqrt{ϵ/m}$.

Mean Field Theory.

We first formulate a qualitative mean field theory to describe the phase behavior of the system, which is essentially a driven-dissipative system (28). At the mean field level, the power of energy per particle for the driven-flow by active collisions and the dissipative-flow by the solvent can be written as $W_{\text{driv}}=f_aϵ$ and $W_{\mathrm{disp}}=\gamma \overline{v^2}$, respectively. We define the kinetic temperature of the system $T_k\equiv m⟨\overline{v^2}⟩/d{k}_B$, with which we can rewrite $W_{\mathrm{disp}}=\gamma d{k}_B{T}_k/m$. $f_a$ is the average active collision frequency per particle, which can be approximated as $f_a\simeq x_a{\overline{v}}_a/(2l_r)$, where $x_a$ and ${\overline{v}}_a$ are the fraction and average speed of active particles, respectively, with $l_r$ the mean free path between active collisions. At low density, we assume $l_r\simeq \sigma /{\tilde{\rho }}_r$ with ${\tilde{\rho }}_r$ the density of effective reactants, i.e., the average density of particles that can be activated at $T_k$. Therefore, the mean field dynamic equation for $T_k$ can be written as

$$\begin{array}{c}\hfill \frac{\partial (d{k}_B{T}_k)}{\partial t}=W_{\text{driv}}-W_{\mathrm{disp}}=\frac{x_a{\tilde{\rho }}_r{\overline{v}}_aϵ}{2\sigma }-\frac{\gamma d{k}_B{T}_k}{m}.\end{array}$$ [2]

Increasing $x_a$ encourages more collisions between active particles to increase ${\overline{v}}_a$, and as a first-order approximation, we assume ${\overline{v}}_a\simeq (1+A{x}_a)v_0$, with $A$ a positive constant. Then, we have $d{k}_B{T}_k\simeq x_{a}m{\overline{v}}_a^2\simeq (x_a+2A{x}_a^2)ϵ$. Accordingly, as a first-order approximation, we have $x_a\simeq d{k}_B{T}_k/ϵ-2A{d}^2{k}_B^2{T}_k^2/{ϵ}^2$. Moreover, increasing the reaction barrier $E_b$ decreases the effective reactant density ${\tilde{\rho }}_r$, and as a first-order approximation, we assume ${\tilde{\rho }}_r/\tilde{\rho }\simeq 1-B(1-x_a)E_b/ϵ$ with $B$ a positive constant. Thus, by keeping only the first three leading terms, Eq. 2 can be written as

$$\begin{array}{c}\hfill \frac{\partial T_k}{\partial t}=a{T}_k-b{T}_k^2-c{T}_k^3,\end{array}$$ [3]

where $a=\frac{\tilde{\rho }}{2{\tau }_0}[(1-B{E}_b/ϵ)-({\tau }_0/{\tau }_d)]$, $b=-\frac{d{k}_B\tilde{\rho }}{2{\tau }_0{ϵ}^2}[(1+A)B{E}_b-Aϵ]$, $c=\frac{d^2{k}_B^2}{2{\tau }_0{ϵ}^3}\tilde{\rho }[4ϵA^2+(3A-4A^2)B{E}_b]$, and ${\tau }_d=m/\gamma$. Eq. 3 with $b<0$ is the simplest equation that was used to investigate catastrophic shifts at a deterministic level (30). The right-hand side of Eq. 3 can be seen as the gradient of an effective potential $V(T_k)$, and $\partial T_k/\partial t=-\partial V/\partial T_k$, in which the sign of the parameter $b$ controls the nature of the transition as shown in Fig. 1B. For $b>0$, one can see that $V(T_k)$ only has one minimum suggesting that the transition from active state ($T_k>0$) to the absorbing state ($T_k=0$) is a continuous transition. When $b<0$, $V(T_k)$ features two local minima, which implies that with decreasing $a$, the system undergoes a discontinuous transition from the active state to the active-absorbing bistable region and then to the absorbing state. This can be also understood as that the existence of reaction barrier delays the dynamic phase transition but also increases the cooperativity during the transition, which sharpens the transition. With a high enough reaction barrier, the absorbing transition becomes discontinuous (28). $b=0$ is the tricritical point that separates the continuous and discontinuous phase transitions (28).

To validate the theoretical prediction on the phase behavior, we perform event-driven simulations for a 2D system of $N=$10,000 reactive hard spheres at various densities $\tilde{\rho }$ and reaction barriers $E_b$. All simulations start from random configurations and velocities of sufficiently large initial kinetic temperature, and we use the kinetic temperature $T_k$ as the order parameter of the system. As shown in Fig. 1C, for low-density systems, $T_k$ decreases smoothly to zero with increasing $E_b$, and with increasing the density of the system, the absorbing transition occurs at higher $E_b$ with the transition becoming sharper. With the finite size scaling of the dynamic evolution of the system, we confirm that at ${\tilde{\rho }}_{\mathrm{tri}}=0.275$ and $E_{b,\mathrm{t}ri}/ϵ=0.1678$, the kinetic temperature decays as $T_k\sim t^{-0.37}$ before reaching the saturated value, which signatures a tricritical point (SI Appendix, Fig. S1). For systems at lower densities, at the transition point, $T_k\sim t^{-0.54}$, which belongs to the conserved directed percolation universality class like the conventional absorbing transition (SI Appendix, Fig. S1). For systems at higher densities, we find that the absorbing transition is discontinuous, and these qualitatively confirm the prediction from the mean field theory.

Phase Diagram.

The phase diagram of barrier-controlled reactive particles is summarized in Fig. 2A, which features both a continuous and a discontinuous absorbing transition separated by a tricritical point located at around ${\tilde{\rho }}_{\mathrm{tri}}=0.275$ and $E_{b,\mathrm{t}ri}/ϵ=0.1678$. We further check the structure of the system at the transition points. As shown in Fig. 2B, at the continuous absorbing transition, $S(k\to 0)\sim k^{0.45}$. At the tricritical point, the structure factor follows the same scaling as the conventional continuous absorbing transition, although the dynamic relaxation is different. For systems at higher densities that undergo discontinuous absorbing transition, we find that at the stability limit, the structure factor of the metastable hyperuniform fluid $S(k\to 0)\sim k^{1.2}$, which is a new hyperuniform scaling in nonequilibrium dynamic systems. We confirm that the location of the stability limit of metastable nonequilibrium hyperuniform fluid does not change visibly with varying the system size from $N=$3,000 to 30,000. The value of 1.2 in the scaling exponent in $S(k)$ is obtained through fitting $S(k)$ in the largest system simulated in this work, i.e., $N=$30,000, and depending on the system, the obtained exponent is different (SI Appendix, Fig. S2). As we do not have any theoretical explanation of the value of the scaling exponent, we note that it is unclear what the exact scaling exponent is in the thermodynamic limit, and in the range of system size $N=$3,000 $\sim$ 30,000 studied in this work, the fitting exponent in $S(k\to 0)$ increases with increasing $N$. Based on the obtained phase diagram, we investigate the kinetic pathway of the discontinuous absorbing transition. We simulate the metastable hyperuniform fluids of different system size starting from random configurations, and for each system, we perform 10 independent simulations, in which the time evolution of the kinetic temperature $T_k$ is shown in Fig. 2C. One can see that $T_k$ reaches a plateau around $ϵ/k_B$ quickly and remains constant for a period of time $t_{\mathrm{wait}}$ before suddenly drops to zero, which is a signature of discontinuous phase transition (31, 32). It is known that the discontinuous phase transition is a barrier-crossing process, of which the barrier height is related to $log(t_{\mathrm{wait}})$. In the inset of Fig. 2C, we plot $⟨log(t_{\mathrm{wait}}/{\tau }_0)⟩$ as a function of system size $N$. We also confirm that there is no glassy dynamics in the system based on the calculated mean squared displacement of the particles (SI Appendix, Fig. S3). Intriguingly, we find that $⟨log(t_{\mathrm{wait}}/{\tau }_0)⟩$ increases monotonically with $N$, which suggests that the kinetic pathway is not nucleation and growth. Because if the kinetic pathway is nucleation and growth, one shall have a well-defined nucleation rate, which is the number of critical nuclei found in a system of unit volume within a unit time period. With increasing the system size $N$, the probability of finding a critical nucleus in the system within a unit time increases, which should make the waiting time $t_{\mathrm{wait}}$ in the metastable state shorter, while in Fig. 2C, $t_{\mathrm{wait}}$ counterintuitively increases with $N$.

Fig. 2.

Phase behavior of barrier-controlled reactive hard spheres. (A) Phase diagram of barrier-controlled reactive hard spheres, where the orange curve indicates the continuous absorbing transition, the red dot indicates the tricritical point, and the black dashed curve indicates the stability limit of hyperuniform fluid in the discontinuous absorbing transition. (B) The structure factor near the critical point of continuous phase transition ($\tilde{\rho }=0.1675,0.25$), the tricritical point ${\tilde{\rho }}_{\mathrm{tri}}=0.275,E_{b,\mathrm{t}ri}/ϵ=0.1678$, and the stability limit of the hyperuniform fluid in the discontinuous phase transition ($\tilde{\rho }=0.35,0.4$). Here, the dashed and dotted dashed lines indicate two hyperuniform scaling $S(k\to 0)\sim k^{0.45}$ and $\sim k^{1.2}$, respectively. (C) The time evolution of the kinetic temperature $T_k$ in the metastable hyperuniform fluid of different sizes at $\tilde{\rho }=0.4,E_b/ϵ=0.753$, in which for each system size, we perform 10 independent simulations. Inset: $⟨log(t_{\mathrm{wait}}/{\tau }_0)⟩$ as a function of the system size $N$, where error bars are SDs.

Kinetic Pathway of Phase Transition.

The kinetic pathway of continuous absorbing transition below the critical density is the same as continuous phase transitions in equilibrium (33): In a finite system, the finite size effect induces a mestastable hyperuniform fluid with a finite $t_{\mathrm{wait}}$, while with increasing the system size $t_{\mathrm{wait}}$ drops quickly to below the equilibration time of the system (SI Appendix, Fig. S4). To understand the physics of the discontinuous absorbing transition, we investigate a typical dynamic trajectory of phase-transforming hyperuniform fluid of $\tilde{\rho }=0.4$ at $E_b/ϵ=0.753$, of which the time evolution of $T_k$ is shown in Fig. 3A. One can see that the kinetic temperature of the system reaches the metastable steady state with $k_B{T}_k/ϵ\approx 1$ within about ${10}^2{\tau }_0$ and drops abruptly to zero at around $t_{\mathrm{wait}}\approx {10}^4{\tau }_0$. In Fig. 3B, we plot the structure factor of the system along the phase transition in comparison with the starting metastable hyperuniform fluid. One can see that at the beginning of the phase transition, e.g., $t=$11,633${\tau }_0$, the structure factor starts to deviate from the metastable hyperuniform fluid in the smallest $k$ region corresponding to the wavelength of system size, while the local structure, i.e., $1<k\sigma <10$, remains intact. This can be also seen in the time evolution of the radial distribution function of the system $g(r)$ as shown in Fig. 3C, in which the short-range correlation does not have any visible change in the beginning of phase transformation. The corresponding snapshots of the systems along the transition are shown in Fig. 3E, and one can see that compared with the typical snapshot of the metastable hyperuniform fluid in Fig. 3D, there is hardly any visible difference at the beginning of the transition. This is due to the fact that the structural difference appears first in the small $k$ region corresponding to the long-range correlation, and the structural difference becomes visible only when the deviation propagates to the intermediate length-scale, e.g., $t=$11,653${\tau }_0$ in Fig. 3E, where multiple clusters of immobile clusters appear simultaneously like spinodal decomposition (34). Afterward, the clusters of immobile particles percolate, and the system transforms into an absorbing state. A movie of the phase transformation can be found in Movie S1. This implies that the phase transition is triggered by long-wavelength fluctuations, and explains why $t_{\mathrm{wait}}$ increases with $N$. Because the probability of having a fluctuation of the wavelength of the system size within a unit time decreases with increasing the system size. This suggests that in the thermodynamic limit, the rate of discontinuous absorbing transition could be zero. Here, we note that $E_b/ϵ=0.753$ is the stability limit of metastable hyperuniform fluid at $\tilde{\rho }=0.4$, and a tiny increase of the reaction barrier $E_b$ makes $t_{\mathrm{wait}}$ drop to zero.

Fig. 3.

Kinetic pathway of the discontinuous absorbing transition. (A) The time evolution of the kinetic temperature of the system of $N=$30,000 reactive particles at $\tilde{\rho }=0.4$ and $E_b/ϵ=0.753$. (B and C) The structure factor $S(k)$ (B) and radial distribution function $g(r)$ (C) for system at various times during the discontinuous phase transition from a metastable hyperuniform fluid to an absorbing state, where the dotted dashed lines indicates the hyperuniform scaling of $S(k\to 0)\sim k^{1.2}$. (D) A typical snapshot of the metastable hyperuniform fluid. (E) The snapshots of the system along a typical dynamic trajectory of the discontinuous absorbing transition. In (D) and (E), the particles are color coded based on their kinetic temperature, and $T_k^{\infty }$ is the kinetic temperature of the metastable hyperuniform fluid.

Conclusion and Discussions

In conclusion, using mean field theory and computer simulation, we have investigated a system of barrier-controlled reactive particles forming a nonequilibrium hyperuniform fluid, which can undergo either continuous or discontinuous phase transitions to an absorbing state depending on the reaction barrier. Intriguingly, we find that the discontinuous freezing of the metastable hyperuniform fluid into the absorbing state does not have the kinetic pathway of nucleation and growth, and the transition rate decreases with increasing the system size. This is surprising, as in most nonequilibrium discontinuous phase transitions, e.g., motility-induced phase separation (35–37), there is always a kinetic pathway of nucleation and growth. By checking the structural change along the phase transformation, we find that the discontinuous absorbing transition is triggered by long-wavelength fluctuations, of which the probability decreases with increasing the system size. This suggests that the fluctuation required to trigger the transition may diverge for an infinitely large system, in which the metastable hyperuniform fluid could be kinetically stable. This challenges the common understanding of metastability in discontinuous phase transitions. Lastly, the “metastable yet kinetically stable” hyperuniform fluid features a hyperuniform scaling $S(k\to 0)\sim k^{1.2}$, which is different from the other dynamic hyperuniform states, i.e., the critical hyperuniform state and the nonequilibrium hyperuniform fluid.

Materials and Methods

In the simulation, we employ square $L\times L$ boxes with periodic boundaries to calculate the structure factor $S(k)$. The structure factor is estimated by the formula $S(\mathbf{k})=\frac{1}{N}〈{\left|\sum _{j=0}^N{\mathrm{e}}^{i\mathbf{k}·{\mathbf{r}}_j}\right|}^2〉$, where $\mathbf{k}=\left[k_x,k_y\right]=\left[i\frac{2\pi }{L},j\frac{2\pi }{L}\right]$ with $i,j=0,\pm 1,\pm 2,\pm 3,...$. except $i=j=0$. The vector $\mathbf{k}$ is then projected onto the scalar $k$. For the finite-size analysis of critical points in the continuous regime, all results are averaged over 10 independent simulations, and we use natural homogeneous initial configurations as done in refs. 17 and 28.

The finite-size analysis at the critical point of continuous phase transition, system size dependence of the structure factor and the mean squared displacement at the stability limit of discontinuous absorbing transition, and the kinetic pathway of the continuous absorbing transition below the critical density can be found in SI Appendix, and a snapshot movie of the phase transformation near the stability limit of a discontinuous phase transition can be found in Movie S1.

Supplementary Material

Appendix 01 (PDF)

Data, Materials, and Software Availability

All study data are included in the article and/or supporting information, and the orignial data for Fig. 1 to Fig. 3 can be found at online data repository (https://doi.org/10.5281/zenodo.10070815) (38).