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# Role of hydrodynamics in liquid–liquid transition of a single-component substance

Edited by Pablo G. Debenedetti, Princeton University, Princeton, NJ, and approved January 15, 2020 (received for review July 5, 2019)

## Significance

Liquid–liquid transition (LLT) in a single-component substance is a counterintuitive phenomenon if we regard a liquid to be isotropic and homogeneous both macroscopically and microscopically. However, we can explain LLT rather naturally once we accept local structural ordering in a liquid. However, there has so far been no study on the role of hydrodynamic interaction, although the fluidity is the most intrinsic feature of liquids. Here we describe a theory of LLT considering the hydrodynamic degree of freedom. Our study not only reveals crucial importance of dynamical coupling between density change upon LLT and hydrodynamic transport but also provides a basis to understand how fluid transport can be coupled to LLT, which may be important for future applications.

## Abstract

Liquid–liquid transition (LLT) is an unconventional transition between two liquid states in a single-component system. This phenomenon has recently attracted considerable attention not only because of its counterintuitive nature but also since it is crucial for our fundamental understanding of the liquid state. However, its physical understanding has remained elusive, particularly of the critical dynamics and phase-ordering kinetics. So far, the hydrodynamic degree of freedom, which is the most intrinsic kinetic feature of liquids, has been neglected in its theoretical description. Here we develop a Ginzburg–Landau-type kinetic theory of LLT taking it into account, based on a two-order parameter model. We examine slow critical fluctuations of the nonconserved order parameter coupled to the hydrodynamic degree of freedom in equilibrium. We also study the nonequilibrium process of LLT. We show both analytically and numerically that domain growth becomes faster (slower), depending upon the density decrease (increase) upon the transition, as a consequence of hydrodynamic flow induced by the density change. The coupling between nonconserved order parameter and hydrodynamic interaction results in anomalous domain growth in both nucleation-growth–type and spinodal-decomposition–type LLT. Our study highlights the characteristic features of hydrodynamic fluctuations and phase ordering during LLT under complex interplay among conserved and nonconserved order parameters and the hydrodynamic transport intrinsic to the liquid state.

- liquid–liquid transition
- phase-ordering kinetics
- hydrodynamic interaction
- critical dynamics
- coarse-grained model

It was believed for a long time that a pure substance has a unique liquid state. This intuitive belief comes from a physical picture that the liquid state is in the same uniform, isotropic, random disordered state as the gas state, and the only difference between the gas and liquid states is the density, which is the order parameter for a liquid–gas transition. Contrary to this intuition, there have been many pieces of experimental and numerical evidence that even a single-component substance may have more than two isotropic homogeneous liquid states. This transition between the liquid states is now widely known as “liquid–liquid transition (LLT).” Examples of liquids exhibiting LLT include water (1⇓⇓⇓⇓⇓–7), carbon (8, 9), phosphorus (10, 11), silicon (12), and

This difficulty has been overcome in LLT in molecular liquids (14, 15) such as triphenyl phosphite (TPP) (16⇓–18), l-butanol (19), D-mannitol (20, 21), and aqueous solutions (22, 23). It is because LLT in these molecular liquids takes place in a supercooled liquid state near the glass transition point, and thus the kinetics are very slow, which makes the time-resolved measurements rather easy. Thus the kinetics of LLT can be followed by various experimental techniques such as optical microscopy, light scattering, X-ray scattering, dielectric spectroscopy, calorimetry, and rheology measurements (22⇓⇓⇓⇓⇓⇓–29).

From a theoretical aspect, on the other hand, the physical nature of this transition on a microscopic level has remained elusive since many-body interactions must play a crucial role in this unconventional phase transition. This problem is related to a fundamental theoretical problem of how we should describe the liquid state physically. To describe this type of phase-transition phenomenon on a phenomenological level, we need to construct a coarse-grained Ginzburg–Landau model (30⇓⇓–33). Thus, to describe LLT, we proposed a two-order parameter model of liquid based on the physical idea that we need at least two order parameters: In addition to the density, we introduced the fraction of locally favored structures as an additional order parameter (14, 15, 34). The former is a conserved order parameter, and thus its fluctuation relaxes diffusively (

However, the hydrodynamic degree of freedom is the essential dynamical feature of a liquid, and thus it is natural to expect that it plays a crucial role in LLT. For example, it is widely known for ordinary critical phenomena that the coupling between the order parameter fluctuation mode and the hydrodynamic mode leads to anomalous enhancement of the transport coefficient in critical fluids near the gas–liquid critical point and binary liquid mixtures near the critical point (33, 35, 44). The coupling between the hydrodynamic mode and the thermal diffusion mode is also an essential factor determining the dynamics of the density field, i.e., the dynamic structure factor (45). However, such mode-coupling effects are mostly unknown for the critical point associated with LLT, which is controlled by the nonconserved order parameter ϕ.

The hydrodynamic degree of freedom should also play a significant role in the nonequilibrium ordering process during LLT. For example, the viscosity difference between the two liquid phases should have a significant dynamical impact on LLT (14). Furthermore, the density is generally different between the two liquid phases. In water, for example, recent molecular simulations imply that the density difference between low-density liquid (LDL) and high-density liquid (HDL) water is larger than

Concerning this point, it is worth noting that hydrodynamic flow also plays an essential role in liquid–solid phase transitions. There have been several attempts to include fluid convection coupled to an energy variable to describe the growth and melting of a solid domain in a simplified condition (46⇓⇓–49), where the release, absorption, and transport of heat govern the kinetics of phase transitions [diffusion-controlled kinetics in model C (35)]. However, the kinetics of LLT in molecular liquids such as TPP take place very slowly, and thus we may assume that they proceed in an isothermal condition (reaction-limited kinetics) (18); in other words, the temperature relaxation is assumed to be much faster than the kinetics of LLT. In such a case, the Ginzburg–Landau model can be significantly simplified since we may neglect the coupling between the order parameter and the energy variable. This simplification allows us to analyze phase-ordering kinetics of LLT in detail.

In this article, thus, we construct two-order parameter kinetics coupled to the fluid velocity field, taking into account the density and viscosity differences between the two phases. We show how critical dynamics near the critical point of LLT are influenced by the couplings among the conserved order parameter (density field), the nonconserved one, the velocity field, and the heat mode. We calculate the dynamical structure factor near the critical point (45, 50, 51). We also study the kinetics of LLT under an isothermal condition, revealing the significance of hydrodynamic flow during nucleation-growth–type (NG-type) and spinodal-decomposition–type (SD-type) LLT induced by the density difference between the two phases.

## Results

### Theoretical Model.

#### Thermodynamic model.

We construct the time-dependent Ginzburg–Landau (TDGL) model of LLT with hydrodynamics. To perform the Landau expansion of the free energy, we first need to specify the reference mass density ρ and fraction of locally favored structures c, which we set to be *SI Appendix*, section A for details).

Since we incorporate ϕ as an additional thermodynamic variable, the differential form of the free energy reads*SI Appendix*, section A for the details and the numerical parameters). We present the result for *A* and *B*, respectively. Since α characterizes the density difference between the two phases below the critical point, the slope of the coexistence curve given by the Clausius–Clapeyron relation strongly depends on α, and thus the supercritical state is realized for *A* and *B*, respectively.

#### Dynamical equations.

Here, we describe the hydrodynamic equations of our model in detail. The free-energy functional is expressed in terms of the relevant thermodynamic variables as**1** and *SI Appendix*, section C).

Since the density is generally different between liquid 1 and liquid 2, we need to solve the following hydrodynamic equations with fluid compressibility,**6** denotes the continuity of the fluid mass density. Eq. **7** represents the momentum conservation, where **4**. In the viscous stress in Eq. **11**, we assume that the shear viscosity **8** represents energy transport, where λ is the thermal conductivity. Eq. **9** represents the time evolution of the order parameter ϕ, where

For the theoretical study of critical dynamics, we use Eqs. **6**–**9** while neglecting the thermal noise terms. On the other hand, in numerical simulations of isothermal NG-type and SD-type phase-ordering kinetics, we integrate Eqs. **6**, **7**, and **9** and add the thermal noise terms, **7** and θ in Eq. **9**, which arise from hydrodynamic fluctuations satisfying the fluctuation–dissipation theorem as (33, 52)*Materials and Methods*).

### Critical Dynamics Associated with LLT.

Here we consider the dynamics of density fluctuation near the critical point of LLT. We calculate the dynamic structure factor

In the case of liquids with LLT, on the other hand, there is an extra thermodynamic variable ϕ with the nonconserved nature. Then, the coupling between the nonconserved (ϕ) and conserved (ρ) order parameter leads to the crossover between diffusive (*A* (see *SI Appendix*, section B for the details of the calculation). Here we choose the values of the thermodynamic quantities similar to those for supercooled water. Specifically, we adopt *A* also leads to an extra central component besides the Rayleigh peak in the dynamic structure factor, as presented in Fig. 2 *B* and *C*. In Fig. 2*B*, the wavenumber q is in the range of visible and infrared light. In this q range, the second central mode is evident, whose line width does not depend on the wavenumber q. In Fig. 2*C*, on the other hand, the slowest relaxation mode with nondiffusive relaxation rate is found, where the q range is close to that of inelastic X-ray scattering experiments (56). These modes arise from the coupling between the thermal diffusion and the relaxation of the nonconserved order parameter and are absent in simple liquids, as shown by the dashed curves. The additional central component is reminiscent of the so-called Mountain mode (57, 58), which is associated with the viscoelastic relaxation mode (50). In our model, however, the origin of this mode stems from the temporal fluctuation of locally favored structures. Note that such nondiffusive relaxation has recently been observed in molecular dynamics simulations for strong glass formers (e.g., silica), where the relaxation of locally favored structures (e.g., tetrahedral structures in silica) plays an essential role in local density relaxation (59, 60). More precisely, local density change can take place by the formation or annihilation of locally favored structures without diffusive mode. Of course, the diffusive nature should appear at very small q because of the conserved nature of the hydrodynamic modes such as density ρ.

### Dynamics of LLT.

Next, we focus on the kinetics of LLT, which take place in a metastable or an unstable state of LLT. As described in the Introduction, we hereafter consider isothermal kinetics for simplicity; that is, we assume that τ is constant in space and time after the quench.

#### Hydrodynamic effects on the growth of a single circular droplet.

First, we consider how the growth of a single droplet is affected by the presence of the hydrodynamic degree of freedom. For this purpose, we derive the domain growth law by adapting the thin interface approach as follows. From the continuity equation across the interface, the fluid velocity inside and outside a liquid 2 droplet satisfies the relation of *SI Appendix*, section C for the derivation)**16**, we can see that the droplet growth becomes faster (slower) when the density inside the liquid 2 droplet is smaller (larger) than that in the metastable liquid 1, yet not changing the growth exponent. Thus, we may conclude that the density difference between the two phases characterized by α strongly affects the droplet growth law in NG-type LLT.

This kinetic correction factor, arising from hydrodynamic flow during droplet growth, indicates that the effect of hydrodynamic flow becomes more significant when the density difference between the two phases becomes larger. We can also see that the transition from a low-density to a high-density state is slower than in the opposite direction. In the case of water, for example, *Materials and Methods* for the details of the numerical simulations).

Here, it is worth noting that this result is in contrast to the droplet growth equation in a heat-coupled system (model C) obtained in ref. 61, in which the growth is suppressed by viscous stress with the correction proportional to

#### Hydrodynamic interaction between a pair of droplets.

Next we consider hydrodynamic interaction between growing droplets during NG-type LLT, i.e., how hydrodynamic interaction between droplets affects their growth rates. For simplicity, we here consider the growth of a pair of equal-size droplets. As described in the preceding section, the fluid flow around a droplet is inward (outward) when the density of the droplet is larger (smaller) than that in the host liquid. For the inward case (**15** for the dilute limit). In the same way, repulsive hydrodynamic interaction emerges for the outward case (

We confirm this tendency by numerical simulations and show the results in Fig. 4. We can see acceleration (deceleration) of the droplet growth at the facing side of the interface for *A* and *C*) outside a droplet corresponds to outward flow for the other droplet, the outward flow at the facing interface enhances the droplet growth in this side more than that at the interface in the opposite side. Thus, the droplet coalescence becomes faster (slower) when the density of the new phase is larger (smaller) than that of the initial phase.

This hydrodynamic interaction caused by longitudinal fluid flow in a “compressible” fluid is markedly different from the standard hydrodynamic theory of droplet coalescence in a phase-separating “incompressible” fluid, where the only transverse component is relevant for the kinetics (62, 63). Here it should be noted that the droplet coalescence is also affected by the Marangoni flow induced by the surface tension gradient at the interface (63), but this effect is negligible in our case.

#### NG-type and SD-type LLT near the spinodal point.

Now we turn our attention to overall phase-ordering kinetics of LLT. We show our numerical simulation results for both NG-type and SD-type LLT, focusing on the pattern evolution and hydrodynamic effects on it. First, we equilibrate the system at a high temperature of

We observe NG-type LLT for *SI Appendix*, section D and Fig. S1). However, they can be distinguished rather clearly by the distribution of the order parameter ϕ. The probability distribution of the order parameter ϕ, *C*). On the other hand, it exhibits the gradual, continuous change in the peak position of *D* and *E*). The results are consistent with the temporal change in the order-parameter distribution function observed experimentally for NG-type and SD-type LLT (64).

As described in the previous section, long-range hydrodynamic interaction via longitudinal fluid flow is induced by the density difference between the two phases. During the phase-ordering kinetics, furthermore, the transverse fluid flow is also induced as in the case of fluid phase separation (model H) (33, 62, 65). To reveal which of the longitudinal (*D*. The transverse velocity field also arises due to the inhomogeneous curvature of the domain interface, but the magnitude of the transverse velocity field is much weaker than that of the longitudinal one.

This hydrodynamic flow also affects the domain morphology, since hydrodynamic interaction changes the growth rate, as described in the previous section. For *SI Appendix*, section E and Fig. S2). To conclude, hydrodynamic interaction stemming from the density change upon the phase transition crucially affects the domain morphology for both NG-type and SD-type LLT.

#### Temporal change of the volume fraction in NG-type LLT.

Here we consider how the volume fraction of the new phase (liquid 2) evolves with time for NG-type LLT from liquid 1 to 2. The nucleation-growth mechanism is often explained by the Kolmogorov–Johnson–Mehl–Avrami (KJMA) theory, where the growth of the fraction of the more stable phase **17** is widely adopted, the constancy of the nucleation and growth rates was not assumed a priori in the original KJMA theory (66⇓⇓⇓–70).

In Fig. 7*A*, the solid curve shows the simulation result of the time evolution of the fraction of the more stable domains of liquid 2, **17**. We can see that the KJMA theory with constant nucleation and growth rates cannot explain the observed NG-type ordering behavior.

To seek the origin of this deviation, we investigate the time dependence of the nucleation (Fig. 7 *A* and *B*) and growth rates (Fig. 7 *C* and *D*) . In Fig. 7*A*, we also show the number of the nucleation events *B*. We can see that the nucleation rate monotonically increases with time for

Now we consider the origin of this increase in the nucleation rate during the transition. In the isothermal nucleation kinetics, it is usually assumed that the metastable distribution of the order parameter, or the quasi-equilibrium situation, is realized before nucleation takes place. This is why we can assume that the nucleation rate is constant with time during the transition (Fig. 7*E*). In the present case, however, the relaxation of the order parameter ϕ toward the metastable distribution is very slow near the spinodal point (*SI Appendix*, section D), and thus nucleation takes place before the metastable order parameter distribution is realized (Fig. 7*F*). Since the time evolution equation of the order parameter ϕ given by Eq. **9** contains the random thermal noise term, as it should be in a real system, the spatially inhomogeneous acceleration of the relaxation toward the metastable distribution induces the inhomogeneous nucleation probability, resulting in the time-dependent nucleation rate. We note that the nucleation barrier is lower in regions of higher ϕ, since a smaller difference in ϕ leads to the lower interfacial tension, i.e., the lower nucleation barrier.

Next, we examine the time dependence of the growth rate. As displayed in Fig. 7*C*, the growth rate strongly depends on the incubation time *SI Appendix*, section C and Eq. **S21**): The domain growth is accelerated when *D*, where we can see that the direction of the fluid flow at the interface changes from inward to outward as the phase ordering proceeds, resulting in the acceleration of the domain growth in the final stage of the ordering for

Thus, we may conclude that the deviation from the KJMA theory (Eq. **17**) is a consequence of slow relaxation near the spinodal point and hydrodynamic interaction leading to the increase of the nucleation frequency and the growth rate with time, respectively.

#### Temporal change of the structure factor during LLT.

Here we focus on the time development of the structure factor during LLT, which can be measured by scattering experiments (see refs. 24, 26, and 27 for such experiments on TPP). We show our simulation results on the time development of the two types of structure factor *A* and *B*, respectively (see *SI Appendix*, section F and Fig. S3 for the temporal change in the structure factor for the density, which has the same wavenumber dependence as

In contrast, the situation is fundamentally different for LLT, which is driven by the ordering of the nonconserved order parameter. Here we study the functional form of the structure factor in LLT. For NG-type LLT (at *SI Appendix*, section G and Fig. S4 for the details). This is particularly the case for SD-type LLT, which can be seen from the fact that the distribution function of ϕ does not have a bimodal shape and instead has a broad unimodal shape.

We note that this result is consistent with the experimental results, where a deviation from Porod’s law is indeed observed around the spinodal point by time-resolved light-scattering experiments (26).

## Summary

We study phase-ordering kinetics of LLT by the time-dependent Ginzburg–Landau model considering the hydrodynamic degree of freedom, which is crucial to describe the dynamics in a liquid state. The model has the conserved (mass density ρ) and the nonconserved order parameter (the fraction of locally favored structures ϕ). The former is the order parameter for gas–liquid transition, whereas the latter is the one for LLT. We discuss hydrodynamic fluctuation modes, density ρ, LLT order parameter ϕ, and internal energy and their couplings for stable and metastable states against LLT. We obtain the dynamic structure factor under the coupling of the relevant modes. We find that the couplings result in the crossover between diffusive and nondiffusive relaxation of hydrodynamic fluctuations.

We have also studied how hydrodynamics affect the kinetics of LLT under the assumption of the isothermal condition. We find that the density decrease/increase upon the phase transition, reflecting the density difference between the two liquid phases, inevitably induces compressible hydrodynamic flow, causing not only acceleration/deceleration of the domain growth kinetics but also long-range hydrodynamic repulsive/attractive interaction among the domains. The nucleation kinetics in the vicinity of the spinodal point exhibit the deviation from the classical KJMA theory since the nucleation and growth rates both increase with time. The structure factor also exhibits the deviation from Porod’s law due to the absence of a sharp interface during the phase transition. The deviation of the structure factor is a characteristic of phase ordering in nonconserved order parameter systems.

Our study reveals the crucial importance of hydrodynamic interaction in the critical dynamics near the critical point of LLT and the pattern evolution and kinetics of liquid–liquid transition. Concerning the latter, the effect of external hydrodynamic perturbation such as shear flow (33, 72) on liquid–liquid transition is an interesting topic for future research. Finally, we note that we can also include the elastic degree of freedom in our model to describe the polyamorphic transition. If LLT is accompanied by solidification, density change during the transition may also induce long-range elastic interaction (33, 46, 73).

## Materials and Methods

Here, we describe our numerical simulation scheme in detail. Since the density is different between the two phases, the isochoric simulation is not suitable to study the phase-ordering kinetics, and we need to perform isobaric simulations. To do so, we consider the Gibbs potential of the following form:**19**, **23**, **24**, and **26** in the numerical simulations.

Next we rewrite the dynamical equations in the dimensionless form by proper scaling. We take the scale units as

### Data Availability.

The data that support the findings of this study are available from the corresponding authors upon reasonable request.

## Acknowledgments

This study was supported by the Mitsubishi Foundation and Grants-in-Aid for Specially Promoted Research (Grant JP25000002), Scientific Research (Grant JP18H03675), and Innovative Areas of Softcrystal (Grant JP17H06375) from the Japan Society for the Promotion of Science. The numerical calculations were partially performed on CRAY XC40 at the Yukawa Institute for Theoretical Physics (YITP) at Kyoto University and on the SGI ICE XA/UV hybrid system at the Institute for Solid State Physics (ISSP) at the University of Tokyo.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: tanaka{at}iis.u-tokyo.ac.jp or takae{at}iis.u-tokyo.ac.jp.

Author contributions: K.T. and H.T. designed research; K.T. performed research; K.T. and H.T. analyzed data; and K.T. and H.T. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1911544117/-/DCSupplemental.

Published under the PNAS license.

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