Liquid-liquid transition and polyamorphism.

Two or more liquid states may exist even for single-component substances, which is known as liquid polymorphism, and the transition between them is called liquid-liquid transition (LLT). On the other hand, the existence of two or more amorphous states is called polyamorphism, and the transition between them is called amorphous-amorphous transition (AAT). Recently, we have accumulated a lot of experimental and numerical evidence for LLT and AAT. These intriguing phenomena provide crucial information on the fundamental nature of liquid and amorphous states. Here, we review the recent progress in this field and discuss how we can physically rationalize the existence of two or more liquids (glasses) for a single-component substance. We also discuss the relationship between liquid-, amorphous-, and crystal-polymorphisms, putting a particular focus on the roles of thermodynamics, mechanics, and kinetics.


I. INTRODUCTION
Liquid is one of the fundamental states of matter. It is much denser than the gas state but can still flow, unlike the solid state. Among the three fundamental states of matter, gas, liquid, and solid (crystal), the state of liquid is most poorly understood because of the difficulty in its physical description. For the gas state, the physics is simplified because the mean-free path is much longer than the particle (atom or molecular) size, and thus, we may basically ignore interparticle interactions and treat them perturbatively. The solid (crystalline) state has a long-range translational order, simplifying the theoretical description by the Fourier (wavenumber) space description. For the liquid state, we can have neither of such simplifications. Although the liquid state theory is a robust theoretical framework to describe the structure and dynamics of the equilibrium state of simple liquids, there remain some poorly understood phenomena.
The famous examples are the problems of glass transition 1,2 and water's anomalies. 3 Furthermore, it is often argued that a singlecomponent substance may have two or more liquid states, which is known as "liquid polymorphism," 4 and the transition between them is called "liquid-liquid transition (LLT)." [5][6][7][8][9][10][11][12][13] In relation to it, some materials such as water have two or more amorphous states, which is known as "amorphous polymorphism" 14 or "polyamorphism," and the transition between them is called "amorphous-amorphous transition (AAT)." One of the most well-studied systems, concerning liquid and amorphous polymorphisms, is water. Here, we take water as an example to show how controversial this field is (see Refs. 3,7,9, and 15-17 for review). Water is known to have more than two distinct amorphous states. Mishima et al. [18][19][20] discovered low-density amorphous ice (LDA) and high-density amorphous ice (HDA) and the first-order-like transition between them. Recent advanced experimental studies made a more detailed classification of the amorphous states that depend upon preparation protocols. 21 The amorphous state generally forms by the rapid cooling of the liquid state, and its structure is believed to be virtually the same as its parent liquid. Thus, a natural expectation is that there may be two liquid states, low-density liquid (LDL) and high-density liquid (HDL), corresponding to LDA and HDA, respectively. Numerical simulations indeed showed that some water models, such as ST2 and TIP4P/2005, have a liquid-liquid transition and the associated second-critical point (CP). 13,22,23 In particular, the second critical points of ST2 water 24 as well as two realistic water models, TIP4P/2005 and TIP4P/Ice, 25 have recently been confirmed unambiguously. It was also shown that the critical point is consistent with the three-dimensional (3D) Ising universality class, 25 as expected from the two-state feature and the resulting scalar nature of the order parameter 26,27 (see below). These findings suggest the presence of the second critical point in real water.
The presence of the second-critical point has also been considered the origin of water's anomalies. 7 Water is known to behave very differently from ordinary liquids; 3 for example, it shows the density maximum at 4 ○ C; the isothermal compressibility and the heat capacity both show unusually steep increases with decreasing temperature T, unlike ordinary liquids, in which they monotonically decrease. These phenomena have been ascribed to the crossover from HDL-like to LDL-like liquid upon cooling and the resulting criticality associated with the second critical point. The crossover was believed to occur at the so-called Widom line, 28 where the amplitude and correlation length of critical fluctuation are maximized. The scenario looks exotic and appealing, thus becoming extremely popular. However, there has been no consensus on its validity, as will be discussed later. Even the connection between LDA and HDA to LDL and HDL, respectively, has been a matter of debate.
Although there is a lot of supporting numerical evidence for LLT in model water, 7,9,12,13,22,23,25,29 the debate is continuing for LLT in real water and its link to water's anomalies. This situation is because the homogeneous ice nucleation prevents us from experimentally accessing to LLT. Nevertheless, there are on-going efforts to access it (see, e.g., Refs. 12 and 30-34 on the recent efforts). For overcoming the intrinsic difficulty to access LLT of pure water, there were also efforts for seeking the signature of LLT by using aqueous solutions, for which water crystallization can be avoided by mixing other components, 35-40 yet which are often controversial. Thus, there is no firm consensus on the presence of LLT in real water at this moment.
Such controversy is not limited to experiments but also involves numerical simulations. The difficulty again comes from the metastability of a supercooled liquid to crystallization. For example, Limmer and Chandler 41 challenged the existence of LLT in ST2 water. After a long debate, it has been confirmed convincingly by Palmer et al. 13,24 However, whether LLT exists or not may sensitively depend on the details of the interaction potential, and thus, simulations cannot provide a definite answer to whether LLT exists in real water or not.
LLT and AAT are exciting and intriguing phenomena. However, the difficulties in the experimental and numerical studies of LLT and AAT in water mentioned above are common to all types of systems, including atomic and molecular systems, as shown below. Because of the confusing situation of experimental studies, we have to admit that its physical understanding is still primitive.
In this Perspective, we review the current situation of research on LLT and AAT and discuss future research directions to overcome the controversial situation. The organization of this article is as follows: In Sec. II, we describe the theoretical basis of LLT, mainly focusing on what the relevant order parameter is and how this order parameter is coupled with the density that is a genuine order parameter of liquid. In the following sections, based on this theoretical model of LLT, we discuss LLTs and AATs of various systems reported by experiments and numerical simulations. In Sec. III, we discuss LLT observed by numerical simulations in model systems, which are free from criticisms. In Sec. IV, we discuss LLT and AAT observed in single-component and multi-component atomic systems both experimentally and numerically. In Sec. V, we review molecular liquid-atomic liquid transition in hydrogen, which is the simplest molecular (or atomic) system. It has recently attracted considerable attention because of its importance in planetary physics. In Sec. VI, we discuss LLT and AAT observed in singlecomponent and multi-component molecular systems both experimentally and numerically. We put special focuses on two liquids, triphenyl phosphite and water, which have been studied most extensively. In Sec. VII, we discuss the relationship between liquid and amorphous polymorphisms, focusing on the difference in the character between LLT and AAT, i.e., thermodynamical vs mechanical transition. In Sec. VIII, we briefly discuss the relationship of crystal polymorphisms to liquid and amorphous ones. In Sec. IX, we mention some predictions and possible future directions of research. In Sec. X, we summarize our paper.

II. THEORETICAL BASIS OF LLT
A. Two-order parameter model of liquid

What is the relevant order parameter of LLT?
First, we consider the fundamental question concerning the nature of the liquid: what the relevant order parameter(s) describing a liquid state is? A classical liquid-state theory describes a liquid state by a single order parameter, density ρ, as for a gas state. This description is natural, considering that the relevant order parameter describing a gas-liquid transition is the density. However, this is not necessarily the case at lower temperatures because any liquids tend to form locally ordered structures favored energetically and/or entropically. [42][43][44] Typical examples of energetically driven local ordering can be seen in so-called tetrahedral liquids such as water, Si, Ge, and SiO 2 . Chalcogenides also tend to form such local structural ordering in a liquid state. Even systems with nearly isotropic interactions, such as metallic systems, tend to form icosahedral structures in their liquid states 42,[45][46][47][48] Furthermore, hardsphere systems whose interactions are purely entropic tend to form locally favored structures such as icosahedral structures. [49][50][51][52][53][54] All these facts indicate that a single order parameter, density, cannot describe the liquid state, and we need an additional order parameter, which describes local structural ordering. The additional order parameter should be a scalar order parameter because of the rotational invariance of a liquid state.  27 LLT can be understood as a transition between a ψ-gas state and a ψ-liquid state. Bottom: schematic representation of the free-energy surface of a system having liquid-liquid transition on the ρ-ψ plane. 60 This is a case for Δv < 0.
Since we have the two scalar order parameters, we can naturally have two gas-liquid-type phase transitions, which involve the breakdown of neither translational nor rotational symmetry. Thus, the two transitions should belong to the 3D Ising universality class, 57,58 which was confirmed by numerical simulations for water. 25,59 We argue that the density is the order parameter relevant for a gasliquid transition, whereas the order parameter ψ is relevant for a liquid-liquid transition (LLT). In this physical picture, we can very naturally accept the counter-intuitive phenomenon of LLT as follows: As the difference between a gas and liquid is described by the density ρ, the difference in liquid I and liquid II is described by the difference in the fraction of locally favored structure, ψ. The density difference between liquids I and II is just a consequence of the dependence of ρ on ψ through their coupling. In other words, we may regard LLT as the gas-liquid-like transition of locally favored structures (see Fig. 1).

Density may not be the order parameter of LLT
Here, we emphasize the essential difference in the physical nature between the two order parameters: The spatial change in ρ should be made only through the material transport, and its change at a specific location r should be coupled with the change in ρ in its surrounding region. In contrast, locally favored structures can be created and annihilated locally without the constraint of such conservation. Thus, we can conclude that the density ρ is the conserved order parameter, whereas the order parameter ψ is the nonconserved one. As will be described below, this difference in the order parameter's character leads to the fundamental difference in the ordering dynamics of the two order parameters, i.e., gas-liquid and liquid-liquid transitions.
Concerning the above, we briefly discuss the nature of the order parameter governing LLT. In the above, we argue that the order parameter controlling LLT should be the fraction of locally favored structures. Since a liquid-liquid transition is also seen in particles interacting with spherically symmetric potentials such as the Jagla model, [61][62][63] it has often been argued that the density is the relevant order parameter of LLT (see, e.g., Ref. 63). Nevertheless, we argue that the local bond orientational order parameter is still a relevant order parameter for this case since it is natural to expect that the presence of the two length scales in the potential leads to the selection of a specific local symmetry or locally favored structures. Theoretically, it is natural to have the two order parameters with different characters (conserved and non-conserved), ρ and ψ, rather than to have only the density as an order parameter: As mentioned above, the presence of the two scalar order parameters allow us to have a gas-liquid-type transition for each order parameter. More importantly, the conserved order parameter, i.e., density ρ, cannot explain the dynamics of LLT.
B. Thermodynamics 1. Two-state model As described above, a liquid is in a disordered state macroscopically; however, it can possess a short-range bond orientational order. Based on this physical picture, we express a liquid state by a simple two-(or multi-)state model with the cooperativity of such bond ordering(s) (see Fig. 2). The phenomenological two-state model of liquid-liquid transition (LLT) was developed by Strässler and where f ρ is the free energy of normal liquid structures, J represents the cooperativity, k B is the Boltzmann constant, T is temperature, and P is pressure. In the two-order-parameter description of liquid, we need to consider the total free energy of the system, including f (ρ) and the coupling between the two order parameters, ρ and ψ, 26,27 which we will discuss later. Equation (1) describes the thermodynamics of LLT, e.g., the phase diagram concerning LLT (see Fig. 1). Then, the corresponding free-energy functional as a function of the space-dependent order parameter ψ(r) is expressed as f {ψ(r)} = dr [−ΔGψ(r) + Jψ(r)(1 − ψ(r)) + k B T(ψ(r) ln ψ(r) + (1 − ψ(r)) ln(1 − ψ(r)))], (2) where ΔG = ΔE − TΔσ + ΔvP. ΔG is the free energy change associated with the formation of a locally favored structure. This free energy and the scalar (or discrete) nature of the order parameter ψ immediately tell us that the critical point associated with the order parameter ψ belongs to the Ising universality class, 27,60 as the gas-liquid critical point associated with the order parameter ρ do. The dynamical critical behaviors of the two transitions are, however, different, reflecting the non-conserved and conserved nature of ψ and ρ, respectively (see below).
Here, we briefly consider the origin of the cooperative formation of locally favored structures, or the non-zero J, which is the most fundamental and essential issue for our understanding of liquid-liquid transition. For the Ising magnet, the interaction energy between spins directly causes the cooperativity. 57,58 The entropic origin was also suggested for water. 74 In our case, thus, the interaction between locally favored structures should be the origin of cooperativity. So, the question is how the atomic (or molecular) interaction controls the interaction between locally favored structures. First, the interaction energy leading to the formation of locally favored structures, which is strong enough to overcome the entropy loss, should also lead to their cooperative formation if there is little geometrical frustration upon their formation. Second, the entropic gain upon the formation of locally favored structures as in hard-sphere liquids can also lead to their cooperative formation if there is little geometrical frustration upon their formation. These two mechanisms are a consequence that the same interaction controls the elementary unit (atom or molecule) and a higher-level structure made of the same unit (locally favored structure). Third, the cooperativity may arise from the van der Waals interactions originating from the local density difference between locally favored and normalliquid structures, δρ. We can estimate the strength of this attractive interaction as where U 11 is the interaction between basic units ("atoms" or "molecules") of size a, ρ is the density, and is the size of a locally favored structure. This interaction itself might be too weak to cause LLT for δρ/ρ ≪ 1 but should affect the value of J. The fact that numerical simulations with classical potentials cause LLT indicates that one of these mechanisms should be responsible for LLTs seen in these simulations. Finally, the cooperativity should also stem from the change in the interparticle interaction, e.g., the cooperative change in the electronic structure, induced by the formation of locally favored structures. It may help the formation of others nearby a locally favored structure. This mechanism may play an essential role in real liquids. Since the origin of the cooperative ordering of ψ is critical for our understanding of LLT, further careful studies are highly desirable. The equilibrium value of ψ is determined by the condition ∂f (ψ)/∂ψ = 0 or where β = 1/k B T. Here, we note that the degeneracy of each state, or the entropy difference between the two states, strongly affects the phase behavior. At the critical point of LLT, the conditions, ψ (ψc) = 0, and f (4) ψ (ψc) > 0 should be satisfied. Thus, we obtain We also obtain the first-order phase-transition line Tt(P) as Here, we note that the first-order transition occurs only if Tt < Tc.
For Tt > Tc, this Tt is a temperature where ΔG = 0, and thus,ψ = 1 2, which gives the so-called Widom line. The maximum of K T is located near Tt, but not necessarily coincides with it since the noncritical part of K T has its own T-P dependence. 83,84 The signs of Δv and Δσ determine the slope of Tt(P) (see Fig. 3). Liquid I and liquid II are defined as the two points of the liquid-state free energy on the 2D order-parameter (ρ-ψ) plane (see Fig. 1). The above phenomenological theory indicates that LLT may not be rare and can occur in any system: The only requirement to have LLT is non-zero J. It is a general conclusion of the two-state 27 or mixture 64,65 models. Franzese et al. 85 also emphasized the generality of LLT, based on a lattice model.

A liquid is neither a mixture of two distinct structures nor a mixture of two liquids
Here, we mention two types of common misunderstandings on the two-state model of LLT. First, we stress that the two-state model describes a liquid as a mixture of two states but not a mixture of two distinct structures. Quite often, it is assumed in the literature that a liquid is composed of two specific distinct structures. For example, it is often assumed that water is a mixture of low-density-amorphous (LDA) and high-density-amorphous (HDA) structures, both of which are low-entropy states. This kind of model fails to consider a significant entropy difference between locally favored and normal-liquid structures, leading to a wrong conclusion. The disordered nature of a liquid state indicates that the normal-liquid state should have a large entropy (σρ ≫ σψ), i.e., a large degeneracy of state (g ρ ≫ g ψ ), in general. For example, for water's anomalies, many mixture models of two distinct structures predict that the fraction of LDA-like structures is as high as ∼50% at ambient condition. On the other hand, considering the significant entropy difference between locally favored structures (LFS) and normal-liquid structures (NLS), we predict that ψ ≪ 1. Thus, the water's density anomaly can be approximated by a simple exponential function (or the Boltzmann factor) at ambient condition (see below). [66][67][68] We also note that in the literature, we quite often see that a liquid is regarded as a mixture of two liquids. In the case of water, for example, water is often described as a mixture of low-density liquid (LDL) and high-density liquid (HDL), and the order parameter is defined as the fraction of LDL (or HDL). However, this description is not correct physically. We can easily recognize this description's unphysical nature from the fact that we do not describe a fluid as a mixture of gas and liquid, and the order parameter of gas-liquid transition is density and not the fraction of gas (or liquid). Thus, we should define the order parameter of LLT as the fraction of locally favored structures. The terms, LDL and HDL, should be used only for describing the macroscopic liquid states.

Free energy including the coupling to the density field
Here, we describe the thermodynamics of liquid in terms of the two order parameters, ψ and ρ, and their couplings, based on the Ginzburg-Landau-type approach. For simplicity, we consider a liquid state near the critical point of LLT (Tc, Pc) [or the mean-field spinodal point (Tsp, Psp)]. Thus, we ignore the gas-liquid critical point. We perform the Landau expansion of the free energy around the critical point of LLT. We set the reference mass density ρ and fraction of locally favored structures ψ, which we set to be ρc and ψc at the critical point. We expand the free energy in terms of δψ = ψ − ψc, δρ = ρ/ρc− 1, and τ = T T 0 c − 1 86,87 (here T 0 c is the bare critical temperature without a coupling between ψ and ρ) as follows: where v 0 is the molecular volume, Here, Δσ is the entropy difference between liquid I and liquid II. Here, α is the coupling parameter determining the density difference between the two liquid phases. We examine how the density and the compressibility depend on α in equilibrium and metastable states below. μ 0 and n 0 are the dimensionless chemical potential and number density at the reference state, respectively, but they do not appear in the thermodynamic derivatives and are thus neglected hereafter. K 0 is the dimensionless isothermal compressibility at the reference state. α T and C 0 denote the thermal expansion coefficient and heat capacity, respectively.
Since we introduce δψ as an additional thermodynamic variable, the differential form of the free energy includes a related term, where σ is the entropy density and μ is the chemical potential. Here, Γ = ∂f /∂δψ = f 0 (δψ, τ)/∂δψ − αδρ and vanishes in equilibrium and metastable states. Then, the thermodynamic pressure is given as which also depends on δψ. Here, β = 1/k B T 0 is the inverse of the thermal energy at the reference state. From the above equations, we can see that the reduced density, δρ, depends on the coupling constant α.

Lattice model
Next, we mention the lattice model, [88][89][90][91][92][93][94][95][96] which was developed to explain water's anomalies and LLT of water. The lattice model generally has a direct relation to the phenomenological model, such as the two-state model described above. 57,58 For example, Poole et al. proposed the van der Waals equation of water that incorporates the effects of the network of hydrogen bonds, based on the lattice model of water, 89 and reproduced the two critical points. 97 Then, along a similar line, analytical theories of liquid water incorporating hydrogen bondings were further developed. 98,99 Sastry et al. 92 introduced the Hamiltonian composed of the contributions from non-hydrogen-bonding (NHB) and hydrogenbonding (HB) parts on the interaction energy. They assume that fluid is composed of N cells of equal size. Each cell i = 1, . . ., N is characterized by a quantity ni, with ni = 1 if the average density of the cell is above a threshold and ni = 0 otherwise. The threshold is set to distinguish liquid and gas densities. The non-hydrogenbonding part of the Hamiltonian, i.e., van der Waals attractive interaction, between the molecules is given by the following lattice-gas Hamiltonian: where > 0 is the strength of the attraction, the first sum is extended to nearest-neighbor cells (i, j), and the second term determines the average density in the system through the chemical potential μ (greater μ favors the liquid state). This Hamiltonian is for the standard lattice-gas model, which describes a simple fluid with the gas-liquid critical point. Since in water, two molecules can form an HB only if they are correctly oriented, Sastry et al. 92 described this intermolecular interaction by introducing an orientational variable per each hydrogenbonding arm. The orientation of the arm of the molecule on site i that faces site j is represented by a (discrete) Potts variable sij = 1, . . ., q, with a finite number q of possible orientational states. The Potts variable sij interacts only with the variable sji of the arm of the molecule (if any) on site j facing site i. Therefore, in addition to H NHB , we should consider the following HB contribution: where −J HB < 0 represents the energy decrease due to the HB formed when two nearest-neighbor cells are occupied by a molecule (ninj = 1) and the two arms sij and sji have the appropriate orientation (i.e., δs ij ,s ji = 1).
Then, Franzese et al. 94,95 added an intramolecular interaction term that, for each occupied cell (ni = 1), gives a negative contribution (−Js < 0) to the energy when two of its arms are in the appropriate orientational state (δs ik ,s il = 1 assuming that they have to be in the same state), where the term is summed over all the cells and over all six different pairs (k,l)i of the four arms belonging to the same molecule i. Therefore, the total Hamiltonian of the system is given by It was shown that this Hamiltonian has two critical points, i.e., the gas-liquid and liquid-liquid critical points. In this model, the presence of LLT is a consequence of Js > 0. This fact implies that the strength of the intramolecular interaction, Js, favoring tetrahedral symmetry is related to the cooperativity strength, J, in the above twostate model. It is interesting to seek the precise relationship between the two approaches.

Relationship between LLT and the liquid-crystal phase diagram
Here, we show possible relationships between LLT and the liquid-crystal phase diagram, 27 including crystal polymorphs, in Figs. 3(a)-3(d). The only necessary conditions to have LLT are (i) the existence of locally favored structures and (ii) their cooperative formation (J > 0). However, the relative locations between the critical point of LLT, Tc = J/2k B , and the melting point, Tm = ΔHm/Δσm, where ΔHm and Δσm are the changes in the enthalpy and entropy of melting, respectively, depend on the relationship between J, ΔHm, and Δσm. For Tc > Tm, LLT can exist in a stable equilibrium-liquid state [see Fig. 3(a)], as observed for some atomic systems. In contrast, LLT exists in a supercooled liquid state for Tc < Tm [see Figs. 3(b)-3(d)], which are typical for molecular liquids. If the LLT is located below the homogeneous nucleation line of crystals, it is hard to access LLT experimentally. LLT might exist even in a glassy state for Tc < Tg. In such a case, it is almost impossible to access LLT. Thus, the relationship between Tc, Tm, and Tg determines how easily we can access LLT.
Liquid with Tc > Tm: The candidates of liquids with Tc > Tm are atomic systems such as hydrogen, carbon, and phosphorus [see Fig. 3(a)]. In this case, the LLT lines and the associated critical points may exist in an equilibrium liquid state. In the case of carbon, for example, there can be a few candidates of locally favored structures, reflecting sp (ψ1)-, sp 2 (ψ2)-, and sp 3 (ψ3)-type bondings. In Fig. 3(a), sp 2 -type bonding is assumed to form at ambient pressure. We call this liquid ψ2-liquid. It might transform into ψ1 liquid at negative pressure. We denote the critical points associated with ψ2and ψ3-type orderings as CPψ 2 and CPψ 3 , respectively. Above the critical points, the type of liquid changes continuously. However, we should note that the sp 2 configuration has a high rotational energy barrier and tends to form the planar structure with smaller entropy. Glosli and Ree 100 suggested that it makes the sp 2 liquid less favorable than the sp or sp 3 liquid. Thus, sp 2 (ψ2)-liquid might not exist.
The sign of the slope of the melting line obeys the Clausius-Clapeyron relation dTm/dP = Δvm/Δσm, where Tm is the melting

FIG. 3. Possible relationships between
LLT and the liquid-crystal phase diagram. (a) Schematic P-T phase diagram for a system with three polymorphs (ψ2-, ψ3-, and ρ-crystals) where Tc > Tm (possibly in carbon). Here, we show the case of Δv/Δσ < 0. In this case, we assume more than two types of locally favored structures in the liquid. (b) Schematic P-T phase diagram for a system with two polymorphs (including ψ-crystal), where Tg < Tc < Tm, as in ST2 and TIP4P waters. ψ-crystal is stable at low pressures, whereas ρ-crystal is stable at high pressures. CPψ is a critical point of ψ ordering. The "ms" stands for "metastable." Here, we show the case of Δv/Δσ < 0. The dashed and dotteddashed lines are the spinodal and firstorder transition lines of LLT, respectively. (c) Schematic P-T phase diagram of a system without polymorphism, where Tc < Tg and Δv/Δσ > 0. CPψ is a critical point of ψ ordering and located at negative pressure. (d) The same as (c) but for Δv/Δσ < 0. The gas-liquid critical point (CPρ) is not shown in panels (a)-(d). This figure is reproduced with permission from Fig. 18  point, and Δσm and Δvm are the changes in entropy and volume upon crystal melting, respectively. Since Δσm > 0 besides hard spheres, the sign of dTm/dP is determined solely by Δvm. Thus, in this figure, the density of each phase is related as follows: ψ2-liquid < ψ2-crystal < ψ3-liquid < ψ3-crystal < ρ-liquid < ρ-crystal. For carbon, ψ2-crystal is graphite, and ψ3-crystal is diamond.
Liquid with Tc < Tm and polymorphs: Figure 3(b) shows the case of Tc < Tm, for which the LLT line and the associated critical point CPψ exist in a supercooled state below the melting line. This system has two polymorphs, ψand ρ-crystals. Water may be such an example. 7,22 For water, for example, ice Ih is identified as the ψ-crystal, whereas ices III, V, . . . are identified as ρ-crystals. 66,68,81 The liquid density should be higher than that of the ψ-crystal but lower than that of the ρ-crystal, consistent with what is known for real water. The situations of Si and Ge may be similar to that of water (see below).
Liquid with Tc < Tm and without polymorphism: Even a liquid with Tc < Tm and without polymorphism may have LLT, whose Pc may be located at either positive  Fig. 3(b) is the absence of the ψ-type crystal at positive pressure, i.e., dTm/dP > 0 for P ≥ 0, which applies to all molecular systems besides water. LLT in triphenyl phosphite (TPP) 101,102 and n-butanol 103 belong to the type of Fig. 3(c), for which the locally favored structures are denser than the normal liquid structures (Δv > 0). We also know that Δσ > 0 for these systems. For example, for TPP, the fact that Δv > 0 has also been confirmed by high-pressure measurements. 104, 105 For the case of Fig. 3(d), we expect the presence of the ψ-crystal at negative pressure because Δv < 0 suggests the presence of the ψ-crystal.

Relationship of LLT with the melting-point maximum and kink
As shown in Fig. 3(a), the presence of LLT in an equilibrium liquid state may cause the melting-point maximum for a crystal. Thus, the melting-point maximum is often regarded as a signature of LLT. For example, Rapoport 65 developed a mixture model to explain this phenomenon. As he already pointed out, we emphasize that the melting-point maximum does not necessarily mean LLT. Recently, this problem was also discussed by Imre and Rzoska, 106 Makov and Yahel, 107 and Fuchizaki. 108 The melting-point maximum without LLT can, for example, be seen in systems interacting with soft potentials such as the Gaussian Core Model (GCM). This unusual behavior stems from the fact that the pair potential between GCM particles is bounded. 109 Because of this feature, the core is unable to provide the excluded volume effects enough to maintain the crystalline order, leading to a The Journal of Chemical Physics PERSPECTIVE scitation.org/journal/jcp re-entrant melting of the crystal at high pressure. It is a typical example of the melting-point maximum without LLT. The presence of the melting-point maximum is a characteristic feature of systems of particles with bounded repulsive interactions. [110][111][112] Moreover, the GCM shows an fcc-to-bcc transition. The fcc phase is stable at low T and P, where the repulsion is strong enough to avoid penetration. In contrast, the core is forced to penetrate at high P, and thus, the bcc phase becomes the stable one. Such behavior was recently reported for a simple atomic system of Na: 113 the screening of interionic interactions by conduction electrons induces softening of the short-range repulsion at high P. These examples indicate that the presence of the melting-point maximum is not necessarily a signature of LLT.
Here, we show that the melting-point maximum can also be induced by the continuous change in the liquid structure in the framework of the two-order parameter model of liquid. 26 We can explain the non-monotonic P-dependence of the melting point, Tm, by the P-dependence ofψ. It predicts the following approximate relation for the P-dependence of Tm [see also Fig. 4(a)], where T b m (P) is the background part, i.e., the melting point of the crystal to a hypothetical liquid made of only normal-liquid structures. ΔT is the strength of the effect of locally favored structures, i.e., ψ ordering, on the melting-point shift. According to the Clausius-Clapeyron relation, dTm/dP = Δvm/Δσm. We assume that both the volume and entropy of a crystal change monotonically with an increase in P, which may be reasonable for ordinary systems besides the systems interacting with the soft potentials mentioned above. On the other hand, the two-state model predicts that the specific volume vs and entropy σ of the liquid depends on P through the following relations: (a) A system without a kink in the melting curve. dTm/dP is also schematically shown. Our twoorder-parameter model predicts the exponential-like functional shape, reflecting the pressure dependence ofψ. (b) A system with a kink of the melting curve, reflecting the presence of LLT. In this case, dTm/dP has a discontinuous jump, reflecting the kink in the pressure dependence of the melting point T. This figure is reproduced with permission from Fig. 19  where v B sp and σ B are the T, P-dependent specific volume and entropy of normal-liquid structures, respectively. Depending upon the competition between these two effects, thus, the melting point can have a broad maximum as a function of P even without LLT. In such a case, the pressure dependence of dTm/dP must be continuous, as shown in Fig. 4(a).
In contrast, the melting-point maximum due to LLT should have the discontinuous jump of dTm/dP across the LLT line, i.e., a sharp kink of the melting-point curve, as shown in Fig. 4(b). Thus, we need special care to identify LLT based on the shape of the melting-point curve.
Here, we note that Mishima and Stanley used the kink of the metastable melting line of ice IV to show the LLT in liquid water. 114 C. LLT in a liquid mixture Later, we discuss LLT in a mixture of two liquids, one of which has LLT and the other does not. Debenedetti and his co-workers developed a microscopic model for aqueous solutions. 115,116 We consider how LLT is affected by mixing a liquid without LLT, based on a phenomenological two-order-parameter model. 36,117,118 Anisimov and his co-workers 119 also studied this problem. Since we mainly consider molecular-liquid mixtures in the following, we assume that the mixture is composed of molecular liquid 1 (with LLT) and molecular liquid 2 (without LLT). However, this discussion can also be applied to atomic-liquid mixtures, as is. We denote the composition of liquid 2 in the mixture as ϕ. According to our two-orderparameter model, we can write down the Gibbs free energy of a binary mixture of liquids 1 and 2 in terms of ϕ and ψ as where ρ and ψ represent the fractions of normal-liquid and locally favored structures of liquid 1 (with LLT), respectively. The Gibbs free energy of liquid 2 is defined as F 0 2 . We set Hj and gj as the enthalpy and degeneracy of each state (j = ρ or ψ). The enthalpy Hj is given by Ej + Pvj, where Ej and vj are the energy and specific volume of state j. We assume that gψ ≪ gρ since locally favored structures have specific structures, whereas normal-liquid structures have many possible configurations. 26,27 Here, χ 12 is the interaction parameter concerning the miscibility, and χψ is the interaction parameter concerning LLT.
The interaction parameter χψ, which governs LLT in our model, has a ϕ-dependence, since the probability that one molecule 1 can interact with nearest-neighbor molecules 1, which is characterized by χ 12 , monotonically decreases with an increase in ϕ. For an ideal mixture, χψ(ϕ) = (1 − ϕ)χ 0 ψ , where χ 0 ψ = χψ(0). Strictly speaking, however, we should use the relation χψ(ϕ) . This relation takes into account the fact that the attractive interactions between molecules 1 and 2 increase the local composition of molecule 2 around molecule 1 than the average composition. Here, Δμ mix = μ(ϕ) − μ(0) is a difference in the chemical potential of molecule 1 between the mixture and pure liquid 1 at the same temperature and pressure. In the above, we used the relation between the activity a 1 (ϕ) and μ mix (ϕ), a 1 (ϕ) = exp(Δμ mix (ϕ)/k B T).
We note that the chemical potential of molecule 1 in the mixture, μ(ϕ), is given by F − ϕ(∂F/∂ϕ). Thus, we can obtain the explicit forms of F(ϕ, ψ) and μ(ϕ) (or Δμ mix ) by solving the differential equation with this expression of χψ(ϕ) in a self-consistent manner. However, for a low ϕ region, we may neglect higher-order terms, ϕ n (n ≥ 3). Once we assume it, we can solve this differential equation analytically and obtain the following relation: In the above, we used the fact that T SD of pure material 1 (ϕ = 0) is given by Thus, the activity of liquid 1, a 1 , is given by Now, we show that both Tm(ϕ) and T SD (ϕ) can be scaled by the activity a 1 (ϕ) = exp(Δμ mix (ϕ)/k B T). At the melting (equilibrium) point, the difference Δμ mix (ϕ) is identical to that between crystal 1 and liquid 1, Δμc− l (ϕ). Thus, the melting point depression is also determined by Δμ mix (ϕ) as follows: where ΔH 0 m and Δσ 0 m are the enthalpy and entropy of melting of crystal 1 in the pure state, respectively, T 0 m is the melting temperature of crystal 1 in the pure state, and Tm(ϕ) is that of crystal 1 in the mixture of the composition ϕ. Here, we used the relation of Δσ 0 m = ΔH 0 m T 0 m . Thus, we obtain the following relation: On the other hand, the spinodal temperature of LLT, T SD , is given from the condition of ∂ 2 F/∂ψ 2 = 0 as According to Eqs. (19) and (20), Tm(ϕ) and T SD (ϕ) should be the functions of only a 1 (ϕ).

D. The hierarchical nature of the two-state model
Here, we discuss the hierarchical nature of the two-state model, 83,84 which is crucial for considering the effect of the twostate feature on the dynamics. At each state point, the strength of thermodynamic anomalies is directly proportional to the order parameter ψ, i.e., the amount of locally favored structures. On the other hand, each molecule's dynamic properties are not determined locally but depend on the local structural environment including its nearest neighbors. Thus, to describe the dynamics based on the two-state model, we need the spatially coarse-grained order parameter ψ D . Here, ψ D is the fraction of dynamic ψ-state after coarse-graining (see Refs. 83 and 84 on the details). In general, ψ D ≤ ψ since the neighbors of a ψ-state molecule are not necessarily in ψ states. We call the line of ψ(T, P) = 1/2 the "static Schottky line," on which the thermodynamic fluctuation associated with the two-state feature is maximized, and the so-called Schottky anomaly has a maximum. We note that this line coincides with the so-called Widom line. On the other hand, we call the line of ψ D (T, P) = 1/2 the "dynamic Schottky line," on which dynamical fluctuation is maximized. If there is criticality, these two lines should meet at a critical point, as shown in Fig. 5. This crossing of the static and dynamic Schottky lines at the LLCP may be used to seek the location of the liquid-liquid critical point (LLCP). 84,120 This method should be useful especially for a system for which we cannot directly access the LLCP experimentally, as in the case of water.
In the hierarchical two-state model, a dynamic quantity X, such as the viscosity η and the structural relaxation time τα, is expressed by the generalized Arrhenius law as a is the activation energy difference between dynamic ψ and ρ states, and X 0 is the prefactor. A hidden assumption behind Eq. (21) is that the lifetime of dynamic ρ and ψ states should be much shorter than the characteristic dynamical timescale of the system. It allows us to treat the average of the activation energies as the effective activation energy. 80 Although this assumption is difficult to verify experimentally, in simulations of water, we directly confirm it by the fact that the lifetime of dynamic states is approximately two orders of magnitude shorter than water's reorientation time. 84

E. Kinetics of LLT
Next, we discuss the kinetics of LLT. Since LLT occurs in a liquid state, the hydrodynamic degrees of freedom should play a critical role in the dynamics. Recently, we developed a dynamic model of LLT, incorporating the hydrodynamic degrees of freedom. 87 Since the density is generally different between liquid I and liquid II, we need to solve the following coupled hydrodynamic equations for compressible fluid: where e = f + Ts is the internal energy density. Equation (22) denotes the continuity of the fluid mass density. Equation (23) represents the momentum conservation, where ↔ Π and ↔ σ represent the reversible and dissipative stress tensor, respectively. They are given by where βv 0p = βv 0 p − D ∇δψ 2 2 is the diagonal part of the reversible stress tensor and p is given by Eq. (12). In the expression of the viscous stress in Eq. (27), we assume that the shear viscosity η(δψ) and the bulk viscosity ζ(δψ) primarily depend solely on the order parameter δψ. Equation (24) represents energy transport, where λ is the thermal conductivity. Equation (25) represents the time evolution of the order parameter δψ, wherê The kinetic coefficient L is assumed to be much smaller than unity to ensure that the phase transition is governed primarily by the time evolution of the order parameter δψ (non-conserved kinetics).

Critical dynamics associated with LLT
First, we consider the dynamics of density fluctuations near the critical point of LLT 87 since scattering experiments can measure it. The dynamic structure factor Sq(ω) is the essential quantity characterizing the liquid dynamics. Here, q is the wavenumber, and ω is the angular frequency.
In pure liquids without LLT, Sq(ω) is generally known to be composed of three distinct peaks: the Rayleigh peak centered at ω = 0 and the Brillouin doublet centered at ω = ±csq, where cs is the speed of sound. [122][123][124] The half widths of Rayleigh and Brillouin peaks are proportional to q 2 , determined by thermal diffusion and sound attenuation, respectively. The diffusive character of the relaxations originates from the conserved nature of these hydrodynamic modes.
For a liquid with LLT, on the other hand, there is an additional thermodynamic variable δψ with the non-conserved nature. Then, the coupling between the non-conserved (δψ) and conserved order parameter (ρ) results in the crossover between diffusive (q 2 ) and non-diffusive (q 0 ) relaxation behaviors. The additional nondiffusive mode appears as an additional central component besides the Rayleigh peak in the dynamic structure factor (see Fig. 2 of Ref. 87). This behavior stems from the coupling between the thermal diffusion and the non-conserved order parameter modes. Thus, it is absent in simple liquids. This additional central component is reminiscent of the so-called Mountain mode, 125,126 which is associated with the viscoelastic relaxation mode in a supercooled liquid. 123 However, we stress that this central peak comes from the temporal fluctuation of locally favored structures for a liquid with LLT. The critical point is that a local density change can occur through the formation and annihilation of locally favored structures without a diffusive mode since locally favored and normal-liquid structures have different local densities. Of course, the diffusive nature should appear in the small q limit because of the conserved nature of the hydrodynamic mode. It is highly desirable to check the above predictions on the dynamical structure factor near the second critical point and the dispersion relations (see Ref. 87 for details) by scattering experiments or numerical simulations.

Nucleation-growth (NG)-type LLT
Now, we turn our attention to the nonequilibrium dynamics of LLT. When we quench and anneal liquid I at a temperature metastable with respect to LLT, nucleation-growth (NG)-type LLT occurs. Our model predicts that droplets of liquid II are randomly nucleated in both space and time in liquid I, and the droplet size R grows with a nearly constant interface velocity as R ∝ t. 27,86,87,[101][102][103] The Journal of Chemical Physics

PERSPECTIVE scitation.org/journal/jcp
Considering the hydrodynamic effect, we obtain the following expression for dR/dt: 87 where ρ I and ρ II are the density of liquids I and II, respectively, Δ is the free-energy difference with respect to the variation of δψ between the two phases, γ is the interface tension, and K is the mean curvature of the interface. This relation tells us that the droplet growth is faster (slower) when ρ II is smaller (larger) than ρ I . In the late stage, droplets of liquid II collide, coalesce, and further grow. This behavior is characteristic of NG-type ordering. We also found that the hydrodynamic degrees of freedom cause interdroplet interactions through the hydrodynamic flow induced by the density change accompanied by LLT. Eventually, the new phase covers the entire system, and the boundary between droplets tends to disappear. Thus, liquid I almost transforms into homogeneous liquid II [see Fig. 10(a)]. This behavior is a consequence of the non-conserved nature of the order parameter ψ.
However, it should be noted that the complete homogenization may not occur if the system is dynamically arrested by glass transition. This situation is indeed the case for TPP and n-butanol, where LLT takes place below the glass-transition point of liquid II. [101][102][103] If a conserved order parameter governed the LLT, the diameter would grow in proportion to R ∼ t 1/3 , and the system would never become homogeneous again. 57,58 The so-called lever rule determines the final volume fractions of the two phases.
Next, we consider the temporal evolution of the average order parameter ψ during the transformation. The volume fraction of the nucleated droplets, Φ(t), is expected to increase, following the Avrami-Kolmogorov equation, 58 Φ(t) = 1 − exp(−Kt n ), where n is the so-called Avrami exponent. For the case of homogeneous nucleation and isotropic linear growth, n = d + 1 (d is the spatial dimensionality). 58 A possible deviation from this prediction is discussed in Ref. 87.
The NG-type transformation is also characterized by the temporal change in the probability distribution function of the bond order parameter ψ, P(ψ): P(ψ) changes from a single Gaussian shape at t = 0 to another Gaussian (t → ∞) through a double-peaked shape for NG [see Fig. 6(a)]. 86,87,102,127 This behavior is general to NG-type transition and a consequence of the fact that the new phase's nuclei already have the final equilibrium value of the order parameter determined by the free energy at that temperature.

Spinodal-decomposition (SD)-type LLT
Next, we consider SD-type LLT, which occurs when a liquid is quenched into the unstable region below T SD . [101][102][103] Below T SD , infinitesimally small fluctuations of ψ grow without decaying. Initially, the amplitude of the ψ fluctuations grow exponentially until the nonlinear terms slow down the growth.
For the late-stage ordering of a non-conserved order parameter, the interface motion is described by the Allen-Cahn equation v = dR dt = −LK, where v is the interface velocity, L is the kinetic coefficient, and K is the mean curvature of the domain interface (∼1/R). This relation yields the domain coarsening law of R ∼ √ Lt. However, this is a case for a symmetric quench where there is symmetry against the order parameter fluctuations, and a sharp interface between the two phases is formed (e.g., the ordering of Ising ferromagnet under zero-field). For an off-symmetric case, a sharp interface is never formed, and the scaling argument is not firmly justified, making the quantitative analysis of the coarsening behavior of SD-type LLT difficult.
The evolution of the average value of ψ for a critical quench can be approximated by the theoretical prediction for the SD-type evolution of a non-conserved order parameter, where Υ is the growth rate of the order parameter. 58 This relation is obtained from the kinetic equation by ignoring the gradient term, the thermal noise, and the couplings between the two order parameters. To derive this relation, we also neglect the linear and cubic terms in the free energy after expanding in terms of δψ, although they may play an essential role in many cases. For more detailed discussions, including the effects of hydrodynamics, see Ref. 87. Finally, we mention the temporal change in the probability distribution function of the order parameter ψ, P(ψ). For SD-type LLT, the distribution transiently becomes broader. Reflecting the nonconserved nature of ψ, the mean value of ψ,ψ, should continuously increase with time for SD-type LLT [see Fig. 6 86,87,102,127 F. Phase transition in liquids other than LLT with the Ising criticality Here, we mention other kinds of phase transitions in a liquid state, for which the relevant order parameter is not the scalar order parameter, unlike LLT we discussed above. This problem The Journal of Chemical Physics PERSPECTIVE scitation.org/journal/jcp was also discussed by Anisimov and his co-workers, 79 focusing on whether the transition is accompanied by phase separation or not. The most well-known examples in phase transition in isotropic single-component liquids are λ-transitions in quantum liquid, helium, [128][129][130][131][132] and classical atomic liquids such as sulfur [133][134][135] and selenium. 136 The order parameters of these transitions are n = 2 and n → 0 in the terminology of the n-vector model. The former order parameter is characterized by its continuous symmetry, whereas the latter is the order parameter for the self-avoiding walk of polymers. In contrast, the order parameter of LLT discussed above is n = 1 and, thus, has discrete symmetry. 27 Thus, LLT should belong to the universality class of three-dimensional (3D) Ising criticality. 57,58 On the other hand, the universality classes of the systems of n → 0 and n = 2 are fundamentally different from the 3D-Ising universality class (n = 1) LLT belongs to (for details on the dynamic universality class, see Ref. 137). Very recently, however, LLT has been suggested for sulfur. 138 We will discuss this point later (see Sec. IV A 1). We also note that there are also phase transitions between liquid and liquid-crystalline phases, for which the liquid-crystalline phase is in a liquid state but has an orientational order. For nematic ordering, for example, the order parameter is tensorial. 57 Thus, we do not discuss these types of phase transitions in liquids in this article, although they are physically attractive.

III. LLT IN MODEL SYSTEMS
First, we focus our attention to numerical studies of model systems since they have shown the most convincing evidence of LLT, for which there has been a firm consensus among researchers. The liquid-liquid transition was first realized in computers by Hemmer and Stell. 61 They introduced the so-called softcore (Stell-Hemmer) potential, which has two characteristic distances: a strict hardcore at a short distance and a softcore at a larger distance. The existence of the two characteristic distances of interparticle interactions allows the system to collapse from the largest to the smallest distance on applying pressure. This model was also used to study the liquidstate anomalies. 139 It was developed to the so-called Jagra model, which mimics the essential feature of liquids like water by monodisperse spherical particles. 62,[140][141][142] The core-softened potential family provided physical insight into the phase behavior, including more than one critical point. [143][144][145][146] For a model fluid forming orientationdependent intermolecular bonds, Roberts et al. also showed by Monte Carlo simulations the presence of both vapor-liquid and closed-loop liquid-liquid equilibrium in a pure substance. 147 This kind of model systems is computationally efficient and useful to understand the basic features of LLT. It has physically demonstrated that the existence of two-liquid states, i.e., LLT, can indeed occur for a single-component system. These models provide physical insight into the origin of LLT and, thus, have conceptual significance. However, people are often interested in whether LLT can happen in a real substance and where its critical point is. To answer such questions, we need more realistic models for each substance. We discuss simulations of realistic model materials later.
Patchy-particle systems developed rather recently are another interesting example. Smallenburg, Filon, and Sciortino 148 recently succeeded in controlling the relationship between the liquid-liquid transition and the liquid-solid transition. They successfully brought LLT in a thermodynamically stable state concerning crystallization by tuning the softness of the interparticle interaction and the flexibility of the bonds of patchy particles. Later, the same idea was applied to a liquid of limited valence particles composed of four singlestranded DNA grafted on a central core in a tetrahedral arrangement 149 as well as the ST2 water model. 150 It was also shown that colloids with four single-strands of DNA with abundant bonding ability can exhibit multiple liquid-liquid critical points with the Ising universality. 151 These studies provide critical information on how the precise nature of the potential affects the liquid-solid transition and the liquid-liquid transition (see also Ref. 152).
These model systems, where we can control the characteristics of the interaction potential arbitrarily, are quite useful to gain a deep insight into how the interaction potential affects the relationships between gas-liquid, liquid-liquid, and liquid-solid transitions.

IV. LLT AND AAT IN ATOMIC SYSTEMS
Now, we turn our attention to LLT in real materials. First, we focus on atomic systems and then discuss molecular systems. The application of pressure can induce changes in the free energy of materials that exceed those of the strongest chemical bonds present at ambient pressure (>10 eV). 153 Thus, pressure can completely redistribute electronic densities and change the nature of the chemical bonds. In this way, pressure leads to drastic changes in materials, e.g., converting insulators into metals and soft chemical bonds into stiff bonds. Such a pressure effect plays a critical role in LLT and AAT, particularly for atomic systems.

A. Pure atomic systems
The physically most simple systems are single-component atomic systems. However, the study of LLT is challenging, both experimentally and numerically. The experimental difficulty comes from the fact that atoms are usually interacting with high energy, e.g., with covalent or metallic bondings, and thus, the second-critical points may be located at high temperatures and pressures, 6,11 as discussed above. Thus, accessing LLT experimentally in these liquids is generally challenging; however, LLT in these systems may take place in an equilibrium liquid phase above the melting point, which is quite attractive.
Numerically, on the other hand, it is crucial to deal with electronic degrees of freedom for an accurate description of the interaction between atoms. The most accurate way to simulate atomic systems is the first-principle ab initio calculation. For example, this type of simulation provides detailed information on the changes in the structure-property relationship, including the change in the electronic state, e.g., the liquid metal-amorphous semiconductor transition. 154 However, they are computationally costly and, thus, suffer from the finite-size effects and limitation of the timescale. So, classical molecular dynamics (MD) simulations have also been widely used to study LLT.

S
Very recently, the existence of LLT has been reported for sulfur. Furthermore, the location of its critical point has been identified in an equilibrium liquid state. 138 It is the first experimental identification of the second critical point. Henry et al. 138 observed the discontinuous jump in density across the LLT line, which has a critical endpoint around Tc = 1100 K-1200 K and Pc = 2.1 GPa-2.2 GPa. The phase diagram of sulfur is schematically shown in Fig. 7.
The positive slope of the LLT line in the T-P phase diagram indicates that Δv/Δσ > 0. This situation is similar to the case of TPP, where the locally favored structures have a smaller specific volume than the normal liquid structures. The identification of the locally favored structures is highly desirable. It has also been shown that LLT is distinct from the long-known λ-transition, which has a feature of second-order transition. The locations of the two transitions in the T-P phase diagrams and the slopes of the transition lines are also different: The λ-transition line has a negative slope, 155 whereas the LLT line has a positive slope. Since the critical point is located in an equilibrium liquid state and its location is identified, sulfur may be an ideal system for studying static and dynamic critical phenomena associated with LLT. It is also quite interesting to study the transition kinetics. For some liquids similar to S, such as Se, we must carefully identify the nature of phase transitions in a liquid state.

Si and Ge
One of the most well-studied atomic systems is silicon. Since the early studies on the unusual relations between liquid and amorphous Si, 156-158 the phase behaviors of Si have attracted much attention. McMillan and his co-workers 159 reported the pressure-induced amorphous-amorphous transition similar to that of water. 18,20 Later, they showed that the transition takes place between semiconducting and metallic forms of solid amorphous silicon and discussed its connection to LLT, based on molecular dynamics (MD) simulations with the Stillinger-Weber potential. 160 Hedler et al. showed that irradiation with high-energy heavy ions induces the plastic deformation of amorphous silicon reminiscent of glass transition and regarded this phenomenon as evidence for the existence of the low-density liquid. 161 Concerning LLT of Si, recently, Beye et al.
experimentally studied how the electronic structure of silicon changes after strong laser excitation and found the two-step transition from the semiconducting through the semi-metallic to the metallic state in the valence electrons. 162 They concluded that silicon melts via the LDL state in a delayed first-order phase transition into the HDL state. Okada et al. 163 applied x-ray Compton scattering combined with Car-Parrinello molecular dynamics simulations to clarify the bonding state of liquid Si. They found evidence for the coexistence of two distinct bonding species, metallic and covalent, which supports the possibility of LLT.
The possibility of LLT in Si has been studied by many researchers, using both first-principle calculations (e.g., Refs. 164 and 165) and classical molecular dynamics simulations. Sastry and his co-workers used the Stillinger-Weber potential to access LLT and the liquid state anomalies of silicon (see, e.g., Refs. 166-168). Molinero et al. suggested the possible existence of LLT in the modified Stillinger-Weber systems. 169 In the course of these simulation studies, the subtlety of the simulation studies was pointed out. Jakse and Pasturel pointed out that the structural and electronic properties of both low-density liquid (LDL) and high-density liquid (HDL) phases are quite different between quantum and classical simulations, which originates from the inability of empirical pseudopotentials to describe changes in chemical bonds induced by density and temperature variations. Beaucage and Mousseau also pointed out that the phase behavior is extremely sensitive to the interaction potential: 152 The Stillinger-Weber potential, 170 which is popularly used to study silicon by classical simulations, consists of the isotropic two-body term and the three-body one favoring the tetrahedral symmetry. They showed that the increase in the magnitude of the three-body term by 5% makes LLT inaccessible. This work points out that the details of the potential could affect strongly the nature and even the existence of the liquid-liquid transition. Furthermore, very recently, Debenedetti and his co-workers showed by the careful analysis of the free-energy landscape that there is no evidence of LLT for Stillinger-Weber silicon as well as the modified Stillinger-Weber potentials with various strengths of the three-body terms (i.e., tetrahedrality). 171 Similarly, we observed no indication of LLT in these systems but, instead, the homogeneous crystal nucleation, 172 suggesting that we need special care to distinguish LLT and crystallization. 173 These results suggest that water-like anomalies may not require the second critical point and can be explained by the simple two-state feature. 26,66,83,84,92,99,174 We note that a very recent density functional theory study showed three sequential LLTs and a possible ionization-driven transition with an increase in density. 175 The existence of polyamorphic transition and LLT has also been suggested for germanium. 157,[176][177][178] The clear experimental evidence of the presence of a metallic amorphous state was reported by Bhat and co-workers. 179

P
LLT of phosphorus has also attracted considerable attention since the direct experimental observation of two fluid phases by Katayama and his co-workers. 180,181 They directly observed the coexistence of the two phases during the transition above the melting point by x-ray scattering measurements. Unlike the above cases of Si and Ge, where the transition happens only in a metastable supercooled liquid state below the melting point, it occurs in the equilibrium fluid state. Later, they reported macroscopic observation supporting the transient two-phase coexistence. 181 Although this transition is a clear first-order transition, it is accompanied by a substantial density change, which is too large for LLT. Later, Monaco et al. 182 pointed out that the location of the phase transition lies above the gas-liquid critical point of the molecular fluid phase of P, i.e., in the supercritical-fluid region, and concluded that the transition is between a dense molecular fluid and a polymeric liquid (see Fig. 8). Since then, this transition has been considered to be a fluid-liquid phase transition. The first-principle and reactive-force-field calculations also confirmed the existence of the transition, [183][184][185][186] and the transition is between the fluid with stable tetrahedral P 4 molecules and the polymeric liquid. The similarity of this transition to the λ-transition in S was also pointed out. 185 It was also shown 184 that a small overlap of the molecular orbitals of the tetrahedral P4 molecules leads to the narrow bandwidths for the low-density fluid, resulting in the nonmetallic character, whereas the dense packing of P ions in the highdensity liquid leads to a large degree of overlap of s and p wave functions of the P atom, leading to the metallic character. The change in the character of the transition along the first-order-transition line was also studied recently. 187 It was also reported that the local bonding geometry changes from tetrahedral to simple-cubic one upon the transition. 185 It is interesting whether the transition is λ-like transition in S or LLT-like. This question is related to the nature of the order parameter of the transition, i.e., n → 0 or n = 1 in the n-vector model (see Sec. II F).

C
The possible existence of LLT in carbon has also attracted much attention (see also a very recent review on the current situation 188 ). Bundy found the maximum of the melting point of graphite. 189,190 It was regarded as an indication of LLT, based on the two-species model by Rapoport. 65 Then, van Thiel and Ree 191 predicted the LLT based on the thermodynamic equation of state data for graphite and diamond. Simulation studies reproduced the melting-point maximum 192 and showed that the liquid can have two-, three-, and fourfold coordinated configurations and the application of pressure increases the coordination number. 192,193 Togaya 194 experimentally suggested the possible existence of LLT based on the discontinuous changes in the slopes of the melting line and the electrical resistivity in the liquid state. Classical MD calculations supported the existence of LLT. 100 On the contrary, Grumbach and Martin 195 employed firstprinciples simulations to investigate the phase diagram of carbon, but the presence of LLT was neither denied nor proved. Wu et al. 196 showed by first-principle molecular dynamics simulations that there is no evidence of LLT in carbon. Wang et al. 197 used first-principles electronic structure theory to calculate the melting curve accurately. They confirmed the presence of a reentrant point in the diamond melting line but found no evidence for LLT near the reentrant point. Ghiringhelli et al. 198 also showed by density-functional based molecular-dynamics simulations that with increasing pressure, liquid carbon undergoes a gradual, continuous transformation from a liquid with local threefold coordination to a "diamondlike" liquid, without LLT.
Recent, classical simulations using the carbon-carbon potential, which was matched with the ab initio MD results for the liquid structure, indicated that a quasi-2D carbon exhibits LLT from the twofold-coordinated to threefold-coordinated liquids with decreasing temperature. 199 So, the situation is still controversial, and further careful studies are necessary. 188

Other pure atomic systems
Many other candidates were suggested to have LLT. 6,200 Many atomic systems that have V-shaped phase diagrams, including group IV elements, which we call water-type systems, 81 are such candidates. They include Si, Ge, Ga, Bi, and Sb. For Ga, the existence of LLT was suggested both experimentally 201,202 and numerically. 203,204 It was also shown that the liquid is composed of two species similar to solid Ga I and Ga II structures. 205 However, there is also a report suggesting the absence of LLT. 206 So, the situation is still controversial. LLT was also suggested for Ce. 207 The situation of Bi is somewhat similar. The existence of LLT in Bi was suggested. 208,209 The nature of a new liquid has been discussed, including its liquidcrystalline nature. 210 Recently, unusual ferromagnetism in pure Bi was reported, and its emergence was ascribed to a structural memory effect in the molten state. 211 Concerning the LLT, it was recently pointed out that the precise determination of the liquid structure needs special care. 212,213 Emuna et al. 214 suggested that the transition may be either a transition between a distorted short-range order and a more close-packed order or the retention of a "solid-state" memory in the liquid. They concluded that the conclusive determination would require additional studies such as pressure-dependent liquid structure determination.
In relation to these works, Li et al. observed humps in the temperature dependence of the resistivity in liquid Bi, Sb, and their alloys upon the first heating, but not on cooling, 215 which is also suggestive of the memory effect in the liquid after the transformation from the solid. This effect is not restricted to Bi and Sb but may be more generic, 216,217 suggesting that special care is necessary to There are also many studies on group VI elements, nontransition elements, such as Te and Se. 6,200 For Te, some reports suggested the discontinuous change in the conductivity, density, and structure in the liquid state. 6,[218][219][220] For Se, the existence of LLT was suggested experimentally, 6,200 but ab initio simulations indicated its absence, although it might be due to the inaccuracy of the interatomic interaction. 221 Further studies may be necessary to prove the existence of LLT and clarify its nature.
For Ni, a melting-curve maximum has been observed at about 50 GPa and 1920 K. 222 Since the kink is sharp, the presence of LLT in the melt was suggested. There are also many other candidates for LLT. For interesting readers, we recommend reading Refs. 6 and 200.

B. Oxides
The existence of LLT has also been suggested for oxides, such as SiO 2 , GeO 2 , and B 2 O 3 . 200 For SiO 2 , polyamorphism was suggested by experimental studies based on pressure-induced irreversible densification of the amorphous state, [223][224][225][226][227] which was also supported by numerical simulations. [228][229][230] A recent experimental study followed the transition process from high-density to low-density amorphous states, observed the inhomogeneous transitional state, and interpreted the phenomenon as the amorphous-amorphous transition between two distinct mega basins in the energy landscape. 231 The presence of polyamorphism looks suggestive of the presence of LLT, as in the case of water. Thus, many simulation works were performed to seek the possibility of LLT, and its presence in a supercooled metastable state at high pressure was suggested. [232][233][234][235][236][237] Fluctuations associated with LLT was also investigated. 238 Nevertheless, there has been no direct experimental confirmation of LLT yet. Here, we note that the local tetrahedral order in silica is more disordered than that in water, 239 which may cause some differences in the character of LLT between silica and water. We also discuss the relationship between polyamorphism and LLT later from a more general context in Sec. VII.
Furthermore, the dynamics of silica changes from super-Arrhenius (fragile) behavior to the Arrhenius (strong-liquid) behavior upon cooling from higher temperatures. This fragile-to-strong transition behavior in simulated silica models was linked to the presence of LLT, as in water in the framework of the glass-transition phenomenology. 228,229 We have recently proposed a different scenario, in which the change in the dynamics originates from the two-state feature (i.e., the Arrhenius-Arrhenius transition). 80,83,84 In this scenario, the fragile-to-strong transition is nothing to do with the glass-transition phenomena. This point needs further careful studies.
The situation is very similar for GeO 2 and B 2 O 3 . Polyamorphism was suggested for GeO 2 experimentally 224,240,241 and numerically. 242 The similarity and difference between SiO 2 and GeO 2 was discussed in detail in Refs. 243 and 244. We also note that LLT was suggested for liquid germanate (Li 2 O-4GeO 2 ), based on the pressure-induced change in the coordination number revealed by x-ray absorption measurements. 245 Polyamorphism was also suggested for B 2 O 3 experimentally 246-252 and numerically. 230,253 The deep link between polyamorphism and polymorphism in B 2 O 3 has also recently been suggested. 254 We note that other oxides such as TiO 2 and mixtures of oxides may exhibit similar behaviors.
For both SiO 2 and GeO 2 , the locally favored structure at low pressure and temperature has tetrahedral order, and its fraction is a natural candidate of the order parameter characterizing the liquid state, as in water. For B 2 O 3 , the relevant order parameter may be the fraction of boroxol ring structures. In these liquids, whether there is only one type of locally favored structure besides random normalliquid structure or another local structure becomes more stable at higher pressure is not clear, which is a fascinating subject for future research.

C. Other multi-component atomic systems
One of the most well-studied multi-component atomic systems, for which LLT was suggested, is the yttrium oxide-aluminum oxide mixture. Aasland and McMillan 255 directly observed the coexistence of the two liquids with microscopy and confirmed that the compositions of the two phases are equivalent within the range of errors, based on which they treated the system as a quasisingle-component system. So, they claimed that the phenomenon should be LLT. The follow-up studies on various physical quantities, including the structural and vibrational properties, supported this claim. [256][257][258][259][260][261][262] However, there have been criticisms on this claim 263,264 and debates on its nature. 265,266 One is the common problem of multicomponent systems: When we observe the coexistence of the two liquid or amorphous phases, the critical point is whether the composition is the same or different between them. If the composition is the same, it might be LLT, whereas if not, it is phase separation. Furthermore, the order parameter of LLT should be non-conserved, whereas that of phase separation should be conserved. In principle, if the experiment is done at constant pressure, the coexistence of the two phases should be observed only transiently, and the fractions of the two phases should change with time. Finally, the newly formed, more stable phase should occupy the whole sample, if there is no dynamic arrest due to glass transition. The other problem is associated with the fact that LLT is observed in a supercooled state. Such situations are common to many systems showing LLT-like behaviors, besides some atomic systems whose LLT is observed in the equilibrium liquid state above the melting point. We argue that the best way to resolve the controversy is to study the kinetics of transition.
Fuchizaki and his co-workers systematically studied LLT in SnI 4 and GeI 4 . 108,[267][268][269][270] LLTs in these systems take place above the melting points and are thus convincing. Thus, they are worth further detailed studies, including the kinetics.
The other systems, whose polyamorphism and LLT have recently attracted attention, are chalcogenide [271][272][273][274] and metallic systems. [275][276][277][278][279]274,[280][281][282] and metallic liquids 277,[283][284][285][286] also exhibit the fragile-to-strong-transition-like behaviors, as we discussed about silica in the above. However, such a fragile-to-strong crossover can be caused by the two-state feature of these liquids (i.e., the Arrhenius-Arrhenius transition) instead of the transition of glassy dynamics. 80,84,287 As we see in the above, there are a plenty of multi-component atomic systems, but none are free from criticisms. So, we need further careful studies on each system to establish the occurrence of LLT.

V. MOLECULAR LIQUID-ATOMIC LIQUID TRANSITION IN HYDROGEN
The properties of hydrogen at high pressure are of crucial importance in astrophysics and high-pressure physics. 290 For example, dense fluid metallic hydrogen is known to occupy the interiors of Jupiter and Saturn. Thus, much attention has been paid to the phase behavior of hydrogen at high pressure. In warm dense liquids, there have been many reports suggesting the liquid-liquid transitions, based on experiments 291-296 and quantum mechanical numerical simulations. 289,[297][298][299][300][301][302][303][304][305][306][307] Most of these simulation studies agreed on the existence of a first-order transition between the molecular and atomic fluids. However, there were debates on its position in the phase diagram since it strongly depends on the particular approximation employed for the exchange-correlation functional. This difficulty comes from many-body quantum interactions. Hydrogen has many polymorphs: free rotating molecules (phase I), broken symmetry due to quadrupole interactions (phase II), packing of weakly bonded molecules (phase III), and "mixed" states (phases IV and V). Even at higher pressure, it is considered to further transform into atomic metal. Thus, LLT is thought to be related to the (molecular) solid-phase I-III-to-(atomic) metal transition. We show a schematic phase diagram of hydrogen in Fig. 9. We note that the liquid-liquid critical point is located at a much higher temperature than the gas-liquid critical point. It is because that the atomic binding energy controls the former, whereas the much weaker van-der-Waals intermolecular force controls the latter. It is an example of how pressure drastically changes the nature of the interaction. 153 The emerging picture is summarized as follows (see recent papers 289,294,296 ). A first-order thermodynamic transition with Polymorphism exists for hydrogen crystals: solid states based around free-rotating molecules (phase I), broken symmetry due to quadrupole interactions (phase II), packing of weakly bonded molecules (phase III), and proposed "mixed" states (phases IV and V). The latter two phases have alternating layers of rotating molecules similar to phase I and weak molecules akin to phase III. The first-order liquid-liquid transition is thought to exist somewhere in a white region of the phase diagram, with the critical endpoint (red filled circle). There is a consensus that LLT, or liquid hydrogen dissociation, is located below the plasma phase transition. We note that the gas-liquid critical point of para-hydrogen (black filled circle) is located around 33 K and 1.2 MPa. 288 This figure is schematically drawn, based on Fig. 6 of Ref. 289. density discontinuity and metallization exists below the critical point, whereas a continuous crossover accompanied by metallization occurs in the supercritical-fluid region. The dissociation and metallization are considered to be caused by orbital overlapping and the subsequent electron delocalization. At very high temperatures, it is expected that another transition with a different origin, i.e., plasma phase transition characterized by the ionization process, occurs. Because of the severe limitation due to heavy quantum calculations, the nature of LLT on a macroscopic scale is mostly unknown. In this LLT, the H 2 dimer fraction may be the candidate of the nonconserved order parameter (ψ in our two-order-parameter model). The LLT in hydrogen is unique since H 2 and H are distinct chemical species, and the chemical reaction between them is very fast. Considering that the transformation of locally favored and normalliquid structures is a kind of chemical reaction, 73,74 the essential feature of LLT in hydrogen should be similar to that of classical systems.

VI. LLT IN MOLECULAR SYSTEMS
Next, we discuss LLT in molecular liquids. There are several candidates of LLT in single-component molecular liquids, but not so many. They include water, triphenyl phosphite (TPP), 101,102 n-butanol, 103 D-mannitol, 308-310 trans-1,2-dichloroethylene, [311][312][313][314][315][316] and molten sodium acetate trihydrate. 317 Numerical simulations also suggested LLT in methanol. 318,319 For the former four and the latter two systems, LLTs are claimed to exist in a supercooled state below the melting point and an equilibrium liquid state above the melting point, respectively.
Hereafter, we mainly discuss the LLTs of TPP, water, and aqueous solutions, which are the most well-studied molecular systems that are expected to have LLTs.
A. LLT in molecular liquids: The case of TPP

Brief research history of the unconventional phase transition in TPP
TPP is one of the most experimentally well-studied molecular systems that may have LLTs. Kivelson and co-workers reported a quite surprising phenomenon in the study of a supercooled state of TPP. 104,320,321 When TPP is cooled rapidly enough, it becomes a glassy state, as ordinary glass-forming liquids. This supercooled liquid shows drastic slowing down of the dynamics, as a typical fragile glass former. On the other hand, if TPP is rapidly quenched to a specific temperature between 213 K and 223 K and then annealed at that temperature, it slowly transforms into the so-called glacial phase while releasing heat. The liquid is initially transparent, becomes turbid during the transition, and finally becomes transparent again. Surprisingly, thus, the glacial phase looks like an optically transparent homogeneous amorphous phase without any sharp x-ray scattering peaks, but it is distinct from the glass state formed by rapid quenching.
This finding has attracted considerable attention and stimulated intensive experimental research on the nature of the glacial phase. Many different interpretations have been proposed: The glacial phase was thought to be a new amorphous phase 321-325 or a highly correlated liquid. 326 However, the glacial phase is often  [331][332][333][334][335][336][337][338] or nano-clustering. 339 Now, we have consensus that all the controversies come from the fact that this transition is accompanied by the formation of nanocrystals, and thus, the final glacial state often contains nano-crystals. The presence of nano-crystals was also detected by heat capacity and thermal conductivity measurements for both TPP 340 and n-butanol. 341 Thus, the only remaining interpretations of the glacial phase are currently either glass II formed by LLT (sometimes with nano-crystals) or glass I with nano-crystals.
Here, we note that the nano-crystal formation may be typical to LLTs existing below the melting point, which inevitably makes the presence of LLT controversial. Thus, at this moment, it is fair to say that there is no molecular system with LLT free from any doubt. For example, similar situations are seen in other candidates of LLTs, such as n-butanol (LLT 103 vs nano-crystal formation [342][343][344][345] or mesophase formation 330 ), D-mannitol (polyamorphism 308-310 vs nano-crystal formation 346 ), and confined water 29 (LLT 347 vs other surface specific phenomena). So, the critical question is how unambiguously we can prove the presence of LLT experimentally.
For TPP, some additional pieces of experimental evidence supportive of the LLT scenario rather than the nano-crystal scenario has recently been reported. 127,[348][349][350][351][352][353][354] Below, we discuss in detail why we believe that the transition phenomenon in a supercooled state of TPP is LLT.

Transition kinetics of LLT
We followed the kinetic process of liquid-liquid transition with phase-contrast microscopy for two pure organic liquids, TPP 101,102 and n-butanol. 103 Here, we show the typical kinetic processes of LLT observed in TPP with phase-contrast microscopy in Fig. 10: (a) NG-type and (b) SD-type LLT. The two types of pattern evolution, NG-and SD-type, strongly support that the transition is LLT, i.e., the ordering of the non-conserved scalar order parameter predicted by the two-order-parameter model (see Sec. II). We have also confirmed this conclusion by numerical simulations. 86,87 The heat evolution was also measured during the LLT by many researchers. According to our model, the heat evolution is proportional to the development of the bond order parameter ψ since the heat is released upon the formation of locally favored structures. 101,356 However, it can also be interpreted as the heat release upon the nano-crystal formation. We indeed confirmed that nano-crystallites form during the transformation when the annealing temperature Ta is higher than 214 K. Thus, above 214 K, the release of heat comes from both the formation of locally favored structures and nanocrystals. However, below this temperature, LLT occurs without crystallization, which is difficult to explain by the nano-crystallization scenario.
For n-butanol, we observed the pattern evolution (NG-and SDtype) similar to that of TPP; however, crystallization always occurs, which makes the interpretation more complicated than in TPP. For example, Ramos and his co-workers claimed that the phenomena observed in n-butanol are aborted crystallization and not LLT, 342,344 but we argue that it is LLT based on the pattern evolution during the transformation process, which is characteristic of LLT.

Glass transitions and fragilities of liquids I and II
We confirmed that in both TPP and n-butanol, liquid I transforms into a glassy state of liquid II, i.e., glass II. The situation can be understood in the schematic state diagram of LLT for TPP on the T-ψ plane (see Fig. 11), which includes the ψ-dependent  The results for the slow heating rate. The gray curve is a heating curve of liquid I without annealing, and the blue curve is a heating curve of liquid II obtained after annealing for 600 min at 216 K. After the complete transformation from liquid I to the glacial phase (glass 2) by constant-temperature annealing, the glass I-to-liquid I transition signal disappears completely. Instead, an endothermic peak appears at a higher temperature, followed by a significant exothermic peak due to crystallization. This exothermic peak makes it challenging to clarify the origin of the endothermic process. (b) The results of flash DSC measurements. The black curve is obtained for a sample without annealing (liquid I), and the blue curve is obtained for a sample after annealing (the glacial phase or glass II). The yellow dashed curve is taken after re-cooled from a temperature Trc in the endothermic peak (see the inset for the temperature protocol). The glasstransition signal of liquid I is observed in the yellow dashed curve around 220 K, indicating that the glacial phase (glass II) has already returned liquid I during the endothermic process before reaching Trc. The gray curve is for a sample fully crystallized. This figure is reproduced with permission from Fig. 1  temperature than liquid II. 101 The width of the glass transition range of glass II (∼23 K) is much broader than that of glass I (∼4 K), suggesting that liquid I is more fragile than liquid II. 101,353,357 This result of the thermodynamic measurements is consistent with that of dynamic ones 322,324,326 that the temperature dependence of the structural relaxation time τα is super-Arrhenius (typical for fragile liquids) for liquid I while more Arrhenius-like (typical for strong liquids) for liquid II.
Next, we show the temporal change in the onset of the glass transition temperature in Fig. 12. 353 There, we can see that for SD-type LLT below T I→II SD ∼ 214 K, the onset temperature of the glass transitions gradually and continuously shifts toward a high The Journal of Chemical Physics PERSPECTIVE scitation.org/journal/jcp temperature with the annealing time t. In contrast, for NG-type LLT above T I→II SD , those of glasses I and II stay almost constant with the annealing time t until the completion of the transformation, whose timing is indicated by the arrows in Fig. 12. These behaviors are entirely consistent with the phenomenology of NG-and SD-type phase transformation (see Fig. 6). We stress that the above glasstransition behaviors are also in agreement with the characteristics of pattern evolution during LLT revealed by optical microscopy (see Fig. 10). The differential scanning calorimetry (DSC) results indicate that SD-type LLT of the continuous nature, indeed, takes place below T I→II SD (see Fig. 11). Thus, this finding strongly supports the physical picture of the two-order-parameter model of LLT. 27

Reversibility of LLT
We can rather easily follow the forward process of LLT, from liquid I to glass II. However, the opposite was challenging since crystallization occurs before the reverse LLT from liquid II to I occurs.
We have recently overcome this difficulty by using flash DSC. 353 The result is shown in Fig. 13(b). We can see a distinct endothermic peak coming from the reverse LLT from liquid II to I, around 250 K. The yellow dashed curve, which is the DSC curve of the sample rapidly cooled from Trc during the reverse LLT, exhibits the same glass-transition signal as liquid I, indicating that liquid II already transforms back almost to liquid I at Trc.
We cannot explain the above behavior if we interpret the glacial phase as a mixture of liquid I and nano-crystals. The reason is as follows: If liquid I remains in the glacial phase, we should observe the glass-transition signal of liquid II in the blue curve. However, we cannot detect any such indication. In the nano-crystal scenario, this absence of the glass transition of liquid I indicates that the glacial phase is made of only nano-crystals, but this contradicts with the small heat of fusion of the glacial phase than crystals [see Fig. 13(b)] and the presence of only weak sharp signals of x-ray scattering. However, one may still argue that it is due to the disordered character of the nano-crystals, and the endothermic peak around 250 K is the melting of such defective nano-crystals. In this interpretation, however, the gradual change in the glass-transition temperature as a function of Trc 353 cannot be explained since the only glass state is liquid I in this scenario. Furthermore, x-ray scattering measurements indicate no formation of a new type of crystal, and no disappearance and position change in the Bragg peaks around 250 K. All these facts support the LLT scenario. The details on the difficulties of the nano-crystal scenario were discussed in the supplementary material of Ref. 353.

The formation of locally favored structures and their character
Theoretically, it is reasonable to assume that the order parameter governing LLT is the fraction of locally favored structures; however, we need experimental evidence. For this purpose, we studied the structural evolution during the LLT of TPP by time-resolved simultaneous small-and wide-angle x-ray scattering (WAXS) measurements. 352 As shown in Fig. 14, we found that locally favored clusters, whose characteristic size is a few nanometers, form, and their number density monotonically increases with time during LLT. This result supports the above assignment of the order parameter and its non-conserved nature. We also showed 352 that there is a distinct difference in the structure between the locally favored structures and the crystal, denying that the glacial phase is a mixture of liquid I and nano-crystals and supporting the LLT scenario.
On the nature of locally favored structures in TPP, there were also a few proposals based on experiments 358,359 and the density functional theory calculations. 360,361 These studies pointed out the importance of hydrogen bonding (C-H⋯O hydrogen bonding) in the formation of the locally favored structures. Wynne and his coworkers 354 very recently reported the presence of two crystalline polymorphs, the equilibrium crystal 2 and a metastable crystal 1. They proposed that liquid I has a local structure similar to a metastable polymorph crystal 1 (parallel-stacked), whereas liquid II has a more stable local structure of T-shape-stacking. 354 The presence of polymorphs was also suggested by crystallization of TPP from a polar ionic liquid solution 362 and spontaneous crystallization from melt. 363 The precise identification of locally favored structures is critical in our understanding and establishment of LLT in TPP and, thus, highly desirable. On the identification of the order parameter, it is worth noting that we need to consider what is described in Sec. II B 2.

Relationship between the molecular mobility and the order parameter during LLT
LLT of TPP is the transformation from liquid I to glass II. Thus, it is accompanied by drastic slowing down of dynamics. The temperature dependence of the complex dielectric constant 326,329 and its temporal change during the transformation from liquid I (normal liquid) to glass II (the glacial phase) 336 were studied. Hédoux et al. 336 found that the structural relaxation time enormously slows down, and its distribution becomes broader during the transition.
We also investigated this process by making time-resolved simultaneous measurements of dielectric spectroscopy and phasecontrast microscopy/Raman spectroscopy. 127 We revealed that the temporal change in dielectric relaxation crucially depends on whether LLT is NG-type or SD-type. We also suggested that the activation energy of the molecular rotation is controlled by the local fraction of locally favored structures, as in the case of water. 67,80,83,84 Thus, the dynamics is also controlled by the order parameter ψ.
One remaining question is why the distribution of the structural relaxation is so broad for liquid II. 127,336 Judging from the glasstransition width 101,353,357 and the degree of the super-Arrhenius character of the dynamical slowing down, 127,322,336 liquid II is much stronger than liquid I. For ordinary glass formers, it is well established that the distribution of the structural relaxation time is narrower for a stronger liquid. The unusually broad distribution for strong liquid II is thus unconventional. 127,322,336 We have ascribed it to the incompleteness of LLT due to the dynamical arrest by glass transition and the resulting broad distribution of the order parameter ψ. 127 This point needs further careful study.

LLT and wetting phenomena
If there are two liquids, it is natural to expect that they interact with a solid surface differently, indicating that LLT may affect the wetting properties of a liquid to the solid substrate. We, indeed, showed that the wettability of liquids I and II to a solid substrate is different. [364][365][366] We found a transition from partial to complete wetting at the wetting-transition temperature Tw for NG-type LLT when approaching the spinodal temperature T SD of LLT (see Fig. 15). This critical-point-wetting-like behavior 367 suggests that the interfacial tension between liquids I and II decreases in a manner consistent with the mean-field criticality, 102 although the extrapolation is too extensive to draw a definite conclusion. Wetting effects of LLT also have a unique feature. We found that for NG-type LLT, wetting effects to a substrate significantly affect the transition kinetics, whereas, for SD-type LLT, there is no influence. 365,366 This behavior is markedly different from that of a system of a conserved order parameter, e.g., phase separation, 368 where SD-type ordering is also affected by the surface field significantly. This unusual feature of the wetting effects on LLT is a consequence of the non-conserved nature of the order parameter governing LLT.
We also revealed by investigating the wetting to various solid substrates 365 that the wetting behavior is not controlled by dispersion forces but by weak hydrogen bonding to a solid substrate. This microscopic nature of the wetting mechanism is fundamentally different from the ordinary macroscopic mechanism responsible for wetting effects on phase separation, 368 where dispersion forces play a dominant role. This fact suggests the critical role of hydrogen bonding in forming the locally favored structure of TPP. We stress that the nano-crystal scenario cannot explain the wetting behaviors described above.
We also found that the rubbing treatment of a substrate, i.e., the introduction of surface roughness, strongly influences the kinetics of LLT of TPP. 366 When TPP is confined between rubbed surfaces, LLT becomes barrierless even above T SD , i.e., in a metastable state, and thus, the kinetics is significantly accelerated compared to LLT of Chemical Physics without rubbed surfaces (see Fig. 16). We revealed that this unusual behavior is induced by a wedge-filling transition and the resulting drastic reduction in the nucleation barrier. However, SD-type LLT is not affected by rubbing since there is no activation barrier for SDtype LLT even in the bulk (see Fig. 16). Similar to the above results on the wetting effects on LLT, this finding proves the presence of NG-type and SD-type LLT, supporting the conclusion that LLT is truly a first-order transition with criticality and not nano-crystal formation. We note that this conclusion was also supported by time-resolved light scattering studies of LLT. 350,351 From the application's viewpoint, our results show that we can use solid substrates or particles as a catalyst to promote LLT in a metastable region above T SD . It may allow us not only to make spatial patterning of liquid I and liquid II using chemically or topologically patterned surfaces but also to control the kinetics of LLT. 366 The wetting mechanism for LLT, especially whether wetting is sensitive to specific interactions or controlled by dispersion forces, in atomic liquids, oxides, and chalcogenides, is also an interesting topic of future study.

Coupling between LLT and crystal nucleation
LLT of TPP occurs in a supercooled state below the melting temperature of the crystal, Tm. Thus, there is a possibility that crystallization is affected by LLT. It is especially the case since, as we described above, the crystal is more wettable to liquid II than liquid I. Some time ago, ten Wolde and Frenkel 369 pointed out that protein crystallization may be enhanced by critical-like fluctuations associated with phase separation. Recognizing the universality of such a scenario to any transition with criticality, we recently found a nontrivial coupling between crystal nucleation and LLT in TPP. 370 We showed that we can increase the crystal nucleation frequency drastically by short-time pre-annealing near but above T SD (Fig. 17). We separated the effects of the thermodynamic and kinetic factors controlling crystal nucleation and found that the reduction in the crystal-liquid interfacial energy due to the presence of critical-like order-parameter fluctuations is responsible for the phenomenon. This method may allow us to not only control the crystal grain size by increasing the crystal nucleation frequency but also seek LLT hidden behind crystallization by probing the effect of short-time annealing on crystal nucleation frequency.

LLT in a mixture of TPP with other liquids
How LLT is affected by mixing with another liquid is of fundamental and technological interest. It is especially the case since, in many systems, LLT can be hidden by crystallization as in water. The mixing of another liquid might avoid crystallization and reveal LLT. This problem was discussed experimentally, 371 theoretically, 116 and numerically. 372 We also studied this problem experimentally and found that a mixture of TPP and toluene still exhibits LLT while mixed, whereas a mixture of TPP and diethyl ether shows LLT but is accompanied by phase separation. 373 This finding indicates that the miscibility of liquid I with another liquid can be different from that of liquid II.
We show LLTs of a mixture of TPP and diethyl ether without and with phase separation in Figs. 18(a)-18(d) and 18(e)-18(h), respectively. The difference in the former and latter is the fraction of diethyl either (2.98% and 4.45%, respectively). The early stage pattern evolution looks the same between the two cases. However, the difference emerges in the late stage: for the former, the system eventually becomes homogeneous as in LLT of pure TPP, whereas for the latter, droplets reappear by phase separation after the homogenization of LLT. We observed such behavior only when the volume fraction of diethyl ether ϕ d ≥ 4%, and not for ϕ d ≤ 3%, and the final volume fraction of droplets [ Fig. 18(h)] increases with an increase in ϕ d .
The occurrence of phase separation was also confirmed by the fact that T II g (ϕ d ) decreases with an increase in ϕ d until ϕ d = 4% but does not depend on ϕ d for ϕ d ≥ 4% [ Fig. 18(i)]. Note that Tg of the diethyl-ether-rich phase is located below the temperature range of measurements.
We observed the spatial distribution of the two components of the mixture directly using micro-Raman spectroscopy measurements. We followed the temporal evolution of the peak intensity at a fixed point of the sample of ϕ d = 8% at 209 K [see Fig. 18(j)]. The intensity is almost constant before t = 450 s but increases continuously after that. We also performed real-time 2D Raman microscopy mapping [the insets of Fig. 18(j)]. We can see that the system is homogeneous at t = 200 s, but the phase separates into TPP-rich and TPP-poor regions at t = 1800 s. This result firmly confirms the LLT-induced phase separation, which is also supported by the temporal change in Tg during this process [see the square symbols of Fig. 18(j)].
We also found that the addition of a small amount of toluene to TPP transforms solid-state glass II to liquid II with fluidity. In such LLT, less viscous liquid I transforms to more viscous liquid II, suggesting a possible control of the fluidity, i.e., the transport property of a liquid, by LLT.
Finally, we note that we can use LLT in a mixture to seek LLT of its component liquid, which is difficult to access experimentally, for example, due to crystallization or glass transition, as in water. We will show such an application to water later in Secs. VI B 3 and VI B 4.
To summarize, liquids I and II differ in the density, the refractive index, the glass-transition temperature, the fragility, the miscibility with other liquids, the wettability to a substrate, and the dynamics.
10. LLT or other phenomena: The nature of the glacial phase in TPP As discussed above, there have been continuous debates on whether the transition in TPP is LLT or nano-crystal formation. The presence of nano-crystals after the transformation of liquid I at least above T SD ≳ 213.5 K is not a matter of debate. Thus, the critical question is whether nano-crystals are embedded in a glassy state of untransformed "liquid I" or transformed "liquid II." Here, we pick up some debates within recent several years. Babkov et al. 374 analyzed the experimental and calculated infrared spectra. They concluded that the ordinary glass and the glacial phase are made of different types of conformers, and the glacial phase consists of crystalline nuclei and a highly transformed supercooled liquid II, whose conformation is somewhat similar to that of the crystal but with a significant difference. This result seems consistent with a previous study on the conformation of TPP molecules by combining nuclear magnetic resonance (NMR) and density functional theory. 375 This work showed that amorphous I has a broad distribution of conformations typical of the ordinary glass, whereas amorphous II consists of a mean preferential conformer (i) ϕ d -dependence of T II g . T II g decreases monotonically with an increase in ϕ d below 4%, whereas T II g keeps almost constant for ϕ d ≥ 4%. This indicates that phase separation occurs for ϕ d ≥ 4%, which is consistent with optical microscopy observation [(a)-(h)]. (j) Temporal change in Tg (squares) and the intensity of Raman peak at 3068 cm −1 (circles) for ϕ d = 8% and at T = 209 K. Note that this peak comes from TPP and does not from diethyl ether. The insets of (j) are the intensity maps for the peak at 3068 cm −1 measured at t = 200 s (left) and at t = 1800 s (right). The area size and the spatial resolution of the left inset are 36 × 36 μm 2 and 6 × 6 μm 2 , respectively. On the other hand, those of the right inset are 100 × 100 μm 2 and 3 × 3 μm 2 , respectively. More reddish (bluish) color means higher (lower) intensity, i.e., a higher (lower) fraction of TPP. This figure is reproduced with permission from Fig. 2  similar to that in the crystal but with the significant conformational distribution.
Tarnackca et al., 338 on the other hand, suggested, based on combining differential scanning calorimetry (DSC), broadband dielectric, and Raman spectroscopies of nano-confined TPP, that the formation of the glacial phase is a result of the crystallites formed within liquid I. Thus, the debates continue. This situation is quite similar to LLT of n-butanol, which was also ascribed to nano-crystal formation. [342][343][344][345] Now, we explain some reasons why we believe that the transition phenomenon in TPP is LLT. We already discussed this problem in detail in Refs. 26 and 355. Thus, we discuss this problem from new angles, focusing on whether the transition is due to LLT or nano-crystal formation.
First, we note that the indication of nano-crystals in TPP exists only for an annealing temperature above 213 K, and below this annealing temperature, there is no indication of the presence of nano-crystals in DSC, 101,353 light scattering, 350,351 and x-ray scattering 352 measurements. In the case of n-butanol, microcrystallites were observed even in the lowest temperature below T SD . Unlike in TPP, in n-butanol, we cannot access a state free from crystallization, probably because T SD is located only slightly above Tg.
When NG-type LLT occurs in TPP, droplets show the Maltese cross under polarizing microscopy observation, 101,330 which is The Journal of Chemical Physics PERSPECTIVE scitation.org/journal/jcp a clear indication of optical anisotropy. It should be due to either nano-crystals aligned along the growth direction 26,101,355 or liquid crystal. 330 The presence of sharp-diffraction peaks in x-ray scattering supports the presence of nano-crystals. Then, the question is how nano-crystals align specific directions relative to the growth direction of glass II droplets. The spherulite formation, or densely branched morphology, itself is quite general for crystallization: It can be induced by the crystal growth instability due to either static (e.g., impurities) or dynamic heterogeneities. 376 What is unusual is that the glass state is the majority of the final state, and the nano-crystal phase is the minority, despite the absence of impurities: the crystal fraction is 50% at 223 K and then monotonically decreases with decreasing Ta toward zero. The spherulite formation in TPP is also unusual since, for non-polymeric pure systems, the crystal fraction is generally very high. This unusual behavior can naturally be explained if we assume glass II droplets are nucleated, and the nano-crystal formation occurs exclusively at their growth front. Nano-crystals cannot be nucleated inside the droplets because of the glassy nature of glass II and the resulting dynamic arrest. As we mentioned, crystals are more wettable to liquid II than liquid I, and thus, it is reasonable to assume that crystal nucleation preferentially takes place only at the interface with particular orientational relation to the interface. The interface is the only region where the high molecular mobility and the low liquid-crystal interface tension can be both satisfied. 26,355 However, nucleated crystals cannot grow if their growth speed is slower than that of glass II droplets: liquid II at the droplet interface quickly transforms into glass II, leading to the formation and stabilization of nano-crystals. This peculiar growth behavior is possible if the crystal nucleation and growth rates are higher and slower than the growth rate of droplets of glass II, respectively. This scenario looks physically reasonable but needs to be confirmed experimentally.
There are other pieces of evidence supporting the LLT scenario.
1. Pattern evolutions characteristic of NG-type and SD-type ordering of a non-conserved order parameter are hard to explain by the nano-crystal scenario, as discussed above. In contrast, the theoretical model of LLT can naturally reproduce all the essential features of pattern evolution 86 . We found that the glass II droplets are preferentially formed on some substrates. 365 In principle, it is possible to explain the wetting behavior by the nano-crystal scenario since the liquid I droplets containing nano-crystals have a different refractive index from the liquid I matrix. However, we showed that the dispersion force cannot explain the wetting behavior of TPP but is specific to substrates having hydrogen-bonding capability. 365 The nano-crystal scenario cannot explain this specificity of the wetting behavior. 5. The change in the miscibility with another liquid (diethyl ether) mixed with TPP is also hard to explain by the nanocrystal scenario (see Sec. VI A 9). Molecular-level interactions determine the miscibility of two liquids. Thus, if there is only liquid I for TPP, it is hard to imagine that the immiscibility is caused by the formation of nano-crystals in the same liquid I. Furthermore, toluene is miscible with TPP in an entire composition range, and LLT is observed between two "liquid" states, liquids I and II, without glass formation. Since there is no nano-crystal formation for this case, the phenomenon can be explained solely by the LLT scenario.
Because of all these reasons, we argue that the transition in TTP is truly LLT.
We infer that the same conclusion may apply to the transitions in n-butanol. It is because the pattern evolution during the transition, which is characteristic of NG-type and SD-type LLT (see above), is hard to explain by the other scenarios. However, experimental studies on n-butanol are much fewer than those on TPP; thus, further careful studies are necessary to settle the controversy.

B. LLT in molecular liquids: The case of water
Water is the essential liquid on earth. Since the first suggestion of LLT in model water by numerical simulations, 22 the presence or absence of LLT and its connection to the anomalous behaviors of water have continued to be one of the most popular topics in liquid science (see the Introduction and the following reviews on recent progress in the field 3,12,13,15,16,377 ). Water is one of the most wellstudied molecular systems. There are three major research topics: polyamorphism, LLT, and water's anomalies. Hereafter, we mainly discuss the polyamorphism and LLT of water.

Polyamorphism
Polyamorphism of water and the first-order transition between the two amorphous ices [i.e., low-density-amorphous (LDA) and high-density-amorphous (HDA) ices] were discovered by Mishima et al. [18][19][20] Mishima and Stanley also discussed its connection to LLT. 7,114 Since then, the nature of amorphous ices has attracted considerable attention. Amorphous ices are now classified into amorphous solid water (ASW), hyper-quenched glassy water (HGW), low-density amorphous ice (LDA), high-density amorphous ice (HDA), very high-density amorphous ice (VHDA), unannealed HDA (uHDA), expanded HDA (eHDA), and relaxed HDA (rHDA). 378 Extensive research on polyamorphism has been performed, and the complex nature of amorphous states of water has been revealed. 31,34,379 The nature of pressure-induced amorphous ices has also been a matter of discussion (see, e.g., Ref. 380): a glassy state of high-pressure water or a collapsed crystalline state. It was concluded 380 that HDA, as prepared by compression of ice I at 77 K, is a collapsed crystalline material that does not have a thermodynamic connection with the liquid under pressure, but this may not apply to the annealed HDA (eHDA) under pressure. The polyamorphism of water is the most well studied one, and thus, it should provide a deep insight into the polyamorphism of other The Journal of Chemical Physics PERSPECTIVE scitation.org/journal/jcp materials. However, all the difficulties come from the mechanical nature of pressure-induced crystal-to-amorphous transformation and polyamorphic transition. The relation between HDA and VHDA, i.e., continuous or discontinuous, is also not so clear. 381 A very recent study 382 suggested that LDA and VHDA are two distinct stable forms of amorphous ices, and HDA may be a structurally unstable intermediate state.
Concerning the relationship between polyamorphism and the existence of two liquids [i.e., low-density liquid (HDL) and highdensity liquid (HDL)], two distinct glass transitions for LDA and eHDA were reported, 383 which supports the presence of two liquids for water, i.e., LLT. Recent studies also showed the diffusive dynamics just above the glass-transition signature 32,34,384 and the presence of HDL. 385 However, the glass-transition characters of LDA and HDA 377,[386][387][388][389][390][391][392] and the link between polyamorphism and LLT have still been a matter of debate. 393 Furthermore, a recent study on the vapor pressure of liquid and solid water showed that supercooled liquid water and low-density amorphous solid water (ASW) do not belong to the same phase, and thus, there is no continuous vapor pressure curve between them. 394 Concerning the nature of polyamorphism, numerical studies of the polyamorphism of model waters, which have LLT, would provide valuable information. 147,[395][396][397][398][399][400][401] It is a promising way to solve the controversies concerning the physical nature of polyamorphism and amorphous-amorphous transition. However, we need to pay special attention to the fact that the nature of a glass state is significantly different between experiments and simulations.

LLT of water
Next, we discuss the LLT of water. Since the discovery of LLT in ST2 water by Poole et al., 22 the possible existence of LLT and its connection to water's anomalies have attracted considerable attention. This topic has been mainly studied by numerical simulations of model waters, such as ST2, TIP4P, TIP5P, and coarse-grained mW water. Then, the consensus that LLT exists in some water models was formed. However, Limmer and Chandler 41,402,403 published a series of papers against it. They calculated the free energies as functions of two order parameters, density ρ and crystalline bond orientational order parameter of six-fold symmetry Q 6 . They found separate free-energy basins for liquid I and crystal, but no basin for another liquid II. From these observations, they claimed that the presence of a second liquid, and thus, LLT, e.g., in ST2 water, might be fake, more specifically, a shadow of crystallization. This investigation of the free-energy structure is beneficial to study the phase transition taking place in the metastable state. However, after serious re-investigations of LLT (see Ref. 13 for the history), the very existence of LLT in ST2 water was firmly confirmed by Palmer et al. 24 It also turned out that there was an error in numerical simulations of Limmer and Chandler. Now, there is a consensus that ST2, TIP4P, and TIP5P have LLT, but mW water does not. This difference in various water models may provide critical information on which feature of the interaction potential can lead to the LLT critical point. Now, we turn our attention to experimental efforts to access the LLT of water. Experimentally, it is challenging to access LLT and its critical point since it exists below the homogeneous nucleation temperature, even if it exists. The temperature region where homogeneous crystal nucleation occurs before a supercooled state is equilibrated is called "no man's land." 7 Thus, it is a real challenge to observe LLT in no man's land experimentally.
Three strategies have so far been proposed to overcome this difficulty. The first is to use spatial confinement in nm-sized pores to prevent crystallization, 29,347 which was introduced and actively used by Chen, Mallamace, and their co-workers. [404][405][406][407][408][409] These works show evidence suggestive of a fragile-to-strong liquid transition and LLT. Although these works provide interesting results, they inevitably suffer from criticisms that water confined into an nm-scale space surrounded by a wall is intrinsically different from bulk water because of the presence of water-wall interactions and the reduced effective dimensionality. [410][411][412][413][414] It was shown, 412 for example, that (i) without special care, we cannot conclude even whether water inside an nmsized pore is liquid or solid or amorphous or crystalline and (ii) the interaction with the wall makes the confined state intrinsically inhomogeneous. It was also pointed out 414 that water in the hydrophilic confinement may be under significant tension, around 100 MPa, inside the pore.
The second is to use a small water droplet combined with an intense pulsed x-ray technique to approach or access no man's land, which was introduced by Nilsson and his co-workers. 415,416 This fascinating method allows for accessing a deeply supercooled liquid state, which has never been visited. However, Goy et al. questioned the temperature estimates for supercooled water droplets fast evaporating in a vacuum. 417 Nevertheless, this new state-of-art technique is a promising method, and its application to the investigation of water at high pressure is very interesting.
The third is to use an aqueous solution to access LLT, which we will discuss in detail below for aqueous salt solutions (see Sec. VI B 3) and for aqueous solutions of organic materials (see Sec. VI B 4).

LLT in aqueous salt solutions
Here, we discuss the third method. First, we focus on a method that mixes a salt with water to prevent crystallization. [418][419][420][421][422][423] This method was applied by Suzuki and Mishima for dilute LiClwater solutions. 37,[424][425][426] They observed two distinctly different OH stretching vibrational modes in the glasses of dilute LiCl aqueous solution formed at 0.5 GPa at low temperature, suggestive of two distinct glassy states, and a discontinuous change in the spectrum upon heating at ambient pressure, which resembles the HDA-LDA transition. On the other hand, Kanno suggested phase separation rather than polyamorphism. 420 Other groups 427 also studied this transition. Since phase separation also occurs in this solution, special care is necessary to identify the nature of the transition. 401 Bove et al. also found that transition occurs between HDA and VHDA 428 but recently showed no corresponding LLT. 393 It indicates that polyamorphism may not necessarily be linked to LLT directly (see Sec. VII).

LLT in aqueous glycerol and polyalcohol solutions
Recently, we took a different strategy: mixing water with glycerol to avoid crystallization of water. Glycerol is a well-known The Journal of Chemical Physics PERSPECTIVE scitation.org/journal/jcp cryoprotectant liquid and can induce strong frustration against water crystallization. In an aqueous glycerol solution, we found the direct experimental evidence for genuine (isocompositional) LLT without accompanying phase separation. 35 We reported that liquid I transforms via two types of kinetic pathways, which are characteristic of the first-order transition of a non-conserved order parameter, i.e., NG-type and SD-type LLT, toward the final homogeneous liquid II (see Fig. 19). The dynamic processes of pattern evolution are the same as those observed in TPP and n-butanol (compare Fig. 19 with Fig. 10). Glycerol is miscible with water for both liquids I and II, and thus, the situation should be equivalent to the case of LLT in a mixture of TPP and toluene 373 (see Sec. VI A 9). We show the state diagram of water/glycerol mixtures in Fig. 20. We can see the similarity between the liquid-solid phase diagram of water/glycerol mixtures and the T-P phase diagram of pure water, both of which have a V-shaped melting curve. 81 Towey and Dougan 429 suggested that glycerol molecules act to "pressurize" water. We found differences in the density, the refractive index, the local structure, the hydrogen bonding state, the glass-transition temperature, and the fragility between liquids I and II. Furthermore, the transition is induced by the local structural ordering of water rather than glycerol, suggesting its connection to LLT in pure water.
We also found that the LLT is not specific to glycerol-water mixtures: it is generally observed in 14 aqueous solutions of sugar and polyol molecules, which form hydrogen bonding with water molecules. 36 Some systems exhibit phase separation. We showed that LLT without demixing only takes place for intermediate molecular weights for which there is a subtle balance between frustration and mixing entropy effects (see Fig. 21). We also revealed that both the melting of ice and liquid-liquid transitions in all these aqueous solutions are controlled solely by the water activity aw, which is related to the difference in the chemical potential between an aqueous solution and pure water at the same temperature and pressure [see Fig. 20(b)]. In Sec. II C, we theoretically showed that the water activity aw is determined by the degree of local tetrahedral ordering, i.e., the order parameter ψ [see Eq. (18)], indicating that both phenomena are driven by local structural ordering toward ice-like local structures (see also the theory of the three-state model 36,117,118 ).
We can apply our theory for LLT in a mixture (see Sec. II C). From the fittings, we determine T 0 SD and χ 12 , where 1 = water and 2 = solute (see Sec. II C for the definition of χ 12 ). Here, we note that Eq. (19) is the same as that in Refs. 431 and 432 and the meltingpoint depression is consistent with that reported there. Figure 20(b) shows the phase diagram scaled by the water activity, aw = a 1 , instead of c. We found that T SD of all the aqueous solutions studied can be collapsed on a single straight line in this plot and well fitted by Eq. (20). The above finding suggests the critical role of water in the LLTs of the aqueous solutions. The water activity aw given by Eq. (18) can be interpreted as the effective concentration of water molecules that are free from hydrations to solute molecules, i.e., the distortion of hydrogen bonding, and thus are capable of tetrahedral ordering (i.e., the formation of ψ structures). The fact that T SD can be scaled by aw independent of the types of solute molecules suggests the presence of LLT even in pure water, but this point needs further careful study to be confirmed. We note that the solute molecules stabilize liquid II by suppressing the formation of cubic ice Ic and hexagonal ice I h and increase the viscosity high enough for the kinetics of LLT to be followed. The above results indicate that LLTs of the aqueous organic solutions are universally controlled by the water activity aw alone. The pressure dependence of the spinodal, T SD (P), is also expected to be scaled by aw. 431 However, whether the change in aw can be "directly" translated to that in P or not is a delicate issue. The reason is that the effects of P on water structures are global (or homogeneous), whereas those of solutes are local. Further study is highly desirable for answering this interesting question.
Here, we should note that other scenarios conflicting with our scenario were also proposed. Suzuki and Mishima studied mixtures of glycerol and other polyols and denied LLT at ambient pressure. 37,433 They thought that it is caused by nano-crystal formation. Feldman and his co-workers [434][435][436] also suggested that what we observed may be due to nano-crystal formation. Loerting and co-workers also draw the same conclusion 40,437 (see also a recent review 401 ).
However, the criticisms may not be justified because of the fundamental difference in the experimental methods. We used a small sample, and the temperature was quenched by 100 K/min. In contrast, in all other experiments, the temperature change speed is far slower since the sample volumes used were much larger. The ice nucleation can be avoided on a significant level only by a rapid quench that is fast enough to bypass the most dangerous temperature region for ice crystal nucleation.
Similar to the LLTs of TPP and n-butanol, there is no doubt that nano-crystals form in the transformation process. Thus, the critical question is whether the observed transition is primarily driven by LLT or nano-crystal formation. Bruijn and his co-workers 438 suggested, based on vibrational spectroscopy measurements, that the LLT occurs but is immediately followed by a rapid formation of small (probably nanometer-sized) ice crystals. Now, we explain the reasons why we believe that the observed transition is LLT rather than nano-crystal formation. First, we consider whether the nano-crystal scenario can explain the presence of NG-type and SD-type orderings. We note that the lines of the homogeneous nucleation temperature T H reported by Miyata et al. 439 and MacKenzie 440 are located near the spinodal line of LLT (see Fig. S1 of Ref. 35). In the crystallization scenario, SD-like LLT may be regarded to be a result of homogeneous nucleation of cubic ices. However, we note that nucleation of spherical droplets formed at T > T SD (see Fig. 21), which consists of a newly emerged liquid phase and nm-sized cubic ices, is difficult to explain by this scenario. Unlike single-component systems such as TPP and n-butanol, the spherulite droplet formation by crystallization may be justified for mixtures. The formation of nano-scale cubic ices proceeds while expelling glycerol, and the amorphous state enriched in glycerol has a higher glass-transition temperature than that of the original solution. In this scenario, the glycerol-rich liquid I phase after cubic ice formation looks as if it were liquid II even though there is no LLT. In the following, we show a little experimental evidence that excludes such a possibility and then demonstrate that the transition we observed is genuine (isocompositional) LLT. Figure 22(a) shows the annealing temperature Ta-dependence of the enthalpy difference, ΔH, between liquids I and II. With decreasing Ta, the total amount of heat release, ΔH, decreases, similar to the case of TPP. 356 This result suggests the decrease in the amount of the cubic ices formed during LLT with a decrease in Ta. Around TL = 162.5 K, the enthalpy change, ΔH, almost reaches the value of the HDA/LDA transition in pure water. 21 This result suggests that pure liquid II without cubic ices is formed below TL, although not proven.
This conclusion is more strongly supported by small (SAXS) and wide-angle x-ray scattering (WAXS) measurements [see Fig. 22(b)]. We found that the volume fraction of cubic ices in the liquid II phase, ϕc, and the scattering intensity at q = 0.15 nm −1 , I(q = 0.15 nm −1 ) both decrease with decreasing Ta and almost disappear around TL = 161 K. This result suggests that the amount of cubic ices in liquid II becomes zero around that temperature, consistently with the above independent estimation of TL = 162.5 K from DSC measurements.
We also measured the glass-transition width of liquid II, ΔT gII , for samples prepared at various Ta [see Fig. 22(c)]. We found that ΔT gII of liquid II significantly increases with a decrease in Ta and finally reaches 13 K (note that ΔT gI = 6 K). This result means that liquid II becomes stronger, i.e., less fragile, with decreasing Ta. In other words, the fragility difference between liquids I and II is more pronounced for more pure liquid II containing fewer cubic ices. In the nano-crystal scenario, however, the system after transformation should approach pure liquid I with decreasing Ta as a consequence of less cubic ice formation. Accordingly, this scenario predicts the increase in fragility, i.e., the decrease in ΔT gII , with decreasing Ta and cannot explain the stronger nature of more pure liquid II.  Based on these results, we argue that the transition we observe should be genuine LLT, although no consensus has been formed yet. The situation looks very similar to the cases of TPP 101,356 and n-butanol 103 (see Sec. VI A).

C. Future direction of LLT research in molecular liquids
In the above, we argued that the transition behaviors observed in TPP, n-butanol, and some aqueous solutions of organic substances are indeed LLT. However, some researchers consider that they are not LLT. The settlement of these arguments may be the first step toward the establishment of LLT.
On noting that the source of these controversies is nano-crystal formation in a supercooled state, the finding of LLT in an equilibrium liquid state is highly desirable. The presence of LLT above the melting point was suggested for trans-1,2-dichloroethylene [311][312][313][314][315][316] and molten sodium acetate trihydrate. 317 If these suggestions are confirmed, the understanding of LLT will be deepened significantly. So, continuous efforts for the firm establishment of LLT in molecular liquids are highly desirable.
Since LLT of molecular liquids may exist in a deeply supercooled state not accessible by ordinal experimental techniques, the application of vapor-deposition may also be an interesting direction of research since it is possible to access the bottom of the free-energy landscape of a disordered state while avoiding crystallization. 441,442 The unusually stable glasses made in this way are widely known as "ultrastable glasses." 441,443 Polyamorphism was indeed suggested by using vapor-deposited samples for toluene 444 and 2-methyltetrahydrofuran. 445 It is a promising way to reveal a second amorphous state hidden behind crystallization. It is particularly interesting to study a vapor-deposited glass of a liquid, for which LLT is suggested, such as TPP, n-butanol, and D-mannitol. However, careful selections of the substrate temperature and deposition rate might be required for forming a glassy state of liquid II since its glass-transition temperature is often quite different from that of liquid I.

VII. RELATIONSHIP BETWEEN POLYAMORPHISM AND LLT
The existence of an amorphous-amorphous transition (AAT) is often considered as a signature of LLT in a supercooled state. However, this connection is not so straightforwardly justified. It is mainly because the transition between two distinct amorphous states is induced "mechanically" in nonergodic states, whereas LLT is induced "thermodynamically" in ergodic states. The entropy difference between normal-liquid and locally favored structures in a liquid state should be significantly larger than that in a glassy state since the translational and rotational degrees of freedom are very small for both types of amorphous states (i.e., small Δσ). Thus, there may be no direct connection between a liquid-state transition and a solid-state transition even if we assume the equilibrium nature for a solid-state transition.
If we compress an open crystal with low coordination, it collapses into a dense amorphous state when the applied pressure exceeds the mechanical stability limit. If we remove the pressure, this state can no longer be stable and thus tends to expand. Since this reverse process usually requires thermal activation, the pressure release is not necessarily enough to induce the reverse transition. Because of the intrinsically nonergodic nature of amorphous states, we cannot use the concept of free energy for the theoretical description of AAT. A mechanical factor, which is absent in LLT, should play a critical role in AAT. Thus, there is no simple way to link AAT to LLT on a phenomenological level, although the interaction potential controlling both phenomena is the same.
A system with LLT that involves a significant volume change upon the transition likely has an AAT. However, how the transition line of the former is related to that of the latter is not so clear. The free energy describing the LLT does not include any contribution of elasticity since it is absent in the liquid state. Since the momentum conservation controls the dynamics of a liquid, viscoelastic effects might play some roles, 446 but not elastic effects. In contrast, the AAT is primarily controlled mechanically. Thus, it can happen even in an athermal situation, i.e., at zero temperature. Now, we consider possible relationships between LLT and AAT qualitatively. The simplest case is that LLT and AAT are closely linked. It may be the case of LLT accompanying a significant volume change. Suppose that there is also a distinct first-order-like transition between LDA and HDA, i.e., polyamorphism. If we heat LDA above its glass transition, it should continuously transform to LDL. If we heat HDA above its glass transition above the pressure, where HDL is more stable than LDL, it should transform to HDL. However, whether the volume change in this HDA-to-HDL transition is continuous or discontinuous may depend on the preparation history of HDA.
Another possibility is that AAT exists, but LLT does not. For example, let us consider a system having two distinct local structures, e.g., normal-liquid structures (ρ) and locally favored structures (ψ), with a significant volume difference. We consider a case that the ψstructure has a much larger volume than the ρ-structure. We further assume that the cooperativity of ψ-structure formation is very weak (J ∼ 0) so that the critical point of LLT is located well below the glasstransition temperature. In this case, if we heat LDA and HDA above their glass-transition temperatures, they will become liquids. However, there should be no discontinuous transition between these two liquids. They are continuously connected.
These considerations lead us to the conclusion that the presence of mechanically induced AAT does not necessarily mean LLT. Thus, we need special care to confirm the presence of LLT convincingly even if AAT exists.
Convincing experimental evidence for polyamorphism has been reported for many liquids including water, 7,19,20,114 Si, 160,447,448 silica, [449][450][451] and metallic glasses. 276 Typically, there are low-density and high-density amorphous states. The low-density one is formed at ambient pressure, whereas the high-density one is formed by applying pressure to either an amorphous or a crystalline state prepared at ambient pressure. When pressure is reduced, a high-density amorphous state transforms back to a low-density one at a specific pressure. This hysteresis behavior is often interpreted as a manifestation of the first-order nature of the transition. However, we point out that this transition between the two nonergodic states is far more complicated than the liquid-liquid transition from the following reasons. It is not easy to figure out the role of the cooperativity (J) in such a solid-state transition because of the nonergodic nature of both states. Furthermore, the link between the transition and the The Journal of Chemical Physics PERSPECTIVE scitation.org/journal/jcp underlying free energy is obscured by the effects of mechanical stress on the transition. It is not easy to separate the thermodynamic factors and the mechanical ones for pressure-induced phase transitions in a nonergodic state.
Here, we consider this problem in more detail. 68 The elastic effects on solid-state phase separation 452 were formulated by Onuki, based on the Ginzburg-Landau-type approach. 58,453 Although the type of the order parameter, conserved vs non-conserved, is different, we may discuss the primary effects of elasticity on the phase transition on the same ground. The standard elastic theory of isotropic matter tells us that the elastic energy is given by where B is the bulk modulus, G is the shear modulus, u is the deformation vector, and d is the spatial dimensionality. The lowest order coupling between the bond order parameter (ψ) and ∇ ⋅ u is given by the following free energy: 58 where λ is the coupling constant. For many systems with AAT, such as water, λ is negative. The total free energy should include these elastic contributions. We also have a condition of mechanical equilibrium δF(ρ, ψ, ui)/δui = 0. The elastic effects cause a marked difference between solid-state and liquid-state transitions.
In addition to the possible differences in the values of ΔE, Δv, and Δσ, and J between these two cases, thus, we need to include the above-described elastic terms into our free energy for a solid case.
In a nonergodic solid-state, the thermodynamic free energy is not useful in reality, but we tentatively assume that it is valid. Then, we can show that the spinodal line of LLT determined by the location where the coefficient of δψ 2 becomes zero is shifted by the elastic effects. 452,453 If we ignore the δψ-dependence of K and G (i.e., K = K 0 = const and G = G 0 = const), the extra bilinear term proportional to δψ 2 is given by −(1/2K L0 )λ 2 δψ 2 , where K L0 = K 0 + (4/3)G 0 is the longitudinal modulus. Thus, the spinodal line is shifted by elastic effects whenever λ ≠ 0. If we include the dependence of K and G on δψ, it results in the additional shift of the spinodal line.
Furthermore, there should be the elastic energy barrier in addition to the thermodynamic barrier coming from the interfacial energy, for a first-order phase-transformation (nucleation) process in a solid-state. The importance of such elastic effects can be recognized from the fact that HDA of water breaks into small pieces upon the transformation into LDA while releasing the elastic energy accumulated. 7,9 The volume difference between the two phases again plays a crucial role in the phase transition. The difference in the shear modulus also affects the domain shape during the phase transformation. From this respect, direct observation of domains in the process of AAT is highly desirable. The phenomenological theoretical description of both LLT and ATT in a consistent manner is challenging but an important direction for future research.
There have been studies on the kinetics of amorphousamorphous transition (AAT). Experimental studies on SiO 2 and GeO 2 showed the slow logarithmic kinetics of the temporal change in the volume, 454 unlike the Avrami-Kolmogorov-type behavior observed for crystal-crystal transition (CCT), which was ascribed to a deep hierarchy of structural process with the uniform distribution of energy barriers. 455 We may also explain such a logarithmic decay if we assume that the activation energy is a linearly decreasing function of the order parameter. [456][457][458] The difference in the pressure dependence of the kinetics between AAT and CCT was also pointed out. 454 Very recently, Sato et al. found that silica glass becomes inhomogeneous during AAT and consists of optically invisible sub-nm-scale domains for the two amorphous polymorphs. 459 The kinetics was also studied for the HDA-to-LDA transition of water. 379,[460][461][462][463][464] The results indicated the macroscopic two-phase coexistence during the intermediate stage of the transformation, which was interpreted as a sign of the first-order nature of the transformation. Such a coexistence was also observed as a function of temperature 32 and pressure. 465 For the different interpretations related to CCT, see Sec. VIII. Here, we note that the mechanical stress must be involved in the solid-state transition accompanied by the volume change, as mentioned above. Thus, it is interesting to study its role in the kinetics.
One of the most promising ways to study the relationship between LLT and AAT may be molecular dynamics simulations. Giovambattista and his co-workers 397,398,[466][467][468][469][470][471][472][473] have recently taken this approach to study the LLT-ATT relationship for water. For example, they reported that the state diagram obtained is consistent with the LLT hypothesis but sensitive to the sample preparation history, reflecting the nonergodic nature of glass. 472 Later, they found that the evolution of each glass sample, during compression/decompression or heating, is controlled by macroscopic properties of the initial glass sample, including the density, the temperature, and the characteristics of the potential energy landscape. 473 Although the glassy states obtained by molecular dynamics simulations are much more liquid-like than experimental glasses, this type of approach would shed fresh light on the relationship between LLT and AAT.

VIII. RELATIONSHIP OF POLYMORPHISM TO LLT AND AAT
It is natural to expect the similarity between local structures in a liquid and the crystal structures (see, e.g., Refs. 26,44,82,474,and 475) since the same interaction potential controls both liquid and solid states. We recently revealed that crystal nucleation is generally assisted by crystal-like preordering in a supercooled liquid state, which also affects polymorph selection. 53,72,172,[476][477][478][479] Similar behaviors are confirmed for a variety of systems (see, e.g., Refs. [480][481][482][483][484][485][486][487][488]. Since LLT may be regarded as the transition between the locally favored structures, it should also have a connection to crystallization. For example, the role of liquid polymorphism on crystallization was indicated for silicon. 481 There are several interesting questions on the relationship of crystal polymorphism to liquid and amorphous ones. Since an amorphous state is metastable against its crystalline counterpart, claimed that the formation of HDA is a consequence of a kinetically arrested transformation between low-density ice I and high-density ice XV 489 rather than the first-order AAT linked to LLT, as proposed previously. 490 Lin et al. showed that pressureinduced amorphous ice is an intermediate state in the phase transition from the low-pressure low-density ices to the high-pressure high-density ices. 491 They also showed that the structural evolution from ice VII to ice I induced by heating takes place in the following sequence: 492 ice VII into HDA, followed by HDA-to-LDA transition, and then crystallization of LDA into ice I. These studies have shown a complex interplay between amorphous and crystal polymorphisms. When LLT exists in an equilibrium liquid state, the first-order transition line of LLT causes a kink in the melting-point curve, reflecting the density and entropy difference between the two liquids [see Figs. 3(a) and 4]. We revealed that preordering in a liquid that is consistent with the crystal structure to be formed leads to the reduction in the liquid-crystal interface energy, promoting crystal nucleation. For example, in the absence of LLT, we find that the crystal nucleation barrier significantly increases near the deep eutectic point, since the liquid structure becomes very disordered around there, due to competing orders. 172 This physical picture suggests that there should be a difference between crystallization behaviors from the two liquids for a system whose LLT exists in an equilibrium liquid state and produces a kink in the melting-point curve. It is an interesting problem for future investigation.
As we discussed above (see Sec. VI A 8), the critical point of LLT can also affect the crystallization behavior. It should be noted that although there is a connection between locally favored structures and crystalline precursors, they are not the same. The order parameter of LLT is scalar and rotationally invariant, but the one of crystallization is bond-orientational order with a specific orientation. 26,56

IX. FUTURE DIRECTIONS OF RESEARCH
Here, we mention some predictions and possible future research directions to deepen our understanding of LLT and AAT.
1. The two-state model predicts that the criticality associated with LLT should belong to the Ising universality class. This prediction has been confirmed numerically for model water systems, 25,59 but there has been no experimental confirmation. It can be accomplished by measuring thermodynamic quantities such as the heat capacity or analyzing the wavenumber-dependent static scattering intensity. The recent finding of LLCP in sulfur 138 may be a very promising system to explore the criticality of LLT (see Fig. 7). 2. The hierarchical nature of the two-state model (see Sec. II D) predicts that the static and dynamic Schottky lines should meet at the LLCP (see Fig. 5). This prediction may be used to predict the location of the LLCP of a system, for which direct experimental access to it is difficult, as in water. 84,120 3. Our dynamical theory 87 described in Sec. II E predicts the presence of an extra mode coming from the coupling between the two order parameters in the dynamic structure factor and the peculiar dispersion relations. These features may be measured by dynamic scattering experiments near the second critical point. Liquid sulfur may again be a good candidate for not only the check of the prediction but also the study of dynamic criticality. Our theory also predicts the coupling between the order parameter and flow during LLT, which may directly be observed by microscopy observation of pattern formation for a system that has a significant density difference between liquids I and II. 4. A critical question concerning the nature of LLT is the origin of the cooperativity of LFS formation. Numerical simulations of a realistic model system, whose potential is systematically tunable, may provide a clear answer. 5. For atomic systems, the precise measurements of the melting point as a function of pressure may provide crucial information on whether LLT exists. The development of experimental techniques for this purpose is highly desirable. This strategy may also be applied to some molecular systems, as was pioneered by Mishima and Stanley. 114 Time-resolved xray/light scattering or turbidity measurements to follow the relaxation after a pressure jump of a liquid are even more promising methods to check whether LLT exists or not since the strong scattering should be observed only when LLT occurs. 6. Experimental techniques that can directly access LFS are highly desirable. Scattering techniques are promising if LFS produces peculiar scattering peaks, such as the so-called First Sharp Diffraction Peak (FSDP) (see, e.g., Ref. 493), which has recently been shown to be a direct measure of the order parameter ψ for tetrahedral liquids such as silica and water. 494,495 Real-space observation of a thin glass film by transmission electron microscopy is another possible method to access LFS directly (see, e.g., Ref. 496). 7. In relation to the above point, the formation of LFS is a minimum requirement to have LLT. The identification of LFS is more difficult for molecular liquids because of the complex shape and interaction of molecules. Accurate computer modeling of molecules is highly desirable from this respect. 8. Once LLT is firmly established, we can study how the glasstransition behavior, including the fragility, slow-β relaxation, and boson peak, is different between the two liquids. Such a study may provide crucial information on these poorly understood problems from a new angle. 9. As we discussed in Sec. VI A 8, the crystal nucleation frequency may show a peculiar temperature dependence near the critical point or the spinodal line of LLT. 370 This fact may be used to detect the LLT hidden in a metastable state below the melting point. 10. Optical/electron microscopy observation of the pattern evolution during AAT may be very useful to identify the nature of the transition. Observation of the domain shape during AAT should provide crucial information on the role of mechanical stress in the transition. Polarizing microscopy observation of the transformation process is also very interesting since it provides information on anisotropic mechanical stress generated by AAT (or crystal formation) via the stress-optical law. These measurements should provide information on the The Journal of Chemical Physics PERSPECTIVE scitation.org/journal/jcp elastic coupling between domains. Information on mechanical stress can also be accessed much easily by numerical simulations. These results may be compared with the theory of the coarse-grained model with elasticity (see Sec. VII). 11. So far, AAT has mainly been triggered by a pressure decrease in experiments to induce the transformation from HDA to LDA. It is interesting to trigger AAT of both directions by a temperature (or pressure) jump, using the slope of the transition line in the P-T state diagram. 12. On the links of LLT with polyamorphism and polymorphism, numerical simulations are quite promising, as discussed in Sec. VII. On the latter relation, the relationship between the symmetries of LFS and the crystal and its impact on the crystal nucleation frequency is an interesting topic of research. 13. At this moment, few studies have considered the effect of external fields, such as optical (see, e.g., Ref. 497 on the light-induced λ transition in S), electric, magnetic, and shear fields, 319,498 on LLT and AAT. Since the two liquids have different refractive indices, dielectric constants, magnetic susceptibilities, and viscosities, these external fields should influence LLT and AAT. It is an interesting direction for future research.

X. SUMMARY
Liquid-liquid transition is an intriguing phenomenon and has attracted considerable attention for a long time. However, as we see above, only a few convincing examples have been reported experimentally. Even for these examples, it is not easy to study the nature of LLT since it exists in temperature-pressure regions, which are hard to access experimentally. On the other hand, the presence of LLT has been firmly established by numerical simulations and theories. It is highly desirable to find an example of LLT whose transition behavior is easy to access experimentally. For studying static and dynamic critical phenomena experimentally, we need a system with LLT in an equilibrium state. There are some atomic systems whose second critical points exist in an equilibrium liquid state but at very high temperatures and pressures. Sulfur is a promising system since the location of the LLCP is determined experimentally. 138 So, the development of experimental techniques to access static and dynamic critical phenomena and the kinetics of LLT is also highly desirable, as mentioned above.
For studying the nature of LLT, including the dynamics, molecular liquids are suitable since the dynamics can be slow due to the influence of glass transitions, as in the case of triphenyl phosphite. However, the process of LLT can be contaminated by nano-crystal formation since it occurs in a state metastable against crystallization. The critical question here is why crystals appear as nano-crystals. We proposed a possible scenario, but this question should be addressed experimentally by direct microscopic observation, such as electron and atomic force microscopy observation. We showed some pieces of evidence and explanations for the presence of LLT in triphenyl phosphite. Even for this most well-studied molecular system, there is still no consensus. We need more effort to establish the consensus by new experiments.
The only requirements to have LLT in a liquid are the formation of locally favored structures and its cooperativity, and it is natural to expect the cooperativity in the ordering. Thus, we expect that LLT should be ubiquitous. However, searching LLT is highly nontrivial because of extreme conditions for atomic systems and interference by other phenomena such as crystallization and vitrification for molecular systems. So far, all LLTs in molecular systems were found accidentally. If the interatomic (or intermolecular) potentials are very accurately known, the prediction of LLT by numerical simulations may be possible. Unfortunately, the presence or absence of LLT and its location sensitively depend on the details of the potential, as we see above. It is an important research topic to elucidate which feature of the potential leads to the cooperativity of the formation of locally favored structures in a liquid. Currently, we do not have a simple way to find a new LLT due to the difficulties mentioned above. Nevertheless, if we can find an example of LLT, for which we can easily access experimentally, this field will develop tremendously.
We also need further study on the relationship between liquid-, amorphous-, and crystal-polymorphs, which provides crucial information on the relationship between liquid, glass, and crystal. From this respect, it is highly desirable to develop a liquid-state theory that can describe the liquid-liquid transition, glass transition, and crystallization by taking many-body correlations into account.
Liquid is one of the principal states of matter: it is dense yet still flows and provides a field for various chemical and biological reactions. Although it may be quite challenging to realize LLT at ambient conditions, the existence of LLT in a stable molecular liquid was reported for trans-1,2-dichloroethylene 311-315 and molten sodium acetate trihydrate, 317 although the former is controversial. In such systems, we may use LLT to control the static and dynamic properties of a liquid, such as the refractive index, fluidity (viscosity), miscibility with other fluids, and wettability to a substrate, without changing molecules. Such switching of liquid properties by external fields may open up new intriguing applications of LLT. For example, the occurrence of an LLT in chalcogenide materials is suggested to enable the fabrication of memory devices combining high speed with excellent data retention. 274 We hope that this Perspective article stimulates further research of LLT and polyamorphism and helps bridge different material communities to promote a deeper understanding of these intriguing phenomena.

ACKNOWLEDGMENTS
The author expresses sincere thanks to Mika Kobayashi, Rei Kurita, Hiroshi Mataki, Ken-ichiro Murata, Ryotaro Shimizu, and Kyohei Takae for their collaboration on the research of LLT. He is also grateful to Mark Ediger for providing the opportunity to write this Perspective. This study was partly supported by Scientific Research (A) (KAKENHI Grant No. JP18H03675) and Specially Promoted Research (KAKENHI Grant Nos. JP25000002 and JP20H05619) from the Japan Society for the Promotion of Science (JSPS) and the Mitsubishi Foundation.

DATA AVAILABILITY
Data sharing is not applicable to this article as no new data were created or analyzed in this study.