Relationship between the Potential Energy Landscape and the Dynamic Crossover in a Water-Like Monatomic Liquid with a Liquid-Liquid Phase Transition

Transcription

1 Relationship between the Potential Energy Landscape and the Dynamic Crossover in a Water-Like Monatomic Liquid with a Liquid-Liquid Phase Transition Gang Sun 1, Limei Xu 1,2,, Nicolas Giovambattista 3,4, 1 International Center for Quantum Materials, School of Physics, Peking University, Beijing , China 2 Collaborative Innovation Center of Quantum Matter, Beijing, China 3 Department of Physics, Brooklyn College of the City University of New York, Brooklyn, New York 11210, United States 4 Ph.D. Programs in Chemistry and Physics, The Graduate Center of the City University of New York, New York, NY 10016, United States (Dated: December 12, 2016) 1

2 Abstract In the case of fragile liquids, dynamical properties such as the structural relaxation time evolve from Arrhenius at high-temperatures to non-arrhenius at low temperatures. Computational studies show that (i) in the Arrhenius dynamic domain, the liquid samples regions of the potential energy landscape (PEL) that are insensitive to temperature (PEL-independent regime) and the relaxation is exponential, while (ii) in the non-arrhenius dynamic domain, the topography of the PEL explored by the liquid varies with temperature (PEL-influenced regime) and the relaxation is non-exponential. In this work we explore whether the correlation between dynamics and PEL regimes, points (i) and (ii), hold for the Fermi-Jagla (FJ) liquid. This is a monatomic model liquid that exhibits many of water anomalous properties, including maxima in density and diffusivity. The FJ model is a rather complex liquid that exhibits a liquid-liquid phase transition (LLPT) and a liquid-liquid critical point (LLCP), as hypothesized for the case of water. We find that, for the FJ liquid, the correlation between dynamics and the PEL regimes is not always present and depends on the density of the liquid. For example, at high density, the liquid exhibits Arrhenius/non-Arrhenius (AnA) dynamical crossover, exponential/non-exponential (EnE) relaxation crossover, and a PELindependent/PEL-influenced regime crossover, consistent with points (i) and (ii). However, in the vicinity of the LLCP, the AnA crossover is absent but the liquid exhibits EnE relaxation and PEL regime crossovers. At very low density, crystallization intervenes and the PEL regime crossover is suppressed. Yet, the AnA dynamical crossover and the EnE relaxation crossover remain. It follows that the dynamics in liquids (AnA and EnE crossovers) are not necessarily correlated with the changes between the PEL regimes, as one could have expected. Interestingly, the AnA crossover in the FJ liquid is not related to the presence of the Widom line. This result may seem to be at odds with previous studies of polymorphic model liquids and a simple explanation is provided. 2

3 I. INTRODUCTION The dynamics of liquids at high temperature is usually characterized by a relaxation time τ(t) that obeys the Arrhenius equation, τ(t) = τ 0 exp(e A /k B T) (1) where the prefactor τ 0 and the activation energy E A are T-independent (but they may depend on the density or pressure of the system); k B is the Boltzmann constant. In many liquids, so-called fragile liquids [1], τ(t) deviates from Eqn. 1 at low temperatures and the liquid becomes non-arrhenius (see, e.g., Refs. [1 4]). There are a few theories that predict a non-arrhenius dynamic behavior in liquids at low temperatures, such as Mode Coupling Theory [5] and the Adam-Gibbs approach [6]; see also Refs. [7, 8]. A theoretical approach to study liquids at low temperatures as well as glasses is the potential energy landscape (PEL) formalism (see, e.g., Refs. [9 16]). For a system of N atoms, the PEL is the hypersurface in (3N + 1)-dimensional space defined by the potential energy of the system as function of the atom coordinates, V ( r 1, r 2,..., r N ) [11]. At any given time t, the system is represented by a single point on the PEL given by the atom coordinates at t. Hence, as the atom coordinates change with time, the representative point of the system moves describing a trajectory on the PEL. In the high-temperature liquid state, the system can visit large regions of the PEL and, as the temperature decreases, the representative point is constrained to move within more localized regions of the PEL. Upon further cooling, in the glass state, ergodicity is broken and the representative point of the system can only move within specific basins of the PEL. The basin minima are called inherent structures (IS). A complete theory based on the topography of the PEL has been developed that allows one to express the free energy of liquids in terms of the IS energy E IS (basin depth), Hessian of the PEL at the IS (basin curvature), and distributions of IS in the PEL (density of IS states) [13]. The PEL has been used extensively to understand the behavior of liquids, the glass transition, and the behavior of glasses. Examples include, the study of amorphous-amorphous transformations [16, 17], aging effects in glasses [18, 19], dynamics in liquids [3, 20 27], and more complex systems, such as protein folding and molecular/atomic clusters (see, e.g., Ref. [28]). Relevant to this work is the relationship between the PEL topography and the dynamics in liquids. In a computational study of a binary Lennard-Jones mixture (BLJM), 3

4 Sastry and collaborators [2] found that, for a selected density, the Arrhenius/non-Arrhenius dynamic crossover in this fragile [29] liquid correlates with a change in the behavior of the PEL properties sampled by the system. Specifically, they found that the high-t Arrhenius behavior corresponds to the system exploring regions of the PEL where E IS (T) is independent of temperature. On the contrary, in the low-t non-arrhenius regime, the liquid samples regions of the PEL where E IS (T) varies with temperature. These findings imply that the Arrhenius dynamic regime occurs at temperatures where the topography of the PEL is not relevant (PEL-independent regime) while the non-arrhenius dynamics is accompanied by explorations of deeper regions of the PEL where the topography of the PEL becomes relevant (PEL-influenced regime). Remarkably, Ref. [2] shows that, at least for the BLJM and for a given density, the Arrhenius/non-Arrhenius dynamic crossover temperature coincides with the temperature at which the system evolves from the PEL-independent to the PELinfluenced regimes. The work of Sastry and collaborators provides a simple explanation of the non-arrhenius relaxation time found in fragile liquids based on the topography of the PEL. The study in Ref. [2] is based on a simple liquid, i.e., a liquid that shows the typical behavior found in most liquids. Studies based on more complex liquids, such as orthoterphenyl (OTP) [30, 31], water [15, 32], and silica [25] also indicate that E IS is constant at high temperatures while E IS decreases non-linearly upon cooling at lower temperatures. In many of these liquids, such as water [15], the correlation between the dynamic crossover and the PEL is also apparent, in agreement with Ref. [2]. A common view is that if a liquid exhibits a dynamic crossover then it is accompanied by a change between the PELindependent/PEL-influenced regimes. However, a quantitative test of the correlation between dynamic crossover and PEL regimes, with the same protocol/definitions employed in Ref. [2] have not been performed in a wide range of densities and for complex (as opposite to simple) liquids. In this work, we investigate whether the Arrhenius/non-Arrhenius dynamic crossover and the PEL-independent/PEL-influenced crossover occur simultaneously in the Fermi-Jagla (FJ) liquid, as found for the BLJM liquid case. The FJ liquid is a monatomic complex liquid that exhibits many water-like anomalous properties, including the presence of density, diffusivity, and compressibility maxima along constant pressure or temperature paths (see the phase diagram in Fig. 2 and Ref. [33]). In addition, as found in computer simulations of ST2 water [34 37], the FJ model exhibits liquid polymorphism and the pres- 4

5 ence of a LLCP [33, 38]. The FJ model also presents glass polymorphism [39, 40], as found in experiments (see, e.g., Ref. [41 45]) and simulations [46 51] of glassy water. We also note that, contrary to the BLJM, the FJ liquid crystallizes at low pressures (see Fig. 2). The rich complexity in the phase diagram of the FJ model makes it an interesting system to explore the correlation between the dynamic and PEL crossovers. We note that in the case of polymorphic liquids, such as water, dynamic crossovers have been reported and correlated with anomalous properties such as the existence of a Widom line at temperatures above the LLCP temperature [52 55]. It follows that in polymorphic liquids, signatures in the PEL are not trivially related to the liquid dynamics. This work is organized as follows. In Sec. II we present the simulation details. The main results are discussed in Sec. III, including the presence of dynamic crossovers and their relation with the PEL properties. A summary and discussion is included in Sec. IV. II. SIMULATION DETAILS We perform molecular dynamics (MD) simulations of an atomistic liquid with pair interactions represented via the FJ pair potential. The FJ potential is a smooth version of the Jagla model [56, 57] originally proposed to model the thermodynamic behavior of water and silica. As shown in Fig. 1, the FJ potential is a core-softened potential with two length scales and an attractive part. In its original form [33], the FJ potential is truncated at the cutoff distance r c = 4.0. In this work, in order to perform PEL calculations, we modify the original FJ potential given in Ref. [33] by adding a switching function which brings the energy and force smoothly to zero between an inner (r 1 = 0.9r c ) and outer cutoff distance (r c = 4.0). The modified FJ pair potential is given by A 0 U(r) = ε 0 [(a/r) n exp[ A 1 A 0 (r/a A 2 )] B exp[ B 1 B 0 (r/a B 2 )] ] + S(r), (2) where the parameters A i and B i are listed in Table I (parameters a and ε 0 define the units of length and energy and hence, are not relevant). S(r) is a switching function defined as C : (r < r 1 ) S(r) = A (r r 3 1) 3 + B(r r 4 1) 4 + C : (r 1 < r < r c ) with A = ( 3U (r c ) + (r c r 1 )U (r c ))/(r c r 1 ) 2 5

6 B = (2U (r c ) (r c r 1 )U (r c ))/(r c r 1 ) 3 C = U(r c ) (r c r 1 )U (r c ) 1 12 (r c r 1 ) 2 U (r c ) MD simulations are performed for up to 10 7 steps at constant volume and temperature for a system of N = 1728 particles. Particles are located in a cubic box and periodic boundary conditions are employed. MD simulation details, such as the simulation time step and thermostat employed, can be found in Ref. [33]; all quantities are given in reduced units [33]. We calculate the IS energy, E IS (T,V ), along isochores as follows. For a given (T,V ), we perform extensive MD simulations in order to obtain independent IS. Specifically, at each state point, we extract 100 independent configurations from the MD simulations. These configurations are separated in time so that the atoms mean-square displacement between two consecutive configurations is at least 10 a 2, corresponding to more than 3 times the atom hard-core radius. Each of these configurations is then used to obtain an IS by performing a potential energy minimization using the conjugate gradient algorithm [58]. The average potential energy over all IS at a given thermodynamic state (T,V ) gives the average IS energy, E IS (T,V ). III. RESULTS A. Regions of the Potential Energy Landscape Sampled by the Liquid In this section, we determine the thermodynamic states of the FJ liquid that correspond to the PEL-influenced and PEL-independent regimes. The behavior of E IS (T) for selected volumes is shown in Fig. 3 and E IS (T) for all the volumes studied is presented in Fig. 4. We find two different behaviors depending on the volume of the system. At v = V/N > 3.4, E IS (T) remains constant at all temperatures accessible to the liquid state, including the lowest temperatures at which crystallization can be avoided (see Fig. 2). This implies that, at these large volumes, the liquid is always in the PEL-independent regime. Instead, at v 3.4, E IS (T) is constant only at high temperature while at low temperatures it decreases non-linearly as the temperature decreases. It follows that the liquid at these small volumes is in the PEL-independent regime at high temperatures and evolves to the PEL-influenced regime at low temperatures, as found in the BLJM [2]. We note that the range of IS energies 6

7 explored by the liquid (δe IS ) in the PEL-influenced regime (v 3.4) varies with v. For example, at v = 2.0, δe IS 0.1 while at v = 3.4, δe IS It follows that as v increases towards the value v 0 3.4, the distinction between PEL-influenced and PELindependent regimes becomes less evident and it vanishes at v > v 0. We stress that the intervals δe IS explored by the FJ liquid in the PEL-influenced regime are not negligible. For comparison, for the BLJM of Ref. [2], δe IS (in reduced energy units given by the LJ parameter ǫ) which is similar to the IS energy range covered in our simulations (in FJ reduced units, ǫ 0 ). Similarly, the ratio δe IS /E IS,0, where E IS,0 is the asymptotic value of E IS (T) at high temperature, in the FJ liquid (v = 2.0) and BLJM of Ref. [2] are also comparable; δe IS /E IS,0 0.1/4.54 = 2.20% and δe IS /E IS,0 0.17/6.87 = 2.47%, respectively. The onset temperature T 0 (v) indicating the crossover from PEL-independent to PELinfluenced regimes, at v v 0, can be obtained from E IS (T) as shown in Figs. 3(a)-(d). The T 0 (v) line together with the behavior of E IS (T) for all volumes studied are shown in Fig. 4. It follows that T 0 (v) 0.35 ± 0.01 and it is roughly v-independent. In particular, T 0 (v) 2 T c where T c 0.18 is the LLCP temperature [33] (see Fig. 4) and hence, the LLCP is located within the PEL-influenced regime. Interestingly, a similar situation seems to hold in ST2 water. Specifically, Fig. 4a of Ref. [15] suggests that the PEL-independent regime corresponds to T > 300 K for P = 0.1 MPa while the LLCP temperature in this model is T c 245 K (P c 190 MPa) [15]. That the LLCP in ST2 water is located in the PEL-influenced regime may not be surprising since for this water model the LLCP occurs in the deeply supercooled region, where the liquid is expected to explore deep regions of the PEL. However, in the case of the FJ liquid, the LLCP occurs in the equilibrium regime and hence, it is not evident that for this model liquid the LLCP should occur within the PEL-influenced regime. The T 0 -line is also included in the phase diagram of Fig. 2. Interestingly, the T 0 -line extends over both LDL-like and HDL-like domains and hence, it is not a property that can be associated to only LDL or HDL. Specifically, the T 0 -line is observed at 1.8 v 3.4 while the LLCP volume is v c 2.9. Accordingly, the T 0 -line and hence, the PEL-independent/PELinfluenced crossover, occur in the HDL-like state for all volumes studied (1.8 v < v c ) and in the LDL-like state for only v c < v <

8 B. Connection between Dynamics and PEL regimes In this section, we identify changes in the dynamics of the FJ liquid and test whether such changes correlate with the system evolving from the PEL independent (at high-t) to the PEL influenced regime (at low-t). We focus on two signatures in the dynamics of the FJ liquid, (1) an Arrhenius/non-Arrhenius crossover exhibited by the relaxation time of the liquid (Sec. III B 1), and (2) an exponential/non-exponential crossover exhibited by the self-intermediate scattering function (Sec. III B 2). Our conclusions are based on the self-intermediate scattering function F s (k,t) of the FJ liquid at constant volume. F s (k,t) is the Fourier transform of the van Hove correlation function [59], G s (r,t) = 1 N N δ( r i (t) r i (0) r) (3) i=0 which gives the probability that a particle displaces over a distance r during time t. Following Ref. [2], we consider the case k = k 0 where k 0 is the wave vector magnitude (at a given volume) corresponding to the first peak of the static structure factor S(k) [59]. F s (k 0,t) is shown in Fig. 5(a), 5(b), and 5(c) for v = 2.2, 3.0, and 3.8, respectively, above and below the LLCP volume v c 2.9. We include all temperatures accessible to the liquid; for example, the lowest temperatures reported at v = 2.2 and 3.8 are, respectively, T = 0.10 and T = 0.17 while the corresponding crystallization is first observed at T = 0.09 and At v = 2.2, F s (k 0,t) shows the behavior commonly observed in liquids at low temperatures (see Ref. [60, 61] for the case of the BLJM). Specifically, at high temperature F s (k 0,t) decays exponentially to zero while, as temperature decreases, F s (k 0,t) starts to develop a plateau at intermediate times, indicating the onset of non-exponential relaxation. The non-exponential relaxation is evident at v = 2.2 because crystallization at this volume can be avoided to very low temperatures. At larger volumes, however, the LLPT intervenes and/or crystallization occurs (see Fig. 2) and the signatures of non-exponential relaxation in F s (k 0,t) become less evident; see Fig. 5(b) and 5(c). 1. Arrhenius/non-Arrhenius Crossover in the Relaxation Time In this section we test whether the PEL-independent/PEL-influenced regime crossover is accompanied by an Arrhenius/non-Arrhenius crossover in the structural relaxation time, 8

9 τ(t). Both of these crossovers were found to occur at close temperatures in the BLJM at ρ = 1.2 [2]. Next, we show that for the FJ liquid, the correlation between the PELindependent/PEL-influenced crossover and the Arrhenius/non-Arrhenius crossover is not trivial and depends on the volume. Only at certain volumes (e.g., v < 2.6) we recover qualitatively the results found in the BLJM. In order to identify the Arrhenius/non-Arrhenius crossover in the FJ liquid dynamics, we calculate τ(t) at constant volume. Following Ref. [2], we define τ(t) as the time at which F s (k 0,t) = 1/e. Figs. 6(a)-(c) show τ(t) at v = 2.2, 3.0 and 3.8 obtained from Fig. 5. At all volumes studied, it is possible to identify a range of temperatures where the relaxation time is Arrhenius, i.e., τ(t) = τ 0 exp ( EA T ). (4) Here, the parameter τ 0 and the activation energy E A depend on v; see Figs. 7(a) and 7(b). The Arrhenius regime is indicated by the dashed-blue line in Figs 6(a)-(c), which is obtained by fitting τ(t) for approximately 1/T < 5 using Eqn. 4. At low temperatures, the dynamics is non-arrhenius. This crossover corresponds to the Arrhenius/non-Arrhenius dynamical crossover that is found in fragile liquids such as BLJM [29, 60, 61] and water [62]. The corresponding Arrhenius/non-Arrhenius crossover temperature, T τ, is indicated by the red arrow in Figs 6(a)-(c). In the low-temperature non-arrhenius regime the relaxation is nonexponential (Sec. III B 2), as one would expect [29, 60, 61]. The Arrhenius/non-Arrhenius crossover is evident at small volumes, see, e.g., Fig. 6(a), due to the low crystallization temperature. The effects of changing the volume on τ(t) is shown in Figs. 8(a) and 8(b). As the volume increases from 1.8 to 3.0 [Figs. 8(a)], close to the LLCP volume, the lowest temperature accessible to the liquid state increases and, in particular, the non-arrhenius regime is suppressed. Close to the LLCP [e.g, v = 3.0], the liquid remains Arrhenius at low temperatures down to the LL coexistence region (Fig. 6). That is, in Fig. 6(b), T τ (red arrow) is lower than the temperature below which the system enters the LL coexistence region, indicated by the black arrow. Interestingly, further increase of the volume from v = 3.0 to 4.0 leads to a re-emergence of a weak Arrhenius/non-Arrhenius crossover; see Fig. 8(b). The behavior of T τ is shown in the phase diagram of Fig. 2, as function of pressure, and in Fig. 9, as function of volume. At v v c 2.9, the T τ -line moves into the LL coexistence 9

10 region indicating that the supercritical equilibrium liquid does not exhibit a low-t dynamical crossover. The Arrhenius/non-Arrhenius crossover exists only in the supercritical liquid at v > v c and v < v c. It is apparent from Figs. 2 and 9 that it is the presence of the LLCP that suppresses the low-t Arrhenius/non-Arrhenius dynamical crossover at intermediate volumes. We conclude this section with a discussion of the relationship between (i) the PELindependent/PEL-influenced regime crossover defined by T 0 and (ii) the Arrhenius/non- Arrhenius crossover in the liquid dynamics defined by T τ. The results presented above indicate that, at the qualitative level, the relationship between (i) and (ii) in the FJ liquid is non-trivial and depends on the density considered. Specifically, at small volumes, we find that the liquid is Arrhenius at high temperatures [see, e.g., Fig. 6(a) for approximately 1/T < 4.5 (0.22 < T)] and it is in the PEL-independent regime [see, e.g., Fig. 3(a) for T > 0.28] while at low temperatures the liquid is in the PEL-influenced regime and its dynamics is non-arrhenius. Hence, at small volumes, the relationship between the dynamical regimes and the PEL regimes, observed in the BLJM, holds for the FJ liquid as well, at least qualitatively [2]. As shown in Figs. 2 and 9, these conclusions hold for v < 2.6, where both T 0 and T τ are located within the high-density liquid state. The same conclusions apply to the low-density liquid at approximately v = (Fig. 9). However, in the range 2.6 v < 3.2 the Arrhenius/non-Arrhenius crossover is suppressed by the LLPT (i.e., the T τ -line is within the LL coexistence region in Fig. 9) while one still observes the PEL-independent/PEL-influenced regime crossover (the T 0 -line in Fig. 9 in located within the supercritical equilibrium liquid region). It follows that for 2.6 v < 3.2, the PELindependent/PEL-influenced regime crossover is not accompanied by the Arrhenius/non- Arrhenius dynamical crossover. At even larger volumes, 3.5 v 4.0, the correlation between the PEL regimes and the dynamics of the FJ liquid also breaks down. At these volumes, however, there is a weak Arrhenius/non-Arrhenius dynamical crossover (i.e., the T τ -line is within the equilibrium liquid domain in Fig. 9) while the PEL-independent/PELinfluenced regime crossover is suppressed due to crystallization (there is no T 0 -line in Fig. 9 for v > 3.4). We note that the PEL-independent/PEL-influenced crossover and the Arrhenius/non- Arrhenius dynamical crossover are expected to occur at close temperatures (for a given density) [2]. We can test this expectation in the FJ liquid for v < 2.6 and 3.2 v

11 where both crossovers are observed. As shown in Figs. 2 and 9, T τ < T 0 at these volumes. Hence, with the definitions of T τ and T 0 employed in this work, the PEL-independent/PELinfluenced crossover precedes the dynamical crossover upon cooling. However, we note that it is possible that alternative definitions of T τ and particularly T 0, may bring these crossover temperatures closer to each other. The precise value of T 0 obtained with the definition employed in this work is indeed sensitive to the specific value of E IS for large T. For example, if in Fig. 3(b) (v = 2.6) the location of the red-dashed line is shifted by, e.g., 0.01 (within the error bars in E IS ), the value of T 0 would change by 0.08 which is considerable. This is due to the slow decay of E IS (T), upon cooling, from its asymptotic value at high temperatures. The temperature T τ corresponds to the strong-to-fragile transition commonly identified in fragile liquids, including BLJM [60, 61] and SPCE water [62]. Hence, the behavior predicted by MCT may hold only at T < T τ. In this case, one would find that the mode coupling temperature obeys T MCT < T τ < T 0, as expected. We also note that crystallization and/or the LLPT occur at low T for all volumes studied and we cannot identify a fragileto-strong transition in the FJ liquid at very low temperatures. Interestingly, a fragile-tostrong transition at very low temperature has been identified in computer simulations of ST2 water [15]. 2. Exponential/non-Exponential Relaxation Crossover of the Self-Intermediate Scattering Function In Ref. [2], it is noted that for the BLJM the PEL-independent/PEL-influenced crossover is also accompanied by a change in the relaxation of the liquid. Specifically, at hightemperature within the PEL-independent regime, F s (k,t) is an exponential decaying function of time while, at low-temperatures within the PEL-influenced regime, F s (k,t) is nonexponential. In this section, we show that such a correlation holds for the FJ model only at v 3.4, within the low-density and high-density liquid domains, but fails at v > 3.4, within the low-density liquid domain. We first identify the exponential/non-exponential crossover in F s (k,t) for the FJ liquid. We follow the procedure employed in Ref. [2] and focus on the long-time behavior of F s (k 0,t), for t > t 0 = 0.7. This removes the short-time Gaussian dependence characteristic of liq- 11

12 uids [59]. The long-time behavior of F s (k 0,t) can be described by a stretched exponential function, F s (k 0,t) exp [ ( ) ] β(t) t τ(t) (5) where 0 < β(t) 1 and may depend on v. In the case β = 1, Eqn. 5 becomes a simple exponential function. Figs. 10(a)-(c) show F s (k 0,t) for v = 2.2, 3.0 and 3.8 [from Figs. 5(a)-(c)] for t > t 0 and normalized by its value at t 0. The lines in the figure are fits of F s (k 0,t)/F s (k 0,t 0 ) using Eqn. 5. The exponents β(t) corresponding to the F s (k 0,t) shown in Fig. 10(a)-(c) are given in Fig. 10(d)-(f). As expected, we find that (i) β(t) 1 at high temperatures for all volumes studied (exponential relaxation); and (ii) the relaxation becomes non-exponential at low temperatures with β(t) < 1. The exponential/non-exponential relaxation crossover temperature, T β (v), is obtained as shown in Fig. 10(d)-(f). Next, we test whether the crossover between the PEL regimes, indicated by T 0 (v), correlates with the exponential/non-exponential relaxation crossover in F s (k 0,t), indicated by T β (v). T 0 (v) and T β (v) are shown in Fig. 9. For all volumes studied, both crossover temperatures occur within the equilibrium low- and high-density liquid domains. However, T 0 (v) only exists for v < 3.4 while T β (v) is defined for all volumes considered. Accordingly, only at v < 3.4, we find that the PEL influenced/pel independent regime crossover is accompanied by an exponential/non-exponential crossover in the relaxation of the FJ liquid. At v > 3.4, the exponential/non-exponential crossover is still present but crystallization suppresses the PEL-influenced regime. We note that T 0 (v) T β (v) only at v < 2.8. As for the case of T 0, the value of T β (v) is sensitive to the specific definition employed. We tested an alternative fit of the F s (k 0,t) to define T β (v). Specifically, we calculated T β (v) by fitting F s (k 0,t) for F s (k 0,t) < 0.6; this removes the short time relaxation for t < t where t depends on the temperature and volume considered. The truncated scattering function is then normalized by its value at t = t and shifted by t t. We find that, while the exponential/non-exponential crossover remains, T β (v) can shift by , which is considerable. 12

13 C. Diffusivity of the FJ Liquid at low Temperatures In this section, we discuss briefly the diffusivity of the FJ liquid at different volumes and temperatures. Our aim is to test whether the diffusion coefficient D(T) exhibits an Arrhenius/non-Arrhenius crossover as well, and whether such a crossover coincides with the Arrhenius/non-Arrhenius crossover in τ(t). For each state (T,V ), we first calculate the atoms mean-squared displacement < r 2 (t) >. Then, we use the Einstein relation to evaluate the diffusion coefficient, D = lim t r 2 (t) 6t (6) where... denotes the average over all molecules and time origins. The values of D as function of volume for T 0.17 are shown in Fig. 11. Our diffusivities for T < 0.3 are in agreement with the values reported in Fig. 6 of Ref. [33] for the original FJ liquid. However, we note that the present diffusivities at T > 0.3 are larger than those reported there. After revisiting the calculations of < r 2 (t) >, using the original data of Ref. [33], we conclude that the values of D at T 0.3 in Ref. [33] are incorrect [63]. Fig. 12 shows D(T) for the case v = 2.2, 3.0 and 3.8, for all temperatures where crystallization can be avoided. At all volumes studied, D(T) is Arrhenius at high temperatures (approximately, 1/T < 5.0), consistent with the T-dependence of τ(t) (Fig. 6). We also test whether the diffusion coefficient shows an Arrhenius/non-Arrhenius crossover at low-temperature, for v > v c and v < v c, consistent with the corresponding Arrhenius/non- Arrhenius crossover of τ(t) [red arrows in Fig. 6]. Such a dynamic crossover in D(T) can be identified only at small volumes; see, e.g., Fig. 12(a). At v v c [Fig. 12(b)] and v > v c [Fig. 12(c)], the Arrhenius/non-Arrhenius crossover in D(T) is absent or cannot be clearly identified; this is partly due to the smooth variation of D(T) upon cooling. The effects of volume on D(T) seem to be consistent with the effects of volume on τ(t) shown in Fig. 8. Specifically, Figs. 13(a) and 13(b) show D(T) for v 3.0 and v 3.0, respectively. A comparison of Figs. 8 and 13 indicates that at v = 3.0, τ(t) is minimum for a fixed temperature while D(T) is maximum. In other words, as v increases or decreases from v 3.0, τ(t) increases monotonically [Fig. 8] while, accordingly, D(T) decreases monotonically [Fig. 13] 13

14 IV. SUMMARY AND DISCUSSION We studied the dynamics of the FJ liquid in a wide range of temperatures and volumes, including the region where the LLPT and LLCP are located. Our aim was to test whether crossovers in the dynamical properties of the liquid are related with the regions of the PEL explored by the liquid. Such correlations were originally reported in Ref. [2] for a BLJM at ρ = 1.2. For the BLJM at ρ = 1.2 [2], the structural relaxation time evolves from Arrhenius at high-temperatures to non-arrhenius at low temperatures and the self-intermediate scattering function exhibits exponential relaxation at high-temperature and non-exponential relaxation at low-temperature. These two crossovers in the dynamical properties of the BLJM occur approximately at the temperature at which the liquid evolves from the PEL-independent to the PEL-influenced regime. We found that in the case of the FJ liquid, the relationship between the dynamical crossovers and PEL regime crossover does not generally hold; the conclusions depend on the density considered. As summarized in Fig. 9, we find that (i) at 1.8 v 2.6 and 3.2 < v < 3.4 (above and below the LLCP volume v c 2.9), the FJ liquid exhibits an Arrhenius/non-Arrhenius (AnA) crossover in τ(t), an exponential/nonexponential (EnE) crossover in F s (k 0,t), and a PEL-influenced/PEL-independent crossover. Hence, at these volumes, we recover qualitatively the picture that emerges from the BLJM study [2]. However, (ii) at 2.6 < v < 3.2, the LLCP suppresses the AnA crossover and only the EnE and PEL regime crossovers are present. (iii) At large volumes, 3.4 < v 4.1, crystallization intervenes and the PEL-crossover is absent; at these volumes, only the AnA and EnE dynamical crossovers are observed. We also evaluated the diffusion coefficient of the FJ liquid at different temperatures and volumes. It was found that the diffusion coefficient varies smoothly with temperature making it difficult to identify the Arrhenius/non-Arrhenius crossover at large and intermediate volumes. An Arrhenius/non-Arrhenius crossover could be identified only at low volumes, approximately v < 2.6. The relationship between dynamics and PEL regimes in polymorphic liquids is not evident; for example, the presence of a phase separation could induce sudden changes in the PEL properties sampled by the system [64]. While in fragile liquids an Arrhenius/non-Arrhenius dynamic crossover occurs at low temperature, in the case of polymorphic liquids, addi- 14

15 tional dynamic crossover may be found. Specifically, computer simulations indicate that an Arrhenius/non-Arrhenius crossover occurs when crossing the Widom line at T > T c [52, 53]. In these studies, the liquid has a LLPT line in the P-versus-T plane with non-zero slope. In the case of the FJ liquid, the slope of the LLPT line in the P-versus-T plane is very small, close to zero [33]; see Fig. 2. This implies that the Arrhenius/non-Arrhenius dynamic crossovers that we discuss in this work is not related to a Widom line. Specifically, linear scaling theory states that for a polymorphic liquid with zero-sloped LLPT line, (i) the Widom line must coincide with the temperature of maximum density (ρ max -line), and (ii) it must also have zero slope [65]. The ρ max -line in Fig. 2 has indeed zero slope in the proximity of the LLCP, as predicted by linear scaling theory. However, the Arrhenius/non-Arrhenius crossover temperature T τ does not coincide with the ρ max -line in the proximity of the LLCP. Our results also indicate that the Widom line of a polymorphic liquid with zero-sloped LLPT line is not accompanied by an Arrhenius/non-Arrhenius crossover. We also note that linear scaling theory predicts that the isobaric specific heat maxima line can only exist at T < T c (for polymorphic liquids with zero-sloped LLPT line) and, accordingly, such a line could not be observed in the FJ model [33]. The PEL formalism is one of the few approaches, based solely on statistical mechanics, that can be used to characterize liquids and glasses [13]. Hence, our results are important in the understanding of liquids at low temperature and particularly, liquid polymorphism. ACKNOWLEDGMENTS We thank F. Sciortino for fruitful discussions. XLM and SG acknowledge the financial support by the National Basic Research Program of China (973 Program, Grant No. 2015CB856801), the National Key Research and Development Program of China (Grant No.:2016YFA ), and the National Natural Science Foundation of China (NSFC Grant No , ). This work was also supported, in part, by a grant of computer time from the City University of New York High Performance Computing Center under NSF Grants CNS , CNS and ACI We are also grateful for computational resources provided by the supercomputer TianHe-1A in Tianjin, China. 15

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19 TABLE I. Pair Interaction Potential Parameters Used in Eqn. 2 n A 0 A 1 A 2 B 0 B 1 B U(r) Ferrmi-Jagla modified model Lennard-Jones 12-6 potential r/a FIG. 1. Fermi-Jagla pair interaction potential (including a switching function at r c 4.0; see text). The FJ potential is characterized by a hard-core radius r a, a core-softened part at approximately a r b 2a, and a weak attractive part. For comparison, we include a Lennard-Jones pair potential with same minimum depth and location. 19

20 P a v=2.03 v=2.11 crystallization v=2.20 LLCP T τ ρ min D min v=2.28 D max κ max v=3.49 κ max v=4.10 v=4.36 v=2.37 v=2.55 v=2.65 v=2.74 v=2.84 v=2.94 v=3.05 v=3.15 v=3.26 v=3.38 ρ max v=3.60 v=3.72 v=3.84 v=3.98 v=4.49 v=2.46 v= T k B T 0 FIG. 2. Phase diagram of the FJ liquid (adapted with permission from Ref. [33]. Copyright (2011) American Chemical Society). The following lines indicate the existence of water-like anomalous properties: ρ max and ρ min (magenta) are the temperatures of maximum and minimum density (along isobars); D max and D min (blue) are the pressures of maximum and minimum diffusivity (along isotherms); κ max (green) is the temperature of maximum compressibility (along isobars). The T 0 (P) line, at v 3.4, is the onset temperature above (below) which the system is in the PELindependent (PEL-influenced) regime; at v > 3.4, the system is always in the PEL independent regime. Error bars in T 0 (v) are ± The liquid-liquid spinodal lines (violet empty triangles), LLPT line (red line), and LLCP (red circle) are also included. Orange asterisks indicate the lowest temperature at which crystallization could be avoided; the orange line indicates the lowest temperature accessible to the liquid from Ref. [33]. T τ is the low-temperature Arrhenius/non- Arrhenius crossover temperature of τ(t). 20

21 E IS (a) v= T k B E IS (b) v= T k B E IS E IS (c) v= (d) v= T k B T k B E IS E IS (e) v= (f) v= T k B T k B FIG. 3. Inherent structure energy per particle, E IS (T), as a function of temperature along selected isochores; see also Fig. 4. Each data point represents an average over 100 independent IS. At v > 3.4, E IS (T) is practically constant at all accessible temperatures (above the crystallization temperature, see Fig. 2) and the system is in the so-called PEL-independent regime. At v 3.4, E IS (T) is constant only at high temperatures (red dashed lines) while, at low temperatures, E IS (T) decreases upon cooling. The deviation of E IS (T) from its high-temperature value (v 3.4) defines the onset temperature, T 0 (P) (arrows), below which the system is in the so-called PEL-influenced regime. Different IS energy intervals are covered in (a)-(f) because E IS becomes less sensitive to T as v increases. Very large fluctuations in E IS occur at v

22 T 0 v=2.0 E IS Crystallization LLCP v= v= T k B FIG. 4. E IS (T) along isochores (top to bottom) v = 2.0, 2.2, 2.4, Included are the LLCP (red circle) and spinodal lines (violet line) from Fig. 2 and onset temperature T 0 (v) (see Fig. 3). The LLCP and LLPT are located at T < T 0 (v), i.e., in the PEL-influenced domain. Solid and empty symbols correspond to liquid and crystalline states, respectively. 22

23 1 1 F s (k 0,t) (a) v=2.2 F s (k 0,t) (b) v=3.0 T=2.5 T=2.0 T=1.5 T=1.0 T=0.5 T=0.34 T=0.25 T=0.22 T=0.20 T=0.18 T=0.17 T= t (ma 2 ) 1/2 F s (k 0,t) t (ma 2 ) 1/2 T=2.5 T=2.0 T=1.5 T=1.0 T=0.5 T=0.25 T=0.22 T=0.2 T=0.18 T= (c) v= t (ma 2 ) 1/2 FIG. 5. Self-intermediate scattering function, F s (k 0, t), at (a) v = 2.2 < v c, (b) v = 3.0 v c, and (c) v = 3.8 > v c for different temperatures [temperatures in (a) are T = 0.10, 0.11, 0.12, , 0.20, 0.22, 0.25, 0.34, 0.5, 1.0, 1.5, 2.0 and 2.5]. The wave vector k 0 corresponds to the location of the first peak in the structure factor S(k) (k 0 = 4.43, 4.72, 4.36 in (a), (b), and (c), respectively; k 0 is practically independent of temperature). For v = 3.8, the liquid crystallizes at T < 0.17 (see Fig. 2). At v = 3.0 and 2.2, the liquid enters the coexistence region at T 0.18 and 0.11, respectively (square symbols), and crystallization occurs at T = 0.15 and T = 0.09, respectively; see Fig. 2. Dashed lines indicate F s (k 0, t) = 1/e which defines the relaxation time (see Fig. 6). 23

24 τ (ma 2 ) -1/ τ (ma 2 ) -1/ (a) v= ε 0 /(k B T) (b) v= ε 0 /(k B T) 100 τ (ma 2 ) -1/ (c) v= ε 0 /(k B T) FIG. 6. Relaxation time τ(t) at (a) v = 2.2 < v c, (b) v = 3.0 v c, and (c) v = 3.8 > v c obtained from the self-intermediate scattering functions shown in Fig. 5. τ(t) is defined as the time at which F s (k 0, t) = 1/e. Black arrows indicate the temperature at which the system enters the liquid-liquid coexistence region; see Fig. 2. Red arrows indicate approximately the Arrhenius/non-Arrhenius crossover temperature T τ (see Fig. 2). 24

25 (a) 0.35 τ 0 (ma 2 ) 1/ V/(Na 3 ) (b) E A V/(Na 3 ) FIG. 7. Fitting parameters (a) τ 0 and (b) E A in the Arrhenius equation for the relaxation time τ(t) (see Eqn. 4) as a function of volume. τ 0 and E A are obtained by fitting τ(t) using Eqn. 4 for approximately 1/T > 5. Both τ 0 and E A exhibit an anomalous (non-monotonic) behavior with an extremum at v This values are close to the LLCP volume and to the location of the diffusivity maxima line (D max -line in Fig. 2) which extends along the v 2.84 isochore for T <

26 τ (ma 2 ) -1/ (a) v=1.8 v=2.0 v=2.2 v=2.4 v=2.6 v=2.8 v=3.0 τ (ma 2 ) -1/ (b) v=4.1 v=4.0 v=3.8 v=3.6 v=3.49 v=3.4 v=3.2 v= ε 0 /(k B T) ε 0 /(k B T) FIG. 8. Relaxation time τ(t) at (a) v = v c and (b) v = Arrows indicate the temperature at which the liquid enters the LLPT region (2.2 v 3.49). We exclude the temperatures at which crystallization occurs. At low temperatures, the Arrhenius/non-Arrhenius dynamic crossover is evident only at small and large volumes (see, e.g., red arrows in Fig. 6); as v 3.0 (close to the LLCP volume v c 2.9), such a crossover vanishes. Indeed, at v = 3, 0, a low-temperature Arrhenius/non-Arrhenius crossover is very weak and can be identified within the LL coexistence region; see Fig. 6(b). 26

27 T o T k B LLCP T β T τ 0.1 T X V/(N a 3 ) FIG. 9. Relative location in the T v plane of the LLCP (red circle), spinodal lines (blue lines), and crossover temperature lines studied in this work. The exponential/non-exponential crossover temperature T β (v) is obtained as shown in Fig. 10; the Arrhenius/non-Arrhenius crossover temperature T τ (v) is determined as shown in Fig. 6 (red arrows); and the onset temperature T 0 (v) is determined as shown in Fig. 3. The crystallization temperatures (maroon line) are included for comparison. 27

28 F s (k 0,t-t 0 )/F s (k 0,t 0 ) (a) v= t-t 0 (ma 2 ) 1/2 T=0.7 T=0.5 T=0.4 T=0.34 T=0.28 T=0.25 T=0.22 T=0.19 T=0.18 F s (k 0,t-t 0 )/F s (k 0,t 0 ) (b) v= t-t 0 (ma 2 ) 1/2 T=0.7 T=0.5 T=0.4 T=0.34 T=0.28 T=0.25 T=0.22 T=0.19 T=0.18 F s (k,t-t 0 )/F s (k,t 0 ) (c) v= t-t 0 (ma 2 ) 1/2 T=0.7 T=0.5 T=0.4 T=0.34 T=0.28 T=0.25 T=0.22 T=0.19 T=0.18 β (d) v= T k B β (e) v= T k B β (f) v= T k B FIG. 10. (a)(b)(c) Self-intermediate scattering function, F s (k 0, t), at v = 2.2, 3.0 and 3.8 from Fig. 5 shifted by time t 0 = 0.7 and normalized to its value at t 0 to show the long time relaxation behavior. Lines are stretch-exponential fits (see text) with exponent β (0 < β 1). (d)(e)(f) β(t) for the F s (k 0, t) s in (a)-(c). At high temperatures, β(t) 1 indicating that F s (k 0, t) is a simple exponential function. The temperature at which β(t) deviates from 1 (arrows) defines the crossover temperature T β (v). 28

29 D (m ) 0.5 /a V/N*a 3 FIG. 11. Self-diffusion coefficient D(T) as function of volume at (top to bottom) T = 0.50, 0.40, 0.34, 0.28, 0.25, 0.22, 0.20, 0.19, 0.18 and The maxima in D(v) at different T s defines the diffusivity maxima line (D max -line) of Fig

30 10 0 D (m ) 0.5 /a (a) v= ε 0 /(k B T) 10 0 D (m ) 0.5 /a (b) v= ε 0 /(k B T) 10 0 D (m ) 0.5 /a (c) v= ε 0 /(k B T) FIG. 12. Self-diffusion coefficient D(T) as a function of temperature at (a) v = 2.2, (b) v = 3.0, and (c) v = 3.8. D(T) is Arrhenius at high temperatures (dashed-lines). At low temperatures, an Arrhenius/non-Arrhenius crossover occurs as v = 2.2; at larger volumes, such a crossover is difficult to identify or absent. Green arrows signal the lowest temperature at which the equilibrium liquid state can be studied before crystallization or the LLPT intervene. 30

31 10 0 (a) 10 0 (b) D (m ) 0.5 /a v=1.8 v=2.0 v=2.2 v=2.4 v=2.6 v=2.8 v= ε 0 /(k B T) D (m ) 0.5 /a v=4.1 v=4.0 v=3.8 v=3.6 v=3.49 v=3.4 v=3.2 v= ε 0 /(k B T) FIG. 13. Self-diffusion coefficient D(T) at (a) v = v c and (b) v = Arrows indicate the temperature at which the liquid enters the LLPT region (2.2 v 3.49). At high temperatures, D(T) is Arrhenius at all volumes studied. At low temperatures, an Arrhenius/non- Arrhenius crossover is observable at v but it becomes difficult to identify or absent at large volumes (see also Fig. 12). 31