Dynamical crossover in supercritical core-softened fluids

Accepted Manuscript Dynamical Crossover in Supercritical Core-Softened Fluids Eu.A. Gaiduk, Yu.D. Fomin, V.N. Ryzhov, E.N. Tsiok, V.V. Brazhkin PII:

S0378-3812(16)30111-X

DOI:

10.1016/j.fluid.2016.02.046

Reference:

FLUID 11035

To appear in:

Fluid Phase Equilibria

Received Date: 23 October 2015 Revised Date:

26 February 2016

Accepted Date: 29 February 2016

Please cite this article as: E.A. Gaiduk, Y.D. Fomin, V.N. Ryzhov, E.N. Tsiok, V.V. Brazhkin, Dynamical Crossover in Supercritical Core-Softened Fluids, Fluid Phase Equilibria (2016), doi: 10.1016/ j.fluid.2016.02.046. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT Dynamical Crossover in Supercritical Core-Softened Fluids Eu. A. Gaiduk, Yu. D. Fomin, V. N. Ryzhov Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk 108840, Moscow, Russia Moscow Institute of Physics and Technology, Dolgoprudny, Moscow Region 141700, Russia

E. N. Tsiok, V. V. Brazhkin

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Institute for High Pressure Physics, Russian Academy of Sciences, Troitsk , Moscow, 108840 Russia (Dated: February 26, 2016)

PACS numbers: 61.20.Gy, 61.20.Ne, 64.60.Kw

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It is well known that some liquids can demonstrate an anomalous behavior. Interestingly, this behavior can be qualitatively reproduced with simple core-softened isotropic pair-potential systems. Although the anomalous properties of liquids are usually manifested at low and moderate temperatures, it has recently been recognized that many important phenomena can appear in supercritical fluids. However, studies of the supercritical behavior of core-softened fluids have been not yet reported. In this work, we study dynamical crossover in supercritical core-softened systems. The crossover line is calculated from three different criteria, and good agreement between them is observed. It is found that the behavior of the dynamical crossover line of core-softened systems is quite complex due to its quasi-binary nature.

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The most fundamental approach to the behavior of matter involves interactions between the particles of the substance. The most accurate approach is based on quantum mechanical treatment of interactions. However, such abinitio methods require a lot of computational resources and cannot be applied to the systems larger than several hundreds of atoms. A great advance was made by application of so-called effective potentials. These potentials are constructed in such a way that they allow obtaining some principal properties of interest with much smaller efforts [1]. In the simplest case, the interaction is approximated by pair interactions 
only.
One  of the most studied systems σ 12 σ 6 − r is the Lennard-Jones (LJ) system which has the potential U (r) = 4ε r . This system demonstrates a generic view of the phase diagram of a substance containing gas, liquid and crystal phases and well describes the behavior of noble gases and some molecular substances [2]. In our recent works, it was shown that supercritical region of the phase diagram can be divided into two parts: rigid liquid and dense gas [3–5]. These regions differ in the microscopic dynamics of particles and are separated by a crossover line called Frenkel line. The difference in microscopic dynamics leads to a number of consequences. In particular, close to the melting line liquids demonstrate transverse excitations and therefore they have non-zero shear rigidity at frequencies below the threshold one [5, 6]. Basing on this we call this regime rigid liquid. On the other hand far from the melting line even in high pressure and high temperature limit the liquids do not demonstrate shear rigidity and behave like a gas. Therefore a term ’dense gas’ is used to them. Later on, the phenomenon of dynamical crossover was studied for a number of other fluids [7–11]. Great attention of researchers is also focused on the anomalous behavior of liquids (see, e.g., [12] for the list of anomalies of water). Although anomalous properties were first discovered in water, many other liquids also demonstrate an anomalous behavior including liquids of very different natures: silicon [13], liquid silica [14], beryllium fluoride [15–17], some liquid metals [18], polymeric solutions [19], etc. Interestingly, it was found that models with isotropic pair core-softened potentials can demonstrate an anomalous behavior [20–27]. Diffusion, density and structural anomalies are widely discussed in the literature [28]. Such systems can also demonstrate numerous structural phase transitions in the solid region. This kind of behavior cannot be obtained in LJ-like systems. These observations make it possible to suppose that the behavior of the Frenkel line can also be more complex in the systems with core-softened potentials. A particular form of core-softened systems studied in our previous works is characterized by the following interaction potential:  n d U (r)/ε = + λ0 − λ1 th(k1 [r − σ1 ]) + λ2 th(k2 [r − σ2 ]), r

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where d and ε spesify the length and energy scales, respectively, and λi and σi are varied. A large set of parameters was considered. It was found that the phase diagram and anomalous behavior of the system strongly depend on the parameters of the potential . An important feature of this system is its quasi-binary behavior, i.e., some features

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FIG. 1: (Color online) Interaction potentials studied in this work (eq. (1)).

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System no. σ1 σ2 λ0 λ1 λ2 1 1.35 0 0.5 0.5 0 2 1.35 1.8 0.5 0.7 0.2

TABLE I: Parameters of potential (1) used in this study.

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commonly observed in binary mixtures [20]. For example, the system can be easily vitrified [20, 29]. The phase diagram of the system consists of a low density close packed FCC phase, a high density FCC phase and a set of intermediate structures, which can be considered as close packing of particles at a large length scale σ1 , a close packed structure at a small length scale d and a set of different structures in the region where the competition between these length scales takes place. Importantly, the low density FCC phase demonstrates a maximum of the melting line, a phenomena observed for many metals [30], but not in simple systems such as rare gases. The interplay between two length scales obviously affects the properties of the liquid too. Therefore, it is of interest to monitor the phenomena of dynamical crossover in the fluid for such a system. Moreover, the dynamic properties of binary mixtures were widely discussed in the literature and a complex behavior was discovered (see, e.g., [31, 32, 34–37] and references therein). This allows assuming that the quasi-binary system studied in this work can also demonstrate an unusual behavior. The system with potential ( 1) has many unusual properties: a complex phase diagram with a maximum on the melting line and many different solid phases, anomalous density, diffusivity and structure, etc. For this reason, we believe that investigation of the dynamical crossover in this system will be helpful for understanding of the behavior of supercritical water and other anomalous fluids. Two sets of the parameters of the potential (1) were studied: a purely repulsive system with the shoulder width σ1 /d = 1.35 and a system with a repulsive shoulder and an attractive well. The parameters of the potentials are given in Table I. The exponent in the first term of the potential is n = 14 for both cases. Below, we refer to the systems with different parameters as ”system 1” and ”system 2”. The potentials are shown in Fig. 1. We measure all properties of the system in reduced units with respect to d and ε: ρ˜ = N/V · d3 , T˜ = kB T /ε, etc. Since only these reduced units are used in this work, we omit the tilde marks. In all cases, systems of 1000 particles in a cubic box with periodic boundaries were simulated by a molecular dynamics method. The timestep was set to 0.0005 reduced units of time. The system was equilibrated by 3.5 · 105 steps followed by more 1.5 · 105 steps for data collection. During the equilibration run, the temperature was held constant by a Nose–Hoover thermostat. The production run was made in the N V E ensemble. All simulations were performed with the LAMMPS simulation package [38]. Several criteria can be used to determine the dynamical crossover line [3–5]. The velocity autocorrelation function (VACF) criterion and the isochoric heat capacity criterion [5] are among the most convenient criteria. P 1 i (0) The velocity autocorrelation function is defined as Z(t) = 3N h ViV(t)V i where Vi (t) is the velocity of the i-th 2 i (0) particle at the time t. The VACFs of rigid fluids demonstrate an oscillatory behavior, whereas the VACFs of dense

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gases decease monotonically. Therefore, the Frenkel line corresponds to the (ρ, T ) points where oscillations disappear [5]. The isochoric heat capacity criterion states that the Frenkel line is a line at which the isochoric heat capacity per particle is cv = 2kB . It is based on the counting inclusion of contributions to the heat capacity of the liquid from longitudinal and transverse excitations. A detailed theory of the heat capacity of the liquid based on excitation spectra was proposed in recent works [39–41]. One can show that the contribution to the heat capacity from the potential energy of transverse modes in the rigid regime is 1 · kB per particle. At the Frenkel line, transverse excitations disappear and, therefore, the heat capacity per particle at theFrenkel line should be close to 2kB . The crossover between different regimes of the fluid can also be identified by the appearance of strong positive sound dispersion (PSD). Positive sound dispersion means that the velocity of excitations in liquids at a certain finite wavelength k exceeds the adiabatic speed of sound cs . Positive sound dispersion was experimentally observed in a number of low-temperature fluids (see, e.g., [42–44] and references therein). As the temperature increases, PSD disappears. Previously, we considered PSD in Lennard-Jones and soft-sphere systems [5] and found that the temperature of disappearance of PSD is consistent in both cases with the Frenkel temperature obtained from the VACF and cv = 2kB criteria. The disappearance of PSD is indeed due to changes in the excitation spectrum, which take place at the Frenkel line [5]. Therefore, it is reasonable to relate the disappearance of PSD to the crossing of the Frenkel line. To test the presence or absence of PSD, we compute the longitudinal autocorrelation function of the velocity current function:

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After the calculation of CL (k, t), we evaluated its Fourier transform C˜L (k, ω). The frequency of the excitations at the wave vector k is given by the position of the maximum of C˜L (k, ω). The adiabatic speed of sound is calculated directly from simulations.

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In theis work, we calculate the Frenkel line from the VACF, cv = 2kB and PSD criteria for system 1 and from the VACF and cv = 2kB criteria for system 2. Let us consider the Frenkel line of system 1. We start with the calculation of the points where cv = 2kB . The positions of these points in the phase diagram are shown in Fig. 2. Further, we calculate the VACFs of system 1. Fig. 3 shows the evolution of VACFs of the system at the density ρ = 0.65 with the temperature. One can see strong oscillations at the lowest temperature. However, these oscillations decrease rapidly with an increase in the temperature and disappear at T = 0.61. Interestingly, with a further increase in the temperature, the oscillations of VACFs appear again and disappear for the second time at T = 1.45. The points where the oscillations of VACFs disappear are shown in Fig. 2. This kind of a reentrant oscillation behavior of VACF is reported for the first time. It can be attributed to the quasi-binary nature of the system [20]. The potential of system 1 contains two length scales: d and σ1 . The behaviors of such systems at low and high temperatures are qualitatively different [20]. Since the potential is purely repulsive, it is energetically favorable for particles to stay far from each other. Therefore, the behavior of the system at low temperatures and densities is determined by the parameter σ1 . If the density and temperature increase, particles penetrate through the soft core and the behavior of the system is more likely determined by the small length scale d. Apparently, there is an intermediate region where the effects of both length scales are comparable and, therefore, competition between local structures occurs. This quasi-binary nature is obviously responsible for the complex behavior of the VACFs of the system. Fig. 4 shows the dispersion curves for system 1 at the density ρ = 0.8 and two temperatures, (a) below and (b) above the Frenkel line, obtained by the VACFs and heat capacity criteria. At low temperature T = 0.4, clear positive dispersion is observed. If the temperature is increased, the excitation frequencies approach the line cs · k. Finally, the PSD disappears. However, it is rather difficult to identify the point of disappearance of PSD because it becomes close to the level of computational errors.

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FIG. 2: (Color online) Phase diagram of system 1 with the Frenkel line obtained by three criteria (see main text). The solid part of the phase diagrams contains numerous phases: face centered cubic (FCC), face centered orthorhombic (c/a = 1.6, FCO), simple cubic (SC) and simple hexagonal (SH).

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Similar to the case of VACFs, we observe some kind of the reversible behavior of PSD. There is a density region where, if the fluid is heated isochorically, PSD disappears, then appears again and disappears for the second time at higher temperatures. At the density ρ = 0.8, the temperatures of the first and second losses of PSD are T1 = 1.6 and T2 = 2.2, respectively. Indeed, the errors in the calculations of PSD are so large that the second appearance of PSD can be attributed to computational uncertainties. However, taking into account a complex behavior of VACFs, one can suppose that PSD also demonstrates a complex behavior. Moreover, one can note that two branches of the disappearance of PSD at low densities correspond to two effects on VACFs. The lower temperature branch is consistent with the disappearance of oscillations of VACFs. Apparently at the temperatures of the second branch of the disappearance of PSD, VACFs are monotonic both at lower and higher temperatures, however, a non-monotonic behavior of the intensity of VACFs is observed, i.e., the VACF curve for T = 2.8 is below that for T = 3.0 but above the curve for T = 3.2 (see Fig. 5). The Frenkel lines of system 1 obtained from all three criteria are summarized in Fig. 2. One can see that at the high density regime (ρ > 0.75) all three curves are in good agreement. The same match was found for Lennard-Jones

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FIG. 4: (Color online) Spectra of longitudinal excitations in system 1 at ρ = 0.8 and T = (a) 0.4 and (b) 2.6. The straight lines are cs · k, where cs is the adiabatic speed of sound.

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n in our previous work [5]. The high density regime corresponds to the major role of the d length scale, i.e., dr term of the potentaial. Therefore, this result seems to be consistent with our previous works. The situation at low densities is more complicated. One can see that the curves obtained from the cV = 2kB criterion, the first disappearance of PSD and the disappearance of oscillations of VACFs are close to each other and can be considered as matching. However, the second temperature of the disappearance of PSD is much higher than all these lines. Moreover, in the region where VACFs demonstrate two points of disappearance of oscillations, the second branch of the lines obtained from the VACF criterion is below the common curve from three criteria. The above discussion shows that the behavior of the Frenkel line in a purely repulsive core-softened system becomes complicated. It is interesting to analyze the behavior of the Frenkel line for systems with both repulsive and attractive parts of the potential. System 2 is an example of a core-softened system with both repulsive and attractive forces. The phase diagram of this system was reported in our previous work [22]. Similar to the case of the purely repulsive system, the phase diagram contains two FCC phases, with low and high densities, and a simple hexagonal phase. However, other crystalline phases are absent in this system. The low-density FCC phase demonstrates a maximum on the melting curve. In view of the similarities of the phase diagrams of the purely repulsive system and the system

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with attraction, we expect to find a complex behavior of the Frenkel line in system 2. The Frenkel lines of system 2 obtained from the VACF and cv = 2kB criteria are shown in Fig. 6. One can see that, similar to system 1, the lines found from the VACF criterion are split into two branches. At low densities, the upper branch of the line obtained from the VACF criterion is in good agreement with the line found from the cv criterion. At higher densities, all lines merge. One can conclude from this picture that the attractive part of the potential does not change the qualitative behavior of the Frenkel line.

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In conclusion, we have studied in detail the dynamical crossover line in the supercritical regime of core-softened fluids. We have used the VACF, cV and PSD criteria and have found good agreement between them. At low densities, the Frenkel line is flat: the temperature is almost independent of the density, while it increases rapidly at higher densities. A similar behavior was recently observed for the Frenkel line of water [9]. In addition, we have observed that the crossover lines obtained from the VACFs and PSD criteria are split into two pseudo branches at low densities. The appearance of the additional branches of the Frenkel line is not observed in simple systems and is caused by the quasi-binary nature of the core-softened fluids.

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Eu.A.G. thanks the Joint Supercomputer Center of the Russian Academy of Sciences. Yu.D.F. is grateful to the Russian Scientific Center at the National Research Centre Kurchatov Institute for computational facilities. This work was supported by the Russian Science Foundation (grant no. 14-22-00093).

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