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of the fluid evolution into gas.
refer to as “hydrodynamic anomalies”.
lines are the dynamical crossover lines suggested in [26, 27].
anomaly in a2 (T, ρ), see Fig. 4d.
up. Verification of the subleading power laws and understanding of an>1 behavior is even more challenging task.
these ranges widely cover the region corresponding to both liquid and fluid states. In the remainder of this paper we use the dimensionless ˜ = U/ε. As we will only use these reduced variables. The straight dashed lines represent a1 t−3/2 asymptotics with appropriate amplitude a1 . 2 h|v(0)| i 2 where h|v(0)| i = 3kB T /m. Calculating VAF of the system with high accuracy we show that amplitude of VAF long-time tails demonstrates non trivial temperature and density dependence (hydrodynamic anomalies). Note. temperature and density. At each thermodynamic point. . we calculated it by using formula (1). Long-time VAF behavior In Fig. and σ is the core diameter. III.84 T = 0.7. SIMUALTION DETAILS To avoid the finite size effects at hydrodynamic scales one should consider large enough volume. see Fig. The result is : a1 (T. It was also shown that the insufficient volume results in not only strong increase of the calculation noise in the correlation function but also on the sudden suppression of t−α1 hydrodynamic tail at time scales where Vh (t) > V . So the simulation volume V should be much larger that Vh (th ). In Fig. we introduce typical Z(t) curves at triple point density ρ = 0. We see that there are clear maxima on a1 (T ) curves in the density region ρ ≥ 0. It is clear from the picture that long-time VAF behavior is well defined by power function a1 t−3/2 which is straight line in logarithmic scale used. U (r) = 4ε[(σ/r)12 − (σ/r)6 ]. This line determines the states corresponding to anomaly of hydrodynamic VAF tails along isochors and so we will refer it to as isochoric hydrodynamic anomalies line.2 0. ρ) = 2 −3/2 [4π(D + ν)] . The basic fluid model used is the one component Lennard-Jones pair potential model. First. where th is the characteristic time when the dynamic correlation functions start exhibiting the hydrodynamic asymptotic scaling. where m and V are the molecular mass and system volume correspondingly. The volume of fluid “disturbed” by the given moving particle grows as Vh (t) ∼ [tν]d/2 . and time t˜ = t/[σ m/ε].84 and different temperatures. 3ρ (1) In order to verify our estimation of tail amplitude a1 . density ρ˜ ≡ N σ /V . It was recently shown in two-dimensional hard disk system at small and intermediate densities that reliable calculation of VAF hydrodynamic tails in fluid requires at least 642 particles.7 − 1.2. II. The theoretical approach based on hydrodynamic approximation makes it possible to estimate VAF tail amplitude a1 . We also see that the tail amplitude a1 depends on the temperature and its change exceeds the error of VAF determination essentially. the minimum volume Vh (th ) is not universal: it depends on temperature and density. where ε is the unit of energy. According to equilibrium temperature-density phase diagram [32– 34]. U p 3 ˜ T = T /ε. temperature quantities for LJ fluid: r˜ = r/σ.7 T = 1. Simulating LJ particle system we apply pair potential in the standard form. that we will always use VAF normalized on its initial value hv(0) · v(t)i Z(t) = .1 T = 2 VAF 1E-3 1E-4 1 time 10 FIG. Even these amount of particles required terabytes of operational memory to process the particle trajectories and calculate VAF at satisfactory A. we discuss long-time properties.01 = 0. moreover. The line joining these maxima on T − ρ plain is drawn in Fig. 2. where ν is the kinematic viscosity.0) at temperatures within the ranges from melting (or liquid-gas) line up to T = 40. changes qualitatively with temperature. we calculate VAF with high accuracy and analyze its behavior at intermediate and long time scales.3a the a1 (T ) dependencies at different densities are shown. (Color online) Z(t) at ρ = 0.2 in Ref. see Fig. To optimize the calculation speed and the operational memory we developed the unique parallel algorithm that was built in the DL POLY Molecular Simulation Package [29–31]. We show that at intermediate time scales behavior VAF in fluid is out of the frameworks of linearised Navier-Stocks hydrodynamics and. We have considered the system of N = 1283 ≈ 2 × 106 particles that were simulated under periodic boundary conditions in 3-dimensional cube in the Nose-Hover (NVT) ensemble. RESULTS We investigate LJ system in the density range of ρ ∈ (0. 1. we omit the tildes.84 and different temperature. We used the data of . Generally. accuracy. 1.
It is well established for simple liquids that the decay of the velocity autocorrelation function at large time scales qualitatively is well described by the integral relation : Z d3 k Zh (t.0 2 0 ry o ry ry o ry 0 . 4b we show the fit of (2) to VAF calculated using molecular dynamics. the long-time t−3/2 behavior of VAF is “universal”. Expression (1) appeared to be very accurate. investigated in detail below. then one can get only the leading asymptotic scaling. 1). ρ) at ρ ∈ (0. 0) ∼ a1 t−3/2 .7 5 M D 0 . That allowed us to extrapolate a1 in the low-density region where VAF was not actually calculated within MD.0 0 8 a a ) 0 .8 th e o 0 . r0 ) ∼ exp (−Ak 2 t) . ρ normalised on the corresponding a1 (T. (Color online) Dependence of VAF long-time tail amplitude a1 for LJ fluid on temperature (a) and density (b). Here we keep finite r0 that helps us to have progress in analytical description of VAF at intermediate time scales. To demonstrate it. 3.0 1 5 1 0 . the values of D were additionally calculated by Green-Kubo relation to control accuracy. Typically r0 is taken to be zero. In Fig. In Fig. where Th is certain characteristic temperature scale.0 0 0 0 . However the intermediate-time properties of VAF “nonuniversally” depend on both the temperature and the density: at intermediate time scales: at T = Th (Th ≈ 10 for ρ = 1) the sign of VAF deviation from t−3/2 changes. Meier [35. Intermediate-time VAF behavior It follows from our calculations. ∼ a1 t−3/2 with a1 defined by (1). It was found that a1 demonstrates areas with non-monotonic behavior along isotherms. B.0 0 4 1 M D 1 th e o ry 0 . The cutoff k0 = 2π/r0 is related to the break up of long-wave length fluctuational hydrodynamics at nanoscales ∼ r0 . Note that this anomaly can be also detected within the mode-coupling theory and it was seen in simulations of the lattice gas .0 0 6 0 .7 th e o b ) T = 6 T = 4 T = 3 T = 2 . r0 ) ≤ Zh (t. T ) we can describe VAF at long-time scales as well as at intermediate ones. However this description breaks up at T ≈ Th .0 0 d e n s ity FIG. 4a).0 2 5 ρ= ρ= ρ= ρ= ρ= ρ= ρ= ρ= ρ= ρ= a 1 0 .2 5 0 . which is the locus of a1 extrema obtained at isothermal conditions (see Fig. 3) that provides the self-consistency of our results.5 T = 1 .8 5 M D 0 . and l is the average distance between molecules. 6). we introduce “master curve” which is a set of VAFs at different T. It should be noted that isochoric and isothermal hydrodynamic anomaly lines are located in essentially different areas of phase diagram and correspond to different physics (see discussion in Section IV).0 0 0 .0 0 5 0 .5. Insert in Fig.5 0 . 1).9 th e o 0 . A deeper inside into nature of crossover temperature Th is afforded by considering the analytical expressions describing VAF at long times. 36].7 5 th e 0 . So keeping finite r0 should always cause negative deviation of VAF from long-time asymptotics.0 0 2 0 . T ∈ (1.0 1 0 ry 0 . We see universal asymptotics at t → ∞.5 0 0 . The calculated tail amplitudes are in excellent agreement with those estimated from VAF directly (see Fig. 4b shows that r0 (and so the negative deviation degree) decreases with temperature up to Th where r0 is of the order of l. 36] to get values of kinematic shear viscosity and self-diffusion coefficient. We see that adjusting the cutoff r0 (ρ.3 0 .7 5 1 . The bullets represent result of direct estimation from VAF obtained by molecular dynamic simulation and the stars are the result of calculation by using formula 1. ρ) (see typical picture for ρ = 1 in Fig. 3b we show a1 (ρ) curves at different temperatures which reveal clear maxima. Here A is estimated like A ∼ ν + D. It follows from (2) that Zh (t.0 0 0 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 te m p e ra tu re 0 . (2) (2π)3 k<2π/r0 originating from long-wave length fluctuational hydrodynamics.7 M D 0 . where D is the self-diffusion coefficient and ν is the kinematic viscosity.02. Using the Meier data [35.8 5 th e 0 . At T > Th the fit (2) strongly contradicts . So there is one more hydrodynamic anomalies line. the isothermal one. we calculated a1 (T.
ρ= 1 0 . T = 20.1 ~ t -7 h c p p e r c o la tio n c lu s te r 1 .1 1 1 t im e 0 . r c ro s s o v e r te m p e ra tu re a 1(T ) m a x c ro s s o v e r 5 0 ρ= 1 . ρ= 0 .9 1 . dashed and dot-dashed straight lines are the subleasing term a2 t−7/4 and leading asymptotics a1 t−3/2 respectively. (3) n>0 The leading term of this expansion at n = 1 is well known a1 t−3/2 and next subleasing one is a2 t−7/4 . It was shown within the frames of both the modecoupling theory  and the Enskog expansion  that non-analytic dispersion relations for hydrodynamic frequencies may take place at intermediate time and spatial scales.1 1 0 0 T = 4 0 T = 1 1 1 0 T = 8 0 6 4 0 . T = 2 0 ~ t -3 /2 0 . ρ= 0 . 4. 8 4 0 . Moreover this series diverges at t → ∞.1 d ) tc t im e 1 0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 te m p e ra tu re FIG.0 0 2 1 E -3 c ) 0 .7 0 . (c) Solid line represents VAF at rho = 1.4 0 . (b) Solid line represents VAF at ρ = 1 T = 4. we note that the expansion (3) is approximate one. (Color online) (a) Master curve of VAFs normalised on corresponding a1 (T ) for ρ = 1.0 d e n s it y V A F V A F n o r m a liz e d T = 1 7 T 8 h y d r o d y n a m ic s c a le . That leads to infinite asymptotic expansion for time correlation functions. the simulation results at intermediate time scales: even the sign of the deviation (positive) cannot be reproduced.6 0 . straight line is the t−3/2 asymptotics. The arrow shows the time tc corresponding to crossover to Navier-Stokes hydrodynamics regime. 4d. This contradiction is caused by the fact that at T > Th Eq. 8 4 a 2 .1 0 . we see that . dashed line is the integral (1) with adjusted cutoff r0 .8 ρ= 1 . (d) Temperature dependencies of amplitude a2 of subleasing term corresponding to n = 2 in the expansion (3) in common with the same dependencies for leading term amplitude a1 . ρ= 1 0 . The dependence of this time on temperature at ρ = 1 is shown in the insert. 7 a 2 . 7 0 . So it is natural to explain the positive deviation of VAF from t−3/2 at intermediate time scales by the influence of these terms.0 1 2 4 0 3 0 2 0 1 0 0 T = 5 h 0 .2 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0 V A F te m p e ra tu re a 1 . The snapshots demonstrate typical spatial distribution of atoms with hcp and tetrahedral local order at temperatures corresponding to their percolation thresholds.0 0 8 a 2 . ρ= 0 .0 0 6 a 1 .8 0 .0 1 0 . In Fig. for VAF that expansion has the form Z(t) ∼ X n an t1/2 −2 . The insert displays the dependence of crossover temperature on density. So we understand these series as just a hint that the term which behaves like ∝ t−7/4 may dominate at intermediate time scales. The arrow shows VAF at crossover temperature Th at which the deviation of VAF from leading asymptotics changes its sign. (2) based on purely Navier-Stokes hydrodynamics fails at intermediate time scales and so we have to use more rigorous expressions arising from generalized hydrodynamics. T = 4 1 0 1 2 3 4 5 6 te m p e ra tu re 7 8 T = 4 1 E -3 T = 3 1 a ) b ) 1 E -4 0 .4 0 . In this connection.0 /4 1 0 t im e ρ= 1 a 1 .0 0 4 V A F t e t r a h e d r a p e r c o la t io n c lu s te r ta il c o e ffic ie n ts tc 0 . ρ= 0 . Particularly.
(Color online) The dependencies of self-diffusion coefficient D and kinematic shear viscosity ν on temperature (a) and density (b). The insert shows difference between density dependence of dynamic and kinematic viscosity. IV. Moreover. 5. the additional study is needed to understand the physics of that effect completely but it is a matter for separate work. we see that these temperatures have almost the same density dependencies. Refs. ρ) and ν(T.0 d e n s it y te m p e ra tu re FIG. ρ= 0 . tail amplitude behaviour is mostly defined by both the self-diffusion coefficient and kinematic shear viscosity. we can estimate its amplitude a2 . ρ= 0 . Moreover. 8 . In the insert of Fig. This fact suggests the underlined physical reasons which cause the isochoric hydrodynamic anomaly and Th crossover are coupled. To understand this effect we should . ρ= 0 . 8 .5 = 2 . The question arises that physics stands behind that behaviour? It is obvious that hydrodynamic anomalies revealed are related to some fundamental changes of fluid collective particle motion. we show these dependencies for the areas of temperature-density phase diagram corresponding to the location of hydrodynamic anomalies.5 6 ρ= 1 . ν D . [35.0 0 . 9 5 ρ= 0 . 5. So. It is important that crossover temperatures Th obtained at different densities are in close correlation with temperature of isochoric hydrodynamic anomaly.3 0 . Of course. The very fact that temperature dependence of viscosity develops a minimum at high enough temperature is well known for a long time. This crossover time has predictable temperature dependence – it increases with T as we see in the insert of Fig.5 1 .1 0 . In Fig. 8 5 ρ= 0 . T D . 0 . ρ) dependencies. T ν. DISCUSSION In previous section we showed that amplitudes of VAF long-time tails demonstrate non monotonic behaviour which is expressed by the existence of isochoric and isothermal hydrodynamic anomalies lines. According to this formula. T ν. 8 5 ρ= 0 . In Fig. T ν. 5 ρ= 0 . We see that at high density region where isochoric anomaly line takes place the self diffusion coefficient is much smaller than the kinematic viscosity and so maxima on a1 (T ) dependencies are mostly induced by minima on ν(T ) curves. 4c for explanation). A. as can be seen from Fig. T ν. Hydrodynamic anomalies and viscosity We see in Fig. ν 3 2 0 . The t−3/2 behavior takes place at larger time scales and so we have a crossover to Navier-Stokes hydrodynamics characterized by a crossover time tc (see Fig.4 ρ= 0 . 9 .g. the deviation of intermediate-time VAF from main t−3/2 asymptotics changes its sign at crossover temperature Th which correlates to temperature of isochoric anomaly. 9 . the part of the VAF corresponding to intermediate time scales is well fitted by this term at T > Th . 7 5 D .1 a ) b ) 2 4 6 0 . the typical temperature dependence a2 (T ) is shown in comparison with a1 (T ). We see that a2 (T ) is non-monotonic and so hydrodynamic anomaly also takes place. 3 that a1 are well described by (1).5 = 2 . 2. 7 5 1 = 1 . The data were taken from [35. T D . 38] and references therein. 4c. adjusting the VAF by this term as pictured in Fig. Thus our results reveal that intermediate-time VAF behavior at high enough (T > Th ) temperatures can be satisfactory described by a2 t−7/4 term. 4d. T 1 0 4 ν D .5 = 4 = 6 = 1 .2 0 . see e.5 ρ= 1 . The concurrent D(T ) contribution only shifts maxima of a1 to smaller temperatures. the temperatures at which subleading terms are firstly detected correlate with those corresponding to a1 (T ) maxima. ρ= 0 . 9 5 ρ= 0 . Here we discuss these issues in more details. 5a) and investigate the nature of the isochoric hydrodynamic anomaly.5 = 4 = 6 D 0 . So one should look for physical explanation of the hydrodynamic anomalies in D(T. T D . First we focus on temperature dependencies of ν(T ) and D(T ) (Fig. 36]. 4d. 4c.
But. 1). 5b). 1. one can easily find using the Boltsman equation within the τ -approximation that η = P τ (for onecomponent gas). This is so for the local structure evolution. This quantity is proportional to dynamic viscosity (resistance to shear stress deformation) and inverse to the density which promotes flowing. CONCLUSIONS In conclusion. That structure demonstrates complex multistage evolution that changes with temperature and density. At very low densities. Pressure P increases with temperature like ρT . quantitatively these lines are located far away from each other (see tau Frenkel line in Fig. Taking the Lennard-Jones fluid as an exemplary system we show that long-time and intermediate-time behavior of VAF changes essentially depending on the temperature and density.6 consider typical temperature dependencies for dense fluids and gases. The simplest model to show it is based on the assumption that the fluid flow obeys the Arrhenius equation for molecular kinetics. This “VAF Frenkel line” joins state points where oscillations of VAF disappear. p. That gives exponential Arrhenius law η ∼ exp(−E/T ) which works well for dense simple liquids and even for many liquid metals and alloys. at intermediate densities. the grow of η dominates and so the ν starts to increase developing the minimum. surprisingly. Later. It should be noted that minimum on ν(ρ) curve is the spatial characteristic of kinematic viscosity. distinct liquid and gas phases do not exist – this is some intermediate state. B. 39]. Chapter 7. shear viscosity decreases with temperature. The wide spread of the hydrodynamic anomaly lines over the phase diagram. it was recently found for simple fluids  that the solid-like close packed (hcp. see Fig.283. These clusters may in turn connect with each other and form the percolation cluster. we investigate fitches of dynamical correlation function at long-time scales. ν has very large value – dilute gas hardly flows due to gravity. In particular. The locations of these lines are also different (see VAF Frenkel line in Fig. In gases. Our working hypothesis that the local order and the (hydro)dynamical lines are somehow coupled with each other. see Fig. The decrease of ν in low density region is mostly due to the ρ increase as far as η is small. Hydrodynamic anomalies and “dynamical” line We must note that crossover from liquid-like to gaslike behaviour expressed by minimum on ν(T ) (or η(T )) has already been discussed in the context of so called dynamical (“Frenkel”) line [26. similar argu- ments have been used in the textbook . 3). We see that the self-diffusion coefficient decreases with density monotonously but viscosity has minimum. that this criterion deals with VAF features at small times and is quite different from hydrodynamic anomaly line reflecting the long-time behaviour. It was written that minimum viscosity locus should coincide with dynamical crossover line defined by the criterion. we should emphasize however that these hydrodynamical lines should not be associated to some “phase transition” or “narrow crossover”. This minimum stands behind maximum on a1 (ρ). V. Hydrodynamic anomalies and local structure evolution It is well known that in liquids and fluids there is no long-range order but there is local order represented by molecule clusters with certain symmetries. Note that this anomaly takes place at low density region. But. C. 4d. 1). It should be stressed. τ ≈ τ0 where τ is the liquid relaxation time and τ0 is the minimal period of transverse quasiharmonic waves. Finally. Despite of qualitative soundness of the arguments presented. Now. and the relatively large width of the hydrodynamic anomalies in a1 and a2 are another arguments about the continuous multistage nature of fluid evolution to gas. In liquids and dense fluids. the same authors proposed another criterium for dynamical crossover line based on features of VAF at atomic time scales . Fluid kinematic viscosity can be qualitatively identified with the resistance to flow and shear under the force of gravity. In supercritical fluid. the dynamic viscosity increases with density monotonously (see the insert in Fig. the fragments of the close packed clusters – tetrahedra – survive at temperatures up to the several orders of magnitude higher than the melting temperature (see color gradient in Fig. So it is rather obvious that there the liquid-like decrease of τ with temperature turns into the increase of η with temperature like in gases. So η increases with temperature in gases at fixed ρ. Two lines on temperature-density plain correspond to maxima of a1 : isochoric and isothermal . we turn to the density dependencies of ν(ρ) and D(ρ) (Fig. however. That is why the kinematic viscosity diverges at ρ → 0 whereas dynamic viscosity tends to zero. We can approximately locate areas of the phase diagram where the percolation clusters formed by the mentioned local order elements disappear . Thus we see that isothermal hydrodynamic anomaly reflects crossover from dilute gas-like to fluid-like behavior in respect to flowing features. In particular. It is well known that the evolution of the supercritical fluid towards the ideal gas is the continuous multistage process  where one liquid properties disappear earlier than others. while τ decreases like p l/ kB T /m. 1 to estimate tetrahedra fractions at different temperatures and densities). 5b). the amplitude a1 of leading longtime asymptotics a1 t−3/2 has non-monotonous behaviour (see Fig.fcc) clusters (and so their percolation cluster) disappears at temperatures several times higher the melting temperature.
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