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Pressure-energy correlations in liquids. III. Statistical mechanics and thermodynamics of liquids with hidden scale invariance
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Figures

Image of FIG. 1.

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FIG. 1.

(a) Scatter plot of total virial and potential energy (in LJ units) for the standard single-component LJ liquid at (argon units) and near-zero pressure, simulated at constant volume (density , argon units, left panel) and constant pressure (, argon units, right panel). (b) Scatter plot of single-particle virial and potential energy for the same simulation as in the left panel of (a). The single-particle correlation is much weaker, , showing that collective effects are crucial for the correlation.

Image of FIG. 2.

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FIG. 2.

Comparison of for simulations using the LJ potential and two IPL potentials: the repulsive term in and the IPL potential that optimizes the agreement in the fluctuations of potential energy and virial by minimizing Eq. (36). The left panel shows these at a density of 0.82 and a temperature of 0.67 (LJ units), the right one at a density of 0.90 and a temperature of 0.80 (where the potential leads to crystallization).

Image of FIG. 3.

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FIG. 3.

(a) Illustration of the difference between the LJ potential , the empirically matched IPL potential with and , and their difference . (b) Linear fit, , to between and , and the remainder (full black curve).

Image of FIG. 4.

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FIG. 4.

Effect on fixed-volume fluctuations of adding a linear term to the IPL potential. The linear term is that shown in Fig. 3(b). Configurations were generated by an simulation using the LJ potential, and the different determinations of energy (LJ, IPL, and eIPL) and virial were computed on these configurations. The dashed lines indicate a perfect match. Including the linear term when computing the energy improves the match to the true (LJ) fluctuations (the correlation coefficient goes from 0.950 to 0.970), while it reduces the match to the virial (the correlation coefficient goes from 0.987 to 0.971, which is probably related to the fact that the pair virial is discontinuous at —we find that smoothing the linear part around restores the match somewhat). The insets show the pair potentials and virials: brown dashed lines, LJ; black lines, IPL; and red lines, eIPL. The overall conclusion from Fig. 4 is that the addition of the linear term induces little change in the fluctuations.

Image of FIG. 5.

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FIG. 5.

Comparison of potential energy calculated using , (black points), and (red points) for configurations drawn from an simulation using the LJ potential for the same state as in Fig. 4. The potential energy, in particular, is very poorly represented by the inverse power-law contribution when the volume is allowed to fluctuate. In fact, the correlation between and is not only weak, it is negative. Including the linear term makes a huge difference here, yielding a correlation coefficient of 0.977 between and (changed from ). The slope is somewhat less than unity, indicating that there are significant contributions from pair distances beyond (i.e., from the “rest” part of the potential). The linear term affects the virial fluctuations much less presumably because the derivative of the potential is dominated by the IPL term.

Image of FIG. 6.

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FIG. 6.

Computer simulations of virial and potential energy during the aging of two strongly correlating molecular liquids following temperature down-jumps at constant volume ( simulations). (a) The asymmetric dumbbell model at a density of . The liquid was first equilibrated at . Here, simultaneous values of virial and potential energy are plotted, producing the green ellipse, the elongation of which directly reflects the strong correlation. Temperature was then changed to where the red ellipse marks the equilibrium fluctuations. The aging process itself is given by the blue points. These points follow the line defined by the two equilibrium simulations, showing that virial and potential energy correlate also out of equilibrium. (b) Similar temperature down-jump simulation of the LW OTP system (Ref. 30). Again, green marks the high-temperature equilibrium , red the low-temperature equilibrium , and blue the aging toward equilibrium. In both (a) and (b), the slope of the dashed line is not precisely the number of Eq. (6) because the liquids are not perfectly correlating; the line slope is (see Paper I, Appendix B), a number that is close to whenever the liquid is strongly correlating. (c) Virial and potential energy for the asymmetric dumbbell model as functions of time after the temperature jump of (a); in the lower subfigure, data were averaged over . Virial and potential energy clearly correlate closely, both on short and long time scales.

Image of FIG. 7.

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FIG. 7.

Virial vs potential energy after a temperature down-jump at constant volume applied to SPC water, which is not strongly correlating (colors as in Fig. 6). (a) SPC water at equilibrated at , subsequently subjected to an isochoric temperature down jump to . Clearly, and are not strongly correlated during the aging process. (b) Same procedure starting from a state point.

Image of FIG. 8.

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FIG. 8.

Crystallization of the supercooled LW OTP liquid where each molecule consists of three LJ spheres with fixed bond lengths and angles (Ref. 30). (a) Pressure (right) and energy (left) monitored as functions of time during crystallization at constant volume. Both quantities were averaged over ; on this time scale the pressure/energy fluctuations directly reflect the virial/potential energy fluctuations (Paper II). The horizontal dashed lines indicate the liquid (upper line) and crystal (lower line), the averages of which were obtained from the simulation by averaging over times 0–2 and , respectively. Both liquid and crystal show strong correlations, and the correlations are also present during the crystallization. Inset: crystal structure from the simulation. (b) Radial distribution functions of liquid and crystalline phases. The two spikes present in both phases come from the fixed bond lengths.

Image of FIG. 9.

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FIG. 9.

plot for the asymmetric dumbbell model for various states at the same volume. The upper right corner shows data for simultaneous values of virial and potential energy for four equilibrium simulations . When quenching each of these to zero temperature in order to identify the inherent states, the crosses are arrived at. The intermediate points are glasses prepared by different cooling rates: out-of-equilibrium systems generated by cooling in from to the temperature in question. This plot confirms the findings of Figs. 6 and 8 that strong virial/potential energy correlations are not limited to thermal equilibrium situations.

Tables

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Table I.

Variances of potential energy and virial , and of various contributions to and , of two different ensembles at the LJ state point given by and (dimensionless units).

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Table II.

Check of relation (47) between and for the LJ and the Kob–Andersen binary Lennard-Jones (KABLJ) liquids. The units for and are defined in terms of the length and energy parameters and for the interactions of the large particles. The excess isochoric heat capacity was calculated from the potential energy fluctuations in the ensemble.

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/content/aip/journal/jcp/131/23/10.1063/1.3265955
2009-12-17
2014-04-24

Abstract

In this third paper of the series, which started with Bailey et al. [J. Chem. Phys.129, 184507 (2008);ibid.129, 184508 (2008)], we continue the development of the theoretical understanding of strongly correlating liquids—those whose instantaneous potential energy and virial are more than 90% correlated in their thermal equilibrium fluctuations at constant volume. The existence of such liquids was detailed in previous work, which identified them, based on computer simulations, as a large class of liquids, including van der Waals liquids but not, e.g., hydrogen-bonded liquids. We here discuss the following: (1) the scaling properties of inverse power-law and extended inverse power-law potentials (the latter includes a linear term that “hides” the approximate scale invariance); (2) results from computer simulations of molecular models concerning out-of-equilibrium conditions; (3) ensemble dependence of the virial/potential-energy correlation coefficient; (4) connection to the Grüneisen parameter; and (5) interpretation of strong correlations in terms of the energy-bond formalism.

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Scitation: Pressure-energy correlations in liquids. III. Statistical mechanics and thermodynamics of liquids with hidden scale invariance
http://aip.metastore.ingenta.com/content/aip/journal/jcp/131/23/10.1063/1.3265955
10.1063/1.3265955
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