
Download
XML
0.00MB

Download
PDF
0.00MB

Read Online
HTML
0.00MB
Abstract
We show that a number of model liquids at fixed volume exhibit strong correlations between equilibrium fluctuations of the configurational parts of (instantaneous) pressure and energy. We present detailed results for 13 systems, showing in which systems these correlations are significant. These include LennardJones liquids (both single and twocomponent) and several other simple liquids, neither hydrogenbonding liquids such as methanol and water, nor the Dzugutov liquid, which has significant contributions to pressure at the second nearest neighbor distance. The pressureenergy correlations, which for the LennardJones case are shown to also be present in the crystal and glass phases, reflect an effective inverse powerlaw potential dominating fluctuations, even at zero and slightly negative pressure. An exception to the inverse powerlaw explanation is a liquid with hardsphere repulsion and a squarewell attractive part, where a strong correlation is observed, but only after time averaging. The companion paper [N. P. Bailey et al., J. Chem. Phys.129, 184508 (2008)] gives a thorough analysis of the correlations, with a focus on the LennardJones liquid, and a discussion of some experimental and theoretical consequences.
Useful discussions with Søren Toxværd are gratefully acknowledged. Center for viscous liquid dynamics “Glass and Time” is sponsored by The Danish National Research Foundation.
I. INTRODUCTION
II. SIMULATED SYSTEMS
III. RESULTS
A. The standard singlecomponent LennardJones system
B. A case with little correlation: The Dzugutov system
C. When competition between van der Waals and Coulomb interactions kills the correlation: TIP5P water
D. Results for all systems
IV. SUMMARY
Key Topics
 Many body systems
 8.0
 Glass transitions
 6.0
 Monte Carlo methods
 6.0
 Computer simulation
 4.0
 Copper
 4.0
Figures
Equilibrium fluctuations of (a) pressure and energy and (b) virial and potential energy , in a singlecomponent LennardJones system simulated in the ensemble at and (argon units). The timeaveraged pressure was close to zero . The correlation coefficient between and is 0.94, whereas the correlation coefficient is only 0.70 between and . Correlation coefficients were calculated over the total simulation time .
Click to view
Equilibrium fluctuations of (a) pressure and energy and (b) virial and potential energy , in a singlecomponent LennardJones system simulated in the ensemble at and (argon units). The timeaveraged pressure was close to zero . The correlation coefficient between and is 0.94, whereas the correlation coefficient is only 0.70 between and . Correlation coefficients were calculated over the total simulation time .
(a) Scatter plot of instantaneous virial and potential energy from the simulation of Fig. 1. The dashed line is a guide to the eyes, with a slope determined by the ratio of standard deviations of and [Eq. (7)]. (b) Example of a system with almost no correlation between and : TIP5P water at and density of (NVT). This system has Coulomb, in addition to LennardJones, interactions.
Click to view
(a) Scatter plot of instantaneous virial and potential energy from the simulation of Fig. 1. The dashed line is a guide to the eyes, with a slope determined by the ratio of standard deviations of and [Eq. (7)]. (b) Example of a system with almost no correlation between and : TIP5P water at and density of (NVT). This system has Coulomb, in addition to LennardJones, interactions.
Illustration of the “effective inverse power law” chosen in this case to match the LennardJones potential and its first two derivatives at the point . The vertical line marks the division into the repulsive and attractive parts of the LennardJones potential.
Click to view
Illustration of the “effective inverse power law” chosen in this case to match the LennardJones potential and its first two derivatives at the point . The vertical line marks the division into the repulsive and attractive parts of the LennardJones potential.
Scatter plots of the configurational parts of pressure and energy—virial vs potential energy—for several state points of the SCLJ liquid (NVT). Each oval represents simulations at one particular temperature and density where each data point marks instantaneous values of virial and potential energy. The dashed lines mark constant density paths with the highest density to the upper left (densities: 39.8, 37.4, 36.0, 34.6, and ). State points on the dotted line have zero average pressure. The plot includes three crystallized samples (lower left corner), discussed at the end of Sec. III A and, in more detail, in Paper II (reproduced from Ref. 10).
Click to view
Scatter plots of the configurational parts of pressure and energy—virial vs potential energy—for several state points of the SCLJ liquid (NVT). Each oval represents simulations at one particular temperature and density where each data point marks instantaneous values of virial and potential energy. The dashed lines mark constant density paths with the highest density to the upper left (densities: 39.8, 37.4, 36.0, 34.6, and ). State points on the dotted line have zero average pressure. The plot includes three crystallized samples (lower left corner), discussed at the end of Sec. III A and, in more detail, in Paper II (reproduced from Ref. 10).
Upper plot, correlation coefficient for the SCLJ system as a function of temperature for several densities (NVT). This figure makes clear the different effects of density and temperature on . Lower plot, effective slope as a function of . Simulations at temperatures higher than those shown here indicate that the slope slowly approaches the value 4 as increases. This is to be expected because as collisions become harder, involving shorter distances, the effective inverse powerlaw exponent approaches the 12 from the repulsive term of the LennardJones potential.
Click to view
Upper plot, correlation coefficient for the SCLJ system as a function of temperature for several densities (NVT). This figure makes clear the different effects of density and temperature on . Lower plot, effective slope as a function of . Simulations at temperatures higher than those shown here indicate that the slope slowly approaches the value 4 as increases. This is to be expected because as collisions become harder, involving shorter distances, the effective inverse powerlaw exponent approaches the 12 from the repulsive term of the LennardJones potential.
Scatter plot of the correlations for a perfect facecenteredcubic (fcc) crystal of LennardJones atoms at temperatures 1, 2, 3, 5, 10, 20, 30, 40, 50, 60, 70, and , as well as for defective crystals (i.e., crystallized from the liquid) at temperatures 50, 70, and (NVT). The dashed line gives the best fit to the (barely visible) lowesttemperature data . The inset shows the temperature dependence of at very low temperatures. The crystalline case is examined in detail in Paper II, where we find that does not converge to unity at , but rather to a value very close to unity. All state points refer to the highest density of Fig. 4, 39.8 mol/l.
Click to view
Scatter plot of the correlations for a perfect facecenteredcubic (fcc) crystal of LennardJones atoms at temperatures 1, 2, 3, 5, 10, 20, 30, 40, 50, 60, 70, and , as well as for defective crystals (i.e., crystallized from the liquid) at temperatures 50, 70, and (NVT). The dashed line gives the best fit to the (barely visible) lowesttemperature data . The inset shows the temperature dependence of at very low temperatures. The crystalline case is examined in detail in Paper II, where we find that does not converge to unity at , but rather to a value very close to unity. All state points refer to the highest density of Fig. 4, 39.8 mol/l.
A plot of the Dzugutov pair potential, with the LennardJones potential (shifted by a constant) shown for comparison.
Click to view
A plot of the Dzugutov pair potential, with the LennardJones potential (shifted by a constant) shown for comparison.
Scatter plot of correlations for the Dzugutov system at density 0.88 and temperatures 0.65 and 0.70 (NVE). The dashed lines indicate the bestfit line using linear regression. These are consistent with the temperature dependence of the mean values of and , as they should be (see Appendix B), but they clearly do not represent the direction of greatest variance. The full lines have slopes equal to the ratio of standard deviations of the two quantities [Eq. (7)]. The correlation coefficient is 0.585 and 0.604 for and , respectively.
Click to view
Scatter plot of correlations for the Dzugutov system at density 0.88 and temperatures 0.65 and 0.70 (NVE). The dashed lines indicate the bestfit line using linear regression. These are consistent with the temperature dependence of the mean values of and , as they should be (see Appendix B), but they clearly do not represent the direction of greatest variance. The full lines have slopes equal to the ratio of standard deviations of the two quantities [Eq. (7)]. The correlation coefficient is 0.585 and 0.604 for and , respectively.
Plot of for TIP5P water in simulations with densities chosen to give an average pressure of . Not only is the magnitude of low (less than 0.2) in the temperature range shown, but it changes sign around the density maximum. The vertical arrow indicates the state point used for Fig. 2(b).
Click to view
Plot of for TIP5P water in simulations with densities chosen to give an average pressure of . Not only is the magnitude of low (less than 0.2) in the temperature range shown, but it changes sign around the density maximum. The vertical arrow indicates the state point used for Fig. 2(b).
correlations for all simulated liquids; plotted vs . Both quantities correspond to a pressure, which is given in units of GPa; for model systems not specifically corresponding to real systems, such as SCLJ, KABLJ, and SQW, argon units were used to set the energy and length scales. If the correlation is perfect the data fall on the diagonal marked by a dashed line. For the TIP5P model of water only temperatures with are included; volumes were chosen to give a pressure close to zero.
Click to view
correlations for all simulated liquids; plotted vs . Both quantities correspond to a pressure, which is given in units of GPa; for model systems not specifically corresponding to real systems, such as SCLJ, KABLJ, and SQW, argon units were used to set the energy and length scales. If the correlation is perfect the data fall on the diagonal marked by a dashed line. For the TIP5P model of water only temperatures with are included; volumes were chosen to give a pressure close to zero.
Illustration of the squarewell potential, indicating the four microscopic processes, which contribute to the virial.
Click to view
Illustration of the squarewell potential, indicating the four microscopic processes, which contribute to the virial.
Energyenergy, virialvirial, and energyvirial correlation functions for SQW at packing fraction and temperature (normalized to unity at ). The cross correlation has been multiplied by . The arrow marks the time , roughly of the relaxation time (determined from the longtime part of the energyvirial crosscorrelation function). This time was used for averaging.
Click to view
Energyenergy, virialvirial, and energyvirial correlation functions for SQW at packing fraction and temperature (normalized to unity at ). The cross correlation has been multiplied by . The arrow marks the time , roughly of the relaxation time (determined from the longtime part of the energyvirial crosscorrelation function). This time was used for averaging.
Illustration of replacement of discontinuous step by a finite slope for the squarewell potential for the purpose of calculating the virial. The limit is taken at the end.
Click to view
Illustration of replacement of discontinuous step by a finite slope for the squarewell potential for the purpose of calculating the virial. The limit is taken at the end.
Tables
Correlation coefficients and effective slopes for the SCLJ system for the state points in Fig. 4. is the thermally averaged pressure. The last five states were chosen to approximately follow the isobar .
Click to view
Correlation coefficients and effective slopes for the SCLJ system for the state points in Fig. 4. is the thermally averaged pressure. The last five states were chosen to approximately follow the isobar .
Correlation coefficients and slopes at selected state points for all investigated systems besides SCLJ. Argon units were used for DZ, EXP, KABLJ, and SQW by choosing the length parameter (of the larger particle when there were two types) to be and the energy parameter to be . The phase is indicated as liquid or glass. SQW data involve time averaging over periods 3.0, 3.0, 8.0, and 9.0, respectively, for the four listed state points. A minus sign has been included with the slope when ; note that the values only really make sense as slopes when is close to unity. The ensemble was NVT except for CU, DZ, MGCU, and SQW, where it was NVE.
Click to view
Correlation coefficients and slopes at selected state points for all investigated systems besides SCLJ. Argon units were used for DZ, EXP, KABLJ, and SQW by choosing the length parameter (of the larger particle when there were two types) to be and the energy parameter to be . The phase is indicated as liquid or glass. SQW data involve time averaging over periods 3.0, 3.0, 8.0, and 9.0, respectively, for the four listed state points. A minus sign has been included with the slope when ; note that the values only really make sense as slopes when is close to unity. The ensemble was NVT except for CU, DZ, MGCU, and SQW, where it was NVE.
Article metrics loading...
Abstract
We show that a number of model liquids at fixed volume exhibit strong correlations between equilibrium fluctuations of the configurational parts of (instantaneous) pressure and energy. We present detailed results for 13 systems, showing in which systems these correlations are significant. These include LennardJones liquids (both single and twocomponent) and several other simple liquids, neither hydrogenbonding liquids such as methanol and water, nor the Dzugutov liquid, which has significant contributions to pressure at the second nearest neighbor distance. The pressureenergy correlations, which for the LennardJones case are shown to also be present in the crystal and glass phases, reflect an effective inverse powerlaw potential dominating fluctuations, even at zero and slightly negative pressure. An exception to the inverse powerlaw explanation is a liquid with hardsphere repulsion and a squarewell attractive part, where a strong correlation is observed, but only after time averaging. The companion paper [N. P. Bailey et al., J. Chem. Phys.129, 184508 (2008)] gives a thorough analysis of the correlations, with a focus on the LennardJones liquid, and a discussion of some experimental and theoretical consequences.
Full text loading...
Commenting has been disabled for this content