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Abstract
We present a detailed analysis and discuss consequences of the strong correlations of the configurational parts of pressure and energy in their equilibrium fluctuations at fixed volume reported for simulations of several liquids in the previous paper [N. P. Bailey et al., J. Chem. Phys.129, 184507 (2008)]. The analysis concentrates specifically on the singlecomponent LennardJones system. We demonstrate that the potential may be replaced, at fixed volume, by an effective power law but not simply because only shortdistance encounters dominate the fluctuations. Indeed, contributions to the fluctuations are associated with the whole first peak of the radial distribution function, as we demonstrate by an eigenvectoranalysis of the spatially resolved covariance matrix. The reason the effective power law works so well depends crucially on going beyond singlepair effects and on the constraint of fixed volume. In particular, a better approximation to the potential includes a linear term, which contributes to the mean values of potential energy and virial, but little to their fluctuations, for density fluctuations which conserve volume. We also study in detail the zero temperature limit of the (classical) crystalline phase, where the correlation coefficient becomes very close, but not equal, to unity, in more than one dimension; in one dimension the limiting value is exactly unity. In the second half of the paper we consider four consequences of strong pressureenergy correlations: (1) analyzing experimental data for supercritical argon we find 96% correlation; (2) we discuss the particular significance acquired by the correlations for viscous van der Waals liquids approaching the glass transition: For strongly correlating viscous liquids knowledge of just one of the eight frequencydependent thermoviscoelastic response functions basically implies knowledge of them all; (3) we reinterpret aging simulations of orthoterphenyl carried out by Mossa et al. [Eur. Phys. J. B30, 351 (2002)], showing their conclusions follow from the strongly correlating property; and (4) we briefly discuss the presence of the correlations (after appropriate time averaging) in model biomembranes, showing that significant correlations may be present even in quite complex systems.
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. ANALYSIS
A. The effective inverse power law
B. Lowtemperature limit: Anharmonic vibrations of a crystal
1. The onedimensional crystal
2. The threedimensional crystal
3. Statistics of and
4. The role of the coefficients
C. Fluctuation modes
D. Synthesis: Why the effective power law works even at longer distances
III. SOME CONSEQUENCES OF STRONG PRESSUREENERGY CORRELATIONS
A. Measurable consequences of instantaneous correlations
B. Time averaging: Pressureenergy correlations in highly viscous liquids
C. Aging and energy landscapes
D. Biomembranes
IV. CONCLUSIONS AND OUTLOOK
Key Topics
 Viscosity
 20.0
 Eigenvalues
 19.0
 Statistical properties
 7.0
 Viscosity measurements
 5.0
 Amorphous solids
 4.0
Figures
Effective powerlaw exponents defined by derivative ratios of different orders [Eqs. (10)] for the standard LennardJones potential LJ(12,6). All converge to 12 at small ; they diverge when the derivative in the denominator vanishes, which happens for larger , the higher the order of this derivative. The term “effective inverse power law” in this paper refers to a power law chosen to match at some point , the potential minimum where diverges. A convenient choice is to match at , giving 18. In Secs. II B and II C we show that plays an important role in the understanding of fluctuations associated with pair distances close to the potential minimum .
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Effective powerlaw exponents defined by derivative ratios of different orders [Eqs. (10)] for the standard LennardJones potential LJ(12,6). All converge to 12 at small ; they diverge when the derivative in the denominator vanishes, which happens for larger , the higher the order of this derivative. The term “effective inverse power law” in this paper refers to a power law chosen to match at some point , the potential minimum where diverges. A convenient choice is to match at , giving 18. In Secs. II B and II C we show that plays an important role in the understanding of fluctuations associated with pair distances close to the potential minimum .
(a) The LennardJones potential fitted by an effective powerlaw potential covering the most important part of the repulsive part of the potential. The exponent was chosen to be 18, which optimizes agreement at , where the effective power law exactly matches not just but also its first two derivatives. Also shown are the Taylor series expansions of about up to third and fourth orders. The RDF (at ,) is also shown as a convenient reference for thinking about where contributions to potential energy and virial fluctuations come from. (b) Error made in approximating with different effective power laws matched at different points and with Taylor expansions up to third order about the same point.
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(a) The LennardJones potential fitted by an effective powerlaw potential covering the most important part of the repulsive part of the potential. The exponent was chosen to be 18, which optimizes agreement at , where the effective power law exactly matches not just but also its first two derivatives. Also shown are the Taylor series expansions of about up to third and fourth orders. The RDF (at ,) is also shown as a convenient reference for thinking about where contributions to potential energy and virial fluctuations come from. (b) Error made in approximating with different effective power laws matched at different points and with Taylor expansions up to third order about the same point.
Scatter plot of true and reconstructed potential energy and virial fluctuations (dimensionless units) for the LJ liquid, where the reconstructed values and were calculated from the true configurations, assuming an inverse powerlaw potential with exponent of 19.2; mean values have been subtracted off. The state point is the same as in Fig. 1 of Paper I (zero average pressure, NVT ensemble). The correlation coefficients are displayed in the figures; the dashed lines indicate slope unity. The fact that actual and reconstructed fluctuations correlate strongly, and with slopes close to unity, support the idea that the correlation is derived from an effective inverse powerlaw potential dominating fluctuations.
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Scatter plot of true and reconstructed potential energy and virial fluctuations (dimensionless units) for the LJ liquid, where the reconstructed values and were calculated from the true configurations, assuming an inverse powerlaw potential with exponent of 19.2; mean values have been subtracted off. The state point is the same as in Fig. 1 of Paper I (zero average pressure, NVT ensemble). The correlation coefficients are displayed in the figures; the dashed lines indicate slope unity. The fact that actual and reconstructed fluctuations correlate strongly, and with slopes close to unity, support the idea that the correlation is derived from an effective inverse powerlaw potential dominating fluctuations.
Plots of predicted correlation coefficient for for a crystal of LJ(12,6) particles for different degrees of correlation between the quantities and , and of lowtemperature simulation data. The first three curves (counting from the bottom) assume that the variances of and are equal, and that their correlation coefficients are 0, 0.5, and 0.75, respectively. The fourth curve (up triangles) results from considering an sc lattice and assuming individual particles have uncorrelated Gaussiandistributed displacements, leading to specific values for the variances and covariance of and . The fifth (left triangles) shows the same estimate for a fcc lattice. The right triangles are data from an simulation of a perfect fcc crystal at . The conclusion from this figure is that does not tend to unity as , although it becomes extremely close. The inset shows the corresponding slopes [Eq. (6)].
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Plots of predicted correlation coefficient for for a crystal of LJ(12,6) particles for different degrees of correlation between the quantities and , and of lowtemperature simulation data. The first three curves (counting from the bottom) assume that the variances of and are equal, and that their correlation coefficients are 0, 0.5, and 0.75, respectively. The fourth curve (up triangles) results from considering an sc lattice and assuming individual particles have uncorrelated Gaussiandistributed displacements, leading to specific values for the variances and covariance of and . The fifth (left triangles) shows the same estimate for a fcc lattice. The right triangles are data from an simulation of a perfect fcc crystal at . The conclusion from this figure is that does not tend to unity as , although it becomes extremely close. The inset shows the corresponding slopes [Eq. (6)].
Representations of the eigenvectors 3, 4, 5, and 6 of the supercovariance matrix. Squares represent variation in values for a mode; diamonds represent variation in values.
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Representations of the eigenvectors 3, 4, 5, and 6 of the supercovariance matrix. Squares represent variation in values for a mode; diamonds represent variation in values.
Intuitive picture of allowed and disallowed fluctuations in : (a) is not allowed because it violates the global constraint ; (b) satisfies the global constraint but not locality; (c) could correspond, for instance, to a single bond becoming shorter, but this is inconsistent with fixed volume (vanishing first moment—such a change cannot happen in isolation); and (d) is allowed—it corresponds, for example, to a single particle being displaced toward one neighbor and away from another. Thus one bond shortens and one lengthens.
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Intuitive picture of allowed and disallowed fluctuations in : (a) is not allowed because it violates the global constraint ; (b) satisfies the global constraint but not locality; (c) could correspond, for instance, to a single bond becoming shorter, but this is inconsistent with fixed volume (vanishing first moment—such a change cannot happen in isolation); and (d) is allowed—it corresponds, for example, to a single particle being displaced toward one neighbor and away from another. Thus one bond shortens and one lengthens.
The true potential , the best effective power law (in the sense that the fluctuations in potential energy and virial and reproduced most faithfully), and their difference . Also shown are the projected versions and where the constant and linear terms (determined over the interval to ) have been subtracted off. It is the projected functions that should be compared in order to make a statement about the smallness of relative to since only the projected functions contribute to fluctuations of total potential energy.
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The true potential , the best effective power law (in the sense that the fluctuations in potential energy and virial and reproduced most faithfully), and their difference . Also shown are the projected versions and where the constant and linear terms (determined over the interval to ) have been subtracted off. It is the projected functions that should be compared in order to make a statement about the smallness of relative to since only the projected functions contribute to fluctuations of total potential energy.
Scatter plot of instantaneous virial and hypervirial (in dimensionless units) for a SCLJ system at , (NVE). The correlation coefficient between these quantities is 0.998. The hypervirial is the main contribution to the configurational part of the bulk modulus; it gives (after dividing by volume) the change in virial for a given relative change in volume. The sizable constant term in the linear fit shows that Eq. (91) is a poor approximation. The slope is 4.9, about 10% smaller than for this state point. The difference reflects the limit of the validity of the powerlaw description—in fact, a more detailed analysis shows that the relation between and is dominated by , which is smaller than (Fig. 1).
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Scatter plot of instantaneous virial and hypervirial (in dimensionless units) for a SCLJ system at , (NVE). The correlation coefficient between these quantities is 0.998. The hypervirial is the main contribution to the configurational part of the bulk modulus; it gives (after dividing by volume) the change in virial for a given relative change in volume. The sizable constant term in the linear fit shows that Eq. (91) is a poor approximation. The slope is 4.9, about 10% smaller than for this state point. The difference reflects the limit of the validity of the powerlaw description—in fact, a more detailed analysis shows that the relation between and is dominated by , which is smaller than (Fig. 1).
Data from the NIST database (Ref. 45) for supercritical argon at three different densities covering the temperature range of showing a strong virialpotentialenergy correlation (reproduced from Ref. 4). Here , , and . The diagonal line corresponds to perfect correlation. The inset shows “unsubtracted” values for and ; the fact that the data do not fall on the solid line indicates that a powerlaw description does not hold for the full thermodynamics.
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Data from the NIST database (Ref. 45) for supercritical argon at three different densities covering the temperature range of showing a strong virialpotentialenergy correlation (reproduced from Ref. 4). Here , , and . The diagonal line corresponds to perfect correlation. The inset shows “unsubtracted” values for and ; the fact that the data do not fall on the solid line indicates that a powerlaw description does not hold for the full thermodynamics.
Upper panel, timeaveraged (over , where is the structural relaxation time) normalized fluctuations of and in NVT simulations of the Kob–Andersen (Ref. 20) binary LennardJones (KABLJ) system, plotted against time in units of . The density was , and the temperature was . Middle panel, imaginary parts of the three response functions , ; and , scaled to the maximum value. Lower panel, dynamic Prigogine–Defay ratio for the same simulation. The approach toward unity at frequencies smaller than the losspeak frequency is exactly equivalent to the correlation between timeaveraged quantities shown in the upper panel (reproduced from Ref. 19).
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Upper panel, timeaveraged (over , where is the structural relaxation time) normalized fluctuations of and in NVT simulations of the Kob–Andersen (Ref. 20) binary LennardJones (KABLJ) system, plotted against time in units of . The density was , and the temperature was . Middle panel, imaginary parts of the three response functions , ; and , scaled to the maximum value. Lower panel, dynamic Prigogine–Defay ratio for the same simulation. The approach toward unity at frequencies smaller than the losspeak frequency is exactly equivalent to the correlation between timeaveraged quantities shown in the upper panel (reproduced from Ref. 19).
Normalized fluctuations of energy (×) and volume (○) for a 1,2dipalmitoylsnglycero3phosphocholine (DPPC) membrane at (Ref. 44). Each data point represents a average. Energy and volume are correlated with a correlation coefficient of ().
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Normalized fluctuations of energy (×) and volume (○) for a 1,2dipalmitoylsnglycero3phosphocholine (DPPC) membrane at (Ref. 44). Each data point represents a average. Energy and volume are correlated with a correlation coefficient of ().
Tables
Statistics of and assuming uncorrelated particle displacements with variance for each Cartesian component, for sc and fcc lattices.
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Statistics of and assuming uncorrelated particle displacements with variance for each Cartesian component, for sc and fcc lattices.
First ten eigenvalues of the supercovariance matrix [Eq. (65)], their fractional contributions to the three (co)variances [Eqs. (62)–(64)], and their effective slopes [Eq. (67)] for the SCLJ liquid with parameters as in Fig. 1 of Paper I (, ). Contributions from the dominant four eigenvectors are in boldface. The last three rows list sums of the third, fourth, and fifth columns over, respectively, the dominant four, the first ten, and all eigenvectors. The sum of the fifth column over all eigenvectors should be compared [see Eq. (64)] to the listed in Table I of Paper I.
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First ten eigenvalues of the supercovariance matrix [Eq. (65)], their fractional contributions to the three (co)variances [Eqs. (62)–(64)], and their effective slopes [Eq. (67)] for the SCLJ liquid with parameters as in Fig. 1 of Paper I (, ). Contributions from the dominant four eigenvectors are in boldface. The last three rows list sums of the third, fourth, and fifth columns over, respectively, the dominant four, the first ten, and all eigenvectors. The sum of the fifth column over all eigenvectors should be compared [see Eq. (64)] to the listed in Table I of Paper I.
Data from simulations of fully hydrated phospholipid membranes of 1,2dimyristoylsnglycero3phosphocholine (DMPC), 1,2dimyristoylsnglycero3phosphoLserine with sodium as counter ion (DMPSNa), hydrated DMPS (DMPSH), and DPPC (Refs. 40 and 44). The columns list temperature, correlation coefficient between volume and energy, average lateral area per lipid, simulation time in equilibrium, and total simulation time.
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Data from simulations of fully hydrated phospholipid membranes of 1,2dimyristoylsnglycero3phosphocholine (DMPC), 1,2dimyristoylsnglycero3phosphoLserine with sodium as counter ion (DMPSNa), hydrated DMPS (DMPSH), and DPPC (Refs. 40 and 44). The columns list temperature, correlation coefficient between volume and energy, average lateral area per lipid, simulation time in equilibrium, and total simulation time.
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Abstract
We present a detailed analysis and discuss consequences of the strong correlations of the configurational parts of pressure and energy in their equilibrium fluctuations at fixed volume reported for simulations of several liquids in the previous paper [N. P. Bailey et al., J. Chem. Phys.129, 184507 (2008)]. The analysis concentrates specifically on the singlecomponent LennardJones system. We demonstrate that the potential may be replaced, at fixed volume, by an effective power law but not simply because only shortdistance encounters dominate the fluctuations. Indeed, contributions to the fluctuations are associated with the whole first peak of the radial distribution function, as we demonstrate by an eigenvectoranalysis of the spatially resolved covariance matrix. The reason the effective power law works so well depends crucially on going beyond singlepair effects and on the constraint of fixed volume. In particular, a better approximation to the potential includes a linear term, which contributes to the mean values of potential energy and virial, but little to their fluctuations, for density fluctuations which conserve volume. We also study in detail the zero temperature limit of the (classical) crystalline phase, where the correlation coefficient becomes very close, but not equal, to unity, in more than one dimension; in one dimension the limiting value is exactly unity. In the second half of the paper we consider four consequences of strong pressureenergy correlations: (1) analyzing experimental data for supercritical argon we find 96% correlation; (2) we discuss the particular significance acquired by the correlations for viscous van der Waals liquids approaching the glass transition: For strongly correlating viscous liquids knowledge of just one of the eight frequencydependent thermoviscoelastic response functions basically implies knowledge of them all; (3) we reinterpret aging simulations of orthoterphenyl carried out by Mossa et al. [Eur. Phys. J. B30, 351 (2002)], showing their conclusions follow from the strongly correlating property; and (4) we briefly discuss the presence of the correlations (after appropriate time averaging) in model biomembranes, showing that significant correlations may be present even in quite complex systems.
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