J. Phys. Chem. B 2005, 109, 21437-21444
21437
Diagrammatic Formulation of the Kinetic Theory of Fluctuations in Equilibrium Classical Fluids. V. The Short Time Approximation for the Memory Function† Madhav Ranganathan‡ and Hans C. Andersen* Department of Chemistry, Stanford UniVersity, Stanford, California 94305 ReceiVed: May 3, 2005; In Final Form: June 18, 2005
The correlation function for density fluctuations in a monatomic fluid obeys a formally exact kinetic equation containing a memory function. A previously derived short time approximation (STA) for this memory function is tested by comparing its predictions with the results of molecular dynamic simulations of a dense LennardJones fluid at a variety of temperatures. This approximation takes into account the contribution to the correlation function of uncorrelated repulsive binary collisions. The qualitative changes of predicted correlation functions with temperature and wave vector are generally correct. The major exception to this is the transverse current correlation function for small wave vector. The quantitative accuracy is better at short times than long times and better at high temperatures than low temperatures. The major failing of the STA is its underestimation of the amplitudes of the negative dips in the current autocorrelation functions and of the temperature dependence of the amplitudes of the dips. Despite its deficiencies in predicting the time dependence of current correlation functions, the STA gives accurate results for the self-diffusion coefficient and the shear viscosity coefficient at the highest temperatures studied.
1. Introduction The kinetic theory of fluctuations in equilibrium liquids has received a significant amount of attention in the last couple of decades because of its importance for understanding the properties of supercooled liquids above the glass transition. Such fluctuations are studied theoretically using computer simulations,1-4 the kinetic theory of liquids,5-11 mode coupling theory,12,13 and simplified theoretical models.14-16 Using a diagrammatic formulation8-10 of the kinetic theory of fluctuations in liquids, we have formulated a “short time approximation” (STA) for the memory function of the time correlation functions of density fluctuations in atomic liquids.11,17 The approximation was derived by summing all diagrammatic contributions to the memory function that are of leading order in a small parameter that is a measure of the softness of the repulsive part of the interatomic potential. The approximation includes the contribution to the memory function from a single binary repulsive collision between two atoms. The use of this approximation for the memory function is equivalent to assuming that the physics important for the time dependence of the correlation function is a series of uncorrelated, repulsive binary collisions. In effect, the STA is a generalization of the Enskog kinetic theory for the hard sphere fluid. The generalization takes into account the facts that (1) the repulsive collisions involve the continuous repulsive part of the interatomic potential (rather than a hard sphere potential) and (2) the other parts of the potential can have an effect on the static equilibrium structure and hence on the collision frequency for the repulsive collisions. In this paper, we present tests of the short time approximation for a standard model of liquids, namely, a one-component Lennard-Jones fluid at a high density and a variety of temper†
Part of the special issue “Irwin Oppenheim Festschrift”. * To whom correspondence should be addressed. ‡ Current address: Institute of Physical Science and Technology, University of Maryland, College Park, MD 20742.
atures. The predictions of the STA for various correlation functions of density fluctuations are compared with molecular dynamics computer simulations. 2. The Short Time Approximation We are interested in the properties of a one-component monatomic fluid at equilibrium. The fluid consists of N particles of mass m in a volume V at temperature T. The state of the system at any time is specified by rNpN, the positions and momenta of all the particles. The particles of the fluid interact through pairwise additive central forces. The Hamiltonian for the system is N
H(rNpN) )
∑ i)1
pi . pi 2m
N
+
∑ u(ri - rj)
i 0.5 is the simulation result for T* ) 1.554, and the lower dashed curve for the same time is the simulation result for T* ) 0.723.
decays more slowly, whereas the simulation curve drops more rapidly and at the lowest temperature has a negative dip indicative of highly damped transverse oscillations. At all but the lowest temperature, the long time decay of both the simulation and STA results is exponential, which allows values of the shear viscosities to be extracted.19 The results are given in Table 2.
negative dips in the current autocorrelation functions and of the temperature dependence of the amplitudes of the dips. Despite its deficiencies in predicting the time dependence of current correlation functions, the STA gives accurate results for the self-diffusion coefficient and the shear viscosity coefficient at the highest temperature studied. The STA describes the dynamical behavior of a fluid as a consequence of uncorrelated repulsive binary collisions. It is presumably true that some of the inaccuracy of the STA derives from its neglect of attractive forces in the dynamics. Preliminary calculations indicate that this is true at least for short times. Because the STA neglects correlations among the collisions, it should not be surprising that the situations in which it is most inaccurate are those in which the physics of certain collective effects are important. This includes sound waves, which cause oscillations in the coherent intermediate scattering function for small wave vector at all temperatures; shear waves, which cause oscillations in the transverse current correlation function for small wave vectors and low temperatures; and the caging of atoms for extended periods of time, which causes oscillations
6. Discussion and Conclusions On the basis of the comparisons above, we can make the following generalizations about the STA results for the correlation functions for the range of temperatures and densities studied: (1) The qualitative changes of predicted correlation functions with temperature and wave vector are generally correct. The major exception to this is the transverse current correlation function for small wave vector. (2) The quantitative accuracy is better at short times than long times and better at high temperatures than low temperatures. (3) The major failing of the STA is its underestimation of the amplitudes of the
21444 J. Phys. Chem. B, Vol. 109, No. 45, 2005
Ranganathan and Andersen (1) If Γµν is in the O(tn) set, then Γµν(t) is calculated numerically by the method discussed above, giving results that are exact except for statistical error. (2) Off-diagonal elements Γµν(t) that are not in the O(tn) set are set equal to zero. (3) Diagonal elements Γµµ(t) that are not in the O(tn) set are approximated as
Γµxµyµzµxµyµz(t) ≈ (constant) × Γµxµyµ′zµxµyµ′z (t)
Figure 11. Transverse current correlation function, normalized to have an initial value of unity, for the LJ fluid at reduced wave vector 6.75 for reduced temperatures of 0.723 and 1.554. The STA calculations are shown as solid lines, and the simulation results are shown as dashed lines. For both the simulation data and the STA results, the curves that are higher at short times (t* < 0.1) correspond to a reduced temperature of 0.723 and the curves that are lower at these short times correspond to a reduced temperature of 1.554.
TABLE 2: Reduced Kinematic Shear Viscosity Coefficients, η*, for Various Reduced Temperatures T*, Obtained from the Short Time Approximation and from Computer Simulationsa T*
η*(STA)
η*(simulations)
0.723 1.000 1.277 1.554
2.1 2.6 3.0 2.8
3.86 3.37 3.09
a
The simulation results do not give exponential behavior for long times at a reduced temperature of 0.723; hence, we do not have a viscosity coefficient for the simulation results at that temperature. The statistical errors in the calculation of the STA transverse current correlation function lead to statistical errors of less than 3% in the calculated viscosities.
in the coherent and incoherent longitudinal currents, both for large and small wave vectors, with amplitudes that become larger as the temperature is lowered. Approximations that include attractive forces and collective effects will be required to obtain theories that are accurate at both short and long times for high temperature liquids as well as low temperature liquids. Acknowledgment. The authors would like to thank Thomas Young for providing the molecular dynamics data used in this work. This work was supported by the National Science Foundation through grants CHE0010117 and CHE0408786. Appendix: Details on the Numerical Solution of the Kinetic Equations In the numerical procedure for obtaining a specific matrix element of the correlation function in the O(tn) approximation for a specific n using a specific finite truncated basis set A, matrix elements of the form Γµν(t) for µ and ν in the truncated set A are required. They are obtained in the following way:
(6)
where µ′z is the largest value for which Γµxµyµ′zµxµyµµ′z (t) is in the O(tn) set. The constant is chosen so that this equality is exact at t ) 0. (The values of Γµν(0) can be expressed simply in terms of g(r).) Since the various diagonal elements of Γ are rapidly decaying on about the same time scale, this provides a reasonable approximation. (See an additional comment below.) This set of choices has the following features: (1) The solution for the correlation function of interest is correct through O(tn) at short times; that is, to that order of time, it gives the same results as solution of the infinite set of coupled matrix equations. (2) The inclusion of nonzero results for all of the diagonal elements of Γ makes the calculated correlation function decay to zero at long times, which is a characteristic of the solution of the infinite set of coupled matrix equations. One comment is in order. For the case of Γµµ(t) matrix elements that are not in the O(tn) set, it is possible that there might not be a µ′z that satisfies eq 8. In that case, however, since all off-diagonal elements of the form Γµν(t) and Γνµ(t) are either equal to zero by symmetry or are set equal zero in the calculation of the O(tn) approximation, the numerical solution for the correlation function of interest using the truncated A basis is not affected by the value of Γµµ(t), and hence, the latter is not needed. References and Notes (1) Boon, J. P.; Yip, S. Molecular Hydrodynamics; McGraw-Hill: New York, 1980. (2) Hansen, J.-P.; McDonald, I. R. Theory of Simple Liquids, 2nd ed.; Academic: London, 1986. (3) Allen, M. P.; Tildesley, D. J. Computer Simulation of Liquids; Oxford: New York, 1989. (4) Balucani, U.; Zoppi, M. Dynamics of the Liquid State, 1st ed.; Oxford: New York, 1994. (5) Mazenko, G. F. Phys. ReV. 1974, A9, 360-387. (6) Mazenko, G. F.; Yip, S. In Statistical Mechanics. Part B: TimeDependent Processes; Berne, B. J. Ed.; Plenum: New York, 1977. (7) Sjo¨gren, L. Phys. ReV. 1980, 22, 2866-2882. (8) Andersen, H. C. J. Phys. Chem. B 2002, 106, 8326-8337. (9) Andersen, H. C. J. Phys. Chem. B 2003, 107, 10226-10233. (10) Andersen, H. C. J. Phys. Chem. B 2003, 107, 10234-10242. (11) Ranganathan, M.; Andersen, H. C. J. Chem. Phys. 2004, 121, 12431257. (12) Go¨tze, W. In Liquids, Freezing and the Glass Transition; Levesque, D., Hansen, J. P., Zinn-Justin, J., Eds.; North-Holland: New York, 1991; Part I, p 287. (13) Go¨tze, W.; Sjo¨gren, L. Transp. Theory Stat. Phys. 1995, 24, 801853. (14) Xia, X.; Wolynes, P. G. Phys. ReV. Lett. 2001, 86, 5526-5529. (15) Ritort, F.; Sollich, P. AdV. Phys. 2003, 52, 219-342. (16) Garrahan, J. P.; Chandler, D. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 9710-9714. (17) Ranganathan, M. Ph.D. Thesis, Stanford University, Stanford, CA, 2003. (18) Young, T.; Andersen, H. C. J. Chem. Phys. 2003, 118, 34473450. (19) Young, T.; Andersen, H. C. J. Phys. Chem. B 2005, 109, 29852994. (20) Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237-5247.
