10484
J. Phys. Chem. 1992, 96, 10484-10487
Molecular Dynamics Simulations of a Tagged Particle Moving in a Solvent: Two-Dimensional Soft-Disk Liquid Jayasankar E.Variyart and Daniel Kivelson* Department of Chemistry and Biochemistry. University of California, Los Angeles, California 90024 (Received: June 30, 1992; I n Final Form: August 12, 1992)
We study, by means of molecular dynamics simulations, a tagged soft disk in a two-dimensional liquid composed of soft disks. We study the velocity autocorrelation function of the tagged particle as a function of its density and radius relative to those of the solvent particles. At low densities, the velocity autocorrelation function decays monotonically in time, but at higher densities, the velocity autocorrelation function dips to negative values at relatively short times. The ‘critical density” at which this qualitative change takes place increases as the tagged-particle density and radius increase. We use these results, in conjunction with a very simple model, to discuss the concept of correlation length in liquids that grows as the viscosity increases.
Introduction As the temperature of a supercooled liquid is lowered, its viscosity (a dynamical quantity) increases catastrophically, its configurational entropy (a thermodynamic property) decreases markedly, but its structure as indicated by the pair correlation function remains relatively unchanged.’ Is there some more complex, perhaps long-range structural order that sets in as the supercooled liquid is cooled that accounts for the viscosity and entropy changes?2 This is the question that motivates this work. In a previous study? we examined by molecular dynamics (MD) simulations the behavior of two-dimensional liquids composed of soft disks. We identified clusters of molecules and then followed the behavior of these clusters, and indeed we found some indications of long-range (or at least intermediaterange) mperativity. We studied the velocity autocorrelation functions (VACFs) of single particles and of clusters, a cluster being an arbitrary designation of a collection of closely spaced particles. We found that at low (gas) densities, the VACFs of single particles decreased monotonically with time, but that in liquids, the VACFs dipped to negative values at times under a picosecond; this negative dip might be envisaged as an indication of highly damped vibrational motion. The same qualitative behavior was detected for the VACFs of clusters, but the density and time at which a negative dip appeared increased with increasing cluster size. The interpretation given to these results suggested the present simulations in which the VACF of a tagged particle is studied as a function of the size and mass of this particle relative to those of the solvent particles. Here, too, we focus on two-dimensional soft disks. Although the tagged particle studied here is a more definitive probe of solvent behavior than are the clusters studied in ref 3, the MD simulations for the tagged particle have much larger statistical uncertainties. Model The present results for a tagged particle are in many ways similar to those obtained previously for cluster^;^ the density and time at which a negative dip occurs increase as the size and mass of the tagged particle increases relative to those of the solvent. In order to discuss these trends coherently, we introduce a very oversimplified model which is a slight variant of that presented in ref 3. If the normalized velocity autocorrelationfunction of the tagged particle is C(t),where C(0) = 1, then the translational diffusion constant, D,is given by the relation
D = 2d(ksT/M)Xmdt c(t)
(1)
where M is the mass of the tagged particle, Tis the temperature, ‘Current address: Department of Chemistry, Indian Institute of Technology, Kharagpur, India 721 302.
kB is the Boltzmann constant, and d is the dimensionality. (Although this integral may not be truly convergent for d < 3, this need not concern us here. See ref 4.) We can write the exact equation of motionSas
where (3) F1 is the force on the tagged particle, and ( ) indicates an equilibrium ensemble average. We will assume that the memory function y ( f ) carries no memory, i.e., that Y(t)
= rNt)
(4)
in which case eq 2 becomes d2C(r) + r -dC(t) dt2 dt
+ wo2C(r)= o
the equation of motion for a damped oscillator. Equations 1 and 5 together yield
D
2d(k~T/M)(r/Oo2)
(6)
We examine the solutions to eq 5 in two limiting cases: >> 1. In this case, C(r) is oscillatory and has the (1) 4w02/r2 general form exp[(iwo - r/2)t]. (2) 4w02/r2
