Volume 115, number 5 PHYSICS LETTERS A 14 April 1986

T H E D I S P E R S I O N RELATION OF T H E DYNAMIC STRUCTURE FACTOR OF O N E - D I M E N S I O N A L LENNARD-JONES FLUIDS

Marvin BISHOP and Mitchell NORCROSS

Department of Mathematics/Computer Science, Manhattan College, Rioerdale, N Y 10471, USA

Received 17 December 1985; accepted for publication 4 February 1985

A molecular dynamics computer simulation of a one-dimensional repulsive Lennard-Jones system shows that "gaps" exist in the dispersion relation of the dynamic structure factor.

In a recent article de Schepper et al. [1] have pre- sented results of molecular dynamics simulations of the intermediate scattering function, F(k, t), of dense Lennard-Jones-like fluids at small wavevectors, k. They found that there was a "gap" in the dispersion relation of the dynamic structure factor, S(k, ~o), which is the Fourier transform o fF(k , t). This "gap" has also been observed in liquid argon [2] and neon [3], in kinetic theory calculations [4,5] for a dense system of hard spheres, in a one-dimensional model [6] of weakly coupled harmonic chains and in a com- puter simulation [7] of a dense hard sphere fluid. In this note we extend the molecular dynamics calcula- tions of Bishop and Berne [8] for one-dimensional Lennard-Jones systems in order to further test for the existence of the dispersion "gap".

If the position of particle j is x/(t) at time t, then the Fourier transform of the particle density, n (k, t), is

N 1 ~expIikx]( t )] t) (1)

where N is the number of particles in the system. The intermediate scattering function is given by

F(k, t) = (n(k, t) n* (k ,0) ) , (2)

where * denotes the complex conjugate and ( ) indi- cates an ensemble average. Here all quantities are in the usual reduced units [9].

Newton's equations of motion for 100 particles at

0.375-9601/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

a reduced density, p, of 0.935 and interacting through a repulsive Lennard-Jones potential were integrated using the Runga-Kutta-Gil l algorithm with step size 0.005 in reduced time units. This density was selected because the study by Bishop [10] demonstrated that there was a propagating collective mode at this densi- ty. The periodic boundary conditions employed in this investigation fix the smallest wavevector, kmin, as 27r/L where L = NiP. The particle velocities were selected from a maxwellian distribution at a tempera- ture of 0.735 in reduced units. After 1000 equilibra- tion steps, 12000 equilibrium steps were generated and placed on disk for further analysis.

F(k, t) was computed from the stored position values by replacing the ensemble average in eq. (2) by a time average. The resulting functions were Fourier transformed to obtain S(k, co). A dispersion relation was determined by plotting the position of the ~ v~ 0 peak (~max) versus k/kmi n. The result is shown in fig. 1. Bishop and Berne [8] found that in the first Brillouin zone the Lennard-Jones data could be represented by the dispersion relation for a perfect one-dimensional harmonic lattice:

~n = A lsin (nn/N)l . (3)

We have used the first three points of ¢Ama x versus k/kmi n to determine A. The agreement between the Lennard-Jones data and eq. (3) is very good for the first few Brillouin zones but gets progressively worse at higher values ofk . This is expected since our mod- el system is not really a perfect harmonic solid.

219

Volume 115, number 5 PHYSICS LETTERS A 14 April 1986

X

39 r- L

3Z :--

3 0 P

za L a

Z6 F t-

Z2 ~--

2 0 -

is L-

31SPERSION RELflTION +

F

/

;t ÷

A l ÷

Z i

÷

÷

÷

÷

! f

i

! I ]

" t / i + / J ' ~ 1

li tL ! -,/ . i

0 L.[_U~L~ L.l I I~_-L ~ L_I-LJ Lt4.1~.LLLJ-I I-t-L-It~-I.I-~I~-J-J--I-I~IJ-LA IJ-_l ,;___';,: ;L-~ l

0 50 i O0 150 200 250 300 350 400 450 500 550

K/KMIN

Fig. 1. The dispersion relation of the collective modes. + and X represent the results of two independent trajectory calculations.

The X's in the figure are values calculated from a

completely independent trajectory and, thus, give an

indication of the numerical errors in the calculations.

"Gaps" are clearly seen at multiples o f k m in which correspond approximately to multiples o f 27tO. More-

over, the "gaps" are wider for higher k values.

This research has been supported by the Donors

of the Petroleum Research Fund, administered by

the American Chemical Society, and the Manhattan

College Computer Center. We wish to thank Drs.

Bruin, de Schepper and Michels for helpful discus-

sions.

[1] I.M. de Schepper, J.C. van Rijs, A.A. van Well, P. Verkerk, L.A. de Graaf and C. Bruin, Phys. Rev. A 29 (1984) 1602.

220

[2] I.M. de Schepper, P. Verkerk, A.A. van Well and L.A. de Graaf, Phys. Rev. Lett. 50 (1983) 974.

[3] A.A. van Well and L.A. de Graaf, Phys. Rev. A 32 (1985) 2396.

[4] I.M. de Schepper and E.G.D. Cohen, Phys. Rev. A 22 (1980) 287.

[5] I.M. de Schepper and E.G.D. Cohen, J. Stat. Ph,s. 27 (1982) 223.

[6] G. Radons, J. Keller and T. Geisel, Z. Phys. B 50 (1983) 289.

[7] C. Bruin, J.P.J. Michels, J.C. van Rijs, L.A. de Graaf and I.M. de Schepper, Phys. Lett. A 110 (1985) 40.

[8] M. Bishop and B.J. Berne, J. Chem. Phys. 59 (1973) 5337.

[9] M. Bishop, M. Dezosa and J. LaUi, J. Stat. Phys. 25 (1981) 229.

[10] M. Bishop, J. Stat. Phys. 29 (1982) 623.