Momentum transfer analysis in Lennard-Jones fluids

Volume 84A, number 3

PHYSICS LETTERS

20 July 1981

MOMENTUM TRANSFER ANALYSIS IN LENNARD-JONES FLUIDS

U. BALUCANI and R. VALLAURI Istituto di Elettronica Quantistica del CNR, Florence, Italy

and C.S. MURTHY Royal Holloway College, Egham, Surrey, TW2O OEX, UK Received 14 April 1981

The transfer of momentum of a test particle to its neighbours is investigated in Lennard-Jones fluids by computer experiments at different densities.

The studies of microscopic motions in fluids have received a great impetus after the molecular dynamics (MD) simulations studies performed on hard spheres [1] and Lennard-Jones systems [2]. In both cases the single-particle velocity autocorrelation function (vacf)

paper we try to clarify this role presenting the results of a MD investigation in Lennard-Jones argon, both at intermediate density and near its triple point. In this respect an important piece of information can be obtained not from the vacf
was evaluated at several thermodynamic points. At iiquid densities the vacf has a negative part, explained

from the cross-correlation function(ccf)(u1(0)u2(t)) which describes how the momentum of particle I is

in terms of a rebound of the test particle from the shell of its neighbours. Most of the subsequent theoretical work has been directed toward a more quantitative explanation ofthese high-density features. Several

distributed among the neighbouring atoms taken at specified initial separations. A convenient procedure to obtain the ccf by a MD experiment is the following. We start analysing pairs of atoms whose initial separa-

theories make use of a memory function formalism, the pioneering work being that of Berne et al. [3].

tions are in the first and the second “shells” of neighbours (henceforth referred as I-SON and Il-SON). I-SON includes particles between a0 = 0.9 a and b0 = 1.5 a whereas for the (I + 11)-SON a0 = 0.9 a and b0 = 2.4 a. We evaluate the correlation function of the relative velocity u12(t) = u1(t) — u2(t), i.e.

The simplest approximations of these theories correctly reproduce the short-time dynamics, but the solid-like features of the liquid at short distances are overemphasized, leading to definite oscillations of the vacf not observed in the simulations. An alternative approach was first pursued by Zwanzig and Bixon

[4] who described the motion in terms of a hydrodynamic picture, i.e. in terms of a collective quasimacroscopic behaviour. The negative tail is fairly well reproduced, but the short time agreement is of course

It is evident that neither the two descriptions can account for the dynamics at intermediate times and one may expect that gradually merge into each other, depending on the increasing number of neighbours affected by the motion of a given atom. In this lost.

p(t)

=

(u12(0). u12(t))

,

(1)

both for I-SON and (I + 11)-SON. In detail, the motion for 108 argon atoms interacting through a LennardJones potential (e = 119.8 K and a = 3.405 A) is obtamed by standard methods, using a time step ~t

=

2

s. On a separate record we store the differences between the velocities of a central atom and those of the particles which at t = 0 are either in the I-SON or in the (1+11)-SON with respect to the central one. The corresponding p(t) as given by eq. (1) is computed X 10—14

0031—9163/81/0000—0000/s 02.50 © North-Holland Publishing Company

133

Volume 84A, number 3

PHYSICS LETTERS

over a time interval of 200 ~t. After that, a new set of pairs is chosen and processed, repeating this procedure twenty times. The total average is finally evaluated over nearly 8000 distinct pairs. Once that p(t) is obtained, the usual single-particle vacf is also evaluated . Taking into account the relation p(t)

=

2(u1(0) •u1(t))

—

2(u1(0)’u2(t))

(2)

,

one readily obtains by difference the ccf(u1(0)u2(t))

in the same range of separations as that of p(t). In such a way the momentum transferred to separate shells of neighbours can be analysed as a function of time. The result of this analysis is reported in fig. 1 for = 0.81). The full line represents the normalized vacf, whereas the ccf(normalized to (u~(0)))obtained for atoms starting

argon near its triple point (p* = 0.83, T*

I

1 0.83 T

0.81

0.8

20 July 1981

from the I-SON and the (1+ 11)-SON are given by the dotted and the dashed lines, respectively. The qualitative behaviour is easily understood. Particle 1 loses momentum in the “forward” direction and this is gained by the surrounding particles. Were the central atom 1 plus its first neighbours an isolated system, the momentum of particle 1 would be entirely distributed in the I-SON when the vacf(u1(0) •u1(t)) vanishes. In this case the ccf in the I-SON should present a maximum near t = 0.3 ps and the value of this maximum should be equal to the inverse of the num-

ber N1 of atoms in the I-SON. At the triple point the actual value of N1 turns out to be 11 .9 and the maximum of the normalized ccf should be I/N1 = 0.084, which is definitely higher than the MD finding 0.055. This suggests that the actual number of particles among which the initial momentum of particle 1 is shared at t ~ 0.3 ps is higher, and indeed a much better agreement is found if both the I and the Il-SON are included. In this case N1+2 turns out to be 49.1, leading to a theoretical value of the maximum of 0.020 to be compared with the MD value of 0.024 (dashed line in fig. 1). The short time dynamics of the ccf (u~(0)-

can be exactly evaluated. After some algebra we find (u1(0)’u2(t)> 0.6

(u~(0))

0.4

~°~n(a0,b0) Bao

:

~

0.2

,‘\

/

\

1/

qxlO —// “/-.~/

.~/‘

----v..

0 •

0.5

•

psec

•

1.5

Fig. 1. Velocity autocorrelation function (full line) and crosscorrelation functions in the I-SON and (1+11)-SON (dotted and dashed lines, respectively) for Lennard-Jones argon near its triple pomt. All correlation functions have been divided by . Note the change in the scale of the cross-correlation functions.

134,

~2

—~-+

(3)

where n(a

~..

0

~[b~g’(b0)-a~g’(a0)]

2 1[IP”(r)+w’(r)]~(r) 2 ~l r drr

0, b0) is the number of particles within the range of separations (a0, b0), w(r) and g(r) are the L-J potential and the pair distribution function, respectively, and the primes denote derivatives. The inverse proportionality to n(a0, b0) is in qualitative agreement with the results of fig. 1 in this time regime. more comparison (3) andA the MDquantitative data is possible only upbetween to timeseq. t 0.1 ps: a very good agreement is found for the I-SON, whereas for the (I + 11)-SON eq. (3) predicts values lower that those observed in the MD simulation. In .

.

.

any case, it must be noted that m tins time regime the valuesof the ccfare very small and accuracy problems

Volume 84A, number 3

PHYSICS LETTERS

probably limit a meaningful comparison,

Similar data for the vacf and the ccf obtained at a = 0.53 and a higher temperature T* = 2.45 are reported in fig. 2. In this case the vacf does not present a negative part, as well as the ccf. Again, lower density p”

20 July 1981

lost and the ccf attain their maximum values. The subsequent decay of the ccf describes the spread of the initial momentum over a larger and larger number of particles, but the features of this decay appear to be

different for the gas and for the liquid. In the first

the data at very shcrt times are semiquantitatively ac-

case the vacf is still positive: particle 1 continues to

counted by eq. (3). The initial momentum of particle 1 is gradually lost to larger and larger number of “neighbours” and after t 0.5 ps the vacf shows a

of the ccf which is relatively slow. In the liquid case the

nearly exponential decay, as expected from a descrip-

negative momentum is transferred to its neighbours,

tion

in terms of a simple Langevin equation in which

the friction coefficient is time independent,

velocity of particle 1 is soon reversed and part of this leading to a faster decay of the ccf. Taken alone, this process would ultimately lead to correlated oscillations

At both densities the time evolution of the various ccf can be qualitatively interpreted considering their features in subsequent time regimes. Initially, the momentum of particle 1 is simply transferred to its irnmediate neighbours, which on the average move “forward” in the same direction. Therefore, here the ccf are positive and increase as these neighbours gain momentum. Afterwards,

lose “forward” momentum, thus delaying the decrease

also the atoms of the Il-SON

gradually begin to be “pushed”, either directly or through the I-SON particles. At this point an important fraction of the momentum of particle 1 has been

both in the vacf and in the ccf. A plateau in both functions is instead observed. Beyond, say, 0.7 ps the number of the involved particles is so high (more than fifty) that a clear phase relationship between the vacf and the ccf is lost. In this regime a hydrodynamic treatment for the vacf is clearly justified, and any difference between the ccf involving atoms initially in the I-SON or in the (I + 11)-SON should indeed disappear as observed in the MD experiments. In fig. 1 another remarkable long-time feature is apparent in the ccf atliquid density,namely a negative peak near t = 1.25 ps. As expected from the previous arguments, this feature is nearly identical for the ccf

in the I-SON and for that in the (I + 11)-SON. Roughly speaking, this negative region indicates an average re-

1

bound of the particles which were initially around

particle 1. Because the number of particles involved 0.8

-0

in the process is so large, this rebound occurs by colli-

T= 2.45

sions with particles other than particle 1 whose velocity on the average does not change sign. The effect appears to be different in more cohesive

fluids, as found by Gaskell and Mason [5]. In a theoretical investigation of cooperative motions in liquid metals they evaluated the time behaviour of momen-

0.6

tum transfer between a test particle and the particles 0.

in the I-SON in liquid rubidium and found an oscilla-

.~-,

tory behaviour at long times in correspondence with a 0.2

.-

——

— — —

similar one in the vacf. The same analysis for rigidsphere fluids led to a cross correlation function which

..

/

•

/ /

“~•. —

0

0.5

psec

1

approaches zero from the positive side even if the vacf shows a shallow negative part. The Lennard-Jones

~

1.5

Fig. 2. The same as in fig. 1, but for Lennard-Jones argon at room temperature and intermediate density, in the gas phase.

liquid presented here

is in be-

tween the two extreme cases studied by Gaskell and Mason. The attractive part of the potential is not strong enough so to allow for an oscillatory type of

motion of a single particle. Instead the velocity field 135

Volume 84A, number 3

built up around the test particle, at t =

PHYSICS LETTERS 1.25

ps, is

likely to sustain its backward motion. In conclusion, the analysis of momentum transfer

between a test particle and its neighbours gives a better insight of the dynamical processes leading to the decay of single particle vacf. The role of the attractive part of the interaction potential is particularly important in the long-time regime and will be the subject of further MD investigations, We gratefully thank Prof. K. Singer for discussions and valuable suggestions during the progress of the

work.

136

20 July 1981

References [1] B.J.AlderandT.E.Wainwright, Phys. Rev. Al (1970) 18. [2] A. Rahman,Phys. Rev. 136 (1964) 405; andBoon L. Verlet, Phys. Rev.J. A2 (1970) 2514. [3] D. B.J.Levesque Berne, J.P. and S.A. Rice, Chem. Phys. 45

(1966) 1086. [4] R. Zwanzig and M. Bixon, Phys. Rev. A2 (1970) 2005. [5] T. Gaskell and P.E. Mason, J. Phys. Cl4 (1981) 561,and

references therein.