# Advances in chemical physics - Prigogine I.

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F. Liquid or Solid

All of these simulations were done at temperatures at or near the melting point of carbon monoxide and nitrogen. Therefore we must show that these simulations represent liquids, not solids. The following characteristics of our results all indicate that we are dealing with liquids.

The coefficients of self diffusion for each of these simulations (see Tables

II, III, IV, and V), are all very close to those measured experimentally for liquid CO.52 If we were dealing with solids these coefficients would be an order of magnitude or more smaller.

Following Verlet,44 consider the function p*,R(0 defined by

Pk b(0 = ; £ {cos KXlt) + cosxr,(0 + cos XZ,(<)} (181)

3 i= 1

where К = 4nN1/3IL, L is the length of a side of the cube enclosing the N molecules, and X^t), Y^t), and Zt{t) are the center of mass coordinates of

TABLE IV

Equilibrium Properties from Stockmayer Simulation of CO with = 1.172 Debyes and from Lennard-Jones Plus Quadrupole-Quadrupole Simulation of N2

Liquid CO N2

N 216 216

TK (71.84 ± 2.99)°K (66.29 ± 2.09)°K

TT (69.94 ± 3.59)°K (68.86 ± 2.69)°K

TR (74.67) ± 6.95)°K (62.43 ± 3.29)°K

<F2> (10.54 ± 1.18) x 10"n (dyne)2 (10.22 ± .96) x 10““ (dyne)2

<Ar2> (19.99 ± 1.50) x 10-28 (dyne-cm)2 (18.29 ± 1.29 x 10~28 (dyne-cm)2

(J2) (29.88 ± 2.78) x 10"54 (g cm2/s)2 (23.99 ± 1.27) x 10"54 (g cm2/s)

Djjf 1.86 x 10~5 cm2/s 1.15 x 10~5 cm2/s

<Vc>

N

<Vt>

N

~ —8.1 x 10“14erg ~ —6.6 x 10“14erg --13 x 10~14 erg ~ —8.4 x 10~14 erg

TABLE V

Data for Approximate Memory Functions

Simulation Stockmayer Modified Stockmayer

1.1503 x 10“13/s 0.9564 x 10~13/s

/>)* 0.5710 x 10"13/s

<a2> <v2> 0.6469 x 1026/s2 0.7406 x 1026/s2

<a2> <v2> (1.050 ± .20) x 1052/s4 (1.4067±.12) x 1052/s4

<N2> <J2> 1.2932 x 1026/s2

<$2> <J2> (3.3249±.20) x 1052/s4

133

134

В. J. BERNE AND G. D. HARP

the /th molecule at time t. For a cubic lattice, which was our initial starting configuration in each of the simulations, the distribution of Xt is given by

P(Xi) dx{ = 5^хг - у m = 1, ..., N1/3 (182)

Using this distribution, the mean and variance of pK>R are N and 0, respectively. Therefore Pk)R(0 should be TV for all times in a solid. On the other hand, for a gas or liquid the Xt are uniformly distributed between

0 and L:

dx-

P(Xi) dxt = -~ 0 < xt < L (183)

The mean and variance of pK>R for the uniform distribution are 0 and (Nj2)1/2, respectively. This implies that for a liquid, Pk,r(0 should oscillate around 0 with an amplitude of oscillation of ~(iV)1/2. Plots of Pk>r(0 for the Stockmayer and the modified Stockmayer simulation of carbon monoxide using 512 molecules are shown in Figure 5. The behavior of Pk,r(0 in each °f these simulations is that of a liquid.

Finally, consider the behavior of the mean square displacement of the center of mass of a molecule, <(AR(/))2>. R(t) is the center of mass of the molecule at time t and AR(?) = R(f) — R(0). Rahman has just completed a dynamics study of liquid and solid argon53 in the neighborhood of 84°K— the melting point of argon. He finds that for short times, 0 < t <, 2.5 x 10"13 s, the mean square displacement of an atom in the solid is identical with that of an atom in the liquid at the same temperature. That is, for short times the atoms in both states behave like free gas particles and their displacements are given by (3KTjM)t2. However, the long time behavior of these functions is quite different. In the liquid <(AR(0)2) increases mono-tonically with time. On the other hand, in the solid <(AR(f))2> reaches a maximum value of ~0.5A2 at f=~6xl0-13s and then decreases slightly in an oscillatory fashion to a value of 0.4 A2 at / = 25 x 10“13 s. The main point of interest here is that the mean square displacement in the solid is bounded, whereas in the liquid it is not. Our mean square displacements behave like Rahman’s: they increase monotonically in the real time interval 0 < t < 25 x 10“13 s. (See Figures 35 and 39.) However, we can not yet conclude from this that we are dealing with liquids because it is possible that for t > 25 x 10“13 s these functions will approach some asymptotic value characteristic of a solid. The following arguments suggest that this is not the case: If we were dealing with solids, then these functions would have approached their asymptotic values long before 25 x 10-13 s. Consider a cubic harmonic Debye solid which is a fair

ON THE CALCULATION OF TIME CORRELATION FUNCTIONS

135

22

20

18

16

14

12

10

8

6

4

_ 2 <(E 0 -2 -4 -6 -8 710 -12 -14 -16 -18 -20 -22

• Modified Stockmoyer Simulation ‘'’к,?»1--7 о Stockmayer Simulation </,K,R>;-2-2

• 0 0 О О о

о о о о

10

20

25

30

t ( in 10 13 s )

Fig. 5. The functions pK,R0) from the Stockmayer and modified Stockmayer simulation

of CO.

approximation to solid nitrogen, carbon monoxide, and argon. Vineyard54 has shown that if 0D is the characteristic temperature of a solid of this type, then the mean square displacement of an atom in the lattice is given by:

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