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Microscopic mechanism of the cage effect in simple liquids
Volume 95A, number 2
PHYSICS LETTERS
18 April 1983
MICROSCOPIC MECHANISM OF THE CAGE EFFECT IN SIMPLE LIQUIDS Y. ENDO
Department of Physics, Tokyo Metropolitan University, Tokyo 158, Japan and H. ENDO
Department of Physics, Hitotsubashi University, Tokyo 186, Japan Received 6 December 1982
Atomic motions in a soft-core liquid are classified by molecular dynamics simulation into subgroups which are characterized by the number of atoms surrounding a central atom. It is found that the rigidity and lifetime of the microstructure of the cage depend on the local environment around the central atom.
In the study of liquids by molecular dynamics (MD) simulations, one generally calculates the averaged velocity autocorrelation function (VAFs) over all atoms in a system. From this study one infers the atomic motions in liquids in the averaged sense. At high density and low temperature, the VAF has a deep negative minimum followed by a shallow secondary minimum or a plateau [ 1,2]. The negative region of the VAF has been explained in terms of reversed motion of the central atom backscattered, on the average, by a cage formed by its surrounding atoms. Recently, Dean and Kushick [3] studied the role of attractive forces in the cage effect in a Lennard-Jones liquid, and pointed out that the principal role of the attractive force is tO enhance the cohesiveness of the cages. Balucani et al. [4] discussed the correlated motions of a central atom with its neighbouring shells by means of the cross correlation functions. The correlation functions averaged over all atoms in the system are rather of a macroscopic nature, and therefore mask the influence of the local microstructures. Fehder [5] presented instantaneous configuration and trajectories of all atoms in dense liquids by graphical displays. These motion pictures give most microscopic information; however, it is difficult to extract quantitative information on the atomic motions. In recent papers [6,7] we resolved the VAF of a 92
soft-core liquid into several components. This was an attempt to classify the average motions of a liquid atom into several basic modes. We develop in the present paper an alternative way of classification of the VAF. In liquids a temporal local structure around a central atom varies in space according to the number and the positions of the surrounding atoms. In order to describe quantitatively the microscopic atomic motions and the microscopic mechanism of the cage effect, we classify the individual VAF of all atoms into subgroups which reflect the local environment of each central atom. The model liquid under study is composed of 500 particles interacting through a soft-core potential, V(r) = e(o/r) n, with n = 15. In the soft-core model the state is identified by the reduced density p* = p ( e / k T ) 3/n (p = p t o 3 ; p i is the number density)• MD simulation is carried out for p -- 0.85, and a state with p = 0•9893 is obtained• This p* is close to the freezing density , o f = 1•06. At first we calculate the radial distribution function (RDF) of this system, and determine the radius r 1 of the first shell from the first minimum of the RDF. The value of r I is estimated to be 1.480. Then, we count the number s of surrounding atoms within a sphere with radius r 1 around each central atom at an initial time t o . Thereafter, we calculate the VAF of individual atoms• This procedure is
0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
Volume 95A, number 2
PHYSICS LETTERS
Table 1 Frequency of subgroups. The local atomic environments are characterized by the number s, the n u m b e r of surrounding atoms within 1.48o.
18 April 1983 ~sH') 0.2 0.1
s
Frequency
s=9
Relative frequency 0
7 8 9 10 11 12 13 14 15 16 17
8 275 3603 21051 56341 68655 38640 10137 1217 72 1
4.00 1.38 1.80 1.05 2.82 3.43 1.93 5.07 6.09 3.60 5.00
x 10 -5 x 10 -3 X 10 -2 × 10 -1 x 10 -1 x 10 -1 x 10 -1 × 10 -2 X 10 -3 × 10 -4 x 10 -6
-0.1
-f
0.5 s:lO
s=11
S=12
5=13 0.1
repeated 400 times, and then 500 X 400 types o f individual VAFs are classified into subgroups characterized by the number s. In table 1 the number s, its frequency and relative frequency are listed. It is seen from table 1 that the average number o f atoms within r 1 is 11.76. Fig. 1 shows the macroscopic V A F xI,(t) averaged over all 500 X 400 atoms. In fig. 2 is shown the microscopic V A F q~s(t) averaged over atoms belonging to each subgroup presented in table l. The frequency in the subgroups s = 7, 8, 15, 16, 17 is not large enough to take an ensemble average, so the ~s(t) belonging to such subgroups are omitted from fig. 2. From q~s(t) we can obtain quantitatively microscopic information about the individual motion of
x~t) 1.0
0.5
.5
Fig. 1. Macroscopic velocity autocorrelation function ~ ( t ) averaged over all. 500 x 400 atoms. The time t is expressed in units zo = p'-l/a(m/kT)l/2 (p' is the n u m b e r density).
0
S=14
-O.I -0.2~
Fig. 2. Microscopic velocity autocorrelation function q~s(t) averaged over atoms belonging to the subgroups s = 9, 10, 11, 12, 13, 14 shown in table 1. Since the short-time features of the q,s(t) are similar to that of q,(t), they are omitted.
each atom embedded in its local environment, and from which we can infer the microscopic mechanism of the cage effect. The time at which ~s(t) first becomes negative is insensitive to the number s. This indicates that the position o f the first shell formed by the surrounding atoms does not change with s. This first negative minimum of ~s(t) becomes deeper with increasing s. This suggests that, when the central atom is scattered in the backward direction by the cage wall, the central atom feels a more steep potential as s increases. A remarkable difference among the ~s(t) is found in the time region after the first minimum. ~s(t) for s = 9 and 10 has a positive maximum around t = 0.25, and changes in sign more than once. In these cases the number of atoms within r I is fewer than the average number of atoms, 11.76, then atoms distribute more densely outside the sphere o f the radius r 1 . In other words, the atoms constructing the cage are enclosed further by a strong outer wall. Consequently, such a
93
Volume 95A, number 2
PHYSICS LETTERS
cage would be hard to deform. This may be the reason why the central atom for s = 9 and 10 is rebounded more than once until the initial correlation decays. The oscillatory motion of atoms in the low local density region persist s longer than in the highly densed local region, though the oscillation is very weak. For s larger than 10, XI's(t) remains negative at intermediate times. ~s(t), for s = 11 and 12, is distinguished by the negative maximum around t = 0.28. Since this negative maximum can be regarded as the remainder of the positive maximum appeared at almost the same position for s = 9 and 10, we can infer the time-dependent microscopic mechanism of the caging effect from the systematic change of these maxima. After the central atom is rebounded firstly by the cage wall, it reverses its direction and then encounters the opposite cage wall. If the opposite cage wall remains rigid, the central atom would be rebounded once more in the same direction as the initial. This is the case for s = 9 or 10. In the case of s = 11 and 12, the opposite cage wall is not so rigid as to be able to rebound the central atom, but is rigid enough to reduce the velocity of the central atom. Since in this case the velocity of the central atom reduced sufficiently, the atoms constructing the cage wall moving in the same direction with the central atom can catch up and push the central atom once again, producing a secondary m i n i m u m around t = 0.39. This secondary minimum corresponds essentially to that observed at s = 9 and 10 in the second period of oscillation. For s = 13 and 14, the shape of the secondary minimum of xI's(t ) becomes like a plateau. In contrast to the case of s = 9 and 10, the number of atoms within r 1 is more than the average number of atoms, 11.76,
94
18 April 1983
then atoms distribute dilutely outside the sphere of radius r 1 . In other words, the atoms constructing the cage are enclosed by a weak outer wall. Consequently, such a cage would be easy to deform, though at the early stage the cage is more rigid than that for smaller s. This may be the reason why the maximum following the first deep negative m i n i m u m disappears at s = 13 and 14. The present investigation is based on the study of the systematic change of the ~s(t). A further investigation on the microkinetics of liquid atoms will be presented in a forthcoming paper from the study of the time-dependent RDF and the correlations between the central atom and its surroundings in each subgroup. We would like to thank Professors P.A. Egelstaff and K. Suzuki for many enlightening discussions. A grant of computing time in FACOM M-180 II AD SYSTEM by the Computer Center of Hitotsubashi University is also acknowledged.
References [1] A. Rahman, Phys. Rev. 136 (1964) 405. [2] D. Levesque and L. Verlet, Phys. Rev. A2 (1970) 2514. [3] D.P. Dean and J.N. Kushick, J. Chem. Phys. 76 (1982) 619. [4] V. Balucani, R. VaUauriand C.S. Murthy, Phys. Lett. 84A (1981) 133; J. Chem. Phys. 77 (1982) 3233. [51 P.L. Fehder, J. Chem. Phys. 50 (1969) 2617. [6] H. Endo and Y. Endo, Prog. Theor. Phys. 66 (1981) 794. [ 7] H. Endo, Y. Endo and N. Ogita, J. Chem. Phys. 77 (1982) 5184.
PHYSICS LETTERS
18 April 1983
MICROSCOPIC MECHANISM OF THE CAGE EFFECT IN SIMPLE LIQUIDS Y. ENDO
Department of Physics, Tokyo Metropolitan University, Tokyo 158, Japan and H. ENDO
Department of Physics, Hitotsubashi University, Tokyo 186, Japan Received 6 December 1982
Atomic motions in a soft-core liquid are classified by molecular dynamics simulation into subgroups which are characterized by the number of atoms surrounding a central atom. It is found that the rigidity and lifetime of the microstructure of the cage depend on the local environment around the central atom.
In the study of liquids by molecular dynamics (MD) simulations, one generally calculates the averaged velocity autocorrelation function (VAFs) over all atoms in a system. From this study one infers the atomic motions in liquids in the averaged sense. At high density and low temperature, the VAF has a deep negative minimum followed by a shallow secondary minimum or a plateau [ 1,2]. The negative region of the VAF has been explained in terms of reversed motion of the central atom backscattered, on the average, by a cage formed by its surrounding atoms. Recently, Dean and Kushick [3] studied the role of attractive forces in the cage effect in a Lennard-Jones liquid, and pointed out that the principal role of the attractive force is tO enhance the cohesiveness of the cages. Balucani et al. [4] discussed the correlated motions of a central atom with its neighbouring shells by means of the cross correlation functions. The correlation functions averaged over all atoms in the system are rather of a macroscopic nature, and therefore mask the influence of the local microstructures. Fehder [5] presented instantaneous configuration and trajectories of all atoms in dense liquids by graphical displays. These motion pictures give most microscopic information; however, it is difficult to extract quantitative information on the atomic motions. In recent papers [6,7] we resolved the VAF of a 92
soft-core liquid into several components. This was an attempt to classify the average motions of a liquid atom into several basic modes. We develop in the present paper an alternative way of classification of the VAF. In liquids a temporal local structure around a central atom varies in space according to the number and the positions of the surrounding atoms. In order to describe quantitatively the microscopic atomic motions and the microscopic mechanism of the cage effect, we classify the individual VAF of all atoms into subgroups which reflect the local environment of each central atom. The model liquid under study is composed of 500 particles interacting through a soft-core potential, V(r) = e(o/r) n, with n = 15. In the soft-core model the state is identified by the reduced density p* = p ( e / k T ) 3/n (p = p t o 3 ; p i is the number density)• MD simulation is carried out for p -- 0.85, and a state with p = 0•9893 is obtained• This p* is close to the freezing density , o f = 1•06. At first we calculate the radial distribution function (RDF) of this system, and determine the radius r 1 of the first shell from the first minimum of the RDF. The value of r I is estimated to be 1.480. Then, we count the number s of surrounding atoms within a sphere with radius r 1 around each central atom at an initial time t o . Thereafter, we calculate the VAF of individual atoms• This procedure is
0 0 3 1 - 9 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
Volume 95A, number 2
PHYSICS LETTERS
Table 1 Frequency of subgroups. The local atomic environments are characterized by the number s, the n u m b e r of surrounding atoms within 1.48o.
18 April 1983 ~sH') 0.2 0.1
s
Frequency
s=9
Relative frequency 0
7 8 9 10 11 12 13 14 15 16 17
8 275 3603 21051 56341 68655 38640 10137 1217 72 1
4.00 1.38 1.80 1.05 2.82 3.43 1.93 5.07 6.09 3.60 5.00
x 10 -5 x 10 -3 X 10 -2 × 10 -1 x 10 -1 x 10 -1 x 10 -1 × 10 -2 X 10 -3 × 10 -4 x 10 -6
-0.1
-f
0.5 s:lO
s=11
S=12
5=13 0.1
repeated 400 times, and then 500 X 400 types o f individual VAFs are classified into subgroups characterized by the number s. In table 1 the number s, its frequency and relative frequency are listed. It is seen from table 1 that the average number o f atoms within r 1 is 11.76. Fig. 1 shows the macroscopic V A F xI,(t) averaged over all 500 X 400 atoms. In fig. 2 is shown the microscopic V A F q~s(t) averaged over atoms belonging to each subgroup presented in table l. The frequency in the subgroups s = 7, 8, 15, 16, 17 is not large enough to take an ensemble average, so the ~s(t) belonging to such subgroups are omitted from fig. 2. From q~s(t) we can obtain quantitatively microscopic information about the individual motion of
x~t) 1.0
0.5
.5
Fig. 1. Macroscopic velocity autocorrelation function ~ ( t ) averaged over all. 500 x 400 atoms. The time t is expressed in units zo = p'-l/a(m/kT)l/2 (p' is the n u m b e r density).
0
S=14
-O.I -0.2~
Fig. 2. Microscopic velocity autocorrelation function q~s(t) averaged over atoms belonging to the subgroups s = 9, 10, 11, 12, 13, 14 shown in table 1. Since the short-time features of the q,s(t) are similar to that of q,(t), they are omitted.
each atom embedded in its local environment, and from which we can infer the microscopic mechanism of the cage effect. The time at which ~s(t) first becomes negative is insensitive to the number s. This indicates that the position o f the first shell formed by the surrounding atoms does not change with s. This first negative minimum of ~s(t) becomes deeper with increasing s. This suggests that, when the central atom is scattered in the backward direction by the cage wall, the central atom feels a more steep potential as s increases. A remarkable difference among the ~s(t) is found in the time region after the first minimum. ~s(t) for s = 9 and 10 has a positive maximum around t = 0.25, and changes in sign more than once. In these cases the number of atoms within r I is fewer than the average number of atoms, 11.76, then atoms distribute more densely outside the sphere o f the radius r 1 . In other words, the atoms constructing the cage are enclosed further by a strong outer wall. Consequently, such a
93
Volume 95A, number 2
PHYSICS LETTERS
cage would be hard to deform. This may be the reason why the central atom for s = 9 and 10 is rebounded more than once until the initial correlation decays. The oscillatory motion of atoms in the low local density region persist s longer than in the highly densed local region, though the oscillation is very weak. For s larger than 10, XI's(t) remains negative at intermediate times. ~s(t), for s = 11 and 12, is distinguished by the negative maximum around t = 0.28. Since this negative maximum can be regarded as the remainder of the positive maximum appeared at almost the same position for s = 9 and 10, we can infer the time-dependent microscopic mechanism of the caging effect from the systematic change of these maxima. After the central atom is rebounded firstly by the cage wall, it reverses its direction and then encounters the opposite cage wall. If the opposite cage wall remains rigid, the central atom would be rebounded once more in the same direction as the initial. This is the case for s = 9 or 10. In the case of s = 11 and 12, the opposite cage wall is not so rigid as to be able to rebound the central atom, but is rigid enough to reduce the velocity of the central atom. Since in this case the velocity of the central atom reduced sufficiently, the atoms constructing the cage wall moving in the same direction with the central atom can catch up and push the central atom once again, producing a secondary m i n i m u m around t = 0.39. This secondary minimum corresponds essentially to that observed at s = 9 and 10 in the second period of oscillation. For s = 13 and 14, the shape of the secondary minimum of xI's(t ) becomes like a plateau. In contrast to the case of s = 9 and 10, the number of atoms within r 1 is more than the average number of atoms, 11.76,
94
18 April 1983
then atoms distribute dilutely outside the sphere of radius r 1 . In other words, the atoms constructing the cage are enclosed by a weak outer wall. Consequently, such a cage would be easy to deform, though at the early stage the cage is more rigid than that for smaller s. This may be the reason why the maximum following the first deep negative m i n i m u m disappears at s = 13 and 14. The present investigation is based on the study of the systematic change of the ~s(t). A further investigation on the microkinetics of liquid atoms will be presented in a forthcoming paper from the study of the time-dependent RDF and the correlations between the central atom and its surroundings in each subgroup. We would like to thank Professors P.A. Egelstaff and K. Suzuki for many enlightening discussions. A grant of computing time in FACOM M-180 II AD SYSTEM by the Computer Center of Hitotsubashi University is also acknowledged.
References [1] A. Rahman, Phys. Rev. 136 (1964) 405. [2] D. Levesque and L. Verlet, Phys. Rev. A2 (1970) 2514. [3] D.P. Dean and J.N. Kushick, J. Chem. Phys. 76 (1982) 619. [4] V. Balucani, R. VaUauriand C.S. Murthy, Phys. Lett. 84A (1981) 133; J. Chem. Phys. 77 (1982) 3233. [51 P.L. Fehder, J. Chem. Phys. 50 (1969) 2617. [6] H. Endo and Y. Endo, Prog. Theor. Phys. 66 (1981) 794. [ 7] H. Endo, Y. Endo and N. Ogita, J. Chem. Phys. 77 (1982) 5184.