- Email: [email protected]
Theoretical studies on the cluster structure in the supercritical area
Fluid Phase Equilibria 144 Ž1998. 279–286
Theoretical studies on the cluster structure in the supercritical area Osamu Kitao a
a, )
, Kazutoshi Tanabe a , Shuichiro Ono a , Sin’ichiro Kumakura b , Koichiro Nakanishi c
Department of Physical Chemistry, National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305, Japan b Graduate School of Engineering, Kyoto UniÕersity, Kyoto 606-01, Japan c College of Science and Industrial Technology, Kurashiki UniÕersity of Science and the Arts, Kurashiki 712, Japan Received 13 January 1997; accepted 22 July 1997
Abstract In supercritical fluid, solvent molecules form some characteristic clusters around solute molecules. The cluster structures were investigated by molecular dynamics simulations using supercritical carbon dioxide fluid containing a naphthalene molecule. Diffusion coefficient of the naphthalene molecule was confirmed to show a large fluctuation when trajectory of the solute was treated by a time scale of 100 ps Žpicosecond.. On the basis of radial distribution function between the solute and solvents, we summarized that the fluctuation of the diffusion coefficient of the solute molecule had a relation to a fluctuation of the solvent fluid density. This investigation gave us an image how we considered the cluster structure in the supercritical area. q 1998 Elsevier Science B.V. Keywords: Supercritical fluid; Carbon dioxide; Naphthalene; Molecular dynamics; Cluster; Diffusion coefficient; Density fluctuation
1. Introduction Supercritical fluid is a fluid heated to above the critical temperature and compressed to above the critical density. This fluid can move between states of high and low density without phase transition. Since the supercritical fluid can change the density continuously, a slight change of temperature or pressure manipulates thermodynamic and transport properties of this fluid. Due to this unique property, many fundamental and applied researches have been carried out on the supercritical fluids for both reaction and extraction processes w1x. Carbon dioxide fluid has been often used as a tractable solvent for the extraction process by the supercritical fluid. At the first step of the extraction, solvent )
Corresponding author.
0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 Ž 9 7 . 0 0 2 7 2 - 0
280
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
molecules form characteristic clusters around the solute molecules. Detailed information on the cluster and solvation structure is extremely important to improve the process. We have already done MO ŽMolecular Orbital. and MC Ž Monte Carlo. simulation studies on this topic w2–4x and here we report the results of MD Ž Molecular Dynamics. simulation studies on a system of carbon dioxide fluid containing one naphthalene molecule. Using the MD results, we calculated diffusion coefficients of the naphthalene molecule in the carbon dioxide fluid. Although our simulation time is too short to compare with experimental data, the computational data give us important information to consider the solvent cluster structures around the solute molecule in the supercritical area.
2. Calculational details MD simulation was done within the NVE ensemble. The system consisted of 300 carbon dioxide molecules and one naphthalene molecule with periodic boundary conditions. The intermolecular potential energy was described by Murthy et al. w5x for carbon dioxide molecules and our work w6x for carbon dioxide and naphthalene molecule. The equation of motion was integrated by the Gear predictor–corrector method with the time step of 1 fs Ž femtosecond. . After checking an equilibration by monitoring change of total energy, several average values were calculated from one million steps. The temperature was set to about 320 K, and the densities were 0.25, 0.50, and 0.75 grcm3.
3. Results and discussion 3.1. Diffusion coefficients by simulations of 1 ns We calculated diffusion coefficients of one naphthalene molecule in supercritical carbon dioxide fluid. The diffusion coefficient D was estimated from gradient of MSD Žmean square displacement. of the solute molecule. The MSD was presented in Fig. 1 for three densities. These plots confirmed that our statistics were enough to estimate the diffusion coefficient of one solute by the Einstein’s relation w7x. We used these diffusion coefficients to investigate characteristic cluster structure of carbon dioxide molecules around a naphthalene molecule. First, we calculated the diffusion coefficient by records of 1 ns Žnanosecond. and summarized them in Table 1. Since an experimental diffusion coefficient is 1.27 = 10y8 m2rs at 109 atm and 3.18 K w8x, our calculation reproduces well the order of magnitude, though the experimental value around the critical point is far from stable one. Pure carbon dioxide fluid has the critical point at 304 K, 73 atm, and 0.466 grcm3, and our system is expected to have a similar value due to the dilute mixture. 3.2. Diffusion coefficients by simulations of 100 ps Next, we recalculated the diffusion coefficients at 100 ps Žpicosecond. intervals and summarized the values in Table 2. We can see several fluctuations of the diffusion coefficients especially near critical density Ž0.50 grcm3 . and at the higher one Ž0.75 grcm3 .. We have proceeded the detailed analyses on the drastic change at 0.50 grcm3, period 6–7 Ž1.5 = 10y8 m2rs. and period 7–8
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
281
Fig. 1. MSD at 0.25 grcm3 Ždotted line., 0.50 grcm3 Žsolid line., and 0.75 grcm3 Žbroken line.. In this simulation, one step is 1 fs.
Ž0.89 = 10y8 m2rs.; in the latter period, the diffusion coefficient was reduced to half of the previous period. Throughout the text, we call the former data 1, and the latter data 2, respectively. 3.3. Trajectories of naphthalene molecule We followed the trajectories of naphthalene molecule concerning data 1 and data 2 and plotted them in Fig. 2. Axes for coordinates x, y, and z were drawn in these figures, and arrows were also drawn for the trajectories of the solute in data 1. We can see that the solute moves in a large area in data 1, and localizes in a small area in data 2. In addition to the fluctuation of the diffusion coefficient for the short term, we can confirm the fluctuation of the diffusion of the solute molecule from these trajectories. 3.4. Lifetime of carbon dioxide cluster around a naphthalene molecule Lifetime was calculated for clusters of carbon dioxide molecules around a naphthalene molecule. The clusters were defined so that the solute molecule interacted with solvent molecules over 1.5 kJrmol in the absolute value. The threshold was determined as follows. Ž1. MC simulations were done for the same system under the similar condition w6x. Ž2. RDF Žradial distribution functions. were calculated between the carbon dioxide molecules and Table 1 Diffusion coefficients D for 1 ns Density Žgrcm3 . D Ž10y8 m2 rs.
0.25 2.32
0.50 1.19
0.75 0.73
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
282
Table 2 Diffusion coefficients for every 100 ps Ž10y8 m2 rs. Density Žgrcm3 . 0.25 0.50 0.75
Period Ž100 ps.
Average
0–1
1–2
2–3
3–4
4–5
5–6
6–7
7–8
8–9
9–10
2.38 1.17 0.68
1.79 1.21 0.83
1.93 1.32 0.62
2.01 1.09 0.75
2.88 1.39 1.27
2.25 1.33 0.52
1.92 1.51 0.63
1.80 0.89 0.54
2.14 1.29 1.26
2.45 1.46 1.27
2.32 1.19 0.73
the naphthalene molecule. The whole space around the solute molecule was divided into three equal parts by viewing from the origin of the naphthalene molecule: top, side, and plane. For each space, coordination number of the solvent molecules was accumulated until the first end of the first peak. These numbers were 1.8 Ž top. , 6.4 Ž side. , and 4.5 Ž plane. . Therefore, total coordination number was 12.7. Ž3. MD simulations were done and coordination number was calculated by simple RDF between the solute and the solvent from the record of 1 ns. The above threshold value was determined so that the coordination number would be 12.7. On the basis of this definition for the coordination, the lifetime of clusters by carbon dioxide molecules around the solute was estimated to be less than 5 ps. The solvent molecules were confirmed not to localize around the solute molecule for a long time as 100 ps. Therefore, the change of the effective mass of the cluster around the solute has little relation to the change of the diffusion coefficient of the solute. 3.5. Radial distribution function In order to investigate the reason that determines the above difference of diffusion coefficients, RDF was calculated for both data 1 and data 2, and summarized in Fig. 3. In these figures, the solid line is due to carbon dioxide molecules, and the dotted line is due to carbon dioxide molecule and naphthalene molecule. Since the RDF by MD has more statistical information than that by MC, we used only MD results here to investigate the characteristic cluster structures around the solute molecule. The RDF between carbon dioxide molecules were also plotted in the same figures. Since the RDF are results from 300 molecules, we cannot see interesting information from the difference between solid line of data 1 and that of data 2. The RDF between carbon dioxide molecule naphthalene molecule should have information to explain the large difference of the diffusion coefficient between data 1 and data 2. First we see the cluster part of these figures. The dotted lines, RDF between the solute and the ˚ and 6 A. ˚ The former peak is due to carbon dioxide solvent, have two broad peaks around 4 A molecules on the top space of the solute molecule, and the latter is due to those on the plane space of the solute molecule. These contents have already been confirmed by MC simulations and analyses on the results by each of the three spaces described in Section 3.4. Next we investigate the outer part of the cluster region. The interesting difference is seen after the main peaks of the RDF between carbon dioxide molecules and naphthalene molecule. The plot of data 1 converges to one just after the main peaks; on the other hand, in the plot for data 2, the curve continues above one. This difference suggests that, in the outer region of the cluster, carbon dioxide molecules exists more in data 2 than in data 1. To confirm these results, statistical analysis was added
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
283
Fig. 2. Trajectories of naphthalene for data 1 Župper. and data 2 Žbottom. in supercritical carbon dioxide fluid of 0.50 grcm3.
to the RDF in Fig. 3. We accumulated the number of carbon dioxide molecules around the solute and ˚ at which the RDF of carbon dioxide got a ratio between data 2. The result was 1.05 before 7.35 A, molecule and naphthalene molecule had a minimum. The value was 1.06 when the accumulation was ˚ These results are consistent with the difference of diffusion coefficients between done before 10 A. two trajectories, although they do not reflect directly a large ratio of the diffusion coefficients: 1.70. Fig. 4 shows one snapshot from data 2. We can see that the naphthalene molecule is located in the dense area of carbon dioxide fluid.
284
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
285
Fig. 4. One snapshot from data 2. The naphthalene molecule is surrounded by a circle.
There may be two possibilities to explain the reason for the drastic change of the diffusion coefficient. One is the change in effective mass of the cluster caused by solvent molecules around the solute molecule, and another is the density fluctuation of the solvent fluid where the solute molecules move. The investigation concerning the lifetime of the cluster denied the first possibility. The difference of the RDF suggested a relationship between the fluctuation of the density of the solvent fluid and that of the diffusion coefficient.
Fig. 3. RDF for data 1 Župper. and data 2 Žbottom.. The solid lines are due to carbon dioxide molecules and the dotted lines are due to carbon dioxide molecule and naphthalene molecule.
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
286
4. Conclusion Carbon dioxide fluid containing a naphthalene molecule was investigated by NVE molecular dynamics simulations in the supercritical area. Diffusion coefficient of the naphthalene molecule was calculated and the computational value by the simulation of 1 ns was confirmed to reproduce well the experimental data with the order of magnitude. On the other hand, the value summarized by the simulation of 100 ps showed a large fluctuation. Moreover, the trajectory of each record reflected the fluctuation of the diffusion coefficient. The lifetime of the cluster of solvent molecules around the solute was estimated to be less than 5 ps and the solvent molecules were confirmed not to locate around the solute molecule for as long as 100 ps. RDF between the solvent molecule and solute molecule had some information to explain the fluctuation. During the slow diffusion period, the solvent carbon dioxide molecules existed a lot around the solute molecule both for cluster region and the outer part of the cluster; namely, the solute moved in the dense area of the solvent fluid. As a conclusion, the fluctuation of the solvent fluid density was confirmed to have some relation to the large fluctuation of the diffusion coefficient of the solute molecule.
Acknowledgements We would like to acknowledge Drs. M. Mikami and H. Tanaka for their useful comments, and Dr. T. Matsumoto for his support in drawing the figures. The present calculations have been carried out at the Supercomputer Laboratory at the Institute for Chemical Research, Kyoto University, the Computer Center of the Institute of Molecular Science, and National Institute of Materials and Chemical Research. This study has been partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture, Japan.
References w1x w2x w3x w4x w5x w6x w7x w8x
S. Saito, J. Supercritical Fluids 8 Ž1995. 177. J.-W. Shen, K.B. Domanski, O. Kitao, K. Nakanishi, Fluid Phase Equilib. 104 Ž1995. 375. J.-W. Shen, O. Kitao, K. Nakanishi, Fluid Phase Equilib. 120 Ž1996. 81. J.-W. Shen, S. Kumakura, O. Kitao, K. Nakanishi, Fluid Phase Equilib. 116 Ž1996. 304. C.S. Murthy, S.F. O’Shea, I.R. McDonald, Mol. Phys. 50 Ž1983. 531. S. Kumakura, O. Kitao, K. Nakanishi, under preparation. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford Univ. Press, Oxford, 1987. M.B. Iomtev, Y.V. Tsekhanskaya, Russ. J. Phys. Chem. 38 Ž1964. 485.
Theoretical studies on the cluster structure in the supercritical area Osamu Kitao a
a, )
, Kazutoshi Tanabe a , Shuichiro Ono a , Sin’ichiro Kumakura b , Koichiro Nakanishi c
Department of Physical Chemistry, National Institute of Materials and Chemical Research, Tsukuba, Ibaraki 305, Japan b Graduate School of Engineering, Kyoto UniÕersity, Kyoto 606-01, Japan c College of Science and Industrial Technology, Kurashiki UniÕersity of Science and the Arts, Kurashiki 712, Japan Received 13 January 1997; accepted 22 July 1997
Abstract In supercritical fluid, solvent molecules form some characteristic clusters around solute molecules. The cluster structures were investigated by molecular dynamics simulations using supercritical carbon dioxide fluid containing a naphthalene molecule. Diffusion coefficient of the naphthalene molecule was confirmed to show a large fluctuation when trajectory of the solute was treated by a time scale of 100 ps Žpicosecond.. On the basis of radial distribution function between the solute and solvents, we summarized that the fluctuation of the diffusion coefficient of the solute molecule had a relation to a fluctuation of the solvent fluid density. This investigation gave us an image how we considered the cluster structure in the supercritical area. q 1998 Elsevier Science B.V. Keywords: Supercritical fluid; Carbon dioxide; Naphthalene; Molecular dynamics; Cluster; Diffusion coefficient; Density fluctuation
1. Introduction Supercritical fluid is a fluid heated to above the critical temperature and compressed to above the critical density. This fluid can move between states of high and low density without phase transition. Since the supercritical fluid can change the density continuously, a slight change of temperature or pressure manipulates thermodynamic and transport properties of this fluid. Due to this unique property, many fundamental and applied researches have been carried out on the supercritical fluids for both reaction and extraction processes w1x. Carbon dioxide fluid has been often used as a tractable solvent for the extraction process by the supercritical fluid. At the first step of the extraction, solvent )
Corresponding author.
0378-3812r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII S 0 3 7 8 - 3 8 1 2 Ž 9 7 . 0 0 2 7 2 - 0
280
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
molecules form characteristic clusters around the solute molecules. Detailed information on the cluster and solvation structure is extremely important to improve the process. We have already done MO ŽMolecular Orbital. and MC Ž Monte Carlo. simulation studies on this topic w2–4x and here we report the results of MD Ž Molecular Dynamics. simulation studies on a system of carbon dioxide fluid containing one naphthalene molecule. Using the MD results, we calculated diffusion coefficients of the naphthalene molecule in the carbon dioxide fluid. Although our simulation time is too short to compare with experimental data, the computational data give us important information to consider the solvent cluster structures around the solute molecule in the supercritical area.
2. Calculational details MD simulation was done within the NVE ensemble. The system consisted of 300 carbon dioxide molecules and one naphthalene molecule with periodic boundary conditions. The intermolecular potential energy was described by Murthy et al. w5x for carbon dioxide molecules and our work w6x for carbon dioxide and naphthalene molecule. The equation of motion was integrated by the Gear predictor–corrector method with the time step of 1 fs Ž femtosecond. . After checking an equilibration by monitoring change of total energy, several average values were calculated from one million steps. The temperature was set to about 320 K, and the densities were 0.25, 0.50, and 0.75 grcm3.
3. Results and discussion 3.1. Diffusion coefficients by simulations of 1 ns We calculated diffusion coefficients of one naphthalene molecule in supercritical carbon dioxide fluid. The diffusion coefficient D was estimated from gradient of MSD Žmean square displacement. of the solute molecule. The MSD was presented in Fig. 1 for three densities. These plots confirmed that our statistics were enough to estimate the diffusion coefficient of one solute by the Einstein’s relation w7x. We used these diffusion coefficients to investigate characteristic cluster structure of carbon dioxide molecules around a naphthalene molecule. First, we calculated the diffusion coefficient by records of 1 ns Žnanosecond. and summarized them in Table 1. Since an experimental diffusion coefficient is 1.27 = 10y8 m2rs at 109 atm and 3.18 K w8x, our calculation reproduces well the order of magnitude, though the experimental value around the critical point is far from stable one. Pure carbon dioxide fluid has the critical point at 304 K, 73 atm, and 0.466 grcm3, and our system is expected to have a similar value due to the dilute mixture. 3.2. Diffusion coefficients by simulations of 100 ps Next, we recalculated the diffusion coefficients at 100 ps Žpicosecond. intervals and summarized the values in Table 2. We can see several fluctuations of the diffusion coefficients especially near critical density Ž0.50 grcm3 . and at the higher one Ž0.75 grcm3 .. We have proceeded the detailed analyses on the drastic change at 0.50 grcm3, period 6–7 Ž1.5 = 10y8 m2rs. and period 7–8
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
281
Fig. 1. MSD at 0.25 grcm3 Ždotted line., 0.50 grcm3 Žsolid line., and 0.75 grcm3 Žbroken line.. In this simulation, one step is 1 fs.
Ž0.89 = 10y8 m2rs.; in the latter period, the diffusion coefficient was reduced to half of the previous period. Throughout the text, we call the former data 1, and the latter data 2, respectively. 3.3. Trajectories of naphthalene molecule We followed the trajectories of naphthalene molecule concerning data 1 and data 2 and plotted them in Fig. 2. Axes for coordinates x, y, and z were drawn in these figures, and arrows were also drawn for the trajectories of the solute in data 1. We can see that the solute moves in a large area in data 1, and localizes in a small area in data 2. In addition to the fluctuation of the diffusion coefficient for the short term, we can confirm the fluctuation of the diffusion of the solute molecule from these trajectories. 3.4. Lifetime of carbon dioxide cluster around a naphthalene molecule Lifetime was calculated for clusters of carbon dioxide molecules around a naphthalene molecule. The clusters were defined so that the solute molecule interacted with solvent molecules over 1.5 kJrmol in the absolute value. The threshold was determined as follows. Ž1. MC simulations were done for the same system under the similar condition w6x. Ž2. RDF Žradial distribution functions. were calculated between the carbon dioxide molecules and Table 1 Diffusion coefficients D for 1 ns Density Žgrcm3 . D Ž10y8 m2 rs.
0.25 2.32
0.50 1.19
0.75 0.73
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
282
Table 2 Diffusion coefficients for every 100 ps Ž10y8 m2 rs. Density Žgrcm3 . 0.25 0.50 0.75
Period Ž100 ps.
Average
0–1
1–2
2–3
3–4
4–5
5–6
6–7
7–8
8–9
9–10
2.38 1.17 0.68
1.79 1.21 0.83
1.93 1.32 0.62
2.01 1.09 0.75
2.88 1.39 1.27
2.25 1.33 0.52
1.92 1.51 0.63
1.80 0.89 0.54
2.14 1.29 1.26
2.45 1.46 1.27
2.32 1.19 0.73
the naphthalene molecule. The whole space around the solute molecule was divided into three equal parts by viewing from the origin of the naphthalene molecule: top, side, and plane. For each space, coordination number of the solvent molecules was accumulated until the first end of the first peak. These numbers were 1.8 Ž top. , 6.4 Ž side. , and 4.5 Ž plane. . Therefore, total coordination number was 12.7. Ž3. MD simulations were done and coordination number was calculated by simple RDF between the solute and the solvent from the record of 1 ns. The above threshold value was determined so that the coordination number would be 12.7. On the basis of this definition for the coordination, the lifetime of clusters by carbon dioxide molecules around the solute was estimated to be less than 5 ps. The solvent molecules were confirmed not to localize around the solute molecule for a long time as 100 ps. Therefore, the change of the effective mass of the cluster around the solute has little relation to the change of the diffusion coefficient of the solute. 3.5. Radial distribution function In order to investigate the reason that determines the above difference of diffusion coefficients, RDF was calculated for both data 1 and data 2, and summarized in Fig. 3. In these figures, the solid line is due to carbon dioxide molecules, and the dotted line is due to carbon dioxide molecule and naphthalene molecule. Since the RDF by MD has more statistical information than that by MC, we used only MD results here to investigate the characteristic cluster structures around the solute molecule. The RDF between carbon dioxide molecules were also plotted in the same figures. Since the RDF are results from 300 molecules, we cannot see interesting information from the difference between solid line of data 1 and that of data 2. The RDF between carbon dioxide molecule naphthalene molecule should have information to explain the large difference of the diffusion coefficient between data 1 and data 2. First we see the cluster part of these figures. The dotted lines, RDF between the solute and the ˚ and 6 A. ˚ The former peak is due to carbon dioxide solvent, have two broad peaks around 4 A molecules on the top space of the solute molecule, and the latter is due to those on the plane space of the solute molecule. These contents have already been confirmed by MC simulations and analyses on the results by each of the three spaces described in Section 3.4. Next we investigate the outer part of the cluster region. The interesting difference is seen after the main peaks of the RDF between carbon dioxide molecules and naphthalene molecule. The plot of data 1 converges to one just after the main peaks; on the other hand, in the plot for data 2, the curve continues above one. This difference suggests that, in the outer region of the cluster, carbon dioxide molecules exists more in data 2 than in data 1. To confirm these results, statistical analysis was added
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
283
Fig. 2. Trajectories of naphthalene for data 1 Župper. and data 2 Žbottom. in supercritical carbon dioxide fluid of 0.50 grcm3.
to the RDF in Fig. 3. We accumulated the number of carbon dioxide molecules around the solute and ˚ at which the RDF of carbon dioxide got a ratio between data 2. The result was 1.05 before 7.35 A, molecule and naphthalene molecule had a minimum. The value was 1.06 when the accumulation was ˚ These results are consistent with the difference of diffusion coefficients between done before 10 A. two trajectories, although they do not reflect directly a large ratio of the diffusion coefficients: 1.70. Fig. 4 shows one snapshot from data 2. We can see that the naphthalene molecule is located in the dense area of carbon dioxide fluid.
284
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
285
Fig. 4. One snapshot from data 2. The naphthalene molecule is surrounded by a circle.
There may be two possibilities to explain the reason for the drastic change of the diffusion coefficient. One is the change in effective mass of the cluster caused by solvent molecules around the solute molecule, and another is the density fluctuation of the solvent fluid where the solute molecules move. The investigation concerning the lifetime of the cluster denied the first possibility. The difference of the RDF suggested a relationship between the fluctuation of the density of the solvent fluid and that of the diffusion coefficient.
Fig. 3. RDF for data 1 Župper. and data 2 Žbottom.. The solid lines are due to carbon dioxide molecules and the dotted lines are due to carbon dioxide molecule and naphthalene molecule.
O. Kitao et al.r Fluid Phase Equilibria 144 (1998) 279–286
286
4. Conclusion Carbon dioxide fluid containing a naphthalene molecule was investigated by NVE molecular dynamics simulations in the supercritical area. Diffusion coefficient of the naphthalene molecule was calculated and the computational value by the simulation of 1 ns was confirmed to reproduce well the experimental data with the order of magnitude. On the other hand, the value summarized by the simulation of 100 ps showed a large fluctuation. Moreover, the trajectory of each record reflected the fluctuation of the diffusion coefficient. The lifetime of the cluster of solvent molecules around the solute was estimated to be less than 5 ps and the solvent molecules were confirmed not to locate around the solute molecule for as long as 100 ps. RDF between the solvent molecule and solute molecule had some information to explain the fluctuation. During the slow diffusion period, the solvent carbon dioxide molecules existed a lot around the solute molecule both for cluster region and the outer part of the cluster; namely, the solute moved in the dense area of the solvent fluid. As a conclusion, the fluctuation of the solvent fluid density was confirmed to have some relation to the large fluctuation of the diffusion coefficient of the solute molecule.
Acknowledgements We would like to acknowledge Drs. M. Mikami and H. Tanaka for their useful comments, and Dr. T. Matsumoto for his support in drawing the figures. The present calculations have been carried out at the Supercomputer Laboratory at the Institute for Chemical Research, Kyoto University, the Computer Center of the Institute of Molecular Science, and National Institute of Materials and Chemical Research. This study has been partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture, Japan.
References w1x w2x w3x w4x w5x w6x w7x w8x
S. Saito, J. Supercritical Fluids 8 Ž1995. 177. J.-W. Shen, K.B. Domanski, O. Kitao, K. Nakanishi, Fluid Phase Equilib. 104 Ž1995. 375. J.-W. Shen, O. Kitao, K. Nakanishi, Fluid Phase Equilib. 120 Ž1996. 81. J.-W. Shen, S. Kumakura, O. Kitao, K. Nakanishi, Fluid Phase Equilib. 116 Ž1996. 304. C.S. Murthy, S.F. O’Shea, I.R. McDonald, Mol. Phys. 50 Ž1983. 531. S. Kumakura, O. Kitao, K. Nakanishi, under preparation. M.P. Allen, D.J. Tildesley, Computer Simulation of Liquids, Oxford Univ. Press, Oxford, 1987. M.B. Iomtev, Y.V. Tsekhanskaya, Russ. J. Phys. Chem. 38 Ž1964. 485.