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A neutron scattering study of the structure of supercritical carbon dioxide
23 June 1995
ELSEVIER
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 240 (1995) 84-88
A neutron scattering study of the structure of supercritical carbon dioxide Ryo Ishii a, Susumu Okazaki a,*, Isao Okada a, Michihiro Furusaka b, Noboru Watanabe b, Masakatsu Misawa c, Toshiharu Fukunaga d a Departmentof Electronic Chemistry, TokyoInstitute of Technology, 4259 Nagatsutu, Midori-ku, Yokohama227, Japan b National Laboratoryfor High Energy Physics, Oho, Tsukuba 305, Japan c Departmentof Chemistry, Faculty of Science, Niigatu University, Igarashi-Ninomachi, Niigatu 950-21, Japan d Departmentof Crystalline Materials Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464, Japan
Received 12 January 1995; in final form 6 April 1995
Abstract
The structure factor S(Q) of supercritical carbon dioxide has been measured over a wide range of the scattering vector Q (0.018-30 ,~-1) using two neutron apparatuses. The measured S(Q) has been Fourier transformed into the neutron weighted radial distribution function, GN(r). This measurement enables us to analyze the short-range and long-range structures simultaneously in the supercritical fluid.
1. Introduction
A supercritical fluid is conjectured to change its physical properties such as thermal conductivity, viscosity, diffusion coefficient, etc., with increasing pressure from an attraction-predominant gas-like property to a repulsion-predominant liquid-like one. This may present a key for a better understanding of the many-body effects of molecules on the structure and dynamics in the condensed phase, e.g. the role of interactions and volume. The structure of supercritical fluids [1-9] has been intensively investigated from this viewpoint, independently of the interest in critical exponent of the correlation length near the critical point. Most of the measurements were based upon neutron diffrac* Corresponding author.
tion, since pressure-resistant containers can easily be used, which is not the case with X-ray diffraction. Since supercritical fluids near the critical point have a long-ranged correlation reflecting large density fluctuation which is not observed for ordinary gases or liquids, measured structure factors S ( Q ) have a large value at small Q. So far, structure factors have only been measured, however, either in the small Q range [3] or in the large Q range [1,2,4-9], because two different types of neutron diffractometers must be used to cover both ranges. If we measure S ( Q ) only in the large Q range, for example, the radial distribution function, GN(r), derived by its Fourier transformation will yield an incorrect baseline because of a lack in the long-range density fluctuation. For such substances, Fourier transformation must be performed including the lower Q part.
0009-2614//95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-2614(95)00519-6
R. Ishii et al. / Chemical Physics Letters 240 (1995) 84-88
In the present work, S(Q) of supercritical CO 2 has been measured over a wide range of Q (0.018-30 ,~-1) at 310 K and at 10.1 MPa using two time-offlight neutron apparatuses; one is for a small and medium Q range (0.015-1.3 ,~-1) and the other for a larger Q range (0.3-30 ~ - 1 ) . In the present measurement such a wide range of Q can completely be covered by the two apparatuses. Further measurements along the isotherms of this fluid will provide useful information on the transition of an attractionpredominant short-ranged structure to a repulsionpredominant one.
85
10
A t2 v
o)
V
Neutron diffraction was performed using the small/medium-angle diffractometer (WINK) and the high-intensity total scattering apparatus (HIT) installed at a pulsed spallation neutron source (KENS) at the National Laboratory for High Energy Physics (KEK), Tsukuba, Japan. The WINK utilizes incident neutrons of wide wavelength band (1.0-16 A) from a solid methane moderator at 20 K to cover a small Q range (0.015-12 ,~-1), while the HIT utilizes short wavelength neutrons (0.3-3 ,~) from a roomtemperature H 2 0 moderator to cover a larger Q range (0.3-30 A -1) than the WINK. A cylindrical sample container made of Ti-Zr null alloy (with 8 mm inner diameter and 0.6 mm wall thickness; sample height 40 mm) was used for the measurement by the HIT to avoid Bragg reflection. For the WINK measurement, a cylindrical sample container made of aluminum alloy (JIS A5052) was used to reduce the small angle scattering from the container. In order to avoid the multiple Bragg scattering from the container, the obandwidth of incident neutron was limited to 4.5-16 A, which could cover a Q range (0.015-1.3 ,~-1). As the neutron intensity at lower Q was estimated to be rather weak, we designed a larger container (with 15 mm inner diameter and 3 mm wall thickness; sample height 17 mm). During the experiments, the temperature and pressure of the sample were kept at 310 K and 10.1 MPa, respectively. The observed intensities by the HIT were corrected for the Ti-Zr container, background, absorption [10], multiple scattering [11] and incoherent scattering. The scattering intensities were normalized
~
w-
r I
2. Experimental
V
O.Ol
o.1
1
I i iiitll
i
ii
10
Ql~.t Fig. 1. Structure factor S(Q) for supercritical CO 2 at T = 310 K and P = 10.1 MPa. (@) WINK, ( Q ) HIT and (A) PVT data [14].
using a standard vanadium rod. Correction for the WINK data was almost the same as for the HIT except that a standard water sample [12] was used at smaller angles instead of vanadium for normalization, and the correction for multiple and incoherent scattering was made so that the value of S(Q) in the Faber-Ziman form [13] was consistent with that obtained by the HIT.
3. Results and discussion Fig. 1 presents S(Q) for supercritical CO 2. The value of S(Q --* 0) agrees excellently with that evaluated from the isothermal compressibility [14]. The radial distribution function GN(r) obtained from Fourier transformation of S(Q) is shown in Fig. 2. Two sharp peaks at r = 1.17 and 2.35 A, correspond to intramolecular C - O and O - O distances, respectively. The distribution at r > 3 ~, shows just intermolecular correlation. The first and second nearest neighbor correlations are centered at about 4 and 8.3 A, respectively. It is noted that the correlation is clearly found up to the second nearest neighbor distance, although the density of this fluid is only two thirds of that near the triple point [14]. The so-called Ornstein-Zernike-Debye (OZD) plot is presented in Fig. 3 for S(Q) observed in the low Q region. As the value of S(Q) is not so large
R. Ishii et a L / Chemical Physics Letters 240 (1995) 84-88
86
1.06
2 o
o
o
J
o o
o
1.04
o
9 z
-o
1.02
z
o
0.98
0 2
4
6
8
lO
0
I
I
I
J
I
5
10
15
20
25
~
I 30
~
I 35
= 40
r/~ Fig. 2. Neutron weighted radial distribution function, GN(r), for supercritical CO 2 at T = 310 K and P = 10.1 MPa.
compared with unity at the measured point which is far from the critical point, 1/[S(Q) - 1] is plotted in Fig. 3 to take account of self-scattering from the atoms. The plot shows nearly a straight line. Thus, the long-range pair correlation of molecules, gl(r), due to the density fluctuation can be written in the same way as for the fluid just near the critical point
[15], g l ( r ) = kTKr exp( - r/~ )/4"tr~ 2r,
(1)
0.5 0
0
0 0
0.4
0 0
0 0
6
0 0
0.3
0 0 0
0
0 0
/
0.1
~ 0
I 0.5
~
where ~: is the correlation length. The value of sc was evaluated to be 8.9 ,~ from the slope of the plot. Fig. 4 shows GN(r) in an enlarged scale over a wider r space, which makes the long-range correlation more visible. The GN(r) has a non-unity baseline up to at least 40 A, which is not found in ordinary liquids. The gl(r) calculated from Eq. (1) is plotted by a broken line. It is interesting to see that the deviation of the baseline of the GN(r) is clearly found still for r smaller than 10 .~. This reveals that both density fluctuation and intermolecular structure contribute to the short-range correlation. It is difficult, however, to separate these two contributions quantitatively by experiment alone, since the exclusive volume of the molecule plays a complicated role in GN(r) in relation to the density fluctuation. On the other hand, the contribution from the density fluctuation becomes predominant in GN(r) for r larger than 10 A; the oscillation of the correlation function caused by packing of molecules has almost disappeared in the long-range structure. Such structural information is one of the most interesting characteristics of a supercritical fluid and is obtained by this measurement, to our knowledge, for the first time. The S(Q) and GN(r) thus obtained enable us to analyze some properties related to the structure of o
o
0 0
0.2
Fig. 4. Neutron weighted radial distribution function, GN(r), of an enlarged scale over a wider r space ( ) as well as its long-range correlation, gl(r), from Eq. (1) ( . . . . . ).
I 1
~
I 1.5 02/1
J
I 2
~
I 2.5
3
O-~A "2
Fig. 3. Ornstein-Zernike-Debye plot of S(Q) for supercritical CO 2 at T = 310 K and P = 10.1 MPa.
R. Ishii et aL/ Chemical Physics Letters 240 (1995) 84-88
87
total density fluctuation in the actual fluid. The distance is about four and a half times longer than the correlation length (~ = 8.9 ~,). In conclusion, S(Q) was measured over a wide Q range using two time-of-flight neutron apparatuses. This measurement enables us to discuss the longrange and short-range structures simultaneously in supercritical CO 2. Measurements for this fluid along the isotherms are in progress. The change in the short-range structure from the attraction-predominant one to the repulsion-predominant one as well as the density fluctuation will be clarified using the abovementioned analysis.
0.8 0.6 0.4
0.2 0 -0.2 -0.4 0
5
10
15
20
25
30
35
40
Acknowledgement Fig. 5. The ratio, ~(r), of density fluctuation produced by the correlation within r to that to infinity, see Eq. (2) in the text.
the fluid. For example, based upon the compressibility equation, a ratio, a(r), of the density fluctuation produced by the correlation within r to that to the infinity may be defined as
= (l + 47rPforr'z[GN( r') -- l] dr')/S(O), (2) where N represents the number of atoms, the suffixes r and ~ stand for the integral ranges of the compressibility equation [16]; p is the number density of the atoms and S(0) is the extrapolated value at Q = 0 of S(Q). The Faber-Ziman type radial distribution function GN(r) presents the density fluctuation in the same way as the usual pair correlation function if the intramolecular part is subtracted in the integration of G~(r). This function is helpful for obtaining a spatial picture of the density fluctuation. The c~(r) obtained for this fluid is presented in Fig. 5. For example, 75% covering distance is about 40 ,~. This presents a size of kernel in which the correlation to the central molecule cannot be neglected. For example, molecular dynamics simulation must adopt the basic cell with each side longer than 40 A in order to reproduce roughly 75% of the
The authors are grateful for invaluable discussions with and suggestions from Professor Y. Taniguchi at Ritsumeikan University, Professor M. Nakahara at Kyoto University and Professor K. Nagahama at Tokyo Metropolitan University. The expenses of this work were partly defrayed by a Grant-in-Aid for Scientific Research on Priority Areas (05222207) from the Ministry of Education, Science and Culture, Japan. One of us (RI) has been supported by the Japan Society for the Promotion of Science for Japanese Junior Scientists.
References [1] A.K. Adya and C.J. Wormald, Mol. Phys. 74 (1991) 735. [2] T. Bausenwein, H. Bertagnolli, D. Gutwerk, K. T6dheide and P. Chieux, Mol. Phys. 76 (1992) 127. [3] J.D. Londono, V.M. Shah, G.D. Wignall, H.D. Cochran and P.R. Bienkowski, J. Chem. Phys 99 (1993) 466. [4] P. Postorino, R.H. Tromp, M.A. Ricci, A.K. Soper and G.W. Neilson, Nature 366 (1993) 668. [5] U. Buontempo, P. Postorino, M.A. Ricci and A.K. Soper, Mol. Phys. 81 (1994) 217. [6] G. Strauss, H. Zweier, H. Bertagnolli, T. Bausenwein, K. TiSdheide and P. Chieux, J. Chem. Phys. 101 (1994) 662. [7] T. Bausenwein, H. Bertagnolli, A. David, K. Goller, H. Zweier, K. T6dheide and P. Chieux, J. Chem. Phys. 101 (1994) 672. [8] R.H. Tromp, P. Postorino, G.W. Neilson, M.A. Ricci and A.K. Soper, J. Chem. Phys. 101 (1994) 6210. [9] K. Nishikawa and M. Takematsu, Chem. Phys. Letters 226 (1994) 359. [10] H.H. Paalman and C.J. Pings, J. Appl. Phys. 33 (1962) 2635.
88
R. Ishii et al. / Chemical Physics Letters 240 (1995) 84-88
[11] I.A. Blech and B.L. Averbach, Phys. Rev. 137 A (1965) 1113. [12] G.D. Wignall and F.S. Bates, J. Appl. Cryst. 20 (1987) 28. [13] T.E. Faber and J.M. Ziman, Phil. Mag. 11 (1965) 153. [14] IUPAC thermodynamic tables of the fluid state; carbon dioxide (Pergamon Press, Oxford, 1976).
[15] H.E. Stanley, Introduction to phase transitions and critical phenomena (Clarendon Press, Oxford, 1971). [16] J.-P. Hansen and I.R. McDonald, Theory of simple liquids (Academic Press, New York, 1986).
ELSEVIER
CHEMICAL PHYSICS LETTERS Chemical Physics Letters 240 (1995) 84-88
A neutron scattering study of the structure of supercritical carbon dioxide Ryo Ishii a, Susumu Okazaki a,*, Isao Okada a, Michihiro Furusaka b, Noboru Watanabe b, Masakatsu Misawa c, Toshiharu Fukunaga d a Departmentof Electronic Chemistry, TokyoInstitute of Technology, 4259 Nagatsutu, Midori-ku, Yokohama227, Japan b National Laboratoryfor High Energy Physics, Oho, Tsukuba 305, Japan c Departmentof Chemistry, Faculty of Science, Niigatu University, Igarashi-Ninomachi, Niigatu 950-21, Japan d Departmentof Crystalline Materials Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464, Japan
Received 12 January 1995; in final form 6 April 1995
Abstract
The structure factor S(Q) of supercritical carbon dioxide has been measured over a wide range of the scattering vector Q (0.018-30 ,~-1) using two neutron apparatuses. The measured S(Q) has been Fourier transformed into the neutron weighted radial distribution function, GN(r). This measurement enables us to analyze the short-range and long-range structures simultaneously in the supercritical fluid.
1. Introduction
A supercritical fluid is conjectured to change its physical properties such as thermal conductivity, viscosity, diffusion coefficient, etc., with increasing pressure from an attraction-predominant gas-like property to a repulsion-predominant liquid-like one. This may present a key for a better understanding of the many-body effects of molecules on the structure and dynamics in the condensed phase, e.g. the role of interactions and volume. The structure of supercritical fluids [1-9] has been intensively investigated from this viewpoint, independently of the interest in critical exponent of the correlation length near the critical point. Most of the measurements were based upon neutron diffrac* Corresponding author.
tion, since pressure-resistant containers can easily be used, which is not the case with X-ray diffraction. Since supercritical fluids near the critical point have a long-ranged correlation reflecting large density fluctuation which is not observed for ordinary gases or liquids, measured structure factors S ( Q ) have a large value at small Q. So far, structure factors have only been measured, however, either in the small Q range [3] or in the large Q range [1,2,4-9], because two different types of neutron diffractometers must be used to cover both ranges. If we measure S ( Q ) only in the large Q range, for example, the radial distribution function, GN(r), derived by its Fourier transformation will yield an incorrect baseline because of a lack in the long-range density fluctuation. For such substances, Fourier transformation must be performed including the lower Q part.
0009-2614//95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-2614(95)00519-6
R. Ishii et al. / Chemical Physics Letters 240 (1995) 84-88
In the present work, S(Q) of supercritical CO 2 has been measured over a wide range of Q (0.018-30 ,~-1) at 310 K and at 10.1 MPa using two time-offlight neutron apparatuses; one is for a small and medium Q range (0.015-1.3 ,~-1) and the other for a larger Q range (0.3-30 ~ - 1 ) . In the present measurement such a wide range of Q can completely be covered by the two apparatuses. Further measurements along the isotherms of this fluid will provide useful information on the transition of an attractionpredominant short-ranged structure to a repulsionpredominant one.
85
10
A t2 v
o)
V
Neutron diffraction was performed using the small/medium-angle diffractometer (WINK) and the high-intensity total scattering apparatus (HIT) installed at a pulsed spallation neutron source (KENS) at the National Laboratory for High Energy Physics (KEK), Tsukuba, Japan. The WINK utilizes incident neutrons of wide wavelength band (1.0-16 A) from a solid methane moderator at 20 K to cover a small Q range (0.015-12 ,~-1), while the HIT utilizes short wavelength neutrons (0.3-3 ,~) from a roomtemperature H 2 0 moderator to cover a larger Q range (0.3-30 A -1) than the WINK. A cylindrical sample container made of Ti-Zr null alloy (with 8 mm inner diameter and 0.6 mm wall thickness; sample height 40 mm) was used for the measurement by the HIT to avoid Bragg reflection. For the WINK measurement, a cylindrical sample container made of aluminum alloy (JIS A5052) was used to reduce the small angle scattering from the container. In order to avoid the multiple Bragg scattering from the container, the obandwidth of incident neutron was limited to 4.5-16 A, which could cover a Q range (0.015-1.3 ,~-1). As the neutron intensity at lower Q was estimated to be rather weak, we designed a larger container (with 15 mm inner diameter and 3 mm wall thickness; sample height 17 mm). During the experiments, the temperature and pressure of the sample were kept at 310 K and 10.1 MPa, respectively. The observed intensities by the HIT were corrected for the Ti-Zr container, background, absorption [10], multiple scattering [11] and incoherent scattering. The scattering intensities were normalized
~
w-
r I
2. Experimental
V
O.Ol
o.1
1
I i iiitll
i
ii
10
Ql~.t Fig. 1. Structure factor S(Q) for supercritical CO 2 at T = 310 K and P = 10.1 MPa. (@) WINK, ( Q ) HIT and (A) PVT data [14].
using a standard vanadium rod. Correction for the WINK data was almost the same as for the HIT except that a standard water sample [12] was used at smaller angles instead of vanadium for normalization, and the correction for multiple and incoherent scattering was made so that the value of S(Q) in the Faber-Ziman form [13] was consistent with that obtained by the HIT.
3. Results and discussion Fig. 1 presents S(Q) for supercritical CO 2. The value of S(Q --* 0) agrees excellently with that evaluated from the isothermal compressibility [14]. The radial distribution function GN(r) obtained from Fourier transformation of S(Q) is shown in Fig. 2. Two sharp peaks at r = 1.17 and 2.35 A, correspond to intramolecular C - O and O - O distances, respectively. The distribution at r > 3 ~, shows just intermolecular correlation. The first and second nearest neighbor correlations are centered at about 4 and 8.3 A, respectively. It is noted that the correlation is clearly found up to the second nearest neighbor distance, although the density of this fluid is only two thirds of that near the triple point [14]. The so-called Ornstein-Zernike-Debye (OZD) plot is presented in Fig. 3 for S(Q) observed in the low Q region. As the value of S(Q) is not so large
R. Ishii et a L / Chemical Physics Letters 240 (1995) 84-88
86
1.06
2 o
o
o
J
o o
o
1.04
o
9 z
-o
1.02
z
o
0.98
0 2
4
6
8
lO
0
I
I
I
J
I
5
10
15
20
25
~
I 30
~
I 35
= 40
r/~ Fig. 2. Neutron weighted radial distribution function, GN(r), for supercritical CO 2 at T = 310 K and P = 10.1 MPa.
compared with unity at the measured point which is far from the critical point, 1/[S(Q) - 1] is plotted in Fig. 3 to take account of self-scattering from the atoms. The plot shows nearly a straight line. Thus, the long-range pair correlation of molecules, gl(r), due to the density fluctuation can be written in the same way as for the fluid just near the critical point
[15], g l ( r ) = kTKr exp( - r/~ )/4"tr~ 2r,
(1)
0.5 0
0
0 0
0.4
0 0
0 0
6
0 0
0.3
0 0 0
0
0 0
/
0.1
~ 0
I 0.5
~
where ~: is the correlation length. The value of sc was evaluated to be 8.9 ,~ from the slope of the plot. Fig. 4 shows GN(r) in an enlarged scale over a wider r space, which makes the long-range correlation more visible. The GN(r) has a non-unity baseline up to at least 40 A, which is not found in ordinary liquids. The gl(r) calculated from Eq. (1) is plotted by a broken line. It is interesting to see that the deviation of the baseline of the GN(r) is clearly found still for r smaller than 10 .~. This reveals that both density fluctuation and intermolecular structure contribute to the short-range correlation. It is difficult, however, to separate these two contributions quantitatively by experiment alone, since the exclusive volume of the molecule plays a complicated role in GN(r) in relation to the density fluctuation. On the other hand, the contribution from the density fluctuation becomes predominant in GN(r) for r larger than 10 A; the oscillation of the correlation function caused by packing of molecules has almost disappeared in the long-range structure. Such structural information is one of the most interesting characteristics of a supercritical fluid and is obtained by this measurement, to our knowledge, for the first time. The S(Q) and GN(r) thus obtained enable us to analyze some properties related to the structure of o
o
0 0
0.2
Fig. 4. Neutron weighted radial distribution function, GN(r), of an enlarged scale over a wider r space ( ) as well as its long-range correlation, gl(r), from Eq. (1) ( . . . . . ).
I 1
~
I 1.5 02/1
J
I 2
~
I 2.5
3
O-~A "2
Fig. 3. Ornstein-Zernike-Debye plot of S(Q) for supercritical CO 2 at T = 310 K and P = 10.1 MPa.
R. Ishii et aL/ Chemical Physics Letters 240 (1995) 84-88
87
total density fluctuation in the actual fluid. The distance is about four and a half times longer than the correlation length (~ = 8.9 ~,). In conclusion, S(Q) was measured over a wide Q range using two time-of-flight neutron apparatuses. This measurement enables us to discuss the longrange and short-range structures simultaneously in supercritical CO 2. Measurements for this fluid along the isotherms are in progress. The change in the short-range structure from the attraction-predominant one to the repulsion-predominant one as well as the density fluctuation will be clarified using the abovementioned analysis.
0.8 0.6 0.4
0.2 0 -0.2 -0.4 0
5
10
15
20
25
30
35
40
Acknowledgement Fig. 5. The ratio, ~(r), of density fluctuation produced by the correlation within r to that to infinity, see Eq. (2) in the text.
the fluid. For example, based upon the compressibility equation, a ratio, a(r), of the density fluctuation produced by the correlation within r to that to the infinity may be defined as
= (l + 47rPforr'z[GN( r') -- l] dr')/S(O), (2) where N represents the number of atoms, the suffixes r and ~ stand for the integral ranges of the compressibility equation [16]; p is the number density of the atoms and S(0) is the extrapolated value at Q = 0 of S(Q). The Faber-Ziman type radial distribution function GN(r) presents the density fluctuation in the same way as the usual pair correlation function if the intramolecular part is subtracted in the integration of G~(r). This function is helpful for obtaining a spatial picture of the density fluctuation. The c~(r) obtained for this fluid is presented in Fig. 5. For example, 75% covering distance is about 40 ,~. This presents a size of kernel in which the correlation to the central molecule cannot be neglected. For example, molecular dynamics simulation must adopt the basic cell with each side longer than 40 A in order to reproduce roughly 75% of the
The authors are grateful for invaluable discussions with and suggestions from Professor Y. Taniguchi at Ritsumeikan University, Professor M. Nakahara at Kyoto University and Professor K. Nagahama at Tokyo Metropolitan University. The expenses of this work were partly defrayed by a Grant-in-Aid for Scientific Research on Priority Areas (05222207) from the Ministry of Education, Science and Culture, Japan. One of us (RI) has been supported by the Japan Society for the Promotion of Science for Japanese Junior Scientists.
References [1] A.K. Adya and C.J. Wormald, Mol. Phys. 74 (1991) 735. [2] T. Bausenwein, H. Bertagnolli, D. Gutwerk, K. T6dheide and P. Chieux, Mol. Phys. 76 (1992) 127. [3] J.D. Londono, V.M. Shah, G.D. Wignall, H.D. Cochran and P.R. Bienkowski, J. Chem. Phys 99 (1993) 466. [4] P. Postorino, R.H. Tromp, M.A. Ricci, A.K. Soper and G.W. Neilson, Nature 366 (1993) 668. [5] U. Buontempo, P. Postorino, M.A. Ricci and A.K. Soper, Mol. Phys. 81 (1994) 217. [6] G. Strauss, H. Zweier, H. Bertagnolli, T. Bausenwein, K. TiSdheide and P. Chieux, J. Chem. Phys. 101 (1994) 662. [7] T. Bausenwein, H. Bertagnolli, A. David, K. Goller, H. Zweier, K. T6dheide and P. Chieux, J. Chem. Phys. 101 (1994) 672. [8] R.H. Tromp, P. Postorino, G.W. Neilson, M.A. Ricci and A.K. Soper, J. Chem. Phys. 101 (1994) 6210. [9] K. Nishikawa and M. Takematsu, Chem. Phys. Letters 226 (1994) 359. [10] H.H. Paalman and C.J. Pings, J. Appl. Phys. 33 (1962) 2635.
88
R. Ishii et al. / Chemical Physics Letters 240 (1995) 84-88
[11] I.A. Blech and B.L. Averbach, Phys. Rev. 137 A (1965) 1113. [12] G.D. Wignall and F.S. Bates, J. Appl. Cryst. 20 (1987) 28. [13] T.E. Faber and J.M. Ziman, Phil. Mag. 11 (1965) 153. [14] IUPAC thermodynamic tables of the fluid state; carbon dioxide (Pergamon Press, Oxford, 1976).
[15] H.E. Stanley, Introduction to phase transitions and critical phenomena (Clarendon Press, Oxford, 1971). [16] J.-P. Hansen and I.R. McDonald, Theory of simple liquids (Academic Press, New York, 1986).