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Short wavelength sound dispersion in liquid argon close to solidification
Physica 136B (1986) 168-171 North-Holland, Amsterdam
SHORT WAVELENGTH SOUND DISPERSION IN LIQUID ARGON CLOSE TO SOLIDIFICATION Peter V E R K E R K and Ad A. VAN W E L L Interuniversitair Reactor Instituut, 2629 JB, Delft, The Netherlands
In addition to previously published neutron scattering experiments on liquid argon at 120 K and four pressures up to 400 bar, more data are obtained at 120 K and 19.5, 266.6 and 844 bar. The data at the two lower pressures have been used to check the consistency of the two sets of measurements. The data at 844 bar have been analyzed in terms of three extended hydrodynamic modes for wave numbers 4.2 ~ k ~<22.2 nm 1. Within the experimental error no anomalous dispersion is visible. The sound propagation gap, previously found in liquid argon, neon and computer simulations at liquid densities, decreases with density and disappears completely close to solidification. This has been confirmed by a computer simulation of a Lennard-Jones liquid.
1. Introduction Recently it has been shown, that the dynamic structure factor S(k, to) can be described in terms of three hydrodynamic modes at wave numbers k far exceeding the hydrodynamic region. This has been demonstrated for the Revised Enskog Theory (RET) for hard spheres [1], for liquid argon [2] and liquid neon [3], and for computer simulations of Lennard-Jones like fluids [4] and of a hard-spheres fluid [5]. One of the parameters in the hydrodynamic description is the sound frequency tos, which approaches c~k in the hydrodynamic limit, with cs the adiabatic speed of sound. Zuilhof et al. [6] show that in R E T for hard-spheres fluids a sound propagation gap (tos = 0) exists around kd = 2~r at reduced densities nd 3 > 0.73 (d is the diameter of the particles, n the number density). Bruin et al. [5] show that in a computer simulation of a hard-spheres fluid at nd 3= 0.813 the sound dispersion is in reasonable agreement with RET. The sound propagation gap has also been seen in liquid argon at 120K and densities n = 17.6-20.1nm -3 [7], in liquid neon at 35 K and n =33.4 and 34.6nm -3 [3], and in computer simulations with Lennard-Jones-like potentials at thermodynamic states corresponding with the argon and neon measurements. However, Bruin et al. [5] show that the shape of the dispersion is
considerably different for liquid argon and for the RET. Moreover, the argon data show a decrease of size of the gap with density [7]. In this paper we present data on the d y n a m i c structure factor obtained from a neutron scattering experiment on liquid argon at 120 K and 844 bar (with n = 21.6 nm-3), in order to study the behavior of the sound dispersion at high density close to solidification.
2. Theory We describe the dynamic structure factor with three hydrodynamic modes [1]
S(k, to)
S(k)
-Re
+~
Aj(k)
l
z + wj(k) z=i,o'
(1)
with A 0 and w0 real and A_+ 1 and w_+t either real or complex conjugated pairs: A _ 1 - A * 1 and * The six parameters in eq. 1 can in W_I = W+I. principle be determined from a fit to experimental S(k, to) data. For k---~ 0 this triplet goes over into the Landau-Placzek triplet of linearized hydrodynamics [2]. The sound dispersion is obtained from eq. (1) as tos(k) = lm W + l ( k ). In the case of a sound propagation gap the coefficients Aj and wj are real for j = 0, -+1 and tos = 0.
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
169
P. Verkerk and A . A . van Well / Sound dispersion in liquid argon
3. Experiment The neutron scattering experiment has been performed at the 2 MW reactor at Delft with the rotating-crystal time-of-flight spectrometer RKS I [15]. The incident wavelength was 0.41 nm, the flux on the sample 3 x 104s -1, the flight path between sample and detectors 120cm, and the scattered neutrons were detected at 17 counter banks at scattering angles between 11.8 ° and 92.2 °. The wavenumber range covered was 4.2 nm -1 ~< k ~< 22.2 nm -1. More details are given in refs. 8 and 9. The sample consisted of 36Ar because of its large completely coherent scattering cross section of 77.9 barn [10] for thermal neutrons. The container was made of 5052 aluminium alloy capillary with inner diameter 0.75 mm and wall thickness 0.25 mm. A detailed description of the container including temperature control is given elsewhere [11]. The temperature in the experiment was 120 _+ 0.05 K, the density was accurate within 1%. The measurements were performed at three different pressures: 19.5,266.6 and 844 bar, corresponding with densities n = 17.6, 19.5 and 21.6 nm -3 [12]. Because the lower two densities are equal to two of the four densities of neutron measurements performed with the time-of-flight spectrometer IN-4 of the Institut L a u e - L a n g e v i n Grenoble on liquid argon at 120K and n = 17.6-20.1 nm -3 [13], the consistency of the two experiments can be checked. Also vanadium and the empty container were measured in order to normalize the argon data, to determine the time-of-flight resolution of the spectrometer and to determine the background scattering. Each measurement took approximately 200 hours.
4. Results The neutron scattering data were corrected for background scattering, multiple scattering, resolution, detector efficiency and self shielding, and converted to a symmetrized S(k, to) on a rectangular (k, to)-grid according to ref. 14. The first two corrections mentioned are particularly impor-
tant at the smaller scattering angles since here the scattering by liquid argon is very low, especially at 844 bar where S(k = 0) = 0.045. The fully corrected data at 19.5 and 266.6 bar are consistent with the corresponding results from the IN-4 experiment [9]. The fully corrected S(k, to) of liquid argon at
k=6.0nm
-I
0.016
0.008
I
I
I
I
k = 1 2 . 0 n m -I 0,016
0.008
¢,.,
I
I
I
I
I
I
I
I
i ............
* ............
3 0.8
O.Lt
.........
k = 22, 2nm -I
0,32
0.16
I
0
.....................
5
10
0J(ps-') Fig. 1. Experimental
S(k, ~o) (error
bars) and fitted sum of
three Lorentzians (solid line) obeying the first and second frequency moment.
170
P. Verkerk and A . A . van Well / Sound dispersion in liquid argon
15 coS
[ps-L]
,i~II
I0
}
/
/ I
0
5
I0
15
2O
25 k(nm "1}
Fig. 2. Dispersion relations in argon at 120 K and 844 bar. Solid line: c~k; the error bars represent the experimental results for oJs.
120 K and 844 bar is given in fig. 1 as function of to at a few values of k. The first frequency moment (to) of the unsymmetrized S(k, to) does not deviate more than 10% from its exact value hk2/2M (M being the mass of an argon atom) in the k-range covered by the experiment. The sound dispersion is determined from least squares fits of eq. (1) to the experimental S(k, to) at k = 4.2(0.6)22.2 n m - [ Examples of the fits are given in fig. 1. The uncertainty in tos has been estimated by first fitting eq. (1) with the exact first and second frequency moments imposed on the model S(k, to) and then repeating the fitting with the first, second and third moments imposed [2]. The resulting oJ~(k) is given in fig. 2, the error bars connect the values for tos from the two different fits. Also given is c~k.
5. Discussion
In comparison with the results obtained earlier, mentioned in the introduction, the absence of a -1 sound propagation gap around k = 2 0 n m (where the main peak of the static structure factor is located) is remarkable. It is noteworthy that accurate computer simulations of a LennardJones fluid confirm this result [17]. We tried to compare the present results with the data obtained by Sk61d et al. [16] on liquid argon at 85 K and a density of 21.2 nm -3 by fitting
eq. (1) to these data, but the results for the fitted parameters (e.g. ws) are very inaccurate and it is impossible to draw meaningful conclusions about the sound propagation gap near the triple point. Together with the previously published neutron scattering results [13], S(k, ~o) is now available at five densities along the 120 K-isotherm between the gas-liquid coexistence region and the point of solidification. It appears that the sound propagation gap decreases in size with density and disappears entirely before solidification is reached. This differs from the RET for hard spheres, and could be due to the differences in interaction potential. This assumption could be verified by means of a computer simulation of a hard-spheres fluid at high density (nd 3= 0.8). Within the experimental error no anomalous dispersion is visible, which is consistent with mode coupling theory [7, 18]. References [1] I.M. de Schepper and E.G.D. Cohen, Phys. Rev. A22 (1980) 287; J. Stat. Phys. 27 (1982) 223. [2] I.M. de Schepper, P. Verkerk, A.A. van Well and L.A. de Graaf, Phys. Rev. Lett. 50 (1983) 974. [3} A.A. van Well and L.A. de Graaf, Phys. Rev. A32 (1985) 2396. [4] I.M de Schepper, J.C. van Rijs, A.A. van Well, P. Verkerk, L.A. de Graaf and C. Bruin, Phys. Rev. A29 (1984) 1602. [5] C. Bruin, J.P.J. Michels, J.C. van Rijs, L.A. de Graaf and I.M. de Schepper, Phys. Lett. l l 0 A (1985) 40. [6] M.J. Zuilhof, E.G.D. Cohen and I.M. de Schepper, Phys. Lett. 103A (1984) 120. [7] I.M. de Schepper, P. Verkerk, A.A. van Well and L.A. de Graaf, Phys. Lett. 104A (1984) 29. [8] P. Verkerk, Proc. Symp. Neutron Inelastic Scattering, Vienna 1977 (IAEA, Vienna, 1978) p. 53. [9] P. Verkerk, Ph.D. Thesis, University of Technology, Delft, the Netherlands (1985). [10] V.F. Sears, Thermal-Neutron Scattering Lengths and Cross Sections for Condensed-Matter Research, AECL8490, Chalk River Nuclear Laboratories, Chalk River, Ontario (1984). [11] P. Verkerk and A.M.M. Pruisken, Nucl. Instr. Meth. 160 (1979) 439. [12] R.B. Stewart, R.T. Jacobson, J.H. Becker, J.C.J. Teng and P.K.K. Mui in: Proc. 8th Symp. Thermophysical Properties I, J.W. Sengers, ed. (Am. Soc. Mechanical Engineers, New York, 1982) p. 97. [13] A.A. van Well, P. Verkerk, L.A. de Graaf, J.-B. Suck and J.R.D. Copley, Phys. Rev. A31 (1985) 3391.
P. Verkerk and A . A . van Well / Sound dispersion in liquid argon
[14] P. Verkerk and A.A. van Well, Nucl. Instr. Meth. Phys. Res. A228 (1985) 438. [15] J. Joffrin, ed., Neutron Beam Facilities in Western Europe, July 1981 (European Science Foundation, Strasbourg, 1981). [16] K. Sk61d, J.M, Rowe, G. Ostrowski and P.D. Randolph,
171
Phys. Rev. A6 (1972) 1107, [17] J.C. van Rijs, 1.M. de Schepper, C. Bruin, D.A. van Delft and A.F. Bakker, Phys. Lett. l l l A (1985) 58. [18] M.H. Ernst and J.R. Dorfman, J. Stat. Phys. 12 (1975) 311.
SHORT WAVELENGTH SOUND DISPERSION IN LIQUID ARGON CLOSE TO SOLIDIFICATION Peter V E R K E R K and Ad A. VAN W E L L Interuniversitair Reactor Instituut, 2629 JB, Delft, The Netherlands
In addition to previously published neutron scattering experiments on liquid argon at 120 K and four pressures up to 400 bar, more data are obtained at 120 K and 19.5, 266.6 and 844 bar. The data at the two lower pressures have been used to check the consistency of the two sets of measurements. The data at 844 bar have been analyzed in terms of three extended hydrodynamic modes for wave numbers 4.2 ~ k ~<22.2 nm 1. Within the experimental error no anomalous dispersion is visible. The sound propagation gap, previously found in liquid argon, neon and computer simulations at liquid densities, decreases with density and disappears completely close to solidification. This has been confirmed by a computer simulation of a Lennard-Jones liquid.
1. Introduction Recently it has been shown, that the dynamic structure factor S(k, to) can be described in terms of three hydrodynamic modes at wave numbers k far exceeding the hydrodynamic region. This has been demonstrated for the Revised Enskog Theory (RET) for hard spheres [1], for liquid argon [2] and liquid neon [3], and for computer simulations of Lennard-Jones like fluids [4] and of a hard-spheres fluid [5]. One of the parameters in the hydrodynamic description is the sound frequency tos, which approaches c~k in the hydrodynamic limit, with cs the adiabatic speed of sound. Zuilhof et al. [6] show that in R E T for hard-spheres fluids a sound propagation gap (tos = 0) exists around kd = 2~r at reduced densities nd 3 > 0.73 (d is the diameter of the particles, n the number density). Bruin et al. [5] show that in a computer simulation of a hard-spheres fluid at nd 3= 0.813 the sound dispersion is in reasonable agreement with RET. The sound propagation gap has also been seen in liquid argon at 120K and densities n = 17.6-20.1nm -3 [7], in liquid neon at 35 K and n =33.4 and 34.6nm -3 [3], and in computer simulations with Lennard-Jones-like potentials at thermodynamic states corresponding with the argon and neon measurements. However, Bruin et al. [5] show that the shape of the dispersion is
considerably different for liquid argon and for the RET. Moreover, the argon data show a decrease of size of the gap with density [7]. In this paper we present data on the d y n a m i c structure factor obtained from a neutron scattering experiment on liquid argon at 120 K and 844 bar (with n = 21.6 nm-3), in order to study the behavior of the sound dispersion at high density close to solidification.
2. Theory We describe the dynamic structure factor with three hydrodynamic modes [1]
S(k, to)
S(k)
-Re
+~
Aj(k)
l
z + wj(k) z=i,o'
(1)
with A 0 and w0 real and A_+ 1 and w_+t either real or complex conjugated pairs: A _ 1 - A * 1 and * The six parameters in eq. 1 can in W_I = W+I. principle be determined from a fit to experimental S(k, to) data. For k---~ 0 this triplet goes over into the Landau-Placzek triplet of linearized hydrodynamics [2]. The sound dispersion is obtained from eq. (1) as tos(k) = lm W + l ( k ). In the case of a sound propagation gap the coefficients Aj and wj are real for j = 0, -+1 and tos = 0.
0378-4363/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
169
P. Verkerk and A . A . van Well / Sound dispersion in liquid argon
3. Experiment The neutron scattering experiment has been performed at the 2 MW reactor at Delft with the rotating-crystal time-of-flight spectrometer RKS I [15]. The incident wavelength was 0.41 nm, the flux on the sample 3 x 104s -1, the flight path between sample and detectors 120cm, and the scattered neutrons were detected at 17 counter banks at scattering angles between 11.8 ° and 92.2 °. The wavenumber range covered was 4.2 nm -1 ~< k ~< 22.2 nm -1. More details are given in refs. 8 and 9. The sample consisted of 36Ar because of its large completely coherent scattering cross section of 77.9 barn [10] for thermal neutrons. The container was made of 5052 aluminium alloy capillary with inner diameter 0.75 mm and wall thickness 0.25 mm. A detailed description of the container including temperature control is given elsewhere [11]. The temperature in the experiment was 120 _+ 0.05 K, the density was accurate within 1%. The measurements were performed at three different pressures: 19.5,266.6 and 844 bar, corresponding with densities n = 17.6, 19.5 and 21.6 nm -3 [12]. Because the lower two densities are equal to two of the four densities of neutron measurements performed with the time-of-flight spectrometer IN-4 of the Institut L a u e - L a n g e v i n Grenoble on liquid argon at 120K and n = 17.6-20.1 nm -3 [13], the consistency of the two experiments can be checked. Also vanadium and the empty container were measured in order to normalize the argon data, to determine the time-of-flight resolution of the spectrometer and to determine the background scattering. Each measurement took approximately 200 hours.
4. Results The neutron scattering data were corrected for background scattering, multiple scattering, resolution, detector efficiency and self shielding, and converted to a symmetrized S(k, to) on a rectangular (k, to)-grid according to ref. 14. The first two corrections mentioned are particularly impor-
tant at the smaller scattering angles since here the scattering by liquid argon is very low, especially at 844 bar where S(k = 0) = 0.045. The fully corrected data at 19.5 and 266.6 bar are consistent with the corresponding results from the IN-4 experiment [9]. The fully corrected S(k, to) of liquid argon at
k=6.0nm
-I
0.016
0.008
I
I
I
I
k = 1 2 . 0 n m -I 0,016
0.008
¢,.,
I
I
I
I
I
I
I
I
i ............
* ............
3 0.8
O.Lt
.........
k = 22, 2nm -I
0,32
0.16
I
0
.....................
5
10
0J(ps-') Fig. 1. Experimental
S(k, ~o) (error
bars) and fitted sum of
three Lorentzians (solid line) obeying the first and second frequency moment.
170
P. Verkerk and A . A . van Well / Sound dispersion in liquid argon
15 coS
[ps-L]
,i~II
I0
}
/
/ I
0
5
I0
15
2O
25 k(nm "1}
Fig. 2. Dispersion relations in argon at 120 K and 844 bar. Solid line: c~k; the error bars represent the experimental results for oJs.
120 K and 844 bar is given in fig. 1 as function of to at a few values of k. The first frequency moment (to) of the unsymmetrized S(k, to) does not deviate more than 10% from its exact value hk2/2M (M being the mass of an argon atom) in the k-range covered by the experiment. The sound dispersion is determined from least squares fits of eq. (1) to the experimental S(k, to) at k = 4.2(0.6)22.2 n m - [ Examples of the fits are given in fig. 1. The uncertainty in tos has been estimated by first fitting eq. (1) with the exact first and second frequency moments imposed on the model S(k, to) and then repeating the fitting with the first, second and third moments imposed [2]. The resulting oJ~(k) is given in fig. 2, the error bars connect the values for tos from the two different fits. Also given is c~k.
5. Discussion
In comparison with the results obtained earlier, mentioned in the introduction, the absence of a -1 sound propagation gap around k = 2 0 n m (where the main peak of the static structure factor is located) is remarkable. It is noteworthy that accurate computer simulations of a LennardJones fluid confirm this result [17]. We tried to compare the present results with the data obtained by Sk61d et al. [16] on liquid argon at 85 K and a density of 21.2 nm -3 by fitting
eq. (1) to these data, but the results for the fitted parameters (e.g. ws) are very inaccurate and it is impossible to draw meaningful conclusions about the sound propagation gap near the triple point. Together with the previously published neutron scattering results [13], S(k, ~o) is now available at five densities along the 120 K-isotherm between the gas-liquid coexistence region and the point of solidification. It appears that the sound propagation gap decreases in size with density and disappears entirely before solidification is reached. This differs from the RET for hard spheres, and could be due to the differences in interaction potential. This assumption could be verified by means of a computer simulation of a hard-spheres fluid at high density (nd 3= 0.8). Within the experimental error no anomalous dispersion is visible, which is consistent with mode coupling theory [7, 18]. References [1] I.M. de Schepper and E.G.D. Cohen, Phys. Rev. A22 (1980) 287; J. Stat. Phys. 27 (1982) 223. [2] I.M. de Schepper, P. Verkerk, A.A. van Well and L.A. de Graaf, Phys. Rev. Lett. 50 (1983) 974. [3} A.A. van Well and L.A. de Graaf, Phys. Rev. A32 (1985) 2396. [4] I.M de Schepper, J.C. van Rijs, A.A. van Well, P. Verkerk, L.A. de Graaf and C. Bruin, Phys. Rev. A29 (1984) 1602. [5] C. Bruin, J.P.J. Michels, J.C. van Rijs, L.A. de Graaf and I.M. de Schepper, Phys. Lett. l l 0 A (1985) 40. [6] M.J. Zuilhof, E.G.D. Cohen and I.M. de Schepper, Phys. Lett. 103A (1984) 120. [7] I.M. de Schepper, P. Verkerk, A.A. van Well and L.A. de Graaf, Phys. Lett. 104A (1984) 29. [8] P. Verkerk, Proc. Symp. Neutron Inelastic Scattering, Vienna 1977 (IAEA, Vienna, 1978) p. 53. [9] P. Verkerk, Ph.D. Thesis, University of Technology, Delft, the Netherlands (1985). [10] V.F. Sears, Thermal-Neutron Scattering Lengths and Cross Sections for Condensed-Matter Research, AECL8490, Chalk River Nuclear Laboratories, Chalk River, Ontario (1984). [11] P. Verkerk and A.M.M. Pruisken, Nucl. Instr. Meth. 160 (1979) 439. [12] R.B. Stewart, R.T. Jacobson, J.H. Becker, J.C.J. Teng and P.K.K. Mui in: Proc. 8th Symp. Thermophysical Properties I, J.W. Sengers, ed. (Am. Soc. Mechanical Engineers, New York, 1982) p. 97. [13] A.A. van Well, P. Verkerk, L.A. de Graaf, J.-B. Suck and J.R.D. Copley, Phys. Rev. A31 (1985) 3391.
P. Verkerk and A . A . van Well / Sound dispersion in liquid argon
[14] P. Verkerk and A.A. van Well, Nucl. Instr. Meth. Phys. Res. A228 (1985) 438. [15] J. Joffrin, ed., Neutron Beam Facilities in Western Europe, July 1981 (European Science Foundation, Strasbourg, 1981). [16] K. Sk61d, J.M, Rowe, G. Ostrowski and P.D. Randolph,
171
Phys. Rev. A6 (1972) 1107, [17] J.C. van Rijs, 1.M. de Schepper, C. Bruin, D.A. van Delft and A.F. Bakker, Phys. Lett. l l l A (1985) 58. [18] M.H. Ernst and J.R. Dorfman, J. Stat. Phys. 12 (1975) 311.