A method for determining the density fluctuations of supercritical fluids absolutely based on small-angle scattering experiments and application to supercritical methanol

Density fluctuation is a vital concept for understanding disordered systems. A supercritical fluid is a typical disordered system having extremely large inhomogeneity. To determine the density fluctuations using a scattering method, the key physical quantities are the fluid density and the normalized scattering intensity, as well as the small-angle scattering signals. Here, we propose a methodology to obtain all of these quantities absolutely from a scattering experiment. Normalization of scattering intensity relating to the number of molecules per unit volume was performed using fluid density evaluated directly from in situ measurements of the X-ray absorption coefficients. Conversion of scattering intensity to absolute value concerning scattering volume was achieved utilizing the value of the density fluctuation in the ideal state. An analysis of supercritical carbon dioxide confirmed the validity of the present method. By applying this method, the density fluctuations of supercritical methanol were quantitatively determined for the first time.


Introduction
A supercritical fluid is a fluid above its critical temperature. 1) Derived from the specific balance of molecular interactions, [1][2][3][4] the density of a supercritical fluid changes drastically without passing through a gas-liquid phase transition. The density change varies continuously through the intermediate-density range from gas-like to liquid-like values. The dependence of the supercritical fluid structure on the fluid density has been intensively investigated from the viewpoints of both scientific interest and fundamental understanding for the sustainable engineering applications. 5,6) In view of its structural disorder, a supercritical fluid is a typical disordered system with extremely large structural inhomogeneity in molecular distribution. [7][8][9] The structural fluctuations are specifically observed even in neat and onecomponent systems. If a histogram of the number of molecules in an aggregate is constructed for a supercritical fluid, the frequency in the histogram follows an exponential function, rather than a Gaussian function. 9) Therefore, as a key concept, information about the average structure does not extract the primary structural knowledge for supercritical fluids.
Investigation of the mesoscopic length scale is an effective approach for understanding disordered systems. This is because the intermolecular aggregation and the disordered molecular distribution are observed on the mesoscale. Smallangle scattering is one of the most effective methods for structure investigation on aggregation and inhomogeneity at the mesoscale lengths. 10) On the basis of the density-density correlation function, the size distribution in aggregates is discussed in terms of the correlation length, ξ. The correlation length is the decay constant in the density-density correlation function, as / / { ( )} x -r r exp . Chu and Lin have performed small-angle X-ray scattering (SAXS) experiments for supercritical carbon dioxide (CO 2 ) in the vicinity of the critical point. 11) The study found that the value of ξ takes a maximum around the critical density and significantly depends on the fluid temperature. Ishii et al. have reported small-angle neutron scattering study for various substances, indicating that the values of ξ universally take maxima around the critical density independent of the substance. 12) Arai et al. have reported the density dependence of the correlation length of benzene in supercritical states. 13) This study found that the reduced correlation lengths normalized by the molecular size are uniformly discussed in the same manner as inhomogeneities in the number density of the molecular distribution, which are defined as the density fluctuations.
The density fluctuation is a physical parameter quantitatively representing the structural inhomogeneity itself. 14,15) Therefore, the density fluctuation is a vital concept for understanding structure in disordered systems. The density fluctuations are thermodynamically related to the isothermal compressibility, which is the second derivative of the Gibbs free energy, G. Using the small-angle scattering method, this higher-order derivative can be obtained directly from the zero-angle scattering-intensity evaluated from the small-angle scattering signals. [16][17][18][19] Density fluctuations in supercritical states have been investigated for various substances. Nishikawa et al. have investigated the density fluctuations for various types of molecular liquids. 13,18,[20][21][22][23] Their systematic investigations discovered the existence of a ridge of density fluctuations (the so-called Nishikawa ridge) 7,[24][25][26][27][28] in the region of the Widom line. 29) On the basis of the concept of the Nishikawa ridge, the transition between gaslike and liquid-like structures in the supercritical state is discussed from the viewpoints of the inhomogeneous bandwidth of Raman scattering 30) and the radial distribution function obtained from X-ray diffraction experiments. 31) Tamura et al. have investigated the static structure and its fluctuations in supercritical fluid metals. 32,33) Inui and coworkers have clarified the structural fluctuations and the dynamics of supercritical metallic fluids. [34][35][36] The primary structural characteristics of neat supercritical fluids have been unveiled in terms of density fluctuations.
Nishikawa and coworkers have applied the isothermal compressibility calculated thermodynamically using the equation of state (EOS) to normalize scattering signals to their absolute values. 18) In general, the accuracy of an EOS drops off in the critical region. To evaluate fluctuations in the mixing state of binary solutions, comparisons with thermodynamic quantities or standard liquid samples have been employed for the scattering-intensity normalization. [37][38][39] Water is commonly used as a standard sample. Considering the application of this procedure to the present experiment, it is concerned that unintentional mixing of the standard liquid with the sample fluids because the present type of experiment has to be performed without disassembling the sample container to keep the path length constant. In addition, a typical technique for determining the absolute value of the scattering-intensity in various scattering experiments has been to utilize the scattering intensity of glassy carbon as a basis. 40) However, a method that uses solid standard samples is not appropriate for the present study due to the inability to fill the sample container precisely with a solid sample.
This issue concerning the normalization of the scattering intensity limits investigations of the dependence of density fluctuations on molecular structure and, correspondingly, on various types of intermolecular interactions. Here, we report an improved method for determining density fluctuations in the supercritical region. The method is based entirely on a small-angle scattering experiment only of the target sample. The normalization procedure reported here has not been performed previously because of its experimental difficulty. The reason for this is as follows: (1) it has not previously been possible to obtain the scattering signals with sufficient signal-to-noise (S/N) ratios in the extremely low-density region, in which the scattering intensity is quite weak. (2) The stability of sample thickness under high-pressure and/or high-temperature conditions, which is related to the size of the scattering volume, has not previously been sufficient for the utilization of the present method.

SAXS measurements
The SAXS intensities of supercritical fluids were measured using the SAXS apparatus set at the BL-15A2 station 41) of the Photon Factory (PF) of the High Energy Accelerator Research Organization (KEK), Tsukuba, Japan. During the SAXS measurements, the X-ray absorption coefficients of the samples were measured simultaneously using an in situ beam-monitoring apparatus with a silicon photodiode device. [41][42][43] The scattering signals were measured using the two-dimensional semiconductor detector of PILATUS 2 M (Dectris). The recorded two-dimensional scattering signals were converted into one-dimensional data by carefully considering and masking parasitic scattering using FIT2D software. 44) The pressure of the sample fluid was measured using an IDOS universal Pressure Module (20 MPa range, Druck) digital pressure sensor backed up by a DPI 150 (Druck) pressure indicator to within ±0.005 MPa. The background scattering was acquired under vacuum conditions at the same temperature studied.
For the CO 2 measurements, the SAXS experiments were performed along an isotherm at 317.50 K. The temperature of the sample fluid was monitored with a platinum resistance thermometer (HAYASHI DENKO) using the four-wire method, backed up by a DMM 2700 digital multimeter (Keithley). The temperature sensor was directly inserted into the sample fluid. The temperature-measurement system was calibrated using a standard Model 5612 (Hart Scientific) thermometer backed up by a 1502 A (Hart Scientific) temperature indicator to within ±0.01 K. The sample temperature was controlled using a K10 (HAAKE) circulator. Other conditions for the CO 2 measurements were as follows: X-ray wavelength, 1.2036 Å; camera length (distance from the sample to detector), 2644 mm; exposure time, 60 s; and typical temperature fluctuation, ±0.02 K.
For the methanol measurements, the SAXS experiments were performed along an isotherm at 533.2 K. The temperature of the sample fluid was monitored with a platinum resistance thermometer of Plamic Pt 100Ω (Netsushin) with 1.6 mm in diameter using the four-wire method. The platinum resistance element (MC-0403, Netsushin) was selected to be a size of 0.4 mm in diameter and 3 mm in length for monitoring at elevated temperatures. The sensor was backed up by a DB1000 (Chino) temperature indicator/controller. The temperature sensor was directly inserted into the sample fluid. The temperature-measurement system was calibrated using a standard Model 5612 (Hart Scientific) thermometer backed up by a 1502 A (Hart Scientific) temperature indicator to within ±0.02 K. Cartridge heaters were inserted into the cell body for controlling the sample temperature. Other conditions for the methanol measurements were as follows:

Sample holder
A high-temperature and high-pressure sample holder was designed and constructed for the present measurements. The key design concepts were (1) to keep the path length constant even under high-pressure and/or high-temperature conditions, (2) to prevent sample convection inside the cell, and (3) to insert a temperature sensor directly and horizontally into the cell to provide an accurate temperature monitor. The cell body was made of titanium alloy (Ti−6Al−4 V). The X-ray windows were single-crystal diamond disks of 5.0 mm in diameter and 0.7 mm in thickness, which were sealed into the central part of the cell body using a handmade gold O-ring and a handmade platinum spacer. The preparation procedure of the gold O-ring is described in Fig. A.1 of the Appendix. The basic design for the window seal has been reported in our previous study. [45][46][47] Figure 1 shows a photograph of the sample holder set at the BL-15A2 station of the PF.

Results and discussion
3.1. Determination method using CO 2 Figure 2 shows the data points measured for supercritical CO 2 in the pressure-temperature and density-temperature planes. The reduced temperature of the selected isotherm was T/T c = 1.04; i.e. 4% above the critical temperature. 48) The critical temperature defined in the EOS literature 48) is used to determine the reduced temperature. The density of supercritical CO 2 was evaluated directly using the X-ray absorption coefficients, which were acquired simultaneously during the measurements of the SAXS signals, as described later. As shown in Fig. 2, the present measurements were performed over a wide density (or pressure) range, including the extremely low-density region. The normalization of the scattering intensities concerning the scattering volume was examined using the data in the extremely low-density region. Figure 3 shows the SAXS profiles of supercritical CO 2 along an isotherm at 317.50 K over a wide density range. In the supercritical region, large forward scattering is observable due to the large structural inhomogeneity of the molecular distribution. The increase in SAXS represents a qualitative enhancement of the structural disorder in the supercritical fluid. From the carefully one-dimensionalized SAXS signals, the scattering intensity in the zero-angle limit was determined using an extrapolation procedure based on the Ornstein-Zernike theory. 15) Figure 4 shows the evaluated density of supercritical CO 2 along the isotherm at 317.50 K. The fluid density was evaluated directly using the X-ray absorption coefficient of the sample, m rℓ m (m : m mass absorption coefficient, ρ: density, and ℓ: sample thickness), which was simultaneously acquired using the in situ direct-beam-monitoring apparatus. [41][42][43] The absorption coefficient was determined using the incident and transmitted X-ray intensities according to the Beer-Lambert relation. To determine the density from the absorption coefficient, we employed the parameter m ℓ m rather than evaluating the two individual parameters m m and ℓ.
The present density data agree well with the density values calculated using the fundamental EOS of CO 2 , 48) which is known to have high accuracy even in the critical region. The comparison, therefore, demonstrates that the present density evaluation was performed with sufficient accuracy. Katayama  has also reported a density-determination method for liquids under high pressure and high-temperature using the Beer-Lambert relation of X-ray absorption. 49) Abedi et al. have proposed a method for determining phase behavior, elemental composition, and density using the Beer-Lambert relation of X-ray imaging. 50) As shown in Fig. 4, the present procedure accurately evaluates the density under supercritical conditions over a wide range of pressures from gas-like to liquid-like regions. The important advantage of the direct evaluation of sample density is the acquisition of fluid density data at the sample position, in which the scattering measurement is undoubtedly examined. This is why supercritical fluids are generally susceptible to macroscopic changes caused by slight differences in temperature or pressure. The density is the order parameter for discussions of physical quantities in critical and supercritical states. 51) According to the Ornstein-Zernike theory, 15) the SAXS profiles shown in Fig. 3 can be represented by the following equation: where I(0) is the zero-angle scattering intensity. The scattering intensity in the lower-density region was also fitted with a Lorentz-type relation. Figure 5 shows the Ornstein-Zernike plots for the SAXS profiles of supercritical CO 2 obtained in the present measurements. The vertical axis was normalized using the evaluated density data shown in Fig. 4.
The reduced X-ray scattering intensities at the zero-degree angle, I(0)/ρ, was estimated using extrapolations to the zeroangle limit. As shown in Fig. 5, the Ornstein-Zernike plots exhibit sufficient linearity in both the lower-and higherdensity regions.
The zero-angle scattering intensity I(0) is related to the density fluctuations as follows: 15,18) where N is the number of molecules in the volume and Z is the number of electrons in a molecule. The brackets 〈〉 denote the averaged value in the scattering volume V. The density fluctuations are defined as the mean-square fluctuation of N normalized by the averaged number of molecules 〈N〉: In the present experiment, the sample thickness was kept sufficiently constant by employing the sample holder shown in Fig. 1. The SAXS intensities were acquired using incident X-rays with the same beam size. In the present experiment, the scattering volume was therefore kept constant throughout the measurement. For this type of measurement, Eq. (2) can be rewritten using the number density ρ′ in each state as follows: 52) In Eq. (3), the scattering volume V is an unknown parameter for the evaluation of density fluctuations. The normalization relative to the scattering volume was achieved using the value of the density fluctuation in the ideal state.
The density fluctuations can also be represented in terms of the isothermal compressibility, k , T as follows: 15,18) where k B and T are the Boltzmann's constant and the absolute temperature, respectively. The isothermal compressibility in the ideal state, k  , T can then be is expressed in the form 14) Note that Eq. (5) uses the ideal-gas relationship PV = nRT, where P is pressure, n is number of moles, and R is the gas constant. Using the relation of Eqs. (4) and (5), the density where N A is the Avogadro's number. In the present method, we achieved normalization relative to the scattering volume using the value for the density fluctuations in the ideal state, / ( ) á D ñ á ñ = N N 1.
2 Figure 6 shows the normalization of the scattering intensities based on Eq. (6). In previous investigations, the number density has been calculated using the EOS. In addition, the scaling factor relating to the scattering volume has been obtained from the second derivative of G using a thermodynamic calculation based on the pressuredensity-temperature relationship from the EOS. 18) On these bases, the density fluctuations were evaluated absolutely from small-angle scattering experiments involving only the target sample. Figure 7 shows the density fluctuations of supercritical CO 2 along an isotherm at 317.50 K. As discussed above, the EOS for CO 2 constructed by Span and Wagner 48) is known to have high accuracy even in the critical region. The present experimental data for the density fluctuations exhibit good agreement with the values calculated using the EOS by Span and Wagner, confirming the accuracy of the evaluation method developed here.

Application to supercritical methanol
The method described above was applied to evaluating the density fluctuations of supercritical methanol. In a previous trial (data not shown), we could not obtain definite results for the density fluctuations of supercritical methanol due to experimental and analytical difficulties. In particular, we found that the experimental difficulty of SAXS measurements for supercritical methanol was caused by convection of the sample fluid in the sample container used. The SAXS signals were affected by very slight changes in the fluid Measured points for supercritical methanol in the pressuretemperature, density-temperature, and density-pressure planes. The solid lines in the pressure-temperature and density-temperature diagrams represent the gas-liquid coexistence curve of methanol. In the density-pressure diagram, the solid line displays density data calculated using the methanol EOS of de Reuck et al. 53) The experimental density was evaluated directly from the X-ray absorption coefficients. The present monitoring system enables us to measure temperature, pressure, and density of the fluid directly and simultaneously during measurements of the scattering signals.
temperature, fluid pressure, and sample heterogeneity on the macroscale. In addition, the accuracy of the EOS generally drops in the critical region. Figure 8 shows the present measured points for supercritical methanol in the pressuretemperature, density-temperature, and density-pressure planes. The density data were evaluated from the X-ray absorption coefficients, in the same manner as for supercritical CO 2 . The reduced temperature normalized by the critical temperature 53) was set to be T/T c = 1.04. In the present study, the critical temperature defined in the EOS literature was used to determine the reduced temperature. Figure 9 shows the isothermal change in the density fluctuations of supercritical methanol. The small-angle scattering method has an advantage in determining the density fluctuations because the scattering approach provides a direct method for evaluating the second derivative of G. On the basis of Figs. 7 and 9, it was found that the density fluctuations of methanol around the critical density (ρ c = 0.276 g cm -3 ) 53) are larger than those of CO 2 by approximately a factor of two in It should be mentioned that the difference in the maximum value of the density fluctuations is likely to be caused by the hydrogen bonding effect in methanol, compared to the quadrupole moment of CO 2 . The dependence of density fluctuations on hydrogen bonding is under investigation using the present determination method.

Conclusions
In this paper, we have reported the development of a method for determining the density fluctuations of supercritical fluids.
This method is based entirely on small-angle scattering experiments involving the target sample alone. The normalization of scattering intensity relative to the number of molecules per unit volume was examined using fluid density evaluated directly from in situ measurements of the X-ray absorption coefficients. The conversion of scattering intensities to absolute values related to the scattering volume was achieved directly using the value of the density fluctuations in the ideal state, / ( ) á D ñ á ñ N N 2 = 1. An analysis of supercritical CO 2 confirmed the validity of the present method. The density fluctuations of supercritical methanol, which was evaluated quantitatively for the first time here, were larger than those of CO 2 , possibly due to the hydrogen bonding ability in methanol.
It is considered that the devised method has been achievable for the following experimental reasons: (1) The constructed sample holder maintains the sample thickness with high stability under high-pressure and/or high-temperature conditions. (2) The X-ray absorption coefficients can be acquired accurately by in situ beam-monitoring apparatus set at the PF. (3) The extremely intense X-ray source at the BL-15A2 station of the PF leads to short exposure times and high S/N SAXS signals even for extremely low sample densities. (4) The usage of a two-dimensional semiconductor detector with low noise enables accurate measurements for the scattering intensities. Using the present method, the density fluctuations will be analyzed more widely and accurately, even having complex intermolecular interactions. By applying the present method, scattering-intensity normalization can be performed in scattering experiments for multi-component systems and in specific scattering techniques, such as the anomalous-dispersion X-ray scattering method.