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Experimental Observation of Maximum Density Fluctuation in Liquid Te

We performed small-angle X-ray scattering measurements of liquid Te using a synchrotron radiation facility and observed the maximum scattering intensity near 620 K in the supercooled region (melting temperature 723 K). This result is an experimental observation of the ridge line of the critical density fluctuation associated with the liquid-liquid phase transition that is argued to exist in the supercooled region and verifies the existence of the transition. Similar results have been reported for supercooled liquid water, demonstrating the universality of the liquid-liquid transition concept (or inhomogeneous model) in explaining the thermodynamics of these"anomalous liquids". Moreover, as a more essential issue, it has been demonstrated that liquids can be inhomogeneous at mesoscopic scales, even in their thermal equilibrium state. This will contribute to a greater understanding of the physics of liquids, including glass-forming liquids.

Liquid Te has long been known as an "anomalous liquid" that exhibits many thermodynamic anomalies: its density (ρ) is greatest around its melting temperature (T m =723 K) [1,2]; the pressure variation of the melting temperature shows a maximum at approximately 1 GPa [3]; the heat capacity (C P ) reaches a maximum in the supercooled region [4]; and the ultrasonic sound velocity (v US ) shows a minimum and maximum at approximately 620 K and 1070 K, respectively [5]. Furthermore, its electrical features are different from those of other liquid metals. Although it is a semiconductor in the solid state, it exhibits (poor) metallic electrical conductivity when melted [6], and its electrical conductivity increases as the temperature increases up to approximately 1070 K in the liquid region [7]. The existence of a continuous liquid-liquid phase transition (LLT) in the high-pressure [3,8] or supercooled [4] regions and the associated mesoscopic inhomogeneity near ambient conditions [9,10] has long been highlighted. This LLT concept was first introduced to explain the electrical properties as a two-species model [11] or inhomogeneous transport model [12]. However, at that time, Mott clearly objected to this inhomogeneous transport model [13], stating that "this regime does not exist in general in disordered materials, though it may in fluids near a critical or convolution point". He did not believe that liquids other than supercritical fluids and phase-separating liquids are inhomogeneous. However, the application of this LLT concept is not limited to the electrical properties of liquid Te; it can explain the melting curve maxima in the high-pressure region [3,8]. This was later extended by Tsuchiya to understand various properties, including thermodynamics and structure [9]. In the case of liquid Te, it has been shown that adding impurities, such as Se, can shift the thermodynamic and electrical properties to the high-temperature side, and these mixtures can be considered as hypothetical supercooled liquid Te [14][15][16]. Tsuchiya also applied his model to many Te mixtures [17][18][19] to confirm its validity. However, mesoscopic inhomogeneity, which is the essence of the model, has not been experimentally proven for a long time, and results are not conclusive. Thirty years later, we succeeded in demonstrating the existence of mesoscopic density inhomogeneity in Te-Se [16] and Te-Ge mixtures [20] by small-angle X-ray scattering measurements (SAXS) using the SPring-8 synchrotron radiation facility in Japan. The temperature dependences of the SAXS intensity and its momentum transfer (Q)dependent slope, which are important parameters regarding the density inhomogeneity, show maxima near the center of the continuous LLT region. The maxima were always near the center of the LLT region, even when the Se concentration and pressure were changed and when the mixing element was changed from Se to Ge. That is, it was shown that this inhomogeneity is not caused by mixing or by the type of mixing element but by the LLT of Te itself. However, the same SAXS measurements were performed for pure liquid Te, but no significant temperature change in the SAXS intensity associated with LLT was observed above T m . We concluded that the density inhomogeneity was too small to be detected at temperatures above T m because the real LLT was approximately 100 K below T m [4].
Meanwhile, liquid water exhibits thermodynamic anomalies (review [21]) similar to those of liquid Te [8,[22][23][24] and is often referred to as an "anomalous liquid". The density is greatest at 277 K, which is slightly above T m (273 K), the pressure variation of T m is negative, the heat capacity shows a divergent-like increase in the supercooled region, and the ultrasonic sound velocity shows a maximum at approximately 350 K. Furthermore, the LLT concept (review [25]) and the critical point of the LLT hypothesis [26,27] have been proposed independently of the aforementioned model of liquid Te. Regarding the LLT concept of liquid water, there has been no clear verification experiment, mainly because the actual LLT is located in the deep supercooled "no-man's land" [27], and an actual experiment is nearly impossible. However, the latest state-of-the-art X-ray free-electron laser (XFEL) technology has recently made it possible to conduct X-ray measurements in this no-man's land [28][29][30]. In particular, SAXS measurements have shown a maximum of the integrated scattering intensity near 235 K in its temperature variation, and they concluded that this demonstrated the maximum isothermal compressibility or a so-called Widom line, which is strong evidence of the LLT concept [30]. Cautious comments have been made as to whether or not this really is the maximum [31], so further verification is necessary.
Therefore, a new high-speed X-ray measuring device was installed at the beamline of SPring-8, where we previously performed SAXS measurements, and we repeated these SAXS measurements of liquid Te in the current study. In the case of Te, the actual LLT region is considered to be a metastable supercooled state rather than an unstable no-man's land like that of water. In fact, Tsuchiya et al. found the region considered to be the actual LLT using a macroscopic sized (sub-millimeter) sample [4]. The new device has a faster effective measuring speed (1 min) compared to the previous one (approximately 30 min) and is expected to complete SAXS measurements before supercooled liquid Te solidifies.
The experiments were conducted at the beamline BL04B2 of SPring-8 in Japan, as for the previous SAXS measurements [16,20]. The incident X-ray energy was 61.7 keV using Si(2 2 0) crystal reflection as a monochromator, but approximately 7 % of the high-harmonics of 123.4 keV were mixed. In this time, this high-harmonic X-ray was used to investigate the behavior of the higher-Q structure factor, as discussed later. Liquid samples were installed in handmade single-crystalline sapphire cells and heated by Mo heaters. The cell was set in a lowpressure vessel, which was filled with He gas at a pressure of approximately 1.5 bar. The details of the experimental conditions are presented in the review paper [20]. In the present experiment, a two-dimensional detector device using CsI scintillation (Varex XRD 1621AN3) was used [32], although we did use an imaging plate (RIGAKU R-AXIS IV++) in the previous experiments [16,20]. The new detector has 2048×2048 pixels, with a spatial resolution (pixel size) of 200 µm. Figure 1(a) shows the time evolution of the temperature during the measurement. The X-ray transmission used to determine ρ is also plotted in Fig. 1(b). After increasing the temperature to approximately 1100 K to melt the sample completely, measurement was performed while lowering the temperature. At temperatures much higher than T m , the detector was exposed for 20 min to capture a scattering image while maintaining constant temperature. The temperature and transmission changes were almost always linked (see Fig. 4(a)), indi- cating that the sample responded sufficiently quickly to the temperature change and was also sufficiently stable. Meanwhile, in the low-temperature region, including the supercooled region (within the rectangular frame indicated by the dashed lines in Fig. 1(a,b)), the scattering images were captured every 1 min while cooling the temperature at an almost constant rate to prioritize speed. Figure 1(c, d) shows an enlarged image of this part. As shown in Fig. 1(c), the cooling was performed at an almost constant rate of -13.1 [K/min]. During this cooling, the transmission ( Fig. 1(d)) increased rapidly just below T m , which corresponds to the fact that ρ reaches its maximum near T m and decreases in the supercooled region [4] (see also Fig. 4(a)). With further cooling, the transmission drops sharply at the red arrow, indicating that the sample has solidified. Before and after this solidification, the scattering pattern changes from a halo pattern, which is characteristic of liquids, to a pattern of many intense concentric spots, which is characteristic of polycrystals, confirming that the sample indeed solidified. Below this temperature, the transmission remains almost constant. The supercooled state of approximately 150 K was maintained until it solidified.
The obtained scattering patterns were carefully analyzed [20] considering scattering from the cell, fluorescent X-ray from the sample, and multiple scattering in the sample, and the Q-dependent scattering intensities from the liquid sample were obtained. Figures 2 and 3 demonstrate the temperature variations of these scattering intensities in the high-Q and low-Q regions, respectively. Note that the main peak observed at approximately 20 nm −1 in Fig. 2 is due to the high-harmonic X-ray, which is contained in approximately 7 % of the incident beam.
Because the absolute value of S(Q) was not experimentally determined in the present measurements, the scattering intensities in the high-Q and low-Q regions were separately normalized to fit the S(Q) obtained by neutron diffraction (ND) measurements [33] and the S(Q → 0) value from our previous SAXS results [16], respectively. Figure 2 shows the result in the high- Q region: Over the entire temperature range from 1122 K to solidification at 583 K, the peak position of S(Q) shifted to smaller Q, reflecting the anomalous behavior of liquid Te, where the distance to the nearest atomic neighbor increases despite cooling. The peak intensity of S(Q) increased gradually up to around the melting point and then decreased slightly. The overall shape and temperature changes above T m are close to those of the neutron results [33], whose S(Q) at 773 K is shown by the open circles in the figure. , However, in the small-Q region, a different behavior can be observed. In the high-temperature region shown in Fig. 3(a), the intensity decreases almost monotonously, while the slope remains almost unchanged. This temperature variation is normal for liquids. However, below this temperature, the slope of the Q dependence changes and S(Q → 0) increases. S(Q → 0) exhibits the highest value at approximately 631 K and decreases again below this temperature. To clarify the temperature variation, the scattering intensity was fitted by the linear function S(Q) = S 0 (1−b q Q). Figure 4(1) represents these parameters: (1b) S 0 and (1c) b q (red open circles). We also plot the previous results [16] above T m as black closed circles. Although the slope parameter b q varies significantly, we believe that our results are mostly consistent. (1a) ρ estimated from the present  (2) for liquid Se50Te50. The red curves are visual guides. The green curves are those from the literature [4,15]. On the right axis of (a), αP calculated from the temperature variation of these densities are plotted by pink marks and lines, respectively. CP and vUS [4,5,34] are plotted in (d) by black and brown curves, respectively. The blue hatched region indicates the LLT region, where αP is negative. The vertical red dot-dashed line indicates the "ridge line" of the LLT, where αP , CP , and vUS exhibit peaks.
X-ray transmission ( Fig. 1(b)) is also plotted by the red open circles with a line, which almost agrees with the values in the literature [4], indicated by a green line, and the previous results [16], indicated by black closed circles. The temperature of 581 K at which the liquid sample solidified is almost the same as in the literature, which indicates that this solidification is due to the spinodal limit or that the sample became unstable with excessive supercooling. In Fig. 4 (1a), the thermal expansion coefficient calculated from the temperature variation of densities ρ as α P = −(1/ρ)(∂ρ/∂T ) from the present experiment and literature values are also plotted by the pink squares and lines on the right axis, respectively. In the figure, a negative thermal expansion region between 581 K and 723 K, which is considered to be the LLT region, is indicated by the blue hatch. With cooling from 1123 K to approximately 750 K (1b), S 0 gradually decreased, which can be interpreted as the liquid becoming more homogeneous because it moved away from the liquid-gas (LGT) critical point, which should be located at a higher temperature. This behavior is more clearly seen for liquid water [24]. In contrast, in the LLT region, S 0 clearly increases to a maximum at approximately 620-630 K. Around this temperature, (1a) α P , (1d) isobaric heat ca-pacity C P , and ultrasonic sound velocity v US [4] all exhibit peak values. These facts clearly indicate that the maximum of S 0 is due to the density fluctuation associated with LLT and can be interpreted as the critical ridge line of the LLT or Widom line. With further cooling, S 0 decreases to a smaller value. Note that this large change in S 0 is observed only in the LLT region below T m , and there is almost no change in the real liquid above T m . It is reasonable that no change was observed in the previous measurement [16] at temperatures higher than T m , as indicated by the black closed circles in the figure. The parameter b q shown in Fig. 4 (1c) shows mostly no significant changes above T m . However, in the supercooled region, it seems to show a maximum at approximately 620-630 K, which is in conjunction with the variation of S 0 , although the linearly rising tendency is relatively large. Regarding this b q parameter, which corresponds to the domain size of the density inhomogeneity, further investigation may be required. In the case of liquid water, it has also been noted that the temperature change in the characteristic size of the density fluctuation is not the usual Ising-like critical behavior [35].
We also conducted SAXS measurements for liquid Se 50 Te 50 , and the results are shown in Fig. 4(2) as red open circles. The marks and lines are the same as those for the results of liquid Te shown in Fig. 4(1). The data in the supercooled region below T m was newly obtained. At temperatures above T m , the results of the current and previous experiments were close, indicating their reliability. (2b) S 0 shows a maximum near the center of the LLT region indicated by the blue hatched region, where (2a) α P , (2d) C P , and v US all show peaks, clearly indicating that the LLT is the cause of the observed density fluctuation. By contrast, in the supercooled region, there is no sign of a significant increase in S 0 , which means that cooling or supercooling do not induce the density fluctuation. In liquid Te, the temperature at which S 0 exhibited its maximum was in the supercooled region. However, we can conclude that the cause of the density fluctuation is not supercooling but instead the LLT.
We performed SAXS measurements for liquid Te and successfully observed that the SAXS intensity shows a maximum, which indicates that the density fluctuation shows a maximum in the supercooled region; its temperature variation is linked to that of ρ, α P , v US , and C P .; and it clearly proves that the origin of this density inhomogeneity is not cooling or supercooling but the LLT, which is proposed to be located in the supercooled region. This result experimentally verifies the LLT concept [3,[9][10][11][12], which has been controversial for a long time. Meanwhile, it also provides an answer to the simple and early and essential question by Mott [13] of whether the liquids can have mesoscopic inhomogeneity except the LGT supercritical region. It has been shown that thermal equilibrium liquids above T m can be universally inhomogeneous at the mesoscopic scale, which is a very important finding. Regarding inhomogeneity in liquids, it has been proposed that glass transition liquids have a "dynamical heterogeneity" [36]. Most researchers on glass transition phenomena seek the origin of inhomogeneity in the supercooling nature, assuming homogeneity in real liquids above the melting temperature. However, this assumption is doubted by the present findings. Further, we believe that important physics is hidden in the "dynamic" feature. In fact, the SAXS measurements implicitly revealed that the "static" density inhomogeneity in both liquid Te and liquid water [30] were too small in quantity to be detectable [24]. However, such a small inhomogeneity in density is accompanied by quantitatively large underlying thermodynamic anomalies. The inhomogeneity in density detected by SAXS measurements is only the tip of the iceberg of the inhomogeneity or fluctuation that truly affects thermodynamics. We recently proposed a method for detecting such true inhomogeneity or "dynamical fluctuation" using sound waves [24]. In fact, we could measure large changes in real liquid regions above T m for both Te [10] and water [24] using this method. By making full use of these static and dynamical fluctuation measurement methods and clarifying the mesoscopic fluctuations in various liquids, we hope that the understanding of liquids, including glass transition liquids, will be significantly advanced.