**2. Density**

Density is a fundamental thermodynamic property of matter, relevant to the determination of other physical parameters, such as viscosity, surface tension, heat capacity, and even to the description of the radial density function (RDF) from which the short-range order can be evaluated. In the lieterature, several experimental techniques have been described for measuring the density of liquid metals. These include the maximum bubble pressure, Archimedean, liquid drop, dilatometric, and gamma radiation attenuation methods [17]. In our research we implemented a high-accuracy measurement based on gamma radiation attenuation.

#### *2.1. Gamma Radiation Attenuation Method*

This method is based on measuring the attenuation of gamma radiation passing through a sample. A γ radiation source is located on one side of the furnace, containing the sample, and a γ radiation detector is placed on the other side. The beam, passing through the liquid sample, is attenuated, and the density can be calculated by:

$$\mathbf{I} = \mathbf{I}\_0 \mathbf{e}^{(-\mu\rho\mathbf{x})} \tag{1}$$

where I0 and I are the intensity of the beam before the sample and the intensity measured at the detector, respectively; μ, ρ, and x are the absorption coefficient, the density, and the optical thickness of the entire path from the source to the detector. The contribution of the liquid can be deduced from premeasuring the transmission in the exact experimental configuration without the sample. To eliminate the thermal expansion of the experimental setup, we measured relative density. This method is mostly used for materials with high density, in which the attenuation due to the liquid is significant as compared with the structural materials, to obtain a large signal-to-noise ratio.

The experimental error is due to the following two main factors: (i) statistical error of the detector counts and (ii) the radiation background intensity. The first is assumed to have Lorentz distribution, and hence follows ΔI = √I. To reduce the statistical error, we significantly increased the measurement time. To reduce background radiation we subtracted reference measurements of the radiation intensity measured without a sample as a function of temperature, with the same temperature profile. As a result, the relative error of the density is proportional to the squared sum of the relative errors of the intensities with and without the liquid sample.

## *2.2. Density Results*

The density of pure lead and bismuth was measured by the gamma radiation attenuation method. The radiation source used was 137Cs with a characteristic energy of 662 Kev. This source is monoenergetic, and has a relatively long half-life and a large cross-section for absorption in the liquid samples. The crucible was made of fused quartz, which has low thermal expansion, with a square cross-section for simple geometry. The detector chosen was a CsI scintillator due to its high sensitivity and dynamic range.

Density measurements in elemental lead (99.999% purity) were carried out in the temperature range of 330 to 950 ◦C every 20 ◦C, as shown in Figure 1. It can be seen that the density decreases monotonically as the temperature is increased, without discontinuities or changes in slope. The uncertainty (standard deviation) in these measurements was estimated to be 1% due to the sampling statistics. The thermal expansion of liquid Pb can be evaluated, and was found to be 1.22 × 10−<sup>4</sup> K−1. Comparing the present data with previous measurements [18] of the density obtained by the Archimedean method shows a small difference that is contained by the experimental error. The difference in the density at melting point was found to be less than 1%.

**Figure 1.** Lead density as a function of temperature compared with data from reference [18].

The density of elemental bismuth (99.999% purity) was measured and reported in [19] in a temperature range of 200–1000 ◦C every 2 ◦C to 5 ◦C (Figure 2). A good agreemen<sup>t</sup> between the previously published data and our measurements can be seen in Figure 2. Similar to liquid Pb, liquid Bi expands as the temperature increases. The errors in the measurements are calculated to be 0.1%. Density changes are observed at ca. 550 ◦C and 720 ◦C, which might point to structural changes in the liquid phase of Bi [19].

**Figure 2.** Bismuth density as a function of temperature reproduced with permission from Ref. [19] compared with data from Ref. [20].

#### **3. Sound Velocity**

Sound velocity is an important thermodynamic quantity, sensitive to changes in the material properties. Tomeasure the velocity of soundinliquids, ultrasonic orlasermethods can be employed [21,22]. In optical systems, stress waves are generated, which result in surface motion. This method is suitable for measuring sound velocity at very high temperatures and pressures. The two systems used in our studies are modifications of the well-known ultrasonic method designed for high-accuracy measurement. The first is based on transmission and includes two transducers to measure the transmitted wave, one to generate the acoustic wave and the other to receive it (i.e., the wave travels only once in the liquid sample). The second apparatus is based on reflection of the emitted wave from the base of the sample container back to the transducer. Both these tabletop systems are presented schematically in Figure 3 and operate at ambient pressure and have a unique high accuracy that is preserved at elevated temperatures.

#### *3.1. Sound Velocity Measurements by Ultrasonic Techniques*

The pulse transmission system is composed of two ceramic buffer rods that serve as waveguides and two electronic ultrasound transducers that are attached to them, the first transmitting an acoustic wave and the second receiving it. The buffer rods allow the transducers to operate below the maximum working temperature. The crucible containing the liquid sample is machined at the top of the lower buffer rod. An ultrasonic elastic wave travels through the upper buffer rod, which is immersed in the liquid metal, traveling through the liquid sample and continuing to the lower buffer rod. The upper rod is attached to an accurate linear motor. By moving this rod a known distance of Δx, the traveling time of the sound wave is increased in Δt, and the sound velocity is deduced from the ratio C = Δx/Δt (i.e., it is a differential measurement). The uncertainty of the measurement estimated to be 0.1% to 0.35%, mainly due to the finite precision of the linear motor. More details on this measurement method are in reference [23].

The pulse-echo ultrasonic technique is, in principle, similar to the pulse transmission method. However, in this method we used a single buffer rod and a transducer that operates as both transmitter and receiver of the acoustic wave. The liquid sample is held in a ceramic crucible, the bottom of which serves as a reflector. To eliminate uncertainty as to the distance the ultrasonic wave travels through the liquid, the rod is translated by a known distance, Δx. As the wave travels a distance of 2Δx in the liquid metal with extra time Δt, the sound velocity is determined by c = 2Δx/Δt. This method has a similar error, as the limited accuracy of the linear motor is the same, and it is estimated to be 0.2%. More details on this measurement technique are in reference [24].

In both cases, the experimental apparatus was placed in a glovebox with a protective gaseous atmosphere of high-purity argon in a constant flow mode.

The transmission technique benefits from a higher signal-to-noise ratio, due to the short distance the sound wave travels through the liquid, and due to the attenuated shear waves in the thicker lower rod. The pulse-echo technique is easier to apply and has a simple setup, however, its main drawback is that the bottom of the crucible needs to be polished to a high quality to minimize losses upon reflection of the sound wave. In addition, the amplitude of the wave is more attenuated since the sound wave travels through the liquid twice.

**Figure 3.** Schematic view of (**a**) the ultrasonic pulse transmission experimental measurement apparatus reproduced with permission from [23] and (**b**) the ultrasonic pulse-echo experimental measurement apparatus reproduced with permission from [24].

#### *3.2. Sound Velocity Results*

#### 3.2.1. Elemental Systems

The sound velocity in pure liquid lead and bismuth has been measured by both of the ultrasonic techniques presented in the previous section as reported in [23,25]. For liquid Pb, the sound velocity decreases as the temperature is increased with a constant rate, as shown in Figure 4a. The difference between the measured and literature data increases with temperature and reaches about 1.5% at ca. 1000 ◦C. Our two methods agree reasonably well within the overlapping measurement range. For the liquid Bi (Figure 4b), the overall tendency is a negative temperature coefficient, but it shows a more complex behavior, namely, that the temperature coefficient changes with temperature. There is a good agreemen<sup>t</sup> between the two techniques.

**Figure 4.** The temperature dependence of the sound velocity in (**a**) pure liquid lead [23,25] compared with data from [26] and in (**b**) pure liquid bismuth [23,25] compared with data from [26].

The velocity of sound was measured for elemental tin and antimony in [23] using the pulse transmission method only. For tin, we observe a normal sound velocity dependency, with an excellent agreemen<sup>t</sup> with previous data, as shown in Figure 5a. Antimony has an anomalous behavior as presented in Figure 5b. Up to a temperature of ~830 ◦C the sound velocity increases with increasing temperature, up to a maximum, then, decreasing nonlinearly.

**Figure 5.** The temperature dependence of the sound velocity in (**a**) pure liquid tin [23] compared with data from [26] and in (**b**) pure liquid antimony [23] compared with data from [26].

The sound velocity of liquid gallium (99.99% purity) and liquid indium (99.999% purity) were measured using the pulse-echo setup and the results are presented in Figure 6 and detailed in the Supplementary Material. Both elements display normal behavior and a good agreemen<sup>t</sup> between the measured values and previously published data.

**Figure 6.** The temperature dependence of the sound velocity measured in (**a**) pure liquid gallium compared with data from [27,28] and in (**b**) pure liquid indium compared with data from [27,29].

## 3.2.2. Binary Systems

The sound velocities in liquid Pb-Sn and Bi-Sn were measured in [15] using the transmission method. Figure 7 presents the sound velocity as a function of temperature for the following four compositions of the Pb-Sn system up to 1000 ◦C: Pb13Sn87, Pb26Sn74, Pb46Sn54, and Pb70Sn30 (at.%). For all the compositions a normal behavior is observed, as in the two component elements.

**Figure 7.** The temperature dependence of the sound velocity in the liquid Pb-Sn system at different alloy compositions adapted with permission from [15].

The sound velocity of binary systems Bi-Pb [25] and Bi-Sb [24] were measured by the pulse-echo technique. In Figure 8, we present some results of the Bi-Sb isomorphous binary alloy. Measurements in the liquid phase were carried up to temperatures of ca. 900 ◦C for the following selected compositions: Bi13Sb87, Bi35Sb65, Bi53Sb47, and Bi70Sb30 (all in at.%). In this system, the Sb-rich alloys, Bi13Sb87 and Bi35Sb65, display anomalous behavior similar to Sb, but with less significant trends. The temperature

at which the sound velocity is maximal decreases from ~830 ◦C to ~700 ◦C at 13% Bi and ~520 ◦C at 35% Bi alloy composition. As the Bi concentration is increased the temperature dependence of the sound velocity becomes more linear, and for the Bi-rich alloy, Bi70Sb30, a semi-normal behavior at the low temperatures near the solidification is observed, similar to Bi.

**Figure 8.** Sound velocity in the liquid Bi-Sb system as a function of temperature at selected alloy compositions, adapted with permission from [24].

#### **4. Modeling Binary Phase Diagrams under Pressure**

Phase diagrams of binary alloys are expected to vary with pressure. Measuring thermophysical properties under pressure is experimentally challenging in addition to the vast amount of data required to construct the diagram as a function of temperature, composition, and pressure. Therefore, we proposed to follow a different route, i.e., to calculate the pressure-dependent phase diagram with input from ambient measurements of sound velocity and density to obtain the variation of interaction parameters with pressure. Lastly, information on the temperature-pressure phase diagram of the elements constituting the binary system is required.

The equilibrium condition to determine phase line is an equality of the chemical potentials of the same component in the two different phases, calculated from the Gibbs free energy:

$$
\mu\_{\rm i} = \left(\frac{\partial \mathcal{G}}{\partial \mathcal{N}\_{\rm i}}\right)\_{\rm P, T} \tag{2}
$$

where μi is the chemical potential of component i, G is the Gibbs free energy, and Ni is the number of particles.

Most binary alloy systems do not behave as ideal solutions. In systems with an asymmetric miscibility gap, the Gibbs free energy can be expressed to the lowest order in composition in the form of a sub-regular solution:

$$\mathbf{G} = \boldsymbol{\aleph}\_{\text{A}} \mathbf{G}\_{\text{A}} + \boldsymbol{\aleph}\_{\text{B}} \mathbf{G}\_{\text{B}} + \text{RT}(\boldsymbol{\aleph}\_{\text{A}} \ln \boldsymbol{\aleph}\_{\text{A}} + \boldsymbol{\aleph}\_{\text{B}} \ln \boldsymbol{\aleph}\_{\text{B}}) + \mathbf{J}\_{0} \mathbf{X}\_{\text{A}} \mathbf{X}\_{\text{B}} + \mathbf{J}\_{1} \mathbf{X}\_{\text{A}} \boldsymbol{\aleph}\_{\text{B}} (\mathbf{X}\_{\text{A}} - \boldsymbol{\aleph}\_{\text{B}}) \tag{3}$$

where XA and XB are the atomic fractions of each component, GA and GB are the partial Gibbs energy, and J0 and J1 are the regular and sub-regular interaction coefficients. The latter depend on pressure, and the variation of those parameters with pressure is a crucial input for the calculated phase diagram under pressure.

The pressure dependence of the interaction coefficient may be expanded to the second order and the deviation of the molar volume from its ideal value can be expressed in the following manner:

$$
\delta\mathbf{V} = \frac{\partial\|\_0}{\partial\mathbf{P}}\chi\_\mathbf{A}\chi\_\mathbf{B} + \frac{\partial\|\_1}{\partial\mathbf{P}}\chi\_\mathbf{A}\chi\_\mathbf{B}(\chi\_\mathbf{A} - \chi\_\mathbf{B})\tag{4}
$$

$$\frac{\partial \delta \mathbf{V}}{\partial \mathbf{P}} = \frac{\partial^2 \mathbf{J}\_0}{\partial \mathbf{P}^2} \mathbf{X}\_{\mathbf{A}} \mathbf{X}\_{\mathbf{B}} + \frac{\partial^2 \mathbf{J}\_1}{\partial \mathbf{P}^2} \mathbf{X}\_{\mathbf{A}} \mathbf{X}\_{\mathbf{B}} (\mathbf{X}\_{\mathbf{A}} - \mathbf{X}\_{\mathbf{B}}) \tag{5}$$

The sound velocity of the liquid at ambient pressure (Cs), is related to the molar volume and the adiabatic compressibility (KS) by:

$$\frac{1}{|\mathcal{C}\_{\sf s}|^{2}} = \rho \mathcal{K}\_{\sf s} = -\frac{\mathcal{M}}{\sqrt{2}} \Big(\frac{\partial \mathcal{V}}{\partial \mathcal{P}}\Big)\_{\sf S} \tag{6}$$

Hence, the sound velocity of an ideal solution, which is the velocity of the elements weighted by the relative composition, can be related to the measured one using Equation (6) to obtain the relation:

$$\left(\frac{\mathbf{C}\_{\rm id}}{\mathbf{C}\_{\rm s}}\right)^{2} - 1 = -\frac{2\delta\mathbf{V}}{\mathbf{V}\_{\rm id}} + \frac{\frac{\partial\delta\mathbf{V}}{\partial\mathbf{P}}}{\chi\_{\rm A}\frac{1}{\mathbf{C}\_{\rm s,A}}\left(-\frac{\mathbf{V}\_{\rm A}}{\mathbf{M}\_{\rm A}}\right) + \chi\_{\rm B}\frac{1}{\mathbf{C}\_{\rm s,B}}\left(-\frac{\mathbf{V}\_{\rm B}}{\mathbf{M}\_{\rm B}}\right)} + 0(\delta\mathbf{V}^{2})\tag{7}$$

The pressure dependence of the interaction parameter, J(P), is derived from measurements of the sound velocity and density performed at ambient pressure to determine the deviation of the molar volume from its ideal values (δV) to estimate ∂δV∂P . Extending the CALPHAD methodology, we calculated the phase diagrams of several binary alloys under pressure, including both isomorphous and eutectic systems which included: Bi-Sb, Bi-Sn, Pb-Sn [15], and Bi-Pb [25]. The model cannot represent the formation of new high-pressure phases in the P-T diagram of the alloy. The limitation on the pressure range arises from the fact that the interaction parameters are expanded only up to the second order.

The phase diagram of the isomorphous binary system Bi-Sb was calculated in [15] up to a pressure of 1.7 GPa and is shown in Figure 9. The solidus decreases significantly with pressure, while the liquidus slightly decreases, mainly due to the anomaly in the melting temperature of the Bi with respect to pressure. The calculated phase diagram of this alloy is limited to a pressure of 1.7 GPa. Extension of this study to higher pressures is a subject for future study.

**Figure 9.** Calculated phase diagram of the isomorphic system Bi-Sb from ambient pressure up to pressure of 1.7 GPa adapted with permission from [15].

A different example is the eutectic phase diagram of Pb-Sn, which has been calculated in [15] up to a pressure of 1.25 GPa and is presented in Figure 10. The model captures the shifts in the eutectic composition and temperature. This useful information is hard to obtain experimentally. Note that the eutectic point shifts to a composition rich in Sn, and the eutectic temperature increases with increasing pressure.

**Figure 10.** Calculated phase diagram of the eutectic system Pb-Sn from ambient pressure up to pressure of 1.25 GPa, adapted with permission from [15].

## **5. Electrical Resistivity**

Electrical resistivity is one of the basic transport properties of a material. It is a useful experimental tool for studying phase transformations in solid and in liquid phases, for example, to identify melting point due to the abrupt change in resistivity resulting from the loss of the long-range order. In the literature, two classes of experimental techniques for resistivity measurements in liquid metals have been proposed, i.e., non-contact and contact methods. In this study, we applied a contact method implemented using a tabletop setup that was designed and built to be simple, modular, and accurate.

The experimental apparatus is based on an alternating current (AC) source with the four-point probe technique commonly used in the literature. The use of AC instead of direct current (DC) reduces the Seebeck effect and, by using a known reference frequency, enables better elimination of external noise [30]. In the present setup, the melt is held in a quartz test tube and is in direct contact with the immersed electrodes. Measurements are carried out under a protective gaseous argon atmosphere in a constant flow mode to avoid the enhanced reactivity of liquid metals at high temperature with the structural materials constructing the experimental chamber, which were chosen to have low reactivity. Figure 11 displays a schematic view of the apparatus.

**Figure 11.** Schematic of the measuring system.

An alternating current is supplied to the sample, and the voltage drop across the sample is measured. The calculated resistivity (ρsample) from the measured voltage drop is as follows:

$$
\rho\_{\text{sample}} = \mathbf{V}\_{\text{meas}} \frac{\mathbf{R}\_{\text{shunt}}}{\mathbf{V}\_{\text{in}}} \mathbf{G} \tag{8}
$$

where G is the geometric constant of the cell, Vmeas the measured voltage, Vin the input voltage, and Rshunt the shunt resistor. The precise cell dimensions are needed to convert the measured voltage to resistivity. However, if only the relative resistivity or the temperature coe fficient is of interest, one can ignore the geometric constant.

The estimated error consists of statistical and systematic errors. The main contribution to systematic error arises from cell dimensions. The measured voltage is averaged over a temperature window of 5 ºC, and the standard deviation of the statistical distribution of the voltage within this window is calculated. The error is, therefore, presented in Equation (9).

$$\frac{\Delta p}{\rho} = \sqrt{\left(2\left(\frac{\Delta \mathcal{V}\_{\text{meas}}}{\mathcal{V}\_{\text{meas}}}\right)^2 + \left(\frac{\Delta \mathcal{R}\_{\text{shunt}}}{\mathcal{R}\_{\text{shunt}}}\right)^2 + \left(\frac{\Delta \mathcal{L}\_{\text{elec}}}{\mathcal{L}\_{\text{elec}}}\right)^2 + \left(\frac{\Delta \mathcal{A}}{\mathcal{A}}\right)^2\right)}}\tag{9}$$

where A and L are the sample's cross-section and length, determined by the voltage electrodes. We assume that only the voltage measurement contains both statistical and systematic errors. Other terms consist of systematic errors only.

The error in determining the absolute value of electrical resistivity is approximately 3%, a major part of which is derived from the uncertainty of the geometric dimensions of the cell. Consequently, the error in determining the relative values of the resistivity is only 0.1%.

#### *5.1. Electrical Resistivity Results*

## 5.1.1. Elemental Systems

The electrical resistivity was measured for elemental bismuth, tin, indium (99.999% purity), and gallium (99.99% purity). The results are presented in Figure 12 and in tabular form in the Supplementary Material. The resistance varies linearly with temperature in the liquid state, upon heating and cooling cycles. Deviation of the resistivity-temperature coefficient from published data can be seen for the temperature coefficient upon heating and cooling cycles for Bi and Ga.

**Figure 12.** *Cont.*

**Figure 12.** Resistivity of pure metals (**a**) bismuth and references [31,32]; (**b**) gallium and references [31,33]; (**c**) indium and references [34,35]; and (**d**) tin and references [36–38]. Cooling and heating rate of 60 ◦C/h.

#### 5.1.2. Binary Systems

The electrical resistance of Bi-Ga and Ga-In alloys (prepared from the same sources as the elemental samples above) was measured as a function of temperature and composition (see Supplementary Material). To ensure reliable data, the resistivity of every composition was measured for two different samples, each undergoing at least three cooling and heating cycles.

The resistivity of the Bi-Ga system was measured for the following compositions: Bi30Ga70, Bi33Ga67, Bi50Ga50, Bi67Ga33, and Bi70Ga70 (in at.%) and the results are presented in Figure 13. A linear trend is found for all measured compositions, in which the resistivity increases with increasing temperature.

**Figure 13.** Electrical resistivity in the liquid Bi-Ga system as a function of temperature at selected alloy compositions. The uncertainty is smaller than the symbol size. The black dots are data from [31].

The temperature coefficients *d*ρ*dT* were calculated by a linear fit to the resistivity data and are presented as a function of Bi concentration in Figure 14. The coefficient values are calculated over the measured temperature range displayed in Figure 13. Our results sugges<sup>t</sup> a possible correlation of the temperature coefficient in the Bi-Ga system to a second-order polynomial as a function of composition. Parabolic dependence is an expected behavior for alloys containing metals with mixed valences [39].

**Figure 14.** Resistivity coefficient of Bi-Ga alloy vs. Bi concentration.

The temperature dependence of the resistivity at different compositions is displayed in Figure 15. This isotherm plot shows a linear correlation between the absolute resistivity of the melt and the Bi concentration. The smooth dependence suggests that no obvious transitions are taking place in the melt with increasing Bi concentration. Furthermore, as the Bi concentration is increased, the slope of the resistivity curve increases as indicated from the distance between the isotherms.

**Figure 15.** Absolute resistivity of Bi-Ga binary alloy vs. Bi concentration. The uncertainty is smaller than the symbol size.

A measurement of the electrical resistivity of liquid Bi33Ga67 (in at.%) alloy near the melting point obtained during slow cooling with a rate of 60 ◦C/h is presented in Figure 16. The resistivity-temperature curve presents an abrupt change at 260 ◦C, about 50 ◦C above the liquidus. This change is correlated with phase separation in the melt [40]. Following this shoulder, a drastic increase of resistivity is observed below 225 ◦C, which is characteristic of the two-phase zone that contains a mixture of liquid and solid states. The difference between the present results and the results reported by Wang et al. [40] may originate due to use of DC vs. AC measurements which may produce an out of phase signal near the solidification or due to degradation of the contacts.

**Figure 16.** Resistivity of Bi33Ga67 liquid alloy compared to data from [40].

The resistivity of pure indium, gallium, and the selected binary alloys Ga86In14, Ga70In30, Ga25In75, and Ga10In90 (in at.%) were measured, and in Figure 17 we summarize the results for these compositions. The results present linear dependence of the resistivity with respect to temperature. No relation between the composition and the absolute resistivity values was found, in contrast to the Bi-Ga system. Two compositions exhibited the following outlying behaviors: the eutectic composition, Ga86In14, presented the lowest resistivity of the measured compositions; and the Ga70In30 had a significantly higher value of the resistivity-temperature slope.

**Figure 17.** Electrical resistivity in the liquid Ga-In system as a function of temperature at selected alloy compositions. The uncertainty is smaller than the symbol size.

The eutectic alloy (Figure 18) displays abnormal behavior at 90 ◦C and 250 ◦C, which might indicate a possible transformation in the liquid phase. Structural transformation at 90 ◦C is seen upon heating and cooling cycles, which sugges<sup>t</sup> a reversible process. A transformation at similar temperatures was reported previously based on XRD measurements [41].

**Figure 18.** Electrical resistivity of eutectic Ga-In. Cooling and heating rate of 60 ◦C/h.

The temperature coefficient of the Ga-In system presents no clear tendency withindium concentration, as can be seen in Figure 19. This result is in contradiction to the theory [42] for alloys with an equivalent amount of valence electrons, in which the change in composition will maintain the electron density unchanged and the trend should be linear.

**Figure 19.** Resistivity coefficient of Ga-In as a function of indium concentration. Bars represent the uncertainty (symbols without bars have an uncertainty smaller than the symbol size).

The resistivity vallues as a function of composition for Ga-In binary alloys at several temperatures are presented in Figure 20. A unique phenomenon can be observed in these data, namely, that the hypereutectic area displays a general parabolic trend with a maximum resistivity between Ga70In30 and Ga25In75, and the hypoeutectic region shows a linear trend where the resistivity decreases until reaching a minimum at eutectic concentration, significantly below the resistivity of either elemental component.

**Figure 20.** Absolute resistivity vs. indium concentration in Ga-In alloys.
