Until now, heat capacity measurements performed with levitation techniques have required accurate knowledge of the sample’s emissivity beforehand. For a sample levitated using an aerodynamic levitator, it experiences both radiative and forced convective heat loss. The sample’s emissivity only allows for the calculation of the radiative heat loss term, and a model has yet to be developed to accurately describe the total combined heat loss for aerodynamic levitation (ADL). In this study, we will introduce a novel multiple-gas cooling method for heat capacity measurement for ADL where two types of inert levitation gases (Ar and Kr) with different thermal conductivities were used to generate two cooling curves for the same sample. For samples being cooled at different cooling rates, the total heat loss is the same. The radiative heat loss was expressed using Stefan–Boltzmann’s law, and the convective heat loss using Ranz–Marshall’s equation. The two independent parameters (i.e., emissivity and heat capacity) of one given sample could then be solved using the two independent cooling curves. The heat capacities of gold, copper, nickel, iron, and palladium around the melting point were measured using this method. The multiple-gas cooling method for heat capacity measurement introduced in this study is the first heat capacity measurement method available for ADL and can be performed for materials with unknown emissivity. This newly developed method is important for the study of the thermophysical properties of high-temperature liquids, especially molten oxides with low electrical conductivity.

## I. INTRODUCTION

Heat capacity is a key thermophysical property that determines the amount of energy transfer with changes in temperature. It is closely related to calculations of thermal conductivity, enthalpy, and Gibbs free energy.^{1,2} Conventionally, the high-temperature heat capacity is measured using drop calorimetry,^{3,4} in which a sample is heated to the target temperature and then dropped quickly into a calorimeter at a known initial temperature. The sample is generally heated in a furnace, and it sometimes needs to be kept in a container. These factors limit the temperatures at which drop calorimetry can be used. In addition, the sample can be contaminated when kept in the container because it requires a long time to reach thermal equilibrium.

In the earlier stages of heat capacity measurement with levitation techniques, only the heat capacity to emissivity ratio could be determined from experiments, and accurate knowledge of the sample’s hemispherical emissivity is required to convert this ratio into heat capacity.^{5–7} In recent years, a blackbody furnace, in addition to the levitator, was used for heat capacity and emissivity measurements.^{8,9} Ishikawa *et al.* had utilized this method to obtain the spectral emissivity and heat capacity of liquid metals such as platinum, nickel, zirconium, rhodium, titanium, and niobium.^{9–13} Similarly, Kobatake *et al.*^{8,14–16} used a blackbody furnace in addition to an electromagnetic levitator (EML) and laser modulation setup to measure the emissivity, heat capacity, and thermal conductivity of molten palladium, titanium, cobalt, and silicon. The maximum temperature of the blackbody furnace used for emissivity measurements is 1773 K, which covers the melting points of most metals. However, when it comes to the heat capacity measurements of molten oxides, a temperature limit of 1773 K is much lower than the melting points of most molten oxides, such as Al_{2}O_{3}, CaO, ZrO_{2}, and UO_{2}. Ishikawa *et al.* had extrapolated the measured temperature to above the limit of the blackbody furnace in previous experiments.^{9,10,12} However, the uncertainty in the extrapolated temperature becomes more challenging to discuss as temperature increases. On the other hand, experiments conducted with a blackbody furnace require the sample to stay within its molten state for a few minutes, making it not applicable to materials with high vapor pressures.

In addition, this measurement design with a blackbody furnace has not been implemented for aerodynamic levitation (ADL) because aside from radiative heat loss, additional forced convection is taking place and complicates the heat transport process between the levitation gas and the levitated sample.^{17,18} Millot *et al.*^{7} used ADL to measure the heat capacity of liquid boron while ignoring the convective heat loss, which is a questionable assumption. Recently, Ushakov *et al.* introduced drop-and-catch (DnC) calorimetry^{19} for fusion enthalpy measurements; however, this method was not applicable to heat capacity measurements as accurate measurement of the sample’s temperature after the drop is currently not feasible.

ADL is a key instrument for studying molten oxides as neither electrostatic levitation (ESL) nor EML can levitate molten oxides. It also offers the unique advantage of separating the levitation and the cooling processes, and it allows for the control of the cooling speed through the use of different types of levitation gases. However, this advantage has been neglected as there is no need to control the sample’s cooling speed for density, surface tension, and viscosity measurements at a specific temperature.^{7,20–22} In this study, utilizing this advantage that ADL has, we introduce a new ADL-based method for measuring the heat capacity of molten materials. This is carried out by cooling the levitated sample using two different types of levitation gas, thereby providing two cooling curves for the same sample. The radiative heat loss contribution of the cooling curve is described using Stefan–Boltzmann’s law, and the convection contribution is estimated using Ranz–Marshall’s equation. The two sets of cooling curves [Eqs. (10) and (11)] enable solving for two independent parameters: emissivity and heat capacity. With the multiple-gas cooling method introduced in this study, the sample’s hemispherical emissivity is no longer required prior to the heat capacity measurement because that value will be obtained simultaneously with the heat capacity value. This is a major breakthrough in the development of thermophysical property measurement methods for levitation techniques as the multiple-gas cooling method is not only the first available heat capacity measurement technique for ADL but it also means that, heat capacity of molten materials with unknown emissivity can be obtained through experiment.

In Secs. II and III, we will first take a look at the experimental setup used in this study and construct a model to describe the heat loss of a spherical droplet levitated with ADL. We will then move on to discuss the fundamental idea behind the multiple-gas cooling method and how to derive the heat capacity from multiple cooling curves of the same sample. Finally, the heat capacity of molten platinum, gold, copper, nickel, iron, and palladium measured with the multiple-gas cooling method is presented and compared to reference values.

## II. EXPERIMENTAL SETUP

The ADL setup used in this study was based on our previous work^{23} with some design modifications. A simplified schematic of the current ADL setup is shown in Fig. 1. Two fiber lasers (976 nm, Pearl) were used to heat the sample from the top and the bottom to minimize the temperature gradient within the sample. The WinVue user interface software installed in a connected personal computer (PC) was used to control the operation of the two fiber lasers and to ensure that the output power was the same during measurements. A long-pass filter of 1400 nm (FEL1400, Thorlabs) was used along with a pyrometer (1500 nm, IR-CAI8CN, CHINO) for temperature measurements, and the signal was recorded every 1 ms with a data logger (GL 900, Graphtec). The sample was levitated within a conical nozzle with a hole of 1.4 mm diameter, and a sapphire window was placed on top of the nozzle to minimize surface oxidation. The levitated sample was observed during the experiment with two charge-coupled cameras, one with a 950 nm short-pass filter for low temperature observations and one with a 550 nm short-pass filter for high temperature observations. When levitated, the sample is contained within the nozzle, below the rim. Due to the force exerted by the levitation gas, the sample’s shape might deviate slightly from that of a perfect sphere. In this study, we assumed that this deviation is negligible and that the sample is spherical before solidification. Background lighting was provided so that the sample was observable at temperatures lower than the melting point. Ar (5N, Air Liquide) and Kr (5N, Tokyo Gas Chemicals Co. Ltd.) were used as levitation gases, and Ar-4 vol. %H_{2} gas (Air Liquide) was used to remove any possible oxide layer that could have formed on the sample’s surface. Finally, cooling water was circulated within the nozzle to maintain it at between 25 to 27 °C.

## III. HEAT LOSS ANALYSIS

The sample is treated as a blackbody in the pyrometer; thus, the obtained temperature signals need to be corrected based on Wien’s law,^{24}

where *T* is the corrected temperature of the sample, *T*_{P} is the as-recorded temperature signal of the pyrometer, *T*_{L} is the liquidus temperature, and *T*_{P,L} is the as-recorded temperature signal by the pyrometer at the sample’s melting point. The melting point was identified by a sudden increase in temperature owing to recalescence after undercooling on the cooling curve. To apply this correction to temperatures other than the melting point when the sample is in the liquid state, we assumed that the emissivity of liquid metals is temperature independent and is equal to that at the melting point.^{25} To minimize the effect a changing emissivity has on temperature-correction, our later calculations on heat capacity is confined to ±50 K around the melting point. Figure 2 shows the temperature curve before and after correction with Wien’s law for a platinum sample cooled using Ar gas.

Figure 3 shows a comparison of the two corrected cooling curves after calibration. As Kr gas has a lower thermal conductivity than Ar gas, sample cooling in Kr gas was slower. In this study, we focused on the heat capacity at temperatures near the melting point (T_{m} ± 50). The heat loss within this temperature range was expressed as the sum of the radiative heat loss *Q*_{rad} and the forced convective heat loss *Q*_{convect}.

### A. Radiative heat loss analysis

The radiative heat loss of the levitated sample was expressed using Stefan–Boltzmann’s law as

where *A* is the surface area of the sample; *ɛ*, the spectral emissivity of the sample; *σ*, Stefan–Boltzmann’s constant; and *T*_{sample}, the temperature of the sample.

### B. Convective heat loss analysis

The convective heat loss is related to the heat capacity of the sample and is calculated using Newton’s law of cooling,

where *A* is the surface area of the sample; *h*, the heat transfer coefficient; *T*_{sample}, the temperature of the sample; and *T*_{gas}, the temperature of the surrounding gas. For a droplet surrounded by gas moving at a certain velocity, the heat transfer coefficient was expressed using Ranz–Marshall’s equation as

Here, *d* is the diameter of the sample, *κ* is the thermal conductivity of the gas, *v* is the relative velocity between the sample and the gas, *μ* is the dynamic viscosity of the gas, *ρ* is the density of the gas, and *C*_{P} is the heat capacity of the gas. As mentioned previously, the original Ranz–Marshall’s equation describes the convective heat loss of a droplet in an open space. Several studies have previously introduced corrections by multiplying the original Ranz–Marshall’s equation with the sample’s and the gas’s density,^{26–32} viscosity,^{26–33} heat capacity,^{27} and specific enthalpy^{28–32} values. However, the convective heat loss values calculated using these modified Ranz–Marshall’s equations are quite different. In addition, there are no suitable models to describe the convective heat loss of a droplet in a confined space. Therefore, to describe the convective heat loss of a droplet within the conical nozzle in this study, we decided to multiply the original Ranz–Marshall’s equation with a coefficient *α*. Here, we assumed that *α* should depend on the gas type and the design of the conical nozzle. In the ideal case where the sample is levitated within an open environment, *α* will be one. Thus, the original Ranz–Marshall’s equation was modified into

where *α* is the introduced coefficient. We also assumed the velocity difference [*v* in Eq. (5)] between the gas and the sample to be the terminal velocity of the sample, which can be calculated based on its Reynolds number as follows:

Here, *d* is the diameter of the molten sample; *ρ*_{s}, the density of the sample, *ρ*_{f} is the density of the gas, and *μ*_{f} is the dynamic viscosity of the gas. In this study, the Prantl number was 0.658 for samples levitated in Ar and 0.672 for samples levitated in Kr. The calculated Reynolds numbers are shown in Table I, and since the diameters of the samples are controlled within 1.2–1.6 mm, the Reynolds number is higher for materials with higher density, such as liquid Au, Pt, and Pd. Overall, we believe that laminar flow prevailed in the current setup and based on the Reynolds numbers, the terminal velocities *v*_{t} should be calculated using the last part of Eq. (8).

. | Reynolds numbers . | |
---|---|---|

Material . | Ar . | Kr . |

Au | 2040–2500 | 2660–3300 |

Cu | 1350–1850 | 1760–2420 |

Fe | 1360–1770 | 1780–2310 |

Ni | 1320–1890 | 1720–2470 |

Pd | 2040–2700 | 2660–3530 |

Pt | 1850–2500 | 2420–3270 |

. | Reynolds numbers . | |
---|---|---|

Material . | Ar . | Kr . |

Au | 2040–2500 | 2660–3300 |

Cu | 1350–1850 | 1760–2420 |

Fe | 1360–1770 | 1780–2310 |

Ni | 1320–1890 | 1720–2470 |

Pd | 2040–2700 | 2660–3530 |

Pt | 1850–2500 | 2420–3270 |

### C. Fitting and calibration process

For the sample, the total heat lost during cooling in argon or krypton gas can be expressed in general as

Here, *T*_{sample} is the sample’s temperature when cooled using argon or krypton gas, *T*_{gas} is the temperature of the cooling gas, *α* is the added coefficient to modify the convective heat loss term, and *t* is the corresponding cooling time. At each temperature step, the following also holds true:

Here, *T*_{(in Ar)} and *T*_{(in Kr)} are the sample’s temperatures when cooled using argon or krypton gas, respectively, and *α*_{Ar} and *α*_{Kr} are the added *α* coefficients used to modify the convective heat loss contribution in argon or in krypton gas, respectively. The thermophysical properties of Ar and Kr gas (Table II) at room temperature and atmosphere pressure were taken from the CRC Handbook of Chemistry and Physics.^{34} Within this study, we assumed that the temperatures of the levitation gases were kept constant at room temperature. On the left-hand side of Eqs. (10) and (11) are the time-dependent sample temperatures, which are the experimentally obtained cooling curves of the samples. On the right-hand side are the modeled cooling curves that depend on two unknown independent variables *C*_{P} and *ɛ*. As there are two unknowns for each equation, a series of *C*_{P}-*ɛ* values can be obtained per equation when least squares fittings are performed, as shown by the black lines in Fig. 4. The intersection of the two *C*_{P}-*ɛ* lines obtained using the two cooling curves (Ar and Kr) gives us the sample’s heat capacity. However, it should be noted here that the two coefficients *α*_{Ar} and *α*_{Kr}, in Eqs. (10) and (11), that modify the convective heat loss terms must be determined first.

. | Thermal conductivity (mW/mK) . | Viscosity (μPa s)
. |
---|---|---|

Ar | 18.3 | 23.1 |

Kr | 9.5 | 25.6 |

. | Thermal conductivity (mW/mK) . | Viscosity (μPa s)
. |
---|---|---|

Ar | 18.3 | 23.1 |

Kr | 9.5 | 25.6 |

To solve for *α*_{Ar} and *α*_{Kr}, we first performed calibrations using the emissivity (*ɛ* = 0.25) and heat capacity (*C*_{P} = 38.8 ± 1.8 J/mol K) values reported for platinum by Ishikawa *et al.*^{10} using electrostatic levitation. This study was chosen because platinum is an inert material, and electrostatic levitation was conducted under vacuum with no convective heat loss. Calibrations are then performed using Eqs. (10) and (11). As mentioned above, the left-hand sides of Eqs. (10) and (11) are the experimentally obtained cooling curves, and the right-hand sides are the least squares fitted cooling curves. Using the *C*_{P} and *ɛ* reference data for platinum by Ishikawa *et al.*,^{10} least squares fitting of the experimentally obtained cooling curves then provides the two parameters *α*_{Ar} and *α*_{Kr}.

For this calibration, we used three platinum samples; the coefficients obtained using the least squares method for the 1.4 mm diameter opening conical nozzle used in this study, for samples with diameters between 1.2 and 1.6 mm, were *α*_{Ar} = 0.751 ± 0.053 and *α*_{Kr} = 0.790 ± 0.067. It was found that with the current setup, the coefficient *α* is the least sensitive to the sample’s diameter within this range, with the uncertainty in *α* heavily affected by the uncertainty in the heat capacity of liquid platinum^{10} used for calibration. Figure 4 shows an example of the relationship between the emissivity and the heat capacity of molten platinum around the melting point before and after calibration. Figures 5 and 6 show that the heat loss and temperature change in the sample in Ar or Kr gas can be well described by the sum of Stefan–Boltzmann’s law and the modified Ranz–Marshall’s equation. The final calculated cooling curves of the sample in both Ar and Kr gases matched well with the experimental cooling curve data, with *r*^{2} values higher than 0.99 for both the Ar and Kr cooling curves shown in Fig. 6. As seen in Fig. 5, when liquid platinum was cooled with argon gas, the cumulative convective heat loss accounted for around half of the cumulative total heat loss. This means that the assumption made by Millot *et al.*^{7} that convective heat loss of liquid boron cooled in argon gas above 2000 °C is negligible is inaccurate. Based on Eq. (10) or Eq. (11), we can see that for the same cooling curve, ignoring the convective heat loss term will lead to a reduced heat capacity value. This suggests that Millot *et al.*^{7} had underestimated the heat capacity of liquid boron.

In this study, the obtained heat capacity of molten platinum from 1992 to 2092 K was 38.7 ± 0.9 J/mol K; this was similar to that mentioned in the report by Ishikawa *et al.*^{10} The main advantage of the proposed analysis method is that, during the initial temperature calibration step, even if an incorrect emissivity value was obtained for the molten sample at 1500 nm owing to problems with the pyrometer’s aim or the lens being partially contaminated with metal vapor, it would not affect the final calculation. As long as the two calibrated cooling curves were accurate, we could solve for emissivity and heat capacity. However, as the current method relied on calibration with a material of known emissivity and heat capacity, the accuracy of the calculated results depended on that of the calibration source, and it could be improved with the use of heat capacity and total emissivity data of inert materials such as liquid gold, whose reference heat capacity values do not deviate significantly from one another.^{35–38} In addition, although the current convective heat loss model is a simple but sufficient modification to the Ranz–Marshall’s equation, it could be improved through flow-field simulation and a more focused discussion on heat loss analysis in ADL. Finally, the structure of the levitation nozzle could be adjusted to more closely represent an open environment while still minimizing sample oxidation.

## IV. CALCULATED HEAT CAPACITY RESULTS

Platinum (3N8, Nilaco), gold (3N5, Nilaco), copper (4N, Kojundo Chemicals), nickel (4N, Kojundo Chemicals), iron (3N up, Kojundo Chemicals), and palladium (3N up, Kojundo Chemicals) spheres with diameters of around 1.5 mm were prepared under a reducing atmosphere with Ar-4%H_{2}. The changes in the sample mass before and after melting were negligible (<0.1%). The cooling curves of the samples were first obtained using Ar; then, the sample was reduced with Ar-4%H_{2}, and finally, the sample was cooled with Kr. For all the six metals measured, samples generally exhibited a mass change of less than −1% (roughly −0.1 to −0.2 mg) before and after the experiment. As the sample was initially heated with a mixture of Ar-4%H_{2} gas, some hydrogen could have dissolved in the molten metals. For example, the hydrogen solubility of liquid iron, copper, and nickel at the melting point has been reported to be between 5.3 and 41 cm^{3}/100 g metal.^{39,40} However, in the current study, the expected amount of hydrogen uptake should be lower than the reported hydrogen solubility values. This is because an Ar-4%H_{2} mixture gas was used in this study, which has a lower hydrogen partial pressure. In addition, the laser heating time in this study is shorter as our goal was not to establish a hydrogen absorption equilibrium. Finally, the sample was heated again under argon or krypton flow, leading to additional hydrogen loss during the heat capacity measurement.

Table III shows the heat capacity values calculated using two cooling curves for each sample. The constant-pressure heat capacity obtained using ADL agreed well with previously published data, thus verifying the validity of the proposed multiple-gas cooling method. For palladium, Paradis *et al.*^{13} reported a much smaller heat capacity value than the one in this study. However, they assumed that the emissivity of liquid palladium is wavelength independent and equal to that at 900 nm when calculating radiative heat loss. This led them to wrongly estimate the normal emissivity and resulted in a much lower heat capacity value.

. | C_{P} (J/mol K)
. | References . |
---|---|---|

Pt, calibration | 38.7 ± 0.9 | This study |

38.8 ± 1.8 | Ishikawa et al.^{10} | |

Au | 30.9 ± 0.6 | This study |

28.3 | Wust et al.^{35} | |

28.9 | Umino^{36} | |

32.1 | Plaza^{37} | |

29.4 | Smithells^{41} | |

33.6 | Tester et al.^{38} | |

Cu | 30.9 ± 0.7 | This study |

36.3 | Chekhovskoi et al.^{42} | |

32.8 | Chase^{43} | |

30.2 | Dokko and Bautista^{44} | |

31.5 | Smithells^{41} | |

Ni | 40.3 ± 1.3 | This study |

38.9 | Chase^{43} | |

39.0 | Vollmer et al.^{45} | |

39.9 | Ishikawa^{11} | |

36.4 | Smithells^{41} | |

43.1 | Geoffray et al.^{46} | |

Fe | 44.5 ± 0.8 | This study |

46.0 | Chase^{43} | |

43.9 | Wilson^{47} | |

46.1 | Beutl et al.^{48} | |

44.4 | Smithells^{41} | |

Pd | 34.4 ± 1.3 | This study |

37.4 | Cagran and Pottlacher^{49} | |

41.2 | Arblaster^{50} | |

41.0 | Guthrie and Iida^{51} | |

40.5 | Watanabe et al.^{14} | |

29.1 | Paradis et al.^{13} | |

34.7 | Barine et al.^{52} |

. | C_{P} (J/mol K)
. | References . |
---|---|---|

Pt, calibration | 38.7 ± 0.9 | This study |

38.8 ± 1.8 | Ishikawa et al.^{10} | |

Au | 30.9 ± 0.6 | This study |

28.3 | Wust et al.^{35} | |

28.9 | Umino^{36} | |

32.1 | Plaza^{37} | |

29.4 | Smithells^{41} | |

33.6 | Tester et al.^{38} | |

Cu | 30.9 ± 0.7 | This study |

36.3 | Chekhovskoi et al.^{42} | |

32.8 | Chase^{43} | |

30.2 | Dokko and Bautista^{44} | |

31.5 | Smithells^{41} | |

Ni | 40.3 ± 1.3 | This study |

38.9 | Chase^{43} | |

39.0 | Vollmer et al.^{45} | |

39.9 | Ishikawa^{11} | |

36.4 | Smithells^{41} | |

43.1 | Geoffray et al.^{46} | |

Fe | 44.5 ± 0.8 | This study |

46.0 | Chase^{43} | |

43.9 | Wilson^{47} | |

46.1 | Beutl et al.^{48} | |

44.4 | Smithells^{41} | |

Pd | 34.4 ± 1.3 | This study |

37.4 | Cagran and Pottlacher^{49} | |

41.2 | Arblaster^{50} | |

41.0 | Guthrie and Iida^{51} | |

40.5 | Watanabe et al.^{14} | |

29.1 | Paradis et al.^{13} | |

34.7 | Barine et al.^{52} |

A common concern with conducting experiments with aerodynamic levitation is the heavy oxidation on the sample’s surface. The use of a sapphire top cover, as mentioned in Sec. II, was extremely effective in preventing sample oxidation. This effectiveness is shown by the negligible mass change of the sample before and after the experiment, as well as the close agreement our data have with the reference data. However, a thin oxide layer could potentially form on the sample’s surface and affect its surface conditions. To ensure the quality of our presented data, although normal emissivity values for each of the molten metals around their melting points were also obtained, these data are not reported here as normal emissivity is highly sensitive to the sample’s surface conditions.

## V. CONCLUSION

This study proposed a new analysis method to measure the heat capacity of molten materials using ADL. The use of two types of cooling gases generated two cooling curves of the same sample and enabled the calculation of the emissivity and heat capacity, which are two independent parameters. The radiative heat loss was described using Stefan–Boltzmann’s law, and the convective heat loss using Ranz–Marshall’s equation with a modified coefficient for a conical nozzle. This coefficient was calculated by calibrating our data on molten platinum with those reported by Ishikawa *et al.*^{10} The obtained heat capacity data of molten Au, Cu, Ni, Fe, and Pd at *T*_{m} ± 50 K agreed well with the reference data. Unlike the heat capacity measurement methods used with EML or ESL, the multiple-gas cooling method introduced here for ADL does not require additional emissivity measurements with a blackbody furnace, which generally has a maximum temperature limit of lower than 2000 K. This limits the ESL + blackbody furnace setup’s applicability mainly because of the unknown uncertainty in extrapolating the measured temperature to values much higher than the temperature limit of the blackbody furnace and the long time required to measure the emissivity of the sample. Therefore, when it comes to the heat capacity measurement of molten materials with much higher melting points or with high vapor pressures, the method proposed in this study is currently the only one available to be used with levitation techniques.

## ACKNOWLEDGMENTS

This study was supported by the Japan Atomic Energy Agency (JAEA) (Grant No. JPJA18B18071972) and by a grant-in-aid from a fellowship from the Japan Society for the Promotion of Science (JSPS) (Grant No. 20J10376). Society for the Promotion of Science (JSPS) (Grant No. 20J10376).

The authors declare that they have no conflicts of interest.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.