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Round-robin measurement of surface tension of high-temperature liquid platinum free of oxygen adsorption by oscillating droplet method using levitation techniques

  • Yusaku Seimiya ORCID logo EMAIL logo , Shuto Tomita , Tohei Kawaguchi , Hidekazu Kobatake ORCID logo , Jürgen Brillo ORCID logo , Suguru Shiratori ORCID logo , Ken-ichi Sugioka ORCID logo , Takehiko Ishikawa ORCID logo and Shumpei Ozawa ORCID logo
Published/Copyright: December 31, 2023
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High Temperature Materials and Processes
From the journal High Temperature Materials and Processes Volume 42 Issue 1

Abstract

Round-robin measurement of surface tension of high-temperature liquid platinum was conducted free of any contamination from the supporting materials and oxygen adsorption, using an electrostatic levitator (ESL), two electromagnetic levitator (EML), and an aerodynamic levitator (ADL). The measured temperature dependences of the surface tension using ESL and two EMLs were in good agreement and were expressed as σ = 1,798 ± 74.3 − ( 0.12 ± 0.0445 ) × ( T − 2,041 ) [10–3 N·m–1] (1,900–2,600 K). However, the surface tension values measured with ADL were slightly lower than those exceeding the uncertainty of the measurement plots at high temperatures.

Keyword: High temperature melt; thermophysical properties; electrostatic levitation; electromagnetic levitation; aerodynamics levitation

1 Introduction

Highly reliable data on the surface tension and its temperature dependence for high-temperature metallic melts are essential for the improvement and optimization of various high-temperature melt processes involving a free surface such as welding, brazing, and thermal spraying. To measure the surface tension of high-temperature melts, container-free measurements with the oscillating droplet method using levitation techniques such as electrostatic levitation (ESL) [1,2,3,4,5], electromagnetic levitation (EML) [5,6,7,8,9,10,11,12,13,14], and aerodynamic levitation (ADL) [15,16,17,18] have been increasingly employed to prevent sample contamination caused by supporting materials at high temperatures. The surface tension of a levitated droplet can be calculated using the following Rayleigh equation [19,20,21,22,23]:

(1) σ = 3 8 π M f R 2 ,

where σ is the surface tension, M is the sample mass, and f R is the frequency of surface oscillation for the l = 2 mode, called the Rayleigh frequency. However, the reported data for the surface tension of liquid metals such as titanium [5] and silicon [11] obtained using levitation techniques often show considerable scatter. This scatter is often attributed to the effects of sample purity and oxygen adsorption from the atmospheric gas without sufficient discussion. It is also possible that inherent errors associated with the measurement method and apparatus affect the results. To ensure the validity of the container-free measurements of surface tension of high-temperature metallic melts using ESL, EML, and ADL, and to guarantee the reliability of the results, conducting a round-robin test is indispensable, especially considering the effects of oxygen partial pressure in the measurement atmosphere.

In this study, we measured the surface tension of liquid platinum using container-free methods with ESL, two different EMLs, and ADL. Liquid platinum is highly chemically stable such that its surface tension is unaffected by oxygen adsorption, even when the measurements are conducted in open air. This study aimed to validate the container-free measurement of surface tension of a high-temperature melt using the three different aforementioned levitation techniques based on the results of round-robin tests. In addition, by using ESL, we measured the density of the liquid platinum required for the surface tension measurement.

2 Experimental procedure

2.1 ESL

The nomenclature used in this study is given in Table 1. Figure 1 shows a schematic diagram of the ESL used in this study, which is installed at the Japan Aerospace Exploration Agency (JAXA). A cubic platinum sample with a purity of 99.95 mass% and a mass of approximately 40 mg was placed between a pair of disk electrodes in a vacuum chamber. After the chamber was evacuated to the order of 10−5 Pa using a turbomolecular pump backed up by a scroll pump, the positively charged sample was electrostatically levitated by applying a high voltage between the electrodes. The levitated sample was heated and then melted by irradiation with 100 W CO2 lasers from three directions [2]. The rotation of the levitated sample was controlled by appropriately aligning the CO2 laser beams or by a rotating magnetic field generated by the four coils beneath the bottom electrode. The sample temperature was measured by monochromatic pyrometers. The emissivity setting of the pyrometers was adjusted so that the plateau temperature of the liquid phase after the recalescence at the cooling stage indicates the equilibrium melting point of platinum based on the valid assumption that the emissivity of the sample does not change with the temperature within the operating wavelength range.

Table 1

Nomenclature

Symbol Description
σ Surface tension [N m ‒1 ]
T Temperature [K]
ρ Density [kg m ‒3 ]
r Radius of the droplet [m]
Q Sample charge [c]
E Electric field applied to levitate the sample [V m−1]
F Correction term for the effects of charge and gravity on the frequency
M Sample mass [kg]
G Gravitational acceleration [m s−2]
U Potential difference between the top and bottom electrodes [V]
L Distance between top and bottom electrodes [m]
ε 0 Permittivity of vacuum [F m−1]
σ 0 Surface tension calculated without considering the correction term F
R(θ ) Length from the center of gravity to the edge for the droplet [m]
θ Polar angle [rad]
V Volume [m3]
f R Rayleigh frequency
l Oscillation mode
m Oscillation index
f 2,m Frequencies of l = 2 mode (m = 0, ±1, and ±2) [Hz]
f 1,m Frequencies of l = 1 for levitated droplet (m = 0 and ±1) [Hz]
ft Frequency of translation oscillation [Hz]
A Projected area of the top-view image of the levitated droplet
Rx Radius of the droplet along the x−axis
Ry Radius of the droplet along the y-axis
f 2,0 Frequency of surface oscillation m = 0 in l = 2 mode, [Hz]
f 2,±1 Frequency of surface oscillation m = ±1 in l = 2 mode, [Hz]
f 2,±2 Frequency of surface oscillation m = ±2 in l = 2 mode, [Hz]
u(i) Standard uncertainty of each source
c(i) Sensitivity coefficient of each source
u ρ (i) Uncertainty contributions of density measurement
u σ (i) Uncertainty contributions of surface-tension measurement
d cal ref Calibration of micrometer
d res ref Resolution of micrometer
d rep ref Repeatability of diameter measurement for reference sphere
d p res ref Resolution of image for reference sphere
d p re p ref Repeatability of contour fitting for reference sphere
rp re s Resolution of image for levitated droplet
rp rep Repeatability of contour fitting for levitated droplet
m cal Resolution of the electronic balances
m res Calibration of electronic balances
m rep Repeatability of mass measurement
f res Frequency resolution of FFT analysis
f rep Repeatability of identifying frequency peaks using FFT spectrum
L res Resolution of scale
L rep Repeatability of interelectrode distance measurement
U res Resolution of electrode potential control
U sta Stability of electrode potential control
u c(ρ) Combined standard uncertainty in the density measurement
u c(σ ESL) Combined standard uncertainty in the surface-tension measurement using ESL
u c(σ EML) Combined standard uncertainty in the surface-tension measurement using EML
u c(σ ADL) Combined standard uncertainty in the surface-tension measurement using ADL
k p Coverage factor
u exp Expanded uncertainty
β ˆ Temperature coefficient
ρ ̅ Mean value of measured density
T ̅ Mean value of temperature
δ ˆ e Residual variance of measured data
Figure 1 Schematic view of the ESL furnace and its diagnostic apparatus [3]: (1) sample, (2) top electrode, (3) bottom electrode, (4) side electrodes, (5) He–Ne lasers, (6) position detectors, (7) CO2 laser beams, (8) pyrometers, (9) ultraviolet light, (10) CCD camera, (11) CCD cameras with telephoto objective lens.
Figure 1

Schematic view of the ESL furnace and its diagnostic apparatus [3]: (1) sample, (2) top electrode, (3) bottom electrode, (4) side electrodes, (5) He–Ne lasers, (6) position detectors, (7) CO2 laser beams, (8) pyrometers, (9) ultraviolet light, (10) CCD camera, (11) CCD cameras with telephoto objective lens.

A small sinusoidal electric field was superimposed on the levitation field to excite the surface oscillation of m = 0 for the l = 2 mode in the levitated droplet while monitoring the oscillation using an oscillation detection system composed of a power meter with a vertical slit at the sampling frequency of 4,096 Hz. The measured signal was analyzed using the fast Fourier transform (FFT) to determine the frequency of the m = 0 oscillation (f 2,0). The surface tension of liquid platinum, σ, was calculated from the frequency of the m = 0 oscillation using the following Rayleigh equation [19] modified for a nonuniform surface charge distribution [25,26]:

(2) σ = r 3 ρ 8 ( 2 π f 2 , 0 ) 2 − Q 2 8 π 2 r 6 ρ ε 0 ( 1 − F ( σ 0 , q , e ) ) ,

where r is the radius of the droplet when it assumed a spherical shape, ρ is the density of liquid platinum, and ε 0 is the vacuum permittivity. Q is the droplet charge that is given by the following equation:

(3) M ∙ G = QU L ,

where G is the gravitational acceleration, U is the potential difference between the top and bottom electrodes, and L is the spacing between the electrodes. F ( σ 0 , q , e ) is the correction term for the effect of the droplet deformation on f 2,0, which is defined as follows:

(4) F ( σ 0 , q , e ) = ( 243.31 σ 0 2 − 63.14 q 2 σ 0 + 1.54 q 4 ) E 2 r ε 0 176 σ 0 3 − 120 q 2 σ 0 2 + 27 σ 0 q 4 − 2 q 6 ,

where σ 0 is the surface tension calculated without considering the correction term F. q and e are defined as follows:

(5) q 2 = Q 2 16 π 2 r 3 ε 0 ,

(6) e 2 = U L 2 r ε 0 .

Since the electrostatically levitated droplet is axisymmetric along the vertical axis, its volume can be evaluated from the fitted data of the side view images of the droplet free of surface oscillation observed by three high-speed video (HSV) cameras in conjunction with the temperature, while cooling the sample according to the following equation:

(7) V = 2 π 3 ∫ 0 π R ( θ ) 3 sin θ d θ ,

where R ( θ ) is the length from the center of gravity to the edge for the droplet and θ is the polar angle. The density of liquid platinum for each temperature was determined from the sample mass M divided by the volume V.

(8) ρ = M V .

A more detailed description of the procedures for the surface tension and density measurements can be found in the previous studies [3,24].

2.2 EML

Two EMLs installed at the Chiba Institute of Technology (CIT) and the German Aerospace Center (DLR), respectively, were used to compare the results. The schematics of these facilities are shown in Figure 2. The major differences between these EML facilities are the shape and size of the EML coil and the frequency of high-frequency AC power; these differences usually affect the size of the metallic sample that can be levitated, the deformation of the levitated droplet, and the droplet rotation, as well as the ability to heat the sample. As a result, the frequencies of the surface oscillations and translational oscillations of the levitated droplet, which are necessary to calculate the surface tension, are altered. Moreover, both the shape and size of the chamber differ for these EML facilities, which can affect the amount of oxygen adsorption from the measurement atmosphere on the levitated droplet that acts to decrease the surface tension.

Figure 2 Schematic view of the EML furnace installed at (a) CIT and (b) DLR, and its diagnostic apparatus [7,8].
Figure 2

Schematic view of the EML furnace installed at (a) CIT and (b) DLR, and its diagnostic apparatus [7,8].

A piece of the platinum sample with a mass of 1,600–3,000 mg was placed on a sample holder and was positioned in the center of the EML coil. The sample was first electromagnetically levitated and then melted under the flow of high-purity commercial helium gas. The temperature of the droplet was controlled by varying the flow rate of the helium gas using a monochromatic pyrometer. The emissivity setting of the pyrometer was adjusted so that the plateau temperature of the liquid phase during the melting of the sample was equal to the equilibrium melting point of platinum. After the droplet temperature became constant, the oscillation of the droplet was monitored from above using a HSV camera at 500 fps for 16.4 s at CIT and at 800 fps 6.4 s at DLR.

The frequencies of the surface oscillations of m = 0, ±1, and ±2 for the l = 2 mode and those of the center of gravity (oscillations of the m = 0 and ±1 for the l = 1 mode) were obtained from time-sequence data of the HSV images using FFT analysis. The frequencies of the m = 0, ±1, and ±2 oscillations can be identified based on the rule shown in Table 2, where A is the area of the projection image of the levitated droplet, R + is the sum of the radii along the x and y axes, R x and R y , and R − is the difference between R x and R y [22,23]. The influence of the apparent droplet rotations, induced by the phase differences between m = +1 and m = –1, and m = +2 and m = –2, was considered, of real rotation in the analysis [23]. The surface tension of liquid platinum was calculated from these frequencies using the following Rayleigh equation [19], calibrated with the Cummings and Blackburn equation [27]:

(9) σ = 3 8 π M 1 5 ∑ m = − 2 2 f 2 , m 2 − f t 2 1.905 + 1.200 × G 8 π 2 f t 2 3 M 4 π ρ −3 2 ,

(10) f t = 1 3 ∑ m = − 1 1 f 1 , m ,

where f 2,m are the frequencies of surface oscillation for m = 0, ±1, and ±2 for the l = 2 mode, and f t are the frequencies of the motion of the center of gravity of the droplet. Detailed descriptions of the measurement procedures and facilities can be found elsewhere [7,8].

Table 2

Relationship among oscillation of m = 0, ±1, and ±2, corresponding to that of area, A, R x , and R y

m = 0 m = ±1 m = ±2
A Yes Yes No
R x + R y Yes Yes No
R x – R y No Yes Yes

2.3 ADL

A small cube of platinum (40 mg) was rapidly melted on a p-BN plate in the air by irradiation with a semiconductor laser beam and was then immediately solidified into a spherical shape with a diameter of 1.6 mm. The spherical sample was placed on a nozzle of the ADL, as schematically shown in Figure 3, and then levitated on an argon gas jet from the bottom at a rate of approximately 0.9 L·min−1 controlled by a digital mass flow controller. The levitated sample was heated and melted by irradiation with the semiconductor laser. The temperature of the levitated droplets was controlled by varying the laser output while simultaneously using two monochromatic pyrometers with different wavelengths, λ 1 = 0.8−1.6 μm and λ 2 = 1.95−2.5 μm, where the emissivity settings of the pyrometers were adjusted to obtain the same temperature under the assumption that the sample is a gray body [28]. When the droplet temperature became constant, a sound wave was applied to the gas jet prior to its introduction into the ADL nozzle using two loudspeakers facing each other so that surface oscillation of m = 0 for the l = 2 mode is excited in the levitated droplet. After turning off the sound wave applied to the gas jet, the behavior of the surface oscillations of the droplet was monitored with an HSV camera at 2,000 fps for less than 1 s until the surface oscillation of the droplet was damped.

Figure 3 Schematic view of the ADL furnace.
Figure 3

Schematic view of the ADL furnace.

The frequency of the surface oscillation of m = 0 for the l = 2 mode was determined from time-sequence data of the droplet diameter using the FFT. The surface tension of liquid platinum was calculated from the frequency of the surface oscillation of m = 0 for the l = 2 mode and the sample mass after the experiment using the Rayleigh equation (equation (1)) [15,16,19].

3 Results

Figure 4 shows the density of liquid platinum as a function of temperature, T, measured with ESL, along with the literature data presented for comparison [4,29,30,31,32,33,34]. The density of liquid platinum was measured over a wide temperature range of approximately 500 K between 1,700–2,200 K, including the undercooling region. Since the measured density decreases linearly with the increasing temperature, it can be described by a linear approximation based on the plotted measurement results as follows:

(11) ρ = 18,906 − 1.0852 ( T − 2 , 041 ) [ kg· m − 3 ] ,

which is in relatively good agreement with the results reported by Gather et al. [31] and Ishikawa et al. [4].

Figure 4 Density of liquid platinum measured using the ESL method together with the literature data [4,29–34].
Figure 4

Density of liquid platinum measured using the ESL method together with the literature data [4,29–34].

Figure 5 shows a typical frequency spectrum for the surface oscillation of the droplet levitated by the ESL, as obtained by the FFT analysis. Only one distinct peak appears in the spectrum, indicating that only the m = 0 oscillation for the l = 2 mode is excited in the droplet by ESL.

Figure 5 FFT results of a typical oscillation signal obtained from a liquid platinum levitated by ESL.
Figure 5

FFT results of a typical oscillation signal obtained from a liquid platinum levitated by ESL.

Figure 6 shows typical frequency spectra of oscillations of A, R +, and R − for the droplets levitated by the EMLs installed at CIT (Figure 6(a)) and DLR (Figure 6(b)). In the EML technique, the single surface oscillation of the l = 2 mode, known as the Rayleigh oscillation, splits into three, corresponding to oscillations of m = 0, ±1, and ±2 because of the droplet deformation due to the gravitational acceleration and the electromagnetic forces from the levitation coil [27]. Furthermore, the frequency peaks of the surface oscillations for m = ±1 and ±2 are symmetrically split into two when the droplet rotates around an axis perpendicular to the projection plane, as is usually observed in an electromagnetically levitated droplet, even though the frequencies of the surface oscillations are actually unchanged. As a result, five peaks are usually observed in the frequency spectra of oscillations of the A, R +, and R − for the droplet levitated by EML, as shown in the spectra obtained in the experiments carried out at CIT and DLR. Since the surface oscillation in the levitated droplet was observed for a slightly longer time in the experiment carried out at CIT (16.4 s) than in the experiment carried out at DLR (6.5 s), the FFT spectra presented in Figure 6(a) that were obtained in the former experiment appear to be somewhat noisy due to the higher resolution. The discrepancies in the frequencies of the surface oscillations for m = 0, ±1, and ±2 between the spectra presented in Figure 6(a) and (b) stem from differences in the droplet size as well as the temperature.

Figure 6 Surface oscillation frequency peaks of m = 0, ±1, ±2 obtained from the results of the FFT analysis of the oscillation behavior of liquid platinum levitated by the EML installed at (a) CIT and (b) DLR. The frequencies of m = ±1 and ±2 split into two peaks due to droplet rotation.
Figure 6

Surface oscillation frequency peaks of m = 0, ±1, ±2 obtained from the results of the FFT analysis of the oscillation behavior of liquid platinum levitated by the EML installed at (a) CIT and (b) DLR. The frequencies of m = ±1 and ±2 split into two peaks due to droplet rotation.

Figure 7 shows the typical examples of the frequency spectra of the oscillation of the droplet diameter in ADL. These were obtained from the experiments in which the surface oscillation of the droplet was repeatedly excited and damped by turning the sound waves of different frequencies applied to the gas jet on and off while the droplet was levitated and maintained at 2,500 K. When the frequency of the sound wave applied to the gas jet is varied from 180 to 210 Hz, the spectra consistently show a peak at approximately 191 Hz along with a peak corresponding to the frequency of the sound wave for all of the sound wave frequencies. When the frequency of the applied sound wave is far from 191 Hz, only a single peak corresponding to the sound wave frequency was detected in the spectra. These results indicate that the peak at 191 Hz corresponds to the natural frequency of the surface oscillation of the levitated droplet.

Figure 7 Two peaks were obtained from the FFT analysis of the surface oscillation behavior when the oscillation was excited by applying sound waves to the liquid platinum levitated by the ADL. The first peak shifted from 180 to 210 Hz according to the sound wave frequency. The frequency of the second peak was constant at 191 Hz, indicating that this was the natural frequency.
Figure 7

Two peaks were obtained from the FFT analysis of the surface oscillation behavior when the oscillation was excited by applying sound waves to the liquid platinum levitated by the ADL. The first peak shifted from 180 to 210 Hz according to the sound wave frequency. The frequency of the second peak was constant at 191 Hz, indicating that this was the natural frequency.

Figure 8 displays the temperature dependence of the surface tension of liquid platinum as measured using ESL, two EMLs, and ADL together with literature data [4,32,33,34,35,36] for comparison. Uncertainty bars for each plot were evaluated on the basis of the ISO Guide to the Expression of Uncertainty in Measurement (GUM) [37], with a selected coverage factor of k p = 2. The evaluation of these uncertainties is described in detail in the next section. Table 3 presents a summary of the measurement results obtained with these four apparatuses. The surface tension of liquid platinum measured using ESL () decreases linearly with the increasing temperature, which is in good agreement with the data reported by Ishikawa et al. [4] and Allen [35].

Figure 8 Comparison of the measurement results of the surface tension of liquid platinum by ESL (), EML (DLR: , CIT: ), and ADL () together with the literature data [4,32−36].
Figure 8

Comparison of the measurement results of the surface tension of liquid platinum by ESL (), EML (DLR: , CIT: ), and ADL () together with the literature data [4,32−36].

Table 3

Results of a round robin measurement of the surface tension of liquid platinum using the droplet oscillation method with ESL, EML, and ADL

Technique Surface tension σ (10–3 N·m–1) Temperature coefficient dσ/dT (10–3 N·m–1·K–1) Temperature range T (K)
ESL 1,795 ± 74.3 –0.1790 ± 0.1288 1,900–2,200
EML (CIT) 1,799 ± 20.4 –0.1253 ± 0.0675 1,950–2,600
EML (DLR) 1,804 ± 32.8 –0.1280 ± 0.0857 2,030–2,240
ADL 1,778 ± 36.0 –0.1606 ± 0.0655 2,080–2,830

Although the measured temperature ranges show a significant discrepancy between the measurements using two different EMLs (, ) due to their different coil shapes, the temperature dependence of the surface tension revealed in these measurements agrees well with the results obtained using ESL, as indicated by the solid line. This suggests the validity of the measurement methods for the surface tension of high-temperature metallic melt using ESL and EML. Consequently, the temperature dependence of the surface tension of liquid platinum can be expressed using a linear least-squares method based on the measurement data obtained using ESL and EMLs as follows:

(12) σ = 1,798 − 0.12 ( T − 2,041 ) [ 10 − 3 N· m − 1 ] .

The intercept of 1,798 × 10−3 N·m−1 corresponds to the surface tension at the melting temperature of platinum (2,041 K).

Conversely, the surface tension values of liquid platinum measured using ADL () appear to be lower than the values measured with ESL and EMLs throughout the entire measurement temperature range. As a result, when using the least-squares method to calculate the temperature dependence of the surface tension based on all of the plots of measurement results obtained with ESL, two EMLs, and ADL, the surface tension gradient with respect to temperature becomes steeper as indicated by the dotted line. Notably, many of the plots of the data obtained at high temperatures with EML and ADL measurements deviate from this approximation line, confirming the deviation of the measurement results with ADL from those with ESL and EMLs.

4 Discussion

The temperature dependence of the density of liquid platinum, which is essential for surface tension measurements, was measured using ESL. The measurement results were in good agreement with those reported by Gathers et al. [31] and Ishikawa et al. [4] (cf. Figure 4). Using this result, the surface tension of liquid platinum unaffected by oxygen adsorption was measured by the oscillating droplet method using four different levitation apparatuses: ESL, two EMLs, and ADL. Although the measurements with ESL and two EMLs demonstrated almost identical surface tension temperature dependence, lower values were obtained with ADL. In this section, to support the validity of our measurements, the uncertainties in the measurements are evaluated based on GUM [30].

4.1 Uncertainty for density measurement using ESL

The mass of the levitated droplet, M, that was used for density calculation was determined from the value measured after the solidification. This value is consistent with the mass measured prior to the experiment due to the absence of evaporation during the experiments. The volume of the levitated droplet used in ESL was determined from its 2D image observed from the horizontal direction by the HSV camera. The droplet diameter (d) was converted from pixels (px) to meters (m) by using the reference sphere, the diameter of which (d ref) was measured using a micrometer. Therefore, the main sources of uncertainty in the density measurement using ESL were the calibration of the electronic balance (m cal), measurement resolution of the balance (m res), repeatability of the sample mass measurements performed using the balance (m rep), calibration of the micrometer ( d cal ref ), measurement resolution of the micrometer ( d res ref ), repeatability of the diameter measurements for the reference sphere using the micrometer ( d rep ref ), resolutions of the HSV camera images for the levitated droplet ( d p res ) and the reference sphere ( dp res ref ), repeatability of the numerical contour fittings for the levitated droplet ( d p rep ) and the reference sphere ( dp rep ref ), and droplet volume conversions from the 2D images (V rep). The combined standard uncertainty in the density measurement [u c(ρ)] can be evaluated using the uncertainty contributions [u ρ (i)] of each source (i), as follows:

(13) u c ( ρ ) = [ u ρ ( m cal ) ] 2 + [ u ρ ( m res ) ] 2 + [ u ρ ( m rep ) ] 2 + [ u ρ ( dp res ref ) ] 2 + [ u ρ ( d cal ref ) ] 2 + [ u ρ ( d rep ref ) ] 2 + [ u ρ ( dp res ref ) ] 2 + [ u ρ ( dp rep ref ) ] 2 + [ u ρ ( rp res ) ] 2 + [ u ρ ( rp rep ) ] 2 + [ u ρ ( V rep ) ] 2 ,

where u ρ (i) is obtained by multiplying the individual standard uncertainty [u(i)] and the sensitivity coefficient [c(i)] for each source. As a representative example, Table 4 shows the uncertainty budget for the density measurement of liquid platinum using ESL when the largest value of the combined standard uncertainty is obtained for all plots. The standard uncertainties for the repeatability of the sample mass measurements using the balance [u(m rep)] and the diameter measurements for the reference sphere using the micrometer [u( d rep ref )] were evaluated based on 10 repeated measurements for the same sample. u(V rep) was calculated as the estimated standard deviation of the volume of the reference sphere estimated from the reference sphere image a few dozen times. When the coverage factor k p = 2 is selected to expand the uncertainty in our measurement plots to satisfy 95.45% confidence, the maximum value was calculated as ±710 kg·m−3, corresponding to approximately ±3.78% for the measurement plots. This indicates that the uncertainty of our density measurement is sufficiently low.

Table 4

Uncertainty budget of density measurements of liquid platinum using ESL for the largest calculated uncertainty

Source of uncertainty Value Divisor Standard uncertainty u(i) Sensitivity coefficient c(i) Uncertainty contribution u ρ (i)
Resolution of electro balance 5 × 10−7 kg 3 2.89 × 10−7 kg 4.77 × 108 m−3 138 kg·m−3
Calibration uncertainty of electro balance 2 × 10−7 kg 2 1 × 10−7 kg 4.77 × 108 m−3 47.7 kg·m−3
Repeatability of mass measurement 4.22 × 10−7 kg 1 4.22 × 10−7 kg 4.77 × 108 m−3 201 kg·m−3
Calibration uncertainty of micrometer 2 × 10−6 m 2 1 × 10−6 m −2.96 × 107 kg·m−4 −29.6 kg·m−3
Resolution of micrometer 5 × 10−6 m 3 2.89 × 10−6 m −2.96 × 107 kg·m−4 −85.3 kg·m−3
Repeatability of diameter measurement 3.68 × 10−6 m 1 3.68 × 10−6 m −2.96 × 107 kg·m−4 −109 kg·m−3
Resolution of HSV image for reference sphere 0.5 px 3 0.289 px 190 kg·m−3·px−1 55 kg·m−3
Repeatability of contour fitting on HSV image of reference sphere 0.613 Hz 1 0.613 Hz 190 kg·m−3·Hz−1 117 kg·m−3
Resolution of HSV image for droplet 0.5 px 3 0.289 px −240 kg·m−3·px−1 −69.2 kg·m−3
Repeatability of contour fitting on HSV image of droplet 0.613 Hz 1 0.613 Hz −240 kg·m−3·Hz−1 −147 kg·m−3
Volume converted from HSV image of droplet 4.24 × 10−12 m3 1 4.24 × 10−12 m3 −8.99 × 1012 kg·m−6 −37.9 kg·m−3
Combined uncertainty, u c(ρ) = 355 kg·m−3
Expanded uncertainty, u exp = 710 × 10−3 kg·m−3 (coverage factor k p = 2)

4.2 Uncertainty for surface tension measurement by ESL

As described in equations (2)−(6), the oscillating droplet method employing ESL enables the calculation of the surface tension of a levitated droplet based on seven parameters: r, ρ, M, f 2,0 , F, U, and L. The sources of uncertainty in the surface tension measurement performed using ESL with respect to r and M are d cal ref , d res ref , d rep ref , dp rep ref , rp rep , dp rep ref , dp res , m cal, m res, and m rep, same as in the case of the density measurement. For f 2,0 , U, and L, the uncertainties that must be considered include the frequency resolution of the FFT analysis (f res), repeatability of the FFT frequency analysis ( f rep ), resolution of the measured electrode potential (U res), stability of the electrode potential (U sta), scale resolution for measuring the distance between the electrodes (L res), and repeatability of these distance measurements (L rep). Therefore, the combined standard uncertainty in the surface tension measurement u c(σ ESL) can be evaluated as follows:

(14) u c ( σ ESL ) = [ u σ ( d cal ref ) ] 2 + [ u σ ( d res ref ) ] 2 + [ u σ ( d rep ref ) ] 2 + [ u σ ( dp res ref ) ] 2 + [ u σ ( dp rep ref ) ] 2 + [ u σ ( rp res ) ] 2 + [ u σ ( rp rep ) ] 2 + [ u σ ( m cal ) ] 2 + [ u σ ( m res ) ] 2 + [ u σ ( m rep ) ] 2 + [ u σ ( f 2 , 0 rep ) ] 2 + [ u σ ( f 2 , 0 rep ) ] 2 + [ u σ ( F ) ] 2 + [ u σ ( ρ ) ] 2 + [ u σ ( L res ) ] 2 + [ u σ ( L rep ) ] 2 + [ u σ ( U res ) ] 2 + [ u σ ( U sta ) ] 2 .

Table 5 presents the uncertainty budget for the surface tension measurement of liquid platinum using ESL when the calculated result of the uncertainty for the measurement plot showed the largest value. u( f 2,0 rep ) was determined based on a sampling rate of 4,096 Hz when monitoring the surface oscillation for 1 s. u( f 2 , 0 rep ) was evaluated based on four repeated identifications of the surface oscillations, using different samples with the same mass and temperature. u σ ( U sta ) was derived from the variations within the indicated values in ±300 V. In this study, with a selected coverage factor k p = 2, the largest uncertainty in the surface tension measurement with ESL was calculated as ±74.3 × 10−3 N·m−3. This represents approximately ±4.1% of the measured values. The droplet density makes the largest contribution to this uncertainty.

Table 5

Uncertainty budget of surface tension measurements of liquid platinum using ESL for the largest calculated uncertainty

Source of uncertainty Value Divisor Standard uncertainty u(i) Sensitivity coefficient c(i) Uncertainty contribution u σ (i)
Resolution of micrometer 5 × 10−6 m 3 2.89 × 10−6 m 4.49 × 10−2 N·m−2 1.3 × 10−7 N·m−1
Calibration uncertainty of micrometer 2 × 10−6 m 2 1 × 10−6 m 4.49 × 10−2 N·m−2 4.49 × 10−8 N·m−1
Repeatability of diameter measurement 3.68 × 10−6 m 1 3.68 × 10−6 m 4.49 × 10−2 N·m−2 1.65 × 10−7 N·m−1
Resolution of HSV image for reference sphere 0.5 px 3 0.289 px −1.77 × 10−2 N·m−1·px−1 5.1 × 10−3 N·m−1
Repeatability of contour fitting on HSV image of reference sphere 0.613 Hz 1 0.613 px −1.77 × 10−2 N·m−1·px−1 1.08 × 10−2 N·m−1
Resolution of HSV image for droplet 0.5 px 3 0.289 px 4.45 × 10−2 N·m−1·px−1 1.28 × 10−2 N·m−1
Repeatability of contour fitting on HSV image of droplet 0.613 px 1 0.613 px 4.45 × 10−2 N·m−1·px−1 2.72 × 10−2 N·m−1
Resolution of electro balance 5 × 10−7 kg 3 2.89 × 10−7 kg −899 N·m−1·kg−1 −2.7 × 10−5 N·m−1
Calibration uncertainty of electro balance 2 × 10−7 kg 2 1 × 10−7 kg −899 N·m−1·kg−1 −9 × 10−5 N·m−1
Repeatability of mass measurement 4.22 × 10−7 kg 1 4.22 × 10−7 kg −899 N·m−1·kg−1 3.79 × 10−4 N·m−1
Resolution of FFT analysis for surface oscillation frequency 0.5 Hz 3 0.289 Hz 1.87 × 10−2 N·m−1·Hz−1 5.4 × 10−3 N·m−1
Repeatability of identification for f 2,0 via FFT analysis 6.96 × 10−2 Hz 1 0.866 Hz 1.87 × 10−2 N·m−1·Hz−1 1.3 × 10−3 N·m−1
Uncertainty of density measurement 351 kg·m−3 1 351 kg·m−3 9.66 × 10−5 N kg·m2 3.4 × 10−2 N·m−1
Resolution of electron potential control 50 V 3 28.9 V 2.09 × 10−9 N·m−1·V−1 6.04 × 10−5 N·m−1
Repeatability of electron potential control 300 V 1 300 V 2.09 × 10−9 N·m−1·V−1 6.28 × 10−4 N·m−1
Resolution of scale 0.25 × 10−3 m 3 1.44 × 10−3 m 3.64 N·m−2 5.26 × 10−4 N·m−1
Repeatability of electrode distance measurements 0.5 × 10−3 m 1 0.5 × 10−3 m 3.64 N·m−2 1.05 × 10−3 N·m−1
Uncertainty of correction term F 6.77 × 10−4 1 6.77 × 10−4 1.8 N·m−1 1.21 × 10−3 N·m−1
Combined uncertainty, u c(σ ESL) = 37.1 × 10−3 N·m−1
Expanded uncertainty, u exp = 74.3 × 10−3 N·m−1 (coverage factor k p = 2)

4.3 Uncertainty for surface tension measurement by EML

In the oscillating droplet method employed using EML, the surface tension of a levitated droplet is calculated from seven parameters: M, f 2,0, f 2,±1, f 2,±2, f 1,0, f 1,±1, and ρ, as described in equations (9) and (10). The combined standard uncertainty in the surface tension measurement u c(σ EML) can be evaluated from the following equation:

(15) u c ( σ EML ) = [ u σ ( m cal ) ] 2 + [ u σ ( m res ) ] 2 + [ u σ ( m rep ) ] 2 + [ u σ ( f 2 , 0 res ) ] 2 + [ u σ ( f 2 , ± 1 res ) ] 2 + [ u σ ( f 2 , ± 2 res ) ] 2 + [ u σ ( f 1 , 0 res ) ] 2 + [ u σ ( f 1 , ± 1 res ) ] 2 + [ u σ ( f 2 , 0 rep ) ] 2 + [ u σ ( f 2 , ± 1 rep ) ] 2 + [ u σ ( f 2 , ± 2 rep ) ] 2 + [ u σ ( f 1 , 0 rep ) ] 2 + [ u σ ( f 1 , ± 1 rep ) ] 2 + [ u σ ( ρ ) ] 2 .

Tables 6 and 7 show the uncertainty budgets in the surface tension measurements of liquid platinum using EMLs installed at CIT and DLR, respectively, when the calculated results of the uncertainties for the measurement plot show the largest values. u( f 2 , m res ) and u( f 1 , m res ) were determined from five repeated identifications of the surface oscillations for a single sample. When the coverage factor k p = 2 was selected, the largest values of uncertainties in the surface tension measurement using EML were ±32.8 × 10−3 and ±20.4 × 10−3 N·m−1 for CIT and DLR, respectively. The measurement using the EML of CIT has a slightly larger uncertainty, which can be attributed to the larger sample size.

Table 6

Uncertainty budget of surface tension measurements of liquid platinum using EML installed at CIT for the largest calculated uncertainty

Source of uncertainty Value Divisor Standard uncertainty u(i) Sensitivity coefficient c(i) Uncertainty contribution u σ (i)
Calibration of electronic balances 1 × 10−8 kg 2 5 × 10−9 kg 685 N·m−1·kg−1 1.97 × 10−5 N·m−1
Measuring resolution of electronic balance 5 × 10−8 kg 3 2.89 × 10−8 kg 685 N·m−1·kg−1 3.42 × 10−6 N·m−1
Repeatability of sample mass measurement 4.22 × 10−7 kg 1 4.22 × 10−7 kg 685 N·m−1·kg−1 2.89 × 10−5 N·m−1
Frequency resolution of FFT analysis for m = 0 in l = 1 3.05 × 10–2 Hz 3 1.76 × 10−2 Hz 5.09 × 10−2 N·m−1·Hz−1 8.96 × 10−3 N·m−1
Repeatability of identification for f 1,0 via FFT analysis 2.66 × 10–2 Hz 1 2.66 × 10−2 Hz 5.09 × 10−2 N·m−1·Hz−1 1.35 × 10−3 N·m−1
Frequency resolution of FFT analysis for m = −1 in l = 1 3.05 × 10–2 Hz 3 1.76 × 10−2 Hz 4.76 × 10−2 N·m−1·Hz−1 8.39 × 10−4 N·m−1
Repeatability of identification for f 1,−1 via FFT analysis 7.63 × 10–2·Hz 1 3.9 × 10−2 Hz 4.76 × 10−2 N·m−1·Hz−1 1.86 × 10−3 N·m−1
Frequency resolution of FFT analysis for m = +1 in l = 1 3.05 × 10–2 Hz 3 1.76 × 10−2 Hz 8.66 × 10−2 N·m−1·Hz−1 1.53 × 10−3 N·m−1
Repeatability of identification for f 1,+1 via FFT analysis 7.63 × 10–2 Hz 1 7.63 × 10−2 Hz 8.66 × 10−2 N·m−1·Hz−1 6.6 × 10−3 N·m−1
Frequency resolution of FFT analysis for m = 0 in l = 2 3.05 × 10–2 Hz 3 1.76 × 10−2 Hz 2.98 × 10−2 N·m−1·Hz−1 5.25 × 10−4 N·m−1
Repeatability of identification for f 2,0 via FFT analysis 1.73 × 10−2 Hz 1 1.73 × 10−2 Hz 2.98 × 10−2 N·m−1·Hz−1 5.15 × 10−4 N·m−1
Frequency resolution of FFT analysis for m = ±1 in l = 2 3.05 × 10−2 Hz 3 1.76 × 10−2 Hz 7.57 × 10−2 N·m−1·Hz−1 1.33 × 10−3 N·m−1
Repeatability of identification for f 2,±1 via FFT analysis 1.47 × 10−2 Hz 1 1.47 × 10−2 Hz 7.57 × 10−2 N·m−1·Hz−1 1.11 × 10−2 N·m−1
Frequency resolution of FFT analysis for m = ±2 in l = 2 3.05 × 10−2 Hz 3 1.76 × 10−2 Hz 7.57 × 10−2 N·m−1·Hz−1 1.33 × 10−3 N·m−1
Repeatability of identification for f 2,±2 via FFT analysis 1.03 × 10−2 Hz 1 1.03 × 10−2 Hz 7.57 × 10−2 N·m−1·Hz−1 7.8 × 10−3 N·m−1
Reported density for liquid platinum 343 kg·m−3 1 343 kg·m−3 −1.58 × 10−5 Nm2·kg−1 −5.4 × 10−3 N·m−1
Combined uncertainty, u c(σ EML): 16.4 × 10−3 N·m−1
Expanded uncertainty, u exp: 32.8 × 10−3 N·m−1 (coverage factor k p = 2)
Table 7

Uncertainty budget of surface tension measurements of liquid platinum using EML installed at DLR for the largest calculated uncertainty

Source of uncertainty Value Divisor Standard u(i) Sensitivity coefficient c(i) Uncertainty contribution u σ (i)
Calibration of electronic balances 1 × 10−8 kg 2 5 × 10−9 kg 1.17 × 103 N·m−1·kg−1 3.39 × 10−7 N·m−1
Measuring resolution of electronic balance 5 × 10−10 kg 3 2.89 × 10−10 kg 1.17 × 103 N·m−1·kg−1 5.87 × 10−6 N·m−1
Repeatability of sample mass measurement 4 × 10−9 kg 2 2 × 10−9 kg 1.17 × 103 N·m−1·kg−1 2.35 × 10−6 N·m−1
Frequency resolution of FFT analysis for m = 0 in l = 1 7.75 × 10−2 Hz 3 4.48 × 10−2 Hz 1.94 × 10−2 N·m−1·Hz−1 8.68 × 10−4 N·m−1
Repeatability of identification for f 1,0 via FFT analysis 1.59 × 10−1 Hz 1 1.59 × 10−1 Hz 1.94 × 10−2 N·m−1·Hz−1 3.08 × 10−3 N·m−1
Frequency resolution of FFT analysis for m = −1 in l = 1 7.75 × 10−2 Hz 3 4.48 × 10−2 Hz 1.18 × 10−2 N·m−1·Hz−1 5.26 × 10−4 N·m−1
Repeatability of identification for f 1,−1 via FFT analysis 4.71 × 10−4 Hz 1 4.71 × 10−4 Hz 1.18 × 10−2 N·m−1·Hz−1 5.54 × 10−6 N·m−1
Frequency resolution of FFT analysis for m = +1 in l = 1 7.75 × 10−2 Hz 3 4.48 × 10−2 Hz 1.18 × 10−2 N·m−1·Hz−1 5.26 × 10−4 N·m−1
Repeatability of identification for f 1,+1 via FFT analysis 9.24 × 10−2 Hz 1 9.24 × 10−2 Hz 1.18 × 10−2 N·m−1·Hz−1 1.09 × 10−3 N·m−1
Frequency resolution of FFT analysis for m = 0 in l = 2 7.75 × 10−2 Hz 3 4.48 × 10−2 Hz 2.43 × 10−2 N·m−1·Hz−1 1.09 × 10−4 N·m−1
Repeatability of identification for f 2,0 via FFT analysis 2.73 × 10−1 Hz 1 2.73 × 10−1 Hz 2.43 × 10−2 N·m−1·Hz−1 6.64 × 10−3 N·m−1
Frequency resolution of FFT analysis for m = ±1 in l = 2 7.75 × 10−2 Hz 3 4.48 × 10−2 Hz 5.29 × 10−2 N·m−1·Hz−1 2.37 × 10−4 N·m−1
Repeatability of identification for f 2,±1 via FFT analysis 9.24 × 10−2 Hz 1 9.24 × 10−2 Hz 5.29 × 10−2 N·m−1·Hz−1 4.89 × 10−3 N·m−1
Frequency resolution of FFT analysis for m = ±2 in l = 2 7.75 × 10−2 Hz 3 4.48 × 10−2 Hz 5.14 × 10−2 N·m−1·Hz−1 2.3 × 10−4 N·m−1
Repeatability of identification for f 2,±2 via FFT analysis 4.6 × 10−2 Hz 1 4.6 × 10−2 Hz 5.14 × 10−2 N·m−1·Hz−1 2.36 × 10−3 N·m−1
Reported density for liquid platinum 350 kg·m−3 1 350 kg·m−3 −6.99 × 10−6 N·m2·kg−1 −2.45 × 10−3 N·m−1
Combined uncertainty, u c( σ EML): 10.2 × 10−3 N·m−1
Expanded uncertainty, u exp: 20.4 × 10−3 N·m−1 (coverage factor k p = 2)

4.4 Uncertainty for surface tension measurement by ADL

In the determination of the surface tension of an aerodynamically levitated droplet from the Rayleigh equation and parameters m and f 2.0 [6], the uncertainty [u c(σ ADL)] can be evaluated as follows:

(16) u c ( σ ADL ) = [ u σ ( m cal ) ] 2 + [ u σ ( m res ) ] 2 + [ u σ ( m rep ) ] 2 + [ u σ ( f 2 , 0 rep ) ] 2 + [ u σ ( f 2 , 0 rep ) ] 2 .

Table 8 presents the uncertainty budget in the surface tension measurement of liquid platinum using ADL when the calculated uncertainty shows the maximum value. u(f res) was determined by recording the droplet images at a rate of 2,000 fps for 0.3 s. u( m rep ) and u( f 2 , 0 rep ) were calculated from seven repeated measurements. When the coverage factor k p = 2 was selected, the maximum value of u c ( σ ADL ) was ±36.0 × 10−3 N·m−1, corresponding to approximately ±2.0% for the measurement plot. The remarkably high value of u( f 2 , 0 rep ) can be attributed to the small sample size, which results in an extremely short decay time for the surface oscillation.

Table 8

Uncertainty budget of surface tension measurements of liquid platinum using ADL for the largest calculated uncertainty

Source of uncertainty Value Divisor Standard uncertainty, u(i) Sensitivity coefficient, c(i) Uncertainty contribution, u σ (i)
Calibration of electronic balances 2 × 10−7 kg 2 1 × 10−7 kg 4.49 × 103 N·m−1 kg−1 1.3 × 10−3 N·m−1
Measuring resolution of electronic balance 5 × 10−7 kg 3 2.89 × 10−8 kg 4.49 × 103 N·m−1 kg−1 4.49 × 10−3 N·m−1
Repeatability of sample mass measurement 4.22 × 10−8 kg 1 4.22 × 10−8 kg 4.49 × 103 N·m−1 kg−1 1.9 × 10−3 N·m−1
Resolution of FFT analysis for surface oscillation frequency 1.18 × 10−3 Hz 1 1.18 × 10−3 Hz 1.8 × 10−2 N·m−1 Hz−1 2.12 × 10−5 N·m−1
Repeatability of identification for f 2,0 via FFT analysis 1.67 Hz 3 9.62 × 10−1 Hz 1.8 × 10−2 N·m−1 Hz−1 1.73 × 10−2 N·m−1
Combined uncertainty, u c(σ ADL) = 18.0 × 10−3 N·m−1
Expanded uncertainty, u exp = 36.0 × 10−3 N·m−1 (coverage factor k p = 2)

These uncertainty evaluations demonstrate that for various plots of the measured surface tension at high temperatures using ADL, the temperature dependence of the surface tension is lower than those measured with ESL and EMLs beyond the uncertainty in the measurement.

When a droplet is levitated with ESL and EML, external forces such as gravitational acceleration, electric charge, and electromagnetic force induce droplet deformation, thereby altering the frequencies of surface oscillations. Therefore, the effects of droplet deformation by such forces are calibrated to ensure accurate surface tension measurements with ESL and EML. However, in the measurement with ADL, the effects of droplet deformation due to external forces such as the gas jet pressure, fluid flow induced by the jet stream, and gravitational acceleration have not been adequately considered. To better understand the discrepancies in the measured surface tension values obtained using ESL, EML, and ADL, a further investigation of how the deformation of the aerodynamically levitated droplet affects the behavior of surface oscillations is required.

4.5 Uncertainty for gradients of density and surface tension relative to temperature

The temperature dependence of density can be expressed in the following form:

(17) ρ = β ˆ ( T − T ̅ ) + ρ ̅ ,

where β ˆ is the temperature coefficient of density, and T ̅ and ρ ̅ are the mean values of temperature and density, respectively. The uncertainty in β ˆ of this regression line can be predicted using the following equations:

(18) u 2 ( β ˆ ) = δ ˆ e 2 + u c 2 ( ρ i ) ∑ ( T i − T ̅ ) 2 ,

(19) δ ˆ e 2 = ∑ ρ − β ˆ ( T i − T ¯ ) + ρ ¯ n − 2 ,

where δ ˆ e is the residual variance of the measured data and n is the number of measuring plots. By using this formula, the uncertainty in the gradient of the measured density with respect to temperature (equation (11)) was evaluated to be ±0.3888 kg·m−3·K−1 with a k p = 2. Similarly, the uncertainty in the gradient of the measured surface tension with respect to temperature (equation (12)), determined using ESL and EMLs, was evaluated to be ±0.0445 × 10−3 N·m−1·K−1.

5 Summary

The surface tension of liquid platinum free of any contaminations from the supporting materials and atmospheric oxygen was measured by the oscillating droplet method using ESL, two EMLs, and ADL. In addition, the density was precisely measured with ESL over a wide temperature range of approximately 500 K, including the undercooling conditions. The temperature dependence of the density for liquid platinum was described by a linear fit of the plotted data given by

ρ = 18,906 ± 715 − ( 1.0852 ± 0.3888 ) × ( T − 2,041 ) [ kg· m − 3 ] ( 1 , 700 – 2 , 200 K ) .

The round-robin measurement of the surface tension of liquid platinum clarified that a nearly identical surface tension-temperature relationship can be obtained using both ESL and EML. When considering the uncertainty in the measurements with ESL and EML, the temperature dependence of the surface tension was described as follows:

σ = 1,798 ± 74.3 − ( 0.12 ± 0.0445 ) × ( T − 2,041 ) [ 10 − 3 N· m − 1 ] ( 1 , 900 – 2 , 600 K ) .

The surface tensions measured in the round-robin test with ADL were slightly lower than those measured with ESL and EMLs, particularly at higher temperatures, with the discrepancy exceeding the measurement uncertainty. To understand the origin of this discrepancy in the measurement results, it is necessary to investigate the effect of the deformation of the aerodynamically levitated droplet on the surface oscillation behavior.

Acknowledgments

This study was financially supported by JSPS KAKENHI under Grant P20H02453. This study was supported by Grant-in-Aid for Front Loading Research from the Advisory Committee for Space Utilization Research in ISAS/JAXA.

  1. Funding information: This research was partially funded by the Japan Society for the Promotion of Science (JSPS) KAKENHI under Grant No. P20H02453, and by the Grant-in-Aid for Front Loading Research, which is supported by the Advisory Committee for Space Utilization Research in the Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency (ISAS/JAXA).

  2. Author contributions: Yusaku Seimiya: writing – original draft preparation, investigation; Shuto Tomita: investigation; Tohei Kawaguchi: investigation; Hidekazu Kobatake: investigation, review and editing; Jürgen Brillo: investigation, review and editing; Suguru Shiratori: investigation, image analysis, review and editing; Ken-ichi Sugioka: review and editing; Takehiko Ishikawa: investigation, review and editing; Shumpei Ozawa: supervision, writing – original draft preparation, writing – review and editing, investigation, resource.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: The data that support the findings of this study are available from the corresponding author upon reasonable request.

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Received: 2023-10-27
Revised: 2023-11-28
Accepted: 2023-12-14
Published Online: 2023-12-31

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Articles in the same Issue

  1. Research Articles
  2. First-principles investigation of phase stability and elastic properties of Laves phase TaCr2 by ruthenium alloying
  3. Improvement and prediction on high temperature melting characteristics of coal ash
  4. First-principles calculations to investigate the thermal response of the ZrC(1−x)Nx ceramics at extreme conditions
  5. Study on the cladding path during the solidification process of multi-layer cladding of large steel ingots
  6. Thermodynamic analysis of vanadium distribution behavior in blast furnaces and basic oxygen furnaces
  7. Comparison of data-driven prediction methods for comprehensive coke ratio of blast furnace
  8. Effect of different isothermal times on the microstructure and mechanical properties of high-strength rebar
  9. Analysis of the evolution law of oxide inclusions in U75V heavy rail steel during the LF–RH refining process
  10. Simultaneous extraction of uranium and niobium from a low-grade natural betafite ore
  11. Transfer and transformation mechanism of chromium in stainless steel slag in pedosphere
  12. Effect of tool traverse speed on joint line remnant and mechanical properties of friction stir welded 2195-T8 Al–Li alloy joints
  13. Technology and analysis of 08Cr9W3Co3VNbCuBN steel large diameter thick wall pipe welding process
  14. Influence of shielding gas on machining and wear aspects of AISI 310–AISI 2205 dissimilar stainless steel joints
  15. Effect of post-weld heat treatment on 6156 aluminum alloy joint formed by electron beam welding
  16. Ash melting behavior and mechanism of high-calcium bituminous coal in the process of blast furnace pulverized coal injection
  17. Effect of high temperature tempering on the phase composition and structure of steelmaking slag
  18. Numerical simulation of shrinkage porosity defect in billet continuous casting
  19. Influence of submerged entry nozzle on funnel mold surface velocity
  20. Effect of cold-rolling deformation and rare earth yttrium on microstructure and texture of oriented silicon steel
  21. Investigation of microstructure, machinability, and mechanical properties of new-generation hybrid lead-free brass alloys
  22. Soft sensor method of multimode BOF steelmaking endpoint carbon content and temperature based on vMF-WSAE dynamic deep learning
  23. Mechanical properties and nugget evolution in resistance spot welding of Zn–Al–Mg galvanized DC51D steel
  24. Research on the behaviour and mechanism of void welding based on multiple scales
  25. Preparation of CaO–SiO2–Al2O3 inorganic fibers from melting-separated red mud
  26. Study on diffusion kinetics of chromium and nickel electrochemical co-deposition in a NaCl–KCl–NaF–Cr2O3–NiO molten salt
  27. Enhancing the efficiency of polytetrafluoroethylene-modified silica hydrosols coated solar panels by using artificial neural network and response surface methodology
  28. High-temperature corrosion behaviours of nickel–iron-based alloys with different molybdenum and tungsten contents in a coal ash/flue gas environment
  29. Characteristics and purification of Himalayan salt by high temperature melting
  30. Temperature uniformity optimization with power-frequency coordinated variation in multi-source microwave based on sequential quadratic programming
  31. A novel method for CO2 injection direct smelting vanadium steel: Dephosphorization and vanadium retention
  32. A study of the void surface healing mechanism in 316LN steel
  33. Effect of chemical composition and heat treatment on intergranular corrosion and strength of AlMgSiCu alloys
  34. Soft sensor method for endpoint carbon content and temperature of BOF based on multi-cluster dynamic adaptive selection ensemble learning
  35. Evaluating thermal properties and activation energy of phthalonitrile using sulfur-containing curing agents
  36. Investigation of the liquidus temperature calculation method for medium manganese steel
  37. High-temperature corrosion model of Incoloy 800H alloy connected with Ni-201 in MgCl2–KCl heat transfer fluid
  38. Investigation of the microstructure and mechanical properties of Mg–Al–Zn alloy joints formed by different laser welding processes
  39. Effect of refining slag compositions on its melting property and desulphurization
  40. Effect of P and Ti on the agglomeration behavior of Al2O3 inclusions in Fe–P–Ti alloys
  41. Cation-doping effects on the conductivities of the mayenite Ca12Al14O33
  42. Modification of Al2O3 inclusions in SWRH82B steel by La/Y rare-earth element treatment
  43. Possibility of metallic cobalt formation in the oxide scale during high-temperature oxidation of Co-27Cr-6Mo alloy in air
  44. Multi-source microwave heating temperature uniformity study based on adaptive dynamic programming
  45. Round-robin measurement of surface tension of high-temperature liquid platinum free of oxygen adsorption by oscillating droplet method using levitation techniques
  46. High-temperature production of AlN in Mg alloys with ammonia gas
  47. Review Article
  48. Advances in ultrasonic welding of lightweight alloys: A review
  49. Topical Issue on High-temperature Phase Change Materials for Energy Storage
  50. Compositional and thermophysical study of Al–Si- and Zn–Al–Mg-based eutectic alloys for latent heat storage
  51. Corrosion behavior of a Co−Cr−Mo−Si alloy in pure Al and Al−Si melt
  52. Al–Si–Fe alloy-based phase change material for high-temperature thermal energy storage
  53. Density and surface tension measurements of molten Al–Si based alloys
  54. Graphite crucible interaction with Fe–Si–B phase change material in pilot-scale experiments
  55. Topical Issue on Nuclear Energy Application Materials
  56. Dry synthesis of brannerite (UTi2O6) by mechanochemical treatment
  57. Special Issue on Polymer and Composite Materials (PCM) and Graphene and Novel Nanomaterials - Part I
  58. Heat management of LED-based Cu2O deposits on the optimal structure of heat sink
  59. Special Issue on Recent Developments in 3D Printed Carbon Materials - Part I
  60. Porous metal foam flow field and heat evaluation in PEMFC: A review
  61. Special Issue on Advancements in Solar Energy Technologies and Systems
  62. Research on electric energy measurement system based on intelligent sensor data in artificial intelligence environment
  63. Study of photovoltaic integrated prefabricated components for assembled buildings based on sensing technology supported by solar energy
  64. Topical Issue on Focus of Hot Deformation of Metaland High Entropy Alloys - Part I
  65. Performance optimization and investigation of metal-cored filler wires for high-strength steel during gas metal arc welding
  66. Three-dimensional transient heat transfer analysis of micro-plasma arc welding process using volumetric heat source models
https://doi.org/10.1515/htmp-2022-0306
Keywords for this article
High temperature melt; thermophysical properties; electrostatic levitation; electromagnetic levitation; aerodynamics levitation
Creative Commons
BY 4.0

Articles in the same Issue

  1. Research Articles
  2. First-principles investigation of phase stability and elastic properties of Laves phase TaCr2 by ruthenium alloying
  3. Improvement and prediction on high temperature melting characteristics of coal ash
  4. First-principles calculations to investigate the thermal response of the ZrC(1−x)Nx ceramics at extreme conditions
  5. Study on the cladding path during the solidification process of multi-layer cladding of large steel ingots
  6. Thermodynamic analysis of vanadium distribution behavior in blast furnaces and basic oxygen furnaces
  7. Comparison of data-driven prediction methods for comprehensive coke ratio of blast furnace
  8. Effect of different isothermal times on the microstructure and mechanical properties of high-strength rebar
  9. Analysis of the evolution law of oxide inclusions in U75V heavy rail steel during the LF–RH refining process
  10. Simultaneous extraction of uranium and niobium from a low-grade natural betafite ore
  11. Transfer and transformation mechanism of chromium in stainless steel slag in pedosphere
  12. Effect of tool traverse speed on joint line remnant and mechanical properties of friction stir welded 2195-T8 Al–Li alloy joints
  13. Technology and analysis of 08Cr9W3Co3VNbCuBN steel large diameter thick wall pipe welding process
  14. Influence of shielding gas on machining and wear aspects of AISI 310–AISI 2205 dissimilar stainless steel joints
  15. Effect of post-weld heat treatment on 6156 aluminum alloy joint formed by electron beam welding
  16. Ash melting behavior and mechanism of high-calcium bituminous coal in the process of blast furnace pulverized coal injection
  17. Effect of high temperature tempering on the phase composition and structure of steelmaking slag
  18. Numerical simulation of shrinkage porosity defect in billet continuous casting
  19. Influence of submerged entry nozzle on funnel mold surface velocity
  20. Effect of cold-rolling deformation and rare earth yttrium on microstructure and texture of oriented silicon steel
  21. Investigation of microstructure, machinability, and mechanical properties of new-generation hybrid lead-free brass alloys
  22. Soft sensor method of multimode BOF steelmaking endpoint carbon content and temperature based on vMF-WSAE dynamic deep learning
  23. Mechanical properties and nugget evolution in resistance spot welding of Zn–Al–Mg galvanized DC51D steel
  24. Research on the behaviour and mechanism of void welding based on multiple scales
  25. Preparation of CaO–SiO2–Al2O3 inorganic fibers from melting-separated red mud
  26. Study on diffusion kinetics of chromium and nickel electrochemical co-deposition in a NaCl–KCl–NaF–Cr2O3–NiO molten salt
  27. Enhancing the efficiency of polytetrafluoroethylene-modified silica hydrosols coated solar panels by using artificial neural network and response surface methodology
  28. High-temperature corrosion behaviours of nickel–iron-based alloys with different molybdenum and tungsten contents in a coal ash/flue gas environment
  29. Characteristics and purification of Himalayan salt by high temperature melting
  30. Temperature uniformity optimization with power-frequency coordinated variation in multi-source microwave based on sequential quadratic programming
  31. A novel method for CO2 injection direct smelting vanadium steel: Dephosphorization and vanadium retention
  32. A study of the void surface healing mechanism in 316LN steel
  33. Effect of chemical composition and heat treatment on intergranular corrosion and strength of AlMgSiCu alloys
  34. Soft sensor method for endpoint carbon content and temperature of BOF based on multi-cluster dynamic adaptive selection ensemble learning
  35. Evaluating thermal properties and activation energy of phthalonitrile using sulfur-containing curing agents
  36. Investigation of the liquidus temperature calculation method for medium manganese steel
  37. High-temperature corrosion model of Incoloy 800H alloy connected with Ni-201 in MgCl2–KCl heat transfer fluid
  38. Investigation of the microstructure and mechanical properties of Mg–Al–Zn alloy joints formed by different laser welding processes
  39. Effect of refining slag compositions on its melting property and desulphurization
  40. Effect of P and Ti on the agglomeration behavior of Al2O3 inclusions in Fe–P–Ti alloys
  41. Cation-doping effects on the conductivities of the mayenite Ca12Al14O33
  42. Modification of Al2O3 inclusions in SWRH82B steel by La/Y rare-earth element treatment
  43. Possibility of metallic cobalt formation in the oxide scale during high-temperature oxidation of Co-27Cr-6Mo alloy in air
  44. Multi-source microwave heating temperature uniformity study based on adaptive dynamic programming
  45. Round-robin measurement of surface tension of high-temperature liquid platinum free of oxygen adsorption by oscillating droplet method using levitation techniques
  46. High-temperature production of AlN in Mg alloys with ammonia gas
  47. Review Article
  48. Advances in ultrasonic welding of lightweight alloys: A review
  49. Topical Issue on High-temperature Phase Change Materials for Energy Storage
  50. Compositional and thermophysical study of Al–Si- and Zn–Al–Mg-based eutectic alloys for latent heat storage
  51. Corrosion behavior of a Co−Cr−Mo−Si alloy in pure Al and Al−Si melt
  52. Al–Si–Fe alloy-based phase change material for high-temperature thermal energy storage
  53. Density and surface tension measurements of molten Al–Si based alloys
  54. Graphite crucible interaction with Fe–Si–B phase change material in pilot-scale experiments
  55. Topical Issue on Nuclear Energy Application Materials
  56. Dry synthesis of brannerite (UTi2O6) by mechanochemical treatment
  57. Special Issue on Polymer and Composite Materials (PCM) and Graphene and Novel Nanomaterials - Part I
  58. Heat management of LED-based Cu2O deposits on the optimal structure of heat sink
  59. Special Issue on Recent Developments in 3D Printed Carbon Materials - Part I
  60. Porous metal foam flow field and heat evaluation in PEMFC: A review
  61. Special Issue on Advancements in Solar Energy Technologies and Systems
  62. Research on electric energy measurement system based on intelligent sensor data in artificial intelligence environment
  63. Study of photovoltaic integrated prefabricated components for assembled buildings based on sensing technology supported by solar energy
  64. Topical Issue on Focus of Hot Deformation of Metaland High Entropy Alloys - Part I
  65. Performance optimization and investigation of metal-cored filler wires for high-strength steel during gas metal arc welding
  66. Three-dimensional transient heat transfer analysis of micro-plasma arc welding process using volumetric heat source models
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