Development of Heat-of-fusion Measurement for Metals Using a Closed-type Aerodynamic Levitator

Abstract

The heats of fusion of Fe, Ni, and Co were measured using a closed-type aerodynamic levitation method to prevent chemical interactions with the container and oxidation of the samples. The hypercooling limits of these metals were experimentally determined using the correlation between the undercooling temperature and thermal plateau time. The heats of fusion of the metals were obtained as the product of the hypercooling limit and the isobaric heat capacity. The experimentally determined hypercooling limits for Fe, Ni, and Co were 280, 414, and 360 K, respectively. Using these hypercooling limits, the heats of fusion of pure Fe, Ni, and Co were determined as 12.7 ± 2.2 kJ mol⁻¹, 16.9 ± 5.6 kJ mol⁻¹, 14.8 ± 2.8 kJ mol⁻¹, respectively. Notably, these experimentally determined heats of fusion using the closed-type aerodynamic levitation method closely align with the literature values within the range of experimental uncertainty, affirming the reliability of this measurement technique.

1. Introduction

The importance of numerical simulations has been rapidly increasing in material processing, such as casting and welding, to improve or develop these techniques. For accurate numerical simulations, the precise thermophysical properties of the materials are crucial input parameters.^1,2) However, the measurement of the thermophysical properties of metals in the liquid phase is challenging owing to the high reactivity of the metallic liquid at elevated temperatures. This high reactivity of metallic samples often leads to chemical contamination from the contact materials. To prevent contamination from the contact materials, containerless processing for the measurement of thermophysical properties using levitation has been developed. The thermophysical properties of metallic liquids and liquid alloys, such as surface tension,^3,4,5,6,7,8) density,^9,10,11,12) viscosity^7,8,13) heat capacity,¹⁴⁾ thermal conductivity,^14,15) and normal spectral emissivity,^16,17,18) have been measured using electromagnetic levitation or electrostatic levitation.

The heat of fusion of metal is a critical thermophysical property that influences the temperature field at the solid/liquid interface during solidification processes, because the temperature field at the interface affects the microstructure of the solid product. Therefore, obtaining reliable data on the heat of fusion is crucial for numerical simulations. Conventionally, the heat of fusion has been measured using thermal analysis methods, such as drop calorimetry, adiabatic calorimetry, and differential scanning calorimetry with containers. However, these conventional measurement methods use containers, and the reactivity of metals in the liquid phase at elevated temperatures and the chemical reaction of metals with the container could result in sample contamination and increase the uncertainty in the measurement. Because of this difficulty in measurement, there is wide scattering of the heat-of-fusion data, even for pure metals. Table 1 shows the heat of fusion of Fe measured by drop calorimetry^{19,20,21,22,23)} and adiabatic calorimetry,^24,25,26) mixture.^27,28,29) And the value of heat-of-fusion summarized in NIST-JANAF varied from 11.6 to 16.2 kJ mol⁻¹ (i.e. by more than 30%).³⁰⁾ To address the aforementioned challenges and improve the measurement accuracy, the heat of fusion of metals should be measured using a containerless method.

Table 1. Heats of fusion of Fe in the literature.

Sample	Year	Calorimetric Method	Heat of fusion, ΔHf [kJ mol−1]	Reference(s)
Fe	1918	Drop	11.6	19
	1926, 1929	Mixture	16.2	27,28,29
	1927	Drop	15.1	20
	1956	Mixture	15.3	29
	1962	Drop	13.8	21
	1966	Drop	13.8	22
	1965–1968	Adiabatic	14.4	24,25,26
	1975	Drop	14.1	23

The electrostatic levitation method has been used by several research groups.^31,32) However, conducting experiments under vacuum conditions introduces difficulties in controlling oxygen partial pressure and preventing metal evaporation. In the present work, we used aerodynamic levitation with controlled oxygen partial pressure under normal pressure to mitigate these challenges and provide accurate heat-of-fusion measurements. To evaluate the applicability of this measurement method for metals, the heats of fusion of Fe, Ni, and Co were measured and compared with reference values in the literature.

2. Principle of Measurement

The heat of fusion, ΔH_f, can be determined from the temperature profile during recalescence considering the heat balance between the released heat as latent heat and the heat loss to the surroundings. When electrostatic levitation in a vacuum is used, the heat of fusion released during the solidification process is lost by radiation.^26,27) Meanwhile, when the sample is levitated using aerodynamic levitation, the heat loss is attributed to the sum of the radiation and forced convection heat transfer by the gas.

In line with the conceptual framework proposed by Jeon et al.³²⁾ for electrostatic levitation, solidification under aerodynamic levitation can be categorized into three distinct types with respect to undercooling, ΔT: no undercooling, undercooling, and hypercooling. Figure 1 depicts the temperature profile during each type of solidification process.

Fig. 1. Three categories of temperature profile during solidification with (a) no undercooling, (b) undercooling, and (c) hypercooling.

2.1. Heat Balance with No Undercooling

Without undercooling, a thermal plateau time, Δt, emerges, during which the sample temperature remains constant at its melting temperature, T_m. This thermal plateau time ends when solidification of the sample is complete. In such instances, the heat of fusion released by solidification is balanced with the radiation and heat transfer by the gas, as delineated by the following equation:

  

$$\mathrm{\Delta }{H}_{\text{f}}=\int \left\{\frac{{\sigma }_{\text{B}}A{\epsilon }_{\text{T}}}{m}\left(T^4-T_{\text{r}}^4\right)+h_{\text{g}}\left(T-T_{\text{r}}\right)\right\}dt.$$(1)

Here, m is the sample mass, σ_B is the Stefan–Boltzmann constant, T is the sample temperature, T_r is the surrounding temperature, and h_g denotes the heat transfer coefficient of the gas.

2.2. Heat Balance in Undercooling

Prior to solidification, the sample undergoes a temperature increase to T_m following nucleation. Subsequently the solidification progresses during Δt. In this type of the temperature profile, the heat of fusion released by solidification is balanced with the recalescence, radiation at the plateau, and heat transfer by the gas. Consequently, the heat of fusion is expressed using the isobaric heat capacity, C_p, and ΔT at nucleation as follows:

  

$$\mathrm{\Delta }{H}_{\text{f}}=C_{\text{p}}\mathrm{\Delta }T+\int \left\{\frac{{\sigma }_{\text{B}}A{\epsilon }_{\text{T}}}{m}\left(T^4-T_{\text{r}}^4\right)+h_{\text{g}}\left(T-T_{\text{r}}\right)\right\}dt.$$(2)

2.3. Heat Balance in Hypercooling

As ΔT increases, Δt decreases. This phenomenon occurs because the heat of fusion is consumed to elevate the temperature up to the T_m of the sample. Ultimately, Δt decreases to zero, which is defined as the hypercooling limit, ΔT_hyp. Therefore, under this undercooling condition, the heat of fusion can be solely expressed in terms of recalescence using ΔT_hyp as follows:

  

$$\mathrm{\Delta }{H}_{\text{f}}=C_{\text{p}}\mathrm{\Delta }{T}_{\text{hyp}}$$(3)

Therefore, the heat of fusion can be determined using C_p and ΔT_hyp.

The heat of fusion can be determined by establishing ΔT_hyp from the correlation between ΔT and Δt using the heat balance in Eqs. (1), (2), (3) as follows. Throughout Δt, the sample temperature remains at T_m. Assuming constant values for the total hemispheric emissivity and the gas flow rate, the terms involving the radiation and the heat transfer coefficient of the gas in Eq. (2) can be approximated as constant. Consequently, by denoting the constant value of both the terms of radiation and heat transfer in Eq. (2) as α, the correlation between ΔT and Δt can be expressed as follows:

  

$$\mathrm{\Delta }t=-\frac{C_{\text{p}}}{\alpha }\mathrm{\Delta }T+\frac{\mathrm{\Delta }{H}_{\text{f}}}{\alpha }.$$(4)

Equation (4) shows that Δt and ΔT have a liner relationship with a slope of $-\frac{C_{\text{p}}}{\alpha }$ and an intercept of $\frac{\mathrm{\Delta }{H}_{\text{f}}}{\alpha }$. Through fitting of Eq. (4) to the experimental data, ΔT_hyp can be obtained corresponding to the point where Δt is zero. Using ΔT_hyp, the heat of fusion can be measured using Eq. (3).

3. Experimental Details

Pure Fe (99.996% purity, Nilaco Corporation), Ni (>99% purity, Nilaco Corporation) and Co (99.995% purity, Nilaco Corporation) wires with 2.0-mm diameter were used for the measurements. These metal wires were cut to a mass of 24–26 mg, which is considered to be optimal for aerodynamic levitation. The samples were positioned on a gas nozzle within a closed chamber and then levitated in a flow of Ar-3 vol.% H₂ gas mixture (Fig. 2). The gas supply to the chamber involved passing it through a magnesium deoxidizer furnace to eliminate the residual oxygen. The oxygen partial pressures of the introduced and exhausted gases were monitored using a zirconia oxygen sensor to ensure that the oxygen partial pressure in the chamber remained sufficiently low to prevent the oxidation of the metal samples. Throughout the measurements, the oxygen partial pressure in the aerodynamic levitator was kept below 10⁻²² Pa.

Fig. 2. Schematic illustration of the closed-type aerodynamic levitator used for heat-of-fusion measurements.

The levitated metals in the Ar–H₂ gas flow were subjected to heating by irradiation from a CO₂ laser (Novanta, SYNRAD100; power rating: 100 W). The sample temperature measurements were conducted from the side of the levitated sample using a two-color pyrometer (CHINO Corporation, IR-CZH7N). The sampling rate of temperature measurement was 0.1 s. This sampling rate cause the uncertainty in Δt, and the half of this value was indicated as error bar in Figs. 5 and 6.

Fig. 5. Correlation between the undercooling and thermal plateau time obtained from the temperature profile during the solidification process of Fe (●), Ni (■), and Co (▲).

Fig. 6. Plot of thermal plateau time and undercooling with a fitting line (solid). The dashed line indicates the uncertainty of the fitting line.

The spectral radiance of the object can be expressed using the Wien’s law:

  

$$L_{\lambda }\left(\lambda ,T\right)=\epsilon \left(\lambda \right)\cdot c_1{\lambda }^{-5}\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left(\frac{-c_2}{\lambda T}\right).$$(5)

Here, L_λ(λ,T) indicates the spectral radiance, λ denotes the measurement wavelength, T is the sample temperature, ε(λ) is the spectral emissivity, c₁ is 1.19×10⁻¹⁶ (W·m²·sr⁻¹), and c₂ is 1.44×10⁻² (m·K).

Using Eq. (5), the radiance ratio, R_L(T), at two different wavelengths λ₁ and λ₂ is expressed as follows:

  

$$R_{\text{L}}\left(T\right)=R_{\epsilon }\cdot R_{\lambda }^{-5}\cdot \mathrm{e}\mathrm{x}\mathrm{p}\left(-\frac{c_2\mathrm{\Lambda }}{T}\right).$$(6)

Here, R_ε is the spectral emissivity ratio: ε(λ₁)/ε(λ₂), $R_{\lambda }^{-5}$ is the wavelength ratio: λ₁/λ₂, and Λ is the composite wavelength: (λ₂−λ₁)/λ₁λ₂.The thermal plateau temperature observed during the melting was used to calibrate the input spectral emissivity ratio of the pyrometer using Eq. (6).

The average loss of sample mass during measurements was limited to less 1% of the initial sample mass.

4. Results

4.1. Hypercooling Limit

Figure 3 illustrates a typical temperature profile of the aerodynamically levitated Fe sample during the measurement. Initially, the sample temperature shows a plateau, at which temperature was maintained at T_m, corresponding to liquid–solid coexistence. Subsequent CO₂ laser irradiation gradually induced melting of the sample. Following the complete melting of the metal sample, the sample temperature increased. The temperature of each liquid sample during the experiment was calibrated using the melting temperature. Upon reaching the maximum temperature in the liquid state, the CO₂ laser was deactivated, and sample was allowed to cool. Nucleation occurred in the undercooled temperature region, leading to recalescence. The sample temperature increased to T_m and then underwent cooling after the solidification was completed.

Fig. 3. A typical temperature profile of the aerodynamically levitated Fe sample during the measurement.

Typical recalescence temperature profiles observed in the solidification processes of Fe are presented in Figs. 4(a) and 4(b). After undercooling to below the T_m of Fe, a rapid temperature increase corresponding to ΔT of 70 K (Fig. 4(a)) was observed. This phenomenon is attributed to the nucleation of solid Fe solidifying from the liquid phase accompanied by the release of the latent heat (heat of fusion). Following recalescence to the T_m of Fe driven by the release of heat of fusion, Δt was observed. As ΔT increased, Δt became shorter and eventually reached zero when ΔT reached 280 K (Fig. 4(b)). This could be a result of the heat of fusion being consumed.

Fig. 4. Typical recalescence temperature profiles observed during the solidification processes of Fe with (a) small under cooling (ΔT = 70 K) and (b) large under cooling (ΔT = 280 K).

The determination of Δt in Fig. 4 was as follows. Since the temperature cooling profile changes before and after the thermal plateau, the deviation of the temperature profile, i.e. the change in cooling rate, dT ⁄ dt, expressed by the following equation was obtainable from the experimental data:

  

$$\frac{dT}{dt}=\frac{T_{t+0.1}-T_t}{0.1}$$(7)

Here, t is measurement time, T_t is the sample temperature at t, and the number of 0.1 is derived from the sampling rate of the temperature measurement of the pyrometer used in this study. The beginning of the thermal plateau was identified as the moment when the temperature returned to the melting point, whereas the ending of the plateau time was determined as the time when the cooling rate reverted to its initial value.

Figure 5 illustrates the correlation between ΔT and Δt obtained from the temperature profile during the solidification process shown in Figs. 4(a) and 4(b). The error bar on the vertical axis for each data shown in Fig. 5 is 0.1 s, determined by the sampling rate for the temperature measurement using the pyrometer. For all metals — Fe (●), Ni (■), and Co (▲) — linear correlations between ΔT and Δt were observed. Linear-fitting functions for the ΔT and Δt data plots for Fe, Ni, and Co were obtained as follows, respectively:

  

$$\text{Fe:}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\Delta }t=-3.7\times {10}^{-3}\cdot \mathrm{\Delta }T+1.0,$$(8)

  

$$\text{Ni:}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\Delta }t=-3.3\times {10}^{-3}\cdot \mathrm{\Delta }T+1.4,$$(9)

  

$$\text{Co:}\mathrm{\hspace{0.17em} }\mathrm{\hspace{0.17em} }\mathrm{\Delta }t=-3.8\times {10}^{-3}\cdot \mathrm{\Delta }T+\mathrm{1.4.}$$(10)

This result suggests that even using aerodynamic levitation, a linear correlation can be obtained, similar to the case of electrostatic levitation³²⁾ and as predicted from the Eq. (4). Using these experimentally determined functions, the ΔT_hyp values of Fe, Ni, and Co were determined to be 280, 414, and 360 K, respectively. The ΔT_hyp value of Ni shows a good agreement with the literature values of 444 K measured using the electrostatic levitation method,³³⁾ but that of Fe in literature value 357 K for Fe is larger higher than our value of 280 K. The discrepancy of ΔT_hyp found in Fe will be discussed in Chapter 5.

4.2. Heat of Fusion

The heats of fusion of Fe, Ni, and Co were determined using Eq. (3) as the product of C_p and ΔT_hyp. We used the values of C_p reported in the literature, which were measured using laser-modulated calorimetry with electromagnetic levitation.^34,35,36) The C_p values are 45.4 ± 3.2 J mol⁻¹ K⁻¹ for Fe,³⁴⁾ 40.8 ± 5.0 J mol⁻¹ K⁻¹ for Ni,³⁵⁾ and 41.2 ± 1.9 J mol⁻¹ K⁻¹ for Co.³⁶⁾ Because the C_p values in the literature^34,35,36) were independent against temperature, the heats of fusion of Fe, Ni, and Co could be simply determined from Eq. (3) as the product of the ΔT_hyp and C_p. The determined heats of fusion of Fe, Ni, and Co were 12.7, 16.9, and 14.8 kJ mol⁻¹, respectively, as summarized in Table 2.

Table 2. Heats of fusion and hypercooling limit of pure metals measured in this study and isobaric molar heat capacity from the literature that were used for calculating the heats of fusion.

Sample	Cp [J mol−1 K−1]	ΔThyp [K]	ΔHf [kJ mol−1]
Fe	45.434)	280	12.7
Ni	40.835)	414	16.9
Co	41.236)	360	14.8

5. Discussion

5.1. Uncertainty Analysis of the Heat-of-fusion Measurement

The uncertainty in the heat-of-fusion measurement was assessed following the principles outlined in the Guide to the Expression of Uncertainty in Measurement (GUM).³⁷⁾ The uncertainty in the heat of fusion, denoted as u(ΔH_f), can be quantified using the following uncertainty propagation equation:

  

$$u^2\left(\mathrm{\Delta }{H}_{\text{f}}\right)={\left(\frac{\partial \mathrm{\Delta }{H}_{\text{f}}}{\partial C_{\text{p}}}\right)}^2{u}^2\left(C_{\text{p}}\right)+{\left(\frac{\partial \mathrm{\Delta }{H}_{\text{f}}}{\partial \mathrm{\Delta }{T}_{\text{hyp}}}\right)}^2{u}^2\left(\mathrm{\Delta }{T}_{\text{hyp}}\right).$$(11)

Here, the u(C_p) and u(ΔT_hyp) represent the uncertainty of the isobaric heat capacity and the hypercooling limit, respectively. For u(C_p), the uncertainties reported in the literature^28,29,30) for the isobaric heat capacities were used.

The uncertainty of the hypercooling limit arises from the uncertainty associated with the regression line (Fig. 5) used to determine it. To determine the uncertainty of the regression line, the linear-fit function is expressed by the following formula using the slope of the regression line, $\hat{\beta }$, the average value of ΔT, $\overline{\mathrm{\Delta }T}$, and the average value of Δt, $\overline{\mathrm{\Delta }t}$:

  

$$\mathrm{\Delta }t=\hat{\beta }\left(\mathrm{\Delta }T-\overline{\mathrm{\Delta }T}\right)+\overline{\mathrm{\Delta }t}.$$(12)

The uncertainty of the regression line can be attributed to $\hat{\beta }$ and $\overline{\mathrm{\Delta }t}$. In this case, the uncertainties of $\hat{\beta }$ and $\overline{\mathrm{\Delta }t}$ are expressed by the following equations:

  

$$u^2\left(\hat{\beta }\right)=\frac{\sum \left[{\left\{\mathrm{\Delta }{t}_i-\left(\hat{\beta }\left(\mathrm{\Delta }{T}_i-\overline{\mathrm{\Delta }T}\right)+\overline{\mathrm{\Delta }t}\right)\right\}}^2\right]}{\left(n-2\right)\sum {\left(\mathrm{\Delta }{T}_i-\overline{\mathrm{\Delta }T}\right)}^2},$$(13)

  

$$u^2\left(\overline{\mathrm{\Delta }t}\right)=\frac{\sum \left[{\left\{\mathrm{\Delta }{t}_i-\left(\hat{\beta }\left(\mathrm{\Delta }{T}_i-\overline{\mathrm{\Delta }T}\right)+\overline{\mathrm{\Delta }t}\right)\right\}}^2\right]}{n-2}.$$(14)

Here, $u\left(\hat{\beta }\right)$ and $u\left(\overline{\mathrm{\Delta }t}\right)$ denote the uncertainty of the slope of the regression line and the standard deviation of Δt from the regression line, respectively. The factors Δt_i and ΔT_i are the experimental values of Δt and ΔT, respectively, and n is the number of data points used for the fitting of the linear function. Using the above Eqs. (12), (13), (14), the uncertainty of the regression line was determined. The results for Fe are indicated in Fig. 6 as dashed lines. The uncertainty of the hypercooling limit was obtained from the uncertainty of the regression line at the x-intercept.

As described in the section 4.1., there was a large discrepancy in ΔT_hyp of Fe between our data and the value reported in Kang et al.,³³⁾ despite the good agreement found in the case of Ni. This difference could be explained by the degree of undercooling achieved during the measurement. In the case of Ni, both our experiment and one reported by Kang et al.,³³⁾ achieved large undercooling up to almost 250 K. However, In the case of Fe, the maximum undercooling of Fe achieved was 280 K in our measurement, while, the maximum undercooling reported in literature was about 200 K. As shown in this chapter, the uncertainty of the regression line to ΔT_hyp and Δt is a key to determine the ΔT_hyp in this method. A larger undercooling range allows us to increase the reliability in ΔT_hyp determination. Therefore, one reason of the discrepancy found in Fe could attribute the difference of the maximum undercooling in the measurement.

The respective contributions of the uncertainty components in the heat-of-fusion measurement for Fe are presented in Table 3. The combined standard uncertainty, denoted as u(ΔH_f), was 1.1 kJ mol⁻¹, and the expanded uncertainty at the 95.45% level of confidence (coverage factor = 2) was 2.2 kJ mol⁻¹. This uncertainty was 17.3% of the heat of fusion (12.7 kJ mol⁻¹). The uncertainty of the heat of fusion of Ni and Co (coverage factor = 2) determined in the same manner was 5.6 and 2.8 kJ mol⁻¹, respectively.

Table 3. Uncertainty in the heat-of-fusion measurement for Fe.

Property	Standard uncertainty	Sensitivity coefficient	Contribution [kJ mol−1]
Cp	3.2	$\frac{\partial \mathrm{\Delta }{H}_{\text{f}}}{\partial C_{\text{p}}}$=280	0.9
ΔThyp	16	$\frac{\partial \mathrm{\Delta }{H}_{\text{f}}}{\partial \mathrm{\Delta }{T}_{\text{hyp}}}$=45.4	0.7
Combined standard uncertainty, u			1.1
Expanded uncertainty (k = 2), 2u			2.2

5.2. Effectiveness of the Aerodynamic-levitation Method for Heat-of-fusion Measurement

Figure 7 displays the heats of fusion of Fe, Ni, and Co measured in this study juxtaposed with the literature values. The heats of fusion determined in this study are plotted as black circles (●). The literature values obtained by drop calorimetry (□),^{19,20,21,22,23)} adiabatic calorimetry (△),^24,25,26) and mixture calorimetry (◇)^27,28,29) are also shown in this figure. In the figure, data without specified measurement method (○)^38,39,40) are also included. The error bars of the heats of fusion measured in this study represent the respective values of u(ΔH_f).

Fig. 7. Comparison of heats of fusion of the metals measured in this study (●) with literature data measured by drop calorimetry (◇), mixture calorimetry (△), and adiabatic calorimetry (□). The data without specified measurement method (○) are also included.

The heats of fusion measured using the aerodynamic levitator in this study exhibit good agreement with those obtained by other measurement methods within the range of experimental uncertainty. Therefore, the heat of fusion of metals can be accurately measured based on the hypercooling limit determined using an aerodynamic levitator. This newly developed technique will enable the measurement of the heats of fusion of highly reactive metals and alloys by preventing reactions with the container and residual ambient gas.

For the alloys or pure metals with unknown isobaric molar heat capacity, it is necessary to measure this parameter independently using different technique such as modulation laser calorimetry.^34,35,36) Meanwhile, this research has shown that the linear relation between Δt and ΔT can be obtained using the aerodynamic levitator. This finding suggests the possibility of experimental determination of the variable α, which is a function of isobaric heat capacity, total hemispherical emissivity and heat transfer coefficient of the gas, from the slope of the linear relation of Δt and ΔT. If the heat transfer coefficient of the gas can be determined through additional measurements, it may be possible to simultaneously measure ΔH_f and C_p as reported in the measurement using electrostatic levitation.^31,32,33)

6. Conclusions

The heats of fusion of Fe, Ni, and Co were successfully measured using a closed-type aerodynamic levitator that prevented chemical reactions with the container and oxidation of the sample. The heats of fusion of Fe, Ni, and Co were determined to be 12.7 ± 1.1 kJ mol⁻¹, 16.9 ± 2.8 kJ mol⁻¹, and 14.8 ± kJ mol⁻¹, respectively, with combined standard uncertainty. The measured heats of fusion were in good agreement with the literature values. Therefore, heat-of-fusion measurements using a closed-type aerodynamic levitator are expected to be effective for metals and alloys with high reactivity.

Acknowledgements

This work was supported by Grant-in-Aid for Scientific Research (B) Grant Number 20H02490, as well as by the 28th ISIJ Research Promotion Grant of Iron and Steel Institute of Japan.

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