Hypercooling limit, heat of fusion, and temperature-dependent specific heat of Fe-Cr-Ni melts

Abstract

Thermophysical properties of Fe-Cr-Ni melts are studied using electrostatic levitation and rapid solidification techniques. Six hypoeutectic Fe_0.72Cr_wNi_(0.28−w) alloys with a Cr/Ni ratio of around 0.8 were melted and solidified at different degrees of undercooling. From the observed relationship between the undercooling and thermal plateau time, the hypercooling limit and heat of fusion of Fe_0.72Cr_wNi_(0.28−w) melts are determined as a function of Cr mass fraction. A ratio of specific heat and total hemispherical emissivity of the Fe-Cr-Ni melts is calculated using the time-temperature profiles. A new method is presented to evaluate the temperature dependence of specific heat for undercooled melts and applied to this alloy family.

1. Introduction

Fe-Cr-Ni systems present the basis for technically important stainless steels. Their phase composition and stability are controlled by varying Cr and Ni contents which stabilize a ferrite (BCC phase) and an austenite (FCC phase), respectively based on the Schaeffler [1] or DeLong [2] diagrams. Fe-Cr-Ni based steels with an iron content of around 0.72 (mass fraction), such as austenitic 304 and 316 steels, are widely used as structural materials for fast breeder fission and light water reactors [3–5]. Inconel alloys X-718 and 750 are used in reactor core components [3,6]. Fe-Cr-Ni based high-entropy alloys such as FeCrNiMnCo [7], FeCrNiAlCo [8], and FeCrNiAlCu [9] also have attracted increasing attention because of their unique microstructures and adjustable properties.

Processing for a specific application for a particular steel often involves melting and solidification. Numerical or analytical approaches for process control have been widely used to simulate the solidification and phase transformation behavior of Fe-Cr-Ni based steels [10–12] during casting and welding manufacturing operations [13,14]. To use such advanced techniques, accurate knowledge of high temperature thermophysical properties of Fe-Cr-Ni are critical as reference data for control of thermal, chemical, and mechanical treatments of its alloys. Understanding the compositional dependence is also essential in practical terms because a small variation of the melt composition due to preferential evaporation of volatile alloying elements may cause significant effects on the thermophysical properties at high temperature [15]. From a fundamental point of view, accurate thermophysical properties of Fe-Cr-Ni are also required for theoretical analysis of the solidification path which may involve competitive nucleation and growth behavior due to undercooling during rapid solidification [16–20].

Among the thermophysical properties, heat of fusion can be regarded as one of the most important properties for practical and theoretical purposes. However, systematic experimental data for heat of fusion of Fe-Cr-Ni have been rarely reported due to difficulties of high-temperature measurements. Rösner-Kuhn et al. reported enthalpy and heat capacity of Fe-Cr-Ni melts using levitation drop calorimetry [21]. Recently, enthalpy of mixing, enthalpy of formation, and the self-diffusion coefficient of FCC and BCC phases for the binary and ternary systems containing Fe, Cr, and Ni were reported based on modified embedded-atom interatomic potentials [22]. In recent decades, containerless processing has emerged as an effective tool for measuring thermophysical properties of metallic melts. A few thermophysical properties such as density and viscosity of Fe-Cr-Ni melts have been reported using electromagnetic levitation (EML) [23] and evaluation of the enthalpy-related properties is still lacking.

In this work, enthalpy-related thermophysical properties for Fe-Cr-Ni melts are presented using electrostatic levitation (ESL). Fig. 1 shows a partial phase diagram of Fe_0.72Cr_wNi_(0.28−w) with a low Cr content highlighted in yellow. At deep undercoolings in this region, the solidification path involves conversion from primary metastable ferrite (δ) to stable austenite (γ) as shown by metastable extensions of the ferrite equilibrium phase diagram [1,2,24]. Six compositions (Cr mass fraction w_Cr:0.07, 0.09, 0.11, 0.12, 0.14, 0.16) were selected in the hypoeutectic region of the phase diagram and melting and re-solidification of the alloys were carried out using ESL. From multiple thermal cycles of each alloy, the hypercooling limit and heat of fusion of Fe_0.72Cr_wNi_(0.28−w) melts are determined as a function of Cr mass fraction. A ratio of specific heat at constant pressure and total hemispherical emissivity of the Fe-Cr-Ni melts is also calculated from the free-cooling curves. Based on the estimated hypercooling limit profiles, we present a new method to predict the temperature dependence for specific heat in this alloy family.

Fig. 1.

Equilibrium phase diagram for the ternary Fe_0.72Cr_wNi_(0.28−w) system calculated using ThermoCalc (2017b) software using FEDEMO database. Dotted lines indicate metastable extensions of the equilibrium phase diagram.

2. Experimental

Fe-Cr-Ni alloys (Fe 99.95%, Ni 99.999% / Alfa Aesar, Cr 99.999% / ESPI) were prepared by arc-melting the pure elements together under an Ar atmosphere (TABLE 1). During arc-melting a zirconium sphere was used as a getter to reduce the remaining oxygen in the arc-melting chamber prior to the melting of the elements. The amount of mass evaporation after arc-melting was less than 0.5% of the initial mass. The electrostatic levitation (ESL) facility at the NASA Marshall Space Flight Center (MSFC) in Huntsville, AL was used for melting and re-solidification of the alloys. In the ESL chamber, coulomb forces introduced by a pair of vertical electrodes counteract the sample’s weight, and the sample’s lateral position is stabilized using another two pairs of horizontal electrodes. The processing atmosphere of the chamber is high vacuum, typically in the 10⁻⁵ Pa range. Samples with a mass of approximately 35 mg (a diameter of ~ 2 mm) were levitated between two electrodes and melted using a 200W neodymium-doped yttrium aluminum garnet (Nd:YAG) laser. The temperature of samples was monitored with a LumaSense Technologies IMPAC IGA 140 single-color pyrometer with a spot size of ~ 0.8 mm, which is operated at a wavelength range of 1.45 − 1.8 μm. After several thermal cycles in the ESL chamber, the amount of mass evaporation was limited to less than 5% of the initial mass.

TABLE 1.

Source and mass fraction purity of samples used in this study.

Chemical name	Source	Initial mass fraction purity	Purification method
Fe	Alfa Aesar	0.9995	none
Cr	Alfa Aesar	> 0.9999	none
Ni	ESPI	> 0.9999	none

3. Results

3.1. Determination of hypercooling limit

The six Fe-Cr-Ni alloys were processed through multiple thermal cycles using ESL. Fig. 2 shows the representative time-temperature profiles measured at Cr concentrations of 0.12 and 0.16. The solid alloy is heated up to its liquidus temperature T_L (dashed lines in Fig. 2), and the liquid is superheated up to a desired temperature to evaporate or dissolve any oxygen-containing phases on the surface of the melt. Since the population of potential heterogeneous nucleation sites are significantly reduced by using containerless processing techniques, such as ESL, the liquid can be undercooled significantly below T_L when the heating laser is turned off. After undercooling, the rapid release of latent heat for crystallization leads to a steep rise of temperature in a process called recalescence. These alloys show a double recalescence due to the sequential formation of BCC δ-phase and FCC γ-phase [24]. At low Cr concentrations the phase diagram shows that the driving force is high and the transformation is too fast to be observed in the thermal profile but at high Cr concentrations the driving force is low and double recalescence is slow enough to be clearly seen (red arrow in Fig. 2b).

Fig. 2.

Time t - temperature T profiles of (a) Fe_0.72Cr_0.12Ni_0.16 and (b) Fe_0.72Cr_0.16Ni_0.12 alloys using ESL. The liquidus temperature T_L (dashed lines) for each composition was determined based on the Fe-Cr-Ni phase diagram in Fig. 1.

Upon solidification, a fraction f_R of the melt solidifies under non-equilibrium (recalescence) condition, and the remaining melt solidifies under near-equilibrium conditions, showing a thermal plateau time. The f_R becomes unity if the undercooling reaches the hypercooling limit ΔT_hyp of the melt as the plateau time goes to zero. Since the ΔT_hyp is difficult to achieve, even using containerless levitation techniques, it is often estimated from the inverse relationship between undercooling and thermal plateau time; as the undercooling goes up the plateau time goes down. Fig. 3a to 3f show changes in the thermal plateau time with undercooling of the six Fe-Cr-Ni melts and the values are tabulated in TABLE 2. The ΔT_hyp of each composition is estimated from linear extrapolation of the experimental results. As shown in Fig. 3g, the ΔT_hyp of Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction shows similar values near 350 K, which is close to that (357 K) of pure Fe using ESL [25]. Hypercooling limit was also reported for pure elements using ESL [25–28].

Fig. 3.

(a to f) Relation between undercooling ΔT and thermal plateau time Δt of the six Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction w_Cr. Solid and dashed lines for each composition indicate a linear fitting of the experimental results and a standard uncertainty of the linear fitting, respectively. (g) Estimated hypercooling limit of Fe_0.72Cr_wNi_(0.28−w) melts as a function of Cr mass fraction w_Cr. Error bars indicate the standard uncertainty.

TABLE 2.

Experimental values of undercooling ΔT and thermal plateau time Δt of the six Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction w_Cr at pressure p = 5.0 × 10⁻⁵ Pa.^a

wCr	ΔT/K	Δt/s	ΔT/K	Δt/s	ΔT/K	Δt/s
0.07	32.38	4.26	112.53	3.17	116.69	3.13
	93.45	3.43	114.15	3.10	125.88	3.04
	108.17	3.33	114.25	3.12	146.37	2.75
0.09	25.35	4.54	76.51	3.78	121.23	3.10
	64.45	3.96	77.52	3.53	129.14	2.98
	64.97	3.96	78.09	3.64	132.58	2.88
	72.30	3.66	108.58	3.32	133.51	2.89
	73.97	3.53	110.03	3.28	140.35	2.88
	75.75	3.66	120.40	3.19
0.11	30.92	4.47	66.06	3.82	100.00	3.46
	57.97	3.90	74.05	3.87	100.81	3.48
	62.31	3.88	76.17	3.71	102.74	3.32
0.12	62.07	3.79	87.07	3.62	110.07	3.32
	63.07	4.10	88.66	3.44	112.07	3.09
	75.48	3.67	89.07	3.72
	82.02	3.48	108.07	3.39
0.14	32.33	4.22	48.38	4.05	63.86	3.64
	32.75	4.08	49.62	4.00	72.33	3.72
	36.40	4.08	51.77	3.87	76.94	3.57
	36.94	4.13	57.98	3.92	78.20	3.60
	39.06	3.96	58.30	3.86	83.91	3.42
	46.48	3.90	58.93	3.90	92.29	3.36
	47.58	3.99	58.98	3.92	97.77	3.23
0.16	35.80	4.09	54.57	4.02	59.68	3.76
	48.64	3.99	57.92	3.79	64.02	3.78
	50.27	3.97	58.07	3.80	92.28	3.37

^aStandard uncertainties u are u(w_Fe) = 0.0019, 0.0013, 0.0009, 0.0014, 0.0015, 0.002 and u(w_Cr) = 0.0012, 0.0032, 0.0009, 0.0009, 0.0013, 0.001 and u(w_Ni) = 0.0013, 0.0015, 0.0009, 0.0011, 0.0016, 0.0017 for each Cr mass fraction of 0.07, 0.09, 0.11, 0.12, 0.14, 0.16 respectively, u(ΔT) = 2 K, u(Δt) = 0.006 s, u(p) = 3.2 × 10⁻⁸ Pa.

According to a simplified heat balance, the heat of fusion ΔH_f is equal to the product of the hypercooling limit ΔT_hyp and specific heat C_p at constant pressure, ΔH_f = ΔT_hyp × C_p, assuming a temperature-independent specific heat from T_L. In the present work, C_p values at T_L of the six Fe-Cr-Ni compositions were obtained in Thermo-Calc (2017b) software using FEDEMO database and thus ΔH_f of the alloys could be calculated. As shown in Fig. 4, the C_p values show a small difference within 44.07 ± 0.062 J K⁻¹ mol⁻¹ over the entire range of Cr concentrations. The weak decrease of ΔH_f of Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction is not statistically significant. These results compare favorably to literature values where the C_p values of Fe_0.7Cr_0.15Ni_0.15, Fe_0.7Cr_0.16Ni_0.14, and Fe_0.702Cr_0.185Ni_0.113 were reported as 44.16 ± 4.71 J K⁻¹ mol⁻¹, 43.46 ± 5.53 J K⁻¹ mol⁻¹, and 43.86 ± 5.84 J K⁻¹ mol⁻¹ at their T_L [21]. The ΔH_f values of Fe_0.705Cr_0.185Ni_0.11 and Fe_0.69Cr_0.185Ni_0.125 were reported to be 13.2 kJ mol⁻¹ and 14.5 kJ mol⁻¹, respectively [29]. By comparison, the ΔH_f of pure Fe is determined to be 16.2 kJ mol⁻¹ by using the ΔT_hyp of 357 K [25] and the C_p of 45.4 J K⁻¹ mol⁻¹ [30] measured by the hypercooling limit method using ESL and the laser modulation calorimetry method using EML, respectively. The thermophysical properties for evaluation of ΔH_f of Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction were summarized in TABLE 3.

Fig. 4.

Specific heat C_p at T_L and heat of fusion ΔH_f of Fe_0.72Cr_wNi_(0.28−w) melts as a function of Cr mass fraction w_Cr. The C_p of each composition was calculated in Thermo-Calc (2017b) software using FEDEMO database. Error bars indicate the standard uncertainty.

TABLE 3.

Experimental values of hypercooling limit ΔT_hyp and heat of fusion ΔH_f of the six Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction w_Cr at pressure p = 5.0 × 10⁻⁵ Pa.^a The ΔH_f is calculated with ΔH_f = ΔT_hyp × specific heat C_p at liquidus temperature T_L.

wCr	TL /K	Cp at TL /J K−1 mol−1	ΔThyp /K	ΔHf /J mol−1
0.07	1746.54b	44.09b	353 ± 10.0c	15563 ± 440.89c
0.09	1745.94	44.11	344 ± 13.5	15173 ± 595.48
0.11	1744.32	44.12	348 ± 24.0	15353 ± 1058.9
0.12	1743.07	44.12	349 ± 36.5	15397 ± 1610.4
0.14	1739.57	44.01	344 ± 18.0	15138 ± 792.11
0.16	1734.57	43.98	341 ± 31.0	14996 ± 1363.3

^aStandard uncertainties u are u(w_Fe) = 0.0019, 0.0013, 0.0009, 0.0014, 0.0015, 0.002 and u(w_Cr) = 0.0012, 0.0032, 0.0009, 0.0009, 0.0013, 0.001 and u(w_Ni) = 0.0013, 0.0015, 0.0009, 0.0011, 0.0016, 0.0017 for each Cr mass fraction of 0.07, 0.09, 0.11, 0.12, 0.14, 0.16 respectively, u(ΔT) = 2 K, u(p) = 3.2 × 10⁻⁸ Pa.

^bThe T_L and the C_p at T_L of each composition were obtained in ThermoCalc (2017b) software using FEDEMO database.

^cStandard uncertainty.

3.2. Specific heat at constant pressure and total hemispherical emissivity

Since levitated samples are cooled down in the ESL chamber by radiation loss only when the heating laser is turned off, the heat balance equation is simply given by

$$m{C}_{\text{p}}dT/dt=-{\sigma }_{\text{B}}A{\epsilon }_{\text{T}}\left(T^4-T_{\text{r}}^4\right)$$ (1)

where m is the sample mass, C_p is the specific heat at constant pressure, dT/dt is the cooling rate at temperature T, T_r is the surrounding temperature, t is the time, ^σB is the Stefan-Boltzmann constant, A is the surface area of the sample, and ε_T is the total hemispherical emissivity. The measured time-temperature profiles (cooling curves) thus provide a measure of the linked property of C_p / ε_T as a function of temperature. Fig. 5 shows C_p / ε_T values of the six Fe-Cr-Ni melts calculated with Eq (1) and the values are tabulated in TABLE 4. For the calculation, raw cooing curves of each composition were processed using adjacent-averaging smoothing and trace interpolation functions in the data analysis software (Origin 8.5, OriginLabs Corp.). The C_p / ε_T values show a linear increase with decreasing temperature and its degree decreases as the Cr concentration increases. Although there are no reported values in the literature for Fe-Cr-Ni, Lee et al. reported that the C_p / ε_T of liquid Fe increases with a temperature dependence of C_p / ε_T (J K⁻¹ mol⁻¹) = 143.8 − 0.1358 × (T – T_m) below its T_m [25]. It was also reported that the C_p / ɛ_T of liquid Ni increases with decreasing temperature in the stable and undercooled liquid regions [31]. Many elemental [27,32] and alloy [33–36] melts exhibit such increasing behavior in C_p / ɛ_T with decreasing temperature, but the opposite has also been reported [37].

Fig. 5.

C_p / ɛ_T of Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction w_Cr as a function of temperature T. The C_p / ɛ_T values of each composition were evaluated by averaging seven results calculated with the smoothing processed cooling curves. Red dashed lines indicate the T_L of each composition. Error bars indicate the standard uncertainty.

TABLE 4.

Experimental values of the ratio C_p / ɛ_T of specific heat C_p and total hemispherical emissivity ɛ_T of the six Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction w_Cr at temperature T and pressure p = 5.0 × 10⁻⁵ Pa.^a

wCr	T/K	Cp ɛT−1/J·K−1·mol−1	T/K	Cp εT−1/J·K−1·mol−1	T/K	CP εT−1/J·K−1·mol−1
0.07	1632	163.22 ± 0.944b	1704	151.50 ± 2.527b	1776	148.08 ± 2.455b
	1640	163.92 ± 4.257	1712	153.88 ± 2.321	1784	148.31 ± 3.072
	1648	158.34 ± 2.015	1720	151.80 ± 2.280	1792	144.07 ± 2.762
	1656	161.21 ± 2.215	1728	154.87 ± 2.954	1800	146.88 ± 4.124
	1664	157.65 ± 2.761	1736	150.96 ± 4.228	1808	144.45 ± 3.268
	1672	157.81 ± 3.934	1744	148.79 ± 2.330	1816	142.42 ± 5.573
	1680	154.21 ± 2.296	1752	147.34 ± 2.916	1824	144.19 ± 1.784
	1688	157.96 ± 3.445	1760	151.31 ± 2.364
	1696	157.47 ± 3.129	1768	149.66 ± 2.413
0.09	1625	161.21 ± 6.725b	1697	153.08 ± 4.167b	1769	149.56 ± 3.279b
	1633	165.13 ± 4.606	1705	155.03 ± 3.398	1777	150.62 ± 5.033
	1641	161.71 ± 3.144	1713	152.85 ± 5.267	1785	148.84 ± 4.569
	1649	163.40 ± 3.541	1721	155.38 ± 3.300	1793	145.93 ± 3.662
	1657	160.54 ± 2.762	1729	150.93 ± 3.211	1801	148.07 ± 3.340
	1665	159.87 ± 2.437	1737	154.60 ± 4.074	1809	147.29 ± 4.473
	1673	157.29 ± 2.920	1745	151.15 ± 5.126	1817	142.95 ± 3.368
	1681	159.76 ± 3.887	1753	154.05 ± 2.838	1825	144.67 ± 5.143
	1689	157.30 ± 3.948	1761	147.43 ± 4.700	1833	142.05 ± 3.696
0.11	1653	162.19 ± 4.607b	1717	152.76 ± 4.064b	1781	148.79 ± 1.173b
	1661	159.80 ± 3.470	1725	156.30 ± 2.396	1789	150.26 ± 2.193
	1669	157.17 ± 3.146	1733	155.25 ± 1.948	1797	146.66 ± 1.458
	1677	157.59 ± 2.009	1741	151.34 ± 1.044	1805	148.18 ± 2.567
	1685	159.46 ± 1.756	1749	153.04 ± 2.130	1813	145.68 ± 0.834
	1693	157.92 ± 2.421	1757	150.66 ± 3.335	1821	147.05 ± 2.671
	1701	154.22 ± 1.860	1765	152.81 ± 2.037	1829	145.54 ± 0.979
	1709	156.70 ± 3.880	1773	150.38 ± 3.878
0.12	1676	159.17 ± 2.730b	1732	153.21 ± 2.213b	1788	150.33 ± 3.863b
	1684	160.55 ± 4.571	1740	155.14 ± 4.128	1796	146.79 ± 6.181
	1692	156.98 ± 2.468	1748	153.61 ± 1.460	1804	148.10 ± 1.252
	1700	153.58 ± 6.193	1756	152.87 ± 4.858	1812	149.07 ± 4.788
	1708	158.76 ± 1.977	1764	150.71 ± 1.252	1820	147.01 ± 5.949
	1716	155.75 ± 1.479	1772	152.54 ± 2.143	1828	145.64 ± 3.044
	1724	155.31 ± 3.190	1780	149.88 ± 1.252	1836	145.49 ± 5.796
0.14	1686	156.74 ± 3.738b	1742	153.31 ± 6.741b	1798	152.91 ± 3.159b
	1694	159.18 ± 6.627	1750	154.23 ± 2.041	1806	154.39 ± 4.401
	1702	156.40 ± 5.402	1758	153.52 ± 5.215	1814	151.45 ± 4.095
	1710	158.16 ± 5.665	1766	153.40 ± 4.874	1822	152.34 ± 3.378
	1718	154.24 ± 4.318	1774	153.94 ± 4.389	1830	149.68 ± 4.696
	1726	156.71 ± 4.493	1782	153.07 ± 3.728	1838	149.08 ± 5.894
	1734	156.57 ± 5.103	1790	151.45 ± 3.175	1846	151.39 ± 4.001
0.16	1690	156.40 ± 2.888b	1738	153.63 ± 3.201b	1786	153.12 ± 0.712b
	1698	157.23 ± 3.451	1746	154.61 ± 3.452	1794	153.79 ± 1.493
	1706	154.84 ± 3.051	1754	153.65 ± 2.194	1802	151.02 ± 3.499
	1714	156.97 ± 4.792	1762	154.87 ± 1.883	1810	152.48 ± 2.660
	1722	155.50 ± 2.073	1770	153.32 ± 3.258	1818	151.73 ± 2.127
	1730	154.19 ± 3.153	1778	154.59 ± 0.896	1826	151.40 ± 2.650

^aStandard uncertainties u are u(w_Fe) = 0.0019, 0.0013, 0.0009, 0.0014, 0.0015, 0.002 and u(w_Cr) = 0.0012, 0.0032, 0.0009, 0.0009, 0.0013, 0.001 and u(w_Ni) = 0.0013, 0.0015, 0.0009, 0.0011, 0.0016, 0.0017 for each Cr mass fraction of 0.07, 0.09, 0.11, 0.12, 0.14, 0.16 respectively, u(T) = 2 K, and u(p) = 3.2 × 10⁻⁸ Pa.

^bStandard uncertainty.

From the measured C_p / ε_T and the C_p obtained in ThermoCalc software, the ε_T of Fe_0.72Cr_wNi_(0.28−w) melts is calculated at their T_L and the results are shown with the C_p / ε_T in Fig. 6. The C_p / ε_T increases almost linearly and the ε_T decreases with increasing Cr concentration. The thermophysical properties for evaluation of C_p / ε_T and ε_T of Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction were summarized in TABLE 5.

Fig. 6.

C_p / ε_T and ε_T of Fe_0.72Cr_wNi_(0.28−w) melts as a function Cr mass fraction w_Cr at their T_L. Error bars indicates the standard uncertainty.

TABLE 5.

Linear expression for the ratio C_p / ε_T of specific heat C_p and total hemispherical emissivity ε_T of Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction w_Cr at pressure p = 5.0 × 10⁻⁵ Pa.^a The C_p / ε_T is expressed by a linear function of temperature T with C_p / ε_T = C_p / ε_T at T_L − k₁ (T − T_L). T_l is the liquidus temperature and k₁ is the temperature coefficient. The ε_T at T_L is calculated from the relation of C_p / ε_T and C_p.

wCr	TL/K	T range /K	Cp / εT at TL /J K−1 mol−1	k1 /J K−2 mol−1	εT at TL
0.07	1746.54b	1632 – 1824	150.68 ± 3.219c	0.101 ± 0.0055c	0.293 ± 0.0063c
0.09	1745.94	1625 – 1833	151.78 ± 4.205	0.098 ± 0.0055	0.291 ± 0.0081
0.11	1744.32	1653 – 1829	152.38 ± 2.831	0.088 ± 0.0053	0.290 ± 0.0054
0.12	1743.07	1676 – 1836	153.51 ± 3.669	0.090 ± 0.0057	0.287 ± 0.0069
0.14	1739.57	1686 – 1846	154.93 ± 4.864	0.042 ± 0.0054	0.284 ± 0.0089
0.16	1734.57	1690 – 1826	154.93 ± 2.554	0.037 ± 0.0050	0.284 ± 0.0047

^aStandard uncertainties u are u(w_Fe) = 0.0019, 0.0013, 0.0009, 0.0014, 0.0015, 0.002 and u(w_Cr) = 0.0012, 0.0032, 0.0009, 0.0009, 0.0013, 0.001 and u(w_Ni) = 0.0013, 0.0015, 0.0009, 0.0011, 0.0016, 0.0017 for each Cr mass fraction of 0.07, 0.09, 0.11, 0.12, 0.14, 0.16 respectively, u(T) = 2 K, and u(p) = 3.2 × 10⁻⁸ Pa.

^bThe T_L of each composition was obtained in ThermoCalc (2017b) software using FEDEMO database.

^cStandard uncertainty.

4. Discussion

C_p / ε_T is a composite quantity that may be evaluated as a function of temperature by experimental results using ESL, but separating these properties is often difficult. For Fe_0.72Cr_0.07Ni_0.21, as an example, the ε_T should decrease from 0.293 to 0.237 at its T_L to T_hyp if the C_p is assumed to be a constant value of 44.09 J K⁻¹ mol⁻¹ with temperatures. It seems that this change is considerably large, but no comparable data has been reported to contradict this extrapolation. In previous work, a noncontact laser modulation calorimetry technique using EML has been developed and applied to measure C_p of metallic melts independently [38]. However, the method is limited in measuring the C_p of deeply undercooled melts because the modulation time required to conduct the measurement is relatively long and to avoid premature solidification the tests are often run above T_L [30,39] or on a solid sample [40]. In recent work, Ishikawa developed an advanced technique for an independent measurement of ε_T of metallic melts by using a combination of ESL and Fourier transform infrared spectrometer (FTIR) [41] and reported that ε_T of molten Zr shows a temperature dependence of ε_T = 0.317 + 1.77 × 10⁻⁵ (T–T_m) [42]. Note that for an undercooling of 300 K this results in a change in emissivity of less than 2% indicating that emissivity can be effectively assumed to be constant. However, it is technically difficult to match temperatures of the sample to that of the blackbody to maintain identical environmental conditions within the experiment.

Here, we suggest a simple method to approximate the temperature dependence in C_p for undercooled melts. Fig. 7 shows a schematic for two paths upon solidification of a melt. First, if there is sufficient undercooling to cause the melt to become fully solid then the latent heat of fusion is equal to the sensible heat removed during hypercooling. Second, if there is no-undercooling for the melt then the latent heat of fusion is equal to the sensible heat removed over time at T_L. The two solidification paths correspond to red and blue lines in Fig. 7 and can be referred hypercooling and no-undercooling conditions, respectively. Then, the heat balance in Eq. 1 can be modified by evaluating the integral form of the equation under the two conditions. These two conditions are described in the heat balance equations below:

$$\mathrm{\Delta }{H}_{\text{f}}=\int C_{\text{p}}dT$$ (2)

$$\mathrm{\Delta }{H}_{\text{f}}=\int -\frac{{\sigma }_{\text{B}}A{\epsilon }_{\text{T}}}{m}\left(T_{\text{L}}^4-T_{\text{r}}^4\right)dt$$ (3)

Fig. 7.

Schematic for solidification paths of a melt. Red and blue lines correspond to hypercooling and no-undercooling conditions upon solidification of the melt.

Note that hypercooling limit profile in Fig. 3 gives two important quantities; one is the hypercooling limit (ΔT_hyp) of a melt, and the other is the thermal plateau time limit (Δt_max) of the melt. This implies that the heat of fusion can be calculated under the hypercooling and the no-undercooling conditions using Eq. (2) and (3) respectively and the results must be the same.

Fig. 8 shows the results of ΔH_f of the six Fe_0.72Cr_wNi_(0.28−w) melts calculated from the two cases. The results for the no-undercooling case are higher than those for the hypercooling case and the deviation decreases with increasing Cr concentration.

Fig. 8.

Comparison of the results of ΔH_f of the six Fe_0.72Cr_wNi_(0.28−w) melts calculated with the hypercooling and the no-undercooling methods. A solid line indicates a linear fitting of the results for the hypercooling case. Errors indicate the standard uncertainty.

Note that in Fig. 8 the ΔH_f for the hypercooling case was calculated using the relation of ΔH_f = ΔT_hyp × C_p assuming a temperature-independent C_p from T_L. Therefore, the observed difference between the two results in Fig. 8 is most likely caused by a temperature-dependence in the C_p of each composition. This temperature-dependence can be simply deduced using Eq. (2) by assuming a linear increase in C_p with decreasing temperature from T_L which results in a matching of the two values.

Using this technique, Fig. 9 shows the projected temperature dependence in C_p for the six Fe_0.72Cr_wNi_(0.28−w) melts. For the calculation, a linearly-fitted value was applied to the hypercooling limit calculation (solid line in Fig. 8). At low Cr compositions the temperature dependence of C_p is highest reaching a maximum for Fe_0.72Cr_0.07Ni_0.21 where the C_p is 44.09 J K⁻¹ mol⁻¹ at T_L and increases to 52.68 J K⁻¹ mol⁻¹ at T_hyp. The slope decreases gradually as the Cr concentration increases and for Fe_0.72Cr_0.16Ni_0.12 the C_p shows only a small temperature dependence of less than 2% in the undercooled liquid region. The thermophysical properties for evaluating the temperature-dependent C_p of Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction were summarized in TABLE 6.

Fig. 9.

Temperature-dependence in the C_p of Fe_0.72Cr_wNi_(0.28−w) melts with Cr mass fraction w_Cr below their T_L. The temperature-dependence was deduced by matching the two results for ΔH_f of the Fe-Cr-Ni alloys in Fig. 8

TABLE 6.

Experimental values of heat of fusion ΔH_f for the no-undercooling case of the six Fe_0.72Cr_wNi_(0.28−w) melts with mass fraction w_cr at pressure p = 5.0 × 10⁻⁵ Pa.^a The temperature-dependence for specific heat C_p is expressed by a linear function of temperature T with C_p = C_p at T_L −k₂ (T − T_L). T_L is the liquidus temperature and k₂ is the temperature coefficient.

wCr	TL/K	ΔHf by no-undercooling /K	Cp at TL /J K−1 mol−1	k2 /J K−2 mol−1	T range/K
0.07	1746.54b	16983 ± 245.65c	44.09b	0.0245 ± 0.0040c	1396 – 1747
0.09	1745.94	16777 ± 283.10c	44.11	0.0227 ± 0.0047	1397 – 1746
0.11	1744.32	16459 ± 135.21	44.21	0.0199 ± 0.0023	1398 – 1744
0.12	1743.07	16206 ± 321.60	44.12	0.0165 ± 0.0054	1398 – 1743
0.14	1739.57	15377 ± 273.55	44.01	0.0048 ± 0.0046	1397– 1740
0.16	1734.57	15187 ± 305.79	43.98	0.0033 ± 0.0053	1394– 1735

^aStandard uncertainties u are u(w_Fe) = 0.0019, 0.0013, 0.0009, 0.0014, 0.0015, 0.002 and u(w_Cr) = 0.0012, 0.0032, 0.0009, 0.0009, 0.0013, 0.001 and u(w_Ni) = 0.0013, 0.0015, 0.0009, 0.0011, 0.0016, 0.0017 for each Cr mass fraction of 0.07, 0.09, 0.11, 0.12, 0.14, 0.16 respectively, u(T) = 2 K, and u(p) = 3.2 × 10⁻⁸ Pa.

^bThe T_l and the C_p at T_L of each composition were obtained in ThermoCalc (2017b) software using FEDEMO database.

^cStandard uncertainty.

No experimental data are available in the undercooled liquid state over the range of Fe_0.72Cr_wNi_(0.28−w) compositions, but the C_p behavior of Fe_0.72Cr_0.16Ni_0.12 is consistent with the result of liquid 304 stainless steel, Fe_0.72Cr_0.18Ni_0.10, which exhibits a constant C_p of 44 J K⁻¹ mol⁻¹ using noncontact modulated laser calorimetry at temperatures between 1750 − 1970 K [39]. Preliminary evaluations show that similar behavior is also found in a Fe_0.60Cr_0.20Ni_0.20 melt using the EML furnace (ISS-EML) on the International Space Station, showing a constant C_p of 43.7 ± 0.5 J K⁻¹ mol⁻¹ over the range 1675 – 1818 K [43]. For comparison, the C_p of liquid Fe from ESL testing increases from 45.1 J K⁻¹ mol⁻¹ to 56.9 J K⁻¹ mol⁻¹ over the range defined by its T_m (1811 K) to T_hyp (1454 K), if the ε_T is assumed to be a constant value of 0.314 with the temperature [25].

In general, the specific heat of a liquid metal is contributed by vibrational and configurational parts [44] and the contributions to the specific heat are decoupled [45,46]. At the liquidus, the vibrational part contributes mainly to the specific heat because the thermal energy exceeding the interaction between clusters increases the energy for vibrational degrees of freedom. However, upon cooling below the T_L the vibrational contribution would become negligible, because the energy for each degree of freedom is proportional to the temperature. On the other hand, the configurational part arises from changes in the number of arrangements of atoms and thus becomes dominant upon cooling below the T_L because new structural configurations in the liquid could be made to decrease its energy. V. Wessels et al. reported that a rapid chemical and topological order in liquid Cu₄₆Zr₅₄ increases the specific heat upon cooling [34]. Therefore, the temperature-dependence in C_p of the six Fe_0.72Cr_wNi_(0.28−w) melts may be caused by structural changes with decreasing temperature. In the future, more elaborate experimental efforts with liquid structure of the melts should be performed for more comprehensive understanding of the temperature-dependence in C_p.

5. Conclusion

The enthalpy-related properties of six Fe_0.72Cr_wNi_(0.28-w) melts (0.07 ≤ w ≤ 0.16) have been studied using ESL. The ΔT_hyp of the Fe_0.72Cr_wNi_(0.28−w) melts was estimated from a relation between undercooling and thermal plateau time, and the ΔH_f of each alloy was calculated with the ΔT_hyp and the C_p obtained from ThermoCalc software. The C_p / ε_T of the Fe_0.72Cr_wNi_(0.28−w) melts was also calculated from the measured time-temperature profiles. In this work, we presented a new method for predicting the temperature-dependence in C_p of metallic melts by using two extreme cases from the integral form of the heat balance; one by assuming the hypercooling limit and the other by assuming no-undercooling. The temperature dependence in C_p of the Fe_0.72Cr_w Ni_(0.28−w) melts was deduced by matching the results from the two cases. The C_p of the Fe_0.72Cr_w Ni_(0.28−w) melts increases with decreasing temperature and the increase decreases as the Cr concentration decreases. This method may be limited for metallic melts showing a non-linear behavior in C_p, but presents still an approximate aspect for the temperature-dependent C_p.