Enthalpy of Mixing in Al–Tb Liquid

Abstract

The liquid-phase enthalpy of mixing for Al–Tb alloys is measured for 3, 5, 8, 10, and 20 at% Tb at selected temperatures in the range from 1364 to 1439 K. Methods include isothermal solution calorimetry and isoperibolic electromagnetic levitation drop calorimetry. Mixing enthalpy is determined relative to the unmixed pure (Al and Tb) components. The required formation enthalpy for the Al₃Tb phase is computed from first-principles calculations. Based on our measurements, three different semi-empirical solution models are offered for the excess free energy of the liquid, including regular, subregular, and associate model formulations. These models are also compared with the Miedema model prediction of mixing enthalpy.

1. Introduction

Phase selection pathways in glass-forming metallic liquids are influenced by the development of short- and medium-range order [1,2,3,4,5,6,7,8,9,10], which contribute to local energetics and dynamics. Even in binary glass-forming systems, clear evidence of the influence of ordering in the liquid state has been reported [3,4,5,11,12,13,14,15,16,17]. For example, the marginal glass-forming ability in the Al–RE (aluminum rare-earth) systems has been associated with ordering in the liquid phase [18,19,20,21,22,23,24,25,26,27,28,29]. In addition, due to their relative simplicity compared to conventional many-component glass-forming alloys, Al–RE and Al–RE–TM (TM denotes transition metal) alloys are excellent model systems for the investigation of phase stability [30,31,32,33], metallic glass formation [34,35,36,37], liquid properties and crystallization [38,39,40], and glassy alloy structure/properties [25,41,42]. As with all glass-forming liquids, however, properties may become increasingly temperature dependent near and below the liquidus [43,44,45], highlighting the need for reliable thermodynamic descriptions of the liquid state. Numerous theoretical/computational studies, thermodynamic assessments, and associated CALPHAD models for the Al–RE systems are available [32,33,46,47,48,49,50], but reported experimental measurements of excess mixing quantities for Al–RE liquids are very limited, as listed for the Al–lanthanides in

Table 1. A listing of thermodynamic mixing property measurements, calculations, and phase diagram assessments reported for the Al–RE (lanthanide) binary systems.

System	CALPHAD Assessments	FP Calculations of Crystal Phase Energies	Experimental Measurement of Liquid Mixing Properties
Al–La	[51,52]	[53,54]	[55,56,57]
Al–Ce	[52,58,59]	[53]	[60,61,62]
Al–Pr	[52,63]	[53]	[63,64]
Al–Nd	[52,58]	[53]	[64]
Al–Pm	-	[53]	-
Al–Sm	[52,65]	[53,66]	[67]
Al–Eu	-	[53]	-
Al–Gd	[68,69,70]	[53]	[71,72]
Al–Tb	[69]	[53]	-
Al–Dy	[69,73]	[53]	-
Al–Ho	[69]	[53]	-
Al–Tm	-	[53]	-
Al–Yb	[74]	[53]	-
Al–Lu	-	[53]	-

.

In the current work, we focus on the Al–Tb binary system (see Figure 1) and make critical measurements of the enthalpy of mixing, $\mathrm{\Delta }{H}_{mix}$, for the liquid phase. Specifically, we perform solution calorimetry and electromagnetic levitation drop calorimetry to measure $\mathrm{\Delta }{H}_{mix}$ in dilute Al–Tb alloys with Tb content up to $x_{Tb} = 0.2$ (mole fraction). In addition, we compare three different semi-empirical solution models for the excess free energy of the liquid, incorporating regular, subregular, and associate [75] formulations for the mixing enthalpy$.$ These are compared with the Miedema model [76,77,78] for this system.

2. Calorimetry Experiments

We employ here two different calorimetric techniques to determine $\mathrm{\Delta }{H}_{mix}$ for the Al–Tb liquid phase in the stable temperature range. Using an isothermal solution calorimetry (ISC) method, we measure the thermal transient associated with the dissolution of a small amount of Tb in a liquid Al solvent, permitting direct determination of $\mathrm{\Delta }{H}_{mix}$ for the liquid phase. Using an electromagnetic levitation drop calorimetry (EMLDC) method, we measure the heat evolved upon rapid cooling of the high temperature equilibrium alloy liquid to a low temperature multi-phase state, from which we determine the high temperature mixing enthalpy, $\mathrm{\Delta }{H}_{mix}$. All test specimens were prepared from the pure elemental components (0.9999 Al and 0.999 Tb, by weight (pure Al and Tb were supplied by Cerac, Inc., Milwaukee, WI, USA)). Alloy specimens of desired composition were prepared for EMLDC by arc melting the pure components five times in succession on a water-cooled copper hearth under a high-purity (ultra-high purity (99.999%) argon, supplied by Matheson Company, Des Moines, IA, USA) argon atmosphere. Test specimens were then cut to the desired size using a diamond saw.

The central components of the custom-built ISC are illustrated schematically in Figure 2. The entire assembly shown in the figure is contained within a vacuum chamber, evacuated to 10⁻⁵ torr, refilled to 760 torr with ultra-high purity argon prior to heating, and maintained at this pressure throughout the experiment. For the measurement, a mass (m_Tb) of pure Tb, initially at ambient temperature (T_a), is introduced into a bath of pure Al liquid of known mass (m_Al), with the system held isothermally at T₀ using an integrated thermocouple-based closed-loop control system. The resulting net thermal exchange between the calorimeter and the bath associated with isothermal control is given by

$${\dot{q}}_{cal} = {\dot{q}}_{heat} + {\dot{q}}_{melt} + {\dot{q}}_{mix} + {\dot{q}}_0$$

(1)

where the first three terms on the right-hand-side (RHS) are exchange fluxes associated with heating, melting, and mixing of the added material, respectively. The fourth term is the heat flux required to maintain isothermal conditions at T₀ in the bath, with no addition of material. Upon the addition of the terbium mass, m_Tb, to the bath, the net exchange gives rise to a sensible thermal transient which is measured using a thermopile in contact with the melt crucible. An example thermal trace is shown in Figure 3. The measured temperature transient is related to the specimen-bath reaction enthalpy according to:

$$\delta q_{meas} = {\int }_0^{\infty }\left({\dot{q}}_{cal} - {\dot{q}}_0\right)dt = \beta {\int }_0^{\infty }\left(T - T_0\right)dt,$$

(2)

and we compute the incremental heat of mixing for this transient as

$$\delta q_{mix} = \delta q_{meas} - \delta q_{heat} - \delta q_{melt},$$

(3)

where

$$\delta q_{heat} = \frac{m_{Tb}}{M_{Tb}}({\int }_{T_a}^{T_{melt}}{c}_p^{sol}dT + {\int }_{T_{melt}}^{T_0}{c}_p^{liq}dT)\mathrm{and}$$

(4)

$$\delta q_{melt} = \frac{m_{Tb}}{M_{Tb}}\left[H_{Tb}^{liq}\left(T_{melt}\right) - H_{Tb}^{sol}\left(T_{melt}\right)\right].$$

(5)

Here, M_Tb and m_Tb denote the molar and total mass of the Tb specimen, respectively, and β is the ISC calorimeter constant. The temperature dependent molar heat capacities (c_p) were determined from SGTE free energy parameterizations [79].

The ISC calorimeter constant, β, was determined using pure Al solid specimens and a pure Al bath. In this case, $\delta q_{mix}=0$, and the left-hand side (LHS) of Equation (2) can computed from the heat capacity and enthalpy of melting for Al, which are explicitly known. The right-hand side (RHS) of Equation (2) is determined from the measured temperature signal, permitting quantitative determination of β for the specific temperature and mass of the bath, using Equations (3)–(5). Calibration measurements used to determine the calorimeter constant are summarized in

Table 2. ISC calibration data.

Parameter	Value
Ta	298 K
T0	1364 K
$m_{Al}^{crucible}$	9.4869 g
mAl	0.7009 g
MAl	26.98 g/mol
MTb	158.93 g/mol
$\delta q_{heat}$	826.05 J
$\delta q_{melt}$	278.26 J
$\delta q_{meas}/\beta$	−13,247.6 sK
$\beta$	−0.08336 ± 5% J/sK

, and the constant was determined as −0.08336 J/sK (The reported uncertainty of ±5% for β is based on 1.5 standard deviations, with respect to pure Al measurements, as described in Appendix A).

Immediately following the calibration measurement, $\mathrm{\Delta }{H}_{mix}$ measurements were performed at T₀ = 1364 K by introducing solid Tb (T_a = 298 K) to the melt in incremental amounts of 1.8528 g, 3.4288 g, and 4.861 g, yielding alloys with mole fractions of Tb ($x_{Tb}$) of 0.030, 0.054, and 0.075, respectively. The results are summarized in

Table 3. Incremental ISC mixing enthalpy measurements for T₀ = 1364 K.

mAl (g)	δmTb (g)	mTb (g)	${\mathit{x}}_{\mathit{T}\mathit{b}}$	δqmix (J/mol)	∆Hmix (J/mol)
10.1878	1.8528	1.8528	0.03	−4955.05	−4955.05
-	1.5676	3.4288	0.05	−3994.04	−8949.09
-	1.4322	4.8610	0.08	−2044.05	−10,993.1
10.85	6.4362	6.4362	0.09	−1074.16	-
-	1.4510	7.8872	0.11	−943.902	-

. Incremental mixing enthalpy (relative to the previous state) is denoted as δq_mix while the mixing enthalpy relative to the unmixed pure Al and Tb components is denoted as ∆H_mix. Attempts to perform ISC measurements with higher Tb content resulted in non-negligible reaction with the ZrO₂ calorimeter crucible, which interfered with the thermal transient signal. Measured quantities are shown in Table 3 for these cases, but no enthalpy of mixing is reported.

For higher solute content, we turn to a container-less approach and employ electromagnetic levitation coupled with drop-calorimetry (EMLDC). The basic setup is shown in Figure 4. In this method, a stable levitation condition is established with a specimen of known mass under vacuum (10⁻⁶ Torr) at a fixed temperature (T_drop). Temperature is controlled by induction heating with forced gas (ultra-high purity (99.999%) He + 5% H₂ gas, supplied by Matheson Company) (He + 5% H₂) and measured using a non-contact infrared pyrometer. After a stable condition is reached at T_drop, the coil is de-energized and the specimen falls by gravity into the calorimeter vessel. The calorimeter vessel is simply a solid copper cup with a conical bottom and a total mass of 167.0 g. The specimen temperature at the time of contact with the calorimeter (T_con) is computed by accounting for heat loss during the fall period (see Appendix B). The temperature transient in the calorimeter is measured using a thermocouple positioned just below the contact surface. A typical thermal trace is illustrated in Figure 5. The measured signal reflects rapid heat transfer from the specimen to the calorimeter upon contact and the evolution of heat during solidification and other phase changes. After the initial local heating, the measured signal reveals a slow cooling effect associated with heat redistribution within the calorimeter and heat loss from the calorimeter to its surroundings as it returns to temperature, T₀.

The measured thermal exchange in the calorimeter is generally given as

$${\dot{q}}_{cal} = {\dot{q}}_{cool} + {\dot{q}}_{fr} + {\dot{q}}_{trans} + {\dot{q}}_0,$$

(6)

where the first three terms on the right-hand-side give contributions from the dropped specimen, including cooling, freezing, and any other solid-state phase transitions that may occur, respectively. The last term is associated with thermal equilibration of the calorimeter, involving heat transfer within the calorimeter itself and thermal exchange with the surroundings. The thermal transient measured in the calorimeter includes all of these contributions, so the specimen contribution is given by

$$\delta q_{spec } = \delta q_{cal} - \delta q_0 = \varphi {\int }_0^{\infty }\left(T - T_0\right)dt.$$

(7)

The enthalpy of mixing in the liquid can be determined as

$$q_{mix} = \delta q_{spec} - n\left({\sum }^{}{f}_p\mathrm{\Delta }{H}_f - {\sum }^{}{x}_i\delta q_i^T\right),$$

(8)

where n is the total number of moles (of atoms) in the specimen, ∆H_f is the molar enthalpy of formation of a given phase at 298 K, f_p is the molar phase fraction of each phase, and the first term in parentheses is summed over all phases present in the quenched droplet. The second term in parentheses reflects the total molar enthalpy change for each elemental component, between the 298 K ground state and the liquid at T_con, and it is summed over all components (Al and Tb in the present case), where x_i is the mole fraction of each component. Explicitly:

$$\delta q_{Al}^T = {\int }_{298}^{T_m}{c}_p^{Al, fcc}dT + \mathrm{\Delta }{H}_{m } + {\int }_{Tm}^{T_{con}}{c}_p^{Al, liq}dT$$

(9)

and

$$\delta q_{Tb}^T = {\int }_{298}^{T_m}{c}_p^{Tb,\eta }dT + \mathrm{\Delta }{H}_{m } + {\int }_{T_m}^{T_{con}}{c}_p^{Tb,liq}dT.$$

(10)

Note that the expression in brackets in Equation (8) gives the total molar enthalpy difference between the quenched droplet at ambient temperature and the unmixed liquid at T = T_con. The required formation enthalpy (for Al₃Tb) is computed using a DFT approach (see Appendix C).

For our experiments, the time scale for the measured local calorimeter heating upon specimen contact is much shorter than the time scale for cooling back to ambient conditions. Accordingly, we model the heating as instantaneous and locally adiabatic followed by Newtonian cooling with a characteristic exponential decay. We note that conditions are not adiabatic, but we define a local adiabatic equivalent temperature, T_A, as the maximum temperature that would be reached in the limit of instantaneous adiabatic heating of a small local (i.e., effective) mass of the calorimeter upon specimen contact. The thermal transient measured upon contact with the non-adiabatic copper calorimeter is modeled as

$$T_0 + \left(T_A - T_0\right)\exp \left(-\frac{t}{\tau }\right),$$

(11)

where τ is a calorimeter relaxation time, and T_A is a local adiabatic equivalent temperature corresponding to the limit of rapid heat extraction from the specimen. This temperature is treated as a fit parameter, along with τ. Equation (11) describes the measured behavior very closely, as shown by the root-mean-square-error (RMSE) values listed in

Table 4. Summary of EMLDC measurements for ∆H_mix for Al–Tb liquid.

#	Type	${\mathit{x}}_{\mathit{T}\mathit{b}}$	Mass (g)	Tdrop (K)	Tcon	T0 (K)	TA (K)	T∞ (K)	RMSE (K)	qcal (J/mol)	$\mathit{\varphi }$	∆Hmix (kJ/mol)
1	cal	0	0.5206	1245.0	1241.8	292.3	304.4	294.9	0.013	744.4	2.692	-
2	cal	0	0.5159	1161.8	1159.1	298.0	309.0	294.7	0.004	684.9	2.710	-
3	cal	0	0.5190	1176.3	1173.5	298.2	309.7	295.2	0.031	697.6	2.657	-
4	cal	0	0.5133	1078.9	1076.5	295.9	306.2	294.6	0.042	632.7	2.693	-
5	cal	0	0.5173	1303.7	1297.7	296.5	309.0	294.7	0.016	773.2	2.717	-
6	cal	0	0.5168	1258.9	1255.7	298.1	310.1	295.3	0.001	744.7	2.718	-
7	meas	0.1	0.4796	1292.0	1286.1	297.5	305.6	296.2	0.015	493.1	2.698	−16.61
8	cal	0	0.2476	1080.3	1076.5	292.9	298.1	293.2	0.001	307.5	2.570	-
9	cal	0	0.2467	1245.6	1240.7	295.4	301.4	293.7	0.024	353.6	2.585	-
10	meas	0.2	0.3476	1406.5	1394.7	294.6	300.4	294.0	0.004	343.2	2.578	−27.08
11	meas	0.2	0.3249	1439.5	1426.7	293.1	298.8	293.2	0.007	332.0	2.578	−26.37

, determined over the temperature range T < T_A − 0.25 K. Once the values of τ and T_A have been determined, the appropriate heat balance can be applied. Moreover, since $q_{cal} = q_{spec}$ for this adiabatic heating, Equations (7)–(10) yield

$$q_{spec} = -\varphi {\int }_{T_0}^{T_A}{c}_P^{Cu\left(s\right)}dT,$$

(12)

where the calorimeter constant, φ, includes the effective calorimeter mass associated with non-uniform rapid heating upon contact, so that it includes the effects of specimen size and calorimeter size and shape.

The time-dependent heat transfer will be influenced by specimen size for a given calorimeter configuration (size, shape, insulation, thermocouple placement, etc.). Accordingly, determination of the calorimeter constant requires a standard specimen of the approximate size/mass of the specimen for which the enthalpy measurement is to be performed. In this work, two sets of calibration measurements were performed using pure Al specimens, with masses of 0.25 g and 0.5 g, nominally. Calibration measurements are listed along with applicable alloy test measurements in Table 4. Underlined values of the calorimeter constant (φ) indicate an average of measured values determined from applicable calibration experiments listed. The stated specimen temperatures correspond to those shown in Figure 5. Pure Al specimens were used to determine φ for the two different nominal specimen sizes, where:

$$\frac{M_{Al}}{m}{q}_{meas} = {\int }_{T_h}^{T_m}{c}_P^{Al\left(L\right)}dT + \mathsf{\Delta }{H}_m + {\int }_{T_m}^{T_A}{c}_P^{Al\left(s\right)}dT$$

(13)

where m is the specimen mass. We note that Equation (13) can be recovered from Equations (7)–(9) for a pure Al specimen, since $q_{mix}$ and $\mathsf{\Delta }{H}_f$ both are equal to zero. Combining Equations (12) and (13), the calorimeter constant was determined using the pure Al samples, and the enthalpies of the alloy specimens were measured using the mean calorimeter constant determined from the calibration runs, as listed (type “cal”) in Table 4 (standard relative uncertainty of 1.3%; see Appendix A).

The EMLDC method described above requires determination of the thermodynamic state of the quenched specimen. This requires knowledge of the phases present in the quenched droplet, their relative amounts, and their formation energies. Based on the equilibrium phase diagram, we expect the quenched droplets to be comprised of the Al (fcc) and Al₃Tb (δ) phases and estimate the expected molar fraction of the δ phase as 0.4 and 0.8 for alloy compositions ($x_{Tb}$) of 0.1 and 0.2, respectively. We verified these estimates experimentally using X-ray diffraction (XRD) and scanning electron microscopy, as shown in Figure 6 (SEM analysis was done using a JEOL JSM-5910LV instrument. XRD was performed with a PANalytical X-Pert Pro diffractometer). Based on the XRD patterns shown in Figure 6a, the phases present in the quenched droplets were identified as Al (fcc) and Al₃Tb (δ) for both compositions ($x_{Tb} = 0.1, 0.2)$. The corresponding microstructures are shown in Figure 6b,c, revealing primary δ phase (light) and a two-phase (fcc + δ) eutectic constituent (dark). To measure the fraction of primary δ, quantitative image analysis was performed using two images (1280 × 960 pixels) for each composition, with pixel sizes of 0.1 and 0.2 mm. Measurements yielding average δ fractions of 0.31 and 0.77 for $x_{Tb}$ = 0.1 and 0.2, respectively. Based on the eutectic composition of $x_{Tb} = 0.03$ (see Figure 1), the two-phase eutectic structure is expected to include 12% δ and 88% Al (fcc). Accordingly, the total molar fraction of the δ phase was determined to be 0.396 and 0.797% for $x_{Tb} =$ 0.1 and 0.2, respectively, in very good agreement with our original estimates of 0.4 and 0.8. Assuming that the formation energy at 0 K is equivalent to the formation enthalpy at 298 K, we computed the required enthalpy of formation for Al₃Tb (δ) from first principles, obtaining a value of −43.2 kJ/mol (see Appendix C), relative to the pure Al and Tb reference states at 0 K [53,80].

3. Analysis and Discussion

To establish additional thermodynamic context for our enthalpy of mixing measurements for the Al–Tb liquid phase, we now consider appropriate models and specifically examine the application of three different semi-empirical treatments, all arising from the general model

$$\mathrm{\Delta }{H}_{mix}^{Liq} = \frac{1}{1 + 2y_{{\mathrm{Al}}_2\mathrm{Tb}}}\left({\sum }_i\left({\sum }_{j>i}{y}_i{y}_j{\sum }_{k=0}^n{}^{k}L_{i,j}^{\phi }\right) + y_{{\mathrm{Al}}_2\mathrm{Tb}}\mathrm{\Delta }{H}_f^{{\mathrm{Al}}_2\mathrm{Tb}}\right)$$

(14)

where $x_i$ are the elemental mole fractions, and $y_i$ are mole fractions of associate species Al, Tb, and Al₂Tb. The model parameters determined on the basis of our measurements are listed in

Table 5. Parameters for thermodynamic models of $\mathrm{\Delta }{H}_{mix}^{Liq}$.

	Regular	Subregular	Associate
i	Al	Al	Al	Tb	Al2Tb
j	Tb	Tb	Tb	Al2Tb	Al
${}^{0}L_{i,j}$ (J/mol)	−160,000	−128,000 + 29.001T	−75,252	−20,342	−35,455
${}^{1}L_{i,j}$ (J/mol)	-	−80,455 + 30.998T	-	-	-
${}^{3}L_{i,j}$ (J/mol)	-	30,342	-	-	-
$\mathrm{\Delta }{G}_{A{l}_{2}Tb}^{°}$ (J/mol)	-	-	−113,233 + 18.904T
Notes	$y_i = x_i$ $\left(y_{{\mathrm{Al}}_2\mathrm{Tb}} = 0\right)$		$y_i$ computed from $\mathrm{\Delta }{G}_{A{l}_{2}Tb}^{°} = -RTln\left(\frac{a_{A{l}_{2}Tb}}{a_{Al}^2{\mathrm{a}}_{\mathrm{Tb}}}\right)$ $2\mathrm{Al} + \mathrm{Tb} \to {\mathrm{Al}}_2\mathrm{Tb}$

. The resulting model descriptions of the mixing enthalpy are plotted in Figure 7 for T = 1364 K and compared with measured values. Corresponding model descriptions of chemical activity and partial molar mixing enthalpy $\left({\overline{\mathrm{\Delta }H}}_{mix}\right)$ are also plotted. In addition, the Miedema model prediction [76,77,78] is included in each figure for comparison.

It is significant to point out that the experimental data show very good agreement with the Miedema [76,77,78] prediction, which involves no adjustable parameters and serves as a baseline model. These figures also illustrate clearly that all three of the CALPHAD formulations (with the parameters listed in Table 5) provide similar descriptions of ∆H_mix for this alloy over the dilute regime. Comparison of the modeled chemical activities suggests relatively small differences in the non-ideality for Al-rich compositions. This behavior is isolated more clearly in the partial molar mixing enthalpy, plotted in Figure 7c, which shows the different values of $\left({\overline{\mathrm{\Delta }H}}_{mix}\right)$ in the limit as $x_{Tb}\to 0$. Generally, Figure 7 shows that multiple model formulations can provide reasonable descriptions of the measured behavior. Further discrimination will require experimental measurements in intermediate and Tb-rich compositional regimes. Considering the challenges related to crucible reactions, this calls for additional container-less experiments.

4. Conclusions

We report here measurements of the enthalpy of mixing in the liquid phase for Al–Tb alloys, for Tb mole fractions up to $x_{Tb} = 0.2$. Specifically, we utilize isothermal solution calorimetry and isoperibolic electromagnetic levitation drop calorimetry to measure the enthalpy of mixing, employing X-ray diffraction, scanning electron microscopy, and first principles calculations to provide necessary quantification of the quenched states and component reference states. Based on our measurements, we offer three different assessed semi-empirical solution models for the excess free energy of the liquid, including regular, subregular, and associate model formulations for the enthalpy of mixing. All three of these model formulations provide very good representation of the measured behavior over the limited range examined. Our results highlight the need for additional measurements targeting the composition range of 0.5 < $x_{Tb}$ < 0.7, which would enable further refinement of the general model.