Effect of Copper on the Formation of L1₂ Intermetallic Phases in Al–Cu–X (X = Ti, Zr, Hf) Alloys

Abstract

Transition metal trialuminides of the Al₃X type of groups 4 and 5 of the periodic system have reduced density, high melting points, and corrosion resistance. Aluminides with a cubic lattice of the Al₃Sc type can be used as a nucleating phase for aluminum alloys. However, low plasticity and a tetragonal lattice limit their application. In this work, we stabilized the metastable cubic lattice of Al₃X-type aluminides by replacing aluminum with copper. The conditions for the formation of L1₂ metastable aluminides in the Al–Cu–TM (TM: Ti, Zr, Hf) alloys were studied using a wide range of copper concentrations. A high concentration of copper (hypereutectic alloys) is the one of the necessary conditions for the formation of (Al_1−xCu_x)₃Ti, (Al_1−xCu_x)₃Zr, (Al_1−xCu_x)₃Hf aluminides. With an increase in the copper concentration, the number of metastable aluminides sharply increased. The process of their formation strongly depended on the sequence of dissolution of the corresponding components in the melts. The low volume fraction of precipitated titanium aluminides was the result of insufficient supersaturation of α-Al with titanium (at the peritectic temperature) compared to that for alloys with zirconium and hafnium. Under identical synthesis conditions in the crystal lattice of metastable aluminides formed in experimental Al–Cu–Ti, Al–Cu–Zr, Al–Cu–Hf alloys, copper was found to substitute up to 8, 10, and 13 at.% of aluminum, respectively. The crystallographic and dimensional similarities of the lattices in metastable transition metal aluminides and in α-Al suggest their usefulness as modifying additions in aluminum-based alloys.

1. Introduction

High-tech industries require the search for new materials with improved technological properties. Group 4 and 5 transition metal (TM) trialuminides have attracted much attention due to their low density, high melting temperatures and oxidation resistance [1,2,3,4,5]. Aluminides of the Al₃X type are considered as independent materials [3,5,6,7,8] or as reinforcing additions in aluminum alloys [4,9,10,11,12,13,14]. The authors of [15,16,17,18] continue to study the synthesis conditions and properties of metal-intermetallic laminate composites, where Al₃X intermetallic compounds serve as one of the layers. Another practical application of Al₃X aluminides is their use as modifying additions for grain refinement of aluminum alloys [19,20,21,22]. The Al₃X intermetallic compounds obtained under equilibrium conditions have a tetragonal structure (D0₂₂ or D0₂₃) and low plasticity, which limits their application. One of the ways to increase the plasticity of aluminides is to transform the tetragonal lattice into a cubic one.

It is known [3,5,7,23,24,25,26] that the cubic lattice of aluminides can be stabilized by adding period 4 elements (Cr, Mn, Fe, Co, Ni, Cu, Zn.). This increases the plasticity of aluminides and, accordingly, improves the elastic properties of aluminide alloys. Such alloys are usually produced either by mechanical alloying or by melting in an electric arc furnace under vacuum with a non-consumable electrode and crystallization on a water-cooled copper hearth [3,23,24,25,26,27]. Cast alloys containing L1₂ type TM aluminides are produced by high-temperature overheating followed by fast melt cooling [22,28,29,30]. Such alloys can be used as modifying master alloys to refine the grain structure of other alloys [31,32], since aluminides with the fcc lattice (L1₂-type) are especially attractive due to their high degree of similarity with the α-Al lattice. In addition, Al₃X aluminides have a low density, a high elastic modulus and high melting temperatures. They are ideal strengthening phases in high-strength, heat-resistant aluminum-based alloys.

The effect of copper addition on the formation of L1₂ aluminides in ternary Al–Cu–X (X: Ti, Zr, Hf) alloys was studied. When analyzing the energy dispersive X-ray microanalysis (EDX) data, for the copper-containing TM aluminides formed in the alloys, the authors relied on calculated and experimental data [8]. In [8], it was shown that the substitution of copper instead of aluminum in the crystal lattice of aluminides is energetically more favorable than the substitution of TM with copper. The EDX analysis results were compared with the calculated and experimental data [8] on the smelting of ternary (Al,Cu)_0.75Zr_0.25 and (Al,Cu)_0.75Ti_0.25 aluminide alloys.

2. Materials and Methods

Smelting of alloys based on aluminum and copper TM additions (Ti, Zr, Hf) was carried out in a graphite crucible in the ERF-93 electric resistance furnace (Ural Scientific Center, Yekaterinburg, Russia) in an argon atmosphere. Synthesis was carried out at temperatures from 1200 to 1300 °C to ensure sufficient overheating of the melt above the liquidus. Commercial purity aluminum (99.7% purity), high purity titanium (99.95%) and zirconium (99.99%) crystal bars grown by the iodide process, as well as electrolytic hafnium (99.68%) and M00 oxygen-free copper (99.96%) were used for synthesis. All the reagents used in the work were produced by REAHIM Trading House LLC, Yekaterinburg, Russia. The smelting process was carried out in two ways.

The first one consisted of the simultaneous addition of M00 copper rods and a pre-synthesized master-alloy (containing TM) to the aluminum melt. In the second case, the copper was dissolved in an aluminum melt and then a master-alloy (containing TM) was added. In both cases, after the dissolution of all components, the melts were stirred for 30 min and poured into a bronze mold. The crystallization rate in the mold (dimensions: 100 × 80 × 10 mm) was ≈200 °C/s. The weight of the resulting ingots was ≈200 g. Samples for chemical and metallographic analysis were taken from the lower part of the ingot.

Metallographic analyses of the samples were performed using an GX-57 optical inverted microscope (OLYMPUS Corporation, Tokyo, Japan) (magnifications from 50× to 1500×) and EVO 40 scanning electron microscopes (Carl Zeiss AG, Oberkochen, Germany). All electron images presented in this work were obtained using a back-scattering electron (BSE) detector (Carl Zeiss AG, Oberkochen, Germany) (with high sensitivity to differences in atomic number). To determine the chemical composition of the matrix and TM aluminides, an energy dispersive X-ray microanalysis (EDX) was carried out using an INCA X-Act system (Oxford Instruments plc, Abingdon, Great Britain).

The chemical composition of the synthesized alloys was determined by inductively coupled plasma optical emission spectrometry using the Optima 2100DV. Diffraction studies were performed using an XRD-7000 X-ray diffractometer (Shimadzu Corporation, Kyoto, Japan) (CuK_α-radiation, graphite monochromator, the accumulation of data during 1.8 s in increments of 0.03 degrees in the range of 20−80 degrees). Unit cell parameters were calculated with the east-square method using the positions (in 2θ) of 10 peaks.

3. Results and Discussion

The compositions of the studied Al–Cu–Ti, Al–Cu–Zr and Al–Cu–Hf ternary alloys varied with a range of copper concentrations, from hypoeutectic to hypereutectic. The main structural components of these alloys were the θ-phase Al₂Cu (indicated by the green arrows in the Figures), the α-Al + θ eutectic (indicated by the blue arrows) and TM trialuminides with a cubic lattice (red arrows). The melting temperature was determined by the at.% of the transition metal in the corresponding Al-TM system and its melting temperature. Depending on the degree of overheating of the melts above the liquidus, either stable aluminides with a tetragonal (D0₂₂ or D0₂₃) lattice structure, metastable with a cubic (L1₂) structure, or both, were formed in the alloys. Since the aim of the study was to obtain metastable aluminides with a cubic lattice having the L1₂ structure in alloys, the table below lists the compositions of only those alloys in which aluminides of the L1₂ structural type were formed. The

Table 1. Compositions of Al–Cu–X (X=Ti, Zr, Hf) alloys and the L1₂ aluminides formed in them.

Alloy№	Alloys Composition, wt.% (at.%), Degree of Supersaturation with TM.	Synthesis Temperature, °C	Composition of Sluminides Having the L12 Crystal Structure	Stable Aluminides Having the D023 and D022 Crystal Structure
T1	Al–41.91Cu–1.64Ti (Al–23.70Cu–1.23Ti), 2 times	1250	Cuboids and dendrites 70.68Al–6.36Cu–22.96Ti or (Al0.92Cu0.08)nTi *	Needles and plates containing 5.71 at.%Cu
T2	Al–40.33Cu–2.75Ti (Al–22.68Cu–2.05Ti), 3.4 times	1300	Cuboids and dendrites 70.68Al–6.36Cu–22.96Ti or (Al0.92Cu0.08)nTi *	Needles and plates containing 5.71 at.%Cu
Z1	Al–38.78Cu–4.56Zr (Al–22.13Cu–1.81Zr), 26 times	1200	Cuboids and dendrites 68.34Al–6.27Cu–25.39Zr or (Al0.92Cu0.08)3Zr	There are no particles having this crystal structure
Z2	Al–31.47Cu–8.63Zr (Al–17.64Cu–3.37Zr) 48 times	1300	Cuboids and dendrites 67.08Al–7.27Cu–25.65Zr or (Al0.90Cu0.10)3Zr	There are no particles having this crystal structure
Z3	Al–36.28Cu–9.02Zr (Al–21.19Cu–3.67Zr) 52 times	1200	Cuboids and dendrites 65.39Al–9.05Cu–25.56Zr or (Al0.88Cu0.12)3Zr	Needles and plates
Z4	Al–31.25Cu–8.37Zr (Al–17.45Cu–3.26Zr) 47 times	1300	Single cuboids, not analyzed	Plates (Al0.97Cu0.03)3Zr
H1	Al–29.73Cu–13.24Hf) (Al–17.63Cu–2.79Hf 174 times	1250	Single cuboids, not analyzed	Plates containing 3.80 at.%Cu
H2	Al–29.47Cu–13.69Hf (Al–17.54Cu–2.91Hf) 182 times	1300	Single cuboids, not analyzed	Plates containing 3.80 at.%Cu
H3	Al–32.83Cu–13.04Hf (Al–19.92Cu–2.82Hf) 176 times	1300	69.03Al–7.67Cu–23.30Hf or (Al0.90Cu0.10)nHf *	Single plates and needles Al3Hf
H4	Al–34.09Cu–13.04Hf (Al–20.91Cu–2.84Hf) 178 times	1250	69.90Al–7.39Cu–22.71Hf or (Al0.90Cu0.10)nHf *	Plates containing 3.55 at.%Cu and needles Al3Hf
H5	Al–35.66Cu–15.36Hf (Al–22.81Cu–3.50Hf) 219 times	1200	67.09Al–10.03Cu–22.88Hf or (Al0.87Cu0.13)nHf *	There are no particles having this crystal structure

* According to EDX analysis, value of n is in the range from 3 to 4.

lists the compositions of the alloys containing L1₂ aluminides. The composition of the aluminides was estimated from the results of 5–10 EDX spectra analyses. The degree of supersaturation of the TM alloys was determined from the ratio of the atomic fraction of the calculated critical C₀ value (TM concentration at the temperature of peritectic transformation) and the TM fraction in the alloy.

3.1. Al–Cu–Ti System

Stable D0₂₂ intermetallic compounds were formed in the alloys synthesized at 1250 and 1300 °C, with a wide range of copper concentrations (from 9.98 to 32.24 wt.%—Hereinafter, unless otherwise indicated, wt.%—) and with a low titanium content. Only in the Al–41.91Cu–1.64Ti and Al–40.33Cu–2.75Ti alloys (T1 and T2), along with stable aluminides (needles and plates), single metastable dendrites and L1₂ cuboids (size is less than 7 μm) were formed; they were visible at high magnifications (see Figure 1). Their average composition (at.%) was 70.68Al–6.36Cu–22.96Ti. The stoichiometric composition of the aluminides can be written in the following form: (Al_0.92Cu_0.08)_nTi. It can be seen that copper substituted 8 at.% of the aluminum in the L1₂ crystal lattice of titanium aluminide.

The compositions of the aluminides formed in the T1 and T2 alloys were compared with that of the L1₂ cast aluminide alloy (70.1Al–6.9Cu–23.0Ti) synthesized in [8], (at.%). It was shown that the differences in their compositions were in the tenths and hundredths of a percent. The stoichiometric composition of the aluminides can be written in the following form: (Al_0.91Cu_0.09)₃Ti. It can be seen that copper substituted 9 at.% of the aluminum.

Note that in [8], two series of ternary aluminide alloys, i.e., Al–Cu–Ti and Al–Cu–Zr, with copper contents of 8 and 12 at.% were synthesized. Melting was carried out in order to determine the copper concentration, which stabilizes the metastable L1₂ lattice of titanium and/or zirconium trialuminides. In cast alloys, the formation of one or two additional phases was identified. Such phases disappeared after homogenizing annealing at 1000 °C for 72 h. In addition, after annealing, the scatter of data on the composition of the aluminide alloy decreased, approaching a stoichiometric one.

3.2. Al–Cu–Zr System

Two hypereutectic alloys, i.e., Al–38.78Cu–4.56Zr and Al–36.28Cu–9.02Zr (Z1 and Z2), differed in their zirconium content and were melted at 1200 °C. In the first alloy (Figure 2a), as a result of significant overheating of the melt, only metastable trialuminides were formed. They had a cubic (L1₂) structure and the following composition: 68.34Al–6.27Cu–25.39Zr. These intermetallics formed large (50 to 150 µm) clusters of fragmented cuboids and dendrites of various growth shapes. Note that the light inclusions visible on the intermetallic compounds were undissolved zirconium particles. In the second alloy, the Zr content was twice as high and, accordingly, the overheating was less. In this case, both structural types of aluminides were formed. Intermetallic particles having the L1₂ lattice and a composition of 65.39Al–9.05Cu–25.56Zr (Z2) formed clusters. The size of the clusters, like that of the aluminides themselves, was an order of magnitude smaller than in the Z1 alloy (see Figure 2).

It can be seen that the higher the atomic fraction of Zr in the alloy, the greater the number of aluminides and the smaller their size. The composition of intermetallic particles formed in two alloys of the Al–Cu–Zr system (Z1 and Z3) was compared. It was shown that with an increase in the concentration of Zr in the alloys from 1.81 to 3.67 at.%, the atomic fraction of Cu in the formed aluminides increased from 6.27 up to 9.05 at.%. The stoichiometric composition of these aluminides (Al_0.92Cu_0.08)₃Zr and (Al_0.88Cu_0.12)₃Zr was also analyzed. It can be concluded that with an increase in the atomic percentage of Zr, the substitution of aluminum with copper in the aluminides crystal lattice increased from 8 to 12 at.%, although the copper content in the alloy (Z3) was less than in (Z1). This once again confirmed that the composition and size of the formed L1₂ aluminides depended mainly on the atomic percentage of TM, in this case, zirconium.

The next two alloys, Al–31.47Cu–8.63Zr and Al–31.25Cu–8.37Zr (Z2 and Z4), having practically the same copper and zirconium compositions, were melted at 1300°C, but with a different sequence of copper and zirconium dissolution in aluminum melt. In the first alloy (the dissolution of copper in an aluminum melt, and addition of master-alloy containing zirconium after that), only metastable L1₂ trialuminides comprising 67.08Al–7.27Cu–25.65Zr or (Al_0.90Cu_0.10)₃Zr were formed. The aluminides formed clusters (up to 100–150 µm in size) of particles of different shapes and sizes (up to 10–20 µm). In the second alloy (Figure 3b), along with a small amount of metastable L1₂ trialuminides, a lot of needles and plates with the D0₂₃ lattice structure and composition (Al_0.97Cu_0.03)₃Zr were formed. In such particles, compared with L1₂ aluminides, copper substituted three times less aluminum. Such a difference in the structure of these alloys indicated a significant effect of the sequence of dissolution of copper and TM in the aluminum melt.

Formed in three cast alloys of the Al–Cu–Zr system, zirconium aluminides with L1₂ structure had an average composition of 66.94Al–7.53Cu–25.53Zr. This was close to the composition of the 67.90Al–7.50Cu–24.70Zr cast aluminide alloy, synthesized in [8]. In both cases, copper substituted up to 10 at.% of the aluminum in the crystal lattice of the aluminides. According to the data presented in [8], the cubic (L1₂) lattice parameter of this phase in the cast state was a = 0.4070 nm. This value is close to that measured in our work, a = 0.40662(7) nm for the 65.39Al–9.05Cu–25.56Zr phase formed in the Z3 alloy.

3.3. Al–Cu–Hf Yystem

Among the five alloys of this system, Al–29.73Cu–13.24Hf and Al–29.47Cu–13.69Hf (H1 and H2) had similar compositions, both in terms of their copper and hafnium concentrations; however, they differed in melting temperature and in the sequence of Cu and Hf dissolution in molten Al. The H1 alloy was synthesized at 1250 °C using the second method (first, copper was dissolved in the aluminum melt, then hafnium), and alloy H2 using the first method (conventional scheme), but at a temperature of 1300 °C (i.e., with greater overheating of the melt). In both alloys, as can be seen in Figure 4, mainly stable (plates and needles) hafnium aluminides and only a small number of metastable ones (clusters of cuboid particles) were formed.

Subsequently, 5 wt.% of Cu was added to the alloys, and melting was carried out at the same temperatures, i.e., 1250 °C and 1300 °C. As a result, stable and metastable hafnium trialuminides were formed in the alloys, but with a different ratio (see Figure 5). An increase in the copper concentration in the alloys changed the ratio of stable and metastable trialuminides in favor of the formation of a larger amount of the latter. This is clearly seen from a comparison of the particle growth shapes and sizes and the volume fraction of intermetallic particles precipitated in the alloys (Figure 4 and Figure 5).

An EDX analysis of the aluminides formed in the H4 and H3 alloys showed that the copper concentration in L1₂ intermetallic compounds was three times higher than that in the aluminides with the D0₂₂ structure. The chemical composition of the metastable aluminides in the investigated alloys differed insignificantly, amounting to 69.90Al–7.39Cu–22.71Hf and 69.03Al–7.67Cu–23.30Hf, respectively. The average stoichiometric composition can be written as (Al_0.90Cu_0.10)_nHf. It can be seen that in the crystal lattice of the aluminides, copper substituted about 10 at.% of aluminum (for both alloys).

Finally, the hypereutectic H5 alloy with a high (3.50 at.%) hafnium concentration was synthesized at 1200 °C. Only metastable aluminides of composition (at.%) 67.09Al–10.03Cu–22.88Hf or (Al_0.87Cu_0.13)_nHf were formed in this alloy. It can be seen that copper substituted 13 at.% of the aluminum. The distribution and growth patterns of L1₂ aluminides are clearly seen in Figure 6. According to the EDX data, Al_2.5Cu_0.5Hf aluminides with a cubic lattice (L1₂) a = 0.40441(5) nm formed in the H5 alloy. The XRD analysis results for the Al–36.28Cu–9.02Zr (Z3), Al–35.66Cu–15.36Hf (H5) alloys are shown in Figure 7.

Thus, in two alloys (Z1, Z2) of the Al–Cu–Zr system and in one (H5) of the Al–Cu–Hf system, only aluminides having the L1₂ structural structure were formed. The aluminides had compositions of (Al_0.92Cu_0.08)₃Zr, (Al_0.90Cu_0.10)₃Zr and (Al_0.87Cu_0.13)_nHf, in which copper substituted 8, 10, and 13 at.% of aluminum, respectively. The composition of the aluminides was determined by the atomic fraction of TM in the alloys.

3.4. Formation of Trialuminides with a Cubic Lattice in Alloys of the Al–Cu–X (X: Ti, Zr, Hf) System

The aluminide phases having the L1₂ structure formed in the Al–Cu–X (X: Ti, Zr) alloys smelted in this work were compared with the L1₂ aluminide alloys (in the cast state) synthesized in [8]. It was shown that they were almost completely identical both in composition and in the percentage of substitution of copper in the aluminum crystal lattice. However, the volume fraction of Al₃Ti particles precipitated from the α-Al solid solution (in two alloys of the Al–Cu–Ti system) turned out to be very low. This was the result of insufficient supersaturation with titanium, according to the classical theory of nucleation. To verify this fact, the following calculations were carried out:

1.

the nucleation rate per unit volume [33,34], which is a function of the diffusion activation energy (Q);

2.

the chemical driving force (${\Delta \mathrm{F}}_{\mathrm{ch}}$);

3.

the elastic strain energy (${\Delta \mathrm{F}}_{\mathrm{el}}$), which reduces the chemical driving force (${\Delta \mathrm{F}}_{\mathrm{ch}}$), preventing nucleation.

According to [35], the chemical driving force ${\Delta \mathrm{F}}_{\mathrm{ch}}$ for nucleation of Al₃X aluminides can be found from the following equation:

$${\Delta \mathrm{F}}_{\mathrm{ch}}=\frac{\mathrm{RT}}{{\mathrm{V}}_{{\mathrm{Al}}_3\mathrm{X}}}\left(\frac{{\mathrm{C}}_{{\mathrm{Al}}_3\mathrm{X}}-{\mathrm{C}}_{\mathsf{\alpha }}}{1-{\mathrm{C}}_{\mathsf{\alpha }}}\right)\ln \left(\frac{{\mathrm{C}}_0}{{\mathrm{C}}_{\mathsf{\alpha }}}\right)$$

(1)

where R = 8.314 (J/mol∙K) is the universal gas constant; ${\mathrm{C}}_{{\mathrm{Al}}_3\mathrm{X}}$ is the atomic fraction of TM in the Al₃X intermetallic compound equal to 0.25; ${\mathrm{C}}_0$ is the atomic fraction of TM in the α-Al solid solution; ${\mathrm{C}}_{\mathsf{\alpha }}$ is the solubility of TM in the solid α-Al; and ${\mathrm{V}}_{{\mathrm{Al}}_3\mathrm{X}}$ is the molar volume of the Al₃X phase, determined by the following equation:

$${\mathrm{V}}_{{\mathrm{Al}}_3\mathrm{X}}=\frac{{\mathrm{N}}_{\mathrm{a}}{\left({\mathrm{a}}_{{\mathrm{Al}}_3\mathrm{X}}\right)}^3}{4}$$

(2)

where ${\mathrm{N}}_{\mathrm{a}}$= $6.022\times {10}^{23}\left({\mathrm{mol}}^{-1}\right)$ is Avogadro constant and ${\mathrm{a}}_{{\mathrm{Al}}_3\mathrm{X}}$ is the lattice parameter of the Al₃X phase.

The values of the cubic lattice parameter of the Al₃Ti, Al₃Zr, and Al₃Hf trialuminides under consideration, obtained experimentally in [36], were 0.3967, 0.4080, and 0.4048 nm, respectively.

The solubility limit of TM in Al could be found from the following equation (according to [37]):

$${\mathrm{C}}_{\mathsf{\alpha }}^{{\mathrm{Al}}_3\mathrm{X}}=\exp \left(\frac{4\Delta \mathrm{G}\left({\mathrm{Al}}_3\mathrm{X}\right)-{\Delta \mathrm{G}}^{\mathrm{X}}}{\mathrm{kT}}\right)=\exp \left(\frac{{\Delta \mathrm{S}}_{\mathrm{vib}}^{\mathrm{X}}-4{\Delta \mathrm{S}}_{\mathrm{vib}}\left({\mathrm{Al}}_3\mathrm{X}\right)}{\mathrm{k}}\right)\times \exp \left(\frac{4\Delta \mathrm{H}\left({\mathrm{Al}}_3\mathrm{X}\right)-{\Delta \mathrm{H}}^{\mathrm{X}}}{\mathrm{kT}}\right)$$

(3)

where $\mathrm{k}=1.38\times {10}^{-23}\left(\mathrm{J}·{\mathrm{K}}^{-1}\right)$ is the Boltzmann constant; $\Delta \mathrm{G}\left({\mathrm{Al}}_3\mathrm{X}\right)$ is the free energy of formation for Al₃X aluminides; ${\Delta \mathrm{G}}^{\mathrm{X}}$ is the free energy of formation of a single TM in Al (fcc); ${\Delta \mathrm{S}}_{\mathrm{vib}}^{\mathrm{X}}$ is the vibrational entropy of 1 atom of TM; ${\Delta \mathrm{S}}_{\mathrm{vib}}\left({\mathrm{Al}}_3\mathrm{X}\right)$—vibrational entropy of Al₃X (per 1 atom); $\Delta \mathrm{H}\left({\mathrm{Al}}_3\mathrm{X}\right)$ is the formation enthalpy for ${\mathrm{Al}}_3\mathrm{X}$ per 1 atom; and ${\Delta \mathrm{H}}^{\mathrm{X}}$ is the enthalpy of formation (per 1 atom) for TM impurity in Al.

Table 2. Mixing enthalpy of Al₃X (X = Ti, Zr, Hf) compounds and TM atoms.

System	$\Delta \mathrm{H}=4\Delta \mathrm{H}\left(\mathrm{A}{\mathrm{l}}_3\mathrm{X}\right),$ eV/atom	$\Delta \mathrm{H}=\Delta {\mathrm{H}}^{\mathrm{X}},$ eV/atom	$\Delta {\mathrm{H}}_{\mathrm{s}\mathrm{o}\mathrm{l}}=4\Delta \mathrm{H}\left(\mathrm{A}{\mathrm{l}}_3\mathrm{X}\right)-\Delta {\mathrm{H}}^{\mathrm{X}},$ eV/atom
	L12	Impurity	L12
Al-Ti	−1.576	−1.197	−0.379
Al-Zr	−1.964	−1.250	−0.714
Al-Hf	−1.609	−0.954	−0.655

and

Table 3. Vibrational entropy of Al₃X (X = Ti, Zr, Hf) compounds and TM atoms.

System	$\Delta {\mathrm{S}}_{\mathrm{v}\mathrm{i}\mathrm{b}}=4\Delta {\mathrm{S}}_{\mathrm{v}\mathrm{i}\mathrm{b}}\left(\mathrm{A}{\mathrm{l}}_3\mathrm{X}\right),$ k/atom	$\Delta {\mathrm{S}}_{\mathrm{v}\mathrm{i}\mathrm{b}}=\Delta {\mathrm{S}}_{\mathrm{v}\mathrm{i}\mathrm{b}}^{\mathrm{X}},$ k/atom	$\Delta {\mathrm{S}}_{\mathrm{v}\mathrm{i}\mathrm{b}}^{\mathrm{X}}-4\Delta {\mathrm{S}}_{\mathrm{v}\mathrm{i}\mathrm{b}}\left(\mathrm{A}{\mathrm{l}}_3\mathrm{X}\right),$ k/atom
	L12	Impurity	L12
Al-Ti	−3.365	−2.518	0.847
Al-Zr	−3.354	−1.000	2.354
Al-Hf	−2.989	−0.837	2.152

present literature data on the enthalpies and vibrational entropies included in Equation (3) for Al₃X (X = Ti, Zr, Hf) compounds and TM atoms, as determined in [37] based on first-principles calculations.

The exponential dependences of ${\mathrm{C}}_{\mathsf{\alpha }}^{{\mathrm{Al}}_3\mathrm{X}}$ on temperature for L1₂ trialuminides were obtained by substituting the tabular values into Equation (3).

$${\mathrm{C}}_{\mathsf{\alpha }}^{{\mathrm{Al}}_3\mathrm{Ti}}=2.334\times \exp \left(\frac{-4398.1}{\mathrm{T}}\right)$$

(4)

$${\mathrm{C}}_{\mathsf{\alpha }}^{{\mathrm{Al}}_3\mathrm{Zr}}=10.528\times \exp \left(\frac{-8285.6}{\mathrm{T}}\right)$$

(5)

$${\mathrm{C}}_{\mathsf{\alpha }}^{{\mathrm{Al}}_3\mathrm{Hf}}=8.602\times \exp \left(\frac{-7600.9}{\mathrm{T}}\right)$$

(6)

The values of chemical driving force ${\Delta \mathrm{F}}_{\mathrm{ch}}$ were calculated by substituting the values determined from Equations (4)–(6) into Equation (1) (for a given T). The results are given in

Table 4. Calculation of the chemical driving force ${\Delta \mathrm{F}}_{\mathrm{ch}}$, which determines the precipitation of the Al₃X phase.

Al3X (L12)	${\mathrm{V}}_{\mathrm{A}{\mathrm{l}}_3\mathrm{X}}, {\mathrm{m}}^3\mathrm{·}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$	${\mathrm{C}}_{\mathrm{\alpha }}^{\mathrm{A}{\mathrm{l}}_3\mathrm{X}},$ Atomic Fraction (T = 698 K)	${\mathrm{C}}_{\mathrm{A}{\mathrm{l}}_3\mathrm{X}},$ Atomic Fraction	$\Delta {\mathrm{F}}_{\mathrm{c}\mathrm{h}}, \mathrm{M}\mathrm{J}\mathrm{·}{\mathrm{m}}^{-3}$ (Peritectic Temperature)
Al3Ti	0.94 × 10–5	0.02151	0.25	193.7 × ln(46.5 × C0)
Al3Zr	1.02 × 10–5	0.00148	0.25	189.3 × ln(675.7 × C0)
Al3Hf	0.99 × 10–5	0.00228	0.25	192.4 × ln(438.6 × C0)

.

In reactions involving a solid solution, it is necessary to take into account the elastic strain energy because of the formation of a second phase, which is fully or partially coherent with the matrix. This causes stress on the lattices of the matrix and the precipitating intermetallic phase. The elastic strain energy, ${\Delta \mathrm{F}}_{\mathrm{el}}$, reduces the chemical driving force, and hence, the nucleation rate. Considering only volumetric strains (for isotropic materials), the strain energy per unit volume for a coherent particle can be calculated using Equation (7) [33,34].

$${\Delta \mathrm{F}}_{\mathrm{el}}=2\mathsf{\mu }\frac{\left(1+\mathsf{\nu }\right)}{\left(1-\mathsf{\nu }\right)}{\mathsf{\delta }}^2$$

(7)

where for Al at ambient temperature the shear modulus µ = 25.4 GPa [38] and Poisson’s ratio ν = 0.345 [39].

The lattice parameters mismatch (δ) of L1₂ intermetallic compounds with the α-Al lattice (absolute value) was determined using Equation (8).

$$\mathsf{\delta }=\left|1-\frac{{\mathrm{a}}_{\mathrm{A}{\mathrm{l}}_3\mathrm{X}}}{{\mathrm{a}}_{\alpha -\mathrm{Al}}}\right|\times 100\%$$

(8)

where ${\mathrm{a}}_{\alpha -\mathrm{A}\mathrm{l}}$= 0.40496 nm [36] is the lattice parameter of α−Al.

The value of δ was much larger for Al₃Ti (δ = 2.04%) intermetallic compounds. For Al₃Zr, it was significantly smaller (δ = 0.75%). The value of the lattice parameters closest to α-Al was characteristic of Al₃Hf (δ = 0.04%) intermetallic compounds [36]. Using the known values of µ, ν and δ in Equation (7), for each of the systems, the values of ${\Delta \mathrm{F}}_{\mathrm{el}}$ were determined to be 43.4, 5.9 and 0.017 $\mathrm{MJ}·{\mathrm{m}}^{-3}$ for Al₃Ti, Al₃Zr and Al₃Hf, respectively. Thus, the elastic strain energy preventing the nucleation for Al₃Ti (L1₂) was 7.4 times higher compared to that for Al₃Zr (L1₂) and was incomparably higher than that for Al₃Hf (L1₂).

The critical value of C₀ from Equation (1) is necessary to overcome the elastic strain barrier. This value can be calculated by equalizing the values of ${\Delta \mathrm{F}}_{\mathrm{ch}}$ and ${\Delta \mathrm{F}}_{\mathrm{el}}$ (at peritectic temperature). For the Al–Ti alloys, the C₀ value was equal to 2.7 at.% Ti; it exceeded the C₀ values for Zr and Hf, equal to 0.15 at.% and 0.23 at.%, by factors of 18 and 11.7, respectively. Thus, the Al–Ti system at the peritectic temperature required a significantly (at least by one order of magnitude) greater supersaturation with TM compared to the Al–Zr and Al–Hf systems.

Previously [40,41], it was shown that higher cooling rates are required to achieve a greater driving force of nucleation, which depends on the value of C₀. Therefore, to achieve a high volume fraction of primary aluminides in the Al-Ti system, higher crystallization rates are required. As noted above, the rate of nucleation, in addition to ${\Delta \mathrm{F}}_{\mathrm{ch}}$ and ${\Delta \mathrm{F}}_{\mathrm{el}}$, strongly depend on the Q value (the activation energy of the TM diffusion process in Al). According to [42], the Q value for Ti in α-Al is 260 kJ/mol, i.e., higher than for Zr (242 kJ/mol) and Hf (241 kJ/mol). Hence, low values of the chemical driving force ${\Delta \mathrm{F}}_{\mathrm{ch}}$, as well as high values of the elastic strain energy ${\Delta \mathrm{F}}_{\mathrm{el}}$ and diffusion activation energy Q inhibit the nucleation process in the Al–Ti system. This fact explains the isolated cases (practically an absence) of the formation of Al₃Ti aluminides having a cubic lattice in the Al–Cu–Ti alloys synthesized in this work in comparison with the formation of Al₃Zr and Al₃Hf in the Al–Cu–Zr and Al–Cu–Hf systems. It is noteworthy that the Al-Zr and Al-Hf systems had similar properties (to each other) in terms of diffusion. However, Al-Zr required almost two times less supersaturation, and hence, lower cooling rates to overcome the energy barrier associated with ${\Delta \mathrm{F}}_{\mathrm{el}}$.

4. Conclusions

Our investigation of the conditions for the formation of metastable aluminides in Al–Cu–Ti, Al–Cu–Zr and Al–Cu–Hf alloys with a wide range of copper concentrations made it possible to draw the following conclusions:

(1)

One of the necessary conditions for the formation of (Al_1-xCu_x)₃Ti, (Al_1-xCu_x)₃Zr, (Al_1-xCu_x)₃Hf aluminides having L1₂ structure is the hypereutectic concentration of copper in the alloys. Ceteris paribus, an increase in the copper concentration (above the eutectic) in the alloys rapidly changed the ratio of stable and metastable trialuminides in favor of the formation of a larger amount of the latter. The sequence of dissolution in liquid aluminum of copper and then of TM (Zr or Hf) significantly accelerated the process of formation of metastable aluminides in the alloys.

(2)

Under identical synthesis conditions in the crystal lattice of metastable aluminides formed in experimental Al–Cu–Ti, Al–Cu–Zr and Al–Cu–Hf alloys, copper substituted up to 8, 10, and 13 at.% of the aluminum, respectively. The amount of aluminum substituted with copper in the crystal lattice of metastable aluminides was three times higher than in stable ones.

(3)

In the studied Al–Cu–Ti alloys, the low volume fraction of precipitating aluminides was the result of insufficient supersaturation of α-Al with titanium at the peritectic temperature (according to calculations based on the classical nucleation theory). The degree of supersaturation of the investigated alloys with titanium was 2 and 3.5 times, with zirconium being in the range of 26 and 53 times, and hafnium 174 to 219 times. As such, the difference in the degree of supersaturation of the alloys was 2–3 orders of magnitude.

(4)

Al_2.5Cu_0.5Zr and Al_2.5Cu_0.5Hf aluminides having the L1₂ structure with lattice parameters a = 0.40662(7) nm (Z3) and a = 0.40441(5) nm (H5), respectively, were synthesized. The high degrees of crystallographic and dimensional similarity between the lattices of aluminides and α-Al (0.4050 nm) suggest the applicability of these intermetallic compounds as modifiers of aluminum alloys.