INTRODUCTION

Great attention has been recently paid to multicomponent solid solutions: medium- and high-entropy alloys (HEAs). This is a fundamentally new class of materials that was first described in the papers [1, 2] in 2003–2004. A high-entropy alloy (HEA) is defined as a single-phase substitutional solid solution containing five or more principle elements, each ranging from 5 to 35 at.%, and stabilized by configuration entropy. High-entropy alloys possess excellent thermal stability, strength, high-temperature oxidation resistance, hardness, high-temperature creep resistance, and corrosion resistance. Solid solutions comprising four principal elements are classified as mediumentropy alloys; the high-entropy effect that stabilizes solid solutions is weaker in such alloys.

The difference in atomic radii and Young’s moduli of constituent elements of a multicomponent solid solution significantly distorts the lattice because atoms shift from ideal positions (lattice sites). The level of such distortions (type III microstresses) can be determined by X-ray diffraction, specifically by measuring the intensity ratio of two lines in the X-ray diffraction pattern that belong to the test and reference samples [3]. Such studies were conducted, for example, in [4] for the high-entropy CrMnFeCoNi alloy.

Type II microstresses are of considerable interest for HEAs: lattice microdistortions that are balanced within individual crystallites or their parts (mosaic blocks) and, along with the coherent scattering domain (CSD) sizes, broaden the diffraction lines compared to the reference sample [3]. Type II microstresses, ε = ∆d/d, where ∆d is the maximum deviation of the interplanar spacing for a given interference line from the average d value.

The objective of this research effort is to examine type II microstresses in multicomponent medium- and high-entropy bcc alloys with electron concentration ranging from 4.6 to 5.47 e/a using X-ray diffraction and analyze how type II microstresses influence the properties of multicomponent solid solutions, particularly Young’s modulus and hardness.

EXPERIMENTAL PROCEDURE

Multicomponent alloys were melted using a starting charge weighing more than 50 g in a MIFI-9 vacuum arc furnace with a nonconsumable tungsten electrode on a water-cooled copper hearth in a purified argon atmosphere. The starting components were at least 99.5 wt.% pure. The ingots were remelted six times.

An acid halide electrolyte with heating was used for electrolytic polishing of the samples. The samples were polished in galvanostatic mode (I = const), the current density being approximately 1 A/dm2.

For X-ray diffraction of multicomponent samples, we employed DRON-3M and DRON-4 diffractometers with a filtered Cu radiation source (Ni filter). The spectrum was cleared with a graphite monochromator placed on the diffracted beam [5]. X-ray diffraction spectra were taken in Bragg–Brentano θ–2θ geometry [6]. Discrete recording was used. Electronic control and data transmission units (BUIP-2, BUIP-3M) developed by the Kharkiv Polytechnic Institute (KhPI) were employed for point-by-point recording. The scan angle was ∆(2θ) = 0.05–0.1° and exposure at one point lasted 10–100 sec (depending on the intensity of diffraction peaks) in discrete recording mode.

The phase composition was analyzed with ASTM data sets (Powder Diffraction File of the International Centre for Diffraction Data]) [7]. To separate the matching diffraction peaks and calculate sublattice parameters, the modified New_Profile software was used (developed by KhPI) [8]. The X-ray diffraction data were preliminary processed in the following sequence: smooth and subtract the background (three different subtraction options are incorporated in the software) and separate Kα2 from the doublet of Kα1 and Kα2 lines (Rechinger method proposed as the main option [6]). The X-ray peak parameters were found with the method of selecting model functions that best described the experimental data. Bell-shaped functions such as parabolas (Cauchy) were the model ones. Complex patterns were processed with full-profile analysis.

Substructural characteristics (CSD sizes and microdistortion <ε>) were determined by fitting [9]. As the main fitting function, we chose the Cauchy equation, I(x) = I0/(1 + kx2), where I0 is the peak intensity and k = = 1/(ω/2)2, with ω being the width of a reflection at half intensity (refection half-width).

The samples were subjected to microindentation employing a Micron Gamma unit [10] at a load on the Berkovich diamond pyramid, with a rake of 65°, ranging from 0.98 to 2.94 N, under automatic loading and unloading. The accuracy of determining the force F was 10–3 N and indentation depth was h ± 2.5 nm. The hardness (HIT) and effective Young’s modulus (Er) were determined in compliance with international standard ISO 14577: 1-2015 [11].

EXPERIMENTAL RESULTS AND DISCUSSION

X-ray diffraction data indicate that all samples of medium-entropy (samples 1, 2, and 6) and high-entropy (samples 3, 4, and 5) alloys whose compositions are summarized in Table 1 are single-phase substitutional bcc solid solutions. The average electron concentration, Csd, of the alloys ranges between 4.6 and 5.47 e/a and corresponds to the electron concentration leading to a stable bcc lattice. According to [12], bcc solid solutions are formed within electron concentrations of 4.2 to 7.2 e/a. Figure 1 shows X-ray diffraction patterns for samples 4–6 as an example (Table 1).

TABLE 1. Average Electron Concentration (Csd) and Concentration (c) of Elements in Medium-Entropy (Samples 1, 2, and 6) and High-Entropy (Samples 3–5) Alloys
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Fig. 1.
figure 1

Fragments of X-ray diffraction spectra for multicomponent alloys 4 (1), 5 (2), and 6 (3) (numbers and chemical compositions of the alloys are indicated in Table 1)

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The lattice parameters (a), hardness (H), and Young’s modulus (E) found experimentally and calculated with the mixture rule (Vegard rule) and lattice microdistortions (<ε>), CSD sizes, and average lattice size mismatch (∆a/a)av of the test alloy samples are summarized in Table 2. The mixture rule calculation was performed as follows: amix = Σciai, Hmix = ΣciHi, and Emix = ΣciEi, where ci, ai, Hi, and Ei are the concentration, bcc lattice parameter, hardness, and Young’s modulus of a constituent element of the alloy (Tables 1 and 3). The Zr and Ti bcc lattice parameters were calculated in [13]. The average relative lattice size mismatch for the alloys was calculated as (∆a/a)av = Σ[ci│(ai – amix)│/amix].

TABLE 2. Lattice Parameters (a), Hardness (H), and Young’s Modulus (E) Determined Experimentally and Calculated with the Mixture Rule, and also Microdistortions (<ε>), CSD Sizes, and Average Relative Mismatch (∆a/a)av of Alloy Samples
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TABLE 3. Lattice Parameter (a), Hardness (H), Young’s Modulus (E) of Alloy Elements [13, 14]
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Table 2 shows that even a small change in the quantitative chemical composition of medium-entropy samples 2 and 6 (44.97 and 42.54 at.% Nb, 16.67 and 18.72 at.% W, 17.47 and 19.82 at.% Mo, 20.89 and 18.92 at.% Ta) leads to a noticeable change in the lattice parameter a (0.32035 and 0.32311 nm), microdistortion <ε> (0.166 and 0.178%), CSD size (47 and 57 nm), microhardness H (6.8 and 6.3 GPa), and Young’s modulus E (160 and 157 GPa).

High microdistortions <ε> (0.21 and 0.22%) and small CSD sizes (23 and 26 nm) are exhibited by samples 3 and 4 that include both atoms with a relatively high lattice parameter (W, Ta, Ti) and atoms with a relatively small lattice parameter (Cr, V); these are alloys with great average relative size mismatch (∆a/a)av: 0.0299 and 0.0437.

Table 2 shows that experimental lattice microdistortions <ε> are almost one order of magnitude lower than the calculated lattice size mismatch (∆a/a)av used in [15] to assess hardening ∆H = kH(∆a/a)avGav (here Gav is the alloy shear modulus and kH is a coefficient of 1.4–1.7). This difference is explained by the fact that (∆a/a)av describes only the lattice size mismatch for elements contained in a multicomponent solid solution. In addition, (∆a/a)av is calculated with tabulated values and <ε> is the actual average lattice strain, where the actual atomic sizes may substantially differ from tabulated ones because of distinct bonding forces between the atoms. For the same reason, the experimental E values are also substantially lower than the ones calculated with the mixture rule, Emix (Table 3).

Figure 2 shows dependences of the experimental lattice parameter a (Fig. 2a), Young’s modulus E, and hardness H (Fig. 2b) on the average electron concentration Csd and dependences of Young’s modulus E and hardness H on the lattice parameter a (Fig. 2c) for test alloys 1–6, whose chemical composition is provided in Table 1. With higher electron concentration Csd, the lattice parameter a clearly tends to decrease and Young’s modulus E and hardness H of alloys 1–6 tend to increase (Fig. 2a, b). Young’s modulus and hardness decrease with increasing lattice parameter (Fig. 2c). Note that these dependences are consistent with those described previously in [12].

Fig. 2.
figure 2

Lattice parameter a (a), Young’s modulus E (1), and hardness H (2) versus average electron concentration Csd (b) and lattice parameter a (c) for alloys 1–6 (Tables 1 and 2)

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Greater lattice microdistortions <ε> lead to higher hardness of the alloys (Fig. 3a). Note that the H/E ratio for the test alloys ranges from 0.04 to 0.06 and, in accordance with [16], is typical of HEAs.

Fig. 3.
figure 3

Hardness H (a), solid-solution hardening ∆H = Hexp – Hmix (1), and εE/10 (2) (b) versus lattice microdistortions for alloys 1–6 (Tables 1 and 2)

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Table 2 indicates that the calculated hardness Hmix of all single-phase multicomponent alloys significantly exceeds experimental H, being evident of substantial solid-solution hardening of the alloys. The difference, ∆H = = H – Hmix, that characterizes solid-solution hardening ranges from 2.9 to 6.4 GPa for the test alloys and increases with greater lattice microdistortions ε (Fig. 3a). For different alloys and thus different <ε> and E, εE can be easily calculated using Table 3. The dependences of ∆H and εE on ε provided in Fig. 3b deserve special attention. It is obvious that

$$ \varDelta H=\upalpha \upvarepsilon E, $$
(1)

then

$$ H={H}_{\mathrm{mix}}+\varDelta H={H}_{\mathrm{mix}}+\upalpha \upvarepsilon E. $$
(2)

According to calculations, average αav defined from experimental H, E, and ε equals approximately 22 for the test alloys. Deviations of αi for individual alloys from average αav can be associated with the accuracy of measured H and E (standard deviation is ±5% [11]) and with calculated Hmix since the tabulated hardness values of constituent elements of the alloy (Table 3) can differ from their actual hardness. Moreover, the change in the CSD size can contribute to hardening.

A substantial difference in the atomic size mismatch, calculated from tabulated lattice parameters (or atomic radii) for pure elements, and microdistortions ε was described above. This difference seems to be quite natural because distortions introduced with a further distance from the atom of a doping element sharply decrease by law 1/r2 and ε characterizes not maximum possible but average changes in the interplanar spacing. It is generally believed [17, 18] that distortions of solid-solution lattices do not noticeably broaden the X-ray lines (and, hence, ε) but reduce their intensity. Accordingly, the root-mean-square statistical deviations can be used for measuring distortions of solid-solution lattices [3, 18].

Nevertheless, the obtained data suggest that ε, if defined precisely (from X-ray line width), can be used for measuring distortions of solid-solution lattices. The same conclusion was made in [19], indicating that solidsolution hardening is characterized by a linear dependence between the root-mean-square atom displacements and distortion ε.

Conclusions

The multicomponent alloys with an average electron concentration ranging from 4.6 to 5.47 e/a, produced by vacuum arc melting, represent substitutional single-phase bcc solid solutions. Even a small change in the alloy quantitative chemical composition (44.97 and 42.54 at.% Nb, 16.67 and 18.72 at.% W, 17.47 and 19.82 at.% Mo, 20.89 and 18.92 at.% Ta) leads to a noticeable change in the lattice parameter a (0.32035 and 0.32311 nm), lattice microdistortions <ε> (0.166 and 0.178%), CSD sizes (47 and 57 nm), hardness H (6.8 and 6.3 GPa), and Young’s modulus E (160 and 157 GPa).

High microdistortions and small CSD sizes are shown by HEAs that include atoms with both a large lattice parameter (W, Ta, Ti) and a small lattice parameter (Cr, V), i.e., by alloys with relatively high average size mismatch (<ε> is 0.21 and 0.22%, CSD size is 23 and 26 nm, and (∆a/a)av is 0.0299 and 0.0437 for alloys 3 and 4).

Higher electron concentration Csd in the test multicomponent alloys increases Young’s modulus and hardness and decreases the lattice parameter. Higher hardness and Young’s modulus are also observed when lattice microdistortions become greater.

The experimental hardness H values significantly exceed those calculated with the mixture rule, Hmix, being indicative of substantial solid-solution hardening (∆H = H – Hmix for the test alloys ranges from 2.9 to 6.4 GPa). A relationship has been established between the magnitude of solid-solution hardening, lattice microdistortions, and Young’s modulus: ∆H = αεE, with αav ≈ 22. Deviations of coefficient αi for individual alloys from average αav can be associated with both the accuracy of measured H and Young’s modulus E and calculated Hmix, as well as with potential failure to incorporate changes in the CSD size.

The data testify that lattice microdistortions (type II microstresses) can, if determined precisely, be used for measuring the distortion of solid-solution lattices and for assessing the magnitude of solid-solution hardening.