Deformation Mechanisms and Processing Maps for High Entropy Alloys (Presentation of Processing Maps in Terms of Zener–Hollomon Parameter): Review

Abstract

In this review paper, the hot compressive deformation mechanisms and processing maps of high-entropy alloys (HEAs) with different chemical compositions and crystal structures are analyzed. The stress exponent (n₁) values measured from the series of compression tests for the HEAs performed at different temperatures and strain rates are distributed between 3 and 35, and they are most populated between 3 and 7. Power law breakdown (PLB) is found to typically occur at T/T_m ≤ 0.6 (where T is the testing temperature and T_m is the melting temperature). In Al_xCrMnFeCoNi (x = 0–1) and Al_xCrFeCoNi (x = 0–1) HEAs, n₁ tends to decrease as the concentration of Al increases, suggesting that Al acts as a solute atom that exerts a drag force on dislocation slip motion at high temperatures. The values of activation energy for plastic flow (Q_c) for the HEAs are most populated in the range between 300 and 400 kJ/mol. These values are close to the activation energy of the tracer diffusivity of elements in the HEAs ranging between 240 and 408 kJ/mol. The power dissipation efficiency $\left(\eta \right)$ of the HEAs is shown to follow a single equation, which is uniquely related to n₁. Flow instability for the HEAs is shown to occur near n₁ = 7, implying that the onset of flow instability occurs at the transition from power law creep to PLB. Processing maps for the HEAs are demonstrated to be represented by plotting $\eta$ as a function of the Zener–Hollomon parameter (Z = $\exp \left(\frac{Q_{\mathrm{c}}}{RT}\right)$, where R is the gas constant). Flow stability prevails at Z ≤ 10¹² s⁻¹, while flow instability does at Z ≥ 3 × 10¹⁴ s⁻¹.

1. Introduction

High-entropy alloys (HEAs) are a new family of solid-solution alloys made of four or more principal alloying elements alloyed in equiatomic or near-equiatomic concentrations (with each constituent element having a concentration from 5 to 35 atomic percent (at.%)) [1,2]. The first approach for designing an HEA was to obtain a single-phase solid solution by maximizing the mixing configurational entropy, but later, new design approaches involving multiple phases and/or intermetallics were explored [1,2,3].

Thermomechanical working at high temperatures is necessary not only to form and shape materials into components but also to produce the desired microstructures and properties of products [4,5]. For this reason, the high-temperature deformation mechanisms and hot workability of HEAs have been studied [6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100]. Finding the constitutive equations that mathematically depict the material response to the applied hot deformation conditions can be useful in predicting the flow stress or strain rates and identifying the rate-determining deformation mechanisms at different temperatures/strain rates and microstructural parameters. Characterization of hot workability using a processing map is important for the optimization of hot working conditions of materials and fabrication of defect-free components [101]. If a material is processed under unstable flow conditions, adiabatic shear banding or cracking can occur, and if a material is processed under optimal conditions, superior microstructure and mechanical properties can be obtained.

In this work, we have reviewed and analyzed the hot compression data of HEAs available in the literature to elucidate their deformation mechanisms and optimal hot working conditions.

2. History and Materials

The first paper reporting the hot compression behavior of HEAs was available in 2011, and the first studied HEAs were Nb₂₅Mo₂₅Ta₂₅W₂₅ and V₂₀Nb₂₀Mo₂₀Ta₂₀W₂₀ alloys [6], where the microstructural change during hot compression at temperatures of 1073–1873 K and at a strain rate of 10⁻³ s⁻¹ was examined. The papers published in the literature from 2011 to the present can be categorized into three groups (Figure 1a). The first group of papers reports the hot compressive deformation data of the HEAs in the limited temperature and strain rate range. The second group of papers reports the hot compression data of the HEAs in a wide range of temperature and strain rates, but processing maps are not constructed. The third group of papers reports the hot compression data as well as the processing maps of the HEAs. The number of publications increases almost exponentially with time, indicating that attention to this academic and engineering field has rapidly increased.

The HEA materials studied by hot compression can be classified into three groups (Figure 1b). The first material group is the Cr-Mn-Fe-Co-Ni series HEAs containing Al, Sn, Zr, Sn, C, and N [9,20,23,27,34,37,38,42,45,60,62,63,64]; the second material group is the Cr-Fe-Co-Ni series HEAs containing Zr, Ta, Nb, Mo, Cu, C, and N [13,25,51,53,72,84,92,96,97,100]; and the third material group is the other composition HEAs, including the materials of TiVNbMoTa, Mn₅Co₂₅Fe₂₅Ni₂₅Ti₂₀, MnFeCoNiCu, etc. [8,11,12,22,29,32,33,40,41,47,52,56,68,71,74,79,91,93,95,99]. Information regarding the chemical compositions of the HEAs, grain sizes, crystal structure, types of phases, temperature and strain rate ranges for hot compression tests are provided in

Table 1. Information regarding the chemical compositions of the HEAs, grain sizes, crystal structure, types of phases, temperature and strain rate ranges for hot compression tests.

Composition	Grain Size (μm)	Major Phase	Minor Phase	T Range (K)	$\dot{\mathit{\epsilon }}$ Range (s−1)	Tm (K)	n1	Qc (kJ/mol)	Material Group
CrMnFeCoNi [9]	300	FCC	-	873–1373	10−4–10−2	1801	4.9–12.1	352.2 (εSS)	1
CrMnFeCoNi [23]		FCC	-	1073–1273	10−3–1	1801	5–8.5	350.5 (ε0.9)	1
CrMnFeCoNi [27]	419	FCC	-	1023–1323	10−3–10	1801	5.8–22.8	410.9 (ε0.6)	1
CrMnFeCoNi [45]	12.8	FCC	-	1023–1423	10−3–10	1801	4.7–11.1	311.9 (ε0.7)	1
Al0.5CrMnFeCoNi [34]	547	FCC	BCC	1023–1323	10−3–10	1722	4.2–10.8	343 (ε0.6)	1
Al0.5CrMnFeCoNi [62]	547	FCC	BCC	1423&1473	10−3–10	1722	3.6–4.1	570.5 (ε0.6)	1
Al0.7CrMnFeCoNi [63]		FCC	BCC	1173–1373	10−3–10	1695	3.2–5.3	309.5 (ε0.7)	1
AlCrMnFeCoNi [42]	118	BCC	-	1173–1373	10−3–10	1657	3.4–5.3	336.3 (ε0.5)	1
CrMnFeCoNiSn0.5 [64]	>100	FCC	L21	1023–1248	10−3–10	1683	4.6–9.4	322.2 (ε0.7)	1
CrMnFeCoNiC0.5 [37]	600	FCC	-	973–1273	10−3–1	1801	8–26	479 (ε0.6)	1
CrMnFeCoNi-1 at.%C [20]	125	FCC	M7C3	973–1273	10−3–1	1801	8.4–33	605.1 (ε0.6)	1
(CrMnFeCoNi)95C5 [38]	50	FCC	M23C6	1073&1273	10−3–10−1	1801	6.3–10.5	424.1 (ε0.6)	1
Cr25Mn15Fe10Co35Ni15 [60]	190	FCC	-	1123–1273	10−3–10−1	1801	5–6.3	310.4 (ε1.0)	1
CrFeCoNi [96]	>100	FCC	-	1173–1373	10−3–10−1	1872	5.8–7	390.1 (ε0.8)	2
Al0.3CrFeCoNi [25]	>100	FCC	-	1023–1223	5 × 10−4–10−1	1806	5.2–9.7	361.4 (ε0.7)	2
Al0.3CrFeCoNi [92]	52	FCC	-	1023–1423	10−3–10	1806	4.5–11.1	320.3 (ε0.7)	2
Al0.5CrFeCoNi [13]	>100	FCC	BCC	1173–1473	10−3–1	1767	4.4–5.8	296.8 (ε0.8)	2
Al0.5CrFeCoNi [13]	>100	FCC	BCC	1223–1373	10−3–1	1767	4.7–4.8	304 (ε0.8)	2
Al0.5CrFeCoNi [84]	>100	FCC	BCC	1173–1373	1.3 × 10−3–10−1	1767	4.7–6.3	361.8 (ε0.7)	2
Al0.6CrFeCoNi [53]	>100	FCC	BCC	1173–1473	10−3–1	1749	4.5–6.4	380 (ε0.6)	2
Al0.6CrFeCoNi [97]	>100	FCC	BCC	1223–1373	10−3–1	1749	3.7–4.5	287.5 (ε0.8)	2
Al0.7CrFeCoNi [72]	>100	FCC	BCC	1173–1373	10−3–10−1	1732	4.1–5.1	375.5 (ε0.6)	2
Al0.7CrFeCoNi [100]	>100	FCC	BCC	1073–1373	10−2–10	1732	4.7–6.5	321.3 (ε0.5)	2
AlCrFeCoNi [51]	194	BCC1	BCC2	1073–1373	10−3–1	1684	3.3–4.2	180.6 (ε0.8)	2
(CrFeCoNi)90Zr10 [54]	>100	FCC	Ni2Zr + Ni7Zr2	1073–1323	10−3–10	1897	3.9–6.6	327.8 (ε0.6)	2
CrFeCoNiTa0.395 [69]	>100	FCC	Laves	1073–1373	10−3–1	1999	4.1–6.9	383.5 (ε0.6)	2
CrFeCoNiNb0.25 [89]	>100	FCC	Laves	1073–1273	10−2–10	1923	5–9.2	431.9 (ε0.8)	2
CrFeCoNiMo0.2 [18]		FCC	-	973–1373	10−3–1	1921	3.4–20	491.2 (ε0.6)	2
CrFeCoNiCu (as-cast) [90]	>100	FCC1	FCC2	1073–1173	10−2–1	1769	3.4–9.7	374.2 (ε0.4)	2
CrFeCoNiCu (solid-solutionized) [90]	>100	FCC1	FCC2	1073–1173	10−2–1	1769	-	-	2
CrFeCoNiCu1.2 [66]	>100	FCC1	FCC2	973–1123	10−3–1	1753	9.4–25	394.9 (ε0.3)	2
Cr25Fe15Co45Ni15–0.1C [61]	>100	FCC	-	1123–1273	10−3–10−1	1828	4.8–7.8	479.6 (ε0.5)	2
Cr25Fe15Co45Ni15–0.05N [61]	>100	FCC	-	1123–1273	10−3–10−1	1828	4.7–6.2	308.5 (ε0.5)	2
Cr10Mn40Fe40Co10–3.3 at.%C [29]	225	FCC	-	1173–1373	10−2–1	1727	5.1–10	466.2 (ε0.6)	3
MnFeCoNiCu [32]	>100	FCC	-	1123–1323	10−3–10	1637	3.2–12	510.2 (ε0.7)	3
CrMn2FeNi2Cu [95]		FCC1	FCC2	873–1273	10−3–10−1	1692	7.2–15.5	363.3 (ε0.7)	3
TiFeCoNiCu [12]	>100	FCC1	FCC2 + BCC + Ti2(Ni,Co)	1073–1273	10−3–10−1	1721	2.2–4.5	426.4 (ε0.7)	3
AlCrFeCoNi2.1 [22]		FCC	BCC	1073–1373	10−3–10	1692	3.3–5.2	295.5 (ε0.6)	3
AlCrFeCoNi2.1 [47]		FCC	BCC	1073–1473	10−3–10	1692	3–5.9	332.9 (ε0.6)	3
AlCrFeNiCu [56]	>100	FCC	BCC	1173–1323	10−3–1	1602	3.3–4.3	154.4 (ε0.6)	3
AlFeCoNiCu [41]	>100	FCC	BCC	1173–1373	10−1–10	1520	5.6–7.2	328.3 (ε0.5)	3
Al5Ti3Cr15Mn10(FeNi)67 [91]	>100	FCC	BCC	1053–1373	10−2–10−1	1769	4.1–5.3	375.5 (ε0.6)	3
Mn5Co25Fe25Ni25Ti20 [52]		FCC	BCC + Ti2Ni + Ti2Co	1073–1273	10−3–1	1791	2.8–4	305.2 (ε0.6)	3
TiZrNbMoHf [11]	>100	BCC	-	1073–1473	10−3–10−1	2444	2.9–6.4	431.4 (ε0.6)	3
TiZrNbMoHf [40]	>100	BCC	-	1373–1523	10−3–5 × 10−1	2444	3.2–5.1	290.7 (ε0.6)	3
Ti29Zr24Nb23Hf24 [74]	361	BCC	-	973–1373	10−3–10	2308	4.1–10	234.6 (ε0.2)	3
TiZrNbHfTa [33]	140	BCC	-	1273–1473	10−4–10−2	2523	2.7–3.3	244.4 (ε0.6)	3
AlCrFeNi [93]	>100	BCC(A2)	BCC(B2)	1073–1373	10−3–1	1663	4.1–8.3	370 (ε0.6)	3
VNbMoTa [68]	>100	BCC	-	1173–1373	10−3–10−1	2780	14.5–19.5	454.2 (ε0.5)	3
AlTiVNb2 [79]	>100	BCC(B2)	-	1273–1473	10−3–10−1	2111	4.1–6.7	391.4 (ε0.6)	3
AlTi3VZr1.5Nb [99]	>100	BCC(B2)	-	1373–1523	10−3–1	1984	2.9–3	210.6 (ε0.6)	3
AlCrFeCoNiCu [8]	>100	BCC	FCC	973–1303	10−3–10−1	1630	2.9–4.4	300.7 (ε0.7)	3
TiVNbMoTa [71]	0.58	BCC	FCC	1373–1573	5 × 10−4–5 × 10−1	2612	2.2–3	291.1 (ε0.3)	3

. Most of the HEAs studied for hot compression are as-cast or heat-treated (homogenized) cast with coarse grain sizes. Among the Cr-Mn-Fe-Co-Ni series HEAs and Cr-Fe-Co-Ni series HEAs, Al_xCoCrFeNi and Al_xCoCrFeMnNi (x = 0–1) HEAs [9,13,23,25,27,34,42,45,51,53,62,63,71,84,92,96,97,100] have been the most studied, where the addition of Al can facilitate the formation of BCC phase from the FCC matrix. At low Al levels corresponding to x = 0–0.4 or 0.5, the alloys have a single FCC phase, but with a further increase in Al content in the range of x = 0.4 or 0.5–0.9, both FCC and BCC phases coexist, and at Al addition beyond x = 0.9–0.95, a BCC single phase is obtained [102,103].

HEAs with a single FCC phase or FCC (major) + BCC (minor) phases [9,13,18,22,23,25,27,29,32,34,37,41,45,47,52,53,56,60,61,62,63,72,84,91,92,96,97,100] have been most extensively studied for hot compression tests (Figure 1c). HEAs with a single BCC phase or BCC (major) + FCC (minor) phases have also been popularly studied [8,11,33,40,42,51,68,71,74,79,93,99]. HEAs with FCC1 and FCC2 or FCC+ intermetallic compound (IMC) precipitates have been recently studied [12,20,38,54,64,66,70,89,90,95].

3. Deformation Mechanisms

The raw data (true stress–true strain curves) in the second and third groups of papers, where a series of hot compression tests were systematically carried out over wide temperature and strain rate ranges, were digitally extracted from the published papers using a software. For example, the true stress–true strain curves for Al_0.7CrMnFeCoNi HEA [62] at different temperatures and different strain rates are shown in Figure 2a–d.

The hyperbolic sine Garofalo equation has been widely used to describe the steady-state relationship between the flow stress, temperature, and strain rate ($\dot{\epsilon }$) over a wide range of temperatures and strain rates where power law creep and power law breakdown (PLB) govern plastic flow [4,104], which is expressed as:

$$\dot{\epsilon }=A \sin \mathrm{h}{\left(\alpha \sigma \right)}^n\exp \left(-\frac{Q_{\mathrm{c}}}{RT}\right)$$

(1)

where A and α are the experimentally determined material constants, n is the stress exponent, and Q_c is the activation energy for plastic flow. The hyperbolic sine function is mathematically reduced to the equation describing power law creep at low stresses (Equation (2)) and to the equation describing power law breakdown (PLB) at high stresses (Equation (3)):

$$\dot{\epsilon }=A_1 {\left(\sigma \right)}^{n_1}\exp \left(-\frac{Q_{\mathrm{c}}}{RT}\right)$$

(2)

$$\dot{\epsilon }=A_2 \exp \left(\beta \sigma \right) \exp \left(-\frac{Q_{\mathrm{c}}}{RT}\right)$$

(3)

where $A_1$, $A_2$, and β are the material constants and n₁ is the stress exponent, which is ideally equal to n, but can be different). The constants α in Equation (1), n₁ in Equation (2), and β in Equation (3) can be related by β = αn₁ [4,104].

Figure 3a,b shows the plots of $\log \dot{\epsilon }-\log \sigma$ and $\log \dot{\epsilon }-\sigma$ at a given strain of 0.7 for the Al_0.7CrMnFeCoNi HEA, where steady-state (SS) flow is attained at almost all temperatures and strain rates. The slopes of the regression lines of the $\log \dot{\epsilon }-\log \sigma$ and $\log \dot{\epsilon }-\sigma$ curves are used to determine the values of n₁ and β at each temperature, respectively. Figure 3c shows the plot of $\log \dot{\epsilon }-\log \left(\sinh \left(\alpha \sigma \right)\right)$, where $\alpha$ is the average of the $\alpha \left(=\beta /n_1\right)$ values measured at all temperatures. According to Equation (1), the slopes (=${\left[\frac{\partial \log \dot{\epsilon }}{\partial \log \left(\sinh \left(\alpha \sigma \right)\right)}\right]}_T)$ of the $\log \dot{\epsilon }-\log \left(\sinh \left(\alpha \sigma \right)\right)$ curves represent the n value at each temperature. Figure 3d shows the plot of $\log \left(\sinh \left(\alpha \sigma \right)\right)-\frac{1000}{T}$, of which the slope (=${\left[\frac{\partial \ln \left(\sinh \left(\alpha \sigma \right)\right)}{\partial \left(\frac{1000}{T}\right)}\right]}_{\dot{\epsilon }})$ represents the s value at each strain rate. The Q_c value can be calculated by using the average (N) of the n values measured at different temperatures and the average (S) of the s values measured at different strain rates as follows:

$$Q_{\mathrm{c}}=RNS$$

(4)

It is well-known that a value of n₁ represents a specific deformation mechanism; Haper–Dorn creep and diffusional creep are associated with n₁ = 1 [105,106,107,108], grain boundary sliding is associated with n₁ = 2 [109,110], solute drag creep is associated with n₁ = 3 [111], dislocation climb creep is associated with 5–7 [112,113], and power law breakdown occurs when n₁ > 7 [4]. Jeong and Kim [62] analyzed the tensile, compressive, and creep data of CrMnFeCoNi and Al_0.5CrMnFeCoNi HEAs with various grain sizes and proposed the constitutive equations that can quantitatively predict their flow stresses as a function of strain rate, temperature, and grain size (

Table 2. The constitutive equations of deformation mechanisms in HEAs.

Creep Process	Equation	ki Value
		Ruano et al. [113]	CrMnFeCoNi [62]	Al0.5CrMnFeCoNi [62]
Diffusional creep
Nabarro–Herring [105,106]	${\dot{\epsilon }}_1=k_1\left(D_{\mathrm{L}}/d^2\right)\left(E{b}^3/kT\right)\left(\sigma /E\right)$	14	14	14
Coble [107]	${\dot{\epsilon }}_2=k_2\left(D_{\mathrm{gb}}b/d^3\right)\left(E{b}^3/kT\right)\left(\sigma /E\right)$	50	50	50
Grain boundary sliding (GBS)
Lattice-diffusion-controlled [109]	${\dot{\epsilon }}_3=k_3\left(D_{\mathrm{L}}/d^2\right){\left(\sigma /E\right)}^2$	6.4 × 109	3.1 × 108	6.7 × 108
Pipe-diffusion-controlled [110]	${\dot{\epsilon }}_4=k_4\alpha \left(D_{\mathrm{p}}/d^2\right){\left(\sigma /E\right)}^4$	3.2 × 1011	1.6 × 1010	3.4 × 1010
Grain-boundary-diffusion-controlled [109]	${\dot{\epsilon }}_5=k_5\left(D_{\mathrm{gb}}b/d^3\right){\left(\sigma /E\right)}^2$	5.6 × 108	1.9 × 107	5.9 × 107
Slip creep
Harper–Dorn [108]	${\dot{\epsilon }}_6=k_6\left(D_{\mathrm{L}}/b^2\right)\left(E{b}^3/kT\right)\left(\sigma /E\right)$	1.7 × 10−11	1.7 × 10−11	1.7 × 10−11
Lattice-diffusion-controlled dislocation climb creep [112]	${\dot{\epsilon }}_7=k_7\left(D_{\mathrm{L}}/b^2\right){\left(\sigma /E\right)}^5$	1 × 1011	2.6 × 109	1.5 × 109
Pipe-diffusion-controlled dislocation climb creep [112]	${\dot{\epsilon }}_8=k_8\left(D_{\mathrm{p}}/b^2\right){\left(\sigma /E\right)}^7$	5 × 1012	1.1 × 109	3.9 × 109
Solute drag creep [111]	${\dot{\epsilon }}_{\mathrm{CJ}}=\frac{2\gamma \tilde{D}kT}{X_s{X}_a{e}^{2}E{b}^5}{\left[2\left(1+\nu \right)\right]}^4\left(\frac{1-\nu }{1+\nu }\right){\left(\frac{\sigma }{E}\right)}^3$

). Figure 4a,b shows the n₁ values of HEAs with different crystal structures as a function of temperature (T) and a homologous temperature (T/T_m), respectively, where T_m is the melting temperature of the HEAs. As the T_m values of most of the HEAs are not available in the literature, we calculated the T_m values of the HEAs using the relation $T_m={\displaystyle \sum }_{i=1}^n{c}_i{\left(T_m\right)}_i$ [114], where $c_i$ is the atomic percentage of the ith component and ${\left(T_m\right)}_i$ is the melting point of the ith component of the alloy. The plots in Figure 4a,b show that the hot compression tests for the HEAs have been conducted in the temperature range between 873 K and 1573 K, corresponding to the T/T_m range between 0.4 and 0.9. The n₁ value is distributed between 3 and 35, and the n₁ values are most populated between 3 and 7. As T and T/T_m decrease, n₁ tends to increase, and at T/T_m ≤ 0.6, the number of the data associated with n₁ ≥ 7 sharply increases. In the T/T_m range between 0.6 and 0.9, the fraction of the data associated with n₁ ≥ 7 is 0.17, while in the T/T_m range between 0.4 and 0.6, the fraction of n₁ ≥ 7 is 0.66. This result indicates that PLB dominates plastic flow below 0.6 T/T_m, while power law creep dominates plastic flow above 0.6 T/T_m. It should be noted that the HEAs that show the lowest n₁ of ~3 are the Al_xCoCrFeNi and Al_xCoCrFeMnNi HEAs with BCC or BCC + FCC (minor) phases, containing an Al element. Figure 4c shows that the n values of the HEAs are slightly smaller than their n₁ values. They mostly range between 2.5 and 5. The n tends to increase as T/T_m decreases, but it is not as sensitive as n₁ to T/T_m variation.

Figure 5a,b shows the $\log \dot{\epsilon }-\log \sigma$ curves for the Al_xCrMnFeCoNi (x = 0–1) [9,27,34,42,63] and Al_xCrFeCoNi (x = 0–1) HEAs [13,25,51,53,72,84,92,96] at a given temperature of 1173 K. The n₁ value decreases from 5 to ~3 as x increases. This result indicates that as the amount of BCC phase (rich with Al) increases, the characteristics of viscous glide creep associated with n₁~3 become more pronounced. Jeong and Kim [42] analyzed the deformation behavior of the AlCrMnFeCoNi HEA and found that aluminum, which is largest in size among the constituent elements [42], acts as a solute that causes solute drag creep and showed that the solute drag creep model proposed by Hong and Weertman [111] for conventional metals can quantitatively explain the deformation behavior of the AlCrMnFeCoNi HEA with reasonable assumptions on the diffusivity of Al in the HEA. From the plots in Figure 5a,b, it should also be noted that flow stress tends to decrease the added amount of Al increases especially at low strain rates, indicating that the BCC phase deforming under solute drag creep is weaker than the FCC phase deforming under dislocation climb creep.

Deformation mechanism maps represent the dominant deformation mechanism for a given metallic material under different conditions. Figure 6a,b shows the deformation mechanism maps as a function of strain rate (10⁻⁵ to 10 s⁻¹) and temperature (873 and 1573 K) at a given (coarse) grain size of 100 μm for CoCrFeMnNi and Al_0.5CoCrFeMnNi. On the maps, the data of the CoCrFeMnNi and Al_0.5CoCrFeMnNi HEAs are loaded. For the CoCrFeMnNi HEA, at high temperatures, dislocation climb creep controlled by lattice diffusivity (D_L) governs plastic flow, but when the temperature is low, the rate-controlling mechanism changes to dislocation climb creep controlled by D_p (or PLB). As the strain rate increases, the region associated with D_p-controlled dislocation climb creep (or PLB) expands to a higher temperature. For the Al_0.5CoCrFeMnNi, solute drag creep appears in the bottom right corner of the map. Jeong and Kim [62] showed that when the grain size is sufficiently small, grain boundary sliding mechanism can play a more important role than solute drag creep in the Al_0.5CoCrFeMnNi HEA.

Figure 7a shows the Q_c values of HEAs calculated using Equation (4). The Q_c value is in the range between 150 and 600 kJ/mol, and the data distribution is most populated in the range between 300 and 400 kJ/mol. The activation energy of the tracer diffusivity of elements in the HEAs ranges between 240 and 408 kJ/mol [115,116], implying that the activation energy of plastic flow for the HEA is related to the atomic diffusivity of elements constituting the HEAs. Figure 7b shows the relation between n₁ and Q_c for the HEAs. A smaller Q_c is obtained at smaller n₁, and this is more apparent near n₁~3, where solute drag creep governs the deformation mechanism. It is worthwhile to note that the activation energy for the solute diffusion (Q_solute) of magnesium in aluminum (136 kJ mol⁻¹) is lower than the activation energy for self-diffusion in pure aluminum (142 kJ/mol) [117].

4. Processing Maps

A processing map, which is useful in finding the optimal condition for hot forging or extrusion, is composed of a power dissipation map and a flow instability map. According to Prasad et al. [101], the total power, P, absorbed in a material is divided into the power dissipation content (G), which represents the power dissipated by plastic deformation giving rise to a temperature increase in the workpiece and the power dissipation co-content (J), which represents the power dissipated by a change in its microstructure, such as dynamic recovery and dynamic recrystallization [101].

$$P=\mathsf{\sigma }\dot{\epsilon }=G+J={{\displaystyle \int }}_0^{\dot{\epsilon }}\sigma d\dot{\epsilon }+{{\displaystyle \int }}_0^{\sigma }\dot{\epsilon }d\sigma$$

(5)

The efficiency of power dissipation, η, which represents the power dissipation efficiency due to a change in microstructure during plastic flow, is defined as [101]:

$$\eta =\frac{J}{J_{max}}=2\left(1-\frac{1}{\sigma \dot{\epsilon }}{{\displaystyle \int }}_0^{\dot{\epsilon }}\sigma d\dot{\epsilon }\right)$$

(6)

where J_max is the maximum J value (=P/2).

The strain rate sensitivity, m, which is equal to 1/n₁, can be calculated by:

$$m=\frac{1}{n_1}=\left[\frac{\partial ln\sigma }{\partial ln\dot{\epsilon }}\right]$$

(7)

When m is assumed to be constant over the investigated strain rate range (as assumed by Prasad et al. [101]), $\eta =\frac{2m}{m+1}$, but when m is not constant (as considered by Murty et al. [118]), η can be directly determined from Equation (6) by calculating ${\int }_0^{\dot{\epsilon }}\sigma d\dot{\epsilon }$ through the numerical integration procedure, using Equation (8):

$${{\displaystyle \int }}_0^{\dot{\epsilon }}\sigma d\dot{\epsilon }={{\displaystyle \int }}_0^{{\dot{\epsilon }}_{min}}\sigma d\dot{\epsilon }+{{\displaystyle \int }}_{{\dot{\epsilon }}_{min}}^{\dot{\epsilon }}\sigma d\dot{\epsilon }={\left(\frac{\sigma \dot{\epsilon }}{m+1}\right)}_{{\dot{\epsilon }}_{min}}+ {{\displaystyle \int }}_{{\dot{\epsilon }}_{min}}^{\dot{\epsilon }}\sigma d\dot{\epsilon }$$

(8)

In drawing the flow instability map, Ziegler’s plastic flow theory is used, and according to Murty et al. [119] (when m is not a constant),

$$\xi =2m-\eta <0$$

(9)

When ξ is negative, deformation in the material is predicted to be unstable, such that localized flow, adiabatic shear banding, or cracking can take place.

Kim and Jeong [120] suggested an empirical equation for $\eta$ by analyzing the behavior of $\eta$ values of many metals calculated by following the numerical method proposed by Murty (Equation (8)) as a function of $n_1$:

$$\eta =\left[\frac{{10}^4}{{\left(n_1-1\right)}^2}\right]\left(\frac{2}{n_1+1}\right)+\left[\frac{\tanh \left(n_1-5\right)+1}{2}·\frac{{\left(n_1-1\right)}^{1.5}}{{\left(n_1-1\right)}^{1.5}+{10}^2}\right]\left(\frac{2}{n_1}·\frac{e^{n_1}-1}{e^{n_1}}\right)$$

(10)

By using this equation, the $\eta$ value can be easily obtained once n₁ is known without numerically solving Equation (8). Kim and Jeong [121] also presented a simple form of the flow instability criterion based on the observation that unstable flow occurs in many metals when $n_1$ is larger than 7 (i.e. when PLB governs plastic flow):

$$\xi =7-n_1<0 \mathrm{or} \eta <0.285$$

(11)

Unlike in the procedure for determining $n_1$ for calculating $Q_{\mathrm{c}}$ (Figure 3a), where a linear fitting is applied to the data in the plot of $\log \dot{\epsilon }-\log \sigma$, the $n_1$ value as a function of strain rate, temperature, and strain, which is necessary for constructing processing maps, has often been determined using a third- or fourth-order polynomial fitting to the data in the plot of $\log \dot{\epsilon }-\log \sigma$. This polynomial fitting curve, however, sometimes has difficulty describing the power law creep. This example is shown in Figure 8. Thus, Kim and Jeong [121] proposed the exponential fitting method for the determination of the m from the plot of $\log \dot{\epsilon }-\log \sigma$. According to the method,

$$\log \dot{\epsilon }=\mathrm{a}+\mathrm{c} \times \exp \left(\frac{\log \sigma -b}{d}\right)+f\times \exp \left(\frac{\log \sigma -b}{g}\right)$$

(12)

When Equation (12) is used, m can be calculated by Equation (13):

$$m={\left[\frac{c}{d}\times \exp \left(\frac{\log \sigma -b}{d}\right)+\frac{f}{g}\times \exp \left(\frac{\log \sigma -b}{g}\right)\right]}^{-1}$$

(13)

As observed in Figure 8, the exponential fitting, where m tends to decrease gradually with increasing strain rate, provides a better fit to the series of data compared with the polynomial fitting, which sometimes creates uncertain fluctuation between the data points.

Figure 9a–c shows the η values of the HEAs calculated by Murty’s method (Equations (6) and (8)) as a function of n₁ for the three material groups of HEAs. It is obvious that the η values of all the three material groups of HEAs follow Equation (10) well in the entire range of n₁, regardless of the differences in composition and crystal structure. Furthermore, it is observed that the flow instability condition determined by Equation (9) occurs at n₁≈7, supporting that the onset of flow instability occurs at the transition from power law creep to PLB (Equation (11)).

Figure 10a–d shows the processing maps for Al_0.5CrMnFeCoNi and Al_0.3CrFeCoNi HEAs constructed based on Murty’s approach and Kim and Jeong’s approach. A good match is observed between the two methods in power dissipation maps as well as flow instability maps. However, some mismatch is observed for the Al_0.3CrFeCoNi HEA in the flow instability at low strain rates and at low temperatures. This occurs because in Murty’s method, the material is assumed to follow the power law creep at low strain rates below ${\dot{\epsilon }}_{min}$ (Equation (8)), but this assumption can be wrong at low temperatures if PLB governs plastic flow below ${\dot{\epsilon }}_{min}$.

Figure 11a,b shows the plots of n₁ as a function of the Zener–Hollomon parameter (Z = $\exp \left(\frac{Q_{\mathrm{c}}}{RT}\right))$ for Al_0.5CrMnFeCoNi and Al_0.3CrFeCoNi. There is a good correlation between n₁ and Z, indicating that as strain rate increases and temperature decreases (i.e., as Z increases), n₁ tends to increase. This occurs because according to the deformation mechanism maps (Figure 6a,b), as strain rate increases and temperature decreases, the deformation mechanism changes from solute drag creep (associated with n₁ = 3) to dislocation climb creep (associated with n₁ = 5) and then power law breakdown (associated with n₁ > 3). $\eta$ is a function of n₁ according to Equation (10). Hence, $\eta$ can also be expressed as a function of Z. Figure 11c,d shows the plot of $\eta$ as a function of Z for Al_0.5CrMnFeCoNi and Al_0.3CrFeCoNi, where a good correlation between $\eta$ and Z is observed. $\eta$ tends to decrease as Z increases. In addition, most of the data belonging to the flow instability condition (determined by Murty’s approach using Equation (9)) are positioned below $\eta =0.285$, supporting the validity of Equation (11). Some mismatches are observed between Equation (9) and Equation (10) and this can be attributed to the aforementioned assumption of power law creep below ${\dot{\epsilon }}_{min}$ in Murty’s method.

The plots in Figure 11c,d represent the “processing maps expressed as a function of Z” because if one knows the temperature and strain rate, the Z value can be calculated, and then the power dissipation efficiency and flow (in)stability can be readily determined from the plot.

Figure 12a–c shows the plots of processing maps for the three material groups of HEAs expressed as a function of Z using each Q_c value of the HEAs (Table 1), and Figure 12d shows the plot where all the data in Figure 12a–c overlap. The η values of each material are well correlated as a function of Z, and the data for each group converge to a common curve. Also, it is observed that all the data of the three groups converge to a single common curve. According to the plot in Figure 12d, flow stability is nearly guaranteed at Z ≤ 10¹² s⁻¹, while flow instability is nearly inevitable at Z ≥ 2 × 10¹⁵ s⁻¹. At 10¹² s⁻¹ ≤ Z ≤ 2 × 10¹⁵ s⁻¹, flow stability and instability conditions coexist, and flow instability becomes more dominant as Z increases.

Figure 13a–c shows the plots of processing maps for the three material groups of HEAs constructed as a function of Z using the average Q_c value of all the HEAs (317.2 kJ/mol), and Figure 12d shows the plot where all the data in Figure 13a–c overlap. Note that all the data lie close to a common curve. According to the plot in Figure 12d, flow stability prevails at Z ≤ 10¹² s⁻¹, while flow instability prevails at Z ≥ 3 × 10¹⁴ s⁻¹. By plotting in this fashion, one can easily compare the η values of the different HEAs at a given temperature and strain rate as well as predict the optimum hot working conditions of the HEAs with unknown Q_c values.

5. Conclusions

The hot compressive behaviors of the HEA materials with different chemical compositions and crystal structures and processing maps were analyzed, and the following observations were made.

• Hot compression tests on many HEAs have been conducted in the temperature range between 873 K and 1573 K, corresponding to the T/T_m range between 0.4 and 0.9. The n₁ values are most populated between 3 and 7.
• As T/T_m decreases, n₁ tends to increase, and power law breakdown typically occurs at T/T_m ≤ 0.6.
• In Al_xCrMnFeCoNi (x = 0–1) and Al_xCrFeCoNi (x = 0–1) HEAs, n₁ tends to increase as the concentration of Al increases, implying that Al acts as a solute atom that exerts a drag force on dislocation slip motion.
• The activation energy for plastic flow (Q_c) in the HEAs is calculated to be in the range between 150 and 600 kJ/mol, and the data distribution is populated in the Q_c value range between 300 and 400 kJ/mol. The average Q_c value for all the HEAs is 317 kJ/mol.
• The $\eta$ value of the HEAs can be expressed as a function of n₁ only. Flow instability is shown to occur near n₁ = 7, implying that the onset of flow instability occurs at the transition from power law creep to PLB.
• Processing maps for all the HEAs are demonstrated to be constructed using the Zener–Hollomon parameter (Z = $\exp \left(\frac{Q_{\mathrm{c}}}{RT}\right))$. According to the analysis result, flow stability prevails at Z ≤ 10¹² s⁻¹ in all HEAs.