University of Ljubljana. Faculty of Mathematics and Physics. Department of Physics. High-entropy alloys. Author: Darja Gačnik

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1 University of Ljubljana Faculty of Mathematics and Physics Department of Physics Seminar I 1 st year of Master study programme High-entropy alloys Author: Darja Gačnik Mentor: prof. Janez Dolinšek Ljubljana, January 2017 Summary High-entropy alloys or HEAs are metallic alloys, composed of at least five chemical elements with concentrations of each element between 35 and 5 atomic percent. Despite their complicated chemical composition, they form simple crystal lattices like body-centered cubic, face-centered cubic and hexagonal close-packed. This is a result of high mixing entropy effect, large lattice distortion and sluggish diffusion. The synthesis, processing and analysis of HEAs is feasible and they form structures with extraordinary properties. The fact that they can be produced from various elements with various percentage ranges broadens the number of possible compositions. HEAs introduce a new concept of developing new materials with unique properties and have great potential in academic studies as well as in commercial applications. It is expected that a lot of scientific research and commercial applications of HEAs will be undertaken in the future.

2 Contents 1 Introduction Core effects High-entropy effect Gibbs free energy Entropy Lattice distortion effect Sluggish diffusion effect Cocktail effect Conclusion Introduction High-entropy alloys or HEAs are metallic alloys composed of at least five chemical elements in equal or near equal atomic percents (at. %). In order for an alloy to be specified as a HEA, the concentrations of components must be between 5 and 35 at. % [1, 2]. Few structure models of HEAs is presented on Figure 1. Figure 1: Structure model of AlCrCuFeTiZn HEA.. The atoms are denoted with different colors: Al - red, Cr - blue, Cu - yellow, Fe - pink, Ti - orange, and Zn - grey. (»Adapted from [3]. «) Commonly used alloys are typically composed of one principal element, with only minor additions of other elements that are added for property enhancement or easier processing. However, it has been considered that alloys composed from a greater number of principal elements will form complicated structures, which are difficult to analyze and engineer. It seemed that they would not have any practical value and therefore research of these multielemental alloys was very limited [1, 4]. Experimental results yielded quite opposite conclusions. Multielemental alloys formed solid solution phases. In 1995, Jien-Wei Yeh [5] suggested that multielemental alloys would possess high mixing entropy, which would have an important role because it would favor formation of simple solid solution 2

3 phases. This was confirmed by several experimental results and these alloys were then named highentropy alloys [5, 6]. 2 Core effects HEAs exhibit some important effects, most of which have only been observed in high-entropy systems and are very useful in various applications. They originate from complex interactions among the elements composing the HEAs [5]. 2.1 High-entropy effect Although theoretically HEAs can form a large number of phases, only few are formed in reality, due to the high-entropy effect. This effect plays an important role because it favors formation of simple solid solution phases with FCC (face-centered cubic cell), BCC (body-centered cubic cell) and HCP (hexagonal closed-packed cell) [4] Gibbs free energy Gibbs free energy G is a thermodynamic potential, which is used for calculation of work that is performed by a thermodynamic system at a constant temperature and pressure. For the Gibbs free energy the following applies: G = H TS, where H is the enthalpy and S is entropy of the system. In stable phase the difference in the Gibbs free energy between the elemental and the mixed state, G mix = H mix T S mix, is minimal. We denoted G mix as the Gibbs free energy of mixing, H mix as the enthalpy of mixing, T as the absolute temperature and S mix as the entropy of mixing. It is obvious that the temperature is of great importance for determining stable phases in HEAs. However, it must be emphasized that it is the competition between the mixing enthalpy and the mixing entropy that determines the formation of phases and is therefore a good parameter for prediction of mutual solubility in solid solution phases [4, 7] Entropy The statistical-mechanics definition of the entropy states that entropy of the system is linearly related to the logarithm of the number w, where w indicates the number of possible micro-states corresponding to the macroscopic state of a system. This definition is written with the equation S = k B ln w, where k B is Boltzmann s constant. The mixing entropy S mix is correlated with the possible atomic arrangements that the system can take. It is the increase in the difference between the total entropy of several separate systems in thermodynamic equilibrium and their partitioned, mixed without any chemical reaction, closed system in a new thermodynamic equilibrium. 3

4 Let s consider an ideal solution, which is an alloy of randomly dispersed different ions, at constant temperature and pressure. In such a system, the forces between every pair of ions are the same. In order to use Boltzmann's formula, we have to calculate the number of ways w of arranging N i ions of i-th component on the lattice. With N = i N i we note the total number of ions and hence the number of lattice sites. The number w is therefore equal to the number of permutations of N objects, corrected by the fact that N i of them are identical for every i-th element, so w = entropy is therefore: S mix = k B ln w = k B ln ( N! i N i! N! N i! i ) = k B ln N! k B ln i N i! = k B ln N! k B ln N i!. The Boltzmann's mixing i. This formula can be written in a more elegant way if one applies Stirling s formula for ln N! and ln N i!: ln N! N ln N N and ln N i! N i ln N i N i, where N and N i are large numbers. If we took into an account the equation N = i N i, the equation for S mix can be simplified as: S mix = k B N ln N k B N k B i N i ln N i k B N i ln N i i N = k B N N i ln N i N N i. + k B i N i = k B N i ln N i k B i N i ln N i = We introduce a new variable x i = N i, which is the atomic fraction of the i-th element or probability of N finding i-th element at a given lattice site. Because the Boltzmann constant is equal to k B = R N A, where R is the gas constant (R = 8.31 J K mol ) and N A is the Avogadro constant (N A = mol), and n = N is the number of moles, the N A equation follows as S mix = R N A N i x i ln x i = nr i x i ln x i. The mixing entropy of an ideal multielement system is therefore S mix = nr i x i ln x i. If the system consists of r elements in equimolar fractions, then x i = 1 and the mixing entropy can be r r written as S mix equimolar = nr 1 i ln 1 = nr r r r (1 ln 1 ) = nr ln 1 = nr ln r. Figure 2 shows r r r a graph of the mixing entropy per mole as a function of the number of elements in an equimolar ideal system, in other words a graph of the function S mix equimolar (r) = R ln r [4, 5, 8, 9]. n 4

5 Figure 2: Graph of the mixing entropy per mole as a function of the number of elements in an equimolar ideal system. (»Adapted from [1]. «) HEAs mostly consist of 5 to 13 different elements. When there are 5 different elements in a HEA, it is predicted that the mixing entropy is already high enough to prevail over the mixing enthalpy in most alloy systems, even if the alloys aren t equimolar. According to the minimization of free Gibbs energy, this ensures formation of solid solution phases. The upper limit is set at 13 elements because there isn t any greater benefit in composing HEAs with more elements due to the logarithmic dependency of the mixing entropy on the number of elements in the alloy. For HEAs to form, the concentration of each element in the alloy system does not need to be equimolar, but can range between 5 and 35 atomic %. This is shown in Figure 3, where the mixing entropy per mole for a ternary alloy system as a function of atomic ratios of all three elements is plotted. It can be seen that the mixing entropy reaches maximum when the alloy system is equimolar, but it doesn t change significantly near the maximum. Widening the range of atomic concentrations broadens the number of possible HEAs. Still, the range is limited, and therefore HEAs do not contain any elements that have atomic concentration over 50%, as it is the case in traditional alloys [1, 4, 10]. Figure 3: Graph of the mixing entropy dependence on the atomic concentration of elements (concentration of C element: c c = 1 c A c B ) in ternary alloy system. (»Adapted from [4]. «) As an example, few calculated values for mixing Gibbs free energy are listed below in Table 1. From it it can be concluded that for CoCrFeMnNi alloy entropy term does not influence the formation of solid solution, because mixing enthalpy is already negative. For other three HEAs it can be seen that mixing 5

6 entropy doesn t depart much near maximum value, when alloy is equimolar, so widening the concentration range of elements in HEAs is indeed justifiable. Table 1: Calculated mixing Gibbs free energy (from calculated mixing enthalpy and entropy and measured melting temperature) for 4 different HEAs [12]. HEA H mix [ J mol ] S J mix [ K mol ] T m [K] G mix [ kj mol ] CoCrFeMnNi , ,40-28,14 Al 0.25CoCrCu 0.75FeNi -0,71 14, ,78-24,90 Al 0.3CoCrCuFeNi 1,56 14, ,22-24,72 AlCoCrCuFeNi -4,78 14, ,25-24,17 However, the mixing entropies for terminal solutions or ordered compound states are smaller than the ones calculated above. This is due to a limited number of ways the compounds can mix. We assumed random dispersion of atoms, which is an approximation. We have to note that random solution states are defined as high-temperature solid solution states, where high thermal energy causes random dispersion of different elements. But even this dispersion is not completely random. The proper approach is by studying the distribution of atoms in HEAs. One must take into account the difference in intermolecular forces between pairs of different kinds. Hence certain atoms are likely to have particular neighbors, which results in lower mixing entropy. The excess in mixing entropy comes also from vibrational, magnetic and electronic distributions [1, 4, 10]. 2.2 Lattice distortion effect HEAs are composed of various elements and therefore form a lattice with/on? a multielement basis. These elements can be of different sizes, which lead to distortion of the lattice. Larger ions need more space, so they push away their neighbours, and small ones are surrounded by extra space. This results in a /causes a strong internal stress-strain field, because large ions cause compression and small ones cause tension in the lattice. In Figure 4, there is a schematic representation of this effect with oneelement, two-element (where elements are very different in atomic sizes) and multielement lattice structure. However, the stress-strain field is nott influenced only by different sizes of compound elements but also by energy of the bonds between them. Stronger bonds tend to have smaller bonding distances than weaker bonds. Because of this effect, the strain energy of the lattice increases and therefore overall free energy of the lattice also increases. Even more, stress field in the lattice is not uniform and therefore HEAs have local stress gradients that slow down the movement of ions and are responsible for sluggish diffusion. Lattice distortion effect is very important because it determines whether the solid solution phases are stable. If HEAs are composed of elements that cause the lattice distortion energy to be too high for retaining the crystal structure, it collapses to an amorphous structure [4, 6]. 6

7 Figure 4: Schematic representation of a BCC lattice with a) one element (Cr), b) two elements (Cr, V) and c) six elements (Cr, Ni, Fe, Co, Al, Ti), where atoms are distributed randomly. (»Adapted from [4].«) This effect influences mechanical, thermal, electrical, optical and chemical behaviour of the materials. It causes a high strength for solid solutions (especially for HEAs with BCC lattice), high thermal and electrical resistance, tensile brittleness and diffuse X-ray scattering. In Figure 5 it is schematically illustrated how optical properties change due to lattice distortion. In a) X-ray beams are being reflected from Braggs planes on one-element alloy, where the direction of diffracted beams from both Bragg planes are the same. In b) diffracted beams from the multielement alloy Bragg planes diffract in various directions [1, 5, 6]. Figure 5: X-ray diffraction on a) one-element lattice and b) multielement lattice. (»Adapted from [4]. «) The more elements that are in the alloy, the more the lattice is distorted and X-rays are diffracted. That is why the X-ray diffraction intensities of alloys with more elements are smaller than those with fewer elements under the same X-ray diffraction experimental conditions and specimen geometry. This can be observed in Figure 6, where measured X-ray diffraction patterns of alloying series with different numbers? of elements, are plotted. The intensities of all peaks are lower especially for alloys with 5 and more elements, which are alloys classified as HEAs. It can be seen that intensities at high angles are lowered much more than those at low angles [12]. However, the reduction in intensities always occurs at nonzero temperature due to the Debye-Waller thermal effect. This effect states that the intensities of X-ray diffraction is? lower with factor e MT, where M T is the Debye-Waller temperature factor. M T is proportional to ( sin ϑ )2, where ϑ is the angle of the incident X-ray beam and λ is its wavelength. This happens because with increasing temperature ions vibrate and deviate from their neutral positions. The thermal moving of ions causes roughening of the Bragg diffraction planes and therefore lowers the X-ray intensities. Still, the measured decrease of X-ray intensities was much lower than the one predicted with taking into account only the temperature effect [12]. λ 7

8 Figure 6: X-ray diffraction patterns of alloys, composed with Cu, Ni, Al, Co, Cr, Fe or/and Si. (»Adapted from [12]. «) Therefore, one must also take into account the effect of lattice distortion. Hence, another modification factor for intensities variation of the X-ray diffraction was proposed as e MD. Analogous to the Debye- Waller temperature factor, the distortion factor is also proportional to ( sin ϑ λ )2. When both factors were included, the predictions were matching with the data. This is schematically illustrated in Figure 7, where intensities of diffraction peaks were plotted as a function of sin ϑ λ [12]. Figure 7: Schematic illustration of temperature and distortion effects on X-ray diffraction intensity. (»Adapted from [12]. «) 2.3 Sluggish diffusion effect One must take into account that in the equation for the Gibbs free energy, the entropy is multiplied by the absolute temperature, which is usually the melting temperature of the alloy. With decreasing temperature, HEAs might undergo phase transformations, such as spinodal decomposition, ordering or precipitation of intermetallic phases during cooling. However, phase separations that depend on atomic 8

9 diffusion during cooling are often inhibited by the sluggish diffusion effect, which states that atoms in HEAs move very slowly. The diffusion in HEAs is slower than in pure metals or even in stainless steels, so that τ HEA > τ ss > τ pm, where τ HEA is the characteristic diffusion time for HEAs, τ ss for stainless steels and τ pm for pure metals. This effect happens due to two contributions [1, 4]. The first contribution to sluggish diffusion is that with moving of atoms their surrounding changes too. The difference in local atomic configuration throughout the movement of atoms leads to different bonding with surrounding atoms and different local potential. Thus, if the atom had jumped into a state with low local potential, the possibility of jumping out would be low. It would be trapped, and the occupation time in this state would be significantly long. In contrast, if the atom had jumped into a state with high local potential, it would have a higher chance to hop back to the initial position. In both cases, the diffusion is slowed down. In conventional alloys this does not occur, because local energy is mostly identical due to the similar surrounding of each element in alloy system [4, 6]. One can even calculate the effect of energy fluctuation on the diffusion. The main parameter that expresses the difficulty in movement of an atom is the mean difference in the potential energy (MD), which is zero for pure metals. This is schematically represented in Figure 8. The diagram shows the lattice potential energy as a function of the distance between two lattice sites for three different alloy systems. Letter L indicates the position of atoms before, and M the position after the migration of atoms. As expected for a pure metal, the potential energy before and after the migration is identical. The activation energy required for an atom in pure metal to migrate from site L to M and reverse is noted as E b. For other two alloys the symmetry is lost and the activation energy E a for the atom to move from site L to M is increased for MD, E 2 a = E b + MD, and for movement of the atom in other way, it is 2 lowered for MD, E 2 a = E b MD. Thus, in those alloys, the position M is metastable and there is higher 2 chance for the atoms to jump to position L, because it is energetically more favorable [6]. Figure 8: Schematic diagram of the potential energy change during the movement of a Ni atom for three different alloys: pure metal (black), Fe-Cr-Ni (blue) and HEA alloy CoCrFeMn0.5Ni (red). (»Adapted from [6].«) 9

10 The second contribution to sluggish diffusion in HEAs is that the diffusion rate of each element is different. Some elements are less active and have lower MD in the local potential energy. However, for a phase transformation to occur, typically many kinds of elements have to move, thus slow-moving elements become the rate-limiting factor that slow down the diffusion. The sluggish diffusion effect allows better high-temperature strength and structural stability. It also contributes for using HEAs as diffusion barrier coatings or as creep resistance structural parts [6, 10]. 2.4 Cocktail effect HEAs are composed from different elements and can therefore exhibit properties that their elements have in a one-element alloy. Properties of HEAs can be, however, more unexpected. Beside the properties of individual composing elements, properties of HEAs are also affected by the interaction among the component elements and lattice distortion [4, 5, 6]. 3 Conclusion To date, the study of HEAs indicated that they could exhibit many new phenomena and properties. The fact that they can be made from various elements with various percentage ranges broadens the number of possible compositions. HEAs introduce a new concept of developing new materials with unique properties and have great potential in academic studies as well as in commercial applications. However, there are still a lot of new materials and new theories to be found and applied in functional use [1, 5]. References [1] J.-W. Yeh et al., Adv. Eng. Mater. 6, (2004). [2] S. Wang, Entropy 15, (2013). [3] S. Wang, AIP Advances 3, (2013). [4] Y. Zhang et al., Progress in Material Science 61, 1-21 (2004). [5] J.-W. Yeh, Annales De Chimie Science des Materiaux 31, (2006). [6] Ming-Hung Tsai, Jien-Wei Yeh, Materials Research Letters 2, (2014). [7] ( ). [8] ( ). [9] B.S. Murty, J.-W. Yeh, S. Ranganathan, High-Entropy Alloys (Butterworth-Heinemann Elsevier, Amsterdam, 2014). [10] H. K. D. H. Bhadeshia, Materials Science and Technology 31, (2015). [11] M. C. Gao, J.-W. Yeh, P. K. Liaw and Y. Zhang, High-Entropy Alloys (Springer International Publishing, Switzerland, 2016). [12] J.-W. Yeh et al., Materials Chemistry and Physics 103, (2007). 10