Entropy Determination of Single-Phase High Entropy Alloys with Different Crystal Structures Over a Wide Temperature Range

Article Entropy Determination of Single-Phase High Entropy Alloys with Different Crystal Structures over a Wide Temperature Range

Sebastian Haas 1, Mike Mosbacher 1, Oleg N. Senkov 2 ID , Michael Feuerbacher 3, Jens Freudenberger 4,5 ID , Senol Gezgin 6, Rainer Völkl 1 and Uwe Glatzel 1,* ID

1 Metals and Alloys, University Bayreuth, 95447 Bayreuth, Germany; [email protected] (S.H.); [email protected] (M.M.); [email protected] (R.V.) 2 UES, Inc., 4401 Dayton-Xenia Rd., Dayton, OH 45432, USA; [email protected] 3 Institut für Mikrostrukturforschung, Forschungszentrum Jülich, 52425 Jülich, Germany; [email protected] 4 Leibniz Institute for Solid State and Materials Research Dresden (IFW Dresden), 01069 Dresden, Germany; [email protected] 5 TU Bergakademie Freiberg, Institut für Werkstoffwissenschaft, 09599 Freiberg, Germany 6 NETZSCH Group, Analyzing & Testing, 95100 Selb, Germany; [email protected] * Correspondence: [email protected]; Tel.: +49-921-55-5555

Received: 31 July 2018; Accepted: 20 August 2018; Published: 30 August 2018

Abstract: We determined the entropy of high entropy alloys by investigating single-crystalline nickel and five high entropy alloys: two fcc-alloys, two bcc-alloys and one hcp-alloy. Since the configurational entropy of these single-phase alloys differs from alloys using a base element, it is important to quantify ◦ the entropy. Using differential scanning calorimetry, cp-measurements are carried out from −170 C to the materials’ solidus temperatures TS. From these experiments, we determined the thermal entropy and compared it to the configurational entropy for each of the studied alloys. We applied the rule of mixture to predict molar heat capacities of the alloys at room temperature, which were in good agreement with the Dulong-Petit law. The molar heat capacity of the studied alloys was about three times the universal gas constant, hence the thermal entropy was the major contribution to total entropy. The configurational entropy, due to the chemical composition and number of components, contributes less on the absolute scale. Thermal entropy has approximately equal values for all alloys tested by DSC, while the crystal structure shows a small effect in their order. Finally, the contributions of entropy and enthalpy to the Gibbs free energy was calculated and examined and it was found that the stabilization of the solid solution phase in high entropy alloys was mostly caused by increased configurational entropy.

Keywords: HEA; entropy; multicomponent; differential scanning calorimetry (DSC); speciﬁc heat

1. Introduction The conventional strategy of alloy design is based on the selection of one base element for the primary properties of a material, e.g., iron in steels or nickel in nickel-based superalloys. This base element dominates the chemical composition, usually representing more than 80 at. %, and relatively small amounts of other elements are added to modify the alloys’ properties [1–3]. Thus the regions next to a single element of multicomponent phase diagrams have been well investigated in the past. A novel way of alloy design concentrates on the unexplored centers of phase diagrams, where the alloys consist of elements in near equiatomic ratios. Such an alloy was pointed out by Cantor et al. [1] in 2004, when the equiatomic system of Cr, Mn, Fe, Co and Ni was found to result in a single-phase material. This simple face-centered cubic microstructure is free of any precipitates and stable over a

Entropy 2018, 20, 654; doi:10.3390/e20090654 www.mdpi.com/journal/entropy Entropy 2018, 20, 654 2 of 12 wide temperature range [4]. Merely after long time annealing for 500 h at intermediate temperatures precipitations may segregate, e.g., a Cr-rich phase at 700 ◦C and three different phases (NiMn-rich, FeCo-rich, Cr-rich) at 500 ◦C[5]. However, a single-phase microstructure is the key factor for a high configurational entropy. In the same year Yeh et al. [3] pointed out the concept of high entropy alloys, independent of Cantor’s work. His definition of high entropy alloys is to consist of at least five or more elements with concentrations of each element between 5 and 35 at. % [6]. Many elements, and therefore a high value of configurational entropy, can lead to a more stable solid-solution phase with randomly distributed atoms [3]. A solid-solution phase with statistically distributed atoms in the crystal lattice is claimed to lead to interesting and outstanding properties, e.g., high hardness, wear resistance, high temperature strength and stability, sluggish diffusion, oxidation and corrosion resistance [6,7]. Yeh et al. [8] named four core effects, which are characteristic for microstructures and properties of high entropy alloys: the formation of one random solid-solution phase to reach a high entropy effect [8]; severe lattice distortion in the random solid-solution [9]; sluggish diffusion kinetics [6,10]; the so called “cocktail effect” [8]. Nevertheless the major part of investigated alloys with compositional requirements of high entropy alloys do not form single solid-solutions, but consist of several, mostly intermetallic phases, which can be brittle, difficult to process. This observation particularly disagrees with the crucial issue of a single-phase microstructure. Numerous examinations deal with the prediction of the conditions when a solid-solution phase is stable or additional intermetallic compounds are forming [7,10–12], but no reliable approaches have yet been proposed. Thus we follow the idea to calculate the total Gibbs free energy of an alloy system in single-phase state to compare it with formation enthalpies of several intermetallic compounds. Therefore the determination of thermal enthalpy and especially of entropy over a wide temperature range is necessary. A short insight in thermodynamics and in the way of calculating different parts of the entropy is given in the following part of the introduction: The terms and definitions of the entropy theory, as well as the basic approach of determinations refer to Gaskell [13]. Changes in Gibbs free energy ∆Gtotal of a system at any state and temperature depends on the entropy Stotal and on the enthalpy Hthermal (see Equation (1)). The consideration of both thermochemical parameters leads to a description of the equilibrium state of an alloy system by minimization of the Gibbs free energy at a fixed temperature:

∆Gtotal = ∆Hthermal − T∆Stotal, (1)

The entropy of an alloy consists of the configurational entropy Sconf [7] and the thermal entropy Sthermal [13]: Stotal = Sconf + Sthermal, (2) = − · n · ∆Sconf R ∑i=1 xi ln xi, (3) The calculation of configurational entropy, also called mixing entropy, is given in Equation (3) with n as the number of elements, xi the concentration of each element i and R as the universal gas constant. The equation is derived from the mixing of noble gases and is adopted to fully disordered solid solution, which are assumed in our work. If elements are distributed non-equally between possible sub-lattices, then a more general equation should be used [14]. In case of an equiatomic alloy, Equation (3) is reduced to ∆Sconf = R ln n. Thus the configurational entropy of 5-component equiatomic alloy is ~1.6·R. At 0 K the resulting total entropy may not be zero. This does not violate the dS third law of thermodynamics ( = 0) which is often misinterpreted as S(T = 0) = 0. Crystalline dT T=0 solids may exhibit a non-zero entropy at the absolute zero point due to a randomly crystallographic orientation. A change of entropy in this point is not possible, because there is no ability of motion or diffusion. Entropy 2018, 20, 654 3 of 12

The thermal entropy Sthermal can be directly determined by measuring the temperature dependent heat capacity at constant pressure cp(T) by differential scanning calorimetry (DSC):

Entropy 2018, 20, x FOR PEER REVIEW Z T 1 3 of 12 ∆Sthermal(T) = cp(T) dT, (4) 0 T The thermal entropy Sthermal can be directly determined by measuring the temperature dependent heatNext capacity to the at entropy,constant pressure there is ancp(T) enthalpy-contribution by differential scanning to calorimetry the Gibbs free(DSC): energy, mentioned in Equation (1). This thermal enthalpy can be calculatedT using1 Equation (5): ∆S (T) = ∫ c (T) dT, (4) thermal 0 p T Z T Next to the entropy, there is an∆ Henthalpythermal(T-contribution) = cp(T to)dT, the Gibbs free energy, mentioned in (5) Equation (1). This thermal enthalpy can be calculated 0using Equation (5):

Therefore, the total change in Gibbs free energyT ( ) at a certain temperature is presented in the ∆Hthermal(T) = ∫0 cp T dT, (5) following Equation (6). Setting the value of configurational entropy to zero, just the thermal entropy is consideredTherefore, and the the result total ischange the change in Gibbs in thermal free energy Gibbs at freea certain energy: temperature is presented in the following Equation (6). Setting the value of configurational entropy to zero, just the thermal entropy is considered and the result is the change in thermalZ T Gibbs free energy: Z T 1 ∆Gtotal = ∆Hthermal − T∆Stotal = cp(T)dT − T ∆Sconf + cp(T) dT , (6) T T 1 T ∆G = ∆H − T∆S = ∫ c0(T)dT − T (∆S + ∫ c (T0) dT), (6) total thermal total 0 p conf 0 p T A schematic drawing of c from 0 K to temperatures in the range of incipient melting after reaching A schematic drawing ofp cp from 0 K to temperatures in the range of incipient melting after thereaching solidus the temperature solidus temperature TS is given T inS is Figure given1 a. in For Figure lower 1a. temperatures For lower temperatures c p(T) can be cp(T) extrapolated can be 3 downextrapolated to 0 K following down to the 0 Debye K following T -law the [13 ]. Debye For temperatures T3‐law [13]. closeFor temperatures to room temperature close to (RT) room the heattemperature capacity is(RT) close the to heat 3R ifcapacity there is is no close change to 3R in if magneticthere is no behavior change in and magnetic no phase behavior transformation. and no Thisphase is known transformation. as the Dulong-Petit This is known law, as which the Dulong‐Petit states that every law, which solid states that consists that every of N solid atoms that has 3Nconsists modes of of N vibration atoms has (corresponding 3N modes of vibration to the freedom (corresponding of motion to the in threefreedom dimensions). of motion in Energetic three considerationsdimensions). Energetic using the considerations equipartition using theorem the equipartition lead to cp = theorem 3·R for lead sufficiently to cp = 3· highR for temperaturessufficiently (RT).high DSC temperatures measurements (RT). DSC are carriedmeasurements out until are the carried solidus out temperatureuntil the solidus is reachedtemperature and is the reached cp-value risesand dramatically. the cp-value rises In Figure dramatically.1b, the In calculated Figure 1b, thermal the calculated entropy thermal and total entropy entropy and total are schematicallyentropy are illustrated.schematically The illustrated. thermal entropy The thermal is shifted entropy vertically is shifted by the vertically configurational by the configurational entropy over entropy the whole temperatureover the whole range, temperature in case there range, are no in contributionscase there are byno phasecontributions changes. by This phase is due changes. to the This low influenceis due ofto various the low alloys influence on the of values various of thermalalloys on entropy, the values while of thethermal configurational entropy, while entropy the configurationa has a high impact.l Theentropy vertical has offset a high between impact. thermal The vertical and totaloffset entropy, between caused thermal by and the total alloys’ entropy, configuration, caused by is the still holdingalloys’ onconfiguration, after melting is instill the holding liquid on state. after melting in the liquid state.

FigureFigure 1. 1.Schematic Schematic drawing drawing ((aa)) ofof thethe heat capacity capacity of of an an ideal ideal metal metal from from 0 Kelvin 0 Kelvin to tothe the liquid liquid state state withwith no no change change in in magnetic magnetic behaviorbehavior andand no phase transition transition over over the the whole whole temperature temperature range range and and (b)(b of) of the the calculated calculated thermal thermal entropy entropy andand totaltotal entropy from from 0 0 K K to to the the liquid liquid state. state.

2. Materials and Methods

Entropy 2018, 20, 654 4 of 12

2. Materials and Methods

The temperature dependent molar heat capacity cp(T) of the specimen was experimentally determined in alumina crucibles using differential scanning calorimetry (DSC 204, Netzsch, Selb, Germany) in the temperature range from −170 ◦C to 600 ◦C under a flushing flow of nitrogen. In this ◦ ◦ temperature range oxidation of the samples is not critical. For T = 20 C − TS (if TS < 1600 C) samples were measured using a Netzsch DSC 404 F1 Pegasus under an Ar 5.0 gas flush with a rate of 70 ml/min in crucibles composed of 80% Pt and 20% Rh. The crucibles are lined with ceramic inlays to prevent interaction of the metallic specimens with the Pt-Rh crucible. The calibration of temperature and enthalpy of the two Netzsch devices was performed using the calibration set 6.239.2–91.3 under the conditions of 10 K/min and a nitrogen flow of 40 mL/min. The two measurements were evaluated for each material and the curves were connected in the common temperature range from room temperature to 600 ◦C. Single-crystalline (SX) nickel and SX-Cantor alloy (favored equiatomic composition of Cr, Mn, Fe, Co, Ni) were cast using a proprietary Bridgman investment casting furnace. Cylindrical specimens (Ø 5 mm, height 1 mm) were then cut out of the rods by electrical discharge machining (EDM). All other alloys were in poly-crystalline state. The bcc-alloys were provided by Senkov et al. [15], the noble metal alloy by Freudenberger et al. [16] and the hcp-alloy by Feuerbacher et al. [17]. The recast layer of the EDM samples was removed by etching and the base of all samples was finely ground with SiC paper up to 2000 grit to ensure good thermal contact to the DSC-sensor. All materials tested several times showed good reproducibility in their DSC-signal, even at different heating rates of 10 K/min and 20 K/min respectively. Except for pure nickel all materials belong to the high entropy alloys group. They form a single-phase microstructure and contain at least four elements in desired, near equiatomic composition. The single-phase solid-solution has been confirmed by the authors using X-ray diffraction experiments. Homogenization of the samples was reached by very slow cooling rates after melting, except of fcc-noble metal that was annealed for 24 h at 1000 ◦C and bcc 5-component was annealed for 24 h at 1200 ◦C. Although the ROM may only represent a very rough estimate of the melting temperature of the alloys, it has been applied for the bcc-5 component and hcp-alloys. The reasons are twofold: (i) the melting temperature of the bcc-5 component alloy cannot be experimentally determined with our set-up and (ii) we are facing significant reactions of the metals with the crucible during investigation and, therefore, the determined values would not reflect the samples under investigation. The Mn content of 11 at. % in the Cantor alloy is due to the single-crystal investment casting process. Mn evaporates from the melt in the vacuum of 10−2 Pa of the Bridgman furnace during the slow withdrawal of the single-crystal.

3. Results and Discussion The speciﬁc heat capacity in units of J/(g·K) has been converted into the molar heat capacity in units of J/(mol·K) by multiplication with the molar mass of the alloys corresponding to their chemical composition. Figure2 shows the molar heat capacity of all materials as a function of temperature. For temperatures below −170 ◦C, the curves were extrapolated using the ﬁt-function a·T3 (parameter a is listed in Table1) and c p(0 K) = 0.

Table 1. Values of the parameter a in J/mol·K4 in the ﬁt-function a·T3.

hcp bcc-4 Comp. bcc-5 Comp. fcc-Cantor fcc-Nickel fcc-Noble 1.8 × 10−5 7.6 × 10−6 8.2 × 10−6 9.8 × 10−6 9.0 × 10−6 1.0 × 10−5 Entropy 2018, 20, 654 5 of 12 Entropy 2018, 20, x FOR PEER REVIEW 5 of 12

S Figure 2. MolarMolar heat heat capacities capacities of of pure pure nickel and five five high entropy alloys. Green Green crosses indicate T S,, redred lines lines show show the the beginning beginning of oxidation of oxidation or reactions or reactions with the with crucible the crucible that make that the make data not the evaluable. data not evaluable. Table 2. Sample labelling, crystal specification, chemical composition (measured by µ-XRF) and solidus Table 2. Sample labelling, crystal specification, chemical composition (measured by µ‐XRF) and temperature. In case of pure Ni, TS is equal to the melting temperature Tm. The rule of mixture (ROM) solidusis used tem for calculation,perature. In ifcase no of literature pure Ni, data TS is is equal available. to the All melting alloys temperature are in a single-phase Tm. The rule state of overmixture the (ROM)entire temperature is used for calculation, region. if no literature data is available. All alloys are in a single-phase state over the entire temperature region. Alloy Name Used Crystal Type Chemical Solidus Alloy Name Used Crystal Type Chemical Composition Sconf in J/(mol·K) Solidus in This Paper and Structure Composition (at. %) Sconf in J/(mol∙K) Temperature TS in This Paper and Structure (at. %) Temperature TS fcc-Ni SX-fcc Pure Ni (>99.99%) 0 1455 ◦C[18] fcc‐Ni SX-fcc Pure Ni (>99.99%) 0 1455◦ °C [18] fcc-Cantor SX-fcc Co22Cr24Fe22Mn11Ni21 1.58·R 1280 C[19] ◦ fcc‐Cantorfcc-noble metal SX- PX-fccfcc Co Au2216CrCu2417FeNi2217MnPd1134NiPt1621 1.561.58∙R·R 11961280 C°C [19] ◦ fcc‐noblebcc-4 component metal PX PX-bcc-fcc Au Mo16Cu27Nb17Ni27Ta17Pd23W34Pt23 16 1.591.56∙R·R 2904 1196C[15 ]°C ◦ bcc-5 component PX-bcc Hf21Mo20Nb21Ti17Zr21 1.38·R 2170 C (ROM) bcc‐4 component PX-bcc Mo27Nb27Ta23W23 1.59∙R 2904◦ °C [15] hcp PX-hcp Ho18Dy16Y28Gd20Tb18 1.61·R 1416 C (ROM) bcc‐5 component PX-bcc Hf21Mo20Nb21Ti17Zr21 1.38∙R 2170 °C (ROM) hcp PX-hcp Ho18Dy16Y28Gd20Tb18 1.61∙R 1416 °C (ROM) The two single-crystal fcc-materials, Nickel and Cantor alloy, show a steady increase in cp until shortlyThe before two single‐crystal their melting fcc‐materials, points. Both Nickel curves and are Cantor very close alloy, to show each othera steady with increase the exception in cp until of shortlythe Curie-peak before their in Ni melting at around points. 356 Both◦C, whichcurves liesare very in excellent close to agreementeach other with the literature exception data of (e.g., the Curie354 ◦C[-peak20]). in The Ni at Cantor around alloy 356 shows °C, which a broad lies plateau-likein excellent peakagreement between with 600 literature◦C and 800data◦ C.(e.g., This 354 peak °C [20]).is most The likely Cantor caused alloy by shows a change a broad in magnetic plateau‐like behavior. peak between Jin et al. [60021] °C have and shown 800 °C. that This Co peak and Feis most shift likelythe Curie caused temperature by a change of Ni in to magnetic higher temperatures behavior. Jin and et al. their [21] investigations have shown onthat Cantor Co and alloy Fe showshift the the Curieexact sametemperature plateau-like of Ni peak to higher starting temperatures at around 600 and◦C. their Incipient investigations melting at on the Cantor solidus alloy temperatures show the exactis indicated same plateau‐like by excessive peak jumps starting in the at curves. around The 600 solidus °C. Incipient temperatures melting ofat fcc-nickel,the solidus fcc-Cantor temperatures and isfcc-noble indicated do by not excessive correspond jumps to the in valuesthe curves. listed The in Table solidus2, but temperatures differ by about of fcc‐nickel, 30 K in case fcc‐Cantor of fcc-Cantor and fccand-noble fcc-nickel do not and correspond by about 70 to Kthe in casevalues of listed fcc-noble. in Table The other2, but three differ alloys by about show 30 decreasing K in case specific of fcc‐ Cantorheat capacities and fcc‐nickel at elevated and by temperatures, about 70 K in which case of indicates fcc‐noble. reactions The other with three the atmospherealloys show and/ordecreasing the specificcrucible. heat High capacities chemical atinteraction elevated and temperatures, oxidation with which the indicates ceramic liners reactions of the with crucibles the atmosphere occur with and/orthe alloys the Hf-Mo-Nb-Ti-Zr crucible. High chemical and Ho-Dy-Y-Gd-Tb, interaction and so theiroxidation curves with are finallythe ceramic cut-off liners at 700 of◦ Cthe and crucibles 900 ◦C occurrespectively. with the Melting alloys intervals,Hf‐Mo‐Nb‐Ti‐Zr starting and from Ho‐Dy‐Y the solidus-Gd‐Tb, temperatures, so their curves are not are evaluable finally cut for-off the at bcc-5 700 °Ccomponent and 900 and °C respectively.hcp alloy. In caseMelting of the intervals, bcc-4 component starting from alloy t thehe solidus curve mistakenly temperatures, suggests are not the evaluablesolidus temperature for the bcc‐5 to becomponent at about 1520 and◦ C.hcp However, alloy. In Senkovcase of etthe al. bcc‐4 [15] expect component a melting alloy temperature the curve mistakenlyof about 2904 suggests◦C, using the the solidus rule of temperature mixture. This to issue be at is aboutlikely to 1520 be due °C. toHowever, chemical Senkov reactions et withal. [15] the expectPt-Rh cruciblea melting and temperature therefore we of about have to 2904 regard °C, using the values the rule of the of mixture. heat capacity This issue of bcc-4 is likely component to be due for to chemical reactions with the Pt‐Rh crucible and therefore we have to regard the values of the heat capacity of bcc-4 component for higher temperatures with caution. For further calculations

Entropy 2018, 20, 654 6 of 12 higher temperatures with caution. For further calculations concerning entropy, enthalpy and Gibbs free energy, the data of bcc- and hcp-alloys respectively are just used in the temperature range until Entropy 2018, 20, x FOR PEER REVIEW 6 of 12 the red lines in Figure2. Tableconcerning2 gives entropy, an overview enthalpy of and all Gibbs tested free samples, energy, theirthe data crystal of bcc structures,- and hcp‐alloys chemical respectively compositions, are conﬁgurationaljust used in entropiesthe temperature and solidus range until temperatures. the red lines Note in Figure that 2. in case of bcc-5 component and the hcp alloy, thereTable was 2 nogives literature an overview data of concerning all tested samples, solidus their temperature crystal structures, available. chemical Therefore, compositions, the possible solidusconfigurational temperatures entropies were calculated and solidus using temperatures. the rule of Note mixture that in (ROM). case of bcc-5 component and the hcp alloy, there was no literature data concerning solidus temperature available. Therefore, the Obvious shifts in the trend of cp indicate phase transitions or changes in the magnetic behavior, e.g., thepossible Curie solidus point temperatures of nickel at 356were◦ Ccalcul as mentionedated using the above. rule of The mixture hcp-alloy (ROM). Ho-Dy-Y-Gd-Tb shows a Obvious shifts in the trend of cp indicate phase transitions or changes in the magnetic behavior, very pronounced peak at −120 ◦C, probably due to a not investigated magnetic phase transformation, e.g., the Curie point of nickel at 356 °C as mentioned above. The hcp‐alloy Ho‐Dy‐Y-Gd‐Tb shows a because oxidation seems to be very unlikely in such a low temperature range. very pronounced peak at −120 °C, probably due to a not investigated magnetic phase transformation, Ofbecause particular oxidation interest seems areto be the very heat unlikely capacities in such ata low room temperature temperature, range. where all curves seem to approachOf a cp particularvalue close interest to 25 are J/(mol the heat·K) = capacities 3R (see Figure at room3a–d), temperature, according where to the all Dulong-Petit-law curves seem to [ 13]. The bcc-5approach component a cp value equiatomic close to 25 alloy, J/(mol∙ Hf-Mo-Nb-Ti-Zr,K) = 3R (see Figure shows 3a–d), the according lowest molarto the heatDulong‐Petit‐law capacity at room temperature[13]. The withbcc-5 aboutcomponent 23.5 J/(molequiatomic·K) (Figurealloy, Hf‐Mo‐Nb‐Ti‐Zr,3b), the hcp-alloy shows Ho-Dy-Y-Gd-Tb the lowest molar shows heat capacity the highest valueat at room about temperature 30.6 J/(mol with·K) about (Figure 23.53 d).J/(mol∙K) The high (Figure value 3b), of the the hcp‐a molarlloy heatHo‐Dy‐Y capacity-Gd‐Tb of shows the hcp-alloy the mighthighest originate value from at about magnetic 30.6 J/(mol∙K) ordering (Figure of the 3d). 4f The electrons. high value This of the is likelymolar heat the casecapacity as pure of the Gd hcp shows- ferromagneticalloy might ordering originate below from magnetic 19.9 ◦C. However,ordering of this the suggestion4f electrons. needsThis is to likely be veriﬁed the case in as future pure Gd studies. shows ferromagnetic ordering below 19.9 °C. However, this suggestion needs to be verified in future The other four alloys show RT cp-values pretty close to 25 J/(mol·K). We applied the rule of mixture studies. The other four alloys show RT cp-values pretty close to 25 J/(mol∙K). We applied the rule of (ROM)mixture to predict (ROM) molar to predict heat capacities molar heat of capacities the alloys of and the alloystheir average and their molar avera massge molar (see mass Equation (see (7)). The calculatedEquation (7)). data The is calculated in very good data is agreement in very good with agreement the experimental with the experimental results and results shows and a shows maximum deviationa maximum of 6%: deviation of 6%: 1 n 1 n c =1 · c , u 1 = · u , (7) cp,alloy = ∙ ∑∑n ic=1 , pui alloy= ∙ ∑n u∑, i=1 i p,alloy nn i=1 pi alloy n i=n1 i (7)

FigureFigure 3. Molar 3. Molar heat heat capacities capacities at roomat room temperature temperature of of the the investigated investigated alloys,alloys, bothboth experimental and and rule of mixturerule of (ROM) mixture data (ROM) for (dataa) fcc-Cantor, for (a) fcc-Cantor, (b) both (b bcc-alloys,) both bcc-alloys, (c) fcc-noble (c) fcc‐noble and (d and) the (d hcp-alloy.) the hcp-alloy. Values for Values for pure elements were taken from [20]. The red line indicates the Dulong-Petit law with cp(RT) pure elements were taken from [20]. The red line indicates the Dulong-Petit law with cp(RT) ≈ 3·R. ≈ 3·R.

FigureFigure4 shows 4 shows the thermalthe thermal entropy entropy S thermal Sthermal determined with with Equation Equation (4). (4). Sthermal Sthermal is equialis equial zero at zero at −273−◦273C andincreases °C andincreases continuously continuously with with increasing increasing temperature temperature having having different different slopes slopes for for different different alloys. The curvesalloys. areThe drawncurves tillare shortlydrawn till before shortly the before solidus the temperature solidus temperature of the alloys, of the except alloys, for except Ho-Dy-Y-Gd-Tb, for Ho‐ Hf-Mo-Nb-Ti-ZrDy‐Y-Gd‐Tb, and Hf‐Mo‐Nb‐Ti‐Zr Mo-Nb-Ta-W where and chemical Mo‐Nb‐Ta reactions-W where with the chemical crucible/environment reactions with make th thee high crucible/environment make the high temperature regions inaccessible. The highest thermal entropy temperature regions inaccessible. The highest thermal entropy appears for the hcp-alloy, caused by the appears for the hcp‐alloy, caused by the highest heat capacity at low temperatures and additionally

Entropy 2018, 20, 654 7 of 12

Entropy 2018, 20, x FOR PEER REVIEW 7 of 12 ◦ highest heat capacity at low temperatures and additionally the early peak in the cp-curve at about −120 C. the early peak in the cp‐curve at about −120 °C. The fcc-nickel, fcc-Cantor alloy and fcc noble metal The fcc-nickel, fcc-Cantor alloy and fcc noble metal alloy have almost the same temperature dependence of Entropyalloy have 2018, 20almost, x FOR the PEER same REVIEW temperature dependence of Sthermal, which is noticeably weaker than7 ofthat 12 Sthermal, which is noticeably weaker than that of the hcp-alloy. These fcc materials have similar Sthermal of the hcp-alloy. These fcc materials have similar Sthermal values at any given temperature. The two valuesthebcc‐alloys at early any givenpeak exhibit in temperature. thethe clowestp‐curve values Theat about two of thermal −120 bcc-alloys °C. entropy. The exhibit fcc -nickel, the lowest fcc-Cantor values alloy of thermaland fcc noble entropy. metal alloy have almost the same temperature dependence of Sthermal, which is noticeably weaker than that of the hcp-alloy. These fcc materials have similar Sthermal values at any given temperature. The two bcc‐alloys exhibit the lowest values of thermal entropy.

Figure 4. Thermal entropy Sthermal of nickel and the five high entropy alloys. Figure 4. Thermal entropy Sthermal of nickel and the ﬁve high entropy alloys.

The total entropy Stotal = Sconf + Sthermal over the whole temperature range starting from absolute Thezero totalis plotted entropy inFigure Figure Stotal 4. Thermal=5. SWhileconf entropy+ the Sthermal curve Sthermalover of of nickel nickel the wholestarts and the at temperature five0 J/(mol∙K) high entropy at range absolute alloys. starting zero, fromthe bcc absolute-4 zero iscomponent plottedin alloy Figure Mo‐Nb‐Ta5. While-W theexhibits curve a configurational of nickel starts entropy at 0 J/(mol of R ln(4)·K) = at 1.38 absolute·R = 11.5 zero, J/(mol∙K), the bcc-4 The total entropy Stotal = Sconf + Sthermal over the whole temperature range starting from absolute componentbecause alloy of its Mo-Nb-Ta-Wfour components. exhibits All other a configurational high entropy alloys, entropy with of five R ln(4)differ =ent 1.38 elements·R = 11.5 show J/(mol an ·K), zero is plotted in Figure 5. While the curve of nickel starts at 0 J/(mol∙K) at absolute zero, the bcc-4 becauseoffset of of its R fourln(5) components.= 1.6·R = 13.4 J/(mol∙K). All other In hightheory entropy the configurational alloys, with entropy five differentis equal for elements all alloys show componentwith the same alloy number Mo‐Nb‐Ta and- W concentrations exhibits a configurational of elements. entropy No considerations of R ln(4) = 1.38 are · doneR = 11.5 with J/(mol∙K), respect an offset of R ln(5) = 1.6·R = 13.4 J/(mol·K). In theory the configurational entropy is equal for all becausetosimilarities of its betweenfour components. participating All otheratoms high, like entropy differences alloys, in the with atomic five differ radiusent or elements crystal structures show an alloysoffsetof with the ofpure the R ln(5) samemetals. = number1.6 In· Rour = 13.4case, and J/(mol∙K). all concentrations elements In intheory the of hcp the elements.- alloyconfigurational promote No considerations a hexagonalentropy is close equal are packed done for all with crystalalloys respect tosimilaritieswithstructure the between same and exhibit number participating quite and concentrations equal atoms, atomic like radii. of differences elements. The bcc No- in5 component the considerations atomic alloyradius are however ordone crystal with consists structures respect of of the puretosimilaritieselements metals. of different Inbetween our case, crystalparticipating all structures elements atoms with in, like the larger differences hcp-alloy atomic promoteinsize the differences. atomic a hexagonal radius Nevertheless, or crystal close packedstructures the same crystal structureofvalue the and pureof S exhibitconf metals. is assumed, quite In our equal case, as stated atomicall elements in radii.the Gibbs in The the paradoxhcp bcc-5-alloy component [22]. promote For dislocation a alloy hexagonal however movement, close consists packed however, ofcrystal elements structure and exhibit quite equal atomic radii. The bcc-5 component alloy however consists of of differentdifferent crystal atom sizes structures in solid with-solution larger crystal atomic structures size differences. indeed play Nevertheless, an important role. the same Consequently, value of Sconf is assumed,elementsthis mixing as of stated paradoxdifferent in will thecrystal Gibbsbe investigatedstructures paradox with thoroughly [22 larger]. For atomic dislocation in a future size differences. work. movement, It can Nevertheless, be however, seen from different the Figure same 5 atom value of Sconf is assumed, as stated in the Gibbs paradox [22]. For dislocation movement, however, sizesthat in solid-solution at any given temperature crystal structures Stotal is the indeed smallest play for fcc an-Ni, important followed role.by the Consequently, bcc alloys, then thisthe fcc mixing differentalloys and atom being sizes the in highest solid- solutionfor the hcp crystal alloy. structures indeed play an important role. Consequently, paradox will be investigated thoroughly in a future work. It can be seen from Figure5 that at any this mixing paradox will be investigated thoroughly in a future work. It can be seen from Figure 5 giventhat temperature at any given S totaltemperatureis the smallest Stotal is the for smallest fcc-Ni, for followed fcc-Ni, byfollowed the bcc by alloys,the bcc thenalloys, the then fcc th alloyse fcc and beingalloys the highest and being for the the highest hcp alloy. for the hcp alloy.

Figure 5. Total entropy as a sum of temperature dependent thermal entropy Sthermal and constant

composition dependent configurational entropy Sconf.

Figure 5. Total entropy as a sum of temperature dependent thermal entropy Sthermal and constant Figure 5. Total entropy as a sum of temperature dependent thermal entropy Sthermal and constant composition dependent configurational entropy Sconf. composition dependent conﬁgurational entropy Sconf.

Entropy 2018, 20, 654 8 of 12 Entropy 2018, 20, x FOR PEER REVIEW 8 of 12

TheThe contribution contribution to the to the Gibbs Gibbs free free energy energy is the is product the product of entropy of entropy and temperature. and temperature. Using Using thermal andthermal configurational and configurational entropy and entropy their sum, and their we can sum, examine we can their examine contributions their contributions to the Gibbs to the free Gibbs energy free energy separately. Figure 6a shows the energy contribution of thermal entropy. We can see a separately. Figure6a shows the energy contribution of thermal entropy. We can see a clear order with the clear order with the alloy crystal structure: the highest contribution by thermal entropy is given by alloy crystal structure: the highest contribution by thermal entropy is given by the hcp-alloy,mainly because the hcp‐alloy, mainly because of the high value of molar heat capacity at low temperatures (−120 °C) of the high value of molar heat capacity at low temperatures (−120 ◦C) and also at room temperature. Pure and also at room temperature. Pure nickel, noble and Cantor, all fcc-structures, overlie each other nickel, noble and Cantor, all fcc-structures, overlie each other over a wide temperature range. The lowest over a wide temperature range. The lowest heat capacities and therefore thermal entropy heatcontributions capacities and are therefore given by thermal the bcc entropy (4 and contributions5 component) are alloys. given Using by the the bcc sum (4 and of thermal 5 component) and alloys.configurational Using the sum entropy, of thermal the total and entropy configurational times theentropy, temperature the total is shown entropy in Figure times the6b. temperatureAlloys with is shownthe same in Figure crystal6b. structure Alloys with and the configurational same crystal entropy structure are and close configurational to each other. entropy For example, are close the to total each other.entropy For example, input to Gibbs the total free entropy energy for input fcc- toNi Gibbs is smaller free than energy for forfcc fcc-NiCantor is and smaller fcc nobel than metal for fcc alloys. Cantor andTo fcc explore nobel the metal difference, alloys. To two explore materials the with difference, fcc‐structure, two materials pure nickel with and fcc-structure, Cantor alloy, pure are nickel drawn and ◦ ◦ Cantorin Figure alloy, 7a are in drawn a temperature in Figure 7rangea in a from temperature 600 °C to range 1600 from °C. Dashed 600 C tolines 1600 showC. Dashedthe configurational lines show the configurationalentropy contribution entropy to contribution Gibbs free toenergy Gibbs of free both energy materials, of both calculated materials, by calculatedEquation (3). by EquationWhile the (3). Whilecontribution the contribution of configurational of configurational entropy rises entropy with riseshigher with temperatures higher temperatures for fcc‐Cantor for alloy, fcc-Cantor fcc‐nickel alloy, fcc-nickeldoes not does exhibit not any exhibit configurational any configurational entropy and, entropy theref and,ore, therefore, has no contribution has no contribution to Gibbs free to Gibbsenergy. free energy.Continuous Continuous lines linesshow show the product the product of the of temperature the temperature and and the thetotal total entropy. entropy. Thus Thus the the influence influence of of configurationconfiguration and and thermal thermal input input can canbe quantified be quantified at certain at certain temperatures. temperatures. At 1000 At ◦ 1000C the °C gap the between gap continuousbetween continuou and dasheds and lines dashed is almost lines similar,is almost meaning similar, thatmeaning at this that temperature at this temperature there is athere negligible is a differencenegligible in thedifference influence in the of thermalinfluence entropy, of thermal but justentropy, the chemical but just compositionthe chemical ofcomposition the solid-solutions of the affectssolid the-solutions varying energy affects contributions. the varying The energy energy contributions. level of total Theentropy energy contribution level of is abouttotal entropy90 kJ/mol contribution is about 90 kJ/mol for fcc-nickel and 108 kJ/mol for fcc‐Cantor. The difference of 18.0 for fcc-nickel and 108 kJ/mol for fcc-Cantor. The difference of 18.0 kJ/mol is very close to fcc-Cantor kJ/mol is very close to fcc‐Cantor configurational entropy contribution of 1273 K∙R∙ln(5) = 17 kJ/mol configurational entropy contribution of 1273 K·R·ln(5) = 17 kJ/mol (T·Sconf) at this energy level, resulting (T∙Sconf) at this energy level, resulting that thermal entropy has an equal impact for materials with the that thermal entropy has an equal impact for materials with the same crystal structures (in this case fcc) same crystal structures (in this case fcc) and thus cannot play role in stabilizing a solid solution phase and thus cannot play role in stabilizing a solid solution phase in multicomponent alloys. Figure7b shows a in multicomponent alloys. Figure 7b shows a bigger section of the area near the melting interval of biggerboth section materials. of the The area two near values the meltingof melting interval enthalpy of both 13.3 materials. kJ/mol and The 15.0 two kJ/mol values have of melting been detected enthalpy 13.3by kJ/mol DSC and and it 15.0is obvious kJ/mol that have differences been detected in entropy by DSC contribution and it is obviousto Gibbs thatfree differencesenergy still inremains entropy contributiondependent to from Gibbs the free configurational energy still remains entropy dependent until the end from of the the configurational solid state. Investigations entropy until on thethe endfcc- of thenoble solid alloy state. instead Investigations of fcc‐Cantor on the yield fcc-noble to similar alloy results instead and of fcc-Cantorare, therefore, yield not to shown similar in results Figure and 7 in are, therefore,detail. not shown in Figure7 in detail.

Figure 6. Contribution of thermal entropy (a) and total entropy (b) of nickel and all high entropy Figure 6. Contribution of thermal entropy (a) and total entropy (b) of nickel and all high entropy alloys to Gibbs free energy. The horizontal arrows refer to Figure 7 with magnified details of a special alloys to Gibbs free energy. The horizontal arrows refer to Figure7 with magniﬁed details of a special temperature range. temperature range.

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Figure 7. Contribution of total and configurational entropy of fcc‐nickel and fcc‐Cantor to Gibbs free Figure 7. Contribution of total and conﬁgurational entropy of fcc-nickel and fcc-Cantor to Gibbs free energy in the range from (a) 600 °C to 1600 °C and from (b) 1100 °C to 1600 °C. energy in the range from (a) 600 ◦C to 1600 ◦C and from (b) 1100 ◦C to 1600 ◦C.

Gibbs energy has another input besides entropy, namely thermal enthalpy, Hthermal. The temperatureGibbs energy dependence has another of input the enthalpy, besides entropy, calculated namely with thermal Equation enthalpy, (5), is displHthermalayed. The in Figuretemperature 8a, dependencewhile Figure of 8b the,c enthalpy,show thermal calculated Gibbs withfree energy Equation and (5), total is displayedGibbs free inenergy, Figure respectively.8a, while Figure Similar8b,c showto Sthermal thermal, Hthermal Gibbs slightly free energy depends and on total the type Gibbs of freethe crystal energy, structure respectively. having Similar smallest to Svaluesthermal,H forthermal the slightlybcc alloys depends and onhighest the type values of thefor crystalthe hcp structure alloy and having shows smallest almost valuesno dependence for the bcc on alloys the number and highest of valuescomponents for the and hcp alloy composition and shows almost(Figure no8a). dependence As a result of on such the numberbehavior, of the components thermal part and of the alloy compositionGibbs free (Figureenergy,8 a).∆G Asthermal a, result also does of such not behavior, depend on the the thermal number part and of the concentration Gibbs free energy, of the alloying∆Gthermal , alsoelements, does not but depend its temperature on the number dependence and concentrationis slightly stronger of the for alloying the hcp elements,alloy and weaker but its temperaturefor the bcc dependencealloys relative is slightly to the strongerfcc materials for the (Figure hcp alloy 8b). andOn the weaker other for hand, the bccthe alloystotal Gibbs relative free to energy, the fcc materials∆Gtotal, which additionally includes the configurational term, has a stronger temperature dependence and (Figure8b). On the other hand, the total Gibbs free energy, ∆Gtotal, which additionally includes the configurationalthus becomes term,noticeably has a smaller stronger at temperature higher temperatures dependence for andthe alloys thus becomes with larger noticeably configurational smaller at higherentropy temperatures (Figure 8c). for These the observations alloys with larger indicate configurational that, although entropy the contribution (Figure8c). of S Thesethermal to observations the Gibbs free energy is much higher than that of than Sconfig, at any given temperature ∆Gthermal is nearly the indicate that, although the contribution of Sthermal to the Gibbs free energy is much higher than that of than same for the simple and complex alloys of the same type of crystal structure, i.e., Sthermal does not play Sconfig, at any given temperature ∆Gthermal is nearly the same for the simple and complex alloys of the same any role in stabilizing a solid solution phase in complex, multicomponent alloys: On the other hand, type of crystal structure, i.e., Sthermal does not play any role in stabilizing a solid solution phase in complex, Sconfig increases with the number of constituents, which noticeably decreases ΔGtotal of a multicomponent alloys: On the other hand, S increases with the number of constituents, which multicomponent solid‐solution relative to thatconfig of pure metals, especially at high temperatures. noticeably decreases ∆Gtotal of a multicomponent solid-solution relative to that of pure metals, especially Moreover, in some specific cases ∆Gtotal of a multicomponent solid‐solution with high Sconfig can at high temperatures. Moreover, in some specific cases ∆Gtotal of a multicomponent solid-solution with become smaller than ∆Gtotal of competing intermetallic phases, resulting in a single-phase solid high S can become smaller than ∆G of competing intermetallic phases, resulting in a single-phase solutionconfig alloy. total solid solution alloy.

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FigureFigure 8. 8.Temperature Temperature dependence dependence ofof ((aa)) thermalthermal enthalpy, enthalpy, ( (bb)) thermal thermal part part of of Gibbs Gibbs free free energy energy and and (c)( totalc) total Gibbs Gibbs free free energy energy for for the the studied studied materials.materials.

4.4 Conclusions. Conclusions In this work the heat capacity, entropy, enthalpy and Gibbs free energy of six different single- In this work the heat capacity, entropy, enthalpy and Gibbs free energy of six different single-phase phase solid solution alloys (three fcc‐alloys, two bcc‐alloys and one hcp‐alloy) were experimentally solid solution alloys (three fcc-alloys, two bcc-alloys and one hcp-alloy) were experimentally determined determined and investigated over a wide temperature range. and investigated over a wide temperature range. • At room temperature (RT), molar heat capacities of the studied alloys are close to 3R, in • Ataccordance room temperature with the (RT),Dulong molar-Petit heat law. capacities of the studied alloys are close to 3R, in accordance • withThe the measured Dulong-Petit RT heat law. capacities of the studied alloys are in good agreement with the heat • Thecapacities measured calculated RT heat using capacities the ruleof of themixture studied of pure alloys elements are in. good agreement with the heat • capacitiesThermal calculatedentropy and using thermal the ruleenthalpy of mixture increase of while pure the elements. thermal part of the Gibbs free energy • Thermaldecreases entropy with an and increase thermal in enthalpytemperature. increase The temperature while the thermal dependence part of of the these Gibbs quantities free energy is decreasesthe strongest with for an increase the hcp alloy in temperature. and weakestThe for temperaturethe bcc alloys, dependence with the fcc of alloys these showing quantities intermediate behavior. For the alloys with the same crystal structure, the thermal contributions is the strongest for the hcp alloy and weakest for the bcc alloys, with the fcc alloys showing to the Gibbs free energy do not depend on the number and concentration of the alloying intermediate behavior. For the alloys with the same crystal structure, the thermal contributions to elements. Therefore, in spite the thermal entropy, Sthermal, is much higher than the configurational the Gibbs free energy do not depend on the number and concentration of the alloying elements. entropy, Sconf, at T > 20 °C, Sthermal does not increase thermal stability of a solid solution phase in Therefore, in spite the thermal entropy, Sthermal, is much higher than the conﬁgurational entropy, complex, multicomponent◦ alloys relative to simple alloys or pure metals. • SconfThe, atcompositional T > 20 C, Sthermal dependdoesence not of the increase Gibbs thermalfree energy stability of the of solid a solid solution solution alloys phase is totally in complex, due multicomponentto the configurational alloys entropy. relative toOf simple the same alloys crystal or pure structure, metals. solid solution alloys with higher • TheSconf compositional have stronger dependence temperature of dependence the Gibbs freeof the energy Gibbs of free the energy solid solution and smaller alloys ΔG istot totallyal values due toat the a given conﬁgurational temperature. entropy. Thus, although Of the same being crystal smaller structure, than Sthermal solid, only solution Sconf contributes alloys with to higher the Sconfthermalhave stability stronger of temperature the complex, dependence multicomponent of the solid Gibbs solution free energy alloys. and smaller ∆Gtotal values at a given temperature. Thus, although being smaller than Sthermal, only Sconf contributes to the thermal stability of the complex, multicomponent solid solution alloys.

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Author Contributions: Conceptualization, U.G., R.V., S.H., M.M.; Methodology, S.H., M.M., S.G.; Software, S.G.; Validation, S.H., M.M., O.N.S.; Formal Analysis, S.H., O.N.S.; Investigation, S.H., M.M., O.N.S.; Resources, O.N.S., J.F., M.F., M.M.; Data Curation, S.H., O.N.S.; Writing-Original Draft Preparation, S.H.; Writing-Review & Editing, S.H., M.M., O.N.S., J.F., M.F.; Visualization, S.H., M.M.; Supervision, U.G. Funding: This research was funded by German Research foundation (DFG), i.e., during the projects GL 181/50, GL 181/56, FR 1714/7 and FE 571/4, the three last within the priority programme SPP2006 “Compositionally Complex Alloys—High Entropy Alloys (CCA-HEA)”. O.N. Senkov acknowledges financial support through the Air Force on-site Contract FA8650-15-D-5230 managed by UES, Inc., Dayton, OH, USA. Conflicts of Interest: The authors declare no conflict of interest.

References

1. Cantor, B.; Chang, I.T.H.; Knight, P.; Vincent, A.J.B. Microstructural development in equiatomic multicomponent alloys. Mater. Sci. Eng. A 2004, 375–377, 213–218. [CrossRef] 2. Hsu, C.; Yeh, J.; Chen, S.; Shun, T. Wear resistance and high-temperature compression strength of Fcc

CuCoNiCrAl0.5Fe alloy with boron addition. Metall. Mater. Trans. A 2004, 35, 1465–1469. [CrossRef] 3. Yeh, J.W.; Chen, S.K.; Lin, S.J.; Gan, J.Y.; Chin, T.S.; Shun, T.T.; Tsau, C.H.; Chang, S.J. Nanostructured high-entropy alloys with multiple principal elements: Novel alloy design concepts and outcomes. Adv. Eng. Mater. 2004, 6, 299–303. [CrossRef] 4. Brocq, L.M.; Akhatova, A.; Perrière, L.; Chebini, S.; Sauvage, X.; Leroy, E.; Champion, Y. Insights into the phase diagram of the CrMnFeCoNi high entropy alloy. Acta Mater. 2015, 88, 355–365. [CrossRef] 5. Otto, F.; Dlouhý, A.; Pradeep, K.G.; Kubˇenová, M.; Raabe, D.; Eggeler, G.; George, E.P. Decomposition of the single-phase high-entropy alloy CrMnFeCoNi after prolonged anneals at intermediate temperatures. Acta Mater. 2016, 112, 40–52. [CrossRef] 6. Tsai, K.Y.; Tsai, M.H.; Yeh, J.W. Sluggish diffusion in Co-Cr-Fe-Mn-Ni high-entropy alloys. Acta Mater. 2013, 61, 4887–4897. [CrossRef] 7. Zhang, W.; Liaw, P.K.; Zhang, Y. Science and technology in high-entropy alloys. Sci. China Mater. 2017, 61, 2–22. [CrossRef] 8. Yeh, J.W. Recent Progress in High-Entropy Alloys. Ann. Chim. Sci. Mater. 2006, 31, 633–648. [CrossRef] 9. Yeh, J.W.; Chang, S.Y.; Der Hong, Y.; Chen, S.K.; Lin, S.J. Anomalous decrease in X-ray diffraction intensities of Cu-Ni-Al-Co-Cr-Fe-Si alloy systems with multi-principal elements. Mater. Chem. Phys. 2007, 103, 41–46. [CrossRef] 10. Vaidya, M.; Trubel, S.; Murty, B.S.; Wilde, G.; Divinski, S.V. Ni tracer diffusion in CoCrFeNi and CoCrFeMnNi high entropy alloys. J. Alloys Compd. 2016, 688, 994–1001. [CrossRef] 11. Yang, X.; Zhang, Y. Prediction of high-entropy stabilized solid-solution in multi-component alloys. Mater. Chem. Phys. 2012, 132, 233–238. [CrossRef] 12. Senkov, O.N.; Miracle, D.B. A new thermodynamic parameter to predict formation of solid solution or intermetallic phases in high entropy alloys. J. Alloys Compd. 2016, 658, 603–607. [CrossRef] 13. Gaskell, D. Introduction to the Thermodynamics of Materials; Taylor & Francis: Washington, DC, USA, 1995. 14. Miracle, D.B.; Senkov, O.N. A critical review of high entropy alloys and related concepts. Acta Mater. 2017, 122, 448–511. [CrossRef] 15. Senkov, O.N.; Wilks, G.B.; Miracle, D.B.; Chuang, C.P.; Liaw, P.K. Refractory high-entropy alloys. Intermetallics 2010, 18, 1758–1765. [CrossRef] 16. Freudenberger, J.; Rafaja, D.; Geissler, D.; Giebeler, L.; Ullrich, A.; Kauffmann, A.; Heilmaier, M.; Nielsch, K. Face Centred Cubic Multi-Component Equiatomic Solid Solutions in the Au-Cu-Ni-Pd-Pt System. Metals 2017, 7, 135. [CrossRef] 17. Feuerbacher, M.; Heidelmann, M.; Thomas, C. Hexagonal High-entropy Alloys. Mater. Res. Lett. 2014, 3, 1–6. [CrossRef] 18. Greenwood, N.N.; Earnshaw, A. Chemistry of the Elements, 2nd ed.; Elsevier: New York, NY, USA, 1997. 19. Gali, A.; George, E.P. Tensile properties of high- and medium-entropy alloys. Intermetallics 2013, 39, 74–78. [CrossRef] 20. Lide, D.R. Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, USA, 2013. Entropy 2018, 20, 654 12 of 12

21. Jin, K.; Mu, S.; An, K.; Porter, W.D.; Samolyuk, G.D.; Stocks, G.M.; Bei, H. Thermophysical properties of Ni-containing single-phase concentrated solid solution alloys. Mater. Des. 2017, 117, 185–192. [CrossRef] 22. Gibbs, J.W. On the Equilibrium of Heterogeneous Substances. Am. J. Sci. Arts 1878, 16, 1820–1879. [CrossRef]

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