Basic and Applied Research: Section I Critical Evaluation and Optimization of the Thermodynamic Properties and Phase Diagrams of the AI-Mg, AI-Sr, Mg-Sr, and AI-Mg-Sr S y s t e m s P. C h a r t r a n d a n d A.D. P e l t o n C e n t r e d e R e c h e r c h e en C a l c u l T h e r m o c h i m i q u e Ecole P o l y t e c h n i q u e P.O. B o x 6 0 7 9 , S t a t i o n "Centre Ville" M o n t r e a l , Quebec, C a n a d a , H3C 3 A 7 (Submitted May 3, 1994; in revisedform October 4, 1994) All available thermodynamic and phase diagram data were critically assessed for all phases in the AIMg, AI-Sr, and Mg-Sr systems at 1 bar pressure from room temperature to above the liquidus temperatures. For these systems, all reliable data were simultaneously optimized to obtain a set of model equations for the Gibbs energy of the liquid alloy and all solid phases as functions of composition and temperature. The modified quasi-chemical model was used for the liquid. The AI-Mg-Sr ternary phase diagram was calculated from the optimized thermodynamic properties of the binary systems. Since no reliable ternary data were available, three assumptions were made: no ternary terms were added to the model parameters for the thermodynamic properties of the liquid, no ternary solid solutions are present in the system, and no ternary compound is present in the system. The calculated ternary phase diagram is thus a first approximation, which can be improved by the addition of new experimental data and can be used as a base for the calculation of phase diagrams of multicomponent systems. [A-A] + [B-BI =2 [A - B I Introduction Strontium is used, like sodium, in aluminum cast alloys containing silicon to modify the acicular structure of the A1-Si eutectic. Timminco Ltd., which produces A1-Sr master alloys for the aluminum industry, has patented master A1-Sr-Mg compositions with an increased Sr content. To control the production of ingots of these master alloys, the A1-Mg-Sr phase diagram is required, but no satisfactory experimental phase diagram is currently available. The prediction of the phase diagram is possible from the thermodynamic optimizations of the three binary systems using appropriate models. [82Mur] reviewed the literature and optimized the A1-Mg system. Other optimizations of this system were performed by [77Sab], [86Lud], [90Saul, and [93Zuo]. [89Alc] studied and optimized the A1-Sr system. [91Sri] also made an optimization of this system. [86Nay] made the only optimization of the Mg-Sr system. Because more recent thermodynamic data are now available, and in order to obtain more complete and precise results, the present authors decided to reoptimize the three binary systems before calculating the ternary phase diagram. A ternary diagram calculation requires that the same model be used in all binary systems for each phase present in the ternary field. For the liquid phase, the present authors used the modified quasi-chemical model [86Pel], which is well adapted to describe the ordered liquid in the A1-Sr system. The model considers the Gibbs energy for the formation of two A-B bonds from one A-A bond and one B-B bond (see Eq 1). (Eq 1) Expressions for enthalpies and entropies of mixing are written in terms of the bond fractions XAA,Xse, and Xae, and in terms of (m-qT) which is the Gibbs energy change of the bond exchange reaction (Eq 1). The equilibrium bond fractions are obtained by setting: | (Eq 2) 3XA8; while taking account of the mass balances. This results in an "equilibrium constant" for the bond exchange reaction (Eq 1). Details of the model were presented by [86Pel]. The fixed parameters of the model in the present evaluation are "coordination numbers" ZA1 = bAiX,ZMg = bMgZ,and Zsr = bsrZ where Z = 6, bAl --- 1.3774, bMg --- 0.9183, and bsr = 2.0661. These parameters are those used in previous evaluations, which are forming a data base on metalhc liquid solutions. The thermodynamic properties of A1, Mg, and Sr used in this evaluation are given in Table 1 and apply to Eq 3: H =A + ; CvdTJ/g-atom 298 15 S=B+;29815 (C[/T~tr J/K . g-atom Cp = a + b (10-3)T+ c (105)T -~+ d (10-6)T2 J/K. g - a t o m Journal of Phase Equilibria Vol. 15 No. 6 1994 (Eq3) 591 S e c t i o n I: B a s i c a n d A p p l i e d R e s e a r c h Table 1 Thermodynamic Properties of Elements and Compounds Relative to Elements at 25 ~ A B a b Elements (fromliterature) AI(L)(298.15to 933 45 K) ................... 10 711.00 AI(L)(>933 45 K) ............................... 8 680.87 Mg/L)1298.15 to923 K) ...................... 847700 MglL) (>923K/ ................................... 5 943.07 Sr(L) (298.15to 2000 K)..................... 1001931 Al(s)(298.15to 933.45 K) . . . . . . . . . . . . 0 Mg(s) (298.15to 923 K) ....................... 0 (c(Srl (298 15to829 3 K) . . . . . . . 0 (TSr)1829.3to 1042 15K) . . . . . . . . . 200668 39 7961 35.2537 41.8612 36.5117 61.1563 28.3215 32.6770 52.3000 56.1759 31.3758 31 7482 21.3886 32.6352 30.9616 31.3758 21.3886 22.2170 12.6775 -16 3929 ... 11.7780 ... .. -16.3929 11.7780 13 8909 26.7776 34.3321 35.7317 34.8738 24.9642 34.5783 39 8740 35.9122 38 3116 38 5300 40.0430 27.5451 25.5212 27.1812 29.5441 24.8229 28.9911 22.8758 21.5472 21.5600 21.6648 5.5876 0.12105 -4.5611 -10.3362 ~5.2983 ~).24156 12.0004 12.1826 12.2151 12.4823 -3.6066 20.7526 -3.6066 20.7526 -2.2233 -1 4924 -2 0918 -2.8853 -2.4044 -1.6831 12.7927 8.5873 12.0365 16.6021 13.8350 9.6850 Compounds(optimized) AI45Mg281298.15to 724.25K) ............. All2Mgt71298.15to 733.65K) .......... R (Alll5sMg~42)1576.14to 674.58 K)... AhSr 1298.15to 1298.25K) . . . . . . . AI_,Sr1298.15to 1195.45K). . . . . . . . A17Sr~1617.97to 937.1 K).................... Mg17Sr2(298.15to 879.24KI . . . . . . Mg~sSr9(298.15to 871.72K)............... Mg23Sr61298.15to 876.2K)................. Mg2Sr(298.15to 953.23 K).................. -306.45 -250.00 153.14 -31 210.0 -28 447.5 -19 279.3 -1 985.68 -3684.85 -4050.00 7 106.34 Note: See Eq 3. For A1, S01298) for the solid phase, A~H0, At~,~S0` and expressions of Cp for solid and liquid were taken from [77Bar], who also fixed the melting point at 660.3 ~ Below the melting point, Cp(L) was set equal to Cp(s). For Mg, Afu~H0and AtusS0 were taken from [85Cha], who also fixed the melting point at 649.85 ~ S0(298) for the solid phase and expressions of Cp for solid and liquid were taken from [77Bar]. Below the melting point, Cp(L) was set equal to Cp(s). For Sr, S01298) for the o~Sr solid phase, Afu,H~ Afu,S0, and expressions of Cp for solid allotropes and liquid were taken from [77Bar], who also fixed the melting point at 660.3 ~ AtrsH~and AusS0 were also taken from [77Bar]. The properties of the compounds listed in Table 1 were obtained from optimizations performed in this work. Equations 4 through 7 provide optimized parameters of the modified quasi-chemical model for the liquid. - b.,x,n (Eq 4) L, b x + b , x , , where m, n = A1, Mg, and Sr (Z = 6, bAl = 1.3774, bMg = 0.9183, and bsr = 2.0661 ). ForA1-Mg: c o = - 1 6 757.8 + 3 395.7 Ys,.-5 632.7 ~ r J/g-atom rl = - 9 + 4 Ysr J/K. g-atom (Eq 7) The AI-Mg System Equilibrium Diagram The AI-Mg system was studied by numerous authors, who reported a great many liquidus, solidus, and solvus points. A review of these data was published by [82Mur]. Most of the present calculated phase diagram (Fig. 1 and 2) agrees with this review. [82Mur] stated that the equilibrium solid phases are: the (A1)fcc solid solution with a m a x i m u m solubility of Mg of 18,9 at.% at the eutectic temperature of 450 ~ the (Mg)cph solid solution with a maximum solubility of AI of 11.8 at.% at the eutectic temperature of 437 ~ the ~fcc solid solution; the 7 solid solution with the c~Mn structure; and the R phase at 42 at.% Mg (also designated e). Results of [81Sch] are not considered in the review of [82Mur]. The (A1)fcc solidus points measured by [81Sch] agree in this region with the experimental results of [36Kaw], [39Sie], [40Kurl], [45But], and [79Sti]. co = - 9 8 1 3 . 4 + 6169.4 YMg- 1536 Y~g J/g-atom r I = 4 + 4 YMgJ/K. g-atom (Eq 5) For A1-Sr: co = - 4 7 938.5 + 2 863,4 Ysr + 20 283 ~ r J/g-atom r I = - 8 J/K- g-atom For Mg-Sr: 592 (Eq 6) In their optimizations, [90Saul and [93Zuo] included another phase named ~ (from [77Sch]), and [86Lud] considered the phase and not the R phase. The results of [89Goe] supported the conclusion of [82Mur], but stated that the R phase is stable between 305 + 5 and 405 + 5 ~ instead of 320 and 370 +_5 ~ proposed by [82Mur]. [89Goe] studied several diffusion couples of Al, Mg, and their alloys in graphite dies between 280 and 435 ~ during 5 to 21 days. The following phases were observed: (Al)fcc, (Mg)cph, [3, 7, and R. Journal of Phase Equilibria Vol. 15 No. 6 1994 Basic a n d A p p l i e d R e s e a r c h : S e c t i o n I <F*A *C'T> 700 9 V T X 0 9 660.3 600 500 - [20Hart] [290~x] [32Sa1] [33Sch] [35Zak] [35Kaw] [3BBun] [3BHum] ! 9 [38Kur] A [39Sle] 9 [40Kur] [] [45But] 9 [72Ten] 9 {77Sch] 0 (79Stl] + [81Sch] LIGUIO 451+~ 0 (A1) fcc f695_~+L_. 450&1 437+I Gamma 4O0 g L I= 111 300 200 100 0 O0 0 20 0.40 0 60 Mo]e F r a c t i o n o f Mg A1 0.80 :1 0 Mg Fig. 1 OptimizedA1-Mgphasediagram. <'s 500 . . . I . ' [['20'Ha'n] I [36Kaw] [37F:m] [38Hum] [3BKun] [40KuP] [45But] Z~ 475 9 [] I . . . . ' I ' I . . . . I X V 0 LIQUID 9 9 + 460 5+--:1 *C'T> [50"Ma~<] [B5Sam] [70Ban] [70Br'a] [77Sch] (8~Scn] X + 437+_1 Gamma 425 0 o.i 401+_5 400 A 375 350 z AV ~ o~ ' 'A' 0 35 '~1 . 0.40 ] 9 , I . . . 0.45 A1 . . . . I 0 50 Mole . . . . I 0 55 Fraction , 0 60 of Mg Fig. 2 35 to 75 at.% Mg region of the optimized AI-Mg phase diagram. Crystal Structures and Lattice Parameters The crystal structures and lattice parameters were discussed by [82Mur], who mentioned an uncertainty concerning the exact number of atoms per unit cell for the [3 phase. In the present evaluation, the hypothetical stoichiometry of the [~compound 0 65 0 70 .7 Mg is A145Mg28as proposed by [65Sam]. Otherwise the evaluation of [82Mur] is accepted. Thermodynamics The enthalpy of mixing of the liquid was determined cap orimetrically by [30Kaw], [71 Bat], and [91Aga]. [76Bha] and Journal of Phase Equilibria Vol. 15 No. 6 1994 593 S e c t i o n I: Basic and Applied Research <F*A *C'T> j ....... % ~N~ I ......... f ......... I ......... V 9 1X 9 ~]~N~ , o ~ I ......... [30Kaw) [69Be]) [71Batt [75Bha] (83Kaz] J ......... 800Oc 800~ 800~C 800OC 700OC -- I ......... I ......... I ......... 9 [B6uun] 800~ ZS [9~Aga) 670~ 0 [s 574~ZC D [91Aga] 5750C 0 [glAga] 700~ / / ." 00%. ") i ~ - /~ Colcolate~ ~75Oc o~,4~ I ...... 7 o ~ x . -2.0 x v x x x ~v @ % ii -3.0 9 v V ..1-, LI.I -4.0 V V -5.0 ........ 0 O0 I ......... O.aO A] I ......... 0 20 I ......... 0.30 1 ......... I ......... I ......... 0 40 0.50 0 60 Mole F r a c t a o n of Mg I ......... 0.70 ] ......... I ........ 0 BO 0 90 Mg Fig. 3 Calculated enthalpy of mixing of AI-Mg liquid alloys at 675 ~ [86Jun] derived the enthalpy of mixing from partial pressure measurements, while [69Bel] and [87Tiw] obtained the enthalpy of mixing from emf measurements. The methods used by [83Kaz] are not clear. The results of [91Aga] appear to be reliable. Systematic errors were limited by the use of three different calorimetric methods. From their results, the liquid is a regular solution with a minimum enthalpy of mixing of -2.2 kJ/mol at 50 at.% Mg. Results are shown in Fig. 3. The activity of Mg in the liquid alloy was determined with emf measurements by [62Ere], [69Bell, [69Tsy], and [87Tiw]. [41Sch], [71Vya], [76Bha], and [86Jun] measured the partial pressure of Mg. All results are scattered but show a small negative deviation from ideality, except for those of [69Tsy]. See Fig. 4. By the integration of DTA curves, [78Prel) obtained the enthalpy required to heat the solid phases from a temperature just below the eutectic or the solidus to a temperature just above the liquidus. [80Tim] obtained the enthalpy of fusion of the 13 phase. Results of [78Pre 1] and [80Tim] are shown in Fig. 5. The activity of Mg in solid phases at 387 and 437 ~ was determined by [70Bro] using emf techniques. See Fig. 6 and 7. The liquid was modeled by the modified quasi-chemical model using the results of [91Aga] for the enthalpy of mixing and experimental liquidus points of the (Al)fcc and (Mg)cph phases. Measurements of the activity of Mg in the liquid were not used directly for the optimization process because of the high dispersion of data but were used for validation of calculated parameters. Figure 8 shows the calculated entropy of the liquid at 675 ~ the excess entropy, Se• is less than -1 J/tool K 9 over the entire composition range. In Fig. 3 and 4, the calculated enthalpy of mixing and Mg activity in the liquid are compared with the experimental data. All the solid phases were modeled by taking into account the experimental results of [70Bro] and [78Pre 1], in concert with the solvus, solidus, and liquidus of the phase diagram. The (Mg)cph solid solution was modeled as a Henrien solution. The calculated parameters are: R T I n ]PAl= 12 740-- l l.2T J/g-atom The (Al)fcc phase was modeled as a Henrien solution w~th an additional regular mixing term as follows: G e x = X A I X M g ( 1 2 0 0 0 - 10.72T)+ 150XMg J/g-atom Optimization o f T h e r m o d y n a m i c and Phase Diagram Properties The calculated optimized A1-Mg phase diagram is shown in Fig. 1. Figure 2 shows the 35 to 75 at.% Mg region. The calculated parameters for the solid phases are presented in Table l and Eq 8, 9, and 11. The calculated parameters for the liquid phase are shown in Eq 5. Calculated invariant points are listed in Table 2. 594 (Eq 8) (Eq9) For these phases, the mutual solid solubilities considered in the present evaluation are those of [82Mur] (-18.9 at.% Mg in A1 and -11.5 at.% A1 in Mg). The atomic radii ratio of A1 and Mg is 1.12, which suggests high mutual solubilities. For calculation, the ~ phase was assumed to be stoichiometric. This assumption has little effect on the solvus of the (Al)fcc and y phases because of the small range of homogeneity of the 13phase. See [37Fin], [38Kur2], [65Sam], and [70Bro]. The en- Journal of Phase Equilibria Vol. 15 No. 6 1994 Basic and Applied Research: Section I <F*A *C'T> ].0 Y~ [415ch] 800Oc 9 [71Luk] 650~ 0 .g0 9 [4cScn] 850~ 9 [6PEre] 650~ 0 [76Bha] 800~ 0.80 O [69Be1] 850Oc [] [85dun] 800~ ~7 [5gTsy] A [B7Taw] 8000C 0.70 x 0.60 ~" o5o "~ 0.40 9 [71Vya] 800Oc B00~ --Calculated A 800~ V 0.30 0 0.20 V / I I F oA 0 10 O 9 O~ 0 O0 0.10 0 40 0 30 0 20 0 50 0 60 O. 70 0 80 0,90 1.0 Mg Mole Fractaon of Mg A1 Fig. 4 Calculated activity of Mg in A1-Mg liquid alloys at 800 ~ <F'A *C'T> <F'A *C'~T> I 0 9 [70Bro} 387~ 0 90 ~0 0 BO [ ~ oI 9 [80T 1m] \ 0 70 o, ..... o o ..... , t / o Go 0 50 l o~ O 40 ....... O 10 ........ 0 oo 0 ~0 AI 0 20 0 30 0 40 0 50 0 60 Mo]e FPectlon of Mg 0 70 0 ao 0 90 ~ c Mg Fig. 5 Calculatedenthalpy to heat AI-Mg solid alloys from a temperature just below eutecUc or solidus to a temperaturejust above liquidus. thalpies of fusion of [78Prel] and [80Tim] were used with the temperature of fusion (estimated at 451 ~ to calculate the entropy of fusion of the compound. The optimized entropy of fusion for the [3phase is 10.94 J/K. g-atom, which is reasonable. The yphase was modeled with a defect model [90Li], which is similar to the Wagner-Schottky model. This model incorporates the Gibbs energy associated with the defects on each side of the stoichiometric composition (in this case A112Mg17). The entropic term for the expression of the Gibbs energy of the phase is expressed in terms of the mole fractions of the majority point defects on the Al-rich (XA) and Mg-rich (XM) sides of the stoichiometric composition. The Gibbs energy of forma- f / 0 30 0 20 o 9 / O oo 0 I0 AI 0 20 0 30 0 4o 0 50 Mole Fraction o( 0 60 Mg 0 70 0 eo 0 90 Ng Fig. 6 Calculated activity of Mg in A1-Mg solid alloys at 387 ~ tion of the majority defects are added to this entropic term. The Gibbs energy of the phase is then given by: G = RT[XAlnX A + (1 - Xz)ln(1 - XA) ] + RT[XMlnX M + (1 - XM)ln(1 -XM)] + GAX z + GMX M (Eq 10) For a fixed deviation from the stoichiometric composition, equilibrium mole fractions of the majority defects X A and XM can be calculated to obtain the Gibbs energy of the phase. The optimized expression for the Gibbs energy at the stoichiometric composition Al12Mg17 is given in Table 1. The optimized expressions for the Gibbs energies of formation of the majority Journal of Phase Equilibria Vol. 15 No. 6 1994 595 Section I: Basic and Applied Research <F*A *C'T> o 90 o 9 [70Bro] 6o ~37~ ~ 50 80 / 40 0 vO o~o 30 o 50 o 20 40 9 9 f 9 / 1 0 ........ <F'A i ......... i . . . . . i ........ i ......... i ......... i ......... i ........ * C ' T > r ......... o 30 F 0 ~0 O 0 10 sE -1 O0 0 ~0 0 20 0 30 AI 0 .40 0 50 Mo]e r m a c t l o n 0 60 o' Mg 0 70 0 80 0 gO 0 defects on the Al-rich and Mg-rich sides of Al12Mg17 are, respectively: (Eq 1 l) The optimized entropy of fusion for the '/phase (at the composition of the maximum of the azeotrope) is 10.27 J/K. g-atom. This value is reasonable. The linear expression for the Gibbs energy of the R phase, which is considered stoichiometric for calculation purposes (42 at.% Mg), is given in Table 1 and was obtained from the formation and decomposition temperatures of [89Goe] (305 + 5 ~ and 405 _+5 ~ In Fig. 5 to 7, the calculated "enthalpy of fusion" of solid phases and the calculated Mg activities in the solid phases are compared with experimental data. The principal differences between the present optimization and those of [90Sau] and [93Zuo] are their inclusion of a phase and their use of an optimized enthalpy of mixing, which is an average of the various measurements in Fig. 3, whereas in the present study the results of [91Aga] are given preference. The AI-Sr System Equilibrium Diagram [89Alc] reviewed the A1-Sr system. According to their evaluation, the stable phases of the system are: the liquid, the (Al)fcc solid solution, the (TSr)bcc solid solution, the (c~Sr)fcc solid solution, and three intermetallic compounds--Al4Sr, A12Sr, and AlvSrs. The solid phases considered in the present evaluation (Fig. 9 and 10) are the same, but the interpretation of the phase diagram is somewhat different. The proposed phase diagram of [89Alc] was largely inspired by the experimental points of [75Bru]. The present evaluation is based on the work of [86Clo], which is more consistent with thermodynamic principles. [75Bru], supported by [79Vak], reported that the melting point of A14Sr is -1040 ~ whereas [86C1o] observed a melting point at 1025 ~ In a first attempt, [74Vak] proposed a melting point at 1000 ~ [75Bru] suggested that Al2Sr is associated with a peritectic reaction at 936 _+2 ~ and their results are supported by [79Vak], who placed the reaction at 940 ~ 596 ........ oo J ......... o Io i ........ o 20 A] Mg Fig. 7 Calculated activity of Mg in A1-Mg solid alloys at 437 ~ G A= 18 200 J/g-atom G M= 30 2 5 0 - 14T J/g-atom 0 i ~ 0 30 ...... J ~ 0 40 .. I ......... J ........ 0 5o 0 60 Mole F r a c t i o n o f Mg t ......... 0 70 I ........ 0 eo i 0 90 Mg Fig. 8 Calculated entropy of AI-Mg liquid alloys at 675 ~ [86Clo] observed a thermal arrest at ~920 _+ 1 ~ and proposed that this was associated with a peritectic reaction. The description of the experiments of [86Clo] is more complete, and their results are self-consistent in the 0 to 70 at.% Sr range of composition. Results of [74Vak], [75Bru], and [79Vak] are not supported by a good description of the experimental methods. The flatness of the liquidus of A12Sr as proposed by [75Bin] requires an unreasonable entropy of fusion of this compound (>25 J/K. g-atom), while the experimental liquidus points of A12Sr of [86C1o] agree with a more reasonable entropy of fusion. The present evaluation considers that A1,Sr is a congruent compound and that there is a eutectic reaction L---~A14Sr+A12Sr at 920 ~ with a eutectic liquid composition at ~32 at.% St. This interpretation agrees better with the experimental points of [86Clo] and is more probable than the proposed peritectic reaction by reason of symmetry of the liquidus of AI4Sr. Interpretations of previous authors show a liquidus of A14Sr very asymmetric on either side of the stoichiometric composition. The peritectic reaction L+A12Sr-+AlvSr8 at N664 ~ is placed at 56 at.% Sr as proposed by [86C1o] (who reported A1Sr instead of AITSrs). Another difference between the present evaluation and the conclusions of [89Alc] is the liquidus of ySr. As demonstrated by [89Alc] and as discussed below, the solubility of A1 in ySr should be very small, so the limiting slope of the liquidus ofySr at Xsr---~l (which is related to the enthalpy of fusion and the temperature of fusion of Sr by Eq 12) must be more negative than the proposed slope of [89Alc]. (Eq 12) In the present evaluation, the experimental liquidus of ],Sr was not used in the optimization of the liquid, but was calculated afterwards from the optimized thermodynamic properties of the liquid (and the properties of ySr from the literature). The calculated temperature of the eutectic L----~AITSrs+'ySris that of [86Clo] at 580 ~ the eutectic liquid composition is more uncertain and was set at -70 at.% Sr from the experimental points of [86Clo]. Journal of Phase Equilibria Vol. 15 No. 6 1994 Basic and Applied Research: 1025+_5 1000 13 900 2 ~~ .~ -~A - ~ 800 ~ 700 [] [74Yak] 0 [75Bru] A [79Yak] V [83Han] + [85Sat] 9 [86C1o) A &' D I <F*A *C'T> 1~.00 U Section O A A Liquid 0 769 0 Z~ kD [] A A ~ rO (_ 580+_2 ~ D 600 /~a ~ 9 L F- D" 556.5 =~- 5O0 C- O0 ? 400 345+_25 300 ~ 0 0O 0 40 0.60 Mole Fraction of Sr O. 20 A1 0 8O .0 Sr Fig. 9 Optimized AI-Sr phase diagram. <F*A*C*T> 680 I [83Han] [85Sat] [86C1o] V + +/ . . . . i . . . . 9 670 o V LIQUID L + A]4Sr V 660 (At) 9 / + L v v 654+:I V V V V (AI) [)50 I , 0 O0 , , , I , , , , I , , , T7 V V V + AI4Sr , 0 005 0 010 A1 Mole Fraction of Sr Fig. 10 Al-rich region of the optimized A1-Sr phase diagram. 0 015 0 020 Sr Thermodynamics The enthalpy of mixing of the liquid was determined cap orimetrically by [83Som] and [85Esi] (see Fig. 11), and as shown in [89Alc], the results are in good agreement for the composition of the minimum of the curve (35 to 40 at.% Sr). The activity of Sr in the liquid alloy was measured by [74Bur], Journal of Phase Equilibria Vol. 15 No. 6 1994 597 Section I: Basic and Applied Research <F*A *C*T> <F*A *C'T> / ~~ ~ o [835oml 852~C o (8350m] 857os z~ [835om] 902~ i D :;2 ...... ~ 7gZOc .I . . . . . . . . . ~ ........ I, o ~o o 20 0 30 o 40 24 __, o oo AI ~Ole I .... o 5o Fraction / ......... J ......... i ....... ; ......... o 6o o zo o 80 o 90 o o f Sr Sr x 0 -2 0 -3 0 -4 0 5 0 60 00 0 10 ^~ 0 20 0 3o 0 4o Mo]e s 0 50 0 6o of 5r 0 7o 0 80 0 90 0 Sr Fig. 13 Calculated entropy of A1-Srliquid alloys at 1050 ~ The activity of Sr in the liquid alloy was measured by [74Bur], [79Vak], and [91Sri]. See Fig. 12 for In Tsr. Experiments of [9 l Sri], who used Knudsen and pseudoisopiestic methods at 1050 ~ were more complete and detailed. [74Bur] also measured activities with the Knudsen weight loss method between 850 and 1100 ~ Estimations of the thermodynamic properties of solids were given by [84Kha] but were judged incomplete and obsolete by [89Alc]: this conclusion is accepted in the present evaluation. Optimization of Thermodynamic Properties and Phase Diagram The calculated optimized A1-Sr phase diagram is shown in Fig. 9 and 10. The optimized properties of compounds are listed in Table 1. The optimized parameters for the liquid phase are listed in Eq 6, and calculated invariant points are presented in Table 2. The liquid phase was modeled using the results of [83Som] and [85Esi] for the enthalpy of mixing and the activity of Sr given by [91Sri]. As discussed earlier, no experimental liquidus points of TSr were used in the optimization process. Figure 13 598 0 20 [91Srl] 1050~ [748ur] 1027OC 0 30 0 ao 0 50 Ho}e F'~actlon 0 60 of qr 0 70 0 ao o 90 Sr Fig. 12 Calculated activity coefficient of Sr in AI-Sr liquid alloys at 1050 ~ shows the calculated entropy of the liquid at 1050 ~ In Fig. 11 and 12, the calculated enthalpy of mixing and In '~Sr are compared with the experimental points. <F*A *C'T> I 0 10 A] Fig. 11 Calculated enthalpy of mixing of AI-Sr liquid alloys at 797 and 1500 ~ / oo 9 D For calculation, all solid phases were presumed stoichiometric, although a very small solubility of Sr in A1 was observed (-0.0077 at.% Sr in AI at 600 ~ as discussed by [89A1c]. [89A1c] rejected the -5.5 at.% solubility of A1 in Sr reported by [79Vak]; this conclusion is accepted in the present work. The calculated eutectic in Fig. 10 is in good agreement with experimental results of [83Han], [85Sat], and [86Clo], so the assumption of only a very small solubility of Sr in A1 is justified. The atomic radii ratio is -1.5 also suggesting a very small solubility of A1 in St. More precise measurements should be made for the determination of the solubility of A1 in Sr. [39Now] reported that the "A1Sr" phase (A17Sr8 as demonstrated by [83For]) decomposed below 300 ~ The calculated entropies of fusion of the intermetallic compounds are 17.7 J/K- g-atom for AI4Sr, t6.1 J/K g-atom 9 for AI2Sr, and 9.0 J/K. g-atom for AlvSr8. These are reasonable values, The Mg-Sr System Equilibrium Diagram [86Nay] reviewed the Mg-Sr system. From their conclusions, the stable phases are: the liquid, the (Mg)cph solid solution, the (TSr)bcc solid solution, the ((~Sr)fcc solid solution, and four intermetallic compounds--MglvSr 2. Mg3sSr~, Mg~3Sr~, and Mg2&; In the present evaluation (Fig. 14 and 15), the region between 10 and 20 at.% Sr and the liquidus of the Sr allotropes differ from the phase diagram proposed by [86Nay]. [86Nay] suggested that the liquidus points of ~Sr and ySr correspond to the experimental points of [47Ray]. In the present evaluation, experimental points of [73Bro] are considered because they respect the theoretical limiting slope of the liquidus of Sr at Xsr---~1 (Eq 12) if a negligible solubility of Mg in Sr allotropes is present; moreover [73Bro] and [47Ray] are both Ph.D. theses produced Journal of Phase Equilibria Vol. 15 No. 6 1994 Basic a n d Applied Research: S e c t i o n I Table 2 Calculated Special Points of the AI-Mg-Sr System Temperature, Reaction ~ Phase AI Mg Sr Reaction type 100 0 0 0 0 100 0 0 0 0 100 100 Melting Melting Melting Allotropic 0 0 EutecUc Eutectic Congruent Eutectic Eutectic Congruent Eutectic Eutectic Eutectic Peritectoid Eutectoid Pure components L <---)(Al)fcc ........................... L ~ (Mg)cph ........................ L <--->ySr ................................. 7Sr~-> ccSr ............................. 660.3 649.8 769.0 556.5 AI-Mg s y s t e m L ~ (A1)fcc + 13..................... 450 + 1 L <--->~ .................................. L+-> [3 + (y) ........................... 451-+1 449_+2 L ~ (T) ................................. L<--> (y) + (Mg)cph ................. 460 + 1 437 -+ 1 [3 + (y) ~--)R ........................... R +--~13+ (y) ............................ 401 -+5 303 -+ 10 L (A1)fcc L L (7) ... L (7) (Mg)cph (y) (y) 63.8 81.8 . . 57.8 53.7 45.4 29.9 37.4 11.3 50.7 46.3 . 36.2 18.2 . . 42.2 46.3 54.6 70.1 62.6 88.7 49.3 53.7 . . . 0 0 0 0 0 0 0 0 AI-Srs y s t e m L ~-r (A1)fcc + A14Sr............... 6 5 4 -+ 1 L +-) A h S r ............................. L ~-~ A14Sr + AI2Sr ................. L ~--~AlzSr ............................. L + AlzSr ~ AIvSr8 ................ L <---)A17Sr8 + ySr ................... AlvSr8 ~ AlzSr + ~ S r ............ 1025 -+ 5 9 2 0 -+ 2 922 -+ 2 664 _+.3 580-+2 345 -+ 25 L (A1)fcc . . . L . . L L . . . 99.13 100 . . . . . 68.4 . _. _ _ . 44.4 31.1 . . . . 0 0 0.87 0 .. 3"1.6 . 0 . . . . . 0 0 . 55.6 68.9 . . Eutectic Eutectic Congruent Eutectic Congruent Pentectic Eutectic Eutectoxd Mg-Sr s y s t e m L ~--) (Mg)cph + MglTSr2 ....... 587-+3 L <-'>M g l 7Sr2 ......................... L ~-> Mg17Sr2 + Mg38Sr9 ........ L + Mg23Sr6 <--->Mg38Sr9 ........ L + Mg2Sr <--->Mg23Sr6 ........... L ~-> Mg2Sr ............................ L <---->Mg2Sr + otSr .................. 606-+2 591 + 2 599 + 3 603 -+ 3 680 + 3 4 2 6 -+ 2 L (Mg)cph . . L L L . . L 0 0 . . . . . . . 0 0 0 . . . . 0 . . . 93.8 100 . . 85.0 82.9 81.3 . . 30.1 6.2 0 69.9 Eutectic Eutectic Congruent Eutectic Peritectic Peritectlc Congruent Eutectic . 15.0 17.1 18.7 . AI-Mg-Sr system L ~-> Mg2Sr + A12Sr ............... L <--->(Mg)cph + A h S r ............ 565 538 L L (Mg)cph 26.1 17.0 4.6 40.6 79.6 95.4 33.3 3.4 0 Saddle Saddle Saddle 521 L 17.2 62.5 20.3 Peritectic 517 509 L L 16.7 15.7 64.3 68.3 19.0 15.9 Peritectxc Saddle 509 501 L L (Mg)cph 15.6 16.4 1.5 67.6 73.4 98.5 16.8 10.2 0 Eutectic Peritectic Peritectlc L <-->(Mg)cph + Mgl7Sr2 + A12Sr ................................. 496 L <---)(y) + A14Sr...................... 460 L ~ 13+ AlnSr ........................ L <--->(A1)fcc + A14Sr + [~......... 451 450 L + A12Sr <-->AI7Sr8 + Mg2Sr.. L +-~A14Sr + (y) + [~................ 450 449 L +-~ (Mg)cph + (y) + A14Sr .... 437 L ~ AI7Sr8 + Mg2Sr + ~ S r .... 412 L (Mg)cph L (7) L L (A1)fcc L L (7) L (Mg)cph (y) L 15.1 1.2 45.4 45.4 61.5 63.8 81.8 17.8 57.8 53.7 29.9 11.3 37.4 4.8 73.9 98.8 54.6 54.6 38.4 36.2 18.2 23.8 42.1 46.3 70.1 88.7 62.6 27.3 11.0 0 0.0306 0 0.0156 0.0147 0 58.4 0.0156 0 0.0354 0 0 67.9 Eutectic Eutectlc Saddle Saddle Saddle Eutectic Eutectic Peritectic Eutectlc Eutectlc Eutectlc Eutectic Eutectlc Eutectic L + Mg2Sr ~ A12Sr + Mg23Sr6 .............................. L + Mg23Sr6 ~ Mg38Sr9 + A12Sr ................................... L <--->MglTSr2 + AI2Sr ............ L <---~Mg38SI9 + Mgl7Sr2 + AI2Sr .................................. L + AlaSr <-->(Mg)cph + AI2Sr Journal of Phase Equilibria Vol. 15 No. 6 1994 599 S e c t i o n I: Basic and Applied Research <F*A *C'T> BOO 769 : 0 [39Vos] LIQUID [47Kle] 0 0 [47Ray]L]qu]dus 680+_3 700 9 [47Ray] S a ] : d u s A [73Bro]L]qulgus 649 8 {~ o 9 [73Bro] [] 0 So]lgus 605+_2 t_ 600 A~ 0 0 ~/~ 4J ro \ % 500 [] 556 5 ~ eJ 2/" : 426+_2 400 300 0 20 0 O0 0 40 0 60 Mole F r a c t i o n Mg of Sr 0.80 1 0 Sr Fig. 14 Optimized Mg-Sr phase diagram. in the same laboratory under the supervision of EA. Kanda. Hence, it may be assumed that [73Bro] is an improvement of the work of [47Ray]. The 10 to 20 at.% Sr region of the phase diagram is subject to more than one interpretation as stated by [86Nay]. The interpretation of [86Nay] shows a large asymmetry for the liquidus of MglvSr 2 on either side of the stoichiometric compound, which is thermodynamically very unlikely. In addition, since all four compounds of this system are close in composition and no decomposition at lower temperature is observed, then their entropies of formation must be approximatively equal. The interpretation of [86Nay] requires that the entropy of formation of MglvSr2 and Mg23Sr6 be high in order to produce flat liquidus curves. However, this results in a calculated eutectoid decomposition at relatively high temperature of one or more of the four compounds. In our interpretation of this region, the assumption was made that all four intermetallic compounds are stable at room temperature because no decomposition was observed experimentally. For these reasons, then, Mg38Sr 9 is associated with a peritectic reaction (L+Mgz3Sr6--~Mg38Sr 9) at 599 ~ and with a eutectic reaction L--->MglvSr2+Mg38Sr9 at 591 ~ [86Nay] proposed this interpretation as an alternative. Crystal S t r u c t u r e s and Lattice Parameters All necessary information is described in the review of [86Nay]. Thermodynamics The available thermodynamic data include the enthalpy ot" mixing of the liquid measured by [77Som] at 807 ~ (Fig. 16). The minimum of the enthalpy of mixing is between 30 and 35 at.% Sr. [80Som] measured the activity of Mg in liquid at 757 600 ~ using a modified Ruff boiling point method (Fig. 17) and estimated that the minimum of the excess entropy of the liquid is approximately -2 J/K. mol at 30 at.% Sr. [64Kin] measured the enthalpy of formation of MgzSr from solid Mg and Sr. A tin solution calorimeter was used to obtain a value of-21.35 kJ/mol. O p t i m i z a t i o n of T h e r m o d y n a m i c and P h a s e Diagram Properties The calculated optimized Mg-Sr phase diagram is shown in Fig. 14 and 15. The optimized Gibbs energies of the compounds are shown in Table 1. The optimized parameters for the liquid phase are listed in Eq 7, and calculated invariant points are presented in Table 2. The liquid was modeled by using the experimental values of enthalpy of mixing of [77Som], the activity of Mg in the liquid of [80Som], and the liquidus curves of Mg and Sr allotropic phases of [73Bro]. Figure 16 shows the calculated enthalpy of mixing at 807 ~ along with the experimental results of [77Som]. The calculated entropy of the liquid at 807 ~ is presented in Fig. 18: the excess entropy curve has a minimum of -2.45 at 32 at.% Sr in agreement with the evaluation of [80Sore]. Figure 18 shows the calculated activity of Mg in liquid alloys at 757 ~ in comparison with results of [80Som]. All solid phases were assumed to be stoichiometric for the calculations. This is valid for intermetallic compounds and for the Mg solid phase where the solubility of Sr is stated to be negligible by [39Vos], [47Kle], and [47Ray]. [47Ray] and [73Bro] observed a nonnegligible solubility of Mg in Sr (14.5 at.% and 6.5 at.% at the eutectic temperature, respectively). Only a very small solubility of Mg in Sr is predicted by the difference in Journal of Phase Equilibria Vol. 15 No. 6 1994 Basic and Applied Research: Section I ooo[ <F*A *C'T> 65O 650 .Z~ [39Vos] [47Kle] 640 I ~ / [47Ray] [73Bro) 630 o [3 Z~ 620 LIQUID A [] 610 C._ 606+2 ~0 0 600 603+_3 & A 587+3 590 580 0 % 570 ' A 0 0 0 0 [~ Z~ 0 0 591+_2 0 o % o~@ A~ o 2 , ' , I . . . . 0 05 013 & 5~ A 0 [] <2 0 A A ] , , , t . . . . ] , 0 10 0 ~5 Mole Fract2on of 5r Mg , [ . , , J . 0 20 , , , 0 25 Sr Fig. 15 Mg-richregion ofthe optimized Mg-Sr phase diagram. <F*A *C'T> 0 ......... r ......... -2 0 i ........ i ......... 9 i ......... [77Som] MD i ........ ~ ......... 807~ Mole F r e c t l O n I ........ I ......... I. . . . 9 Of Sr Sr Fig. 16 Calculated enthalpy of mixing of Mg-Sr liquid alloys at 807 ~ atomic radii (more than 34%) and by observation of Mg solubility in similar systems (Mg-Ca and Mg-Ba). The calculated parameters for the solid intermetallic phases give no decomposition at room temperature, and the calculated entropies of fusion of these phases are 7.87 J/K g-atom 9 for Mg17Sr2, 9.31 J/K 9g-atom for Mg38Sr~, 9.51 J/K g-atom 9 for Mg23Sr6, and 11.34 J/K. g-atm for Mg2Sr, which are reasonable values. The AI-Mg-Sr System Equilibrium Diagram [80Mak] studied solid solubility at 400 ~ in the Mg-rich region of the system by X-ray analysis, micrography, and microhardness. A ternary compound was reported and named "X,'" of unknown stoichiometry, in equilibrium with (Mg)cph, "MggSr" (MglTSr2), and Al12Mg17 (7). [81Makl] performed micrography, X-ray analysis, and microhardness tests on 200 alloys of A1, Mg, and Sr quenched from 400 ~ Large solubilities were observed in the solid phases, Another ternary compound was reported (different from "X') with a stoichiometry of"A134Mg6Sr60" (A16MgSrm). [80Mak2] studied the liquidus surfaces in Al-rich and Mg-rich regions of the A1-A14SrAt3Mgz([3 ) and Mg-MgjvSQ-AIt2Mgl7(7) systems. The authors considered MglvSr2-Alj2Mg17(7) and A14SrA13Mg2(I3) as simple eutectic quasi-binary systems. A ternary eutectic was reported at 71 at.% Mg, 27 at.% A1, and 2 at.% Sr at 430 ~ and another at 35 at.% Mg, 63.6 at.% AI, and 1.4 at.% Sr at 445 ~ [82Makl] studied pseudobinary sections in the ternary system. The system AlnMglT(7)-Mg17Sr 2 was considered quasi-binary with a eutectic at 32 tool% Mg17Sr2 at 438 ~ with a solubility at TE of MglvSr2 in Al12Mg17 o f - 3 tool% and of All2Mgl7 in Mgl7Sr2 of 46 mol%. The system Al3Mg2([~)-MglvSr2 was reported to be a simple eutectic type with eutectic at 38 mol% MglTSr~ at 439 ~ with a solubility at TE of Mgl7Sr 2 in A13Mg2 of -5 tool%, and of AI3Mg 2 in Mg 17Sr2 of 36 mol%. The system AI4Sr-Mg2Sr was considered to be a simple eutectic type with eutectic at 40 tool% A14Sr at 610 ~ with a solubility at TE of Mg2Sr in AI4Sr of 48 mol%, and of A14Sr in Mg2Sr of 25 mol%. The system A14Sr-MglySr2 was considered quasi-binary with a eutectic at 46 tool% A14Sr at 571 ~ with a solubility at TE of A14Sr in MglvSQ of 26 mol%, and of Mgj 7Sr2 in A14Sr of 35 tool%. The system A14SrA13Mge(~3) was reported to have one eutectic at 85 tool% A13Mg 2 at 446 ~ with a solubility at Te ofAl3Mg 2 in A14Sr of 18 mol%, and of AlaSr in A13Mg2 of 3 mol%. There is no mention of the "S" phase of [81Mak 1] in this study. [82Mak2] used Journal of Phase Equilibria Vol. 15 No. 6 1994 601 S e c t i o n I: B a s i c a n d A p p l i e d R e s e a r c h < F * A * C ' T > < F * A 9C 60 0 9C ........ q ......... i ........ i ......... t ................. i ......... r ......... i ......... * C ' T > i ....... 5o ~ " q 8C 9 {80Soml "t 0 7C 75voc \ 30 20 0 5C I 0 0 40 0 0 30 0 20 0 q 10 2 0 0 oo o ~o o 2o Mg o 30 o 40 o 5o ~ole f r a c t i o n o 60 of Sr o 70 o eo o 90 o Sr 0 oc - - - - SE 0 10 Mg 0 2o 0 3d 0 4o 0 5o 0 60 Mole rrectlon of Sm 0 7o 0 ao 0 90 UP 0 ~1 % Fig. 17 Calculated activity of Mg in Mg-Sr liquid alloys at 757 Fig. 18 Calculated entropy of Mg-Sr liquid alloys at 807 ~ eC. their previous results to approximate the liquidus surface of the Sr-AI4Sr-Mg2Sr system by a simplex method from 19 judiciously chosen liquidus points. The ternary "'S" phase was considered in this study. Thermodynamics No data are available for the liquid or the possible ternary phases. Calculated Phase Diagram The calculated AI-Mg-Sr phase diagram is shown in Fig. 19 and 20 considered are: the liquid, the (A1)fcc binary solid solution, the (Mg)cph binary solid solution, c~Srand ]'Sr allotropic solid phases, the binary y solid solution, and the stoichiometric intermetallic compounds--l, R, A14Sr,AlzSr, A1jSrs, Mg 17Sr2, Mg38Sr9, Mg23Sr6, and Mg2Sr. Calculated ternary uninvariant points are listed in Table 2. The thermodynamic properties of the liquid were estimated from the optimized binary parameters by the modified quasichemical model for the ternary liquid phase with the symmetric approximation [86Pel, 93Eri] with no adjustable ternary terms added since no ternary thermodynamic data are available. No ternary solid phase was considered in the present evaluation because of the uncertainties related to them (existence, stability and homogeneity range, melting or decomposition temperature, etc.). The extensive solubilities between solid phases reported by [80Mak], [81Makl], [81Mak2], and [82Makl] seem unlikely for solids of such different crystal structures and stoichiometry. The three solid solutions in the AI-Mg system--(A 1)fcc, (Mg)cph, and ]'--were not extended into the ternary field due to a lack of pertinent information. Because of these uncertainties and because no measured ternary phase equilibrium data are available for comparison, no error estimates for the calculated ternary invariant temperatures are given in Table 2. The liquidus surface of the calculated phase diagram is divided into thirteen pnnlary crystallization fields: (A1)fcc, (Mg)cph, ocSr, ySr, (]'), 13, A14Sr, AI2Sr, AIvSr~, Mg]7Sr2, Mg38Sr 9, Mg23Srr, and MgxSr. Liquidus surfaces of A14Sr, A12Sr, and 602 Mg2Sr dominate the liquidus surface. Three quasi-binary systems are observed: AI4Sr-]3 (it must be noted that the 13phase was supposed stoichiometric: thus this system might actually not be quasi-binary), A12Sr-Mg2Sr, and A12Sr-Mgl7Sr2. Two systems, A14Sr-(y) and Al4Sr-(Mg)cph, are not quasi-binary but have a maximum on their common univariant line. Six ternary eutectic invariants and four ternary peritectic invariants are observed on the calculated liquidus surface. The 400 ~ isothermal section of [81 Mak 1] shows a triangulation involving Mg, MglySr2, and the y-phase. Given the low thermodynamic stabilities of these compounds relative to A14Sr and A12Sr, this seems unlikely. The 400 ~ section of [81 Mak 1] also shows triangulations involving A14Sr in equilibrium with Mg]7Sr2, Mg23Sr 6, and Mg2Sr. In the present optimization, these compounds are calculated to be in triangulations with A12Sr. This seemingly large difference actually only requires a moderate change to the ternary phase diagram. If the univariant line on Fig. 19 and 20, which starts in the A1-Sr binary and descends to point Pc, were to descend instead to a point on the univariant line between the MgzSr and A12Sr fields just to the right of point P4, then the triangulations of [81Makl] would be reproduced. However, given the questions raised above regarding the results of [81Makl ] and the inconsistencies among the various publications of the same group [80Mak, 81Makl, 81Mak2, 82Makl, 82Mak2], it was decided not to attempt to reproduce the reported ternary sections by introducing ternary parameters to the liquid model or ternary solid solubilites. Rather, the calculated ternary phase diagram is presented as a starting point for future experimental verification and refinement. Conclusions A self-consistent set of model equations for the Gibbs energy of the phases is required to permit the calculation of the ternary AI-Mg-Sr phase diagram. Hence, the present evaluators critically evaluated and reoptimized the thermodynamic properties of the binary AI-Mg, AI-Sr, and Mg-Sr systems with all available data in order to calculate the A1-Mg-Sr phase diagram Journal of Phase Equilibria Vol. 15 No. 6 1994 Basic and Applied Research: Section I Mg (650} 4510C El 412 E2 437 450Oc E3 449 Ea 450 509~ 5380C E5 496 555Oc E5 50s - 450 P2 - PI 501 P3 - 517 P4 - 521 A] s~4 AI sr ( t 0zl25) Sr AI7Sr 8 664 g2OA]2Sr (922} (660) (759) Mole Fractlon Fig. 19 AI-Mg-Sr calculated phase diagram in atomic percent. Mg (550) - 41; 451OC E2 - El 43 460Oc E3 - 44! 509Oc E4 - 451 538Oc E5 - 491 565Oc E6 - 50! Pl - 451 P2 - 50 P3 - 511 P4 - 52 A] 654 AI4Sr HOa5) (660) 9eoAl 19~2) Sr A17SrB554 Welght Percent 58D SP (769) Fig. 20 A1-Mg-Sr calculated phase diagram in weight percent. from these binary parameters. Available thermodynamic data in the binary systems were reproduced within experimental errors, with a minimum number of coefficients. For the ternary system, this work is a first step to a more complete evaluation that could be extended by the availability of new experimental results and by the addition of other components like silicon, manganese, etc. (for example, the AI-Mg-Si-C has already been optimized [94Ber)). Acknowledgment The authors are indebted to Dominique B~rub6 for her help in the calculation of the binary AI-Mg system. Journal of Phase Equilibria Vol. 15 No. 6 1994 603 Section I: B a s i c a n d A p p l i e d Research Cited References 20Han: D. Hanson and M.L.V. Gayler, "The Constitution of the Alloys of Aluminium and Magnesmm,"J. Inst. Met, 24, 201-232 (1920). 29Dixl: E.H. Dix and E Keller, "Equilibrium Relations in AluminiumMagnesium Alloys of High Purity," American Institute of Mining Metallurgical Engineering Tech. Publ., 187, 1- 17 ( 1929); J. Inst. Met, 41,488 (1929). 29Dix2: E.H. Dix and E Keller, "Equilibrium Relations in AluminiumMagnesium Alloys,"Z. 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