Modeling of Thermodynamic Properties and Phase Equilibria for the Cu-Mg Binary System

Transcription

1 Materals Scence and Engneerng Publcatons Materals Scence and Engneerng Modelng of Thermodynamc Propertes and Phase Equlbra for the Cu-Mg Bnary System Shhua Zhou Iowa State Unversty, Y Wang Pennsylvana State Unversty Frank G. Sh Unversty of Calforna - Irvne Ferdnand Sommer Max-Planck-Insttute Long-Qng Chen Pennsylvana State Unversty See next page for addtonal authors Follow ths and addtonal works at: Part of the Metallurgy Commons The complete bblographc nformaton for ths tem can be found at mse_pubs/165. For nformaton on how to cte ths tem, please vst howtocte.html. Ths Artcle s brought to you for free and open access by the Materals Scence and Engneerng at Iowa State Unversty Dgtal Repostory. It has been accepted for ncluson n Materals Scence and Engneerng Publcatons by an authorzed admnstrator of Iowa State Unversty Dgtal Repostory. For more nformaton, please contact dgrep@astate.edu.

2 Modelng of Thermodynamc Propertes and Phase Equlbra for the Cu- Mg Bnary System Abstract The phase equlbra assocated wth the bnary Cu-Mg system are analyzed by applyng results from frstprncples calculatons to a general soluton thermodynamcs treatment. Dfferng from prevously reported models, we employ a four-speces assocaton model for the lqud, whle the termnal and ntermedate sold phases are modeled as substtutonal solutons wth one or two sublattces, respectvely. The zero-kelvn enthalpes of formaton for the ntermedate compounds, Cu 2 Mg-C15 (cf24) and CuMg 2 -C b (of48) are computed usng the Venna Ab-nto Smulaton Package (VASP). The Gbbs free energy functons for the ndvdual phases are evaluated, and the resultng bnary phase dagram s presented over the full composton range. Whle the phase dagram we propose exhbts only modest devaton from prevously reported models of phase equlbra, our treatment provdes better agreement wth expermental reports of heat capacty and enthalpy of mxng, ndcatng a more self-consstent thermodynamc descrpton of ths bnary system. Keywords Ames Laboratory, Materals and Engneerng Physcs Dscplnes Metallurgy Comments Ths artcle s from Journal of Phase Equlbra and Dffuson 28 (2007): 158, do: /s Posted wth permsson. Rghts Copyrght 2007 ASM Internatonal. Ths paper was publshed n Journal of Phase Equlbra and Drstrbuton, Vol. 28, Issue 2, pp and s made avalable as an electronc reprnt wth the permsson of ASM Internatonal. One prnt or electronc copy may be made for personal use only. Systematc or multple reproducton, dstrbuton to multple locatons va electronc or other means, duplcatons of any materal n ths paper for a fee or for commercal purposes, or modfcaton of the content of ths paper are prohbted. Authors Shhua Zhou, Y Wang, Frank G. Sh, Ferdnand Sommer, Long-Qng Chen, Z-Ku L, and Ralph E. Napoltano Ths artcle s avalable at Iowa State Unversty Dgtal Repostory:

3 Secton I: Basc and Appled Research Modelng of Thermodynamc Propertes and Phase Equlbra for the Cu-Mg Bnary System Shhua Zhou, Y Wang, Frank G. Sh, Ferdnand Sommer, Long-Qng Chen, Z-Ku Lu and Ralph E. Napoltano (Submtted November 2, 2006) JPEDAV (2007) 28: DOI: /s ÓASM Internatonal The phase equlbra assocated wth the bnary Cu-Mg system are analyzed by applyng results from frst-prncples calculatons to a general soluton thermodynamcs treatment. Dfferng from prevously reported models, we employ a four-speces assocaton model for the lqud, whle the termnal and ntermedate sold phases are modeled as substtutonal solutons wth one or two sublattces, respectvely. The zero-kelvn enthalpes of formaton for the ntermedate compounds, Cu 2 Mg-C15 (cf24) and CuMg 2 -C b (of48) are computed usng the Venna Ab-nto Smulaton Package (VASP). The Gbbs free energy functons for the ndvdual phases are evaluated, and the resultng bnary phase dagram s presented over the full composton range. Whle the phase dagram we propose exhbts only modest devaton from prevously reported models of phase equlbra, our treatment provdes better agreement wth expermental reports of heat capacty and enthalpy of mxng, ndcatng a more self-consstent thermodynamc descrpton of ths bnary system. Keywords 1. Introducton thermodynamcs, phase dagram, phase equlbra Gven the promsng mechancal, chemcal, and magnetc propertes of varous metallc glasses and the potental wdespread use of the so-called bulk metallc glasses, [1-4] much recent attenton has been gven to the development of our general understandng of the thermodynamcs and knetcs for glass formaton and glass formng ablty. [5] Whle alloys exhbtng promsng glass formaton tendency are frequently ternary, quaternary, or hgher order systems, [1-4] several bnary glass-formng metallc alloys have been dentfed. Of these, the Al-based alloys have been nvestgated most thoroughly. [6] In addton, glass formaton has been dentfed n several other bnary systems, [7-12] most notably ncludng Cu-Mg, [7] Pd-S, [8,9] and Cu-Zr. [8] By vrtue of ther smplcty compared wth ther manycomponent counterparts, these bnary alloys present us wth an opportunty to nvestgate the detaled relatonshp between the thermodynamc descrptons of relevant phase Shhua Zhou, Y Wang, Long-Qng Chen, and Z-Ku Lu, Department of Materals Scence and Engneerng, The Pennsylvana State Unversty, Unversty Park, PA 16802, USA; Frank G. Sh, Department of Chemcal Engneerng and Materals Scence, Unversty of Calforna, Irvne, CA 92697, USA; Ferdnand Sommer, Max-Planck-Insttute for Metals Research, Hesenbergstraße 3, D Stuttgart, Germany; Shhua Zhou and Ralph E. Napoltano, Materals & Engneerng Physcs Program, Ames Laboratory, USDOE, Ames, IA, USA; Ralph E. Napoltano, Department of Materals Scence and Engneerng, Iowa State Unversty, 116 Wlhelm Hall, Ames, IA, USA. Contact e-mal: ralphn@astate.edu equlbra and observed glass-formng behavor. It s n ths ven that the present thermodynamc analyss of the Cu-Mg bnary system s motvated. In a subsequent paper, we wll model the thermodynamc propertes of the undercooled lqud and examne more closely the mplcatons regardng glass formaton tendency, wth respect to the competng crystallne phases. Thermodynamc models for the bnary Cu-Mg system have been offered by Coughanowr et al. [13] and by Zou and Chang. [14] Whle these treatments are well posed and have resulted n phase dagrams that agree well wth expermental reports, there are three specfc features of the modelng approaches that lmt ther potental applcablty to more general phase stablty problems. Frst, n each of these treatments, the lqud phase s modeled as a regular soluton, a model that cannot descrbe the chemcal orderng observed n ths system. For example, -ray dffracton experments reported by Lukens and Wagner [15] ndcate the exstence of chemcal short range order n the lqud phase wth Cu 2 Mg and CuMg 2 stochometres. Ths dscrepancy s clearly observed n the nablty of these models to accurately descrbe the heat capacty of the undercooled lqud, as shown n Fg. 1. Second, these prevous models treat the Mg 2 Cu-C b as a stochometrc compound, precludng any descrpton of nonequlbrum compostons for these phases. Fnally, the models constran the temperature dependence of the Gbbs free energy of ths phase to be lnear, mplyng that there s no contrbuton to the heat capacty from chemcal mxng. Consequently, whle the equlbrum phase dagram produced from these models may be useful, more fundamental thermodynamc quanttes assocated wth these models, such as the heat capacty, do not agree well wth expermental measurements, such as those reported by Feufel and Sommer. [16] In the work presented here, we address each of these lmtatons wthn the framework of a general CAL- 158 Journal of Phase Equlbra and Dffuson Vol. 28 No

4 Basc and Appled Research: Secton I The thermodynamc propertes of pure Cu and Mg n varous structures are computed usng the parameters from the SGTE database. [41] The lqud phase s descrbed wth an assocaton model. [42,43] The termnal fcc and hcp sold solutons are treated wth a sngle lattce whle ntermedate phases are descrbed usng a two-sublattce model, [44] where each sublattce s treated as a regular soluton. In the followng sectons, the thermodynamc treatment of each phase s descrbed n detal. Fg. 1 The heat capacty of the lqud Cu 14.5 Mg 85.5 alloy calculated usng the parameters n Tables 2-4 PHAD [17,18] formulaton by () employng an assocaton model capable of descrbng the nonlnear temperature dependence of chemcal short range order n the lqud phase and () treatng the ntermetallc phases as substtutonal sold solutons on two sublattces, permttng quantfcaton of the Gbbs free energes of the ntermetallc phases at non-stochometrc compostons. In addton, we ncorporate frst-prncples calculatons to compute the zero- Kelvn energes of end-member phases n unstable structures, snce expermental data are not avalable for these phases. Remanng model parameters, descrbng the Gbbs free energes assocated wth the formaton of compounds or soluton phases, are determned through a systematc sememprcal optmzaton, employng avalable expermental data from calormetry, -ray dffracton, electron-probe mcrochemcal analyss, and optcal mcrography. [7,16,19-40] The resultng thermodynamc propertes and the assocated equlbrum phase dagram are compared wth the pror thermodynamc modelng reported by Coughanowr et al. [13] and by Zou and Chang. [14] 2. Thermodynamc Models We descrbe the phase equlbra n the Cu-Mg bnary system by modelng the Gbbs free energy for each relevant phase over the approprate range of composton at constant pressure (1 atm). In any case where a temperature dependent parameter (P) s requred, we use the general form, PðTÞ ¼a þ bt þ ct ln T þ dt 2 þ et 1 þ ft 3 þ gt 7 þ ht 9 ðeq 1Þ 2.1 The Lqud Phase We employ a four-speces assocaton model, [42,43] where Cu 2 Mg and CuMg 2 are chosen as the relevant ntermedate chemcal assocates. [15] Thus, the Gbbs free energy of the lqud phase s gven by G lq m ¼ x G lq þ RT x ln x þ ð; j ¼ Cu; Cu 2 Mg; CuMg 2 ; MgÞ j> x x j L lq ;j ; ðeq 2Þ where x and G lq denote the mole fracton and molar Gbbs free energy for the th speces n the lqud phase, respectvely, and where we have consdered only par-wse nteracton between speces. The Gbbs free energy of the ntermedate assocates, Cu m Mg n, are gven as G lq Cu mmg n ¼ m G lq Cu þ n G lq Mg þ DGlq Cu mmg n ; ðeq 3Þ where DG lq Cu m Mg n represents the Gbbs free energy of formaton. Substtutng Eq 3 nto Eq 2, the Gbbs free energy of the lqud phase s wrtten as G lq m ¼ðx Cu þ2x Cu2Mg þx CuMg2 Þ G lq Cu þðx Mg þx Cu2Mg þ2x CuMg2 Þ G lq Mg þx Cu2 MgDG lq Cu þx 2Mg CuMg 2 DG lq CuMg 2 þrtðx Cu lnx Cu þx Cu2Mglnx Cu2Mg þx CuMg2 lnx CuMg2 þx Mg lnx Mg Þþx Cu x Cu2Mg L lq Cu;Cu þx 2Mg Cux CuMg2 L lq Cu;CuMg 2 þx Cu x Mg L lq Cu;Mg þx CuMg 2 x Cu2Mg L lq CuMg 2;Cu 2Mg þx Mg x Cu2 Mg L lq Cu 2 Mg;Mg þx Mgx CuMg2 L lq CuMg 2 ;Mg ðeq 4Þ The nteracton energes ð L lq ;j Þ are descrbed here as constants, whle the Gbbs free energes of formaton ðdg lq Cu m Mg n ) are descrbed as functons of temperature, usng the form gven n Eq 1. Where suffcent heat capacty data are avalable, the parameters a)e are used. Otherwse, we employ only parameters a and b. Ths wll be dscussed n further detal n a subsequent secton. 2.2 Termnal fcc and hcp Phases The fcc and hcp phases are treated as smple bnary substtutonal solutons wth Gbbs free energes expressed as G U m ¼ x GU þ RT x ln x þ xs G U m ðeq 5Þ ¼Cu;Mg ¼Cu;Mg Journal of Phase Equlbra and Dffuson Vol. 28 No

5 Secton I: Basc and Appled Research where x denotes mole fracton of element, and G U ð ¼ Cu, MgÞ denotes the molar Gbbs free energy of the pure element wth the structure UðU ¼ fcc or hcpþ. The excess Gbbs free energy xs G U m s expressed as, xs G U m ¼ x Cux Mg n j¼0 j L U Cu;Mg ðx Cu x Mg Þ j ðeq 6Þ where the j L U Cu;Mg coeffcents are left as nteracton parameters to be evaluated wth expermental data. Here, we consder only the j = 0 term and assume that L U Cu;Mg s constant (.e., a regular soluton). 2.3 Intermedate Cu 2 Mg-C15 and Mg 2 Cu-C b Phases Usng a two-sublattce model, we descrbe each ntermedate compound as a sold soluton of the form (Cu,Mg) 2 (Cu,Mg) 1. The Gbbs free energy s gven as G h m ¼ ¼Cu;Mg ð2y I ¼Cu;Mg y I j¼cu;mg ln y I þ y II y II j G h :j þ RT ln y II Þþ xs G h m ðeq 7Þ where the colon separates the components on dfferent sublattces, and y I and y II are the sublattce ste occupancy fractons. The superscrpt, h, represents the Cu 2 Mg-C15 or Mg 2 Cu-C b structure. Wth the sublattce descrpton we have adopted, each of the two structures (C15 and C b ) can assume four dfferent stochometres (Cu 2 Cu, Mg 2 Cu, Cu 2 Mg, and Mg 2 Mg), wth only one beng that of the stable compound. These are Cu 2 Mg-C15 and Mg 2 Cu-C b, and, for these compounds, we usng the form gven n Eq 1 and evaluate the related coeffcents usng avalable enthalpy of formaton, meltng temperature, and heat capacty data, as dscussed n a subsequent secton. For the unstable compounds (.e., Cu 2 Cu-C15, Mg 2 Cu-C15, Mg 2 Mg-C15, Cu 2 Cu-C b, Cu 2 Mg-C b, and Mg 2 Mg-C b ), there are no avalable heat capacty data wth whch to evaluate the coeffcents c-g. Therefore, we express the Gbbs free energy as express G C15 Cu:Mg and G C b Mg:Cu G h :j ¼ 2 G ref þ G ref j þ DG h :j ; ð; j ¼ Cu; MgÞ ðeq 8Þ where G ref and G ref j are the molar Gbbs free energy of ether fcc-cu or hcp-mg, and DG h :j s the Gbbs free energy of formaton for the compound () 2 (j) 1 from the 2() +(j) mxture, whch we treat as a constant. The excess Gbbs free energy term n Eq 7 s modeled as, xs G h m ¼yI Cu yi Mg y II k L h Cu;Mg: ðyi Cu yi Mg Þk þ y II Cu yii Mg ¼Cu;Mg k¼0 y I ¼Cu;Mg k¼0 k L h :Cu;Mg ðyii Cu yii Mg Þk ðeq 9Þ Once agan, we consder only the k = 0 term n each nner sum, and assume that L h s a constant. 2.4 Calculaton of Zero-Kelvn Enthalpes from Frst Prncples To facltate the determnaton of the Gbbs free energy of formaton for the ntermedate compound phases, we compute the enthalpy of formaton for the end-members at zero Kelvn. For these calculatons, we employ the VASP [45] mplementaton of the plane wave method usng the Vanderblt ultrasoft pseudopotental [46] wth a generalzed gradent approxmaton (GGA). [47] The Monkhost k ponts are employed for hgh precson calculatons. The 3s3p and 3d4s4p shells are treated as valence states wth core rad of 2.88 a.u. and 2.48 a.u. for the hcp- Mg and fcc-cu, respectvely. To ensure that the unt cell corresponds to a stable structure, we fully relax the cell shape and the nternal atomc coordnates of the stable endmembers of the compounds and relax only the cell volumes of those end-members whch are unstable. The enthalpy of formaton, DH / f, of a compound u s calculated as the dfference between the energy of the compound and the lnear combnaton of the energes of the pure elements n ther reference states, DH / f ¼ E / x / Cu Efcc Cu x/ Mg Ehcp Mg ; ðeq 10Þ where x / s the mole fracton of component n the / structure. The values of E / ; ECu fcc ; and Ehcp Mg are the computed zero-kelvn energes of the ndcated phases, each consdered here to be stochometrc. The calculated results are lsted n Table 1 and compared wth expermental data. Agan, we note here that for the stable phases (Cu 2 Mg-C15 and Mg 2 Cu-C b ), both expermental and frst-prncples data are consdered n our evaluaton of thermodynamc parameters. For the unstable ntermetallc phases (D H f > 0), however, expermental data are not attanable and the zero- Kelvn energes become essental n assessng the relatve stablty of these compounds. The determnaton of model parameters s dscussed n the next secton. 3. Determnaton of the Thermodynamc Model Parameters Expressons for the standard Gbbs free energy ð G / Þ of each pure component n the relevant phases are taken from the SGTE database [41] as lsted n Table 2. In addton, based on the models descrbed n the precedng sectons, we compute a standard Gbbs free energy for each of the two ntermedate compounds and evaluate the parameters, as lsted n Table 3. For the relevant excess Gbbs free energes, we employ a total of eght Gbbs free energy of formaton terms and 12 nteracton parameters (four of whch we assume to be equal to zero). These are lsted n Table 4, along wth the results from our parameter evaluaton. In the present secton, the methodology used for determnaton of the parameters lsted n Tables 3 and 4 s dscussed. 160 Journal of Phase Equlbra and Dffuson Vol. 28 No

6 Basc and Appled Research: Secton I Table 1 A summary of the results from the frst-prncples calculatons Phase Formula DH, kj/mol of atoms Frst prncples Modelng Experment 0 K 298 K Cu-fcc Cu 0 Mg-hcp Mg 0 Cu 2 Mg-C 15 Cu 2 Cu Cu 2 Mg )15.72 )11.4 )11.3 [36] )8.04 [37] CuMg Mg 2 Mg CuMg 2 -C b Cu 2 Cu Cu 2 Mg CuMg 2 )13.20 )9.6 )9.55 [36] Mg 2 Mg Table 2 The thermodynamc parameters for pure Cu and Mg [41] Cu phases G lq Cu G fcc Cu G hcp Cu T mn, K T max, K G ref G fcc fcc Cu G Cu a ) ) ) b ) c )31.38 ) ) d ) )3 e f )7 g ) )21 h Mg phases G lq Mg G fcc Mg G hcp Mg T mn T max G ref G hcp Mg 0 G hcp Mg 0 0 a ) ) ) b ) ) c ) ) ) d )4 e f ) )6 g ) )20 h Note: G h ¼ G ref þ a þ bt þ ct ln T þ dt 2 þ et 1 þ ft 3 þ gt 7 þ ht 9 (J/mol) For the lqud phase, we evaluate the parameters DG lq Cu m Mg n and L lq ;j, from Eq 2-3, usng reported values of actvty, [33] chemcal potental, [30-35] and enthalpy of mxng. [29,30] Intally we model DG lq Cu m Mg n usng only the frst two terms n Eq 1 (.e., a þ bt). Ths treatment, shown n Fg. 1, yelds a heat capacty for Cu 15.5 Mg 84.5 lqud that s slghtly hgher than the models of Coughanowr et al. [13] and Zou and Chang, [14] but one that remans much lower than the reported expermental data. [7] We assert that ths apparent excess heat capacty may be due to chemcal (and perhaps structural) orderng n the lqud phase over ths range of temperatures. Accordngly, we descrbe the assocated nonlnear temperature dependence by fttng the parameters c-e n Eq 1 for DG lq CuMg 2. The evaluated results are lsted n Table 4 and the correspondng heat capacty s also plotted n Fg. 1, where a maxmum n the C p (T) curve s exhbted. We note that such maxma n C lq p (T) have been observed n a number of systems (e.g., toluene, Au-S, Journal of Phase Equlbra and Dffuson Vol. 28 No

7 Secton I: Basc and Appled Research Table 3 Ground state standard Gbbs free energy parameters for the ntermetallc phases Cu 2 Mg-C15 and Mg 2 Cu-C b h G :j G C15 Cu:Mg, J/mol G Cb Mg:Cu, J/mol T mn, K T max, K G ref a )58201 ) )53491 ) b c )76.1 ) ) ) d ) ) )3 e f ) )6 ) )6 g h Table 4 Excess Gbbs free energy parameters (all temperatures) Phase Parameters Value, J/mol Lqud hcp fcc CuMg 2 -C15 CuMg 2 -C b DG lq Cu2Mg DG lq ) T ) T ) CuMg2 T ln (T) T T )1 L lq Cu;Cu2Mg )20012 L lq Cu;CuMg2 )24230 L lq Cu;Mg )22611 L lq Cu2Mg;CuMg2 0 L lq Cu2Mg;Mg )25845 L lq CuMg2;Mg 0 L hcp Cu;Mg:Va L fcc Cu;Mg:Va )19345 DG C15 Cu:Cu DG C15 Mg:Cu DG C15 Mg:Mg L C15 Cu:Cu;Mg )27868 L C15 Cu;Mg:Mg 3521 DG Cb Cu:Cu DG Cb DG Cb Mg:Mg L Cb Cu:Mg Cu:Cu;Mg 0 L Cb Cu;Mg:Mg 0 Regardng the ntermedate compounds and the evaluaton of coeffcents n the expressons for G C15 Cu:Mg and G C b Mg:Cu, as n Eq 1 and 7, we requre expermental or theoretcal values for heat capacty, enthalpy of formaton, and meltng temperature. The parameters that contrbute to the heat capacty (.e., coeffcents c-f n Eq 1 are evaluated usng the expermental results of Feufel and Sommer, [16] recognzng that, above ther meltng temperatures, the heat capactes of the Cu 2 Mg-C15 and Mg 2 Cu-C b phases should approach that of the lqud phase, as gven by the SGTE model. [41] The avalable enthalpy of formaton data nclude only the stable Cu 2 Mg-C15 and Mg 2 Cu-C b phases. They are lsted n Table 1 along wth our frst prncples calculatons, whch nclude these phases as well as the unstable compounds for whch expermental data are not avalable. In addton, we consder the electromotve force (EMF) measurements of actvty and chemcal potental, reported by Arta et al. [38] and Eremenko et al. [39] for the Cu-Mg bnary system. These are shown n Fg. 2 and compared wth the values of chemcal potental assocated wth our frst prncples calculatons and wth expermental measurements of DH f. [36,37] In ths fgure, the fcc/cu 2 Mg-C15 coexstence curve suggests that the expermental values reported by Kng and Kleppa [36] may be the most accurate, snce they exhbt excellent agreement wth the EMF data of Eremenko et al. [39] Based on ths selecton and the heat capacty consderatons dscussed above, the coeffcents lsted n Table 3 for G C15 Cu:Mg and G C b Mg:Cu are evaluated wth the correspondng expermental data. [16,21,22,24,36,38,39] Fnally, the two nteracton parameters for the Cu 2 Mg-C15 phase ð L C15 Cu:Cu;Mg and L C15 Cu;Mg:MgÞ are evaluated usng avalable phase equlbrum data. [20-24] The Gbbs free As 2 Se 3, [48] La 20 Mg 50 N 30, Al 30 La 50 N 20, Al 7.5 Cu N 10 Zr 65, [49] Al 7.5 Cu 27.5 Zr 65, [50] ) agan suggestng some type of clusterng/orderng reacton n the lqud phase. The nteracton parameter for each of the termnal sold soluton phases s determned from expermental estmates of the respectve soldus and solvus boundares, where the solublty of Cu n the hcp-mg phase has been reported to be 0.15, 0.23, and 0.23 at.% by Hansen, [25] Stepanov and Kornlov, [26] and Yue and Perre, [27] respectvely. The resultng nteracton parameter for the hcp soluton phase s hgh and postve, as lsted n Table 4. For the fcc sold soluton, the nteracton parameter lsted n Table 4 s determned from the fcc phase boundary data reported by Jones [22] and Rogelberg. [23] Fg. 2 The chemcal potental of Mg assocated wth the ndcated two-phase equlbrum, plotted over the relevant temperature range for the Cu-Mg system. The sold curves are computed and compared wth pror modelng and experment. The reference states are fcc-cu and hcp-mg 162 Journal of Phase Equlbra and Dffuson Vol. 28 No

8 Basc and Appled Research: Secton I energy of formaton, DG h :j, for each unstable end-member n Eq 8, s treated as a constant and determned usng the frstprncples data n Table 1. The parameter assessment practce s teratve and culmnates wth an optmzed ft nvolvng all selected data and calculaton results. The fnal parameters from the overall optmzaton are lsted n Tables 3 and Phase Equlbrum Results Fg. 3 The Cu-Mg phase dagram calculated usng the parameters lsted n Tables 2-4, shown wth the relevant expermental data The phase dagram yelded by our modelng effort s shown n Fg. 3 along wth prevously proposed phase dagrams and relevant expermental data. In Table 5, the nvarant reactons are compared wth expermental reports [20-22] and wth the models of Coughanowr [13] and Zou and Chang. [14] We note that the equlbrum phase boundares produced by our model do not dffer sgnfcantly from pror reports, except that our results may exhbt slghtly better agreement wth expermental lqudus and soldus data for the termnal soluton phases. However, there are several mportant features of our thermodynamc descrpton of ths bnary system that do dffer substantally from prevously reported models but are not dramatcally evdent n the equlbrum phase dagram tself. These dfferences prmarly arse from () the use of chemcal assocates n our treatment of the lqud, () our two-sublattce soluton descrpton of the ntermedate phases, and () ncluson of non-lnear temperature dependent free energes of formaton, all of whch offer mprovements over pror treatments Table 5 Invarant reactons n the Cu-Mg system Reacton Calculated results Ths work Ref. [13] Ref. [14] Expermental data Lq f Cu 2 Mg-C15 T, K [20] 1070 [21] 1092 [22] 1066 [24] x (lqud, Mg) [24] x (C15, Mg) [24] Lq f CuMg 2 -C b T, K [22,24] x (lqud, Mg) [24] x (Cb, Mg) [24] Lq f fcc + Cu 2 Mg-C15 T, K [22] 998 [20,24] x (lqud, Mg) [20] x (fcc,mg) [24] x (C15,Mg) [24] Lq f CuMg 2 -C b +Cu 2 Mg-C15 T, K [20,21] 825 [22,24] x (lqud, Mg) [20] x (Cb,Mg) [20] x (C15,Mg) [20] Lq f hcp + CuMg 2 -C b T, K [20] 758 [21,22] x (lqud, Mg) [21,22] x (hcp,mg) [22] x (Cb,Mg) [22] Journal of Phase Equlbra and Dffuson Vol. 28 No

9 Secton I: Basc and Appled Research Fg. 4 The chemcal potental of Mg as a functon of composton for the Cu-Mg lqud phase calculated for 1,100 K and compared wth pror modelng and experment. The reference states are lq-cu and lq-mg Fg. 6 The enthalpy of mxng of the lqud phase calculated for 1,120 K and compared wth pror modelng and experment. The reference states are lq-cu and lq-mg Table 6 The meltng enthalpes of the compounds Cu 2 Mg-C15 and Mg 2 Cu-C b Compound Calculated results, kj/mol Ths work Ref. [13] Ref. [14] Expermental data, kj/mol [16] Cu 2 Mg-C ± 1.5 Mg 2 Cu-C b ± 1.5 Fg. 5 The actvty of Cu and Mg calculated for 1,200 K and compared wth pror modelng and experment. The reference states are lq-cu and lq-mg and permt the quanttatve descrpton of thermodynamc parameters for both undercooled lquds and nonequlbrum compostons of the ntermetallc phases. Fgures 4 and 5 show that there s lttle dfference between pror models and ours wth regard to quantfcaton of chemcal potental of the lqud phase over the full range of composton at 1100 K and 1200 K. However, Fg. 6 shows that our model provdes much better agreement wth the expermental measurements of enthalpy of mxng for the lqud phase, as reported by Sommer et al. [29] Ths feature s essental f drvng forces for varous phase transtons nvolvng both equlbrum and nonequlbrum compostons of the relevant phases are to be computed accurately and compared. In addton, we employ the temperature-dependent descrpton gven n Eq 1 for the Gbbs free energy of Cu 2 Mg-C15 and Mg 2 Cu-C b rather than the smpler descrpton used n earler reports. [13,14] As a result, t s clear from Fg. 7 that our descrpton reproduces the expermental heat capacty data [16] far better than prevously reported models. [13,14] In addton, the modeled meltng enthalpes for Cu 2 Mg-C15 and Mg 2 Cu-C b are lsted n Table 6, where our results show better agreement wth the expermental data for these phases [16] than those reported by Coughanowr et al. [13] and Zou and Chang. [14] Fnally, for the purpose of further drect comparson, we nclude n Fg. 1 our modelng results for the heat capacty of the Cu 14.5 Mg 85.5 lqud, showng the dramatc mprovement n low-temperature heat capacty compared to the models by Coughanowr et al. [13] and Zou and Chang. [14] 164 Journal of Phase Equlbra and Dffuson Vol. 28 No

10 Basc and Appled Research: Secton I C b phase, () the use of frst prncples calculatons to obtan the zero-kelvn energes for the unstable compounds, and () the use of an assocaton model for the lqud phase. As a result, the current model provdes a more realstc quantfcaton of heat capacty for the relevant phases and a better descrpton of the composton-dependent enthalpy of mxng for the lqud. Acknowledgments Work at the Ames Laboratory was supported by the U.S. Department of Energy, Basc Energy Scences, under Contract No. DE-AC02-07CH References Fg. 7 The heat capacty (a) Cu 2 Mg-C15 and (b) Mg 2 Cu-C b computed usng our model parameters n Tables 1-3 and compared wth prevously reported models and data 5. Summary and Conclusons Employng a soluton thermodynamcs approach, the Gbbs free energy vs. composton curves for all relevant phases n the Cu-Mg bnary system were estmated, and the assocated bnary phase dagram s reported here. Whle the resultng equlbrum phase dagram exhbts only modest devaton from prevously reported dagrams, the current thermodynamc descrpton exhbts several key dfferences from earler thermodynamc models. These dfferences arse from () the two-sublattce soluton treatment of the Mg 2 Cu- 1. A. Inoue and A. Takeuch, Recent Progress n Bulk Glassy Alloys, Mater. Trans., JIM,, 2002, 43, p A. Inoue, H. Kmura, and A. Takeuch, n Thermec-2003, TRANS TECH PUBLICATIONS LTD, Zurch-Uetkon, 2003, vol , p J.F. Loffler, Bulk Metallc Glasses, Intermetallcs, 2003, 11, p A. Inoue and A. Takeuch, Recent Progress n Bulk Glassy, Nanoquascrystallne and Nanocrystallne Alloys, Mater. Sc. Eng., A, 2004, , p S.H. Zhou and R.E. Napoltano, Phase Equlbra and Thermodynamc Lmts for Parttonless Crystallzaton n the Al-La Bnary System, Acta Mater., 2006, 54, p A. Inoue, Amorphous, Nanoquascrystallne and Nanocrystallne Alloys n Albased Systems, Prog. Mater. Sc., 1998, 43, p F. Sommer, G. Bucher, and B. Predel, Thermodynamc Investgatons of Mg-Cu and Mg-N Metallc Glasses, J. Phys. Colloq., 1980, C8, p T. Masumoto and R. Maddn, Structural Stablty and Mechancal Propertes of Amorphous Metals, Mater. Sc. Eng., 1975, 19, p T. Masumoto and R. Maddn, Mechancal Propertes of Pd-20 At. Percent S Alloy Quenched from the Lqud State, Acta Met., 1971, 19, p W.L. Johnson, In: H. Beck, H.J. Guntherodt, Eds., Glassy Metals, Sprnger-Verlag, Berln, F.P. Messel, S. Frota-Pessoa, J. Wood, J. Tyler, and J.E. Keem, Electronc Densty of States n Amorphous Zrconum Alloys, Phys. Rev. B, 1983, 27, p M. Tenhover and W.L. Johnson, Superconductvty and the Electronc Structure of Zr- and Hf-based Metallc Glasses, Phys. Rev. B, 1983, 27, p C.A. Coughanowr, I. Ansara, R. Luoma, M. Hamalanen, and H.L. Lukas, Assessment of the Cu-Mg System, Z. Metallkd., 1991, 82, p Y. Zou and Y.A. Chang, Thermodynamc Calculaton of the Mg-Cu Phase Dagram, Z. Metallkd., 1993, 84, p W.E. Lukens and C.N.J. Wagner, The Structure of Lqud Cu- Mg Alloys, Z. Naturforsch., 1973, 28, p H. Feufel and F. Sommer, Thermodynamc Investgatons of Bnary-Lqud and Sold Cu-Mg and Mg-N Alloys and Ternary Lqud Cu-Mg-N Alloys, J. Alloy. Compd., 1995, 224, p L. Kaufman and H. Bernsten, Computer Calculaton of Phase Dagrams. Vol. 4, Academc Press Inc., New York, 1970 Journal of Phase Equlbra and Dffuson Vol. 28 No

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