Phase stability in nanoscale material systems: extension from bulk phase diagrams

Transcription

1 Electonic Supplementay Mateial (ESI) fo Nanoscale. This jounal is The Royal Society of Chemisty 2015 Phase stability in nanoscale mateial systems: extension fom bulk phase diagams Sauabh Bajaj a,, Michael G. Havety b, Raymundo Aóyave c,d, William A. Goddad III FRSC a, Sadasivan Shanka e a Depatment of Applied Physics and Mateials Science, Califonia Institute of Technology, Pasadena, CA 91125, USA b Pocess Technology Modeling, Design and Technology Solutions, Technology and Manufactuing Goup, Intel Copoation, Santa Claa, CA 95052, USA c Depatment of Mateials Science and Engineeing, Texas A&M Univesity, College Station, TX 77843, USA d Depatment of Mechanical Engineeing, Texas A&M Univesity, College Station, TX 77843, USA e School of Engineeing and Applied Sciences, Havad Univesity, Cambidge, MA 02138, USA S1. Computational methods S1.1. The CALPHAD method The method of CALculation of PHAse Diagams (o CALPHAD fo shot) has been widely utilized to calculate bulk phase diagams and themodynamic popeties of multi-component systems [S1]. It involves the use of Gibbs fee enegy models developed fo vaious types of phases, such as andom solutions (gases, liquids, and solids), sublattice phases, ionic phases, etc. Vaiables used in these models ae calculated by fitting eithe to expeimental data o ab initio calculations. The CALPHAD method [S1] is well established fo calculating bulk phase diagams, and is a good stating point fo calculating phase equilibia in nanoscale systems. In this wok, since we have consideed suface effects on binay solution phases, themodynamic models fo only these types of phases will be descibed. The Gibbs fee enegy of a bulk andom solution phase φ is given by, G φ,bulk m = G ef + G ideal mix + G xs mix, (S1) Coesponding addess: Depatment of Applied Physics and Mateials Science, Califonia Institute of Technology, Pasadena, CA 91125, USA, Tel.: addess: sbajaj@caltech.edu (Sauabh Bajaj) Pepint submitted to Elsevie May 6, 2015

2 whee, G ef is the sum of standad Gibbs enegies of each component, G ideal mix is the ideal mixing configuational entopy contibution to the Gibbs fee enegy of the solution phases, and G xs mix, called the excess enegy of mixing, takes into account all the non-ideal tempeatue dependent effects such as inteaction between components, non-ideal configuational entopy, vibational and electonic entopy, etc. Expanding each tem, Eqn. (S1) becomes, G φ,bulk m = i=a,b = x A o G bulk A x i o G bulk i + RT i=a,b x i log e x i + G xs mix + x B o G bulk B + RT (x A log e x A + x B log e x B ) + G xs mix, (S2) whee, x A and x B ae mole factions of components A and B of the phase, espectively, and o G φ,bulk A and o G φ,bulk B ae the standad Gibbs enegies of the phase containing only the pue component A and B, espectively. These ae obtained fom the Scientific Goup Themodata Euope (SGTE) database [S2]. R is the gas constant, and T is the tempeatue. The excess Gibbs fee enegy of mixing G xs mix is expanded accoding to the Redlich-Kiste fomalism [S3] as, G xs mix = x A x B L φ v (x A x B ) v, (S3) whee, v is the ode of expansion (v = 0 fo egula solution phases and v = 1 o above fo non-egula solution phases), and L φ v is called the non-ideal inteaction paamete. As the excess Gibbs fee enegy of mixing must include tempeatue dependency of othe souces of entopy (non-ideal configuational, vibational, and electonic) apat fom ideal configuational entopy, these paametes ae futhe expanded as, v L φ v = A φ v + Bv φ.t, (S4) whee, A φ v and Bv φ ae use-defined paametes that ae calculated and optimized in the CALPHAD method with available expeimental data on positions of equilibium lines in the phase diagam, phase themodynamic popeties such as enthalpy and entopy of mixing, etc., and/o simila data calculated fom ab initio calculations, which is paticulaly useful in cases whee no expeimental data is available. S2

3 Refeences [S1] N. Saundes, A.P. Miodownik, Pegamon Mate. Se. (1998). [S2] A.T. Dinsdale, CALPHAD 15 (1991) [S3] O. Redlich, A.T. Kiste, Ind. Eng. Chem. 40 (1948) [S4] J. Lee, M. Nakamoto, T. Tanaka, J. Mate. Sci. 40 (2005) [S5] L.E. Mu, Intefacial Phenomena in Metals and Alloys, Addison-Wesley Publishing Company, London, 1975, p [S6] F. Millot, V. Saou-Kanian, J.C. Rifflet, B. Vinet, Mat. Sci. Eng. A 495(2008) 813. [S7] R.J. Jaccodine, J. Electochem. Soc. 110 (1963) [S8] T. Iida, R.I.L. Guthie, The Physical Popeties of Liquid Metals, Oxfod Science Publications, (1993). [S9] I. Sa, B.-M. Lee, C.-J. Kim, M.-H. Jo, B.-J. Lee, CALPHAD 32 (2008) [S10] K.C. Mills: Recommended values of themophysical popeties fo selected commecial alloys, Woodhead Publishing Ltd., Cambidge, UK, [S11] J. Schmitz, J. Billo, I. Egy, R. Schmid-Fetze, Int. J. Mate. Res. 100:11 (2009) [S12] W.R. Tyson, W.A. Mille, Suf. Sci. 62 (1977) [S13] L.Z. Mezey, J. Gibe, Jap. J. App. Phys. 21:11 (1982) [S14] G. Gazel, J. Janczak-Rusch, L. Zabdy, CALPHAD 36 (2012) [S15] J. Billo, I. Egy, J. Westphal, Int. J. Mat. Res. 99 (2008) [S16] W. Gasio, Z. Mose, J. Pstus, J. Phase Equilib. 21 (2000) [S17] M. Gündüz, J.D. Hunt, Acta Metall. 33 (1985) [S18] K. Dick, T. Dhanasekaan, Z. Zhang, D. Meisel, J. Am. Chem. Soc. 124 (2002) [S19] P.R. Couchman, W.A. Jesse, Natue 269 (1977) S3

4 [S20] Ph. Buffat, J.-P. Boel, Phys. Rev. A 13 (1976) [S21] A.N. Goldstein, Appl. Phys. A: Mate. Sci. Pocess. 62 (1995) [S22] F.G. Meng, H.S. Liu, L.B. Liu, Z.P. Jin, J. Alloys Cmpds. 431 (2007) [S23] R.W. Olesinski, G.J. Abbaschian, Bull. Alloy Phase Diagams 5 (1984) [S24] Y.-B. Kang, C. Aliavci, P.J. Spence, G. Eiksson, C.D. Fuest, P. Chatand, A.D. Pelton, JOM 61 (4) (2009) [S25] V.T. Witusiewicz, U. Hecht, S.G. Fies, S. Rex, J. Alloys Cmpds. 385 (2004) [S26] N. Saundes, Al-Cu system, in: I. Ansaa, A.T. Dinsdale, M.H. Rand (Eds.), COST-507: Themochemical Database Fo light Metal Alloys, Euopean Communities, Luxembug (1998) [S27] A.F. Lopeandía, J. Rodíguez-Viejo, Themochim. Acta 461 (2007) [S28] V.I. Akhaov, L.M. Magat, Phys. Met. Metallog. 6(5) (1958) [S29] M. Ellne, K. Kolatschek, B. Pedel, J. Less-Common Met. 170 (1991) [S30] C. Kittel, Intoduction to Solid State Physics, sixth edition, John Wiley, (1986). [S31] A. Meetsma, J.L. De Boe, S. Van Smaalen, J. Solid State Chem. 83 (1989) [S32] J. Sun, S.L. Simon, Themochim. Acta 463 (2007) [S33] D. Mott, J. Galkowski, L. Wang, J. Luo, C-J. Zhong, Langmui 23 (2007) [S34] P. Laty, J.C. Joud, P. Desé, Suf. Sci. 69 (1977) S4

5 Table S1: Suface aea to volume atio of diffeent shapes of nano-paticles. Geometical Size Suface Aea Volume Ratio of Coefficient of atio nanostuctue (a o ) Suface Aea/Volume Cube a 6a 2 a 3 6 a 6 Regula Tetahedon a 3a 2 a Hexahedon a,l; a/l = 1 (6a a 2 ) 3 3a 3 Regula icosahedon a 5 3a 2 5(3+ 5)a 3 Sphee 4π 2 4π a a ( ) (3+ 5)a Cylinde,H; /H = 1 4π 2 π S5

6 Table S2: Themodynamic and physical popeties used in the calculation of alloy suface tensions and phase diagams of the Au-Si, Ge-Si, and Al-Cu nanoscale systems (L: Liquid, S: Solid) Vaiable Function Refeence Suface tension (J/m 2 ) σau L = x 10 4 T [S4] σau S = x 10 4 T [S5] σsi L = x 10 5 (T ) [S6] σsi S = x 10 4 (T ) [S7] σge L = x 10 4 (T ) [S8] σge S = x 10 4 (T ) [S9] σal L = x 10 4 (T - 933) [S10, S11] σal S = x 10 4 T [S12, S13] σcu L = x 10 4 T [S8] σcu S = x 10 4 T [S14] Mola volume (m 3 /mol) V L Au = x 10 5 T x T [S8] V S Au = x 10 5 [S8] V L Si = 11.1 x 10 6 [ x 10 4 (T ) V S Si = x 10 5 [S9] V L Ge = 13.2 x 10 6 [1 + 8 x 10 5 (T ) V S Ge = x 10 5 [S9] VAl L = x 10 6 [1 + 9 x 10 5 (T - 933) VAl S = x 10 6 [ x 10 4 (T - 933) VCu L = 7.94 x 10 6 [1 + 1 x 10 4 (T ) VCu S = 7.01 x x T x T 2 VAl S = 9 x 2Cu 10 6 [S8] [S8] [S15] [S16] [S8] [S14] [S17] S6

7 Liquid Bulk 5 nm (sphee) 5 nm (icosahedon) T (K) Dia-GeSi 200 Dia-Ge + Dia-Si Ge x(si) Si Figue S1: (Colo online) Phase diagam of the Ge-Si alloy system calculated fo two paticle shapes - sphee and a egula icosahedon, both fo a paticle size of 5 nm, and compaed with the bulk phase diagam. S7

8 1400 Au 1350 bulk 1300 Dick (2002) Calc. (this wok) (K) (nm) Figue S2: (Colo online) Calculated melting points of Au as a function of paticle size compaed with expeimental data fom Ref. [S18]. S8

9 (a) Si bulk 1550 (b) Goldstein (1996) Calc. (this wok) (a) Iida, Guthie (1993) & Mezey (1982) (K) (d) (b) (c) Couchman, Jesse (1977) Uppe limit Lowe limit (d) Buffat, Boell (1976) 1250 (c) (nm) Figue S3: (Colo online) Calculated melting points of Si, using suface tension data fom Mallot et al [S6] fo liquid and fom Jaccodine et al [S7] fo the solid phase, as a function of paticle size compaed with expeimental data fom Iida & Guthie [S8], Mezey et al [S13], Couchman & Jesse [S19], Buffat & Boell [S20], and Goldstein [S21]. S9

10 Table S3: Themodynamic functions used in the calculation of phase diagams in this wok (in J mol 1 and K). All bulk and size-independent functions ae obtained fom (a) Au-Si: SGTE database [S2] and Ref. [S22], (b) Ge-Si: SGTE database [S2] and Refs. [S23] and [S24] fo the liquid and diamond phases, espectively, and (c) Al-Cu: Ref. [S25] fo the liquid and γd8 3 phases, and the COST-507 database [S26] fo the est of the phases. Standad element efeence Gibbs enegies o G fcc,nano Au o G dia,nano Si o G dia,nano Ge o G fcc,nano Al o G fcc,nano Cu Liquid phases G liq,nano Au G liq,nano Si G liq,nano Ge G liq,nano Al G liq,nano Cu = o G fcc,bulk Au = o G dia,bulk Si = o G dia,bulk Ge = o G fcc,bulk Al = o G fcc,bulk Cu = G liq,bulk Au = G liq,bulk Si = G liq,bulk Ge = G liq,bulk Al = G liq,bulk Cu Intemetallic compounds G Al2Cu,nano Al:Cu = 2 o G fcc,nano Al Inteaction paametes 1. Au-Si L liq,nano T T T T T T T T T T T T T T T T T 2 + o G fcc,nano Cu 0 = ( L liq,nano 1 = ( L liq,nano 2 = ( Ge-Si L liq,nano 0 = ( L dia,nano 0 = ( Al-Cu L liq,nano 0 = ( L liq,nano 1 = ( L liq,nano 2 = ( L liq,nano 3 = T + ( ) + (6.75).T ) + ( ).T ) + ( ).T ) ( ).T ) + ( ).T ) + ( ).T ) + ( ).T ) + ( ).T ) + ( ).T S10

11 1250 Ge bulk 1200 (K) 1150 Lopeandia (2007) Calc. (this wok) (nm) Figue S4: (Colo online) Calculated melting points of Ge as a function of paticle size compaed with expeimental data fom Ref. [S27]. S11

12 Figue S5: (Colo online) Unit cell of the Al 2 Cu compound. S12

13 Table S4: Calculated lattice constants (in Å) and cohesive enegy, E coh (in ev/atom) of Al, Cu, and the Al 2 Cu phase fom DFT using the LDA appoximation. Expeimental data ae shown in paentheses. Phase Space goup Peason symbol Lattice constants E coh (ev/atom) Al Fm 3m (no. 225) cf4 a = (4.047 a ) (3.39 c ) Cu Fm 3m (no. 225) cf4 a = ( b ) (3.49 c ) Al 2 Cu I4/mcm (no. 140) ti12 a = (6.063 d ) c = (4.872 d ) a: Ref. [S28] b: Ref. [S29] c: Ref. [S30] d: Ref. [S31] S13

14 950 Al bulk 925 Sun (2007) Calc. (this wok) (K) (nm) Figue S6: (Colo online) Calculated melting points of Al as a function of paticle size compaed with expeimental data fom Ref. [S32]. S14

15 Cu bulk (K) Mott (2007) Calc. (this wok) (nm) Figue S7: (Colo online) Calculated melting points of Cu as a function of paticle size compaed with expeimental data fom Ref. [S33]. S15

16 Al-Cu 1375 K Calc. (this wok) Schmitz (2009) Laty (1977) σ s liq (J.m -2 ) x Al bulk (at.%) Figue S8: (Colo online) Calculated suface tension of the liquid phase in the Al-Cu system compaed with expeimental data fom Refs. [S11, S34]. S16