Atomic Transport & Phase Transformations Lecture 7

Transcription

1 Atomic Transport & Phase Transformations Lecture 7 PD Dr. Nikolay Zotov zotov@imw.uni-stuttgart.de

2 Lecture I-7 Outline Limited solubility in the solid state Effect of the maximum temperature of the miscibility gap CALPHAD Eutectic phase diagrams Eutectoid phase diagrams Peritectic phase diagrams Metatectic phase diagrams

3 Classification Unlimited Solubility in Liquid and Solid Limited Solubilty W L = 0 W S = 0 With intermetallic Phases W < 0 in Solid in Liquid in Liquid W S > 0 W L > 0 and in Solid W L > 0 W S > 0

4 Unlimited Solubility in the Liquid Unlimited Solubility in the Solid (W S = 0) Temperature (K) Ge-Si 0,0 0,2 0,4 0,6 0,8 1, X Ni 0,30 WIDTH of TWO PHASE REGION 0,25 0,20 0,15 0,10 0,05 0, Temperature (K)

5 Unlimited Solubility in the Liquid Limited Solubility in the Solid (W S > 0) T MA and T MB (relatively) high The contribution of Entropy >> DH mix ; T L T 5 << T M A for T > T max solid-solution T M A Tmax L+a T M B a T max T 5 a +a A X B B

6 Unlimited Solubility in the Liquid /Limited Solubility in the Solid Miscibility Gap Regular Solution model for the Solid: G reg = (1 X B )G AS + X B G B S + RT[(1-X B )ln(1-x B ) + X B ln(x B )] + W S (1-X B )X B DG mix reg = RT[(1-X B )ln(1-x B ) + X B ln(x B )] + W(1-X B )X B Condition for minimum: DG/ X B = 0 and 2 DG/ X B2 > 0 min DG/ X B = 0 = W S (1 2X B ) + RTln(X B /(1 X B ) T = W S /R(2X B 1) / (ln(x B ) ln(1 X B )) Equation for the Miscibility-gap line (co-existance curve)

7 Miscibility Gap T = (2 W S /RX B W S /R) / (ln(x B ) ln(1 X B )) Miscibility gap (Binodal curve) 2 DG/ X B2 = 2 W S RT/X B (1-X B ); > 0 min = 0 inflection point(s) < 0 max T = (2 W S /R)X B (1-X B ) (Spinodal curve) T T SC / X B = 2 (W S /R) (1 2X B ); Extremum at X B = ½ 2 T SC / X B2 = -4 W S /R < 0 T gm spinodal W S = 2RT gm T > T gm homogeneous solution binodal T < T gm phase separation A X B B

8 Unlimited Solubility in the Liquid - Limited Solubility in the Solid Miscibility Gap A = Pt T M = 1768 o C DH M = 20 kj/mol B = Ir T M = 2466 o C DH M = 26 kj/mol W S ~ 2RT gm = J/mol Temperature (C) ,0 0,2 0,4 0,6 0,8 1,0 X Ir The real missibility gap is not symmetric sub-regular solution

9 Unlimited Solubility in the Liquid Limited Solubility in the Solid G Pt = J/mol G Ir = J/mol T = 750 o C G (J/mol) T = 1400 o C ,0 0,2 0,4 0,6 0,8 1,0 X B

10 Unlimited Solubility in the Liquid - Limited Solubility in the Solid Missibility Gap A = Pt T M = 1768 o C DH M = kj/mol B = Rh T M = 1963 o C DH M = kj/mol W S ~ 2RT gm = J/mol

11 Unlimited Solubility in the Liquid Limited Solubility in the Solid (W S > 0) W 1 T gm ~ W/2R W 2 W increases further, T gm increases and both melting temperatures decrease further. W 1 << W 2

12 Unlimited Solubility in the Liquid Limited Solubility in the Solid W S increases further and the melting temperatures decrease!!! The Gibbs energy of the solid-solution flattens at higher temperature, compared to the Gibbs energy of the (ideal) liquid solution; Presence of negative Enthalpy of mixing in the liquid (W L < 0) further enhances this effect. T 1 >> T MA and T M B 2 common tangents!!!

13 Unlimited Solubility in the Liquid Limited Solubility in the Solid Regular solution model: k max = -2W + 4RT at X B = ½ If the liquid has a negative Interaction parameter W L < 0, while the Solid has W S > 0 k L = 2W L + 4RT > k S = -2W S + 4RT S DG L 0,0 0,2 0,4 0,6 0,8 1,0 XB

14 Unlimited Solubility in the Liquid Limited Solubility in the Solid At T = T 2 (point (Y) X 2 ) G L = G S Minimum of the Solidus Minimum of the Liquidus X 2 Prince (1966)

15 Unlimited Solubility in the Liquid Limited Solubility in the Solid Estimation of the X 2 point At T = T 2 (point X 2 ) both curves have a minimum G BS G AS + W S (1-2X 2 ) + RT 2 ln(x 2 /(1-X 2 )) = 0 G BL G AL - W L (1-2X 2 ) + RT 2 ln(x 2 /(1-X 2 )) = 0 X 2 = ½ + (DG B (T 2 ) DG A (T 2 ))/2 <W>; <W> = ½ ( W L + W S ); DG A ~ DH ma (1 T 2 /T MA ); DG B ~ DH mb (1 T 2 /T MB ) X 2 ~ ½ + [DH mb - DH ma -T 2 (DH mb /T M B - DH A m /T MA )]/2 <W>;

16 Unlimited Solubility in the Liquid Limited Solubility in the Solid W S ~ 2RT gm = J/mol A B C D a X alloy = 60 wt% Ni T M (Ni) = 1450 o C T M (Au) = 1064 o C Both Ni and Au are fcc A (T ~ 1300 o C) 100% Liquid X Ni = 60 wt% B (T ~ 1100 o C) X(l) ~ 39.5 wt% Ni, X(a) ~ 67 wt% Ni f(a) ~ 73%, f(l) ~ 27% C (T ~ 900 o C) 100% solid-solution with X Ni = 60 wt% D (T ~ 500 o C) X(a 1 ) ~ 6 wt% Ni, X(a 2 ) ~ 93 wt% Ni f(a 1 ) ~ 37%, f(a 2 ) ~ 63% X E = ½ / <W>; <W> ~ J/mol

17 Unlimited Solubility in the Liquid Limited Solubility in the Solid Equilibrium conditions G BL - W L (1 X BL ) 2 + RT m lnx BL = G BS + RT m lnx B S + W S (1 X BS ) 2 G AL W L X B L 2 + RT m ln(1 - X BL ) = G AS + RT m ln(1 - X BS ) + W S (X BS ) 2

18 Unlimited Solubility in the Liquid - Limited Solubility in the Solid Numerical Calculations Au T M = 1336 K, DH M = 2955 cal/g.at Ni T M = 1725 K, DH M = 4210 cal/g.at DH mix S = X Au X Ni [5770 X Au X Ni 3400 X Au X Ni ](1 T/2660) W S = [5770 X Au X Ni 3400 X Au X Ni ](1 T/2660) DH mix L = X Au X Ni [1040 X Au + 460X Ni ] ~ 1040 X Au X Ni [1 0.5X Ni ] W L = 1040[1 0.5X Ni ] ( sub-regular model for the Liquid) E Gaye and Lupis (1975)

19 Computer Coupling of Phase Diagrams and Thermochemistry CALPHAD Thermo-Calc PANDAT MTDATA

20 Unlimited Solubility in the Liquid Limited Solubility in the Solid (W S > 0) # Flattening of the liquidus and the solidus # T gm = 1260 o C = 1533 K very high # W S = 2RT gm ~ J/mol W S = J/mol W L = J/mol DeHoff (2008)

21 Unlimited Solubility in the Liquid Limited Solubility in the Solid (W S > 0) For even higher W S : T gm # Melting of the phase mixture a 1 + a 2 takes place before reaching the hypothetical maximum of the miscibility gap T gm. # Mixing of the two components A and B becomes possible only in the liquid state.

22 Unlimited Solubility in the Liquid - Limited Solubility in the Solid (W S >> 0) Melting temperatures of A and B similar!!! Liquid DG curve for ideal solution (or W L < 0) DG S bimodal at all temperatures, because W S very large. Euthectic phase Diagram T 4 Eutectic Temperature X E Eutectic composition Prince (1966) X E

23 Eutectic T 2 = T E Eutectic temperature T B CD Invariant tie line T A Composition of a 1 phase = X B 1 C E D Composition of a 2 phase = X B 2 At X E takes place: L a 1 + a 2 Eutectic reaction Al-Cu X B 1 X B 2 X E Easterling (2009)

24 Eutectic (1) (2) (3) Solidification modes: (1) L solidifies as a 1 solid solution at T 1 ; (2) L solidifies as a 1, but a 2 precipitates at lower temperature (T 3 ); (3) L solidifies as a 1 (T 1 ), followed by an a 1 + a 2 eutectic mixture at T 2.

25 Euthectic Pb F m -3 m fcc structure T M = 327 o C; DH M = 4.77 kj/mol Sn I 4 1 /a md tetragonal structure T M = 232 o C; DH M = 7 kj/mol Very low melting temperatures!!!! A E C X E = 61.9 wt% Sn, X E ~ 73 at% Sn A (T = 230 o C) 100 % Liqiud 61.9 wt% Sn E (T = 183 o C) 43.8 wt % a 18.3 wt% Sn 56.2 wt% ß 97.8 wt% Sn C (T = 100 o C) 38.6 wt% a 5.00 wt% Sn 61.4 wt% ß 98.2 wt% Sn

26 Euthectic A X Sn (l) = 0.4 X Sn (a) ~ A B C D E B X Sn (l) = 0.63 X Sn (a) ~ 0.26 fraction(a) = 65% C X Sn (l) = X Sn (a) ~ 0.28 fraction(a) = 76% D X Sn (a) ~ 0.22; X Sn (ß) ~ 0.97 fraction(a) = 83% E X Sn (a) ~ 0.065; X Sn (ß) ~ 0.97 fraction(a) = 73% The slope of the phase boundary between a + ß and ß is almost vertical because the solubility of Pb in Sn is very small

27 Euthectic Cu F m -3 m fcc structure T M = 1358 o C; DH M = 13.1 kj/mol Ag F m -3 m fcc structure T M = 1234 o C; DH M = 11.3 kj/mol # similar melting temperatures and enthalpies of fusion

28 Simple Calculation of Liquidus Lines Gibbs Helmholz Equation: (G/T)/ T = - H/T 2 ; (DG/T)/ T = - DH/T 2 ; during a phase transition Euthectic Partial molar Gibbs energy of the A component in the Liquid phase µ AL = G AL = G AL + RTln(a AL ) ; DG AL = RTln(a AL ) ; DG AL /T = Rln(a AL ) ; At the liquidus Solid and Liquid are in equilibrium (DG AL /T)/ T = Rln(a AL )/ T = -DH melta / T 2 ; ln(a AL ) = DH melta /R [-1/T + 1/T MA ] Ideal Solution model for the Liquid: ln(1 - X BL ) = DH melta /R [1/T MA - 1/T ]

29 Eutectic Simple Calculation of the Solvus: I/ The Solvus line(s) are just the missibility lines on the a 1 and a 2 sides (regular solution). T E T = W/R (2X B 1)/ ln (X B /(1 X B )) II/ X Ba / T = DH mix ba / T(X Bß X Ba ) 2 G ma / (X Ba ) 2 (Lecture 6); Ideal solution for a phase ; 2 G ma / (X Ba ) 2 = RT ([1/X Ba 1/(1-X Ba )] = RT/X Ba (1-X Ba ) X Ba 0 X B 1 X Ba / T ~ X Ba DH mix ba /RT 2 ; T ~ - DH mix ba /Rln(X Ba )

30 Eutectic # The ideal solution model gives the right trend but the slope of the liquidus is bigger! Temperature ( o C) # Empirical phenomenological models: T liq A = T MA PX B QX B2 ; T X Sn (wt %) or T liq A = T MA P*W B Q*W B2 ; Pb Sn T liq A = W Sn ; 20 0,00 0,05 0,10 0,15 0,20 0,25 X Sn T liq B = W Pb ; X Sn1 ~.28 T E = 183 o C; W S ~ 3250 J/mol

31 Eutectic Al T M = 660 o C fcc structure DH M = 10.7 kj/mol R = 1.18 Å c = 1.61 Si T M = 1414 o C diamond structure DH M = 50.2 kj/mol R = 1.11 Å c = 1.9 W S ~ J/mol x E ~ ½ /<W> ; (<W> ~ J/mol)

32 Eutectic T liq A = W Si 0.15 (W Si ) 2 ; Hernandez (2005)

33 1000J Eutectic J X Cu1 ~.125 T E = 778 o C; W S ~ J/mol ,000 0,025 0,050 0,075 0,100 0,125 XB

34 Limiting cases of eutectic phase diagrams Sn - Zn Shift of the Eutectic composition in the direction of the pure component with lower melting temperature

35 Eutectoid phase diagrams # Partial immisiblity between a and a in the solid state; # Formation of Miscibility gap at low temperatures; # Increase of the Interaction parameter and the maximum of the missibility gap; a + a # Cross-section of the 2-Phase regions a + ß and a + a.

36 Eutectoid phase diagrams Eutectoid tie-line (c d) Eutectoid point (e) Solid phase in equilibrium with two solids Eutectoid reaction g a + ß

37 Eutectoid Reaction Eutectic point Eutectoid Point g (Austenite) a (Ferrite) + Fe 3 C (Cementite) Eutectoid reaction

38 Melting temperatures of A and B very different Peritectic Phase diagrams P Peritectic point Peritectic reaction L + a 1 a 2 ; P

39 Peritectic Phase diagrams A Ag T M = 1234 K DH = 11.3 kj/mol Pt T M = 1768 K DH = 20.0 kj/mol B P a 1 X Pt (P) ~ 0.43 A T = 2000 K, X Pt (l) ~ wt% Liquid a 2 C B T ~ 1700 K, X Pt (l) = 0.43, ~ 100 wt% Liquid X Pt (a 1 ) = 0.83; ~ 0 wt% a 1. P T ~ 1450 K; X Pt (a 2 ) = 0.43, 100 wt% a 2 phase C T ~ 1000 K; X Pt (a 1 ) ~ 0.95, 35 wt% a 1 phase X Pt (a 2 ) ~ 0.18, 65 wt% a 2 phase

40 Peritectic Phase diagrams Ag T M = 1234 K DH = 11.3 kj/mol a 2 A B C D P a 1 Pt T M = 1768 K DH = 20.0 kj/mol X Pt (P) ~ 0.43 A X Pt (l) = wt% Liquid B C X Pt (l) = 0.16, ~ 100 wt% Liquid X Pt (a 2 ) = 0.23; ~ 0 wt% a 1. X Pt (l) ~ 0.06, ~ 0 wt% liquid X Pt (a 2 ) ~ 0.16, ~ 100 wt% a 2 phase D X Pt (a 2 ) ~ 0.16, 100 wt% a 2 phase

41 Metatectic Phase diagrams # Point M Metatectic Point # Metatectic reaction: ß l + a ; # Alloy ß at M is partially remelted at T < T M Prince (1996)

42 Metatectic Phase diagrams # M Metatectic Point 1605 o C Hf(High-tem) L + Hf(Low-temp) # E Eutectic point M L Hf(low-temp) + Th(High-temP) E # E* - Eutectoid Point E* Th(High-temp) Hf(Low-Temp) + Th(Low-temp)