δf = δn I [ ( FI (N I ) N I ) T,V δn I [ ( FI N I ( ) F N T,V ( ) FII (N N I ) + N I ) ( ) FII T,V N II T,V T,V ] ] = 0 = 0 (8.2) = µ (8.3) G

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8 ( ) 8. 1 ( ) F F = F I (N I, T, V I ) + F II (N II, T, V II ) (8.1) F

δf = δn I [ ( FI (N I ) N I 8. 1 111 ) T,V δn I [ ( FI N I ( ) F N T,V ( ) FII (N N I ) + N I ) ( ) FII T,V N II T,V T,V ] ] = 0 = 0 (8.2) = µ (8.3) G F G(N, T, p) = F (N, T, V ) + pv (8.4) T, p N ( ) ( ) ( ) ( ) ( ) G F F V V = + + p N T,p N T,p V N,T N T,p N T,p (8.5) p = ( F/ V ) N,T ( ) G = µ (8.6) N T,p µ G = µn (8.7) 1 T, p N 2 G 2 G(αN, T, p) = αg(n, T, p) (8.8)

112 8. α 1 G(αN, T, p) G(αN, T, p) = αn G(N, T, p) α αn = N (8.9) α=1 α N µn G (8.7) i G = i µ i N i (8.10) N i x i G = i µ i x i (8.11) G 8. 2 T p n m f = 2 m + n (8.12) mn 1 m 2 + mn m (8.13) i

j µ (j) i 8. 3 113 µ (1) 1 = µ (2) 1 = = µ (m) 1 = µ 1 µ (1) 2 = µ (2) 2 = = µ (m) 2 = µ 2 µ (1) n = µ (2) n = = µ (m) n = µ n (8.14) n(m 1) f f = 2 + mn m n(m 1) = 2 m + n (8.15) f = 0 1 m + n 8. 3 I II (E I 0 < E II 0 ) (K I > K II ) 8.1 I II I II (phase transition) T tr G I (T tr ) = G II (T tr ) (8.16)

114 8. Phase I Free Energy Ttr Phase II Temperature 8.1 (I) (II) G G = H T tr S = 0 (8.17) H II H I = T tr (S II S I ) (8.18) S II S I > 0 I II II I (latent heat) 1 (first-order transition) 2 (second-order transition) 1 1 2 2 II

8. 4 115 Fe bcc fcc Debye 8. 4 C p H T H(T ) = H 0 + C p (T )dt (8.19) T 0 H 0 T 0 T 0 S T C p (T ) S(T ) = S 0 + dt (8.20) T 0 T S f 2ncal/K/mol 8.4nJ/K/mol Richards rule n 1 NaCl n = 2 (9 11J/K/n-mol) ( 14J/K/n-mol) ( 30 J/K/n-mol)

116 8. 8. 5 8.2 2 2 I (undercooled or supercooled liquid) (metastable equilibrium) Fe-Fe 3 C Tm[I] Ttr Tm[II] 8.2 T tr I II T m[i] I T m[ii] II 8. 6 H 2 O 8.3 H,O 2 H 2 + 1/2O 2 H 2 O

8. 6 117 1 22.064 [MPa] 611 [Pa] 273.16 647.10 8.3 H 2O - (p T diagram) f = 3 m (8.21) 1 m = 1 f = 2 2 2 2 2 - f = 3 2 = 1 1 3 m = 3 f = 0 3 (triple point) 2 3 (critical point) A,B 2 2

118 8. (Clausius-Clapeyron) dp dt = S V (8.22) dg = SdT + V dp (8.23) 2 S A dt + V A dp = S B dt + V B dp (8.24) dp/dt p T p T 8. 7 8. 7. 1 1 (homogeneous nucleation) σ (driving force) r (embryo) (droplet model) G = G v 4πr 3 /3 + 4πr 2 σ (8.25) G v (II ) (I ) G v = G I G II = H T S (8.26)

8. 7 119 4e -10 G * r * 5e -07 Gsurface r -4e -10 Gtotal Gvolume 8.4 H S T H tr = H II H I = H T = T tr G = 0 S = H tr T tr (8.27) S, H tr G v = H tr + T H tr T tr = H tr T T tr (8.28) 8.4 (critical radius) r dg/dr = 0 r = 2σ G v = 2σT tr H tr T (8.29) (8.28) (energy barrier or activation barrier G ) G = 16π 3 σ 3 G 2 v = 16π 3 σ 3 T 2 tr (H tr T ) 2 (8.30) Cu Cu 1356 1.44 10 2 erg/cm 2 1.88 10 10 erg/cm 3 8.4 100K

120 8. (over heating) I Z N n I = N nz (8.31) ) Nn exp ( G kt (8.32) G d ( ) I exp ( G + G d ) kt (8.33) exp ( 1/T T 2) exp ( 1/T ) TTT (Time- Temperature-Transformation diagram) 8.5 (amorphous) 8. 7. 2 (inhomogeneous nucleation) 8.6 (substrate:s) (crystal:c) (liquid:l) (contact angle)θ

8. 7 121 Temperature: T T m time: t 8.5 1 TTT T m (melting temperature) liquid σ ls σ lc θ crystal σ cs substrate 8.6 σ ls = σ cs + cos θσ lc (8.34) G hetero = G homo f(θ) = ( G v 4πr 3 /3 + 4πr 2 σ lc ) 2 3 cos θ + cos 3 θ 4 (8.35) (Appendix )

122 8. θ (wet) 8. 7. 3 (Jackson model) smooth surface facet rough surface non-facet Jackson N N A one layer Z S ( N NA N ) Z S (8.36) N A ɛ ( H = N A 1 N ) A Z s ɛ (8.37) N N N A W = N! N A!(N N A )! (8.38) S = k B ln W (Stirling s approximation) ln N! = N ln N N γ = N A /N S = k B N {(1 γ) ln(1 γ) + γ ln γ} (8.39) Z c L 0

8. 7 123 0.4 α=4 0.2 0-0. 2-0. 4 α=3 0.2 0.4 0.6 0.8 1 γ α=2 α=1 8.7 L 0 = Z c ɛ (8.40) G Nk B T tr = αγ(1 γ) + {(1 γ) ln(1 γ) + γ ln γ} (8.41) α = L 0 k B T tr Z s Z c (8.42) α 8.7 α 2 KurzFisher) W. Kurz and D. J. Fisher, Fundamentals of Solidification, Trans Tech Publications, 1984, Switzerland. Chalmers) Bruce Chalmers, Principles of Solidification, John Wiley & Sons, Inc., 1964, New York. 1971 Flemings) Merton C. Flemings, Solidification Processing, McGraw-Hill, 1974, New York.

124 8.. 1 A G interface = A lc σ lc + A cs σ cs A cs σ ls (43) G interface = A lc σ lc + πr 2 (σ cs σ ls ) (44) R = r sin θ σ lc = σ cs + σ ls cos θ (45) G interface = A lc σ lc πr 2 cos θσ ls (46) G interface = G volume + G interface = v c G v + (A lc πr 2 cos θ)σ ls v c (47) v c = πr3 (2 3 cos θ + cos 3 θ) 3 (48) A lc = 2πr 2 (1 cos θ) (49) G hetero = G homo f(θ) = ( G v 4πr 3 /3 + 4πr 2 σ lc ) 2 3 cos θ + cos 3 θ 4 (50) f(θ) = 2 3 cos θ + cos3 θ 4 = (2 + cos θ)(1 cos θ)2 4 (51)

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