(Jackson model) Ziman) (fluidity) (viscosity) (Free v

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2 (Jackson model) Ziman) (fluidity) (viscosity) (Free volume)( 0.1 ) (random structure) 12 ( (short range order) ) (random close-packed) Bernal

3 Metal Shrinkage [%] Fe 4.0 Al 6.6 Cu 4.9 Mg 4.2 Flemings, Appendix B). Temperature: T time:t (melting temperature: ) (super cooling or under cooling) (heat of fusion, latent heat: H m ) 0.2 G L G S 2

4 4 Free energy: G G L G S Temperature: T 0.2 G = 0 G = G S G L G = G S G L = H T S (1) H S T H m = H L H S = H T = G = 0 S = H m (2) S, H m G = H m + T H m = H m T (3) S f 2ncal/K/mol 8.4nJ/K/mol (Richards rule) n 1 NaCl n = 2 (9 11J/K/n-mol)

5 e -10 G * r * 5e -07 Gsurface r -4e -10 Gtotal Gvolume 0.3 ( 14J/K/n-mol) ( 30 J/K/n-mol) (homogeneous nucleation) σ (driving force) r G = G v 4πr 3 /3 + 4πr 2 σ (4) 0.3 (critical radius) r dg/dr = 0 r = 2σ G v = 2σ H m T (5) (3) (energy barrier or activation barrier G ) G = 16π 3 σ 3 G 2 v = 16π 3 σ 3 T 2 m (H m T ) 2 (6) Cu Cu 1356

6 erg/cm erg/cm K (over heating) I Z N n I = N nz (7) ) Nn exp ( G kt G d ( ) I exp ( G + G d ) kt exp ( 1/T T 2) exp ( 1/T ) TTT (Time- Temperature-Transformation diagram) 0.4 (amorphous) (8) (9) (inhomogeneous nucleation) 0.5 (substrate:s) (crystal:c) (liquid:l) (contact angle)θ

7 Temperature: T time: t 0.4 TTT liquid σ ls σ lc θ crystal σ cs substrate 0.5

8 8 σ ls = σ cs + cos θσ lc (10) G hetero = G homo f(θ) = ( G v 4πr 3 /3 + 4πr 2 σ lc ) 2 3 cos θ + cos 3 θ 4 (11) (Appendix ) θ (wet) ( Si ) (inoculation) (Jackson model) smooth surface facet rough surface non-facet Jackson N N A one layer Z S ( N NA N ) Z S (12) N A ɛ ( H = N A 1 N ) A Z s ɛ (13) N

9 α= α= γ α=2 α=1 0.6 N N A W = N! N A!(N N A )! (14) 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} (15) Z c L 0 L 0 = Z c ɛ (16) G Nk B = αγ(1 γ) + {(1 γ) ln(1 γ) + γ ln γ} (17) α = L 0 k B Z s Z c (18) α 0.6 α 2

10 System Orientation fcc < 100 > bcc < 100 > hcp < > bct < 110 > diamond < 112 > Chalmers) 0.2 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 Flemings) Merton C. Flemings, Solidification Processing, McGraw-Hill, 1974, New York. Ziman) J. M. Ziman, Models of disorder, Cambridge University Press, A G interface = A lc σ lc + A cs σ cs A cs σ ls (19) G interface = A lc σ lc + πr 2 (σ cs σ ls ) (20) R = r sin θ σ lc = σ cs + σ ls cos θ (21)

11 G interface = A lc σ lc πr 2 cos θσ ls (22) G interface = G volume + G interface = v c G v + (A lc πr 2 cos θ)σ ls v c (23) v c = πr3 (2 3 cos θ + cos 3 θ) 3 (24) A lc = 2πr 2 (1 cos θ) (25) G hetero = G homo f(θ) = ( G v 4πr 3 /3 + 4πr 2 σ lc ) 2 3 cos θ + cos 3 θ 4 (26) f(θ) = 2 3 cos θ + cos3 θ 4 = (2 + cos θ)(1 cos θ)2 4 (27)

知能科学：ニューラルネットワーク

知能科学：ニューラルネットワーク 2 3 4 (Neural Network) (Deep Learning) (Deep Learning) ( x x = ax + b x x x ? x x x w σ b = σ(wx + b) x w b w b .2.8.6 σ(x) = + e x.4.2 -.2 - -5 5 x w x2 w2 σ x3 w3 b = σ(w x + w 2 x 2 + w 3 x 3 + b) x,