Cahn-Hilliard Cahn-Hilliard 4 Cahn-Hilliard 5 x F (x F v F µ x (- µ v (- v M F M µ (- M F J v (- J v M µ (- Fik (-4 (- (-5 J D D µ M (-4 (-5 µ µ + RT ln a γ a γ (-5 (-6 D D * (ln γ * (ln γ D D +, D D + (ln (ln (-6 * * D D (-7-78-
* * D M RT, D M RT (-7 R T D Darken D D + D D 6 D Gibbs-Duhem Gibbs-Duhem ( (-9 (-8 dµ + dµ (-9 (-9 µ µ + RT ln γ µ µ + RT ln γ d d (ln γ (ln γ (ln (ln D (-8 (-6 (- D (- * * (ln D γ ( D + D + (ln (- (- G D G G µ + µ (- (- Gibbs-Duhem (- (- G (ln γ RT + (ln (- (- (-7 D (-4 (- G D M(, (-4 M (, ( M + M (-5 (-4 G Ω G (-5 / Ω + RT /( (-4 D ( M + M ( RT Ω D M R * D M *, ( T D, ( RT D (-4 D Fik (-6-79-
G µ J D M(, M(, G µ (-6 (-7 (-7 G G sys χ J M(, x (-8 δgsys δ( G + E + Estr δg δe δestr χ + + χ + χ + χs tr δ δ δ δ δ (-9 G sys G E str χ,, E χ χ χ str χ (-7 µ (-8 (- M (, M( χ J M( + ξ ( x, t x (- ξ( xt, t x (- xt (, χ( xt, M{(,, xt T} + [ Mxt {(,, T}(, ξ xt t ] (- (- (- r (, r t t { } div M{ ( r, t, T}[grad{ χ( r, t} + ξ( r, t] (- Gibbs-Duhem Gibbs-Duhem (- Cahn-Hilliard -8-
Cahn 7 G sys Gsys G ( η Y< hkl> ( κ L + + dx (- x G G( L dx, x Estr Y hkl ( L η < > dx, x E L κ dx (- x L η Y (-,, < hkl> κ F x,,( / χ δ Gsys F d F G ( χ + η Y< hkl> ( κ δ dx ( / (-4 (- G ( M( χ M( + η Y ( < hkl> M κ (-5 t G ( D M( + η Y, ( < hkl> K M κ (-6 Cahn-Hilliard D K (-7 t ( M ( K (-6 D (-5 χ χ M( χ M( M( + t (-8 χ (-4 χ t / t (-8 t F HG I KJ t ( + t t ( + t (-9 t -8-
t t ( + t t ( t.% t t (- (-4 (- (-4 G (, T G ( T( + G ( T +Ω(, T ( + RT{ ln + ( ln( } (4-8 G ( T X T Ω (, T X (4- (4- G Ω χ G( T G( T + ( +Ω(, T( + RT{ln ln( } (4- (- - - Cahn x, pq, f ( x, p, q x x -8-
f f f f( x, p, q f( x, p, q + ( x x + ( x x + ( x x + (4-! x! x x x x x x x i x ( x x p, q p q (4- p f f f( x, p, q f( x, p, q + ( p p + ( p p px x! p x x p p p p + i ( p p q q q f f f( x, p, q f( x, p, q ( q q ( q q + + q x x! q x x p p p p q q q q f f f + ( q q x x x x x x x x q x p p q q p p p p q q + + f f f f( x, p, q f( x, p, q + ( x x + ( p p + ( q q x x x p x x q x x p p p p p p q q q q q q f f f + + + ( x x ( p p x x x p x x q x x p p p p p p q q q q q q ( q q f f f x x p p x x q q q q p p x p x x x q x x q p x x p p p p p p q q q q q q + ( ( + ( ( + ( ( + (4-4 x p, q f f f( x, p, q f( x, p, q + ( p p + ( q q pp p q p p q q q q f f f + ( p p + ( q q + ( p p ( q q + (4-5 p p p q p p q p p p q q q q q q f( x, p, q + L( x( p p + K ( x( q q + K ( x( p p + K ( x( q q + K ( x( p p ( q q + 4 Lx (, Ki ( x x -8-
(4-5 ( x, pq, (,, (,, f ( x, p, q (,, (,, (( f f + L + K ( ( + K ( ( + K ( ( + K ( ( ( + 4 f(,, + L( ( + K ( ( + K ( ( (4-6 ( x f x x x f x x x { (, (, ( } { (, (, ( } (4-6 (4-6 f{ x (,,} + Lx { ( }{ x ( } + K{ x ( }{ x ( } + K{ x ( }{ x ( } f { ( x,,} + L{ ( x}{ ( x} + K { ( x}{ ( x} + K { ( x}{ ( x} f{ x (,,} + Lx { ( }{ x ( } + K{ x ( }{ x ( } + K{ x ( }{ x ( } Lx {( }{ x ( } Lx { ( } L( f (,, f(,, + K (( + K (( (4-7 f(,, E E [ K ( ( + K ( ( ] dv K ( ( dv + K ( ( dv V V V (4-8,j j r s r f g dr f g n ds f gdr,j K ( ( dv K ( {( n} ds ( K ( dv V S V K K K( {( n} ds ( dv ( dv S V V (4-9 n E -84-
K E K( ( dv + K( ( dv K( ( dv V V V (4- K ( K( K (4- E K( ( dv V (4- K ( K ( χ (4- x x+ + + x x ( x x x x x x (4- E Ω Ω x x x E Ω x x+ (4-4 (4- (4-4 E Ω( x ( x Ω( x( x + Ω ( x 4 x 4 x x x x (4-5 (4-7 K( Ω( x, K( Ω ( x (4-6 4 4 (4- K K K x ( ( Ω( (4-7 x -85-
(- Y (hkl < hkl> eigen 9 (pure dilatation eigen e ij T F H G I K J (4-8 Vegard η( η eigen e ij T C ijkl C C C C C C C C C C C C C C C C C C C C44 C C44 C C 44 (4-9 Eshelby eigen (4-8(4-9 E T T T Ce + Ce + Ce T T T T T T T T E Cijkleij ekl + Cee+ Cee+ Cee ( C+ C T T T T T T + Cee + Cee + Cee (4- x < hk l > ( hkl x y z xyz C x ( σ x ij T T T σ x Ce + Ce + Ce ( C + C (4- eigen (4- σ x x x -86-
σ C + C + C C y z y x x y z x (4- z x σ x ( C + C C C (4- x E E ( C + C σ C x x (4- E E (E E (4-(4- E str ( C C C C C C + + + Estr E E ( C + C C C (4-4 C C, C, C ( xyz ( xyz 44 C l l l l C ijkl ip jq km ln pqmn l ij C C l l l l C p q m n pqmn l C + l C + l C + l l C + l l C + l l C + 4l l C + 4l l C + 4l l C 4 4 4 ( l + l + l C + ( l l + l l + l l 4 4 4 C ( n + n + n + C ( n n + n n C 4 4 4 C ( n + n + n + (C C + C 44 C + ( C C + C ( n n + n n 44 + 4( l l + l l + l l C 44 + n n + 4 C ( n n + n n + n n ( n n 44 + n n + n n + n n (4-5 l n l n l n n n + + 4 4 4,, n + + ( n + n + n ( nn n n nn (4-5 (4-4 E str C + C C + C C + ( 44 C + C ( nn + nn + nn (4-6 C Y < hkl> C + C + C C Y < hkl > + ( C + C ( nn + nn + nn C C (4-7 44 ( n, n, n ( hkl,, n h l k, k, n nn + nn + nn h k + l + l h n h + k + l h + k + l h + k + l ( h + k + l eigen η( -87-
E Y str ( η < hkl > ( (4-8 < hkl >< > < hkl >< > Y < > ( C + C ( C C 6C Y 44( C + C < > (4-9 C C + C + 4C 44 (4-7 [ ] ( C C + C 44 (Y ( C C + C > < < hkl> 44 > (C C + C < > 44 < C /(C C 7 44 Cahn Khahaturyan (( ( n, n, n Y < hkl> (4-8 (4-8 Cahn-Hilliard Phase-field J.W.Cahn: "The Seleted Works of J.W.Cahn", ed. by W.C.Carter and W.C.Johnaon, TMS, (998, p.9-5. J.E.Hilliard: "Phase Transformation", ed. by H.I.aronson, SM, Metals Park, Ohio, (97, p.497-56. " ",,(997, 7 4 " ",,(998, 8 5,(996, pp.9-4. 6 P.G.Shewmon,,, (976, 4 7 J.W.Cahn : ta Metall., (96, pp.79-8. 8 N.Saunders and.p.miodownik : "CLPHD",,Pergamon, (998. 9,, (976,.Khahaturyan:"Theory of Strutural Transformations in Solids.", Wiley, New York, NY, (98 T.Koyama and T.Miyazaki ; Mater. Trans. JIM., 9(998,pp.69-78. -88-