2007 5

iii 1 1 1.1.................... 1 2 5 2.1 (shear stress) (shear strain)...... 5 2.1.1...................... 6 2.1.2.................... 6 2.2....................... 7 2.2.1........................ 10 2.2.2 Bingham..................... 10 2.2.3 Herschel-Bulkley................. 11 2.2.4 Casson...................... 11 2.3......................... 13 2.3.1.................... 13 2.3.2...................... 16 3 21 3.1........................ 21 3.2.......................... 23 3.3.......................... 24 3.3.1................... 25 3.3.2..................... 27 3.3.3..................... 29 4 33 4.1............................ 33 4.1.1...................... 34 4.1.2 /.................. 34 4.1.3 /................... 36

iv 4.2 /.......................... 36 4.2.1.................... 36 4.2.2................. 37 4.2.3................... 38 4.2.4..................... 40 4.2.5.................... 41 4.2.6.......................... 42 4.3........................ 45 4.3.1................... 45 4.3.2 Cahn-Hilliard................. 46 4.3.3..................... 49 4.3.4................ 50

v 2.1................................. 5 2.2................................. 8 2.3................................. 8 2.4................................. 9 2.5................................. 10 2.6................................. 11 2.7................................. 12 2.8................................. 13 2.9................................. 14 2.10................................. 17 2.11................................. 18 2.12................................. 20 3.1................................. 21 3.2................................. 23 3.3................................. 25 3.4................................. 26 3.5................................. 27 3.6................................. 28 4.1................................. 34 4.2................................. 35 4.3................................. 35 4.4................................. 36 4.5................................. 37 4.6................................. 39 4.7................................. 41 4.8................................. 41 4.9................................. 42

vi 4.10................................. 43 4.11................................. 44 4.12................................. 48 4.13................................. 49 4.14................................. 51

1 1 1.1 2,3 5000 18

2 1 18 19 1928 80 1990 1970 DNA

1.1. 3 (?)

4 1

5 2 ) ) 2.1 (shear stress) (shear strain) l x Q P F l H u 0 Y u Z y O P Q 2.1:

6 2 2.1.1 (Hookean solid) τ γ τ 2.1 2 P Q F A τ = F/A (2.1.1) 2 l F x γ = x/l (2.1.2) (Hooke s law) τ = Gγ (2.1.3) (shear modulus) G G (tensile stress) T (tensile strain) s E = T/s (Young s modulus) 2.1.2 2.1 Q P u 0 P Q P Couette P Q OH O 0 H u 0 OH 1 Y u O y D y = l u = u 0 u = Dy (2.1.4)

2.2. 7 D = u 0 (2.1.5) l D Y YZ YZ YZ YZ τ x ( ) γ = γ/ t (Newtonian fluid) 2.1 u 0 F F = Aη u 0 l (2.1.1), (2.1.5) D = τ η (2.1.6) (2.1.7) η (viscosity) u 0 = x/ t u 0 l = x l 1 t = γ t = γ (2.1.8) τ = η γ (2.1.9) τ y 2.2

8 2 2.2: elastic viscous t 0 t 2.3: γ τ γ = f(τ) (2.2.1) γ τ 2.2 Bingham 2.3 t 0 τ = G 0 γ τ = η B γ

2.2. 9 (a) (b) (c) eff eff eff 2.4: G 0 = η B t 0 (2.2.2) (instantaneous modulus) η τ/ γ η eff = τ γ (2.2.3) η eff η eff γ γ = τ/η η = dτ/d γ η diff = dτ d γ (2.2.4) η diff 1 η eff γ 2.4 3 (a) η eff γ (b) η eff γ (shear thinning) (c) η eff γ (shear thickening)

10 2 (a) (b) 2.5: 2.2.1 k n γ = τ n (2.2.5) k n > 1 2.5(a) n < 1 2.5(b) n = 1 2.2.2 Bingham τ f B f B γ τ f B γ = { τ fb η B (τ > f B ) 0 (τ < f B ) (2.2.6) Bingham Bingham Bingham (2.2.6) 2.6 f B Bingham η B (plastic viscosity) Bingham (plastic flow)

2.2. 11 f B 2.6: 2.2.3 Herschel-Bulkley τ f H f H (τ f H ) n γ γ = { (τ fh ) n k (τ > f H ) 0 (τ < f H ) (2.2.7) Herschel-Bulkley Bingham (2.2.7) 2.7 n = 1 Bingham f H = 0 n = 1, f H = 0 2.2.4 Casson k 0 k 1 Casson τ = k0 + k 1 γ (2.2.8) k 0 k 1

12 2 f H 2.7: (2.2.8) f C = k 2 0, η C = k 2 1 (2.2.9) γ = τ fc ηc (2.2.10) 2.8 γ τ (2.2.10) τ f C f C η C Casson Casson Casson (2.2.8) Casson (2.2.8) 1) 2)

2.3. 13 2.8: 2.3 2.3.1 2.9 Ω M 1. 2.

14 2 b a h 2.9: 3. 4. 2 3 Ω 2 r r + r 2 h r τ 2πhr 2 τ 2πr 2 τ + d dr (2πhr2 τ)dr (2.3.1) d dr (2πhr2 τ)dr (2.3.2)

2.3. 15 r M M = 2πhr 2 τ (2.3.3) a b τ a τ b M = 2πha 2 τ a = 2πhb 2 τ b (2.3.4) Ω r ω(r) u = rω du dr = r dω dr + ω (2.3.5) ω r du dr = ω (2.3.6) du dr ω = r dω (2.3.7) dr ω r dω/dr < 0 γ γ = r dω dr (2.2.1) (2.3.8) r dω dr (2.3.3) r τ = f(τ) (2.3.9) r dω dr = M dω πhr 2 dτ = 2τ dω dτ (2.3.10)

16 2 2τ dω dτ = f(τ) (2.3.11) ω = 1 τ f(τ) dτ + const (2.3.12) 2 τ Ω = 1 τa f(τ) dτ + const (2.3.13) 2 τ 0 = 1 τb f(τ) dτ + const (2.3.14) 2 τ Ω f(τ) Ω = 1 2 τa τ b f(τ) dτ (2.3.15) τ (2.3.3) M γ = f( 2πhr ) (2.3.16) 2 r b a 1 (2.3.17) a γ r 2.3.2 f(τ) = τ/η Ω = 1 2 τa 1 τ b η dτ = 1 2η (τ a τ b ) = 1 ( M 2η 2πha M ) 2 2πhb 2 (2.3.18)

2.3. 17 M 2.10: Ω = M ( 1 4πhη a 1 ) 2 b 2 (2.3.19) Margules Ω M Ω M 2.10 η (2.2.5) Ω = 1 τa 1 2 τ b k τ n 1 dτ = 1 2kn (τ a n τb n ) = 1 [( ) n ( ) n ] M M 2kn 2πha 2 2πhb 2 (2.3.20) [ log Ω = n log M + log 1 2n(2πh) n ( 1 a 2n 1 b 2n ) ] 1 k log Ω log M 2.11 (2.3.21)

18 2 log logm 2.11: Bingham Bingham (2.2.6) Ω M Bingham f B 3 1. τ a < f B f B Ω = 0 2. τ b < f B < τ a f B < τ < τ a f B < τ (2.3.3) f B < r c M 2πhr 2 (2.3.22) ( ) 1/2 M r c = (2.3.23) 2πhf B r < r c r > r c τ < f B (2.3.15) Ω = 1 2 τa f B f(τ) dτ (2.3.24) τ

2.3. 19 f(τ) = (τ f B )/η B τ b < f B < τ a Ω = 1 τa 2η B f B τ f B dτ = 1 [ τ a f B f B log τ ] a τ 2η B f B (2.3.25) Bingham f B M c f B = M c 2πha 2 (2.3.26) τ a = M Ω M 2πha 2 Ω = 1 4πha 2 η B [M M c M c log MMc ] (2.3.27) 3. f B < τ b (2.3.15) Ω = 1 τa 2η B τ b (2.3.4) a 2 τ a = b 2 τ b τ f B dτ (2.3.28) τ Ω = 1 2η B [(1 a2 b 2 ) τ a 2f B log b ] a (2.3.29) τ a f B M M c Ω M Ω = 1 4πha 2 η B [(1 a2 b 2 ) M 2M c log b ] a (2.3.30) Bingham Ω M 2.12 M = M c M > (b/a) 2 M c η B

20 2 M C M 2.12:

21 3 3.1 3.1 a k F r F = k(r a) (3.1.1) 1 a 2 (tensile stress)t 3.1: a

22 3 T = (tensile strain)s k(r a) a 2 (3.1.2) s = r a a (Young s modulus) E = T s = k a (3.1.3) (3.1.4) U(r) r = a U(r) = U(a) + 1 2 (r a)2 d 2 U dr 2 + (3.1.5) r=a = 1 2 k (r a)2 + const. (3.1.6) k = d2 U dr 2 (3.1.7) (3.1.8) r=a ( r U(r) = εf a) (3.1.9) r = a ε U(a) = ε f(x) f(1) = 1 k = d2 U dr 2 = ε r=a a f (1) (3.1.10) 2 f (1) C E = C ε a 3 (3.1.11)

3.2. 23 3.2 ) ) 3.2(b) ε ν (relaxation time) t 0 ( t 1 0 ν exp ε ) k B T (3.2.1) (a) (b) 3.2:

24 3 ν 10 12 Hz ε 1 ε ε 0.4ε t 0 = 10 12 10 10 t 0 t 0 (2.2.2) G 0 (3.2.1) η = G ( ) 0 ε ν exp k B T (3.2.2) (Arrhenius behavior) 3.3 (glassy state)

3.3. 25 3.3.1 t vib t config t 0 3.3 t config η T 0 Vogel-Fulcher η = η 0 exp B T T 0 (3.3.1) T 0 Vogel-Fulcher (2.2.2) η G 0 t config t config = η 0 B exp (3.3.2) G 0 T T 0 t exp t config > t exp T g (glass transition temperature) T m V 3.4 T g log t 1/t vib 1/t exp 1/t config 1/T g 1/T 3.3:

26 3 1 T g t config t exp t exp T g (kinetic transition) 3.5(a) C p = T ( ) S T p 3.5(b) T = 0 (residual entropy) (broken ergodicity) (excess V liquid glass(1) glass(2) crystal T (2) T (1) g g T m T 3.4:

3.3. 27 configurational entropy) 3.5(b) S C T g Kauzmann T k T k Vogel-Fulcher T 0 3.3.2 (free volume theory) (cooperatively rearranging region theory) v f v v f v = f g + α f (T T g ) (3.3.3) f g α f (a) C p (b) S S 2 (2) S C S 2 (1) T g T T k T (2) T (1) g g T m T 3.5:

28 3 { } b η = a exp f g + α f (T T g ) ( ) bv η = a exp v f { = a exp b/α f T (T g f g /α f ) } (3.3.4) (3.3.5) T 0 = T g f g /α f Vogel-Fulcher (3.3.1) (3.3.4) (3.3.3) 3.3 (a) 1 (b) 1 Adam and Gibbs 1965 (a) (b) 3.6:

3.3. 29 (cooperatively rearranging region=crr) Vogel-Fulcher T 0 1 µ CRR z ( ) t 1 config ν exp z µ k B T (3.3.6) Arrhenius (3.2.2) ε T z S C C ( t 1 config ν exp C ) T S C (3.3.7) S C T T k Vogel-Fulcher 3.3.3 1 λ d θ Bragg 2d sin θ = λ d r F

30 3 F = = dv n(r) exp [i(k k ) r] (3.3.8) dv n(r) exp [iq r] (3.3.9) n(r) 1 k k q = k k m f m = dv n m (r r m ) exp [ iq (r r m )] (3.3.10) r m m F (q) = m f m exp( iq r m ) (3.3.11) I F 2 I(q) = m q r r m α I(q) = m f m f n exp( iq (r m r n )) (3.3.12) n f m f n exp( iqr mn cos α) (3.3.13) n q = q r mn = r m r n exp(iqr cos α) = 2π 1 d(cos α) exp(iqr mn cos α) (3.3.14) 4π 1 = sin qr mn qr mn (3.3.15) I(q) = m (f m f n sin qr mn )/qr mn (3.3.16) n

3.3. 31 f = f m = f n N I(q) = Nf 2 [ 1 + (sin qr mn )/qr mn ] m m (3.3.17) r ρ(r) I(q) = Nf 2 [1 + R 0 dr4πr 2 ρ(r) ] sin qr qr (3.3.18) R ρ 0 I(q) = Nf 2 [1 + R 0 dr4πr 2 [ρ(r) ρ 0 ] sin qr qr + ρ 0 q R 0 ] dr4πr sin qr (3.3.19) (3.3.19) R 3 I(q) = Nf 2 [1 + S(q) 0 dr4πr 2 [ρ(r) ρ 0 ] ] sin qr qr (3.3.20) S(q) I(q) Nf 2 = 1 + 0 dr4πr 2 [ρ(r) ρ 0 ] sin qr qr (3.3.21) ρ(r) g(r)ρ 0 (3.3.22) sin qr/qr exp(iq r) 2 sin qr S(q) = 1 + 4πρ 0 dr [g(r) 1] r (3.3.23) 0 qr = 1 + ρ 0 dr [g(r) 1] exp(iq r) (3.3.24) 0

32 3 Fourier g(r) 1 = = 1 dq [S(q) 1] exp( iq r) (3.3.25) 8π 3 ρ 0 1 dq [S(q) 1] q sin qr (3.3.26) 2π 2 ρ 0 r S(q)

33 4 T = 1618 (order parameter) / 4.1 0 4.1(a)

34 4 (a) (b) T C T T C T 4.1: 1 4.1(b) 2 4.1.1 T ρ gas ρ liq ρ = ρ liq ρ gas T c 4.1.2 / 0 T C 770 354 M σ i = ±1 N M = 1 N N σ i (4.1.1) i

4.1. 35 T C T 4.2: M T T C 4.3:

36 4 T T C 4.4: 4.1.3 / NH 4 Cl + 2 NH 4 Cl + + + N + φ = 2N + N 1 (4.1.2) 4.2 / 4.2.1 / Curie- Weiss / Bragg-Williams (mean-field approximation)

4.2. / 37 A B A+B Temperature 4.5: (regular solution model) 2 A B 4.5 F A+B F A + F B (mixing free energy)f mix F A+B (F A + F B ) F A+B (mixing entropy)s mix (mixing energy)u mix F mix = U mix T S mix (4.2.1) S mix U mix 4.2.2 2 z A B φ A φ B

38 4 φ A + φ B = 1 (4.2.2) φ A = V A V, φ B = V B V (4.2.3) A B S = k B p i ln p i (4.2.4) p i A B 2 φ A φ B i S mix = k B (φ A ln φ A + φ B ln φ B ) (4.2.5) 1 φ A = 1 φ B = 1 S mix = 0 4.2.3 A B ε AA, ε BB A B ε AB zφ A A zφ B B 1 z ( ) φ 2 2 A ε AA + φ 2 Bε BB + 2φ A φ B ε AB (4.2.6) 1/2 z 2 (φ Aε AA + φ B ε BB ) (4.2.6)

4.2. / 39 0 >2.0 F k B T =2.0 <2.0-1 0 0.5 1 4.6: U mix = z [ ] (φ 2 2 A φ A )ε AA + (φ 2 B φ B )ε BB + 2φ A φ B ε AB (4.2.7) = z 2 [ φ Aφ B ε AA φ A φ B ε BB + 2φ A φ B ε AB ] (4.2.8) = z 2 φ Aφ B (2ε AB ε AA ε BB ) (4.2.9) χ χ = z 2k B T (2ε AB ε AA ε BB ) (4.2.10) A B χ U mix = χφ A φ B k B T (4.2.11) F mix k B T = φ A ln φ A + φ B ln φ B + χφ A φ B (4.2.12) φ = φ A (= 1 φ B ) F k B T = φ ln φ + (1 φ) ln (1 φ) + χφ(1 φ) (4.2.13)

40 4 4.6 χ 2.0 2 χ 2.0 φ = 0.5 1 4.2.4 A φ 0 V 0 A B V 1 V 2 A φ 1 φ 2 A φ 0 V 0 = φ 1 V 1 + φ 2 V 2 (4.2.14) φ 0 = V 1 V 0 φ 1 + V 2 V 0 φ 2 (4.2.15) α 1 = V 1 V 0 α 2 = V 2 V 0 α 1 + α 2 = 1 φ 0 = α 1 φ 1 + (1 α 1 )φ 2 = φ 2 + (φ 1 φ 2 )α 1 α 1 = φ 0 φ 2 φ 1 φ 2 φ 0 = (1 α 2 )φ 1 + α 2 φ 2 = φ 1 + (φ 2 φ 1 )α 2 α 2 = φ 0 φ 1 φ 2 φ 1 2 F sep = α 1 F mix (φ 1 ) + α 2 F mix (φ 2 ) (4.2.16) = φ 0 φ 2 φ 1 φ 2 F mix (φ 1 ) + φ 0 φ 1 φ 2 φ 1 F mix (φ 2 ) (4.2.17) 4.7(a) φ 0 φ 1 φ 2 F sep > F 0 φ 1 φ 2 4.7(b) F sep < F 0

4.2. / 41 (a) F (b) F F mix ( 1 ) F mix ( F sep 2 ) F 0 F 0 F sep F mix ( 1 ) F mix ( 2 ) 1 0 2 1 0 2 4.7: F F b F b ' F a ' F a a b 4.8: 4.2.5 4.8 φ a F a F a (metastable) φ b F b F b (unstable) 1. d 2 F dφ 2 > 0:

42 4 F >2.0 =2.0 <2.0 4.9: 2. d 2 F dφ 2 < 0: d2 F = 0 dφ 2 (spinodal point) d2 F > 0 dφ 2 d2 F < 0 (critical dφ 2 point) 3 d3 F dφ 3 4.2.6 (4.2.13) χ 4.9 χ > 2 2 χ = 2 2 2 1 df = 0 dφ d2 F = 0 φ = 0.5 dφ 2 χ < 2 φ = 0.5 4.10 χ

4.3. 43 spinodal line unstable binodal line metastable c 4.10: (binodal line) (spinodal line) χ ε AA ε BB ε AB T (4.2.10) χ 1 T 4.11 4.3 (spinodal decomposition) enhance (nucleation and growth)

44 4 T gas liquid 4.11: 4.3.1 enhance (uphill diffusion) µ = ( ) F φ T,V d 2 F dφ 2 = dµ dφ < 0 F φ x

4.3. 45 φ dφ dx 1 ( dφ(x) dx κ [ ( ) ] 2 dφ(x) F = A f 0 (φ(x)) + κ dx (4.3.1) dx Fick dφ J D dx ) 2 J = D dφ(x) dx dφ dt = dj dx (4.3.2) (4.3.2) (4.3.3) φ t = D 2 φ (4.3.4) x 2 J A B J A = M d dx (µ A µ B ) (4.3.5) M Onsager µ A µ B A B 4.3.2 Cahn-Hilliard enhance

46 4 f µ f φ (4.3.1) φ δφ F δf [( ) df0 δf = A δφ + 2κ dφ ] d dφ dx dx δφ dx (4.3.6) [ ] df0 = A dφ φ 2κd2 δφdx (4.3.7) dx 2 (4.3.5) µ = df 0 dφ 2κd2 φ dx 2 (4.3.8) J A = Mf 0 φ x φ 2Mκ 3 (4.3.9) x ( ) 3 f 0 = d2 f 0 dφ 2 (4.3.3) [ φ t = M f 0 2 φ x + f 2 0 ( φ x ) ] 2 2κ 4 φ x 4 (4.3.10) φ [ φ t = M f 0 ] 2 φ x φ 2 2κ 4 x 4 (4.3.11) Cahn-Hilliard 2 D eff = Mf 0

4.3. 47 R R(q m ) 0 q m q c q 4.12: M f 0 f 0 < 0 D eff < 0 Cahn-Hilliard (4.3.11) φ Fourier φ = q φ q cos qx (4.3.11) φ q t = ( q 2 D eff 2q 4 Mκ ) φ q (4.3.12) ) ] φ q = exp [ D eff q (1 2 + 2κq2 t (4.3.13) f 0 φ(x, t) = φ 0 + A q cos qx exp [ D eff q (1 2 + 2κq2 f 0 ) ] t (4.3.14) ( ) R(q) = D eff q 2 1 + 2κq2 f 0 (4.3.15)

48 4 (amplification factor) 4.12 R(q) R(q) = 0 q R q m = 1 2 ( f ) 1/2 0 (4.3.16) κ R(q m ) = Mf 0 2 q c q c = q m = q c / 2 8κ (4.3.17) ( f ) 1/2 0 (4.3.18) 2κ 4.3.3 v F v r F (r) = 4 3 πr3 F v + 4πr 2 γ (4.3.19) γ F (r) r 4.13 r = 2γ F v

4.3. 49 F(r) 0 r* r 4.13: r F = F (r ) = 16πγ3 3 F 2 v (nucleation rate) exp ( F ) k B T (4.3.20) (4.3.21) (homogeneous nucleation) (heterogeneous nucleation) 4.3.4 driving force (Ostwald ripening)

50 4 (a) φ φ 1 R(t) φ 0 (b) φ 2 φ R(t) x φ 1 φ 0 w(t) (c) φ 2 φ R(t) x φ 1 φ 0 w(t) φ 2 x 4.14: 1/3 (Lifshitz-Slyozov ) R(t) w(t)?? (early stage) 4.14(a) R(t) w(t) early stage (intermediate stage) R(t) w(t) (late sage) R(t) late stage (dynamical scaling) R(t)

4.3. 51 G(r, t) x G(r, t) = G(x), x = r R(t) (4.3.22) G(x) R(t) R(t) Lifshitz-Slyozov