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- あきみ わにべ
- 5 years ago
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3 iii (shear stress) (shear strain) Bingham Herschel-Bulkley Casson / / /
4 iv Cahn-Hilliard Flory-Huggins
5 v Bingham (a) (b) (c) Bingham Herschel-Bulkley Casson Ω M Ω M Bingham Ω M (a) (b) t vib t config T m T g (a) T m (b) (a) 1 (b) (a) (b)
6 vi 4.5 A, B (a) (b) φ a φ b χ r R.A.L.Jones Soft Condensed Matter Fig (a) (b) C-C R.A.L.Jones Soft Condensed Matter Fig (a) (b) (c) (d) (a) (b) (c) (d)
7 vii 6.9 (a) (b) (c) (a) (b) (c) (a) A (b) C (a) (b) (a) (b) (a) (b) n u 1, u (7.3.17) (7.3.18) (a) S (b) S T c1 T c (a) (splay) n 0 (b) (twist) n 0 (c) (bend) n ( n) (a) H (b) H < H c x H > H c (a) van der Waals (b) van der Waals h (i)(ii)(iii) (a) (b) (c)
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9 ,
10 DNA
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13 5 2 ) ) 2.1 (shear stress) (shear strain) (Hookean solid) τ γ τ P Q F A τ = F/A (2.1.1) l x Q P F l H u 0 u Y Z y O P Q 2.1:
14 6 2 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) 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) D = u 0 l D (2.1.5) Y YZ YZ YZ YZ τ x ( ) γ = γ/ t (Newtonian fluid) 2.1 u 0 F
15 : F = Aη u 0 l (2.1.1), (2.1.5) (2.1.6) D = τ η (2.1.7) η (viscosity) u 0 = x/ t u 0 l = x l 1 t = γ = γ (2.1.8) t τ = η γ (2.1.9) τ y 2.2 γ τ γ = f(τ) (2.2.1)
16 8 2 elastic viscous t 0 t 2.3: Bingham (a) (b) (c) eff eff eff 2.4: (a) (b) (c) γ τ 2.2 Bingham 2.3 t 0 τ = G 0 γ τ = η B γ 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
17 (a) (b) 2.5: γ = τ/η η = dτ/d γ η diff = dτ d γ (2.2.4) η diff 1 η eff γ (a) η eff γ (b) η eff γ (shear thinning) (c) η eff γ (shear thickening) k n γ = τ n (2.2.5) k n > 1 2.5(a) n < 1 2.5(b) n = 1
18 10 2 f B 2.6: Bingham 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) 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
19 f H 2.7: Herschel-Bulkley f H = 0 n = 1, f H = Casson k 0 k 1 Casson τ = k0 + k 1 γ (2.2.8) k 0 k 1 (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)
20 : Casson 2) Ω M 1. 2.
21 b a h 2.9: 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) r M M = 2πhr 2 τ (2.3.3)
22 14 2 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 dω πhr 2 = 2τ dτ dτ (2.3.10) 2τ dω dτ = f(τ) (2.3.11) ω = 1 τ f(τ) dτ + const (2.3.12) 2 τ
23 Ω = = 1 2 τa τb f(τ) dτ + const τ (2.3.13) f(τ) dτ + const τ (2.3.14) Ω f(τ) Ω = 1 2 τa τ b f(τ) dτ (2.3.15) τ (2.3.3) M γ = f( 2πhr 2 ) (2.3.16) r b a a 1 (2.3.17) γ r f(τ) = τ/η Ω = 1 2 τa 1 τ b η dτ = 1 2η (τ a τ b ) = 1 ( M 2η 2πha 2 M ) 2πhb 2 (2.3.18) Ω = M ( 1 4πhη a 2 1 ) b 2 (2.3.19) Margules Ω M Ω M 2.10 η (2.2.5)
24 16 2 M 2.10: Ω M log logm 2.11: Ω M Ω = 1 2 τa 1 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)
25 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 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 2πha Ω M 2 ] 1 Ω = [M 4πha 2 M c M c log MMc η B (2.3.27)
26 18 2 M C M 2.12: Bingham Ω M 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 ) 1 Ω = [(1 4πha 2 a2 η B 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
27 a k F r F = k(r a) (3.1.1) 1 a 2 (tensile stress)t T = (tensile strain)s k(r a) a 2 (3.1.2) a 3.1:
28 20 3 s = r a a (Young s modulus) (3.1.3) E = T s = k a (3.1.4) U(r) r = a U(r) = U(a) (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 2 f (1) (3.1.10) f (1) C E = C ε a 3 (3.1.11) 3.2 )
29 ) 3.2(b) ε ν (relaxation time) t 0 ( t 1 0 ν exp ε ) k B T (3.2.1) ν Hz ε 1 ε ε 0.4ε t 0 = 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) (a) (b) 3.2: (a) (b)
30 (glassy state) t vib t config t 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
31 T g 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 configurational entropy) 3.5(b) S C T g Kauzmann T k T k Vogel-Fulcher T (free volume theory) (cooperatively rearranging region theory)
32 24 3 v f v v f v = f g + α f (T T g ) (3.3.3) f g α f ( ) bv η = a exp v f (3.3.4) { } b η = a exp f g + α f (T T g ) { = a exp b/α f T (T g f g /α f ) } (3.3.5) T 0 = T g f g /α f Vogel-Fulcher (3.3.1) (3.3.4) (3.3.3) 3.6 (a) 1 (b) 1 Adam and Gibbs 1965 (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)
33 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 λ d θ Bragg 2d sin θ = λ d r F 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
34 26 3 F (q) = m f m exp( iq r m ) (3.3.11) I F 2 I(q) = f m f n exp( iq (r m r n )) (3.3.12) m n q r r m α I(q) = f m f n exp( iqr mn cos α) (3.3.13) m 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 I(q) = m = sin qr mn qr mn (3.3.15) (f m f n sin qr mn )/qr mn (3.3.16) n f = f m = f n N I(q) = Nf 2 [ 1 + (sin qr mn )/qr mn ] m m r ρ(r) I(q) = Nf 2 [1 + R 0 dr4πr 2 ρ(r) ] sin qr qr 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 (3.3.17) (3.3.18) ] (3.3.19)
35 (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 = 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) S(q) = 1 + 4πρ 0 Fourier 0 2 sin qr dr [g(r) 1] r qr (3.3.23) = 1 + ρ 0 dr [g(r) 1] exp(iq r) (3.3.24) 0 g(r) 1 = = 1 8π 3 dq [S(q) 1] exp( iq r) (3.3.25) ρ 0 1 2π 2 dq [S(q) 1] q sin qr (3.3.26) ρ 0 r S(q)
36 28 3 log t 1/t vib 1/t exp 1/t config 1/T g 1/T 3.3: t vib t config V liquid glass(1) glass(2) crystal T (2) T (1) g g T m T 3.4: T m T g
37 (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: (a) T m (b) (a) (b) 3.6: (a) 1 (b) 1
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39 31 4 T = 1618 (order parameter) / (a) 1 4.1(b) 2
40 32 4 (a) (b) T C T T C T 4.1: (a) (b) T ρ gas ρ liq ρ = ρ liq ρ gas T c / 0 T C M σ i = ±1 N M = 1 N N σ i (4.1.1) i / NH 4 Cl + 2 NH 4 Cl N +
41 T C T 4.2: M T T C 4.3:
42 34 4 T T C 4.4: φ = 2N + N 1 (4.1.2) 4.2 / / Curie-Weiss / Bragg-Williams (mean-field approximation) (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 )
43 4.2. / 35 A B A+B Temperature 4.5: A, B2 F mix (mixing entropy)s mix (mixing energy)u mix F mix = U mix T S mix (4.2.1) S mix U mix z A B φ A φ B 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
44 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) U mix = z 2 [ (φ 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) 4.6 χ χ 2.0 φ = 0.5 1
45 4.2. / 37 0 >2.0 F k B T =2.0 < : 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
46 38 4 (a) F (b) F F mix ( 1 ) F mix ( F sep 2 ) F 0 F 0 F sep F mix ( 1 ) F mix ( 2 ) : (a) (b) F F b F b ' F a ' F a a b 4.8: φ a φ b 4.7(b) F sep < F φ a F a F a (metastable) φ b F b F b (unstable)
47 4.2. / 39 F >2.0 =2.0 < : χ 1. d 2 F dφ 2 2. d 2 F dφ 2 > 0: < 0: d2 F dφ = 0 2 (spinodal point) d2 F dφ > 0 2 d2 F dφ < 0 (critical point) 2 3 d3 F dφ (4.2.13) χ 4.9 χ > 2 2 χ = df dφ = 0 d2 F dφ 2 = 0 φ = 0.5
48 40 4 spinodal line unstable binodal line metastable c 4.10: χ < 2 φ = χ (binodal line) (spinodal line) χ ε AA ε BB ε AB T (4.2.10) χ 1 T (spinodal decomposition) enhance (nucleation and growth)
49 T gas liquid 4.11: enhance (uphill diffusion) µ = ( ) F φ T,V d 2 F dφ 2 = dµ dφ < 0 F φ x 1 ) 2 φ dφ dx ( dφ(x) dx κ
50 42 4 [ ( ) ] 2 dφ(x) F = A f 0 (φ(x)) + κ dx (4.3.1) dx Fick dφ dx J D J = D dφ(x) dx dφ dt = dj dx (4.3.2) (4.3.2) (4.3.3) φ t = D 2 φ x 2 (4.3.4) J A B J A = M d dx (µ A µ B ) (4.3.5) M Onsager µ A µ B A B Cahn-Hilliard enhance f µ f φ (4.3.1) φ δφ F δf
51 [( ) df0 δf = A δφ + 2κ dφ ] d dφ dx dx δφ dx (4.3.6) [ ] df0 = A dφ φ 2κd2 dx 2 δφdx (4.3.7) (4.3.5) (4.3.3) [ φ t = M µ = df 0 dφ 2κd2 φ dx 2 (4.3.8) J A = Mf 0 φ x φ 2Mκ 3 x 3 (4.3.9) ( ) f 0 = d2 f 0 dφ 2 f 0 2 φ x 2 + f 0 ( φ x ) ] 2 2κ 4 φ x 4 (4.3.10) φ [ φ t = M f 0 2 ] φ x 2 φ 2κ 4 x 4 Cahn-Hilliard 2 (4.3.11) D eff = Mf 0 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 = ( q 2 D eff 2q 4 Mκ ) φ q (4.3.12) t ) ] φ q = exp [ D eff q ( κq2 f 0 t (4.3.13)
52 44 4 R R(q m ) 0 q m q c q 4.12: φ(x, t) = φ 0 + A q cos qx exp ) ] [ D eff q ( κq2 f 0 t (4.3.14) ( ) R(q) = D eff q κq2 f 0 (4.3.15) (amplification factor) 4.12 R(q) R(q) q = 0 R q m = 1 2 ( f ) 1/2 0 (4.3.16) κ R(q m ) = Mf 0 2 8κ q c q c = q m = q c / 2 (4.3.17) ( f ) 1/2 0 (4.3.18) 2κ
53 F(r) 0 r* r 4.13: r 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 r F = F (r ) = 16πγ3 3 F 2 v (4.3.20) (nucleation rate) exp ( F ) k B T (4.3.21)
54 46 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: 3 (homogeneous nucleation) (heterogeneous nucleation) driving force (Ostwald ripening) 1/3 (Lifshitz-Slyozov )
55 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) G(r, t) x G(r, t) = G(x), x = r R(t) (4.3.22) G(x) R(t) R(t) Lifshitz-Slyozov
56
57 (polymer) A-A-A- (monomer) (degree of polymerisation)n (CH 2 ) ) PMMA (Polymethyl methacrylate) PDMS Polydimethyl siloxane N
58 : R.A.L.Jones Soft Condensed Matter Fig.5.1
59 : (a) (b) C-C (ideal chain) 1 C-C (segment) (bond) b z N random walk r n
60 : 1-2 R N R = r n (5.2.1) n=1 R = 0 R 2 R 2 N N = r n r m (5.2.2) n=1 m=1 n m r n r m = r n r m = 0 R 2 N = r 2 n = Nb 2 n=1 (5.2.3) R N 1 2 R R P (R, N) b i (i = 1, 2,...z) N R R b i 1 z R P (R, N) = 1 z z P (R b i, N) (5.2.4) i=1 N 1, R b i P (R b i, N 1) = P (R, N) P N P b iα P b iα b iβ (5.2.5) R α 2 R α R β
61 b iα, R iα b i R (5.2.5) (5.2.4) 1 z P N = b2 6 z b iα = 0, 1 z i=1 2 P R 2 (5.2.6) z b iα b iβ = 1 3 δ αβb 2 (5.2.7) N = 0 R (5.2.6) ( ) 3 ) 3 2 P (R, N) = exp ( 3R2 2πNb 2 2Nb 2 i=1 R (5.2.8) U chain = n U (r n, r n+1,..., r n+nc ) (5.2.9) r n r m n m random walk R 2 = Nb 2 eff (5.2.10) b eff 5.3 r Gauss (Gaussian chain) p(r) = ( ) 3 ) 3 2 exp ( 3r2 2πb 2 2b 2 (5.3.1) R n r n = R n R n 1 (5.3.1) {R n } (R 0, R 1,..., R N ) ( ) 3N ) 3 2 P ({R n }) = exp ( 3 N 2πb 2 2b 2 (R n R n 1 ) 2 (5.3.2) 0 n=1 k U = 1 N 2 k (R n R n 1 ) 2 (5.3.3) n=1
62 : 1-3 exp( U/k B T ) k = 3k BT b 2 (5.3.4) (5.3.2) 5.4 random walk self-avoiding walk R R+dR W (R)dR W 0 (R)dR N z N (5.2.8) W 0 (R)dR = P (R, N)4πR 2 dr = ( ) 3 ) 3 z N 4πR 2 2 exp ( 3R2 2πNb 2 2Nb 2 (5.4.1) R 3 1 v c R 3 R 3 /v c 1 1 v c /R 3 N(N 1)/2
63 p(r) ( R 3 v c ln 1 v ) c R 3 ( p(r) = 1 v ) N(N 1) c 2 [ R 3 1 ( = exp 2 N(N 1) ln 1 v )] c R 3 (5.4.2) v c 2 N 1 N(N 1) N R3 p(r) = exp ( N 2 ) v c 2R (5.4.3) W (R) = W 0 (R)p(R) R 2 exp ( 3R2 2Nb 2 N 2 ) v c 2R 3 (5.4.4) (5.4.1) W 0 (R) R 0 = (2Nb 2 /3) 1/2 W (R) (5.4.4) R 3R 2 R 0 R 0 2Nb 2 + 3N 2 v c + 1 = 0 (5.4.5) 4R 3 ( ) R 5 ( ) R 3 = 9 6 v c N (5.4.6) 16 b 3 R 0 N 1 2 R = R 0 ( N 1 2 v c b 3 ) 1 5 N 3 5 (5.4.7) (R N 1 2 ) N R g N b (5.4.8) R g R 5.5
64 : polymer polymer : ε pp polymer solvent : ε ps (5.5.1) solvent solvent : ε ss i N pp (i) N ps (i), N ss (i) E i = N (i) pp ε pp N (i) ps ε ps N (i) ss ε ss (5.5.2) ( R) [ P (R) W (R) exp E(R) ] k B T E(R) R (5.5.3) R 3 1 φ = Nv c /R 3
65 5.6. Flory-Huggins 57 N pp N pp (i) 1 2 znφ N ps (i) zn(1 φ) Nss 0 N (i) ss [ 1 znφ + zn(1 φ) 2 ] (5.5.4) N 0 ss (5.5.2) E(R) 1 2 znφ(ε pp + ε ss 2ε ps ) + (φ independent) = zn 2 v c R 3 ε + (R independent) (5.5.5) (5.5.6) ε = 1 2 (ε pp + ε ss ) ε ps (5.5.7) χ = z ε k B T (5.5.8) (4.2.10) (5.5.5) (5.5.3) P (R) R 2 exp [ 3R2 2Nb 2 N 2 ] v c (1 2χ) 2R3 (5.5.9) (5.4.4) ( v = v c (1 2χ) = v c 1 2z ) k B T ε (5.5.10) 5.6 Flory-Huggins F mono k B T = φ A ln φ A + φ B ln φ B + χφ A φ B (5.6.1) N 1 N F mol poly k B T = φ A ln φ A + φ B ln φ B + Nχφ A φ B (5.6.2)
![T = 1 [φ ln φ + (1 φ) ln(1 φ)] + χφ(1](/docs-images/91/107651681/images/66-3.jpg "φ) (5.6.")
66 : 2 χ 1 F site poly k B T = φ A N ln φ A + φ B N ln φ B + χφ A φ B (5.6.3) Flory-Huggins χ χn χ c = 2 N χ > χ c (5.6.4) χ < χ c (5.6.5) N χ c χ 0 2 A B N A = N B = N φ A = φ, φ B = 1 φ F k B T = 1 [φ ln φ + (1 φ) ln(1 φ)] + χφ(1 φ) (5.6.6) N φ = 1 2 F (φ) F (φ) F φ = 0 (5.6.7) 1 ln φ1 φ = Nχ (5.6.8) 1 2φ
67 5.6. Flory-Huggins 59 Nχ 1 φ b = exp( Nχ) (5.6.9) 1 φ b = exp( Nχ) (5.6.10) A B 2 sharp A, B
68
69 (hydrophilicity) (hydrophobicity) (lipophilicity) (hydrophobic interaction) 4 2
70 (hydrophilic head-group) (hydrophobic tail) (amphiphilic molecule) (surface activity) (surfactant) 6.2 (micelle) 1 1 N N E N = k B T Nϵ N ϵ N 1 1 f f f = N P N N [ ( k B T log P ) ] N N 1 + E N (6.2.1) P N N / N
71 P N /N φ φ = N P N (6.2.2) (6.2.2) P N µ µ = ϵ N + k BT N log P N N (6.2.3) 1 N 2 (6.2.3) ( ) N(µ ϵn ) P N = N exp k B T (6.2.4) N = 1 µ 1 P 1 P N = N [ P 1 exp (ϵ ] N 1 ϵ N ) (6.2.5) k B T ϵ 1 < ϵ N ϵ 1 ϵ N ϵ N N (6.2.4) µ < min{ϵ N } φ P N N N = 1 µ ϵ N (critical micelle concentration=cmc) φ c φ c P 1 = P 1 (6.2.6) N > 1 φ φ c φ ϵ N N N r = (3Nv/4π) 1/3 v γ
72 : R.A.L.Jones Soft Condensed Matter Fig.9.1 4πr 2 γ ϵ ϵ N ϵ N = ϵ + 4π N ( ) 2/3 3Nv (6.2.7) 4π = ϵ + αk BT N 1/3 (6.2.8) αk B T = 4πγ(3v/4π) 2/3 (6.2.5) P N = N { [ ( P 1 exp α 1 1 )]} N (6.2.9) N 1/3 N [P 1 exp(α)] N (6.2.10) P 1 P 1 exp(α) < 1 P 1 exp( α) P N 1 ϵ N N Israerachivilli
73 : (critical packing parameter) 3 (optimum head-group area)a 0 (critical chain length)l c (hydrocarbon volume)v v l c a a 0 M r 4πr 3 /3 = Mv 4πr 2 = Ma 0 M r = 3v/a 0 r l c v l c a N s = (6.3.1) v l c a 0 (6.3.2) a 0 v
74 : l M r πr 2 l = Mv 2πrl = Ma 0 r = 2v/a < N s 1 2 (6.3.3) 6.3 N s 6.4 N ϵ N N = M N < M N > M ϵ N N = M ϵ N = ϵ M + Λ(N M) 2 (6.4.1)
75 (6.2.4) P N P M (6.4.1) [ ( )] N/M PM P N = N M exp M(ϵM ϵ N ) (6.4.2) k B T [ ( )] PM MΛ(M N) 2 N/M P N = N M exp (6.4.3) k B T M < N M 2 >= k BT 2MΛ (6.4.4) k B T M ϵ = ϵ 1 ϵ M P M = M [ ( )] M ϵ P 1 exp (6.4.5) k B T φ = P 1 + P M P 1 < exp( ϵ/k B T ) φ c = exp( ϵ/k B T ) 6.5 ϵ N N N (endcap) E endcap 1 α ϵ N = ϵ + αk BT N (6.5.1) ϵ 1 (6.2.4) P N = N [P 1 e α ] N e α (6.5.2)
76 : N=1 NxN = x/(1 x) 2 φ = N = N P N (6.5.3) N(P 1 e α ) N e α (6.5.4) = P 1 [1 P 1 e α ] 2 (6.5.5) P 1 = 1 + 2φeα 1 + 4φe α 2φe 2α (6.5.6) φe α 1 P 1 φ φ c e α φ c P 1 e α φ c P N = N ( 1 φ 1/2 e α/2) N e α P N / N = 0 N (6.5.7) N max = φe α (6.5.8) P N Ne N/M (6.5.9)
77 (bilayer) 2 CMC 1 (6.2.4) ϵ N = ϵ + αk BT N (6.6.1) P N = N[P 1 e α ] N e α N (6.6.2) e α e α N (vesicle) 6.7 (bending energy) 2 (principal curvature)c 1 c c 1, c 2 c 1 c 2 da c 1 c 2 [ ] k E el = 2 (c 1 + c 2 2c 0 ) 2 + kc 1 c 2 da (6.7.1)
78 : c 0 (spontaneous curvature) k k (bending modulus) (saddle-spray modulus) k (mean curvature)(c 1 +c 2 )/ (lyotropic liquid crystal) (lyo-) N s < 1/3
79 : (a) (b) (c) (d) 4.24 L 1 I H I L α L 1 H I L α L 1 H I H I L α a b H I I 6 L 2
80 : (a) (b) (c) (d) : 4.26
81 : (a) (b) (c) 4.27 c V 2 d I 2 1 hydrophile-lipophile balance = HLB HLB
82
83 Reinitzer Kristalline Fluessigkeiten (=Liquid Crystal) Fergason RCA Reinitzer (N)
84 第 7 章 液晶 76 (a) (b) (c) 図 7.1: サーモトロピック液晶の温度による相転移の模式図 (a) は等方相で 重心も配向もランダム (b) は液晶相で 重心位置はランダムだが配向は揃っ ている (c) は結晶相で重心も配向も揃っている 7.1 の (b) スメクチック (Sm) 相 方向が揃っているだけでなく 分子が層状に並 んでいる相 図 7.2 層平面内では重心位置に秩序はなく液体的だが 分子の軸方向に沿った秩序がある また分子の配向が層の積層方向に 対して垂直なスメクチック A(Sm A) 相 図 7.2(a) の他に 配向が層 に対して傾いているスメクチック C(Sm C) 相 図 7.2(b) や その分 子の配向方向がピッチ p で周期的に変化しているスメクチック C (Sm C ) 相 図 7.3(b) など様々なバリエーションがある カイラルネマチック (N ) 相 ネマチック相と同様に分子の重心位置に 秩序はないが 分子の配向方向がある距離 d ピッチ で周期的に変化 する このピッチ p は 100nm から無限大を取りうる また d が光の波 長程度の時に可視光を散乱して色が付いて見える p が温度に敏感なこ とから これを用いて thermotropic device 温度計 が作られる 図 7.3(a) ディスコティック相とカラムナー相 ディスコティック相は円板状の分 子が同じ方向を向いている相で 並進秩序がなく長距離配向秩序があ るという点でネマチック相と同様である 従ってネマチック D 相と呼 ぶこともある カラムナー相は円板状の分子が積み重なって柱のよう になっている相で 分子の積層方向に対して垂直な方向への秩序と配向 秩序を持っている 図 7.4 液晶は方向秩序を持っているため マクロな性質にも異方性が現れる 例 えば配向方向に平行方向の屈折率 n と垂直方向の屈折率 n が違う場合には 複屈折 と呼ばれる現象が見られる また電場や磁場を加えることによって 分子に電気双極子モーメントや磁気双極子モーメントが誘起され 電場や磁 場の方向に沿って配向する 更にガラス平面などに垂直に分子が揃う ホメ オトロピック配向 や 平行に分子が揃う プレーナー配向 ホモジニアス 配向 などの力学的配向が起こる 液晶ディプレイはこれらの性質を利用し
85 7.1. 液晶とは 77 (a) (b) 図 7.2: (a) スメクチック A 相 (b) スメクチック C 相 (b) (a) p 図 7.3: (a) カイラルネマチック相 (b) カイラルスメクチック相 (a) (b) 図 7.4: (a) ディスコティック相 (b) カラムナー相
86 78 7 (a) (b) 7.5: (a) (b) u 1 u r 12 n 7.6: n u 1, u 2 ON/OFF 7.2 n u du u du ψ(u)du ψ(u) (orientation distribution function) ψ(u) duψ(u) = 1 (7.2.1) u ψ(u) = 1 4π (7.2.2) u n
87 P = n u = ψ(u) (7.2.3) u -u n u = 0 (n u) 2 u ux 2 = u 2 y = u 2 z u 2 x + u 2 y + u 2 z = 1 u 2 z = 1 3 n z (n u) 2 = 1 3 (7.2.4) n u 2 z > 1 3 (n u) S = 3 2 (n u) (7.2.5) S = 0 n (n u) 2 = 1 S = 1 S 7.3 r 12 u 1, u 2 2 w(r 12, u 1, u 2 ) w(r 12, u 1, u 2 ) = w(r 12, u 1, u 2 ) = w(r 12, u 1, u 2 ) (7.3.1) w w i w a w(r 12, u 1, u 2 ) = w i (r 12 ) + w a (r 12 )(u 1 u 2 ) 2 (7.3.2)
88 80 7 u w mf (u) w mf (u) = dr 12 du 2 w(r 12, u 1, u 2 )g 2 (r 12, u 2 ) (7.3.3) g 2 (r 12, u 2 ) r 12 u 2 u 2 r 12 ψ(u) g 2 (r 12, u 2 ) = g 2 (r 12 )ψ(u 2 ) (7.3.4) w mf (u) = + dr 12 du 2 w i (r 12 )g 2 (r 12 )ψ(u 2 ) dr 12 du 2 (u u 2 ) 2 w a (r 12 )g 2 (r 12 )ψ(u 2 ) (7.3.5) 1 u 2 r 12 w mf (u) = const U du 2 (u u 2 ) 2 ψ(u 2 ) (7.3.6) U = dr 12 w a (r 12 )g 2 (r 12 ) (7.3.7) U U > 0 (7.3.7) ψ(u) = exp( βw ( mf (u)) β = 1 ) du exp( βwmf (u)) k B T (7.3.8) u 3 u α, (α = x, y, z) (7.3.7) w mf (u) = U (u u ) 2 = Uu α u β u α u β (7.3.9)
89 ψ(u ) u 2 u n z ψ(u ) z u 2 x = u 2 y = 1 2 u (1 2 z ) (7.2) S = 3 2 u 2 z 1 3 (7.3.10) u 2 z u 2 x u x u y = 1 (2S + 1) (7.3.11) 3 = u 2 y = 1 ( S + 1) (7.3.12) 3 = u xu z = u zu x = 0 (7.3.13) (7.3.12) (7.3.13) (7.3) w mf (u) = U [ 1 3 ( S + 1)(u2 x + u 2 y) + 1 ] 3 (2S + 1)u2 z = USu 2 z + const (7.3.14) (7.3) (7.3) ψ(u) ψ(u) = exp(βusu2 z) du exp(βusu 2 z ) (7.3.15) (7.3) S = 3 2 du exp(βusu 2 z )(u 2 z 1 3 du exp(βusu 2 z ) (7.3.16) t = u z, x = βus (7.3) 2k B T x = I(x) (7.3.17) 3U 1 0 I(x) = dtext (t ) 1 (7.3.18) 0 dtext2 y = I(x) y = (2k B T/3U)x x T S T > T c1 x = 0 S = 0 T = T c1
90 82 7 y T = T c1 T = T c2 y = I(x) x T > T c1 7.7: (7.3.17) (7.3.18) x = T = T c2 x 1 I(x) I(x) = 4 ( ) 45 x + (7.3.19) T = T c2 y = (2k B T/3U)x y = I(x) 2k B T c2 /3U = 4/45 T c2 = 2 U (7.3.20) 15 k B 7.8 (a) S F S = 0 (7.3.21) T > T c1 (7.3) S = 0 F S = 0 T < T c1 (7.3) 3 F 2 1 T = T c1 2 T = T c2 (7.3) 1 S = 0 T = T c2 T e S = 0 S 0
91 : (a) S (b) S T c1 T c2 5 F T c2 < T e < T c1 7.4 S n n 7.9: (a) (splay) n 0 (b) (twist) n 0 (c) (bend) n ( n) 0 5
92 第 7 章 液晶 84 図 7.10: フレデリクス転移 (a) 液晶に磁場 H を加えて液晶層を通過する光 の強さを制御する (b) 配向ベクトルの空間変化 H < Hc では配向ベクトル は x 軸を向いたままだが H > Hc では磁場方向を向くようになる 土井正 男 ソフトマター物理学入門 第 5 章より ギーは線形のバネを仮定すると空間変化の 2 乗に比例する 例えばフックの 法則では f = kx, E = 12 kx2 液晶の弾性エネルギーは次の形に書く事ができる fel = K1 ( n)2 + K2 (n ( n))2 + K3 (n ( n)) (7.4.1) ここで K1, K2, K3 は [J/m] の単位を持つ定数で フランク弾性定数 (Frank elastic constant) と呼ばれる これらはそれぞれ splay, twist bend の 変形が起きた場合のエネルギー変化に対応する 外場がある時はここに外場との相互作用項が加わる 例えば磁場 H に対し ては 1 fh = x(h n)2 + (n independent) 2 (7.4.2) ここで x = αd nseq で αd は分子長軸方向の帯磁率と短軸方向の帯磁率の 差 また Seq は S の平衡値である 7.5 フレデリクス転移 2 枚の平行基板に液晶が挟まれていて 基板上では液晶は x 軸方向を向いて いるものとする この時 z 軸方向に磁場 H をかけた場合を考える nは x z 面内にあり その向きは z 座標のみに依存する と仮定する するとnの各 成分は
93 n z (z) = cos θ(z), z y (z) = 0, n z (z) = sin θ(z) (7.5.1) n = n x x + n y y + n z z = n z dθ = cos θ z dz ( nz n = y n y z, n x z n z x, n y x n ) x y ( = 0, n ) x z, 0 ( = 0, sin θ dθ ) dz, 0 (7.5.2) (7.5.3) (7.4) (7.4) F tot = = dz(f el + f H ) (7.5.4) [ ( ) 2 1 dθ dz 2 K 1 cos 2 θ + 1 ( ) ] 2 dθ dz 2 K 3 sin 2 θ 1 dz 2 xh sin2 θ θ(0) = θ(l) = 0 (7.5.5) F tot θ(z) = θ 0 sin( πz L ) (7.5.6) θ 1 (7.5) (7.5) θ 0 F tot = 1 4 [ K1 π 2 L ] xh2 L θ0 2 = 1 4 xl(h2 c H 2 )θ0 2 (7.5.7) H c = K1 π 2 xl 2 (7.5.8)
94 86 7 H < H c θ 0 = 0 H > H c (Fredericks transition) ON/OFF
95 m 8.1 1nm 1µm van der Waals k B T 1 k B T
96 : (a) (b) r-r' r h 8.2: (a) van der Waals (b) van der Waals h 8.2 van der Waals 8.2(a) r 2 van der Waals
97 U atom (r) = c ( a0 ) 6 (8.2.1) r a 0 c k B T 1 van der Waals 2 h van der Waals U particle (h) = dr 1 r 1 V 1 = dr 1 r 1 V 1 dr 2 n 2 ca 2 0 r 2 V 2 r 1 r 2 6 A H dr 2 r 2 V 2 π r 1 r 2 6 (8.2.2) n A H = π 2 n 2 ca 6 0 n 1/a 3 0 A H c S S h 2 S U particle (h) = Sw(h) (8.2.3) w(h) (8.2) r 1 V 1 dr 1 A H dr 2 r 2 V 2 π r 1 r 2 6 = A H 12πh 2 (8.2.4) R h S Rh R h U particle Sw(h) A H R h (8.2.5) van der Waals R/h R = 0.1µm h 1nm R/h = 100 van der Waals
98 90 8 h 8.3: (1) (2) van der Waals z ψ(z) ψ(0) = ψ s > 0 ψ(z) = ψ s exp( κz) (8.2.6) 1/κ κ = ( i n iqi 2 ) 1/2 (8.2.7) ϵk B T n i i q i i ϵ NaCl 1mM 1/κ 10 nm (M= mol/l) 1M 1/κ 0.3 nm
99 w tot (h) (i) (ii) (iii) h 8.4: (i)(ii)(iii) w charge (h) DLVO van der Waals w tot (h) = w(h) + w charge (h) (8.2.8) 8.4 w tot (h) h van der Waals w charge (h) (i) (ii)(iii) (a) h p 2
100 92 8 (a) (b) (c) Π h h h 8.5: (a) (b) (c) (b) (a) (c) R g h < 2R g Π = n p k B T n p Π 0 (h > 2R g ) w dep (h) = n p k B T (h 2R g ) (h < 2R g ) R U dep (h) = πr h 0 (h > 2R g ) dxw(x) = 1 2 πrn pk B T (h 2R g ) 2 (h < 2R g )
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........................
80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t = 0 i r 0 t(> 0) j r 0 + r < δ(r 0 x i (0))δ(r 0 + r x j (t)) > (4.2) r r 0 G i j (r, t) dr 0
79 4 4.1 4.1.1 x i (t) x j (t) O O r 0 + r r r 0 x i (0) r 0 x i (0) 4.1 L. van. Hove 1954 space-time correlation function V N 4.1 ρ 0 = N/V i t 80 4 r ˆρ i (r, t) δ(r x i (t)) (4.1) x i (t) ρ i ˆρ i t
1 (Berry,1975) 2-6 p (S πr 2 )p πr 2 p 2πRγ p p = 2γ R (2.5).1-1 : : : : ( ).2 α, β α, β () X S = X X α X β (.1) 1 2
2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re
![II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re](/thumbs/94/118770263.jpg "II ( ) (7/31) II ( [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Re")
II 29 7 29-7-27 ( ) (7/31) II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ (6/29)] Navier Stokes 3 [ (6/19)] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I Euler Navier
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II 14 14-7-8 8/4 II (http://www.damp.tottori-u.ac.jp/~ooshida/edu/fluid/) [ (3.4)] Navier Stokes [ 6/ ] Navier Stokes 3 [ ] Reynolds [ (4.6), (45.8)] [ p.186] Navier Stokes I 1 balance law t (ρv i )+ j
3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C
![3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C](/thumbs/94/119172213.jpg "3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C")
3 3.1 R r r + R R r Rr [ ] ˆn(r) = ˆn(r + R) (3.1) R R = r ˆn(r) = ˆn(0) r 0 R = r C nn (r, r ) = C nn (r + R, r + R) = C nn (r r, 0) (3.2) ( 2.2 ) C nn (r r ) = C nn (R(r r )) [2 ] 2 g(r, r ) ˆn(r) ˆn(r
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(1.2) T D = 0 T = D = 30 kn 1.2 (1.4) 2F W = 0 F = W/2 = 300 kn/2 = 150 kn 1.3 (1.9) R = W 1 + W 2 = = 1100 N. (1.9) W 2 b W 1 a = 0
1 1 1.1 1.) T D = T = D = kn 1. 1.4) F W = F = W/ = kn/ = 15 kn 1. 1.9) R = W 1 + W = 6 + 5 = 11 N. 1.9) W b W 1 a = a = W /W 1 )b = 5/6) = 5 cm 1.4 AB AC P 1, P x, y x, y y x 1.4.) P sin 6 + P 1 sin 45
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19 σ = P/A o σ B Maximum tensile strength σ % 0.2% proof stress σ EL Elastic limit Work hardening coefficient failure necking σ PL Proportional
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