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Transcription

1 21 7 2

2

3 i Flory-Huggins van t Hoff ( )

4 ii

5 iii ( ) A Flory-Huggins Flory-Huggins A B C ( )

6 iv 3.A Gibbs B C A B C D

7 1 1 2 Einstein 1 A theory is the more impressive the greater the simplicity of its premises, the more varied the kinds of things that it relates and the more extended the area of its applicability. Therefore classical thermodynamics has made a deep impression on me. It is the only physical theory of universal content which I am convinced, within the areas of the applicability of its basic concepts, will never be overthrown. - A. Einstein (1949) - 2, 3, 4, 5, ( ) 2 1 (system) (surroundings)

8 Albert Einstein (1879/3/ /4/18) 3 (closed system) (open system) (isolated system) 1 2 component) (state propery) 2 1 (intensive property) 1 ( ) (extensive property) 2

9 1.1 3 ( ) (single phase system) 2 (multiphase system) (homogeneous system) (heterogeneous system) (phase) 2 1 (equilibrium state) (state variable) Gibbs (Gibbs phase rule) Φ = C + 2 P (1.1) Φ C P Gibbs Φ Φ + 1 (external constraints)

10 James Prescott Joule (1818/12/ /10/11) 1.2 (internal energy) U (heat) Q W du = dq + dw (1.2) (thermal energy) ( ) ( ) E = mc 2 (E: m: c: ) c 2 (1.2) U du d (exact differential total diferential)

11 Julius Robert von Mayer (1814/11/ /3/20) Q W dq dw d (inexact differential) (1.2) W p V dv dw = p dv (1.3) p (1.3) (1.2) du = dq p dv (1.4) p p p (isobaric process) (1.4) p = p du = dq pdv (1.5)

12 Rudolf Julius Emanuel Clausius (1822/1/2 1888/8/24) Clausius Kelvin(Thomson) ( )( ) 2 1 (spontaneous process) (irreversible process) 1 (reversible process) 2 S

13 William Thomson (Lord Kelvin) (1824/6/ /12/17) ds > dq T (1.6) ds = dq T (1.7) T dq T T = T (1.6) Clausius (1.7) Clausius 2 clausius Kelvin S (Carnot ) Q/T Clausius (1.7) S ( T ) (1.6) (1.4) du < T ds p dv (1.8)

14 8 1 (1.7) (1.5) du = T ds pdv (1.9) du = 0 dv = 0 (1.8) ds > 0 (1.10) 1.4 δ δ Clausius (1.6) δs δq T (1.11) δu = δq p δv (1.12) T δs δu + p δv (1.13) δu = 0 δv = 0 (1.13) δs 0 (1.14)

15 Hermann Ludwig Ferdinand von Helmholtz (1821/8/ /9/8) T = T δv = 0 (1.13) T δs δu (1.15) A U T S (1.16) Helmholtz (Helmholtz free energy) δa 0 (1.17) T V Helmholtz Helmholtz A T = T p = p (1.13) δ(u + pv T S) 0 (1.18)

16 Josiah Willard Gibbs (1839/2/ /4/28) (enthalpy)h Gibbs (Gibbs free energy)g H U + pv (1.19) G U + pv T S = H T S (1.20) Gibbs G (1.18) δg 0 (1.21) T p Gibbs H Gibbs G [ ] T p 1 T = p = (1.21) n l Gibbs G l Gibbs Ḡl n g Gibbs G g Gibbs

17 1.5 ( ) 11 Ḡg G l = n l Ḡ l (T, p), G g = n g Ḡ g (T, p) (1.22) δn l G l δn l Ḡ l (T, p) G g δn l Ḡ g (T, p) Gibbs G δg = [Ḡg(T, p) Ḡl(T, p)]δn l (1.23) (1.21) [Ḡg(T, p) Ḡl(T, p)]δn l 0 (1.24) δn l Ḡ g (T, p) = Ḡl(T, p) (1.25) 1 Gibbs 1.5 ( ) T V T p Helmholtz A Gibbs G A G (T, V ) A (T, p) G T V A(T, V ) T p G(T, p) T V T p (T, V ) A (T, p) G (characteristic function)

18 12 1 (generating function) (S, V ) U(S, V ) (S, p) H(S, p) 1. (T, p) G(T, p) (T, V ) A(T, V ) (S, p) H(S, p) (S, V ) U(S, V ) Gibbs G (1.20) dg = du + pdv + V dp T ds SdT (1.26) d 2 (1.9) dg = V dp SdT (1.27) ( ) G V = p T ( ) G S = T p (1.28) (1.29) (1.29) G (1.20) H H = G T ( ) G = T p [ ] (G/T ) (1/T ) p (1.30) Gibbs - Helmholtz

19 1.5 ( ) James Clerk Maxwell (1831/6/ /11/5) H H U + pv (1.9) dh = T ds + V dp (1.31) ( ) H p p T = V + T ( ) S p T (1.28) (1.29) ( ) ( ) V = 2 G S T p T = p Maxwell (1.32) (1.33) ( ) ( ) H V = V T p T T p T (1.32) (1.33) (1.34) ( ) H = V (1 αt ) (1.35) p T α α 1 ( ) V V T p (1.36)

20 14 1 (1.9) ( ) U V T = p + T ( ) p T V (1.37) ( U/ V ) T p i (internal pressure) ( p/ T ) V β (thermal pressure coefficient) p i = p + T β (1.38) (1.34) (1.37) (thermal equation of state) r r + 1 r (principal solvent) i (i = 0, 1,, r) n i n n = r n i (1.39) i= T p (T, p) Y Y T p n 0 n 1 n r

21 n i (i = 0, 1,, n r λ Y λ Y Y (T, p, λn 0, λn 1,, λn r ) = λy (T, p, n 0, n 1,, n r ) (1.40) (partial molar quantity)y i ( ) Y Y i n i T,p,n j(j i) (1.41) Y = r n i Y i (1.42) i=0 Y i Y dy = (1.42 ( ) ( ) Y Y dt + dp + T p,n i p T,n i dy = 2 r Y i dn i (1.43) i=0 r (n i dy i + Y i dn i ) (1.44) i=0 ( ) ( ) Y Y dt + dp T p,n i p T,n i T p n r n i dy i = 0 (1.45) i=0 r n i (dy i ) T,p = 0 (1.46) i=0 r x i (dy i ) T,p = 0 (1.47) i=0 x i ( n i /n) ( )

22 16 1 Y i (Y i ) T,p x i x 2 x r ( r i=0 x i = 1 x i r ) (dy i ) T,p = r ( ) Yi dx j (1.48) x j T,p,x k (k j) j=i (1.47) r [ r ( ) ] Yi x i dx j = 0 (1.49) x j T,p,x k (k j) j=1 i=0 dx j r i=0 ( ) Yi x i = 0 (j = 1, 2,, r) (1.50) x j T,p,x k (k j) i Y i (S, V ) U du = T ds pdv + r µ i dn i (1.51) µ i i (chemical potential) ( ) U µ i n i i=0 S,V,n j (j i) (1.52) du dn i (i = 0, 1, 2,, r) Gibbs (1.51) µ i ( ) U T = (1.53) S V,n i

23 (1.51) ( ) U p = (1.54) V S,n i ds = 1 T du + p T dv r = ( ) S du + U V,n i i=0 U µ i ( S V T dn i ) U,n i dv + r ( ) S dn i n i U,V,n j (j i) i=0 (1.55) ( ) S µ i = T (1.56) n i U,V,n j (j i) ( ) 1 S T = (1.57) U V,n i ( ) S p = T (1.58) V U,n i U = T S pv + du = T ds + SdT V dp pdv + (1.51) SdT V dp + r µ i n i (1.59) i=0 r (µ i dn i + n i dµ i ) (1.60) i=0 r n i dµ i = 0 (1.61) i=0

24 Pierre Maurice Marie Duhem (1861/6/ /9/14) T p r [ r ( ) ] µi n i dn i = 0 (1.62) n j T,p i=0 j=0 dn i r i=0 ( ) µi n i = 0 (1.63) n j T,p (1.45) (1.50) (1.61) (1.63) Gibbs - Duhem (S, p) H dh = T ds + V dp + ( ) H µ i = n i V = ( H p r µ i dn i (1.64) i=0 S,p,n j(j i) ) (1.65) S,n i (1.66)

25 T = ( ) H (1.67) S p,n i (T, V ) A da = SdT pdv + r µ i dn i (1.68) i=0 ( ) A µ i = (1.69) n i T,V,n j (j i) ( ) A S = (1.70) T V,n i ( ) A p = (1.71) V T,n i (T, p) G dg = SdT + V dp + r µ i dn i (1.72) i=0 ( ) G µ i = (1.73) n i T,p,n j (j i) ( ) G S = (1.74) T p,n i ( ) G V = (1.75) p T,n i Gibbs G G G = U + pv T S U (1.59) G = r µ i n i (1.76) i=0

26 20 1 G = µn Gibbs G m ( G/n) µ r G m = µ i x i (1.77) i=0 µ i T p x i (i = 1, 2,, r) µ i = µ i (T, p, x 1, x 2,, x r ) (mole fraction)x i n i n x i n i (1.78) n r i=0 x i = 1 r 2. (weight fraction)w i i q i q w i q i q = q i r j=0 q j (1.79) r i=0 w i = 1 r x i i (molar mass) M i q i = n i M i w i = x i M i w i /M r j=0 x, x i = i r jm j j=0 (w j/m j ) (1.80) 3. (volume fraction)ϕ i i (molar volume) Vi (specific volume)v i (v i V i /M i) x i w i ϕ i x i Vi w i vi r j=0 x jvj = r j=0 w jvj, r ϕ i = 1 (1.81) i=0

27 r V i v I (partial molar volume)v i (partial specific volume)v i (v i V i /M i ) V i v i Vi = V i 4. (molarity)m i ( ) m i m 0 M 1 0 n i = x i (1.82) n 0 M 0 x 0 M 0 x i = m i M r j=1 m j = m i r j=0 m j (1.83) 5. (volume molarity)c i i ( ) C i n i V = x i V m = m i v M (1.84) V m ( V/n) v M v M V n 0 M 0 (1.85) V V i V = r n i V i, i=0 ( ) V V i n i n 0 M 0 v M = v 0 + T,p,n j(j i) (1.86) r m i V i (1.87) i=1

28 22 1 C i r C i V i = 1 (1.88) i=0 6. (mass concentration)c i i ρ c i n im i V ρ = r ρ i = i=0 = x im i V m = m im i v M = C i M i (1.89) r r i=0 c i = n im i V i=0 (1.90) r i=0 c i V i M i = r c i v i = 1 (1.91) i=0 Gibbs-Duhem (1.50) r ( ) w i Yi = 0 (j = 1, 2,, r) (1.92) M i w j i=0 r i=0 r i=0 ( ϕ i Yi Vi ϕ j m i ( Yi m j ) T,p,w k (k j) ) T,p,ϕ k (k j) T,p,m k (k j) = 0 (j = 1, 2,, r) (1.93) = 0 (j = 1, 2,, r) (1.94)

29 r ( ) Yi C i = 0 (j = 1, 2,, r) (1.95) C j T,p,C k (k j) r ( ) c i Yi = 0 (j = 1, 2,, r) (1.96) M i c j i=0 i=0 T,p,c k (k j) Gibbs-Duhem 1.7 r (α β ) T α = T β = T (1.97) p α = p β = p (1.98) α β i δn i (i = 0, 1,, r) δg 0 α G G α β G G β δg α = δg β = r µ α i δn i (1.99) i=0 r µ β i δn i (1.100) i=0 µ α i µβ i α β i δg = δg α +δgβ δg = r i=0 (µα i µβ i )δn i r (µ α i µ β i )δn i 0 (1.101) i=0 δn i µ α i = µ β i (i = 0, 1,, r) (1.102)

30 24 1 T p {x} ({x} x 1, x 2,, x r ) (1.97) (1.98) (1.102) µ α i (T, p, {x α }) = µ β i (T, p, {xβ }) (1.103) P µ α 0 (T, p, {x α }) = µ β 0 (T, p, {xβ }) = = µ P 0 (T, p, {x P }) µ α 1 (T, p, {x α }) = µ β 1 (T, p, {xβ }) = = µ P 1 (T, p, {x P }) µ α 2 (T, p, {x α }) = µ β 2 (T, p, {xβ }) = = µ P 2 (T, p, {x P }) µ α r (T, p, {x α }) = µ β r (T, p, {x β }) = = µ P r (T, p, {x P }) (1.104) ( ) ( ) ( ) Gibbs-Duhem

31 1.A Nicolas Léonard Sadi Carnot (1796/6/1 1832/8/24) 1.A 7 Joule Mayer Helmholtz abcd cd A B A B 1. A abcd A cd

32 26 1 g e h f A c i a d k b B 1.11 ( ) 2. ef A A 3. A ef gh B gh 4. B gh cd B B 5. B

33 1.A James Watt (1736/1/ /8/25) A cd ik 6. A ik ef 7. (3) (4) (5) (6) (3) (4) (5) (6) (3) (4) (5)

34 28 1 ef A ik A A A ik cd B A B B A A B B A A B B A B A

35 1.A Benoit Paul Emile Clapeyron (1799/2/ /1/28) 1.14 Count von Rumford (Benjamin Thompson)(1753/3/ /8/21) V p Clapeyron Clapeyron Rumford Joule

36 ( ) 2 ( ) Kelvin(William Thomson) Joule ( ) 1 ( 1 ) 1 1 Clausius ( 1 ) ( ) Clausius Clausius 2 8 8

37 1.A Walther Hermann Nernst (1864/6/ /11/18) Kelvin(William Thomson) Clausius (0) 0

38 32 3 Nernst ( 2 ) ( ) 1. A. Einstein, Autobiographical Notes in Albert Einstein: Philosopher - Scientist, P. A. Schilpp, Ed., Cambridge University Press, London, M. Planck, Treatise on Thermodynamics, Dover, New York, Kenneth Denbigh ( ), III

39 , 2, 3 T p 1 Gibbs G 2 1 x 1 Gibbs G m G m x T 1 (1.77) x 1 = 0 0 µ 0 x 1 = 1 1 µ 1 2 G m 2 P P P P x 1 x 1 x 1 = 0 x 1 = 1 µ 0 (x 1) = µ 0 (x 1) (2.1) µ 1 (x 1) = µ 1 (x 1) (2.2) 2 (1.102) P P 2 G m x 1 x 1 = 0 µ 0 x 1 = 1 µ 1 x 1 2 (1.77) G m (x 1 ) = (1 x 1 )µ 0 (x 1 ) + x 1 µ 1 (x 1 ) (2.3)

40 34 2 T c T 2 G m µ 1 o µ 0 o T 1 µ 0 (x 1 ) µ 0 (x 1 ") P N N" P" µ 1 (x 1 ) µ 1 (x 1 ") 0 x 1 x 1 " x Gibbs G m ( ) x 1 Y i µ i (1.50) ( ) Gm x 1 T,p = µ 1 (x 1 ) µ 0 (x 1 ) (2.4) µ 0 (x 1 ) = G m (x 1 ) x 1 ( Gm x 1 ) T,p ( ) Gm µ 1 (x 1 ) = G m (x 1 ) + (1 x 1 ) x 1 T,p (2.5) (2.6) ( G m / x 1 ) T,p x 1 G m (2.5) x 1 = 0 µ 0 (x 1 ) (2.6) x 1 = 1 µ 1 (x 1 ) 2.1 T 1 G m x 1 N N (spinodal point)

41 35 Q G m G m * G m 0 Q Q" (a) x 1 G m 0 Q G m G m * Q Q" (b) x 1 G m G m 0 Q (c) Q" G m * Q x G m ( 2 ) ( ) G m µ1 = = 0 (2.7) x 1 x 2 1 T,p T 1 x 1 = 0 P P x 1 = 1 P N N P N N 2.2 Q Q Q 2 Q Q Q n n n T,p

42 x 0 1 x 1 x 1 1 n = n + n (2.8) nx 0 1 = n x 1 + n x 1 (2.9) n n = x 1 x 0 1 x 1 x 1 (2.10) n n = x0 1 x 1 x 1 x 1 (2.11) Q Gibbs G m Q Gibbs G m 2 Gibbs G m (2.10) (2.11) ng m = n G m + n G m (2.12) G m = G m + x0 1 x 1 x 1 x 1 (G m G m) (2.13) 2.2(a) G m x 1 G m (2.13) Q Q Q G m Q Gibbs G0 m 1 (1.21) Q 1 2.2(b) G m x 1 G m Q Q Q (2.13) G m Q G0 m (1.21)

43 Q T 1 P N P N (metastable) 2.2 Q Q 2.2(a) Q 1 2.2(c) Q Q Q Q 2 G m Q 1 G0 m 2.1 T 1 G m x 1 (critical solution point) (2.7) ( 3 ) G m x 3 1 T,p ( 2 ) µ 1 = x 2 1 T,p 37 = 0 (2.14) 2.1 P P N N T 1 x (phase diagram) T P P ( 2.3 ) (coexistence curve) (binodal) T 1 P 1 P P 2 P P n /n (2.10) (2.11) PP P P (lever rule) 2 P P (tie line) 1 (cloud point) (precipitation point) (cloud point curve) 2 (

44 38 2 T c C T T 1 P P P" N N" 0 x 1 x 1 0 x 1 x 1 " ) T N N ( 2.3 ) (spinodal) T c (upper critical solution point) 2 (lower critical solution point) (2.7) (2.14)

45 Gibbs G m Gibbs G G m ( G) 2 1 G m ( G) 1 G m G G m (2.7) (2.14) G m Flory-Huggins Flory-Huggins 4, 5 Huggins Flory (Flory Huggins Huggins ) V 0 V 1 P 1 P 1 V 1 V0 (2.15) V 0

46 V 0 N N 0 N 1 V N = N 0 + P 1 N 1 (2.16) V = (N 0 + P 1 N 1 )V 0 (2.17) V 0 V 1 T ϕ 1 ϕ 0 ϕ 1 = ϕ 0 = 1 ϕ 1 = P 1 N 1 N 0 + P 1 N 1 (2.18) N 0 N 0 + P 1 N 1 (2.19) Ω 1 2 N N 1

47 P N z 2 z 3 z 1 z 1 1 P 1 ν 1 ν 1 = Nz(z 1) P1 2 (2.20) i 1 N P 1 (i 1) i ν i [ ] P1 1 N P1 (i 1) ν i = [N P 1 (i 1)] z(z 1) P 1 2 N (2.21) [N P 1 (i 1)]/N i 2 N 1 N 0 N 0 N 0! N 0!N 1! 2 N1 1 N 1 Ω = ν i (2.22) N 1!σ N1 σ 1 2 (2.22) (2.21) Ω = i=1 ( ) (P1 1)N N! P1 N 1 1 ω N 1 (2.23) N 0!(P 1 N 1 )! N ω P 1z(z 1) P 1 2 σe P 1 1 (2.24)

48 42 2 (2.23) Stirling ( ) x x x! = (2.25) e S Boltzmann S = k B ln Ω (2.26) S S = k B (N 0 ln ϕ 0 + N 1 ln ϕ 1 ) + k B N 1 ln ω (2.27) S S = S(N 1 = 0) + S(N 0 = 0) = k B N 1 ln ω (2.28) (mixing entropy) S S = S S = k B (N 0 ln ϕ 0 + N 1 ln ϕ 1 ) = R(n 0 ln ϕ 0 + n 1 ln ϕ 1 ) (2.29) n 0 n 1 ( ) 2 2 u (a a)+(b b) (a b)+(b a) u 00 u 11 u 01 u u (u 00 + u 11 ) (2.30)

49 a b zx (mixing energy) U U = zx u (2.31) zn/2 ϕ 0 ϕ 1 zx 2 zx = znϕ 0 ϕ 1 (2.32) U = Nz uϕ 0 ϕ 1 (2.33) U (mixing enthalpy) H (2.33)( U = H ) (2.29) 1 (1.20) Gibbs (mixing Gibbs free energy) G G = RT [n 0 ln ϕ 0 + n 1 ln ϕ 1 + (n 0 + P 1 n 1 )χ H ϕ 0 ϕ 1 ] (2.34) χ H z u k B T (2.35) χ H chi χ χ S χ χ χ H + χ S (2.36) χ T χ χ H χ S ( ) χ χ H = T (2.37) T p,ϕ 1

50 44 2 χ S = χ + T ( ) χ (2.38) T p,ϕ 1 2 Gibbs G G = n 0 µ 0 + n 1 µ 1 + RT [n 0 ln ϕ 0 + n 1 ln ϕ 1 + (n 0 + P 1 n 1 )χϕ 0 ϕ 1 ] (2.39) µ i i Gibbs G m G m = ϕ 0 µ 0 + ϕ ( 1 µ 1 + RT ϕ 0 ln ϕ 0 + ϕ ) 1 ln ϕ 1 + χϕ 0 ϕ 1 (2.40) P 1 P 1 (2.39) (1.73) µ 0 µ 1 [ ) ] µ 0 = µ 0 + RT ln(1 ϕ 1 ) + (1 1P1 ϕ 1 + χϕ 2 1 (2.41) µ 1 = µ 1 + RT [ln ϕ 1 (P 1 1)(1 ϕ 1 ) + P 1 χ(1 ϕ 1 ) 2 ] (2.42) (1.102) µ 0 (ϕ 1) = µ 0 (ϕ 1), µ 1 (ϕ 1) = µ 1 (ϕ 1) (2.43) (2.41) (2.42) (2.7) ( ϕ P ) 1 1 ϕ P 1 χ 2P 1 χ = 0 (2.44) T ϕ 1 χ T (2.44) (1.81) x 1 ϕ 1 ϕ 1c χ c (2.7) (2.14) 1 1 ϕ 1c + (1 1P1 ) + 2χ c ϕ 1c = 0 (2.45)

51 (1 ϕ 1c ) 2 + 2χ c = 0 (2.46) (1.81) 1 ϕ 1c = (2.47) 1 + P 1/2 1 χ c = 1 ( ) 2 (2.48) 2 P 1/2 1 P 1 ϕ 1c = 0 χ c = 1/2 (theta temperature) Θ χ T χ = 1 2 ψ + Θ T ψ (2.49) T = Θ χ = 1/2 Flory-Huggins

52 , 7 Θ = 307.2K ψ = M v ( ) ( ) ϕ Flory-Huggins Flory-Huggins S H χ Flory-Huggins S Gibbs G S G χ χ T ( )

53 χ Flory-Huggins 8, 9 χ ( ) χ χ ( ϕ 1 ) Gibbs G 0 1 µ 0 µ 1 µ 0 (2.41) χ µ 0 χ [ ) ] µ 0 = µ 0 + RT ln(1 ϕ 1 ) + (1 1P1 ϕ 1 + χϕ 2 1 (2.50) χ T p ϕ 1 P 1 1 (2.42) µ 1 = µ 1 + RT [ln ϕ 1 (P 1 1)(1 ϕ 1 ) + P 1 χ P (1 ϕ 1 ) 2 ] (2.51) Y i µ i Gibbs-Duhem (1.93) χ χ P χ P (1 ϕ 1 ) 2 = χϕ 1 (1 ϕ 1 ) + 1 χdϕ 1 ϕ 1 (2.52) (1.76) 2 Gibbs G [ G = n 0 µ 0 + n 1 µ 1 + nrt ϕ 0 ln ϕ 0 + ϕ ] 1 ln ϕ 1 + gϕ 0 ϕ 1 (2.53) P 1 n n 0 + P 1 n 1 (2.54) g 1 ϕ 0 1 ϕ 1 χdϕ 1 (2.55)

54 48 2 χ g ( ) g χ = g ϕ 0 ϕ 1 T,p (2.56) χ g 1 ϕ 1 = 0 χ = χ 0 + χ 1 ϕ 1 + χ 2 ϕ (2.57) g = g 0 + g 1 ϕ 1 + g 2 ϕ (2.58) g i = j=i χ j j + 1 (2.59) χ i = (i + 1)(g i g i+1 ) (2.60) χ g T ϕ g χ Koningsveld 10 (2.58) g ϕ 2 1 χ g = g 00 + g 01 T + g 1ϕ 1 + g 2 ϕ 2 1 (2.61) χ = g 00 + g 01 T g 1 + 2(g 1 g 2 )ϕ 1 + 3g 2 ϕ 2 1 (2.62) g 00 g 01 g 1 g 2 T P 1 g χ ϕ 2 1

55 (2.7) (2.14) Y c X c = 2χ c + 1 ϕ 1c P 1 ϕ 1c 1 (1 ϕ 1c ) 2 1 ( ) χ P 1 ϕ 2 = 3 + 1c ϕ 1 c ( χ ϕ 1 ) ( 2 χ ϕ 2 1 ϕ 1c (2.63) c ) (2.64) c (2.62) ( X c = 2 g 00 + g ) 01 2g 1 (1 3ϕ 1c ) 6g 2 ϕ 1c (1 2ϕ 1c ) (2.65) T c Y c = 6(g 1 g 2 ) + 24g 2 ϕ 1c (2.66) X c Y c T c ϕ 1c P 1 Y c χ 1c ϕ 1c = 0 (2.66) g 1 g 2 (2.65) X c T c g 00 g 01 Konigsveld 10 + g g = T χ χ = T ϕ ϕ 2 1 (2.67) ϕ ϕ 2 1 (2.68) Koningsveld kleintjens 11 g = α + β 0 + β 1 /T 1 γϕ 1 (2.69) + α = β 0 = β 1 = K γ = (2.68) (2.69 Flory-Huggins 2

56 50 2 g χ ϕ 1 g χ ϕ 1 T ϕ 1 g χ B 0 µ 0 ( )( ) K ϕ 1 µ0 = R 0 RT ϕ ϕ K K ϕ 4π2 ñ 2 V 0 N A λ 4 0 ( ) ñ ϕ T,p T,p (2.70) (2.71) R 0 θ = 0 Rayleigh µ 0 0 ( µ 0 µ 0 µ 0) ñ λ 0 1 ϕ 1 ϕ ϕ = 1 ( ) (2.72) 1 + v 0 1 vp w 1 w v 0 v p ϕ (2.50) (2.70) Z χ + 1 χ 2 ϕ ϕ

57 = 1 2 [ 1 1 ϕ + 1 P ϕ K ] ϕ R 0 (2.73) Z P M P v p V 0 M (2.74) ϕ (1/RT )( µ 0 / ϕ) T,p + ( M w = 43600)

58 Z ϕ = 0 P (1/RT )( µ/ ϕ) T,p ϕ (1/RT )( µ 0 / ϕ) T,p = 0 ( ) 2 ϕ sp ϕ (1/RT )( µ 0 / ϕ) T,p (1/RT )( µ 0 / ϕ) T,p = 0 ϕ T sp 2.6 (2.73) Z 2.7 T C Z 1. (ϕ > 0.1) Z 1/2

59 Z Z 2 χ 2 χ(t, ϕ; P ) = χ conc (T, ϕ; P ) + [χ dil (T, ϕ; P ) χ conc (T, ϕ; P )]Q(T, ϕ; P ) (2.75) [] χ dil χ conc χ dil (T ; P ) χ conc(t ; P ) Q (overlap concentration) ϕ ϕ/ϕ ϕ ϕ = D P 1/2 (2.76) D Q ϕ 0 1 ϕ (2.75) Z (2.73) Z(T, ϕ; P ) = Z conc (T, ϕ; P ) + [χ dil(t ; P ) χ conc(t ; P )]R(ϕ/ϕ ) (2.77) Z conc = χ conc ( χconc ϕ ) ϕ (2.78) T,p R(ϕ/ϕ ) = Q(ϕ/ϕ ) + 1 ϕ dq(ϕ/ϕ ) 2 ϕ d(ϕ/ϕ ) (2.79) (2.78) lim Z conc = χ conc(t ; P ) (2.80) ϕ 0 ϕ 0 R(ϕ/ϕ ) 1 (2.77) lim Z = ϕ 0 χ dil(t ; P ) (2.81) χ dil Z ϕ 0

60 Y Z ( P 4 ) 1 Z conc Z conc (T, ϕ; P ) = χ conc(t ; P ) + 1 ϕ + f(t, ϕ; P ) (2.82) 2 Y Y = Z 1 2 ϕ (2.83) (2.82) (2.88) Y (T, ϕ; P ) = Y dil (T, ϕ; P ) + Y conc (T, ϕ; P ) (2.84) Y dil Y conc Y dil (T, ϕ; P ) = [χ dil(t ; P ) χ conc(t ; P )]R(ϕ/ϕ ) (2.85) Y conc (T, ϕ; P ) = χ conc(t ; P ) + f(t, ϕ; P ) (2.86) 2.7 (2.83) Y 2.8 T C Y T (ϕ = 0.1)

61 χ conc Y ϕ 2 Y dil Y conc Y dil ϕ 0 Y Y conc Y conc Y conc (T, ϕ; P 4 ) = χ conc(t ; P 4 ) + 10ϕ b(t ; P 4 )ϕ 2 (2.87) χ conc(t ; P 4 ) b(t ; P 4 ) χ conc ( :M w = :M w = ) 2.9 χ conc(t ; P 4 ) b(t ; P 4 ) ( ) [ ( )] Θ Θ χ conc(t ; P 4 ) = T exp 30 T 1 (2.88)

62 χ dil [ ( )] Θ b(t ; P 4 ) = 50.5 exp 18 T 1 (2.89) Y conc (T, ϕ; P 4 ) Y (2.84) Y dil (T, ϕ; P 4 ) [χ dil (T ; P 4) χ conc(t ; P 4 )] R(ϕ/ϕ ) P 4 R = exp( 20ϕ 2150ϕ 3 ) (2.90) χ dil (T ; P ) 2.10 P T P ( ) Θ χ dil(t ) = T 1 ( ) 2 Θ T 1 (2.91) P Z ( ) Θ χ conc(t ; P ) = T 1

63 ( 0.075P 1/2 45P ) [ ( )] Θ exp (40 520P 2/3 ) T 1 (2.92) R(ϕ/ϕ ) = exp( P 1/2 ϕ 0.3P 3/2 ϕ 3 ) (2.93) Z Z(T, ϕ; P ) = χ conc(t ; P ) ϕ + A(P )ϕ B(T ; P )ϕ 2 +[χ dil(t ) χ conc(t ; P )]R(P 1/2 ϕ) (2.94) χ dil (T ) χ conc(t ; P ) R(P 1/2 ϕ) (2.91) (2.92) (2.93) A(P ) B(T ; P ) Z (2.73) χ A(P ) = 1.4P 1/3 (2.95) [ ( )] Θ B(T ; P ) = 7P 1/3 exp 18 T 1 (2.96) (2.94) χ = 2 ϕ ϕ 2 Zϕdϕ (2.97) χ(t, ϕ; P ) = χ conc(t ; P ) ϕ + A(P ) B(T ; P ) { ϕ B(T ; P ) + ln[1 + B(T ; P } )ϕ2 ] B(T ; P ) 2 ϕ 2 0 +[χ dil(t ) χ conc(t ; P )]Q(P 1/2 ϕ) (2.98) Q(x) x 2 [1 ( x x x 3 ) exp( 1.875x 0.432x x 3 )] (2.99)

64 ϕ c (2/x 2 ) x R(x)xdx 0 Z = 2Z c 1 ϕ c P ϕ c (2.100) ( ) 1 (1 ϕ c ) 2 1 Z (P ϕ c ) 2 = 2 ϕ (2.101) c (2.94) ϕ 2 c 2(1 ϕ c ) + 1 2P ϕ c = χ conc(t c ; P ) 1 2 A(P )ϕ 4 c B(T c ; P )ϕ 2 c +[χ dil(t c ) χ conc(t c ; P )]R(P 1/2 ϕ c )(2.102) c ϕ c (2 ϕ c ) 2(1 ϕ c ) 2 1 2P ϕ 2 c = 2A(P )[2 + B(T c; P )ϕ 2 c]ϕ 3 c [1 + B(T c ; P )ϕ 2 c] 2 P 1/2 [χ dil(t c ) χ conc(t c ; P )] ( P ϕ 2 c)r(p 1/2 ϕ c ) (2.103)

65 T c P Flory-Huggins (2.47) (2.48) (2.49) (2.97) χ 2.13 ( ) M w (2.97) χ µ M w

66

67 (1/RT )( µ 0 / ϕ) T,p ϕ 13 P-5 M w C P-13 M w C P-5 P-13 T (1/RT )( µ 0 / ϕ) T,p ϕ P-13 T = 26.0 C Z ϕ Z Z(T, ϕ; P ) = χ conc(t ; P ) ϕ + A(P )ϕ B(T ; P )ϕ 2 +[χ rmdil(t ) χ conc(t ; P )]R(ϕ; P ) (2.104)

68 Z χ χ(t, ϕ; P ) = χ conc(t ; P ) ϕ + A(P ) { } ϕ 2 B(T ; P ) 2 1 B(T ; P ) ln[1 + B(T ; P )ϕ2 ] +[χ dil(t ) χ conc(t ; P )]Q(ϕ; P ) (2.105) χ dil ( ) ( ) 2 Θ Θ χ dil(t ) = T T 1 (2.106) χ conc A(P ) conc(t ; P ) = ( ) Θ P 1/2 T 1 χ (2.107) A(P ) = 2P 1/3 (2.108) B(T ; P ) ( ) Θ B(T ; P ) = 8.73P 1/3 600 T 1 (2.109)

69 R R = exp( 3.3P 1/3 ϕ) (2.110) Q Q(ϕ; P ) = 2 (3.3P 1/3 ϕ) 2 [1 ( P 1/3 ϕ) exp( 3.3P 1/3 ϕ)] (2.111) Z χ (1) ϕ χ 2 ϕ (2) χ

70 µ 0 ( + ) (2.98) χ µ 0 (= µ 0 µ 0) 2.18 M = Flory-Huggins µ 0 Flory-Huggins µ 0 Θ ( ) µ 0 χϕ 2 χ van t Hoff µ 0 = µ 0 µ 0 = RT ϕ P (2.112) µ 0 µ 0 van t Hoff van t Hoff

71 µ 0 [ ] ϕ µ 0 = µ 0 RT + Γ(T, ϕ; P )ϕ2 P (2.113) 2 Γ Gamma Flory- Huggins χ Γ = χ ln(1 ϕ) + ϕ ϕ 2 (2.114) 2.18 Γ χ 2 Flory-Huggins χ Flory-Huggins (2.113) Gibbs-Duhem (1.93) 1 µ 1 µ 1 = µ 1 + RT [ln ϕ ϕ + ΓP ϕ(1 ϕ) + P ϕ 0 Γdϕ] (2.115) µ 1 lim ϕ 0 (µ 1 RT ln ϕ) (2.116) (2.113) (2.115) Gibbs G { G = (n 0 + P n 1 ) (1 ϕ)µ 0 + ϕ P µ 1 [ +RT ϕ P + ϕ ln ϕ ϕ ]} P + ϕ Γdϕ 0 (2.117) J J Γ ( ) Γ ϕ (2.118) ϕ

72 J J Z J = Z + 1 2(1 ϕ) (2.119) 2.7 Z J 2.19 Z J 2 J = J conc + (Jdil Jconc)Q (2.120) 3 Jdil J conc Q Jdil (2.91) χ dil Jdil = 1 2 χ dil (2.121) ( ) ( ) 2 Θ Θ Jdil = 0.26 T T 1 (2.122) J conc Q J conc = J c0 + J c1 ϕ 2 (2.123)

73 Q = exp( P 1/2 ϕ) (2.124) Jconc = J c0 J c0 = ( ) Θ 0.23 P 1/3 T 1 ( ) Θ J c1 = T 1 (2.125) (2.126) 2.19 (2.123) J conc (2.120) (2.120) Γ = J c J c1ϕ 2 Γ = 1 ϕ ϕ 2 Jϕ d ϕ (2.127) 0

74 (J dil J c0 ) 1 (1 + P 1/2 ϕ) exp ( P 1/2 ϕ) pϕ 2 (2.128) µ 0 (2.113) µ 1 (2.115) 2.20

75 2.A 69 2.A Maxwell H 1 c D t = 4π c j E + 1 c B t = 0 D = 4π ρ B = 0 (2.A.1) (2.A.2) (2.A.3) (2.A.4) H E D B j ρ c D, B, j D = ϵe (2.A.5) B = µh j = σe (2.A.6) (2.A.7) ϵ µ σ µ = 1 σ = 0 q r P P = qr (2.A.8) R A ϵ R r R r (r r A Vector) j = P (2.A.9) t ρ = P (2.A.10) (2.A.5) (2.A.10) (2.A.1) (2.A.4) H = Ṗ ϵ cė 4π c (2.A.11)

76 70 2 E + 1 c Ḣ = 0 E = 4π ϵ P H = 0 (2.A.12) (2.A.13) (2.A.14) = / t A H = A (2.A.15) (2.A.14) (2.A.12) (E + Ȧ) = 0 1 c (2.A.16) ϕ E = Ȧ ϕ 1 c (2.A.17) (2.A.12) (2.A.11) (2.A.13) 2 A ϵ c 2 Ä ( A + ϵ c ϕ) = 4π c Ṗ (2.A.18) 2 ϕ ϵ c ϕ c t ( A + ϵ c ϕ) = 4π ϵ P (2.A.19) (2.A.11) (2.A.14) H, E Lorentz A + ϵ c ϕ = 0 (2.A.20) A ϕ 2 A = Ṗ ϵ cä 4π c (2.A.21) 2 ϕ ϵ c ϕ = 4π ϵ P (2.A.22) A Π (2.A.23) 1 c

77 2.A 71 ϕ 1 ϵ Π (2.A.24) Hertz Π Lorentz (2.A.20) (2.A.21) (2.A.22) 2 Π ϵ c Π 2 = 4πP Π = P (t r/ c ) e r e P (2.A.25) (2.A.26) Hertz (2.A.26 P P P (t)δ(r)e (2.A.27) Π e x ψ(r, t) (2.A.25) 2 ψ(r, t) ϵ c 2 2 Fourier (2.A.28) ψ(r, ω) = P (ω) = ψ(r, t) = 4πP (t)δ(r) t2 (2.A.28) + + ψ(r, t)exp( iωt)dt P (t)exp( iωt)dt 2 ψ(r, ω) + k 2 ψ(r, ω) = 4πP (ω)δ(r) (2.A.29) (2.A.30) (2.A.31) k 2 ϵω 2 / c 2 (2.A.32) ψ (2.A.31) 1 r d 2 dr 2 (rψ) + k2 ψ = 4πP (ω)δ(r) (2.A.33)

78 72 2 r 0 0 rψ = Af(r) f(r) f(r) = exp( ikr) ψ = A exp( ikr) r (2.A.34) A ψ (2.A.33) A = P (ω) ψ(r, ω) = P (ω) exp( ikr) r (2.A.35) Fourier ψ(r, t) = 1 2π = 1 2π + + P (ω) exp( ikr) exp(iωt)dω r P (ω) exp[iω(t r/ c )] dω r ψ(r, t) = P (t r/ c )/r (2.A.36) (2.A.37) e y, e z e x 3 (2.A.26) (2.A.26) Π = {P } {P} e or Π = r r (2.A.38) c ϵ ñ c = c/ñ, ϵ = ñ 2 (2.A.39) (2.A.15) (2.A.17) (2.A.23) (2.A.24) H = Π (2.A.40) 1 c

79 2.A 73 E = 1 ϵ Π 4π ϵ P (2.A.41) (r 0) P = 0 E = 1 ( Π) Π (2.A.42) ϵ 1 c 2 (2.A.26) (2.A.40) (2.A.42) r λ E = 1 [ r (r { P}) r (r {Ṗ} + 2r(r {Ṗ}) ϵ c 2 r 3 + c r ] 4 r (r {P}) + 2r(r {P}) + r 5 H = 1 c (λ: ) ( r { P} E = 1 ϵ c r 2 + r {Ṗ} ) r 3 r (r { P}) c 2 r 3 (2.A.43) (2.A.44) (2.A.45) H = r { P} c c r 2 (2.A.46) Rayleigh R θ P x r e r x r θ x r = re r (2.A.47) e = cosθ x e r sinθ x e θx (2.A.48) (2.A.45) E = { P }sinθ x ϵ c 2 e θx E θx e θx (2.A.49) r E θx = { P }sinθ x c 2 r (2.A.50)

80 74 2 (2.A.46) H = { P }sinθ x c c e ϕ H ϕ e ϕ r (2.A.51) H ϕ = { P }sinθ x c c r = ñe θx (2.A.52) r Poynting S S c 4π E H (2.A.53) S = ñ c 4π E 2 θ x e r = ñ{ P } 2 sin 2 θ x 4π c 3 r 2 e r (2.A.54) S = ñ c 4π E θ 2 x ñ c 8π I (2.A.55) I I = 2Eθ 2 x E θx = E 0θx exp(iωt) I = E 2 0θ x (2.A.56) x E 0 = E 0 0 exp(iωt) (z ) P = αe 0 0 exp(iωt) (2.A.57) α (2.A.50) E θx = sinθ x c 2 r ω2 αe 0 0 exp[iω(t r/ c )] (2.A.58) (2.A.56) I = α 2 (E0 0 ) 2 16π4 sin 2 θ x λ 4 r 2 (2.A.59)

81 2.A 75 I 0 = (E 0 0 ) 2 I = 16π4 I 0 λ 4 α2 sin2 θ x r 2 (2.A.60) ( ) z E 0 P x y S = ñ c 4π (E 2 θ x + E 2 θ y )e r S = ñ c 4π (E 2 θ x + E 2 θ y ) ñ c 8π I I = 2(E 2 θ x + E 2 θ y ) = (E 2 0θ x + E 2 0θ y ) (2.A.61) (2.A.62) (2.A.63) I 0 = 2(E 0 0 ) 2 I = 8π4 α 2 I 0 λ 4 r 2 (sin2 θ x + sin 2 θ y ) = 8π4 α 2 λ 4 r 2 (1 + cos2 θ) (2.A.64) θ r z ( ) V N I I 0 = 8π4 α 2 λ 4 r 2 N(1 + cos2 θ) (2.A.65) Rayleigh R θ R θ I r 2 V I cos 2 θ (2.A.66) R θ = 8π4 α 2 λ 4 ρ N/V N V = 8π4 α 2 λ 4 ρ (2.A.67)

82 B V V ϵ ϵ = ϵ ϵ (2.B.1) V ϵ 2.A α = α α α = V 4π ϵ = V 2π ñ ñ (2.B.2) (2.A.67) α 2 ( α) 2 ( ) ρ V 1 =Ensemble Rayleigh R θ = 2π2 ñ 2 V λ 4 ( ñ) 2 (2.B.3) Rθ R θ ñ Rθ T p (Species 0) E V r (N 1, N 2,, N r ) E, V, N 1,, N r T p N 0 µ 1, µ 2,, µ r Hybrid Ensemble N = N 0, N 1,, N r N = N 1, N 2,, N r µ = µ 1, µ 2,, µ r (2.B.4) Hybrid Ensemble Γ(T, p, N 0, µ ) Γ(T, p, N 0, µ ) = e pv/kt e N µ /kt Q(T, V, N) V N 0 (2.B.5)

83 2.B 77 Q N (Canonical Ensemble) Q Helmholtz A A = kt ln Q (2.B.6) T p N 0 µ V N P (V, N ; T, p, N 0, µ ) r P (V, N ) = Γ 1 exp[( pv + N i µ i A)/kT ] (2.B.7) A A(T, V, N) (2.B.8) ( pv + N i µ i A) V N i V = V V N i = N i N i V N i (most probable value) Gauss i=1 P (V, N ) = Cexp( φ/kt ) (2.B.9) φ 1 2 ( 2 ) A r ( V 2 ( V ) 2 2 ) A + V N i T,N V N i=1 i T,V,N k + 1 r r ( 2 ) A N i N j 2 N i N j T,V,N k i=1 j=1 (2.B.10) C ( ) ( ) A A = p, = µ i V T,N N i T,V,N k (2.B.11) ( 2 ) ( ) A p V 2 = = 1 V κ V (2.B.12) T,N T,N ( 2 A V N i ) T,V,N k ( ) p = N i T,V,N k = ( V/ N i) T,p,Nk ( V/ p) T,N = V i κ V (2.B.13)

84 78 2 V i i κ κ 1 ( ) V (2.B.14) V p T,N ( 2 ) A ( 2 A N i N j N i N j ) T,V,N k = ( ) µi N j ( ) µi = N j ( ) µi + p T,V,N k = T,p,N k T,N ( p N j ( ) µj N i ) ( ) µi = V i, m i = M in i p T,N M 0 N 0 T,V,N k = V iv j κ V + M j M 0 N 0 ( ) µi m j T,V,N k T,p,m k T,V,N k (2.B.15) (2.B.16) (2.B.17) M 0 M j 0 j ξ V V + r x i N i N i = m i m i i=1 V i N i V (2.B.18) (i = 1, 2,, r) (2.B.19) (2.B.13) (2.B.14) (2.B.17) (2.B.10) φ = V 2κ ξ2 + M 0N 0 2 r r i=1 j=1 ( ) m i m j µi x i x j M i m j T,p,m k (2.B.20) V = V m j = m j P (V, N ) V N i ξ x i ( P (ξ, x 1,, x r ) = Cexp V 2κkT ξ2 M 0N 0 2 r i=1 j=1 r ) ψ ij x i x j (2.B.21)

85 2.B 79 ψ ij = m ( ) im j µi = m ( ) im j µj = ψ ji (2.B.22) M i kt m j T,p,m k M j kt m i T,p,m k ξ 2 x i x j ξ 2 x i x j C ψ ij ψ Q Q 1 = Q T (2.B.23) Q 1 ψq = Λ (2.B.24) Λ λ i ψ = Λ Q x ξ x = Qξ ξ = ξ 1, ξ 2,, ξ r (ξ = Q 1 x) (2.B.25) x T ψx = (Qξ) T ψ(qξ) = ξ T (Q T ψq)ξ = ξ T Λξ (2.B.26) P ( P = Cexp V 2κkT ξ2 M 0N 0 2 r ) λ i ξi 2 i=1 (2.B.27) P dξdξ 1dξ 2 dξ r = 1 [ V (M0 N 0 /2) r C = 2π r+1 κkt ψ ] 1/2 (2.B.28) (2.B.27) (2.B.28) ξ 2 = κkt V (2.B.29) ξi 2 1 =, ξ i ξ j = 0 (2.B.30) M 0 N 0 λ i

86 80 2 x i x j = k l Q ikq jl ξ k ξ l (2.B.30) x i x j = k k Q ik Q jk ξ 2 k = 1 M 0 N 0 Q ik Q jk λ k k Q ik Q jk λ k = (QΛ 1 Q T ) ij = (ψ 1 ) ij = ψij ψ (2.B.31) (2.B.32) x i x j = ψ ij M 0 N 0 ψ (2.B.33) ψ ij ψ ij x i x j x i ( ) ξ 2 ( N i = 0 ) (2.B.18) ρ = N V = r i=0 N i = N V V ξ = V V V V (2.B.34) ln ρ = ln N ln V (2.B.35) = ρ ρ ξ 2 = ( ρ)2 ρ 2 (2.B.36) (2.B.37) ξ 2 Hybrid Ensemble ñ ( ) ñ r ( ) ñ ñ = V + N i V T,N N i T,V,N k i=1 (2.B.38)

87 2.B 81 T p N ( ) ( ) ñ ñ r ( ) ñ dñ = dp + dt + dn i p T,m T p,m N i T,p,N k i=0 (2.B.39) ( ) ( ) ñ ñ = V p ( ñ N i T,N ) T,V,N k = T,m ( ) ñ ( ) p V T,N N i = M ( ) i ñ M 0 N 0 m i T,p,N k + = 1 κv ( ) ñ p T,m ( ) ( ñ p p T,m N i T,p,m k + V i κv ( ñ p ) ) T,V,N k T,m (2.B.38) (2.B.40) (2.B.41) ñ = V ( ) ñ + 1 ( ) ñ r V i N i κv p T,m κv p T,m i=1 + 1 r ( ) ñ M i N i M 0 N 0 m i=1 i T,p,m k = 1 ( ) ñ r ( ) ñ ξ + m i x i κ p T,m m i T,p,m k i=1 (2.B.29) (2.B.33) ( ñ) 2 = 1 ( ñ κ 2 p r r + = kt κv ) 2 i=1 j=1 ) 2 ( ñ p + 1 M 0 N 0 ξ 2 T,m m i m j ( ñ m i T,m r r i=1 j=1 ) T,p,m k ( ñ m j ) m i m j ( ñ m i ) T,p,m k T,p,m k x i x j (2.B.40) (2.B.41) (2.B.42)

88 82 2 ( ñ )T,p,mk ψ ij m j ψ (2.B.43) (2.B.3) R θ = 2π2 ñ 2 ( ) 2 kt ñ λ 4 κ p T,m + 2π2 ñ 2 V r r ( ) ñ λ 4 m i m j M 0 N 0 m i=1 j=1 i ( ñ ψ m j )T,p,mk ij ψ T,p,m k (2.B.44) 1 R θ,0 2 R θ R θ = R θ,0 + R θ (2.B.45) 0 c 0 = M 0 N 0 /N A V (g/ml) R θ R θ = 2π2 ñ 2 N A λ 4 c 0 r r i=1 j=1 ( ) ( ñ ñ ψ m i m j m i T,p,m k m j )T,p,mk ij ψ (2.B.46) i (i=1, 2,, r) µ i µ i = µ 0 i (T, p) + kt ln γ i m i (2.B.47) γ i γ i ln γ i = M i ( r B ij m j + j=1 r j=1 k=1 r B ijk m j m k + ) (2.B.48) m 1, m 2,, m r 0 γ i 1 (2.B.47) (2.B.22) ψ ij (2.B.46) R θ

89 2.B 83 [ ] (2.B.46) ψ ij / ψ R θ ψ ij ψ = M ( ) 1 1kT µ1 m1 2 m 1 T,p (2.B.49) R θ = 2π2 ñ 2 M 1 kt N A λ 4 c 0 ( ñ m 1 ) 2 T,p ( µ1 m 1 Gibbs-Duhem (1.94) ( ) µ1 = N ( ) 0 µ0 m 1 N 1 m 1 T,p ) 1 T,p T,p (2.B.50) (2.B.51) c ( c 1 ) c = M 1N 1 N A V = M 0N 0 N A V m 1 (2.B.52) m 1 = M 1N 1 M 0 N 0, V = N 0 V 0 + N 1 V 1 (2.B.53) ( m 1 ) T,p = c ( ) 0N 0 V 0 V c T,p (2.B.50) R θ = 2π2 ñ 2 ( ) 2 /( ) RT V 0 c ñ µ0 N A λ 4 c T,p c Kc = 1 R θ V 0 RT K 2π2 ñ 2 N A λ 4 ( ) µ0 c T,p ( ) 2 ñ c T,p T,p (2.B.54) (2.B.55) (2.B.56) (2.B.57)

90 84 2 K µ 0 π ( ) c µ 0 µ 0 = V0 π = V0 RT M + A 2c 2 + A 3 c 3 + (2.B.58) A 2 2 A 3 3 ( ) µ0 c T,p = V 0 ( ) ( π 1 = V0 RT c T,p (2.B.56) ) M + 2A 2c + 3A 3 c 2 + (2.B.59) Kc R θ = 1 M + 2A 2c + 3A 3 c 2 + (2.B.60) V 0 V 0 [Flory-Huggins ] Flory-Huggins 0 mu 0 1 µ 1 (2.50) (2.51) 1 ϕ 1 (2.B.56) c ϕ 1 K ϕ K ϕ V 0 ϕ 1 R θ = 1 RT ( ) µ0 ϕ 1 T,p (2.B.61) K ϕ = 2π2 ñ 2 ( ) 2 ñ N A λ 4 (2.B.62) ϕ 1 (2.B.61) (2.50) χ ϕ χ 1 = 1 ( K ) ϕv 0 ϕ ϕ 1 P 1 ϕ 1 R θ (2.B.63)

91 2.C ( ) Maurice Loyal Huggins (1897/9/ /12/17) 2.22 Paul John Flory (1910/6/ /9/8) ( µ 0 / ϕ 1 ) T,p ( ( µ 1 / ϕ 1 ) T,p ) χ ( 2 G/ ϕ 2 1) T,p = µ 1 / ϕ 1 ) T,p 0 (Spinodal) T sp T = T sp 1/R θ = 0 ϕ 1 T R θ (K ϕ V 0 ϕ 1 /R θ ) 0 T sp 2.C ( )

92 Joseph Edward Mayer (1904/2/5 1983/10/15) 2.24 John Gamble Kirkwood (1907/5/ /8/9)

93 2.C ( ) Walter Hugo Stockmayer (1914/4/7 2004/5/9) 2.26 Bruno Hasbrouck Zimm (1920/10/ /11/26)

94 88 1. III R. Koningsveld, W. H. Stockmayer, and E. Nies, Polymer Phase Diagrams, Oxford University Press, Oxford, M. L. Huggins, J. Chem. Phys., 9, 440 (1941); Ann. New York Acad. Sci., 43, 1 (1942). 5. P. J. Flory, J. Chem. Phys., 9, 660 (1941); ibid., 10, 51 (1942). 6. A. R. Shultz and P. J. Flory, J. Am. Chem. Soc., 74, 4760 (1952). 7. H. Fujita and Y. Einaga, Makromol. Chem., Macromol. Symp., 12, 75 (1987). 8. H. Tompa, Polymer Solutions, Butterworths Scientific Publications, London, R. Koningsveld, J. Polym. Sci., Part A-2, 6, 325 (1968). 10. R. Koningsveld, L. A. Kleintjens, and A. R. Shlutz, J. Polym. Sci., Part A-2, 8, 1261 (1970). 11. R. Koningsveld and L. A. Kleintjens, Macromolecules, 4, 637 (1971). 12. Y. Einaga, S. Ohashi, Z. Tong, and H. Fujita, Macromolecules, 17, 527 (1984). 13. N. Takano, Y. Einaga, and H. Fujita, Polym. J., 17, 1123 (1985). 14. Y. Einaga, Z. Tong, and H. Fujita, Macromolecules, 18, 2258 (1985).

95 (quasibinary system) q ( 1 2 q) i P i V i V i /V 0 V 0 V i V 0 T ( ) V q V = V 0 (n 0 + P i n i ) (3.1) n 0 n i 0 i ( ) i ϕ i i=1 ϕ i V in i V (3.2) q ϕ i (i = 1, 2,, q) ϕ i ϕ i = n i P i n 0 + q j=1 P jn j (3.3)

96 90 3 ϕ q ϕ = ϕ i (3.4) i ξ i i=1 ξ i = ϕ i ϕ (3.5) q i=1 ξ i = 1 q ϕ i ϕ q 1 ξ i [ϕ i ] [ξ i ] 2 1 (1.104) µ i (T, p, [ϕ i]) = µ i (T, p, [ϕ i ]) (i = 0, 1, 2,, q) (3.6) µ i (T, p, ϕ, [ξ j]) = µ i (T, p, ϕ, [ξ j ]) (i = 0, 1, 2,, q) (3.7) Gibbs G µ i (i 0, 1, 2,, q) G 11 G 12 G 1q J sp G 21 G 22 G 2q G q1 G q2 G qq µ 11 µ 12 µ 1q = µ 21 µ 22 µ 2q µ q1 µ q2 µ qq = 0 (3.8)

97 ( 2 ) G G ij ϕ i ϕ j ( ) µi µ ij ϕ j T,p,ϕ k (k i,k j) T,p,ϕ k (k j) (3.9) (3.10) (3.8) 1 3.B (3.8) J sp ϕ 1 J sp ϕ 2 J sp ϕ q G 21 G 22 G 2q G q1 G q2 G qq = 0 (3.11) gibbs G G G 2 ( B D ) 2 (ϕ 1, ϕ 2) (ϕ 1, ϕ 2) 2 µ i (i = 0, 1, 2) 2 µ i (ϕ 1, ϕ 2) = µ i (ϕ 1, ϕ 2) (i = 0, 1, 2) (3.12) (3.6) ( A B C D E )

98 Gibbs

99 G ( K C L ) C 2 1 T CC 5 C (critical line) 1 2 X 0X T

100 ( AA 2 C 5 B) A 2 ( ) B 2 A K ( ) 0X T 2 Q Q Q 2 Q Q 0X 0X T 2 Q Q 0X w 2

101 (mother solution) 2 (shadow curve) 2 1 T c 3.2 Flory-Huggns 0 j(j = 1, 2,, q) Gibbs G G = n 0 µ 0 + q n j µ j j=1 { +RT (1 ϕ) ln(1 ϕ) + q j=1 } +g(t, p, ϕ, [ξ j ]; [P j ])ϕ(1 ϕ) ϕξ j P j ln(ϕξ j ) (3.13) [P j ] P 1, P 2,, P q µ 0 µ 0 = µ 0 + RT [ ln(1 ϕ) + ( 1 1 ] )ϕ + χ(t, p, ϕ; f(p ))ϕ 2 P n (3.14) f(p ) [ξ j ] [P j ] P n 1 P n q j=1 ξ j P j (3.15)

102 96 3 i µ i ( µ i = µ i + RT [ln(ξ i ϕ) (P i 1) + P i 1 1 P n { [ ( ) ] g +P i (1 ϕ) (1 ϕ) g + ϕ ϕ [ ( ) g q 1 ( )]}] g +ϕ m i ξ j ξ i ξ j j=1 ) ϕ (3.16) i q m i = 1 i = q m i = 0 χ ( ) g χ = g (1 ϕ) ϕ T,p (3.17) χ f(p ) (3.8) (3.13) 1 1 ϕ + 1 P w ϕ 2χ ( ) χ ϕ = 0 (3.18) ϕ T,p P w q P w ξ j P j (3.19) j=1 (3.18) 1 (1 ϕ) 2 P ( ) ( z χ 2 ) (P w ϕ) 2 3 χ ϕ T,p ϕ 2 ϕ = 0 (3.20) T,p P z P z q j=1 ξ2 j P j P w (3.21) g χ ϕ f(p ) ϕ c χ(t c ) ϕ c = Pw P 1/2 z (3.22)

103 χ(t c ) = 1 2 (1 + P z 1/2 )( ) P w Pz 1/2 (3.23) (3.18) χ f(p ) T sp ϕ P w ( M w ) M w PS166 M w = PSM5 M w = z M z M w M z /M w M w χ (separation factor)σ i ϕ i ϕ i σ i ln(ϕ i /ϕ i ) P i (i = 1, 2,, q) (3.24) i

104 Breitenbach-Wolf ( ) (3.24) ( ) W ln i = ln r σ i M i (3.25) W i W i W i i M i i r ϕ i W i P i M i ln(w i /W i ) M i Breitenbach-Wolf 5 g f(p ) µ i = µ i (i = 1, 2,, q) [ ( )] g σ i = ln(1 ϕ) + 2(ϕ 1)g ϕ(1 ϕ) ϕ (3.26) X X X X g [ξ i ] σ i i Breitenbach-Wolf 3.5

105 M n = M w = M z = g f(p ) g f(p ) (3.26) [ ( ) g σ i = ln(1 ϕ) + (2ϕ 1)g ϕ(1 ϕ) ϕ { ( ) g q 1 ( )}] g (1 ϕ) m i ξ j ξ i ξ j j=1 (3.27) i q m i = 1 i = q m i = 0 f(p ) g/ ξ i 2 i σ i P i Breienbach-Wolf 7, i P i (= V i /V 0 ) ϕ i (i = 1, 2) V i V 0 i 0 ϕ ϕ 1 + ϕ van t Hoff 0 µ 0 ( ) ϕ µ 0 = µ 0 RT + Γ(T, p, ϕ 1, ϕ 2 ; P 1, P 2 )ϕ 2 P n P n (3.28) P 1 n = ϕ 1P1 1 + ϕ 2 P2 1 ϕ (3.29)

106 100 3 Gibbs-Duhem ( ) 1 ϕ µ0 + ϕ ( ) 1 µ1 + ϕ ( ) 2 µ2 = 0 V 0 ϕ 1 V 1 ϕ 1 V 2 ϕ 1 (3.30) ( ) 1 ϕ µ0 + ϕ ( ) 1 µ1 + ϕ ( ) 2 µ2 = 0 V 0 ϕ 2 V 1 ϕ 2 V 2 ϕ 2 (3.31) ϕ 1 ϕ 2 ϕ 1 ξ ( ϕ 1 /ϕ) ( ) µ0 (1 ϕ) + ϕξ ( ) ( ) µ1 ϕ(1 ξ) µ2 + = 0 ϕ P 1 ϕ P 2 ϕ (3.32) ( ) µ0 (1 ϕ) + ϕξ ( ) ( ) µ1 ϕ(1 ξ) µ2 + = 0 ξ P 1 ξ P 2 ξ (3.33) 3.A 1 2 µ 1 µ 2 [ µ 1 = µ 1 + RT ln(ϕξ) ϕ + 0 ( 1 P 1 P 2 ϕ { ( )} ] Γ +P 1 Γ + (1 ξ) dϕ ξ ) (1 ξ)ϕ + ΓP 1 ϕ(1 ϕ) (3.34) [ ( ) µ 2 = µ P2 2 + RT ln[ϕ(1 ξ)] ϕ + 1 ξϕ + ΓP 2 ϕ(1 ϕ) P 1 ϕ { ( )} ] Γ +P 2 Γ ξ dϕ (3.35) ξ 0 µ i lim ϕ 0 (µ i RT ln ϕ i ) (3.36) µ i (i = 1, 2) µ 0 Gibbs G [ G = (n 0 + n 1 P 1 + n 2 P 2 ) (1 ϕ)µ 0 + ϕ 1 µ 1 + ϕ 2 µ 2 P 1 P 2 + ϕ 1 P 1 ln ϕ 1 + ϕ 2 P 2 ln ϕ 2 + ϕ ϕ 0 Γ(T, p, ϕ 1, ϕ 2 ; P 1, P 2 )dϕ { + RT ϕ P n }] (3.37)

107 Γ J J Γ + 1 ( ) γ ϕ (3.38) 2 ϕ 3 J J = J c0 + J c1 ϕ 2 + (J dil J c0 ) exp( ϕ/b) (3.39) ( ) + ( ) Θ J c0 = 0.036P 1/ T 1 (3.40) J c1 = ( Θ T 1) (3.41)

108 102 3 P P ( ) ( ) 2 Θ Θ Jdil = 0.26 T T 1 (3.42) b = P 1/2 (3.43) P [ξp 1/3 1 + (1 ξ)p 1/3 2 ] 3 (3.44) P [ξp 1/2 1 + (1 ξ)p 1/2 2 ] 2 (3.45) (3.44) (3.45) 3 J c0 b (1 M w = 43600(f 4 ) M w = (f 40 )) C f 40 2 f 4 2 ϕ C f 40 2 f 4 2 f 40 f 4 ϕ 40 ϕ ξ 4 f 4 f G G 11 G 22 (G 12 ) 2 = 0 (3.46) G ϕ 1 G ϕ 2 G 21 G 22 = 0 (3.47)

109 ( )

110 ( ) f 4 M w = (f 10 ) T c ϕ c ϕ c T c

111 T c ξ ϕ c ξ 4

112 ( ) ( )

113 Gibbs 3 T ϕ 3 Flory-Huggins J Γ M = 43600(f4) M = (f128) 3.12 T ϕ 3.13 T f128 ξ T u (= C) T l (= C) T S d S m S c 3 ϕ = ξ 2 = KL LM MN T u L N 2 L L N ϕ ξ 2 N M T l ( ) T = 13.8 C 3

114 ( )

115 ( )

116 ξ 2 = van t Hoff q 0 Gibbs G G = (n 0 + +RT +ϕ q i=1 [ n i P i ) (1 ϕ)µ 0 + { ϕ P n + ϕ P n 0 q i=1 ϕ i P i ln ϕ i q i=1 ϕ i P i µ i }] Γ(T, p, ϕ 1,, ϕ q ; P 1,, P q )dϕ P n [ (3.48) q ξ i P i ] 1 (3.49) i=1 2 Γ Γ = Γ c Γ c1ϕ 2 +2(Γ d0 Γ c0 ) [1 e (1 ϕ/ϕ + ϕϕ )]( ) ϕ 2 (3.50) ϕ Γ c0 ϕ i Γ c0i ϕ i q Γ c0 = ξ i Γ c0i (3.51) i=1

117 ϕ = q ξ i ϕ i (3.52) i=1 ( ) ( ) 2 Θ Θ Γ d0 = 0.26 T T 1 (3.53) Γ c0i = 0.03 ( ) Θ 0.23 P 1/3 T 1 i ( ) Θ Γ c1 = T 1 ϕ i = 1 P 1/2 i 0 µ 0 (3.54) (3.55) (3.56) µ 0 µ 0 RT = ϕ P n Γϕ 2 (3.57) i µ i µ i µ i RT 0 ϕ = ln(ξ i ϕ) P i ϕ + P i ϕ(1 ϕ)γ + P i P n 0 ϕ [ +P i (1 ξ i ) Γ q 1 ] Γ ξ j dϕ ξ i ξ j j i Γdϕ (i = 1, 2,, q 1) (3.58) µ q µ q RT q 1 = ln(1 ξ i )ϕ P q ϕ + P q ϕ(1 ϕ)γ P n 0 i=1 ϕ ϕ q 1 Γ +P q Γdϕ P q ξ j dϕ (3.59) ξ j 0 j=1

118 112 3 σ i [ ϕ σ i = Γϕ + Γdϕ + ϕ 0 0 { (1 ξ i ) Γ q 1 } ] Γ ξ j dϕ ξ i ξ j j i (i = 1, 2,, q 1) (3.60) [ σ q = Γϕ + ϕ 0 Γdϕ ϕ q 1 0 j=1 ] Γ ξ j dϕ ξ j (3.61) [X] X X X σ i Γ

119 3.A Gibbs A Gibbs 0 µ 0 2 Γ ( µ 0 = µ ϕ1 0 RT + ϕ ) 2 + Γϕ 2 P 1 P 2 (3.A.1) ϕ i P i i ϕ Gibbs-Duhem 1 ϕ V 0 µ 0 ϕ 1 + ϕ 1 V 1 µ 1 ϕ 1 + ϕ 2 V 2 µ 2 ϕ 1 = 0 1 ϕ V 0 µ 0 ϕ 2 + ϕ 1 V 1 µ 1 ϕ 2 + ϕ 2 V 2 µ 2 ϕ 2 = 0 (3.A.2) (3.A.3) V i i 1 ξ( ϕ 1 /ϕ) = ϕ 1 ϕ + 1 ξ ϕ ξ, = ϕ 2 ϕ ξ ϕ (3.A.2) ξ (3.A.4) 1 ϕ ϕ µ 0 V 0 ϕ + ξ ϕ 2 µ 1 V 1 ϕ + 1 ξ ϕ 2 µ 2 V 2 ϕ = 1 ϕ (1 ξ) µ o V 0 ξ ϕ ξ(1 ξ) µ 1 V 1 ξ ϕ V 2 (1 ξ) 2 µ 2 ξ (3.A.5) (3.A.3) 1 ϕ ϕ µ 0 V 0 ϕ + ξ ϕ 2 µ 1 V 1 ϕ + 1 ξ ϕ 2 µ 2 V 2 ϕ = 1 ϕ ξ µ o V 0 ξ + ϕ ξ 2 µ 1 V 1 ξ + ϕ ξ(1 ξ) µ 2 V 2 ξ (3.A.6) 2 1 ϕ V 0 µ 0 ξ + ϕ V 1 ξ µ 1 ξ + ϕ V 1 (1 ξ) µ 2 ξ = 0 (3.A.7)

120 114 3 (3.A.5) (3.A.6) ξ 0 ξ 1 1 ϕ V 0 µ 0 ϕ + ϕ V 1 ξ µ 1 ϕ + ϕ V 2 (1 ξ) µ 2 ϕ = 0 )(3.A.1) ϕ (3.A.8) 1 µ 0 RT ϕ = 1 P n ξ µ 1 V 1 ϕ + 1 ξ µ 2 V 2 ϕ [ 1 = RT V 0 P n ( 1 ϕ 1 ( 2Γϕ + ϕ 2 Γ ) ϕ ) ( ) ] 1 Γϕ 2 + ϕ 1 ϕ ϕ ξ µ ξ [ µ 2 = RT C+ 1 (ln ϕ ϕ)+ϕ(1 ϕ)γ+ P 1 P n P 2 ϕ 0 ] Γdϕ (3.A.8) (3.A.9) (3.A.10) (3.A.11) C P i = V i /V 0 ξ 1 µ 2 RT P 2 ξ = 1 [ 1 RT P 1 (1 ξ) 2 µ 1 + ξ 1 ξ [ 1 +ϕ(1 ϕ) (1 ξ) 2 Γ ξ ] µ 1 + ξ Γ ξ ] + 1 (ln ϕ ϕ) P 1 (1 ξ) 2 1 (1 ξ) ϕ Γ 1 ξ 0 ξ dϕ + 1 (1 ξ) 2 C + 1 C 1 ξ ξ (3.A.1) ξ ( 1 µ 0 1 RT ξ = 1 ) ϕ ϕ 2 Γ P 1 P 2 ξ (3.A.7) 1 µ 1 = 1 ( 1 (ln ϕ ϕ) 1 RT P 1 P 1 P 1 P 2 ϕ [ + Γ + (1 ξ) Γ ] dϕ + ξ 0 ϕ 0 Γdϕ (3.A.12) (3.A.13) ) (1 ξ)(1 ϕ) + ϕ(1 ϕ)γ [ C + (1 ξ) C ] ξ (3.A.14)

121 3.A Gibbs 115 (3.A.11) 1 µ 2 = 1 ( 1 (ln ϕ ϕ) + 1 RT P 2 P 2 P 1 P 2 ϕ [ + Γ ξ Γ ] dϕ + ξ 0 ) ξ(1 ϕ) + ϕ(1 ϕ)γ [ C ξ C ξ ϕ = ] (3.A.15) µ 1 RT = µ 1 RT + ln(y 1 ϕ 1 ) (3.A.16) µ 2 RT = µ 2 RT + ln(y 2 ϕ 2 ) (3.A.17) y 1 y 2 ϕ 0 y 1 = 1 y2 = 1 (3.A.14) (3.A.15) ϕ 0 µ 1 RT +ln ϕ 1 = lim ϕ 0 (ln ϕ) ( 1 P 1 µ 2 RT + ln ϕ 2 = lim ϕ 0 (ln ϕ) P 2 [ )(1 ξ)+p 1 C+(1 ξ) C ξ ( 1 P 2 P 1 )ξ + P 2 [ C ξ C ξ 2 C ] ] (3.A.18) (3.A.19) C = µ 1 RT P 1 ξ + µ 2 RT P 2 (1 ξ) + ξ P 1 ln ξ + 1 ξ P 2 ln(1 ξ) (3.A.20) (3.A.14) (3.A.15) µ 1 µ 1 = 1 ( 1 (ln ϕ 1 ϕ) + 1 ) (1 ξ)ϕ + ϕ(1 ϕ)γ RT P 1 P 1 P 1 P 2 ϕ [ + Γ + (1 ξ) Γ ] dϕ (3.A.21) ξ 0 µ 2 µ 2 = 1 ( 1 (ln ϕ 2 ϕ) + 1 ) ξϕ + ϕ(1 ϕ)γ RT P 2 P 2 P 1 P 2 ϕ [ + Γ ξ Γ ] dϕ (3.A.22) ξ 0

122 116 3 Gibbs G G RT G G RT (1 ϕ) µ 0 µ 0 RT + ϕ 1 µ 1 µ 1 P 1 RT = ϕ 1 P 1 ln ϕ 1 + ϕ 2 P 2 ln ϕ 2 ϕ P n + ϕ + ϕ 2 µ 2 µ 2 P 2 RT ϕ 0 Γdϕ (3.A.23) Gibbs G m µ 1 µ 2 (3.A.21) ϕ = 1 ξ = 1 µ 1 RT P 1 = µ 1 RT P 1 1 P Γ(ξ = 1)dϕ (3.A.22) ϕ = 1 ξ = 0 µ 2 RT P 2 = µ 2 RT P 2 1 P Γ(ξ = 0)dϕ (3.A.24) (3.A.25) (3.A.21) (3.A.22) (3.A.24) (3.A.25) µ 1 µ 2 µ 1 µ 1 = 1 ( 1 (ln ϕ 1 ϕ) + 1 ) (1 ξ)ϕ + ϕ(1 ϕ)γ RT P 1 P 1 P 1 P 2 ϕ [ + Γ + (1 ξ) Γ ] Γ(ξ = 1)dϕ (3.A.26) ξ P µ 2 µ 2 RT P 2 = 1 ( 1 (ln ϕ 2 ϕ) P 2 + ϕ 0 [ Γ ξ Γ ξ 1 P 1 P 2 ] + 1 P 2 1 ) ξϕ + ϕ(1 ϕ)γ 0 Γ(ξ = 0)dϕ (3.A.27) (3.A.1) Gibbs G m G m RT G G RT

123 3.A Gibbs 117 (1 ϕ) µ 0 µ 0 RT + ϕ 1 µ 1 µ 1 P 1 RT = ϕ 1 P 1 ln ϕ 1 + ϕ 2 P 2 ln ϕ 2 + ϕ ϕ 0 Γdϕ + ϕ 2 µ 2 µ 2 P 2 RT 1 1 ϕ 1 Γ(ξ = 1)dϕ ϕ 2 Γ(ξ = 0))dϕ 0 0 (3.A.28) G Gibbs G G G RT = G RT + ϕ P n ϕ Γ(ξ = 1)dϕ ϕ 2 Γ(ξ = 0)dϕ 0 (3.A.29)

124 B T p q + 1 n 0 n 1 n q ( mol) n q i=0 n i n α β ( ) Gibbs G α G G α β G G β Gibbs α β 0( ) n α 0 nβ 0 n α 0 + n β 0 = n 0 q q n α ( n α i ) n β ( n β i ) i=0 i=0 (3.B.1) q = 1 2 [2 ] 1 2 gibbs G β α 1 δn α 1 ( ) (δ 2 G) T,p,n0,n 1 > 0 (3.B.2) (δ 2 G) T,p,n0,n 1 α δ 2 G α ) T,p,n α 0 β (δ 2 G β ) T,p,n β (3.B.2) 0 (δ 2 G) T,p,n0,n 1 = (δ 2 G α ) T,p,n α 0 + (δ 2 G β ) T,p,n β 0 = 1 2 (µα 11 + µ β 11 )(δnα 1 ) 2 > 0 (3.B.3) ( µ α 2 G α ) 11 n α2 1 T,p,n α 0 ( ) µ α = 1 n α 1 T,p,n α 0

125 3.B 119 [ = n α 1 = 1 x 1 n α ( n α 1 n α 0 + nα 1 ( ) µ1 x 1 )]( ) µ1 T,p x 1 T,p (3.B.4) µ β 11 = 1 x 1 n β ( ) µ1 x 1 T,p (3.B.5) x 1 1 x 1 µ 1 α β (3.B.1) (3.B.4) (3.B.5) (3.B.3) µ α 11 µ β 11 (3.B.6) (δ 2 G) T,p,n0,n 1 = 1 2 µα 11(δn α 1 ) 2 = 1 x 1 2n α ( ) µ1 x 1 (δn α 1 ) 2 > 0 T,p (3.B.7) 2 1 ( ) µ1 > 0 (3.B.8) x 1 T,p ( ) µ1 < 0 (3.B.9) x 1 2 (3.B.8) (3.B.9) x 1 µ 1 x 1 ( ) µ1 = 0 (3.B.10) x 1 T,p T,p

126 (3.B.10) x 1 2 T 2 x 1 2 T c x 1 µ 1 (3.B.10) ( 2 ) µ 1 x 2 = 0 (3.B.11) 1 T,p (3.B.10) (3.B.10) (3.B.11) gibbs G ( 2 ) G = 0 (3.B.12) [ ] x 2 1 ( 3 ) G x 3 1 T,p T,p = 0 (3.B.13) r + 1 (δ 2 G) T,p,n0,,n q = 1 q q µ α 2 ijδn α i δn α j > 0 (3.B.14) µ α ij ( µ α 2 G α ) ij n α i nα j δn α i i=1 j=1 T,p,n α 0 ( ) µ α = i n α j T,p,n α 0 (3.B.15) (3.B.14) µ α ii > 0 (i = 1, 2,, q) (3.B.16) µ 11 µ 12 µ 1q µ 21 µ 22 µ 2q > 0 (3.B.17) µ q1 µ q2 µ qq

127 3.B µ 11 µ 12 µ 1q µ 21 µ 22 µ 2q = 0 (3.B.18) µ q1 µ q2 µ qq µ i G G 11 G 12 G 1q G G 21 G 22 G 2q G q1 G q2 G qq = 0 (3.B.19) ( 2 ) G G ij x i x j T,p,x k (3.B.20) G x 1 G x 2 G x q G 21 G 22 G 2q = 0 (3.B.21) G q1 G q2 G qq G x i = ( ) G x i T,p,x j(j i) (3.B.22) x i ϕ i Gibbs G (ϕ) G (ϕ) G q i=0 n iv i (3.B.23)

128 122 3 V i i G (ϕ) ( ) ( G (ϕ) 2 G (ϕ) ) G (ϕ) i ϕ i T,p,ϕ j, G (ϕ) ij ϕ i ϕ j (3.B.19) (3.B.21) G ij G (ϕ) ij T,p.ϕ k (3.B.24)

129 3.C C (UCST) [ ] UCST 1 T 3.17 ( ) C [ ] (3.8) Rayleigh

130 I 1 30 ( 30 )

131 3.C 125 (2.B.46) T I I 1 T I 1 0 T s 1 T s PICS (Pulse Induced Critical Scattering) (C D) (A B) I T s [ ] f 4 (M w = 45300) f 40 (M w = ) ξ 4 f 4 2 r( ) T r T T 0 r 1 T 0 r T 0 r 0

132 f 4 (M w = 45300)+f 40 (M w = ) r (ξ = 0.500)

133 r (ξ = 0.950) 3.21 ϕ c ϕ c ϕ c ϕ r T r = 1 T r = 1 T T ϕ 3.22 ϕ c T c III R. Koningsveld, W. H. Stockmayer, and E. Nies, Polymer Phase Diagrams, Oxford University Press, Oxford, G. Rehage, D. Moeller, and O. Ernst, Makromol. Chem., 88, 232 (2965).

134 J. W. Kennedy, M. Gordon, and R. Koningsveld, J. Polym. Sci., C39, 71 (1972). 5. J. W. Breitenbach and B. A. Wolf, Makromol. Chem., 108, 263 (1967). 6. L. A. Kleintjens, R. Koningsveld, and W. H. Stockmayer, Brit. Polym. J., 8, 144 (1976). 7. H. Fujita and Y. Einaga, Makromol. Chem., Macromol. Symp., 12, 75 (1987). 8. Z. Tong, Y. Einaga, and H. Fujita, Macromolecules, 18, 2264 (1985). 9. Y. Einaga, Z. Tong, and H. Fujita, Macromolecules, 18, 2258 (1985). 10. H. Tompa, Trans. Faraday Soc., 45, 1142 (1949). 11. Y. Einaga, Y. Nakamura, and H. Fujita, Macromolecules, 20, 1083 (1987). 12. K. W. Derham, J. Goldsbrough, and M. Gordon, Pure & Appl. Chem., 38, 97 (1974). 13. M. Tsuyumoto, Y. Einaga, and H. Fujita, Polym. J., 16, 229 (1984).

135 (PS) (PBD) 2 PS M w PBD M w C PS-PBD 1 PS-PBD 4.1 (PS) (PVME) PS M w 2100 PVME M w C 28 C 30 C (C 2 HCl 3 ) 14 C PS+C 2 HCl C 30 C 3 2

136

137 (PS) (PIP) (CH) PS M w PIP M w PS-PIP 1 PS PIP 15 C Φ PS T PS+CH2 PS+PIP PS ξ PS ( ) PS-PIP ξ PS ( T ) 4.4 ξ PS (PS ) ξ rmp S (PS ) PS+CH2 2 ( ) (

138

139 ( ) ( ) ) 1 (PS) (PIB) (CH) (Benzene) ξ PS M w 2 ξ PS (PS ) 4.4 PS+PIP+CH PS+CH2 PIB PIB+Benzene2 PS 4.6 PS+PIB+CH3 PS+PIB+Benzene3 PS

140 ( ) ( ) PIB PS-PIB PS PIB PS+PIB+CH3 CH PS PS+CH2 Φ PS T PS+PIB+Benzene3 Benzene PIB PIB+CH2 Φ PIB T 3 30 C ( ) ( )

141 M w PS PIB PS+PIB+CH3 Φ PS PS PS PS+PIB+Benzene3 Φ PIB PIB PIB PS+CH2 PIB+Benzene2

142 van t Hoff µ 0 i µ i van t Hoff Flory-Huggins Gibbs G 2 ( 2 ) ϕ i G = RT (N 0 + N i P i ) ϕ 0 ln ϕ 0 + ln ϕ i + h P i i=1 i=1 (4.1) N i ϕ i P i i h Flory-Huggins h h = χ 01 ϕ 0 ϕ 1 + χ 02 ϕ 0 ϕ 2 + χ 12 ϕ 1 ϕ 2 (4.2) 6, 7 χ ij i j 2 3 Koningsveld 8 h h = g 01 (ϕ 1, ϕ 2 ) + g 02 (ϕ 1, ϕ 2 ) + g 12 (ϕ 1, ϕ 2 ) (4.3) 3 g ij h 9 µ 0

143 χ µ 0 µ o RT ( = ln ϕ ) + χϕ 2 (4.4) P n ϕ = ϕ 1 + ϕ 2 P n 1 P n = ξ 1 P 1 + ξ 2 P 2 (4.5) ξ i = ϕ i /ϕ χ (4.1) h [ ] (h/ϕ) χ = ϕ T,p,ξ 1 (4.6) (4.4) Gibbs-Duhem i µ i µ i µ ( i = ln ϕ i + P i 1 1 ) ϕ RT P n [ ( ) χ P i {χϕ 0 ϕ + ϕ χ + (1 ξ i ) ξ i 0 T,p,ϕ ] } dϕ (i = 1, 2) (4.7) µ i = lim ϕ 0 (µ i RT ln ϕ i ) (i = 1, 2) (4.8) (4.7) i µ i h h = ϕ ϕ 0 χdϕ + ϕ 2 i=1 [ µ ξ i µ 0 i P i RT 1 ] + 1 P i (4.9) µ i µ 0 Flory-Huggins 1 2

144 138 4 T µ 0 3 π 11 π RT = µ 0 µ 0 V 0 RT 2 c i = + 1 M i 2 i= i=1 j=1 k=1 i=1 j=1 2 B ij c i c j 2 B ijk c i c j c k + (4.10) (4.4) 4.A χ χ = ξ 2 1χ 11 (ϕ 1 ) + ξ 2 2χ 22 (ϕ 2 ) + 2ξ 1 ξ 2 χ 12 (ϕ 1, ϕ 2 ) (4.11) χ ii χ 12 χ ii (ϕ i ) = E ii + E iii ϕ i + E iiii ϕ 2 i + (i = 1, 2) (4.12) E B E ij = 1 ( 1 V 0B ij 2 v i v j χ 12 (ϕ 1, ϕ 2 ) = E (E 112ϕ 1 + E 122 ϕ 2 ) + (4.13) ), E ijk = 1 3 ( 1 V ) 0B ijk, (4.14) v i v j v k v i i (4.11) χ ii (ϕ i ) (4.12) 0 i 2 2 χ 12 (ϕ 1, ϕ 2 ) χ 11 (ϕ 1 ) χ 22 (ϕ 2 ) 3 χ 12 (ϕ 1, ϕ 2 ) χ ij 2 (4.14) 1/2 1/3 V 0 (4.11) χ

145 (4.4) (4.7) 2 2.B (2.B.46) 4.B Rayleigh R 0 KV 0 ϕ = 1 + ϕ(1 ϕ) 1 P w ϕ(p w L + Y ) R 0 W X (4.15) K 2πn 2 /N A λ 4 0 (n λ 0 ) W X Y W = γ2 1P 1 ξ 1 + γ 2 2P 2 ξ 2 (1 ϕ) 2 (4.16) [ ( ) ] ξ 1 ξ 2 1 X = 1 + {( γ γ 1 2P 1ξ 1 + γ 2 2P 1 γ 2 ) 2 P 1 P ϕ ϕ 2ξ 2 P n 2( γ 1 γ 2 )( γ 1 P 1 γ 2 P 2 )ϕ ( γ 1 γ 2 ) 2 P 1 P 2 (1 ϕ) 2 ϕl +2( γ 1 γ 2 )( γ 1 ξ 1 + γ 2 ξ 2 )P 1 P 2 (1 ϕ)ϕq } +( γ 1 ξ 1 + γ 2 ξ 2 ) 2 P 1 P 2 S [ ( 1 Y = ξ 1 ξ 2 2(P 1 P 2 )Q P 1 P 2 1 ϕ + 1 ] )S P n ϕ L +P 1 P 2 ϕq 2 (4.17) (4.18) P w = P 1 ξ 1 + P 2 ξ 2 (4.19) γ i = γ i 2 ϕ j γ j (4.20) j=1 ( ) n γ i = (4.21) ϕ i ϕ k ( ) χ L = 2χ + ϕ (4.22) ϕ ξ 1

146 140 4 ( ) χ Q = (4.23) ξ 1 ϕ ϕ ( 2 ) χ S = dϕ (4.24) 0 γ i ξ 2 1 (4.15) χ χ (4.11) (4.15) 3 Rayleigh R 0 3 χ 11 (ϕ 1 ) χ 22 (ϕ 2 ) χ 12 (ϕ 1, ϕ 2 ) χ 11 (ϕ 1 ) χ 22 (ϕ 2 ) R 0 (4.15) χ 12 (ϕ 1, ϕ 2 ) R 0 χ 12 (ϕ 1, ϕ 2 ) χ 12 (ϕ 1, ϕ 2 ) 3 R 0 (4.15) R 0 χ 12 (ϕ 1, ϕ 2 ) 0 i 2 (4.15) L ii χ ii (ϕ i ) KV 0 γ i R 0 = 1 1 ϕ + 1 P i ϕ L ii (4.25) χ ii (ϕ i ) = 1 ϕ 2 i ϕi 0 L ii ϕ i dϕ i (4.26) 2 χ ii L ii (4.15) ϕ KV 0 ϕ = 1 P + 1 P [ (γ 1 P 1 ξ 1 + γ 2 P 2 ξ 2 ) 2 (1 2χ 0 ) R 0 ( ) χ0 2ξ 1 ξ 2 (γ 1 P 1 ξ 1 + γ 2 P 2 ξ 2 )(γ 1 P 1 γ 2 P 2 ) ξ 1 ( (ξ 1 ξ 2 ) 2 (γ 1 P 1 γ 2 P 2 ) 2 2 )] χ 0 ξ1 2 ϕ + (4.27)

147 P γ 2 1P 1 ξ 1 + γ 2 2P 2 ξ 2 (4.28) χ 0 1 χ(ϕ, ξ 1 ) = χ 0 (ξ 1 ) + χ 1 (ξ 1 )ϕ + (4.29) (Θ) 12 γ 1 P 1 ξ 1 + γ 2 P 2 ξ 2 = 0 (4.27) KV 0 ϕ = 1 P (ξ 1ξ 2 ) 2 (γ 1 P 1 γ 2 P 2 ) 2 ( 2 ) χ 0 R 0 P 2 ξ1 2 ϕ + (4.30) (4.2) χ χ = χ 01 ξ 1 +χ 02 (1 ξ 1 )+χ 12 ξ 1 (1 ξ 1 ) χ 12 (PS) (PIP) (CH) L 11 (ϕ 1 ) ( 2 Z 2Z = L 11 ) [ L 11 (ϕ 1 ) = 2 χ conc + ϕ Aϕ Bϕ 2 1 ( ) Θ χ conc = T 1 [ ( exp P 2/3 1 +(χ dil χ conc)r(p 1/2 1 ϕ 1 ) ( )( Θ T 1 P 1/2 1 )] ] 45 P 2 1 ) (4.31) (4.32)

148 ( ) ( ) 2 Θ Θ χ dil = T T 1 (4.33) A = 1.4P 1/3 1 (4.34) [ ( )] B = 7P 1/3 Θ 1 exp 18 T 1 (4.35) R(x) = exp( x 0.3x 3 ) (4.36) (M w = 53300) L 22 (ϕ 2 ) 2 L 22,dil (ϕ 2 ) L 22,conc (ϕ 2 ) L 22 (ϕ 2 ) = L 22,dil (ϕ 2 ) + L 22,conc (ϕ 2 ) (4.37) 2 L 22,dil (ϕ 2 ) = exp( 47ϕ 2 ) (4.38)

149 ( L 2,conc (ϕ 2 ) = ) ϕ 2 (4.39) T 4.8 (4.26) χ 22 (ϕ 2 ) χ 22 (ϕ 2 ) = ( ) ϕ 2 3 T [1 (1 + 47ϕ 2) exp( 47ϕ 2 )] (47ϕ 2 ) 2 (4.40) (PIP+CH PIP ) 4.9 PS+PIP+CH3 3 ξ PS (=ξ 1 )= ϕ KV 0 ϕ/ R 0 P 1 χ 12 (ϕ, ξ 1 ) ϕ ϕ 1

150 144 4 χ 12 (ϕ, ξ 1 ) = k 0 + (k 1 ξ 1 + k 2 ξ 2 )ϕ (4.41) k 0 k 1 k 2 (4.31) (4.40) (4.41) (4.15) KV 0 ϕ/ R k 0 k 1 k 2 KV 0 ϕ/ R 0 k 0 k 1 k 2 k 0 = 0.44, k 1 = T, k 2 = T (4.42) L ii (ϕ i ) χ ii (ϕ i ) χ 12 (ϕ, ξ 1 ) 4.C L ii (=2Z) (PS)+ (PIP)+ (CH)3 PS+ (PIB)+CH 3 χ ii χ 12 ϕ ξ 1 (=ξ PS ) χ 11 χ 22 χ 12 ξ PS (=ξ 1 ) 0 1 χ 12 ξ PS ϕ PIP+CH2 PIB+CH2 χ 22 PS+CH2 χ 11 ϕ=0 χ 12 χ 11 χ 22 χ 12 ξ PS = 0 PIP+CH2 PIB+CH2 χ 22 ξ PS PS+CH2 χ 11

151

152 µ 0(ϕ, ξ 1 ) = µ 0(ϕ, ξ 1 ) (4.43) µ i(ϕ, ξ 1 ) = µ i (ϕ, ξ 1 ) (i = 1, 2) (4.44) (4.44) σ i σ i = 1 ( ) ϕ i ln (i = 1, 2) (4.45) P i ϕ i (4.4) (4.7) σ i { ϕ [ ( ) ] } χ σ i = ln(1 ϕ) + χϕ + χ + (1 ξ i ) dϕ (4.46) ξ i 0 T,p,ϕ {X} X X X (4.43) (4.45) T 4 ϕ i ϕ i (i = 1, 2) (ϕ, ξ 1) (ϕ, ξ 1 ) 3 T (M w = 43900)+ (M w = 53300) C

153 ( )

154 ξ PS (= ξ 1 ) 4.4 ϕ ξ PS ξ PS 0.50 ϕ ξ PS (M w =53600)+ (M w =154000) (M w =53600)+ (M w =152000) T =30 C T =20 C ( ) N, N D

155

156 A T µ 0 π V 0 π = (µ 0 µ 0) (4.A.1) 3 π 11 π RT = 2 i= c i M i i=1 j=1 k=1 i=1 j=1 2 B ij c i c j 2 B ijk c i c j c k + (4.A.2) 1 2 V 0 π RT = ϕ P n i=1 j=1 2 i=1 j=1 k=1 2 D ij ϕ i ϕ j 2 D ijk ϕ i ϕ j ϕ k + (4.A.3) D ij = V 0B ij v i v j, D ijk = V 0B ijk v i v j v k, (4.A.4) (4.4) (4.A.1) (4.A.3) χ χ = ϕ 1 ( E ij ϕ i ϕ j + E ijk ϕ i ϕ j ϕ k + ) i=1 j=1 i=1 j=1 k=1 (4.A.5) E ij = 1 2 (1 D ij), E ijk = 1 3 (1 D ijk), (4.A.6)

157 4.A 151 E ij E ijk (4.A.5) χ = ξ1(e E 111 ϕ 1 + E 1111 ϕ ) +ξ2(e E 222 ϕ 2 + E 2222 ϕ ) +2ξ 1 ξ 2 [E (E 112ϕ 1 + E 122 ϕ 2 ) +2E 1112 ϕ E 1122 ϕ 1 ϕ 2 + 2E 1222 ϕ ] (4.A.7) ξ = χ χ 11 (ϕ 1 ) (4.A.7) χ 11 (ϕ 1 ) = E 11 + E 111 ϕ 1 + E 1111 ϕ (4.A.8) ξ 2 = χ = χ 22 (ϕ 2 ) χ 22 (ϕ 2 ) = E 22 + E 222 ϕ 2 + E 2222 ϕ (4.A.9) (4.A.7 ξ1χ 2 11 (ϕ 1 ) + ξ2χ 2 22 (ϕ 2 ) 2ξ 1 ξ 2 χ 12 (ϕ 1, ϕ 2 ) χ 12 (ϕ 1 ϕ 2 ) χ 12 (ϕ 1 ϕ 2 ) = E (E 112ϕ 1 + E 122 ϕ 2 ) +2E 1112 ϕ E 1122 ϕ 1 ϕ 2 + 2E 1222 ϕ (4.A.10) χ = ξ 2 1χ 11 (ϕ 1 ) + ξ 2 2χ 22 (ϕ 2 ) + 2ξ 1 ξ 2 χ 12 (ϕ 1, ϕ 2 ) (4.A.11)

158 B T p Rayleigh R 0 11 ( 2 2.B (2.B.46) ) R 0 = Kv M RT r i=1 j=1 r ( ) ñ m i T,p,m k ( ñ m j )T,p,mk ψ ij ψ (4.B.1) K (K = 2π 2 n 2 /N A λ 4 0) m i i v M v M = V 0(n 0 + r i=1 n ip i ) n 0 M 0 V 0 = M 0 (1 ϕ) (4.B.2) n i i ( ) ψ µ ij = ( µ i / m j ) mk ψ ij ψ ij 3 i ϕ i m i ϕ i = M 0 m i P i 1 + M 0 2 i=i m ip i (4.B.3) ϕ i m i = M 0 P i (1 ϕ i )(1 ϕ) (4.B.4) ( m i ) m k ( ) n m i ϕ j = M 0 P i ϕ j (1 ϕ) m i [( ) 2 ( ) ] = M 0 (1 ϕ)p i ϕ j ϕ i ϕ k ϕ j ϕ k T,p,m k = M 0 (1 ϕ)p i [( ) n ϕ i j=1 ϕ k 2 j=1 ( ) ] n ϕ j ϕ j ϕ k (4.B.5) (4.B.6) = M 0 (1 ϕ)p i γ i (4.B.7)