Flory- Huggins theory( Lattice model(格子模型) 就找体系中有N个溶剂分子+N2个链段数为的高分子 合时8总格子数.NN+xN2肉+x角+N 回a 已有个高分子放入,剩下Nxj个空格,求第计1个高分子的放 置方式W1??? 放置第计1个高分子的第1个链段的概率 2放置第计1个高分子的第2个链段的概率 Z(N- j-1)/N 3放置第汁1个高分子的第3个链段的概率 (∠z-1)(Nxj2)N x.放置第x个链段的概率 (2-1)(-xj-x+1)/N
Flory-Huggins theory (Lattice Model (Ṭᆀ⁑ර)) փ㌫ѝᴹN1њⓦ࠶ࡲᆀ + N2њ䬮⇥ᮠѪxⲴ儈࠶ᆀ ᙫṬᆀᮠ: N=N1+xN2 ᐢᴹjњ儈࠶ᆀ᭮ޕ ,࢙лN-xjњオṬ, ≲ㅜj+1њ儈࠶ᆀⲴ᭮ 㖞ᯩᔿWj+1 ??? 1. ᭮㖞ㅜj+1њ儈࠶ᆀⲴㅜ1њ䬮⇥Ⲵᾲ⦷ N-xj 2. ᭮㖞ㅜj+1њ儈࠶ᆀⲴㅜ2њ䬮⇥Ⲵᾲ⦷ Z 3. ᭮㖞ㅜj+1њ儈࠶ᆀⲴㅜ3њ䬮⇥Ⲵᾲ⦷ (Z-1) …. (Z-1) x. ᭮㖞ㅜxњ䬮⇥Ⲵᾲ⦷ 15 (N-xj-1)/N (N-xj-2)/N (N-xj-x+1)/N 1 1 1 2 N N xN I 2 2 1 2 xN N xN I
Entropy of mixing from FH theory Wm=(N-x)×2 Z-1 N rn segment Z-1 ≈ W N X/-x 总方式 N2-1 (x-1) ∏W (N=x)( (N=2x)∠(N-xN2) N 1(Z-1 N N,st chain (N-xN2) Entropy of solution: kIn22=k N2(x-1)In N +InN|-In N,1-In(N-xN,)
2 2 1 1 2 1 0 22 2 1 11 ! ! ! ... ! ! ! 2! ! N x N j j Z N N x N xN x W N N N N x N x N xN § · : 3 ¨ ¸ © ¹ 1 st 2 st N2 st chain Entropy of mixing from FH theory 1 123 1 11 1 j N xj N xj N xj N xj x W N xj Z Z Z Z NNN N § ·§ ·§ · § · u u u ¨ ¸¨ ¸¨ ¸ ¨ ¸ © ¹© ¹© ¹ © ¹ x th 1 segment st 4 th 3 rd 2 nd Z | Z – 1 2 ( 1) 2 2 11 ! ! ! N x Z N N N N xN § · ¨ ¸ © ¹ 2 22 1 ln ( 1)ln ln ! ln ! ln( )! solution Z S k k N x N N N xN N : ª º § · « » ¨ ¸ ¬ ¼ © ¹ Entropy of solution: ᙫᯩᔿ 16 1 1 1 ! ! x j Z N xj W N N xj x § · ¨ ¸ © ¹
Entropy of mixing from FH theory Using Stirlings approximation (Inr! a xlnr-x), we have: Z-1 kIN Ir +N In N2(x-1)n N,+xN N,+xM Entropy of the pure solvent Z-1 kN, Inx+(x-D)Ir (N,=0)and Solvent =0 and pure polymer: Therefore mning solution-( 4、Nx+NNM+xNy N,+xN where D=-N 小2 xN2__Xn2__xN, V Nxn,v N+xN n,+xi N+xM,n,+ ASmuving=-KN, n +N2 In] =-Rn, Ind,+n2, Ind2] k In4+Ind2
Entropy of mixing from FH theory » ¼ º « ¬ ª e Z N x N xN N N N xN N Ssolution k N 1 ln ln ( 1)ln 2 1 2 2 2 1 2 1 1 Using Stirling’s approximation (lnx! | xlnx –x), we have: Ssolvent 0 ( ) ' SS SS mixing solution solvent polymer Entropy of the pure solvent and pure polymer: 2 1 ln ( 1)ln polymer Z S kN x x e ª º « » ¬ ¼ (N1 = 0) and Therefore, 1 1 1 1 1 1 21 2 s s m m N n N V NnV N xN n xn V V I 2 2 2 2 2 1 21 2 s s m m xN xn xN V Nxn V N xN n xn V V I where 17 1 2 1 2 12 12 ln ln N xN kN N N xN N xN ª º « » ¬ ¼ ' S kN N mixing > 11 2 2 ln ln I I @ Rn n > 112 2 ln ln I I @ 2 11 2 ln ln m s V k V x I II I ª º « » ¬ ¼ 0 conf ' S
Free Energy of FH Theory Huggins Enthalpy: AH miting KTXM 2=RTxn, -kTxp% N Nx, n, L x,n, x2N2v Nx2n,y x, M+x2 n,txn x, N+x,, x,,+x,n Gibbs Free Energy△Gmg=△Hmw-7△s △ G=k7(Nm+N2ln+xxN)分子数 RT(n1ln+n2n+xxn1)摩尔数 kT 呐+加n+x通式 △F △Gm=(鸟 6+In+2p 2 kT For Polymer Solutions x,=l
Free Energy of FH Theory 12 12 12 = m mixing s V H kT N RT n kT V Huggins Enthalpy: ' F F II I FI Gibbs Free Energy 1 1 2 2 112 RT n n x n ln ln II I F 1 1 2 2 1 12 ln ln ' G kT N N x N mixing I I I F 1 2 1 2 12 1 2 ln ln m s V kT Vx x I I I I II F § · ¨ ¸ © ¹ ᪙ቄᮠ ᆀᮠ࠶ 䙊ᔿ 1 1 11 1 1 11 1 1 1 2 2 11 2 2 s s m m xN xn x N V Nx nV xN x N xn xn V V I 2 2 22 2 2 22 2 1 1 2 2 11 2 2 s s m m xN xn x N V Nx n V xN x N xn xn V V I For Polymer Solutions x1=1 ' ' ' G H TS mixing mixing mixing s m m m V G F V kT ' ' 1 2 1 2 12 1 2 ln ln x x I I I I II F § · ¨ ¸ © ¹
Chemical potentials(化学位): =RT Inp,+ (for solvent) T P n, △G △F (4Gm-中2an)=Rr(4m92) kT 2 n中 +x中2 a(4G RT[n-(x-1).十x9 (for polymer) T P V oAG aAFm 0中1 =RT(AFm-中1 0中1 In the case of 2 <<1, InP,=In(1-p2)*-P2-92 RT 2+x-2 2+yx=12,有热效应的“理想溶液”?? x<1/2, good solvent x=1/2, theta O solvent x>1/2, poor solvent
Chemical potentials (ॆᆖս): | 2 1 2 2 2 2 1 lnI ln 1 I I I 2 3 1 2 22 1 1 2 RT w x 'P I F I I ª º § · « » ¨ ¸ ¬ ¼ © ¹ 2 2 1 1 22 1 , , 1 ln 1 m T Pn G RT n x ' 'P I I FI ª º w ª º § · « » « » ¨ ¸ ¬ ¼ w ¬ ¼ © ¹ 1 2 2 2 11 2 , , ln 1 m T Pn G RT x x n ' 'P I I FI ª º w ª º « » ¬ ¼ w ¬ ¼ (for solvent) (for polymer) In the case of I << 1, F < 1/2, good solvent F = 1/2, theta 4solvent F > 1/2, poor solvent F=1/2, ᴹ✝᭸ᓄⲴĀ⨶ᜣⓦ⏢”??? 19 s m m m V G F V kT ' ' 2 1 1 2 12 ln ln x I II I I FI § · ¨ ¸ © ¹ = ഥೞ ܩ߂ െ ߶ଶ డ௱ீ డథమ =ܴܶ ܨ߂ െ ߶ଶ డ௱ி డథమ = ܸത ௦ ܸ ܩ߂ െ ߶ଵ ܩ߂߲ ߲߶ଵ = ܴܶ ܨ߂ െ ߶ଵ ܨ߂߲ ߲߶ଵ