1 高分子鎖の統計的性質� 不可転載・不可他目的使用� 分子内相互作用と理想鎖� lj v(ri, j ) 遠隔相互作用� € € j� 2 ボンドベクトル� j-1� ρ(x j ;x j−1 ) ≡ 1 δ (li − a)e−u(φ )/ kB T 4 πa 2 近接内部回転� 末端間ベクトルが値Rを取る確率:� Z(R,T) = € ∫ ... ∫ dx ...dx 1 n e−[U ({x})+V ({x})]/ kB T ∏ ρ(x j ;x j−1 )δ (∑ l j − R) n−1 j=1 U = ∑ u(φ j ) + ... j j V = ∑ v(ri, j ) i< j € € 1 3 f� R € 固定� S = kB lnW (R) = nk B ln ω (R) ∂ (−TS(R)) f= ∂R € 張力と伸長の関係� 理想鎖� Gauss鎖とLangevin鎖� U =V =0 fa R = L na kB T ランダムフライト鎖� n € W (R) = 4 ∫ ... ∫ ∏ ρ(l )dl dl ...dl i 1 2 L(x) = coth x −1/ x n i=1 R / na = exp− ∫ L−1 (y)dy 0 3R 2 = exp− 1+ ...) 2 ( 2na € 伸びきり効果� 0.3� € Gauss分布� 2R 2 3 3 / 2 P0 (R) = exp − 2 2 2πna 2na φ 0 (R) = −TS = € 近似式� fa 2 r˜ 2 τ ( r˜ ) ≡ = 3˜r{1+ A } kB T 3 1− r˜ 2 3k B T 2 R 2na 2 < R 2 > 0 = na 2 fa R =3 kB T na 線型バネ� A=1 Langevin鎖� € € € 2 直接測定例� 5 AFM (W.Zhang et al, J. Phys. Chem. B104, (2000) 10258)� 250 FORCE f(R/na)a/kBT τ ( r˜ ) ≡ 200 150 T = 300 fa 2 r˜ 2 = 3˜r{1+ A } kB T 3 1− r˜ 2 PNIPAM PEO A=0.2 A=2.2 € 100 50 0 0.2 0.4 0.6 0.8 1.0 END-TO-END DISTANCE R/na MD (D.Bedrov and G.D.Smith, J.Chem.Phys. 118, (2003) 6656) AFM(W.Zhang and X.Zhang, Prog. Polym. Sci. 28, (2003) 1271) ガウス鎖の性質� 任意のモノマー対(i,j)の分布関数� 3 P0 (rij ) = 2 2 π a i− 6 2 3rij exp − 2 j 2a i − j rij 平均2乗末端間距離と慣性半径� € < R 2 > 0 = na 2 1 1 2 < s2 > 0 ≡ ∑ < si >= na 2 n i 6 € i� R 相関関数� € j� 1 −iq⋅ r g˜ 0 (q) ≡ ∑ < e ij > = nD(< s2 > q 2 ) n i, j D(x) ≡ 2 −x x (e −1+ x ) ≈ 1− 3 + ... (x << 1) x2 ≈ 2x (x >> 1) € Debye関数� € 3 内部回転ポテンシャルの影響� C∞ = 平均2乗末端間距離� < R 2 >= ∑ < l i ⋅ l j >= Cn na 2 1+ cosθ < cos φ > (1− cosθ )(1− < cos φ >) lp ≡< R ⋅ l1 /a >= C'n a € x = L /lp Kuhnステップ数� < R 2 >= L2 D( L /lp ) ≈ 2l p L(L >> l p ), L2 (L << lp ) L = na, lp = const € 岡の式� € 持続長� € 1+ cosθ 1+ < cos φ > ⋅ 1− cosθ 1− < cos φ > C'∞ = i, j 7 n → ∞, a → 0, θ → 0 D(x) ≡ 固い鎖(stiff chain)� 2 −x € (e −1+ x ) x2 € € 排除体積効果(遠隔相互作用の効果)� P(r) = 1 r f , RF d RF f (x) ~ e −xδ 8 r 2 ∫ g(r)4πr dr = n r 0 (δ = ν −1) 平均2乗末端間距離と慣性半径� € € € ν a( n r ) = r 1/ 2 RF ≡< R 2 > F = an ν ν = 3/5 1/ 2 < s2 > F ≡ Can ν Flory則� 相関関数� € g(r) ≈ R € 5/3 nr 1 r ≈ 3 3 4 πr /3 r a Floryの考え方� φ (R) = g˜ (q) ≈ ( aq) € € −5 / 3 n 2 3k€ BT R 2 + v 0 d R d = min 2 R 2na R ≈ an ν , Edwards則� ν = 3/(d + 2) € 4 スケーリング則と熱ブロブ模型� thermal blob ξ = agν θ ガウス鎖のブロブ� 9 2 τ gτ ≈ 1, v 0 ≈ τ , τ ≡ 1− Θ /T ξ3 gτ ≈ 1/ τ 2 ( € RF = agτ € νθ ν ) gn ブロブの膨潤鎖� ν = 3/5 € = aτ 1/ 5 n 3 / 5 τ Rθ = an ν θ , θ = 1/2 € ( RG = agτ ν c = 1/3 νθ ν c ) gn = aτ −1/ 3 n1/ 3 τ ブロブの最密充填� ガウス鎖のブロブ� € 高温コイル-グロビュール転移 RH /RG = 0.69 € c=1.264x10-3 wt% Mw= 8,400,000 Globular aggregation 10 Mw=615,500 Coil-globule transition spinodal 2Φ? RH /RG = 1.50 ○: RG ●: RH RH /RG = 1.50 (C.P.=1.58)� € (S.Fujishige et al, J.Phys.Chem. 93 (1989) 3311) cloud points € (R.G. de Azevedo et al, Fluid Phase Eq. 185 (2001) 189) 5 グロビュール構造とハミルトン酔歩� 水素結合� van der Waals� 11 疎水性凝集� single-chain network core-shell structure n W (n) ≈ ω H S0 = nk B ln ω H Hamilton Walk € コイル-へリックス転移(ZIMM-BRAGG) 12 …ccchhhcccchhhhcccccchhhhhccc… σ sss σ σ ssss σ Z(T,n) = ∑ { j} σ sssss σ (n −Σς jς )! jς ∏η ς n (∏ jς !)(n − Σς jς − Σ jς )! ς =1 = max ς ης = σ sς (ς ≥ 1) σ = 0.01 1.0 € t t, θ, ν, ζ 0.8 0.6 θ ≡ ∑ ς jς / n jς / n = (1− θ − ν ) ης t ς ν ≡ ∑ jς / n n = 100 σ = 0.01 ζ θ 0.4 0.2 t ≡1− ν /(1− θ ) ν 0.0 -2 -1 0 TEMPERATURE € € ln s 1 2 ≈ ε H /k B T € 6
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