体积排除修正 (1)_2(h) 链包含的体积h3链段的体积v=B h XXV 个链段占有的体积分数v/3如2 h 其他链段不与之发生重叠的概率(1-h3) 整条链两两链段都不发生重叠的总概率 g2()=l1 h 0 第一个第二个第三个 ≈ exp r =0 ex\Yo Inl 1 ≈exp =ex h3 p 2h23
ࠔҒୣܭѼऴ 25 䬮वਜ਼Ⲵփ〟 h 3 䬮⇥Ⲵփ〟 vc=l 3 (1) :(h) m n h ањ䬮⇥ঐᴹⲴփ〟࠶ᮠvc /h 3 ަԆ䬮⇥ноѻਁ⭏䟽ਐⲴᾲ⦷(1-vc /h 3 ) ᮤᶑ䬮єє䬮⇥䜭нਁ⭏䟽ਐⲴᙫᾲ⦷ 1 33 3 0 1 1 1 2 ... 1 x cc c i vv v h i hh h : § ·§ · § · | ¨ ¸¨ ¸ ¨ ¸ © ¹© ¹ © ¹ 1 1 3 3 0 0 exp ln 1 exp ln 1 x x c c i i v v i i h h ª ºª º §· §· | « »« » ¨¸ ¨¸ ¬ ¼¬ ¼ ©¹ ©¹ ¦ ln 1 iu iu V V § · ¨ ¸ | © ¹ 1 2 3 3 0 exp exp 2 x c c i v vx i h h ª º ª º | « » « » ¬ ¼ ¬ ¼ ¦ ㅜањ ㅜйњ ㅜҼњ 2 3 c xv h I
溶液中真实单链构象的径向分布函数Wh2x (2)E()H=mxVm≈h n kT V E(h)hi x(1-)=-x2-x2=xx conSt -x W(h,x)=Wo(h,x)2(h)exp/-(h) kT 链内两体 相互作用 3h 能的总和 W(h, x)oc h' expl 2x122h3 包括了链段的排 斥体积和溶剂 链段相互作用 当x=1/2,理想高斯链 当x<1/2,???
ⓦ⏢ѝⵏᇎঅ䬮ᶴ䊑Ⲵᖴੁ࠶ᐳ࠭ᮠW(h,x) (2) E(h) 2 2 2 2 3 3 , exp 1 2 2 2 c h xv W hx h xl h F ª º v « » ¬ ¼ ᖃF=1/2, ⨶ᜣ儈ᯟ䬮 26 2 3 c xv h I ᖃF 1/2, ??? 䬴њ։ ӈ֒⭞ 㜳Ⲻᙱૂ ऻᤢҼ䬴⇫Ⲻᧈ -ࡸ12ૂ։〥ᯛ 䬴⇫ӈ֒⭞ െ exp ݄ ߗ ݔ ,݄ ܹ = ݔ ,݄ ܹ ݄ തܧ ݇ܶ 1 2 m s H V kT V FI I E h kT 3 2 2 1 c h v F II 3 3 2 2 2 c c h h v v FI FI 3 22 6 c c h xv x v h F F 2 3 c x v const h F 3 V h m | V v s c
1. Polymer chain in good solvents- method I 3h x x)∝h2exp2xl22h3 aw(h 0 h aw(h,x 3h = exp 3h 3x Vo ah 2xF-2(1-2x)2h+h2 x 3h23 h h 2x2 )()16P Wa(h-x)的极值ho=(2xP/3)2 最可几尺寸≈根均方尺寸 hxl。(1-2x)∝x (1-2x)7 7=1-2x l∝x' u=v X)=VT X<12,v3/5 x12,v=12x12,v=
1. Polymer chain in good solvents – method I , 0 W hx h w w 2 2 2 3 , 3 exp 1 2 2 2 W hx c h xv h xl h F w ª º « » w ¬ ¼ *2 2 2 *3 3 3 1 12 0 2 4 c h x v xl h F W0 (h,x)Ⲵᶱ٬h0 *=(2xl2 /3)1/2 5 1/ 2 3 * * ** 3 0 0 9 6 1 2 16 c h h v x hh l F §·§· ¨¸¨¸ ©¹©¹ F Q F , Q=1/2 F!Q "" 1/5 1/5 1/2 * * 0 3 3 1 2 1 2 c c x v v hh x l l l Q F F § · ª º v v ¨ ¸ « » © ¹ ¬ ¼ 27 2 2 2 2 3 3 , exp 1 2 2 2 c h xv W hx h xl h F ª º v « » ¬ ¼ ᴰਟࠐቪረ|ṩ൷ᯩቪረ 1 2 c c uv v F W W F 1 2 2 2 2 4 3 3 2 12 2 c h xv h h xl h F ½ ª º ® ¾ « » ¯ ¿ ¬ ¼ v 1/5 v x W
1.Polymer chain in good solvents-method II 3h 单链自由能:G~-k1n(h,x) oc exp 2x22h3 X)+ <1/2 2x122h G/kt 2 exch G second virial coefficent conformation two body interaction: F品 ton volume hxx3(1-2x) 1/5 solvent-segment Interaction Two body interactions are important in good solvents
1. Polymer chain in good solvents - method II Two body interactions are important in good solvents! ~ ln G kT B ) 2 2 3 2 , exp 1 2 ... 2 3 2 c x v x l h h h x F ª º ) v « » ¬ ¼ 2 2 3 2 ~ 12 2 3 ... 2 c B x v G kT h h xl F § · ¨ ¸ © ¹ 0 G h w w F<1/2 two body interaction: excluded volume repulsion solvent-segment interaction / G kTB second Virial coefficent 2 3 c xv h I conformation entropy + 28 অ䬮㠚⭡㜭˖ 1/5 3/5 h x v 1 2F ൎ 3 2ݔ݈ଶ݄ + 1 2 െ ߯ ݒ ߶ଶ ଶ + ݓ߶ଶ ଷ ݄ ଷ
0 Temperature 0 Solvents X=1/2 →h2~M x (z-2)△2_T(a-02) RT o Solution 2(2-2)AE1227 O Temperature k R
4Temperature & 4Solvents F h 2~1 2 12 1 2 ( 2) 1 2 Z Vs kT RT T T T H G G F ' 4 Solution 29 2 12 1 2 2 2 2 s Z V T k R T H G G ' 4 Temperature T TT