The statistical mechanics of rubber elasticity ro=(ojO 0) ∑G(,N)=3k2∑n+∑G(N、) =(xy2)l=1 ∑ △G(h,M)=7张可 N∴ Number of segment etween Crosslink △2=x2+y2+=2-(x2+y2+=2) 入2x2+2y+22-(x+y2+=2 y 航∑M(2)+2-)+(2-)3吹 h2=N2 y
The statistical mechanics of rubber elasticity 16 2 2 1 11 3 (, ) ( ) 2 n nn i ii c B i c i ii c G N kT G N N l ¦ ¦¦ h h 2 2 1 1 3 (, ) 2 n n i ii c B i i c G N kT N l ' ¦ ¦ ' h h 2 2 22 2 2 2 i i ii i i i 0 00 ' hxyz x y z 2 22 22 2 0 00 111 x iy iz i OOO xyz 22 2 2 22 2 2 2 xi yi zi i i i 0 0 0 0 00 OOO x y z xyz 2 22 2 0 00 0 1111 nnnn i ii i ¦¦¦¦ xyzh 2 2 22 2 0 0 00 0 11 1 1 1 3 3 nn n n i ii i n h ¦¦¦ ¦ xyz h 2 22 0 00 11 1 nnn i ii ¦¦¦ xyz 22 2 2 2 0 1 1 111 3 n ix y z i OOO n § · ' ª º¨ ¸ ¬ ¼© ¹ ¦ h h 2 2 0 c h Nl Nc : Number of segment between Crosslink
The equations of state of rubber elasticity △G=∑AG(,N)==m2(2+42+2-3) For uniaxial stress,n,=2,Ay=n=a, n 入3,2=1,2=1, (volume remains constant a(△G) a(△G) al 0(1 aI nkT knT22+=-3 a(2 al True stress f AonkBT λ)b 2-12(4/4 Modulus" increase in a linear manner with increasing Ao: cross section area of the unstrained rubber temperature. This is a typical entropy-elastic Vo: the volume of the rubber
The equations of state of rubber elasticity O O x OOO yzt O O 1 t 0 l l O , ,, () () TV TV TV G G f l l O O § · § ·§ · w' w' w ¨ ¸ ¨ ¸¨ ¸ © ¹ © ¹© ¹ w ww 2 2 0 0 12 1 3 2 B B l nk T f nk T ll l O O OO O w w § · § · §· § · ¨ ¸ ¨ ¸ ¨¸ ¨ ¸ w w © ¹ © ¹ ©¹ © ¹ 0 2 0 0 1 / // A nk TB f A l V OO O O §· § · ¨¸ ¨ ¸ ©¹ © ¹ For uniaxial stress, (volume remains constant) , 1 OOO xyz , 1 2 OOt , , 2 0 1 B nk T V V O O § · ¨ ¸ © ¹ True stress “Modulus” increase in a linear manner with increasing temperature. This is a typical entropy-elastic. A0 : cross section area of the unstrained rubber V0 : the volume of the rubber True stress 17 222 1 1 (, ) 3 2 n ii c B x y z i G G N nk OOO ' ' ¦ h
The equations of state of rubber elasticity True stress(真应力) ORT Nominal stress(习用应力) Rubber with high crosslinking density, i.e. low Me behave stiff a(1 nk,T nkrI n PRT M Small strain limit :2-1 Note: 2=lll=1+a RT ORT E→>0 solids Ea=E E( E=3 M
The equations of state of rubber elasticity 0 00 0 1 1 mole c a a a c cc n n MN m N N V V M VM M § ·§ · § ·§ · U ¨ ¸¨ ¸ ¨ ¸¨ ¸ © ¹© ¹ © ¹© ¹ ¸ ¹ · ¨ © § O O U V 2 1 Mc RT Rubber with high crosslinking density, i.e. low Mc , behave stiff 2 1 c RT M U V O O § · ¨ ¸ © ¹ True stress (ⵏᓄ࣋( Nominal stress (Ґ⭘ᓄ࣋( 2 2 0 0 12 1 3 2 B B l nk T f nk T ll l O O OO O w w § · § · §· § · ¨ ¸ ¨ ¸ ¨¸ ¨ ¸ w w © ¹ © ¹ ©¹ © ¹ 0 f A V 0 0 solids 0 3 1 1 c RT l l EE E M l H U V O VH O o Small strain limit: Oo1 3 c RT E M U Note: O=l/l0=1+H 18
Gel Swelling Swelling of rubbers in solvents AF=I 3oRTV NkT(32-3)= 2/3 2M 入, a△ F aAFm,OMF102 n22 -1/3 △L1= do, an0 入=,=22==训4 △ 。+nV Mhe= pRT VoA M (22: volume fraction of polymer Flory-Huggins solution theory component in swollen gel) (91+6 ≈RT(m(1-.)++x92) At swelling equilibrium △41=△41a1+△1m=0 Isotropic swelling of a network polymer 1-g)+g+xg2+,1g=0 ener equation
Gel Swelling ¾ Swelling of rubbers in solvents Isotropic swelling of a network polymer 1/3 1, 1 2 el c RT V M U ' P I 2 1, 2 2 2 2 22 2 1 ln(1 ) 1 ln(1 ) mix RT x RT P I I FI I I FI § · § · ' ¨ ¸ ¨ ¸ © ¹ © ¹ | 0 'P1 'P1,el 'P1,m * * *2 *1/3 1 22 2 2 ln(1 ) 0 c V M U I I FI I 0 2 1 xyz V V V OOO O I 3 OOOO O xyz V 1/3 O I2 2 2/3 0 2 1 3 33 1 2 2 el c RTV F NkT M U O I ' (I2 : volume fraction of polymer component in swollen gel) ¾ Flory-Huggins solution theory: ¾ At swelling equilibrium (Flory-Rehner equation) 0 2 0 11 V V nV I 2 1 1 1 21 0 F F F m el nn n I P I w' w' w' w ' w w ww 19
Deviations from classical statistical theories Crosslinking of a polymer with finite MW indicating the formation of loose chain ends “ Chain defects” in network:(a) permanent physical crosslink;(b) temporary physical crosslink;(c)intramolecular crosslink Segment vectors do not follow Gaussian statistics when rubbers are highly stretched At high extension ratio, crystallization may occur in rubbers
Deviations from classical statistical theories ¾ Crosslinking of a polymer with finite MW indicating the formation of loose chain ends. ¾ “Chain defects” in network: (a) permanent physical crosslink; (b) temporary physical crosslink; (c) intramolecular crosslink. (a) (b) (c) ¾ Segment vectors do not follow Gaussian statistics when rubbers are highly stretched. ¾ At high extension ratio, crystallization may occur in rubbers. O=l/l0 20