Outline. (strain) !!! Rouse ! PASTA

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1 Outline!!! Rouse! : Zimm ( )!!! PASTA (strain)

2 (shear deformation) x! h! h! h shear strain! γ = x h x/h! x/h h 1! x 1! x 1 h 1 = x x! h! x 1! h 1! h (uniaxial elongation) L 0! ΔL! Cauchy strain! ε C = ΔL L 0 = L L 0 L 0! ΔL/L L! ε C = L 0! ΔL! L 0 ΔL! L! L

3 L 0! L 0! ΔL! ΔL! 1 L! L! L 0! 1 L 0! ΔL 1+ =3L 0! L ΔL 1 =L ε 0! C1 = 0! = 1! L0! 3L ε C1+ = 0! = 3! L0! ε C = ΔL L 0 = ΔL L 0 ΔL! L 0! εc L 0! = = 1! L 0! ΔL =L 0! ε C1 + ε C L! 4L 0! L 0! Hencky strain! ε = ln L L 0 L! ε = ln 1 L 0! ε L 0! 1 = ln = ln! L 0! 1 4L ε 1+ = ln 0! = ln 4! L0! = ln! ln log e natural logarithm! L 0! ε 4L = ln 0! = ln! L 0! = ε 1 + ε ln xy = ln x + ln y 4L 0!

4 1 L 0! L 1! L! ε 1 = ln L 1 L 0 ε = ln L L 1 ε 1+ = ε 1 + ε 1 ε 1+ = ln L L 0 ε 1 + ε = ln L 1 L 0 + ln L L 1 = ln L 1 L 0 L L 1 = ln L L 0 y = e x ε C = ΔL L 0 ΔL = L L 0 ε = ln L L 0 = ln L 0 + ΔL L 0 ε = ln( 1 + ε C ) ε C = e ε 1 x = ln y = ln 1+ ΔL L 0 1+ ε C = e ε Taylor e ε = 1 + ε + 1! ε + 1 3! ε 3 + ε 1 e ε 1+ ε ε C = e ε 1 ( 1+ ε) 1 = ε ε 1 ε C ε Cauchy Hencky Cauchy strain ε c ε C = e ε Hencky strain ε ε C = ε D 0! D! (Poisson s ratio)! z-x,y- L 0! L! ε = ln L L 0 > 0 ( L > L 0 ) ε = ln D D 0 < 0 ( D < D 0 ) ν = ε ε ν > 0

5 L 0 D 0 = LD ε = ln L = ln D 0 L 0 D L = D 0 L 0 D = ln D 0 D = ln D D 0 = ε L L 0 L = L 0 e ε = L 0 (1+ ε C )! λ L L 0 = e ε = 1 + ε C ε = lnλ ν = ε ε = ν 1 y = ln x x = e y ln xy = ln x + ln y ln x < 0 0 < x < 1 ln x > 0 x > 1 lne = 1 ln 1 = 0 strain rate! = [1/s]! h! x! shear rate! γ dγ dt γ = x γ = v w h h [1/s]! x! v w x dx dt

6 1cm 1mm/s h = 1 [cm]! x = v w t = 5 [mm] h! v w = 1 [mm/s] γ = v w h = 1 [mm/s] = 1 [mm/s] 1 [cm] 10 [mm] γ = 0.1 [1 / s]! t = 5 ( ) ( 5 [s]) = 0.5 γ = γ t = 0.1 [1/s] h! 0! y! v w v x (y) x! γ = v w γ = v x y y y v x (y) = ay h a v x (h) = v w v x (y) = v w h y v x (y) = γ y = ε dε dt ε(t) = ln L(t) ε(t) = 1 dl(t) L 0 L(t) dt stress ε L(t) = L 0 e εt 1 4!

7 S! F! S! S! F! F! : S! : F! F! F! F! σ = F S shear stress! = [Pa] = [N/m ]! σ = F S σ = F S = F S σ = F S S! F! σ E = F S S S true stress S 0! engineering stress! Hencky true strain! Cauchy engineering strain! σ xy = F x S y y! x σ yz, σ zx, σ yx, σ zy, σ xz σ xy = σ yx, σ zx = σ xz, σ zy = σ yz σ zx = σ xz = 0, σ zy = σ yz = 0 σ xy = σ yx 0 σ xx 0 y! x! S y! F x!

8 y! F! σ xx = x x σ yy, σ zz F! p x! σ xx = σ yy = σ zz = p x F σ xx = F x S p σ E = F x S = σ xx σ yy! t σ(t) γ(t) σ(t) = Gγ (t) σ E (t) = Eε(t) G: E: G, E [Pa]! shear modulus! Young s modulus! E = G( 1+ ν) ν = ν = 1/ E = 3G! t σ(t)! γ (t) σ(t) = η γ (t) σ E (t) = η E ε(t) η: shear viscosity! η E : (viscosity) [Pa s]! η E = 3η

9 !! ~00 GPa!! ~80 GPa! ~3 GPa!!! ~1 MPa! stress relaxation ~ 10-3 Pa s = 1 mpa s! γ(t)! γ 0! 0! σ(t)! 0! t! t! t=0 γ 0 γ 0! γ 0! σ(t) γ 0! γ 0 σ(t) σ(t) γ 0 G(t) σ(t) γ 0 γ 0! relaxation modulus!

10 G(t)! 0! t! G(t)! 0! t! γ 0! γ(t)! σ (t)! 0! T! t! γ (t) = γ 0 cosωt ω = π T σ(t) = Gγ (t) = Gγ 0 cosωt 0! t! T = γ 0 = σ(t) = η γ (t) = ηωγ 0 sinωt σ 0! γ(t) or σ(t)! δ γ (t) = γ 0 cosωt 0! T! t [s]! δ& σ(t) = σ 0 cos( ωt + δ ) π! ωt [radian]! δ = 0! δ = π/! 0 < δ < δ/!

11 ( ) σ(t) = σ 0 cos ωt + δ ( ) = σ 0 cosδ cosωt sinδ sinωt [ ] σ(t) = γ 0 G (ω)cosωt G (ω)sinωt G (ω) σ 0 cosδ γ 0 G (ω) σ 0 sinδ γ 0! storage modulus!! loss modulus! [Pa]! γ 0! γ 0 σ 0 G (ω), G (ω) γ 0 ω σ(t) = Gγ (t) = Gγ 0 cosωt σ(t) = ηγ (t) = ηωγ 0 sinωt [ ] σ(t) = γ 0 G (ω)cosωt G (ω)sinωt (loss tangent)! tanδ G (ω) G (ω) G (ω) = G, G (ω) = 0 δ = 0 G (ω) = 0, G (ω) = ηω δ = π / tanδ < 1: G (ω) < G (ω) tanδ > 1: G (ω) > G (ω) ω G' (ω), G"(ω)! log G(ω)! G (ω) G (ω) e iθ cosθ + isinθ e i ( θ 1 +θ ) = e iθ 1 e iθ Euler -1! i! sin θ! 0! e iθ! θ cos θ! 1! tan δ = 1! tan δ > 1! tan δ < 1! log ω! d dt eiωt = iωe iωt -i!

12 ( )( cosθ + isinθ ) e iθ 1 eiθ = cosθ 1 + isinθ 1 = cosθ 1 cosθ sinθ 1 sinθ +i( sinθ 1 cosθ + cosθ 1 sinθ ) = cos( θ 1 + θ ) + isin( θ 1 + θ ) d dt eiωt = e i ( θ 1 +θ ) ( cosωt + isinωt ) = d dt = ω sinωt + iω cosωt = iω(cosωt + isinωt) = iωe iωt γ (t) = γ 0 cosωt γ (t) = Re γ * (t) ( ) σ(t) = σ 0 cos ωt + δ γ * (t) γ 0 e iωt σ * i(ωt +δ ) (t) σ 0 e σ(t) = Re σ * (t) σ * (t) = σ 0 e i(ωt +δ ) = σ 0 e iδ e iωt = σ 0 e iδ γ 0 e iωt γ 0 σ * (t) = G * (ω)γ * (t) G * (ω) σ 0 γ 0 e iδ G (ω) σ 0 cosδ G (ω) σ 0 sinδ γ 0 γ 0 G * (ω) σ 0 γ 0 e iδ = σ 0 γ 0 ( cosδ + isinδ ) G * (ω) = G (ω) + i G (ω) σ * (t) = G * (ω)γ * (t) = G * (ω)γ 0 e iωt = ( G (ω) + i G (ω))γ 0 cosωt + isinωt ( ) [ ] σ(t) = Re σ * (t) = γ 0 G (ω)cosωt G (ω)sinωt

13 Maxwell σ 1 (t) = Gγ 1 (t) γ 1 (t) = 1 G σ 1(t) = 1 G σ(t) γ 1 (t) = 1 G σ(t) σ 1 σ σ (t) = η γ (t) γ (t) = 1 η σ (t) = 1 η σ(t) G! η& σ 1 σ σ γ 1 γ σ 1 (t) = Gγ 1 (t) σ (t) = η γ (t) G! η& γ 1 γ γ σ γ (t) = γ 1 (t) + γ (t) σ(t) = σ 1 (t) = σ (t) γ (t) = γ 1 (t) + γ (t) dσ(t) dt γ (t) = γ 1 (t) + γ (t) + 1 dγ (t) σ(t) = G τ dt = 1 G σ(t) + 1 η σ(t) G τ η G γ 1 (t = +0) = γ 0 γ (t = +0) = 0 γ(t)! 0! σ(t = +0) = σ 1 (t = +0) = Gγ 1 (t = +0) = Gγ 0 t > 0 γ (t) = γ 0 dσ(t) dt + 1 τ σ(t) = 0 dσ(t) dt t /τ t > 0 σ(t) = Gγ 0 e γ 0! dγ (t) = 0 dt = 1 τ σ(t) t! t /τ σ(t) e G(t) = σ(t) γ 0 = Ge t /τ G(t) = Ge t /τ t > 0 0 t < 0 τ = η G G(t)! G! 0! η = Gτ G(t)! G! G/e! 0! τ& G(t)! t! t! 0! t!

14 γ 1 (t)! γ 0! γ (t)! γ 1 (t)! 0! t! γ (t)! γ 0! σ 1 (t) = Gγ 1 (t) =! σ(t) =! σ (t) = ηγ (t) log G(t) 10G G 10 1 G 10 G 10 3 G 10 4 G G(t) 10 5 G 10 3 τ 10 τ 10 1 τ τ 10τ 10 τ 10 3 τ log t G(t) G 0.8G 0.6G 0.4G 0.G G(t) 0! t! 0 0 τ τ 3τ 4τ 5τ 6τ t Maxwell! γ (t) = γ 0 cosωt dσ(t) dt + 1 dγ (t) σ(t) = G τ dt [ ] σ(t) = γ 0 G (ω)cosωt G (ω)sinωt (1) (1) G (ω), G (ω) (1) dσ * (t) dt γ * (t) γ 0 e iωt + 1 τ σ * (t) = G dγ * (t) dt γ (t) = Re γ * (t) (1*) * (t) (1*) d Re σ * (t) dt + 1 τ Re σ * (t) σ(t) Re σ * (t) = G d Re γ * (t) dt (1*) σ * (t) (1) (1 * )

15 (1*) γ * (t) γ 0 e iωt dσ * (t) dt (1*) + 1 τ σ * (t) = Gγ 0 iωe iωt σ * (t) = σ 0 * e iωt σ 0 * iω + 1 τ eiωt = Gγ 0 iωe iωt σ * 0 = G iω iω + 1 γ 0 = G iωτ 1 + iωτ γ 0 τ (1*) σ * (t) = σ * 0 e iωt = G iωτ 1+ iωτ γ 0e iωt = G * (ω)γ * (t) G * (ω) = G iωτ 1 + iωτ = G iωτ ( 1 iωτ ) ( 1 + iωτ )( 1 iωτ ) = G iωτ + ω τ 1 + ω τ = G (ω) + i G (ω) G (ω) = G ω τ G (ω) = G ωτ 1+ ω τ 1+ ω τ G'(ω), G''(ω) G 0.8G 0.6G 0.4G 0.G τ τ G'(ω) G''(ω) 3 τ 4 τ ω 5 τ 6 τ 7 τ 8 τ G'(ω)/G, G''(ω)/G log ωτ log G'(ω), log G"(ω) 10G G 10 1 G 10 G 10 3 G 10 4 G 10 3 τ G''(ω) 10 τ 10 1 τ 1 τ log ω 10 τ G'(ω) 10 τ 10 3 τ

16 T T 0 G(t,T ) = G(t / a T,T 0 )! T > T 0 a T < 1! T < T 0 a T > 1 Langevin Einstein b b 1 R b N N R = b j + b j b k j=1 b R = b 1 + b + b b N b j = 0 b j = b b j b k = 0 j k j k R = Nb Δt t N N = t Δt R(t) = Nb = 6Dt D = b 6Δt

17 k ' ' = 0 ζ x kx + f (t) = 0 f (t 1 ) f (t ) = Aδ(t 1 t ) lim t x = 1 τ x + 1 ζ f (t) x(t = 0) = 0 x(t) = 1 ζ 0 t e (t t )/τ f ( t )d t x(t) = 1 dt ζ 1 dt 0 0 = A ζ 0 ---(*) dt 1 e (t t 1)/τ e (t t 1)/τ (t t )/τ = A τ ζ τ ζ k (*) f (t 1 ) f (t ) x(t) x = A τ 0 ζ = A k kζ x = A 0 4ζ k x = 1 0 k BT A = ζ k B T f (t 1 ) f (t ) = ζk B Tδ(t 1 t ) : k=0 ζ x + f (t) = 0 x(t = 0) = 0 x(t) = 1 f ( t )d t ζ 0 x(t) = 1 t t dt ζ 1 dt f (t 1 ) f (t ) 0 0 = A t dt ζ 1 = A 0 ζ t = k T B ζ t Dt D = k B T ζ t x = 1 ζ f (t) D ζ

18 l 110 l 0 M = 10 n ~ 7000 L max ~ m 1mm 3m or 50:50 l R! R 0 = C nl n n= C = 6.7 l =1.54 A n 7000 R 300A R!

19 10 cm mm 3 m 9000 D=300 4π D ρ 3 = ~ 80 M / N A 3 r 1! r 3!! r! l 0 = lcos! r j r l j+k 0 0 e k /m n λ 110 l = 1.54A! R = r r 0 i j n r i r i+k 0 ij 0 + k= = nl 0 1+ e 1/m 1 e 1/m R nl0 m C 0 = m cos θ λ ml 0 = C cosθ l b R = C 0 nl Nb L max = nl 0 Nb Kuhn b R 0 L max Kuhn = C l cosθ = λ n K n N = C cos θ n K ~ 10 (PE) n K ~ 15 N b (PS) n K

20 r! k = 3k BT r 0 3 r P(r) exp r r 0 P(r) exp U(r) k B T U(r) = 3k BT r = 1 r kr 0 = 3k BT b f = U r = kr r! k! r! 3kBT 1 k = r n r! r! K k = 3k B T b ζ r j k ( r j+1 r j ) k ( r j 1 r j ) + f j (t) = 0 f iα (t) f jβ ( t ) = ζk B Tδ αβ δ(t t )

21 k N f! R! R ζ N Nζ ζ N R k N R + f(t) = 0 k N = 3k B T R f = k N R k N = 3k B T Nb 1 N k N! x! d x dt ζ N dx dt k N x + f (t) = 0 = k N x x exp ζ N t τ ζ N Nζ τ R = ζ N k N k N k B T R 0 Nζ Nb k B T = k B T Nb τ ζ N k N τ R ζb N k B T N

22 G! ν = G νk B T ν = σ αβ = νk N R α R β Rouse ν = R R y SR y νr y S σ xy = νr y F x σ xy = νk N F x = k N R x R x R y G R σ xy = νk N R x R y = R y R x γr y R x = R x +γr y R y R y = νk N (R x +γr y )R y 0 = ν 3k BT R R γ y 0 0 = νk B Tγ Gγ R R x R y 0 = 0 k N = 3k BT R G = νk B T R

23 G η 0 & G νk B ρ ν = M / N A ρrt T = M η 0 ~ Gτ R N 1 M 1 N Einstein τ R ζb N k B T D G ~ k BT ζ N D G τ R ~ Nb = R 0 R 0 N τ R N ~ k BT Nζ D G 1 N R 0! N 1/N D G 1/ N logτ τ R N log M logη 0 η 0 1 N log M log DG -1 1 D G N log M

24 Θ M < M e! M > M e! Zimm Rouse R 0 R 0 N α b Θ 1/ = 0.5 α = ~ 3 / 5 = 0.6 Θ ( ) k N k BT R 0 ζ N η s R 0 τ ζ N ~ η R 3 s 0 k N k B T

25 η = η s +η p G ~ νk B T η p ~ Gτ ~ η s νr 0 3 ~ η s φ [ η] η η s ~ R 3 0 cη s N η s = 3 φ ~ νr 0 c = = Nν D G ~ k BT ζ N D G τ ~ R 0 ~ k BT η s R 0 τ ~ η 3 sr 0 k B T N 3α = N 3/ N 1.8 [ η] ~ R 3 0 N N 3α 1 = N 1/ N 0.8 D G ~ k T B N α = N 1/ η s R 0 N 0.6 Θ Θ Θ

26 Rouse logτ logη 0 log D G ~3.5 log M 1 ~3.5 log M -1 -? log M M e

27 a! a! M < M e M > M e M M e a Me N e b N e = ( ) a ~34 a ~8 Z Z M M e L = Za a R 1 Z! R 0 = Nb = Za a

28 τ e τ e Me = Rouse t t = 0 =τ e τ e ζb N e k B T ~ ζa4 k B Tb 1 1 D c τ d L L = Za N ~ L τ d D c ~ k BT Nζ 1 N τ d N 3

29 D c τ d τ d ~ L τ d ~ ζb k B T N 3 N e ~ τ e Z 3 D c ~ k T B Nζ L = Za τ R ~ ζb k B T N ~ τ e Z D G τ d ~ R 0 3 τ d N R 0 = Nb D G ~ k B T ζ t = 0! N e N 1 N ~ R 0! t ~ τ d R 0! Me G η 0 & ν = ρ ν = M e / N A G ν k T = B ρrt M e η 0 ~ Gτ d N 3

30 σ αβ = G λ n α n β n λ L L 0 = L Za τ d ~ ζb k B T G ~ ρrt M e M 0 η ~ Gτ d M 3 D G ~ k B T ζ N 3 N e ~ τ e Z 3 M 3 N e N 1 M logτ ~3.5 3 log M logη 0 1 log M log DG -1 ~ ? log M Rouse 3 τ R N << τ d N M e

31 h(γ)! G( t, γ ) γ ~ 0 h(γ ) γ : t ~ = τ0! R d! t! γ& η 0 M 3 η 0 M 3.5

32 CLF (Contour Length Fluctuation) primitive path η 0 M 3 η 0 M 3.5 CR (Constraint Release) CCR (Convective CR) CCR CCR CCR CCR CCR

33 CLF ( CR () PASTA CLF CR Slip-link! CLF CR Virtual slip-links! each polymer moves in its own virtual space!

34 (1)Afine deformation! ()Contour Length Fluctuation! (3)Reptation! (4)Constraint Renewal (CR)! Assumptions" Binary entanglement! Entanglement points move affinely! Higher order Rouse modes are ignored! Z M M e τ R = τ Z e τ d ~ τ ez 3 stress σ Z=60 Z=30 Z=10 Z= shear rate γ 0.13! (MLD)! stress 10 1 Linear Z= N σ shear rate Mead, Larson, Doi, Macromolecules,31, 7895 (1998)!

35 7000 多分散試料での予測 weight = (number of chains)*z PS686 Mw = 80 k Mw/Mn = Zw = 0.4 Zw/Zn = Z=M/M e G', G" [Pa] G' (sim.) G" (sim.) G' (exp.) G" (exp) PS C τ e =. ms G e = 0.5 MPa ωa T [rad/s] つのモデルパラメタを決定 η E + (t) [Pa s] フィッティングパラメタ無しで定量的に予測可能 PS C t [s] (1/s) 0.13(1/s) 0.055(1/s) 0.011(1/s) 3η 0 (exp.) simulation simulation simulation simulation 3η 0 (sim.) A. Minegishi et al.,! Rheol. Acta, 40(4), 39 (001)!

36 10 7 PS C 10 7 PS C η B + (t) [Pa s] [1/s] 0.05 [1/s] [1/s] 0.5 [1/s] sim /s sim /s sim /s sim /s t [s] A. Nishioka et al.,! J. Non-Newtonian Fluid! Mech. 89, p.87 (000).! η P + (t) [Pa s] [1/s] 0.03 [1/s] [1/s] 0.3 [1/s] sim /s sim /s sim /s sim /s t [s] A. Nishioka et al.,! J. Non-Newtonian Fluid! Mech. 89, p.87 (000).! weight = (number of chains)*z PS686 30k 1.5wt% Z=M/M e η E + (t) [Pa s] k 1.5wt% / PS C 0.57(1/s) 0.097(1/s) 0.047(1/s) 0.013(1/s) 3η 0 simulation simulation simulation simulation t [s] ひずみ硬化性の増強を定量的に予測 A. Minegishi et al.,! Rheol. Acta, 40, 39 (001)!

37 直鎖高分子と星形高分子の比較 shear viscosity Za=36 Za=30 Z=30 Za=0 Z=80 Z=60 Z=0 Za=10 Z Za dominated by CCR! Z= shear rate zero-shear viscosity η Star Linear η 0 Z Z linear or Z arm η η 0 exp(βz a ) β ~ Z a

38 G(t, γ) Linear polymer! Star polymer! 10 0 γ= γ= γ= γ= γ=8 Linear Z=0 γ= t G(t,γ) 10 0 γ= γ=0.5 γ= 10 - γ= γ=8 Star Za=10 γ= t G(t,γ) γ=16 γ=8 γ=4 γ= γ=0.5 γ=1 Linear Z= t G(t,γ) γ=16 γ=8 γ=4 γ= γ=0.5 γ=1 Star Za= t h(γ) Linear Z=0 Star Za=10 DE γ η + E (t) Linear Z=0 1e- 4e-3 e-3 1/τ R =.5e t 1e-3 τ R =400 1e-4 e-4 4e-4

39 η + E (t) Star Za=10 1e- 4e-3 e t e-5 e-4 1e-4 1e-3 τ R =400 1/τ R =.5e-3 4e-4 直鎖高分子と星形高分子の! 線形レオロジーは大きく異なるが! 非線形レオロジーはきわめて類似非線形粘度! ダンピング関数! 一軸伸長粘度の非線形性 ( ひずみ硬化 ) CLF と CCR は直鎖と星形に同等に働く Reptation, CLF, CR の 3 つの運動 " を考慮した stochastic simulation" " 直鎖星形高分子について多くのの線形非線形レオロジーを定量的に予測可能 " " 超高分子量成分によるひずみ硬化性の増強 " " 直鎖と星形の非線形レオロジーの類似 star! linear! H! pompom! comb! general!

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