Outline!!! Rouse! : Zimm ( )!!! PASTA (strain)
(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
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!
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 4.0 3.5 3.0.5.0 1.5 1.0 0.5 ε C = e ε 1 0.0 0.0 0.5 1.0 1.5.0 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
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 = 1 0.5 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
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! 3 8 4 18.
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!
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
!! ~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!
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/!
( ) 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!
( )( 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
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!
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 * )
(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 0 0 1 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 1 0.8 0.6 0.4 0. 0-3 - -1 0 1 3 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
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 3 + + 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
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
3 1.6 1.54 l 110 l 0 M = 10 n ~ 7000 L max ~ 9000 0.9 m 1mm 3m 1.54 110 or 50:50 l R! R 0 = C nl n n= C = 6.7 l =1.54 A n 7000 R 300A R!
10 cm 300 3 1 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
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 )
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
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
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
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
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
Rouse logτ logη 0 log D G ~3.5 log M 1 ~3.5 log M -1 -? log M M e
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
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
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
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 ~3.5 3 -? log M Rouse 3 τ R N << τ d N M e
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
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
CLF ( CR () PASTA CLF CR Slip-link! CLF CR Virtual slip-links! each polymer moves in its own virtual space!
(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 10 0 10-1 10-10 -3 Z=60 Z=30 Z=10 Z=0 10-7 10-6 10-5 10-4 10-3 10-10 -1 shear rate γ 0.13! (MLD)! stress 10 1 Linear Z=0 10 0 N 1 10-1 10-10 -3 σ 10-6 10-5 10-4 10-3 10 - shear rate Mead, Larson, Doi, Macromolecules,31, 7895 (1998)!
7000 多分散試料での予測 weight = (number of chains)*z 6000 5000 4000 3000 000 1000 PS686 Mw = 80 k Mw/Mn = Zw = 0.4 Zw/Zn = 1.7 0 10-1 10 0 10 1 10 10 3 Z=M/M e G', G" [Pa] 10 6 10 5 10 4 10 3 G' (sim.) G" (sim.) G' (exp.) G" (exp) PS686 160 C τ e =. ms G e = 0.5 MPa 10 10-3 10-10 -1 10 0 10 1 10 ωa T [rad/s] つのモデルパラメタを決定 η E + (t) [Pa s] 10 7 10 6 10 5 フィッティングパラメタ無しで定量的に予測可能 PS686 160 C 10 4 10-1 10 0 10 1 t [s] 10 10 3 0.564(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)!
10 7 PS686 160 C 10 7 PS686 160 C η B + (t) [Pa s] 10 6 0.01 [1/s] 0.05 [1/s] 10 5 0.1 [1/s] 0.5 [1/s] sim. 0.01 1/s sim. 0.05 1/s sim. 0.1 1/s sim. 0.5 1/s 10 4 10-1 10 0 10 1 10 10 3 t [s] A. Nishioka et al.,! J. Non-Newtonian Fluid! Mech. 89, p.87 (000).! η P + (t) [Pa s] 10 6 0.01 [1/s] 0.03 [1/s] 10 5 0.1 [1/s] 0.3 [1/s] sim. 0.01 1/s sim. 0.03 1/s sim. 0.1 1/s sim. 0.3 1/s 10 4 10-1 10 0 10 1 10 10 3 t [s] A. Nishioka et al.,! J. Non-Newtonian Fluid! Mech. 89, p.87 (000).! weight = (number of chains)*z 7000 6000 5000 4000 3000 000 1000 0 PS686 30k 1.5wt% 10-1 10 0 10 1 10 10 3 Z=M/M e η E + (t) [Pa s] 10 8 10 7 10 6 10 5 30k 1.5wt% / PS686 160 C 0.57(1/s) 0.097(1/s) 0.047(1/s) 0.013(1/s) 3η 0 simulation simulation simulation simulation 10 4 10-1 10 0 10 1 10 10 3 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)!
直鎖高分子と星形高分子 の比較 shear viscosity 10 5 10 4 10 3 10 Za=36 Za=30 Z=30 Za=0 Z=80 Z=60 Z=0 Za=10 Z Za dominated by CCR! Z=10 10 1 10-8 10-7 10-6 10-5 10-4 10-3 10 - shear rate zero-shear viscosity η 0 10 5 10 4 Star 10 3 10 Linear η 0 Z 3.45 10 1 10 0 10 0 10 1 10 Z linear or Z arm η 0 10 5 10 4 η 0 exp(βz a ) β ~ 0.4 10 3 8 10 1 14 16 18 0 Z a
G(t, γ) Linear polymer! Star polymer! 10 0 γ= γ=1 10-1 γ=0.5 10 - γ=4 10-3 γ=8 Linear Z=0 γ=16 10-4 10 1 10 10 3 10 4 10 5 t G(t,γ) 10 0 γ=1 10-1 γ=0.5 γ= 10 - γ=4 10-3 γ=8 Star Za=10 γ=16 10-4 10 1 10 10 3 10 4 10 5 10 6 t G(t,γ) 10 10 1 10 0 10-1 10-10 -3 γ=16 γ=8 γ=4 γ= γ=0.5 γ=1 Linear Z=0 10-4 10 1 10 10 3 10 4 10 5 t G(t,γ) 10 10 1 10 0 10-1 10-10 -3 γ=16 γ=8 γ=4 γ= γ=0.5 γ=1 Star Za=10 10-4 10 1 10 10 3 10 4 10 5 10 6 t h(γ) 10 0 10-1 10-10 -3 Linear Z=0 Star Za=10 DE 10-1 10 0 10 1 10 γ η + E (t) 10 4 10 3 10 Linear Z=0 1e- 4e-3 e-3 1/τ R =.5e-3 10 1 10 1 10 10 3 10 4 10 5 t 1e-3 τ R =400 1e-4 e-4 4e-4
η + E (t) 10 4 10 3 10 Star Za=10 1e- 4e-3 e-3 10 1 10 1 10 10 3 10 4 10 5 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!