! ! ! Rouse ! : Zimm ( ) !

! !PASTA

Outline

(strain)

(shear deformation)

h!

x!

h

= xh

h!

shear strain!

x/h !

x/h

h2!x2!

h1!

x1! x1h1

=x2h2

h1!x1!

(uniaxial elongation)

L0!

L!

L!

C =LL0

=L L0L0

!

C =

Cauchy strain!

!

L/L

L0!

L!

L!L0

L

L

L0!

L!

L!

L0!

L!

L!

2L!

2L0!

2L!C =LL0

=2L2L0

2 2

1

2

L0!

2L0!

4L0!

C1 + C2

L1=L0! C1 = L0!L0! = 1!

L2=2L0!2L0!2L0!C2 = = 1!

L1+2=3L0!

C1+2 = L0!3L0! = 3!

1

= ln LL0L0!

L! = ln

ln loge natural logarithm!

ln xy = ln x + ln y

Hencky strain!

2 2

1

2

L0!

2L0!

4L0!

= 1 + 2

L0!2L0!1 = ln = ln 2!

2L0!4L0!2 = ln = ln 2!

1+2 = ln L0!4L0! = ln 4!

1

= 2 ln 2!

2

L0!

L1!

L2!

1

2

1 = lnL1L0

2 = lnL2L1

1

1+2 = lnL2L0

1 + 2 = lnL1L0

+ ln L2L1

= ln L1L0

L2L1

= ln L2L0

1+2 = 1 + 2 = ln L

L0= ln L0 + L

L0= ln 1+ L

L0

L = L L0C =LL0

1+ C = ey = ex x = ln y

= ln 1+ C( )

C = e 1

e = 1+ + 1

2! 2 + 1

3! 3 +

Taylor

1 e 1+

C = e 1 1+ ( ) 1 =

1 C 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0

Cau

chy

stra

in c

Hencky strain

C =

Cauchy Hencky

C = e 1

(Poisson s ratio) !

z- x,y-

L0!

L!

D0!

D!

= ln L

L0> 0(L > L0 )

= ln

DD0

< 0(D < D0 )

= > 0

L0D02 = LD2

LL0

=D0D

2

= ln LL0

= ln D0D

2

= 2 ln D0D

= 2 ln DD0

= 2

=

=12

0.5

12

L L0

L = L0e = L0 (1+ C )

ln1 = 0ln xy = ln x + ln ylne = 1

ln x < 00 < x < 1ln x > 0x > 1

y = ln xx = ey

!

LL0

= e = 1+ C = ln

strain rate!

=

[1/s]!

d

dtx!

x!

vw x

dxdt

= xh

= vwh

h!

shear rate! [1/s]!

1cm 1mm/s vw = 1[mm/s]

h = 1 [cm]! = vw

h=1[mm/s]1[cm]

=1[mm/s]10[mm]

= 0.1[1 / s]

h!

!t = 5

= t = 0.1[1/s]( ) 5[s]( )=0.5

x = vwt = 5[mm]

h!

0!

y!

x!

vx (y)

vw vx (y) = ayvx (h) = vw

vx (y) =vwhy

y y a

vx (y) = y = vw

h

= vx

y =

d

dt

L(t) = L0et

1 2 2 4 !3 8 4 18 .

(t) = ln L(t)L0

(t) = 1L(t)

dL(t)dt

stress

F!S!

= FSshear stress!

[Pa] = [N/m2]!

=

2

= FS

S!F!

2

= FS

F!

F!

F!

= 2F2S

=FS

: 2F!: 2S!

2F!S!

F!

S!

E =FS

S S

true stressS0 !

engineering stress !

Hencky true strain !Cauchy engineering strain !

x!

y!

Sy! xy =

FxSy

y !x

yz , zx , yx , zy , xz xy = yx , zx = xz , zy = yz

zx = xz = 0, zy = yz = 0 xy = yx 0

xx 0

Fx!

xx = x x yy , zz

E =FxS

= xx yy

F!F!

xx =FxS p

x F

p

xx = yy = zz = p

x!

y!

(t) = G (t)

E (t) = E(t)

!t (t) (t)

G:

E:

G, E [Pa]!

E = 2G 1+ ( ) = = 1/2 E = 3G

shear modulus!

Young s modulus!

!t (t) !

(t) = (t)

(t)

E (t) = E (t)

:

E:

E = 3

(viscosity) [Pa s]!

shear viscosity!

! ! ~200 GPa!! ~80 GPa!

~3 GPa!!

! ~1 MPa!

~ 10-3 Pa s = 1 mPa s!

stress relaxation

t!

(t)!

0!

0!

t!

(t)!

0!

t=0 0

0 !0 2 !(t) 2

0 !0 2 (t) 2

(t) 0

0

G(t) (t) 0

!relaxation modulus!

t!

G(t)!

0!

t!0!

G(t)!

(t) = G (t) = G 0 cost

0! t!

(t)!

(t)!

T!

(t) = 0 cost

= 2T

T =

0 =

0!

0! t!

(t) = (t)= 0 sint

(t) or (t)!

0! t [s]!

(t) = 0 cost

T!

2! t [radian]!

(t) = 0 cos t + ( )&

0!

= 0! = /2!0 < < /2!

0 !0 2 0 2

(t) = 0 cos t + ( )

G ( ), G ( ) 0

(t) = 0 G ( )cost G ( )sint[ ]= 0 cos cost sin sint( )

G ( ) 0 0cos

G ( ) 0 0sin

!storage modulus!

!loss modulus!

[Pa]!

(t) = (t) = 0 sint (t) = G (t) = G 0 cost

(t) = 0 G ( )cost G ( )sint[ ]G ( ) = G, G ( ) = 0 = 0G ( ) = 0, G ( ) = = / 2

(loss tangent)!

tan G ( )G ( )

tan < 1 : G ( ) < G ( )tan > 1 : G ( ) > G ( )

G' (), G"() !

log !

log G()!

tan = 1!tan > 1! tan < 1!

G ( )

G ( )

Euler

ei cos + i sin

1!-1!

i!

-i!

cos !

sin !

0!ei 1 +2( ) = ei1ei2

ddtei t = iei t

ei!

= cos 1 +2( ) + i sin 1 +2( )

= cos1 cos2 sin1 sin2+i sin1 cos2 + cos1 sin2( )

ei1ei2 = cos1 + i sin1( ) cos2 + i sin2( )

ddtei t = d

dtcost + i sint( )

= sint + i cost= i (cost + i sint)= iei t

= ei 1 +2( )

(t) = 0 cost

(t) = Re *(t)

(t) = 0 cos t + ( )

(t) = Re *(t)

*(t) 0ei t *(t) 0e

i( t+ )

*(t) = 0ei( t+ ) = 0e

iei t = 0 0

ei 0ei t

*(t) = G*( ) *(t)

G*( ) 0 0

ei

G ( ) 0 0cos G ( )

0 0sin

G*( ) 0 0

ei = 0 0

cos + i sin( )

G*( ) = G ( ) + i G ( )

*(t) = G*( ) *(t) = G*( ) 0ei t

(t) = Re *(t) = 0 G ( )cost G ( )sint[ ]= G ( ) + i G ( )( ) 0 cost + i sint( )

Maxwell

1(t) = G 1(t)

2 (t) = 2 (t)

G!

&

1

2

2

1

1

2

G!

&

1

2

(t) = 1(t) = 2 (t)

(t) = 1(t) + 2 (t)

1(t) =1G1(t) =

1G (t)

2 (t) =

1 2 (t) =

1 (t)

1(t) =

1G (t)

(t) = 1(t) + 2 (t)

=1G (t) + 1

(t)

1(t) = G 1(t)

2 (t) = 2 (t)

(t) = 1(t) + 2 (t)

d (t)dt

+1 (t) = G d (t)

dt

G

G

1(t = +0) = 0

2 (t = +0) = 0

t!

(t)!

0!

0!

(t = +0) = 1(t = +0) = G 1(t = +0) = G 0

t > 0 (t) = 0d (t)dt

= 0

(t) e t /d (t)dt

+1 (t) = 0 d (t)

dt=

1 (t)

(t) = G 0e t /t > 0 G(t) =

(t) 0

= Ge t /

G(t)!

t!0!

G!G(t) =Ge t / t > 00 t < 0

= G

&

G/e!

G(t)!G(t)!

G!

t! t!0!0!

= G

1(t)!

t!0!

0!

1(t)!

2(t)!

2(t)!

t!

0!

0!

1(t) = G 1(t)

2 (t) = 2 (t)

=!

(t)

=!

log t

101G

102G

103G

104G

105G

10G

G

10 102 103101102103

log G

(t)

G(t)

G( t)

t

G(t)

0.2G

0.4G

0.6G

0.8G

G

00 2 3 4 5 6

d (t)dt

+1 (t) = G d (t)

dt(1)Maxwell !

(t) = 0 cost

(t) = 0 G ( )cost G ( )sint[ ] (1)

G ( ), G ( )

d *(t)dt

+1 *(t) = G d

*(t)dt

(1*)(1)

(t) = Re *(t) *(t) 0ei t

(1*) *(t)

dRe *(t) dt

+1Re *(t) = G

dRe *(t) dt

(1*)

(t) Re *(t)

(1*) *(t)

(1)

*(t) = 0*ei t

(1*) *(t) 0ei t

d *(t)dt

+1 *(t) = G 0ie

i t

(1*)

0* = G i

i + 1

0 = Gi1+ i

0

0* i + 1

ei t = G 0ie

i t

*(t) = 0*ei t = G i

1+ i 0e

i t

= Gi 1 i( )1+ i( ) 1 i( ) = G

i + 2 2

1+ 2 2

(1*)

= G*( ) *(t)

= G ( ) + i G ( )

G ( ) = G 2 2

1+ 2 2G ( ) = G

1+ 2 2

G*( ) = G i1+ i

G'(

) , G

''(

)

G''()

G'()

1

2

3

4

5

6

7

8

0

G

0.8G

0

0.6G

0.4G

0.2G

0

0.2

0.4

0.6

0.8

1

-3 -2 -1 0 1 2 3

log

G'(

)/G, G

''(

)/G log G

'() ,

log G

"(

)

log

1

10

G

10G

102G

103G

104G102

103

103

102

101

101G

G'()

G''()

G(t,T ) = G(t / aT ,T0 )

!

!

T T0

T > T0 aT < 1

T < T0 aT > 1

Langevin

Einstein

b1

bN

b2

R

R = b1 + b2 + b3 ++ bN

b j = 0 b j2 = b2

R2 = b j2

j=1

N

+ b j bkjk

R2 = Nb2

b j bk = 0 j k

b

t

t N N =tt

R(t)2 = Nb2 = 6Dt

D = b2

6t

x kx + f (t) = 0 f (t1) f (t2 ) = A (t1 t2 )

k ' '

= 0

x = 1

x + 1

f (t)

x(t) = 1

e(t t )/ f ( t )d t0

t

limt

x(t)2 = 1 2

dt1 dt20

e(tt1 )/(tt2 )/ f (t1) f (t2 )0

= A 2

dt1e2(tt1 )/

0

=A 2

2

k

x(t = 0) = 0 (*)

---(*)

x(t)2 x20= A 2

2= A2k

k2x2

0= 12kBT

A = 2 kBT

f (t1) f (t2 ) = 2kBT (t1 t2 )

k2x2

0= A4

2 x(t)2 = 1

2dt10t

dt2 f (t1) f (t2 )0t

x(t) = 1

f ( t )d t0

t

x + f (t) = 0: k=0

= A 2

dt10t

=A 2

t = 2 kBT

t 2Dt

D = kBT

x(t = 0) = 0

x = 1

f (t)

D

1.26

3

M = 10 n ~ 7000

Lmax ~ 9000 0.9 m

1.54

110

1mm 3m

l l0

1.54

110

50:50

2 or

l

R20=Cnl

2 n

A300R

7.6=C l =1.54A

R!

7000n

R!

n=

9000 300

D=300

=

43

D2

3

M / NA~ 80

3 1 mm

3 m 10 cm

l = 1.54!

110

l0 = lcos !

! r2!r1! r3!

rj rj+k 0 l02e k /m

R20= ri rj 0

ij

n

n ri ri+k 0k=

+

= nl02 1+ e1/m

1 e1/m

R20 nl0

2 2m C = 2mcos2

ml0 =C2cos

l

bR2

0=Cnl

2 Nb2

Lmax = nl0 Nb

b R2

0

Lmax= Clcos

= 2

nK nN

= Ccos2

nK ~10 nK ~15

Kuhn

N b

Kuhn

(PE) (PS)

nK

r P(r) exp 32 r2

0

r2

r!

P(r) exp U(r)kBT

U(r) = 3kBT2 r2

0

r2 = 12kr2

k = 3kBTr2

0

= 3kBTb2

f = Ur

= kr

r!K

213n

Tkk B =r

k!r!

r! r!

k = 3kBTb2

rj k rj+1 rj( ) k rj1 rj( )+ f j (t) = 0

fi (t) f j ( t ) = 2kBT(t t )

R

N R kNR+ f (t) = 0

N N

kN

f = kNR

f! R!

kN =3kBTR2

= 3kBTNb2

1N

kN

Ndxdt

kN x + f (t) = 0

d xdt

= kN N

x

tx exp

NkN

kN!x!

R =

NkN N Nb

2

kBT

N N kN

kBTR2

0

= kBTNb2

R

b2N 2

kBT N 2

G!

TkG B

=Rouse

=

= kN RR =

Fx = kNRx

R xy = RyFx

S Ry RyS

xy = kN RxRy

Ry

GRy = RyRx = Rx +Ry Ry

Rx Ry

R R

xy = kN Rx Ry= kN (Rx +Ry )Ry 0

R

G = kBT

kN =3kBTR2

= kBT

RxRy 0 = 0

= 3kBTR2

0

Ry20

G

G 0&

ANM / =

NMMRTTkG B

11 =

0 ~G R N

DG ~kBT N

~ kBTN

R0!

Einstein

R

b2N 2

kBT

DG R ~ Nb2 = R0

2

DG 1N

R N2R0

2 N

N 1/N

NDG /1N

DG1N02NR

MlogMlogMlog

log GDlog0log

1 2

-1

M < Me! M > Me!

Zimm Rouse

R0 R

2

0 Nb

=1 / 2 = 0.5~ 3 / 5 = 0.6

( )

N sR0

kN

kBTR02

N

kN~ sR0

3

kBT

G ~ kBT =s +p

p ~G ~sR03 ~s

[ ] scs

~ R03

N

c = = N

~R03

s = DG ~kBTN

~ kBTsR0

DG ~ R02

~ sR03

kBT N 3 = N

3/2

N1.8

[ ] ~ R03

N N 31 = N

1/2

N 0.8

DG ~kBTsR0

N = N1/2

N 0.6

Rouse

MlogMlogMlog

log GDlog0log

1 2

-1 ~3.5 ~3.5

-2 ?

eM

eMM

a!

Me a

a ~34 a ~82

Neb2

Ne = ( )

Z

eMMZ

L = ZaR2

0= Nb2 = Za2

a

a

1 2

Z!

R

Me Rouse=e

e

e

b2Ne2

kBT~ a

4

kBTb2

0=t

e=t

1

Dc ~kBTN

1N

2~ LD dcL = Za N

d N3

1

L

d

d2~ LD dc Dc ~

kBTN

d ~b2

kBTN 3

Ne~ eZ

3

L = Za

R ~b2

kBTN 2 ~ eZ

2

DG d ~ R02

3Nd R0

2 = Nb2

DG ~kBT

NeN 2

1N 2

R0!t = 0!

t ~ d~ R0!

Me

G 0&

=ANM /e

=

eMRTTkG B

=

0 ~G d N3

= G 2nn

n L

L0= LZa

d ~b2

kBTN 3

Ne~ eZ

3M 3

~G d M3

DG ~kBT

NeN 2

1M 2

G ~ RTM e

M 0

MlogMlogMlog

log GDlog0log

1 2

-1 ~3.5 ~3.5

-2 ?

eM

3 3 2

Rouse32 NN dR

: t = 0!t ~ R!t ~ d!

h()!

),( tG

t! &

)(h

~ 0

0 M3 0 M

3.5

CLF (Contour Length Fluctuation) primitive path

0 M3

0 M3.5

CR (Constraint Release)

CCR (Convective CR)

CCR

CCR

CCR CCR

CCR

CLF ( CR ( )

PASTA

CLF CR

CLF CR

Slip-link!

Virtual slip-links!

each polymer moves in its own virtual space!

(1)Afine deformation!(2)Contour Length Fluctuation!(3)Reptation!(4)Constraint Renewal (CR)!

Binary entanglement!Entanglement points move affinely!Higher order Rouse modes are ignored!

Assumptions"

eMMZ

3e~ Zd

2eZR =

10-3

10-2

10-1

100

10-7 10-6 10-5 10-4 10-3 10-2 10-1

stres

s

shear rate

Z=60

Z=30Z=20

Z=10

0.123!(MLD)!

10-3

10-2

10-1

100

101

10-6 10-5 10-4 10-3 10-2

stres

sshear rate

Linear Z=20

N1

Mead, Larson, Doi, Macromolecules,31, 7895 (1998)!

0

1000

2000

3000

4000

5000

6000

7000

10-1 100 101 102 103

PS686

wei

ght =

(num

ber o

f cha

ins)

*Z

Z=M/Me

Mw = 280 kMw/Mn = 2

Zw = 20.4Zw/Zn = 1.7

2

102

103

104

105

106

10-3 10-2 10-1 100 101 102

G' (sim.)G" (sim.)G' (exp.)G" (exp)

G',

G"

[Pa]

aT [rad/s]

160CPS686

e = 2.2 ms

Ge = 0.5 MPa

A. Minegishi et al., !Rheol. Acta, 40(4), 329 (2001)!

104

105

106

107

10-1 100 101 102 103

0.564(1/s)0.123(1/s)0.055(1/s)0.011(1/s)30(exp.)simulationsimulationsimulationsimulation30(sim.)

E+ (

t) [P

a s]

t [s]

PS686160C

104

105

106

107

10-1 100 101 102 103

0.01 [1/s]0.025 [1/s]0.1 [1/s]0.25 [1/s]sim. 0.01 1/ssim. 0.025 1/ssim. 0.1 1/ssim. 0.25 1/s

B+ (

t)

[Pa

s]

t [s]

PS686 160C

A. Nishioka et al., !J. Non-Newtonian Fluid!Mech. 89, p.287 (2000).!

104

105

106

107

10-1 100 101 102 103

0.01 [1/s]0.03 [1/s]0.1 [1/s]0.3 [1/s]sim. 0.01 1/ssim. 0.03 1/ssim. 0.1 1/ssim. 0.3 1/s

P+ (

t)

[Pa

s]

t [s]

PS686 160C

A. Nishioka et al., !J. Non-Newtonian Fluid!Mech. 89, p.287 (2000).!

0

1000

2000

3000

4000

5000

6000

7000

10-1 100 101 102 103

PS686

wei

ght =

(num

ber o

f cha

ins)

*Z

Z=M/Me

3220k1.5wt%

104

105

106

107

108

10-1 100 101 102 103

0.572(1/s)0.097(1/s)0.047(1/s)0.013(1/s)3

0simulationsimulationsimulationsimulation

E+ (

t) [P

a s]

t [s]

3220k 1.5wt% / PS686160C

A. Minegishi et al., !Rheol. Acta, 40, 329 (2001)!

101

102

103

104

105

10-8 10-7 10-6 10-5 10-4 10-3 10-2

shea

r visc

osity

shear rate

2Za=362Za=30

2Za=20

2Za=10

Z=80 Z=60

Z=30

Z=20

Z=10

dominated by CCR!

2Za

Z

100

101

102

103

104

105

100 101 102

zero

-she

ar v

isco

sity

0

Zlinear or 2Zarm

Linear

Star

0 Z3.45

103

104

105

8 10 12 14 16 18 20

0

Za

0 exp(Za) ~ 0.4

10-4

10-3

10-2

10-1

100

101 102 103 104 105

G(t,)

t

Linear Z=20

=0.5

=1=2

=4

=8=16

10-4

10-3

10-2

10-1

100

101 102 103 104 105 106

G(t,)

t

Star Za=10

=0.5

=1

=2

=4

=8

=16

Linear polymer! Star polymer!

10-4

10-3

10-2

10-1

100

101

102

101 102 103 104 105

G(t,)

t

Linear Z=20

=0.5

=1

=2

=4

=8

=16

10-4

10-3

10-2

10-1

100

101

102

101 102 103 104 105 106

G(t,)

t

Star Za=10

=0.5

=1

=2

=4

=8

=16

10-3

10-2

10-1

100

10-1 100 101 102

h( )

DE

Linear Z=20

Star Za=10

101

102

103

104

101 102 103 104 105

+E(

t)t

Linear Z=202e-4

1e-32e-3

4e-31e-2

4e-4

1e-4

R=400

1/R=2.5e-3

101

102

103

104

101 102 103 104 105

+E(

t)

t

Star Za=102e-4

2e-5

1e-3

2e-34e-3

1e-2

4e-4

1e-4

R=400

1/R=2.5e-3

!!

CLF CCR

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Reptation, CLF, CR 3" stochastic simulation""""""

linear! star!

H! pompom!

comb!general!