! ! ! 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
!!
Reptation, CLF, CR 3" stochastic simulation""""""
linear! star!
H! pompom!
comb!general!