Hierarchy of models for entangled polymers€¦ · Hierarchy of models for entangled polymers...
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Hierarchy of models for entangled polymers
Alexei E. Likhtman
Department of Mathematics, University of Reading
With Sathish K. Sukumaran, Ponmurugan Mani, Jing Cao
Hierarchical modelling in polymer dynamics
• Constitutive equations
– Tube theories
• Single chain models
?),(σσ
fDt
D=
Tube
Model?jm
slip-spring
• Single chain models
– Coarse-grained
many-chains models
» Atomistic simulations
> Quantum mechanics simulations
bead-spring
molecular dynamics
Well established coarse-
graining procedures,
force-fields,
commercial packages
CR
Model?ja
slip-spring
model
40 years of the Tube theory 1967-2007
• Original assumptions
– Independent motion of each chain
– along the primitive path like a 1-D Rouse chain
– purely entropic stress (Kramer’s)
• Predictions
– Stress plateau
–
– Two stage relaxation and damping function
– Mean square displacement
– for stars
4.3
wd M∝∝τη
)exp( Mνη ∝
• Problems
– constraint release vs. tube dilution
– branch point motion
– tube diameter under deformation
– tube field and its stress contribution
– how to find the tube in MD?
– for stars
– Contour length fluctuations
)exp( wMνη ∝
•many contradicting theories coexist
•many proven results remain ignored
•one theory postdoc per
new experiment
20 instantaneous trajectories
20 mean paths
20 mean paths
Construction of the slip-spring model
ja
jm
timeelementary
size coil
re temperatu
parameters model Rouse
0
2
−
−
−
τ
gR
T
chain thealonglink -slip offriction -
chain) anchoring in the monomers ofnumber effective(or link -slip ofstrength -
links-slipbetween beads ofnumber average -
parameters New
s
s
e
N
N
ζ
slip_links19.exeA.L., Macromolecules, 2005, 38 (14), 6128
Rubinstein, Panyukov, Macromolecules 2002, 35, 6670-6686
Construction of the slip-spring model
ja
jm
timeelementary
size coil
re temperatu
parameters model Rouse
0
2
−
−
−
τ
gR
T
chain thealonglink -slip offriction -
chain) anchoring in the monomers ofnumber effective(or link -slip ofstrength -
links-slipbetween beads ofnumber average -
parameters New
s
s
e
N
N
ζ
slip_links19.exeA.L., Macromolecules, 2005, 38 (14), 6128
Rubinstein, Panyukov, Macromolecules 2002, 35, 6670-6686
Constraint release
Hua and Schieber 1998
Shanbhag, Larson, Takimoto, Doi 2001
Outline
• Molecular dynamics as an experimental tool
• Molecular dynamics fitted by slip-springs• Molecular dynamics fitted by slip-springs
• Microscopic definition of entanglements
Molecular Dynamics -- Kremer-Grest model
• Polymers – Bead-FENE
spring chains
0
2 2
2
0
( ) ln 12
FENE
kR rU r
R
= − −
• With excluded volume – Purely
• k = 30ε/σ2
• R0=1.5σ
Density, ρ = 0.85• With excluded volume – Purely
repulsive Lennard-Jones
interaction between beads
otherwise 0
2 r 4
14)( 61
612
=
<
+
−
=
rrrUrLJ
σσε
Density, ρ = 0.85
Friction coefficent, ζ = 0.5
Time step, dt = 0.012
Temperature, T = ε/k
K.Kremer, G. S. Grest
JCP 92 5057 (1990)
Stress relaxation for N=350
0
30
60
1
10
100
G(t
)
G(t
)
bond length
relaxation0.1 1 10
0.01 0.1 1 10 100 1000 10000 1000001E-3
0.01
0.1 t
G(t
)
t
GN
(0)
τe
relaxation
collisions
"Rouse" dynamics
entangled
Multiple tau correlator
G(t
)
1e-3
1e-2
1e-1
1e0
1e1
G(t
)
1e-3
1e-2
1e-1
1e0
1e1
Magatti and Ferri, Applied optics, 40, 4011 (2000)
t0.1 1 10 100 1,000 10,000 100,000
t0.1 1 10 100 1,000 10,000 100,000
Stress autocorrelation
N=1,2,5,10,25,50,100,200,350
1
10 N=350
N=200
N=100
N=50
N=25
N=10
N=5
N=2
G(t
)
0.1 1 10 100 1000 10000 100000
0.01
0.1
N=2
N=1
G(t
)
tA.E.Likhtman, S.K.Sukumaran, J.Ramirez, Macromolecules, 2007, 40, 6748-6757
1.0
1.2
1.4
1.6
1.8
Normalized G(t)
1 10 100 1000 10000 1000000.0
0.2
0.4
0.6
0.8
G(t
)t1
/2
t
Rouse
Entanglements
Fit with slip-springs model as an extrapolation tool
A.E.Likhtman, S.K.Sukumaran, J.Ramirez, Macromolecules, 2007, 40, 6748-6757
Varying chain stiffness
1E-3
0.01
0.1
1
10
50100
2030
50 70
50100
200
kb=3, ev2, x10
-2
k =1.1, x10-3
kb=1.5, x0.1
kb=3, x10
kb=5,ev4
G(t
)
200
0.003
0.004
0.005
0.006
1.5 1.13odd
0
-3rho06
rho0.4
1 10 100 1000 10000 100000
1E-8
1E-7
1E-6
1E-5
1E-4
300100
50200100
100
350
50
100 150
200
200
ρ=0.6, x10-5
kb=-3, x10
-4
kb=1.1, x10
-3
t
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.11.21E-3
0.002
0.003
3
5ev4
G~p-3
(Fetters)
GN
(0) p
3
p
slip-spring fit with free G0 and M0
same with fixed M0=rho/G0
PPA
G~p-7/5
(Semenov)
A.E.Likhtman, S.K.Sukumaran, J.Ramirez, Macromolecules, 2007, 40, 6748-6757
g1(t) – monomer mean square displacement
100
1000
(t)
N=50,100,200,350
10 100 1000 10000 100000 10000001
10
g1(t
)
t
g1(i,t) – monomer mean square displacement
0.2
0.4
0.6
0.8
1.0
1.2
1.4
0.2
0.4
0.6
0.8
1.0
1.2
1.4g
1i/t
1/2
g1i/t
1/2
N=50 N=100
10 100 1000 10000 100000 10000000.2
10 100 1000 10000 100000 10000000.2
10 100 1000 10000 100000 10000000.2
0.4
0.6
0.8
1.0
10 100 1000 10000 100000 10000000.2
0.4
0.6
0.8
1.0t t
g1i/t
1/2
t
g1i/t
1/2
t
N=200N=350
Slip-links vs MD
101
102
103
104
105
106
0.4
0.6
0.8
1
(g1(t
)/σ
2)/
(t/τ
)0.5
101
102
103
104
0.4
0.6
0.8
1
1.2
(a) (c)
Central
Peripheral
N=200 N=50
101
102
103
104
105
106
t/τ
10-1
100
(G(t
)/ε/
σ3)(
t/τ)
0.5
101
102
103
104
t/τ
10-1
100
(b) (d)
τe
S.K. Sukumaran and AEL, Macromolecules 2009, 42, 4300–4309
Mismatch between static properties in MD and slip-springs
)( 2RR ji >−<
||
)(
ji
RR ji
−
>−<
Additional interactions in slip-spring model
Static properties of the melt in MD
||
)( 2
ji
RR ji
−
>−<
>−−=< ++++ ))(()( 11 sisiii RRRRsP
N=50, slip-spring model with additional potential
Simulations of Langevin equations with memory
N=200, slip-spring model with additional potential and with memory
S.K. Sukumaran and AEL, Macromolecules 2009, 42, 4300–4309
Stress relaxation for N=200, slip-spring model with additional potential and with memory
)(tGt
S.K. Sukumaran and AEL, Macromolecules 2009, 42, 4300–4309
Slip-springs model summary
•Large deformation / fast flows
•Slip-spring strength should change accordingly
(at least in networks: Panyukov-Rubinstein Macromolecules 2002)
Advantages comparing to the tube model:•one simple consistent model with specified assumptions and few parameters
to describe many experiments simultaneously.
Problems
• but what about newly created slip-springs,
• and how do existing ones relax back to equilibrium
•In flows much faster than reptation time all slip-springs will disappear,
but the chains still do not obey Rouse – creation algorithm should
change in non-linear regime
•Branch point motion, especially in non-linear regime
Reason: slip-springs is still an empirical model
Theory of mean paths
Free energy of the mean path
010=avτ
110=avτ
210=avτ
310=avτ
410=avτ
510=avτ
Minimal distance between mean paths
sq
ua
re m
inim
al d
ista
nce
time
sq
ua
re m
inim
al d
ista
nce
E. W. Dijkstra, Numerische
Mathematik, 1, 269 (1959).
Period of Entanglement for
pair of chain
MD time step
Number of Entanglem
ents
300280260240220200180160140120100806040200
3.0E+0
2.0E+0
1.0E+0
0.0E+0
3
2
1
Period of Entanglement for
pair of chainNumber of Entanglem
ents
Number of Entanglem
ents
300280260240220200180160140120100806040200
3.0E+0
2.0E+0
1.0E+0
0.0E+0
3
2
1
MD time step
Entangled mean paths
Distribution of entanglement strengths
<d> is the average distance fluctuations
during entanglement
sNd ~><
Cut-off
Entanglement survival probability
1
G(t
) an
d P
(t)
G(t)
d2<1
100,3 == Nkb
100 1000 10000 100000
0.1
G(t
) an
d P
(t)
t
d <1
1<d2<1.5
1.5<d2<2
2<d2<2.5
2.5<d2<3
d2>3
(arbitrary vertical shift applied)
0.1
1G
(t)
an
d P
(t)
G(t)
Entanglement survival probability
350,0 == Nkb
1000 10000 100000
0.01
G(t
) a
nd P
(t)
t
G(t)
d2<9
d2<6
d2<5
d2<4
d2<3
(arbitrary vertical shift applied)
Entanglement density along the chain
1.0
1.1
1.2
1.3
1.4
1.5norm
aliz
ed d
ensity o
f enta
ngle
ments 150,3 == Nkb
0 20 40 60 80 100 120 1400.5
0.6
0.7
0.8
0.9
1.0
norm
aliz
ed d
ensity o
f enta
ngle
ments
monomer number
d2<2
2<d2<3
3<d2<4
Segment survival function
0.6
0.8
1.0(s
,t)
t0=1; (psi0)
t1=16; (psi1)
t2=32; (psi2)
t3=64; (psi3)
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4ψ(s
,t)
s=|i/n-0.5|
Fit with Doi-Edwards solution
6.5E-1
6.0E-1
5.5E-1
5.0E-1
4.5E-1
4.0E-1
3.5E-1
0.0E+0 2.7E-1
psi
4.5E-14.0E-13.5E-13.0E-12.5E-12.0E-11.5E-11.0E-15.0E-26.9E-17
psi
3.5E-1
3.0E-1
2.5E-1
2.0E-1
1.5E-1
1.0E-1
5.0E-2
1.4E-17
-5.0E-2
0.0E+0 2.7E-1
No constraint release
0.50
0.45
0.40
0.35
0.30
5% of free chains
in the 95% fixed ends environment
psi
0.450.400.350.300.250.200.150.100.050.00
psi
0.25
0.20
0.15
0.10
0.05
0.00
s
Mean-square displacementof entanglement point
100
mea
n-s
qua
re d
ispla
cem
ent
100<t<120, da<1.5
100<t<120, any da
200<t<220, da<1.5
- slope 1/4
- msd of the middle monomer
- (msd of the middle monomer)/3
- g1 of difference between r2 and r1
order of magnitude for the
100 1000 10000 100000 1000000
1
10
τe
mea
n-s
qua
re d
ispla
cem
ent
t
order of magnitude for the
square of tube diameter
τd
Conclusions.
•Slip-springs model agrees with
molecular dynamics very well
•BUT to describe early time you
probably need excluded volume
and memory functions in slip-springsand memory functions in slip-springs
•Long lived tight contacts definitely
exist – experimental fact.
•One needs to study their dynamics
and feed the results to slip-links or
tube model.