Pipeline Ecologies: Rural Entanglements of Fiber-Optic Cables
Entanglements and stress correlations in coarsegrained molecular dynamics
description
Transcript of Entanglements and stress correlations in coarsegrained molecular dynamics
Entanglements and stress correlations in coarsegrained
molecular dynamics
Alexei E. Likhtman, Sathish K. Sukumuran,
Jorge RamirezDepartment of Applied Mathematics,
University of Leeds, Leeds LS2 9JT, [email protected]
Hierarchical modelling in polymer dynamicsHierarchical modelling in polymer dynamics
• Constitutive equations
– Tube theories
• Single chain models
– Coarse-grained many-chains models
» Atomistic simulations
> Quantum mechanics simulations
?),(f
Dt
D
Kremer-Grest MD, Padding-Briels Twentanglemets,
NAPLES
Well established coarse-graining procedures,
force-fields, commercial packages
Traditional rheology
Traditional physics
CR
TubeModel?
The weakestlink
The missing linkThe missing link
Many chains system
+ self-consistent field + self-consistent field
One chain model
The ultimate goal: Stochastic equation of motion
for the chain in self-consistent entanglement field
Is there a tube model?Is there a tube model?
Best definition of the tube model:one-dimensional Rouse chain projected onto three-dimensional random walk tube.
Open questions:
•Can I have expression for the tube field, please?•How to “measure” tube in MD?•Is the tube semiflexible?•Diameter = persistence length?•Branch point motion•How does the contour length changes with deformation?•Tube parameters for different polymers?•Tube parameters for different concentrations?
Rubinstein-Panyukov network modelRubinstein-Panyukov network model
Rubinstein and Panyukov, Macromolecules 2002, 6670
Construction of the modelConstruction of the model
timeelementary
size coil
re temperatu
parameters model Rouse
0
2
gR
T
timeelementary
size coil
re temperatu
parameters model Rouse
0
2
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
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
ja
jm
Constraint releaseConstraint release
Hua and Schieber 1998Shanbhag, Larson, Takimoto, Doi 2001
1 10 1000.2
0.4
0.6
0.8
1.0
0.1 1 10 100 1,000103
104
105
1 10 1000.2
0.4
0.6
0.8
1.0
1k 10k 100k
6x10-5
1.2x10-4
1.8x10-4
1k 10k 100k
1E-12
1E-11
101 102 103 104 105 106 107 108 109
104
105
106
1k 10k 100k 1M
1E-11
1E-10
1k 10k 100k 1M
2x10-5
4x10-5
6x10-58x10-510-4
10k 100k 1M
5E-12
1E-11
1.5E-11
2E-11
1k 10k 100k 1M 10M
1E-10
1E-9
1k 10k
4x10-5
6x10-5
8x10-5
102 103 104 105 106 107
104
105
106
10-2 10-1 100 101 102 103 104 105 106104
105
106
1k 10k 100k 1M
1E-11
1E-10
1k 10k 100k
5x10-5
10-4
1.5x10-42x10-4
G(0)
N=2.2MPa
by extrapolation
too slow
too unstable?
experiments needed
experiments needed
q=0.05A-1
q=0.077A-1
12.4K 24.7K 190K q=0.115A-1
125K 61K 34K
s-1
q=0.03A-1
q=0.05A-1
q=0.068A-1
q=0.076A-1
q=0.096A-1
q=0.115A-1
/M
3 w
PS
PBd
PI
PEP
PE
DiffusionDM2
w (m2
/s)(g/mol)2
ViscosityG'/G''(Pa vs (s-1
))
NSES(q,t)/S(q,0) vs t (ns)
/M3
w (Pa*s/(g/mol)
3)
/ M
3 w
A.E.Likhtman, Macromolecules 2005
t, ns0,1 1 10 100
S(q
,t)/S
(q,0
)
1
0,95
0,9
0,85
0,8
0,75
0,7
0,65
0,6
0,55
0,5
0,45
0,4
0,35
0,3
0,25
0,2
0,15
0,1
0,05
2k
6k
12k
Mwmat
Rouse
Relaxation of dilute long chains (36K) in a short matrix: constraint release
Relaxation of dilute long chains (36K) in a short matrix: constraint release
M.Zamponi et al, PRL 2006
labeled
Molecular Dynamics -- Kremer-GrestMolecular Dynamics -- Kremer-Grest
• Polymers – Bead-FENE spring chains
0
2 2
20
( ) ln 12FENE
kR rU r
R
• With excluded volume – Purely repulsive Lennard-Jones
interaction between beads
otherwise 0
2 r 4
14)( 61
612
rrrU rLJ
• k = 30/2
• R0=1.5
Density, = 0.85
Friction coefficent, = 0.5
Time step, dt = 0.012
Temperature, T = /k
K.Kremer, G. S. Grest
JCP 92 5057 (1990)
g1(t) from MD for N=100,350g1(t) from MD for N=100,350
t10 100 1,000 10,000 100,000
g1(t)
1e0
1e1
1e2
1e3 1
0.5 1/4
0.5
21 , ( , ) ( ,0)g i t i t i r r
1
11( ) 1 ,
N
i
g t g i tN
e
d
R
t10 100 1,000 10,000 100,000
g1(t)
1.1e0
1e0
9e-1
8e-1
7e-1
6e-1
5e-1
4e-1
3e-1
2e-1
g1(i,t)/t0.5 from MD for N=350g1(i,t)/t0.5 from MD for N=350g
1(i,t
)/t0
.5
ends
middle
t
t0.1 1 10 100 1,000 10,000 100,000
G(t)
1e-4
1e-3
1e-2
1e-1
1e0
1e1
G(t) from MD for N=50,100,200,350 (Ne~50)G(t) from MD for N=50,100,200,350 (Ne~50)
e
( ) ( ) (0)V
G t tkT
t1 10 100 1,000 10,000 100,000
G(t)
1e0
G(t) from MD for N=50,100,200,350 (Ne~70)
G(t) from MD for N=50,100,200,350 (Ne~70)
e
ttG )(
G(t) from MD for N=50,100,200,350 (Ne~50)G(t) from MD for N=50,100,200,350 (Ne~50)
t
10 100 1,000 10,000 100,000
g1(t)/t^0.5
1e0
9.5e-1
9e-1
8.5e-1
8e-1
7.5e-17e-1
6.5e-1
6e-1
5.5e-1
5e-1
4.5e-1
4e-1
3.5e-13e-1
g1(i,t) -- MD vs sliplinks mapping 1:1 (N=200)g1(i,t) -- MD vs sliplinks mapping 1:1 (N=200)g
1(i,t
)/t0
.5
t
1 1
0
e
d
Lines - MDPoints - slip-links
Lines - MDPoints - slip-links
t
10 100 1,000 10,000 100,000
G(t)*t^0.5
1e0
G(t) -- MD vs sliplinks mapping 1:1 (N=200)G(t) -- MD vs sliplinks mapping 1:1 (N=200)G
(t)*
t1/2
t
1 50
e
d
Lines - MDPoints - slip-links
Lines - MDPoints - slip-links
)0()()0()( virtualchainchainchain tt
)0()()0()(
)0()()0()(
virtualvirtualchainvirtual
virtualchainchainchain
tt
tt
)0()( chainchain t
Questions for discussionQuestions for discussion
• Binary nature of entanglements?– Can one propose an experiment which contradicts
this?
• Non-linear flows: – do entanglements appear in the middle of the
chain?
• Is there an instability in monodisperse linear polymers?
Log(gamma)210-1-2
Log(Sxy) 5e0
4e0