Transcript Entanglements and stress correlations in coarsegrained molecular dynamics Alexei E. Likhtman, Sathish K. Sukumuran, Jorge Ramirez Department of Applied Mathematics, University of Leeds, Leeds LS2 9JT, UK [email protected].

Entanglements and stress
correlations in coarsegrained
molecular dynamics
Alexei E. Likhtman,
Sathish K. Sukumuran,
Jorge Ramirez
Department of Applied Mathematics,
University of Leeds, Leeds LS2 9JT, UK
[email protected]
Hierarchical modelling in polymer dynamics
D
 f ( , ?)
Dt
• Constitutive equations
– Tube theories
Tube
Model?
• Single chain models
The weakest
link
– Coarse-grained
many-chains models
CR
Kremer-Grest MD,
Padding-Briels Twentanglemets,
NAPLES
» Atomistic simulations
> Quantum mechanics simulations
Well established coarsegraining procedures,
force-fields,
commercial packages
The missing link
The ultimate goal:
Stochastic equation of motion
for the chain in
self-consistent entanglement field
Many chains system
One chain model
+ self-consistent field
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 model
Rubinstein and Panyukov, Macromolecules 2002, 6670
Construction of the model
mj
aj
Rouse model paramet ers
T  temperatu
re
Rg  coil size
2
 0  element arytime
New paramet ers
N e - averagenumber of beads bet ween slip - links
N s - st rengt hof slip - link (or effect ivenumber of monomersin t heanchoringchain)
 s - frict ionof slip - link along t hechain
Constraint release
Hua and Schieber 1998
Shanbhag, Larson, Takimoto, Doi 2001
1.0
G'/G''
S(q,t)/S(q,0) vs t (ns)
(Pa vs  (s ))
w
Diffusion
3
(Pa*s/(g/mol) )
2
2
DMw (m /s)(g/mol)
2
-1
1E-11
-4
1.8x10
(0)
GN =2.2MPa
-1
q=0.077A
0.6
-4
1.2x10
by extrapolation
0.4
0.2
/M
-1
q=0.05A
0.8
PE
Viscosity
3
NSE
-5
12.4K
24.7K
190K
6x10
1E-12
q=0.115A-1
1
10
1k
100
10k
100k
1k
10k
100k
1.0
1E-10
6
PEP
3
5
0.6
10
-1
q=0.03A
-1
q=0.05A
-1
q=0.068A
-1
q=0.076A
-1
q=0.096A
-1
q=0.115A
0.4
0.2
experiments
needed
/Mw
10
0.8
1
4
1E-11
10
10
1
2
3
4
5
6
7
8
1k
9
10 10 10 10 10 10 10 10 10
100
10k
100k
10
experiments
needed
/ Mw
1E-10
PI
1M
-4
2x10
-4
1.5x10
6
-4
3
5
10
10
1E-11
-5
5x10
4
10
-2
-1
0
1
2
3
4
5
6
1k
10 10 10 10 10 10 10 10 10
10k
100k
1M
1k
10k
100k
-5
8x10
10
6
-5
6x10
1E-9
PBd
too unstable?
10
5
10
4
-5
4x10
1E-10
10
2
10
3
10
4
10
5
10
6
10
7
1k
10k
100k
1M
10M
PS
too slow
10-5
8x10
-5
6x10
1.5E-11
5
10k
-4
2E-11
10
1k
1E-11
4x10
5E-12
2x10
-5
4
10
125K
61K
34K
-5
3
100.1
1
10
100
s
-1
1,000
10k
100k
1M
A.E.Likhtman, Macromolecules 2005
1k
10k
100k
1M
Relaxation of dilute long chains (36K) in
a short matrix: constraint release
1
Mwmat
0,95
0,9
0,85
12k
6k
0,8
0,75
0,7
0,65
0,6
0,55
0,5
2k
0,45
0,4
0,35
0,3
0,25
0,2
labeled
0,15
Rouse
0,1
0,05
0,1
M.Zamponi et al, PRL 2006
1
10
t, ns
100
Molecular Dynamics -- Kremer-Grest
• Polymers – Bead-FENE
spring chains

r 
U FENE (r )  
ln 1  2 
2
 R0 
kR02
2
• k = 30/2
• R0=1.5
• With excluded volume – Purely
repulsive Lennard-Jones
interaction between beads
    1 
16


U rLJ ( r )  4      
r2
 r   r  4 


0
otherwise
12
6
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=100,350
1
1e3
g1 i, t    r(i, t )  r(i,0) 
2
d
1 N
g1(t )   g1 i, t 
N i 1
g1(t)
1e2
0.5
0.5
1e1
1/4
R
e
1e0
10
100
1,000
10,000
t
100,000
1.1e0
1e0
ends
9e-1
g1(i,t)/t0.5
g1(t)
g1(i,t)/t0.5 from MD for N=350
8e-1
7e-1
6e-1
5e-1
4e-1
middle
3e-1
2e-1
10
100
1,000
t
t
10,000
100,000
G(t)
G(t) from MD for N=50,100,200,350 (Ne~50)
1e1
e
1e0
1e-1
1e-2
1e-3
V
G (t ) 
  (t )  (0)
kT
1e-4
0.1
1
10
100
t
1,000
10,000
100,000
G(t) from MD for N=50,100,200,350
G(t) from MD for N=50,100,200,350 (Ne~50)
(Ne~70)
e
1e0
G(t ) t
1
10
100
1,000
t
10,000
100,000
g1(i,t)/t0.5
g1(i,t) -- MD vs sliplinks mapping 1:1 (N=200)
1e0
9.5e-1
9e-1
8.5e-1
8e-1
7.5e-1
7e-1
6.5e-1
6e-1
5.5e-1
5e-1
4.5e-1
4e-1
3.5e-1
3e-1
Lines - MD
Points - slip-links
1
d
1
0
e
10
100
1,000
10,000
tt
100,000
G(t) -- MD vs sliplinks mapping 1:1 (N=200)
  chain (t ) chain (0)     chain (t ) virtual (0) 
  virtual (t ) chain (0)     virtual (t ) virtual (0) 
5
01
d
G(t)*t1/2
1e0
Lines - MD
Points - slip-links
  chain (t ) chain (0) 
  chain (t ) chain (0)     chain (t ) virtual (0) 
e
10
100
1,000
t
t
10,000
100,000
Questions 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?
Log(Sxy)
• Is there an instability in monodisperse linear
polymers?
5e0
4e0
-2
-1
0
Log(gam m a)
1
2