1.
A Coarse-grained Molecular Model
of Stress-Strain and Birefringence-
Strain Properties of glassy
Polystyrene
Kapileswar Nayak1, Daniel J. Read1, Manlio
Tassieri2,Peter J. Hine2
1 School of Applied Mathematics, Leeds
2 School of Physics and Astronomy, Leeds
2.
Bead Spring Model
Bead size, Mw=2000 (~20 monomer unit)
Random walk chain model
Entanglement Mw=18,000 (9 segment)
Point like slip-links imposed to capture the entanglement topology
During strong deformation (when the chains pull tightly across the
entanglement) the springs ‘bend’.
3.
Nonlinear Spring Potentials
2-bead potential (Finitely Extensible) 3-bead potential (Finitely Extensible)
x2
x2 1 x2
3
( ) f ln1 − 2 2
u3−bead = − 1.5α − 2 + 32( f − 0.5)
4
= − α ln1 − 2 + ln1 + 2
u 2−bead fL
L 2 L
2
x2
3 bead potential (f =0.5) gives the 2 bead + 8( f − 0.5) ln1 + − ln a + b + c + df 4
3
4
f
L2 2 3/ 2
f f
potential to within a few percent.
L=Maximum extension
Variation of ‘f’ (fraction of spring) give
x=distance between two beads
same results with <5% error.
b2=square of average bead size
2-bead potential used for most springs. f =fraction of spring
3 bead potential used for springs with a
slip-link.
3 bead potential
with variation of ‘f’
4.
Dynamic Monte Carlo Simulation Scheme
Glassy Dynamics (Plastic events): (a) Activated hopping (b) Strain-induced
hopping
Fluctuation of beads (fig a) are based on
Monte Carlo hopping scheme.
Springs fluctuate through the slip-links (fig b),
again using a Monte Carlo scheme.
(a) Hopping of single (b) Bead pass
MC hopping scheme based on ‘Detail bead through the slip-link
balance’:
p0 ( ) gives larger barrier height( ). Enhance the “glassiness” in the system
5.
Simulation Results: Monomer diffusion constant
Monomer diffusion constant (Dmon) calculated without imposing ‘slip-links’
Calculation of time scales from simulation
(chain of 72 bead)
Dmon τe (steps) τR (steps) τd (steps)
p0
4.45×104 3.4×105
0.5 0.037 696
New Results:
We have done simulations for poly-disperse (PDS) systems ((i)Shrinkage,
(ii) annealing, (iii) Further stretch, (iv)Superimposition of stress-strain curves,
(v) Stretching of 3:1 frozen samples and comparison with 4:1 annealed
samples, (vi) All the simulations (i-v) carried out with two assumptions: (a)
addition of extra slip-links (b) No addition of extra slip-links)
Extensive study of ‘distribution of internal chain lengths during stretching’
6.
Simulation Results: Shrinkage
3τe
Expt result
Birefringence fall immediately
during annealing
Initial fall in shrinkage delayed
by 3τe (~2000 time steps in
simulation)
Characteristic annealing time for
fall in shrinkage is approximately
τR.
Qualitative correspondence with
Assumption: Addition of extra slip-links
experimental results.
7.
Simulation Results: Further Stretch
125 Bead
PDS
40
Annealed samples used for further stretch.
Expt result
Simulation time of stretching (equivalent to expt.) 30
True Stress (MPa)
depends upon p0
20
There is fall in initial stress
10
Strain hardening curves moves to true strain axis o f fse t b ire f rin g e n c e
0 0 .0 0 8 8 0 .2 7
0 0 .0 1 0 .3 1 5
-1 0 .0 0 6 4 4 0 .1 8 5
0 0 .0 0 5 6 7 0 .1 4
0 0 0 .0 0 6 4 2 0 .1 8 2
-1 0 .0 0 4 0 4 0 .0 8
Simulation results correspond well with experiment 0 0 .2 0 .4 0 .6 0 .8
T ru e S tra in
8.
Superimposition of Stress-Strain curves
PDS 125 Bead
Superimposition of curves is very
good for lower annealed samples Expt result
Longer annealed samples no longer
superimpose
This corresponds well with experiment
9.
Comparison of 3:1 frozen, 4:1 annealed samples
PDS 125 Bead
40
4:1 drawn samples, annealed for 30,000 (time
Expt result
steps), with shrinkages (PDS:3.04, 125-beads: 30
True Stress (MPa)
3.1) respectively, used for further stretch.
3:1 drawn sample, frozen immediately, used 20
for further stretch.
10
Compare 4:1 and 3:1 samples by shifting the 3:1 frozen
4:1 67mins
curves so that, the strain hardening region 0
0 0.4 0.8 1.2
superposed. True Strain
10.
Variation of simulation parameters
Hop length variation:Decreasing the hop
length delay the strain hardening process
Strain-induced hop variation:
More strain induced hops delays the strain
hardening
Decreasing p0 (hop probability) does not
change the curves for the majority of the strain
Hop length p0=0.005 p0=0.001 p0=0.0001
τe
0.5 696 (steps) 1590 (steps) 8077 82157
0.1 13000 31657 166523 -
11.
Variation of cut-off parameter, initial configuration
Break down of simulation,
sudden rise high stress
Results strongly depends upon the
nature of the spring potential (cut-off
parameter in spring potential near the
finite extensibility limit)
Strong strain hardening region is
sensitive to the initial chain configuration
Increasing size of the chain (72 to
125), initial configuration has small effect
on stress-strain curves
12.
Distribution of Internal chain lengths during stretching
ε=0.40 ε=0.53 ε=0.66
ε=0.80 ε=0.93
ε=0.87
Contour plots
represents the end-
to-end probability
distribution of
segments with
increasing size (at
different strain)
There is a growth
of peak at finite
extensible limit (near
end-to-end
distance=1.0),
indicates some of the
bonds are highly
stressed, contributes
90% of the overall
ε=1.1
ε=1.0 ε=1.06 stress.
13.
Conclusions & Future Works
Present model qualitatively corresponds with experiment.
However, optimization of the simulation of simulation parameters
required.
Checks on parameter sensitivity.
Look for quantitative match with experimental results.
Compare with detailed molecular simulation?
Aim to construct simplified constitutive model, to feed into Oxford
work.
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