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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
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’.
Nonlinear Spring Potentials
2-bead potential (Finitely Extensible)         3-bead potential (Finitely Extensible)
                                                                                                   x2 
                      x2  1        x2 
                3
                                                              (                        )   f ln1 − 2 2 
                                                 u3−bead = − 1.5α − 2 + 32( f − 0.5)
                                                                                   4
             = − α ln1 − 2  + ln1 + 2 
  u 2−bead                                                                                        fL
                      L 2          L                                                               
                2                     
                                                              x2 
  3 bead potential (f =0.5) gives the 2 bead + 8( f − 0.5) ln1 +  − 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’
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
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’
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.
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
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
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
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     -
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
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.
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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Bradford 2nd April

  • 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 ln1 − 2 2  u3−bead = − 1.5α − 2 + 32( f − 0.5) 4 = − α ln1 − 2  + ln1 + 2  u 2−bead  fL  L 2  L   2      x2  3 bead potential (f =0.5) gives the 2 bead + 8( f − 0.5) ln1 +  − 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.