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Bradford 2nd April Bradford 2nd April Presentation Transcript

  • 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.