1. Microstructure of a polymer glass subjected to instantaneous shear strains Matthew L. Wallace and Béla Joós Michael Plischke

2. UBC Vancouver, July 2007 Introduction Model: a short chain polymer melt (10 monomers) Different types of rigidity transitions The glass transition and the onset of rigidity Shearing the glass: the elastic and plastic regimes Microstructure of the deformed glass: displacements, stresses,

3. UBC Vancouver, July 2007 The issues Polymer glass under deformation Glasses are heterogeneous What happens to the glass when deformed: a lot of questions from aging, mechanical properties, and thermal properties Which properties are we interested in this study? We will focus on the microstructure as a first step in understanding the effect of deformation on the properties of the glass. Main message: deformation reduces heterogeneity

4. UBC Vancouver, July 2007 Outline Our way of preparing the polymer melt near the glass transition: pressure quench at constant temperature to improve statistics Onset of rigidity in the glass: a new angle on the glass transition Deforming the glass below the rigidity transition: the elastic and plastic regime Macroscopic signatures Changes in the microstructure What is learned, what needs to be learned.

5. UBC Vancouver, July 2007 MD Polymer Glass Polymer “melt” of ~1000 particles with chains of length 10. LJ interactions between all particles + FENE potential between nearest neighbours in a chain ( Kremer and Grest, 1990) Competing length scales prevent crystallization FENE L-J

6. UBC Vancouver, July 2007 Approaching the Glass Transition Instead of approaching the final states along isobars by lowering T (very high cooling rates) We propose an isothermal compression method (blue curves) for better exploration of phase space System gets “stuck” in wells of lower P.E. Below T G, the system is closer to equilibrium (less aging)

7. UBC Vancouver, July 2007 Equilibrate in the NVT ensemble with Brownian dynamics as a thermostat Apply a steady compression rate of 0.015 Final volume realized in the NPT ensemble with a damped-force algorithm external “piston” force regulates pressure Numerical algorithms

8. UBC Vancouver, July 2007 The glass transition temperature T G Φ: Packing Fraction At T G, there is kinetic arrest, the liquid can no longer change configurations (expt. time scale issue). T G determined by a change in the volume density. We obtain T G = 0.465 + 0.005 But we cannot assume T G to be the rigidity onset: the viscosity does not diverge at T G.

9. UBC Vancouver, July 2007 Rigidity of Mechanical Structures

10. UBC Vancouver, July 2007 Onset of mechanical rigidity Triangular lattice: geometric percolation at p=p c (0.349), rigidity percolation p= p r > p c (p r = 0.66). Multiple connectivity required for mechanical rigidity in disordered systems

11. UBC Vancouver, July 2007 Entropic rigidity At T>0 K, rigidity sets in at the onset of geometric percolation, through the creation of an entropic spring Plischke and Joos, PRL 1998 Moukarzel and Duxbury, PRE 1999

12. UBC Vancouver, July 2007 The entropic spring force = It is a Gaussian spring (zero equilibrium length) whose strength is proportional to the temperature T

13. UBC Vancouver, July 2007 The onset of rigidity in melts With permanent crosslinks, at a fixed temperature: Well defined point of onset of the entropic rigidity : It is geometric percolation p c where there is a diverging length scale (such as in rubber)

14. UBC Vancouver, July 2007 Rigidity in melts without crosslinks Not clear where the onset is Is it at T G that we have percolating regions of “jammed” or immobile particles that can carry the strain? Wallace, Joos, Plischke, PRE 2004

15. UBC Vancouver, July 2007 Calculating the shear viscosity Using the intrinsic fluctuations in the system: The shear viscosity equals:

16. UBC Vancouver, July 2007 Viscosity diverges at onset of rigidity Empirical models of  : VFT (Vogel-Fulcher-Tamann) (T 0 associated with an “ideal” glass state) T 0 = 0.41 + 0.02 T c =0.422 + 0.006 dynamical scaling (Colby, 2000)  measured to T=0.49 > T G =0.465 extrapolation required

17. UBC Vancouver, July 2007 Calculating the shear modulus Two ways: Applying a finite affine deformation Using the intrinsic fluctuations in the system driven by temperature to obtain its shear strength, as the limit to ∞ of G(t) called G eq where

18. UBC Vancouver, July 2007 G eq or extrapolating G(t) to infinity Power law fit of tail: G(t) = G eq + A t -  G' eq = G(t=150) G eq = G(t=  )

19. UBC Vancouver, July 2007 The shear modulus : G eq vs  s  s (  =0.1) < < G eq These µ’s are the response of the system to the finite deformation and not the shear modulus of the deformed relaxed system

20. UBC Vancouver, July 2007 The shear modulus G' eq, G eq, and μ s G' eq : short time (t=150) G eq : extrapolated to infinity* μ s : applied shear Rigidity onset at T 1 =0.44 < T G = 0.465 * using distribution of energy barriers observed during first t=150

21. UBC Vancouver, July 2007 Meaning of T 1 : the onset of rigidity T1T1 T 0 (0.41) and T c (0.422) gave extrapolated values for the onset of rigidity. Measurement of  stopped at 0.49 (T G = 0.465) T 1 = 0.44 is the onset of G eq and  s, and the cusp in C P, the heat capacity (is it the appearance of floppy modes with rising T ?)

22. UBC Vancouver, July 2007 Issues on rigidity in the polymer glass T G is the temperature at which the melt stops flowing. It is not a point of divergence of the viscosity (For glass makers:  s = 10 12 Pa ·s or  =  s / G  = 400 s for SiO 2 In simulations:  s = 10 7 or  =  s / G  = 10 5 (simulations  10 3, unit of time:  2 ps) (issues of time scale and aging) Comparison with gelation due to permanent crosslinks: no clearly defined length scale, but there could be a dynamical one Onset of rigidity: divergence of viscosity, onset of shear modulus, cusp in heat capacity (disappearance of floppy modes)

23. UBC Vancouver, July 2007 Properties of the deformed “rigid” glassy system Glassy system just below a temperature T 1 (“rigidity threshold”): very little cooperative movement (except at long timescales) Previous study: examining mechanical properties of a polymer glass (e.g. shear modulus) across T G. TGTG T1T1 T MC Samples used to investigate effects of shear (present work) Wallace and Joos, PRL 2006

24. UBC Vancouver, July 2007 Plastic and elastic deformations Glassy systems have a clear yield strain What specific local dynamical and structural changes occur? Pressure variations in an NVT ensemble Plastic

25. UBC Vancouver, July 2007 Decay of the shear stress after deformation Shows both the initial stress and the subsequent decay in the system

26. UBC Vancouver, July 2007 Structural changes (1) Changes in the energy of the inherent structures (e IS ) are relevant to subtle structural changes Initial decrease / increase in polymer bond length for elastic / plastic deformations Plastic deformations create a new “well” in the PEL – different from those explored by slow relaxations in a normal aging process In “relaxed”, deformed system, changes in the energy landscape are entirely due to L-J interactions Immediately after deformation After t w =10 3 time units

27. UBC Vancouver, July 2007 Local bond-orientational order parameter Q 6 Order parameter proposed by Steinhardt, Nelson and Ronchetti (1983) Used by Torquato et al. on disordered materials to study packing

28. UBC Vancouver, July 2007 Structural changes (2) Q 6 measures subtle angular correlations (towards an FCC structure) between particles at long time t w after deformations We can resolve a clear increase in Q 6 for elastic deformations, but limited impact on system dynamics

29. UBC Vancouver, July 2007 Diffusion Effect of "caging" observed near the transition (T G = 0.465). At T G, still possibility to rearrange under deformation.

30. UBC Vancouver, July 2007 Glasses are heterogeneous Widmer-Cooper, Harrowel, Fynewever, PRL 2004 The propensity reveals more acurately the fast and slow regions than a single run Propensity: Mean squared deviation of the displacements of a particle in different iso- configurations

31. UBC Vancouver, July 2007 Mobility and “sub-diffusion” Initially, plastic shear forces the creation of “mobile” regions of mobile particles Once the system is allowed to relax, cooperative re-arrangements remain possible Rearrangements from plastic deformations allow cage escape in more regions In the case of elastic deformations, new mobile particles can be created, but only temporarily

32. UBC Vancouver, July 2007 Heterogeneous dynamics The non-Gaussian parameter α 2 (t) indicates a decrease in deviations from Gaussian behavior Deviations from a Gaussian distribution become less apparent for plastic deformations

33. UBC Vancouver, July 2007 Cooperative movement The dynamical heterogeneity is spatially correlated The peak of α 2 (t) coincides with the beginning of sub-diffusive behavior – can indicate a maximum in “mobile cluster” size Snapshots of dynamically heterogeneous systems. Left: the clusters are localized. Right: as cluster size increases, significant large-scale relaxation is possible.

34. UBC Vancouver, July 2007 Structural changes (3) Based on changes in L-J potentials and the formation of larger mobile clusters, plastic deformations must induce substantial local reconfigurations

35. UBC Vancouver, July 2007 Fraction of nearest neighbours which are the fastest 5% the slowest 5% ε = 0, reference system, ε = 0.2, smaller domains of fast and slow particles

36. UBC Vancouver, July 2007 Fraction of n-n’s on the same chain which are the fastest which are the slowest 5% This means that the islands of fast particles are getting smaller

37. UBC Vancouver, July 2007 Average distance between fast particles Evidence of reduction in size of mobile regions and increase in size of jammed regions with increasing deformation Increasing jamming in elastic region, as seen in slowest particle fast particles slow particles

38. UBC Vancouver, July 2007 Distances between particles There is homogenization with applied deformation, most evident with the fast particles

39. UBC Vancouver, July 2007 Glasses age! Glasses evolve towards lower energy states: consequently longer relaxation times Incoherent intermediate scattering function: B ouchaud, 2000 Kob, 2000

40. UBC Vancouver, July 2007 On route to irreversible changes Statistics of big jumps show accelerated equilibrium for large ε, but also that fast regions become smaller. More stable glass, less aging?

41. UBC Vancouver, July 2007 Irreversible microstructural changes Polymers shrink after deformation Reduction in grain size or correlations in inhomogeneities

42. UBC Vancouver, July 2007 Conclusion We have presented attempts to characterize the effect of deformations on the structure of the glass that did not require huge computing times The net effect of deformations appears to be connected to general “jamming” phenomena, and what the deformations can do to un-jam the structure What they reveal is a more homogeneous glass with a smaller “grain” structure More studies are required (highly computer intensive) Currently working on applying oscillating shear to the glass, and monitoring the aging of the glasses prepared by shear deformation

43. UBC Vancouver, July 2007 Heterogeneous dynamics The non-Gaussian parameter α 2 (t) indicates a decrease in deviations from Gaussian behavior Deviations from a Gaussian distribution become less apparent for plastic deformations

44. UBC Vancouver, July 2007 Conclusion The location of the onset of rigidity is well-defined in networks with permanent links. In networks with permanent links, the percolation model is as credible, if not more, than any other. Experimental and theoretical issues such as effects of the hard core to be resolved With permanent crosslinks Temperature driven system Location of the onset of rigidity determined to be below the glass transition, no clearly defined length scales. Questions of time scales and definition Under applied stress, permanent changes can occur, notions of “overaging” and “rejuvenation”. What are the structure and the properties of the “overaged” glass?

45. UBC Vancouver, July 2007 Discussion on “overaging” Evidence that the phenomenon is universal (Experiments on colloids, computer simulations on a polymer glass, similar results on LJ binary mixtures) Shear increases ordering Two distinct regimes: elastic and plastic Repeated applications of plastic deformation, in particular, yield increasingly longer relaxation times Is this a mean to achieve more homogeneous glasses? )changes in relaxation times not significant)

46. UBC Vancouver, July 2007 Increase in pressure The increase in order is at the expense of the potential energy Note again the two regimes, elastic and plastic

47. UBC Vancouver, July 2007 Viassnoff and Lequeux Phys. Rev. Lett. 89, 065701 (2002) Experiments on dense purely repulsive colloids

48. UBC Vancouver, July 2007 Mechanical vs entropic rigidity Rigidity at T=0K (Rigidity Theory and Applications, Thorpe and Duxbury eds., Plenum 1999) In essence, in unstressed systems, multiple connectivity is required for rigidity Mean field model (Maxwell counting), the onset of rigidity occurs at the point where the number of degrees of freedom equals the number of constraints (stretching and bending)

49. UBC Vancouver, July 2007 Affine deformation

50. UBC Vancouver, July 2007 Two regimes: elastic and plastic Effect of repeated deformations Blue: first t w =0: solid line t w =10 3 : dashed line Red: second Main curves: Plastic  (rejuvenation + overaging) Inset: Elastic   overaging) Observe increasingly longer relaxation times

51. UBC Vancouver, July 2007 To calculate µ We subject the glass to instantaneous, affine shear deformations (ε) These deformations can be repeated in the same or different directions (giving identical results) after letting the sample equilibrate for a waiting time t w each time Process repeated for different values of ε time ε tot (1 direction) twtw twtw 2ε ε

52. UBC Vancouver, July 2007