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Published on January 9, 2008

Author: Virginia

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Microstructure of a polymer glass subjected to instantaneous shear strains:  Microstructure of a polymer glass subjected to instantaneous shear strains Matthew L. Wallace and Béla Joós Michael Plischke Introduction:  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, The issues:  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 Outline:  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. MD Polymer Glass:  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 L-J L-J L-J Approaching the Glass Transition:  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 TG, the system is closer to equilibrium (less aging) Numerical algorithms:  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 Numerical algorithms The glass transition temperature TG:  The glass transition temperature TG Φ: Packing Fraction At TG, there is kinetic arrest, the liquid can no longer change configurations (expt. time scale issue). TG determined by a change in the volume density. We obtain TG = 0.465 + 0.005 But we cannot assume TG to be the rigidity onset: the viscosity does not diverge at TG. Rigidity of Mechanical Structures:  Rigidity of Mechanical Structures Onset of mechanical rigidity:  Onset of mechanical rigidity Triangular lattice: geometric percolation at p=pc (0.349), rigidity percolation p= pr > pc (pr = 0.66) . Multiple connectivity required for mechanical rigidity in disordered systems Entropic rigidity:  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 The entropic spring:  The entropic spring force = It is a Gaussian spring (zero equilibrium length) whose strength is proportional to the temperature T The onset of rigidity in melts:  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 pc where there is a diverging length scale (such as in rubber) Rigidity in melts without crosslinks:  Rigidity in melts without crosslinks Not clear where the onset is Is it at TG that we have percolating regions of “jammed” or immobile particles that can carry the strain? Wallace, Joos, Plischke, PRE 2004 Calculating the shear viscosity:  Calculating the shear viscosity Using the intrinsic fluctuations in the system: The shear viscosity equals: Viscosity diverges at onset of rigidity:  Viscosity diverges at onset of rigidity  measured to T=0.49 > TG=0.465 extrapolation required Calculating the shear modulus:  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 Geq where Geq or extrapolating G(t) to infinity:  Geq or extrapolating G(t) to infinity Power law fit of tail: G(t) = Geq + A t- G'eq = G(t=150) Geq = G(t=) The shear modulus : Geq vs s:  The shear modulus : Geq vs s s (=0.1) < < Geq These µ’s are the response of the system to the finite deformation and not the shear modulus of the deformed relaxed system The shear modulus G'eq , Geq , and μs:  The shear modulus G'eq , Geq , and μs G'eq : short time (t=150) Geq : extrapolated to infinity* μs : applied shear Rigidity onset at T1 =0.44 < TG = 0.465 * using distribution of energy barriers observed during first t=150 Meaning of T1: the onset of rigidity :  Meaning of T1: the onset of rigidity T1 T0 (0.41) and Tc (0.422) gave extrapolated values for the onset of rigidity. Measurement of  stopped at 0.49 (TG = 0.465) T1 = 0.44 is the onset of Geq and s, and the cusp in CP, the heat capacity (is it the appearance of floppy modes with rising T ?) Issues on rigidity in the polymer glass:  Issues on rigidity in the polymer glass TG is the temperature at which the melt stops flowing. It is not a point of divergence of the viscosity (For glass makers: s= 1012 Pa ·s or  = s / G = 400 s for SiO2 In simulations: s= 107 or  = s / G = 105 (simulations  103, 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) Properties of the deformed “rigid” glassy system:  Properties of the deformed “rigid” glassy system Glassy system just below a temperature T1 (“rigidity threshold”): very little cooperative movement (except at long timescales) Previous study: examining mechanical properties of a polymer glass (e.g. shear modulus) across TG . TG T1 TMC Samples used to investigate effects of shear (present work) Wallace and Joos, PRL 2006 Plastic and elastic deformations:  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 Decay of the shear stress after deformation:  Decay of the shear stress after deformation Shows both the initial stress and the subsequent decay in the system Structural changes (1):  Structural changes (1) Changes in the energy of the inherent structures (eIS) 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 tw=103 time units Local bond-orientational order parameter Q6:  Local bond-orientational order parameter Q6 Order parameter proposed by Steinhardt, Nelson and Ronchetti (1983) Used by Torquato et al. on disordered materials to study packing Structural changes (2):  Structural changes (2) Q6 measures subtle angular correlations (towards an FCC structure) between particles at long time tw after deformations We can resolve a clear increase in Q6 for elastic deformations, but limited impact on system dynamics Diffusion:  Diffusion Effect of "caging" observed near the transition (T G = 0.465). At TG, still possibility to rearrange under deformation. Glasses are heterogeneous:  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 Mobility and “sub-diffusion”:  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 Heterogeneous dynamics:  Heterogeneous dynamics The non-Gaussian parameter α2(t) indicates a decrease in deviations from Gaussian behavior Deviations from a Gaussian distribution become less apparent for plastic deformations Cooperative movement:  Cooperative movement The dynamical heterogeneity is spatially correlated The peak of α2(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. Structural changes (3):  Structural changes (3) Based on changes in L-J potentials and the formation of larger mobile clusters, plastic deformations must induce substantial local reconfigurations Fraction of nearest neighbours which :  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 Fraction of n-n’s on the same chain:  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 Average distance between fast particles:  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 Distances between particles:  Distances between particles There is homogenization with applied deformation, most evident with the fast particles Glasses age!:  Glasses age! Glasses evolve towards lower energy states: consequently longer relaxation times Incoherent intermediate scattering function: Bouchaud, 2000 Kob, 2000 On route to irreversible changes:  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? Irreversible microstructural changes:  Irreversible microstructural changes Polymers shrink after deformation Reduction in grain size or correlations in inhomogeneities Conclusion:  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 Heterogeneous dynamics:  Heterogeneous dynamics The non-Gaussian parameter α2(t) indicates a decrease in deviations from Gaussian behavior Deviations from a Gaussian distribution become less apparent for plastic deformations Conclusion:  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? Discussion on “overaging”:  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) Increase in pressure:  Increase in pressure The increase in order is at the expense of the potential energy Note again the two regimes, elastic and plastic Viassnoff and Lequeux:  Viassnoff and Lequeux Phys. Rev. Lett. 89, 065701 (2002) Experiments on dense purely repulsive colloids Mechanical vs entropic rigidity:  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) Affine deformation:  Affine deformation Two regimes: elastic and plastic :  Two regimes: elastic and plastic Effect of repeated deformations Blue: first tw=0: solid line tw=103: dashed line Red: second Main curves: Plastic e = 0.2 (rejuvenation + overaging) Inset: Elastic e =0.05 (overaging) Observe increasingly longer relaxation times To calculate µ :  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 tw each time Process repeated for different values of ε time εtot (1 direction)

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