Microstructure of a soft glass Béla Joós Matthew L. Wallace Michael Plischke (SFU)

Queen's CSE Colloquium, October 2007 Glass is a phase of matter Glasses are ubiquitous in nature A glass is a phase such as the solid or gaseous, or liquid phases as opposed to a type of material It is a disordered phase, amorphous, like a liquid frozen in time

Queen's CSE Colloquium, October 2007 Phase Transitions In nature there are various kinds of transitions: First order: ex: solid -> liquid, jump in physical observables such as volume, or energy Continuous: ex: Gelling transition as an example of percolation transition (the gel is rigid, i.e. resists shear)

Queen's CSE Colloquium, October 2007 Mechanical vs entropic 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

Queen's CSE Colloquium, October 2007 The Glass transition The glass-maker’s viewpoint: at T G viscosity= Pa s A continuous transition characterized by a divergence in viscosity As to what really happens microscopically, there is really no consensus. There are a number of competing pictures Conference: Mechanical Behaviour of Glassy Materials (UBC, July 2007)

Queen's CSE Colloquium, October 2007 The three viewpoints 1.A transition to an ideal zero entropy state 2.A dynamical transition resulting from the jamming of particles together 3.Not a transition but a cross-over where there is a rapid change in viscosity (critical slowing down)

Queen's CSE Colloquium, October 2007 Some facts to illustrate the issues Heat capacity: heat transferred into object as its temperature is raised In experiments: T raised by increments ΔT during time Δt Drop in C p, critical slowing down The three viewpoints have common features (slowing down), but very different views of the glass. How to distinguish them?

Queen's CSE Colloquium, October 2007 The challenges As T decreases, slowing down in the system, increasing run times to simulate anything (also an issue experimentally) Configuration space very complex often represented as an energy landscape Glasses age: they continuously evolve Glasses evolve towards lower energy states: consequently longer relaxation times Bouchaud (2000) Have to find clever ways to characterize the glass

Queen's CSE Colloquium, October 2007 Our perspective Model: a short chain polymer melt (10 monomers) (e.g. plastic) The glass transition and the onset of rigidity Shearing the glass: the elastic and plastic regimes Microstructure of the deformed glass: displacements, stresses,

Queen's CSE Colloquium, October 2007 Molecular Dynamics of a 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

Queen's CSE Colloquium, October 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 potential energy Below T G, the system is closer to equilibrium (less aging)

Queen's CSE Colloquium, October 2007 Equilibrate in the NVT ensemble with Brownian dynamics as a thermostat Apply a steady compression rate of Final volume realized in the NPT ensemble with a damped-force algorithm external “piston” force regulates pressure Numerical algorithms

Queen's CSE Colloquium, October 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 = But we cannot assume T G to be the rigidity onset: the viscosity does not diverge at T G.

Queen's CSE Colloquium, October 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.

Queen's CSE Colloquium, October 2007 Rigidity of Mechanical Structures

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 2007 The entropic spring force = It is a Gaussian spring (zero equilibrium length) whose strength is proportional to the temperature T

Queen's CSE Colloquium, October 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)

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 2007 Calculating the shear viscosity Using the intrinsic fluctuations in the system: The shear viscosity equals:

Queen's CSE Colloquium, October 2007 Viscosity diverges at onset of rigidity Empirical models of  : VFT (Vogel-Fulcher-Tamann) (T 0 associated with an “ideal” glass state) T 0 = T c = dynamical scaling (Colby, 2000)  measured to T=0.49 > T G =0.465 extrapolation required

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 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=  )

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 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 = * using distribution of energy barriers observed during first t=150

Queen's CSE Colloquium, October 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 ?)

Queen's CSE Colloquium, October 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 = 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)

Queen's CSE Colloquium, October 2007 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

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 2007 Decay of the shear stress after deformation Shows both the initial stress and the subsequent decay in the system

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 2007 Local bond-orientational order parameter Q 6 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

Queen's CSE Colloquium, October 2007 Diffusion Effect of "caging" observed near the transition (T G = 0.465). At T G, still possibility to rearrange under deformation.

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 2007 Heterogeneous dynamics The non-Gaussian parameter α 2 (t) measures deviations from Gaussian behavior Deviations from a Gaussian distribution become less apparent for plastic deformations

Queen's CSE Colloquium, October 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.

Queen's CSE Colloquium, October 2007 Effect of shear on the microstructure Based on changes in L-J potentials and the formation of larger mobile clusters, plastic deformations must induce substantial local reconfigurations

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 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

Queen's CSE Colloquium, October 2007 Distances between particles There is homogenization with applied deformation, most evident with the fast particles

Queen's CSE Colloquium, October 2007 Glasses age! Glasses evolve towards lower energy states: consequently longer relaxation times Incoherent intermediate scattering function: B ouchaud, 2000 Kob, 2000

Queen's CSE Colloquium, October 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?

Queen's CSE Colloquium, October 2007 Irreversible microstructural changes Polymers shrink after deformation Reduction in grain size or correlations in inhomogeneities

Queen's CSE Colloquium, October 2007 Conclusion (1) Real glasses versus glasses on the computer: time scales and a better grasp on the computer of the microstructure At the latest conference at UBC on glasses, there was a growing consensus that this is really an issue of critical slowing down, in other words not a real transition

Queen's CSE Colloquium, October 2007 Conclusion (2) 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