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Microstructure of
a soft glass
Béla Joós
Matthew L. WallaceMichael Plischke (SFU)
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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
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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)
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Queen's CSE Colloquium, October 2007
Mechanical vs entropic rigidity
• Triangular lattice: geometric percolation at p=pc (0.349), rigidity percolation p= pr > pc (pr = 0.66) .
• Multiple connectivity required for mechanical rigidity
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Queen's CSE Colloquium, October 2007
The Glass transition
• The glass-maker’s viewpoint:
at TG viscosity= 1012 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)
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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)
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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 Cp, critical slowing down
The three viewpoints have common features (slowing down), but very different views of the glass. How to distinguish them?
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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
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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,
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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
L-J
L-J
L-J
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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 TG, the system is closer to equilibrium (less aging)
P
T
Initial state
TG
Final States
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Queen's CSE Colloquium, October 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
)(2
2
tWdt
dxm
x
U
dt
xdm i
i
i
ii
external “piston” force regulates pressure
Numerical algorithms
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Queen's CSE Colloquium, October 2007
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.
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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.
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Queen's CSE Colloquium, October 2007
Rigidity of Mechanical Structures
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Queen's CSE Colloquium, October 2007
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
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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
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Queen's CSE Colloquium, October 2007
The entropic spring
RNa
TkB2
3force =
It is a Gaussian spring (zero equilibrium length) whose strength is proportional to the temperature T
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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 pc where there is a diverging length scale (such as in rubber)
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Queen's CSE Colloquium, October 2007
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
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Queen's CSE Colloquium, October 2007
Calculating the shear viscosity
• Using the intrinsic fluctuations in the system:
The shear viscosity equals:
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Queen's CSE Colloquium, October 2007
Viscosity diverges at onset of rigidity
Empirical models of :• VFT (Vogel-Fulcher-
Tamann)
(T0 associated with an “ideal” glass state)
T0 = 0.41 + 0.02 Tc=0.422 + 0.006
• dynamical scaling (Colby, 2000)
0
exp~TT
Eact
kT
E
T
TT
C
C exp~9
measured to T=0.49 > TG=0.465 extrapolation required
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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 Geq
where
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Queen's CSE Colloquium, October 2007
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=)
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Queen's CSE Colloquium, October 2007
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
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Queen's CSE Colloquium, October 2007
The shear modulus G'eq , Geq , and μs
G'eq : short time(t=150)
Geq : extrapolatedto infinity*
μs : applied shear
Rigidity onset at T1 =0.44 < TG = 0.465
* using distribution of energy barriers observed during first t=150
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Queen's CSE Colloquium, October 2007
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 ofGeq and s, and the cusp in CP, the heat capacity(is it the appearance of floppy modes with rising T ?)
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Queen's CSE Colloquium, October 2007
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)
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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
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Queen's CSE Colloquium, October 2007
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 TMCSamples used to investigate effects of shear (present work)
Wallace and Joos, PRL 2006
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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?
0,00 0,05 0,10 0,15 0,20 0,25
0,55
0,60
0,65
0,70
0,75
0,80
0,85
0,90
P
Pressure variations in an NVT ensemble
Plastic
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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
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Queen's CSE Colloquium, October 2007
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
0,00 0,08 0,16 0,24
11,05
11,10
11,15
Total Energy
e IS
15,956
15,960FENE potential
0,00 0,08 0,16 0,24
-4,92
-4,86
-4,80LJ potential
Immediately after deformationAfter tw=103 time units
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Queen's CSE Colloquium, October 2007
Local bond-orientational order parameter Q6
• 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
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Queen's CSE Colloquium, October 2007
Diffusion
N
i
ii rtrNdt
d
1
2)0()(
1
6
1D
Effect of "caging" observed near the transition (T G = 0.465).
At TG, still possibility to rearrange under deformation.
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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
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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
100 1000 10000
0,1
plastic
elastic / reference
beginning of sub-diffusion
<r
2 (t)>
t
100 10001E-4
1E-3
0,01
0,1
plastic
elastic / reference
fra
ctio
n o
f m
ob
ile p
art
icle
s
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Queen's CSE Colloquium, October 2007
Heterogeneous dynamics
• The non-Gaussian parameter α2(t) measures deviations from Gaussian behavior
• Deviations from a Gaussian distribution become less apparent for plastic deformations
0,0 0,5 1,0 1,50,0
0,1
0,2
0,3
0,4
0,5
long times
short times
fixed mobile
P(
r)
r
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Queen's CSE Colloquium, October 2007
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.
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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
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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
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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
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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
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Queen's CSE Colloquium, October 2007
Distances between particles
There is homogenization with applied deformation, most evident
with the fast particles
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Queen's CSE Colloquium, October 2007
Glasses age!
Glasses evolve towards lower energy states: consequently longer relaxation times
N
jwjwj
wwq
trtrqiN
ttC
1
)(exp1
),(
Incoherent intermediate scattering function:
Bouchaud, 2000
Kob, 2000
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Queen's CSE Colloquium, October 2007
On route to irreversible changes
Statistics of big jumpsshow accelerated equilibrium for large ε, but also that fast regions become smaller.
More stable glass, less aging?
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Queen's CSE Colloquium, October 2007
Irreversible microstructural changes
Polymers shrink after deformation
Reduction in grain size or correlations in inhomogeneities
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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
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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