Transcript Dynamical and structural changes in a polymer glass

Microstructure of a
polymer glass subjected
to instantaneous shear
strains
Matthew L. Wallace and Béla Joós
Michael Plischke
Introduction
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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,
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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
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Outline
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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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MD Polymer Glass
L-J
• 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)
L-J
L-J
L-J
• Competing length scales prevent
crystallization
FENE
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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)
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P
Final States
Initial state
TG
T
Numerical algorithms
• Equilibrate in the NVT
ensemble with Brownian
dynamics as a thermostat
d 2 xi
U i
dxi
m 2 
 m
 Wi (t )
xi
dt
dt
• Apply a steady compression
rate of 0.015
• Final volume realized in the
NPT ensemble with a
damped-force algorithm
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external “piston” force
regulates pressure
The glass transition temperature TG
• 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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Φ: Packing Fraction
Rigidity of Mechanical Structures
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Onset of mechanical rigidity
in disordered systems
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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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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The entropic spring
force =
3k BT
R
2
Na
It is a Gaussian spring (zero equilibrium length) whose strength is
proportional to the temperature T
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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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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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Calculating the shear viscosity
• Using the intrinsic fluctuations in the system:
The shear viscosity equals:
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Viscosity diverges at onset of rigidity
• measured to T=0.49 > TG=0.465
extrapolation required
Empirical models of  :
• VFT (Vogel-Fulcher-Tamann)
 Eact 

 ~  exp
T “ideal”
T0 
(T0 associated withan
glass state)
T0 = 0.41 + 0.02 Tc=0.422 + 0.006
• dynamical
scaling
 T  TC
 ~ 
 TC
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9

E
 exp  (Colby, 2000)
 kT 

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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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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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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The shear modulus G'eq , Geq , and μs
G'eq : short time
(t=150)
Rigidity onset at T1 =0.44 < TG = 0.465
Geq : extrapolated
to infinity*
μs : applied shear
* using distribution
of energy barriers observed
during first t=150
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Meaning of T1: the onset of rigidity
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 ?)
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T1
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)
•Onset of rigidity: divergence of viscosity, onset of shear
modulus, cusp in heat capacity (disappearance of floppy modes)
•Comparison with gelation due to permanent crosslinks: no
clearly defined length scale, but there could be a dynamical one
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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 .
Wallace and Joos, PRL 2006
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Samples used to
investigate effects
of shear (present
work)
T1
TG
TMC
Plastic and elastic deformations
•
Glassy systems have a clear yield
strain
What specific local dynamical and
structural changes occur?
0,90
0,85
0,80
0,75
P
•
0,70
0,65
0,60
Plastic
0,55
0,00
0,05
0,10
0,15
0,20
0,25

Pressure variations in an NVT
ensemble
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Decay of the shear stress after deformation
Shows both the initial
stress and the
subsequent decay in
the system
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Structural changes (1)
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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
Total Energy
FENE potential
-4,80
LJ potential
11,05
-4,86
-4,92
0,00
0,08

0,16
0,24 0,00
0,08

0,16
0,24
Immediately after deformation
After tw=103 time units
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15,960
15,956
11,10
eIS
•
11,15
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
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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
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Diffusion

Effect of "caging" observed near the transition (T G = 0.465).
At TG, still possibility to rearrange under deformation.
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
2
1 d 1 N
D
 ri (t )  r i (0) 


6 dt  N i 1

Glasses are heterogeneous
Propensity: Mean squared
deviation of the displacements of
a particle in different isoconfigurations
The propensity reveals more
acurately the fast and slow regions
than a single run
Widmer-Cooper, Harrowel, Fynewever, PRL 2004
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Mobility and “sub-diffusion”
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2
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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
0,01
plastic
1E-3
elastic / reference
1E-4
<r (t)>
•
fraction of mobile particles
0,1
100
1000
beginning of sub-diffusion
plastic
0,1
elastic / reference
100
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t
1000
10000
Heterogeneous dynamics
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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
0,5
0,4
short times
0,3
P(r)
•
fixed mobile
0,2
long times
0,1
0,0
0,0
0,5
r
1,0
1,5
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Cooperative movement
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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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Structural changes (3)
• 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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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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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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Average distance between fast particles
fast particles
slow 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
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Distances between particles
There is homogenization with applied deformation, most evident
with the fast particles
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Glasses age!
Kob, 2000
Bouchaud, 2000
Glasses evolve towards lower
energy states: consequently
longer relaxation times
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Incoherent intermediate
scattering function:
Cq (t w ,  t w ) 
1
N



exp
i
q
 r j t w     r j (t w ) 

N
j 1
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?
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Irreversible microstructural changes
Polymers shrink after deformation
Reduction in grain size or correlations in
inhomogeneities
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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
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Heterogeneous dynamics
0,5
elastic / reference
0,4
short times
0,3
2,4
fixed mobile
2(t)
•
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
P(r)
•
0,2
1,6
plastic
long times
0,1
0,8
0,0
0,0
0,5
r
1,0
1,5
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100
t
1000
Conclusion
With permanent crosslinks
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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
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?
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Discussion on “overaging”
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Evidence that the phenomenon is universal (Experiments on colloids, computer simulations
on a polymer glass, similar results on LJ binary mixtures)
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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)
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Increase in pressure
The increase in
order is at the
expense of the
potential energy
Note again the
two regimes,
elastic and
plastic
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Viassnoff and
Lequeux
Experiments on dense purely repulsive
colloids
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Phys. Rev. Lett. 89, 065701 (2002)
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)
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Affine deformation
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Two regimes: elastic and plastic
Main curves:
Plastic   0.2
(rejuvenation +
overaging)
Inset:
Elastic  0.05
overaging)
Effect of
repeated
deformations
Blue: first
tw=0: solid line
tw=103: dashed
line
Red: second
Observe increasingly longer relaxation times
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To calculate µ
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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
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εtot
(1 direction)
2ε
ε
tw
tw
time
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