Anomalous Diffusion in Heterogeneous Glass-Forming Materials J.S. Langer University of California, Santa Barbara

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Anomalous Diffusion in Heterogeneous
Glass-Forming MaterialsJ.S. Langer University
of California, Santa Barbara
Workshop on Dynamical Heterogeneities in Glasses,
Lorentz Center, Leiden, August 2008
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Outline

• Continuous-time random walks (CTRWs) in
  heterogeneous glassy systems
• JSL and S. Mukhopadhyay PRE 77, 061505
  (2008), JSL arXiv0806.0958
• Predictions of the excitation-chain (XC) theory
  for temperature dependent parameters
• JSL PRE 73, 041504 (2006), PRL 97, 115704
  (2006)
• Comparisons with data for ortho-terphenyl (OTP)
  Diffusion, viscosity, neutron scattering
• Length scales, Stokes-Einstein violation,
  stretched exponentials

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Continuous-time random walks in heterogeneous
systems
Montroll and Shlesinger, Studies in Stat. Mech.
XI (1984) Bouchaud and Georges, Physics Reports
195, 127 (1990) Chaudhuri, Berthier, and Kob, PRL
99, 060604 (2007)
Basic assumptions to be made here
Glass-forming materials consist of glassy domains
in which a tagged molecule is frozen, and mobile
regions (of high propensity) in which it can
move.
A molecule in a glassy domain remains fixed until
it is encountered by a mobile region (as in
kinetically constrained models). The boundaries
between glassy and mobile regions diffuse on
alpha time scales.

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Two-component, two-dimensional, Lennard-Jones
glass with quasi-crystalline components (Y. Shi
and M. Falk). Blue molecules are frozen in
low-energy environments. Red molecules have
higher propensity for motion.
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Continuous-time random walks in heterogeneous
systems, contd.
The glassy waiting time distribution is
is the alpha relaxation time.
is the characteristic size of a glassy
domain. is a characteristic molecular
length arising in the XC theory. This result is
derived by averaging over a Gaussian distribution
of domain sizes. Note that is a
stretched exponential with index ½.
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Glassy waiting time distribution
scaled size of a glassy domain
lowest eigenvalue of the diffusion
kernel in a domain of size R.
Note that the undirected distance between the
tagged molecule and the domain boundary is the
diffusing variable. The domain sizes are
distributed Gaussianly, with scale size R.
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Continuous-time random walks, contd.
The mobile waiting time distribution is
where is the time scale for
diffusive jumps in a mobile region -- related to,
but not exactly the same as the beta relaxation
time.
The mobile jump-length distribution is
where r is the jump length in units R, and a is
a dimension-less parameter to be determined.
The fraction of the system occupied by mobile
regions is PM(T). The glassy fraction is P G(T)
1 - PM(T). After a mobile jump, the
probability that the molecule is still in a
mobile region is PM(T). The probability that it
has jumped into a glassy region where it has
become immobilized is P G(T).
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Continuous-time random walks mathematical results
Define distribution functions
for molecules starting
at r t 0 in glassy and mobile regions.
Their Fourier-Laplace transforms are
where
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More mathematics
has an essential singularity at u0 and a branch
cut along the negative u axis.
The self intermediate scattering function is
which requires computing the inverse Laplace
transform of a function of
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k0.8
k10
Theoretical low-temperature intermediate
scattering function for PG.5 and k 0.8
10.0 (from top to bottom). The initial
intra-cage (ballistic) behavior is not resolved
in this two-time CTRW approximation.
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slope - 1
k0.7
k10
slope -1/2
Double logarithmic plot of the low-temperature
scattering function at long times, for k 0.7
10.0 from top to bottom. Slopes -
stretched-exponential indices. Note crossover
from diffusive behavior (slope - 1) to
anomalous behavior (slope - ½) with increasing
time and/or increasing k.
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t10
t0.03
Low-temperature displacement distributions for t
0.03 10.0. Note the crossover from
exponential to Gaussian behavior at long
times and large displacements. The peak near
x0 is really a delta function in this
approximation.
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XC predictions for T-dependent parameters
kBTZ bare activation energy for density
fluctuations, STZs, etc.
for T near Tg. (Vogel-Fulcher)
R(T) l spatial extent of an excitation chain
that can activate a stable molecular
rearrangement maximum size of a stable glassy
domain.
describes chainlike molecular
displacements that enable stable transitions
between inherent states. Unlike TZ,
should be mechanism independent.
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Tg
TMC
Theoretical R(T) and glassy fraction PG(T) for
OTP
XC theory needs a correction for vanishingly
short chains at high T.


surface-to-volume ratio at low T
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First estimates of parameters from neutron
scattering data Kiebel et al, PRB 45
(1992) Wittke et al, Z.Phys B 91 (1993)
The lowest temperature reported is T 293 K
the mode-coupling temperature (only marginally
within the activation region). Data shown here
are for k 2 (red circles) and 1.2 (blue
triangles) inverse Angstroms.
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k 1.2
Fplateau
k 2.0
tß
Near tß 10-12 sec.
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k 1.2
k 2.0
tß
ta
But fits to the viscosity imply TZ 3000 K.
Values of k provide initial information about
length scales.
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Diffusion (from CTRW analysis)
In dimensional units
at low T

Viscosity Shear-transformation-zone
(STZ) theory, JSL PRE (2008)
kBTZ STZ activation energy ?E (T) inverse
Eyring rate of STZ shear transitions.
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Simultaneous fits to diffusion and viscosity data
using combined CTRW, STZ, and
XC theories
Diffusion data from Mapes, Swallen and Ediger, J.
Phys. Chem. B 110 (2006) Visocosity data from
Laughlin and Uhlmann, J. Phys. Chem. 76 (1972)
and Cukierman, Lane, and Uhlmann, J. Chem. Phys,
59 (1973)
The solid curves are CTRW-STZ-XC fits. The
dashed curves are mode-coupling, power-law fits.
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Parameter values for OTP
TZ 2000 K (for the diffusion constant) TZ
3000 K (for the viscosity)
The size of the OTP molecule is about 3 Å.
The length of a link in an excitation chain l
0.7 Å.
In the low-T limit of the diffusion constant, the
length a l 0.05 Å !
Stokes-Einstein ratio
At the glass temperature
Small length scales imply collective
rearrangements in activation and diffusion
mechanisms. The Stokes-Einstein violation
implies different mechanisms for diffusion and
viscosity in solidlike materials near the glass
temperature.
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Scattering function at k 2 Å-1 for T 327 K
(blue), 306 K (green), 293 K
(red), and 280 K (dashed line, no data)
The two higher temperatures are beyond the range
of the theory the activation barriers are too
low and the CTRW analysis is inadequate. But the
time scales are roughly correct.
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T293 K
b.69
-- F10-5
b.73
b.5
T327 K
Stretched-exponential fits to CTRW results
The crossover to b1/2 occurs well beyond the
observable range. The SE fitting function seems
exact over at least three decades in t. The
effective b seems to approach unity with
increasing T and decreasing k.
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Conclusions

• Heterogeneous length scales seem remarkably
  small, almost independent of the theoretical
  uncertainties. The most serious experimental
  uncertainties are in the scattering data. Can
  these be taken to longer times and lower
  temperatures?
• The Stokes-Einstein violation occurs because the
  molecular mechanisms for diffusive and viscous
  relaxation become different from each other in
  the solidlike material near a glass transition.
• Stretched-exponential behavior is directly
  related to heterogeneity, but the observable
  indices may be curve-fitting artifacts rather
  than intrinsic, universal properties of
  glass-forming materials.