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Anomalous Diffusion in Heterogeneous Glass-Forming Materials
J.S. Langer University of California, Santa Barbara
Workshop on Dynamical Heterogeneities in Glasses, …Lorentz Center, Leiden, August 2008
Outline
• Continuous-time random walks (CTRW’s) in heterogeneous glassy systems
JSL and S. Mukhopadhyay PRE 77, 061505 (2008), JSL arXiv:0806.0958
• 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
Continuous-time random walks in heterogeneous systems
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.
Basic assumptions to be made here:
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)
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.
Continuous-time random walks in heterogeneous systems, cont’d.
The glassy waiting time distribution is:
222
)T(*R)T(
t*t);*texp(*)t(G
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.
)T( )T(*R
*)t(G
Glassy waiting time distribution
*texp*t)(/expd*)t(G 2220 0
2
*R/R = scaled size of a glassy domain
20 1 /)( = 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*.
Continuous-time random walks, cont’d.
The mobile waiting time distribution is
)T(
)T(*);t(exp*)t(
MM
where is the time scale for diffusive jumps in a mobile region -- related to, but not exactly the same as the beta relaxation time.
)T(M
The mobile jump-length distribution is
2
2
232 22
1
a
*rexp
)a(*)r(f
/
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).
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:
*)t*,r(n*),t*,r(n MG
)u*,k(W
)u*,k(N
u)u*,k(n;
)u*,k(W
)u*,k(N
u)u*,k(n~ M
MG
G
11
where
;/u)u(~*)k(fP)u*,k(N GMG 11
;/u)u(~*)k(f̂P)u*,k(N GGM 1
.)u(~*)k(f̂P/u*)k(f̂P)u*,k(W GGM 1
More mathematics:
*dtee*dt*)t(e)u(~ *t*tuG
*tuG
2
002
has an essential singularity at u=0 and a branch cut along the negative u axis.
The self intermediate scattering function is
)u*,k(n~P)u*,k(n~Pei
du*)t*,k(F̂ MMGG
*tui
iS
2
which requires computing the inverse Laplace transform of a function of )u(~
G
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.
k*=10
k*=0.8
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*.
k*=10
k*=0.7
slope -1/2
slope - 1
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 x*=0 is really a delta function in this approximation.
t*=0.03
t*=10
XC predictions for T-dependent parameters
)T(T
Tln Z
0
Alpha relaxation time
kBTZ = bare activation energy for density fluctuations, STZ’s, etc.
1
0TT)T(*R)T(
for T near Tg. (Vogel-Fulcher)
0)T( for T ~ TMC ~ upper end of super-Arrhenius region
)T( describes chainlike molecular displacements that enable stable transitions between inherent states. Unlike TZ, should be mechanism independent.
)T(
R*(T) l = spatial extent of an excitation chain that can activate a stable molecular rearrangement ~ maximum size of a stable glassy domain.
Theoretical R*(T) and glassy fraction PG(T) for OTP
XC theory needs a correction for vanishingly short chains at high T.
~
3
)T(*R*h
)T(*R)T(PG ~ ~ surface-to-volume ratio at low T
Tg TMC
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.
Near tβ ~ 10-12 sec. *twMMG eA*)t*,k(n̂;*)t*,k(n̂ 01
110 MM /;*)k(fPw
*)k(f̂P
*)k(f̂A
M
M
1
1MMG
plateau P)A(PF 1
Fplateauk = 2.0
k = 1.2
tβ
tβ tα
.1021 14
00
sec)T/T(expw
*t MM
MCM TKT.sec 50010 150
KTT
TTexpt/t Z
MZ 2500103
But fits to the viscosity imply TZ ~ 3000 K.
k = 2.0
k = 1.2
Values of k* provide initial information about length scales.
Diffusion (from CTRW analysis)
*tG*k
S
/P)/(
*ta
*k
*)t*,k(F̂*)t(*r
123
33
2
0
2
22
In dimensional units t)T(D)t(r 2
)T(
a
)T()T()T(P)/(
a)T(D
MG
2222 2
23
3
at low T
Viscosity
Shear-transformation-zone (STZ) theory, JSL PRE (2008)
)T(T
Tln;)T()T(
T
T)T( Z
EE
N
00
kBTZ’ = STZ activation energy
E (T) = inverse Eyring rate of STZ shear transitions.
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.
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.
)T(
a)T(D
222 In the low-T limit of the diffusion constant,
the length a l ~ 0.05 Å !
Stokes-Einstein ratio
210~T
TTexp
)T(
)T(
T
)T()T(D
g
ZZ
g
g
g
gNg
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.
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.
b=.69
b=.5
b=.73-- F~10-5
Stretched-exponential fits to CTRW results: bplatSS t.constexpFF
The crossover to b=1/2 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.
T=327 K
T=293 K
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.