Microscopic Picture of Aging in Silica Glass:
A Computer Simulation
Katharina Vollmayr-Lee, Robin Bjorkquist, Landon M. Chambers
Bucknell University
∆tj
7
∆tdr
6
rn (t)
5
∆Rj
4
3
2
1
0
1
0
waiting time tw
0
waiting time windows
t [ns]
tik
10
Ne/∆t [1/ns]
10000
Tf=3250 K
1000
Tf=3000 K
Tf=2750 K
100
O-atoms
Tf=2500 K
10
0.01
0.1 t [ns]
w
1
10
Acknowledgments: J. Horbach & A. Zippelius
What is a Glass?
Examples:
◮
Drinking Glass, Pyrex, Schott
Ceran Cooktop
◮
Fiber-Optics, Telescope Lense,
Reading Glasses
◮
Solar Panel Glass
◮
Vulcanic Glass
◮
Golf-Club
What is a Glass?
V
Glass:
liquid
system falls
glass
out of equilibrium
crystal
Tm
Crystal
Glass
T
Liquid
Structure: discordered
What is a Glass?
V
Glass:
liquid
system falls
glass
out of equilibrium
crystal
Tm
Crystal
Glass
T
Liquid
Structure: discordered
Dynamics: frozen in
What is a Glass?
SiO2
log η [Poise]
12
−2
Dynamics:
strong
Viscocity η as function of inverse
temperature T
GeO2
Glycerol
0.4
LJ
◮
slowing down of many decades
fragile
◮
strong and fragile glass formers
◮
SiO2 strong glass former
4 Ca(NO3)26KNO
Tg/T
1.0
[C.A. Angell and W. Sichina, Ann. NY Acad. Sci. 279, 53 (1976)]
−→ very interesting dynamics
System: SiO2
lt
me
◮
rich phase diagram
◮
similar to water (H2 O)
pressure
[S. Stoeffler and J. Arndt, Naturwissenschaften 56, 100 (1969)]
Model: BKS Potential
[B.W.H. van Beest et al., PRL 64, 1955 (1990)]
φ(rij ) =
qi qj e2
Cij
+ Aij e−Bij rij − 6
rij
rij
112 Si & 224 O
Tc = 3330 K
ρ = 2.32
g/cm3
O
Si
Numerical Solution: Euler Step
Initialize:
x(t0 ), v(t0 ) , a(t0 )
x(t 0 + ∆ t), v(t0 +∆ t), a(t 0+ ∆ t)
x(t0 +2 ∆ t), v(t0 +2 ∆ t), a(t0 +2 ∆ t)
etc.
= Iteration Step:
x(t+ ∆ t)=x(t)+v(t)∆ t
v(t+ ∆ t)=v(t)+ a(t)∆ t
a(t)=F(t)/m=−(dU/dx)(t)
Molecular Dynamics Simulation
Initialize:
xi (t0 ), vi (t0 ,) ai (t0 )
particles i=1,...,N
three dimensions
xi (t0 +∆ t), vi (t0 +∆ t), ai (t0 + ∆ t)
xi (t0 +2 ∆ t), vi (t0 +2 ∆ t), ai (t0 +2 ∆ t)
etc.
= Iteration Step:
(Velocity Verlet)
xi (t+∆ t)=xi (t)+vi (t)∆ t+ai (t)(∆ t)2 /2
vi (t+∆ t)=vi (t) +(ai (t)+ai(t+∆ t)) ∆ t/2
ai(t)=Fi (t)/mi = − i U(t)/mi
∆
Simulations
T
5000 K
3760 K
3250 K
3000 K
2750 K
2500 K
Simulation Runs
20 initial configurations
Ti
Tf
0.33 ns
NVT
(Nose Hoover)
33 ns
NVE
(Velocity Verlet)
dynamics
time
Aging
T
5000 K
3760 K
3250 K
3000 K
2750 K
2500 K
Simulation Runs
20 initial configurations
Ti
Tf
aging:
0.33 ns
NVT
(Nose Hoover)
baby
33 ns
NVE
(Velocity Verlet)
teenager
adult
time
Aging
T
5000 K
3760 K
3250 K
3000 K
2750 K
2500 K
Simulation Runs
20 initial configurations
Ti
Tf
aging:
0.33 ns
33 ns
NVT
NVE
(Nose Hoover)
(Velocity Verlet)
baby
waiting time
teenager
tw
adult
time
Aging Example: Mean Square Displacement
∆r 2 (tw , tw + t) =
1
N
N
P
(ri (tw + t) − ri (tw ))2
i=1
tw
10
Ti=5000 K
1
tw=0 ns
aging:
2
10
10
baby
teenager
adult
0
waiting time tw
tw=24.0 ns
2
∆r (tw,tw+t) [Å ]
10
tw+t
-1
◮
-2
mean square displacement
∆r 2 depends on waiting
time tw (colors)
[KVL et al., 2010]
10
-3
◮ similarly for Cq (tw , tw + t) and
χ4 (tw , tw + t) (see talk of Chris
Gorman)
Tf=2500 K
O-atoms
10
-4
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
t [ns]
Goal: Single Particle Picture (not
1
N
N
P
)
i=1
Goal: Single Particle Picture
Cage Picture
1
∆r 2 (tw , tw + t) = N
10
Ti=5000 K
1
(ri (tw + t) − ri (tw ))2
tw=0 ns
0
tw=24.0 ns
2
∆r (tw,tw+t) [Å ]
10
N
P
i=1
2
10
10
-1
-2
10
-3
Tf=2500 K
O-atoms
10
-4
-6
10
-5
10
-4
10
-3
10
-2
10
-1
10
0
10
1
10
t [ns]
ballistic motion
∆r2 ∼ t2
cage
jump
study jumps
diffusion
∆r2 ∼ t
Jump Definition
∆R
0.00327 ns
t
{rn } , σ
{rn } , σ
{rn } , σ
{rn } , σ
7
6
ri,z (t) [Å]
5
4
3
2
1
0
3
4
5
6
ri,y (t) [Å]
∆R
7
6
∆R
∆R > 3 σ
rn,z (t)
5
4
3
σ
2
1
0
1
t [ns]
10
Jump Definition: Aging Dependence
7
6
rn (t)
5
4
3
2
1
0
1
0
waiting time tw
0
waiting time windows
t [ns]
tik
10
Number of Jumping Particles per Time
10000
7
6
rn (t)
5
4
3
2
Ne/∆t [1/ns]
1
0
Tf=3250 K
1000
1
0
waiting time tw
0
waiting time windows
t [ns]
tik
10
∆t
Tf=3000 K
Tf=2750 K
100
O-atoms
Tf=2500 K
10
0.01
0.1 t [ns]
w
1
10
Ne /∆t = Number of jump
events occuring in time
interval ∆t
◮ equilibration time
consistent with Cq , ∆r 2 ,
and χ4 (see arrows)
Ne /∆t decreasing with
increasing tw
−→ Ne /∆t depends strongly on waiting time tw
◮
Average Jump Length
7
6
O-atoms:
∆Rj
5
Tf=3250 K
rn (t)
2.5
4
3
∆Rj [Å]
Tf=3000 K
2
1
2
0
Tf=2750 K
1
Tf=2500 K
0
waiting time tw
0
waiting time windows
Si-atoms:
1.5
1
Ti=5000 K
0.01
t [ns]
tik
10
◮
O-atoms jump farther
than Si-atoms
◮
compare:
dOSi = 1.608Å, dOO = 2.626Å,
0.1 t [ns]
w
1
10
−→ ∆Rj is mostly independent of tw
dSiSi = 3.077Å
Jump Length Distribution
O-atoms
P(∆Rj) [1/Å]
7
tw=0.00 ns
tw=0.02 ns
tw=0.16 ns
tw=1.31 ns
tw=10.5 ns
tw=29.5 ns
Ti=5000 K
Tf=2500 K
0.1
6
∆Rj
5
rn (t)
1
4
3
2
1
0
1
1
0.01
0
waiting time tw
0
waiting time windows
t [ns]
tik
10
0.1
◮
peak at ∆Rj = 0:
reversible jumps
◮
exponential decay
◮
compare LJ & Polymer
0.01
Si-atoms
0.001
0
0.001
0
1
1
2
3
2
4
5
3
4
∆Rj [Å]
5
6
7
[Warren & Rottler, EPL 2009]
−→ P (∆Rj ) is independent of tw (colors)
Time Averages: Jump Duration ∆tj & Time in Cage ∆tdr
7
trCq
Tf=2750 K
1
jump duration
∆tj
∆tdr
8
Tf=2500 K
6
rn,z (t)
∆tdr [ns]
10
time in cage
5
4
3
2
Tf=3000 K
1
0
1
Tf=3250 K
O-atoms
Ti=5000 K
∆tj [ns]
0.1
0
waiting time tw
0
waiting time windows
0.1 t [ns]
w
1
10
−→ ∆tdr is independent of tw !
tik
10
◮
∆tdr ≫ ∆tj
◮
no artifact due to finite
time-window
◮
artifact due to finite
simulation run
0.01
0.01
t [ns]
Distribution of Time in Cage P (∆tdr )
P(∆tdr) [1/ns]
-4
10
Tf=2500 K
-5
10
-4
-6
10
P(∆tdr)
10
8
10
10
Tf=3250 K
-5
0.01
4
3
2
1
0
1
0
waiting time tw
0
waiting time windows
◮
-7
t [ns]
tik
10
2500 K: powerlaw
compare LJ & Polymer
-9
[Warren & Rottler, EPL 2009]
0
-8
∆tdr
5
-8
10
10
10
time in cage
7
6
-6
10
-7
tw=0.00 ns
tw=0.02 ns
tw=0.16 ns
tw=1.31 ns
tw=10.5 ns
tw=29.5 ns
rn,z (t)
10
2
∆tdr [ns]
0.1
4
6
O-atoms
1 ∆t [ns]
dr
Ti=5000 K
10
−→ P (∆tdr ) is independent of tw !
◮
3250 K: exponential
Distribution of Time in Cage P (∆tdr )
8
-5
10
Tf=2500 K
Tf=2750 K
Tf=3000 K
Tf=3250 K
time in cage
7
∆tdr
6
rn,z (t)
tw=1.31 ns
5
4
3
2
-6
P(∆tdr)
10
1
-5
10
0
1
-6
10
-7
-7
10
10
waiting time tw
0
waiting time windows
t [ns]
tik
10
-8
10
0
10
20
30
-5
10
-8
10
◮
power law to exponential
◮
compare:
1 d FA-& East-model
(dynamic fascilitation)
-6
10
-7
10
-8
10
-9
10
-9
10
0
0.01
0
5
10
0.1
15
1
∆tdr [ns]
10
[Jung et. al., JCP 2005]
Summary
∆tdr
Main waiting time tw -dependence due to
number of jump events per time Ne /∆t.
(not ∆Rj , ∆tj , ∆tdr )
∆Rj
5
rn (t)
Aging:
∆tj
7
6
4
3
2
1
0
1
0
waiting time tw
0
waiting time windows
t [ns]
tik
10
Ne/∆t
−→ Cooperative Processes:
◮
◮
space-time clusters introduce tw -dependence
of length & time
dynamic heterogeneity [Donati et al.,PRL 1998]
◮ avalanches [KVL,Baker, EPL 2006]
◮ dynamic fascilitation [Jung et al.,JCP 2005]
Acknowledgments: Supported by NSF REU grants PHY-0552790
& REU-0997424. Thanks to J. Horbach, A. Zippelius & University
Göttingen.