“Multiscale Molecular Modeling”
Edinburgh, June 30 - July 3, 2010
Two Coarse Graining Studies
(1) density fluctuations in water
(2) excitations in glass formers
David Chandler
UC Berkeley
1. Density fluctuations in Water
Liquid close to vapor-liquid coexistence
log Pv(N) , v small
log Pv(N) , v large
observed volume v
0
<N>
N
Perturbation ~ ε N
0
<N>
ε < 0 (attract)
ε > 0 (repel)
Solvent density = ρ n(r) + δρ(r)
Ising field
• G. Hummer, S. Garde, A. Garcia, A Pohorille, & L. Pratt, PNAS ’96.
• K. Lum, DC & J. Weeks, J.Phys. Chem B ‘99
N
Gaussian field
Between hydrated melittin tetramer
M. Hagan, P.Varilly, K, Tay & DC, in progress ‘09
mean separation of dimer surfaces
0.25
Pv(N)
0.20
0.15
0.10
0.05
0
0
10
20
30
40
Number of waters, N
50
60
70
observed volume
1
0.1
PV (N )
0.01
0.001
0.0001
1e-05
1e-06
1e-07
d = 6.5Å
d = 6.75Å
d = 7.0Å
0
10
20
30
40
50
60
N in cavity
4
Water density coarse grained over 0.4 nm, n(r) ,
with hydrated hard sphere (radius R)
R = 0.25 nm
R = 1.0 nm
5
Instantaneous interface
() ( )

ρ (r
) = 21 ρ ,
NO

 
ρ ri = ∑ f r − ri ,
i =1
surface
bulk
()
f r ∝ e −r
2
σ ≈ 0.2nm
/ 2σ 2
Instantaneous water-vapor interface
1s ~ 1.2 ps
Grid spacing ~ 1Å
Evaporation of water.
Patrick Varilly & D Chandler
work in progress (2010)
event frequency ~ 10-2 / ns-nm ; event duration ~ 1 ps
harvested by transition path sampling
Curvature at evaporation site
Evaporation transition state at t=75fs (third frame)
Soft interface in protein self-assembly
A. Willard & D. Chandler, J.Phys.Chem. 2010
10
2. Elementary excitations in structural glass formers
Below onset temperature,
To , a mobility field : n(r )
= 1, mobility at r
n(r ) = 0, no mobility at r
Facilitated dynamics
c = �n(r )�
space
Super-Arrhenius growth
of relaxation time
∆F� ∼ ln �
�eq = c −1/d
time
τ ∼ c ln c
11
Excitations from
atomic motion
Sample P ⎡⎣ x(t) ⎤⎦ Cexcitation ⎡⎣ x(t) ⎤⎦
Cexcitation ⎡⎣ x(t) ⎤⎦ = 1, if at least once in trajectory of length Δt
Δr(t) = |r1(quenched) (t) − r1(quenched) (0)|>σ
AND Δr(t + Δt) > σ ,
= 0, else.
Single particle displacement not
discriminating
x(quenched) (t) = inherent structure
≈ x(coarse) (t) =
δt
1
dt ' x(t '+ t)
δ t ∫0
q
r1
δ t = coarse graining time < Δt = time to detect excitation
Δt  τ = structural relaxation time
Excitations (mobility) atoms packed & disordered
d=2 WCA mixture, To=1.5
T=0.1
Blue, Δri (t) = |ri (coarse) (t) − ri (coarse) (0)|=0; red, Δri (t) ≥ σ
T=0.5
T=1.0
Keys, Hedges, Chandler &
Glotzer, in preparation
(2010)
Excitation density in transition path ensemble
( )
µ r,t = density of other displaced particles
with tagged particle at origin
Keys, Hedges, Chandler &
Glotzer, in preparation (2010)
d=2 WCA mixture, To=1.5.
Spatial patterns of elementary excitations like
those of a one-component dilute gas.
Temperature dependence of transport & excitations
Mean excitation concentration
()
c∝ µ r , r→∞
Arrow model (Garrahan & Chandler, PNAS '03)
⎡ ⎛ 1 1⎞⎤
c ∝ exp ⎢ −J ⎜ − ⎟ ⎥
⎢⎣ ⎝ T To ⎠ ⎥⎦
⇒
⎡ ⎛ 1 1 ⎞2⎤
τ
= exp ⎢J 2 ⎜ − ⎟ ⎥ , T ≤ To
τo
⎢ ⎝ T To ⎠ ⎥
⎣
⎦
Kob-Andersen 80:20 LJ mixture, d = 3.
MD simulation with N = 10,000, ρσ 3 = 1.2
Elmatad,
Chandler &
Garrahan,
J.Phys.
Chem.B’09
Keys, Hedges, Glotzer & Chandler,
in prep, ‘10
J = 1.5
To = 0.9
⎡⎣J ⎤⎦
≈ 1.45, ⎡⎣To ⎤⎦ from c ≈ 0.90
from c
15
Equilibrium & transport J’s and To’s
⎡
⎛1
1 ⎞⎤
c ∝ exp ⎢ −Jeq ⎜ −
⎟⎥,
To, eq ⎠ ⎥
⎢⎣
⎝T
⎦
eq
⎡
⎛1
1 ⎞
2
τ ∝ exp ⎢Jtrans
−
⎜
⎟
⎢
To, trans ⎠
⎝T
⎣
2
⎤
⎥
⎥
⎦
eq
eq
eq
Keys, Hedges, Glotzer & Chandler,
in prep, ’10
16
Transport data quadratic data collapse below onset temperature
Elmatad, Chandler & Garrahan, J. Phys. Chem. B (2009)
normal
supercooled
67 data sets
(103 data points)
onset temperature
397
Material properties J and To
not so for other functional form parameters (Elmatad et al, in prep. ’10)
Example: OTP and double exponential
τ ∝ exp [(K/T ) exp(C/T )]
high T
low T
all T<To
To
351
350
352
J/To
8.0
8.1
7.9
K
1.34
203
15.7
C
1.96
×103
769
1.32
×103
1/T
Example: PP and non-analytic exponent
high T
low T
all T<To
To
392
397
392
J/To
5.8
5.7
5.8
Tc
370
322
346
X
29
67
45
�
�
τ ∝ exp X(Tc /T − 1)1.57
18
Water density at small and large length scales
Lum, DC & Weeks, J. Phys. Chem. B ‘99; Ten Wolde, Sun & DC,
Phys. Rev. E ’01; Ten Wolde & DC, PNAS ‘02
Fields:

ρ(r) = ni ρ + δρ(r), r ∈v i
⎛ 1
⎞
⎛ 1
exp ⎜ −
H ⎡⎣ ni ⎤⎦⎟ = ∫ D ρ(r) exp ⎜ −
H ⎡ ni , δρ(r)
⎝ kBT
⎠
⎝ kBT ⎣
{ }
{
(
}
)
vex
⎞
⎤⎟
⎦⎠
nj =0
≈ − ∑ µ − Δµ(v ex ) ni − ε ∑ ni n j
i
ij
continuous function
of solute coord’s RN
δρ(r) = 0
ni =1
δρ(r) ≠ 0 and Gaussian
Dynamics:
ni (t) → ni (t + δ t) , Monte Carlo
{
} {
}
R N (t) → R N (t + Δt) = R N (t) − Δt Γ
⎤
∂ ⎡
N
N
⎢V R (t) + ∑ ni (t) Δµ(v ex ) ⎥ + Δf (t)
N
∂R (t) ⎣
⎦
i
(
)
19