6/3/2008
Molecular Dynamics Simulations and
Neutron Scattering:
Polymer Melt and Solution Dynamics
Grant D. Smith
Department of Materials Science and Engineering
University of Utah
Polymer Dynamics and Relaxation
Richard H. Boyd and Grant D. Smith
Cambridge University Press (2007)
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6/3/2008
Bill Gate’s Opinion
MD simulation = dissimulation
the act of deceiving deception, dissembling, deceit
misrepresentation, falsification - a willful perversion
of facts
fakery - the act of faking (or the product of faking)
indirection - deceitful action that is not
straightforward
Outline
Molecular Dynamics Simulations
Force Fields and Force Field Parametrization
MD Simulations and Neutrons: The Connection
Local Melt Dynamics: Polyethylene
Local Solution Dynamics: PEO/water Solutions
Chain Dynamics: Poly(butadiene)
Glass and Sub-glass Processes: Poly(butadiene)
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6/3/2008
MD Background:
Scales of Modeling
mesoscale
Time Scale
-4
10
-6
10
-8
10
-12
10
S
Monte Carlo
S
S
continuum
quantum
chemistry
molecular
dynamics
domain
exp [-∆E/kT]
F = MA
S
Ηψ = Εψ
-10
10
-8
-6
M 10 M 10 M
Length Scale
-4
10 M
MD Background:
Classical Equations of Motion
NVE
H = K + V = total energy = constant
K=
1
U (rij )
∑2mv , V = ∑
i, j
2
i i
i
∂H
∂V
dpj
drj ∂H
v
F
dt = ∂pj = j , dt = - ∂rj =- ∂rj = j
NVT
H = K + V + Ks + Vs = constant
Ks = ½ Qps2
Vs = (f + 1)kBT ln(s)
ps = conjugate momentum of thermostat
Q = conjugate mass
s = generalized coordinate of the thermostat
f
= number of degrees of freedom
5000 atoms, 4 nm
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6/3/2008
Force fields
Intramolecular and Intermolecular Interactions
bond stretch
torsional
intermolecular
interactions
valence angle
bend
r
V (r ) = V
NB
intramolecular
nonbonded
r
(r ) +
∑V
BOND
( rij ) +
bonds
V
DIS − REP
(rij ) =
Aij
rij12
 σ
− 6 = 4ε 
rij
 rij

Cij
∑V
BEND
(θ ijk ) +
bends
∑V
TORS
(ϕ ijkl )
dihedrals
12
6

σ  
 −  

r  

 ij  
C ij
qi q j
r
r 1 N
V NB ( r ) = V POL ( r ) + ∑ Aij exp( − Bij rij ) − 6 +
2 i , j =1
4πε 0 rij
rij
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6/3/2008
Force fields: Where do they come from?
“Generic” (e.g., Dreiding):
Roughly describe a wide range of materials, not
parameterized or validated
“ Trained” (e.g., AMBER, COMPASS, CHARMM, OPLS)
Parameterized to reproduce the properties of a broad set of
molecules such as small organics, peptides or amino acids
“Specialized” (e.g., Atomistic Polarizable Potential for
Liquids, Electrolytes and Polymers (APPLE&P))
Carefully parameterized and validated potentials designed to
reproduce properties of a small class of specific molecules
Force Field Parametrization:
Intermolecular Interactions
binding (kcal/mol)
hydrophobic path
quantum chemistry
force field
3
0
-3
-6
3
hydrophilic path
4
5
r(C-OW) (Å)
6
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6/3/2008
Force Field Parametrization:
Intramolecular Interactions
∆E (kcal/mol)
6
t-φ2-t, force field
t-φ2-t, quantum chemistry
4
2
0
0
40
80
120
160
φ2, degrees
conformational energy (kcal/mol)
Poly(butadiene)
7
QC
FF
6
H2
H
C
C C
C
H
H2
5
4
3
H2
H2
C
C
C C
H H
2
1
0
0
60
120
180
240
β dihedral angle
300
360
CH2
HC
CH
C
H2
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6/3/2008
Force Field Parametrization:
Validation: Static Structure Factor
S(q)-1
1.0
Neutron Diffraction
MD
0.5
0.0
-0.5
1 2 3 4 5 6 7 8 9 10
q, Å -1
Force Field Parametrization:
Validation: Time Correlation Functions
TACF (t ) =
θ (t ) θ (0) − θ (0)
θ (0)
2
− θ ( 0)
2
2
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Force Field Parametrization:
Validation: Dynamic Structure Factor for PEO Melt
I(q,t)
1
QENS, q=0.718 Å -1
QENS, q=0.913Å -1
QENS, q=1.095 Å -1
0.1
0.01
0.001
0
50
100 150 200 250
time (ps)
Incoherent Dynamic Structure Factor, Coherent Dynamic Structure Factor,
QENS and MD
NSE and MD
MD-Neutron Connection: Dynamic Structure Factors
Simulation yields intermediate dynamic structure factors (time
domain) directly
Coherent scattering
scoh ( q, t ) =
N
N
r
r
r
r
1
< ∑ exp iq • ( rk (t ) − rj (0)) >=< ∑ sin q | rk ( t ) − rj (0) | / q | rk (t ) − rj (0) |>
N j ,k
j ,k
Incoherent scattering
sinc (q, t ) =
N
r r
r
r
r
r
r
1 N
< ∑ expiq • ( rk (t ) − rk (0)) > =< ∑sin q | rk (t ) − rk (0) | / q | rk (t ) − rk (0) |>
N k
k
Simulation time scales from femtoseconds to microseconds
Experimental time scales from picoseconds to nanoseconds
A Fourier-Laplace to frequency domain (or an inverse transform of
QENS data) is required for comparison
Direct comparisons can be made with NSE data
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6/3/2008
Example: Local Dynamics is a Polyethylene Melt:
MD simulations provide data over a wider range of time
MD simulations are in good agreement with QENS
Librations or Conformational Transitions?
 q 2 < ∆rH2 (t ) > 
s inc ( q, t ) = exp −

6


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6/3/2008
PEO/Water Solutions: Local Dynamics
Polymer Dynamics
Mw = 500
q = 0.4 A-1
q = 2.4 A-1
10000
τDSF(q=0.58 Å-1 )
τDSF(q=0.91 Å-1)
This is due to slowing
conformational
dynamics and
increasing
translational/rotational
dynamics with dilution
τDSF(q=1.57 Å-1)
100
10
1
0.1
0.0
0.2
0.4
0.6
0.8
1.0
wP
102
τTRANS(C-C)
τTRANS(C-O)
time (ps)
Low molecular weight
PEO exhibits a
minimum in local
dynamics with
concentration
time (ps)
1000
τ2(CH vector)
101
0.0
0.2
0.4
0.6
0.8
1.0
wP
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6/3/2008
PEO/Water Solutions: Local Dynamics
Water Dynamics
1.0
w P =0.53, 298 K
0.8
I(q,t)
q=0.57 Å -1
0.6
q=0.90 Å -1
0.4
q=1.62 Å
0.2
0.0
-1
q=2.24 Å -1
0
10
20
30
time (ps)
Water quasi-confinement
wP=0.17, qc=0.95 Å-1
wP=0.17
wP=0.52, qc=1.34 Å-1
1.0
1.0
wP=0.52
wP=0.78
gO-O(r)
Γ (ps-1)
wP=0.78, qc=1.58 Å-1
0.5
0.0
0
1
2
3
-2
q2, Å
4
5
0.5
0.0
2
4
6
8
10
separation (Å)
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6/3/2008
Large Scale Dynamics: Single chain coherent
dynamic structure factor for PBD (353 K)
S'(q,t) = S(q,t)/S(q) =
∑<sin[qRmn(t)]/qRmn(t)> / ∑<sin[qRmn(0)]/qRmn(0)>
(m,n)
(m,n)
H2
H
C
C C
C
H
H2
1.0
-1
q = 0.05 Å
S'(q,t)
0.8
0.6
H2
H2
C
C
C C
H H
-1
q = 0.08 Å
-1
q = 0.10 Å
-1
0.4
q = 0.24 Å
-1
q = 0.20 Å
-1
q = 0.14 Å
0.2
-1
0.0
q = 0.30 Å
0
5
10
15
time (ns)
20
CH2
HC
CH
C
H2
Why does the Rouse model provide such
a poor description of S(q,t)?
1.0
Chain stiffness?
0.8
S'(q,t)
Internal viscosity?
0.6
Non-Gaussian
displacements?
0.4
0.2
0.0
0.1
1
time (ns)
10
Figure 5
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6/3/2008
Can dynamic neutron scattering tell us
about the bifurcation of the glass and subglass processes?
dielectric relaxation
10
α−β
-1
log (1/τ) (s )
8
6
β
4
2
α
0
-2
Ts
-4
3
4
Tg
5
6
7
8
1000/T
MD Simulations and Dynamic Neutron
Scattering in Polymers
--Natural partners (overlapping time and length scales)
--DNS experiments are important for validation of
simulations
--Simulations can direct experiment
--Simulations can help provide mechanistic insight into
experimental results
--Simulations can help provide sanity checks for
conclusions based on limited (but expanding!) DNS
capabilities
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6/3/2008
14