quantifying fluctuation/correlation effects in inhomogeneous polymers by fast monte carlo...
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Quantifying Fluctuation/Correlation Effects in Inhomogeneous Polymers
by Fast Monte Carlo Simulations
Department of Chemical & Biological Engineeringand School of Biomedical Engineering
David (Qiang) Wang
Laboratory of Computational Soft Materials
Laboratory of Computational Soft Materials
Jing Zong, Delian Yang, Yuhua Yin, and Pengfei Zhang
Coarse-Grained Simulations of Multi-Chain Systems
• Conventional Monte Carlo (MC) simulations:
Hard-core excluded-volume interactions:
u(r→0)→∞+
Model of chain connectivity
or SMAW on lattice
1. Orders of magnitude faster (better) sampling of configuration space;
2. All advanced MC techniques can be used;
Advantages:
• Fast MC Simulations: Finite u(r→0)
Q. Wang and Y. Yin, JCP, 130, 104903 (2009); Q. Wang, Soft Matter, 5, 4564 (2009); 6, 6206 (2010).
2
3,0
3,0
e
e
n
V R
R
b
bN
V nN
N
N
N
3. Much wider range of controlling system fluctuations can be studied;
4. No parameter-fitting when compared with polymer field theories.
Part 1: Fast Off-lattice Monte Carlo (FOMC) Simulations
System I – Compressible Homopolymer Melts
,
12
, 1 ,
1
21 1
1
ˆ ˆ( ) ( )
ˆMicroscopic segmental
3,
2
1( ) d d (| |) (0)
2
density ( )
2
n NC E C
k s k s
n N
k s
Eij
i j
k sk s
H H H Ha
nNH u r u u
R R
r r rr r r
r r R
,0
2
3,0 ,0
,0 3Normalization: , , , .
6g gg g
NR R
vN nB Ca
R V R r r
0 0
0 if | |(| |)
0 otherwise
(| |), d (| |) 1
u
uv u
r rr r
r r r r,0 2gR
n: number of chains;N: number of segments
on each chain.
2
1 ,1 1
11-chain structure factor ( ) exp 1 , | | .
n N
k sk s
S q qnN
q R q
N64, B25
System I – Compressible Homopolymer Melts
2
,1 1
1Total structure factor ( ) exp 1 .
n N
t k sk s
S qnN
q R
1
0R
22
2
,0
PA1
For our system, the random-phase approximation (RPA) gives
, where for discrete Gaussian chain
1 exp 2 2exp exp 1( ) with
1 e
1ˆ ( )
( ) (
x
.
)
p
t
g
N x N x N xS q
N
N Nu q
BC S q
N x N
x R
S q
q
Since RPA includes fluctuations at the Gaussian level, deviations
from it are due to non-Gaussian fluctuations in the system.
System I – Compressible Homopolymer Melts
N64, B25
Q. Wang and Y. Yin, JCP, 130, 104903 (2009)
System I – Compressible Homopolymer Melts
System II – Compressible Symmetric Diblock Copolymers
System II – Compressible Symmetric Diblock Copolymers
A
A
A B A00
00
B
A B
A , B ,1 1 1
01
ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )
ˆ ˆ( )
1d d (| |)
2
d d (| |)
, ,
( )
ˆ ( ) ) ˆ (Nn n N
k s k sk s k s
C E
N
E
H H H
H u
u
nN
V
r r r r
r
r r r r
r r
r r R R
r
r r
r r
2
30
151 if | |
( ) 2
0 otherwise
rr
u r
r r
,0 ,0Normalization: , 1.e eR R a N r r
,00.1 ,
, ,
,
e
N
R
N
N
N
System II – Compressible Symmetric Diblock Copolymers
• Canonical-Ensemble Simulations
• Replica Exchange (RE)
Exchange configurations between simulations at different N to greatly improve the sampling efficiency.
• Multiple Histogram Reweighting (HR)
Interpolate at any point within the simulation range;
Minimize errors using all the information collected;
Accurately locate the order-disorder transition.
Trial moves: Hopping, Reptation, Pivot, and Box-length change.
System II – Compressible Symmetric Diblock Copolymers
6
,0cp 2
2 eR
N
N
2
2 01
d ( )3 Continuous Gaussian chains: d
2 d
ˆ ˆ Dirac -function interactions: d d ( ) ( ) ( )2
n NC k
k
E
sH s
a s
vH
Standard Fi
R
r r r r r
eld The r
r
o y
Field Theories vs. Particle-Based MC Simulationsof Multi-Chain Systems
Discrete chains
Finite-range interactions (in continuum)
Particle - Based MC Simulation
Direct Comparison Based on the SAME Hamiltonian
(not vs. )
Discrete chains
Finite-range interactions
Particle - Based MC Simulation
Discrete chains
Finite-range interactions
Field Theory
No parameter-fitting
Lattice chains with MOLS
Kronecker- interactions
Fast Lattice MC Simulation
Lattice chains with MOLS
Kronecker- interactions
Lattice Field Theories
No parameter-fitting
Direct Comparison Based on the SAME Hamiltonian
(not vs. )
• Kronecker -function interactions are isotropic on a lattice (while nearest-neighbor interactions are anisotropic) and straightforward to use;
• Lattice simulations are in general much faster than off-lattice simulations. FLMC simulation is very fast due to the use of Kronecker -function interactions and multiple occupancy of lattice sites (MOLS).
Advantages of FLMC Simulations:
Lattice chains with MOLS
Kronecker- interactions
Fast Lattice MC Simulation
Lattice chains with MOLS
Kronecker- interactions
Lattice Field Theories
No parameter-fitting
Part 2: Fast Lattice Monte Carlo (FLMC) Simulations and Direct Comparison with Lattice Self-Consistent Field (LSCF) Theory
System III – Compressible Homopolymer Melts in 1D
20 0
2
0
0
1
enforces chain connectivity on a lattice,
1,
2 2
where , , 40 is the chain length, is the number
of chains, and is the total numbe
ˆ(
r of l
)
C EB
C
E
H H H k T
H
H
nN nC N n
V
C
V
N E
V
r
r
attice sites.
x
s2 s1,3 s4,6 s5
0
( ) ( ) exp ( ) ln ( ) .
( ) exp( ) .
2 ( )
Ec
E
E
Ec
E
Z N g E H f N Z N n
g E HEu N H n
n Z N
Density of States g(E)Wang-Landau – Transition-Matrix MC
2vN C N
LSCF ,LS
0
CFLSCF predictions: 0, random walk
, ( ) ( 0) .
.
, ( )2c c c B c c
c
c c
E
ENu f f N f N s k u
N
f
fn
FLMC LSCF @ finite C En0 fc sc/kB R2e,g
N0 (no correlations) 0 0 0
finite N0 N→∞ (no fluctuations) 0
0 ,FLMC ,FLMC
2,
At , FLMC results ( , , ,
) approach LSCF predictions at a rate of 1 .
c c B
e g
C E n f s k
R C
large
P. Zhang, X. Zhang, B. Li, and Q. Wang, Soft Matter, 7, 4461 (2011).
System III – Compressible Homopolymer Melts in 1D
260, , ,
Lattice type Simple Cubic Latti
12
c
5
e.x
Nn
CL
NL
System IV – Confined Compressible Homopolymers in 3D
Lx10
L
(C→∞)
Closest to wall
Middle of film
Lx10
L
System IV – Confined Compressible Homopolymers in 3D
2FLMC LSCF1
1( ) ( )
xL
xx
x xL
Q. Wang, Soft Matter, 5, 4564 (2009); 6, 6206 (2010).
System IV – Confined Compressible Homopolymers in 3D
6 2 5 2,0 2 10eR C C N
41.6 10 N65.1 10 N
158N
Q. Wang, Soft Matter, 5, 4564 (2009); 6, 6206 (2010).
System IV – Confined Compressible Homopolymers in 3D
Coarse-Grained Simulations of Multi-Chain Systems
• Conventional Monte Carlo (MC) simulations:
Hard-core excluded-volume interactions:
u(r→0)→∞+
Model of chain connectivity
or SMAW on lattice
• Fast MC Simulations: Finite u(r→0)
1. Orders of magnitude faster (better) sampling of configuration space;
2. All advanced MC techniques can be used;
Advantages:
Q. Wang and Y. Yin, JCP, 130, 104903 (2009); Q. Wang, Soft Matter, 5, 4564 (2009); 6, 6206 (2010).
2
3,0
3,0
e
e
n
V R
R
b
bN
V nN
N
N
N
3. Much wider range of controlling system fluctuations can be studied;
4. No parameter-fitting when compared with polymer field theories.
A
1D Profile along :
( ) 1 2 ( ) 1 .
4d exp ( )
( )( )
d ( )
max ( ) ,
:
1.
f t t
tt i f t
L
tf t
i
Scalar Order Parameter
n
nn
n
L(n)
j (x,y,z)
n
t
1 2
2
Periodic boundary conditions require
( ), 0,1,2,
( ) .
j j j
j jj
L n L n
L n L
j n n
n
System II – Compressible Symmetric Diblock Copolymers
SCFT, Incompressible, CGC, Dirac .