Doros N. Theodorou
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Transcript of Doros N. Theodorou
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PREDICTION OF POLYMER PHYSICAL PROPERTIES
THROUGH NEW, CONNECTIVITY-ALTERING MONTE CARLO ALGORITHMS
Doros N. Theodorou
Department of Chemical Engineering, University of Patras and ICE/HT-FORTH, GR-26500 Patras, Greece and Institute of Physical Chemistry, NRCPS “Demokritos”,
GR-15310 Ag. Paraskevi, Athens, Greece.
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PROBLEM
Dense, long-chain polymer systems are very difficult to equilibrate with conventional simulation methods
Longest relaxation time of polymer melt:s – s
Longest time that can be simulated with atomistic MD: ~ 10 ns
SOLUTION
Develop “bold” Monte Carlo algorithms that can quickly sample distant regions in configuration space
Use moves that modify connectivity among polymer segments
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UNITED ATOM LINEAR POLYETHYLENE
C1000, 24000 interacting sites, flat MW distribution (I=1.05)
T=450 K, P = 1 atm
Atomistic model:
•Lennard-Jones interaction sites
• Constant bond lengths (l=1.54Å)
• Flexible bond angles
• Torsional potential
Mavrantzas, V.G. et al., Macromolecules 32, 5072 (1999)
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CONCERTED ROTATION MONTE CARLO
L. R. Dodd, T.D. Boone, DNT, 1993
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CONCERTED ROTATION MONTE CARLO
“driver” angle
“driver” angle
L. R. Dodd, T.D. Boone, DNT, 1993
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CONCERTED ROTATION MONTE CARLO
“driver” angle
“driver” angle
L. R. Dodd, T.D. Boone, DNT, 1993
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CONCERTED ROTATION MONTE CARLO
“driver” angle
“driver” angle
L. R. Dodd, T.D. Boone, DNT, 1993
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END-BRIDGING MONTE CARLO
P.V.K. Pant & DNT, 1994
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END-BRIDGING MONTE CARLO
P.V.K. Pant & DNT, 1994
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Convenient Ensemble:
Fixed N total number of chains
n total number of mers
P pressure
T temperature
k*
relative chemical potentials for all k-mer species but two
END-BRIDGING MONTE CARLO
i j kj i
j k
i j
i k
k=1,…,m, ki, j
Chain length distribution controlled through k* profile
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EBMC PERFORMANCE AS A FUNCTION OF CHAIN LENGTH
rcm
t0=CPU time for <[rcm(t)-rcm(0)]2> to reach <R2>
R
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EQUILIBRATION OF CHAIN CONFORMATIONS
0e+00 1e+05 2e+05 3e+05 4e+05CPU (secs)
4000
6000
8000
10000
12000
<R2 >
(A2 )
C500: <R2> versus CPU time
I=1.09, T=450 K, b=1 atm
bulk CUC
200 300 400 500 600 700 800X
0
5000
10000
15000
20000
25000
30000
<R2>
(A2 )
C400: <R2> versus chain length
I=1.085, T=450K, b=1atm
bulkCUCsbest linear fit to bulk
o
0e+00 1e+05 2e+05 3e+05 4e+05CPU (secs)
4000
6000
8000
10000
12000
<R2 >
(A2 )
C500: <R2> versus CPU time
I=1.09, T=450 K, b=1 atm
bulk CUC
200 300 400 500 600 700 800X
0
5000
10000
15000
20000
25000
30000
<R2>
(A2 )
C400: <R2> versus chain length
I=1.085, T=450K, b=1atm
bulkCUCsbest linear fit to bulk
o
Mavrantzas, V.G., Boone, T.D., Zervopoulou, E., DNT, Macromolecules 32, 5072 (1999)
R C500:
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END-BRIDGING MONTE CARLO OF cis-1,4 POLYISOPRENE MELTS
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END-BRIDGING MONTE CARLO OF cis-1,4 POLYISOPRENE MELTS
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END-BRIDGING MONTE CARLO OF cis-1,4 POLYISOPRENE MELTS
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Combination with greatly facilitates equilibration at lowtemperatures.
parallel tempering
-1100 -1000 -900 -8000
10
20
30
40cis-1,4 PI 303K
318K 333K 353K 373K 393K 413K 438K 463K 493K 523K 553K
P
op
ula
tio
n D
istr
ibu
tio
n
U+PV (kcal/mol)
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VOLUMETRIC PROPERTIES OF cis-1,4 POLYISOPRENE
T=413K
3o 1.178 cm / gX
Literature values:N.Nemoto et al., Macromolecules, 1971: υ = 1.1964 cm3/gC.D. Han et al., Macromolecules, 1989: υ = 1.183 cm3/g
T=413K:
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END-BRIDGING IN ATACTIC POLYPROPYLENE
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END-BRIDGING IN ATACTIC POLYPROPYLENE
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END-BRIDGING IN ATACTIC POLYPROPYLENE
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END-BRIDGING IN ATACTIC POLYPROPYLENE
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END-BRIDGING IN ATACTIC POLYPROPYLENE
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CHARACTERISTIC RATIOS OF PPR
2
2
limnl
RC
n
n skeletal bonds, each of length l
m
isotactic (iPP) mmm…
r
syndiotactic (sPP) rrr…
atactic (aPP) rmr…
(random)
[1] Ballard et al., Polymer 19, 379 (1978); Zirkel et al., Macromolecules 52, 6148 (1992)
[2]Suter, U.W. and Flory, P.J. Macromolecules 8, 765 (1975)
[3]Ryckaert, J.-P., in Binder and Ciccotti (Eds)
PT EBMCPPtype Melt CUC
Experiment[1]
RISmodel[2]
chains[3]
aPP 6.21.0 5.20.4 5.5 5.5 6.1
sPP 8.51.1 9.60.1 11.0 8.0
iPP 6.60.3 6.2 4.2 6.1
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N,~,TV
Ab
c
MC simulations performed at given b, T, . Resulting c() dependence integrated to yield A/N as a function of at given b, T.
~
SAMPLING ORIENTED POLYMER MELTS
Conformation tensor:R
oR2
~
3RR
c average over all chains
unperturbed
A/N(,T,c)~Helmholtz energy function in flowing melt:
In quiescent, underformed melt, c = I~
with N=number of chains, =mass density
Introduce thermodynamic “fields”
][c~,,TB
)~,T,(N
A
c~Tk
1
c (1,3)
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xx
yy = -P
zz = -P
PE MELT UNDER UNIAXIAL EXTENSIONAL FLOW
maximalrelaxation time
xx ( ) .
dtx
dL
xL
ε1
Helmholtz energy, energy, and entropy of oriented melt
Mavrantzas, V.G. and DNT, Macromolecules, 31, 6310 (1998)
0.00 0.10 0.20 0.30 0.40xx
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Ene
rgy
Cha
nge
(J/g
)
TSUA
C78
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xx
yy = -P
zz = -P
PE MELT UNDER UNIAXIAL EXTENSIONAL FLOW
Mavrantzas, V.G. and DNT, Comp.Theor.Polym.Sci., 10, 1 (2000)
Cpredicted=(2.350.10)10-9 Pa-1 (C200 melt)
Cexperimental= 2.20 10-9 Pa-1
(Janeschitz-Kriegl)
Birefringence
0 1 2 3 4xx-yy (MPa)
0.000
0.002
0.004
0.006
0.008
0.010
nxx
-nyy
C200
C78
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SOLUBILITY OF OLIGOMERS IN POLYMER MELTS
s1-mer
polymer
polymer
SCISSION
FUSION
[f1'Npn0PT*] statistical ensemble
f1' f1/exp[(s1+3)(n)/(kBT)]
f1= oligomer fugacity
(n)=(i- j)/(si-sj), polymer chemical potential per segment
Np: total number of polymer chains. n0 : number of polymer segments if all oligomers were connected to chains. P : pressure. T : temperature. * : profile of relative chemical potentials controlling polymer chain length distribution.
Zervopoulou, E., Mavrantzas, V.G., DNT J.Chem.Phys. 115, 2860 (2001)
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SOLUBILITY OF C10 and C20 IN PE (NERD force field)
Method 1: Insertion-deletion moves in the f1NpnPT* ensemble
Method 2: Fusion-scission moves in the f1'Npn0PT* ensemble
0.0 0.2 0.4 0.6 0.8 1.0fugacity of C
10 (atm )
wei
ght f
ract
ion
of C
10
Experim enta l
M ethod 2M ethod 1
Solubility o f C10
and C20
in PE (N ER D force fie ld)(un iform m olecular w eight d istribution, I=1.08)
T=458K
0.000 0.005 0.010 0.015fugacity of C
20 (atm )
wei
ght f
ract
ion
of C
20 M ethod 2
T=474K
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SWELLING OF PE UPON SORPTION OF C10
0.0 0.2 0.4 0.6 0.8 1.0fugacity of C
10 (atm )
(V-V
0)/V
0 (%
) M ethod 1M ethod 2
Sw elling and density o f the system as a function o f C10
fugacity(un iform m olecu lar w eight d istribution, I=1.08,T = 458K)
0 0.2 0 .4 0 .6 0 .8 1fugacity o f C 10 (atm )
0
0.2
0 .4
0 .6
0 .8
1
den
sity
(g
r/cc
)
M ethod 1M ethod 2
T=458K
Method 1: Insertion-deletion moves in the f1NpnPT* ensemble
Method 2: Fusion-scission moves in the f1'Npn0PT* ensemble
T=458K
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Double Bridging(Karayiannis et al., 2001)
i“predator” mer i of ich
j attacks “prey” mer j of jch trimer (ja, jb, jc) adjacent to jja
jbjc
is excised from jch
SIMULATION OF STRICTLY MONODISPERSE MELTS: DOUBLE BRIDGING MONTE CARLO
N. Karayiannis, V.G. Mavrantzas, DNT, 2001
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Double Bridging(Karayiannis et al., 2001)
ij
“predator” mer j2 of jch
j2
attacks “prey” mer i2 of ich
i2
trimer (ia, ib, ic) adjacent to i2
iaib
ic
is excised from ich
SIMULATION OF STRICTLY MONODISPERSE MELTS: DOUBLE BRIDGING MONTE CARLO
N. Karayiannis, V.G. Mavrantzas, DNT, 2001
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Double Bridging(Karayiannis et al., 2001)
ij
j2
i2
trimer (ja’,jb’,jc’) connects i and j ja’
jb’jc’
ia’
ib’ic’
trimer (ia’,ib’,ic’) connects j2 and i2
SIMULATION OF STRICTLY MONODISPERSE MELTS: DOUBLE BRIDGING MONTE CARLO
N. Karayiannis, V.G. Mavrantzas, DNT, 2001
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Double Bridging(Karayiannis et al., 2001)
new chain jch’ is formed
new chain ich’ is formed
SIMULATION OF STRICTLY MONODISPERSE MELTS: DOUBLE BRIDGING MONTE CARLO
N. Karayiannis, V.G. Mavrantzas, DNT, 2001
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INTRAMOLECULAR DOUBLE REBRIDGING(N. Karayiannis, V.G. Mavrantzas, DNT, 2001)
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DB & IDR: MONODISPERSE LINEAR PE at 450K,1atm
0 2 0 0 4 0 0 6 0 0 8 0 0 1 0 0 0C h ain L en gth
1 .2 8
1 .2 9
1 .3 0
1 .3 1
1 .3 2
1 .3 3
1 .3 4
1 .3 5
1 .3 6S
pec
ific
Vol
um
e (c
m3 /g
)
ex p erim en ta l d a tasim u la tio n resu lts
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DB & IDR: MONODISPERSE LINEAR C1000 MELT 8000 atoms, T=450K, P=1atm
0 2 4 6 8 1 0 1 2 1 4k (A
-1)
-1 .0
-0 .5
0 .0
0 .5
1 .0
1 .5
2 .0S
(k)-
1
s im u la tion re su ltsX -ra y d iffrac tion d a ta
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SUMMARYAlgorithms based on End-Bridging Monte Carlo (EBMC) equilibrate atomistic models of polymer melts of average molecular weight 104-105 g/mol at all length scales.
Free energy and birefringence of oriented melts under steady-state processing flows can be obtained through EBMC in the presence of orienting fields.
Variable connectivity MC schemes allow prediction of sorption isotherms of oligomers in polymer melts without the need to insert/delete or exchange molecules between phases.
Performance at low temperatures can be enhanced by combining EBMC with parallel tempering.
Double Bridging and Intramolecular Double Rebridging equilibrate monodisperse melt systems with precisely defined molecular architectures.
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ACKNOWLEDGMENTSCollaborators
Dr. Vlasis Mavrantzas
Dr. Manolis Doxastakis Dr. Vagelis Harmandaris Mr. Nikos Karayiannis Dr. Christina Samara Dr. Vanessa Zervopoulou
Sponsors
DG12 of the European Commission, Brite-EuRam and GROWTH programmes (projects MPFLOW, PERMOD, DEFSAM)
DG12 of the European Commission, TMR programme (NEWRUP Research Network)
Greek GSRT, PENED programme, contracts 218-95E, 95-99E
SIMU Network