Doros N. Theodorou

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.

[email protected] [email protected]

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

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)

CONCERTED ROTATION MONTE CARLO

L. R. Dodd, T.D. Boone, DNT, 1993


“driver” angle

“driver” angle



“driver” angle

“driver” angle



“driver” angle

“driver” angle


END-BRIDGING MONTE CARLO

P.V.K. Pant & DNT, 1994


P.V.K. Pant & DNT, 1994

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


i j kj i

j k

i j

i k

k=1,…,m, ki, j

Chain length distribution controlled through k* profile

EBMC PERFORMANCE AS A FUNCTION OF CHAIN LENGTH

rcm

t0=CPU time for <[rcm(t)-rcm(0)]2> to reach <R2>

R

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:

END-BRIDGING MONTE CARLO OF cis-1,4 POLYISOPRENE MELTS



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)

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:

END-BRIDGING IN ATACTIC POLYPROPYLENE





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

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)

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

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

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)

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

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

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


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




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




new chain jch’ is formed

new chain ich’ is formed



INTRAMOLECULAR DOUBLE REBRIDGING(N. Karayiannis, V.G. Mavrantzas, DNT, 2001)

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

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

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.

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

Documents

Doros N. Theodorou