rotational isomeric states model of erythro diisotatic poly(norbornene)

8/14/2019 ROTATIONAL ISOMERIC STATES MODEL OF ERYTHRO DIISOTATIC POLY(NORBORNENE)

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ROTATIONAL ISOMERIC STATES MODEL OF ERYTHRO DI-ISOTATIC POLY(NORBORNENE)

Journal: Macromolecules

Manuscript ID: ma-2009-01319b

Manuscript Type: Article

Date Submitted by theAuthor:

19-Jun-2009

Complete List of Authors: Chung, Won Jae; Georgia Institute of Technology, School ofChemical & Biomolecular EngineeringHenderson, Clifford; Georgia Institute of Technology, School ofChemical & Biomolecular Engineering

Ludovice, Peter; Georgia Institute of Technology, School ofChemical & Biomolecular Engineering

ACS Paragon Plus Environment

Submitted to Macromolecules


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ROTATIONAL ISOMERIC STATES MODEL OFERYTHRO DI-ISOTATIC POLY(NORBORNENE)

Won J. Chung, Clifford L.Henderson and Peter J. Ludovice*

School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA

30332-0100, USA

*[email protected]

RECEIVED DATE (to be automatically inserted after your manuscript is accepted if required

according to the journal that you are submitting your paper to)

ABSTRACT

A new rotational isomeric states (RIS) model was developed that accurately predicts the unique

conformation of erythro di-isotactic poly(norbornene), a polymer with various important applications in

microelectronics. This model reflects the helix-kink morphology previously observed for this particular

polymer synthesized via a vinyl-like mechanism using a Pd catalyst. The model is based on

conformations observed in Monte Carlo simulations of oligomers containing 11 to 19 repeat units.

These simulations indicated the origin of the kinks in the helix-kink morphology is a reversal of the

helix symmetry, and this was incorporated into the RIS model. These kinks are kinetically trapped and

can move along the polymer chain and are created and destroyed at the chain ends. This model predicts

a rigid rod conformation that eventually transitions to a random coil at a degree of polymerization of

approximately 500, as these kinks disrupt the helical order. The resulting RIS model also reproduced the

extended chain behavior predicted by viscometry experiments.

KEYWORDS: RIS, poly(norbornene), helix, kink, simulation

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INTRODUCTION

Poly(norbornene) (PNB) produced via a vinyl-like polymerization is a new high performance

polymer that is useful as a new photoresist material in the microelectronics industry. This particular

PNB polymer has unique properties such as a low dielectric constant (2.2 < < 2.4), low absorption

coefficient at ultraviolet wavelengths, and low optical birefringence that make it suitable for use as an

interlayer dielectric, a deep UV photoresist, and a waveguide material. [1] Recent advances in

polymerization catalyst technology have allowed norbornene monomer to undergo a metal catalyzed

vinyl addition polymerization that retains the bicyclo-heptane group in the polymer backbone.[1-4]

This retention of the bicyclo-heptane group is in contrast to the Ring Opening Metathesis

Polymerization (ROMP) mechanism for norbornene which retains only a single cyclopentyl-ring in the

polymer backbone is shown in Figure 1. [5,6] As seen in Figure 1, there are a variety of stereochemical

configurations that can result from this polymerization depending on the exact nature of the catalyst

used and resulting monomer enchainment mechanism. First, addition across the norbornene double

bond could result in the monomer being enchained in either an endo-endo, endo-exo, or an exo-exo

form. Previous NMR studies of metal catalyzed vinyl addition polymerization, and in particular the Pd

catalyzed materials discussed in detail in this work, suggest that the exo-exo form is the only

enchainment product of the reaction.[7] Second, the monomer stereochemical orientations with respect

to one another can result in either an erythro di-isotactic form, an erythro di-syndiotactic form, or some

more random intermediate form (see Figure 2).

The structural complexity of the bicyclo-heptane group in the polymer backbone makes it difficult

to experimentally characterize the stereochemical isomerism. [8-10]. Howevever, previous simulations

and their comparison to viscometry experiments indicate that the Pd-catalyzed polymer is likely the

erythro di-isotactic configuration.[3] The simulation of only this configuration produces a polymer with

a highly extended conformation where the polymer dimensions scale as nearly the square of the

molecular weight. Of the various catalysts studied, only the Pd catalyst produces a similar scaling.[3] In

the same simulation study, the scaling of intrinsic viscosity with molecular weight of the non-

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stereoregular polymer was slighter over 1/2 indicating a random coil conformation. This same scaling

was observed experimentally for the Naked Ni-catalyzed polymer indicating that this catalyst

produced an atactic polymers. A separate study shows random coil behavior in similarly Ni catalyzed

PNB.[11]

More detailed modeling of the erythro di-isotactic configuration produces a helix kink

conformation, where the helix is occasionally interrupted by a kink.[4] The kink appears to be a

reversal of the helix symmetry from left to right handed. This helix-kink morphology results in a set of

unusual physical properties for the polymer that can be important to consider in its various applications.

For example, the helix-kink morphology is believed to be a critical underlying cause for the unusual

dissolution behavior exhibited by functionalized polynorbornenes that have been investigated as matrix

materials for UV sensitive photoresists that are used in the photolithography processes that are critical

for microelectronics manufacturing. [12] This work has focused on the use of molecular modeling and

and its comparison to experiment to determine the conformational behavior of erythro di-isotactic PNB

and develop a rotational isomeric states (RIS) model. This model will allow better prediction of this

polymers unique conformational behavior.

Figure 1. Poly(norbornene) produced via the vinyl-like polymerization (top) and ring opening

metathesis polymerization method (bottom).

n

n

n

n

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Figure 2. Structure of PNB. The erythro di-isotactic (left) and erythro di-syndiotactic (right) trimers are

shown to illustrate possible stereochemical variation.

This aforementioned helix-kink morphology present in erythro di-isotactic PNB induces a level

of structural order in PNB that is intermediate between a crystal and a truly amorphous glass.[4,13] The

PNB helical conformation is similar to substituted poly(acetylenes) and this is presumably due to the

alternating double bonds these polymer share.[4] This intermediate order appears as a split in the

amorphous halo that is typically observed in the wide angle x-ray diffraction (WAXD) for these

polymers.[4,14,13] The splitting of the first peak in the amorphous halo into two peaks in the Wide

Angle X-ray Diffraction Scattering (WAXD) pattern for PNB is indicative of the formation of

intermediate range order due to strong intramolecular interactions [4]. This extended chain behavior

that mimics the conformation of a rigid-rod appears to be unique to the erythro di-isotactic isomer of

PNB. Simulation of PNB without this alternating bridge-head carbon showed a much less extended

conformation that was similar to a random coil conformation.[3]

In this work we develop an RIS model that includes a more accurate description of this unique

helix-kink conformation that was described in the previous work.[4] This work uses Monte Carlo

models to characterize the conformation of the kink as a switching of the symmetry of the helices over a

few repeat units. This more accurate RIS model is validated against experimental viscometry results.

Such an RIS model is useful in calculating the dimensions of chains in solution and for generating initial

conformational guesses for bulk simulations for poly(norbornene). The framework of the RIS model

also has application for other polymers that may adopt the helix-kink framework such as various

1 2

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poly(acetylenes) and poly(norbornene) produced by Ring Opening Metathesis Polymerization (ROMP)

catalysts.

SIMULATION METHOD

Force Field

Previous modeling work done on PNB used ab initio calculations to customize bond stretch and

bond angle parameters to include the effect of the ring strain present in bicyclo-heptane groups in

PNB.[2] Such ring strain is not typically included in generic force fields. The bond angle parameters

from this work were unrealistically high, relative to generic force fields, even accounting for the ring

strain in bicyclo-heptane. We believe that these angle bend force constants are artificially high due to

the additional coupling of bonded force terms in the bicyclo-heptane groups in PNB. Quantum

mechanics programs simply estimate the second derivative of the energy with respect to a particular

bond angle. In simple hydrocarbons this derivative is independent from other bonded force field terms.

However, these derivatives are highly dependent on bond angle and bond length terms in a bicyclic

group such as those found in PNB. This second derivative estimate actually includes some of these

additional bond angle and bond length terms and is overestimated by using the second derivative from

quantum calculations. Therefore the force field used in this work utilized the bond stretch parameters

from these ab initio calculations, but the bond angle terms used were taken from a generic force field.

The force field used in this work utilizes bond stretch parameters taken from the customized force

field from ab initio calculations, the angle bending parameters from CHARMM 3.03,[15] and the non-

bonded parameters from the Dreiding 2.21 force field.[16] This composite force field with the bond

stretch, angle bending, and non-bonded parameters is compared against the MNDO calculated energy

values for the erythro di-isotactic dimer seen in Figure 3.[17] The difference between the two was then

fitted as the intrinsic torsion potential also shown in Figure 3. Here, the intrinsic torsional potential

includes both the intrinsic energy of the rotation of the sigma bonding orbitals as well as some quantum

correction of the torsional potential. MNDO semi-empirical quantum calculations were used because

their simplicity allowed the easy calculation of the potential energy as a function of the central torsional

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angle for numerous angles. However, the MNDO results in Figure 3 also compared very well to more

rigorous Density Functional Theory calculations.[2] The results are also essentially identical to

previous semi-empirical calculations using the AM1 model.[2]

-10

-5

0

5

10

15

20

0 50 100 150 200 250 300 350

Energy(kcal/mole)

Torsion Angle (Degrees)

Figure 3. Torsion potential of erythro di-isotactic norbornene dimer. The force field with the

customized bond stretch terms, CHARMM 3.03 angle bend terms, along with the non-bonded

parameters from Dreiding 2.21 is compared against the MNDO values. The dots represent the MNDO

values, the short dashed line represents the intrinsic torsion potential, the short-long dashed line

represent potential without the intrinsic torsion potential, and the solid line represent the final force field

with the intrinsic torsion potential.

The shift dihedral functional form was selected to describe the intrinsic torsion potential for the

erythro di-isotactic PNB in Equation 1.

( )[ ]= +=6

1nn,0n, ncos1K21E [1]

The parameters K,n, , and 0,n in Equation 1 are the dihedral force constant, torsion angle, and phase

shift, respectively. This equation was used to fit the MNDO results seen in Figure 3 and the optimized

parameters are listed in Table 1.

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Table 1: Intrinsic torsion parameters for erythro di-isotatic PNB using the shift dihedral form (0,n =180 to reflect the intrinsic torsional energy maximum at 180).

n K,i n,i1 -3.6002 0.1547

2 0.5303 -0.6704

3 -2.7103 -0.0344

4 -0.0377 10.1365

5 -2.0476 0.28656 -0.9161 0.6188

Monte Carlo Model

Metropolis Monte Carlo (MC) [18] simulations with the pivot algorithm [19] followed by energy

minimization were used to simulate isolated chains of PNB. The intramolecular interactions in PNB

dominate over the intermolecular interactions because of the bulky backbone structure of this polymer.

Therefore, the isolated polymer is a reasonable model for the polymer conformation. Primarily, the

intramolecular torsional barriers and backbone steric hindrance determines the polymer conformation

while the surrounding polymer produces only minor structural perturbation [3,4]. Similar isolated chain

simulations have been used to accurately reproduce experimental conformations in other polymer

systems that do not have strong inter-chain interactions. [20-22].

The modeled erythro di-isotactic PNB chain began in its extended conformation (torsion angles at

180o) or at random conformations. Next, a movable backbone torsion angle is randomly selected and

randomly perturbed followed by energy minimization. Energy minimization began with the conjugate

gradient [23] algorithm followed by a quasi-Newton method with a Broyden, Fletcher, Goldfarb,

Shanno (BFGS) update of the Hessian [24]. Each minimization step, included in the MC moves,

proceeded until the potential energy gradient was less the 0.001 kcal/(mole ). Single chains of 11, 15,

and 19 repeat units of erythro di-isotatic PNB were simulated. These chains were simulated for 2000

MC moves and resulted in 3 typical conformations with an acceptance ratio of approximately 15%. MC

simulations were performed on four different initial conformations that included isolated PNB chains in

the extended conformation and three additional conformations that used backbone torsion angles in a

random uniform distribution. Five MC simulations were performed on each of the four initial

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conformations, totaling 20 MC simulations for each chain length. Each of these were run for a total of

2000 MC moves which was sufficient to converge to a local energy minimum.

Rotational Isomeric States Model

Rotational Isomeric States (RIS) [25,26] models are first order Markov models that describe a

polymer conformation in terms of a torsion angle distribution, which assumes that each torsion angle is

determined only by the one previous backbone torsion angle. Typically this distribution is extracted

from a model of a small oligomer of the polymer that includes only two adjacent torsion angles. The

long-range effects of the bulky bicyclo-heptane ring in PNB make this assumption unreasonable; and

larger structures that include more than just neighboring torsion angles must be used to produce these

probabilities [4,9,10]. RIS states maps were calculated previously for a trimer, pentamer, and heptamer

for the erythro di-isotatic PNB that showed drastically different behavior as a function of molecular

weight.[4]

The aforementioned MC model was used to sample the conformational space of erythro di-isotatic

PNB and determine the probability distribution of backbone torsion angles using larger oligomers to

include long-range effects. This torsioinal angle distribution, that includes these long-range effects, is

then used in a reduced RIS model that models the distribution of a torsion angle as a function of its

adjacent torsion angle only. The major advantage of the RIS model is that it can be used to efficiently

model dilute solution behavior or to generate the initial conformations for bulk modeling in periodic

boundary conditions.[27-33] Using the RIS matrix generation scheme [25,26] the unperturbed mean

square of the end-to-end distance,0

2r , as well as the unperturbed mean square of the radius of

gyration,0

2s , can also be quickly determined as a function molecular weight.

RESULTS AND DISCUSSION

Monte Carlo Simulation

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Metropolis MC simulation with the pivot algorithm was conducted for erythro di-isotactic PNB

with chain lengths of 11, 15, and 19 repeat units corresponding to molecular weights of 848, 1412, and

1788 g/mole, respectively at a temperature of 298 K. The energy evolution of the MC simulation on a

chain length of 11 repeat units for 2000 MC moves equilibrated after only 250 MC moves for several

different starting configurations where the initial chain was in its extended or random conformations and

all had the same equilibrium energy of 432 kcal/mole with an acceptance ratio of approximately 15%.

Due to the fact that MC simulation involves abrupt transitions, the effect of the number of MC moves

was examined. For the 11 repeat unit erythro di-isotactic PNB, 2000, 5000, and 10,000 total MC moves

were compared. For all three cases, the equilibrium conformation is achieved after about 250 MC

moves with a resulting energy of 432 kcal/mole and no abrupt transitions or new minima were observed.

Therefore, 2000 MC moves on the Metropolis MC simulation are sufficiently long enough to reach

equilibrium.

Independent of the starting conformation, the final conformation always consisted of repetitive

torsion angles of either ~120o or ~240o, and both configurations had the same equilibrium energy of 432

kcal/mole. With the exception of the torsion angles near the ends of the polymer chain, all initial

conformations produced helices with repeating angles of ~240

, or 120

. Both helices have

approximately 16 repeat units per turn and they are mirror images of each other since their backbone

torsion angles are symmetric about 1800. No kinks were observed in these helices because this 11

repeat oligomer is simply not long enough to show any kinks in the helices.

Previously, an RIS model was developed by calculating a RIS state map for the heptamer erythro

di-isotactic PNB [4]. The energies were calculated by rotating the two middle adjacent torsion angles in

increments of 10o with the customized force field [2]. The energy was minimized at each point with

respect to the 4 external torsion angles. The RIS state map is only an approximate characterization of

the energetic states because the 4 external torsion angles are in local minima. This model produced

helical regions with occasional interruptions of the helices with kinks with the probability of a helix

angle followed by a kink of approximately 3.27% as determined from the Boltzmann factor using the

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0

50

100

150

200

250

300

350

0 500 1000 1500 2000

Torsion

Angle(Degrees)

Monte Carlo Move #

Figure 4. MC simulation on erythro di-isotatic PNB with 15 repeat units. Torsion angle distribution as

a function of MC move is shown with the two torsion angles to each of the ends excluded. The

equilibrium conformation has an energy of 612.0 kcal/mole that consists of repetitive torsion angle of

120o or 240o.

The torsion angles that make up the kink are identical in a relative sense but have different

absolute values depending on the symmetry of the helix transition. Going from a helical region

consisting of repeated torsion angles of 120 to a helical region made up of 240 repeated torsion

angles, the kink is in the order of 100, 200, and 260 torsion angles. However, going from a helical

region consisting of repeated torsion angles of 2400 to a helical region of repeated torsion angles of 120

the transition exhibits the following torsion angles in sequence of 260, 160, and 100. The sequential

transition angles between helices consisting of repeated helical angles of 120 or 240as seen below.

1 2 3

120 100 200 260 240 260 160 100 120n n n

kink kink

An abrupt transition is seen around MC move number 1450 and 1750 in Figure 4. However, there

are no significant energetic transitions associated with this torsion angle change as seen in the energy

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The same distribution after MC move 1750, where the kink and the associated transition angles have

shifted to the right of the oligomer chain as seen below.

0000000000 240260200100120120120120120120

kink

The energy barrier for shifting of the kink, moving down or up along the chain is approximately 4.0

kcal/mole which is the maximum in the energy difference between the energy between moves 1450 and

1750 and the average energy before and after this period from Figure 5.

In order to determine if this behavior observed in the oligomer of 15 repeat units was

representative of the polymer chain, an even longer oligomer of 19 repeat units was also simulated. Thisoligomer showed essentially the same behavior observed in the oligomer of 15 repeat units. The exact

same 3 possible conformations were observed, which includes helical conformations consisting of either

repeating torsion angles of ~1200 or ~2400 with an equilibrium energy of 785.2 kcal/mole and

conformations that have a helical segment that is interrupted with the kink transition angles followed by

a helical segment with an equilibrium energy of 789.0 kcal/mole, as shown in Figure 6.

780

800

820

840

860

880

0 500 1000 1500 2000

Energy(kcal/mole)

Monte Carlo Move #

Figure 6. Potential energy from the MC simulation of erythro di-isotatic PNB with 19 repeat units.

Solid line represents final conformation consisting of repeated torsion angle of 240o, the dotted line

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represents the conformation with 120o repeated torsion angles, and the short-long dashed line represents

the conformation that has the repeated torsion angle of 120o interrupted with the kink transition angles

then back in to helical region consisting of repeated torsion angle of 240 o, with equilibrium energies of

785.2, 785.2, and 789.0 kcal/mole, respectively.

Independent of the starting conformation, the final conformations that consisted of repetitive

torsion angles of either ~120 or ~240, which both had the same potential energy of 785.20 kcal/mole

after energy minimization. The torsion angle distributions are shown in Figure 7a and7b. Deviations

from 120 or 240 in Figures 7a and 7b adopt values similar to the transition angles above. The angle in

Figure 7a that transitions from ~120 to ~100 and then later to ~200 is the third angle from the end of

the oligomeric chain (the two end angle are are not included in these figures). This illustrates the

formation of a kink from the end of the PNB chain. As this angle is third from the end transitions from

~100 to ~200 the adjacent angle that is fourth from the end transitions from ~120 to ~100 as seen in

Figure 7a. A similar phenomenon is observed in Figure 7b as the torsion angle located 3rd from the end

transitions from ~240 to ~260. These transitions are the formation of a kink at the chain end. Since

these kinks are a change in the symmetry of the helices their net formation or destruction can only occur

at the chain end.

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15

50

100

150

200

250

300

350

0 500 1000 1500 2000

TorsionAngles(Degrees)

Monte Carlo Move #

100

150

200

250

300

350

0 500 1000 1500 2000

TorsionA

ngle(Degrees)

Monte Carlo Move #

(a) (b)


a function of MC move is shown with the two torsion angles to each of the ends excluded for (a) helix

with 1200 torsion angles with an equilibrium energy of 785.2 kcal/mole. (b) helix with 240 torsion

angles with an equilibrium energy of 785.2 kcal/mole.

The 3rd type of conformation included the helix-kink morphology where the helical region is

interrupted by kink transition torsion angles where the handedness of the helicex was changing. The


120 followed with the kink transition angles in the order of 100, 200, and 260 followed by another

set of repetitive torsion angle of 240 is shown in Figure 8. Assuming a Boltzman distribution for this

observed energy difference between the kinked and un-kinked helix suggests that the probability of the

chain possesing a kink is approximately 0.17%. This low probability indicates that observing a kink in

the simulation of our oligomers should be highly unlikely. However, as with the smaller oligomers, this

energy difference does not reflect the fact that kinks are formed at the chain ends and therefore appear

with a greater frequency in the simulations here.

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This kinked helix structure, comprised of two angles of 240 with the remaining non-transition

angles at 120 forms after approximately 1400 MC moves. Prior to this no angles of 240 existed, and

these are formed as a kink which starts near the end of the chain moves two positions into the 120

angles eventually forming two 240 angles in the wake of this shift. Note the angle distribution at the

end of the chain does not match the angle distribution above exactly because of angle variation at the

chain end. However, after the transition only the angles in this distribution are observed (120, 240,

100, 200 and 260).

0

50

100

150

200

250

300

350

0 500 1000 1500 2000

TorsionAngle(Degre

es)

Monte Carlo Move #


a function of MC move is shown with the two torsion angles to each of the ends excluded. The


120 followed with the kink transition angles of 100, 200, and 260 followed by another set of

repetitive torsion angle of 240.

From the MC simulation above the mean squared unperturbed radius of gyration0

2s averaged

over simulated chains varies as a power law function with molecular weight. The value of0

2s scales

with molecular weight to the 1.85 power from a non-linear fit of MC simulations of the three different

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molecular weights (R2=0976). An identical trend vs. molecular weight is obtained for the mean squared

unperturbed end-to-end distance0

2r where the scaling exponent is 1.89 with molecular weight. Both

scalings are indicative of a highly extended chain similar to the rigid-rod behavior for the erythro di-

isotactic PNB. This exponent should vary from for a random coil in the condition to a value of 2

for a rigid rod. These scaling values compare well to their experimental scaling of approximately 2

determined from intrinsic viscosity.[3,4].

Rotational Isomeric States Model

RIS models are used to describe the microstructure of isolated polymer chains in detail as a

function of their torsional angle distributions.[34] Additionally, RIS models are used to generate initial

conformations in periodic boundary conditions where combinations of subsequent minimizations and or

dynamics are used to relax the structure to simulate bulk behavior.[26,33] Because polymer glasses are

highly constrained they relax only small amounts under typical phase space sampling techniques. For

this reason, RIS models are particularly important in producing accurate initial conformations because

the relaxed models do not depart significantly from the vicinity of the initial conformation generated

[27,29-32].

Previously, preliminary RIS descriptions for the erythro di-isotactic PNB were developed based

on potential energy contour maps for much smaller oligomers than the ones simulated here. [4,8].

These models looked at only two adjacent torsion angles. While they both characterized the elongated

nature of the polymer chain, they did not include the detailed nature of helices interrupted by kinks

caused by a helix symmetry reversal. This level of order is not revealed with such small models and is

the basis of the RIS model developed here. The statistical weight matrix, U, for the RIS model for

erythro di-isotactic PNB is given in equation 2

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=

==

=

=

=

=

=

=

=

00000001

1000000001000000

00002.010000

00010000

00001000

00000100

000000002.01

100

160260

240

260

200

100

120

100160260240260200100120

1

1

1

1

1

1

1

1

2

b

b

a

a

bbaa

U

[2]

where the state of bond 2 is defined according to statistical weights, which depend on the state of only

the previous bond 1. Some of the torsion angles repeat but the transition of one helix to another is path

dependent. The superscript a and b represents the path of going from a ~120 helix to that of ~240

helix and ~240 helix to that of ~120 helix, respectively. Based on the aforementioned Boltzmann

probability using the energy difference from the MC simulations of the oligomer with 15 and 19 repeat

units, the probability of a kink (i.e. going from one helix to another) is ~0.2%.

0

2

4

6

8

10

12

0 0.001 0.002 0.003 0.004 0.005

/

1/(# bonds)

Figure 9. The ratio of the unperturbed mean square end-to-end distance to the unperturbed mean square

radius of gyration plotted against inverse number of bonds from the RIS model. The value of this ratio

is approximately constant at a value of 11.5 and decreases rapidly after passing a certain chain length.

In the limit, the value of this ratio is 12 for a rigid rod and a value of 6 for random coil.

ExperimentalRegion

Rigid Rod

Random Coil

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This RIS model was used to calculate the scaling of the mean square radius of gyration and mean

square end-to-end distance as a function of molecular weight by using the matrix generation scheme of

the RIS model [25,26]. The ratio of the unperturbed mean square end-to-end distance 20r to the

unperturbed mean square radius of gyration 20s is plotted as a function of the inverse of the degree of

polymerization (number of bonds) from the RIS model is shown in Figure 9. The value of this ratio is

approximately constant at 11.5 until a polymer containing approximately 1000 bonds is reached and the

value begins to decreases to 6. Experimentally, a transition from rigid to random coil behavior was

observed at a similar chain size using the ratio of the radius of gyration to the hydrodynamic radius for a

Pd catalyzed polymer.[9] The value of this ratio varies from 12 in the rigid rod limit to 6 in the random

coil limit. These results indicate that PNB behaves like a highly extended chain at low molecular weight

which transitions to a random coil in the very high molecular weight limit as the kinks become more

prevalent. The molecular weight of the erythro di-isotactic PNB used in the previous viscometry

experiments are in the low molecular weight range so they should exhibit the rigid rod scaling in

viscometry [3,4]. This extended chain conformation is close to the rigid rod limit because the kink

causes a deviation of the chain persistence direction by only ~3 at 0K as shown in Figure 10. The

conformation pictured in Figure 14 is an energy minimized structure and consequently represents the

polymer chain in the 0K temperature limit. The polymer is certainly not this rigid, as even the helix is

not perfect at typical temperatures as scene in the previous bulk simulations.[4] Since every other bond

is non-rotatable, the turns of each repeat unit in the helix are very slow and require 16 repeat units to

make 1 turn in the helix.

~30~30

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Figure 10. The angle created by the kink that changes the helical symmetry changes the persistence

direction by ~3. The helices are perfect helices at 120 and 240, where in realistic bulk structures the

helices will not be perfect and the angle created would be slightly different.

The values of 20s from the RIS model and the MC simulations used to develop the RIS model

are plotted as a function of molecular weight shown in Figure 11. The scaling of 20s with molecular

weight is 1.96 from the RIS model and 1.85 from the MC simulations. This agreement indicates that the

RIS model has captured the appropriate scaling observed in the atomically-detailed model used in the

MC simulations. Slightly better agreement between the RIS and MC models is obtained for 20r where

the RIS and MC scaling are 1.92 and 1.89 respectively. However 20s is generally more reliable as it is

based on all the atoms in the oligomer chain rather than the end atoms.

10

100

1000

104

105

106

1000 104

105

MW

Figure 11. Comparison of MC results for erythro di-isotactic PNB (circles) and RIS model calculations

(triangles) for the unperturbed mean square radius of gyration.

The intrinsic viscosity [ ] is calculated from the universal viscosity law to relate the [ ] scaling

with molecular weight. While this law assumes a random-coil conformation, a proportionality between

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[h] and a function of the molecular weight (M) and the unperturbed radius of gyration 20s .At the

condition, the following scaling law is derived from the universal viscosity law [4]

[ ] 2123

20

MM

s

. [3]

This proportionality is valid regardless of whether the polymer conformation is a random coil or a more

elongated chain as observed in PNB. However, the proportionality constant in the relationship above

does vary depending on the polymer conformation, so the actual universal viscosity law cannot be used

given that the ratio of 20r to20s changes with molecular weight. Using the 1.98 power scaling of

20s

with molecular weight and equation 6 we obtain a 1.97 power scaling of intrinsic viscosity with

molecular weight. This is consistent with the experimentally determined value of this scaling of

approximately 2.[3,4]

While the RIS model does reproduce the appropriate scaling from equation 3, it does not take into

account the dynamic variation about the torsion angle used in the RIS model. The force field employed

here possesses a slightly flatter energy curve in the vicinity of the RIS angles relative to many vinyl

polymers typically modeled by RIS models as seen in Figure 3. This means that there is some variation

about these angles of 120 and 240 and this warrants an investigation of the effect of this flexibility.

Previous bulk simulations showed a variation of 20 and resulted in a much less regular polymer

conformation than the one pictured in Figure 11.[4] To determine the effect of this torsional state

flexibility, the RIS statistical weight matrix was incorporated into the amorphous builder module in the

Cerius2 software developed by Accelrys. The combination of the van der Waals energy and the RIS

statistical weight matrix is utilized in order to prevent the chain being built from overlapping with itself.

Chains were generated randomly using the RIS model above with a specified tolerance on the torsional

angles and if the next added monomer increased the potential energy by more than 10 kcal/mole the unit

was rejected and a new torsion angle was chosen. If this procedure failed 20 times another monomeric

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0

100

200

300

400

500

600

700

800

0 500 1000 1500

1/2

# bonds

Figure 12. Simulation of 21

20s of PNB determined from RIS model at various degrees of

polymerization. Simulations include a window of 0, 10, and 20 variation about the local energy

minima of the backbone torsion angles represented by,, and , respectively. All error bars are the

95% confidence interval about the mean.

CONCLUSIONS

A new RIS model was developed for the erythro di-isotactic isomer of poly(norbornene) that

reflects the helix-kink morphology previously observed for this particular polymer in these and previous

simulations. Monte Carlo simulations indicate that the origin of the kink is the reversal of the helical

symmetry, which occurs in approximately 0.2% of the monomeric units. These kinks can form or be

destroyed at the polymer ends and each one of these kinks adds approximately 4 kcals/mole to the

energy of the helical structure. Similarly, the barrier to the movement of the kink down the chain also

appears to be approximately 4 kcals/mole, suggesting that these kinks are kinetically trapped in the

polymer chain and their distribution may be modified by high temperature annealing. The RIS model

and the MC simulations, on which this model is based, predict a polymer that has rigidrod like

behavior that persists until a transition to random coil behavior that begins when the number of

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backbone bonds exceeds 1000. The RIS model accurately reproduces the rigid-rod scaling behavior of

this polymer extracted from viscometry experiments for a Pd catalyzed polymer that is assumed to be

erythro di-isotactic polymer. A sensitivity analysis of including flexibility about the RIS torsional states

was conducted and this found that the rigid rod scaling observed in the viscometry experiments

disappeared from the model when such flexibility was incorporated. This RIS model reproduces the

unique conformation of this industrially important polymer that has a molecular weight dependent

conformation.

ACKNOWLEDGMENT

The authors gratefully acknowledge financial support from NSF award DMR-000309236 and Promerus

Corporation. We also wish to thank Larry Rhodes, Larry Seger, and Robert Shick of the Promerus

Corporation for helpful input and discussions.

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ge 27 of 27 Submitted to Macromolecules

rotational isomeric states model of erythro diisotatic poly(norbornene)

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