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
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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
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)
Figure 7. MC simulation on erythro di-isotatic PNB with 19 repeat units. Torsion angle distribution as
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
equilibrium conformation has an energy of 789.0 kcal/mole that consists of repetitive torsion angle of
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 #
Figure 8. MC simulation on erythro di-isotatic PNB with 19 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 789.0 kcal/mole that consists of repetitive torsion angle of
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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