microstructure of 2,3 erythro di-isotactic polynorbornene ... · markov model in which the...

Microstructure of 2,3 erythro di-isotactic polynorbornenefrom atomistic simulation

S. Ahmed1, P.J. Ludovice* , P. Kohl

School of Chemical Engineering, Georgia, Institute of Technology, Atlanta, GA 30332-0100, USA

Received 16 April 1999; received in revised form 8 September 1999; accepted 9 September 1999

Abstract

A new RIS model was developed for erythro di-isotactic polynorbornene that included long-range steric interactions that are needed toproperly model its conformation. These interactions were included by extracting RIS states from an approximate energy state map for theheptamer of this polymer. The RIS model predicted a novel helix-kink conformation for this polymer in which helices were occasionallydisrupted by a backbone kink and a value of approximately 1.9 for the exponent “a” from the Mark–Houwink–Sakurada equation at theucondition. These RIS results were consistent with independent single chain Monte Carlo simulations and were also used to generate bulkperiodic structures of this polymer. The results of these simulations compare well to both experimental viscometry and wide angle X-raydiffraction results for two polymer samples synthesized using Pd and zirconocene homogeneous polymerization catalysts. These resultsindicated that the likely stereochemical configuration for polymers produced using these catalysts is the erythro di-isotactic form. They alsosuggested that polymers of similar structure such ascispoly(t-butyl acetylene) may also adopt this kink-helix conformation.q 2000 ElsevierScience Ltd. All rights reserved.

Keywords: Polynorbornene; Atomistic simulation; Rotational isomeric states

1. Introduction

The bi-cyclic variation of polynorbornene is a polymercurrently under investigation for a number of applications,including deep ultra-violet photoresists and interleveldielectrics in microelectronics applications [1–6]. The 2,3-norbornene monomer undergoes a vinyl-like polymeriza-tion that retains the bi-cyclic conformation in the backboneof the resulting polymer as seen in Fig. 1. This is unlike thecommonly employed Ring Opening Metathesis Polymeriza-tion (ROMP) mechanism which retains only a single ring inthe polymer backbone [7,8]. This polymer is currently beingdeveloped for inter-level dielectric applications by the BFGoodrich Company under the trade name Avatrelw dielec-tric polymer. Polynorbornene has excellent dielectric prop-erties and both processing and potential cost advantagesover other materials currently being used as inter-leveldielectrics [1,9]. The high barrier to backbone rotationproduces a high glass transition temperature which allowsthe use of this polymer in the high temperature processingsteps employed in microelectronics fabrication. This lack of

segmental mobility coupled with a lack of a strong dipoleproduces a low dielectric constant, which makes this mate-rial a good dielectric insulator. However, the bi-cyclic back-bone structure makes it difficult for direct experimentalcharacterization of the polymer microstructure using 2DNMR [1,10,11]. Therefore, we used a combination of mole-cular modeling and experiment to determine the likelymicrostructure of this polymer. This work utilized a custo-mized force field for poly(norbornene) developedpreviously using semi-empirical, Hartree–Fock, andDensity Functional Theory quantum calculations [12].This force field was developed to account for the highlevel of ring strain and steric hindrance of the backbonethat occurs in poly(norbornene). Generic force fields arenot parameterized for bi-cyclic structures, and the optimizedparameters for both the bicycloheptane geometry as well asthe backbone torsional barrier differed from generic forcefield parameters.

Based on NMR experiments carried out by researchers atthe BF Goodrich Corporation, poly(norbornene) is believedto undergo 2,3 exo–exo polymerization (Fig. 1), regardlessof the homogeneous polymerization catalyst used [1]. Thehydrogens that occupy the endo positions on carbons 2 and 3extend below the plane of the polymer backbone and are notpictured in Fig. 1. We hypothesize that different catalyst

Computational and Theoretical Polymer Science 10 (2000) 221–233

1089-3156/00/$ - see front matterq 2000 Elsevier Science Ltd. All rights reserved.PII: S1089-3156(99)00083-5

* Corresponding author.E-mail address:[email protected] (P.J. Ludovice).1 Current address: Merck & Company, Whitehouse Station, NJ, USA.

systems produce different stereochemical configurationsand this gives rise to different properties [13,14]. It ispresumed that the zirconocene catalyst analyzed byKaminsky and co-workers [15,16], and the palladiumcatalyst developed by the BF Goodrich corporation[17] both produce the highly stereo-regular 2,3 erythrodi-isotactic polynorbornene. Our hypothesis is based onthe fact that both these polymers have identical WideAngle X-ray Diffraction (WAXD) patterns and they areboth intractable or insoluble in organic solvents undertypical conditions. In the 2,3 erythro di-isotactic config-uration, the bridging carbons (carbon #7) point in alter-nating directions when the polymer is in the allcisextended conformation (see Fig. 1). Fig. 2 contains aheptamer of the erythro di-isotactic configuration, whichis analyzed in this work. Preliminary WAXD results forthe Pd catalyst are identical to Kaminsky’s WAXD forpolymer produced from the ziconocene catalyst thatindicates that poly(norbornene) produced from thesecatalysts have the same or similar structure [18].Although other current work is focused on the connec-tion between various stereochemical configurations andcatalysts [13], the microstructure of the erythro di-isotactic isomer of polynorbornene, in the glassy state,is investigated here. The comparison of simulation

results to experiment is used to test our hypothesisthat this is the likely stereochemical configuration forpoly(norbornene) obtained with the Pd and zirconocenecatalysts above.

2. Results and discussion

2.1. Rotational isomeric states model

A Rotational Isomeric States [19,20] (RIS) model wasdeveloped for 2,3 erythro di-isotactic polynorbornene. RISmodels approximate the conformation space of the polymerchain as a set of low energy conditional torsional states. TheRIS model, in a typical implementation, is a first-orderMarkov model in which the probability of any backbonetorsional angle occurring is dependent on the state of theprevious rotatable bond. Using matrix techniques, thesemodels can be used to elucidate the microstructure ofisolated polymer chains [19,20] and can also be used togenerate the starting conformations for subsequent minimi-zation or dynamics studies in periodic boundary conditions[21–23]. This model assumes a first-order Markov Processin which the occupation of a given torsional state is a func-tion of only the previous state. This means that the energetic

S. Ahmed et al. / Computational and Theoretical Polymer Science 10 (2000) 221–233222

5

6

7

4

1

2

3

Fig. 1. Structure of polynorbornene showing 2,3 exo–exo enchainment. This particular trimer is erythro di-isotactic isomer.

φ1

φ2

φ3

φ4

φ5

φ6

Fig. 2. 2,3 erythro di-isotactic polynorbornene heptamer. The numbering scheme for the rotatable backbone torsions in the heptamer is shown.

states are determined by specifying two adjacent torsionalstates and described by a second-rank transition probabilitytensor. The interactions that determine these states arecalled second order interactions within the RIS frameworkbecause they involve two adjacent torsion angles. While theformulation of a second order Markov model that involvesthird-order RIS interactions is possible, it is impracticalgiven the mathematical complexity involved for polymers.The typical RIS model incorporates only second-order inter-

actions by using a serial product of large second-ranktensors that produce an ensemble average of geometricproperties over all the rotational isomeric states. To ourknowledge explicit inclusion of third order RIS interactionshas not been used for large polymers because this wouldinvolve a product of generator tensors of rank three. Rather,we have included these higher order interactions in anapproximate fashion as described below.

The bulky nature of the bicycloheptane backbone groupmakes these long range intramolecular effects significant.This is clear from a comparison of conformational energystate maps for a trimer and heptamer of polynorbornene.Fig. 3 shows a potential energy contour as a function oftwo backbone torsion angles using a trimer of di-isotacticpolynorbornene. A trimer of polynorbornene contains tworotatable backbone torsion angles and is the smallest oligo-mer that can be used to characterize the RIS states. Theenergies used to produce the contours in Fig. 3 were calcu-lated using version 1.5 of the Cerius2w program from Mole-cular Simulations Inc. [24]. Energies were calculated atincrements of 108 and utilized the aforementioned customforce field [12]. The internal energy was minimized withrespect to all other degrees of freedom in the trimer otherthan the two torsion angles. In this RIS treatment we assumethat the bond connecting atoms 2 and 3 is non-rotatable. Thehighly strained bicycloheptane group makes this a reason-able assumption although this is not always the case formore flexible rings such as those found in the amino acidproline. Fig. 3 illustrates the classic symmetry of the typicalRIS model. In this particular case the symmetry is about thenormal to the diagonal�F1 � 2F2� as opposed to the diag-onal �F1 � F2� simply because of the anti-symmetric defi-nition of the torsional angles used. We used the samereference for the definition of the torsional angles as thatused in the Cerius2w program. For more flexible polymers,such as poly(olefins), the energy map of a segment contain-ing two rotatable bonds is sufficient to characterize theconformational behavior of the entire polymer chainthrough the RIS model. This occurs because long rangeenergetic interactions along the polymer chain are relativelysmall and do not significantly change the RIS state map likethe one seen in Fig. 3. However the bulky nature of thebicycloheptane group in the backbone of poly(norbornene)suggests that long-range energetic effects may be moresignificant. Calculations of the same RIS state map picturedin Fig. 3 for the poly(norbornene) pentamer and heptamershow drastically different behavior. Fig. 4 is the RIS statemap for the heptamer of erythro di-isotactic poly(norbor-nene), whose structure is pictured in Fig. 2. These energycalculations were carried out in a similar fashion to thetrimer calculations except that the energy was minimizedat each point with respect to the four external torsion angles(angles 1, 2, 5 and 6 in Fig. 2). Angles 1 and 2 on the axes inFig. 4 are the internal angles only, and correspond directlyto angles 3 and 4 from Fig. 2. Because the four additionalinternal angles are in local minima only, the energy map in

S. Ahmed et al. / Computational and Theoretical Polymer Science 10 (2000) 221–233 223

0 60 120 180 240 300 3600

60

120

180

240

300

360

Φ2 (degrees)

Φ1

(degrees)+

+

Fig. 3. Potential energy contour plot for torsion anglesf1 and f2 in atrimer. The minima are marked with a1 symbol. Contour levels are at4,7,10 and 13 kcal/mol above the minimum.

0 60 120 180 240 300 3600

60

120

180

240

300

360

Φ1

(degrees)

Φ2 (degrees)

+

+

+

+

Fig. 4. Potential energy contour plot for torsion anglesf3 and f4 in aheptamer. The minima are marked with a1 symbol and the group ofdots indicates the region in which all the Boltzmann Jump minima wereclustered. Contour levels are at 3,6,9 12 and 14 kcal/mol above the mini-mum.

Fig. 4 is only an approximate characterization of theenergetic states. This approximation is responsible for theloss of the typical symmetry in this plot. A more appropriatemap is obtained by finding the global energy minima forstructures with fixed internal angles (3 and 4). Better still,integration of the Boltzmann factor for all conformationswith fixed internal angles would account for the entropyof these additional degrees of freedom. These approacheswould require exhaustive searches and enumerations overall conformational variations of these four angles. If localminima were obtained for conformations involving theadditional four angles at the same resolution (108 incre-ments) as the two internal angles, this would require 1:67×106 times the computational resources of the approximateapproach used in Fig. 4. Currently the calculation in Fig. 4requires approximately one day on a workstation so themore detailed calculation was infeasible. However, we arecurrently investigating the symmetry of this molecule in

order to reduce the resolution of such a search withoutneglecting important conformations.

Despite the approximate nature of the energy map in Fig.4, various structural features are readily apparent. The mostprominent feature is the lowest energy state atF1 < 1208and F2 < 2008: Despite the fact that this is the lowest-energy state, this state is an artifact of the approximatemethod used to obtain this map. This state indicates that ifa backbone torsion adopts an angle of approximately 1208the following angle is very likely to adopt a value of 2008.The map shows that essentially no angle is likely to producea subsequent angle of 1208 as seen in the absence of anyreasonable low energy minima atF1 < 1208: This energymap contains these artifacts because any state is dependenton the local minimum in which the other four externalangles reside. This state occurs because the other four anglesassociated with these values of the internal angles happen tobe in a very low energy minimum, while the other statescould have found higher energy minima.

The next lowest state is atF1 < 200 andF2 < 2008:Given the similarity of these two angles this state involvesthe repetition of approximately the same angle over andover again. Minimized structures with these angles suggestan incommensurate helix that adopts a repeated angle in thevicinity of 2008. We confirm that this helical structure is inthe vicinity of this state by carrying out a Monte Carlo (MC)search on the stereo-regular heptamer as explained below.The heptamer conformations obtained from this simulationare all clustered in the vicinity of the helical state seen inFig. 4.

There are two other relatively low-energy states thatoccur on the energy map in Fig. 4. The first occurs atF1 <200 andF2 < 3008 which is a state that represents a devia-tion from the helical conformation to a larger angle. Theother state occurs at approximatelyF1 < 300 andF2 <2008 which represents the chain jumping back into the heli-cal conformation. These low energy states in the materialindicate a polymer chain that adopts a helical conformationthat is occasionally disrupted by a kink. This kink consistsof the state atF1 < 200 andF2 < 300 followed by the stateat F1 < 300 andF2 < 2008:

By integrating the Boltzmann factor, in the vicinity of thethree local energy states described above, we can obtainprobability weighting for an RIS model. Because the energymap on which this model is based is approximate, we willuseF1 � 200 andF2 � 3008 as the energy states. The tran-sition state matrix (U) for this RIS model is:

==

===

01

03385.01

300

200300200

U

1

1

22

o

o

oo

φφ

φφ�1�

indicating that 3.385% of the angles at 2008 will jump out ofthe helix conformation. This RIS model can be used with thegeometry of the polymer chain determined from previousquantum calculations [12] to obtain the scaling of radius ofgyration and end-to-end distance as a function of molecular


6

7

8

9

10

11

12

0 0.01 0.02 0.03 0.04 0.05 0.06

1/n

<r2

>/<

s2 >

Fig. 5. The ratio of end–end distance to radius of gyration plotted againstinverse number of bonds from the RIS model.

1000

104

105

106

10 100 1000Number of Bonds

<r2

>

Fig. 6. Radius of Gyration plotted against the number of bonds from the RISmodel on a double logarithm scale.

weight by using the matrix generation scheme of the RISmodel [19,20]. Note that this RIS scheme assumes that thebond in the bicycloheptane group is non-rotatable andmakes no contribution to the chain flexibility. This assump-tion is justified by the rigid nature of the bicycloheptanegroup as stated above.

Fig. 5 is a plot of the ratio of the unperturbed meansquared end-to-end distancekr2

0l to the unperturbed radiusof gyrationks2

0l vs. the inverse of the degree of polymeriza-tion �1=n� from the approximate RIS model. The value ofthis ratio is approximately constant at a value of 11.5 andthen decreases as the chain approaches infinite length. In thelimit of infinite chain length this ratio approaches a value of12 for a rigid rod, 6 for a random coil or 2 for a collapsedcoil that can occur in globular proteins. Given the presenceof the kink we postulate that this polymer will behave like arandom coil in the infinite molecular weight limit. The linein Fig. 6 simply connects the last simulated point with thislimit. Although our calculations have not reached this limit-ing behavior, the line suggests that this postulate is notunreasonable. Based on the RIS model, the polymer ispresumed to be a helix with the occasional kink. A perfecthelix would have a ratio of slightly less than 12 at lowmolecular weights due to the finite cross-section of the poly-mer caused by the helical windings. However in the limit oftruly infinite molecular weight the cross-section of the helixwould be infinitesimal compared to the length, so this wouldapproach an infinitely thin rod which has a value of 12 forthis ratio. The stereo-regular poly(norbornene) pictured inFig. 5 is a helix with the occasional kink. At low molecularweight the polymer is predominantly helical. Even the aver-aging of the occasional kink does not reduce the ratio tomuch below 12 because the kink has a limited range ofangles such that a few kinks do not significantly deviatethe polymer chain from its initial persistence direction.However, as more than a few kinks are incorporated thechain begins to behave like a random coil. From calcula-tions described below using the RIS model (see Table 1), theaverage number of kinks that occur in 100 repeat units isslightly less than three. This is not sufficient to cause signif-icant deviation from an elongated conformation. However,at higher degrees of polymerization, the polymer chainbegins to slowly transition to random coil behavior. Thisis why the ratio in Fig. 5 begins to decrease at 1=n� 0:01�n� 100�: It will require a very large degree of polymeriza-tion for this ratio to approach the theoretical value of six.

Fig. 6 contains a log–log plot of the ratio of the radius ofgyration vs. the degree of polymerization calculated fromthe RIS model. The slope of the best-fit line through thepoints in this plot is 1:927^ 0:014 which indicates thatthe radius of gyration scales with the degree of polymeriza-tion to approximately the 1.9 power. The reported range ofthis slope is the 90% confidence interval for this valueassuming a Normal distribution of the error about thebest-fit line. This scaling exponent approaches two as thepolymer geometry approaches a rigid rod shape wherethe dimension of the polymer chain scales linearly withthe degree of polymerization. This limit occurs with rigidrods, but this case illustrates that it can occur with lessregulars geometric shapes, such as the kinked-helixproduced by these calculations. For a perfect helix thevalue of this exponent would be two, but this value is lessthan two due to the presence of the kinks in the chain back-bone. However, since the range of the degree of polymer-ization (n) is the same for both Figs. 5 and 6 the exponent isonly slightly less than two. This occurs because the ratio ofmean squared end-to-end distance to the mean squaredradius of gyration only varies be from approximately 11.5to 10.5, indicating that the polymer still adopts a relativelyelongated average conformation over this range ofn.

Using the universal viscosity law we may relate the scal-ing of the radius of gyration with molecular weight obtainedfrom Fig. 6 to a similar scaling of the intrinsic viscosity [h ]with molecular weight. This can provide us with someexperimental validation. However, there is some questionas the applicability of the universal viscosity law since it isbased on the Einstein viscosity model [25,26], whichassumes spherically shaped particles in the solution. Toaddress this we will assume that polymers, most of whichdo not adopt an instantaneous shape similar to a sphere cansweep out an effective sphere as they rotate in solution.From the Einstein model, one can express the intrinsic visc-osity [h ] as a function of the effective volume of the sphereVe. This relationship is given Eq. (2)

�h� � 2:5NVe

M�2�

where N is Avogadro’s number andM is the molecularweight. The universal viscosity law is typically derived byassuming that the sphere volume can be expressed using aneffective radiusRe, that is proportional to the root-mean-squared end-to-end distance of the polymer chain. Herewe assume such a relationship in Eq. (3)

Re � agkr20l1=2 �3�

wherea is the solvent expansion parameter that describesthe expansion of the chain above itsu condition value due tointeraction with solvent. The parameterg is the ratio of theradius of the effective sphere swept out by the rotating poly-mer to the actual average end-to-end distance that the poly-mer adopts instantaneously. We assume here thatg is not afunction of molecular weight and only depends on the


Table 1Probability of one of more kinks occurring in a poly(norbornene) oligomerbased on RIS model

Degree of polymerization Probability of one or more kinks

7 0.180211 0.282415 0.371919 0.4502

conformational behavior of the chain. One might expect thatg for a random coil is slightly larger than that of a rigid rodbecause the rigid rod most likely does not sweep out all ofthe three-dimensional (3D) space available to it. We alsoassume thata is independent of molecular weight. This isreasonable because Flory Krigbaum theory can be used toestimate thata scales asM0.1 at high molecular weight [27].Assuming that the volume in Eq. (2) is given by the volumeof a sphere with the effective radius given in Eq. (3), weobtain Eq. (4)

�h� � 2:5g3 4pN3

kr20l

M

!3=2

M1=2a3 �4�

By combining some of the constants we obtain thecommonly used form of the universal viscosity law [19]given in Eq. (5)

�h� � Fkr2

0lM

!3=2

M1=2a3 �5�

whereF is an empirical constant. In this case the effect ofthe non-spherical shape of the chain (contained ing ) doesnot effect the scaling of the intrinsic viscosity with molecu-lar weight, it only affects the pre-factorF . At the u condi-tion wherea � 1; or in the high molecular weight limitwhere the dependence ofa on M is very weak, we havethe following scaling law

�h� / kr20l

M

!3=2

M1=2 �6�

Empirical observation has validated this scaling for mole-cules of arbitrary shape [28]. This probably occurs becausenon-spherical molecules sweep out an effective sphere asassumed above. We typically assume that random coils arespherical in shape, but this is only true for an external refer-ence frame. Even random coils appear to be non-sphericalwhen observed in their internal reference frame. The instan-taneous shape of a random coil polymer is more like anoblate or prolate spheroid [29,30]. Given this, we will useEq. (6) across the entire range of polymer shapes fromrandom coils to rigid rods. For a random coil the meansquared radius of gyration is proportional to the molecularweight and for a rigid rod it is proportional to the square ofthe molecular weight. From this we obtain the limits inscaling of intrinsic viscosity with molecular weight at theu condition.

�h� / M1=2�random coil�: �7�

�h� / M2

�elongated structure where size increases linearly withn��8�

This exponent, typically referred to as “a” in the Mark–Houwink–Sakurada empirical relationship between intrin-sic viscosity and molecular weight, varies between 1/2 for a

random coil in au solvent and approximately 0.8 in a goodsolvent. Eq. (8) is dependent on our previous justification forusing Eq. (6) with rigid elongated structures. Additionalvalidation of the scaling obtained in Eq. (8) is seen in theresults of Prager, who found that intrinsic viscosity of rigiddumbbell shaped molecules scaled in the same quadraticfashion [31]. After substituting the slope of 1.927 fromFig. 6 into Eq. (6) we obtain an exponent of 1:9^ 0:2 forthe scaling of intrinsic viscosity with molecular weight. Thiscan be compared to the experimental results below.

2.2. Monte Carlo simulations

Two types of MC simulations were carried out on 2,3-erythro di-isotactic polynorbornene to validate the confor-mational behavior derived from the RIS model above. Thisvalidation is important because the RIS model is based onan approximate energy map that does not completelysample the conformational space of the heptamer of poly-norbornene. First, a conformational search was conductedon a di-isotactic polynorbornene heptamer. Using the Boltz-mann Jump algorithm developed by Venkatachalam [32].The Boltzmann Jump algorithm is particularly useful forsampling local conformational minima. It carries out anumber of random torsion angle perturbations using theMC Metropolis algorithm [33] until the molecule is suffi-ciently far from its previous point in conformation space topotentially be in another local energy minimum. At thispoint an energy minimization is carried out to produce acollection of local energy minima. We employed this algo-rithm to produce 3000 different energy minimized structuresusing 50 MC steps each before energy minimization. Theconformers were energy minimized to a root mean square(rms) force of 0.1 kcal/mol/A˚ . We employed a temperatureof 5000 K in the MC search to ensure a broad sampling ofphase space. The high temperature is employed to assist thestructure in moving to completely different areas of phasespace, but since the final structures undergo energy mini-mization, they in no way reflect the conformational distri-bution at this artificially elevated temperature. Despite thehigh temperature employed, all of the conformationsproduced resided in the vicinity of the helical conformationas indicated by the darkened region in Fig. 4. What appearsto be a large dot in Fig. 4 is actually a tight cluster of pointsgenerated from the Boltzmann Jump simulation. The Boltz-mann Jump calculations verify the occurrence of the helicalstate in Fig. 4. This is essentially the same helical structurethat may occur in substituted polyacetylenes as we discussbelow.

Second, a Metropolis-MC [33] simulation incorporatingthe pivot algorithm [34] followed by energy minimizationwas used to simulate isolated chains of polynorbornene. Nocorrection was applied for the potential conformational biascaused by the minimization step, but without the minimiza-tion step the acceptance ratio was prohibitively low. This isto be expected at the low temperature (300 K) used in these


simulations for sterically hindered polymers such as poly-(norbornene) [35]. With the minimization step, the accep-tance ratio was approximately 2%. Starting from an initialextended conformation, a random torsional perturbation,using a random uniform distribution, is applied to one rota-table bond, this is followed by 4000 steps of energy mini-mization using the BFGS [36] algorithm. The assumptionthat isolated chains are sufficient to describe the structure ofthis polymer is justified by the fact that the bulky backboneof the polymer makes the intramolecular interactions muchstronger than the intermolecular ones. It is assumed that thepolymer conformations are determined primarily by theintramolecular torsional barriers and steric hindrance,while the surrounding polymer produces a minor structuralperturbation. Similar simulation approaches have beenshown to faithfully reproduce experimental data, albeit forsimpler polymers [35,37,38]. This single-chain assumptionis justified by the extremely high torsional barriers for thispolymer. Due to the computational requirements of theenergy minimization in this simulation, small oligomerswere simulated corresponding to degrees of polymerizationof 11, 15 and 19. These three oligomers were simulated for1000 total steps, and resulted in 15–20 accepted structures.A direct comparison can also be made between the results ofthe MC and RIS simulations and this is seen in Fig. 7. Goodagreement is observed between the MC and RIS resultsfurther validating the approximate RIS model. Fig. 7 indi-cates that both the RIS and MC models suggest that theradius of gyration scales with the molecular weight toapproximately the 1.9 power. The slope obtained for theMC simulation is 1:95^ 0:219 where this range is the90% confidence interval on the slope in Fig. 7. After substi-tuting this value of the slope into equation two, the MCmodels predict that intrinsic viscosity varies with the mole-

cular weight to the 1:9^ 0:3 power, which compares well tothe value obtained from the RIS model.

Since this is quite close to the elongated rod limit of two,the overall dimensions of this polymer increase almost line-arly with degree of polymerization. A perfect helix wouldapproach this limiting exponent of two as well, however, theconformations sampled in the MC model were not perfecthelices and appeared to have helical character with kink-likeconformations suggested by the RIS model. This occurreddespite the failure of the Boltzmann Jump calculations tosample the kink architecture. The better sampling of thisMC procedure is due in part to the longer chains in thesimulation. The probabilities from the RIS statistical weightmatrix in Eq. (1) can be used to estimate the probability ofoccurrence of one more kinks in a given oligomer. This canbe calculated from the conditional probabilities of a helixangle�f1 � 2008� transitioning to a kink angle�f2 � 3008�:This probability (p23) is 0.032745 as seen in equation one,and the corresponding probability that a helix angle isfollowed by another helix angles is 0.96755 and is denotedp22. We may estimate thea priori probabilities that a givenangle is at the helix or kink value by recalling that all kinkangles must follow a helix angle. This allows us to equatethe a priori probability of the occurrence of a kink angle(P300) with the fractions of diads (or adjacent pairs of mono-meric units) that contain a helix angle followed by a kinkangle. This equality is expressed in Eq. (9)

P300� 1 2 P200� P200p23 �9�where P200 and P300 are the a priori probabilities of theoccurrence off � 200 andf � 3008 in the backbondtorsion angle distribution. These are also equal to the frac-tion of helix and kink angles, respectively. Solving Eq. (5)we obtainP200� 0:9683 andP300� 0:0317 for the a prioriprobabilities. By employing the first order Markov modelinherent in the RIS model we may calculate the probabilityof one of more kinks (Pkinks) occurring in an oligomer ofdegree of polymerizationn that containsn 2 1 angles. Thiscan be done by subtracting the probability of no kinks occur-ring from 1 as in Eq. (10)

Pkinks � 1 2 Pno kinks� 1 2 P200pn2222 : �10�

These values are listed in Table 1 and they indicated that thelikelihood of even one kink occurring in the heptamer isrelatively low. This explained the fact that the BoltzmannJump simulation did not encounter any kinks while thelarger oligomers did.

2.3. Viscometry experiments

Validation that the intractable polynorbornene producedvia the zirconocene or Pd catalyst can, in principle, be madeby comparing the results of the RIS and MC calculationswith viscometry measurements made on these polymers.Recall that we are presuming that the zirconocene and Pdcatalysts produce essentially the same stereochemical


10

100

1000

104

105

10 100 1000Number of Bonds

<s2 >

Fig. 7. Comparison of Monte Carlo results for erythro di-isotactic polymer(diamonds) and RIS calculations for the same stereochemical isomer (filledcircles).

structure because of their similar WAXD results and the factthat they are both intractable. However, because these poly-mers are intractable, they will not dissolve in most solvents.In order to make a qualitative comparison we carried outintrinsic viscosity measurements on the Pd catalyzed poly-mer. However, direct comparison is complicated by the factthat this particular polymer is not soluble in any knownsolvent at reasonable temperatures. Additional difficultiesarise from the rapid polymerization kinetics obtained withthe Pd catalyst developed by BF Goodrich. The polymeriza-tion proceeds rapidly to a relatively high molecular weight,so accurate systematic variation of the average molecularweight is somewhat difficult. To overcome the solubilityissue, this Pd catalyst was used to polymerize the butylderivative of norbornene where ann-butyl group wasattached randomly to either carbons 5 or 6 of the norbornenemonomer. Synthesis using this monomer produced a poly(norbornene) derivative soluble in 1,2,4-trichlorobenzene(TCB). To obtain a qualitative comparison for this polymer,viscometry on the alkane derivative was performed in 1,2,4Tricholorobenzene (TCB) at the 708C. This has been deter-mined to be theu condition for other polynorbornenesamples using other polymerization catalysts in TCB fromlight scattering [18]. Its sparing solubility implies that thesolvent swelling effect is small so this may be near theucondition, however, this was not confirmed as theu condi-tion for this polymer made using the Pd catalyst. We shallassume that this polymer is near theu condition because ofits high barriers to backbone torsional rotation [12]. Thesehigh barriers prevent the polymer from undergoing confor-mational changes due to changes in solvation resulting fromchanges in solvent or temperature. At 708C the polymer islocked in a particular conformation that is insensitive to thesolvent and temperature (as long as the temperature is wellbelowTg where torsional rotation does not occurs). The fact

that the chain conformation is relatively fixed implies thatthe polymer is always near itsu condition. This fact isconsistent with the aforementioned discovery that twoother isomers of poly(norbornene) with drastically different“a” exponent values both have the sameu condition (708Cin TCB) [18]. These other isomers are presumed to havedifferent stereochemical configurations than the Pd cata-lyzed samples and are soluble in TCB above room tempera-ture. TCB solutions were heated to 708C to dissolve thesepolymers and light scattering coincidentally showed that thesecond viral coefficient was negligible indicating that thiswas theu condition. The fact that this initial temperatureattempted just happened to be theu condition for two differ-ent stereoisomers implies that these polymers may be neartheu condition for a variety of temperatures.

It was also assumed that the short hydrocarbon sidechains do not affect the conformation significantly becausethey have a minimal effect on the severe steric hindrancedue to the bicycloheptane backbone. The primary effect ofalkane chains appears to be to make the primary torsionalminima more predominant [10]. This would suggest that then-butyl derivative would have slightly higher scaling expo-nent of intrinsic viscosity with molecular weight than thepolymer without then-butyl side group. Fig. 8 illustratesthat the intrinsic viscosity ofn-butyl derivative scales withthe molecular weight to approximately the power two.Using the covariance matrix from the linear regression onthe plot in Fig. 8, we obtain values of 2:0^ 0:7 and 2:0^

1:5 for the 80 and 90% confidence intervals on the slope,respectively. These samples were supplied by BF Goodrich,and were closely spaced due to the aforementioned difficultyin controlling the kinetics of this polymerization.

Although the comparison to experimental data is onlyqualitative, it suggests that the Pd catalyzed polymerizationproduces an erythro di-isotactic polymer because they bothhave an intrinsic viscosity scaling with molecular weightnear the extreme value of two. Results using the MC pivotalgorithm described above indicate that this scaling expo-nent is much lower for other stereochemical isomers ofpoly(norbornene) [13,14]. This discounts the possibilitythat all stereochemical isomers of poly(norbornene) havean “a” value near two, and qualitatively validates thehypothesis that this Pd catalyzed polymer adopts the erythrodi-isotactic form.

2.4. Bulk simulations and X-ray diffraction

In molecular modeling, a polymeric glass is conceptua-lized as an ensemble of microscopic structures in detailedmechanical equilibrium, each microscopic structure beingrepresented by a model cube with periodic boundaries, filledwith segments from a single “parent chain” [21]. Each of themicroscopic structures is in a state of mechanical equili-brium, which is a local minimum in potential energy. Mole-cular mechanics and molecular dynamics methods can thenbe used to search for these local minima. The use of local


0.3

0.4

0.5

0.6

0.7

6 104 7 104

Mw

-η'

Fig. 8. Intrinsic viscosity (in dl/g) plotted against the molecular weight (ing/mol) for experimental viscometry results on a double logarithm scale.

minima assumes that glasses are not in thermodynamicequilibrium. Since local minima are sought, the minimamay be expected to depend strongly on the starting confor-mation. Therefore, it becomes necessary to generate thestarting conformation using the aforementioned RISmodel. This assures that the initial conformation is consis-tent with the torsion angle distribution from the RIS model.This approach has been successfully used in the past togenerate initial starting conformations for polymer glasses[21–23].

At present, a priori knowledge about whether the struc-ture has been relaxed enough to be representative of the realpolymer is not available, therefore, comparison with experi-ment is essential before the model can be used for predictivepurposes. Comparing the predicted X-ray diffraction patternto the experimental X-ray diffraction pattern allows valida-tion of the model results. This comparison not only validatesthe model but also provides information about the modelstructure and any order or lack thereof that exists in it.Molecular mechanics methods have been successfullyused to investigate the structure of a number of polymerglasses including atactic polypropylene [21], polyvi-nylchloride (PVC) [22] and polycarbonate (PC) [23]. Onthe other hand, it has been demonstrated that alternatelyusing molecular mechanics and molecular dynamics duringminimization achieves a more realistic energy minima forrigid polymers like polysulfone [39]. Therefore this is theapproach used in modeling di-isotactic polynorbornene. Theuse of molecular dynamics only helps the minimizationprocedure to find a slightly improved local energy mini-mum, it does not accurately sample phase space in theglassy phase. It is presumed that the initial conformation

is reasonable because the RIS model was used to bias theinitial bulk conformation. Therefore the local energy mini-mum found is representative of the intermediate order in thisglass.

The RIS statistical weight matrices were incorporatedinto the Amorphous Builder module in the Cerius2w soft-ware to generate isolated chains as well as chains in periodicboundary conditions. An experimentally determined densityof 1.05 g/cm3 was used for the bulk models [40]. All theresults presented below represent an average taken overthree different starting conformations. Four chains of 30monomer units each constituted a single model. No statis-tical difference was observed within the three startingconformations at the end of the simulations. These threemodels were averaged to produce the dashed line in Fig.9. Further, one model with two chains of 200 monomer unitswas also simulated to study the effects of model size. Theisolated chains provide information about intramolecularcontributions to structure, and combined with the bulkmodels can help isolate intermolecular contributions to thestructure.

All the models were subjected to molecular mechanicsminimization for 5000 steps using a conjugate gradientminimization algorithm [41], 10 ps of constant NVTdynamics at 300 K, and then subjected to minimization toa final rms force of 0.1 kcal/mol/A˚ . WAXD diffractionpatterns simulated from the final structure, using Cerius2w,were then compared to experimental X-ray diffractionpatterns (Fig. 9). The experimental data has been takenfrom the work of Noll and Kaminsky using zirconocenecatalyzed polymer [42]. A good comparison to experimentwas observed after eliminating the small-angle componentof the scattering calculated for the models using a sphericalsmear with a model size correction factor of 1.700 [43].Since intensity is in arbitrary units, the experimental curvehas been off-set for clarity. The important points of compar-ison are the relative heights of the peaks and the angle atwhich the peaks occur. Both sizes of poly(norbornene)models gave essentially identical results indicating thatthere is little sensitivity to model size.

As can be seen from the X-ray diffraction pattern forpolynorbornene (Fig. 9) the first peak in the amorphoushalo splits into two peaks. This corresponds to the formationof intermediate range order, which is characterized by asplitting, that occurs in the amorphous halo, typically at awave vector magnitude ofQ� 1 �A21

: This has beenpreviously observed in polymethacrylates [43,44] andPVC [22].In PVC the origin of intermediate range order isintermolecular, and is due to the polar interactions of thechloromethine groups [22]. Conversely, the origin of thisorder is more likely to be the intramolecular interactionsin poly(norbornene) given the relative weakness of the inter-molecular interactions.

Typically the first, or lower angle, peak in the WAXDpattern is predominantly intermolecular in nature becausethis corresponds to a larger interatomic distance (at 2u �


10 15 20 25 302θ (degrees)

Inte

nsit

y

Fig. 9. Graph comparing the simulated diffraction pattern to the experi-mental Wide Angle X-ray Diffraction Scattering (WAXD) pattern for poly-norbornene. The bold line represents experimental WAXD data [42]. Thedashed line is the four chain model and the solid line is the two chain largermodel. The dotted line represents simulation data for polynorbornene–polyethylene 50: 50 random copolymer. Cuka(l � 1:54178 �A� is usedas a X-ray source for experiment and simulation.

11; d � 8 �A�: Likewise, the second, or higher angle peak, ispredominantly intramolecular in origin and is associatedwith a smaller interatomic distance (at 2Q � 188; d �4:8 �A�: The radial pair distribution functionsg�r� containa large number of peaks and do not clearly indicate thatany particular atom pair is responsible for these reflections.The radial distribution functions times their differentialvolume element are seen in Figs. 10 and 11 for the bridgingcarbon (#7) in the bulk polymer and the isolated chains,respectively. It is difficult to see any distinct pattern inthese or other general radial distribution functions due tothe broad distributions of atomic spacings caused by therotation of the bicycloheptane group in the polymer.However the peak at approximately 4.8 A˚ in Fig. 11 indi-cates that at least some of the low angle peak is do to spacing

within the polymer chain. Jacobson has simulatedcispoly(t-butyl acetylene) in the same helical conformation predictedby our RIS model, and that produced the same two WAXDpeaks seen in Fig. 9 [45]. This polymer is a useful analogueof poly(norbornene) because it has very similar conforma-tional energetics. In its elongated form it has the identicalconformation as thecis form of poly(norbornene) seen inFig. 2. Also, every other backbone bond in both these poly-mers is non-rotatable due to the double bond in poly(acety-lene) and the bicycloheptane group in poly(norbornene).The WAXD pattern for the isolated poly(norbornene) chainsis contained in Fig. 12 and provides insight into whatportion of the WAXD peak is intermolecular vs. intramole-cular in nature. The peak at 2u � 188 in the isolated chainWAXD pattern clearly indicates that part of this high anglepeak is intramolecular in nature. The lower angle peak in thebulk WAXD pattern at 2u � 11 has been replaced by abroader peak at an even lower angle in the isolated chainWAXD pattern. This new lower angle peak is due to therelative increase in larger distance spacing that naturallyoccurs in isolated chains. In the bulk, these large distancesalong the polymer chains are eclipsed by the more homo-geneously distributed shorter distances. The disappearanceof the bulk peak at 2u � 11 indicates that the lower anglepeak is predominantly intermolecular in nature. This sameconclusion was reached by Jacobson for his simulations ofcompletely helical poly(t-butyl acetylene) by carrying outWAXD at different densities [45]. Because Jacobson’ssystem was a symmetrically packed unit cell of alignedhelices, different densities could be easily simulated withoutextensive regeneration of the initial chain conformation andsubsequent energetic relaxation. Changes in bulk density,while preserving the helix structure resulted in significantchanges to the low angle peak and negligible changes in thehigh angle peak. This indicated that the low angle peak ispredominantly intermolecular, while the high angle peak ispredominantly intramolecular. Since both Jacobson’spoly(t-butyl acetylene) structures and our poly(norbornene)structures are dominated by the same helical conformation


0

200

400

600

800

1000

0 2 4 6 8 10

R (Angstroms)

g(r)

4π

r2

Fig. 10. Radial distribution function times the spherical volume element forbridging carbons in bulk polynorbornene models.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 2 4 6 8 10

R (Angstroms)

g(r)

4π

2

Fig. 11. Radial distribution function times the spherical volume element forbridging carbons in isolated polynorbornene chains.

0 5 10 15 20 25 30 35 402θ (degrees)

Inte

nsity

Fig. 12. WAXD pattern for isolated chains and bulk models. Solid linerepresents data for bulk models. Dashed line represents data for isolatedchains.

S.

Ah

me

de

ta

l./

Co

mp

uta

tion

ala

nd

Th

eo

retica

lPo

lyme

rS

cien

ce1

0(2

00

0)

22

1–

23

3231

Fig. 13. Periodic and parent-chain images of the bulk simulation of poly(norbornene) with two chains of 200 repeat units. Note the helix and kink regions in the parent chain that appeared in all of the simulatedchains.

this result is consistent with the above interpretation of theWAXD pattern.

In addition to the various sizes of poly(norbornene), abulk simulation of a random copolymer of ethylene andnorbornene was also simulated. The copolymer contained50% ethylene monomers and was simulated using the sameapproach used for the poly(norbornene) systems. The initialguess used the aforementioned RIS model in the initialconformation generation, but the ethylene unit torsionangles were chosen randomly. The WAXD pattern resultingfrom an average over three simulations of four chains eachwith 30 monomer units for this copolymer is seen in Fig. 9.The introduction of the ethylene comonomer reduces theheight of the low angle peak relative to high angle peak.The relative reduction in the low angle peak observed in Fig.12 is approximately the same as that observed by Kaminskyin experimental WAXD patterns on such random copoly-mers [42].This adds additional verification that the simu-lated poly(norbornene) structure is reasonable.

Although many polymers form helical conformations, thehelix-kink conformation proposed for erythro di-isotacticpoly(norbornene) is relatively new. Some additionalevidence of the accuracy of this suggested conformation isalso provided by Jacobson’s results on poly(t-butyl acety-lene). While our helix-kink model accurately reproducedthe experimental WAXD results, the aligned pure helicalconformation suggested by Jacobson for poly(t-butyl acet-ylene) over-predicted the low angle peak [45]. The experi-mental WAXD pattern of poly(t-butyl acetylene) is almostidentical to Kaminsky’s measurement of the zirconocenecatalyzed poly(nobornene). The ratio of the areas of thelow to high angle peaks is of order 1.25 for both Kaminsky’spoly(norbornene) data in Fig. 9 and for the experimentaldata for poly(t-butyl acetylene). While Jacobson’s modelreproduces the peak positions well, his ratio of the relativeareas of the peaks is well above two. We attribute this highprediction to the fact that the pure helix model contains toomuch intermolecular order. The pure helix model consists ofaligned helices packed closely together in a parallel fashion.Whereas the helix kink model has noticeably less intermo-lecular order because the kinks reduce the ability of thechain to pack in a regular fashion. Relative to the intermo-lecular order, the intramolecular order is not significantlyreduced by the kink because most of the chain has a generalhelical conformation. Recall that the RIS model suggeststhat almost 97% of the backbone torsion angles were inthe helical conformation. This helix-kink conformationcan be seen in the simulated bulk poly(norbornene) systemsby observing individual chains. The large, two chain, simu-lation is pictured in Fig. 13. The helical and kink conforma-tions are easily seen, and the structure is qualitativelyconsistent with the RIS model in that much of the chain ishelical in conformation. The number of kinks in all of thebulk chains occurred at a greater frequency than thatsuggested in the Table 1. This occurs because the effectivepersistence length of the polymer is much larger than the

edge of the periodic cell used in the bulk simulation. There-fore, the bulk chains may be somewhat perturbed by the factthat a kink may be generated when the polymer chain runsinto another periodic image of itself in this small unit cell.The persistence length cannot be quantitatively use toconfirm this because of the scaling of the chain dimensionswith degree of polymerization observed in this polymercauses the persistence length to diverge.

3. Conclusions

A new RIS model was developed for erythro di-isotacticpolynorbornene that included long-range steric interactionsthat are needed to properly model its conformation. Theseinteractions were included by extracting RIS states from anapproximate energy state map for the heptamer of this poly-mer. The RIS model predicted a novel helix-kink conforma-tion for this polymer in which helices were occasionallydisrupted by a backbone kink. The same RIS modelpredicted a value of approximately 1.9 for the exponent“a” from the Mark–Houwink–Sakurada equation at theucondition, and thatkr2

0l=ks20l � varied from 10.5 to 11.5 over

a wide range of molecular weight. These RIS results wereconsistent with independent MC simulations. These simula-tions were compared with experimental results for twodifferent poly(norbornene) polymers synthesized by Pdand zirconocene homogeneous polymerization catalysts,respectively. These two samples were presumed to havethe same structure because they are both insoluble andhave identical WAXD patterns. Other homogeneous cata-lysts produce poly(norbornenes) with very different solubi-lities and WAXD patterns. Viscometry experimentsindicated a scaling parameter “a” very near to two for thePd-catalyzed polymer suggesting that this polymer may beerythro di-isotactic. Although there was a fair amount oferror in this value. The RIS model was utilized to createinitial conformations of bulk polymers in spatially periodiccells that were energetically relaxed to produce bulk modelsof poly(norbornene). Accurate prediction of the WAXDpattern from these models for both poly(norbornene) andpoly(ethylene-co-norbornene) synthesized from the zirco-nocene catalyst provides further evidence that these catalystsystems may produce erythro di-isotactic poly(norbornene)structures. The WAXD peak positions for these poly(nor-bornene) samples were very similar to those of a stereo-regular poly(t-butyl acetylene) sample. Comparison ofthese simulations and poly(t-butyl acetylene) simulationscarried out by Jacobson indicate similar conformationalbehavior for these two polymers. Over-prediction of theintermolecular order in poly(t-butyl acetylene) may occurbecause this polymer also has a helix-kink architecture asopposed to a perfect helix conformation used in the model.Similarities between these polymers may occur becausethey both have the same basic structure. Every other bond


in their backbones is essentially non-rotatable and they bothcontain bulky side groups.

Acknowledgements

We gratefully acknowledge the financial support of theBF Goodrich Corporation as well as the Georgia TechMolecular Design Institute funded by the Office of NavalResearch. The substantial contributions of samples andexperimental analysis from Robert Schick, Brian Goodalland George Benedikt of the BF Goodrich are gratefullyacknowledged. Insightful discussions with A. Abhiramanof Georgia Tech were also most valuable.

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