Molecular Configurations of Elastomers Via Solid State NMR - ACS

14 Molecular Configurations of Elastomers Via Solid State NMR Y . - Y . LIN, JAMES E . MARK, and JEROME L . A C K E R M A N

Downloaded by AUBURN UNIV on November 17, 2016 | http://pubs.acs.org Publication Date: July 19, 1982 | doi: 10.1021/bk-1982-0193.ch014

University of Cincinnati, Department of Chemistry, Cincinnati, O H 45221

A theoretical investigation of the use of NMR lineshape second moments in determining elastomer chain configura tions has been undertaken, Monte Carlo chains have been generated by computer using a modified rotational isomeric state (RIS) theory in which parameters have been included which simulate bulk uniaxial deformation. The behavior of the model for a hypothetical poly(methylene) system and for a real poly(p-fluorostyrene) system has been examined. Ex cluded volume effects are described. Initial experimental approaches are discussed. The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (1), by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation: namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chainjunction entanglements. Ullman has summarized three levels of network affine deformation: I) affine deformation of junction coordinates; 2) of mean (i.e., rms) junction coordinates, with concomitant fluctuations about mean positions; 3) of statistical chain segments; as well as: 4) nοn-affine behavior at the molecular level. Each of these possibilities engenders different behaviors of the network in response to bulk deformation. Yet it has proved notoriously difficult to obtain a direct experimental categorization of the affineness of macromolecular deformation. This is primarily due to the lack of experimental modalities which can probe molecular configurations, and yet which are sensitive to relative configurations of topologically close (and, potentially, spacially remote) segments, while discriminating against those of topologically remote, spacially close segments. Small angle neutron scattering (SANS) of "marked" (isotopically labeled) chains in unmarked networks has proven to be of potential value in 0097-6156/82/0193-0279$06.00/0 © 1982 American Chemical Society Mark and Lal; Elastomers and Rubber Elasticity ACS Symposium Series; American Chemical Society: Washington, DC, 1982.


280

ELASTOMERS

A N D RUBBER

ELASTICITY

determining which affine model is most appropriate for a given system. The theoretical scattering laws for the alternative models have been derived (6). Initial experimental results (7,8) have suggested that the true description of the deformation of a network may involve a combination of the four possibilities, and that the description can depend on the thermal and mechanical history of the sample. In addition, models incorporating entanglements have been evaluated using neutron scattering (9). Because of the grandiose scale of the apparatus involved, SANS facilities are few in number worldwide; access to them is limited and expensive. We have attempted to devise an experiment which employs solid state nuclear magnetic resonance to examine some aspects of affine deformation. We have completed a series of computer experiments which indicate that it should be possible to test the validity of segmental affine behavior with our technique. NMR Second Moments and Segment Configurations In molecularly rigid systems, the direct (through-space) dipole-dipole interaction between nuclear spins I = h is normally the dominant source of broadening of the NMR lineshape. For a pair of similar nuclear spins i and j in a magnetic field H the dipolar splitting in their spectrum is given by q

p

2.

ωρ

2

= γ , *

( c o s 9

li

)

- J - L

P (x) = fe(3x - I) 2

2

where Ύ and I are the magnetogyric ratio and spin quantum number respectively. As illustrated in Figure I, incorporates geometric information as Θ.., the angle between the internuclear vector and the magnetic field, and as rj., the length of the internuclear vector. When a large nurWber of spins interact, the numerous lines of the splitting pattern overlap and merge into a continuous lineshape whose functional form cannot be obtained from theory. Yet this shape contains useful geometrical information. One means of representing this dipolar lineshape g(u>) is by a moment expansion Μ

η

=

/

(

ω

-

ω

ο

)

η

9

(

ω

)

ά

ω

about the nominal resonance frequency ω . Because of the properties of the dipolar interaction, the lineshape is symmetric about ω , and therefore only even moments are required. The great utility of moments is that, although the lineshape cannot be calculated analytically for an arbitrary configuration of nuclear spins, any moment may in principle be calculated to arbitrary precision from first principles (10). In practice, only the lowest moments are calculable because of computer time and precision constraints. In particular, the second moment M is the lowest moment containing spacial information ?

Mark and Lal; Elastomers and Rubber Elasticity ACS Symposium Series; American Chemical Society: Washington, DC, 1982.

14.

281

Molecular Configurations of Elastomers

L I N ET A L .

and is given by Μ

2

= θΣ!

[Ρ (οο5θ )Γ " ] 3

2

η

J


2

π

i >j The sum is carried out over all pairs of nuclear spins ij, counting each pair exactly once. C is a collection of fundamental constants and also contains the total number of spins. If we were to have an isolated polymer chain with a single nuclear spin attached to each segment (the marked chain) crosslinked into an unmarked network, the second moment of the N M R line of that spin species would carry information relating to the separation of chain segments, and to their relative orientation with respect to the field direction. If the network were to be subjected to a bulk deformation, these geometrical parameters would be altered, and hence we would expect a corresponding change in the value of the experimentally measured Μ · Two limiting geometrical cases illustrate this nipely. For a uniaxial elongation λ, transverse separations decrease by λ" at constant mass density. If the magnetic field is col linear with the elongation axis the contribution to M 2 of a spin pair with internuclear vector parallel to the axis changes as λ " . The contribution of a spin pair oriented transversely would follow λ . For other orientations, there will be changes in both separation and angular factor. Clearly, these results are anticipated for spins on segments embedded in an elastic continuum, i.e., for the segmental affine case. Other microscopic segmental changes would be expected to follow different M ^ versus elongation behaviors. We note here that the use of N M R in determining orientational information is not new; second and fourth moments of the resonance line in bulk materials have been used to obtain second and higher moments (more precisely, Legendre polynomial averages Ρ (cosO..)) of segmental orienta tional distribution functions (II, 12, 13,"14). 'We are interested more particularly in changes in relative segment orientations (configurations). These moment studies have been performed on polymer systems such as polyethylene (or on penetrants in polymer systems) in which the interacting spins (protons or fluorines) reside on the same or on adjacent atoms. This allows essentially no freedom of variation in the internuclear vectors upon deformation of the network. The primary informational content therefore relates to independent segmental orientation distributions. By placing single spins on alternate segments, there should be much greater sensitivity to changes in the chain extension upon bulk deformation. This effect was confirmed in our early computer results. Placing spins on every third segment was also attempted. Because of the strong inverse sixth power dependence on r, the second moment values, although strongly dependent on chain dimensions, were of a magnitude too small to be reliably measured experimentally against a background of other broad ening effects (i.e., linewidths of a few hundred Hz). J

2

2

6

3

Rotational Isomeric State Model with Deformation In order to develop a molecular model of elongation from which M~*s can be calculated, we have chosen the rotational isomeric state (Rfe)


282

ELASTOMERS A N D RUBBER

ELASTICITY

model (15). Consider the Monte Carlo generation of a polymethylene chain via the RIS scheme. Each new methylene residue may be appended to the growing chain in one of the three rotational states: trans(t), gauche plus (g ) or gauche minus (g"). The statistical weight (SW) of each conformer is given by the corresponding Boltzmann factor. This procedure generates an essentially isotropic chain. If, however, we modify the statistical weights by an angular factor SW •> (I + aP (cose))SW


2

with Θ being the angle between an elongation axis and the potential new bond, we can cause the chain to orient (See Figure 2). The Ρ2 function has been chosen as it is the lowest order uniaxial term in the expansion in spherical harmonics of any generalized angular distribution. We note parenthetically that the Ρ y function possesses the precise differential behavior corresponding to small (λ I) uniaxial de formations at constant bulk density: namely that along ζ (PJicosQ) ~ I) a small deformation άλ requires that transverse to ζ (P (s0)~-!fe) the deformation must be -ifedA. The parameter a alters the intensity of the bias, and in fact may be thought of as an order parameter. We can further bias the chain by altering the trans/gauche energy difference. This allows us to manipulate the trans/gauche population ratio, and therefore alters the isotropic elongation of the chain. For each chain generated, we can calculate M^, and then take averages over many chains. A p r o é l e m arises, in that the strong r dependence of M requires that close overlap of spins be prevented. Thus, even though excluded volume interactions have no effect on chain dimensions in the bulk amorphous phase, it is important in the present application to build in an excluded volume effect (simulated with appropriate hard sphere potentials), so that occasional close encounters of the RIS phantom segments do not lead to unrealistically large values of M z

2

2

2 #

Computer Method As a simplified initial approach to the problem, we chose a polymethylene chain with a single spin residing at every other carbon center, to allow sufficient flexibility between spins. The calculational procedure is based on that of Flory and Mark (16), with conformer parameters taken from Abe, et a l . (17) and Yoon, et a l . (18, 19) The elongation and field directions were col linear. For each methylene segment to be appended to the chain, the bond directions corresponding to the three possible rotamers are calculated from the transformation matrices in the usual way. The angular weighting factors are then determined and are multiplied with the statistical weights. These weights are normalized to probabilities, and a random number is used to select one rotamer. The new backbone atom must now be checked for overlap with other atoms on the chain. The distance from every other atom is then calculated. If there is a spin on this atom, terms for the second moment summation are accumulated in this program loop as well.


14.

LIN ET AL.

283



H»

θ„

Figure 1.

Spherical coordinates used to calculate the dipolar interaction between two nuclear spins in a strong magneticfieldH . 0

These variables, connecting all possible pairs of spins in a multispin system, are used for the evaluation of the second moment M . 2

H

0

s Deformation A x i s

c,

Ν- I

Figure 2.

Angle variable used to bias a polymer chain for orientation.

The Boltzmann statistical weight for each rotamer is multiplied by (1 + aP (cos Θ)), with a being an adjustable parameter, before a particular rotamer is selected. Values of a, between zero and unity, will give rise to an order parameter for segment orientations. Average of P (cos Θ) of 0.0 to 0.5. 2

2



284

ELASTOMERS A N D R U B B E R ELASTICITY

It is at this point that the bulk of the CPU time is utilized, since the number of distance calculations increases with the square of the number of chain segments. To reduce this time, an abbreviated check is made to determine if a given atomic cartesian coordinate is within a specified distance (e.g., 12 &) of the newly appended atom. Thus, the vast majority of existing segments are disposed of with one, or at most three, subtraction calculations rather than full calculations of the exact distance. Terms in the second moment from atoms outside this cubic volume are not included in the summation. This is valid because of the rapidity at which the magnitude of a term decreases with increasing separation. If an overlap is found, the rotamer selection procedure is restarted, but using a new statistical weight matrix in which the disallowed rotamer is totally excluded. If the newly chosen rotamer also overlaps with an existing atom the procedure is attempted again using the remaining possibility. If an overlap is once again found, the chain is denoted as terminated. Further calculations on this chain are abandoned, and the properties of this chain are not included in the averages. In the pf luorostyrene case, a set of several exclusion distances was required in this vinyl system to attain the experimental characteristic ratios. All of these geometric parameters are summarized in Table I. The algorithm also incorporates tacticity control for vinyl chains. The random number generator is used to choose between d- and l-versions of the transformation matrix. A single parameter controls the relative probability of d- and l-residues. The poly(p-fluorostyrene) results present ed here are for atactic (stereochemically irregular) chains. Results All results were obtained from chains of 200 backbone atoms, averaging 100 chains to obtain M~, axial extensions, end-to-end distances, and segmental and chain order parameters and directors. For poly(methylene), an exclusion distance (hard sphere diameter) of 2.00 R was used to prevent overlap of methylene residues. The calculation reproduced the accepted theoretical and experimental characteristic ratios (mean square unperturbed end-to-end distance relative to that for a freely jointed gaussian chain with the same number of segments) of 5.9. This was for zero angular bias and a trans/gauche energy separation of 2.09 kJ mol" . Figure 3 shows the variation of chain dimensions expressed as the averaged ζ component (stretch direction) of the end-to-end distance as a function of the angular bias a and the trans/gauche energy difference ΔΕ. As expected, for a given angular bias, increasing Δ Ε causes an isotropic extension of the chains. Fixing ΔΕ and increasing a causes the chains to orient, producing some net elongations along z, but also a contraction in the transverse direction (the orientational properties are not apparent in the figure). It is interesting to note that except at high elongations (-100%), hard sphere overlaps still occur to the level of 1.0 - 2.5% in the polymethylene chain for the range of averaged segmental order parameters between 0.0 and 0.5 (produced by 0 < a £ |), although chain dimensions are unaffected by these encounters. At the high elongations generated by increased ΔΕ, the volume density of segments becomes small, and there fore the probability of encounters drops off.


14.

LIN ET A L .


285

Table I. Geometrical Parameters


poly(methylene) C-C bond length

I.53Â

CCC bond angle

112°

trans/gauche energy (nominal)

2.09 kJ

pentane interaction energy

8.37 kJ

CH2-CH2 exclusion limit (nonbonded)

2.00Â

poly(p-fluorostyrene) C-C backbone bond length

I.53Â

CCC backbone bond angle

112°

trans/gauche energy (nominal)

2.09 kJ

pentane interaction energy

8.37 kJ

C(backbone)-C( 1 -phenyl) bond length

I.5IÂ

CC(backbone)-C( 1 -phenyl) bond angle

120°

all phenyl bond angles

120°

all phenyl C-C bond lengths

I.39Â

C-F bond length

I.30Â

all C-H bond lengths

I.I0Â

backbone C-C exclusion limit (nonbonded)

4.60Â

non-neighboring F-F exclusion limit

2.70Â

neighboring F-F exclusion limit

6.39Â


286

ELASTOMERS AND RUBBER ELASTICITY

Figure 4 displays the corresponding M results of the chains whose dimensions were shown in Figure 3. There is a clear correlation of the calculated N M R linewidth (as represented by M ^ with the a and Δ Ε parameters. A t very low angular bias, increasing Δ EL causes a decrease in segment volume density (larger r-), causing M ^ to decrease. A t higher angular bias, the chain segments tend to orient along the stretch direction, disposing the nearest neighbor spins with their internuclear vectors nearly parallel with H (cose^ ~ I, a high M situation). This effect is most pronounced for The chains with the largest proportion of trans rotamers, since gauche sequences would tend to divert the chain at least temporarily from the stretch direction. Thus, the apparent crossover region in the M map is expected from qualitative arguments. The closest physical realization of the system examined above would be a polymer such as poly(vinyl fluoride) doped into poly(ethylene) (observ ing the F resonance) or poly(trifluoroethylene) doped into poly(tetrafluorethylene), observing H. Neither of these systems is likely to be sufficiently miscible, and there is the possibility of significant proportions of head-to-head linkages, which would dominate the experimental M because of the proximity of the spins on adjacent backbone atoms. A preferable system is poly(p-fluorostyrene) doped into poly(styrene). Since rotations about the 1,4 phenyl axis do not alter the position of the fluorine, the F spin may be regarded as being at the end of a long "bond" to the backbone carbon. In standard RIS theory, this polymer would be treated with dyad statistical weights to automatically take into account conformations of the vinyl monomer unit which are excluded on steric grounds. We have found it more convenient to retain the monad statistical weight structure employed for the poly(methylene) calculations. The calculations reproduce the experimental unperturbed dimensions quite well when a reasonable set of hard sphere exclusion distances is employed. Figure 5 is an O R T E P computer plot of the first 50 backbone carbons in a typical chain. Only the fluorine atoms of the sidechains are shown. The various hard sphere exclusions conspire dramatically to keep the fluorines well separated and the chain highly extended even without introducing any external perturbations. The characteristic ratio from the computer calculations is about 11.6; from data for poly(p-chlorostyrene), C R = I I.I, poly(p-bromostyrene), C R = 12.3, and poly(styrene), C R = 10.3 (all in toluene at 30°C), we expect the experimental value for the fluoropolymer to be in the range of 10 to 12. The hard sphere exclusion distances for this system were chosen as follows: nonbonded backbone carbons, 4.60 R (twice the C - H bond distance plus van der Waals radius for H); non-neighboring fluorines, 2.70 R (twice van der Waals radius for F); neighboring fluorines, 6.39 R (distance across phenyl ring using dimensions of Table I and van der Waals radius for H of 1.20 R). Figure 6 shows the importance of having separate exclusion distances for neighboring and non-neighboring fluorines. The neighbor fluorine exclusion is the main determinant of overall chain dimensions, serving to keep phenyl groups separated in the same manner as dyad statistics would have done. 2

2


2

2


LIN ET AL.


287


no F

ANGULAR BIAS a Figure 3. Average values of the z-component of the end-to-end distance as a func tion of angular bias, a, and trans/gauche energy separation ΔΕ. Each point represents an average of 100 chains of 200 backbone atoms each. In this calculation, F spins reside at the coordinates of alternate backbone atoms. 19

ANGULAR BIAS a Figure 4. Average values of M for the same chains represented in Figure 3. 2



Figure 5. ORTEP computer plot showing the positions of thefluorineatoms of thefirst50 backbone carbons of a poly(p-fluorostyrene) chain. The excluded volumes used in the calculation keep thefluorineswell separated.



14.

LIN ET AL.


289

8 C-C, Long Range F-F Exclusion

h

2

KH

2 Z

4

C-C, Long Range F-F, Neighbor F-F Exclusion

•
> | | Η c i _ j l ^ ~ resonance R F is a p p l i e d t o t h e s a m p l e f o r an i n t e r v a l τ. Tne spin s y s t e m e v o l v e s under t h e s c a l e d spin-spin i n t e r a c t i o n . T h e c o r r e s p o n d i n g f r e e i n d u c t i o n d e c a y (FID), unobservable d u r i n g t h e i r r a d i a t i o n , is m a p p e d out by m e a s u r i n g t h e i n i t i a l F I D a m p l i t u d e as a f u n c t i o n of τ . Certain a r t i f a c t s m a y be c a n c e l l e d w i t h phase a l t e r n a t i o n o f t h e N M R s i g n a l : a l t e r n a t e i r r a d i a t i o n s a r e p r e c e d e d w i t h α π pulse and those signals a r e subtracted from the computer summation. D

n

S D

n

o n

Acknowledgements T h e authors w i s h t o a c k n o w l e d g e t h e D o n o r s o f T h e P e t r o l e u m R e s e a r c h F u n d , a d m i n i s t e r e d by t h e A m e r i c a n C h e m i c a l S o c i e t y ( J L A ) , and the N a t i o n a l S c i e n c e F o u n d a t i o n ( J E M ) f o r p a r t i a l support o f t h i s w o r k .

Literature Cited 1. 2. 3. 4. 5. 6. 7.

Kuhn, W. Kolloid Ζ. 1936, 76, 258. Guth, E.; James, Η. M. Ind. Eng. Chem. 1941, 33, 624. Flory, Ρ. J.; Rehner, J., Jr. J. Chem. Phys. 1943, 11, 512. Flory, P. J. J. Chem. Phys. 1977, 66, 5720. Ronca, G.; Allegra, G. J. Chem. Phys. 1975, 63, 4990. Ullman, R. J. Chem. Phys. 1979, 71, 436. Picot, C.; Duplessix, R.; Decker, D.; Benoit, H.; Boue, F.; Cotton, J. P.; Daoud, M . ; Farnoux, B.; Jannink, G.; Nierlich, M . ; de Vries, A . J.; Pincus, P. Macromolecules 1977, 10, 436. 8. Clough, S. B.; Maconnachie, Α.; Allen, G. Macromolecules 1980, 13, 774. 9. Ullman, R., presented at the ACS Symposium on Elastomers and Rubber Elasticity, New York, August, 1981. 10. Abragam, A. "The Principles of Nuclear Magnetism", Oxford Univer sity Press: London, 1970.


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11. 12. 13. 14. 15.


16. 17. 18. 19. 20. 21.

McBrierty, V. J.; Ward, I. M . J. Phys. D 1968, 1, 1529. McBrierty, V. J.; McDonald, I. R.; Ward, I. M . J. Phys. D 1971, 1, 88. Bower, D. I. J. Poly. Sci. Poly. Phys. Ed. 1981, 19, 93. von Meerwall, E.; Ferguson, R. D. J. Poly. Sci. Poly. Phys. Ed. 1981, 19, 77. Flory, P. J. "Statistical Mechanics of Chain Molecules", Interscience: New York (1969). Flory, P. J. Proc. Roy. Soc. Lond. 1976, A.351, 351. Abe, Α.; Jernigan, R. L.; Flory, P. J. J. Amer. Chem. Soc. 1966, 88, 631. Yoon, D. Y.; Sundararajan, P. R.; Flory, P. J. Macromolecules 1975, 8, 776. Yoon, D. Y.; Flory, P. J. Macromolecules 1977, 10, 562. Yannoni, C. S., private communication. Yannoni, C. S.; Wind, R. A. J. Magn. Resonan. 1980, 38, 493.

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March 22, 1982.


Molecular Configurations of Elastomers Via Solid State NMR - ACS

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