A Comparison of Models and Model Parameters for the Interpretation

13 A Comparison of Models and Model Parameters for the Interpretation of Carbon-13 Relaxation in Common Polymers

Downloaded by UNIV OF ARIZONA on May 7, 2017 | http://pubs.acs.org Publication Date: July 2, 1979 | doi: 10.1021/bk-1979-0103.ch013

A L A N A N T H O N Y J O N E S , G A R Y L . R O B I N S O N , and F R E D R I C E . G E R R Jeppson Laboratory, Department of Chemistry, Clark University, Worcester, M A 01610

Carbon-13 s p i n r e l a x a t i o n i n polymers i s now a common probe of chain dynamics r e s u l t i n g i n a p r o l i f e r a t i o n o f polymers studied and models used t o r e l a t e s p i n r e l a x a t i o n t o polymer motion ( l ) . P r e s e n t l y i t appears u s e f u l t o draw comparisons both among the dynamics o f various polymers and among the i n t e r p r e t a t i o n a l models. For these comparisons, i t i s necessary t o center a t t e n t i o n on p o l ymers which have been thoroughly i n v e s t i g a t e d experimentally. By t h i s we mean i t i s d e s i r a b l e t o have measurments o f s e v e r a l d i f ferent r e l a x a t i o n parameters i n c l u d i n g the s p i n - l a t t i c e r e l a x a t i o n time T i , the nuclear Overhauser enhancement NOE, and t o a l i m i t e d extent the s p i n - s p i n r e l a x a t i o n time T . Although carbon-13 NMR i s a most u s e f u l probe o f chain dynamics, i t i s a l s o important t o measure r e l a x a t i o n parameters o f other n u c l e i i n the same polymer. Both proton and f l u o r i n e n u c l e i are e x c e l l e n t candidates which add information on motions a t other frequencies not a v a i l a b l e from carbon-13 r e l a x a t i o n alone (2). An approach somewhat s i m i l a r t o observing two types o f n u c l e i i s the observation of one nucleus at several magnetic f i e l d strengths since t h i s a l s o provides informat i o n on dynamics i n other frequency domains. L a s t l y i t i s i n f o r mative t o vary such non-spectroscopic parameters as polymer molecular weight, concentration, temperature, and solvent. If a model can c o n s i s t e n t l y i n t e r p r e t a large amount o f spin r e l a x a t i o n data under a v a r i e t y of experimental conditions, then i t i s worthwhile t o consider i n some d e t a i l the s i g n i f i c a n c e o f the model parameters. 2

Survey o f I n t e r p r e t a t i o n a l Models Relaxation parameters are u s u a l l y w r i t t e n i n terms o f t r a n s i t i o n p r o b a b i l i t i e s , W, and s p e c t r a l d e n s i t i e s , J . For carbon-13 n u c l e i under c o n d i t i o n o f proton decoupling (^) l/Ti = W

0

+ 2W

+ W

2

0-8412-0505-l/79/47-103-271$05.00/0 © 1979 A m e r i c a n C h e m i c a l Society

Pasika; Carbon-13 NMR in Polymer Science ACS Symposium Series; American Chemical Society: Washington, DC, 1979.

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CARBON-13 NMR IN POLYMER SCIENCE (NOE)

1

-

=

Y

S

( W

-

2

W

0

= T^giWs -

) / Y

W

0

( W

I

2W

+

0

lI

+

W

2

)

) / Y J

[1] W

W


=

1I =

2

3V

2 I

Y

2

2

2 s

^J (co )AOr 1

2

3Y Y ^ J ((D )/I0r I

S

6

I

2

6

2

The carbon n u c l e i are t o be i d e n t i f i e d with I, the proton n u c l e i with S, and the carbon-proton i n t e r n u c l e a r distance with r. The s p e c t r a l density i s the F o u r i e r transform o f a c o r r e l a t i o n func t i o n which i s u s u a l l y based on a p r o b a b i l i s t i c d e s c r i p t i o n o f the motion modulating the d i p o l e - d i p o l e i n t e r a c t i o n s . The s p i n - s p i n r e l a x a t i o n time, T , i s u s u a l l y w r i t t e n d i r e c t l y as a f u n c t i o n o f spectral densities Q ) . 2

1/T

2

=( l A 0 ) » W ( J 3J (w ) 1

1

hj

+ 6J (u) 2

k )

+ (Uj)

s

+

+

[la]

(U)J).

(0) + 6J Ο

n

1

Ο

The simplest motional d e s c r i p t i o n i s i s o t r o p i c tumbling char a c t e r i z e d by a s i n g l e exponential c o r r e l a t i o n time (k). This model has been s u c c e s s f u l l y employed t o i n t e r p r e t carbon-13 r e l a x a t i o n i n a few cases, notably the methylene carbons i n p o l y i s o butylene among the w e l l studied systems However, t h i s model i s unable t o account f o r r e l a x a t i o n i n many macromolecular sys tems, f o r instance polystyrene (6) and poly(phenylene oxide)(7^ 8). I n the l a t t e r case, the estimate of the c o r r e l a t i o n time var i e s by an order o f magnitude between an i n t e r p r e t a t i o n based on T i and an i n t e r p r e t a t i o n based on the NOE (7). The f a i l u r e o f t h i s model l e d t o the a p p l i c a t i o n of motional d e s c r i p t i o n s i n v o l v i n g s e v e r a l c o r r e l a t i o n times. The simplest of these, a two c o r r e l a t i o n time model, was developed by Woessner (£) and suggested f o r macromolecular systems by Allerhand, D o d r e l l , and Glushko (^). The model considers two motions modulating the dipole-dipole interaction: a n i s o t r o p i c i n t e r n a l r o t a t i o n about an axis which a l s o undergoes o v e r a l l r o t a t o r y d i f f u s i o n . This model can s u c c e s s f u l l y account for the carbon-13 T i and NOE values ob served f o r the methyl carbons i n PIB (jj). The methyl group i s



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p i c t u r e d as r o t a t i n g about the t h r e e - f o l d symmetry a x i s which i s r i g i d l y attached t o a backbone bond undergoing i s o t r o p i c tumbling caused by backbone rearrangements. On the other hand, t h i s model f a i l e d t o account f o r r e l a x a t i o n of n u c l e i i n the phenyl group of polystyrene (6). I t was soon r e a l i z e d that a d i s t r i b u t i o n of exponential cor r e l a t i o n times i s r e q u i r e d to c h a r a c t e r i z e backbone motion f o r a s u c c e s s f u l i n t e r p r e t a t i o n of both carbon-13 T i and NOE values i n many polymers ( l , 10). A c o r r e l a t i o n f u n c t i o n corresponding to a d i s t r i b u t i o n of exponential c o r r e l a t i o n times can be generated i n two ways. F i r s t , a convenient mathematical form can serve as the b a s i s f o r generating and a d j u s t i n g a d i s t r i b u t i o n of c o r r e l a t i o n times. Functions used e a r l i e r f o r the a n a l y s i s of d i e l e c t r i c r e l a x a t i o n such as the Cole-Cole ( l l ) and Fuoss-Kirkwood (12) de s c r i p t i o n s can be a p p l i e d to the i n t e r p r e t a t i o n of carbon-13 r e l a x a t i o n . Probably the most p r o f i c i e n t of the mathematical form models i s the l o g - X d i s t r i b u t i o n introduced by Schaefer ( l O ) . These models are able t o account f o r carbon-13 T i and NOE data a l though some authors have questioned the p h y s i c a l i n s i g h t provided by the f i t t i n g parameters ( l l , 13). The second method used to generate c o r r e l a t i o n functions which r e s u l t i n a d i s t r i b u t i o n of e x p o n e n t i a l c o r r e l a t i o n times i s to s t a r t with a l a t t i c e model and consider rearrangements caused by a crankshaft motion, the three-bond jump. There are now at l e a s t three m o d i f i c a t i o n s of t h i s model a l l based on the approach introduced by Valeur, J a r r y , Geny and Monnerie (VJGM)QA, 15, l6). The fundamental c o r r e l a t i o n f u n c t i o n i s i d e n t i c a l with one devel oped by Glarum ( l ? ) , and Hunt and Powles (l8) from a d i f f e r e n t p h y s i c a l p i c t u r e . This second type of model i n v o l v i n g a d i s t r i b u t i o n o f c o r r e l a t i o n times has been s u c c e s s f u l i n i n t e r p r e t i n g carbon-13 r e l a x a t i o n data ( l l ) , and i t has a l s o been s u c c e s s f u l i n i n t e r p r e t i n g proton r e l a x a t i o n data or proton and carbon r e l a x a t i o n data which were not i n t e r p r e t e d as w e l l by the d i s t r i b u t i o n s of c o r r e l a t i o n times generated from mathematical forms (2, 8). In a d d i t i o n to these i n t e r p r e t a t i o n a l advantages, the l a t t i c e models may provide a b e t t e r b a s i s f o r p h y s i c a l i n s i g h t since they are based on a s p e c i f i c motion p o s s i b l e i n l i n e a r polymers ( l ^ ) and since the dependence of model parameters on c o n c e n t r a t i o n and tem perature seems reasonable ( L l ) . Because of these p o t e n t i a l advan tages, we s h a l l t u r n our a t t e n t i o n to a more d e t a i l e d consideration of the l a t t i c e models. 2

L a t t i c e Models The s t a r t i n g point of a l l three models i s the equation n

n

dP*/dt = w(-2P + P " a a a /

v

2

+ Ρ a

Ω + 2

)


[2]

274

CARBON-13 NMR IN POLYMER SCIENCE

which expresses the time dependence of the p r o b a b i l i t y Ρ of bond η i n the a d i r e c t i o n . The p r o b a b i l i t y per unit time that any p a r t i c u l a r three-bond segment with the proper gauche conformation undergoes rearrangement i s w. For a very long chain, a continuous s o l u t i o n was produced by VJGM (lA, 15) which served as a b a s i s for a c o r r e l a t i o n f u n c t i o n and, by F o u r i e r transformation, a s p e c t r a l density. However, t h i s r e s u l t alone d i d not provide a consistent i n t e r p r e t a t i o n of spin r e l a x a t i o n u n t i l the c o r r e l a t i o n f u n c t i o n was modified by the i n c l u s i o n of an a d d i t i o n a l exponential decay (l6, 19)» The modified s p e c t r a l density, most simply w r i t t e n as


1/2

has two adjustable parameters. The r a t e of occurrence of the three-bond jump i s governed by the choice of T-Q, and the added ex p o n e n t i a l decay i s governed by TQ. The p h y s i c a l s i g n i f i c a n c e of TQ i s of some i n t e r e s t since i t was not introduced from the funda mental l a t t i c e equation, Eq. 2. In i n t e r p r e t a t i o n a l a p p l i c a t i o n s i t i s found that and TQ have d i f f e r e n t apparent a c t i v a t i o n energies (2, 20). The a c t i v a t i o n energy f o r TQ i s lower than f o r τ-ρ and has values nearly equal to the a c t i v a t i o n energy derived from solvent v i s c o s i t y (2, 20). The l a r g e r a c t i v a t i o n energy of τ-ρ has been associated with backbone rearrangements w h i l e that of TQ has been associated with long range tumbling. Going back to the l a t t i c e equations, i t i s not easy to support such a d i s t i n c t i o n , but on the other hand TQ does enter the c o r r e l a t i o n f u n c t i o n i n the same manner as an en t i r e l y independent o v e r a l l r o t a t o r y d i f f u s i o n c o r r e l a t i o n time.


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A second s o l u t i o n to Eq. 2 i s derived from a d i f f e r e n t p h y s i c a l p i c t u r e . Jones and Stockmayer ( l ^ ) solved the l a t t i c e equa t i o n f o r a f i n i t e segment length. Rearrangements caused by the three-bond jump are considered i n a segment c o n t a i n i n g 2m-1 bonds with complete neglect of d i r e c t i o n a l c o r r e l a t i o n s of bonds outside the segment. T h i s y i e l d s a dynamic d e s c r i p t i o n f o r the c e n t r a l bond i n the segment which has been c a r r i e d through to produce the s p e c t r a l d e n s i t y (13),


J(u>) = 2

- 1 T

k

=

w X

Σ k=l 1

> k

λ, = k s i n

G

= l / s + (2/s)

k

2

+

2

U) T|

m + 1 S

=

—2~

((2k-l)n/2(m+l))

s=l Σ exp(-yq) c o s ( ( 2 k - l ) TTq/2s) q=l

γ = 1η9· In t h i s model the parameters are the segment length governed by the choice of 2m - 1, and the r a t e of occurrence of the three-bond jump governed by the choice of w but u s u a l l y expressed by which equals (2W)" . For the f i r s t time, the form of the model given here allows f o r any choice of m. A p p l i c a t i o n s of t h i s model have y i e l d e d s e n s i b l e values f o r an apparent a c t i v a t i o n energy f o r backbone rearrangements based on (6, 8, 2 l ) . Over temperature i n t e r v a l s of 50° t o 100°C, the segment length remains constant with values of the order of 5 to 15 bonds f o r polymers i n s o l u t i o n , and segment length has been considered as a measure of length of chain involved i n cooperative or coupled motions (6, 8, 13j 2 l ) . The Jones and Stockmayer model i s not a continuous s o l u t i o n and becomes cumbersome f o r segment lengths of the order of hundreds which i s encountered i n some but not a l l s o l i d rubbers. The t h i r d l a t t i c e s o l u t i o n presented by Bendler and Y a r i s (22) allows the number of bonds t o become a continuous v a r i a b l e with the parameters of the model being both a short and long range cut o f f of motions expressed as an upper and lower frequency, and W , r e s p e c t i v e l y . The s p e c t r a l d e n s i t y expression i s 1

.D


276


J(ci)) = ( 2 / ( o )

l/2

1

(W /

2

1

2

- W / )))

B

(WB

A

f

)

2

4

X /(l + X )

(W /o))

l/2

A

1

X = k(W /o)) /

2

2

[5] W

2

= 2w/5


w = three-bond jump r a t e . When t h i s model i s a p p l i e d t o the i n t e r p r e t a t i o n o f s p i n r e l a x a t i o n o f polymers i n s o l u t i o n , the extent o f cooperation motion can be measured by a parameter R = (W^/Wg) / which i s found t o take on values from 1 t o 50· I f a bond i s the smallest moving u n i t , then R, c a l l e d the "range", corresponds approximately t o the num ber o f bonds involved i n cooperative or coupled motion (22). Both Wg and W^ are s t r o n g l y temperature dependent and vary non-monotoni c a l l y w i t h temperature which appears t o complicate t h i s simple i d e n t i f i c a t i o n o f R. 1

2

Table I . COMPARISONS OF INTERPRETATIONS OF METHINE CARBON RELAXATION IN DISSOLVED POLYSTYRENE T i (ms)

T

2

(ms)

NOE

Experiment (10)

65

26

1.8

VJQA

66

25

1.8

(22)

Model Parameters

8

τ = 3 · 5 χ 10- s 0

T

Jones and Stockmayer

65

28

2.07

T

D

h

=

1.9 χ Ι Ο " s 9

= 1.0 χ 1 0

-9

s

(2m - l ) = 17 ι 65

26

2.1

w

6

A

= 6.3 χ 10 Hz

W-, = k χ 10 Hz Β 9

R = 16Λ



13.

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Following Bendler and Y a r i s (22), i t seems f r u i t f u l t o com pare the r e s u l t s o f f i t t i n g the three l a t t i c e models t o a common data set. A standard choice would he the experimental values obtained by Schaefer (lO) on i s o t a c t i c polystyrene i n o - d i c h l o r o benzene a t 35°C. Table I contains the f i t t i n g parameters f o r the VJGM and Bendler and Y a r i s models (22) combined with the Jones and Stockmayer approach. A l l three models can account f o r the data w i t h i n experimental e r r o r , and the only immediate comparison o f model parameters i s between the range*of the Bendler and Y a r i s model and the segment length of the Jones and Stockmayer model. Given the r a t h e r d i f f e r e n t mathematical approaches of the two mod e l s , the range and segment length are s u r p r i s i n g l y s i m i l a r . To compare the time scales o f the dynamics c h a r a c t e r i z a t i o n produced by each model, the s p e c t r a l d e n s i t y or c o r r e l a t i o n func t i o n can be w r i t t e n as a d i s t r i b u t i o n o f exponential c o r r e l a t i o n times. For a c o r r e l a t i o n f u n c t i o n , $ ( t ) , the general expression is oo $(t) = J G ( T ) e x p ( - t / ) d [6] ο T

T

except i n the Jones and Stockmayer model where the i n t e g r a l i s replaced by a sum. With expressions given by the authors o f each model, one can c a l c u l a t e the weighted inverse f i r s t moment from

ο which corresponds t o the weighted harmonic average c o r r e l a t i o n time τ^"*. P r e v i o u s l y i n the Jones and Stockmayer model, the un weighted harmonic average c o r r e l a t i o n time was l a b e l e d τ^· For the VJGM model, the inverse f i r s t moment i s i n f i n i t e which i n d i cates the presence o f very many short c o r r e l a t i o n times produced by the continuous s o l u t i o n . For the Jones and Stockmayer model, a summation corresponding t o Eq. 7 y i e l d s a value o f 0.68 ns f o r based on the polystyrene i n t e r p r e t a t i o n . I n the case o f the Bend l e r and Y a r i s model, Eq. 7 reduces the simple expression

1

^h* = 3/(w + W A

2

+ (W^) / ).

B

[8]

The value of c a l c u l a t e d from W and W^ corresponding t o the polystyrene i n t e r p r e t a t i o n i s 0.72 ns which i s i n good agreement with the time s c a l e of the Jones and Stockmayer model. One can a l s o compare the s p e c t r a l d e n s i t i e s f o r the three models as was done by Bendler and Y a r i s f o r t h e i r model r e l a t i v e to the VJGM model (22). The comparison o f the s p e c t r a l d e n s i t i e s produced by the three models using the parameters given i n Table I B



278


i s shown i n F i g . 1. A l l three s p e c t r a l d e n s i t i e s are of rather s i m i l a r shape i n the region of common M R measurements which leads to the conclusion that i t may not be p o s s i b l e t o d i s t i n g u i s h between the three l a t t i c e models on the b a s i s o f i n t e r p r e t i v e ability. The models may be b e t t e r d i s t i n g u i s h e d through ease o f u t i l i z a t i o n and p h y s i c a l s i g n i f i c a n c e o f model parameters. The VJGM model i s e a s i e s t t o apply r e q u i r i n g adjustment of only two continuous parameters i n a concise equation. The Bendler and Y a r i s model involves a numerical i n t e g r a t i o n while the Jones and Stockmayer model, a summation over d i s c r e t e values. As mentioned, the p h y s i c a l s i g n i f i c a n c e o f and TQ i n the VJGM model i s comp l i c a t e d , and the use o f a continuous frequency d i s t r i b u t i o n with an upper and lower c u t o f f i s rather d i f f e r e n t from common dynamic characterizations. The concept o f a "range" from the Bendler and Y a r i s model or segment length from the Jones and Stockmayer model i s appealing but more a p p l i c a t i o n s are r e q u i r e d t o t e s t any r e a l significance. Comparison o f Polymer Dynamics Since none o f the l a t t i c e models i s now c l e a r l y superior, the choice for i n t e r p r e t a t i o n o f s p i n r e l a x a t i o n i n polymers i s a r b i t r a r y . F a m i l i a r i t y leads us t o s e l e c t the Jones and Stockmayer model so we w i l l now consider a p p l i c a t i o n of t h i s model t o s e v e r a l w e l l studied polymer systems i n order t o compare dynamics from polymer t o polymer. A l s o the equations required t o consider a n i s o t r o p i c i n t e r n a l r o t a t i o n of substituent groups and o v e r a l l molecular tumbling as independent motions i n a d d i t i o n t o backbone rearrangements caused by the three-bond jump are a v a i l a b l e f o r the Jones and Stockmayer model (13). Some o f the best studied d i s s o l v e d polymers are polystyrene (6, 10, 23-28), polyisobutylene (£, 1J, 2£, ^O), and poly(phenylene oxideJ(7, 8). We w i l l a l s o add t o these systems polyethylene which i s an i n t e r e s t i n g reference point f o r polymers i n t e r p r e t e d from a t e t r a h e d r a l l a t t i c e viewpoint. The four systems w i l l be abbreviated as PS, PIB, M PP0, and PE i n r e s p e c t i v e order o f i n t r o d u c t i o n here. The f i r s t o f these systems, PS, has data a v a i l able as a f u n c t i o n o f molecular weight, temperature, f i e l d strength, concentration and from three d i f f e r e n t types of n u c l e i . S i m i l a r large data bases are a v a i l a b l e f o r PIB and MpPPO, but the PE r e s u l t s are quoted from a proton and carbon-13 study as a funct i o n o f molecular weight and temperature ( ^ l ) . The PE data are complicated by an unusual temperature dependence o f the r e l a x a t i o n which precluded any estimation of an apparent a c t i v a t i o n energy. W i t h i n experimental e r r o r , both the proton and carbon-13 s p i n l a t t i c e r e l a x a t i o n times o f PE d i s s o l v e d i n p-xylene are independent of temperature between 90 and l80°C. The proton and carbon-13 r e l a x a t i o n as a f u n c t i o n o f molecular weight can be c o n s i s t e n t l y i n t e r p r e t e d i n terms o f the model employed here, but no account o f the temperature behaviour i s o f f e r e d . 2



13. JONES ET AL.

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Relaxation

10

LOG Figure 1. Logarithm of the spectral density vs. logarithm of the frequency, ω. The lines are the fit of the spectral density for the various models to relaxation observed in an o-dichlorobenzene solution of isotactic PS: ( ), spectral density derived from the Bendler and Yaris model; ( ), spectral density derived from the VJGM model; and (- · ·), spectral density derived from the Jones and Stock mayer model.



280

-Ρ

-ρ

CA


CO

+1 *1

οο

+1

ο CvJ

+1 OJ

ο CO

o 1-3

•s

ο d

H PI

oo

6

H CO Ο

03

H

LT\

OJ

CA

Ο

-4"

ο

Ο

C\J

CA

Ο

Ο

ο

Eh

Ο

ê

Β CO H

«

ω

-p

fH (D

Ο

ο

Ο H

LT\

ω EH

ο ο -Ρ (U

Ο Ο CO

ω Η

? s?

ο ο

ο ο

ο Ρ ο


13.

JONES ET AL.

Carbon-13

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Relaxation

The i n t e r p r e t a t i o n o f the four systems i s summarized i n Table I I , and the a p p l i c a b i l i t y of a l a t t i c e model i s reasonable since a l l four backbones can be considered as being at l e a s t approxi mately c h a r a c t e r i z e d by a t e t r a h e d r a l l a t t i c e . The c o r r e l a t i o n time f o r the three-bond jump, = (2w) , v a r i e s c o n s i d e r a b l y among the four systems i n general i n c r e a s i n g w i t h the s i z e o f the three-bond unit. The simple average c o r r e l a t i o n time, τ , a l s o follows the same general trend. Apparent a c t i v a t i o n energies do not vary so g r e a t l y w i t h the s i z e and complexity o f the three-bond unit. The segment length f o r cooperative or coupled motion i s r e l a t i v e l y short f o r these simple backbones i n d i l u t e s o l u t i o n . Note that i n a d i f f e r e n t solvent, a somewhat longer segment length of IT bonds produces the best f i t f o r polystyrene (Table i ) . From another p e r s p e c t i v e , short segment lengths correspond t o rather narrow d i s t r i b u t i o n s o f c o r r e l a t i o n times. -1


a

Table I I I . COMPARISON OF SUBSTITUENT GROUP ROTATION IN DILUTE SOLUTION

τ

(°c)

E (kj)

Polymer

Group

PIB (21)

methyl

50

0.21

18 + 5

PS (28)

phenyl

50

1

20+5

M PP0 (8)

phenyl

50

0.23

2

T

I

(ns)

R

a

5± 2

In Table I I I , the c h a r a c t e r i z a t i o n o f a n i s o t r o p i c r o t a t i o n o f substituent groups i s compared f o r t h i s same c o l l e c t i o n o f poly mers. I n d i l u t e s o l u t i o n , a l l three polymers have s i m i l a r time scales f o r t h i s motion as i n d i c a t e d by the value o f the i n t e r n a l r o t a t i o n c o r r e l a t i o n time, T ^ . However, the a c t i v a t i o n energy f o r substituent group r o t a t i o n i n PIB and PS i s much higher than the a c t i v a t i o n energy f o r phenyl group r o t a t i o n i n M PP0. This i n d i c a t e s some r e a l d i f f e r e n c e s i n the nature of the two l o c a l motions. The c o r r e l a t i o n time f o r phenyl group r o t a t i o n i n poly styrene i s rather u n c e r t a i n since i t i s c a l c u l a t e d from a very small d i f f e r e n c e between T i values. This same problem precludes a very accurate estimate of the a c t i v a t i o n energy although some attempts have been made (2k, 27, 28). The r e l a t i o n s h i p between i n t e r n a l r o t a t i o n o f substituents and backbone rearrangements can be considered from the i n t e r p r e t a tion. The time scales of a n i s o t r o p i c i n t e r n a l r o t a t i o n and back bone rearrangements are w e l l separated i n M PP0. I n a d d i t i o n , the concentration and temperature dependences o f these two q u a n t i t i e s are quite d i f f e r e n t l e a d i n g us t o conclude that the motions are independent. I n PIB and PS, i n t e r n a l r o t a t i o n and backbone R

2

2


Pasika; Carbon-13 NMR in Polymer Science ACS Symposium Series; American Chemical Society: Washington, DC, 1979. backbone rearrangement backbone rearrangement

phenyl group r o t a t i o n phenyl group r o t a t i o n

Phenyl carbons

Phenyl carbon

Methyl protons

carbon

Methylene

backbone rearrangement backbone rearrangement

carbon

Methylene

backbone rearrangement

Methyl carbon

carbon

Methylene

Nuclei

Dominant M o t i o n a l Source of Spin R e l a x a t i o n

independent

independent

R e l a t i o n s h i p between Backbone Rearrangement and I n t e r n a l R o t a t i o n

COMPARISON BETWEEN BACKBONE DYNAMICS AND SUBSTITUENT GROUP ROTATION

Table IV.



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rearrangements are close i n time s c a l e , and the a c t i v a t i o n energy for methyl group r o t a t i o n i n PIB i s equal t o the a c t i v a t i o n energy f o r backbone rearrangement. However, a n a l y s i s o f the concentra t i o n dependence o f motions i n PIB p o i n t s t o the independence of the two motions (2l). The time scale o f backbone rearrangements changes by orders o f magnitude i n t r a v e r s i n g the concentration range from d i l u t e s o l u t i o n t o the bulk while the time scale of methyl group r o t a t i o n remains almost constant. T h i s i n t e r p r e t a t i o n i s at odds w i t h one presented by Heatley (^0) but data at two f i e l d strengths support the i n t e r p r e t a t i o n v o i c e d here. Table IV attempts t o summarize these comparisons between backbone r e a r rangements and s u b s t i t u e n t group r o t a t i o n . I t a l s o l i s t s the motion which i s the major source o f s p i n r e l a x a t i o n through modu l a t i o n o f the d i p o l e - d i p o l e i n t e r a c t i o n . I n only M PP0 does i n t e r n a l r o t a t i o n o f a s u b s t i t u e n t group, the phenyl group, become the dominant source o f s p i n r e l a x a t i o n . 2

Table V. COMPARISON OF BACKBONE REARRANGEMENTS IN SOLIDS

Polymer

Temperature

PE (amorphous) (32) PIB (rubber)

(jO

(ns)

.060

.030

45° 4 5 0

(ns)

11

Segment Length 2m - 1

22

C i s - P o l y i s oprene (rubber) ( l O )

35°

0.35

5.25

57

Cis-Polybutadiene (rubber) (lO)

35°

0.0065

3· Τ

23x10*

Only r e c e n t l y have the l a t t i c e models been a p p l i e d t o s o l i d amorphous and rubbery polymers (21-22). Table V contains a sum mary of the i n t e r p r e t a t i o n s of s e v e r a l s o l i d polymers. I n general, l e s s extensive data are a v a i l a b l e on these systems. For PE (^2) and PIB (^), the i n t e r p r e t a t i o n i s based on carbon-13 T i and NOE values. For c i s - p o l y i s o p r e n e and c i s polybutadiene, the i n t e r p r e t a t i o n i s based on T i , NOE and T values (lO) although u t i l i z a t i o n of T i s complicated because of the p o s s i b l e presence o f systemat i c e r r o r s p a r t i c u l a r l y i n spectrometers employing superconducting magnets. The i n t e r p r e t a t i o n f o r PE and t o a l e s s e r extent PIB i s not unique. S e v e r a l choices of segment length are p o s s i b l e and the parameters l i s t e d i n Table V are f o r a minimum length which 2

2


284


accounts f o r T i and the NOE. The other p o s s i b l e choices of segment length are not much longer since a maximum NOE i s observed i n PE, and the PIB data are c o n s i s t e n t with a very narrow d i s t r i b u t i o n of c o r r e l a t i o n times (5., 2 l ) . Of course coupling between chains i s completely neglected i n t h i s l a t t i c e model so the s i g n i f i c a n c e of segment lengths i n v o l v e d i n cooperative l o c a l motions i s unclear. For c i s - p o l y i s o p r e n e and cis-poiybutadiene, the choice of parameters accounting f o r T i , the NOE, and T i s unique. However, the segment length i s determined e n t i r e l y by matching T which i s the l e a s t r e l i a b l e piece of input information although great care was taken by Schaefer (lO) to l i m i t systematic e r r o r s . In PE, the NOE and apparent T depend on the degree of c r y s t a l l i n i t y (^2) and mostly amorphous PE i s considered here. As c r y s t a l l i n i t y increases, the NOE and T decrease which corresponds to an increase i n c o u p l i n g segment length. However, the decrease i n T may w e l l r e f l e c t macroscopic and microscopic inhomogeneities i n magnetic f i e l d strength across the sample caused by instrument a l problems and sample preparation. These kinds of complications l e d us and others (2) to doubt the s i g n i f i c a n c e of dynamic i n t e r p r e t a t i o n s which rest strongly on the observed values of T . It is p r e f e r a b l e to develop another measure s e n s i t i v e to low frequency motions which i s l e s s s u s c e p t i b l e to systematic e r r o r s . P o s s i b l y Tj_p could p l a y t h i s r o l e i n viscous s o l u t i o n and bulk m a t e r i a l s as i t i s now p r o v i d i n g i n s i g h t i n t o g l a s s y m a t e r i a l s (33)One l a s t comment on the i n t e r p r e t a t i o n of c i s - p o l y i s o p r e n e and cis-polybutadiene i s appropriate. Models based on a t e t r a h e d r a l l a t t i c e of equivalent bonds are c l e a r l y not s t r i c t l y a p p l i c a b l e t o a polymer c o n t a i n i n g both s i n g l e and double bonds. Thus, model parameters i n c l u d i n g the c o u p l i n g segment length should not be taken too s e r i o u s l y although i t i s very n e a r l y equal t o the ranges obtained by Bendler and Y a r i s employing t h e i r model. One can be somewhat hopeful about the a p p l i c a t i o n of the l a t t i c e mode l s to non-tetrahedral backbones i f the c o n n e c t i v i t y o f the chain i s the major f a c t o r determining the c o r r e l a t i o n f u n c t i o n and not the t e t r a h e d r a l geometry (jk). 2

2


2

2

2

2

Summary The l a t t i c e models provide u s e f u l i n t e r p r e t a t i o n s of s p i n r e l a x a t i o n i n d i s s o l v e d polymers and rubbery or amorphous bulk p o l y mers. Very l a r g e data bases are r e q u i r e d to d i s t i n g u i s h the i n t e r p r e t i v e a b i l i t y of l a t t i c e models from other models, but as yet no important d i s t i n c t i o n between the l a t t i c e models i s apparent. In s o l u t i o n , the s p e c t r a l d e n s i t y at s e v e r a l frequencies can be determined by observing both carbon-13 and proton r e l a x a t i o n processes. However, a l l the frequencies are r a t h e r high unless T data are a l s o i n c l u d e d which then i n v o l v e s the prospect of systematic e r r o r s . I t should be mentioned that only e f f e c t i v e r o t a t i o n a l motions of e i t h e r very l o c a l or very long range nature are r e q u i r e d t o account f o r s o l u t i o n observations. The l o c a l 2



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motions which r e s u l t i n r o t a t i o n a l averaging of d i p o l a r i n t e r a c t i o n s are the backbone rearrangements caused by the three-bond jump or a n i s o t r o p i c r o t a t i o n o f s u b s t i t u e n t groups. The only long range motion i n c l u d e d i s o v e r a l l r o t a t o r y d i f f u s i o n . Together these account f o r the molecular weight dependence o f s p i n r e l a x a t i o n although mid-range motions corresponding t o Rouse-Zimm modes of an order higher than one are not included. The e f f e c t s o f l o c a l motions are seen i n the data at high molecular weight and the e f f e c t s o f o v e r a l l r o t a t o r y d i f f u s i o n , at low molecular weights. However, i t has not been necessary t o e x p l i c i t l y i n c l u d e other types o f cooperative, long range motions. P o s s i b l y Tip would be s e n s i t i v e t o motions corresponding t o the higher Rouse-Zimm modes, and t h i s would a i d i n the e l u c i d a t i o n o f the r o l e o f these motions i n spin relaxation. The other c l a s s o f motion only now being introduced i n t o i n t e r p r e t i v e models i s o s c i l l a t o r y motion. A n i s o t r o p i c o s c i l l a t o r y motions o f s u b s t i t u e n t groups have been considered by Chachaty ( l 2 ) but not i n conjunction with a l a t t i c e d e s c r i p t i o n o f backbone motion. No attempt t o develop a model based on o s c i l l a t o r y backbone rearrangements i s known t o these authors, and t h i s avenue may be very important f o r the i n t e r p r e t a t i o n o f concentrated s o l u t i o n s , rubbery or amorphous s o l i d s , and e s p e c i a l l y g l a s s y polymers

(22). Acknowledgments Many h e l p f u l d i s c u s s i o n s with W. H. Stockmayer are g r e a t l y appreciated. The research was c a r r i e d out with f i n a n c i a l support of the N a t i o n a l Science Foundation, Grant DMRTTI6088, Polymers Program. This research was supported i n p a r t by a N a t i o n a l Science Foundation Equipment Grant No. CHE77-09059-

Abstract A brief survey of models for the interpretation of spin relaxation in polymers suggests models based on the occurrence of the three-bond jump on a tetrahedral lattice are capable both of accounting for observations and providing some physical insight. The lattice model of Valeur, Jarry, Geny and Monnerie is compared with the more recent revisions of Jones and Stockmayer, and Bendler and Yaris. Since i t is found that a l l three lattice models have comparable interpretive ability and produce very similar descriptions of the spectral density when applied to the same data, one model, the Jones and Stockmayer version, was used to interpret several well studied polymers. The resulting characterization of motions in dissolved polyethylene, polyisobutylene, polystyrene and poly(phenylene oxide) are reviewed for trends in time scale, apparent activation energy and the extent of cooperative motion. Time scales varied from picoseconds to nanoseconds, the activation energies for backbone rearrangements areallabout


286



20kJ, and the length of chain involved in cooperative motion is only 5 to 15 bonds. Spin relaxation in four solid polymers was also interpreted with the model. Amorphous polyethylene and polyisobutylene rubber undergo motion nearly as rapid as dissolved polymers and the segment length for cooperative motion is not ap preciably longer either. Cis-polyisoprene and cis-polybutadiene are also very mobile as solid rubbers but the apparent segment length for cooperative motion is much longer than for simple dis solved polymers. Of course, a tetrahedral lattice model is not strictly applicable to these last two polymers, and interchain cooperativity was not properly considered for any of the solid polymers. Literature Cited 1. For a recent overview, Schaefer, J. in "Topics in Carbon-13 NMR Spectroscopy," Vol. 1, G. C. Levy, Ed. (Wiley-Interscience, New York, 1974). 2. Heatley, F. and Cox, Μ. Κ., Polymer, (1977), 18, 225. 3. Doddrell, D., Glushko, V. and Allerhand, Α., J. Chem. Phys., (1972), 56, 3683. 4. Bloembergen, Ν., Purcell, Ε. M. and Pound, R. V., Phys. Rev., (1948), 73, 679. 5. Komoroski, R. A. and Mandelkern, L., J. Polym. Sci., (1976), Sym. No.54,201. 6. Matsuo, K., Kuhlmann, K. F., Yang, H. W.-Η., Geny, F., Stock mayer, W. H. and Jones, Α. Α., J. Polym. Sci., Polym. Phys. Ed., (1977), 15, 1347. 7. Laupretre, F. and Monnerie, L., Eur. Polym. J., (1974), 10, 21. 8. Jones, A. A. and Lubianez, R. P., Macromolecules, (1978), 11, 126. 9. Woessner, D. E., J. Chem. Phys., (1962), 36, 1. 10. Schaefer, J., Macromolecules, (1973), 6, 882. 11. Heatley, F. and Begum, Α., Polymer, (1976), 17, 399. 12. Ghesquiere, D., Ban, B. and Chachaty,C.,Macromolecules, (1977), 10, 743. 13. Jones, A. A. and Stockmayer, W. H., J. Polym. Sci., Polym. Phys. Ed., (1977), 15, 847. 14. Valeur, Β., Jarry, J. P., Geny, F. and Monnerie, L., J. Polym. Sci., Polym. Phys. Ed., (1975), 13, 667. 15. Valeur, Β., Monnerie, L. and Jarry, J. P., J. Polym. Sci., Polym. Phys. Ed., (1975), 13, 675. 16. Valeur, Β., Jarry, J. P., Geny, F. and Monnerie, L., J. Polym. Sci., Polym. Phys. Ed., (1975), 13, 2251. 17. Glarum, S. H., J. Chem. Phys., (1960), 33, 639. 18. Hunt, Β. I. and Powles, J. G., Proc. Phys.Soc.,(1966), 88, 513. 19. DuBois-Violette, Ε., Geny, F., Monnerie, L. and Parodi, O., J. Chem. Phys., (1969), 66, 1865. 20. Schilling, F.C.,Cais, R. E. and Bovey, F. Α., Macromole-



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cules, (1978), 11, 325. 21. Jones, Α. Α., Lubianez, R. P., Hanson, M. A. and Shostak, S. L., to appear J. Polym. Sci., Polym. Phys. Ed., (1978). 22. Bendler, J. and Yaris, R., to appear Macromolecules, (1978). 23. Allerhand, A. and Hailstone, R. K., J. Chem. Phys., (1972), 56, 3718. 24. Laupretre, F., Noel, C. and Monnerie, L., J. Polym. Sci., Polym. Phys. Ed., (1977), 15, 2127. 25. Inoue, Y. and Konno, T., Polym. J., (1976), 8, 457. 26. Heatley, F. and Begum, Α., Polymer, (1976), 17, 399. 27. Jones, Α. Α., J. Polym. Sci., Polym. Phys. Ed., (1977), 15, 863. 28. Matsuo, K. and Stockmayer, W. H., private communication. 29. Inoue, Y., Nishioka, A. and Chujo, R., J. Polym. Sci., Polym. Phys. Ed., (1973), 11, 2237. 30. Heatley, F., Polymer, (1975), l6, 493. 31. Gerr, F. E. and Jones, Α. Α., unpublished results. 32. Komoroski, R. A., Maxfield, J., Sakaguchi, F. and Mandelkern, L., Macromolecules, (1977), 10, 550. 33. Schaefer, J., Stejskal, E. O. and Buchdahl, R., Macromole cules, (1977), 10, 384. 34. Stockmayer, W. H., private communication. Discussion D. Axelson, F l o r i d a State U n i v e r s i t y , F l o r i d a : One comment and then one question. The polyethylene sample you showed i n the l a s t s l i d e I've taken below i 5° down t o -^4-0°C. The sample develops a broad d i s t r i b u t i o n or non-exponential auto c o r r e l a t i o n f u n c t i o n at low temperatures. One o f the giveaways i s that i n going through the T i minimum i f i t i s not 110 ms at 67 MHz and i f a d i s t r i b u t i o n e x i s t s , i t w i l l be higher, and i n t h i s case the m i n i mum i s almost at 300 ms. The system very q u i c k l y develops a d i s t r i b u t i o n and the T i ' s v i r t u a l l y l e v e l o f f f o r the l a s t 20 or 35 degrees. Even t h i s macromolecule sample i s tremendously com plicated. The other p o i n t i s r e l a t e d t o the methyl group r o t a t i o n you were t a l k i n g about. Some o f the polymers we have been l o o k i n g at i n s o l u t i o n as w e l l as i n bulk have long side chains. The end methyl w i t h a s i x or eight carbon a l k y l side chain has a T i value anywhere from twenty seconds on down. The T i value i s frequency dependent. I t shouldn't be frequency dependent over 200 or 3°0 ms. I was wondering i f you have any f e e l f o r what p h y s i c a l l y i s t a k i n g place or i f you have had any opportunity t o i n v e s t i g a t e models that would t r y t o a l l e v i a t e t h i s p e c u l i a r i t y , A. Jones, C l a r k U n i v e r s i t y , Massachusetts: When the methyl group i s attached t o the backbone or even t o a side chain i t has a whole d i s t r i b u t i o n o f c o r r e l a t i o n times a s s o c i a t e d e i t h e r w i t h the backbone or the side chain i n a d d i t i o n t o i t s methyl group rotation. Our assumption i n these models i s that they are inde pendent so that t h e methyl group T i would be c a l c u l a t e d on the


288



b a s i s o f a very large d i s t r i b u t i o n o f c o r r e l a t i o n times even though there probably i s a f a s t c o r r e l a t i o n time that i s dominant. A l l those other times are i n the c o r r e l a t i o n f u n c t i o n and that may be what you are d e t e c t i n g . I'd l i k e t o point out that i n some polyethylene data the molecular weight dependence i n d i c a t e d that at a molecular weight o f 150, h a l f of the T i was s t i l l a s s o c i a t e d with backbone rearrangements which i n v o l v e s a d i s t r i b u t i o n o f c o r r e l a t i o n times. Even a short side chain may have a d i s t r i b u t i o n o f c o r r e l a t i o n times a s s o c i a t e d w i t h i t which you do not observe u n t i l the data i s c l o s e r t o the T i minimum than i n the polyethylene data considered here. I t i s transparent u n t i l you get a s u f f i c i e n t data base. D. Axelson: We p u b l i s h e d a paper (G. C. Levy, D. E. Axelson, R. Schwartz and J . Hochmann, J . Am. Chem. Soc., 100, klO (l9ï8)) l a t e l a s t year on methacrylates i n which we t r i e d t o i n t e r p r e t the side chains i n terms o f the e f f e c t o f the backbone d i s t r i b u t i o n . The side chain was governed by a combination o f the d i s t r i b u t i o n and the m u l t i p l e i n t e r n a l r o t a t i o n s . The problem we had i s that when you get t o the end o f the chain you s t i l l can't account f o r the frequency dependence. We have not proceeded t o the stage o f doing d i l u t e s o l u t i o n s t u d i e s because as you mentioned, there may be an i n t e r m o l e c u l a r e f f e c t with other chains. A. Jones: I don't f e e l the m u l t i p l e i n t e r n a l r o t a t i o n s model i s a p p l i c a b l e f o r side chains any longer than 3 or k carbons. As soon as i t i s longer than J-h carbons, a crankshaft type o f mot i o n dominates i n s t e a d o f m u l t i p l e i n t e r n a l r o t a t i o n s . The viewp o i n t probably r e f l e c t s some b i a s on my part. J. Prud* homme, U n i v e r s i t y o f Montreal, Que.: What i s the p h y s i c a l e x p l a n a t i o n f o r the l e v e l i n g ? When we see the curve o f T i as a f u n c t i o n o f molecular weight there i s a plateau. This appears at a molecular weight over 1000 and sometimes 10,000. What i s the p h y s i c a l e x p l a n a t i o n f o r t h i s ? A. Jones: With low molecular weights, the o v e r a l l molecular tumbling i s at a r a t e not f a r from the Larmour frequency, the b a s i c frequency of the experiment. As molecular weight increases these o v e r a l l tumbling motions become v e r y slow and very f a r r e moved from the Larmour frequency. The motion c l o s e s t t o the Larmour frequency comes from backbone rearrangements, very l o c a l motions, which do not depend upon molecular weight. That three bond jump i n the model does not depend on chain length, i t only i n v o l v e s three bonds i n the middle o f a long chain. When those motions are dominant, the r e l a x a t i o n c l o s e s t t o the Larmour f r e quency stems from l o c a l motions y i e l d i n g molecular weight independent r e l a x a t i o n . J. Prud* homme: What i s amazing i s that f o r some substances t h i s happens at a very h i g h molecular weight.


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A. Jones: I f the backbone motion i s very slow then higher molecular weights are r e q u i r e d before the o v e r a l l tumbling motion becomes much slower than the r e l a t i v e l y slow backbone motions. Dr. Bovey mentioned p o l y s u l f o n e s . I would expect T i to have an extended molecular weight dependence because the backbone motions are r e l a t i v e l y slow i n t h i s case. W. G. M i l l e r , U n i v e r s i t y of Minnesota, Minnesota: In v i s c o e l a s t i c studies the monomeric f r i c t i o n c o e f f i c i e n t i s used to describe motion. The same parameter i s used t o look at t r a n s l a t i o n a l d i f f u s i o n of solvent and i t s c o n c e n t r a t i o n dependence. Is there any r e l a t i o n s h i p between t h i s parameter and your three bond motion or i s the c o r r e l a t i o n length way too long? A. Jones: I don't t h i n k I can give you a d e f i n i t i v e answer. The three-bond jump should be experiencing a f r i c t i o n p a r t i a l l y determined by i t s environment, i . e . , solvent or other chains around i t , but most probably i t i s determined by the r o t a t i o n a l p o t e n t i a l a s s o c i a t e d with conformational changes w i t h i n the chain. For many systems we see r a t h e r l i t t l e concentration dependency of the a c t i v a t i o n energy f o r these motions. For instance, the phenyl group r o t a t i o n which has a very low b a r r i e r appears t o be a good d e a l more solvent dependent than some of the other backbone rearrangements i n v o l v i n g three bond jumps. Most others seem t o be mostly i n f l u e n c e d by the shape of p o t e n t i a l energy surface assoc i a t e d with the chain conformations. I. C. P. Smith, NEC - Ontario: When you are t e s t i n g those models, crankshafts, cut o f f and so on, there was a p a u c i t y of data on frequency dependence. Have you t r i e d t o f i t the v a r i o u s models to say three magnetic f i e l d s ? A. Jones: Yes, I didn't show a l l the data. Most of the data i s from the l i t e r a t u r e . Polystyrene data and p o l y i s o b u t y l e n e data at two f i e l d strengths can be accounted f o r by these l a t t i c e mode l s based upon the three bond jump. Some recent data by Dr. Bovey on the polybutene was more d i f f i c u l t to f i t with regard to the frequency dependence. (F. C. S c h i l l i n g , R. E. Cais and F. A. Bovey, Macromolecules 11, 325 (1978).) Frequency dependent data I t h i n k i s important information to acquire when t r y i n g to understand the dynamics. I. C. P. Smith: But does i t d i s t i n g u i s h between the models? I t would be n i c e to f i n d the best of the three. A. Jones: I t does not d i s t i n g u i s h between the l a t t i c e models. I don't t h i n k we are going to e a s i l y d i s t i n g u i s h between the l a t t i c e models. The same b a s i c equations are employed i n each case. The three l a t t i c e models I mentioned have s p e c t r a l d e n s i t i e s of the same shape. P. Sipos , Dupont, Ontario: Using the concept of a chain segment you have shown l a r g e d i f f e r e n c e s between p o l y i s o b u t y l e n e



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and other polymer systems. Is i t that the b a s i c r e l a x a t i o n mode occurs on a long chain because a very short segment o s c i l l a t e s independently or i s i t a concerted phenomenon? To get agreement must a large number o f segments be considered together? A. Jones: We would l i k e t o know whether i t i s a long range o s c i l l a t i o n or whether i t i s a complete t u r n i n g around o f a s e r i e s of units. There are measures o f whether i t i s a complete r o t a t i o n such as the values o f NTi f o r d i f f e r e n t carbons. I f NTi i s a constant, the motion i s probably r o t a t i o n a l . In solution, I think these things tend t o be r o t a t i o n a l . As you go t o bulk systems and i n p a r t i c u l a r g l a s s y systems, J . Schaefer ( j . Schaefer, E. 0. S t e j s k a l and R. Buchdahl, Macromolecules 10, 384 (1977) ) f e e l s very s t r o n g l y that o s c i l l a t o r y motion predominates. I would agree, although I can't produce an o s c i l l a t o r y model t o i n t e r p r e t the data. RECEIVED M a r c h 13, 1979.


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