Probing the Real Structure of Chain Molecules by Vibrational

29 Probing the Real Structure of Chain Molecules by Vibrational Spectroscopy GIUSEPPE ZERBI

Downloaded by CORNELL UNIV on May 29, 2017 | http://pubs.acs.org Publication Date: June 1, 1983 | doi: 10.1021/ba-1983-0203.ch029

Istituto Chimica Industriale, Politecnico, Piazza L. Da Vinci, 32, Milan, Italy The field of vibrational spectroscopy and molecular dynamics is presented and discussed with a review of the basic concepts and results of the more recent techniques developed. Emphasis is placed on the possibilities vibrational spectroscopy offers for providing detailed qualitative and quantitative information on polymer structure, in particular, its use to detect disorder in polymeric materials. The advantages vibrational spectroscopy offers over other physical techniques are mentioned. The motions of the CH group are dealt with primarily. Examples are given of the usefulness of defect calculations for structural analysis. Measurement of the chain length of ordered chains now becomes possible. By selectively deuterating one end of the molecule and by using end-group resonance modes together with CD gap modes, the structural evolution of both ends and tail of a hydrocarbon chain can be followed. Various aspects of vibrational spectroscopy, for example, factor group splitting and the effect of Fermi resonances on Raman spectra are discussed. The state-of-the-art interpretation of the Raman spectrum of polymethylene chains in the all trans structure is given. Specific phenomena such as the physical transport of matter across crystal boundaries during annealing of polymer solids can be studied. 2

2

HEMISTS WHO NEED

to characterize a given p o l y m e r sample or w i s h to probe the structure of a p o l y m e r i c material are asked generally to use a l l the physical or p h y s i c o c h e m i c a l techniques presently a v a i l able that may h e l p i n their endeavor. E a c h technique has advantages and disadvantages because it may provide very detailed information 0065-2393/83/0203-0487$ 12.25/0 © 1983 American Chemical Society

Craver; Polymer Characterization Advances in Chemistry; American Chemical Society: Washington, DC, 1983.


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for some samples i n a p h y s i c a l condition, yet it may be useless for other kinds of samples i n a different p h y s i c a l state. Often, limitations and advantages are discussed i n terms of a technical vocabulary w i t h w h i c h the specialists assume everyone is familiar. T h u s , a barrier often is b u i l t b e t w e e n the community that develops methods a n d those w h o may use such methods. T h e purpose of this chapter is to focus on the f i e l d of vibrational spectroscopy and molecular dynamics of polymers and to try to b u i l d a bridge between specialists and users. W e purposely shall a v o i d lengthy mathematical developments and w i l l make an effort to present a n d discuss the basic concepts and results of the recent techniques that have b e e n d e v e l o p e d i n the f i e l d of vibrational spectroscopy. W e hope our efforts may be of help i n the structural characterization of polymeric materials. T h i s study w i l l emphasize the possibilities vibrational spectroscopy offers for the understanding of the structure of real polymers. T h e w o r d " r e a l " describes the p o l y m e r i c materials as they actually are prepared i n a university or industrial laboratory and w h i c h may contain structural as w e l l as c h e m i c a l disorder directly connected w i t h the chemical or p h y s i c a l treatments that were used d u r i n g their preparation. O b v i o u s l y , the reality of a p o l y m e r i c material as described b y a given technique depends on the capability of a g i v e n technique to go from overall information to deeper and deeper detail on the molecular scale. W e w i s h here to (1) analyze how far vibrational spectroscopy and molecular dynamics can go at present i n p r o v i d i n g detailed qualitative and quantitative structural information on p o l y m e r s ; a n d (2) illustrate w h e n and how vibrational spectroscopy can be of h e l p i n structural analysis w h e n and where other physical techniques fail. Infinite Perfect Polymers Polymers as O n e - D i m e n s i o n a l Crystals. T h e vibrational analysis of polymer molecules considered as objects w i t h perfect structures has b e e n discussed elsewhere (I - 5 ) . W e shall point out only a few aspects later i n this chapter w h e n d e a l i n g w i t h p o l y m e r s w i t h nonperfect structures. L e t us assume that d u r i n g the synthesis of the polymer: 1. T h e c h e m i c a l reaction performs the desired polymerization w i t h 100% y i e l d , and the l i n k i n g of the c h e m i c a l units i n the polymer c h a i n (head-to-tail, etc.) occurs without mistakes. 2. I f a stereospecific polymerization is taking place, it yields a polymer w i t h perfect tacticity. 3. T h e intramolecular a t o m - a t o m interactions i n such a


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c h e m i c a l l y perfect molecule generate a m i n i m u m - e n ergy structure that corresponds to a perfectly regular chain w i t h a g i v e n overall shape (planar zig-zag, h e l i c a l , etc.). 4. T h e p o l y m e r chains pack w i t h a perfect order i n a tridimensional lattice. A t present, problems of c h a i n folding i n single crystals, and other problems are ne glected intentionally. Because intramolecular forces and couplings i n organic m o l e cules are m u c h stronger than intermolecular forces, the w h o l e v i b r a tional spectroscopy of polymers first considered the p o l y m e r m o l e c u l e as prepared i n Steps 1 through 3 just listed. T h e decision to study first the polymer m o l e c u l e as a single c h a i n i n an empty space is justified because the intermolecular interactions that occur w h e n the m o l e c u l e is e m b e d d e d i n a t r i d i m e n s i o n a l lattice are very weak and act as small perturbations on the normal vibrations of the isolated c h a i n where covalent b o n d i n g is strongly m a k i n g up the w h o l e structure. Such a structurally perfect, infinite, a n d isolated p o l y m e r c h a i n is c a l l e d a one-dimensional crystal. I n d e e d , i f an operation of rototranslation or screw motion Κ(0,δ) (where θ is a rotation about the chain axis and δ is a translation along the same axis) is a p p l i e d to the starting c h e m i c a l repeating unit, we generate an infinite oned i m e n s i o n a l crystal i n w h i c h a one-dimensional unit c e l l w i t h repeat distance d can be located. T h e one-dimensional unit c e l l may contain η c h e m i c a l units organized i n a helix w i t h m turns. Geometrical relationships among the structural parameters 0, δ, and d were fully discussed by structural chemists (6). A c c o r d i n g to the values of θ and δ, w e may f i n d that the one-dimensional crystal is a planar zig-zag c h a i n ( θ = π, δ = d/2, polyethylene) or a slightly t w i s t e d helix (0 « 247Γ/15, polytetrafluoroethylene) or a h i g h l y t w i s t e d h e l i x (e.g., 0 = π/2, δ = d/2, orthorhombic polyoxymethylene; 0 = 2ττ/3, δ = d/3, isotactic polypropylene). T h e c h e m i c a l repeat u n i t may not coincide w i t h the crystallographic repeat unit even i n the one-dimensional crystal. T h e simplest example is single-chain polyethylene, w h i c h consists of a planar z i g zag chain (point group D ) i n w h i c h the single C H unit is the c h e m i cal repeat unit, and the one-dimensional crystallographic repeat u n i t is made up of two C H groups i n trans conformation. A more complex case is that of hexagonal polyoxymethylene, whose one-dimensional crystallographic repeat unit contains nine c h e m i c a l units c o i l e d i n a chain w i t h five turns. T h e vibrational analysis of a perfect p o l y m e r c h a i n considered as a one-dimensional crystal has to account for the very large n u m b e r of vibrations that occur i n such a system. F o r an isolated m o l e c u l e , i f Ν is m

2

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the n u m b e r of atoms then 3 N — 6 degrees of vibrational freedom or normal modes must be described. Analogously, for an infinite p o l y m e r chain, an infinite n u m b e r of normal modes that occur simultaneously must be accounted for. A s i n a small finite m o l e c u l e , the overall v i b r a tional motion of the p o l y m e r c h a i n can be described as the superposi tion of an infinite number of normal modes. T h e fact that the observed I R and Raman spectra are generally very simple suggests that very few of these innumerous normal modes contribute to the absorption of I R light or to the Raman scattering process, even i f p h y s i c a l l y a l l the other modes do exist a n d perform their duties i n m a k i n g the w h o l e molecule vibrate a n d i n h e l p i n g to determine several p h y s i c a l prop erties (7). W e l l - e s t a b l i s h e d theories o f molecular dynamics (8) show that i f the assumption that atoms vibrate i n a harmonic potential is made, the equation of motion c a n be solved e n d i n g w i t h the solution of an eigenvalue equation: G FL =LA

(1)

T h e vibrational coordinates chosen to set u p E q u a t i o n 1 are the tradi tional vibrational internal displacement coordinates (8). I n E q u a t i o n 1, the matrix G contains a l l the information o n the geometry of the m o l e c u l e and the mass of the atoms, F contains the information o n the intramolecular potential i n the harmonic approxi mation, a n d A is a diagonal matrix g i v i n g the vibrational fre quencies k . T h e solution of the secular determinant K

\G F - Ε Λ| = 0

(2)

provides the vibrational frequencies ^ ( c m ) = [\ /(47r c )] (where c is the speed o f light) and finally, w i t h some more calculations, one can determine the vibrational displacements, L , of each atom, thus b e i n g able to k n o w the shape of each o f the 3 N — 6 normal vibrations (8). T h e solution of Equations 1 or 2 requires knowledge of the geom etry of the molecule a n d o f the vibrational potential. A l t h o u g h the geometrical parameters may be d e r i v e d from other experiments, the vibrational force f i e l d is d e r i v e d from the analysis of the vibrational spectra of similar molecules (9, 10). H o w e v e r , a discussion o n this important point is outside the scope of this study. T h e reader is re ferred to the relevant literature o n molecular potentials (8-11) for d e r i v i n g a critical evaluation of the reliability of the various intramo lecular potentials proposed i n the literature. A s a w o r k i n g hypothesis, w e assume for the time b e i n g that for polymers the vibrational poten- 1

K

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2

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K

t


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tial is k n o w n to such an extent as to a l l o w reliable calculations to be performed (9, 12-15). F o r the calculation of the vibrational normal modes, Q of a p o l y m e r c h a i n , one must be able to treat the vibrations of an infinite one-dimensional crystal that can generate an infinite number of nor mal modes. L e t Ν be the n u m b e r of atoms i n the c h e m i c a l repeat u n i t and Ρ the number of units, Ρ b e i n g very large. T o make mathematics more directly understandable, the physics of the phenomenon of the vibrations of an infinite p o l y m e r c h a i n are examined briefly. T a k i n g the nine normal vibrations of the — C H unit (three masses, nine vibrational degrees of freedom), let us focus on the C H symmetric stretching mode, v ( C H ) . If a long chain of C H units is constructed w i t h a w e l l - d e f i n e d and translationally regular geometry of the chain, by exciting the v ( G H ) mode of the first unit such a motion is transmitted to the next C H a n d to the next one, and so on along the c h a i n (16). B y i m p o s i n g a certain phase relationship, φ, between the vibrational displacements of the first u n i t w i t h respect to the nearest neighbor a n d then to the second nearest one and so on, a cooperative normal mode, Q CH2S? is established throughout the w h o l e chain. S u c h a mode is a standing w a v e - l i k e mode w i t h wavelength λ = 2(p/j)d (where j = CV/2 . . . P" ) during w h i c h a l l atoms move around their e q u i l i b r i u m position w i t h the same frequency, ν(φ), such as i n any normal mode of a s m a l l molecule (2, 3). T h e m a i n variable i n this p r o b l e m is the phase difference, φ, between n e i g h b o r i n g units; φ can take any value from 0 to 2π. T h e corresponding frequencies, ^ C H S W > of the C H symmetric stretching mode, taken before as an example, w i l l vary smoothly d e p e n d i n g on the intramolecular c o u p l i n g be t w e e n oscillators. S u c h a c o u p l i n g is described again by the G and F matrices as for small molecules. T h e plot of ν vs. φ collects the infinite normal modes of v ( C H ) into a curve called a dispersion curve. Because nine degrees of vibrational freedom exist for the single C H group, i n a similar manner nine dispersion curves w i l l be con structed throughout the same range of values of φ. T h e actual calculation of the dispersion curves can be made by using an eigenvalue equation similar to that of E q u a t i o n 1, where each of the quantities now becomes phase dependent, that is, they become modulated by the phase factor φ (14-17). Ky

2

s


2

2

s

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2

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1

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2

s

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G (φ) F (φ) L (φ) - L (φ) Λ (φ)

(3)

I n this case s

s

G (φ) = G + Σ G* e*** + Σ & e-** 0

S

F (φ) = F" +

Σ

(4a)

S

Ρ e** + Σ Ρ ^


(4b)

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i n E q u a t i o n s 4a and 4b, matrices G° a n d F° correspond to those o f the isolated — C H — unit, a n d G and F describe the interactions o f each C H w i t h its neighbor s units away o n both sides. D i s p e r s i o n curves were calculated for several o n e - d i m e n s i o n a l perfect polymer chains (14,15,18,19) and they p r o v i d e d basic infor mation on the c o u p l i n g of intramolecular vibrations. I n turn, the ex perimental k n o w l e d g e of the dispersion curves provides the informa t i o n necessary for the precise d e f i n i t i o n o f the intramolecular m e chanics of these types of molecules. I n E q u a t i o n s 4a a n d 4b, the index, s, accounts for the distance of intramolecular interactions b e t w e e n c h e m i c a l repeating units. T h e precise knowledge of s so far has b e e n neglected generally because for most of the covalent s i m p l e and classical polymers considered, s does not extend further than the second nearest neighbor (12). T h i s p h e n o m e n o n means that the intramolecular interatomic interactions for most of the atoms organized i n a network of covalent bonds d o not extend too far, but fall off very q u i c k l y , as expected. T h i s behavior is not the case w i t h more recently prepared organic polymers, such as polyacetylene, that become c o n d u c t i n g or semiconducting w h e n suit ably d o p e d w i t h different substances. E x p e r i m e n t s show that the d e r e a l i z a t i o n o f the 7r-electrons i n these systems extends very far along the c h a i n a n d that small pertur bations at one site of the c h a i n are felt at large distances further along the molecule. F o r practical reasons, both spectroscopy (21 -24) a n d quantum mechanics (25) i n their calculations must truncate t h e i r i n teractions at some value of s. F o r these systems, the actual extent of interaction is not yet k n o w n ; its knowledge should be very relevant to the understanding o f the electrical properties of these systems a n d to the related materials i n physics and biology. T o be more precise i n the d e f i n i t i o n of the dispersion curves for a one-dimensional crystal (2), one c a n state that i f Ν is the n u m ber o f atoms p e r c h e m i c a l repeat unit, 3N roots o f the secular deter minant are calculated for each value of φ u s i n g E q u a t i o n 3. T h e n , 3N branches c a n be d r a w n i n the dispersion curve w h e n plotted as a function of the phase shift. T w o of these branches tend to ν = 0 c m w h e n φ tends to 0. D e p e n d i n g o n the geometry of the p o l y m e r c h a i n , one o f these branches m a y again reach ν = 0 for — π < φ < π. T h e dispersion curves are p e r i o d i c functions of φ w i t h p e r i o d 2π. H e n c e , i t is only necessary to consider the dispersion curves i n the interval —π < φ < π. Approximately h a l f o f these solutions are enough and are plotted 2

s

s

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1

- 1

' E x p e r i m e n t a l d i s p e r s i o n curves generally are d e r i v e d from neutron scattering e x p e r i m e n t s ; d i s c u s s i o n o f this t e c h n i q u e is o u t s i d e t h e scope o f this study.



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generally i n the interval 0 < φ < π (14,15,18). T h e t w o branches i n the dispersion relation that reach zero at φ = 0 are c a l l e d acoustical branches because at φ = 0 they describe a r i g i d motion of the c h a i n w i t h the center of mass o f the u n i t c e l l m o v i n g i n phase w i t h a l l the others. F o r the normal modes (generally c a l l e d phonons) along the 3N 2 optical branches, the center of mass remains u n d i s p l a c e d for a l l values of φ, i n c l u d i n g φ = 0. T h e origin of the name " o p t i c a l b r a n c h e s " is because some phonons may interact w i t h the electromagnetic wave thus absorbing and/or scattering light of proper frequency. O n c e the infinite normal frequencies (phonon frequencies) of the one-dimensional crystal are organized along the many branches o f the dispersion relation, one must f i n d w h i c h of these phonons interact w i t h an electromagnetic f i e l d thus absorbing I R light or scattering v i s i b l e light i n Raman experiments. Selection rules were w o r k e d out (16) and state that i f θ is the angle w i t h w h i c h units are rotated i n the generation of the w h o l e c h a i n b y the screw motion Β.(θ,δ), t h e n the normal modes w i t h phase difference φ = 0 and φ = θ are IR-active; motions w i t h φ =0,9, and 20 are Raman-active. U s i n g the language of solid-state physics, normal modes corresponding to φ = 0 , 0, a n d 2Θ i n the most classical cases possess wave vector k = 0 i n reciprocal space, (k = φ/d). F o r brevity, these normal modes w i l l be c a l l e d "fc = 0 phonons." If a stretch-oriented sample is examined w i t h p o l a r i z e d I R light, phonons w i t h φ = 0 have a transition moment p a r a l l e l to the d r a w i n g direction that generally coincides w i t h the c h a i n direction. Phonons w i t h φ = θ show a transition moment generally p e r p e n d i c u l a r to the chain axis (1 -2). N o r m a l l y b y extrusion or b y stretching, molecular chains are oriented predominantly parallel to the c h a i n axis b u t are oriented randomly i n a plane orthogonal to the c h a i n axis. F u l l o r i e n tation along the three crystallographic planes for I R studies only has b e e n achieved o n samples of syndiotactic polypropylene (26). A fully t r i d i m e n s i o n a l l y oriented specimen of single crystal mats of polyethylene w i t h a very small mosaic spread was reported (27), but this f i n d i n g has not b e e n repeated b y others. O r i e n t e d specimens as extruded rods also were e x a m i n e d b y Raman spectroscopy i n different scattering geometries (28-30). B y using these experimental techniques, detailed experimental informa tion o n the symmetry species of the spectroscopically active phonons can be obtained. T h e classification into symmetry species makes pos sible an improvement i n the matching between frequencies p r e d i c t e d by theories and those observed i n actual experiments. IR and Raman experiments and calculations have b e e n performed by numerous investigators i n the last twenty years o n many p o l y m e r i c



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π Figure 1. Dispersion curves of single-chain polyethylene in the 1500— 0-cm~* range, calculated with the force field of Ref 88. Spectro scopic activity for k = 0 modes (φ = 0, π) is indicated. Direction of transition moments for IR-active modes in a stretch-oriented sample are given. The symbols \\ andljrnean transition moments parallel and perpendicular to the stretching directions, respectively. Key: O , IR; and •, Raman. systems considered to be perfect, one-dimensional crystals. C a l c u l a tions and experiments were i m p r o v e d along w i t h the technological development of computers and of I R and Raman spectrometers (2, 31 ) (Figure 1). A comment on the relevance of such types of studies i n p o l y m e r



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science seems appropriate. I n particular, we w o u l d l i k e to examine briefly the types of information d e r i v e d from the study of the v i b r a tional spectra of polymers as perfect, one-dimensional crystals a n d compare them w i t h the information d e r i v e d from other p h y s i c a l tech niques. W e neglect i n this discussion the relevance of these k i n d s of studies to the specialized basic field of molecular a n d crystal d y n a m ics but w i l l analyze their relevance for p o l y m e r chemists. T h e first observation of practical importance is that theories of p o l y m e r dynamics p r o v i d e d the way to interpret i n detail the ob served spectra of polymers that otherwise c o u l d not be interpreted i n terms of the w e l l - k n o w n spectroscopic correlations (32,33). I n d e e d , as seen before, the few spectroscopically active phonons along the d i s persion branches may occur i n spectral regions that are away from those predicted by spectral correlations. T h e i r frequencies d e p e n d on the extent of intramolecular couplings, w h i c h are, i n turn, related to the geometrical shape of the p o l y m e r c h a i n . H e n c e , those spectroscopically active phonons generate bands i n IR and/or Raman spectra that are very characteristic of a p o l y m e r c h a i n c o i l e d i n a specific conformational structure, w h i c h is supposed to extend for a considerable length i n space. T h e IR and/or Raman bands observed thus provide direct infor mation that the c h a i n has taken up a conformationally regular struc ture i n space (one-dimensional translation periodicity) (34) as gener ated by a rototranslational operator w i t h w e l l - d e f i n e d values for 0 a n d δ. T h e so-called bands due to k = 0 phonons that appear i n the spec trum on careful slow solidification from melt or crystallization from solution are c a l l e d " r e g u l a r i t y b a n d s " (34) because they refer to a specific operator R(0,o). I n a d d i t i o n , k = 0 modes of stretch-oriented samples can provide symmetry species and, possibly, information on δ and 0, hence, on the shape of the molecule. O n m e l t i n g , these bands disappear almost completely because most of the conformational reg ularity collapses (34, 35). If some weak bands still r e m a i n , it means that sections of chains i n the melt (or solution) are still h e l i c a l w i t h the same R(0, δ) still operating on a short range (36). D e t e r m i n a t i o n of the length of short regular segments i n a l i q u i d p o l y m e r (or i n solution) is still under investigation. O n m e l t i n g or d i s s o l v i n g , Step 3 i n the formation of the p o l y m e r is removed and the spectrum of a substance formed d u r i n g Steps 1 and 2 remains. T h i s spectrum provides information on c h e m i c a l and stereochemical structure. I n such a case, classical spectroscopic cor relation tables again may be of h e l p . A f u l l analysis of this p r o b l e m has b e e n carried out recently for two samples of isotactic and syndiotactic polypropylene i n the melt a n d s o l i d state (30, 37). T h e use of regularity bands (or k = 0 modes) for a q u i c k d e t e r m i -



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nation of the geometry of a p o l y m e r chain i n the one-dimensional state is not straightforward and easy. T h e choice of a m o d e l that w i l l be consistent w i t h the observed spectra must be based on a careful comparison of experiments and calculations that use carefully selected vibrational potentials. x-Ray diffraction studies p r o v i d e d more detailed information on the structure of many polymers (31). I n a few cases, however, the shape of the molecular c h a i n first was d e t e r m i n e d from vibrational studies (38). T h e p o l y m o r p h i c form of p o l y v i n y l i d e n e fluoride was detected b y IR spectroscopy, and the shapes of the two modifications first were determined from I R spectra (39). T h e most recent example of the application of I R spectroscopy to the structural studies of polymers is that of polyacetylene either i n the cis or trans form. T h e structures of both modifications were d e t e r m i n e d entirely from I R and R a m a n studies (40); however, x-ray diffraction data are h i n d e r e d greatly by the unusual nature of the sample. A l t h o u g h x-rays may be used to measure interatomic distances more easily, vibrational spectroscopy finds shapes of molecules that are consistent w i t h the experimental spectrum. Calculations of normal modes may come later based on some assumed geometry. I f the overa l l shape of the molecule has a certain evidence, refinement of force f i e l d and geometrical parameters to fit the experimental spectrum may provide somewhat better geometrical parameters. Polymers i n T h r e e D i m e n s i o n s . Step 4 i n the b u i l d i n g of the perfect p o l y m e r material consists of p a c k i n g the molecular chains into a tridimensional lattice. W h e n organic molecules crystallize, the v i b r a tional spectrum shows n e w bands i n the I R or Raman spectra that are related directly to the arrangements of the molecules i n the crystal. T h e considerations previously made for one-dimensional crystals must now be extended to three dimensions. T r i d i m e n s i o n a l p e r i o d i c i t y is reached by m i n i m i z a t i o n of intermolecular interactions, and the molecules w i l l adjust to form the crystal generating a t r i d i m e n s i o n a l pattern that can be described by its s y m metry properties i n space and by the n u m b e r of molecules i n the repeating unit c e l l . P o l y m e r molecules, by controlled experiments, can crystallize i n t r i d i m e n s i o n a l networks thus a d d i n g to the already discussed one-dimensional periodicity along the chain and the p e r i odicity across chains. T h e vibrational motions propagating along the c h a i n w i l l then be perturbed slightly by the weak intermolecular forces a n d n e w v i b r a tions (phonons) w i l l be generated transverse to the chain axis, w i t h molecules b o u n c i n g or s l i d i n g r i g i d l y against each other. D i s p e r s i o n curves still can be calculated for phonons propagating w i t h i n the crystal along a l l possible directions (2, 41).



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A g a i n , only a few phonons w i l l be spectroscopically active according to the symmetry of the crystals and according to the n u m b e r of molecules i n the t r i d i m e n s i o n a l crystallographic unit c e l l . T h e concepts developed for small molecules fully apply to crystalline p o l y mers (Figure 2). L e t k = 0 modes of the one-dimensional crystal be considered, i n a first approximation, internal modes of the m o l e c u l e . Because of intermolecular (interchain) c o u p l i n g , each internal mode splits into m u l t i p l e t s , w i t h the m a x i m u m degree of m u l t i p l i c i t y b e i n g equal to the n u m b e r of chains per unit c e l l . T h e symmetry of the crystal may reduce the m u l t i p l i c i t y of the corresponding bands observed i n the spectra either because some transitions may be symmetry forbidden or because vibrations may be degenerate (42, 43). Such a splitting is generally c a l l e d "factor group s p l i t t i n g , " w i t h the extent of splitting b e i n g due to the strength of the intermolecular interactions (43). These intermolecular interactions are different for different normal modes because the extent of interaction depends on the vibrational amplitude for each normal mode. T h e larger the a m plitude (atoms move nearer to each other), the larger the splitting. Because intermolecular forces are weak, splittings w i l l be generally small and, for the reasons just mentioned, may be unobserved for many normal modes that do not b r i n g atoms of one c h a i n i n close contact w i t h atoms of the n e i g h b o r i n g chains. " E x t e r n a l modes," o w i n g to the motions of the r i g i d chains (or quasi-rigid chains) one against the other, give rise to external lattice modes that occur i n the very l o w frequency region of the spectrum (42, 43). 2

Because the concept of factor group splitting is of great h e l p i n understanding the p h y s i c a l properties of p o l y m e t h y l e n e chains (discussed later), a description of the same p h e n o m e n o n from another v i e w p o i n t is given. If two energy levels of proper symmetry and described b y the corresponding wavefunctions accidentally c o i n c i d e or occur very near i n energy, the wavefunctions may m i x and the levels may r e p e l each other. T h e extent of r e p u l s i o n depends on the extent of m i x i n g of the starting wavefunctions. T h e extent of m i x i n g (hence of the splitting) slowly decreases and disappears w h e n energies no longer match and move away from each other. Factor group splitting originates from precisely this situation. T h e vibrational energy l e v e l for a g i v e n normal mode for one isolated molecule (natural frequency) coincides i n energy w i t h that of the neighboring unit(s) i n the unit c e l l . I f symmetry allows, r e p u l s i o n takes place and a doublet (or multiplet) occurs. T h e c o u p l i n g of the A s an example, i n crystalline orthorhombic polyethylene, only C H r o c k i n g s h o w a n o n n e g l i g i b l e s p l i t t i n g (—25 c m ) . 2

2

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- 1


498

POLYMER CHARACTERIZATION SINGLE CHAIN

ORTHORHOMBIC CRYSTAL (+) Bau

B3u

(+) B2u


B2u

(-) B

3 u

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(inactive)

(+) Ag

Figure 2. Scheme for factor group splitting in poluethylene. Key: top, IR; and bottom, Raman spectra. (See Refs. 28 and 29 for definition of reference axes.)



29.

zERBi


499

levels is related to the d y n a m i c a l c o u p l i n g (F and G matrices of E q u a tion 3) w i t h i n the crystal. O n e of the ways to check whether an observed splitting originates from intermolecular interactions i n crystals is to look at the spectra of m i x e d crystals of isotopic species. G e n e r a l l y , m i x e d crystals of h y d r o genated and deuterated materials were studied because they are more easily available (42, 43). T h u s , by isotopic d i l u t i o n preparation of crystals is possible i n w h i c h the molecule under study can become a guest i n a lattice of the isotopic derivative. T h e interatomic potential i n isotopic derivatives is the same as that of the parent m o l e c u l e (44); hence, changes i n frequencies are related only to changes of the masses of the atoms (G matrix of E q u a tion 1). F o r the isotopically d i l u t e d molecule, a g i v e n energy l e v e l (natural frequency) of the guest molecule does not match w i t h that of the molecules i n the host lattice a n d thus remains unperturbed. A single b a n d w i l l then appear somewhere i n the m i d d l e of the observed doublet (or multiplet), its intensity b e i n g proportional to the amount of isolated guests i n the host lattice. A t very large d i l u t i o n s , the doublet (multiplet) disappears because only isolated islands i n the host lattice exist. T h e method of isotopic d i l u t i o n has b e e n used w i d e l y i n solid-state vibrational spectroscopy of simple organic molecules (42, 43) and has received a great deal of attention i n recent years i n the f i e l d of polymers, m a i n l y by K r i m m a n d coworkers (45, 46). T h e concept of guest and host molecules w i l l be taken up again later i n a somewhat different context. In the f i e l d of p o l y m e r spectroscopy, factor group splitting a n d external lattice modes are of particular importance for the determination of the crystallinity of a p o l y m e r sample. T h e determination of the crystallinity of polymers has been r e v i e w e d extensively, a n d a reanalysis of the p r o b l e m is outside the scope of this work (2, 34). T h e only important aspect is that vibrational spectroscopy allows the differentiation b e t w e e n one-dimensional a n d t r i d i m e n s i o n a l order. I n fact, k = 0 modes from an isolated c h a i n (i.e., regularity bands) inform us that the polymer c h a i n has c o i l e d up i n a g i v e n conformation. W h e n regularity bands split into a m u l t i p l e t and, possibly, lattice modes are observed (crystallinity bands), w e are informed that a t r i d i m e n s i o n a l p e r i o d i c i t y has b e e n formed. T h e concept of regularity and crystallinity bands is not n e w (34), but it seems not yet w e l l accepted i n practical p o l y m e r chemistry and physics where often regularity bands still are used to study and measure crystallinity. A particular feature of the s o l i d state of a p o l y m e r i c material has to be pointed out. Researchers report (47) that b y suitable thermal treatments p o l y m e r solids do not immediately pack i n t r i d i m e n s i o n a l lattices but may form aggregates of p o l y m e r chains w i t h o n e - d i m e n -



500


sional longitudinal order but w i t h no lateral periodicity. T h i s phase generally has been named " s m e c t i c . " A c c o r d i n g to the previous discussion, the spectrum of the smectic form of a p o l y m e r clearly w i l l only show regularity bands w h i l e crystallinity bands w i l l not occur. B y proper annealing of the material, chains move inside the crystal and adjust to form a tridimensional arrangement. A t this stage, crystall i n i t y bands (factor group splitting and lattice modes) w i l l occur. E x p e r i m e n t a l l y , for most of the polymers studied, regularity bands have been observed. A textbook case for vibrational spectroscopy of polymers is that of isotactic polypropylene (Figure 3). F o r such a polymer, Steps 1 through 4 i n the formation of the p o l y m e r can be followed separately. Tacticity bands are observed for the spectrum i n melt and solution and a l l o w for differentiation between the syndiotactic derivative (37). T h e c o i l i n g of the c h a i n i n a three-fold h e l i x isolated from the neighboring c h a i n is f o l l o w e d b y observing the appearance of regularity bands w h e n a l i q u i d sample is q u e n c h e d into the smectic form (47). Factor group splittings testifying to the formation of an ordered tridimensional arrangement recently were observed i n I R and Raman spectra after suitable annealing of the smectic sample (48) (Figure 3).

Disordered

Polymers

T h e vibrational structural analysis of polymers, w h e n considered as an infinite and perfect system, showed q u i c k l y its weakness because a l l polymers i n their real structure contain a large amount of matter i n a state of disorder. A n understanding of the erroneously c a l l e d " a m o r p h o u s " bands then becomes essential. E v e n single crystals of polymers contain an intrinsic disorder because the very l o n g molecular chain has to fold back and forth to crystallize (7). Whatever may be the type of f o l d i n g (49), a conformational disorder certainly must be considered because the molecule i n a single crystal is different from the regular one generated by the operation R(0,3). T h i s s y m metry operation still acts i n the region of the crystallites where chains can still be considered as one-dimensional crystals. As discussed previously ( 5 0 - 5 4 ) , a l l four steps (1 through 4) i n the make-up of a p o l y m e r i c material generally fail i n performing their duties. I n reference to the four steps discussed earlier, the consequences of a n o n i d e a l p o l y m e r i z a t i o n are the f o l l o w i n g : 1. 2. 3. 4.

Inversion of type of l i n k i n g occurs. Stereoregularity is affected. Conformational disorder is introduced. Stacking faults, dislocations, etc., easily occur.



Figure 3. Raman spectrum of isotatic polypropylene from smectic (one-dimensional) to crystalline (tridimensional) orders. Raman regularity bands split into doublets due to crystallinity.


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Numerous studies of these types of errors were reported u s i n g different p h y s i c o c h e m i c a l techniques. T h e purpose of this chapter is to b r i n g to the attention of p o l y m e r chemists those aspects of vibrational spectroscopy that may be useful i n the qualitative a n d quantitative study of the disorder i n p o l y m e r i c materials. T h e m a i n goal of the researchers i n the f i e l d of vibrational spectroscopy is to develop vibrational spectroscopy i n such a way as to provide information that other techniques cannot give. I n d e e d , c o m petition, for example, w i t h N M R spectroscopy i n the study of stereoregularity of polymers i n solution is unnecessary. Q u i c k , inexpensive, a n d very detailed qualitative and quantitative information is prov i d e d by N M R spectroscopy on p o l y m e r tacticity and type of l i n k i n g w h e n polymers are d i s s o l v e d i n a suitable solvent. T h e study of a p o l y m e r i n the s o l i d state by N M R may not be so q u i c k and i n e x p e n sive e v e n w i t h the most recent developments (55). Analogously, x-ray and neutron diffraction techniques on solids provide overall information that must be interpreted b y suitable molecular m o d e l i n g (49). T h e s e techniques, however, do not provide direct signals that arise from specific atomic groups i n a specific structure (49). O f more importance is the study of the detailed conformation of a p o l y m e r i c material, especially i n the s o l i d state. I n this case, the c o n tributions from N M R , x-ray, and/or neutron scattering are not as specific. Therefore, w e w i s h to show here that vibrational spectroscopy can provide detailed information on a m u c h smaller molecular scale. Finite Chains: Determination of Chain Length. A l l models that try to describe the reality of a s o l i d p o l y m e r consider the existence of segments of supposedly perfect chains a n d domains of disordered material. D o m a i n s of disordered material may be of different type, but generally one may conceive the existence (1) of disordered sections that interrupt regular chains; and (2) of domains where disordered matter surrounds crystallites w i t h ordered structure (7). O r d e r e d p o l y m e r chains are interrupted b y small or large sections of disordered material. T h e question we w i s h to answer is h o w to measure the c h a i n length of the ordered chains. Recent studies of Raman spectroscopy have opened the way to d e t e r m i n i n g the existence of such finite chains and to measuring their length either i n the solid or i n the l i q u i d phase. T h e original i d e a by M i z u s h i m a a n d S h i m a n o u c h i (56), f o l l o w e d by the p i o n e e r i n g work of S h i m a n o u c h i et a l . (57) on the l o w frequency " a c c o r d i o n m o t i o n " of finite molecular chains, opened up a new f i e l d of R a m a n spectroscopy. T h e simplest case of p o l y m e t h ylene chains that comprise a l l normal hydrocarbons w i t h increasing


29.

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Vibrational

Spectroscopy

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length and that e n d w i t h polyethylene w i l l be considered first. T h e second to the last acoustical branch (ω ) i n the dispersion curve of single-chain polyethylene (14,15, 58) refers to the normal modes that are related to the cooperative stretching of the C — C bonds and the b e n d i n g of the C C C angles. F o r finite chains of various lengths, the normal mode w i t h only one node located i n the m i d d l e of the c h a i n produces the lowest frequency l y i n g on this branch (Figure 4). T h e plot of the corresponding eigenvectors shows that this mode is just that of an accordion that extends and compresses the molecule along the c h a i n axis l e a v i n g its center unshifted. It is then a totally symmet ric Raman-active mode. If the m o d e l of the additivity of b o n d polarizabilities is accepted (59, 60), the sum of the contributions from each C - C b o n d predicts that the change i n p o l a r i z a b i l i t y , hence the Raman intensity, must be very large. T h i s prediction is experimentally v e r i f i e d (57, 58). I n d e e d , for polymethylene chains such a low-frequency mode scatters l i g h t more than any other normal mode. Because the slope of the acoustical branch is directly related to the l o n g i t u d i n a l elastic coefficient of the system, S h i m a n o u c h i (58) compares the normal hydrocarbon chain to an elastic rod, w i t h Young's modulus E, uniform density p, and length L . T h e frequency of the longitudinal acoustic mode ( L A M ) is then related to its length by the f o l l o w i n g relationship:


5

T h e observation of such a strong Raman l i n e at l o w frequencies for polymethylene chains allows the direct determination of their length, L, once reasonable values for Ε and ρ are assumed. S u c h a discovery has made L A M spectroscopy a very powerful tool for the determina tion of c h a i n lengths i n molecules or polymers of p o l y m e t h y l e n e . C o m p a r i s o n between c h a i n lengths from L A M spectroscopy a n d small angle x-ray or neutron scattering allows the measurement of the angle of tilt of the molecular chains w i t h respect to the l a m e l l a surface i n single crystals (61 ). T h e direct use of the straightforward formula by S h i m a n o u c h i p r o v i d e d important information i n morphological stud ies of polyethylene (62, 63). Moreover, study of the profile of the Raman l i n e made possible the determination of the chain-length d i s tribution i n samples of polyethylene (64). T h e extension of Shimanouchi's method from η-hydrocarbons to polyethylene i m p l i e s that the disordered matter at both ends of the perfect stems i n polyethylene crystals does not affect both frequencies and intensities of the L A M mode. T h i s p r o b l e m is not yet fully solved. Polymethylene chains may be terminated by different kinds of atomic


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groups for w h i c h mass and conformation may be relevant i n the perturbation of L A M modes. F o r s o l i d polyethylene or l i q u i d or amorphous substances w i t h a p o l y m e t h y l e n e residue, conformational i r regularities at both ends may again perturb the L A M mode. F o l l o w i n g the initial i d e a of Strobl (65), K r i m m and coworkers (66-68) considered the p r o b l e m of the L A M mode of an elastic r o d whose ends were perturbed symmetrically b y masses and/or forces. T h e effect of conformational disorder o n a realistic molecular m o d e l was first considered b y Z e r b i et a l . (69) w h o showed that the introduction of conformational defects at the e n d of the p o l y m e t h y l e n e chain introduces a transversal component i n the l o n g i t u d i n a l motion, thus affecting frequencies and intensities. C o n s e q u e n t l y , more than one normal mode i n the low-frequency region acquires character of L A M thus g i v i n g intensities to other modes l y i n g nearby. T h i s issue has b e e n restated and expanded b y Snyder et a l . w h o actually c a l c u lated the intensity o f L A M modes i n l i q u i d n-alkanes a n d d e t e r m i n e d the concentrations of rotamers i n the same l i q u i d compounds (70). T h e p r o b l e m of the perturbation b y the end-groups is not yet completely solved because the contribution of mass and force constant defect at the ends of p o l y m e t h y l e n e chains are not understood fully. Rabolt discussed long-chain alcohols (71) and recently M i n o n i a n d Z e r b i (72-74) discussed, i n a more general way, fatty acids and the case where molecules j o i n at both ends w i t h some sort of b o n d , such as i n phospholipids and membranes. L A M spectroscopy m a i n l y has b e e n restricted to n-alkanes because these systems are easier to h a n dle. H o w e v e r , L A M modes were observed for isotatic p o l y p r o p y l e n e (75) and polyoxymethylene (76). T h e theory of L A M of h e l i c a l p o l y mers still, however, is u n d e v e l o p e d . Conformational

Disorder

G e n e r a l Concepts. T h e discussion p r i m a r i l y w i l l focus o n the possibility of detecting the conformational disorder i n a p o l y m e r c h a i n using vibrational spectroscopy. U s e of the spectrum i m p l i e s detailed k n o w l e d g e , as far as possible, of the extent to w h i c h the introduction of some k i n d of conformational disorder perturbs the normal vibrations of the polymer. T h e first duty is then to discuss the dynamics of conformationally disordered chains and t h e n to predict, i f possible, the observations expected i n the I R and/or R a m a n spectrum. Just the same concepts, however, can be a p p l i e d to the study o f c h e m i c a l (77), stereospecific (78, 79), and mass defects (80). A s stated earlier, however, c h e m i c a l a n d stereospecific defects are studied more easily b y N M R methods because these two structural characteristics of the p o l y m e r chain also are maintained i n solution, thus making N M R an easier and cheaper method of analysis. I R and Raman



506


spectroscopy again become very useful w h e n polymers cannot be dis solved, thus m a k i n g N M R analysis less effective. Mass defects are particularly interesting w h e n hydrogen atoms are replaced b y d e u terium atoms i n a preassigned way and site i n the m o l e c u l e . (This case w i l l be treated i n detail later.) F r o m a theoretical v i e w p o i n t , several investigators tackled the various problems and t r i e d to calculate the shape of the normal modes and the vibrational frequencies of conformationally disordered p o l y mers. Several kinds of conformational disorder are possible, such as the existence of a few well-characterized kinks (jogs, folds, etc.) (7) e m b e d d e d i n a host lattice of translationally regular p o l y m e r c h a i n or instead, a distribution of conformational disorder w i t h varying geom etry, concentration, and distribution along the c h a i n (7). T h e discussion that follows again w i l l center on the simplest case of polymethylene chains that comprises a large class of compounds that has been the subject of extensive research (81). Attempts to ex t e n d such kinds of studies to other polymers are still few a n d very tentative. W e first w i l l consider a few, well-isolated conformational defects i n a host lattice of translationally regular one-dimensional crystals. T y p i c a l kinks such as G(T) G' or GGTGG (where G and Τ refer to gauche and trans, respectively), for example, are energetically l i k e l y to occur i n a p o l y m e r (82). T h e former introduces only a local distor tion i n the c h a i n , a n d i f η is odd, the trajectory of the c h a i n continues to be straight i n the same direction as before, w i t h the c h a i n only l y i n g on a different, but parallel, plane. T h e latter defect describes the trajectory for a tight fold re-entry of the polyethylene c h a i n along the [200] crystallographic plane. T h e perturbations by these kinds of conformational distortions i n an otherwise regular one-dimensional lattice were treated m a i n l y i n the f o l l o w i n g ways: n

1. B y n u m e r i c a l techniques (50 -54, 78 -80, 83 -85). 2. B y application of Green's function techniques (86, 87). 3. B y studying the spectra of short m o d e l compounds (88). Details of the techniques are discussed i n various specialized articles and were r e v i e w e d recently (81). W e feel that the n u m e r i c a l technique is best suited for treating realistic cases of real polymers that may contain different types of disorder w i t h any type of distribution and concentration. O u r efforts were made along these lines and we shall report here on the basic concepts of these studies. A very long, one-dimensional lattice that happens to host a defect consisting of a few well-characterized con formational defects w i l l be considered. E a c h defect has its o w n natu-



29.

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ral frequency (or frequencies). I f such frequency (ies) happen to occur w i t h i n the b a n d of frequencies of the host lattice (i.e., they occur w i t h i n the frequency b a n d spanned b y the dispersion curves of the perfect, one-dimensional crystal) strong or weak c o u p l i n g w i t h the other modes may occur, w i t h large or small perturbations of frequencies and displacements. E a c h type of vibrational mode thus formed is c a l l e d a"resonance m o d e " and the corresponding frequency is c a l l e d a "resonance frequency" because the proper frequency of the defect is resonating w i t h those of the perfect lattice that l i e nearby i n energy. T h e resulting atomic displacements of the defect almost may be washed out b y the phonons of the host lattice (Figures 5a and 5b) thus i n t r o d u c i n g a very small perturbation or may instead generate normal modes i n w h i c h the atomic displacements mostly are clustered around the defect, l e a v i n g the motions of the n e i g h b o r i n g unit almost unperturbed. Resonance modes (or i n - b a n d modes) t h e n can be d i s t i n guished from quasi-localized, or pseudoresonance modes. T h e i r occurrence i n the spectrum depends on the intrinsic spectroscopic intensity associated w i t h these normal modes. T h e p r o b l e m of the intensities i n I R and R a m a n spectroscopy w i l l be dealt w i t h i n detail later. I f the proper or natural frequency of the defect occurs i n an e n ergy region of the spectrum where no other modes l i e nearby w i t h w h i c h to couple, the vibrational mode of the defect cannot be transmitted b y the lattice, and its displacements are d a m p e d i m m e d i a t e l y .

Phonon mode a

Resonance mode b

Gap mode c Figure 5. Examples of resonance and gap modes in a one-dimensional lattice modeling a simple, single-chain polymer. Displacements are orthogonal to the chain axis, but the phonon propagates along the axis.



508


It then follows that the mode is h i g h l y l o c a l i z e d at the defect (Figure 5c), and the natural frequency is not changed greatly from the hypothetical case of an isolated state. I n such a case, one deals w i t h a gap mode or an out-of-band mode because the frequency occurs either i n the gap b e t w e e n two dispersion curves i n the dispersion relation or above the first dispersion curve of the perfect host lattice. These concepts are not n e w because they are d e r i v e d from s o l i d state physics (89). T h e only difference is that solid-state physics generally has dealt w i t h very s i m p l e types of impurities that c o u l d be treated easily. Organic polymers instead usually contain defects w i t h each one contributing many characteristic frequencies, and the n u m ber and distribution of defects may be such that n o n n e g l i g i b l e d y n a m ical perturbations a n d c o u p l i n g b e t w e e n defects may occur throughout the host lattice. F o r this reason, we think that the understanding of real polymers requires the use of a n u m e r i c a l method that is capable of treating realistic cases m a k i n g analytical methods elegant, but unusable. I n deed, the c h e m i c a l reality of a p o l y m e r must be kept without introd u c i n g into the calculations c h e m i c a l l y and p h y s i c a l l y unacceptable simplifications. F o r this reason, w e b e l i e v e that the method of the negative eigenvalue theorem ( N E T ) proposed by D e a n (90) a n d heavi l y used by our group is best suited for the study of the spectra and structure of disordered polymers. D e t a i l s of the method are discussed elsewhere (52, 53). N E T just solves secular equations of very large sizes, thus a l l o w i n g treatment of disordered polymers as very large molecules. B a sically, the same secular equation as i n E q u a t i o n 2 is solved. T h e extent of intradefect or interdefect c o u p l i n g is just determined b y the shape of the defect and the masses of the atoms ( G matrix), a n d b y the intramolecular potential (F matrix). T h e shape of the normal modes (hence the extent of localization on the defect) is j u d g e d after calculation of the vibrational displacements, w h i c h can be made by suitable n u m e r i c a l methods (91, 92). D e t a i l e d Conformational Information. W e report here a few detailed examples of frequencies due to specific conformational defects i n a frans-polymethylene chain. T h i s information may be of general use for a l l molecules that contain polymethylene residues. G G defects: CH wagging. T h e simplest case of a defect mode that occurs w i t h i n the frequency b a n d spanned by the wagging motions of the trans host lattice, but w h i c h shows a h i g h l y l o c a l i z e d character, is the C H wagging motion of the C H group isolated between two GG structures. T h e localization can be understood b y the fact that the GG structure introduces a strong deformation of the p o l y m e t h y l e n e c h a i n thus d e c o u p l i n g the wagging of

RESONANCE MODES.

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ZERBi


509

the trans sections from the one l o c a l i z e d on the defect. T h i s mode represents the t y p i c a l case of a pseudolocalized mode generated mostly by molecular geometry. These cases are considered as t y p i c a l situations of organic solid-state physics. T h e frequency calculated w i t h the force f i e l d (88) is at 1359 c m " . G T G defects: CH wagging. T h i s defect introduces two wagging modes because two C H groups are j o i n e d b y a b o n d i n trans conformation and are d e c o u p l e d partly from the host lattice by gauche conformations. The intradefect coupling generates out- and in-phase modes calculated near 1375 and 1309 c m , respectively. T h e calculated v a l ues d e p e n d slightly on the force f i e l d used (93). Based on the force field used by Snyder (88), their values are calculated at 1375 and 1309 c m , respectively w h e n the defect is e m b e d d e d i n a l o n g trans lattice. G G T G G defects: CH wagging. T h i s defect, as already m e n tioned, is the one that may exist on the surface of the single crystal of polyethylene i f the c h a i n re-enters i n the crystal w i t h a tight fold re-entry along the [200] crystallographic plane (7,49). Z e r b i et al. have tried to predict where this defect, i f it exists, may absorb i n the I R or scatter i n the Raman. These modes may h e l p eventually i n the study of the structure of the surface of single crystal polyethylene w h i c h is, at present, a matter of strong debate (49). Calculations were made and several characteristic resonance modes were described. I n the C H wagging region, the calculated modes are practically those calculated for the GG and GTG defects w h e n c o m b i n e d i n the GGTGG defect by taking the sum and differences (symmetry combinations) w i t h respect to the local two-fold axis crossing the C - C b o n d (94). Other resonance modes for such a defect were calculated i n other regions of the spectrum, but they were not as characteristic, w i t h the exception of the C H rocking, w h i c h w i l l be discussed i n the next section. T h e experimental test of the calculated numbers of this section can only be made i f one studies a molecular system that is k n o w n for certain, from other independent techniques, to contain the desired defects r i g i d l y h e l d by the structure of the molecule. x-Ray diffraction work demonstrated that the c y c l i c molecule C H contains two GGTGG defects j o i n i n g two trans planar sections (95). T h e I R spectrum shows absorption bands just near where calculations predict the wagging modes of the GGTGG defect to occur. F r e q u e n c y matching, however, may not be such good p h y s i c a l evidence because p o l y m e r molecules generally are so complicated that a b a n d can be found i n the spectrum wherever one wishes. A p p l i cation of recent theories to the p r e d i c t i o n of I R intensities has pro1

2

2

- 1


- 1

2

2

2

3 4

6 8


510


v i d e d the prediction of an intensity pattern i n very good agreement w i t h the observed one (96) thus g i v i n g more selective information and adding more faith i n these kinds of calculations. CH rocking. A c c o r d i n g to the previous discus sion, gap modes occur i n a frequency region of the spectrum where no other vibrational modes of the perfect host lattice occur. C o n f o r m a tional and configurational gap modes were calculated a n d observed by R u b c i c et a l . for configurationally and conformationally disordered p o l y v i n y l chloride i n the frequency gap b e l o w the C C I stretching (77-79). T h i s calculation was performed to h e l p understand the c o m plex spectrum i n this frequency range, w h i c h has b e e n the subject of great interest among various groups. Because this chapter deals m a i n l y w i t h the motions of C H groups, the t y p i c a l gap mode calculated w h e n a head-to-head defect is introduced i n a p o l y v i n y l chloride single c h a i n w i l l be mentioned. W h e n the sequence - C H - C H - between two heavy boundaries occurs i n a molecule, a r o c k i n g mode is calculated near 850 c m (77). T h i s information, however, is not new, because the study of copoly mers of ethylene a n d propylene already e m p i r i c a l l y showed the exis tence of an absorption o w i n g to such a group (97). G G T G G defects: CH rocking. M o r e interesting for structural diagnosis are the somewhat complex vibrational motions that can be described p r i m a r i l y as C H r o c k i n g of several C H groups on and near the GGTGG mode. S u c h a vibrational mode is calculated to occur at 714 c m , just b e l o w the l i m i t i n g k = 0 of the C H r o c k i n g mode of the all trans sequence, i n the gap b e t w e e n C H rocking and C - C b e n d i n g branches (ω and ω ) , respectively (94). T h i s mode was observed i n the spectrum of the c y c l i c hydrocarbon C H mentioned earlier for the C H wagging modes (94). As an example of the usefulness of defect calculations for struc tural diagnosis, we quote the case of such c y c l i c molecules. T h e s i multaneous occurrence of the resonance C H wagging modes near 1350 c m and of the gap mode near 714 c m reportedly c o u l d be taken as a strong i n d i c a t i o n of a GGTGG defect (94). T h e i r relative intensity should d e p e n d on the number of C H groups i n the trans planar host lattice. T h i s fact was verified for a group of cyclic paraffins (98). C D R o c k i n g M o d e s as Probes for the L o c a l Conformation of P o l y m e t h y l e n e C h a i n s . T h e p r i n c i p l e of the technique may be ex p l a i n e d as follows. I f i n a polymethylene c h a i n one C H is replaced b y a C D group, the C D r o c k i n g mode occurs near 620 c m i n the gap between the C H r o c k i n g and the top of the ω branch. Calculations show that the motion is h i g h l y l o c a l i z e d on the C D and that the vibration extends slightly only to the nearest C H group on both sides. Moreover, this mode is strongly conformation dependent.

GAP MODES.

2


2

2

2

- 1

2

2

2

- 1

2

2

9

5

3 4

6 8

2

2

- 1

- 1

2

2

2

2

- 1

2

2

5

2

2


29.

511


ZERBi

T h e first calculation a n d application of this concept were pre sented by Snyder (99), w h o tried to understand the structure of par tially deuterated polyethylene. W e repeated the calculation for l o n g hydrocarbon chains (100) a n d long-chain fatty acids (72—74) w h e r e C D groups were selectively located at different sites along the polymethylene chain. Calculations show that the C D r o c k i n g frequency strongly de pends on the conformation about the C - C b o n d to w h i c h it is attached o n both sides. T a b l e I shows the results of calculations for n nonadecane w h e r e the C D group has b e e n p l a c e d i n p o s i t i o n C-2 (thus l a b e l i n g the head of the molecule) and i n position C-10 (thus l a b e l i n g the m i d d l e of the molecule. F o r - C D - , — C D — , a n d - C D - , frequencies are p r e d i c t e d to occur near 620, 650, and 670 c m , respectively where no other ab sorption occurs. Observations on n-nonadecane fully support the theoretical predictions (100). T h e G G structure was not observed be cause of its small population, according to the rotational isomeric state model. U s i n g these labels, the e v o l u t i o n of the conformation of the ends and of the i n n e r part of the n-nonadecane m o l e c u l e c o u l d then be f o l l o w e d from the s o l i d state through phase transition a n d m e l t i n g . A 2

2

2


2

2

2

- 1

Table I. Calculated C D 2 Rocking Gap Frequencies for a Polymethylene Molecule as a Function of Local Conformation Used as Markers for Conformational Mapping of Polymethylene Chains Frequency (cm )

Chain

1

{-CH -CH -CD Τ Τ G Τ Τ G {-CH -CH -CD -CH Τ (Τ Τ) G (Τ Τ) G (Τ Τ) G' (Τ Τ) 2

2

2

2

CH }

2

2

3

2

615 658 658

-CH -} Τ Τ G G' 2

Τ G G G'

(Τ (Τ (Τ (Τ

G) G) G) G)

Τ Τ G Τ

Τ Τ G

(G G) (G G ) (G G )

T G G

-615

-650

-677


512


mechanism for phase transition and m e l t i n g was proposed on the basis of the observation of these gap modes together w i t h the occurrence i n IR and/or Raman of other resonance modes (100). W i t h n-hydrocarbons w i t h a l i m i t e d c h a i n length, such as n-nonadecane, no evidence was found for the existence or formation of conformational kinks l i k e those c l a i m e d b y others to be the origin o f the m e l t i n g o f the crystal. C D rocking modes selectively placed i n a polymethylene c h a i n thus become very useful probes for m a p p i n g the conformational to pology of such chains i n various physical states. W e believe that this method is u n i q u e l y important because it can be a p p l i e d to molecules i n the solid as w e l l as the l i q u i d or solution states. W e recently a p p l i e d this method to the m a p p i n g of the conforma tional topology of some fatty acids i n the solid a n d solution state (72-74). A l t h o u g h C D groups placed at sites situated away from the carboxyl group behave i n the same w a y as hydrocarbons, some other efforts must be made w h e n , b y approaching the carboxyl group, C D rocking may couple w i t h the vibrational motion l o c a l i z e d o n the C O O H group (72-74).


2

2

2

Resonance Modes as Markers Polymethylene Chains

of Conformational

Mobility

of

Structurally useful defect modes were calculated a n d observed that can be used for the study of the conformational m o b i l i t y of l o n g a l k y l residues such as are found i n fatty acids, p h o s p h o l i p i d s , m e m brane systems, and other systems. W e now w i l l focus o n a few internal modes of the C H group, namely the internal deformation (umbrella motion, C H C7) w i t h natural frequency near 1375 c m and the exter nal deformation, δ C H , w i t h natural frequency near 890 c m ( F i g u r e 6). T h e former mode shows a w e l l - k n o w n characteristic m e d i u m i n tensity b a n d i n the I R spectrum; the latter shows a clear peak i n the Raman spectrum. T h e natural frequencies of both these modes occur just i n the m i d d l e of the frequency b a n d spanned b y t w o dispersion curves. C o u p l i n g thus is expected to occur. F r o m the v i e w p o i n t of dynamics, these modes actually should be considered end-group modes (or surface modes of one-dimensional crystals) that were s h o w n to be damped q u i c k l y along the crystal extending for a few C H groups along the c h a i n (J 00). 3

- 1

3

- 1

3

2

If the geometry of the c h a i n changes near the C H group, the c o u p l i n g changes and the frequencies as w e l l as the vibrational a m plitudes change. T h i s result is exactly what was p r e d i c t e d from cal culations and tested experimentally o n η-hydrocarbons (100,101). I f a structure such a s — C H — C H - C H — C H is generated i n the system, modes are generated at 1342 a n d 866 c m . I f a structure such as— C H — C H — C H — C H occurs, n e w modes are generated at 1355 a n d 3

2

2

2

3

- 1

2

2

2

3



29. ZERBI

Vibrational

Spectroscopy

513

Figure 6. Dispersion curve of single-chain polyethylene in the 1500—0cmr range. Location of CD rocking gap modes and CH deformation resonance modes. Arrows for the latter two modes indicate which range of phonons of the host lattice are likely to couple with them. 1

2

3

844 c m . Other resonance modes are generated i n the spectrum as indicated i n T a b l e I I . Calculations were tested on η-hydrocarbons i n the α-phase and i n the melt (100). O f particular importance are the n e w modes that occur on the l o w frequency side of δ C H and that are easily observable i n the R a m a n spectrum. A l t h o u g h the absorptions i n the I R region may be over l a p p e d partially b y absorptions of other defects, the Raman-active - 1

3


514

POLYMER CHARACTERIZATION Table II. Calculated Resonance Frequencies as Markers for Conformational Mobility of Chain Ends in Molecules with Polymethylene Residues Frequency (cm ) -1

Chain End

Calculated

T G -CH -CH -CH -CH 2

2

2

G Τ -CH -CH -CH -CH


2

2

2

3

3

Observed

1348 874 769

1342 (IR) 866 ( R , IR) 766 (IR)

1367 851

- 1 3 5 5 (IR) 844 (IR, R")

N o t e : O b s e r v e d data are f r o m n - n o n a d e c a n e {100, R refers to R a m a n .

e

101).

a

modes near the δ C H are isolated and may be used as h i g h l y selective markers i n structural studies of complex molecules (101). These signals clearly were observed i n n-nonadecane through its phase transition a n d m e l t i n g thus a l l o w i n g attention to focus, o n a molecular scale, o n the structural evolution of c h a i n ends w i t h t e m perature (100). U s i n g intensity spectroscopy (102, 103) (see later dis cussion), quantitative determinations also were attempted. I n d e p e n dent information o n c h a i n ends was obtained from the study of C D rocking gap modes that were p l a c e d selectively i n position C-2 i n the molecule of n-nonadecane. T h e information was i n f u l l qualitative and quantitative agreement. B y such selective deuteration o n one e n d o f the molecule, and b y u s i n g end-group resonance modes together w i t h C D gap modes, the structural evolution o f both ends a n d tail of a hydrocarbon c h a i n can be followed. 3

2

2

The Raman Spectrum as a Sensitive Probe for the Overall Conformation of a Polymethylene Chain Because k n o w l e d g e of the structure a n d structural evolution of the p o l y m e t h y l e n e c h a i n i n several p o l y m e r i c systems or b i o l o g i c a l molecules is of great importance, many attempts were made i n the past few years to correlate the observed changes o f the Raman spec trum w i t h the conformational changes of the molecules (104). Disagreements can be found among various researchers w h o m a i n l y base their interpretation o n qualitative intuitions o n the v i b r a tional assignment of the R a m a n spectrum. T h e R a m a n spectrum of these systems can be obtained easily from samples i n a w k w a r d p h y s i cal states such as an aqueous solution, molecules spread o n a surface as t h i n layers, and other states. T h u s , the Raman spectrum has become an essential source of data o n these complex organic a n d b i o l o g i c a l systems.


29.

ZERBi

Vibrational

Spectroscopy

515

A substantial improvement i n the interpretation of the observed spectra i n terms of molecular structure has come about w h e n the existence of F e r m i resonances between the vibrational levels of C H sequences were p o i n t e d out. T h e historical development of these n e w ideas are presented here. O u r discussion is l i m i t e d to the Raman spectrum, even i f the I R spectrum also is affected. F i r s t Snyder et a l . presented the idea that both the C H stretching and b e n d i n g regions of the Raman spectrum are affected by F e r m i resonances (105). T h e e v i dence they gave was still qualitative, but was supported by some experiments as w e l l as by the results of previous dynamic calculations on perfect p o l y m e t h y l e n e chains. T h e m a i n concepts are the f o l l o w i n g (see F i g u r e 7). F o r an a l l trans polymethylene c h a i n , the vibrational frequencies l y i n g on the lower portion of the dispersion branch of the C H rocking (Figure 7) have their first overtone levels (—720 X 2 = —1440 c m ) just f a l l i n g close to the fundamental levels of the C H b e n d i n g motions. O n their turn, the first overtones of the levels along the C H dispersion curve occur just i n the frequency region where the fundamental C H stretchings occur (2 x —1440 = —2880 c m ) . W h e n symmetry allows, F e r m i resonance can take place (in such a case many F e r m i resonances) and levels r e p e l each other w i t h mutual b o r r o w i n g of i n t e n sity. T h e issue is complicated further by the fact that polymethylene chains normally crystallize i n an orthorhombic lattice. Intermolecular interactions i n the u n i t c e l l generate n e w levels thus p r o v i d i n g more levels that may participate i n the F e r m i resonances. These concepts were translated i n a more qualitative way b y a mathematical treatment for the F e r m i resonance interactions i n n alkanes restricted to the F e r m i interactions of the first overtones of the C H b e n d i n g modes w i t h the C H stretching fundamentals of a single chain (106). A t the same t i m e , a very similar and parallel dynamic treatment that i n c l u d e d the interactions of C H rocking w i t h b e n d i n g and b e n d i n g w i t h C H stretching for single c h a i n as w e l l as for chains organized i n a t r i d i m e n s i o n a l lattice was reported. Moreover, i n t e n sity calculations also were i n c l u d e d (107, 108). O f particular interest to the latter group was not just the interpretation of the spectrum originating from the perfect, crystalline, a l l trans structure of the polymethylene c h a i n but was the fact that n e w spectral features c o u l d be predicted w h e n the F e r m i resonance scheme of the trans structure is m o d i f i e d by the introduction of gauche conformers. T h i s information may be relevant for a l l researchers interested i n the study of the structure of actual or real p o l y m e t h ylene systems w i t h a partially ordered and a partially eonformationally disordered structure.


2

2

- 1

2

2

- 1

2

2

2




516


29.

ZERBi

Vibrational

Spectroscopy

517

T h e investigation by Abbate et a l . (108) currently is b e i n g fol l o w e d by others (109, 110). Without going into the mathematical details, w e w i l l briefly sketch the state of the art i n the interpretation of the Raman spectrum of polymethylene chains i n the a l l trans structure. 1. W e w i l l first describe a polymethylene c h a i n i n an orthorhombic lattice w i t h two molecules per crystallo graphic u n i t c e l l . N o t only the IR-active doublet of the and B^u species of the C H rocking modes, but also a l l the IR-inactive phonons along the dispersion branch es, ω , generate two phonon states of proper symme try and phase to enter i n F e r m i resonance w i t h the C H b e n d i n g motions near 1400 c m . F i g u r e s 8a and 8b 2

9

2


- 1

a

b

ι

1500

1480

L

ι ll.il»

1420

1440

1460 Frequency, c m

- 1

Figure 8. Fermi resonance between the fundamental of δ CH of species Ag and the first overtone of CH rockings, (a), and Fermi reso nances between the fundamental δ CH of species B and the first over tones of CH rockings (b). 2

2

2

lg

2


518

POLYMER CHARACTERIZATION show the calculated sequence of A and Β modes aris i n g from the C H b e n d i n g fundamentals that entered i n F e r m i resonance w i t h the overtones of the C H rockings, l e n d i n g intensities to them (spectroscopically, i n the double harmonic approximation, intensities of overtone and combination levels should be zero). F i g u r e 9 shows the calculated R a m a n spectrum obtained by convolution of L o r e n z i a n bands of both sequences w e i g h e d by the intensity factor predicted by intensity calculations (108). T h e calculated spectrum agrees w i t h the observed one. T h e peaks near 1420 and 1440 c m form the factor group doublet due to the orthorhombic lattice w h i l e the broad scattering near 1460 c m is only due to the superposition of two phonon levels of a trans c h a i n that have borrowed intensity from the two fundamentals near 1420 and 1440 cm . g

w

2

2

- 1


- 1

- 1

2. T h e same reasoning can be a p p l i e d to the region of the C H stretchings. A single a l l trans isolated p o l y m e t h ylene c h a i n w i l l be considered first for clarity. F i g u r e 10a shows the calculated spectrum (with intensities i n cluded) w h e n intramolecular F e r m i resonance b e t w e e n various C H groups has not yet been s w i t c h e d on. T h e 2

1500

1480

1460 1440 Frequency, c m

1420

-1

Figure 9. Calculated Raman spectrum in the ltôO-cm' range obtained as convolution with Lorenzian bands of the transitions of Figures 8a and 8b. 1


29.

zERBi

519


7-5 »


r

2950

2900 Frequency, c m

2850 -1

Figure 10a. Calculated Raman spectrum in the CH stretching region taking into account the Fermi resonance only within an isolated CH group. 2

only F e r m i resonance is that w h i c h exists w i t h i n the single and isolated C H group for w h i c h the first over tone of the C H b e n d i n g (Αχ) interacts w i t h the symmet ric C H stretching (Αχ), w h i l e the antisymmetric C H stretching (B ) remains unaffected. T h i s resonance was first pointed out b y L a v a l l a y and Sheppard i n 1,1,1,3,3,3hexadeuteropropane (111). I f the intramolecular C H F e r m i resonance is switched o n , the manifold of over tones from C H b e n d i n g modes occurs and spreads just above the C H symmetric stretching. A strong repulsion b e t w e e n the overtone levels and the C H fundamental takes place (see F i g u r e 10b). T h e fundamental moves toward 2850 c m a n d lends intensity to the manifold of overtone levels thus decreasing its apparent intensity. T h e background scattering is raised substantially from 2850 to 2990 c m ' . T h e B C H antisymmetric stretch remains unperturbed a n d floats o n top of the A back ground scattering. 2

2

2

2

x

2

2

2

2

- 1

1

w

2

g

3. I f a t r i d i m e n s i o n a l orthorhombic lattice n o w is consid ered, n e w overtone levels are generated and the density of overtone states i n the C H stretching region increases, a l l possibly i n F e r m i resonance. Calculations show [in agreement w i t h the original idea of Snyder et a l . (105)] that these levels cluster m a i n l y b e l o w and b e t w e e n the peak near 2880 c m and the A peak near 2850 c m thus again m a i n l y affecting the intensity and frequency of the A fundamental. 4. W e w i l l n o w move backward from the perfect structure - 1

g

g


- 1

520



7.5 k

2950

290Ô Frequency, c m

2850 -1

Figure 10b. Calculated Raman spectrum in the CH stretching region for the all trans-polymethylene chain when Fermi resonances are switched on and inserted in the resonance scheme of Figure 10a. toward a more disordered structure. First, the intermolecular interactions of the orthorhombic lattice w i l l be removed b y p l a c i n g the molecule: a, as a guest i n a host lattice of perdeuterated material (104); and b, i n a cage of a clathrate (112). T h e t r i d i m e n s i o n a l order c a n b e d e stroyed b y m a k i n g some sort of smectic structure as i n the α-phase observed for n-nonadecane (J00). W e expect the v a l l e y between the peaks near 2850 and 2880 c m " to become deeper or emptier because i n t e r m o l e c u l a r F e r m i resonances were s w i t c h e d off. T h e A 2 8 5 0 - c m b a n d is expected eventually to shift slightly toward higher frequencies and to regain the i n tensity that it lent to the levels that were removed. T h e m u c h questioned lateral interactions proposed b y G a b e r and Peticolas for phospholipids a n d measured b y the same group b y t a k i n g the ratio of the intensities i n the C H stretching region b y changing the environment of the trans chains (104) finds i n the facts just presented a possible explanation. 5. T h e l o n g trans sequences w i l l n o w be interrupted b y gauche distortions. Because both the C H r o c k i n g and b e n d i n g modes i n gauche conformations occur at some what different frequencies, the population of vibrational levels participating i n the trans interaction scheme de creases a n d the spectral distribution both i n the C H b e n d i n g and C H stretching region is m o d i f i e d . Q u a l i t a tively, the vibrational spectrum should change from that of a f u l l , a l l trans single-chain molecule toward that of a 1

g

-1

2

2

2


29.

521


ZERBi

fully d e c o u p l e d structure, thus t e n d i n g toward the situ ation described by the spectrum of F i g u r e 10. O f partic ular importance is that the 2 8 5 0 - c m peak should move toward higher frequencies close to the always unper turbed b a n d near 2880 c m " and the 2 X C H δ peak near 2930 c m should gain more intensity because it tends to acquire its l o c a l i z e d identity and does not share w i t h the two p h o n o n m a n i f o l d the intensity b o r r o w e d from the C H symmetric stretching. I f the p r e d i c t e d behavior ac tually occurs, the peak near 2930 c m should increase its intensity w h e n gauche conformations are generated w i t h respect to the b a n d near 2850 c m , w h i c h repre sents the a l l trans segments. T h e relative ratio hsso/l2940 then is expected to change, decreasing by increasing the concentration of conformationally distorted non-trans C - C bonds. -1

1

2

- 1

2

- 1


- 1

These predictions were verified i n a series of experiments by W u n d e r and Murajver on several m o d e l systems. L i q u i d n-hexadecane at different temperatures (109), solutions of hexadecane at various concentrations and temperatures (109), polyethylene from the s o l i d to the melt at various temperatures (110), and s o l i d samples of p o l y e t h ylene w i t h varying content of an amorphous substance (108) a l l show the same k i n d of predicted behavior. T h e same fact also was observed for l i q u i d fatty acids at different temperatures (J 13). W u n d e r and co workers (J09, J JO) point out that such a ratio is very sensitive to small changes i n the trans-gauche ratio. It becomes then possible to follow the thermal conformational evolution of a p o l y m e t h y l e n i c system. A parallel, but not yet quantitatively f o l l o w e d , behavior is observed for the Raman l i n e near 1460 c m , w h i c h again is characteristic of l o n g sequences of trans structures and decreases w h e n the population of trans conformers decreases. O n e of the advantages of the method just proposed is that it can be adopted for numerous polymethylene systems i n a l l p h y s i c a l c o n d i tions and phases. Because C H stretching modes are strong scatterers and scatter i n a region generally free from overlapping, the method can be adopted for a large variety of substances i n extreme e x p e r i mental conditions. F r o m just the quick observation of a spectrum of a l i q u i d n-alkane or a p h o s p h o l i p i d system, we learn that a fraction of the substance is i n the trans conformation a n d another is i n a distorted structure whose relative ratio changes according to temperature, solvent, c h e m i c a l or physical treatment, and for other reasons. T h e p r o b l e m still u n s o l v e d is the precise accounting of the Raman intensities for an absolute or relative quantitative determination of the populations of conformers. A n interesting attempt was made (109) to relate the observed ratio of - 1



522


the intensities I2930/J2840 w i t h the translgauche population as determ i n e d from the statistical isomeric state m o d e l . Because our proposed interpretation of the spectral changes observed i n the Raman spectra as just discussed may f i n d some i m m e d i ate applications i n various fields, a w o r d of caution needs to be said on these matters. T h e observed changes of the Raman spectra i n the C H stretching region of systems containing n-alkane residues i n going from ordered to disordered phases may be the result of a combination of several phenomena, one of w h i c h may be the simple F e r m i resonance interactions just discussed. M o r e complicated phenomena may be envisaged (such as intermolecular effects, b a n d shape changes, changes i n anharmonicity of the C H bonds, changes i n harmonic diagonal or/end off diagonal force constants w i t h torsional angles or w i t h the environment, etc.). T o our knowledge, neither theory nor experiments have shown whether these additional phenomena concomitantly exist and w h i c h one plays the dominant role (if any) i n m a k i n g up the observed spectrum. M o r e work is n e e d e d on this i m portant issue. Lattices with Tridimensional

Disorder

T h e appearance and disappearance of a t r i d i m e n s i o n a l order i n crystalline materials and specifically of crystalline polymethylene molecules received little attention i n the past, although, as shown here, most of the attention was focused on intramolecular order and disorder. F o r polymers, the existence of the factor group splitting (noted earlier) helps i n the precise definition of crystallinity bands and informs us that molecular chains are packed w i t h a t r i d i m e n s i o n a l order. T h e study of the extent of intramolecular c o u p l i n g i n p o l y methylene chains was carried out extensively by K r i m m and coworkers i n an attempt to understand the organization of molecular chains i n single crystals grown from solution or from the melt (45, 49). T h e spectra of isotopically d i l u t e d crystals were used as a tool and very small frequency changes on isotopic d i l u t i o n were rationalized i n terms of the organization and regular clustering of c h a i n stems w i t h i n the crystal (49). T h i s issue is at present a key p r o b l e m i n p o l y m e r morphology and p o l y m e r solid-state physics (49). As already discussed, the factor group splitting arises because of the c o u p l i n g of the natural frequency of the oscillators of the isolated chain w i t h the i d e n t i c a l ones of the chains i n the surrounding crystal. T h e possibility of r e m o v i n g the intermolecular c o u p l i n g of a g i v e n vibrational l e v e l , w h i l e leaving some others practically unperturbed recently was considered (114, 115). I n a m i x e d crystal of n alkanes where the two components differ slightly i n length, f u l l m i s c i b i l i t y i n solids without any phase separation was shown to occur for


29.

ZERBi


523

n-alkanes of the type C + C (116). If η is sufficiently large, the sequences of normal modes l a b e l e d w i t h the index j along the disper sion curves of the infinite polymethylene c h a i n generally show differ ent frequencies for the two molecules. T h e sequences of the C H rocking modes that show the largest dispersion (all l y i n g approxi mately on the ω branch of polyethylene) n o w w i l l be considered. A l t h o u g h for large j values the frequency differences are large for the two molecules (tending toward small j and to the l i m i t i n g j = 0 modes), the frequency mismatch of the two frequencies decreases (Figure 11). F o r chains longer than ~ 5 C H units, the in-phase j = 0 mode always occurs near 720 c m for all trans planar n-hydrocarbons. I n a rough approximation, one may first expect that i n a m i x e d crystal of n-alkanes of the type just mentioned, where molecules can pack w i t h h i g h intermolecular order, the splitting of the C H normal modes depends on the frequency mismatch. Singlets w i l l appear for modes w i t h many nodes (large j ) , and splittings may appear for normal modes w i t h a s m a l l n u m b e r of nodes (small j ) t e n d i n g to the w e l l k n o w n splitting for the l i m i t i n g in-phase motion (j = 0) near 720 c m . These expectations were v e r i f i e d i n the I R spectra of several m i x e d crystals of n-alkanes w i t h lengths slightly differing (114, 115), where practically only the l i m i t i n g r o c k i n g mode near 725 c m shows split ting. T h e I R spectrum of m i x e d crystals of n-alkanes w i t h different lengths then can become a tool for the study of molecular motions i n the s o l i d state. I n d e e d , the concepts just described and the e x p e r i ments that were d e r i v e d a l l o w e d detailed information to be gained on the capability of hydrocarbon molecules to migrate w i t h i n a crystal and from one crystal to another w h e n p l a c e d i n p h y s i c a l contact but separated by w e l l - d e f i n e d crystal boundaries. T h e p h e n o m e n o n first reported by Ungar a n d K e l l e r (117) can be f o l l o w e d i n detail (114,115, 118) w i t h the spectroscopic techniques described i n this chapter. I n a mechanical mixture of crystalline c o m p o u n d A + crystalline c o m p o u n d Β (A and Β b e i n g two n-alkanes of general formula A = C H and Β = Ο η Η , η either even or odd); η must be some integer such that the structure w i l l be orthorhombic or m o n o c l i n i c (119). T h e spectrum of the mechanical mixture A + Β shows through out the w h o l e spectrum the factor group splitting of each c o m p o u n d independent from the other, each one h a v i n g its o w n orthorhombic structure, thus a l l o w i n g normal i n - and out-of-phase c o u p l i n g of each vibrational l e v e l . B y annealing at different temperatures b e l o w , u p to, and above the s o l i d - s o l i d phase transition, each doublet of the modes at large j becomes a singlet, although the region near 720 c m " still shows the normal doublet. T h e disappearance of the doublet on an nealing of the mechanical mixture occurs because molecules of type A n

n±2

2

9

2


- 1

2

- 1

- 1

n

2 n + 2

± 2

2 ( η ± 2 ) + 2

1


524

POLYMER CHARACTERIZATION 950h R17

«15

cm"


900

«13

850 R13

Ru

800 R11

750

R

700 L

Ri,R7

R

1» 7

'23

'21

Figure 11. Scheme of the observed frequency mismatch for a few components of CH rocking for C ^ H ^ ana C2 É . Factor group splitting was removed from the figure for simplicity. [Values of j + 1 are reported here because fixed boundary conditions are imposed for both ends ( see Ref. 3).] 2

3

4g



29.

ZERBi

525


migrate from their o w n crystal into the crystal of C o m p o u n d Β and vice versa. Islands of A into Β and of Β into A then are formed. A t the e n d of the process, the c o m p o u n d obtained is just that w h i c h can be obtained i n d e p e n d e n t l y b y co-crystallization of A and Β from a suitable solvent. T h e rate of migration and the energetics of the p h e n o m e n o n were studied (114, 115,118) u s i n g the vibrational spectrum and calorimetric measurements. M o r e o v e r , u s i n g many of the spectroscopic signals d e r i v e d from the study of the dynamics of disordered materials dis cussed previously, one can then state that molecules migrate at larger distances i n proportion to their length a n d at different rates according to the temperature and time of annealing. M o l e c u l e s migrate substan tially as r i g i d bodies without i n t r o d u c i n g noticeable conformational defects. W h e n m i x e d crystals are formed, the geometrical mismatch of chain lengths w i t h i n the crystal forces a few molecules to tilt their heads w i t h GT or TG structure at either end. T h e p h e n o m e n o n , however, is not restricted to molecules w i t h u n e q u a l length. Self-diffusion or self-mixing b e t w e e n molecules of the same length was observed and studied by u s i n g solid crystalline C H 4 and its perdeuteroderivative C D . D e t a i l s of the p h e n o m e n o n as studied from spectroscopy (114, 115) and a discussion o n its possible origin are g i v e n elsewhere (118). W i t h the techniques presented i n this chapter, it becomes possi ble to study a p h e n o m e n o n that gives a positive answer to the still open question (7) of whether d u r i n g annealing of p o l y m e r i c solids a physical transport of matter across crystal boundaries occurs. A t least for polymethylene chains i n the s o l i d state, our studies seem to prove that matter is transported across crystal boundaries on annealing and that chain migration can take place i n solids also without the existence of amorphous or conformationally irregular domains i n the material through w h i c h chains s h o u l d move (or do move) supposedly w i t h a snake-like motion (120). T h e most relevant information d e r i v e d from these studies is that hydrocarbon chains move w i t h i n a n d across crys tallites substantially as r i g i d bodies. Snake-like h o p p i n g from site to site w i t h i n the b u l k should not occur i n this case because molecules are crystalline and no sizable conformational deformations were yet detected spectroscopically at temperatures just b e l o w the m e l t i n g point where migration is very fast. 3 6

7

3 6

7 4

Vibrational Intensities: A New Future for IR and Spectroscopy of Polymers

Raman

Recent basic studies of vibrational spectroscopy tackled the p r o b l e m of p r e d i c t i n g the absorption coefficients i n the I R region or the scattering power i n the Raman region for various molecular species (121 ). U n t i l a few years ago, the study of vibrational intensities



526


was conducted by a few specialists of molecular spectroscopy w h o dealt w i t h the basic measurements o n several m o d e l molecules a n d discussed the various theoretical techniques for accounting for the observed spectrum. Recently, efforts have b e e n made, a n d are b e i n g made, to apply these techniques to molecules of practical importance for d e r i v i n g quantitative information on molecular species or molecular confor mations of more general interest to chemists. T h e attempts made by a few researchers i n the f i e l d of vibrational (IR and Raman) intensities just parallel the efforts made by numerous groups d u r i n g the last thirty years to calculate the vibrational frequencies from force constants. Vibrational intensities are related to the electrical properties of molecules (change i n the total electric d i p o l e moment or polarizability d u r i n g the vibrational motion). I f a m o d e l of such electrical properties can be found, it can be made parametric u s i n g the experimentally observed intensities. S u c h parameters (called intensity parameters) may be used to predict the u n k n o w n intensities of another similar molecule. A m o n g the various models proposed and tested (122, 123), the valence electrooptical scheme (124) seems w e l l suited for intensity calculations. T h e corresponding parameters are c a l l e d I R or R a m a n electrooptical parameters (eop). C o n s i d e r a b l e effort has b e e n made recently to develop such a detailed m o d e l and to carry out a leastsquares fitting of the eop values to the experimental intensities of many molecules. Unfortunately, the n u m b e r of molecules for w h i c h reliable ex perimental measurements are available is still very l i m i t e d , b e i n g acceptable for I R spectroscopy, but extremely l i m i t e d for R a m a n spectroscopy. F o r η-hydrocarbons, I R measurements are available for methane, ethane, propane, a n d several de utero derivatives i n the gas phase. D a t a on l i q u i d η-butane and n-pentane were collected recently (125). S o l i d η-hydrocarbons from η-butane to η-heptane were mea sured by Snyder (126). M o s t of these experimental data were i n c l u d e d i n least-squares fittings (127) of a set of eop values after the necessary analyses of the data i n terms of the various theoretical technicalities were carried out. These eop values accurately predict the spectra of n-decane (128), of polyethylene, and of perdeuteropolyethylene (127, 129). Analogously, the R a m a n spectra of polyethylene a n d perdeutero polyethylene were p r e d i c t e d accurately w i t h the eop values d e r i v e d from smaller n-hydrocarbons (28, 29, 129). Attempts to j u m p ahead and use the eop values obtained for the p r e d i c t i o n of the absorption coefficients of defects i n p o l y m e t h y l e n e molecules are still tentative, but nevertheless, are of exploratory relevance. A n example of the ap-


29.

ZERBi


527

plication of intensity calculations i n conformationally disordered p o l y methylene chains is the p r e d i c t i o n of the intensity pattern for the group of resonance modes near 1350 c m generated b y the GGTGG defect i n a c y c l i c C H m o l e c u l e as discussed (96) previously. F i g u r e 12 shows the pattern calculated from group intensities compared w i t h the experimental spectrum. T h e agreement is very good a n d shows again the usefulness of these k i n d s of work. F o r the vibrational spectra of perfect, single-chain p o l y e t h y l e n e , intensity calculations have h e l p e d i n the detailed interpretation of the R a m a n spectrum i n the C H b e n d i n g a n d stretching regions w h e r e F e r m i resonances occur (J08) as discussed earlier. F i n a l l y , for the Raman spectrum of perdeuteropolyethylene, i n tensity calculations were decisive i n s o l v i n g the controversy of the vibrational pattern near 980 c m . T h e pattern near 980 c m can be described (28, 29) as a strong doublet w i t h a very small and almost unnoticeable shoulder on the high-frequency side of the strongest peak. T h e peak near 970 c m was assigned to the C D deformation mode, and the peak near 990 c m was taken to be the C D r o c k i n g mode. Calculations indicate that the C D r o c k i n g mode contributes an - 1

3 4

6 8


2

- 1

- 1

- 1

2

- 1

2

2

ο ζ < m (Ζ ο CO m

0.018

1380

0.066

0169

1300

1340

cm"

1260

Figure 12. Comparison between the observed IR spectrum of cyclic CH solid in the 1260-1380-cm~ region and the intensities calcu lated for the CH wagging resonance modes of the G G T G G defect. Numbers on the bottom refer to calculated intensities. No simulation of the spectrum was made because the spectrum is overlapped with other bands. 34

1

68

2


528


extremely weak scattering i n this region, a n d the relative integrated area of the strong doublet approaches the value calculated for the C D b e n d i n g motion. A s already suggested b y others (130), the w h o l e d o u blet ( 9 8 9 - 9 7 0 c m " ) must then be assigned to a factor group splitting of the C D deformation mode w h i l e the very weak shoulder is instead the scattering due to the C D rocking mode. G e n e r a l l y , intensity studies also can h e l p greatly i n the characterization of p o l y m e r i c materials i n the perfect or disordered state i f the spectral intensities of many more m o d e l molecules are measured carefully. W e hope to convince more people to carry out such measurements. 2

1

2


2

Literature Cited 1. Krimm, S. Adv. Polym. Sci. 1960, 2, 51. 2. Zerbi, G. In "Applied Spectroscopy Reviews"; Brame, E. G., Ed.; Dekker: New York, 1969; Vol. 2. 3. Zbinden, R. "Infrared Spectroscopy of High Polymers"; Academic: New York, 1964. 4. Koenig, J. L. In "Applied Spectroscopy Reviews"; Brame, E .G.,Ed.; Dekker: New York, 1971; Vol. 4. 5. Koenig,J.L.J.Polym. Sci., Part D 1972, 59. 6. De Santis, P.; Giglio, E.; Liquori, A. M.; Ripamonti, A.J.Polym. Sci., Part A 1963, 1, 1383. 7. Wünderlich, B. "Macromolecular Physics"; Academic: New York, 1976, Vols. 1 and 2. 8. Wilson, E. B.; Decius, J. C.; Cross, P. C. "Molecular Vibrations"; McGraw-Hill: New York, 1955. 9. Zerbi, G. In "Modem Methods in Vibrational Spectroscopy"; Orville-Thomas, W.J.;Barnes, A.J.,Eds.; Elsevier: Amsterdam, 1977. 10. Shimanouchi, T. In "Physical Chemistry, An Advanced Treatise"; Eyring, H.; Anderson, D.; Jost, W.; Eds.; Academic: New York, 1970; Vol. 4. 11. Califano, S. "Vibrational States"; Wiley: London, 1976. 12. Schachtschneider, J. H.; Snyder, R. G. Spectrochim. Acta 1963, 19, 17. 13. Snyder, R. G.; Zerbi, G. Spectrochim. Acta 1967, 23A, 391. 14. Piseri, L.; Zerbi, G.J.Chem. Phys. 1968, 48, 3561. 15. Zerbi, G.; Piseri,L.J.Chem. Phys. 1968, 49, 3840. 16. Higgs, P. W. Proc. R. Soc. (London) 1953, A220, 472. 17. Piseri, L.; Zerbi, G.J.Mol. Spectrosc. 1968, 26, 254. 18. Tasumi, M.; Shimanouchi, T.; Miyazawa, T. J. Mol. Spectrosc. 1962, 9, 261. 19. Piseri, L.; Powell, B. M.; Dolling, G.J.Chem. Phys. 1973, 57, 158. 20. Chiang, C. K.; Drug, M. A.; Gau, S. C.; Heeger, A. J.; Shirakawa, H.; Louis, E .J.;Mac Diarmid, A. G.; Park, Y. W.J.Am. Chem. Soc. 1978, 100, 1019. 21. Gavin, R. M.; Rice, S. A.J.Chem. Phys. 1971, 55, 2675. 22. Kakitani, T. Prog. Theor. Phys. 1974, 51, 656. 23. Zerbi, G.; Zannoni, G. Chem. Phys. Lett. 1982, 1982, 50. 24. Zannoni, G.; Zerbi, G. Chem. Phys. Lett. 1982, 1982, 55. 25. Suhay, S.J.Chem. Phys. 1980, 73, 3843. 26. Peraldo, M.; Cambini, M. Spectrochim. Acta 1965, 21, 1509. 27. White,J.W. In "Structural Studies of Macromolecules by Spectroscopic Methods"; Irvin, K.J.,Ed.; Wiley: London, 1976.



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