Use of Raman Spectroscopy in Characterizing the Structure and

5

Downloaded by UNIV LAVAL on July 14, 2016 | http://pubs.acs.org Publication Date: May 5, 1993 | doi: 10.1021/ba-1993-0236.ch005

Use of Raman Spectroscopy in Characterizing the Structure and Properties of Crystalline Polymers Leo Mandelkern and Rufina G . Alamo Department of Chemistry and Institute of Molecular Biophysics, Florida State University, Tallahassee, FL 32306

An overview of some of the applications of Raman spectroscopy to the understanding of the structure of crystalline, flexible chain polymers is presented. Selected physical properties are discussed in terms of these structural parameters. The major emphasis is on the polyethylenes, but the relationships between structure and properties are easily generalized to other polymer systems. The three main spectral regions of interest in the study of the polyethylenes are the 5-50-cm longitudinal acoustic mode (LAM) region, the DLAM region in the vicinity of about 200 cm , and the 900-1500-cm internal mode region. -1

-1

-1

JALN OVERVIEW OF T H E APPLICATIONS OF

RAMAN SPECTROSCOPY to the under

standing o f the structure o f crystalline, flexible c h a i n polymers is given i n this chapter. T h e s e structural parameters

are t h e n related to selected physical

properties. Polyethylenes are u s e d as the vehicle for o u r discussion for several reasons: These polymers have a richness o f spectroscopic i n f o r m a t i o n a n d have admirably served as m o d e l systems for crystalhne polymers. W e

find

that the key structural factors that are f o u n d i n the polyethylenes are also present i n other crystalline systems. A basic understanding o f the thermodynamics o f fusion o f crystalline polymers as w e l l as their crystallization kinetics is c r u c i a l to understanding the crystallization behavior o f polymers (J). H o w e v e r , to understand the p r o p e r ties that actually develop i n a system, it is necessary to identify a n d quantify 0065-2393/93/0236-0157$09.50/0 © 1993 American Chemical Society

Urban and Craver; Structure-Property Relations in Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1993.

158

STRUCTURE-PROPERTY RELATIONS IN POLYMERS


the k e y structural quantities that describe the crystalline state because polymers c a n only crystallize i n a reasonable time scale at temperatures w e l l b e l o w their m e l t i n g points. F o r kinetic reasons, the crystallization process is very rarely o r never close to completion. F o r homopolymers, the level o f crystalhnity that c a n b e attained ranges f r o m 40 to greater than 9 0 % d e p e n d i n g o n the described nature o f the repeating unit, the molecular weight, a n d the crystallization conditions. H e n c e , a n o n e q u i h b r i u m state invariably develops. T h e resulting system is thus polycrystalline a n d m o r p h o logically complex. Extensive experimental studies have identified some o f the key i n d e p e n dent structural variables that describe the crystalline state (2-4). These variables are the degree o f crystallinity, the structure o f the residual noncrys talline region, the crystallite thickness distribution, the structure a n d relative amount o f the interfacial region, details o f the crystallite structure b e y o n d that o f the lamellar habit, a n d the supermolecular structure (the arrangement of the crystallites). These quantities can b e v a r i e d over w i d e limits b y control of molecular weight a n d crystallization conditions ( 2 , 4-6). A l t h o u g h R a m a n spectroscopy cannot identify a n d quantify all o f these structural variables, this technique c a n make a significant contribution. T h e spectral regions that c a n be used to describe specific aspects o f the crystallization behavior have b e e n s u m m a r i z e d ( 3 ) , b u t f o r present purposes, it w i l l b e h e l p f u l to briefly describe t h e m once again. T h e r e are three m a i n spectral regions that give structural i n f o r m a t i o n about the crystalline state o f the polyethylenes. T h e internal modes, w h i c h give quantitative information w i t h respect to the elements o f phase struc tures, are i n the region o f approximately 9 0 0 - 1 5 0 0 c m (7-9). T h e longitu d i n a l acoustic m o d e ( L A M ) , w h i c h gives the o r d e r e d sequence length distri b u t i o n , lies i n the range o f about 5 - 5 0 c m (10-17). T h e disordered L A M ( D L A M ) is i n the region o f 200 c m a n d gives a measure o f the long-range conformational disorder (18-21). - 1

- 1

- 1

Structure Phase Structure. A quantitative description o f the phase structure o f semicrystalline polymers is o f p r i m a r y importance i n understanding p r o p e r ties ( 2 , 4). T h r e e distinct structural regions are involved: the o r d e r e d crystalline region that, f o r polyethylene, represents the orthorhombic unit cell, the isotropic, fiquidlike o r disordered region, a n d the interphase that is c o m p r i s e d o f chain units that connect these t w o conformationally very different structural regions. I n p o l y m e r crystallization, a chain can traverse all three regions or, i n some cases, c a n b e restricted to the crystalline a n d interfacial regions. T h e participation o f a given molecule i n all three regions is a u n i q u e a n d important feature o f p o l y m e r crystallization. I n 1949, F l o r y


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159


p o i n t e d out that a demarcation between the o r d e r e d crystalline region a n d the disordered l i q u i d l i k e region is not sharply d e f i n e d because for most polymers the flux o f chains that emerges f r o m the crystallite cannot be i m m e d i a t e l y dissipated i n the liquidlike region (22). Consequently, an inter phase or interfacial region develops that involves a partially o r d e r e d set o f chain units. Theoretical analyses o f the interfacial structure are i n general agreement w i t h one another ( 2 3 - 2 9 ) . It is apparent f r o m these studies that the detailed structure and extent o f the interphase are specific to a given polymer. M a n y different kinds o f measurement can b e u s e d to quantitatively describe the phase structure. U t i l i z i n g the p r i n c i p l e o f the additivity o f a particular property o f the p u r e l i q u i d a n d crystalline states, measurements o f the density a n d enthalpy o f fusion can be used to determine the degree o f crystallinity. W e shall designate these levels o f crystallinity as (1 — X) a n d (1 — λ ) , respectively, w h e r e λ is the noncrystalline fraction, d is the density, a n d Δ Η is the enthalpy o f fusion. W i d e - a n d small-angle X - r a y diffraction, as w e l l as several different types o f N M R measurements, can also be profitably used for this purpose. d

Δ Η

Analysis o f the R a m a n internal modes for the purpose o f quantitative elucidation o f the phase structure was originally given b y Strobl a n d H a g e d o r n (7). T h e core degree o f crystallinity, a , is obtained b y this m e t h o d because the only contribution to this quantity comes f r o m the structure o f the o r t h o r h o m b i c unit cell. a is calculated f r o m the C H b e n d i n g b a n d at 1416 cm a n d is the component o f this vibration that is split b y the crystal field. T h e degree o f l i q u i d l i k e material, a , is obtained f r o m the twisting region at 1303 c m . T h e total integrated intensity o f the twisting region, 1 2 9 5 - 1 3 0 3 c m " , is independent o f the phase structure (7, 8). M e t h o d s o f analyzing the spectra were given i n detail b y Strobl a n d H a g e d o r n (7), as w e l l as i n reports f r o m this laboratory (8, 9). T h e analysis o f a large amount o f experimental data y i e l d e d a significant finding: a + a 1 ( 2 , 8). T h e spectra o f the completely l i q u i d l i k e a n d the completely crystalline polymers cannot be superposed a n d p r o p o r t i o n e d to represent an observed spectra; a partially ordered, p r i m a r i l y trans, anisotropic region must also be i n c l u d e d . T h i s contribution has b e e n defined as the interfacial region a . c

c

2

- 1

a

- 1

1

a

c

b

F i g u r e 1 is a plot o f the core crystallinity a against (1 — λ ) for an extensive set o f data o f linear polyethylenes, b r a n c h e d polyethylenes, a n d r a n d o m ethylene copolymers. T h e data points for the linear polymers fall o n the 45° straight line over the very large range o f 0 . 4 - 0 . 9 i n the level o f crystallinity. W e can therefore conclude that the core crystallinity and (1 — λ) are identical to one another. F o r the b r a n c h e d polymers a n d copoly mers, a is about 5 % less than (1 — λ ) . T h i s small disparity can be attributed to the b r o a d m e l t i n g range of the structurally irregular chains. (1 — λ ) includes the contribution o f a small amount o f crystallinity that has already disappeared at r o o m temperature. Because a is measured at a m b i c

Δ Η

Δ Η

c

Δ Η

Δ Η

c



160


(Ι-Χ)ΔΗ

Figure 1. Plot of degree of crystallinity, a , as determined from the Raman internal modes, against (l — λ) for linear (—) and branched (- -) polyethylene and ethylene copolymers. (Reproduced with permission from reference 2. Copyright 1985 Society of Polymer Science, Japan.) c

ΔΗ

ent temperature, analysis.

this contribution is not i n c l u d e d i n the internal

mode

I n contrast to the concordance between these t w o methods, there is a significant quantitative difference between (1 — X ) a n d (1 — λ ) . A s is illustrated i n F i g u r e 2, (1 — X ) for linear polyethylene is always greater than (1 — λ ) . A similar result is also observed w i t h structurally irregular copoly mers ( 2 ) . d

Δ Η

d

Δ Η

T h e basis f o r the discrepancy between (1 — X ) a n d (1 — λ ) c a n be f o u n d i n the plot o f F i g u r e 3, w h e r e (1 — X ) is plotted against the s u m ( a + a ) . A l l the data fall quite w e l l o n the 4 5 ° straight line, w h i c h indicates a one-to-one correspondence between the t w o quantities. Because c t mea sures the core crystallinity a n d a measures the interfacial content, a self-consistent a n d physically satisfying interpretation o f these results is f o u n d i n the conclusion that the density measures b o t h the core crystallinity a n d the partially o r d e r e d anisotropic interfacial region. O n the other hand, the enthalpy o f fusion measures only the core crystallinity. d

Δ

Η

d

c

b

c

b



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MANDELKERN & ALAMO


161

Figure 2. Plot of degree of crystallinity obtained from density, (l — K) , against the value obtained from the enthalpy of fusion, (l — λ) , for linear polyethylene. (Reproduced with permission from reference 2. Copyright 1985 Society of Polymer Science, Japan.) d

Διι

A l t h o u g h the degree o f crystalhnity is a well-established quantitative concept, different methods o f measurement y i e l d different results. These methods have the same functional dependence w i t h respect to molecular constitution a n d crystallization conditions. H o w e v e r , there

are small b u t

significant differences that are due to the contributions f r o m the different elements o f phase structure. F o r example, M a g i l l a n d co-workers have f o u n d that for poly(tetramethyl-p-silphenylene siloxane), (1 — X ) is slightly higher d

than (1 - λ )

Δ Η

(30).

Studies w i t h other polymers, as w e l l as other methods o f measurement, give similar results a n d give strong support to the quantitative analysis o f the R a m a n internal modes that has b e e n

given for the polyethylenes. F o r

example, using the methods o f h i g h resolution solid-state carbon-13 Ν M R , it is possible to decompose the T

l

relaxation decay curves into three c o m p o

nents that correspond to the crystalline, liquidlike, a n d interfacial regions (31, 32). A plot o f a , c

d e t e r m i n e d f r o m the R a m a n internal modes, against the

degree o f crystallinity d e t e r m i n e d b y Ν M R is given i n F i g u r e 4 f o r a variety



162


a + ab c

Figure 3. Plot of (l — k) against the sum of (a + ct )for linear and branched polyethylenes and ethylene copolymers. [Reproduced with permission from reference 2. Copyright 1985 Society of Polymer Science, Japan.) d

c

b

o f polyethylene samples. W e find a one-to-one correspondence between the two methods similar to that f o u n d between a a n d (1 — λ ) . F r o m the decomposition o f the T magnetization decay curves o f a set o f polyethylenes the interfacial components w e r e f o u n d to b e quantitatively identical to a obtained f r o m the R a m a n internal modes (31). c

Δ Η

x

b

T h e line shapes o f the carbon-13 resonances i n the solid state o f polyethylene ( 3 3 ) , polypropylene (34), a n d poly(tetramethylene oxide) ( 3 5 ) have b e e n d e c o m p o s e d into three components that correspond to the c o m p o nents u n d e r discussion. T h e results are s u m m a r i z e d i n T a b l e I. Several important features are s u m m a r i z e d i n this table. T h e fractions o f the interfa cial regions are quite similar for the three polymers; a l l are i n the range 0 . 1 6 - 0 . 3 0 . W e c a n conclude, therefore, that c h a i n molecules, i n general, are characterized b y significant interfacial contents. F o r each o f the two p o l y (propylene) specimens (the same sample crystallized i n t w o different



5.

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163


Figure 4. Plot of Raman-determined degree of crystallinity a against NMR degree of crystallinity OS MR- (Reproduced with permission from reference 31. Copyright 1983 John Wiley & Sons, Inc.) c

N

Table I. Phase Structures from Carbon-13 Line Shapes Polymer

d -

Poly (ethylene) Poly (propylene) Sample A Sample Β Poly(tetramethylene oxide) ·Δ-(1-λ) -(1-λ) Directly observed. Λ

Α Η

*)

d

d -

(l-\)

λ)

ΔΗ

0.454 0.540

b

a

0.70

0.761 0.718 0.828

A

N M R

0.480 0.570 0.60

0.26 0.29

b(NMR)

a

Réf.

0.16

34

0.30 0.27 0.22

34 34 35

.

ways) there is very g o o d agreement between (1 — λ )

Δ

d

a n d (1 — X )

Η

M o r e o v e r , the difference between (1 — X ) a n d (1 — λ )

Δ

Η

N M R

.

corresponds, i n

each case, to the interfacial fraction d e d u c e d f r o m N M R . T h u s , the N M R results f o r polypropylene parallel the R a m a n internal m o d e analysis o f the polyethylenes. Studies w i t h other polymers w o u l d further clarify the general ity o f these results. T h e analysis o f the b r o a d line p r o t o n N M R resonance also requires a three-component phase structure f o r its d e c o m p o s i t i o n ( 3 6 ) . T h e magnitude


164


o f the interfacial content (and its molecular weight dependence) is similar to that d e d u c e d f r o m the R a m a n analysis ( 3 7 ) . T h e interfacial content o f linear polyethylene is very dependent o n chain length for rapidly crystallized molecular-weight fractions (38).

A t low molecu

lar weights, the interfacial content is relatively s m a l l — a b o u t 5 % . H o w e v e r , an appreciable interfacial content, o n the order o f 1 5 - 2 0 % , is observed at m u c h higher molecular weights. T h e change

i n interfacial content

with

molecular weight parallels the variation i n the interfacial free energy that is associated w i t h the basal plane o f the lamellae. This significant p r o p o r t i o n o f interphase that is d e d u c e d f r o m the analysis o f the experimental data has important ramifications i n understanding certain macroscopic properties. Downloaded by UNIV LAVAL on July 14, 2016 | http://pubs.acs.org Publication Date: May 5, 1993 | doi: 10.1021/ba-1993-0236.ch005

T h e phase structure o f r a n d o m copolymers depends not only o n the molecular weight, but also o n the co-unit content and, i n certain specific cases, o n the c h e m i c a l nature o f the co-unit itself. T h e introduction o f noncrystallizing co-units into the chain leads to a r a p i d a n d c o n t i n u i n g decrease i n the crystallinity level w i t h increasing side-group content.

The

level o f the core crystallinity varies f r o m about 4 8 % for 0.5 m o l % of branches to about 7 % for 6 m o l % o f b r a n c h points. T h e c h e m i c a l nature of a specific b r a n c h type has virtually no influence o n the crystallinity level as l o n g as it does not enter the lattice. I n cases w h e r e the branches enter the lattice, such as C H a n d C l , higher levels o f crystallinity w i l l be observed. T h e changes i n 3

a

c

w i t h molecular weight, at a fixed co-unit content, follow a pattern that is

similar to homopolymers, although the level of crystallinity is m u c h r e d u c e d (39). T h e interfacial content o f r a n d o m copolymers at a fixed co-unit content is independent o f molecular weight. T h i s conclusion is evidenced b y studies o f a set o f ethylene-hexene

copolymers that have a most probable molecular-

weight distribution w i t h b r a n c h contents that range f r o m 1.2 to 1.7 m o l % (39)

a n d a set o f hydrogenated polybutadienes (randomly ethyl-branched

ethylene copolymers) o f slightly higher b r a n c h content (39). I n contrast, the interfacial content o f homopolymers increases substantially w i t h molecular weight (8).

T h e interfacial content o f r a n d o m copolymers is a strong m o n o -

tonically increasing function o f the co-unit content. T h e value o f a proaches 1 5 - 2 0 % , w h i l e at the same time a

c

is r e d u c e d to as l o w as

b

ap 5-8%

(40) . Therefore, the interfacial region can represent an appreciable p o r t i o n o f the total crystallite structure. M o r e extensive experiments are necessary to determine whether details o f the interfacial structure change as the c h e m i c a l nature o f the co-unit is varied. Analysis o f the R a m a n internal modes has p i o n e e r e d the

quantitative

determination o f the p r o p o r t i o n o f interphase i n crystalline polymers: T h e existence a n d importance o f an interfacial region is n o w w e l l established. Analysis of this spectral region also allows for the quantitative establishment o f the complete phase structure. O t h e r experimental methods, such as N M R


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165

previously cited, have quantitatively c o n f i r m e d the results obtained b y R a m a n spectroscopy for polyethylene and other crystalline polymers.

Ordered Sequence Length. T h e o r d e r e d sequence length can b e obtained f r o m the R a m a n low-frequency longitudinal acoustic vibrational m o d e ( L A M ) b y using the relation given b y S h i m a n o u c h i and co-workers ( 1 0 , 11). T h e appropriate relation can b e expressed as


Δν =

m

(E^

— I— I 2cL

1/2

(1)

\ρ /

w

H e r e Δ ν is the mode frequency, m is the m o d e order (m = 1, 3 , 5 , . . . ), c is the velocity o f light, ρ is the density o f the vibrating sequence, a n d Ε is the Young's modulus i n the chain direction. T h e o r d e r e d sequence length L is thus inversely related to the m o d e frequency. T h i s length can b e related, i n a straightforward manner, to the lamellar thickness. T h e L A M b a n d has b e e n observed i n many crystalline polymers a n d has b e e n w i d e l y used i n studying their structure (12-17, 41) . W e focus o u r attention o n the use o f this b a n d i n studying the structure of the polyethylenes and the n-alkanes. A n abundance o f experimental data is available for these systems. H o w e v e r , before analyzing experimental data, the question o f the direct applicability a n d validity o f e q 1 must b e addressed. E q u a t i o n 1 was d e r i v e d o n the basis o f a u n i f o r m elastic r o d w i t h the n-alkane as the p r o p e r molecular analogue (10, 11). T h e applicability o f e q 1 to real p o l y m e r systems can thus be p r o p e r l y questioned (41). F o r crystalline polymers, w h i c h characteristically possess a lamellar morphology, there has b e e n the concern that the noncrystalline units attached to the crystalline sequences o r stems w i l l require serious modification o f e q 1. T h e n-alkanes themselves, i n either extended o r f o l d e d forms (42, 43), also have a lamellar habit. I n this case, the question arises as to what influence the molecular e n d groups have o n e q 1 a n d if, i n fact, the o r d e r e d sequence lengths c a n b e obtained f r o m the observed L A M (31, 44). O n e matter o f immediate concern f o r any system (the polyethylenes a n d n-alkanes, i n particular) is the correct value o f E. F o r polyethylene the values used range f r o m Ε = 2.9 Χ 1 0 d y n e s / c m (14) to 3.6 Χ 1 0 d y n e s / c m (10, 11). Because the modulus enters e q 1 as the square root, the uncertainty 1 2

2

1 2

2

i n L w i l l b e 5 % f o r this range o f Ε values. I n some applications, this uncertainty w i l l not b e o f serious concern, whereas i n others it w i l l b e important (44). T h e r e are at least two methods b y w h i c h to test the validity o f e q 1 a n d to assess the conditions o f molecular a n d crystallite structure u n d e r w h i c h corrections, i f any, are necessary. O n e m e t h o d is to measure the m o d e orders and see i f the p r o p e r frequency ratio is observed. T h e other m e t h o d is to


166


measure the crystallite thickness b y a n independent m e t h o d a n d relate it to the o r d e r e d sequence length obtained b y the R a m a n L A M . T h e analysis o f the

small-angle X - r a y scattering

calculate

the crystallite thickness.

m a x i m a has b e e n

a traditional w a y to

T o obtain the thickness,

the directly

observed periodicity has to b e corrected f o r the level o f erystallinity. T o compare this d i m e n s i o n w i t h the o r d e r e d sequence length d e t e r m i n e d b y LAM,

a correction must b e made f o r chain tilt (45).

T h u s , a series o f

corrections, each subject to experimental error, must b e made to enable a comparison o f the small-angle X - r a y scattering result w i t h those f r o m R a m a n . Therefore, i n many cases w h e n the n e e d f o r the r e q u i r e d corrections is not recognized, it is not surprising that agreement is not obtained a n d modifica Downloaded by UNIV LAVAL on July 14, 2016 | http://pubs.acs.org Publication Date: May 5, 1993 | doi: 10.1021/ba-1993-0236.ch005

tion is made to e q 1. H o w e v e r , w h e n care is taken i n m a k i n g the corrections to p r o p e r l y compare crystallite thicknesses, e q 1 is shown to b e valid (46). (If the data i n reference 45, w h e r e t h e chain tilt was taken into account, were also corrected

f o r the crystallinity level, then good agreement is f o u n d

between the L A M a n d small-angle X - r a y diffraction m a x i m u m for t h e l a m e l lar thickness.) T h e analysis o f m o d e orders has b e e n shown to b e effective i n assessing e q 1 as is illustrated i n T a b l e I I (16). H e r e , several molecular weight fractions o f linear polyethylene were crystallized i n various ways to obtain different

Table II. Longitudinal Acoustical Mode Frequencies and Properties of Crystalline Polyethylene Samples (16) Crystallization Conditions 93,000 120

Degree of Crystallinity

Solution crystal T = 85.8 °C Solution crystal T„ = 87.2 °C Solution crystal T = 89.5 °C Quenched dry ice-isopropanol Isothermal crystallization T = 125 °C 30 days Isothermal crystallization T = 126 °C 30 days

a

Av (cm- ) 3

(cm~ ) 1

1

β = Av / Avj 3

3

0.72

23.6 ± 0.2 68.0 ± 0.5

2.88

0.78

22.8 ± 0.2 64.5 + 0.5

2.83

0.81

21.0 + 0.2 59.5 + 0.5

2.83

0.62

15.0 + 0.2 44.0 ± 0.5

2.93

0.94

8.5 + 0.1 24.0 ± 0.5

2.82

0.89

7.0 ± 0.2 20.5 ± 0.5

2.93

c

93,000 124 93,000 135

c

11,500 189 8,400 334

c

27,800 405

c

a

Determined from enthalpy of fusion.


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167

levels o f crystallinity a n d the first a n d t h i r d m o d e orders, Δν\ a n d Δ ν , were d e t e r m i n e d . A s indicated i n the table, the ratio Δ ν / Δ ν is close to the theoretically expected value for this range i n o r d e r e d sequence lengths, crystallinity levels, a n d molecular weight. H e n c e , w e can conclude f r o m these data that for the different crystallization modes, a n d L = 100 Â or more, e q 1 gives a very good representation o f the experimental data. Therefore, e q 1 can be used i n its simple f o r m to determine the o r d e r e d sequence length. O t h e r kinds o f experiments have also shown that there is no appreciable perturbation o f the L A M frequencies b y the noncrystalline portions o f the molecule (47, 48). T h e r e is, therefore, no n e e d to develop complex theories to account for corrections to e q 1 that do not, i n fact, exist or are negligible (49-55). 3


1

3

T h e r e is, however, one situation for polymers where the application o f e q 1 i n its simple f o r m fails. A study of the thermodynamic a n d structural properties o f a series o f molecular weight fractions o f hydrogenated polybutadienes w i t h a fixed value o f 2 . 3 - m o l % b r a n c h points p o i n t e d out this failure (56). H e r e , the observed small-angle diffraction m a x i m u m , f o u n d to be 155 Â, is independent o f molecular weight. W h e n this m a x i m u m is corrected b y the level o f core crystallinity, the crystallite or lamellar thickness is obtained. T h e thickness is about 60 A for M < 1.6 Χ 1 0 (where M is the weightaverage molecular weight) a n d monotonically decreases to 30 + 5 Â for M = 4.2 Χ 1 0 , the highest molecular weight studied. T h i n - s e c t i o n electron microscopy studies c o n f i r m these lamellar thicknesses (56). T h e most p r o b a ble o r d e r e d sequence length, L , obtained f r o m the L A M , for the same set o f fractions crystallized i n the same manner, is about 70 Â a n d independent o f chain length. W h e n corrected w i t h a chain tilt angle o f 30°, the crystallite or lamellar thickness is f o u n d to be 61 Â for all o f the fractions. C o m p a r i s o n o f the experimental results shows that the L A M , small-angle diffraction maxima, a n d electron microscopy give excellent agreement i n the crystallite thickness o n the order of 60 Â or higher. H o w e v e r , w h e n L decreases f r o m the 60 to 30 Â, there is a serious error i n the L A M - d e t e r m i n e d values. I n this range o f extremely small crystallite thickness, the L A M values are about a factor o f 2 too large. Clearly, e q 1 is not applicable i n this case. It must be recognized, however, that these results present a very u n i q u e situation. N o t only are w e dealing w i t h small crystallite sizes, b u t concomitantly the thick ness o f the interphase is about one-quarter to o n e - t h i r d o f the crystallite thickness (56). W e thus have an example w h e r e small o r d e r e d sequence w

w

4

w

5

R

c

lengths, about 2 0 - 5 0 carbon atoms, are j o i n e d w i t h a partially o r d e r e d interphase. I n addition, the interfacial region w i l l contain a higher than n o m i n a l concentration o f b r a n c h points. I n this special case, the pertinent vibrations between the crystalline sequences are not u n c o u p l e d . U s u a l l y for p o l y m e r systems, the vibrations between the o r d e r e d sequences are u n c o u p l e d a n d e q 1 can be p r o p e r l y applied. It is clear that theory w i l l have to be m o d i f i e d to account for this very special situation.


168



A n o t h e r example w h e r e there are major complications i n relating the L A M frequencies to o r d e r e d sequence lengths is i n interpreting the structure o f the n-alkanes i n the solid state. Recently, n-alkanes containing u p to 400 carbon atoms have b e e n synthesized (42, 43). T h e r e is interest i n quantita tively describing their structures a n d using L A M i n this endeavor. E l u c i d a t i n g the details o f the f o l d e d structures that can develop (42, 44, 57, 58) is o f particular importance. T h e p r o b l e m to be resolved is whether sharp folds that lead to integral-valued f o l d lengths are f o r m e d or whether, i n contrast, a disordered overlayer is present. T h i s is the same p r o b l e m that has already b e e n resolved for polymers crystallized f r o m dilute solution (46, 59). I n this case, it has b e e n established that a significant interfacial and disordered overlayer is present. T h e p r o b l e m that exists w i t h the n-alkane can be briefly o u t l i n e d as follows. I n the extended f o r m (i.e., for molecular crystals) the m o d e orders are not i n the ratio o f 5:3:1 (11, 14, 44). T h e deviations decrease w i t h increasing chain length a n d b e c o m e very small at carbon numbers greater than about 200. T h e r e is, therefore, a definite influence o f the e n d group o f the molecule o n the observed frequency. H o w e v e r , a calibration can be established between Δν a n d η that allows for the determination o f the o r d e r e d sequence lengths o f extended n-alkanes (44). T h e p r o b l e m w i t h f o l d e d structures arises because the frequencies o f the o r d e r e d sequence lengths are i n the region w h e r e there is a serious discrepancy i n the ratio o f m o d e orders. H e n c e , an u n k n o w n correction must be made. E v e n i f the assumption is made for the purpose o f this p r o b l e m that the o r d e r e d sequences are u n c o u p l e d , their determination f r o m àv is ambiguous at best (44). Therefore, u n t i l the basic theoretical problems are solved, o r d e r e d sequence lengths of the f o l d e d n-alkanes cannot be obtained w i t h the certainty necessary to resolve the structural p r o b l e m o f concern. 1

T h e L A M frequency not only allows for the determination of the most probable o r d e r e d sequence length i n crystalline polymers; the distribution o f these lengths can also b e obtained. (The true position o f the L A M b a n d is not only displaced b y the distribution o f o r d e r e d sequence lengths, b u t also b y temperature a n d frequency factors.) F o l l o w i n g Snyder et al. (60, 61) the distribution f u n c t i o n f(L) can b e directly related to the observed intensity, Z ° , at frequency Δν b y the relation bs

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Use of Raman Spectroscopy in Characterizing the Structure and

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