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Applications of laser spectroscopy in polymer science
PolymerTesting3 (1983) 249-266
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
D. B. DuPR~
Departmentof Chemistry, Universityof Louisville, Louisville, Kentucky 40292, USA
SUMMARY Applications of Rayleigh, Brillouin and Raman spectroscopy in polymer science are outlined. Since the advent of the laser and the new optical technology that it has spawned, new uses of light scattering have developed, particularly in the determination of dynamic properties of materials. The formalism of time correlation functions best describes the evolution of these microscopic processes. The physical basis of the three major light-scattering spectroscopies is first discussed within this general theoretical framework. A selection of practical applications will then be discussed that includes information obtainable about local polymer chain motion, large-scale diffusion, relaxation behavior, phase transitions and ordered states of macromolecules.
1.
INTRODUCTION
Light scattering in polymer science has moved beyond its historical function in the determination of weight-average molecular weights, radii of gyration, solution virial coefficients and gross polymer morphology. With the advent of the laser, more detailed structural and dynamic information on polymers in solution and bulk became available. Laser Raman spectrometers with sophisticated data handling units are close to the status of routine supplements to IR instrumentation in the laboratory. A few notable polymer programs utilize optical interferometry to probe high frequency acoustical fluctuations and relaxations in the analysis of Rayleigh-Brillouin spectra of polymers. More detailed information imbedded in the Rayleigh line can be uncovered by new 249 Polymer
Testing
0142-9418/83/$03.00 (~) Applied
Printed in Northern Ireland
Science
Publishers
Ltd,
England,
1983.
250
D.B. DuPI~
techniques of quasielastic or p h o t o n - c o u n t spectroscopy. T h e success of all these methods depends on the high intensity, directionality and coherency of the laser beam. T h e purpose of this article is to sketch what can be done and learned about polymers with these three m o d e r n light-scattering techniques. A n outline of the general a r r a n g e m e n t of a light-scattering experiment will be followed by a theoretical discussion of a unifying concept in spectroscopy: time correlation functions. T h e use of time correlation functions in magnetic resonance has been well developed for some time. T h e formalism has m a d e impressive inroads m o r e recently in light-scattering theory and is now having an impact on the p o l y m e r literature that cannot be ignored. T h e essential features behind the formulation will be explained in a pedogogical m a n n e r and it will b e shown how Rayleigh, Brillouin and R a m a n scattering m a y be viewed as special cases. T h e mathematics will not be overly involved. It is h o p e d that this truncated discussion will not do too much injustice to the well-developed physics of light scattering but serve as an introduction to one entering the literature at this level. A few examples of the use of each procedure in p o l y m e r science will be presented in the last section along with some critical discussion of advantages and disadvantages. This is not a review in any sense; examples have b e e n chosen in an attempt to clarify in a simple m a n n e r how each light-scattering m e t h o d fits into problems in p o l y m e r science. 2.
GEOME-q~YOF THE BASIC LIGHT-SCATI~RINGEXPERIMENT
Figure 1 illustrates the g e o m e t r y of a typical light-scattering experiment. A strong light s o u r c e - - n o w a d a y s a laser of high monochromaticity, h q i s directed n i
Laser/
I
ki
ki
c, " - . x j , Fig. 1. Geometry of a typical light-scattering experiment. A laser beam of wave vector /~i and polarization state hi is directed through a sample S. A portion of the beam is scattered in the volume V through an angle e. The scattered beam is specified by a wave vector/~ and polarization hr. The inset shows the relation between /~i, /~f and t], the scattering vector. The laser beam may also suffer a frequency change to = toi - tof.
A P P L I C A T I O N S O F L A S E R S P E C T R O S C O P Y IN P O L Y M E R SCIENCE
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through a suitably positioned sample, S, of refractive index n. The propagation of the incident light beam may be described by a vector/~i of magnitude 2"rm/~ which points along the forward axis. In encounter with the material in the scattering volume V, a portion of the laser beam is scattered through an angle 0. T h e scattered beam may be described by a similar vector kt of magnitude 2wn/~f directed a l o n g t h e axis of observation. As shown in the inset of Fig. 1, a scattering vector ~ = k i - kf describes this diversion of the beam with a possible frequency change of to = toi-tof. The initial polarization of the laser, fii, may also be altered in the scattering process to a condition, fir. Scattering events thus take (/~i, hi, toi) into (/~, ~e, tot). Usually the wavelength of the beam is changed very little so that Ikil Ik l. From Fig. 1 and the law of cosines it is easy to show that: q = Iql = 2ki sin 0/2 = - ~ - sin 0/2
(1)
A
3.
an/VIE C O R R E L A T I O N FUNCTIONS IN L I G H T - S C A T ] ? E R I N G F O R M U L A T I O N
Having set up the general geometry of a light-scattering experiment we now consider the physical mechanism that results in the diversion of the beam in the first instance. Light is scattered due to fluctuations in the refractive index (or alternately viewed, fluctuations in the dielectric constant or polarization) within the medium in the volume V. Such fluctuations, in turn, are due to the dynamics of the motion of molecules, segments of molecules or collections of molecules therein. These motions are usually correlated with one another to some extent over short or long time scales (as short a s 1 0 - 1 3 S, as long as --100 s in polymers). The intensity of light scattered through ~ and shifted in frequency by to = toi-tot will be denoted as I(~, to) and we will focus our interpretation of light scattering on fluctuations in the polarization, or. In general, ot is a tensor quantity (i.e. it has fluctuating components in directions other than that of the exciting beam polarization). T o simplify this discussion, we will neglect all matrix subscripts of ot and polarization states of the incoming and outgoing beam. (See Refs 1 and 2 for the complete treatment.) Modern light-scattering theory is largely written in terms of a correlation function, (ot(O)ct(t)), which is a statistical average over the product of a physical quantity (here a) at some time (say t = O) and its value at some later time t. The function mathematically describes the recall or decay of m e m o r y of the parameter as time evolves. The correlation of a property with itself is intimately related to the dynamics that governs motion within the medium. It turns out that the evolution in time of the polarizability is related to the spectral
252
D.B. DuPRI~
C(t)
L (W)
I
o 1~'1~ o 2 .
/
t
0
60
A single exponentialtime correlation function, C(t) and its Lorentzian Fourier transform, L(to). ~"is the decay time (1/e point of the exponential).
density, I(q, to) through:
I(q, to) = ~
dte-~'(a(O)a(t))
(2)
which is mathematically the familiar procedure of the Fourier transform. In principle the knowledge of the correlation function (a(O)~(t)) can determine I(q, to) through eqn. (2) and vice versa through the inverse transform: (ot(0)t~(t)) = I~= dtoe+i°'I(q' to)
(3)
Most polymer scientists are more familiar with the frequency side of the above relations from their experience with dispersive instrumentation such as IR, UV-VIS, and fluorescence spectrometers. Modern computing devices have made the measurement of time correlation functions rapid and more sensitive to details of the scattering process, molecular motion and relaxation mechanisms. As an example of eqn (2), suppose the correlation function, which we will also denote as C(t), is a single exponential with a decay time (1/e value) of -r, i.e. C(t)~ e -'/'. The Fourier transform of this simple function is a Lorentzian L(to; "/'-I)--'T-1/[to2-~-('T-1)2], of half width, F = ' r -1. The time and frequency behavior of such a single relaxation process is shown in Fig. 2. We will now consider in this abbreviated fashion, Raman, Brillouin and Rayleigh scattering from the time correlation point of view. 3.1 Raman effect Consider first an otherwise immobile molecule, molecular segment, or functional group undergoing a simple vibration of the nuclear framework in a normal mode, Q = Q0 cos fit. In the classical point of view, an entering beam of light may be Doppler shifted to higher and lower frequencies (+1"~) by
APPLICATIONS OF LASER SPEC'rROSCOPY IN POLYMER SCIENCE
t]
t
0
~-fl
0
253
~+fl
Fig. 3. The correlation function expected for a sustained vibration of frequency l-l. The Fourier transform of this sinusoidal function results in two 8-functions at to :ell on the frequency scale. The lower frequency spike corresponds to the Stokes component of a Raman spectrum; the higher frequency spike is the anti-Stokes component.
interaction with the electronic polarizability change that this vibration induces. Quantum mechanically, a photon of the entering beam may gain or lose a quantum of energy hI~ in the scattering process with the resultant energy shift, hto~ = htol + hfl. From either point of view, scattering from a molecular vibration will result in a frequency shift of the light of the order of 102-103 cm -1, in the units usually used in experiment. Shifts toward higher frequencies (antiStokes components) are much less intense as the higher energy vibrational states are less populated. In terms of the correlation function of the polarizability, a normal vibration is highly correlated with itself and (a(O)a(t)> should go as e ±in', where we use the Euler relationt to express the sinusoidal character of a sustained vibration of frequency f~. T h e exponential notation is also useful in evaluating the transforms of eqns (2) and (3). Substituting$ (ct(0)ot(t))-e ±mr into eqn (2), we see that
I(q, to) ~ I
dfe-~"e:~m' = 8(to + ll)
(4)
T h e integration above produced two &functions,§ the Stokes and anti-Stokes spikes at ( t o - O ) and (to+O), respectively; cf. Fig. 3. As a vibration of one molecule has little effect on that of another molecule (even when the frequencies are the same), correlations exist only within a given molecule. Each vibration is effectively independent and the time scale of the vibration is also very fast ( r - 1 0 -13 s). For the above reasons there is little practical information in the angular dependence of a Raman spectrum. Raman measurements are thus usually performed at a fixed angle (90°). ? e ±~u = cos u + i sin u. We will also drop a number of constants and sometimes important physical terms to expedite this discussion. For example, here the significant term (8a/80)0 that determines whether a vibrational mode is Raman-active has been deleted. § A 8-function may be written as 8(to)= S ~ e u°' dr.
254
D. B. DuPRI~ R
B
B
I
I
I
(~i -- ~B
O) i
(.~ i -4- (.~ B
Fig. 4. Features of a typical Rayleigh-Brillouin spectrum. The central component, R, is flanked
by two equally spaced Brillouin side bands, B, of halfwidth, F. 3.2. BriUouinscattering In a light-scattering spectrum there are also frequency c o m p o n e n t s shifted to a significantly lesser degree ( - 0 . 1 - 1 0 cm -1) which are due to acoustical motions in the sample. T h e rarefactions and compressions of thermally induced sound waves cause changes in the polarizability and dielectric constant of the m e d i u m and thus scatter light. Acoustical disturbances (called phonons in the q u a n t u m mechanical parlance) are correlated over m u c h longer ranges but eventually d a m p out (in times - r - 1 0 - 1 2 - 1 0 -8 s) due to the viscoelastic nature of the material. O n e can envisage a D o p p l e r shift of light radiation bouncing off these sound waves. As a particular sound wave is equally likely to be moving toward as away f r o m the direction of the incoming b e a m , one would expect shifts to both higher and lower frequencies. These shifts, called Brillouin doublets, are depicted in Fig. 4 as side bands very near the central Rayleigh c o m p o n e n t . t If Vs represents the velocity of a sound wave of wavelength A and frequency ~ , the fractional shift in the frequency of the Brillouin doublets should b e °')/3__ £O0
+2(V~n sin 0/2 \C/
(5)
where n is the refractive index of the medium. Account has b e e n t a k e n in eqn (5) for the c o m p o n e n t of the sound propagation sampled in the g e o m e t r y of the scattering experiment, cf. Fig. 5. T h e Brillouin shifts are also easily related to t Brillouin doublets may also be regarded as being caused by inelastic photon-phonon collisions with resultant Stokes and anti-Stokes components as a phonon is created or annihilated in the encounter, i.e. htof= htoi+h~ s.
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
255
A
$55 flsA = Vs Fig. 5. Schematic representation of the m e c h a n i s m of Brillouin scattering from propagating s o u n d waves of velocity, V s. Light m a y be regarded as being Doppler shifted to higher and lower frequencies, ±I)s, as it is b o u n c e d off the crests of the moving s o u n d waves. T h e c o m p o n e n t of the s o u n d propagation observed d e p e n d s on the scattering vector ~].
the scattering vector ~ through: ~oB = : ~ V s . q
(in cm -1)
(6)
These shifts range in frequency from - 0 for forward scattering to - 1 0 GHz for backward scattering. The ~ vector, or scattering angle, selected, thus determines the wavelength or frequency of the sound wave monitored. For most situations Brillouin spectroscopy is concerned with hypersonic disturbances. The Brillouin doublets in Fig. 4 are shown to be broader than the incident laser profile. This is to account for the finite lifetime of the sound disturbances that engender the displacements. The spectral width, F, of each of the doublets can be related to the (hypersonic) sound attenuation coefficient, o~, through F = aVJ2~r
(7)
The attenuation coefficient in turn is governed by various relaxation mechanisms in the material. In general F is a complicated function of elastic constants of the material, structural mode and volume relaxations, shear and volume viscosities, and specific heat contributions. The viscoelastic parameters are functions of frequency (lying in the hypersonic region); multiple relaxation processes occur particularly in polymeric substances. The method is a potential probe of relaxations that occur in the vicinity of 10 -1° s. Returning to the time correlation formulation, damping of the propagating sound disturbances can be included by multiplying by an additional, dissipative term: (ot(O)ot(t))
~ e ±itas' . e -r~'
(8)
256
D.B. t)uPR~
k L
(~)
C(t) r--> I
t Fig. 6.
-fls
0
+9. !
Time correlation function for a damped oscillation of frequency 1"~s. The Fourier transform is two Lorentzians of halfwidth F s centered at ±1"~s.
The Fourier transform of the above is two Lorentzians of halfwidth Fs displaced from the central laser frequency by +~s = Vs. q (cf. Fig. 6). 3.3. Quasi-elastic Rayleigh scattering Consider now a collection of large molecules isolated from one another (dilute solution) and undergoing overall unimpeded translation and rotation. An impinging laser beam will be scattered by fluctuations in the polarizability that these motions induce. The motions in this case are purely diffusive or dissipative in nature. In the frequency domain, translation and rotation of the molecule will lead to a Doppler broadening of the beam otherwise elastically scattered. The resultant breadth of this Rayleigh component is often referred to as quasielastic light scattering. If the translational motion is due to isotropic diffusion of the macromolecule, described by a diffusion constant Dr, a portion of the Rayleigh line will be broadened by DTq 2. Rotation of the molecule about its center of mass contributes an amount DR to the half-width, where DR is the rotational diffusion constant. Hence the spectral halfwidth of the scattered beam due to these processes is:
F ~ DTq2+ 6DR
(9)
The appearance of the scattering vector in the translational term is due to the fact that only a certain component of the translational motion of a given molecule is seen in the scattering geometry defined by ~. No angular dependence is expected in the rotational term; a molecule rotating about its center of mass looks the same on a time average no matter what the angle of view. Intramolecular bending modes will contribute additional terms to the spectral profile with halfwidth contributions in proportion to their inverse relaxation times, -r~1. Very large molecules will also exhibit intramolecular interference effects due to constructive/destructive interactions of light scattered from different parts of the molecule. These complications along with depolarization effects are discus-
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
257
DR
C(tl Dr
~r-l-F= o
Drq2+6DR t
Fig. 7. Representation of the time correlation function of a rod-shaped macromolecule undergoing unimpeded translational and rotational diffusion.
sed in detail in Ref. 1. They are generally not important when qL <-3, where L is the long dimension of the macromolecule. From the time correlation point of view: (a (0)or (t)) - exp [-(Dr(/2
+
6DR)t]
(10)
at least in dilute solution. The Fourier transform of the above is of course a Lorentzian of halfwidth F centered at the incident laser frequency; (cf. Figs 7 and 1). The rotational component in eqns. (9) and (10) is detectable only in the depolarized scattering mode and is difficult to extract experimentally. The correlation function may be directly measured by performing a statistical analysis on intensity fluctuations coming from the detector of photons arriving from the scattering volume. Suitably time-delayed versions of the photon density may be correlated with one another for different lag times in hardwired computing devices known as autocorrelators. These instruments currently have minimum delay times as short as 100 ns and can compute a full correlation function, averaging over several millions of statistical samples, in a matter of minutes. A correlation time for a polymer may lie in the range t - 1 0 - 6 - 1 0 0 s depending on the size of the macromolecule and the viscosity of the medium. At higher concentrations one would expect translational diffusion to become highly anisotropic, and rotational motion to be highly hindered. For example, diffusion perpendicular to a long rod-shaped macromolecule would be prohibitive in concentrated solution although the molecule may easily slip along its long axis in a tube formed by its neighbors. Rotation into angles past this hypothetical and constantly changing tube boundary would be very restricted and become more so as the polymer concentration increases. The dynamics of polymer in concentrated and semi-dilute regions has been discussed in recent
258
D . B . DoPRI~
theoretical papers 3"4 and some experimental tests s of the predictions of these theories are now appearing in the literature. The wide range of relaxation times that this newest of light-scattering methods encompasses makes it potentially the most powerful for examination of the dynamics of polymers in bulk and solution.
4.
APPLICATIONS IN POLYMER SCIENCE
4.1. Raman scattering The complementary nature of IR and Raman spectroscopy is best appreciated in molecules with high degrees of symmetry where IR and Raman activities are mutually exclusive. IR absorption depends on the presence of an oscillating permanent dipole whereas Raman activity is due to polarizability changes. Nonpolar functionalities invisible in the IR are frequently easily polarized and couple with the electric field of the light beam. Homonuclear molecular links in the backbone of chain molecules (e.g. C--C bonds in polyethylene and its derivatives) are thus good candidates for study by Raman spectroscopy. Polar pendents may or may not have Raman activity according to their symmetry (cf. polystyrene and polymethylmethyacrylate). The sensivity of the Raman effect to the polymer backbone structure indicates an important application in the detection and study of configurational states and conformational changes in macromolecules. Cis-trans isomerization and conjugation states may be specified by Raman studies of polymers with ethylinic main chain linkages (e.g. isoprene, butadiene polymers). The stereoregularity of substituted vinyl polymers may be deduced from the spectroscopic selection rules and polarization states of Raman lines and Ir bands. Locked-in successions of trans-gauche configurations may also be determined for polymers in the solid state from Raman lines originating from the chain backbone. Upon melting the polymer conformation becomes appreciably disordered as the main chain bonds assume different rotational isomeric states. New frequencies then appear and some (associated with crystalline lattice modes) disappear in the Raman spectrum. Noteable line broadenings are also a feature of the spectrum in the melt. An example of the difference in the Raman spectrum of a polymer in the molten and solid states is shown in Fig. 8. Similar effects occur when a polymer is transferred into solution but in some cases there is evidence that no major conformational change of the polymer occurs upon dissolution. Backbone C - - C vibrational features may be followed in the study of conformational transitions of polymers in solution induced by changes of pH, ionic strength, salt content or temperature. An example of this is a transition of polymethylacrylic acid (PMMA) in aqueous solution analogous to protein
846~
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1448
292
834
12i 261
/
524
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558
200
500
800 WAVE
Fig. 8.
NUMBER
1100
1400
(_M -1
Raman spectrum of polyethyleneoxide in the solid (top) and molten (bottom) states. [After J. L. Koenig, 1° with permission.]
260
D.B. DtJPRI~
denaturation that is brought on by neutralization of the polyelectrolyte by N a O H . T h e transition was shown to be due to a gradual elongation of the macromolecule by randomization of the backbone structure through a critical neutralization range. 6 Raman evidence speaks against a cooperative phase transition in this case. T h e ability to carry out Raman spectroscopy of polymers in aqueous solutions is one of the major advantages of the m e t h o d over its IR counterpart. Water is a weak Raman scatterer (except in the region of 1650 cm -1 and 3600 cm -1) and thus does not obscure vibrational information about the solute as it does in I R spectroscopy. This feature is particularly useful in the study of biological macromolecules whose natural state is in water solution. Further noteable advantages of Raman spectroscopy in polymer science include the following. 1. Raman features are generally quite sharp whereas I R vibrational bands are typically broad. It is thus easier to resolve nearby or overlapping features in a Raman spectrum. 2. In a Raman spectrometer a single instrument covers the entire 10 cm -~ to 4000 cm -~ region of the vibrational spectrum. Access to this entire region by I R spectroscopy requires several instruments and different radiation sources and detectors. 3. Owing to the narrow beam and high intensity of the laser source only small samples are required ( - 1 mg). T h e laser may also be focused down for further power density enhancement or for work on microspecimens. Samples thus may be examined in their natural condition or environment by the laser method. Fibers, coatings, molded plastic articles may be studied in situ either by transmission or reflection of the scattered laser radiation. 4. T h e polarization state of a vibration is easily determined in a Raman apparatus by the flip of an analyzer by 90 ° . Important information lies in depolarization ratios of Raman vibrations that are polarized in a locatable direction with respect to a draw or major crystallographic axis of an ordered polymer. In particular the average of the second and fourth I_~gendre polynomials, /52, and/54, which quantitatively specify uniaxial ordering, is available from Raman studies on stretched polymers. 7 T h e major difficulties with Raman applications are the possibilities of sample fluorescence which may override the weak Raman process and thermal degradation of the specimen due to laser heating. Catalogs of Raman assignments are not now as extensive as those in the IR. T o the author's knowledge no correlation function has been obtained by Fourier inversion of a Raman spectral feature of a polymer as has been done with low-molecular-weight liquids. 8
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
261
Further information on applications of Raman spectroscopy in polymer science is available in the reviews of Hendra, 9 Koenig, 1° Boerio and Koenig, 11 Fanconi 12 and Synder. 13 4.2. Brillouin scattering and hypersonic relaxation in polymers Brillouin studies on a series of homologous n-alkanes reveal that hypersonic relaxation of chain molecules in the fluid state is dominated by local translational motion. Large scale molecular reorientations and trans-gauche isomerizations which relax at times greater than 10-1°s can be ruled out in the frequency regime detectable by Brillouin scattering. Polymeric relaxation behavior has also been studied by the method. Most significantly, Brillouin spectrosco1~y extends relaxation measurements into the 'gigahertz' region and thus supplements mechanical and dielectric studies of lower frequency. Relaxations may be conveniently spotted by varying the temperature of a spectrum and observing the change in the spectral halfwidth, F, of the Brillouin components. Figure 9 is the Rayleigh-Brillouin spectrum at
23" C
100 °
C
Fig. 9. Rayleigh-Brillouin spectra of O M N at two temperatures, 100 °C and 23 °C. A t 23 °C t h e t e t r a m e r is near a hypersonic loss m a x i m u m . [After G. D. Patterson, 15 with permission.]
262
D.B. DuPRI~
tWO temperatures of 2,2,4,4,6,6,8,8,-octamethylnonane (OMN), a tetramer of polyisobutylene, which can be prepared in high optical quality. The spectrum recorded at the lower temperature has much broader BriUouin doublets and a more intense central peak. The broadening is an indication that some type of relaxation mode with ~-- 1/l~s = 10 -1° s has come into play. Relaxation modes are alternately identified by a maximum in a parameter, tan 8, which is related to F through: tan 8 = (2F/oJB)/[1 - (F/2a~B)2] ~ 2F/O)B
(11)
The frequency and temperature of the maximum loss observed by Brillouin. scattering frequently extrapolate well to lower frequency data in In oJm~xversus 1/T plots. Secondary relaxations have also been observed in some polymer melts and are evidenced by an additional resolved peak in tan/t versus temperature. The first Brillouin examination of polymers reported in the literature (in the late 1960s) concerned the glass transition of polyethylmethacrylate (PEMA) and polymethylmethacrylate (PMMA). A change in the slope of the hypersonic velocity obtained from eqn. (6) was observed around the accepted T~ of PMMA. More striking was an abrupt, almost discontinuous change in a quantity related to the central and Brillouin side band integrated intensities (the Landau-Placzek ratio = I¢/2IB) at the Tg of PEMA. The initial observation seemed to presage extraordinarily accurate identifications of the glass transition in amorphous polymers. These features were not always reproducible, however, in subsequent work on other polymers and other preparations of the same polymer. Discrepancies have now been related to differences in the thermal history and memory effects built into the samples by strain. Valuable information about glass transition behavior has, however, been obtained from Brillouin scattering. In particular, studies show that the glass transition as sensed in the hypersonic region is controlled by local structural rearrangements in the fluid and is largely unrelated to the long range character of the polymer or to rotational isomerizations of the chain molecules. The glass transition of polymers is thus physically very similar to that manifested by other glassforming materials. Other applications of Brillouin spectroscopy in polymer science include the problems of polymer crystallinity and compatibility. The presence of crystallites in an amorphous polymer matrix or component segregation in a polymer blend generates microscopic inhomogeneities and hence a mechanism of enhancedhypersound propagation and attenuation. Brillouin linewidths should increase under these influences. In the case of very large crystallites additional Brillouin components due to transverse sound waves would be expected. In an oriented polymer, the sound velocity and thus the Brillouin spectrum
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
263
depends on the orientation of the sample with respect to the laser beam. The effect has been studied in films and drawn fibers. There are some significant theoretical and experimental problems associated with Brillouin studies in polymers. Phonon velocity and attenuation depends on many important elastic constants, viscosities, and thermodynamic and relaxation properties of the material. A spectrum is frequently difficult to interpret in a quantitative fashion as a host of mechanisms may be contributing to the effect at the same time. Impurities, residual monomer and other heterogeneities in polymer samples further complicate the analysis. A number of typically polymeric processes lying outside the relaxation window of a few time decades, about 10 -a° s, are invisible to the method. These include effects of long-range reorientations, chain entanglements and crosslinking, and trans-gaucheconformational changes. Only relaxations of the local structure of the fluid (governed by small-scale chain flexibility and torsional freedom) and specific heat relaxations (there are many low lying vibrational states in polymers) have been shown to be important in the structuring of Brillouin linewidths and shifts in polymers. The chain character of the macromolecule does influence the spectrum as mobility and flexibility of small segments are very anisotropic even at the local level. The reader should consult the reviews of Patterson 14-16 for further details. 4.3. Rayleigh scattering Rayleigh scattering has been historically a major tool in the characterization of polymers. Static measurements of the angular and concentration dependence of the total light-scattering intensity of a polymer solution can be used to extract the weight-average molecular weight, radius of gyration, and solution virial coefficients from Zimm plots. The discussion of Section 3.3 of this article suggests a new use of dynamic light scattering in the determination of translational and rotational diffusion constants of large molecules. No artificial concentration, field or flow gradients need to be set up for such measurements. The molecules are in natural Brownian motion under thermal influence. The light beam merely senses fluctuations that these diffusive movements induce. Figure 10 shows the correlation function (on a logarithmic plot) obtained from intensity fluctuation spectroscopy of a dilute solution of polystyrene. The emulsion technique used to polymerize the monomer results in highly uniform, spherical, polymer particles of controllable diameter. So uniform are these spheres that such latexes are used as calibration standards in electron microscopy. As the conformation of the macromolecule is spherical, there is no rotational component to F in eqn. (9). The 1/e point of C(t) is thus a direct measure of DT. The radius of the isolated polystyrene particle obtained from the Stokes-Einstein relation (D = kT/6~r~a, where rl is the solvent viscosity) is
D.B. D~PR~
264
0.0
-0.5
t--
1.0
I
"1.5
0
I
I
I
0.8
I
I
1.6
I
I
2.4
3.2
t Imsecl Fig. 10. Logarithmic plot of the correlation function observed for polystyrene spheres of diameter 0-176 ~zm. The straight line indicates a single exponential decay with time constant, ~-= 2.46 ms.
1200
I
I
I
I
I
I
I
I
I
I
800
~l sec-'l 400
0
I
0.0
0.2
0.4
I
0.6
sin 2 le/2]
Fig. 11. The angular dependence of the reciprocal relaxation time, ~.-1 for polystyrene is in accord with eqn. (9) when only translational diffusion is possible (i.e. ~.-1 goes as q2).
A P P L I C A T I O N S O F L A S E R S P E C T R O S C O P Y IN P O L Y M E R SCIENCE
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in good agreement with the value obtained from electron microscopy. In general, translational diffusion constants so measured are accurate to 4-3%. Figure 11 illustrates the predicted angular dependence of "r-1 for this polystyrene sample. A rod-shaped macromolecule such as polybenzylglutamate (PBG) in its a-helical conformation exhibits both translational and rotational diffusion contributions to F. DT and D R have different angular dependences in eqn. (9) that, in principle, allow their separation in depolarized scattering. Diffusive motion may also be used to follow conformational changes of macromolecules. In the case of PBG a helix to random coil transition may be induced by changing the temperature and/or solvent composition of a binary solvent system containing a denaturant acid. The helix-coil transition of PBLG in a dichloroacetic acid/dichloroethane mixture has been studied using this technique. 17 DT and DR of a number of macromolecules, many with biological significance, have been obtained by inelastic light scattering. Light scattering from polymeric liquid crystals is engendered by fluctuations in three major modes of deformation, corresponding to splay, twist and bending motions of molecules about the local ordering axis (the director, in liquid crystal terminology). Study of the dynamic light-scattering spectra in selected geometries of an aligned liquid crystal can give quantitative information about elastic constants and viscosity coefficients of these materials. TM Polymer liquid crystals have become of more concern recently because of practical applications that the phase has in the formation of ultra-high strength fibers from ordered fluids of highly elongated macromolecules. Photon-correlation spectroscopy has also been applied, with varying degrees of success, to the study of bulk polymers. Information about slower relaxing modes in the time range 1 0 - 6 tO 100S is theoretically available by this technique. As light scattering from bulk polymers is extremely weak, it is difficult to remove non-intrinsic factors from the scattered field. The presence of impurities, inhomogeneities, and unrelaxed strains may easily obscure the polymer mechanisms that one seeks to study. Nevertheless very careful measurements have been made on a few polymers and the method has been shown to be of value in the study of the glass transition. 19 Applications of dynamic Rayleigh scattering in polymer Science are reviewed in Refs 19-23. ACKNOWLEDGEMENT
Portions of this work were supported by the National Science Foundation under grant DMR 81 12975. The author would also like to thank Drs L. L.
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C h a p o y , J. R. F e r n a n d e s a n d G. D . P a t t e r s o n f o r h e l p f u l c o m m e n t s o n a n d criticisms o f t h e m a n u s c r i p t .
REFERENCES 1. BERNE, B. J. and PECORA, R. (1976). Dynamic Light Scattering with Applications to Chemistry, Biology and Physics, New York, John Wiley. 2. CHu, B. (1974). Laser Light Scattering, New York, Academic Press. 3. EDWARDS,S. F. and DoI, M. (1982). J. Chem. SOc. Faraday, Trans. 2, 78, 113. 4. DoI, M. and EDWARDS,S. F. (1978). J. Chem. SOc. Faraday Trans. 2, '74, 560. 5. ZERO, K. M. and PECORA, R. (1982) Macromolecules, 15 87. 6. KOE~G, J. L., ANGOOD,A. C., SEI~m~, J. and ~ , J. B. (1960), J. Am. Chem. SOc. 91, 7250. 7. WARD, I. M.Ed. (1975). Structure and Properties of Oriented Polymers, New York, John Wiley. 8. GORDOrq,R. G. (1968). Adv. Magnetic Resonance, 3, 1. 9. ~2,~RA, P. J. (1969). Adv. Polymer Sci., 6, 151. 10. KOENIO,J. L. (1977). Appl. Spectros. Reviews, 4, 233. 11. BOERtO,F. J. and KOE~rtG, J. L. (1972). Rev. Macro. Chem., 8, 209. 12. FArqCOr,a,B. (1980). Ann. Rev. Phys. Chem., 31, 265. 13. SYrCDER,R. G. (1980). Infrared and Raman spectra of Polymers, In: Methods of Experimental Physics, Vol. 16A, (Ed. R. A. Fava), New York, Academic Press, p. 73. 14. PA'rrERSOIq,G. D. and LATrtAM,J. P. (1980). J. Polym. Sci: Macromol. Rev., 15, 1 15. PATTERSON,G. D. (1980). CRC Crit. Rev. Sol. State and Mat. Sci., 9, 373. 16. PATrERSOrq,G. D. (1980). Rayleigh-Brillouin scattering in Polymers, In: Methods of Experimental Physics, Vol 16A, (Ed. R. A. Fava) New York, Academic Press, p. 170. 17. FORD, N. C., LEE, W. and KARASZ,F. E. (1969). J. Chem. Phys., 50, 3098. 18. DuPRfl, D. B. (1982). Techniques for the Evaluation of Material Constants in Lyotropic systems and the study of Pretransitional Phenomena in Polymeric Liquid Crystals, In: Polymer Liquid Crystals, (Ed. A. Cfferri, W. R. Krigbaum and R. B. Meyer) New York, Academic Press, pp. 165-200. 19. PATrERSOrq,G. D. (1983). Adv. Polym. Sci., 48, 125. 20. CHtJ, B. (1970). Adv. Macromol. Chem., 2, 89. 21. PE'nCOLAS,W. L. (1972). Adv. Polym. Sci., 9, 285. 22. BLAGROBE,R. J. (1973). Rev. Macromol. Chem., 10, 71. 23. BERNE,B. J. and PECORA, R. (1974). Ann. Rev. Phys. Chem., 2S, 233.
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
D. B. DuPR~
Departmentof Chemistry, Universityof Louisville, Louisville, Kentucky 40292, USA
SUMMARY Applications of Rayleigh, Brillouin and Raman spectroscopy in polymer science are outlined. Since the advent of the laser and the new optical technology that it has spawned, new uses of light scattering have developed, particularly in the determination of dynamic properties of materials. The formalism of time correlation functions best describes the evolution of these microscopic processes. The physical basis of the three major light-scattering spectroscopies is first discussed within this general theoretical framework. A selection of practical applications will then be discussed that includes information obtainable about local polymer chain motion, large-scale diffusion, relaxation behavior, phase transitions and ordered states of macromolecules.
1.
INTRODUCTION
Light scattering in polymer science has moved beyond its historical function in the determination of weight-average molecular weights, radii of gyration, solution virial coefficients and gross polymer morphology. With the advent of the laser, more detailed structural and dynamic information on polymers in solution and bulk became available. Laser Raman spectrometers with sophisticated data handling units are close to the status of routine supplements to IR instrumentation in the laboratory. A few notable polymer programs utilize optical interferometry to probe high frequency acoustical fluctuations and relaxations in the analysis of Rayleigh-Brillouin spectra of polymers. More detailed information imbedded in the Rayleigh line can be uncovered by new 249 Polymer
Testing
0142-9418/83/$03.00 (~) Applied
Printed in Northern Ireland
Science
Publishers
Ltd,
England,
1983.
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D.B. DuPI~
techniques of quasielastic or p h o t o n - c o u n t spectroscopy. T h e success of all these methods depends on the high intensity, directionality and coherency of the laser beam. T h e purpose of this article is to sketch what can be done and learned about polymers with these three m o d e r n light-scattering techniques. A n outline of the general a r r a n g e m e n t of a light-scattering experiment will be followed by a theoretical discussion of a unifying concept in spectroscopy: time correlation functions. T h e use of time correlation functions in magnetic resonance has been well developed for some time. T h e formalism has m a d e impressive inroads m o r e recently in light-scattering theory and is now having an impact on the p o l y m e r literature that cannot be ignored. T h e essential features behind the formulation will be explained in a pedogogical m a n n e r and it will b e shown how Rayleigh, Brillouin and R a m a n scattering m a y be viewed as special cases. T h e mathematics will not be overly involved. It is h o p e d that this truncated discussion will not do too much injustice to the well-developed physics of light scattering but serve as an introduction to one entering the literature at this level. A few examples of the use of each procedure in p o l y m e r science will be presented in the last section along with some critical discussion of advantages and disadvantages. This is not a review in any sense; examples have b e e n chosen in an attempt to clarify in a simple m a n n e r how each light-scattering m e t h o d fits into problems in p o l y m e r science. 2.
GEOME-q~YOF THE BASIC LIGHT-SCATI~RINGEXPERIMENT
Figure 1 illustrates the g e o m e t r y of a typical light-scattering experiment. A strong light s o u r c e - - n o w a d a y s a laser of high monochromaticity, h q i s directed n i
Laser/
I
ki
ki
c, " - . x j , Fig. 1. Geometry of a typical light-scattering experiment. A laser beam of wave vector /~i and polarization state hi is directed through a sample S. A portion of the beam is scattered in the volume V through an angle e. The scattered beam is specified by a wave vector/~ and polarization hr. The inset shows the relation between /~i, /~f and t], the scattering vector. The laser beam may also suffer a frequency change to = toi - tof.
A P P L I C A T I O N S O F L A S E R S P E C T R O S C O P Y IN P O L Y M E R SCIENCE
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through a suitably positioned sample, S, of refractive index n. The propagation of the incident light beam may be described by a vector/~i of magnitude 2"rm/~ which points along the forward axis. In encounter with the material in the scattering volume V, a portion of the laser beam is scattered through an angle 0. T h e scattered beam may be described by a similar vector kt of magnitude 2wn/~f directed a l o n g t h e axis of observation. As shown in the inset of Fig. 1, a scattering vector ~ = k i - kf describes this diversion of the beam with a possible frequency change of to = toi-tof. The initial polarization of the laser, fii, may also be altered in the scattering process to a condition, fir. Scattering events thus take (/~i, hi, toi) into (/~, ~e, tot). Usually the wavelength of the beam is changed very little so that Ikil Ik l. From Fig. 1 and the law of cosines it is easy to show that: q = Iql = 2ki sin 0/2 = - ~ - sin 0/2
(1)
A
3.
an/VIE C O R R E L A T I O N FUNCTIONS IN L I G H T - S C A T ] ? E R I N G F O R M U L A T I O N
Having set up the general geometry of a light-scattering experiment we now consider the physical mechanism that results in the diversion of the beam in the first instance. Light is scattered due to fluctuations in the refractive index (or alternately viewed, fluctuations in the dielectric constant or polarization) within the medium in the volume V. Such fluctuations, in turn, are due to the dynamics of the motion of molecules, segments of molecules or collections of molecules therein. These motions are usually correlated with one another to some extent over short or long time scales (as short a s 1 0 - 1 3 S, as long as --100 s in polymers). The intensity of light scattered through ~ and shifted in frequency by to = toi-tot will be denoted as I(~, to) and we will focus our interpretation of light scattering on fluctuations in the polarization, or. In general, ot is a tensor quantity (i.e. it has fluctuating components in directions other than that of the exciting beam polarization). T o simplify this discussion, we will neglect all matrix subscripts of ot and polarization states of the incoming and outgoing beam. (See Refs 1 and 2 for the complete treatment.) Modern light-scattering theory is largely written in terms of a correlation function, (ot(O)ct(t)), which is a statistical average over the product of a physical quantity (here a) at some time (say t = O) and its value at some later time t. The function mathematically describes the recall or decay of m e m o r y of the parameter as time evolves. The correlation of a property with itself is intimately related to the dynamics that governs motion within the medium. It turns out that the evolution in time of the polarizability is related to the spectral
252
D.B. DuPRI~
C(t)
L (W)
I
o 1~'1~ o 2 .
/
t
0
60
A single exponentialtime correlation function, C(t) and its Lorentzian Fourier transform, L(to). ~"is the decay time (1/e point of the exponential).
density, I(q, to) through:
I(q, to) = ~
dte-~'(a(O)a(t))
(2)
which is mathematically the familiar procedure of the Fourier transform. In principle the knowledge of the correlation function (a(O)~(t)) can determine I(q, to) through eqn. (2) and vice versa through the inverse transform: (ot(0)t~(t)) = I~= dtoe+i°'I(q' to)
(3)
Most polymer scientists are more familiar with the frequency side of the above relations from their experience with dispersive instrumentation such as IR, UV-VIS, and fluorescence spectrometers. Modern computing devices have made the measurement of time correlation functions rapid and more sensitive to details of the scattering process, molecular motion and relaxation mechanisms. As an example of eqn (2), suppose the correlation function, which we will also denote as C(t), is a single exponential with a decay time (1/e value) of -r, i.e. C(t)~ e -'/'. The Fourier transform of this simple function is a Lorentzian L(to; "/'-I)--'T-1/[to2-~-('T-1)2], of half width, F = ' r -1. The time and frequency behavior of such a single relaxation process is shown in Fig. 2. We will now consider in this abbreviated fashion, Raman, Brillouin and Rayleigh scattering from the time correlation point of view. 3.1 Raman effect Consider first an otherwise immobile molecule, molecular segment, or functional group undergoing a simple vibration of the nuclear framework in a normal mode, Q = Q0 cos fit. In the classical point of view, an entering beam of light may be Doppler shifted to higher and lower frequencies (+1"~) by
APPLICATIONS OF LASER SPEC'rROSCOPY IN POLYMER SCIENCE
t]
t
0
~-fl
0
253
~+fl
Fig. 3. The correlation function expected for a sustained vibration of frequency l-l. The Fourier transform of this sinusoidal function results in two 8-functions at to :ell on the frequency scale. The lower frequency spike corresponds to the Stokes component of a Raman spectrum; the higher frequency spike is the anti-Stokes component.
interaction with the electronic polarizability change that this vibration induces. Quantum mechanically, a photon of the entering beam may gain or lose a quantum of energy hI~ in the scattering process with the resultant energy shift, hto~ = htol + hfl. From either point of view, scattering from a molecular vibration will result in a frequency shift of the light of the order of 102-103 cm -1, in the units usually used in experiment. Shifts toward higher frequencies (antiStokes components) are much less intense as the higher energy vibrational states are less populated. In terms of the correlation function of the polarizability, a normal vibration is highly correlated with itself and (a(O)a(t)> should go as e ±in', where we use the Euler relationt to express the sinusoidal character of a sustained vibration of frequency f~. T h e exponential notation is also useful in evaluating the transforms of eqns (2) and (3). Substituting$ (ct(0)ot(t))-e ±mr into eqn (2), we see that
I(q, to) ~ I
dfe-~"e:~m' = 8(to + ll)
(4)
T h e integration above produced two &functions,§ the Stokes and anti-Stokes spikes at ( t o - O ) and (to+O), respectively; cf. Fig. 3. As a vibration of one molecule has little effect on that of another molecule (even when the frequencies are the same), correlations exist only within a given molecule. Each vibration is effectively independent and the time scale of the vibration is also very fast ( r - 1 0 -13 s). For the above reasons there is little practical information in the angular dependence of a Raman spectrum. Raman measurements are thus usually performed at a fixed angle (90°). ? e ±~u = cos u + i sin u. We will also drop a number of constants and sometimes important physical terms to expedite this discussion. For example, here the significant term (8a/80)0 that determines whether a vibrational mode is Raman-active has been deleted. § A 8-function may be written as 8(to)= S ~ e u°' dr.
254
D. B. DuPRI~ R
B
B
I
I
I
(~i -- ~B
O) i
(.~ i -4- (.~ B
Fig. 4. Features of a typical Rayleigh-Brillouin spectrum. The central component, R, is flanked
by two equally spaced Brillouin side bands, B, of halfwidth, F. 3.2. BriUouinscattering In a light-scattering spectrum there are also frequency c o m p o n e n t s shifted to a significantly lesser degree ( - 0 . 1 - 1 0 cm -1) which are due to acoustical motions in the sample. T h e rarefactions and compressions of thermally induced sound waves cause changes in the polarizability and dielectric constant of the m e d i u m and thus scatter light. Acoustical disturbances (called phonons in the q u a n t u m mechanical parlance) are correlated over m u c h longer ranges but eventually d a m p out (in times - r - 1 0 - 1 2 - 1 0 -8 s) due to the viscoelastic nature of the material. O n e can envisage a D o p p l e r shift of light radiation bouncing off these sound waves. As a particular sound wave is equally likely to be moving toward as away f r o m the direction of the incoming b e a m , one would expect shifts to both higher and lower frequencies. These shifts, called Brillouin doublets, are depicted in Fig. 4 as side bands very near the central Rayleigh c o m p o n e n t . t If Vs represents the velocity of a sound wave of wavelength A and frequency ~ , the fractional shift in the frequency of the Brillouin doublets should b e °')/3__ £O0
+2(V~n sin 0/2 \C/
(5)
where n is the refractive index of the medium. Account has b e e n t a k e n in eqn (5) for the c o m p o n e n t of the sound propagation sampled in the g e o m e t r y of the scattering experiment, cf. Fig. 5. T h e Brillouin shifts are also easily related to t Brillouin doublets may also be regarded as being caused by inelastic photon-phonon collisions with resultant Stokes and anti-Stokes components as a phonon is created or annihilated in the encounter, i.e. htof= htoi+h~ s.
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
255
A
$55 flsA = Vs Fig. 5. Schematic representation of the m e c h a n i s m of Brillouin scattering from propagating s o u n d waves of velocity, V s. Light m a y be regarded as being Doppler shifted to higher and lower frequencies, ±I)s, as it is b o u n c e d off the crests of the moving s o u n d waves. T h e c o m p o n e n t of the s o u n d propagation observed d e p e n d s on the scattering vector ~].
the scattering vector ~ through: ~oB = : ~ V s . q
(in cm -1)
(6)
These shifts range in frequency from - 0 for forward scattering to - 1 0 GHz for backward scattering. The ~ vector, or scattering angle, selected, thus determines the wavelength or frequency of the sound wave monitored. For most situations Brillouin spectroscopy is concerned with hypersonic disturbances. The Brillouin doublets in Fig. 4 are shown to be broader than the incident laser profile. This is to account for the finite lifetime of the sound disturbances that engender the displacements. The spectral width, F, of each of the doublets can be related to the (hypersonic) sound attenuation coefficient, o~, through F = aVJ2~r
(7)
The attenuation coefficient in turn is governed by various relaxation mechanisms in the material. In general F is a complicated function of elastic constants of the material, structural mode and volume relaxations, shear and volume viscosities, and specific heat contributions. The viscoelastic parameters are functions of frequency (lying in the hypersonic region); multiple relaxation processes occur particularly in polymeric substances. The method is a potential probe of relaxations that occur in the vicinity of 10 -1° s. Returning to the time correlation formulation, damping of the propagating sound disturbances can be included by multiplying by an additional, dissipative term: (ot(O)ot(t))
~ e ±itas' . e -r~'
(8)
256
D.B. t)uPR~
k L
(~)
C(t) r--> I
t Fig. 6.
-fls
0
+9. !
Time correlation function for a damped oscillation of frequency 1"~s. The Fourier transform is two Lorentzians of halfwidth F s centered at ±1"~s.
The Fourier transform of the above is two Lorentzians of halfwidth Fs displaced from the central laser frequency by +~s = Vs. q (cf. Fig. 6). 3.3. Quasi-elastic Rayleigh scattering Consider now a collection of large molecules isolated from one another (dilute solution) and undergoing overall unimpeded translation and rotation. An impinging laser beam will be scattered by fluctuations in the polarizability that these motions induce. The motions in this case are purely diffusive or dissipative in nature. In the frequency domain, translation and rotation of the molecule will lead to a Doppler broadening of the beam otherwise elastically scattered. The resultant breadth of this Rayleigh component is often referred to as quasielastic light scattering. If the translational motion is due to isotropic diffusion of the macromolecule, described by a diffusion constant Dr, a portion of the Rayleigh line will be broadened by DTq 2. Rotation of the molecule about its center of mass contributes an amount DR to the half-width, where DR is the rotational diffusion constant. Hence the spectral halfwidth of the scattered beam due to these processes is:
F ~ DTq2+ 6DR
(9)
The appearance of the scattering vector in the translational term is due to the fact that only a certain component of the translational motion of a given molecule is seen in the scattering geometry defined by ~. No angular dependence is expected in the rotational term; a molecule rotating about its center of mass looks the same on a time average no matter what the angle of view. Intramolecular bending modes will contribute additional terms to the spectral profile with halfwidth contributions in proportion to their inverse relaxation times, -r~1. Very large molecules will also exhibit intramolecular interference effects due to constructive/destructive interactions of light scattered from different parts of the molecule. These complications along with depolarization effects are discus-
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
257
DR
C(tl Dr
~r-l-F= o
Drq2+6DR t
Fig. 7. Representation of the time correlation function of a rod-shaped macromolecule undergoing unimpeded translational and rotational diffusion.
sed in detail in Ref. 1. They are generally not important when qL <-3, where L is the long dimension of the macromolecule. From the time correlation point of view: (a (0)or (t)) - exp [-(Dr(/2
+
6DR)t]
(10)
at least in dilute solution. The Fourier transform of the above is of course a Lorentzian of halfwidth F centered at the incident laser frequency; (cf. Figs 7 and 1). The rotational component in eqns. (9) and (10) is detectable only in the depolarized scattering mode and is difficult to extract experimentally. The correlation function may be directly measured by performing a statistical analysis on intensity fluctuations coming from the detector of photons arriving from the scattering volume. Suitably time-delayed versions of the photon density may be correlated with one another for different lag times in hardwired computing devices known as autocorrelators. These instruments currently have minimum delay times as short as 100 ns and can compute a full correlation function, averaging over several millions of statistical samples, in a matter of minutes. A correlation time for a polymer may lie in the range t - 1 0 - 6 - 1 0 0 s depending on the size of the macromolecule and the viscosity of the medium. At higher concentrations one would expect translational diffusion to become highly anisotropic, and rotational motion to be highly hindered. For example, diffusion perpendicular to a long rod-shaped macromolecule would be prohibitive in concentrated solution although the molecule may easily slip along its long axis in a tube formed by its neighbors. Rotation into angles past this hypothetical and constantly changing tube boundary would be very restricted and become more so as the polymer concentration increases. The dynamics of polymer in concentrated and semi-dilute regions has been discussed in recent
258
D . B . DoPRI~
theoretical papers 3"4 and some experimental tests s of the predictions of these theories are now appearing in the literature. The wide range of relaxation times that this newest of light-scattering methods encompasses makes it potentially the most powerful for examination of the dynamics of polymers in bulk and solution.
4.
APPLICATIONS IN POLYMER SCIENCE
4.1. Raman scattering The complementary nature of IR and Raman spectroscopy is best appreciated in molecules with high degrees of symmetry where IR and Raman activities are mutually exclusive. IR absorption depends on the presence of an oscillating permanent dipole whereas Raman activity is due to polarizability changes. Nonpolar functionalities invisible in the IR are frequently easily polarized and couple with the electric field of the light beam. Homonuclear molecular links in the backbone of chain molecules (e.g. C--C bonds in polyethylene and its derivatives) are thus good candidates for study by Raman spectroscopy. Polar pendents may or may not have Raman activity according to their symmetry (cf. polystyrene and polymethylmethyacrylate). The sensivity of the Raman effect to the polymer backbone structure indicates an important application in the detection and study of configurational states and conformational changes in macromolecules. Cis-trans isomerization and conjugation states may be specified by Raman studies of polymers with ethylinic main chain linkages (e.g. isoprene, butadiene polymers). The stereoregularity of substituted vinyl polymers may be deduced from the spectroscopic selection rules and polarization states of Raman lines and Ir bands. Locked-in successions of trans-gauche configurations may also be determined for polymers in the solid state from Raman lines originating from the chain backbone. Upon melting the polymer conformation becomes appreciably disordered as the main chain bonds assume different rotational isomeric states. New frequencies then appear and some (associated with crystalline lattice modes) disappear in the Raman spectrum. Noteable line broadenings are also a feature of the spectrum in the melt. An example of the difference in the Raman spectrum of a polymer in the molten and solid states is shown in Fig. 8. Similar effects occur when a polymer is transferred into solution but in some cases there is evidence that no major conformational change of the polymer occurs upon dissolution. Backbone C - - C vibrational features may be followed in the study of conformational transitions of polymers in solution induced by changes of pH, ionic strength, salt content or temperature. An example of this is a transition of polymethylacrylic acid (PMMA) in aqueous solution analogous to protein
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Fig. 8.
NUMBER
1100
1400
(_M -1
Raman spectrum of polyethyleneoxide in the solid (top) and molten (bottom) states. [After J. L. Koenig, 1° with permission.]
260
D.B. DtJPRI~
denaturation that is brought on by neutralization of the polyelectrolyte by N a O H . T h e transition was shown to be due to a gradual elongation of the macromolecule by randomization of the backbone structure through a critical neutralization range. 6 Raman evidence speaks against a cooperative phase transition in this case. T h e ability to carry out Raman spectroscopy of polymers in aqueous solutions is one of the major advantages of the m e t h o d over its IR counterpart. Water is a weak Raman scatterer (except in the region of 1650 cm -1 and 3600 cm -1) and thus does not obscure vibrational information about the solute as it does in I R spectroscopy. This feature is particularly useful in the study of biological macromolecules whose natural state is in water solution. Further noteable advantages of Raman spectroscopy in polymer science include the following. 1. Raman features are generally quite sharp whereas I R vibrational bands are typically broad. It is thus easier to resolve nearby or overlapping features in a Raman spectrum. 2. In a Raman spectrometer a single instrument covers the entire 10 cm -~ to 4000 cm -~ region of the vibrational spectrum. Access to this entire region by I R spectroscopy requires several instruments and different radiation sources and detectors. 3. Owing to the narrow beam and high intensity of the laser source only small samples are required ( - 1 mg). T h e laser may also be focused down for further power density enhancement or for work on microspecimens. Samples thus may be examined in their natural condition or environment by the laser method. Fibers, coatings, molded plastic articles may be studied in situ either by transmission or reflection of the scattered laser radiation. 4. T h e polarization state of a vibration is easily determined in a Raman apparatus by the flip of an analyzer by 90 ° . Important information lies in depolarization ratios of Raman vibrations that are polarized in a locatable direction with respect to a draw or major crystallographic axis of an ordered polymer. In particular the average of the second and fourth I_~gendre polynomials, /52, and/54, which quantitatively specify uniaxial ordering, is available from Raman studies on stretched polymers. 7 T h e major difficulties with Raman applications are the possibilities of sample fluorescence which may override the weak Raman process and thermal degradation of the specimen due to laser heating. Catalogs of Raman assignments are not now as extensive as those in the IR. T o the author's knowledge no correlation function has been obtained by Fourier inversion of a Raman spectral feature of a polymer as has been done with low-molecular-weight liquids. 8
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
261
Further information on applications of Raman spectroscopy in polymer science is available in the reviews of Hendra, 9 Koenig, 1° Boerio and Koenig, 11 Fanconi 12 and Synder. 13 4.2. Brillouin scattering and hypersonic relaxation in polymers Brillouin studies on a series of homologous n-alkanes reveal that hypersonic relaxation of chain molecules in the fluid state is dominated by local translational motion. Large scale molecular reorientations and trans-gauche isomerizations which relax at times greater than 10-1°s can be ruled out in the frequency regime detectable by Brillouin scattering. Polymeric relaxation behavior has also been studied by the method. Most significantly, Brillouin spectrosco1~y extends relaxation measurements into the 'gigahertz' region and thus supplements mechanical and dielectric studies of lower frequency. Relaxations may be conveniently spotted by varying the temperature of a spectrum and observing the change in the spectral halfwidth, F, of the Brillouin components. Figure 9 is the Rayleigh-Brillouin spectrum at
23" C
100 °
C
Fig. 9. Rayleigh-Brillouin spectra of O M N at two temperatures, 100 °C and 23 °C. A t 23 °C t h e t e t r a m e r is near a hypersonic loss m a x i m u m . [After G. D. Patterson, 15 with permission.]
262
D.B. DuPRI~
tWO temperatures of 2,2,4,4,6,6,8,8,-octamethylnonane (OMN), a tetramer of polyisobutylene, which can be prepared in high optical quality. The spectrum recorded at the lower temperature has much broader BriUouin doublets and a more intense central peak. The broadening is an indication that some type of relaxation mode with ~-- 1/l~s = 10 -1° s has come into play. Relaxation modes are alternately identified by a maximum in a parameter, tan 8, which is related to F through: tan 8 = (2F/oJB)/[1 - (F/2a~B)2] ~ 2F/O)B
(11)
The frequency and temperature of the maximum loss observed by Brillouin. scattering frequently extrapolate well to lower frequency data in In oJm~xversus 1/T plots. Secondary relaxations have also been observed in some polymer melts and are evidenced by an additional resolved peak in tan/t versus temperature. The first Brillouin examination of polymers reported in the literature (in the late 1960s) concerned the glass transition of polyethylmethacrylate (PEMA) and polymethylmethacrylate (PMMA). A change in the slope of the hypersonic velocity obtained from eqn. (6) was observed around the accepted T~ of PMMA. More striking was an abrupt, almost discontinuous change in a quantity related to the central and Brillouin side band integrated intensities (the Landau-Placzek ratio = I¢/2IB) at the Tg of PEMA. The initial observation seemed to presage extraordinarily accurate identifications of the glass transition in amorphous polymers. These features were not always reproducible, however, in subsequent work on other polymers and other preparations of the same polymer. Discrepancies have now been related to differences in the thermal history and memory effects built into the samples by strain. Valuable information about glass transition behavior has, however, been obtained from Brillouin scattering. In particular, studies show that the glass transition as sensed in the hypersonic region is controlled by local structural rearrangements in the fluid and is largely unrelated to the long range character of the polymer or to rotational isomerizations of the chain molecules. The glass transition of polymers is thus physically very similar to that manifested by other glassforming materials. Other applications of Brillouin spectroscopy in polymer science include the problems of polymer crystallinity and compatibility. The presence of crystallites in an amorphous polymer matrix or component segregation in a polymer blend generates microscopic inhomogeneities and hence a mechanism of enhancedhypersound propagation and attenuation. Brillouin linewidths should increase under these influences. In the case of very large crystallites additional Brillouin components due to transverse sound waves would be expected. In an oriented polymer, the sound velocity and thus the Brillouin spectrum
APPLICATIONS OF LASER SPECTROSCOPY IN POLYMER SCIENCE
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depends on the orientation of the sample with respect to the laser beam. The effect has been studied in films and drawn fibers. There are some significant theoretical and experimental problems associated with Brillouin studies in polymers. Phonon velocity and attenuation depends on many important elastic constants, viscosities, and thermodynamic and relaxation properties of the material. A spectrum is frequently difficult to interpret in a quantitative fashion as a host of mechanisms may be contributing to the effect at the same time. Impurities, residual monomer and other heterogeneities in polymer samples further complicate the analysis. A number of typically polymeric processes lying outside the relaxation window of a few time decades, about 10 -a° s, are invisible to the method. These include effects of long-range reorientations, chain entanglements and crosslinking, and trans-gaucheconformational changes. Only relaxations of the local structure of the fluid (governed by small-scale chain flexibility and torsional freedom) and specific heat relaxations (there are many low lying vibrational states in polymers) have been shown to be important in the structuring of Brillouin linewidths and shifts in polymers. The chain character of the macromolecule does influence the spectrum as mobility and flexibility of small segments are very anisotropic even at the local level. The reader should consult the reviews of Patterson 14-16 for further details. 4.3. Rayleigh scattering Rayleigh scattering has been historically a major tool in the characterization of polymers. Static measurements of the angular and concentration dependence of the total light-scattering intensity of a polymer solution can be used to extract the weight-average molecular weight, radius of gyration, and solution virial coefficients from Zimm plots. The discussion of Section 3.3 of this article suggests a new use of dynamic light scattering in the determination of translational and rotational diffusion constants of large molecules. No artificial concentration, field or flow gradients need to be set up for such measurements. The molecules are in natural Brownian motion under thermal influence. The light beam merely senses fluctuations that these diffusive movements induce. Figure 10 shows the correlation function (on a logarithmic plot) obtained from intensity fluctuation spectroscopy of a dilute solution of polystyrene. The emulsion technique used to polymerize the monomer results in highly uniform, spherical, polymer particles of controllable diameter. So uniform are these spheres that such latexes are used as calibration standards in electron microscopy. As the conformation of the macromolecule is spherical, there is no rotational component to F in eqn. (9). The 1/e point of C(t) is thus a direct measure of DT. The radius of the isolated polystyrene particle obtained from the Stokes-Einstein relation (D = kT/6~r~a, where rl is the solvent viscosity) is
D.B. D~PR~
264
0.0
-0.5
t--
1.0
I
"1.5
0
I
I
I
0.8
I
I
1.6
I
I
2.4
3.2
t Imsecl Fig. 10. Logarithmic plot of the correlation function observed for polystyrene spheres of diameter 0-176 ~zm. The straight line indicates a single exponential decay with time constant, ~-= 2.46 ms.
1200
I
I
I
I
I
I
I
I
I
I
800
~l sec-'l 400
0
I
0.0
0.2
0.4
I
0.6
sin 2 le/2]
Fig. 11. The angular dependence of the reciprocal relaxation time, ~.-1 for polystyrene is in accord with eqn. (9) when only translational diffusion is possible (i.e. ~.-1 goes as q2).
A P P L I C A T I O N S O F L A S E R S P E C T R O S C O P Y IN P O L Y M E R SCIENCE
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in good agreement with the value obtained from electron microscopy. In general, translational diffusion constants so measured are accurate to 4-3%. Figure 11 illustrates the predicted angular dependence of "r-1 for this polystyrene sample. A rod-shaped macromolecule such as polybenzylglutamate (PBG) in its a-helical conformation exhibits both translational and rotational diffusion contributions to F. DT and D R have different angular dependences in eqn. (9) that, in principle, allow their separation in depolarized scattering. Diffusive motion may also be used to follow conformational changes of macromolecules. In the case of PBG a helix to random coil transition may be induced by changing the temperature and/or solvent composition of a binary solvent system containing a denaturant acid. The helix-coil transition of PBLG in a dichloroacetic acid/dichloroethane mixture has been studied using this technique. 17 DT and DR of a number of macromolecules, many with biological significance, have been obtained by inelastic light scattering. Light scattering from polymeric liquid crystals is engendered by fluctuations in three major modes of deformation, corresponding to splay, twist and bending motions of molecules about the local ordering axis (the director, in liquid crystal terminology). Study of the dynamic light-scattering spectra in selected geometries of an aligned liquid crystal can give quantitative information about elastic constants and viscosity coefficients of these materials. TM Polymer liquid crystals have become of more concern recently because of practical applications that the phase has in the formation of ultra-high strength fibers from ordered fluids of highly elongated macromolecules. Photon-correlation spectroscopy has also been applied, with varying degrees of success, to the study of bulk polymers. Information about slower relaxing modes in the time range 1 0 - 6 tO 100S is theoretically available by this technique. As light scattering from bulk polymers is extremely weak, it is difficult to remove non-intrinsic factors from the scattered field. The presence of impurities, inhomogeneities, and unrelaxed strains may easily obscure the polymer mechanisms that one seeks to study. Nevertheless very careful measurements have been made on a few polymers and the method has been shown to be of value in the study of the glass transition. 19 Applications of dynamic Rayleigh scattering in polymer Science are reviewed in Refs 19-23. ACKNOWLEDGEMENT
Portions of this work were supported by the National Science Foundation under grant DMR 81 12975. The author would also like to thank Drs L. L.
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D.B. DuPRI~
C h a p o y , J. R. F e r n a n d e s a n d G. D . P a t t e r s o n f o r h e l p f u l c o m m e n t s o n a n d criticisms o f t h e m a n u s c r i p t .
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