Richard 5. Stein Polymer Research Institute Universitv of Massachusetts
I I
Studies of
In studying low molecular weight materials, the refraction and absorption of light are useful observations which provide information about molecular structure. Refraction is a consequence of molecular polarizability, a measure of the ease of displacement of electrons in the field of a light wave; whereas absorption is a consequence of the light wave inducing molecular transitions from lower to higher energy levels. Ordinarily, these properties do not depend upon the direction of polarization of the light and the system is said to be isotropic. In strong electric or magnetic fields, molecules become oriented and the optical properties become dependent upon the relationship between the field direction and the polarization direction of the light. For example, the Kerr effect is the anisotropy in refractive index (birefringence) resulting from an electric field. For low molecular weight materials with ordinary fields, such effects are usually small. For polymers, however, considerable orientation of long chain molecules can be produced by simply stretching their solids and the observation of the resulting birefringence provides us with technologically interesting information about the molecular orientation. Similarly, dichroism. or deoendence of ootical ahsorotion on the ~ o l a r ization direction reveals useful information about the orientation of a articular arts of molecules which absorb the light. Interference effects are not usually observed when examining low molecular weight molecules with visible light, since their size is small as compared with its wavelength. Studies with much shorter wavelength X-rays are required for obtaining molecular information. However, polymer molecules, or superstructures composed of them, often approach the wavelength of visible light in size, so that the scattering of such light can provide information about the size, shape, and orientation of these structures. Optical methods are useful in that they are non-destructive and can be applied to the study of commercial samples as they are made, and they are rapid so that changes with time can he followed.
Solid Polymers with Light
tively, more precise measurements may be made using compensator techniques. The birefringence can be related to the orientation of the constituent units by use of the equation A
=
z in> + 2)?m, - b A f , gK
n
(3)
,
where n is the average refractive index, bl and bz are the polarizahilities along and perpendicular to the axes of such units and f, is an "orientation function" of such units defined as
where Bi is the angle that the symmetry axis of the unit makes with respect to the stretching direction. The function f j is defined such that it is zero for random orientation and approaches unit as the units line up parallel to the stretching direction. These "units" may be crystals, molecules, segments of molecules, or even chemical bonds. For example, a carbon dioxide molecule is anisotropic and has a higher polarizability along its axis than perpendicular to it (Fig. 1). However, if the molecules are randomly oriented (Fig. Za), the value of / = 0 and the principal refractive indices are equal. If the molecules can be aligned (Fig. Zb), then f approaches one and nl > nz so the birefringence becomes positive. Similarly, an unstretched ruhher consists of randomly arranged long chain molecules. Each of these molecules can he thought of as being composed of anisotropic segments which can he taken as the unit of summation in eqn. (3). When the ruhher is stretched, the segments orient (Fig. 3) and their f increases in a manner describable by the kinetic theory of rubber elasticity ( 3 )which gives
Birefringence
The anisotropy of refractive index can be demonstrated by stretching a thin film of a polymer between crossed polaroids. Before stretching, the field of view is dark, but as one stretches the sample, light of vivid colors is transmitted. The fraction transmitted (for stretching a t 45" to the polarization directions) is given by equation (1) (1, 2)
Figure 1. The principal poiarizabiiities at a carbon dioxide molecule
where d is the thickness, X is the wavelength of light in vacuum, and A is the birefringence defined by
where nl and nz are the refractive indices for light polarized parallel and perpendicular to the stretching direction. ~ - ~ ~ - . By quantitatively measuring the transmission, using ohototuhes. the hirefrinwnce may be obtained. AltemaSupported in part hy grants from the National Science Foundation and the Army Research Office (Durham). 748
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n~
n8
Figure 2. The refractive indices of carbon dioxide when (a) randomly oriented and ( b ) aligned.
where z is the number of segments between crosslinking points (where the molecules are chemically connected together) and ol is the elongation ratio (stretched length/ unstretched length). This theory leads to the result that the ratio of birefringence to stress, a, of a stretched rubber is
where T is the temDerature and b~ and bz are the ~ o l a r i z abilities of the molecular segments. ~ e i e n ttheoiies (4) ~ e r m i the t calculation of these Dolarizahilities in terms of &olecular structure, and the vaiues of A / a may be used to obtain information about such useful molecular parameters as the ootential of rotation about carbon-carbon bonds. These parameters affect molecular flexibility and hence influence the elastic pro~ertiesof the rubber. In general the value of (bl-- ba) depends on the anisotropy of molecular structure, and it becomes less, for example, in going from amorphous polyethylene to poly(viny1 chloride) to poly(vinylidene chloride) as one introduces polarizable bonds oriented perpendicularly to the polymer chain. Polymer molecules can be oriented by shearing them in solution as well as by stretching, and the observation that the birefringence obtained by orienting polymers in these very different ways can he quantitatively related gives one faith in the correctness of the theory and of our structural concepts. Upon rapidly stretching a polymer, molecular orientation is not instantaneous but is time dependent. This time dependence determines whether or not a polymer is brittle when it is rapidly stretched. At low temperatures, molecular orientation is slow and, when the molecules cannot move as fast as a sample is stretched, breakingoccun. The observation of the time dependence of birefringence is a good way of following this rate of molecular orientation. This can be done by observing the change of birefringence with time following the rapid stretching of a sample, or by the dynamic birefringence technique in which the oscillation of birefringence is observed when a sample is subject to an oscillating strain (1, 5, 6). When the frequency of vibration becomes sufficiently great that molecular orientation can no longer follow it, the birefringence change becomes less and it lags behind the stress change in phase. Many polymers, such as polyethylene, Nylon, and poly(ethylene terephthalate) (Dacron or Mylar) are partly crystalline. The molecules are sufficiently regular so that a t least, in part, they may enter a crystalline lattice to form small, submicroscopic crystals. These crystals mechanically strengthen the polymer and bind molecules together. When a crystalline polymer is stretched, both the
Figure 3. The refractive indices of an (a) unstretched and (b) stretched rubber.
cwstals and the amomhous regions orient such that the &ins align in the stretching ckrection. Such orientation is desirable for some a ~ ~ l i c a t i o nFor s . examole. in textile fibers, Mgh orientation in the fiber direction results in a high modulus and strength in this direction a t the expense of strength in a direction perpendicular to the fiber where i t does not matter. For such a crystalline polymer, it is a good approximation to write for the total birefringence
where @,, is the volume fraction of crystalline material and A,, and A,, are the contributions to the birefringence from the crystalline and amorphous phases. The form birefringence, A,,,,, represents a small contribution resulting from the distortion of the electric field of the light wave by the phase boundary. It may be estimated by swelling the amorphous phase with solvents of different refractive index and is usually about 5-1090 of the contribution. The crystalline birefringence depends on the intrinsic birefringence of the crystal A",, and the crystalline orientation function according to the equation
where uniaxial crystal symmetry is assumed. In some cases such as polyethylene, A",, can be obtained from microscopic measurements on larger crystals of low molecular weight analogs (n-paraffins). The crystalline orientation function may he measured from X-ray diffraction, so that A,, can be obtained. Thus, by inserting this along with ,@, obtained from X-ray density or calorimetric studies into eqn. (7), the crystalline contribution to A may be calculated and compared with the total birefringence ( l , 2 )(Fig. 4). In this wav. the orientation of a crvstalline nolvmer may be resol;& into the orientation of the crystaliine and amor~hous Darts. Such a division denends uoon the amount, size, and perfection of the crystals. since the amomhous material is the lower modulus contributor. the more-it deforms, the greater the compliance of the polymer. Thus, a study of this relationship between the structure and morphology of a crystalline polymer and the dis-
Figure 4. The contribution to the birefringence of the crystalline and form birefringence upon stretching branched polyethylene (From Ref. ( I ) ) .
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where A,,, and Aper are ahsorbances for light polarized parallel and perpendicular to the stretching direction. The dichroism may be related to the orientation function, fa, of the absorbing part of the molecule by (8)
D - 1
o+i = C o f a
I
I
0
I
0
2
3
Log t (sec.) Figure 5 . The variation of the change in crystalline and amorphous birefringence with time following the rapid stretching of branched poiyethylene (From Ref. (7)).
tribution of local strain between the crystalline and amorphous phases can be quite useful in understanding the effect of structure on mechanical properties. By combining measurements of the time dependence of birefringence with those of the time dependence of X-ray diffraction, it is possible to study the change in deformation of the two phases with time. For example, i t has been shown, as seen in Figure 5, that with polyethylene (7),initial rapid deformation of the sample occurs by deforming the amorphous phase. Within a few tenths of a second following stretching, the crystals orient and relieve the amorphous strain. The rate a t which this strain redistribution occurs determines whether or not the sample will be brittle. If the sample is studied in a vibration experiment as a function of frequency (6),it is often found that as the frequency increases, the crystals orient less and the amount of the strain home hy the amorphous phase increases. This effect becomes greater with annealed samples for which the crystals are more perfect and less deformable. An important property of natural rubber is that it is not crystalline (and hence very deformable) in the unstretched state, hut it rapidly crystallizes upon stretching. Thus, the rubber becomes stronger as i t is stretched. Birefringence, combined with X-ray diffraction has proved quite useful for the study of the amount and rate of this "stress-induced crystallization." Dichroism
Birefringence is a consequence of the overall orientation of a material. A companion technique, dichroism, while somewhat more difficult to use experimentally, has the advantage that i t may characterize the orientation of a selected part of a structure. An example (8) of the use of dichroism is the study of the orientation of double bonds produced in poly(viny1 chloride) by dehydrohalogenation according to the equation H H H H H H H H
I I I I -c-c-c-cI
I
I
KOH
--t
(10)
where C, is a constant deoendent uoon the nature of the absorbing group. It is sometimes pos'sible by making measurements a t different waveleneths to observe the orientation of different parts of the mol&ule. Only certain molecules which absorb in the visible can he studied in this way with visible light. Sometimes, an absorbing dye is added to the polymer in the hope that it will orient in the same way as the polymer itself, hut this is often an uncertain assumption. A better technique is to use infrared radiation. since all organic molecules have characteristic infrared absorhances characteristic of normal modes of vibration of certain parts of their structures. For example, polyethylene absorbs a t 720 cm-1 as a consequence of the rocking motion of CHz groups. This absorption is also dependent upon the polarization of the radiation and is described by eqn. (10) where now the constant C, is given by
where 6. is the angle between the molecular axis and the transition moment direction for the particular mode of vibration which is involved in the absorption. It is evident that by selecting the wavelength a t which an absorption measurement is made, the orientation of different parts of a structure may be determined (9), For example, Figures 6 and 7 show a comparison of the change in the orientation of the crystal a axis (found from measurement a t 730 cm-1) and that of the amorphous regions (from measurement at 1352 cm-') following the stretching of bonded polyethylene (10). The crystalline orientation is seen to increase with time whereas the amorphous orientation decreases as had been previously concluded from the birefringence and X-ray studies. For copolymers, it is possible to follow individually the orientation of the constituents by measurement a t the wavelengths of their respective absorption hords. An apparatus has been recently described which permits the measurement of infrared dichroism in times of onlv a few milliseconds followine the raoid stretchine of a sample (11). Raman spectroscopy has the potential of offering another optical technique for studying the orientation of particular parts of an oriented polymer structure. As is shown in a recent theoretical analysis (12), the polarization of the Raman scattering depends upon the orientation of the v
1 1 1 1 -c=c-c=c-
I
&lkhA When the polymer possesses a small fraction of such conjugated double bonds, it becomes colored. The absorption of light by these structures is greatest when the light is polarized along the axis of the double bonds. Consequently, when a polymer containing these units is stretched, the absorption of light is greatest when the light is polarized along the stretching direction. This principle is used to prepare one type of commercial polarizing material. The dichroism, D, is defined as
D 750
=
A,,A,
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(9)
I
2 3 LOG( tloT). (SEC)
4
Figure 6. The variation in the orientation function of the crystal a-axis with time is determined from the infrared dichroism at 730 cm-' foilowing the rapid stretching of branched polyethylene (From Ref. ( 1 0 ) ) .
Figure 9.Typical light scattering phl poiyethyiene film with (a) V, polariri
graphs from a spherulitic branched ,n, and (b) H , polarization.
LOG t , (SEC) Figure 7. The variation in the orientation of the amorphous regions with time following as determined from the infrared dichroism at 1352 e m - ' the rapid stretching of branched polyethylene (From Ref. ( 1 0 ) ) .
transition moment for the transition of concern. I t could provide additional information over that obtained from infrared in that i t is dependent upon both the second and fourth moment of the orientation distribution, whereas infrared (and birefringence) depend upon only the second moment. Light Scattering
The cloudiness of crystalline polymers such as polyethylene is well known. The polymer becomes clear when the crystals are melted. The observation that the scattering is largely in the forward direction leads to the conclusion that it predominately arises from structures larger than the wavelength of light. Since the crystals themselves are of the order of 0.05 wavelengths in size, the scattering units must he superstructures involving the crystals. The observation of the scattering is facilitated by the use of a laser as schematically shown in Figure 8. The parallel, monochromatic polarized light beam from the laser is passed through a film of the polymer, then through an analyzing polaroid after which the resulting scattered rays are photographed. Two types of scattering patterns are obtained depending upon polarization conditions as shown in Figure 9. The V, type pattern is obtained when the polarization of the incident beam and that of the analyzer is vertical, whereas the H , pattern is ohtained with vertical incident radiation but horizontal orientation of the polarization direction of the analyzer. These Datterns are now known to arise from the spherulitic structure of the polymer, as shown in Figure-10 for ~olvhutene-1.These s~herulitesconsist of both crystals and amorphous material and arise because the growth is
Fgure 10 Spherul~tlcstructure of polybutene 1 as seen in a mcrascope between crossed polarolds in the unstietched state and when stretched 20% (From Ref ( 2 2 ) )
nucleated, often by a foreign particle a t the center. The crystals then grow radially outward with preferential orientation of some particular crystalline direction with respect to the radius. With polyethylene, the b crystal axis is directed along the radius so that the c axis which is the chain axis is perpendicular to the radius and rotates about it, usually with some degree of regularity. These roughly spherically symmetrical structures grow until they impinge. As a consequence of the preferential orientation of the crystals within these spherulites, they are optically anisotropic with the radial and tangential refractive index being different. This anisotropy leads to the maltese cross appearance between crossed polaroids as seen in Figure 10.
The scattering from such a complex structure can be predicted surprisingly well on the basis of the scattering from an internally uniform anistropic sphere imbedded in a uniform isotropic medium of different refractive index. One must add the waves scattered from all parts of the spherulite to find the scattered wave amplitude a t a given point. For example, Figure 11 shows a predicted Hu pattern which is quite similar to the experimental pattern in Figure 9h (13).Similar agreement is found for the Vu pattern. The H , pattern shows a four-leaf clover appearance having maximum intensity at values of the azimuthal angle, p , (Fig. 8), which are odd multiples of 45". A maximum occurs in the radial direction such that the variable U is
U ,,,,,
Figure 8. A schematic diagram of a photographic laser light scattering experiment (From Ref. ( I ) ) .
=
( 4 r R / h ) sin(% ,, 12)
=
(12)
4.1
where R is the radius of the spherulite, h is the wavelength of light in the medium and Om,, is the angle (Fig. 8) between the incident and scattered ray a t which the intensity maximum occurs. A consequence of this equation is that the larger the radius of the spherulite, the smaller the angle 8,,, a t which the scattering maximum occurs. This is a familiar result Volume 50, Number 11, November 1973
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Figure 11. A calculated ti, light spherulite (From Ref. (23)).
scattering pattern far
an idealized
common to all scattering and diffraction experiments in that large objects scatter a t small angles and small ones do so a t large angles. (Bragg's law is like this.) I t arises because the fall off of intensity results from the increasing path difference between rays from different parts of the particle occurring as the scattering angle increases. The bigger the particle, the more rapidly will this path difference become greater as the scattering angle increases, and hence the scattering will occur at smaller angles. The scattering occurs at the convenient angle because the size of the spherulite is comparahle with the wavelength of visible light. It is a cardinal rule in all scattering studies that one should choose a wavelength comparahle with the size of the structure being studied. Hence, for measuring much smaller colloidal particles or molecules, one uses X-ray scattering. The physicist uses the much shorter wavelength de Broglie waves of high energy protons and neutrons for measuring nuclei and their suhstructures. The size of the spherulite may he readily ohtained from the value of Om,, and amees well when the size can also he measured microscopically. The light scattering techniaue has three imnortant advantaees: (1) it mav be utilized for smaller stkctures than c a h h e r&olved dy a light microscope; (2) i t gives a rapid average for a collection of spherulites, and (3) like other optical measurements, it is very fast and can be used to follow rapid changes in structure with time. This principle has been recently used in a commercial device for following the rate of growth of s~herulitesof nolvlethvlene tere~hthalate)under com.. mercially interesting conditions (14). When a s~herulitic~ o l v m e ris stretched. the s~herulites deform, as may he seen Figure 10, and may he approximated as ellipsoids. The calculated scattering from such isolated ellipsoids is found to agree well with experimental scattering patterns (15, 16) (Fig. 12), and may he used to measure the degree of deformation. I t is interesting to note that while the sample and the spherulites are extended in the vertical direction, the light scattering pattern is extended horizontally. This again illustrates the reciprocity between the size of the structure and the scattering angle. The use of light scattering movies to follow the spherulite deformation accom~anvinp the raoid . . . stretching of a sample has hcen described ( I 7 ) . Thc ~hservarionof the oscillating light scarrerlng partern accompanying the oscillating strain of a sample is employed in the dynamic light scattering technique (18, 19). The amplitude and phase of the scattered intensity
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, D " " ~ CLON~A,ION
Figure 12. Experimental scattering patterns for isotactic polypropylene Stretched in t h e vertical direction to different elongations as compared With theoretical calculations (From Ref. ( 7 6 ) ) .
change depends upon the mobility of the scattering structure which depends upon frequency and temperature. The model of isolated perfect and uniform sphemlites is, of course, an oversimplification and does not exactly amee with ex~eriment.The differences can he accounted " for by taking into account: (1) their irregular boundaries; (2) inters~heruliticinterference: and (3) . . the irreeular internal st;ucture. In fact, in disordered polymers, formed, for example, by rapidly quenching from the melt, spherulites are often not observed and the smaller amount of scattering results entirely from this more local structure. In such cases, the scattering is best descrihed by a statistical theory such as obtained by Stein and Wilson (20) hv eeneralization of the Dehve-Bueche theorv . .(21). of scatteling by heterogeneous medLa. In these theories, the scattering is described hv correlation functions which describe, for example, the probability that pairs of crystals are parallel as a function of their separation. The scattering arises because the crystals tend to he parallel over distances comparahle with the wavelength of light.
;cry
Conclusions
Polymers interact uniquely with light in such a way as to illustrate the directional nature of refraction, ahsorption, and scattering. The strong effects of polarization demonstrate that electronic motion in molecules is anisotropic and may he utilized to learn about molecular shape, orientation, and size. The ability of these techniques to follow rapid changes gives rise to the science of "rhea-optics," the study of flow by optical techniques. Literature Cited
S.. in Ke, 8.."Never Methods of Polymer Characterization." John Wilcy &Son%New York. 1964. Chapter IV. 121 Wilker. G . L..Aduan. PdvmarSci ,. 8.91 . 119111. ,~ ~. 13) Treloar. L. d. G.. " ~ h & i c s of Rubber Elasticity, "Oxford U n i v Press, h d Ed.. (1) Stein, R.
~
1967.
J.,
P. "Stafi3ticsl Moehsnics of Chain Molecules." Wiley-lnfe~seienee,Now York, 1969,Chapter IX. ( 5 ) Stein, R. S.. Onwi 3.. Sssewri. K.. and Keedy, A.. J Appl Pkys., 34. 80 (19631. (6) Sfein.R. S.,Accounlr ofcham. Re$. 5.121 11972). (4) Flow,
D.
(71 Oda. T , snd Stein. R. S.. J. Pohm. Sei. A2. 10.685 (19721.
E d l . 10.?409~197?1. (151 Sfein,R.S.. Clough. S.,andvsnAartsen,J.J., J A p p l . Phys., 36.3072(19621 (161 Samuols,R. J.,J Polym. Sri.. C. Na. 13. 37119661.
(17) Erhardt, P. F.,and Stein. R. S., "Appltd Polymer Symposium. High S p e d Teatin& Vol. IV: Tho R h d o m of Solids, Wilcy-Inferscience. New York, 1967. Val. 5 . p 113. (181 8tein.R. S.,Polym. Em. S c i . 9. 320(19691. (191 Hashimoto. T., "Static and Dynamic Light Sc~tfaringStudy of Crystalline Polymer Filma." Ph.D. Thesis. Univenity of Marsaehusafu, Amhent, Maasaehusetta, 197% Hashimoto, T., Prud'hamme. R. E.. and Stein, R. S.. J Palym. Sei. (Phva.Ed.1. 11.709~19731. 1201 Stein, R. S. and Wilson, P. R.,J. Appl. Phw., 33.1914 (1962). I211 Debye, P. andBueche,A.M., J. Appl Phya., M,518(19491. 1221 Sasawi.K.,Rhodes. M . B . , s n d Stein,R. S.,J Polvm. Sci., fl, 1,571 (1963). (231 Stein, R. S. io "Rheology, Theory and Applications." (Editor: Eitieh, F. K.1 Vol. 5, Chapter 6. 1969.
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