Dynamic light scattering from polymers

Makromol. Chem., Suppl. 2, 73-80 (1979)

DYNAMIC LIGHT SCATTERING FROM POLYMERS

73

R. Pecora

Department of Chemistry, Stanford University, Stanford, California 94305, USA

Abstract - Applications of both polarized and depolarized dynamic light scattering to the study of polymers are described. Polarized light scattering is used to study translational diffusion, diffusion virial coefficients, rotational motion of long rod-shaped polymers, long- wavelength intramolecular motions, dynamics of pseudogels and gels and density fluctuations in bulk polymers. Depolarized scattering is used to study rotational diffusion of rigid macromolecules, local and long-wavelength intramolecular motions, dynamics in semi-dilute solutions, rotational motion of small molecules in glassy polymers and optical anisotropy fluctuations of bulk polymers.

INTRODUCTION

Static light scattering from macromolecules was first developed in the 1940 's and 50's to measure equilibrium properties of polymers both in solution and in bulk (1). In this technique the average photon flux scattered from a macromolecular system at a given scattering angle is measured. Molecular weights, radii of gyration, solution virial coefficients, molecular optical anisotropies and sizes and structures of heterogeneous regions in bulk polymers are routinely obtained from this type of experiment. Static light scattering is now a relatively mature field although continued improvements in instrumentation (mainly the use of lasers and associated techniques) are steadily increasing its reliability and range of application.

Dynamic light scattering from macromolecules started its development after the introduction of the laser in the 1960's and has progressed greatly since then (2). It has yet, however, to reach its full potential. New applications are constantly arising as the instrumental techniques advance and more polymer scientists become aware of its possible uses.

The difference between static and dynamic light scattering is that, instead of measuring merely the average scattered photon flux (intensity), dynamic light scattering measures either the fluctuations in time of this intensity or its spectral distribution. Since the intensity fluctuations and the spectral distribution arise from motions of the macromolecules doing the scattering, dynamical properties of macromolecules such as translational and rotational diffusion coefficients and intramolecular relaxation times may be obtained.

In this short review only some selected topics are discussed. No attempt is made to be comprehensive either in the topics covered or the literature cited. For instance, an important field, electrophoretic light scattering will not be discussed at all ( 3 ) . In many cases literature citations are only to a recent work which contains references to previous publications.

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EXPERDENTAL METHODS

Dynamic light scattering methods employ a laser source, usually a relatively low power continuous gas laser (Argon ion, Krypton, Helium-Neon) and a photomultiplier as the detector. The choice of pre- and post-detection systems depends upon the time scale of the dynamic quantity to be measured. For "slow" processes (7 2 loU6 some optics) impinges directly upon the photomultiplier. The output of the photomultiplier is proportional to the scattered photon intensity which fluctuates in time since it arises from the thermal motions of the scat- terers. This output is then fed into a device which computes the time autocorrelation function of this scattered intensity (or a quantity equivalent to it). This device is usually an autocorrelator although in some cases it is a spectrum analyzer or a computer. The resulting autocorrelation function is then usually fed into a computer for further analysis. Most dynamic light scattering experiments on macromolecules in solution have in fact been of this type-variously called optical-mixing spectroscopy, intensity-fluctuation spectroscopy or photon correlation spectroscopy.

In the case of faster fluctuations ( T s lo-' tor is placed before the photomultiplier. The filter is usually a piezo- electrically swept Fabry-Perot interferometer, although pressure-swept Fabry-Perot's are sometimes used. The D.C. output of the photomultiplier is measured for each filter setting as the filter is swept. This technique gives the spectrum of scattered light directly. On-line computers are usually available to collect and further analyze the scattered spectrum.

In both the optical-mixing and Fabry-Perot methods the polarization of the incident and scattered light may be selected. The plane of the incident and scattered beams define the scattering plane. The polarization of the incident light is usually chosen to be perpendicular to this plane. The component of scattered light with polarization also perpendicular to the scattering plane is called the "polarized component" and the component of scattered light with polarization in the scattering plane is called the "depolarized component.Il contain different information about polymer dynamics. Most dynamic light scattering experiments performed on macromolecular solutions measure only the polarized component. The depolarized component is usually much weaker in intensity than the polarized component (about of the polarized intensity), and often relaxes on a faster time scale than the polarized component, necessitating the use of the Fabry-Perot method. However, as we shall describe, the depolarized component contains much information not obtainable by polarized scattering. In the following discussion, polarized and depolarized scattering are discussed separately. In this discussion the small contribution to the polarized scattering of terms similar in form to the depolarized scattering is neglected since they are usually not easily detected.

6 ), the scattered light (after traversing

s ), a filter or monochroma-

The polarized and depolarized components usually

POLARIZED SCATTERING

Translational Diffusion Coefficients The reciprocal of q, the magnitude of the scattering vector, is an important characteristic length in scattering experiments. The scattering vector length depends upon h the wavelength of the incident light in the scattering medium and the scattering angle 8 ,

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For dilute macromolecular solutions for which qL s 1, where L is a characteristic length of the molecule, it may be shown (2,4) that the time autocorrelation function of the polarized scattered intensity, &, is given by

where D is the macromolecular translational diffusion coefficient and A and B are constants. Optical-mixing experiments measuring D are the most common and routine of the dynamic light scattering experiments on macromolecules. It is currently the best method of measuring diffusion coefficients. Furthermore, since the translational diffusion coefficient changes upon structural change of the macromolecule, light scattering measurements of D may be used to study such processes as helix-coil transi- tions (19) and protein denaturation (20). The thermal denaturation of ribonuclease has been studied by measuring D throughout the transition region by dynamic light scattering (21). The method is rapid (usually requiring less than a minute), nondestructive, and be performed on very small sample volumes (volumes ~0.01ml).

There is now a large literature on this subject, including theoretical and experimental studies of the concentration dependence of D in what is called the "dilute region" for both rigid, spherical and flexible macromolecules (5-13). Electric charge and ionic strength effects on D have also been investigated (11-18). There is still, however, considerable controversy about the correct form of the concentration dependence of D for the rigid spherical polymers studied, since it is difficult to separate the effects of charge, solution nonideality and hydrodynamic friction from each other. For nonspherical, rigid polymers there is as yet no theory for the second virial coefficient and few experiments.

In a polydisperse polymer solution, the diffusion coefficient will be a function of molecular weight, so that the time-autocorrelation function (Eq. 2) becomes a superposition of exponentials. By analyzing the form of the experimentally measured correlation functions, useful measures of the polydispersity may be obtained (22,23).

Intramolecular and Rotational Relaxation Times If the product of the scattering vector and a characteristic length of a molecule is of order one or greater, intramolecular interference can affect the dynamic polarized light scattering spectrum. For instance, for a thin- rod shaped macromolecule of length L, the time correlation function of the polarized intensity is

where 0 is the rotational diffusion coefficient of the long axis of the rod (24). The coefficients in Eq. (3) are functions of qL. S1(qL) vanishes as qL - 0 and So(qL) approaches 1 as qL - 0. angle), D may be obtained. This value of D and theoretical expressions for SO and S 1 may then be used with data obtained at high q to obtain values of the rotational diffusion coefficient @ (25-26). observe both 0 and D byopolarized scattering the rod-shaped molecule must be at least about 1000 A in length. Such long rigid molecules are usually biological structures such as viruses and structural proteins.

For flexible molecules obeying Rouse-Zimm dynamics, Pecora (27,28) has shown that the time-autocorrelation of the polarized scattered light has a form similar to that given in Eq. 3 , except that in place of 3 times the

Thus, at low q (low

In order to

rotational diffusion coefficient, one finds the reciprocal of the relaxation time for the longest wavelength intramolecular chain mode. If the product of q and the radius of gyration RG of the chain is small compared to 1, the term containing the intramolecular relaxation is negligible and the measured time correlation function will be given by Eq. ( 2 ) . Perico -- et al, obtain similar results using a Rouse-Zh model including hydrodynamic interaction (29). If 4% >.> 1, then the simple form of Eq. ( 3 ) is no longer adequate to describe the spectrum. DuBois-Violette and deGennes have, using the Rouse-Zimm model, derived analytical forms for this case ( 3 0 ) . Fujime has modified these theories to take account of chain stiffness ( 3 1 , 3 2 ) .

Several groups have performed polarized scattering experiments to observe intramolecular modes of large molecules. Most of these experiments have been performed on polystyrenes ( 3 3 - 3 7 ) and various DNA's ( 3 8 - 4 3 ) . Some as yet unpublished work on 1-phage DNA performed by C.-C. Wang in my labora- tory illustrates some of the major features of polarized scattering from these systems.

Semi-Dilute Solutions and Gels Perhaps the most promising recent application of polarized dynamic light scattering is to the study of flexible polymer chains in the "semi-dilute" solution, that is, in solutions in which the polymers overlap strongly but in which there remain significant amounts of solvent. The theory of light scattering from such systems is based largely on scaling concepts intro- duced by deGennes ( 4 4 ) . In semi-dilute solutions, motion of the molecule as a whole is severely restricted since there are many contact points between chain segments. According to deGennes, there is an important characteristic length, 5 , giving the mean distance between contact points. Different dynamical behavior is observed depending upon q5. If q5 k 1, the system is described as a "pseudogel" and the light scattering experiment probes motion between contact points. These theoretical ideas are discussed elsewhere in this symposium. Experiments on polystyrenes have shown that the theory is in qualitative accord with experiment ( 4 5 - 4 7 ) .

Gels with chemical cross-links have been studied by several groups, both from the hydrodynamic point of view ( 4 8 - 5 2 ) and a microscopic point of view considering gel cross-links as harmonically bound "particles" execut- ing Brownian motion about an equilibrium position ( 5 3 - 5 5 ) .

Bulk Polymers Brillouin spectra as measured by Fabry-Perot interferometry provide a con- venient means of measuring the dispersion of sound velocities in the hypersonic region. These studies have yielded a wide variety of thermo- dynamic and kinetic parameters ( 5 6 ) .

Optical-mixing studies of the polarized light scattering frompoly(methy1 methacrylate) in the amorphous state have shown that the correlation functions measured are not simple exponentials and that more adequate methods of correlation function characterization must be found before much progress can be made with this method ( 5 7 - 5 9 ) .

DEPOLARIZED SCATTERING

Rotational Relaxation Times For dilute solutions of rigid cylindrically-symmetric molecules with optical anisotropy 8, the spec<rum of depolarized scattered light, IW, is a single Lorentzian,

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where w is the frequency s h i f t of the s c a t t e r e d l i g h t from the inc iden t l a s e r frequency and 0 i s the r o t a t i o n a l d i f fus ion c o e f f i c i e n t of the rod. In most cases where the Fabry-Perot technique is appl icable the q2D con t r i - bu t ion i n Eq. 4 i s neg l ig ib l e so t h a t the s p e c t r a l width depends so l e ly pn the r o t a t i o n a l d i f fus ion c o e f f i c i e n t @.

optical-mixing experiments which a l s o give @ c a n be performed. molecules with molecular weights l e s s than about lo6, the Fabry-Perot technique i s usua l ly u t i l i z e d .

A frequent procedure f o r cha rac t e r i z ing r i g i d macromolecules i n so lu t ion i s t o combine polarized l i g h t s c a t t e r i n g measurements o f D wi th depolarized measurements of 0. The r o t a t i o n and t r ans l a t iona1 ,d i f fus ion c o e f f i c i e n t s obtained can then be used along with t h e o r e t i c a l r e l a t i o n s between these q u a n t i t i e s and so lu t ion dimensions t o obta in the s o l u t i o n dimensions (60). We have used t h i s procedure fo r Bovine Pancreatic Trypsin Inh ib i to r , a p ro te in molecule wi th molecular weight 6 5gO. using the Pe r r in r e l a t i o n s a r e 32 A by 16 A (61)

Intramolecular Relaxation Times of F lex ib le Polymers i n Di lu te Solu t ion Depolarized spec t r a of f l e x i b l e and semi- f lex ib le polymers y i e ld i n t r a - molecular r e l axa t ion times. The mechanism f o r t h e i r appearance i s d i f f e r e n t from t h a t f o r po lar ized spec t r a where the s t r u c t u r a l f l uc tua t ions must be of order l / q i n order t o be observable. In the depolarized case , a l oca l change i n the p o l a r i z a b i l i t y due to tumbling of some o p t i c a l l y - an iso t ropic group could a f f e c t the depolarized spectrum. The depolarized spectrum then i s capable of observing " loca l motions."

Bauer, Brauman and Pecora have observed the depolarized spec t r a of poly- s ty renes i n d i l u t e so lu t ion (62). The spec t r a e x h i b i t a t l e a s t two components-a slow component whose r e l axa t ion time i s propor t iona l t o molecular weight and a f a s t component whose r e l axa t ion time is independent of molecular weight. For d i l u t e polystyrenes i n CCl4, the f a s t r e l axa t ion time i s 4.5 f 1.0 ns . The slow r e l axa t ion t i m e i n these systems i s i d e n t i f i e d with h a l f the r e l axa t ion time of the longes t Rouse-Zim mode for the chain, and the f a s t , molecular-weight-independent t i m e is assigned t o a l o c a l , co r re l a t ed motion o f phenyl groups about the main cha in back- bone. Monte Carlo simulations of t he dynamics of chains on a t e t r ahedra l l a t t i c e with o p t i c a l l y an i so t rop ic s i d e groups g ive depolarized spec t r a t h a t a r e q u a l i t a t i v e l y s imi l a r t o those observed i n polystyrenes (63). Moro and Pecora ( 6 4 ) have attempted t o expla in the dominance of t h e longes t chain mode i n the low frequency p a r t of the depolarized spectrum by i n t r o - ducing l o c a l t ransverse s t i f f n e s s i n t o the chain. Experiments from my labora tory on polyamic ac id and D N A ' s i n so lu t ion ind ica t e t h a t l oca l motions con t r ibu te t o the depolarized spec t r a of these polymers.

Semi-Dilute Solutions To the b e s t of my knowledge, no depolarized s c a t t e r i n g experiments have been published on f l e x i b l e polymers i n the semi-dilute reg ion i n which polymer chains overlap s i g n i f i c a n t l y bu t i n which la rge amounts of so lvent a r e s t i l l present . S tudies have, however, been performed on r ig id- rod polymers i n t h i s region. Time cons tan ts from depolarized s c a t t e r i n g f o r semi-dilute so lu t ions of l i g h t mesomyosin ( 6 5 ) agree only q u a l i t a t i v e l y with the pred ic t ions of Doi (66) f o r r o t a t i o n a l motion of so lu t ions of long t h i n rods . by Doi. a f lex ing time of t he molecule i n the semi-dilute reg ion where ove ra l l

I f r o t a t i o n is slow (p/6)3 5 10- s ), the Fabry-Perot technique is no longer appl icable , bu t depolarized

For macro-

Solu t ion dimensions obtained

The time cons tan ts a r e much f a s t e r than those pred ic ted It i s poss ib le t h a t t he l i g h t s c a t t e r i n g technique i s measuring

molecular rotation may be highly hindered.

Bulk Polymers and Solutions of Small Molecules in Glassy Polymers Few depolarized light scattering experiments have been performed on bulk polymers. Patterson a. ( 6 7 ) have studied depolarized scattering from polystyrene near the glass-rubber transition using optical-mixing spectroscopy. They find strongly non-exponential correlation functions that are difficult to interpret.

Ouano and Pecora ( 6 8 ) have studied the rotational motion of chlorobenzene in pol$nethyl methacrylate) glasses. They find that the depolarized spectrum consists of a fast component (7 - picoseconds) and a slow component (T =nanoseconds). A possible interpretation of these results is that the chlorobenzene in these glasses finds itself in different environments, one of which is similar to that of bulk chlorobenzene.

CONCLUSION

A wide variety of problems in polymer science have been studied by dynamic light scattering. Major areas awaiting further development in the next few years include the polarized scattering from semi-dilute and concentra- ted polymer solutions and bulk polymers, as well as the whole field of de po lar ized scattering .

Acknowledgement - This work was supported by grants from the National Institutes of Health and the National Science Foundation (USA).

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Dynamic light scattering from polymers