Polymer Molecular Weight Methods - American Chemical Society

4 Self-Beat Spectroscopy and Molecular Weight

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N. C. FORD, JR., R. GABLER, and F. E. KARASZ University of Massachusetts, Amherst, Mass. 01002

Self-beat spectroscopy is a newly developed optical technique in which the spectral distribution of light scattered from solute molecules undergoing Brownian motion is analyzed to allow calculation of the diffusion coefficient of the molecules. A brief review of the literature shows that this technique has been applied to many types of solutions containing molecules whose molecular weights range from 10 to 10 . Using proper experimental procedure and solutions in thermal equilibrium, diffusion coefficients can be obtained in several minutes with accuracies typically on the order of 1%. Diffusion coefficients can be related to molecular weights by the Stokes-Einstein equation, by combination with sedimentation velocity data, or by the consideration of homologous polymer solutions. 4

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'T^he interaction of light with matter has long provided important clues into the mysteries of molecular structure and behavior. The informa tion obtained from such interactions is almost as varied as are the different parts of the electromagnetic spectrum itself. The phenomena of light scattering, absorption, fluorescence, rotation, reflection, and refraction have been used to add to our knowledge of molecular systems. Tech niques utilizing these phenomena can yield information on a gross level in which parameters such as the general shape, conformation, and molec ular weight are important; they can also help elucidate details of microstructure i n which the nature and strengths of chemical and physical forces are important. Each technique gives a unique type of information about the system being studied because of the nature of the interaction process with it. In this paper we describe a recently developed technique utilizing the interaction of light with matter—i.e., self-beat spectroscopy. In particular, we consider the light scattered from a solution of molecules A

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which are i n a state of macroscopic equilibrium and show that diffusion coefficients of these molecules may be readily obtained by studying the spectral distribution of this scattered light. When light traverses a perfectly homogeneous transparent solution, it w i l l not be scattered although it may be refracted at the entrance and exit interfaces. If, however, the solution contains inhomogeneities, which all real solutions do because of thermal excitations, the incident light w i l l also be scattered. L o r d Rayleigh ( J ) was the first to obtain a quantita tive expression relating the intensity of the scattered light of incident wavelength λο to the scattering angle Θ. His results pertained to dilute gases containing molecules small compared with λ , and his work also provided a theoretical explanation for the blue color of the sky. ( It has also been reported (2) that Leonardo da V i n c i around 1500 thought the color of the sky was caused by scattered light.) In 1935, Putzeys and Brosteaux (3) reasoned that Rayleigh's equations would also describe the excess fight scattered by a dilute solution of macromolecules over that of the solvent, and they used this fact to examine the molecular weights of several small proteins. Thus began the use of light scattering as an analytical tool i n physics and chemistry. Analysis also showed that the light scattered from particles whose dimensions were on the order of the wavelength of the incident light contained information regarding their size and shape (4-6). This result becomes plausible when one con siders that the light scattered from different parts of a particle with a fixed geometry w i l l interfere constructively and destructively in a manner de pendent on that geometry, thus giving rise to an angular distribution of intensity. Zimm (7) utilized these concepts to develop a technique which would give both the molecular weight of a scattering molecule and its radius of gyration R . This method has since become a standard way to determine molecular weights. 0

G

However, Rayleigh's work failed to describe adequately the effects of scattering from concentrated solutions. Experimentally, one observes a diminution of scattering intensity from the expected Rayleigh intensity as the solution concentration increases. This is caused by stronger inter actions between scattering molecules and a loss of the spatial domain in which the molecules may move. Concentrated solutions have neigh boring molecules which scatter light whose phases are related, and the Rayleigh assumption of independent (or incoherent) scatterers is no longer valid. A theory describing all situations was advanced by Einstein (8) who asserted that the scattering was caused by fluctuations in the local dielectric constant. Debye (9, JO) then interpreted these fluctua tions of the dielectric constant in solutions to be caused by fluctuations in the local concentration of the scattering molecules; it is this approach which has laid the theoretical foundation for macromolecular light-scat-


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tering experiments. The classical technique of light scattering, where the scattered intensity is measured as a function of scattering angle, can provide the molecular weight, information about the gross shape, and, although it is not discussed here, information pertaining to the thermo dynamic interaction with the solvent i n which the molecules are dispersed (virial coefficients). There are several reviews concerning this type of scattering and its applications (6, 11, 12, 13). In the application of Einstein's theory to conventional light scattering, only the time-averaged dielectric constant fluctuation < ( 8 c ) > was con sidered and not its time dependence. The time dependence of these fluctuations would be expected to contain additional information. M i c r o scopic fluctuations in concentration would not only cause the dielectric constant to deviate from an average value but would also shift the frequency of the scattered light in a manner directly related to the time dependence of the fluctuations. Because these fluctuations grow and decay in time, the scattered light is modulated, and consequently the frequency of the scattered light is shifted from that of the incident. Alter natively, it is perfectly correct to consider the frequency shifts of the scattered light to be a Doppler effect initiated by the Brownian motion of the individual molecules. The light scattered from a macroscopic solu tion illuminated by a monochromatic beam then would display a spec trum centered around the incident frequency. The width of this spectrum would contain information related to the statistical motion of the mole cules. This implies that the light scattered by concentration fluctuations in a solution exhibits a spectrum characteristic of the time dependence of these concentration fluctuations. The width of the expected spectrum, however, is much too small to be observed by conventional spectrometers. However, in 1947, Forrester, Parkins, and Gerjouy (14) proposed a tech nique (optical mixing) where the beat frequencies of the interfering optical waves might be measured. Using this idea, Forrester, Gudmundsen, and Johnson (15) did observe beat frequencies between the split Zeeman lines of a mercury light source although their signal-to-noise ratios were low; because of this, the method was not directly applicable to molecular light scattering. W i t h the development of the laser, a suffi ciently powerful light source with precise monochromaticity became available, and measurements of the beat frequencies contained in the scattered light became possible. Both Pecora (16) and Komarov and Fisher (17) adapted van Hove's space-time correlation function approach for neutron scattering (18) to the light-scattering problem to calculate the spectral distribution of the light scattered from a solution. Using a molecular analysis, Pecora assumed the scattering particles to be undergoing Brownian motion, and predicted a Lorentzian line shape for the spectral distribution of the

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scattered light. H e also predicted that the width of the Lorentzian would be directly related not only to the translational diffusion coefficient but also in some cases to the rotational diffusion coefficient and the frequen cies of the normal modes of vibration for flexible molecules. In later work he considered the effects of solutions containing polymers and other types of molecules with a variety of different conformations and flexi bilities on the spectral profile (19-21). In 1964, Cummins, Knable, and Yeh (22) observed the spectral broadening of the scattered light from the Brownian motion of solute molecules for the first time. Using dilute solutions of polystyrene latex spheres ( P S L ) and the technique of optical mixing, they achieved a resolution ( ω / Δ ω ) on the order of 10 which was much greater than any other form of optical spectroscopy. It remained, however, for D u b i n , Lunacek, and Benedek (23) and Arecchi, Giglio, and Tartari (24) to make quantitative measurements on solutions of P S L to determine that the observed spectral profile agreed with Pecora's theory. In their work, D u b i n et al. (23) also showed the scattered profile to be Lorentzian for solutions of several small proteins and non-Lorentzian for scattering from solutions of calf thymus deoxyribonucleic acid ( D N A ) and tobacco mosaic virus ( T M V ) , respectively. W i t h these initial successes, the tech nique of optical mixing rapidly became a means of measuring molecular parameters not easily observed with other methods. 14

A survey of the literature shows that the technique of optical mixing has been used to obtain the spectrum of the scattered light from many types of solutions containing biologically interesting molecules whose molecular weights have ranged from 10 to 10 daltons. As stated before, initial measurements on solutions of T M V showed a non-Lorentzian line shape (-23) although later analysis (25) demonstrated that the observed shape was a sum of superimposed Lorentzians whose individual widths were related to the translational diffusion coefficient (D ) and the rota tional diffusion coefficient (D ). Wada, Suda, Tsuda, and Soda (26) also measured the scattered depolarized spectral profile of T M V solutions to determine D while D u b i n , Clark, and Benedek (27) performed a similar measurement on solutions of the protein lysozyme. Other macromolecules whose spectral widths have been measured are the coliphages T4, T5, T7, and λ (28); the blood protein haemocyamin (29); the poly(«-amino acid) poly(y-benzyl L-glutamate) (30); polystyrene in 2-butanone (31); the copolymer p o l y - d ( A T ) (32); the enzyme RNase (33); the viruses M S 2 (34) and R17 (35); the proteins lysozyme (23, 36), bovine serum a l bumin (23, 37), ovalbumin (23), and casein (38); and the contractile muscle protein myosin (39, 40). The effect of motile organisms such as sperm and bacteria on the scattered spectrum has also been considered (41^43). Finally, it has been reported that the spectral profile of light 4

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scattered from solutions may be used as a probe to investigate the kinetics of chemical reactions which are occurring within a solution (32, 44-48). A comprehensive treatment of many phases of light scattering is found in the text by Fabelinskii (49). A treatment of the specific topic of optical mixing techniques may be found i n reviews by C h u (50), C u m mins and Swinney (51), Benedek (52a), and F o r d (52b). From this brief review of light-scattering techniques, it can be seen that this approach is one of the most powerful methods for molecular investigation. Not only is it profitable to measure the distribution of scattered intensities as a function of scattering angle, but it is also prac tical and advantageous to measure the spectral distribution as well. W i t h i n the last several years, investigators using optical mixing tech niques have progressed from considering simple systems such as solutions of P S L spheres to pursuing the complicated dynamics of myosin and probing the kinetics of the helix-coil transformation i n polymers. F r o m a biochemical viewpoint, optical mixing techniques represent a powerful investigative tool, both for the type of information obtained and for the rapidity with which information is obtained. W h e n used i n conjunction with already established physicochemical techniques, this should provide a more detailed description of molecular systems. Theory W h e n an electromagnetic wave of angular frequency ω is incident upon a dilute solution, it causes electrons to oscillate with a frequency equal to that of the initial wave. The magnitude of these displacements depends on the polarizability of the media, which we w i l l assume to be isotropic, and the magnitude of the electric field. Since it is a funda mental principle of electrodynamics that an oscillating electric charge w i l l radiate energy, the vibrating electrons reradiate energy in the form of spherical electromagnetic waves of frequency ω . These waves com bine through interference to form the resultant wave in the liquid. This then, is a brief qualitative sketch of the phenomenon responsible for light scattering as well as refraction. W e now present a mathematical description of the process and later consider the statistical nature of the light scattered from a group of randomly moving particles. Consider a planar electromagnetic wave i n a medium with index of refraction n, having the form 0

0

E

L

= Ε exp[i(k · r -

orf)]

(!)

where Ε is the vector magnitude, k is the wave vector with |k| = 2πη/λ , α>ο is the angular frequency, and r is the position vector. The medium 0

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is then polarized according to the relation P(r, t) = a(r, t)E

(2)

L

where « ( r , t) is local polarizability. O n a microscopic level a ( r , t) w i l l not actually be a constant but w i l l vary because of thermal fluctuations; hence we can write


a(r, t) = a

with a being the time-averaged of Equation 3 is responsible for the second term and its relation fluctuation of local polarization tion vector of the form 0

0

+ 8a(r, 0

(3)

polarizability. The first term on the R H S refraction although we w i l l consider only to the light-scattering phenomenon. This then gives rise to a fluctuating polariza

8P(r, t) = 8a(r, t)E

(4)

L

It is these fluctuations that cause the inhomogeneities in the fluid the reradiated waves. The total amplitude of the electric field at a tance R from the scattering volume is given by the summation of contributions of the infinitesimal scattering elements and can be pressed as

and dis the ex

where V is the illuminated volume, t = t — ( |R — r\/c ) is the retarded time, c and c are the velocity of light in the liquid and in vacuo, respec tively, and Κ = ( R — r ) / | R — r| is a unit vector in the direction of the scattered light. If it is assumed that the magnitude of the first two time derivatives of 8a(r, t) is negligible compared with the frequency of the incident radiation ω , then after some manipulation and, remembering that for a nonmagnetic media R

L

L

0

ε = 4χα + 1

(6)

where c is the dielectric constant, Equation 5 reduces to E.(R, t) = A k x ( k x E ) exp[i(k. · R s

8

corf)] f

8e(r, t)

(7)

exp[t'(k - k.) · r]dV where now |k | = n|ko|, and A represents constant factors. A t this point it is possible to gain some physical insight into the scat tering process. To do this, we w i l l consider that the fluctuations i n the s

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local dielectric constant can be represented mathematically by an infinite number of weighted sine and cosine functions, each pair having a differ ent frequency (or wave vector) and phase argument. This is merely saying that the complicated fluctuations in the dielectric constant may be decomposed into an infinite number of well-defined periodic functions —i.e., the fluctuations may be represented by a Fourier decomposition (54). Using complex notation, this concept may be written: 8e(r, t) = C f

8ε(Κ, t) exp[iK · r]d*K

0

(8)

where C is a constant and Κ is a dummy variable. Substituting E q u a tion 8 into Equation 7 yields 0

E ( R , t) = Ak x(k xE)

f

8

8

8

8e(K, t)\fvexp[i(k

exp[i(k · R -

ω at)]

s

(9)

- k . + K ) · r]dv\d*K

where now we need only integrate over the illuminated volume as all other space w i l l give negligible contributions to the integral. The ex pression i n braces is a three-dimensional Dirac delta function δ[Κ — ( k — k ) ] which has the property of being zero for all conditions except when Κ = k — k . This in effect is telling us that the scattering is caused by one particular component of the dielectric constant fluctuation, namely that fluctuation whose wave vector Κ is equal to the difference between the incident and scattered wave vectors. Fixing the scattered angle θ and the wavelength of the incident light λ w i l l determine the particular scattering wave vector ( K ) from which the scattering is being observed. From Figure 1 we see that 8

s

0

|K| = 2k sin (19/2)

(10)

where we have assumed |k| = |k | which is justified since ΔΚ/Κ — 10" . Interpreting this in terms of wavelength shows 12

e

λ = 2Λ sin ((9/2)

(11)

where Κ = 2ττ/Λ. Equation 11 is completely analogous to the Bragg equation for x-ray scattering from a crystal of lattice spacing Λ (55), and shows the similarity between the two processes. Equation 9 now becomes E.(R, t) = A k x ( k x E ) exp[i(k · R s

e

8

= < E(t)E*(t)E(t

+

τ) E*(t + τ) >

(26)

where the photocurrent is proportional to the incident intensity I(t). C (r) and Ci(r) must now be related. This relationship may be calcu lated if it can be assumed that the scattered field is a Gaussian random variable (59). Stating the result, we have E

Polymer Molecular Weight Methods - American Chemical Society

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