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DYNAMIC LIGHT SCATTERING
WITH APPLICATIONS TO CHEMISTRY, BIOLOGY AND PHYSICS
By Bruce J. Berne, Robert Pecora Dover Publications, Inc.
Copyright © 2000 Bruce J. Berne and Robert Pecora
All rights reserved.
ISBN: 978-0-486-32024-3
CHAPTER 1
INTRODUCTION
1 · 1 HISTORICAL SKETCH
Electromagnetic radiation is one of the most important probes of the structure and dynamics of matter. The absorption of ultraviolet, visible, infrared, and microwave radiation has provided detailed information about electronic, vibrational, and rotational energy levels of molecules and has in some instances enabled the chemist and physicist to determine the structure of complex molecules. Radiofrequency spectroscopy has had an enormous impact on solid-state and molecular physics and physical, inorganic, and organic chemistry. The structure of solids and biological macromolecules has been elucidated by x-ray diffraction experiments. Raman scattering is another spectroscopic technique that provides information similar to that of absorption spectroscopy. When photons impinge on a molecule they can either impart energy to (or gain energy from) the translational, rotational, vibrational, and electronic degrees of freedom of the molecules. They thereby suffer frequency shifts. Thus the frequency spectrum of the scattered light will exhibit resonances at the frequencies corresponding to these transitions. Raman scattering, therefore, provides information about the energy spectra of molecules. This book deals only with the characteristics of the light scattered from translational and rotational degrees of freedom, that is, with what is now commonly called Rayleigh scattering.
Recent advances in laser techniques have made possible the measurement of very small frequency shifts in the light scattered from gases, liquids, and solids. Moreover, because of the high intensities of laser sources, it is possible to measure even weakly scattered light. Thus the main difficulties in performing light-scattering experiments encountered in the past are eliminated when lasers are used. This explains the rather remarkable proliferation of laser light-scattering experiments in recent years. The structure and dynamics of such diverse systems as solids, liquid crystals, gels, solutions of biological macromolecules, simple molecular fluids, electrolyte solutions, dispersions of microorganisms, solutions of viruses, membrane vesicles, protoplasm in algae, and colloidal dispersions have now been studied by laser light-scattering techniques.
When light impinges on matter, the electric field of the light induces an oscillating polarization of the electrons in the molecules. The molecules then serve as secondary sources of light and subsequently radiate (scatter) light. The frequency shifts, the angular distribution, the polarization, and the intensity of the scattered light are determined by the size, shape, and molecular interactions in the scattering material. Thus from the light-scattering characteristics of a given system it should be possible, with the aid of electrodynamics and the theory of time-dependent statistical mechanics, to obtain information about the structure and molecular dynamics of the scattering medium.
The basic theory of Rayleigh scattering was developed more than a half century ago by Rayleigh, Mie, Smoluchowski, Einstein, and Debye. It is well worth summarizing some of the high points in the history of the field.
Since the experimental studies by Tyndall (1869) on light scattering from aerosols and the initial theoretical work of Rayleigh (1871, 1881), light scattering has been used to study a variety of physical phenomena. These studies concerned scattering from assemblies of noninteracting particles sufficiently small compared to the wavelength of the light to be regarded as point–dipole oscillators. In his 1881 article Rayleigh also presented an approximate theory for particles of any shape and size having a relative refractive index approximately equal to one. Rayleigh (1899) explained the blue color of the sky and the red sunsets as due to the preferential scattering of blue light by the molecules in the atmosphere. In subsequent papers (Rayleigh 1910, 1914, 1918) Rayleigh derived the full formula for spheres of arbitrary size. For these larger particles there are fixed phase relations between the waves scattered from different points of the same particle, but each scattering element of the particle is regarded as an independent dipole oscillator. Debye (1915) made further contributions to the theory of these large particles and extended the calculations to particles of nonspherical shape.
Gans (1925) also contributed to the theory of large particles of relative refractive index approximately equal to one. The theory of such particles is often referred to as the Rayleigh-Gans theory. According to Kerker (1969), "Gans' contribution to this method was hardly significant and it seems more appropriate to call it Rayleigh-Debye scattering."
In large particles of relative refractive index much different from one there are not only fixed spatial relations between the scattering elements, there is also a strong dependence of the electric field amplitude on the position in the particle. There are formidable theoretical problems associated with the treatment of these large particles. Only for the case of spheres does there exist a complete solution. Mie (1908), and independently Debye (1909) solved this problem. This type of scattering is now referred to as Mie scattering. These problems are discussed at great length in the monographs by Van de Hulst (1957) and by Kerker (1969), and are consequently not considered in this book. Studies of the angular dependence and polarization of the scattered light are now routinely used to study the shapes and sizes of large particles.
Although Rayleigh had developed a theory of light scattering from gases with some success, it was soon found that the intensity of scattering by condensed phases (molecule per molecule) was less than that predicted by his formula by more than one order of magnitude. This effect was correctly attributed to the destructive interference between the wavelets scattered from different molecules, but unfortunately the means of calculating the extent of this interference were not known at that time. Smoluchowski (1908) and Einstein (1910) elegantly circumvented this difficulty by considering the liquid to be a continuous medium in which thermal fluctuations give rise to local inhomogeneities and thereby to density and concentration fluctuations. These authors developed a fluctuation theory of light scattering.
According to this theory, the intensity of the scattered light can be calculated from the mean-square fluctuations in density and concentration which in turn can be determined from macroscopic data such as the isothermal compressibility and the concentration-dependence of the osmotic pressure. The intensity of the light is thus obtained without considering the detailed molecular structure of the medium. This phenomenological approach to light scattering has continued to play a very important role in the theory of light scattering, although profound questions regarding the validity of this approach have been raised [see, for example, Fixman (1955), and more recently Felderhof (1974) and references cited therein].
The scattering from a system of particles whose positions are correlated (governed by a pair-correlation function) was investigated by Zernike and Prins (1927) in connection with the theory of x-ray diffraction of liquids. The same theory applies to light scattering from liquids. This theory was developed by Ornstein and Zernike (1914, 1916, and 1926), who extensively applied it to the study of the intense scattering of light that occurs in the fluid critical region (critical opalescence). The marked increase in the turbidity of the fluids near the gas-liquid critical point is a consequence of the fact that the pair-correlation function in a system near its critical point becomes infinitely long-ranged.
In the foregoing phenomenological theory no attempt was made to describe the effects of molecular optical anisotropy on the intensity, angular dependence, and polarization characteristics, of the scattered light. Subsequent work dealt with a molecular theory of independent optically anisotropic scatterers (Cabannes, 1929; Gans, 1921, 1923). Debye and Zimm and co-workers synthesized the Rayleigh-Debye and the phenomenological points of view in the 1940s and developed light scattering as a method for studying molecular weights, sizes, shapes, and interactions of macromolecules in solution. The classic papers on the subject are reprinted in McIntyre and Gornick (1964).
All these studies treated only the intensities of the scattered light. There was, however, a parallel development in light scattering which started with the work of Leon Brillouin (1914, 1922), who predicted a doublet in the frequency distribution of the scattered light due to scattering from thermal sound waves in a solid. This doublet is now known as the Brillouin doublet.
In the early 1930s Gross conducted a series of light-scattering experiments on liquids observing the Brillouin doublet and a central or Rayleigh line whose peak maximum was unshifted. Landau and Placzek (1934) gave a theoretical explanation of the Rayleigh line using a quasi-thermodynamic approach. They showed that the ratio of the integrated intensity of the central line to that of the doublet is given by the heatcapacity ratio (now known as the Landau-Placzek ratio):
Ic/Id = [cP - cV]/cV
This field was carried on by only a few workers, mainly in the Soviet Union and India (see, for example, Fabelinskii, 1968 and references cited therein), but it was not until the development of the laser in the early 1960s that these measurements of frequency changes became a major tool for the study of liquids. The modern hydrodynamic theory of light scattering from liquids is described in Chapters 10, 11, 12, and 13.
With the advent of the laser, another type of experiment became possible. In 1964, Pecora showed that the frequency distribution of light scattered from macromolecular solutions would yield values of the macromolecular diffusion coefficient and under certain conditions might be used to study rotational motion and flexing of macromolecules. These frequency changes are so small that conventional monochromators (or "filters") could not be used to resolve the frequency distribution of the scattered light. In 1964, Cummins, Knable, and Yeh used an optical-mixing technique to spectrally resolve the light scattered from dilute suspensions of polystyrene spheres. Since this pioneering work applications have proliferated, and optical-mixing spectroscopy has become a major field of research for workers in chemistry, physics, and biology.
It is the purpose of this book to describe the theory of light-scattering spectroscopy experiments and its applications to major topics of interest to chemists, physicists, and biologists. The older theories concerned with integrated intensities are described in detail only where they are of importance in understanding spectral distribution experiments. The emphasis throughout is on the use of light scattering to study the dynamics of fluctuations in fluids and not on the electrodynamical theory of the interaction of radiation with matter.
1 · 2 SYNOPSIS
In a light-scattering experiment, light from a laser passes through a polarizer to define the polarization of the incident beam and then impinges on the scattering medium. The scattered light then passes through an analyzer which selects a given polarization and finally enters a detector. The position of the detector defines the scattering angle θ. In addition, the intersection of the incident beam and the beam interecepted by the detector defines a scattering region of volume V. This is illustrated in Fig. 1.2.1 Prelaser light-scattering experiments usually used mercury sources. The detector used in these experiments was normally a phototube whose dc output was proportional to the intensity of the scattered light beam. In modern light-scattering experiments the scattered light spectral distribution (or the equivalent) is also measured. In these experiments a photomultiplier is the main detector, but the pre- and postphotomultiplier systems differ depending on the frequency change of the scattered light. The three different methods used, called filter, homodyne, and heterodyne methods, are schematically illustrated in Fig. 1.2.2 Note that homodyne and heterodyne methods use no monochromator or "filter" between the scattering cell and the photomultiplier. These methods are discussed in Chapter 4.
The spectral characteristics of the scattered light depend on the time scales characterizing the motions of the scatterers. These relationships are discussed in Chapter 3. The quantities measured in light-scattering experiments are the time-correlation function of either the scattered field or the scattered intensity (or their spectral densities). Consequently, time-correlation functions and their spectral densities are central to an understanding of light scattering. They are, therefore, discussed at the outset in in Chapter 2.
The theory of light scattering from the simplest systems—dilute solutions or gases composed of spherical molecules—is presented in Chapter 5. This chapter includes discussions of the applications of light scattering to the study of macromolecular diffusion, electrophoretic motions, and the motility of microorganisms. In Chapter 6, a theory of light scattering from a simple model system in chemical equilibrium is presented. Conditions are given under which it might be possible to measure rate constants for chemical reactions by this method, although there have as yet been no unequivocal experimental results that report measurements of rate constants. An important new technique, fluorescence fluctuation spectroscopy (FFS), is also discussed in this chapter. This technique has been successfully used to measure rate constants for binding of small molecules to macromolecules as well as the diffusion of molecules in membranes. It was thus felt that a treatment of chemical kinetics would be of value to workers in these related areas.
Light scattering can be used to measure rotational time constants for nonspherical molecules in gases and solutions. The theory of scattering from these systems is somewhat more complicated than that from spherical molecules, so that in Chapter 7 several alternative procedures for arriving at some of the results are presented. The mathematical techniques presented in this chapter are useful also for treating related problems such as fluorescence depolarization (Appendix 7.B), electron-spin resonance (ESR), nuclear magnetic relaxation (NMR), and neutron scattering.
When molecules are no longer small compared to the wavelength of light, intramolecular interference becomes important in light-scattering experiments. Since this interference depends on the mass distribution in the molecule, this phenomenon forms the basis for measurements of radii of gyration of macromolecules from integrated intensity measurements. Chapter 8 reviews the theory of light scattering from polymer solutions and also shows how intramolecular interference affects the scattered light frequency dependence and the integrated intensity.
Chapters 9-14 treat systems composed of interacting molecules and the collective modes in these systems. Chapter 9 shows how the long-range Coulomb forces affect light-scattering spectra from solutions. Chapter 10 gives a short treatment of the phenomenological basis of hydrodynamics and then applies it to the calculation of light-scattering spectra. The Brillouin doublet and central line described in Sec. 1.1 as well as the Landau-Placzek ratio are all predicted by this theory.
Chapter 11 reviews the statistical mechanical basis of hydrodynamics and discusses theories that may be used to extend hydrodynamics beyond the "classical" equations discussed in Chapter 10. Chapter 12 applies the statistical mechanical theory to the calculation of depolarized light-scattering spectra from dense liquids where interactions between anisotropic molecules are important.
(Continues...)
Excerpted from DYNAMIC LIGHT SCATTERING by Bruce J. Berne, Robert Pecora. Copyright © 2000 Bruce J. Berne and Robert Pecora. Excerpted by permission of Dover Publications, Inc..
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