7.3 APPLICATIONS OF PHOTON CORRELATION SPECTROSCOPY TO POLYMERS 7.3.1 Scope 7.3.2 Polymer and Colloid Characterization 7.3.3 Polymer Dynamics 7.3.4 Gels 7.3.5 Bulk Polymers 7.3.6 Liquid Crystals 7.3.7 Polymer Adsorption 7.3.8 Phase Transitions 7.3.9 Electrophoretic Scattering and Application to Poly electrolytes
924 924 924 925 927 928 928 929 930 930
7.4
931
7.1
REFERENCES
INTRODUCTION
The light-scattering technique of photon correlation spectroscopy is now firmly established as a valuable tool for the study of the structure and dynamics of macromolecules. Over the last 25 years there has been steady development of the technique and a growing, and now extensive, range of applications to polymer systems. The determination of translational and rotational diffusion coefficients and internal molecular flexing provides a means of polymer characterization for molecular weight and molecular weight distributions, giving size and structural information. Study of polymer dynamics in isolated or weakly interacting molecules in dilute solutions, and strongly interacting molecules in semidilute and concentrated solutions, has provided experimental data concurrent with the extensive development of theoretical models of polymers using renormalization group and scaling theories. Photon correlation spectroscopy is also an important technique in the study and characterization of crosslinked and physical gels, solid and melt bulk polymers, colloids, liquid crystals, aspects of polymer phase transitions and polyelectrolytes. As well as the application to synthetic polymers the technique has found broad application to biopolymers; these applications are not addressed here. Molecular scattering of light has been investigated over the last 100 years, and scattering at the incident wavelength, Rayleigh and Rayleigh-Debye scattering, has been used extensively in more recent times for characterization of molecular weight, size, shape and interactions. The treatment of total intensity light scattering and its application to static molecular and particle characterization is CPC-DD
911
Solution Methods
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well documented. 1 " 5 Photon correlation spectroscopy is a more recently developed inelastic lightscattering process, involving frequency broadening of the central Rayleigh line about the incident light wave frequency. The earlier discoveries of the inelastic light-scattering processes of Raman and Brillouin scattering, in which there is a frequency shift between the incident and scattered light, have also been applied extensively to polymer systems. Combined Rayleigh-Brillouin scattering has proven fruitful in the study of bulk systems. We concentrate here only on the technique and application of photon correlation spectroscopy. As well as the name 'photon correlation spectroscopy' many other names have been used in the literature to describe the method, in which broadening of the central Rayleigh line is measured, dependent on the way in which the scattered signal is detected or processed. These names include: light-beating spectroscopy, self-beat spectroscopy, quasielastic light scattering, Rayleigh linewidth spectroscopy, laser Doppler spectroscopy, intensity fluctuation spectroscopy and optical mixing spectroscopy. For standardization we adopt here the general name of photon correlation spectroscopy (PCS). This name has recently come into more general use since processing of the scattered signal is usually done by digital time autocorrelation of detected photons. Light incident on a medium induces an oscillatory polarization of the medium which in turn acts as a secondary source of radiation in the form of scattering. Fluctuations in the dielectric constant of the medium induce net scattering of radiation. The intensity, angular distribution, polarization and frequency shift of the scattered light are determined by the size, shape and molecular interactions of the scattering centres. With this process information on the structure, interactions and molecular dynamics within the scattering medium can be derived. The frequency distribution of light scattered from macromolecules was investigated by Pecora (1964 and subsequently) 6-8 who showed that the macromolecular translational diffusion coefficient, the rotational diffusion coefficient and internal motion dynamics may be derived. Also in 1964, the previously demonstrated 9 technique of optical mixing was used 10 to measure the small frequency broadening of the Rayleigh-scattered peak from dilute suspensions of polystyrene latex particles. The subsequent growth of photon correlation spectroscopy and its application to polymers and biopolymers is described in a number of books and reviews. 11-30 For early and general introduc tions see the books of Berne and Pecora 17 and Chu 12 and to two review proceedings. 1 3 1 8 , 2 0 - 2 2 Attention is also drawn to Chu's account of the application of PCS to polymer solutions in Chapter 8 of this volume. 7.2 7.2.1
PHOTON CORRELATION SPECTROSCOPY TECHNIQUES Summary of Light Scattering
Light is scattered by a molecule in solution if the molecule has a polarizability different from its surroundings. 31-33 The molecular polarizability difference gives a spatial inhomogeneity to the dielectric constant of the medium or equivalently a refractive index difference. The classical mechanism of light scattering involves the electric field of the incident light inducing an oscillating dipole moment in the molecule which re-radiates to form the scattered radiation. Inhomogeneities of the medium may arise from spontaneous thermal fluctuations or concentration fluctuations, for example for a polymer solution or colloidal dispersion. The intensity of the scattered light depends on the intensity of the incident light, the scattering angle, the solution parameters and the light polarization. Figure 1(a) illustrates a form of the scattering geometry. A linearly polarized, monochromatic plane wave of wavelength A is incident on the scattering medium and an optical detector is at the point P. For Rayleigh scattering the molecules have dimensions <^ A, so that the molecules sense a uniform field E0 and the scattering is not sensitive to the shape of the molecule. Conservation of momentum in the scattering process leads to a wavevector diagram as shown in Figure 1(b). The quantity q = ks — k{\s the difference between the wave vectors of the scattered and incident waves. Since in Rayleigh scattering the wavelength of the scattered light is very near to that of the incident light, ks~k{= 2njX>x. Then 2/c;sin
l_2j
— — sin
-
(1)
A, l_2j where 6 is the angle of scattering, n0 is the medium refractive index and Lx the vacuum wavelength of the light. If a is the largest dimension of the scattering molecules, when qa <^ 1 they behave as point scatterers.
Photon Correlation Spectroscopy: Technique and Scope
q
913
Ifl-^sin(f)
Figure 1 (a) Light-scattering geometry; and (b) wavevectors and conservation of momentum diagram with definition of wavevector q
The electric field of the scattered light is in general Es{R,t)
=
ks A ( * s A E 0 ) a —
4neR
^-Qi(co0t-ks.R)eiq.r
(2)
for a molecule of polarizability a and where e is the relative dielectric constant of the medium surrounding the molecule. For molecules of size <^ X (Rayleigh scattering) the polarized scattered light intensity / s at position distance R is
- - r-i
I4
/0
V2
Ln 0 J
sin 2 0
[n 2
16TT 2 K 2 L
-
n2]2 °
(3)
where V is the molecular volume and n0 ( = y/s) is the refractive index of the surrounding medium. When the polarizability of the molecule a is isotropic such that the molecule a is homogeneous with refractive index n, then a = (n2 — n%) V. The scattered light intensity is seen to be proportional to /cj (i.e. \/Xf), to I0 and inversely proportional to R2. The total intensity of scattered light contains information on the static properties of the scattering medium, i.e. the size and shape of the scattering molecules and the thermodynamic quantities. The other basic quantity which can be measured by light scattering is the frequency distribution of the scattered light; this carries information on dynamical quantities, such as diffusion coefficients, internal motion and molecular velocity. There light is scattered from refractive index fluctuations, which for a polymer solution arise from the polarizability difference between the solute and solvent.
7.2.2
Dynamic Light Scattering
The phase ^ s of the scattered field £ s is made up of the phase of the incident field at position r, co0t — k{r plus a phase shift due to propagation of the scattered field from position r to position /?, —ks(R — r\ so that \j/s = (co0t — ks'R) + (&s — k^-r. The scattered electric field from a single molecule is Esj(t)
Aj(t)e-i
=
(4)
where Aj(t) is the amplitude of the scattered light from molecule j and \l/j(t) is the phase difference from the optical path of light scattered from position r} compared to that at the origin, ij/j(t) = q*r(t\ i.e. the phase of the scattered field depends on the position r of the scattering molecule. The total scattered field is N
EMt)
=
I
Aj(t)Qi
(5)
j
Since the molecules are moving, r is a function of time and ij/s has a dependence determined by the molecular dynamics. The scattered field varies in time due to translational diffusional motion or to changes in Aj(t) induced by rotational or internal motion. A less common case is for smaller values of molecular occupational number N, where fluctuations in N lead to scattered light fluctuations;34 polymer solutions at typical concentrations provide many molecules in the scattering volume. For
Solution Methods
914
example a laser beam focused into a polymer solution illuminates a sample volume defined by a 100 jam focal spot diameter, with detection optics collecting from that volume; the sample volume is ~ 10" 6 cm 3 . A polymer of molecular weight M ~ 105 at a concentration of 1 mgcm" 3 is equivalent to about 10 10 molecules in the sample volume. When the scattering molecule is undergoing Brownian motion, r is a random variable and Es has a randomly modulated phase. The scattered light is broadened in frequency with an optical frequency distribution S(a>) as illustrated in Figure 2(a). Since the particle motion contains no preferred direction, the spectrum contains a continuous distribution of frequencies centred around co0. The correlation function of the electric field G (1) (T) is also a measure of the frequency distribution and contains information on the molecular motion. 3 5 - 4 0 It is the Fourier transform of the power spectrum S(a>) G^d)
=
+
T)>
(6)
where < > denotes a time or ensemble average and T is the correlation time S(co)
1
=
— 271
f00
G ( 1 ) (T)e i < a t di
(7)
J0
Discussions of the nature of correlation functions and molecular dynamics can be found in refs. 41 and 42. (a)
Df- W
Figure 2 (a) Illustration of an optical spectrum of scattered light; and (b) electric field correlation function
An illustration of a scattered optical spectrum and its normalized field correlation function is shown in Figure 2. The broadening of the Rayleigh-scattered light spectrum contains information on the motion of the scattering molecules. We are concerned here only with the broadening of the central Rayleigh component of the scattered spectrum; the other types of inelastic scattering, the Brillouin doublet and Stokes and anti-Stokes Raman scattering, both occur at much greater frequency shifts. The normalized electric field correlation illustrated in Figure 2(b) is g(1)W =