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B r i l l o u i n Scattering and P o l y m e r Science G. D. PATTERSON Bell Laboratories, Murray Hill, NJ 07974
Brillouin scattering measures the velocity and attenuation of hypersonic thermal acoustic phonons using light scattering. This technique has been applied now to many problems in polymer science. This chapter describes the theory and experimental procedures used in Brillouin scattering. Many examples are presented of the types of information that can be obtained. These include the adiabatic and isothermal compressibility, the volume and shear viscosity, and the ratio of specific heats for low-viscosity fluids. When the viscosity increases, the hypersonic glass-rubber relaxation is observed and it correlates well with other dynamic mechanical and dielectric data. Near the glass transition one can obtain the high-frequency limiting moduli for compression and shear, the Poisson ratio; and from measurements of the attenuation one can determine the homogeneity of the sample.
Τ ight scattering i n dense media is caused b y fluctuations i n the local dielectric tensor c ( I ) . In 1922 Brillouin (2) predicted that thermal acoustic phonons would lead to such fluctuations and hence to light scattering. In addition, the scattered light should be shifted i n frequency because the phonons are moving. The frequency shift is given by: ± Δ ω / ω = 2 n ( V / C ) sin 0/2 0
(1)
where ω is the incident frequency, η is the refractive index, V is the phonon velocity, C is the speed of light i n a vacuum, and θ is the scatter ing angle i n the scattering plane. Because the acoustic phonons i n fluids and amorphous solids are damped, the shifted Brillouin peaks have a half-width at half-height given b y ( 3 ) : 0
Γ = ^ 0-8412-0406-3/79/33-174-141$05.50/l © 1979 American Chemical Society Koenig; Probing Polymer Structures Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
(2)
142
PROBING
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Β
POLYMER
STRUCTURES
Β
Figure 1. Rayleigh-Brillouin Spectrum of n-hexadecane at 120 C. Only the central peak caused by thermal expansion is clearly shown in this spectrum. e
where a is the phonon attenuation coefficient and Γ is measured i n Hertz. In addition there w i l l be several unshifted peaks with finite width because of thermal expansion, mechanical and thermal relaxation, and optical anisotropy fluctuations. The Rayleigh-Brillouin spectrum of n-hexadecane at 120°C is shown i n Figure 1. In this chapter we w i l l describe the theory and experimental tech niques of Brillouin scattering. Applications w i l l be made to several polymeric systems and the type of information that can be obtained w i l l be discussed.
Theory The propagation of acoustic phonons i n amorphous media depends on the mechanical and thermal moduli. W e w i l l denote the modulus of compression by K , the shear modulus by G , the longitudinal modulus by Μ = Κ -f- 4/3G, the thermal conductivity by #c, the thermal expansion coefficient by a, and the ratio of specific heats b y γ = C / C . P
V
Koenig; Probing Polymer Structures Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
9.
143
Brillouin Scattering
PATTERSON
The dispersion equation for longitudinal phonons is:
(Mq* -
J)
K
+
P
W = 0
(3)
where q = 2ττ/λ is the magnitude of the propagation vector for phonons with wavelength λ . Brillouin scattering is caused by the creation or annihilation of an acoustic phonon. The value of q is determined by the scattering angle and the wavelength of the light i n the medium: Ρ
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Ρ
0/2
q = ^ s m Λ
(4)
where λ is the wavelength of the light i n a vacuum. The longitudinal Brillouin splitting is given by the real part of the two complex roots of Equation 3 and the linewidth by the imaginary part of the complex roots. Pure imaginary roots correspond to peaks centered at the incident fre quency. The dynamic mechanical loss associated with the longitudinal waves is .given b y : - „ *
2Γ ΐ)Δω ΐ) (
(
2
tan δ = Δω ΐ) (
F~2
() 5
— Γ ΐ) (
The corresponding dispersion equation for the transverse phonons is : Gg
2
-
ω —0
Ρ
(6)
2
The spectrum of amorphous bisphenol-A polycarbonate showing both longitudinal and transverse Brillouin peaks is shown i n Figure 2. The solutions to Equations 3 and 6 depend on the form adopted for the moduli. The moduli are i n general complex and can be represented as: Μ (ω,ρ,Τ)
-
M ' (ω,ρ,Γ) + i M " (ω,ρ,Γ)
(7)
where Μ' and Μ " are the real and imaginary parts, respectively. T h e results appropriate to each relaxation time regime w i l l be given i n the discussion section. Typical Brillouin splittings are i n the range 10 -10 H z . T h e fre quency of the acoustic phonons being studied is given directly b y Δ ω . This can be seen easily from the relation for the velocity: 8
Δω =
10
qV = ω
ρ
Koenig; Probing Polymer Structures Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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144
PROBING
POLYMER
STRUCTURES
Figure 2. Rayleigh-Brillouin spec trum of amorphous bisphenol-A polycarbonate at 25°C showing both longitudinal (L) and trans verse (T) peaks where ω is the phonon frequency. Although high-resolution optical spectroscopy is used as the tool, Brillouin scattering should be considered a dynamic mechanical technique i n the hypersonic frequency range. ρ
Experimental The apparatus necessary to measure the Brillouin spectrum of poly mers is now well developed. T h e first requirement is an intense source of collimated light with a narrow frequency distribution. A n argon ion laser provides a bright incident beam. If the laser has a cavity length of 1 m , the output w i l l consist of a series of lines separated by 150 M H z . A single cavity mode may be selected b y introducing an étalon into the laser cavity with a free spectral range of 10 G H z . The resulting single line has a w i d t h of approximately 10 M H z . Over 1 W has been obtained under these conditions for the 5145-A line. The longitudinal Brillouin spectrum appears when both the incident and scattered light are polarized vertically with respect to the scattering plane (Zvv). The transverse Brillouin spectrum results when the incident polarization is perpendicular to the scattered polarization ( 7 v = J V H ) . H
Koenig; Probing Polymer Structures Advances in Chemistry; American Chemical Society: Washington, DC, 1979.
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9.
PATTERSON
145
Brillouin Scattering
Thus, the polarization of both the incident and scattered light must be controlled. A typical argon laser produces linearly polarized light. The incident polarization can be adjusted to be either vertical or horizontal with respect to the scattering plane with a double Fresnel rhomb polari zation rotator. The scattered light can be analyzed with a Glan-Thompson prism polarizer. In order to increase the power density i n the scattering volume, the incident light can be focussed. Beam diameters of 100-200 μ. are optimal for most work. A variety of sample configurations is possible. F o r most amorphous samples above the glass transition a square quartz cuvette is convenient for observation at 90°. If the sample is to be cooled below r , a round test tube w i l l withstand more strain and resist cracking. M a n y polymer samples can be prepared most easily as films. As long as the film is thick enough to have a measurable scattering volume, Brillouin spectra may be obtained. Successful spectra also have been collected from tensile bars, extruded rods, and other common mechanical specimens. The main requirement is that the scattering geometry be known and that the incident and scattered beams be well defined. The scattered light is collimated and analyzed for polarization. The instrument most commonly used to resolve the Brillouin spectrum is the Fabry-Perot interferometer. This device consists of a pair of highly reflective, optically polished mirrors. The transmission function for a plane parallel Fabry-Perot interferometer is: ( g )
IM h
_
(7^/(1 - i ^ ) ) , 4F . / nd\ 2
Λ
2
2
.
( w
where Τ is the transmission of the mirrors, R is the reflectivity, F is called the finesse, η is the refractive index of the medium between the plates, d is the distance between the plates, and C is the speed of light i n a vacuum. W h e n ω = (vCN/nd), where Ν is an integer, the interferometer w i l l have its maximum transmission:
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