Light scattering dielectric constant fluctuations

Equation (3.2.13) is an expression for the scattered light spectral density in terms of dielectric constant fluctuations. Nowhere in this treatment was it necessary to determine the explicit dependence of these fluctuations on molecular properties. In fact, this theoretical expression is purely phenomenological, ny attempt to write this formula in molecular terms will necessarily involve some degree of approximation. Nevertheless, a molecular formulation will contribute much to our intuitive understanding of light scattering and will also be useful for practical application. 

Since in the homodyne method only the scattered light impinges on the photocathode, E(t) in Eq. (4.3.1) is equal to the scattered field Es(t), so that is proportional to h(t)—which is consequently sometimes called the homodyne correlation function. The amplitude of Ea(t), the scattered field, is proportional to the instantaneous dielectric constant fluctuations in the scattering volume and, of course, fluctuates in the same manner. In certain circumstances the homodyne correlation function may be simply expressed in terms of h(t) or equivalently / (q, t) or / (q, t) of Eqs. (3.2.15) and (3.3.3b), respectively, as we now discuss. 

There are five linear hydrodynamic equations containing the seven fluctuations (pi, u x,ii y, uu,pi,si, 7i). The local equilibrium thermodynamic equations of state can be used to eliminate two of the four scalar field quantities (pi, si, Ti, pi). In this chapter we chose the temperature and number density as independent variables, although we could equally well have chosen the pressure and entropy. One useful criterion for choosing a particular set is that the equilibrium fluctuations of the two variables be statistically independent. The two sets (pi = dp, T = ST) and (pi = Sp, si = Ss) both involve two variables that are statistically independent, that is, statistical independence of the two variables simplifies our analysis considerably. It is particularly convenient to chose the set (Pi,Ti) over the set (pi, si) because the dielectric constant derivatives (de/dp)T and (de/dT) are more readily obtained from experiment (other than light scattering) than are (ds/dS)p and (ds/dp)s-... 

Consider light scattering in liquid with due u count of the time dependence of the fluctuations in dielectric constants c(f,i) = t + <5t(r, <). 

Light scattering brings detailed information on the dielectric constant s fluctuations if and only if their scale and the wavelength of light are of the same order of magnitude (i.e. 1000 A or more). 

Fluctuations of such extent involve collective motion of a great number of molecules and therefore can be described by the laws of macroscopic physics, namely, thermodynamics and hydrodynamics. Thus, small parts of the system where fluctuations of the macroscopic values manifest themselves in the properties of scattered light (the Fourier transform) contain rather many molecules that enables one to speak of local values of such macroscopic terms as entropy, enthalpy, and pressure. Every point f corresponding to a small space element in liquid at an instant i can be ascribed some values of entropy density a(r,i), of molecule number density p(r,l), of energy e(f,l), of pressure P(r,i), and of the dielectric constant e(f,t). 

The most commonly used technique for determining 5 is photon correlation spectroscopy (PCS) [also known as quasi-elastic light scattering (QELS)]. PCS has become one of the standard tools of the trade for the colloid chemist. In this technique concentration fluctuations arising from the diffusive motion of the dispersion particles give rise to fluctuations in the dielectric constant of the medium are monitored photometrically. These fluctuations decay exponentially with a time constant related to the diffusion coefficient, Ds, of the scatterer, which can in turn be related to its hydrodynamic radius through the Stokes-Einstein equation ... 

The isothermal time dependence of relaxation and fluctuation due to molecular motions in liquids at equilibrium usually cannot be described by the simple linear exponential function exp(-t/r), where t is the relaxation time. This fact is well known, especially for polymers, from measurements of the time or frequency dependence of the response of the equilibrium liquid to external stimuli such as in mechanical [6], dielectric [7, 33], and light-scattering [15, 34] measurements, and nuclear-magnetic-resonance spectroscopy [14]. The correlation or relaxation function measured usually decays slower than the exponential function and this feature is often referred to as non-exponential decay or non-exponentiality. Since the same molecular motions are responsible for structural recovery, certainly we can expect that the time dependence of the structural-relaxation function under non-equilibrium conditions is also non-exponential. An experiment by Kovacs on structural relaxation involving a more complicated thermal history showed that the structural-relaxation function even far from equilibrium is non-exponential. For example (Fig. 2.7), poly(vinyl acetate) is first subjected to a down-quench from Tq = 40 °C to 10 °C, and then, holding the temperature constant, the sample... 

Let us consider for a moment the light scattered from a pure liquid or amorphous solid. Einstein 66) extended the treatment of Rayleigh by suggesting that for pure liquids the polariiability, a, which occurs in Eq.(18) must be replaced with fluctuations in the polarizability. Act which occur in the medium. Since the polarizability is related to the optical dielectric constant approximately by the relation... 

A (r, t)l8Q represents local fluctuations of the relative permittivity (dielectric constant) in the target. The task of classical light-scattering theory has been reduced to solving Equation [34]. The only role of quantum theory, therefore is to calculate the atomic polarizability a - m other words, the microscopic properties of the scattering medium. Once this is achieved, the polarizability can then be related to the dielectric constant through the well-known Clausius-Mossotti relation. 

Scattering of light in a medium is caused by fluctuations of the optical dielectric constants 5s(f, t). In isotropic liquids 58(f, t) are mainly due to density fluctuations caused by fluctuations in the temperature. For liquid crystals in their ordered phases, an additional and important contribution to 5 (f, t) arises from director axis fluctuations. 

The orientation of the director is not constant in time and fluctuates due to thermal excitations. Because the energy associated with the local perturbation of the director is small, the amplitude of the thermal fluctuations is large. With the director, also the optic axis fluctuates and with it the optical dielectric constant, which results in strong scattering of light. This feature is most prominently observed in the opaque appearance of liquid crystals. 

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