Frequency-time correlation function molecular liquids

As we have seen so far in this section, in the theoretical development for the time-correlation functions of liquids at wavelengths and frequencies of a molecular scale, the memory-function formalism based on projection operators has played a key role. By combining this convenient framework and the interaction-site representation of molecules. 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

Experimentalists often rely on motional models, based on hydrodynamics, in order to interpret their liquid state spectra. MD simulations, can be considered as model-free in the sense that they do not assume the molecular motion to be in any specific regime. MD can be used to evaluate the motional models and even replace them. MD simulations can be used to calculate both the correlation times and the whole correlation functions. This is useful in those cases when correlation times cannot be deduced from measurements of other isotopes in the same molecule or when there is no method available at all. Correlation functions give information about intermolecular interactions and reveal cases when several motional modes are contributing to relaxation mechanisms at slightly different time scales. This can be observed as multiple decay rates. Time correlation functions from MD simulations can be Fourier transformed to power spectra if needed to provide line shapes and frequencies. 

In liquids and dense gases where collisions, intramolecular molecular motions and energy relaxation occur on the picosecond timescales, spectroscopic lineshape studies in the frequency domain were for a long time the principle source of dynamical information on the equilibrium state of manybody systems. These interpretations were based on the scattering of incident radiation as a consequence of molecular motion such as vibration, rotation and translation. Spectroscopic lineshape analyses were intepreted through arguments based on the fluctuation-dissipation theorem and linear response theory (9,10). In generating details of the dynamics of molecules, this approach relies on FT techniques, but the statistical physics depends on the fact that the radiation probe is only weakly coupled to the system. If the pertubation does not disturb the system from its equilibrium properties, then linear response theory allows one to evaluate the response in terms of the time correlation functions (TCF) of the equilibrium state. Since each spectroscopic technique probes the expectation value... 

In the above discussion of the frequency dependent permittivity, the analysis has been based on either the single particle rotational diffusion model of Debye, or empirical extensions of this model. A more general approach can be developed in terms of time correlation functions [6], which in turn have to be interpreted in terms of a suitable molecular model. While using the correlation function approach does not simplify the analysis, it is useful, since experimental correlation functions can be compared with those deduced from approximate theories, and perhaps more usefully with the results of molecular dynamics simulations. Since the use of correlation functions will be mentioned in the context of liquid crystals, they will be briefly introduced here. The dipole-dipole time correlation function C(t) is related to the frequency dependent permittivity through a Laplace transform such that ... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

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... 

Zardalidis et al. [53] investigated the molecular dynamics of PEO/PVAc blends as a function of composition, temperature and pressure. For blends with a high PVAc content, all short-range correlations were dominated by the PVAc component. The fact that an invariant frequency dispersion was observed for different blend compositions under isochronal conditions suggested a covariance of the width of the distribution with stmctural relaxation time, as found in other glassforming liquids. 

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