Correlation and Memory Function Analysis of Molecular Motion in Fluids

BY M. W. EVANS

1 Introduction

The foundations and framework of this review have been established in two previous articles of this series, the first by Wyllie, who discussed the broad theoretical concepts necessary for the interpretation of far i.r. and lower frequency spectroscopic data; and the other by Brot, whose main contribution lay in annotating and evaluating the recent attempts at solving the problem of the dynamic internal field. In both these articles several models of fluid dynamics were discussed on the molecular scale, and related to bulk properties such as the dielectric loss and dispersion with classical fluctuation-dissipation theory. This third report on advances within six years aims to be selective and complementary rather than comprehensive. The natural complement is found in fields of work covering the dynamical processes in fluids that give rise to absorption in the far i.r. region of the electromagnetic spectrum and which represent different kinds of experimentally observable phenomena.

The past twenty years have seen the emergence of Kubo's generalizations of classical fluctuation-dissipation theory, enabling a coherent analysis of superficially different bulk transport properties to be made in terms of the motions of vectors defined in the molecular frame. Zwanzig, Gordon, and Berne have made significant advances in isolating some fundamental statistical theorems pertaining to the fluid state, as well as in emphasizing the role of the most important concept of the correlation function. This is fonnally the ratio of the covariance of stationary (time independent) random process to its variance. Here 'correlation' takes on the narrow statistical sense of being the relationship between two or more measurable random events occurring in a temporal sequence. The frequencies of such events brings us into the domain of spectroscopy, since a spectral function, being a distribution of probabilities of events occurring with given frequencies, is itself statistical in nature. Fourier1 2 provided the link between temporal and frequency domains in his integral theorem, so that the correlation and spectra) functions, C(t) and [??](iω) respectively, are Fourier transform pairs.

Molecular fluctuations in phase space may be correlated statistically and related to many different kinds9 of absorptions and dispersions, the former being the real and the latter the imaginary parts of the Fourier transform of C(t). The correlated random variable is conveniently a vector, or tensor trace, whose magnitude and direction may be evolving in time. To discuss the absorption of energy by molecules from an electromagnetic field (a stream of photons), it is sufficient8 to define such a vector u in the molecular frame (x,y,z). The direction may be that of the permanent dipole (µ)if this exists, so that u may be normalized to unity (u =µ/|µ|). It is generally true that any molecule (i), whether dipolar or not, in a fluid of finite density, made up of a finite ensemble of N molecules, will experience the resultant electrostatic field of the (N - l) others at the instant t. This means that (i) will carry a small interaction-induced, temporary dipole [(i)M]. To correlate its value at an initial instant t = 0 with that at time t later it is sufficient to take the projection (i)M(O)·(i)M(t). This can be repeated for all the (N-1) other molecules [by making the cross-correlations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] so that the spectrum of induced dipolar fluctuations is the Fourier transform of all such projections averaged over phase space. The spectrum from the permanent dipole is similarly a Fourier transform of auto-and cross-correlation functions of u, which unlike M, does not fluctuate in magnitude. The total spectrum is a sum of these two parts, with a contribution from correlations between u and M.

The far i.r./microwave spectrum of a fluid is composed mainly of such orientational contributions, but collision-induced dipole moments are dependent also on the centre-of-mass co-ordinates of a molecule in the external (laboratory) frame, that is, on the relative positions of the interacting molecules. Consequently, part of the energy of the photon which is absorbed may be converted into molecular translational energy after collision, and it is possible to observe pure translational absorption in such cases as binary mixtures of rare gas atoms. The separated translational and rotational components can be observed in the spectrum of compressed or liquid hydrogen, but for heavier molecules such as oxygen and nitrogen the translational component is not separable, and the broad bands observed are roto-translational in origin. In fluids such as CH4 and CD4 the induced far i.r. bands peak at frequencies which indicate I1/2 (rotational) rather than M1/2 (translational) dependence. In what follows, we treat the far i.r. absorptions of molecular fluids exclusively in terms of orientational correlation functions since the translational intensity is so much weaker.

The scattering of electromagnetic or neutron radiation can be related to correlations of the vectors u and r, respectively, the latter being the position of the i'th nucleus in the fluid. In principle, the inelastic scattering of neutrons, being much heavier than photons, can be analysed for change of angle and speed. An incoming wave of length γ is characterized by a vector k0 which specifies its direction. The wave scattered with a vector k1 has an intensity proportional to S(k, ω), where k = k0 - k1 and w is the change in angular frequency. The structure factor S(k, ω) is the three-dimensional Fourier transform of the van Hove correlation function. The scattering can be either incoherent (from one centre) or coherent (from a pair of centres) according to the nature of the atomic nuclei in the molecule. The former arises from the 'self part of the correlation function and the latter from the distinct. Thermal neutrons have a wavelength of about 10 nm, and so k is comparable with the intermolecular spacing, and the coherently scattered beam yields the distinct part of the van Hove correlation function upon Fourier transformation.

Visible light (nowadays from a laser) is scattered coherently with negligible change of momentum, and the spectrum is thus nearly S(O,...