CHAPTER 1
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, ω). The change of frequency is small, but observable if the incident light is from a laser and so highly monochromatic. Measurement of the intensity and angle, but not the spectrum of the scattered light tells us only about the static properties: in particular, S(k =0) is related to the compressibility, and so such scattering is intense near the critical point in a fluid, where this tends to infinity. The spectrum of scattered light is a more useful observable and it has three distinct peaks, a Rayleigh line centred at ω = 0 (the zero frequency displacement) and two Brillouin lines at ω = ±Wvω, where W is the speed of sound and vω the wave number of the particular phonon mode responsible for the scattering. The Rayleigh line arises from density (or more immediately refractive index) fluctuations corresponding to local entropy fluctuations at constant pressure. Such fluctuations do not propagate through the fluid and so the Rayleigh feature is centred on ω = 0. It contains a depolarized component of significant breadth centred on the incident frequency and known as the Rayleigh wings. In molecular fluids such as benzene the spectrum is thus composed of a relatively narrow diffuse superimposed on a much broader background. High-resolution, high-power measurements have disclosed some additional features, notably a very narrow central doublet: in its general contour and dependence on the scattering angle and polarizations, the diffuse band and central doublet fit the theory of Rytov, which phenomenologically associates the diffuse band with scattering by transverse shear waves. However, the absolute integrated intensities, which can be obtained by comparison with intensities of the polarized Rayleigh and Brillouin spectrum (see below), are very close to those predicted from theoretical expressions obtained for the reorientations of single vectors (such as u). Furthermore, the inverse half-width of the diffuse band (the rotational relaxation time), fits reasonably well in magnitude, temperature dependence, and activation energy into the broad scheme of fluctuation-dissipation theory mentioned above, in which other phenomena such as far i.r. and microwave absorption/dispersion, nuclear spin-rotation relaxation, Raman scattering, and near i.r. vibration-rotation absorptions are described by temporal correlations of selected vectors. The close relation between induced and permanent far i.r. absorptions and the Rayleigh bands are described below in some detail, and the conclusions obtained on the same fluid by different authors using some of the different experimental techniques described above, are checked for consistency.
The Brillouin lines arise from fluctuations of density due to fluctuations of pressure at constant entropy. These form the acoustic mode spectrum of the condensed phase and they are present in all liquids at equilibrium, and they diffract light at the appropriate Bragg angle. The frequency shift is a Doppler effect of the moving 'grating', and, since the sound wave of appropriate length and orientation can be moving in either direction, a pair of lines is produced, one on each side of the incident frequency. Polarized Brillouin bands can be described in terms of a molecular correlation function related to the trace of the polarizability tensor. Depolarized bands are Fourier transforms of correlation functions related to its xy'th (off-diagonal) elements. The total intensity of the Rayleigh and Brillouin lines yields the compressibility, the ratio of intensities yields the ratio of specific heats at constant pressure and constant volume, the width of the Rayleigh line yields the thermal diffusivity, and the displacement and width of the Brillouin lines yield the speed and coefficient of absorption of sound at frequencies above 10 GHz, i.e. above the range of mechanically generated sound waves.
Correlations of the vector u are involved also in the determination of n.m.r. lines broadened by spin-spin coupling and relaxation and the only currently known experimental source of information about fluctuation of J, the total molecular angular momentum vector, is spin-rotation relaxation. In a linear molecule, u and J are related by: J = Iω, u = u x ω, where ω is the total angular velocity vector, perpendicular to u.
2 General Formalism in Classical Mechanics: Memory Functions
The major problem in building up an analytical description of the dense fluid is that of describing the complex many-body interactions in a realistic fashion. To do this the intermolecular potential must be defined. On the one hand, recent articles have exposed the shortcomings of the classical Lennard-Jones potential even for rare-gas atomic interactions, but on the other several recent numerical solutions of the Newton equations for groups of up to 864 basically Lennard-Jones potentials have had a marked degree of success in reproducing macroscopic equilibrium properties and also their corresponding spectra.
This section describes the recent attempts at bypassing the intermolecular potential problem by using the very general statistical properties of a canonical ensemble of particles to derive a series of integro-differential equations linking the correlation function of u or other vectors to that of its n'th derivatives (u(n)) or memory function. It is hoped that the latter might have a simpler analytical dependence upon time, and so might be simulated empirically with a reasonable degree of success. A straightforward derivation of these equations was given in classical mechanics by Berne, Boon, and Rice, and the same theorem was later proven by Berne in quantum notation. The classical derivation is briefly recalled here in order to emphasize how generally applicable is this formalism. Subsequently, several currently popular models of the fluid state can take their place in this scheme (Table 1), which is essentially an expression in La place space as a quotient of polynomials whose coefficients are equilibrium averages having the units of s-2 ,s-4, and so on, for example zero-time averages of derivatives of u. This is consistent with the fact that any classical auto-correlation function has a series expansion (1), where the brackets <> denote the canonical average (2) in N particle phase-space ΓN. Here H(N) and ZN are the hamiltonian and the canonical partition function. Therefore, in general, u(ΓN) may be any vector property of phase space. (In quantum mechanics, correlation functions have a real, even part and an odd, imaginary part, so that a quantum correction is made by replacing t with [t - ih/2kT].)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
The classical auto-correlation function, C(t), may be written as equation (3), where L(N) is the Liouville operator. It is assumed that <u> = 0 and (u • u) = 1. Differentiation of equation (3) twice, followed by partwise integration, leads directly to equation (4), the Laplace transform of which, together with an algebraic identity for C(p), yields the fundamental relation (5), where the kernel K is the memory function, defined in Laplace space by equation (6).
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)
It is important to note that equation (5) embodies no assumptions other than those of equation (3), that is that the N particle ensemble is canonical, and obeys the Llouville equation of motion. Equation (5) is true for any vector whose expectation vanishes and whose auto-correlation function is even under time reversal. In ref. it is shown how it can be rederived using linear regression theory and by means of the properties of transport coefficients. It is a fundamental theorem of molecular statistical mechanics.
Defining a projection operator P onto a well-behaved function, G(ΓN), of the phase space as in (7), where[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], equation (8) can be written, showing that the kernel K is related to the dynamical coherence (or memory) of the N particle ensemble. It is possible subsequently to define a set of kernels, or memory functions, K0(t), ..., Kn(t), ... giving equation (9), where the dynamical quantities fn are defined so that f0 = u and fn =[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. This leads directly to the sequence (10) first derived by Mori. In Laplace space, this is a continued fraction; linking C(p) to Kn(p) as in equation (11).
If a property F(t) is defined as in equation (12), t hen we ca n write equation (13). This is widely known as the generalized Langevin equation, since it reduces to the classical equation when K is a delta function, that is when the system for Brownian motion lacks dynamical coherence and has no memory for past events, so that F(t) is a Gauss/Markov random variable. It is important to note that equation (13) may be derived (as above) with the assumptions inherent in equation (3), but without any about the nature of Brownian motion as such. Equations (14) and (15) hold by definition; the latter is generally known as the second fluctuation-dissipation theorem of Kubo.
