J. Phys. Chem. B 2005, 109, 6879-6883
6879
A Mode-Coupling Theory of Vibrational Line Broadening in Near-Critical Fluids† S. A. Egorov* Department of Chemistry, UniVersity of Virginia, CharlottesVille, Virginia 22901
C. P. Lawrence and J. L. Skinner* Theoretical Chemistry Institute and Department of Chemistry, UniVersity of Wisconsin, Madison, Wisconsin 53706 ReceiVed: NoVember 2, 2004; In Final Form: January 7, 2005
We present a fully microscopic mode-coupling theory of near-critical line broadening. All the structural and dynamical input required by the theory is calculated directly from intermolecular potentials. We compute vibrational frequency time-correlation functions and line shapes as the critical point is approached along both the critical isochore and the liquid-gas coexistence curve. Theory is shown to be in good agreement with simulation.
Vibrational spectroscopy provides a powerful tool for studying structural and dynamical properties of fluids. This is because the vibrational frequency of a solute (chromophore) in solution fluctuates, reflecting the dynamics of its local environment.1-3 While the vibrational line shift is determined entirely by the local microstructure, the line width generally reflects both the structure and dynamics of the surrounding solvent. In particular, an increase in the line width arises due to the growth of meansquare frequency fluctuations and/or due to the increasing correlation time of these fluctuations. Since the pioneering work of Clouter and Kiefte,4 it has been well established that numerous simple molecular liquids, such as N2, O2, CO, CH4, H2, and HD, exhibit anomalous broadening of vibrational line shapes as the critical point of the fluid is approached.5-10 A particularly extensive set of measurements has been reported recently by Musso et al.,11 who measured the Raman line shift and width of N2 as its critical point is approached from above along the critical isochore, and from below along the liquid-gas coexistence line. The major trends observed in the experiments can be summarized as follows. First, the line shift does not exhibit any near-critical anomalies, while the line width increases dramatically (but does not diverge) as the critical point is approached. Second, the increase of the line width is faster (as a function of temperature) along the coexistence curve than along the critical isochore. Regarding theoretical analysis of near-critical line broadening, Mukamel, Stern, and Ronis (MSR) have developed a hydrodynamic approach based on mode-coupling theory (MCT).12 Using this method, they were able to obtain an adequate fit of the line widths measured by Clouter and Kiefte along the coexistence lines for oxygen and nitrogen. Subsequently, Musso et al.11 employed the MSR theory to fit their data for nitrogen along the critical isochore. To obtain good agreement between theory and experiment, it was found necessary to separate the line widths into critical and noncritical contributions, and only use MCT for the critical part. Recently, two of us13 performed a detailed analysis of the MSR theory and found that the aforementioned good agreement †
Part of the special issue “David Chandler Festschrift”.
between theory and experiment could to some extent be fortuitous. Specifically, regarding the coexistence curve results,4 the application of the MSR theory did not account for the density variation along this curve, which is quite substantial. If this density variation had been included, the agreement between theory and experiment would likely have been worse, since the line width given by the MSR approach is quite sensitive to the density. As far as the data analysis along the critical isochore is concerned,11 the critical and noncritical contributions to the line width are probably not additive, and in any case the separation is not unique. It is clear from the above that performing a comparison of MSR theory with experiments involves some ambiguities. A more direct test of the accuracy of the theory that does not involve any adjustable parameters is provided by comparing it with computer simulations. This was performed for several microscopic models,13 and it was found that MSR theory was not in quantitative agreement with simulations. Possible reasons for the discrepancies between the MSR theory and the simulation results involve certain approximations employed in the theory. Namely, MSR used the MCT formalism to construct the time-correlation function (TCF) of the solute vibrational frequency fluctuations, from which they obtained the vibrational line shape and line width. Quite generally, in the MCT approach the dynamics of the variable of interest (in the present case, the fluctuating frequency) is coupled to the appropriately chosen modes (MSR selected the solute selfdensity and the solvent collective density). In deriving the MCT equation for the fluctuating frequency TCF, MSR expressed the wavevector-dependent coupling between the frequency and the density modes in terms of the direct Fourier transform of the frequency. However, it has been shown14,15 that more accurate results are obtained by employing the projection operator formalism16,17 to compute the wave-vector-dependent vertex that couples the dynamical variable and the density modes. Another approximation employed by MSR involves calculating the dynamical input for the MCT equations (solute and solvent dynamic structure factors) in the hydrodynamic limit. As a result, the expression for the line shape given by the MSR theory depends neither on the form of the intermolecular interaction
10.1021/jp0449861 CCC: $30.25 © 2005 American Chemical Society Published on Web 02/25/2005
6880 J. Phys. Chem. B, Vol. 109, No. 14, 2005
Egorov et al.
potential nor on the local microstructure of the solvent. At the same time, for nonpolar fluids such as nitrogen, the range of the intermolecular interactions that produce the fluctuations of the chromophore’s vibrational frequency is on the order of a few angstroms, while the relevant time scales for the frequency fluctuations range from a few hundred femtoseconds to tens of picoseconds. Hence, the adequacy of the hydrodynamic approach, which is valid only at long distance and time scales, could be questionable. The above discussion suggests that a more satisfactory approach would involve a fully microscopic MCT treatment, where short times and small distances are described more correctly. Such a theory for the frequency TCF has been recently developed and successfully applied to a Lennard-Jones (LJ) fluid near the triple point.14 The structural and dynamical input required by the theory is obtained directly from intermolecular potentials. Furthermore, the theory is based on the projection operator formalism, and the coupling between the fluctuating frequency and the density modes is expressed in terms of the properly calculated wave-vector-dependent vertex. The theory was also successful in calculating the frequency TCF for a supercritical LJ fluid.15 In the present work we apply a closely related theory to the problem of near-critical vibrational line broadening, and compare our results to simulation results for the line width as a function of temperature. We consider a fluid comprised of N + 1 spherical particles, which interact via a pairwise additive potential φ(r). One tagged particle (the solute) has a single vibrational degree of freedom. The Hamiltonian for the system with the solute in its ground vibrational state is
H0 )
p02 2m
N
+
pi2
N
N
r0 - b r i|) + ∑φ(|b ri - b r j|) + ∑φ(|b ∑ i)1 2m i)1 i
