J. Phys. Chem. 1996, 100, 18327-18334
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Infrared and Molecular Dynamics Study of D2O Rotational Relaxation in Supercritical CO2 and Xe Lawrence E. Bowman, Bruce J. Palmer, Bruce C. Garrett, John L. Fulton,* Clement R. Yonker, David M. Pfund, and Scott L. Wallen Pacific Northwest National Laboratory,† Richland, Washington 99352 ReceiVed: April 17, 1996; In Final Form: September 9, 1996X
Supercritical fluid solvents allow continuous tuning of solvent density and therefore continuous control of the number and distance of solvent-solute interactions. Such interactions exert torque on molecules in solution and thereby influence the way in which the rotational orientation of a solute molecule varies with time. In this work, we examine the rotational relaxation of D2O in two supercritical fluid solvents: Xe and CO2. In the former case, there are no electrostatic interactions between the solute and solvent. Since the nonelectrostatic interactions of D2O are very nearly centrosymmetric about the center of mass, Xe is expected to exert little torque on the rotating solute until liquid densities are reached. For D2O dissolved in CO2, the dipolequadrupole interaction can exert torque on the rotating D2O, thereby hindering its rotation. Molecular dynamics simulations of these systems were used to calculate dipole autocorrelation functions containing only contributions from the solute rotational motion. These were then used to predict the rotation wings of the asymmetric stretching band, ν3, of D2O. Infrared spectra for this band were obtained in both solvents at 110.0 °C as a function of density. In Xe, the experimental and simulated correlation functions indicated that the solute was free to rotate, even at the highest densities examined. In CO2, however, the rotation of D2O was increasingly hindered as the solvent density was increased from gaslike to liquidlike densities. At low densities the correlation functions had negative minima, indicating a partial reversal in the direction of the average transition dipole moment vector as the molecules freely rotated. In CO2, at high densities the functions exhibited simple, monotonic decay, indicating a persistence in the direction of the vector with time and substantially hindered rotation.
Introduction The supercritical fluid state offers one of the best opportunities to study solvation phenomena since physical properties such as density, dielectric constant, solvating power, and viscosity are continuously variable by simply adjusting the pressure.1 For instance, above the critical temperature, Tc, the density of a fluid can be varied from gaslike to liquidlike values without a change in phase, leading to an absence of the usual discontinuity in density that occurs during a typical gas-to-liquid phase transition. In this respect, investigations of supercritical fluids provide an improved understanding of condensed phase interactions by bridging the gap between the gaseous and liquid states. The solvation environment of a molecule dissolved in a supercritical fluid changes dramatically as the density is continuously increased from gaslike to liquidlike values. One may follow the evolution of a coupled rotational-vibrational band through a broad range of densities and thereby elucidate, in detail, the effect on molecular level dynamics as the transition from gaslike to liquidlike solvation environments is made. Over this density range, a transition occurs from a state of nearly free rotation, where a solute molecule undergoes many full rotations before collision with a solvent molecule, to a more hindered rotational state approaching rotational diffusion of the molecular orientation, where multiple, momentum-altering collisions occur on the time scale of a single rotation of a solute molecule. These rotational relaxation processes can be followed experimentally through the analysis of the spectral line shapes measured by infrared (IR) and Raman spectroscopies.2,3 * Corresponding author. † Operated by Battelle Memorial Institute. X Abstract published in AdVance ACS Abstracts, November 1, 1996.
S0022-3654(96)01117-3 CCC: $12.00
Concurrently, there is a good opportunity to use classical trajectories generated from molecular dynamics simulations to predict spectra for comparison with experiment since the energies of individual rotational states are relatively low. Lowenergy rotational states are easily populated in a collection of solute molecules at room temperature or above, so that the motion of the rotational wave packets are well approximated by classical mechanics. Molecular dynamics simulations can be performed to obtain the classical dipole autocorrelation function, which can subsequently be transformed to obtain the rotational contribution to the line shape. In the present case, the resulting line shape is determined solely by the rotational relaxation of the dilute solute molecule in the supercritical fluid solvent since the vibrational motions are not considered in the molecular dynamics simulation. Previously, a molecular dynamics study of the rotational relaxation properties of pure, supercritical CO2 showed that the quadrupolar interactions of CO2 slow down the rate of rotational relaxation appreciably.4 More recent experimental work has examined the rotational and vibrational Raman spectrum of H2 dissolved in supercritical CO2.5 All of the bands broadened as the pressure was increased. These results were interpreted as being caused by changes in the local solvent environment of the hydrogen molecules with increases in density, however, a more detailed analysis concerning this dependence was not pursued. For this study we have chosen to investigate D2O, a solute with a permanent dipole, dissolved in two simple supercritical solvents, CO2 (Tc ) 31.1 °C) and Xe (Tc ) 16.6 °C). CO2 is a small, linear, and symmetric molecule which, because of its low polarizability, is a relatively weak solvent for highly polarizable or polar solutes. Although CO2 has no permanent © 1996 American Chemical Society
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dipole moment, the relatively large, electronegative oxygen atoms impart to the molecule an especially large quadrupole moment. It is this quadrupole moment that gives CO2 unusual spectroscopic6 and solvent properties which greatly enhances the solubility of certain polar solutes in the fluid relative to other nonpolar fluids.7 In contrast, xenon is a spherical, monatomic fluid having no dipolar or quadrupolar interactions with the solute. Although there are some complications, as discussed below, in the choice of an asymmetric top molecule such as D2O in terms of the experimental analysis, the ultimate goal is to interpret the IR data within the framework of the results from the molecular dynamics simulations. The selection of these systems and the means chosen to study them permit an investigation of the effect of dipole-quadrupole interactions on the rotational relaxation of the solute. Many water models, such as the simple point charge model of Straatsma et al., picture the water molecule as a single Lennard-Jones site centered on the oxygen atom with a distribution of partial electrostatic charges.8 One may justify the assumption of spherical symmetry in the repulsive exchange forces between water molecules by the small extent of the bonded hydrogens and the short length of the O-H bond (≈1 Å) relative to the strongly repulsive region of the Lennard-Jones potential. This effectively screens the hydrogens from exchange interactions with other atoms. Adopting such a picture, the interactions between such a model molecule and an uncharged Lennard-Jones particle, such as the Xe atom in the present case, are also Lennard-Jones in nature. Straatsma et al.8 and Guillot et al.9 have successfully used such models of water-inert gas interactions to estimate the hydration free energies for these inert gases. Within this model the oxygen atom is the center of the force exerted by Xe on the water, and we expect that negligible torque is exerted on such a water molecule during collisions with Xe. We will determine whether this model of D2O-Xe interactions, which successfully predicted the above thermodynamic properties of water-Xe mixtures, can predict the dynamics of this system as well. The choice of Xe and CO2 as solvent molecules makes possible a comparison of the differing effects of simple Lennard-Jones collisions versus dipole-quadrupole interactions on the rotational relaxation of the D2O molecule. Both types of solute-solvent interactions are present in CO2, but only Lennard-Jones collisions are present in the case of Xe. In this study, D2O was chosen rather than H2O in order to avoid interfering IR bands of the CO2 solvent and to eliminate interference from atmospheric moisture. The differences in the thermodynamic properties of D2O and H2O dissolved in supercritical fluids are expected to be small. Infrared Spectra and Rotational Relaxation
C(t) ) C(t)VC(t)1R
(1)
where ω is the frequency displacement from the shifted vibrational band center, I(ω) is the absorption at this frequency, u(t) is the unit vector in the direction of the transition dipole moment (i.e., the dipole induced by the vibration) of a molecule for the vibration being probed at time t, and the brackets specify averaging over all possible initial conditions (t ) 0). With the assumption of uncorrelated vibrational and rotational motions,
(2)
A common method to compute the individual correlation functions is to perform isotropic Raman experiments on highly polarized (symmetric) vibrations in order to determine C(t)V and thereby allow one to determine C(t)1R. A number of mechanisms can contribute to the spectral line shape resulting in C(t)V, including, but not limited to, population relaxation, lifetime broadening, vibrational dephasing or environmental fluctuations, and resonance energy transfer.12 The predominant vibrational relaxation mechanism in this system is believed to be pure dephasing. The relative contribution of the reorientational and vibrational terms to the dipole autocorrelation function is complicated for even the simplest molecules and is more problematic in the present case, considering the asymmetry of the molecule.13 However, molecular dynamics simulations provide a means of assessing the dynamics due to purely reorientational motions, and therefore, the relative contribution of reorientational and vibrational relaxation can be determined from the IR measurements. One can test the applicability of the separation of C(t) into vibrational and reorientational components by examining the time scales of the two relaxation processes.8 Another such test of separability and a check of the main broadening mechanism for an infrared band is the comparison of the experimental second moment, M(2), to the classically calculated second moment, M(2)CL. If M(2) , M(2)CL, it indicates that the primary broadening mechanism is due to nonorientational relaxation processes.14,15 The behavior of C(t) at small times can be related to the even moments of the spectrum, M(n), defined as
M(n) ) ∫dω ωnI(ω) )
dn [C(t)]t)0 dtn
(3)
where M(2) describes the correlation function at very short times, prior to any “collisions” of the solute with its neighbors.16 The classically calculated second moment, M(2)CL, depends only on the average rotational kinetic energy, which is a function of molecular geometry and temperature when rotation is the main broadening mechanism.14 Therefore, it is a purely kinetic quantity. In general, when inhomogeneous broadening may also be present, M(2) is
M(2) ) kT
In the absence of inhomogeneous broadening, IR spectral bands may be converted to the normalized, dipole autocorrelation function, C(t), via Fourier analysis as described by Gordon10 according to
∫bandI(ω)eiωt dω C(t) ) 〈u(0)‚u(t)〉 ) ∫bandI(ω) dω
the dipole autocorrelation function can be written in terms of the purely vibrational, C(t)V, and infrared (l ) 1) reorientational, C(t)1R, correlation functions as11
( )
1 1 + + 〈(V1 - V0)2〉 - 〈V1 - V0〉2 Iy Iz
(4)
where the above equation is written without the inclusion of quantum mechanical effects.2 The first term on the right-hand side is M(2)CL. The moments of inertia Iy and Iz are calculated in a coordinate system centered on the oxygen, and the x direction lies along u(0) (parallel to D-D). The terms k and T are the Boltzmann constant and temperature while V0 and V1 are the respective intermolecular potential energies for a molecule in the ground and excited vibrational states. The terms in angular brackets are a positive contribution resulting from fluctuations in the frequency shift, where the shift in the band center is 〈V1 - V0〉/hc. The results presented below will show that the primary broadening mechanism in the D2O vibrational band is rotational and that the analysis of the D2O IR correlation functions is possible through a careful consideration of the above complications and limitations.
D2O Rotational Relaxation in CO2 and Xe
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A qualitative description of the behavior of C(t)1R at the extremes of gaslike and liquidlike densities can be used to describe the effect of rotational interactions between the solvent and the D2O molecule on the spectral line shape. At low solvent coupling and for sufficiently short times, the D2O molecule rotates freely, and C(t)1R can be crudely approximated in the form of a damped oscillation written as -t/τR
C(t)1R ≈ cos(Ωt)e
(5)
where Ω is a thermally weighted rotational frequency and τR is the rotational relaxation time. This form does not reproduce the free rotor behavior exactly but qualitatively matches the short time behavior of the free rotor correlation function. The damping is due to the fact that there is a distribution in the frequency of rotation, and these change with energy such that the correlation function decays. When the rotation of D2O is strongly coupled to the solvent, the rotational relaxation approaches the diffusive regime and C(t)1R ≈ e-t/τD where τD is the diffusional correlation time. From eq 1, the corresponding spectral line shapes are described by
Ifree rotor(ω) ≈
1 1 + (ω - Ω)2 + 1/τ2R (ω + Ω)2 + 1/τ2R
(6)
for the free rotor and
ID(ω) ≈
1 ω + 1/τ2D 2
(7)
for the rotational diffusion regime. While the models are only rough approximations, they do describe the transition from free rotation to rotational diffusion. The spectral line shape for free rotation consists of two peaks offset from the center of the vibrational band (corresponding to the P and R branches observed in the IR spectra) while the line shape for the rotational diffusion consists of a single peak coincident with the vibrational band center. An investigation of the changes in the spectral line shape can then be used to investigate the transition from one rotational regime to another. Simulation Methods Simulated correlation functions provide average descriptions of the reorientation of molecules with time. In the classical approximation, the correlation functions are real-valued, even functions of time. Fourier transformation of a correlation function then provides a calculated rotational spectrum (to within a normalization factor) with frequencies shifted by the band center frequency of the vibrational transition. The above treatment neglects frequency shifts; however, since the frequency range used in the transform is offset by the mean shift, only the fluctuations in the shift degrade the comparison with experiment. The simulations reported here were performed using one D2O in 255 solvent molecules. The solvent was either CO2 or Xe. The Murthy-Singer-McDonald.17,18 (MSM) model was used for CO2, while the parameters for Xe are those of McDonald and Singer.19 The extended simple point charge (SPC/E) model of Berendsen et al.20 was used for water, except that the charges were scaled to reproduce D2O’s gas phase dipole moment (1.85 D). The charges in the SPC/E model, as in most models of water, have been optimized to reproduce the behavior of liquid water under standard conditions. The dipole moment of individual water molecules in the liquid is substantially enhanced due to the polarizability of water and the close proximity of
TABLE 1: Pair Potential Parameters for MD Simulations atom
/k (K)
σ (Å)
q
OD2O DD2O OCO2 CCO2
78.2 0 83.1 29.0
3.166 0 3.014 2.785
-0.6576 0.3288 -0.2980 0.5960
other water molecules. However, because neither CO2 nor Xe has a permanent dipole moment, the dipole moment of water in these solvents is expected to remain relatively close to the gas phase value. The molecular potential functions all consisted of pairwise additive interactions of the form
φij(rij) ) 4ij
)
[( ) ( ) ] σij rij
12
-
σij rij
6
+
qiqj + aij + rij bij(rc - rij); rij < rc (8)
qiqj ; rij g rc rij
The constants aij and bij were chosen so that both the potential and the force due to the Lennard-Jones interactions vanish at the truncation distance rc. The long-range Coulombic interactions are handled via an Ewald summation. This is only necessary for the D2O-CO2 system since Xe has no permanent charges. Each individual atom type has several parameters associated with it. They are i, the Lennard-Jones well depth, σi, the separation at which the potential energy equals zero, and qi, the fixed partial site charge. The parameters ij and σij for the cross Lennard-Jones interactions were then obtained from the combination rules ij ) xij and σij ) (σi + σj)/2. A list of the parameters used in the simulations is given in Table 1. All simulations were performed at constant volume and temperature using Nose´’s isothermal molecular dynamics algorithm.21 The volumes were chosen to match the experimental densities. The equations of motion were integrated using the velocity Verlet algorithm recast as a three-point predictorcorrector.22 The rigid molecular geometries of both D2O and CO2 were maintained using a variant of the SHAKE algorithm.23,24 The D-O bond length was 0.9572 Å, and the D-O-D angle was 104.52°. The C-O bond length was 1.16 Å. The time step was set at 2.5 fs. The scaling variable mass in the Nose´ algorithm was chosen so that fluctuations in the scaling variable were on a time scale of 0.5 ps or longer. Previous simulations have shown that the use of Nose´’s algorithm does not greatly effect the dynamics of single molecule correlation functions.22 For the D2O-CO2 system, a minimum of four simulations, each lasting 50 ps, were performed at each density. Run in this manner, the lowest density case gave very poor statistics. For this case, the run time was increased to 500 ps, and no Ewald summation was performed. Instead, all interactions were truncated at a cutoff distance of 10 Å. The D2O-Xe system was much easier to simulate, so four simulations, each lasting 125 ps, were performed at each density. For each solvent-solute system the individual simulations were combined to give an average correlation function as well as a rough estimate of the uncertainty. Because coupling between the rotational and translational motion appeared to be fairly weak, the thermalization of D2O rotational rates was poor, resulting in a large variation in the dipole-dipole correlation function from one run to the next. To improve the statistics, the entire system was rethermalized every 2.5 ps during the simulation by resetting the velocities of all atoms from a Maxwell-Boltzmann distribution. The cor-
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relation functions were calculated in between thermalizations using the sliding window technique, and the results were combined at the end of the simulation to give a single correlation function. The rethermalizations improved the run-to-run reproducibility of the functions, particularly for the D2O-Xe simulations. Experimental Section The details of the high-pressure apparatus have been given previously.25 The sample was contained in a magnetically stirred, high-pressure transmission cell having a path length of 5 cm. The cell had a volume of 7.3 mL with a maximum working pressure of 1300 bar. The beam port windows were sapphire. A long path transmission cell was required for this study due to the very dilute solutions used and the necessity of obtaining sufficient signal-to-noise ratio for accurate determination of the experimental correlation functions. The sapphire windows act as a filter to block infrared radiation below about 2500 cm-1. This allows the detector gain to be increased, improving the signal-to-noise ratio which ranged between 50 and 100 over the spectral region of interest. Fluids were pressurized with a syringe pump (Isco, Model 260D) and plumbed to the high-pressure cell using 1/16 in. stainless steel tubing and stainless steel fittings. The temperature of the transmission cell was maintained at 110 ( 0.1 °C using a threemode controller and a platinum resistance temperature probe (Omega, No. N2001). The pressure of the IR cell was monitored using a pressure transducer (Precise Sensors, Model No. C451-10,000-01-G-15F-PE10) with an uncertainty of 0.1 bar. Construction details of a cell similar to the one used in this work can be found elsewhere.26 For the D2O solute molecules, the hydrogen-bonding interactions provide the strongest attractive component. However, in these studies we introduce the D2O probe molecule near the limit of infinite dilution in order to eliminate the potential for intermolecular hydrogen bonding. Additionally, we chose to run these experiments at 110 °C to reduce the possibility of intermolecular hydrogen bonding between solute molecules, which allowed us to ensure homogeneous solute-solvent systems. Previous researchers have shown that solvent clustering in supercritical fluids can become extensive in the region just above the critical point.27-30 This clustering phenomenon will lead to a wider distribution of rotational energy states as well as a broader distribution of local solute environments, resulting in increased inhomogeneous broadening of the purely vibrational bands. This effect can be eliminated by working relatively far (i.e., 110 °C) from the solvent’s Tc (31.1 and 16.6 °C for CO2 and Xe, respectively). The procedure for solution preparation was as follows. First, air was purged from the IR cell and associated transfer lines by flushing with CO2 or Xe. For spectra of D2O solutions, 1.0 µL of D2O was introduced into the IR cell using transfer techniques developed to minimize contamination of the D2O by atmospheric moisture. The concentration of the D2O in the cell was 7.5 mM. The cell was then heated to 110 °C and pressurized with the fluid in stepwise fashion. The contents of the cell were vigorously stirred for 10-15 min to ensure complete mixing and thermal equilibration prior to the spectral measurements. D2O (99.9 atom % D, Aldrich) and research grade Xe (99.999%, Spectra Gases) were used as received. CO2 was SFC grade (>99.98%, Scott Specialty Gases) and was flowed through a 1 L bed of 3.0 Å molecular sieves (EM Science) to remove residual water. All densities of CO2 were obtained through application of the equation of state developed by Huang31 while the densities of Xe were calculated from corresponding states
Figure 1. IR spectra of the ν3 mode of D2O dissolved in Xe as a function of the system pressure and density at 110 °C.
with an accurate equation of state for CH432 and published critical constants of Xe and CH4. A nitrogen-purged Nicolet 740 FT-IR spectrometer (Nicolet Analytical Instruments) equipped with a Ge-on-KBr beam splitter and a MCT-B (mercury cadmium telluride) detector was used to obtain all infrared spectra. A total of 1024 scans were coadded at 4 cm-1 resolution to obtain the desired signal-tonoise ratio. IR spectra for D2O were obtained by subtracting the detector response for the pure solvent from those for the solutions. After background subtraction the raw spectral data were Fourier transformed by numerical integration using the trapezoidal rule. In order to check the validity of the procedure, the C(t)’s were back-transformed and compared to the original IR spectra. The agreement between the raw data and these backtransformed spectra was excellent. Results and Discussion Infrared spectra of the ν3 mode of D2O dissolved in Xe and CO2 at several pressures are shown in Figures 1 and 2, respectively. The behavior of the spectral line shapes of the Xe and CO2 solutions with increasing fluid density can be described as follows. At lower pressures, the solute is gaslike and experiences approximately free rotation in both solvents. The multitude of peaks present in the atmospheric pressure spectra are the individual rotational-vibrational lines. These are present at pressures up to approximately 40 bar in CO2 and nearly 90 bar in Xe. At intermediate pressures the rotational fine structure broadens and coalesces into broad peaks on either side of the 0-0 transition (Q branch). Increasing the pressure of CO2 beyond 60 bar results in the P and R rotationalvibrational branches gradually merging into the Q branch until at the highest pressures a single Lorentzian-like peak is obtained. The observed P and R branches are comparable to the two peaks seen in eq 6 and gradually merge into a single peak described by eq 7. This phenomenon is not observed for D2O dissolved in supercritical Xe. In this simpler solvent, possessing no dipolar or quadrupolar interactions with D2O, the P and R branches are clearly observed even at the highest pressure of 517 bar. The densities of Xe and CO2 at the highest pressures
D2O Rotational Relaxation in CO2 and Xe
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Figure 3. Plot of the peak centers of D2O’s ν3 mode as a function of density when dissolved in Xe and CO2 at 110 °C. Lines are leastsquares fits to data points.
Figure 2. IR spectra of the ν3 mode of D2O dissolved in CO2 as a function of the system pressure and density at 110 °C.
(517 and 345 bar) examined were 0.0156 and 0.0153 mol/cm3, respectively. Therefore, at these high pressures, with approximately the same density, the IR spectra of D2O appear to indicate that the rotational relaxation of D2O dissolved in supercritical CO2 is severely hindered relative to D2O dissolved in supercritical Xe. As described previously, the difficulties implicit in the interpretation of IR band shapes in any molecule, let alone an asymmetric molecule such as D2O, require further consideration of the mechanisms responsible for the band broadening. In this respect the molecular dynamics simulations provide a means of estimating the purely rotational contribution to the band shapes. However, first we further analyze the IR data. As a first check of the separability of the vibrational/reorientational processes and to identify whether orientational relaxation is a primary mechanism of band broadening, we have computed the values of M(2) for D2O from the raw spectral data using eq 3 for all conditions studied. In CO2 the experimental values of M(2) ranged from 218 to 242 ps-2 while in Xe M(2) ranged from 232 to 264 ps-2. The value of M(2)CL for D2O is 226 ps-2 at 110 °C and indeed M(2) ≈ M(2)CL. This is indicative of the validity of separating the vibrational/reorientational parts of the experimentally determined C(t) and implies that the primary broadening mechanism is reorientational in nature. Another important consideration is the extent to which inhomogeneous broadening of the vibrational mode contributes to the overall bandwidth. One measure of this is to determine the change in the frequency of the vibrational mode as the transition from gaslike to liquidlike densities is made. In Figure 3, a plot of the IR frequency of the dissolved D2O’s ν3 mode versus density is presented for both supercritical solvents. The frequencies of the peak maxima are red-shifted by approximately 14 cm-1 when the density is increased from gaslike to liquidlike values. This change in the shift provides us with an approximate measure of the maximum possible variation in local solvent environments, which reflects the amount of inhomogeneous broadening contributing to the ν3 mode’s line shape. Since the change in shift with solvent density is relatively small, we conclude that the inhomogeneous contribution to broadening
contributes only a small part to the observed spectral broadening for the densities investigated. The red shifts in the vibrational frequency of the ν3 mode for D2O dissolved in both Xe and CO2 indicate that in both fluids attractive interactions dominate the solute molecule’s intermolecular potential over the density ranges studied. This is generally observed with most fluids. Repulsive interactions do not dominate the frequency shift until liquid densities are reached. In the two solvent systems, the frequency shift and its dependence on density are indistinguishable, within the experimental uncertainty. A number of theories concerning vibrational frequency shifts have been proposed ranging from the Kirkwood-Bauer-Magat (KBM) dielectric continuum model33,34 to a molecular-based, perturbed hard-sphere theory developed by Schweizer and Chandler (S-C)35 that describes how the interplay of attractive and repulsive interactions contributes to frequency shift. Both the KBM36,37 and S-C38 theories have been successfully used to describe the densityinduced frequency shifts in fluids. In light of the simple KBM relationship and the similar values for the dielectric constant of CO2 and Xe over this density range, the similarity of the frequency shifts in Xe and CO2 solutions is not unexpected. Experimentally determining the contribution of pure vibrational dephasing to the line shape is important for a strict determination of C(t)1R; however, in the present case only a qualitative description of this term is possible when solely considering the IR experimental data. For the lowest pressure data, the full width at half-maximum (fwhm) line widths are approximately 100 cm-1, and it is estimated that the majority of this width is attributable to reorientational processes, since the vibrational relaxation should occur on a much longer time scale, generally on the order of tens of picoseconds, which corresponds to line widths of approximately 0.5-1.0 cm-1. Although a separation of the vibrational and reorientational contributions to the line shape should be possible according to theory, there is currently no means of predicting the exact contribution from vibrational dephasing in the present system. Several useful theoretical descriptions, namely the isolated binary collision model,39 the hydrodynamic model,40 and the Schweizer-Chandler model,33 have been put forth concerning the “pressure” dependence of the dephasing line width. In the first two models the vibrational line width is proportional to ηT/F and ηT, respectively, where η is the viscosity and in general line widths increase with increasing pressure. The SchweizerChandler model accounts for attractive and repulsive contributions to the line width, implying that a narrowing or broadening
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Figure 4. Experimentally determined dipole autocorrelation functions, C(t), for the ν3 mode of D2O dissolved in Xe at 110 °C and pressures of (A) 1, 10, and 58 bar and (B) 69, 138, 244, and 517 bar.
Figure 5. Experimentally determined dipole autocorrelation functions, C(t), for the ν3 mode of D2O dissolved in CO2 at 110 °C and pressures of (A) 1, 11, and 58 bar and (B) 68, 138, 244, and 345 bar.
can occur with increasing pressures. In this regard we are unable to accurately predict the contribution of vibrational dephasing to the IR line widths without the aid of the molecular dynamics simulations as given below. Figure 4 A,B is the experimentally determined C(t)’s for the ν3 mode of D2O dissolved in Xe, and Figure 5 A,B is the experimentally determined C(t)’s for the ν3 mode of D2O dissolved in CO2. Immediately these figures allow one to see the spectral manifestations of the differing solvation environments in Xe and CO2. In the former the correlation function decays below 0 in approximately 0.13 ps. This reversal in sign implies that there is, on average, a partial reversal in the direction of the transition dipole moment with the molecular rotation being relatively free. This feature is reproduced by the approximation to the free rotor correlation function represented by eq 5. This sign change of the correlation function in the Xe solution persists up to the highest pressures studied, implying that even at relatively high densities D2O rotates freely. Qualitatively, all of the correlation functions for the Xe solvent appear quite similar, except that as the pressure is increased the minimum of the correlation function increases from -0.25 at 1 bar to -0.15 at 517 bar. On the other hand, the C(t)’s for D2O dissolved in CO2 change dramatically in going from 1 to 345 bar. The minimum of the correlation function gradually increases, and above 86 bar the correlation function no longer changes sign. The correlation function appears to be gradually approaching a simple, monotonic decay. This is indicative of a more hindered rotation resulting from the increased torque on the solute due to intermolecular interactions of the solvent.
However, one must use caution in solely interpreting the experimental correlation functions in terms of rotational relaxation since, as described in eq 2, there is obviously a vibrational dephasing contribution that is certainly present and is expected to be pressure dependent. The use of molecular dynamics simulations allows us another means to estimate the relative contributions of the reorientational and vibrational relaxation. The simulated dipole moment autocorrelation functions, C(t)MD 1R , of D2O dissolved in Xe and D2O dissolved in CO2 systems are given in Figures 6 and 7, respectively. The three (calculated) pressures investigated for each solution are 36.5, 86, and 345 bar. Estimated uncertainties at the 95% confidence level are plotted along with the correlation functions at 36.5 bar. These uncertainties were obtained as the standard errors of the four simulations performed at this condition. At the higher pressures the uncertainties were smaller. Also plotted in Figure 7, for comparison, is the experimental correlation function, C(t), of the ν3 mode of D2O dissolved in CO2 at 345 bar. At each condition, the simulated correlation function, C(t)MD 1R , decayed to essentially zero within approximately 1 ps, indicating that the direction of the dipole had been completely randomized by thermal motion. Within the experimental uncertainty, correlation functions are independent of pressure in the Xe system over the entire pressure range in agreement with the experimentally determined C(t)’s. This lack of density dependence results from the nature of the interaction between the D2O molecule and Xe atoms which are primarily van der Waals attractions and repulsions involving
D2O Rotational Relaxation in CO2 and Xe
MD Figure 6. Simulated dipole autocorrelation functions, C(t)1R , for D2O dissolved in Xe at pressures of 36, 86, and 345 bar and a temperature of 110 °C. The uncertainties (- - -) of the simulation at 36 bar determined as the standard error of multiple runs is also plotted. For clarity the arrows demarcate the standard errors at arbitrary points (b) along the correlation function.
MD Figure 7. Simulated dipole autocorrelation functions, C(t)1R , for D2O dissolved in CO2 at pressures of 36, 86, and 345 bar and a temperature of 110 °C. The uncertainties (- - -) of the simulation at 36 bar determined as the standard error of multiple runs is also plotted as well as the experimentally determined dipole autocorrelation function, C(t), at a pressure of 345 bar (O). For clarity the arrows demarcate the standard errors at arbitrary points (b) along the correlation function.
oxygen and Xe. Due to the nature of these purely van der Waals collisions and the small moment on the D2O molecule, the Xe is incapable of exerting large torques on the D2O molecule. Results for the CO2 solutions reflect the effects of the intermolecular interaction between the dipole of the D2O molecule and the quadrupole of the CO2. As evinced in both the simulated (345 bar) and experimental correlation functions at higher pressures (>86 bar), this interaction was capable of exerting a significant torque on the D2O solute molecule. The transition dipole moments tended to remain pointed in their original direction, their rotation hindered by interactions with the solvent. A logarithmic plot of C(t) for D2O dissolved in CO2 at 345 bar provides an estimate of the reorientational correlation time under the most hindered conditions. This time was 0.2 ps, which is significantly shorter than can be expected for vibrational relaxation, lending confidence to the original assumptions concerning the separability of these two contributions. There is excellent agreement between the simulated C(t)MD 1R and the experimental C(t) for the highest molecular
J. Phys. Chem., Vol. 100, No. 47, 1996 18333
Figure 8. Experimental (- - -) and simulated (s) IR rotation wings for D2O dissolved in Xe at pressures of 36, 86, and 345 bar and a temperature of 110 °C.
Figure 9. Experimental (- - -) and simulated (s) IR rotation wings for D2O dissolved in CO2 at pressures of 36, 86, and 345 bar and a temperature of 110 °C.
densities of CO2. The results calculated from the molecular dynamics simulations for CO2 at 345 bar were within 0.08 (with C(0) normalized to 1) of the experimental data over the full 1 ps time interval. Thus, we conclude that for these systems C(t)V is not a significant component in the measured C(t)’s. For the highest density studied, a molecular dynamics simulation of D2O dissolved in CO2 was run with the electrostatic interaction of the quadrupole moment turned off. The resulting C(t)MD 1R was quite similar to the result for Xe with a change in the sign of the correlation function occurring at 0.10 ps. This allows us to unequivocally attribute the more hindered rotation, in the D2O/CO2 system at high density, to the electrostatic interactions of D2O with CO2’s quadrupole moment. Although in the experimental system other mechanisms of rotational hindrance are possible, our results indicate that interaction of the D2O with the quadrupole of CO2 is the predominant mechanism. It was of interest to further analyze the results from the simulations for comparison to experiment, and so inverse Fourier transforms were performed on the simulated correlation functions. The results of these transforms yielded simulated infrared spectra of D2O dissolved in Xe and D2O dissolved in CO2. These simulated spectra along with the corresponding experimental IR spectra are plotted in Figures 8 and 9 for the two respective solvent systems. The simulated spectra were shifted by the experimental central band frequencies, and the amplitudes were
18334 J. Phys. Chem., Vol. 100, No. 47, 1996 scaled appropriately to enable comparison with experimental data. The shapes of the experimental rotation wings were well reproduced by the simulations for both supercritical solvent systems. As previously noted, the simulated spectra take into account only rotational broadening and allow us to conclude that this is the predominant mechanism of line broadening in these systems. Therefore, the above analysis of our experimental IR data, in which we neglect the contribution of vibrational relaxation, is plausible and allows us to more conclusively interpret the differing degrees of line broadening as a function of increasing pressure for D2O dissolved in Xe and D2O dissolved in CO2. It is clear that in the case of the Xe solvent the spectra are of the form expressed in eq 6 for the free rotor even at the highest density studied while the spectra for the CO2 solvent evolve from a free rotor form at lower density to a more “diffusional” character expressed through eq 7. This transition from a free rotor to a more hindered rotor with increasing density can be attributed to the fundamental difference between CO2 and Xe, which is the presence of a quadrupole-dipole interaction with the solute in the latter. Conclusion The rotational relaxation of D2O dissolved in both supercritical Xe and supercritical CO2 has been examined as a function of density at 110 °C using IR spectroscopy and molecular dynamics simulations. The unique combination of these two techniques has allowed us to analyze the IR spectra of an asymmetric rotor, D2O, determine the contribution of reorientation to band shape, and qualitatively describe the differences between the two solvent systems in regards to the molecular rotation. In Xe discrete rotational-vibrational lines were observable at the relatively high pressure of 86.7 bar, and the IR spectra still exhibited P and R branches at 517 bar, which was the maximum pressure examined. Rotation appears to be essentially unhindered even at these high Xe densities. In CO2 the rotational-vibrational fine structure coalesced into broad P and R envelopes at pressures above approximately 40 bar. As pressure was further increased, the spectra became increasingly symmetric with the P and R branches disappearing well before 345 bar. Dipole moment autocorrelation functions containing no contributions from vibrational relaxation were calculated from NPT molecular dynamics simulations for D2O dissolved in Xe and CO2. The simulations were performed at 110 °C and at pressures of 36.5, 86, and 345 bar. Simulated results of Xe solutions exhibited a lack of pressure dependence, demonstrating that D2O behaved like a free rotor even at a fairly high density. Hindered rotation was observed in CO2 at 345 bar. At this pressure the correlation function, C(t)MD 1R , was positive for all times and was in good agreement with the experimental C(t). The results were compared to experimental correlation functions obtained from Fourier inversion of the IR band, allowing increased confidence in the analysis of the IR data. The experimentally derived correlation functions pointed to a tremendous difference in the two solvent systems. In Xe, even at the highest densities, the correlation function resembled a damped oscillation (eq 5), indicating that the rotation of D2O was relatively free and unhindered. In CO2 at the lowest densities, the same damped oscillation is observed; however, above 86 bar the correlation functions decreased monotonically, leading to the conclusion that the rotation of D2O is more hindered in CO2 as the density is increased. The differences in the two solvent systems are attributable to the differences in their intermolecular interactions. In Xe the interactions are predominantly van der Waals attractions and repulsions while in CO2 the presence of a quadrupole allows electrostatic
Bowman et al. interaction with the dipole of D2O in addition to the van der Waals and repulsive interactions. These additional interactions exert torque on the solute, resulting in more hindered rotations with increasing density. Combining results from molecular dynamics simulations and IR spectroscopy has allowed us to conclude that IR line broadening in these supercritical fluid systems of small rotors is predominantly due to rotational relaxation, whereas this is clearly not the case in much higher density liquid systems. Acknowledgment. This research was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, Chemical Sciences Division of the U.S. Department of Energy, under Contract DE-AC06-76RLO 1830. References and Notes (1) Chemical Engineering at Supercritical Fluid Conditions; Paulaitis, M. E., Penninger, J. M. L., Gray, R. D., Davidson, P., Eds.; Ann Arbor Science: Ann Arbor, MI, 1983. (2) Gordon, R. G. J. Chem. Phys. 1964, 41, 1819-1829. (3) Gordon, R. G. J. Chem. Phys. 1966, 44, 1830-1836. (4) Chialvo, A. A.; Heath, D. L.; Debenedetti, P. G. J. Chem. Phys. 1989, 91, 7818. (5) Howdle, S. M.; Bagratashvili, V. N. Chem. Phys. Lett. 1993, 214, 215. (6) Yoon, J.-H.; Hacura, A.; Baglin, F. G. J. Chem. Phys. 1989, 91, 5230. (7) Fulton, J. L.; Yee, G. G.; Smith, R. D. J. Am. Chem. Soc. 1991, 113, 8327. (8) Straatsma, T. P.; Berendsen, H. J. C.; Postma, J. P. M. J. Chem. Phys. 1986, 85, 6720. (9) Guillot, B.; Guissani, Y.; Bratos, S. J. Chem. Phys. 1991, 95, 3643. (10) Gordon, R. G. J. Chem. Phys. 1965, 43, 1307. (11) Bratos, S.; Rios, J.; Guissani, Y. J. Chem. Phys. 1970, 52, 439. (12) Oxtoby, D. W. AdV. Chem. Phys. 1979, 40,1. (13) Nafie, L. A.; Peticolas, W. L. J. Chem. Phys. 1972, 57, 3145. (14) Rothschild, W. G. J. Chem. Phys. 1976, 65, 455. (15) Jones, D. R.; Andersen, H. C.; Pecora, R. Chem. Phys. 1975, 9, 339. (16) Clarke, J. H. R. In AdVances in Infrared and Raman Spectroscopy; Clarke, J. H. R., Hester, R. E., Eds.; Heyden: London, 1978; Vol. 4, Chapter 4. (17) Murthy, C. S.; Singer, K.; McDonald, I. R. Mol. Phys. 1981, 44, 135. (18) Geiger, L. C.; Ladanyi, B.; Chapin, M. E. J. Chem. Phys. 1990, 93, 4533. (19) McDonald, I. R.; Singer, K. Mol. Phys. 1972, 23, 29-40. (20) Berendsen, H. J. C.; Grigera, J. R.; Straatsma, T. P. J. Phys. Chem. 1987, 91, 6269. (21) Nose´, S. Mol. Phys. 1984, 52, 255. (22) Computer Simulation of Liquids; Allen, M. P., Tildesley, D. J., Eds.; Clarendon: Oxford, 1987. (23) Palmer, B. J. J. Comput. Phys. 1993, 104, 470. (24) Palmer, B. J.; Garrett, B. C. J. Chem. Phys. 1993, 98, 4047. (25) Yee, G. G.; Fulton, J. L.; Blitz, J. P.; Smith, R. D. J. Phys.Chem. 1991, 95, 1403-1409. (26) Blitz, J. P.; Fulton, J. L.; Smith, R. D. Appl. Spectrosc. 1989, 43, 812-816. (27) Debenedetti, P. G. Chem. Eng. Sci. 1987, 42, 2203. (28) Eckert, C. A.; Ziger, D. H.; Johnston, K. P.; Ellison, T. K. Fluid Phase Equilib. 1983, 14, 176. (29) Kim, S.; Johnston, K. P. Ind. Eng. Chem. Res. 1987, 26, 1206. (30) Kim, S.; Johnston, K. P. AIChE J. 1987, 33, 1603. (31) Huang, F. H.; Li, M. H.; Lee, L. L.; Starling, K. E.; Chung, F. T. H. J. Chem. Eng. Jpn. 1985, 18, 490. (32) McCarty, R. D. Cryogenics 1974, 276. (33) Kirkwood, J. G. J. Chem. Phys. 1934, 2, 351. (34) Bauer, E.; Magat, M. J. Phys. Radium 1938, 9, 319. (35) Schweizer, K. S.; Chandler, D. J. Chem. Phys. 1982, 76, 2296. (36) Yee, G. G.; Fulton, J. L.; Smith, R. D. J. Phys.Chem. 1992, 96, 6172. (37) Akimoto, S.; Kajimoto, O. Chem. Phys. Lett. 1993, 209, 263. (38) Ben-Amotz, D.; LaPlant, F.; Shea, D.; Gardecki, J.; List, D. In Supercritical Fluid Technology; Bright, F. B., McNally, M. E. P., Eds.; ACS Symposium Series 488; American Chemical Society: Washington, DC, 1992. (39) Fischer, S. F.; Laubereau, A. Chem. Phys. Lett. 1975, 35, 6. (40) Oxtoby, D. W. J. Phys. Chem. 1983, 87, 3028.
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