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V. COMPARISON OF THEORY AND EXPERIMENT

V. COMPARISON OF THEORY AND EXPERIMENT

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656



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Figure 10 (a) T1 , T vs. density for the solvent ethane at 34° C and the best fit

theoretically calculated curve. The theory was scaled to match the data at the critical

density, 6.88 mol/L. The best agreement was found for ω D 150 cm 1 . (b) T1 , T

vs. density for the solvent ethane at 50° C and the theoretically calculated curve.

The scaling factor, frequency ω, and the hard sphere diameters are the same as

those used in the fit of the 34° C data. Given that there are no free parameters, the

agreement is very good.



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Figure 11 Plots of the ethane solvent parameters vs. density for the thermodynamic properties that enter the theory at the two temperature, 34° C (solid lines)

and 50° C (dashed lines), the temperatures for the data and theory curves in Fig. 10.

Most of the parameters change dramatically with temperature.



fluoroform, 2 mol/L is the lowest density for which W(CO)6 has sufficient

solubility to perform the experiments at 28° C.) ω is not adjusted, but is

set equal to 150 cm 1 , the value obtained in the fit of the ethane data. An

accurate value of the effective hard sphere diameter is not available for

fluoroform. Hard sphere diameters tend to be slightly temperature dependent, becoming smaller as the temperature is increased (104). X-ray data

are available at 70 K (105), well below the experimental temperature. The

˚ The best agreement

70 K x-ray data yield a hard sphere diameter of 4.60 A.

˚ The hard sphere diameter

with the T1 , T at 28° C is obtained with 3.28 A.

used is ¾30% smaller than the 70 K x-ray determined value; it is not known

how this value would compare to a good value of the effective hard sphere

diameter at 301 K. The theory does a remarkably good job of reproducing

the shape of the data with only the adjustment in the solvent size as a fitting

parameter that affects the shape of the calculated curve. Figure 12b shows

data taken at 44° C, which is the equivalent increase in temperature above

Tc as the higher temperature data taken in ethane (Fig. 10). The calculated



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curve (solid line) is obtained using the thermodynamic/hydrodynamic input

parameters for the higher temperature and using the same scaling factor as

˚ Therefore,

at 28° C. ω is 150 cm 1 and the fluoroform diameter is 3.28 A.

there are no free parameters in the calculation of the lifetimes along this

isotherm. As with the data taken in ethane, the theory does a very good job

of reproducing the density-dependent data.

As shown in Fig. 9 and discussed in the theory section, the calculations are quite sensitive to the choice of ω. The fact that the same ω

gives good theoretical agreement for the T1 , T data in both ethane and

fluoroform suggests that this is not an arbitrary value, but, rather, it reflects

the energy deposited to the solvent upon vibrational relaxation. A number

of calculations were performed to examine the influence of ω on the calculation of T1 , T . The curves in Fig. 9 are all for single values of ω.

Calculations were also performed in which distributions of ω were used. It

was found that ω need not be a single frequency. For example, an ensemble

average over relaxation occurring with a Gaussian distribution of ω centered

about 150 cm 1 with standard deviation of, e.g., 20 cm 1 , gave the identical

results. We also tried to reproduce the data with various bimodal distributions. Such distributions were not able to reproduce the data. ω D 150 cm 1

may be a single frequency or the average of a spread of frequencies about

150 cm 1 . This frequency is most likely located within the single “phonon”

density of states (DOS) of the continuum of low-frequency modes of the

solvents. Instantaneous normal mode calculations in CCl4 , CHCl3 , and CS2

show cut-offs in the DOS at ¾150, ¾180, and ¾200 cm 1 , respectively

(106,107). The higher frequency portions of the DOS are dominated by

orientational modes. Since ethane, fluoroform, and CO2 are much lighter

Figure 12 (a) T1 , T data measured in fluoroform on the near critical isotherm

28° C , 2 K above Tc , and the calculated curve, which is scaled to match the data at

the critical density, 7.56 mol/L. The fluoroform hard sphere diameter was adjusted

since a good value at experimental temperatures is not available. A diameter of

˚ yielded the optimal fit. ω is not adjustable. It is set equal to 150 cm 1 ,

3.28 A

the value obtained in the fit of the ethane data. The theory does a very good job

of reproducing the shape of the data with only the adjustment in the solvent size

as a fitting parameter that affects the shape of the calculated curve. (b) T1 , T

data taken at 44° C, which is the equivalent increase in temperature above Tc as the

higher temperature data taken in ethane (Fig. 10b). The theory curve is calculated

using the same scaling factor, frequency, and solvent hard sphere diameter as at the

lower temperature. Considering that there are no free parameters, the theory does

an excellent job of reproducing the higher temperature data.



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than the molecules that comprise the liquids mentioned above, it is expected

that their DOS will extend to higher frequencies.

In the gas phase, the asymmetric CO stretch lifetime is 1.28 š 0.1 ns.

The solvent can provide an alternative relaxation pathway that requires

single phonon excitation (or phonon annihilation) (102) at 150 cm 1 . Some

support for this picture is provided by the results shown in Fig. 8. When

Ar is the solvent at 3 mol/L, a single exponential decay is observed

with a lifetime that is the same as the zero density lifetime, within

experimental error. While Ar is effective at relaxing the low-frequency

modes of W(CO)6 , as discussed in conjunction with Fig. 8, it has no

affect on the asymmetric CO stretch lifetime. The DOS of Ar cuts off

at ¾60 cm 1 (108). If the role of the solvent is to open a relaxation

pathway involving intermolecular interactions that require the deposition

of 150 cm 1 into the solvent, then in Ar the process would require the

excitation of three phonons. A three-phonon process would be much less

probable than single phonon processes that may occur in the polyatomic

solvents. In this picture, the differences in the actual lifetimes measured

in ethane, fluoroform, and CO2 (see Fig. 3) are attributed to differences in

the phonon DOS at ¾150 cm 1 or to the magnitude of the coupling matrix

elements.

Figure 13 shows a comparison of T1 , T data with calculated curves

for the CO2 solvent at two temperatures, 33° C (2 K above Tc ) and 50° C.

The calculated curves are scaled to the data at 33° C and 2 mol/L. Again,

ω is set to 150 cm 1 . The solvent hard sphere diameter is the literature

˚ (104). The agreement between theory and experiment, while

value, 3.60 A

not poor, is clearly not as good as that displayed for ethane and fluoroform.

Adjusting ω does not improve the agreement between theory and data, nor

does a further variation of the solvent diameter. The T1 , T data appear

to differ from the data in the other two solvents. After the initial rapid

decrease in the lifetime at low densities, the data curve with CO2 as the

solvent is much flatter than the data obtained in the other two solvents. The

theory does well up to ¾6 mol/L but then overestimates the decrease in

lifetime at higher densities.

An interesting question arises as to why the theory does not do as

good a job of reproducing the data with CO2 as the solvent. The CO2 data

becomes much flatter above ¾6 mol/L than the data taken in the other two

solvents. The theoretical curves for the CO2 solvent have shapes that are

similar to the theoretical and data curves for the other two solvents. This

might suggest that the CO2 solvent data is modified by some special chemical interaction. The source of the discrepancy between theory and data does



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Figure 13 (a) T1 , T data with calculated curves for the CO2 solvent at

33° C (2 K above Tc ). The calculated curve is scaled to the data at 2.0 mol/L.

˚ The agreement

ω D 150 cm 1 , and the solvent hard sphere diameter is 3.60 A.

between theory and experiment, while not terrible, is clearly not as good as that

displayed for ethane (Fig. 10) and fluoroform (Fig. 12). The data in CO2 appear to

be different than in ethane and fluoroform, as discussed in the text. (b) T1 , T data

with calculated curves for the CO2 solvent at 50° C. The scaling factor, frequency

ω, and the hard sphere diameters are the same as those used in the fit of the 33° C

data. Again, the theory drastically overestimates the slope at higher densities.



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not appear to be associated with near-critical phenomena since the discrepancy for the near-critical isotherm is not reduced for the higher temperature

isotherm. There may be a specific interaction between CO2 and W(CO)6 .

Supercritical CO2 tends to form T-shaped structures with the oxygen of

one CO2 coordinated with the carbon of another CO2 , forming a T (109).

It is possible that CO2 has a tendency to coordinate with the oxygens of

W(CO)6 in the same manner. Such a specific interaction could modify the

density dependence of the vibrational lifetime. Substantial association of

CO2 with the COs of W(CO)6 would not be reflected in the hard sphere

direct correlation function used in the calculations, nor would it occur in

a Lennard-Jones description of the interactions. The specific W(CO)6 /CO2

chemical interaction is an appealing explanation given the good agreement

between theory and experiment for the other solvents. On the other hand, the

reduced quality of the agreement for the CO2 data may reflect inadequacies

in the theory.

Figure 4 shows the density dependence of the vibrational absorption line shift in the three solvents at the same temperatures used in the T1

measurements. The line shift behavior is seen to be similar to the vibrational

lifetime trends. Notice that there are no substantial density-independent

plateaus like those frequently observed in solvatochromic shifts of electronic transitions.

B. Temperature Dependence



The isothermal data in the three supercritical solvents all show similar

trends. For the near-critical isotherm, the lifetime initially decreases rapidly

with density. At an intermediate density less than c , the slope changes and

develops a much weaker density dependence. Along the higher temperature isotherm, the density dependence is somewhat smoother and does not

exhibit as substantial a change of slope. The isopycnic (constant density)

data, however, display an interesting solvent dependence.

Figure 14 shows the lifetime dependence on temperature in ethane

at a constant density equal to c . The data have a region, extending well

above the critical temperature, in which the lifetime becomes longer as

the temperature is increased. This behavior in ethane is unexpected and

is in contrast to theoretically proposed trends of vibrational lifetimes with

temperature at constant density (102,110). We will refer to this temperature

range as inverted (16,111).

In general, an increase in temperature (at constant density) should

decrease lifetimes. Vibrational relaxation is caused by fluctuating forces in



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Figure 14 T1 , T vs. temperature for the solvent ethane at fixed density (the

critical density) and the theoretically calculated curve. The scaling factor, frequency

ω, and the hard sphere diameters are the same as those used in the fit of the 34° C

density-dependent data, i.e., there are no free parameters in this calculation. Notice

the presence of an inverted regime, i.e., a range of temperatures for which the lifetime increases with temperature, contrary to expected behavior. The lifetime peaks

at ¾375 K before decreasing with temperature. Remarkably, the theory captures

this phenomenon, though it overestimates the drop in the lifetime with temperature

after 375 K.



the bath acting on the vibrational oscillator, i.e., by the Fourier component of the force-force correlation function at ω, the frequency associated

with energy deposition into the bath. Since the bath is incoherent, there

are no well-defined phase relationships between modes that could lead to

destructive interference at some frequencies. Thus, when the temperature

is increased, all Fourier components of the force-force correlation function

might be expected to increase in amplitude because all modes increase in

population. The Fourier component of the force at ω would then increase

with temperature, and vibrational relaxation, at constant density, would be

expected to become faster.

It is important to emphasize again that the data displayed in Fig. 14

are at fixed density. At each temperature, the pressure was adjusted to bring



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the density back to 6.88 mol/L, the required pressure being determined

from an accurate equation of state for ethane (74,75). Had the density not

been held constant, the increase of the lifetime with temperature would not

have been as remarkable. For instance, the lifetime of the CO asymmetric

stretching mode of W(CO)6 in liquid CHCl3 has been observed to lengthen

as the temperature is increased from the melting point to the boiling point

of the liquid (16). But in this system, as the temperature increases, the

liquid density decreases significantly. For the experiments in CHCl3 liquid,

the increase in lifetime with temperature was ascribed to effects arising

from the decrease in density (16,106).

The solid line in Fig. 14 is the theoretically calculated temperature

dependence of the lifetime using Equation (21). The frequency, the hard

sphere diameter, and the scaling factor used in the calculation are those

obtained from the fitting of the density dependent data in ethane at 34° C.

Therefore, there are no adjustable parameters in the calculation of the theoretical curve in Fig. 14. The theory does a good job of predicting the overall

temperature dependence of the lifetime. The inverted region between Tc and

¾375 K is reproduced, though the predicted lifetimes drop too rapidly with

temperature past the inverted region.

The calculations include many detailed temperature-dependent

properties of ethane (see Fig. 11). Many of these properties are highly

temperature dependent in the range from the critical temperature to ¾70 K

above Tc , and they come into the calculation in a complex manner.

The temperature also enters explicitly in the Egelstaff quantum correction

[Equation (27)] and in the Kawasaki expression for 1/ 1 k . It is the

complex interplay of these changing parameters, which are determined by

the equation of state of ethane, that is responsible for the initial increase

of T1 with temperature. By ¾70 K above the critical point, however, the

various physical properties of ethane are no longer changing as rapidly. It

is at this point that the lifetime decreases with increasing temperature.

Once the last vestiges of the influence of the critical point are gone,

the expected constant density temperature dependence is manifested, i.e.,

the lifetime decreases with increasing temperature. From a microscopic

perspective, when the temperature is increased, the occupation numbers of

all modes of the solvent increase. The amplitude of all solvent modes will

increase including the Fourier component of the bath spectral density at ω.

However, vibrational relaxation depends on the spectrum of forces, not on

the spectral density of the solvent fluctuations. As the temperature is initially

increased above the critical temperature, both the data and the theory show

that the relevant Fourier component of the force-force correlation function



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initially decreases, leading to an increase in the lifetime. In ethane, once the

temperature is well above the critical temperature, the physical properties of

the fluid are not changing rapidly, and the normal temperature dependence

(decreasing lifetime with increasing temperature) occurs.

Figure 15 shows the lifetime as a function of temperature at the

critical density of carbon dioxide. With CO2 as the solvent there is no

inverted region in which the lifetime becomes longer as the temperature is

increased. Instead, the lifetime decreases approximately linearly. Thus the

inverted behavior is not universal but is specific to the properties of the

particular solvent. The fact that the nature of the temperature dependence

changes fundamentally when the solvent is changed from ethane to CO2

demonstrates the sensitivity of the vibrational relaxation to the details of

the solvent properties. The solid line is the theoretically calculated curve.

The calculation of the temperature dependence is done with no adjustable



Figure 15 T1 , T vs. temperature for the solvent carbon dioxide at the critical density and the theoretically calculated curve. The frequency ω and the hard

sphere diameters are the same as those used in the fit of the 33° C data. The theory

is scaled to match the data at 33° C and the critical density, 10.6 mol/L. Unlike

ethane at the critical density, there is no inverted region, and the vibrational lifetime decreases nearly linearly with temperature. The theory does not quantitatively

fit the data, but it does show the correct general behavior. Most importantly, the

hydrodynamic/thermodynamic theory shows the existence of the inverted region in

ethane and the lack of one in carbon dioxide.



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parameters other than the scaling factor used to match the theory with the

data at 33° C. The input parameters are taken from the physical properties of

the solvent and the parameters obtained from fitting the isothermal density

dependent data at 33° C. The theory does a reasonable job of reproducing

the data. The most important feature of the theoretical calculation is the

absence of an inverted region. Aside from the hard sphere diameters, the

only differences between the ethane calculations and the CO2 calculations

are the physical properties of the solvents obtained from the equations of

state of the two fluids. The interplay of parameters that give rise to an

inverted region in ethane and no inverted region in CO2 are complex. The

fact that the theory can reproduce this qualitative difference in the vibrational relaxation temperature dependence in the two solvents shows that it

is capturing the essential features of vibrational relaxation as a function of

temperature and density in supercritical fluids.

The theory reflects the solvent properties through the thermodynamic/hydrodynamic input parameters obtained from the accurate equations

of state for the two solvents. However, the theory employs a hard sphere

solute-solvent direct correlation function C12 , which is a measure of the

spatial distribution of the particles. Therefore, the agreement between theory

and experiment does not depend on a solute-solvent spatial distribution

determined by attractive solute-solvent interactions. In particular, it is not

necessary to invoke local density augmentation (solute-solvent clustering)

(31,112,113) in the vicinity of the critical point arising from significant

attractive solute-solvent interactions to theoretically replicate the data.



VI. CONCLUDING REMARKS



We have presented experimental and theoretical results for vibrational relaxation of a solute, W(CO)6 , in several different polyatomic supercritical

solvents (ethane, carbon dioxide, and fluoroform), in argon, and in the

collisionless gas phase. The gas phase dynamics reveal an intramolecular

vibrational relaxation/redistribution lifetime of 1.28 š 0.1 ns, as well as the

presence of faster (140 ps) and slower (>100 ns) components. The slower

component is attributed to a heating-induced spectral shift of the CO stretch.

The fast component results from the time evolution of the superposition

state created by thermally populated low-frequency vibrational modes. The

slow and fast components are strictly gas phase phenomena, and both disappear upon addition of sufficiently high pressures of argon. The vibrational



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