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D. A Viscoelastic Theory of Vibrational Dephasing

D. A Viscoelastic Theory of Vibrational Dephasing

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Vibrational Dephasing in Liquids



421



Figure 17 Schematic illustration of the viscoelastic (VE) model of dephasing.

The vibrating molecule (toluene here) occupies a cavity within the solvent with a

certain size in D 0. In D 1, the radius of the cavity is slightly larger, because

of vibrational anharmonicity. This effect couples shear fluctuations of the solvent

to the vibrational frequency. See also Fig. 18.



displaces the repulsive wall of the vibrator outward, forcing the solute

cavity to expand.

The energetics of this model are shown in Fig. 18. The solute-cavity

radius has a minimum energy at a radius r0 when the vibrator is in D 0 and

a minimum at a larger radius r0 C dr when it is in D 1. When the vibrator

is in D 0, there will be a distribution of cavity sizes at equilibrium due

to thermal excitations in the solvent. These different cavities have different

vertical transition energies, and the width of the size distribution maps into

the width of transitions ω .

The frequency correlation time ω corresponds to the time it takes

for a single vibrator to sample all different cavity sizes. The fluctuationdissipation theorem (144) shows that this time can be found by calculating

the time for a vertically excited D 0 vibrator to reach the minimum in

D 1. This calculation is carried out by assuming that the solvent responds

as a viscoelastic continuum to the outward push of the vibrator. At early

times, the solvent behaves elastically with a modulus G1 . The push of

the vibrator launches sound waves (acoustic phonons) into the solvent,

allowing partial expansion of the cavity. This process corresponds to a

rapid, inertial solvent motion. At later times, viscous flow of the solvent

allows the remaining expansion to occur. The time for this diffusive motion

is related to the viscosity Á by G1 and the net force constant at the cavity



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Berg



Figure 18 Free-energy potentials corresponding to the VE theory of dephasing.

The equilibrium cavity radius is larger in D 1 than in D 0 by and amount

υr (see Fig. 17). As a result, the energy of the vibrational transition depends on

the cavity radius. At thermal equilibrium, shear fluctuations in the liquid will cause

fluctuations in the radius, which in turn cause fluctuations in the transition frequency

ω. (Adapted from Ref. 8.)



boundary KS :

ω



D



4

1

C

Á

G1

3KS



37



With this information, we can calculate the separate contributions of inertial

and diffusive solvent motion to the linewidth.

The first result of this calculation is that the inertial motion causes

almost no dephasing. This result is a direct contrast to models like the

IBC theory, which attribute all the dephasing to collisional, i.e., inertial,

dynamics. The difference between these theories lies in their assumptions

about correlations in the solvent motion. The IBC explicitly assumes that

the collisions are independent, i.e., the solvent motion has no correlations.

As a result, the collisions are an effective sink for phase memory from

the vibration. On the other hand, within the VE model the solvent motions

appear as sound waves. Their effect on the vibrational frequency decays

as they propagate away from the vibrator, but they remain fully coherent

at all times. Because they remain coherent, they cannot destroy the phase



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Vibrational Dephasing in Liquids



423



memory of the vibration. The degree to which inertial solvent motion is

coherent is an important and unresolved issue. The IBC and VE models

represent extreme viewpoints on this issue, and the truth may lie somewhere

in between. However, measuring the extent of vibrational dephasing due to

inertial motion can provide incisive information on the degree of coherence

in inertial solvent motion.

For the moment, assume that the VE picture is correct and inertial

solvent motion causes negligible dephasing. Diffusive motion must be the

primary cause of coherence decay. In the VE theory, the diffusive motion is

the relaxation of stress fluctuations in the solvent by viscous flow. The VE

theory calculates both the magnitude ω and lifetime ω of the resulting

vibrational frequency perturbations. A Kubo-like treatment then predicts

the coherence decay as a function of the viscosity of the solvent. Figure 19

shows results for typical solvent parameters. At low viscosity, the modulation is in the fast limit, so the decay is slow and nearly exponential.

Under these conditions, the dephasing time is inversely proportional to the

viscosity, as in previous theories [Equation (19)]. As the viscosity increases,

the modulation rate slows. The decay becomes faster and approaches a



Figure 19 The VE prediction of the diffusive component of the vibrational coherence decay CFID as a function of the solvent viscosity (Á D 1, 2, 4, 8, 16, 32 and

1 cP) for typical parameters. At low viscosity, the decay is exponential, its rate is

inversely proportional to the viscosity, and the corresponding Raman line is homogeneously broadened. At high viscosity, the decay becomes Gaussian, its decay

time reaches a limiting value, and the Raman line is inhomogeneously broadened.

(From Ref. 8.)



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Berg



Gaussian shape. In the limit of infinite viscosity, e.g., in a glass, the line

shape approaches a well-behaved limit. These qualitative features are the

same as were seen experimentally in supercooled toluene (Section IV.C).

A quantitative fit of the VE theory to the toluene data is shown

in Fig. 20. In addition to the VE dephasing, an additional temperatureindependent, fast-modulation dephasing process had to be included.

Several reasonable mechanisms exist for this additional dephasing: the

population lifetime and inertial dynamics acting through phonon scattering

or through imperfect correlations in the solvent. The theory reproduces



Figure 20 The VE theory of dephasing fit to Raman-FID data on toluene (see

Fig. 15). The entire range of data is fit with only three temperature-independent

parameters: the solvent’s high-frequency elastic modulus, a solvent-solute coupling

constant, and a homogeneous dephasing time. A temperature-independent homogeneous dephasing process is assumed in addition to the VE mechanism.



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Vibrational Dephasing in Liquids



425



the temperature-dependent changes of the decay shape very well. Only

three parameters are used to fit the entire temperature range: the solvent

modulus G1 , the coupling strength between the solvent and vibration, and

the dephasing time of the viscosity independent process T2 . (In contrast, the

Kubo analysis (Fig. 15) uses two fitting parameters for each temperature.)

The temperature-dependent changes in the coherence decay are determined

by the independently measured viscosity.

The ability to correctly reproduce the viscosity dependence of the

dephasing is a major accomplishment for the viscoelastic theory. Its significance can be judged by comparison to the viscosity predictions of other

theories. As already pointed out (Section II.C 22), existing theories invoking

repulsive interactions severely misrepresent the viscosity dependence at

high viscosity. In Schweizer-Chandler theory, there is an implicit viscosity

dependence that is not unreasonable on first impression. The frequency

correlation time is determined by the diffusion constant D, which can

be estimated from the viscosity and molecular diameter by the StokesEinstein relation:

2



D



D



3



Á

38

6D

2kT

In comparison, the correlation time in the viscoelastic theory is given by

Equation (37). For typical parameters, ωSC ¾ 50 ωVE . Thus, the SchweizerChandler predicts frequency perturbations in the slow-modulation limit at

low viscosity. In the VE theory, the slow-modulation limit is only reached

at high viscosity. The fundamental difference is in the interaction assumed

in each theory. Schweizer-Chandler theory assumes coupling through longrange attractive forces; VE theory assumes that short-range repulsive forces

dominate. Thus, the toluene results are in accord with our earlier conclusion

that repulsive interactions are primarily responsible for dephasing in pure

liquids.

SC

ω



E. Solvent-Assisted IVR in Ethanol



The bandwidth of the symmetric methyl stretch in ethanol-1,1-d2 is

unusually wide 15 cm 1 compared to other methyl-containing liquids,

e.g., acetonitrile 6.5 cm 1 and methyl iodide 5.7 cm 1 (Table 1).

(The deuteration prevents mixing of the methyl and methylene hydrogen

vibrations.) Within the context of the dephasing ideas already presented, a

number of possible reasons for this difference present themselves. However,

a closer experimental examination of the system shows that none of these

are correct, and a new dephasing mechanism must be considered (5).



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426



Berg



The first experiment to show that this system is unusual is the Raman

FID as a function of temperature (Fig. 21). Ethanol supercools easily

and forms a glass at 97 K. But unlike the experiments in supercooled

toluene (Fig. 15), the decay in ethanol at 80 K remains exponential, and

the dephasing rate is changed only slightly at low temperature. This result

eliminates shear fluctuations, density fluctuations, and other mechanisms

with an explicit viscosity dependence.

Theories based on the Enskog collision time (84) or other solid-like

approaches do not have a strongly temperature-dependent frequency correlation time. But they do have a temperature-dependent factor resulting from

the need to create the solvent fluctuations in the first place. Thus, all fastmodulation theories predict that the dephasing rate will go to zero at 0 K.



Figure 21 Raman FID (points) of the sym-methyl stretch in CH3 CD2 OH in the

liquid (295 K), in the high-temperature glass (80 K), and in the low-temperature

glass (12 K). Fits (solid curves) are based on exponential decays. The increase in

dephasing rate at low temperatures is unexpected. (Adapted from Ref. 5.)



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