1996
J. Phys. Chem. 1996, 100, 1996-2000
ARTICLES Rotational Relaxation in Fluid Nitrogen: An Ambiguity in the Interpretation of the Q-Branch Collapse? Simon I. Temkin and William A. Steele* Chemistry Department, 152 DaVey Laboratory, Penn State UniVersity, UniVersity Park, PennsylVania 16802 ReceiVed: June 27, 1995; In Final Form: October 21, 1995X
A molecular dynamics simulation of nitrogen at 295 K and at densities ranging from 400 to 1000 amagat has been carried out. Relaxation times which give the rotational contribution to isotropic Raman Q-branch halfwidths have been determined from the simulations. A comparison of these simulations with experiment leads to the conclusions that (1) extreme motional narrowing theory is applicable to the calculation of bandwidths for densities not less than 700 amagat; (2) the nonlinear density dependence of rotational energy relaxation times should be incorporated in the scaling laws to obtain dependable calculations of bandwidths; (3) by rescaling molecular dynamics results of Levesque et al. (J. Chem. Phys. 1980, 72, 2744) for liquid nitrogen we show that negative rotational relaxation-vibrational dephasing cross correlation is not negligible for fluid nitrogen at high density and 295 K; and (4) calculations of the collision frequency for a model ensemble of hard spheres with density-dependent diameters (Ben-Amotz, D.; Hershbach, D. R. J. Phys. Chem. 1993, 97, 2295) yield good estimates for the density dependence of the simulated relaxation times.
Introduction Recently we have reported1 a molecular dynamics (MD) simulation of liquid nitrogen along the liquid-vapor coexistence line. Correlation times for rotational energy relaxation τE were calculated, and the validity of the criterion of Q-branch motional narrowing theory
ωQτE , 1
(1)
was shown to hold for all temperatures studied. Here, ωQ is the average frequency of the Q-branch vibration-rotation band evaluated relative to the vibrational frequency (see also (4)). Recently, the spectroscopic measurements along the coexistence curve were complemented by isothermal high-pressure experiments where much higher densities were reached.2-4 Numerous applications were proposed (in particular, remote thermometry of flames and gas mixtures5), with temperature and other parameters manifested indirectly in the band shape. The methods developed to calculate temperature from band shape are based on the impact approximation for collisional broadening and therefore are reliable only for low densities. Nevertheless, the theory has often been used beyond its rigorous limits, based on the supposition that the binary collision model is effectively valid at higher densities. The general quantum mechanical treatment of overlapping spectral components is complicated, and scaling and fitting laws6 offer tractable representations for the spectral exchange operator. In these theories, parameters are extracted from the low-pressure fits to the resolved spectra. In exceptional cases, not only j-dependent widths of individual components but off-diagonal state-to-state matrix elements have been measured7 so that comparative analyses of different scaling models can be done in great detail.8,9 However, it has been shown that extrapolations of these calculations to higher * Corresponding author. X Abstract published in AdVance ACS Abstracts, January 1, 1996.
0022-3654/96/20100-1996$12.00/0
densities result in contours that are wider than experimental.10 At densities for which τE obeys eq 1, the spectral contours are Lorentzian and their half-widths are directly proportional to τE. Since the scaling laws were derived from binary impact theory, they result in a linear density dependence for τE-1. In view of the initial disagreement between experiment and model, experimental data have recently been fitted using a nonlinear density dependence for τE.2, 3 A method for the physical measurement for this rotational energy relaxation time does not exist for high pressure, although a few data are available at low pressures from acoustics. An alternative is given by MD simulation, which provides relaxation times that can be used to check the value of density when Q-branch collapse starts and how best to estimate the half-width in the context of impact theories. Such calculations are the main objectives of the present paper. It will be shown that at room temperature the motional narrowing criterion for the Q-branch is well fulfilled for densities equal to or greater than about 700 amagat (an amagat equals 2.69 × 1019 mol‚cm-3). (For vapor-liquid coexistence, motional narrowing is valid along the whole curve1). This means that extreme narrowing expressions should be replaced by more sophisticated theoretical expressions for band shapes at densities lower than 700 amagat. Experiment versus Theory Based on MD Simulation Results The MD simulations were performed using the same parameters as in ref 1. At T ) 295 K, density was varied between 400 and 1000 amagat to span the entire region for which the Q-branch half-width has been observed.2,3 The correlation time τE for rotational energy relaxation can be defined either as the area (as in this paper) or the slope of the exponentially decaying part (as in ref 11) of the rotational energy time correlation function KE(t) equal to [〈E(t) E(0)〉 - 〈E(0)〉2]/[〈E2(0)〉 〈E(0)〉2], where E is the molecular rotational energy. This time is of primary interest in view of the fact that its product with © 1996 American Chemical Society
Rotational Relaxation in Fluid Nitrogen
J. Phys. Chem., Vol. 100, No. 6, 1996 1997
Figure 1. Comparison of the calculated rotational contribution to the Q-branch half-width with the Raman/CARS data: (1) MD-calculated rotational width in the extreme narrowing limit; (2) calculated, SPEG and SC models; (3) calculated, ECS-EP model; (4) Raman measurements;2 (5) CARS measurements;4 (6) CARS measurements;15 (7) timeresolved measurements;3 (8) a sum of pure dephasing and cross term contributions.
ωQ, the average rotational frequency of the Q-branch, is the key parameter for band shape changes with density. If the strong inequality
τEωQ . 1
(2)
is valid, the band contour is an asymmetric envelope of either resolved or unresolved rotational lines. When the medium is pressurized, the product in eq 2 starts to decrease. If eq 1 is valid, the Q-branch contour is Lorentzian and the rotational contribution to its half-width is defined entirely by τE through the extreme narrowing result: hwhm ) ωQ2τE ∆ωrot
(3)
where the frequency ωQ is a molecular constant and was introduced12 through the rotation-vibration coupling constant R e:
ωQ ) Re
2IkBT p2
(4)
(I is the moment of inertia for linear rotators.) With Re ) 0.0174 cm-1 13 ωQ (T)295 K) ) 1.752 cm-1 ) 3.3 × 1011 s-1. For the characterization of experimental half-widths, the dimensionless variable ∆ω* ) ∆ωhwhm/ωQ is useful.14 The rotational contribution to this dimensionless half-width while eq 1 holds becomes ωQτE. In these terms, eq 1 may be rewritten as ∆ω* rot , 1, where ∆ω* rot is the rotational half-width. Figure 1 shows a collection of experimental ∆ω* obtained from stimulated Raman measurements,2 high-pressure CARS data,4,15 and timeresolved measurements of dephasing rates3 for nitrogen at room temperature. As curve 1 of Figure 1 shows, use of the simulated τE causes the inequality of eq 1 to hold strongly at densities in the vicinity of 700 amagat and higher. For lower densities the rotational half-width does not obey this extreme narrowing inequality. Alternative calculations of ∆ω* rot are shown in Figure 1, based on the three most exploited impact models: ECS-EP (energy-corrected sudden exponential polynomial) scaling and SPEG (statistical polynomial energy gap) fitting laws and the quasiclassical SC (strong collision) model6,8 together with the simulated τE. In ref 8 the dependence of the dimensionless rotational half-width on the dimensionless rotational energy
relaxation time ωQτE was calculated for all models (see Figure 5 in ref 8). These results were used to give curves 2 and 3 in Figure 1. For ωQτE lower than 0.4, the SPEG and SC models coincide8 to give curve 2, while the ECS-EP model gives slightly different values (curve 3, Figure 1). It can be shown16 that for ωQτE -∆ωcross > ∆hwhm dph
(11)
In these terms the former fitting (10) corresponds to hwhm ≈ ∆hwhm -∆ωcross dph
Figure 3. Ratios of isothermal collision frequencies: (1) effective collision frequency calculated for HS diameter d ) 3.68 Å3 divided by that for density-dependent d ) σHS from ref 31. (2) Effective collision frequency calculated for fused HS33 divided by that for densitydependent d ) σHS from ref 31; (3) the same ratio as for 2, but with variable diameters of fused HS as in ref 31.
with density dependent-diameters. Finally, curve 3 in Figure 3 shows the frequencies of the fused sphere model with densitydependent size parameters (calculated from eq 9) divided by those obtained from the density-dependent hard sphere calculation. It appears that the best coincidence between calculation and experiment is obtained if a density variation of the hard molecule size is used in eq 7. Finally, the angular momentum relaxation time τJ was also calculated from the MD simulations. These values are shown by the squares in Figure 2. For the τJ-1 shown by curve 5 in Figure 2, eq 8 was used with the substitution of σE for σJ and an angular momentum cross section of 10 Å2 that was taken from ECS-EP calculations.8,9 These relaxation times are shown to make the point that τJ e τE for all densities, thus ruling out the Langevin description of rotational relaxation for fluid nitrogen at room temperature. This was previously shown to also be the case for liquid nitrogen.1
We conclude that the theory can be compatible with either contradictory experimental set, because a calculation of hwhm ∆ωcross is not available beyond the limits VτV , 1 (see the into Appendix). This lack makes subdivision of ∆ωhwhm exp hwhm and ∆hwhm ∆ωrot dph ambiguous. In addition, it was shown that at moderate densities (less than 700 amagat for nitrogen at room temperature), the extreme narrowing formulas for the Q-branch contour are not applicable. Thus, some particular modeling of the relaxation must be employed. If impact theory is chosen, the input τE(n) can be provided by simulations. If suitable simulations are not available, collision frequencies that are nonlinear functions of density (eq 8) calculated using a variable effective HS diameter31 might be a good alternative. For evaluations either of the crosscorrelation contribution or for the dependence of relaxation time on density, MD calculations are highly recommended if spectroscopic probing is to be dependable. Appendix To explain qualitatively the cross-correlation between rotational relaxation and pure vibrational dephasing in dense media, we generalize a fluctuating liquid cage model23,24 to incorporate vibrations. In the single-particle approximation we have an effective Hamiltonian
H(t) ) H0 + V(t) H0 ) Hvib +
Conclusions Until recently, it appeared that the nitrogen Q-branch collapse was well understood. The narrowing had been studied by different experimental methods over a wide range of temperatures and densities, and sophisticated theoretical models had been devoted to the problem.24 As for MD calculations, their capabilities certainly were not used in full. It was our aim in the preceding1 and present papers to bring together a complete collection of experimental data and to check theoretical models against them, having the simulation output as a mediator. We have shown here that the contradiction between the different sets of experimental data (refs 2, 4, and 15) is not simply in numbers. When analyzed using the MD calculations of rotational relaxation, the different data sets result in qualitatively different conclusions. For the older Raman2 and CARS4 data, our approach based on the MD calculations of τE for density in the range 400-700 amagat gives hwhm ∆ωhwhm exp ≈ ∆ωrot
(10)
in agreement with previous fittings. On the contrary, the measured Q-branch half-widths for the latest CARS data15 are significantly smaller than the rotational contribution alone that has been estimated from the simulations. Thus, to successfully
(12)
(13)
Jˆ 2 2I
where H0 is the Hamiltonian of the freely vibrating and rotating molecule and V(t) stands for the effective potential of the instantaneous liquid cage interacting with the spectroscopically active molecule inside.24,34,35 The Hamiltonian (eq 13) governs the time evolution of the scalar part of the polarizability tensor dQ(t) via the stochastic Liouville equation (p ) 1):
∂ Q d (t) ) iH×(t) dQ(t) ∂t
(14)
Assuming a rapidly fluctuating V(t), eq 14 is averaged23 to give
∂ Q d (t) ) iωvibdjQ(t) + iωjdjQ(t) + ∑j′Γj′j(0) dj′Q(t) ∂t j
(15)
(Γj′j(iω) is expressed through Laplace transform of a correlation function of V(t) in the interaction representation, so that ||Γj′j|| ∝ V2τV.24,36) In eq 15, vibrational population changes were neglected and the rotational component dQj was introduced through which the whole band shape is expressed:
I(iω) )
Re π
∑j ∫dt exp(-iωt) djQ(t)
(16)
2000 J. Phys. Chem., Vol. 100, No. 6, 1996
Temkin and Steele
As has been pointed out,23, 36 eq 15 has common mathematical features with the impact theory master equation derived for the gas phase. Thus, this gives another way for treating relaxation in liquids in terms of collisions. We now split the relaxation operator in eq 15 into two parts by subtracting the operator resulting from the anisotropic interaction fluctuations: 1 2 Γjj′ ) Γjj′ + Γjj′
References and Notes
(17)
1 || ∝ Van2τVan ||Γjj′
Γ1 has a known eigenfunction with zero eigenvalue.23,36
∑j′ Γj′j1 dj′Q ) 0
(18)
where dQj ) dQj(t ) 0). We assume that the Γ1 operator is corrected24,37,38,39 for the detailed balance so that
∑j′ Γjj′1 ρj′B ) 0
(19)
where the Boltzmann distribution over rotational levels is designated by FBj. The Q-branch collapse condition (eq 1) is to be rewritten as
Van2τVan . ωQ In this limit all contributions to the half-width (eq 5) are determined by perturbation theory. The first term from eq 5 can be shown24,40,41 to coincide with eq 3, since the operator Γ1j′j determines the rotational energy relaxation or, equivalently, the spectral exchange inside the Q-branch.24,36 To obtain the second and the third terms we use the impact operator eigenfunction expansion from ref 22. For the pure dephasing term, the representation is 2 ∆hwhm dph ≈ [Γj′j]00
(20)
The cross correlation term is 2 ]0n/λn ∆ωcross ) 2∑[Ωj′j]0n Im[Γj′j
(21)
n*0
where square brackets designate the matrix elements of, correspondingly, the diagonal operator of rotational frequencies and the “residual” operator from eq 17, which are defined over the set of eigenfunctions of the Γ1 operator. The λn designate the eigenvalues of the latter. It follows from eqs 15 and 17 that 2 ]0n/λn Im[Γj′j
∝
V2τV - Van2τVan Van2τVan
analytically known sets of eigenfunctions of model integral operators.24) Scaling by ωQ suppresses the temperature dependence of the cross term so that its dimensionless ordinate in Figure 1 is the same for all thermodynamic states where VτV , 1 and equation (15) is valid.24,36
(22)
This factor in the cross term depends only weakly on temperature and density because both the dispersions V2 and Van2 and relaxation times τV and τVan vary similarly to one another. With temperature increase, the modulus of the matrix element [Ωj′j]0n in eq 21 increases because the spreading of the Boltzmann distribution. Roughly, this temperature dependence coincides with that of the ωQ. (At least, this is the case for all
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