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Raman study of vibrational relaxation in liquid cyclohexane at high pressure
(Shemica Physics 38 (1979) 131-l 37 0 North-Holland Publishing Company
RAMAN STUDY OF MBRATIONAL AT HIGH PRESSURE
RELAXATION IN LIQUID CYCLOHEXANE
Kazutoshi TANABE National Chemical Laboratory for Industry, Honnmhi.
Shibuyaku. Tokyo ISI. Japan
and Jiri JONAS Department of Chemistry, School of Chemical Sciences, University of Illinois, Urbana. Illinois 61601, USA Received 13 November 1978 The Raman lint shapes of six totally symmetric bands of liquid cyclohexane have been measured as a function of pressure at 0.1-1.5 kbar and of temperature at 25-175’C. Densities and shear viscosities have also beeen measured, and the hard sphere diameter of the cyclohexane molecule has been evaluated from the analysis of the viscosity data. The observed isotropic line width for all modes studied increases with increasing pressure and increasing temperature. The essential features of the experimental data for vt (2938 cm-‘) and ~2 ( 2852 cm-‘) bands are interpreted in terms of the vibrational dephasing model. However, for the lower frequency ~3, ~4, us and ~6 bands (1465 cm-l to 388 cm-t), hw = kTand, therefore, only qualitative discussion of the experimental data is possible. The observed isotropic line widths 6&s for the lower frequency bands show stronger temperature dependence than that found for 6,bs of the ut and ~2 bands, and lower sensitivity to volume changes. Phenomenological discussion of the experimental data for the 6&s of the low frequency bands is presented.
1. Introduction The mechanism of the broadening of isotropic Raman lines in liquid has recently been investigated from
experimental and theoretical viewpoints [ 11. In a preceding work [2], we have successfully interpreted the pressure and temperature dependence of the vibrational relaxation widths of Raman lines in liquid benzeneds. We found that the vibrational relaxation broadening contributes differently to the Raman line widths of individual vibrational modes and that the vibrational dephasing collinear binary collision model as proposed by Fischer and Laubereau [3] and Oxtoby and Rice [4] explains the general features of the vibrational relaxation data. In our earlier paper [Z], an approximate procedure was applied to estimate the vibrational relaxation widths for degenerate Raman lines, and, therefore, this approach introduced some ambiguousness in the results obtained. in this respect, cyclohexane is a more
desirable molecule for studying the dependence of the vibrational relaxation broadening of various Raman lines because it has six totally symmetric (a,& vibrational modes. This work has the following goals: first, to determine the temperature and pressure dependence of the Raman line widths ul, ~2, v3, ~4, v5 and vg in liquid cyclohexane; second, to obtain the density, viscosity and hard sphere diameter data for liquid cyclohexane necessary for analysis; third, to attempt to interpret the experimental line width data in terms of the dephasing model; and, fourth, to discuss the dependence of the observed Raman line widths on temperature, pressure and vibrational modes. For this purpose, the Raman band shapes of six totally symmetric modes v1 (2938 cm-‘), ~2 (2852 cm-‘), v3 (1465 cm-‘), t4 (1157 cm-‘), us (802 cm-‘) and vg (383 cm-‘) in liquid cyclohexane have been measured as a function of temperature and of pressure. Density and shear viscosity have also been
K. Tanabe, J. Jonas / Vibrational relaxation in liquid cyclohexane
132
measured over the same temperature and pressure range.
2. Experimental Experimental details of the Raman measurements at high pressure were described elsewhere [S]. Densities were measured with an accuracy of 50.5% using a densitometer with a linear variable differential transformer [6], and shear viscosities were measured with an accuracy of 23% using a high pressure rolling ball viscometer [7]. Samples of cyclohexanedra for the isotopic dihtion study were purchased from the Merck Sharp Dohme of Canada Ltd., and used without purification. The vr mode of cyclohexane splits due to the Fermi resonance into 2938 and 2923 cm-‘, and the width of the 2938 cm-’ line was chosen for ur.
3. Results and discussion From the observed spectra of the parallel and perpendicular components of the Raman bands, the isotropic component spectra were obtained 4,o(V)
=rVV(u)
-
!j~Vli(V)
7
(1)
and the isotropic line widths, which are attributed
to the vibrational relaxation on the assumption that both
0
a5
ls P(kr) Fig. 1. Observed isotropic Raman line width B,bs(fwhh) of vt. Lines are 25,50,75,100,125,150 and 175’C from bottom to top.
0
cl5
lo
15 Pwar) Fig. 2. Observed isotropic Raman line width Eobs(fwhh) of q. Lines are 25,50,75,100,125,150 and 175’C from bottom to top.
the vibrational and reorientational relaxations do not correlate with each other, were evaluated. The experimental isotropic widths (fwhh) as a function of temperature and pressure are plotted in figs. l-6. The line widths increase with increasing temperature and pressure for all bands studied. Such a dependence is consistent with that observed for benzene-de [Z], where, for some bands, the widths showed a slight decrease with increasing temperature. This unusual behavior may have resulted from the approximate procedure *usedfor estimating the isotropic line widths.
.%
to
05
IO P (Id=)
15
Fig. 3. Observedisotropic Ranian ‘tie widthsobs(fwhh) of rra. Lines are 25,50,7.5,100,125,150 and 17’5’C from bottom to top.
K. Tanabe. J. Jonas / Vibrational relaxation in liquid cyclohexane
133
Fig. 4. Observed isotropic Raman line width 6obs(fwhh) of ~4. Lines are 25,50, 75, 100, 125,150 and 175’C from bottom to top.
Fig. 6. Observed isotropic Raman line width 6,bs(fwhh) of ~6. Lines are 25, SO,75, 100, 125,150 and 17S°C from bottom to top.
Inspection of these figures indicates that the line widths broadened due to the vibrational relaxation differ considerably from one band to the other. In some previous studies [8], it was assumed that the vibrational relaxation width is independent of vibrational modes. On this basis, reorientational relaxation widths for observed infrared bands were obtained from the vibrational relaxation width determined by Raman spectra. But, the present results defiiitely demonstrate that generally the vibrational relaxation width strongly depends on the vibrational modes, and
the above method [8] is invaIid for separating the vibrational and reorientational relaxation widths. The experimental data of densities and shear viscosities measured in this study are summarized in table 1. Our values are in good agreement with the density and viscosity data at 1 bar reported in the literature [9]. From the density and viscosity data, the hard sphere diameter of liquid cyclohexane molecule was, evaluated using the usual procedure [9,10]. The obtained value of the diameter as a function of temperature is plotted in fig. 7, and the value at 25°C ccmpares favorably with that previously reported [9,11]. A detailed discussion of various possible mechanisms responsible for vibrational broadening of Raman line shapes was presented [5]. Fischer and Laubereau [3] and Oxtoby and Rice [4] used a collinear isolated binary collision model to calculate the vibrational dephasing relaxation rate. They considered the repulsive part of intermolecular potential in the exponential form, and derived for the rate &J-l=
P (Hz&
Fig. 5. Observedisotropic Raman line width 6,bs(fWhh) ~5. Lmes are 25,50,75, tom to top.
100, 125,150
of and 17S°C from bot-
kBT 1 M2 w2L2 Tc ’
4/-q’
(2)
where ?-pj,is the dephasing relaxation time, p the reduced mass for colliding molecular pairs, y a parameter defiiing vibrational amplitude of atoms in a vibrating molecule, M the reduced mass for vibrating oscillator, kB the Boltzmann constant, w the angular vibrationa frequency (=2ncv), L a parameter measur-
K. Tanabe. J. Jonas / Vibrational relaxation illliquid cyclohexane
134
Table 1 Densities and shear viscosities of liquid cyclohesane
T (“Cl
P (kbar)
Density (g cmd3)
Viscosity
0.001
0.8909
0.250
0.7751 0.7830 0.7933
50
0.001 0.100 0.300 0.500
0.7493 0.7589 0.7755 0.7891
0.5989 0.6781 0.8561 1.0606
75
0.001 0.100 0.500 1.000
0.7261 0.7369 0.7719 0.8015
0.4213 0.4746 0.7318 1.1699
tures.
100
0.100 0.500 1.000 1.500
0.7160 0.7550 0.7887 0.8123
0.3221 0.5455 0.8479 1.2546
the elastic collision time rc was approximated Enskog relaxation time rE,
125
0.100 0.500
0.6949 0.7377 0.7756 0.8015 0.8213
0.2994 0.4380 0.6568 0.9397 1.301G
0.6727 0.7196 0.7619 0.7907 0.8111 0.8281 0.8467
0.2450 0.3453 0.5117 0.7275 0.9972 1.3249 1.7150
0.6482 0.7002 0.7477 0.7801 0.8021 0.8184 0.8336 0.8524
_
25
0.100
1 .OOD 1 so0 2.000.
150
0.100 0.500
1.000 !.SOO 2.000 2.500 3.000 175
0.100 0.500 1.000 1.500 2.000
2.500 3.000 3.500
(cP)
1.0141 1.2158
Fig. 7. Hard-sphere
(TJ-* =$(@r
diameters
calculated
= 4(rkBT/m)l/’
at
various tempera-
by the
02pg(o) ,
(3)
where p is the number density and&a) is the Enskog correction factor (the radial distribution function at contact). The Carnahan-Starling approximation [13] was used for estimatingg(o). The line width broadened due to the vibrational dephasing was calculated by 6 ph = (rmrph)--* .
(4)
The basic condition in the theoretical work dealing with dephasing is that fiw > kT and this condition is fulfilled for the u1 and us modes. There is a small
20
-
t T
E
15-
2
-
0
ing the range of intermolecular potential between colliding molecules, and rC the average time.between two consecutive collisions (collision time). The parameters p, y and M are defined in the paper of Fischer and Laubereau [3], but the procedure for estimating these is not straightforward for so complex a molecule as cyclohexane. We proposed an approximate method which was used in our study of benzene4 PI. Analogously as in the preceding studies [2,5,12],
co” IO-
51 5
I
I
IO
15
I
20
25
S ph
Fig. 8. Observed
Hidths for VI
(Cm-‘) line widths versus theoretical
and ~2.
dephasing
K. Tanabe. J. Jonas / Vibrational relerotion in liquid cyclohernne
contribution (10%) due to the resonant energytransfer process as obtained from a dilution study in CeDe, but this process is also a result of quasielastic cotlisions. Since we attempt a qualitative interpretation of the experimental data for vl and vz bands, we assume, in the following treatment, that 8obs is due to pure dephasing and that anharmonicity correction [l] can be neglected. The calculated line widths for v1 and v2 obtained from the dephasing model are plotted against the observed isotropic line widths in fig. 8. The agreement can be regarded as Eair, particularly in view of the scatter of the points at narrower line widths. However, one may conclude that the dephasing process seems to represent the major broadening mechanism for the V, and uz bands and that the binary collision dephasing model reproduces the general behavior of the experimental data. A simple inspection of figs. 3 to 6 indicates that the interpretation of the nature of broadening of the line widths for the vs, v4, vs and ve wih not be straightforward. The important fact is that for these bands the basic condition that tzw >>kT is no longer valid, in particular for the low frequency bands vs and ve. If Frlw= kT, then the vibrational motion is no longer weakly coupled to the bath and consequently, it cannot be treated as a degree of freedom which is loosely coupled to the bath. ,Therefore, we discuss the experimental results for the pressure and temperature dependences of the isotropic line widths, sobs (fwhm) only in qualitative terms.
T
E
0
x
so”
T3” x lO-3 Fig. 9. Temperature dependence of the observed isotropic he width sob (fwhh) of u1 and “6 at constant density p = Cl.7800 g cm- s : lines denoted by q(p) and q,(p); and at constant pressure of 100 bar: lines denoted by VI(P) and 4(p).
135
The first striking difference ,betweeti the. behavior -. of Sobs for v3, v4, us and ve with temperature and pres-
sure when compared to analogous behavior for the u1 and v2 bands is the weaker dependence on volume changes. To emphasize this experimental fact we present fig. 9 which compares the observed isotropic line width &,b,(fWhm) of v, and ve at COnStant density and constant pressure. The reason why we use TV* scale becomes clear by inspect&g eqs. (2) and (3), which show that at constant density Sph a: Tq* because the radial distribution function at contact g(u) changes less than 10% over the temperature range measured. We use such a plot for the ve band only for comparison with the behavior of the vr band and do not imply that eqs. (2) and (3) describe the line width of this band. One readily concludes that 6obs for the ve band is only weakly dependent on density..Otherwise, the behavior of h&s for ve at constant pressure will resemble more that of the behavior of f&s for the vr mode. Because the decreasing density and ;ncreasing temperature have just the opposite effects: on s,,&, one readily understands why Sobs for the ~1 mode shows a small dependence on temperature at constant pressure. However, at constant density the temperature dependence of the ve mode is only about 17% larger than that observed for the ur band at-constant density. Another distinct behavior of the line width for u3, v4, us and ve is their stronger temperature dependence at constant density than that observed for the high frequency vr and u2 bands. One illustrates this difference in the temperature behavior by calculating
where i = 2-6 and all values are normalized to the value observed at 2.5OC(fig. IO). The v2 band, w&h as mentioned above, can be interpreted in terms of the dephasing model, has, as expected, the same tkmperature dependence at contrast density as the vr band. In contrast, the IJ~,v4, us, and ne modes exhibit a stronger temperature dependence than that observed. for the it and y2 bands. All these experimental facts indicate that the .dephasing process could be only one. of the. broadening mechanisms for .the lower frequency bands v3, v4, vs and ve. At this point, it seems mean;. ingless to speculate why, e.g., the u4 band shows the strangest temperature dependence.
136
K. Tanabe, J. Jonas / Vibrationa! relaxation in Zquid cyclohexane
T (OK) Fii. 10. Schematic diagram for relative temperature change under constantdensity of the ratio (SVi/SV,)N= (Svi/6vl)T/ (W~q)25.
One may mention several possible broadening mechanisms which may affect the sobs for the lower
frequency bands in cyclohexane. (1) Since ho 2 kT, one may expect that the relationship between the population relaxation and dephasing is no longer simple [ 11. (2) In deriving eq. (I), it is a priori assumed that the vibrational and rotational motion of molecules do not couple with each other. Recently, Miller and Clarke [ 141 studied this problem, and indicated that the coupling may account for an apparent difference of the reorientational line widths obtained from Raman and Rayleigh spectra. It is believed that the coupling may also broaden the isotropic Raman line widths estimated using eq. (1). Since, as reported in the paper by Amorim da Costa et al. [ 1.51, the reorientational line width from the RayIeigh measurement is not certain, the contribution of the vibration-rotation coupling to the isotropic Raman line widths cannot be estimated. At present, we feel that, although the exact mechanism contributing the line widths of the ug, u4, us and v6 bands is not determined definitely, the vibration-rotation coupling can be an imp.ortant factor.
4. Conclusions The results of the experimental study of the temperature and pressure effects on the isotropic line
widths in cyclohexane can be summarized as follows: (1) The isotropic Raman line widths of the six alg bands of liquid cyclohexane increase with increasing temperature and pressure. (2) The value of 5.52 (A) (25°C) for the hard sphere diameter was obtained from the analysis of the viscosity data. (3) The dephasing theory predicts semi-quantitatively the temperature and pressure dependence of the line widths for the v1 and v2 vibrational modes. (4) For the lower frequency z+, v4, LJ~and vg bands, for which tzw = kT, the line width shows less sensitivity to volume changes when compared to analogous behavior of the higher frequency u1 and v2 bands. At the same time, Sobs for v3, v+ us and vg exhibits stronger temperature dependence at constant density than 6obs for the v1 and v2.bands. (5) The observed isotropic line width varies widely for individual bands within a range of 1.8 cm-’ to 10 cm-‘. (6) The experimental data for the lower frequency bands cannot be interpreted in terms of the simple dephasing model.
Acknowledgement This work was supported in part by the National Science Foundation under Grant NSF CHE 77-07621. The authors express their thanks to Dr. J. Schroeder for his valuable discussions, and to Dr. T. DeFries, Mr. P.T. Shark0 and Mr. W. Lamb for their assistance in density and viscosity measurements.
References [l] D-W. Oxtoby, Advan. Chem. Phys., to be published. [2] K. Tanabe and J. Jonas, J. Chem. Phys. 67 (1977) 4222. [3] S.F. Fischer and A. Laubereau, Chem. Phys. Letters 35 (1975) 6. [4] D.W. Oxtoby and Stuart A. Rice, Chem. Phys. Letters 42 (1976) 1. [S] J. Schroeder, V.H. Schicmann, P.T. Shark0 and J. Jonas, J. Chem. Phys. 66 (1977) 3215. [6] J.A. Akai, Ph.D. Thesis, University of Illinois (1977). [7] R.A. Assink, Ph.D. TIthesis, University of Illinois (1972). [S] A.E. Boldenskuland V.E. Pogorelev,Opt. Scpectry. 28 (1970) 248. [9] D. Chandler, I. Chem. Phys. 62 (1975) 1358.
K. Tanabe.J_ Jonas / Vibrationalrelaxationin liquid cyclohexane [IO] H.J. Parkhurst Jr. and I. Jonas, J. Chem. Phys. 63 (1975) 2706. [ll] E. Wilhelm, I. Chem. Phys. 58 (1973) 3558. 1121 J. Schroeder, V.H. Schiemann and J. Jonas, Mol. Phys. 34 (1977) 1501. [13] N.F. Carnahan and K.E. Starling, J. Chem. Phys. 51 (1969) 635.
137
[ 141 S. Miller and J.H.R. Clarke, Chem. Phys. Letters 56 (1978) 235. [IS] A.M. Amorim da Costa, M.A. Norman and J.H.R. Clarke, Mol. Phys. 29 (1975) 191.
RAMAN STUDY OF MBRATIONAL AT HIGH PRESSURE
RELAXATION IN LIQUID CYCLOHEXANE
Kazutoshi TANABE National Chemical Laboratory for Industry, Honnmhi.
Shibuyaku. Tokyo ISI. Japan
and Jiri JONAS Department of Chemistry, School of Chemical Sciences, University of Illinois, Urbana. Illinois 61601, USA Received 13 November 1978 The Raman lint shapes of six totally symmetric bands of liquid cyclohexane have been measured as a function of pressure at 0.1-1.5 kbar and of temperature at 25-175’C. Densities and shear viscosities have also beeen measured, and the hard sphere diameter of the cyclohexane molecule has been evaluated from the analysis of the viscosity data. The observed isotropic line width for all modes studied increases with increasing pressure and increasing temperature. The essential features of the experimental data for vt (2938 cm-‘) and ~2 ( 2852 cm-‘) bands are interpreted in terms of the vibrational dephasing model. However, for the lower frequency ~3, ~4, us and ~6 bands (1465 cm-l to 388 cm-t), hw = kTand, therefore, only qualitative discussion of the experimental data is possible. The observed isotropic line widths 6&s for the lower frequency bands show stronger temperature dependence than that found for 6,bs of the ut and ~2 bands, and lower sensitivity to volume changes. Phenomenological discussion of the experimental data for the 6&s of the low frequency bands is presented.
1. Introduction The mechanism of the broadening of isotropic Raman lines in liquid has recently been investigated from
experimental and theoretical viewpoints [ 11. In a preceding work [2], we have successfully interpreted the pressure and temperature dependence of the vibrational relaxation widths of Raman lines in liquid benzeneds. We found that the vibrational relaxation broadening contributes differently to the Raman line widths of individual vibrational modes and that the vibrational dephasing collinear binary collision model as proposed by Fischer and Laubereau [3] and Oxtoby and Rice [4] explains the general features of the vibrational relaxation data. In our earlier paper [Z], an approximate procedure was applied to estimate the vibrational relaxation widths for degenerate Raman lines, and, therefore, this approach introduced some ambiguousness in the results obtained. in this respect, cyclohexane is a more
desirable molecule for studying the dependence of the vibrational relaxation broadening of various Raman lines because it has six totally symmetric (a,& vibrational modes. This work has the following goals: first, to determine the temperature and pressure dependence of the Raman line widths ul, ~2, v3, ~4, v5 and vg in liquid cyclohexane; second, to obtain the density, viscosity and hard sphere diameter data for liquid cyclohexane necessary for analysis; third, to attempt to interpret the experimental line width data in terms of the dephasing model; and, fourth, to discuss the dependence of the observed Raman line widths on temperature, pressure and vibrational modes. For this purpose, the Raman band shapes of six totally symmetric modes v1 (2938 cm-‘), ~2 (2852 cm-‘), v3 (1465 cm-‘), t4 (1157 cm-‘), us (802 cm-‘) and vg (383 cm-‘) in liquid cyclohexane have been measured as a function of temperature and of pressure. Density and shear viscosity have also been
K. Tanabe, J. Jonas / Vibrational relaxation in liquid cyclohexane
132
measured over the same temperature and pressure range.
2. Experimental Experimental details of the Raman measurements at high pressure were described elsewhere [S]. Densities were measured with an accuracy of 50.5% using a densitometer with a linear variable differential transformer [6], and shear viscosities were measured with an accuracy of 23% using a high pressure rolling ball viscometer [7]. Samples of cyclohexanedra for the isotopic dihtion study were purchased from the Merck Sharp Dohme of Canada Ltd., and used without purification. The vr mode of cyclohexane splits due to the Fermi resonance into 2938 and 2923 cm-‘, and the width of the 2938 cm-’ line was chosen for ur.
3. Results and discussion From the observed spectra of the parallel and perpendicular components of the Raman bands, the isotropic component spectra were obtained 4,o(V)
=rVV(u)
-
!j~Vli(V)
7
(1)
and the isotropic line widths, which are attributed
to the vibrational relaxation on the assumption that both
0
a5
ls P(kr) Fig. 1. Observed isotropic Raman line width B,bs(fwhh) of vt. Lines are 25,50,75,100,125,150 and 175’C from bottom to top.
0
cl5
lo
15 Pwar) Fig. 2. Observed isotropic Raman line width Eobs(fwhh) of q. Lines are 25,50,75,100,125,150 and 175’C from bottom to top.
the vibrational and reorientational relaxations do not correlate with each other, were evaluated. The experimental isotropic widths (fwhh) as a function of temperature and pressure are plotted in figs. l-6. The line widths increase with increasing temperature and pressure for all bands studied. Such a dependence is consistent with that observed for benzene-de [Z], where, for some bands, the widths showed a slight decrease with increasing temperature. This unusual behavior may have resulted from the approximate procedure *usedfor estimating the isotropic line widths.
.%
to
05
IO P (Id=)
15
Fig. 3. Observedisotropic Ranian ‘tie widthsobs(fwhh) of rra. Lines are 25,50,7.5,100,125,150 and 17’5’C from bottom to top.
K. Tanabe. J. Jonas / Vibrational relaxation in liquid cyclohexane
133
Fig. 4. Observed isotropic Raman line width 6obs(fwhh) of ~4. Lines are 25,50, 75, 100, 125,150 and 175’C from bottom to top.
Fig. 6. Observed isotropic Raman line width 6,bs(fwhh) of ~6. Lines are 25, SO,75, 100, 125,150 and 17S°C from bottom to top.
Inspection of these figures indicates that the line widths broadened due to the vibrational relaxation differ considerably from one band to the other. In some previous studies [8], it was assumed that the vibrational relaxation width is independent of vibrational modes. On this basis, reorientational relaxation widths for observed infrared bands were obtained from the vibrational relaxation width determined by Raman spectra. But, the present results defiiitely demonstrate that generally the vibrational relaxation width strongly depends on the vibrational modes, and
the above method [8] is invaIid for separating the vibrational and reorientational relaxation widths. The experimental data of densities and shear viscosities measured in this study are summarized in table 1. Our values are in good agreement with the density and viscosity data at 1 bar reported in the literature [9]. From the density and viscosity data, the hard sphere diameter of liquid cyclohexane molecule was, evaluated using the usual procedure [9,10]. The obtained value of the diameter as a function of temperature is plotted in fig. 7, and the value at 25°C ccmpares favorably with that previously reported [9,11]. A detailed discussion of various possible mechanisms responsible for vibrational broadening of Raman line shapes was presented [5]. Fischer and Laubereau [3] and Oxtoby and Rice [4] used a collinear isolated binary collision model to calculate the vibrational dephasing relaxation rate. They considered the repulsive part of intermolecular potential in the exponential form, and derived for the rate &J-l=
P (Hz&
Fig. 5. Observedisotropic Raman line width 6,bs(fWhh) ~5. Lmes are 25,50,75, tom to top.
100, 125,150
of and 17S°C from bot-
kBT 1 M2 w2L2 Tc ’
4/-q’
(2)
where ?-pj,is the dephasing relaxation time, p the reduced mass for colliding molecular pairs, y a parameter defiiing vibrational amplitude of atoms in a vibrating molecule, M the reduced mass for vibrating oscillator, kB the Boltzmann constant, w the angular vibrationa frequency (=2ncv), L a parameter measur-
K. Tanabe. J. Jonas / Vibrational relaxation illliquid cyclohexane
134
Table 1 Densities and shear viscosities of liquid cyclohesane
T (“Cl
P (kbar)
Density (g cmd3)
Viscosity
0.001
0.8909
0.250
0.7751 0.7830 0.7933
50
0.001 0.100 0.300 0.500
0.7493 0.7589 0.7755 0.7891
0.5989 0.6781 0.8561 1.0606
75
0.001 0.100 0.500 1.000
0.7261 0.7369 0.7719 0.8015
0.4213 0.4746 0.7318 1.1699
tures.
100
0.100 0.500 1.000 1.500
0.7160 0.7550 0.7887 0.8123
0.3221 0.5455 0.8479 1.2546
the elastic collision time rc was approximated Enskog relaxation time rE,
125
0.100 0.500
0.6949 0.7377 0.7756 0.8015 0.8213
0.2994 0.4380 0.6568 0.9397 1.301G
0.6727 0.7196 0.7619 0.7907 0.8111 0.8281 0.8467
0.2450 0.3453 0.5117 0.7275 0.9972 1.3249 1.7150
0.6482 0.7002 0.7477 0.7801 0.8021 0.8184 0.8336 0.8524
_
25
0.100
1 .OOD 1 so0 2.000.
150
0.100 0.500
1.000 !.SOO 2.000 2.500 3.000 175
0.100 0.500 1.000 1.500 2.000
2.500 3.000 3.500
(cP)
1.0141 1.2158
Fig. 7. Hard-sphere
(TJ-* =$(@r
diameters
calculated
= 4(rkBT/m)l/’
at
various tempera-
by the
02pg(o) ,
(3)
where p is the number density and&a) is the Enskog correction factor (the radial distribution function at contact). The Carnahan-Starling approximation [13] was used for estimatingg(o). The line width broadened due to the vibrational dephasing was calculated by 6 ph = (rmrph)--* .
(4)
The basic condition in the theoretical work dealing with dephasing is that fiw > kT and this condition is fulfilled for the u1 and us modes. There is a small
20
-
t T
E
15-
2
-
0
ing the range of intermolecular potential between colliding molecules, and rC the average time.between two consecutive collisions (collision time). The parameters p, y and M are defined in the paper of Fischer and Laubereau [3], but the procedure for estimating these is not straightforward for so complex a molecule as cyclohexane. We proposed an approximate method which was used in our study of benzene4 PI. Analogously as in the preceding studies [2,5,12],
co” IO-
51 5
I
I
IO
15
I
20
25
S ph
Fig. 8. Observed
Hidths for VI
(Cm-‘) line widths versus theoretical
and ~2.
dephasing
K. Tanabe. J. Jonas / Vibrational relerotion in liquid cyclohernne
contribution (10%) due to the resonant energytransfer process as obtained from a dilution study in CeDe, but this process is also a result of quasielastic cotlisions. Since we attempt a qualitative interpretation of the experimental data for vl and vz bands, we assume, in the following treatment, that 8obs is due to pure dephasing and that anharmonicity correction [l] can be neglected. The calculated line widths for v1 and v2 obtained from the dephasing model are plotted against the observed isotropic line widths in fig. 8. The agreement can be regarded as Eair, particularly in view of the scatter of the points at narrower line widths. However, one may conclude that the dephasing process seems to represent the major broadening mechanism for the V, and uz bands and that the binary collision dephasing model reproduces the general behavior of the experimental data. A simple inspection of figs. 3 to 6 indicates that the interpretation of the nature of broadening of the line widths for the vs, v4, vs and ve wih not be straightforward. The important fact is that for these bands the basic condition that tzw >>kT is no longer valid, in particular for the low frequency bands vs and ve. If Frlw= kT, then the vibrational motion is no longer weakly coupled to the bath and consequently, it cannot be treated as a degree of freedom which is loosely coupled to the bath. ,Therefore, we discuss the experimental results for the pressure and temperature dependences of the isotropic line widths, sobs (fwhm) only in qualitative terms.
T
E
0
x
so”
T3” x lO-3 Fig. 9. Temperature dependence of the observed isotropic he width sob (fwhh) of u1 and “6 at constant density p = Cl.7800 g cm- s : lines denoted by q(p) and q,(p); and at constant pressure of 100 bar: lines denoted by VI(P) and 4(p).
135
The first striking difference ,betweeti the. behavior -. of Sobs for v3, v4, us and ve with temperature and pres-
sure when compared to analogous behavior for the u1 and v2 bands is the weaker dependence on volume changes. To emphasize this experimental fact we present fig. 9 which compares the observed isotropic line width &,b,(fWhm) of v, and ve at COnStant density and constant pressure. The reason why we use TV* scale becomes clear by inspect&g eqs. (2) and (3), which show that at constant density Sph a: Tq* because the radial distribution function at contact g(u) changes less than 10% over the temperature range measured. We use such a plot for the ve band only for comparison with the behavior of the vr band and do not imply that eqs. (2) and (3) describe the line width of this band. One readily concludes that 6obs for the ve band is only weakly dependent on density..Otherwise, the behavior of h&s for ve at constant pressure will resemble more that of the behavior of f&s for the vr mode. Because the decreasing density and ;ncreasing temperature have just the opposite effects: on s,,&, one readily understands why Sobs for the ~1 mode shows a small dependence on temperature at constant pressure. However, at constant density the temperature dependence of the ve mode is only about 17% larger than that observed for the ur band at-constant density. Another distinct behavior of the line width for u3, v4, us and ve is their stronger temperature dependence at constant density than that observed for the high frequency vr and u2 bands. One illustrates this difference in the temperature behavior by calculating
where i = 2-6 and all values are normalized to the value observed at 2.5OC(fig. IO). The v2 band, w&h as mentioned above, can be interpreted in terms of the dephasing model, has, as expected, the same tkmperature dependence at contrast density as the vr band. In contrast, the IJ~,v4, us, and ne modes exhibit a stronger temperature dependence than that observed. for the it and y2 bands. All these experimental facts indicate that the .dephasing process could be only one. of the. broadening mechanisms for .the lower frequency bands v3, v4, vs and ve. At this point, it seems mean;. ingless to speculate why, e.g., the u4 band shows the strangest temperature dependence.
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K. Tanabe, J. Jonas / Vibrationa! relaxation in Zquid cyclohexane
T (OK) Fii. 10. Schematic diagram for relative temperature change under constantdensity of the ratio (SVi/SV,)N= (Svi/6vl)T/ (W~q)25.
One may mention several possible broadening mechanisms which may affect the sobs for the lower
frequency bands in cyclohexane. (1) Since ho 2 kT, one may expect that the relationship between the population relaxation and dephasing is no longer simple [ 11. (2) In deriving eq. (I), it is a priori assumed that the vibrational and rotational motion of molecules do not couple with each other. Recently, Miller and Clarke [ 141 studied this problem, and indicated that the coupling may account for an apparent difference of the reorientational line widths obtained from Raman and Rayleigh spectra. It is believed that the coupling may also broaden the isotropic Raman line widths estimated using eq. (1). Since, as reported in the paper by Amorim da Costa et al. [ 1.51, the reorientational line width from the RayIeigh measurement is not certain, the contribution of the vibration-rotation coupling to the isotropic Raman line widths cannot be estimated. At present, we feel that, although the exact mechanism contributing the line widths of the ug, u4, us and v6 bands is not determined definitely, the vibration-rotation coupling can be an imp.ortant factor.
4. Conclusions The results of the experimental study of the temperature and pressure effects on the isotropic line
widths in cyclohexane can be summarized as follows: (1) The isotropic Raman line widths of the six alg bands of liquid cyclohexane increase with increasing temperature and pressure. (2) The value of 5.52 (A) (25°C) for the hard sphere diameter was obtained from the analysis of the viscosity data. (3) The dephasing theory predicts semi-quantitatively the temperature and pressure dependence of the line widths for the v1 and v2 vibrational modes. (4) For the lower frequency z+, v4, LJ~and vg bands, for which tzw = kT, the line width shows less sensitivity to volume changes when compared to analogous behavior of the higher frequency u1 and v2 bands. At the same time, Sobs for v3, v+ us and vg exhibits stronger temperature dependence at constant density than 6obs for the v1 and v2.bands. (5) The observed isotropic line width varies widely for individual bands within a range of 1.8 cm-’ to 10 cm-‘. (6) The experimental data for the lower frequency bands cannot be interpreted in terms of the simple dephasing model.
Acknowledgement This work was supported in part by the National Science Foundation under Grant NSF CHE 77-07621. The authors express their thanks to Dr. J. Schroeder for his valuable discussions, and to Dr. T. DeFries, Mr. P.T. Shark0 and Mr. W. Lamb for their assistance in density and viscosity measurements.
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K. Tanabe.J_ Jonas / Vibrationalrelaxationin liquid cyclohexane [IO] H.J. Parkhurst Jr. and I. Jonas, J. Chem. Phys. 63 (1975) 2706. [ll] E. Wilhelm, I. Chem. Phys. 58 (1973) 3558. 1121 J. Schroeder, V.H. Schiemann and J. Jonas, Mol. Phys. 34 (1977) 1501. [13] N.F. Carnahan and K.E. Starling, J. Chem. Phys. 51 (1969) 635.
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[ 141 S. Miller and J.H.R. Clarke, Chem. Phys. Letters 56 (1978) 235. [IS] A.M. Amorim da Costa, M.A. Norman and J.H.R. Clarke, Mol. Phys. 29 (1975) 191.