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Raman scattering investigation of the orientational dynamics of methyl iodide
391
Chemical Physics 110 (1986) 391401 North-Holland, Amsterdam
RAMAN SCATTERING INVESTIGATION OF THE ORIENTATIONAL DYNAMICS OF METHYL
IODIDE
SK. DEB, M.L. BANSAL and A.P. ROY Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India Received 30 June 1986
A Raman scattering study of the us vibration-rotation band in methyl iodide as a function of temperature and dilution (in cyclohexane) has been performed. All the data satisfy the second moment criterion. Detailed information about rotational correlation function, angular velocity correlation function, various correlation times and mean-square torque has been obtained. The correlation function, in the pure liquid, decays slowly with decrease in temperature whereas it decays faster with decreasing concentration in cyclohexane. It has been shown, from a consideration of the second moment as a function of concentration, that the contribution of collision-induced scattering to the vs band of methyl iodide is negligible. Applicability of a simple “damped librator model”, with a view to understanding certain aspects of the rotational motion in methyl iodide, is discussed.
1. Introduction
Raman band profile analysis is being extenused for obtaining the single particle, pure reorientation correlation function (C*(f)) for a wide variety of molecules, in different environments [1,2]. Several experimental difficulties viz. the presence of overlapping bands, either due to natural isotopic component or other impurity molecules, and large fluorescent background, severely limit the accuracy of C2(t) thus obtained. The central portion of the band profile which determines the long-time decay of C*(t) can however be obtained fairly accurately even in the presence of the above disturbing factors. The line shape near the band centre has mostly been found to be lorentzian [3] and hence C*(t) at long times decays exponentially with slope determined by the inverse of the width of the band profile. The short-time behaviour of C*(t) on the other hand is sensitive to the wing of the band profile and is most difficult to determine. In an earlier paper [4] we have demonstrated the importance of the careful recording of the band profile far into the wing, satisfying the rotational second moment sum rule for accurate determination of Cz(t). Further, the sively
importance of this sum rule in determining an accurate value of mean-square torque has also been highlighted. Recently, it has been realized that the fluctuations in angular velocity and angular momentum play a very important role in molecular reorientation processes in dense phases. This is because the fluctuation of angular momentum and velocity is related to the intermolecular torque. Thus determination of angular momentum correlation function (ACF) is essential for a thorough understanding of the dynamics of liquids. It has been shown that an estimate of decay of the angular velocity correlation function (AVCF), under certain conditions, can be obtained from the orientational correlation function itself [5-71. This is based on the fact that the second cumulant expansion is a good approximation for the orientational correlation function in the liquid and the reduced angular velocity correlation time is much smaller than unity. The AVCF has been found to decay on a much faster time scale and hence is determined by the short-time behaviour of C*(t). This also requires that the short-time decay of C2(t) be determined as accurately as possible. The determination of C*(t), AVCF and mean-square torque as a function of
0301-0104/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
392
S. K. Deb et al. / Ramun stuttering
temperature as well as at various concentrations in mixture with some “inert” solvent is expected to provide valuable information on the role played by different intermolecular forces in the reorientation process. In particular, we will show that information about contribution of “collision-induced scattering” to the band profile can be obtained from the solution studies. The choice of a suitable solvent is however far from trivial because of the presence of solvent bands which may interfere with the band under investigation. Methyl iodide (CH,I) is a symmetric top molecule with permanent dipole moment = 1.6 x lo-l8 esu and it remains in liquid form over a large temperature range. The totally symmetric v3(A1) band at 522 cm-’ is very intense, has a large depolarization ratio and is free from neighbouring interfering bands apart from a weak contribution due to a hot band on the low-frequency side [g]. Analysis of the polarized and the depolarized profiles provides the correlation function (C*(t)) for reorientation of the symmetry axis (C-I axis) of the CH,I molecule. A large number of investigations using a variety of techniques have been carried out on this molecule. These have been reviewed recently by Evans and Evans [9]. It is evident from the literature that a consistent picture of reorientation of CH,I molecules in liquids is yet to emerge in spite of the large number of studies. This is partly due to the lack of availability of very accurate data. We have presented in an earlier paper [4] the Raman correlation function C,(t), the correlation time and mean-square torque at room temperature for this molecule in pure liquid and indicated the inadequacies of the earlier works. In this paper, we report a Raman scattering investigation of the orientational motion of CH,I in neat liquid as a function of temperature below the room temperature. The usefulness of low-temperature studies lies in the fact that the mean kinetic energy decreases and the intermolecular interaction is expected to play a more dominant role. We also present results for CH,I in solution of cyclohexane (C,H,,) at various molar concentrations. Cyclohexane is a globular molecule and it has no prominent Raman active band in the 500-700 cm-’ range [lo]. This makes it an ideal
study of CH, I
solvent for studying the v3 band profile of CH,I. The dipole moment of C,H,, is zero and when CH,I is dissolved in C,H,, at very low concentration, the strong dipole-dipole interaction between the CH,I molecules is reduced to a large extent due to the presence of C,H,* molecules. Thus the study of orientational dynamics of neat liquid at low temperature and in solution of C,H,, at different concentrations is expected to provide valuable information about the role of various electric multipolar interactions, in particular dipoledipole interaction between the CH,I molecules, on its reorientation dynamics. The various experimental quantities reported in this paper are the Raman correlation function C,( t ), the angular velocity correlation function (AVCF), the meansquare torque and the various correlation times. We would like to mention that the rotational second moment criterion has been satisfied in all the cases and hence the reported experimental quantities are accurate. We have also shown from a consideration of second moment as function of concentration that the contribution of “collisioninduced scattering” to Raman band profile in CH,I is negligible.
2. Experimental details Spectral grade CH,I was treated with fresh NaOH solution to remove traces of free iodine. The separated CH,I was then vacuum distilled over molecular sieves in three stages directly into a Pyrex capillary cell. The mixture samples were also prepared by direct vacuum distillation of iodine free, spectral grade CH,I and C,H,, into the same capillary cell. The exact molar fraction of the prepared samples were determined using the integrated intensity of proton NMR signals of CH,I and C,H,, obtained in a 500 MHz high-resolution NMR machine. Low-temperature measurements were made in a cold-finger-type liquid-nitrogen cryostat. The cold finger is made of high-purity (OFHC) copper and is specially designed so that it surrounds the sample capillary. The sample capillary was held to the cold finger by a thin layer of low-temperature varnish GE7031. The thermal contact has been
S.K. Deb et al. / Raman scattering study of CHJ
checked by monitoring the melting point of methyl iodide. The temperature stability and accuracy are estimated to be f 0.5 and f 1 K respectively. The experimental arrangement for recording the band profiles is the same as that described in our earlier paper [4], except that the laser power was about 80 mW (X = 4416 A). The instrumental resolution for pure liquid and mixture samples were kept at 2.6 and 3.5 cm-’ respectively. The polarized (I,,) and depolarized (I,) line profiles for the V~ mode have been recorded over 175 cm-’ on the high-frequency side for all the samples. A typical I, band profile for pure CH,I has been given in the earlier paper [4]. The Z,, and I, profiles for 10% CH,I in &Hi, are shown in fig. 1. The contribution from two weak C,H,, bands around 610 and 630 cm-’ have been subtracted from the knowledge of the relative concentration and intensities. The as recorded line profiles on the high-frequency side were digitized manually at intervals of 0.77 cm-‘. Only the Stokes side of the I,, and I, profiles were considered to avoid hot-band contamination. Different line profiles were analysed independently and the values reported in the following represent averages over all the runs.
393
3. Analysis of the experimental data The rotational correlation function C*(t) can be obtained using the Fourier transform of the isotropic and anisotropic band profiles defined as
Z;,(w) = [Z,,(w) - 4Z,(a)] X
Z,,b) - S,b>)du (j(
-I, )
Li,b)=Lb)(jr,b)
da)-'.
0)
(2)
Then C2(t) is given by C,(t) = NGN&
(3)
where N(t) = /Z,,(o) O(t) = jZi,(o)
cos ot dw, cos ot dw.
C,(t) represents the reorientation of the symmetry axis (C-I axis) of the CH,I molecule. The orientational correlation time is defined as Tc= jgmCz(t) dt.
(4)
The long-time decay of Cz(t) in most cases is found to be exponential, i.e. at large times it decays as 295K
Raman
600 shift
(cm-‘)
Fig. 1. The polarized (a) and depolarized (b) band profiles in the rj mode for CH,I diluted in cyclohexane. Relative intensities of the bands are indicated by the various scale factors. To avoid overlap, the spectra have been relatively displaced; horizontal lines indicate the zero-intensity level. Contributions from weak cyclohexane bands around 610 and 630 cm-’ have been subtracted to yield the dashed lines (see text).
G(t)
=A exp(-t/T,),
(5)
where A is a constant and we define the relaxation time TVas the slope of the In Cz(t) curve at large time. The time is expressed in picoseconds or alternatively in reduced units Q-u), i.e. in units of free rotor time (Z/kT)‘12, where Z is the moment of inertia about an axis normal to the symmetry axis, k is Boltzmann’s constant and T is the absolute temperature. The rotational second and fourth moments are defined as M2 (rot) = M, (anis) - M2 (iso), M,(rot)
(6)
= M,,(anis) - M,(iso) -6M,(rot)M,(iso),
(7)
S. K. Deb et al. / Roman scattering stuc+ of CH, I
394
where M,(iso) and M,,(anis) are the n th moment of the isotropic and the anisotropic band profiles defined above in eqs. (1) and (2). The mean-square torque (( OV)‘) acting on the molecule can be obtained from &(rot) using the relation M,(rot) = 96(1 + 1,,/161)
+ 3((OV)‘),
(8)
where I,, is the moment of inertia about the symmetry axis. M4(rot) and ((C)V)*) are expressed in their respective reduced units, i.e. in units of (M/Z)* and (kT)2 respectively. The absolute units of ((OV)*) and &(rot) are dyne* cm* and cm-* respectively. The angular velocity correlation function (AVCF) denoted by GUI(t) can be calculated by using the relation [6] Fig. 2. In C,(
= id*[ln
C,(t)]/dt*,
(9)
where wI denotes the component of angular velocity o perpendicular to the symmetry axis in the molecular frame. It can however be obtained directly from the recorded line profiles using the expression G&)=
-t{N”(t)/N(t)-D”(t)/D(t)
The primes on N and D denote first and second time derivatives respectively.
4. Results and discussion 4. I. Correlation junction, mean-square torque
correlation
time
and
The Raman correlation functions C,(t) for pure CH,I at 295, 258, 241 and 215 K are shown in fig. 2 by plotting In C*(t) and C*(t) as a function of reduced time t. C, (t ) at 295 K compares very well with that reported earlier [4]. At short times (t < 0.2 ru), as expected from short-time expansion of C*(t) up to the t2 term, C*(r) follows the free rotor curve quite closely for all temperatures. As t
t) for CH,I as a function of t in reduced units of (I/kT)
‘I2 for various temperatures.
increases the effect of intermolecular interactions becomes important resulting in a deviation from the free rotor behaviour and at very large times (t > 1.4 ru) all of them decay exponentially. The overall decay becomes slower with decrease in temperature and it is seen from fig. 2 that C*(f) decays to 0.585 at 215 K as against 0.353 at 295 K after t = 2.0 ru. The second moment M2(rot) and the different correlation times are shown in table 1. The experimental M,(rot) at all temperatures is close to the theoretical value of 6 ru within fO.l ru. Thus the second moment criterion is satisfied without having to take recourse to a line-fitting procedure or base-line adjustment, and the shorttime decay of C*(t) is fairly accurate. This enabled us to obtain the mean-square torque from the experimental M4(rot) and these are also shown in table 1. The correlation times rc and 7S have also been determined and these increase with a decrease in temperature. It is well known that if C*(t) decays exponentially for all times, as is expected from Debye’s rotational diffusion model, then the correlation times rc and 7S would be identical. But it is seen that rS> T, at all temperatures and this is a manifestation of departure from exponential decay of C*(t) at short and inter-
S.K. Deb et al. / Raman scattering study of CH,I
Table 1 Correlation T
times and mean-square
torque
(( 0Y)2)
for CH,I
as a function
395
of temperature
wu2>
w
M2 (rot) 0-4
7c (PSI
ZS,
295 258 241 215
5.90 5.94 5.95 6.08
1.14 1.68 2.08 3.60
1.30 2.16 2.53 4.65
mediate times. The slower decay of C,( t ) and increase in correlation time with decrease in temperature shows that the rotation of molecule becomes progressively hindered at low temperatures. It is seen from table 1 that ((Ok’)‘) in ru increases with decrease in temperature and at 215 K has a value which is 1.7times the value at 295 K. On the other hand ((OV)2) in dyne2 cm2 and.in cme2 remains approximately constant to within &5%. It is to be noted that the ((OV)2) expressed in ru is a measure of the torque relative to the mean kinetic energy and its increase implies a predominant role is played by the intermolecular potential in the reorientation of CH,I molecule at lower temperatures. But the mean-square torque ((OV)2} in absolute unit remains constant and this shows that the intermolecular potential as well as its anisotropy does not change appreciably when the temperature changes from 295 to 215 K. This is because the intermolecular potential for dipolar molecules like CH,I is mostly of electric multipolar in origin. It is long-range and hence relatively less sensitive to small changes in the mean intermolecular separation. Similar variation of torque has also been observed for CH,Cl, CH,F and CHF, from far-infrared (FIR) measurements WIThe Raman correlation functions C,(t) for CH,I mixed in C,H,, at 6546, 45%, 20% and 10% molar fractions are shown in fig. 3. C2(t) for pure liquid is shown as 100% solution. The second moment and different correlation times are given in table 2. The second moment criterion is also satisfied in this case and hence the short-time behaviour of Cz( t ) should be quite accurate. ((OV)2> calculated from the M4(rot) are also presented in table 2. At short time, C*(t) follows
(N
(cmw2)
(10m2” dyne’ cm2)
90 107 129 154
384 347 365 346
15.0 13.6 14.3 13.6
the free rotor decay curve quite closely and at long time (t > 1 ru) the decay is exponential. C,(t) decays faster with decrease in concentration. ((OV)2> and the different correlation times also decrease with decrease in concentration. Thus, the hindrance to reorientation of CH,I molecules decreases in solution of C,H,, and this can be explained as being due to reduction of net dipolar interaction on a CH,I molecule when dissolved in C,H,, at very low molar fraction, resulting in progressively less hindered rotation of CH,I molecules. The strong hindrance to rotation at low temperature and much faster rotation when dissolved in cyclohexane shows that the strong dipole-dipole interaction plays a predominant role in reorientation of CH,I molecules in liquid. t(psec)
1
0.6
-1.6
!
-2.0 t_-
‘*O t(r.u)
2*o
3.0
Fig. 3. ln C,(r) as a function of t (ru) for CH31 in cyclohexane. Dilution is indicated as percentage of molar fraction.
3%
S.K. Deb et al. / Raman xattering
Table 2 Correlation times and mean square torque ((0V)2)
study of CH,I
for CH31 as a function of dilution in C6H12
X (49)
M2(rot) (m)
7c (ps)
is)
lco 65 45 20 10
3.90 5.98 5.96 5.94 5.93
1.14 0.88 0.81 0.73 0.70
1.30 1.04 0.96 0.84 0.80
((OU2)
4.2. Collision-induced scattering In our data analyses, no correction has been applied for collision-induced scattering. Though for Rayleigh line profile, elusion-educe scattering contributes appreciably, the induced effects are generally believed to be less significant for vibration-rotation bands since it involves fluctuations of the derivatives of the molecular polarizability. We show that our measurements of M,(rot) for methyl iodide as a function of inflation in cyclohexane provides direct evidence that the contribution of collision-induced scattering to the vj band of CH,I is negligible. The collision-induced contribution to pair polarizability for vibrational Raman scattering under the dipole-educespole (DID) appro~mation can be written as 1121
where a,, (r; and a2, cy; denote the Rayleigh and Raman polarizability tensors of molecule 1 and 2 respectively and T12 is the dipole moment operator. For pure liquid the molecules 1 and 2 are identical and hence the contributions from both terms are around the same vibrational band profile. But in mixtures with a small concentration of the solute molecules, 1 and 2 are different and only one term contributes to a particular vibrational mode. Further, the contribution to vibrational line of CH,I will depend on the Rayleigh polar&ability of C,H,,. The latter is known to be small 1131 and hence the contribution from collision-induced scattering will be small for dilute solution of CH,I in GH,,. In other words, contribution of collision-induced scattering to M,(rot) is expected to reduce progressively with decreasing
0-u)
(cm-‘)
(1Kz6 dyne’ cm’)
90 69 68 62 61
384 291 286 259 251
15.0 11.4 11.2 10.2 10.0
concentration of CH,I in C,H,,. It ‘is seen from table 2 that M*(rot) has a value close to 6 III for the whole range of concentrations (ZOO-lo%), implying that even for the neat liquid collision-induced scattering contributes negligibly to M.(rot)_ 4.3. Angular velocity correlation function (A VCF) We have seen that our data for pure liquid as well as solution satisfies the second moment sum rule and the #~sion-indu~d scattering contribution is negligible. Hence the short-time behaviour of C*(t) is quite accurate and it is meaningful to determine the AVCF (GUl(t)). We have obtained AVCF for pure liquid CH,I as a function of temperature and also for solution of CH,I in cyclohexane. The Gel(t) for pure CH,I at 295 and 215 K are shown in fig. 4. The decay of AVCF is much faster than C,(t) and it exhibits a distinct negative portion and decays to zero when the decay of C2(t) becomes exponential (as can be seen from eq. (9)). The time at which AVCF becomes negative decreases from 0.136 ps at 295 K to 0.116 ps at 215 K. The magnitude of the negative minimum increases as the temperature decreases. The AVCF for solution of CH,I in C,H,, also shows a monotonic variation; the time at which AVCF becomes negative increases and the magnitude of the negative minimum decreases with decrease in concentration (fig. 5). It is to be mentioned here that o1 is the component of angular velocity w normal to the symmetry axis and CH,I being a molecule with a large moment of inertia about an axis normal to the symmetry axis (asymmetry parameter = 20), has its angular momentum vector (J) almost normal to the sym-
S.K. Deb et al. / Raman seuttering stdy
t (r.u) Fig. 4. The angular velocity correlation function (GwI( t)) of CH,I at 295 and 215 K. Results obtained by Kocot et al. (171 for 298 K and Klti et al. [6] are shown as filled circles and crosses respectively.
metry axis. So, AVCF also represents the behaviour of the angular momentum correlation function (ACF) to a good approximation. The negative going feature of AVCF and ACF indicates that, on the average there is a reversal of the angular velocity and angular momentum vector
TIME ( psec)
I-
I 1.0
I
0.5
397
during the reorientation process. The changes in angular momentum and angular velocity vectors are related to the intermolecular torque which in turn is related to the anisotropy of the intermolecular potential. The reversal of these vectors thus indicates strong non-central character of the potential as has been revealed by molecular dynamics calculations [14]. We note here that, in the extended diffusion model (EDM) it is assumed that the ACF and AVCF decay exponentially with time and hence always remain positive. Our results thus show that the orientational dynamics of methyl iodide as a function of temperature as well as in dilute solution of cyclohexane cannot be described within the framework of EDM.
---295K
I-
of CH, I
1 (T.U) Fig. 5. The angular velocity correlation function (AVCF) for pure CH,I (100%) and 10% molar fraction in cyclohexane.
5. Comparison with other experiments We have compared our C*(t) at room temperature with other works in our earlier paper [4]. Although considerable work on CH,I exist at room temperature the determination of C*(t) at low temperature is very rare. The C*(t) at 195 K reported by Wright et al. [15] differs from our value at 215 K to a large extent, although the slopes are more or less similar. As has been pointed out [4] the large discrepancy can be attributed to a premature truncation of the experimental profile. The C*(t) for CH,I mixed in C,H,, has not been reported earlier, but C2(f) for 20% CH,I in hexane by Constant and Fauquembergue [16] shows much faster decay of C*(t) compared to pure CH,I. This is similar to what has been,observed by us for CH,I in C,H,,. Recently Kocot et al. [17] have investigated the orientational dynamics of CH,I by FIR absorption at various temperatures. They have measured absorption up to 200 cm-’ and obtained an integrated absorption value which is about 10% higher than the theoretical value. We have compared their AVCF at 298 K with our value at 295 K in fig. 4. Considering that the techniques are entirely different, the agreement is excellent, except that the negative region in the FIR measurement is slightly broader. Kluk et al. [6] have also obtained an AVCF from the Raman band profile of the v3 mode at 300 K. Their AVCF is in poor
398
S. K. Deb et al. / Raman stuttering studv
agreement with our data (fig. 4) and this may be due to premature truncation and consequent low value of &(rot) (= 1.14 ru) obtained by these authors. Dill et al. [18] have obtained AVCFs using depolarized Rayleigh scattering at various pressures. The AVCF at 1 bar and 296 K compares quite well with ours as well as the data of Kocot et al. but for the fact that the negative dip in ref. [18] is much smaller.
The infrared and Raman correlation functions (C,( 1) and C,( t ) respectively) in this model can be calculated using Kubo line shape theory and assuming w(t) to be a gaussian random process [23]. The C,( f ) has been calculated by Rothschild et al. [22] and is given by C,(t) = exp[ -f(t)1
model
The AVCF for pure liquid as well as for mixture suggests that, on the average there is a reversal of the angular velocity and angular momentum vector due to the intermolecular torque. This feature is characteristic of libratory type of motion and hence there have been attempts to describe orientational dynamics in terms of libration [19-221. However the absence of any sustained oscillation in AVCF (except perhaps at 215 K) and also the absence of any oscillatory behaviour in C*(t) indicate that these librations have to be highly damped and hence can be termed at best as “quasi-libration”. We, therefore, attempt to describe the behaviour of AVCF in terms of a simple “damped librator” model. The molecule is assumed to be a linear librator performing libration in the potential cage produced by the neighbouring molecules. The fluctuation in this potential due to the changes in molecular environment gives rise to damping which is proportional to the angular velocity, but independent of position. The AVCF for this motion can be shown to be given by WI
Cl(t)
= exp(- tvf)[ c43sJ2’t+ (y/252’) sin Pt] , (11)
where
and f2 is the libration frequency and y is the damping constant. This expression for GUl(t) is for the underdamped case (SE> $y) and shows damped oscillation whereas for the overdamped case (a < $y) Gal(t) exhibits monotonic decay.
y
(12)
where f(t)
6. Theoretical description: damped lib&or
of CH, I
= (2/Q2){
Yt + (1/Q2)
X{(y2-Q*)[exp(-+yt)cosPt-l] + (y/2P)(
y2 - 31(12)
Xexp( - $yt) sin Pt}}. C2(t) for the underdamped be given by (see appendix) C,(t) = t{l+
case can be shown to
3 exp[ -4f(t)]}.
(13) Both C,(t) and C2(t) exhibit correct short-time behaviour up to t2 term. At long times, Cl(t) decays exponentially with a slope given by &?*/2y, whereas C, (t ) decays to 0.25 at long time. Thus, these expressions are expected to be valid only for very hindered rotation. Let us now compare this model with experimental data for CH,I. The Gw, (t) in this model exhibits damped oscillations depending on the relative magnitude of D and y. The experimental AVCFs do not show such strong oscillatory behaviour and decay to zero reasonably fast after the first negative cycle. Thus it will be impossible to fit the calculated GUl(t) to the entire experimental AVCF and hence we determine the parameters Jz and y from least-squares fitting only up to the first negative lobe. The fits to experimental AVCFs at 295 and 215 K are shown in fig. 6. The calculated AVCF exhibits damped oscillation even after the experimental AVCF has decayed to zero. The values of the libration frequency D (in cm-‘) and y (in cm-‘) obtained from least-squares fit to AVCF at different temperatures are given in table 3. The libration frequency increases with decrease in temperature and at 215 K its value is 89.2 cm-‘. This value may be compared with the libration frequencies 94 cm-’ (R,) and 117 cm-’ (R,, R,,) observed for solid CH,I [24,25]. In fig. 7 we have
399
S. K. Deb et al. / Raman scattering study of CH, I Table 3 Parameters of the damped hbrator model as obtained by fitting to the experimental
T (K)
D
295 258 241 215
AVCF
Y
(4
(cm-‘)
(fu)
(cm-t)
8.22 9.11 9.74 10.32
83.2 86.2 89.1 89.2
6.35 6.17 6.22 5.64
64.3 58.4 56.9 48.7
1.0
0.5
z z 0.0
-0.5 0.0
0.5
1.0
t(r.u.1 Fig. 6. Comparison between experimentally determined AVCF ) using the “damped and calculated AVCF ((0 -0) librator model”. Parameters of the model are given in table 3.
shown the comparison with C2(t) at 295 and 215 K for values of 52 and y obtained from fit to AVCF. The calculated C*(t) corresponding to 215 K exhibits a hump characteristic of libration which however is not observed experimentally. The C, (t ) for 295 K always remains higher than the experimental value. This is because the C,(t) and Cr(t) given in eqs. (12) and (13) are applicable for hindered rotation and therefore may not be applicable to the data at 295 K. Rothschild et al. [22] have used this model to explain the infrared dephasing of the vs mode of CH,I and obtained s2 and y as a function of temperature. Over the temperature range 297-210 K, they find 52 to vary from 63 to 68 cm-’ which may be compared with our values of 83-89 cm-’ (table 3). More importantly, however, they conclude that damping increases with decreasing temperature whereas our analysis suggests the reverse trend which is consistent with the physical picture that as the system becomes more solid-like with cooling, libratory motion becomes progressively less damped and eventually it becomes underdamped in the solid phase.
7. Conclusion
1.0
t
(r.u)
2.0
Fig. 7. Cz(t) as function of t. The solid lines are obtained using the “damped librator model”. The experimental results (-0) are also shown for comparison.
We have obtained accurate Cr(t) for CH,I at different temperatures and also in solution of C,H,, at different concentrations. The data have been shown to satisfy the rotational second moment sum rule and this has enabled us to determine the mean-square torque. The variation of C,(t), different correlation times and mean-square torque as a function of temperature and concentration indicate the importance of electric mul-
400
S. K. Deb et al. / Raman scattering study of CH, I
tipolar, particularly that of dipole-dipole interaction in the orientational dynamics of CH,I in liquid. The AVCFs obtained in pure liquid as well as in solution reflect the inadequacy of models for molecular reorientation in liquid based on the extended diffusion model. A simple model based on damped libration in a harmonic well has been shown to provide a reasonable description of certain features of the experimental correlation functions.
given by C,(r)
= a(1 + 3 exp[ -4f(r)]}.
(A-4)
We also obtain the limiting expression for C,( r ) as r --, 0 and r --, co respectively. As r -+ 0, f( r ) + r * and thus C*(r)
= 1 - 6r2/2!
And as r --j 00, f(r) C,(r)
=
1 +
+ .... + 2yr,G2
3 exp( -8yr/L?‘).
(A-5) and hence 64.6)
Acknowledgement
The authors gratefully acknowledge Dr. R.V. Hosur of Tata Institute of Fundamental Research, Bombay, India for providing the NMR data on mixture samples for determining concentrations. Appendix
Here we obtain an expression for the Raman correlation function C,( f ) for a linear molecule undergoing damped libration. This is accomplished by using the Kubo line shape theory and assuming that the angular momentum is a gausSian random process [23]. Then following a procedure similar to that outlined by Dattagupta and Sood [26] and Deb and Bhagwat [27] the Raman correlation function can be written as G(t)
(A.11
= [exp Al,,
where A=
-L;f(t).
(A.2)
Here L, is a 5 X 5 matrix representing the x component of the angular momentum operator in an I= 2 spherical harmonic basis [28] and f(t)
= Jd’t - ~)(w(O)*W(~))
dr.
(A-3)
The expression for f(t) has already been given along with eq. (12) in the text. [ loo denotes the m = m’ = 0 element of the matrix. C*(t) can be obtained by diagonalizing the matrix given in eq. (A.2) using a unitary matrix formed from its eigenvectors. Then it can be shown that C*(t) is
References [l] W.A. Steele, Advan. Chem. Phys. 34 (1976) 1. [2] J. Vincent-Geisse, in: Vibrational spectroscopy of molecular liquids and solids, eds. S. Bratos and R.M. Pick (Plenum Press, New York, 1980) p. 117. [3] F.J. Bartoli and T.A. Litovitz, J. Chem. Phys. 56 (1972) 404. [4] M.L. Bansal, SK. Deb and A.P. Roy, Chem. Phys. Letters 83 (1981) 83. [5] W.A. Steele, J. Chem. Phys. 38 (1963) 2404, 2411. [6] E. Kluk, T.W. Zedra and J. Zedra, Acta Phys. Polon. A56 (1979) 121. [7] B. Janik, J. Sciesinski, M.W. Evans, E. Kluk and T. Grochulski, Chem. Phys. 70 (1982) 77. [8] J.H. Campbell, J.F. Fisher and J. Jonas, J. Chem. Phys. 61 (1974) 346. [9] G.J. Evans and M.W. Evans, J. Mol. Liquids 25 (1983) 177. [IO] L.M. Sverdlov, M.A. Kovner and E.P. Krainov, Vibrational spectra of polyatomic molecules (Wiley, New York, 1974) p. 238. [ll] A. Gerschel, C. Brot, I. Dimicoli and A. Riou, Mol. Phys. 33 (1977) 527. 1121G.C. Tabiz, in: Specialist periodical reports. Molecular spectroscopy, Vol. 6 (Chem. Sot., London, 1979) p. 136. 1131 A.M. Amorim da Costa, M.A. Norman and J.H.R. Clarke, Mol. Phys. 29 (1975) 191. ]141 B.J. Beme and G.D. Harp, Advan. Chem. Phys. 17 (1970) 63. ]151 R.B. Wright, M. Schwartz and C.H. Wang, J. Chem. Phys. 58 (1973) 5125. and R. Fauquembergue, Advan. Raman [161 M. Constant Spectry. 1 (1972) 413. J. Sciesinski, T. Grochulski, L. [171 A. Kocot, S. Gandor, Pszczolkowski and K. Leibler, Mol. Phys. 53 (1984) 1481. [181 J.F. Dill, T.A. Litovitz and J.A. Bucaro, J. Chem. Phys. 62 (1975) 3839. ]191 N.E. Hill, Proc. Phys. Sot. 82 (1963) 723. f201 B. Lassier and C. Brot, Chem. Phys. Letters 1 (1968) 581. 1211 J.N. Kushik, J. Chem. Phys. 67 (1977) 2068.
S. K. Deb et al. / Raman scattering study of CH,I (221 W.G. Rothschild, J. Soussen-Jacob, J. Bessiere and J. Vincent-Geisse, J. Chem. Phys. 79 (1982) 3002. [23] R. Kubo, in: Fluctuation, relaxation and resonance in magnetic systems, ed. D. ter Haar, (Oliver and Boyd, Edinburgh, 1962) p. 23. [24] M. Ito, J. Chem. Phys. 41 (1964) 1255.
401
[25] J.A. Janik, J.M. Jan&, A. Bajorek, K. Parlinski and M. Sudnik-Hrynkiewin, Physica 35 (1967) 451. [26] S. Dattagupta and A.K. 8ood, Z. Physik B44 (1981) 85. (271 SK. Deb and K.V. Bhagwat, Chem. Phys. 98 (1985) 251. [28] R.H. Dicke and J.P. Wittke, Introduction to quantum mechanics (Addison-Wesley, Reading, 1963) p. 191.
Chemical Physics 110 (1986) 391401 North-Holland, Amsterdam
RAMAN SCATTERING INVESTIGATION OF THE ORIENTATIONAL DYNAMICS OF METHYL
IODIDE
SK. DEB, M.L. BANSAL and A.P. ROY Nuclear Physics Division, Bhabha Atomic Research Centre, Bombay 400 085, India Received 30 June 1986
A Raman scattering study of the us vibration-rotation band in methyl iodide as a function of temperature and dilution (in cyclohexane) has been performed. All the data satisfy the second moment criterion. Detailed information about rotational correlation function, angular velocity correlation function, various correlation times and mean-square torque has been obtained. The correlation function, in the pure liquid, decays slowly with decrease in temperature whereas it decays faster with decreasing concentration in cyclohexane. It has been shown, from a consideration of the second moment as a function of concentration, that the contribution of collision-induced scattering to the vs band of methyl iodide is negligible. Applicability of a simple “damped librator model”, with a view to understanding certain aspects of the rotational motion in methyl iodide, is discussed.
1. Introduction
Raman band profile analysis is being extenused for obtaining the single particle, pure reorientation correlation function (C*(f)) for a wide variety of molecules, in different environments [1,2]. Several experimental difficulties viz. the presence of overlapping bands, either due to natural isotopic component or other impurity molecules, and large fluorescent background, severely limit the accuracy of C2(t) thus obtained. The central portion of the band profile which determines the long-time decay of C*(t) can however be obtained fairly accurately even in the presence of the above disturbing factors. The line shape near the band centre has mostly been found to be lorentzian [3] and hence C*(t) at long times decays exponentially with slope determined by the inverse of the width of the band profile. The short-time behaviour of C*(t) on the other hand is sensitive to the wing of the band profile and is most difficult to determine. In an earlier paper [4] we have demonstrated the importance of the careful recording of the band profile far into the wing, satisfying the rotational second moment sum rule for accurate determination of Cz(t). Further, the sively
importance of this sum rule in determining an accurate value of mean-square torque has also been highlighted. Recently, it has been realized that the fluctuations in angular velocity and angular momentum play a very important role in molecular reorientation processes in dense phases. This is because the fluctuation of angular momentum and velocity is related to the intermolecular torque. Thus determination of angular momentum correlation function (ACF) is essential for a thorough understanding of the dynamics of liquids. It has been shown that an estimate of decay of the angular velocity correlation function (AVCF), under certain conditions, can be obtained from the orientational correlation function itself [5-71. This is based on the fact that the second cumulant expansion is a good approximation for the orientational correlation function in the liquid and the reduced angular velocity correlation time is much smaller than unity. The AVCF has been found to decay on a much faster time scale and hence is determined by the short-time behaviour of C*(t). This also requires that the short-time decay of C2(t) be determined as accurately as possible. The determination of C*(t), AVCF and mean-square torque as a function of
0301-0104/86/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
392
S. K. Deb et al. / Ramun stuttering
temperature as well as at various concentrations in mixture with some “inert” solvent is expected to provide valuable information on the role played by different intermolecular forces in the reorientation process. In particular, we will show that information about contribution of “collision-induced scattering” to the band profile can be obtained from the solution studies. The choice of a suitable solvent is however far from trivial because of the presence of solvent bands which may interfere with the band under investigation. Methyl iodide (CH,I) is a symmetric top molecule with permanent dipole moment = 1.6 x lo-l8 esu and it remains in liquid form over a large temperature range. The totally symmetric v3(A1) band at 522 cm-’ is very intense, has a large depolarization ratio and is free from neighbouring interfering bands apart from a weak contribution due to a hot band on the low-frequency side [g]. Analysis of the polarized and the depolarized profiles provides the correlation function (C*(t)) for reorientation of the symmetry axis (C-I axis) of the CH,I molecule. A large number of investigations using a variety of techniques have been carried out on this molecule. These have been reviewed recently by Evans and Evans [9]. It is evident from the literature that a consistent picture of reorientation of CH,I molecules in liquids is yet to emerge in spite of the large number of studies. This is partly due to the lack of availability of very accurate data. We have presented in an earlier paper [4] the Raman correlation function C,(t), the correlation time and mean-square torque at room temperature for this molecule in pure liquid and indicated the inadequacies of the earlier works. In this paper, we report a Raman scattering investigation of the orientational motion of CH,I in neat liquid as a function of temperature below the room temperature. The usefulness of low-temperature studies lies in the fact that the mean kinetic energy decreases and the intermolecular interaction is expected to play a more dominant role. We also present results for CH,I in solution of cyclohexane (C,H,,) at various molar concentrations. Cyclohexane is a globular molecule and it has no prominent Raman active band in the 500-700 cm-’ range [lo]. This makes it an ideal
study of CH, I
solvent for studying the v3 band profile of CH,I. The dipole moment of C,H,, is zero and when CH,I is dissolved in C,H,, at very low concentration, the strong dipole-dipole interaction between the CH,I molecules is reduced to a large extent due to the presence of C,H,* molecules. Thus the study of orientational dynamics of neat liquid at low temperature and in solution of C,H,, at different concentrations is expected to provide valuable information about the role of various electric multipolar interactions, in particular dipoledipole interaction between the CH,I molecules, on its reorientation dynamics. The various experimental quantities reported in this paper are the Raman correlation function C,( t ), the angular velocity correlation function (AVCF), the meansquare torque and the various correlation times. We would like to mention that the rotational second moment criterion has been satisfied in all the cases and hence the reported experimental quantities are accurate. We have also shown from a consideration of second moment as function of concentration that the contribution of “collisioninduced scattering” to Raman band profile in CH,I is negligible.
2. Experimental details Spectral grade CH,I was treated with fresh NaOH solution to remove traces of free iodine. The separated CH,I was then vacuum distilled over molecular sieves in three stages directly into a Pyrex capillary cell. The mixture samples were also prepared by direct vacuum distillation of iodine free, spectral grade CH,I and C,H,, into the same capillary cell. The exact molar fraction of the prepared samples were determined using the integrated intensity of proton NMR signals of CH,I and C,H,, obtained in a 500 MHz high-resolution NMR machine. Low-temperature measurements were made in a cold-finger-type liquid-nitrogen cryostat. The cold finger is made of high-purity (OFHC) copper and is specially designed so that it surrounds the sample capillary. The sample capillary was held to the cold finger by a thin layer of low-temperature varnish GE7031. The thermal contact has been
S.K. Deb et al. / Raman scattering study of CHJ
checked by monitoring the melting point of methyl iodide. The temperature stability and accuracy are estimated to be f 0.5 and f 1 K respectively. The experimental arrangement for recording the band profiles is the same as that described in our earlier paper [4], except that the laser power was about 80 mW (X = 4416 A). The instrumental resolution for pure liquid and mixture samples were kept at 2.6 and 3.5 cm-’ respectively. The polarized (I,,) and depolarized (I,) line profiles for the V~ mode have been recorded over 175 cm-’ on the high-frequency side for all the samples. A typical I, band profile for pure CH,I has been given in the earlier paper [4]. The Z,, and I, profiles for 10% CH,I in &Hi, are shown in fig. 1. The contribution from two weak C,H,, bands around 610 and 630 cm-’ have been subtracted from the knowledge of the relative concentration and intensities. The as recorded line profiles on the high-frequency side were digitized manually at intervals of 0.77 cm-‘. Only the Stokes side of the I,, and I, profiles were considered to avoid hot-band contamination. Different line profiles were analysed independently and the values reported in the following represent averages over all the runs.
393
3. Analysis of the experimental data The rotational correlation function C*(t) can be obtained using the Fourier transform of the isotropic and anisotropic band profiles defined as
Z;,(w) = [Z,,(w) - 4Z,(a)] X
Z,,b) - S,b>)du (j(
-I, )
Li,b)=Lb)(jr,b)
da)-'.
0)
(2)
Then C2(t) is given by C,(t) = NGN&
(3)
where N(t) = /Z,,(o) O(t) = jZi,(o)
cos ot dw, cos ot dw.
C,(t) represents the reorientation of the symmetry axis (C-I axis) of the CH,I molecule. The orientational correlation time is defined as Tc= jgmCz(t) dt.
(4)
The long-time decay of Cz(t) in most cases is found to be exponential, i.e. at large times it decays as 295K
Raman
600 shift
(cm-‘)
Fig. 1. The polarized (a) and depolarized (b) band profiles in the rj mode for CH,I diluted in cyclohexane. Relative intensities of the bands are indicated by the various scale factors. To avoid overlap, the spectra have been relatively displaced; horizontal lines indicate the zero-intensity level. Contributions from weak cyclohexane bands around 610 and 630 cm-’ have been subtracted to yield the dashed lines (see text).
G(t)
=A exp(-t/T,),
(5)
where A is a constant and we define the relaxation time TVas the slope of the In Cz(t) curve at large time. The time is expressed in picoseconds or alternatively in reduced units Q-u), i.e. in units of free rotor time (Z/kT)‘12, where Z is the moment of inertia about an axis normal to the symmetry axis, k is Boltzmann’s constant and T is the absolute temperature. The rotational second and fourth moments are defined as M2 (rot) = M, (anis) - M2 (iso), M,(rot)
(6)
= M,,(anis) - M,(iso) -6M,(rot)M,(iso),
(7)
S. K. Deb et al. / Roman scattering stuc+ of CH, I
394
where M,(iso) and M,,(anis) are the n th moment of the isotropic and the anisotropic band profiles defined above in eqs. (1) and (2). The mean-square torque (( OV)‘) acting on the molecule can be obtained from &(rot) using the relation M,(rot) = 96(1 + 1,,/161)
+ 3((OV)‘),
(8)
where I,, is the moment of inertia about the symmetry axis. M4(rot) and ((C)V)*) are expressed in their respective reduced units, i.e. in units of (M/Z)* and (kT)2 respectively. The absolute units of ((OV)*) and &(rot) are dyne* cm* and cm-* respectively. The angular velocity correlation function (AVCF) denoted by GUI(t) can be calculated by using the relation [6] Fig. 2. In C,(
= id*[ln
C,(t)]/dt*,
(9)
where wI denotes the component of angular velocity o perpendicular to the symmetry axis in the molecular frame. It can however be obtained directly from the recorded line profiles using the expression G&)=
-t{N”(t)/N(t)-D”(t)/D(t)
The primes on N and D denote first and second time derivatives respectively.
4. Results and discussion 4. I. Correlation junction, mean-square torque
correlation
time
and
The Raman correlation functions C,(t) for pure CH,I at 295, 258, 241 and 215 K are shown in fig. 2 by plotting In C*(t) and C*(t) as a function of reduced time t. C, (t ) at 295 K compares very well with that reported earlier [4]. At short times (t < 0.2 ru), as expected from short-time expansion of C*(t) up to the t2 term, C*(r) follows the free rotor curve quite closely for all temperatures. As t
t) for CH,I as a function of t in reduced units of (I/kT)
‘I2 for various temperatures.
increases the effect of intermolecular interactions becomes important resulting in a deviation from the free rotor behaviour and at very large times (t > 1.4 ru) all of them decay exponentially. The overall decay becomes slower with decrease in temperature and it is seen from fig. 2 that C*(f) decays to 0.585 at 215 K as against 0.353 at 295 K after t = 2.0 ru. The second moment M2(rot) and the different correlation times are shown in table 1. The experimental M,(rot) at all temperatures is close to the theoretical value of 6 ru within fO.l ru. Thus the second moment criterion is satisfied without having to take recourse to a line-fitting procedure or base-line adjustment, and the shorttime decay of C*(t) is fairly accurate. This enabled us to obtain the mean-square torque from the experimental M4(rot) and these are also shown in table 1. The correlation times rc and 7S have also been determined and these increase with a decrease in temperature. It is well known that if C*(t) decays exponentially for all times, as is expected from Debye’s rotational diffusion model, then the correlation times rc and 7S would be identical. But it is seen that rS> T, at all temperatures and this is a manifestation of departure from exponential decay of C*(t) at short and inter-
S.K. Deb et al. / Raman scattering study of CH,I
Table 1 Correlation T
times and mean-square
torque
(( 0Y)2)
for CH,I
as a function
395
of temperature
wu2>
w
M2 (rot) 0-4
7c (PSI
ZS,
295 258 241 215
5.90 5.94 5.95 6.08
1.14 1.68 2.08 3.60
1.30 2.16 2.53 4.65
mediate times. The slower decay of C,( t ) and increase in correlation time with decrease in temperature shows that the rotation of molecule becomes progressively hindered at low temperatures. It is seen from table 1 that ((Ok’)‘) in ru increases with decrease in temperature and at 215 K has a value which is 1.7times the value at 295 K. On the other hand ((OV)2) in dyne2 cm2 and.in cme2 remains approximately constant to within &5%. It is to be noted that the ((OV)2) expressed in ru is a measure of the torque relative to the mean kinetic energy and its increase implies a predominant role is played by the intermolecular potential in the reorientation of CH,I molecule at lower temperatures. But the mean-square torque ((OV)2} in absolute unit remains constant and this shows that the intermolecular potential as well as its anisotropy does not change appreciably when the temperature changes from 295 to 215 K. This is because the intermolecular potential for dipolar molecules like CH,I is mostly of electric multipolar in origin. It is long-range and hence relatively less sensitive to small changes in the mean intermolecular separation. Similar variation of torque has also been observed for CH,Cl, CH,F and CHF, from far-infrared (FIR) measurements WIThe Raman correlation functions C,(t) for CH,I mixed in C,H,, at 6546, 45%, 20% and 10% molar fractions are shown in fig. 3. C2(t) for pure liquid is shown as 100% solution. The second moment and different correlation times are given in table 2. The second moment criterion is also satisfied in this case and hence the short-time behaviour of Cz( t ) should be quite accurate. ((OV)2> calculated from the M4(rot) are also presented in table 2. At short time, C*(t) follows
(N
(cmw2)
(10m2” dyne’ cm2)
90 107 129 154
384 347 365 346
15.0 13.6 14.3 13.6
the free rotor decay curve quite closely and at long time (t > 1 ru) the decay is exponential. C,(t) decays faster with decrease in concentration. ((OV)2> and the different correlation times also decrease with decrease in concentration. Thus, the hindrance to reorientation of CH,I molecules decreases in solution of C,H,, and this can be explained as being due to reduction of net dipolar interaction on a CH,I molecule when dissolved in C,H,, at very low molar fraction, resulting in progressively less hindered rotation of CH,I molecules. The strong hindrance to rotation at low temperature and much faster rotation when dissolved in cyclohexane shows that the strong dipole-dipole interaction plays a predominant role in reorientation of CH,I molecules in liquid. t(psec)
1
0.6
-1.6
!
-2.0 t_-
‘*O t(r.u)
2*o
3.0
Fig. 3. ln C,(r) as a function of t (ru) for CH31 in cyclohexane. Dilution is indicated as percentage of molar fraction.
3%
S.K. Deb et al. / Raman xattering
Table 2 Correlation times and mean square torque ((0V)2)
study of CH,I
for CH31 as a function of dilution in C6H12
X (49)
M2(rot) (m)
7c (ps)
is)
lco 65 45 20 10
3.90 5.98 5.96 5.94 5.93
1.14 0.88 0.81 0.73 0.70
1.30 1.04 0.96 0.84 0.80
((OU2)
4.2. Collision-induced scattering In our data analyses, no correction has been applied for collision-induced scattering. Though for Rayleigh line profile, elusion-educe scattering contributes appreciably, the induced effects are generally believed to be less significant for vibration-rotation bands since it involves fluctuations of the derivatives of the molecular polarizability. We show that our measurements of M,(rot) for methyl iodide as a function of inflation in cyclohexane provides direct evidence that the contribution of collision-induced scattering to the vj band of CH,I is negligible. The collision-induced contribution to pair polarizability for vibrational Raman scattering under the dipole-educespole (DID) appro~mation can be written as 1121
where a,, (r; and a2, cy; denote the Rayleigh and Raman polarizability tensors of molecule 1 and 2 respectively and T12 is the dipole moment operator. For pure liquid the molecules 1 and 2 are identical and hence the contributions from both terms are around the same vibrational band profile. But in mixtures with a small concentration of the solute molecules, 1 and 2 are different and only one term contributes to a particular vibrational mode. Further, the contribution to vibrational line of CH,I will depend on the Rayleigh polar&ability of C,H,,. The latter is known to be small 1131 and hence the contribution from collision-induced scattering will be small for dilute solution of CH,I in GH,,. In other words, contribution of collision-induced scattering to M,(rot) is expected to reduce progressively with decreasing
0-u)
(cm-‘)
(1Kz6 dyne’ cm’)
90 69 68 62 61
384 291 286 259 251
15.0 11.4 11.2 10.2 10.0
concentration of CH,I in C,H,,. It ‘is seen from table 2 that M*(rot) has a value close to 6 III for the whole range of concentrations (ZOO-lo%), implying that even for the neat liquid collision-induced scattering contributes negligibly to M.(rot)_ 4.3. Angular velocity correlation function (A VCF) We have seen that our data for pure liquid as well as solution satisfies the second moment sum rule and the #~sion-indu~d scattering contribution is negligible. Hence the short-time behaviour of C*(t) is quite accurate and it is meaningful to determine the AVCF (GUl(t)). We have obtained AVCF for pure liquid CH,I as a function of temperature and also for solution of CH,I in cyclohexane. The Gel(t) for pure CH,I at 295 and 215 K are shown in fig. 4. The decay of AVCF is much faster than C,(t) and it exhibits a distinct negative portion and decays to zero when the decay of C2(t) becomes exponential (as can be seen from eq. (9)). The time at which AVCF becomes negative decreases from 0.136 ps at 295 K to 0.116 ps at 215 K. The magnitude of the negative minimum increases as the temperature decreases. The AVCF for solution of CH,I in C,H,, also shows a monotonic variation; the time at which AVCF becomes negative increases and the magnitude of the negative minimum decreases with decrease in concentration (fig. 5). It is to be mentioned here that o1 is the component of angular velocity w normal to the symmetry axis and CH,I being a molecule with a large moment of inertia about an axis normal to the symmetry axis (asymmetry parameter = 20), has its angular momentum vector (J) almost normal to the sym-
S.K. Deb et al. / Raman seuttering stdy
t (r.u) Fig. 4. The angular velocity correlation function (GwI( t)) of CH,I at 295 and 215 K. Results obtained by Kocot et al. (171 for 298 K and Klti et al. [6] are shown as filled circles and crosses respectively.
metry axis. So, AVCF also represents the behaviour of the angular momentum correlation function (ACF) to a good approximation. The negative going feature of AVCF and ACF indicates that, on the average there is a reversal of the angular velocity and angular momentum vector
TIME ( psec)
I-
I 1.0
I
0.5
397
during the reorientation process. The changes in angular momentum and angular velocity vectors are related to the intermolecular torque which in turn is related to the anisotropy of the intermolecular potential. The reversal of these vectors thus indicates strong non-central character of the potential as has been revealed by molecular dynamics calculations [14]. We note here that, in the extended diffusion model (EDM) it is assumed that the ACF and AVCF decay exponentially with time and hence always remain positive. Our results thus show that the orientational dynamics of methyl iodide as a function of temperature as well as in dilute solution of cyclohexane cannot be described within the framework of EDM.
---295K
I-
of CH, I
1 (T.U) Fig. 5. The angular velocity correlation function (AVCF) for pure CH,I (100%) and 10% molar fraction in cyclohexane.
5. Comparison with other experiments We have compared our C*(t) at room temperature with other works in our earlier paper [4]. Although considerable work on CH,I exist at room temperature the determination of C*(t) at low temperature is very rare. The C*(t) at 195 K reported by Wright et al. [15] differs from our value at 215 K to a large extent, although the slopes are more or less similar. As has been pointed out [4] the large discrepancy can be attributed to a premature truncation of the experimental profile. The C*(t) for CH,I mixed in C,H,, has not been reported earlier, but C2(f) for 20% CH,I in hexane by Constant and Fauquembergue [16] shows much faster decay of C*(t) compared to pure CH,I. This is similar to what has been,observed by us for CH,I in C,H,,. Recently Kocot et al. [17] have investigated the orientational dynamics of CH,I by FIR absorption at various temperatures. They have measured absorption up to 200 cm-’ and obtained an integrated absorption value which is about 10% higher than the theoretical value. We have compared their AVCF at 298 K with our value at 295 K in fig. 4. Considering that the techniques are entirely different, the agreement is excellent, except that the negative region in the FIR measurement is slightly broader. Kluk et al. [6] have also obtained an AVCF from the Raman band profile of the v3 mode at 300 K. Their AVCF is in poor
398
S. K. Deb et al. / Raman stuttering studv
agreement with our data (fig. 4) and this may be due to premature truncation and consequent low value of &(rot) (= 1.14 ru) obtained by these authors. Dill et al. [18] have obtained AVCFs using depolarized Rayleigh scattering at various pressures. The AVCF at 1 bar and 296 K compares quite well with ours as well as the data of Kocot et al. but for the fact that the negative dip in ref. [18] is much smaller.
The infrared and Raman correlation functions (C,( 1) and C,( t ) respectively) in this model can be calculated using Kubo line shape theory and assuming w(t) to be a gaussian random process [23]. The C,( f ) has been calculated by Rothschild et al. [22] and is given by C,(t) = exp[ -f(t)1
model
The AVCF for pure liquid as well as for mixture suggests that, on the average there is a reversal of the angular velocity and angular momentum vector due to the intermolecular torque. This feature is characteristic of libratory type of motion and hence there have been attempts to describe orientational dynamics in terms of libration [19-221. However the absence of any sustained oscillation in AVCF (except perhaps at 215 K) and also the absence of any oscillatory behaviour in C*(t) indicate that these librations have to be highly damped and hence can be termed at best as “quasi-libration”. We, therefore, attempt to describe the behaviour of AVCF in terms of a simple “damped librator” model. The molecule is assumed to be a linear librator performing libration in the potential cage produced by the neighbouring molecules. The fluctuation in this potential due to the changes in molecular environment gives rise to damping which is proportional to the angular velocity, but independent of position. The AVCF for this motion can be shown to be given by WI
Cl(t)
= exp(- tvf)[ c43sJ2’t+ (y/252’) sin Pt] , (11)
where
and f2 is the libration frequency and y is the damping constant. This expression for GUl(t) is for the underdamped case (SE> $y) and shows damped oscillation whereas for the overdamped case (a < $y) Gal(t) exhibits monotonic decay.
y
(12)
where f(t)
6. Theoretical description: damped lib&or
of CH, I
= (2/Q2){
Yt + (1/Q2)
X{(y2-Q*)[exp(-+yt)cosPt-l] + (y/2P)(
y2 - 31(12)
Xexp( - $yt) sin Pt}}. C2(t) for the underdamped be given by (see appendix) C,(t) = t{l+
case can be shown to
3 exp[ -4f(t)]}.
(13) Both C,(t) and C2(t) exhibit correct short-time behaviour up to t2 term. At long times, Cl(t) decays exponentially with a slope given by &?*/2y, whereas C, (t ) decays to 0.25 at long time. Thus, these expressions are expected to be valid only for very hindered rotation. Let us now compare this model with experimental data for CH,I. The Gw, (t) in this model exhibits damped oscillations depending on the relative magnitude of D and y. The experimental AVCFs do not show such strong oscillatory behaviour and decay to zero reasonably fast after the first negative cycle. Thus it will be impossible to fit the calculated GUl(t) to the entire experimental AVCF and hence we determine the parameters Jz and y from least-squares fitting only up to the first negative lobe. The fits to experimental AVCFs at 295 and 215 K are shown in fig. 6. The calculated AVCF exhibits damped oscillation even after the experimental AVCF has decayed to zero. The values of the libration frequency D (in cm-‘) and y (in cm-‘) obtained from least-squares fit to AVCF at different temperatures are given in table 3. The libration frequency increases with decrease in temperature and at 215 K its value is 89.2 cm-‘. This value may be compared with the libration frequencies 94 cm-’ (R,) and 117 cm-’ (R,, R,,) observed for solid CH,I [24,25]. In fig. 7 we have
399
S. K. Deb et al. / Raman scattering study of CH, I Table 3 Parameters of the damped hbrator model as obtained by fitting to the experimental
T (K)
D
295 258 241 215
AVCF
Y
(4
(cm-‘)
(fu)
(cm-t)
8.22 9.11 9.74 10.32
83.2 86.2 89.1 89.2
6.35 6.17 6.22 5.64
64.3 58.4 56.9 48.7
1.0
0.5
z z 0.0
-0.5 0.0
0.5
1.0
t(r.u.1 Fig. 6. Comparison between experimentally determined AVCF ) using the “damped and calculated AVCF ((0 -0) librator model”. Parameters of the model are given in table 3.
shown the comparison with C2(t) at 295 and 215 K for values of 52 and y obtained from fit to AVCF. The calculated C*(t) corresponding to 215 K exhibits a hump characteristic of libration which however is not observed experimentally. The C, (t ) for 295 K always remains higher than the experimental value. This is because the C,(t) and Cr(t) given in eqs. (12) and (13) are applicable for hindered rotation and therefore may not be applicable to the data at 295 K. Rothschild et al. [22] have used this model to explain the infrared dephasing of the vs mode of CH,I and obtained s2 and y as a function of temperature. Over the temperature range 297-210 K, they find 52 to vary from 63 to 68 cm-’ which may be compared with our values of 83-89 cm-’ (table 3). More importantly, however, they conclude that damping increases with decreasing temperature whereas our analysis suggests the reverse trend which is consistent with the physical picture that as the system becomes more solid-like with cooling, libratory motion becomes progressively less damped and eventually it becomes underdamped in the solid phase.
7. Conclusion
1.0
t
(r.u)
2.0
Fig. 7. Cz(t) as function of t. The solid lines are obtained using the “damped librator model”. The experimental results (-0) are also shown for comparison.
We have obtained accurate Cr(t) for CH,I at different temperatures and also in solution of C,H,, at different concentrations. The data have been shown to satisfy the rotational second moment sum rule and this has enabled us to determine the mean-square torque. The variation of C,(t), different correlation times and mean-square torque as a function of temperature and concentration indicate the importance of electric mul-
400
S. K. Deb et al. / Raman scattering study of CH, I
tipolar, particularly that of dipole-dipole interaction in the orientational dynamics of CH,I in liquid. The AVCFs obtained in pure liquid as well as in solution reflect the inadequacy of models for molecular reorientation in liquid based on the extended diffusion model. A simple model based on damped libration in a harmonic well has been shown to provide a reasonable description of certain features of the experimental correlation functions.
given by C,(r)
= a(1 + 3 exp[ -4f(r)]}.
(A-4)
We also obtain the limiting expression for C,( r ) as r --, 0 and r --, co respectively. As r -+ 0, f( r ) + r * and thus C*(r)
= 1 - 6r2/2!
And as r --j 00, f(r) C,(r)
=
1 +
+ .... + 2yr,G2
3 exp( -8yr/L?‘).
(A-5) and hence 64.6)
Acknowledgement
The authors gratefully acknowledge Dr. R.V. Hosur of Tata Institute of Fundamental Research, Bombay, India for providing the NMR data on mixture samples for determining concentrations. Appendix
Here we obtain an expression for the Raman correlation function C,( f ) for a linear molecule undergoing damped libration. This is accomplished by using the Kubo line shape theory and assuming that the angular momentum is a gausSian random process [23]. Then following a procedure similar to that outlined by Dattagupta and Sood [26] and Deb and Bhagwat [27] the Raman correlation function can be written as G(t)
(A.11
= [exp Al,,
where A=
-L;f(t).
(A.2)
Here L, is a 5 X 5 matrix representing the x component of the angular momentum operator in an I= 2 spherical harmonic basis [28] and f(t)
= Jd’t - ~)(w(O)*W(~))
dr.
(A-3)
The expression for f(t) has already been given along with eq. (12) in the text. [ loo denotes the m = m’ = 0 element of the matrix. C*(t) can be obtained by diagonalizing the matrix given in eq. (A.2) using a unitary matrix formed from its eigenvectors. Then it can be shown that C*(t) is
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[25] J.A. Janik, J.M. Jan&, A. Bajorek, K. Parlinski and M. Sudnik-Hrynkiewin, Physica 35 (1967) 451. [26] S. Dattagupta and A.K. 8ood, Z. Physik B44 (1981) 85. (271 SK. Deb and K.V. Bhagwat, Chem. Phys. 98 (1985) 251. [28] R.H. Dicke and J.P. Wittke, Introduction to quantum mechanics (Addison-Wesley, Reading, 1963) p. 191.