J. Phys. Chem. 1994,98, 482-4488
4482
Vibrational Modes of Liquid +Alkanes: Simulated Isotropic Raman Spectra and Band Progressions for C S H ~ ~ - C ~and O HC16Du ~~ D.A. Cates, H.L. Straws, and R. G. Snyder’ Department of Chemistry, University of California, Berkeley, California 94720 Received: November 22, 1993; In Final Form: February 3, 1994”
The isotropic Raman spectra of the liquid n-alkanes C5H12 through C20H.42 along with ClaD34 were calculated using a simple intensity model. The calculated spectra closely resemble the observed spectra. Well-defined patterns in the spectra are revealed when the frequencies of the Raman bands are assembled together for the liquids C 4 2 0 . The patterns observed are associated with the existence of band progressions. These resemble the progressions found in the spectra of the all-trans chains in the crystal. The introduction of conformational disorder into an assembly of ordered chains affects the frequencies of the bands much less than the distribution of intensity. A rationale for this behavior is presented. The vibrational modes of Cl6H34 and Cl6D34 have been characterized, both for the crystal and for the liquid, in terms of the frequency distribution of the potential energy for specific kinds of vibrational modes.
I. Introduction Thevibrational spectra of assemblies of flexible chain molecules in the liquid state, though complex and not well understood,remain a virtually untapped source of information about conformational statistics and structure. To develop methods for analyzing the spectra of such systems, we have focused on the n-alkane~l-~ From the standpoint of vibrational spectroscopy, these chains are among the simplest. In addition,many derivatives of then-alkanes have major importance in biology and material science. Earlier, we presented methods for simulating the isotropic Raman spectra of liquid n-alkanes and demonstrated the application of these methods to (212,c16, and C20.l In the present paper, we have extended this analysis to include the complete set of liquid n-alkanes Cs through CZO. The availability of an uninterrupted series of spectra is critical in revealing spectral patterns that define the chain-length dependencyof the frequencies of the vibrational modes as well as the character of their motion. Patterns of this sort have been previously established for the ordered all-trans n-alkanes that constitute the ~ r y s t a l .A~ major finding of the present work is the surprising similarity of the spectral patterns for the liquid n-alkanes to those for the ordered chains. Liquid C16D34has been also been included so as to provide a test of our methods and because the vibrational spectra of conformationally disordered,deuterated n-alkanes of chain lengths of more than a few carbons have not previously been analyzed.
11. Methods: Simulation and Measurement of Spectra We have used computational procedures similar to those in our earlier studies.lJ Therefore, only a brief account of these will be given. The isotropic Raman spectrum S(v) of a liquid n-alkane is assumed to be the sum of the spectra of its constituent conformers. The ensemble representation of S ( v ) is then MT
where S(v) is the Raman spectrum whose intensity is given in terms of Raman scattering activity, SM(u) is the spectrum of conformer M,and MT is the total number of conformers. a Abstract published in
Advance ACS Abstracts, March 15, 1994.
0022-3654/94/2098-4482%04.50/0
The simulated spectrum is derived from an ensemble made up of conformers generated using a Monte Carlo procedure with conditional probabilitiescalculated on the basis of the rotational isomeric state model.5 The value of E,, the gauche-trans energy difference,was taken as 800 cal/mol, the value determined in our earlier ca1culations.l The significance of the fact that this value is higher than that normally used has been discussed in ref 1. In the present work, we again explored the effect of using values of E, that differed from 800 cal/mol by more than 100 cal/mol. In all cases, the overall fit to the observed spectra was found to be inferior, in agreement with our earlier finding.’ The “pentane interaction” energy ( E d ) was assumed to be 2000 cal/mol as before.l The small fraction of chains that self-overlap was excluded. Each torsion angle defining a trans or a gauche CC bond was allowed to fluctuate independently and randomly about its equilibrium value. The amplitude of the fluctuations was constrained to follow a Gaussian distribution with a root-meansquare deviation of flOO.l It was found necessary to include these fluctuations in order to simulate the combined effects of thermal motion and packing constraints. Their inclusion has a significant effect in broadening the conformationally sensitive bands, so as to bring the shapes of certain spectral features much more into line with the observed shapes. The isotropic Raman intensitieswere calculated using a simple bond polarizability model that included contributions primarily from CC stretchingand CCC bending coordinates. In our version ofthismodel, thescatteringactivityofthekthmodeof aconformer is given by
where ti and L$ are normal-coordinate elements associated with CC stretching (R) and CCC bending ( w ) , respectively. D, is the ratio of the derivative of the mean polarizability for CCC bending to that for CC stretching; that is, D, = a u / &where ~, the & are the mean polarizability derivatives. For the value of this ratio we used 0.295.2 It was necessary to include two additional parameters to reproduce the low values of the observed intensities of the HCH methylene bending ( 6 ) and the HCH methyl umbrella (U) bands.’ (The methylene group vibrations aredefined in Figure 1.) Thevaluesof these parameters, expressed in terms of the ratios &&/&R and (Y’u/&R, are -0.04 and -0.08. Their inclusion has a negligible effect on the intensities of other bands. 0 1994 American Chemical Society
The Journal of Physical Chemistry, Vol. 98, No.16, 1994 4483
Vibrational Modes of Liquid n-Alkanes
+ BEND (SCISSORS)
TWIST
v+ WAG
u
ROCK
Fipw 1. Definition of methylene group vibrations.
The frequencies and normal coordinates were computed for each conformer using a force field determined expressly for disordered n-alkanes.6 The number of conformers used in an ensemblewas varied inversely with the chain length of the n-alkane so that the total number of carbons remained constant, around 2.5 X 104. Under this condition,the average error in thecalculated intensity due to ensemble averaging was estimated to be less than 5%. The simulated Raman spectra were first expressed as 2-cm-linterval histograms, in which intensity was expressed in terms of scatteringactivity S(v). To simulatethe experimentallymeasured spectrum, Z(v), the intensity in each interval was converted to a Lorentzian band centered within the interval. All bands were assigned widths (full width at half-maximum) of 8 cm-1.' Finally, the spectrum was converted from an S(v) to an Z(v) spectrum through the relation
where T1 is the temperature of the sample. (Note that two separate temperatures are involved in the simplations. One is the temperature ( T I )at which the sample is maintained while its Raman spectrum is measured; the other is the temperature (Tz) that characterizes the distribution of conformers. Normally, T1 and T2 are the same, but they need not be.) The observed isotropic Raman spectra of liquid C I ~C, ~ Zand , c 1 6were measured, as described in ref 7, by Dr. James R. Scherer at Western Regional Research Laboratory and were redigitized for display here. All other spectra, including that of C16D34, were measured in our laboratory as described in ref 2.
i
E
100 200 300 400 500 800 100 200 FREQUENCY (cm")
300 400
500
600
CALCULATED
HI. Mscuesion A. Comparison of Simulated and Observed Spectra. The calculated and observed isotropic Raman spectra of the liquid n-alkanes C S - C ~are O shown in Figure 2 for the region 100-600 cm-1 and in Figure 3 for the 600-1400-~m-~region. The agreement between the calculated and observed spectra is generally excellent. All the major features are well reproduced, as are most of the weaker features. The match is especially good for n-alkanes longer than about Cs.The Raman intensity in the lo(MOO-cm-l low-frequency region has its origin in both CC stretching and CCC bending motions, while that in the 6001400-cm-1 region comes almost entirely from CC stretching. A more detailed discussion of the intensity distribution and other features of these spectra may be found in ref 1. The quality of the fit shows no significant trend as the length of the n-alkanes increases. The conformational statistics for the n-alkanes C S - C ~thus O appear to be nearly independent of chain length, in keeping with what we found in our earlier, more limited analyses.l.2 The quality of the match between the observed and calculated spectra varies between different frequency regions. The lowfrequency region 100-175 cm-' is one of two especially troublesome regions. A broad band is observed here that is not present in the calculated spectrum. Its intensityincreaseswith increasing chain length. Since this band appears in a frequency region dominated completely by torsional modes, we explored the
100
200 300 400
500 FREQUENCY (cm")
Figure 2. Observed and calculated isotropic Raman spectra of the n-alkanl ClHlz through C&z in the rcgion 100400 cm-1: ,(The asterisks denote spectra that have been redigitid from the ongmal.) Spectral resolution in cm-1: CS.4.0; C6-Clh 3.0; C11 and CIS,2.0; C I ~ cis, 2.5; cis, 2.0; Ci&z& 2.5.
4484 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994
I
CALCULATED
JJJ( ) ,
u
600
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1400 so0 400 FREQUENCY (cm )
1000
1200
1400
CALCULATED
600
800
1000
1200
1400 600 800 FREQUENCY (cm”)
1000
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1400
Figure 3. Observed and calculated isotropic Raman spectra of the n-alkanes CsH12 through CmH42 in the region 600-1400 an-’.(An asterisk denotes a spectrum that has been redigitizedfrom the original.) The spectral resolution is the same as that indicated for Figure 1 .
Cates et al. possibility that it might be associated with these modes. Torsion about a gauche CC bond (but not about a trans bond) is allowed by the local symmetry of this bond to contribute intensity to the isotropic spectrum. The introductionof a polarizability parameter associated with torsion about gauche bonds did not, however, reproduce the observed band in the calculated spectrum. Another possibility is that the intensity observed in the 100175-cm-l region arises from more or less localized interchain modes having the character of chain translation perpendicular to the chain direction. The isotropic scattering activity developed from fluctuations in the lateral distance between neighboring chain segments would, of course, be expected to be intrinsically small. However, as eq 3 shows, the low frequency of these modes would greatly enhance their observed intensity. In support of this assignment, we note that those lattice modes of crystalline polyethylene that involve lateral translation of the chains-modes therefore closely related to the type of vibrations proposed to occur in the liquid-have frequencies of the same order of magnitude as that associated with the unexplained feature in the spectra of the liquid n-alkanes. The translational bands of crystalline polyethylene are found near 80 and 109 cm-1 for this polymer at 2 K.8 The second problem area is the region 1175-1 300 cm-1, where calculated intensities tend to be relatively too high. Our model in its simplest form allows intensity in this region to come only from C-C stretching and CCC bending. However, methylene wagging, rocking, and twistingmotions are also present and could in principle contribute intensity. In an attempt to improve the fit, we included methylene wagging. Local symmetry forbids methylenes flanked by trans bonds from contributing, but not methylenes adjoined by one or two gauche bonds. With appropriate wagging polarizability parameters, we were able to improve the fit in this region, but only at the expense of making it worse elsewhere. We did not explore the second possibility, that of including contributions from the methylene rocks. The observed and calculated Raman spectra of liquid C16D34 are shown in Figure 4. Overall, the agreement is excellent, though it may not be quite as good as for the hydrogenated chain. This would not be surprising since the valence force field used for C16D34is derived from the vibrational frequenciesof hydrogenated n-alkanes6 and is therefore more suited to the latter. The spectrum of C16D34 is discussed further in section 1II.C. B. Band hogressions. The isotropic Raman spectra of the liquid n-alkanes show band progressions. For any given spectrum, however, the evidence for progressionsis rather tenuous, consisting of an occasional complex of a few more or less evenly-spaced bands. When the spectra of a series of liquid n-alkanes are assembled together, as shown in Figures 5 and 6,well-defined patterns become apparent. These patterns come about because a progressionband for some given n-alkaneundergoes a systematic shift in frequency in going to the next longer or next shorter n-alkane! Band patterns indicating the existence of progressions for the liquid n-alkanes have also been observed in the infrared.6 In the case of the all-trans chain in crystalline n-alkanes, almost all the bands in the infrared and Raman spectra are associated with progressions. The analysisof these bands has led to detailed assignments, which in turn have led to the determination of dispersion curves499 and vibrational force fields.10.11 The existence of progressions in the spectra of the liquid may at first sight seem surprising. Their origin lies in corresponding progressions in the density of vibrational states. The latter progressions tend to persist in going from the crystalline to the liquidstate, in spite of a high degreeof conformationalrandomness in the liquid. It turns out, as we will now see, that much of the intrachain coupling is maintained in going from the all-trans chain to a chain that represents an average of the chains in the liquid. To illustrate this for a simple system, we will consider an ensemble of model chains that are capable of mimicking conformationaldisorder. Earlier, we have employed such a chain
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4485
Vibrational Modes of Liquid n-Alkanes
A, = a
+ 26 cos &
(5)
where Ak = V k Z , & = ku/(n + I), and k = 1, 2, 3, ..., n, with n being the number of oscillators. Except for the lowest-frequency modes, we can assume a >> b, so that to a good approximation
vk = vo(l
+ b/a cos 4&)
(6)
This equation represents a progression whose bands are centered near VO. If n is not too small, say >lo, the progression extends from approximately YO( 1- b / a ) to yo( 1 b/a), a frequency range of about 2bvo/a. The situation for an ensembleof disorderedchains is somewhat similar if we consider averages. For a disordered chain, the value of br in eq 4 is stochastically determined for each bond. We will assume that b, can have one of two values, which corresponds to a trans or a gauche bond. The dihedral angle, 71, defining the relative orientation of adjoining oscillators is assumed to be 180° or &6O0, so as to correspond to a trans or gauche bond in an n-alkane. The value of bi is proportional to cos T ~ .The cosine dependence corresponds to the dihedral angle dependence of the inverse kinetic energy interaction term between nearest-neighbor internal coordinates.2.13 The kinetic energy interactionis generally much larger than the correspondingpotential energy interaction, so we can neglect the latter. The ratio b,/bt therefore equals -1/2. We will now replace the bi in the H matrix (eq 4) with 6, an average value given by
+
100
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Frequency (cm”)
Calculated
1
600 700 800 900 1000 1100 1200 1300 1 0
Frequency (cm”) Figure 4. Observed and calculated isotropic Raman spectra of liquid ClsD34 at rwm temperature (spectral resolution 2.4 cm-l). to characterize the distribution of infrared intensity in an assembly of disordered chains.12 This model chain consists of a collinear sequence of identical harmonic oscillators with nearest-neighbor coupling. The degree of coupling between adjoining oscillators is determined by their relative orientation. The frequenciesof this chain are obtained from the eigenvalues of the matrix H defined
0
1
‘J
L O * . . The diagonal element, a, is equal to VOZ, where vo is the frequency of an uncoupled oscillator. For an ordered chain, that is, for a chain in which br= b for all i bonds, the frequency V k of the kth vibrational mode is given by
wherept and p , are the probabilities for the Occurrence of trans and gauche bonds and where bt and b, are the corresponding interaction constants. We have used the absolute values of the interactions in eq 7 to obtain the 5, since the values of the eigenvalues of H are independent of the signs of the bi0l2J4To estimate 6 for the liquid, we have assumed the value ofp, of 0.65 calculated using E, and EBBequal to 600 and 2000 cal/mol, respectively. According to our model, the value of the ratio 6/lbtl is reduced from 1.00 to about 0.82 in going from the crystal to the liquid. As a result, although the center of the progression remains unchanged in frequency, the total frequency span of the progression is reduced by about 20%. The contraction in the experimentally observed frequency span is, however, generally less than that predicted by the model. For example,the frequency range of the wag progression for the ordered chain is about 200 cm-I. The model predicts that, in going to the liquid, the range would be reduced by about 40 cm-l, with each end shifting symmetrically inward about 20 cm-I. The observed reduction is near zero (see below). Paramount among a number of possible reasons for this discrepancy is the neglect in the model of any statistical fluctuations in the distribution of trans/gauche sequences. Such fluctuations would allow appreciable concentrations of trans-bond runs long enough to havevibrational properties similar to those of the ordered chain. The presence of these sequences would have the effect of extendingthe frequencyrange of the density of states to near that of the ordered chain. To describethe changes in the frequency distributionof a progression band in going from an ordered to a disordered system, it would be more appropriate to do so in terms of the second moment of the distribution rather than a simple reduction of span. Unlike band frequencies,band intensities may undergo drastic changes in going from the all-trans chain to an assembly of disordered chains. The reason is that, in going from a trans to a gauche bond, the sign of the coupling constant is reversed and, as a result, the signs of many of the eigenvector elements are also reversed. This tends to scramble the phase relations between the local dipole moment derivatives for the infrared spectrum or, likewise, between the polarizability derivatives for the Raman
4486
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 -
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there isa redistributionof intensitydependson the typeofvibration involved. The critical factor is how the direction of the local dipole moment derivative in the infrared case (or the orientation of the local polarizability derivative in the Raman case) is affected by a change in the local conformation. This matter is discussed in some detail in ref 12. In interesting and useful papers relevant to the present one, Mendelsohn and c o - w ~ r k e rreported s~~ on the use of the infrared wagging band progression to estimate conformational disorder in the acyl chains in phospholipid bilayers. They showed that with increasing disorder, induced by heating the sample, the intensities of the wagging bands diminish, while their frequencies are nearly unaffected. A question of importance is whether the progression bands represent all-trans chains, as the authors assumed, or perhaps also include a significant concentration of conformationally disordered chains. That the latter might be the case is suggested by the following. The wagging progressions that appear in both theinfrared and Raman spectra are a reflection of a wagging mode progression in the density of states. The density-of-states progression persists even when the assembly of all-trans chains becomes conformationally disordered. Consequently, the progressions in the observed spectra also tend to persist. This behavior is evident in the wagging mode region of the infrared spectra of the liquid n-alkanes. In Figure 7, the frequencies of the first four wagging modes (k = 1-4) of the all-trans chain are found to coincide or nearly coincide with the frequencies of the most intense bands in the spectrum of the liquid. In this case, the wagging bands making up the progression must be attributed entirely to disordered chains since there is a negligible concentration of all-trans chains in the liquid. This analogy between the lipids and n-alkanes may be complicated by the fact that the origin of the intensities of the wagging modes is probably different for the two kinds of molecules. Further study is needed to clarify this matter. The isotropic Raman spectra of the liquid n-alkanes show a constant frequency band near 1374 cm-l. This band appears to represent one of the constant-frequency, localized wagging modes that have been assigned to specific conformational sequences, such as gtg, gtg', and gg.6 To date, bands associated with these sequences have been identified in the infrared spectrum, but not in the Raman. These infrared bands have been widely used to diagnose the conformation of the polymethylene chains that are found in many types of systems. The observation of a similar band in the Raman is therefore significant. By virtue of its frequency, the Raman band a t 1374cm-1 appears to beclosely related to the infrared 1370-cm-1 band. In an attempt to assign the Raman band, a need to clarify our earlier assignment
The Journal of Physical Chemistry, Vol. 98, No. 16, 1994 4487
Vibrational Modes of Liquid n-Alkanes
I
All-trans Cl,”,
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were sufficiently large. In this case the inversion center would be, in effect, partially removed. In recent calculationsaimed at simulating the infrared spectrum of conformationally disordered polymethylene chains, we found that the amplitudes of the C-C torsions are too small to allow the gtg’ wagging mode to appear in the infrared with any significant intensity.16 In our present isotropic Raman simulations, we found that thegtg wagging mode is not likely tocontribute significantly to the Raman spectrum for the same reason. Thus, selection rules based on local symmetry arguments appear to work well and to lead to a straightforward assignment of the Raman (1374 cm-l) and infrared (1 370 cm-1) bands to different sequences: to gtg’ and gtg, respectively. That these bands have different origins is also suggested by their having somewhat different frequencies. In conclusion, then, contrary to what has been sometimes assumed in the literature, the 1370-cm-linfrared band is a measure of gtg, not of gtg’ (kink), sequences. It is the Raman 1374-cm-1 band that is associated with kinks. It should be mentioned that, in application to recently reported infrared spectra of phospholipidbilayers in their disorderedstate,” our assignments indicate a relatively high concentration of gtg sequences, although from packing considerations alone, it would seem that gtg’ (kink) sequences should be favored. Finally, we consider a special class of low-frequency modes that respond to changes in conformational disorder differently from the modes so far discussed. These modes are closely related to longitudinal acoustic modes (LAM). They behave differently because the condition that a >> b in eq 4 is no longer met. The behavior of their frequencies with respect to chain length changes much in going from the crystal to the liquid. These bands are marked in Figure 5, where, for the all-trans chains, the most intense band of this type is designated LAM- 1 and, for the liquid, D-LAM. The frequenciesof the LAM- 1bands are linearlyrelated to the inverse of the chain length; that is, v(LAM-1) = A/n, where A is a constant related to the longitudinal elastic modulus of the extended chain. The band for the liquid that is the equivalent of the LAM-1 band is the highly inhomogeneous D-LAM band.18J9 The frequency of D-LAM is related to the inverse of the square of the chain length; that is, v(D-LAM) = YO + A’/n2. As n becomes larger, v(D-LAM) approaches a constant value. In this respect the D-LAM bands, in contrast with the LAM-1 bands, mimic the behavior of optic modes, not acoustic modes.19 C. Frequency Distribution of the Potential Energy: CE and CyQ As we have previously noted, it appears that the vibrations of long perdeuterated n-alkanes have not been previously described in the literature in any detail. This is the case for both the all-trans chain and the liquid. A convenient and succinct description of the normal vibrations of C&4, which we have taken to represent perdeuterated n-alkanes, is provided by the frequency distribution of potential energy (FDPE). The FDPE consists of the potential energy-separated into contributions from each of the various kinds of internal coordinates and summed over all conformers-plotted as a function of frequency. To generate the FDPE, we first evaluate the potential energy of the individual modes. The potential energy distribution of the kth mode of a conformer is given by
H
0
200
400
600
600
1000
1200
1400
1600
Frequency (cm”)
Figure 8. Frequency distribution of the potential energy, Pj(vr),for the vibrational modes of all-trans (crystalline) C1&4 and C16D34. In each distribution the pcak heights shown have been multiplied by the indicated scale factor.
of the 1370-cm-I infrared band became apparent. The infrared band was originally attributed to both gtg and gtg’ (kink) sequencesbecause the frequenciesof the waggingmodes calculated for these two sequences were found to be essentially the same (1369 f 4 cm-1) and very near to the observed infrared frequency (1360 cm-l).6 It is important to note, however, that the assignment was originally presented with the caveat that, strictly speaking, the wagging mode associated with the gtg‘ “kink” should not appear in the infrared spectrum, since the gtg’ sequence has a local center of symmetry that forbids it from appearing. However, it seemed that such a band might appear if the amplitude of torsional motion about the central C-C band of the gtg’ sequence
where i refers to individual internal coordinates, Lik is the ith element of the kth normal coordinate, Fir is the diagonal force constant associated with i, and Vk is the frequency of the mode. The quantity pl;f is then defined to represent, for mode k of conformer M , the sum of the whose internal coordinate i belongs to a class of coordinates of typej , for example, methylene wagging. To obtain the frequency distribution of the F$, the
e
Cates et ai.
44811 The Journal of Physical Chemistry, Vol. 98, No. 16, 1994
Figures 8 and 9 show for CE(C,,H,,) and Cf,(C,,D,,) the FDPE for the major coordinate types, plotted for width intervals of 2 cm-I. Much of what we found in our earlier analyses of the all-trans hydrogenated n-alkanes is summarized in Figure 9 in the FDPE for CE.10,’4 Likewise, the FDPE for Cy6, in the form of the all-trans chain and as a liquid, will serve to characterize the vibrational modes of long perdeuterated n-alkanes in general. We limit our discussion here to a brief comparison of the FDPE of Cy, and C z .
Scissors
Liquid
I
x2
Twist
First, for the all-trans chain, we note that for some types of modes the FDPE for Cy6 is quite similar to that for C i . This is true for CCC bending and CC torsion and to some extent for methylene twisting and rocking. However, the FDPE of C z and Cy, differ significantly for methylene scissoring, wagging, and CC stretching. These modes increasingly mix as their frequencies come into proximity. This occurs because all these modes are symmetric with respect to the skeletal plane defined by the carbon atoms; that is, all are in-plane. The mixing is much greater for Cy, than for C z . The scissors, wagging, and CC stretching modes (ordered here in terms of decreasing frequency) are for C z reasonably separated in frequency. In going from CE to Cy,, the scissors and wagging modes move to lower frequencies and, as a result, now overlap, or nearly overlap, the CC stretches, whose frequencies undergo a much smaller downward shift. The FDPE for Cy, is therefore more complex than for C z , This is evident in Figure 8.
x4
h
>
Y
d’
CC Tors
0
200
400
600
800
x33
1000
1200
1400
1
Frequency (cm”)
Twist
A
x4
The FDPE for liquid and c:,, displayed in Figure 9, are similar to the corresponding FDPE for the ordered, all-trans chain. This is consistent with the similarities found in our earlier comparison of the vibrations of ordered and disordered chains. The frequency regions in which modes of a given type occur are much the same for the crystal and liquid, progressions are clearly in evidence in the FDPE of the liquid, and the band spacings in these progressions are similar to those for the ordered chain.
I
Acknowledgment. We gratefully acknowledge support of this work by the National Institutes of Health (Grant GM 27690). References and Notes
CC Tors
0
200
400
600
800
1000
1200
1400
1600
Frequency (cm”)
Figure 9. Frequency distribution of the potential energy, P , ( Y ~ )for , the vibrational modes of liquid Cl6H34 and C&4. In each distribution the peak heights shown have been multiplied by the indicated scale factor.
spectrum is divided into equal frequency intervals YJ (C = 1 , 2, 3, ...), and the $for all modes from all conformers are disposed into their appropriate intervals according to the frequency V& of each mode. The resulting FDPE may be expressed (9)
where NM is the total number of conformers.
(1) Snyder, R. G. J . Chem. Soc., Faraday Trans. 1992,88, 1823. (2) Snyder, R. G.; Kim, Yeswk J . Phys. Chem. 1991, 95,602. (3) Hallmark, V. M.; Bohan, S. P.; Strauss, H. L.; Snyder, R. G. Macromolecules 1991, 24, 4025. (4) Snyder, R. G.; Schachtschneider, J. H. Spectrochim. Acta 1963,19, 85. (5) Flory, P. J. Statistical Mechanics of Chain Molecules; Wiley-Interscience: New York, 1969. (6) Snyder, R. G. J . Chem. Phys. 1967,47, 1316. (7) Scherer, J. R.; Snyder, R. G. J. Chem. Phys. 1980, 72, 5798. (8) Dean, G. D.; Martin, D. H. Chem. Phys. Lett. 1967, 1 , 415. (9) Tasumi, M.;Shimanouchi,T.;Miyazawa,T. J. Mol.Specrrosc. 1962, 9, 261. (10) Schachtschneider, J. H.; Snyder, R. G. Spectrochim. Acta 1963,19, 117. ( 1 1) Harada, I.; Takeuchi, H.; Sakakibaro, M.; Matsuura, M.; Shimanouchi, T. Bull. Chem. SOC.Jpn. 1977,50, 102. (12) Snyder, R. G. Macromolecules 1990, 23,2081. (13) Wilson, E. B.; Decius, J. C.; Cross, P. C. Molecular Vibrations; McGraw-Hill: New York, 1955. (14) Parlett, B. N. The Symmetric Eigenualue Problem; Prentice-Hall: Englewood Cliffs, NJ, 1980. (15) Chia,N.-C.;Mendelsohn,R. J . Phys. Chem. 1992,96,10543. Senak, L.; Moore, D.; Mendelsohn, R. J . Phys. Chem. 1992, 96, 2749. (16) Cates, D. A.; Strauss, H. L.; Snyder, R. G. To be published. (17) Senak, L.; Davis, M. A.; Mendelsohn, R. J . Phys. Chem. 1991.95, 2565. Casal, L. H.; McElhaney, R. N. Biochemistry 1990, 29, 5423. (18) Snyder, R. G. J . Chem. Phys. 1982, 76, 3921. (19) Snyder, R. G.; Strauss, H. L. J . Chem. Phys. 1987,87, 3779.
