J . Phys. Chem. 1986, 90, 5623-5630
5623
Temperature and Phase Behavior of Infrared Intensities: The Poly(methy1ene) Chain R. G. Snyder,* M. Maroncelli? H. L. Strauss, and V. M. Hallmark Department of Chemistry, University of California, Berkeley, California 94720 (Received: March 27, 1986)
The infrared band intensities of crystalline n-alkanes and polyethylene decrease nonlinearly with increasing temperature. For all modes except the C-H stretches, the decrease is large and far exceeds that expected from density and refractive index effects. Similar anomalous decreases occur for the odd n-alkanes at their principal solid-solid (orthorhombic-to-hexagonal) phase transition. Further decreases occur in going to the liquid and gas phase. The intensities of the methylene bending and rocking fundamentals for the gas at 300 K are about I / , those of the crystalline solid at 77 K. A good correlation between the temperature coefficients of the intensity and of the lateral expansion is found for the crystal. This relation suggests that low-frequency modes play an important role in determining the temperature behavior of intensities. A mechanism involving low-frequency modes is proposed that appears to qualitatively explain our experimental results. The sensitivity of intensities to temperature and phase must be taken into account in infrared studies of poly(methy1ene) chain systems and in the transfer of observed and calculated gas-phase intensities to the condensed state. Similar temperature behavior is expected for Raman intensities and for other flexible chain molecules.
Introduction Infrared intensities are defined experimentally by 1
I =
bp
SbandAVdv
(1)
where b is the pathlength in the sample, p is the sample density, and A, is the absorbance (to the base e ) at frequency v in wavenumbers. The measurement of infrared intensities has been largely confined to small molecules in the gas phase. Large molecules have received relatively little attention and the temperature behavior of their intensities even less. Yet, in an obvious way the temperature behavior of infrared intensities in the condensed state bears directly on the use of infrared spectroscopy for quantitative analysis and structural investigation in cases where temperature is a variable. The present work concerns the poly(methylene) chain and thus is relevant to studies dealing with how temperature affects the structure of chain-molecule assemblies, such as lipid bilayers, micelles, and hydrocarbon polymers. Such studies have become common by virtue of modern FTIR instrumentation, but the question of how intensities are intrinsically affected by temperature has remained essentially unaddressed. The paucity of experimental studies on solids is probably partly accounted for by the expectation of only a small intrinsic temperature effect. A priori, such an expectation might seem well founded since the Einstein coefficients, which determine the intensity of a spectral line associated with the transition between two isolated energy levels, do not change with temperature. In the near- and mid-infrared regions, small intensity changes with temperature would be expected from the Boltzmann factors and from changes in the density of the sample. However, as we shall show, these factors fall far short of accounting for the large effects that are observed. The present study concerns the temperature and phase behavior of the infrared intensities of the prominent bands of the poly(methylene) chain. The subject attracted our attention during the course of work on phase transitions and conformational disorder in crystalline n-alkanes.'12 We noted what seemed to be surprisingly large changes in the intensities of the methylene rocking-mode bands when the temperature of the solid n-alkane was changed. In addition, unexpectedly large intensity changes occurred at the principal solid-solid phase transition. The magnitude of these intensity changes seemed especially surprising in view of the nonpolar nature of the molecules. Subsequently, we extended our solid-state measurements to cover a much wider temperature range and in addition have determined intensities for the gas phase. This collection of intensity data, combined with similar data for the liquid that is available from the literature, Current address: Department of Chemistry, University of Chicago, Chicago, IL 60637.
0022-3654/86/2090-5623$01.50/0
provides a uniquely comprehensive picture for a single system. In what follows, we review briefly some results from earlier studies on the temperature behavior of infrared intensities for the condensed state and describe our own measurements on solid n-alkanes and polyethylene (PE). Then we discuss an observed relation between the temperature dependencies of infrared intensities and the lateral density of the crystalline solid. After comparing the infrared intensities for the solid, liquid, and gas phases of the polymethylene chain, we discuss the difficulty of accounting for our findings in terms of current theory. Finally, we propose a mechanism that appears to account at least qualitatively for the observed sensitivity of infrared intensities to temperature and physical state. We will be primarily concerned with three regions of the infrared spectrum of the poly(methy1ene) chain. These regions, in which the most intense bands occur, are indicated in Figure 1 for PE. Each is associated with a different type of methylene vibration: C-H stretching, 3000-2750 cm-'; H C H bending, 1500-1420 cm-'; C H 2 rocking, 750-690 cm-I. In referring to individual bands as well as to the modes that give rise to them, it is convenient to follow the notation of ref 3. Thus d", 6, and P are used to indicate C-H stretching (which includes both the symmetric and antisymmetric stretching modes), bending, and rocking, respectively. In the case of the crystalline n-alkanes, a subscript k may be attached to the P to indicate that the mode is the kth member of the rocking-mode progression. For example, P1, refers to the intense fundamental band at 720 cm-l (see ref 3).
Background The temperature dependence of the infrared intensities of small molecules has been reviewed by Person and Steele.4 Gases and liquids, but not solids, are discussed. For liquids, infrared intensities usually decrease with increasing temperature. The changes are reported to be linear with temperature, but the intensity measurements are usually confined to a relatively small temperature range. The observed temperature coefficients can be dramatically different for bands from different molecules as well as for bands from the same molecule. If the intensity change is relatively small, it can sometimes be explained by the change in the refractive index, which determines the internal field of the m o l e c ~ l e s .However, ~ in many cases, the refractive index change accounts for less than one-third of the observed intensity change. Other factors are therefore also involved. (1) Snyder, R. G.; Maroncelli, M.; Qi, S. P.; Strauss, H. L. Science Wushington, D.C. 1981, 214, 188-190. ( 2 ) Maroncelli, M.; Qi, S. P.; Strauss, H. L.; Snyder, R. G. J . Am. Chem. SOC.1982, 104, 6237-6247. (3) Snyder, R. G.; Schachtschneider, J. H. Spectrochim. Acta 1963, 19, 85-116. (4) Person, W. B.; Steele, D. Mol. Spectrosc. 1974, 2, 357-438.
0 1986 American Chemical Society
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986
known pressure by evacuating the cell and then placing it in pressure equilibrium with the liquid n-alkane. Sample pressure was then determined by adjusting the temperature of the liquid. Gas-phase intensities were obtained from the slope of a plot of observed intensity vs. pathlength for a given sample vapor pressure. Unless otherwise indicated, the integration was extended over the entire region of measurable absorption associated with the given type of vibration. For example, the intensity of the C-H stretches was obtained by integrating from 2750 to 3000 cm-’. In some instances, well-resolved bands of the crystal were measured individually.
CH2
Bending
CH2
Rocking
-\
I I
I
I
I
I
1400 Frequency (cm-’) Figure 1. Infrared spectrum of polyethylene at 300 K. 3000
2200
6
For solids, studies on the temperature dependence of intensities are practically nonexistent. Most of the data that does exist has come from infrared studies on polymers, whose relatively high melting points and low vapor pressures enable measurements to be made over a significantly wider temperature range than is usually possible for small molecules. These studies have revealed significant temperature effects, although in nearly all cases intensity changes are measured in terms of changes in the peak absorbance. Okada and Mandelkerq5 for example, have reported for PE a large negative temperature coefficient for the absorptivity of the 1894-cm-I band6 that is commonly used to measure the crystallinity of solid PE. These authors noted that the intense 720-cm-l fundamental of n-C94Hl, has a similarly large negative temperature dependence. Zackmann and Stuart7 also found this to be the case for the same band in the spectrum of n-C28HS8. Recently, the temperature dependence of the peak intensities of many of the bands of semicrystalline PE in the temperature range 100-300 K have been reported.8 Finally, we note that the temperature dependence of infrared intensities for polymers has been utilized to detect solid-solid transitions and to establish glass transition temperatures, since at the glass transition temperature there is a change of slope in a plot of peak intensity vs. temperature. Transitions in poly(ethylene polystyrene,I0 polyacrylonitrile,’ and poly(viny1 chloride)’* have been studied in this way.
,
Experimental Procedures All infrared spectra were measured at a resolution of 1 cm-I with an evacuatable Nicolet Model 8000 FTIR spectrometer equipped with TGS and cooled MCT infrared detectors. Two methods of temperature control for the solids were used. For temperatures in the range from about 0 to 100 OC, the sample was housed in a copper-block assembly through which a thermostated liquid was circulated (see ref 2 for details). For the 7-300 K range, the sample was cooled in a Lake Shore Cryotonics CTI Model 21 closed-cycle helium refrigerator equipped with a silicon diode thermometer and a Lake Shore temperature controller. Gas-phase spectra were measured with a Wilkes variable pathlength multipass cell. The cell was filled with vapor at a (5) Okada, T.; Mandelkern, L.J . Polym. Sci., Part A-2 1967, 239-262. (6) Snyder, R. G. J. Chem. Phys. 1978, 68, 4156-4166. (7) Zachmann, H. G.; Stuart, H. A. Macromol. Chem. 1%1,44,622-642. (8) Dahme, A.; Dechant, J. Aria Polym. 1982, 33, 490-494, 546-549. (9) Hannon, M. J.; Koenig, J. L. J . Polym. Sci., Parr A-2 1969, 7 , 1085-1 099. (IO) Huang, Y . S.; Koenig, J. L. J . Appl. Polym. Sci. 1971, 15, 1237-1 245. (1 l)-Ogura, K.; Kawamura, S.; Sobue, H. Macromolecules 1971,4,79-81. (12) Ogura, K. Er. Polym. J . 1975, 7 , 221-223. ~
Snyder et al.
Tbe Solid State The temperature behavior of some infrared intensities for the solid n-alkanes and polyethylene is examined in this section. Two kinds of intensity changes were measured. One is the change with temperature, which is expressed through the temperature coefficient a In I a=dT where I is the integrated intensity and T the temperature. The second kind of change is that which occurs in going through a solid-solid phase transition. This is measured by d, defined d = -AI
(3)
10
where I, is the intensity that is observed just below the transition temperature and IIjust above, so that AI = I, - I , . We will be concerned only with the principal solid-solid transition, which occurs when an odd-numbered n-alkane goes from phase I, the low-temperature orthorhombic form, to phase 11, the high-temperature, so-called “hexagonal” or “rotator” form.2 In the second part of this section, we discuss a relation between the thermal coefficients of intensity and lateral expansion for the crystal. The latter is expressed by the temperature coefficient p=- d In ab (4) aT where a and b are the dimensions of the orthorhombic unit cell. Observed Temperature Behavior of Infrared Intensities. The accuracy of the measured intensities and of the temperature coefficients derived from them varies but is sufficient to establish trends with temperature. In general, accuracy is less at low temperatures because the narrowness (fwhh < 2.0 cm-’) of the methylene bend and rock bands makes it difficult to maintain a g o d signal-to-noise ratio and at the same time provide the high spectral resolution necessary for accuracy. PE from 7 to 300 K . The sample was prepared directly on a KBr window by evaporation from a hot p-xylene solution of linear PE of molecular weight around lo5. Spectra were measured every 50 K over temperatures ranging from 7 to 300 K. Intensities of the entire C-H stretching region (3000-2750 cm-I), the bending bands near 1465 cm-’, and the rocking bands near 720 cm-] were obtained by direct integration. Their values are plotted against temperature in Figure 2. The values of the temperature coefficient, a,are given in Figure 3. The latter may be in error by 50% or more in the region below 200 K and 30% or more above 200 K. It is apparent from Figure 3 that a is a nonlinear function of temperature and that its value for the C-H stretches is significantly less than for the bends and rocks. The values of a for the bends and rocks are similar. For both modes, there is a rapid change in the temperature region near 250 K. n-C,, from 7 to 300 K. The n-C21H44sample used was of high purity, 99.9% as determined by gas chromatography and mass spectral a n a l y ~ i s . ~ .The ’ ~ solid-solid transition temperature and the melting point are 32.6 and 40.2 “C, respectively.2 The spectrum of Czl was measured with the sample in a pressed KBr (13) Schaerer, A. A,; Busso, C. J.; Smith, A. E.; Skinner, L. B. J. Am. Chem. SOC.1955, 77, 2017-2019.
Temperature and Phase Behavior of Infrared Intensities
.
O
t
D
C-H
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986 5625
I
IOOSS O 3 9
C-H
Stretching (total1
Stretching (total)
80
.f L
-ao
1 ,
a IOOS
0
-
0 ’
IO0
X)O
300
Temperoture ( K )
,
,
,
,
,
u
80 -
Figure 2. Infrared intensities of the methylene C-H stretching, bending, and rocking bands of polyethylene from 7 to 300 K. 60 0
100
200
3 00
Temperature (K)
Figure 4. Infrared intensities of the methylene C-H stretching, bending, and rocking bands of n-C2,HUfrom 7 to 300 K.
Temperature (K) Figure 3. Temperature coefficient, a, of the C-H stretching (d*), bending (a), and rocking (P) bands of polyethylene from 50 to 370 K based on intensity data from 7 to 380 K. Two sets of data are represented. The low-temperature data points are connected by solid lines, the high-temperature by dashed lines (see text).
disk. Temperature plots of the integrated intensities for phase I and the values of a derived from them are shown in Figures 4 and 5, respectively. Overall, the temperature behavior of a for CZlis similar to that for PE: the dependence on T i s nonlinear and a is smallest for the C-H stretching bands. On the other hand, the values of a for CZltend to be significantly larger than those for the corresponding bands of PE, especially at higher temperatures. CZlalso differs from PE in that for CZIthe value of a for the rocking mode is significantly larger than that for the bending mode at temperatures above 175 K. PE from 280 to 375 K. The PE film used in these measurements was prepared in essentially the same way as for the film used in the low-temperature meaurements described above, except that the solvent was tetrachloroethylene. Spectra were measured at 10-20 K intervals. The C-H stretches were not measured. The values of a for the bends and rocks, which are plotted against temperature in Figure 3, were derived from intensities obtained through band-fitting procedures. However, direct integration gave similar values. Near 300 K, the values of a are similar to those found in the low-temperature measurements on PE. It should be mentioned that we found a complication for P E that does not exist for the n-alkanes. For both P E and the nalkanes in their orthorhombic crystalline form, the bending and
Temperature (Kl Figure 5. Temperature coefficient, a, of the C-H stretching (d*), bending ( 6 ) , and rocking (P) bands of n-C21H44 from 50 to 300 K.
rocking fundamentals appear as narrow doublets because of interchain coupling. However, in the case of PE, the existence of a third, somewhat broader component was revealed in the curve-fitting procedure used to determine intensities. This third component, which is shown in Figure 6, exists for both the bending and rocking fundamentals. In our intensity measurements we have included all three components. However, the temperature dependence of the third component is different from that of the two crystal-split components. The intensity of the third component
5626 The Journal of Physical Chemistry, Vol, 90, No. 22, 1986
Snyder et al.
1
1
: I
i
0
.
\,
1500
1480
1460
1440
1420
Rocking
760
740
720
700
Phose II 36.5'C
680
Frequency (cm-')
Figure 6. Components of the observed infrared bending and rocking fundamental bands of polyethylene at 97 "C. The two crystalline components are marked by c; the third component is marked by an asterisk. 0.35-
,z 0.30z-
co C
I E
0
c
.- 0.25c
u
l i
0.20-
-20
0
20
40
60
80
100
Temperoture ("c)
Figure 7. Temperature dependence of the fractional intensity of the 3rd component (1465 cm-I) of the bending mode of semicrystalline poly-
ethylene. The numbers indicate the order in which the measurements were made. (Fractional intensity = Il4,5/(I)4,5 + 11463 + See Figure 6.) increases with increasing temperature, in contrast to what we have observed for all bands associated with the crystalline state. The relative intensity of the third component of the bending band (1465 cm-') is plotted against temperature in Figure 7. The temperature of the PE sample was changed between measurements in the chronological order indicated in the figure. As the temperature is increased to near 270 K, the intensity of the third component band begins to increase at the expense of the crystal doublet, and at 371 K. the hiehest temDerature attained, it has become 34% of the total intenity of all t'hree bands. Lowering the temperature
1
1000
,
900
800 700 Frequency (cm-')
600
Figure 8. Infrared spectra of crystalline n-C21H44in the rocking-mode region for the n-alkane in phase I and phase 11. The absorptivity scale is the same for both spectra. All the bands in this region of the spectrum of phase I are methylene rocking modes except for the weak band near 975 cm-I and the band near 890 cm-l, the latter of which is an in-plane methyl rocking mode. The weak bands in the phase I1 spectrum that do not appear in phase I are mostly defect-induced methylene rocking modes.2
results in an apparently reversible intensity decrease. A similar behavior is observed for the third component of the rocking-mode band (720 cm-I). The third component is apparently unique to PE since similar measurements on the temperature behavior of n-C94H190 failed to reveal a third component under either the bend or the rock doublets. Additional measurements are needed to determine the origin of this band. n-Alkanes from about 10-30 K below T, Up to T,. The most accurate data on the temperature behavior of infrared intensities were measured for the rocking modes of the odd-numbered crystalline n-alkanes ( n = 17-29) in the relatively narrow temperature range beginning about 10-30 K below T, and ending just below the melting point. The spectra upon which these data are based were measured in our earlier investigation on phase behavior and conformational disorder.',2 Because the n-alkane films used in that work were thick, the intensities of the C-H bands or the k = 1 bend or rock fundamentals near 1465 and 720 cm-I could not be determined. What was measured were the rocking-mode bands near 800 cm-'. These are listed in Table I. As we will show, the temperature behavior of these bands is very similar to that of the 720-cm-' fundamental. In fact, it appears that all rocking bands behave similarly. This similarity may be seen from the spectra in Figure 8, which shows n-C21H44in phases I and 11. The phase I spectrum undergoes a nearly uniform attenuation in intensity as the solid n-alkane is transformed to phase 11. A representative plot of intensity vs. temperature for a rocking-mode band is shown in Figure 9. This particular plot is for the P9 rocking band of n-CI9. We note that the value of a for phase I is about -0.55 X deg-', but for phase I1 it is much larger, about -2.2 X lo-* deg-'. The intensity loss at the solid-solid transition is about 30%. The Dhase I values of a, measured just below T,, are nearly deg-I j for all the nyalkanes that the same ((-0.55 f 0.04) X
Temperature and Phase Behavior of Infrared Intensities
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986 5627
I
TABLE I: Observed Values of the Temperature Coefficients (a)of Some Rocking-Mode Intensities and Lateral Expansion Coefficients ((3) for Crystalline n-Alkanes" phase n k vb. cm-' 10za, des-' 102B,bdeK1 I
17 19
21
760 720 782 831 760 720 782 762
19 19
1 1
520 520
23 17 19
I1
I I1
n-CnHz*+z
9 1 9 13 9 1 9 9
-0.42 f 0.06 -0.50 f 0.06 -0.54 f 0.06 -0.51 f 0.06 -1.03 f 0.2 -2.2 f 0.3 -2.5 f 0.3 -3.0 f 0.4
D I
0.058 f 0.05 0.049 f 0.05 0.049 f 0.05 0.051 f 0.05 0.11 f 0.01 0.10 f 0.01 0.11 f 0.01 0.13 f 0.01
n-CnDzn+2
-0.26 f 0.04 -1.39 f 0.15
" k indicates the rocking mode and vt its frequency. Values derived from ref 15.
01
TABLE II: Observed Values of the Intensity Loss, d , That Occurs at the n-Alkane Phase I to Phase I1 Transition and Calculated Values of d Obtained with 6" db obsd' calcd 102a: deg-I 102(3,ddeg-I &/Ac
C17 C19
Czl
-0.36 -0.30 -0.49
-0.38 -0.31 -0.38
-0.580 -0.401 -0.438
0.058 0.049 0.051
Phose I
n
'
17
I
19
I
I
21
1
I
23
25
I
29
27
Number of Corbons
Figure 10. Chain-length dependence of the intensity temperature-coefficient, a,for rocking modes near 800 cm-I in the infrared spectra of phase I and I1 n-alkanes.
0.0376 0.0384 0.0442
3.0-
Values of the parameters used in eq 6 are also tabulated. Defined in eq 3. CDerivedfrom /3 with eq 5. dDerived from data in ref 15. 'Estimate error is about +25%. &ci =-
2 2.0-
0.55 %/deg
T
0
d
d -30%
I
-
W c U C 0,
.-c
0
I
0
20
24 28 Tern p e ro tu re ("C)
32
Figure 9. Temperature dependence of the intensity of the 782-cm-I rocking band (P,) of n-CI9H4 in the temperature region around the phase I to phase I1 transition.
were carefully determined (n = 19, 23, 27, and 29) and are near deg-' found for n-Czl in KBr the value (Figure 5 ) of -0.50 X at 300 K, a temperature about 6 K below the solidsolid transition temperature. We note from Figure 3 that it appears reasonable that a value of this approximate magnitude would be obtained for PE by extrapolation to a temperature just below the melting point (-415 K). As noted above, a for the phase I1 n-alkanes is considerably larger than for phase I. It is also dependent on chain length as may be seen in Table I and in Figure 10, where a for both phases is plotted against the number of carbons. The value of a for phase I1 increases approximately linearly with the number of carbons. There is a large loss in intensity at T, in going from phase I to phase I1 (Figure 9) that tends to be greater for the longer chains. The losses for some n-alkanes are listed in Table I1 in terms of d (eq 3), and these range from -35% to -49%. We note that for odd n-alkanes longer than CZ3there are other solidsolid transitions in addition to the a transition. As noted in ref 2, intensities of the rocking-mode bands decrease at these transitions also.
0.04
I
0.00
I
I
0.1 2
fi (deg''al@) Figure 11. Temperature coefficient of intensity (a)plotted against the temperature coefficient of lateral expansion ((3). The data are for polyethylene (A),phase I n-alkanes (0),and phase I1 n-alkanes ( 0 ) .
A direct comparison between the temperature coefficients of the P1 and P9 rocking bands can be made in the case of ~ C 1 9 . The intensities of the Pl band were measured for solid n-ClgH40 diluted (1:24) in n-C19D40.14The intensities of P9 are from the present study. From Table I, we see that, within experimental error, the values of CY for the P1 and P9 bands are the same both for phase I and for phase 11. The value of d is also the same. Thus it is likely that the temperature and phase behavior of different methylene rocking bands is essentially the same for a given nalkane. A second conclusion that follows from the above data is that intermolecular vibrational coupling does not play an important role in the temperature or phase-change behavior of intensities. Measurements on n-C19D40 indicate the existence of a large isotope effect on the temperature dependence. Thus, for n-CI9D40, the values of the a associated with the intense P I band near 520 cm-' are (-0.26 f 0.4) X 10-* deg-' for phase I and (-1.39 f 0.15) X deg-l for phase I1 (Table I). For d the value is about -0.17 f 0.05. These three values are roughly half those observed for (14) Casal, H. L.;Mantsch, H. H.; Cameron, D. G.; Snyder, R. G. J . Chem. Phys. 1982, 77, 2825-2830.
Snyder et al.
5628 The Journal of Physical Chemistry, Vol. 90, No. 22, 1986
F
TABLE III: Infrared Intensities Per Methylene for the Poly(methy1ene) cham in the Crystalline (77 K), Liquid (300 K), and Gas (300 K) Phaseso
P
r:
0.05
int, cm/mmol Dhase
T. K
C-H stretch
CH2 bend
CH2 rock
solidb liquidC gasf
I1 300 300
6261
760
210
830od
413c 25 1
115’ 82
6520
“Errors estimated to be about fZO%,. *These intensities were computed with the parameters derived from n-alkanes given in ref 22 (see text). CFrom ref 23 (see text). CCI4. According to eq 7, this solvent increases the intensity about 1% over the value of the neat liquid. eIn C2CI4. From ref 7, we estimate a 4% increase due to this solvent. /In cyclo-C6H12. Our measurements on n-C9HZo. f
kz is f 10%. The value of kl is determined largely by the polyethylene data points, which have large uncertainties along the a coordinate. Consequently the error associated with the value of kl is large, perhaps f100% so that k l = 0 is possible. Equation 5 forms a basis for calculating the intensity change that follows from the change in the lateral dimensions of the unit cell when an n-alkane undergoes a solid-solid phase transition. An appropriate expression, derived from eq 5 , is Temperoture (K)
Figure 12. Temperature dependence of the temperature coefficient,
for lateral expansion in crystalline polyethylene.
p,
((O), (A),and (0)are
data points taken from ref 16, 17, and 18, respectively. See text.).
the undeuterated molecule, n-CI9H4,,. Relation between the Temperature Coefficients of Intensity and Lateral Expansion. The large intensity loss observed for the n-alkanes at the solidsolid phase transition points to a possible connection between intensities and lateral density. Such a connection is supported also by a correlation between the temperature coefficients of intensity and those of lateral expansion. This correlation is shown in Figure 11. In this figure, the observed values of a for the methylene rocking modes of polyethylene and the n-alkanes in both phase I and I1 are plotted against the corresponding values of p, the expansion coefficient defined in eq 4. Before discussing this relation, it is appropriate to comment on the origin of the values of /3 used in Figure 1 1 . The values of p for the n-alkanes, which are listed in Table I, were estimated from the X-ray measurements reported by Ungar.15 The values of p for PE over the temperature range 0-420 K were obtained by combining the results of three separate studies.I6-ls These values are shown in Figure 12. Shen et a1.I6 found that their /3 values, which covered the range 0-250 K, could be reproduced with the Griineisen equation of state.lg-zl The values of the heat capacity and thermal energy needed for the Griineisen equation were calculated by Shen et al. from the Debye mode1.20~21The solid line in Figure 12 represents the values calculated from this equation. The other data points in Figure 12 are from Davis et a].” and Swan,ls through which we have drawn a (dashed) line, shaped by eye to connect smoothly with the results of Shen et al. The intensity coefficient is very nearly a quadratic function of the expansion coefficient. The solid line in Figure 11 is given by the equation cy = k l @+ k2P2 (5) where k , = 1.67 and k2 = -2.01 X lo2. The estimated error in
( 1 5) Ungar, G. J. Phys. Cfiem. 1983,87,689-695. (16) Shen, M.; Hansen, W. N.; Romo, P. C. J . Cfiem. Phys. 1969, 51, 425-430. (17) Davis, G. T.; Eby, R. K.; Colson, J. P. J. Appl. Phys. 1970, 41, 4316-4326. (18) Swan, P. R. J . Polym. Sci. 1962, 56, 403-407. (19) Griineisen, E. In Hand6ucfi der Pfiysik,Geifer, H., Scheel, K., a s . ; Springer-Verlag: Berlin, 1926; Vol. 10, pp 1-59.
(20) Slater, J. C. Introduction to Chemical Physics; McGraw-Hill: New
York, 1939. (21) Kittel, C. Introduction to Solid Sfare Physics; Wiley: New York, 1956.
where AA is the increase in cross-sectional area A in going from phase I to phase I1 and where the values of a and p are for phase I at a temperature just below the transition. Equation 6 has been used to compute d for the phase I to phase I1 transition for C1,, CI9,and CZ1.The calculated and observed values listed in Table I1 show excellent agreement.
Comparison of Solid-, Liquid-, and Gas-Phase Intensities Infrared intensities for the methylene C-H stretch, bend, and rock fundamentals of the solid at 77 K and of the liquid and gas near 300 K are listed in Table 111. The solid-state values at 77 K were obtained from ref 22, which reports infrared intensities of solid n-butane through n-octane at 77 K together with a group-moment analysis. The parameters derived in that study were used to calculate intensities for the infinite poly(methy1ene) chain. These are listed in Table 111. The liquid-state intensities are from the measurements of W e ~ l e and r ~ ~were obtained from the slope of a plot of Wexler’s values against the number of methylenes for a series of n-alkanes ( n = 5-8, 10, 16, and 22) in solution. The solvents used are indicated in Table 111. The gas-phase intensities are for n-C9H20vapor near 300 K (see Experimental Procedures). The C-H stretching intensity was corrected for the contribution from the methyl groups. This contribution was estimated by assuming that the intensity per C-H bond is the same for methyls and methylenes, an assumption supported by intensity measurements on gas-phase n - p r ~ p a n e . ~ ~ For correcting the bending intensity, it was similarly assumed that the per-mode bending contribution is the same for methyls and methylenes. Figure 13 shows how the intensities of the rocking mode bands behave for the solid over a wide temperature range and at the solid-solid phase transition. This plot is based primarily on the solid-state intensities measured for CZ1. The temperatures along the abscissa have been scaled by dividing by the melting point (313 K) of C21.It is probable that the bending-mode intensities display a behavior similar to that of the rocking mode, and it also appears that a plot of intensity vs. a normalized temperature for polyethylene would resemble Figure 13, except, of course, phase I1 would be absent. Included in Figure 13 for comparison are (22) Snyder, R. G. J . Cfiem. Phys. 1965, 42. 1744-1763. (23) Wexler, A. S. Spectrochim. Acta 1965, 21, 1725-1742. (24) Kondo, S.; Saeki, S.Spectrocfiim. Acta, Part A 1973, 29, 735-751. (25) Maroncelli, M.; Strauss, H. L.; Snyder, R. G. J. Cfiem. Phys. 1985, 82. 281 1-2824.
The Journal of Physical Chemistry, Vol. 90, No. 22, 1986 5629
Temperature and Phase Behavior of Infrared Intensities the intensities of the liquid and gas near 300 K. The magnitude of the dependence of intensity on temperature and physical state is apparent in Table 111 and Figure 13. W e note that the intensities of the bending and rocking modes are reduced by about two-thirds in going from the solid near 0 K to the gas at 300 K. One of the striking aspects of Figure 13 is the magnitude of the intensity loss that occurs in phase I in going from near 0 K up to the solid-solid transition at T,. This loss, which occurs in the absence of a phase transition, constitutes most of the total intensity loss that is incurred in going from the solid near 0 K to the gas a t 300 K. Another remarkable aspect of Figure 13 is that the rocking-mode intensity for phase I1 is less than that of the liquid and is comparable to the gas.
Discussion of Possible Intensity-Loss Mechanisms First we will discuss factors that are known to affect intensities in the condensed state. In the second part of this section, we suggest that low-frequency modes may affect intensities. In this discussion the focus is mainly on the intensities of the bending and rocking bands since their temperature and phase dependence is greater and better determined than for the C-H stretches. Density, Refractive Index, and Conformational Effects. We will demonstrate that these factors, which are the ones normally invoked to account for the effect of temperature on infrared intensities, are inadequate to account for the temperature effects observed for the poly(methy1ene) chain. We have not included temperature per se because in general its effect through the Boltzmann factor is small for the bands were are considering. It is largest for the 720-cm-’ band, whose intensity is increased about 3% by going from 0 to 300 K. Crystalline Solid. Changing the temperature of crystalline n-C21in phase I from 7 to 300 K results in an intensity decrease of about 35% for the rocking-mode bands. The magnitude of this change should be kept in mind in considering the effects of density, refractive index, and conformation on the intensities of the crystalline solid. First, we consider density. The change of density of n-CZ1with temperature affects intensities in two different ways. To estimate the magnitude of the resultant intensity changes, we need to know how temperature affects the density. W e have used the density-temperature data obtained by Cole and Holmes26 for n-C36, since data for n-Czl are not available. They found that the density of n-C36 decreased by about 5% in going from 123 to 300 K, a change that is probably close to what would be observed for n-C2, over the same temperature range. An increase in the density of the sample tends to decrease the amount of sample in the spectrometer beam. The maximum effect occurs when the thickness of the film is kept constant. In that case the decrease in the amount of sample in the beam equals the increase in the sample density, namely, about 5% over the temperature range 0-300 K. However, since the thickness of the film is not constrained, the actual change is probably considerably less than 5%. A second, more subtle effect of a density change is through its effect on the refractive index, which in turn affects intensities. The dependence of intensities on refractive index can be estimated from the Polo-Wilson equation2’ (7) The refractive index change for a 5% density change is about 2% as calculated from the Lorentz-Lorenz equation for n = 1.44. Then eq 7 predicts an intensity decrease of about 2%. Conformational disorder in this solid can be ruled out as a significant factor. The concentration of conformational disorder in phase I is known to remain low over the entire temperature range. In terms of gauche-bond concentration, it is significantly less than 1% for n-C21.25 (26) Cole, E. A.; Holmes, D.R. J . Polym. Sci. 1960, 46, 245-256. (27) Polo, S. R.; Wilson, M. K. J . Chem. Phys. 1955, 23, 2316-2377.
Thus, we find that the total combined contributions from density, refractive index, and conformational effects are a factor of 5 short of accounting for the observed changes. A similar discrepancy is also found for the intense loss that occurs at the phase I to phase I1 transition. As we have noted, there is at least a 40% intensity decrease in the rocking-mode intensities for n-C2, in going through this transition. From the density change at the transition, which has been measured by Ungar,I5 we find that the total effect is again at most about 7%. We note that there is a significant increase in conformational disorder in going to phase 11, but the effect on the rocking-mode intensities is estimated from ref 2 to be at most 10% and is probably much less. Thus, we are again left with a large unaccounted for intensity loss. Gaseous and Liquid States. Now we turn to the intensity changes observed in going from the solid to the gas. For reference, we note in Table I11 that the intensities of the bends and rocks for the crystalline solid at low temperature are reduced by about 70 f 5% in going to the gas phase at 300 K. Equation 7 indicates that the refractive index effect will reduce the intensity only by about 25%, far short of the observed reduction. However, in this case we must consider the effects of conformational disorder. Conformational change can affect intensities both through a frequency redistribution of intensity due to changes in normal coordinates and in a more basic way through changes in charge distribution. In going from a conformationally ordered chain to a disordered chain, a mixing of normal coordinates results and there is a shift of intensity out of the normally dominant k = 1 band of an ordered system and into modes of the disordered system that are displaced away from the k = 1 band. However, in the case of the C-H stretching and bending modes, intensity cannot be lost in this way since we integrate over a frequency region that is sufficiently wide to include all the displaced bands.28 In the case of the rocking-mode fundamental near 720 cm-I, integration is between 750 and 690 cm-I so that for the disordered chain some intensity is lost in the region above 750 cm-’. However, results from recent measurements and theory indicate that this loss is less than 20% of the total integrated intensity.29 We now consider to what extent intensities are dependent on conformation through the conformational dependence of the charge distribution. The effect of conformational change on the intensity of an isolated methylene rocking mode has been found to be quite small.30 This finding is based on measurements of the intensity of the localized rocking band that appears in the region 660 to 620 cm-I for an isotopically isolated CD2 group. These measurements indicate that within an experimental error of f7% the intensity of the CD2 rocking mode is unchanged in going from the case where the CC bonds adjoining the CD2 group are both trans to the case where one C C bond is trans and the other is gauche. In conclusion, the combined effects of density, refractive index, and conformation are much too small to account for the observed temperature and phase sensitivities. Effect of Low-Frequency Modes. The correlation that exists between intensities and lateral expansion in the crystal suggests a mechanism quite different from those we have just considered. It is well-known that thermal expansion results from the vibrational anharmonicity that is associated primarily with low-frequency lattice vibrations. In fact, the temperature dependence of lateral expansion of many solids, including polyethylene, can in large part be accounted for on this b a ~ i s . ’ ~ , ~ ’ We propose therefore that the temperature and phase dependencies of intensities are also linked to low-frequency modes. There are several different ways that low-frequency, high-amplitude vibrations could affect the infrared bands associated with modes at higher frequencies. One possible way is through (28) Snyder, R. G. J . Chem. Phys. 1967, 47, 1316-1360. (29) Snyder, R. G., unpublished results. (30) Maroncelli, M.; Strauss, H. L.; Snyder, R.G. J . Phys. Chem. 1985, 89, 4390-4395. (31) Broadhurst, M. G.; Mopsik, F. I. J . Chem. Phys. 1971, 54, 4239-4246.
5630 The Journal of Physical Chemistry, Vol. 90, No
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Snyder et al. counts for this since the value of a/d.r(ap’/as( depends on the type of group coordinates. Conversely, in keeping with observation, the model indicates that the intensities of modes having the same type of motion will tend to exhibit a similar temperature behavior. On the basis of this model, intermolecular coupling effects are not expected to play a significant role. They are not observed. However, an isotope effect is predicted because the amplitudes of the torsional modes are affected by mass changes. Amplitudes will decrease in going to the deuterated chain, leading both to smaller values of the temperature coefficient and to smaller intensity changes at the solid-solid phase transition. As we have noted, a large isotope effect for n-C,, is observed.
solid
hose I phose II
Summary
0 0
0.2
04 0.6 0.0 Tempe r o I u re, T/T,
1.0
Figure 13. Relative intensities of the rocking modes of n-alkanes in the solid, liquid, and gas phases. For the solid phase, intensity is plotted against a reduced temperature TIT,, where T, is the melting point.
band-shape changes, for which we consider a hypothetical case. If, in going to higher temperatures, the wings of a band became extended, an intensity decrease might seem to occur because the wings tend to get lost in the background. However, in the present case we know this does not happen because the “lost” intensity would in large part reappear in the gas-phase spectra since for this phase the background can be accurately determined. In our view a more likely mechanism is one in which low-frequency “lattice-like” modes affect intensities directly through their large amplitude. The term “lattice-like” is appropriate to use in referring to the low-frequency modes of a crystal comprised of flexible chains such as the poly(methy1ene) chain because the distinction between lattice modes and low-frequency internal modes is blurred. The torsional modes of the carbon skeleton are the most probable participants because their rotatory character would seem to couple well to the lateral expansion while their intramolecular component, Le., internal torsion about C C bonds, can affect the transition moment through higher order dipole-derivative terms. To facilitate discussion, it is convenient to express infrared intensities in terms of group coordinates, si,where si = d*, 6, or P depending on the type of vibration. The intensity of the kth mode may then be expressed
where p’ is the dipole moment of the chain, Qkis the kth normal coordinate, dp/ds is the “group moment” derivative, and L i k is an eigenvector element associated with the group coordinate, s, of the ith methylene.22 Our model departs from that expressed by eq 8 in that we now include terms of the type a/arlap’/asl, which allow for a dependence of the local derived moment on the internal torsional coordinates r associated with the CC bonds adjoining the group coordinates. It appears that our experimental findings can be qualitatively accounted for by this model. The influence of temperature is small at low temperatures because internal torsional amplitudes AT are small. With increasing temperature, the effect on intensity increases nonlinearly as more torsional modes are activated. The nonlinear relation between the temperature coefficients of intensity and lateral expansion also seems understandable since torsional/rotatory motion, in contrast to translational-like motion, approaches an unconstrained state in a nonlinear manner as the lattice expands; that is, a unit of lattice expansion increases torsional amplitude more effectively as the lattice expands. It is observed that different types of modes differ in their intensity behavior with respect to temperature. The model ac-
The infrared intensities of the major bands of the poly(methylene) chain have been determined as a function of temperature and physical state. The three types of bands studied were the C-H stretches (3000-2750 cm-’), the HCH bends (1 500-1420 cm-I), and the C H 2 rocks (750-690 cm-’). The intensities of semicrystalline linear polyethyene and of the crystalline n-alkane n-C21H44were measured in the temperature range from 7 K to near the melting points. The intensities of the odd-numbered crystalline n-alkanes, C17-C23,were measured in the temperature region around the solidsolid transition from phase I to phase 11. Gas-phase intensities were measured for n-C, at 300 K. Surprisingly large intensity changes with temperature were observed in the crystalline state. Intensities decreased nonlinearly with increasing temperature. The magnitude of the change observed for the methylene bending and rocking modes greatly exceeds that expected solely from density and refractive index effects. Anomalously large intensity decreases were also observed for crystalline n-alkanes undergoing the solidsolid (phase I to phase 11) transition. Intensities decrease in going from the solid to the liquid and to the gas. For the bending and rocking modes the those of the solid gas-phase intensities at 300 K are only about at 77 K. The models commonly employed to explain intensity changes with temperature and physical state greatly underestimate the observed changes. However, a strong correlation that was found to exist between the temperature coefficients of the rocking-mode intensities and the lateral expansion of the crystal suggests that low-frequency skeletal modes are involved. We propose that low-frequency torsional modes of the chain can affect intensities through intramolecular twisting about C-C bonds. This mechanism seems to account qualitatively for the following observations: (i) nonlinear changes with temperature; (ii) different sensitivities for different types of modes but similar sensitivities for the same type; (iii) chain-length-independent correlation between the tempeature coefficients of intensities and lateral density; (iv) insensitivity to intermolecular coupling; (v) the existence of an isotope effect for both the temperature coefficient of intensities and the intensity loss at the solid-solid phase transition. Our experimental findings suggest that in quantitative infrared determinations involving chain-molecule systems the neglect of temperature effects on intensities can lead to serious errors. As we have shown, this must certainly be the case for those ubiquitous and widely studied systems in which poly(methy1ene) chains are a major component. Similarly large temperature effects are likely for Raman intensities as well. Finally, our results dramatically underscore the dangers in transferring measured or calculated gas-phase intensities to the condensed state. Acknowledgment. We gratefully acknowledge support by the National Institutes of Health (Grant GM 27690) and the National Science Foundation (Grant DMR-84-037-11). We are indebted to Dr. Goran Ungar of the Rudjer BoskoviE Institute, Zagreb, Yugoslavia for furnishing us with X-ray data on the unit-cell dimensions of phase I and I1 n-alkanes. Thanks also are due to Dr. Richard A. MacPhail, who made the low-temperature intensity measurements on polyethylene, and to Dr. Hans Hagemann for his help in various aspects of the intensity measurements. Registry No. Polymethylene, 25038-57-7.
