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Vibrational spectra of crystalline n-paraffins
JOURNAL
OF
MOLECULAR
Vibrational
4, 411-434 (1960)
SPECTROSCOPY
Spectra
Part I. Methylene
of Crystalline
n-Paraffins
Rocking and Wagging
Modes*
ROBERT G. SNYDER Shell Development Company,
Emeryville,
California
Infrared spectra of crystalline n-paraffins from n-CzoHaz through n-CaoHsz are reported. It is found that to an excellent approximation, the frequencies of methylene rocking and wagging modes are a function of a single parameter. This parameter is related to the phase difference, ‘p, in the motion between two adjacent methylene groups of a given chain. From the infrared frequencies, explicit equations have been derived which give methylene rocking and wagging frequencies as a function of ‘p. Extrapolation to infinite chain length indicates some revisions in earlier infrared and Raman assignments in the vibrational spectrum of polyethylene. INTRODUCTION
Few classes of molecules have been the subject of as much detailed study as the n-paraffins. Yet, the most recent and complete analyses of their molecular vibrational spectra leave many important questions unanswered concerning both the lower and higher members. Complete and reliable vibrational assignments have been made only for methane, ethane and possibly propane, and at the other chain length extreme, uncertainties still exist regarding the interpretation of the spectrum of polyethylene. This paper is the first of a series which approach this problem via a detailed study of the infrared spectra of a series of n-paraffins of intermediate chain length. These compounds, running continuously from n-C&H42 through n-C&Hsz were prepared and purified in these laboratories (1). The advantages of studying a homologous series of very pure and fully extended n-paraffins are several. Opportunities are afforded to observe the effect upon the vibrational modes, of systematically varying boundary conditions (chain length) at chain lengths long enough that end effects are relatively unimportant and yet short enough to allow measurable optical activity to most vibrations. Then there exists the possi* A brief account of this work was given at the Symposium on Molecular Structure and Spectroscopy, Ohio State University, June, 1958 and at the Conference of the Western Spectroscopy Association, Asilomar, California, February, 1958. 411
412
SNYDER
bility of extrapolating this information to shorter or longer chain lengths with the hope of elucidating the spectra of these molecules. With n-paraffins in the molecular weight range of those used in this research, experimental difficulties of sample preparation are minimized since these compounds are all solids at room temperature. This paper will confine itself to methylene rocking and wagging modes. To be considered in the future are: methylene twisting and CC stretching modes; the overtone and combination spectra; and intermolecular effects. EXPERIMENTAL
The normal paraffins studied in this research were prepared in these laboratories by Schaerer and Busso (1). Isolation of the n-paraffins from lubricating oil distillate was carried out first by urea extraction to eliminate the more highly branched isoparaffins and cyclic compounds. Purification was by repeated vacuum distillation with treatment with sulfuric acid and crystallization from toluene between each distillation. The process was repeated until impurities amounted to less than one per cent (1, 2). All spectra reported here were obtained with a Beckman IR-7 flicker-beam infrared recording spectrophotometer. Most samples were prepared by the potassium bromide pressed disk technique. The paraffin was dissolved in a few milliliters of solvent (usually toluene) and poured onto about two grams of powdered potassium bromide. The resulting paste was worked in a mortar until the solvent had evaporated completely. The dry powder was then placed in a one-inch diameter die and pressed under evacuation to produce a translucent disk. Although the disk was sometimes quite opaque t’o visible light, it was surprisingly transparent to infrared radiation. There was nothing to indicate from the spectra obtained or from observations made with a polarizing microscope that the samples prepared as described had anything but a wholly random orientation of the crystallites in the matrix. Although any orientation other than random would seem unlikely, it was thought wort’hwhile to investigate the possibility of differences between the spectra of samples prepared in this manner and the spectra of those prepared by another method. To this end, samples of n-paraffins were melted and rapidly resolidified between salt plates. The spectra obtained from t,he resultant sandwiches were identical to those obtained from the potassium bromide disks, thus showing that neither orientation or change in crystal structure had occurred. Spectra of the samples at low temperature were obtained using a cell of t’he genre of Wagner and Hornig (3). During the course of this work it was desirable to examine the spectra of oriented crystals of the n-paraffins. Since single crystals of proper dimensions are extremely difficult to obtain, recourse was made to oriented polycrystalline films. Such a film can be obtained by slow cooling and pressing of a melt of t,he
SPECTRA OF CRYSTALLINE
n-PARAFFTNS
413
paraffinbetween two ~t~~~~ bromide windows (8). The ~rys~al~tesof s-par&ffins ~~v~~ an odd comber of carbon atoms are then oriented so that the direction of the chains is ~er~e~d~~ularto the faces of the windows. This orientation was confirmed by noting that when the sample is insertedinta the spectrophotometer so that the faces of the sample windows are perpe~dieu~arto the incident radiation, the bands whicthhave been shown by other workers (2) to be pa~I~e~to the chain axis, are greatly ~v~ake~edin i~t~~~~ty~~~~~rne~s prewarm in this manner were fixed in the cold cell describe above and cooled $0 liquid ~~trog~~terngesture before their spectra were reworded*Only odd-numbered ~-parades (specifically wX&+H~~ and ,n-C2~H60) were oriented because ,samplesof even numbered members prepared in this manner have their chains tilted with respect to the cr@X%lfaces (4). F’igures 1-11 show the infrared spectra of ra~~dom~yoriented s~rnp~esat -180°C. The percertf;t~~srni~ta~~e scales cannot be cornp~~d because different amounts of compensation were provided in the referencebeam for different spectra depending on the tra~srn~sio~~ of the samples,Table Tlists the frequencies of observed absorption bar& together with ass~~~rne~~ts for which ~~st~~catiun
414
SNYDER
lc
75-
u‘ 0
9
.s E
50_
: ;:
25_
I
O_ 1500
1400
1300
1200
I I_
IO
1110
Frequency. cm-’ FIG. 3. Infrared
absorption
spectrum
1000
I 900
800
of n-CzzHas at -180°C.
25
1400
1300
1200
1100 Frequency,
FIG.
0 1500
A/ 1400
4.
Infrared
I 1300
absorption
I 1200
spectrum
absorption
900
PO0
700
BJO
700
of n-C&HA8 at -180°C.
1100 Frequency,
FIG. 5. Infrared
1000 cm-’
1000
<
cm-’
spectrum
of n-Cz4Hsa at -180°C.
SPECTRA OF CRYSTALLINE
0 1500
1300
1400
izoo
1100 Frequency.
415
n-PARAFFINS
1000
900
800
7
BOO
?OO
cm-’
FIG. 6. Infrared absorption spectrum of n-C&Hsn at -180°C.
L””
c
75
t d +
50
@ 3 ;:
25
0 1500
1300
1400
WJO
1100 Frequency,
1000
900
cm-’
FIG.
7. Infrared absorption spectrum of n-C&Hra at -180°C.
FIG.
8.
Frequency,
cm-’
Infrared absorption spectrum of n-C2?Hss at -180°C.
416
0
1500
SNYDER
N
I
1400
1300
1200
1100 Frequency,
FIG. 9. Infrared
absorption
FIG. 10. Infrared
absorption
FIG. 11. Infrared
absorption
1000
900
800
&
cm-’
spectrum
of n-C&Hss at -180°C.
25
Frequency,
cm-’
spectrum
Frequency,
of n-C&Hco at -180°C.
cm-’
spectrum
of n-C30H62 at -180°C.
will be given later. Certain regions of these spectra were recorded using greater sample thickness so that more frequencies are listed in Table I than there are absorption peaks shown in the spectra. Included in Table I are the results of the polarization measurements for n-C26H52and n-&,HGo.
TABLE
I
OBSERVED
FREQUENCIES OF ABSORPTION BANDS IN THE INFRARED SPECTRA (1500-700 -180°C CM-')OFTHE ?&-PARAFFINS FROM12-C 20H ~~THROUoH~-Cx,H62AT ASSIGNMENTS OFFUNDAMENTALVIBRATIONS-POLARIZATION OF BANDS IN THE SPECTRA OF n-CzsHsz AND n-CLHso Assignment0 n-CLoH42 721(s) 737(m) 767(m) 817(m) 880(m)
893(m) 945(m) 973(vw) 994(m) 1006 (w) 1012(w) 1040 (w) 1049 (w) 1053 (VW) 1061 (sh) 1065(m) 1124(m) 1179(w) 1186(w) 1211 (VW) 1224(w) 1248(vw) 1259(w) 1279 (VW) 1294(w) 1299 (w) 1306(vw) 1326(w) 1352(w) 1367(m) 1371 (w) 1387(w) 1401(?) 1437(vw) ~1465(s)~
RI R7 Rg
761 (m) 765(m) 807 (m)
R11 R13 RCH~
862 (m) 865(m) 891 (m) 1
RI&
Wt ws w7
WP Wll
B&Q
894(m) ( 922(m) 927 (m) 1 975(w) 980(w) 986 (w) 1 998(w) 1022 (VW) 1029(w) 1033(vw) 1036 (VW) 1044(w) 1049(w) 1061 (w) 1068(w) 1096(m) 1125(m) 1137(m) 1173(vw) 1178(vw)
R9
R11 R13 RCH~
Rls
Rl7
129O(vw) 1297 (VW) 1304(vw) 1307(vw) 1319(w) 1337(vw) 1344(w) 1362 (w) 1367 (w) 1376(m) 1386 (w) 1399(vw) 1415(vw) 1420 (VW) 1426 (VW) 1434(w) 1444(m) ~1465(s)~
893(m) 900(m) 958(w) 97l(vw)
1192(vw) 1203 (w)
993(w) W2
1007 (w) 1014(w)
1220 (w) B, B&a
(Continued)
721(s) 731(m) 752(m) 788(m) 842(m)
1198(w)
1223 (w)
Cz&
ws w9
Wll WlZ
B&I3
2 X 722 B, B&s
n-CzzHae
1185(vw)
1205 (w)
Wa
1027 (w) 1050 (w)
1237 (w)
1059 (VW)
1239 (w)
Wa
1254 (w)
W6
1071 (m)
R1
1269 (VW)
W6
1126(m)
R7
1287 (w)
n-CzJLa 722(s) 735(s) i 743(m)
n-CzlHar (Continued)
1178(w)
1274(vw)
W? 417
1185(w)
RI RT Rs Rn RI3 RCH~ R16 R17
418
SNYDER TABLE
I-continued
~Freque$lcp~*b
AssignmentC
-1
n-C&Ha 1206(vw) 1219(w) 1238(?) 1250(w) 1268(vw) 1282(w) 1291 (w)
W3 WS W7
1299(w) 1306(w) 1313(w)
WS
1339(w)
Wt1
1355(sh) 1368(m) 1376(sh) 1385(?) 1397(?) l437(8h) -1465(s)d
B&3
*~-GuHG
740(m)
831 (m) 883 (m) 887(m)
SQlb) 894(m) i YlO(?) 938(w) 943(w) 966 (VW) 971 (VW) 989(sh) 990(w) 996 (w)
1004(vw) 1012(w) 1020 (VW)
R,
R13
RS R
CH3
RI7
RI9
1032(vw) 1036(vw) 1041 (w) 1045(w) 1055(w) 1064(w) 1075(w) 1083(vw) 1101 (m) 1127(m) 1137(m) 1173(vw) 1177(vw) 1184(vw) 1188(vw) 1195(w) 1200(w) 1217(w) 1232(w) 1247(w) 1261 (VW) 1265(vw) 1276(w) 1283 (VW) 1291 (VW) 1296(vw) 1305(w) 1331 (w) 1349 (VW) 1355 (VW) 1365(w) 1373 (m) 1374(sh) 137fqvw) 1385(vw) 1403 (VW) 1418(vw) 1423(vm) 1430(w) 1434(vw) 1438(sh) 1444(m) 1446(sh) 1458(w)
1462(s) 1465(m) 1475(s)
(Continued) B B& B
n-Cz4H.w 721(s) 742(m) 771 (m) 812(m) 863 (m> 891 (m) 917(m)
wz Wa :s Wg W,
W$l Wll Wd?) Wn B,&
971 (w) 973 (w) 1 1011 (w) 1018(ur) 1038(w) 1052 (w) 1054(sh) 1061 (VW) 1078 (m) 1127(m) 1158(?) 1177(w) 1183(w) 1202(w) 1215(w) l232(vw) 1245(w) 1261 (VW) 1273(w) 1287 (w) 1297(sh) 1302(w) 1304(vw) 1314(vw) 1328(w) 1351 (w) 1368(m) 1373 (w)
2 X 722
1385 (w) 1401 (VW) 1423 (VW)
RI Rs RI1 RU RI5 RCH,
RI7 RI9
W, Ws W,
Ws
W11 WI3 GH~
SPECTRA
OF CRYSTALLINE TABLE
Frer~ny&,~
I-continued
Assignmentc
(Continued)
n-C&& 1437(vw) 1447(w) 1453(m) 1456(w) 1462(w) 1474(s)
B&
B ?Z‘GSH62
1173(vw, 1) 1177(vw) 1185(w, I) 1194(w) 1198(w, I) wz 1213(w, II?) Wa 1227(w, I) Wd 1229 (WV) 124O(w, 11) W6 1254(vw,
728(m)
RI
74O(m) 747(ml)
Rs i
I)
ws
1257(vw) 1263 (VW) 127O(w, 11) W7 128O(vw) 1285(vw) 1294 (VW) 1295(w, II) 1303(vw,l)
ws
1321(w, 11) Wil 1336(vw)
891 (m) 894 (m) 900(m) 905 (m) 951 (m) 956(m)
&HZ Rx7
RI9
968(vw, b, II) 973 (VW) 98O(vw) 989(vw, b, II) 998(w, J-1 Rz, 1004(w, If 1 1023(w, II) 1035(vw, II) IO39(vw, I) f046(w, 105O(w, 1059(w, 1065(w,
419
n-PARAFFINS
ID I) I) I)
1081 (w, II) 1105(m, I) 1128(m, II) 1136(m, I)
1343(w, II> 1359(w) N1374(m)d 1386(w) 1402(vw) 1415(vw) 1421 (VW) 1427(vw) 1432(vw) 1434(vw) 1443 (m) N1465(S)d
w1a WI6 B&a
2 x 722 B, %H,
n-‘C26HS4 721(s) 736(m) 756(m) 788(m) 833(m) 882 (m) 888 (m) 893 (m)
RI RS RI1 RI3 RI& R17 RCH
933 (m) 969 (VW) 981 (w) 984 (w) 996 (VW) 1024(w) 1029 (w) 1045 (w) 1052 (w) 1057 (w) 1062(w) 1084(m) 1129(m) 1177(w) 1182(w) 1198(w) 1211(w) 1225 (VW) 1238(w) 1252(vw) 1265(w) 1277(vw) 1290(w) 1~3(w) 1298(vw) 1302(w) 1304(sh) 1308(vw) 1316(w) 1339 (w) 1356 (w) 1368(m) 1386(w) 14lO(vw) 1419(vw)
I
~1465(s)~
R19
Rll
w3 Ws W7
ws
Wll WI3 BZH~
B, E?H~ d327H5~
743(m) 753 (m)
Rs
SNYDER
420
TABLE Assignmentc
d&H66 758(m)
(Continued)
RI1
783 (m) 786(m) 1
RI3
824 (m)
Rl5
868 (m) 871 (m) I
R17
892 (m) 895 (m)
RCA~
916(m) 921 (m) 1 961 (w) 967 (w) 1 971 (VW) 986 (w) 1004(w) 1009(w) 1 1033 (w) 1037(?) 1043(vw) 1047(w) 1057(vw) 1062(w) 1066 (w) 1086 (w) 1109(w) 1129(w) 1137(w) 1183(vw) 1195(w) 1210(w) 1222(vw) 1225(vw) 1234(w) 1246 (VW) 125O(vw) 1255(vw) 1260(w) 1267(vw) 1276(vw) 1286(w) 1292(vw) 1297(vw)
R
19
Rt1
R23
I-continued
Frequeue&.
1
b
Assignment’:
1301 (VW) 1310(w) 1326(vw) 1333(w) 1350(w) 1362(sh) 1375(m) 1385 (w) 1403(w, b) 142O(vw) 1425(vw) 1431 (w) 1435 (vw) 1444(w) 1459(s) 1465(m)
W11 W13 WlS B&
2 x 722 B B& B
n-C%Hss 720(s) 733(s) i 748(m) 750(m) 1 773 (m) 810(m) 853 (m)
Ws
WP
Ws
AssignmentP
1
n-CsTHss (Continued)
1473(s)
wz wa wq
Frequeue~);s~ b
889(m) 893 (m) i 899 (w) 901 (w) 944 (w) 947 (w) 1 983(vw) 987(w) 991 (w) 1 1012(w) 1024(w) 1027(w) 1034(vw) 1039(w) 1046 (w) 1053 (w)
Rt R11 R13 RI5 R17 R
CH3
R19
RZl
1061 (w) 1087(m) 1128(m) 1175(vw) 1180(w) 1184(vw) 1192(vw) 1198(vw) 1207(vw) 1218(vw) 1233(vw) 1258(vw) 1266(vw) 1284(w) 1287 (w) 1300(w) 1308(w) 1332(vw) 1365(m) 1373 (w) 1379(w) 1382(sh) 1411 (VW) 1417(vw) 1424(vw) 1429 (sh) 1437(w) 1444(w) 1453(w) 1457(sh) 1462(s) 1465(m) 1476(s)
R23
Ws Ws W, Wg
WI1 B&I
B B& B
n-CzsH99
722(s) 731(m) )1 734(s) j 1 740(m)
744(m) 750(m) 767 (m)
RI R9
J
R11
TABLE
RI3
771(m)
970(w) 977 (w) 1 980(w)
R23
997(w, II?) 1005 (VW) 1009(w, 1015(w,
I) R I) 1 % I)
1018(w, 1024(vw) 1030 (VW) 104l(w,
I)
1047(w, II) 1054(vw) 1059(vw) 1063(sh) 1068(w, I)
1254(w) 126O(vw) 1265(vw) 127O(vw) 1278(w) 1279 (VW) 1295 (VW) 1301 (w) 1323 (w) 1336(sh) 1342(w) 1357(w) 1365(sh) 1375(m) 1385(w) 1400(w) 1418(vw) 1423 (VW) 1428(w) 1432 (VW) 1435(sh) 1438(sh) 1444(w) 1458(w) 1462(s) 1466(m) 1475(s)
1082(vw, II) 109O(w, II) llll(w, I) 113O(w, II) 1136(w, I) 1173(vw) 1181 (w) 1188(sh) 1192(w) 1206(w) 1218(w) 1221 (VW) 1230(w) 1238(vw) 1242(vw) 1244(vw) 1249 (VW)
I-continued
W?
994(w) 999(w) WS
2 X 722 B BEH~ B
n-Go&z
WP Wa W4
722(s) 735 (9) 1 742&j
Rll
762(m)
Ri3
R1
15 829 (m) 870 (m)
R17 R19
Ws we
913(w). 915(w) 954(w) 958(w) I
97O(vw) 977 (VW)
RZl R23
1
1026 (w) 1030 (w) 1042(sh) 1047 (m) 1057 (w) 1063 (w) 1092 (m) 1129(m) 1175(vw) 1177(vw) 1180(w) 1185(w) 1191 (VW) 1196(?) 1206(w) 1214(vw) 1230(w) 1244(vw) 1253(w) 126O(vw) 1275(sh) 1282(w) 1297 (w) 1302(w) 1305 (w) 1310 (VW) 1324(w) 1344(w) 1361 (sh) 1367(m) 1375(w) 1382(w) 142O(vw) 1425 (VW) 1431 (VW) 1438(vw) 1445 (VW) 1461(s) 1466(sh) 1474(s)
R26
Wa Ws Wr Ws(?) Wll
B&
B BOHR B
a w, m, s, v, sh, and b indicate weak, medium, strong, very, shoulder, and broad, respectively. b Frequencies to i 1 cm-l. c B, W, and R are methylene bending, wagging, and rocking, respectively. Numbers as as rymmetric and subscripts are the assigned k values. B& and B &n3 are,espectively, symmetric methyl bending. Ron* is out-of-plane methyl rocking. d Unresolved. 421
SPECTRA SELECTION
OF CRYSTALLINE
n-PARAFFINS
RULES AND NORMAL MODES n-PARAFFIN MOLECULES
423
OF ISOLATED
Selection rules for n-paraffins having odd and even numbers of carbon atoms are given in Table II. These selection rules are for the fully extended form of the molecules, which have point group symmetries Czv and &, depending on whether the number of carbon atoms in the molecule is odd or even, respectively. experimental (b,5) and theoretical (6,7) studies of the spectra of ~-paraffil~s by earlier workers have shown the very useful fact that the d~tributions of frequencies for modes of the same basic motion are confined to relatively well-defined regions. Although there is some overlapping, at least part of it is between modes which belong to different symmetry species. There is, however, still considerable uncertainty as to the limits of the regions occupied by the different types of modes. Estimates of the location of these regions by others are given in Table III t.ogether with estimates based on the present work in the cases of methylene wagging and rocking modes and on unpublished results in the cases of methylene testing and CC stretching. SIMPLE
VIBRATIONAL
THEORY
FOR TRE
FINITE
CHAIN
Theoretical treatments of the problem of vibrations of the n-paraffin molecule which attempt to account for interaction between different groups of fundamentals, such as between methylene rocking modes and methylene twisting modes, have been published (6’, 7). The sensitivity of the numerical results of these calculations to certain interaction constants whose values are not known, and to perturbations from the methyl end groups, which are not taken into account, do not permit quantitative agreement between calculated and observed frequencies of normal modes. However, as will be seen later, qualitatively the solutions agree with the experimental findings presented here. TABLE
REGIONS OF THE SPECTRAOF
III
CRYSTALLINE
~-PARAFFINS
CHARACTERISTIC
OF SPECIFIC VIBRATIONS Brown et nl. (3, 5)
.Mode
This research
-~
CHZ racking
72%1050
72(rm1050
CC stretching CHJ twisting
884-1165 1176-1243
890-1150 1100-1350
970-1140” 1170-1300’
CH2 wagging
1280-1445 1480-1640
1200-1350 1470
1175-1415 A470
CHZ bending
a Conclusions drawn from work to be published.
424
SNYDER
No attempt will be made here to improve the theoretical treatments existant (F-8). Rather, the simple hypot,hesis is proposed and found to be adequate that the frequencies of the methylene rocking and wagging modes are functions of a single parameter which can be readily calculated for a given mode of any n-paraffin. In establishing the empirical relation between this parameter and the normal mode frequencies, a number of coefficients are involved which contain implicitly kinetic and potential energy terms. The numerical values of the coefficients are compared to the same coefficients calculated from theory. It will be helpful to describe in some detail a very simple model of the vibrational problem in order to establish some perspective for later arguments. A simple model is described by the following assumptions: (1) All methylene groups and CC bonds are considered to be entirely equivalent regardless of their position in the chain. (2) The methyl groups at the ends of the molecule are not considered. (3) Co~~pling between different types of methylene and skeletal motions are not considered. (4) No interaction force constants between different methylene and/or skeletal coordinates are considered. These assumptions permit an easy zero order approximation to the solution of the problem. Considering the coordinates Sl , S, , * * * S, for a particular vibration of a system of m oscillators (i.e., methylene groups, CC bonds), which are numbered consecutively beginning at one end of the molecule, we find, using the GF notation (9), a diagonal F matrix with a constant element f on t,he diagonal and a G matrix which has a constant diagonal element ga , and may or may not have constant nonvanish~g nearest off-diagonal and next nearest off-diagonal elements $6 and gc , respectively. All other G elements are zero. Then the ordinary form of the secular equation IGF-XE)=O
is found to have the simple form a0 -
X
ai
al
a0 -
a2
al
0
... ...
a2
X
a2
0
al
a2
a0 al
..‘..I.....................
x
i ii ;j =O
al a() .
x : i
_ .
.
(1)
.
.
.
.
.
*I .
Fith X = 4*‘~’ 1 and a0 = gaf, al = $bf, and a2 = g,J. All other elements vanish. 1The units of X used in the text following are such that force constants are given as millidynes/angstrom and masses and lengths in gram atomic units and angstroms, respectively, so that x = 5.889 x for w in cm-l.
10-T w*
SPECTRA
OF CRYSTALLINE
425
n-PARAFFINS
The solution is given closely by ?Xk,m
=
a0
-t
cPk,rn
=
h/(m
2%
co8 +
cPk,m
$
2a2
l)(?C = 1,2,3,
co8
2pk,m
,
(2)
* * * m).
(3)
Physically, ~h,~ is the phase difference between adjacent oscillators in the system of m osc~tors, for the kth normal mode. The integer k: characterizes each normal mode. Thus, assigning a band to a normal mode means both ident~ying it with regard to what kind of motion is involved and then associating a particular h: with it. Eigenvectors are given by ak,rn
=[z/(m
+
l)l”‘“[s~
wk,m
,
sh(m
-
l)(Ok,m , *’
’
Sk
pk.m]
(4)
from which approximate normal coordinates and intensities can be found. When coupling of various modes (rocking with twisting modes, for example) is considered, Eq. (2) must be replaced with the more general expression
The magnitude of the higher terms is a measure of the coupling. The usefulness of Eq. (5) is based on the fact that the constants aa depend on the geometry of the n-paraffin and the force constants and consequently are independent of the chain length m. Hence, once these constants are determined from the spectra of some selected n-paraffins, Eq. (5) can be used to predict the frequencies of normal modes for any other unbranched n-paraffin having a long chain length. Further, when the chain becomes infinitely long, the only modes which can be optically active are Q = 0 and cp = a. Hence, the frequencies of these fundamentals can be estimated. The convention as to what constitutes in-phase and out-of-phase motion between two methylene groups is arbitrary and must be defined for the purposes of the discussion following. The phase angle cpwill be related to the gerade (g) or ungerade (u) property of the motion of adjacent oscillators with respect to the symmetry operation of inversion through the midpoint of the CC bond joining the two groups. Then we define for methylene groups, u:cp = 0; g:p = r. TEE ~ETHYLENE
ROCKING
MODES
Of the series listed in Table III, the methylene rockmg modes are the easiest to analyze. The rocking modes develop dipole moments perpendicular to the plane of the molecule. Although it is possible to factor these modes into active and inactive symmetry species, it is more convenient for our purposes to note in Eq. (3) that for infrared active vibrations, k must be an odd integer (see Table IV). Identification of the rocking mode absorption bands and assignment of these to the particular normal modes are not difficult. Early in the study of n-paraffins
SNYDER
426
TABLE RELATION
BETWEEN POLARIZATION,
IV INFRARED ACTTVITY, AND k FOR
ROCKING AND WAGGING k
CHz rocking
MODES CHz wagging
odd
odd even
active inactive
pm3lle1, actrive perpendicular, active
even
odd even
active inactive
active inactive
the strong absorption band at 720 cm-’ was identified as the rocking mode, k = 1 (JO). The remaining modes are seen to account for most of the stronger absorption bands in the region from 700 to 1000 cm-l, except for the constant frequency one at 890 cm-l, which belongs to a methyl rocking mode. The splitting of many of the absorption bands is an intermolecular effect (11) which will be considered in a separate publication. Kear the 720 cm-’ end of the series there is a decided convergence onto the very strong limiting mode resulting in a number of absorption bands being submerged under this strong band. In order to determine how many bands are so submerged, it was necessary bo trace the pattern back into t)he lower n-paraffins (9). At the high-frequency end of the spectrum the pattern becomes confused with the carbon-carbon stretching modes which appear in t#his region. On the assumption that there is no abrupt change either in the distribution pattern of rocking modes or in the regular manner in which doubling occurs for the odd members, it was possible to identify absorption bands through fi = m - 2 for the odd numbered members and through k = m - 3 for the even numbered ones, where m is the number of methylene groups in the molecule. However, because of the rapidly diminishing intensities with larger k’s and the appearance of unrelated absorption bands in this same frequency region where the rocking mode bands are expected, the assignment of the highest frequency infrared rocking mode was not attempted. Figure 12 shows the positions in frequency of all the observed rocking modes together wit!h their assignments ( ICvalues). When the peak was a doublet, the lower frequency band was arbitrarily chosen. Since the bands have been assigned, it is now possible to test their dependence upon cp. To this end in Fig. 13 is shown a plot of the frequency of each rocking mode versus (OIP calculated from k and m (Eq. 13)). It. is seen that all points fall very nearly on a smooth curve. There is no apparent evidence in Fig. IS that the chain length influences the frequencies other than by entering into cp. It is, t#herefore, concluded that tjhe frequencies of the methylene rocking modes are t,o an excellent approximation, a function of cp only. The scat,ter in Fig. 13 is a little greater than what could be expected from ex-
SPECTRA OF CRYSTALLINE
21 20
0
k=7 k=l ?OO
I 750
FIG.
n-PARAFFINS
I 000
I 850
a 950
I 900
IO00
II $50
12. Observed array of methylene rocking modes
1050
0% O0
1000
(D
00
oB”
0 950
@JP@
Z'E 900 Y 3 850
800
7001 0.3
0.4
, 0.5
1 0.6
8 0.7
0.8
!t n FIG.
13. Frequency-phase curve for methylene rooking modes
I 0.9
0
428
SNYDER
perimental error alone and arises at least in part, from cooperative effects between molecules and from the static crystalline field, both of which depend on crystal str~~cture. This dependence is clearly exhibited in Fig. 12 where modes having the same &, especially when k is small, appear alternately on each side of the smooth curve drawn empirically through the data points. The zig-zagging arises from the fact, that the odd numbered n-paraffins have a different crystal structure than the even n~~mbered ones (4). No attempt was made t.o allow for t,his varia~~i~)lI in frequency. In order to obtain experimental values for the constants in Eq. (5), relating the rocking mode frequency, W, to the parameter, P, the data points shown in Fig. 13 were fit,ted by least squares to the funct’ion
Omission of the constant term permitted adjustment of the function so that at for n-paraffins IL-C&I~~ cp = 0, w = 720 cm-’ . All rocking mode frequencies through ,&Z&Hso in the region T20 to 1015 cm-‘, where unc~rtaillty as to assignments is least, were used. The calculation was performed on the Electrodata computer using a rout& curve fitting program. Rdjustment of the coefficients calculated for Eq. (5a) as indicated above gives wz = (61.~1 -
21.70 cos $7 + 10.72 cosf 9 -
{6)
5.29 COBB $0 + 6.85 cos4 rpo10*
for w in cm-‘. Finally the coe~cient,s~ ai , of Eq. (5) can he readily obtained from those of Eq. (6). These are listed in Table V in the first row. For comparison there are included in Table V along with the experimentally found values of ai , values calc~llated from theory. One set of t,hese (second row of the table) is calculated on the basis of the model for which it is assumed that (1) methylene groups are equivalent, (2) end groups (met’hyl groups) can be ignored, and (3) coupling between rocking modes and other vibrations is unimportant). Numerical values are based on a diagonal P matrix whose diagonal element, is a met,hylerlc rocking force constant from malonic nitrile (12). TABLE V COEFFICIENTS
I_-
FOR THE ROCKING-MODE
Coelkient
.”
Observed CaIculatedb-no interaction Calculatedb-interaction with twisting modes (7) 8 Coefficients are for Ey. (5). tifa = 0.70 X 10FL erg&adz.
80 0.409 0.490 0.462
FREQUENCY-PHASE
a1 -0.153 -0.217 -0.201
** --
EQUATION* fza
u4
-
0.053
-0.008
0.005
0.037 0.050
0 -0.010
0 0.002
SPECTRA OF CRYSTALLINE n-PARAFFINS
429
In the third row of Table V are listed a set of coefficients based on the more sophisticated model of Primas and Gtinthard (6). This model differs from the above one in that no longer is interaction between methylene rocking modes and twisting modes neglected. The same rocking force constant from malonic nitrile is used together with a methylene twisting force constant from the same compound. The appearance of significant although small values of a3 and a4 among the experimentally observed coefficients, shows the necessity of including interactions with other vibrations. Then it is not surprising that the more elaborate model, which includes the twisting interaction yields values for these coefficients which are in better agreement with the observed ones than the simple model. It is pleasing that for all coefficients including a2 and a4 , the observed values agree reasonably well with the theoretical values (row 3 of Table V). In all cases the signs of the coefficients agree. Unquestionably a better numerical agreement could be obtained using a better F matrix. Substitution of Q = ?r into Eq. (6) gives 1033 cm-r for the frequency of the Raman active rocking mode of an infinitely long n-paraffin. In view of this limiting frequency it is difficult to accept the assignment of a line in the Raman speetrum near 1168 cm-’ or even 1130 cm-’ to this vibration (13, 14). It is of course possible that the out-of-plane methyl rocking or the methylene twisting vibrations might interact to disturb drastically the frequency-phase curve between cp = 0.94~ and cp = ‘or.However, this is deemed rather unlikely. The frequencyphase curve shows a definite trend towards leveling off near cp = 0.94a. Further, Primas and Gtinthard (6) have shown at least qualitatively that the inclusion of methylene wagging interaction, affects the frequency-phase curve in producing some over-all displacemellts, but no sharp breaks. Finally, there is good evidence (3) that the methyl out-of-phase rocking mode is near 890 cm-r and consequently strong interaction near 1100 cm-’ would not be expected. Only one line exhibiting appreciable intensity and having a frequency less than 1100 cm-’ is present in the Raman spectrum of polyethylene (14). This line corresponds to a frequency of 1060 cm-‘. Since it is unlikely that this mode has a very low intensity (15)) the 1060-cm-’ line is probably to be assigned to the Raman active methylene rocking mode. THE ~ETHYLE~E
WAGGING MODES
Unfortunately, the infrared intensities of these modes are much less than those of the rocking modes. Substitution of polar groups for the terminal methyl groups can enhance their intensity greatly. However, since the primary concern of this report is the unperturbed crystalline n-paraffins, the intensity problem will have to be lived with. The absorption bands usually attributed to wagging modes can be seen in the spectra in the region 1400 to 1175 cm-“.
430
SNYDER 30 29 20 27 26
n
25 24 23 22 21 20
1175
1200
1225
1250
1275
1300
1325
1350
fd(cni’) FIG.
14. Observed
array of methylene
wagging modes
Even numbered molecules have pi(n - 2) infrared active wagging modes, whose resultant dipole is in the plane of the carbon skeleton. The odd numbered n-paraffins have n - 2 infrared active wagging modes all of which also give dipoles in the skeletal plane. These latter bands are, however, alternately parallel and perpendicularly polarized. In the region 1340-1200 cm-’ even numbered molecules have four or five absorption bands regularly spaced with intervals of about 35 cm-’ for n-CBOH42 and decreasing to about 22 cm-’ for n-C28H58 , which can safely be assigned to wagging vibrations (9). Since the odd numbered ones have approximately twice as many absorption bands permitted, it is to be expected that separations should be about half that observed for the even numbered ones. Indeed, such is the case as seen in Fig. 14 where the observed array of wagging modes is shown. Polarization measurements have been made on oriented films (see Experimental) of n-C&H,, in the wagging mode region of the spectrum. The results are given in Table I where bands are designated as being polarized perpendicular (I ) or parallel ( /1) to the chain axis (2). With the wagging modes for the odd n-paraffins identified, the array shown in Fig. 14 can be constructed. The question to be considered now is the assignment of the proper k to each absorption band. An answer to the problem of finding all bands having the same k can be found in a detailed look at the predicted wagging mode pattern. Table IV gives in summary the relation between k, infrared activity, and polarization which characterizes the pattern of wagging modes. A change of one unit in n will affect (P~,~ less t,han a change of one unit in 1; (Eq. (3)), since
SPECTRA
OF CRYSTALLINE
n-PARAFFINS
431
Thus the frequency change in following the kth mode from wC,H~~+~ to nC,L+1H28+4must be smaller than the spacing of the wagging mode series for a given n-paraffin. These considerations dictate the lines drawn through the absorption bands as shown in Fig. 1.4 Numerical values of k will now be considered. Assuming that the function w = f(u)) is mo~lotonic, the lines through absorption bands of given k values must, with increasing n tend toward frequencies associated with smaller ‘p. Thus the low-frequency region must be for the smallest k’s. Numbering can be established from the array by selecting two absorption bands of different n-paraffins having nearly equal frequencies. For these bands
and consequently k ---=m+Am+I’ m+1
k + Ak
(7)
where m is the number of methylene groups of the shorter ~-para~n. Then, for example, taking the band near 1220 cm-’ for n-C&H*s and estimating Ak = 1.0 for a band of n-CBHso near this frequency, gives k = 3. This then could be taken as a starting point for the assignments shown in Fig. 14. Further support for this assignment is provided by the fact that the predicted polarization of bands (Table IV) is in agreement with that observed for n-C&Hr,z (Table I). Some complexity in the spectra occurs in the region 1180-1205 cm-‘, particularly for the members having shorter chain lengths. Identification of the band corresponding to k = 2 in this region is not difficult by virtue of its inactivity for even members and its expected position relative to the k = 3 band. A more formidable task is the location of k = 1. Two bands active in the spectra of most odd and even members are found. These are separated by about 6 cm-‘. It is possible that both are associated with the k = 1 mode. The mechanism of the splitting could not be intermolecular in origin, however, since for the odd numbered members the selection rules for the crystal allow infrared activity for only one component of a parallel band. That part of the complexity in this region is due to bands accountable to another type of methylene motion is suggested by certain regularities in the positions and activities of the absorption bands which remain after the wagging modes are el~i~ted from the spectra. It is found that the overall pattern of these residual bands is consistent with selection rules for twisting modes. Others (2) have, on the basis of a study of shorter n-paraffins placed the methylene twisting modes in this region. Thus the pattern and the position of these bands seem to indicate that they are to be assigned to twisting modes. The frequency-phase curve for methylene wagging modes based on the above
482
SNYDER 137500 1350-
09 a*O
1325-
00
1300whli')
o@ 8@
1275-
1200 0
,p?+@@ , 0.1 0.2
I 0.3
I 0.4
I 0.5
I 0.6
7
+ i? FIG. 15. Frequency-phase
curve for methylene
wagging
modes
is shown in Fig. 15. The scatter of the points from a smooth curve can be attributed to experimental error in determining the frequencies. The excellence of the fit again seems to speak well for the assignments. A least-squares analysis of the data gives the equation:
assignments
w’ = (177.6 -
21.7 cos P -
4.4 cos’ cp -
8.5 cos3 cp)104.
(8)
The data did not justify determining more than the first four constants. The constants in Eq. (8) together with constants derived for wagging modes uncoupled from all other modes, for which a force constant from malonic nitrile (12) is used, are given in Table VI. The agreement is probably as good as could be expected for the assumptions made in calculating these for the simple model. There is a serious difference between the location of the infrared active wagging mode in polyethylene, drawn from the assignments of wagging modes made here and what has been concluded from other evidence by other researchers. This asTABLE
VI
COEFFICIENTSFOR THE WAGGING-MODE FREQUENCY-PHASEEQUATIONS Coefficient
Observed Calculatedb
(no interaction)
8 Coefficients are for Eq. (5). b fw = 0.663 X lo-” ergs/rad2.
I
a0
a,
a2
a
1.040 1.036
-0.165 -0.079
-0.013 -0.004
0.012 -
SPECTRA
OF CRYSTALLINE
n-PARAFFINS
4x3
signment has been argued over the last several years. Recently there has been attention given to this problem by Krimm et al. (16) who conclude that this mode appears near 1370 cm-l as one component of a complex absorption band. The evidence cited for this assignment will be reviewed brieffy. This fundamental cannot be recognized on the basis of intensity since all the strong infrared bands in polyethylene have been accounted for already. Thus, what is sought is a weak band of parallel polarizatio1~. Two such bands have been considered. However, one, at 1304 cm’-“, is ruled out as being characteristic of only the amorphous form of the polymer. This leaves the 1370-cmS-1component which meets the polarization requirement and is characterist,ic of methylene groups as proved by its persistence in the case of end substituted n-paraffins (11). Also it does appear in the spectrum of the crystalline polymer. Against this assignment is the argument based on the observed distribution of wagging modes and the agreement of the experimentally found constant, al , with that predicted from the simple model (Table VI). Putting into Eq. (8) the limit cp = 0, one obtains for the infrared fundamental a value of 1196 cm-‘. It is suggested that possibly this mode is the 1175-cm-’ band found in the spectrum of polyethylene by Nielsen and Woollett (14). These authors report this weak band has parallel polarization, is characteristic of the crystalline polymer, and is not readily interpretable otherwise. However, more to be stressed here than finding a new assignment, is the fact that it is inappropriate to look near 1370 cm-l for this fundamental. Substituting cp = rr into Eq. (9) gives 1425 cm-l for the Raman active methylene wagging fundamental. Recently, evidence for assigning the Raman active mode in polyethylene to a line at 1415 cm-’ has been presented (24). The assignment is in accord with Raman intensities calculated by Theimer (15). Taken together with the results here, the Raman fundamental seems established. RECEIVED: August 11, 1959 REFERENCES 1. A. A. SCHAERER,C. J. BUSSO, A. E. SMITH, AND L. B. SKINNER, J. Am. Chem. Sot. 77, 2017 (1955).
8. J. K. BROWN, N. SEEPPARD, AND D. M. SIMPSON, Trans. Roy. Sot. A247,35 (1954). 8. E. L. WAGNER AND D. F. HORNIO, 2. Chem. Phys. 18, 296(1950). 4. A. E. SMITH, J. Chem. Phys. 21, 2229 (1953). 6. J. K. BROWN, N. SHEPPARD,ANDD. M. SIMPSON,Disc. ~ura~a~ See. 9,261 (1950). 6. H. PRIMAS AND H. H. G~NTHARD, H&I. Chim. Aeta 36, 1659 (1953). 7. H. PRIMAS AND H. H. G~NTKABD, Helv. Chim. Acta 36, 1791(1953). 8. 0. THEIMER, J. Chem. Phys. 2’7, 408 (1957). 9. E. B. WILSON, JR., J. C. DECIUS, AND P. C. CROSS, “Molecular Vibrations.” McGrswHill, New York, 1955. 10. N. SHEPPARDAND G. B. B. M. SUTHERLAND,Nature 169, 739 (1947).
4x4
SNYDER
11. R. S. STEIN AND G. B. B. M. SUTHERLAND,J. Chem. Phys. 22 1993 (1954). 12. F. HALVERSONAND R. J. FRANGEL,J. Chem. Phys. 17,694 (1949). IS. T. SHIXANOUTI AND S. MMIZUSHI~IA, J. Chem. Pkys. 17, 1102 (1949). 14. J. R. NIELSEN AND A. H. WOOLLETT,J. Chem. Phys. 26, 1391 (1957). 15. 0. TWEIMER, J. Chem. Phys. 27, 1041 (1957). 16. S. KEIMM, C. Y. LIANG, AND G. B. B. M. SUTHERLAND,J. Chew Phys. 26, Ii49 (1956) 17. G. C. PIMENTELAND W. A. KLEMPERER,J. Chem. Phys. 23,376 (1955).
OF
MOLECULAR
Vibrational
4, 411-434 (1960)
SPECTROSCOPY
Spectra
Part I. Methylene
of Crystalline
n-Paraffins
Rocking and Wagging
Modes*
ROBERT G. SNYDER Shell Development Company,
Emeryville,
California
Infrared spectra of crystalline n-paraffins from n-CzoHaz through n-CaoHsz are reported. It is found that to an excellent approximation, the frequencies of methylene rocking and wagging modes are a function of a single parameter. This parameter is related to the phase difference, ‘p, in the motion between two adjacent methylene groups of a given chain. From the infrared frequencies, explicit equations have been derived which give methylene rocking and wagging frequencies as a function of ‘p. Extrapolation to infinite chain length indicates some revisions in earlier infrared and Raman assignments in the vibrational spectrum of polyethylene. INTRODUCTION
Few classes of molecules have been the subject of as much detailed study as the n-paraffins. Yet, the most recent and complete analyses of their molecular vibrational spectra leave many important questions unanswered concerning both the lower and higher members. Complete and reliable vibrational assignments have been made only for methane, ethane and possibly propane, and at the other chain length extreme, uncertainties still exist regarding the interpretation of the spectrum of polyethylene. This paper is the first of a series which approach this problem via a detailed study of the infrared spectra of a series of n-paraffins of intermediate chain length. These compounds, running continuously from n-C&H42 through n-C&Hsz were prepared and purified in these laboratories (1). The advantages of studying a homologous series of very pure and fully extended n-paraffins are several. Opportunities are afforded to observe the effect upon the vibrational modes, of systematically varying boundary conditions (chain length) at chain lengths long enough that end effects are relatively unimportant and yet short enough to allow measurable optical activity to most vibrations. Then there exists the possi* A brief account of this work was given at the Symposium on Molecular Structure and Spectroscopy, Ohio State University, June, 1958 and at the Conference of the Western Spectroscopy Association, Asilomar, California, February, 1958. 411
412
SNYDER
bility of extrapolating this information to shorter or longer chain lengths with the hope of elucidating the spectra of these molecules. With n-paraffins in the molecular weight range of those used in this research, experimental difficulties of sample preparation are minimized since these compounds are all solids at room temperature. This paper will confine itself to methylene rocking and wagging modes. To be considered in the future are: methylene twisting and CC stretching modes; the overtone and combination spectra; and intermolecular effects. EXPERIMENTAL
The normal paraffins studied in this research were prepared in these laboratories by Schaerer and Busso (1). Isolation of the n-paraffins from lubricating oil distillate was carried out first by urea extraction to eliminate the more highly branched isoparaffins and cyclic compounds. Purification was by repeated vacuum distillation with treatment with sulfuric acid and crystallization from toluene between each distillation. The process was repeated until impurities amounted to less than one per cent (1, 2). All spectra reported here were obtained with a Beckman IR-7 flicker-beam infrared recording spectrophotometer. Most samples were prepared by the potassium bromide pressed disk technique. The paraffin was dissolved in a few milliliters of solvent (usually toluene) and poured onto about two grams of powdered potassium bromide. The resulting paste was worked in a mortar until the solvent had evaporated completely. The dry powder was then placed in a one-inch diameter die and pressed under evacuation to produce a translucent disk. Although the disk was sometimes quite opaque t’o visible light, it was surprisingly transparent to infrared radiation. There was nothing to indicate from the spectra obtained or from observations made with a polarizing microscope that the samples prepared as described had anything but a wholly random orientation of the crystallites in the matrix. Although any orientation other than random would seem unlikely, it was thought wort’hwhile to investigate the possibility of differences between the spectra of samples prepared in this manner and the spectra of those prepared by another method. To this end, samples of n-paraffins were melted and rapidly resolidified between salt plates. The spectra obtained from t,he resultant sandwiches were identical to those obtained from the potassium bromide disks, thus showing that neither orientation or change in crystal structure had occurred. Spectra of the samples at low temperature were obtained using a cell of t’he genre of Wagner and Hornig (3). During the course of this work it was desirable to examine the spectra of oriented crystals of the n-paraffins. Since single crystals of proper dimensions are extremely difficult to obtain, recourse was made to oriented polycrystalline films. Such a film can be obtained by slow cooling and pressing of a melt of t,he
SPECTRA OF CRYSTALLINE
n-PARAFFTNS
413
paraffinbetween two ~t~~~~ bromide windows (8). The ~rys~al~tesof s-par&ffins ~~v~~ an odd comber of carbon atoms are then oriented so that the direction of the chains is ~er~e~d~~ularto the faces of the windows. This orientation was confirmed by noting that when the sample is insertedinta the spectrophotometer so that the faces of the sample windows are perpe~dieu~arto the incident radiation, the bands whicthhave been shown by other workers (2) to be pa~I~e~to the chain axis, are greatly ~v~ake~edin i~t~~~~ty~~~~~rne~s prewarm in this manner were fixed in the cold cell describe above and cooled $0 liquid ~~trog~~terngesture before their spectra were reworded*Only odd-numbered ~-parades (specifically wX&+H~~ and ,n-C2~H60) were oriented because ,samplesof even numbered members prepared in this manner have their chains tilted with respect to the cr@X%lfaces (4). F’igures 1-11 show the infrared spectra of ra~~dom~yoriented s~rnp~esat -180°C. The percertf;t~~srni~ta~~e scales cannot be cornp~~d because different amounts of compensation were provided in the referencebeam for different spectra depending on the tra~srn~sio~~ of the samples,Table Tlists the frequencies of observed absorption bar& together with ass~~~rne~~ts for which ~~st~~catiun
414
SNYDER
lc
75-
u‘ 0
9
.s E
50_
: ;:
25_
I
O_ 1500
1400
1300
1200
I I_
IO
1110
Frequency. cm-’ FIG. 3. Infrared
absorption
spectrum
1000
I 900
800
of n-CzzHas at -180°C.
25
1400
1300
1200
1100 Frequency,
FIG.
0 1500
A/ 1400
4.
Infrared
I 1300
absorption
I 1200
spectrum
absorption
900
PO0
700
BJO
700
of n-C&HA8 at -180°C.
1100 Frequency,
FIG. 5. Infrared
1000 cm-’
1000
<
cm-’
spectrum
of n-Cz4Hsa at -180°C.
SPECTRA OF CRYSTALLINE
0 1500
1300
1400
izoo
1100 Frequency.
415
n-PARAFFINS
1000
900
800
7
BOO
?OO
cm-’
FIG. 6. Infrared absorption spectrum of n-C&Hsn at -180°C.
L””
c
75
t d +
50
@ 3 ;:
25
0 1500
1300
1400
WJO
1100 Frequency,
1000
900
cm-’
FIG.
7. Infrared absorption spectrum of n-C&Hra at -180°C.
FIG.
8.
Frequency,
cm-’
Infrared absorption spectrum of n-C2?Hss at -180°C.
416
0
1500
SNYDER
N
I
1400
1300
1200
1100 Frequency,
FIG. 9. Infrared
absorption
FIG. 10. Infrared
absorption
FIG. 11. Infrared
absorption
1000
900
800
&
cm-’
spectrum
of n-C&Hss at -180°C.
25
Frequency,
cm-’
spectrum
Frequency,
of n-C&Hco at -180°C.
cm-’
spectrum
of n-C30H62 at -180°C.
will be given later. Certain regions of these spectra were recorded using greater sample thickness so that more frequencies are listed in Table I than there are absorption peaks shown in the spectra. Included in Table I are the results of the polarization measurements for n-C26H52and n-&,HGo.
TABLE
I
OBSERVED
FREQUENCIES OF ABSORPTION BANDS IN THE INFRARED SPECTRA (1500-700 -180°C CM-')OFTHE ?&-PARAFFINS FROM12-C 20H ~~THROUoH~-Cx,H62AT ASSIGNMENTS OFFUNDAMENTALVIBRATIONS-POLARIZATION OF BANDS IN THE SPECTRA OF n-CzsHsz AND n-CLHso Assignment0 n-CLoH42 721(s) 737(m) 767(m) 817(m) 880(m)
893(m) 945(m) 973(vw) 994(m) 1006 (w) 1012(w) 1040 (w) 1049 (w) 1053 (VW) 1061 (sh) 1065(m) 1124(m) 1179(w) 1186(w) 1211 (VW) 1224(w) 1248(vw) 1259(w) 1279 (VW) 1294(w) 1299 (w) 1306(vw) 1326(w) 1352(w) 1367(m) 1371 (w) 1387(w) 1401(?) 1437(vw) ~1465(s)~
RI R7 Rg
761 (m) 765(m) 807 (m)
R11 R13 RCH~
862 (m) 865(m) 891 (m) 1
RI&
Wt ws w7
WP Wll
B&Q
894(m) ( 922(m) 927 (m) 1 975(w) 980(w) 986 (w) 1 998(w) 1022 (VW) 1029(w) 1033(vw) 1036 (VW) 1044(w) 1049(w) 1061 (w) 1068(w) 1096(m) 1125(m) 1137(m) 1173(vw) 1178(vw)
R9
R11 R13 RCH~
Rls
Rl7
129O(vw) 1297 (VW) 1304(vw) 1307(vw) 1319(w) 1337(vw) 1344(w) 1362 (w) 1367 (w) 1376(m) 1386 (w) 1399(vw) 1415(vw) 1420 (VW) 1426 (VW) 1434(w) 1444(m) ~1465(s)~
893(m) 900(m) 958(w) 97l(vw)
1192(vw) 1203 (w)
993(w) W2
1007 (w) 1014(w)
1220 (w) B, B&a
(Continued)
721(s) 731(m) 752(m) 788(m) 842(m)
1198(w)
1223 (w)
Cz&
ws w9
Wll WlZ
B&I3
2 X 722 B, B&s
n-CzzHae
1185(vw)
1205 (w)
Wa
1027 (w) 1050 (w)
1237 (w)
1059 (VW)
1239 (w)
Wa
1254 (w)
W6
1071 (m)
R1
1269 (VW)
W6
1126(m)
R7
1287 (w)
n-CzJLa 722(s) 735(s) i 743(m)
n-CzlHar (Continued)
1178(w)
1274(vw)
W? 417
1185(w)
RI RT Rs Rn RI3 RCH~ R16 R17
418
SNYDER TABLE
I-continued
~Freque$lcp~*b
AssignmentC
-1
n-C&Ha 1206(vw) 1219(w) 1238(?) 1250(w) 1268(vw) 1282(w) 1291 (w)
W3 WS W7
1299(w) 1306(w) 1313(w)
WS
1339(w)
Wt1
1355(sh) 1368(m) 1376(sh) 1385(?) 1397(?) l437(8h) -1465(s)d
B&3
*~-GuHG
740(m)
831 (m) 883 (m) 887(m)
SQlb) 894(m) i YlO(?) 938(w) 943(w) 966 (VW) 971 (VW) 989(sh) 990(w) 996 (w)
1004(vw) 1012(w) 1020 (VW)
R,
R13
RS R
CH3
RI7
RI9
1032(vw) 1036(vw) 1041 (w) 1045(w) 1055(w) 1064(w) 1075(w) 1083(vw) 1101 (m) 1127(m) 1137(m) 1173(vw) 1177(vw) 1184(vw) 1188(vw) 1195(w) 1200(w) 1217(w) 1232(w) 1247(w) 1261 (VW) 1265(vw) 1276(w) 1283 (VW) 1291 (VW) 1296(vw) 1305(w) 1331 (w) 1349 (VW) 1355 (VW) 1365(w) 1373 (m) 1374(sh) 137fqvw) 1385(vw) 1403 (VW) 1418(vw) 1423(vm) 1430(w) 1434(vw) 1438(sh) 1444(m) 1446(sh) 1458(w)
1462(s) 1465(m) 1475(s)
(Continued) B B& B
n-Cz4H.w 721(s) 742(m) 771 (m) 812(m) 863 (m> 891 (m) 917(m)
wz Wa :s Wg W,
W$l Wll Wd?) Wn B,&
971 (w) 973 (w) 1 1011 (w) 1018(ur) 1038(w) 1052 (w) 1054(sh) 1061 (VW) 1078 (m) 1127(m) 1158(?) 1177(w) 1183(w) 1202(w) 1215(w) l232(vw) 1245(w) 1261 (VW) 1273(w) 1287 (w) 1297(sh) 1302(w) 1304(vw) 1314(vw) 1328(w) 1351 (w) 1368(m) 1373 (w)
2 X 722
1385 (w) 1401 (VW) 1423 (VW)
RI Rs RI1 RU RI5 RCH,
RI7 RI9
W, Ws W,
Ws
W11 WI3 GH~
SPECTRA
OF CRYSTALLINE TABLE
Frer~ny&,~
I-continued
Assignmentc
(Continued)
n-C&& 1437(vw) 1447(w) 1453(m) 1456(w) 1462(w) 1474(s)
B&
B ?Z‘GSH62
1173(vw, 1) 1177(vw) 1185(w, I) 1194(w) 1198(w, I) wz 1213(w, II?) Wa 1227(w, I) Wd 1229 (WV) 124O(w, 11) W6 1254(vw,
728(m)
RI
74O(m) 747(ml)
Rs i
I)
ws
1257(vw) 1263 (VW) 127O(w, 11) W7 128O(vw) 1285(vw) 1294 (VW) 1295(w, II) 1303(vw,l)
ws
1321(w, 11) Wil 1336(vw)
891 (m) 894 (m) 900(m) 905 (m) 951 (m) 956(m)
&HZ Rx7
RI9
968(vw, b, II) 973 (VW) 98O(vw) 989(vw, b, II) 998(w, J-1 Rz, 1004(w, If 1 1023(w, II) 1035(vw, II) IO39(vw, I) f046(w, 105O(w, 1059(w, 1065(w,
419
n-PARAFFINS
ID I) I) I)
1081 (w, II) 1105(m, I) 1128(m, II) 1136(m, I)
1343(w, II> 1359(w) N1374(m)d 1386(w) 1402(vw) 1415(vw) 1421 (VW) 1427(vw) 1432(vw) 1434(vw) 1443 (m) N1465(S)d
w1a WI6 B&a
2 x 722 B, %H,
n-‘C26HS4 721(s) 736(m) 756(m) 788(m) 833(m) 882 (m) 888 (m) 893 (m)
RI RS RI1 RI3 RI& R17 RCH
933 (m) 969 (VW) 981 (w) 984 (w) 996 (VW) 1024(w) 1029 (w) 1045 (w) 1052 (w) 1057 (w) 1062(w) 1084(m) 1129(m) 1177(w) 1182(w) 1198(w) 1211(w) 1225 (VW) 1238(w) 1252(vw) 1265(w) 1277(vw) 1290(w) 1~3(w) 1298(vw) 1302(w) 1304(sh) 1308(vw) 1316(w) 1339 (w) 1356 (w) 1368(m) 1386(w) 14lO(vw) 1419(vw)
I
~1465(s)~
R19
Rll
w3 Ws W7
ws
Wll WI3 BZH~
B, E?H~ d327H5~
743(m) 753 (m)
Rs
SNYDER
420
TABLE Assignmentc
d&H66 758(m)
(Continued)
RI1
783 (m) 786(m) 1
RI3
824 (m)
Rl5
868 (m) 871 (m) I
R17
892 (m) 895 (m)
RCA~
916(m) 921 (m) 1 961 (w) 967 (w) 1 971 (VW) 986 (w) 1004(w) 1009(w) 1 1033 (w) 1037(?) 1043(vw) 1047(w) 1057(vw) 1062(w) 1066 (w) 1086 (w) 1109(w) 1129(w) 1137(w) 1183(vw) 1195(w) 1210(w) 1222(vw) 1225(vw) 1234(w) 1246 (VW) 125O(vw) 1255(vw) 1260(w) 1267(vw) 1276(vw) 1286(w) 1292(vw) 1297(vw)
R
19
Rt1
R23
I-continued
Frequeue&.
1
b
Assignment’:
1301 (VW) 1310(w) 1326(vw) 1333(w) 1350(w) 1362(sh) 1375(m) 1385 (w) 1403(w, b) 142O(vw) 1425(vw) 1431 (w) 1435 (vw) 1444(w) 1459(s) 1465(m)
W11 W13 WlS B&
2 x 722 B B& B
n-C%Hss 720(s) 733(s) i 748(m) 750(m) 1 773 (m) 810(m) 853 (m)
Ws
WP
Ws
AssignmentP
1
n-CsTHss (Continued)
1473(s)
wz wa wq
Frequeue~);s~ b
889(m) 893 (m) i 899 (w) 901 (w) 944 (w) 947 (w) 1 983(vw) 987(w) 991 (w) 1 1012(w) 1024(w) 1027(w) 1034(vw) 1039(w) 1046 (w) 1053 (w)
Rt R11 R13 RI5 R17 R
CH3
R19
RZl
1061 (w) 1087(m) 1128(m) 1175(vw) 1180(w) 1184(vw) 1192(vw) 1198(vw) 1207(vw) 1218(vw) 1233(vw) 1258(vw) 1266(vw) 1284(w) 1287 (w) 1300(w) 1308(w) 1332(vw) 1365(m) 1373 (w) 1379(w) 1382(sh) 1411 (VW) 1417(vw) 1424(vw) 1429 (sh) 1437(w) 1444(w) 1453(w) 1457(sh) 1462(s) 1465(m) 1476(s)
R23
Ws Ws W, Wg
WI1 B&I
B B& B
n-CzsH99
722(s) 731(m) )1 734(s) j 1 740(m)
744(m) 750(m) 767 (m)
RI R9
J
R11
TABLE
RI3
771(m)
970(w) 977 (w) 1 980(w)
R23
997(w, II?) 1005 (VW) 1009(w, 1015(w,
I) R I) 1 % I)
1018(w, 1024(vw) 1030 (VW) 104l(w,
I)
1047(w, II) 1054(vw) 1059(vw) 1063(sh) 1068(w, I)
1254(w) 126O(vw) 1265(vw) 127O(vw) 1278(w) 1279 (VW) 1295 (VW) 1301 (w) 1323 (w) 1336(sh) 1342(w) 1357(w) 1365(sh) 1375(m) 1385(w) 1400(w) 1418(vw) 1423 (VW) 1428(w) 1432 (VW) 1435(sh) 1438(sh) 1444(w) 1458(w) 1462(s) 1466(m) 1475(s)
1082(vw, II) 109O(w, II) llll(w, I) 113O(w, II) 1136(w, I) 1173(vw) 1181 (w) 1188(sh) 1192(w) 1206(w) 1218(w) 1221 (VW) 1230(w) 1238(vw) 1242(vw) 1244(vw) 1249 (VW)
I-continued
W?
994(w) 999(w) WS
2 X 722 B BEH~ B
n-Go&z
WP Wa W4
722(s) 735 (9) 1 742&j
Rll
762(m)
Ri3
R1
15 829 (m) 870 (m)
R17 R19
Ws we
913(w). 915(w) 954(w) 958(w) I
97O(vw) 977 (VW)
RZl R23
1
1026 (w) 1030 (w) 1042(sh) 1047 (m) 1057 (w) 1063 (w) 1092 (m) 1129(m) 1175(vw) 1177(vw) 1180(w) 1185(w) 1191 (VW) 1196(?) 1206(w) 1214(vw) 1230(w) 1244(vw) 1253(w) 126O(vw) 1275(sh) 1282(w) 1297 (w) 1302(w) 1305 (w) 1310 (VW) 1324(w) 1344(w) 1361 (sh) 1367(m) 1375(w) 1382(w) 142O(vw) 1425 (VW) 1431 (VW) 1438(vw) 1445 (VW) 1461(s) 1466(sh) 1474(s)
R26
Wa Ws Wr Ws(?) Wll
B&
B BOHR B
a w, m, s, v, sh, and b indicate weak, medium, strong, very, shoulder, and broad, respectively. b Frequencies to i 1 cm-l. c B, W, and R are methylene bending, wagging, and rocking, respectively. Numbers as as rymmetric and subscripts are the assigned k values. B& and B &n3 are,espectively, symmetric methyl bending. Ron* is out-of-plane methyl rocking. d Unresolved. 421
SPECTRA SELECTION
OF CRYSTALLINE
n-PARAFFINS
RULES AND NORMAL MODES n-PARAFFIN MOLECULES
423
OF ISOLATED
Selection rules for n-paraffins having odd and even numbers of carbon atoms are given in Table II. These selection rules are for the fully extended form of the molecules, which have point group symmetries Czv and &, depending on whether the number of carbon atoms in the molecule is odd or even, respectively. experimental (b,5) and theoretical (6,7) studies of the spectra of ~-paraffil~s by earlier workers have shown the very useful fact that the d~tributions of frequencies for modes of the same basic motion are confined to relatively well-defined regions. Although there is some overlapping, at least part of it is between modes which belong to different symmetry species. There is, however, still considerable uncertainty as to the limits of the regions occupied by the different types of modes. Estimates of the location of these regions by others are given in Table III t.ogether with estimates based on the present work in the cases of methylene wagging and rocking modes and on unpublished results in the cases of methylene testing and CC stretching. SIMPLE
VIBRATIONAL
THEORY
FOR TRE
FINITE
CHAIN
Theoretical treatments of the problem of vibrations of the n-paraffin molecule which attempt to account for interaction between different groups of fundamentals, such as between methylene rocking modes and methylene twisting modes, have been published (6’, 7). The sensitivity of the numerical results of these calculations to certain interaction constants whose values are not known, and to perturbations from the methyl end groups, which are not taken into account, do not permit quantitative agreement between calculated and observed frequencies of normal modes. However, as will be seen later, qualitatively the solutions agree with the experimental findings presented here. TABLE
REGIONS OF THE SPECTRAOF
III
CRYSTALLINE
~-PARAFFINS
CHARACTERISTIC
OF SPECIFIC VIBRATIONS Brown et nl. (3, 5)
.Mode
This research
-~
CHZ racking
72%1050
72(rm1050
CC stretching CHJ twisting
884-1165 1176-1243
890-1150 1100-1350
970-1140” 1170-1300’
CH2 wagging
1280-1445 1480-1640
1200-1350 1470
1175-1415 A470
CHZ bending
a Conclusions drawn from work to be published.
424
SNYDER
No attempt will be made here to improve the theoretical treatments existant (F-8). Rather, the simple hypot,hesis is proposed and found to be adequate that the frequencies of the methylene rocking and wagging modes are functions of a single parameter which can be readily calculated for a given mode of any n-paraffin. In establishing the empirical relation between this parameter and the normal mode frequencies, a number of coefficients are involved which contain implicitly kinetic and potential energy terms. The numerical values of the coefficients are compared to the same coefficients calculated from theory. It will be helpful to describe in some detail a very simple model of the vibrational problem in order to establish some perspective for later arguments. A simple model is described by the following assumptions: (1) All methylene groups and CC bonds are considered to be entirely equivalent regardless of their position in the chain. (2) The methyl groups at the ends of the molecule are not considered. (3) Co~~pling between different types of methylene and skeletal motions are not considered. (4) No interaction force constants between different methylene and/or skeletal coordinates are considered. These assumptions permit an easy zero order approximation to the solution of the problem. Considering the coordinates Sl , S, , * * * S, for a particular vibration of a system of m oscillators (i.e., methylene groups, CC bonds), which are numbered consecutively beginning at one end of the molecule, we find, using the GF notation (9), a diagonal F matrix with a constant element f on t,he diagonal and a G matrix which has a constant diagonal element ga , and may or may not have constant nonvanish~g nearest off-diagonal and next nearest off-diagonal elements $6 and gc , respectively. All other G elements are zero. Then the ordinary form of the secular equation IGF-XE)=O
is found to have the simple form a0 -
X
ai
al
a0 -
a2
al
0
... ...
a2
X
a2
0
al
a2
a0 al
..‘..I.....................
x
i ii ;j =O
al a() .
x : i
_ .
.
(1)
.
.
.
.
.
*I .
Fith X = 4*‘~’ 1 and a0 = gaf, al = $bf, and a2 = g,J. All other elements vanish. 1The units of X used in the text following are such that force constants are given as millidynes/angstrom and masses and lengths in gram atomic units and angstroms, respectively, so that x = 5.889 x for w in cm-l.
10-T w*
SPECTRA
OF CRYSTALLINE
425
n-PARAFFINS
The solution is given closely by ?Xk,m
=
a0
-t
cPk,rn
=
h/(m
2%
co8 +
cPk,m
$
2a2
l)(?C = 1,2,3,
co8
2pk,m
,
(2)
* * * m).
(3)
Physically, ~h,~ is the phase difference between adjacent oscillators in the system of m osc~tors, for the kth normal mode. The integer k: characterizes each normal mode. Thus, assigning a band to a normal mode means both ident~ying it with regard to what kind of motion is involved and then associating a particular h: with it. Eigenvectors are given by ak,rn
=[z/(m
+
l)l”‘“[s~
wk,m
,
sh(m
-
l)(Ok,m , *’
’
Sk
pk.m]
(4)
from which approximate normal coordinates and intensities can be found. When coupling of various modes (rocking with twisting modes, for example) is considered, Eq. (2) must be replaced with the more general expression
The magnitude of the higher terms is a measure of the coupling. The usefulness of Eq. (5) is based on the fact that the constants aa depend on the geometry of the n-paraffin and the force constants and consequently are independent of the chain length m. Hence, once these constants are determined from the spectra of some selected n-paraffins, Eq. (5) can be used to predict the frequencies of normal modes for any other unbranched n-paraffin having a long chain length. Further, when the chain becomes infinitely long, the only modes which can be optically active are Q = 0 and cp = a. Hence, the frequencies of these fundamentals can be estimated. The convention as to what constitutes in-phase and out-of-phase motion between two methylene groups is arbitrary and must be defined for the purposes of the discussion following. The phase angle cpwill be related to the gerade (g) or ungerade (u) property of the motion of adjacent oscillators with respect to the symmetry operation of inversion through the midpoint of the CC bond joining the two groups. Then we define for methylene groups, u:cp = 0; g:p = r. TEE ~ETHYLENE
ROCKING
MODES
Of the series listed in Table III, the methylene rockmg modes are the easiest to analyze. The rocking modes develop dipole moments perpendicular to the plane of the molecule. Although it is possible to factor these modes into active and inactive symmetry species, it is more convenient for our purposes to note in Eq. (3) that for infrared active vibrations, k must be an odd integer (see Table IV). Identification of the rocking mode absorption bands and assignment of these to the particular normal modes are not difficult. Early in the study of n-paraffins
SNYDER
426
TABLE RELATION
BETWEEN POLARIZATION,
IV INFRARED ACTTVITY, AND k FOR
ROCKING AND WAGGING k
CHz rocking
MODES CHz wagging
odd
odd even
active inactive
pm3lle1, actrive perpendicular, active
even
odd even
active inactive
active inactive
the strong absorption band at 720 cm-’ was identified as the rocking mode, k = 1 (JO). The remaining modes are seen to account for most of the stronger absorption bands in the region from 700 to 1000 cm-l, except for the constant frequency one at 890 cm-l, which belongs to a methyl rocking mode. The splitting of many of the absorption bands is an intermolecular effect (11) which will be considered in a separate publication. Kear the 720 cm-’ end of the series there is a decided convergence onto the very strong limiting mode resulting in a number of absorption bands being submerged under this strong band. In order to determine how many bands are so submerged, it was necessary bo trace the pattern back into t)he lower n-paraffins (9). At the high-frequency end of the spectrum the pattern becomes confused with the carbon-carbon stretching modes which appear in t#his region. On the assumption that there is no abrupt change either in the distribution pattern of rocking modes or in the regular manner in which doubling occurs for the odd members, it was possible to identify absorption bands through fi = m - 2 for the odd numbered members and through k = m - 3 for the even numbered ones, where m is the number of methylene groups in the molecule. However, because of the rapidly diminishing intensities with larger k’s and the appearance of unrelated absorption bands in this same frequency region where the rocking mode bands are expected, the assignment of the highest frequency infrared rocking mode was not attempted. Figure 12 shows the positions in frequency of all the observed rocking modes together wit!h their assignments ( ICvalues). When the peak was a doublet, the lower frequency band was arbitrarily chosen. Since the bands have been assigned, it is now possible to test their dependence upon cp. To this end in Fig. 13 is shown a plot of the frequency of each rocking mode versus (OIP calculated from k and m (Eq. 13)). It. is seen that all points fall very nearly on a smooth curve. There is no apparent evidence in Fig. IS that the chain length influences the frequencies other than by entering into cp. It is, t#herefore, concluded that tjhe frequencies of the methylene rocking modes are t,o an excellent approximation, a function of cp only. The scat,ter in Fig. 13 is a little greater than what could be expected from ex-
SPECTRA OF CRYSTALLINE
21 20
0
k=7 k=l ?OO
I 750
FIG.
n-PARAFFINS
I 000
I 850
a 950
I 900
IO00
II $50
12. Observed array of methylene rocking modes
1050
0% O0
1000
(D
00
oB”
0 950
@JP@
Z'E 900 Y 3 850
800
7001 0.3
0.4
, 0.5
1 0.6
8 0.7
0.8
!t n FIG.
13. Frequency-phase curve for methylene rooking modes
I 0.9
0
428
SNYDER
perimental error alone and arises at least in part, from cooperative effects between molecules and from the static crystalline field, both of which depend on crystal str~~cture. This dependence is clearly exhibited in Fig. 12 where modes having the same &, especially when k is small, appear alternately on each side of the smooth curve drawn empirically through the data points. The zig-zagging arises from the fact, that the odd numbered n-paraffins have a different crystal structure than the even n~~mbered ones (4). No attempt was made t.o allow for t,his varia~~i~)lI in frequency. In order to obtain experimental values for the constants in Eq. (5), relating the rocking mode frequency, W, to the parameter, P, the data points shown in Fig. 13 were fit,ted by least squares to the funct’ion
Omission of the constant term permitted adjustment of the function so that at for n-paraffins IL-C&I~~ cp = 0, w = 720 cm-’ . All rocking mode frequencies through ,&Z&Hso in the region T20 to 1015 cm-‘, where unc~rtaillty as to assignments is least, were used. The calculation was performed on the Electrodata computer using a rout& curve fitting program. Rdjustment of the coefficients calculated for Eq. (5a) as indicated above gives wz = (61.~1 -
21.70 cos $7 + 10.72 cosf 9 -
{6)
5.29 COBB $0 + 6.85 cos4 rpo10*
for w in cm-‘. Finally the coe~cient,s~ ai , of Eq. (5) can he readily obtained from those of Eq. (6). These are listed in Table V in the first row. For comparison there are included in Table V along with the experimentally found values of ai , values calc~llated from theory. One set of t,hese (second row of the table) is calculated on the basis of the model for which it is assumed that (1) methylene groups are equivalent, (2) end groups (met’hyl groups) can be ignored, and (3) coupling between rocking modes and other vibrations is unimportant). Numerical values are based on a diagonal P matrix whose diagonal element, is a met,hylerlc rocking force constant from malonic nitrile (12). TABLE V COEFFICIENTS
I_-
FOR THE ROCKING-MODE
Coelkient
.”
Observed CaIculatedb-no interaction Calculatedb-interaction with twisting modes (7) 8 Coefficients are for Ey. (5). tifa = 0.70 X 10FL erg&adz.
80 0.409 0.490 0.462
FREQUENCY-PHASE
a1 -0.153 -0.217 -0.201
** --
EQUATION* fza
u4
-
0.053
-0.008
0.005
0.037 0.050
0 -0.010
0 0.002
SPECTRA OF CRYSTALLINE n-PARAFFINS
429
In the third row of Table V are listed a set of coefficients based on the more sophisticated model of Primas and Gtinthard (6). This model differs from the above one in that no longer is interaction between methylene rocking modes and twisting modes neglected. The same rocking force constant from malonic nitrile is used together with a methylene twisting force constant from the same compound. The appearance of significant although small values of a3 and a4 among the experimentally observed coefficients, shows the necessity of including interactions with other vibrations. Then it is not surprising that the more elaborate model, which includes the twisting interaction yields values for these coefficients which are in better agreement with the observed ones than the simple model. It is pleasing that for all coefficients including a2 and a4 , the observed values agree reasonably well with the theoretical values (row 3 of Table V). In all cases the signs of the coefficients agree. Unquestionably a better numerical agreement could be obtained using a better F matrix. Substitution of Q = ?r into Eq. (6) gives 1033 cm-r for the frequency of the Raman active rocking mode of an infinitely long n-paraffin. In view of this limiting frequency it is difficult to accept the assignment of a line in the Raman speetrum near 1168 cm-’ or even 1130 cm-’ to this vibration (13, 14). It is of course possible that the out-of-plane methyl rocking or the methylene twisting vibrations might interact to disturb drastically the frequency-phase curve between cp = 0.94~ and cp = ‘or.However, this is deemed rather unlikely. The frequencyphase curve shows a definite trend towards leveling off near cp = 0.94a. Further, Primas and Gtinthard (6) have shown at least qualitatively that the inclusion of methylene wagging interaction, affects the frequency-phase curve in producing some over-all displacemellts, but no sharp breaks. Finally, there is good evidence (3) that the methyl out-of-phase rocking mode is near 890 cm-r and consequently strong interaction near 1100 cm-’ would not be expected. Only one line exhibiting appreciable intensity and having a frequency less than 1100 cm-’ is present in the Raman spectrum of polyethylene (14). This line corresponds to a frequency of 1060 cm-‘. Since it is unlikely that this mode has a very low intensity (15)) the 1060-cm-’ line is probably to be assigned to the Raman active methylene rocking mode. THE ~ETHYLE~E
WAGGING MODES
Unfortunately, the infrared intensities of these modes are much less than those of the rocking modes. Substitution of polar groups for the terminal methyl groups can enhance their intensity greatly. However, since the primary concern of this report is the unperturbed crystalline n-paraffins, the intensity problem will have to be lived with. The absorption bands usually attributed to wagging modes can be seen in the spectra in the region 1400 to 1175 cm-“.
430
SNYDER 30 29 20 27 26
n
25 24 23 22 21 20
1175
1200
1225
1250
1275
1300
1325
1350
fd(cni’) FIG.
14. Observed
array of methylene
wagging modes
Even numbered molecules have pi(n - 2) infrared active wagging modes, whose resultant dipole is in the plane of the carbon skeleton. The odd numbered n-paraffins have n - 2 infrared active wagging modes all of which also give dipoles in the skeletal plane. These latter bands are, however, alternately parallel and perpendicularly polarized. In the region 1340-1200 cm-’ even numbered molecules have four or five absorption bands regularly spaced with intervals of about 35 cm-’ for n-CBOH42 and decreasing to about 22 cm-’ for n-C28H58 , which can safely be assigned to wagging vibrations (9). Since the odd numbered ones have approximately twice as many absorption bands permitted, it is to be expected that separations should be about half that observed for the even numbered ones. Indeed, such is the case as seen in Fig. 14 where the observed array of wagging modes is shown. Polarization measurements have been made on oriented films (see Experimental) of n-C&H,, in the wagging mode region of the spectrum. The results are given in Table I where bands are designated as being polarized perpendicular (I ) or parallel ( /1) to the chain axis (2). With the wagging modes for the odd n-paraffins identified, the array shown in Fig. 14 can be constructed. The question to be considered now is the assignment of the proper k to each absorption band. An answer to the problem of finding all bands having the same k can be found in a detailed look at the predicted wagging mode pattern. Table IV gives in summary the relation between k, infrared activity, and polarization which characterizes the pattern of wagging modes. A change of one unit in n will affect (P~,~ less t,han a change of one unit in 1; (Eq. (3)), since
SPECTRA
OF CRYSTALLINE
n-PARAFFINS
431
Thus the frequency change in following the kth mode from wC,H~~+~ to nC,L+1H28+4must be smaller than the spacing of the wagging mode series for a given n-paraffin. These considerations dictate the lines drawn through the absorption bands as shown in Fig. 1.4 Numerical values of k will now be considered. Assuming that the function w = f(u)) is mo~lotonic, the lines through absorption bands of given k values must, with increasing n tend toward frequencies associated with smaller ‘p. Thus the low-frequency region must be for the smallest k’s. Numbering can be established from the array by selecting two absorption bands of different n-paraffins having nearly equal frequencies. For these bands
and consequently k ---=m+Am+I’ m+1
k + Ak
(7)
where m is the number of methylene groups of the shorter ~-para~n. Then, for example, taking the band near 1220 cm-’ for n-C&H*s and estimating Ak = 1.0 for a band of n-CBHso near this frequency, gives k = 3. This then could be taken as a starting point for the assignments shown in Fig. 14. Further support for this assignment is provided by the fact that the predicted polarization of bands (Table IV) is in agreement with that observed for n-C&Hr,z (Table I). Some complexity in the spectra occurs in the region 1180-1205 cm-‘, particularly for the members having shorter chain lengths. Identification of the band corresponding to k = 2 in this region is not difficult by virtue of its inactivity for even members and its expected position relative to the k = 3 band. A more formidable task is the location of k = 1. Two bands active in the spectra of most odd and even members are found. These are separated by about 6 cm-‘. It is possible that both are associated with the k = 1 mode. The mechanism of the splitting could not be intermolecular in origin, however, since for the odd numbered members the selection rules for the crystal allow infrared activity for only one component of a parallel band. That part of the complexity in this region is due to bands accountable to another type of methylene motion is suggested by certain regularities in the positions and activities of the absorption bands which remain after the wagging modes are el~i~ted from the spectra. It is found that the overall pattern of these residual bands is consistent with selection rules for twisting modes. Others (2) have, on the basis of a study of shorter n-paraffins placed the methylene twisting modes in this region. Thus the pattern and the position of these bands seem to indicate that they are to be assigned to twisting modes. The frequency-phase curve for methylene wagging modes based on the above
482
SNYDER 137500 1350-
09 a*O
1325-
00
1300whli')
o@ 8@
1275-
1200 0
,p?+@@ , 0.1 0.2
I 0.3
I 0.4
I 0.5
I 0.6
7
+ i? FIG. 15. Frequency-phase
curve for methylene
wagging
modes
is shown in Fig. 15. The scatter of the points from a smooth curve can be attributed to experimental error in determining the frequencies. The excellence of the fit again seems to speak well for the assignments. A least-squares analysis of the data gives the equation:
assignments
w’ = (177.6 -
21.7 cos P -
4.4 cos’ cp -
8.5 cos3 cp)104.
(8)
The data did not justify determining more than the first four constants. The constants in Eq. (8) together with constants derived for wagging modes uncoupled from all other modes, for which a force constant from malonic nitrile (12) is used, are given in Table VI. The agreement is probably as good as could be expected for the assumptions made in calculating these for the simple model. There is a serious difference between the location of the infrared active wagging mode in polyethylene, drawn from the assignments of wagging modes made here and what has been concluded from other evidence by other researchers. This asTABLE
VI
COEFFICIENTSFOR THE WAGGING-MODE FREQUENCY-PHASEEQUATIONS Coefficient
Observed Calculatedb
(no interaction)
8 Coefficients are for Eq. (5). b fw = 0.663 X lo-” ergs/rad2.
I
a0
a,
a2
a
1.040 1.036
-0.165 -0.079
-0.013 -0.004
0.012 -
SPECTRA
OF CRYSTALLINE
n-PARAFFINS
4x3
signment has been argued over the last several years. Recently there has been attention given to this problem by Krimm et al. (16) who conclude that this mode appears near 1370 cm-l as one component of a complex absorption band. The evidence cited for this assignment will be reviewed brieffy. This fundamental cannot be recognized on the basis of intensity since all the strong infrared bands in polyethylene have been accounted for already. Thus, what is sought is a weak band of parallel polarizatio1~. Two such bands have been considered. However, one, at 1304 cm’-“, is ruled out as being characteristic of only the amorphous form of the polymer. This leaves the 1370-cmS-1component which meets the polarization requirement and is characterist,ic of methylene groups as proved by its persistence in the case of end substituted n-paraffins (11). Also it does appear in the spectrum of the crystalline polymer. Against this assignment is the argument based on the observed distribution of wagging modes and the agreement of the experimentally found constant, al , with that predicted from the simple model (Table VI). Putting into Eq. (8) the limit cp = 0, one obtains for the infrared fundamental a value of 1196 cm-‘. It is suggested that possibly this mode is the 1175-cm-’ band found in the spectrum of polyethylene by Nielsen and Woollett (14). These authors report this weak band has parallel polarization, is characteristic of the crystalline polymer, and is not readily interpretable otherwise. However, more to be stressed here than finding a new assignment, is the fact that it is inappropriate to look near 1370 cm-l for this fundamental. Substituting cp = rr into Eq. (9) gives 1425 cm-l for the Raman active methylene wagging fundamental. Recently, evidence for assigning the Raman active mode in polyethylene to a line at 1415 cm-’ has been presented (24). The assignment is in accord with Raman intensities calculated by Theimer (15). Taken together with the results here, the Raman fundamental seems established. RECEIVED: August 11, 1959 REFERENCES 1. A. A. SCHAERER,C. J. BUSSO, A. E. SMITH, AND L. B. SKINNER, J. Am. Chem. Sot. 77, 2017 (1955).
8. J. K. BROWN, N. SEEPPARD, AND D. M. SIMPSON, Trans. Roy. Sot. A247,35 (1954). 8. E. L. WAGNER AND D. F. HORNIO, 2. Chem. Phys. 18, 296(1950). 4. A. E. SMITH, J. Chem. Phys. 21, 2229 (1953). 6. J. K. BROWN, N. SHEPPARD,ANDD. M. SIMPSON,Disc. ~ura~a~ See. 9,261 (1950). 6. H. PRIMAS AND H. H. G~NTHARD, H&I. Chim. Aeta 36, 1659 (1953). 7. H. PRIMAS AND H. H. G~NTKABD, Helv. Chim. Acta 36, 1791(1953). 8. 0. THEIMER, J. Chem. Phys. 2’7, 408 (1957). 9. E. B. WILSON, JR., J. C. DECIUS, AND P. C. CROSS, “Molecular Vibrations.” McGrswHill, New York, 1955. 10. N. SHEPPARDAND G. B. B. M. SUTHERLAND,Nature 169, 739 (1947).
4x4
SNYDER
11. R. S. STEIN AND G. B. B. M. SUTHERLAND,J. Chem. Phys. 22 1993 (1954). 12. F. HALVERSONAND R. J. FRANGEL,J. Chem. Phys. 17,694 (1949). IS. T. SHIXANOUTI AND S. MMIZUSHI~IA, J. Chem. Pkys. 17, 1102 (1949). 14. J. R. NIELSEN AND A. H. WOOLLETT,J. Chem. Phys. 26, 1391 (1957). 15. 0. TWEIMER, J. Chem. Phys. 27, 1041 (1957). 16. S. KEIMM, C. Y. LIANG, AND G. B. B. M. SUTHERLAND,J. Chew Phys. 26, Ii49 (1956) 17. G. C. PIMENTELAND W. A. KLEMPERER,J. Chem. Phys. 23,376 (1955).