Vibrational dynamics of syndiotactic poly(1-butene)

Vibrational dynamics of syndiotactic poly(1-butene)

Poonam Sharma, Poonam Tandon, V.D. Gupta *

Department of Physics, University of Lucknow, Lucknow 226 007, India

Received 10 August 1999; received in revised form 2 November 1999; accepted 26 January 2000

Abstract

The normal modes and their dispersions for form 1 of syndiotactic poly(1-butene) have been obtained using WilsonÕsGF matrix method as modi®ed by Higgs. Optically active frequencies corresponding to the zone center and zone

boundary are identi®ed and discussed. The Urey Bradley potential ®eld is obtained by least square ®tting of the ob-

served Fourier transform infrared absorption bands. Speci®c heat has been obtained from dispersion curves via density

of states. A comparative study of isotactic and syndiotactic forms of poly(1-butene) is presented. Ó 2000 Elsevier

Science Ltd. All rights reserved.

Keywords: Phonons; Dispersion curves; Density of states; Syndiotactic; Urey Bradley ®eld; Heat capacity

1. Introduction

Poly(1-butene) exists in two tactic forms, syndiotactic

and isotactic. Syndiotactic poly(1-butene) crystallizes in

two di�erent crystalline forms, namely form 1 and form

2 [1], whereas the isotactic form exists in three crystalline

forms. Form 1, obtained by cooling from the melt, has a

hexagonal unit cell containing six 3/1 helices; form 2 has

a tetragonal unit cell with four 11/3 helices, and form 3

has an orthorhombic unit cell having two 4/1 helices [2±

6]. Isotactic poly(1-butene) (iPB) is prepared by casting

from decalin, benzene, p-xylene, toluene or carbon tetra-

chloride. Di�erent forms were obtained by di�erent prep-

aration methods [7]. Syndiotactic poly(1-butene) (sPB) is

obtained by syndiospeci®c polymerization of 1-alkenes

in the presence of homogenous catalysts based on group

4A metallocene/methylalumoxane systems [8±12]. Both

the forms, form 1 and form 2 of sPB are characterized

by helical chain conformation of the TTGG kind. sPB

goes into s(2/1) helical conformation packed in an or-

thorhombic unit cell with a � 16:81 �A, b � 6:06 �A and

c � 7:73 �A [13]. A symmetry of the kind s(2/1)2 with a

chain repetition of 7.73 �A has been found for form 1

while form 2 has a (5/3) helical symmetry with a chain

repetition of 20.0 �A [1]. The crystal structure of form

2 has been presented by De Rosa and Scaldarolla

[14]. Form 1 is found to be thermodynamically stable,

whereas form 2 exists only under tension [1,14]. The

infrared spectra of form 1 of sPB cast from ®ve di�erent

solvents, chloroform, benzene, toluene, xylene and car-

bon tetrachloride reveal the same spectral pattern which

coincides with the spectrum of form 1 of sPB obtained

by cooling from the melt. These di�erent solvents pro-

duce the same structural form and hence the same

spectra [7].

Normal coordinate calculations and assignment for

various vibrational modes for form 1 of sPB based on

valence force ®eld of Holland-Moritz and Sansen [2]

have been reported earlier by Ishioka et al. [7]. However,

for full understanding of vibrational spectra, it is nec-

essary to know the dispersive nature of normal modes.

Such studies also provide information on the degree of

coupling and dependence of the frequency of a given

mode on the sequence length of the ordered conforma-

tion. These curves are also useful in calculating the

density of vibrational states which in turn can be used

for obtaining thermodynamic properties such as speci®c

heat, entropy, enthalpy and free energy.

European Polymer Journal 36 (2000) 2629±2638

* Corresponding author. Fax: +91-522-223405/+91-522-

223938.

E-mail address: [email protected] (V.D. Gupta).

0014-3057/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S0 01 4 -3 05 7 (00 )0 0 04 9 -5

Recently, much work has been done on phonon

dispersion and heat capacity of a variety of polymeric

systems and the present authors have demonstrated its

potential in the full interpretation of the observed

spectra [15±17]. In continuation of this work, we report

here calculations of normal modes and their dispersion

within the ®rst Brillouin zone for form 1 of sPB. The

calculations are based on Urey Bradley force ®eld, which

in addition to valence force ®eld accounts for the intra-

unit nonbonded and tension terms. The density of states

obtained from dispersion curves is used to calculate the

heat capacity of sPB in the temperature range 50±500 K.

2. Theory

2.1. Calculation of normal mode frequencies

The calculation of normal mode frequencies has been

carried out according to WilsonÕs GF matrix method

[19±21] as modi®ed by Higgs [22] for an in®nite poly-

meric chain. In brief, the vibrational secular equation,

which gives the normal mode frequencies, has the form

jG�d�F �d� ÿ k�d�I j � 0; 0 6 d 6 p; �1�

where G is the inverse kinetic energy matrix, F is the

potential energy matrix and d is the vibrational phase

di�erence between the corresponding modes of adjacent

residue units.

The vibrational frequencies mi(d) (in cmÿ1) are related

to the eigenvalues ki(d) by the following relation:

ki�d� � 4p2c2m2i �d�: �2�

A plot of mi(d) versus d gives the dispersion curve for

the i-th mode.

2.2. Calculation of heat capacity

Dispersion curves can be used to calculate the heat

capacity of a polymeric system. For a one-dimensional

system, the density-of-state function or the frequency

distribution function is calculated from the relation

g�m� �X

j

�dmj=dd�ÿ1jmj�d��m: �3�

The sum is over all branches j. Considering a solid as

an assembly of harmonic oscillators, the frequency dis-

tribution g(m) is equivalent to a partition function.

The constant volume heat capacity Cv can be calcu-

lated using DebyeÕs relation

Cv �X

j

g�mj�kNA�hmj=kT �2 exp�hmj=kT ��exp�hmj=kT � ÿ 1�2 �4�

with

Zg�mj�dmj � 1: �5�

3. Results and discussion

A chemical repeat unit of sPB is shown in Fig. 1. sPB

has 24 atoms in a two residue repeat unit which give rise

to 72 dispersion curves. The frequencies of vibrations

have been calculated for phase values ranging from 0 to

p at an interval of 0.05p. The calculated frequencies at

d � 0 and d � p are optically active. Initially, approxi-

mate force constants were transferred from isotactic

poly(4-methyl,1-pentene) (PMP) [18]. These force con-

stants are then modi®ed to obtain the ``best'' ®t between

the calculated frequencies at d � 0 and d � p and the

peaks observed in the recorded FTIR spectra. The ®nal

force constants along with the internal coordinate are

given in Table 1. Since all the modes above 1350 cmÿ1

are nondispersive in nature, the dispersion curves are

plotted only for the modes below 1350 cmÿ1. The lowest

two branches of the dispersion curves correspond to the

four acoustic modes (m � 0 at d � 0 and d � p). Out of

these, three are due to the translation (one parallel and

two perpendicular to the chain axis) and one due to

rotation around the chain. For the sake of simplicity, the

modes are discussed under three separate sections viz

backbone, side chain and mixed modes.

3.1. Backbone modes

Modes involving the motion of the skeletal atoms are

termed as backbone modes. These modes along with

their assignments at both d � 0 and d � p are given in

Table 2. The calculated frequencies at 2920, 2915, 2912

and 2909 cmÿ1 having a mixed contribution of asym-

metric stretches of methylene and methine bond stretch

are assigned to the peak observed at 2918 cmÿ1 in the

FTIR spectra (Fig. 2). The width of this peak and its

asymmetric nature justi®es the assignment. The sym-

metric stretch of methylene group calculated at 2861

cmÿ1 corresponds to the observed frequency at 2866

Fig. 1. One chemical repeat unit of spoly(1-butene).

2630 P. Sharma et al. / European Polymer Journal 36 (2000) 2629±2638

cmÿ1. The CH2 scissoring modes calculated at 1467 and

1463 cmÿ1 correspond to the observed peak at 1463

cmÿ1. Our assignments are supported by the fact that

these modes are observed in a similar range in iPMP and

other synthetic polymers. Being highly localized, all

these modes are nondispersive.

3.2. Side chain modes

The side chain of sPB consists of a methyl and a

methylene group. Pure side chain modes and their as-

signments are given in Table 3. The infrared spectra of

sPB exhibit strong peaks at 2959 and 2874 cmÿ1 which

are assigned to the asymmetric and symmetric stretches

of methyl group calculated at 2963 and 2870 cmÿ1, re-

spectively. The calculated frequencies at 2911 and 2862

cmÿ1 which are due to the asymmetric and symmetric

stretch vibrations of the CH2 group correspond respec-

tively to the peaks observed at 2918 and 2866 cmÿ1.

These assignments are reported in a similar range in iPB

[4]. The asymmetric bending of CH3 calculated at 1463

cmÿ1 matches with the observed frequency at the same

value. The peak observed at 1463 cmÿ1 is also assigned

to the modes calculated at 1474 and 1454 cmÿ1 which

have a mixed contribution of the CH2 and CH3 scis-

soring modes. No other peak is observed around these

values and since the half-width at half-maximum is ap-

proximately 22 wave numbers, and in addition, it has an

asymmetric pro®le; hence, the 1463 cmÿ1 peak may

contain the peaks observed at 1474 and 1454 cmÿ1. The

second derivative of the spectra in this region is clearly in

support of it. Mixing of CH3 group bend and Cb±Cc±H

in opposite phases is observed in the mode calculated at

1390 cmÿ1 which corresponds to the observed peak at

1388 cmÿ1 in the IR spectra. These assignments are in

good agreement with the normal mode analysis reported

earlier by Ishioka et al. [7]. The peak observed at 781

cmÿ1 is assigned to the calculated frequencies at 778 and

773 cmÿ1. Both these modes are pure side chain modes

with mixed contributions of CH2 and CH3 rocking.

3.3. Mixed modes

The modes which involve the motion of back bone as

well as side chain atoms are termed as mixed modes. The

mode at 1312 cmÿ1 mainly involves the wagging of the

methylene groups present in both backbone (bb) and

side chain (sc) and is assigned to the peak observed at

1313 cmÿ1. The major contribution of the bb CH2

twisting is contained in the mode calculated at 1163 and

1095 cmÿ1. These modes are highly coupled with the side

chain vibrations and are assigned to the observed peaks

at 1161 and 1074 cmÿ1, respectively. The mode at 1123

cmÿ1 has a major contribution coming from the side

chain CH2 twisting and 20% contribution of CH bend-

ing. This mode is assigned to the observed peak at 1108

cmÿ1 and does not show much dispersion. The calcu-

lated frequency at 1027 cmÿ1 with major contributions

from CH3 and CH2 (side chain) rockings, matches well

with the observed frequency. The two methyl group

rocking from the two units which constitute a repeat

unit of sPB are calculated at 1008 and 997 and 965 and

960 cmÿ1. These are assigned to the peak appearing at

1000 cmÿ1 (Raman spectra) [6] and 972 cmÿ1 (IR spec-

tra). In the case of PMP [18], the corresponding fre-

quencies are calculated at 1007 and 986 cmÿ1. Another

mixed mode at frequency 935 cmÿ1 has contributions of

CH3 rocking and C±C stretches from both the main and

side chains. This is assigned to the observed peak at 924

cmÿ1. The modes calculated at 798 and 725 cmÿ1 are the

other two mixed modes assigned to the observed fre-

quencies at 807 and 725 cmÿ1. These modes show very

little dispersion.

Comparison of the observed frequencies at d � 0 for

syndiotactic and isotactic poly(1-butene) is shown in

Table 1

Internal coordinates and Urey Bradley force constants (md/�A)

Internal coordinates Force constants

m[Ce±H] 4.210

m[Ce±Ca] 3.220

m[Ca±H] 4.377

m[Ca±Cb] 2.860

m[Ca±Ce] 2.050

m[Cb±H] 4.210

m[Cb±Cc] 2.100

m[Cc±H] 4.140

s[Ca±Cb] 0.022

s[Cb±Cc] 0.007

s[Ce±Ca] 0.024

s[Ca±Ce1] 0.220

/[H±Ce±H] 0.414(0.31)

/[H±Ce±Ca] 0.4235(0.2)

/[Ca±Ce±Ca] 0.540(0.17)

/[Ca±Ce±H] 0.4235(0.2)

/[Ce±Ca±Ce1] 0.50(0.175)

/[Ce1±Ca±Cb] 0.50(0.175)

/[Cb±Ca±H] 0.372(0.20)

/[H±Ca±Ce] 0.414(0.20)

/[Ce±Ca±Cb] 0.54(0.175)

/[Ca±Cb±H] 0.4667(0.2)

/[H±Cb±Cc] 0.393(0.20)

/[Ca±Cb±Cc] 0.35(0.175)

/[H±Cb±H] 0.405(0.295)

/[Cb±Cc±H] 0.441(0.22)

/[H±Cc±H] 0.384(0.32)

/[H±Ca±Ce1] 0.345(0.20)

Note: m, /, x, s denote stretch, angle bend, wag and torsion,

respectively. Stretching force constants between the nonbonded

atoms in each angular triplet (gem con®guration) are given in

parentheses.

P. Sharma et al. / European Polymer Journal 36 (2000) 2629±2638 2631

Table 4. Except for a few, most of the bands observed in

the IR spectra appear in the same range for both the

tactic forms. The di�erence in the observed frequency

arises mainly because of the placement of the side

groups in di�erent lateral positions which in turn brings

about the change in interaction constants. It is these

constants which are responsible for the frequency shifts

in the two tactic forms. A greater change is expected in

the low frequency region, but because of the nonavail-

ability of the spectra below 700 cmÿ1, comparison is not

possible.

3.4. Dispersion curves

Dispersion curves provide information on the extent

and the degree of coupling. Dispersion curves also help

in the understanding of both the symmetry-dependent

and symmetry-independent spectral features. In general,

the IR absorption spectra and Raman spectra from

polymeric systems are complex and cannot be unraveled

without the full knowledge of the dispersion curves. The

regions of high density of states, which are observable in

the IR and Raman spectra under suitable conditions,

depend on the pro®le of the dispersion curves which play

an important role in the thermodynamical behavior of

the system. Mixed modes are the ones which show the

maximum dispersion. These modes are given in Table 5.

Zone center modes at 1335 and 1312 cmÿ1 have a con-

tribution from the CH2 wag present in the main and side

chains, respectively. On moving away from the zone

center, both the modes have an increasing contribution

from the main chain CH2 and have a tendency to bunch

near the zone boundary. The mode at 1295 cmÿ1, at

d � 0 has 29% and 18% contribution coming from CH2

wagging of the main and side chains. Besides this, it has

20% contribution from the C±C stretch of bb and 12%

from CH bending. This mode disperses to 1287 cmÿ1 at

d � p with 48% contribution coming from CH2 wagging

of side chain. In the mode calculated at 1163 cmÿ1 (at

d � 0), on further increase of d, the contribution of side

chain twisting becomes dominant at d � 0:9p. This

Table 2

Pure back bone modes

Frequency (cmÿ1) Assignments % PED at d � 0

Calculated Observed

2921 2918 m[Ca±H]�76� � m[Ce±H](20)

2920 2918 m[Ca±H]�50� � m[Ce±H](48)

2861 2866 m[Ce±H](95)

1467 1463 /[H±Ce±H]�75� � /[H±Ce±Ca]�9� � /[Ca±Ce±H](6)

1340 1353 /[Ca±Ce±H]�23� � m[Ce±Ca]�19� � /[H±Ce±Ca]�19� � m[Ca±Ce]�9� � m[Ca±Cb]�7� � /[H±Ca±Ce](5)

49 ± s[Ca±Ce1]�25� � s[Ce±Ca]�20� � /[Ce±Ca±Ce1]�19� � /[Ca±Ce±Ca]�10� � /[Ce±Ca±Cb]�6� � /[Ce1±

Ca±Cb](5)

0 ± s[Ca±Ce1]�29� � s[Ce±Ca]�27� � /[H±Ce±Ca]�13� � /[Ca±Ce±H]�13� � /[H±Ce±H]�5� � /[Ca±Ce±

Ca](5)

0 ± /[Ca±Ce±H]�35� � /[H±Ce±Ca]�33� � /[H±Ce±H]�15� � /[Ca±Ce±Ca](14)

Assignments % PED at d � p

2921 2918 m[Ca±H]�59� � m[Ce±H](37)

2920 2918 m[Ca±H]�60� � m[Ce±H](39)

2861 2866 m[Ce±H]�94� � m[Cb±H](5)

1466 1463 /[H±Ce±H]�77� � /[H±Ce±Ca]�10� � /[Ca±Ce±H](6)

1336 1353 m[Ce±Ca]�27� � /[Ca±Ce±H]�24� � /[H±Ce±Ca]�16� � /[H±Ca±Ce]�9� � m[Ca±Ce](6)

47 ± s[Ce±Ca]�34� � s[Ca±Ce1]�15� � /[Ca±Ce±Ca]�10� � s[Ca±Cb]�8� � /[Ce±Ca±Ce1]�7� � /[Ce1±Ca±

Cb]�7� � /[Ce±Ca±Cb](6)

0 ± s[Ca±Ce1]�29� � /[H±Ce±Ca]�17� � /[Ca±Ce±H]�15� � s[Ce±Ca]�12� � /[Ce±Ca±Ce1]�10� � /[Ca±

Ce±Ca]�8� � /[H±Ce±H](5)

0 ± /[Ca±Ce±H]�35� � /[H±Ce±Ca]�33� � /[H±Ce±H]�15� � /[Ca±Ce±Ca](14)

Fig. 2. FTIR spectra of sPB (3000±700 cmÿ1).


mode disperses by 14 wave numbers and tends to come

close to 1123 cmÿ1 wherein the contribution of the side

chain CH2 twisting decreases from 71% to 22% over the

full d range. At d � p, this mode with 41% contribution

from bb CH2 twist, appears at 1149 cmÿ1.

The mode at 1095 cmÿ1 is a highly mixed mode at

both the zone center and the zone boundary. At the zone

center, this mode has a mixed contribution of CH2 (bb)

twisting and CH bending but CH2 twisting (sc) starts

mixing with it as d progresses. On reaching the zone

boundary, it shows a dispersion of 16 wave numbers.

The mode calculated at 1020 cmÿ1 has the zone center

PED as CH3 rocking 24%, m(Ce±Ca) 20%, /(H±Ca±Ce)

14%, m(Ca±Cb) 7%. As the d value is increased, the

contribution of CH3 rocking increases and the mode

comes very close to the mode at 1011 cmÿ1 (at d � p)

which has dominant contribution from CH3 rocking.

Modes calculated at 1229, 884, 755, 489, 213 and 170

cmÿ1 are some other mixed modes which disperse by 15±

20 wave numbers and are shown in Figs. 3±6.

Table 3

Pure side chain modes


Calculated Observed

2963 2959 m[Cc±H](99)

2963 2959 m[Cc±H](99)

2911 2918 m[Cb±H](94)

2870 2874 m[Cc±H]�94� � m[Cb±H](5)

2870 2874 m[Cc±H]�95� � m[Cb±H](5)

2869 2874 m[Cc±H](100)

2869 2874 m[Cc±H](100)

2862 2866 m[Cb±H]�90� � m[Cc±H](5)

1474 1463 /[H±Cc±H]�45� � /[H±Cb±H]�37� � /[Ca±Cb±H](7)

1474 1463 /[H±Cc±H]�47� � /[H±Cb±H]�37� � /[Ca±Cb±H](7)

1464 1463 /[H±Cc±H](93)

1454 1463 /[H±Cc±H]�47� � /[H±Cb±H]�40� � /[Ca±Cb±H](8)

1454 1463 /[H±Cc±H]�45� � /[H±Cb±H]�41� � /[Ca±Cb±H](7)

1390 1388 /[Cb±Cc±H]�49� � /[H±Cc±H]�45� � m[Cb±Cc](5)

1390 1388 /[Cb±Cc±H]�49� � /[H±Cc±H](45)

1276 1265 /[H±Cb±Cc]�35� � /[Ca±Cb±H]�31� � m[Ca±Cb]�10� � m[Cb±Cc](7)

997 1000 /[Cb±Cc±H]�50� � /[Ca±Cb±H]�11� � m[Ca±Cb]�9� � /[Cb±Ca±H]�7� � /[H±Ca±Ce1](5)

773 781 /[H±Cb±Cc]�23� � /[Ca±Cb±H]�20� � /[Cb±Cc±H]�11� � m[Cb±Cc]�10� � /[H±Ce±Ca]�6� � m[Ca±

Cb]�6� � m[Ca±Ce](5)

Table 4

Comparison of modes of sPB and iPB

Assignments Frequency (cmÿ1)

Syndiotactic Isotactic

CH3 asymmetric stretch 2959 2961, 2958

CH3 symmetric stretch 2874 2874

CH stretch 2918 2914

CH2 asymmetric stretch 2918 2914

CH2 symmetric stretch 2866 2851

CH3 scissoring 1463 1463, 1458

CH2 scissoring (backbone) 1463 1441

CH2 scissoring (side chain) 1463 1439

CH3 symmetric deformation 1388 1380

CH2 wagging (backbone) 1353, 1286 1342, 1302

CH2 twisting (backbone) 1163, 1074 1222, 1207

CH2 rocking (backbone) 883, 807 816, 798

CH2 wagging (side chain) 1313 1366, 1321

CH2 twisting (side chain) 1265, 1108 1263

CH2 rocking (side chain) 781, 753 764, 758

CH3 rocking 1027, 972, 924 1062, 972, 924

CH bending 1176 1331


Table 5

Mix modes


Calculated Observed

2915 2918 m[Ce±H]�79� � m[Ca±H](18)

2912 2918 m[Cb±H]�46� � m[Ce±H]�32� � m[Ca±H](22)

2909 2918 m[Cb±H]�48� � m[Ca±H]�34� � m[Ce±H](17)

2862 2866 m[Cb±H]�69� � m[Ce±H](26)

2861 2866 m[Ce±H]�74� � m[Cb±H](25)

1464 1463 /[H±Cc±H]�94� � /[Cb±Cc±H](5)

1463 1463 /[H±Ce±H]�79� � /[H±Ce±Ca]�9� � /[Ca±Ce±H](7)

1312 1309 /[H±Ce±Ca]�13� � /[Ca±Ce±H]�13� � /[H±Cb±Cc]�12� � /[Ca±Cb±H]�11� � m[Ca±Ce]�11� � m[Ce±

Ca]�9� � /[H±Ca±Ce1]�8� � m[Ca±Cb](8)

1295 1286 m[Ce±Ca]�20� � /[Ca±Ce±H]�18� � /[H±Ca±Ce]�12� � /[H±Ce±Ca]�11� � /[Ca±Cb±H]�10� � /[H±Cb±

Cc]�8� � m[Ca±Cb](7)

1249 1265 /[Ca±Cb±H]�18� � /[H±Cb±Cc]�16� � /[H±Ca±Ce1]�13� � /[H±Ca±Ce]�13� � m[Ca±Cb]�7� � m[Ca±

Ce](7)

1229 1238 /[H±Ce±Ca]�24� � /[Ca±Ce±H]�20� � /[Cb±Ca±H]�16� � /[H±Ca±Ce]�14� � m[Ca±Cb]�6� � /[Ca±Cb±

H](5)

1183 1176 /[H±Ca±Ce1]�25� � /[Ca±Ce±H]�18� � /[H±Ce±Ca]�13� � /[Ca±Cb±H]�11� � /[H±Cb±Cc]�9� � /[H±

Ca±Ce](8)

1173 1176 /[Cb±Ca±H]�29� � /[H±Ce±Ca]�15� � /[Ca±Ce±H]�15� � /[Ca±Cb±H]�11� � /[H±Ca±Ce](9)

1163 1161 /[H±Ce±Ca]�32� � /[Ca±Ce±H]�29� � /[Ca±Cb±H]�10� � /[H±Ca±Ce](5)

1123 1108 /[H±Cb±Cc]�37� � /[Ca±Cb±H]�34� � /[H±Ca±Ce]�12� � /[H±Ca±Ce1](9)

1118 1108 /[Ca±Cb±H]�32� � /[H±Cb±Cc]�31� � /[H±Ca±Ce]�12� � /[Ca±Ce±H]�9� � /[H±Ce±Ca](7)

1095 1074 /[Ca±Ce±H]�30� � /[H±Ce±Ca]�28� � /[Cb±Ca±H]�19� � /[H±Ca±Ce1](12)

1027 1027 /[Cb±Cc±H]�26� � m[Ce±Ca]�17� � /[H±Cb±Cc]�12� � /[H±Ca±Ce]�11� � /[H±Ce±Ca]�8� � m[Ca±

Cb](6)

1020 1027 /[Cb±Cc±H]�24� � m[Ce±Ca]�20� � /[H±Ca±Ce]�14� � m[Ca±Cb]�7� � /[H±Cb±Cc](6)

1008 1000 /[Cb±Cc±H]�52� � m[Ca±Cb]�13� � /[Ca±Cb±H]�11� � /[H±Cb±Cc](6)

965 972 /[Cb±Cc±H]�54� � m[Ce±Ca]�19� � /[Ca±Cb±H](6)

960 972 /[Cb±Cc±H]�55� � m[Ce±Ca](19)

935 924 /[Cb±Cc±H]�18� � m[Ca±Cb]�14� � m[Ce1±Ce]�14� � /[Ca±Ce±H]�11� � /[H±Ce±Ca]�6� � /[H±Ca±

Ce1]�6� � m[Cb±Cc](5)

926 924 /[Cb±Cc±H]�32� � m[Ca±Cb]�17� � m[Cb±Cc]�7� � m[Ca±Ce]�7� � /[Ca±Ce±H]�7� � /[Cb±Ca±

H]�6� � /[H±Ce±Ca](6)

884 883 m[Ca±Ce]�21� � /[Ca±Ce±H]�18� � /[H±Ce±Ca]�17� � /[Cb±Cc±H]�11� � /[H±Ca±Ce1]�9� � m[Ca±

Cb](6)

839 ± m[Cb±Cc]�42� � m[Ca±Ce]�19� � /[H±Cb±Cc]�5� � /[Ca±Ce±Ca](5)

833 ± m[Cb±Cc]�68� � /[H±Ce±Ca]�8� � /[Ca±Ce±H](5)

798 807 m[Cb±Cc]�19� � /[Ca±Ce±H]�19� � /[H±Ce±Ca]�10� � /[H±Cb±Cc]�8� � /[Cb±Cc±H]�7� � m[Ca±

Ce]�7� � m[Ca±Cb](6)

778 781 /[H±Cb±Cc]�29� � /[Ca±Cb±H]�21� � /[Cb±Cc±H]�12� � /[Ca±Ce±H]�8� � /[H±Ce±Ca]�7� � m[Ce±

Ca](5)

755 753 m[Ca±Ce]�13� � m[Cb±Cc]�13� � /[H±Cb±Cc]�13� � /[Ca±Cb±H]�12� � m[Ca±Cb]�9� � /[H±Ce±

Ca]�8� � /[Cb±Cc±H]�6� � /[Ca±Ce±Ca](5)

725 725 m[Ca±Ce]�26� � m[Ca±Cb]�23� � m[Ce±Ca]�10� � /[Ca±Ce±H]�9� � /[H±Ce±Ca](9)

489 492 /[Ce±Ca±Cb]�18� � /[Ca±Ce±Ca]�17� � /[Ca±Cb±Cc]�13� � /[Ce1±Ca±Cb]�7� � m[Ca±Cb]�6� � /[Cb±

Ca±H](5)

445 ± /[Ce±Ca±Ce1]�17� � /[Ca±Ce±Ca]�13� � /[Ce1±Ca±Cb]�10� � s[Ce±Ca]�8� � s[Ca±Ce1]�7� � /[H±Ca±

Ce1]�7� � /[Ce±Ca±Cb]�6� � /[Cb±Ca±H]�6� � /[H±Ca±Ce](5)

380 ± /[Ce±Ca±Cb]�24� � /[Ce±Ca±Ce1]�17� � /[Ce1±Ca±Cb]�17� � /[Ca±Cb±Cc]�16� � m[Ca±Ce](6)

328 ± /[Ce±Ca±Ce1]�19� � /[Ca±Cb±Cc]�18� � /[Ce±Ca±Cb]�16� � m[Ce±Ca]�6� � /[Ca±Ce±Ca]�6� � /[Ce1±

Ca±Cb](5)

313 ± /[Ce1±Ca±Cb]�20� � /[Ca±Cb±Cc]�17� � /[Ce±Ca±Cb]�17� � /[Ca±Ce±Ca]�9� � m[Ca±Ce]�8� � s[Ca±

Cb](5)

287 ± /[Ce1±Ca±Cb]�24� � /[Ce±Ca±Ce1]�17� � s[Ca±Ce1]�8� � s[Ce±Ca]�7� � m[Ca±Ce]�7� � /[Ca±Cb±

Cc]�6� � /[Ce±Ca±Cb](6)

214 ± s[Cb±Cc]�46� � s[Ca±Cb]�13� � /[Ca±Cb±Cc]�10� � s[Ce±Ca]�7� � /[Ce±Ca±Cb]�6� � /[Ce1±Ca±Cb](5)


Table 5 (continued)


Calculated Observed

182 ± s[Cb±Cc](82)

170 ± /[Ca±Cb±Cc]�28� � /[Ca±Ce±Ca]�14� � /[Ce±Ca±Cb]�10� � s[Cb±Cc]�9� � s[Ca±Ce1](7)

163 ± s[Cb±Cc]�25� � /[Ca±Cb±Cc]�19� � s[Ca±Cb]�9� � /[Ce1±Ca±Cb]�8� � /[Ce±Ca±Ce1]�5� �/[H±Ca±Ce](5)

154 ± s[Ca±Cb]�26� � s[Cb±Cc]�19� � /[Ca±Ce±Ca]�13� � /[Ce±Ca±Cb]�12� � /[Ca±Cb±Cc](7)

111 ± s[Ca±Cb]�65� � /[Ce1±Ca±Cb]�6� � s[Ca±Ce1]�6� � /[H±Cb±Cc](5)

98 ± s[Ca±Cb]�18� � /[Ca±Ce±Ca]�17� � s[Ce±Ca]�15� � /[Ce1±Ca±Cb]�12� � /[Ce±Ca±Ce1]�10� �s[Ca±Ce1](7)

70 ± s[Ca±Ce1]�20� � s[Ce±Ca]�14� � s[Ca±Cb]�10� � /[Ce1±Ca±Cb]�10� � /[Ca±Cb±Cc]�7� � /[Ce±Ca±

Cb]�7� � /[Ce±Ca±Ce1]�6� � /[H±Ce±Ca](5)

58 ± s[Ce±Ca]�29� � s[Ca±Cb]�19� � s[Ca±Ce1]�15� � /[Ce±Ca±Cb]�10� � /[Ca±Ce±H]�7� � /[H±Ce±Ca](6)

Assignments % PED at d � p

2914 2918 m[Ce±H]�54� � m[Ca±H]�23� � m[Cb±H](22)

2913 2918 m[Ce±H]�56� � m[Ca±H]�35� � m[Cb±H](9)

2910 2918 m[Cb±H]�73� � m[Ca±H]�19� � m[Ce±H](8)

2862 2866 m[Cb±H]�69� � m[Ce±H](26)

2861 2866 m[Ce±H]�73� � m[Cb±H](25)

1464 1463 /[H±Cc±H]�77� � /[H±Ce±H](15)

1464 1463 /[H±Ce±H]�64� � /[H±Cc±H]�19� � /[H±Ce±Ca]�7� � /[Ca±Ce±H](6)

1330 1309 /[H±Ce±Ca]�18� � /[Ca±Ce±H]�18� � m[Ca±Ce]�14� � m[Ca±Cb]�10� � m[Ce±Ca]�9� �/[H±Ca±Ce1]�7� � /[Ca±Cb±H]�6� � /[H±Cb±Cc](6)

1287 1286 /[H±Cb±Cc]�25� � /[Ca±Cb±H]�23� � m[Ca±Cb]�11� � /[Ca±Ce±H]�6� � /[H±Ca±Ce1]�6� �m[Ce±Ca]�6� � /[H±Ce±Ca]�5� � m[Cb±Cc](5)

1248 1265 /[H±Ca±Ce]�26� � /[Ca±Ce±H]�16� � /[Cb±Ca±H]�14� � /[H±Ce±Ca]�13� � /[Ca±Cb±H]�7� �m[Ce±Ca]�6� � m[Ca±Cb](6)

1209 1238 /[H±Ce±Ca]�21� � /[H±Ca±Ce1]�17� � /[Ca±Ce±H]�14� � /[H±Ca±Ce]�10� � /[H±Cb±

Cc]�9� � /[Ca±Cb±H]�8� � /[Cb±Ca±H](7)

1190 1176 /[H±Ca±Ce1]�16� � /[H±Ca±Ce]�16� � /[Cb±Ca±H]�15� � /[Ca±Cb±H]�13� � /[H±Ce±

Ca]�10� � /[Ca±Ce±H](9)

1180 1176 /[Ca±Ce±H]�28� � /[H±Ce±Ca]�23� � /[Cb±Ca±H]�16� � /[H±Ca±Ce1]�8� � /[H±Cb±Cc](5)

1149 1161 /[Ca±Cb±H]�22� � /[H±Ce±Ca]�21� � /[Ca±Ce±H]�20� � /[H±Cb±Cc]�11� � /[Cb±Ca±H]�8� �/[H±Ca±Ce](7)

1127 1108 /[Ca±Ce±H]�26� � /[H±Ce±Ca]�26� � /[Ca±Cb±H]�11� � /[H±Cb±Cc]�11� � /[Cb±Ca±

H]�10� � /[H±Ca±Ce1](8)

1122 1108 /[H±Cb±Cc]�29� � /[Ca±Cb±H]�26� � /[H±Ca±Ce1]�12� � /[H±Ca±Ce]�11� � /[H±Ce±

Ca]�6� � /[Ca±Ce±H](5)

1111 1074 /[H±Cb±Cc]�25� � /[Ca±Cb±H]�21� � /[Ca±Ce±H]�19� � /[H±Ce±Ca]�12� � /[H±Ca±

Ce]�10� � /[Cb±Ca±H](6)

1031 1027 m[Ce±Ca]�20� � /[Cb±Cc±H]�17� � /[H±Ca±Ce]�12� � /[H±Cb±Cc]�10� � /[H±Ce±Ca]�10� �/[Cb±Ca±H]�7� � m[Ca±Cb](5)

1013 1027 /[Cb±Cc±H]�46� � m[Ca±Cb]�15� � /[Ca±Cb±H]�11� � /[H±Cb±Cc]�5� � /[Cb±Ca±H](5)

1011 1000 /[Cb±Cc±H]�33� � m[Ce±Ca]�18� � /[H±Ca±Ce]�13� � m[Ca±Cb]�9� � /[H±Cb±Cc](5)

966 972 /[Cb±Cc±H]�68� � m[Ce±Ca]�11� � /[Ca±Cb±H](5)

954 972 /[Cb±Cc±H]�38� � m[Ce±Ca]�29� � /[H±Ca±Ce1](5)

939 924 /[Cb±Cc±H]�34� � m[Ca±Cb]�10� � m[Ce1±Ce]�10� � /[Cb±Ca±H]�9� � /[H±Ca±Ce1]�8� �/[Ca±Ce±H](5)

927 924 /[Cb±Cc±H]�20� � m[Ca±Ce]�13� � m[Ca±Cb]�11� � /[Ca±Ce±H]�11� � m[Ce±Ca]�9� � /[H±Ce±

Ca]�7� � m[Cb±Cc](6)

866 883 m[Cb±Cc]�16� � m[Ca±Cb]�16� � /[Ca±Ce±H]�15� � /[H±Ce±Ca]�15� � /[Cb±Cc±H]�10� � m[Ce1±Ce](6)

848 ± m[Cb±Cc]�33� � m[Ca±Ce]�21� � /[H±Ca±Ce1]�8� � /[Ca±Ce±H]�8� � /[H±Ce±Ca]�5� � /[H±Cb±Cc](5)

823 ± m[Cb±Cc]�52� � m[Ca±Ce]�24� � /[H±Ca±Ce1](5)

800 807 m[Cb±Cc]�39� � /[Ca±Ce±H]�16� � /[H±Ce±Ca]�15� � m[Ca±Cb]�9� � /[Cb±Cc±H]�5� � s[Ce±Ca](5)

(continued on next page)


The 380 cmÿ1 mode has a major contribution of Ce±

Ca±Cb bending mixed with the other bendings of bb and

sc as Ce±Ca±Ce1 [17%] and Ce1±Ca±Cb [17%]. The

contribution of side chain mode decreases until the helix

angle and the energy drops to 402 cmÿ1. It becomes

purely a bb bending mode. The mode at 328 cmÿ1 shows

a maximum dispersion of 35 wave numbers to reach 363

cmÿ1 at the zone boundary. The mode at 313 cmÿ1 is

Table 5 (continued)


Calculated Observed

780 781 /[H±Cb±Cc]�35� � /[Ca±Cb±H]�27� � /[Cb±Cc±H]�15� � s[Ca±Cb](6)

740 753 m[Ca±Cb]�21� � m[Ca±Ce]�14� � /[H±Ce±Ca]�13� � m[Ce±Ca]�10� � /[Ca±Ce±H]�10� � m[Cb±

Cc]�10� � /[Ca±Ce±Ca](5)

725 725 m[Ca±Ce]�25� � /[Ca±Ce±H]�5� � /[H±Ce±Ca]�15� � m[Ca±Cb]�10� � /[Ca±Ce±Ca](5)

474 492 /[Ce±Ca±Cb]�17� � /[Ce1±Ca±Cb]�16� � /[Ca±Ce±Ca]�12� � /[Ca±Cb±Cc]�10� � /[Ce±Ca±Ce1](9)

452 ± /[Ce±Ca±Cb]�32� � /[Ca±Cb±Cc]�17� � /[Ca±Ce±Ca]�10� � /[Cb±Ca±H](5)

402 ± /[Ce±Ca±Ce1]�38� � /[Ce1±Ca±Cb]�11� � /[Ca±Ce±Ca]�10� � /[Ca±Cb±Cc]�6� � m[Ca±Ce](5)

363 ± /[Ce1±Ca±Cb]�27� � /[Ca±Ce±Ca]�16� � m[Ca±Cb]�8� � s[Ce±Ca]�8� � /[H±Ca±Ce1]�7� � /[Ce±Ca±

Cb]�6� � m[Ca±Ce]�5� � /[Cb±Ca±H](5)

285 ± /[Ce±Ca±Ce1]�18� � /[Ce1±Ca±Cb]�15� � /[Ca±Cb±Cc]�15� � s[Ca±Cb]�10� � s[Ca±Ce1]�9� � s[Ce±

Ca]�8� � /[Ce±Ca±Cb](5)

264 ± /[Ca±Cb±Cc]�30� � /[Ce±Ca±Ce1]�14� � m[Ca±Ce]�10� � /[Ce±Ca±Cb]�9� � s[Ca±Ce1]�9� �s[Cb±Cc](6)

232 ± s[Ce±Ca]�22� � /[Ce±Ca±Cb]�19� � s[Cb±Cc]�14� � /[Ca±Cb±Cc]�11� � s[Ca±Cb](8)

205 ± s[Cb±Cc]�43� � /[Ca±Cb±Cc]�11� � /[Ca±Ce±Ca]�10� � s[Ca±Ce1]�9� � s[Ca±Cb](7)

189 ± s[Cb±Cc]�62� � /[Ce±Ca±Ce1]�10� � /[Ca±Ce±Ca](7)

161 ± /[Ca±Cb±Cc]�24� � s[Cb±Cc]�16� � /[Ce±Ca±Ce1]�12� � /[Ce±Ca±Cb]�11� � /[Ca±Ce±Ca](10)

157 ± s[Cb±Cc]�32� � /[Ce1±Ca±Cb]�17� � s[Ca±Cb]�16� � /[Ca±Cb±Cc](9)

141 ± s[Ca±Cb]�42� � s[Cb±Cc]�15� � /[Ce±Ca±Cb]�10� � /[Ca±Ce±Ca]�9� � s[Ca±Ce1](7)

82 ± s[Ca±Cb]�35� � /[Ce1±Ca±Cb]�16� � /[Ce±Ca±Cb]�13� � /[Ca±Cb±Cc]�7� � /[Ca±Ce±Ca]�6� �s[Ce±Ca](6)

75 ± s[Ca±Cb]�42� � /[Ce1±Ca±Cb]�20� � /[Ca±Ce±Ca]�9� � /[Ce±Ca±Cb]�8� � /[H±Cb±Cc](5)

56 ± s[Ca±Ce1]�41� � s[Ce±Ca]�19� � /[Ce±Ca±Ce1]�12� � /[Cb±Ca±H](6)

Fig. 3. (a) Dispersion curves of sPB (1350±950 cmÿ1) and (b)

density of states g(m) (1350±950 cmÿ1).




another dispersive mode with contributions from the

bendings of �Ce1±Ca±Cb� 20%� �Ca±Cb±Cc� 17% ��Ce±Ca±Cb� 17%� �Ca±Ce±Ca� 9% and stretch of (Ca±

Ce) 8% which appears at 285 cmÿ1 at d � p. The mode at

182 cmÿ1 has 82% contribution from methyl torsion

vibration at d � 0. As the d value is increased, the per-

centage contribution of methyl group torsion vibration

goes on decreasing upto 40% at d � 0:65p. On further

increase in d value, the methyl torsion increases again

and the mode with 43% contribution of methyl torsion,

Fig. 7. Variation of heat capacity Cv with temperature.






reaches 205 cmÿ1 at d � p. The mode at 287 cmÿ1 dis-

perses by 23 wave numbers at d � p.

3.5. Density of states and heat capacity

As explained in the theory, the inverse of the slope of

the dispersion curves leads to the density of states which

indicate how the energy is partitioned in various normal

modes. These are shown in Figs. 3±6. The peaks in the

frequency distribution curves compare well with the

observed frequencies. The frequency distribution func-

tion can also be used to calculate the thermodynamical

properties such as heat capacity, enthalpy changes, etc.

It has been used to obtain the heat capacity as a function

of temperature. The predictive values of heat capacity

have been calculated and plotted within the temperature

range 50±500 K (Fig. 7). The contribution to the heat

capacity due to purely skeletal, purely side chain and the

mixture of two are calculated separately and plotted in

Fig. 6. The maximum contribution comes from the

mixed modes. However, the contribution from the lat-

tice modes is bound to make a di�erence to the heat

capacity because of its sensitivity to low frequency

modes. At present, the calculation of dispersion curves

for a three-dimensional crystal is extremely di�cult be-

cause of the large matrix size and enormous number of

interactions which are di�cult to visualize and quantify.

Inspite of several limitations involved in the calculation

of speci®c heat and absence of experimental data, the

present work would provide a good starting point for

further basic studies on the thermodynamical behavior

of the polymer.

Although the spectra are not available below 450

cmÿ1, calculations are expected to be correct for more

than one reason. First, most of the frequencies occur in

the same range as the corresponding ones in other syn-

thetic polymers. Second, since the force constants which

provide good matching in the higher frequency region

are also involved in the lower frequency region, rea-

sonable values are expected in this region as well.

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Vibrational dynamics of syndiotactic poly(1-butene)