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Energetic calculations of the chain conformation of isotactic polymers of olefins in the crystalline state
European Polymer Journal, Vol. 12, pp. 831 to 836. Pergamon Press 1976. Printed in Great Britain.
ENERGETIC CALCULATIONS OF THE CHAIN CONFORMATION OF ISOTACTIC POLYMERS OF OLEFINS IN THE CRYSTALLINE STATE P. CORRADINI, V. PETRACCONE and B. Pmozzi Istituto Chimico dell'Universit~ di Napoli, Via Mezzocannone 4, 80134 Napoli, Italy (Received 1 April 1976)
Abstract---Energetic calculations have been carried out on a correlated series of isotactic aliphatic polymers (polypropylene (PP), poly-:~-butene (PB), poly-3-methylbutene (P3MB), poly-(S)-3-methylpentene-1 (P(S)3MP) and isotactic polystyrene (PS)). The possible variation of the C---C--C bond angles and of all the internal rotation angles (no group of atoms being taken as a unit) was considered. The possibility to predict the experimental helical parameters without any previous assumption is discussed. The differences in the shape of the minima for polymers with aliphatic and aromatic branched chains have been critically evaluated. For the P(S)3MP the results have shown how the asymmetric configuration of the side group influences the chain conformation of the polymer.
The chain conformation of isotactic vinyl polymers the approximation of using constant bond angles and in the crystalline state is always helical, and corre- considering methyl groups as single-atom units. In sponds in all known cases to successions of nearly the present paper, we show the results of energetic trans and nearly gauche isomorphous internal calculations on a correlated series of isotactic alipharotation angles [1]. Different unit heights c / M and tic polymers (polypropylene(PP), poly-~t-butene (PB), unit twists 2nN/M for the various studied poly-3-methylbutene-I (P3MB), poly-(S)-3-methylpenpolymers [2] (where c is the identity period, M the tene-1 (P(S)3MP)) and isotactic polystyrene (PS). At number of configurational units and N the number variance with previous calculations, we include the of pitches) arise mainly from deviations of the internal possible variation of all C---C---C bond angles of the rotation angles from exactly trans and exactly gauche main chain and of all the internal rotation angles (no local conformations. group of atoms being taken as a unit) making use Qualitative correlations between chain conforma- of the parameters for non-bonded interactions and tion and bulkiness of the side-group were put forward torsional potentials as given by Flory [10]. first by Natta, Corradini and Bassi [3] and by Bunn Thus, for the case of PP, P3MB and PS, calculaand Holmes [4]; it was noted that intramolecular in- tions represent a further refinement over those of teractions are predominant in determining the helical Natta et al. (for PP) [5], Yadokoro et al. (for PP and structure[I] and its characteristic parameters, the P3MB) [9], Liquori and De Santis (for PS)[11]. The ratio M/N always being between 3 and 4. Intermol- calculation of energetic maps of the optically active ecular packing interactions may lead to the stabiliza- polymer P(S)3MP was never performed previously, tion of helices, near to an intramolecular energy mini- while some preliminary results on PB, relevant to its mum, where the integers M and N are small. polymorphic behaviour, were anticipated by us quite Quantitative energetic calculations of the most recently [12]. stable conformations of an isolated chain under the restriction of periodic repetition were performed first M E T H O D OF COMPUTATION for isotactic polypropylene by Natta, Corradini and The conformational energy per constitutional Ganis [5] with the approximation of constant bond angles; as far as the non-bonded interactions were repeating unit (CRU) has been calculated, as in a preconcerned, the calculations were simplified by taking vious paper [12], as the sum of 3 terms: the methylene and methyl groups as single-atom E = Etor + Eben + Enb units. The minima of the potential energy as a function of the internal rotation angles of the chain were Etor represents the sum over all the C - - C bonds of found very near to those present in the crystals [6]. terms Eio r = 2.8 (1 + cos 3 ai)/2~kcal tool 1 The result was confirmed by a subsequent calculation of De Santis, Giglio, Liquori, Ripamonti [7]. relative to the internal rotation angles a~ which are The techniques and input data for energetic confor- comprised within each CRU. Ebe n represents the sum mational calculations have been improved lately, over all the valence angles zj of the deformation enermainly as a result of work in the field of bio- gies polymers [8]. As a result, calculations with better E~e. = K~ (z2 - "ro)2/2 potential functions for some isotactic polymers of olefins (under the restriction of Periodic repetition) were comprised within a single CRU (the constants K~ are performed quite recently by Tadokoro et al. [9] with given in Table 1). We took Zo = 109.5° for all kinds 831
832
P. CORRADINI,V. PETRACCONEand B. Pmozzi
Table 1. Parameters used in the energy calculations Bond angle
K~ (kcal mol-1 deg-z)
< CCC
0.044
< CCH < HCH
0.029 0.024
Interacting pair
a o x 10- 3 (kcal mol- 1 A12)
c~j (kcalmol- 1 A6)
"Crnin
398 57 7.3
366 128 47
3.6 3.1 2.6
C, C C, H H, H
(A)
of angles [13], with the obvious exception of bond angles for the benzene ring. E,b represents a sum of Lennard-Jones terms, truncated for internuclear distances rij > rmin =
-
al/rmi,)
-
(ci/r~
-
(I)
Cu/rmin) 6
for r o ~< rmi. E.~{ = 0
for r u > rmi,
(2)
where a~j, co and rmi. are constants (see Table 1). The sum has been performed over all the distances rij (taken only once) between atoms i and j, separated by three or more bonds and included between the atoms present in one single CRU or in the successive ones, provided they were comprised within the following section of chain: f I I I
CRU
i i t I
As far as the internal rotation angles are concerned, they have been varied in intervals of 5° in the regions of minimum energy and of 10° in other regions.
The rotation angles kept as variables in our calculations are defined as in Fig. 1. As far as the valence angles are concerned, the approximation was used of assuming local Cav symmetry for the methyl and methine groups and local C2h symmetry for the methylene groups; furthermore, for the methylene groups the H---C--H angles were always taken to be 108°. In this way, only one angle zj for each carbon atom needs to be defined; the C---C--C angles along the chain were kept as variable in the calculations (and varied in intervals of 1° in the intervals 110-114 ° and 110-116 °, respectively, for the methine (zl) and methylene (r2) groups) while the C---C--C angles in the lateral groups were given the values of 112° for the methylene groups and of 111° for the methine groups. For the C----C--H angle at the methyl groups we took the value zj = 111 °. The angle C----C--C in the benzenic ring was assumed equal to 120°. As far as the bond lengths are concerned, all the C---C single bond lengths were assumed equal to 1.53 A. For the C--C bonds in the benzene ring, a distance of 1.39 A was assumed. All the C - - H bond lengths were assumed to be 1.10A. The reason for the form given to the Lennard-Jones potential has been discussed in a previous paper [12]. The summation of terms [1] also for rij > rmi,, while appropriate for an "isolated" chain, would tend to give lower energies for compact conformations; however, higher intramolecular energies for more extended conformations would be compensated, in the crystalline state, by a decrease of the intermolecular energy through contacts with the surrounding molecules. Thus, the form given to the Lennard-Jones potential tends to avoid spurious energy minima, eventually due to attractive interactions between atoms separated by many bonds and allows us to take into account only the interaction distances of a relatively low number of atoms. However, since the sum E.b has been restricted to
CH3
CH 3
CH2
H "I"
C ~
C6
H
C
,,,I,
\/
..,-\ H PB 0"3 0"4
H
C2-C3-C6-CH 3
,,,-\ H P3MB C2-C3-C6-H
H
,,,\ H P(S}3MP
C2-C3-C6-CH 2
H PS C2-C3-C(=,)-C(=r)
C3-C6-CH~CH 3
Fig. 1. The parameters which have been kept as variable in the calculations for P(S)3MP are shown in the model. For the other polymers with simpler side-chains, the symbols have analogous significance. The atoms used as a reference to measure aa and a4 are also given for the various polymers (below).
Energetic calculations
833
o, I
I
>2.5
>2.5 E .
100"
100'
80,
80'
60"
60,
160 °
180"
200"
220 °
~ 160"
Fig. 2. Internal energy of an isotactic PP chain for different helicoidal conformations. For each pair of internal rotation angles 01, 02, the reported energy corresponds to the minimum obtained by varying 03, zl, z2 as indicated in the text. The curves are reported at intervals of 0.5 kcal(mol of CRU)- 1. a rather short section of chain, the calculated energies represent a good approximation of the conformational energy per C R U of an infinite chain only when the unit height (h) and unit twist (t) are such as not to lead to repulsive interactions between n o n - b o n d e d atoms, separated by m a n y b o n d s a n d thus not included in the considered section of chain. This is so in all the regions of m i n i m u m energy, where the unit heights are greater than 1.50A, and the unit twists are between 3 and 4. O n the other hand, as shown by T a d o k o r o et al. [9], in the regions where the unit heights are less than ~ 1.50 A a n d the unit twists are less than 4, the energy may be higher than that calculated.
I
0.6
o-,
180,
200"
220 °
Fig. 4. Internal energy of an isotactic P3MB chain for different conformations. For each pair of internal rotation angles al, a2, the reported energy corresponds to the minimum obtained varying 03, a4, ~5, q , ~2 as indicated in the text. The curves are reported at intervals of 0.5 kcal (mol of CRU) -1. The values assumed by o3 (see Fig. 1) are approximately trans. RESULTS The results for the various polymers are represented in Figs. 2, 3, 4, 5 and 6 as isoenergetic curves vs 01 and a2 in the regions 140 ° < a t < 2 4 0 °, 40 ° < a2 < 120 ° for the values o f the other parameters that minimize the energy. O t h e r results are reported, because relevant for the discussion, in Tables 2, 3, 4, and 5. In the case of isotactic PB (Fig. 3) we reported moreover in the graph the curves that represent the loci of points which correspond to the three conformations, experimentally observed, in the crystalline state. For isotactic P(S)3MP, three graphs are reported since the right-handed helix, because of the chirality of the lateral group, is not equivalent to the lefth a n d e d one, as for the other polymers. In particular,
=,.2.5
/"~_\
100'
I ='2.5
80 °
100'
60"
80 °
o', 160 °
180 °
200 °
220"
Fig. 3. Internal energy of isotactic PB chain as a function of the internal rotation angles of the chain a I and ~2 (a3 is always in approximately trans conformation). For each pair of internal rotation angles ol, a2, the reported energy corresponds to the minimum obtained by varying ~3, ¢r4, q , ~2 as indicated in the text. The curves are reported at intervals of 0.5 kcal(mol of CRU)- t The open curves are the loci of the points which correspond to the unit twists of the three conformations, experimentally observed, in the crystalline state.
60 °
o-, 160 °
180"
200 °
220 °
Fig. 5. Internal energy of an isotactic PS chain for different conformations. For each pair of internal rotation angles al, 02, the reported energy corresponds to the minimum obtained by varying a3, q , ~2 as indicated in the text. The curves are reported at intervals of 0.5 kcal (mol of CRU)-1. The values assumed by o3 are near to 60 °.
P. CORRADINI,V. PETRACCONE and B. PrRozzI
834 0"1: 0"C1_C2- C3 _ ¢ 4
0"3-- C r c 2 _ c 3 _ c 6 _ c - r
%=o'c~_c3_c4_¢5
Cr4= 0"C3 _C6 _ C7 _ CB
80*
05
c6
%
cl
Lo
C7
J
,,e,8o°
zoo*-~
,,;-,8o"
o,-,ooo
,o.2/~
~+~ 80`
80'
%
.
O •
o#c2
60*
----r--
2.0
\
60'
60*
>2.5 I
) 180*
200*
m (o)
220*
N?
L_
180*
I 200*
i._..
220 °
o "I
(c)
(b)
Fig. 6. Internal energy of an isotactic P(S)3MP chain for both the left-handed (part A and B) and the right-handed helix (part C). For each pair of internal rotation angles (rt, o2, the reported energy corresponds to the minimum obtained by varying as, a4, 0-5, 06, ~1, "c2 as indicated in the text. The curves are reported at intervals of 0.5 kcal (mol of C R U ) - t. Table 2. Values assumed by some parameters, which were kept as variables, for the conformations of minimum energy. The AE values (kcal/mol of CRU) refer to the absolute minimum of each polymer Polymer
ax
a2
PP PB PB P3MB P3MB P(S)3MP P(S)3MP P(S)3MP P(S)3MP P(S)3MP P(S)3MP PS
176 ° 177.5 ° 204 ° 205 ° 175 ° 176 ° 205 ° -205 ° -174 ° 205 ° 177 ° 176 °
59.5 ° 54 ° 79 ° 78 ° 57.5 ° 58 ° 77.5 ° -78 ° -59 ° 79.5 ° 53.5 ° 64 °
o3
a4
180 ° -160 ° 165 ° 180 ° 180 ° -160 ° 45 ° 65 ° -160 ° -175 ° 60 °
160 ° 170 ° 175 ° 180 ° 75 ° 60 °
~2
AE
116 ° 116 ° 113 ° 113 ° 116 ° 116 ° 113 ° 113 ° 116 ° 113 ° 116 ° 114 °
--0.90 -0.60 -0.20 0.40 0.80 1.00 2.00 --
for the l e f t - h a n d e d helix, we have c o m p a r a b l e values o f the energy for regions o f a4 very different b e t w e e n themselves. P a r t C of Fig. 6 s h o w s the c o n f o r m a t i o n a l m a p o f the r i g h t - h a n d e d helix, while p a r t s A a n d B c o r r e s p o n d to the c o n f o r m a t i o n a l m a p o f the left-
h a n d e d helix for values o f a4 in the respective regions: a4 ~ 60°; a4 ~ 180 °. T h e values a s s u m e d by the p a r a m e t e r s , t a k e n as variable at the calculated m i n i m u m points, are r e p o r t e d in Table 2 with the exclusion o f the internal r o t a t i o n angles a r o u n d m e t h y l g r o u p s w h i c h are always within 10 ° of the staggered c o n f o r m a t i o n , a n d o f the angle 21, o n the m e t h i n e group, which assumes always the value o f 111 °. DISCUSSION T h e calculated values o f h (unit height) a n d t (unit twist) c o r r e s p o n d i n g to the p o i n t s o f energy m i n i m u m are r e p o r t e d in T a b l e 3 in c o m p a r i s o n to the values of h a n d t experimentally observed. Table 4 s h o w s the hcalc values that we w o u l d foresee u n d e r the restriction that t = texv c o m p a r e d with the e x p e r i m e n t a l values hexp a n d the difference in energy (AE) b e t w e e n the c o n f o r m a t i o n c o r r e s p o n d i n g to hcalc a n d that o f the n e a r e s t energy m i n i m u m . The AF. values are representative o f the o r d e r o f m a g n i t u d e o f the extent to which i n t e r m o l e c u l a r forces are able
Table 3. Calculated values of the unit heights h and unit twist t, corresponding to the energy minima for the various polymers. They are compared with the values of h and t, experimentally observed Polymer PP PB (form I) PB (form II) ~ PB (form III)J P3MB P(S)3MP* PS
Our calculations h(/~) t 2.176 2.166 1.802 1.791 f 1.794 I 1.782 2.155
2.95 2.83 3.83 3.80 3.78 } 3.86 3.06
X-ray results h(/~,) t 2.17 2.17 ~ 1.91 . 1.89 1.71 1.70
+_ 0.02 + 0.02 + 0.02 + 0.02 _ 0.02 + 0.02 2.22 + 0.02
Ref.
3 3 3.65 + 0.03 4 4 4
[14] [15] [12] [16] [17] [18]
3
[19]
* The two values of h and t calculated for P(S)3MP refer to the two minima of the nearly fourfold left-handed helix (Figs. 6A and B).
Energetic calculations Table 4. Calculated values of the unit height, which minimize the energy under the constraint that the helical symmetry is that observed in the crystalline state. The corresponding values of the energy (in respect to the unconstrained minimum) are reported as AE's Polymer PP PB (form I) PB (form II) PB (form III) P3MB P(S)3MP* PS
Symmetry 3/1 3/1 11/3 4/1 4/1 4/1 3/1
hexp (~-)
hc,j¢.
AE
2.17 2.15 1.85 1.80 1.76 fl.79 1.70 + 0.02 /.1.79 2.22 _ 0.02 2.16
0.05 0.60 0.10 0.10 0.30 0.20 0.10 0.05
2.17 2.17 1.91 1.89 1.71
___0.02 + 0.02 + 0.02 + 0.02 + 0.02
* The two values of hca~c and AE reported for P(S)3MP refer to the two minima of the nearly fourfold left-handed helix (Fig. 6A and B)
to modify the energetic m i n i m a calculated for the isolated chain. Table 5 shows the parameters which correspond to the m i n i m u m of conformational energy, under the restriction that h = hexp a n d t = fexpThe prediction of the unit height for isotactic polypropylene (without any previous assumption concerning valence angles) is excellent, while for the other polymers may be considered as good. In accordance with experimental observations on various polymers [2], the energetic maps of polymers with aliphatic b r a n c h e d chains, show elongated m i n i m a corresponding to pairs of internal rotation angles almost equally "distorted" from staggered conformations. The distortion from staggered values applies as well to the internal rotation angles of the lateral groups. A principle of "isodistortion" was put forward by Natta, Corradini and Bassi [3] on the basis of qualitative energetic considerations, and finds quantitative justification in our results. The experimental observation that isotactic substituted polystyrene having fourfold helices need not have isodistorted chain conformations [20, 21] finds justification in the form of the energetic m i n i m u m for polystyrene, which, differently from the case of P3MB, expands itself also in the region of cr2 = 80-90 °, while a 1 = 180 °. Another point to be noted is the presence of two m i n i m a for PB, P 3 M B a n d P(S)3MP, one corresponding to a threefold helix and one corresponding to the region of a nearly fourfold helix. For the threefold helical conformation, while the torsional potentials are at a minimum, the n o n - b o n d e d interactions force the valence angle T2 to be distorted to 116 °. For the other conformations, while the torsional
* Allegra, Corradini and Ganis [22] have shown the dependence of the optical activity on the ratio E between the internal partition functions, relative to the conformational states of the side groups for one monomer unit in a (TG) vs (GT)-chain. From the calculated energetic maps of P(S)3MP, it is seen that e is of the order of 0.6 in the range of temperature 300-400 K. According to Fig. 2 of the above quoted paper, this would give rise in a solution to a pronounced prevalence of the left-handed sense of spiralization.
835
Table 5. Some parameters corresponding to the minimum of conformational energy under the constraint that h = he~pand t = texp. The values of AE (kcal/mol of CRU) represent, for each polymer, the difference in energy between the corresponding conformation and that of the nearest minimum. The other parameters assume the same value as those reported in Table 2 for the nearest energetic minima Polymer
rz
ax
~r2
PP PB (form I) PB (form II) PB (form III) P3MB
116C' ll& 115 ° 116° 113 °
177 ° 177 ° 200 ° 199.5° 208 °
62 ° 62 ° 75.5 ° 83.5 ° 82.Y
P(S)3MP*
113 °
208"5°
82"5c
PS
116 °
169 °
66 °
AE 0.05 0.80 0.20 0.30 0.35 f 0.40 ( 0.30 0.50
* The two values of AE reported for P(S)3MP refer to the two minima of nearly fourfold left-handed helix (Figs. 6A and B). potentials are no longer at a minimum, the valence angle ~2 is less distorted (f,i. v2 = 113 ° for the fourfold almost "isodistorted" conformations). Finally, the results obtained for P(S)3MP show the way in which the asymmetric configuration of the lateral group influences the chain conformation of the polymer. The conformations of m i n i m u m potential energy are two for the left-handed helix and only one for the righth a n d e d helix. In the solid polymer [18], the conformation of the lateral groups may take statistically b o t h the conformations of m i n i m u m energy found in the present paper (calculated to differ less than 1 kcal/mole of CRU). A requirement for good packing is that the two conformations are represented in equal a m o u n t s in the crystalline lattice. In the melt or in solution, left-handed local helical conformations should prevail in length for entropic reasons* over the right-handed local helical conformations.
REFERENCES
1. P. Corradini, The Stereochemistry of Macromolecules, (Edited by A. D. Ketley) Part III, pp. 1-60, Marcel Dekker, New York (1968). 2. R. L. Miller, Crystallographic data for various polymers, in Polymer Handbook, 2nd edition, John Wiley, New York (1974). 3. G. Natta, P. Corradini and I. W. Bassi, Gazz. Chim. Italiana 89, 784 (1959). 4. C. W. Bunn and D. R. Holmes, Discuss. Faraday Soc. 25, 95 (1958). 5. G. Natta, P. Corradini and P. Ganis, J. Polym. Sci. 58, 1191 (1962). 6. G. Natta and P. Corradini, Nuovo Cim. Suppl. 15, 40 (1960). 7. P. De Santis, F. Giglio, A. M. Liquori and A. Ripamonti, J. Polym. Sci. A1, 1383 (1963). 8. A. J. Hopfinger, Conformational Properties of Macromolecules, Academic Press, New York (1973). 9. H. Tadokoro, K. Tai, M. Yokoyama and M. Kobayashi, J. Polym. Sci., Polym. Phys. ed. 11, 825 (1973). 10. U. W. Suter and P. S. Flory, Macromolecules 8, 765 (1975). 11. A. M. Liquori and P. De Santis, J. Polym. Sci. C16, 4583 (1969).
836
P. CORRADINI,V. PETRACCONEand B. Pmozzl
12. V. Petraccone, B. Pirozzi, A. Frasci and P. Corradini, Europ. Polym. J. 12, 323 (1976). 13. R. G. Snyder and G. Zerbi, Spectrochim. Acta 23A, 391 (1967). 14. G. Natta and P. Corradini, Nuovo Cim. Suppl. 15, 40 (1960). 15. G. Natta, P. Corradini and I. W. Bassi, Nuovo Cim. Suppl. 15, 52 (1960). 16. G. Cojazzi, V. Malta, G. Celotti and R. Zannetti, to be published in Makromolek. Chem. 17. P. Corradini, P. Ganis and V. Petraccone, Europ. Polym. J. 6, 281 (1970).
18. V. Petraccone, P. Ganis, P. Corradini and G. Montagnoli, Europ. Polym. J. 8, 99 (1972). 19. G. Natta, P. Corradini and I. W. Bassi, Nuovo Cim. Suppl. 15, 68 (1960). 20. P. Corradini and P. Ganis, Nuovo Cim. Suppl. 15, 96 (1960). 21. P. Corradini and P. Ganis, Nuovo Cim. Suppl. 15, 104 (1960). 22. G. Allegra, P. Corradini and P. Ganis, Makromolek. Chem. 90, 60 (1966).
Riassunto----Sono stati eseguiti calcoli energetici di conformazioni della catena allo stato cristallino di una serie di polimeri alifatici isotattici (polipropilene (PP), poli-~-butene (PB), poli-3-metilbutene (P3MB), poli-(S)-metilpentene-1 e del polistirene isotattico (PS)). Sono state considerate le possibili variazioni degli angoli di legame C---C----42 della catena principale e di tutti gli angoli di rotazione interna; nessun gruppo di atomi ~ stato considerato come un'unitL Viene discussa la possibilit~ di prevedere i parametri sperimentali dell'elica (h, t) senza nessuna assunzione a priori. Per il P(S)3MP i risultati mostrano come la configurazione asimmetrica del gruppo laterale influenzi la chiralit/t dell'elica nel polimero.
ENERGETIC CALCULATIONS OF THE CHAIN CONFORMATION OF ISOTACTIC POLYMERS OF OLEFINS IN THE CRYSTALLINE STATE P. CORRADINI, V. PETRACCONE and B. Pmozzi Istituto Chimico dell'Universit~ di Napoli, Via Mezzocannone 4, 80134 Napoli, Italy (Received 1 April 1976)
Abstract---Energetic calculations have been carried out on a correlated series of isotactic aliphatic polymers (polypropylene (PP), poly-:~-butene (PB), poly-3-methylbutene (P3MB), poly-(S)-3-methylpentene-1 (P(S)3MP) and isotactic polystyrene (PS)). The possible variation of the C---C--C bond angles and of all the internal rotation angles (no group of atoms being taken as a unit) was considered. The possibility to predict the experimental helical parameters without any previous assumption is discussed. The differences in the shape of the minima for polymers with aliphatic and aromatic branched chains have been critically evaluated. For the P(S)3MP the results have shown how the asymmetric configuration of the side group influences the chain conformation of the polymer.
The chain conformation of isotactic vinyl polymers the approximation of using constant bond angles and in the crystalline state is always helical, and corre- considering methyl groups as single-atom units. In sponds in all known cases to successions of nearly the present paper, we show the results of energetic trans and nearly gauche isomorphous internal calculations on a correlated series of isotactic alipharotation angles [1]. Different unit heights c / M and tic polymers (polypropylene(PP), poly-~t-butene (PB), unit twists 2nN/M for the various studied poly-3-methylbutene-I (P3MB), poly-(S)-3-methylpenpolymers [2] (where c is the identity period, M the tene-1 (P(S)3MP)) and isotactic polystyrene (PS). At number of configurational units and N the number variance with previous calculations, we include the of pitches) arise mainly from deviations of the internal possible variation of all C---C---C bond angles of the rotation angles from exactly trans and exactly gauche main chain and of all the internal rotation angles (no local conformations. group of atoms being taken as a unit) making use Qualitative correlations between chain conforma- of the parameters for non-bonded interactions and tion and bulkiness of the side-group were put forward torsional potentials as given by Flory [10]. first by Natta, Corradini and Bassi [3] and by Bunn Thus, for the case of PP, P3MB and PS, calculaand Holmes [4]; it was noted that intramolecular in- tions represent a further refinement over those of teractions are predominant in determining the helical Natta et al. (for PP) [5], Yadokoro et al. (for PP and structure[I] and its characteristic parameters, the P3MB) [9], Liquori and De Santis (for PS)[11]. The ratio M/N always being between 3 and 4. Intermol- calculation of energetic maps of the optically active ecular packing interactions may lead to the stabiliza- polymer P(S)3MP was never performed previously, tion of helices, near to an intramolecular energy mini- while some preliminary results on PB, relevant to its mum, where the integers M and N are small. polymorphic behaviour, were anticipated by us quite Quantitative energetic calculations of the most recently [12]. stable conformations of an isolated chain under the restriction of periodic repetition were performed first M E T H O D OF COMPUTATION for isotactic polypropylene by Natta, Corradini and The conformational energy per constitutional Ganis [5] with the approximation of constant bond angles; as far as the non-bonded interactions were repeating unit (CRU) has been calculated, as in a preconcerned, the calculations were simplified by taking vious paper [12], as the sum of 3 terms: the methylene and methyl groups as single-atom E = Etor + Eben + Enb units. The minima of the potential energy as a function of the internal rotation angles of the chain were Etor represents the sum over all the C - - C bonds of found very near to those present in the crystals [6]. terms Eio r = 2.8 (1 + cos 3 ai)/2~kcal tool 1 The result was confirmed by a subsequent calculation of De Santis, Giglio, Liquori, Ripamonti [7]. relative to the internal rotation angles a~ which are The techniques and input data for energetic confor- comprised within each CRU. Ebe n represents the sum mational calculations have been improved lately, over all the valence angles zj of the deformation enermainly as a result of work in the field of bio- gies polymers [8]. As a result, calculations with better E~e. = K~ (z2 - "ro)2/2 potential functions for some isotactic polymers of olefins (under the restriction of Periodic repetition) were comprised within a single CRU (the constants K~ are performed quite recently by Tadokoro et al. [9] with given in Table 1). We took Zo = 109.5° for all kinds 831
832
P. CORRADINI,V. PETRACCONEand B. Pmozzi
Table 1. Parameters used in the energy calculations Bond angle
K~ (kcal mol-1 deg-z)
< CCC
0.044
< CCH < HCH
0.029 0.024
Interacting pair
a o x 10- 3 (kcal mol- 1 A12)
c~j (kcalmol- 1 A6)
"Crnin
398 57 7.3
366 128 47
3.6 3.1 2.6
C, C C, H H, H
(A)
of angles [13], with the obvious exception of bond angles for the benzene ring. E,b represents a sum of Lennard-Jones terms, truncated for internuclear distances rij > rmin =
-
al/rmi,)
-
(ci/r~
-
(I)
Cu/rmin) 6
for r o ~< rmi. E.~{ = 0
for r u > rmi,
(2)
where a~j, co and rmi. are constants (see Table 1). The sum has been performed over all the distances rij (taken only once) between atoms i and j, separated by three or more bonds and included between the atoms present in one single CRU or in the successive ones, provided they were comprised within the following section of chain: f I I I
CRU
i i t I
As far as the internal rotation angles are concerned, they have been varied in intervals of 5° in the regions of minimum energy and of 10° in other regions.
The rotation angles kept as variables in our calculations are defined as in Fig. 1. As far as the valence angles are concerned, the approximation was used of assuming local Cav symmetry for the methyl and methine groups and local C2h symmetry for the methylene groups; furthermore, for the methylene groups the H---C--H angles were always taken to be 108°. In this way, only one angle zj for each carbon atom needs to be defined; the C---C--C angles along the chain were kept as variable in the calculations (and varied in intervals of 1° in the intervals 110-114 ° and 110-116 °, respectively, for the methine (zl) and methylene (r2) groups) while the C---C--C angles in the lateral groups were given the values of 112° for the methylene groups and of 111° for the methine groups. For the C----C--H angle at the methyl groups we took the value zj = 111 °. The angle C----C--C in the benzenic ring was assumed equal to 120°. As far as the bond lengths are concerned, all the C---C single bond lengths were assumed equal to 1.53 A. For the C--C bonds in the benzene ring, a distance of 1.39 A was assumed. All the C - - H bond lengths were assumed to be 1.10A. The reason for the form given to the Lennard-Jones potential has been discussed in a previous paper [12]. The summation of terms [1] also for rij > rmi,, while appropriate for an "isolated" chain, would tend to give lower energies for compact conformations; however, higher intramolecular energies for more extended conformations would be compensated, in the crystalline state, by a decrease of the intermolecular energy through contacts with the surrounding molecules. Thus, the form given to the Lennard-Jones potential tends to avoid spurious energy minima, eventually due to attractive interactions between atoms separated by many bonds and allows us to take into account only the interaction distances of a relatively low number of atoms. However, since the sum E.b has been restricted to
CH3
CH 3
CH2
H "I"
C ~
C6
H
C
,,,I,
\/
..,-\ H PB 0"3 0"4
H
C2-C3-C6-CH 3
,,,-\ H P3MB C2-C3-C6-H
H
,,,\ H P(S}3MP
C2-C3-C6-CH 2
H PS C2-C3-C(=,)-C(=r)
C3-C6-CH~CH 3
Fig. 1. The parameters which have been kept as variable in the calculations for P(S)3MP are shown in the model. For the other polymers with simpler side-chains, the symbols have analogous significance. The atoms used as a reference to measure aa and a4 are also given for the various polymers (below).
Energetic calculations
833
o, I
I
>2.5
>2.5 E .
100"
100'
80,
80'
60"
60,
160 °
180"
200"
220 °
~ 160"
Fig. 2. Internal energy of an isotactic PP chain for different helicoidal conformations. For each pair of internal rotation angles 01, 02, the reported energy corresponds to the minimum obtained by varying 03, zl, z2 as indicated in the text. The curves are reported at intervals of 0.5 kcal(mol of CRU)- 1. a rather short section of chain, the calculated energies represent a good approximation of the conformational energy per C R U of an infinite chain only when the unit height (h) and unit twist (t) are such as not to lead to repulsive interactions between n o n - b o n d e d atoms, separated by m a n y b o n d s a n d thus not included in the considered section of chain. This is so in all the regions of m i n i m u m energy, where the unit heights are greater than 1.50A, and the unit twists are between 3 and 4. O n the other hand, as shown by T a d o k o r o et al. [9], in the regions where the unit heights are less than ~ 1.50 A a n d the unit twists are less than 4, the energy may be higher than that calculated.
I
0.6
o-,
180,
200"
220 °
Fig. 4. Internal energy of an isotactic P3MB chain for different conformations. For each pair of internal rotation angles al, a2, the reported energy corresponds to the minimum obtained varying 03, a4, ~5, q , ~2 as indicated in the text. The curves are reported at intervals of 0.5 kcal (mol of CRU) -1. The values assumed by o3 (see Fig. 1) are approximately trans. RESULTS The results for the various polymers are represented in Figs. 2, 3, 4, 5 and 6 as isoenergetic curves vs 01 and a2 in the regions 140 ° < a t < 2 4 0 °, 40 ° < a2 < 120 ° for the values o f the other parameters that minimize the energy. O t h e r results are reported, because relevant for the discussion, in Tables 2, 3, 4, and 5. In the case of isotactic PB (Fig. 3) we reported moreover in the graph the curves that represent the loci of points which correspond to the three conformations, experimentally observed, in the crystalline state. For isotactic P(S)3MP, three graphs are reported since the right-handed helix, because of the chirality of the lateral group, is not equivalent to the lefth a n d e d one, as for the other polymers. In particular,
=,.2.5
/"~_\
100'
I ='2.5
80 °
100'
60"
80 °
o', 160 °
180 °
200 °
220"
Fig. 3. Internal energy of isotactic PB chain as a function of the internal rotation angles of the chain a I and ~2 (a3 is always in approximately trans conformation). For each pair of internal rotation angles ol, a2, the reported energy corresponds to the minimum obtained by varying ~3, ¢r4, q , ~2 as indicated in the text. The curves are reported at intervals of 0.5 kcal(mol of CRU)- t The open curves are the loci of the points which correspond to the unit twists of the three conformations, experimentally observed, in the crystalline state.
60 °
o-, 160 °
180"
200 °
220 °
Fig. 5. Internal energy of an isotactic PS chain for different conformations. For each pair of internal rotation angles al, 02, the reported energy corresponds to the minimum obtained by varying a3, q , ~2 as indicated in the text. The curves are reported at intervals of 0.5 kcal (mol of CRU)-1. The values assumed by o3 are near to 60 °.
P. CORRADINI,V. PETRACCONE and B. PrRozzI
834 0"1: 0"C1_C2- C3 _ ¢ 4
0"3-- C r c 2 _ c 3 _ c 6 _ c - r
%=o'c~_c3_c4_¢5
Cr4= 0"C3 _C6 _ C7 _ CB
80*
05
c6
%
cl
Lo
C7
J
,,e,8o°
zoo*-~
,,;-,8o"
o,-,ooo
,o.2/~
~+~ 80`
80'
%
.
O •
o#c2
60*
----r--
2.0
\
60'
60*
>2.5 I
) 180*
200*
m (o)
220*
N?
L_
180*
I 200*
i._..
220 °
o "I
(c)
(b)
Fig. 6. Internal energy of an isotactic P(S)3MP chain for both the left-handed (part A and B) and the right-handed helix (part C). For each pair of internal rotation angles (rt, o2, the reported energy corresponds to the minimum obtained by varying as, a4, 0-5, 06, ~1, "c2 as indicated in the text. The curves are reported at intervals of 0.5 kcal (mol of C R U ) - t. Table 2. Values assumed by some parameters, which were kept as variables, for the conformations of minimum energy. The AE values (kcal/mol of CRU) refer to the absolute minimum of each polymer Polymer
ax
a2
PP PB PB P3MB P3MB P(S)3MP P(S)3MP P(S)3MP P(S)3MP P(S)3MP P(S)3MP PS
176 ° 177.5 ° 204 ° 205 ° 175 ° 176 ° 205 ° -205 ° -174 ° 205 ° 177 ° 176 °
59.5 ° 54 ° 79 ° 78 ° 57.5 ° 58 ° 77.5 ° -78 ° -59 ° 79.5 ° 53.5 ° 64 °
o3
a4
180 ° -160 ° 165 ° 180 ° 180 ° -160 ° 45 ° 65 ° -160 ° -175 ° 60 °
160 ° 170 ° 175 ° 180 ° 75 ° 60 °
~2
AE
116 ° 116 ° 113 ° 113 ° 116 ° 116 ° 113 ° 113 ° 116 ° 113 ° 116 ° 114 °
--0.90 -0.60 -0.20 0.40 0.80 1.00 2.00 --
for the l e f t - h a n d e d helix, we have c o m p a r a b l e values o f the energy for regions o f a4 very different b e t w e e n themselves. P a r t C of Fig. 6 s h o w s the c o n f o r m a t i o n a l m a p o f the r i g h t - h a n d e d helix, while p a r t s A a n d B c o r r e s p o n d to the c o n f o r m a t i o n a l m a p o f the left-
h a n d e d helix for values o f a4 in the respective regions: a4 ~ 60°; a4 ~ 180 °. T h e values a s s u m e d by the p a r a m e t e r s , t a k e n as variable at the calculated m i n i m u m points, are r e p o r t e d in Table 2 with the exclusion o f the internal r o t a t i o n angles a r o u n d m e t h y l g r o u p s w h i c h are always within 10 ° of the staggered c o n f o r m a t i o n , a n d o f the angle 21, o n the m e t h i n e group, which assumes always the value o f 111 °. DISCUSSION T h e calculated values o f h (unit height) a n d t (unit twist) c o r r e s p o n d i n g to the p o i n t s o f energy m i n i m u m are r e p o r t e d in T a b l e 3 in c o m p a r i s o n to the values of h a n d t experimentally observed. Table 4 s h o w s the hcalc values that we w o u l d foresee u n d e r the restriction that t = texv c o m p a r e d with the e x p e r i m e n t a l values hexp a n d the difference in energy (AE) b e t w e e n the c o n f o r m a t i o n c o r r e s p o n d i n g to hcalc a n d that o f the n e a r e s t energy m i n i m u m . The AF. values are representative o f the o r d e r o f m a g n i t u d e o f the extent to which i n t e r m o l e c u l a r forces are able
Table 3. Calculated values of the unit heights h and unit twist t, corresponding to the energy minima for the various polymers. They are compared with the values of h and t, experimentally observed Polymer PP PB (form I) PB (form II) ~ PB (form III)J P3MB P(S)3MP* PS
Our calculations h(/~) t 2.176 2.166 1.802 1.791 f 1.794 I 1.782 2.155
2.95 2.83 3.83 3.80 3.78 } 3.86 3.06
X-ray results h(/~,) t 2.17 2.17 ~ 1.91 . 1.89 1.71 1.70
+_ 0.02 + 0.02 + 0.02 + 0.02 _ 0.02 + 0.02 2.22 + 0.02
Ref.
3 3 3.65 + 0.03 4 4 4
[14] [15] [12] [16] [17] [18]
3
[19]
* The two values of h and t calculated for P(S)3MP refer to the two minima of the nearly fourfold left-handed helix (Figs. 6A and B).
Energetic calculations Table 4. Calculated values of the unit height, which minimize the energy under the constraint that the helical symmetry is that observed in the crystalline state. The corresponding values of the energy (in respect to the unconstrained minimum) are reported as AE's Polymer PP PB (form I) PB (form II) PB (form III) P3MB P(S)3MP* PS
Symmetry 3/1 3/1 11/3 4/1 4/1 4/1 3/1
hexp (~-)
hc,j¢.
AE
2.17 2.15 1.85 1.80 1.76 fl.79 1.70 + 0.02 /.1.79 2.22 _ 0.02 2.16
0.05 0.60 0.10 0.10 0.30 0.20 0.10 0.05
2.17 2.17 1.91 1.89 1.71
___0.02 + 0.02 + 0.02 + 0.02 + 0.02
* The two values of hca~c and AE reported for P(S)3MP refer to the two minima of the nearly fourfold left-handed helix (Fig. 6A and B)
to modify the energetic m i n i m a calculated for the isolated chain. Table 5 shows the parameters which correspond to the m i n i m u m of conformational energy, under the restriction that h = hexp a n d t = fexpThe prediction of the unit height for isotactic polypropylene (without any previous assumption concerning valence angles) is excellent, while for the other polymers may be considered as good. In accordance with experimental observations on various polymers [2], the energetic maps of polymers with aliphatic b r a n c h e d chains, show elongated m i n i m a corresponding to pairs of internal rotation angles almost equally "distorted" from staggered conformations. The distortion from staggered values applies as well to the internal rotation angles of the lateral groups. A principle of "isodistortion" was put forward by Natta, Corradini and Bassi [3] on the basis of qualitative energetic considerations, and finds quantitative justification in our results. The experimental observation that isotactic substituted polystyrene having fourfold helices need not have isodistorted chain conformations [20, 21] finds justification in the form of the energetic m i n i m u m for polystyrene, which, differently from the case of P3MB, expands itself also in the region of cr2 = 80-90 °, while a 1 = 180 °. Another point to be noted is the presence of two m i n i m a for PB, P 3 M B a n d P(S)3MP, one corresponding to a threefold helix and one corresponding to the region of a nearly fourfold helix. For the threefold helical conformation, while the torsional potentials are at a minimum, the n o n - b o n d e d interactions force the valence angle T2 to be distorted to 116 °. For the other conformations, while the torsional
* Allegra, Corradini and Ganis [22] have shown the dependence of the optical activity on the ratio E between the internal partition functions, relative to the conformational states of the side groups for one monomer unit in a (TG) vs (GT)-chain. From the calculated energetic maps of P(S)3MP, it is seen that e is of the order of 0.6 in the range of temperature 300-400 K. According to Fig. 2 of the above quoted paper, this would give rise in a solution to a pronounced prevalence of the left-handed sense of spiralization.
835
Table 5. Some parameters corresponding to the minimum of conformational energy under the constraint that h = he~pand t = texp. The values of AE (kcal/mol of CRU) represent, for each polymer, the difference in energy between the corresponding conformation and that of the nearest minimum. The other parameters assume the same value as those reported in Table 2 for the nearest energetic minima Polymer
rz
ax
~r2
PP PB (form I) PB (form II) PB (form III) P3MB
116C' ll& 115 ° 116° 113 °
177 ° 177 ° 200 ° 199.5° 208 °
62 ° 62 ° 75.5 ° 83.5 ° 82.Y
P(S)3MP*
113 °
208"5°
82"5c
PS
116 °
169 °
66 °
AE 0.05 0.80 0.20 0.30 0.35 f 0.40 ( 0.30 0.50
* The two values of AE reported for P(S)3MP refer to the two minima of nearly fourfold left-handed helix (Figs. 6A and B). potentials are no longer at a minimum, the valence angle ~2 is less distorted (f,i. v2 = 113 ° for the fourfold almost "isodistorted" conformations). Finally, the results obtained for P(S)3MP show the way in which the asymmetric configuration of the lateral group influences the chain conformation of the polymer. The conformations of m i n i m u m potential energy are two for the left-handed helix and only one for the righth a n d e d helix. In the solid polymer [18], the conformation of the lateral groups may take statistically b o t h the conformations of m i n i m u m energy found in the present paper (calculated to differ less than 1 kcal/mole of CRU). A requirement for good packing is that the two conformations are represented in equal a m o u n t s in the crystalline lattice. In the melt or in solution, left-handed local helical conformations should prevail in length for entropic reasons* over the right-handed local helical conformations.
REFERENCES
1. P. Corradini, The Stereochemistry of Macromolecules, (Edited by A. D. Ketley) Part III, pp. 1-60, Marcel Dekker, New York (1968). 2. R. L. Miller, Crystallographic data for various polymers, in Polymer Handbook, 2nd edition, John Wiley, New York (1974). 3. G. Natta, P. Corradini and I. W. Bassi, Gazz. Chim. Italiana 89, 784 (1959). 4. C. W. Bunn and D. R. Holmes, Discuss. Faraday Soc. 25, 95 (1958). 5. G. Natta, P. Corradini and P. Ganis, J. Polym. Sci. 58, 1191 (1962). 6. G. Natta and P. Corradini, Nuovo Cim. Suppl. 15, 40 (1960). 7. P. De Santis, F. Giglio, A. M. Liquori and A. Ripamonti, J. Polym. Sci. A1, 1383 (1963). 8. A. J. Hopfinger, Conformational Properties of Macromolecules, Academic Press, New York (1973). 9. H. Tadokoro, K. Tai, M. Yokoyama and M. Kobayashi, J. Polym. Sci., Polym. Phys. ed. 11, 825 (1973). 10. U. W. Suter and P. S. Flory, Macromolecules 8, 765 (1975). 11. A. M. Liquori and P. De Santis, J. Polym. Sci. C16, 4583 (1969).
836
P. CORRADINI,V. PETRACCONEand B. Pmozzl
12. V. Petraccone, B. Pirozzi, A. Frasci and P. Corradini, Europ. Polym. J. 12, 323 (1976). 13. R. G. Snyder and G. Zerbi, Spectrochim. Acta 23A, 391 (1967). 14. G. Natta and P. Corradini, Nuovo Cim. Suppl. 15, 40 (1960). 15. G. Natta, P. Corradini and I. W. Bassi, Nuovo Cim. Suppl. 15, 52 (1960). 16. G. Cojazzi, V. Malta, G. Celotti and R. Zannetti, to be published in Makromolek. Chem. 17. P. Corradini, P. Ganis and V. Petraccone, Europ. Polym. J. 6, 281 (1970).
18. V. Petraccone, P. Ganis, P. Corradini and G. Montagnoli, Europ. Polym. J. 8, 99 (1972). 19. G. Natta, P. Corradini and I. W. Bassi, Nuovo Cim. Suppl. 15, 68 (1960). 20. P. Corradini and P. Ganis, Nuovo Cim. Suppl. 15, 96 (1960). 21. P. Corradini and P. Ganis, Nuovo Cim. Suppl. 15, 104 (1960). 22. G. Allegra, P. Corradini and P. Ganis, Makromolek. Chem. 90, 60 (1966).
Riassunto----Sono stati eseguiti calcoli energetici di conformazioni della catena allo stato cristallino di una serie di polimeri alifatici isotattici (polipropilene (PP), poli-~-butene (PB), poli-3-metilbutene (P3MB), poli-(S)-metilpentene-1 e del polistirene isotattico (PS)). Sono state considerate le possibili variazioni degli angoli di legame C---C----42 della catena principale e di tutti gli angoli di rotazione interna; nessun gruppo di atomi ~ stato considerato come un'unitL Viene discussa la possibilit~ di prevedere i parametri sperimentali dell'elica (h, t) senza nessuna assunzione a priori. Per il P(S)3MP i risultati mostrano come la configurazione asimmetrica del gruppo laterale influenzi la chiralit/t dell'elica nel polimero.