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“Machine” experimental study of the structure of amorphous polymers
Study of structure of amorphous polymers
1169
mic method with those which had been calculated (samples 12-17 in Table 4) indicates that ~he calculation sctieme is valid. Where deviations existed or where there were no peaks present on the diagram of [v]] against J1, one must give preference to the calculated values. Translated by K. A. ALLEN REFERENCES 1. P. A. SMALL~ J. Appl. Chem. 3: 71, 1953 2. J. L. GARDON, Encyclopedia for Polymer Science and Technology, vol. 3, N. Y., 853, 1965 3. R. F. FEDORS, Polymer Engng. and Sci. 14: 147, 1974 4. A. I. KITAIGORODSKII, Organicheskaya Kristallokhimiya (Organic Crystal Chemistry). Izd. Akad. Nauk SSSR, 1953 5. A. A. ASKADSKII, Deformatsiya polimerov (Polymer Deformations). Iz4. " K h l m i y a " , 1973 6. A. WEISSBERGER, E. PROSKAUER, J. RIDDICK and E. TUPPS. Organieheskic rastvoritcli (Organic Solvents). Izd. inostr. Lit., 1958 7. A. A. TAGER, L K. KOLMAKOVA, G. Ya. SHEMY~t{]NA, Ya. S. VYGODgKII a n d S. N. SALAZKIN, Vysokomol. soyed. BI8: 569, 1976 (Not translated in Polymer Sci.
U.S.S.R.)
"MACHINE" EXPERIMENTAL STUDY OF THE STRUCTURE OF AMORPHOUS POLYMERS * A. ~v[. SKVORTSOV,A. A. SARIBAN and T. M. BmSHTEI~ High Polymers Institute, U.S.S.R. Academy of Sciences
{Received 18 June 1976) The dependence of the intra- and intermolecular structure of polymer chains on concentration have been calculated for model chains on a cubic lattice b y "machine experiment". The proportion of chains with intramolccular contact has been found to cease to depend on solvent quality a n d to approach the value equalling t h a t for the separate chain in a O solvent. The primary result will be a change in conformation of those parts of the chains in which the n u m b e r of internal contacts deviates most from Gaussian. No significant differences have been found from random strueturation in t h e intra- a n d inter-molecular ones at all the investigated concentrations with c ~<0.7. A * Vysokomol. soyed. A19: No. 5, 1014-1021, 1977.
1170
A. M. SKVORTSOVet al.
model of the amorphous state of polymers, with a random chain distribution, and each chain of Gaussian type, is proposed. The dimensions of the folded, structured parts in such a model are examined, also the causes of crystallinity or the opposite. It is suggested that the results arequalitatively correct with regard to the behaviour of noncrystal-line polymers in the melt or in bulk.
CLARIFICATION of the high density of amorphous polymers in bulk or in the melt (around 0-8-0-9 of that in the crystalline state) was sought b y Kargin and co-work,:r~ [1] in the folded model of amorphous polymers. According to this or its modification [2, 3], amorphous bulk polymers have partly an orientated state in a considerable region of parallel folding of the chains, the so-called packets, fibrils and associates. A typical feature of such models is the assumed substantial orderly arrangement of the macromolecules inside as molecules with folded structures, as well as between them, due to considerable correlation in position and the conformation of adjacent chains. More recent neutron scattering experiments and X-ray studies of deuterium and iodine labelled macromolecules in concentrated solutions [4, 5] and in bulk [6, 7] showed however that the mean square radius of inertia ( R ~) of amorphous polymer chains coincides with that in a 0 solvent, (R2}0, and that its dependence on the molecular weight (mol. wt.) is a straight line. From this it follows that t h e conformation of amorphous polymer chains is the random Gaussian coil without there being any particular intramolecular structure present. It is stressed t h a t these are the only direct experimental results about the macromolecular structure present in the amorphous state. The experimental light scattering [8] and small angle X-ray scattering studies [9, 10] of amorphous polymers also indicate the absence of any substantial orderly arrangement of zones, although it does not give any information of what the structure is in the amorphous state. It still remains unclear to what extent the fairly dense packing in amorphous bulk polymers hinders the mutual distribution and orientation of individual chains. In other words, is a larger density feasible b y filling the space of uncorrelated Gaussian coils, or can one expect some more or less longer zones to be present in which ~he chains remain parallel while bending simultaneously. Earlier studies [11, 12] dealt with "concentrated solutions"~f model polymer chain lattices in machine experiments, i.e. using the Monte Carlo method. In our approach we used only the polymer chain model without making any assumptions about the properties of the system as a whole. The method therefore can be regarded as an "exp.eriment" giving results which were not obvious earlier. The variations of the model enabled us to clarify what features of the reactions are responsible for any of the properties of the condensed state. In this study we deal with the structure of m~)del lattice chains of polymers in the amorphous state using the Monte Carlo method in a machine experiment a n d p a y particular attention to the mutual packing of the chains.
Study of structure of amorphous polymers
1171
EXPERIMENTAL
The model and the method of calculation were the same as previously described [12], so that we give here only the main outlines. The flexible polymer chain was modelled by a sequence of N chain units (cubes) on a simple cubic lattice: the probability of trans- (forward series) and each of the four cis- (lateral series, or row) isomers was the same. The specific volume of each chain unit equalled that of the unit cell and was non-permeable by other chain units; the blank unit cells modelled the solvent. The chain units of a single or of various chains reacted with each other at energy ~ (the energy value is expressed in units of/~T). The contact energy characterized the solvent quality like Flory's 7. parameter. An important feature of the model under investigation is the independence of the contact energy of the polymer structure at the contact point. The energy e was attributed to chains coming in contact with each other at any orientation and regardless of the presence (or absence) of a bend in the region of contact. We shall return to the problem once more in the concluding part of this study. The chains were plotted at random inside the selected volume, i.e. the cube of edge l, using periodical limits. The particular cube volume c was filled by plotting cl~/N chains. The differing degrees of filling the cube thus represented a collection of systems from which the averaging of all the values was produced. Values l and N were varied within the limits of: l ~ 5-21, N = 16-121. The maximum filling of the space attained in our calculations was about 0.7. The results were virtually independent of l and were exclusively determined by the value of e (the solvent quality), chain length N and concentration c. RESULTS
The state of individual chains. The c o n c e n t r a t i o n d e p e n d e n c e of the a v e r a g e n u m b e r o f i n t r a m o l e c u l a r a n d i n t r a m o l e c u l a r contacts, a n d of t h e t o t a l average n u m b e r of c o n t a c t s =d-, for a chain consisting o f N = 6 1 chain units, is r e p r o d u c e d in Fig. 1. The results are given for good solvents (--~=-0), p r c c i p i t a n t s ( - - ~ = 0 . 5 ) a n d conditions ( - - c = 0 . 3 ) . A detailed e v a l u a t i o n of the c o n c e n t r a t i o n dependences of the geometrical characteristics o f separate chains a n d of their changes as a f u n c t i o n of mol. wt. is given in [11, 12]. W e o n l y note t h a t t h e final state of the chains at high c o n c e n t r a t i o n s is t h e Gaussian coil, as one can g a t h e r f r o m Fig. 1, a n d t h a t this applies to a n y solvent, w h e t h e r good or bad. W e shall m a k e a more detailed e x a m i n a t i o n of the changes in i n t r a m o l e cular c o n t a c t s as ~ f u n c t i o n of concentration. E a c h chain u n i t could h a v e simult a n e o u s contacts, in our model, w i t h 1, 2, 3 or 4 o t h e r chain units. ~Ve shall call (vt ( I ) ) t h e n u m b e r of u n i t s which m a k e a single i n t r a m o l e c u l a r c o n t a c t ,
cular c o n t a c t s will o b v i o u s l y be:
=N/2~k(ri(k)>. k=l
The c o n c e n t r a t i o n dependences of for a chain 61 units long in a good, 0 solvent or a precipitant, are illustrated in Fig. 2, w h i c h shows t h a t the n u m b e r of chain units p a r t i c i p a t i n g in i n t r a m o l e c u l a r reactions (contacts) of v a r i o u s t y p e s r e m a i n s u n c h a n g e d w i t h c o n c e n t r a t i o n u n d e r 0 conditions, w h i c h reflects t h e c o n s t a n c y of t h e internal chain structure. The i n t r a m o l e c u l a r c o n t a c t s are in this case d o m i n a n t l y " s i n g u l a r " a n d t h e n u m b e r of chains p a r t i c i p a t i n g
1172
A . M. SXVORTSOV e~ a/.
s i m u l t a n e o u s l y in/~ n u m b e r s o f contacts diminishes as n u m b e r k becomes larger. One gathers f r o m ~ig. 2 t h a t for k = l , 2, 3 will be satisfactorily approxim a t e d by: ~- ~.
(me) 60
(m i )
Cl
~0 ~
qO
20
10
0.2
o.q
06
I 0.2
0.8 c
I o.q
I o.g
I o.e e
tO0 80 60 qO
20 I
.1
I
0"2
O'q
O'g
I
0"8c
FzG. 1. The concentration dependences of." a--the average number of intramolecular contacts; b--the average number of intermolecular contacts; c--the total average number of contacts, of chains consisting of 61 chains units. 1--in a good solvent, --8-----0; 2--in a 0 solvent, --8=0.3; 3--in a precipitant, --e=0.5. T h e p o l y m e r - p o l y m e r contacts do n o t give a n e n e r g y yield in a good solvent; t h e n u m b e r o f chain units involved in a singular c o n t a c t is t h e r e f o r e n a t u r a l l y smaller t h a n urider 0 conditions. A n y multiple contacts are entirely absent.
Study of structure of amorphous polymers
1173
The initially swelling chain contracts as the concentration increases and will "lengthen" to the 0 condition; here we have a small number of double, treble, etc. contacts which are typical of a separate chain under 0 conditions.
O' q I v': (_T)); (y~ (E)) ; ('v~f.~)), ('v~[~))
0"05 - ~~3
/ /7"3
~
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O"005
1
0"2
I
O'q
I
0"8 c
0'2
O'q
0.6
0.8 c
Fxo. 2. The concentration dependence of the proportion of chain units participating in intramolecular reactions with one (I), two (II), three (III), or foLtr (IV) other chain units present in a chain consisting of 61 units. 1 - 3 - - a s in Fig. 1. I n a precipitant, in contrast, where the polymer-polymer contacts are more advantageous t h a n between polymer and solvent, the number of intramolecular contacts of all types is higher in a dilute solution t h a n under 0 conditions. I t is true to say t h a t the proportion o f chain units completely surrounded by other units of the same chain is small under these conditions, but the number of double contacts equals t h a t of the singular, while t h a t of treble ones is only h a l f that. The coiled macromolecules start to uncoil as the concentration increases and there is a drop of the proportion of chain units in contact with e a c h other. The extent of this reduction differs for various conbact types until t h e
1174
A . M . SKVORTSOVet a~.
0 level is reached (The slope of the descend to the 0 level of numbers). The proportion of singular contacts changes as a convex curve, while t h a t of the chain units participating in multiple contacts simultaneously decreases more rapidly with concentration. The most compact parts of the chain differing in structure from the Gaussian will therefore resolve with increasing concentration in the first instance. The detailed characteristics examined here (Fig. 2), as well as
Ix._l,/.x"
x i..-e a
xJ' x~X~ (
b
~"
I
FIo. 3. The conformation of chain parts resulting in: a - - a single; b - - a triple contact. Tho contacting chain mlits are indicated b y crosses.
the more general ones (Fig. la), thus show that the chain structure is similar to the Gaussian coil in concentrates. The tendency to reach the latter state will be all the greater when chains are longer [11, 12], but the final "forgetting" of its pre-history, i.e. of the swollen or globular state, will only occur at concentration c ~ 0.7, which is close to the density of the polymer in the condensed state. Using our model, generally speaking one can fill the space, to a considerable density, with flexible Gaussian chains; even when the chains in a separate state were non-Gaussian, e.g. swelling coils or the opposite, namely globular structures (compressed), the chains will attempt to conform into chaotic coils as the packing density increases (especially on evaporation of the solvent), and this will correspond with macromolecules in 0 conditions. Intra- and inter-molecular vhain packing. We now go over to an examination of the values which directly characterize the mutual positions of the polymer chains. The parts which come close together and into contact can collide a t separate and singular chain units (lattice joints); such a part we shall call singular while two consecutive collisions will be twins, three will be triplets, etc. Figure 3 shows examples of such contacts at single and triple points. One can see that some of the contacts will not be just at singular points and t h a t they will be determined by the mutual orientation and position in space of the parts nearest to each other. The contacts of a singular flexible chain have been shown to be dominantly by singular parts (between singular parts) [13]; the proportion of contacting chain units present in the singular parts will be about 70%. T h i s value remains almost constant regardless of concentration.
Study of structure of amorphous polymers
1175
Figure 4a illustrates the concentration dependence of the number of intramolecular contacting units O~which form a twin and therefore result in a parallel or anti-parallel position of short chain lengths. One can see that the proportion of such units is about 20% and is practically independent of c or --~. The value of 8~ indicates the existence of occasional bends in the chain which result in intramolecular folds identical with those present in a separate chain under 0 conditions. After the first contact is made, actually realized between two consecutive chain bends, the probability of a twin appearing, i.e. of intramolecular "parallelism" appearing in the cubic lattice chain, is about ¼, i.e. 25%.
b
O'q
0"2
x
x
0
0
0
0"3
Zx3I ol
0
x x
x
x
oo
o
o
2
x
o
o
ol
0"2 0"I
I
I
I
0"2
o'q
0"6
]
[
[
[
O'#
0.2
0.4;
O.G
I
0.8 c
C
Fxo. 4. The concentration dependence of the proportion of orderly arranged, parallelized chain units making: a--intramolecular contacts; b--intermolecular contacts, in a chain consisting of 61 chain units. (1-3=as in Fig. 1.) Calculations showed that the probability of longer folds, containing triplets or quadruplets rapidly drops in the separate chain [13]. The proportion of chain units incorporated in such "extended" intramolecular parallel parts was found go be small even in the condensed state, i.e. of the order of 5 0 , so that any significant differences between intramolecular structuration and the random one were not detected (a simpler assessment o f the probability of an "extended" chain with 3 units was found to be (¼)3--0.06) There was no marked intramolecular structuration either (Fig. 4b). 'i'he latter diagram shows the proportion of chain units ~ present as twin parallel parts. The number of such units, compared with the total number of ir~h'amolecular contacts, was independent of c and about 0.27, which corresponds to a random parallelization of individual chain units (the probability of intermolecular parallel contacts forming in a random approach of the chain units, gave 0.25 as for an intramolecular parallelism). The comparison of Fig. 4a with 4b shows that the probability of parallel chains existing is the same in intra- as in inter-molecular contacts according to random statistics. It also follows from our calculations that the proportion of inter- to intra-nmbcular contacts occurring in parallel parts, i.e. b y means of three or more chain units, is smaller b y one order of magnitude than that of twin parallel parts (about 5 o/o compared with 27%). The absence of any specific "parallelizatien" of adjacent chains means that there will not be such an occurrence as regards the more r e m o ~ chains.
1176
A, M. SKVORTSOV et ~ .
One can therefore state t h a t there is no intra- and intermolecular orientation differing from the random one over all the calculated concentration range in our model. The flexible chain model which assumes the Gaussian coil conformation as the concentration increases (regardless of the state in which the chains existed in dilute solutions) are therefore capable of filling the space to a density comparable with that of the amorphous state without any specific occurrence of orientation.
FIG. 5. A two dimensional diagram of the structure of an amorphous polymer having a random arrangement of the polymer chains of Gaussian type.
The struv2ure of the amorphous state of polymers. One can gather from the above that the structure of the condensed state in our model depends significantly on the structure, of separate chains in a 0 solvent. The number of intramolecular contacts and the local density associated with it for chain units around a n y one separate unit will depend on the chain rigidity, b u t also on the mechanism of this rigidity, i.e. on the possibility of folding as a result of large bends in the chain [13]. In the case of flexible chains which easily bend as in our model, the number of intramoleeular contacts in a 0 solvent is typically larger and the local chain unit density consequently also larger. It follows from Fig. la that the proportion of linked chain unit contacts, a : 2 (m~)/N, is ~,ery large in separate chains, i.e. of the order of one (or in other words, one chain unit participates on average in an intra-chain contact). The reason is the relatively easy collision of chain units in our model and the fairly large free volume. A fairly large number o f energetically favourable interactions must exist for its concentration at the O point. The intramolecular contacts are realized chiefly b y the adjacent chain parts folding over each other, so t h a t any loop formation is minimized [13]. A large extent of folding is therefore typical of our flexible chain model under conditions. The dimensions of these folded and structured parts are small, i.e. about several chain units, which amoun¢~ to 20-30 A in the ease of about 10 A thick chains. The estimate is correct for flexible polymers capable of folding
Study of structure of amorphous polymers
1177
easily on each other and possessing a low local asymmetry, i.e. chains with a persistent length close to their specific width. In the case of more rigid chains of a persistent length 2-3 times larger than their width, the dimensions of the folded part will be larger and can reach about 100 A. A two dimensional structural model of the flexible chain shown in Fig. 5 is based on our results for an amorphous polymer; its chain arrangement is random and each chain Gaussian. In contrast to the model of the amorphous state suggested b y other investigators [1-3], we did not examine the larger areas of 10~-105/~ b u t restricted ourselves to the structure present in a smaller volume, which had the linear dimension of about 108 _~. A typical feature of the described model is particularly the ease with which intermolecular contacts occur in any one chain with several others. The separate chains do not fold into globules b y folding over themselves b u t neither do they produce bundles b y a long contact with any one chain. The chain model remains a Guassian coil in the condensed state at any contact energy, whether positive or negative. In other words, our system has no transition to the crystalline state on varying E. As all the energies were expressed in kT units, the polymers represented b y this model remain amorphous when condensed from the solvent or cooled from the melt, and the reasons are purely thermodynamic and not kinetic. Let us now examine the main characteristics of our model chains and their correlations with real chains. We examined chains which have a rotation-isomeric mechanism of flexibility and we assumed that each row is principally capable of realizing several energetically equivalent rotation isomers. The flexibility of such chains therefore does not depend on the temperature and remains constant. Reduction in temperature does not increase their rigidity due to freezing out some of the less favourable rotation isomers, and there is therefore no crystallization which is specific for rigid chain polymers. The folding also did not result in crystallization of the chains in the examined model. The reason is the ease of intramolecular contact formation (Fig. 5), b u t also the lack of any chain unit reaction in the contact as a function of structure. The adjacent chains had accesss to any unit of the particular chain from all the sides, e.g. at the point of the bend, and could yield (or lose) the same energy. The particular model thus lacks the mechanism which could cause crystallization or at least some orderly arrangement of the system (such a mechanism could of course be easily introduced into the model, and we intend to do so). The main reason for the polymer model not crystallizing is the absence of an orderly arrangement of the branches present; atactic polymers do not crystallize. In other words, if the chain structure differs at the bend from that of the linear (rigid) part in iso- or syndio-tactic crystallizing polymers, one must expect some defects to be present and therefore a smaller energy yield, while the position at the bend of atactic, non-crystallizing polymers and of the linear chain pa1~ is rather similar. The atactic polymer chain is a "flat defect" and the bending
1178
A.M. S x ' v o ~ s o v et al.
positions, as well as t h e r e m a i n i n g chain p a r t s , c o n t a i n c h a o t i c a l l y a r r a n g e d , p r o t r u d i n g b r a n c h e s w h i c h h i n d e r t h e i n t r a - a n d i n t e r - m o l e c u l a r packing. T h e a t a c t i c p o l y m e r c a n be q u a l i t a t i v e l y p r e s e n t e d for this r e a s o n as consisting o f chains h a v i n g identical c o n t a c t energies of a n y of t h e i r chain units, e v e n if t h e r e are no contacts. T h e linear chain s t r u c t u r e in t h e s a m e a t a c t i c p o l y m e r , in w h i c h t h e energet i c a l l y m o s t f a v o u r a b l e isomers will be retained, is also irregular a n d t h e n a t u r e o f t h e i r r e g u l a r i t y differs in t h e i n d i v i d u a l chain p a r t s . T h e a n a l o g y w i t h t h e m o d e l e x a m i n e d b y us is e v i d e n t in this respect. W e t h e r e f o r e s a y t h a t o u r results are q u a l i t a t i v e l y correct a n d describe t h e b e h a v i o u r o f non-crystallizing p o l y m e r s in t h e m e l t or in bulk. T h e a u t h o r s t h a n k V. A. K a b a n o v for v a l u a b l e advice a n d useful criticism. Translated by K . A. ALLE~
REFERENCES
1. V. A. KARGIN, A. I. KITAIGORODSKII and G. L. SLONIMSKII, Kolloid. Zhur. 19: 131, 1957 2. J. CLEMF_aNTand P. GELL, J. Maeromol. Sci. B5: 505, 1971 3. S. A. ARZHAKOV, N. F. BAKEYEV and V. A. KABANOV, Vysokomol. soyed. 15: 630, 1973 (Not translated in Polymer Sei. U.S.S.R.) 4. H. HAYASHI, F. HAMADA and A. NAKAJIM~, Maeromoleeules 7: 959, 1974 5. R. G. KIRSTE, W. A. CRUSOEand K. IBEL, Polymer 16: 120, 1975 6. J. P. COTTON, D. DECKER, H. BENOIT, B. FARNOUX and Others, Maeromoleeules 7: 863, 1974 7. G. D. WIGNALL, J. SCHELTEN and D. G. BALLARD, J. Appl. Crystallogr. 7: 190, 1974 8. G. D. PATTERSON, Polymer Preprints 15: 14, 1974 9. A. L. RENNINGER, G. G. WICKS and D. R. UHLMANN, J. Polymer Sei., Polymer Phys. Ed. 13: 1247, 1975 10. A. L. RENNINGER and D. R. UHLMANN, J. Polymer Sci., Polymer Phys. Ed. 18: 1481, 1975 11. A. A. SARIBAN, T. M. BIRSHTEIN and A. M. SKV0RTSOV, Dokl. Akad. Nauk SSSR 229: 1404, 1976 12. T. M. BIRSHTEIN, A. M. SKVORTSOV and A. A. SARIBAN, Vysokomol. soyed. AIg: 63, 1977 (Translated in Polymer Sei. U.S.S.R. 1~ 1, 1977) 13. T. M. BIRSHTEIN, A. A. SKVORTSOV and A. A. SARIBAN, Vysokomol. soyed. AI8: 1978, 1976 (Translated in Polymer Sci. U.S.S.R. 18: 9, 2260, 1976)
1169
mic method with those which had been calculated (samples 12-17 in Table 4) indicates that ~he calculation sctieme is valid. Where deviations existed or where there were no peaks present on the diagram of [v]] against J1, one must give preference to the calculated values. Translated by K. A. ALLEN REFERENCES 1. P. A. SMALL~ J. Appl. Chem. 3: 71, 1953 2. J. L. GARDON, Encyclopedia for Polymer Science and Technology, vol. 3, N. Y., 853, 1965 3. R. F. FEDORS, Polymer Engng. and Sci. 14: 147, 1974 4. A. I. KITAIGORODSKII, Organicheskaya Kristallokhimiya (Organic Crystal Chemistry). Izd. Akad. Nauk SSSR, 1953 5. A. A. ASKADSKII, Deformatsiya polimerov (Polymer Deformations). Iz4. " K h l m i y a " , 1973 6. A. WEISSBERGER, E. PROSKAUER, J. RIDDICK and E. TUPPS. Organieheskic rastvoritcli (Organic Solvents). Izd. inostr. Lit., 1958 7. A. A. TAGER, L K. KOLMAKOVA, G. Ya. SHEMY~t{]NA, Ya. S. VYGODgKII a n d S. N. SALAZKIN, Vysokomol. soyed. BI8: 569, 1976 (Not translated in Polymer Sci.
U.S.S.R.)
"MACHINE" EXPERIMENTAL STUDY OF THE STRUCTURE OF AMORPHOUS POLYMERS * A. ~v[. SKVORTSOV,A. A. SARIBAN and T. M. BmSHTEI~ High Polymers Institute, U.S.S.R. Academy of Sciences
{Received 18 June 1976) The dependence of the intra- and intermolecular structure of polymer chains on concentration have been calculated for model chains on a cubic lattice b y "machine experiment". The proportion of chains with intramolccular contact has been found to cease to depend on solvent quality a n d to approach the value equalling t h a t for the separate chain in a O solvent. The primary result will be a change in conformation of those parts of the chains in which the n u m b e r of internal contacts deviates most from Gaussian. No significant differences have been found from random strueturation in t h e intra- a n d inter-molecular ones at all the investigated concentrations with c ~<0.7. A * Vysokomol. soyed. A19: No. 5, 1014-1021, 1977.
1170
A. M. SKVORTSOVet al.
model of the amorphous state of polymers, with a random chain distribution, and each chain of Gaussian type, is proposed. The dimensions of the folded, structured parts in such a model are examined, also the causes of crystallinity or the opposite. It is suggested that the results arequalitatively correct with regard to the behaviour of noncrystal-line polymers in the melt or in bulk.
CLARIFICATION of the high density of amorphous polymers in bulk or in the melt (around 0-8-0-9 of that in the crystalline state) was sought b y Kargin and co-work,:r~ [1] in the folded model of amorphous polymers. According to this or its modification [2, 3], amorphous bulk polymers have partly an orientated state in a considerable region of parallel folding of the chains, the so-called packets, fibrils and associates. A typical feature of such models is the assumed substantial orderly arrangement of the macromolecules inside as molecules with folded structures, as well as between them, due to considerable correlation in position and the conformation of adjacent chains. More recent neutron scattering experiments and X-ray studies of deuterium and iodine labelled macromolecules in concentrated solutions [4, 5] and in bulk [6, 7] showed however that the mean square radius of inertia ( R ~) of amorphous polymer chains coincides with that in a 0 solvent, (R2}0, and that its dependence on the molecular weight (mol. wt.) is a straight line. From this it follows that t h e conformation of amorphous polymer chains is the random Gaussian coil without there being any particular intramolecular structure present. It is stressed t h a t these are the only direct experimental results about the macromolecular structure present in the amorphous state. The experimental light scattering [8] and small angle X-ray scattering studies [9, 10] of amorphous polymers also indicate the absence of any substantial orderly arrangement of zones, although it does not give any information of what the structure is in the amorphous state. It still remains unclear to what extent the fairly dense packing in amorphous bulk polymers hinders the mutual distribution and orientation of individual chains. In other words, is a larger density feasible b y filling the space of uncorrelated Gaussian coils, or can one expect some more or less longer zones to be present in which ~he chains remain parallel while bending simultaneously. Earlier studies [11, 12] dealt with "concentrated solutions"~f model polymer chain lattices in machine experiments, i.e. using the Monte Carlo method. In our approach we used only the polymer chain model without making any assumptions about the properties of the system as a whole. The method therefore can be regarded as an "exp.eriment" giving results which were not obvious earlier. The variations of the model enabled us to clarify what features of the reactions are responsible for any of the properties of the condensed state. In this study we deal with the structure of m~)del lattice chains of polymers in the amorphous state using the Monte Carlo method in a machine experiment a n d p a y particular attention to the mutual packing of the chains.
Study of structure of amorphous polymers
1171
EXPERIMENTAL
The model and the method of calculation were the same as previously described [12], so that we give here only the main outlines. The flexible polymer chain was modelled by a sequence of N chain units (cubes) on a simple cubic lattice: the probability of trans- (forward series) and each of the four cis- (lateral series, or row) isomers was the same. The specific volume of each chain unit equalled that of the unit cell and was non-permeable by other chain units; the blank unit cells modelled the solvent. The chain units of a single or of various chains reacted with each other at energy ~ (the energy value is expressed in units of/~T). The contact energy characterized the solvent quality like Flory's 7. parameter. An important feature of the model under investigation is the independence of the contact energy of the polymer structure at the contact point. The energy e was attributed to chains coming in contact with each other at any orientation and regardless of the presence (or absence) of a bend in the region of contact. We shall return to the problem once more in the concluding part of this study. The chains were plotted at random inside the selected volume, i.e. the cube of edge l, using periodical limits. The particular cube volume c was filled by plotting cl~/N chains. The differing degrees of filling the cube thus represented a collection of systems from which the averaging of all the values was produced. Values l and N were varied within the limits of: l ~ 5-21, N = 16-121. The maximum filling of the space attained in our calculations was about 0.7. The results were virtually independent of l and were exclusively determined by the value of e (the solvent quality), chain length N and concentration c. RESULTS
The state of individual chains. The c o n c e n t r a t i o n d e p e n d e n c e of the a v e r a g e n u m b e r o f i n t r a m o l e c u l a r
cular c o n t a c t s will o b v i o u s l y be:
The c o n c e n t r a t i o n dependences of
1172
A . M. SXVORTSOV e~ a/.
s i m u l t a n e o u s l y in/~ n u m b e r s o f contacts diminishes as n u m b e r k becomes larger. One gathers f r o m ~ig. 2 t h a t
(me) 60
(m i )
Cl
~0 ~
qO
20
10
0.2
o.q
06
I 0.2
0.8 c
I o.q
I o.g
I o.e e
tO0 80 60 qO
20 I
.1
I
0"2
O'q
O'g
I
0"8c
FzG. 1. The concentration dependences of." a--the average number of intramolecular contacts; b--the average number of intermolecular contacts; c--the total average number of contacts, of chains consisting of 61 chains units. 1--in a good solvent, --8-----0; 2--in a 0 solvent, --8=0.3; 3--in a precipitant, --e=0.5. T h e p o l y m e r - p o l y m e r contacts do n o t give a n e n e r g y yield in a good solvent; t h e n u m b e r o f chain units involved in a singular c o n t a c t is t h e r e f o r e n a t u r a l l y smaller t h a n urider 0 conditions. A n y multiple contacts are entirely absent.
Study of structure of amorphous polymers
1173
The initially swelling chain contracts as the concentration increases and will "lengthen" to the 0 condition; here we have a small number of double, treble, etc. contacts which are typical of a separate chain under 0 conditions.
O' q I v': (_T)); (y~ (E)) ; ('v~f.~)), ('v~[~))
0"05 - ~~3
/ /7"3
~
0'0101 Z7-2
O"005
1
0"2
I
O'q
I
0"8 c
0'2
O'q
0.6
0.8 c
Fxo. 2. The concentration dependence of the proportion of chain units participating in intramolecular reactions with one (I), two (II), three (III), or foLtr (IV) other chain units present in a chain consisting of 61 units. 1 - 3 - - a s in Fig. 1. I n a precipitant, in contrast, where the polymer-polymer contacts are more advantageous t h a n between polymer and solvent, the number of intramolecular contacts of all types is higher in a dilute solution t h a n under 0 conditions. I t is true to say t h a t the proportion o f
1174
A . M . SKVORTSOVet a~.
0 level is reached (The slope of the descend to the 0 level of numbers). The proportion of singular contacts changes as a convex curve, while t h a t of the chain units participating in multiple contacts simultaneously decreases more rapidly with concentration. The most compact parts of the chain differing in structure from the Gaussian will therefore resolve with increasing concentration in the first instance. The detailed characteristics examined here (Fig. 2), as well as
Ix._l,/.x"
x i..-e a
xJ' x~X~ (
b
~"
I
FIo. 3. The conformation of chain parts resulting in: a - - a single; b - - a triple contact. Tho contacting chain mlits are indicated b y crosses.
the more general ones (Fig. la), thus show that the chain structure is similar to the Gaussian coil in concentrates. The tendency to reach the latter state will be all the greater when chains are longer [11, 12], but the final "forgetting" of its pre-history, i.e. of the swollen or globular state, will only occur at concentration c ~ 0.7, which is close to the density of the polymer in the condensed state. Using our model, generally speaking one can fill the space, to a considerable density, with flexible Gaussian chains; even when the chains in a separate state were non-Gaussian, e.g. swelling coils or the opposite, namely globular structures (compressed), the chains will attempt to conform into chaotic coils as the packing density increases (especially on evaporation of the solvent), and this will correspond with macromolecules in 0 conditions. Intra- and inter-molecular vhain packing. We now go over to an examination of the values which directly characterize the mutual positions of the polymer chains. The parts which come close together and into contact can collide a t separate and singular chain units (lattice joints); such a part we shall call singular while two consecutive collisions will be twins, three will be triplets, etc. Figure 3 shows examples of such contacts at single and triple points. One can see that some of the contacts will not be just at singular points and t h a t they will be determined by the mutual orientation and position in space of the parts nearest to each other. The contacts of a singular flexible chain have been shown to be dominantly by singular parts (between singular parts) [13]; the proportion of contacting chain units present in the singular parts will be about 70%. T h i s value remains almost constant regardless of concentration.
Study of structure of amorphous polymers
1175
Figure 4a illustrates the concentration dependence of the number of intramolecular contacting units O~which form a twin and therefore result in a parallel or anti-parallel position of short chain lengths. One can see that the proportion of such units is about 20% and is practically independent of c or --~. The value of 8~ indicates the existence of occasional bends in the chain which result in intramolecular folds identical with those present in a separate chain under 0 conditions. After the first contact is made, actually realized between two consecutive chain bends, the probability of a twin appearing, i.e. of intramolecular "parallelism" appearing in the cubic lattice chain, is about ¼, i.e. 25%.
b
O'q
0"2
x
x
0
0
0
0"3
Zx3I ol
0
x x
x
x
oo
o
o
2
x
o
o
ol
0"2 0"I
I
I
I
0"2
o'q
0"6
]
[
[
[
O'#
0.2
0.4;
O.G
I
0.8 c
C
Fxo. 4. The concentration dependence of the proportion of orderly arranged, parallelized chain units making: a--intramolecular contacts; b--intermolecular contacts, in a chain consisting of 61 chain units. (1-3=as in Fig. 1.) Calculations showed that the probability of longer folds, containing triplets or quadruplets rapidly drops in the separate chain [13]. The proportion of chain units incorporated in such "extended" intramolecular parallel parts was found go be small even in the condensed state, i.e. of the order of 5 0 , so that any significant differences between intramolecular structuration and the random one were not detected (a simpler assessment o f the probability of an "extended" chain with 3 units was found to be (¼)3--0.06) There was no marked intramolecular structuration either (Fig. 4b). 'i'he latter diagram shows the proportion of chain units ~ present as twin parallel parts. The number of such units, compared with the total number of ir~h'amolecular contacts, was independent of c and about 0.27, which corresponds to a random parallelization of individual chain units (the probability of intermolecular parallel contacts forming in a random approach of the chain units, gave 0.25 as for an intramolecular parallelism). The comparison of Fig. 4a with 4b shows that the probability of parallel chains existing is the same in intra- as in inter-molecular contacts according to random statistics. It also follows from our calculations that the proportion of inter- to intra-nmbcular contacts occurring in parallel parts, i.e. b y means of three or more chain units, is smaller b y one order of magnitude than that of twin parallel parts (about 5 o/o compared with 27%). The absence of any specific "parallelizatien" of adjacent chains means that there will not be such an occurrence as regards the more r e m o ~ chains.
1176
A, M. SKVORTSOV et ~ .
One can therefore state t h a t there is no intra- and intermolecular orientation differing from the random one over all the calculated concentration range in our model. The flexible chain model which assumes the Gaussian coil conformation as the concentration increases (regardless of the state in which the chains existed in dilute solutions) are therefore capable of filling the space to a density comparable with that of the amorphous state without any specific occurrence of orientation.
FIG. 5. A two dimensional diagram of the structure of an amorphous polymer having a random arrangement of the polymer chains of Gaussian type.
The struv2ure of the amorphous state of polymers. One can gather from the above that the structure of the condensed state in our model depends significantly on the structure, of separate chains in a 0 solvent. The number of intramolecular contacts and the local density associated with it for chain units around a n y one separate unit will depend on the chain rigidity, b u t also on the mechanism of this rigidity, i.e. on the possibility of folding as a result of large bends in the chain [13]. In the case of flexible chains which easily bend as in our model, the number of intramoleeular contacts in a 0 solvent is typically larger and the local chain unit density consequently also larger. It follows from Fig. la that the proportion of linked chain unit contacts, a : 2 (m~)/N, is ~,ery large in separate chains, i.e. of the order of one (or in other words, one chain unit participates on average in an intra-chain contact). The reason is the relatively easy collision of chain units in our model and the fairly large free volume. A fairly large number o f energetically favourable interactions must exist for its concentration at the O point. The intramolecular contacts are realized chiefly b y the adjacent chain parts folding over each other, so t h a t any loop formation is minimized [13]. A large extent of folding is therefore typical of our flexible chain model under conditions. The dimensions of these folded and structured parts are small, i.e. about several chain units, which amoun¢~ to 20-30 A in the ease of about 10 A thick chains. The estimate is correct for flexible polymers capable of folding
Study of structure of amorphous polymers
1177
easily on each other and possessing a low local asymmetry, i.e. chains with a persistent length close to their specific width. In the case of more rigid chains of a persistent length 2-3 times larger than their width, the dimensions of the folded part will be larger and can reach about 100 A. A two dimensional structural model of the flexible chain shown in Fig. 5 is based on our results for an amorphous polymer; its chain arrangement is random and each chain Gaussian. In contrast to the model of the amorphous state suggested b y other investigators [1-3], we did not examine the larger areas of 10~-105/~ b u t restricted ourselves to the structure present in a smaller volume, which had the linear dimension of about 108 _~. A typical feature of the described model is particularly the ease with which intermolecular contacts occur in any one chain with several others. The separate chains do not fold into globules b y folding over themselves b u t neither do they produce bundles b y a long contact with any one chain. The chain model remains a Guassian coil in the condensed state at any contact energy, whether positive or negative. In other words, our system has no transition to the crystalline state on varying E. As all the energies were expressed in kT units, the polymers represented b y this model remain amorphous when condensed from the solvent or cooled from the melt, and the reasons are purely thermodynamic and not kinetic. Let us now examine the main characteristics of our model chains and their correlations with real chains. We examined chains which have a rotation-isomeric mechanism of flexibility and we assumed that each row is principally capable of realizing several energetically equivalent rotation isomers. The flexibility of such chains therefore does not depend on the temperature and remains constant. Reduction in temperature does not increase their rigidity due to freezing out some of the less favourable rotation isomers, and there is therefore no crystallization which is specific for rigid chain polymers. The folding also did not result in crystallization of the chains in the examined model. The reason is the ease of intramolecular contact formation (Fig. 5), b u t also the lack of any chain unit reaction in the contact as a function of structure. The adjacent chains had accesss to any unit of the particular chain from all the sides, e.g. at the point of the bend, and could yield (or lose) the same energy. The particular model thus lacks the mechanism which could cause crystallization or at least some orderly arrangement of the system (such a mechanism could of course be easily introduced into the model, and we intend to do so). The main reason for the polymer model not crystallizing is the absence of an orderly arrangement of the branches present; atactic polymers do not crystallize. In other words, if the chain structure differs at the bend from that of the linear (rigid) part in iso- or syndio-tactic crystallizing polymers, one must expect some defects to be present and therefore a smaller energy yield, while the position at the bend of atactic, non-crystallizing polymers and of the linear chain pa1~ is rather similar. The atactic polymer chain is a "flat defect" and the bending
1178
A.M. S x ' v o ~ s o v et al.
positions, as well as t h e r e m a i n i n g chain p a r t s , c o n t a i n c h a o t i c a l l y a r r a n g e d , p r o t r u d i n g b r a n c h e s w h i c h h i n d e r t h e i n t r a - a n d i n t e r - m o l e c u l a r packing. T h e a t a c t i c p o l y m e r c a n be q u a l i t a t i v e l y p r e s e n t e d for this r e a s o n as consisting o f chains h a v i n g identical c o n t a c t energies of a n y of t h e i r chain units, e v e n if t h e r e are no contacts. T h e linear chain s t r u c t u r e in t h e s a m e a t a c t i c p o l y m e r , in w h i c h t h e energet i c a l l y m o s t f a v o u r a b l e isomers will be retained, is also irregular a n d t h e n a t u r e o f t h e i r r e g u l a r i t y differs in t h e i n d i v i d u a l chain p a r t s . T h e a n a l o g y w i t h t h e m o d e l e x a m i n e d b y us is e v i d e n t in this respect. W e t h e r e f o r e s a y t h a t o u r results are q u a l i t a t i v e l y correct a n d describe t h e b e h a v i o u r o f non-crystallizing p o l y m e r s in t h e m e l t or in bulk. T h e a u t h o r s t h a n k V. A. K a b a n o v for v a l u a b l e advice a n d useful criticism. Translated by K . A. ALLE~
REFERENCES
1. V. A. KARGIN, A. I. KITAIGORODSKII and G. L. SLONIMSKII, Kolloid. Zhur. 19: 131, 1957 2. J. CLEMF_aNTand P. GELL, J. Maeromol. Sci. B5: 505, 1971 3. S. A. ARZHAKOV, N. F. BAKEYEV and V. A. KABANOV, Vysokomol. soyed. 15: 630, 1973 (Not translated in Polymer Sei. U.S.S.R.) 4. H. HAYASHI, F. HAMADA and A. NAKAJIM~, Maeromoleeules 7: 959, 1974 5. R. G. KIRSTE, W. A. CRUSOEand K. IBEL, Polymer 16: 120, 1975 6. J. P. COTTON, D. DECKER, H. BENOIT, B. FARNOUX and Others, Maeromoleeules 7: 863, 1974 7. G. D. WIGNALL, J. SCHELTEN and D. G. BALLARD, J. Appl. Crystallogr. 7: 190, 1974 8. G. D. PATTERSON, Polymer Preprints 15: 14, 1974 9. A. L. RENNINGER, G. G. WICKS and D. R. UHLMANN, J. Polymer Sei., Polymer Phys. Ed. 13: 1247, 1975 10. A. L. RENNINGER and D. R. UHLMANN, J. Polymer Sci., Polymer Phys. Ed. 18: 1481, 1975 11. A. A. SARIBAN, T. M. BIRSHTEIN and A. M. SKV0RTSOV, Dokl. Akad. Nauk SSSR 229: 1404, 1976 12. T. M. BIRSHTEIN, A. M. SKVORTSOV and A. A. SARIBAN, Vysokomol. soyed. AIg: 63, 1977 (Translated in Polymer Sei. U.S.S.R. 1~ 1, 1977) 13. T. M. BIRSHTEIN, A. A. SKVORTSOV and A. A. SARIBAN, Vysokomol. soyed. AI8: 1978, 1976 (Translated in Polymer Sci. U.S.S.R. 18: 9, 2260, 1976)