Faraday Discuss. Chem. SOC., 1985,79, 73-84 X-Ray Analysis of the Structure of Liquid-crystalline Copolyesters BY JOHN BLACKWELL,* AMIT BISWAS, GENARO A. GUTIERREZ AND ROBIN A. CHIVERS Department of Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 44106, U.S.A. Received 10 th December, 1984 The X-ray analyses of the structures of a group of aromatic copolyesters that form liquid-crystalline melts is described. X-ray patterns of melt-spun fibres of wholly aromatic copolymers prepared from p-hydroxybenzoic acid (HBA) and 2-hydroxy-6-naphthoic acid (HNA) show a high degree of axial orientation and there is also evidence for three-dimensional order. The meridional maxima are aperiodic, and it is shown that their positions and intensities are predicted by a model consisting of an assembly of parallel chains of completely random monomer sequences.Analyses of the scattering by aperiodic chains are summarized, first for point monomers and subsequently for an atomic model. From the breadth of some of the meridional maxima it is.possible to estimate the persistence or correlation length for the stiff-chain conformation, which is found to be between 9 and 13 monomer units, depending on the HBA/ HNA ratio. Determination of the three-dimensional structure is initiated via calculation of the cylindrically averaged transform for the random chain, followed by consideration of the interference effects caused by chain packing. The patent literature describes a number of aromatic copolyesters [see ref. (1) for a review] prepared from monomers such as p-hydroxybenzoic acid (HBA) and 2-hydroxy-6-naphthoic acid (HNA), that form mesomorphic melts and have poten- tial applications as, for example, high-strength fibres and novel moulded plastics.Homopolymers of HBA and HNA are crystalline, infusible materials, but copoly- merization breaks up the ordered structure and nematic melts occur as a result of the extended conformation. The mechanical properties of the different copolymers2 are very dependent on the chemistry of the monomers and this has prompted the study of their physical structure described below. Analyses of their morphology3 and thermal properties4 have been reported recently by other groups. X-ray diffraction patterns of melt-spun fibres of copoly( HBA/ HNA) at three comonomer ratios’ are shown schematically in fig.1. The polymer specimens were prepared6 at Celanese Research Co., Summit, N.J. The diffraction patterns reveal a high degree of axial orientation of the molecules, and the observation of sharp equatorial and off-equatorial Bragg maxima point to the existence of some three- dimensional order. Blundell’ used the intensity of these reflections to estimate a ‘degree of crystallinity’ of ca. 21% for a preparation of the 40/60 copolymer. Our attention was first focused on the meridional maxima. The d spacings of these maxima for five comonomer ratios’ are listed in table 1, where it can be seen that they are aperiodic, i.e. they are not orders of a simple repeat, and also shift progressively with the monomer ratio. There are no analytical data presently available on the monomer sequence distribution: n.m.r.methods, for example, have not been successful because of the low solubility and the chemical similarity of the 7374 X-RAY ANALYSIS OF LIQUID-CRYSTALLINE COPOLYESTERS Fig. 1. Schematic representations of the X-ray fibre diagrams of melt-spun fibres of copoly(HBA/HNA) for three comonomer ratios: (a) 30/70, (b) 58/42 and (c) 75/25 [from ref. ( 5 ) ] . monomers. However,- these X-ray data argue against the existence of extensive block-copolymer structure, which would lead to periodic meridional maxima charac- teristic of one or both types of block. We have therefore analysed the data in terms of a completely random monomer sequence. DIFFRACTION BY APERIODIC POLYMER CHAINS Fig.2 ( a ) shows a projection of a model of a typical random sequence of copoly( HBA/ HNA). The monomers were constructed' using standard bond lengths and angles, with planar aromatic and carboxy groups. The only conformational freedom is due to torsional rotation about the aromatic-carboxy linkage bonds; these angles were set so that the mutual inclination of the planes of the aromatic and carboxy groups was 30°, consistent with the results of conformational analysis.' The meridional intensity I ( 2 ) depends on the Fourier transform of the projection of the structure onto the chain axis, z. Since the aromatic-carboxy bonds areJ. BLACKWELL, A. BISWAS, G. A. GUTIERREZ AND R. A. CHIVERS 75 Table 1. Observed and calculated meridional intensity maxima for copoly( HBA/ HNA)" monomer experimental d spacings/A calculated d spacings/A mole ratio, HBA/HNA film diffractometer atomic model point model 25/75 8.09 f 0.07 4.08 f 0.04 2.77 f 0.03 2.05 f 0.03 30/70 7.95 4.1 1 2.83 2.06 50/50 7.43 2.84 2.02 58/42 7.35 2.98 2.05 75/25 6.78 3.03 2.03 8.1 1 f0.07 4.15 f 0.02 2.85 f 0.01 2.09 f 0.01 7.89 4.09 2.87 2.09 7.49 2.95 2.09 7.19 2.96 2.08 6.70 3.09 2.09 8.09 4.20 2.85 2.10 8.01 4.2 1 2.86 2.10 7.6 1 4.3 1 2.95 2.10 7.45 4.40 2.99 2.1 1 7.04 3.09 2.1 1 7.98 4.17 2.84 2.10 7.88 4.17 2.85 2.10 7.41 4.1 1 2.93 2.10 7.19 4.0 1 2.98 2.10 6.75 3.09 2.1 1 a From ref.( 5 ) . ( b ) * B o B * N - N * B * N . B * N * B * B * Fig. 2. ( a ) Projection of a model of a typical random sequence of copoly(HBA/HNA) and (b) point model for the same sequence.approximately parallel to the fibre axis, this z projection will be largely independent of the conformation. In particular the axial advance will be approximately constant for each monomer type, Le. approximately equal to the residue length. Based on the molecular models, the lengths of the HBA and HNA residues were taken as 6.35 and 8.37 A, respectively. As a first approximation to the structure of wholly aromatic copolyester chains" we represented each residue by a point placed for convenience at the ester oxygen and separated from its neighbours by the appropriate residue lengths, as in fig. 2( 6 ) . The Fourier transform F,(Z) of a chain of N point monomers is given by N F J Z ) = C exp 2niZzj j = 176 X-RAY ANALYSIS OF LIQUID-CRYSTALLINE COPOLYESTERS where zj is the coordinate of the jth point.If we assume there is no axial register of the chains and neglect for the moment the Lorentz and polarization effects, then the intensity is obtained by averaging the square of the modulus of Fc(Z) over all n possible monomer sequences: where pc is the probability of the cth chain and depends only on the monomer ratio in the case of a completely random copolymer. I ( 2 ) was first evaluated approxi- mately by use of Monte Carlo methods to set up random chain sequence^.^ Sub- sequently this was replaced' ' by an exact calculation via the autocorrelation function of the chain, Q ( z ) : I ( 2 ) = 1 Q(z,) exp 2.rriZz1. 1 (3) Q ( z ) is the probability of first, second, third etc. nearest neighbours along the random chain and is zero except at specific values of z = zl.Q ( z ) for the 58/42 copoly(HBA/HNA) is shown in fig. 3 out to the 14th (positive) nearest neighbour in an infinite chain. The summation in eqn (3) has a closed solution'* which can be derived by treatment of the chain as a one-dimensional paracry~tal'~ with bimodal statistics. In fig. 3 the positive terms in Q ( z ) are separated into components labeled H,, HI, H2, H3 etc. for the zeroth, first, second, third etc. nearest neighbours. It can be seen that H2(z) is the self-convolution of Hl(z): and in general H n ( z ) = H n - , ( z ) * H d z ) . Thus I ( Z ) , the Fourier transform (a) of Q(z), can be written as +W +W I ( z ) = flQ(z)I = C F[Hrn(z)l= C Frn(Z) -W -W where The summation of the power series in eqn (5) can be written as F ( Z ) + F*(Z) I ( 2 ) = 1 + 1 - F ( 2 ) 1 - F * ( Z ) 1 + F ( 2 ) = R e ( l - F ( z ) ) for an infinite chain.F*(Z) is the complex conjugate of F ( Z ) . For a limited chain of N monomers, I ( Z ) is given by l - F ( Z ) - "1-F(Z)I2 1 + F ( Z ) 2F(Z)[l- FN(Z)] I ( 2 ) =Re (7)J. BLACKWELL, A. BISWAS, G. A. GUTIERREZ AND R. A. CHIVERS h 0.8- 0, t.3 v 77 p Y 0.4 -s 2 0.2 0 c M ._ 0 0 20 40 6-0 SO 100 120 z l A Fig. 3. Autocorrelation function Q( z) for 58/42 random copoly( HBA/HNA) plotted against z out to the 14th nearest neighbour of an infinite chain. H,, H I , H2 etc. indicate the terms for the zeroth, first, second etc. nearest neighbours [from ref. (12)]. It will be shown below that the above analysis predicts the positions of the aperiodic maxima for copoly(HBA/ HNA).This approach was derived indepen- dently, but note that similar analyses have been reported previously for inorganic layer structure^'^ and deformed a-keratin fibres.15 However, to our knowledge this has not been done before for polyatomic monomers, which is essential if we are to compare the intensities as well as the positions of the meridional maxima. Conversion into atomic monomers is achieved by separating Q(z) into its components : (9) Q(z) = Q(0) + C C QAB(z) A B where QAB( Z) describes the probability of sequences beginning with monomer A and ending in monomer B. [There will be four such QAB(z) series for copoly- (HBA/HNA).] Since Q ( O ) =C (10) A where PA is the mole fraction of monomer A, we can derive the I ( 2 ) for the atomic model by analogy with eqn (3): I ( z ) = C P A E u ( z ) + C C C QAB(zi ) h B ( Z ) ~ X P 2rizzi (11) A A B 1 where F A B ( 2 ) is the Fourier transform of the convolution of residue A with residue B: F A B ( 2 ) = C C &,,kfsB ~ X P 2 ~ i z ( z k , - zj.A) (12) j k where f is the atomic scattering factor, z is the axial atomic coordinate and the subscripts j,A and k,B denote the jth atom of residue A and the kth atom of residue78 X-RAY ANALYSIS OF LIQUID-CRYSTALLINE COPOLYESTERS B, respectively.Closed forms analogous to eqn (7) for an infinite atomic model are derivedI6 by separating H,(z) into its AB components: QAA(z) can be written as for which the Fourier transform is and so on for the other components of Q ( z ) . These expressions are then multiplied by the respective F A B ( 2 ) terms in eqn (1 1). Expressions for finite chains analogous to eqn (8) can be derived in the same way.The above analyses have been for the meridional data. Extension to the entire fibre diagram requires calculation of I( R, Z), where R is the radial polar coordinate in reciprocal space. The meridional intensity derived above now becomes I ( 0 , Z ) . We have approached this problemi7 by calculation of the cylindrically averaged intensity transforms for the random chain. This requires definition of the conforma- tion, and for this we have considered two extremes: an extended random conforma- tion, Le. a straight chain with random torsion angles, and a rigid conformation in which all the planes of the aromatic rings are parallel to each other.I(R,Z) is derived by modification of eqn (1 1) to consider a three-dimensional structure. For the random conformation where AAB(R, z, =C C J ; , ~ ~ B J O ( 2 7 T ~ r A j ) J O ( 2 7 T R ~ B , k ) cos 27Tz(zk,B-zj,A) (17) j k and BAB(R, 2) is the equivalent sine term. For the rigid conformation, I ( R , 2) is calculated via eqn (16) except that AAB(& z, C CJ;,kfiSBJn(27TR~,A)Jn(27T~rk,B) n j k xcos r n ( 4 j , A - 4k,B)+2n2(2j,A- 2k,B)1 (18) and BAB(R, 2) in the equivalent sine term. r, 4 and z are the atomic polar coordinates. Application of the above analyses in investigations of the structure of copoly( HBA/HNA) are described below. RESULTS AND DISCUSSION Fig. 4 and 5 compare the observed and calculated meridional intensity for 30/70 and 75/25 copoly(HBA/HNA).* The positions of the observed and calcu- lated maxima are given in table 1.The observed data, shown as the dashed lines,J. BLACKWELL, A. BISWAS, G. A. GUTIERREZ AND R. A. CHIVERS 79 , 10 20 30 40 : 2 e p 0 Fig. 4. Observed and calculated meridional intensity data for 30/70 copoly(HBA/HNA). Dashed line, observed 8/28 diffractometer scan; upper solid line, calculated data for atomic model; lower solid line, calculated data for point model [from ref. (S)]. 3 Fig. 5. Observed and calculated meridional intensities for 75/25 copoly( HBA/ HNA). Dashed line, observed 8/28 diffractometer scan; upper solid line, calculated data for atomic model: lower solid line, calculated data for point model [from ref. (S)].80 X-RAY ANALYSIS OF LIQUID-CRYSTALLINE COPOLYESTERS were recorded by Dr J.B. Stamatoff of Celanese Research Co., as 8/28 diffractometer scans.' The solid lines are calculated data for point and atomic models using eqn (3) and (1 1). The models were for chains of limited length, and in order to minimize the subsidiary maxima the data were averaged over a normal distribution of chain lengths centred on 11 monomers (with (T I= 1 monomer). The calculated data have also been corrected for the Lorentz and polarization effects. It can be seen that the point model predicts the positions of the observed maxima to within CQ. 0.1 A; this agreement could be improved by refinement of the residue lengths. At high proportions of HNA, four aperiodic maxima are predicted. As the HBA content increases, the first maximum moves outwards (to lower d spacings), the third moves inwards and the second gets progressively weaker until it disappears at about the 50/50 residue ratio.Meanwhile the strong maximum at d = 2. 1 A is unchanged in position across the entire range of composition. The latter effect arises because the 2.1 A maximum corresponds to the third order of the HBA length (6.35 A) and the fourth order of the HNA length (8.37 A). The point model can be seen as a lattice with a repeat of ca. 2.1 A, but with low occupancy, and hence the 2.1 A maximum is a Bragg peak occurring at all compositions. However, the occupancy is not random but is as defined by Q ( z ) ] , with the result that aperiodic The point model predicts the positions of the maxima but cannot be expected to give good agreement for the intensities because intraresidue interferences have been ignored.Conversion into an atomic model leads to prediction of maxima in approximately the same positions: the F A B ( 2 ) terms in eqn (1 1) vary relatively slowly with 2 and are sampled by interference functions analogous with that for the point model. However, the intensity agreement is now reasonably good. In fig. 4 and 5 the peak heights at d = 2.1 A have been set equal, and it can be seen that there is good agreement for the other peaks except that the first peak is predicted to be too weak. This is a general feature across the entire composition range and points to several defects in the model. In particular, the model for the copolymer is defined as a straight chain in which all residues have their ester oxygen-ester oxygen vectors parallel to the z axis. This can only be an approximation, as is apparent from fig.2(a): rather there must be a distribution of these vectors about the z axis. The F A B ( 2 ) terms' have minima in the region d = 6-8 A, which are smoothed when a distribution of residue orientations is considered, with the effect that the intensity of the first maxima for the atomic model is increased. In separate work'* on copolymers of HBA, 2,6-dihydroxynaphthalene (DHN) and terephthalic acid (TPA) we have refined the residue orientations to obtain a match between the observed and calculated maxima. A second factor that can be expected to change the meridional intensities is interference caused by preferred axial stagger of the chains.Our calculations so far have been for nematic structures with random axial stagger, but the presence of off-equatorial Bragg maxima points to the existence of some three-dimensional order, which will be expected to affect the meridional intensities. Calculations incorporating preferred axial stagger for the chains are now in progress and will be described in a future paper. Thus it can be seen that we can reproduce the meridional scattering for copoly( HBA/ HNA) using a completely random monomer sequence. This approach has also been applied to copoly(HBA/DHN/TPA) "J' and to several other copolyesters with equal success. This leads to the question of the sensitivity of this type of X-ray data to non-random sequence distribution.This can be modelled by variation of Q ( z ) to take account of different neighbour probability in blocky structures. We have studied this in detail for copoly( HBA/DHN/TPA) and have maxima occur at d > 2.1 d .J. BLACKWELL, A. BISWAS, G. A. GUTIERREZ AND R. 6 8 10 15 A I I 10 20 30 LO E 2ei0 4. CHIVERS 81 2e/c Fig. 6. ( a ) Calculated meridional intensities for 58/42 copoly( HBA/HNA) for atomic models of different chain lengths (marked on the curves). ( b ) Observed meridional intensity data for 58/42 copoly(HBA/HNA). been able to show that all but minimal blockiness can be ruled out for these copolymers in favour of the completely random m0de1.I~ Thus, in this relatively special case of a stiff copolymer of monomer units of different lengths, we can use X-ray methods to investigate sequence distribution, i.e. microstructure. The calculated data shown in fig. 4 and 5 were obtained using models of chains with an average degree of polymerization (d.p.) of 11. The actual d.p. for these polymers is ca. 150, based on a reported molecular weight of 25 000.6 Nevertheless, the main features of the meridional data can be predicted for chains of 6 or more monomers. Fig. 6 shows the meridional intensity for 58/42 copoly( HBA/HNA) calculated for chains of 6, 8, 10 and 15 monomers.8 The data have been normalized to have the same peak height at d == 2.1 A, for ease of comparison. This copolymer ratio gives three maxima at d = 7.2, 3.0 and 2.1 A; the other peaks are subsidiary maxima caused by the monodisperse short chain length.It can be seen that the82 X-RAY ANALYSIS OF LIQUID-CRYSTALLINE COPOLYESTERS first two peaks are approximately independent of chain length. However, the peak at d == 2.1 A gets progressively sharper as the chain length incremes, as expected for a Bragg peak. Fig. 6( 6) shows the observed data for this monomer ratio: following correction for instrumental broadening the peak width at d = 2.1 A is reproduced by a chain of 11 monomers. Given the higher d.p. for the actual molecules, this ‘chain length’ must correspond to a persistence or correlation length for the stiff-chain conformation: it is the distance beyond which our approximation of a straight chain with residues parallel to the chain axis is no longer adequate. Comparison of the linewidth data for other HBA/HNA ratios from 30/70 to 75/25 shows that the correlation length increases from 9 to 13 residues as the HBA content increases.* Non-linearity of the chain is due in large part to the offsetting 2,6-saphthylene linkages, and hence it i s to be expected that the chains will tend to be straighter as the HBA content increases.The results in fig. 6 were obtained for finite chains using eqn (1 1). In more recent work we have approached this via the infinite chain with H , ( z ) defined as the sum of distribution functions for the possible HBA and HNA lengths. The above calculations yield a model for the chain with which it is possible to proceed to consider the three-dimensional structure. Our first step in this direction has been to calculate the cylindrically averaged transforms for (extended) random and rigid chain^.'^ Fig.7 shows these data for 58/42 copoly(HBA/HNA), together with a schematic representation of the observed data. The calculated data are for a single (average) random chain, i.e. no allowance is made for interchain interference effects. Note also that fibre disorientation causes arcing of the observed data. The two models each have the same axial projection and thus predict the meridional data I( 0 , Z ) as described above. There is high intensity along the equator that declines steadily with R. No maxima are seen corresponding to the equatorials at d = 4.6, 2.6 and 2.3 A, but these can be expected to arise when the chains are packed on a hexagonal lattice with a = 5.2 A.There is intensity in the region of the off-equatorial at d = 3.3 A. This is weaker than the lowest intensity contour shown in fig. 7 (a) and ( 6 ) but is probably sufficient to account for the observed sampling. The ‘layer line’ at 2 = 1/2.1 k’ is more extensive (in the radial direction) than those for the other meridionals, as is observed. The major differences between the data for the random and rigid models are in the extent of the 2 = 1/7.5 A-’ layer line and the intensity of the diffuse non- meridional layer line at 2 = 1/4.5 A-’. In this respect the random model gives significantly better agreement with the observed data and is considered to be preferable to the rigid conformation in which the aromatic rings are parallel to one another. Similar conclusions were obtained in studies of the 30/70 and 75/25 copolymers.Random torsion angles will lead to chains with a cylindrical cross- section, which will tend to pack hexagonalty. Nevertheless, the existence of three- dimensional order points to axial register for some of the chains, which may require a regular conformation in certain parts of the structure. The calculations so far represent only a starting point for these investigations of the three-dimensional structure and further work is in progress in this area. This work is being supported by NSF Grant no. DMR8 1-07130 from the Polymer Program. ’ J-I. Jin, S. Antoun, C . Ober and R. W. Lenz, Br. Polym. 1, 1980, 12, 132. G. W. Calundann and M. Jaffe, in Roc. Robert A. Welch Found. Symp. XXVI, Synthetic Polymers, 1982, p. 247.+., 0 . 5 p tan (28) 1 .o h 8 0.5 cd c, 0:5 tan (28) 1 :o tan (28) Fig. 7. ( a ) I ( & 2) intensity data for 58/42 copoly(HBA/HNA) in the random conformation, ( b ) cylindrically averaged I ( R, 2) intensity data for 58/42 in the rigid conformation (aromatic groups parallel to each other) and (c) schematic representation of one quadrant of observed data [from ref. (17)].84 X-RAY ANALYSIS OF LIQUID-CRYSTALLINE COPOLYESTERS A. M. Donald, C. Viney and A. H. Windle, Polymer, 1983, 24, 155. M. Y. Cao and B. Wunderlich, Macromolecules, in press. G. A. Gutierrez, R. A. Chivers, J. Blackwell, J. B. Stamatoff and H. Yoon, Polymer, 1983, 24,937. G. W. Calundann, U.S. Patent, 4 161 470,1979. D. J. Blundell, Polymer, 1982, 23, 359. R. A. Chivers, J. Blackwell and G. A. Gutierrez, Polymer, 1984, 25, 435. J. P. Hummell and P. J. Flory, Macromolecules, 1980, 13, 479. J. Blackwell and G. A. Gutierrez, Polymer, 1982, 23, 671. J. Blackwell, G. A. Gutierrez and R. A. Chivers, Macromolecules, 1984, 17, 1219. 1984,23, 1343. l 2 J. Blackwell, G. A. Gutierrez, R. A. Chivers and W. Ruland, J. Polym. Sci., Polym. Phys. Ed., l 3 R. Hosemann, 2. Phys., 1950, 128, 465. l4 S. Hendricks and E. Teller, J. Chem. Phys., 1942, 10, 147. l 5 R. Bonart, Prog. Colloid Polym. Sci., 1975, 58, 36. l 6 J. Blackwell, A. Biswas and R. Bonart, Macromolecules, submitted for publication. l7 R. A. Chivers and J. Blackwell, Polymer, in press. l 8 G. A. Gutierrez, R. A. Chivers and J. Blackwell, Polymer, in press. l9 G. A. Gutierrez and J. Blackwell, Macromolecules, in press.