Introductory Lecture : Levels of Order in Amorphous Polymers BY PAUL J. FLORY Department of Chemistry, Stanford University, Stanford, California 94305, U.S.A. Received 21st September, 1979 Early theoretical predictions that amorphous and liquid polymers shouId be devoid of significant order down to dimensions commensurate with the diameter of the chain are supported by an extensive body of evidence from X-ray scattering at small and at wide angles, neutron scattering at small and at intermediate angles, recent electron microscopic investigations, analysis of rubber elasticity including, especiaIly, its dependence on temperature and dilution, and cyclization equilibrium constants. If the nodular domains previously observed in electron micrographs are dismissed as artifacts, as studies documented in this Discussion indicate, then depolarized Rayleigh scattering (DRS) offers the only substantial evidence for a detectable degree of order in liquids consisting of long-chain molecules.This evidence is examined in detail. The analysis presented leads to the conclusion that the enhancement of the DRS of higher n-alkanes by factors of 2-2.5 over the values that would be observed for uncorrelated molecules is attributable to: (i) steric constraints on relative orientations of neighbouring chains which restrict them to orientations generated by rotation about one axis only, this axis being normal to their plane of contact, and (ii) a small preference in the align- ment of neighbouring chains. These correlations do not imply order in the sense in which this term customarily is used.Rather, they complement the distance correlations manifested in the radial distribution functions deduced from wide-angle X-ray scattering; like them, they are of very short range. The arrangment of long-chain molecules in condensed phases is a subject that has evoked intense controversy. Two divergent viewpoints have been propounded with regard to the liquid or amorphous state, including glasses. The magnitude of the efforts that have been committed to their resolution is extraordinary. The issues at stake are of strategic importance ; molecular interpretation of the properties of poly- mers obviously requires an understanding of morphology at the molecular level. the polymeric chains in so-called amorphous polymers are organized in small bundles, nodules, meander arrays, or paracrystals.The sizes of these ordered domains2-5 generally are believed to be in the range 50-100 A. In a strict sense, therefore, the system is not amorphous. The alternative view holds that amorphous polymers are virtually devoid of all vestiges of order, even at a level approaching the diameter of the ~hain.~-ll The state of disorder is considered to resemble that in a low molecular-weight liquid; hence, the term " amorphous " is altogether appropriate. In the nascent era of polymer science, when the primordial scene was uncluttered by conflicting experiments and opposing theories, the prevalence of randomness in amorphous polymers seemed eminently reasonable. Theories of polymer properties and behaviour founded on this premise met with gratifying success, and this seemed to provide sufficient confirmation of the hypothesis concerning molecular disorder.In the late 1950s and the 1960s, however, this hypothesis came to be eclipsed by various conceptions of ordered domains considered to pervade the " amorphous ", or non- crystalline, polymer. Electron micrographs offered the primary evidence for " no- dules '' in the size range menti~ned.~'~ They were forthcoming in profusion from many laboratories, and for virtually all polymers of note. These micrographs are According to one body ofP. J . FLORY 15 now securely interleaved in journal volumes occupying the shelves of libraries around the world. As if these inexorable revelations visible to the unerring eye were not enough, results of X-ray scattering, both small-angle (SAXS) and wide-angle (WAXS), and electron diffraction as well, were presented in support of bundle model^.^-^ Pos- sibly no less persuasive than these experimental observations were conceptual diffi- culties of envisaging random packing of linearly connected units at the densities typi- cal of the condensed state without organizing them in some fashion.It was argued that a random arrangement is inc~nceivable.~ IMPLICATIONS OF THEORY Contrary to the contention recited above, the hypothesis of random disorder has been consistently affirmed, implicitly if not explicitly, by statistical thermodynamic theories of polymers and their mixtures. These theories invariably yield partition functions that are separable into intra- and inter-molecular parts, each independent of the other.12-14 The derivation of the partition function for such a system rests on the formulation of the expectancy of x vacant sites contiguously situated in the man- ner required to accommodate the x-meric chain in a specified configuration.Although this expectancy becomes very small at high densities, it is sensibly the same for the overwhelming majority of the configurations that the chain may assume. It follows at once that the configurations of the chains should be unaltered by the strictures attending their coexistence within the volume of the condensed phase. Bundles or other non-random arrangements would entail substantial alterations of the chain configurations, in direct violation of this principle.Partial ordering of any kind must increase the free energy for a system of flexible c h a i n ~ . l ~ v ~ ~ Only for chains of limited flexibility is an ordered arrangement preferred.12 The demarcation between the two categories of behaviour is discrete; chains having a flexibility greater than a critical limit should gravitate to a state of total disorder (in absence of forces promoting crystallization). As these studies make clear, the arrangement of chain segments within small '' domains ", or regions of space, and the configurational characteristics of the poly- mer chains, expressed, for example, in their mean dimensions, are inseparably related. Order in the former would connote perturbation of the latter, and vice versa. EXPERIMENTAL EVIDENCE FOR RANDOM DISORDER Experiments have provided abundant evidence supporting the prevalence of randomness in amorphous polymers.This evidence has been reviewed elsewhere. Investigations on rubber elasticity, especially concerning effects of dilution and tem- perature on the stress, are particularly compelling. Equally so are the studies of Semlyen and co-workers I5 on cyclization equilibria which depend directly on the configurational characteristics of the chains. Careful measurements of activities in polymer-diluent systems fail to reveal any indication of dissipation of the ~ r d e r , ' ~ , ~ ~ implicit in bundle models, upon addition of the diluent to the amorphous polymer. The most striking evidence bearing on the issue has been provided by small-angle neutron scattering (SANS).16-22 The radius of gyration of the deuterated polymer dispersed in the unlabelled host invariably turns out to be in close agreement with the value for the isolated, unperturbed chain, as determined in a 0-solvent. Polymers investigated include poly(methy1 methacrylate) (PMMA),16 p o l y ~ t y r e n e , ~ ~ ' ~ ~ poly- ethylene (PE),19920 polypropylene 21 and poly(oxyethylene).** Local ordering of units would be difficult to reconcile with these findings. Such ordering must bias the local16 ORDER IN AMORPHOUS POLYMERS conformation, and this in turn should be reflected in the dimensions of the chain as a whole, i.e., in its radius of gyration. To be sure, the effect of rearranging the con- formation within a sequence of units, in order to render it compatible with other chains in an ordered domain or bundle, could be compensated by stretching or compressing the portions of chain reaching from one bundle to the next, but that such compensa- tions should be executed faithfully by inanimate chain molecules in five polymer series without exception, and for all molecular weights within a given series, is scarcely con- ceivable.Even this incredibly remote possibility is ruled out by neutron scattering data at intermediate angles that probe distances between pairs of atoms (e.g., deuterium) separated by comparatively short sequences of units.I8 Of particular significance are the results of Kirste et aZ.16' on syndiotactic poly(methy1 methacrylate) (PMMA). The scattering functions for the protonated polymer dispersed in the deuterated host polymer matrix are in good agreement with SANS for dilute solutions in acetone as well as with earlier SAXS results23 on solutions of PMMA in the same solvent.They are concordant also with scattering functions computed on the basis of realistic treat- ment of the structures and conformational energies of these molecules using con- figuration statistical Any perturbation of the conformation affecting intramolecular distances down to 15-20 8, would be manifested in the observed scat- tering functions, causing departures from the calculations and from experimental results for the polymer in dilute solutions. The neutron scattering results of Dettenmaier 2s on n-CJ6H,, dispersed in the (liquid) deuterated alkane are in excellent agreement with calculations based on the rotational isomeric state treatment of unperturbed alkane chains.26 This agreement obtains at small angles where the data yield the radius of gyration25 and also at higher angles where the relevant correlation lengths are on the order of 10 If the alkane chains aggregated to yield bundles to an appreciable degree, their conformations al- most certainly would have been altered, and this should have been apparent in the scattering function.The body of evidence in support of prevalence of randomness in molten and amorphous states (including glasses) is formidable. The earlier evidence to the con- trary, stemming principally from electron microscopy, SAXS and WAXS, has there- fore been called into question.*-ll The papers contributed to this session of the Discussion extend previous investigations by these methods.The SAXS results of Uhlmann" decisively refute earlier claims that features ob- served by this technique (and by electron diffraction as well) confirm the presence of nodules or bundles in amorphous polymers. He and his collaborators showed previously that the manifestations of these features in electron micrographs recede into the unresolved background when observed by refined techniques. They are not revealed even at the highest resolution attainable. A different explanation of the earlier results showing nodular features has been offered recently by Thomas and R o ~ h e . ~ ~ They concur with Uhlmannl' in the conclusion that the nodules identified in countless electron micrographs are artifacts.Love11 et aZ.lo present an incisive analysis of WAXS results on molten polyethylene (PE) which is irreconcilable with a parallel arrangement of the chains. The molecular morphology in PMMA is similarly shown to be random. Parallelity of short range is found, however, for the more rigid polymer, poly(tetrafluoroethylene), in its molten state. In the same vein, Longman et aZ.28 have applied WAXS to the determination of the radial distribution functions in liquid n-alkanes of various lengths and in molten PE. They observe short-range order only; correlations do not extend beyond E 20 8, in any of these liquids, according to their analysis of the WAXS measurements.P. J . FLORY 17 Results for PE resemble those for the higher alkanes (n 3 16), apart from an intimation of somewhat greater disorder in PE.This may, however, be due to the higher temperature required to melt PE. A recent computer simulation investigation of systems of n-alkane chains carried out by Corradini and co-workers 29 is especially illuminating in this connection. Choosing n-C30H62 as a prototype of PE and using Monte Carlo techniques, they induced worm-like rearrangements of 31 such chains confined to a cubic space cor- responding to the volume of the liquid alkane. Periodic boundary conditions were imposed in order to eliminate extraneous surface effects. Actual structural para- meters, including bond angles, were employed. Hindrances affecting bond rotations and intermolecular forces were realistically approximated.After a steady state had been reached, the mean-squared radius of gyration of the chains corresponded closely to that for free, unperturbed chains. The radial density distribution function (r.d.f.) thus computed exhibits three maxima in good agreement with the WAXS results of Longman et al? at 400 K. The orientation correlation function (3 (cos2~) - 1)/2, where t , ~ is the angle between an intermolecular pair of C-C bonds, is negligible for distances >5 A according to the computations carried out by Corradini and his colla- b o r a t o r ~ . ~ ~ This investigation provides a most convincing demonstration of the propensity for randomness in a system of flexible long chains. SHORT- RANGE ORDER The investigations using wide-angle scattering (WAXS) cited aboves-11*28 and the computer simulation study of Corradini and coworkers 29 direct attention to the short- range order necessarily present in any condensed fluid, amorphous polymers included. In brief, the intermolecular contributions to the r.d.f.for polymers appear to resemble those for low molecular liquids, if the difference in packing densities is taken into ac- count. The short range radial correlations manifested in the r.d.f.s for liquids enter- tain the possibility that orientational correlations of short range may conceivably occur as well. The question thus raised is this: are orientational correlations, con- fined perhaps to first neighbours, necessarily ruled out altogether by the results sum- marized above? Should one not expect some degree of correlation between the directions of neighbouring chains, just as the volume exclusion between segments leads to periodicity of short range in the r.d.f.? The depolarized Rayleigh scattering (DRS) by n-alkane liquids is indicative of orientational correlation^,^^-^^ and, presumably, in other liquids consisting of long- chain molecules.If the molecules are independently oriented at random, then the DRS intensity for the neat liquid should equal the sum of contributions of the indivi- dual molecules as evaluated from measurements on dilute solutions in an isotropic solvent such as CC1,. The intensities IHv of the DRS for the higher n-alkanes appre- ciably exceed I& calculated from the molar DRS at high dilution (cf. seq). Bothorel and co-workers 34*35 have introduced the quantity J = IHv/I&v - I (1) as a measure of “ correlated molecular orientation,” CMO.Patterson and his have attempted to relate the CMO thus evaluated to the thermodynamics of mixing on the premise that the ordering presumed to be present in the neat liquid n- alkane should be dissipated by dilution. The CMO indicated by DRS appears to be at variance with the compelling array of evidence recited above. It stands as the principal pocket of evidence precluding un- qualified acceptance of the assertion that liquids comprising long-chain molecules are18 ORDER IN AMORPHOUS POLYMERS 1.5 W 2 rr c ,h g1-0- A" N U totally disordered (apart from the short-range order implicit in their r.d.f.s). As such, it deserves critical examination. The remainder of this paper is devoted to an inquiry into the implications of results of DRS measurements on the higher n-alkanes.I 1 I - - 4 . 0 * + - " A0 3.0 -* 4 Y \ a n w h Y - 2.0 OPTICAL ANISOTROPY AND INTERMOLECULAR CORRELATIONS Fig. 1 shows unpublished results of C a r l ~ o n ~ ~ on the DRS of two n-alkanes plotted against temperature. The ratio scaled on the right-hand ordinate corresponds to J + 1 in the terminology of Bothorel and his c011aborators.~~-~~ The mean-squared optical anisotropy ( y 2 ) is defined formally as the configurational average of y2 = (3/2)Tr(&&) (2) I I 1 I, 0 50 100 150 T/*C FIG. 1 .-Mean-squared " apparent " optical anisotropies < y2 >app for two n-alkanes (0, n-C16H34; 0, obtained from depolarized scattering measurements on the neat and plotted against the temperature of measurement. The right-hand ordinate scale expresses the ratio of ( y2 )app [to the value <y2 >o obtained from measurements on dilute solutions in CC14.32*33 Plotted on the left is the ratio of <y)app to the number n of C-C bonds.ti being the traceless part of the molecular polarizability tensor. Stated otherwise, (2/3)y2 is the sum of the squares of the elements of B. The quantity (y2)o appearing in the ratio plotted in fig. 1 refers to the molecule at infinite dilution, and (y2)app, the apparent optical anisotropy, to the neat liquid. The latter quantity is proportional to the depolarized intensity I& divided by the number density. Measured intensities were corrected for collison-induced depolarized scattering32 by employing narrow band-pass filters as described by Carlson and Florym3' The increase in ( y 2 ) as the melting point is a p p r ~ a c h e d ~ ~ .~ ~ is attributable to heterophase fluctuation^.^^ This explanation is confirmed by analysis of the depend- ence of viscosity on ternperat~re.~~ The aberration caused by a small fraction of the unstable (crystalline) phase can be avoided by reliance on results at higher tempera- tures. Here, the ratio ( i e . , J + 1) approaches 2.0-2.5. Similar values have been obtained by Quinones and B0therel~O9~~ and by Fischer et aL31 The ratio (y2)app/(y2)0 = J + 1 has been offered as a measure of " orientational order," or CMO, on the premise that it reflects a preference for parallel arrangements of neighbouring chain^.^^.^^ This preference should be most marked for sequences of bonds in the trans conformation.The optical polarizability tensors for such se- quences have been assumed implicitly to be cylindrically symmetric35 about the longi- tudinal axis of the trans planar zig-zag. In fact, the polarizability tensor for a trans alkane sequence is decidedly acylindrical. This may be shown as follows.P. J . FLORY 19 We assume the polarizability tensor to be formulable as the sum of contributions from its constituent bonds or, preferably, of groups of Then, if bond angles are taken to be tetrahedral, the tensor Br for an n-alkane, regardless of its conformation, depends only on the quantity40 where the Act’s are differences between parallel and transverse polarizabilities of the indicated bonds.Each methylene group, comprising one C-C and two C-H bonds, contributes a tensor expressed in a reference frame with the C-C bond as x-axis. Transformed to the XYZ reference frame defined in fig. 2, this becomes r = Aacc - 2Aa,, (3) I‘ diag(2/3, - 1/3, - 1/3) Y I 1 1 I Y FIG. 2.--Carbon skeleton of a trans sequence, and the principal axes for an even number of bonds. with signs of the off-diagonal elements alternatingly positive and negative for succes- sive bonds. Hence, for a sequence of n, bonds in planar trans array where is alternately & fi and 0. For sequences with n, even, eqn (5) simplifies to For long sequences of odd n, this expression is an acceptable approximation. The sum of squares of its elements, (and hence y2) is affected equally by rotations about 2 and X .Thus, the CMO parameter J is not unambiguously diagnostic of axial order. a12 = -j= [I - (-1)~t1/2/2 ant = (n,/3)r diag(1, 0, - 1). (6) (7) Obviously, ant is not cylindrical. The square of the optical anisotropy for a sequence of n, trans bonds is Y.”, = n,2(r2/3)(1 + 42/n,2) (8) yf, = n:(r2/3) (9) as follows from eqn (2) and (5). For even values of n, [see eqn (7)], which can be employed also as an approximation for odd values of n, $ 1.20 ORDER IN AMORPHOUS POLYMERS A long polymethylene chain may be regarded as a succession of sequences of trans bonds, each sequence being separated from the next by one or more bonds in gauche conformations. For a tetrahedrally bonded chain, two such sequences joined by a gauche bond (120" torsion) contribute additively to the mean-squared optical aniso- tropy ( y 2 ) for the chain as a whole, as is shown in the Appendix.Their resultant consequently is the same as would be obtained if the directions of the two sequences were uncorrelated. In fact, ( y 2 ) for the molecule can be approximated as the sum of contributions of these sequences (see Appendix). It is appropriate therefore to regard the DRS for the liquid n-alkane as the sum of contributions from trans sequences. The enhancement of the apparent optical anisotropy is attributable on this basis to correlations between neighbouring sequences, which we now undertake to consider. The distribution of sequence lengths may, for the purposes at hand, be suppressed; it will suffice to assume that the distribution can be replaced by an average, the actual value of which is unimportant. Ignoring off-diagonal elements in eqn (5), we simply take (10) where the numeral subscript denotes a single sequence of average length (unspecified), and a is a constant related to y l for the sequence by 6i1 = a diag( 1, 0, - 1) y f = 3a2.(1 1) Consider a set of N sequences. sequence and its immediate neighbours. the set is These may, for example, comprise a central The average squared optical anisotropy of (Y&) = (3/2)Tr 2 2 {aiej) i j = (3/2)Tr(Z &,ai + 2 2 <&,&,)) (12) I i # j where i andjindex the N sequences and the angle brackets denote statistical mechanical averages over all orientations of sequences of the set. The trace of &,ai being in- variant to rotations, angle brackets are omitted.The first term in eqn (12) is just Nyt. It follows that the second term in eqn (12) divided by the first may be identified with Bothorel's parameter J. In the sequel we adopt a " mean field " approximation allowing Bi and Gj for members i and j of the set of sequences to be averaged separately. This approxima- tion seems warranted in view of the small degrees of correlation that will be required. On this basis the second term in eqn (12) gives JN = (3/2yf)(N - 1)Tr(6Q2 where the subscript 1 designates a single representative sequence of the set. Sym- metry of rotations about principal axes of a chain sequence assures that off-diagonal elements of (a,} vanish. Hence, Tr(6,)2 may be evaluated as the sum of the squares of the diagonal elements of (a,), in which each element is averaged over the set of N sequences.For perfect order, (a,) is given by eqn (10). Hence, J N z N - 1 (14) Correlation of rotations about the X-axis (fig. 2) of neighbouring sequences prob- ably is small (but not necessarily nil) owing to the similarity of transverse dimensionsP . J . FLORY 21 of the sequence: 254.5 and 4.15 A in the Y and 2 directions, respectively. rotation about X converts eqn (10) to Random = adiag(1, -3, -3) from which it follows that JN,X = (3/4)(N - l>* Random rotation about 2, instead of X , leads to the same result; i.e., JN,z = JN,x. Random rotation of 8, about Y renders the tensor null; hence, JN,y = 0. Rotations of the X-axes of immediate neighbour sequences having lengths appre- ciably greater than their diameters * are restricted by steric interactions.Such rota- tions can occur freely only about the axis that is normal to their mutual tangent plane at the point of contact. Rotations about the axis that is perpendicular to this normal and to the X axis are sterically obstructed. Accordingly, we consider rotations I,U about 2 following initial rotations x about X . If the rotations x are random, the tensor becomes axially symmetric and the same results are obtained by rotations (v) about Y instead of 2. Taking the first rotation (x) to be random, one readily obtains JN,xz = ( 3 / 2 ) 2 [ ( ~ ~ ~ 2 ~ ) 2 - (COS~I,U) + 1/3](N - 1). (17) If the rotation t , ~ is also random (cos2y/) = 3, giving JN = (3/16) (N - 1).Thus, for a pair of sequences in contact and subject only to the steric inhibition limiting rotations transverse to the X axis to rotation about a single axis, the DRS for the pair is en- hanced by a factor 1 + J2 = 1 + 3/16. Inasmuch as each sequence impinges on a number of neighbours, the total effect must be considerably greater. Eqn (17) refers to a set of sequences all rotated about the same transverse axis and, hence, parallel to a plane perpendicular to that axis. This unnatural constraint may be removed by performing an additional set of rotations about the axis X of the initial (fixed) reference frame. Letting these latter rotations be random, we establish the fixed X-axis as an axis of symmetry. Placement of one of the N sequences with its backbone along the X-axis completes the model of a sequence and the N - 1 neighbours with which it is correlated.This central sequence may be assigned the initial orientation. Its anisotropy tensor, denoted Gc below, is given by eqn (10). Eqn (13) is then replaced by where (C;,) is the averaged tensor for one of the N - 1 neighbour sequences. Let the initial rotations about the X-axis of the N - 1 neighbours be executed at random, possible correlations between the transverse directions, Y or 2, being ig- nored. Next, the N - 1 sequences are subjected to arbitrary rotations v about one transverse axis. Finally, random rotations about the initial (laboratory) X-axis are introduced. The result of these operations substituted in eqn (1 8) yields * Inasmuch as y2 for a trans sequence depends on the square of the number of bonds conformed in the planar array, it is appropriate to consider the r.m.s.sequence length. For an alkane chain at ordinary temperature ~ ' ( n : ) FZ 3 bonds (see Appendix). A succession of n, trans bonds disposes n, + 2 bonds in the planar array. The length of a sequence of the r.m.s. length comprising % five coplanar bonds and the pendent atoms extending beyond them is at least twice the mean breadth. Hence, the stated condition requiring the length to exceed the diameter appreciably appears to be fulfilled.22 ORDER IN AMORPHOUS POLYMERS Designation of 2 as the transverse axis is arbitrary and inconsequential. The second term in the brackets occurs as a consequence of fixing the central sequence. If this sequence is allowed to assume an arbitrary orientation, subject only to the stipulations for other sequences, the second term in eqn (19) disappears. The parameter JN for a set of N sequences oriented with respect to an axis ( X ) but cylindrically distributed about that specified axis is given therefore by the first term alone.If the sequences were directed at random over solid angle, then (cos2y/) would equal one-third and JN would vanish, as obviously it must. If they are rotated at random about one axis only, (cos2y/) = 3 and JN,xzx = (N - 1)(3/64)(1 + 6 / N ) . (20) For N = 7, i.e., for an average of six neighbours correlated with a given sequence, JN,xzx = 0.522. Thus, nearly half of the CMO indicated by the DRS measurements may be taken into account without postulating preferential axial orientation.The " correlation " arising from the steric exclusion of rotation about one of the transverse axes for a given pair of incident sequences is solely responsible for the calculated en- hancement of (f). The estimates of JN based on the primary steric constraint identified above may be enhanced by (i) correlation of directions of the transverse axes of neighbouring sequences, i.e., by constraints on rotations x; (ii) by a larger value of N ; and (iii) by correlations of the directions of longitudinal axes ( X ) . For reasons stated earlier, the contribution of (i) probably is small. A substantial increase in N would imply that correlations extend to second neighbours and beyond. Although these factors may increase JN, (iii) seems fie most likely cause for the major contribution.In fig. 3, JN calculated according to eqn (19) is plotted against (cos2y/) for N = 7. A mere increase of (cos2y/) from its random value of 0,5 to 0.60-0.65 raises JN to 1.03-1.34, which is in the range observed for the higher n-alkanes;30*33*36 see fig. 1. 2.0 c 2 3 1.5 N a a s, h hl W II 1.0 0.5 goo - (I' 1 oo zoo 3 0' 1 I I I I I I I 0.5 0.6 0.7 (cos2 Q ) FIG. 3.-Enhancement factor JN calculated according to eqn (19) as a function of < cos2v/ ) for a set of N sequences consisting of a central member and its N - 1 neighbours; v/ is the angle between one of the latter and the former; the average comprehends all configurations of the set of sequences. The upper abscissa scale expresses the restriction on the range of v/ required to reproduce the value of (cos'ty ) on the lower abscissa scale, a uniform distribution of v/ being assumed within the range O < v/ < v/*.P. J .FLORY 23 That the implied degrees of orientation are small is illustrated by the magnitude of the effective restriction on t , ~ plotted along the upper abscissa of fig. 3. A uniform distri- bution of I,Y is assumed over the range 0 < ly < v/*, with angles y > v/* forbidden. The quantity 90" - v/* plotted in fig. 3 expresses the restriction on t , ~ that is required to reproduce the value of (cos2y) on the lower abscissa. Exclusion of ly from only ~ 2 0 % of the full range 0-90" suffices to account for the experimental observations. Even this small bias on the orientations should be regarded as an upper bound, inasmuch as other contributions, e.g., (i) and (ii) above, have not been taken into account.Although definitive DRS investigations have been confined principally to n-al- kanes, effects of dilution on the stress-optical coefficients for various cross-linked polymer^,^^-^^ polyethylene i n ~ l u d e d , ~ ~ ? ~ ~ are indicative of local correlations in the undiluted state. The magnitudes of these effects are comparable with those mani- fested in the DRS of the n-alkanes. CONCLUSIONS Values of J departing from zero imply intermolecular correlations. The magni- tudes observed for J in liquid n-alkanes at temperatures well above their melting points are too small, however, to support the inference that these systems are ordered, much less that their chains are arranged in bundles.According to the analysis presented, the enhancements of the optical anisotropies of the magnitudes observed in these liquids represent at most only small degrees of mutual orientation of the axes of neighbouring chain sequences. Steric inhibitions effectively limiting relative orientations of neigh- bouring sequences to those accessible through rotation about a single transverse axis may account for a large part of the enhancement observed. This effect is akin to the volume exclusion that causes the radial distribution function to be periodic over the short range of near neighbours. The results of depolarized Rayleigh scattering experiments are not, therefore, at variance with the weight of evidence from other methods and approaches.As Uhlmannl' suggests in his contribution to this Discussion, the time has come when controversies on the morphology of amorphous polymers should " be laid to rest " and '' attention be directed to more fruitful areas." Amongst the latter are many aspects of polymer properties and behaviour that have fallen into neglect because of preoccupation of investigators with the issues of this long-lived controversy. The prevalence of randomness in amorphous polymers and the concomitancy of unperturbed configuration for their chains present circumstances peculiarly propitious for achievement of a deeper, more realistic understanding of the properties of poly- mers in condensed phases. The considerable knowledge and information potentially available on the conformations and configurations of polymer chains is directly trans- ferable to the amorphous state.It bears importantly also on the morphology of semicrystalline polymers. Thus, experiments conducted on dilute solutions in the interests of expediency and theories addressing isolated chains are at once applicable to condensed phases. APPENDIX Consider two trans sequences like the one shown in fig. 2, these sequences being joined by a bond in a gauche conformation generated by a rotation &27t/3 about the connecting bond. The anisotropy tensor 8' for the second sequence in its reference frame X'Y'Z' is to be expressed in the reference frame XYZ of the first sequence,24 ORDER IN AMORPHOUS POLYMERS The required transformation R, is the resultant of three operations: (i) a rotation of 5912 about 2' to align the X' axis with the axis of the gauche bond where 8 is the sup- plement of the tetrahedral bond angle, (ii) a rotation 2x13 or -2n/3 about the bond axis, and (iii) a rotation T9/2 about the 2 axis.Symbolically, R, = Rz(-~9/2)Rx(g2~~/3)Rz(s0/2). ( A 9 The factor s is + 1 or - 1, depending on the direction of the bond subject to rotation and, hence, on the signs applicable of 9 / 2 in steps (i) and (iii) ; g is + 1 or - 1, depend- ing on the sign of the gauche rotation. The result is Transforming 8' expressed by eqn (10) in X' Y Z ' to coordinate system XYZ, we ob- tain Since none of the non-zero elements of B and 6' combine upon forming their sum, the anisotropy yg for the pair is just the sum of their separate anisotropies, y2 and f 2 .Thus, the contributions of successive sequences of the tetrahedral chain combine as if independent of one another. The number of trans sequences consisting of one or more trans bonds is npgt in a long chain of n bonds,p,, being the fraction of bond pairs in g + t or g - t conformations. The conditional expectation q = q ( t / t ) that a trans bond is followed by another trans bond may be treated as constant, to a good approximation. It follows that the average number of sequences comprising n, trans bonds in a molecule is vnt z np,,q""-l(l - q). (A41 A gauche bond between two trans sequences can be considered to be in the planar array of either sequence. Including it with one or the other, we set y2 for a sequence of n, trans bonds equal to yf where m = n, + 1 .Then, the mean-squared anisotropy per bond is a, with yk given by eqn (8) with nt replaced by m. For a temperature of 300 K, pgr = 0.26 and q = 0.58 for n-alkane chains. Substitution of yf in eqn (A5) yields (y2)/n E 0.39 A6, which compares favourably with more accurate calculation^^^ giving 0.37 A6 for this ratio. Approximation of the optical anisotropy of a system of n-alkane chains as the sum of contributions of trans sequences seems therefore to be justified. This work was supported by the National Science Foundation, Grant No. DMR- 76-20638-A02. V. A. Kargin, A. I. Kitaigorodskii and G. L. Slonimskii, Kolloid-Z., 1957, 19, 131. Yu. K. Ovchinnikov, G. S. Markova and V. A. Kargin, Vysokomol. soedin., 1969, 11, 329. (Translated in Polymer Sci.U.S.S.R., 1969, 11, 369.) Yu. K. Ovchinnikov, E. M. Antipov, G. S. Markova and N. F. Bakeev, Macromol. Chem., 1976,177,1567. P. H. 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