摘要:

520 GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Date Subject Volume 1955 1956 1956 1957 1958 1957 1958 1959 1959 1960 1960 1961 1961 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 Physical Chemistry of Enzymes 20 Membrane Phenomena 21 Physical Chemistry of Processes at High Pressures 22 Molecular Mechanism of Rate Processes in Solids 23 Interactions in Ionic Solutions 24 Configurations and Interactions of Macromolecules and Liquid Crystals 25 Ions of the Transition Elements 26 Energy Transfer with special reference to Biological Systems 27 28 Oxidation-Reduction Reactions in Ionizing Solvents 29 The Physical Chemistry of Aerosols 30 Radiation Effects in Inorganic Solids 31 The Structure and Properties of Ionic Melts 32 Inelastic Collisions of Atoms and Simple Molecules 33 High Resolution Nuclear Magnetic Resonance 34 The Structure of Electronically-Excited Species in the Gas-Phase 35 Fundamental Processes in Radiation Chemistry 36 Chemical Reactions in the Atmosphere 37 Dislocations in Solids 38 The Kinetics of Proton Transfer Processes 39 Intermolecular Forces 40 The Role of the Adsorbed State in Heterogeneous Catalysis 41 Colloid Stability in Aqueous and Non-Aqueous Media 42 43 44 45 Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Oxidation 46 Crystal Imperfections and the Chemical Reactivity of Solids The Structure and Properties of Liquids Bonding in Metallo-Organic Compounds 47 Motions in Molecular Crystals 48 Polymer SoIuZions 49 The Vitreous State 50 Electrical Conduction in Organic Solids 51 Surface Chemistry of Oxides 52 Reactions of Small Molecules in Excited States 53 The Photoelectron Spectroscopy of Molecules 54 Molecular Beam Scattering 55 Intermediates in Electrochemical Reactions 56 Gels and Gelling Processes 57 Photo-effects in Adsorbed Species 58 Physical Adsorption in Condensed Phases 59 Electron Spectroscopy of Solids and Surfaces 60 Precipitation 61 Potential Energy Surfaces 62 Radiation Effects in Liquids and Solids 63 Ion-Ion and Ion-Solvent Interactions 64 Colloid Stability 65 Structure and Motion in Molecular Liquids 66 Kinetics of State Selected Species 67 Organization of Macromolecules in the Condensed Phase 68 For current availability of Discussion volumes, see back cover.OFFICERS AND COUNCIL OF THE FARADAY DIVISION 1979-80 President Vice-presidents who have held ofice as President Prof Sir George Porter MA SCD CCHEM FRIC FRS Dr T. M.Sugden CBE MA SCD CCHEM FRIC FRS Prof R. P. Bell MA CCHEM FRIC FRSE FRS Vice-presidents Prof A. D. Buckingham MA PHD CCHEM Prof P. Gray MA SCD CCHEM FRIC FRS Prof G. J. Hills DSC CCHEM FRIC Prof N. Sheppard MA PHD CCHEM Ordinary Members Prof W. J. Albery MA DPHIL Prof J. H. Baxendale DSC Prof F. Franks DSC CCHEM FRIC Prof D. A. King PHD SCD CCHEM FRIC Prof G. R. Luckhurst PHD FRIC FRACI FRS FRIC FRS Prof P. Meares MA SCD CCHEM FRIC FRSE Honorary Secretary Honorary Treasurer Secretary Faraday Division Members on the Primary Journals Committee Prof N.M. Atherton BSC PHD Dr M. A. D. Fluendy MA DPHIL Dr D. Husain MA PHD DSC SCD CCHEM FRIC FINSTP Editor Deputy Editor Editorial Ofice Prof J. s. Rowlinson MA DPHIL CCHEM FRIC FI CHEME FENG FRS Prof D. H. Everett MBE MAD SC C CHEM FRIC FRSE Prof F. C. Tompkins DSC CCHEM FRIC FRS Prof H. A. Skinner BA DPHIL CCHEM FRIC Prof H. Gg. Wagner Prof D. H. Whiffen MA DPHIL DSC CCHEM FRIC FRS Prof J. H. Purnell MA PHD SCD Dr J. P. Simons SCD Prof F. S. Stone DSC CCHEM FRIC Dr D. A. Young PHD DSC MINSTP CCHEM FRIC Prof G. J. Hills DSC CCHEM FRIC Prof P. Gray MA SCD CCHEM FRIC FRS Mrs Y. A. Fish BA Dr G. Saville MA DPHIL Prof H. R. Thirsk PHD DSC DIC Dr D. A. Young PHD DSC MINSTP D. A. Young PHD DSC MINSTP Ms S. J. Williams BA MSC The Faraday Division of the Chemical Society, previously The Faraday Society, founded in 1903 to promote the study of Sciences lying between Chemistry, Physics and Biology Burlington House, London WlV OBN telephone 01-734-9864 CCHEM FRIC Q The Royal Society of Chcmistry

ISSN:0301-7249    

DOI:10.1039/DC97968FX001

出版商:RSC    

年代:1979

数据来源: RSC

摘要:

520 GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Date Subject Volume 1955 1956 1956 1957 1958 1957 1958 1959 1959 1960 1960 1961 1961 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 Physical Chemistry of Enzymes 20 Membrane Phenomena 21 Physical Chemistry of Processes at High Pressures 22 Molecular Mechanism of Rate Processes in Solids 23 Interactions in Ionic Solutions 24 Configurations and Interactions of Macromolecules and Liquid Crystals 25 Ions of the Transition Elements 26 Energy Transfer with special reference to Biological Systems 27 28 Oxidation-Reduction Reactions in Ionizing Solvents 29 The Physical Chemistry of Aerosols 30 Radiation Effects in Inorganic Solids 31 The Structure and Properties of Ionic Melts 32 Inelastic Collisions of Atoms and Simple Molecules 33 High Resolution Nuclear Magnetic Resonance 34 The Structure of Electronically-Excited Species in the Gas-Phase 35 Fundamental Processes in Radiation Chemistry 36 Chemical Reactions in the Atmosphere 37 Dislocations in Solids 38 The Kinetics of Proton Transfer Processes 39 Intermolecular Forces 40 The Role of the Adsorbed State in Heterogeneous Catalysis 41 Colloid Stability in Aqueous and Non-Aqueous Media 42 43 44 45 Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Oxidation 46 Crystal Imperfections and the Chemical Reactivity of Solids The Structure and Properties of Liquids Bonding in Metallo-Organic Compounds 47 Motions in Molecular Crystals 48 Polymer SoIuZions 49 The Vitreous State 50 Electrical Conduction in Organic Solids 51 Surface Chemistry of Oxides 52 Reactions of Small Molecules in Excited States 53 The Photoelectron Spectroscopy of Molecules 54 Molecular Beam Scattering 55 Intermediates in Electrochemical Reactions 56 Gels and Gelling Processes 57 Photo-effects in Adsorbed Species 58 Physical Adsorption in Condensed Phases 59 Electron Spectroscopy of Solids and Surfaces 60 Precipitation 61 Potential Energy Surfaces 62 Radiation Effects in Liquids and Solids 63 Ion-Ion and Ion-Solvent Interactions 64 Colloid Stability 65 Structure and Motion in Molecular Liquids 66 Kinetics of State Selected Species 67 Organization of Macromolecules in the Condensed Phase 68 For current availability of Discussion volumes, see back cover.520 GENERAL DISCUSSIONS OF THE FARADAY SOCIETY Date Subject Volume 1955 1956 1956 1957 1958 1957 1958 1959 1959 1960 1960 1961 1961 1962 1962 1963 1963 1964 1964 1965 1965 1966 1966 1967 1967 1968 1968 1969 1969 1970 1970 1971 1971 1972 1972 1973 1973 1974 1974 1975 1975 1976 1977 1977 1977 1978 1978 1979 1979 Physical Chemistry of Enzymes 20 Membrane Phenomena 21 Physical Chemistry of Processes at High Pressures 22 Molecular Mechanism of Rate Processes in Solids 23 Interactions in Ionic Solutions 24 Configurations and Interactions of Macromolecules and Liquid Crystals 25 Ions of the Transition Elements 26 Energy Transfer with special reference to Biological Systems 27 28 Oxidation-Reduction Reactions in Ionizing Solvents 29 The Physical Chemistry of Aerosols 30 Radiation Effects in Inorganic Solids 31 The Structure and Properties of Ionic Melts 32 Inelastic Collisions of Atoms and Simple Molecules 33 High Resolution Nuclear Magnetic Resonance 34 The Structure of Electronically-Excited Species in the Gas-Phase 35 Fundamental Processes in Radiation Chemistry 36 Chemical Reactions in the Atmosphere 37 Dislocations in Solids 38 The Kinetics of Proton Transfer Processes 39 Intermolecular Forces 40 The Role of the Adsorbed State in Heterogeneous Catalysis 41 Colloid Stability in Aqueous and Non-Aqueous Media 42 43 44 45 Molecular Dynamics of the Chemical Reactions of Gases Electrode Reactions of Organic Compounds Homogeneous Catalysis with Special Reference to Hydrogenation and Oxidation 46 Crystal Imperfections and the Chemical Reactivity of Solids The Structure and Properties of Liquids Bonding in Metallo-Organic Compounds 47 Motions in Molecular Crystals 48 Polymer SoIuZions 49 The Vitreous State 50 Electrical Conduction in Organic Solids 51 Surface Chemistry of Oxides 52 Reactions of Small Molecules in Excited States 53 The Photoelectron Spectroscopy of Molecules 54 Molecular Beam Scattering 55 Intermediates in Electrochemical Reactions 56 Gels and Gelling Processes 57 Photo-effects in Adsorbed Species 58 Physical Adsorption in Condensed Phases 59 Electron Spectroscopy of Solids and Surfaces 60 Precipitation 61 Potential Energy Surfaces 62 Radiation Effects in Liquids and Solids 63 Ion-Ion and Ion-Solvent Interactions 64 Colloid Stability 65 Structure and Motion in Molecular Liquids 66 Kinetics of State Selected Species 67 Organization of Macromolecules in the Condensed Phase 68 For current availability of Discussion volumes, see back cover.

ISSN:0301-7249    

DOI:10.1039/DC97968BX003

出版商:RSC    

年代:1979

数据来源: RSC

摘要:

General Introduction BY F. C . FRANK €€. H. Wills Physics Laboratory, University of Bristol, Royal Fort, Tyndall Avenue, Bristol BS8 1TL Received 25th September, 1979 In this general introduction to the Discussion I look on it as my task to highlight those points where it seems to me that we can reach decision, or at least clarify the lilies of enquiry which can lead us to decision. In doing this, I see it as no part of my task to adopt a neutral stance as between right and wrong. Basically, this is a discussion on certain aspects of the fascinating general topic of restricted randomness, particularly in application to long-chain polymers. In the first section, up to Uhlmann’s paper, we are concerned with polymer melts in equilibrium. In dilute solution, the dominant restriction on randomness, apart from the very fact that the polymer molecule is a chain, is self-exclusion, and Paul Flory taught us how to cope with that many years ago.In the interior (and I em- phasize interior) of the pure amorphous phase. tnutual exclusion has a large effect on the total entropy, but its effect on molecular conformations is the relatively minor one of virtually cancelling the effects of self-exclusion. That knowledge we also owe to Flory. Hence a limited number of parameters suffice both to describe and explain the conformations in this case. Uhlmann’s paper dismisses for us the aberrant nodu- lar structures which have been proposed: there only remains to ask how in certain circumstances the appearance of nodular structure can be produced. What I have said about the pure amorphous phase is not, logically speaking, con- tradicted by the paper by Pechhold and Grossman, because according to their theory the polymer melt is not an amorphous phase but rather a remarkable kind of cubic mesophase.To justify that they have to use a modification of the rules of statistical mechanics, the cluster-entropy-hypothesis, and how they can justify that I do not know. No doubt we shall come to that in the discussion. For myself I will take one valuable point from that paper, a warning to show what bizarrely different models can be deemed consistent with the same diffraction evidence. Many more subtleties arise when we have to deal with phase boundaries, and par- ticularly the boundary between crystalline and disordered polymer, whether the latter is the phase from which a crystal grows, or the disordered layer at the surface of crystal lamellae in the final state.This, in one way or another, is what we are concerned with in much the largest section of our Discussion, from Point’s paper almost to the end. 1 would like to emphasize that the most important restriction on randomness in these situations, whether we are considering statistical equilibrium or kinetic restrictions on the attainment of equilibrium, is not self-exclusion but mutual exclusion. In this connection we have become accustomed to the famous switchboard analogy. I always have a great deal of sympathy for that poor girl, the telephone operator trying to get all the sockets plugged up on Flory’s switchboard, when the plugs and cables have the same thickness as the centre-spacing between plug-holes which are in close- packed array, and the cables are required to run in a way which is in some sense ran- dom, and she is told she may use only a very few double plugs connecting adjacent holes.Of course she’ll never do it, poor girl, and I’m surprised that after twenty years or so of trying she hasn’t gone on strike.8 GENERAL INTRODUCTION I said that was an impossible model at the Cooperstown Conference on Crystal Growth in 1958.' Explicitly, we were talking about the nucleus for polymer crystalli- zation, but the point at issue was the same. I called it an impossible model, and I quote, " because if these fringing chains are not in crystalline packing they need more cross-sectional area per chain than they do in the crystal ".Flory made the curious response that all models which had been presented could be reasoned to be impossible. I think that is an inadequate excuse for going on presenting impossible models. Fig. 3 of the paper by Flory and Yoon in this Discussion is a classic example of the impos- sible pictures which have continued to appear in the polymer literature, presented by a wide variety of authors, from then to now. Guttman et al. show, in the next paper, that the computer-generated model of Yoon and Flory, which fits the neutron-scatter- ing evidence, is likewise sterically impossible. Let me attempt a quantitative estimate of the overcrowding factor which makes these models impossible. In doing that I shall be covering much the same ground as DiMarzio, but I prefer to do it in my own words; after all, I did get my foot into this doorway first, only I failed to perceive the necessity of belabouring what seemed an obvious point.I start with the assumption that, at an interface normal to the chain directions in the crystal, all chains enter the disordered region, as represented in fig. 3 of Yoon's paper. Then I see the overcrowding as the product of three factors, D, A and B, (fig. 1). D > 1 A > 1 8 >1 FIG. 1.-Density factors at a crystal-amorphous boundary with through-going chains. D > 1 : directional randomization; for anisotropy, D = 2 . A > 1 : avoidance; say, A FZ 1.5. B > 1 : backtracking; B % N * (what is N?). Product BAD x 3. D > 1 : the factor from randomization of direction. If the chains all parallel in the crystal are still straight in the amorphous region, but have an isotropic distribution of directions, then D = 2, a result first obtained in the polymer context by Flory, I be- lieve.Note in passing that if the emerging chains randomized their directions over the hemisphere we should have D = co, but the distribution would not be isotropic: to produce isotropy the probability of changing direction by 0 from the normal must be weighted by a factor cos 8 ; that's how the sun comes to look about equally bright across the whole disc. A > 1 : the avoidance.factor. Straight chains in random directions will intercept each other: to avoid this they must make deviations increasing the required length by a factor A which I will estimate as z 1.5.B > 1 : the back-tracking factor. If the path is anything like a random walk it makes decreases as well as increases in all coordinates. This increases the chain density by a factor B, which for truly random walks would be of order N3 where N is the number of persistence lengths in a loop or tie-chain. We could estimate N* as the thickness of the disordered layer measured in persistence lengths, but I will just leave it as B > 1.F. C . FRANK 9 Hence on the initial assumption, the density of the disordered layer exceeds that of the crystal by the product BAD, in which all three factors are certainly > 1, and the product is > 3 if it is remotely reasonable to apply the word " amorphous " in descrip- tion of that disordered layer.I think there will be universal consent that the density of the disordered region, so far from being more than 3 times greater than that of the crystal should instead be something like 10% less. Where do we find a countervailing factor of at least 10/3? Putting aside crystal chains which terminate at the interface (only significantly available in low molecular weight material) there are two possibilities (fig. 2). The answer is very simple. FIG. 2.-Alternative resolutions of the density paradox (1 - p - 2 1 / ~ ) COS e = O . ~ / B A D g 3/10. One, if the interface is normal to the crystalline chains, is that at least (3 - 0.9)/3 of them, i.e. 70%, fold back immediately, at, or even before, the interface. If the fold occurs before the interface the resulting two-chain space near the surface will draw in chains from the disordered region, but I think that makes an energetic defect which will not be very common.The alternative, if there is no backfolding, as I said at Cooperstown in 1958, is to have an interface oblique to the chain axes. The required obliquity is at least arc sec (3/0.9) = 72.5'. Lamellae with tilted chains are familiar, but never with a tilt nearly as large as this. The general formula, with a back-folding probabilityp and an obliquity 0 is (1 - p - 21/L) cos 0 = 0.9/BAD ;I< 3/10 where the term 21/L, with 1 the crystalline stem-length, L the full length of a chain, makes the greatest possible allowance for chain ends at the interface. One may enquire whether back-folding can be avoided, with a mean interface per-10 GENERAL INTRODUCTION pendicular to the crystal chains, by giving it a steep city-scape profile of roofs and steeples.The answer is No : overcrowding is relieved on the roofs, but is worse than ever in the valleys, the streets of the city. That, I think, is about as much as purely steric arguments can tell us about our problem. No kind of microscopy can help us, at least until someone invents a neu- tron microscope : only for neutrons can we label individual molecules to make them distinguishable, without excessively modifying the intermolecular forces, and even in that case, using deuteration as the label, that difficulty is still with us. X-rays and electrons will distinguish crystalline and non-crystalline regions for us, and indeed tell us the structure in the crystalline regions, but to get at the configuration of single whole molecules we have to make use of neutron scattering with deuteration-labelling, despite its faults.To avoid segregation of labelled from unlabelled material one resorts to rapid quenching. Until we fully understand the mechanism of the crystal- lization process we cannot know how that affects the resulting conformations, and in any case it limits the range of conditions in which conformations can be studied. The*simplest mode of analysis of neutron scattering data gives us a radius of gyra- tion. That is essentially the square root of the ratio of the second to the zeroth mo- ment of a distribution function. If you have ever looked at the question how well is a function of any complexity defined by knowledge of its first few moments, you will be aware that that is very limited knowledge indeed. Nevertheless a result has emerged from these analyses, that for a variety of different polymers, of various molecular weights, the radius of gyration is nearly the same in the crystallized state as in the melt: a result of sufficient universality to indicate that it must mean something.But the meaning is not necessarily, I think, that which it suggests to Fischer, his Erstarrungs- modell. He calls it soZidiJication model here, but I think the German word gives me a slightly more definite idea of what he means, which seems to accord also with Flory’s beliefs : but in either language that is a very brief specification of what has to be a very complicated process.I would like to see a much fuller analysis of the necessary de- tails of its mechanism. The flat uniformity of the experimental result is enlivened by the results reported by Guenet et al. in their paper for isotactic polystyrene which is found fortunately free from the complication of segregation of the deuterated species. The radius of gyration increases on crystallization to only a small extent when the molecular weight is 2.5 x lo5, increases by 40% for molecular weight 5 x lo5, both in a matrix of molecular weight 4 x lo5, and decreases about 10% for molecules of molecular weight 5 x lo5 in a matrix of molecular weight 1.75 x lo6. Something more interesting than simple rigor mortis is happening there.When we use the neutron-scattering data more completely, what we can obtain is the mean square Fourier transform of the distribution functions for individual molecules: and there is no uniqueness theorem for the problem of inverting that. Even if there was, knowing the distribution functions would not tell us the conforma- tions. All we can do is to make models and see whether they will fit the scattering data within experimental error. If they don’t, they are wrong. If they do, they are not necessarily right. You must call in all aids you can to limit the models to be tested. It is essential that they should pass tests of steric acceptability, as everyone who uses the corresponding trial-and-error method for X-ray crystal structure determination knows. Keller and Sadler’s model fits the shape of the scattering curve, but fails by a factor of two in absolute intensity: but how reliable are the absolute intensities in neutron- scattering measurements? A factor of two is quite a lot to laugh off.Yoon and Flory produce a computer-generated stochastic model which buys agreement in shape and intensity at the expense of unacceptably large variation in real space density, asF. C . FRANK 11 Guttman shows. It must be rejected. We have a good many other models before us in this Discussion, and I must leave it to the experts in the subject to try and thrash out the question of which of them are not demonstrably wrong. In making models, we must respect the principles of equilibrium statistical mecha- nics, but cannot wholly rely on them, since we have every reason to believe that in polymer crystallization we have only a frustrated approach towards thermodynamic equilibrium.Most of the models are in some way or another founded on their authors’ conceptions of the nature of the process of high-polymer crystallization from the melt. That is a process on which direct detailed information is hard to get, and some imaginative extrapolation from what one knows about related problems is al- most unavoidable. In the work of Kovacs on low molecular weight poly(ethy1ene oxide) the com- bination of a favourable material and brilliant technique tells us how complicated a real polymer crystallization process can be, and gives us much insight into what, at any rate, chains of modest length can do.It would be helpful for the problem of melt crystallization if we fully understood crystallization of polymers from dilute solution, but we don’t. I think we still don’t understand why the rate of crystal growth is pro- portional to a fractional power of the concentration, as shown by Keller and co- w o r k e r ~ . ~ ~ Mandelkern showed more than twenty years ago that the growth kinetics implied repeated surface nucleation. Lauritzen showed that that implied a transition between what he called regime I and regime I1 kinetics, and there I myself finished off the solution of the problem which Lauritzen had started using tricks first taught me by Burton and Cabrera (and, pace Hoffman, my result is not ~nwieldy).~ Hoffman et al. appear to have found that transition, but the puzzle then is to identify the defects which by implication dissect the growth front into half-micron segments. I feel that there is some key idea still missing from the picture, and perhaps Point’s new look at the problem of crystallization from dilute solution will point the way forward.These, however, are all laboratory problems. What matters practically is crystal- lization from the melt. Flory has put forward a kinetic argument to show that close folding is impossible. If it were true, one would wonder how crystallization of any kind is possible, but his argument is refuted both by Hoffman et al. and by Klein, pointing out that he has overlooked de Gennes’ process of reptation: the ability of a chain to worm its way longitudinally through the tangle.I like pictorial analogues which aid thought, like the telephone switchboard which helps me to perceive the impossibility of some of Flory’s models. My analogue for the polymer melt is a pan of spaghetti. If I can shake the spaghetti pan three times a second, and a characteris- tic frequency of thermal agitation is 3 x 10’l Hz, then a millisecond for the polymer melt is equivalent to 3 years for the spaghetti. I doubt whether 3 years shaking is long enough for standard length unbroken spaghetti, which corresponds to a medium high molecular weight polymer, in length to diameter ratio, to reach an equilibrium degree of entanglement: but if I get one piece of spaghettti between my lips, and gently suck while I go on shaking, I think I shall have it out within a minute: and the free energy of crystallization provides just that suction.The resistance to extraction rises initially as we pull out un-pinned loops further and further along the chain, and if the molecular weight is too high the crystal may lose patience, stop pulling on that chain and go to work on another. The first one remains attached to the crystal, the tension in it relaxes, but it cannot start crystallizing again except by a surface nucleation event or the arrival of another growth step. If it utilizes the latter it leaves another chain dangling, and so on. Klein’s estimates put the typical experiment in polymer melt crystallization rather nearer to his transition to high molecular weight behaviour than Hoffman’s estimates do. Guenet’s remarkable change in behaviour between iso-12 GENERAL INTRODUCTION tactic polystyrene matrices of 4 x lo5 and 1.75 x lo6 molecular weight may indicate passage beyond this transition limit.Equilibrium statistical mechanics may still have something useful to teach us in this problem. At that same discussion at Cooperstown in 1958 Flory made the per- spicacious remark " (the crystal) surface presents an impenetrable barrier to the ran- dom coil, and this restricts the statistical possibilities of the coil. The problem re- sembles that encountered in treating the surface tension of a dilute polymer solution ". I do not believe the consequences of that remark have been sufficiently followed up. Z O -h. L cos l T z / h cos2 TCz/h cos K r / h const FIG. 3.-Distribution of end-points and of points not near ends, for a random chain between im- penetrable walls; h g (LA)+.Let me consider a simple problem which may have some relevance to the state of affairs in the disordered layer between crystal lamellae. Consider first a long-chain polymer molecule in dilute solution, confined in the gap of width h between impene- trable walls at z = +h/2. Let the persistence length be J. and the full length of the chain be NA. Now, neglecting self-exclusion corrections, which should make no drastic difference, it is easy to show that the distribution function in z for either end point of the chain is proportional to cos m / h . It is maximum in the middle of the gap, because configurations which end there suffer least rejection of phantom con- figurations, those calculated without regard for the presence of the boundary walls, which must be rejected for infringement of the non-crossing boundary conditions; and it is zero at the walls because the probability of such infringement, high everywhere, is virtually infinite for configurations terminating there.Now consider a point in the mid-range of the chain, further, along the chain, from either end than h2/11. Such a point is the end-point of two long part-chains, and the configuration number is the product of the configuration numbers of the two parts. The distribution of such points in z is therefore proportional to cos2 nz/h. By my initial assumption the greater part of the chain belongs to this mid-range and should have this distribution. Now, instead of dilute solution, let us have molten polymer in the-gap.If chains still behaved in the same way that would be the full density of matter. We should have twice the mean density in the middle, and virtual emptiness near the walls. But intui- tion, which I strongly trust in this case, tells me the density should be much more nearly uniform. Evidently, where boundaries are present, mutual exclusion does something much more drastic than just to cancel the effects of self-exclusion. How does one modify the theory to make it talk sense? Formally, the answer is to bring in Boltzmann factors : to assign a potential energy Vi(r) to each segment of a chain and hence factors exp [- Vi(r)/kT] for each segment in the statistical weight for any configuration. The consequences are not simple, especially as usually presented, and a nice little recent paper by DiMarzio and Gutt- man5 attempts to set them out in simple terms.It is an instructive paper, but I think it is partly wrong. In explaining the use of a potential to represent mutual exclusion, they equate exp (- VJkT) to the " fraction of emptiness ". I think that cannot be right. Space is equally occupied by separate molecules of solvent, orF . C. FRANK 13 similar molecules tied together as polymer chains, but the consequences are entirely different, as my example shows. Statistically, of course, the effect of chain connection comes from the fact that if a chain is excluded from one point, it has a correlated par- tial exclusion from neighbouring points: but I would like to think of it from another point of view, namely from the fact that chains can transmit non-hydrostatic stress and so maintain pressure gradients.After all, I can discuss the free energy of defor- mation of rubber entirely in terms of constraints on the randomness of chain con- figuration, without ever mentioning stress : but stress is still a valid concept in rubber, and has to obey the laws of stress continuity known to us in elasticity theory. In my example there is certainly a z-wise compressive stress, because the walls constrain the chain configurations more the closer they are together. That stress, -ozz, should be uniform in z. I would like to think there are transverse tensile stresses oxx and o,,,, in thin layers near the walls sucking the segments down towards those regions. If I may think of it that way those potential energies V, become more real, much less of a formal device. Will somebody tell me whether I may? What we actually need to flatten this density distribution is an attractive potential Vi = -kT In 2, for a mono- layer of segments, adjacent to the wall. Di Marzio will recognize that. Several other themes of considerable interest emerge from papers in this Discussion which I have not found time to mention but I hope I have said enough now to start a few arguments. I expect to be told where I have been mistaken. Only Gibbs and God made no mistakes, as the Russians say, though they don’t like to be quoted when they say it. Let the arguments begin! Growth and Perfection of Crystals (Proceedings of an International Conference on Crystal Growth held at Cooperstown, New York, August 27-29, 1958), ed. Doremus, Roberts and Turnbull (John Wiley, New York, 1958), discussion remarks of F. C. Frank and P. J. Flory, pp. 529 and 530. D. J. Blundell and A. Keller, Polymer Letters, 1968, 6, 433. A. Keller and E. Pedemonte, J. Crystal Growth, 1973, 18, 11 1. F. C. Frank, J. Crystal Growth, 1974, 22, 233. E. A. DiMarzio and C. M. Guttman, J . Res. Nut. Bur. Standards, 1978, 83, 165.

ISSN:0301-7249    

DOI:10.1039/DC9796800007

出版商:RSC    

年代:1979

数据来源: RSC

摘要:

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. Geil, J . Macromol. Sci., Phys. Ed., 1976, 12, 173.P. J . FLORY 25 C. S. Wang and G. S. Y. Yeh, J. Macromol. Sci., Phys. Ed., 1978, B15, 107. M. R. Gupta and G. S. Y. Yeh, J. Macromol. Sci., Phys. Ed., 1978, B15, 119. W. R. Pechhold and H. P. Grossman, Faraday Disc. Chem. Soc., 1979,68,58; W. R. Pechhold and S. Blasenbrey, Angew. Makromol. Chem., 1972, 22, 3. P. J. Flory, Pure Appl. Chem., Macromol. Chem., 1972, 8, 1 ; reprinted in Rubber Chem. Tech., 1975, 48, 513. 1. Voight-Martin and F. C. Mijlhoff, J. Appl. Phys., 1975, 46, 1165. See also R. Lovell and A. H. Windle, Polymer, in press. D. R. Uhlmann, Faraday Disc.Chem. SOC., 1979, 68, 87. See also A. L. Renninger, G. G. Wicks and D. R. Uhlmann, J . Polymer Sci., Polymer Phys. Ed., 1975, 13, 1247; M. Meyer, J. Vander Sande and D. R. Uhlmann, J. Polymer Sci., Polymer Phys. Ed., 1978, 16, 2005. * G. W. Longman, G. D. Wignall and R. P. Sheldon, Polymer, 1976,17,485. lo R. Lovell, G. R. Mitchell and A. H. Windle, Faraday Disc. Chem. SOC., 1979, 68,46. l 2 P. J. Flory, Proc. Roy. SOC. A.,.1956, 234, 60. l3 P. J. Flory, Disc. Faraday SOC., 1970, 49, 7; Ber. Bunsenges. phys. Chem., 1977, 81, 885; l4 P. J. Flory, J . Macromol. Sci., Phys. Ed., 1976, B12, 1. l5 J. A. Semlyen, Adv. Polymer Sci., 1976, 21,41. l6 (a) R. G. Kirste, W. A. Kruse and K. Ibel, Polymer, 1975, 16, 120; (b) R. G. Kirste, W. A. l7 J. P. Cotton, D. Decker, H.Benoit, B. Farnoux, J. Higgins, G. Jannink, R. Ober, C . Picot and l 8 G. D. Wignall, D. G. H. Ballard and J. Schelten, Eur. Polymer J., 1974, 10, 861. l9 J. Schelten, G. D. Wignall and D. G. H. Ballard, Polymer, 1974, 15, 682. 2o E. W. Fischer, G. Leiser and K. Ibel, Polymer Letters, 1975, 13, 39. 22 G. A. Allen and T. Tanaka, Polymer, 1978, 19, 271 ; E. W. Fischer, Pure Appl. Chem., 1980, 23 R. G. Kirste, Makromol. Chem., 1967, 101, 91. 24 D. Y. Yoon and P. J. Flory, Macromolecules, 1976, 9, 299. 25 M. Dettenmaier, J. Chem. Phys., 1978, 68, 2319. 26 D. Y. Yoon and P. J. Flory, J. Chem. Phys., 1978,69, 2536. ’’ E. L. Thomas and E. J. Roche, Polymer, 1979,20, 1413. 28 G. W. Longman, G. D. Wignall and R. P. Sheldon, manuscript in press. 29 M. Vacatello, G. Avitabile, Paolo Corradini and A. Tuzi, J . Chem. PhyJ., in press. 30 H. Quinones and P. Bothorel, Compt. rend., 1973, 277, 133; B. Lemaire, G. Fourche and P. Bothorel, Compt. rend., 1972, 274, 1481. E. W. Fischer, G. R. Strobl, M. Dettenmaier, M. Stamm and N. Steidle, Faraday Disc. Chem. SOC., 1979, 68, 26. J. Polymer Sci., 1961, 49, 105. Kruse and J. Schelten, Makromol. Chem., 1972, 162, 299. J. des Cloizeaux, Macromolecules, 1974, 7, 863; Nature (Phys. Sci.), 1973, 245, 13. D. G. H. Ballard, P. Cheshire, G. W. Longman and J. Schelten, Polymer, 1978, 19, 379. in press. 32 G. D. Patterson and P. J. Flory, J.C.S. Faraday IZ, 1972, 68, 1098. 33 C. W. Carlson, Ph.D. Dissertation (Stanford University, 1975). 34 P. Bothorel, C. ClCment and P. Maraval, Compt. rend., 1967, 264, 658; P. Bothorel, C. Such 35 S. Kielich, Chem. Phys. Letters, 1971, 10, 516. 36 V. T. Lam, P. Picker, D. Patterson and P. Tancrede, J.C.S. Faraday ZZ, 1974, 70, 1465; P. Tancrede, D. Patterson and V. T. Lam, J.C.S. Faraday ZZ, 1975, 71, 985; P. Tancrede, P. Bothorel, P. de St. Romain and D. Patterson, J.C.S. Faraday ZZ, 1977,73, 15; P. Tancrede, D. Patterson and P. Bothorel, J.C.S. Faradar IZ, 1977, 73, 29. 37 C. W. Carlson and P. J. Flory, J.C.S. Faruday ZZ, 1977, 73, 1505. 38 G. D. Patterson, Ph.D. Dissertation (Stanford University, 1972). 39 J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, Oxford, 1946), pp. 382-390. 40 R. A. Sack, J. Chem. Phys., 1956, 25, 1087. 41 M. Fukuda, G. L. Wilkes and R. S. Stein, J. Polymer Sci A-2, 1971, 9, 1417. 42 T. Ishikawa and K. Nagai, J. Polymer Sci. A-2, 1969, 7, 1123; Polymer J., 1970, 1, 116. 43 G. Rehage, E. E. Schafer and J. Schwarz, Angew. Makromol. Chem,, 1971,16117, 231, 44 A. N. Gent, Macromolecules, 1969, 2, 262; A. N. Gent and T. H. Kuan, J. Polymer Sci. A-2, 45 M. H. Liberman, Y. Abe and P. J. Flory, Macromolecules, 1972, 5, 550. and C. ClCment, J. Chim. phys., 1972, 69, 1453. 1971, 9, 927.

ISSN:0301-7249    

DOI:10.1039/DC9796800014

出版商:RSC    

年代:1979

数据来源: RSC

摘要:

Molecular Orientational Correlations and Local Order in n-Alkane Liquids BY E. w. FISCHER," G. R. STROBL, M. DETTENMAIER,? M. STAMM$ AND N. STEIDLE~ Institut fur Physikalische Chemie, Universitat Mainz, and Sonderforschungsbereich 41, Federal Republic of Germany Received 12th September, 1979 The molecular orientation correlation, the short-range order and the chain conformation in liquid n-alkanes have been studied by various experimental methods supposing that the nature of " order " in these simple systems will reflect the typical features of the organization of polymer molecules. The orientational correlation has been investigated by means of the dependence of depolarized light scattering and magnetic birefringence on concentration of diluents, on chain lengths and on temperature.There exists a very weak orientational correlation, which is far from a nematic-like state and which can be characterized by a correlation length -=c 1OA. The orientational correlation shows a temperature dependence as predicted by de Gennes' theory of pretransitional ordering in the isotropic phase of a liquid. Alternatively the temperature dependence of order in the n-alkanes can be described in terms of a heterophase fluctuation (Frenkel), but again the number of CH2 units in an " ordered phase " is extremely small (z 8 to 9 units). The conformation of the chains has been studied by small-angle neutron scattering and Raman spectroscopy. In spite of the local segmental orientation correlation it was found that the conforma- tion of the single chains behaves as expected from Flory's theory: no conformation changes occur in the melt. The problem of local order and " structure '' within the amorphous state of poly- mers has attracted much attention in recent years.Two kinds of models have been proposed: (i) In the " coil model " it is assumed that the material is homogeneous in structure and that the configuration statistics of a single molecule in the melt or glassy state are the same as those of an unperturbed molecule in so1ution.1'2 (ii) The various " bundle models " are based on the assumption of domains with nematic liquid- crystal like arrangements of the macromo1ecules.3-5 The structure of polymeric glasses and melts has been studied by means of various methods [for recent reviews see, e.g., ref. (6)-(9)].The evaluation of the experimental data has often been impeded, however, by difficulties characteristic of polymer samples, e.g., incomplete information about chemical structure and molecular weight distribution, artificial heterogeneities being present in the sample and other sources of possible errors. Therefore it may be advisable to attack the problem of the organiza- tion of macromolecules in the condensed state by studying the simplest systems of chain molecules, namely the n-alkanes. We suppose that the nature of " order " in these systems will reflect the typical features of organization of polymer molecules. 7 Ecole Polytechnique FtdCrale de Lausanne, Dtpartment des MatCriaux, Laboratoire des $ Institut fiir Festkorperforschung der Kernforschungsanlage Jiilich, Postfach 19 13, 5 170 Jiilich, 0 Universitat Ulm, Abt.Exp. Phys. 1, Oberer Eselsweg 9,7900 Ulm, Federal Republic of Germany. polymkres 132, Ch. de Bellerive, 1007 Lausanne, Switzerland. Federal Republic of Germany.FISCHER, STROBL, DETTENMAIER, STAMM AND STEIDLE 27 So, for example, one of the main arguments for assuming a “ bundle ” structure is based on packing c o n ~ i d e r a t i o n s ~ ~ ~ * ~ ~ which should be applicable to paraffin melts as well as to polymers. Fig. 1 shows schematic drawings of the two extreme cases of models for the struc- ture of a melt of short-chain molecules. The problem of “ order ” in paraffin melts FIG. 1 .-Models of the structure of n-alkane melts. (a) Unperturbed molecular conformations as in solution, (b) liquid-crystal-like structures.l’ has already been studied by many authors both experimentally and from a theoretical point of view.l2-I7 We used the experimental techniques summarized in Table 1, and the results of these experiments will be reported and discussed in the following sec- tions, where we also will refer to previous investigations utilizing these methods.The problem has also been attacked by other types of measurements such as diamagnetic susceptibility,ll Brillouh calorimetric 25 n.m.r. relaxa- tion2”28 and the Kerr effect.29 We will discuss some of these results later. TABLE 1 .-EXPERIMENTAL METHODS USED methods informat ion obtainable depolarized light scattering (d.p.s.) magnetic birefringence (m.b.) small angle neutron scattering (SANS) Raman scattering (R.s.) wide-angle X-ray scattering (WAXS) segmental orientation correlation segmental orientation correlation conformation of single chains t- and g-population short-range order 1.DEPOLARIZED LIGHT SCATTERING (D.P.S.) Light scattering is a sensitive probe with regard to optical inhomogeneities in the sample under investigation. The polarized component of the scattered light is effected by density fluctuations, whereas the depolarized Rayleigh ratio H, is due to fluctua- tions in the optical anisotropy tensor. If one only takes into account the elastically28 LOCAL ORDER IN N-ALKANE LIQUIDS scattered radiation and if the internal field is approximated by the Lorenz-Lorentz relation, the depolarized component is given by 30-33 where & is the wavelength of light in vacuo, n is the refractive index, Ncm3 is the number density of molecules.d2 is the effective mean-squared optical anisotropy of the molecule, which in a liquid is in general different from the intrinsic mean-squared optical anisotropy 8; of a single isolated molecule. The differences are due to inter- molecular orientational correlations and therefore the ratio p = d2/d; (2) yields a convenient quantity for describing the extent of the orientational " order '' in the n-alkane liquids. The meaning of the order parameter p can be further elucidated if (under some simplifying assumptions) an orientation correlation function fe(r) = ( 3 (3 cos20ij - 1) ) (3) is introduced, where Oij is the angle between the axes of the scattering units i and j , which are a distance r apart.The depolarized component H, is proportional to the Fourier transform off0 (r). In the case that the correlation length 4 is much smaller than i, i.e. if no angular dependence of the scattered light is found, one can still measure the value Vc of the so-called correlation volume (4) which is related to the order parameter p by: In the case of flexible chain molecules it may sometimes be convenient to use a com- bined order parameter describing the orentational correlation of monomer units with an optical anisotropy r2. Then pll describes the intramolecular correlation and p12 is related to the correla- tion between units in different chains. One problem in measuring the correlation volume Vc arises from the fact that the depolarized scattering also contains an inelastic component which is due to (i) inter- molecular collisions (AHvl), (ii) correlated collision induced anisotropies (AH,"), and (iii) conventional Raman-scattering (AHV1I1). One example of the spectral decompo- iti ion^^ is shown in fig.2. In fig. 3 the integrated values of the inelastic ~cattering~~ are plotted. They depend strongly on both the temperature and the chain length and have to be subtracted from the measured depolarized intensity. The correction can be up 30% of the measured intensity. The effective anisotropy d2 of n-alkanes in solution exhibits a strong increase with in- creasing c~ncentration~~ as already described by several a ~ t h o r s . ~ ~ ~ ~ ~ So, for example, in the case of C16H34 in CC14 d2 increases from 5.8 A6 at x = 0.5 to 27.8 A6 at x = 1.0.This effect clearly indicates that there exists an orientational correlation between the axes of the polarizability tensors of neighbouring molecules. In order to determine the order parameter p defined by eqn (2) and (5) liquid n-alkanes were investigated by d.p.s. to determine their dependence on temperature and chain length.33.34 Fig. 4 P = P11 PlZ = d 2 K 2 (6)FISCHER, STROBL, DETTENMAIER, STAMM AND STEIDLE 1.6 - -i 1 . 4 - 5 r. k , 1 . 2 - h - a, t .- u 1 . 0 - =Ti' a 29 80 60 40 20 0 20 40 60 80 FIG. 2.-Spectral decomp~sition~~ of depolarized scattered light due to intermolecular collisions. Broad component AH:, narrow component AH:'. 3 / c m-1 0 20 40 60 80 100 120 T /'C FIG. 3.-Integrated inelastically scattered intensity AH: + AH:' + AHFaman as a function oo temperature for various chain lengths.34 @, C12; 0, c16; A, C18; 0, G o ; A, (224; ., c36, shows the results obtained from the suitably calibrated intensities and after subtraction of the inelastic components.Similar results were obtained by Patterson et aZ.36 and C a r l ~ o n . ~ ~ In the case of C36H74 an increase in scattering intensity at small angles was found, indicating that some heterogeneities remain after p~rification.~~ There- fore for the further analysis of the data a constant background scattering of 1.4 x cm6 was subtracted. This value was estimated from the data of Carlson, but using our own methods of evaluation.* * The methods differ mainly with regard to refractive index and the amount of collision-induced depolarization.30 LOCAL ORDER I N N-ALKANE LIQUIDS First we discuss the effects of the number n, of C-C bonds in the n-alkane. The effective anisotropy d2/n, increases with nc for all temperatures but there seems to be a saturation effect so that the order parameter p tends towards a constant value.For a temperature of 80°C the ratios d2/St are plotted against l/nc in fig. 5, which clearly indicates a chain-end effect. The extrapolation to p = 1 (no orientational correlation 9 - 8 . 7.6 5 4 lD E r. * 9 3 2 .m 2 1 0 .i; A c 24 c20 C1B c16 c12 I I I I I I I I 1 I 1 1 . 0 10 20 30 40 50 60 70 80 90 100 120 140 T / O C FIG. 4.-Temperature dependence of the effective optical anisotropy d2 of various n-alkane in the melt) leads to a critical chain length no ~8 9.For the case of C6HI4 the effective anisotropy S2 in the melt 33 agreed completely with the value measured in solution.30 Thus one has to conclude from fig. 5 that the chain ends act as a perturba- tion of the orientational ordering and that a minimum chain length of x8 9 bonds is required in order to establish some weak intermolecular orientational correla- tions. Remarkably, the same chain length is found using Raman spectros~opy~~ as a limit above which the abundance of an all-trans conformation is vanishingly small. The consequences of this observation will be discussed later. The excess anisotropy due to intermolecular order depends not only on n, but also on the temperature, as one would expect from general thermodynamic reasons. The temperature dependence of d.p.s.as demonstrated in fig. 4 cannot be explained by changes in S,", which are much smaller in the temperature range under investigation. On the contrary, one must conclude that the decrease in the effective anisotropy d2FISCHER, STROBL, DETTENMAIER, STAMM AND STEIDLE 31 with rising temperature is due to a decreasing orientational correlation, i.e. the corre- lation volume Ve or the order parameter p depend on temperature. In order to find a suitable quantitative description of the experimental results the similarity between the behaviour of the paraffin melts and the isotropic phase of 0 0.05 0.1 1.5 1 /nc FIG. 5.-Ratio of effective anisotropy d2 in the melt at 80°C to the calculatedJo anisotropy 8% of the single chain as a function of l/nc.nc = number of C-C-bonds. nematic liquid-crystal systems can be used. As an example fig. 6 shows the results 39940 of d.p.s. for the case of 4-butyl-N-(p-methoxy-benzylidene) aniline (MBBA) in the isotropic phase, i.e. a temperature above the nematic-isotropic transition. The changes in p for this case are much larger, of course, i.e. at p E 500 just above the nematic-isotropic transition temperature and p = 80 at 55°C. Because of the similarity it has been proposed 9*33934736 that the temperature effect in alkane melts can be interpreted along the same lines as in the case of MBBA, where the temperature dependence of p can be explained in terms of the general theory of orientational cor- relations in isotropic liquids developed by de C e n n e ~ .~ ~ The application of such a theory to paraffin melts has also been treated by Bend1er.l’ 40 45 50 55 60 6 5 T / O C FIG. 6.-Vv and H, components for MBBA as a function of temperat~re.~32 LOCAL ORDER I N N-ALKANE LIQUIDS It has been shown by de Gennes that in the case of a short-range orientational order above a nematic-isotropic transition point T, the intensity of d.p.s. is propor- tional to kT 2A H, w - (7) where A is the coefficient of the quadratic term in the Landau expansion of the free energy in powers of the order parameter. A can be written as A w (T - T+)Y (8) where y is a critical exponent, which is y = 1 in a mean field theory, and T+ is a tem- perature slightly below T,. Physically T+ is the temperature at which the correlation length 5 would become infinite, if the isotropic phase were still stable below T,.Neglecting this small difference the mean field approximation yields that the order parameter p in eqn (2) is given by T p = - T - T, (9) where T, is a hypothetical transition temperature for the transition from the isotropic to the nematic state. Naturally, T, is located below the melting temperature T, of the n-alkanes, since in the whole temperature range of the melt no indication of a nematic transition exists. Pure speculations ''J* about such a transition for T > T, are proved to be wrong by the results of d.p.s. Applying de Gennes' theory to polymethylene chains the questions arises, what are the units which are subjected to orientational correlation? Clearly, in the case of long chains the optical anisotropy y2 of the units correlated to each other is not identi- cal with the intrinsic anisotropy & of a whole single molecule.Due to the rotational freedom of the members of a chain, the intermolecular ordering is not expected to per- sist for the entire chain length. Therefore we adopt the concept of the " segmental orientation correlation " introduced by Bendler l2 and leave the apparent anisotropy { y i ) of such a correlated segment as an adjustable parameter which can be determined experimentally. Because of the perturbing effect of the chain ends, ( y i } depends on nc. According to eqn (9) one may assume This equation was tested by plotting nJd2 against 1/T and the results of a linear re- gression calculation are given in table 2. TABLE 2.-HYPOTHETICAL TRANSITION TEMPERATURES T, AND APPARENT ANISOTROPIES ( y: ) ACCORDING TO EQN (10) melting point transition temp.regression coefficient n-a1 kane Tm/K Tc/K < 3, )/A6 r2 ClZ 263.6 21 1 0.339 0.990 C16 291.3 223 0.408 0.984 C18 301.3 21 4 0.527 0.963 C 2 0 309.8 218 0.594 0.999 c 2 4 323.8 23 1 0.622 0.980 C36 349.1 226 0.720 0.990FISCHER, STROBL, DETTENMAIER, STAMM A N D STEIDLE 33 0.5- oz 0.4- 20.3- 0.2- 0.1- I D " h N v The values of T, are far below the melting temperature as already indicated quali- tatively by the temperature dependence shown in fig. 4. With regard to chain length, the temperature T, is almost constant; there is only a slight increase observed, see fig. 7. The apparent anistotropy { 7: ) depends strongly on chain length, however, and seems to approach a limiting value { & ) .One obtains (1 1) with { &, ) = 0.903 & 0.05 A6 no = 7.16 & 0.8. Again no is a minimum chain length below which no critical behaviour occurs; that means n-alkanes shorter than about C8H18 should not show a temperature dependent intermolecular correlation. 0.8 J h Y 230 2 22 0 21 0 O ! I I 1 I b 0 10 20 30 40 nC FIG. 7.-Transition temperatures T, and apparent segmental anisotropies < y t ) as a function of chain length. The value of ( & ) can be compared with the optical anisotropy per unit of an isolated chain as calculated by Patterson and F10ry.~' The ratio of these quantities is ~ 2 . 5 and tells us that, on average, segments of 2-3 monomer units are correlated with regard to their orientation.Just above the melting point the number of cor- related segments (or the correlation volume Vt divided by segment volume) is given by Tm/(Tm - T,) according to eqn (9) and amounts to z3 at most. So the orienta- tional correlation in the melt is extremely weak compared with that for nematic systems and the correlation length 4 is certainly < 10 A. The absolute value of T, is questionable, of course, since the validity of eqn (10) over a large extrapolation range is assumed. Thus it is remarkable that, in spite of this uncertainty, a quite different method, namely magnetic birefringence, see fig. 10, yields the same kind of results, with a value of T, which is not very different. Both methods clearly demonstrate that near their melting points all the n-alkanes under investigation are far above a hypothetical nematic-isotropic transition temperature and that such a transition does not take place in the melt.Conclusions from Bril- louin spectroscopy about such a transition18 are erroneous because they are based on34 LOCAL ORDER I N N-ALKANE LIQUIDS the assumption that there must be a linear temperature dependence of the hypersonic velocity u,(T). There are good reasons to believe that us is coupled to the order parameter p in the isotropic liquid and therefore the continuous change of p leads naturally to a slightly curved behaviour. 2. MAGNETIC BIREFRINGENCE (M.B.) When an external magnetic field B is applied to an isotropic liquid consisting of anisotropic molecules, these molecules will be slightly aligned producing a bire- fringence An which can be measured : An = nil - n 1 = A CmB2 (12) where A is the wavelength of light and Cm is the Cotton-Mouton constant.The effect is known to be extremely sensitive with regard to an orientational correlation of the m~lecules-~ and therefore it can be applied to the structure problem of liquid n-alkanes. The experimental set-up will be described in another paper.44 The measurements are rather difficult since the optical and diamagnetic anisotropies of paraffins are very Nevertheless the advantage of m.b. is that small amounts of heterogeneities in the samples do not play such an important role as in the d.p.s. measurements. If an order parameter p according to eqn (6) is introduced, the Cotton-Mouton constant of n-alkanes can be written to a first approximation as O6 cm = - 2n(n2 + 2)2 p L Actm kT AXm P 135 a M, where p is the density, L Avogadro’s number and Mm the molecular weight of a monomer unit.Am, and Axm are the optical and diamagnetic anisotropy of these units, respectively. jT describes both the intra- and the inter-molecular order. The linear dependence on p” is due to the fact that both Act and Ax per correlated segment are proportional to p , and also M = M,. The dependence of Cm on the volume fraction VJV of n-alkane in solution was first studied for C12H26, C16H34 and C20H42. A typical result is shown in fig. 8. According to theory,47 the n-weighted Cm-constants should be additive : and without intermolecular orientation correlation a straight line is expected. Fig.8 shows that in agreement with the results of d.p.s. measurements a dependence of C, on concentration is observed indicating an increase in segmental correlation with increasing concentration. According to the light scattering experiments we should expect that C , of the paraffin melts depends much more strongly on temperature than E l/Taccording to eqn (13). This is indeed observed, as shown in fig. 9 for the longer chains. However, in the case of CSHIS, Cm can be approximated by C, FZ 1/T. The results are similar to those of fig. 4 and one may assume that the temperature dependence can be again described by eqn (9). In order to prove this hypothesis some of the temperature dependent quantities of eqn (13) were eliminated by the use of Cm(T) = [n2(T) + 212 p ( T )FISCHER, STROBL, DETTENMAIER, STAMM AND STEIDLE 35 and one obtains, with eqn (6), K T - T, c,' = - where 2nNAu,Axm 135 A M,k '''* K = 3.0 f 0 0.2 0 .4 0.6 0.8 1.0 V l / v FIG. 8.-Concentration dependence of the Cotton-Mouton constant C,,, of C16H34 in CC14 solution; (-) expected behaviour without intermolecular correlations. FIG. 9.-The a function of \c - 4 . 0 1 I E N I ; -2.04 - - 1 . o - - t -n k I ] ? , , , I , , , , b 0 40 80 120 160 200 T/OC Cotton-Mouton constant of various 9-alkanes and of a polyethylene (M, temperature. 0 , PE; .A, c 3 6 €374; A , c 2 4 H ~ o ; 0, CI, H34; 0, CIZ H26; c5 Hl2. 53 000) as , C8H18 a, The quantity K is supposed to depend only slightly on temperature and the intra- molecular correlation pI1 involves (as does ( y ; } ) information on the size of the segments which are subjected to orientational correlations. Eqn (1 6) has been tested for the case of CI6Hs4 and the results are shown in fig.10. A hypothetical transition temperature of T, = 215 K was obtained in good agreement with the results of d.p.s., see table 2.36 LOCAL ORDER IN N-ALKANE LIQUIDS 7 6 5 Y 4 E u wl t- u) " 3 -1.2 \ 2 1. 0 -60 0 60 120 r/ OC FIG. 10.-Reciprocal value of the (negative) adjusted Cotton-Mouton constant C& of CI6H3., as a function of temperature. A more detailed analysis of the quantity K will be presented elsewhere.44 Pre- sently we are only interested in the chain length dependence of K, which is plotted in fig. 11. It shows behaviour very similar to (7:) in fig. 7 and proves that the concept of a critical behaviour according to eqn (16), combined with a perturbation effect of the chain ends, is well suited to the description of the m.b.results. The extent of orientational correlations is again very small. For example, in the case of CI6Hs4 at 20°C, the intra- and the inter-molecular correlation amounts to p" = 8.3, whereas a bundle of only 5 chains would result in a value of f j ~ 7 5 . It may be mentioned that the pretransitional ordering should also be reflected in the dynamic properties. According to de Gennes41 the relaxation time z of the order parameter varies as Where v is a transport coefficient and A is the quadratic term in the Landau expansion mentioned above. The results of flow birefringence and viscosity measurements 48 indicate such a behaviour at least qualitatively.A similar result is observed from Brillouin scattering2I by n-alkanes. The linewidth of the Brillouin peak which is proportional to the absorption coefficient of the liquid shows a temperature depen-FISCHER, STROBL, DETTENMAIER, STAMM A N D STEIDLE T 37 0 10 20 30 "C FIG. 1 1.-Chain length dependence of the proportionality constant K of eqn (16). dence which can be explained by eqn (18), just as in the case of MBBA observed by ultrasonic measurements .49 3. SMALL-ANGLE NEUTRON SCATTERING (SANS) In the light of the observations reported so far the very important question arises whether the n-alkane molecules in the bulk liquid state possess their unperturbed dimensions or whether the intermolecular interactions lead to a deviation from their 8-conformations. This problem can be studied either by small-angle neutron scatter- ing or by spectroscopic methods.SANS investigations of mixtures of deuterated and protonated n-alkanes have been carried out by Dettenmaier.50 The differential scattering cross-section of radia- tion scattered by N molecules per unit volume dispersed in a medium of different scattering power is given by dE/dQ = N(Ab)2P(K). (19) P(K) is the form factor of a molecule [ K = (4@)sin(B/2)] and Ab the difference in scattering lengths between the molecules and the surrounding medium. For small K the inverse of the form factor takes the form where R is the radius of gyration of an allowed conformation of the molecule and the average is taken over all conformations.According to this equation a plot of P" against ?c2 should give a straight line. From its slope ( R 2 ) can be evaluated, which is an important parameter in characterizing the conformation of the molecule. From experiment the following conclusions may be drawn:50 The radii of gyra- tion of the hexatriacontane and liexadecane molecules in the melt and in cyclohexane solution are close to each other, the difference being within experimental error. Both are in good agreement with the theoretical values calculated on the basis of rotational isomeric state theory,51 but are in strong disagreement with those for extended chains in an all-trans conformation. In addition the whole shape of the scattering curve is in good agreement with the theoretical results.In fig. 12 the experimental 50 and theoretical 52 scattering CUFVeS38 LOCAL ORDER IN N-ALKANE LIQUIDS 0.31 f/ 0 !8' (a) v I I 1 I * 0 0.1 0;2 OI3 OI4 K / P 0.94 0 7 0.4 0*3 0 0.1 0.2 FIG. 12.-Scattering functions F"(K) = nP(rc)lc2 for (a) n-C36H74 and (b) lt-C16H34 in the meIt (0) and in the cyclohexane solution (0) according to Dettenmaier." The drawn lines represent calculations per- formed by Yoon and FloryS1 for conformations with Ea = 600 cal mol-' and all-trans ones. are plotted against K. from the unperturbed state of the chain molecules. Within experimental error there is no indication of deviations 4. RAMAN SPECTROSCOPY The question of the effect of intermolecular forces on the conformations of indivi- dual chains can also be treated by Raman vibrational spectroscopy.Experiments can make use of the fact that the vibrational behaviour of a chain generally shows a sensitive dependence on its rotational isomeric state, being on the other hand only weakly affected by the molecular surroundings. The Raman spectrum of an n-alkane in the liquid state can thus be regarded as reflecting the statistical distribution of single chain conformations, Any modification of this distribution should lead to changes in the spectral shape. In considering the effect of intermolecular forces one can compare the spectra measured for the n-alkane melts with those of solutions in low molecular-weight com- pounds. Experiments of this type have been performed on mixtures of n-alkanes with carbon tetrachloride. Spectra were registered using an argon ion laser and triple monochromator (Coderg T 800).In order to locate the spectral ranges withFISCHER, STROBL, DETTENMAIER, STAMM A N D STEIDLE 39 high conformational sensitivity a temperature dependent measurement on pure n- hexane was first performed. Fig. 13 shows two spectra obtained at -90 and 77 "C, respectively. Changes in shape are clear in the range of the C-C stretching vibra- tions, 1000-1200 cm-l, and around 900 cm-l. The assignments included are those of Snyder .62 I I I I I 1 I I 1500 1400 1300 1200 1100 1000 900 800 V/c m-1 FIG. 13.-Raman spectra of n-hexane measured at (a) -90 and (b) 77°C. Assignments according to Snyder.62 These changes can be compared with the behaviour observed at room temperature for a series of mixtures with carbon tetrachloride.The volume content of n-hexane has been varied between C = 1 and C = 0.125. Fig. 14 shows the spectra in the sensitive region around 900 cm-'. No modification at all can be detected. The occupation numbers of the different rotational isomeric states thus appear to be un- affected by the dilution process. A similar observation has been made for a dilution series of n-hexadecane (C16H34) in carbon tetrachloride. Results are shown in fig. 15. Again spectra do not show a concentration dependence. The conclusion to be drawn is similar to that derived from the SANS experiments : If there are any modifications in chain conformations initiated by intermolecular forces, they are very small and lie below the error range of the measurement.Com- pared to the effect of temperature, this influence is negligible. There is no indication for the occurrence of bundle-like structures with chain conformations different from a random coil. Another spectral region of interest is that below 500 cm-l. Fig. 16 shows mea- surements on melts of different n-alkanes with carbon numbers between n = 6 and 16. A prominent feature in this range is the chain-length dependent " longitudinal acoustical mode " (LAM) associated with an accordion-like deformation of all those chains which are in the stretched all-trans conformation. As is evident from fig. 16, this band becomes very weak above n z 9 indicating that for longer chains the all-40 LOCAL ORDER I N N-ALKANE LIQUIDS 900 850 (C 1 ( d 1 FIG. 14.-Raman spectra measured for various mixtures of n-hexane [volume content C = (a) 1.0 (b) 0.5, (c) 0.25 and ( d ) 0.1251 and carbon tetrachloride (2,4 and 8 spectra have been accumulated for C = 0.5, 0.25 and 0.125, respectively, in order to compensate for the increasing dilution).l ' ' ' ' l " ~ ~ l ~ " ~ 1 " ' ~ l ~ ' ~ ~ l ~ 1300 1200 1100 1000 900 800 ?/ern-' FIG. 15.-Concentration dependence of Raman spectra measured for C16H34 + eels mixtures. trans conformation becomes highly improbable. There remains only a very broad band, here called " pseudo-LAM ". It also appears in the spectrum of polyethylene with a shape very similar to that observed for CI6H3,, and seems to be characteristic for a random coil. It is interesting to note that the intermolecular orientation corre- lation effects become obvious only for chain lengths for which no measurable all-trans population is found.This clearly indicates that the orientational correlation is not caused by a nematic-like state of stretched molecules or larger segments.FISCHER, STROBL, DETTENMAIER, STAMM AND STEIDLE -LHM /\,,LAM LAM pseudo-LAM 41 I . I S * * ' I " ' " ' ' - , , * m , 500 400 300 200 v /c m-' - FIG. 16.--Chain-length dependent low-frequency Raman spectra of liquid n-alkanes, the accordeon- mode (LAM) being indicated. 5. WIDE-ANGLE X-RAY SCATTERING (WAXS) Several authors5s56 have claimed that the analysis of the X-ray or electron intensi- ties scattered by a polyethylene melt shows lateral order and therefore proves the validity of some kind of bundle model. In a careful study by Voigt-Martin et aZ.57 it was shown that, at least in the case of electron diffraction, this conclusion is untenable.Paraffin melts have been also studied by means of X-ray ~ c a t t e r i n g . ~ ~ . ~ ~ As an example in fig. 17(a) the reduced intensity curves si(s) for melts of polyethylene (at 160°C) and n-C12H26 (at 25 "C) are plotted.58 Only small differences are detected and42 LOCAL ORDER I N N-ALKANE LIQUIDS this is also true for other liquid paraffin^.^^.^^ On the other hand electron diffraction studies of gaseous n-C16H34 have been performed;59 the intensity curve used to cal- culate the atomic pair distribution function is plotted in fig. 17(b). There are still two peaks at 2.9 and 5.2 A-1, only the first peak at 1.4 A-l is missing.So there is no doubt that this peak is due to intermolecular interferences, as is also shown by the - 3 1 0 1 2 3 4 5 6 7 SIB;’ FIG. 17.--Seattering curves si(s) in arbitrary units: (a) measured from C12Hz6 (25°C) (i) and from polyethylene (160°C) (ii)”; (b) calculated for C16H34 in the gaseous state.” temperature dependence of the position of this peak.56*58 However, the two models of fig. 1 differ with respect to the long range correlation of segmental orientation and not with regard to packing density. For rather general reasons9 it is therefore quite obvious that WAXS cannot contribute to the solution of our problem in a straight- forward manner. The pair distribution function gcc(r) of C-atoms is rather insensitive with regard to the orientational correlation.6 . DISCUSSION The results of depolarized light scattering and magnetic birefringence showed the existence of a local orientational correlation of segments of the n-alkane molecules in the bulk liquid state. The extent of this correlation is very weak, the correlation volume Vc is certainly < (10 A)’. In spite of this correlation effect there is no change inFISCHER, STROBL, DETTENMAIER, STAMM AND STEIDLE 43 the average chain conformation, so far as neutron small-angle scattering and Raman spectroscopy can detect changes of conformation. The analogy between the isotropic phase of a liquid-crystal system and the alkanes has led us to the introduction of a hypothetical transition temperature T, far below the melting point of the alkanes. The temperature dependence of depolarized light scattering and magnetic birefringence can be described in terms of de Gennes’ theory41 of pretransitional orientation ordering in isotropic phases of anisotropic molecules. Some of the calorimetric data12*25 suggest a similar analysis, and the values of T, reported by Heintz et aZ.25 agree surprisingly well for C12 and C16 with our results.Two alternative explanations of the segmental orientation correlation may be mentioned. Lemaire and Bothorel 6o found by Monte Carlo calculations two ther- modynamically stable states of a C17H36 alkane. State I is composed of totally coiled conformations without molecular correlations ; state I1 is partially ordered and exhi- bits marked molecular orientational correlations between nearest-neighbour chains.The mean optical anisotropy of the two states was calculated and, so far as we can see, a temperature dependence of the value of d2 must be expected, in contrast to the observed behaviour. Since state I1 has a higher free energy ( ~ 6 0 0 cal mol-l) d2 should increase with rising temperature and not decrease, as it was observed to do. A second alternative approach is the explanation of ordering by the assumption of a heterophase fluctuation as has been discussed from a general point of view by Frenkel? In de Gennes’ treatment it was assumed that in a homogeneous phase only the correlation length increases with decreasing temperature without any phase boundaries. In contrast Frenkel treats the pretransitional ordering as a thermody- namic equilibrium in a system consisting of a large phase A and very small “ embryos ” of phase B, which are separated by a phase boundary with surface energy CT.The number N, of embryos consisting of z molecules can be written under some symplifying assumptions as where H, is the heat of fusion, T, is the melting point and 0’ is related to the surface energy 0. t 0 0.1 0.2 0.3 0.4 ( 7, - T, 1 / r , FIG. 18.-The excess anisotropy Aa2 = a2 - & at To = 80°C for various n-alkanes as a function of the relative superheating (To - Tm)/Tm.44 LOCAL ORDER I N N-ALKANE LIQUIDS For the case of the n-alkanes Carlson3’ and Fischer 34 tried an analysis based on these assumptions. A crude test of eqn (21) can be carried out by assuming that the excess depolarized scattering is given by Ad2 = d2 - 6; = N<,> ((~)6,)~ where 6, is the optical anisotropy of a segment incorporated in an average embryo of ( 2 ) segments.A plot of the values In AS2(To) measured at a constant temperature To(=8O0C) as a function of (To - Tm)/Tm where T, is a variable due to various chain lengths, yields surprisingly reasonable results, see fig. 18. The measured values are well represented by a straight line with a slope H, ( 2 ) = 5200 cal mol-I. This is approximately H, - nc x 600 cal mol-l and therefore the constant number of mono- mer units in an embryo is nc ( 2 ) - 8.6. We do not want to extend this analysis fur- ther at the present time. The important point is that again the number of “ corre- lated ” units is very small, if the pretransitional ordering is described in terms of a heterophase fluctuation.The “perturbing role ” of the chain ends is expressed by the chain-length dependence of T,, whereas in the treatment described above it is expressed by the chain-length dependence of the apparent optical anisotropy ( y i ) of the orienting segments. P. J. Flory, Principles of Polymer Chemistry (Cornell University, Ithaca, 1953). V. A. Kargin, J. Polymer Sci., 1958, 30, 247. G . S . Y . Yeh, J. Macromol. Sci., 1972, B6, 451,465. R. E. Robertson, Ann. Rev. Mat. Sci., 1975, 5, 173. G. S. Y . Yeh, Crit. Rev. Macromol. Sci., 1972, 1, 173. E. W. Fischer and M. Dettenmaier, J. Non-cryst. Solids, 1978, 31, 181. F. J. Balta-Callega, K. D. Berling, H. Cackovic, R. Hosemann, and J. Loboda-Cackovic, J. Macromol. Sci. Phys., 1976, B12, 383.’ P. J. Flory, Statistical Mechanics of Chain Molecules (John Wiley, New York, 1969), p. 35. ’ W. Pechhold and S . Blasenbrey, Kolloid-Z., 1970, 241, 955. ’ P. H. Geil, Polym. Materials, ASM, 1977. lo R. E. Robertson, J. Phys. Chem., 1965,69, 1575. l2 J. T. Bendler, Macromolecules, 1977, 10, 162. l3 A. Wulf and A. G. de ROCCO, J. Chem. Phys., 1971,55, 12. l4 B. Lemaire and P. Bothorel, J. Polymer Sci., Polymer Letters Ed., 1978, 16, 321. l5 H. Kimura and H. Nakano, J. Phys. Soc. Japan, 1977,43, 1477. l7 A. Tonelli, Macromolecules, 1976, 9, 863. l9 G. R. Alms and G. D. Patterson, J. Colloid Interface Sci., 1978, 63, 184. 2o G. D. Patterson and J. P. Latham, Macromolecules, 1977, 10, 736. 21 J. V. Champion and D. A. Jackson, Molecular Motions in Liquids, ed. J.Lascombe (D. Reidel, 22 J. V. Champion and D. A. Jackson, Mol. Phys., 1976,31, 1 169. 23 G. D. Patterson, C. P. Lindsey and G. R. Alms, J. Chem. Phys., 1978, 69, 3250. 24 V. T. Lam, P. Picker, D. Patterson and P. Tancrede, J. C. S. Faraday II, 1974, 70, 1465, 1479. 2s A. Heintz and R. N. Lichtenthaler, Ber. Bunsenges. phys. Chem., 1977, 81, 921. 26 K. J. Lim, Polymer, 1969, 10, 951. 27 W. L. F. Golz and H. G. Zachmann, Makromol. Cham., 1975, 176,2721, 28 F. HoriC, R. Kitamaru and T. Suzuki, J. Polymer Sci. Letters, 1977, 15, 65. 29 J. V. Champion, Trans. Faraday Soc., 1970, 66, 2671. 30 G. D. Patterson and P. J. Flory, J.C.S. Faraday II, 1972, 68, 1098. 31 R. S . Stein and P. R. Wilson, J. Appl. Phys., 1962, 33, 1914. 32 J. J. v. Aartsen, Polymer Networks, ed.A. J. Chompff and S . Newman (Plenum Press, N.Y., 33 M. Dettenmaier, S. Fischer and E. W. Fischer, CoIloid Polymer Sci., 1977, 62, 37. 34 S. Fischer, Diplom-Thesis (Mainz, 1976). 35 G. S. Y . Yeh, Pure Appl. Chem., 1972, 31, 65 and T. G. F. Schoon, Brit. Polymer J., 1970, 2, J. G. Curro, J. Chem. Phys., 1974, 61, 1203. J. K. Kruger, Solid State Comm., 1979, 30, 43. Dordrecht, 1974), p. 585. 1971). 86.FISCHER, STROBL, DETTENMAIER, STAMM AND STEIDLE 45 36 G. D. Patterson, A. P. Kennedy and J. P. Latham, Macromolecules, 1977, 10,667. 37 C. Carlson, Thesis (Standford University, 1976). 38 R. F. Schaufele, J. Chem. Phys., 1968, 49, 4168. 39 M. Dettenmaier and E. W. Fischer, Makromol. Chem., 1976, 177, 1185. 40 J. W. Stinson and J. D. Lister, Phys. Rev. Letters, 1970, 25, 503. 41 P. G. de Gennes, Mol. Cryst. Liq. Cryst., 1971, 12, 193. 42 J. C. Filippini, Y. Poggi and G. Maret, Colloques internationaux C.N.R.S., 1976, No. 24. 43 J. G. J. Yama, G . Vertogen and H. T. Koster, Mol. Cryst. Liq. Cryst., 1976, 37, 57. 44 M. Stamm, G. Maret, E. W. Fischer and K. Dransfeld, in preparation. 45 G. H. Meeten, J. Chim. phys., 1972, 1175. 46 M. Stamm, Thesis (Mainz, 1979). 47 H. A. Stuart and A. Peterlin, J. Polymer Sci., 1950, 5, 551. 48 J. V. Champion and P. F. North, Trans. Faraday SOC., 1968, 64,2287. 49 D. Eden, C. W. Garland and R. C. Williamson, J . Chem. Phys., 1973, 58, 1861. M. Dettenmaier, J. Chem. Phys., 1978, 68, 2319. P. J. Flory and R. L. Jernigan, J . Chem. Phys., 1965, 42, 3509. 52 D. Y. Yoon and P. J. Flory, personal communication, paper to be published in J. Chem. Phys. 53 J. K. Ovchinnikov and G. S. Markova, Vysokomol. Soedin, 1967, 9 (A), 449; Polymer Sci. 54 A. Odajima, S. Yamane, 0. Yoda and I. Kuriyama, Rep. Progr. Phys. Japan, 1975,18,207. 55 J. Petermann and H . Gleiter, Phil. Mag., 1973, 28, 271. 56 K. Ovchinnikov, E. M. Antipov, G. S. Markova and N. F. Bakeer, Makromol. Chem., 1976, 57 I. Voigt-Martin and F. C. Mijlhoff, J. Appl. Phys., 1975, 46, 1165; J. Appl. Phys., 1976, 47, U.S.S.R., 1969, 11, 329. 177, 1567. 3492. R. Hoffmann, D@lomarbeit (Mainz, 1978). 59 S. Fitzwater and L. S . Bartell, J. Amer. Chem. SOC., 1976, 98, 8338. 6 o B. Lemaire and P. Bothorel, J. Polymer Sci. Letters, 1978, 16, 321. 61 J. I. Frenkel, Kinetische Theorie der Fliissigkeiten (Deutscher Verlag der Wissenschaften, Berlin, 1957), also Kinetic Theory of Liquids (Oxford University Press). 62 R. G. Snyder, J. Chem. Phys., 1967, 47, 1316.

ISSN:0301-7249    

DOI:10.1039/DC9796800026

出版商:RSC    

年代:1979

数据来源: RSC

摘要:

Wide-angle X-ray Scattering Study of Structural Parameters in Non-Crystalline Polymers BY RICHARD LOVELL, GEOFFREY R. MITCHELL AND ALAN H. WINDLE Department of Metallurgy and Materials Science, University of Cambridge, Pembroke Street, Cambridge Received 1st May, 1979 The paper presents an analysis of WAXS (wide-angle X-ray scattering) data which aids an under- standing of the structure of non-crystalline polymers. Experimental results are compared with calculations of scattering from possible models. Evidence is presented which supports the view that the chains in molten PE do not lie parallel but have a conformation in accord with the predictions of energy calculations. However, the evidence indicates that in " molten " PTFE the chains lie parallel over distances well in excess of their diameters.WAXS- based proposals are made for the conforma- tions of a-PMMA and a-PS. The structure of polymer glasses and melts at a molecular level has been the subject of detailed debate for some years; even so, various strikingly different models are still seriously being discussed. Careful SAXS (small-angle X-ray scattering) measurements have revealed an ab- sence of density variations on a scale required by some bundle models. More detailed information on the local molecular structure has mainly been derived from measure- ment of the dimension of the molecular trajectory combined with conformational energy calculations ; whereas wide-angle X-ray scattering (WAXS), which should, on account of its greater resolution, be capable of providing the extra detail required, has not hitherto been applied with as much success.A main objective of this paper is to demonstrate that, although the main features of WAXS may not directly answer the outstanding structural questions, by careful analysis of the scattering and comparison with sterically viable molecular models, WAXS can make a substantial contribution to detailed structural understanding of non-crystalline polymers. The analysis is made by comparing experimental and calculated scattering by means of reduced intensity functions,lV2 si(s), rather than total intensity, since the features are more evenly weighted in the former. For the experi- mental function i(s) = kI&) - Zf2(s) where s = (4n sin O)/A, k is a scaling factor to electron units and Ef2 is the independent scattering from a repeat unit.The intensity function for models is calculated using the Debye scattering equation : where rjk is the distance between thejth and kth atoms. We have separated polymer glasses and melts into three broad overlapping groups on the basis of their inherent chain flexibility and taken as examples: (a) flexible:R . LOVELL, G . R . MITCHELL AND A . H . WINDLE 47 polyethylene (PE) melt at 140 "C, (b) rigid: polytetrafluoroethylene (PTFE) at 340 "C, (c) semi-rigid : atactic polymethylmethacrylate (a-PMMA) and atactic polystyrene (a-PS) both at room temperature. The analysis leads to proposals of persistent conformations (in terms of both type and length) which are in general accord with predictions based on conformational energy calculations and measurements of characteristic ratio.Evidence is also presented which supports the view that there is little if any correlation of segmental orientation (CSO) in molten PE, whereas a model of PTFE in which chains are sub- stantially parallel leads to WAXS which is similar to that observed. Since this paper is presented in terms such that the main issues can be approached in the context of work on several polymers, some of the detailed argument is in outline form only and will appear more fully elsewhere. MEASUREMENT OF STRUCTURAL PARAMETERS Molecular organization in non-crystalline polymers can be conveniently con- sidered in terms of conformational, spatial and orientational order. Conformational order is described by any regularity of conformation present and the typical sequence lengths.We use the term " persistent conformation " to refer to both these aspects. Spatial order is seen to be a measure of the regularity with which chain segments are packed together. Since even molten polymers have packing densities (ratio of van der Waals volume to total volume) of ~ 0 . 6 , some degree of spatial order must occur where segments are in contact. Wide-angle X-ray scattering is sensitive to levels of both conformational and spatial order, and can be used to measure these parameters. However, the informa- tion it gives concerning the degree of orientational order, probably the most con- troversial aspect for non-crystalline polymers, is not as direct. The general features of WAXS patterns are similar for many polymers and not very different from those of low molecular weight organic liquids.For example, fig. 1 shows the broad similarity for polyethylene and neopentane [C(CH,),]. Hence conclusions from WAXS about the structure of these polymers can only be derived by detailed analysis of the data supplemented with information from conformational energy calculations and spectroscopy. The analysis is made easier if the scattering pattcrn can be separated into those features that arise principally from interference within the chain segments (intra- segmental scattering) and those due to intersegmental scattering. The terms intra- segmental and intersegmental are used deliberately, since even a semi-rigid chain can have contacts with itself that are intermolecular in nature.(By " segment " we mean that portion of a chain about each atom over which an underlying conformation persists.) The separation is relatively easily achieved for solid amorphous polymers where alignment induced by deformation shows that certain features are principally meri- dional (intrasegmental) whereas others are equatorial (inter~egmental).~.~ This identification is further aided by observing the effect of temperature: the intraseg- mental peaks are significantly less affected than are the intersegmental ones. In molten polymers the identification of the peaks relies on this effect of temperature or possibly on analogy with a similar solid (e.g., comparison of molten PE with oriented, " amorphized " PE).6 Since there is no straightforward way of separating intra- segmental and intersegmental peaks in an RDF, we have adopted the method of com- paring models by calculating the reduced intensity functions, i.e., the comparison has been made in reciprocal space.48 WAXS STUDY OF POLYMERS 0 1 2 3 s A-1 FIG.1.-WAXS plots for (a) molten PE at 140 "C (this work) and (b) liquid neopentane, C(CH,),, at - 17 "C (Narten).lo The curves become virtually indistinguishable at higher values of s. In general it appears that the scattering beyond s = 3.0 A-1 is purely intraseg- mental and hence the persistent conformation can be determined by comparing this region of the data with scattering calculated for single chains. PERSISTENT CONFORMATIONS The bond lengths and bond angles of a polymer molecule are almost independent of whether it is in the crystalline or non-crystalline state, the difference in conforma- tion being due to the much less regular sequence of backbone bond rotation angles in the non-crystalline state.The best approach to modelling such a conformation de- pends on the degree of irregularity. For rigid chains, in which quite long regular sequences predominate, a good approximation is to use a length of the regular con- formation equal to the weighted-average sequence length. For highly irregular chains, a statistical description based on conditional probabilities of rotations about individual bonds must be used. The scattering is then calculated for chains built using these statistics. For both types of chain, the WAXS analysis is eased by starting from likely conformations suggested by energy calculations, spectroscopy or molecu- lar model building.Comparison of the scattering calculated for different chains shows its sensitivity to the type of conformation and to the length of regular sequences present. In fact, WAXS can give a more direct and detailed indication of conformational regularity than can be inferred from measurements of radius of gyration. This is not only true for melt crystallized polymers such as PE and PP where the radius of gyration is unchanged by regularity of conformation7V8 but also, as we set out to demonstrate in this paper, for polymer glasses and melts.R . LOVELL, G . R . MITCHELL AND A . H . WINDLE 49 CHAIN SPACING In many polymers (all but a-PS of those considered here), the most prominent WAXS peak is principally intersegmental and its position has often been used to give a value for the chain ~ p a c i n g .~ . ~ More correctly, the peak arises from the spherically averaged segment separation which, particularly for substituted chains, may differ from the backbone separation. For parallel chains the interpretation of the peak position in terms of lateral spacing of the chains is more straightforward. By modelling the position and width of inter- chain scattering peak(s), (within the constraints of the known polymer density and chain diameter) it is possible to determine the extent of packing disorder. Where there is no orientational order, the precise nature of the intersegmental peak is not so clear. For very “flexible” chains such as PE, we have interpreted the intersegmental scattering along lines similar to those developed for low molecular weight liquids and glasses.This new approach is described in the next main section. ORIENTATION CORRELATION A narrow intersegmental peak in si(s) (or its corollary, a slowly damped oscillation in the RDF) has often been taken as a sign of orientational ordering, and more specific- ally chain parallelism, in non-crystalline polymers. 1,6 However, comparison of WAXS data for polyethylene and neopentane (fig. 1) shows this inference to be dubious since neopentane has roughly spherical molecules with no possibility of parallelism. A narrow peak is, to a much greater degree, evidence for a narrow distribution of molecular or intersegmental separation. Regions of parallel ordering must contain segments with a fairly high aspect ratio (length/diameter > 3) and so a better test for parallelism is to look at the intrasegmen- tal scattering for evidence of such segments.This is the method we have applied to the estimation of orientational order, as will be detailed below. MOLTEN PE (FLEXIBLE CHAINS) Although PE and its oligomers are the simplest form of hydrocarbon chain, there is considerable disagreement on the interpretation of experimental data from the melt. A random chain based on three rotational states has been found to model adequately the characteristic ratio and its temperature coefficient in solution and more recently in the melt.7 However, a significant portion of the published wide-angle diffraction data has been interpreted in terms of relatively straight chain segments organized into bundles, 6 ~ 1 2 although this view has since been ~ha1lenged.l~ Bundles of parallel chains of PE in the melt are unlikely to consist of non-straight segments so that we only consider the energetically most favourable straight segment, the all-trans sequence.Other types of straight segments have been considered else- where,I4 together with a much more detailed version of the following approach. Theories of the nematic mesophase and rigid and semi-rigid rod packing indicate that ordered regions may occur for segments with an aspect ratio which is greater than three, which gives the minimum length of PE chain segments in ordered regions as 15 A; the bundle models, however, predict considerably longer segments.The calculated si(s) curves for single all-trans chains of length 15 and 30 A are presented in fig. 2 and compared with experimentally determined functions. Even if the possible torsional variations in rotation angles are c~nsidered,’~ the si(s) curves for scattering vector >4.0A are not compatible with the experimental curve. In order to account50 WAXS STUDY OF POLYMERS for the intersegmental component of the scattering which may affect this region, a large number of parallel chain models have been constructed and the si(s) curves ~alcu1ated.l~ The models have been built to the density of PE melt which at 140 "C is 0.783.15 For hexagonal packing of parallel chains to be in accordance with this density the interchain spacing is 5.2 A; for a more disordered five-fold packing,14 it is 5.0 A, which is not very different from the diameter of an isolated chain.si (s) 2 4 6 a 10 12 &' FIG. 2.-Comparison of reduced intensity function for molten PE with that calculated from straight chains. (a) Experimental at 140 "C. (6) All-trans: sequence length 15 8, (12 bonds). (c) All-trans: sequence length 30 A (24 bonds). ( d ) Parallel chains in a model 24 8, diameter with a five-fold disordered packing. (SAXS due to finite model size subtracted). We found that even for the most disordered model built (at the right density): the match in the si(s) curves [fig. 2(d)] at values >4.0 A-' is not improved; the first peak (intersegmental) comes consistently at 1.4 A-l or greater compared with the experimental position of 1.3 A-1; the first peak is too intense and too narrow in all but unreasonably small models.Thus for molten PE the WAXS data are not compatible with a model of extended chains in " bundles ". The bundle models described appear to be much more appro- priate to amorphized PE data,16 in which, as has been pointed the main peak is narrower and at z 1.5 A-l. In order to consider the scattering from " random " chains, long and hence more representative chains are required. The contacts of an intersegmental nature may be excluded by calculating only the scattering from each atom and its neighbours within the chain separated by up to n skeletal bonds (typically 10) away. The si(s) curve for a chain based on the statistical weights of Abe et al.ll is shown in fig.3(a). The calcu- lated and experimental curves are in general agreement at values >4.0 A-1. Small variations in the weights used gave improvement in some areas but there was no consistent improvement. Energy calculations for pentane"*ls and higher alkanes l9 indicate that some plurality of rotation states is required, since the rotational angles are dependent on the neighbouring rotation states. We built chains allowing for this dependence of the rotation angle. The si(s) curve for one such chain based on the three state model [fig. 3(b)] shows better agreement.R. LOVELL, G. R. MITCHELL AND A . H. WINDLE 51 We must now consider the packing arrangements which are a consequence of " random " chains. For such chains described above the weighted average sequence length of all-trans is 3-4 bonds and so the typical correlation within the chain is normally restricted to 4-5 bonds, which gives a region with an aspect ratio z 1. The statistical weights used of course do not exclude more anisotropic regular seg- ments, but the volume fraction of such segments is small.These may account for the CSO effect observed by other techniques. The globular nature of the average seg- ments suggests a possible representation of such a structure for PE melt to be a mass of segments with a typical intrasegmental correlation of 4-5 bonds, the centroids of the segments distributed as the centres of randomly close packed spheres. si ( S ) I I I I I I I 0 2 4 6 a 10 12 s / i-' FIG. 3,-Comparison of reduced intensity for molten PE with that calculated from " random " chains.(a) 'Random chain with three rotation states, 0", f120". (6) Random chains with dependence of rotation angle on neighbour states, based on 3 rotation states (a). (c) Representation of the complete scattering for an assemblage of random chains (b) based on uncorrelated segments distributed as close packed random spheres. ( d ) Experimental [as fig. 2(a)]. By assuming there is no orientational correlation of neighbouring segments, the scattering intensity is the product of the scattering from an " average " segment and the structure factor for an assemblage of random spheres. The structure factor may be obtained from models or by using the hard-sphere solution of the Percus-Yevick theory,20 the structure factor for which has been given analytically by Wright.21 The solution only requires packing density and radius of spheres.A single rigid inter- action between segments would be unrealistic and so the structure factor for a distri- bution of intersegmental distances centred at 5.5 A was used. This strictly only applies to a multiphase system but does seem a reasonable approach to a satisfactory repre- sentation. The si(s) curve for such a model with a packing density of PE melt (0.575 at 140 "C) using the intrasegmental scattering of fig. 3(b) is presented in fig. 3(c). This is in agreement with the experimental curve in terms of both the first peak posi- tion, height and width and for the rest of the scattering vector values. We conclude that a " random " chain structure, in which there is effectively zero correlation of segmental orientation, satisfactorily reproduces the observed scattering.52 WAXS STUDY OF POLYMERS MOLTEN PTFE (RIGID CHAINS) '' The high melting point (327-340 "C) of PTFE, its low solubility and the high viscosity of its melt are generally construed to be indicative of extraordinary stiffness of the PTFE chain."22 The crystalline conformation is a 13/6 helix with a pseudo-hexagonal chain pack- ing.23 There is a reversible first order transition at w0.5Tm, which is attributed to frequent changes of the sense of spiralization within each chain.24 The experimental si(s) curve is presented in fig.4(a), the raw data being in agree- ment with those of Kilian and J e n ~ k e 1 . ~ ~ The triangular nature of the peak at % 3.0 A-1 is indicative of long segments in a trans-type conformation [cf.PE fig. 2(b) and (c)]. The semi-empirical energy calculations and evaluation of solution properties for si (s) 0 2 4 6 5 A-' FIG. 4.-Comparison of reduced intensity for molten PTFE with that calculated for a correlated chain model. (a) Experimental at 350 "C. (b) Disordered helices arranged as five-fold packing in a 24 A diameter sphere. (SAXS due to finite model size subtracted). perfluoralkanes 26 predict a chain based on four states (t +, t -, g , g> which has a weighted- average aspect ratio of all-trans segments of % 3. Since both the WAXS and solution data point to anisotropic segments which are likely to pack parallel, we now consider the degree of order of such a structure.The packing density of a cylinder to enclose such an all-trans disordered segment is E 0.8. The two dimensional packing fraction of the cylinders of course depends on the " softness " assumed, but a reasonable value, taking the packing density of the melt as ~ 0 . 5 2 , would be 0.52/0.8 (=0.65). A com- parison of this with values derived from the computer simulations of disc packing2' indicates a non-crystalline structure. When we make similar calculations for PE, the two-dimensional packing fraction required is % 0.87, which could only be accounted for by some form of defective crystalline packing. In order to evaluate the possibility of parallel packing in the " melt " we have built a number of PTFE parallel chain models2* to the correct density with various types of chain packing and chain disorder.In contrast to PE the calculated si(s) curves consistently have the first peak in the correct position and of reasonable width although too intense; in addition a satisfactory agreemznt at >2.0 A-' is obtained [fig. 4mi.R. LOVELL, G. R . MITCHELL A N D A . H. WINDLE 53 Thus we conclude that a parallel chain model for molten PTFE is in broad agree- ment with the experimental scattering, although the exact nature of the packing and chain disorder is still to be evaluated. SEMI-RIGID CHAINS In general terms, most polymers give WAXS patterns closer to that of PE than of PTFE, although they usually have a broader main peak. In this section we sum- marize studies on a-PMMA and a-PS which are representative of solid glassy polymers and which have been used in many other investigations.However, both are chemically more complex than PE and PTFE, have substantial sidegroups and are therefore more difficult to analyse. Neither polymer is stereoregular but a-PS has at least 65% racemic dyads29 and a-PMMA z 80z3O and so we have considered their conformations in terms of syndiotactic chains (s-PMMA and s-PS). ATACTIC PMMA The scattering from unoriented and oriented a-PMMA is shown in fig. 5(a) and fig. 6(a). The peaks near s = 2.2 and 3.0 A-1 are purely intrasegmental whereas that near 1.0 A-l has an intersegmental component at w0.95 A-l and an intrasegmental one at z 1.2 A-l. We have ~ h o w n ~ ' ? ~ ~ that the persistent conformation is close to an all-trans chain.However, as shown in fig. 5(b) the calculated scattering only agrees si (5) 0 1 2 3 4 5 6 S/ii--' FIG. 5.-Comparison of reduced intensity for unoriented a-PMMA with that calculated for s-PMMA (a) Experimental. Intrasegmental contribution to first peak is marked at about 1.2 A-l [see fig. 6(a)l. (6) - , (ffff)5 with O1 = 110", 62 = 128". (b) - --, (tttf)5 with 8, = e2 = 116". (c) , (lo", lo", -lo", with = 110", O2 = 128"; best agreement. (c) ---, (lo", lo", with O1 = 110", O2 = 122"; this is the lowest energy conformation of Sundararajan -lo", and F10ry.~~54 WAXS STUDY OF POLYMERS 0 1 2 3 1 2 3 1 2 3 A-1 FIG. 6.-Comparison of reduced intensity (s weighted) for oriented a-PMMA with that calculated for s-PMMA. (a) Experimental. (b) (20", 20", 20", 20°)5 with = 110", 8, = 122".(c) (lo", lo", -lo", with = l l O o , 8, = 128" [i.e., same as fig. 5(c)]. with experiment if the backbone bond angle at the methylene groups (&) is larger than at the &-carbon atoms (&), so that the backbone is curved. The existence of two intra- segmental maxima (s = 2.2 and 3.0 A-') for PMMA is apparently a direct result of the unequal bond angles. The difference between the two angles (18") seems, in fact, to be larger than suggested by energy calculations ( A regular conformation of 16-20 backbone bonds long is needed to give the observed WAXS pattern, in good agreement with Yoon and Flory's prediction for s-PMMA.~~ WAXS analysis can not only provide direct experimental evidence that the per- sistent conformation is near to all-trans, but it can also give more detailed indications as to the conformation.In fig. 6, two-dimensional scattering maps are used to com- pare two distinct conformations near to all-trans: (lo", lo", -lo", -10") which is favoured by energy calculation^^^ and (20", 20", 20", 20") which gives a helix ( z 5/1). Both models predict the meridional maxima (s = 2.2 and 3.0 A-1); however, the non- equatorial component in the data at w 1.2 A-' suggests that the (lo", lo", -lo", -10') conformation is to be preferred. ATACTIC Ps The scattering from unoriented a-PS is shown in fig. 7(a). The peak at s = 0.7 A-l is purely intersegmental, that at 1.4 A-l is predominantly intrasegmental but with an intersegmental component at z 1.35 Hi-' and that at 3.1 A-l is intraseg- mental.4 9 5 Energy calculations for s-PS35*36 have shown that the lowest energy conformation is close to an all-trans chain, (tttt), whereas the next lowest energy is for a (ttgg) chain. As can be seen in fig. 7, (ttgg) gives scattering which is in very good agreement with experiment beyond s = 2.0 A-l. Chains containing a much higher proportion of trans bonds give much poorer agreement, as do chains containing random tt and gg dyads in equal amounts. The best agreement for (ttgg) is obtained for regular se- quences z 12 backbone bonds long. As with the study of a-PMMA we have not yet managed to model the packing of a-PS. Although the intrasegmental scattering dominates above s = 2.0 A-', it may be influenced by the intersegmental component so that some appreciation of the packing is needed before we can confidently propose a conformation.R.LOVELL, G . R . MITCHELL AND A . H . WINDLE 55 0 2 4 6 8 s/ i-' FIG. 7.-Comparison of reduced intensity for unoriented a-PS with that calculated for s-PS. perimental. (a) Ex- Position of intrasegmental contribution to main peak is marked.5 (b) (tttt)3 with Ol = 0, = 116". (c) (ttgg)3 with O1 = O2 = 116". SUMMARY AND CONCLUSIONS This work has demonstrated that WAXS is a sensitive means of investigating the structure of non-crystalline polymers. However, to obtain detailed results, the scattering data must be fully corrected, properly normalized and presented as the reduced intensity function, si(s). The interpretation of the intensity functions has been enhanced by their separation into intersegmental and intrasegmental com- ponents using information derived either from scattering from oriented specimens (glasses) or from the relative change in position of different maxima with changing temperature.The separation step enables us initially to consider isolated molecules : the calculated scattering for different model conformations is compared with the intrasegmental components of the reduced experimental scattering. For the ex- treme cases of non-parallel chains with short sequences on the one hand, and straight, parallel but randomly packed ones on the other, it has been possible to calculate complete scattering curves which take into account the intersegmental components. For the non-parallel chains, this has been achieved by representing the spatial distri- bution of the short sequence lengths by an arrangement derived from the random packing of spheres.The structural information obtained does not depend on con- formational energy calculations, although only models that are sterically reasonable have been considered. The conclusions of the study are: (I) It is possible to analyse WAXS to give direct structural information on non- crystalline polymers which complements that inferred from conformational energy calculations and measured perturbations from random chain dimensions. (2) The following conformations are proposed: (a) Molten PE: A modified three state chain with an average trans sequence length of 3-4 backbone bonds. (b) Mol- ten PT'E: Molecules which are disordered between the two most likely rotational56 WAXS STUDY OF POLYMERS states (ti-, t ') and more or less straight for at least 24 A.(c) a-PMMA: The model is in close agreement with that proposed by Sundararajan and F10ry~~ for s-PMMA on the basis of energy calculations. The backbone bond rotations are (lo", lo", -lo", -loo) which, when combined with the unequal backbone bond angles, leads to curved molecular segments. On the basis of the X-ray work, we suggest that the bond angles should differ by a larger amount than previously proposed (i.e., 1 lo", 128" as opposed to 1 lo", 122°).33 The conformation persists for sequences of 16-20 backbone bonds, which supports predictions made on the basis of measurements of the characteristic ratio and SANS.34 (d) a-PS: On the basis of the scattering at high values of s, where it is definitely intrasegmental, the conformation (ttgg) is preferred over (tttt) which is generally held to be of slightly lower energy.The sequence lengths are of the order of 12 backbone bonds. (3) We have found that the segmental aspect ratio is insufficient for parallelism in PE, sufficient in PTFE and on the borderline in a-PMMA and a-PS. (4) For molten PE, two extreme models for packing have been compared: (a) Disordered packing of straight parallel segments. In this case, the scattering from an isolated molecule is not consistent with experiment at high s; however, we examined this possible packing mode in case it modifies the scattering. Not only did it fail to reduce the significant discrepancy at high s, but also the main intersegmental peak consistently appeared at too high a value of s for the known density of the melt.(b) A packing model appropriate to the preferred " flexible " chains [cf. 2(a) above] has been developed. It is based on the product of the structure factor for random packed spheres and the scattering from a representative length of the chain, which for PE is of the same order as its diameter. The model accounts quantitatively for all the features in the WAXS pattern. (5) For molten PTFE, a model of parallel straight chain segments packed in a dis- ordered fashion is strongly supported by the WAXS data. (6) Thus for the two extremes, of non-parallel chains with short conformational sequences on the one hand and of straight parallel ones on the other, it has been possible to account for the complete WAXS pattern.However, for semi-rigid chains such as PMMA and PS, modelling of the packing has not yet been achieved. The provision of facilities by Prof. Honeycombe and funds by the S.R.C. made this work possible, and the authors are most grateful. U. W. Arndt and D. P. Riley, Phil. Trans. A, 1955,247,409. ' R. Adams, H. H. M. Balyuzi and R. E. Burge, J. Mater. Sci., 1978,13,391. B. E. Warren, X-ray Difrmtion (Addison-Wesley, 1969), p. 117. A. Colebrooke and A. H. Windle, J . Macromol. Sci.-Phys., 1976, B12, 373. R. Lovell and A. H. Windle, Polymer, 1976, 17,488. Yu. K. Ovchinnikov, G. S. Markova and V. A. Kargin, Polymer Sci. U.S.S.R., 1969,11,369. J. Schelten, D. G . H. Ballard, G. D. Wignall, G. W. Longman and W.Schmatz, Polynrer, 1976, 17, 751. D. G. H. Ballard, P. Cheshire, G. W. Longman and J. Schelten, Polymer, 1978, 19, 379. H. P. Mlug and L. E. Alexander, X-ray Difraction Procedures (John Wiley, N.Y. 1974), pp. 847- 850. lo A. H. Narten, J . Chem. Phys., 1979, 70, 299. A. Abe, R. L. Jernigan and P. J. Flory, J . Amer. Chem. Soc., 1966, 88, 631. Yu. K. Ovchinnikov, Ye. M. Antipov and G. S. Markova, Polymer Sci. U.S.S.R., 1975, 17, 2081. l3 I. Voigt-Martin and F. C . Mijlhoff, J. App. Phys., 1976, 47, 3942. l4 G. R. Mitchell, R. Lovell and A. H. Windle, Polymer, submitted for publication. l5 C . W. Stewart and C . A. von Frankenberg, J . Polymer Sci., Part A-2, 1967, 5, 623. l6 0. Yoda. I. Kuriyama and A. Odajima, J. Appl. Phys., 1978, 49, 5468.R . LOVELL, G . R . MICHELL AND A . H. WINDLE 57 l7 E. W. Fischer and M. Dettenmaier, J . Non-Cryst. Solids, 1978, 31, 181. l9 R. A. Scott and H. A. Scheraga, J. Chem. Phys., 1966,44, 3054. ’O E. Thiele, J . Chem. Phys., 1963, 39, 474. ’’ A. C. Wright, Disc. Ekraday Soc., 1970, 50, 11 1. ” P. J, Flory, Statistical Mechanics of Chain Molecules (Interscience, N.Y., 1969), p. 153. 23 C. W. Bunn and E. R. Howells, Nature, 1954, 174, 549. 24 P. Corradini and G. Guerra, Macromolecules, 1977, 10, 1410. 25 H. G. Kilian and E. Jenckel, 2. Elektrochem., 1959, 63, 308. 26 T. W. Bates and W. H. Stockmayer, Macromolecules, 1968, 1, 12, 17. ” G. Mason, J . Colloid Interface Sci., 1976, 56, 483. 28 G. R. Mitchell, R. Lovell and A. H. Windle, to be published. 29 K. Matsuzaki, T. Uryu, K. Osada and T. Kawamura, Macromolecules, 1972, 5, 816. 30 J. S. Higgins, G. Allen and P. N. Brier, Polymer, 1972, 13, 157. 31 R. Lovell and A. H. Windle, Diffraction Studies on Non-Crystalline Substances, ed. I. Hargit- tai and W. J. Orville-Thomas (Akademiai Kiado). 32 R. Lovell and A. H. Windle, Polymer, submitted. 33 P. R. Sundararajan and P. J. Flory, J . Amer. Chem. SOC., 1974,96, 5025. 34 D. Y. Yoon and P. J. Flory, Macromolecules, 1976, 9, 299. 35 D. Y. Yoon, P. R. Sundararajan and P. J. Flory, Macromolecules, 1975, 8, 776. 36 L. Beck, Dissertation (University of Ulm, 1976), p. 39. M. La1 and D. Spencer, Mol. Phys., 1971,22, 649.

ISSN:0301-7249    

DOI:10.1039/DC9796800046

出版商:RSC    

年代:1979

数据来源: RSC

摘要:

Meander Model of Condensed Polymers BY WOLFGANG R. PECHHOLD AND HANS P. GROSSMANN Abteilung Angewandte Physik, Universitat Ulm, Oberer Eselsberg, D 7900 Ulm, W. Germany Receiued 30th May, 1979 On the basis of the cluster-entropy hypothesis (CEH) the short-range order of polymer melts can be described by clusters of nearly parallel chain segments having similar conformations and holes in between these clusters, This bundle model allows the quantitative discussion of the melt transition as well as of the thermal properties of the melt. Within these bundles the molecules are assumed to fold back and forth in a one-dimensional statistical manner defined by the free energy of a tight chain fold. The bundle diameter, r, and the superfolding of bundles are derived by applying the CEH again.As primary blocks (5-30 nm, depending on the kind of polymer and molecular weight) coupled meander cubes are most probable. These are linked via their cube diagonals which serve as the axis of statistical rotation. Appropriate interaction between adjacent cubes may account for a liquid- crystalline transition, in which the paracrystalline but isotropic grains (0.3-3 ,urn) become nematically ordered. The diameters of these coarse grains (found in most amorphous polymers) depend on their grain boundary interaction. The shear deformations of the coupled cubes explain quantitativeIy the rubber-elastic compliances, JoeN, of uncrosslinked polymer melts. Additionally, the unfolding of layers of meander-cubes into shear bands describes the stress-strain relation of high molecular weight polymer melts.1. INTRODUCTION Our current knowledge about the level of order in amorphous polymers should stimulate further development of competing molecular models, by making their suppositions more precise in order to provide a bridge between their microscopic structure description and the understanding of macroscopic properties, thereby predicting effects which might be proved experimentally. No property or correlation should be excluded from consideration. The model theory presented here is based on the cluster-entropy hypothesis (CEH) and additionally assumes that clusters of nearly parallel chain segments and, therefore, chain bundles are energetically favoured even though some conformations have to be excluded. are improved by taking into account topological aspects and all kinds of energetic and entropic contributions to the Gibbs free energy of the equilibrium superstructure.A more detailed check and derivation of some of the results is given el~ewhere.~ For systems in or not far from equilibrium, entropy is defined in thermodynamic statistics by Boltzmann's formula or via the partition function, both of which have to be described in I'-space. For condensed phase systems a factorization into sub- space coordinates is generally impossible, but is frequently assumed without proof. A projection into a subspace must include all other states as well as those of special interest. To take this into account we use the following hypothesis (CEH, which has still to be proved) : clustering in subspace (e.g., conformational, orientational or describing deformation) of m equivalent elements (e.g., segments, segment-lines or In this paper earlierW .R. PECHHOLD AND H. P . GROSSMANN 59 layers of molecules), each havingf accessible states (including vibrational states), does not reduce entropy as long as rn <f, i.e., multiple occupation in the p-space can be neglected (or the dimensionality of the r-space is unaltered). There are several applications of this hypothesis in various fields, for example phase transitions in paraffins, membranes, liquid crystals, the melt transition of close-packed crystals, and particularly amorphous structures and related properties of condensed polymers and biopolymers. 2. SHORT-RANGE ORDER CONCEPT Whereas perfect polymer crystals consist of parallel chains of only one low energy conformation, high temperature phases exist (e.g.in PE and t-1,4-PBD) that have accumulated more than half of the total entropy and enthalpy of melting, but which still appear crystalline in WAXS. Transition data and thermal properties as well as their pressure dependences can be quantitatively described, assuming a mixture of ener- getically favoured chain conformations. On the basis of CEH this is realized in conformational c l u ~ t e r s . ~ - ~ Taking a rotator (C2H4 in PE) as one statistical element in the chain direction, a one-dimensional cooperativity leads to the chain segment (x 4 rotators), the mean distance between conformational changes [indicated as slight bends in fig. l(a)].These segments (as new elements) now cluster laterally, retaining the full conformational entropy ( z R In 5 per mol rotator) even though the elements in one cluster show the same conformation at any one time. The actual conformation of an element changes with time and cluster position, i.e., by fluctuations. The subsequent melting is described by the cooperative formation of holes accumulated in the centres of numerous dislocations nearly parallel to the chain direc- tions. The quasi-crystalline clusters (of 10-30 segments) are thereby broken into short fibrils [fig. I@)]. The additional entropy of this transition is caused by the variety of differently shaped holes and their distribution. The chain segment is reduced to x2 rotators (in PE melt), representing a length, s, which compares with the chain distance, d.The assumption s z d will therefore be made for melts of other flexible chain polymers. The Gibbs free energy of a chain segment referred to the melting temperature amounts to g , = h,( 1 - T/Tm), h, ( z 4 kcal mol-I for PE) being the heat of melting per segment. 3. ONE-DIMENSIONAL FOLD-STATISTICS I N BUNDLES OF MOLECULES This concept of short-range order favours chain parallelism across longer distances, because mutual penetration of neighbouring chains should give rise to a greater increase in energy (larger voids or chain deformations) than the respective gain of entropy. In addition to segment clusters and hole distribution, any types of defects which leave the chains nearly parallel and which do not require too much energy must be con- sidered.It has been emphasized already in early papers that torsional defects, jogs and folds may be incorporated into a bundle of molecules. Their concentrations can be esti- mated by their Boltzmann factors. Small clusters formed by these defects (e.g., a pair of folds) should not reduce their probability according to CEH. With regard to the overall geometry of a (labelied) macromolecule in the melt, tight chain folds are most important. Let n be the number of chain atoms (or units after which folding can take place) and Agf = gf - nfgs * the free energy of formation * Forming a tight chain fold, e.g., in PE-melt, the main energetic contribution gf deduced from semi empirical potential calculations is 3.9 or 3.2 kcal mol-', depending on whether one starts with a60 MEANDER MODEL of a tight chain fold, the free energy 6G of one-dimensional folding of a long enough chain is 6G = npfAgf - kWpflnpf + (1 - p f ) M -pf)], pf = e-A.g‘/kT/(l + e-Agf/kr) (6G/RT),q = nln(1 - pf) N” -npf per molecule.(3.1) (3.2) (3.3) pf denoting the fold probability. In equilibrium pf becomes and ( C ) FIG. 2.-(u) Labelled folded chain in a straight melt bundle. (6) Space-filling superfolding into 9-fold meander cubes. (c) Lamella, superfolded, of melt-crystallized bundles. To calculate, e.g., the mean square of the radius of gyration {z;) of a (labelled) folded chain within a narrow straight bundle of molecules [fig. 2(a)] one has to consider the mean square of distances zl between pairs i andj if k = j’ - i’ andfk denotes the number of chain folds between units i’ andj’, Z i F Zjt = (- l)fk and its conformational average can be written trans-chain or tgtg-conformations.But within a fold nf % 1 segment (2 rotators) will be effectively immobilized (similar to the crystallized segment). Their free energy of melting must therefore be subtracted. So Agr = gf - g, = gf - h, + h,T/T, becomes approximately proportional to T.FIG. 1.-(a) Stuart model for the high-temperature phase in PE. (6) Stuart model for the PE melt containing holes between small clusters of segments. [To face page 60FIG. 5.--(a) Equilibrium topology of 9-fold coupled meander cubes linked via their cube diagonals. (b) Possible topology of a cube diagonal linkage. [To face page 61W. R .PECHHOLD AND H . P . GROSSMANN 61 Carrying out the twofold sum one arrives at l 2 l2 2Pf 2Pf (3.6) ( z 5 ) x - ( j - i ) - -2 [I - exp[ - 2 p f ( j -i)]. The mean square of the radius of gyration (zf) is related to (z&) by <z:> = (n + w2 2 2 <Zi.j> (3-7) O < i < j < n and becomes (2:) N" { 1 - [ 1 - 2 (1 - i)]} = a(npf). (3.8) 6Pf 2nPf 2npf 2npf 6Pf This approximate formula is valid for n @ 1 and np, > 1. A representative folded chain labels a piece of a bundle with the length A* x ( I ~ ( z : ) ) ' / ~ . (3.9) The chain has shortened by folding to A*/nl of its extended length nl and its effective cross-section contains on average q = nl/A* = (npf/4a)'I2 (3.10) From eqn (3.6) the scattering behaviour of a folded labelled chain in parallel stems.the straight bundle can also be easily derived. 4. SUPERFOLDING OF BUNDLES So far two main questions remain unanswered: (i) how large is the diameter, r, of the bundle and (ii) how far does it extend in one direction. Both problems can be solved simultaneously by using equilibrium statistics and essentially CEH, if one adopts the concept of superfolding of bundles, i.e. the meander model. The basic idea is to gain orientational entropy if bundles change their directions. The most simple topology in which a melt-bundle can tightly superfold closely to fill 3-dimensional space is an arrangement of meander cubes [fig. 2(b)]. To enable these cubes to occupy all 3 accessible respective sites, the meandering bundle is assumed to link them uia their cube diagonals serving as axis of statistical rotation.Besides superfolding and rotation a statistical description must also take into account the possible shear deformations of a meander cube. For the quantitative treatment statistical elements must be defined which are adapted to control superfolding, rotation and deformation, respectively. These elements should be as small as possible but must possess enough accessible states that their clustering does not violate CEH. As such elements we have chosen (fig. 3): (i) the segment-line of the bundle containing rld segments of length s (for superfolding), (ii) the segment-line across the meander cube consisting of (r + x)/d segments (for cube rotation) and (iii) molecule-layers of the cube which are composed of (r + x ) ~ / sd segments (for shear deformation).The excess free energy per segment due to the superfolding of a bundle into a meander cube is where Agfold refers to a half meander layer of r(r + x)/ds segments and is determined by the free energies of formation of one superfold (Ag:), 2r/d chain bends62 MEANDER MODEL the orientational entropy of segment-line clusters which differ in chain direction (fig. 3) Ag; ban in principle be deduced from potential calculations and probably will be about twice the value of Agf (the free energy of formation of a tight chain fold, which actually is the core of the superfold at least for a high degree of polymerization). According to similar immobilization effects * Ag: will also be approximately propor- FIG. 3.-Representative superstructural unit and definition of the statistical elements.tional to the absolute temperature. The contributions Ag,,, and Agdef in eqn (4.1) will be discussed in the next two sections. Depending on the kind of polymer, AgJ kT has values in the range -In 3 to -0.13, whereas 2 Agdef/kT w -9 for the un- deformed melt. By inserting (4.2) into eqn (4.1) Ag,/kT can be written d s r (1 ++o +:) +;( s Agdef/kT l+x,r)+*]. (4.3) This excess Gibbs free energy per segment in a meander must become a minimum if the independent variables r/s and x / r t take up their (most probable) equilibrium values. From follow the equations -f From topological arguments in ref. (3). x / r has been fixed at a value of 2 but coupling of adjacent cubes (fig. 4) allows some fluctuation of x / r around this mean value.W .R . PECHHOLD A N D H . P . GROSSMANN 63 2Agdef/kT] + :iF A8rot and (1 + $ln( 1 + 5) + In :-= $2$ + + xlr kT ' the difference of which yields -= r A g W s (1 + x/r)ln(l + r/x)' In table 1 the mean values of x/r and r/s are calculated for possible energies of superfold formation using in addition the approximations TABLE 1 .-x/r AND r/s VALUES FOR DIFFERENT Ag; gi/kT VALUES < X I 0 1.36 1.76 2.01 2.17 2.29 2.39 <rls> 3.85 4.83 5.76 6.66 7.55 8.44 According to semi-empirical potential calculations of tight folds by Grossmann and Becks who found 3.2-3.9 kcal mol-1 for PE and 4.2 kcal rno1-l for PS, we assume the corresponding free energies of superfolding to lie in the ranges 6 < Agr/RT < 8 and 7 < Ag;/kT < 9 for PE and PS, respectively.The mean bundle diameters (in high molecular weight melts) should probably be 25-30 for PE (d z 4.8 A) and 50-60 A for PS (d x 8.5 A). The equilibrium Gibbs free energy of superfolding becomes [using eqn (4.3) and (4.5) 1 Fluctuations in x/r and r/s should be allowed for but should not violate topology. Relying on CEH these fluctuations can be described by reversible coupling of any two adjacent cubes in suitable relative orientation (x/r-fluctuation, fig. 4) and by changing the contour of the bundle cross-section without varying its area. The latter cooperative change corresponds to a fluctuation of r/s which is the number of chains in a segment-line, but holds the numbers of chains and stems within the bundle con- stant. FIG. 4.- -fluctuation within coupled meander cubes.64 MEANDER MODEL 2 0 - 10 The proposed topology [fig.5(a)] of %fold meander cubes, i.e. ( x l r ) = 2, is in accordance with equilibrium statistics (table 1). Therefore the concept of a statistical rotation of these cubes about one of their cube diagonals which enables the primary blocks to compose an isotropic grain is also justified. Below a certain temperature however, (if the melt can reach it) the grains should become anisotropic: this will be discussed briefly in section 5. Before going on to this point we still have to answer the question whether or not the bundle diameter, r, depends on chain molecular weight. Because x / r = 2 for topological reasons, r/s will change proportionally to the free energy of a superfold according to eqn (4.6).The idea now is that pairs of chain ends (from different chains) will be substituted for the tight chain folds within the cores of the superfolds, thereby strongly reducing Agi. Assuming the lower limit (probably valid for PS) for r in the case of finite chains I - I I I I I I l l l I I l 1 1 1 1 l 1 I I I I 1 1 1 1 r(r + x) r = r , (1 - w), with w = ~ nld (4.8) being the probability for tight fold substitution (approximately equal to the ' geo- metrical concentration c ' of pairs of chain ends per half meander layer) one finds "[(l +i)1'2- l]whereA=- nld c"2 3r$' (4.9) From this formula the molecular weight dependence of the mean side length 3r(M) of a meander cube in PE and PS melts is obtained and shown in fig. 6. The curve lo3 104 105 M 106 FIG.6.-Molecular weight dependence of the meander size 3r and the radius of gyration R, for PE- and PS-melt. (a) R,PE, (b) R,PS, (c) 3rps, (d) 3rPE. I'M Rf x 4(3r)' + ~ - R TIn pr/ M, 11%. dlA r,/A kcal mol-I PS 52 1 . 2 8 . 7 60 4 . 7 PE 14 1 . 2 4 . 8 25 4 . 4W. R. PECHHOLD A N D H . P . GROSSMANN 65 for PS fits (without further assumptions) the SANS maxima from ref. (9), which we interpret as the scattering of a paracrystalline cubic superlattice labelled with deuter- ated chain ends preferentially gathered in the superfold lines. Finally the radius of gyration R,(M) of completely labelled chains in the coupled- cube meander topology must be estimated. From eqn (3.9) we know the mean length, A*, of a straight bundle that is labelled by the tightly folded chain under consideration.This piece now labels its part of the meander superstructure [e.g. the upper bold path in fig. 5(a)] which in the first approx- imation can be considered as a tetragonal block with dimensions 3r x 3r x A. A is connected with A* by a %fold reduction in length and subsequent stacking of meander layers along cube diagonals (4.10) Using the R,-formula for a tetragonal block the squared radius of gyration of a labelled chain within the superstructure becomes A E (A*/9r)rd%= A*d?19 = (~;)~'~2/3. (4.1 1) with u(npf) taken from eqn (3.8). The molecular weight dependence of R, for PE and PS according to eqn (4.1 1) is plotted in fig. 6 together with some SANS data from various authors.10 The scattering behaviour of labelled chains can also be treated approximately in closed form or by computer simulation.This will be discussed in a later paper,ll together with model considerations for paraffin melts, the superfold cores of which are made up by pairs of chain ends only. At the highest molecular weights the q21(q) against q plot should exhibit a maximum near 5/Rg according to the meander model. Maxima have so far been found in the case of strong clustering, which we interpret on a similar basis : if N labelled molecules are jointly packed within a bundle (on thermodynamic or kinetic grounds) they appear as if they were one molecule having a radius of gyration RgaPp FS R, dz Such a relation has been found to be true by Schelten et all2 5. ORIENTATION OF SUPERSTRUCTURAL UNITS-LIQUID CRYSTALLINE POLYMERS To describe the mutual arrangement of adjacent meander cubes and the overall orientation-as far as it exists-within a grain we write down the Gibbs free energy Grot of hindered statistical rotation assuming 3 equivalent orientational positions (k = 1,2,3) around the linking diagonal of each cube and taking into account the various boundary free energies G:, between adjacent cubes (pair approximation).For a system of N cubes (x/r = 2) composed of m = (r + x)~/s d segment-lines con- taining (r + x)/d segments each, GJN becomes c:, denote pair concentrations along direction i of adjacent cubes in k and I orient- ation and p:, their a priori probabilities. For simplicity it has been assumed that rotational symmetry exists around a possible " director " (z-axis), z 3 1/3 being the concentration of cubes with fold directions parallel to it.The CEH factor rn in front66 MEANDER MODEL of the entropy term accounts for the fact that segment-lines are the statistical elements and clustering into cubes does not reduce orientational entropy (only if rn < f). This fact must be kept in mind if one deals, for example, with magnetic birefringence or with depolarized light scattering in the meander model. Introducing GLl/m = gLl, the free energy per segment on the boundary between cubes, the rotational free energy per segment-line Ag,,, = Gr,,/mN can be written Agrot/kT = 241 [gL/kT + In (cil/~L)I i k l + zlnz + (1 - z)ln(l - z) - (1 - z)ln2 + CA'(2 - cfz - c& - Ct,) + Cp'(1 - z - cfex - c;, - CiX - CiY - 24,). (5.2) The last two terms represent the 6 necessary relations among the ckI, the A', p i being Lagrange multipliers.The subsequent minimization referred to ckl and to the order parameter q = 22 - 1 is carried out el~ewhere.~ We simplify further by assuming gmax being the mean excess free energy per segment of a crossed-chain boundary which will be calculated in the near future using semiempirical potentials. (The shaded areas in fig. 7 indicate such energetically unfavoured crossing of chains.) For this simplified case the results of the minimization are a + 2 aAgrot'kT = -2ln2 + 31n(l + E) + 21n 3 + 31n - = 0 (5.4) ar 1 + r l a - From the equilibrium condition (5.4), plotted in fig. 8, the equilibrium values of are determined by the points of intersection with the zero line for any temperature: for T > T,,, q = - 1/3 (Pz = 0), for T < T,,, q increases from 1/3 to 1 at low temper- atures.For T = T,, q = &1/3 are equally probable and the system undergoes a phase transition which corresponds to the nematic/isotropic transition of low molecular weight liquid crystals. The transition temperature T,, follows according to point symmetry of the S-shaped curves Therefore isotropic polymer melts with an excess free energy g,,,., (e.g. of 0.8 kcal mol-l) in the crossed-chain boundaries should change over to a nematic phase at T,, z 400 K, as long as neither crystallization nor glass transition occur at still higher temperatures. Associated with this hypothetical liquid-crystalline transition would be a latent heat, 6AhrOt, which can be calculated from Agr,,/kT as a function of T/T, [plotted in fig.9(a)] viaW. R . PECHHOLD AND H. P. GROSSMANN 67 Z b X I / FIG. 7.-Table of all possible boundaries between adjacent cubes indicating crossed- chain areas. the enthalpy function [fig. 9(b)], which in the example chosen exhibits a discontinuity 6AhrOt x 0.25 kT, z 0.2 kcal mo1-I of segment lines. This would correspond to 13 cal mo1-1 segment or 0.25 ca1g-I in the case of PE and to 10 cal mol-I segment or 0.03 cal g-I for PS, respectively. Notwithstanding these small transition heats, the mere occurrence of a liquid-crystalline transition (to be observed optically or by electron diffraction) in one or other flexible chain polymers will strongly support the meander concept. We believe that there is no fundamental difference in melt- superstructures between stiffer chain polymers (which more often show a liquid crystalline phase) and flexible chain polymers.So far polydiethylsiloxane (PDES),13*14 which is definitely a flexible chain polymer (T, x 140 K, T, x 268 K), has been shown to exhibit a liquid crystalline phase up to 25 K above T,. 6. SHEAR FLUCTUATION AND DEFORMATION OF THE MEANDER SUPERSTRUCTURE The coupled meander cubes may perform shear fluctuations which should be re- stricted to a displacement between adjacent layers of molecules such that their close packing or superfolding is not violated. This topological assumption compares with that of entanglements in the coil model. The two main intra-meander shear modes are visualized in fig.10, for a single cube: the cross-sectional shear yZl (or y12) and68 MEANDER MODEL the intra-bundle shear ygtra (or yztra). They give rise to an ideal paracrystalline appearance of a polymer melt (fig. 1 1 ) . The mutual glide motions between ad- jacent molecule-layers are probably performed by at least one dislocation per layer, the centres of which can be considered as a linear array of segmental holes of energy &h that are much smaller than a missing segment. -1 0 1 FIG. 8.-Rotational free energy derivative to determine equilibrium values of the order parameter. T/T, = (a) 0.8, (b) 0.85, (c) 0.9, (d) 0.95, (e) l.O,( f ) l . l , (g) 1.2. The anisotropic mechanical compliances corresponding to these fluctuations are essentially of entropic nature and compare to the rubber-elasticity JoeN of the topological " network " of coupled meander cubes.A straightforward statistical derivation on the basis of CEH leads to feat- cross-sectional and intra-bundle shear, respectively, x NN l/qT, accord- ing to the above mentioned restriction. These anisotropic compliances (which are hardly influenced by a low degree of chemical crosslinking) can be measured at higher frequencies and small amplitudes, for example on a stretched rubber sheet (fig. 15).16 For larger extension ratios, A, one measures 2 5 3 1 3 1 (or 2&32) w 41 w 4.8 x m2 N-l and S12,, (or S2121) w JI w 3.5 x m2 N-l, the ratio S12,2/S3131 beingW . 1 0.8 0.6 0.4 0.2 PZ 0 R . PECHHOLD A N D H . P . GROSSMANN 69 . 2 .4 .6 .8 .o -0.2 I1I-. 2 0.5 I I i 1.5 5 I lb T / 7, A9mt - kT 0.5 1 1.5 T I Tu FIG. 9.-(a) Equilibrium free energy Ag,,,/kT of hindered cube rotation and the order parameter t] as function of the reduced temperature T/T,.(6) Rotational enthalpy Ahro,/kTu as function of the reduced temperature T/T,. 1.46 experimentally compared with 1.50 from eqn (6.1) and (6.2). For isotropic samples the rubber-elastic compliance JoeN can be calculated using Reuss’ averaging (* * * a) (Agrot/kT)leq; (-) Pz = &(32 - 1) = t ( 3 q + 1). These compliances are therefore a sensitive measure for the size of superstructural units in polymer melts, to be characterized by 3r/d. There are two more possibilities of obtaining information on 3rtd from molecular motion:17 (i) an analysis of the activation curve of the glass-process, the freezing-in of these shear fluctuations. In the meander model thef,,,.(103/T) dependence of the main relaxation measured by G’, G” can be described by the first part of which is the jump frequency of a segment, the latter the probability of finding at least one representative small hole (of energy ch) within each of 3 ( 3 r / ~ i ) ~ d/s70 MEANDER MODEL 3 1 2 '1212 } = [?'('') 1' cross -sectional shear max 2kT s2121 intra intra FIG. 10.-The two main intra-meander shear modes demonstrated on a single meander cube. segment-lines, i. e. of having available enough dislocations to enable shear fluctua- tions: (ii) whereas the G"-maximum is due to the shortest shear fluctuations (or Rouse modes) the displacements of whole layers of molecules contribute most of all to the glass relaxation measured in the compliance J ( u , T).The rubber-elastic shear displacements ( y I 2 or y3J of a coupled meander cube require about 2 ( 3 r / ~ I ) ~ jumps of segments in one direction, i.e. 4(3r/d)4 jumps during diffusional motion. The frequency shift between the G"- and J"-maxima should therefore depend on the size of superstructural units according to f $klf;l;Hx. = 4(3r144. (6.5) In table 2 the 4 experimental and theoretical methads having been used so far for getting information on 3r/d are listed, together with their application to polyiso- butylene (PIB).W . R . PECHHOLD A N D H . P . GROSSMANN 71 FIG. 1 1 .-Paracrystalline meander topology of the polymer melt induced by shear fluctuations. TABLE 2.-uPPER AND LOWER LIMITS FOR FIVE DIFFERENT DETERMINATIONS OF 3r/d lower upper method for determination of 3r/d limit limit meander theory [eqn (4.6)] 10 30 JoeN = 3.0 - - * 3.7 x 21 activation diagram analysis [eqn (6.4)] 15 21 ratio of relaxation frequencies [eqn (6.5)] 18 24 label experiments 15 24 m2 N-I [eqn (6.3)] 19 The theoretical uncertainty arises from the lack of a reliable tight fold energy, which we are about to calculate atomistically.With the aid of dipolar label molecules which deposit (energetically favoured) into the meander folds at low label concentration the number of these folds and hence 3r/d has been resolved from an anomalous ccn- centration dependence of the dielectric relaxation strength.'* In table 3 similar analyses for various polymers are listed which so far are not as complete as for PIB.Additionally included in this table have been lower limiting values for the enthalpy relaxation calculated according to which can be compared with experimental data already known. Beyond these intra-meander shear fluctuations large scale deformations become possible by inter-bundle displacements, i.e., by unfolding of suitably arranged meander cubes.* To this purpose whole layers of meander cubes must cooperate and form one shear band. An intermediate type of superstructure being thereby produced is indicated by the file of sheared cubes in fig. 12. Fig 13(a) shows the stress-strain behaviour 031 (ygter) of an already developed shearband, the probability P(031) of which depends strongly on the rotational free energy per segment-line Ag,,, [fig.9(a)] that is lost in the shear band. In fig. 13(b) the fully drawn curves give the true shear- *A still further reversible extension can be achieved by expanding the tightly folded chains within the bundle, a process which probably has taken place in ultra-highly-drawn fibres.72 MEANDER MODEL FIG. 12.-Shear band type of deformation shown on a single file of meander cubes. stress-shear-strain dependence for superimposed intra- and inter-bundle shear ~ 3 1 = ygtra + /?ygter. Fromt his molecular deformation theory [completely derived in ref. (3)] the stress-strain behaviour of any type of deformation can be evaluated if the meander size and the rotational free energy AgJkT are known for a given polymer. In fig. 13(d) the theoretical curves for uniaxial tension are compared with measure- ments on high molecular weight PE.Fig 13(c) shows the dependence of the mean distance d, between shear bands (fig. 14) on the extension ratio A. dz can probably be determined by measuring the spacings of the line-by-line structure revealed after oxygen ion etching by electron microscopy of stretched p01ymers.l~ These line structures together with an orientational analysis of the anisotropy of mechanical compliances (fig. 15) support strongly the shear band concept. 7. FINAL REMARKS It is beyond the scope of this paper, which should present the current status of the meander model, to go into all the details of explaining magnetic birefringence, de- polarized light scattering or neutron scattering results on the basis of the meander model.It may be noted, however, that further development of the meander model is in progress in order quantitatively to account for important properties like the dependence of rubber-elasticity on chemical crosslinking or the rheological behaviour of polymer melts. Shear bands, for example, may serve as nuclei for the high tem- perature phase transition and hence also for extended chain crystallization. Finally we will try to answer the often-repeated question whether or not theW. R. PECHHOLD AND H . P . GROSSMANN 73 N 5 0.6 b 0.4 0.2 cr) Ocr) 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x p (O31) FIG. 13.-Deformation behaviour of a high molecular PE melt (a) anisotropic stress-strain curves of intra- and inter-bundle shear. x / r = 2, s = d = 4.8 8, r/d = 5.~2::: = 1/.t/ 3, ~2::: = 9, T = 433 K. (i) 031 (Y:!'~'); (ii) aZ1 (yZ1); (iii) a31 (y:?'"'); (b) serial superposition of intra- and inter- bundle shear (fully drawn curves) and concentration of shear bands (dashed curves): -Agr,t/kT = (i) 1.0, (ii) 0.5, (iii) 0.13; (c) mean distance of shear bands as a function of extension ratio [conditions as for (a)] (i) 1 .O, (ii) 0.5, (iii) 0.13 ; (d) uniaxial stress-strain curves for PE melt (the dashed curve has been measured on high molecular weight PE) (i) 1.0, (ii) 0.5, (iii) 0.13. - meander model, which must be considered as a topological framework wherein the macromolecules are free to reptate, may be valid in concentrated or even in semi- dilute solutions.Our proof for its validity comes from a quantitative understanding of the conversion factors of P, + Py crosslinking reaction (in a 1 : 1 mixture) carried out by Vo11mert20 and coworkers: assuming that the bundle diameter, r, and the bundle superfolding are independent of dilution (below 80% of polymer content) a decreasing amount of back-and-forth folded chains will be incorporated within the bundles with decreasing polymer content. To find the probability, P, of adjacent chain stems which may undergo a crosslinking reaction one easily calculates across one lateral direction where c is the polymer concentration and 4 is the mean number of parallel stems per macromolecule, as given in eqn (3.10). This result would only be true in a parallel array of different bundles.Taking into account superfolding, the stems of the same macromolecule (of sufficient high M ) become adjacent to each other and must therefore be excluded from P'. Only every third contact in one lateral direction occurs74 MEANDER MODEL FIG. 14.--Geometry of deformation for uniaxial extension at different draw ratios. FIG. 15.-Five anisotropic compliances of a stretched rubber sheet measured at small amplitudes as a function of the extension ratio at 100 “C, 10 Hz.” +, 41; 0, JL ; x , DI ; 0, DII ; A, JT. 2S313, =2S3232 = 4.8 x m2 N-I, 2S2121 = 2S1z12 = 7.0 x m2 N-l, all other Srlcr ~ 0 .W. R . PECHHOLD A N D H. P. GROSSMANN 75 between different bundles and contributes to the crosslinking probability. Therefore The factor 1/2 accounts for the probability of finding two adjacent molecules with different reactive groups (in a 1 : 1 mixture).The mean conversion factor now is P divided by the crosslinking probability of statistically distributed reactive groups, i.e. FIG. 16.- 2b i o 60 80 160 polymer concentration / w t O/O -Experimental conversion data on cis-1 ,Cpolybutadiene (0) and by Vollmert 2o explained by the meander model (theoretical polybutylacrylate (0) curves). 1 by .\/c. In fig. 16 two theoretical curves according to the formula 1 2d 1 conversion = [G - 3 - have been fitted to Vollmert’s results. The fit parameters used are q = 3,3, r/d = 5 for cis-l,4polybutadiene and q = 2,8, r/d = 3 for polybutylacrylate, which are reason- able values. For high polybutadiene content the measurements indicate an increase of the bundle diameter r/d from 5 to 8 which corresponds to the value determined by our analysis (table 3).The observable deviation of the measurements from the theoretical curves at low polymer content is certainly due to a premonitory mutual penetration of different molecules and probably defines the concentration region in which the coil model will be responsible. Support by the Deutsche Forschungsgemeinschaft and by the Fonds der Chemie is gratefully acknowledged. We thank our colleagues for helpful discussions and Mrs. Schiffner for preparing the manuscript.76 MEANDER MODEL TABLE 3.-vALUES OF VARIOUS PARAMETERS FOR DIFFERENT POLYMERS polymer (amorphous) n c c 8 3 h .- W h a 3 e x h a 8 B 3 - ~ ~ ~~~ ~mOllOmer/g mol-' 254 104 62 192 86 86 56 68 54 28 74 pjyDh./g cm-3 1.26 1.05 1.39 1.34 1.19 1.22 0.92 0.91 0.94 0.86 0.97 LmOnom,rl+ 10.5 2.22 2.2 10.8 2.1 2.1 2.3 4.05 4.3 2.4 2.8 meanchaindistance, d/A 5.7 8.7 5.8 4.7 7.6 7.5 6.7 5.5 4.7 4.8 6.7 bl Agflkcal mol-l r/d 8 t (r + x ) / A a% 8 7 180 6.4 5 75 .-.2 TglK 412 363 351 343 302 278 200 203 167 150 2 (r + x ) / A 170 150 290 140 150 140 135 135 115 150 3 ch/kcalmol-' 1.60 1.22 1.51 1.32 1.03 0.96 0.63 0.72 0.60 0.55 x (f= 1 0 - 3 ~ 4 g g, Qy/kcalmol-' 7.5 7.5 11.3 7.5 10 6.2 9.7 7.0 6.3 3 es vol10'3H~ 4 4 4 4 4 4 5 0 2 1 0 4 ~~ 5.2 4.1 3.3 3.3 2.4 1.4 0.50 4.1 43 3 335 373 298 333 353 433 298 5' 4 ' 4 f 3.5d 2 b 1.2' 0.6b 3.5g ?, c, AH:hcor/cal g-l 0.63 1.0 0.73 0.69 0.91 0.90 0.84 1.29 1.36 0.50 9.2 AH:XPlcal g-l O.gh 0.8 ' 0.8 h*k B 5 monomers per 1 2 2 1 2 2 2 1 1 2 2 < segment a - k See ref.(15). W. Pechhold, Polymer Sci., C, Polymer Symp., 1971,32, 123; W. Pechhold and S. Blasenbrey, Kaut., Gummi, Kunstst., 1972, 25, 195; W. Pechhold and S . Blasenbrey, Angew. Makromol. Chem., 1972, 22, 3. W. Pechhold, M. E. T. Hauber and E. Liska, Kolloid-Z., 1973, 251, 818. W. Pechhold, Colloid Polymer Sci., in press. W. Pechhold, E. Liska, H. P. Grossmann and P. C. Hagele, Pure Appl. Chem., 1976,46,127. G. Bautz, Thesis (Ulm, 1978). D. Eggert, H. P. Grossmann and W. Pechhold, in preparation. ' H. P. Grossmann, Thesis (Ulm, 1977). * L. Beck, Thesis (Ulm, 1976). H. Benoit, D. Decker, R. Duplessix, C. Picot, P. Rempp, J. P. Cotton, B. Farnoux, G. Jannink and R. Ober, J . Polymer Sci., Polymer Phys. Ed., 1976, 14, 21 19. lo J. P. Cotton, D. Decker, H. Benoit, B. Farnoux, J. Higgins, G. Jannink, R. Ober, C. Picot and J. de Cloizeaux, Macromolecules, 1974, 7, 863; E. W. Fischer and G. Lieser, J . Polymer Sci., Polymer Letters Ed., 1975,13, 39; J. Schelten, D. G. H. Ballard, G. D. Wignall, G. Longman and W. Schmatz, Polymer, 1976,17,751. R. Genannt, W. Pechhold and H. P. Grossmann, Colloid Polymer Sci., 1977, 255, 285; R. Genannt, W. Pechhold and E. Sautter, in preparation. l2 J. Schelten, G. D. Wignall, D. G. H. Ballard and G. W. Longman, Polymer, 1977, 18, 11 11. l3 C. L. Beatty, J. M. Pochan, M. F. Froix and D. F. Hinman, Macromolecules, 1975, 8, 547. l4 J. M. Pochan, D. F. Hinman and M. F. Froix, Macromolecules, 1976,9,611. Is (a) S. Onogi, T. Masuda and K. Kitagawa, Macrumolecules, 1970,3,109; (b) A. Zosel, Kulloid- Z., 1971,246,657; J. R. Richards, K. Ninomiya and J. D. Ferry, J . Phys. Chem., 1963,67,323; (e) Y . Oyanagi and J. D. Ferry, J. Colloid Sci., 1966,21,547; (f) K. Fujino, K. Senshu and H. Kawai, Rep. Prugr. Polymer Phys. (Japan), 1962,5, 107; (8) N. R. Langley and J. D. Ferry,W. R . PECHHOLD AND H. P . GROSSMANN 77 Macromolecules, 1968, 1, 353; (h) S. E. B. Petrie, in Physical Structure of the Amorphous State, ed. G. Allen and S. E. B. Petrie (Dekker, N.Y., 1974); ( i ) K. H. Illers, Makromol. Chem., 1969, 127, 1 ; (k) K. H. Illers, Kolloid-Z., 1971, 245, 393. l6 W. Konig, Thesis (Ulm, 1979). l7 W. Pechhold, E. Sautter, W. V. Soden, B. Stoll and H. P. Grossman, Makromol. Chem. Suppl. 1979, 3, 247. G. Winkler, B. Stoll, E. Sautter and W. Pechhold, Colloid Polymer Sci., in press. l9 K. P. GroDkurth, Colloid Polymer Sci., 1977, 255, 120. 2o B. Vollmert, Pure Appl. Chem., 1975, 43, 183.

ISSN:0301-7249    

DOI:10.1039/DC9796800058

出版商:RSC    

年代:1979

数据来源: RSC

摘要:

Heterogeneity in Polymers as Studied by Nuclear Magnetic Resonance BY VINCENT J. MCBRIERTY Department of Pure and Applied Physics, Trinity College, Dublin 2, Ireland Received 1st May, 1979 A knowledge of the extent to which polymers are heterogeneous is fundamental to a full under- standing of their properties. This paper discusses the role of n.m.r. in probing spatial heterogeneity in polymers using two characteristic features of n.m.r. : the short-range nature of the dipole-dipole interaction and the transport of spin energy (spin diffusion) that may occur over many tens of nano- metres. The contribution of cross-relaxation is also examined briefly. Several illustrative examples are discussed. The growing conviction that most, if not all, polymeric systems are heterogeneous is rapidly becoming accepted.Aside from the obvious forms of heterogeneity associ- ated, for example, with morphological differences in partially crystalline polymers, other more subtle manifestations are now emerging. In the context of n.m.r., heterogeneity finds its origins in complexities of both molecular structure and motion. Anisotropy in the molecular motions of polymers can lead to a spread of many orders of magnitude in the distribution of correlation times that characterize the motions. The mere fact that polymer molecules are flexible implies that many degrees of freedom are coupled into the motion of any one segment. Considerable effort has been directed towards the elucidation of the details of these anisotropic motions and while it is virtually impossible to treat all the complexities in any one model, a number of important approaches have been formulated.'-'l They subdivide crudely into those that represent the gross features of a distribution of correlation times via empirical distributions 9-11 and those derived from a more molecular approach.1-8 In this paper attention is directed towards the way in which characteristic motions in a polymer may be used to derive information on spatial inhomogeneity.The latter may range from molecular dimensions of nanometres or less (where in fact all materials are heterogeneous) to microns or larger, as in phase separated polymer blends. The contribution of n.m.r. in probing such spatial inhomogeneity has been significant and relies upon two characteristic features of n.m.r. relaxation.The first is the short range nature of the dipole-dipole interaction which predominantly moni- tors near-neighbour interactions and the second is the transport of spin energy (spin diffusion) that may occur over many tens of nanometres. An extension of the second category, appropriate in a few cases, is the cross-relaxation which occurs be- tween spin systems of different resonant nuclei.12-14 In the following sections a number of representative examples that illustrate the role of n.m.r. in the study of heterogeneity in polymers are presented.V. J . MCBRIERTY 79 PARTIALLY CRYSTALLINE POLYMERS The three n.m.r. relaxation times, Tl (spin-lattice), TIP (rotating frame) and T2 (spin-spin), are capable, at least in principle, of distinguishing between local en- vironments characterized by different molecular motions.Under favourable cir- cumstances, Tl and TIP decays are non-exponential while multiple components appear in spectral linewidths. Fig. 1 exhibits typical two-component linewidth and TIP I 4-e + H 0 2 4 6 8 1 0 1 2 r / m s FIG. 1 .-Typical two-component linewidth and TIP decay observed in a partially crystalline polymer. In part (c) 0 = crystalline, 0 = amorphous. Inset shows values of z of (i) 0, (ii) 4, (iii) 6 and (iv) 8 ms. (or Tl) decay spectra observed in polymers with crystalline and amorphous regions. Note that it is often possible to assign the TIP components to one or other of the regions by observing the way in which the shape of the decay changes as a function of the length of the sustaining r.f. pulse in the usual 90"-90" (90" phase shift) pulse sequence used to determine TlP.15 In the data of fig.1, the long T2 (mobile component) is ob- served to decay first which indicates that the short TIP manifests amorphous behaviour. The separation of spectra into components that characterize crystalline and amorphous regions constitutes one of the most powerful advantages of n.m.r. over other methods in establishing the overall relaxation behaviour of polymer systems. Following the first two-component lineshape analysis of Wilson and Pake,16 many studies of increasing sophistication have been carried out to account for the increasing number of situations for which a description in terms of two components is inadequate. In particular, there are many polymers that require the introduction of a third component, in addition to the usual crystalline and amorphous contributions, in order to achieve a satisfactory interpretation of the data.7*'7-22 This third contri- bution of intermediate linewidth has been assigned to material at the interface between the crystalline and amorphous regions and may appear explicitly in linewidth or TIP80 HETEROGENEITY I N POLYMERS AS STUDIED BY N.M.R.data, or may be unresolved in a two component spectrum. Poor resolution can arise in cases where the fraction of interfacial material is small or when its characteristic motions are not appreciably different from those of the truly amorphous or crystalline components. Rotating frame and linewidth data for a high molecular weight linear polyethylene are presented in fig.2.23 The presence of three distinct types of material is demon- 1 6' lo-* 1 o-4 I I I I -200 -120 -40 40 120 t em per a t u re / "C FIG. 2.--T'p(H1 = 10 G) and T2 (A) data for a high molecular weight linear polyethylene. TIP decays are resolved into long (O), intermediate (8) and short (0) components as indicated. strated explicitly in TIP and implicitly in T2. The observed plateau of w70 p s in the long T2 component near 90 "C represents a weighted average of amorphous material that is near liquid-like in character with interfacial molecules executing fairly vigorous motions that are, nevertheless, constrained compared with the truly amorphous com- ponent. Folds and cilia are deemed to be the principal morphological entities in the inter- facial region. Models developed to describe their constrained motions treat the fold or cilium as a chain of N connected segment~.~ Specific character is injected into the model by specifying the mutual orientation of successive polar axes of the chain segments in a particular way.By way of illustration, if one assumes that a cilium is internally rotating and that successive elements of the molecule can rotate about the previous element, then T2 for each element, neglecting inter contributions, takes the simple form : (1) T2 - e3NB2 T2 2 - 2RL ;V . J . MCBRIERTY 81 T2RL is the intra rigid lattice T,; is the bend angle between successive elements and N is the element number where N = 1 denotes the element anchored to the crystal surface.For N,B2 small, T2 does not differ appreciably from T2RL. As Np2 increases, however, T2 correspondingly increases, implying the development of considerable flexibility for elements progressively removed from the point of constraint. The model developed for fold motions adopts a configurational probability approach based upon Gaussian statistics. From the total ensemble of unconstrained configurations of an assumed wormlike chain,24 only those that are consistent with the imposed constraints of the fold, such as specification of the points of entry and exit from the crystal surface, are selected. The constrained probabilities so derived are used to compute the averages required in the calculation of the overall T2 for the fold. Both models provide adequate descriptions of the available linewidth data as indeed is the case with other molecular From a macroscopic viewpoint, lineshape studies have also resulted in gratifying consistency with experiment.Not only have the lineshapes been computed with remarkable accuracy, but inconsistencies in the determination of the degree of crystallinity by n.m.r. have been reconciled with independent assessments over a wide temperature range.22 While interfacial studies such as these have found primary motivation in the area of partially crystalline polymers, it is pertinent to mention that the model calculations discussed above also provide a description of n.m.r. data from a totally different class of heterogeneous material, namely, carbon black-filled elastomers.26 N.m.r.reveals the presence of three distinct types of rubber distinguishable by their characteristic motion^.^^.^^ There is a layer of relatively immobile rubber in the immediate vicinity of the filler particle surrounded by an annular region of considerably more mobile, though still constrained, material. This structure is consistent with the distribution of T2 values predicted by eqn (1). The third unconstrained component exhibits motions comparable with the pure gum. T2 component intensity data can provide estimates of the relative amounts of each type of material present.16 The following caveat must, however, be borne in mind: while it is customary to discuss complex lineshapes or decays in terms of discrete components this does not necessarily imply a sharp discontinuity between regions ; there may well be a more gradual transformation from one type of material to an- other, described by a distribution of relaxation times.Although the decay is formally resolved into two or three discrete components, in such cases this only represents an approximation to the true physical state of the polymer. POLYMER BLENDS The range of properties achieved in blends of two or more component polymers is controlled in large measure by the extent of micro- or macro-phase separation in the composite. Blends of chemically dissimilar polymers have been fabricated such that intimate mixing occurs on a scale of molecular dimension^.^^-^' On the other hand, there are polymer systems where phase separation occurs on a scale of microns or larger.28*32 The full exploitation of the properties of these heterogeneous systems requires a knowledge of the degree of compatability achieved and in this respect n.m.r.has been particularly revealing. Consider the polymer pair, poly(styrene-co-acrilonitrile) (PSAN) and poly(methy1 methacrylate) (PMMA), for which T1 and TIP data are presented in fig. 3.31 The broad TI minimum in neat PMMA centred at 0 "C manifests relaxation due to a-methyl reorientation. At these temperatures the resonant protons in PSAN are only weakly coupled to the lattice. Note, however, that the blends of PSAN and PMMA82 HETEROGENEITY I N POLYMERS AS STUDIED B Y N . M . R . all exhibit exponential decay rather than a weighted superposition of component decays. Thus, within the experimental limits of the measurement, it may be deduced that the a-methyl protons are not only relaxing other protons in PMMA but also those in PSAN through the mechanism of spin diffusion.33 The maximum diffusive path length ( r ) to the relaxation sites may be estimated from the approximate expression ( r ) = [60rI3.too lo-' 10-2 UI \ .- E Y 10-3 L 10-4 1 I I I -160 -80 0 80 FIG. 3 . 4 ~ ) TI (40 MHz) and (b) Tlp (Hl = 10 G) data for PMMA (0), PSAN (a) and PSAN/ PMMA blends in the following proportions by weight : 75/25 (A), 50/50 (A) and 25/75 (a). The time over which diffusion occurs, z, is of the order of 0.5 s (fig. 3). The diffusion coefficient is typically cm2 s-l and therefore ( r ) w 17 nm, which represents an upper limit to the size of PSAN or PMMA aggregates, if present in the blend.The results of simple spin diffusion, in addition, predict the magnitudes of the Tl minima for the three blends on the assumption of strong coupling between the spins. TIP also exhibits exponential decay, with the exception of the 75/25 PSAN/ PMMA blend which shows marginal, though significant, departure from exponential behaviour. Furthermore, the magnitudes of the Tlp minima for the blends are not in accord with the results of simple spin diffusion theory, in contrast to the Tl data. These differences in the TIP response may be rationalized on the basis that the spin systems are not tightly coupled on the much shorter timescale, lo-' s, of the TIP ex- periment. The maximum diffusive path length in this case in w2 nm. Thus it may be concluded from these, albeit approximate, calculations that there is spatial inhomo- geneity in the PSAN/PMMA blends on a scale between 2 and 17 nm.31 temperature/*CV .J . MCBRIERTY 83 Detailed n.m.r. studies have also been carried out on blends of polyvinylidene fluoride (PVF,) and PMMA.I4 In addition to conventional T,, T2 and T,,, cross- relaxation measurements between the I9F and lH nuclei may also be performed in this blend through application of the transient Overhauser One of the pro- cesses that leads to an Overhauser effect involves an exchange of energy between the 19F (I) and 'H (S) spin systems as evidenced by the spectral density term -J0(cq - ms) in the expression for the cross relaxation rate, 6. Such an exchange will only conserve energy, however, when there is additional interaction with a lattice undergoing motions at the difference frequency (@I - ms).These motions provide the neces- sary energy ba1an~ing.l~ Comparison of 6 as a function of temperature in neat PVF, and a 40/60 blend of PVF,/PMMA indicates that thzre is additional cross- relaxation near -40 "C in the blend. It follows that there are additional motions in the blend at the difference frequency (1.6 MHz) and reference to the transition map for PMMA35 shows that a-methyl motion is the most likely candidate. From these observations, supported by T,, T, and T,, data, it is concluded that a substantial fraction of the amorphous PVFz molecules are in intimate contact with PMMA in the blend. While this section has concentrated exclusively upon composites of essentially compatible polymers, it is pertinent to recall again that there are many examples in the literature such as SBS block cop01ymers~~ and polyethylene/polypropylene blended fibres prepared by the surface growth technique,23 to name but two, where n.m.r.has demonstrated almost complete segregation of the component polymers. PLASTICIZED POLY(VINYL CHLORIDE) Considerable insight into the heterogeneous nature of PVC has been derived through observation of the effects of plasticization. Fig. 4 presents T,, and T2 data for neat material, designated PVC(O), and for samples with 1 % PVC(l), 5% PVC(5) and 17% PVC(17) by weight of the plasticizer di-isodecylphthalate (DIDP).36 The T, data support the view that there are three types of material distinguished by their characteristic motions ; a mobile fraction (M), a less mobile, intermediate fraction (I) and a fraction which remains essentially rigid (R) below z 120 "C on the T2 time- scale. The M and I components are both contained in the long T2 component; the initial rise corresponds to the M fraction and the subsequent rise, following the point of inflection, corresponds to the I fraction.Following the procedures described in fig. 1 for the assignment of TlP components, it may be concluded that the minima in the short T,, in the ranges 85-95 "C and 95- 1 15 "C are associated with M and I material, respectively. While the R material has a long intrinsic T,,, its relaxation rate is indistinguishable from the I component due to the effects of spin diffusion.The addition of plasticizer produces dramatic effects on the relaxation behaviour of the I and M fractions. The I fraction shows a somewhat greater dependence on the presence of plasticizer than the M fraction while the R material remains almost totally insensitive to it. This behaviour again demonstrates the heterogeneous nature of the system. Returning to the TIP data, the appearance of the high temperature minimum in the long component manifests partial spin diffusion coupling between the I and M regions.37 Note that the depth of the minimum decreases with increasing plasticizer content since the spin diffusion coupling becomes less efficient with increasing amounts of plasticizer: T,, and TZM increase and therefore the effects of spin diffusion decrease.Since the T,, and T2 data demonstrate that there is some fundamental inhomo-84 HETEROGENEITY I N POLYMERS AS STUDIED BY N.M.R. 20 40 60 80 100 120 140 160 temperature /OC FIG. 4:.-(a) Tlp(Hl = 10 G) and (b) T2 data for PVC (a) and samples containing A, 1 ; 0, 5 and v , 17 % plasticizer by weight. geneity in PVC, it is desirable to determine the extent of these regions in the polymer. Goldman and Shen3* have developed the pulse sequence shown in fig. 5(a) that may be used to monitor spin diffusion between regions of a heterogeneous system de- scribed by significantly different T2 values. The first 90" pulse orients the total mag- netization transverse to the steady laboratory magnetic field. After a time z1 = 30 ,us, the short T2 component has decayed essentially to zero (see fig.4) and leaves the long T2 little altered. The second 90" pulse, out of phase with the first, returns the remaining magnetization along the laboratory field direction. A third 90" pulse is used to examine subsequent effects of spin diffusion at suitably chosen intervals z2. It is convenient to label the decay following the third pulse as S(z2). When r2 is short, say 0.1 ms, there is insufficient time for spin diffusion to equilibrate the rigid and mobile regions and one observes the long component shown in fig. 5(b) (ii). When spin energy transfers out of the short into the long component the shape of S(z2) changes. This change is conveniently monitored as the difference signal S(z2) - S(O.l), [fig. 5(b)]. The change in magnetization of the mobile and rigid material due to spin diffusion redistribution of energy within the spin system is easily visualized by the graphic decomposition of this difference signal [fig.5(c)]. For times <TI, the absence of a redistribution would in fact lead to a zero difference signal. Returning to the experimental data, the difference signal of fig. 5(b) (iv) shows aV. J . MCBRIERTY 85 ( C ) (iii) partial recovery of the less mobile material, with a corresponding heating of the mobile component for r2 = 1 ms. At z2 = 10 ms it is obvious that the entire mobile com- ponent is communicating with the less mobile material and that recovery of a uniform spin temperature is more or less complete. 10 ms, in conjunction with eqn (l), it is concluded that the regions of inhomogeneity are of the order of =lo nm.Using this time of - ( i i i ) ..... . ...... (iv) ( V ) ( v i ) ( v i i ) ..- ,,, ..W--..<-.-.r. .,........ .:. .d ...... :...:. . ..*....*.. . .... .Y .-..- . . ....... . . . . . . . . . . . ...... .,;C.l. . . . . . >.5.---4-’-” --:- I . . _ _ , . - (ii) goo -goo goo FIG. 5.-(a) Goldman-Shen pulse sequence (see text). (b) Diagrams which illustrate the change in shape of the decay curve S(zz) in the G-S experiment on PVC(5) at 85 “C. (iii)-(vii) show the be- haviour of the difference signal S(rz)-S(O.l). Note that (ii) and (iii) denote the mobile and rigid components, respectively. (i) FID, (ii) S(O.1); (iii) S(4OO)-S(O.l); (iv) 4 x [S(1) - S(O.l)]; (v) 4 x [S(10) - S(O.l)]; (vi) 4 x [S(30) - S(O.l)]; (vii) 4 x [S(70) - S(O.l)].(c) Diagram which presents the difference signal in terms of the (i) short and (ii) long component change in time rZ; (iii) shows the combined difference signal. One interpretatioD. of the observed inhomogeneity in PVC suggests the presence of ordered or “ paracrystalline ” regions associated with short segments of syndio- tactic There are a number of other possible candidates. Structures may be present analogous to those induced in i o n o r n e r ~ ~ ~ that may conceivably result from the presence of stabilizers such as lead phthalate or zinc stearate commonly used in these systems. Though less likely in this polymer, molecular weight distributions may also contribute. It is a pleasure to acknowledge the participation of my colleague D. C.Douglass throughout the course of this joint research. P. A. Rouse, J . Chem. Phys., 1953, 21, 1272. D. E. Woessner, J. Chem. Phys., 1962, 37, 647. E. A. DiMartzio and R. J. Rubin, J. Chem. Phys., 1971, 55,4318. J. P. Cohen-Addad, J. Chem. Phys., 1974,60,2440. P. G. deGennes, Macromolecules, 1976, 9, 587; J . Chem. Phys., 1971, 55, 572. R. Kirnmich, Polymer, 1977, 18,233.86 HETEROGENEITY IN POLYMERS AS STUDIED BY N.M.R. ’ D. C. Douglass, V. J. McBrierty and T. A. Weber, J. Chem. Phys., 1976,64,1533: Macromole- cules, 1977, 10, 178. B. Valeur, J. P. Jarry, F. Geny and L. Monnerie, J, Polymer Sci., Polymer Phys. Ed,, 1975, 13, 667. T. M. Connor, Trans. Faraday SOC., 1963, 60, 1574. lo J. Schaefer, Macromolecules, 1973, 6, 882. l1 F. Heatley and A. Begun, Polymer, 1976, 17, 399.I. Solomon, Phys. Rev., 1955, 99, 559. l3 V. J. McBrierty and D. C. Douglass, Macromolecules, 1977, 10, 855. l4 D. C. Douglass and V. J. McBrierty, Macromolecules, 1978, 11, 766. S. R. Hartmann and E. L. Hahn, Phys. Rev., 1962, 128, 2042. l6 C. W. Wilson and G. E. Pake, J. Polymer Sci., 1953, 10, 503. l7 E. W. Fischer and G. Schmidt, Angew. Chem., 1962, 74, 551. l9 D. W. McCall, D. C. Douglass and D. R. Falcone, J. Phys. Chern., 1967, 71,998. ’ O A. Keller, E. Martuscelli, D. J. Priest and Y . Udagawa, J . Polymer Sci. A, 1971, 2, 1807. K. Fujimoto, T. Nishi and R. Kado, Polymer J., 1972, 3,448. 22 K. Bergmann, J . Polymer Sci., Polymer Phys. Ed., 1978, 16, 161 1. 23 V. J. McBrierty, D. C. Douglass and P. J. Barham, to be published. 24 L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon, New York, 1958). ’’ P. Schmedding and H. G . Zachmann, Kolloid-Z. , 1972,250, 1105. ’‘ J. O’Brien, E. M. Cashell, G. E. Wardell and V. J. McBrierty, Macromolecules, 1976,9, 563. 27 S. Kaufman, W. P. Slichter and D. D. Davis, J . Polymer Sci., Part A-2, 1971, 9,829. 28 T. K. Kwei, T. Nishi and R. F. Roberts, Macromolecules, 1974, 7 , 667. 29 T. Nishi, T. T. Wang and T. K. Kwei, Macromolecules, 1975, 8, 227. 30 K. Naito, G. E. Johnson, D. L. Allara and T. K. Kwei, Macromolecules, 1978, 11, 1260. V. J. McBrierty, D. C. Douglass and T. K. Kwei, Macromolecules, 1978, 11, 1265. 32 G. E. Wardell, V. J. McBrierty and D. C . Douglass, J. Appl. Phys., 1974, 45, 3441. 33 D. C. Douglass and G . P. Jones, J . Chem. Phys., 1966, 45, 956. 34 A. Overhauser, Phys. Rev., 1953, 92,477. 35 D. W. McCall, Nat. Bur. Stand. Spec. Pub., 1969, 301, 475. 36 V. J. McBrierty and D. C. Douglass, to be published. 37 D. C. Douglass and V. J. McBrierty, J. Chern. Phys., 1971, 54, 4085. 38 M. Goldman and L. Shen, Phys. Reu., 1966, 144, 321. 39 D. D. Davis and W. P. Slichter, Macromolecules, 1973, 6, 728. H. G. Olf and A. Peterlin, J . Appl. Phys., 1964, 35, 3108. A. Eisenberg and M. King, Ion Containing Polymers (Academic Press, New York, 1977).

ISSN:0301-7249    

DOI:10.1039/DC9796800078

出版商:RSC    

年代:1979

数据来源: RSC

摘要:

Electron Microscopy and SAXS Studies of Amorphous Polymers BY D. R. UHLMANN Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. Received 7th June, 1979 Small angle X-ray scattering (SAXS) and high resolution electron microscopy have been used to characterize the structure of glassy polymers. The SAXS from polycarbonate, poly(methy1 meth- acrylate), poly(ethy1ene terephthalate), poly(viny1 chloride) and polystyrene is inconsistent, both in the form and magnitude of the scattered intensity, with the presence of nodules as representative of the bulk structure. The electron microscope results provide no evidence for heterogeneities on a scale and volume fraction of the reported nodules. Only the pepper and salt features characteristic of microscoge operation near the resolution limit are seen.It is suggested that the structures of these amorphous thermoplastics be regarded as random arrays. 1. INTRODUCTION The question of local order in nominally glassy polymers has been the subject of considerable controversy during the past decade. A sizeable number of investigations, based primarily on electron microscope observations, have cast doubts upon the utility of the random coil model for representing the structure of these polymers. The results of the investigations are well summarized in ref. (1) and (2). The essential and initially surprising feature of these results has been the observa- tion of heterogeneities, typically on a scale of ~50-100 A, in a number of polymers. Among the polymers in which such nodular structures have been observed are polycar- bonate, poly(ethy1ene terephthalate), natural rubber, isotactic and atactic polystyrene, poly(viny1 chloride), and poly(methy1 methacrylate).Investigations of nodular struc- tures have included the following types of observations : (1) the nodular structures have been observed in both direct transmission and replication electron microscopy; (2) dark-field electron microscopy has indicated the presence of ordered regions of approximately the same size as the nodular regions; (3) the nodular structures have been observed to change with changes in the process history of the samples; in particu- lar, they have been observed to increase in size and/or rearrange upon annealing and to align upon stretching; in some cases, the nodules have been suggested to merge on annealing into patches, which in turn aggregate to form lamellar crystalline structures ; (4) when etched by ion bombardment, glassy polymers do not thin down uniformly; some at least develop a granularity on the scale of the nodular structures; (5) the size of the nodules varies from one polymer to another, but does not differ significantly from one form of a given polymer to another (as isotactic vis-Ci-vis atactic); and (6) at least in the case of polystyrene, electron irradiation has a pronounced effect on a dif- fraction halo corresponding to a Bragg law d-spacing which is identified as correspond- ing to an intermolecular rather than an intramolecular distance.88 ELECTRON MICROSCOPY AND SAXS STUDIES OF AMORPHOUS POLYMERS The accumulated weight of these observations has been taken as strong evidence against the random coil model for representing the structure of glassy polymers and support for the existence of regions of local order in the materials.Electron micro- scope studies of thermosetting polymers such as epoxy resins have indicated hetero- geneities on similar scales to those seen in glassy thermoplastics. In these cases, the technique of replication electron microscopy was employed; and the form and scale of the structures were found not to depend on the curing conditions. In addition to such direct observations of heterogeneities in nominally glassy poly- mers, a number of measurements, ranging from X-ray diffraction to mechanical relaxa- tion, have been taken as inconsistent with the random network model.In addition to the work discussed in ref. (I), particular note should be made of the relaxation data obtained on amorphous polystyrene using a torsional braid techniq~e.~ Such data have indicated a transition occurring at temperatures above the glass transition tem- perature, Tg. Such transitions have been termed TI, transitions, and taken by some workers as evidence for local order in the polymers in their glassy state. In contrast to these suggestions of local order in the materials, many properties of amorphous thermoplastics are well described by the random coil model. Many of the results of relevance here were reviewed by Flory? In addition to these, the results of studies using small angle neutron scattering6 and wide angle X-ray scattering’ have provided important insight into the structure of amorphous polymers.These will be considered by others at the present Discussion, and discussed briefly in section 4 below. The present paper will describe small angle X-ray scattering (SAXS) and electron microscopy studies of a number of amorphous thermoplastics. It will also present results of SAXS studies of cured epoxy resins. Some of the results have been reported previously;8$ the interested reader is referred to these papers for details, where appropriate, which are omitted here. 2. SAXS STUDIES SAXS data on polycarbonate (PC), poly(methy1 methacrylate) (PMMA), poly- (ethylene terephthalate) (PET), poly(viny1 chloride) (PVC), polystyrene (PS) and a number of cured epoxy resins were obtained using a Bonse-Hart SAXS system.The system incorporates slotted germanium single crystals in both the incident beam and diffracted beam in order to eliminate the effects of slit-width smearing while still maintaining usable intensities. After correcting for background and absorption, the data were desmeared using a weighting function which was determined e~perimentally.~ In accomplishing this desmearing, an iterative deconvolution procedure was em- ployed, using a multiplicative correction to obtain the trial function at each step. This has been taken as the lower limit of the present measurements. In carrying out the desmearing procedure, it was assumed that the scattering from the specimens was isotropic.This assumption was verified experimentally by rotating the specimen and investigating the effect on the scattered intensity. The intensity was found not to change upon sample rotation. The measured intensities, corrected for background and absorption and desmeared, were placed on an absolute basis by comparison with the scattering measured for a known standard material, obtained under similar diffraction conditions. In all cases, a colloidal silica suspension (Du Pont Ludox IBD 1019-69) diluted to 1.46 yo by volume was used for this purpose. For all the glassy thermoplastics, samples of commercial materials were employed (Lexan, Plexiglas G, PET from Du Pont chill roll, American Hoechst PVC, and lens-D. R. UHLMANN 89 grade PS). The epoxy resins included EPON 828 cured with triethylene tetramine (TETA) for 4 days at 100 "C; EPON 828 cured with nadicmethyl anhydride (NMA) and benzyldimethyl amine (BDMA) for 14 days at 175 "C; and EPON 812 cured with dodecenylsuccinic anhydride (DDSA) together with NMA and BDMA for 24 h at 130°C. A b A A A A 0 0 1 0 - 1 1 0 10 20 30 40 50 60 70 28/min FIG.1.-Variation of absolute SAXS intensity with scattering angle for amorphous PET: 0, de- smeared experimental data; 0, calculated intensity for thermal density fluctuations + distribution of heterogeneities; A, calculated intensity for a 50 vol. % concentration of 100 8, heterogeneities characterized by the crystal excess density. Representative results obtained on the amorphous thermoplastics are shown in fig. 1 and 2 for glassy PET and PVC, respectively.For these and all the other glassy polymers, the intensity decreases markedly with increasing scattering angle at very small angles and reaches or approaches a nearly constant value over a range of inter- mediate small angles. A nearly constant (asymptotic) SAXS intensity at small diffraction angles (26 < 1") is expected for thermal fluctuations in density in an otherwise homogeneous material such as an ideal liquid or glass. In the case of glasses, Weinberglo has sug- gested that the configurational fluctuations (but not the vibrational fluctuations) present in the liquid at the glass-transition temperature, To, should be retained in the90 ELECTRON MICROSCOPY AND SAXS STUDIES OF AMORPHOUS POLYMERS glassy material. The magnitude of the asymptotic zero-angle scattering may then be expressed as : I(0) = v < @.PI2 > = ~TgKT(T.)P2 (1) here KT (T,) is the isothermal compressibility at T .; k is Boltzmann’s constant; p is the average electron density; and ((AP)~) is the mean-square density fluctuation in a region of volume V. A A A A A A I I 1 I I I 0 10 20 30 40 50 60 2 8/min FIG. 2.-Variation of absolute SAXS intensity with scattering angle for PVC: 0 , desmeared ex- perimental data; 0, calculated intensity for frozen-in thermal density fluctuations plus scattering from heterogeneities listed in table 1 ; A, calculated intensity from heterogeneities 200 8, in diameter, having the crystal excess density, and occupying 50 % by volume; (- - -), intensity due to thermal density fluctuations.For each of the glassy thermoplastics, the measured SAXS intensity reaches or approaches an asymptotic value which is close to that predicted for thermal density fluctuations frozen-in at the glass transition. (The difference between the measured and predicted intensities varies from < 10 % of the predicted value for PC to x 50 % for PMMA). Considering the possible uncertainty in the values used for the high temper- ature compressibility, the agreement here must be regarded as impressive. The occurrence of scattering in the very small angle region above the level corres- ponding to thermal density fluctuations indicates the presence of heterogeneities in the material. Using the exact expression for the SAXS from spheres, distributions of heterogeneities such as those shown in table 1 for PVC were obtained by fitting the experimental data.The points labelled by open circles in fig. 2 show the scattering from the particles in table 1 superimposed on that from thermal density fluctuations. The agreement between measured and calculated intensities is seen to be excellent; and comparable agreement could be obtained for each of the other polymers investi- gated. In estimating the actual concentrations of the heterogeneities which are indicated by the increase in SAXS intensity at very small angles, it is necessary to assume a value for the density difference Ap between heterogeneities and matrix. In the case of PET, for example, if the scattering is assumed to be associated with microvoids with a density difference from the matrix equal to the bulk density, (AP)~ = 0.20 electron2 A-6, the values of C(AP)~ obtained from fitting the experimental data would indicateD. R.UHLMANN 91 volume concentrations of (4 x 10-6)-(2 x loe5), and the total concentration of heterogeneities estimated from the intensity invariant (see below) would correspond to a volume fraction of z7 x If, in the more likely case for this material, the scattering is associated with crystallites, with a density difference from the matrix of (AP)~ = 2 x electron2 A-6, the values of C(Ap)' would indicate volume con- centrations of 0.04-0.2 %. The total concentration of heterogeneities estimated from the intensity invariant in this case would correspond to a volume fraction of ~ 0 . 7 %. Even smaller total concentrations of heterogeneities were estimated for PC and PMMA.TABLE 1 .-SIZES AND CONCENTRATIONS OF HETEROGENEITIES DESCRIBING THE SAXS FROM PVC C(AP)~/~O~ electron2 C/% (Ap M 0.034 C/% (Ap = 0.45 radius/A A-6 electron A-3) electron A-3) 4500 3500 2000 lo00 700 500 150 50 5.78 2.89 1.33 1.91 3.24 15.0 8.44 23.1 0.05 0.025 0.01 15 0.0165 0.028 0.13 0.073 0.20 0.000 29 0.000 14 0.000 066 0.000 094 O.OO0 16 0.000 74 0.000 42 0.000 14 A check on the estimated concentrations of heterogeneities can be obtained by evaluating the contribution of the heterogeneities to the mean-square density fluc- tuations, ((AP)~). The magnitude of this quantity for the scattering above that due to thermal density fluctuations may be determined from the invariant integral : lX21'(h) dh = 2 ~ ~ ( ( A p ) ~ ) where I'(h) is the intensity above that due to thermal density fluctuations. In all cases, the magnitude of ((AP)~) determined in this way is close to the sum of the con- tributions from the heterogeneities inferred from fitting the intensity data.In the case of PVC, table 1 shows the concentrations which are implied by assuming the crystal excess density (Ap = 0.034 electron A-3) and also those implied by assum- ing that the heterogeneities are microvoids (Ap = 0.45 electron A-3). It is important to note the concentration of very small (50 A) particles in the indicated distribution. The inclusion of these particles 50 A in radius serves two purposes. First, it helps to account fur the measured intensity in excess of the thermal fluctuation scattering at angles between 20 and 30 min, and it accounts for the slight negative slope observed for this part of the curve.Second, it is needed to make up the difference in the mean- square fluctuation density which the larger particles cannot represent by themselves. It should be noted that particular distributions of heterogeneities, such as that shown in table 1, are by no means unique. Other models with somewhat different values for the sizes and relative contributions could undoubtedly provide an equivalent description of the experimental data. Any such model must, however, associate the scattering at very small angles with small concentrations of rather large particles (as concentrations in the range of 0.04 ~ 0 1 % of particles several thousand A in diameter in the case of PC). Any implication that the heterogeneities are characterized by the crystal excess density is not intended; and it seems likely that many of the heterogeneities are not92 ELECTRON MICROSCOPY AND SAXS STUDIES OF AMORPHOUS POLYMERS characterized by such a difference in density relative to the matrix.In particular, it seems reasonable that the larger heterogeneities are not large crystals but rather are heterogeneities extrinsic to the polymer (such as dirt or perhaps stabilizers or process- ing aids which have precipitated during processing). These should have a larger A p than crystallites, and would be present in concentrations smaller than those indicated for heterogeneities with the crystal excess density. It is also possible that the heterogeneities are characterized by A p values which are smaller than the crystal excess density, in which case the volume fractions would be larger.For PVC with A p = 0.0045 electron A-3 (Ap/p = 1 %), for example, the in- dicated concentration of 500 A heterogeneities would be = 7 %. Such heterogeneities could represent locally dense regions containing relatively high concentrations of chain entanglements, probably given some permanence by the presence of a small con- centration of small crystallites. The forms of the scattering expected for heterogeneities having the size of the nodules reported in electron microscope studies, occupying a volume fraction of 50 % and characterized by the crystal excess density are shown by the filled triangles in fig.1 and 2. It is seen that the experimental data are inconsistent, both in the form and magnitude of the scattered intensity, with the scattering expected for such an assemblage of nodules. Use of the invariant integral to estimate the mean square density fluctuation in the materials, beyond that expected for thermal density fluctuations, also presents problems for the nodule hypothesis. In particular, density differences between heterogeneities and matrix in the range of 1 % or less of the bulk density are indicated. Such small density differences would not produce perceptible contrast in the electron microscope, either by amplitude contrast or by Fresnel type phase contrast; and it seems unlikely that heterogeneities differing from the amorphous density by as little as 1 % could be characterized by sufficient order that perceivable diffraction contrast would be noted.Taken in toto the SAXS results provide strong evidence that the nodular structures seen in the electron microscope are highly unlikely to be representative of the bulk structure. It is suggested, therefore, that the nodules may be associated with surface effects, and that the structures of the polymers be regarded as random amorphous arrays, with small concentrations of heterogeneities superimposed on thermal density fluctuations frozen-in at the glass transition. While most of the controversy surrounding the existence of nodules has centred about thermoplastic polymers, many studies have been conducted on network-forming thermosetting polymers as well.Heterogeneities, generally on a scale of 50-3000 A and present in large volume fractions, have been observed in transmission electron microscope studies of a number of cured epoxy resins. Among such studies, that of Racich and Koutsky3 has provided the most clear-cut evidence for the presence of structural inhomogeneities. These workers observed heterogeneities (which they termed nodular structures) on free surfaces, fracture surfaces and etched surfaces of epoxy resins of widely different cures and chemistries. The SAXS intensity from the three cured epoxy resins are illustrated by the data in fig. 3 for the Epon 8 1 2 / N M A / D D S A material. The intensity is seen to decrease with increasing scattering angle out to = 24 min, beyond which the intensity remains almost constant.The asymptotic level of scattering is close to that expected for thermal density fluctuation (0.3 electron' A-3 as compared with 0 . 1 8 electron2 A-3). The pronounced rise in scattered intensity at small angles (28 < 24') and the excess asymptotic scattering at larger angles (over that predicted by fluctuation theory) can be associated with heterogeneities in the sample, The sizes and concentrations of heterogeneities needed to describe the observed scattering are shown in table 2. TheD . R. UHLMANN 93 I I I 1 I I 1 0 10 20 30 40 50 60 28/min FIG. 3.-Variation of absolute SAXS intensity with scattering angle for Epon 812/NMA/DDSA : 0, desmeared experimental data; 0, calculated intensity for thermal density fluctuations + distri- bution of heterogeneities shown in table 2.presence of large heterogeneities (R > 250 A) is required to explain the large increase in intensity at very small angles. It is likely that these heterogeneities are extrinsic to the polymers. While adventitious impurities such as dirt may be responsible for some of this scattering, the most likely source is gas (air) bubbles incorporated in the resin dur- ing its cure. As seen in table 2, concentrations of gas bubbles in the range of are sufficient to account for the observed scattering (above fluctuation scattering) in the very low angle region. TABLE 2.-cALCULATED SIZES AND CONCENTRATIONS OF HETEROGENEITIES IN EPON 8 12/NMA/ DDSAfo RIA ~ ( l - c)(Ap)’ C(AP = 1 %) c ( A ~ = 10 %) c(Ap = 100 %) 150 3.18 x 0.266 1.95 x 10-3 1.95 x 1 0 - 5 500 1.91 x 0.135 1.17 x 1 0 - 3 1.17 x 10-5 3000 1.77 x 0.124 1.08 x 10-3 1.08 x 1 0 - 5 5000 9.55 x _ _ 5.88 x 1 0 - 3 5.88 x lo-’ 6000 5.53 x _ - 3.38 x 1 0 - 3 3.38 x 10-5 a a a Gives c(1 - c) > 0.25.The scattering at larger angles (still in the small angle region) is associated with smaller heterogeneities. The excess scattering (above that due to thermal fluctuations) indicates the presence of small (< 100 A) inhomogeneities in the materials. Again, concentrations of gas bubbles in the range of would be sufficient to account for the observed excess scattering, although the presence of other structural inhomogenei- ties could also account for the scattering. If the densities of these heterogeneities differ only slightly from that of the matrix (as Ap/p = 1 %), they could be present in sizeable concentrations.For electron microscope studies carried out using direct transmission on thin sec- tions, heterogeneities in a 1000 A thick sample of epoxy would provide discernible94 ELECTRON MICROSCOPY AND SAXS STUDIES OF AMORPHOUS POLYMERS contrast only if they differed in density from the bulk by z 5-10%. Examination of table 2 shows, however, that heterogeneities differing from the bulk density by 10 % or more can only be present at concentrations in the range zO.l%, a range which is much smaller than the concentrations indicated for the nodular features. For studies carried out using electron microscopy of surface replicas, which repre- sent the largest fraction of the observations of nodular features in epoxy resins, a different question is posed by the present results.The volume fractions of hetero- geneities seen in the electron microscope studies are consistent with the SAXS results only if they differ in density by perhaps 1 % or less from the bulk. In light of the modest differences in density between cured and uncured epoxy resins, differences in density of 1 % or less in the cured resin seem not unreasonable. It remains to be established, however, how regions with such differences in density become visible on fracture surfaces. Certainly there is good reason for expecting regions of different crosslink density in cured epoxy resins. What remains to be clarified is the relation between such regions and the features seen in electron microscope studies.The present SAXS results should be viewed as providing data with which any proposed structural model must be consistent. 3 . ELECTRON MICROSCOPE STUDIES The SAXS results discussed in the preceding section cast strong doubts on the vali- dity of the electron microscope observations which suggested the occurrence of nodular structures as essential features of amorphous polymers. It seemed highly desirable, therefore, to re-examine the structure of glassy polymers using the technique of high- resolution electron microscopy. Appropriately thin samples of PC, PET, PVC and PS were cast from solutions using the same techniques as employed by previous investigators ; the specimens were viewed in both bright- and dark-field using a high resolution electron microscope which provided magnifications of 500000 x or more on the photographic plates when desired.For all four polymers, the structures were seen to be featureless down to the limit of resolution of the electron microscope. Only the “pepper and salt” structure characteristic of electron micrographs taken at high resolution is seen. The “ pepper and salt ” structure observed is a result of the use of a finite objective aperture to limit the amount of information (in the form of transmitted or scattered electrons) from the object that is used to construct the image. If the object is considered to be an array of atoms or molecules, then the convolution of the object with an Airy disc (the aperture transform) will yield a blurred image of the array observed as the “pepper and salt ” structure.No evidence was found in either bright-field or dark-field for nodular features as reported by others. The combination of featureless bright-field and dark-field micro- graphs provides strong evidence for a highly homogeneous structure. Since PET was the first polymer for which distinct nodular structures in the glassy state were reported, and since the nodules in this material appear with greater clarity than those in other polymers, it was decided to subject PET to even closer scrutiny. Several through- focus series of micrographs were taken, a representative set of which is shown in fig. 4. This series shows the absence of perceivable structure in the in-focus micrograph, and illustrates how apparent structure can be developed in the micrographs by going to either an underfocus or an overfocus condition.The change in the scale of the “ salt and pepper ” noted with change in focus can be considered as an additional defocus convolution with the original object.FIG. 4.-Through-focus series of electron micrographs of PET showing the absence of structural features when in focus and the development of apparent structural features when out of focus. [To face ptzge 94D. R. UHLMANN 95 These results suggest that the fine-scale (<20 A) apparent structure seen in some previous investigations may simply reflect the use of electron microscopes of insufficient resolution, or may result from the lack of proper focus in taking the micrographs. Neither of these possibilities can be confirmed at present nor can alternative explana- tions based on other effects, such as radiation damage from the electron beam or the possibility that the observed nodules represent surface features of the polymers. 4.GENERAL DISCUSSION The combined results of the SAXS and electron microscope studies suggest that nodular features should not be taken as representative of the bulk structure of thermo- plastic polymers. These findings are in accord with the results of recent light scatter- ing and small angle neutron scattering (SANS) investigations of glassy polymers. Among the former, Patterson1' investigated the effects of cooling on light scattering from PMMA and PC. No evidence was found for an increase in the number density or size of heterogeneities in the polymers as the temperature was lowered, as would be expected for nodular structures as regions of local order.Among the SANS studies, that of Benoit12 can be taken as representative. The radii of gyration of eight molecular weight polystyrene polymers were found to be the same, within experimental error, as those of the polymers in a theta solvent (where the random coil is widely acknowledged as providing a useful representation of the chain conformation). The combined weight of all these studies, together with those cited in ref. ( 5 ) , leads to the conclusion that the structure of amorphous thermoplastics should be represented by random array models such as the random coil, rather than by models such as the nodule hypothesis, whose essence involves regions of locally high order. When examined in detail, all the evidence advanced in support of such local-order models seems subject to question. Even the observation of TI, transitions seems to involve the interaction of the liquid with the braid on which it is supported, and cannot be taken as unequivocal support for regions of local order. It is suggested, therefore, that the controversy be laid to rest, and that attention be directed to more fruitful areas such as the structural changes involved in crystallization and the structural differences between surfaces and bulk material. Financial support for the present work was provided by the U.S. Air Force Office of Scientific Research. This support is gratefully acknowleged, as is the previous support of the National Science Foundation. The direct contributions of Drs. Renninger, Wicks, Straff and Vander Sande, as well as Mr. Matyi and Miss Meyer, made this presentation possible. G. S. Y. Yeh, Crit. Rev. Macromol. Chem., 1972, 1, 173. P. H. Geil, Morphology of Amorphous Polymers, Final Report USAROD, Contract DAHCOC J. L. Racich and J. A. Koutsky, Bull. Amer. Phys. Soc., 1975, 20, 456, J. K. Gillham and R. F. Boyer, J. Macromol. Sci. Phys., 1977, B13, 497. P. J. Flory, J . Macromol. Sci. Phys., 1976, B12, 1 . D. G. H. Ballard, G. D. Wignall and J. Schelten, Europ. Polymer J . 1973, 9 965. G. D. Wignall and G. W. Longman, J. Mater. Sci., 1973, 8, 1439. A. L. Renninger, G. G. Wicks and D. R. Uhlmann, J . Polymer Sci. Phys., 1975, 13, 1247. M. Meyer, J. Vander Sande and D. R. Uhlmann, J. Polymer Sci. Phys. 1978,16,2005. 7OC-0032. lo D. L. Weinberg, Phys. Letters 1963, 7, 324. l1 G. Patterson, J . Macromol. Sci. Phys., 1976, B12, 61. l2 H. Benoit, J. Macromol. Sci. Phys., 1976, B12, 27.

ISSN:0301-7249    

DOI:10.1039/DC9796800087

出版商:RSC    

年代:1979

数据来源: RSC

摘要:

Theory of Long-range Correlations in Polymer Melts BY P. G. DE GENNES Coll6ge de France, 75231 Paris Cedex 05, France Received 22nd October, 1979 We discuss first the small-angle scattering of neutrons by a homopolymer melt, when each chain is tagged at certain sites by deuteration. We then extend these considerations to block copolymers AB, assuming an AB interaction which is weakly repulsive. Starting from the homogeneous melt, we lower the temperature and determine first the locus of the spinodal instability towards microphase separation, and in particular the period of the incipient periodic structure. L. Leibler has recently constructed a Landau theory for the microphases, and a theoretical phase diagram (restricted to the vicinity of the melt) showing regions with lamellar, hexagonal or cubic structures.Finally we discuss the effect of a certain amount of disorder in the chemical sequence on the spinodal instability: in the most interesting case, where the disorder is the same in all chains, we find that disorder increases the spatial period and ultimately suppresses the trend toward microphases. These last considerations may be relevant for the design of certain nematic polymers. CORRELATION I N HOMOPOLYMER MELTS From the theoretical ideas of Floryl and from neutron experiments,2 we know that the chains in a homopolymer melt are essentially ideal. It is important to realize, however, that there remain some interesting correlation effects in the melts, which can be seen in diffraction experiments provided that one uses tagged molecule^.^ This can be understood most directly for the case of homopolymer chains which are all tagged at one end only.Fig. 1 gives us a qualitative picture of the correlation y(r) between tagged sites: apart from normalization factors, we can think of y(r) as the concentra- tion of tagged molecules at point r, when we have put one tagged molecule at the ori- - Ro FIG. 1 .--Qualitative plot of the correlation between " heads " of different molecules in a homo- polymer melt. When one " head " is fixed at the origin, the space available for other coils around it is slightly decreased by the presence of the first coil: there is a correlation hole.P. DE GENNES 97 gin. At large r, y(r) becomes constant and equal to the average number of tags per cm3. But when r is smaller than the average chain size Ro = N3a (N = number of monomers per chain) a new effect appears: we know that one particular chain is present near the origin; thus the concentration of other chains must be slightly de- pleted.There is a correlation hole; the relative shift of y(r) in this region is rather small, since the density associated with the first chain is of order N/R& N-*. However, the correlation hole is visible in neutron experiments, because at small wavevectors (qRo - 1) there are no other physical phenomena contributing to the intensity: in particular the fluctuations of the overall density in the melt (controlled by a compressibility modulus) are very small when compared with the effects of the correlation hole. and we shall summarize it only briefly here.The starting point is a set of identical chains, each with a sequence of monomers 1,2, . . . n, . . . N. Some of the monomers are deute- rated, but they are all assumed to be identical as regards their mutual interactions: this appears to be a good approximation for most melts, although deviations may be found in polyethylene of very high molecular weight. We call c,(r) the concentration of monomers of rank n at point Y, and we are interested in the correlation functions A detailed theory of these correlation effects was constructed long Note that the functions xnm introduced here have nothing to do with a Flory interaction parameter.l The notation expresses the fact, first shown by Y v o ~ , ~ that xnm(r) may be visualized as a non-local susceptibility, giving the shift in cm(r) if a small perturba- tion, acting only on species (n), is applied at point 0.In the factor kT, Tis the tem- perature and k is Boltzmann’s constant. The neutron experiments measure the Fourier transforms Xnm(q) through a formula of the form : where M, is the coherent amplitude for species n, and q is the scattering wavevector. (In practice we control M,, by selective deuteration). The calculatipn of the correlations xnm is based on the very same principles which were used by Debye and Huckel for polyelectrolytes. One puts one monomer (n) at the origin, and then looks at the self-consistent field built up in its neighbourhood. This idea was first applied to polymer solutions by Edwards.6 One central assump- tion of the calculation is to assume that each chain is nearly ideal (just as Debye and Huckel assumed that in zero order, their ions behaved as an ideal gas).This assump- tion of ideality is not accurate for solutions, but is indeed correct for melts. Another interesting feature of the melts is that we do not need to specify the inter- actions between monomers in any detail! At the long wavelengths of interest here, we may in fact describe them simply by imposing a constraint of constant density 2 c,(r) = c = constant. n (3) What is nice in this approach is that one then obtains a formula for the correlations which contains no adjustable parameter (except for the r.m.s. end-to-end distance R,, = N*a of one chain), namely398 LONG-RANGE CORRELATIONS IN POLYMER MELTS where XnOrn(q) is the intrachain correlation calculated for a gaussian coil NkT&(q) = exp {- q2a2 In - ml 16) and the functions Sn and D are sums of x0 functions (5) n J In particular D(q) is the ideal chain scattering function first computed by D e b ~ e .~ Although eqn (4) was derived from a specific Debye-Huckel type of calculation, it is slightly more general : the main assumption is that the self term x:, in eqn (4) is of the ideal chain type. For instance, with a chain where the first half (1 < n < N/2) is labelled and the second half is not, Benoit* has proven recently that the formula for Z(q) resulting from eqn (4) and (2) holds automatically if xo is gaussian: the proof is based only on the incompressibility sum rules derived from eqn (3), namely: m plus the symmetry between the two parts of the chain.What are the typical conclusions from eqn (4) and (2)? Let us start from the same simple case, where only one monomer (n = 1) is deuterated on each chain. The plot of Z(q) is then simple. At low q we have no intensity: the concentration of deuterated sites c1 is simply 1/N times the overall concentration c, and the latter has no fluctuations in our model. At higher q values we do get a signal because, although c is constant, the chains can choose, in a given fluctuation, to put their end-to-end vectors (or “ dipoles ”) parallel: this creates a non-zero polarization P(Y) and c1 = -div P(Y) can be finite. Ultimately, when qRo becomes much larger than unity, interference terms between different chains drop out from Z(q), and we are left with the scattering by an individual point-like object, which is constant.Let us investigate next the case where the chain is tagged on a short sequence near the origin n = 1,2, . . . n,. Then the low-q part of I(q) is still exactly the same (except for a normalization factor) whenever qRe < 1 where Re = ne*a is the size of the tagged portion. Thus Z(q) increases with q in this region. However, when we go to large q values (qR, > 1) the scattering now arises from independent particles of finite size (Re) and the form factor of such a particle drops for qR, > 1 : thus we finally have a decrease in Z(q) at high q. We conclude that there is a peak at intermediate q values. Peaks of this type have indeed been observed in many neutron experiment^.^ Originally, some experimenta- lists were tempted to associate them with an “incipient order’’ in the solution.However, this is not correct: the intensity at the peak is smaller than the intensity which would be due to uncorrelated scatterers with the same chemical sequence. The peak reflects the existence of a hole in the spatial correlations, and not of a hump. We needed to recall this point before embarking upon the more complex case where our molecules are block copolymers, and where real segregation effects may become important: in this latter case, we shall find intensity peaks which are larger than the scattering due to independent objects, and which do represent incipient order. To conclude with the homopolymer case: the neutron data on various types of partly labelled chains in melts do show that the Debye-Huckel formula (4) is a reasonable approximation in 3 dimensions. It should be emphasized, however, that in 2 dimensions (for chains confined to thin sheets in a lamellar system) the situationP.DE GENNES 99 could be much worse : deviations from ideality might be more important. I do hope that some 2 dimensional situations will be studied in future experiments. BLOCK COPOLYMERS We assume now that all our chains are identical block copolymers made (for in- The chemi- stance by anionic polymerization) with two types of monomers A and B. cal sequence will be described by indices a,, a, . . . a, . . . cN where: I. a n = + 1 (A) an= - 1 (B) There is a certain interaction between n and m which we shall write in the form: Here xF is a Flory interaction parameter between A and B.Eqn (9) ensures that V A A = VBB = 0 while V,, = kTXF. In the simplest situations (to which we shall adhere here) V,, is temperature independent and xF is proportional to T - l . We assume xF to be positive and rather small (weak trend towards segregation). The Debye-Huckel (or " random phase approximation ") result for the correla- tions in this case, has the form:3 [xcij],, = [x;:O]nm + vnm. Here [X-'(4)lnm is the inverse matrix to xnm(q) and Xhomo is the correlation matrix for homopolymers, given by eqn (4). It can be checked that eqn (10) is still compatible with the incompressibility sum rule. Eqn (10) is less secure than eqn (4), because it implies assumption of ideal gaussian behaviour which might fail when the trend towards segregation is strong (xF large).However, certain qualitative arguments (restricted to the homogeneous melt, with A and B sequences of comparable length) do suggest that the assumption is still tolerable at temperatures just above microphase separation. We shall first apply eqn (10) to the discussion of a relatively simple case, namely a periodic multiblock copolymer . . . A B A B . . ., each block having the same num- ber d of monomers, and the same r.m.s. size R A = R B = d*a. The complete chains are assumed to be very long, and this simplifies the analysis because end effects can be ignored: we may just as well replace the chain by a cyclic structure, and impose periodicity on the matrices xnm. In this case it is rather simple to invert the matrix xhomo(q) and to write the inverse operator in differential form : where x = q2a2/6.Eqn (1 1) is easily checked on Fourier transforms with respect to the variable n, which here we treat as continuous. In general, eqn (1 1) would have to be supplemented by rather delicate boundary conditions for the operator on the right: but in our cyclic case this complication is removed. Also note that eqn (1 1) holds for all the vector space on which the matrix znm acts, except for one vector (1 . . . 1 . . . 1): because of the incompressibility sum rule, xg:mo acting on this vector gives a divergent result.100 LONG-RANGE CORRELATIONS IN POLYMER MELTS We shall mainly focus our discussion here on the search for an instability pre- monitoring microphase separation : this will correspond to one particular eigenmode (Vn) of the matrix xnm, such that Cxnmrym = co or equivalently 2 h-l)nm~m = 0. Using eqn (10) this gives: m m rn Inserting eqn (9) for the interaction matrix, and noting also that cry,, = 0 by the incompressibility sum rule, we can simplify eqn (12) greatly and obtain : where the parameter is independent of n. For a given sequence o1 .. . 0, . . . crN it is not hard to solve eqn (13a), find the structure of ry, for a given r, and finally impose the self-consistency condition (13b). For the periodic chemical sequence chosen in our example: on = + 1 for -d/2 < n < d/2 - 1 d/2 < n < 3d/2 etc. (14) the adequate solution y, is periodic and vanishes at the ends of each block.In the central block (A): For the B blocks, ryn has a similar structure, but is reversed in sign: d/2 < n < 3d/2 }. (15b) I+Y, 2 -cash (xd/2) + cash X(U - d ) From eqn (13b) one then finds and comparing with eqn ( 1 3a) we arrive at the instability condition : r = N[cosh (xd/2) - 2(xd)-l sinh (xd/2)] xF = (x/2)[1 - tanh (xd/2)/(~d/2)]-'. (16) (17) When plotted against x , the right-hand side of eqn (17) shows a maximum for x = x* - 3.21d-'. This means that the microphase period which tends to occur corres- ponds to a wavelength 2n/q* which is just slightly smaller than the r.m.s. end-to- to end distance of one AB sequence. For xF smaller than the corresponding thres- hold, we expect a peak in the neutron scattering intensity at q E q*. We do not have many data on periodic multiblock systems to compare with these predictions.A different molecule which is easier to produce experimentally is the simple diblock (AB) copolymer. But here, on the theoretical side, the situation is slightly more complex. It has been worked out, however, in some detail by Leibler.g He gives plots of the incipient periodicity 2n/q* for all values of the molar fractionf corresponding to ANPBN(l-P). Furthermore Leibler did not only compute the pair correlation functions (cncm) but also some higher correlations ( c c c ) and ( c c c c ) .P. DE GENNES 101 Starting from this, he can construct a Landau theory of the microphase separation of AB diblocks, at least for temperatures which are not too far below the liquidus. For f - 0.5 he predicts the onset of a simple lamellar phase.But for more dissymetric cases (f# O S ) , he finds a cascade of phases (isotropic, hexagonal, lamellar . . .). These predictions are interesting: they supplement the earlier analysis of Helfand, which was concerned with a low-temperature regime where the spatial boundaries (between A rich and B rich portions) are rather sharp. EFFECTS OF PARTIAL RANDOMNESS IN THE CHEMICAL SEQUENCE The chemical sequence of a block copolymer is always expected to show some statistical deviations from its nominal structure. We can think of two main types of chemical disorder, which we call I and 11. Type I : here different chains in the same sample have different chemical sequences. This implies that the index a,, defined in eqn (7) now has a certain average value en which is intermediate between - 1 and + 1.In many practical cases the deviations o,, - 6,, and om - 5, are expected to be uncorrelated. Then we need simply to replace the interaction V,,, of eqn (7) by an averaged interaction Vnm = RTxF( 1 - 8,pm). The rest of the analysis is unchanged. Type 11: here all chains in the sample have the same sequence, but this sequence has statistical deviations from the model copolymer. This case is conceivably present in a periodic multiblock AB structure, prepared by anionic polymerization, if the suc- cessive time intervals for insertion of A or B monomers have a certain random distri- bution in length. We shall discuss here the effects of type I1 disorder on one particularly simple case : namely we assume that we are in a nearly periodic multiblock structure, with indices a,, which are random stationary variables, and a correlation function onam = cos[A(n - m)] exp(--ln - ml).(1 8) Here 3, = n/d is related to the average periodicity (2d) of the chemical sequence, and 01 measures the randomness. We shall now see that eqn (18) suffices to give us the optimal wavevector q* : returning to eqn (1 3) we find that: m and inserting this into the definition of condition [eqn (13b)l we arrive at the self-consistency On the r.h.s. of eqn (20) it appears plausible to decouple r from onom (since r is a macroscopic quantity with fluctuations of relative magnitude -N-*). This then leads to m and (always ignoring end effects) we obtain explicitly:102 CORRELATIONS I N POLYMER MELTS LONG-RANGE xF'1= low dt cos(At) exp{ -(a + x ) t } 22 (a) For a = 0 (no disorder) this spinodal condition is qualitatively similar to eqn (17): the quanfitative differences reflect the fact that on is a succession of alternate steps for eqn (17) while it is a sinusoidal curve for eqn (22).The minimum in this case is obtained for x = x* = ;1 = 3.14 d - l . (b) For a > 0 (increasing disorder) the whole curve (22) is shifted and the minimum is at x* = ;1 - a. When t( becomes larger than ;1 the minimum remains at 4 = 0 (fig. 2). A I I 1 x x=q2a2/6 FIG. 2.-Qualitative plot of the Flory interaction parameter xF against wavevector squared at the onset of the microphase instability [eqn (22) of text]. The calculation is for a long sequence of equal blocks ABAB.. . on each chain. The parameter a measures the statistical departures from this ideal structure. Note that there is a minimum q = q* only when the disorder js not too strong (a < A). We conclude that microphase separation can be suppressed by a certain amount of type I1 disorder. This observation may be of use in a slightly different field, namely for the preparation of nematic polymers with nematogenic portions A inserted in part of the backbone, and flexible chains B linking them together. When such systems are prepared, it often happens that the result at the temperatures of interest is not a nematic, but rather a smectic A (i.e., a lamellar structure with alternating A rich and B rich portions). Eqn (22) suggests that, if a certain amount of randomness is intro- duced in the length of the B chains, we may well eliminate the spurious smectic phase. Of course in this case, the A sequence is not flexible, and the statistics may be more complex than in eqn (18), but the qualitative trends should be the same. I am indebted to H. Benoit and L. Leibler for various discussions on the Debye- Huckel approach. P. Flory, Principles of Polymer Chemistry (Cornell Univ. Press, Ithaca, N.Y., 1953). ' D. G. Ballard, J. Schelten and G. D. Wignall, Eur. Polymer J . , 1973,9,965; J. P. Cotton et al., Macromolecules, 1974, 7 , 863; R. G. Kirste, V. A. Kruse and K. Ibel, Polymer, 1975, 16, 120.P . DE GENNES 103 ’ P. G . De Gennes, J. Physique, 1970, 31, 235. J. Yvon, Rev. Scientifique ( D k 1939), p. 662. F. Boue et al. in Neutron Inelastic Scattering I977 (Int. Atomic Energy Agency, Vienna, 1978), vol. 1, p. 563. ti S . F. Edwards, Proc. Phys. Soc., 1966, 88, 265. P. Debye, J . Phys. Colloid Chem., 1947, 51, 18. H. Benoit, personal communication, Sept. 1979. L. Leibler, to be published.

ISSN:0301-7249    

DOI:10.1039/DC9796800096

出版商:RSC    

年代:1979

数据来源: RSC