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.