GENERAL DISCUSSION Prof. A. Silberberg (Israel) said: It is in principle impossible for a true homo- polymer to form a gel. If the groups which characterize the polymer can interact to give rise to an energetically favourable, i.e., attractive situation, the homopolymer could not be soluble in the solvent medium. For a homopolymer to be soluble the net interaction between polymer segments must be repulsive. If, therefore, gels are formed it is safe to assume that the polymer possesses some copolymer character. For example, polymethacrylic acid, although chemically uniform, is a stereo copolymer and we may assume that the energetic interactions between two polymer segments in this and similar cases can move from repulsive to attractive, depending upon the nature of the stereo regular environment of the two interacting segments.While the attractive configurations may perhaps be only slightly more rare than the repulsive one, leaving the polymer with an overall repulsive character, a cross-link of some perma- nence can arise in these cases if a whole sequence of attractively interacting segments, along one section of one chain, interacts with another such sequence on another chain and builds up a cooperative unit of sufficient strength and sufficiently long chemical relaxation time. Dr. 0. Smidsr#d (Trondheim) (communicated): I should like to comment on a statement made by Silberberg : " It is in principal impossible for a true homopolymer to form a gel ". This is of course true for equilibrium gels, but in general it may not be so for non-equilibrium gels where there may be a kinetic hindrance to the formation in a given situation, of the thermodynamically favoured precipitate.All experience with fractions of agarose, rc-carrageenan and alginate suggest that the modulus gets higher the closer you get to a true homopolymer in composition ((AB),-polymers are then regarded as homopolymers). Dr. G. S. Park (UWIST, Cardzfl) said: In agreement with the remarks of Prof. Silberberg we attributed gel formation in poly(viny1 chloride) to its stereocopolymeric character.2 It is not necessary for the linking points to be crystalline for a gel to be formed and interesting examples of non crystallized linked gels are given by ABA block-copolymers in media that are solvents for B units but not for the A units.The correlation between X-ray crystallinity and the possibility of gel formation in our copolymers, however, has led us to conclude that the linking points in these gels are crystallites. Prof. F. Franks (Unilever) said: Implicit in the use of eqn (4) of Park's paper is the assumption that the copolymer gel can be equated to crystalline polyvinylidene chloride and presumably the enthalpy of gel melting is equated to the latent heat of fusion of the crystalline homopolymer. It is found that in polysaccharide gels, crystalline remnants can exist well above the observed gel melting temperature and the authors mention that this is also a possibility with their copolymer. If no estimate can be obtained of the fraction of crystallites which persist in the sol phase, then what is the quantitative significance of AHv, or indeed AH,? A.Silberberg and P. F. Mijnlieff, J. Polymer Sci., A2 1970, 8, 1089. M. A. Harrison, P. H. Morgan and G. S. Park, European Polymer J., 1972, 8, 1361. 80GENERAL DISCUSSION a1 Dr. G. S . Park (UWIST, Card@) said: As Franks remarks, the persistence of crystallites at temperatures above T, creates a problem. A similar problem is found when other methods are used to detect the ultimate melting point in copolymers. Flory has pointed out that eqn (4) is applicable to the disappearance of those crystal- lites that have been made from the coming together of very long sequences of repeating crystallisable units. Since the probability of occurrence of these long units is very small, the concentration of crystallites at the ultimate melting point of the polymer is minute and so X-ray methods, specific volume and heat content measurements tend to underestimate the true crystalline melting point.The underestimation that comes from equating T, to the ultimate melting point may be somewhat greater than is found in the other method. An attempt to deal with this situation has been made by Taka- hashi et aL2 They have equated the gel melting point with the point at which each polymer chain goes through at least two crystalline regions. For multifunctional junction points this considerably overestimates the critical number of crystallites required for network formation. Nevertheless, it is interesting to look at the relation- ship that they obtain for the gel melting point.This can be expressed in the form 1 1 -ln(l-x)--ln(l-$)--ln N + - r V V 5 V V where the degree of polymerization is N , 0 is the excess surface free energy per mole associated with the terminal unit at the end of the crystallite (the surface free energy per unit), 5 is the crystallite length and the other symbols have the same meaning as in our paper. For small values of x we can now rewrite eqn (1) in the following way Here, A4 is the mean molecular weight, per monomer unit of the polymer chain, v is the specific volume of the polymer, and the other symbols have the same meaning as in our paper. Eqn (2) can be compared with the various relationships used in our paper and when 0 is small it can be reduced to eqn (5) of our paper and hence gives the values of AHv as predicted by that equation. Comparison with eqn (2) of our paper gives which gives an energy having the same meaning as AHx.The ratio of alnC/a(l/T,) to VvAHv is 5 even if 0 is not small so the crystallite lengths given in table 2 are also those that would be predicted from the relationship of Takahashi et al. Prof. C. A. Smolders (Enschede) said: The dependence of gel melting points on polymer concentration at constant copolymer composition could, in principle, also be treated on the basis of a concentration dependence of x. In a comparable T,(c) study of the system poly(2,6 dimethyl-l,4 phenylene oxide) P. J. Flory, Trans. Faraday Soc., 1955, 51, 848. A. Takahashi, T. Nakamura and I. Kagawa, Polymer J., 1972,3,207 ; A. Takahashi, Polymer J., 1973, 4, 379.P. T. van Emmerik and C. A. Smolders, European Polymer J., 1973, 9,293.82 GENERAL DISCUSSION in toluene, where indeed the concentration dependence of x is noticeable, a theory of melting point depression could be worked out quantitatively and checked by experi- ments. Dr. G. S. Park (UWIST, Card@) said: The correct prediction of the effect of toluene on the melting point of 2,6-dimethyl-1,4 phenylene oxide from x values obtained from cloud point determinations is an interesting demonstration of the validity of eqn (4) of our paper for the melting of homopolymers. In our studies of poly(viny1 chloride)-dioxan gels we made an attempt to obtain the concentration dependence of the gel melting point by assuming a constant x in eqn (4). This gave the wrong sign to By substituting the experimental values of T, and 4 in eqn (4) we were able to calculate the x values at each pair of T' and 4 values but x values obtained by other methods were not available for comparison.A similar situation would hold for the copolymer gels. Prof. C. A. Smolders (Enschede) said: Gel melting points of thermoreversible gels can be nicely determined by differential scanning calorimetry. When using this technique one should keep the system at a temperature below 7" for a sufficiently long time for the gelling process to proceed. In a specific system ~tudied,~ viz. poly(2,6-dimethyl-l,4phenylene oxide) in toluene, this meant that we had to keep the sample at the low temperature for at least one hour, before the endothermic peak on reheating showed up, that enabled the determination of T,.Dr. D. S . Reid (Unilever) said: Application of the van't Hoff isochore assumes that junctions are formed by a two-statey all-or-none process, and also that the junc- tions are monodisperse (i.e., the crystallites are all of the same length ). If these conditions are not met, the enthalpies estimated by this type of approach can be seriously in error. An example which shows just how wrong van't Hoff enthalpies can be is contained in the paper by Reid et al. where, due to the polydisperse nature of the species associating to form junctions, the vant' Hoff enthalpy corresponds to a junction size which is only about 30 % of M, of a junction. In view of this, I wonder why the authors have not carried out more extensive calorimetric studies in order to measure directly the enthalpy changes which occur during the melting of the gel, and correlate these with the composition of the polymer, and the nature of the solvent.Dr. G. S . Park (UWIST, Card$) said: I was surprised that Reid et al. found such large differences between the van't Hoff and the calorimetric enthalpies. It would be interesting if differences of this size occur in our AH values but it does not appear possible to obtain calorimetric values in our system. In the study of helix-coil transmissions made by Reid et al. it is assumed that the whole of the polymer under- goes the transition and so it is easy to calculate the enthalpy change per polymer mole- cule or per polymer unit from the total heat change that takes place in the system.The crystalline-amorphous transition that produces gel melting in our copolymer gels or for instance in poly(viny1 chloride) gels involves only a portion (and probably a very minor portion) of the polymer. Neither the number of network junction points nor the number of monomer units involved are known and so measurements of total heat changes would not allow the required values of the enthalpy per junction point to be calculated. P. T. van Emmerick and C. A. Smolders, European Polymer J., 1973, 9, 157. M. A. Harrison, P. H. Morgan and G. S. Park, European Polymer J., 1972, 8, 1461. P. T. van Emmerik and C. A. Smolders, European Polymer J., 1973, 9, 293.GENERAL DISCUSSION 83 Dr. I. Tar (Lorand Eiitv6s University, Hungary) said : The Harrison-Morgan-Park method (1971) was applied to our model, gelatin+water in 1972.Gelatin does not belong to the same group of gels as Park’s vinylidene chloride copolymers according to the system introduced by Flory in his Introduction. In spite of this, it could be of interest to show some of our results here, because we worked with exactly the same method and looked at a similar problem, the effect of solvent properties on the gel melting point and the heat of gelation, as did Park and his coworkers. We achieved melting points (T,) for isoelectric gelatin gels (pH = 5.33) of various concentrations using the H-M-P microcapillary viscometer and worked out results for AHusing the method of Eldridge and Ferry (1954). It was our aim to show if well known phenomena, such as the strong salting-out effect of Me” chlorides, or the structuring effect due to nonionic surfactants manifest themselves in reasonable AH changes.This being the case, it is our opinion that AH data are useful to detect changes brought about by co-solved substances in the gelling system. heat of gelation AHlkcal mol-1 character and concentration of the co-solved substance (g/dl) hydrophile ionic NaCl BaCI2 Na2S04 5.0 - 68.4 5.0 -61.4 5.0 - 62.9 nonionic PEG M = 550 2.0 - 67.1 PEG M = 2000 1 .o - 60.4 amphiphle ionic NaDS c 1 2H2 5 S04Na 0.056 - 63.4 a nonionic BEROL 043 C18H37E01 oOH 1.0 - 68.6 b M = 711,04 2.0 - 74.5 BEROL 08 c 1 8 H3 7E05 0 OH 0.5 - 74.5 M = 2473, 2 1 .o - 75.6 Rousselot photogelatin in dist. water pH = 5 , 33 - 67.45 2.4 a M.Kustos, Thesis, 1974 (Budapest) ; b Z s . Fodor, Thesis, 1972 (Budapest). Dr. G. S. Park (UWLW, Cardif) said: It is gratifying that the values Dr. Tar obtains for AH correlate with other known effects such as structuring and salting out of the gelatin system. It would be interesting to investigate parallel changes in rigidity as we are now doing in poly (vinyl chloride)-plasticizer systems. Dr. M. Pyrlik and Prof. G. Rehage (Clausthal) (partly communicated) : We wish to comment on the method of determining gel melting points used by Park et al., and, as a consequence, on the evaluation of the results. The method is able to determine a certain state of fluidity of the system, in other words, one measures an iso-viscous state of the melting gel. We believe that it is hardly possible to get the melting point of the largest crystallites (which clearly melt at I.Tar, Zs. Fodor and E. Wolfram, Magy. Ke’m. Foly’oirat, 1973, 79, 532.84 GENERAL DISCUSSION the highest gel temperatures of the melting range) by measuring a viscosity property of the system. This can be proved by the fact that melting of the crystallites, which have to be regarded as the junction points of the network, is not indicated solely by the complete dissociation of the network. Even at the temperature of the commencement of flow, larger aggregates-parts of the network structure-are present in the melt indicating that only a certain proportion of all the crystalline junctions are already molten. We have performed dynamic mechanical measurements on thermally reversible gels with stereospecific PMMA and have found that even at very low rates of temperature rise (3”/h) the temperature dependence of the loss modulus G” (and, consequently, of the viscosity) is by no means a step function but-over a temperature of some 40°C-a gliding Also, far below the commencement of flow, which is measured in the apparatus of Park et al., the slope of the (G”, temperature) curve is considerable, according to our results.0,2 0,4 0,s 0,8 Y z FIG. 1.-Phase diagram of the system gelatine+water. Ts = melting temperature; y2 = weight fraction of gelation. Method : DSC. The disappearance of the temperature dependence of the storage modulus G’ is a better criterion, because “ normal ” macrornolecular solutions, in most cases, do not show a strong temperature dependence of the storage modulus.Also in a melting gel small parts of the dissociating network network structure contribute to Gr ; only after complete melting of the largest crystallites does the storage modulus reach a minimum value. This is the case usually above the temperature, at which the gel is molten. Determinations of gel melting points by measurement of viscous properties can, therefore, not yield a thermodynamic equilibrium value for the melting point of the crystallites. Evaluation of these melting temperatures according to an existing theory, for example the Flory theory of melting point depression or the copolymer theory, is not really possible. Evaluation of melting points using the Ferry-Eldridge relation- M. Pyrlik, W. Borchard, G.Rehage and E. P. Uerpmann, Angew. Makromol. Chem., 1974,36, 133. M. Pyrlik and G. Rehage, Rheol. Acta, in press. M. Pyrlik and G. Rehage, Kolloid-2. 2. Polymere, in press.GENERAL DISCUSSION 85 ship seems to be possible, if the slopes of the concentration dependence of the measured and the equilibrium melting pionts are the same. However, it must be stated, that the Ferry-Eldridge relationship can describe the concentration dependence of the melting points of the crystallites only over a certain concentration range. Most of the solvent-polymer systems have a phase diagram showing a eutectic. Consequently, the gel melting points of gels with crystalline junction points must have similar concentration dependences showing a point of inflection. The figure shows the melting points of an aqueous gelatin gel against the concen- tration, measured using D.S.C.techniques.l The curve is identical with the right hand side of a common eutectic phase diagram. It is clear that the Ferry-Eldridge equation is only applicable to the very low concentration part of the curve, which shows a distinct increase of gel melting points with concentration. Application of the relationship to the main part of the curve (with low concentration dependence) yields results for the crystallization enthalpy AHcr, which are apparently too high. Dr. G. S . Park (UWIST, Cardifl) (communicated) : Rather than giving a measure of an isoviscous state, we believe that our technique gives a good approximation to the temperature at which transition from completely recovei able elastic deformation to a state of non-recoverable viscous flow occurs.This is the sol/gel transition temperature and is independent of the actual value of the viscosity. For this reason the shape of the (G”, temperature) plot does not appear to have relevance for our measurements. Nevertheless, I would agree that unmelted crystallites are still present at temperatures above our gel melting points and the suggestion of Pyrlik and Rehage that the disappearance of temperature dependence in the storage modulus, G’, could give a better measure of the disappearance of the last network traces in gelling systems is useful. Our measurements determine the point at which the number of crystallites reaches such a critical value that the continuous network disappears and so the kind of treatment referred to in our reply to Franks should be applicable.Prof. M. Gordon (University of Essex) : Edwards’s paper presents a fundamental and comprehensive physical theory of polymer dynamics. As in his equilibrium theories, Edwards uses a framework of continuum mathematics. Not only is the polymer chain smoothed to a continuous line, but an infinite number of integrations (Wiener integrals) may be carried out on suitable functions. Now the elementary graph theory, inherent in the usual discrete descriptions of chains and networks (consisting of point-atoms and line-bonds) ought to be contained in Edwards’s more sophisticated formalism, presumably by virtue of an isomorphism of the underlying operator algebras of the continuum and graph-theoretical operators, which is largely unexplored. I have two specific questions : (a) Edwards’ formulation covers the whole range from liquid solutions, through the critical entanglement point, to covalently cross- linked gels, and should, therefore, deal with the transition regions. Near the gel point, long-range effects embodied in the “ extinction probability ” u (see paper by Burchard et al., this Discussion) dominate the properties. For instance, in reversible gelation, the gel point is an Ehrenfest transition of order five.Part of this order comes from the fact that the modulus is a second derivative of the free energy, but three further (temperature) differentiations of the modulus are required before we encounter a discontinuity at the gel point.This is due to the fact that the modulus (or concentra- tion of active network chains) is proportional to (1 - v ) ~ , and (1 - u) vanishes at the gel K. Bergmann, Diplomarbeit (TH Clausthal, 1974).86 GENERAL DISCUSSION point (cf. Burchard et al., this Discussion, eqn (39), using eqn (37) and (38)). Thus the high order of the transition derives from the unprecedented effect of long-range cor- relations embodied in the extinction probability, and this is well supported by experiments on widely different amorphous gelling systems. Edwards’s formulation appears on the surface to be entirely a local one (a primary chain wriggling in a tube). How does a segment of the primary chain know that it is connected by covalent paths to the surface, and how is the transition covered in the continuum formulation? (b) Gel formation is not dependent on pre-existing primary chains.In an f- functional polycondensate, active network chains can be traced after the gel point, but are only brought about by the linking process itself. The discrete (“ graph-like- state ”) formalism covers all gelling systems with the same equations, when suitable parameters and functions are substituted (see DuSek, this Discussion, eqn (5)-(7)). Thus thef-functional condensates also have a fifth-order gel transition, as can be derived from Dobson and Gordon’s eqn (12) and (13).3 Can Prof. Edwards generalise his formulation to free it from the restriction to pre-existing primary chains? Prof. S . F. Edwards (Science Research Council) said: The use of functional integration methods is merely a convenient notation.The content is entirely con- ventional. Thus the discrete nature of atoms and bonds can be built in if one wishes. This puts in a lot of detail however which is unnecessary to describe long range effects, and since it is these long range effects which differentiate polymer physics from other areas, a formalism which permits the discarding of unwanted detail seems worth pursuing. This comment covers Gordon’s point (b) : there is no difficulty in writing the statistical mechanics of any system in these terms, it is indeed standard practice in all of normal solid or liquid state physics in the second quantization formalism (which need have nothing to do with quantum theory). As regards (a), since the formalism is merely a representation of the usual set up, when one puts in the same assumption, one gets out the same conclusions.I must confess some surprise at Gordon’s confidence that the transition is exactly of order five, since I would have thought an exact answer would imply a solution of well known but hitherto insoluble problems such as the percolation problem. However the ferroelectric transition is soluble whilst the ferromagnetic is not, so perhaps that is right. I am not an expert in this matter, and my contribution is confined to study on either side of the transition except in as much as I and Grant have studied the dynamics of diffusion through the transition (4.v.). However it is clear how the transition would arise: the radius of the confining pipe tends to infinity at the transition.Dr. B. Warburton (London University) said: Some gelling processes in practice take place in a surface.4* I would like to ask Edwards if his theory is sufficiently flexible in its generality to be applicable to two dimensional behaviour of chains. At first sight it would appear that equations of the type (1.1) and (1 -2) of his paper have an equivalent form in two dimensions but when complex topological considerations arise, such as those discussed at the end of the present paper and in the author’s M. Gordon, Trud, 1969, Meshdinarodnoi Conf. Kautsh. i. Resine Chemia, Moscow, 1971. see e.g., S. Strella and A A Bibeau, J. Macrornol. Sci., 1966, 1,417. G. R. Dobson and M. Gordon, J. Chem. Phys., 1965,43,705 ; Rubb.Chem. and Technol., 1966, 39, 1472. K. Wibberley, A study of some surface properties of solutions of salts of arabic acid, Ph. D. Thesis (University of London, 1963). E. Shotton, K. Wibberley and A. Vazin, Proc. 4th Znt. Congr. Surface Act., 1967, 11, 1211.GENERAL DISCUSSION 87 it would seem that any theory for the surface would need previous publications,l* to start ab initio. E.g., the invariant expression, I = II(dAxdB)-V(A-B)-l (eqn (2.22) reference (3)) for the condition of encirclement of one curve by another would be meaningless in two dimensions. We have been studying the formation of gel films at aqueous/air and aqueous/oil interfaces in connection with a programme of research into the action of polysacch- aride emulsifying agent^.^'^ It would appear that good emulsifyers, judged by the mechanical stability of the oil-in-water emulsion formed, form a viscoelastic solid or gel film at the interface but in order to allow the emulsion to have complete mobility in its continuous phase no gelation must occur in the bulk solution. Prof.S. F. Edwards (Science Research Council) said : The surface gelation, like all surface phenomena, cannot be a truly two dimensional phenomenon, but must correspond to a state of affairs which decays away quickly as one moves away from the surface. A combination of the theory of surface tension effects with the normal gelation should be possible, but I have not attempted it. The invariant still has a meaning under these conditions. In a mathematically exact two dimensional problem it becomes the angle swept out by a surface chain about a point (say the end of a chain at right angles to the surface).Entropic problems can be solved exactly in that limit, but I suspect that is not really helpful for the three dimensional problems, consequently the thickness of the surface is essential. Prof. A. Silberberg (Israel) said: Many years ago it occurred to me that the single effect on which flow in condensed polymer systems depends is a concertina-like contraction and expansion of the macromolecular chain along sections of its length. Brownian motion of the segments in this direction should be less inhibited and thus relative sliding of neighbouring parallel chain sections is enabled. Developing this model one can immediately derive the 3.5 power dependence of steady state viscosity on polymer molecular weight.These results were never published since I discovered that Eyring had years earlier already written a short paper in which similar ideas were outlined in their essentials. Prof. P. J. F'lory (Stanford) said : A note of caution should be recorded on the use of the freely-jointed model for real chains of limited length. Detailed calculations have been carried out by Drs. Yoon, Conrad and Chang in our laboratory on poly- methylene, poly(dimethylsi1oxane) and polypeptide chains using a three-dimensional expansion in Hermite tensor polynomials. These calculations show the density distributions for chain vector r for fewer than about 50 bonds to be highly anisotropic. In keeping with Monte Carlo calculations of Semlyen and coworkers and of Fixman and Alben on the distribution of scalar r, the probability density exhibits a minimum S.F. Edwards, The statistical mechanics of rubbers in Polymer Networks, Structural and Mech- anical Properties, ed. A. 3. Chompff and S. Newman (Plenum Press, New York, London 1971), p. 83. S, F. Edwards, Disc. Faraday SOC., 1970, 49, 43. E. Shotton and R. F. White in Emulsion Rheology, ed. P. Sherman (Pergamon Press, London, B. Warburton, The properties and kinetics of formation of some hydrocolloid surface films, Ph. D. Thesis (University of London, 1972). M. S h e d and B. Warburton, Polymer, 1974, 15,253. 1963), p. 59.88 GENERAL DISCUSSION near r = 0 where the usual formulation in terms of a Gaussian predicts a maximum. Thus, the departures from the Gaussian distribution are severe, this being in addition to inadequacies of the widely used relation < r 2 ) = n1b2 where n1 is the number of hypothetical bonds and b is the length of one of them. Calculations of ring closure probabilities that (i) rely on this approximation, (ii) involve assumption of Gaussian density at r = 0, and (iii) take no account of the possible effects of angular correlations as noted by Stepto, may be subject to considerable error for short chains.On the other hand, (i) and (ii) affect such calculations oppositely, and may therefore be partially compensatory. The approximate correlation of b values in table 1 of Stepto’s paper with those derived from measurements of the unperturbed dimensions of longer chains may perhaps be explained in this way.If so, the agreement must be regarded as fortuitous. Secondly, I should like to suggest that revision of Kilb’s theory to take account of the array of all isomeric species containing x branches may lead to a lower, rather than a larger A’. Thus, in the simple case of a single kind of reacting group A, only the linear isomer A* -AA-AA-A, etc. 1 A I A I A commencing with the chosen group A* and having exactly one unreacted A group at each “ generation ” is considered. It we take the average over all of the ramified “ x-mers ” that can be constructed about an arbitrarily selected root A*, then the expected number of unreacted A groups at the first generation obviously will be < 1 ; in the limit x+ co it is 3. At the second generation, it turns out to be 1 .O in the same limit.Since the ring closures of short length make the largest contribution, it does not seem to follow that elaboration of Kilb’s theory to take account of the full array of isomeric structures would necessarily raise 2‘. For higher generations, the expectancy must become > 1. Dr. G. J. Howard (UMIST) said: An experimental technique which may be used to examine the extent of intramolecular reaction in reactions of the type discussed by Stepto is to locate the critical condition at which gelation will just fail to occur : this may be carried by adjustment of the dilution of the system or by alteration of the ratio of the reacting functional groups. Some 10 years ago, Bates measured the time to gel in the reaction between a low molecular weight styrene/allyl alcohol copolymer and some aliphatic di-acid chlorides.The figure shows a plot of the reciprocal gel times against the acyl chloride/hydroxyl ratio at constant polyol concentration. [Curves 1, 2, 3 are for one crosslinker at three (increasing) concentrations of the styrene/allyl alcohol copolymer. Curve 4 is the same concentration of polyol as 2 but with a shorter crosslinking reactant (adipoyl chloride rather than suberyl chlor- ide)]. As may be seen, it is relatively easy to locate the limiting reactant ratios. These are the conditions whereby the minor component is just consumed at the critical con- dition for gelation. The intercepts on the axis represent the critical branching coefficient, or its reciprocal. The main point I wish to make is that it is possible to learn something of the intramolecular wastage without the necessity of determining extents of reaction by functional group analysis of the reacting systems.The rather ill-defined polyfunctional component in these reactions perhaps makes these results unsuitable for a critical assessment of the theoretical approaches dis- R. F. Bates, Ph.D. Thesis (Manchester, 1965) ; R. F. Bates and G. J. Howard, J. Polymer Sci., C, 1967, 16, 921.GENERAL DISCUSSION 89 cussed by Stepto. However, it should be noted that, within experimental error the intercepts are reciprocally related. This implies that the term Cext is satisfactorily expressed by the initial polyol concentration alone in this system ; CA + C,, whether taken as initial or final concentrations, varies by a factor of about 2, typically, from one intercept to the other.Within the concentration range studied, the Kilb parameter II’ of Stepto’s eqn (2) is linearly related to the reciprocal concentration of the polyol, but the plot does not go to the origin-indeed it lies rather distant from the origin. The parameter ;1 from Stepto’s eqn (10) is, however, directly related to the reciprocal polyol concentration, i.e., by lines similar to those in Stepto’s fig. 5 but with (CA); as the axis. Dr. R. F. T. Stepto (UMIST) said : Howard’s reanalysis of the results of Bates and Howard appears to give support to the use of the Frisch expression in the analysis of gel-point data. The precise definition of cext to be used in the expression for 3, is still unresolved. The gelation data discussed in the paper (particularly those of Peters and Stepto) show that a, depends on the concentrations of both reactants, and c,, + cbo and c,, + cbc were chosen for text as the simplest expressions displaying such a dependence.The apparent insensitivity of ;1 to c,, (initial concentration of difunctional component) shown by the data of Bates and Howard may be due to the high functionality of the polyol employed by them. In other words, text could depend not only on the con- centrations but also on the functionalities of the reactants. Certainly, more compli- cated expressions than c,, + cbo and c,, + cbc are required to bring the data of Peters and Stepto onto single curves. The method used by Bates, which obviated the necessity of determining extents of reaction at gelation, does have the disadvantage that several gel times have to be determined for each a, evaluated.Dr. S. B. Ross-Murphy (University of London) said: I would like to ask Stepto how a, the gel point conversion, was measured in the cyclisation experiments mentioned in his paper? Further, with reference to fig. 2 of this paper, (which shows plotted against dilution) to what accuracy could a, be measured? One would expect that the parameter 2 and the “ effective bond length ” b of table 1 would be very sensitive to small errors in a,.90 GENERAL DISCUSSION Dr. R. F. T. Stepto (UMIST) said: The method of determining a, was through the extrapolation of (conversion, time) curves to the gel time.The gel time was measured as that time at which the reaction mixture climbed the stirrer in a sealed reaction flask. A test of the reliability of this method in absolute terms has been reported by Bates and Howard,' who showed it corresponded to the maximum time of zero gel fraction. The accuracy to which a;' was typically determined may be seen from the results given by Peters and Stepto.2 In general, a;' was determined to better than & 1 %, and duplicate experiments, not reported in the literature, confirm this. The effects of the uncertainty in a, on the values derived for A and b are negligible. This applies particularly to b which is directly proportional to A-+. Dr. J. A. Semlyen (University of York) said : Cyclization processes frequently take place in gelation reactions and have been mentioned already by Flory in his Introduction and in several papers presented at this Discussion. In Stepto's paper, reference was made to the Jacobson and Stockmayer expression for estimating intramolecular reactions in his systems.In particular, chains were assumed to obey the Gaussian expression for the probability of intramolecular cyclization. He emphasised that v (the number of bonds in the smallest ring that can form) must be sufficiently large for Gaussian statistics to apply as the first term in the sum @(l, 3) (of eqn (5)) constitutes 38 % of its value. Values of v for his polyesters and polyurethanes range from 34 to 136. Very briefly, I would like to present evi- dence from a single aliphatic polyester system which suggests that his values of v may indeed be sufficiently large.Recently, Dr. Jones in our laboratory has measured the molar cyclization equi- librium constants for rings in poly(decamethy1ene adipate) and compared these values with those calculated by the Jacobson and Stockmayer theory, which were made by assuming that chains adopt random-coil conformations and obey Gaussian statistics. The discrepancy between experiment and theory is small for the cyclic dimer ((O~(CH2)lo-O~CO-(CH2),CO)2) with 36 skeletal bonds and it is negligible for the larger cyclics, with 54, 72 and 90 skeletal bonds respectively. Therefore, on the basis of these results, Stepto's use of eqn ( 5 ) appears to be valid for the systems he has studied. Dr. I(. D&ek (Czechoslovak Academy of Sciences) said: It should be pointed out that the present theories of cyclization in crosslinking reactions are based on the conformational statistics of linear sequences of bonds connecting the reacting func- tionalities and neglect correlations due to already existing cycles.Therefore, the theories are asymptotically exact at zero degree of cyclization. The cycles already present influence the probability for the reacting functionalities to come into close contact : if the chain stiffness remained constant the already existing cycles would increase the cyclization probability. This follows also from the application of the random flight statistics to simple cyclic structures carrying reacting functionalities. However, the cycles may increase also the chain stiffness which has an opposite effect on cyclization.According to the results of our statistical treatment of the crosslinking polymerization of diallyl phthalate, the neglect of correlations due to existing cycles seems to underestimate cyclization, especially in the early stages of polymerization. When the degree of cyclization is increased by carrying out the polymerization in the presence of diluents, the fraction of crosslinks wasted in cycles does not grow and R. F. Bates and G. J. Howard, J. Polymer Sci. C, 1967, 16, 921. R. H. Peters and R. F. T. Stepto, S.C.I. Monograph, No. 20, 1966, p. 164.GENERAL DISCUSSION 91 unsaturation does not fall so fast as expected and the stiffening (and possibly the excluded volume) effect seems to become more important. Dr.R. F. T. Stepto (UMZST) (communicated) : Flory’s note of caution concerning the use of Gaussian statistics for short chains finds support not only from the results of the calculations he mentions, but also from the earlier experimental work of Semlyen and co-workers on ring-chain equilibria, involving oligomers of substituted siloxanes and other monomers. However, the results just quoted by Semlyen on ring concentrations in poly(decaniethy1ene adipate) indicate that, for linear chains which are similar in flexibility and numbers of bonds to the chains forming the smallest rings in our systems, the assumption of Gaussian statistics near Y = 0 is justified, at least numerically. The use of Gaussian statistics for gelling systems can also be criticised on the grounds that one is concerned with the mutual concentrations of pairs of chain ends, not of linear chains, but of branched chains.In this case, given one chain end, the configurational behaviour of the other ends is interdependent as they are connected by sequences of skeletal bonds. This is mentioned in the paper following eqn (5) and is emphasised by DuSek’s remarks. However, the approach has been taken that the errors inherent in the Frisch-Kilb approach are more serious than those resulting from the use of these statistics. The point raised by Flory that Kilb’s counting may overestimate, rather than underestimate A’, is an interesting one. Extension of his calculations to higher generations is required before any definite conclusions can be reached. The difficulty of enumerating the isomeric species and the possible sites for intramolecular reaction of course increases with the generation number, but such enumerations could lead to alternative expressions for the gelation condition. The resulting expressions, however, would still refer only to ring formation in infinite species at a particular point in the reaction. The calculations given show clearly that the probabilities r, of eqn (3) should contain weighting factors which are a function of n. In general, such factors will depend also on$ Indeed, a similar calculation for f= 4 shows that at gelation, and in the limit of zero rings, the expected number of unreacted A groups at the first generation is > 1. Gaussian statistics are only a first approximation. Prof. M. Gordon (University of Essex) said: The paper by Burchard and co-workers illustrates how well correlations in structural statistics can be dealt with by the graph- theoretical formalism which lies at the root of the cascade method. Effects due to local correlations are transmitted over long distances here; their fig. 1 shows how a difference in reactivity of two neighbouring functionalities, a local feature, has conse- quences a long distance from this locality. The necessary information is carried from generation to generation up the “ family tree ”, by an elegant implementation of the facility for labelling of the auxiliary variables of the generating functions. In this way, exact calculation of many measurable statistical parameters is achieved despite the correlation trouble. One aspect makes the advance in theory specially important : chemical reactions affecting a gel molecule are always subject to correlations of indefinitely long range, owing to packing restrictions among other reasons, and further development of mathematical tools to cope with this situation is urgent. see M. Gordon and T. C . Parker, Proc. Roy. SOC. Edin. A , 1970/71, 69, 181.