Dependence of the Gel Point on Molecular Structure and Reaction Conditions BY R. F. T. STEPTO Department of Polymer and Fibre Science, The University of Manchester Institute of Science and Technology, Manchester, M60 1 QD Received 2nd January, 1974 The interpretation of gelation in irreversible polymerisations is discussed in terms of a combination of the Frisch and Kilb theories. According to this a branching parameter, A, is determined from the equation A = (acCf- 1)- 1)/((1- ac)a&- l)), and is interpreted using the relationship A = ($rub2)* ~$(l,~)/iVc~~~. In these expressions ccc is the critical value of the branching coefficient, f is the branch-pomt functionality, u is the number of bonds in the smallest ring which can form, b is the effective bond length of these bonds, d(l, t) is a constant, and text is a measure of the average concen- tration of a reaction system.The interpretation is applied to data from polyurethane and polyester forming systems in which f = 3. Two expressions for c,,t are tried-the average initial and average gel-point concentration of reactive groups, (cao+ cb0)/2 and (cacf ct,,J/2, respectively. The polyester results in particular are satisfactorily interpreted in this way, and over-a11 the results indicate a preference for the use of ( ~ a c + q,,-)/2. With this expression the correct limiting behaviour of X at small c;; is found, and the values of b derived from the preceding equations are in better accord with those expected from solution properties. Although the classical theory of gelation due to Flory and to Stockmayer provides the basic explanation of the occurrence of gelation in non-linear polymerisa- tions, it is well-known that the gel point often occurs at extents of reaction larger than those predicted.Such excess reaction can result from the unequal reactivities of like functional groups, and from the formation of ring structures through intramolecular reaction. The former cause is particularly prevalent in reaction mixtures used in commercial resin preparation. However, even when equal reactivity is assured intramolecular reaction occurs to an appreciable extent, even in the absence of solvent, as shown by the data of Stanford and Stepto.2 General formulations of polymerisation statistics to include both unequal reactivity and intramolecular reaction have been described by Gordon and co- workers.3-5 Here, two cases can be distinguished, the equilibrium and non-equili- brium. The non-equilibrium case is the more difficult to treat but is perhaps the more relevant as in many polymerisations based on reversible reactions chemical equilibrium is not allowed to establish itself, and in addition many polymerisations employ irreversible reactions. Descriptions of the non-equilibrium process are essentially in terms of sets of rate equations which describe the interconversion of molecular In addition, the equations can be defined so as to allow computation of the gel point without detailed calculations of the concentrations of all the molecular species. The sets of equations so far developed for irreversible non-linear polymerisations are accurate only at small extents of r e a ~ t i o n , ~ ~ and because of the algebraic and computational effort required in their extension it is pertinent to enquire how far gel points in such polymerisations can be predicted and explained in terms of simple 6970 GEL POINTS analytical expressions.The factors which influence intramolecular reaction are dilution of the reaction system, the distribution of displacement lengths between reactive groups on the same molecule, and in a non-linear polymerisation the numbers of pairs of groups on a given molecule which can react together. Ideally, an expres- sion describing the dependence of the gel point on intramolecular reaction should relate to all three factors. If, in addition, unequal reactivity is to be accounted for, the work of Gordon and Scantlebury indicates that it is doubtful whether such an expression can be found.Case has developed equations for the gel point in the absence of intramolecular reaction, which account for what may be termed the intrinsic unequal reactivities rather than the induced unequal reactivities (or sub- stitution effects 4, of like functional groups. Several approximate theories of gelation which relate the gel point to the amount of intramolecular reaction have been published. Of these, the theories of Frisch l 1 and Kilb l3 lead to expressions which may be simply applied to gelation data from the polymerisations offfunctional reactants (RA,), and from the polymer- isations of f-functional and difunctional reactants (RA2 + RB,). However, in order to examine critically the performance of these expressions it is necessary to have data from systems in which like groups are known to have equal reactivity, and in which the previously mentioned factors affecting intramolecular reaction have been varied systematically.Such data on polyurethane and polyester forming systems 6* 9* 14-16 exist, and the aim of this paper is to use this data and the Frisch and Kilb theories to interpret the dependence of the gel point on molecular structure (chain lengths, and chain structures of reactants), and on reaction conditions (dilution, and ratios of reactants). Initially, the Frisch and Kilb theories are discussed and compared. KILB’S THEORY OF GELATION For RA2 +RB, and RA, polymerisations, the classical theory of gelation gives aJf-1) = 1 (1) at the gel point.a is the branching coefficient, it being the probability of passing along one sequence of bonds from a randomly chosen branch unit to another branch unit, given that the chosen unit has already one of its groups reacted, i.e., that the chosen unit is already part of a chain. Given one group reacted, there are f- 1 possible routes to the next unit so that aCf- 1) is the probability that the next unit is reached. When this probability is equal to unity unlimited molecules are formed, and the gel point is reached. As indicated in eqn (1) this occurs when a = a,. Given equal reactivities, a = PaPb for RA2 + RB, polymerisations, and for RA, polymerisations a = pa, with pa and Pb the (fractional) extents of reaction of A- and B-groups, respectively.Henceforth, the symbol a, will be used merely to represent PaPb or pa at the gel point, rather than the critical value of the branching coefficient. Kilb’s theory l3 shows that at gelation, with small amounts of intramolecular reaction, a,(f- 1)(1 -A’) = 1, where A‘ is an intramolecular branching parameter. A‘ 4 1, and is the probability that a sequence of bonds from a chosen branch unit in a chain leads not to a new branch unit but back to one already in the chain. Eqn (2) was derived for RA2 + RB, polymerisations. However, it holds true for RA, reactions as there is still a fixed number (zero) of reactant units between branch points. (A more complicated expression l 3 is required for RA2+RB2 +RB, reactions, as here the number of units between branch points is of infinite range).R.F. T. STEPTO 71 In detail, 3,’ is the total probability of intramolecular reaction, and may be written as the sum of the probabilities over all sizes of rings, i.e., A’ = C r, . (3) n= 1 In this sum, rl is the probability that a ring forms with the preceding branch point (the chosen branch point), r2 the probability that one forms with the branch point preceding that, and so on. Remembering that the molecules are highly branched, this expression will underestimate the amount of intramolecular reaction as it assumes only one possible ring structure of each size. In other words, it assumes a linear sequence of branch units each with one free functionality, which in RA,+RB, systems can be an A- or a B-group.Kilb gives a molecular interpretation to A‘ by an approximate application of Jacobson and Stockmayer’s expression for the equilibrium ring and chain con- centrations in linear polymers. An expression for A’ similar to Kilb’s may be derived by more straightforward arguments, and these show clearly the approximations involved. Consider a terminal group on a growing chain (in the RA2 + RB, case this could be A- or B-), and assume that the rate of reaction intramolecularly or intermolecularly is proportional to the concentration of groups belonging to the chain (tint) or external to the chain (text), respectively. This assumption requires that the intrinsic rate of reaction is insensitive to whether intra- or intermolecular reaction is occurring.It neglects orientation effects which may be prevalent in intramolecular reacti0n.l The probability of intramolecular reaction (A’) is then (4) clnt is defined by the chain statistics of the growing molecule. Taking the position of the terminal group as reference, the concentration at this position of reactive groups separated by vn bonds may be written as Cint Cint + Cext Cext N - Cint - . A’ = ~ 3 1 - (moles functional groups per unit volume). (A) N v is the number of bonds in the smallest ring which can form, and summing over all ring sizes (n) 4(1,+) = X:=l 1%-3, as defined by Truesdell.18 The value of 4(1, 4) is 2.612. Eqn (5) assumes Gaussian chain statistics. This is inexact not only because the chain is not linear, but also because the simultaneous concentrations due to different points of the same chain (i.e., with n = 1,2,.. .) are assumed independent. In addition, v must be sufficiently large for Gaussian statistics to apply as the first term in the sum +(l, +) constitutes 38 % of its value. The final expression for A‘ is Apart from the fact that molecular complexity will change with extent of reaction and will affect the value of b, cine is to a first approximation independent of extent of72 GEL POINTS reaction. In contrast, text is the instantaneous concentration of external groups, so that I’ will increase as the reaction proceeds. The use of a single value of A’ implies that some average value of text is employed, and at present there is no inde- pendent way of determining this.For RA2 + RB, polymerisations Kilb proposed cext proportional to c,,, the initial concentration of A-groups. However, the data of Peters and Stepto l4 on polyurethane forming systems show clearly that the amount of intramolecular reaction is not a function of the initial concentration of difunctional reactant alone. In general, results on poly~ethane,~* l4 and polyester l5 forming systems, which contain data with r(=c,,/cb0) not equal to 1, show that text is best taken as proportional to Cao + cbo, and not to c,, or cbo alone. In the present paper the average initial concentration, (cao+ cbo)/2, will be used as one estimate of text. This retains the symmetry of the Kilb theory with respect to A- and B-groups, and the interpretation of text as the concentration of external A-groups around a B-group, or vice-versa.An alternative expression is the average concentration at gel, (c,, + cbc)/2. The use of these two expressions implies that one is seeking to characterise the total intramolecular reaction up to the gel point in terms of the ring-chain competition occurring at single points in the reaction, namely at the start of the reaction and at the gel point. The corresponding expressions for RA, polymerisations are c,, and cat. FRISCH’S THEORY OF GELATION Frisch l1 accounts more completely than Kilb for the molecular complexity of the growing chains, and hence for the numerous opportunities for intramolecular reaction, but still employs a single intramolecular branching parameter, A. Frisch’s expression, which is valid for small I, is a,(f- 1)(1-(1 -ac)A+O((l -a,)2)) = 1.(7) Rearrangement, and the neglect of terms O((1 -a,)2) to give a, in terms of ;t yields (for R # 0) = --f(-&+(j-l)(I-I)’ (1-2) 2;t 1-2 4a )’ At I = 0, a, = df- 1)-l. Expansion of the binomial in eqn (8) with retention of the first two terms yields for small 1. The identity of eqn (9) and Kilb’s expression, eqn (2), enables the same molecular interpretation to be placed on I as on A’, at least for small A. The difference between the two theories is the functional dependencies predicted for a, upon 1 and A’. These are shown in fig. 1 for the case f = 3. The expansion of the binomial in eqn (8) is valid for 0 < A < (f+ 1 - 2 J f ) / ( f - 1) and for A > df+ 1 +2df)/cf- l), that is, for f = 3 , O < A < 0.268 and R > 3.732.However, fig. 1 shows that the Kilb approxi- mation will be inaccurate at values of a, sensibly distinct from 2. In contrast, the combination of the Frisch and Kilb theories, taking the functional dependence of a, upon ;1 from the former, and the molecular interpretation of ;t (or A’) from the latter, gives an expression which may more reasonably be applied to gelation data.g* Note that in eqn (7), 1 is not restricted to the range 0 < 3, < 1, but that in the limit a, equal to I, A is infinite. (In eqn (8), the range of values of a, is covered by taking the positive root for R < 1 and the negative root for I > 1.) This behaviour can ac(f- 1)(1-A) = 1, (9)R. F. T. STEPTO 73 arise because of neglect of higher terms in eqn (7).It means that although values of ;1 > 1 may be given by experimental data, any molecular interpretation of A must be characteristic of the behaviour near 3, = 0. FIG. 1.-Relationship between ct;' and the ring-forming parameters A and A' of Frisch and Kilb theories for f = 3. COMPARISON WITH EXPERIMENTAL DATA The experimental data to be considered are from polyurethane and polyester forming reactions of the type RA,+RB,. In all cases the reactions were with trio1 prepolymers having oxypropylene chains. The polyurethane forming reac- tions 6* 9* 14* l6 used hexamethylene diisocyanate (HDI) in benzene at 70", in nitro- benzene at 105", 80" and 50", and decamethylene diisocyanate (DDI) in nitrobenzene at 80". The polyester forming reactions used adipoyl and sebacoyl chlorides in diglyme (diethylene glycol dimethyl ether) at 60".No density corrections were made to the data and concentrations are expressed in units of mol kg-l. 1 . KILB'S THEORY AND cert According to eqn (2) and (6), a,' should be a linear function of A', with A' pro- portional to the dilution of a reaction mixture (c&t). With regard to the two alternatives for text, namely, (cao + Cba)/2 and (cac + cbc)/2, the presentation of most of the data with text proportional to ca0 + cbo has been given before 9* 14* l5 and will not be repeated here. However, fig. 2 shows a;' as a function of (cat + Cbc)-' for some of the polyurethane l4 and polyester l5 forming systems. In general, with respect to both average initial and average gel dilutions, linear behaviour is not observed, and in view of the subsequent interpretation according to Frisch's expression this indicates that the data lie outside the range of a,' to which Kilb's theory may be74 GEL POINTS applied (see fig.1). (It may be noted that the use of (C,,+C~~)-~ as abscissa gave apparent linear behaviour for the polyurethane data.14 In the light of the more recent results 6* 9* 15* l6 this appears to be because the data covered only a small range of values of I I I I I I 4 2 3 5 6 0 (cat+ %)-'/kg mol-I FIG. 2.-Polyurethanes l4 and polyester^.'^ a, against dilution at gel. X , + - polyurethanes (HDI+ OXTMP*)/nitrobenzene/l05" ; v = 34 ; (a) Cao = 1.77 mol kg-' ; (b) Cao = 1.18 mol kg-' ; (c) Cao = 0.89 mol kg-'. 0, O-polyesters/diglyrne/6O0. 1 .-adipoyl chloride + LHT240* ; v = 37 2.-sebacoyl chloride + LHT240 ; v = 41 3.-adipoyl chloride+ LHT112* ; v = 66 4.-sebacoyl chloride+ LHT112 ; v = 70 5.-adipoyl chloride + LG56* ; v = 132 6.-sebacoyl chloride+ LG56 ; v = 136 *OFTMP--oxypropylated trimethylol propane *LHT240, LHTl12-oxypropylated 1,2,6 hexane trio1 *LG56-oxypropylated glycerol In fig.2, the curves for the polyester systems show that the excess reaction at gelation is a decreasing function of v, as required by the expression for tint, eqn (5). This also occurred with (cao + cb0)-l as abscissa. The polyurethane results come from three series of experiments, each at an approximately constant initial isocyanate concentration (cao). Each of the series defines a separate curve, and the same behaviour was found with (ca,+ cb0)-l as abscissa.The results show that at a given text the number of rings formed increases with initial isocyanate concentration. This is consistent with other ring-data on this reaction in both linear l9 and non- linear polymerisations,2 and is probably due to peculiarities of the reaction.lg However, it may be that the average initial or gel concentration is too simple an expression to use for c,,,. This can be resolved if more systematic data, with respect to ratios of components, were obtained for other systems. For the majority of the polyester reactions, r s 1, and no systematic variation in the ratio of reactive groups was undertaken.R. F. T . STEPTO 75 2 . THE FRISCH-KILB THEORY AND EVALUATION OF b As stated previously, the combined theories of Frisch and Kilb provide a more useful basis for the interpretation of gelation data.mental values of ct, via eqn (7), i.e., 2 is evaluated from and is interpreted according to eqn (6), with 1. cc c&t. j0/* I. 6 4 ./’ 0.2 0.4 0.6 0.8 1.0 the experi- (10) - 1.2 (cao 4- cbo)-’ /kg m01-l FIG. 3.-Polyurethanes and polyesters. h against initial dilution. Legend as for fig. 2. Plots of 2 against ( C , , + C ~ ~ ) - ~ and (caC+cbc)-I are presented in fig. 3 and 4 for the same systems as in fig. 2. The polyester results follow essentially the behaviour predicted. Some curvature is present when ( C , , + C ~ ~ ) - ~ is used as abscissa, but this may be expected in view of the previously discussed approximations in the derivations of eqn (6) and (7).For the polyurethane systems (caC+cbc)-l would appear to be the better approximation to c&:, as direct proportionality between 2 and (c,,+ Cbo)-’ is not observed. (The lines a, b, c in fig. 3 for the polyurethane results are merely straight lines through the sets of points. Deviations from linearity must occur at smaller values of A in order that the results extrapolate to the origin). Fig. 5 and 6 show further results 2 s 9* l6 for polyurethane forming systems. These generally follow Frisch-Kilb behaviour at the smaller values of 2, although the scatter of points about the curves drawn is larger than with the previous results. This again may be due to the additional dependence of ring formation on isocyanate concentration. Most of the reactions refer to r E 1, and the effect of isocyanate concentration was not systematically investigated. Again more deviations from linearity occur when (c,,+cbC)-l is used as abscissa. The plots in fig.3 to 6 indicate that, provided ;1 is not too large, the results are in accord with the functional dependence of A upon a, predicted by eqn (lo), with text76 2.0 1.5 2.c I 4 1.; h 0.E 0.4 - - GEL POINTS b AC I 2 3 4 5 (Cat + &)-'/kg mol-' FIG. 4.-Polyurethanes and polyesters. A against dilution at gel. Legend as for fig. 2. X x I/ / / / 0 0.5 1.0 1.5 2.0 2.5 (cao + cb~)-l /kg mol-' FIG. 5.-Poly~rethanes.~* 9* l6 h against initial dilution. 1 .-HDI + LHT240*/nitrobemene/8Oo ; v = 36 2.-DDI+ LHT240/nitrobenzene/80° ; v = 40 3.--HDI+ LHTl12*/nitrobenzene/8O0 ; v = 66 4.-HDI + LG56*/nitrobenzene/8O0 ; v = 112 5.-HDI + LHT240/nitrobenzene/50° ; v = 36 6.-HDI + LG56/benzene/7O0 ; v = 1 15 * see legend to fig.2.R. F. T. STEPTO 77 taken to be proportional to the initial or the gel point concentration of reactive groups. In addition, from the polyurethane results at 105", there is some indication that use of the gel point concentration is more generally valid. (cat+ &-'/kg mol-' FIG. 6.-Polyurethanes. h against dilution at gel. Legend as for fig. 5. TABLE VA VALUES OF b ACCORDING TO FRISCH-KILB THEORY system V (0 b l m (ii) blnm polyesters (diglyme/60°) adipoyl+ LHT240 37 0.235 0.403 sebacoyl+ LHT240 41 0.240 0.408 adipoyl + LHTll2 66 0.207 0.343 sebacoyl+ LHTll2 70 0.227 0.368 adipoyl+ LG56 132 0.202 0.320 sebacoyl+ LG56 136 0.208 0.324 polyurethanes (nitrobenzene/8O0) HDI + LHT240 36 0.22 0.370 DDI+ LHT240 40 0.22 0.370 HDI+ LHTll2 66 0.20 0.310 HDI+ LG56 112 0.16 0.260 polyurethanes (nitrobenzene/50") HDI+ LHT240 36 0.20 0.350 polyurethanes (benzene/70") HDI+ LG56 115 0.16 0.260 polyurethanes l4 (nitrobenzene/l05") HDIf OPTMP 34 - 0.30-0.3578 GEL POINTS It was found previously 9* that the slopes of lines obtained with (cao+ cb0)-l as abscissa were not in proportion to v-*.This can be due to variations in b with chain structure. However, if cext is equated with (c,,+cbo)/2, and with (ca,+cbc)/2, b can be evaluated from eqn (6) and the slopes of the lines, or the slopes of the curves at small 2, and a quantitive assessment of this equation made. Values of b so derived are shown in table 1.Both sets show the same internal consistency with respect to the polyester results and the polyurethane results at 80". Here, for a given difunctional reactant an increase in v means a larger proportion of oxypropylene units as compared to methylene units in a ring, and results in a decrease in the value of b. This is in accord with the known more flexible nature of the polyoxypropylene as compared to the polymethylene chain. The values of b for these chains are approximately 20* 21 0.34 and 0.40 nm, respectively. In addition, amongst the polyesters, the change from adipoyl to sebacoyl chloride produces an increase in b in accord with an increased proportion of methylene units in the smallest ring. The range of values of b from the polyurethane forming reactions at 105" are the maximum and minimum values given by the three lines in fig.4. No values were deducible from fig. 3 as Frisch-Kilb behaviour was not observed with (cao + cbO)-l as abscissa. In absolute terms the values of b derived with (cac + cbc)/2 equal to c,,, are in the better agreement with the values of b expected from solution properties. This indicates that a majority of the rings are formed near the gel point in accord with the rapid increase in the numbers of functional groups per molecule at this point in the reaction. CONCLUSIONS Particularly for the polyester forming systems, the Frisch expression, eqn (lo), provides a satisfactory explanation of the observed gel points, given that 2 is inter- preted according to eqn (6). The unknown quantities in A, namely, b and text, cannot be determined separately from gelation data.The structure of the theory is such that an average value of cext is required. Apart from the polyurethane results at 105" the data indicate that either the average initial or the average gel-point concentration of functional groups can be used for small values of A. Support for the latter expression is found in that the values of b so derived are in better agreement with values from solution properties.20* 21 The simple description used here, eqn (10) and (6), may be contrasted with the more rigorous description of irreversible non-linear polymerisations in terms of sets of rate equation^.^^ In the latter, the quantity text varies automatically as the polymerisation proceeds, and the ring-forming parameter is essentially tint.Thus, the uncertainty attached to text does not exist. The author wishes to acknowledge helpful discussions with Dr. J. L. Stanford during the preparation of this paper. P. J. Flory, Priticiples of Polymer Chemistry (Cornell Univ. Press, Ithaca, 1953), chap. IX. J. L. Stanford and R. F. T. Stepto, IUPAC International Symposium of Macromolecules, Univ. of Aberdeen, 1973, abstract E32. M. Gordon and W. B. Temple, Makronrol. Chem., 1972,160,263. M. Gordon and G. R. Scantlebury, J. Chem. SOC. B, 1967, 1. W. B. Temple, Makromol. Chern., 1972, 160, 277. J. L. Stanford, Ph.0. Thesis (Univ. of Manchester, 1972). H. Jacobson and W. H. Stockmayer, J. Clrem. Phys., 1950, 18, 1600. L. C. Case, J. Polytner Sci., 1957, 26, 333. W. Hopkins, R. H. Peters and R. F. T. Stepto, Polymer, 1974, 15, 315.R. F. T. STEPTO 79 lo F. E. Harris, J. Chem. Phys., 1955, 23, 1518. l 1 H. L. Frisch, paper presented to 128th Meeting Amer. Chem. SOC., Polymer Division, Minne- apolis, 1955. C. A. J. Hoeve, J. Polymer Sci., 1956, 21, 11. l3 R. W. Kilb, J. Phys. Chem., 1958,62,969. l4 R. H. Peters and R. F. T. Stepto, in The Chemistry of PoZymerisation Processes (Monograph No. 20, SOC. Chem. Ind., London, 1965), p. 157. R. S. Smith and R. F. T. Stepto, Makromol. Chem., 1974, 175, 2365. l6 W. 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