A New Technique for Studying Gelation in Polyesters

by Rick Martinelli, M.A.

Haiku Laboratories Technical Memorandum 041201

December, 2004

Notice: this paper has been published in the Journal of Polymer Science, Polymer Physics Edition, Vol 16, 1519-1527 (1978).  The author would like to thank Dr. J.H.Hodgkin of CSIRO, Austrailia for his contributions to this work, and Professor C.W.Macosko of the University of Minnesota for his helpful comments.  This paper presents the results of research carried out by the author at Stanford Research Institute under contract NAS7-689, sponsored by NASA.



A new method for studying polymer network formation has been devised.  Crosslinking reactions are carried out in a recording viscometer, which provides accurate determination of incipient gel points and also serves as a high-speed stirrer.  The molten, nonstoichiometric mixtures are reacted to completion to eliminate the inaccuracies inherent in the determination of reaction extent and this, together with the use of esterification reactions with minimal side reactions, reduces many of the problems of previous methods.  The experimental results for the reactions of simple model compounds are in very close agreement with Flory’s network theory.  A system containing crosslinking reagents with unequally reactive groups has also been considered and the accuracy of the method enables the reactivity ratios of the different groups to be calculated.



The first theoretical treatments of polymer gelation were made by Carothers[1], Flory[2], and Stockmayer[3], who used a combinatorial methods to obtain the distribution of species as a function of reaction extent, the gel point and postgel relationships.  Their work has since been generalized in different directions[4,5] and new, simpler methods have been found [6,7] to deal with more complicated systems.  For the simple systems considered in this paper, Case’s method [5] is the easiest to apply.  The main problems in experimentally dealing with gelling polymers are the inaccuracies in determining the gel point and the extent of reaction.  In this paper we describe a sensitive, accurate method for detecting the gel point and an application of the method to reactions that were driven to completion to eliminate the need to determine reaction extent.  The systems studied each contain two difunctional and one trifunctional monomer where the trifunctional unit may have groups of different reactivity.  Theoretical relationships between stoichiometric parameters of the system at gelation are computed and compared with experimental values.  In the case of unequal reactivity, stoichiometric considerations alone do not suffice and we resort to kinetic theory to obtain the relationships.  Some of this work has been described in a preliminary communication[8] and the method has been applied to more practical systems[9].  



The model systems used in this work all include a diacid (A-A), a dialcohol (B-B), and a triacid which will be represented by







if its groups are equally reactive and by







if one of its groups is assumed to have a lower reactivity than the others.  We will assume that B groups react only with A or a groups and conversely, that no side reactions will take place, and the no cyclic structures are formed prior to gelation.  The gelation criterion is defined as follows: A given structural unit (such as a B¾B) is selected at random from the reacting mixture and the sum of the probabilities that the unit will reappear in any of the branches of the chain to which it is attached is calculated.  If this sum is denoted by π, equals or exceeds one, then gelation occurs; if π is exactly one then the system is said to be at an incipient gel point. (This is equivalent to Flory’s condition for gelation[2].)  Thus, if the system’s parameters imply π < 1, only finite species will be present in the reaction mixture, while if π > 1, there is a positive probability of selecting a unit that is part of an “infinite chain.”  Since, however, the experimental system is finite in nature, we must regard the unit as belonging to a species that is sufficiently crosslinked with itself to make π > 1 (a “wall-to-wall” species).  It is assumed that these highly crosslinked species are responsible for a drastic rise in the viscosity of the mixture, indicating the gel point. Applying the gelation criterion to an equally reactive system, there are two ways in which a B-B unit can be propagated, namely, B-BA-AB-B and








with probabilities PA PB (1 - ρ) and 2 PA PB ρ, respectively, where PA and PB are the fractions of A and B that have reacted, and ρ is the ratio of A contributed by the branch unit to total A. Then in this case

                                                                           π = PAPB (1 + ρ)                                                   (1)

Letting r = 1 / R = ∑B / ∑A be the ratio of all B to all A, it follows from material balance that

r = 1 / R = PA / PB

and eq. (1) may be written

                                                                          π R = PB2 (1 + ρ)                                                   (2)


                                                                           π r = PA2 (1 + ρ)                                                   (3)


Equations (2) and (3) are used to describe mixtures at a critical point which contain an excess of acid and hydroxyl groups, respectively.  It is convenient to consider two types of systems, designated Y and Z and defined, respectively, as one having the groups attached to the trifunctional unit in the excess and one where that group is not in excess.  At present we are dealing with mixtures containing tri-A so that a type-Y system is one where A groups are in excess.  Assuming now that the system has reacted completely and that it is at an incipient gel point, we set π = 1 and either PA = 1 or PB = 1 depending on which group is in excess.  The eqs. (2) and (3) reduce, respectively, to

Rc = 1 + P


rc = 1 + P

where the subscript c means that the system is at a critical point.  Equations (4) and (5) are plotted in Figure1; according to theory, any mixture whose parameters lie inside the “V” will gel, while mixtures with parameters outside the “V” will not gel. The lines themselves represent all points of incipient gelation.  We now consider systems of unequal reactivity. One of the branch compounds used in this study (1,3,5-pentanetricarboxyl acid) has one secondary and two primary acid groups.  In systems having excess hydroxyl groups, this is of no consequence since the reactions are forced to completion.  However, if acid groups are in excess, preferential reaction at the primary groups decreases the number of completely reacted branch units at any given extent of reaction and so lowers the value of π.  There are four ways in which a B-B unit can propagate in this system.  Each of these reactions is shown below together with its probability of formation:

                                                 B-BA-AB-B      PBPA2 (1 - ρ) / P



                                                            AB-B         2 PBPA2ρ / 3P






                                                            AB-B         2 PBPAPa ρ / 3P






                                                            aB-B         2 PBPAPa ρ / 3P




Where PB and ρ are as before, P is the fraction of acid groups that have reacted, and PA, Pa are the fractions of primary and secondary acid groups, respectively, that have reacted. The sum of these probabilities is

                                                              π = PBPA[(1 - ⅓ρ)PA + 4/3ρPa] / P                                      (6)

Note that eq. (6) reduces to eq. (1) if Pa = PA = P.  Assuming that acid groups are in excess (R>1), that the system has reacted completely (PB = 1), and that it is at an incipient gel point (π = 1) yields

                                                               1 = Rc PA [(1 - ⅓ρ) PA + 4/3 ρPa]                                       (7)

Since PA and Pa are not easily obtained, we will employ kinetic data to estimate Rc.  Assume that at equilibrium

~~ A + B ~~ ~~ AB ~~ + H2O


~~a + B ~~ ~~ aB ~~ + H2O

k - 2



k1[A][B] = k – 1 [AB],     k2[a][B] = k – 2[aB]

and we define β by

                                             β = k1 k – 2 / k2 k – 1 = PA(1 - Pa) / Pa(1 - PA)                                        (8)

Also, from material balance,

                                                      1/Rc = P = (1 - ⅓ρ) PA +  ⅓ρPA                                                 (9)

Therefore, if the rate constants k1, k – 1, k2, k – 2 are available and if the system is at a critical point, eqs. (7)-(9) may be solved simultaneously for the Rc, PA, and Pa given ρ.  This was done for various values of β and the results shown in Figure 1.  Since the reactivity of secondary acid groups was assumed to be lower than that of primary, only values of β greater than one are used.  Thus, in this case, incipient gelation curves lie inside the “V”.



Fig. 1. rc vs. r and Rc vs. r for various values of b





Sebacic acid and 1,3,5-pentanetricarboxyl acid (PTA) were obtained from Aldrich Chemical Co. and used as received.  Decanediol was obtained from Matheson, Coleman, and Bell and was recrystallized twice from chloroform. Acid numbers and hydroxyl numbers were used as criteria for purity, which was approximately 99%.  The 1,3,5 - benzenetriacetic acid (BTA) was prepared by the method of Newman and Lowrie[10].



The apparatus is essentially a recording viscometer in which the torque required to rotate one concentric cylinder within another is continuously monitored.  The reaction vessel is a cylindrical glass container approximately 4 in. long by ½ in. i.d. which is maintained at ~140˚C by a xylene vapor bath.  Concentrically located in the reactor is a solid Teflon cylinder approximately 11/2 in. long by 1 in. diameter which is attached to a dc motor via a vertical steel shaft and which also serves as a stirrer.  The field of the motor is held at a constant voltage and rotational velocity is adjusted by varying the voltage supplied to the armature.  A small direct current generator is attached to the motor shaft and serves as a tachometer.  The current required to maintain the motor at a fixed speed is obtained by reading the voltage drop across a 1 Ω resistor in series with the armature.  Calibration of the viscometer with fluids of known viscosity at 500 rpm revealed a fairly linear relationship between current and viscosity over a fairly wide range.



A mixture of the ingredients at known R or r and ρ values is prepared by weighing each ingredient (±0.01%) directly into the reaction cylinder; total mass is approximately 18g.  After the mixture has at least partially melted, the Teflon cylinder is inserted and stirring begun.  The viscosity of the mixture decreases as it comes to temperature; when a constant value is reached, the stirring rate is set to 500 rpm and a “zero torque” reading is taken.  The basic experimental approach is to prepare a nongelling mixture and allow the reaction to proceed until a constant viscosity reading is reached, at which point complete reaction is assumed.  The mixture is then brought nearer to the theoretical gel point by adding a calculated amount of one of the ingredients and the reaction again allowed to proceed to a constant viscosity.  This procedure is repeated until the mixture gels.  For a type-Y system, diol is added to the mixture, thereby decreasing R and bringing the mixture closer to gelation.  For a type-Z system triacid is added in order to decrease r.  However, addition of triacid also increases ρ so that the gel point in this case is approached more rapidly.  The amount of ingredients necessary to achieve the desired ρ, R, or r values for a predetermined total weight of mixture were determined from a computer program, which listed the amount of diol and corresponding R values for a type-Y system and the amount of triacid with  corresponding ρ and r values for a type-Z system.  The proximity of the incipient gel point was estimated by the increase of viscosity with each amount of material added.  The reproducibility of this procedure was confirmed by preparing a different nongelling mixture and proceeding to the gel point by adding different quantities of the reactant.




Results and Discussion


The theoretical treatment is predicated on a number of assumptions that place severe requirements on the chemical reaction and experimental procedure used to verify the theory.  We have found that an esterification reaction, when allowed to proceed long enough so that chain extension and branching linkages can equilibrate by means of transesterification reactions, meets the requirements of the theory.  Thus, esterification can proceed with almost no side reactions, can be driven to completion, and does not appear to be significantly afflicted with cyclic structures prior to gelation.  A number of reactions other than simple esterification were tried, primarily to eliminate the necessity of removing a reaction product in order to get complete reaction[8,11].  None of the alternative reactions, however, was found to be reversible or as free from side reactions. 

Early experiments[11] indicated that proper choice of trifunctional crosslinking agents is very important, and a number of compounds were tried and found unsuitable.  For example, glycerol, tricarballylic acid, and 1,2,6-hexanetriol decomposed; and trimethylolpropane sublimed out of the reaction mixture.  The compound eventually chosen for most of this work was PTA; however, BTA was used in a few type-Y experiments as an equal-reactivity control. 

Data obtained from type-Y systems containing PTA differed considerably from those obtained from type-Z systems and from type-Y systems containing BTA.  With the last two systems, the viscosity increase was usually negligible as the gel line was approached, but when the actual gel point was exceeded, the viscosity either increased beyond the limits of detection of the instrument, or the gelled mixture agglomerated around the stirrer shaft.  Because of this abrupt change, the value of critical r could be located with a 0.2% precision; reported values are the midpoint between a gelling and nongelling mixture. 

To verify the change in viscosity used for detecting the gel point actually represented gelation, several of the reaction products were extracted with benzene in a Soxhlet extractor.  Each was found to be only partially soluble leaving behind between 3 and 33% by weight insolubles, depending on the proximity of the mixture to the gel line.  As a control, several mixtures were prepared with parameters lying outside the “V”; each of these was completely soluble. Results obtained with type-Z systems are shown in Table I and plotted in Figure 2.  Calculated rc values are from eq. (5); percent deviations are reported as

{[(rc)calc – (rc)obs]/[(rc)calc - 1]} x 100

Type –Y systems containing PTA, on the other hand, did not exhibit an abrupt change in viscosity and curves similar to those reported by Flory[2] were obtained.  The difference in behavior between type-Y and type-Z systems can be attributed to the lower reactivity on the secondary carboxyl in PTA.  In the case treated here, a type-Y system contains an excess of carboxyl acid groups so that the mixture is brought to the gel point by addition of diol.  Because acid groups are in excess, reaction of a primary carboxyl group will occur in preference to a secondary group, and formation of unbranched high-molecular-weight chains will be favored.  Gelation will occur only after the majority of the primary carboxylic acid groups have reacted, the branching reactions involving the less reactive secondary carboxyl acid groups become significant.  Experimentally, therefore, one notes a gradual increase in viscosity until the gel point in reached. 



Table I

Critical r-Values for Type Z Experiments




Percent Deviation











































rc                                                     Rc

Fig. 2. Incipient gel points for PTA and BTA systems


Table II

Critical R-Values for Type Y Experiments




Percent deviation











































Type –Y systems containing PTA, on the other hand, did not exhibit an abrupt change in viscosity and curves similar to those reported by Flory[2] were obtained.  The difference in behavior between type-Y and type-Z systems can be attributed to the lower reactivity on the secondary carboxyl in PTA.  In the case treated here, a type-Y system contains an excess of carboxyl acid groups so that the mixture is brought to the gel point by addition of diol.  Because acid groups are in excess, reaction of a primary carboxyl group will occur in preference to a secondary group, and formation of unbranched high-molecular-weight chains will be favored.  Gelation will occur only after the majority of the primary carboxylic acid groups have reacted, the branching reactions involving the less reactive secondary carboxyl acid groups become significant.  Experimentally, therefore, one notes a gradual increase in viscosity until the gel point in reached. 

In the type-Z system treated here, hydroxyl groups are in excess so that the mixture is brought to the gel point by addition of triacid.  Since the reaction is forced to completion after each addition, the secondary carboxylic acid groups are forced to react, and differences in reactivity are of no consequence.  Highly branched low-molecular-weight structures form first, and these will not significantly alter the viscosity of the reaction mixture, the apparatus will detect an abrupt viscosity change only after the composition is such that gelation can take place.

Experimental verification of the hypothesis that the different behavior of Y and Z systems is due to unequal reactivity of the primary and secondary carboxyl groups on the trifunctional unit was achieved by using BTA in which all three primary carboxyl groups are equally reactive.  Thus, in type-Y experiments with BTA carried out at ρ = 0.15, 0.3 and 0.45, not only was there an abrupt increase in viscosity at the gel point characteristic of type-Z systems, but the data fit eq. (4), thus confirming equal reactivity of all carboxylic acid groups.  The results are shown in Figure 2 as solid circles.

Because of the absence of an abrupt change in viscosity in type-Y experiments, critical R values were obtained by fitting R vs. viscosity data to an equation of the form

η = a / (R – Rc) + b

where η is viscosity (in arbitrary units) and a, b are constants, using standard linear least-squares methods.  Rc values thus obtained correspond to infinite viscosity of the mixture.  Using these values, together with their corresponding ρ values, eqs. (7)-(9) were solved simultaneously for β (see Appendix).

Results of this treatment are shown in Table II and are plotted in Figure 2; The points fall roughly on the incipient gel line corresponding to β = 23.39, which is the average of the estimated β values (excluding the value at  ρ = 0.1).  Percent deviations are reported as in Table I.

An analysis of the uncertainties caused by slight impurities in the starting material was carried out.  The maximum error in Rc and rc remains relatively constant at 1-2% over their entire range.  However, the maximum error in ρ increased from about 1% for ρ > 0.5 to greater than 5% for ρ < 0.2.  Thus, the observed deviations in Rc at lower ρ values could be due to impurities.

For type-Z systems, the fact that critical r values are all smaller than predicted by theory, rather than lying on either side of the gel line, suggests that some intramolecular side reactions (cyclization) are taking place.  For type-Y systems, cyclization is probably reduced since linear species are more likely to form than small branched species, but this cannot be deduced from the data.

  1. W.H. Carothers, Trans. Faraday Soc., 32, 39 (1936).
  2. P.J. Flory, J.Am.Chem.Soc., 3083, 3097(1941).
  3. W.H. Stockmayer, J. Polym. Sci., 9, 69, (1952).
  4. Kahn, J. Polym. Sci., 26, 283 (1961).
  5. L.C. Case, J. Polym. Sci., 26, 333 (1961).
  6. C.W. Macosko and D.R. Miller, Macromolecules, 9, 199, 206 (1976).
  7. M. Gordon, Proc. R. Soc. London Ser A, 268, 240 (1962).
  8. J.H. Hodgkin, F.J. Martinelli, J. Heller and H.E. Marsh, Polym. Prepr., 12, 593 (1971).
  9. J.H. Hodgkin and F.J. Martinelli, J. Macromol. Sci Chem., A6, 789 (1972).
  10. M.S. Newman and H.S. Lowrie, J. Am. Chem. Soc., 76, 6196 (1954).
  11. J.H. Hodgkin, F.J. Martinelli and J.Heller, Stanford research Institute, Menlo Park, Calif., Quart. Rept., 1,2 and 3, NAS7-689, 1969.

Appendix (Added 2004)

In the absence of the rate constants k1, k – 1, k2, k – 2 , another approach was used to calculate β values in Table II.  Equations (7) and (9) may be written

                                                                   3/Rc = (3 – ρ)PA2 + 4ρPaPA                                        (A1)


                                                                      3/Rc = (3 – ρ)PA + ρPa                                            (A2)

respectively.  Letting X = Pa/PA < 1 be the reactivity ratio of secondary to primary A groups on the tri-functional unit, these equations may be written as 

                                                                   3/(Rc PA2) = (3 – ρ) + 4ρX                                         (A3)


                                                                      3/(Rc PA) = 3 – ρ + ρX                                            (A4)

respectively.  Squaring A4 gives

                                                      9/(Rc2 PA2) = (3 – ρ)2 + ρ2X2 + 2 ρX(3 – ρ)                         (A5)

and multiplying A3 by the reciprocal of A5 gives


Finally, cross-multiplying in A6 yields

                                                          ρ2 X2 + 2[(3 – ρ)ρ - 6ρ/Rc] X + (3 – ρ) (3 – ρ – 3/Rc) = 0                 (A7)

Uusing the quadratic formula we find


where we’ve chosen the result with the negative square root term.   Once X is known, PA may be calculated from A4, Pa from Pa = X PA and β from equation (8).