Effects of Functionality and Temperature on Gel Points in Random Polymerisation Zahoor Ahmad*, Robert F. T. Stepto and Richard H. Still

Department of Polymer Science and Technology, The Universiry of Manchester Instiiuie of Science and Technology, Sackville Street, Manchester M60IQD, U K

First, the relationship between gel point and average functionality for an RAZ+R‘B, type polymerisation has been investigated by measuring the product of extents of reaction at gelation (a,) for sebacoyl chloride (SC)/polyoxyproplyene (POP)diol/POP triol mixtures reacting at various initial dilutions in diglyme as solvent at 60°C. Intramolecular reaction always delays gelation and a generalisation of the Ahmad-Stepto gel point expression has been used to interpret the gelation data in terms of the ring-forming parameter A&,. From the variation of lib with initial dilution, values of b, the effective bond length of the chain forming the smallest ring structure, have been derived. It is found that b decreases with average polyol functionality (f,,,). This decrease apparently compensates for the increase in Aib with f,,,, so that ring formation is less sensitive to functionality than may have been expected. It is found that the Ahmad- Stepto expression does not predict a consistent relationship betweenA:,,f. and the gel dilution of reactive groups.

Second, the effect of temperature on the gel point has been studied by measuring a, for SC/POP triol mixtures reacting at various initial dilutions in diglyme as solvent at 27“C, 40°C and 60°C. At a given temperature, b decreases as triol molar mass increases, indicating that the POP residue of the chain forming the smallest ring structure is more flexible than the SC residue. This result is in keeping with those from previous investigations.’,? For a given triol, a, decreases as temperature increases, indicating an increase in chain stiffness with temperature. Values of d In (rZ)/dT are derived and found to be larger than those for linear chains.

Keywords gelation,

random polymerisation, polyesters

1 INTRODUCTION

It is well known that, even in the absence of effects of unequal reactivity, gelation in random polymerisations is delayed by intramolecular reaction beyond the Flory-Stockmayer gel point.1,2 The interpretation of such delay in terms of monomer or prepolymer structures and reaction conditions is important for the prediction of gel points and also for the understanding of the random polymerisation process itself, of which intra- molecular reaction is an integral part. In addition, although the delay in terms of extents of reaction may be small for bulk reactions it can be characteristic of marked reductions in the moduli of the (endlinked) networks formed at complete reac- tion.*’ Thus, the properties of endlinked networks cannot be interpreted fully in molecular terms unless the competition between intermolecular and intramolecular reaction which occurs during network formation can be characterised. The experimental study of gel points represents a relatively easy method to achieve such a characterisation.

Interpretations of gel points may be attempted through theo- ries or simulations of polymerisations which seek to describe the interconversion of molecular species as a polymerisation proceed^,',^'^ or by approximate theories which focus on the gel point. The latter use an average probability of intramolecu- lar reaction and simplify the molecular structures involved leading to analytical expressions for the gel point in terms of extents of reaction. The present paper employs the approxi- mate expression for the gel point due to Ahmad and SteptoZ to interpret extents of reaction at gelation in sebacoyl chloride (SC)/polyoxylpropylene (POP) diol/POP triol reactions at 60°C and in SC/POP triol reactions at various temperatures. The expression has been shown previously* to account for

Present address: Department of Chemistry, Quaid-i-Azam University, Islamabad. Pakistan.

intramolecular reaction more completely than those derived by FrischI6 and Ki1b.I’

The Ahmad-Stepto expression2 relates to an RA,+R’B, polymerisation and may be written

ac(f-1) ( d a b - 1 > = 1 I (1) where a c = ( P $ b ) c , the product of extents of reaction at gela- tion, and dab is a ring-forming parameter. Equation (1) may be rearranged to allow determination of L a b , or more directly A:,=dab/(l -dab), from experimentally measured values of a,, with

A:b=(f- 1)l’*a:”- 1. ( 2 ) In Eqn (21,

5 = (f- 2) . { 3/(kvbZ)} 3’2 * @( 1, 3/2) Cext N * (C:+C;)

(3)

Here, cext=(c~+c~) is an average external concentration of A and B groups around a given molecule, that is, the average concentration from all other molecules in the system. It must be chosen arbitratrily and may be equated to either the initial molar concentration of reactive groups, (cao+cw), or the gel point concentration, (cac+cbc) , as an extreme value. c,,~ is the (internal) concentration of B groups around an A group on the same molecule, or vice versa. In the full expression for c,,,, (f-2) is the number of opportunities for intramolecular reac- tion for each size of ring; the factor {3/(2,Cvb,)}3’2 is the pro- bability (assuming Gaussian statistics) that the groups at the ends of a sub-chain of v bonds, forming the smallest ring, are coincident; the function

m

@(1,3/2)=~l’i-3’2=2.612 i=l

BRITISH POLYMER JOURNAL, VOL. 17, NO. 2 1985 14

205

is the sum due to ring structures of all sizes, and N is the Avogadro constant. The chain of v bonds comprises two arms of an R'Brmolecule reacted at one end with an RA2 molecule. The mean-square end-to-end distance of the chain, ( r i ) , is equal to vb2, where b is the effective bond length. The com- bined factor {3/(2nvb2)}3'Z/N may be denoted Pab, and is discussed elsewhere in this issue.Is

Equations (1)-(3) show how the gel point depends on reaction conditions, and reactant molar masses and chain structures. The scale of cCxt in a given polymerisation depends on the dilution of the reaction mixture with solvent. The maximum value of c,,~ is related to the molar masses and densities of the reactants. Thus, even in a reaction in bulk, reactive groups are diluted and intramolecular reaction will occur. Through the various factors in c,,,, intramolecular reaction is seen to increase with functionality (f), and to decrease as the molar masses of the reactants (or v) and the chain stiffness ( b ) increases.

Recent p~blications'-~ have examined the preceding points and shown they are generally valid, although the form of the dependence of &, on dilution of a reaction mixture (cz,) is not always linear, as predicted by Eqn (3). In addition, gel points in systems of mixed functionality (RA2+R'B2+RBf type) have been measured.'* Analysis of the results focused on the limiting behaviour at infinite concentration (c;'xt-+O) when it was shown that

a,(&- I )= 1, (4) consistent with Stockmayer's generalised form of the gelation condition in the absence of intramolecular reaction.' fw is the so-called weight-average functionality of the R'B2/R"Bf mix- ture used, or effectively the functionality averaged in propor- tion to reactive groups. The present publication considers the further analysis of the mixed functionality data" in terms of the generalised form of Eqns (1)-(3), with f w replacing f, and v, replacing v. As withf,, v, is evaluated by averaging in propor- tion to the R'B? and R B f chain ends in a given reaction mixture. The systems used were SC reacting with mixtures of a POP diol and POP triol in diglyme as solvent at 60°C.

New data are also presented showing the variation of gel point with temperature for SC/POP triol systems. In this case, varia- tion in the gel point comes from changes in chain flexibility (b).

2 SYSTEMS STUDIED AND EXPERIMENTAL PROCEDURES

2.1 Functionality studies

Two samples of POP triol, LHTl12 (ex Union Carbide), of M,=1433 g mol-l and 1478 g mol-' were used. They were characterised by end-groups determinations and cryoscopy.I9 The samples had negligible unsaturated chain ends and fn=3.O0+0.0l. They contributed (number-average) values to v, of 50.1 and 51.7 bonds, respectively. Account was taken of these slightly different values when evaluating v, for a given diol/triol mixture.

The POP diol was PPG1025 (ex Union Carbide) of M,=1034 g mol-' contributing 53.3 bonds to v,.

Table 1 gives the values of e, f w , v, and the derived values of b (see later) for the various SC/POP diol/POP triol mixtures used. Q (=fw-2) is the proportion of OH groups belonging to triol.20

A series of reactions at different initial dilutions in diglyme at 60°C was studied for each of the reaction mixtures, which were equimolar in COCl and OH groups. Subsidiary kinetics experi-

ments were to select the diol/triol pair as one having OH groups with equal reactivities towards SC. The equal reactivity, or random reaction of like functional groups is assumed implicitly in the equations discussed so far.

2.2 Temperature studies

Three POP triols, LHT240, LHTl12 and LG56 (ex Union Carbide), with M,=705, 1478 and 2924 g mol-' were used. Reaction mixtures equimolar in COCl and OH groups were again employed and reactions were carried out at different initial dilutions in diglyme as solvent a t 27, 40 and 60°C.

2.3 Experimental procedures

The purification and characterisation of polyols, SC and diglyme for gelation and kinetics studies have been described in detail e l s e ~ h e r e . ~ , ' ~ . ~ ' Stringent precautions were taken to ensure the polyols and diglyme were dry. The polymerisations were followed by determining loss of COCl groups by titration. ac was determined by extrapolation of conversion-time curves to the gel time. The procedures were again as described p r e v i o u ~ t y . ~ ~ ~ ~ . ~ ~ ~

z'oko 1.9

0 0.2 0.4 0.6 0.8 1.0 I

([COCU, + [OH],)-' Ikg mol-')

Fig. 1 Reciprocal of products of extents of reaction at gelation (0;') versus initial dilution of reactive groups (([COCl],+[OH],,)-I). Curves 1-6 are for the SC/POP diol/POP triol reaction systems specified in Table I . Intercepts are equal to (fw-l).

3 FUNCTIONALITY AND GEL POINT

Figure 1 shows a;' versus initial dilution of reactive groups, ( C , ~ + C ~ ~ ) - ~ (=([COCl]o+([OH]o)-l). This representation of the data has been given before,18 except that system 6 and additional data points a t higher dilutions for systems 1-5 have been added in Fig. 1. As expected, a;' decreases with dilution and, in accord with Eqn (4) the curves have been extrapolated to fw- 1 at zero dilution of reactive groups.

The slopes of the curves in Fig. 1 d o not decrease markedly with average functionality so that the probability of intramolecular reaction is apparently not sensitive to f,. The reasons for this behaviour can be explored by analysing the values of ac in accordance with the generalised form of Eqns

206 BRITISH POLYMER JOURNAL, VOL. 77, NO. 2 1985

( 2 ) and (3), with Eqn (3) becoming

0.2 n

x - 0

& (3/2rt~,b~)"~~~(1,3/2) 1 ( 5 ) .- --

( f w - 2 ) - N (c; + CL) '

Aib/(fW-2) is the ring-forming parameter per pair of A and B groups, and is shown in Fig. 2 versus (ci+c$'. The latter quantity is taken as the initial dilution of reactive groups, as in Fig. 1. Deviations form the linear behaviour predicted by Eqn ( 5 ) are apparent, but initial slopes may be compared.

-

0.3 6 P' I I l r a

- 0.2

<=

N I

- A

- 0 4

0. I

0

1 I/ / '

0 0.2 0.4 0.6 0.8 1.0 I

([COCl]o+ (kg mol-'1

Fig. 2 Ringforming parameter normalised for functionality (A:b / ( fw-2) ) versus initial dilution of reactive groups. Curves 1-6 as in Fig. 1.

The values of v, vary little between the six systems. Thus, the different initial slopes of the curves in Fig. 2 correspond to different values of b, as listed in Table 1. The decrease in b asfw decreases shows that reductions in functionality towards linear chains give more flexible chains and lead to more intramolecu- lar reaction per opportunity for such reaction.

Table 1 Values of p, fw and vw, and derived values of b for the SC/POP diol/POP triol reac- tion mixtures investigated in diglyme at 60°C

System e r, Vw blnm

1 1 3.00 62.1 0.262 2 0.81 2.81 62.5 0.240 3 0.65 2.65 62.9 0.221 4 0.50 2.50 63.1 0.186 5 0.35 2.35 63.5 0.179 6 0.19 2.19 63.9 0.173

0.2 n

4 .o

0. I

0

Fig. 3 Ring-forming parameter versus initial dilution of reactive groups. SC/POP triols in diglyme: (a) LHT240: (b) LHTl12: (c) LGS6. 0 27°C; 0 40°C; 60°C.

As in previous use of the initial concentration of reactive groups for c;+c,', gives absolute values of b which are smaller than those expected from solution properties. Use of the other extreme value, c,,+chc, gives values which agree much better. Such behaviour is consistent with the majority of ring structures being formed near the gel point. Unfortunately, for the present results, which are the first using mixed func- tionalities, the use of gel point concentrations for c:+cA did not yield a clear relationship between b andf,". The reason for this has yet to be clarified, but obviously represents a serious limitation in the applicability of present approximate gel point expressions.

4

Figure 3 shows A:, plotted versus initial dilution in accordance with Eqn (3) with c~+c~=c,o+c,,o. In agreement with previous results on acid chloride/POP triol approximately linear behaviour is obtained with initial dilution as abscissa. Plots with somewhat more curvature are obtained if gel point dilution is used for c:+cA. Generally, the curvature is less than

VARIATION OF GEL POINT WITH TEMPERATURE

Table 2 Chain parameters and dimensions characterising pregel intramolecular reaction for SC and POP triols in diglyme ((rz)=vb2; v=number of bonds in the chain forming the smallest ring structure; vAc/v=fraction length of SC (acid chloride) residue in the chain of v bonds: b=effective bond length)

System V

SC/LHT240

SC/LHT112

SC/LG56

36

63

112

T I T

27 0.306 40

60 27

0.175 40 60 27

0.098 40 60

~~~

VacIV

b Jnm (rz)/nm' 103(d In (rZ)/dT)/K- l

liP (ii) lip ( i i ) b li) (ii)

0.268 0.439 2.59 6.94 0.289 0.462 3.01 7.67 8.3 6.7 0.307 0.491 3.39 8.66 0.230 0.376 3.32 8.90 0.242 0.394 3.68 9.79 8.7 6.3 0.265 0.418 4.42 11.0 0.227 0.355 5.76 14.1 0.241 0.372 6.50 15.5 9.0 6.2 0.263 0.392 7.72 17.2

~~

a ( i ) C,,,=C,~ +cw. (ii ) c,, =ca,+cbc.

BRITISH POLYMER JOURNAL, VOL. 17, NO. 2 1985 207

that found for some polyurethane-forming systems, especially those using POP tetrols.',5-7

The plots in Fig. 3 show that slopes decrease as v increases from LHT240 to LG56. In addition, for a given triol, slopes decrease as temperature increases, or, from Eqn (3), as b increases. Plots having the same relative behaviour result with dilution at gel as abscissa.

Analysis of the slopes or limiting slopes as (C:~+CL)-~-O according to Eqn (3) gives the results listed in Table 2. The values of ( r 2 ) for the chain forming the smallest ring structure were used in Fig. 4, where In ( r 2 ) is plotted versus T. The slopes of the lines in Fig. 4 are the values of d In (r*)/dTlisted in Table 2.

Considering the plots in Fig. 3, the main change in A:, is its increase with dilution. However, at a given dilution, it decreases as triol molar mass, or v increases. Thus, values of cant, or the slopes of lines in Fig. 3 decrease generally form (a) to (c). Equation (3) shows that c,,, varies in direct proportion to

0

2.5

I 1.51

T ( ' C )

Fig. 4 In ( r 2 ) vcrsus temperature for SC/POP triol suh-chains: (a) LHT240; (b) LHTl12; (c) LG56. 0 using (i) re3,=c;so+u,,,,; 0 using (ii) cc,t =cac+chc.

(149) 3i?. At a given temperature, b decreases as v increases or vAc/v decreases. Hence, the oxypropylene residue is more flexible than the acid chloride residue in the chain of v bonds. The absolute values of b derived using C ~ + C ~ = C , ~ + C ~ ~ (gel concentration) are in better accord with those expected for linear chains than those derived using c ~ + c ~ = c a 0 + c b ~ , indicat- ing that ca,+cbc is a more appropriate average value to use for probabilities of intramolecular reaction.

The preceding behaviour is in keeping with that found pre- viously for diisocyanate/POP triol, diisocyanate/POP tetrol and diacid chloride/POP triol system^.^-'.*^ In addition, the present results show that for the three triol systems studied b increases and, hence, intramolecular reaction decreases as temperature increases. The effect is not negligible as the ranges of values of b found for a given triol are about the same as those found by changing triol (or v) at a given temperature. The increase in b with temperature indicates lower energy gnuche conformational states and is probably characteristic of the conformational statistics of the POP section of the chain

structure. The resulting values of d In (r2)/dT are generally larger than those found from solution properties or stress/tern- perature measurements on networks.22 Also, conflicting trends in d In ( r2)/dT with v are found using the two extreme values of cC+c;. These inconsistencies can have various origins relating to approximations inherent in Eqns (2) and (3); namely, the molecular structure assumed to derive Eqn (2) is oversimplified, there is no correct single value of ci+cL (it decreases as a polymerisation proceeds), and the assumption of independent Gaussian statistics in describing cyclisation probabilities of sub-chains in branched molecules, presumably under non-0 conditions.

5 CONCLUSIONS

The paper extends the interpretation of gel points given by the Ahmad-Stepto equation to systems in which functionality and temperature are varied. In the limit of zero dilution the pro- duct of extents of reaction at gelation (a,) may be extrapolated (fw- l ) - l as expected for no intramolecular reaction. Interpretation of the variation of a, with dilution in terms of the ring-forming parameter ,lib shows a definite correlation with functionality (fw) in that systems of lower functionality have more flexible chains. The correlation reduces the apparent sensitivity of a, to fw. Such a correlation is not expected if sub-chains of the same chemical structure follow independent Gaussian statistics. Therefore this assumption is brought into question by the present results. In addition, the effects of assuming a single value of A;b or c;+ci for a given system are apparent in the lack of correlation obtained bet- ween &, and gel dilution.

The variation of extents of reaction with dilution for SC/POP triol systems follows the behaviour found previously for other RA,+R'B, systems. A:, is proportional to gel or initial dilution, at least for the more concentrated systems, and the values of b derived reflect changes in the flexibilities of the chains forming the smallest rings. Less intramolecular reaction occurs as tem- perature increases, which again may be interpreted in terms of changes in b. The temperature coefficients of chain dimensions (d In (r2)/dT) are larger than those expected for linear, unperturbed chains and show different trends with v, depend- ing on the value of c:+c[, employed. In this behaviour, one is again seeing the effects of using approximate, analytical expressions for predicting gel points.

6 ACKNOWLEDGEMENT

One of us (Z.A.) would like to thank The British Council lor support during the period of this work.

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11 Ross-Murphy, S.B . , J . Polym. Sci., Polym. Symp., 1975, 53, 11. 12 Cawse, J.L., Stanford, J.L. & Stepto, R.F.T., Proc. 26th IUPACInt .

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