EDITOR’S NOTE-THINKING AHEAD The following paper is printed in the cadre o f providing a two-way channel for the exchange of technical ideas and opinions. The reader m a y be aroused by the judgment that
the argument is not completely supported by experimental results. The authors accept the opportunity for comment. The editor affirms a desire to provide a medium to “talk in print” (see editoria&PRODUCT R&D, December 1969). H . L . G.
Effect of Additive Functionality in Radiation Crosslinking of Polymers A Simple Mathematical Model Larry D. Nichols and George R. Berbeco Moleculon Research Corp., 139 M a i n St., Cambridge, Mass. 02142 A theoretical mathematical model of the effect of additive functionality on gel formation in an irradiated polymer i s presented. The simplified statistical approach i s consistent with the intuitive thought that there is an optimum additive functionality. The model demonstrates this optimum to be 5 to 6 (analogous to 2% or 3 vinyl groups). Similarly, an optimum concentration i s found for initiation of gel formation. This simplified model may be further expanded tu incorporate further constraints and more realistic assumptions.
Mm,
commercial applications of radiation to polymers involve the production of crosslinking, and this often requires the use of some crosslink-promoting additive. Curing of organic coatings and crosslinking of electrical insulation are two such applications. Even after the limitations imposed by biendability and thermal stability, the choice of additives includes several alternatives in most cases. If this selection is not to be done purely on the basis of off-the-shelf costs, some technical guidelines are needed in choosing materials for study. One key question involves the possible advantages of highly polyfunctional additives, with many graftable sites, and whether these are of sufficient interest to justify the added difficulty of making them even in the laboratory. Theoretical studies of gelation of pure polymers have been presented by Flory (1953) and Chapiro (1962), and a statistical approach of free radical events has been made by Henley and Johnson (1969). However, molecular interpretation of irradiative polymer gelation in the presence of a reactive additive has not been quantitatively treated. Although this question is often answered rather intuitively, some significant conclusions can be drawn from a very simple mathematical model similar to those employed in the past additiveless systems. The simplest form of these calculations has been carried out to aid in our own research. I n could readily be made more sophisticated, at the expense of additional parameters, but even in the form presented here it provides substantial added insight into crosslinking phenomena involving polyfunctional additives.
Assumptions
The system contains only two components, the substrate polymer and the additive. The direct effect of the radiation is solely to produce activated sites (radicals) on the substrate polymer. These radicals are unable to react directly with each other to produce crosslinks, but can react with additive molecules. Each additive molecule has some limiting functionality, F , which represents the maximum number of active substrate siZes to which it can bond. The distribution of bonds between substrate molecules and additive molecules is entirely random except for the restriction to no more than F per additive molecule. Interactions between different additive molecules can be ignored. (The system contains a low concentration of additives.) The only effect of the resultant crosslinking to be measured and predicted is the production of gel-i.e., a single macromolecular network of almost infinite molecular weight. The reaction of activated substrate sites with additive molecules is first-order in the total remaining concentration of bondable additive sites. Derivation of Mathematical Model
Several parameters are needed to describe this radiation polymer system. Ind. Eng. Chem. Prod. Res. Develop., Vol. 9 , No. 2, 1970
147
c = number of additive molecules initially present per polymer molecule-i.e., a measure of additive concentration F = functionality of the additive molecules-i.e., the maximum number of bonds that each can form to polymer m o1ecu 1es n = initial number of available additive bonding sites per polymer molecule, which is just c F n* = number of active substrate sites, per original polymer molecule, produced by the radiation dose; this is assumed to be strictly proportional to dose and thus serves as a measure of total dose (it would in fact be related to total dose by the G value for site production) n’ = number of active sites, per original polymer molecule, which actually form bonds to additive molecules g = weight-fraction of gel finally produced I n the above definitions, and elsewhere in this paper, all concentrations (except g ) are expressed in number per original polymer molecule. The derivation proceeds by first calculating the final concentration of species PaAthat is, precisely k polymer molecules bonded to the same additive molecule. Given the assumed random nature of the bonding process, this is readily seen to be:
Equation 1 is the familiar binomial distribution for random events where (i) is a binomial coefficient and the other terms have been defined (Peters and Summers, 1968). Now every polymer molecule will have some specific number of bonds through groups of the type PbA; let the probability that this number will be N be p ( k , N ) . At each bond to a group PkA there will also be some probability p , ( k ) that none of the other K - 1 polymer molecules bonded to the same additive molecule is part of the gel. I t is easily shown that: p ( h , N ) = p2e-Nk A’! where m h
= h(Pk.4)
(2)
and
p , ( h ) = (1 - g ) k (3) thevalue for (PkA) is that given I n the definition of in Equation 1. Now the probability that a given polymer molecule will not be linked to the gel through any groups of the type PkA is just the sum of p ( k , N ) . [ p , ( k ) I Nover all N from 0 to infinity. Without detailing all of the algebra, this sum works out to be ~
m,
e-ma[l - (1 - g)‘
~
l]
Clearly Equations 5 and 6 are complicated and do not allow a simple expression for gel fraction as a function of the other parameters. Nevertheless they contain a lot of information, some of which can be extracted as follows. If we assume some desired gel fraction, g, and some additive functionality, F , it is possible to insert various values for n (additive concentration) in Equation 5 and find the corresponding n f. Then from Equation 6, n* can be found, which is a measure of the total dose required to produce that gel fraction with the specified concentration of an additive of functionality F. One finds on doing this that there is a value of n, for any given g and F, which minimizes the required n*. While in practice the minimum dose may not always be the best dose, this is nevertheless a good way of evaluating the effects of increasing functionality. Table I gives some representative minimum n* values, and the corresponding values of n, for different values of F. Only even values for F a r e listed, since a single double bond is really difunctional and the use of free-radical additives, corresponding to odd F’s, is unlikely. Another way of treating the above equations is by using them to determine the value of n giving a minimum gel threshold value of n*-i.e., the value of n which minimizes the dose required to produce for each F a trace of gelation (Table 11). ResuIts
I n Table I is included the absolute minimum dose required to produce any desired level of crosslinking, even with an additive of infinite functionality. The ratio of the required dose to this absolute minimum is given in the last column-Le., the ratio of n* to n* = 2.30. Table I is based on a desired gel fraction of 90%. The optimum concentration of bondable groups on additive molecules lies in the range of interest, and the dose needed to produce 90% gel can be reduced by almost a factor of 2 in going from a monoolefin to a triolefin, but by only another 44% in going to much higher functionalities. I t is significant that the use of too much additive can lead to less crosslinking (as can be shown by putting a large n in Equations 5 and 6), suggesting that experiments in which “a reasonable amount” of additive is employed should be viewed with caution until other concentrations
Table I. Combinations of Parameters leading to 90% Gel at Minimum Dose for Any Given F
(4)
Finally, the probability that a given polymer molecule will not be linked to the gel at all can be found by multiplying together all terms of the type given by Equation 4 for all K from 2 to F . This product can be written in closed form, and must just be equal to the sol fraction 1 - g. This leads, after some rearrangement, to the important equation:
F
n
n*
Concn. of Additive
Dose Absolute Minimum Dose
2 4 6 Large
5.00 5.04 5.95 Large
6.30 3.81 3.31 2.30
2.50 1.26 0.90 Small
2.74 1.65 1.44 1.oo
Table II. Combinations of Parameters Leading to Minimum Gel Threshold Dose for Any Given F
Equation 5 does not include the important parameter n*, which is our measure of total dose. Because of the assumed first-order relationship between n and n* to yield n’, this lack is filled by the following expression: n f / n = 1 - e-n*/n (6) 148
Ind. Eng. Chem. Prod. Res. Develop., Vol. 9, No. 2, 1970
F
n
n*
Concn. of AdditiLe
Dose Absolute Minimum Dose
2 4 6
1.95 0.65 0.49
2.46 0.82 0.49
0.975 0.162 0.065
1.069 0.356 0.212
have been tried. This observation has been explained by other authors in terms of free radical polymerization kinetics in a polymer plus additive system (Hoffman et al., 1969). Table I1 shows, for any given functionality, the additive concentration which minimizes the dose necessary to initiate gelation. I n this table the doses in the last column are given in the same units used for the last column of Table I, merely to facilitate comparisons. As is apparent, the change in the additive concentration for minimum dose is less over the range F = 4 to F = 6 than from F = 2 to F = 4. This is consistent with the results above in terms of the optimal functionality and suggests the most practical additives to be used. Conclusions
Additive functionality has been demonstrated mathematically to have an optimum value for production of gel. A simplified mathematical model of a polymer system under irradiation has confirmed some of the intuitive approaches to the effect of monomer functionality. There are many ways in which this treatment could be utilized in practice, and even more ways in which it could be generalized and made less dependent upon several of the assumptions made. The effect of dose rate and the kinetics of monomer polymerization could be incorporated to
expand this treatment further. Currently work is under way in these directions. literature Cited
Chapiro, A., “Radiation Chemistry of Polymeric Systems,” pp. 364-77, Interscience, New York, 1962. Flory, P. J., “Principles of Polymer Chemistry,” pp. 37677, 576-89, Cornel1 University Press, Ithaca, N. Y., 1953. Henley, E . J., Johnson, E. R., “Chemistry and Physics of High Energy Reactions,” pp. 375-423, 433-7, University Press, Washington, D. C., 1969. Hoffman, A. S., Jameson, J. T., Salmon, W. A., Smith, D. E., Trageser, D. A., “Electron Radiation Curing of Styrene Polyester Mixtures,” Vol. 11, “Effect of Backbone Reactivity and Dose Rate,” Division of Organic Coatings and Plastics Chemistry, 157th Meeting, ACS, Minneapolis, Minn., April 1969. Peters, W. S.; Summers, G. W., “Statistical Analysis for Business Decisions,” p. 68, Prentice-Hall, Englewood Cliffs, N. J., 1968. RECEIVED for review July 17, 1969 ACCEPTED December 10. 1969 Symposium on Radiation Curing, Division of Organic Coatings and Plastics Chemistry, 157th Meeting, ACS, Minneapolis, Minn., April 1969.
Effect of Composition in Radiation Curing of Unsaturated Polyester Coatings Gerhard J. Pietsch Plastics Division, American Cyanamid Co., Stamford, Conn. 06904 Unsaturated polyester resins can be cured satisfactorily with high energy electrons. Styrene
is a more efficient crosslinking monomer for unsaturated polyester molecules than ringsubstituted styrenes, methacrylate, or dimethacrylates. The degree of unsaturation in the polyester chain, as well as the ratio of monomer to polyester unsaturation, i s critical in obtaining a satisfactory cure at low dose. A styrenated 1,2-propyIene glycol-maleic anhydride alkyd with an alkyd unsaturation to monomer ratio of 1 to 0.6 gave satisfactory cure at
5 megarads.
Dose rates above 2 megarads per second resulted in a lower conver-
sion, at a given total dose, than lower dose rates. Dose fractionation (administering the total dose in two steps instead of one) gave a higher degree of conversion. These effects are explained in terms of competition among recombination, termination, and “chain” propagation mechanisms.
I N RECENT years, polyester resins have gained commercial acceptance as coatings for metal surfaces, masonry, and wood and woodlike substrates. Although unsaturated polyesters comprise only a small fraction of the total coating and finishes market, their annual consumption is continuously expanding. Curing of polyester resins with high energy electrons has received some attention in the past 12 years. Charlesby et al. (1958) and later Hoffman and Smith (1966) established that polyester resins cured by electrons are
comparable in their properties with those cured by peroxides. I t was the objective of the present study to determine the cure and performance of unsaturated polyester coatings on wood and other porous substrates under irradiation with fast electrons. Experimental
Formulation of Unsaturated Polyester Resins. As prototypes for this study, two alkyds were selected: the condensation product of maleic anhydride, phthalic anhydride, Ind. Eng. Chem. Prod. Res. Develop., Vol. 9, No. 2, 1970
149
