Stochastic Description of Copolymerization and Network Formation in

Chapter 19

Stochastic Description of Copolymerization and Network Formation in a Six-Component, Three-Stage Process

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Boudewyn J . R. Scholtens and Truus P. J. M . Tiemersma-Thoone DSM Research, P.O. Box 18, 6160 MD Geleen, Netherlands Numerical results are presented of a stochastic description of branching phenomena in a six-component, three-stage process. The effect of variations in the concentration of trifunctional monomer added in the first stage is remarkably different from the effect of similar variations in the third stage. Tetra-functional monomers added in the first stage shift the gel point to an unexpectedly low conversion. Small variations in the conversion in the first stage affect the network formation in the third stage drastically. Substition effects in the second or third stage may change the pre- and post-gel properties as well. Copolymerization and c r o s s l i n k i n g processes are very important in the paint and coating industry. The t h e o r e t i c a l description of such processes i s helpful in understanding the essential parameters in these complex operations and in improving process conditions and product s p e c i f i c a t i o n s with a minimum of time-consuming systematic experiments . The basis of model c a l c u l a t i o n s for copolymerization, branching and c r o s s - l i n k i n g processes is the stochastic theory of Flory and Stockmayer (l-_3). This c l a s s i c a l method was generalized by Gordon and coworkers with the more powerful method of p r o b a b i l i t y generating functions with cascade substitution for describing branching processes (4-6). With t h i s method i t is possible to treat much more complicated reactions and systems (7-9). In many cases branching leads to gelation and network formation, but not always at the desired moment. Resin manufacturers want to synthesize products with prevention of g e l a t i o n . On the other hand, the resins thus obtained frequently need to form permanent networks a f t e r t h e i r f i n a l application in a paint or coating. As a consequence, a subtle game is played with monomer mixtures which have an end-group average f u n c t i o n a l i t y , T ( i . e . the second moment of the f u n c t i o n a l i t y d i s t r i b u t i o n ) , higher than two. This is the case part i c u l a r l y in multi-stage processes, i . e . polymerization routes in which not a l l monomers are added at once, but in stages (batch-wise). e

0097-6156/89/0404-0213$06.00/0 © 1989 American Chemical Society

Provder; Computer Applications in Applied Polymer Science II ACS Symposium Series; American Chemical Society: Washington, DC, 1989.


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COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE II

Moreover, in such operations the degrees of freedom for synthesizing and c r o s s l i n k i n g are increased, which makes these processes very complicated and d i f f i c u l t to predict. As discussed by Du?ek (10), a two-stage process of macrodiisocyanate synthesis usually r e s u l t s in a wider molecular mass d i s t r i b u t i o n . For resins in which monomers with a f u n c t i o n a l i t y higher than two are used, a two-stage process may influence the f u n c t i o n a l i t y d i s t r i b u t i o n and the gel point. Unequal r e a c t i v i t i e s of equal functional groups and/or substitution effects may further complicate such a multi-stage process. In t h i s study computational results are presented for a s i x component, three-stage process of copolymerization and network f o r mation, based on the stochastic theory of branching processes using p r o b a b i l i t y generating functions and cascade substitutions (11,12). The Three-Stage Process A schematic description of the three-stage process of step reactions is given in Figure 1 and further explained below. Stage 1: Difunctional monomers A, with functional groups c a l l e d c, react by an a l t e r n a t i n g polyaddition reaction with an excess mixture of difunctional D and t r i f u n c t i o n a l T monomers, which a l l have the same functional groups, c a l l e d h (and thus are equally r e a c t i v e ) , to (mainly) h-terminated prepolymer PI. In some calculations t e t r a functional Q monomers with equally reactive h functional groups were present as wel1. Stage 2: Prepolymer PI i s modified with an excess of difunct i o n a l C monomers, also with functional groups c, to (mainly) cterminated prepolymer P2. Unreacted functional groups of the A monomers are assumed not to react further in t h i s stage. Stage 3: The unreacted functional c groups of P2 react in t h i s l a s t stage with a mixture of difunctional E and t r i f u n c t i o n a l F monomers, which have the same functional groups c a l l e d e (and thus are equally r e a c t i v e ) . The h groups are assumed not to react in t h i s stage. In the stochastic theory of branching processes the r e a c t i v i t y of the functional groups i s assumed to be independent of the size of the copolymer. In a d d i t i o n , c y c l i z a t i o n i s postulated not to occur in the sol f r a c t i o n , so that a l l reactions in the sol f r a c t i o n are intermolecular. Bonds once formed are assumed to remain stable, so that no randomization reactions such as t r a n s - e s t e r i f i c a t i o n are incorporated. In our opinion t h i s model i s only approximate because of the necessary simplifying assumptions. The numbers obtained w i l l be of limited value in an absolute sense, but very useful to show patterns, s e n s i t i v i t i e s and trends. With the method applied i t i s possible to take into account substitution effects both in the A and C monomers, as explained elsewhere (12). The substitution effect factor K J J indicates the factor by which the reaction rate between monomer I and any other monomer L i s m u l t i p l i e d for each previous bond formed between monomers I and J ( i . e . f i r s t - s h e l l substitution effects ( 7 ) ) . For p o s i t i v e substitution effects K J J i s larger than 1, for negative effects i t i s smaller than 1.


19. SCHOLTENS & TIEMERSMA-THOONE

Stochastic Description

215


Computer Programs Several computer programs were written in Fortran for a mainframe IBM 4381. In KINREL the various bond p r o b a b i l i t i e s or reaction states are calculated as functions of the conversions by solving a set of 63 coupled d i f f e r e n t i a l equations using Gear's s t i f f method (12). In t h i s program the substitution effects and the addition of the various monomers in the three stages in a sequential order are taken into account. More d e t a i l s on t h i s program and the equations solved are given elsewhere (12). In POLYM the output data of KINREL are used with compositional information to calculate the number and mass average molecular masses (R and M , respectively) and number and end-group average funct i o n a l i t i e s ( f and ? , respectively) in the pre-gel region in a l l stages. In a d d i t i o n , the network c h a r a c t e r i s t i c s such as sol f r a c t i o n , m , and the number of e l a s t i c a l l y active network chains per monomer ( 5 ) , N , are calculated in the post-gel regime of stage 3. POLYMQ is s i m i l a r to POLYM, but with the additional tetrafunct i o n a l Q monomers in stage 1. These two programs contain the formulae derived with the stochastic theory of branching processes which are also specified elsewhere (12). Recently, POLYM and KINREL were also made suitable for c a l c u l a tions on an IBM PS-2 personal computer with a mathematical coprocessor (with about ten times longer elapse times). The two programs are proprietary of DSM and w i l l not be made a v a i l a b l e . n

m

n

e

s

e

Results and Discussion In a l l calculations the molar masses given in the top of Table I were used. F i r s t of a l l , the effects of variations in the concentration of t r i f u n c t i o n a l monomers were determined, as exemplified by the nine formulations of Table I and the r e s u l t i n g prepolymer c h a r a c t e r i s t i c s a f t e r f u l l conversion given in Table II. Formulations F10 to F40 r e s u l t in branched prepolymers, which are cured in the t h i r d stage by difunctional monomers. On the other hand, formulations F00 to F04 r e s u l t in the same l i n e a r prepolymer, which is subsequently cured with various mixtures of d i - and t r i f u n c t i o n a l monomers. The number average f u n c t i o n a l i t i e s of PI (and P2) and of the mixtures of E and F monomers are varied systematically between 2.0 and 2.4. Therefore, the only difference between formulations FjO and FOj is the stage in which the branching units are added. The prepolymer c h a r a c t e r i s t i c s at the end of stages 1 and 2 are collected in Table II. Both the molecular mass and f u n c t i o n a l i t y d i s t r i b u t i o n s become wider in the second stage, in p a r t i c u l a r upon increasing the content of t r i f u n c t i o n a l monomer in the prepolymer. The r e s u l t s of the c r o s s l i n k i n g reaction in stage 3 are presented in Figures 2-7. Figures 2 and 5 show the increase of the mass average molar mass, R , with conversion for the systems with branched and l i n e a r prepolymers, respectively. These results indicate that addition of the branching monomer in the f i r s t stage y i e l d s much higher values of R and the gel point is reached at lower conversion than addition in the t h i r d stage. Translated into p r a c t i c a l properties t h i s means that the processing and application q u a l i t i e s (e.g. flow) of a paint based on formulation F40 w i l l be i n f e r i o r to those of one on the basis of f o r m

m



216


Figure 1. Schematic representation of the three-stage process.

Figure 2. Variation of R with the conversion in stage 3 for f o r mulations F00 to F40; the value of T of P2 ranges from 2.0 to 2.4 (see also Tables I and I I ) . m

n



217


mulation F04. With formulation F50 (T = 2.5) gelation occurs already in the second stage, which is of course highly undesirable for a resin producer. n


Table I. Typical formulations for a three-stage process of network formation monomer

A

D

T

C

E

F

molar mass (kg)

0. 2

0. 2

0.,3

0. 2

0.,2

0. 3

2.,00 2..10 2..20 2..30 2..40

1..00 1..05 1..10 1..15 1,.20

_

0..10 0,.20 0,.30 0,.40 -

2,.00 2,.00 2,.00 2,.00

0,.86 0,.73 0 .61 0 .50

0..10 0..18 0,.26 0,.33

formulation

moles

F00 F10 F20 F30 F40

8..98 8..90 8..83 8..75 8..68

9. 98 9. 80 9. 63 9. 45 9. 28

FOl F02 F03 F04

8,.98 8..98 8,.98 8,.98

9. 98 9. 98 9.,98 9.,98

_

-

-

-

Table II. Prepolymer c h a r a c t e r i s t i c s a f t e r f u l l conversion in stages 1 and 2 P2 (stage 2)

PI (stage 1) M M (kg/mo1) n

m

T

n

T

e

R fyn (kg/mo1) n

^n

^e

formulation F00-F04

3.,79

7.,57

2.,00

2.,00

4.,19

15..22

2.00

2.,00

F10

3..77

8.,28

2..10

2..16

4,.19

19..91

2.10

2..20

F20

3..75

9.,12

2..20

2..36

4,.19

28..67

2.20

2..56

F30

3,.73

10..12

2..30

2..60

4,.19

50,.67

2.30

3,.49

F40

3 .71

11,.34

2,.40

2,.89

4,.19

210 .77

2.40

10,.23

As might be expected, these differences in the pre-gel propert i e s are also r e f l e c t e d in the post-gel regime. The sol f r a c t i o n varies more smoothly with conversion for the branched prepolymer comp o s i t i o n s , c f . Figures 3 and 6. But because the gel point is at (so




218

Figure 4. See legend of Figure 2, but now the v a r i a t i o n of the number of EANC's per monomer.




219


10 : 1

— ^ - conversion (stage 3) 0.0

0.2

0.4

0.6

0.8

1.0

Figure 5. Variation of H with the conversion in stage 3 for f o r mulations F00 to F04; the value of f* of the mixtures of E and F ranges from 2.0 to 2.4 (see also Tables I and I I ) . m

n

F04F03F02F01F00

Figure 6. See legend of Figure 5, but now the v a r i a t i o n of the sol f r a c t i o n .



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COMPUTER APPLICATIONS IN APPLIED POLYMER

SCIENCE II

much) lower conversion in these FjO compositions they possess a lower sol f r a c t i o n over almost the entire range of conversion beyond the gel point. In Figures 4 and 7 the numbers of e l a s t i c a l l y active network chains (EANC's) per monomer are plotted as functions of the conversion (beyond 0.5). Apparently the network build-up is smoother in the compositions with branched prepolymers. As a consequence an almost complete conversion is more c r i t i c a l for formulations F01-F04 than for F10-F40, provided a permanent network is required for good properties such as solvent resistance and adhesion. The e f f e c t of modifying PI by adding a tetrafunctional instead of a t r i f u n c t i o n a l monomer in the f i r s t stage is s i g n i f i c a n t . For a good comparison the concentrations of end groups were kept constant, so the molar concentration of Q was taken as 0.75 times that of T. After t h i s modification formulations F20 to F40 gelled already in stage 2. Figure 8 compares the curves of M versus the conversion of h in stage 2 for the modified (dotted curves) and the o r i g i n a l f o r mulations F20 and F40 ( s o l i d curves). It must therefore be concluded that a small f r a c t i o n of monomers with a f u n c t i o n a l i t y higher than three in stage 1 interferes with a trouble-free resin production (gelation taking place too r e a d i l y ) . Figures 9-11 show what e f f e c t t h i s same Q-modification (Q instead of T) has on the pre- and postgel properties of formulation F10 in the l a s t stage. The M is increased by at least 60 %, and the gel point is decreased from 91 to 78 % conversion. As a consequence, the network build-up s t a r t s at lower conversions for the compositions containing the Q-modified prepolymers. Small variations in the conversion of A in the f i r s t stage (but a complete conversion of h in stage 2) have a very s i g n i f i c a n t impact on the c h a r a c t e r i s t i c s of P2 and on the pre- and post-gel properties in stage 3. This is exemplified in Figures 12-14. In Figure 12 i t can be seen that a decrease of only 2 % in the conversion in stage 1 decreases R by 82 % and s h i f t s the gel point from 0.32 to 0.59. In a d d i t i o n , the sol f r a c t i o n is s i g n i f i c a n t l y higher over almost the entire conversion range beyond the gel point, as shown in Figure 13. These effects are considerably stronger than for s i m i l a r variations in the conversion but now in stage 2, as reported previously (13). Because of the complete conversion of the h groups in stage 2 in the present c a l c u l a t i o n s , the ultimate concent r a t i o n of EANC's (and the sol f r a c t i o n ) is the same for these four examples, as shown in Figure 14. This finding is in contrast with r e s u l t s of s i m i l a r variations in the conversion in stage 2; in that case some h groups remain unreacted, as a r e s u l t of which a d r a s t i c reduction in N (and gel f r a c t i o n ) is observed (13). Therefore i t may be concluded that a minor degree of incompleteness in the conversion of stage 1 only a f f e c t s R of the prepolymer and the gel point in stage 3 - so the flow w i l l be better. At (almost) complete conversion the variations in network structure and solvent resistance are expected to be n e g l i g i b l e . The l a s t topic to be treated is unequal r e a c t i v i t y by s u b s t i t u t i o n e f f e c t s . As a f i r s t example, the effect of an i n f i n i t e l y negat i v e substitution e f f e c t in C due to a reaction with an h group (so CD CT °) compared with the case of equal (random) r e a c t i v i t y of the two functional groups in C for formulation F40. This is suggested as an example of p o l y e s t e r i f i c a t i o n with an anhydride and a carboxylic a c i d , respectively. Figure 15 gives the dramatic effect on m

m

m

e

m

K

=

K

=

i s





221


Figure 8. mulations mulations reduction

Variation of H with the conversion in stage 2 for f o r F20 and F40 - the s o l i d curves - and the same f o r but now with monomer Q instead of T and a simultaneous in molar concentration by 25 % - the dotted curves. m



222


Figure 9. Variation of M with the conversion in stage 3 for f o r mulation F10 - the s o l i d curve - and the same formulation but now with monomer Q in stead of T and a simultaneous reduction in molar concentration by 25 % - the dotted curve. m

1.0

A

|

\

I

I

V

\

0.6

0.4

0.2

I

T

\ I \ I \ I \ I \ I \ I \ I \I

\ \ \ \ \\ \\ \\

^ - c o n v e r s i o n (stage 3)\




SCHOLTENS & TIEMERSMA-THOONE


223


Figure 12. Effect of a p a r t i a l conversion of A in stage 1 (indicated) on M in stage 3 for formulation F40. m




224





225


F40

10'

10

1

- conversion (stage 3) 10

u

0.0

0.2

0.4

0.6

0.8

1.0

Figure 15. Variaton of R with the conversion in stage 3 for f o r mulation F40; 1 means no substitution e f f e c t s , 0 means KcD = K d °' m

=

M and the gel point, Figure 16 concerns the sol f r a c t i o n and Figure 17 the number of EANC's per monomer. The differences are most pronounced near the gel points and they vanish with completion of the reaction. Another example is given in the l a s t three f i g u r e s . For the same formulation F40 the influence of substitution effects in C due to a reaction with E are given. The negative s u b s t i t u t i o n e f f e c t in C gives a preference for reactions of C monomers connected once to D and T over those connected once to E. As a r e s u l t the already bigger molecules containing the D and T monomers grow faster than the smaller molecules containing only C and E monomers, so that the gel point i s shifted to lower conversion by a negative substitution e f f e c t , as shown in Figure 18. This trend i s continued beyond the gel point as shown in Figure 19 for the sol f r a c t i o n and in Figure 20 for the number of EANC's per monomer. Although the present r e s u l t s are only t h e o r e t i c a l and not yet v e r i f i e d by experiments, i t is f e l t that t h i s approach, although very much s i m p l i f i e d with respect to p r a c t i c e , can be used at least as a q u a l i t a t i v e guideline in choosing new experiments for product development. These r e s u l t s are therefore useful in a q u a l i t a t i v e sense to show s e n s i t i v i t e s , trends and d i r e c t i o n s . Experimental v e r i f i c a t i o n may improve the a p p l i c a b i l i t y to ( s e m i - Q u a n t i t a t i v e p r e d i c t i o n s , or indicate shortcomings in the present approach, which may be accordingly adjusted. m


COMPUTER APPLICATIONS IN APPLIED POLYMER SCIENCE


226






111

Figure 18. Varation of H with the conversion in stage 3 for f o r mulation F40; Kce 4, 1 and 1/4, respectively. m

=




228



Acknowledgments The authors are grateful to the management of DSM Resins for t h e i r permission to publish t h i s work, and to Dr. R. van der Linde, Dr. K. Dusek and Prof. M. Gordon for the many stimulating discussions; Mr. E. Peters i s g r a t e f u l l y acknowledged for his programming assistance and Mr. G. Schuler for drawing the f i g u r e s . Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Flory, P.J. J . Amer. Chem. Soc., 1941, 63, 3083, 3091 and 3096. Flory, P.J. Principles of Polymer Chemistry; Ithaca, NY, Cornell University Press, 1953, Ch. IX. Stockmayer, W.H. J . Chem. Phys., 1943, 11, 45; 1944, 12, 125; J . Polym. S c i . , 1952, 9, 69; 1954, 11, 424. Gordon, M. Proc. Roy. Soc. (London), 1962 A268, 240. Dobson, G.R. and Gordon, M. J . Chem. Phys., 1965, 43, 705. Gordon, M. and Malcolm, G.N. Proc. Roy. Soc. (London), 1966, A295, 29. Gordon, M. and Scantlebury, G.R. Trans. Faraday Soc., 1964, 60, 604; Proc. Roy. Soc. (London), 1966, A292, 380; J . Chem. Soc. B, 1967, 1. Burchard, W. Adv. Polym. S c i . , 1983, 48, 1. Dušek, K. Adv. Polym. S c i . , 1986, 78, 1. Dušek, K. Rubber Chem. Technol., 1982, 55, 1. Dušek, K., Scholtens, B.J.R. and Tiemersma-Thoone, G.P.J.M. Polym. B u l l . , 1987, 17, 239.





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12. Tiemersma-Thoone, G.P.J.M., Scholtens, B.J.R. and Dušek, K. In Proceedings of the 1st International Conference on Industrial and Applied Mathematics (ICIAM 87), Contributions from the Netherlands, ed. by van der Burgh, A.H.P. and Matthey, R.M.M., CWI Tract 36, 1987, p. 295. 13. Scholtens, B.J.R., Tiemersma-Thoone, G.P.J.M. and Dušek, K. In Proceedings of the Rolduc Polymer Meeting-2, 1987, Integration of Fundamental Polymer Science and Technology, ed. by Lemstra, P.J. and Kleintjens, L . A . , Elsevier Appl. Sci. Publ., London, 1988, p. 283. RECEIVED

February14,1989


Stochastic Description of Copolymerization and Network Formation in

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