Molecular weight distribution of products of radical polymerization initiated using polyfunctional initiators

Products of radical polymerization initiated using polyfunetional initiators

2141:'

2. G. S. KOLESNIKOV, O. Ya. FEDOTOVA, O. I. PERESISHVILI and S. F. BELEVSKII, Vysokomol. soyed. A12: 317, 1970 (Translated in Polymer Sei. U.S.S.R. 12: 2, 362, 1970) 3. A. I. KOL'TSOV, N. G. BEL'NIKEVICH, V. M. DENISOV, L. N. KORZHAVIN, N. V.

MIKHAILOVA and V. N. NIK TIN, Vysokomol. soyed. A16: 2507, 1974 (Translated in Polymer Sei. U.S.S.R. 16: 11, 2912, 1974) 4. I. Ye. KARDASH, A. Ya. ARDASHNIKOV, F. S. YAKUSHN and A. N. PRAVEDNEKOV,

Vysokomol. soyed. A17: 598, 1975 (Translated in Polymer Sci. U.S.S.R. 17: 3, 689, 1975~ 5. V. P. PSHENITSYNA, L. G. KAZARYA_N, Ye. P. LUR'YE, M. L. LEBEDINSKAYA and

V. V. KOVRIGA, Vysokomol. soyed. A14: 628, 1972 (Translated in Polymer Sci. U.S.S.R. 14: 3, 702, 1972) 6. Ye. V. KAMZOLKINA, P. P. NECHAYEV, V. S. MARKIN, Ya. S. VYGODSKII, T. V. GRIGOR'YEVA and G. Ye. ZAIKOV, Dokl. AN SSSR 219: 650, 1974 7. V. Ye. SMIRNOVA, L. A. LAIUS, M. I. BESSONOV, V. S. BUSKIN, T. I. GARMONOVA,

M. M. KOTON, V. S. SKAZKA and L. M. SHCHERBAKOVA, Vysokomol. soyed. A17: 2210, 1975 (Translated in Polymer Sei. U.S.S.R. 17: 10, 2549, 1975) 8. A. M~ BOGOMOLOV, Optika i spektroskopiya 12: 186, 1962 9. N. P. KULIKOVA, M. V. SHABLYGIN and L. Ye. UTEVSKII, Khimieh. volokna, 1~o. 3, 24, 1973

MOLECULAR WEIGHT DISTRIBUTION OF PRODUCTS OF RADICAL POLYMERIZATION INITIATED USING POLYFUNCTIONAL INITIATORS* S. I. KUCHAlVOV, X. G. IVAlVOVAa n d S. S. IVA~CHEV Scientific-Industrial Association "Plastpolimer", Okha

(Received 20 November 1975) A method is proposed for the first time using kinetic process parameters and results concerning the heat resistance of initiators for the quantitative evaluation of MWD, the variation of the coefficient of po]ydispersion and numerical average molecular weight of products of radical polymerization, initiated using polyfunetional initiators containing peroxide groups of the same heat resistance, which decompose independently. The method can be used not only for peroxide, but also for any fllnctional groups forming radicals, if they react independently and have the same heat resistance. IT HAS been s h o w n in m a n y p a p e r s t h a t p o l y m e r s of high molecular w e i g h t s are o b t a i n e d with high initiator c o n c e n t r a t i o n s a n d therefore high rates o f initiation [1-4] b y radical p o l y m e r i z a t i o n initiated b y p o l y f u n c t i o n a l initiators, p a r t i c u l a r l y oligomer peroxide c o m p o u n d s o f different types. M W o f the p o l y m e r * Vysokomol. soyod. A18: No. 8, 1870-1877, 1976.

2142

S. I . KUCHANOV ¢t al.

increases considerably in these systems with an increase of the degree of conversion and depends on the type of polyfunctional peroxide and the proportion of peroxide groups in the molecule (functionality) [5, 6]. This type of initiator is promising not only from the point of view of possible increase in efficiency of polymerization processes, but also when high molecular weight products cannot be <)btained by radical polymerization. We tried to calculate MWD of a polymer formed in a system containing polyinitiators with peroxide groups of the same heat resistance and decomposing independently. It is the first time that the mathematical simulation of this polymerization process is undertaken and although we are ~concerned with peroxide groups, the approach proposed can be used for any functional groups provided they react independently and have the same heat resistance. The main feature of the polymerization mechanism in the presence of these initiators is the formation during polymerization of macromolecules containing peroxide groups in the main chain, which will then be re-converted into macroradicals capable of continuing polymerization [7-9]. The structure of macromolecules formed during radical polymerization by the action of oligomer or polymer initiators is complex. Figure 1 illustrating this structure indicates diagrammatically that macromolecules consist of a certain number of elementary polymer chains separated by peroxide clusters. By clusters of n length we imply subsequently a sequence consisting of n peroxide groups which is restricted either by two elementary chains (closed clusters), or an elementary chain and the end of the macromolecule (semi-enclosed clusters). When the efficiency of initiation f~- 1, each cluster only consists of a peroxide unit. When ] ~ 1 the clusters may contain a few peroxide blocks separated by fragments from each other, formed by the recombination of primary radicals in the "cell" as a res u l t of the ineffective breakdown of peroxide groups. When using conventional initiators with one peroxide group, all macromolecules will only consist of an elementary chain and will be free from clusters. When using initiators with a number of peroxide groups in the molecule more than one the polymer molecule may consist of an arbitrary number of elementary chains separated by clusters. The number of these chains in the macromolecule m, the same way as the length of the macromolecule L, are random values characterized by distribution functions. I f we ignore the effect of peroxide clusters on MW of macromolecules, the distribution of MWD of the polymer will be described by the superimposition of distributions of random values of m and L. Let us denote by Pm (t) the concentration of macromolecules, containing m elementary chains at t moment of time and by Wm(L)dL the probability of the macromolecule consisting of m chains of this kind having a length in interval L--L~dL. Then, the following formula may be written for the MWD function Of products of polymerization:

F(L, t)= ~ P,n(t) t?/,--1

W,,,(L)

(1)

Products of radical polymerization initiated using polyfunctional initiators

2143

It should be emphasized that this formula is only valid when kinetic chain rupture occurs by the mechanism of addition of ra~licals. Results concerning a more general case, bearing in mlnd disproportionation, are given below. In order to simplify further calculations using the Laplace transform we should change o v e r from the distribution F (L, t) to its characteristic function G co

G(p, t ) = ~ e-~'LF(L,t)dL= 0

oo

~ P~(t) ~ e-~LWm(L) dL, m=l

(2)

0

which is equivalent to MWD, but more convenient for the calculation of stat&tical moments [10]. We will use the probability theorem [11], according to which the characteristic function of distribution of the total of several independent random values is equal to the product of characteristic functions of distribution of each component.

J

'

~

I

I

J

•

~

FIG. 1. Structural layout of polymer macromoleculesformed during polymerization by t h e a c t i o n of polyperoxide: /--elementary polymer chain; 2 and 3--closed and semienclosed clusters; 4--peroxide units. In our case the entire length of the macromolecule is added up of the total of independent random values l--the lengths forming this macromolecule of elementary chains and the theorem indicated is therefore applica,ble. I f ~ (1) is the distribution function according to elementary chain lengths and g (p) is a characteristic function corresponding to this distribution, bearing in mind the foregoing, formula (2) takes the form

m--I

m--I

0

The calculation of MWD of polymer molecules is thus formed of two parts and involves the independent calculation of values of g and Pro. As far as the first is concerned, it does not involve any specific features due to the use of poly-initiators and is determined exclusively by the type of kinetic chain extension and rupture. It is easy to note that distribution according to elementary polymer chain lengths conincides with the M-WD known from the literature [12] for macromolecules obtained during polymerization with monofunctional initiators. Using this condition we obtain for g

g(Io)_ ] e_~Z_:_l e_~/~-dl__o

(1)5

1

(4)

(1+p~)2'

where l=keM/ICrRis the average length of kinetic chMns, ke and/cr rate constants

::21~14

S.: I., K~cm~ov et aJ.

o f ~xtension and rupture by recombination and M and R. are the concentrations ot the monomer and radicals in the system. I t should be noted that formula (4) is only valid for unchanged process conditions, when l i s independent of time (for example, with low conversions). As polymerization goes on, the consumption of monomer or initiator becomes significant and the value of 1 varies with conversion and instead of function g in formula (3) its average value should be used *

I(t')dr

= ~

t

o [I÷p~-(¢)F

I j x(r)dr, 0

,((~)

wh~re ] (t) is the rate of initiation. Hwe and further, angular brackets show average distributi~n~ 0f the time of formation of elementary chains. The second, more complex problem involved in calculating MWD is the calculation of distribution Pm of macromolecules according to m, the number of elementary chains contained. A specific feature of this process is the fact that macromolecules Containing c~usters may undergo degradation as a consequence o f the breakdown of some peroxide group contained in one of these clusters. By joining the monomer radicals formed during similar processes of breakdown grow until '.kinetic chain rupture occurs as a result of addition. If the concentration of radicals contairgng m elementary chainsis denoted by Rm and the concentration of the initiator by P0, the kinetic equation system which describes the process in question takes the form:. x~ dPtn/dt------[2Jllq-(m--yl)~]P,~-t- k__r 2,]o

R ,~m_t_~, D Pro(0)=0, at m~>l

dPo/dt-=--2oPo, dR,n/dt=21~Prn+2,~2 ~, P~--k~Rm O0

Po(0)=P~o R~,

(6) (.7)

R~(0)=0, at m i> 1

(8)

O0

dRo/dt=22oPo-62]tl~= P,--k, Ro,.oZR,,

Rb(0)= 0

(9)

Coefficients 20, 21 and 22 in equation (6)-(9) are equal to probabilities of the effective breakdown during unit time of open clusters, i.e. initiator molecules, semi-enclosed and closed clusters respectively. These coefficients are proportional to the averaq~e length of corresponding clusters no, nl, n2 and the product of the rate constant of decomposition of the peroxide group k and the efficiency of initiation f Xo=//~o,

~l=k/~,

~,2=fk~2

(10)

Since the decomposition of clusters is independent of polymerization, the problem concerning distribution according to the number of peroxide groups may be solved separately. The solution of this problem with arbitrary initial distribution

Products of radical polymerization initiated using polyfunctional initiators

2146

of initiator molecules according to the number of peroxide groups contained takes the form:

n0(z)=

~jaf ~(1--f -6fe~) 1-1 ~a~(l _ f q_fe_~)t

~l(z)---- ~f-~{l"(i--f+fe-~)Y--ff(1--e-')(1--f+feT)J-a} f(1 --e-')~y{ 1 --(1 --f+fe-') y} ~(~)=

(11)

~f-"{jf( 1--e-')[1 q- (1 --f+fe-')~-a] - 2 [1 -- (1 -f+fe~) J]} f(l_e~)~j{ff(l_e-D_lq_(1.f+fe-,)J]} '

where aj is the initial part of initiator molecules with j peroxide groups amd z----kt is dimensionless time. With conventional mono-functionM initiators (when ~x----1 and ~11 other aj----0) formulae (10)and (11)produce 21----~2----0, 20=¥. It follows from equation (8) that the concentration of all radicMs with m/> 1 will be zero which, according to formula (6), results in the inversion to zero of all values of P~n, except for Pi- In this case th e general system of (6)-(9) is reduced to three equations for initiator P0, radicals Ro and inactive polymer chMns Px. These equations are well known in the theory of radical polymerization [12]. In processes with polyfunctional initiators, whese 21 and 22 differ from zero, it becomes necessary to solve a whole chain of differential equations (6)-(9), for which it is advisable to introduce generating functions

z)= VMues of U and V being known from formulae co

P=

R= m--I

(13) fa--i}

the entire concentrations of radicals R and polymer molecules P may be easily calculated. Multiplying term by term equations (6)-(9)by x m and summarizing them, considering equations (12) and (13) we obtain the following pair of equations for generating functions kr d U /av= --2]~,U--f n, \., ~x O____F= 2j~opi}nt_2f~,ent_ 2j.~, Uq- 2f~,,(~e-- U) _ k_r/~V Ov 1--x k

(15)

Using the prineil~le of stationary state, as is normally the ease in calculations Of radieM concentrations [12], the right hand side of equation (15) m~y be adjusted to zero and after 'the elimination Of V system (14)'(15) brought to a differentiM equation for function U, the solution of which

U=P , - (1-")x -

p-----l--ZaP[l--(l--f-{-fe~)']'

(16)

S.I. Kvc~-~ov et al.

3i46

oej(1--f-i-fe-r) J is the overall concentration of polymer

where P = P o° 1--

molecules. Formula (16) enables us to derive the characteristic function of MWD G ~ , z). For this, as follows from equations (3) and (12), it is sufficient that in U(x, z) instead of variable x the function 9(P), more precisely the average value of (g(p)) be substituted. Statistical moments of MWD sad therefore, the numerical average value ~f P . and the weight average value Pw of the degree of polymerization and the coefficient of polydispersion K=P,dPn are easily expressed by the characteristic function and its derivatives at point I0----0 [10].

Pn =

G~,(0, z ) ----2(1")zV~(1, . _ . . ) =2(~)(l__p)_X=Pn(l__p),t G(O, z) U(1, "c)

(17)

G1,~,(0, z) 2 ( l ) Ux~(1, z) + 3 (~'z) = 2 ( T ) 1--~

(18)

a,,(1, ,)

,)

+ 3 (~2)

__

-

K = 1.5 (l"~) (l)-~(1 --p)+2p=K'(1 --p) + 2p,

(19)

where P~ and K' correspond to polymerization under similar conditions but by the action of a mono-initiator with the same concentration of peroxide groups

cpa=Po° o (0). Formulae (17) and (19) indicate that Pn and K increase from P~ and K' to some limiting values dependent on the type of polyperoxide uj and the efficiency of initiation f. Let us illustrate the general ratios obtained using two examples of polymerization by the action of poly-initiators with the following distribution of molecules according to the number of peroxide groups: a) ~z,,.-~-l, ~j=0 at j # 2 ; b) ~xa----~(1--~)~'-I

(20)

Using equation (16) we derive from formulae (17)-(19) characteristics P . and K

Pn = 1 + f(1--e-") ;

a) p--7

2--f+fe-"

b) p--TnP__ n 1+ f(1--at); (1--e"') ~

K=K,+(2_K,)f (

2

e )

(21)

K=:K,+(2_K,)l_(l_tx)(l_f+fe_.,)

If up to now it was considered that kinetic chain rupture only takes place by recombination, we will now examine a more general case considering the possibility of this rupture by disproportionation. If the rate constant of the latter reaction is k'r, then it is evident that the proportion of all radicals, which is ~'--~'r/~r~-~'r), will decrease with disproportionation, whereas the proportion of 1 --2----kr/(kr+kr), with recombination. We will now distinguish between macro-

Products of radical polymerization initiated using polyfuuctional initiators

2147

molecules not only according to the number of internal elementary chains m, but also according to the number i of outer chains which are formed by the mechanism of disproportionation of radicals. If Pint (t) is the concentration of these molecules at t moment of time, ratio ¢o

2

G(p, t)---- ~ ~ Prm(t)gm(p)g~(p),

(22)

m ~ 0 tffi0

is the analogue of formula (3), in which g(p) is determined by formula (4) and the characteristic function for distribution according to lengths of these outer chains will be

gl(P) = ~ e-~'q -le-tfi d[= (1 ~-pl)-i

(23)

0

For concentrations Pm~ a kinetic equation system similar to (6)L(9) may be given, the solution of which should be sought, as before, by introducing generating function oo

2

for which the differential equation is a generalized equation derived from system (14)-(15). The solution of this equation takes the form

oo., . . . . vI

,y,

.-.t,7] 1--p [[1--[--2p(1--p)-ly] ~

l=ro°

} -1

(25)

Parameter p is determined as in formula (16), in which ratio (25) is changed when 2=0. Using equation (25) by calculating corresponding derivatives j t ia easy to obtain a generalization of formulae (17)-(19) considering disproportionation P;,(2+1) 2<1> Pn-= 2-4-1--p ----2 + l - - p (26~

K=K,(2+I--p)-~ p(2--2)'(2+1--p) ; 2+1

K'----(2+1)(3-2)

2(1--p+2p)

2

<~>~

(27)

From ratios (25), (4) and (22), (23) when <~>2= <~>=T~ the following equation may be derived for MWD

F(£)=P where

(1--b~) 2 ~(1 b) 2 ~(l+b) }, 2b (a2~-b2-]-2a) {(a+b) e- - --(a--b) e-

(28~

a=2p(1--p) -1, b----~/(1--2)p, ~-~L/l Theoretical calculations were Compared with experimental.results obtained by studying polymers formed under conditions which approximate ~o model

$1¢g

S . I. Kvc~.wov et ~.

conditions. S t y r e n e was t h e r e f o r e p o l y m e r i z e d in b u l k in t h e presence o f t h e following peroxide compounds: CHs(CH=) 10--C--O--O--C(CH=)I0--CHs

(compoud 1)

CHs(CH=)8--G---O--O---C(CH=)~--C--O~O~C(CH=)8--CH=

L

.

v

~-,,

(compoud 2)

.jlo

P e r o x i d e c o m p o u n d s were o b t a i n e d b y we]l k n o w n m e t h o d s [2, 3, 5]; active c o n t e n t d e t e r m i n e d iodometrically w a s 98-99 ~ . K m e t m p a r a m e t e r s o f polymerization are shown in Table 1. O

o

I

•

•

' " 2~ ' 2O0 28O Time, m/n

Fig. 2. Kinetic curves of styrene polymerization at 80° and a concentration- of peroxide groups of 0.0.025 mole/1, in initiation with mono- (1), di- (2) and poly, peroxides (3). Conditions of polymerization were selected to ensure the same corcentration of peroxide groups c ~ in the polymerization system, independent of the type of initiator used; this produced an agreement of rates of initiation (at a given temperature) within the range of experimental error. Initiator concentration and process temperature zn~e it possible to exclude secondary reactions. The polymers formed were separated by precipitation into ethanol and dried at room temperature in vacuum to constant weight. ~umber average -~n, weight average -~w and viscosizy average molecular weights _71~ were determined for the samples separated by osmometry, light scattering and viscometry. Constants of decomposition and e~RciencJes of fuitiation of the peroxides used (Table 2) [3, 13-15] were chosen for ~he calculations. T a b l e 1 indicates t h a t initial rates o f p o l y m e r i z a t i o n a t a given t e m p e r a t u r e a n d overall c o n c e n t r a t i o n o f peroxide groups are practically i n d e p e n d e n t o f t h e n u m b e r o f peroxide g r o u p s in t h e initiator molecule a n d are almost identical for mono-, di- a n d polyperoxide. F u r t h e r m o r e , according t o Fig. 2, kinetic curves agree for all t h e t h r e e peroxides indicated in t h e conversion range o f . ~ 2 0 % within t h e r a n g e o f e x p e r i m e n t a l error, a l t h o u g h t h e molecular weights

Products of radical polymerization initiated using polyfunctional initiators

2149

o f p o l y m e r s o b t a i n e d (Table 2) m a y differ severall t i m e s for initiation w i t h peroxides of different functionalities. E x p e r i m e n t a l results p o i n t t o t h e a b s e n c e of a n y noticeable relation between polymer molecular weight and constants of all e l e m e n t a r y r e a c t i o n s in t h e r a n g e of v a r i a t i o n o f conversion studied. T ~ L E 1. KL~-~IC ~ ~ , ~ s o~ s~r~.~-~ e O ~ . V ~ I Z ~ I O ~ ImTL~TED BY COMPOUNDS 1--3 Com-

G ° PG ' .

pound

mole/1.

I

wp × 104 (mole/1. •sec) at T, °C 70 80

0.025 0.0025

2.4

1.2

2.4

--

0.025 0.0025

0.9 1.3

2.5 --

0.025 0.0025

106 (mole/1..see) at T, °C 70 80

×

2-4

0-9

1-2

2.4 --

0-9

2.4

E x p e r i m e n t a l results in T a b l e 2 confirm t h e t h e o r e t i c a l conclusions concerning t h e increase in t h e m o l e c u l a r w e i g h t o f p o l y m e r d u r i n g t h e process w i t h p o l y peroxides; t h e higher t h e f u n c t i o n a l i t y of i n i t i a t o r molecules, t h e m o r e m a r k e d TABLE

2. RESULTS

OF INVESTIGATIlqG

PS

SAMPLES

T=70 ° C

T=80

I

P:/P;

°

determined from the value of

. ,ff

K O

×

X

× i

O

q~

¢)

60 280 60 280 60 280

10 20 10 20 10 20

70 80 102 --

C

PJP;

100 100

1.5

102 101

1.3

175 167 I 1.7 ]

--

L.O~,

1,1,L

-ol;

b6(

1.4;¢

•7~;

140

--

204i

-

1.01 148i 295 1.67 500 850

1.07 1.14 2-54 4-30

--

-

1.06 1.44 4-00 4"17

is this increase. Considering t h e a c c u r a c y of t h e e x p e r i m e n t , t h e q u a n t i t a t i v e a g r e e m e n t b e t w e e n t h e o r e t i c a l a n d e x p e r i m e n t a l results s h o w n in T a b l e 2, m a y be r e g a r d e d as s a t i s f a c t o r y . A s t u d y of p r o d u c t s of p o l y m e r i z a t i o n b y gel chrom a t o g r a p h y considering t h e peroxides e x a m i n e d suggests t h a t in a c c o r d a n c e w i t h t h e o r y , p o l y m e r s h a v e a M-WD c u r v e w i t h o n l y one m a x i m u m in e v e r y case.

Translated by E. S ~ . ~ . ~

2150

S.I. Kucm~ov REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

N. S. TSVETKOV, Dissertation, 1972 R. F. MASKO,VSKAYA, Dissertation, 1969 S. G. YERIGOVA, Dissertation, 1969 Yu. L. ZHEREBIN, Dissertation, 1972 N. S. TSVETKOV and Ye. S. BELETSKAYA, Ukr, khim. zh. 33: 380, 1967 N. S. TSVETKOV and R. F. MARKOVSKAYA, Vysokomol. soyed. A16: 1936, 1974 (Translated in Polymer Sci. U.S.S.R. 16: 9, 2238, 1974) V. V. KORSHAK, S. V. ROGOZHIN and T. A. MAKAROVA, Izv. AN SSSR, Otd. kl~im, n., 1958, 1482 N. A. SHAN, F. LEONARD and A. V. TOBOLSKY, J. Polymer Sei. 7: 537, 1951 N. S. TSVET]KOVand R. F. MARKOVSKAYA, Vysokomol. soyed. 7: 169, 1965 (Translated in Polymer Sci. U.S.S.R. 7: 1, 185, 1965) S. I. KUCHANOV and L. M. PIS'MEN, Vysokomol. soyed. A13: 2035, 197r (Translated in Polymer Sei. U.S.S.R. 18: 9, 2288, 1971) B. V. GNEDENKO, Kurs. teorii veroyatnostei, (Course on the Theory of Probabilities). Izd. "Nauka", 1965 Kh. S. BAGDASAR'YAN, Teoriya radikal'noi polimerizatsii (Theory of Radical Polymerization). Izd. "Nauka", 1966 V. L. ANTONOVSKII and L. D. BEZBORODOVA, Zh. fiz. khimii 42: 351, 1968 V. I. GALIREI, Dissertation, 1965 S. S. IVANCHEV, L. V. SKUBILINA and Ye. T. DENISOV, Vysokomol. soyed. B9: 706, 1967 (Not translated in Polymer Sei. U.S.S.R.)

THEORY OF HOMOGENEOUS IRREVERSIBLE COPOLYCONDENSATION* S. I. KUCHA~OV (Received 3 December 1975)

A quantitative theory was developed for irreversible homogeneous inter-eopolycondensation, when the inter-monomer has dependent functional groups. It was shown that this process may be described by a Markoff chain; parameters of this chain were derived as functions of activities and proportions of initial monomers. The function of the composition distribution of polyeonden~ation eopolymers was calculated for the first time, which characterizes composition heterogeneity. I~XTENSIVE e x p e r i m e n t a l s t u d y a n d establislfing m a i n relations of irreversible (non-equilibrium) c o p p l y c o n d e n s a t i o n in h o m o g e n e o u s systems, carried o u t r e c e n t l y using processes of a c c e p t e r c a t a l y t i c polyesterifieation [1] gave a considerable i m p e t u s to t h e d e v e l o p m e n t o f t h e q u a n t i t a t i v e t h e o r y of thi~

* Vysokomol, soyed. A18: No. 8, 1878-1884, 1976.