The two types of gels in the polymerization of bifunctional monomers

2738

V . A . SAL'Nntov and N. M. BoL'arr

REFERENCES 1. T. P. MATVEYEVA, Yu. I. MATVEYEV and A. A. ASKADSKI~ Mekhanika kompozit. materialov, 2, 201, 1986 2. A. A. ASKADSKII and Yu. L MATVEYEV, Khimicheskoye stroyenie i fizicheskie svoistva polimerov (Chemical Structure and Physical Properties of Polymers). Moscow, 1983 3. G.L.SLONIMSKll; A. A. ASKADSKII and A. I. KITAIGORODSKII~ Vysokomol. soyed. AI2: 494, 1970 (Translated in Polymer Sci. U.S.S.R. 12: 3, 556, 1970) 4. F. Kh. DZHEIL, Polimernye monokristally (Polymer Single Crystals). Leningrad, 1968 5. L. D. LANDAU and Ye. M. LIFSHITS, Statisticheskaya fizika (Statistical Physics). Moscow, 1964 6. M. V. VOLKENSHTEIN, Konfiguratsionnaya statistika polimernykh tsepey (Configurational Statistics of Polymer Chains). Moscow, 1959 7. P. FLORY, statisticheskaya mekhanika tsepnykh molekul (Statistical Mechanics of Chain Molecules). Moscow, 1971 8. A.A. ASKADSKII, Yu. I. MATVEYEV, A. V. PASTUKHOV, B.A. ROZENBERG, T.I. PONOMAREVA, N. A. SHCHEGOLEVSKAYA and A. S. MARSHALKOVICH, Vysokomol. soyed. A25: 56, 1983 (Translated in Polymer Sci. U.S.S.R. 2.';: 1, 64, 1983) 9. A. V. PASTUKHOV, Diss... Cand Chem Sci., L. Ya. Karpov Physicochemical Research Institute, Moscow, 1985 10. J. E. MARK and J. L. SULLIVAN, J. Chem. Phys. 66: 1006, 1977

PolymerScienceU.S.S.R. Vol. 30, No. 12, pp. 2738-2744,1988 Printed in Poland

0032--3950/88 $10.00+ .00 0 1990Pezpmon Preu pie

THE TWO TYPES OF GELS IN THE POLYMERIZATION OF BIFUNCTIONAL MONOMERS* V. A. SAL'NIKOV a n d N. M. BOL'BIT Branch of the L. Ya. Karpov Scientific Research Institute for Physical Chemistry

(Received 30 June 1987) It has been shown by mo d©lling the process of radical polymerization of bifunctional monomers in a cubic lattice by using random walk of the "propagating chain end without return and without self-int©rsection that the structure of the gels in the early stages of conversion of the double bonds ( < 20~o) depends on the activity of the double bonds pendant to the chain. With an increase in activity, there is a transition from a system of open interpenetrating chains to a system of compact coils linked by tie chains. The heterogeneity of the gel, as assessed by the radius of correlation, is thus a minimum at a certain intermediate activity. IDEAS f r o m percolation theory have been used recently in the solution o f problems in gel,formation ~I]. He~tman's [2] kinetic percolation model, which postulates the simultaneous g r o w t h o f m a n y clusters (branched polymeric coils) o n a lattice, has been f o u n d * VYsokomol. soycd. A30: No. 12, 2551-2555. 1988.

Two types of gels in polymerization of biftmctional monomers

2739

to be espec/ally fruitful for processes of radical three-dimensional polymerization and copolymerization. The authors obtained the dependence of the gel-formation point, Xc, on a number of parameters

Xc'c°'t2ft-o"5/(1 -ca),

(1)

w h e r e cb, ct a n d c, a r e the initial c o n c e n t r a t i o n s o f t h e m o n o - a n d o f t h e h i - f u n c t i o n a l m o n o m e r s a n d t h e solvent respectively, so t h a t cb+ct+c~ffil, ftfct/(Cb+Ct), a n d c~ TAe~.S I. VALUES O~ THI C O S V n S I O N OF THE VOUaL~ SONDS AND M D ~ U ~ OF C'YC-TUZA~ON AT THI POINT OF THROUGH"OItOWTH ]FOR VARIOUS VALUJW OF THE PARAMBTER ~0 AND I~TTI~ DIMIBNSION$ ( ~ N ~ Isrr~os)

l

100 8o

70 40 32 100 8o

7o 40 32 100 8o

7o 40 32 leo 8o

70 40 32 leo 8O

7O 40 32 100 8O

7O 40 32

po

0"001 0"001 0"001 0"001 0"001 O.Ol 0.01 0.01 0"01 0-01 0.1 0.1 0.1 0.1 0"1 1 1 1 1 1

10 10 10 10 10 leo leo leo leo leo

x~,% 3"6±0.6 5.3+0.8 5"7±0'5 9"8± 1"0 10.9 ± 1.0 2.2 + 0.5 3.7+0.5 3"9 ± 0-5 8"6+0"9

9"5±0"7 1"69 + 0.22 1.85 ± 0"21 2"38 ± 0"21 4"7 + 0"4 6"0:t:0"4 1"81 ± 0"17 2'24+0.09 2"61 +0"10 4"59 ± 0-17 5"48±0"19 3"5±0"4 3.93±0.18 4.40+0.16 7.09±0"25 8.5±0.3 3"9+0"4 5"6+0"3 5.96 + 0-27 10"8 + 0"4 14"3 ± 0"6

A 0.0014 _+0.0002 0.0019 + 0.0002 0.0022 + 0.0002 0.0047 5:0-0005 0.0062 + 0.0005 0.0046 + 0.0002 0.0048 + 0.0003 0.0049 _+0-0002 0-0076 ± 0.0006 0-0090 ± 0.0006 0.0357 ± 0.0004 0"0352 ± 0.0007 0"0358 + 0.0004 0"0377 ± 0.0007 0"0391 ± 0.O006 0"2311 +0"0008 0"2300 + 0"0005 0"2299 + 0"0O05 0"2267 + 0"0008 O'2260 + O'O009 0.4446 ± 0.0005 0"4440 ± 0.0002 0.4432 ± 0'0002 0.4400 + 0"0003 0"4374 + 0.0004 @4879 + 0"0003 0"4871 ±0"0001 0.4864 + 0"0001 0-4816±0.0002 0.4789 ± 0.0003

N 50 50 100 leo 250 50 50 leo 100 2OO 50 50 150 150 4O0 leo 450 550 6OO 65O 5O 40O 550 65O 6O0 5O 250 450 65O 65O

is t h e c o n c e n t r a t i o n o f t h e initiating radicals. I t h a s b e e n s h o w n [2] t h a t eqn. (1) miiy b e q u a f i t a t i v e l y e x p l a i n e d f r o m t h e p o i n t o f view o f F i e r y a n d S t o c k m a y e r ' s self-consistent field t h e o r y . T h e effect o f diffusion h a s b e e n i n v e s t i g a t e d within t h e f r a m e w o r k o f this class o f mod~l~ [3].

2740

V. A. SAL'~ov and N. M. BOL'm'r

The present paper is devoted to an explanation of the role, during gel-formation, of the relative reactivity of the free double bond, that is, that belonging to the monomer, and of the pendant double bond, that is, that belonging to the polymer, and is a development of Herrman's model. The existing experimental results regarding this question are unclear [4-8] and it is tentatively assumed in the majority of cases that the reaetivities of these and other bonds are the same. The following computational scheme was adopted, which approximates to that used in [9]. The monomers containing two double bonds are placed at the nodes of an I x l x l cubic lattice. The initiation of the polymeric chain was carried out by two methods. In the first version, a randomly selected node is first transformed into a radical and subsequent propagation oc¢urs by the walk of the radical to nearest-neighbour nodes containing at least one reactive bond. Each node may thus be visited twice. AnnihilaTABLe 2. VALUESo F

'me

CONVERSION OF THB DOUBLE. BONDS AT THE ]POINT OF THROUGH-GROWTH

]FOR VARIOUS VALUES OF THE PARAMBTI~

100 70 40 32 lo0 70 40 32 100 70 40 32 100 70 48

40 32

0"001 0"001 0"001 0"O01 0"01 0"01 0"01 0"01 0"1 0"1 0"1 0"1

2-89±0"21 4"4±0"3 7"3±0'5 9"3±0"5 1"97±0.15 3-23±0"22 6"5±0"4 8"2±0"4 1"31±0-10 2"~±0"12 3"74± 0"24

Po

AND LATTICE DIMENSIONS (INTBRNAL INITIA'rlON)

lo0 lo0 IO0 70 150 40 200 32 1O0 lo0 150 70 150 40 200 32 lo0 lo0 200 70 150 40 200 32 4.86 ± 0"~ Walling-in of radicals not permitted 1.45__.0.11 100 lo0 1.99±0.19 1O0 70 -0 48 3-7 + 0.5 50 40 '4.5 ± 0.5 50 32

1 1 1 1 10 10 10 10 lo0 lo0 lo0 !00

1"38±0.18 2.15±0-12 3"59±0"26 4.50±0.23 2.40 ± 0"26 3"27± 0"21 5.2±0-4 6.1±0"4 3.4±0.4 4.67 ± 0"28 ~5±0.5 10.4 ± 0"6

50 200 150 200 50 150 150 200 50 150 150 200

1O0 100 100 lo0 lo0

2"71±0"17 4"19±0"28 7"1±0-7 7"6±0"6 10.3±0"7

150 lo0 50 lo0 150

tion of a radical occurs either by walling-in between completely reacted nodes or the radical may be annihilated upon its exit to a face of the lattice. The next radical can initiate a chain at any node in the lattice having a free double bond, and so on. The calculation is terminated when the process has occurred from a selected face of the cube to the opposite face. We call this version internal initiation. In the second version (external initiation), the radicals are generated exclusively at a lattice face, which is equivalent to entry of the propagating polymeric chain into the specified volume from without. Apart from initiation, the versions are identical. The effect of diffusion on chain propagation has been assessed by modelling two extreme eases. I n o n e , the walling-in

Two types of gels in polymerizationof bifunctional monomers

2741

e r a radical is permitted and in the second, the computation is annulled on each walling-in event. The following characteristics of the process were ascertained. The threshold for through-growth is given by ~ = (Nt +N2)/(2l 3) where Nt + Nz is the total number of polymeric bonds formed at the instant of through-growth; 21s is the maximum possible number of bonds in the lattice; N1 is the number of nodes

Q q

0"8

~ -2

] 2 lo9 Po

-2

Fie. 1

2 lo9 l~o. 2

Fzo. 1. Dependence of the indices ~. (1) and v (2) on the parameter Po. Fie. 2. Depeadence of the coetticient Q o n the parameter Po for external (a) and internal (2) initiation.

that have been visited by radicals and N2 is the number of nodes that have been visited twice. The degree of cyclization at the moment of through-growth is given by t = N2/ /(N1 +N~). We note that fl=0 for a linear polymerand t = 1/2 for a completely crosslinked polymer. The parameter in the model is given by the quantity Po = w2/wt where w2 and wl are the relative probabilities of reaction with a radical of a pendant double bond and of a free double bond respectively. It was assumed that, if there is a single free functionality amongst the five neighbouring nodes adjacent to the radical, then the formation of a bond would occur. When the radical is located on a face of the cube, steps outside the lattice and those within the lattice are assumed to be equally probable. For a lattice of size l, the procedure for the calculation with specified characteristics was repeated many times (50-650 times) and the mass of'data obtained was used for statistical processing. The values of X, for a lattice with I in the range/=32-100 was extrapolated to I--.oo. The basic initial data is shown in Tables 1 and 2. A relationship of the type [10]:

X,=X,+A1-1/~,

X#ffi0,

(2)

was used for the extrapolation, where X¢ is the limiting value of X~ as l-+ oo. Since there is only a single radical at any moment in time with the present model in a lattice of size 1, the extrapolation to l--, oo means, at the same time, extrapolation to zero concentration of radicals. Substituting c~ffi0 into eqn. (1) gives Xc=0. Fixing the value of ~ such that ~ffi I led, in a number of variants, to the meaningless value Xc<0, which is outside the

2742

V.A. SAL'mEOVand N. M. BOL'm'r

80

3 E

I

I

-z

o

I

2 to9

Fxo. 3. Dependence of the radius of correlation on the parameter Po for X.=0.05 (1, 3, 5) and X=0.15 (2, 4, 6); I and 2-external initiation; 3 and 4-internal initiation; 5 and 6-the walling-in of radicals forbidden (curves 4 and 6 coincide). limits of error of the calculation. Regression analysis with non-uniformity in the variances confirmed that the selected approximations were reasonable and made it possible to find the values of ~ and A. A confidence level of 0.95 was adopted to determine the confidence limits. The exponent v of the radius of correlation, L, in an approximation of

/3 0"5

--o--..---..o-3 -

T=ll

=--2

0.03

0.#

0.3

O.Ot I

4

I

I

12 X,%

I

#

12 X,%

1=to. 4. Dependence of the degree of cycUzation on the conwz~ion of double bonds. Valces of Po: a - l ; 2-10; 3-100; 4-0.001; 3-0-01 and 6-0.1. External initiation. the type L , ~ X -v was also calculated. Generally L lX-xc[ -v [11], but in the present case Xc=O. The power law W l ~ l =11v was used, where W~=((XI-(X,>)2). The standard error Sm was selected as an estimate of Wz. The error in the estimate of W~ was calculated by standard methods [12]. The dependence of t h e indices 2 and v on the parameter Po of the model is not monotonic in character (Fig. 1). Since A and v are not statistically different in practice over the entire range of P0 studied, it is permissible to interpret eqn. (2) as an interrelationship hetwcen the radius

Two typos of ~

in polymerizationof bifunctional monomers

2743

of correlation and the conversion of the double bonds .1, by replacing .1~ and l by .1 and L respectively: LfQX- ~

Q=A ~ and, like A, it does not change monotonically with an increase in Po (Pig. 2). By using the values of Q and A obtained, one may calculate the dependence of the radius of correlation, L, on Po for various degrees of conversion X (Pig. 3). The fact that the dependence of L on Po passes through an extremum with a minimum at

<0p/0x>

~_L ±

I I

I I

/

I

/

-0.1 I

-2

o

z

toSp0

1~o. 5. Dependence of the mean value of OPLOXon the patametei Po. External initiation. P o " 1 may be interpreted as a transition from homophase (the descending branch) to heterophase (the ascending branch) gel-formation. In the region Po < 1, the polymerizate is represented by a system of open interpenetrating clusters. A reduction in Po is similar in the way it acts on the structure of the gel to the introduction of an inert solvent or a monofunctional comonomer into the system. With P o > l , compact isolated coils with a high degree of cyclization, p, approaching the limiting value for the given Po are formed (the grain model) [13]. The walling-in of pendant double bonds within the coil thus leads to a decrease'in p with an increase in .t; the situation is reversed for Po < 1 (Fig. 4). The principal difference between the two mechanisms of gel-formation appears in the discontinuity in the mean value of the derivative O~/Ox close to Po = 1 (Fig. 5). But in the range of conversion .1~<0.15, the gel is non-unifoim for any given Po, since the radius of correlation in the neighbourhood of generation of a chain (internal initiation) is less than the analogous quantity at the boundary of a cluster (external initiation) (Fig. 3). Diffusion exerts an effect only for internal initiation and at large values of Po when the walling-in of macroradicals markedly affects the structure of the gel. Since the rate of diffusion depends on temperature, the possibility arises of experimentally determining the region of Po from the temperature dependence of the radius of correlation by use of light-scattering data; for Po < 1, L should hardly depend at all on T but for Po > 1, L decreases with an increase in T. The analysis that has been made gives evidence that the formation from bifunctional monomers of two types of polymeric chains, the nodes of which in one case are formed

2744

v. A. SAL'Nn~ov and N. M. BoL'arr

by tetrafunctional chemical nodes of the network and in the other by compact coils of strongly cyclized macromolecules, may be cansedby a difference in reactivity between free double bonds and those pendant to the gel. It may be stated that the specific features of each o f these processes predetermine the kinetic conditions such that the initial structural difference becomes even more accentuated in view of concentration effects during the process. If these considerations are valid, the autocatalytic character of the mechanism [5] clearly reduces the rigidity of the requirement with respect to the absolute deviation of the value of Po from unity in one or other direction, so that both types of kinetics may be observed with monomers of a single homologous series. Translated by G. F. MODI.~N REFERENCES

1. P. de GENNES, Idei skeilinga v fizlke polimerov (Scaling Concepts in Polymer Physics). p. 368, Moscow, 1982 (Russian translation) 2. R. BANSIL, H. J. HERRMAN and D. STAUFER, J. Polymer Sci. Polymer Syrup., 3, 175, 1985 3. R. BENSIL, H. J. HERRMAN and D. STAUffER, Macromolecules 17: 998, 1984 4. K. ISHIZU, M. NUNOMURA and T. FORUTOMI, J. Polymer Sci. Polymer Letters 24: 607, 1986 5. V. I. IRZHAK, B. A. ROZENBERG abd N. S. YENIKOLOPYAN, Setchatyye polimery (Network Polymers). p. 248, Moscow, 1979 6. R. STADLER, Macromolek. Chem. 187: 723, 1986 7. V. N. IGNATOV, V. A. VASNEV and S. V. VINOGRADOVA, Vysokomol. soyed. A29: 899, 1987 (Translated in Polymel Sci. U.S.S.R, 29: 5, 993, 1987) 8. N. E. PLATE, A. D. LITMANOVICH and O. V. NOA, Makromolekulyarnyyereaktsii (Macromolecular Reactions). p. 348, Moscow, 1977 9. J. G. KLOOSTERBOER, G. M. M. van de HEI and H. H. BOOTS, Polymer Commun. 7,5: 354, 1984 10. B. I. SHKI.~VSKiI and A. L. EFROS, Uspekhi fir.. nauk. 117: 401, 1975 11. B. I. SI-IgLOVSKII and A. L. EFROS, Elektronnyye svoistva legirovannykh poluprovodnikov (Electronic Properties of Atioyed Semiconductors). p. 416, Moscow, 1979 12. A. N. ZAIDEL', Elementarnyye otsenki oshibok izmerenii (Elementary Estimation of Errors in Measurements). p. 96, Leningrad, 1978 13. A.A. BERLIN, G. V. KOROLEV, T. Ya. KEFELI and Yu. I. SIVERGIN, Akrilovyye oligomery i materialy na ikh osnove (Acrylic Oligomers and Materials Based on them), p. 232, Moscow, 1983