Intramolecular crosslinking of macromolecules

2088

V. P. BUDTOV and S. I. KOGAN

10. T. E. LIPATOVA, Kataliticheskaya polimerizatsiya oligomerov i deformirovaniye polimernykh setok (Catalytic Polymerization of Oligomers and Deformation of Polymer Networks). p. 206, N auk. dumka, Kiev, 1974 11. Yn. A. LIPATOV, Yu. Yn. KERCHA and L. M. SERGEYEVA, Struktura i svoistva poliuretanov (Structure and Properties of Polyurethanes). p. 215, Nauk. dumka, Kiev, 1974

0032-3950/85 $10,00+.00 © PergamonJournals Ltd.

Polymer ScienceU.S.S.R.Vol. 27, No. 9, pp. 2088-2096, 1985 :Printed in Poland

INTRAMOLECULAR

CROSSLINKING

OF

MACROMOLECULES* V. P. BUDTOV and S. I. KOGAN Okhtinsk "Plastpolimer" Sciences Production Association

(Received26 January 1984) Analytical relations have been derived for the fraction of k times crosslinked macromolecules and its changes with the time of the process. The influence of the size distribution of the coil and the "pulsations" of the dimensions of the chain is taken into account both with use of the results of computer modelling and the model of "normal" coordinates.

THE processes of crosslinking of polymer materials have been studied long and closely [1, 2]. The processes of intlamolecular crosslinking of macromolecules have been fat less studied although the structure of closslinked macromolecules largely determines both the kinetics of the general plocess of crosslinking of a polymer material and its physicochemical and physicomechanJcal properties. There ale also some instances when intramolecular crosslinking is important in itself. Thus, for example, in study of the intramolecular crosslinking of flexible chain polymers [3] an attempt is made to solve the problem of the folm of the macromolecules in solution: does it represent a Ga,ussian chain or an ordered structure? Some aspects of intramolecular crosslinking were studied in [4, 5]. The main results were obtained with use of a computer (Monte Carlo method). It was found that the size of the macromolecules decreases with rise in the number of intramolecular cross-links and it was shown that the kinetics of the process of intramolecular crosslinking I o r a small number of crosslinks weakly depends oa the character of the distribution of the reactive groups and is determined only by the length of the chain and the degree of ,filling b~ active groups. * Vysokomol. goyed. A27: No. 9, 1858-1864, 1985.

Intramolecular crosslinkingof macromolecules

2089

However although the use of a computer in many .cases helps to understand the characteristic features of the behaviour and structure of the macromolecules the results given by the analytical relations have far greater generality. Therefore, we attempted to obtain some analytical results for the change in time in the number of macromolecules crosslinked to different degrees and also to allow for the fact that the real polymer chain pulsates (periodically changes its dimensions) which was not taken into account in any previous study. Concentration of macromolecules with a different number of cross-links. Let us denote by ~k the fraction of chains in ~oluUon with k cross-links. Let us also introduce the probability ~k that a chain with k cross-links receives an additional cross-link i.e. passes into the category 9k+~" Then fall in the fraction of non-crosslinked chains with increase in time is desclibed* thus dgo dt

Cto ~ o

The natural solution of this equation has the form tpo(t) = exp ( - ¢to t) The rate of change in the fraction of macromolecules with one crosslink is determined, on the one hand, by how many macromolecules by the moment of time t have one crosslink ato~o and on the other by how many macromolecules with one cross-link pass into the category of macromolecules with two cross-links-0q~0t dfPx -=c% 9o--Ctl (a1 dt Likewise for other maclomolecules with k cross-links d~k dt The solution of such differential equations is given by the expressions ~°1=c%[exp(-cqL Cto-~l t)~ e xlp~( l-_~aott o) F

exp(--ct2 t)

~'~=~'°~'L(-~.~-~,,)(~=-~,o)

t_ctexp(--~lt'

( ~- o)( ,-

+

exp(--ct2 t)

]

~) (~,o--S~,)(~,~--~,~)J

and for the fraction of macromolecules with k cross-links one may obtain

~ exp ( - ~t t) i=0

(1)

H (O~i__O~j) j=O

Here H' means that in the product i # j . * A task of such a type was solvedfor the process of comminutionof particles on cavitation [6].

2090

V. P. BUDTOV and S. I. KOGAN

Thus, knowing ~+ it is also possible to calculate the dependence of ~k on time. It is clear that at first the fraction of ~k (k>0) rises, leaches a maximum and then falls. In the case of identical gk the solution of the equations presented gives the known result [ 1 ] - a Poisson distribution

(o:t)k ~ k = - ~ .I exp (-0~t)

(2)

The maximum of ¢0k is observed at (~t)m,x = k. The mean number of cross-links Z for such a system Z=~t, i.e. the number of cross-links, grows linearly with time. In the case ~k >~k-1 Z rises in time with acceleration (autocatalytically) and the law of rise is set by the relation between ~k and ~k-1. Calculation of ~o~for a concrete model of the structure of the chain. We shall consider that ~ is inversely proportional to (/~2)a/2 [7] and crosslinked macromolecules may be regarded as a branched chain with functionality of the branching node equal to 4. Then using the relation for/72 of such a macromolecule [7] we get: (/~2)+= -~=~k 1"4k / ( / ~ ) , gk ~o=~-~+~/1 +k) +

(3)

It is interesting to note that change in/~k2 given by the formula (3), in fact, agrees (Fig. 1) with change in/~k2 calculated by the Monte Carlo method [4]. The results of the computer calculations of ~k (X=eot) from the formulae (1)-(3) for k = 0 - 4 are given in Fig. 2. It will be seen that ~0k depends on X=eot extremally. The maximum of ~0k for the ~ase of unequal ei is observed at Xmaxk (k>0)

Xmaxk~k0"67,

(4)

while for equal 0~X~ax~ k.

1

j

5

k

FIo. 1. Relative mean dimensions of chain as a ftmetion of the number of cross-links from the results of [4] (1) and formulae (3) (2). From analysis of Fig. 2 and relation (4) it will be seen that allowance for the real structure (and hence the more precise value of Ctk) of the macromolecules leads to a sharper dependence of ~ on x and to quickening of the whole process of crosslinking.

Intramolecular crosslinking of macromolecules

2091

Using the results obtained it is possible to calculate the character istic viscosity of strongly diluted polymer solutions. In fact let [q]b and [q]bk be respectively the intrinsic viscosities of the solution as a whole and the k times crosslinked macromolecule present in this solution. Then k

In the case of branched macromolecules the ratio G of the viscosity of a branched and linear [q]~ polymer of the same MM is usually considered. Bearing in mind that

I.o~

cc

0,SF#

Z

6 x

0,3 ¸

\ x/~"

0.2 I

/

~./z

\ Z \

\x\

I

\ "~,.~

\

\

\\

0,I

\

';

;/

~-x

r"

f £

5

"--o.,

IO

o_

a:

FzG. 2. Fraction ~ of chains with k cross-links as a function of time (x = ~to (t)). Continuous lines, various = ~ values; broken, cq = ~ , a, k-= 0; b, k = 1 (1), 2 (2) and 4 (3).

2092

V.P.

BtroTov and S. I. KOOAN

this paper considers the case of a monodisperse polymer and in the general case [8, p. 281] Gk=g[ 1 (81=0"5, 1 and 1"5 for different models) we have

(5) k

The calculations from folmula (5) for el = 1.5 are given* in Fig. 3 which presents the values o f the mean number of crosslinks per macromolecule k = ~ k ~ o k. It will be seen that at k ~ 2 the specific characteristic viscosity falls ,,~ 1.5 times which qualitatively well agrees with the experimental data of [3].

l'O

0.65

1,85

I

f

1

2

3.05K I

0"5 3x

Fio. 3. Specific intrinsic viscosity as a function of the specific time. Upper part of graph, mean number of cross-links. Thus the correspondence of the relations obtained for tpk and ~k to the data obtained by the direct modelling of the process with a computer and the experimental data points to the soundness of the assumptions advanced. The ma#nitude ~o. Let us consider in more detail the parameters influencing the magnitude Cto. The magnitude ~o characterizes the probability ot appearance of one cross-link in the maciomolecule and, in fact, fixes the whole series of values of ~k" In the simplest case we shall disregard pulsation of the dimensions of the macromolecule in time. Since the probability of a cross-link ot the macromolecule with the size R is proportional to the mean density of segments in the coil [9, 10] we have N O~o= a ~ 3 ,

where the magnitude a depends on the intensity of the external agent and the nature of the monomer unit; N is the number of monomer units (here and hereafter only monodisperse macromolecules are considered). Usually for R 2 the equilibrium value ( R 2) i s used, then we get * The values [r/]b/[t/]=for x=3 respectively for e~=0.5 and 1 are equal to 0.637 and 0.507.

Intramolecular crosslinking of macromolecules

2093

N
\z<,K->I ~,0 = f 7t~o dR =
1"97
i.e. the real value of ~,o is greater than ~oN2 times because of allowance for the size distribution of the macromolecule. It is interesting to note that the mean dimensions of the macrochain with one cross-link are

dR =_4_ ~o j

7

'

i.e. the square of the dimensions of the macrochain with one cross-link is ~ 2 times less than that of the non-crosslinked chain. Thus chains in line with the size distribution with smaller dimensions cross link more rapidly [12] and the mean dimensions of the crosslinked chain are smaller. Now let us take into account the fact that the dimensions of the macromolecule pulsate. In explicit form the pulsations of the dimensions of the macrochain were observed on computer modelling [12]. Then the mean value of the magnitude ~0 in the period of pulsation T is given by

'f

T

czr= ~--

o~(t)dt

(6)

0

The pulsation of the magnitude R 2 may be considered starting from two models. Considering the pulsations of a freely articulated chain similar to the vibration of a spring with a fixed mass we get

6/19"6 where t is time; 1 is the size of the segment; m is the mass of the chain A = 1 -- ~/- N " We t o o k into account the fact that the minimum size of the chain is determined by the fact that all the units are located in a densely packed medium. For this model the maximum size of the chain

R~ax=212N-~-------12and

the minimum size R~i.=2"29 7 7 N -113 i.e. R~i,=(0.6--0"2 ) for the usual values of N (N=100-1000).

2094

V.P. BUDxov and S. I. KOGAN

Substituting equation (7) in equation (6), we get 2+A 2 ~r = (~}2 [1--A212'5

(8)

For large N the relation (8) has the folm 5

~r ~0.3Ni ~

(8a)

and for N= 100-1000 ST=(2--5"2) (~). For dimensions of the chain crosslinked in such conditions we have 2(1-A2) 2 ( R 2 ) = ( R 2) ~ ~- ~2.7N -÷

(9)

It is interesting to compare the value of (R 2) (flora (9)) with direct computer calculation [4]. Taking N ~ 100 we get (R2)/(R 2)..,0.58, which is close to the value obtained in [4]. Thus the relations (7)-(9) obtained by us and the proposed model are clos~ to the model of [4]. Change in the dimensions of the chain with time may be calculated by also using the Kargin-Slonimskii-Rause model i.e. ragarding the movement of the chain as the result of nolmal viblations. For the Gaussian chain we have "/R2X"

1

~-~ hE

IV i , j = l

l n .~ f

~ 2.~ ~i--

1¥

h~'~

(10)

i=lk=l

where h 2 is the square of the distance between the i and i-k elements of the chain. Since we are interested in the dependence of R 2 on t, then in general form h~=kl2fk (t), fk (t) is a function of time and is the result of summation of the normal coordinates. Transforming relation (10) and making elementary calculations we obtain R2=

N,2 12 ~ : . , ~t -- ~k~ltn--tc) kfk

(11)

Let us evaluate the dependence of the effect discussed. Since the process of crosslinking is slow* an essential contribution to the effect discussed is made by the low-frequency harmonics i.e. the vibrations of the large subchains ot the macromolecule. To evaluate the function R e (t) one may further simplify relation (11). In fact the maximum effect in the function R 2 (t) is given by the item with the value k=n]2 (the maximum of the value k ( n - k ) . The slowest vibration for k = n/2 will have the frequency w2. All the other vibrations of the subchains forf~ may be disregarded and already the steady values may be considered. * The time in which new cross-links appear in the chain is longer than the relaxation time of

~the chain.

Intramolecular crosslinking of macromolecules

2095

R2,,,7~N2412(1-6sin2co2t)

(12)

Then we get

The mean value sin 2 aJ2t=½ and then

(R2)=NI/6, i.e.

maximum size of the chain Rmax 2 274 N I 2 = ~7- ( R 2) =--

we get the known result. The and the minimal

NI2

2 Rmi"- 24

= 4- (R2)" These results insignificantly differ from those given by relation (7). Using expressions (6) and (12) we get for such a model ~ . = (c~) 1.74

and for the dimensions of the chain R 2 = ( R 2 ) 0.533 The results obtained are close to those given by the model of a chain with free vibrations (formulae (8a) and (10)). Thus, allowance fol the size dist~bution and pulsation of macromolecules give a 0 = ( 2 - 5 . 2 ) ( ~ ) and the size of the chain ( R 2 ) = ( 0 . 8 - 0 . 5 )

(g~).

Of~k,

The calculations made (calculation the mutual link of ~k and ak÷ 1, and allowance for the effects of size pulsations and distribution) give a closed system of equations and relations allowing one to calculate change in the properties of the solutions of the chains as a result of intramolecular crosslinking. In particular, it has been shown that the viscosity of the solutions decreases with the course of the process. This helps to explain the experimental data without invoking the concept of the domain structure of the solution [3]. A. CROZY

Translatedby

REFERENCES

1. A. CLARLESBY, Yadernye izlucheniya i polimery (Nuclear Radiations and Polymers). p. 513, Inozt. Lit., Moscow, 1962 2. V. I. IRZHAK, B. A. ROZENBERG and N. S. YENIKOLOPYAN, Setchatye polimery (Network Polymers). p. 243, Nauka, Moscow, 1979 3. S. M. AHARONI, Angew. Macromolec. Chem. 63: 115, 1177 4. I. I. ROMANTSOVA, O. V. NOA, Yu. A. TARAN and N. A. PLATE, Vysokomol. soyed. A21: 1176, 1979 (Translated in Polymer Sci. U.S.S.R. 21: 5, 1293, 1979) 5. I. I. ROMANTSOVA, O. V. NOA, Yu. A. TARAN, A. M. YEL'YASHEVICH, Yu. Ye. GOTLIB and N. A. PLATE, A19: 2800, 1977 (Translated in Polymer Sci. U.S.S.R. 19: 12, 3232, 1977) 6. G. A. SEDUKHA, F.K. BUDAGOV and V.A. BUDTOV, Issledovaniye rabochikh protsessov stroitel'nykh mashin (Study of the Working Processes of Construction Machines) No. 65, p. 137, Inzhenerno stroitel'nyi inst., 1971 7. B. ZIMM and W. STOCKMAER, J. Chem. Phys. 17: 1301, 1949 8. S. R. RAFIKOV, V. P. BUDTOV and Yu. B. MONAKOV, Vvedeniye v fizikokhimiyu rastvorov polimerov (Introduction to the Physicochemistry of Polymer Solutions). Nauka, Moscow, 1978

Ibid.

2096

A. D. LITMANOVICH and V. O. CnERKEZYAN

9. S. EDWARDS, J. Phys. C. 3: 739, 1970 10. Idem., Proc. Soc. A. 334: 447, 1973 11. P. FLORY, Statisticheskaya mekhanika tsepnykh molekul (Statistical Mechanics of Chain Molecules). p. 440, Mis, Moscow, 1971 12. N. K. BALABAYEV, Dissert. Cand. Phys. Math. Sci., IVS Akad. Nauk SSSR, Leningrad, 1982

Polymer ScienceU.S.S.R. Vol. 27, No. 9, pp. 2096-2103, 1985 Printed in Poland

0032-3950/85 $10.00+ .00 © Pergamon Journa Is Ltd.

INTERMOLECULAR EFFECTS IN THE THERMAL DEGRADATION OF POLY-tert-BUTYLACRYLATE* A . D . LITMANOVICH a n d V. O. CHERKEZYAN Topchiyev Institute of Petrochemical Synthesis, U.S.S.R. Academy of Sciences

(Received26 January 1984) The degradation of poly-tert-butylacrylate at 165-200°C occurs with sharply marked acceleration. Comparison of the initial rates of degradation of poly-tert-butylacrylate in the pure state and in composites with polyacrylic acid revealed the considerable contribution of the intermolecular interactions to acceleration. The proposed mechanism of acceleration includes the interaction of the tert-butylacrylate group with two carboxyl groups simultaneously with at least one of these C O O H groups not being the closest neighbour of the ester group along the macromolecular chain. The previously proposed mathematical model of the macromolecular reaction in the polymer melt is developed allowing one to describe quantitatively the kinetics of degradation of poly-tert-butylacrylate. The rate constant s and activation energies of the thermal and autocatalytic stages of the reaction are determined. The results point to the decisive contribution of the intermolecular interactions to the degradation of poly-tertbutylacrylate.

RECENTLY [1] the piinciple was formulated that the reactivity of a polymer changes in the course of the macromolecular reaction as a result of change in the microsurroundings of the reacting gloups of the polymer. In accold with this principle the kinetics of the macromolecular reaction is influenced both by the intta and intermolecular interactions. The corresponding kinetic equations were derived in [1]. It must be expected that the intermolecular effects will be substantial in the reaction in concentrated solutions and especially in the polymer melt. The last case represents a relatively simple model for experimental investigation if the reaction so proceeds that a fragment of the initial side group splits off and the low molecular weight product formed leaves the melt and the reaction medium consists exclusively of maciomolecules. In this case the microsurrounding of any reacting groups are made up of the non-reacting and reacting poly* Vysokomol. soyed. A27: No. 9, 1865-1870, 1985.