Variation in the average degree of polymerization of a polymer during the course of polymerization with chain transfer to impurities

VARIATION IN THE AVERAGE DEGREE OF POLYMERIZATION OF A POLYMER DURING THE COURSE OF POLYMERIZATION WITH CHAIN TRANSFER TO IMPURITIES* A. A. SHAGINYAN a n d N. S. Y E N I K O L O P Y A N Institute of Chemical Physics, U.S.S.R. Academy of Sciences (Received 4 December 1964)

IN THE s t u d y of polymerization processes determination of the number-average and weight-average degrees of polymerization and the ratio of the two is of great importance. Study of the dependence of these characteristics on the yield of polymer is of particular interest. The following special cases have been discussed theoretically in the literature: a) polymerization in the absence of impurities and without chain transfer [1]; b) polymerization with chain transfer to monomer [2]; c) inhibition by impurities [3, 4]. All these problems were solved with the assumption that the initiator is activated instantaneously. The results are approximate 4oecause approximate methods of calculation were used. The aim of the present communication is to determine the nature of the variation in the number-average and weight-average degrees of polymerization during the course of reaction with chain transfer to impurities. The following kinetic scheme was taken for the present problem: ki kp

kt

(l)

rn + M--~rn+ 1, r~-]-X---,Qn +C where C, M and X are respectively the concentrations of initiator, monomer and transfer agent in the system at time t;/ci, kp and kt are the initiation, propagation and transfer rate constants, and r n and Q. are the concentrations of "living" and "dead" polymer molecules containing n monomer units. The system of differential equations corresponding to kinetic scheme (1) takes ~he form: dC --R(v). C0--[J+R(v)] c(v) dv * Vysokomol. soyed. 7: No. 11, 1866-1871, 1965. 2048

The average degree of polymerization of a polymer

dr"=r~ dr

2049

l(T)--[l+R(v)].r,(r)

(2)

~= R (~).r.(~), dM dr

=-Co+(1-~)

c(r)

dX ------R(T). Co+R(T)"C(T), dr where t

= f kpMdt,

ktX

R(~)- kpM "

(3)

0

The initial conditions are:

Ct=o----Co; Xt=o=Xo; Mt=o=Mo; (r,~)t=o=O;

(Qn)t=o=O.

An exact solution of system (2) is possible in special cases. Let us consider the case when J = k J k p = 1. Then from (2) and (3) we obtain

d i (~"n2r"+~" n 2 Q " ) - - - 1 - 2 d~nr,

E nr. Co

(4)

ktXo(M)(k'/k,)-l~nrn Co

Solution of system (4) gives the number-average and weight-average degrees of polymerization

F=Enr.+EnO. Er.+EQ.

~ ~w

M0

Co

q l+(Xo/Co) [1--(1--q) k'/k~] f e(zJco)w (f e -(xo/c°)~

Z n2rn+ ~ n2Qn- - - - 1 + 231o 1 -~, nrn+Z nQa Co

dx)dy

(5)

1

q

where q=(Mo--M)/M ois the degree od conversion, and x and y are the integration variables. When e----kt/kp----1 or < 1 integration in (5) gives (~w)._l:l+2Mofl

-

Xo\

Co l--e; (x°/c°)q) Xo

(6)

2050

A.A.

SHAOI~r~.~ a n d N. S. YENIKOLOPYA~

Co 1-e(Xo/Co)

×

f 1 1--(l--q) *¢xolco)+l t × .l+8(Xo/Co) q +q--2 •

(7)

Equations (5), (6) and (7) are plotted in Figures 1 and 2.

pwXlO-4

pn,lO-~

10 I0-

I 2

8

8

6

6

5

4

4

G

2

7

G

2

I t~2

0"4

~8

08

T

I

I

I

l.Oq

FzG. 1

FIG. 2

FIe. 1. Dependence of number-average degree of polymerization on degree of con, version (according to equation (5)): 1--p.=0, 2--e----10-=, 3--e=10 -z, 4--8----1, 5--8=2,

6--8=5,

7--s=7.

FIe. 2. Dependence of weight-average degree of polymerization degree of conversion (according to equations (6), (7) and (16)): 1--8=0, 2--8=10-=, 3--8=10-t, 4 - 8=1, 5--8=2,

6--8=5.

I n order to obtain the molecular distributions and to find the dependence of Pn and Pw on k I and kt in the most general form it is necessary to assume that t h e ratio

(s) is constant.

This means that the ratio of the probability of an act of transfer in the interval from time t to t+dt, to the probability of an act of propagation, is independent of time. The condition that (8) is constant is satisfied for transfer to monomer, and for polymerization with continuous feeding of monomer and with transfer to impurities. With the condition (8), b y solving the system of differential equations (2) it is possible to obtain the distribution of the ,'living" and "dead" components of the polymeric system.

The average degree of polymerization of a polymer

2051

For "living" polymer: rn__Cos__~.R ( l{ ( 1g~R

_ e_(1+R),)_t_( ~~5 ) n -(e_(~+R) _ e_(l+R)~)

R ~I[v(1-I-R)] i e_(i +.), 6 n~a [v(l__g)]i e_(a+it).} (l~-R)"i~l i! " (1--5)" i~, 7 "7 •

(9)

For "dead" polymer: T

I + R (1 +--R)" o

~ (l--a) -I- ( 1 -~_)1J (1-e-(~+R)')-} a-FR ~ 1 (1--e-(~+R)') ,.

(I+R)"+'L,:, X e-(l¢R)'"~

i!

~

~,u+~j-4 x

[--~ {T(1-~)] ~ (1_~).+, | , ~ i! F

+ ,,-, g .(1-<~)' ~ ~:1

,:,

,'-~k ~lo_<,+,,>, } ~,(1-~)~f

(1o)

"

With instantaneous initiation and the absence of transfer (5-->o% R = 0 ) equation (9) reduces to the expression for a Poisson distribution z--i r~ = Co (n-- 1) !

"

Q.=O From (9) and (I0) we fund the following expression for the weight-average degree of polymerization

ew--

(11)

For the number-average degree of polymerization 5--1

5..pR{R~-4-~-~(1--e-(6+m~)}

(12)

2052

A. A. SHAOr~YA-~and N. S. YE~OLOt'Y~

Let us now examine some more interesting special case arising from equations (11) and (12)

1) J = ki/kp -~ ~-- instantaneous initiation

Pw=I+2

1+(R+1)~ l+Rz

2) J-~ or, R=0--instantaneous initiation and absence of transfer

~w_____l+ (2+z__)) -~n= 1+~ l+z '

This case has been discussed elsewhere in the literature [1].

3) ~=1 (kl=kp) ~(

1 R1 1---_e_.-R~) , ' --P'=

lZ(l+R'

R*+ 1 ~

(1--e-(l+R)*)

4) J = l , R = 0 - -

- -

Pw=l+r'

P"-~ l--e-"

T

The greatest interest is in finding the dependence of Pw and Pn on the yield of polymer

E nz,,+E nQ,~ q= Me From (9) and (10) we obt;ain J--1 Co {Rz+_ 6__J (l_e_(~+m,)~+ Co q=J+R M o Jt~ J M--~oT"

(13)

Equation (13) can be solved with respect to z in the following cases:

1 /Me b) J = l ,

Me T-'~'-~o q

)

(14)

(15)

Putting (14) and (15) into the expressions for Pn(z) and Pw(z) we obtain:

The average degree of polymerization of a polymer

2053

Is) J--->~

l_~-_l~O q_l)2f_2F l~_(a ORLR2[ q--l)--~ _I~CO 1(I'~-eRI(R+I"[(M°/C°'(1q-I']' Pw=

Mo

l+v0q

io q Pn =

R

{M o

2a) J-+c~, R = 0 -

Mo

Pw=l+~oq, 3b) J = l

2

Mo

P,=~oq

_1 Co. 1--e-q Mdco)q]

Me (l+R)q Pn-- Co Mo 1 . _(Mo/Co)(l+R)q ~ o R q + ~ (1--e )

(16)

4b) J----I, R----0 -

M o

Pw----l+~c~q, Pn:

M0

Co

q

l__e-(MjVo)q"

In these expressions R----kt X 0/k~M0. The dependence of the weight-average degree of polymerization on yield of polymer found by means of equation (16) is plotted in Figure 2. It is seen from Figure 2 that it coincides precisely with expressions (6) and (7) for the case when J--ki/kp=l and kt/kp~l. Consequently the dependence of Pwon the yield of polymer is described satisfactorily by equation (16) for kt/k~l with ki/kp--1 in the case of polymerization with transfer to impurities. CONCLUSIONS

(1) The variation in the weight-average and number-average degrees of polymerization during the course of polymerization in which the ratio R:ktX/1% M is constant, and where transfer to impurities occurs, has been calculated. (2) The variation of the number- and weight-average degrees of polymerization with yield of polymer, assuming the above conditions, has been found for the following two cases: 1) instantaneous initiation (ki/kp-->~) and 2) the rate constants of initiation and propagation are equal (kl=kp).

2054

A, A. SKA(~r~YA_~ et al.

(3) The variation in the number- and weight-average degrees of polymerization with yield of polymer has been found for the conditions that the ratio R ktX/k~M is variable and the ratio of the transfer constant to the propagation constant is less than or equal to unity. Translated by E. O. PHILLIPS

REFERENCES 1. 2. 3. 4.

L. GOLD et al., ~. Chem. Phys. 28: 91, 1958 M. LITT and M. SZWARC, J. Polymer Sci. 42: 159, 1960 T. A. OROFINO et al., J. Chem. Phys. 35: 537, 1961 D. BERNARD et al., J. Chem. Phys. 39: 3233, 1963

SOME SPECIFIC FEATURES OF THE KINETICS OF POLYMERIZATION OF FORMALDEHYDE IN THE PRESENCE OF DIETHYLAMINOETHANOL AS CATALYST* Jr A. A. SHAGINYAN, V. A. MININ, N. F. KEDRINA and N. S. Y E N I K O L O P Y A N Institute of Chemical Physics, U.S.S.R. Academy of Sciences (Received 4 December 1964)

A NUMBER of papers has appeared recently dealing with the kinetics of polymerization of formaldehyde [1-6, 8]. However the general kinetic relationships of the polymerization process have not been studied sufficiently. This applies particularly to the kinetics of polymerization of formaldehyde (FA) in the presence of amine catalysts. The process is complicated in this case by participation of impurities, always present in monomerie FA, in the initiation, transfer and termination steps [3, 5]. Data in the literature show that the over-all order of reaction for polymerization of FA in the presence of catalysts such as dibutylamine [3] and triethylamine [4] is greater than two. In catalysis by dibutylamine the order of reaction with respect to catalyst is unity and with respect to monomer it is greater than two. It is evident that it would be difficult to obtain a comprehensive view of the reaction mechanism from these data. The aim of the present work was to study some specific features of the kinetics of polymerization of FA in concentrated solution in toluene in the presence of * Vysokomol. soyed. 7: No. 11, 1872-1876, 1965. t Communication V I in the series "The polymerization of formaldehyde".