A theoretical study of the effect of spontaneous transfer on the size distribution in living polymers

European Pol?mer Journal, I970, Vol. 6. lap. 1605-1617. Pergamoa Press. Printed in England.

A THEORETICAL STUDY OF THE EFFECT OF SPONTANEOUS TRANSFER ON THE SIZE DISTRIBUTION IN LIVING POLYMERS V. S. NASDA and S. C. JAIN Centre for Advanced Study in Physics, University of Delhi, Delhi-7, India (Received 7 October 1969; revised 11 April 1970)

Abstract--Comprehensive theoretical studies have been made on the effect of initiation and transfer to monomer on the molecular size distribution in living polymers. However, no detailed theoretical investigation involving spontaneous transfer has been reported so far. In the present paper, a study is made of the effect of spontaneous transfer on the molecular size distribution in living polymers for the case when the initiation is infinitely fast. The expressions for the number-average chain length and the weight-average chain length are obtained from the appropriate generating functions which are based upon the differential rate equations governing the polymer synthesis. However, the expression for the size distribution is deduced by following the continuum approach. The stationary state solution of the problem is also investigated. This gives the expression for the size distribution in a simple form which is adapted for numerical computations. Finally, the statistical nature of the polymer in a few typical cases is discussed. 1. I N T R O D U C T I O N IN LIVING polymerization, polymer chains continue their growth until the m o n o m e r is completely consumed. Under ideal conditions when initiation is instantaneous and transfer is absent, the resulting material has a Poisson distribution (1~ of chain lengths. The broadening of the Poisson distribution as a result of transfer to m o n o m e r has been discussed by Kyner et al. C2~ and N a n d a / 3 ) Recent experimental results of Pepper and Reilly ~*) for the polymerization of styrene using perchloric acid as initiator in various solvents indicate the absence of any termination mechanism while the reaction seems to be dominated by spontaneous transfer and transfer to monomer. So far no detailed theoretical investigation involving spontaneous transfer has been reported for living polymers. It seems, therefore, worthwhile to discuss this problem here. The kinetic scheme is similar to that of Pepper and Reilly (~ (see Section 2 where expressions for the number-average chain length and the weight-average chain length are deduced directly from the appropriate generating functions). In Section 3, using the continuum approach, the expression for size distribution is obtained. This, however, involves an integral which can be evaluated only numerically. It is shown in Section 4 that, under conditions when the stationary state approximation can be made, the size distribution expression can be put in a simple form which is adapted for numerical computations. Section 5 considers the statistical nature of the resultant polymer in some typical cases. 2. K I N E T I C S C H E M E A N D T H E A V E R A G E S The kinetic scheme corresponds to time independent rate of transfer, analogous to that proposed by Pepper and Reilly. (*~ It involves the following three stages: Initiation:

I + lVl

ki >

1605

N'I

1606

V.S. NANDA and S. C. JAIN

Propagation:

N ~+ M

Spontaneous transfer:

N'~

k2

k3

~ N ~+

~ Nt + L

Here M, L N*t and N~ denote at time t the concentrations of the m o n o m e r , the initiator, the active i-met and the inactive i-met; kl, k2 and k3 represent the rate constants for initiation, propagation and spontaneous transfer respectively. We shall subsequently assume that k~ --~ oo meaning that the whole of the initiator is converted into active propagating centres as soon as the reaction begins. Further after chain transfer, the infinitely fast initiation step results in practically instantaneous rebirth o f an active chain. Thus the n u m b e r of active chains is constant at the value Io throughout the reaction. The differential rate equations appropriate to the above kinetic scheme are dN*l dt dN*l dt

--

(k2 M + k3) N ' I + k3 ~ N't ~=1

--

(k2 M + k3) N*l + k2 M N ' l - l , i >~ 2

dNi_k3N.l,i dt

(1)

t> 1

The m o n o m e r consumption is given by (2)~" dM - - = -- (k~ M + k3) Io. dt

(2)

On integrating equation (2), the m o n o m e r concentration at time t is Nven by =

where

e-~'~o ' - - k2'

(3)

Io = ~ N*l and M ' o = Mo -- Io.

According to equation (3)

tra = k2 Io In

1 r

(4)

is the time in which the whole of the m o n o m e r is consumed. In order to find the expressions for various averages, one may define the following generating functions

G*= ~ N * ~ y ~ l=l

I"It may be noted that the term k3 Io is included even though spontaneous transfer as such does not involve any monomer consumption. This term, however, has to be taken as the chain transfer is followed by an infinitely fast initiation step in which monomer is consumed.

Effect of Spontaneous Transfer on the Size Distribution in Living Polymers

1607

and

G = ~ N i y ~. i=l.

From the set of differential rate equations (1), we find dG* =-dt

{(1 - - y ) k z M ÷ k a } G q -

kaloy

(5)

and dG = k3 G'. dt

(6)

Applying the operator y O/3y successively to equations (5) and (6) and putting y = 1 in the result, we obtain the following expressions for the number-average and the weight-average chain lengths: i(N, ÷ U',)

(N~ + N ' 0 ' i

and i 2 (Nl -- N'l) i

Z i(N~ ÷ N'~)' l

= 1 + Ifl2y(1 + a ) ( 2 - -

~ Y )--2aYfl2÷2(l-l-fl){1--(1 1-~- a

2/3(1+fl)(1 +a){l_(1 (1 + ~3)

1 ~ a

[(1 + flY') (a/3 -- 1)] -~,

where k3

k2 M'o' 5 = M'offo

and Y = (M'o -- M)/M'o. ~,P.J. 6 / 1 2 - - E

l÷a

(8)

1608

v.S. NANDA and S. C. JAIN 3. M O L E C U L A R SIZE DISTRIBUTION FUNCTION

In principle, the expression for the size distribution could be obtained by solving the appropriate differential rate equations but this is difficult in the present case. Therefore, we employ the continuum (5) approach according to which the active centres which enter the reaction at the same time are visualized as capable of continuous growth at the same rate. As the initiation is assumed to be infinitely fast, all the initiator enters the reaction before propagation effectively begins. This also means that after spontaneous transfer the new centre immediately starts a new chain. Thus in the time interval t to (t + dt), the amount dMp of monomer consumed by propagation will produce an increase in length of gro~4ng chains given by d/=

dMp = k., M dt.

(9)

Io Therefore, at any time t, the maximum possible chain length becomes +1,

l=fi[aln\l._ra/--kc,-b

(I0)

where Y' -----(Y -- c~).The unity on the right side of equation (10) takes account of the monomer used in the initiation process. It is convenient to consider the individual contributions of the chains arising from the direct initiation process (called direct chains) and those arising after spontaneous transfer (called transfer chains). Thus at any stage the total number of active chains N" = N*a -b N*~,

where N'a and N*~ are the respective contributions coming from the direct and transfer chains. In the present problem, at t = 0, N*a ---- Io and N'~ = 0. The rate at which the direct chains become inactive is given by d.N*a dt

=

- - k3 N*d.

Integrating this and using equation (3), we obtain N ' a ---- Io e -k3~

--= Io \ l + ~z ]

•

The number of direct dead chains having len~h between 1 and (l + dO is given by (dt) U~,.dt = k3 U'd ~5 dr. Using equations (3) and (10), one obtains dt dl

1 k2 Mo (1 -- a -- Y')"

(12)

Eff~t of Spontaneous Transfer on the Size Distribution in Living Polymers

1609

Substituting this and the value of N'~ from equation (11) in equation (12), we have finally

Y'I" N a " d l = I° (\ 1l - -+-a /

1 - a y'

1

dl.

(13)

In this equation the appropriate value of Y' corresponding to each l is to be obtained from equation (10). The determination of the size distribution for the transfer chains is more complicated. The set of chains starting due to transfer in the interval dt~ at time tt would acquire a length l to (l ,_a d/) at time t where t~ and t are connected by the relation i

l = } k2 M" dtt.

(14)

tI

The number of transfer chains starting between time tt and (tt ÷ dtt) is Io k3 dtt. Out of this set of chains, a fraction f will survive till time t. It is easily seen that f

~--- e - k 3 ( ~ - t l ) .

Thus the number of active transfer chains with length between l and (l q- dO (called henceforth active l-chains) it is given by N*,, dI = [ok3e-k3(c-q'(dt--~-lt \ d l / dl.

(15)

Changing the variables from t to Y', this equation becomes - - Y' 1] ~

N*,l dl = I o

- - -a"Y ' I

1

dl,

(15a)

where Y' and Y', are connected by the relation (obtained from equation (14)) y,

l=fi

i

(

1

y'j.

1

°

y,

t

dY'

÷

1.

(16)

Next, the contribution arising from the dead transfer chains wiil be considered We can find the number of such /-chains, which started growing at time tl, from equation (15) by multiplying it by k3 dt2. Thus the total number of dead transfer /-chains t

$i=0

161o

v.S. NAN~DA and S. C. JAIN

This, in terms of variable Y', becomes

Nt, d l = [ I - - Y'= L

II ]

-,

Y',!

(i-

Y',) dL

(17)

From equations (12) and (15), it can be verified that the sum of direct and transfer active chains

,f.

N'd-r-

Nttdl=Io.

(18)

Similarly from equations (13) and (17), we can verify the sum of direct and transfer inactive chains

fN~dl+fN,,dl=Iok3t ---- -- Io c~3 In { -}T -' ~1aY'

(19)

4. S T A T I O N A R Y STATE SOLUTION OF THE PROBLEM In the previous section, the size distribution was obtained on the basis of continuum approach which is applicable when the average chain lengh is large. It is possible to derive the size distribution expression directly from the differential rate equations under suitable conditions which we shall now consider. Since a growing chain is terminated by spontaneous transfer, its life time is of the order t* = From equation (3) it can be shown that the concentration of the monomer remains effectively constant for periods of about t* for a3 >> 1 (i.e. k2 Io < k3). Under this assumption, equation (1) #ves

1/k3.

N ' I = -to e -~t +

k3 Io

(1 -- e-"'),

(20)

3' where 7' = (k2 M + k3). Here, because of infinite rate of initiation, the concentration of N ' t is taken as -to at t = 0. For t >> 1/y log equation (20) gives

"//k3,

N ' t --

k3 Io -/

(21)

In this derivation, even though y has been taken constant, it is a slowly varying function of time due to decrease in the monomer concentration. This means that N ' I , having attained the value according to equation (21), increases slowly due to decrease in monomer concentration. Using equation (21) and solving equation (1) for N°_,, we find N'2 =

k2 M Io k3 y

Y

(1 -- e-V~).

Effect of Spontaneous Transfer on the Size Distribution in Living Polymers Since we are considering the solution for

./t

1611

>> 1, it follows that effectively~"

Similarly it follows that in general

(k, M)' :o k3 --_ [o N', =

k,

~

(23)

(t + .

where k 3

ct

tX=k2M

1- Y

F o r iN N 1, this can be written in the form N ' i = Io ~ e - " q

(23a)

It m a y be noted that this expression for N ' l could have been directly obtained from the set of equations (1) by assuming dN'ddt = 0. Thus the condition a~3 > 1, invoked for obtaining equation (23), is precisely the condition according to which the concentration of each of the active molecular species does not vary appreciably with time (strictly over the period of the life time of an active centre). In order to obtain the expression for dead i -mers, we substitute equation (23) in the last of the set of equations (1). This gives

dNl d/.,

=

k3

(24)

k2 (1 + / x ) ~+ t'

Integrating equation (24)

N,=-£~-,_i 1 +a)'

(1 + / z ) '

"

(25)

For in > I, equation (25) can be written k3 {e_~ _ e_,~ } N; = k-~-~i

(25a)

To deduce the expressions for the m o m e n t s and averages, we evaluate~ the summations of the type Z~ ," N "~ and Z~ i" N, for different values of r. Using equations (1), we get the following expressions for such s u m m a t i o n s in the case of active chains:

N'~ = I0

(26)

i

This also justifies the use of equation (21) in solving equation (1) for N'2. For completeness, we give here the full expressions. In the last section, we show that, under the stationary state approximation, contribution of the active chains is negligibly small in comparison with the inactive chains.

1612

V.S. NANDA and S. C. JAIN

i

iN°t:

/o

( ') 1+

(27)

/z,

=Io(1 , 1_o Y) and

~i2N~=Io

1+

1÷ (1 ÷ Y) (2 ÷ 2 Y --

-----Io 1 ÷

a2

3=)l f.

(28)

Similarly, the corresponding expressions for the case of dead chains are

y ,~)=~,

= -- Io In 1 - - 1 +

(29)

= M'o Y

(30)

=..o YE1 +1~2_o+ Y,l.

~31,

and

The expressions for the various averages, following the usual procedure, are obtained as

i (w, + N',) i-~

t (N, + N ' 0 ' i

Ifl Y ÷ 1 -- Y]

1 -- aft In (I

Y1)]

(32)

Effect of Spontaneous Transfer on the Size Distribution in Living Polymers

1613

and i 2 (N~ -- N ' t ) iw =

l

i (N, + N ' , ) '

[ ( 1)( l--

,

1--~. B

(33) 1 -- ~ - - - -

: ' - O.

We shall now determine the condition under which the stationary state expressions can be used for the overall polymer. For long chains, comparing equation (7) with equation (32), we find that for ::$ Y >>, 1, which implies kat >> 1, the two equations give Y i, -~ /

Y)

--aln

1[ -- 1 + a

Similarly from equations (8) and (33) with the same approximation, one gets 1 i,~ _ ~ - ( 2 Ct

Y).

--

To show the validity of this approximation numerically, we give (in Table 1) the values of in, iw and R ( = iw/i,) computed from the set of equations (7) and (32) and the set o f equations (8) and (33) at different conversions. It is observed that, when at3 Y exceeds fifty, the stationary state approximation is very good. TABLE 1. CO,',CPARISONOF

THE VALUES OF in,

i~

AND R FROM THE EXACT R E L A T I O N S ( E )

OF SECTION 2 AND THE STATIONARY STATE ( S S ) RESULTS OF SECTION 4 FOR a =

i. Y

E

i,~ SS

0'1 0"3 0"5 0.7 0"9 0"99

62.186 73"327 67.575 56-562 39"591 24"892

94"167 83"141 71-454 57"954 39.844 24"907

R

E

=

1.74

SS

×

103-587 142-712 136.925 122-352 104.827 96.454

0"01

E

SS

103 187-566 167.408 148.231 129-262 110.372 101.885

1.666 1.946 2"026 2"163 2.648 3-875

1"992 2.013 2.074 2-230 2.770 4-090

1"984 2.007 2"061 2"206 2.725 4"023

1-991 2.009 2-065 2.212 2.738 4-046

= 1.74 x 10" 0"1 0"3 0.5 0.7 0"9 0"99

90"955 83.771 72-573 58-983 40-489 25.207

95'656 84"894 72"990 59"128 40"515 25"208

180.500 168-119 149-551 130-099 110"352 101.417

190.503 170-597 150.709 130-822 110"937 101.998

1614

V.S. NANDA and S. C. JAIN 5. D I S C U S S I O N O F T H E R E S U L T S

In the previous section, we deduced the expressions for the number-average and the weight-average chain lengths. The size distribution function in the general case was obtained for the condition i, >> 1. Unfortunately this result is not convenient for numerical computations. If, however, the stationary state can be assumed for the overall polymers, the convenient equations (23) and (25) can be used for calculating the size distribution. It is interesting to investigate the statistical nature of the resultant polymer in a few typical cases. I

I

I

2

3

I

I

1

I

,

¢

4

5

6

7

s

9

5 t-

o

1

o

1

lo

1 / y _.,.,..

F/G. I. Variation of 1/i, vs. 1/Yfor Io/M'o = fl-t = 7"31 x 10-~. Curve 1 represents the case involving only spontaneous transfer for a = 6"0 x 10 -~. Curve 2 is drawla for the case involving only transfer to monomer for ~ = 6-0 x 10-'t We first consider the number-average chain length. The experimental data generally give number-average chain length as a function of fractional conversion. These results are then plotted in the form o f 1/i, vs. 1/Y curves. In Fig. 1, a typical theoretical plot of 1/i, vs. 1 / Y using equation (7) is shown as curve 1. This shows a trend which is convex upwards. F o r the case when initiation is infinitely fast and transfer to m o n o m e r occurs, the expression corresponding to equation (7) has the f o r m (3~

Y# i. =- 1 -k- ~ (Yfl -- 1)'

(34)

where e = k4/(k2 ÷ k4) and k4 stands for rate constant for the transfer t h r o u g h the m o n o m e r . Curve 2 in Fig. 1 is drawn using equation (34) for e -- 6 × 10 . 4 and/3 -1 = 7"31 × 10 -4. This choice is made so that the rate of chain transfer in the early stages

Effect of Spontaneous Transfer on the Size Distribution in Living Polymers

1615

o f this reaction is the same in the two cases considered. This ~ a p h is a straight line n o t passing t h r o u g h the origin. I f the effect of slow initiation is taken into account, the c u r v a t u r e becomes convex d o w n w a r d s as recently discussed by N a n d a a n d Jain. ~ This shows that the e x p e r i m e n t a l 1/i~ vs. 1 / Y p l o t s can be used to t h r o w some light on the n a t u r e o f the chain transfer. The results of P e p p e r (6~ in such a plot for vinyl mesitylene show a trend which is convex upwards. P e p p e r has suggested the possibility o f transfer to m o n o m e r in his system. The results given are subject to r a t h e r large e x p e r i m e n t a l errors. If, however, the general t r e n d is accepted, one should examine the possibility o f a transfer process with rate not varying with time.~ N e x t we consider the m o l e c u l a r size d i s t r i b u t i o n f r o m our general results o f Section 3. In Fig. 2 the v a r i a t i o n o f l.n, with l/i, is shown, using equations (12), (13), (15) a n d (17) t h r o u g h curve 1 for ~ = 10 -3, ~ = 103 at full conversion, when n~ stands for the n u m b e r fraction o f chains o f length l. A c c o r d i n g to e q u a t i o n (7), this choice c o r r e s p o n d s to is = 126. Curve 2 represents the F l o r y distribution.

0.4

o7I 0.6

0.3

0.4

i

2

°sf

0.2 .

0.1

C

J

I I

0-2

°S

7

t/in

0.1

0.0 0

r

T

r

r

z

1

2

3

4

5

6

~lin FiG. 2. Variation of l.nt with length l/i,, using equations (15) and (17). Curve i is t'or a = 10 -3, /3 = 103 at full conversion; curve 2 represents Flory distribution. "~In this connection it may be noted that the mathematical formulation of this paper is applicable to any living polymerization process in which rate of chain transfer is independent of time. Apart from spontaneous transfer, such a situation effectively exists for example when the chain transfer occurs to solvent.

1616

V.S. NANDA and S. C. JAIN

The appearance of a pair of maxima at the two extremes of curve 1 may be readily explained. It is well known that, for infinitely fast initiation in the absence of termination or chain transfer, the resulting polymer has a Poisson distribution. The effect of chain transfer is to broaden the distribution. For ~ --~ 0, the polymer formed would have an almost Poisson distribution~ and the material involved in very short chains would be practically negligible. With increase in the value of this ratio, the distribution would broaden as progressively more and more chains would undergo chain transfer without attaining the maximum chain len~h~, or a value close to it. The transfer chains would naturally be relatively much shorter. With increasing effect of chain transfer, the material involved in these chains becomes more appreciable and a second peak in the size distribution curve at the shorter chain length side also becomes prominent. With increasing effect of chain transfer, the first peak near the maximum chain l e n ~ h becomes lower while the second peak gradually gets more pronounced. Finally we consider the form of the molecular size distribution when the stationary state approximation is valid. Taking e = 10 -2 and fl = I0 ~, the plots of i'n~ vs. i/iN

I 0"4i

03

" ! 0"2

0"1

0.00

l I

i 2

(i/in)

t 3

4

5

-

FIG. 3. Variation of (i.n,) with (ill,) in the case of stationary state analysis. Curve I represents the Flory distribution. Curves 2 and 3 correspond to Y = 0" 5 and 0" 9 respectively with a = 10-2 and ,/3 = 10-'~. ]"For details, in a parallel situation when chain transfer occurs to monomer, see for example Nanda.(3) :~This is obtained by putting t =tm in equation (i0).

Effect of Spontaneous Transfer on the Size Distribution in Living Polymers

1617

are sho~vn in Fig. 3 for different c o n v e r s i o n s for the overall p o l y m e r . C u r v e 1 represents the F l o r y d i s t r i b u t i o n while c u r v e s 2 a n d 3 are d r a w n using e q u a t i o n s ( 2 3 ) a n d (25), for Y = 0 . 5 and 0- 9 respectively. T h e r e is a g a i n p r e p o n d e r a n c e of tonger c h a i n s as c o m p a r e d to c u r v e I, i m p l y i n g g r e a t e r i n h o m o g e n e i t y t h a n for the F l o r y distrib u t i o n . T h i s is u n d e r s t a n d a b l e , as the d i s t r i b u t i o n o b t a i n e d r e p r e s e n t s the superp o s i t i o n o f F l o r y d i s t r i b u t i o n s for w h i c h the a v e r a g e chain l e n g t h varies slowly w i t h time. REFERENCES (1) (2) (3) (4) (5) (6)

P. J. Flory, d. Am. Chem. Soc. 62, 1561 (1940). W. T. Kyner, J. R. M. Radok and M. Wales, J. chem. Phys. 30, 363 (1959). V. S. Nanda, Trans. Faraday Soe. 60, 949 (1964). D. C. Pepper and P. J. Reilly, Proc. R. So¢. 291-A, 41 (1966). V. S. Nanda and R. K. Jain, J. chem. Phys. 48, 1858 (1968). D. C. Pepper, Europ. Polym. J. 1, 41 (1965).

R6sum6---De nombreuses 6tudes th6oriques ont 6t6 faites dans le but de comprendre les effets de l'amor~age et du transfert au monom~re sur la distriution des masses mol6cttlaires des polym~res vivants. Jusqu"A pr&ent, cependant, aucune investigation th~orique tenant compte d'un transfert spontan~ n'a &6 indiqu6e. Dans Ie present travail, on a &udi6 l'influence du transfert spontan~ sur la distribution des masses mol&ulaires des polym~res vivants dans le cas ou l'amor~age est infi.niment rapide. Les expressions des moyennes en hombre et en poids des Iongueurs de ehalne sont obtenues A partir des fonctions de distribution ad~quates bas&s sur les ~quations diff&entielles des vitesses gouvernant la synth~se du poIyrn~re. De route fa~on, l'expression de la distribution est deduite en poursuivant une approche continue. La solution du probt6me dans le cas d'un &at statio~,maire est 6galement envisag6e. Cela donne une expression de la distribution sous une forme simple bien adapt& aux caiculs num&iques. Finalement, la nature statistique du polym&e est discu~Ae dans quelques cas typiques. Sommario--Si sono eseguiti esaurien:i studi teorici circa l'effetto di iniziazione e trasfe.~maento a monomero sulla distribuzione dimensionale della molecola di polimeri in corso di polimerizzazione. Tuttavia, finora non 6 stato presentato alcun studio circa particolareggiate indagini teoriche sul transferimento spontaneo. Nella presente relazione si conduce uno studio sull'effetto di transferimento spontaneo sulla distribuzione dimensionale della molecola di polimeri in corso di polimerizzazione, nel caso in cui l'iniziazione sin infinitamente veloce. Le espressioni per il numero--tunghezza media di catena e per peso--lunghezza media di catena sono ricavate dalle relative funzioni generatrici che si basano sulle equazioni differenziali governanti la sintesi di polimeri. L'espressione per la distribuzione dimensionale viene per6 dedotta per approssimazione continua. Si indag-a pure ta soluzione del problema in condizioni di staticit/L Ci6 cb.e fornisce l'espressione per la dist,.'-ibuzione dimensionale in forma semplice, adatta per il calcolo numerico. Si discute infine della natura statistica del polimero in alcuni casi tipici. Zusammenfassung--Eingehende Untersuchungen wurden durchgef/~hrt tiber den EinfluB der Initiierung und der Monomerfibertragung aug die Molekulargewichtsverteilung von lebenden Polymeren. Es liegen aber bisher noch keine genauen theoretischen Untersuchungen/~ber die spontane ~bertragung vor. In der vorliegenden Arbeit wird der EinfluB der spontanen ~bertragung auf die Molekulargewichtsverteilung von lebenden PolyTneren untersucht for den Fall, dab die Initiierung unendlieh rasch verl~iuft. Die Beziehungen for den Zahlenmittelwert und den Gewichtsmittelwert tier Ketten1/inge werden erhalten aus den entsprechenden erzeugenden Funktionen, die aug den differentiellen Geschwindigkeitsgleichungen basieren, die ftir die Polymersynthese g~ltig sind. Die Beziehung for die Molekulargewichtsverteilung wird dutch das Kontinuum Verfahren abgeleitet. Die L6sung des Problems far den station~.ren Zustand wird auch untersucht. Diese liefert die Beziehung ft~r die Gr613enverteilung in einer einfachen Form, die for zahlenm~Bige Berechnungen verwendbar ist. SchlieBlich wird an wenigen t vpischen Beispielen die statistische Natur des Polymeren diskutiert.