A simple method of determining copolymerization reactivity ratios by means of a computer

European Polymer Journal, 1973, Vol. 9, pp. 1007-1012. Pergamon Press. Printed in England.

A SIMPLE M E T H O D OF D E T E R M I N I N G C O P O L Y M E R I Z A T I O N REACTIVITY RATIOS BY MEANS O F A C O M P U T E R D. BRAUN, W. BRENDLEIN and G. MOTT Deutsches Kunststoff-Institut, 61 Darmstadt, Germany (Received 8 February 1973)

Abstract--A Fortran IV program for determining copolymerization reactivity ratios is proposed. The program is based on the curve-fitting method and has the advantage of delivering values free of personal judgement. To check its validity the system benzylacrylate (BeA)/methylmethacrylate (MMA) was investigated. The reactivity ratios obtained from the Fineman-Ross plots (r~ = 0"34 410 per cent and r2 = 1-7 ± 10 per cent) are in good agreement with values obtained by using the proposed method (rx = 0-36 and r2 ~ 1"78). INTRODUCTION THE APPLICATION o f computers in polymer chemistry is particularly successful in copolymerization calculations, where the copolymerization behaviour at high degrees o f conversion can be calculated relatively quickly by an analytical or numerical integration o f the copolymerization equation. (1-~) There have also been some investigations on the determination o f copolymerization reactivity ratios by means o f a computer, in some publications the authors started f r o m the differential, in other cases f r o m the integrated f o r m o f the copolymerization equation.
/

R I~ 0'355

R2-1.782

FIG. 1. ~.v.J. 9/10--c

1007

1008

D. BRAUN, W. BRENDLEIN and G. MOTT

Mortimer tg) enumerated four possibilities for the determination of the reactivity ratios: (1) approximation-, (2) curve-fitting-, (3) intersection (Mayo-Lewis"))- and (4) linearization (Fineman-Ross~l#))-method. The curve-fitting method has been proposed already by Alfrey e t aL t~s) They chose r-values by trial and error and adapted the calculated copolymerization curve to the measured values. Unfortunately, the reactivity ratios obtained by this method involved some personal judgement, so that different operators can produce different results. In this paper a computer program (Fortran IV) is proposed; the best possible adaption of the curve to the measured values by previously chosen criteria is carried out solely by the computer. PROCEDURE* The computer program is based upon the equation Y = (rl +

r2

--

( r l - - 1) x 2 + x 2) x 2 + (2 -- 2r2) x + r2'

(1)

which is obtained from Lewis and Mayo's differential copolymerization equation ml m2

--

~

dM1 . . dM2

.

.

M1 M2

Ml rt + M2 M2 r2 + MI'

and

x -~

(2)

by the transformations mt Y=mt +m2

M1 MI + M 2 "

Noting that the composition of a copolymer is mainly determined by r2 at low concentrations of M1 and by rt at high concentrations of Mr, the copolymerization diagram and accordingly the measured values were divided into two parts A and B with 0 < A < 50 mole 9/0Mland50 ~< B < 100mole ~oM1. The number of measured values below 50 mole ~ M~ equals K A , the number above 50 mole ~o M1 equals K B . The ordinate y and the abscissa x of the copolymerization diagram appear in the program as indicated variables with the dimension of 1000, i.e. the range of the monomer and polymer composition is divided into 1000 segments, or segments of 0"l mole ~o. The sample input is immediately followed by associating the measured values (xm and ym in mole 9/0) with their corresponding segments, e.g. the measured value 14.3 mole ~o M1 in the monomer and 23.5 mole ~o Mt in the polymer is associated with the segments x~(143) and y~(235). In this way the scheme of the copolymerization diagram is in fact present in the computer. The first estimate of the r-values, which Alfrey e t al. did by trial and error, is carried out in this program by the approximation method by calculating the reverse slopes: r2 ~ x ~ / y ~

from the first measured value and

r l ~ 100 - - x m / l O 0 - - y =

from the last measured values.

* Copies of the program are available on request.

Copolymerization Reactivity Ratios by Computer

1009

With this first approximation, according to Eqn. (1), the polymer compositions y(i) are calculated for the corresponding monomer compositions xm(i) and according to KA

A(x)

1 F = --

KA / , [ym(i) --

y(i)]

i=l

and KB

1Z

B(K) = "-K--B

[Ym(i)-- y(i)],

i=l

the average differences between the measured and the calculated values for the two parts of the diagram A and B are determined. K here means the number o f steps o f iteration. According to the sign of A(K) and B(K), r2 and rl are enlarged or reduced by a certain amount a or fl: for A(K) <~ 0

r2(K + 1) = r2(K) + a

for A(K) > 0

r2(K -? 1) = r2(K) -- a

for B(K) <<.0

r~(K ÷ 1) = r~(K) + fi

for B(K) > 0

r~(K ÷ 1) ---- ra (K) -- ft.

The values of a and fl are subjected to the values of the reactivity ratios in the first step and are always reduced if A(K) or B(K) change their sign, i.e. if the calculated copolymerization curve has reached the measured values from one side and has passed over them. The decrease of a and fl is calculated according to

a(K-}- 1) = a ( K ) . x/l____0 and 10

f l ( K + 1) = fl(K). ~/10 10

With the improved values for the reactivity ratios, the program returns to the beginning of the loop, i.e. to the calculation of the copolymerization curve according to Eqn. (I). These iterations are carried out until one of the following termination criteria is fulfilled: (a)

a < 0.0003

(b)

K---- 100

and

K+4

(c) ~--~A(i)----A(i)and ~=K

fi < 0.0003

K+4

~ ~ B(i) ----B(i). I=K

(a) is the main criterion and (b) and (c) are only security criteria. With the finally calculated reactivity ratios, the copolymerization diagram including both the measured values and the calculated curve is drawn by a plotter.

1010

D. B R A U N , W. B R E N D L E I N and G. MOTT TABLE 1. SAMPLEOUTPUT FOR THE SYSTEMB e A / M M A Mean differences Reactivity ratios rt ra 0.434783

1.666667

0.334783

1.766667

0.366405

1-735044

0.356405

1-745044

0'346405

1"755044

0'349568

1"765044

0'352730

1"775044

0.355892

1"785044

0"354892

1.781882

0"353892

1.782882

0"354208

1"782565

0"354525

1"782249

0"354841

1.781933

0"354741

1"782033

B(K)

A(K)

Iterations

--0.028350

--0.010599

1

0.004959

0.000061

2

--0'006059

--0.003244

3

--0"002628

--0"002193

4

0"000852

--0"001147

5

0-000492

--0'000726

6

0"000135

--0"000309

7

--0"000218

0"000105

8

--0.000107

--0.000026

9

0.000239

0"000077

10

0.000130

0.000044

11

0.000020

0.000012

12

--0"000089

--0-000021

13

--0.000054

--0.000010

14

Final reactivity ratios rt = 0"355

r~ = 1"782

Input measured values Monomer Polymer composition composition 10.000 20"000 30"000 40"000 50-000 60-000 70-000 75"000 90"000

6.000 12.000 17.000 26.000 33"000 40-000 52.000 57-000 77.000

Calculated polymer composition 5"750 11"812 18.257 25-186 32-749 41 "184 50"891 56.427 77.776

Calculated copolymerization curve 5. 2"8393 55. 36-8376 10. 5.7497 60. 41.1843 15. 8.7379 65. 45"8441 20. 11"8119 70. 50.8910 25. 14.9812 75. 56.4268 30. 18.2569 80. 62-5949 35. 21.6528 85. 69-6052 40. 25.1855 90. 77.7764 45. 28"8756 95. 87.6182 50. 32"7487

Copolymerization Reactivity Ratios by Computer

101l

DISCUSSION The validity of this program was checked with m a n y copolymerization systems taken from the literature and was confirmed. As an example, the system benzylacrylate(BeA)/methylmethacrylate(MMA) is chosen; according to the classic Fineman-Ross method the values rl = 0.34 ~ 10 per cent

and

r 2 = 1"7 ~: 10 per cent

had been found/16) Table 1 shows the sample output and Fig. 1 the copolymerization diagram drawn by the plotter. The calculated values rl = 0 . 3 6

and

r2 = 1.78

are in good agreement with those taken from the Fineman-Ross plots. Table 1 also shows the number of iterations to be 14. In all systems calculated up to now, this number was between 10 and 30, indicating that the termination of the program is always caused by the criterion (1) a < 0.0003

and

/3 < 0.0003.

For the differences between the measured and calculated values of the copolymerization diagram, this usually implies a mean deviation of minimum 0.0001 to maximum 0.01 mole ~ which is regarded as sufficiently exact. These mean deviations cause a change of the r-values in the magnitude of 10 -4 , too small to influence the actual values. Compared with the usual procedures, this method has the advantage of yielding the copolymerization reactivity ratios more quickly and easily, with less labour and without the influence of the operator. Moreover it requires no linearization and graphic evaluation of the copolymerization equation, the disadvantages of which have been discussed already in detail. (15) Compared with recent publications (8,9,12) the method described here requires no estimate of the r-values in advance, which had been considered necessary to calculate an optimum m o n o m e r composition for precise determination of the reactivity ratios. Certainly the authors achieve some increasing in accuracy by choosing optimum monomer compositions or by applying the integrated form of the copolymerization equation, but it is considered questionable whether and how far the use of these methods is really worthwhile because finally the determination of the copolymerization reactivity ratios is dependent on the precision of the copolymer analyses. Acknowledgements--The authors thank the Arbeitsgemeinschaft Industrieller Forschungsvereinigungen "Otto von Guericke" E. V. for supporting this work.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

REFERENCES F. R. Mayo and F. M. Lewis, J. Am. chem. Soc. 66, 1594 (1944). D. J. Kahn and H. H. Horowitz, J. Polym. ScL 54, 363 (1961). F. Friedman, J. Polym. Sci. B3, 815 (1965). J. A. Seiner, J. Polym. Sci. A3, 2401, 3025 (1965). V. E. Meyer, J. Polym. ScL A3, 2843 (1965). R. L. Kruse, J. Polym. Sci. 135,437 (1967). R. K. Chan and V. E. Meyer, J. Polym. Sci. C25, 11 (1968). D. W. Behnken, J. Polym. Sci. A2, 645 (1964). P. W. Tidwell and G. A. Mortimer, J. Polym. Sci. A3, 369 (1965). H. J. Harwood, N. W. Johnston and H. Piotrowski, J. Polym. Sci. C25, 23 (1968). D. R. Montgomery and C. E. Fry,,/. Polym. ScL C25, 59 (1968).

1012

D. B R A U N , W. B R E N D L E I N and G. MOTI"

(12) L. I. Batik, M. Y. Basner, B. K. Basov, V. L. Tsailinggol'd and V. I. Korobov, Vysokomolek. Soedin. A13, 1133 (1971). (13) R. M. Joshi and S. G. Joshi, J. MacromoL Sci. Chem. A5, 1329 (1971). (14) M. Fineman and S. D. Ross, J. Polym. Sci. 5, 259 (1950). (15) T. Alfrey, J. J. Bohrer and H. Mark, Copolymerization, p. 16. Interscience, New York (1952). (16) D. Braun and G. Mott, Angew. makromolek. Chem. 18, 183 (1971). R6sum6--On propose un programme en Fortran IV pour la d6termination des rapports de r6activit6 de copolym6risation. L'avantage de ce programme, bas6 sur la m6thode d'ajustement de courbe, et de donner des valeurs libres de jugement personnel. Pour tester sa validit6, o n a 6tudi6 le syst6me acrylate de benzyle (Be A)/m6thacrylate de m6thyle (MMA). Les rapports de r6activit6 obtenus ~t partir des diagrammes de Fineman-Ross (rx = 0,34 4- 10 pour cent et r2 = 1,7 4- 10 pour cent) sont en bon accord avec les valeurs obtenues eta utilisant la m6thode propos6e (rx = 0,36 et r2 = 1,78). Sommario---Si propone un programma Fortran IV per determinare i rapporti di reattivitb, di copolimerizzazione. Esso si basa sul metodo di approssimazione per punti e presenta il vantaggio di fornire valori privi di discernimento personale. Per controllare la sua validit~ si 6 investigato il sistema benzilacrilato (BeA)/metilmetacrilato (MMA). I rapporti di reattivit~t ottenuti con i tracciati di Fineman-Ross (r~ = 0,34 4- 10 percentuale e r2 = 1,7 4- 10 percentuale) concordano con i valori ottenuti impiegando il metodo proposto (r~ = 0,36 e rz = 1,78). Zusammenfassung--Ein Fortran IV Rechenprogramm zur einfachen Bestimmung der Copolymerisationsparameter rl und r2 wird vorgeschlagen. Dieses Programm basiert aufder Methode der Kurvenanpassung und hat den Vorteil, Parameterwerte zu liefern, die v~llig frei von pers/Jnlicher Einflussnahme sind. Die Oberpriifung erfolgte am System Benzylacrylat (BeA), Methylmethacrylat (MMA) fiir das nach der Methode von Fineman-Ross die Parameterwerte rl = 0,34 4- 10 Prozent und r2 = 1,7 4- 10 Prozent angegeben wurden. In guter Obereinstimmung damit wurden mit Hilf¢ dieses Programmes die Parameter rl = 0,36 und r2 = 1,78 gefunden.