Distribution functions by size and friction coefficients of randomly branched macromolecules

Polymer Science U,S.S.R. Vol. 24, 1~o. 6, pp. 1289-1299, 1982 Printed in Poland

0032-3950/82]061289-11507.50]0 ©1983 Pergamon Press L t d .

DISTRIBUTION FUNCTIONS BY SIZE AND FRICTION COEFFICIENTS OF RANDOMLY BRANCHED MACROMOLECULES* S. I. KOGA~ and V. P. BUDTOV Okhtinsk Plastopolimer Research Association

(Received 18 November 1980) The work considers the problem of introducing for a real randomly branched chain the distribution function over the configurational {morphological) parameters. The method of the "machine experiment" with a computer is used to model ensembles of randomly branched macromoleculcs. The use of statistical methods showed that the distribution functions by size and the Stokes hydrodynamic radii for the experimental ensembles of the maeromolecules considered agree with the normal distribution. It is shown that the distribution functions for randomly branched chains are determined only by such parameters of the chain as the number of statistical segments of the chain and the number of points of branching. WITH the brisk development of research in the field of physiochemistry of polymers growing interest is being aroused b y the establishment of quantitative correlations between the structure and properties of the macromolecules. Of fundamental interest in this respect are the investigations of branched macromoleeules. As pointed out earlier [1, 2] existing models of the statistically branched chains can serve only as a first approximation to the actually obtained branched macromolecules. In a previous work [2] we proposed a model of the randomly branched macromolecule, derived theoretical relations for calculating the size of such a macromo]ecule and presented the results of computer calculation of the mean square radii of inertia <_~2>p, the Stokes hydrodynam i c radii p and the parameters of long-chain ramification. This work analyses the results of additional calculations and presents t h a t distribution functions b y radii of inertia and the :Stokes h y dr odyna m i c r~dii obtained from statistical t r e a t m e n t of the results of computer modelling of ensembles of randomly branched macromolecules. Distribution function by size (description of the model). One of the characteristie features of polymers is the existence of the distribution function by size for a given molecular mass M. For linear chain molecules for given M the size distribution function W1 is the consequence of the large number of chain conformations when the spatial disposition of the elements changes without change in the stereochemical disposition of the atoms. The value W1 is given by*

W I = A ' R 2 exp { - - ~R2~ * Vysokomol. soyed. A24: No. 6, 1148-1156, 1982. t There are a number of more exact distribution functions for R but this is ~ot of fundamental importance in this work. 1289

1290

S . I . IKOGAN and V. P. BW)TOV

where R is the radius of inertia of the chain; A' and a are numerical coefficients. l~or such a chain the square of the radius of inertia / ~ averaged over all the conformations is equal to R1 = -6-,

(2)

w h e r e n is the number of segments in the chain. The length of the segment was taken as equal to unity. It should be noted that these relations are also valid for a portion of a chain macromolecule. For branched chains the situation becomes more complicated. As well as the set of the conformations for the branched chain macromolecule also fundamental is the set of configurations when change in the spatial arrangement of the atoms is possible for a different stereochcmical disposition of the atoms. As an example one m a y refer to the different points of attachment of the branches, their different lengths, etc. I t is, therefore, necessary to introduce for the actual branched chain a distribution function over the configurational (morphological) parameters. Let us consider an ensemble of randomly branched macromolecules each of which consists of n elements (n>>l) and has m nodes of branching. The distribution function b y radii of inertia for such an ensemble will be

W(R2)= W(n, m, ~, z),

(a)

where y--the conformational parameters; z--the morphological (configurational) parameters. On fixing z we have distribution function Wbr b y the radii of inertia of the branched chain with fixed points of branching and lengths of the branchings W b r = g (R 9") ] z~const ~---W (~, m, ?)

(4)

Similar distribution functions were considered in [3, 4]. I f W(R*) is averaged over all the conformational set we get

W~= = W (n, m, z),

(5)

where here and hereafter the < > brackets means averaging over the conformational set of the chain macromolecule. Such a distribution function m a y be calculated b y a complex of programmes described b y us earlier [2]. In fact, as the distance between any elements ri, 2 of the chain (including between the branching nodes) we used the ratio _ - 1 / 6 l i - k I (cf. equation (2)). We would recall that the model described in [2] for calculating the randomly branched macromolecule reflecting the real process of growth of the macromolecular chain adds up to the following procedure. On the original linear chain, let us call it the zero generation structure at a random site, one branching centre appears i.e. a first generation branched chain forms. Further growth of the macromolecule occurs with the appearance of one branching centre each on the linear chain and on the first generation branch at a random site. Then on the original

Distributior~ functions of rmldomly branched maeromoleeules

1291

branch and the 1,2,3,... (N--l)generation branches at the random sites branching eentres appear and the randomly branched structure of the Nth general.ion forms. The number of branching eentres is m=2iv--1 and the number of "subchains" ~0=2m-kl. The scheme of formation of the third generation m~eromoleeule is presented below.

Here the fundamental difference of our model from the models of bran¢l~ed macromolecules considered in [5-7] must be stressed. We would reca.ll tha.t ~I~ese studies considered models of statistically branched maeromoleeules having on average identical lengths of the subehains. This sets strict limitatio~s o~ rite ratios of the lengths of the branches of the branched macromolecule, l~videatly, even in the maeromoleeule with one branching point there cannot, b~ bral~,,ttes of equal length. The process of growth of the randomly branelled macromole~ule considered for our model in Fig. 1 cannot occur for the models of the statistie~lly branched macromoleculcs [6-7]. In fact, increase in the number of branelfing points in the statistically branched maeromoleeule will inevitably (in view of the requirement for equality of lengths of the subchains) ellange ~lle lengths of the branches, i.e. the law of change in lengths is not constant from generation to generation. Table 1 for comparison presents statistically and randomly branehed macromoleeules with an identical nmnber of statistical segments and branciting points per molecule. Let us consider for example the second generation maeromoleeules presented in Table 1. In line with the proposed model such macromoleeules have three branching points, four branches and seven subchains (segments between the branching points or between the end of the branch and the point of branching). We shall assume that both the statistically and randomly branched maeromoleeules contain 140 statistical segments each. For the statistical model the mean length of the subehains is hence 20 segments and the lengths of the branches 60, 40, 20 and 20 segments. For the random model in view of the absence of the requirement for equality of lengths of the subchains the limitations on the length of the branches (for a set number of segments per macromolecule) are set only by the character of the growth of the macromolecule (declining, rising or not depending on the generation number lengths of the branches). Table 1 presents tlle random macromolecules with equal (not depending on the generation number) lengths of the branches. For the second generation the length of the branch is equal to 35 segments. Thus, it is clear that the difference between the proposed model of the randomly branched macromolecule and the model of the statistical macromolecule is of a fundamental character.

1292

S . I . KOaAN and V. P. BUDTOV

Results of calculation. For the model described a computer was used to make t h e calculations for six ensembles of the macromolecules differing in the lengths of the generations (Table 2). The data in Table 2 are given in the units of K u h n uegments. As Table 2 shows, change in the lengths of the branches from generation to generation lies within the limits of 1 (ensemble I; identical lengths) to 3 (ensemble VI). The published experimental data [8, 9] for real polymers give values of the parameter of change in the length of the branch over the range from 1 to 1.5. From this we shall below discuss in the main the results obtained for the ensembles I to IV. The sets of values (R~}p, (Rc}p calculated for these ensembles, the ramifica-

(R~bv

tion factors a : - -

(R~)l

(R~}v (here (Ro}I

and h : - -

(R~}I and (Rc}I are respectively the

%/N (~

O.Z --

J

Wul ~

FxG. 1. Experimental distribution functions of g-factor (a) and h-factor (b) for 2nd (1), 3rd (2), 4th (3) and 5th (g) generations of ensemble II.

Distribution functions of randomly branched TABLE WITH

1. C O M P A R I S O N AN

IDENTICAL

OF STATISTICA.LLY AND HAPHAZARDLY NU1VIBER

OF

STATISTICAL

SEGMENTS

maeromolecul~s BRANCHED AI~D

1293

~ACR0~IOLECLrLES

BRAI~CHING

POINTS

PER

I~IACROMOLECULE

Length of branches for m o d e l

Number 5 Z ,Z

T y p e o f m a c r o m o l e c u l e for m o d e l

i

statistical

random

statistical

random

:5 3

2

60

20

40. 20

30

7

4

140

20

60.40 20- 20

35

l: T,

15

7 J

8

300

20

30.60 ~-0.20 20. 30 20

I

37"5 I

mean square of the radius of inertia and the effective hydrodynamic radius for the linear macromolecule containing the same number of elements as the branched) were considered as empirical distributions. Using the Kolmogorov-Smirnov criterion [10] we checked the hypothesis on the agreement of the empirical distribution with certain theoretical distributions* taking as the latter the normal, log-normal, Laplace-Charlet and gamma distribution. Table 3 presents the data of computer treatment of the second to fifth generations of ensemble II. (Tlm number of macromolecules in each generation is equal to 100). The values of the mean arithmetical g-factor go, dispersion ag, coefficients of asymmetry A and excess E for the sampling of k-values of the parameter g were calculated from the formulae k

go=----

k

(6)

* W e w o u l d r e c a l l t h a t w i t h use of t h e K o l m o g o r o v - S m i r n o v c r i t e r i o n for t h e s a m p l i u g o f k v a l u e s o f t h e r a n d o m m a g n i t u d e x t h e m a x i m u m v a l u e of t h e e x p r e s s i o n , y.~k°'51Fl(x) --2'~(x)l is e v a l u a t e d w h e r e _F1 a n d _F~ a r e r e s p e c t i v e l y t h e e m p i r i c a l a n d t h e o r e t i c a l d i s t r i b u t i o n f u n c t i o n s o f t h e r a n d o m m a g n i t u d e x.

1294

S.I. Koo~

and V. P. BUDTOV

T A B L E 2. I~ENG~IvJ~ OF BRAI~CHES AS A F U N C T I O N OF ".C.B.~ (]EN~.~ATION N U M B E R FOR ".C.UL~

ENSEMBLESOF T~[..~ MACROMOLECUI,ESCONSIDERED Generation number

Ensemble I II III IV V VI Tt~.BIm 3.

0

1

2

3

4

5

50 40 90 100 80 120

50 27 60 90 40 40

50 18 40 80 20 12

50 12 27 70 10 5

50 9 18 60 5 3

50 5 12 50 3 1

I ~ E S U L T S OF COMPUTER CALCULATION OF T H E STATISTICAL PARAMETERS OF T H E D I S T R I B U T I O N OF g-FACTORS FOR T H E E N S E M B L E I I

Gener- [ ation, m No. ~2 3 4 5

3 7 15 31

go 0.707 0.564 @427 0.341

A

o'g

go--go[a-o P~

E

PG

PT,N'

0.0662 --0.575 0.002 1 0-0723 1 0.0592 --@204 ]--0.199 --0.006 @884 @1683 O. 569 0.0417 --0.265 @0628 -- O.005 1 @ 983 0.945 0.140 --0.003 @952 0.97 0.919 @0338 --0.181

PLc 1 1 1. @882

(7)

ag= /c--I,_~ (g~--go)' k

A-

b-- 2

,-,Zka 8 k

E=

k--I (k--2) ( k - - 3 )

(b+Z)

¢=z

(g,-g# ka 4

(8)

3k+3]

(9)

T h e values o f P ~ , PG, PLN a n d PLC p r e s e n t e d in T a b l e 3 are t h e probabilities o f accepting t h e h y p o t h e s i s t h a t t h e e x p e r i m e n t a l (i.e. o b t a i n e d as a result o f t h e " m a c h i n e e x p e r i m e n t " ) distribution agrees r e s p e c t i v e l y w i t h normal, g a m m a , log-normal a n d L a p l a c e - C h a r i e r theoretical distributions. As T a b l e 3 shows, t h e d i s t r i b u t i o n o f the g-factor for the macromoleeules o f t h e ensemble I I w i t h sufficiently high p r o b a b i l i t y (especially for t h e 4th a n d 5th generations) m a t c h all four theoretical distributions considered. This f a c t is a direct consequence o f t h e narrowness of the empirical distributions obtained. F o r example, for t h e f o u r t h g e n e r a t i o n all t h e values o f the f a c t o r g o f the sampling considered are w i t h i n the range (--2-3 ~g t o + 2 . 3 ~g) while according to t h e t h e o r y for the n o r m a l distribution o n l y 98 per cent of t h e values o f t h e r a n d o m m a g n i t u d e [10] m u s t

Distribution functions of randomly branched macromolecules

1295

b e in the range (--2.3 qg to -}-2.3 qg) [10]. The ratio g w / g ~ l . O 0 2 estimated for this generation (here ga and gw are respectively the mean numerical and mean weight values of the p a r a m e t e r g). A similar p a t t e r n was also observed for the other generations. Thus, it m a y be not ed t h a t in view of the narrowness of the experimental distribution it is impossible to give preference to this or t h a t theoretical distribution for describing the distribution of the factor g. I n such cases the problem of choosing the distribution is solved from the value of the moments of the asymm e t r y and excess. As is known [10] for the normal distribution A----0, E----1. As Table 3 shows, the values of A for all generations differ from zero. Is the a s y m m e t r y of the empirical distribution characterized by such values of the coefficient A significant? I t turns out t h a t it is not. Taking the value A in formula (8) as equal to zero and calculating the value golAffiowe see (Table 3) t h a t the shift in the value go does not exceed 1%. Consequently, one m a y consider the difference in the p a r a m e t e r A from zero insignificant and adopt the hypothesis t h a t the values of the factor g for the macromolecules of the second to fifth generations of the ensemble I I are distributed b y a normal law with the parameters go and ag, i.e.

WM(g~)---~ .____--exp

f

--

,, I

/"2"

(10)

where the subscript 1 denotes the generation number. TABLE 4. VALUES OF THE PARAMETERS OF ~l'Jtl~ I~ORMAL DISTRIBUTIOI~r CALCULATED WITH A COMPUTER FOR ThU~ ENSEMBLES I-IV (g-FACTOR)

Generation, No. 2

Ensemble

go

ag

I II III IV

0"707 0.707 0'698 0'707

0.0723 0-0723 0.0732 0.0746

II III IV

0"564 0"555 0"527

I II III IV I II Ill IV

A

E

~g/go

0"066 0"066 1"089 0"158

--0.575 --0"575 2-203 --0.496

0.1023 0-1023

0"0592 0.0607 0"0577

--0"204 0'650 --0"194

--0.199 1.180 --0.646

0"1049 0"1095 0.1093

0.429 0"427 0.433 0.366

0"0435 0-0417 0.0477 0-0376

--0.203 --0"265 --0.35i --0"064

--0.080 --0.062 1.354 --0.639

0.1016 0'0975 0.1101 0.1026

0-341 0.341 0"335 0"249

0.0336 0.0338 0.0343 0"0236

--0'167 --0"181 0'393 0.074

--0.075 --0.140 1.381 --0-689

0.0987 0-0990 0.1024 0.0947

0.1046 0-1055

1296

S . I . KOGA~ and V. P. BUDTOV

T A B L E 5. V A L U E OF TiLE PARAMETERS OF T H E I~ORMAL D I S T R I B U T I O N CALCULATED W I T H A COMPuTJ~R I~OR ANSEMBLES

Generation No. 2

Ensemb. le No.

I-IV

(~-FACTOR)

ho

ah

A

I II. III IV

0-902 0.902 0.904 0.809

0.0224 0.0224 0.0230 0-0234

--0-0871 --0.120 0.827 --~759

--~505 --0.416 1.762 --0.506

0-0248 0.0249 0.0254 0"0258

II III. IV

0-836 0.873 0.831

0.0239 0.0247 0.0277

--0.400 --0.352 --~429

0.182 1.283 --0.328

0.0285 0.0283 0.0334

I II III IV

~748 0.748 O.765 0.732

0.0224 0.0223 0.0247 0.0249

--0-640

0.0299

--0.054

0.764 0.767 2.424 --0-719

I II

0.686 0.685 0"698 0.610

0.0220 0.0226 0-0232 0.0208

--0.415 --0.677 --0.266 --0.147

0.558 1.126 1.205 --0.839

0.0320 0.0330 0.0332 0.341

ifi IV

--0-641 --0.661

E

¢h/ho

0.0298 0.0323 0.0341

F i g u r e l a presents t h e e x p e r i m e n t a l distribution functions o f t h e values of the f a c t o r g for the second to fifth generations for the macromolecules of the ensemble I I . F r o m the F i g u r e it is clear t h a t the distributions o f t h e f a c t o r g w i t h rise in the g e n e r a t i o n n u m b e r (and hence respectively t h e n u m b e r o f b r a n c h i n g points per macromolecule) become n a r r o w e r and more symmetrical. T h e p r o c e d u r e described for t r e a t i n g the results of t h e calculations for t h e ensemble I I o f the macromolecules was also applied to the ensembles I, I I I a n d IV. Similar results were o b t a i n e d for t h e m namely, the distribution functions over the p a r a m e t e r s g are Gaussian w i t h a n insignificant difference of the coefficient o f a s y m m e t r y f r o m zero. T h e value of the p a r a m e t e r s o f the Gaussian distributions for the second to fifth generations of all ensembles are given in Table 4. T h e calculations for the f a c t o r h were m a d e b y the above described scheme a n d gave in the m a i n similar results. Table 5 gives the values of t h e p a r a m e t e r s o f t h e Gaussian distributions of the f a c t o r h for the second to fifth generations for the ensembles considered. DISCUSSION OF RESULTS

Tables 2-5 give the results of the calculations. I t is interesting to n o t e t h a t w i t h i n the limits o f a 2 % error the ratio (ag/go) is c o n s t a n t for the second to fifth generations of all t h e ensembles c o n s i d e r e d . (ag/go)=0.103,

(11)

Distribution functions of randomly branched macromoleeules

1297

i.e. strict proportionality between go and a is observed. This greatly simplifies the form of the distribution function l ° exp

-

02)

This fact is uncommonly important since it shows that the function WM depends only on go. For the randomly branched chains (ensembles I-VI) we get g o : 1-- 0.I98m°'5~ O.O13m

(13a)

ffo-~1 -- 0.091m-~- 0.00247m 2

(13b)

or

Then WM has the form W•(g):l.243

(go)--~ e x p { - - 6 . 8 5 ( ~ ° - - 1 ) 3}

(12a)

The following should be noted. Analysis of the data obtained in [3, 4] shows t h a t ~¥br (n, m, 7) is practically the function W1 ((R2), 7) where is taken

as equal to p (Fig. 2). Thus, for the complete size distribution function of the branched chain we get (14)

(

Rtg)

(

~o

The ratio (an/ho) does not remain constant with change in the number of branching points per macromolecule but is described by the simple formula

(aa/ho)~O'0177m °'5_0.00223m

(15)

The mean value of the factor h h 0 is also determined only by the number of branching points and may be described as follows: h o : 1-- 0.0296m~- 0.000662m 2

(16)

The distribution function of the factor h is 1

WM(h)-----/-

[

=exp]--

l

1\2-,

\v24/J

(17)

The distribution functions of the values of factor h for the second to fifth generations of ensemble II are shown in Fig. lb. The transition from the distribution by the factors g and h to the distributions respectively for p :gp=h
S. I. KooA~ and V. P. BUDTOV

1298

The distribution functions for (R*>p and will be determined b y the same parameters as the distribution functions of the factors g and h. Comparing formula (4) with the expressions (14) and (17) we see that the set of the parameters in formula (4) is reduced to n and m. Thus, the distribution functions for the randomly branched chains arc determined only b y such parameters of the chain as n and m (Fig. 2). In conclusion, we shall consider the dependence of the g-factor on m for ,different models of the statistically branched chains and the different laws o f decrease in the chains for the randomly branched chains. As noted, the results o f the calculations [5-7] must deviate from the dependence of g on m obtained b y us a result of the fundamentally different morphology of the chains. In fact, in the region 0 < m < 1 0 0 a difference is observed in the values of the g-factors, b u t it lies within the limits of the differences given b y the different models for t h e statistically branched chains i.e. 15-20°/o .

•//•Vmfl

X

I

9'0 1

.0"5 0'6

0 -2

0

2

q

0"2 0

Fro. 2

I

i

i

u,O

I

80

I

f"1

120

p=2m÷l

Fzo. 3

FIe: 2. Configurational distribution functions for linear (1) and branched (2) chains obtained in [4]. Fro. 3. Dependence of go on m: data of [7] (1), [5] (2), [6] (3) and this work (4, 5). Change in lengths of branches ~<1-5 (4) and change in lengths of branches 3 (6). I t is interesting to note the following. I f the real process of polymerization is conducted in conditions such that with time of the polymerization the molecular weight of the newly formed kinetic polymer chain sharply decreases, t h e n characteristic of the branched chain molecule is a decrease in the lengths o f the branches with increase in time. This means that although the number o f branchings with increase in time grow, the total g-factor of such a chain practically remains unchanged (Fig. 3, curve 5). Thus, the scheme evolved for calculating the molecular characteristics of the randomly branched chain yielded the distribution function over the radii of inertia and the coefficients of forward friction of the chains when the molecular

Thermal degradation of copolymers of VC with VA

1299

weight of the branch changes insignificantly with time. We also obtained data showing sharp change in the dependence of go and h0 on m when the lengths of the branches change with time. The complex ofprogrammes evolved makes it possible t,o calculate go and h0 for any real processes of polymerization. REFERENCES 1. S. R. RAFIKOV, V. P. BUDTOV and Yu. V. MONAKOV, Vvodeniye fizikokhimi3~l rastvorov polimerov (Introduction to the Physicochemistry of Polymer Solutions). p. 328, Nauka, Moscow, 1978 2. V. P. BUDTOV, S. L KOGAN and G. A. OTRADINA. Vysokomol. soyod. A22: 360, 1980 (Translated in Polymer Sci. U.S.S.R. A$2: 2, 399, 360) 3. S. K. GUPTA and W. C. FORSMAN, J. Chem. Phys. 55: 2594, 1973; Macromoloeulos 5: 779, 1972 4. W. C. FORSMAN a n d R. E. HUGHES, J. Chem. Phys. 38: 2118, 1963; S. D. HOWELL and B. E. EICHINGER, Macromolecules 11: 276, 1978 5. M. KURATA and M. FUKATSU, J. Chem. Phys. 41: 2934, 1964 6. B. ZIMM and W. STOCHMAYER, J. Chem. Phys. 17: 1301, 1949 7. V. N. TSVETKOV, Dokl. Akad. Nauk SSSR 78: 1123, 1951 8. V. P. BUDTOV and L. I. GODUNOVA. Vysokomol. soyed. B15: 604, 1973 (Not translated in Polymer Sci. U.S.S.R.) 9. V. M. BELYAYEV, V. P. BUDTOV, L V. DANIEL', A. P. KIEV, G. A. OTRADINA, Z. R. USPENSKAYA and S. Ya. FRENKEL', Karbotsepnye soyedineniya (Carbochain Compounds). p. 184, Nauka, Leningrad, 1977 10. D. NIMMELBLAU, Analiz protsessov statistichoskimi metodami (Analysis of Processes by Statistical Methods). p. 812, Mir, Moscow, 1973

Polymer ScienceU.S.S.R. Vol. °4, To. 6, lap. 1299-1306, 1982 Prin~edin Poland

0032-3950/82/061299-08507.50/0 © 1983 PergamonPress Ltd.

EFFECT OF ADJACENT UNITS DURING THERMAL DEGRADATION OF COPOLYMERS OF VINYL CHLORIDE WITH VINYL ACETATE* 1(. S. MINSKER, V. V. LISlTSKII a n d N. V. DAVYD~.I~KO 40th October Bashkir State University

(Received 23 November 1980) I t is shown that the introduction of vinyl acetate units into the PVC chain raises the random elimination of HC1 since the groups--OCOCH3 exert a destabilizing influence on the adjacent vinyl chloride units. At the initial stages of degradation, the rate of formation of HC1 and CH3COOH is a function of three individual rate * Vysokomol. soyed. A24: No. 6, 1157-1162, 1982.