JOURNAL OF POLYMER SCIENCE VOL. XLIV, PAGES 107-116 (1960)

Molecular Weight Distributions of Vinyl Polymers Grafted to a Solid Polymeric Substrate by

Irradiation (Theoretical)

JOSEPH ZIMMERMAN, Carothers Research Laboratory, Textile Fibers Department, E. I . d u Pont dc Nemours and Company, Inc., Wilmington 98,

Delaware

INTRODUCTION

The grafting of vinyl monomers to a solid polymeric substrate involves vinyl polymerization in a viscous medium where the rate of termination is markedly reduced. There has been a considerable amount of study of radi- ation induced graft polymerization with some recent comprehensive reports by Henglein et al.,l Chapiro,2 Hoffman et aL13 Ballantine et al.,4 Cooper et al.,5 and Worrall and Pinner.6 Various grafting methods are described in which the sequence of irradiation and exposure of the substrate to a vinyl monomer are varied. In this paper we will be concerned with only one particular sequence of events. It will be assumed that the polymeric substrate in any given physical form is soaked in a solution of a vinyl mono- mer until equilibrium solubility is obtained. All excess monomer is then removed from the surface of the polymeric object and the system is frozen to immobilize the monomer. The polymer is then subjected to a given irradiation dose in essentially zero time. The system is then brought to some given higher temperature (e.g., 25OC.) in essentially zero time, and the polymerization is then allowed to proceed without further irradiation. Oxygen is excluded from the environment to avoid termination from ex- traneous sources.

KINETICS OF POLYMERIZATION

The conventional assumptions for vinyl polymerization are made in describing the variation in free radical concentration [R] and in monomer concentration [MI :

- d [ R ] / d t = 2kt[RI2 (1)

[Rol ''I 1 + 2kt[Ro1t

108 J. ZIMMERMAN

where k t and k , are the termination and propagation rate constants, re- spectively, and r = k,/2kt.

With a relatively rigid semicrystalline polymer such as nylon 66 and a dose of 1 Mrad which gives a radical concentration of ca. 7 X 1 0 - 6 moles/ C C . , ~ the conversion of a number of vinyl monomers to polymer is 90% complete in less than 5 minutes of post-irradiation time. If we assume (reasonably) that the propagation rate constant in the viscous matrix is not greater than the normal value in conventional vinyl polymerization, the value of k , at 25OC. cannot be appreciably greater than lo6 cc./mole- see. Thus, from eq. (4) we can deduce that k,/2kt must be greater than 0.25.

In a semicrystalline polymer, it is possible for the activation energy of diffusion to exceed the activation energy of the propagation reaction so that both k , and k, can be diffusion-controlled. In such cases the k,/2kt ratio can be of the order of unity, with k, less than its normal value. It should be pointed out here that in our discussion we are assuming that the vinyl monomers are in the amorphous regions of the semicrystalline polymers and do not have access to the free radicals in the crystallites except a t very long times, when these may L1diffuse7) by hydrogen abstraction into the amorphous regions. We are assuming further in our treatment that the radicals produced in the amorphous regions of the substrate diffuse rapidly throughout the amorphous regions as a result of the propagation reaction. Thus, they tend to become homogeneously distributed even though they are initially produced in “spurs.” We have not attempted to introduce the complications of nonhomogeneous radical distribut,ion a t this time.

It is interesting to consider the effect of the value of r on the conversion of monomer to polymer in some particular case (e.g., 2kt = 5 X lo4 cc./ mole-see. and [Ro] = 7 X 10+ moles/cc.). These calculations are sum- marized in Table I.

TABLE I Time Required to Attain a Given Conversion for Various Valucs of k,/2k,

(2k , = 5 x lo4 cc./mole-see.; [no] = 7 X 1 0 - 6 moles/cr.)

Time to attain conversion of r

(= L, /Zk , ) 50% 90% 95% 99 yo 0.1 49 min. 900 yr. 1 .0 X 1 0 6 yr. 9 X 10l2 yr. 0 . 5 8 . 6 see. 4 . 7 min. 19 min. 7 . !I hr. 1 .o 2.9 see. 26 sec. 54 see. 4 . 7 min. 2 .0 1 . 2 see. 6 .2 see. 9 . 9 see. 26 see.

The extreme sensitivity of time required to attain a given conversion on T , particularly for higher conversions, is noteworthy and will be referred to again later.

MOLECULAR WEIGHT DISTRIBUTION 109

MOLECULAR WEIGHT DISTRIBUTION

Radicals Terminating by Disproportion ; No Significant Chain Transfer ; Complete Conversion

Let us assume that all free radicals which are still alive have an equal

( 5 )

opportunity to react with monomer. The rate of chain growth is then

d x / d t = (l/[R])(d[M]/dt) = k,[Mo](l + 2kt[Ro]t)-'

where 2 is the number of monomer units per chain. For r # I :

x = r[Mo]{(l + 2kt[Ro]t)'-' - 1 )/[R01(1 - r ) (6)

(Gal

while, for r = 1 :

2 = ([MoI/[RoI) 111 (1 + 2kt[Rolt)

The mole fraction of chains which terminate in the time interval t to (t + dt) is

dnt = -d[R]/[Ro] = 2kt[Ro]dt/(l + 2kt[R0]1)~ (7) From eqs. (5 ) and (6) we can solve for dt and t as a function of dx and x;

inserting these into eq. (7) gives us the mole fraction distribution. F o r r # 1:

and for T = 1:

n, = ([RoI/[MoI exp { -x[RoI/[MoI] (84

Equation (8a) is identical to the normal Flory distribution. The weight fraction distribution is

wZ = xnz [Ro I/ [Mol (9)

In normalizing eqs. (8) and @a), we see that if T < 1, 2 for the longest However, if chains may become infinitely large a t complete conversion.

r > 1, the longest chains have a finite length, i.e.,

x m a x = r[MoI/[Ro](r - 1) (10)

Thus, for r = 2, zmaX = 2[MO]/[Ro]

by integrating the product mz from zero to xmax. result (which is rather obvious)

The number-average molecular weight can be obtained in the usual way This gives us the general

P , = [MoI/[Rol (11)

The weight-average molecular weight is given by integration of the product xw, from zero to zma,. This gives us, for r > 0.5

pw 1 ([MoI/[Ro1)(2r/2r - 1) (12)

110 J. ZIMMERMAN

and P , = a for r 5 0.5. The ratio of the two averages is (when r 2. 0.5)

These results apply only to complete conversion.

Pm/Pn = ( 2 r / 2 ~ - I) 0 3 )

Thus, we see that the distribution becomes broader as r (i.e., kp /2k t ) For very high values of r , we get a very narrow (Poisson) becomes smaller.

molecular weight distribution.

DEGREE OF POLYMERIZATION ( x )

Fig. 1. Weight fraction distribution for grafted vinyl pdymers (no chain transfer, 100% conversion); [Ro] = 7 X 10-6 moles/cc., [MLI] = 10-3 moles/cc.: (1) k,/2kt = 0.5; (2) k,/2kt = 1.0; (3) k,/2k, = 2.0.

In Figure 1 we have shown the weight fraction distribution for r = 0.5, It is interesting that when r = 2, the mole fraction is the same 1, and 2.

for all 17: while the weight fraction increases linearly with 5 (up to z,,,,).

No Chain Transfer ; Incomplete Conversion

Obviously, in any real case, complete conversion will never be realized. As a result, the surviving chains will actually never grow infinitely long, and this will have an important narrowing effect on the distribution for low values of kp/2kt.

In Table I1 we have listed values for xmax [Ro]/ [Mo] a t several conversions of monomer to polymer for various values of r calculated from eqs. (4) and (6). The times to achieve these conversions have been previously sum- marized in Table I, and the same values of Ic, and [I<,] are assumed.

MOLECULAR WEIGHT DISTRIBUTION 111

In the following discussion, it is assumed that a t some given time, the reaction is stopped by washing out the unreacted monomer in a time which is relatively very short as compared with the reaction time.

TABLE I1 Length of Longest Chains as a Function of Conversion and k,/2k,

[Mo] for conversions of r ( = kJ2kO 50% goyJ 95 yo 99 y*

0.1 57 1.1 x 108 5 . 7 x 10'C 1.1 x 10" 0 .5 1 9 19 99 1 .0 0.69 2 . 3 3 . 0 4 . 6 2 .0 0.50 1 . X 1.55 1.80

For cases of incomplete conversion, we may consider the vinyl polymer to consist of two parts. One (Polymer 1) has a distribution as described by eq. (S), except that the distribution stops a t a maximum value of z as indicated by Table I and eq. (6). The other (Polymer 2 ) consists of chains all of equal size (smax).

The mole fraction of 'chains which are not terminated a t time t is equal to [R]/ [Ro] = ([MI/ [Mo]) These chains all have a length zmax and a tot.al weight [Rlz,,, = [RO]zmax( [M]/[Mo])l/r. The total weight of vinyl poly- mer is then [Mo] - [MI (in terms of the weight of the repeating unit). Therefore, the weight fraction of chains which are not terminated when the experiment is stopped and the residual monomer is washed out is

The mole fraction distribution for the normally terminated chains (Polymer I ) under conditions of incomplete conversion is

The total number of moles of Polymer 1 is

NI = [Rol - [Rl = [Ro](l - ( [M]/ [Mo]) ' /~] (16)

The number of moles of z-mer is N z , 1 = nz, 1 N1; the weight of z-mer is W z , 1 = N z , s; the total weight of Polymer 1 is W1 = [Mo](l - fz) (1 - [MI/ [Mo]) ; the weight fraction distribution is:

w,, 1 = { [R0]~z/r[Mo]~(1 - fz)

The number-average molecular weight of Polymer 1 is

112 J. ZIMMERMAN

while the number average molecular weight of Polymer 2 is, of course, xmax. P , for the entire polymer is obtained in the usual way:

l / P n = [(I - j*)/Pn, 11 + (fZ/xmax)

P n = [M01(1 - [MI/WOI)/[ROI

(19)

This gives the obvious result

(20)

The m-eight-average molecular weight of Polymer 1 is

([Mo]/[Ro]) ( 2 ~ / 2 ~ - 1) { 1 - ([MI/ [ M O ] ) ( ~ ~ - ' ) ' ~ ~

(1 - fJ(1 - [MI/[MoI) - zmax((2[MI/[MoI) + f Z ( 1 - [MI/[MoI)l _-__ (21)

P W , l =

for r # zmax.

The weight-average molecular weight of Polymer 2 is again Pw for the entire polymer is obtained as usual by the relation

P , = (1 - . f 2 ) P w , 1 + fz xmax (22)

or

- P, =

([Mo]/[Ro])(2~/2~ - 1)( 1 - ( [MI/[MO])(*~-')") - (2[M]zmax/[Mo]) 1 - [MI/[Mol

(224 If r = 1/2, we have

Pw.1 = -

2([MoI/[RoI) In ([MoI/[MI) - zmax{(2[MI/[MoI) + f 2 ( 1 - [MI/[MoI)] (1 - f2) (1 - [MI/[MoI)

(23)

and P , can again be found with the aid of eq. (28):

For the case of T = 1, zmax is deduced from eq. (6a) :

zmax = [Mo]/[RoI In (1 + 2kt[R0lt) = [MoI/[RoI In ([MoI/[MI) (25)

The weight fraction of chains which are not terminated a t time t is de- rived in a manner analogous to that for eq. (14)

s z = In ([Mol/[MI)/([MoI/[Ml~ - 1 (26)

The mole fraction distribution for normally terminated chains (Polymer 1) is

n,. I = [RoI exp { - ~ R O I / [ M O I ) / [ M O I ( ~ - [MI/[Mol) (27)

MOLECULAR WEIGHT DISTRIBUTION 113

while the weight fraction distribution is

w, = [ROI’Z exp { - ~ [ R O I / [ M O I ) / [ M O I ~ ( ~ - fd(1 - [MI/[MoI) (28)

The number-average molecular weight of Polymer 1 is

p z , = ([MoI/[Rol)(~ - f2) (29)

when P , for the entire polymer is again given by eq. (20). The weight-average molecular weights of Polymer 1 and the entire poly-

mer are given by eqs. (21) and (22) with r = 1. Let us now consider a grafting process for a number of monomers in

which we keep [Ro] constant a t 7 x 10-6 moles/cc., 21it constant a t 5 x lo4 cc./mole-sec. and polymerization time constant a t 1000 sec. In Table I11 we have listed the per cent conversion, P , , Pm, and their ratio for several values of r (i.e., k,/2kt).

TABLE I11 Effect of k,/2k, Ratio on Width of Molecular Weight Distribution of Grafted Polymers

(Constant Dose, Time, and k,)

0 .1 44.3 21.6 0.44 6 . 6 15.0 0 .3 82.8 24.9 0.83 6.97 8 .4 0.5 94.7 17.7 0.95 4.0 4.22 0.9 99.53 7.17 0.995 2.18 2.19 1 .o 99.72 5.86 0.997 1.97 1.97 2.0 99.999 1.994 1 .oo 1.33 1.33

From Table I11 we can see that even a t constant polymerization times there is a broadening of the molecular weight distribution as r gets smaller, in spite of the fact that the conversion also gets smaller. In these calcula- tions we have included the small amount of ungrafted homopolymer and not differentiated i t from grafted polymer.

EFFECT OF CHAIN TRANSFER TO THE SUBSTRATE

If chain transfer to the substrate occurs, it may have a marked effect on the molecular weight distribution since i t results in the termination of chains which might otherwise become very long.

Let us consider the grafted polymer as consisting of two parts: Polymer 1 is grafted on original radicals R, produced by irradiation, while Polymer 2 is grafted to new radicals Rz produced by chain transfer to the sub- strate. The rate of disappearance of original radicals is:

-d[Ri]/dt = 2ki[Ril[K.] + ktr[Ri]S (30)

where the total radical concentration [R] is [Ro]/(1 + 2kl[Ro]t) and S is the concentration of C-H (or other) bonds susceptible to chain transfer.

114 J. ZIMMERMAN

S is presumed to be present in excess and thus is a constant. of eq. (36) is:

The solution

= [RoI exp { -kt,St}/(l + 2h[R0lt) (31)

The concentration of new radicals [Rz] is, of course, equal to [R j - [R1] or

[Rz] = [Rol(l - exp { -kt,St])/(l + 2kt[Ro]t) (32)

The fraction of the total radical concentration which is stJill on original chains is equal to exp { -kt ,St]. Thus, if k t , = 4 cc./mole-sec. (for ex- ample) and S = lo-* moles/cc., only 5y0 of the total radical concentration will be on original chains after 75 seconds.

In a manner analogous to that used in deriving the molecular weight distribution in the absence of chain transfer, we can obtain the distribution for “original” polymer (Polymer 1) by setting dnt = d[Rl]/ [Roldt. This gives for r # 1:

n,, = { ([RO]B(T-z)/(l--7) / r [Moll + (CSB-’/ [Moll ] exp [ - k t r ~ ( ~ l / l - r - 1)/2kt[Ito]] (33)

where r = kp/2kt, C, = kt,/k,, and

B = 1 + (1 - r ) [Ro j~ /~ [Mo] (33a)

For T = 1, we get

n,. 1 = 1 [Rol/[M01 exp { - [R~lz/[Mol} + (C,S/[Mol))

exp -kt,S exp { [RoIz/[MoI - 1]/2kt[R01] (34)

The weight, fraction distribution is

ws. I = n,, ~ [ R o I ~ / [ p i j (35)

where [PI] is the concentration of polymer formed on chains initiated from original radicals.

The rate of formation of [PI] is

d [PI Ildt = k p [M 1 [Ri I (36)

(364 = ~, [MoI[RoI exp { -kt,St}/(l + 2kt[R01t)’+~

and the value of [PI] for eq. (35) can be obtained by numerical integration of eq. (36a).

In many cases [PI] will reach a maximum value in experimentally acces- sible times, so that eqs. (34) and (35) should be valid even for incomplete conversion. The rate of formation of “original” polymer [PI] will become less than 1% of the rate of total monomer consumption when [Rlj/[R] becomes less than 0.01 or when kt,St > 4.6. If k t , = 4 cc./mole-see. (for example) and S = lop2 moles/cc., this will occur after 115 sec. If we further assume that 2kt = 5 X lo4 cc./mole-sec. and [no] = 7 X moles/cc., then we will reach this limiting value of [PI] a t 97.5y0 conver-

MOLECULAR WEIGHT DISTRIBUTION 115

sion when r (i.e., k,/2kt) = 1 and a t 84% conversion when r = 0.5. For lower radiation doses (and [Ro]), the limiting value of [PI] will be reached at lower conversions.

For the values of kt,, S , k t , and [Ro] given above, if r = 0.9, a maximum of 83y0 of the monomer will be on “original” chains. If [Ro] is reduced by a factor of 10 to 7 X lop7 moles/cc., a maximum of only 35% of the monomer will be on “original” chains.

The number-average degree of polymerization (P , ) for “original” poly- mer is equal to [PI]/[R,]. The weight-average can be obtained graphi- cally by integrating xwz, 1 from 0 to zmnx. For the constants given just above ([Ro] = 7 x lop6 moles/cc.) we get a value for pm, l / p n , 1 1.55. This shows the extent of narrowing of the distribution of “original” polymer resulting from termination by chain transfer. However, we have up to 17% of the polymer grafted to ‘hew” radicals. The rate of production of these radicals is

d [ R z ] / d t = ktr[R]S (37)

[Rzl = (kt,X/21ct) In (1 + 2kt[Ro1t) = C,S In ([MoI/[MI) (38)

The number-average degree of polymerization for ‘hew” polymer (Polymer 2) is [P&‘[R,,,] where [Pz], the concentration of polymer on “new” radicals, is [Mo] - [MI - [PI]. For the constants we have been using, after 115 sec. we would have 96.5% conversion with 83% of the monomer on “orig- inal” radicals and 13.5y0 on “new” radicals. moles/cc. (as compared with 7 x lop6 for [no]), and p,, 2 would be pn, 1/2.64. Although we have not made an exact calculation for P,, 2, i t would almost certainly be somewhat greater than 2P,, 2, in view of the continually changing monomer concentration. Averaging over the whole grafted polymer for this set of assumed constants, we get for the number- average degree of polymerization :

Therefore, insertion of eqs. ( 2 ) and (4) gives

[R2] would be 3.0 X

P n = [PI] + [P~I/[ROI + [Rnewl

= 0.965[Mo]/l.O X lo+ (39)

and, roughly, for weight-average degree of polymerization, P , N 1.8 P,. In general, one can say that chain transfer will tend to broaden the dis-

tribution when k,/2kt is large and make it narrower when k,/21it is small.

References 1. Henglein, A,, W. Schnabel, and K. Heine, Angew. Chenz., 70, 461 (19%). 2. Chapiro, A,, J . Polymer Sci., 34, 439, 481 (1959). 3. Hoffman, A. S., E. R. Gilliland, E. W. Merrill, and W. H. Stockmayer, J . Polymer

4. Ballantine, D., A. Glines, G. Adler, and D. J. Metz, J . Polymer Sci., 34,419 (1959). 5. Cooper, W., G. Vaughan, S. Miller, and M. Fielden, J . Polymer Sci., 34, 651 (1959). 6. Worrall, R., and S. H. Pinner, J . Polymer Sei., 34, 229 (1959). 7. J. Zimmerman, J . A p p l . Polynzer Sci., 2, 181 (195‘3).

SCi., 34,461 (1959).

116 J. ZIMMERMAN

Synopsis The molecular weight distributions of vinyl polymers grafted by irradiation have been

deduced theoretically. The derivations are rigorous for cases where chain transfer is not important or in all cases where the radiation dose is high (>1 Mrad). In the absence of chain transfer, the width of the distribution depends mainly on conversion of monomer to grafted polymer and on the ratio of the rate constant for propagation to that for ter- mination (kp /2k , ) . As this ratio gets smaller, the molecular weight distribution becomes broader. For relatively small values of k p / 2 k , (e.g., 0.5 or less), the width of the dis- tribution increases markedly with increasing conversion. In such cases, the occurrence of chain transfer tends to make the distribution narrower. For a monomer grafted to a substrate where kp/2k1 = 0.9 (for example), the ratio of weight to number average molecular weight is ca. 2. In general, monomers which have lower rates of propagation should form grafted polymers with broader molecular weight distributions.

Rbum6 La distribution du P.M. de polymkres vinyliques greffbs par irradiation a Btb d6duite

thboriquement. Des dbrivations sont rigoureuses dans les cas ou le transfert de chaine n’est pas important, ou bien dans tous les cas oh la dose de radiation est blevbe (> 1 Mrad). En l’absence de transfert de chaine, l’btendue de la distribution depend surtout de la conversion du monomkre en polymkre greffB et du rapport de la constante de vitesse de propagation B celle de terminaison (kp/2k,) . Si ce rapport devient petit, la distribu- tion du poids molbculaire devient plus large. Pour des valeurs relativement petites de kp/2k, (par exemple 0,5 au moins) l’btendue de la distribution augmente nettement avec un degrb de conversion croissant. Dans de tel cas, l’occurence de transfert de chaine tend B rendre la distribution plus btroite. Pour un monomkre greffB B un substrat oh kp/2k, = 0,9 (par ex) le rapport du poids mol6culaire moyen en poids au poids molbcu- laire moyen du nombre est environ 2. En gbnbral, les monomeres qui ont des vitesses de propagation faibles formeraient des polymhres greff6s avcc une distribution large du P. M.

Zusammenfassung Die Molekulargewichtsverteilung von Strahlungs-gepfropften Vinylpolymeren wurde

theoretisch hergeleitet. Die Ableitungen gelten streng fur alle Falle, bei denen Ketten- ubertragung unbedeutend oder die Strahlungsdosis hoch (> 1 Mrad) iet. Bei Abwesen- heit von Kettenubertragung, hangt die Breite der Verteilung hauptsachlich von der Um- wandlung von Monomerem zu aufgepfropftem Polymerem und vom Verhaltnis der Geschwindigkeitskonstanten des Wachstums zu der des Abbruchs (kp /2k , ) ab. In dem Masse, wie dieses Verhaltnis kleiner wird, wird die Molekulargewichtsverteilung breiter. Bei verhiiltnismassig kleinen Werten von kp/2k1 (2.B. 0,5 oder weniger) nimmt die Verteilungsbreite merklich mit steigendem Umsatz zii. In solchen Fiillen bewirkt das Auftreten einer Ketten-ubertragung eine Verengerung der Verteilung. Bei cler Aufpfrop- fung eines Monomeren auf ein Substrat, wo lcp/2k, = 0,9 (als Beispiel), betragt das Ver- haltnis von Gewichts- zu Zahlenmittel des Molekulargewichts etwa 2. Im allgemeiuen werden Monomere mit niedrigerer Wachstumsgcschwindigkeit Pfropfpolymere mit breiterer Molekulargewichtsverteilung hilden.

Received August 19, 1959