Branched polymers. II—Dimensions in non-interacting media

Branched Polymers. H Dimensions & Non-&teracting Media T. A. OROFINO Theoretical expressions /or the mean square radff of some random flight 'comb" type branched molecules are derived. The structures considered consist o / l i n e a r backbone chains to which linear side chains are attached by one end at various positions along the backbone. Expressions for the (light scattering) z-average radii o/ some mixtures of star and comb molecules likely to be obtained in the syntheses of these materials are also presented.

IN THE preceding paper, expressions for the average molecular weights and molecular weight distributions of a particular class of branched polymers, termed 'comb molecules', were presented. It is the purpose of the present communication to derive expressions for the mean square radii of comb molecules obeying random flight statistics. The particular structures treated correspond, in part, to those examined in Part I. The present selection of specific branched systems encompassed in the general category of 'comb molecules', as in the previous instance, has been made on the basis of the practicality of synthesis of such materials, their importance in systematic studies of branching and the relative ease with which they may be treated thoeretically. The expressions derived here, applicable to various distinct branched structures and to mixtures of branched molecules, should find use in studies of the O-point dimensions of such materials, as obtained for example from intrinsic viscosity or light scattering investigations. RADII

OF

GYRATION MOLECULE

OF

SPECIFIC

BRANCHED

STRUCTURES

The mean square radius ~ of any polymer molecule may be defined

s~=<(1/n) i z~ s~> =1

(1)

where si is the vector distance from the centre of mass to the ith segment of the chain in one particular conformation of the molecule and n is the total number of segments. The angular brackets denote a linear average over all possible conformations of the molecular chain. The computation of the mean square radius of any polymeric structure whose constituent elements obey random flight statistics (which we denote by a subscript zero on pertinent quantities) may often be simplified through application of the Kramers relationship 1

~=(b/n)2Xj(n-j); 305

n>~l

(2)

T. A. OROFINO where b is the length of one freely jointed chain element. The summation is extended over all possible divisions of the molecule into two parts, one containing j and the other n - j segments. Either of the equations (I) or (2) yields for long chain linear polymers the well-known random flight result*

s o-~-nb2/6;

n>~l

(3)

The particularly convenient equation (2) will be employed as an operational definition of the mean square radius in subsequent calculations for various branched structures. (1) Star molecules Expressions for the random flight dimensions of star molecules (and other branched polymer structures) were presented more than ten years ago in a pioneer paper by Z i m m and Stockmayer 3, ~. For the sake of continuity, however, we shall derive equivalent relationships for this type of branched structure which, in our arbitrary system of classification, m a y be considered as a special case of the more general comb molecules. Application of equation (2) to a star molecule consisting of p linear chains, each containing y units, yields s 2o_-_ [(3p - 2) / 6p] b~y = [(3p - 2) / p] s 2O,y

(4)

y>>l where s~, ~ =yb~/6 is the mean square radius of one isolated linear chain of y units. For a star molecule consisting of p branches of lengths Yl, Y2. . . . yp units, application of equation (2) yields .= b 2=

(5) y~>~ 1

where P

y=(1/p) i~=l

.

"=

f=1

and p

p

= ( 3,Y])[( ~ Y ~ ) are the number weight and z-average lengths, respectively, of the collection of p linear chains constituting the branched molecule. *The form of the relationship (3) is applicable to real polymer chains in media where the net segment-solvent interaction is nil (O-solvent), even though the chemical repeating units are not freely jointed. It is merely necessary to redefine n and b in terms of the number of Kuhn statistical elements ~. For convenience, we shall assume throughout that linear components of the branched structures treated are chemically identical and that 'segments' referred to are always freely jointed. 306

BRANCHED

POLYMERS---II

(2) Regular comb molecules The relationship in equation (2) may be applied to regular comb molecules, each consisting of a linear backbone chain of x units to which p linear chains of uniform length y units are attached at regular intervals to form the repeating unit m

LI-

x/p

L1

Figure

1

-- p

The mean square radius so formulated is __

p-1

s,~=(b/Z)2{ ~

i=o

~/p-1

y

~ [i(x/p+y)+]][Z-i(x/p+y)-]]+p

j=0

~ ](Z-j)}

i=o

(8)

where Z = x + py is the total number of segments in the molecule. For x and y large, equation (8) may be simplified to 2__ 2 2 so-so, ~ (1 + N J p 2 )+so, ~ (3 - 2 / p ) N , x,y>~l; p~l

(8a)

where the subscripts x and y on the symbol for mean square radius denote the values of this quantity for the denuded linear backbone and one detached side chain, respectively, and N~ = p y / Z is the number fraction of branch units in the comb molecule. We note that in the special cases y = 0 and y >~ x, equation (8a) can be reduced to the expression for the linear backbone chain and to equation (4) for a p-functional star molecule, respectively. Theoretical expressions for the mean square radii of similar regular comb molecules have been given by Wales, Marshall and Weissberg ~, by BerryL and, for small values of p, by Zimm and Stockmayer~. In principle, these structures can be obtained from the complete coupling reaction of side chains with one species of repeating unit in a regularly alternating copolymer of uniform chain length. Z-AVERAGE

RADII

OF

POLYMER

GYRATION

OF

BRANCHED

MIXTURES

The z-average mean square radius, as obtained experimentally from light scattering measurements, is defined for any collection of polymer molecules by the relation

<~>z= ( ~ n Z ~ ) / ( Z n Z~) 307

(9)

T. A. O R O F I N O

where n~ is the number of molecules containing Z~ segments and possessing 2 mean square radius ~s~.Expressions for ( S 0)~ pertaining to some branched polymer mixtures of interest are presented below. (1) Random star mixtures We consider a collection of random star molecules (see Part I) for which the branch length y is fixed. The mean square radius of a member of the mixture containing p branch chains (and hence, degree of polymerization py) is given by equation (4). If the number of branches per molecule is randomly distributed, with a mean value ~,, application of equation (9) yields <~o>~= (Y b2 / 6) [3 - 2 / (1 + ~,)1

(10)

y>>l (2) Regular comb mixtures A collection of regular comb molecules for which the branch length y is fixed and for which the molecular weight distribution of the constituent backbone molecules is characterized by the three molecular weight averages z x,,, x w and ~=(2~ n ,xi)/(Nn~x~), where n~ is the number of backbone chains containing x~ units, yields according to equations (9) and (Sa) (~>, = (b ~/ 6) [~ + 3 y N + (N u-x,/-p~w) ~,/-fi,, - 2y)] x, y >~ l ;

(11)

"fi, si= O

where N~ =~,y/(-2,~ + ~ y ) is the number fraction of segments belonging to branch chains in any given molecule. We note that equation (11) can be reduced to equation (8a) when the backbone molecular weight averages are equal. (3) Random comb mixtures We consider here a distribution of backbone chains to which branches of fixed length y units are attached at random. Individual members of the final mixture will thus vary in backbone length, in the number of branches attached, and, for specified values of these two variables, in the particular arrangement of the branches along the backbone. In a mixture of these molecules, the distribution of the number of branches attached to a backbone of specified length may be represented by the Bernoulli function used previously in the treatment of the molecular weight distribution for the same system (see Part I). The effect of variations in the positions of attached chains on the z-average mean square radius of the mixture may be taken into account by suitable averaging of the Kramers expression [equation (2)] over all species. The details of this calculation are given in Appendix A. The final result is (s~)=(b~/6ZZ~c',,){Z]~w-~.+-~,y 2 [32jg, + y ( ~ 308

-- 3 P n ) ] }

(12)

B R A N C H E D POLYMERS--1I

where Z . and Zw are the number and weight average degrees of polymerization, respectively, of the final branched polymer mixture; in the preceding paper, these quantities are expressed in terms of the chain length averages of the component linear backbones and branches. For the special case of backbone molecules homogeneous in chain length, equation (12) becomes

<~>=(b'~/6Z,.Z,){Z (xZ.+ 3~,y2)+ ff.y3 (1 - 3~./x)}

(13)

Random comb mixtures of the type treated would be obtained, for example, in the complete or partial coupling reaction of homogeneous side chains with a collection of linear backbone molecules containing reactive sites randomly distributed among the backbone segments. DISCUSSION

AND

CONCLUSIONS

In the preceding sections expressions for the mean square radii of a number of branched molecule structures and mixtures of structures are given. These relationships are applicable to materials which are likely to be obtained in syntheses designed to provide polymers for systematic studies of the effect of branching on molecular dimensions. With this ultimate utilization in mind, it may be desirable to examine briefly the foregoing relationships in regard to (1) the range of structural variables best suited for observations of the effects of branching and (2) the influence of molecular heterogeneity on the mean square radii of mixtures. The dependence of s~ for regular comb molecules on the number of branches attached is shown in Figure 2, where values of the mean square radii, divided by ~0. ~ of the parent backbone molecule, are plotted versus lip. The various curves are parametric in the branch-to-backbone chain length ratio q =y/x and were constructed with the aid of equation (8a). The lowermost, corresponding to a zero y/x ratio, represents the linear molecule; as q increases, the successive curves exhibit increasing degrees of star molecule character. Thus, for example, the curve for q = 1 0 may also be used to describe a star molecule (x= 1) containing branches of length l0 units. At fixed branch and backbone lengths, the mean square radii of regular comb and star molecules approach asymptotic limits with increasing p. These limiting values are easily computed from pertinent equations of preceding sections; for comb molecules, we find lim (~)oomb=(b2/6) (x+ 3y - 2 y / x )

(14)

p--)x

x,r>~l

while for star structures lim (s~) .... = ½b~y

p - - ) ¢~

(15)

From data used in the construction of the curves of Figure 2, one may deduce as a rough guide that for comb molecules this upper limit is essentially attained (within ca. 5 per cent) at p values of the order 25 q_l/~; for star molecules, equation (15) is applicable when p is about fifteen or greater. 309

T. A. OROFINO

3O

L~

0

I

1

i

I

L

05

/

I

I

L

Wp

Figure 2--Plots of the ratio of ~0 for regular comb molecules to s~ for the constituent linear backbones, versus I/p, for various values of the parameter q=y/x T h e progressive a d d i t i o n of b r a n c h chains to a given linear b a c k b o n e or star centre is thus seen to involve, initially, a substantial a u g m e n t a t i o n of the m e a n square radius of the resulting b r a n c h e d molecule; this effect diminishes as the n u m b e r of branches attached increases, however, a n d eventually contributes insignificantly to s~0 u p o n further a d d i t i o n of b r a n c h chains*. W e m a y therefore conclude that in studies where the distinguish*The insensitivity of ~0 to p for relatively large values of this parameter presents some interesting possibilities in regard to the controlled variation of molecular parameters in O-solvent solutions or in bulk polymers. An observable phenomenon in one of these media may, for example, be suspected to depend, to some degree independently, upon the polymer molcular weight M and upon either the molecular size, as characterized by s~, or, the density of segments p within the domain of an isolated polymer coil. For a given random flight linear polymer, M, ~0 and p are interrelated and cannot therefore be independently varied. In a series of comb or star molecules of moderately large and varying p values, however, polymers comprising a wide range of molecular weights, but differing insignificantly in mean square radii, could be made available for study. Likewise, systematic variations in s~ (or p) at constant molecular weight could be realized through alteration of comb or star structure subject to the condition that the total degree of polymerization should remain constant. 310

BRANCHED POLYMERS---II able effects of branch addition in such molecules are of interest, samples for which p varies from zero to about 25 q_X/~ (see above) would appear to be most suitable. The effects of variations in x at fixed y and p/x (fixed branch spacing), and variations in y at fixed x and p on the mean square radii of regular structures more or less parallel those which would be observed for these linear components if detached from the branched molecule, i.e. constancy of the ratios (s~)/x and (~)/y [see equation (3)1, provided that the x or y components of the molecule, as the case may be, represent the major constituent of the branched structure. This is illustrated by the behaviour of the curves in Figure 2. In regard to variations in x at fixed y and p/x, we may compare the ordinate values of two curves, q] and q:, at abscissa values I/p~ and q~/qlPl, respectively. This amounts to comparing the two ratios of mean square radius of the branched molecule to that of the backbone when x assumes two values at the same y and p/x. The two ratios are identical for the respective, isolated, backbone chains; they do not differ greatly for the branched molecules in the range where py is of the order of magnitude of x, or less, i.e. wherever the backbone constitutes the major portion of the branched molecule. Variations in y at fixed x and p may be similarly analysed from Figure 2. At q values greater than unity, the approximate agreement at given p of the ordinate values for two curves q~ and q2, divided by ql and q o, respectively, is in accord with the constancy of such ratios for the isolated branch chains. An indication of the effect of heterogeneity in the number of branches attached to comb and star molecules may be obtained from the curves of Figure 3, where the z-average mean square radii of some random structures (x and y constant) divided by s~ for the corresponding regular structure of the same ~, are plotted versus p. These values were calculated

Figure 3--Plots of the ratio of ~ f o r random structures to s~ for the corresponding regular systems, versus p (/gregular~ Pn, random)"

Curve

a,

star

molecules; b, comb molecules with uniform backbones x=y=100; c, comb molecules with uniform backbones, x= 100, y=10

.gl

2

4

P 311

T. A. OROFINO with the aid of equations (4), (8a), (10) and (13), together with pertinent molecular weight relationships from Part I. As would be expected, the effect of this type of heterogeneity becomes vanishingly small as p increases. In studies where p values of about five or greater are chosen for investigation, regular structures, presumably synthesized with greater difficulty, would appear to afford no important advantages in regard to elimination of the effects of this kind of heterogeneity. As mentioned previously, however, in connection with the asymptotic behaviour of s~ with increasing branch chain content, if the above mentioned range of p values is to be investigated, some thought should be given to selection of appropriate y/x ratios in order to render significant effects of the number of branches attached on the mean square radius of the branched molecule. The effects of molecular weight heterogeneity in backbone and branch chain length are more difficult to assess in detail. From an analysis of the theoretical expressions for the z-average mean square radii of mixtures exhibiting such heterogeneity, however, it appears that the effects are similar to those which would be shown if the same heterogeneity were present in the corresponding linear components, particularly of course when the linear component considered constitutes the major portion of the branched chain. Expressions for the mean square radii of gyration of the various branched structures considered are summarized in Appendix B. Here, these are expressed in terms of the dimensionless parameter g, the ratio of mean square radius of the branched molecule to that of a linear molecule of the same molecular weightL In the case of mixtures, g is expressed in terms of the z-average mean square radius of the branched molecule and the z-average square radius of a mixture of linear molecules each of which has the same molecular weight as its counterpart in the branched sample. APPENDIX THE

Z-AVERAGE OF

MEAN SQUARE RANDOM COMB

A RADIUS FOR MOLECULES

MIXTURES

We consider a collection of backbone molecules, with chain length distribution represented by the set In,; x:], to which branches have been attached at random positions. The z-average radius of the mixture is given by the right hand side of equation (9), the numerator of which is the sum over all species of the product of the square of the degree of polymerization and the mean square radius (averaged, for a given molecule, over all molecular conformations). For each member of the collection of branched molecules, this product, which we designate P~.~with the subscripts denoting the tth member of that portion of the sample whose members possess a common backbone length x~ units, may be expressed in accordance with equation (2) of the text Xi

P.=b~ [ 2~ Kj.+K~.]

(16)

i=1

Here. Ks,: is the Kramers product for the division of a selected molecule at backbone segment }. i.e. the product of the numbers of segments in the 312

BRANCHED POLYMERS--II two portions of the molecule resulting, and Kz,u is the total Kramers product for all divisions along the branches of the molecule. The numerator of the defining equation (9) may now be expressed as the double sum of P , over all values of t and i. The members of the sum may be grouped into terms pertaining to branched molecules of common xi and we may accordingly write xi

:

(17)

= b 2 [ ~ ~n,K'j,+ ~ n,/~] i

j

i

where K'j~ is the mean (number average) Kramers product for division of molecules of backbone length, x~ at a given backbone segment j, averaged over all (nO species, and Ky~ is the analogous mean total product for divisions of the molecule at its branch segments. The mean quantities defined by the foregoing equations are easily evaluated. We consider first the computation of Kj~. If the average number of branches per molecule in the entire branched polymer mixture is ~,, then the probability that a backbone segment selected at random is the origin of a branch chain is -~,,I-2,,. In a molecule of backbone length xi severed at the jth backbone segment, the combined probability that one portion contains r~ :~< j and the other r_o~ x~- j branch chains is P~(r,.r~,j)=

() J \ r2 ! r~

x~-q--~,,

j=1,2,3 ....

(18)

x~

The number average Kramers product for separations at backbone segment j is thus J

Kji = ~

x i -i

1~ P~ (r~, rz, j)" (j + YG) (x, - i + r2y)

(19)

rI=O r2=O

= (z./y.)~ (/x,

-

i ~)

We consider next the evaluation of the quantity K~. In an x~ molecule containing p branch chains, the total Kramers product for divisions along the branch segments is

K~,, = (py~/ 6) (3Z, - 2y)

(20)

where Z~=x~+py. The probability distribution of the number of branches attached to x~ molecules is given in Part I. That expression, in conjunction ~vith equation (20). yields for the desired mean product

K.~, =(y2p.x d 6~.) [3x, + y + (3~.y/~.) ( x , - 1)]

(21)

Insertion of equations (19) and (21) into equation (17) and performance of the required summation yields after simplification (~)~ = (b z/ 6Z.~Z x.) {2.Zx.~x + p.y' [ 3 x Z + y (x,, - 3p.)]} the result given in equation (12) of the text. 313

(22)

T. A. O R O F I N O APPENDIX THE

g FOR

RATIO

B

VARIOUS BRANCHED AND MIXTURES

STRUCTURES

Branched moiecule

g Regular structures

(1) (2) (3)

3/p--2/p 2 3yw/py--2~,/p2y ~ 1--Nu+NJp2+3N~/p--3N~/p2

Star, y constant Star, y~ arbitrary Regular comb

Branched molecules

g Mixtures

(1) (2)

Random star, y constant Regular comb, y constant, xi arbitrary

(3)

Random comb, x, y constant

(3/7. + 1 ) / ( ~ + 3~, + 1)

~,~l-2z) [-2zN ~/~,~ y + 3N~ / ~,, + (-2nN~I-2, ~ y ) (-2 / ~. -- 2y)] Z n (xZ.+ 3-ff,~y2)+ ffny~ (i -- 3 ~ {x) -Z~F 3-Z.p,~yz (1---Pn/x)+(-pny3 /x 2) (x--f),,) (x--2pn) (4)

Random comb, y constant, x~ variable

2fiu,2~/2. + 3 ~ 2 n ~ n y 2 (1 -- P n / ~ 1 +

~ y a (1 --~n/-2.) (-;.--2~.)

This work has been supported in part by the Office of Naval Research. Mellon Institute, Pittsburgh, Pa (Received February 1961)

REFERENCES 1 KRAMERS,H. A. J. chem. Phys. 1946, 14, 415 2 KUHN, W. Kolloidzschr. 1939, 87, 3 :~ ZIMM, B. H. and STOCKMAY~R,W. H. J. chem. Phys. 1949, 17, 1301 STOCKMAVER,W. H. and FIXMAN, M. Ann. N.Y. Acad. Sei. 1953, 57, 334 5 WALES, M., MARSHALL,P. A. and WEISSBERG, S. G. J. Polym. Sci. 1953, 10, 229 BERRY, G. C. Thesis. University of Michigan, Ann Arbor, Michigan, U.S.A. 1960

314