Characterization of Branched Polymers by Size Exclusion

6 Characterization of Branched Polymers by Size Exclusion Chromatography with Light Scattering Detection R. C. JORDAN and M . L . McCONNELL

Downloaded by UNIV LAVAL on June 16, 2014 | http://pubs.acs.org Publication Date: November 26, 1980 | doi: 10.1021/bk-1980-0138.ch006

Chromatix, 560 Oakmead Parkway, Sunnyvale, C A 94086

Size exclusion chromatography (SEC) separates molecules of a polymer sample on the basis of hydrodynamic volume. When the chromatograph is equipped only with a concentration-sensitive detector, i.e. conventional SEC, a molecular weight distribution (MWD) can be obtained from the chromatogram only through use of a calibration function relating molecular weight and elution volume V (1). The calibration technique used in conventional SEC does not always give the correct MWD, however. The molecular size of a dissolved polymer depends on its molecular weight, chemical composition, molecular structure, and experimental parameters such as solvent, temperature, and pressure (2). If the polymer sample and calibration standards differ in chemical composition, the two materials probably will feature unequal molecular size/weight relationships. Such differences also will persist between branched and linear polymers of identical chemical composition. Consequently, assumption of the same molecular weight/V relation for dissimilar calibrant and sample leads to transformation of the sample chromatogram to an apparent MWD. In some cases the r e l a t i o n s h i p between polymer intrinsic v i s c o s i t y ([η]) and molecular weight (M) has been e s t a b l i s h e d f o r the SEC s o l v e n t and temperature c o n d i t i o n s ; i . e . , the e m p i r i c a l Mark-Houwink c o e f f i c i e n t s (2)(K,a) i n the equation

[η>

KM

a

(1)

have been determined. Under these circumstances the " u n i v e r s a l " c a l i b r a t i o n approach can be u t i l i z e d to c a l c u l a t e the c o r r e c t MWD from the sample chromatogram. However, K,a values are not a v a i l a b l e f o r many samples, p a r t i c u l a r l y those with polymer chain branching. A number o f the l i m i t a t i o n s o f conventional SEC can be overcome through use o f a low angle l a s e r l i g h t s c a t t e r i n g (LALLS) d e t e c t o r attached i n s e r i e s with a concentration

0-8412-0586-8/80/47-138-107$05.75/0 © 1980 American Chemical Society

In Size Exclusion Chromatography (GPC); Provder, T.; ACS Symposium Series; American Chemical Society: Washington, DC, 1980.


108

S I Z E

E X C L U S I O N

C

H

R

O

M

A

T

O

G

R

A

P

H

Y

( G P C )

d e t e c t o r (SEC/LALLS). The p r i n c i p l e s and methodology o f the technique are described i n d e t a i l elsewhere ( 4 - 7 ) . Data from both detectors are used to obtain the absolute molecular weight a t each point i n a sample chromatogram. The SEC/LALLS technique i s capable o f q u i c k l y y i e l d i n g the c o r r e c t MWD o f l i n e a r and branched samples without recourse to the approximate column c a l i b r a t i o n methods used i n conventional SEC. The hydrodynamic volume separation mechanism o f SEC, along with the d i f f e r e n t molecular size/weight r e l a t i o n s h i p s o f branched and l i n e a r polymers o f i d e n t i c a l chemical composition, can be e x p l o i t e d with the SEC/LALLS method to gain information about polymer branching. In the s t u d i e s described i n t h i s paper both conventional SEC and SEC/LALLS a r e used to obtain data about branching i n samples o f p o l y v i n y l acetate) (PVA) and p o l y c h l o r o prene (PCP). Theoretical The d i s c u s s i o n and experimental approach presented here r e l y on the p r i n c i p l e s and use o f universal c a l i b r a t i o n f o r SEC analysis. For a review o f t h i s method the reader i s r e f e r r e d to several useful a r t i c l e s (1_,3,8,9). For i l l u s t r a t i o n considfer SEC chromatograms obtained f o r two polymers on the same chromatographic system. One sample i s a l i n e a r homopolymer while the o t h e r i s a branched polymer with the same chemical composition. In the l a t t e r sample assume that the polymer components o f d i f f e r e n t molecular weight have uniform branching c h a r a c t e r i s t i c s so that a l l have s i m i l a r molecular size/weight r e l a t i o n s h i p s . Compare molecular size/weight c h a r a c t e r i s t i c s o f branched and l i n e a r species e l u t i n g a t V i n each chromatogram. Under the universal c a l i b r a t i o n formalism branched and l i n e a r components have the same hydrodynamic volume a t V: (Cn] M ) = b

b

v

([η] Μ ) 1

1

γ

(2)

where s u b s c r i p t s b and 1 denote branched and l i n e a r polymer components, r e s p e c t i v e l y . I f the f r a c t i o n a t V i s homogeneous with respect to hydrodynamic volume, the polymer molecules a t V w i l l be monodisperse with respect to molecular weight. The r a t i o o f the i n t r i n s i c v i s c o s i t i e s o f branched and l i n e a r species at V i s obtained by rearranging eq 2

(3)


6.

J O R D A N

A N D

M

C

C

O

N

N

E

L

Characterization of Branched Polymers

L

109

However, the q u a n t i t y which i s f r e q u e n t l y discussed and r e l a t e d to s p e c i f i c branching models i s the r a t i o o f i n t r i n s i c v i s c o s i t i e s at constant molecular weight (2)


(4)

For polymer samples o f the type considered here t h i s parameter r e f l e c t s the reduction i n molecular s i z e o f branched, r e l a t i v e to l i n e a r , material o f i d e n t i c a l molecular weight. The r e l a t i o n s h i p between g and g^ can be found through use o f the Mark-Houwink r e l a t i o n s h i p . n"he i n t r i n s i c v i s c o s i t y o f a l i n e a r polymer o f the same molecular weight (M. ) as the branched polymer i s

D

(5)

where Κ and a are the Mark-Houwink constants f o r the l i n e a r polymer. Equation 3 gives the r e l a t i o n o f M. to M, a t V, so t h a t we can r e l a t e the i n t r i n s i c v i s c o s i t y o f the l i n e a r polymer at V to the i n t r i n s i c v i s c o s i t y o f l i n e a r polymer having the same molecular weight as the branched polymer a t V:

(6)

S u b s t i t u t i o n o f eq 1 i n t o eq 6 gives

(7)

Use o f eq 7 i n eq 4 gives


110

SIZE EXCLUSION CHROMATOGRAPHY (GPC

Since we have e x p l i c i t l y s p e c i f i e d that branched species o f molecular weight M. a r e contained a t volume V, eq 8 can be written:

(Cn] ) b

9

= M

M

—

~

v

a

,10). Values shown in Tables II-IV cover data c o l l e c t e d between acceptable s i g n a l / noise l i m i t s o f the LALLS and DRI detectors a t the low and high molecular weight ends, r e s p e c t i v e l y , o f the chroma tograms. Consequently i n the high molecular weight p o r t i o n o f chromatograms data were processed when the concentration exceeded 1% o f the peak c o n c e n t r a t i o n , whereas i n the low molecular weight region the lowest concentration used represented 10% o f the maximum. For the NBS standard polystyrene SRM 706 Table II presents SEC/LALLS values o f (M^y as well as band-spreading c o r r e c t e d (MjJ) values obtained from processing data i n the conventional sense. The dependences on V o f l o g Î M ^ v and l o g i M ^ y are p l o t t e d i n Figure 1. In a d d i t i o n to the g* values c a l c u l a t e d from ( M Î ) and ( M ) * v a l u e s o f (M*/Mw)y are reported i n Table I I . The parameter (M*) represents the apparent molecular weight o f l i n e a r polymer a t V, given by the GPC1 program, but without a p p l i c a t i o n o f the band-spreading c o r r e c t i o n (Appendix). T h e r e f o r e , the d i f f e r e n c e between g ' and ( M * ^ ) ' at V shows the e f f e c t o f a p p l y i n g the band-spreading c o r r e c t i o n to data processed i n the conventional sense. In a d d i t i o n , Table II n

v

a

V

w

v

v

+ a


J O R D A N

A N D

M

C

C

O

N

N

E

L

L



Table I

Data from the SEC/LALLS A n a l y s i s o f Narrow MWD Polystyrene Standards

SAMPLE

3 R

U\P

w

n

PC600K

6.65xl0

PC390K

3.76x10

PC233K

2.50x10

NBS705

1.73xl0

PC100K

9.08x10

PC50K

4.87x10

PC37K

3.73x10

(corr)]

5

6.54x10

5

3.72xl0

5

2.45x10

5

1.72xl0

4

8.99X10

4

4.79x10

4

3.66x10

5

5

5

5

4

4

4

a

P r e f i x e s PC and NBS r e f e r to the s u p p l i e r s , Pressure Chemical Co. and National Bureau o f Standards, r e s p e c t fully.

b

The quantity R ( c o r r ) i s the SEC/LALLS sample number average molecuTar weight, c o r r e c t e d f o r band-spreading, while 1^ i s the sample weight-average molecular weight.



116

SIZE EXCLUSION

CHROMATOGRAPHY

(GPC)

6.4

4.2 I

27

I

I

I

ι

I

I

29

31

33

35

37

39

V(ml) m

Figure 1. Dependence on V of \MJA (corr)} for polystyrene calibrants (Table I), and dependence on V of (M„) and (Μ„*) for polystyrene SRM706 (Table II): (A) M -= [MvMJcorr)]" ; (+) M = (M *) ; (O) M = (MJ . The straight line corresponds to Equation 15. n

v

υ

2

w

v

V


6.

JORDAN


A N D M iti sample \ and band-spreading c o r r e c t e d f L v a l u e s . Data from Table IV are p l o t t e d i n Figure 4 as l o g ( l $ ) and l o g (Μ^γ v s . V; a p l o t o f g ' v s . l o g ( Μ ^ f o r PCP i s shown i n Figure 5. a

n

d

d

a

t

a

f

o

r

P C P

i

n

a

d

d

o

n

t

0

t

h

e


ν

ν

Discussion Data i n Figures 3 and 5 show an increase i n g' with de c r e a s i n g molecular weight f o r each o f the p o l y d i s p e r s e PVA and PCP samples. However, the data f o r PCP and one o f the PVA samples (0106) a l s o e x h i b i t g ' values g r e a t e r than unity a t the low molecular weight ends o f the molecular weight d i s t r i butions. Since by i t s d e f i n i t i o n g i s expected to equal o r be l e s s than u n i t y , such values are anomalous. A l s o , some g values are s u s p i c i o u s l y low i n the high molecular weight r e gions o f the PVA samples. Examination o f data f o r the NBS standard polystyrene SRM 706 gives i n s i g h t i n t o the behavior o f g' data i n the molecular weight extremes o f the PVA and PCP samples. The SEC/LALLS data i n the l o g i M ^ v vs V p l o t (Figure 1) e x h i b i t l i n e a r b e h a v i o r , along with curvature i n the high and low molecular weight r e g i o n s . This i s i n c o n t r a s t to the l i n e a r dependence on V o f l o g i R ^ ) ' ' which i s shown by the narrow MWD polystyrene c a l i b r a n t s . Diminished molecular s i z e r e s o l u t i o n i n the molecular weight extremes o f SRM 706 i s a probable cause f o r t h i s discrepancy. High molecular weight, unresolved species serve to increase SEC/LALLS ( ^ ) values a t low V, whereas unresolved low molecular weight components decrease (Mu,) a t high V. The other n o t i c e a b l e feature i n Figure 1 i s the dependence o f log(Mj$) on V, obtained by conventional processing o f the d a t a ; log(MjJ) shows the expected l i n e a r v a r i a t i o n with V throughout the chromatogram, except f o r downward curvature in the high molecular weight r e g i o n . These l a t t e r data correspond to points i n the extreme high molecular weight end o f the chromatogram, where the concentration i s only a few percent o f the peak ( R e s u l t s ) . I t i s doubtful t h a t material at points i n the extreme t a i l s o f the chromatogram has a Gaussian d i s t r i b u t i o n o f molecular weights, thus i n v a l i d a t i n g use o f the band-spreading c a l c u l a t i o n (Appendix). 1

1

2

ν

v

v

v


6.

J O R D A N

A

N

D

M

C

C

O

N

N

E

L

L


121


logM

Β

logM

Figure 2. Dependence on V of (MJ and (M *) for (A) PVA 0106 and (B) PVA 0107-066 (data from Table III: (+) Μ = (M *) ; (Ο) M = (MJ ) V

w

v

w

v


V

SIZE EXCLUSION CHROMATOGRAPHY


122

6.8

6.4

6.0

5.6

(GPC)

4.4

5.2

log (M )v w

Figure 3. Dependence of g on (MJ for PVA samples (data from Table III: (+) = g for PVA 0106; (O) = g' [= gYU] for PVA 0107-066; note that for PVA 0106, g' = 4.18 has been deleted from the plot) V

logM

Figure 4.

Dependence on V of (MJ and (M *) for PCP (data from Table IV: (+) M = (M *) ; (O) M = (MJ ) V

w

w

v

v

V



J O R D A N

A

N

D

Figure 5.

M < C O N N E L L


Dependence of g on (MJ for PCP (data from Table IV) V


124

SIZE

EXCLUSION

CHROMATOGRAPHY

On the other hand concentration detector (DRI) data represent at l e a s t 10% o f the maximum on the low molecular weight s i d e o f the chromatogram ( R e s u l t s ) , and the Gaussian approximation i s probably a p p l i c a b l e . Since polystyrene SRM 706 i s supposedly a l i n e a r polymer sample, g' i s not expected to deviate s t r o n g l y from u n i t y . I n spection o f Table II shows that g* values c l u s t e r about unity throughout most o f the SRM 706 molecular weight d i s t r i b u t i o n . In the t a i l s o f the d i s t r i b u t i o n , however, decreased r e s o l u t i o n and i n a p p l i c a b i l i t y o f the band-spreading c o r r e c t i o n serve to make g' behave anomalously. * For PVA and PCP the dependences on V o f log(M^) and log ( M ) (Figures 2a, 2b and 4) show s i m i l a r trends as f o r SRM 70o. Noticeable downward curvature o f log(M^)y at low molecular weight occurs i n each c a s e , i n c o n t r a s t to r e l a t i v e l y l i n e a r behavior o f l o g ( M ^ ) . Anomalously l a r g e values o f g r e s u l t (Figures 3 and 5 ) . In the highest molecular weight r e g i o n , a sharp downturn i n the l o g (M^) dependence on V i s apparent which i s accompanied by very low g' v a l u e s . The r e s u l t s found i n t h i s work i n d i c a t e that SEC/LALLS can be used to obtain q u a l i t a t i v e data about polymer branching. As suggested e a r l i e r in the d e r i v a t i o n o f the expression f o r g ' , the assumptions inherent i n using t h i s parameter as an approximation to g should not be overlooked. Caution should be e x e r c i s e d f o r polymers with unknown branching c h a r a c t e r i s t i c s , e . g . , where branching frequency i s suspected to vary with molecular weight. In t h i s case the molecular weight h e t e r o geneity a t V can be s i g n i f i c a n t l y more than allowed f o r by the l i n e a r polymer band-spreading c o r r e c t i o n used here (18). The g' data f o r the PVA samples (Table I I I , Figure 3) i n d i c a t e that branching a f f e c t s the molecular volume o f PVA 0107-066 to a g r e a t e r extent than the lower molecular weight PVA 0106. Except f o r the suspect points i n the extremes o f the MWD the g* values f o r PVA 0106 are l a r g e r than f o r PVA 0107-066, and f o r each sample the data show a decrease i n g with i n c r e a s i n g molecular weight. (The r e p r o d u c i b i l i t y o f the t e c h nique i s shown by the agreement o f g' values obtained f o r successive runs of PVA 0107-066.) Studies (19) have shown t h a t , f o r PVA samples wherein c e r t a i n branching c h a r a c t e r i s t i c s ( e . g . branching frequency) increase with the extent o f r e a c t i o n a general decrease i n g (eq 4) with i n c r e a s i n g molecular weight was found. A s i m i l a r dependence o f g on molecular weight i s found f o r the PCP sample (Table IV, Figure 5 ) . The data suggest the presence o f branching throughout the MWD, which i s c o n s i s t e n t with the known p r o p e n s i t y o f t h i s polymer to e x h i b i t branching under most r e a c t i o n c o n d i t i o n s ( V7)· Anomalous g values in the molecular weight extremes again r e f l e c t the dependence on V o f log(M^) and l o g ( M ) (Figure 4 ) . v


(GPC)

w

y

1

v

v

m

1

m

1

1

w

v


6.

JORDAN

AND

Characterization

MCCONNELL

of Branched

Polymers

125

Since the SEC/LALLS technique always yields a weightaverage molecular weight (M^y for the slightly polydisperse fraction at V, a small overestimation of the sample R is expected (5., 10). As noted previously (Results) a 1% to 4% decrease in the narrow MWD polystyrene M values (Table I) accompanied application of the band-spreading correction; Table II shows that for SRM 706 good agreement^*s obtained between SEC/LALLS and conventional SEC sample Pi^ and R values when the band-spreading correction was used. However, the NBS 706 polydispersity index (FL/R ) given by the supplier (ca. 2.1) does not agree with that U . ο ) found here using the SEC/LALLS and conventional SEC techniques. Insensitivity of the LALLS detector to a small amount of low molecular weight material may account for a larger sample M ; however, this is not supported by the conventional SEC data. The reason for the discrepancy remains unclear. n

n


n

n

Appendix Yau, et. al_., (11 ) described a computation method, named GPCV2, which corrects for chromatographic dispersion (bandspreading) in the determination of the MWD using SEC. Although developed for use with a single broad standard calibration scheme, the fundamental equations are also valid for a multiple narrow standard calibration. In this study we have composed a minor variation in GPCV2 to f a c i l i t a t e its use with a universal calibration scheme. Also, we have derived a computational method analogous to GPCV2 which can be used to correct for chromatographic dispersion in the determination of the MWD by SEC/LALLS. To employ GPCV2 in SEC, the column calibration is expressed as M* = D ~ 2 (Al) D

V

i e

where M* is the apparent linear (peak position) molecular weight at retention volume V. Because of chromatographic dispersion, the sample fraction in the detector cell is polydisperse. The weight-average and number-average molecular weights of the polydisperse fraction are calculated as =

KK w

D 2 g 2 )

"

· exp[(D c^/2] · M* 2

(A2)

~T(Tj

v

M

( „)

F ( V

=

F

(

V

)

e x

v

2

P [-(0 σ) /2]. M* 2

(A3)

2

F(V+D o ) 2


126

SIZE

EXCLUSION

CHROMATOGRAPHY

(GPC)

where F(V) i s the normalized ( E F ( V ) = l ) chromatogram under the conditions o f chromatographic d i s p e r s i o n ( . . F(V) i s the d e t e c t o r response a t r e t e n t i o n volume V ) . The term σ i s the d i s p e r s i o n constant. The RJJ and P§ o f the t o t a l sample are c a l c u l a t e d as


y

fl*

= exp[-(D a) /2] -l[(F(v)D

%

= exp[(D a) /2]/E

2

2

2

2

D i e x p

^)

]

exp(-D V)J

(A4)

2

(A5)

v )

Combination o f eqs (A4) and (A5) y i e l d s

%

= -(D2a) e

2

v)D e- 2 ] · Z[F(V)/D - 2 ] D

E [ F (

v

D

1

v

(A6)

i e

1

Η η

T h e r e f o r e , i f M^/M^ i s known, σ = D"

1

^ ln|(fl*/R*)

σ can be c a l c u l a t e d as

* E[F(V)D -¥] i e

· Z[F(V)/D " 2 ]j D

v

i e

(A7)

In t h i s study we have used an Mj£/M c a l c u l a t e d from s u p p l i e r ' s specifications. The GPCV2 equations were developed f o r conventional log(MW) v s . r e t e n t i o n volume c a l i b r a t i o n s . When used i n conjunction with a universal c a l i b r a t i o n , the slope term (Dp) must be c o r r e c t e d f o r the d i f f e r e n t molecular size/weight r e l a t i o n s h i p s o f the c a l i b r a n t s and the samples as derived i n the f o l l o w i n g equations. To understand t h i s c o r r e c t i o n , c o n s i d e r the con ventional c a l i b r a t i o n curve that could be created from the universal c a l i b r a t i o n d a t a . The slope term, Dp, f o r a c a l i b r a t i o n with polymer 1 i s defined a t points a and b on the c a l i b r a t i o n curve as n


6.

JORDAN

AND

Characterization

MCCONNELL

of Branched

Polymers

127

the slope term that must be used to c o r r e c t f o r d i s p e r s i o n o f sample polymer 2 can be s i m i l a r l y defined

,

D

_

1

n

"

2

2 ^

Μ

Ί

" 2,a

.

M

V a V

According to the universal


= K

2

2,a

M

"

V a V

concept

(a,+l)

?

a

(A9)

( - f e - )

n

calibration

(a +l) J

1

=

l l,a

K

(A10)

M

where K , a and K] ,a-j are the Mark-Houwink c o e f f i c i e n t s o f polymer 2 and polymer 1, r e s p e c t i v e l y , and J i s the hydrodynamic volume a t r e t e n t i o n volume a . Similarly, 2

2

a

*AT - AT ]

b

j

=

K

{A11)

]

Reconfiguring A10 and A l l we o b t a i n

2,b

M

=

ά

λ)

η

2+1)

ai2

ι*Α χ /^ *

< >

and

2,a

M

=

λ)/Κ

η&2+1)

( A 1 3 )

ίκΑ*Χ Ζ?

therefore 2,b/ 2,a

M

ί ^ / ^ Ψ ^ )

=

M

-

[M , /M 1

b

l i a

3

( a

l

+ 1 ) / ( a 2 + 1 )

(AH)

and In ( M

2 f b

/M

2 > a

)

=

[(a +l)/(a +l)] 1

2

In ( M

l f b

/M

1 > a

)

(A15)

Combination o f eqs A8 and A9 y i e l d s D

2

=

D

2

l

n

( 2,b M

/ M

2,a>

/ l n

( l,b M

/ M

l,a)

(A16)

S u b s t i t u t i o n o f A15 i n t o A16 y i e l d s D' 2

=

D ( +l)/(a +l) 2

a i

2

(A17)


128

SIZE

EXCLUSION

(GPC)

CHROMATOGRAPHY

Therefore to obtain (M*) and (M*) from M* data obtained by universal c a l i b r a t i o n , eqs A2 ana A3 are employed, but ΌΧ c a l c u l a t e d by A17 i s s u b s t i t u t e d f o r the D terms i n A2 and A3. Note a l s o that the value of σ obtained f o r a given l i n e a r polymer c a l i b r a n t i s an approximation to the true value f o r a branched polymer o r a polymer of d i f f e r i n g monomeric composition, since the dispension f u n c t i o n i s 1 i k e l y to vary f o r the various sample types. Under these c o n d i t i o n s , the d i s p e r s i o n c o r r e c t i o n i s a somewhat poorer approximation than the standard GPCV2 corrections. In SEC/LALLS, the molecular weight measured a t any i n s t a n t i s (M ) . Thus the sample M can be c a l c u l a t e d by the standard definition


2

w

% = *[F(V) ( M ) w

y ]

(

A

1

8

)

Using gq A2 to e l i m i n a t e M* i n eq A3, and s u b u s t i t u t i n g (M ) f o r (M ) a d i s p e r s i o n - c o r r e c t e d (M ) can be determined i n the StC/LALLS experiment: v

F(V) (M ) n

n

v

V

=

2 2

2

T~ · e x p [ - ( D a r ] * 2

Z

2

F(V+D a ) · F(V-D a ) 2

1

(MJ W

V

(A19)

V

2

then f o r the t o t a l

sample

M (corr) n

= l/Z[F(V)/(H ) ] n

Y

(A20)

where M ( c o r r ) denotes the o v e r a l l sample number-average molecular weight from SEC/LALLS, c o r r e c t e d f o r band-spreading. The 1 i m i t a t i o n s on the a p p l i c a t i o n o f eq A19 to branched p o l y mers and the use o f a constant σ f o r various polymers i n SEC/ LALLS a r e i d e n t i c a l t o the l i m i t a t i o n s c i t e d above f o r GPCV2. Acknowledgements: The authors wish to acknowledge the valuable t e c h n i c a l a s s i s t a n c e o f P h i l i p J . C h r i s t and Frank Chambers.


6.

JORDAN AND McCONNELL

Characterization of Branched Polymers 129

LITERATURE CITED 1. 2. 3.


4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

Yau, W. W., Kirkland, J. J., and Bly, D. D., "Modern Size Exclusion Chromatography", John Wiley & Sons, New York Ν. Y., 1979. Flory, P. J., "Principles of Polymer Chemistry", Cornell University Press, Ithaca, Ν. Υ., 1953. Grubisic, Ζ., Rempp, P., and Benoit, H., J. Polym. Sci. B, (1967), 5, 753. Application Notes LS-2, LS-3, LS-5, LS-10, Chromatix, 1977-1979. McConnell, M. L . , Am. Lab., (1978), 10 (5), 63. Ouano, A. C., and Kaye, W., J. Polym. Sci. Polym. Chem. Ed., (1974), 12, 1151. MacRury, Τ. Β., and McConnell, M. L . , J. Appl. Polym. Sci., (1979), 16, 2829. Dubin, P., and Kronstadt, M., in "Plastic Materials Science and Technology", Baijal, M. J., ed., Wiley-Interscience, New York, In press. Weiss, A. R., and Cohn-Ginsburg, Ε., J. Polym. Sci. B, (1969), 7, 379. Hamielec, A. E., Quano, A. C., and Nebenzahl, L. L . , J. Liq. Chromat., (1978), 1 (4), 527. Yau, W. W., Stoklosa, H. J., and Bly, D. D., J. Appl. Polym. Sci., (1977), 2 1 , 1911. Huglin, Μ. Β., in "Light Scattering from Polymer Solutions", Huglin, Μ. Β., ed., Academic Press, New York, Ν. Υ., 1972 p. 165 ff. Application Note LS-7,Chromatix, 1979. Ouano, A. C., J. Chromatogr., (1976), 118, 303. Hellman, Μ. Υ., in "Liquid Chromatography of Polymers and Related Materials", Cazes, J., ed., Marcel Dekker, New York, Ν. Y., 1977, p. 31. Park, W. S., and Graessley, W. W., J. Polym. Sci. Polym. Phys. Ed., (1977), 15, 71. Coleman, Μ. Μ., and Fuller, R. E . , J. Macromol. Sci.-Phys., (1975), B11 (3), 419. Ambler, M. R., Mate, R. D., and Purdon, J. R., Jr., J. Polym. Sci. Polym. Chem. Ed., (1974), 12, 1759. Park, W. S., and Graessley, W. W., J. Polym. Sci. Polym. Phys. Ed., (1977), 1 5 , 85.

RECEIVED May 7, 1980.


Characterization of Branched Polymers by Size Exclusion

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