Multiple Detectors in Size Exclusion Chromatography: Signal Analysis

Chapter 13

Multiple Detectors in Size Exclusion Chromatography: Signal Analysis L. H. Garcia-Rubio

Downloaded by EAST CAROLINA UNIV on March 23, 2016 | http://pubs.acs.org Publication Date: October 2, 1987 | doi: 10.1021/bk-1987-0352.ch013

College of Engineering, Department of Chemical Engineering, University of South Florida, Tampa, FL 33620

The characterization of complex polymers using size exclusion chromatography often requires the use of multiple detectors for the measurement of the polymer concentration and for the estimation of molecular properties such as composition and molecular weight. Depending upon the complexity of the analytical problem, a variety of detectors are used. Typical detection systems include refractometers, infrared and ultraviolet spectrophotometers, densitometers, on-line viscometers and low angle light scattering photometers. All of these detectors have different response factors and their measurement equations propagate the experimental error differently. Because of these differences, direct use of the synchronized signals from a system of detectors often leads to errors in the measurements and/or to partial interpretation of the data. This paper reports on the effect that the differences in detector sensitivity and measurement errors have on the quantitative interpretation of the data from multiple detector systems. As a case study, the signals from a detection system composed of a d i f f e r e n t i a l refractometer, a spectrophotometer and viscosity measurements on collected fractions are presented and discussed.

Complex polymers are those having a joint distribution of molecular properties. Branched polymers, copolymers and stereoregular polymers fall within this category. For example, branched polymers have a joint 0097-6156/87/0352-0220$06.00/0 © 1987 American Chemical Society

Provder; Detection and Data Analysis in Size Exclusion Chromatography ACS Symposium Series; American Chemical Society: Washington, DC, 1987.


13.

GARCIA-RUBIO

Multiple Detectors in SEC: Signal Analysis

221

d i s t r i b u t i o n of m o l e c u l a r w e i g h t ? and b r a n c h i n g f r e q u e n c i e s , copolymers have a j o i n t d i s t r i b u t i o n of c o m p o s i t i o n / m o l e c u l a r weight/sequence l e n g t h and t a c t i c i t y . Complex polymers may r e s u l t from homopolymerizations and c o p o l y m e r i z a t i o n s . Thus, p o l y b u t a d i e n e s a r e complex polymers formed from t h e c i s , t r a n s and v i n y l a d d i t i o n s o f b u t a d i e n e . I n a d d i t i o n , s i n c e p o l y b u t a d i e n e s a r e known t o undergo b r a n c h i n g r e a c t i o n s , p o l y b u t a d i e n e s a r e c h a r a c t e r i z e d by a j o i n t c o m p o s i t i o n / m o l e c u l a r w e i g h t / b r a n c h i n g f r e q u e n c y d i s t r i b u t i o n . In g e n e r a l , complex polymers do not e x h i b i t a unique r e l a t i o n s h i p between the s i z e of the m o l e c u l e s i n s o l u t i o n and the m o l e c u l a r weight as most l i n e a r homopolymers do. T h e r e f o r e , t h e c h a r a c t e r i z a t i o n o f c o m p l e x p o l y m e r s u s i n g s i z e e x c l u s i o n chromatography (SEC) o f t e n r e q u i r e s the use of m u l t i p l e d e t e c t o r s f o r t h e m e a s u r e m e n t o f t h e p o l y m e r c o n c e n t r a t i o n and f o r the e s t i m a t i o n o f m o l e c u l a r p r o p e r t i e s such as c o m p o s i t i o n , m o l e c u l a r weight and b r a n c h i n g f r e q u e n c y (J_2JJ5) · D e p e n d i n g upon the c o m p l e x i t y o f the a n a l y t i c a l problem, a v a r i e t y o f d e t e c t o r s a r e used. T y p i c a l d e t e c t i o n systems i n c l u d e r e f r a c t o m e t e r s ( w h i c h h a v e become the s t a n d a r d d e t e c t o r s i n commercial i n s t r u m e n t s ) , infrared (2-_4) a n d u l t r a v i o l e t s p e c t r o p h o t o m e t e r s (5^1.6), d e n s i t o m e t e r s (17-19), o n - l i n e v i s c o m e t e r s (7,10,20,21) and low a n g l e l i g h t s c a t t e r i n g p h o t o m e t e r s ( 2 2 - 2 5 ) . A l l of t h e s e d e t e c t o r s have d i f f e r e n t r e s p o n s e f a c t o r s and t h e i r measurement e q u a t i o n s p r o p a g a t e the e x p e r i m e n t a l errors d i f f e r e n t l y (26-28). Because of d i f f e r e n c e s i n s e n s i t i v i t y and n o i s e l e v e l , d i r e c t use o f the s y n c h r o n i z e d s i g n a l s from a system of d e t e c t o r s o f t e n l e a d s t o e r r o r s i n t h e m e a s u r e m e n t s a n d / o r t o p a r t i a l i n t e r p r e t a t i o n of the d a t a (11,14,16). T h i s paper r e p o r t s on the e f f e c t t h a t the d i f f e r e n c e s i n d e t e c t o r s e n s i t i v i t y and measurement e r r o r s have on the q u a n t i t a t i v e i n t e r p r e t a t i o n o f the d a t a from m u l t i p l e d e t e c t o r systems. As a c a s e s t u d y , t h e s i g n a l s f r o m a d e t e c t i o n system composed of a d i f f e r e n t i a l r e f r a c t o m e t e r , a s p e c t r o p h o t o m e t e r and v i s c o s i t y measurements on c o l l e c t e d f r a c t i o n s a r e p r e s e n t e d and d i s c u s s e d .

Problem F o r m u l a t i o n and A n a l y s i s The c h a r a c t e r i z a t i o n o f c o m p l e x polymers r e q u i r e s at l e a s t one d e t e c t o r per d e s i r e d p r o p e r t y . O b v i o u s l y , some o f t h e m o l e c u l a r p r o p e r t i e s w i l l be i n t e r - r e l a t e d and t h e r e s p o n s e f r o m any g i v e n d e t e c t o r w i l l i n c l u d e , i n g e n e r a l , c o n t r i b u t i o n s f r o m one o r more p r o p e r t i e s . T h u s , t h e r e s p o n s e f r o m u l t r a v i o l e t and i n f r a r e d s p e c t r o p h o t o m e t e r s a r e known t o c o n t a i n i n f o r m a t i o n on the c o m p o s i t i o n and t h e m i c r o s t r u c t u r e o f t h e polymer c h a i n s (3,4,1 4, 15,28,29), the i n t r i n s i c v i s c o s i t y and l i g h t s c a t t e r i n g m e a s u r e m e n t s , on t h e o t h e r h a n d , r e s p o n d t o a t l e a s t t h e c o m p o s i t i o n and the m o l e c u l a r weight (20,24,25) . The d e t a i l e d d i s c u s s i o n on the i n t e r p r e t a t i o n o f c o m p l e x s i g n a l s has been p r e s e n t e d elsewhere (26-29) and i t w i l l be d i s c u s s e d i n the c o n t e x t of 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 i n a s e p a r a t e p u b l i c a t i o n . In what f o l l o w s i t w i l l be assumed t h a t each d e t e c t o r i s t r a i n e d t o e i t h e r a unique p r o p e r t y or a l i n e a r c o m b i n a t i o n o f polymer properties. The m i n i m u m s e t o f p r o p e r t i e s t o be m o n i t o r e d in a c h r o m a t o g r a p h i c e f f l u e n t a r e t h e polymer c o n c e n t r a t i o n , the polymer



222

DETECTION AND DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

c o m p o s i t i o n and t h e m o l e c u l a r w e i g h t . The c o m p o s i t i o n and p o l y m e r concentration detectors provide information that i s , i n general, i n d e p e n e d n t o f t h e m o l e c u l a r w e i g h t s . The i n t e r p r e t a t i o n o f t h e s i g n a l s f r o m m o l e c u l a r w e i g h t d e t e c t o r s ( i e : L L A L S and o n - l i n e v i s c o m e t e r s ) r e q u i r e s p r i o r knowledge o f t h e polymer c o n c e n t r a t i o n . A s i g n i f i c a n t f r a c t i o n of the e r r o r i n the i n t e r p r e t a t i o n of molecular weight d e t e c t o r s i s l a r g e l y t h e r e s u l t of p r o p a g a t e d measurement e r r o r s a r i s i n g from t h e e s t i m a t i o n o f both t h e polymer c o m p o s i t i o n and the polymer c o n c e n t r a t i o n (14-16,27). F i g u r e 1 shows the e f f e c t of t h e d i f f e r e n c e i n s e n s i t i v i t y f o r t h e c o n c e n t r a t i o n and c o m p o s i t i o n d e t e c t o r s . As i t can be s e e n i n t h e r e g i o n s o f l o w s i g n a l t o n o i s e r a t i o ( i : t h e t a i l s o f t h e c h r o m a t o g r a m ) , t h e r e i s an a p p a r e n t l y d r a m a t i c i n c r e a s e i n t h e content of one o f t h e comonomers, s t y r e n e i n t h i s c a s e . C l e a r l y , i t i s i m p o r t a n t t o assess how much o f t h e s i g n a l r e p r e s e n t s a c t u a l changes i n c o m p o s i t i o n and how much i t i s due t o t h e d i f f e r e n c e s i n s e n s i t i v i t y and n o i s e l e v e l of t h e d e t e c t o r s i n v o l v e d . The types of d e v i a t i o n s shown i n F i g u r e 1 a r e t y p i c a l of m e a s u r e m e n t s c o n d u c t e d w i t h d e t e c t o r s h a v i n g d i f f e r e n t s e n s i t i v i t i e s ( see r e f e r e n c e s c i t e d i n 2 and 14-16). The d e v i a t i o n s o b s e r v e d have been a t t r i b u t e d t o t h e e f f e c t of t h e m i c r o s t r u c t u r e on t h e d e t e c t o r r e s p o n s e , however, no a t t e m p t has been made t o e s t i m a t e t h e e r r o r bounds o f the measurements and thus e l u c i d a t e t h e e x t e n t t o w h i c h t h e d e v i a t i o n s o b s e r v e d c a n be a t t r i b u t e d t o m i c r o s t r u c t u r e o r c o m p o s i t i o n a l e f f e c t s on t h e polymer f r a c t i o n a t i o n process and on t h e detector responses. If a chromatograph equipped with a refractometer, a s p e c t r o p h o t o m e t e r , a v i s c o m e t e r and a LLALS photometer i s assumed f o r the a n a l y s i s of a two component polymer system, t h e e q u a t i o n s r e l a t i n g the c o m p o s i t i o n and t h e polymer c o n c e n t r a t i o n t o t h e r e f r a c t o m e t e r and the s p e c t r o p h o t o m e r a r e g i v e n by (1,14-16) η = [viPi + v (1 - Pi)] c

(1)

A = ε (1 - P J C

(2)

2

2

Where η i s t h e r e f r a c t i v e i n d e x s o l u t i o n and t h e s o l v e n t , increment or r e f r a c t o m e t e r weight

fraction

d i f f e r e n c e between t h e polymer i n i s the s p e c i f i c r e f r a c t i v e index

response t o the i t h s p e c i e s ,

P^ i s t h e

o f t h e i t h s p e c i e s , A i s t h e absorbance,

i s the

a b s o r p t i o n c o e f f i c i e n t or t h e s p e c t r o p h o t o m e t e r r e s p o n s e t o t h e i t h s p e c i e s and C i s t h e t o t a l c o n c e n t r a t i o n o f polymer i n t h e e f f l u e n t s t r e a m . I n t h i s c a s e , i t h a s been assumed t h a t o n l y o n e o f t h e components absorbs at the frequency selected f o r the spectrophotometer. The e q u a t i o n s r e l a t i n g s i g n a l s from the molecular weight d e t e c t o r s t o t h e c o n c e n t r a t i o n and t h e m o l e c u l a r w e i g h t s a r e g i v e n by (22)

KC/R(0) = 1/Mw + 2A C 2


(3)

13.

GARCIA-RUBIO

u


—ι—ι—ι—ι—ι—1—ι—ι—ι—ι—ι _

ι—ι

ι—I

I

I

I

223

I Γ


1.2 L

RETENTION VOLUME

(COUNTS)

F i g . 1. R e s u l t s from the d i r e c t a p p l i c a t i o n of the s i g n a l s from a SEC d e t e c t i o n s y s t e m composed o f a d i f f e r e n t i a l r e f r a c t o m e t e r and uv spectrophotometer (254 nm).


224


W h e r e Mw i s t h e w e i g h t a v e r a g e m o l e c u l a r w e i g h t o r t h e a p p a r e n t m o l e c u l a r weight i n the case of c o p o l y m e r s , A i s t h e s e c o n d v i r i a l c o e f f i c i e n t , and Κ i s an o p t i c a l c o n s t a n t d e f i n e d as 2

UîJ

N

IdcJ

R(6) i s t h e excess R a y l e i g h r a t i o , n and λ are the r e f r a c t i v e index of the s o l v e n t and the wavelength o f the l i g h t s o u r c e . 0

If

the second v i r i a l

0

c o e f f i c i e n t i s n e g l i g i b l e , e q u a t i o n 3 becomes


R ( 6 ) / K = MwC

(5)

In the case o f i n t r i n s i c v i s c o s i t y measurements, the measurements w i l l be i n the form of p r e s s u r e drop or a r a t i o of e f f l u x t i m e s . I n e i t h e r c a s e , the i n t e r p r e t a t i o n e q u a t i o n i s of the form Q = Cn]C

(6)

Where Q = l n ( Δ Π / Δ Π ) when p r e s s u r e t r a n s d u c e r s are used (20) and i t w i l l be d e f i n e d as Q = ( t - t ) / t when c a p i l l a r y v i s c o m e t e r s a r e used ( 7 ) . x

0

0

0

The d e s i r e d p o l y m e r p r o p e r t i e s e q u a t i o n s 1,2,5 and 6 Pi = [ ε

2

- ν (Α/η)]/[ε 2

C = [ ε η - (v 2

2

2

- (v

2

c a n be o b t a i n e d

directly

- vJU/n)]

(7) (8)

- ν )Α]/(ε ν ) 1

2

from

1

Mw = R ( 9 ) / ( K C )

(9)

[η]

(10)

= Q/C

Note t h a t a l l the d e s i r e d p r o p e r t i e s depend d i r e c t l y or i n d i r e c t l y on t h e r a t i o A / n and t h e r e f o r e , on t h e r e l a t i v e r e s p o n s e f a c t o r s and s i g n a l t o n o i s e r a t i o s of the spectrophotometer and t h e d i f f e r e n t i a l r e f r a c t o m e t e r . I n g e n e r a l , s p e c t r o p h o t o m e t e r s are more s e n s i t i v e than d i f f e r e n t i a l r e f r a c t o m e t e r s , t h e r e f o r e , the r a t i o A / n at the t a i l s o f t h e c h r o m a t o g r a m , t h a t i s , the r e f r a c t o m e t e r s i g n a l w i l l be z e r o w h i l e t h e r e s t i l l be a s i g n a l from the s p e c t r o p h o t o m e t e r . I n a d d i t i o n , e q u a t i o n s 9 and 10 a r e h y p e r b o l i c f u n c t i o n s o f t h e c o n c e n t r a t i o n . Thus, as the c o n c e n t r a t i o n decreases the apparent v a l u e s of Mw and [ η ] w i l l i n c r e a s e , i n c r e a s i n g t h e u n c e r t a i n t y i n t h e e s t i m a t e s of the m o l e c u l a r w e i g h t s and the i n t r i n s i c v i s c o s i t i e s ( s e e A p p e n d i x I and r e f e r e n c e s 2 6 - 2 9 ) . I n the l i m i t when A / n •* (or n/A -** 0 ) , the polymer c o m p o s i t i o n and the t o t a l p o l y m e r c o n c e n t r a t i o n have f i n i t e v a l u e s t h a t a r e d i f f e r e n t from the l i m i t s expected on the b a s i s of the mass balances 00

00

lim P = 1/(1 ( A/n)->°° l

-

vj/v ) 2


(11)

13.

GARCIA-RUBIO


lim C = (A/n)-»°°

(v 2

2

1

T h a t i s , depending on the magnitude and s i g n o f the r e s p o n s e f a c t o r s , the weight f r a c t i o n can be g r e a t e r than one or i t can be n e g a t i v e . For e x a m p l e , i f v < v , i n t h e l i m i t where the r e f r a c t o m e t e r s i g n a l has approached z e r o , the f r a c t i o n of monomer 1 i n t h e c o p o l y m e r w i l l be g r e a t e r t h a n 1. T h e same a p p l i e s t o t h e c o n c e n t r a t i o n , i t c a n be g r e a t e r than the c o n c e n t r a t i o n i n j e c t e d i n the chromatograph or i t can a l s o be n e g a t i v e d e p e n d i n g on the v a l u e s o f the r e s p o n s e f a c t o r s . I t i s i m p o r t a n t t o r e a l i z e t h a t the e f f e c t s s u g g e s t e d by e q u a t i o n s 11 and 12 w i l l a p p e a r as d e t e r m i n i s t i c t r e n d s i n t h e d a t a , thus m i s l e a d i n g the e s t i m a t i o n of the m o l e c u l a r weights and the i n t e r p r e t a t i o n o f t h e c o m p o s i t i o n d a t a (26,27). I t i s c l e a r t h a t the l i m i t i n g f a c t o r i n the i n t e r p r e t a t i o n of the s i g n a l s from a m u l t i d e t e c t o r system w i l l be t h e d e t e c t o r w i t h the l e a s t s e n s i t i v i t y . When t h i s s i g n a l becomes z e r o o r i t i s confounded w i t h the i n s t r u m e n t n o i s e , the system of e q u a t i o n s 710 becomes u n d e r - d e t e r m i n e d . I n a d d i t i o n t o the problems a s s o c i a t e d w i t h the d i f f e r e n c e s i n d e t e c t o r s e n s i t i v i t y , t h e m e a s u r e m e n t n o i s e a l s o p l a y s an i m p o r t a n t r o l e . As i t can be s e e n i n F i g u r e 2, t h e i n s t r u m e n t n o i s e p r o p a g a t e d t h r o u g h e q u a t i o n s 7-10 c a n c a u s e w i l d o s c i l l a t i o n s i n t h e r e g i o n s o f low s i g n a l - t o n o i s e r a t i o ( i e : t h e t a i l s o f the chromatograms). C l e a r l y , w i t h o u t g o o d e s t i m a t e s o f t h e p o l y m e r c o n c e n t r a t i o n and t h e polymer c o m p o s i t i o n , the e s t i m a t e s o f both the p o i n t v a l u e s and t h e a v e r a g e m o l e c u l a r p r o p e r t i e s w i l l be c o n s i d e r a b l y b i a s e d . In some cases i t i s p o s s i b l e t o s o l v e the problem e x p e r i m e n t a l l y by c o l l e c t i n g f r a c t i o n s a n d a n a l y z i n g t h e m independently f o r c o n c e n t r a t i o n and composition (JJi-J.6) , u n f o r t u n a t e l y , t h i s i s not a s a t i s f a c t o r y s o l u t i o n . I t i s always p o s s i b l e , h o w e v e r , t o o b t a i n good e s t i m a t e s o f the errors a s s o c i a t e d with the measured polymer p r o p e r t i e s ( i e : c o m p o s i t i o n , m o l e c u l a r w e i g h t s , e t c . ) The a p p l i c a t i o n of s i m p l e e r r o r p r o p a g a t i o n t e c h n i q u e s ( 2 6 , 2 7 , 3 0 ) t o e q u a t i o n s 7-10 y i e l d s g o o d a p p r o x i m a t i o n s to the v a r i a n c e o f the d e s i r e d p r o p e r t i e s x


(12)

ν )Α/(ε ν ) 1

225

2

2

(A/n) var(n)

var(Pi) =

L (^T

1

£ var(n) + (v 2

2

2

2

var(C) =

(v

2

(ε Vj)

- v) 1

+ var(A)

- v )(A/n))

(13) M

x

var(A)

(14)

2

2

var(Mw) =

R(e)

var(R(6)) KC

var(η) =

var(C)

(15)

2

\ KC ""

var(Q)

var(C) lc

(16)

2

From e q u a t i o n s 13-16, the s t a n d a r d e r r o r f o r e a c h m e a s u r e m e n t a s a f u n c t i o n o f t h e e l u t i o n time can be o b t a i n e d . A d d i t i o n a l p r o p a g a t i o n of these e r r o r s through the i n t e g r a t i o n a c r o s s the chromatogram r e s u l t s i n e s t i m a t e s o f the errors a s s o c i a t e d w i t h the SEC c a l c u l a t i o n o f the average p o l y m e r p r o p e r t i e s . T h e r e f o r e , i t e n a b l e s r e l i a b l e s t a t i s t i c a l comparisons between SEC e s t i m a t e s and s t a t i c measurements



226

D E T E C T I O N A N D DATA ANALYSIS IN SIZE E X C L U S I O N C H R O M A T O G R A P H Y

RETENTION VOLUME (COUNTS) Fig.

Effect

2.

of

the

molecular

weight

polystyrene

standard.

using

u n i v e r s a l

the

collected

as

uv

detector

f u n c t i o n The s y m b o l s

of

noise

represent

c a l i b r a t i o n

on

e l u t i o n and

the

estimation

volume

molecular viscosity

for

weights

a

of

obtained

measurements

fractions


the

narrow on

13.

GARCIA-RUBIO

on the t o t a l used and the

Mw


227

polymer. For e x a m p l e , i f t h e U n i v e r s a l C a l i b r a t i o n i s chromatogram i s i n t e g r a t e d u s i n g the t r a p e z o i d a l r u l e ,

= I M ( i ) [ w ( i ) + w(i+1)]

Δν/2

(17)

1

M(i)

=

(18)

J(i)/[n]i


where Δν r e p r e s e n t s t h e e l u t i o n v o l u m e i n c r e m e n t b e t w e e n s a m p l e p o i n t s , w ( i ) i s t h e w e i g h t f r a c t i o n o f p o l y m e r a t t h e i t h sample p o i n t , J ( i ) i s the p o l y s t y r e n e - b a s e d u n i v e r s a l c a l i b r a t i o n c u r v e and [ n ] i i s t h e i n t r i n s i c v i s c o s i t y at the i t h sample p o i n t . The v a r i a n c e f o r the weight average m o l e c u l a r weight can be approximated as

var(Mw) =

M(.i) v a r ( w ( i ) ) +

( w ( i ) + w(i + 1)) [

J

(

i

)

[η]ί

I

var([n]i)

(19)

2

The v a r i a n c e f o r t h e w e i g h t f r a c t i o n w ( i ) c a n be o b t a i n e d f r o m e q u a t i o n 14 and the v a r i a n c e f o r M ( i ) can be c a l c u l a t e d d i r e c t l y f r o m equation 18. N o t e t h a t t h e p r o p a g a t i o n o f e r r o r a n a l y s i s can be r e a d i l y e x t e n d e d t o o t h e r a v e r a g e s and and i t can a l s o be u s e d t o a c c o u n t f o r the e r r o r s a s s o c i a t e d w i t h the c a l i b r a t i o n of columns and detectors.

Experimental

Methods

S t y r e n e - a c r y l o n i t r i l e copolymers were s y n t h e s i z e d by b u l k f r e e - r a d i c a l p o l y m e r i z a t i o n at 60°C and 40°C u s i n g AIBN as i n i t i a t o r . The Copolymer c o m p o s i t i o n s were d e t e r m i n e d by gas chromatography and v e r i f i e d by H NMR spectroscopy. The c o p o l y m e r s were p u r i f i e d p r i o r t o t h e SEC a n a l y s i s by p r e c i p i t a t i o n from a b s o l u t e m e t h a n o l . The 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 i s a m o d i f i e d room t e m p e r a t u r e W a t e r s h i g h p r e s s u r e l i q u i d chromatograph e q u i p p e d w i t h a d i f f e r e n t i a l r e f r a c t o m e t e r and a W a t e r s 440 d u a l uv d e t e c t o r . The i n j e c t i o n v a l v e , d e t e c t o r s and volume counter were i n t e r p h a s e d w i t h a Data G e n e r a l Nova mini computer t h a t was u s e d p r i m a r i l y f o r d a t a l o g g i n g , base l i n e c o r r e c t i o n s , and data s y n c h r o n i z a t i o n . The f i r s t d e t e c t o r downstream from t h e c o l u m n s was used as r e f e r e n c e f o r s y n c h r o n i z a t i o n p u r p o s e s . The r e f r a c t o m e t e r and uv s p e c t r o p h o t o m e t e r s were c a l i b r a t e d t o a b s o l u t e u n i t s u s i n g b e n z e n e - C C l s o l u t i o n s . F r a c t i o n s were c o l l e c t e d a t r e g u l a r i n t e r v a l s f o r i n t r i n s i c v i s c o s i t y m e a s u r e m e n t s . I n t r i n s i c v i s c o s i t i e s were m e a s u r e d w i t h C a n n o n U b b e l o h d e v i s c o m e t e r s a t 2 5 ° C . The SEC was equipped w i t h a s e t of s i x μ - S t y r a g e l c o l u m n s ( 1 00, 500, 1 0 , 1 0 *, 1 0 , and 1 0 A). [ μ - S t y r a g e l i s a r e g i s t e r e d t r a d e m a r k of Waters Associates.] F o r 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 a n a l y s i s a 200 uL s a m p l e v o l u m e o f 0.5% (w/v) c o n c e n t r a t i o n was i n j e c t e d o n t o t h e columns. The s o l v e n t f l o w r a t e was s e t a t 1 m l / m i n . THF was u s e d as X

4

3

5

6


1

228


m o b i l e p h a s e . The columns were c a l i b r a t e d u s i n g 18 narrow p o l y s t y r e n e s t a n d a r d s [ P r e s s u r e C h e m i c a l s ] . The m o l e c u l a r w e i g h t s were c a l c u l a t e d u s i n g t h e u n i v e r s a l c a l i b r a t i o n c u r v e p r i n c i p l e and r e p o r t e d MarkHouwink p a r a m e t e r s . Whenever f r a c t i o n s were c o l l e c t e d t h e m o l e c u l a r w e i g h t s were c a l c u l a t e d d i r e c t l y from the u n i v e r s a l c a l i b r a t i o n and i n t r i n s i c v i s c o s i t y measurements.


R e s u l t s and D i s c u s s i o n Polymers produced a t v a r i o u s i n i t i a l monomer c o m p o s i t i o n s and c o n v e r s i o n l e v e l s were a n a l y s e d u s i n g e q u a t i o n s 7,8 and 10. The 95% c o n f i d e n c e i n t e r v a l (2a) c a l c u l a t e d f r o m t h e c o r r e s p o n d i n g e r r o r p r o p a g a t i o n e q u a t i o n s has b e e n s u p e r i m p o s e d on t o t h e m e a s u r e d p r o p e r t y v a l u e s . The r e s u l t s a r e shown i n f i g u r e s 3~5. F i g u r e 3 shows a low c o n v e r s i o n copolymer s y n t h e s i z e d w i t h a f e e d c o m p o s i t i o n c o n t a i n i n g 60% s t y r e n e ( i e : f = 0.6). I t i s e x p e c t e d , from the t h e o r y of c o p o l y m e r i z a t i o n , t h a t t h i s polymer would have a uniform c o m p o s i t i o n a c r o s s the chromatogram. The d i r e c t i n t e r p r e t a t i o n o f t h e s y n c h r o n i z e d s i g n a l s s u g g e s t s t h a t t h i s i s not the c a s e . As i t can be seen at the t a i l s of the chromatogram ( i e : a t t h e low and high m o l e c u l a r w e i g h t e n d s ) t h e r e a r e s i g n i f i c a n t changes i n c o m p o s i t i o n s u g g e s t i n g t h a t the Mayo-Lewis k i n e t i c s are not v a l i d . H o w e v e r , a p p l i c a t i o n of the e r r o r propagation equations i n d i c a t e s t h a t , given the v a r i a n c e o f the measurements, t h e r e i s no e v i d e n c e t o s u g g e s t t h a t t h e c o m p o s i t i o n s a t the t a i l s of the chromatogram a r e d i f f e r e n t from the v a l u e s at the m i d d l e o f t h e c h r o m a t o g r a m , w h i c h a g r e e w i t h t h e e x p e c t e d c o m p o s i t i o n f r o m c o p o l y m e r i z a t i o n k i n e t i c s . F i g u r e 4 shows the d a t a f o r a h i g h c o n v e r s i o n h i g h s t y r e n e c o n t e n t copolymer (ie: f 0«9, ? = 0.85). In t h i s case i t appears t h a t t h e r e i s e v i d e n c e of the i n t e r m e d i a t e s i z e c h a i n s h a v i n g statistically s i g n i f i c a n t d i f f e r e n c e s i n c o m p o s i t i o n . C o m p o s i t i o n a l l y heterogeneous p o l y m e r c h a i n s c a n be e x p e c t e d a t h i g h c o n v e r s i o n s s i n c e h i g h c o n v e r s i o n c o p o l y m e r s may not f o l l o w the c o p o l y m e r i z a t i o n t h e o r y due t o t h e g e l e f f e c t . F i n a l l y , F i g u r e 5 shows a low c o n v e r s i o n low s t y r e n e copolymer ( f = 0 . 4 , P = 0.55) f o r which the m o l e c u l a r weights were determined from i n t r i n s i c v i s c o s i t y m e a s u r e m e n t s on c o l l e c t e d f r a c t i o n s . C l e a r l y , the c a l c u l a t e d e r r o r bars ( i e : the 95% c o n f i d e n c e i n t e r v a l ) a t each s a m p l i n g p o i n t a l l o w s a r e a l i s t i c i n t e r p r e t a t i o n o f t h e SEC d a t a . The c a l c u l a t i o n of the average Mn, Mw and c o m p o s i t i o n and t h e i r c o r r e s p o n d i n g 95% c o n f i d e n c e i n t e r v a l f o r the whole p o l y m e r a l l o w s t h e s t a t i s t i c a l comparison between the SEC e s t i m a t e d averages and the s t a t i c measurements on the whole polymer. The e r r o r a n a l y s i s on t h e s t a t i c measurements was done i n accordance w i t h the e q u a t i o n s i n A p p e n d i x I and r e f e r e n c e 27. As i t can be s e e n i n T a b l e I , i f a l l o w a n c e i s made f o r t h e p r o p a g a t e d e x p e r i m e n t a l e r r o r s , t h e r e i s good agreement between the SEC and the s t a t i c measurements a t t h e 95% confidence l e v e l . 1 0

=

1

0

1

1 0

Summary and

1

Conclusions

The d i r e c t i n t e r p r e t a t i o n of the s i g n a l s from s e v e r a l d e t e c t o r s i n SEC experiments has been a n a l y s e d . I t i s c l e a r t h a t t h e direct i n t e r p r e t a t i o n of the s y n c h r o n i z e d s i g n a l s from d e t e c t o r s having


13.

GARCIA-RUBIO


~T—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—I

[ 1.2

I

I

I

229

r

!


I I 1

R E T E N T I O N VOLUME(COUNTS)

F i g . 3. I n t e r p r e t a t i o n o f t h e s i g n a l s f r o m a m u l t i d e t e c t o r SEC: a. d i r e c t c a l c u l a t i o n s ; b. 95% c o n f i d e n c e i n t e r v a l i n c l u d e d . Low c o n v e r s i o n , i n t e r m e d i a t e s t y r e n e content SAN copolymer.


230


I I I I I I I I I I I I I I I I 1.4 L


1.2

I

I I I I

I

RETENTION

V O L U M E (COUNTS)

F i g . 4. I n t e r p r e t a t i o n o f t h e s i g n a l s f r o m direct c a l c u l a t i o n s ; b. 95% c o n f i d e n c e conversion s t y r e n e - r i c h SAN copolymer.

a multidetector SEC: a . i n t e r v a l i n c l u d e d . High


13.

GARCIA-RUBIO


ΓΤ—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι—ι


1.2

ι

231

r~p

I

RETENTION

VOLUME

(COUNTS)

F i g . 5. I n t e r p r e t a t i o n o f t h e s i g n a l ? from a m u l t i d e t e c t o r SEC: direct calculations; 95% c o n f i d e n c e i n t e r v a l included. Low c o n v e r s i o n , low s t y r e n e c o n t e n t SAN c o p o l y m e r : φ s t y r e n e content,© molecular weight.


232

DETECTION A N D DATA ANALYSIS IN SIZE EXCLUSION CHROMATOGRAPHY

TABLE I COMPARISON BETWEEN STATIC MEASUREMENTS AND SEC ESTIMATES


Sample

S t a t i c Measurements P [n](dl/g) Mw GC THF 25°C V i s c o s i t y l

SEC R e s u l t s Mw [η]

standard d e v i a t i o n 0.01

0.025

2.0%

3.0%

0.05

R0209 R0204 R0206 R0208 R0210 RÔ211

0.406 0.533 0.588 0.790 0.963 1.048

80031 92955 120151 124305 127264 141323

81190 99061 109240 114680 136300 153060

0.500 0.510 0.725 0.874 0.948 1.053

0.87 0.84 0.80 0.74 0.69 0.68

d i f f e r e n t s e n s i t i v i t i e s and n o i s e l e v e l s c a n l e a d t o a m b i g u i t i e s i n t h e i n t e r p r e t a t i o n o f SEC d a t a . The a p p l i c a t i o n o f s i m p l e e r r o r propagation techniques r e s u l t s i n equations f o r the e s t i m a t i o n o f t h e v a r i a n c e s o f t h e d e s i r e d p o l y m e r p r o p e r t i e s . The p r o p a g a t e d e x p e r i m e n t a l e r r o r s can then be used t o e l i m i n a t e a m b i g u i t i e s i n t h e d a t a a n d f o r t h e s t a t i s t i c a l comparison o f s t a t i c measurements on t h e w h o l e p o l y m e r and t h e e s t i m a t e s f r o m SEC e x p e r i m e n t s . The e r r o r p r o p a g a t i o n a n a l y s i s c a n be r e a d i l y a p p l i e d t o more c o m p l i c a t e d measurement e q u a t i o n s ( i e : e q u a t i o n 3) p r o v i d e d t h a t t h e r e a r e s u f f i c i e n t measurements t o e s t i m a t e a l l t h e parameters i n v o l v e d . Furthermore, i f d e t e c t o r s l i k e t h e diode a r r a y s p e c t r o p h o t o m e t e r s a r e used, s e v e r a l measurements on t h e same v a r i a b l e w i l l be o b t a i n e d , thus allowing f o r the s t a t i s t i c a l e s t i m a t i o n of the d e s i r e d polymer properties. Acknowledgments T h i s r e s e a r c h was s u p p o r t e d by t h e NSF I n i t i a t i o n Grant RII-8507956 The author i s i n d e b t e d t o D r . T. Provder from SCM C o r p o r a t i o n f o r h i s c o n t i n u o u s encouragement and s u p p o r t .

REFERENCES : 1. 2.

Yau, W.; Kirkland, J.; Bly, D., "Modern Size-Exclusion Liquid Chromatography", 1979, J. Wiley, New York. Terry, S. L.; Rodriguez, F. J. Polym. Sci., Part C 1968, 21, 103.


13. GARCIA-RUBIO 3. 4. 5. 6. 7. 8. 9. Downloaded by EAST CAROLINA UNIV on March 23, 2016 | http://pubs.acs.org Publication Date: October 2, 1987 | doi: 10.1021/bk-1987-0352.ch013

10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.


Mirabella, T.; Barral, Ε.; Johnson, J. J. Appl. Polym. Sci. 1975, 19. Mirabella, T.; Barral, Ε.; Johnson, J . J . Appl. Polym. S c i . , 1976, 20. Runyon, J. R. et al, J. Appl. Polym. Sci.,1969, 13, 2359. Adams, H. E. In "Gel Permeation Chromatography"; Altgelt, Κ.; Segal, L . , Eds.; Dekker: New York, 1971. Grubisic-Gallot, Z. et al. J. Appl. Polym. Sci. 1972, 16, 2933. Probst, J. J.; Unger, K.; Cantow, H. J., Angew. Makromol. Chem., 1974, 35, 177. Elgert, K. F.; Wohlschiess, R., Angew. Makromol. Chem., 1977, 57, 87. Teramachi, S.; Hasegawa, Α.; Yoshida, S., Macromolecules., 1983, 16, 542. Balke, S. T.; Patel, R. D. In "Size Exclusion Chromatography (GPC)" Provder, Theodore, Ed.; ACS SYMPOSIUM SERIES No. 138; ACS; Washington, D.C., 1980; pp. 149-82. Inagaki, H.; Tanaka, T., Pure and Appl. Chem., 1982, 54(2), 302. Teramachi, S. Hasegawa, Α.; Yoshida, S., Macromοlecules, 1983, 16, 842. Mori, S.; Suzuki, T., J. Liq. Chrom., 1981, 4(10), 1685. Garcia-Rubio, L. H.; MacGregor, J . F . ; Hamielec, A. E . , ACS SYMPOSIUM SERIES, T. Provder, Ed., 1982, 197, 151. Garcia-Rubio, L. H.; Hamielec, A. E . ; MacGregor, J . F . , ACS SYMPOSIUM SERIES, C. D. Craver Ed., 1983, 202, 310. Francois, J.; Jacob, M.; Grubisic-Gallot, Z.; Benoit, H., J. Appl. Polym. Sci., 1978, 22, 1159. Trathing, B.; Jorde, C., J. Chromatogr., 1982, 24, 147. Boyd, D.; Narasimhan, V.; Huang, R.; Burns, C.M., J. Appl. Polym. Sci., 1984, 29, 595. Ouano, A. C., J. Polym. Sci: Symp. Series., 1973, 43, 299. Lecheaux, D. et a l . , J Appl. Polym. Sci., 1982, 27, 4867. Ouano, A. C.; Kay, W., J. Poly. Sci., Part A-1, 1974, 12, 1151. Hamielec Α. Ε., J. Liq. Chrom., 1980, 3, (3), 381. MacRury, T. B.; McConell, M. L . , J. Appl. Polym. Sci., 1979, 24, 651. Malihi, F. B.; Kuo, C. Y.; Provder, T., J . Appl. Polym. S c i . , 1984, 29, 925. Garcia-Rubio, L. H., J. Appl. Polym. Sci., 1982, 27, 204. Garcia-Rubio, L. H.; Talatinian, Α. V.; MacGregor, J. F., Proc. Symp. on Quantitative Characterization of Plastics and Rubber, 1984, McMaster University, 37-55. Garcia-Rubio, L. H.; Ro, N.; Patel, R., Macromolecules, 1984, 17, 1998. Garcia-Rubio, L. H.; Ro, Ν., Can. J. Chem, 1985, 63(1), 253. Himmelblau, D. M., "Process Analysis by Statistical Methods"., 1970, John Wiley and Sons, New York. Sutton, P. M.; MacGregor, J . F . , Can. J. Chem. Eng., 1977, 55, 602. Reilly, P. M.; Patino-Leal, H., Technometrics, 1981, 23(3), 221.


233

234


APPENDIX I

ERRORS IN THE ESTIMATION OF POLYMER PROPERTIES


In t h i s appendix, a summary o f t h e e r r o r p r o p a g a t i o n e q u a t i o n s and o b j e c t i v e f u n c t i o n s used f o r s t a n d a r d c h a r a c t e r i z a t i o n t e c h n i q u e s a r e p r e s e n t e d . These e q u a t i o n s a r e i m p o r t a n t f o r t h e e v a l u a t i o n o f t h e e r r o r s a s s o c i a t e d w i t h s t a t i c measurements on t h e whole polymers and f o r t h e subsequent s t a t i s t i c a l comparison w i t h t h e SEC e s t i m a t e s ( s e e r e f e r e n c e s 26 a n d 27 f o r a more d e t a i l e d d i s c u s s i o n o f t h e e q u a t i o n s ) . Among t h e models most w i d e l y used t o c o r r e l a t e m e a s u r e d v a r i a b l e s and polymer p r o p e r t i e s i s t h e t r u n c a t e d power s e r i e s model

k.x , ι

(ID

where y i s t h e m e a s u r e d q u a n t i t y ( e f f l u x t i m e , i n t e n s i t y o f l i g h t etc.) and χ i s t h e independent v a r i a b l e , n o r m a l l y c o n c e n t r a t i o n . The parameters a r e g e n e r a l l y i n t e r p r e t e d i n terms of one o r s e v e r a l molecular

properties.

The f i r s t - o r d e r

effects

(k^) represent

" s p e c i f i c polymer p r o p e r t i e s " such a s , i n t r i n s i c v i s c o s i t i e s , s p e c i f i c r e f r a c t i v e index increments, f i r s t v i r i a l c o e f f i c i e n t s e t c . which a r e r e l a t e d d i r e c t l y o r i n d i r e c t l y t o primary molecular p r o p e r t i e s l i k e the number and w e i g h t , average m o l e c u l a r w e i g h t s , c h e m i c a l c o m p o s i t i o n etc. Second and h i g h e r - o r d e r e f f e c t s ( i e : i>1) a r e r e l a t e d t o i n t e r m o l e c u l a r e f f e c t s s u c h as h y d r o g e n b o n d i n g a n d e n d - t o - e n d m o l e c u l a r d i s t a n c e s . The s e c o n d - o r d e r e f f e c t s a r e o f t e n u s e d t o e x p l a i n t h e behaviour of p o l y m e r s o l u t i o n s and p o l y m e r m e l t s . The p r o b l e m s a s s o c i a t e d w i t h t h e e s t i m a t i o n o f polymer p r o p e r t i e s u s i n g e q u a t i o n 11 c a n be d i v i d e d i n t o two c l o s e l y - r e l a t e d s t a t i s t i c a l problems: t h e p r o p a g a t i o n o f e x p e r i m e n t a l e r r o r s through the m e a s u r e m e n t s and measurement e q u a t i o n s and t h e p r o b l e m o f m o d e l d i s c r i m i n a t i o n and parameter e s t i m a t i o n . In o r d e r t o a n a l y z e t h e r e s u l t s f r o m p o l y m e r c h a r a c t e r i z a t i o n measurements, e q u a t i o n 11 i s , n o r m a l l y , t r a n s f o r m e d i n t o i t s " s p e c i f i c form" i . e . :

-

Ν i-1 Σ .k.x ι

(12)

The r a t i o n a l e behind t h i s t r a n s f o r m a t i o n has been t h a t , f o r most c a s e s , t h e f i r s t two terms i n t h e s e r i e s a r e s u f f i c i e n t t o a d e q u a t e l y r e p r e s e n t t h e measurement as a f u n c t i o n o f t h e i n d e p e n d e n t variable i.e.: ζ = y = k χ

x

+ k x 2


(13)

13.


GARCIA-RUBIO

235

N o t e t h a t , t h e t r a n s f o r m a t i o n y/x i m p l i e s an e r r o r t r a n s f o r m a t i o n . T h e r e f o r e , t h e v a r i a b l e ζ may show t r e n d s w h i c h a r e due t o t h e e x p e r i m e n t a l e r r o r . I n a d d i t i o n , p l o t s of ζ (y/x) vs χ have i m p l i c i t c o r r e l a t i o n s s i n c e ζ w i l l v a r y w i t h x. T h i s c o r r e l a t i o n may a l s o a p p e a r i n t h e form of t r e n d s , which can be p a r t i c u l a r l y m i s l e a d i n g i n the assessment o f t h e number o f t e r m s r e q u i r e d by e q u a t i o n 1-1 t o represent the data. I f o y i s t h e v a r i a n c e o f t h e measurement y and σχ i s t h e v a r i a n c e o f t h e i n d e p e n d e n t v a r i a b l e x, t h e n f o r t h e case of independent measurements i n χ and y, t h e v a r i a n c e o f t h e c o r r e l a t i o n v a r i a b l e s y or ζ can be approximated by


2

2

var

(y) = o y

2

var

(z) = oy

2

(14) 2

2

j _ + σχ

2

y

(15)

from e q u a t i o n s 14 and 15, i t can be shown t h a t : var

( z ) > v a r (y)

f o r a l l χ such t h a t

0 < |x| ^ 1

var

( z ) = v a r (y)


χ = x

var

( z ) < var (y)


ay

2

y

2

0

and

(16)

|x|

E q u a t i o n s 14 t h r o u g h 16 i n d i c a t e t h a t c a r e f u l c o n s i d e r a t i o n must be g i v e n t o t h e a n a l y s i s of t h e e r r o r s t r u c t u r e and t o t h e c o n f i d e n c e i n t e r v a l s of t h e c o r r e l a t i o n v a r i a b l e s , o t h e r w i s e t r e n d s due t o e r r o r s w i l l appear as r e a l when, i n f a c t , t h e y may be c o n t a i n e d w i t h i n t h e e r r o r bands of e i t h e r y o r z. Sources of E x p e r i m e n t a l E r r o r T h e r e a r e two main s o u r c e s of e r r o r p r o p a g a t i o n i n s t a t i c measurements, e r r o r s due t o s u c c e s s i v e d i l u t i o n s and e r r o r s due t o i n i t i a l i n s t r u m e n t o f f s e t . Other e r r o r s which a r e a l s o a p p l i c a b l e t o SEC a n a l y s i s a r e d i s c u s s e d i n (J_) . T h e s e e r r o r s c a n be p r o p a g a t e d u s i n g t h e c r i t e r i a p r e s e n t e d h e r e . I f w i s the i n t i a l mass of polymer and V i s t h e amount o f s o l v e n t a d d e d t o o b t a i n t h e d e s i r e d c o n c e n t r a t i o n C i , t h e d i l u t i o n p r o c e s s can be r e p r e s e n t e d by t h e f o l l o w i n g s e t of e q u a t i o n s : x

C, = w Vi C

2

=

w V, + V

(17)

(18) 2



236

... C

= η

η

l v. J-i J

In t h e same d i l u t i o n i s u s e d constant % e r r o r ε i s used).


C

=

η

C

w

Vi.nO+ε)

η times,

( i . e . t h e same

pipette

l

with

(19)

n(1 + ε)

where η i s now t h e d i l u t i o n f a c t o r . The e r r o r a s s o c i a t e d w i t h t h e d i l u t i o n p r o c e s s i s a f u n c t i o n o f t h e i n i t i a l e r r o r i n w and t h e c u m u l a t i v e e r r o r due t o v o l u m e t r i c a d d i t i o n s . From e q u a t i o n 19 i t f o l l o w s d i r e c t l y t h a t t h e v a r i a n c e and t h e c o v a r i a n c e o f C c a n be η approximated by var

+ I

(C ) = var(w)

w

η

(110)

var(V,)

n 2

η ν^

cov

(C , C.) = f 1 W J

1 \ var(w) + f

u v j IjvJ

C

C

i \ f j ) i var(V ) x

(111)

\ W I \ WI

I t can be seen form e q u a t i o n s 19 t o 111 t h a t both t h e v a r i a n c e and t h e c o v a r i a n c e t e r m s d e c r e a s e w i t h i n c r e a s i n g d i l u t i o n (n) k e e p i n g a p p r o x i m a t e l y a c o n s t a n t e r r o r f o r s m a l l d i l u t i o n e r r o r s (ε). A s o u r c e of e r r o r s e l d o m c o n s i d e r e d i s t h e i n i t i a l o f f s e t ( ε ) . T h i s i n i t i a l b i a s may be due t o an i n s t r u m e n t o f f s e t , a c h a n g e i n t h e r e f e r e n c e v a l u e s o r t o an e r r o r i n t h e i n i t i a l c o n c e n t r a t i o n and i t causes a c o n s t a n t d i s p l a c e m e n t a l o n g t h e χ or y a x i s , i . e . : 0

y = k x x

+ k x

2

2

or f o r t h e s p e c i f i c

ζ = y-ερ = k

x

+ ε

0

(112)

model: + k x 2

(113)

χ The i n i t i a l o f f s e t can g i v e r a i s e t o d e t e r m i n i s t i c t r e n d s s u c c e s s i v e d i l u t i o n s a r e employed ( 2 7 ) . Choice

of O b j e c t i v e

whenever

Functions

S i n c e both χ and y c o n t a i n e x p e r i m e n t a l e r r o r , the parameter e s t i m a t i o n problem s h o u l d be t r e a t e d as an e r r o r i n v a r i a b l e s p r o b l e m (3,32). The o b j e c t i v e f u n c t i o n t o be m i n i m i z e d i s g i v e n by:


13.


GARCIA-RUBIO

237

M

=

(114)

I var(ε )

j = 1

jj where

η I

ε . = J

k.x. ι J

i =1


and / \ ? var(ε ) = a y = J

9ε . j 3x. J

2

Similar objective function? model ( e q u a t i o n 1 2 ) .

can

be

constructed

for

the

"specific"

APPENDIX I I CALIBRATION OF

THE

SEC

DETECTION SYSTEM

The main o b j e c t i v e s i n c a l i b r a t i n g t h e SEC d e t e c t i o n s y s t e m i n a b s o l u t e r e f r a c t i v e i n d e x and a b s o r p t i o n u n i t s a r e the e s t i m a t i o n of ν and ε a t t h e n o r m a l f l o w c o n d i t i o n s and the s t a n d a r d i z a t i o n of the measurement e r r o r s . The f i r s t s t e p i n the c a l i b r a t i o n p r o c e s s i s t h e e s t i m a t i o n o f t h e i n s t r u m e n t s c o n s t a n t s t o t r a n s f o r m the computer u n i t s i n t o absorbances and r e f r a c t i v e i n d e x u n i t s . The W a t e r s 440 UV s p e c t r o p h o t o m e t e r d i s p l a y s absorbance u n i t s . T h e r e f o r e , s t e p changes i n the i n s t r u m e n t ' s b a l a n c e and s a m p l i n g o f t h e s i g n a l p r o v i d e t h e n e c e s s a r y d a t a f o r the c a l i b r a t i o n . The e q u a t i o n s o b t a i n e d a r e : 3

ΔΑ!

= (0.1 18685 χ 10

ΔΑ

= (0.122639 x 10~ )

3

2

) . cu

x

(111)

. cu

2

(112)

where ΔΑ(ΐ) i s the a b s o r p t i o n d i f f e r e n c e b e t w e e n t h e s a m p l e and the base l i n e f o r the i t h d e t e c t o r and c u ( i ) i s the a b s o l u t e d i f f e r e n c e i n computer u n i t s (409.5 c u = 1 v o l t ) b e t w e e n t h e b a s e l i n e and the s i g n a l f o r the i t h d e t e c t o r . The c a l i b r a t i o n o f the d i f f e r e n t i a l r e f r a c t o m e t e r r e q u i r e s s o l u t i o n s of known r e f r a c t i v e i n d e x d i f f e r e n c e . Benzene/CCI s o l u t i o n s a r e known t o f o l l o w t h e i d e a l b e h a v i o u r (volumes are a d d i t i v e ) . In a d d i t i o n , t h e r e f r a c t i v e i n d e x o f t h e s e m i x t u r e s i s p r o p o r t i o n a l t o the volume f r a c t i o n of benzene, t h e r e f o r e : 4

n ?

ol

Δη

n

" B

= (n

X

s o l

B -

+

(

n

1

-V

c c l

n

CCU

^ ) = (η β

n ^ )

*

x

E


238


η

. = r e f r a c t i v e i n d e x o f the s o l u t i o n sol

n

D

= r e f r a c t i v e i n d e x o f benzene (1.5020)

D n

CCl

x

D

=

r

e

f

r

a

c

t

i

v

e

i n d e x of CC1

H

(1.4596)

= volume f r a c t i o n of benzene


Using step changes i n the c o n c e n t r a t i o n of benzene, the r e s p o n s e was c o r r e l a t e d w i t h t h e c a l c u l a t e d r e f r a c t i v e i n d e x d i f f e r e n c e (ARI) a t an a t t e n u a t i o n of 4x. 8

ARI = (2.537534 χ 10~ ) .eu

(115)

U s i n g benzene as a t r a c e r , i t i s p o s s i b l e t o e s t i m a t e t h e volume o f the i n j e c t i o n l o o p under t h e assumption t h a t a l l the m a t e r i a l i n j e c t e d l e a v e s the columns i n t h e i n t e r v a l ( t o - t f ) . The mass o f m a t e r i a l i n j e c t e d can be c a l c u l a t e d by i n t e g r a t i n g t h e c o n t e n t s of the d e t e c t o r cell. A s i m p l e mass balance g i v e s rtf

M

= C. V. = I A n ( t ) . v ( t ) dt ο ι loop to—^-^

(116)

J

I f an average v a l u e of f l o w r a t e i s used (v) e q u a t i o n 116 becomes rtf

M

ο

- C. V. = ν J A n ( t ) dt ι loop - \ ν to

(117)

κ

where C^ = c o n c e n t r a t i o n o f the i n j e c t e d

"ccO

v

=

(

M

q

= mass i n j e c t e d

V

solution

The volume V. was e s t i m a t e d u s i n g e q u a t i o n 117 and loop i n j e c t i o n s of v a r i o u s c o n c e n t r a t i o n s i s V. = 2.22 ± 0.12 loop

replicated

3

cm ,

Once t h e volume of the i n j e c t i o n l o o p i s known, the o v e r a l l e x t i n c t i o n c o e f f i c i e n t (ε) and r e f r a c t i v e i n d e x i n c r e m e n t ( v ) f o r an unknown s a m p l e c a n be c a l c u l a t e d d i r e c t l y f r o m e q u a t i o n 116 and t h e c o r r e s p o n d i n g e q u a t i o n s f o r t h e uv s p e c t r o p h o t o m e t e r a t t h e r e q u i r e d wavelengths


13.

GARCIA-RUBIO


239

rtf

i An(t).dt

(118)

t Q

C. V. ι loop

i

C.V. ι loop

w

t f

ΔΑ(t) . dt


The average f l o w r a t e was e s t i m a t e d f r o m a v e r a g i n g between c o n s e c u t i v e counter s i g n a l s . R E C E I V E D May

(119)

the time

4, 1987


elapsed

Multiple Detectors in Size Exclusion Chromatography: Signal Analysis

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