Chromatographic Characterization of Polymers - American Chemical

6 Computer Simulation Study of Multidetector Size-Exclusion Chromatography Downloaded by EAST CAROLINA UNIV on January 2, 2018 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/ba-1995-0247.ch006

Flory-Schulz Molecular Weight Distribution Christian Jackson and Wallace W . Yau

1

Central Research and Development, DuPont, Experimental Station, Wilmington, DE 19880-0228

A computer simulation of size-exclusion chromatography-viscometry-light scattering is described. Data for polymers with a FlorySchulz molecular weight distribution (MWD) are simulated, and the features of the different detector signals are related to the molecular weight and polydispersity of the distribution. The results are compared with previously reported simulated results using a Wesslau MWD.

T H E A C C U R A C Y O F MEASUREMENTS of p o l y m e r m o l e c u l a r w e i g h t dis-

tribution ( M W D ) b y size-exclusion chromatography (SEC) can be i m p r o v e d b y the addition of a molecular-weight-sensitive detector, such as an on-line viscometer or light-scattering (LS) detector. These d e tectors measure solution properties related to molecular weight of the fractionated p o l y m e r . C o u p l i n g b o t h of these detectors i n one S E C instrument potentially offers i m p r o v e d accuracy, p r e c i s i o n , a n d d y namic range for S E C p o l y m e r conformation studies (1-5). H o w e v e r , the increased complexity of these experiments and the subsequent data handling introduce a number of problems not present i n conventional S E C (6-10). A computer simulation o f m u l t i p l e detector S E C was d e veloped to study these effects i n detail. T w o models o f the M W D were used: the Wesslau logarithm to the base 10 (log) normal M W D and the F l o r y - S c h u l z most probable M W D (11-14). T h e models are described 1

C u r r e n t address: C h e v r o n C h e m i c a l C o m p a n y , P . O . B o x 7 4 0 0 , Orange, T X 7 7 6 3 1 .

0065-2393/95/0247-0069$12.00/0 © 1995 American Chemical Society

Provder et al.; Chromatographic Characterization of Polymers Advances in Chemistry; American Chemical Society: Washington, DC, 1995.

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CHROMATOGRAPHIC CHARACTERIZATION OF POLYMERS

and the simulated data are used to illustrate the features of S E C w i t h multiple detectors.

Methodology Wesslau M W D . The model based on the Wesslau M W D has been described previously (15). The weight fraction distribution of x-mer, where χ is the degree of polymerization (DP), measured as a function of the log arithm of the degree of polymerization, is given by

Downloaded by EAST CAROLINA UNIV on January 2, 2018 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/ba-1995-0247.ch006

In (10)

h W

2 2 l n

( x / x o ) )

(

1

)

where = In

β

2

ffi

(2)

where x is the number-average D P , x is the weight-average D P , and x is the peak value. n

w

0

Flory-Sehulz M W D . The weight fraction of polymer at each degree of polymerization, x, at extent of reaction, p, described by the Flory-Schulz distribution (13, 14) is given by χ Γ(α + 1) \ x / η

n

where a is related to the molecular weight polydispersity by x

w

x

n

a+ 1 a

(4)

O n a logarithmic molecular weight scale based on S E C separation, equation 4 becomes T(a + 1) \ x / n

which corresponds to the concentration detector signal from the S E C experiment. The L S detector signal at 0° is proportional to the concentration mul tiplied by the molecular weight at each elution volume and is given by Z 0 o) = M xw (

=

0

l

x

(6)

where M is the monomer molecular weight. The viscometer signal is pro portional to the intrinsic viscosity multiplied by the concentration. The i n trinsic viscosity is given by the M a r k - H o u w i n k equation 0

[η] = K(xM r o


(7)

6.

JACKSON & YAU

71

Computer Simulation Study of SEC

The specific viscosity at each slice is then = K(xM Yw 0

Vsp

(8)

l

x

In the model there is no interdetector volume difference between the three detector signals. The SEC has a calibration curve, relating elution volume, V, to molecular weight, M , of the form M(V) = D ^ -

where D and D were given values of 15 Χ 1 0 and 0.62, respectively. The two weight-fraction M W D s are illustrated in Figure 1 in which they are plotted as a function of the logarithm of molecular weight. Both distributions shown have a number-average molecular weight of 10,000 g/ mol and a polydispersity, M / M , of 2. x


(9)

D 2 V

8

2

w

n

Results and Discussion P e a k P o s i t i o n s . T h e main results for the Wesslau M W D are sum marized for comparison w i t h the results from the F l o r y - S c h u l z M W D . T h e most notable feature is that the tracings from the three detectors (the concentration detector, L S detector, and the viscometer) are all symmetrical Gaussian distributions of equal variance. T h e only differ ences between the signals are the relative heights, corresponding to the weight-average molecular weight and intrinsic viscosity of the polymer, and the peak positions. F o r the L S detector the peak maximum, V , is shifted to lower elution volume, corresponding to higher molecular weight, than the concentration signal peak, V . T h e magnitude of the volume shift depends on the sample polydispersity and the slope of the S E C calibration curve, D , L

R

2

1000

10000

100000

Molecular weight

Figure 1. Wesslau and Flory-Schulz differential weight-fraction MWDs on a logarithmic scale, where W is the weight fraction and M is the molecular weight. Both distributions are for M„ = 10,000 g/mol and M /M = 2.0. w

n


72


v =vL

ln

R

(

* * w /

where x is weight-average degree of polymerization and x is the n u m ber-average degree of polymerization. T h e viscometer peak maximum, Vy, is also shifted to a lower elution volume. F o r a flexible polymer the shift is less than that for the L S peak. T h e magnitude of the shift is determined by the value of the M a r k - H o u w i n k exponent, a, as w e l l as the polydispersity and the calibration curve slope. T y p i c a l signal tracings for the F l o r y - S c h u l z M W D are shown i n Figure 2. T h e tracings have similar shapes but different peak positions. F r o m equation 5 it can be shown that the elution fraction at the maximum in the concentration detector signal has a D P of n

w


(10)

n )

^RI

max

=

(11)

x

xv

F r o m equation 6, the L S signal maximum corresponds to the elution fraction w i t h ι '

•*-LS max

%n

a

=

(12)

and from equation 8 the viscometer peak is at XVisemax = ( «

=

+

xw +

« +

1)/β*η

— a

(13)

Light Scattering Intensity ο Q.

Concentration

Specific Viscosity

ω φ CC u.

ο IS

Β Φ a

Elution V o l u m e

Figure 2. Signal tracings from the three detectors showing excess LS in tensity, specific viscosity, and concentration signals, for a sample with a Flory-Schulz M W D , polydispersity of 2, and a Mark-Houwink exponent of 0.725.


6.

73


JACKSON & YAU

In terms of relative peak positions, the L S intensity peak is shifted to an elution volume lower than the concentration detector peak, given by V.= V

R

-

l

n

(

(

a

+

2 ) / ( a

+

1 ) )

(14)

and the viscometer peak is also shifted b y a volume given by _ V =V Downloaded by EAST CAROLINA UNIV on January 2, 2018 | http://pubs.acs.org Publication Date: May 5, 1995 | doi: 10.1021/ba-1995-0247.ch006

V

In ((α + 1 + a)/(a + 1))

v R

n

^ (15)

These volume shifts can be rewritten i n terms of the polydispersity, F = % / x , as w

n

V

_

i

V

t

. i . w - h ( « p - i )

( 1 6 )

L>2

and v

>

=

V

y

=

In (f) - 1 ° (P(«

+ D-«)

( 1 7 )

Rearranging equations 16 and 17 gives the polydispersity i n terms of the volume shifts

2 - e~

F

~

D2(VR

F

=

I + a - ~ e

vu

~

D2(VR

Vy)

^^ 18

^^ 19

The relative shifts i n peak positions thus depend on the polydisper sity of the M W D and the slope of the S E C calibration curve. T h e shift in the viscometer peak additionally depends on the M a r k - H o u w i n k ex ponent. W h e n the sample is monodisperse, Ρ = 1, the signals from all three detectors have the same peak elution volume. If there is any mo lecular weight polydispersity i n the sample, the L S and specific viscosity peaks are shifted to lower elution volumes (higher molecular weight values). T h e amount of this shift i n the L S signal is the measure of the sample polydispersity. In the case of the viscometer, the volume shift depends additionally on the M a r k - H o u w i n k exponent. T h e difference between the viscometer volume shift and the L S volume shift is the measure of the M a r k - H o u w i n k exponent. This is the same as for the Wesslau distribution, although the magnitudes of the shifts are different. If the resolution of the chromatograph is increased, the slope of the calibration curve, D , w i l l decrease and all the volume differences w i l l 2


74



increase proportionally. T h e main information obtained b y using a molecular-weight-sensitive detector is the weight-average molecular weight or intrinsic viscosity and the sample polydispersity as shown b y the r e l ative position of the detector peaks. A s a result, it is critical that the actual physical volume difference that exists between detectors is correctly compensated before data are analyzed. P e a k S h a p e s . In the case of the Wesslau M W D , the shapes of the peaks from the three detectors are always the same. F o r the F l o r y Schulz distribution, the peak shapes are slightly different and the differences increase w i t h increasing polydispersity. A s the polydispersity increases, the L S and viscosity signals become narrower relative to the concentration detector signal and they also become less skewed. F i g u r e 3 shows the peak variance of the viscosity and L S signals relative to the concentration detector peak variance as a function of polydispersity. The concentration detector peak variance increases from 0.25 m L when the polydispersity is 1.1 to 3.65 m L w h e n the polydispersity is 3.3. T h e L S peak variance increases more slowly. T h e viscometer variance is i n between the two but closer to the L S peak behavior. F i g u r e 4 shows the relative skew of the peaks compared w i t h the refractometer, where the skew is defined as 2

2

Polydispersity

Figure 3. Variance of the viscosity and LS peaks relative to the variance of the concentration peak as a function of molecular weight polydispersity MjM . n



6.

JACKSON & YAU

75


0.4 I 1

' 2

' 3

" 4

Polydispersity

Figure 4. Skew of the viscosity and LS peaks relative to the skew of the concentration peak as a function of molecular weight polydispersity Μ„;/Μ . η

where μ and μ are the second and third moments of the peak, respec tively. T h e behavior is similar to that of the peak variance. T h e skew of the L S and viscosity peaks increases less w i t h polydispersity than the skew of the concentration signal. Figures 5 and 6 show the difference i n peak shapes i n more detail. T h e L S and concentration tracings are shown as a function of elution volume for distributions w i t h polydispersities of 1.1 (Figure 5) and 2 2

3



76

CHROMATOGRAPHIC CHARACTERIZATION O F POLYMERS

Elution Volume (mL)

Figure 6. Difference between the normalized LS and concentration (RI) signals as a function of elution volume for a MWD with polydispersity 2.

(Figure 6). T h e signals are normalized so that the two peaks have equal areas, and the difference between these two signals is also plotted. T h e signals have equal intensity at the peak of the concentration detector response, corresponding to the weight-average molecular weight. There is a maximum in the difference at the number-average molecular weight, and there is a minimum at a molecular weight equal to 4 M corresponding to the ζ + 1 average. T h e L S peak is higher than the concentration detector peak, and the difference increases w i t h increasing molecular weight. T h e viscosity peak behaves i n a similar way to the L S peak, but w i t h the differences reduced by the exponent of the M a r k - H o u w i n k equation. n

Conclusions The computer models described provide a functional simulation of S E C v i s c o m e t r y - L S analysis of linear polymers. T h e results for the F l o r y Schulz M W D are i n qualitative agreement w i t h previous results for the Wesslau M W D . B o t h models emphasize the importance of determining the correct volume offset between the concentration detector and mo lecular weight-sensitive detectors. F o r the F l o r y - S c h u l z model, the peak shape, as well as the peak elution volume, can provide information about molecular weight polydispersity. F u t u r e work w i l l extend the model to incorporate peak skew and polymer branching.

References 1. Jackson, C.; Barth, H . G. In Size Exclusion Chromatography Handbook; Wu, C. S., E d . ; Marcel Dekker: New York, 1995; p 103.



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2. Yau, W . W . Chemtracts 1990, 1, 1. 3. Lesec, J.; Volet, G. J. Liq. Chromatogr. 1990, 13, 381. 4. Yau, W . W.; Jackson, C.; Barth, H . G. In Proceedings of the International Gel Permeation Chromatography Symposium; Waters Associates: Amherst, MA, 1993. 5. Jackson, C.; Barth, H . G . ; Yau, W . W . In Proceedings of the International Gel Permeation Chromatography Symposium; Waters Associates: Amherst, MA, 1993. 6. Mourey, T. H . ; Balke, S. T. In Chromatography of Polymers: Characterization by SEC and FFF; Provder, T., E d . ; ACS Symposium Series 521; American Chemical Society: Washington, D C , 1993; p 180. 7. Balke, S. T. In Modern Methods of Polymer Characterization; Barth, H . G . , Mays, J. W., Eds.; John Wiley and Sons: New York, 1991. 8. Mourey, T. H . ; Miller, S. M . J. Liq. Chromatogr. 1990, 13, 693. 9. Jackson, C.; Barth, H . G. Trends Polym. Sci. 1994, 2, 203. 10. Yau, W . W.; Kirkland, J. J.; Bly, D . D. Modern Size Exclusion Liquid Chromatography; John Wiley and Sons: New York, 1979. 11. Wesslau, H . Makromol. Chem. 1956, 20, 111. 12. Lansing, W . D.; Kraemer, E . O. J. Am. Chem. Soc. 1935, 57, 1369. 13. Flory, P. J. Principles of Polymer Chemistry; Cornell University: Ithaca, NY, 1953. 14. Rodriguez, F. Principles of Polymer Systems; McGraw-Hill: New York, 1982. 15. Jackson, C.; Yau, W . W . J. Chromatogr. 1993, 645, 209.

RECEIVED for review January 6, 1994. A C C E P T E D revised manuscript November 12, 1994.


Chromatographic Characterization of Polymers - American Chemical

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