Application of the Summative-Fractionation Method to the

2 Application of the Summative-Fractionation Method to the Determination of M /M w

n

Downloaded by UNIV OF CALIFORNIA SAN DIEGO on June 2, 2015 | http://pubs.acs.org Publication Date: June 1, 1973 | doi: 10.1021/ba-1973-0125.ch002

for Narrow-Distribution Polymers FRED W. BILLMEYER, JR. and LEONARD R. SIEBERT

α

Rensselaer Polytechnic Institute, Troy, Ν. Y. 12181

The summative-fractionation method was extended to apply to narrow-distribution polymers with polydispersity ( M / M ) less than 1.12. A fractionation parameter H, previously defined and calculated for theoretical molecular weight dis tributions for normal polymers, was computed for narrow -distribution polymers. The calculations were made both with and without correction for fractionation errors, using the Flory-Huggins treatment. The method was applied to a well-characterized anionic polystyrene with Mw = 97,000, for which the polydispersity was estimated by this technique to be 1.02 (in the range 1.014-1.027, 95% confidence limits). w

n

"Precise measurement of some parameter describing the breadth of the molecular weight distribution ( M W D ) has become particularly i m portant for narrow-distribution polymers in recent years. The commercial availability ( J ) of narrow-distribution anionic polystyrenes (2, 3) has provided valuable standards for the calibration of gel permeation chro matography ( G P C ) and other characterization techniques. Although it is generally accepted that these polymers have approximately the Poisson distribution (4) of molecular weights, the measurement of the breadth of their M W D remains a difficult task. In terms of the ratio M /M , hereafter called the polydispersity, taken as the common measure of distribution breadth, these polymers usually have nominal polydispersities below 1.05 or so. Classical methods of determining polydispersity are not accurate in this range. In the sepa rate determinations of M and M , combined experimental errors are A

w

w

n

n

aPresent address: Rochester Products Division, General Motors Corp., Rochester, Ν. Y.

9 In Polymer Molecular Weight Methods; Ezrin, M.; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.


10

P O L Y M E R

M O L E C U L A R

W E I G H T

M E T H O D S

estimated to range between ± 5 % and ± 1 5 % , with the smaller figure likely too optimistic in our opinion. Fractionation by conventional solu bility-based techniques and ultracentrifugation is likewise subject to significant uncertainties in absolute results and is relatively tedious, diffi cult, or expensive to perform. While one might have thought that G P C would provide an easy solution to the problem, the errors derived from peak broadening (5) are comparable with the broadening caused by polydispersity in this range; i n fact, accurate independent knowledge of the polydispersity of these calibrating standards would greatly alleviate the difficulty of correcting G P C results (6). Only the recycle technique proposed by Waters (7) but not yet widely adopted offers promise of overcoming this problem. For these reasons, we sought a fresh approach to this problem through the application of the summative fractionation technique as described by Billmeyer and Stockmayer in 1950 (8). (In this paper we follow a conservative treatment (8), feeling that less conservative treatments (9) can easily lead to overinterpretation of the data.) This method is rela tively rapid and inexpensive and would seem to be particularly suitable for determining polydispersities in the range of interest since the experi mentally determined parameter H experiences its greatest fractional change per unit change in polydispersity as the polydispersity approaches unity. Countering this advantage are increased experimental error i n H at low polydispersities and the importance of the correction for imperfect fractionation. This paper presents a progress report on the application of the summative technique to determining the polydispersity of narrowdistribution anionic polystyrenes. Theory The Summative-Fractionation Method. To review briefly (8), i n the summative-fractionation method one performs a series of small-scale single-step fractional precipitations i n which increasing percentages of the total polymer are precipitated. The weight fractions χ and average molecular weights M of a l l the precipitates are determined. One calcu lates and plots against χ a parameter w

ζ = x(M'

x

-

M )/M w

w

= (1/M ) W

/

(M -

M )f(M)w(M)dM w

(1)

in which iv(M) is the M W D of the sample and f(M) is the fraction of the polymer of molecular weight M appearing i n the precipitate. For an infinitely sharp fraction, ζ is always zero while for a polydisperse polymer ζ is finite and positive (except at χ = 0 and χ = 1 , where ζ = 0) and is a measure of the M W D .

In Polymer Molecular Weight Methods; Ezrin, M.; Advances in Chemistry; American Chemical Society: Washington, DC, 1973.

2.

BiLLMEYER

A N D

Summative-Fractionation Method

SIEBERT

11

If fractionation were perfect, f(M) would be zero below some value of M representing the transition between supernatant and precipitate and unity for all higher values of M . It is shown (8) that the most readily determined value of the z(x) curve is its maximum, H, occurring at the point where this transition value of M is M . Adopting the use of an asterisk to denote quantities calculated with the above assumption of perfect fractionation


w

H* =

f

H = (1/M )

(M -

f°°

W

(M -

00

M )w(M)

(2)

dM

w

M )f(M)W(M)

(3)

dM

w

JM w

If the form_of w(M) is known, H * can be related explicitly to the polydispersity M /M . It was shown that the shape of the ) curve is virtually independent of the nature of the distribution, so that no further information about the M W D , beyond the polydispersity, can be obtained from the experiment. The magnitude of the error from imper fect fractionation, expressed as the difference between H * and H, was also explored and shown to be small for polydispersities of the order of 2 or greater. The Poisson Distribution. W e have extended the earlier work (8) to obtain the relation between H * and M /M for the Poisson distribution. u Me~ u ... w

n

w

n

u

»M

=

MJM

^7+1 =

n

1 +

M 2

(

i

f

(

^ - ^ p

4

)

(5)

The calculations were performed by numerical integration, after replacing the factorial by a lower end Stirling approximation Ml

=

(2%My'*(M/e)

M

(6)

which is within 1% of the true value at M = 10 and closer for all higher values of M . The results are summarized i n Table I and are plotted i n Figure 1. The calculations were carried only up to a polydispersity of 1.122, corresponding to u = 6; lower values of u are not realistic since any sample with this high polydispersity can probably be described better by a distribution other than the Poisson. Other Distributions. W e have recalculated earlier results (8) for several distributions useful for describing samples with higher polydis-


12

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M E T H O D S

persity. These include the Schulz-Zimm form of the familiar exponential distribution (JO, 11) w(M)dM Table I.

6

η > 0

ρ = (η + l)M/M

Ρ

Λλ

w

(7)

Summative-Fractionation Parameter Η for the Poisson Distribution

M /M„

H*

Ho.ooi

Ho.oi

1.001 1.002 1.003 1.005 1.008 1.010 1.014 1.019 1.024 1.031 1.045 1.083 1.099 1.122

0.0126 0.0178 0.0220 0.0280 0.0352 0.0392 0.0465 0.0546 0.0605 0.0691 0.0827 0.1071 0.1165 0.1291

0.0071 0.0105 0.0135 0.0179 0.0236 0.0270 0.0333 0.0405 0.0460 0.0541 0.0673 0.0918 0.1013 0.1139

0.0068 0.0100 0.0127 0.0167 0.0218 0.0248 0.0304 0.0368 0.0418 0.0490 0.0611 0.0840 0.0928 0.1048

w


= ^ ~^

τ

1

Γ

Figure 1. The summative-fractionation parameter H * (ideal-fractionation assumption) as a function of polydispersity Kï /Sî for the Poisson, SchulzZimm exponential, Lansing-Kraemer logarithmic-normal, and rectangular box distributions w

n


2.

BiLLMEYER A N D

siEBERT

13

Summative-Fractionation Method

and the Lansing-Kraemer logarithmic normal distribution (12) y = ( Ι / β ) In (M/M )

w(M) = Ker*

1/K

(8)

0

= &M (x) 0

1 / 2

^

2 / 4


In addition, to demonstrate the lack of dependence of H * on the form of the distribution function, we included the calculation for a highly artificial rectangular box distribution for ~M (1 - a) < M < ~M (1 + a)

w(M)

= l/(2aM )

w(M)

= 0 elsewhere

w

W

(9)

W

The results of these recalculations are listed i n Table II and plotted i n Figure 1. The curves for all the distributions studied differ insignificantly at polydispersities below, say, 1.2. Table II.

Summative-Fractionation Parameter H for Other Distribution Functions

Exponential MJM

n

1.001 1.002 1.005 1.010 1.020 1.050 1.10 1.20 1.33 1.50

H* 0.0126 0.0178 0.0281 0.0397 0.0599 0.0871 0.120 0.163 0.199 0.230

Lansing-Kraemer

M /M w

Rectangular

M /M

H*

n

w

0.0282 0.0564 0.0846 0.113 0.140 0.168 0.196 0.223 0.250

1.005 1.020 1.046 1.083 1.133 1.197 1.277 1.377 1.499

1.001 1.002 1.005 1.015 1.020 1.050 1.10 1.20 1.33 1.50

H*

n

0.015 0.020 0.030 0.052 0.060 0.0925 0.126 0.165 0.194 0.215

Fractionation E r r o r . The effect of imperfect fractionation was studied by integrating Equation 3 numerically with f(M) given by the Flory-Huggins theory f(M)

= Re /{\ aM

+

Re ) aM

(10)

where R is the ratio of the volumes of precipitate and solution, and a can be eliminated by noting that at M = M , f(M) = 0.5 when H is calculated. These calculations were performed with the Poisson distribu tion for w(M) and for R = 0.01 and R = 0.001, covering the range ex pected i n application of the summative-fractionation technique. These results are also listed i n Table I and are plotted together with values of H * for the Poisson distribution i n Figure 2. The differences, though not insignificant, are relatively small and decrease with decreasing poly dispersity. w



14

P O L Y M E R

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M E T H O D S

Figure 2. The summative-fractionation parameter H as a function of polydispersity for the Poisson distribution. H * is calculated for the ideal-fractionation assumption while Ho.ooi d H-o.oi calculated including the FloryHuggins correction for imperfect fractionation, with the volume ratio R equal to 0.001 and 0.01, respectively. an

a r e

Experimental The summative technique was applied to the fractionation of a poly styrene standard, catalog No. 500-16, molecular weight 97,200, A r R o Laboratories. This sample was prepared by Pressure Chemicals Co. and is identified by them as Polymer 4A. As measured at the Mellon Institute and reported by Pressure Chemicals it is characterized as follows: lightscattering M = 96,200 ± 2 % ; osmometry M = 97,600 ± 5% ; viscosity (Θ cyclohexane) M = 98,200 ± 3 % ; M W D J b y 20-cut fractionation, i n cluding maximum opposite error (plus for M and minus for M ) < 1.06:1. The fractionation was carried out i n general as described i n Ref. δ. The solvent was benzene (reagent, redistilled), and the precipitant was hexane with mixed isomers (Fisher H-291, A C S reagent grade, bp, 68.7°C). The fractionation flasks were roughly conical with ground glass stoppers and an 8 X 60 mm tube extending down from the narrow end into which the precipitated phase settled. Capacity was approximately 40 ml. Polymer concentration at precipitation was about 0.07 gram per liter. F i n a l equilibration was for 24 hr at 25.0 °C. The supernatant phase was drawn off with a hypodermic syringe, and the polymer was recovered from both phases by freeze drying. Molecular weights were determined from intrinsic viscosities in benzene using the relation w

n

v

w


n

2.

B I L L M E Y E R

A N D

SIEBERT

[] η

15

Summative-Fractionation Method = 8.5 Χ Wr*M ™*

(11)

v

for benzene at 25°C (3). In replicate series of experiments, the following values of Η were obtained: 0.0425, 0.0423, 0.0415. The corresponding values of polydis persity are in the range 1.014-1.027, or an average value of M /M = 1.02. w


Table III.

n

Upper and Lower 9 5 % Confidence Limits ( C . L . ) on Η and on the Polydispersity

Η

Lower C. L.

0.028 0.040 0.056 0.078 0.087

0.021 0.032 0.047 0.068 0.076

Μ,/Μ,,

Lower C. L.

1.005 1.01 1.02 1.05 1.10

1.0029 1.0065 1.014 1.038 1.078

Upper C. 0.036 0.048 0.065 0.089 0.098 Upper C. 1.0080 1.0146 1.027 1.065 1.13

Independent of these results, an error analysis was carried out using data_obtained for the reproducibility of the precipitation step, determin ing M . The results, summarized in Table III, were calculated using the exponential M W D , but should apply equally well to the Poisson distribu tion at these low values of polydispersity. The experimental results fall close to the calculated 9 5 % confidence limits for H at the appropriate level. x

Discussion The argument for using the summative-distribution method for de termining the polydispersity of narrow-distribution polymers is far from complete, but it has been demonstrated for a single sample that a result with usable precision can be obtained. In fact, no other method known to us combines this order of precision with low cost and short time of the analyses. More work is needed, however, i n several areas, including (1) improvement of experimental techniques, (2) application to samples at different levels of molecular weight (clearly, poorer precision can be expected for high molecular weight samples), and (3) comparison with recycle G P C results (7) on the same sample. Acknowledgments This research was taken in part from the M.Sc. Thesis of L . R. Siebert, Materials Division, Rensselaer Polytechnic Institute. The work was sup-


16

P O L Y M E R

M O L E C U L A R

W E I G H T

M E T H O D S

ported by the General Motors Institute and carried out i n Rensselaer's Materials Research Center, a facility supported by the National Aero nautics and Space Administration. W e thank Howard L o y for experi mental assistance, A r R o Laboratories for the donation of the standard polystyrene, and Robert T. Marcus for assistance i n programming the numerical integrations for digital computer solution.


Literature Cited 1. ArRo Laboratories, Joliet, Ill.; Pressure Chemicals Co., Pittsburgh, Pa.; Waters Associates, Inc., Framingham, Mass. 2. Altares, T., Jr., Wyman, D. P., Allen, V. R., J. Polym. Sci., Part A (1964) 2, 4533. 3. Wyman, D. P., Elyash, L. J., Frazer, W. J., J. Polym. Sci., Part A (1965) 3, 681. 4. Flory, P. J., J. Amer.Chem.Soc. (1940) 62, 1561. 5. Kelley, R. N., Billmeyer, F. W., Jr., Sep. Sci. (1970) 5, 291. 6. Duerksen, J. H., Hamilec, A. E., J. Polym. Sci., Part C (1968) 21, 83. 7. Waters, J. L., J. Polym. Sci., Part A-2 (1970) 8, 411. 8. Billmeyer, F. W., Jr., Stockmayer, W. H., J. Polym. Sci. (1950) 5, 121. 9. Coppick, S., Battista, Ο. Α., Lytton, M. R., Ind. Eng. Chem. (1950) 42, 2533; see also Battista, Ο. Α., in "Polymer Fractionation," Chapter C.4, Academic, New York, 1967. 10. Schulz, G. V., Z. Phys. Chem., Abt. Β (1939) 44, 227. 11. Zimm, Β. H., J. Chem. Phys. (1948) 16, 1099. 12. Lansing, W. D., Kraemer, E. O., J. Amer. Chem. Soc. (1935) 57, 1369. RECEIVED January 17, 1972.


Application of the Summative-Fractionation Method to the

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