JOURNAL OF POLYMER SCIENCE: Polymer Physics Edition VOL. 11 (1973)

Fitting Polymer Distr ibut ion Data to a Schulz -Zimm Function

A function commonly used to represent polymer molecular weight distributions is the Schulz-Zimm function182

f ( M ) = M.-lyze--YM/r(z) (1)

for which number- and weight-average molecular weights are given, respectively, by an = z/y and M , = (z+l)/y, and r(z) is the gamma function. The above form gives the number distribution; the corresponding weight distribution function is

-

w ( M ) = M.yZt1e-~M/r(z+1) ( 2 )

Often data are presented in cumulative form: r M

W ( M ) = Jo w(M’)dM’

When z is an integer, eqs. ( 2 ) and (3) can be combined and integrated in closed form to yield

In our attempts to fit data on the molecular weight distribution of milk casein micelless to eq. (4), it rapidly became apparent that a method, more convenient than laborious curve-fitting, for estimating the parameters z and y would be highly desirable. This note describes such a method. We first estimate z by measuring the ratio of molecular weights corresponding to given cumulative weight fractions, and then find y from z and one molecular weight.

Further, we define

Let M p (0 5 p 5 1) be the molecular weight at which W(Mp) = p.

which is a function only of z. Then eq. (4) can be written

This equation is solved numerically for gp as a function of p and z, with results listed in Table I for B = 0.1, 0.25, 0.5, 0.75, and 0.9 and z between 1 and 30. We then form the ratios gplg3-p = M ~ / M L ~ , which are plotted as functions of z in Figure 1 for p = 0.1 and 0.25. From this graph, the value of z corresponding to the observed ratio of molecular weights can be determined. The parameter y is then obtained from the relation

= Zgs/MB (7)

since gp is known as a function of z from Table I. The reliability of this procedure for determining z and y depends, of course, on the

accuracy with which the Mp can be measured. In particular, it is seen from Figure 1 that the curve of MpIM1-p versus z flattens out considerably at high z, so that small errors in molecular weight ratio will be translated into large errors in z in this region. For example, if Mo., and Mo.15 can each be measured to +5%, the range in possible values of z will be from 3 to 5 at z = 4, and from 5 to 11 at z = 7. If the error in Mp is f270, z will be estimated correctly up to z = 5, will be uncertain by f l up to z = 8, and by f 2 at z = 12. Better estimates of z can be obtained from molecular weights at the ends of the distribution curve. Thus, if M o . ~ and Me.s can each be measured to

1855

@ 1973 by John Wiley & Sons, Inc.

1856 NOTES

-.. 0 5 10 15 20 25 x)

L

Fig. 1. MB/MLB as a function of z: (-) p = 0.25; (--) p = 0.1.

TABLE I Values of ga as a Function z, Calculated from Equation (6)

z 90.1 80.26 g0.s g0.75 80.8

1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30

0.532 0.551 0.582 0.608 0.630 0.649 0.665 0.679 0.691 0.702 0.712 0.720 0.728 0.736 0.742 0.748 0.754 0.760 0.764 0.769 0.778 0.785 0.792 0.798 0.804

0.961 0.864 0.845 0.842 0.844 0.847 0.851 0.855 0.858 0.862 0.865 0.868 0.871 0.874 0.877 0.879 0.882 0.884 0.886 0.888 0.891 0.895 0.898 0.900 0.903

1.678 1.337 1.224 1.168 1.134 1.112 1.096 1.084 1.074 1.067 1.061 1.056 1.051 1.048 1.044 1.042 1.039 1.037 1.035 1.033 1.030 1.028 1.026 1.024 1.022

2.693 1.960 1.703 1.569 1.485 1.426 1.384 1.350 1.324 1.302 1.284 1.268 1.255 1.243 1.232 1.223 1.215 1.207 1.200 1.194 1.183 1.174 1.165 1.158 1.152

3.890 2.662 2.227 1.998 1.855 1.756 1.682 1.624 1.578 1.541 1.509 1.482 1.458 1.438 1.420 1.403 1.389 1.375 1.363 1.352 1.333 1.316 1.301 1.288 1.277

NOTES 1857

f5’%, z will be estimated correctly up to z = 4, will be uncertain by f l up to z = 7 and by f 2 up to z = 10. However, accurate data at the ends of the distribution are difficult to obtain.

This discussion neglects the problems of molecular weight distribution within fractions, which has been discussed by Kotliar.‘ However, for the relatively narrow distributions discussed here, Kotliar’s results indicate that intrafraction polydispersity should not provide a serious hindrance to the determination of distribution parameters.

In conclusion, it is evident tht, given realistic error limits in the determination of molecular weights corresponding to given points in the cumulative distribution, the method described here will not by itself give precise values for z and y . However, Table I and Figure 1 should assist in obtaining a rapid initial estimate of these parameters, which can be refined by subsequent detailed curve fitting.

This research was supported in part by grants from the ACS Petroleum Research Fund and from the National Institutes of Health.

References

1. G. V. Schulz, 2. Physilc. Chem., B43, 25 (1939). 2. B. H. Zimm, J . Chem. Phys., 16, 1099 (1948). 3. J. A. Benbasat and V. A. Bloomfield, J . Polym. Sei., Polym. Physics Ed., 10,

4. A. M. Kotliar, J . Pdym. Sci. A, 2, 1373 (1964). 2475 (1972).

JOHN WELCE VICTOR A. BLOOMFIELD

Department of Biochemistry University of Minnesota St. Paul, Minnesota 55101

Received April 10, 1973