282
M. WALES
SEDIMENTATION EQUILIBRIA OF POLYDISPERSE NONIDEAL SOLUTES. V USES AND LIMITATIONS OF
THE
EQUILIBRIUM ULTRACENTRIFUCIE~
M. WALES Department of Chemistry, University of Wisconsin, Madison, W k o n s i n Received February 20, 1060 INTRODUCTION
Previous work (13, 15) has shown that several reasonably reliable average molecular weights can be determined by the sedimentation equilibrium method, even when the solvent-solute system deviates from ideality. The labor involved is far less than for even a rough fractionation, and very small quantities of material are required. Furthermore, the solutions used do not have to be optically void, as with light-scattering measurements, nor is there any question of diffusion through a membrane. The time required for equilibrium is from 5 days to 2 \reeks. Such a period of time for the attainment of equilibrium has also been found by others (10). The practicability of the method is now considered in some detail. It should be mentioned here that we are eliminating from any consideration materials which have a distribution of chemical composition, since in such systems there also will be a distribution of refractive indices and partial specific volumes. Erroneous values for the several average molecular weights ~vouldbe obtained for certain copolymers, for inhomogeneously substituted polymers, and with mixed solvents. I. THE UNCERTAINTY IN THE NON-IDEALITT CORRECTION FACTOR
In the previous work (13, 15) a value for the non-ideality constant 13 (14) occurring in the expression for weight-average molecular weight H as calculated from osmotic pressure measurements for the system polystyrene-butanone (6). A careful consideration of some results obtained with a series of synthetic mistures of polystyrenes in butanone, and of results obtained previously in comparison with corrected and uncorrected light-scattering results (E), led to the conclusion that the value of B used had been slightly low. Therefore, it was decided to calculate back to the true value by trial and error, using three sedimentation equilibrium esperiments on a polystyrene of high molecular weight at varying concentrations. The calculation can be greatly simplified if the dependence with concentration of the weight-average molecular weight as calculated ideally ( B = 0) can be obtained. For ideal solutions, in the special case of a Schuls distribution (3) of molccular weights, coA(b2 - a')(@ 2)@+3 1 c, = (1) *If:+;" 1-1 ( K j - AzZ)fl+J
+
2
IThis work was carried out under contract No. SS-onr-76300 betwern the Office of Naval Research, United States Kavy Department, nnd the University of Wisconsin
USES AND LIMITATIONS OF EQUILIBRIUM ULTRACEXTRIFUGE
+
where K j = jAb2 (1 - j)Aa2 from the general relation
283
+ ( p + 2 ) / & f w . This relationship was obtained
fi = the partial specific volume of the solute, solution density, w = angular velocity, c g = initial concentration of polymer, cz = concentration a t distance z from the center of rotation, at equilibrium, b, a = distances from center of rotation to meniscus and bottom of sectorshaped sedimentation cell, p = constant in Schulz distribution function, Mw = weight-average molecular weight of the whole polymer, and /(&I) = n-eight-molecwlar weight distribution function. p =
Equation 2 can be used to calculate the molecular weight distribution function at any distance 2' in the cell and to calculate all moments and their x-derivatives in ideal solutions, at equilibrium. Space does not permit detailed presentation of this material. Using the above relations we obtain
(ds) =
-B.lf;,
where the subscript zero refers to infinite dilution
for an infinitely small speed and any molecular weight distribution; -BAM3(b2 - a2)(e2Axb1 - eZAzMa) ( d- k l=~ " ' )0 z(e.4"' - e A H o l ) 2
(3)
(4)
for a monodisperse solution and finite speed; and
for a Schulz distribution, where $0
(Kj-
1 Az2)8+1
For the conditions used in this work, equation 3 applies quite well. I t also can be shown that in this case
and
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M. WALES
Hence, on plotting l/Mkd"' asa function of concentration one obtains a straight line with slope B. However, we have a more powerful check on the correctnew of the value of B than a three-point straight line. The values of M,, M., and M.+I calculated for three M e r e n t experiments at three different concentrations, using a given B value, must be independent of concentration if the choice of B is correct. Furthermore, this value of B must agree reasonably well with equation 7 and the values of M, obtained by direct calculation and by equation 7 must agree. This whole set of calculations involves 120 experimental measurements (scale-line displacements). The first experiments were carried out with a polystyrene of M , = 300,000 and at a speed of 3000 R.P.M. A point of inflection was found in the observed concentration gradient-distance curve near the top of the cell in all of these experiments. This abnormality has been mentioned previously (15) and has since been found to be characteristic of all solutions investigated a t this centrifuge TABLE 1 CaEculation of eonstant B from sedimentption equilibria Experiment. . . . . . . . . . . . . . . . . . 1 Concentration, g./lOO ml.. . . . 0.2373 Speed, R.P.M.. . . . . . . . . . . . . . . . 3450 M,. . . . . . . . . . . . . . . . . . . . . . . . . . 155,000 M.. . . . . . . . . . . . . . . . . . . . . . . . . . 293,000 M,I . . . . . . . . . . . . . . . . . . . . . . . . 440,000 Midea'. . . . . . . . . . . . . . . . . . . . . . . 134,000
2 3 0.3491 0.4968 3480 3460 157,000 158,000 272,000
280,000
420,000 128,000
390,000 118,000
157,000' 214,000* 420.000*
1
B = 3.9 X 10-6 -/g./lOo ml. g.
M . from direct extrapolation = 154,000
* Mean of
three experiments.
speed, shifting further into the solution with increasing solute concentration and molecular weight. The disturbance disappears as the speed is changed, and is undoubtedly caused by resonance vibration in the centrifuge. Previous identification with a solvent impurity was coincidental (15). The next higher speed (3600 R.P.M.) was far too high to obtain good results with this sample because of depletion of polymer near the meniscus and overconcentration near the cell bottom (12). However, approximate agreement with the data could be obtained with these experiments for a value of B = 3.9 X in contrast to B = 2.97 X lo-' as previously calculated from osmotic pressure experiments (6). Another group of three experiments was then attempted with a sample of expected molecular weight 180,000, a t 3600 R.P.M. (mixture of 30 per cent whole I11 and 70 per cent fraction 6W; see table 3). In this case the fit of the data for B = 3.9 X was excellent. The data are summarized in table 1 and figure 1. If the apparent drift in M , and M*+1 values is real, it is evidence for a slight increase of p (the Flory-Huggins interaction constant) with molec-
USES AND LIMITATIONS OF EQUILIBRIUM ULTRACENTRIFUGE
Q'l.
0.2
0.3
c , GArnML.
0.4
285
0.5
FIG.1. Effect of solute concentration on apparent weight-average molecular weight. Curve I, M Y ' t'8. concentration, slope of line 3.9 X 1W6.Curve 11, M. calculated aa-
1
suming B = 3.9 X 10-' -/g./lOo ml. g.
TABLE 2 Comparison of osmotic and ultracentrifugal values of d:/dr*
P
p
for the system polystyrenebutanone
yz-
-
--
mwsnamx
%.
0.47 0.44
30
0.254
27
(7)
0.479t
0.62
25
Value from sedimentation equilibrium
0.471 0.474
0.71s 0.68s
28
(4) (4)
o.484t
0.485 0.490
B=~
~
2
n
52
(6) (11)
Emulsion-polymerized at 60°C. Emulsion-polymerized a t 63°C. Emulsion-polymerized a t 63'C. f 2O Bulk-polymerized a t 60°C. Bulk-polymerized at 130°C. Bulk-polymerized a t 130°C.
x lO*P$ (1/2 - p ) , where P9 = partial specific volume of polymer 9. P, = Dartial molal volume of solvent
and
~~
Units of cm. solvent/(g./lOo ml.)? t Calculated using for polystyrene = 0.906 cc./g. $ Estimated from graphs in reference 4.
ular weight or concentration. The previously mentioned group of experiments showed a trend in M , us. concentration in the opposite direction.
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M. WALES
The weight-average and z-average molecular weights were considerably lower than expected: 157,000 us. 182,000 and 272,000 us. 311,000, respectively. This sample had been recovered from butanone solution from which it had been standing, exposed to light and air, for a year since the earlier experiments (experiments 2 and 3, table 3). The polymer itself was four years old a t the time of the last experiment. A recheck of the intrinsic viscosity in benzene yielded the value 0.605 instead of the former 0.671. This accounts for the decrease in .If, to within 5 per cent. A comparison of values of the constant f i , calculated from B = 3.9 x 10-8, with other values obtained from various sources is given in table 2. The agreement is satisfactory. The difference between the sedimentation value and that obtained by osmotic pressure ( 5 ) may be caused by the variation of B with molecular weight. 11. POLYDISPERSITY AND MOLECULAR WEIGHT AVERAGES;
REPRODUCIBILITY OF THE RESULTS
I n theory, a differential molecular weight distribution curve for a polymer sample may be determined from sedimentation equilibrium data by using the solution of an integral equation. The precision of this curve, a t least for the system polystyrene-butanone, is limited only by the precision of the experiments. However, the calculation of this curve is such that it must reproduce the average molecular weights M,, W,,M,, and M,+1 (12) determined by another type of calculation and by osmotic pressure experiments. The question of calculating ill, directly from sedimentation experiments will be discussed later. In view of the fact that unattainably high precision would be needed to reproduce minor details of the distribution curve, it is suggested that all distribution curves which fit the average molecular weights (“moments”) are equally good approximations to the true distribution curve. It may, of course, be so hard to tit such a curve to the several moments that the integral equation solution may be preferable. The use of a Gram-Charlier series to fit moments* appears to be a more promising approach to this problem as compared to the method adopted by Jullander (8),who uses a generalized three-parameter curve, since the Gram-Charlier series is not restricted to three parameters. Its use appears to involve far less labor than is required for the integral equation solution (14) even by improved methods involving a Laplace inversion (1). We can assume that the uncertainties in the observed average molecular weight ratios reflect the uncertainties in the estimated molecular weight distribution curve. .4ny error in these ratios represents a change in shape of the curve. If two experiments on the same sample happened to yield different average molecular weights but the same ratios, this would mean that the shapes of the distribution curves deduced from the two experiments would be the same, with the curves shifted along the ill-axis. In table 3 the results of six experiments on synthetic mixtures of polystyrenes 2
This suggestion wa8 made by Mr. R . L. Baldwin of this laboratory.
28i
USES A S D LIMITATIONS O F EQUILIBRIUM ULTRACEXTRIFUGE
0
3
N
,'II
1 I
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M. WALES
in butanone are shown. It can be seen that the standard deviation in the ratio of M J M , is considerably lees than the sum of the deviations of the two quantities. Thm reflects the fact that in any one experiment the experimental errors (in integration over the cell, in speed measurement, in comparating reference lines, etc.) are constant over the sedimentation cell. There is hence a tendency to displace all averages by the same amount. I n general, a t least three moments should be available before any curve fitting is attempted. If we assume that the uncertainty in the curve obtained in this way is of the same order as that required to produce the standard deviation in M , / M , which was obtained experimentally, some visualization of these uncertainties can be obtained. If the error in M J M , is assumed to be 10 per cent instead of 7 per cent, as found, it turns out that this causes a negligible error in a distribution curve such that M.: M,:M.: :1:2:3. Furthermore, this leads to the conclusion that a Schulz (2) distribution curve having p = 10 and M,/M, = 1.09 can be just distinguished from a perfectly monodisperse sample, if all four moments are available. Such a curve for M , = 300,000 nevertheless has appreciable quantities of material having molecular weights below 200,000 and above 400,000. Thus, in general, very broad distributions are least sensitive to errors in the molecular weight ratios, as would be expected. 111. CALCULATION OF NUMBER-AVERAGE MOLECULAR WEIGHTS FROM SEDIMENTATION EQUILIBRIUM DATA
It can be shown that the number-average molecular weight at distance c is given by the expression
where K , is a constant of integration. It can further be shown that m f(&f)e(MA"*-MBe.d dM (9)
If, in equation 9, it is assumed that cz
for correction purposes, then
where
=
a1x2
+ a2
USES AND LIMITATIONS OF EQUILIBRIUM ULTRACENTRIFUGE
289
It is thus obvious that an unambiguous calculation of M,, involves either knowledge of f ( M ) in advance or approximation of equation 9 in terms of known integrals over f(M). The latter has not been successful, since one of the integrals necessary in this procedure is 11.1,. Different attempts, using somewhat different experimentally available functions, led to cancellation of terms in M,. In the past (9, 14, 15) attempts have been made to assign values to K. by various means. Essentially, these attempts are equivalent to extrapolating to M,, from M,, M,, etc., so they are unsatisfactory for the determination of distribution curves. The alternative left is to solve for f ( M ) by means of an integral equation (14). While this gives a distribution curve which has to fit the data, the labor involved is great. Actually, it probably would be easier to perform a series of osmotic pressure determinations for the evaluation of M , , and then to fit the four moments to the proposed curve. In the two cases where the integral equation solution was carried out, the M , value calculated from it agreed with an TABLE 4 Average molecular weights of a “pseudo-homogeneous” polymer This hypothetical material is made by combining 14 parts of a monodisperse fraction, M = 100,000, and 1 part of a material having M . = 5,000, M, = 10,000, M . = 20,000, etc. (a Lansing-Kraemer distribution (9)).
M. M,
= 44,000 = 94,000
I/
M.
=
M+I
99,500
100,000
osmotic molecular weight within 10 per cent (1, 12). In addition to the disadvantage of the labor necessary, a question of precision is involved in the integral equation solution for M,, since this procedure weights heavily parts of the molecular weight distribution curve which contribute the least to the observed concentration gradients. Actually, a value of M , calculated from an empirical curve which fits M,, M,, and M , + 1 might differ little from that obtained from an integral equation solution. In most cases, with M , = M , / 2 , this would probably not be a serious consideration. However, cases can arise where the calculation of M , is impossible for all practical purposes. IV. THE PSEUDO-HOMOQENEOUS CASE
An example of this case is given in table 4. It can be seen that from a sedimentation experiment alone, it would be considered that this material was practically monodisperse. Actually, an osmotic pressure determination (with a good membrane) would yield a number-average molecular weight of the order of 50,000. I n this case a combination of osmotic and ultracentrifugal data would be necessary for the satisfactory characterization of the material. It also should be pointed out that from light-scattering and osmotic pressure experiments
290
M. WALES
alone it would be concluded that illn:df,: M,: :1:2:3, a completely false result as far as the nature of the molecular weight distribution is concerned (unless M, were determined from dissymmetry data (5, 16)). Sedimentation velocity experiments would be very valuable in such a case. V. POLYDISPERSITY O F FRACTIONS
A “pseudo-homogeneous” polymer already has been encountered in fraction 6”W (13), although not to such an extreme degree. For this fraction all the higher average molecular weights appeared to be equal after two cuts had been taken from the constituents of high molecular weight. The weight-average molecular weight and higher averages were around 120,000 and the number average was 90,OOO. This means that the distribution has a sharp peak a t )If = 120,000 and a broad low-molecular-weight tail. More information can be deduced about the original polystyrene fraction 6W from which the pseudo-homogeneous fraction was obtained (see table 3). For fraction GW, M , = 90,000, M w = 130,000, M , = 144,000, M,+I = 200,000. It can be seen that this fraction contains at least some material of 90,000 > M 2 200,000, although possibly not very much. Furthermore,
where Mnk = numbcr-average molecular weight a t 5 = x h and M, = numberaverage molecular weight of the whole polymer. At Ck = Co, n 1 , k *Ifn. Letting dMuz/dx approach zero as a first approximation:
From this equation we find for fraction G”W that AInzat the meniscus is 75,000. Hence, the parent fraction must have actually contained material of molecular weight 75,000 > JI > 200,000. While this fraction was the worst offender in this respect, the above behavior is typical of all fractions studied, prepared either at the IJniversity of Wisconsin or at the General Laboratories of the U. S. Rubber Company (15). VI. FRACIIONATION US. ULTRACENTRIFUG.4L ANALYSIS
The above data, together with other results which have been previously obtained (15), clearly show that most polymer fractionations may be expected to yield fractions which are not very homogeneous. Therefore, the cumulative distribution curve obtained will depend on what sort of average molecular weight 8
The equations in footnote 6 of reference 14 are in error and should be ignored. At . . . are always less than M,, M z . . . .
cu, M., S M. but MUr, .Vz.
C,
=
USES .4SD LIMITATIONS OF EQUILIBRIPM ULTRACENTRIFUGE
291
is measured for the fractions, and on how much material is ‘Lmisplaced”(low material in high fractions, etc.). The differential distribution curve obtained from the cumulative curve will reflect even more any errors produced by incorrect weighting of the fractions. To obtain a reasonable reliable differential curve a very great number of fractions should be taken and carefully reprecipitated, preferably in such a manner as to remove material from both high and low ends of the fractions. This material will have to be recycled, and so forth. The ultracentrifugal or ultracentrifugal plus osmotic investigation gives directly three moments (and better than the order of magnitude of a fourth) of the differential distribution curve with much less labor than fractionation. Considering all the sources of error inherent in fractionation (2), it appears that the size-frequency curve obtained by a sedimentation equilibrium method will bein many cases, at least-as reliable as one constructed from measurements made on any but the most carefully prepared fractions. VII. THE NON-IDEALITY FACTOM
All of the preceding discussion applies to solutions where the concentration and the deviations from ideality are small enough to permit the sedimentation to depend chiefly on molecular weight. For cases where the B factor (proportional to 1/2 - p ) , the concentration, and the molecular weight are such that BMC >> 1, dc - 2 A x
z-3-
and is independent of molecular weight. For polystyrene-butanone a t 0.3 g./100 cc., B N X = 1 at M = 847,000. This means that the non-ideality is responsible on the average for about 50 per cent of the sedimentation. There is very little redistribution of the components under these conditions. I t is recognized that the upper limit of sedimentation equilibrium experiments for non-ideal solutions is about JI, = loe in favorable cases. As the B value increases, this value, of course, will decrease. As stated previously, the constant B should not vary with molecular weight and molecular weight distribution. SUMMARY
The non-ideality factor, 13, related to the interaction constant, p, was measured for the system polystyrene-butanone, by using an ultracentrifugal method. The result is consistent nith B values computed from osmotic pressure measurements on the same system. The ultracentrifugal value of B is now being used t o correct for deviations from ideal behavior in this system when calculating molecular weights from sedimentation equilibrium data. Six experiments on synthetic mixtures of separately characterized polystyrene samples gave weight-average molecular weights with a standard deviation of 5 per cent between the expected and obtained results. The values of the standard deviation of the z-average and (z 1)-average molecular weights were 8 per cent and 25 per cent, respectively. The ratios M J M , and M.+l/M. showed standard
+
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M. WALES
deviations considerably less than the sum of the standard deviations of the numerator and denominator. The possibility of calculating number-average molecular weights from sedimentation equilibrium experiments has been investigated. This calculation is possible in most cases, but the osmotic methods may be preferred. A differential molecular weight distribution curve obtained by fitting a curve to the molecular weights M,, Mm, M,, and M,+' is believed t o yield at least as good an approximation to the actual curve for a high polymer as that obtained from all but the most precise fractionations. In the case where association occurs, the distribution curve found will be characteristic of the micelles existing in solution. The author wishes to express his gratitude to Professor J. W. Williams for his interest and encouragement in this work. He also wishes to thank Professor F. T. Adler of the Department of Physics, University of Wisconsin, for helpful discussions. The assistance of Mr. D. L. Swanson in some of the experiments is also gratefully acknowledged. REFERENCES WALES,M.: Unpublished work. (2) BEALL,G.: J. Polymer Sci. 4, 483 (1949). R . F.: Ind. Eng. Chem., Anal. Ed. 18, 342 (1942). (3) BOYER, (4) DOTY,P. M., BROWNSTEIN, M., AND SCHLENER, W.: J. Phys. & Colloid Chem. 63,213 (1949). (5) EWART, R.H.:Unpublished work. (6) EWART, R . H., TINGEY, H. C., AND WALES,M.: Unpublished work. (7) GOLDBERG, A. I., HOHENSTEIN, W. P., AND MARK,H.: J. Polymer Sci. 2, 503 (1947). (8) JULLANDER, I.: Arkiv. Kemi., Mineral. Geol. 21, 1 (1945). (9) LANSING, W D . , AND KRAEMER, E. 0.: J. Am. Chem. SOC.67, 1369 (1935). (10) KICHOLS, J. B., AND BAILEY, E. D . : In Physical Methods of Organi&Chemialry, edited by Arnold Weissberger, 2nd edition, P a r t I. Interscience Publishers, New York (1949). (11) SCHICK, M. J., DOTY,P. M., AND ZIMM, B. H.: J. Am. Chem. Soo. 72, 530 (1950). T . , AND PEDERSEN, K . 0.: The Ultracentrifuge. Oxford University Press, (12) SVEDBERG, London (1940). (13) THOMPSON, J. 0.:J. Phys. & Colloid Chem. 64, 338 (1950). (14) WALES,M.: J. Phys. & Colloid Chem. 62, 235 (1948). (15) WALES,M., WILLIAMS,J. W., THOMPSON, J. O., AND EWART, R. H. J. Phys. & Colloid Chem. 62, 983 (1948). (16) ZIMM,B. H.: J. Chem. Phys. 16, 1093 (1948). (1) ADLER,F. T.,
AND
