H. UTIYAMA,N. TAGATA, AND M. KURATA
1448
Determination of Molecular Weight and Second Virial Coefficient of Polydisperse Nonideal Polymer Solutions by the Sedimentation Equilibrium Method by Hiroyasu Utiyama, Nobuo Tagata,’ and Michio Kurata Department of Industrial Chemistry, Kyoto University, Sakyo-ku, Kyoto, and Institute for Chemical Research, Kyoto University, U j i , Kyoto, Japan (Received October 0 , 1 0 6 8 )
Sedimentation equilibrium experiments have been carried out in order to establish the experimental procedure and the method to determine the weight-average molecular weight and the light-scattering second virial coefficient on polydisperse nonideal polymer solutions. The experimental condition was chosen so that the most accurate results may be obtained under the requirement that the equilibrium is established within 20 hr with B sample of molecular weight of several millions. Two samples of monodisperse polystyrene SA (mol w t = 1.97 X loe) and Sg (mol wt = 0.162 X loe) and four polydisperse samples made up of SA and Sg were examined in 2-butanone at 25 0’. With the Rayleigh interference optical system and the column height of 0.15 cm, the weight-average molecular weight is obtained correct to within f201, irrespective of the degree of polydispersity. The accuracy of the determination of the light-scattering second virial coefficient, on the other hand, depends very much on the detail of the molecular weight distribution. If the molecular weight of the minor component is very large compared with that of the major component, the experimental error amounts to as large as 10%. The uncertainty becomes greater as the second virial coefficient increases. I t has been found that for evaluating the apparent molecular weight in the limit that X approaches zero, the plot of Mapp-lus. X2 is the most appropriate. I t has also been concluded that the weight-average molecular weight and the light-scattering second virial coefficient for polymers with unkiiown molecular weight distribution should be evaluated from the plot of lim Map,’ US. CO, instead of the plot of lim Mapp us. CO. x -rO
Introduction Although sufficient theoretical relations have been worked out and a variety of techniques have been developed for the sedimentation equilibrium method,2-s a standard experimental method for polymer solutions is not yet available. It is important to establish such a method in order to use the sedimentation equilibrium method, rather than light-scattering photometry and osmometry, when the molecular weight or the second virial coefficient can be determined more accurately with a simpler procedure. It is also important to remember that the sedimentation equilibrium method has such distinct features as (1) the amount of material required is very small; (2) the experiment can be carried out routinely once the optical system of the ultracentrifuge is correctly aligned; (3) no essential difficulty is encountered on aqueous systems; and (4) samples of widely different molecular weight can be studied. The present paper is concerned with the establishment of a standard experimental method for the sedimentation equilibrium. It is proposed for the widest application of the method that the method is applicable to both polydisperse and monodisperse nonideal polymer solutions of molecular weights ranging from several tens of thousands to several millions. For such purposes, two monodisperse polymers having widely different molecular weight and some polydisperse polymers prepared by blending them at various weight ratios may be most appropriate as the The Journal of Physical Chemistry
x -0
samples to be investigated. The molecular weight distribution of these samples can be defined accurately, and the evaluated results such as the weight-average molecular weights may be compared with each other. Moreover, we can prepare at will a polydisperse sample with a desired molecular weight. The obtained lightscattering second virial coefficients for the blended samples may be examined for their accuracy by considering their dependence on composition. I n carrying out the sedimentation equilibrium experiments special attention should be paid to (1) the effects of omitting an inert liquid a t the bottom; (2) the smallest possible column height of solution without sacrificing the experimental accuracy; (3) the appropriate procedure for reducing the time required for the attainment of sedimentation equilibrium; and (4) the method for analyzing the experimental data to obtain the weight-average molecular weight and the lightscattering second virial coefficient. (1) On leave of absence from the Research Center, Japan Synthetic Rubber Co., Kawasaki, Kanagawa, Japan. (2) J. W. Williams, K. E. Van Holde, R. L. Baldwin, and H. Fujita. Chem. Rea., 5 8 , 715 (1958). (3) H. Fujita, “Mathematical Theory of Sedimentation Analysis,” Academic Press, Inc., New York. N. Y.,1962,Chapter V. (4) K. E. Van Holde and R. L. Baldwin. J . Phys. Chem., 6 2 , 734 (1958). (5) D.A. Yphantis, Ann. N . Y.Acad. Sci., 8 8 , 586 (1960). ( 6 ) P. E. Hexner, L. E. Radford. and J. W. Beams, Proc. Nat. Acad. Sc6. U.S . , 47, 1848 (1961). (7) F. E. LaBar, ibid., 54, 31 (1965). (8) Q. M. GrifBth, Anal. Biochem., 19, 243 (1967).
SEDIMENTATION EQUILIBRIUM IN POLYMER SOLUTIONS Experimental Section Polymer. Two samples of monodisperse polystyrene were purchased from Pilot Chemical Co. , Pittsburgh, Pa. They were designated as l a and 14a by the manufacturer, and their catalog molecular weights were 160,000 and 1,800,000, respectively. Sample l a was purified several times by dissolving in 2-butanone and reprecipitating into ethanol. The purified sample so obtained is designated here as SB. Because sample 14a was found to have a little broader distribution of molecular weight (M,,./M, N 1.2), it was subject to column fractionation with 2-butanone as the solvent and ethanol as the precipitant, and the middle fraction of about 63% of the total was collected for the present study. It was freeze-dried from a 2butanone solution. This sample is designated as SA. Its homogeneity was checked on gel permeation chromatograms taken with a GPC Model 200, Waters Associate, Inc. Four polydisperse samples were prepared by blending SA and Sg a t four different weight ratios. They are designated here by the symbol M,, with subscript x representing the approximate weight fraction of the sample SA in each of them. The exact values of the weight fraction of SA were 0.802, 0.583, 0.405, and 0.190. Solvent. Reagent grade 2-butanone (Nakarai Chemical Co., Kyoto) was purified by washing five times with saturated aqueous solution of potassium carbonate, drying over anhydrous sodium sulfate, and before use, it was distilled in a column. Its ns0Dwas 1.37363, which is compared to the literature value 1.37374.9 Preparation of Solution and Determination of Concentration. The original solution of SA or S B was prepared by mixing thoroughly dried polymer with the solvent in a sealed tube and keeping the mixture overnight a t 60‘ for complete dissolution. Its concentration in grams per milliliter was calculated as the weight of the polymer times the density of solvent (pzSo = 0.79945‘O) divided by the total weight of the mixture. The solutions of different concentrations were made up by diluting the original solutions volumetrically. Ultracentrifugation. Sedimentation equilibrium experiments were carried out in a Beckman Spinco Model E ultracentrifuge (No. 1045) equipped with the regular accessories supplied by the company. The Rayleigh interference optical system was aligned according to the procedure described by LaBar and Baldwin.1’ The procedure by Gropper12 was also taken into consideration. The schlieren light source was used with the narrowest slit width by rotating by 90’. The front-to-rear position of the light source was adjusted considering the effect of the cell tilt.la The camera lens was focused a t the center of the 12-mm cell. The magnification factor was 2.178, and the
1449 vertical spacing of the fringes on the plate was 0.0271 em. Rayleigh fringe patterns a t sedimentation equilibrium were photographed on Fuji spectroscopic plates of the panchromatic process type, and for the synthetic boundary experiments, Fuji neopan SSS plates were used. The fringe patterns were read using a Nikon profile projector Model L-16 equipped with the zeroadjusting collars and a 1OX objective lens. Determinations of both the vertical and horizontal distances between fringes or that of the distance between the meniscus and the reference wire were reproducible to better than &0.0005 cm on the photographic plates. The readings of the fringe displacement on Fuji neopan SSS plates, however, were reproducible only to =k0.0008 em. The necessary experimental accuracy can be secured by using a somewhat more concentrated polymer solution. For the sake of economy we employed the multicell operation using the An-G rotor for the speed above 3000 rpm and the An-J rotor for the speed below it, with the laterally wedged double-sector cells. The thickness of the double-sector centerpiece in the direction of the optical path was measured with an appropriate micrometer, and the necessary correction for the fringe spacing was applied. The temperature of the rotor was maintained a t 25.0’ by the RTIC temperature control system. Sedimentation Equilibrium Experiment. The column height to be used in sedimentation equilibrium experiments is essentially determined by the time accessible to the experiment and the experimental error allowable for the measurement. The time required for the attainment of equilibrium may be reduced by a factor of a t least by using t,he overspeeding technique described by Hexner, et To determine the distance between the meniscus and the bottom to an accuracy better than &0.3%, the column height cannot be shorter than 0.15 em. With this column height, the time required for the solute of molecular weight 2 X lo6 (the sedimentation coefficient is approximately 33 S for the system polystyrene-2-butanonej to attain the sedimentation equilibrium becomes a t least 28 hr according to the equation of Van Holde and Baldwin.4 Once the column height is determined, the choice of the operating speed of rotation depends only on the experimental accuracy required for the evaluation of the difference in concentration between the bottom and (9) R. R. Dreisbach and R. A. Martin, I n d . E n g . Chem., 41, 2878 (1949). (10) J. A. Riddick and E. E . Toops, Jr., “Organic Solvents,” Interscience Publishers, Inc.. New York, N. Y., 1955. (11) F. E. LaBar and R . L. Baldwin, J . Phys. Chem., 66, 1952 (1902). (12) L. Gropper. A n a l . Biochem., 7, 401 (1904). (18) L. Gropper, (bid., 6, 170 (1963). (14) D. A. Albright and J. W. Williams, J . Phys. Chem., 7 1 , 2780 (1967). Volume Yb, Number 6 May 1969
9450
H. UTIYAMA, N. TAGATA, AND M. KURATA
the meniscus. Setting the experimental error for ( c b - c,) less than &0.3%, the displacement of a fringe between the bottom and the meniscus should be greater than 0.15 cm. Therefore the following relation, which holds for this fringe displacement for polydisperse ideal solutions, is useful to estimate the operating speed of rotation 0.15/K~o= M J
(1)
Here K is the fringe displacement per unit concentration expressed in grams per milliliter, which was 118 for the system polystyrene-2-butanone. The parameter X is defined later by eq 4. The attainment of sedimentation equilibrium was assured by plotting against time the horizontal distance between two fringes separated as distant as possible on the photograph. The centrifugation was not stopped before the distance was observed constant at least for 2 hr. An inert liquid was not inserted a t the cell bottom. The first photograph was taken immediately after the meniscus was formed to check the occurrence of the leak of liquid. Several of the final photograph were taken with different exposure ‘times to select the best one for reading the fringe patterns. The synthetic boundary experiments were carried out with a capillary type double-sector synthetic boundary cell. I n all the cases, the base line was determined by an appropriate blank experiment which was carried out before or after the run under otherwise the same experimental condition. The buoyancy factor (1 - Vopo) for polystyrene in 2-butanone was calculated to be 0.275 by using the figures 0.908 ml/g and 0.799 g/ml for io and po, respectively.
Results and Discussion The Detefmination of the Bottom Position
r b . It is usually recommended that a “false bottom” be synthesized using an immiscible transparent heavy liquid in order to help the determination of rb and to provide a sector-shaped bottom interface. Such a bottom liquid, however, is not generally available for organic solutions, and for a routine equilibrium experiment we prefer to omit the “bottom liquid,” if no serious error is introduced. The following is the empirical procedure we employed to determine the position r b when the “bottom liquid” is not available. When the solution is monodisperse and ideal, the apparent molecular weight Map,,defined by Mapp
= (cb
- ca)/cOX
(2)
can be estimated also from the plot of In e os. r2 with Map,
=
2RT (1
u-0 p0)wz
I
d In c ( r ) dr2
Here co is the solute concentration of the solution before The Journal of Physical Chemistru
02oy
, 505
, 51.0
, -j 515
r2 (cm2) Figure 1. Molecular weight determjnstion of monodisperse polystyrene SO2 from sedimentation equilibrium experiments. The ordinate gives the logarithm of the solute concentration (arbitrary unit), and the abscissa represents the square of the distance (square centimeters) from the axis of rotation. The two filled circles correspond to the data a t meniscus and bottom. The initial concentration was 0.262 g/dl, and the solution was sedimented for 21 hr at 6995 rpm in cyclohexane a t 35.0”, which is the 0 temperature for the system.
centrifugation and X is a parameter defined by = (1 - fiopo)w2(Tb2- r S z ) / 2 R T
(4)
with io being the partial specific volume of the solute a t infinite dilution, po the density of solvent, w the angular speed of the rotor, r , and r b the radial distances from the center of rotation to the meniscus and bottom of the solution column, R the gas constant, and T the absolute temperature. Therefore we can determine the right position of the bottom by trials and errors so that the molecular weights evaluated by the two methods coincide with each other. Once the bottom position is determined, we may use it as long as the same cell assembly is used. The correctness of the bottom position so obtained may be checked by comparing the molecular weights of another sample obtained by the above two methods. For this purpose we chose a monodisperse polystyrene in cyclohexane a t 35.0°, which is the 0 temperature for the system. This polymer sample16 was prepared by the ionic polymerization and purified by repeated reprecipitation. Four solutions ranging in concentration from 0.174 to 0.523 g/dl were used for the experiments. A typical result is shown in Figure 1, where In c is plotted against r2. The molecular weight estimated from the slope of this plot and that (16) Generous gift from Mr. M. Fukuda of our group. lecular weight was 14.7 X 104.
The mo-
1451
SEDIMENTATION EQUILIBRIUM IN POLYMER SOLUTIONS Table I: The Molecular Weights and the Second
0.0
Virial Coefficients Determined from the Sedimentation Equilibrium Experiments on the Monodisperse Polystyrene SAand Sg in 2-Butanone a t 25.0"
70
Molecular Sample
' X
1.97f0.04 0.162 f0.001
SA
(I)
Second virial coefflcient B X lo'
weight, M X
SB
1.70f0.08 2.71 f 0 . 1 3
60
c .
Q
-z
5
+
1.5
I.o
0.5
Figure 2. Molecular weight determination of monodisperse polystyrene samples SAand SBfrom sedimentation equilibrium experiments in 2-butanone at 25.0". The reciprocal of the apparent molecular weights are plotted against the average of the concentrations at the meniscus and at. the bottom. Symbols of the plotted points represent the column height used in the experiments as shown in the figure.
calculated from the values for ca and cb are in agreement within 6 2 % . Therefore we conclude that no bottom liquid need be inserted in agreement with the conclusion obtained by other investigator^.'^ Monodisperse Polystyrenes SAand SB. It has been shown by Williams, et u Z . , ~ that for monodisperse solutions the data can be best analyzed in terms of an equation of the form Mapp-'
means that eq 5 is applicable irrespective of the values of w and the height of solution column (rb - r a ) . Figure 2 represents plots of Mapp-' vs. (c, Cb)/2 for the samples SAand Sg in butanone a t 25'. Various column heights ranging from 1.0 to 3.5 mm were used to obtain plotted points. For either sample the data points fall on a straight line with a deviation of &2% in agreement with the prediction of eq 5 . The accuracy of the determination of the apparent molecular weight is not seriously affected even if we decrease the column height to as low as 1.0 mm (see the data for SA in Figure 2 ) . The values of molecular weight and second virial coefficient deduced from these straight lines are summarized in Table I. Dependence of the Apparent Molecular Weight on X for Polydisperse Polymers. It must be recognized that the treatment of the data for polydisperse solutions is not as straightforward as in light-scattering photometry and osmometry. This disadvantage stems from the fact that here we have to deal with a solution in which the spatial distribution of solute molecules is not uniform. The way in which the solute is redistributed depends on such factors as the speed of the rotor, the polydispersity of the solute component, and the thermodynamic nonideality of the solution. Therefore the light-scattering second virial coefficient B L is~ obtained only when the experimental results are extrapolated in an appropriate way to the condition where either the speed of the rotor w or the column height (rb - r , ) approaches zero. The appropriate method of extrapolation can be derived from the equation developed by F ~ j i t a ,in~ which M app-l is expressed in powers of co as
=
M-'
+B + /2 + higher term (Ca
cb)
in c , and cb
(5)
Here M is the molecular weight of the solute, and B is twice the second virial coefficient of the solution. The higher terms in eq 5 are associated with the third and higher virial coefficients of the solution and may be ignored unless the initial concentration is too high and the solvent is too good for the solute considered. Thus for monodisperse solutions plots of MaPp-' vs. (ea cb)/2 may be expected to follow a straight line over a fairly wide range of the abscissa, and we may take the ordinate intercept and slope of the line as equal to M-l and B, respectively. It is important to note that eq 5 does not contain the parameter X explicitly. This
Mapp-' = M,-'
+ f(A)RLsco+ higher terms in co
(6)
where f (A) is given by f(X) = 1
+ (XM,)2/12
+ [ ( M ~ + I / M-~2(Ma+1Mz+z)~/M,4] )~
+
X (AM,)4/720
+-
(7)
I n these equations, M,, M,, M.+I, and M,+z are the weight, x , x 1, and z 2 are average molecular weights of the solute, and BLSis twice the light-scattering second virial coefficient of the system. Equation 6 indicates that plots of MaPp-lus. co give M , from their
+
+
Volume 75, Number 6 May 1969
H. UTIYAMA, N. TAGATA, AND M. KURATA
1452
Table 11: Various Average Molecular Weights of the Blended Polymer Samples Calculated by Using the Molecular Weights of SAand S g given in Table I and the Composition. The Parameter X Was Estimated from E q 7 MW
x
Sample
Mi 10-8
Calculated values of the second term of the function f(X).
+
3.5
I
I
I
I
I
I
2.0:
2
4
6
8
10
I
3rd termb
1.97 1.97 1.97 1.97
0.143 0,236 0.388 0.882
-0.0004 -0.015 -0.054 -0.624
Calculated value of the third term of the function f(X).
ordinate intercept but their limiting slope a t co = 0 differs from B L by ~ the factor f(X). Because the values of M , and higher average molecular weights are not available in advance in general, f ( X ) cannot be evaluated and so is BLSfrom the data for a single A. This difficulty can be circumvented if the data are obtained for a series of different X a t each co. In this case, MaPp-lfor each fixed co can be plotted against X2, and, according to eq 6, the resulting plots should the ordinate intercept when exgive MW-l B L ~asc ~ trapolated linearly to X2 = 0 in the region of small values of X2. The intercept value is then plotted against CO, and we can take the intercept and slope of the plot at co = 0 as the desired values for MW-l and BLS,respectively. A recent paper of Albright and WilliamsI4 appears to indicate that the experimental condition f ( X ) N 1 can easily be realized for polydisperse polymeric solutes by lowering the speed of rotation. However, we wish to point out that this technique may introduce some disadvantage from the following consideration. I
2nd term5
x
1.97 1.96 1.95 1.92 b
10-8
Ms + 2
Mi + I 10-0
x
1.93 1.87 1.77 1.50
1.61 1.21 0.89 0.50
Mo.8 Mo.8 MO.4 Mo.2 a
x
10-8
If we make use of eq 1 with Kco = 0.136, the paranieter X and the second term of eq 7 are estimated approximately from the equations X = l.lO/Mw;
2nd term = 0.101(M,/Mw)2 (8)
Therefore, the parameter X for the oligostyrene examined On the by Albright and Williams is 1.9 X contrary, the range of X used by them was between 3 X 10-6 and 9 X which is one-sixth to ca. one-half as large as the magnitude estimated above. For these X and the M , they gave, the second term of f(X) amounts to smaller than 0.05, which is small enough to be neglected in agreement with their conclusion. However, these X correspond to cb/ca approximately from 1.19 to 1.69, which are too close to unity to be determined with sufficient accuracy. To show the magnitude of the terms in the series for f (A) for the polymer blends under study, the second and third terms of eq 7 were estimated using the calculated values of the average molecular weights (Table 11). It is obvious that for all the samples the appropriate correction with respect to X is required, but f ( X ) may
I
A2 x 10''
Figure 3. Dependence of the reciprocal of the apparent molecular weight on the square of the parameter X for the blended polystyrene sample M0.z in 2-butanone at 25.0'. The initial solute concentrations are given in the figure. The lines through the experimental points are drawn to extrapolate l/MsDDto vanishing X*. The speed of rotation was varied to cover the necessary range of Xa from 2994 to 4908 rpm. The Journal of Phyeical Chemistry
I
0
w
a? Co
(gJd1.1
Figure 4. Plot of the reciprocal of the apparent molecular weight for Xa = 5 X 10-12 (filled circles) and for Xa = 0 (open circles) us. the initial solute concentration on the blended polystyrene sample M0.z. The data are taken from Figure 3.
1453
SEDIMENTATION EQUILIBRIUM IN POLYMER SOLUTIONS be estimated in good approximation by neglecting the terms higher than the second except for the sample M0.z.
The following is a summary of the above considerations. (1) The parameter X is smaller for a sample of larger weight-average molecular weight. (2) f (A) is larger if a small amount of contaminating component has a molecular weight much larger than that of the principal component, because the ratio (M , / M w ) and hence the second term of f(X) are both increased. (3) If the weight fraction of the component of smaller molecular weight is less than or comparable to that of the component of larger molecular weight, f (A) can be given in good approximation by the first two terms. I n obtaining data for a series of different X a t each co, the speed of rotation, not the column height, may be varied for saving the experimental time and for the convenience of the experimental procedure. Since X2 is proportional to w4, it is sufficient if the experiments are performed at w ranging from 0.8 to 1.5 times as large as that calculated from eq 1. The corresponding change of cb/ca is then from 2.8 to 4.4. The experimental results for the sample MW are illustrated in Figure 3, where l/M,,,, not Map, itself, is plotted against X2 to facilitate the extrapolation to X2 = 0. It is noted that the plotted points for each co follow a curve concave downward with a tendency being more pronounced for larger co. This curvature may be due either to the fact that the series for f ( A ) is slowly convergent for Mo.2 and the terms are of the alternating sign or to the contribution from higher terms in CO, or to both. I n any case, the extrapolation of MaPp-lto X2 = 0 may become more difficult when ( M , / M w ) or B L g is larger.
( I + 8Mtp/12)C. Figure 5. Concentration dependence of the apparent molecular weight on the blended polystyrene sample M0.8in 2-butanone at 25.0". The initial solute concentration eo is multiplied by (1 X2M?/12) in order to compensate the effect of X on MsDp. The z-average molecular weight M. was calculat,edby using the molecular weights of the two monodisperse polystyrene SA and SB and the composition. The solutions were centrifuged for 24 hr at 2233 rpm.
+
I 0
I
I
I
02
04
cL6
11 + XM ';
1 I
1121 CO
Figure 6. Linear dependence of the reciprocal of the apparent molecular weight on (1 X2MS2/12)co. The data are taken from Figure 5.
+
The other important point worth mentioning here is that the weight-average molecular weight can be estimated correctly in the usual manner if we carry out the sedimentation equilibrium experiment a t a fixed value of A, 8s shown in Figure 4. This procedure is equivalent to using the same column height and the same speed of rotation for different initial concentration. The difficulty in the extrapolation of l / M , , to co = 0 reported by Mandelkern, et al.,16 for a highmolecular-weight sample of polyisobutylene in isooctane is simply due to the fact that the value of MX used in their experiment was too large. Concentration Dependence of Apparent Molecular Weight. I n a recent publication Albright and william^'^ have used the apparent molecular weight rather than its reciprocal to extrapolate to zero concentration for estimating the molecular weight and the second virial coefficient. They considered that this extrapolation procedure is easier and not limited to systems for which the nonideality correction is small. We cannot accept their proposal for the following reasons. We limit our consideration here to Mapp(cO) in the limit that X approaches zero. It is necessary to calculate the co2 term in the series expansion for Ma,, in order to know which of the two functions Mapp(cO) and l/M,,,(co) varies linearly with co over a wider range of CO, but we do not need to carry out the complicated calculation because we can make use of the theoretical expression for the reduced scattered intensity for polymer solutions." The quantity R ( 9 , c ) / k c in the limit that 9 approaches zero and (16) L. Mandelkern, L. C. Williams, and 8. G . Weissberg, J. Phys. Chem., 61, 271 (1957). (17) B. H.Zimm, J. Chem. Phys., 16, 1099 (1948). Volume 79,Number 6 May 1969
1454
H. UTIYAMA, N. TAGATA, AND M. KURATA
the apparent molecular weight in the limit of vanishing Therefore, from what we know from the light-scattering study, we may expect that the reciprocal plot is better for the present purpose for monodisperse polymer solutions. This expectation will be equally applicable to polydisperse polymer solutions. Next we compare the ways in which the experimental points follow both plots. The experimental results on the sample M o . are ~ shown in Figures 5 and 6, where co is multiplied by (1 A2M,2/12) to compensate for the effect of f ( A ) . The plotted points in Figure 5 follow a curve which is concave upward, and we cannot find the region, even a t small concentrations, where M changes linearly with CO. Hence, in terms of this plot, neither the apparent molecular weight a t zero concentration nor the limiting tangent can be evaluated without uncertainty. I n the reciprocal plot (Figure 6), on the other hand, the plotted points fall on a straight line a t least in the concentration range studied, and the extrapolated M apB, i.e., the weight-average molecuIa,r weight, stands in good agreement with the calculated molecular weight (see Table 111), the deviation being less than &2%. Figure 7 shows how BLSevaluated from the slope of l/MsPp us. co(1 A2M,2/12) varies with the solute composition. The vertical segments drawn across the experimental points indicate the estimated range of experimenta.1 uncertainty. All available theories for the second virial coefficient of blended polymer solutions18J9indicate that the light-scattering second virial coefficient should be almost independent of the solute composition between 0.5 and 1.0, and it is seen that this prediction is well borne out by the present data. for the samples SA and SB Finally the values of B I , ~ are in fair agreement with those obtained by Outer, et C L Z . , ~ ~from light-scattering measurements.
3.0
X are considered to be physically the same.
w
s X
Y
m
2.0
+
+
~
~~
Table 111: The Weight-Average Molecular Weights and the Light-Scattering Second Virial Coefficients for the Blended Polystyrene Samples Obtained from Sedimentation Equilibrium Experiments X
--------A!f,
Sample
SA Mo.8 M0.s
MO.4
Mo.2
SB
Obsd
CalcdQ
1.97 1.64 1.27 0.87 0.501 0.162
1.61 1.21 0.89 0.50
...
...
BLS x
io4
1.70f0.08 1.76 f 0 . 0 6 1.67 f 0 . 1 2 1.77 f0.12 2.32 f0.20 2.71 f 0 . 1 3
The weight-average molecular weight calculated by using the molecular weight of SA and SB. 5
The Journal of Ph~laicalChsmistrt~
I .o
0
02
0.4
0.6
Q8
1.0
WEIGHT FRACTION OF SA Figure 7. Dependence of twice the light-scattering second virial coefficient BLS for the blended polymer solution on the solut,e composition, B L on ~ the ordinate axis was obtained from sedimentation equilibrium experiments. Mixture of the monodisperse polystyrene SA and Sg was used as the polymer sample and 2-butanone as solvent. The abscissa represents weight fraction of SA, The solution was centrifuged at 25.0'.
Conclusion From the experimental results presented above, the following conclusion may be drawn. When we carry out the sedimentation equilibrium experiment on polydisperse nonideal polymer solutions using the procedure given in the present paper, the weightaverage molecular weight of the dissolved polymer can be obtained within experimental error of &2% irrespective of the degree of polydispersity. The error involved in the determination of the second virial coefficient, on the other hand, is very much affected by the type of molecular weight distribution. If the molecular weight of the minor component is very large compared with that of the major component, it may amount to as large as lo%, and will be even greater for a system of larger second virial coefficient. However, in many cases with usual polydisperse polymers the error will not be greater than 5%.
Acknowledgment. We should like to thank Dr. Hiroshi Fujita for many helpful suggestions both during the course of this work and in the preparation of the manuscript. We thank Mr. Takashi Kageyama of the Institute for his technical assistance. (18) H. Yamakawa and M. Kurata, J . Chem. Phys., 32,1852 (1960). (19) E.F. Casassa, Polymer, 3, 625 (1962). (20) Pt Outer, C. I. Cam, and B. H. Zimm, J. Chcm. Phys.. 18, 880 (1950).
