1050
H. W. OSTERHOUDT AND J. W. WILLIAMS
Sedimentation Equilibria in Polydisperse Pseudo-Ideal Solutions and at Low Centrifugal Fields
by H. W. Osterhoudt and J. W. Williams Laboratory of Physical Chemietry, University of Wisconsin, Madison, Wisconsin
(Received .\ionember 18,1964)
Equations which provide an accurate description of the concentration gradients for a polydisperse component a t sedimentation equilibrium under pseudo-ideal conditions are presented in part I. Consideration of the centrifugal field as a quantity to be varied from experiment to experiment suggests a new method for measuring the “weight,” “z,” and “Z 1” average molecular weights. The quantities M , and M znow appear as the slopes of linear plots of experimentally measurable quantities us. the magnitude of the field. The utility of these equations was tested experimentally by the performance of two series of sedimentation equilibrium experiments a t several low rotor speeds with two different polystyrenes in solution a t the Flory temperature. The data confirm the predictions as to the shapes and intercepts of the several plots. The application of the Schulz function to describe the actual differential molecular weight distribution of the two polystyrenes is also considered. I n part 11,an examination of the equations which describe the sedimentation equilibrium of a polydisperse polymer in a “good solvent” indicates that series of experiments performed at several low centrifugal fields and at several low concentrations ought to permit the evaluation of the light scattering second virial coefficient.
+
Introduction The essential working equations for the sedimentation equilibrium experiment derive from classical thermodynamics. As they pertain to the ultracentrifuge, contributions of Rinde,3 and Lansing and I(raemer4 provide a substantial introduction to the subject. The equations of Rinde for the sedimentation equilibrium of a polydisperse solute were given in a form similar to the ones used today. Furthermore, it is evident from these equations that a description of the molecular weight distribution may be drawn from them. Lansing and Kraemer demonstrated how the results of an experiment could be interpreted to provide several average molecular weights which they had defined. Although there was a t once considerable interest in this experiment with t,he advent of the ultracentrifuge there had been little concern with regard to the effect of nonideal solution behavior on the establishment of equilibrium. In independent endeavors and with but slightly different assumptions Schulz5 and Wales, Bender, Williams, and Ewart6 extended sedimentation equilibrium theory by allowing for the effects of the apThe Journal of Phyeical Chemistry
parent gradients of osmotic pressure in the ultracentrifuge cell. With this beginning Wales and associates went, on to describe several systems in considerable detail, indeed to the extent that it might be said their equations are the precursors of the more elegant relationships which are to be found in the Fujita monograph,’ ones which we shall adopt. The report is divided into two main parts. The first has to do with the determination of molecular weight distributions in an organic high polymer system by the construction of a curve which fits the observed moments M,, M,, and M,,,. This method, an extension ~~
(1) T. Svedberg, Kolloid Z . Erg. Bd., 36, 53 (1925). (2) T . Svedberg, Z. physik. Chem., 121, 65 (1926). (3) H . Rinde, Dissertation, Uppsala, 1928. (4) W. D . Lansing and E. 0. Kraemer, J . Am. Chem. SOC.,57, 1369 (1935). (5) G. V. Schuls, 2. physik. Chem., A193, 168 (1944). (6) M. Wales, M. ,M.Bender, J. W. Williams, and R. H. Ewart, J . Chem. Phys., 14, 353 (1946); cf. also JZ. Wales, et al., J . Phys. Chem., 52, 235, 983 (1948). (7) H. Fujits, “Mathematical Theory of Sedimentation Analysis,” Academic Press, Inc., New York and London, 1962, Chapter V.
SEDIMENTATION EQUILIBRIA I N POLYDISPERSE PSECDO-IDEAL SOLUTIONS
of the Lansing-Kraenier treatment, is based upon the assumption that an exponential distribution which contains these moments as parameters is a good approximation to the true distribution curve. In arriving a t the several moments we present a new approach which makes use of a series of experiments performed a t several low centrifugal fields. In the second part we indicate that a conclusion of llandelkern, Williams, and Weissberg* to the effect that interpretable nieasurenients on high polymers can be made only at the Flory temperature is probably incorrect. An examination of the equations which describe the sedimentation equilibrium in a “good” solvent indicates that series of experinients performed a t several low centrifugal fields and several low concentrations ought to allow the evaluation of the light scattering second virial coefficient.
Part I. Pseudo-Ideal, Incompressible Solutions Theorg. A toniporient of a solution is at sedinientatioti equilibriuni when its total potential is constant from phase to phase as distance in the cell varies from the center of rotation. If the polymeric solute coinponerits arid thf. mediuiu are adjusted to the Flory temperature, if the initial solute concentration is low, and if the solution is incompressible, one finds
AC
t=-
62
=
-
(4)
eo
Here Ac is the difference in equilibrium cv”ntration at the top and bottom of the cell, and co is the original solute concentration in grams per nil. Equations 2, 3, and 4 are iniportaiit for us in that they provide a nieans by which several average iiiolecular weights for a given polymer sample may be coniputed. The first two of them appear in the I>ujita book as eq. 5.171 arid 5.172, where the assumptions under which they have been derived arc c.learly delitieated. The quantity f(ilI)dN is the weight fraction of the total sample which lies between niolccular weights M and JI ddI. In terms of the continuous distributiori funcdtioii, the 1” average niolccular weights weight, “z,” arid “ X are defined as
+
+
/--
JIM=
J
Jfj(XjdJ2
(5)
0
rm
M3f(dfjdLlf
Mz+l Here
10.51
=
0
(7)
[ M2f(MjdJZ
-
For their evaluations we describe three quantities
- r2
b2 - a2
The quantities a, b, and r a r e the radial distances of the top, the bottom, and any arbitrary intermediate cell position. The quantity A, to which frequent reference will be made, is defined as ,\
=
(1 - g p o ) u 2 ( b 2- a’) -
2RT
19)
in which the several quantities have their usual significance. I t has the units moles per gram. Making use Iof this basic statement it can be shown that
arid by differentiation with respect to - dc
Integration of eq. 3 yields
- dM
(3)
(10) These equations should be adequate descriptions of the respective gradients if X 5 1 ’X,.It will be noted that (8) L Mandelkern, L C Williams, and S G Weissberg J C h e m , 61, 271 (1957)
Volume 69, .\-umber
3
Phys
.Ilarch 1.965
H. W. OSTERHOUDT AND J. W. WILLIAMS
1052
the quantities P(X), &(A), and R(X) have the units grams per mole. The expression which involves the concentration gradient a t the cell meniscus, R(X),may be subtracted from or added to the corresponding equation descriptive of the condition a t the bottom of the cell, P(X), to give
So, if the solution is subjected to relatively low centrifugal fields, the quantity X can be kept within the range 0 < X 2 l/Afw. Equations 4 and 8-12 then should furnish methods for obtaining M,, M,,and IW,+~. The average, MI”,is evaluated from the slope of the plot Acjco us. A. The product of M , and M , is obtained by way of the slope of a P(X) - R(X)vs.X graph, while the R(X) us. X 2 yield curves of &(A) us. X 2 and P(X) M,M,M,+I values as limiting slopes. Experimental. A Spirico Nodel E ultracentrifuge was employed in the performance of the sedimentation equilibrium experiments. For them rotor speeds of 4500 r.p.m. or less were required, so that it was necessary to use the 22-pound rotor to reduce effects of precession and fluctuations in speed. Cells were of the double sector type, with solvent in one of the conipartments to provide the reference experiment. The location of the bottom of the solution column was achieved by way of the “false bottom” technique with an inimiscible heavy liquid, in these experiments anhydrous glycerol. The schlieren optical system measured the concentration gradients directly. The experiments were carried out with the system polystyrene-cyclohexane a t the Flory temperature, 35.0’. The solvent, a Baker reagent grade chemical, was distilled from lithium aluminum hydride to remove traces of water just prior to use. Polymer and solvent were weighed into glass-stoppered flasks for the dissolution process, which required warming a t 45’ for 72 hr. Loading of the centrifuge cell was achieved in the usual way; the temperature control circuit of the instrument niaintained Lhe proper temperature. The camera was loaded with Kodak Type 11-G spectroscopic plates. A Iiodak 77A filter was placed between the light source and the rotor chamber. The value of the quantity X was fixed for each experiment but the height of the solution column was arbitrarily varied. With 2-mm. columns the equilibrium time was generally less than 1 day. After the schlieren
+
The Journal of Physical Chemistry
image appeared to be unchanged with time for some 12 hr. a series of photographs was taken in which different orientations of the schlieren phase plate were used. After the image had been recorded the rotor speed was adjusted to a new (higher) X value and a new equilibrium established. Thus, as many as four experiments, each involving a different X value, were performed in a single series. Two different polystyrenes were involved. One of these, designated as 19F, originally distributed by the Dow Chemical Co., has been the subject of several physical chemical characterizations. The second sample, PS-706, was provided by the U. S. National Bureau of Standards, also with certain molecular characteristic constants. Data and Results. The concentration of any given solution was computed by making use of the relationship p = po (1 - 8po)c. In it, p is the solution density in g./nil., po is the solvent density as taken from the International Critical Tables, 5 is the partial specific volume of the polystyrene, taken to be 0.930 ml./g., and c is the solute concentration in grams per ml. The density calculated from this relationship by using a concentration based upon the original solvent and solute weights permitted an estimate of the total volume of the solution and therefore a revised concentration. By successive approximation a final value of the concentration was made available. The schlieren patterns, two photographs for each sedimentation equilibrium experiment, were analyzed by nieans of a Gaertner plate reader. Displacements, Az, were plotted against comparator readings, R, as measures of distances from the center of rotation. Then, a smooth curve was drawn and extrapolated to the meniscus and cell bottom positions. The area under this curve is related to the concentration change over the cell. Thus
+
Ac
=
c(b) - c(a)
=
lb$dT
=
F dn - G A B tan 0 dc
lb
AzdR
(13)
where dR is the differential radial distance as read from the photographic plate, the factors P , G, A , B, arid tan e are the constants of the schlieren optical system, and 10-2 is the transformation constant for the conversion of mm. to em. The refractive index increment, dnldc, for polystyrene in cyclohexane was taken to be 0.1705
(9) J. E. Blair and J. W. Williams, J . Phys. Chem., 68, 161 (1964).
SEDIMENTATION EQUILIBRIA IN POLYDISPERSE PSEUDO-IDEAL SOLUTIONS
1053
ml./g., the datum reported by McIntyre and co-workers. Calculations for the quantities P(X), &(A), and R(X) require the transformation
-dc- -_ dt
exp - ( b
+ 2)M/Mw
(15)
It contains two adjustable parameters, b and M,.
(b2 - a') dc 2r dr
(14)
The slopes of the linear plots of Ac/co us. X and P(X) R(X) us. X were determined by least-squares analysis. I n each case the intercept was taken to be the origin, as required by eq. 4 and 11. The nonlinear plots, such as &(A) us. X2, were fitted where possible so that the sum of the deviations of the individual points was approximately zero. They were extrapolated to an intercept which had been previously determined by a leastsquares analysis of a linear plot. For example, M,, the intercept of the &(A) us. X 2 curve, was accurately known from the slope of the Ac/co us. X plot. For the polystyrene PS-706 there are presented in Figures 1-5 plots of the several functions of X us. X or
For the polystyrene PS-706 and 19F we have made use of the equations
f(M)
1.82 X 10-13M1.5 exp[-8.99 X 10-6M]
=
b
=
0.5; M ,
=
2.78 X lo5
and
f ( M ) = 6.89 X 10-9M0.5exp[-3.34 X 10-6M] b = -0.5; M , = 4.49 X 106 There have been assembled in Table I the several moments for the two samples. Compared are the I
I
I
4.0 0 I
2
-a -
?? 3.0 1.2
I
4
1
2.0
5
1.0 1.0 0.8
A%,
0
I
1.0
0.6
Figure 2.
[ P ( x )- R ( A ) vs. ]
I
2.0 3.0 X(I0')
I
4.0
for PS-706.
0.4
0.2
0
-1.0
2.0
3.0
4.0
X(I0')
Figure 1. Ac/co 1;s. X for PS-706.
A2. From their intercepts and limiting slopes and based on eq. 4 and 8-12, values of M,, M,M,, and M,M,M,+I were obtained. In this way, the successive moments of the molecular weight distribution curve are supplied. Entirely analogous graphs for the polystyrene 19F lead to similar data. This inforniation has been utilized in connection with the well-known Schulz distribution functionl' to describe the heterogeneity of the two polymers. The differential weight distribution may be expressed as
moments which were obtained from the sedimentation equilibrium experiments performed a t different values of X and those coniputed froiii the Schulz function with the parameters indicated. Also included in this table are the two number-average molecular weights, At,, both obtained by osmoinetry, in one case a t the National Bureau of Standards and in the other at the University of Wisconsin.12 I t is seen that observed and calculated values of At, do riot show good agreement. Discussion. There have been presented equations which provide an accurate description of the coiicentration gradients of a polydisperse solute a t sedinientatiori equilibrium under pseudo-ideal conditions and when low centrifugal fields are employed. Consideration of the (10) D. McIntyre, A. Winis, L. C Williams, and L Mandelkern,
J. Phys. Chem , 66, 1932 (1962). (11) G. V. Schulz, Z physik Chem., B43, 25 (1939) (12) L D. Grandine, Jr , Dissertation, 1952
Volume 69,Number 3
March 196'5
1054
H. W. OSTERHOUDT AXD J. W. WILLIAMS
Table I : Moments for Polystyrene PS-706 and 19F ,-----pS-706------. Experimental
Mw M,M, M,M,M,+Ib 'Vf
(2.78 f 0 . 0 5 ) X 105 ( 1 . 0 9 i 0 . 0 2 ) x 10" (5.32 f 0 . 3 9 ) X 1OI6 1.365
n
x
105
---
lQF-----------
,
Schulz distribution
Schulz distribution
Experimental this research
A.iM.L.'
2 . 7 8 X 105 1 . 0 8 x 10" 5 . 4 2 X 10'6
(4.49 f 0 . 0 8 ) x 105 ( 3 . 2 5 f 0 . 0 7 ) X 10" (3.77 f 0.41) x 1017
4.56 X 105 3.47 X 10" 3 . 4 0 x 1017
1.67 X lo6
(1.97 f 0 . 1 0 ) X 105
4.49 3.37 3.53
x x x
105 10" 1017
1 . 5 X 10s
' Results of Mrs. A. M. Linklater, Communicated in Progress Report, Aug. 1, 1957-Oct. 31, 1958 to Department of the Navy, Bureau of Ordnance, Contract N 123-605305-1979A. Sedimentation equilibrium experiments a t a single rotor speed in cyclohexane a t 34.2'. For each polystyrene the figure presented for the moment M,M,M,+~ is an average of the values obtained from the limiting slopes of eq. 9 and 12 and the value obtained by using eq. 5.175 of Fujita.7 Data from the experiments in which X E l / M w were used.
I
I
I
I
I
1
I
3.0k7
6.5
-
s! n 1
+
Y
KEY:
P I U t RIM *
P
2.0
t 0
Figure 3.
1 1.0
2.0 3.0 A(I0')
Figure 5.
4.0
P i x ) , &(A), and R(X) us. A for PS-706.
1
-I I A2 (IO"
Figure 4.
& ( A ) us. h2 for PS-706.
field as a quantity to be varied from experiment to experiment has suggested a new method for measuring ill,,.,AI,, and 111,+1in which the first two now appear as the slopes of linear plots, while the third one is obtained from the liiniting slopes of certain quantities plotted The Journal of Physical Chemietry
I
I
I
I
I
I
2.0
4.0
6.0
0.0
10.0
12.0
X'(IO'*)
2.9
2.4
0
2YW.0
+ @A)]
[P(h)
us. X 2
for PS-706.
against the square of the field strength. The utility of the plots arid the equations upon which they are based have been amply tested. The Schulz function to describe the actual molecular weight distribution seems to provide a useful specification. However, for the polystyrene 19F, a distribution somewhat different from the one previously reported9 from sedimentation velocity data was provided by this analysis. It is believed that this inconsistency arises froni uncertainties in the several constants employed in the equilibrium and in the transport equations. As regards the latter, the doubt lies in the quantities which are used in the equation which relates sedimentation coefficient to molecular weight, riainely s = KM". Thus, the reports of Cantow13 and McCormick1* give for a the values 0.48 and 0.51; but Blair and Williams adopted the theoretical value 0.50 for the pseudo-ideal system. Had the latter taken a = 0.49 for their coniputations the two distribution curves would have been in quite good agreement. (13) H. J. Cantow, MakTomoZ. Chem., 30, 1S9 (1959). (14) H.W.McCormick. J . Polymer Sci., 36, 341 (1959).
SEDIMENTATION EQUILIBRIA IN POLYDISPERSE PSEUDO-IDEAL SOLUTIONS
The value of M , for polystyrene PS-706 obtained by sedimentation equilibrium at the National Bureau of Standards is M , = 2.88 X lo5, as compared to our value M , = 2.78 X lo5. Fractionation of the sample a t NBS has provided the ratio M z / M w = 1.4. The ratio from this investigation is 1.41. As ordinarily performed only a single value of X is used in a sedimentation equilibrium experiment; on occasion this datum is missing from a published article. It would appear that the selection of a value for X should be dictated by the kind of data sought in the experiment. If the quantities M,, M,, and the light scattering second virial coefficient are desired, X should be ideally 0.5,’Mw. This condition (1) allows M, to be readily computed from the concentration gradient a t the top and at the bottom of the cell, and (2) provides insurance that a small correction only is to be applied to the sedimentation equilibrium second virial coefficient to obtain the corresponding light scattering quantity (cj. part 11). If the experiment is to be performed a t the Flory temperature and the quantities M,, M,, and M,+l are sought, X should be approximately 1/M, because the calculation of requires an appreciable difference between the slopes of the dc/df vs. f curve a t the two ends of the cell.
Part 11. Nonideal Systems Theory and Discussion. With relatively few exceptions the polydisperse polymer-solvent systems encountered in practical laboratory work are nonideal in behavior. For them the equations are more cumbersome because the distribution of the several species a t each radial distance will differ from that of the original solute, and the correction for the nonidealities depends both on the total concentration and on the molecular weight distribution at any given distance from the center of rotation. While there may be doubt on the part of some investigators whether the experiment being considered may be used to study solute-solvent interactions, the problem has attracted attention in this laboratory. Thus, by way of improving the situation as it stood in 1957,8Fujita has derived expressions for the intercept and the limiting slope of a plot of 1/ (Mw)app vs. co for systems not too far removed from the Flory temperature and as co becomes very small. It was shown’ that this intercept permits evaluation of the weight-average molecular weight of the solute and the limiting slope can be correlated with the second virial coefficient obtained from light scattering measurements, BLS. The quantity (Mw)appis given experfmentally by the use of the equation (M‘*)BPP
=
E(c>t=o - (c)c=ll/xCo
1055
or its equivalent. For the low concentrations the re sulting equation of Fujita is 1 ~( M A P P
1
- -
M,
+ B”c~
(16)
where
The term X2MZ2/12is a correction term, the magnitude of which may be controlled by adjustments of the quantity X, or its equivalent, the rotor speed, W . We shall prefer to write eq. 16 in the form
Here, BLS’is the product of M w 2and BLS. The higher terms which are neglected involve both X and co. This equation can be derived by considering the expression which describes the concentration gradient of a simple polymer species a t sedimentation aq~ilibrium.’~ It is
XM,cf8
5 ck
(18)
k= 1
In order now to simplify the derivation we assume that we consider very dilute solutions at low centrifugal fields. Then, two approximations may be made
The results upon insertion of these into the expression for the concentration gradient is
(15) This expression, except for a single substitution, is equation 5.103 of the Fujita monograph.
Volume 69,Number 3 March I966
NOTES
1056
If the exponents in this equation are expressed a,s a series expansion it is seen that
alMJ
+ 002)
(20)
Integration over the entire cell, remembering that f, = c,O/co, yields -XM,ftco
=
Ci(1) - Ci(0) -
c f,M,(B,k + a/M,) + . . . P
XM*frCo2
k-1
(21)
Equation 17 is obtained from eq. 21 by summation over all components and rearrangement. Thus, an examination of the equations which describe the sedimentation equilibrium of a polydisperse solute in a better than “poor” solvent indicates that series of
experiments performed a t several low concentrations and a t several low centrifugal fields ought to permit the evaluation of the light scattering second virial coefficient. The approximations made in the derivation become better as co and X decrease. The double ‘plot which is indicated is quite analogous to the Zimm plot of the light scattering experiment in which a quantity related to the intensity of the scattered light is plotted in terms of both the initial polymer concentration and the angle to the incident beam at which the scattered light is observed. Acknowledgments. The authors are indebted to Drs. E. T. Adams, Jr., and V. J. MacCosham for many fruitful discussions. The work itself was supported in part by the U. S. Army Research Office, under Contract DA-ORD-11, and in part by the University Research Committee, with funds supplied by the Wisconsin Alumni Research Foundation.
N OTES
Effect of Additives in Radiolysis of Ethane at High Densities by Catherine M. Wodetzki, P. A. McCusker, and D. B. Peterson’” Department of Chemistry and the Radiation Laboratory,tb Unirersity of Notre Dame, Notre Dame, Indiana 46666 (Received Octaber 86, 1964)
pressures near or below 1 atm.3-b For a given hydrocarbon, G(H2) is reduced to the same plateau value by all three additives and it is generally assumed that this reduction is solely the result of scavenging of thermal hydrogen atoms. However, evidence supporting this assumption is rather limited and there is very little information concerning the use of these sdditives at high gas densities.
Experimental In the course of an investigation2 of the effect of density on the y-radiolysis of gaseous ethane, the thermal hydrogen atom scavengers ethylene, propylene, and nitric oxide were employed as additives. In order to interpret results of such studies properly, it was necessary to examine critically the role of these additives in radiolysis and how this role depends upon density in the range 0.001 (atmospheric pressure) to 0.30 g. cc. - I . The three additives under consideration have frequently been employed as thermal hydrogen atom scavengers in studies of the mechanism of hydrogen formation in the radiolysis of gaseous hydrocarbons at The Journal of Physical Chemistry
Experimental details are described in an a,ccompsnying paper.2
(1) (a) The authors acknowledge with thanks helpful discussions with Professor Milton Burton. (b) The Radiation Laboratory of the University of Notre Dame is operated under contract with the U. S. Atomic Energy Commission. This is A.E.C. Document No. C00-38361. (2) C. M.Wodetski, P. A. McCusker, and D. B. Peterson, J. Phys. Chem., 69, 1046 (1966). (3) L. Dorfman, ibid., 60, 826 (1956). (4) K.Yang and P. J. Manno, J. A m . Chem. SOC.,81, 3507 (1959). (5) R. A. Back, J . Phys. Chem., 64, 124 (1960). (6) K.Yang and P. Gant, ibid., 6 5 , 1861 (1961).
