4274
A. SODA, T. FUJIMOTO, AND M. NAGASAWA
The Johnston-Ogston Effect in Sedimentation
by Atsuhiko Soda, Basic Research Laboratories, Tokyo Rayon Company, Ltd., Kamakura, Japan
Teruo Fujimoto, and Mitsuru Nagasawa Department of Synthetic Chemistry, Nagoya University, Nagoya, Japan
(Receiced April 10, 1967)
Two samples of poly-a-methylstyrene having very narrow molecular weight distributions are centrifuged in poor and good solvents. Their schlieren peaks are observed separately, and the apparent sedimentation coefficient and the area of schlieren peak of the individual solute are determined. It is concluded that the theory of Trautman, et al., on the JohnstonOgston effect is applicable when the solvent is a poor solvent or at infinite dilution if the solvent is a good solvent. The reason why the theory is not applicable a t finite concentrations in good solvent is that the concentration dependencies of the sedimentation coefficients of fast and slow components in mixture are not the same. It is pointed out that the sedimentation coefficients of both slow and fast components deviate from the concentration dependencies predicted from the simple additivity of the effects of both components but the deviations cancel each other if we calculate the sedimentation coefficient of the local center of mass. Consequently, the reciprocal weight-average sedimentation coefficient of the mixture is linear with respect to the total concentration, despite the fact that the sedimentation pattern is bimodal.
Introduction Several researchers' -4 have clearly demonstrated that in a three-component system (one solvent and two solutes) in which one solute sediments faster than the other, the concentration of the slow component computed from the area of the schlieren peak is higher than its true concentration, whereas that of the fast component is lower than its true one. Such phenomena are caused by the hydrodynamic interaction between two solutes. According to Trautman, Schumaker, Harrington, and Schachman,j the concentration distributions of solutes in the cell are supposed to be shown in Figure 1. The effect is now called the JohnstonOgston effect. The original idea of Johnston-Ogston' was extended by Trautman, et aZ.,j who derived the following equations.
The Journal of Physical Chemistry
where # is given by
$'Ss'
# =
dt
0
(3)
J'SF' 0 dt Here, the superscript obsd denotes quantities which may be observed on the schlieren pattern and y denotes the region which is shown in Figure 1, as named by Trautman, et al. The subscript S and F denote the slow and fast components, respectively, and zo are the radical distances measured from the center of rotation to the boundary position of the i component (I) J. P.Johnston and A. G. Ogston, Trans. Faraday Soc., 42, 789 (1946). (2) W. F. Harrington and H. K. Schachman, J . Am. Chem. Soc., 7 5 , 3533 (1953). (3) H. K. Schachman, "Ultracentrifugation in Biochemistry," Academic Press Inc., New York, N. Y., 1959. (4) H. Fujita, "Mathematical Theory of Sedimentation Analysis," Academic Press Inc., New York, N. Y., 1962. (5) R. Trautman, V. N. Schumaker, W. F. Harrington, and H. K. Schachman, J . Chem. Phys., 2 2 , 555 (1954).
JOHNSTON-OGSTON EFFECT I N SEDIMENTATION
4275
V
T
n n 4 in
-
X
Figure 1. Conuntmtion clislribulionr of the slow and fwt components i n B mll.*-5
and to the meniscus, C:"*l and C," arc the observed and original concentrations of the i component, and S,? is its sedimentation coefficient in the y region. Since 1 > $ > 0, it follows that Cs"""(Zs"""/zo)2/C~a > 1 and c,0~~1(~F"'e'//z,)2/CFn
< 1.
If the concentration dependence of sedimentation coefficient is the same for both the slow and fast components in y region, $ may be approximated %+ $ =
SS.$/sF,O'
(4)
where S,.$ is the sedinientatiou coefficient of the i component at the limit when the concentration of the i component nppro:rchcs zero in the absence of the other solute and at atmospheric pressure, that is, the intrinsic sedimentation coefficient of the i component. Using the data of biological polymers in bufler solutions by Harrington and Schachninn,2 Trautman, el al., showed that eq 4 is a pertiuent approximatioil for these biopolymers. As waq pointed out by Trautman, el al., and also by Schachman,' and Fujita,' however, the approximations in cq 4 could not be always valid so that the applicability of the theory with the use of eq 4 seems to be limited. The purpose of this paper is to present data and to show that in which $ is not equal to SS.$/SF.$ the theory is applicable when the concentration dependencies of the sedimentation Coefficients are not so marked or at the limit of infinite dilution of solutes. Moreover, a discussion is given to the concentration dependencies of the sedimeutation Coefficients of both components in y region. The experiments were carried out by using poly-a-methylstyrenes which have very narrow molecular weight distributions. Experimental Section Samples. The samples, poly-a-methylstyrenes, were prepared by the method previously reported.' Those samples have such narrow molecular weight distribu-
tions that the schlieren peaks of the samples can be distinguished separately BS shown in Figure 2, when two samples are centrifuged together in a solvent. The molecular weights ( M ) and the intrinsic sedimentation coefficients (SJ) of the samples are summarized in Table I. Sample A, which WILS used for the experiments in cyclohexane, was prepared by polymerization of a-methylstyrene with a sodium salt of a-methylstyrene tetramer anion which contains two kinds of active species; one has two active ends and the other has one active end. Therefore, sample A is a mixture of two kinds of molecules whose molecular weights are different by the factor of 2. Figure 2 is a schlieren pattern of sample A. Sample A-S and A-F are the EION and fast components which were separated from original sample A by successive fractionations according to the decreming temperature method in cyclohexane. Sample9 B and C, which were used for the experiments in toluene, were prepared by polymerization of a-methylstyrene with n-butyllithium in tetrahydrofuran so that they may have narrow molecular weight distributions of a single peak. The molecular weights of samples A-S and A-P were determined by a lightscatteriug method in toluene with the use of an instrument of the Brice type manufactured
Table I :
I h t n for the Samples (35.W) .Mol
S?.rnple
(X
W l
IO-.)
Si.0.
S
AS
0.73
13.89
A-F
1.52
1!1.47 11.37 17.i~
n
0.4G
C
1.38
ki, dl/i
0.489 0.048 1.98 4.01
Voltime 71. Number IS
Solvcnl
Cyclohexane Cyclohexane Toluene Toluene
December 1867
A. SODA, T. FUJIMOTO, AND 31. NAGASAWA
4276
by Shimadzu Seisakusho Ltd., Kyoto. All procedures were the same as reported previously.6 The molecular weights of samples B and C were determined from their intrinsic viscosities in toluene a t 25” using the following equation.’ =
7.81 x 10-5
~ 0 . 7 3
The mo1t:cular weight distributions of these samples were not determined, but it is clear from their schlieren patterns that they have very narrow molecular weight distributions. (The ratio of the weight average to the number-average molecular weights of these samples is probably close to 1.01.) Solvents used were of special grades of Wako Pure Chemical Inc., Osaka. Cyclohexane used as a poor solvent was washed several times in the cold with a mixture of concentrated nitric and sulfuric acids, followed by repeated washing with distilled water, and finally was fractionally distilled over sodium. Toluene used as a good solvent was fractionally distilled, after successive shaking with concentrated sulfuric acid, aqueous sodium hydroxide, and distilled %later, followed by drying over phosphorus pentoxide. Solutions were prepared by weight. Sedimentation Experiments. Ultracentrifugation was carried out in a Spinco Model E ultracentrifuge with a schlieren optical system. Experiments on solutions of higher concentration were conducted in 3- or 12-mm cells, while those on solutions of lower concentration were done in 12- or 30-mm cells. For the experiments with cyclohexane, three different speeds of rotation (59,780, 42,040, and 29,500 rpm) were used to confirm that the results obtained are independent of the speed of rotation. For the experiments with toluene, only one speed of rotation (42,040 rpm) was used to reduce the pressure effect and to prevent peaks from becoming too sharp or too broad. The temperature used was 35.0” both in cyclohexane and in toluene. Although the 0 temperature of poly-a-methylstyrene in cyclohexane was found to be within the range from 35 to 39°,835” is close enough to the 0 temperature for the purpose of the present work. The sedinientation coefficient was calculated from the position of the maximum of schlieren peak when centrifugation was done a t 59,780 and 42,040 rpm. It was confirmed that this method and the second-moment methodg gave no appreciable difference, since each peak in the sedimentation patterns was not so much skewed and it was sharp. If the 07 boundary was considerably apart from the base line in the patterns taken a t early stages of Centrifugation, the patterns were not used for calculation. When centrifugation was done a t 29,500 rpm, however, the second-moment method was used The J
O U T ~of~ Physical
Chemistry
because the patterns were fairly broad. Measurements of the distance were carried out with a two-dimensional microcomparator manufactured by Shimadzu Seisakusho Ltd. The area of peak was measured with a planimeter after enlarging the pattern by 20 or 50 times on the viewing screen of a universal contour projector of Nippon Kogaku Co., Tokyo, and tracing it on millimeter graph paper. Computation of Sedimentation Coeflcient. It is well known that the pressure gradient produced in the solution a t high speed of rotation has an appreciable effect on the viscosity and the density of the solvent, and on the partial specific volume of the solute as well, which in turn has an effect on the sedimentation coefficient obtained.’O-l? I n this work, the pressure effect was eliminated by extrapolating the observed sedimentation coefficients to the meniscus using the Fujita equation12
s, =
In
(z’To) =
W2(t
- to)
si”(1 - p[(2/20)2 -
l]]
(5)
where (J is the angular velocity, t is the time as observed, to is the zero time correction, p is a parameter independent of x and t, and S,O is the sedimentation coefficient a t atmospheric pressure. Therefore, S t can be obtained as the intercept of the S1vs. ( Z / Z ~ )-~ 1 plot a t ( ~ / x o)~ 1 = 0. An example of the plot is shown in Figure 3 . The values of S t thus determined are found to be independent of the speed of rotation in all cases. An ordinary graph of a log ( x / z o ) vs. t plot is shown in Figure 4. It is observed that the plots are clearly curved because of the pressure effect. The asymptotes in Figure 4 are drawn with the values of X,O determined in Figure 3. The most difficult problem in fitting eq 5 to experimental results lies in the determination of to,to zero time correction. First, the time when the speed of rotation reached 2/3 of its final value is chosen as t0,14,1s and then (6) T. Fujimoto, N. Ozaki, and hl. Kagasan-a, J . Polymer Sei.,A3, 2259 (1965). (7) H. W. hIcCormick, ibid., 41, 327 (1959) (8) hI. Nagasawa, et aZ.,to be published. (9) R. J. Goldberg, J . Phys. Chem., 57, 194 (1953). (10) H. Mosiman and R. Singer, Helo. Chim. Acta, 27, 1123 (1944). (11) J. 0th and V. Desreux, Bull. Sac. Chim. Belges, 63, 133 (1954). (12) H. Fujita, J . Am. Chem. SOC.,78,3598 (1956). (13) H. G. Elias, MakromoZ. Chem., 24, 30 (1959). (14) I. H. Billick, J . Phys. Chem., 66, 1941 (1962). (15) G. hfeyerhoff in “Ultracentrifugal Analysis in Theory and Experiment.” J. W. Williams, Ed., Academic Press Inc., New York, N. Y., 1963, p 47. (16) J. E. Blair and J. W. Williams, J . Phys. Chem., 68, 161 (1964). (17) M.J. R. Cantow, R. S. Porter, and J. F. Johnson, X a h o m o E . Chem., 87, 248 (1965). (18) E. G. Pickels in “Methods of Medical Research,” Vol. 5, A. C. Corcoran, Ed., Tear Book Publishers, Chicago, Ill., 1953.
JOHNSTON-OGSTON EFFECT IN SEDIMENTATION
4277
0.09
0.08 I
ra
$ 0.07 2. 0.08
I
I
0
0.1
0.3
0.2 (z/zo)*
- 1.
0.4
Figure 3. An example of extrapolation of apparent sedimentation coefficient to meniscus by eq 6 : sample, A: solvent, cyclohexane; temperature, 3.5.0'; CFO/CSO = 1.00; COO= 0.107 g/dl; speeds of rotation, 0,42,040 rpm; 0, 59,780 rpm. I
1
I
I
I
h
{ 0.18 v
3
4
0.09
0
IO
20
40
30
50
t , min.
Figure 4. Plot of log (z/zo) against time. Experimental points correspond to the filled circles in Figure 3. The origin, t = 0, is the time when the speed of rotation reaches */a of the final value and the asymptotes are drawn using 8: determined in Figure 3.
Sfo and p in eq 5 are determined by the least-squa.res method. Next, u of eq 6 u =
(An
l/Z
1
I
0
0.2
I
0.4 C, d d l .
0.27
0
0.05
Figure 5. Concentration dependencies of the sedimentation coefficients of samples A-S and A-F in cyclohexane. The filled circles denote the experimental results in the absence of the other solutes, while the open circles with segment denote the experimental results in the mixture of CFO/CSO = 1.00. Speeds of rotation (in rpm): 0, 59,780; 7, 42,040; -0, 59,780; 0,42,040; 0-, 29,500.
is calculated using those values of to, S?, and p.I0 By repeating the above calculation with changing to, a value of to is chosen so that u may become a minimum, and the value of Sfo associated with the value of to is calculated. The zero time correction, to, thus determined of is different from the time when the speed reached the final value by less than a few minutes, but even such a small correction to tois important for determining S p accurately. It was reportedm that such an extrapolation as eq 5 is not good if standard errors in measuring x are considered. I n Figure 3, however, the uncertainty in S1caused by an error of 0.001 em in locating the position of the boundary is within 1%, whereas an error of 30 see in to may cause the uncertainty as much as 10%. Since the apparent sedimentation coefficients, plotted against ( Z / X ~ )-~ 1 fall on a straight line even to the points far from meniscus, and the intercept a t ( X / X ~ )-~ 1 = 0, Sp is independent of the speed of rotation within experimental error as shown in Figures 3 and 5, the procedure may be reasonably accepted though it is somewhat arbitrary. The calculations were carried out, using a program written in FORTRAN IV, on an IBM 7040 electronic computer.
s
