Oct., loci2
1941
iTETAICITY f?EDIMENTATION 8Tt-DIES
VELOCITY SEDIMEKTATIOX STUDIES ON PRESSURE ASD COSCENTRATION DEPENDENT SYSTEJIS BY IRWIN13. BILLICK Analytical Research Division, Esso Research and Engineering Company, Linden, ;Yew Jersey Received March 6 , 1968
The sedimentation behavior of a system in which pressure and concentration dependence were present has been studied as a function of molecular weight, solvent, and speed of centrifugation. Anionically polymerized polvstyrene was investigated in cyclohexane, a poor solvent, at 35", and in toluene at 25'. .'nalysis of the data according to the series solution of Fujita's differential equation yielded values for the sedimentation coefficient, st zero concentration and atmospheric pressure, the concentration dependence parameter, and the pressure dependence parameter. I t was found that concentration does not contribute greatly t o the slope of the sedimentation coefficient LS. time curve within experimental precision. The value fourid for the pressure dependence coefficient agrees with the literature and was not found to be influenced by molecular weight or speed of centrifugation.
Introduction The simultaneous dependence of the sedimentation coefkient on hydrostatic pressure and solute concentration has been investigated by several workers in the past,lV2both experimentally and theoretically. However, the most fundamental approach, from a mathematical point of view, was given by FujitaS3 A differential equation of flow was proposed for the sedimentation behavior of a monodispersed, non-diffusing solute, explicitly in terms of the Sedimentation coefficient, a t zero concentration, a pressure dependence parameter, and a concentration dependence parameter. 'The equation which was set forth was solved by Wales4 both analytically and with a digital computer. Evaluation of the sedimentation coefficient a t zero concentration and one atmosphere pressure from experimental data is possible using Wales' analytical solution only if the values of the pressure and concentration dependence parameters are known, or by a curve fitting technique.6 In a previous publication,€ we have given a solution to lTujitaJs equation which would permit the direct determination of all the unknown parameters. The purpose of this article is to show that velocity sedimentation data can be evaluated adequately using these solutions. An investigation has been carried out using near monodispersed polymer under varying conditions of molecular weight, solvent, concentration, and centrifuge velocity. Experimental
TABLE I POLYMER MOLECULAR WEIGHTS Sample .If," x 10-5 MW/N* s-57 0 975 .. 9-107 s-1118
2.51 2.39
1.04 1.08
automatic camera was started when the speed had reached "3 its final value and the zero time of sedimentation was taken a t this point.gJ0 Sedimentation rates were studied as a function of solute concentration and in the ca8e of one polymer sample, 5-111, as a function of angular velocity. In studying the effect of velocity, only the behavior in a poor solvent was investigated. Solutions were prepared and the run a t highest speed usually was made first. Upon completing a run, the cell was removed from the rotor and placed on a wrist-action shaker. The cell was shaken very gently under a heat lamp until the polymer was all dispersed and then run a t another speed. This procedure was repeated until all the runs a t that particular concentration were carried out. The position of the maximum of the refractive index gradient was measured using a Gaertner toolmaker's microscope. Although some diffusion was apparent during sedimentation, it has been assumed" that the maximum may be nsed in place of the boundary position measured from the second moment.
Calculations and Results The solutions to Fujita's equation6were expressed in the form of' a series, in which the independent variable was either time or position of the boundary from the center of rotation. In this article we will deal only with the form given by eq. 1, although the results are the same when the other variable is used.
x = - -In- r/ro -
Three anionically polymerized polystyrene samples were used. Two of these, S-5 and S-10, were furnished through the courtesy of Dr. H. W. McCormick of the Dow Chemical Co. The other sample, 5-111, was distributed by H. Mark under the auspices of the Commission on Macromolecules of the ICPAC. All the samples used had a very narrrow molecular weight distribution as indicated by the low values of M,/iM,, given in Table I. l'ltracentrifugation was carried out in a Spinco Model E centrifuge equipped with an RTIC temperature control unit. Polymer solutions using cyclohexane a t 35", a poor solvent, and toluene a t 25", a good solvent, were investigated. During a given run, acceleration was carried out a t constant cunent until the desired speed was reached. The
In eq. 1, r is the radial distance of the boundary, is t'he radial distance of t'he meniscus, t is tthe time, w is the angular velocity, and Soois the sedimentation coefficient at. one at'mosphere pressure and infinite dilution. The definit'ion for a , the concentration dependence parameter, is given by
(1) J. 0th and V. Desreux, Bull. 808. chsm. Belges, 68, 133 (1954). (2) H. G. Elias, Makromol. Chem., 84, 80 (1959). ( 3 ) H. Fujita, J . Am. Chem. Sac., 18, 3S98 (1956). (4) M. Wales, ibid.> 81, 4758 (1959). (3) R l . Wales, J . PoZUmer Sei., in press. ( 6 ) I. H. Rillick, J . Phw Chem., 66, 565 ( 1 9 6 2 ) .
(7) H. W. McCormick, J. Polymer Sci., 36, 341 (1959). (8) H. Mark, data accompanying distribution of sample. (9) E. G . Pickels in "Methods of Medical Research," 5701. 5 , A. C. Coreoran, Ed., Year Book Publishers, Chicago, Ill., 1953. ( I O ) R. Trautman, J . Phys. Chem., 60, 1211 (1956). (11) H. Fujita, J. Chem. Phys., 24, 1084 (1956).
W2t
1
+
a!
IRWIN TI. I~ILLICIC
1942
Vol. 66
0.42 0.40
-
0.38 0.36 0.34
-
0.32 -
7
d
0.30
-
0.28-
VI
o*
0.26
-
VI
4-l \
0.24 0.22
/
/ 5-5
-
0.20
0.181
p' s-111
0.14 0.12
o'lo' TIME, (MIN.)
Fig. 1.-Change of apparent sedimentation coefficient with time of centrifugation of polystyrene S-10 in cyclohexane, 35", a t various concentrations; ultracentrifuge speed was 59,780 r.p.m.
where IC is a constant and Cois the initial concentration. Like the value of a, m, the pressure dependence parameter, varies from experiment to experiment and is defined as
m = 1/2pw2r02po (3) where pa is the density of the solvent and p is a constant characteristic of the solute-,solvent system. Valuesfor XoO/(l +a) could beobtained by plotting S us. d t and extrapolating to zero. However, this method of calculation suffers from a lack of precision for two reasons. First, the experimental setting of zero time suffers from lack of precision. Second, a decrease in precision results from the use of a plot of In r/rg us. d t and extrapolating to zero. The diffjculty in a procedure of this type has been amply discussed by Baldwin12 for the case of concentration dependence alone, but the situation in our case is the same. Figure 1 illustrates a plot of S 2,s. 0 2 t , but only serves to illustrate the magnitude of the change in X with time. (12) R. L Thldsin, Bzochern. J . . 66, 503 (1057).
Oll
012
01.3 014 OI.5
d.6 OI.7 OI.8
I
9
CONCENTRATION, g/lOO ml.
Fig. 2.-Concent,ration dependence of the pressure corrected sedimentation coefficient: solid lines for polystyrene in cyclohexane a t 35"; dotted h e for polystyrene S-10 in
toluene.
To increase the precision, we have rewritten eq. 1in the form
+
+
In r = In ro Sow21 - B(u2t)2 ---- (4) and have calculated the values of 80 and B by a least squares procedure. The calculations were carried out, using a program written in Fortran, on an IBM 7090 computer. The values of SO and B do not involve the measured value of ro and are not influenced by small errors in the absolute value of t . Details of this method of calculation will be presented elsewhere. Typical results obtained by this procedure are given in Table IIa for cyclohexane solutions and in Table I I b for toluene solutions. As a result of the least squares calculations, we also have obtained values for the standard errors of estimate of 8') and B. These figures, which give a rough estimate of the precision, also are included in the tables. If the original assumption regarding the concentration dependence holds, a plot of l/So us. CO should be linear and independent of velocity. Figure 2 illustrates the concentration dependence and it is apparent that this assumption holds for both solvent systems. Only points obtained a t 59,780 and 50,740 r.p.m. are included in Fig. 2;
Oct., 19ci2
lTELOCITY SEDIhXENTATION STUDIES
TABLE IIa Speed,
SEDIMENTATION DATAFOR POLYSTYRENE IN CYCLOHEXANE Concn., SQ, Stand. Stand.
Sain1)le
r.p.m.
g./100 ml.
s-10
59 ,780
5-111
59,780
0.78 .38 .15 .09 .44 .27
.I1 44 ,770
44 .24
'
.11 29,500
Speed,
.44 .16 * 11
sved.
error
6.70 7.27 7.92 7.66 6.51 7.02 7.05 6.71 7.06 7.33 6.85 7.30 7.59
0.01 .Ol .17 .10 01 .01 .02 * 01 .Ol .08 .08 .21 .12 I
B
34.6 37.2 51.6 38.6 29.2 40.1 33.8 20.6 23.1 27.4 10.7 14.6 25.8
x
error
B/(So)g
m
0.4 .4 6.2 4.8 0.5 .4 .5
0.77 .70 .82 .66 69
0.80 .73 .83 .66 .72 .83 .69 .51 -50 .52 .30 .30 .46
I
.81 .68 .46 .46 .51 .23 .27 .45
.5 .4 4.0 4.7 1.6 7.0
r.p.m.
g./100 ml.
sved.
error
s-IO
59,780
0.78 * 53 _,.27 ,a19
3.00 3.55 4.55 5.48
0.02
.Ol .05 .ll
however, values of the sedimentation coefficient of 5-111 obtained a t lower velocities also fall on the curve shown, within experimental error, but they have been omitted for the sake of clarity. The values for the sedimentation coefficient a t zero concentration and pressure obtained by a least squares analysis of the data given in Fig. 2 are listed in Table 111. As has been reported previ0usly,7,~3the value for IC in a poor solvent is much less lhan that found in a good solvent.
B
5.6 7.1 12.2 13.4
x
erpor
B/(S0)2
m
0.3 0.1
0.63 .56 .59 .45
0.76 .69 .68 .50
1.1 3.6
,
1.3-l
,
,
P
109
1.2 1.1 1.1 0.8
,
I
I
108
1.4 1.7 1.5 1.2 1.3 1.4 1.2 1.7 1.7 1.6 2.3 2.3 3.4
TABLE IIb SEDIMENTATION DATAFOR POLYSTYBENE IN TOLUEXE Stand. SO, Stand. Conon.,
Sample
s
I
TABLE I11 CONCENTRATION DEPENDENCE, OF POLYSTYRENE Solvent
Cyclohexane
Toluene
Sample
k 100 ml./e.
5 01 7.38 7.93 6.19
5-5 s-111 s-10 s-10
I
SO',
wed.
0.19 .27 .24 1.37
Once a value of k is known, the slope of eq. 2 can be corrected for radial dilution effects and m or p can be calculated. The contribution of the concentration dependence to the slope is not large as can be seen from the data giwn in Tables IIa and IIb. The values for B/(Su)2are a function of both pressure and concentration dependence according to eq. 5. These data can be compared with
+
m(2a 1) - a! B/(S0)' = l-ka
(5)
values for m which are characteristic of the pressure dependence alone. Since m varies from experiment to experiment, values of p calculated according to eq. 3 are also given. A summary of the values of p obtained under the varyiing experimental conditioiis of molecular weight, solvent, and speed of centrifugation is (13) 13. J. Cantow, Makromol. Chem., 30, 169 (1959).
4.9
5.1
5.3
5.5
5.7
5.9
log M,
Fig, 3.-Log-log plot of S&sx, in cyclohexane, us. M,: filled circles, data of Wales5; open circles, this work.
presented in Table IT:. The results for y were obtained by averaging the data a t constant speed, but varying concentrations. The standard deviat,inn, C ) of the data is included in the table to indicate the precision.
PRESSVRE
TABLE IV CORRECTION COEFFICIENT FOR
Solvent
Sample
Toluene
S-10 s-10 s-5
s-111
Speed, r.p.m.
59,780 59,780 59,780 59,780 50 ,740 44,770 38,460 35,600 29,500 24, 630
POLYSTYRENE
J !
x
109
1.05 1.37 1.48 1.33 1.54 1.60 1.68 1.78 1.78 2 25
C
x
10'
0.13 .19 14 .09 .14 I
.14 .14
.32 .86 1.94
1944
IRWINH.BILLICK
Discussion of Results Using previously derived equations, we hare calculated p, the pressure dependence coefficient, directly from sedimentation measurements. The final values of p were obtained after correction for concentration dependence, which tends to increase the sedimentation coefficient with time, an effect opposite to that of pressure. The magnitude of the contribution of the concentration dependence to deviation of the sedimentation coefficient a t zero time can be seen from a comparison of columns 8 and 9 in Tables IIa and IIb. In cyclohexane a t 3 Z 0 , a poor solvent, the concentration dependence is small. At a concentm tion of 0.44 g./lOO ml., B / ( S o ) is 2 only about 4% lower than m, a t the highest velocity. The influence of concentration increases, as one would expect, with decreasing speed and with an increase in the value of k . Thus, in cyclohexane a t 29,500 r.p.m. and a concentration of 0.44 g./lOO ml., m is 13% higher than B,/(X0)2. In toluene, a good solvent, m is greater than B / ( S 0 ) by 2 12% even a t top speed. Although this effect of concentration on the slope is apparent in each individual case, it should be noted that the correction is of the order of the precision of the determination. In fact, at high speed, where the pressure correction is most important, the correction for concentration can be ignored, within experimental error, for the poor solvent system and probably in the good solvent, as well. If there is any effect of speed of centrifugation, or, to a limited extent, molecular weight on p, the data presented in Table JV do not show any indication of a trend, It is probably safe to conclude from our results that the values of p obtained for polystyrene in cyclohexane agree within experimental precision regardless of the conditions of speed or molecular weight. Although the definition3 of p has been given in terms of the variation of the frictional coefficient with preswre, it has been the practice’ to use the
VOl. 66
coefficient describing the pressure change of the solvent viscosity in its place. Following this procedure and using the values for this coefficient and the compressibilities of the solvent summarized by Baldwin and van Holde,14and a value for the compressibility of polystyrene found by Cheng and Schachman,15p has been calculated. For the polystyrene-toluene system, a value of p = 1.0 X lo-$ was calculated and for polystyreneThe value which cyclohexane,-p = 2 X Wales foundJ by his curve fitting technique for polystyrene-cyclohexane was p = 2.2 X loF9. In light of the precision of our data, the agreement may be considered to be very good. The compatibility of our approach with that of obtained by Wales5is shown by plotting log Soo, both techniques us. the log X,,as shown in Fig. 3 . Even though Wales’ raults were obtained a t 34O, Fig. 3 will serve to illustrate that the two methods appear to be consistent. Finally, it can be seen from eq. 4 that a knowledqe of p is not needed to obtain the sedimentation coefficient independent of pressure, However. knowledge of p is necessary to calculate So at some point in the cell, such as in boundary spreading analysis. In such a case, p can be obtained directly from the ultracentrifuge experiment with sufficient accuracy to correct the data for the effects of pressure. DISCUSSION R. T R A U T M(Plum A ~ Island Animal Disease Laborator) , USDA).-Did the true zero sedimentation time show a systematic deviation from the time a t 2 / 3 operating speed? Would this be less in a non-pressure dependent system? I. H. BILLIcK.-The zero time correction was randomly scattered both positively and negatively about the time a t 2/a operating speed. For the most part the correction was less than 1 min. I doubt that the fact the system is pressure deDendent would have any effect. (14) R. L. Baldwin and K. E. van Holde, Portsci~r. Nochpolgmei.. Foiech., 1, 451 (1960). (15) P. Y. Cheng and €1. K. Schachman, J . Am. Chem. Soc., ‘7‘7, 1488 (1955).
