1932
D.
JfCISTYRE,
A. J Y I M S , L.
c. W I L L I A M S , A N D L. ?(/IANDELKERN
layer in the presence of oil molecules produces a monolayer penetration and pressure rise a t constant area. As i n d i ~ a t e d , duplex ~ formation corresponds to the disappearance of minute droplets of the now spreading hydrocarbon. When spreading on the monolayer occurs, penetration of the hydrocarbon molecules into the cephalin monolayer also takes place. Since the surface tensioii of the hydrocarbons decreases with diminishing chain length (hexadecane 29 dyneslcm.; octane 20.5 dynes,'cm.), enhanced spreading and penetration likewise tqke place on decreasing the hydrorarbon chain length. A similar phenomenon is observed in the formation of microemulsions where it is necessary for the
Vol. 66
oil phase molecules to have a smaller chain length than the interphase forming molecule^.^ Likewise, the aromatic hydrocarbons (m-xylene) have high surface tensions and will not spread or penetrate into the cephalin or stearic acid monolayers a t the air /water and oillwater interfaces. Presumably, the non-interaction of Mg ions with the cephalin duplex films is due to their inability to break the internal salt linkage (phosphoric acidamine). Acknowledgments.-This investigation was supported by P. H. s. Research Grant No. B-2067 (C) from the Xational Institute of Eeurological Diseases and Blindness. (9) J H Schulman, W. Stoeokenius, and L. 11. Prince, J Chem., 63, 1677 (1959).
I'hys
CONFORMATION ASD FRICTIOSA4LPROPERTIES OF POLYSTYRENE IK DILUTE SOLUTIOSS BY D. MCIKTYRE, A. WINS, L. C. WILLIAMS,AND L. MANDELKEKN Polvmer Structure Section, Xational Bureau of Standards, Washington $5, D. C. Received March 16, 196%
A high molecular weight fraction of polystyrene was studied by light scattoering, sedimentstion, and viscosity. Measurements were made in cyclohexane at 35,45, and 55", and also in benzene at 40 . It was found that the intrinsic viscosity and frictional coefficient of the polymer were exponential functions of the expansion factor. The equations that fit the dats are: The relations are in good agreement with recently published theories of Kurata[ q ] / [ q ] s = 2 . 3 3 and (f/q)/(f/q)s = 010.6. Yamakawa and Kurata-Stockmayer-Roig.
The hydrodynamic properties of polymer solutions have been studied intensively in the past fifteen years in order to assess the yalidity of the many theoretical descriptions of the conformation of the polymer chain, its variation with solvent iiiteraction, and its effect on the molecular frictional coefficient. To examine these effects systematically requires the concurrent measurement of the radius of gyration of a polymer fraction by light scattering, the virial coefficient by light scattering or osmotic pressure, and the frictional coefficient by viscosity, sedimentation, or diffusion. The measurements ought to be made on several fractions of known molecular weight and molecular weight distribution over a large molecular weight range. There are irery few such comprehensive studies reported in the literature. Perhaps the most thorough study is that of 0 t h and Desreux,lt2who examined a series of fractions of polystyrene from 100,000 to 3,000,000 molecular weight by light scattering, viscosity, and sedimentation. Another less extensive study of the polystyrene system was made by Krigbaum and Carpenter3 and earlier by Outer, Carr, and Zimm.4 The latter studied several fractions of polystyrene in diff erent solvents by light scattering and viscosity measurements; the former studied a single high molecular weight fraction, molecular weight 3.2 X IO6. (1) J. 0 t h and 1'. Desreux, Bull. soc. c h ~ m Belyes, . 63, 285 (1964). (2) J. 0 t h and V. Desreux, tbzd., 66, 303 (1957). (3) IT. R. Krigbaum and D . K Carpenter, J . Phys. Chem., 69, l l G 6 (195.5). (4) P. Outer, C. I. Carr, and B. H. Zimm, J . Chem Phlis. 18,830 tl9iO)
In one respect these two studies differ from the study of 0 t h and Desreux. In going from a poor solvent' (a theta solvent) to a good solvent the polymer coil in solution expands. 0 t h and Desreux's work indicat'es that the expansion remains gaussian whereas the other work indicat,es t8hatit does not. A very recent study by Kirst'e and Schulz5,6 on polymethyl methacrylate fractions also shows a non-gaussian expansion. Theoretical work has made very rapid progress during this int'erval of experimental work, and there is a pressing need for more extensive and reliable experiment'al data. The theoretical as lye11 as experiment'al situation has been summarized in two recent r e v i e ~ s . The ~ ~ ~ hydrodynamic relations proposed by Florygsl/l for the present sample over a very large virial coefficient range. Although it appears on this scale to be almost linear, on a larger scale the benzene value definitely falls below the line that satisfies the cyclohexane data. The experimental value for the slope of the plot of (S2)z'/g against A 2 is 0.366 X IO4. If the experimental value for is put into eq. 18 on the assumption of a monodisperse polymer (or h = 00 in eq. 19), the theoretical value is calculated to be 0.328 X lo4. A polydispersity corresponding to h = 4 would decrease this slope by less than 5%. It is interesting to compare the other published data. 011 replotting the data of (@)z'/s against A2 for references 3 and 4 the slopes are calculated to be 0.63 X lo4 and 0.255 >( lo4, respectively. These slopes are based upon ploints very close to theta and should correspond closely to the theoretical calculations. Since the value of ( R ) , 2 / M , in Table VI1 is the same for thicr work and that of reference 16, the ratio of the experimental slopes should be proportional to the square root of the molecular weights, or 0.63. The actual ratio of the experimental slopes is 0.76. Since the points of reference 3 are on a gradual curve it is possible to take the portion representing the higher virial coefficients and compute a slope of approximately 0.4 x 104. The virial coefficients in this work seem high, especially when they are compared to the values of reference 3 for a comparable molecular weight and temperature in cyclohexane. The refractive index increment, in reference 3 is considered to be constant, whereas in this work it is not. Correspondingly, the light scattering constant K increases in this work by about 6%. This would make higher virial coefficients if applied to Fig. 3 of reference 3 with a fixed intercept, but it is not known whether this can totally account for the difference. Intrinsic Viscosity.-Figure 4 shows the variation of the intrinsic viscosity a t zero shear rate with the virial coefficient. Figure 5 presents the dependence of the intrinsic viscosity on the experimental radius of gyration. The data also may be expressed in the form shown in eq. 20. If the cor-
[aI/[als
=
Q2.33
(20)
rected radii of gyration in Table VI are used instead of the values in Table V, the exponent in the relation can be changed to 2.53. Kurata, Yamakawa, and Utiyama' have shown theoretically that the exponent for this equation in the vicinity of theta is 2.43, when the polymer is monodisperse. When the sample is polydisperse, the exponent, n, would be given by eq. 21. (eq. 27 in reference 7) for an exponential distribution.
1039
The data in the vicinity of theta from the present work would give an exponent of 2.26. The dotted line in the graph indicates the slope for an exponent of 3. The relation given by Flory in eq. 2 is based upon the use of a number average radius and molecular weight in a polydisperse system.21 In order to calculate the parameter 9 it is necessary to correct the measured weight average molecular weight and x-average radius. Krigbaum and Carpenter have given corrections for this q ~ a n t i t y . ~ For corrections near the theta temperature for a polydispersity parameter h = 4, CP can be corrected to give a theta value for these data of 2.5 X 1021,which decreases with A2. This decrease in 9 was noted earlier by Krigbaum and C a r ~ e n t e r . Recently ~ some additional data on polymethyl methacrylate have been presented by Schulz.6 Taking the values from their graph and normalizing their theta value of 9 to 2.5 X loz1the plot in Fig. 6 is obtained. It should be noted that the curve for these polystyrene results shows a monotonically decreasing function for 9 for the entire solvent range. Sedimentation Frictional Coefficient.-It is well known that the frictional coefficient of a molecule is related to its radius. In the model proposed by Flory eq. 22 was derived. Kurata and Yamakawa,13 by considering the effect of the nongaussian expansion of the polymer molecule, arrived at eq. 23.
f/v0
(23) The frictiona,l coefficient is related to the sedimentation constant, So, by eq. 24. Thus it is necessary to know several additional factors besides
flao
=
p(~)01"cyo.7652
=
M(l - ap)
aoso
the sedimentation constant to calculate the frictional coefficient. Even the sedimentation coefficients determined during a run must in principle be corrected for pressure effects on the viscosity and density. Since the effect of pressure on the partial specific volume is unknown, this factor is considered constant. I n these experiments, in which the temperature of the solvent was varied, no pressure corrections were applied to the density and viscosity because accurate corrections are not known, The only data available on the pressure dependence of viscosity in cyclohexane are those of Bridgman.29 The data are given at 30 and 75'. At 75' both cyclohexane arid benzene have about the same pressure dependence. The corrections made for cyclohexane at 35' increase the sedimentation coefficient by lVo. Since the exact corrections a t other temperatures are unknown, and the data were obtained under the same conditions, it is felt that all of the data should be left uncorrected. The partial specific volume was considered to be constant (0.93) over the various temperatures in cyclohexane, and was set equal to 0.92 for benzene at 40'. The results of the partial specific volume (29)
P. W. Bridgman, Proc. Am. Acad. Arts Sci., 61, 67
(1920).
determination are very interesting in themselves, but the data are not sufficiently accurate to establish any information about the enthalpies of dilulion. They strongly suggest that the simple explanation of Streeter and Boyer30 is not correct. They had found that when therc were either very strong solvent-polymer interactions (good solvents) or very strong polymer-polymer interactions the partial specific volume had a maximum value. At the intermediate cases there was a decrease of the partial specific volume, Cyclohexane at 55' is an intermediate case although there is some expansion of the polymer size simply due to an increase of temperature, yet the e does not change. The data are in good agreement with the data of Rosen31and Griffel, Jessup, Cogliano, and Parks.32 Whether the change in the apparent specific volume at low concentrations is real can be settled only by more refined work. Roseii found just the opposite trend, and his differential measurements were far more sensitive to the difference of densities needed to calculate the apparent specific volume. The dependence of the frictional coefficient on the polymer size and on the virial coefficient is shown in Fig. S and 4, respectively. It should be noted that the dotted line of Fig. 5 has a slope of unity. The frictional coefficient follows eq. 23, where an exponent of 0.6 is used because the uncertainty in the data does not allow an accuracy better than 10.05 to be assigned to the slope. The value of 0.6 for the slope agrees closely with the value given by Kurata.
(f/d/(f/d8
=
r-lo*6
(25)
There are in conclusion several experimental studies including this study of polystyrene that indicate that the intrinsic viscosity and the frictional coefficient increase less than would be predicted on the basis of the original theory of Flory. This study has shown that the intrinsic viscosity and the frictional coefficient are related to the expansion factor, 01, by an exponent that agrees well with the theories of Kurata and Yamakawa. Also, the increase of the radius of the polymer molecule with the virial coefficient agrees well with recent theories of the excluded volume. These recent theories are valid at temperatures not far from theta. (30) D. J. Streeter and R. F. Boyer, Ind. Eng. Chem., 48, 1790 (1951). (31) B. Rosen, tbad., 27, 559 (1956). (32) &I. Gnffel, R. 8. Jessup, J . C. Cogllano, and R. P. Parks, J. Res. Natl. Bur. Std., 52, 217 (1954).
It therefore is surprising that the effects seem to be nearly the same in solvents that are far from theta solvents. However, Knrata, Stockmayer, and Roig found that the molecular weight dependence of the intrinsic viscosity waq ill better agrecmrnl with cxperimmtal rwult i in good iolveritq when they used the ratio of intrinsic viscositieq under any solvent condition to that at theta as ~ 1 and~ con. sidered an ellipsoidal expansion of the molecule far from theta conditions.
DISCUSSION H lVT'aRD.-This comment concerns the problem of determining number average molecular weights from ultracentrifuge data. In principle it seems feasible t o measure number average sedimentation constants quite directly by light absorption provzded that the sedimenting species all carry the same number of light absorbing groups per molecule. This could be effected, for example, by tagging the ends of a linear polymer with a chromophore. D. >fCINTYRE.-The determination of the number-average molecular weight is possible if at any point in the cell either Xn is known or the concentration is zero. As Dr. TTJard indicates, the performance of an end group analysis and the simultaneous determination of the total concentration at any point in the cell would allow the number-average molecular weight of the sample to be determined.
S. R. ERLAKDER (Iiorthern Regional Research Laboratory).-JT7ith regard to Dr. Ward's statement, work has been done on amylose by tagging the end group of each molecule with a radioactive unit. I see no reason why the tagging method cannot be applied to ultracentrifuge work to obtain JI,. In addition, one can for practical purposes assume that the concentration at the meniscus (C,) is equal to zero if th? schlieren pattern coincides with the base line at the meniscus. Under the conditions that C, = 0, the numberaverage molecular weight at the cell bottom equals the total weight-average molecular weight as given by Yphantis. One can use the previously determined M," t o obtain a more accurate extrapolation of the pattern at the cell bottom. Hence, the number-average molecular weight could be obtained from sedimentation-equilibrium studies by first obtaining *If,+and then obtaining a pattern where the value of C, is essentially zero. D. hfcIxTYRE.-The earlier experimental work of Rilliams and Wales used the above method to determine the number-average molecular weight. The problem of establishing a point in the cell where the concentration is definitely known to be zero is, of course, difficult. H. A. ESDE(Chemstrand Research Center).-You point out the importance of knowing the molecular weight distribution of the polystyrene samples in order t o interpret more accurately the results of your measurements and that neither in the present work nor in previously published studies has the distribution been definitely knomm. Recently J. J. Hermans and I have used the density gradient technique t o determine Xn and ii,", or any other niolecular weight average. This method utilizes the moments of molecular weights which can easily be related to moments of the concentration distribution in the polymer band formed by density gradient centrifugation.
~
~
