KOTES
348
For polystyrene' and polyisobutylene,6 h has been found to decrease with increasing M and/or c, and it is of interest to compare the behavior of poly-amethylstyrene with that of these other polymers under corresponding conditions. For this purpose, the mean square end-to-end distance ( r 2 ) is a more rational measure of molecular size than M , and so this has been calculated for all the solutions concerned from the relation ( r 2 ) = a2((r02)/M)M. The values of ( r o z ) / M were taken from the review of Stockmayer and Kuratas; the expansion factor a was taken as
Table I : Parameters and Derived Calculations"
x
1M 10-8
349 630
Coucn., c wt. x 102,
log
g./cc.
poises
2 3 I
2.88 4.32 1.45 2.88 4.32
20.4 m 42.6 8.9 m 25.0 m 57.6 15
:!
2
log
Mve/
S
IM,,
M
2.368 2.368 2.368 2.368 2.236
5.73 5.82 5.82 5.97 6.00
0.19 0.28 0.02 0.17 0.20
7,
7;
h
log
0
v)
o
D
o
0
'.i
-
15
e
a
eo
0
0
0
OD
2
99
Q 00
63-
0 0 .
2.5 0
I5 0
:',:
,
1
TI
-2.24 -1.98 -2.30 -1.90 -1.62
All a t 25".
unity for poly-a-methylstyrene in A-1248; and for the other solutions a was calculated from intrinsic viscosity data,',6 using eq. 30, 34, and 38 of ref. 4. The comparison is shown in Figure 2 where values of h are located on a logarithmic map of c vs. (r2). The polystyrene and polyisobutylene data are consistent in representing a nionotonic shift of h from m toward 0 (Zinim-like toward Rouse-like behavior) with increasing c and (P). The poly-a-methylstyrene has somewhat higher vaIues of h than does the polystyrene under comparable conditions. Thus, it is necessary to go to slightly higher concentrations or coil sizes to achieve a given change in effective hydrodynamic interaction corresponding to a shift toward Rouse-like behavior. However, there is no striking manifestation of internal stiffness or steric hindrance of the poly-a-methylstyrene, The logarithms of the molecular weights, M,,, obtained from the cross positions in Figure 1 and similar plots for the other solutions are also given in Table I. As usual, the apparent molecular weight from the viscoelastic measurements is somewhat too large, to a degree which increases with increasing concentration. Values of log T ' , the terminal relaxation time, calculated from the relatioq 7 1 = (7 - v17,)M,,/cRTS, are also given in Table I. Recent investigations by Lamb9 and Philippoff l o have shown that at very high frequencies 7' approaches a limiting value qp which is somewhat higher than T h e Journal of Physical Chemistry
w l
-s -1.4
-2.2 -12
-1 1
-10
Log < r 2 > Figure 2. M a p of values of parameter h as a function of concentration and mean square end-to-end distance for three polymers. Open circles, polystyrene' i+n A-1248; black circles, poly-a-methylstyrene in A-1248; crossed circles, polyisobutylene in Primol D.6
vS, suggesting that the frequency-dependent contribution of the polymer to the viscosity should appear as 7 - 7- rather than 7 - v1q8. For most of the solutions described here, the difference is probably relatively small. This question will be considered in a later communication. Acknowledgment. This work was supported in part by the U. S.Public Health Service under Grant GM10135. (8) W. H. Stockmayer and M. Kurata, A d e a n . Polymer Sci., 3, 196, (1963). (9) J. Lamb and A. J. Matheson, Proc. Roy. Soc. (London), A281, 207 (1964), and personal communication. (10) W. Philippoff, T r a n s . SOC.Rheol., in press.
Theoretical Refinement of the Pendant Drop Method for Measuring Surface Tensions by David Winkel Department of Chemistry, T h e University of W y o m i n g , Laramie, W y o m i n g (Received J u l y 27, 1964)
At present there are relatively few static methods suitable for measuring surface tensions under orthobaric conditions. Two examples are the sessile drop method1a2
NOTES
349
cascade. A snialler source of error arises because d, is measured on a sloping portion of the drop.
m
,
I
I
1
I
- p g b 2 / y , p is density, g is gravity, y is the surface tension, R/b is the reduced principal radius of curvature, x / b and y/b are reduced x- and y-coordinates, and 4 is shown in Figure 1. It is readily seen that the parameter controlling the shape of the drop is p while b controls the size of the drop. Since y = - p g b 2 / p , both b and p must be measured in order to obtain the surface tension. Any characteristic ratio of drop dimensions will be a function of p only, and conversely. Fordham4 has calculated d,/d, as a function of p, which as noted above is not the optimum ratio. The author has calculated dmaJdmin as a function of p. Their relative merits can be seen by referring to Figure 2 , which shows both d,/d, and dmaxldmin as functions of 6. As is readily seen, a given error in dmaJdmin causes a much smaller error in p than the same error in d,/d,. The method for determining b is the same as that of Andreas et aL3 Since X e / b is known from the calculations and
Figure 1.
There is no reason why other ratios cannot be used in the pendant drop method. The present paper describes a method using the ratio of the maximum to minimum diameter. This is the optimum method for two reasons: (1) the two diameters involved are independent, as are errors in each measurement; and ( 2 ) both diameters are measured where they are nearly independent of height. The method can best be understood by referring to the fundamental equation for drop shape.6
where b is the radius of curvature at the vertex, ,8
1
=
2Xe
=
dmax
(1) F. Bashforth and J. C. Adams, “An Attempt to Test the Theories of Capillary Action,!! University Press, Cambridge, England, 1883. (2) A. M.Worthington, PhiE. Mag., 20, 51 (1885). (3) J. M.Andreas, E. A. Hauser, and W. B. Tucker, J . Phys. C h a . , 42, 1001 (1938). (4) S.Fordham, Proc. R o y . SOC. (London), A194, 1 (1948). (6) D. 0. Niederhauser and F. E. Bartell, “Report of ProgressFundamental Research on the Occurrence and Recovery of Petroleum,” 1948-1949, p. 114. (6) A. W.Adamson, “Physical Chemistry of Surfaces,” Interscience Publishers, Inc., New York, N. Y., 1960.
Volume 69,Number 1
January 1966
NOTES
350
therefore
(4) where
H
=
-46
(5)‘
Since p does not appear explicitly, only tables of H us. dmax/drni,, = R have been calculated. The method of solution was in principle somewhat similar to a graphical method due to Lord Kelvin7 except for several important modifications. Assume X r / b and Y,/b to be known. Trial values of X,+,/b and Y,+l/b (X’{+Iand Y’t+l)can be found by
by a grant. Dr. R. S. Hansen contributed helpful discussion during the course of the work. Indiana University supplied time on their IBM 709 for an independent recalculation which confirmed the G-1.5 calculation. (7) W. Thomson, Nature, 34, 290 (1886).
Interaction of Alkali Metal Cations with Silica Gel1 by H. Ti Tien Department of Chemistry, Northeastern University, Boston, Massachusetts (Received September 24,1964)
where S/b is the independent variable and A@ = AS/R,. A trial value of Rr+l/b,R’i+l/b, can be calculated from
_b
~
b
A corrected (unprimed) value of l/(Rt/b) is obtained from
and from this corrected (unprimed) values of Xf+l/band Y,+Jb can be readily calculated by repeating the entire calculation once using the corrected value of R,/b. Convergence was good enough so that increments of 0.01 could be used for AS/b. Furthermore, this method is well adapted for an electronic computer. The above procedure was programmed in machine language on a Bendix G-15 computer and solutions obtained for -0.35 < p < -0.60 in steps of 0.001. From these, tables of X,,, and Xminus. p were obtained by second-order interpolation using Bessel’s formula. Consistency was checked by noting the constancy of higher order differences. From these tables R and H us. p were in turn calculated. Again consistency was checked from higher order differences. A table of H us. R was next obtained by linear interpolation and is to be published separately. Acknowledgments. The author is grateful to the Petroleum Research Fund, which supported this work T h e Journal of P h y W Chembtrg
The equilibrium selectivity order of an ion-exchanging system for the alkali metal ions is usually either that of the Hofmeister series2 or the sequence which follows the crystal radii.3z4 In cases in which these series are not observed, there are two theories which have been advanced recently explaining their exi~tence.5-~Maatman, et al., reported equilibrium exchange studies between alkali metal cations (Li, K, Xa) and silica gel.* They found that the lithium ion is less preferred than either Ka+ or K+, whereas no difference in selectivity coefficients was observed between Na+ and K+. They estimate the pK value of the silanol group of silica gel to be about 6-8. Further, Maatman, et al., compare the environment of the oxygen atoms of -0SiO- and OH- groups in aqueous solution. From their results they conclude that the reactions of these groups with the metal ions are similar. Previously, Dalton, McClanahan, and Maa tmang measured the equilibrium ~~
(1) The experimental work was done at Department of Basic Research, Eastern Pennsylvania Psychiatric Institute, Philadelphia, Pa., while the author was on the staff, 1957-1963. (2) 0. D. Bonner, J . Phys. Chem., 59, 719 (1955). (3) H.P. Gregor, M. J. Hamilton, R. J. Oaa. and F. Bernstein, ibid., 60,266 (1956). (4) C. E. Marshall and G. Garcia, ibid., 63, 1663 (1959). (5) (a) D. 0. Rudin and G. Eisenman, Abstr. C o m m u n . e l s t Congr. Physiol., Buenos Aires, 237 (1959); (b) G. Mattock, “ p H Measurement and Titration,” Macmillan Co., New York, N. Y., 1961, pp. 130-134. (6) G. N. Ling, J . Gen. Physiol., 43, 149 (1960). (7) H.T.Tien, J. P h y s . Chem., 68, 1021 (1964). (8) D. L. Dugger, J. H. Stanton, B. N. Irby, B. L. McConnell, W. W. Cummings, and R. U‘.Maatman, ibid., 68,757 (1964). (9) R. R. Dalton, J. L. McClanahan, and R. W. Maatman, J . Colloid Sci., 17, 207 (1962).
