1818
-
NOTES
.,-
0
8-
n
P
0
.
SILOXANE DIMER
w
W _I
-a
LY
L
1
0
6 -
c. v
0
v1
An aggregation number containing 18 acid residues appears to be the ultimate micelle size reached in low polarity solvents for both the bivalent and monovalent soaps since the micellar phase begins to separate as a liquid-like phase below a solvent solubility parameter of 6.0.’ An ultimate aggregation number of 18 residues appears to be consistent with the most probable spatial geometry of the dinonylnaphthalenesulfonates. Rotation of representative molecular models in space indicate that an average solid angle of 26” is generated by tangents between the polar head group and the bulkiest section of the hydrocarbon residue. As an exercise, one can construct spheres from cones of various solid angles and thus count the number of cone units which make up the sphere. From simple geometry, however, one may
8 w,,,,,,,. CYCLOHEXANE
E !A
4 -
m
I z 3
z
0
2
0--.--02-
W 0
U
0 .
0
I
I
I
I
0 05
0 IO
0 15
0 20
and a correction for those areas of the sphere not covered by conical curved surfaces is as follows
w
-I
16
L
N = 1.05128 4 8 ? 4
0.
$6I
z
-
I
r
fz .
-
z
0
2 w
8 2 4
I
I
I
I
1
following conditions are met: (1) that the micelles be essentially monodisperse in the given solvent ; ( 2 ) that the micelles be spherical in structure and contain a polar core; (3) that the condensed micellar phase have an amorphous or liquid-like character; and (4) that the micellar concentration be at least an order of magnitude greater than the monomer concentration. Figure 2 summarizes the experimentally observed effect of solvent on barium sulfonate micelle size. Aggregation number in terms of monomers per micelle is plotted against the solubility parameter of the solvent. The values of & = 11.3 and K = 0.690 for the barium soap obtained from the plot compare with 8 2 = 10.5 and K = 0.295 for the alkali sulfonates. The greater solubility parameter value for the barium soap monomer reflects the increased polarity of its ionic head while the -greater K value is primarily a result of the larger degree of shielding per monomer unit by the bdrocarbon residues in the bivalent soap. T h e Journal of Physical Cherni&u
+ 0.487
Substituting 8 = 26” (0.436 radian) into the formula yields N = 18 acid residues for the fully packed micelle--in good keeping with the experimental results obtained in the polymethylsiloxane dimer (6 = 6.5) in which 8 monomer units or 16 acid residues are counted. I n addition, extension of the aggregation number os. solubility parameter plot of Figure 2 t o 6 = 6.0-
O n t h e Liquid Film Remaining in a Draining Circular Cylindrical Vessel by Paul Concus Lawrence Radiation Laboratory, Uniuersity of CaZijorniu, Berkeley, Calijornia 94720 (Received October 2S, 1969)
It is of interest to have a theoretical estimate for the thickness of the film that remains on the wall of a right circular cylindrical vessel as it is being drained of a wetting liquid. Such an estimate, which includes the effects of surface tension and viscosity, has been given by Levichl for the case where the contact angle is zero and the static meniscus away from the wall becomes essentially a horizontal plane. I n this note we
v. G , Levi&, “physicochemical Hydrodynanlic~,”PrenticeHall, Englewood Cliffs, N. J., 1962, p 681, eq 133.26.
NOTES
1819
describe how, with the aid of recently published graphs,2 Levich’s technique can be extended to obtain estimates for the cases where the static meniscus away from the wall may be curved, such as in a vessel of small radius or in a reduced gravitational field. The analysis presented for the flat plate film behavior on p 678, 679, and the top half of p 680 of ref 1 carries over unchanged for our problem. (The assumption that the film thickness be small compared with the radius of the vessel is, of course, needed.) The resulting expression for the constant asymptotic film thickness, ho, for a withdrawal velocity that is not too large can be written as ho = ~ . ~ ~ ( ~ u o / c J ) ” / ” / K
=
1
(Bfw
+
lw
(1)
where p is the liquid viscosity, vo is the bulk velocity of the fluid, CJ is the surface tension, and K is the curvature at the wall of a meridian of the static meniscus. (The coefficient 1.34 in eq 1was calculated using a computed value of 0.643 for 01, the asymptotic dimensionless film curvature, rather than the less accurate value of 0.63 given in eq 133.24 of ref 1.) The appropriate value of K to insert into eq 1 for any cylinder radius and gravitational field can be found from ref 2. From eq 1 of ref 2 one obtains that for a perfectly wetting liquid (0’ contact angle) K
f
- 1)
where a is the radius of the vessel and B = pga2/u is the bond number (a dimensionless parameter) where p is the liquid density and g is the gravitational acceleration; the values of f W , the dimensionless meniscus height at thewall (r = 1) and A, twice the dimensionless mean curvature a t the meniscus center ( r = 0), can be read directly as a function of B from the graphs for zero contact angle in Figures 2 and 5 of ref 2, (see Figure 1). Limiting values for K and ho for very large B (essentially horizontal meniscus away from the wall) and very small B (essentially hemispherical meniscus) can be obtained using the asymptotic expressions for fn. and X given in ref 2 . For very large B, eq E a , 18, and 23 of ref 2 yield K = (l/a) 42, so that
r =I Figure 1. Static meniscus meridian. The variables r and f, which are the ones used in ref 2 , are dimensionless. The radius of the cylinder is a. The quantity K denotes the meridian’s curvature at T = 1, and X/2a its curvature at r = 0.
tional to the vessel radius and depends more strongly on the surface tension.
Acknowledgment. This note is an outgrowth of a discussion with G. D. Bizzell, G. E. Crane, and H. :\I. Satterlee in which the author mas made aware of the problem and the pertinent contents of ref 1. The work mas performed in part under Contract KAS 311526 of Lockheed Research Laboratory with NASALewis Research Center and in part under the auspices of the U. S. Atomic Energy Commission. (2) P. Concus, J . Fluid Mech., 34, 481 (1968).
The Absolute Reactivity of the Oxide Radical Ion with Methanol and Ethanol in Water1 by R. Wander, Bonnie L. Gall, and Leon M. Dorfman Department of Chemistry, T h e Ohio State University, Columbus, Ohio 4 8 H O (Received Sovember 349 1969)
which is the same as the expression given by eq 133.26 of ref 1 (after the value of a is corrected). For very small B, eq 9 of ref 2 yields K = l/a, so that ho = 1.34~(p~o/a) ‘Ia
for this case. Note that for large B the film thickness is independent of the vessel radius and, as noted on p 681 of ref 1, is only weakly dependent on the surface tension. For small B , on the other hand, ho is propor-
The reactivity of the basic form of the hydroxyl radical, o-, formed in the radiolysis of water by the ionic dissociation283 of OH
OH
0-
+ €I+
(1)
(1) This work was supported by the U. S. Atomic Energy Commission. (2) J. Rabani and M. S. Matheson, J . P h y s . Chem., 70, 761 (1966). (3) J . L. Weeks and J. Rabani, ibid., 70, 2100 (1966). Volume r4>Number 8
April 1 6 , 1970
