280 Chapter 8. Chemical forces and self-assembly[[Student version, January 17, 2003]]
00.20.40.60.810 0.02 0.04 0.06 0.08 0.1relative osmotic pressurepotassium chloridepotassium oleate√
c,withcinmMCMCFigure 8.6:(Experimental data, with fits.) Comparison of the osmotic behavior of a micelle-forming substance
with that of an ordinary salt. The relative osmotic pressure is defined as the osmotic pressure divided by that of an
ideal, fully dissociated solution with the same number of ions. To emphasize the behavior at small concentration,
the horizontal axis shows√c,wherecis the concentration of the solution. Solid symbols are experimental data for
potassium oleate, a soap, while the open symbols are data for potassium chloride, a fully dissociating salt. The solid
line shows the result of the model discussed in the text (Equation 8.33 withN=3 0 and critical micelle concentration
- 4 mM). For comparison the dashed line shows a similar calculation withN=5.The model accounts for the sharp
kink in the relative osmotic activity at the CMC. It fails at higher concentrations, in part because it neglects the
fact that the surfactant molecules’ headgroups are not fully dissociated. [Data from McBain, 1944.]
something gentler: the hydrophobic effect, an entropic force.
As early as 1913 J. McBain had deduced the existence of well-defined micelles from his quantita-
tive study of the physical properties of soap solutions. One of McBain’s arguments went as follows.
Weknow how many total molecules are in a solution just by measuring how much soap we put in,
and making sure that none of it precipitates out of solution. But we can independently measure
how many independently moving objects the solution contains, by measuring its osmotic pressure
and using the van ’t Hoff relation (Equation 7.7 on page 220). For very dilute solutions McBain
and others found that the osmotic pressure faithfully tracked the total number of amphiphilic ions
(solid symbols on the left of Figure 8.6), just as it would for an ordinary salt like potassium chloride
(open symbols in Figure 8.6). But the similarity ended at a well-defined point, now called thecrit-
ical micelle concentrationorCMC.Beyond this concentration,the ratio of independently moving
objects to all ions dropped sharply(solid symbols on the right of the graph).
McBain was forced to conclude that beyond the CMC his molecules didn’t stay in an ordinary
solution, dispersed through the sample. Nor, however, did they aggregate into a separate bulk
phase, as oil does in vinaigrette. Instead, they were spontaneously assembling into intermediate-
scale objects, bigger than a molecule but still microscopic. Each type of amphiphile, in each type
of polar solvent, had its own characteristic value of the CMC. This value typically decreases at
higher temperature, pointing to the role of the hydrophobic interaction in driving the aggregation
(see Section 7.5.2 on page 243). McBain’s results were not immediately accepted. But eventually,
as a large number of physical quantities were all found to undergo sharp changes at the same
critical concentration as the osmotic pressure (for instance, electrical conductivity), the chemical
community realized that he was right.