282 Chapter 8. Chemical forces and self-assembly[[Student version, January 17, 2003]]
get the relative osmotic activity in terms of the total concentration of amphiphiles. Looking at the
experimental data in Figure 8.6, we see that we must takec∗to be around 1mM;the fit shown used
c∗=1. 4 mM.Twocurves are shown: The best fit (solid line) usedN=30, whereas the poor fit of
the the dashed line shows thatNis greater than 5.
Certainly more detailed methods are needed to obtain a precise size estimate for the micelles
in the experiment shown. But we can extract several lessons from Figure 8.6. First, we have
obtained a qualitative explanation of the very sudden onset of micelle formation by the hypothesis
that geometric packing considerations select a narrow distribution of “best” micelle sizeN.Indeed
the sharpness of the micelle transition could not be explained at all if stable aggregates of two,
three,... monomers could form as intermediates to full micelles. In other words, many monomers
must cooperate to create a micelle, and thiscooperativitysharpens the transition, mitigating the
effects of random thermal motion.Wewill revisit this lesson repeatedly in future chapters. Without
cooperativity, the curve would fall gradually, not suddenly.
8.5 Excursion: On fitting models to data
Figure 8.6 shows some experimental data (the dots), together with a purely mathematical function
(the solid curve). The purpose of graphs like this one is to support an author’s claim that some
physical model captures an important feature of a real-world system. The reader is supposed to
see how the curve passes through the points, then nod approvingly, preferably without thinking too
muchabout the details of either the experiment or the model. But it’s important to develop some
critical skills to use when assessing (or creating) fits to data.
Clearly the model shown by the solid line in Figure 8.6 is only moderately successful. For one
thing, the experimental data show the relative osmotic activity dropping below 50%. Our simplified
model can’t explain this phenomenon, because we assumed that the amphiphiles remain always fully
dissociated. For example, in the model, Na+ions always remain nearly an ideal solution. Actually,
however, measurements of the electrical conductivity of micellar solutions show that the degree of
dissociation goes down as micelles are formed. We could have made the model look much more
like the data simply by assuming that each micelle has an unknown degree of dissociationα,and
choosing a value ofα< 1 which pulled the curve down to meet the data. Why not do this?
Before answering, let’s think about the content of Figure 8.6 as drawn. Our model has two
unknown parameters, the numberNof particles in a micelle and the critical micelle concentration
c∗.Tomake the graph, we adjust their values to match two gross visual features of the data:
- There is a kink in the data at around a millimolar concentration.
- After the kink the data start dropping with a certain slope.
So the mere fact that the curve resembles the data is perhaps not so impressive as it may seem at
first: We dialed two knobs to match two visual features. The real scientific content of the figure
comes in two observations we made:
- Asimple model, based on cooperativity, explains the qualitativeexistenceof a sharp
kink, which we don’t find in simple two-body association. The osmotic activity of
aweak, ordinary acid (for example, acetic acid) as a function of concentration has
no such kink: The degree of dissociation, and hence the relative osmotic activity,
decreases gradually with concentration.