Micellar Shapes and Shape Transitions

The aggregation of surfactants in solution is governed by the ordinary laws of chemical thermodynamics, or mass action, which dictate that at equilibrium the chemical potential ij, of aggregates containing n surfactant molecules is the same for all n, including n = 1—that is, isolated surfactants, or unimers. Thus (Tanford 1980 Israelachvili 1992) 

Now let no be the value of n that minimizes /r , and assume that o 1. As we shall soon see, we can also assume that /z — is greater than ks T for n significantly less than nQ. Aggregates of size n or greater will then tend to be favored over aggregates of other sizes. Then we define 

From Eq. (12-5), we then find that X g — rio(Xi/ X ) . Since no is large, it follows that as X increases, X will remain very small until X approaches X, and then X g will grow rapidly. Since X is a mole fraction, it cannot exceed unity, and therefore X must remain less than X. Hence, as the total inventory of surfactant increases from zero, most of it remains isolated as unimer until the mole fraction of surfactant approaches X, and abruptly thereafter almost all additional surfactant will aggregate into micelles of size o or so (see Fig. 12-3). The concentration at which most additional surfactant forms large aggregates is called the critical micelle concentration, or CMC, and its value is close to X. The detailed distribution of micelle sizes is controlled by the dependence of on n. 

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