Estimation of Micelle Aggregation Number and CMC

In one simple model of micellization (Israelachvili et al., 1976), the standard free energy difference pa- surfactant molecule between N-mer and monomer is given by 

As the area a per smfactant molecule increases, interfadal energy increases, but repulsive energy decreases. It is easily shown that the right-hand side of Equation 4.E1.1 is minimized for 

This is the preferred head group area based on raiergetic considerations. With the above values for y and Cj, the resulting value of a is 0.57 mn.  

For small values of N, this equation yields a a, which is energetically unfavorable. When N reaches a particular value a becomes equal to a. For larger values of N, a a , which is also unfavorable energetically. (A permissible solution in this case is a nonspherical shape, such as elhpsoidal, for which a = a .) Since entropic considerations favor small values of N, we anticipate that micelles will be distributed around a mean aggregation munber sUghdy less than N. With the values of various parameters given above, we find = 63. 

Tanford (1980) has a different expression for repulsion in Equation 4.E.1 given by cja due to their eleetrostatic charges. Note the different dependraice on a. His surface energy term is also proportional to a - a, and not just a, where is the area covered by the polar group of the surfactant molecule. 

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