Spherical balancing aggregate

At the simplest level, the rate of flow-induced aggregation of compact spherical particles is described by Smoluchowski s theory [Eq. (32)]. Such expressions may then be incorporated into population balance equations to determine the evolution of the agglomerate size distribution with time. However with increase in agglomerate size, complex (fractal) structures may be generated that preclude analysis by simple methods as above. 

The solution behavior of low molecular weight amphiphilic molecules has been intensively investigated in the past (12-16) with respect to the formation of liquid crystalline phases. In very dilute aqueous solutions, the amphiphiles are molecularly dispersed dissolved. Above the critical micelle concentration (CMC), the amphiphiles associate and form micelles (Figure 4) of spherical, cylindrical or disc-like shape. The shape and dimension of the micelles, as a function of concentration and temperature, are determined by the "hydrophilic-hydrophobic" balance of the amphiphilic molecules. The formation of spherical aggregates is preferred with increasing volume fraction of the hydrophilic head group of the amphiphile, because the... 

The spontaneous aggregation of ionic surfactants produces an array of structures, whose form depends on the surfactant structure and concentration, oil type and concentration, and temperature (Figures 3 and 4). A completely mixed, singlephase micellar solution is at its minimum free energy. This free energy is determined by multiple intermolecular and ionic interactions that include electrostatic, dipole-dipole, ion-dipole, ion-ion, and dispersion (van der Waals) interactions. At equilibrium, the opposing forces are in balance and the net force is zero. The balance determines the size and shape of the micelle. Figure 5(a) is an older conceptual representation of a spherical micelle in aqueous solution... 

By substituting i = 1 in eqn [122], one finds the line of coexistence between lamella and cylindrical micelles, whereas substitution of i = 2 specifies conditions of coexistence between cylindrical and spherical micelles. The lengths Na and Nb of the soluble and insoluble blocks control the intramolecular solvophiUt solvophobic balance and the morphology of the equilibtirrm aggregates. 

Other types of aggregates such as spherical or cylindrical micelles or bilayers can be formed depending on the balance of the sizes of the hydrophobic tail and the hydrophilic head. 

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