Free energy of aggregation

If the head groups have a permanent dipole moment they can interact at the aggregate surface. The dipoles at the surface of the micelle are oriented normal to the interface so that the poles of the dipoles are located on parallel surfaces. The dipole-dipole interaction for such orientation provides a repulsive contribution to the free energy of aggregation. The latter for the spherical micelles can be estimated by considering that the poles of the dipoles create a capacitor, which has the distance between two planes equal to the distance of charge separation or the dipole length d... 

AB diblock copolymers in the presence of a selective surface can form an adsorbed layer, which is a planar form of aggregation or self-assembly. This is very useful in the manipulation of the surface properties of solid surfaces, especially those that are employed in liquid media. Several situations have been studied both theoretically and experimentally, among them the case of a selective surface but a nonselective solvent [75] which results in swelling of both the anchor and the buoy layers. However, we concentrate on the situation most closely related to the micelle conditions just discussed, namely, adsorption from a selective solvent. Our theoretical discussion is adapted and abbreviated from that of Marques et al. [76], who considered many features not discussed here. They began their analysis from the grand canonical free energy of a block copolymer layer in equilibrium with a reservoir containing soluble block copolymer at chemical potential peK. They also considered the possible effects of micellization in solution on the adsorption process [61]. We assume in this presentation that the anchor layer is in a solvent-free, melt state above Tg. The anchor layer is assumed to be thin and smooth, with a sharp interface between it and the solvent swollen buoy layer. 

There is a point at which these aggregates reach a critical size of minimum stability r and the free energy of formation AG is a maximum. Further addition of material to the critical nucleus decreases the free energy and produces a stable growing nucleus. The nucleation rate is the product of the concentration of critical nuclei N given by... 

Polymorphism occurs whenever a given component exists under different aggregation states as a function of P and T. The stable state requires that the chemical potential of the component (hence, the Gibbs free energy of the phase for a phase composed of a single component) be at minimum at equilibrium. Figure 2.4 shows examples of G-T plots for Al2Si05 in various P conditions. 

The Gibbs free energy of phase y is represented by a straight line connecting the standard state potentials of the two end-members in the mixture. Because we use the term mixture, it is evident that the standard state of both end-members is the same and is that of pure component. The two components are totally immiscible in any proportion and the aggregate is a mechanical mixture of the two components crystallized in form y ... 

A simple mean field theory for micelle formation by a diblock copolymer in a homopolymeric solvent was developed by Leibler et al. (1983). This model enables the calculation of the size and number of chains in a micelle and its free energy of formation. The fraction of copolymer chains aggregating into micelles can also be obtained. A cmc was found for low copolymer contents even for weak incompatibilities between components. Leibler et al. (1983) emphasize that fora finite aggregation number p, the cmc is a region rather than a well-defined concentration and some arbitrariness is involved in its definition. 

Existing Ag aggregates. By adding Ag atoms, the aggregate increases in size and reduces the free energy of the system. 

Aggregates of other impurities, such as chemical sensitization centers. The addition of Ag atoms to them will also reduce the free energy of the whole system. 

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