Equilibrium micellar phases, composite

The mixture CMC is plotted as a function of monomer composition in Figure 1 for an ideal system. Equation 1 can be seen to provide an excellent description of the mixture CMC (equal to Cm for this case). Ideal solution theory as described here has been widely used for ideal surfactant systems (4.6—18). Equation 2 can be used to predict the micellar surfactant composition at any monomer surfactant composition, as illustrated in Figure 2. This relation has been experimentally confirmed (ISIS) As seen in Figure 2, for an ideal system, if the ratio XA/yA < 1 at any composition, it will be so over the entire composition range. In classical phase equilibrium thermodynamic terms, the distribution coefficient between the micellar and monomer phases is independent of composition. 

The maximum additive concentration (MAC) is defined as the maximum amount of solubilisate, at a given concentration of surfactant, that produces a clear solution. Different amounts of solubilisates, in ascending order, are added to a series of vials containing the known concentration of surfactant and mixed until equilibrium is reached. The maximum concentration of solubilisate that forms a clear solution is then determined visually. This same procedure can be repeated for the different concentrations of surfactant in a known amount of solubilisate in order to determine the optimum concentration of surfactant (Figure 4.24). Based on this information, one can construct a ternary phase diagram that describes the effects of three constituents (i.e., solubilisate, surfactant, and water) on the micelle system. Note that unwanted phase transitions can be avoided by ignoring the formulation compositions near the boundary. In general, the MAC increases with an increase in temperature. This may be due to the combination of the increase of solubilisate solubility in the aqueous phase and the micellar phase rather than an increased solubilization by the micelles alone. 

Analogously, at point C the phase separation into a microemulsion of the composition D and a micellar solution of inverse micelles containing solubilized water takes place. This is a Winsor I (WI) type equlibrium. At the intermediate point E the system consists of a single microemulsion phase (ME). At even lower surfactant concentrations (line c), depending on the water/hydrocarbon ratio, the system will either separate into one of the two-phase systems (Winsor I or Winsor II), or a three-phase system may form at point F. This is a Winsor III (Will) equilibrium with an aqueous phase at the bottom, a microemulsion phase in the middle and a hydrocarbon phase at the top. 

Whenever a system has a composition that lies in the polyphasic region, it will generally separate (at equilibrium) into two phases, the representative points of which are located at the two extremes of the tie-line (see Fig. 3). In most cases the tie-hne is inchned i.e., one of the phases is rich in surfactant because it is located relatively near A or far from the OW side. If it is also located far from the AW and AO sides, then it contains both W and O in sizable amounts and fits the definition of a microemulsion (shaded region). Near the upper end of the tie-line in Fig. 3, it is an O/W type microemulsion. The other extreme of the tie-line is located near the OW side and near one of the component vertices (O in Rg. 3) and thus contains essentially one of the components. It is called an excess phase, in this case an oil excess phase. In most cases, particularly with ionic surfactant, the excess phase does contain a very small concentration of amphiphile, about the critical micelle concentration (cmc). In other words, the excess phase does not contain micelles, and as a consequence no micellar solubilization of the other phase can occur in the excess phase, an important feature when the mass balance is to be discussed. 

As a result of change in pressure, composition, and temperature to a lesser degree, the micellar size shown in Figs. 5.10 and 5.11 may change (Nielsen, et al., 1994). The equilibrium between the precipitated phase and the bulk-liquid phase will be effected as a result of the micellar size. This is one main reason that cubic equations such as the PR-EOS, which have been used so successfully in vapor-liquid equilibria, may not be suitable for asphaltene-precipitation and asphaltene-precipitation inhibition calculations. This point will become clear later. 

The minimization of Gibbs free energy of the total system with respect to the independent variables subject to the constraints of Eqs. (5.85) and (5.86) provides the amount and composition of each phase. In general there are eight variables for the calculation of an equilibrium state at constant Tand P. Those are, n y D, n j., n , na and n -Once these variables are obtained, the micellar size can be readily determined. Pan and Firoozabadi (1997a) have used the feasible sequential quadratic programming (FSQP) to minimize G (Zhou, Tits, and Lawrence, 1996). 

Retsos et al. [55, 56] made an attempt to provide a semiquantitative analysis of the interfacial activity of block copolymers at the polymer-polymer interface the emphasis was on understanding the nonmonotonic dependence of the interfacial tension reduction on diblock molecular weight as well as the effects of macromo-lecular architecture and composition when graft copolymers were utilized as additives. The attempt was based on a modification of the analysis of Leibler [75], where the possibility of micellar formation was also taken into account. The thermodynamic equilibrium under consideration was, thus, that between copolymer chains adsorbed at the interface, chains homogeneously distributed in the bulk homopolymers, and chains at micelles formed within the homopolymer phases. 

An ultrafiltration technique, first described by Hutchinson and Schaffer [52] has been used by several workers [38, 53-56]. In a typical experiment a portion of the equilibrated surfactant solution containing a known amount of solubilizate and surfactant is passed through a membrane which is impermeable to micelles but which allows free passage of solubilizate molecules. Filtrand and filtrate are then analysed to determine the composition of the micellar and free phases. As with the equilibrium dialysis technique the selection of a suitable membrane is essential for the success of the technique. 

Self-assembly and morphology of block copolymers depend on their architecture and composition [3]. Several equilibrium phases like lamellae, gyroid, hexagonal-packed cylinders, and body-centered cubic phases were observed in melts. In thin films, microphase separation resulted in formation of lamellae, stripes, and circular domains. Various types of micellar structures and arrangements were seen in dilute solutions [4], These phase behaviors were dictated by Flory-Huggins interaction parameter (/), copolymer degree of polymerization N), and composition (/) in melts and thin films. In addition to these parameters, amphiphilicity was the most important property of block copolymers enabling them to self-assemble into various stmctures in dilute solutions [3]. 

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