Analytical Expressions to Calculate Relative Amounts of Phases in a Three-Component Phase Diagram

Abstract

The relative amounts of different phases in multiphase regions in the common three-component phase diagrams are usually estimated graphically using geometrical features of the diagram. The present contribution introduces algebraic expressions for the lines in the diagram, which allow these numbers to be calculated directly from the experimental results. The method is an extension of the classical methods, which were translated into convenient computer programs [Laughlin, R. G. The Aqueous Phase Behavior of Surfactants; Academic Press: New York, 1994] to calculate the amount of different phases for individual compositions. The introduction of analytical geometry allows simplified expressions to be used for the calculations and the amounts to be presented as continuous functions of the total composition.

A comparison between the experimental and estimated evaporation paths from emulsions

“…In that range, both L E and the linalool vapour pressure approach zero and the dependence of dL/dS on L E is not trivial. However, as demonstrated by equation (14) and the numerical results in Figure 4, dL/dS → 0 as L E → 0. Furthermore, equation (14) shows dL/dS (L E ) to be linear in this range, as:…”

“…[13] This result opened a novel area of research in which straightforward phase diagram studies were combined with algebraic calculations to extract precise information from the diagrams. [14,15] This was then exploited in a series of studies showing the effect of the initial emulsion composition on the evaporation path [16,17] and, unexpectedly and most significantly, by the relative humidity of the surrounding atmosphere. [18] These results indicate such calculations to be potentially useful to predict changes in emulsion phases during evaporation.…”

Evaporation from an ionic liquid emulsion

“…This fact in combination with the results of recent investigations of complex emulsion systems [20,21] encouraged an investigation into the conditions during evaporation of a nontraditional emulsion containing an ionic liquid. The chosen emulsion contains two phases; one of which is a ionic liquid-inwater microemulsion stabilized by a nonionic surfactant and the second phase a lamellar liquid crystal formed by the same surfactant and water with solubilized ionic liquid.…”

Constant vapour pressure evaporation from a fragrance emulsion—effect of solubility of surfactant in the fragrance compound

“…[23] Such a system will form traditional emulsions as well as ones with vesicles in the three-phase part of the system, the ramifications of which have been analysed. [14] The oil and the surfactant are mutually completely soluble, but neither has significant solubility in water, neither is the water soluble in the fragrance/ surfactant solution in the part of it that is relevant to the investigation. The lamellar liquid crystal is formed with approximately equal fractions of water and surfactant but with significantly less of the fragrance compound.…”

“…[13] It is vital to emphasize that the information in this special case is reliable as far as the evaporation path is concerned, but cannot be expected to provide information about the evaporation rate. [4,5,7] The apparent success of the approach in the initial investigation [13] indicated the potential use of the algebraic system to extract information from phase diagrams [14,15] to delineate the evaporation path in selected emulsion systems. With the qualifications about the equilibrium condition noted, the approach has revealed potentially decisive effects on the evaporation direction by initial water:oil ratio.…”