26 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS 1,5' '1.5 Propyl 1.0- 0.5- Methyl .1.0 0.5 0 0'.5 Weight Fraction of Oil 10 Figure 1. Solubility of methyl and propyl parabens in water/peanut oil mixtures. emulsion--an oil/water mixture in which the only deactivation mechanism is the extraction of each paraben into the oil phase according to its own partition coefficient. Evans argued from the Ferguson principle which can be approximately paraphrased as stating that all homologous antimicrobials are equally effective, not at equal concen- trations but at concentrations corresponding to the same fraction of their solubility in the medium in which the antimicrobial activity is being measured. Evans' algebraic treatment of oil/water mixtures is re-presented in graphical form in Figure 1 which shows the apparent solubilities of methyl and propyl parabens in peanut oil/water mixtures as a function of the oil fraction. In the oil-saturated water phase, methyl paraben is much more soluble, but this is reversed in the water-saturated oil phase. Since the solubilities in all mixtures must fall on straight lines connecting the terminal points, one can calculate what Evans calls the "crossover point"--the oil fraction at which the apparent solubilities are equal. Below the crossover point it takes less propyl paraben than methyl paraben to saturate the system and since according to the Ferguson principle the two are equally effective at saturation regardless of the concentration difference, the propyl ester is the more efficient preservative for these mixtures. By the same argument, the methyl ester is

PARABENS 27 0'30 l 0 25 Methyl 0.20 0.15 0.10' Ethyl Propyl 0.05 Butyl o o'.'5 Weight Fraction of Oil Figure 2. Solubility of parabens in water/mineral oil mixtures. more efficient at high oil fractions. In either region--high or low oil fraction-- mixtures of the two esters must have a lower cumulative saturation fraction than an equal concentration of the less soluble homologue and must be less efficient. The only exception occurs at the crossover point where the solubilities are equal here all mixtures are as good as, but no better than, either component. Figure 2 shows similar solubility plots in mineral oil/water mixtures for the first four alkyl parabens. Without discussing each of the six crossovers in detail, one can see that at low oil fractions the butyl ester should be the most efficient preservative and that the methyl ester is best at very high fractions while the propyl ester is best at oil fractions from 0.3 to 0.9. COMPLEX SYSTEMS The rule emerging from Evans' analysis of simple emulsions, that the paraben of lowest apparent solubility in the emulsion is the most efficient one, can be extended to systems of any degree of complexity on grounds of the thermodynamic principle that at equilibrium the chemical potential of every component of the system is the same in all accessible phases in particular, if the system is saturated with one of the parabens,