28 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS the chemical potential of that paraben is equal in all phases to that of the pure crystal phase at the given temperature and pressure. Since, according to the Ferguson principle, the thermodynamic activity relative to the saturated solution is the appropriate measure of antimicrobial potency, all saturated systems are equally antimicrobial. To paraphrase this statement in practical terms, if the unpreserved product can support microbial growth it will be rendered equally antimicrobial by saturating it with any one of the alkyl parabens regardless of big differences in their saturation concentrations. It follows that the least soluble paraben is the most efficient one. If it can be assumed that the thermodynamic activity of each paraben is equal to its saturation fraction (the ratio of concentration to saturation concentration) and that these fractions are additive in applying the Ferguson principle to mixtures, then it also follows that at any concentration the least soluble paraben is superior to all equal weight mixtures. For example, if the methyl ester is the one least soluble in the product to be preserved and if it is measurably effective at, say, 0.75 of its saturation concentration, the same efficacy will be observed with a combination of methyl paraben at half saturation plus propyl paraben at 0.25 saturation or with any three parabens at 0.25 saturation, and so forth all such mixtures will constitute a greater total weight concentration than methyl paraben alone and will, in this sense, be less efficient. The conclusion that at saturation the paraben least soluble in the product is the most efficient preservative follows directly from the Ferguson principle, but the argument regarding mixtures is not rigorous since thermodynamic activity is only roughly proportional to concentration and at high concentrations the deviations from proportionality can be large. It seems unlikely, however, that the forms of the deviations should differ so much from one homologue to another as to cause inversions in the order of efficacy predicted rigorously at saturation. We propose then, as a working hypothesis, that the least soluble paraben is the most efficient one over its entire range of solubility. ANTIMICROBIAL CAPACITY Thus far we have discussed the efficiency of preservatives only in terms of their initial potency, but the capacity to resist many challenges or a massive inoculation must be an equally important factor because it is known that antimicrobials are decomposed or sequestered by the organisms they attack. Apart from this consideration, our working hypothesis would lead to the absurdity of recommending an infinitesimally soluble paraben such as a long-chain fatty alcohol ester for use in a simple aqueous system (although the Ferguson principle itself breaks down when applied to high molecular weight homologues, perhaps, in part because of the capacity factor). Obviously, the concentration of the antimicrobial in the phases where the microbes grow must be high enough to permit transport of a damaging dose to the surfaces of the microbes in a time comparable to their vegetative period. Within this conservative limit, is it likely that the least soluble homologue is consumed so much more rapidly and ineffectively than a more soluble homologue or mixture of homologues as to vitiate its advantage in initial potency? We try here to answer this

PARABENS 29 question by deriving a capacity function from the Ferguson principle as interpreted in terms of the partitioning of the antimicrobial between the medium and the microbe (see, for example, Reference 9). In its simplest form, this interpretation would predict that at any given thermodynamic activity (as defined above), all members of a homologous series should be present within the microbe "phase" at the same concentration. The data reported by Lang and Rye in 1972 (10) support this interpretation very nicely. Their radiochemical measure- ments show that the intracellular concentrations of methyl, ethyl and propyl parabens are the same when Escherichia co/i are in contact with equipotent solutions. The external concentrations are quite different on a weight percentage basis but they correspond to about the same saturation fraction. The theory and these observations can be expressed as: c[ = k's, where c[ is the concentration in the microbe "phase" of the ith paraben and si is its equilibrium saturation fraction in the medium. The Ferguson principle is manifested here in the absence of a subscript on the proportionality constant, k*, which has the same value for all homologues (for a given medium and for a particular species of microbe). On the untested but plausible assumption that the partitioning of each homologue is independent of the presence of other homologues in both phases, we can write: C/ = k* • s i = k*S (2) where C' is the total concentration of all parabens in the microbe phase and S is the equilibrium cumulative saturation fraction in the medium. Generalizing again from the Lang and Rye data, we suppose that there is some internal cumulative concentration C' at which the microbes are incapacitated to a degree that satisfies the criteria of preservation. We can now define the capacity of any mixture of homologous preservatives as the size of the inoculum which reduces S from its initial value to that corresponding to the critical value of C' in eq 2. Assuming first that the process leading to eq 1 is reversible, as though the inoculum were an inert "oil" phase into and out of which the chemically unchanged preservative freely diffuses, the capacity of a single preservative is given by subsiitution of eq 1 into the conservation equation: ci'V = c,V + (3) where c•' and ci are the initial and equilibrium values of the preservative concentration in the bulk volume, V, and v is the volume of the microbe phase. This substitution gives the decay equation: s i = s• 1 q--- (4) where tr• is the apparent solubility of the paraben in the system. The capacity function for a single paraben is given by rearrangement of eq 4 to: