32 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS 5o I Pr Me 0.5 • 1.0 • 1.5 2.0 2.5 ' Concentration g/L. ! / ! ! ! ! ! ! ! ! ! ! ! / ! ! ! / ! ! ! ! ! Figure 4. Capacity of parabens to reduce to zero the growth rate constant of E. Coli in water at 25øC. capacity lines are about half those in Figure 3, but note that for each paraben the line comes from the same negative intercept of o-i/k* on the capacity axis. As the critical saturation fraction approaches unity, the capacities of all the homo- 1ogues vanish simultaneously at any point short of this, the generalization holds that the least soluble homologue has the greatest capacity at any initial concentration within its solubility range. The mathematics of the capacity of mixtures is more complex because substitution of eq 4 in eq 2 gives a decay function of the same order as the number of components. It can be shown by consideration of the reciprocal decay function: •__ k • V 1/s, ci-•----- (7) O O ci c• V that the saturation fraction function of all binary mixtures in a plot of 1/s i against v/V is bounded by the parallel straight lines representing an equal weight of the individual components. Such mixtures must then be inferior to their less soluble component for all levels of inoculation. The reversible partitioning model predicts that the least soluble homologue is not only

PARABENS 33 superior in initial potency and capacity to all other homologues but to all possible binary mixtures as well. In a previous publication (11) on this subject, we discussed an irreversible binding mechanism expressed as: dv = -- Vdc (8) Here it is assumed that the inoculum is added incrementally and that each increment, dv, consumes an increment dc i of the ith paraben. Substitution in eq 1 and integration gives the decay equation: si = si exp -- •-i (9) (see Footnote 1) which rearranges to the capacity function: V c O' i C 7 -- In -- (10) V k* Sctr, Because the decay plots for two homologues at the same initial concentration intersect as shown in Figure 5, it cannot be said unequivocally that the less soluble homologue maintains the higher saturation fraction for all degrees of inoculation. It appears, however, that unless the solubility difference is negligible, intersection occurs only after massive inoculations corresponding to near exhaustion of the preservative and for all practical cases the same solubility rule holds: the less soluble homologue has the greater capacity. In any case, as demonstrated graphically in Figure 5, no mixture of the two homologues can have a greater capacity than the better of the two, because the decay plots of all possible mixtures pass through the intersection point mixtures can be dismissed again on the grounds that they are, at best, as good as the better of any two homologues. PRACTICAL APPLICATION In describing the optimum preservative system based on parabens of known solubility, we assume that the reversible partitioning model is valid. This eliminates consideration of the ambiguities in the capacities which result from the intersection of the irreversible curves. This is probably not a serious omission because even if the irreversible model is correct, it predicts different rankings only for small differences in solubility and only at the low saturation fractions corresponding to large inoculations. From our own experience in product preservation, it appears that the parabens are effective only near saturation and,this is supported by Mitchell's data (12) showing that ch10roxylenol is ineffective at saturation fractions below 0.9. Having put aside the complication of intersections in the decay functions, the problem of selecting the best preservative system is reduced to that of selecting the paraben or paraben mixture with the highest initial cumulative saturation fraction in the absence of intersections, this system will maintain the highest saturation fraction for all levels of inoculation and, therefore, it must have the greatest capacity as well. •This equation is obtained from the first term of the series expansion of eq 4.