KINETICS OF HAIR REDUCTION 285 reaction. Meanwhile, for reaction-controlled processes, the maximum rate would also be expected to be in the early stages, as this represents the point where each reactant has its highest concentration. The first order model is a deceleratory mechanism, however, as seen from Figure 2, by its very definition, the decrease in rate is linear with respect to the fraction or reaction. Wickett's moving boundary model, however, represents a sigmoidal type behavior, as it has a maximum rate at •30% of the way through the reaction. This suggests the presence of an induction period during which the rate gradually increases to a given point, followed by a drop off, presumably due to one of the reasons previously outlined. One of the heterogeneous kinetic models listed in Table I that would also be expected to be relevant in hair/reducing agent interactions is Mampel's contracting area model (see ref. 7), which describes the two-dimensional propagation of an interface into a cylinder. Considering the integral form: 1 - (1 - (•)v2 = kt (15) and carrying out the normalization process gives: 1 - (1 - (•)•/2 = 0.293- t (16) t0.5 The theoretical reduced-time plot for this model is shown in Figure 3 along with that for the first order expression. It can be seen that, although one expression describes a diffusion-controlled process while the other represents a reaction-controlled process, these two models give rise to theoretical curves that are in close proximity. However, the derivatives shown in Figure 4 indicate that the contracting area models shows a more reasonable deceleratory behavior. 0.9 - - 0.8 - .o• 0.7 F - 4--) • 0.6 - ß • o.$ - .•o 0.4 • 0.3 0.2 // Contracting area 0 1 // First order ß // 0.0 / I I I I I I I 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 t/t0. 5 Figure 3. Theoretical reduced-time plots for the first order model and the contracting area expression.

286 JOURNAL OF THE SOCIETY OF COSMETIC CHEMISTS 0.7 o.s - 0.4 - 0.3 - 0.2 Contracting area 0.1 - ?irst order 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 ?taction of Reaction Figure 4. First derivatives of the reduced-time plots for the first order and contracting area models, expressed as a function of the fraction of reaction. In this particular instance the specific reaction rate constant can be equated to the velocity with which the interface is moving (v) and the radius of the cylinder (r) by the equation v k - (17) r To help distinguish between theoretical curyes of close proximity it is often useful to normalize at times other than to. 5. For example, normalizing at to. 9 will allow greater spacing between curves at lower values of o•, while normalizing at to. • allows greater spacing between curves at higher values (see Figure 5). However, experimentally, longer-range extrapolation will, of course, give rise to an increased source of error. Another heterogeneous kinetic model that may be potentially applicable to hair reduc- tion is the Avrami-Erofeev equation. (see ref. 7). This expression is so called as it was simultaneously and independently derived by both authors while working in different fields. In its general form the equation is expressed as [--In(1 - ot)] •/" = kt (18) where n is the summation of the number of stages involved in nucleation, [3, and the number of dimensions of growth, k, i.e., n = [3 + k (19) In our hair reduction experiments we can assume that nucleation is instantaneous (i.e., [3 = 0). Therefore, for one-dimensional growth (i.e., branching chains, no interface), n = 1 and the Avrami-Erofeev expression reduces to the first order equation. However, for two-dimensional growth of an interface, n = 2 and the reaction can be described by the A2 expression: