PERMANENT WAVING AND PERM CHEMISTRY 117 halftime (t0.5) as a normalizing condition, that is, at α = 0.5, the previous equation re- duces to the followin g equation: 0.5 0.693 . kt (17) Normaliza tion then involves dividing the integral equation by this specifi c condition: 0.5 In 1 , 0.693 kt kt α (18) which r edu ces to the following form: 0.5 In 1 0.693 . t t α (19) In short, t ime units cancel out (i.e., t/t0.5) and we are left with a means of creating a time- independent, theoretical master plot of α versus t/t0.5 for this fi rst-order expression. Fig- ure 12 shows master plots for both the fi rst-order mechanism and Wickett’s moving boundary model. Therefore, the applicability of a given model can be identifi ed by treat- ing experimental data in this same manner and comparing the shape of the resulting plots with those for theoretical models. To illustrate th is process, Figure 13 shows experimental data from eight separate SFTK experiments involving the interaction of single-source Asian hair with a 0.42 M, Figure 12. Reduced time plots for the theoretical fi rst-order and moving boundary models.

JOURNAL OF COSMETIC SCIENCE 118 pH 9.3 cysteamine solution. The reduced time approach involves evaluating the half- time for each individual experiment and then creating a new x axis by dividing the experiment time by the halftime for each test (i.e., t/t0.5). Figure 14 shows the result of performing this same analysis on the data from all eight individual experiments. Despite some experimental variability in Figure 13, once the normalization step ren- ders the x axis independent of time unit, all data reduce to a single curve that indi- cates the applicability of the same kinetic mechanism. However, Figure 15 shows these experimental data do not superimpose over either curve for the two theoretical models described earlier. It is therefore concluded that neither model applies in this case. Figure 16 s hows the result of performing this same reduced time analysis on SFTK experimental data for the interaction of this single-source Asian hair with both cysteamine and ATG solutions under identical conditions. Clearly, both data sets reduce down to differently shaped curves. In short, these two solutions not only do produce markedly different reaction rates (see Figure 7) but also two distinctly different kinetic mechanisms also appear to be present. (It will be recalled from the fl uorescence microscopy images shown in Figures 10 and 11 that these two treat- ments also gave rise to different behaviors during dye staining experiments.) This said neither of the two theoretical models described earlier match the experimental data and the need for further mathematical modeling is needed in an attempt to Figure 13. Raw SFTK data for reaction of single source Asian hair with 0.42M, pH 9.3 cysteamine.